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Report ARD 057 October 2005
ADINA R & D, Inc.
Volume I: ADINA Solids & Structures
Theory and Modeling Guide
UTOMATIC
YNAMIC
NCREMENTAL
ONLINEAR
NALYSIS
ADINA
Theory and Modeling Guide Volume I: ADINA Solids & Structures
October 2005 ADINA R & D, Inc. 71 Elton Avenue Watertown, MA 02472 USA tel. (617) 9265199 telefax (617) 9260238
www.adina.com
Notices
ADINA R & D, Inc. owns both this software program system and its documentation. Both the program system and the documentation are copyrighted with all rights reserved by ADINA R & D, Inc.
The information contained in this document is subject to change without notice.
Trademarks
ADINA is a registered trademark of K.J. Bathe / ADINA R & D, Inc.
All other product names are trademarks or registered trademarks of their respective owners. Copyright Notice
8 ADINA R & D, Inc. 1987  2005 October 2005 Printing Printed in the USA
Table of contents
ADINA R & D, Inc. iii
Table of contents
1. Introduction ......................................................................................................... 1 1.1 Objective of this manual.................................................................................. 1 1.2 Supported computers and operating systems .................................................. 2 1.3 Units ................................................................................................................ 2 1.4 ADINA System documentation....................................................................... 4
2. Elements ............................................................................................................... 7 2.1 Truss and cable elements................................................................................. 7
2.1.1 General considerations ............................................................................. 7 2.1.2 Material models and formulations............................................................ 9 2.1.3 Numerical integration............................................................................... 9 2.1.4 Mass matrices ......................................................................................... 10 2.1.5 Element output ....................................................................................... 11 2.1.6 Recommendations on use of elements.................................................... 11 2.1.7 Rebar elements ....................................................................................... 12
2.2 Twodimensional solid elements................................................................... 15 2.2.1 General considerations ........................................................................... 15 2.2.2 Material models and formulations.......................................................... 21 2.2.3 Numerical integration............................................................................. 22 2.2.4 Mass matrices ......................................................................................... 24 2.2.5 Element output ....................................................................................... 25 2.2.6 Recommendations on use of elements.................................................... 30
2.3 Threedimensional solid elements................................................................. 34 2.3.1 General considerations ............................................................................ 34 2.3.2 Material models and nonlinear formulations.......................................... 42 2.3.3 Numerical integration............................................................................. 44 2.3.4 Mass matrices ......................................................................................... 47 2.3.5 Element output ....................................................................................... 47 2.3.6 Recommendations on use of elements.................................................... 53
2.4 Beam elements .............................................................................................. 54 2.4.1 Linear beam element .............................................................................. 58 2.4.2 Large displacement elastic beam element .............................................. 64 2.4.3 Nonlinear elastoplastic beam element................................................... 65 2.4.4 Momentcurvature beam element........................................................... 70 2.4.5 Beamtied element.................................................................................. 80 2.4.6 Bolt element ........................................................................................... 82
2.5 Isobeam elements; axisymmetric shell elements.......................................... 84 2.5.1 General considerations ........................................................................... 84
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iv ADINA Structures ⎯ Theory and Modeling Guide
2.5.2 Numerical integration............................................................................. 89 2.5.3 Linear isobeam elements....................................................................... 91 2.5.4 Nonlinear isobeam elements ................................................................. 94 2.5.5 Axisymmetric shell element ................................................................... 95 2.5.6 Element mass matrices ........................................................................... 96 2.5.7 Element output ....................................................................................... 96
2.6 Plate/shell elements ........................................................................................ 99 2.6.1 Formulation of the element .................................................................. 100 2.6.2 Elastoplastic analysis .......................................................................... 104 2.6.3 Element mass matrices ......................................................................... 105 2.6.4 Element output ..................................................................................... 106
2.7 Shell elements.............................................................................................. 107 2.7.1 Basic assumptions in element formulation........................................... 107 2.7.2 Material models and formulations........................................................ 115 2.7.3 Shell nodal point degrees of freedom................................................... 118 2.7.4 Transition elements .............................................................................. 122 2.7.5 Numerical integration.......................................................................... 124 2.7.6 Composite shell elements ..................................................................... 127 2.7.7 Mass matrices ....................................................................................... 132 2.7.8 Anisotropic failure criteria ................................................................... 133 2.7.9 Element output ..................................................................................... 140 2.7.10 Selection of elements for analysis of thin and thick shells................. 147
2.8 Pipe elements............................................................................................... 151 2.8.1 Material models and formulations........................................................ 158 2.8.2 Pipe internal pressures.......................................................................... 159 2.8.3 Numerical integration........................................................................... 162 2.8.4 Bolt element ......................................................................................... 165 2.8.5 Element mass matrices ......................................................................... 166 2.8.6 Element output ..................................................................................... 167 2.8.7 Recommendations on use of elements.................................................. 168
2.9 General and spring/damper/mass elements ................................................. 169 2.9.1 General elements .................................................................................. 169 2.9.2 Linear and nonlinear spring/damper/mass elements ............................ 171 2.9.3 Springtied element .............................................................................. 176 2.9.4 Usersupplied element .......................................................................... 177
2.10 Displacementbased fluid elements ........................................................... 182 2.11 Potentialbased fluid elements................................................................... 184
2.11.1 Theory: Subsonic velocity formulation.............................................. 187 2.11.2 Theory: Infinitesimal velocity formulation ........................................ 194 2.11.3 Theory: Infinite fluid regions ............................................................. 195 2.11.4 Theory: Ground motion loadings ....................................................... 200
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2.11.5 Theory: Static analysis ....................................................................... 201 2.11.6 Theory: Frequency analysis ............................................................... 204 2.11.7 Theory: Mode superposition .............................................................. 205 2.11.8 Theory: Response spectrum, harmonic and random vibration analysis207 2.11.9 Modeling: Formulation choice and potential master degree of freedom........................................................................................................................ 209 2.11.10 Modeling: Potential degree of freedom fixities................................ 210 2.11.11 Modeling: Elements ......................................................................... 210 2.11.12 Modeling: Potentialinterfaces ......................................................... 211 2.11.13 Modeling: Interface elements ........................................................... 213 2.11.14 Modeling: Loads .............................................................................. 220 2.11.15 Modeling: Phi model completion ..................................................... 222 2.11.16 Modeling: Considerations for static analysis ................................... 237 2.11.17 Modeling: Considerations for dynamic analysis .............................. 238 2.11.18 Modeling: Considerations for frequency analysis and modal participation factor analysis............................................................................ 238 2.11.19 Modeling: Element output................................................................ 239
3. Material models and formulations................................................................. 241 3.1 Stress and strain measures in ADINA ......................................................... 241
3.1.1 Summary ............................................................................................... 241 3.1.2 Large strain thermoplasticity and creep analysis ................................ 246
3.2 Linear elastic material models..................................................................... 251 3.2.1 Elasticisotropic material model........................................................... 253 3.2.2 Elasticorthotropic material model ....................................................... 253
3.3 Nonlinear elastic material model (for truss elements)................................. 262 3.4 Isothermal plasticity material models.......................................................... 265
3.4.1 Plasticbilinear and plasticmultilinear material models ...................... 266 3.4.2 Mrozbilinear material model............................................................... 272 3.4.3 Plasticorthotropic material model ....................................................... 278 3.4.4 Ilyushin material model........................................................................ 282 3.4.5 Gurson material model ......................................................................... 284
3.5 Thermoelastic material models .................................................................. 286 3.6 Thermoelastoplasticity and creep material models................................... 290
3.6.1 Evaluation of thermal strains................................................................ 295 3.6.2 Evaluation of plastic strains ................................................................. 295 3.6.3 Evaluation of creep strains ................................................................... 299 3.6.4 Computational procedures.................................................................... 306 3.6.5 Shifting rule for the irradiation creep model ........................................ 308
3.7 Concrete material model.............................................................................. 309 3.7.1 General considerations ......................................................................... 309
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vi ADINA Structures ⎯ Theory and Modeling Guide
3.7.2 Formulation of the concrete material model in ADINA....................... 310 3.8 Rubber and foam material models............................................................... 328
3.8.1 Isotropic hyperelastic effects................................................................ 329 3.8.2 Viscoelastic effects............................................................................... 348 3.8.3 Mullins effect ....................................................................................... 358 3.8.4 Orthotropic effect ................................................................................. 362 3.8.5 Thermal strain effect ............................................................................ 366 3.8.6 TRS material......................................................................................... 368 3.8.7 Temperaturedependent material........................................................... 368
3.9 Geotechnical material models ..................................................................... 370 3.9.1 Curve description material model ........................................................ 370 3.9.2 DruckerPrager material model ............................................................ 376 3.9.3 Camclay material model ..................................................................... 380 3.9.4 MohrCoulomb material model............................................................ 385
3.10 Fabric material model with wrinkling ....................................................... 389 3.11 Viscoelastic material model ...................................................................... 390 3.12 Porous media formulation ......................................................................... 394 3.13 Gasket material model............................................................................... 399 3.14 Shape Memory Alloy (SMA) material model ............................................ 403 3.15 Usersupplied material model.................................................................... 409
3.15.1 General considerations ....................................................................... 409 3.15.2 2D solid elements.............................................................................. 410 3.15.3 3D solid elements.............................................................................. 421 3.15.4 Usersupplied material models supplied by ADINA R&D, Inc. ........ 422
4. Contact conditions........................................................................................... 437 4.1 Overview ..................................................................................................... 438 4.2 Contact algorithms for implicit analysis....................................................... 446
4.2.1 Constraintfunction method.................................................................. 447 4.2.2 Lagrange multiplier (segment) method ................................................ 448 4.2.3 Rigid target method .............................................................................. 449 4.2.4 Selection of contact algorithm.............................................................. 450
4.3 Contact algorithms for explicit analysis ....................................................... 450 4.3.1 Kinematic constraint method................................................................ 451 4.3.2 Penalty method..................................................................................... 452 4.3.3 Rigid target method .............................................................................. 453 4.3.4 Selection of contact algorithm for explicit analysis ............................. 453
4.4 Contact group properties ............................................................................. 454 4.5 Friction ........................................................................................................ 462
4.5.1 Basic friction models............................................................................ 462 4.5.2 Usersupplied friction models (not available in explicit analysis) ....... 464
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ADINA R & D, Inc. vii
4.5.3 Frictional heat generation...................................................................... 467 4.6 Contact definition features .......................................................................... 468
4.6.1 Analytical rigid targets .......................................................................... 468 4.6.2 Contact body ........................................................................................ 468 4.6.3 Draw beads ........................................................................................... 469
4.7 Contact analysis features ............................................................................. 470 4.7.1 Implicit dynamic contact/impact .......................................................... 470 4.7.2 Explicit dynamic contact /impact ......................................................... 471 4.7.3 Contact detection.................................................................................. 472 4.7.4 Suppression of contact oscillations ...................................................... 473 4.7.5 Restart with contact .............................................................................. 474
4.8 Modeling considerations ............................................................................. 474 4.8.1 Contactor and target selection .............................................................. 474 4.8.2 General modeling hints ........................................................................ 476 4.8.3 Modeling hints for implicit analysis..................................................... 478 4.8.4 Modeling hints for explicit analysis ..................................................... 480 4.8.5 Convergence considerations (implicit analysis only)........................... 482
5. Boundary conditions/applied loading/constraint ......................................... 485 5.1 Introduction ................................................................................................. 485 5.2 Concentrated loads ...................................................................................... 490 5.3 Pressure and distributed loading.................................................................. 493
5.3.1 Two and threedimensional pressure loading ..................................... 496 5.3.2 Hermitian (2node) beam distributed loading ...................................... 499 5.3.3 Isobeam and pipe distributed loading ................................................. 501 5.3.4 Plate/shell pressure loading .................................................................. 503 5.3.5 Shell loading......................................................................................... 504
5.4 Centrifugal and mass proportional loading ................................................. 504 5.5 Prescribed displacements............................................................................. 510 5.6 Prescribed temperatures and temperature gradients .................................... 512 5.7 Pipe internal pressure data........................................................................... 514 5.8 Electromagnetic loading.............................................................................. 515 5.9 Poreflow loads............................................................................................. 516 5.10 Phiflux loads.............................................................................................. 516 5.11 Contact slip loads ...................................................................................... 517 5.12 Surface tension boundary .......................................................................... 518 5.13 Initial load calculations ............................................................................. 519 5.14 Usersupplied loads ................................................................................... 519
5.14.1 General considerations ....................................................................... 519 5.14.2 Usage of the usersupplied loads........................................................ 519 5.14.3 Example: Hydrodynamic forces ......................................................... 522
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viii ADINA Structures ⎯ Theory and Modeling Guide
5.15 Constraint equations .................................................................................. 525 5.15.1 General considerations ....................................................................... 525 5.15.2 Rigid links .......................................................................................... 526
5.16 Mesh Glueing ............................................................................................. 528 6. Eigenvalue problems ....................................................................................... 531
6.1 Linearized buckling analysis ....................................................................... 531 6.1.1 General considerations ......................................................................... 531 6.1.2 Introduction of geometric imperfections .............................................. 534
6.2 Frequency analysis ...................................................................................... 534 6.2.1 Use of determinant search method ........................................................ 536 6.2.2 Use of subspace iteration method......................................................... 536 6.2.3 Use of Lanczos iteration method.......................................................... 538 6.2.4 Modal stresses ...................................................................................... 539
7. Static and implicit dynamic analysis ............................................................. 541 7.1 Linear static analysis ................................................................................... 541
7.1.1 Direct solver; sparse solver .................................................................. 542 7.1.2 Iterative solvers; multigrid solver......................................................... 544
7.2 Nonlinear static analysis.............................................................................. 546 7.2.1 Solution of incremental nonlinear static equations .............................. 547 7.2.2 Line search ........................................................................................... 550 7.2.3 Convergence criteria for equilibrium iterations ................................... 557 7.2.4 Selection of incremental solution method ............................................ 561 7.2.5 Example................................................................................................ 565
7.3 Linear dynamic analysis.............................................................................. 570 7.3.1 Stepbystep implicit time integration .................................................. 571 7.3.2 Time history by mode superposition .................................................... 572 7.3.3 Lumped and consistent damping; Rayleigh damping .......................... 573
7.4 Nonlinear dynamic analysis ........................................................................ 574 7.4.1 Stepbystep implicit time integration .................................................. 575 7.4.2 Mode superposition .............................................................................. 577
7.5 Choosing between implicit and explicit formulations................................. 578 8. Explicit dynamic analysis ............................................................................... 581
8.1 Formulation ................................................................................................. 583 8.1.1 Mass matrix .......................................................................................... 584 8.1.2 Damping ............................................................................................... 585
8.2 Stability ....................................................................................................... 585 8.3 Time step management.................................................................................. 588
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ADINA R & D, Inc. ix
9. Frequency domain analysis ............................................................................ 591 9.1 Response spectrum analysis ........................................................................ 591 9.2 Fourier analysis ........................................................................................... 602 9.3 Harmonic vibration analysis........................................................................ 605 9.4 Random vibration analysis .......................................................................... 612 9.5 SDOF system response................................................................................ 617
10. Fracture mechanics ....................................................................................... 629 10.1 Overview ................................................................................................... 629 10.2 The line contour integration method ......................................................... 631 10.3 The virtual crack extension method .......................................................... 635
10.3.1 Jintegral and energy release rate ....................................................... 635 10.3.2 Virtual material shifts ......................................................................... 640 10.3.3 Results ................................................................................................ 642
10.4 Crack propagation ..................................................................................... 644 10.5 Recommendations on the use of elements and meshes ............................. 650 10.6 Usersupplied fracture mechanics ............................................................. 653
10.6.1 General considerations ....................................................................... 653 10.6.2 Usage of the usersupplied fracture mechanics subroutines............... 653
11. Additional capabilities .................................................................................. 655 11.1 Substructuring ........................................................................................... 655
11.1.1 Substructuring in linear analysis ........................................................ 655 11.1.2 Substructuring in nonlinear analysis .................................................. 657
11.2 Cyclic symmetry analysis.......................................................................... 659 11.3 Reactions calculation................................................................................. 663 11.4 Element birth and death option ................................................................. 664 11.5 Element "death upon rupture" ................................................................... 671 11.6 Initial conditions........................................................................................ 673
11.6.1 Initial displacements, velocities and accelerations ............................. 673 11.6.2 Initial temperatures and temperature gradients .................................. 673 11.6.3 Initial pipe internal pressures ............................................................. 674 11.6.4 Initial strains ....................................................................................... 674 11.6.5 Initial stresses ..................................................................................... 677
11.7 Strain energy densities............................................................................... 680 11.8 Element group inertial properties .............................................................. 681 11.9 Restart option ............................................................................................ 681 11.10 Parallel processing................................................................................... 684 11.11 Usage of memory and disk storage ......................................................... 685 11.12 Remeshing options .................................................................................. 688 11.13 Analysis zooming.................................................................................... 690
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x ADINA Structures ⎯ Theory and Modeling Guide
11.14 Stiffness stabilization .............................................................................. 691 12. Postprocessing considerations ...................................................................... 695
12.1 Calculation of results within the AUI........................................................ 695 12.1.1 What results are available................................................................... 701 12.1.2 Where results can be obtained............................................................ 735 12.1.3 How results are evaluated (including smoothing) .............................. 745 12.1.4 When (for which load step, etc.) are results available........................ 748
12.2 Evaluation of meshes used ........................................................................ 752 Index ...................................................................................................................... 753
1.1 Objective of this manual
ADINA R & D, Inc. 1
1. Introduction
1.1 Objective of this manual
The objective of this manual is to give a concise summary and guide for study of the theoretical basis of the finite element computer program ADINA Solids & Structures (ADINA in short). The program ADINA is used for displacement and stress analysis.
Since a large number of analysis options is available in this computer program, a user might well be initially overwhelmed with the different analysis choices and the theoretical bases of the computer program. A significant number of publications referred to in the text (books, papers and reports) describe in detail the finite element analysis procedures used in the program. However, this literature is very comprehensive and frequently provides more detail than the user needs to consult for the effective use of ADINA. Furthermore, it is important that a user can identify easily which publication should be studied if more information is desired on a specific analysis option.
The intent with this Theory and Modeling Guide is
! To provide a document that summarizes the methods and assumptions used in the computer program ADINA
! To provide specific references that describe the finite element procedures in more detail.
Hence, this manual has been compiled to provide a bridge
between the actual practical use of the ADINA system and the theory documented in various publications. Much reference is made to the book Finite Element Procedures (ref. KJB) and to other publications but we endeavored to be specific when referencing these publications so as to help you to find the relevant information.
ref. K.J. Bathe, Finite Element Procedures, Prentice Hall,
Englewood Cliffs, NJ, 1996.
Following this introductory chapter, Chapter 2 describes the
Chapter 1: Introduction
2 ADINA Structures ⎯ Theory and Modeling Guide
elements available in ADINA. The formulations used for these elements have been proven to be reliable and efficient in linear, large displacement, and large strain analyses. Chapter 3 describes the material models available in ADINA. Chapter 4 describes the different contact formulations and provides modeling tips for contact problems. Loads, boundary conditions and constraints are addressed in Chapter 5. Eigenvalue type analyses such as linearized buckling and frequency analyses are described in Chapter 6. Chapter 7 provides the formulations used for static and implicit dynamic analysis, while Chapter 8 deals with explicit dynamic analysis. Several frequency domain analysis tools are detailed in Chapter 9. Fracture mechanics features are described in Chapter 10. Additional capabilities such as substructures, cyclic symmetry, initial conditions, parallel processing and restarts are discussed in Chapter 11. Finally, some postprocessing considerations are provided in Chapter 12.
We intend to update this report as we continue our work on the ADINA system. If you have any suggestions regarding the material discussed in this manual, we would be glad to hear from you.
1.2 Supported computers and operating systems
Table 1.21 shows all supported computers/operating systems.
1.3 Units When using the ADINA system, it is important to enter all
physical quantities (lengths, forces, masses, times, etc.) using a consistent set of units. For example, when working with the SI system of units, enter lengths in meters, forces in Newtons, masses in kg, times in seconds. When working with other systems of units, all mass and massrelated units must be consistent with the length, force and time units. For example, in the USCS system (USCS is an abbreviation for the U.S. Customary System), when the length unit is inches, the force unit is pound and the time unit is second, the mass unit is lbsec2/in, not lb.
Table 1.31 gives some of the more commonly used units needed for ADINA input.
1.3: Units
ADINA R & D, Inc. 3
Table 1.21: Supported computers/operating systems Platform/operating system
32bit version 64bit version Parallelization options1
HP/HPUX 11, PARISC computers
No Yes, ADINAM Assembly, solver
HP/HPUX 11.22, Itanium computers
No Yes, ADINAM Assembly, solver
Linux kernel 2.4.0 and higher, x86 computers2,3
Yes, ADINAM No Assembly, solver
Linux kernel 2.4.0 and higher, Itanium computers, (including SGI Altix)4
The x86 version of the AUI is provided so that ADINAM can beused.
Yes Assembly, solver
Linux kernel 2.4.21 and higher, Intel x86_64 computers and AMD Opteron computers5
No Yes, ADINAM Assembly, solver
IBM/AIX 5.1 No Yes, ADINAM Solver SGI/IRIX 6.5.16m and higher
No Yes, ADINAM Assembly, solver
Sun/SunOS 5.8 (Solaris 8)
No Yes, ADINAM Solver
Windows 98, NT, 2000, XP6
Yes, ADINAM No Solver (not for Windows 98)
ADINAM is supported only for the indicated program versions.
1) Only ADINA and ADINAT have parallelized assembly. 2) Users of the usersupplied options for Linux must use the Intel compilers when using ADINA system 8.0 and higher. 3) The Linux version for x86 computers requires glibc 2.3.2 or higher (Red Hat 9.0 or higher, or equivalent). 4) The Linux version for Itanium computers requires glibc 2.2.4 or higher. SGI Altix computers require glibc2.3.2 or higher (SGI ProPack 3). 5) The Linux version for x86_64 and AMD Opteron computers requires glibc 2.3.2 or higher. 6) The 3GB address space option is supported.
Chapter 1: Introduction
4 ADINA Structures ⎯ Theory and Modeling Guide
Table 1.31: Units
SI
SI (mm)
USCS (inch)
USCS (kip)
length
meter (m)
millimeter (mm)
inch (in)
inch (in)
force
Newton (N)
Newton (N)
pound (lb)
kip (1000 lb)
time
second (s)
second (s)
second (sec)
second (sec)
mass
kilogram (kg) = Ns2/m
Ns2/mm
lbsec2/in (note 1)
kipsec2/in
pressure, stress, Young=s modulus, etc.
Pascal (Pa) = N/m2
N/mm2
psi = lb/in2
ksi = kip/in2
density
kg/m3
Ns2/mm4
lbsec2/in4
(note 2)
kipsec2/in4
1) A body that weighs 1 lb has a mass of 1/386.1 = 0.002590 lbsec2/in. 2) A body with weight density 1 lb/in3 has a density of 0.002590 lbsec2/in4.
1.4 ADINA System documentation
At the time of printing of this manual, the following documents are available with the ADINA System:
Installation Notes
Describes the installation of the ADINA System on your computer.
ADINA User Interface Command Reference Manual
Volume I: ADINA Solids & Structures Model Definition, Report ARD 052,
October 2005
1. 4 ADINA System documentation
ADINA R & D, Inc. 5
Volume II: ADINA Heat Transfer Model Definition, Report ARD 053,
October 2005 Volume III: ADINA CFD Model Definition, Report ARD 054,
October 2005 Volume IV: Display Processing, Report ARD 055,
October 2005 These documents describe the AUI command language. You use the AUI command language to write batch files for the AUI.
ADINA User Interface Primer, Report ARD 056, October 2005
Tutorial for the ADINA User Interface, presenting a sequence of worked examples which progressively instruct you how to effectively use the AUI.
Theory and Modeling Guide
Volume I: ADINA Solids & Structures, Report ARD 057, October 2005 Volume II: ADINA Heat Transfer, Report ARD 058, October 2005 Volume III: ADINA CFD & FSI, Report ARD 059, October 2005 Provides a concise summary and guide for the theoretical basis of the analysis programs ADINA, ADINAT, ADINAF, ADINAFSI and ADINATMC. The manuals also provide references to other publications which contain further information, but the detail contained in the manuals is usually sufficient for effective understanding and use of the programs.
ADINA Verification Manual, Report ARD 0510, October 2005
Presents solutions to problems which verify and demonstrate the usage of the ADINA System. Input files for these problems are distributed along with the ADINA System programs.
TRANSOR for PATRAN Users Guide, Report ARD 0514,
October 2005 Describes the interface between the ADINA System and MSC.Patran. The ADINA Preference, which allows you to
Chapter 1: Introduction
6 ADINA Structures ⎯ Theory and Modeling Guide
perform pre/postprocessing and analysis within the Patran environment, is described.
TRANSOR for IDEAS Users Guide, Report ARD 0515, October 2005 Describes the interface between the ADINA System and UGS Ideas. The fully integrated TRANSOR graphical interface is described, including the input of additional data not fully described in the Ideas database.
ADINA System 8.3 Release Notes, October 2005
Provides update pages for all ADINA System documentation published before October 2005 and a description of the new and modified features of ADINA System 8.3.
2.1 Truss and cable elements
ADINA R & D, Inc. 7
2. Elements
2.1 Truss and cable elements
• This chapter outlines the theory behind the different element classes, and also provides details on how to use the elements in modeling. This includes the materials that can be used with each element type, their applicability to large displacement and large strain problems, their numerical integration, etc.
2.1.1 General considerations
! The truss elements can be employed as 2node, 3node and 4node elements, or as a 1node ring element. Fig. 2.11 shows the elements available in ADINA.
Z
Y
X
wi
vi
ui
i
(a) 2, 3 and 4node elements
(u, v, w) are global displacementdegrees of freedom (DOF)
One DOF per node
(b) 1node ring element
Figure 2.11: Truss elements available in ADINA
Z
Y
vii
Chapter 2: Elements
8
! Note that the only force transmitted by the truss element is the longitudinal force as illustrated in Fig. 2.12. This force is constant in the 2node truss and the ring element, but can vary in the 3 and 4node truss (cable) elements.
Z
YX
(a) 2 to 4node elements
P
P
area A
Stress constant over
crosssectional area
ref. KJBSections
5.3.1,6.3.3
ADINA Structures ⎯ Theory and Modeling Guide
Z
Z
Y
Y
v
X
(b) Ring element
P
1 radian
R
(hoop stress)
Figure 2.12: Stresses and forces in truss elements
Young's modulus E
area A
R
Hoop strain = v/R Elastic stiffness = EA/R (one radian)
Circumferential force P = A
(as is the axisymmetric 2D solid element, see Section 2.2.1), and this formulation is illustrated in Fig. 2.12.
! The local node numbering and natural coordinate system for the elements are shown in Fig. 2.13.
! The ring element is formulated for one radian of the structure
2node, 3node and 4node truss
2.1 Truss and cable elements
ADINA R & D, Inc. 9
r r
r
1
1
2node element 3node element
4node element
2
23
Figure 2.13: Local node numbering; natural coordinate system
1 2 3
4
r=1 r=+1 r=1r=0
r=+1
r=1
r=1/3
r=1
r=1/3
2.1.2 Material models and formulations
! The truss element can be used with the following material models: elasticisotropic, nonlinearelastic, plasticbilinear, plasticmultilinear, thermoisotropic, thermoplastic, creep, plasticcreep, multilinearplasticcreep, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable, viscoelastic, shapememory alloy.
! The truss element can be used with the small or large displacement formulations. In the small displacement formulation, the displacements and strains are assumed infinitesimally small. In the large displacement formulation, the displacements and rotations can be very large. In all cases, the crosssectional area of the element is assumed to remain unchanged, and the strain is equal to the longitudinal displacement divided by the original length.
All of the material models in the above list can be used with either formulation. The use of a linear material with the small displacement formulation corresponds to a linear formulation, and the use of a nonlinear material with the small displacement formulation corresponds to a materiallynonlinearonly formulation.
2.1.3 Numerical integration
! The integration point labeling of the truss (cable) elements is given in Fig. 2.14.
Chapter 2: Elements
10 ADINA Structures ⎯ Theory and Modeling Guide
r r
r r
1 1
1 1
Onepoint integration Twopoint integration
Threepoint integration Fourpoint integration
2
2 23 3 4
Figure 2.14: Integration point locations
! Note that when the material is temperatureindependent, the 2l
ay be priate wdependent material is used because, due to a varying temperature,
terial properties can vary alon ent.
2.1.4 Mas
The consistent mass matrix is calculated using Eq. (4.25) in ref. which accurately computes the consistent mass
node truss and ring elements require only 1point Gauss numericaintegration for an exact evaluation of the stiffness matrix, because the force is constant in the element. However, 2point Gauss numerical integration m appro hen a temperature
the ma g the length of an elem
s matrices
• KJB, p. 165,distribution.
The lumped mass matrix is formed by dividing the elements mass
among its nodes. The mass assigned to each node is iM ⎛ ⎞⋅⎜ ⎟!L⎝ ⎠
,
e where M = total mass, L = total element length, i! = fraction of the total element length associated with element node i (i.e., for th
2node truss element, 1 2L
=! and 2 2L
=! , which for the 4node
truss 1 2 6L
= =! ! and 3L
4 3= =! ! ). The element has no
rotational mass.
2.1 Truss and cable elements
ADINA R & D, Inc. 11
2.1.5 Elem
t its integration points, the following information to the porthole file, based on the material model. This
RR,
TRESSBRR,
IN,
RR,
lasticmultilinear: PLASTIC_FLAG,
INRR,
EP_STRAINRR, THERMAL_STRAINRR, ELEMENT_TEMPERATURE,
,
LEMENT_TEMPERATURE
2.1.6 Recommendations on use of elements
! Of the 2, 3 and 4node truss elements, the 2node element is s,
ent output
! Each element outputs, a
information is accessible in the AUI using the given variable names. Elasticisotropic, nonlinearelastic: FORCER, STRESSBSTRAINRR
Elasticisotropic with thermal effects: FORCER, SSTRAINRR, THERMAL_STRAELEMENT_TEMPERATURE
Thermoisotropic: FORCER, STRESSRR, STRAINELEMENT_TEMPERATURE
lasticbilinear, pPFORCER, STRESSRR, STRAPLASTIC_STRAINRR
Thermoplastic, creep, plasticcreep, multilinearplasticcreep: PLASTIC_FLAG, NUMBER_OF_SUBINCREMENTS, FORCER, STRESSRR, STRAINRR, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, PLASTIC_STRAINRR, CRE
FE_EFFECTIVE_STRESS, YIELD_STRESS, EFFECTIVE_CREEP_STRAIN
Viscoelastic: PLASTIC_FLAG, STRESSRR, STRAINRRTHERMAL_STRAINRR, E
See Section 12.1.1 for the definitions of those variables that are not selfexplanatory.
usually most effective and can be used in modeling truss structure
Chapter 2: Elements
12 ADINA Structures ⎯ Theory and Modeling Guide
EA/L where L is the length of the element) and nonlinear elastic gap elements (see Section 3.3).
! The 3 and 4node elements are employed to model cables and steel reinforcement in reinforced concrete structures that are modeled with higherorder continuum or shell elements. In this case, the 3 and 4node truss elements are compatible with the continuum or shell elements. ! The internal nodes for the 3node and 4node truss elements are usually best placed at the mid and thirdpoints, respectively, unless a specific predictive capability of the elements is required (see ref. KJB, Examples 5.2 and 5.17, pp. 348 and 370).
2.1.7 Rebar elements
! The AUI can generate truss elements that are connected to the 2D or 3D solid elements in which the truss elements lie. There are several separate cases, corresponding to the type of the truss element group and the type of the solid element group in which the truss elements lie. Axisymmetric truss elements, axisymmetric solid elements: The AUI can connect axisymmetric truss elements that exist in the model prior to data file generation to the axisymmetric 2D elements in which the truss elements lie. The AUI does this connection during data file generation as follows. For each axisymmetric truss element that lies in an axisymmetric 2D element, constraint equations are defined between the axisymmetric truss element node and the three closest corner nodes of the 2D element.
(a) Before data file generation
Axisymmetric 2D solid element
Axisymmetric truss element Constraint equation
(b) After data file generation
Figure 2.15: Rebar in axisymmetric truss, 2D solid elements
2.1 Truss and cable elements
ADINA R & D, Inc. 13
the rebar line and the sides f the 2D elements. The AUI then generates nodes at these
intersections and generates truss elements that connect the uc es ns etween the generated nodes and the corner nodes of the 2D
3D truss elements, planar 2D solid elements: The AUI can generate 3D truss elements and then connect the truss elements to the planar 2D elements in which the truss elements lie. The AUIdoes this during data file generation as follows. For each rebar line, the AUI finds the intersections of o
s c sive nodes. The AUI also defines constraint equatiobelements.
(a) Before data file generation
Planar 2D solid element
Rebar line Truss element
Constraint equation
(b) After data file generation
Figure 2.16: Rebar in 3D truss, planar 2D solid elements
ments: The AUI can generate 3D
g .
so ns between the generated nodes and three
3D truss elements, 3D solid eletruss elements and then connect the truss elements to the 3D elements in which the truss elements lie. The AUI does this durindata file generation as follows. For each rebar line, the AUI findsthe intersections of the rebar line and the faces of the 3D elements The AUI then generates nodes at these intersections and generatestruss elements that connect the successive nodes. The AUI al
efines constraint equatiodcorner nodes of the 3D element faces.
Chapter 2: Elements
14 ADINA Structures ⎯ Theory and Modeling Guide
(a) Before data file generation
3D solid element
Rebar line
Truss element
Constraint equation
(b) After data file generation
Figure 2.17: Rebar in 3D truss, 3D solid elements ! The rebar option is intended for use with lowerorder solid elements (that is, the solid elements should not have midside nodes). ! For 3D truss element generation, if the rebar lines are curved, you should specify enough subdivisions on the rebar lines so that the distance between subdivisions is less than the solid element size. The more subdivisions that are on the rebar lines, the more accurately the AUI will compute the intersections of the rebar lines and the solid element sides or faces.
ent Group icon to open the Define t
ct
S
! To request the connection: AUI: Click the Define ElemElement Group window. In this window, check that the ElemenType has been set to Truss. Click on the Advanced tab. Selethe Use as Rebars option from the Element Option dropdown, and then fill in the RebarLine Label box below the Element Option dropdown box. Command Line: Set OPTION=REBAR in the EGROUP TRUScommand. For 3D truss elements, use the REBARLINE command and RBLINE parameter of the EGROUP TRUSS command to define the rebar lines.
2.2: Twodimensional solid elements
ADINA R & D, Inc. 15
2.2 Twod
2.2.1 General considerations
os, plane strain,
axisymmetric , generalized plane strain and 3D plane stress
the assumptions used in the formulations.
imensional solid elements
! The following kinematic assumptions are available for twdimensional elements in ADINA: plane stres
(membrane). Figures 2.21 and 2.22 show some typical 2D elements and
L
h
(a) 8 & 9node elements
Recommendation:
h/L > 1/10
(b) 3 & 4node elements
(c) 6 & 7node triangles
Figure 2.21: 2D solid elements
! The 3D plane stress element can lie in general threedimensional space.
! The plane stress, plane strain, generalized plane strain, and axisymmetric elements must be defined in the YZ plane. The axisymmetric element must, in addition, lie in the +Y half plane. However, these elements can be combined with any other elements available in ADINA.
! The axisymmetric element provides for the stiffness of one radian of the structure. Hence, when this element is combined with other elements, or when concentrated loads are defined, these must also refer to one radian, see ref. KJB, Examples 5.9 and 5.10, p. 356.
ref. KJBSections 5.3.1
and 5.3.2
Chapter 2: Elements
16 ADINA Structures ⎯ Theory and Modeling Guide
b) Plane strain elementa) Plane stress element
e) 3D plane stress element
(Stresses computed
in element local
coordinate system)
xy = 0
xx = 0
xz = 0
xy = 0
xx = 0
xz = 0
Figure 2.22: Basic assumptions in 2D analysis
xy = 0
xz = 0
xx = 0
X Y
Z
xx y
y
zz
d) Generalized plane strain
xy = 0
xz = 0
= constant ij
x
x y
z
c)Axisymmetric element
xy = 0
xz = 0
xx = u/y
(linear analysis)
x y
z
! he plane strain element provides for the stiffness of unit
! used in ADINA are isoparametric displacementbased finite elements, and their formulation is described in detail in ref. KJB, Section 5.3.
! The basic finite element assumptions are, see Fig. 2.23, for the
y
Tthickness of the structure.
he elements usuallyT
coordinates
1
q
i ii
y h=
= ∑ ; z1
q
i ii
z h=
= ∑
2.2: Twodimensional solid elements
ADINA R & D, Inc. 17
a) Plane strain, plane stress, and axisymmetric elements
w4
v4
z4
y4
Local displacement DOFr
s
Y
Z
12
3 4
5
6
7
9 8
X
b) Generalized plane strain element
X, u
Y, v
Z, w
Section of the structure(must lie in YZ plane) z
p
ypNp
Up
4 3
21
The coordinates of the auxiliary node Np are (xp, yp, zp).
The element has unit length in the X direction.
w4 w3
w2w1 v1
v2
v3v4
Figure 2.23: Conventions used for the nodal coordinates anddisplacements of the 2D solid element
Chapter 2: Elements
18 ADINA Structures ⎯ Theory and Modeling Guide
in plane stress, plane strain, and axifor the displacements
symmetric elements
1v v
q
i ii
h=
= ∑ ; w1
q
i ii
w h=
= ∑
for the displacements in generalized plane strain elements
( ) ( )p p p p
z p yu xU x y y x z z= − − Φ + − Φ
21v vq
pi i zh x
1 2i== + Φ∑
21q
pw h w x1 2i=
i i z= − Φ∑
strain
i i
U
where
hi(r,s) = interpolation function corresponding to node i (r,s) = isoparametric coordinates q = number of element nodes, 4 # q # 9, excluding auxiliary node in generalized plane
y , z = nodal point coordinates vi, wi = nodal point displacements
p p py z, ,Φ Φ = degrees of freedom of the auxiliary node
coordinates of the auxiliary node
! interpolated elements are also available, in which the displacements and pressure are interpolated separately. These elements are effective and should be preferred in the analysis of incompressible me Poisson's ratio is close to 0.5, for rubberlike materials and for ela  of freedom) and 9node element (3 pressure degrees of freedom) are
xp, yp, zp =
In addition to the displacementbased elements, special mixed
dia and inelastic materials (specifically for materials in which
sto plastic materials). The 4node element (1 pressure degree
recommended for such analyses. Note that these mixedinterpolated elements are useful (and available in ADINA) only for plane strain, generalized plane strain, or axisymmetric analysis.
2.2: Twodimensional solid elements
ADINA R & D, Inc. 19
!
m
are introduced. These additional displacem
ver the additional displacement degrees of the element, especially in
odes
ref. T. Sussman and K.J. Bathe, "A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis," J. Computers and Structures, Vol. 26, No. 1/2, pp. 357409, 1987.
In addition to the displacementbased and mixedinterpolatedelements, ADINA also includes the possibility of including incompatible modes in the formulation of the 4node element. Within this element, additional displacement degrees of freedo
ent degrees of freedomare not associated with nodes; therefore the condition of displacement compatibility between adjacent elements is not satisfied in general. Howeof freedom increase the flexibilitybending situations.
Figure 2.24 shows a situation in which the incompatible melements give an improved solution.
4node elements,
no incompatible modes
4node elements,
incompatible modes
8node elements
Plane strain conditions,
beam length = 5 m
beam height = 0.5 m
beam width = 1.0 m
E=2.07 1011 N/m2
= 0.29
= 7800 kg/m3
134.5 Hz
108.6 Hz
107.1 Hz
First bending mode Second bending mode
358.0 Hz
294.0 Hz
279.9 Hz
Third bending mode
674.8 Hz
566.3 Hz
513.9 Hz
First axial mode
540.6 Hz
540.5 Hz
537.9 Hz
Figure 2.24: Frequency analysis of a freefree beam
with three types of finite elements
Chapter 2: Elements
20 ADINA Structures ⎯ Theory and Modeling Guide
4.4.1.
ent performance. in conjunction
ode e is
element.
ode number to all the nodes located on one side. It is rec
For theoretical considerations, see reference KJB, Section
Note that these elements are formulated to pass the patch test. Alsonote that element distortions deteriorate the elem
The incompatible modes feature cannot be usedwith the mixedinterpolation formulation.
The incompatible modes feature is only available for the 4nelement; in particular note that the incompatible modes featurnot available for the 3node triangular
! The elements can be used with 3 to 9 nodes. The interpolationfunctions are defined in ref. KJB, Fig. 5.4, p. 344.
! Triangular elements are formed in ADINA by assigning the same global n
ommended that this "collapsed" side be the one with local endnodes 1 and 4.
The triangular elements can be of different types:
< A collapsed 6node triangular element with the strain
singularity 1
(see Chapter 9) r
< A collapsed 8node element with the strain singularity 1r
(see Chapter 9)
isotropic triangular element
e ctions.
The 6node spatially isotropic triangle is obtained by correcting ctions of the collapsed 8node element. It then
uses the same interpolation functions for each of the 3 corner nodes ections
ref. KJBSection 5.3.2
< A collapsed 6node spatially isotropic triangular element
< A collapsed 7node spatially
These element types are described on pp. 369370 of ref. KJB. The strain singularities are obtained by collapsing one side of an 8nodrectangular element and not correcting the interpolation fun
the interpolation fun
and for each of the midside nodes. Note that when the corrto the interpolation functions are employed, only the element side with end nodes 1 and 4 can be collapsed.
The 3node triangular element is obtained by collapsing one
2.2: Twodimensional solid elements
ADINA R & D, Inc. 21
side of the 4node element. This element exhibits the constant strain conditions (except that the hoop strain in axisymmetric analysis varies over the element) but is usually not effective.
! Linear and nonlinear fracture mechanics analysis of stationary or propagating cracks can be performed with twodimensional elements (see Chapter 9).
2.2.2 Material models and formulations
! The 2D elements can be used with the following material models: elasticisotropic, elasticorthotropic (with or without wrinkling), plasticbilinear, plasticmultilinear, DruckerPrager, Mrozbilinear, plasticorthotropic, thermoisotropic, thermoorthotropic, thermoplastic, creep, plasticcreep, multilinearplasticcreep, concrete, curvedescription, Ogden, MooneyRivlin, ArrudaBoyce, hyperfoam, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable, Gursonplastic, Camclay, MohrCoulomb, viscoelastic, creep irradiation, gasket, shapememory, alloy usersupplied.
! In plane strain, generalized plane strain and axisymmetric analysis, you should use the mixed interpolation formulation whenever using the plasticbilinear, plasticmultilinear, Mroz bilinear, plasticorthotropic, thermoplastic, creep, plasticcreep, multilinearplasticcreep, Ogden, MooneyRivlin, ArrudaBoyce, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable or viscoelastic material models; or when using the elasticisotropic material and the Poisson=s ratio is close to 0.5.
The AUI automatically chooses the mixed interpolation formulation whenever the Ogden, MooneyRivlin or ArrudaBoyce material models are used. For the other material types listed in the previous paragraph, you should explicitly select the mixed interpolation formulation.
! The twodimensional elements can be used with a small displacement/small strain, large displacement/small strain or a large displacement/large strain formulation.
The small displacement/small strain and large displacement/small strain formulations can be used with any material model, except for the Ogden, MooneyRivlin, Arruda
Chapter 2: Elements
22 ADINA Structures ⎯ Theory and Modeling Guide
Boyce and hyperfoam material models. The use of a linear material with the small displacement/ small strain formulation corresponds to a linear formulation, and the use of a nonlinear material with the small displacement/small strain formulation corresponds to a materiallynonlinearonly formulation. The program uses the TL (total Lagrangian) formulation when you choose a large displacement/small strain formulation.
The large displacement/large strain formulations can be used with the plasticbilinear, plasticmultilinear, Mrozbilinear, thermoplastic, creep, plasticcreep, multilinearplasticcreep, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable, viscoelastic and usersupplied material models. The program uses a ULH formulation when you choose a large displacement/large strain formulation with these material models.
A large displacement/large strain formulation is used with the Ogden, MooneyRivlin, ArrudaBoyce and hyperfoam material models. The program uses a TL (total Lagrangian) formulation in this case.
! The basic continuum mechanics formulations are described in ref. KJB, pp. 497537, and the finite element discretization is given in ref. KJB pp. 538542, 549555.
! Note that all these formulations can be mixed in one same finite element mesh. If the elements are initially compatible, then they will remain compatible throughout the complete analysis.
2.2.3 Numerical integration
! For the calculation of all element matrices and vectors, numerical Gauss integration is used. You can use from 2×2 to 6×6 Gauss integration. The numbering convention and the location of the integration points are given in Fig. 2.25 for 2×2 and 3×3 Gauss integration. The convention used for higher orders is analogous. The default Gauss integration orders for rectangular elements are 2×2 for 4node elements and 3×3 otherwise.
ref. KJBSections 5.5.3,
5.5.4 and 5.5.5
ref. KJBSections 6.2 and
6.3.4
ref. KJBSection 6.8.1
2.2: Twodimensional solid elements
ADINA R & D, Inc. 23
r r
ssINS=2INR=2
INS=1
INS=3
INS=1
INS=2INR=1
INR=1INR=2
INR=3
a) Rectangular elements
1point
s
r
1
4point
1 43
2
b) Triangular elements
1 7
65
4
3
7point
2
13point
2 12
1395
6
7
1
10
11
8
43
Figure 2.25: Integration point positions for 2D solid elements
uss e
sis of oneelement cases, see ref. KJB,
! For the 8node rectangular element, the use of 2×2 Gaintegration corresponds to a slight underintegration and onspurious zero energy mode is present. In practice, this particular kinematic mode usually does not present any problem in linear
alysis (except in the analyanFig. 5.40, p. 472) and thus 2×2 Gauss integration can be employed with caution for the 8node elements.
Chapter 2: Elements
24 ADINA Structures ⎯ Theory and Modeling Guide
The 3node, 6node and 7node triangular elements are spatially
ends upon lar
s ents use 7point Gauss
e 4×4 Gauss integration gration.
es s used.
esponse
lacement
used r
model
2.2.4 Mas
oint Gauss
atrix of an element is formed by dividing ss r
ref. KJBSection 6.8.4
! isotropic with respect to integration point locations and interpolation functions (see Figure 2.25). ! The integration order used for triangular elements depthe integration order used for rectangular elements as follows.
hen rectangular elements use 2×2 Gauss integration, trianguWelements use 4point Gauss integration; when rectangular elementuse 3×3 Gauss integration, triangular elemintegration; when rectangular elements usor higher, triangular elements use 13point Gauss inte
However, for 3node triangular elements, when the axisymmetric subtype is not used and when incompatible modare not used, then 1point Gauss integration is alway ! Note that in geometrically nonlinear analysis, the spatial positions of the Gauss integration points change continuously as the element undergoes deformations, but throughout the rthe same material particles are at the integration points.
! The order of numerical integration in the large dispelastic analysis is usually best chosen to be equal to the orderin linear elastic analysis. Hence, the program default values shouldusually be employed. However, in inelastic analysis, a higher ordeintegration should be used when the elements are used to thin structures (see Fig. 6.25, p. 638, ref. KJB), or large deformations (large strains) are anticipated.
s matrices
• The consistent mass matrix is always calculated using either 3×3 Gauss integration for rectangular elements or 7pintegration for triangular elements. The lumped mass m•
the elements mass M equally among its n nodes. Hence, the maassigned to each node is M / n. No special distributory conceptsare employed to distinguish between corner and midside nodes, oto account for element distortion.
2.2: Twodimensional solid elements
ADINA R & D, Inc. 25
n the
mong the
2.2.5 Elem
s.
he
e material e given
est ed in the material coordinate
ith indices YY, ZZ, tc are ystem (see Figure
• Note that n is the number of distinct nonrepeated nodes ielement. Hence, when a quad element side is collapsed to a single node, the total mass of the element is equally distributed afinal nodes of the triangular configuration.
ent output
You can request that ADINA either print or save stresses or force
Stresses: Each element outputs, at its integration points, tfollowing information to the porthole file, based on thmodel. This information is accessible in the AUI using thvariable names.
Notice that for the orthotropic material models, you can requhat the stresses and strains be savt
system. For the 3D plane stress elements, results w
saved in the element local coordinate se2.26).
Figure 2.26: Local coordinate system for 3D plane stress element
X
Y
Z
12
34
y
z
x
The local y directionis determined by localnodes 1 and 2.
The local z directionlies in the plane definedby local nodes 1, 2, 3,and is perpendicular to thelocal y direction.
asticisotropic, elasticorthotropic (no wrinkling, results saved in Elglobal system): STRESS(XYZ), STRAIN(XYZ) Elasticisotropic with thermal effects: STRESS(XYZ), STRAIN(XYZ), THERMAL_STRAIN, ELEMENT_TEMPERATURE
Chapter 2: Elements
26 ADINA Structures ⎯ Theory and Modeling Guide
m):
RAIN(XYZ), ATURE
,
oncrete:
inear:
LASTIC_STRAIN(XY
rozbilinear:
EFORMATION_GRADIENT(XYZ), E_EFFECTIVE_STRESS, YIELD_STRESS,
ults saved in global system):
RAIN, PERATURE
BC), _STRESS,
LEMENT_TEMPERATURE
Elasticorthotropic (no wrinkling, results saved in material systeSTRESS(ABC), STRAIN(ABC) Thermoisotropic, thermoorthotropic (results saved in global system): STRESS(XYZ), STTHERMAL_STRAIN(XYZ), ELEMENT_TEMPER Curve description: CRACK_FLAG, STRESS(XYZ), STRAIN(XYZ), FE_SIGMAP1, FE_SIGMAP2, FE_SIGMAP1_ANGLE, GRAVITY_INSITU_PRESSUREVOLUMETRIC_STRAIN C CRACK_FLAG, STRESS(XYZ), STRAIN(XYZ), FE_SIGMAP1, FE_SIGMAP2, FE_SIGMAP1_ANGLE, GRAVITY_INSITU_PRESSURE, ELEMENT_TEMPERATURE, THERMAL_STRAIN(XYZ) Small strains: Plasticbilinear, plasticmultilinear, MrozbilPLASTIC_FLAG, STRESS(XYZ), STRAIN(XYZ), P Z), FE_EFFECTIVE_STRESS, YIELD_STRESS, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE Large strains: Plasticbilinear, plasticmultilinear, MPLASTIC_FLAG, STRESS(XYZ), DFACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE
lasticorthotropic (resPPLASTIC_FLAG, STRESS(XYZ), STRAIN(XYZ), PLASTIC_STRAIN(XYZ), HILL_EFFECTIVE_STRESS, IELD_STRESS, Y
ACCUMULATED_EFFECTIVE_PLASTIC_STTHERMAL_STRAIN(XYZ), ELEMENT_TEM
lasticorthotropic (results saved in material system): PPLASTIC_FLAG, STRESS(ABC), STRAIN(APLASTIC_STRAIN(ABC), HILL_EFFECTIVEYIELD_STRESS, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, THERMAL_STRAIN(ABC), E
2.2: Twodimensional solid elements
ADINA R & D, Inc. 27
IN(XYZ), LOCATION,
lasticcreep, multilinearreepvariable, multilinear
le: PLASTIC_FLAG,
STCRELEMENT_TEMPERATURE, ACFEEFFECTIVE_CREEP_STRAIN
eepvariable: PLASTIC_FLAG, U (XYZ),
DEFE_EFFECTIVE_STRESS, YIELD_STRESS, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, EF
,
Small strains: DruckerPrager: PLASTIC_FLAG, PLASTIC_FLAG2, STRESS(XYZ), STRAPLASTIC_STRAIN(XYZ), CAP_IELD_FUNCTION Y
Large strains: DruckerPrager: PLASTIC_FLAG, PLASTIC_FLAG2, STRESS(XYZ), DEFORMATION_GRADIENT(XYZ), CAP_LOCATION, YIELD_FUNCTION Small strains: Thermoplastic, creep, p
asticcreep, creepvariable, plasticcplplasticcreepvariabNUMBER_OF_SUBINCREMENTS, STRESS(XYZ),
RAIN(XYZ), PLASTIC_STRAIN(XYZ), EEP_STRAIN(XYZ), THERMAL_STRAIN(XYZ),
CUMULATED_EFFECTIVE_PLASTIC_STRAIN, _EFFECTIVE_STRESS, YIELD_STRESS,
Large strains: Thermoplastic, creep, plasticcreep, multilinearplasticcreep, creepvariable, plasticcreepvariable, multilinear
lasticcrpN MBER_OF_SUBINCREMENTS, STRESS
FORMATION_GRADIENT(XYZ),
FECTIVE_CREEP_STRAIN
MooneyRivlin , Ogden, ArrudaBoyce, hyperfoam (strains saved): STRESS(XYZ), STRAIN(XYZ) MooneyRivlin, Ogden, ArrudaBoyce, hyperfoam (deformation gradients saved): STRESS(XYZ), DEFORMATION_GRADIENT(XYZ).
Small strains: Usersupplied: STRESS(XYZ), STRAIN(XYZ)USER_VARIABLE_I
Chapter 2: Elements
28 ADINA Structures ⎯ Theory and Modeling Guide
GRADIENT(XYZ), USER_VARIABLE_I
Elasticorthotropic (wrinkling, results saved in material system): WRINKLE_FLAG, STRESS(ABC), STRAIN(ABC) Thermoorthotropic (results saved in material system):
Sm
TRESS,
TIC_FLAG, _GRADIENT(XYZ),
FE_EFFECTIVE_STRESS, YIELD_STRESS,
(XYZ), ELEMENT_TEMPERATURE, VOID_VOLUME_FRACTION
(XYZ), TRAIN(XYZ), PLASTIC_STRAIN(XYZ),
YIELD_SURFACE_DIAMETER_P, YIELD_FUNCTION, MEAN_STRESS, DISTORTIONAL_STRESS, VOLUMETRIC_STRAIN, VOID_RATIO, EFFECTIVE_STRESS_RATIO, SPECIFIC_VOLUME La DEYIELD_SURFACE_DIAMETER_P, YIELD_FUNCTION,
EFFECTIVE_STRESS_RATIO, SPECIFIC_VOLUME
Large strains: Usersupplied: STRESS(XYZ), DEFORMATION_
Elasticorthotropic (wrinkling, results saved in global system): WRINKLE_FLAG, STRESS(XYZ), STRAIN(XYZ)
STRESS(ABC), STRAIN(ABC), THERMAL_STRAIN(ABC), ELEMENT_TEMPERATURE
all strains: Gurson plastic: PLASTIC_FLAG, STRESS(XYZ), STRAIN(XYZ), PLASTIC_STRAIN(XYZ), FE_EFFECTIVE_SYIELD_STRESS, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, VOID_VOLUME_FRACTION Large strains: Gurson plastic: PLASSTRESS(XYZ), DEFORMATION
ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, THERMAL_STRAIN
Small strains, Camclay: PLASTIC_FLAG, STRESSS
rge strains, Camclay: PLASTIC_FLAG, STRESS(XYZ), FORMATION_GRADIENT(XYZ),
MEAN_STRESS, DISTORTIONAL_STRESS,VOLUMETRIC_STRAIN, VOID_RATIO,
2.2: Twodimensional solid elements
ADINA R & D, Inc. 29
Mo  S(XYZ), STRAIN(XYZ), PLASTIC_STRAIN(XYZ), YI
T_TEMPERATURE
STRESS(XYZ), STRAIN(XYZ), PLASTIC_STRAIN(XYZ),
AIN, IRRADIATION_STRAIN(XYZ). Large strains: creepirradiation: STRESS(XYZ), DEFORMATION_GRADIENT(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, EFFECTIVE_STRESS, YIELD_STRESS, EFFECTIVE_CREEP_STRAIN, IRRADIATION_STRAIN(XYZ). In the above lists,
STRESS(XYZ) = STRESSYY, STRESSZZ, STRESSYZ, STRESSXX
STRESS(ABC) = STRESSAA, STRESSBB, STRESSAB, STRESSCC
with similar definitions for the other abbreviations used above. But the variable DEFORMATION_GRADIENT(XYZ) is interpreted as follows:
DEFORMATION_GRADIENTYY, DEFORMATION_GRADIENTZZ,
hr Coulomb: PLASTIC_FLAG, STRES
ELD_FUNCTION
Small strains, viscoelastic: STRESS(XYZ), STRAIN(XYZ),THERMAL_STRAIN(XYZ), ELEMEN
Large strains, viscoelastic: STRESS(XYZ), DEFORMATION_GRADIENT(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE Small strains: creepirradiation:
CREEP_STRAIN(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, EFFECTIVE_STRESS, YIELD_STRESS, EFFECTIVE_CREEP_STR
DEFORMATION_GRADIENT(XYZ) =
Chapter 2: Elements
30 ADINA Structures ⎯ Theory and Modeling Guide
_GRADIENTXX, DEFORMATION_GRADIENTYZ,
STRETCH(XYZ)DEFORMATION_GRADIENT(XYZ) in the above lists. See Section 12.1.1 for the definitions of those variables that are not selfexplanatory.
! In ADINA, the stresses are calculated using the strains at the point of interest. Hence, they are not spatially extrapolated or smoothed. However, the AUI can be employed to calculate smoothed stresses (see Section 12.1.3).
! You can also request that strain energy densities be output along with the stresses.
Nodal forces: The nodal forces which correspond to the element stresses can also be requested in ADINA. This force vector is calculated in a linear analysis using
where is the element strain displacement matrix, and is the
DEFORMATION
DEFORMATION_GRADIENTZY
Also note that you can request stretches instead of deformation gradients, and in this case replaces
T
V
dV= ∫F B τ
B τstress vector. The integration is performed over the volume of the element; see Example 5.11 in ref. KJB, pp. 358359.
The nodal forces are accessible in the AUI using the variable names NODAL_FORCEX, NODAL_FORCEY, NODAL_FORCEZ.
! The same relation is used for the element force calculation in a nonlinear analy
2.2.6 Reco
! The 9node element is usually most effective (except in explicit dynamic analysis).
ref. KJBSection 4.2.1,
6.6.3
sis, except that updated quantities are used in the integration.
mmendations on use of elements
2.2: Twodimensional solid elements
ADINA R & D, Inc. 31
! The 8node and 9node elements can be employed for analysis of thick structures and solids, such as (see Fig. 2.27):
< Thick cylinders (axisymmetric idealization)
< Dams (plane strain idealization)
< Membrane sheets (plane stress idealization)
< Turbine blades (generalized plane strain idealization)
and for thin structures, such as (see Fig. 2.28):
< Plates and shells (axisymmetric idealization)
< Beam )
iangular elements should
o ant.
t, consider the use of the incompatible modes .
! When the structure to be modeled is axisymmetric and has a dimension which is very small compared with the others, e.g., thin plates and shells, the use of axisymmetric shell elements is more effective (see Section 2.5).
< Long plates (plane strain idealization)
s (plane stres idealizations
! The 8 and 9node elements are usually most effective if the element is rectangular (nondistorted).
! The 4node rectangular and 3node trnly be used in analyses when bending effects are not signific
If the 4node rectangular elements must be used when bendingeffects are significanelement (see above)
Chapter 2: Elements
32 ADINA Structures ⎯ Theory and Modeling Guide
a) Thick cylinder (axisymmetric idealization)
Axis of revolution
CL
1 unit
b) Long dam (plane strain idealization)
c) Sheet in membrane action (plane stress idealization)
p
t
Figure 2.27: Use of 2D solid element forthick structures and solids
2.2: Twodimensional solid elements
ADINA R & D, Inc. 33
d) Turbine blade (generalized plane strain idealization)
YX
Np
Fp
Mpy
Mpz
Z
Figure 2.27: (continued)
1 radian
Circular plate
t
L
1 radian
Spherical shell
a) Axisymmetric idealizations
b) Plane strain idealization (long plate)
1 unit
P
L
t
c) Plane stress idealization (cantilever)
Figure 2.28: Use of 2D solid element for thin structures
p
Chapter 2: Elements
34 ADINA Structures ⎯ Theory and Modeling Guide
2.3 Threedimensional solid elements
2.3.1 Gene
onal (3D) solid element is a variable 4 to 20node or a 21 or 27node isoparametric element applicable to
n
ral considerations
! The threedimensi
general 3D analysis. Some typical 3D solid elements are showin Fig. 2.31.
(a) 20, 21 & 27node elements
(b) 8node elements
(c) 4, 10 and 11node tetrahedral elements
Figure 2.31: Some 3D solid elements
The 3D solid element should be employed
! in analyses in which the threedimensional state of stress (or strain) is required or in whSe
ich special stress/strain conditions, such as those given in ction 2.2, do not exist (see Figs. 2.32 and 2.33).
2.3: Threedimensional solid elements
ADINA R & D, Inc. 35
(d) 6 and 15 node prisms
(e) Elements for transition zones
Figure 2.31: (continued)
Chapter 2: Elements
36 ADINA Structures ⎯ Theory and Modeling Guide
1,2,3,4,9,10,1 1,12
19 18
20
1714
15
7
8
16
13
5
6
1,2,3,4,9,10,1 1,12
1918
20
1714
15
7
816
13
5
6
27
(f) 13node and 14node pyramid elements
Figure 2.31 (continued)
! The elements usually used are isoparametric displacementbafinite elements, and the formulation of the elements used in AD
sed
INA is described in ref. KJB, Ch. 5.
! The basic finite element assumptions are (see Fig. 2.34):
for the coordinates
q q q
1 1 1i i i i i i
i i ix h x y h y z= = = h z
= = =∑ ∑ ∑
ref. KJBSection 5.3
2.3: Threedimensional solid elements
ADINA R & D, Inc. 37
P
P
E,A
L
L
L
(a) Physical problem considered
(c) 3D model
A
P
B
B
A
A B
(b) 1D model
Figure 2.32: Illustration of use of 3D elements
Plane sections donot remain plane.
Plane sections remain
plane. is constantfor the entire section.
e i
point coordinates ui, vi, wi = nodal point displacements
and for the displacements
1 1 1v v
q q q
i i i i i ii i i
h u h w h w= = =
= = =∑ ∑ ∑
where
h
u
i (r, s, t) = interpolation function corresponding to nod r, s, t = isoparametric coordinates
q = number of element nodes, 8 # q # 27 xi, yi, zi = nodal
Chapter 2: Elements
38 ADINA Structures ⎯ Theory and Modeling Guide
Line loads
(uniform)
No axial displacement
should be possible
X
Y
Z
X
Y
Z
ZY
One quarter of
the structure
is modeled
2D model
a) Plane strain 2D analysis applicable
Fixed end
One quarter of the
structure is modeled
3D model
b) Threedimensional model is used
Figure 2.33: Examples of 3D solid modeling versus
2D solid modeling
! In addition to the displacementbased elements, special mixedinterpolated elements are also available, in which the dis
placementsd
mended in this case.
an pressure are interpolated. These elements are effective and should be preferred in the analysis of incompressible media and inelastic materials (specifically for materials in which Poisson's ratio is close to 0.5, for rubberlike materials and for elastoplastic materials). The use of the 8node (one pressure variable) element or 27node (4 pressure variables) element is recom
2.3: Threedimensional solid elements
ADINA R & D, Inc. 39
Axisymmetric model
c) Axisymmetric analysis applicable
d) Threedimensional model is used
Axisymmetric elements
1 radian
of structure
is modeled
Y
Z
One quarter
of structure
is modeled
Structure and loading are axisymmetric
3D elements
Structure axisymmetric,
loading nonaxisymmetric
Figure 2.33: (continued)
X
Y
Z
X
Y
Z
ition to the displacementbased and mixedinterpolated elements, ADINA also includes the possibility of including
t. The e flexibility of the
ement, especially ent is ent
for
! In add
incompatible modes in the formulation of the 8node elemenaddition of the incompatible modes increases th
in bending situations. This elemelanalogous to the 4node incompatible modes 2D solid elemdiscussed in Section 2.2.1, see the comments in that sectiontheoretical considerations.
The incompatible modes feature cannot be used in conjunction with the mixedinterpolation formulation.
Chapter 2: Elements
40 ADINA Structures ⎯ Theory and Modeling Guide
Z
X
Y
W4
U4
V44
8
t
s
rZ8
Y8
X8
Figure 2.34: Conventions used for the nodal coordinates
and displacements of the 3D solid element
6
13
516
15
714
17
91 18
2
103
20
11
1912
21
22
23
24
25
26
27
ompatible modes feature is de elements or the 4node
terpolate the coordinates functions, resulting in a
ctangular parallelepipeds in Fig. 2.35 have identical geometrical
x
see Example 5.16 on pp. # 20 are
The incompatible modes feature iselement; in particular note that the incnot available for the degenerated 8notetrahedral element.
! It may sometimes be effective to inusing a set of lower order interpolationsubparametric element. For example, the 20node and 8node
only available for the 8node
reinterpolation, i.e.,
20 8* * * * * * ' * * *
1 1( , , ) ( , , ) ( , , )i i
p i ii i
r s t h r s t h r s t= =
= =∑ ∑x x
! The elements can be used with 4 to 20 or with 21 or 27 nodes (tetrahedra, pyramids or prisms are derived from the degeneration of the 4 to 20node rectangular elements,366367, ref. KJB). The interpolation functions for qshown in Fig. 5.5, ref. KJB, p. 345.
2.3: Threedimensional solid elements
ADINA R & D, Inc. 41
a) 20node element
b) 8node element
t
t
r
r
s
s
(r*,s*,t*)
(r*,s*,t*)
All midside nodes are
equidistant from corner
nodes on same side.
Z
YX
Z
YX
Figure 2.35: Same geometry interpolation
using 8 and 20 nodes
t (see Fig.
larity
! Degenerated elements such as prisms, pyramids or tetrahedraare formed in ADINA by assigning the same global node to the local element nodes located along the same side or on the same face.
The triangular prism as a degenerated 20node elemen2.31(d)) can be used in ADINA in three different ways:
< As a 15node triangular prism with the strain singu1r
. This is an extension of the techniques described on pp.
larity
368376 of ref. KJB (see also Chapter 9);
< As a 20node triangular prism with the strain singu 1r
(see also Chapter 9);
Chapter 2: Elements
42 ADINA Structures ⎯ Theory and Modeling Guide
< As a spatially isotropic triangular prism where corrections are applied to the interpolation functions of the collapsed 20node element.
The 10node tetrahedron (see Fig. 2.31(c)) is obtained by
collapsing nodes and sides of rectangular elements. A spatially isotropic 10node tetrahedron is available in ADINA. Similarly, an 11node spatially isotropic tetrahedron is available in ADINA.
The 13node pyramid and the 14node pyramid (see Fig. 2.31(f)) are degenerate solid elements from 20node and 21node bricks, respectively. They can be used as spatially isotropic elements.
The 4node tetrahedron (see Fig. 2.31(c)) is obtained by collapsing nodes and sides of the 8node rectangular element. This element exhibits constant strain conditions.
The elements in Fig. 2.31(e) are not spatially isotropic and should usually only be employed in transition regions.
! Linear and nonlinear fracture mechanics analysis of stationary cracks can be performed with threedimensional elements (see Chapter 9).
2.3.2 Material models and nonlinear formulations
! The 3D elements can be used with the following material models: elasticisotropic, elasticorthotropic, plasticbilinear, plasticmultilinear, DruckerPrager, Mrozbilinear, plasticorthotropic, thermoisotropic, thermoorthotropic, thermoplastic, creep, plasticcreep, multilinearplasticcreep, concrete, curvedescription, Ogden, MooneyRivlin, ArrudaBoyce, hyperfoam, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable, Gursonplastic, Camclay, MohrCoulomb, viscoelastic, irradiation creep, gasket, shapememory alloy, usersupplied.
! You should use the mixed interpolation formulation whenever using the plasticbilinear, plasticmultilinear, Mrozbilinear, plasticorthotropic, thermoplastic, creep, plasticcreep, multilinearplasticcreep, Ogden, MooneyRivlin, ArrudaBoyce, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable or viscoelastic material models; or when using the elasticisotropic
2.3: Threedimensional solid elements
ADINA R & D, Inc. 43
material and the Poisson=s ratio is close to 0.5. The AUI automatically chooses the mixed interpolation
formulation whenever the Ogden, MooneyRivlin or ArrudaBoyce material models are used. For the other material types listed in the previous paragraph, you should explicitly select the mixed interpolation formulation.
! The 3D elements can be used with a small displacement/small strain, large displacement/small strain or a large displacement/ large strain formulation.
The small displacement/small strain and large displacement/small strain formulations can be used with any material model, except for the Ogden, MooneyRivlin, ArrudaBoyce and hyperfoam material models. The use of a linear material with the small displacement/ small strain formulation corresponds to a linear formulation, and the use of a nonlinear material with the small displacement/small strain formulation corresponds to a materiallynonlinearonly formulation. The program uses the TL (total Lagrangian) formulation when you choose a large displacement/small strain formulation.
The large displacement/large strain formulations can be used with the plasticbilinear, plasticmultilinear, Mrozbilinear, thermoplastic, creep, plasticcreep, multilinearplasticcreep, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable, viscoelastic and usersupplied material models. The program uses a ULH formulation when you choose a large displacement/large strain formulation with these material models.
A large displacement/large strain formulation is used with the Ogden, MooneyRivlin, ArrudaBoyce and hyperfoam material models. The program uses a TL (total Lagrangian) formulation in this case.
! The basic continuum mechanics formulations are described in ref. KJB, pp. 497568. The finite element discretization is summarized in Table 6.6, p. 555, ref. KJB.
! Note that all these formulations can be used in one finite element mesh. If the elements are initially compatible, then they will remain compatible throughout the analysis.
ref. KJBSections 6.2
and 6.3.5
ref. KJBSection 6.8.1
Chapter 2: Elements
44 ADINA Structures ⎯ Theory and Modeling Guide
2.3.3 Num
e c erical tegration is used. The same integration order (NINT) is always
assigned to the r and sdirections, and can be from 2 to 6. The integration order in the tdirection (NINTT) can, however, be ass rom 2 to 6. The default Ga e 8node (cube or prism) ele tetrahedra.
! node tetrahedra, the conoutput is as follows: The first integration point is the point with the mo ext integration points are
um .36).
erical integration
! For th alculation of element matrices, Gauss num
ref. KJBSections 5.5.3,
5.5.4 and 5.5.5in
igned independently, and can also be fuss integration orders are 2×2×2 for thments and 3×3×3 otherwise, except for
Except for the 4node, 10node and 11vention for the integration point numbering used in the stress
negative location in r, s and t. The nstlocated by increasing t (and label INT) successively up to its maximum positive value, then increasing s (and label INS) one position towards positive and varying t again from its maxim
egative to its maximum positive values, and so on (Fig. 2n
a) All elements except tetrahedra
t
s
r
INR=1,INS=1,INT=1
INR=2,INS=1,INT=1
INR=1,INS=1,INT=3
Integration order in rs plane=2
Integration order in tdirection=3
Figure 2.36: Example of integration point labeling
for 3D solid elements
2.3: Threedimensional solid elements
ADINA R & D, Inc. 45
! s (see Figure
e 10node and 11node tetrahedral
ration order used for tetrahedral elements depends upon the integration order used for hexahedral (brick) elements as
program computes the total number of integration points INTRST=NINT*NINT*NINTT. If
, then
tetrahedral elements use 5 point Gauss integration; otherwise
Tetrahedral elements are spatially isotropic with respect to integration point locations and interpolation function2.36). For the 4node tetrahedral element, 1point Gauss integration is used. For thelements, 17point Gauss integration is used. ! The integ
follows. Given NINT, NINTT, the
INTRST is equal to 1, then tetrahedral elements use 1 point Gaussintegration; otherwise, if INTRST is less than or equal to 8
tetrahedral elements use 17 point Gauss integration. However, for 4node tetrahedral elements, 1point Gauss
integration is always used.
Chapter 2: Elements
46 ADINA Structures ⎯ Theory and Modeling Guide
b) Tetrahedral elements
Figure 2.36: (continued)
17point (unfolded) (Point 1 at centroid not shown)
8
135
5
1014
22
7
4
4
15
4
3
1
5
4
3
t
r
s1
1point
t
r
s2
5point
1716
5
12
s
6
3 2
t
9
113
t
r
Chapter 2: Elements
46 ADINA Structures ⎯ Theory and Modeling Guide
b) Tetrahedral elements
Figure 2.36: (continued)
17point (unfolded) (Point 1 at centroid not shown)
8
135
5
1014
22
7
4
4
15
4
3
1
5
4
3
t
r
s1
1point
t
r
s2
5point
1716
5
12
s
6
3 2
t
9
113
t
r
! Note that in geometrically nonlinear analysis, the spatial positions of the Gauss integration points change continuously as the element undergoes deformations, but throughout the responsethe same material particles are at the integration points.
! The order of numerical
integration in large displacement elastic
analysis is usually best chosen to be equal to the order appropriate in linear elastic analysis. Hence the default values are usually appropriate. However, in certain inelastic analyses, a higher order should be used (see Fig. 6.25, p. 638, ref. KJB).
ref. KJBSection 6.8.4
2.3: Threedimensional solid elements
ADINA R & D, Inc. 47
2.3.4 Mass matrices
• The consistent mass matrix is always calculated using 3×3×3 Gauss integration except for the tetrahedral 4node, 10node and 11node elements which use a 17point Gauss integration. • The lumped mass matrix of an element is formed by dividing the elements mass M equally among each of its n nodal points. Hence the mass assigned to each node is M / n. No special distributory concepts are employed to distinguish between corner and midside nodes, or to account for element distortion.
• Note that n is the number of distinct nonrepeated nodes in the element. Hence, when an element side or face is collapsed to a single node, the total mass of the element is divided among the unique nodes in the element.
2.3.5 Element output
! You can request that ADINA either print or save stresses or forces.
Stresses: Each element outputs, at its integration points, the following information to the porthole file, based on the material model. This information is accessible in the AUI using the given variable names.
Notice that for the orthotropic material models, you can request that the stresses and strains be saved in the material coordinate
s
Elasticisotropic, elasticorthotropic (results saved in global s
in material system): STRESS(ABC), STRAIN(ABC)
sy tem.
sy tem): STRESS(XYZ), STRAIN(XYZ) Elasticisotropic with thermal effects: STRESS(XYZ), STRAIN(XYZ), THERMAL_STRAIN, ELEMENT_TEMPERATURE Orthotropic (results saved
Chapter 2: Elements
48 ADINA Structures ⎯ Theory and Modeling Guide
Thermoisotropic, thermoorthotropic (results saved in global system): STRESS(XYZ), STRAIN(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE Curve description: CRACK_FLAG, STRESS(XYZ), STRAIN(XYZ), FE_SIGMAP1, FE_SIGMAP2, FE_SIGMAP3, GRAVITY_INSITU_PRESSURE, VOLUMETRIC_STRAIN, FE_SIGMAP1_DIRECTIONX, FE_SIGMAP1_DIRECTIONY, FE_SIGMAP1_DIRECTIONZ, FE_SIGMAP2_DIRECTIONX, FE_SIGMAP2_DIRECTIONY, FE_SIGMAP2_DIRECTIONZ, FE_SIGMAP3_DIRECTIONX, FE_SIGMAP3_DIRECTIONY, FE_SIGMAP3_DIRECTIONZ
FEP3_DIRECTIONY,
FE_SIGMAP3_DIRECTIONZ, THERMAL_STRAIN(XYZ) Small strains: Plasticbilinear, plasticmultilinear, Mrozbilinear: PLASTIC_FLAG, STRESS(XYZ), STRAIN(XYZ),
, YIELD_STRESS, ACTH LaPLDEFE_EFFECTIVE_STRESS, YIELD_STRESS, ACTH
Concrete: CRACK_FLAG, STRESS(XYZ), STRAIN(XYZ), FE_SIGMAP1, FE_SIGMAP2, FE_SIGMAP3, GRAVITY_INSITU_PRESSURE, ELEMENT_TEMPERATURE,FE_SIGMAP1_DIRECTIONX, FE_SIGMAP1_DIRECTIONY, FE_SIGMAP1_DIRECTIONZ, FE_SIGMAP2_DIRECTIONX, FE_SIGMAP2_DIRECTIONY, FE_SIGMAP2_DIRECTIONZ,
_SIGMAP3_DIRECTIONX, FE_SIGMA
PLASTIC_STRAIN(XYZ), FE_EFFECTIVE_STRESS
CUMULATED_EFFECTIVE_PLASTIC_STRAIN, ERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE
rge strains: Plasticbilinear, plasticmultilinear, Mrozbilinear: ASTIC_FLAG, STRESS(XYZ), FORMATION_GRADIENT(XYZ),
CUMULATED_EFFECTIVE_PLASTIC_STRAIN, ERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE
2.3: Threedimensional solid elements
ADINA R & D, Inc. 49
Plasticorthotropic (results saved in global system): PLPLASTIC_STRAIN(XYZ), HILL_EFFECTIVE_STRESS, YIELD_STRESS,
FFECTIVE_PLASTIC_STRAIN,
LASTIC_FLAG, STRESS(ABC), STRAIN(ABC),
STRAIN(ABC), ELEMENT_TEMPERATURE SmPLASTIC_FLAG2, STRESS(XYZ), STRAIN(XYZ),
STIC_FLAG,
C S(XYZ), Z), CAP_LOCATION,
plasticcreep, multilinear
NUMBER_OF_SUBINCREMENTS, STRESS(XYZ),
FECTIVE_PLASTIC_STRAIN, TRESS, YIELD_STRESS,
reep, creepvariable, plasticcreepvariable, multilinear
la
ASTIC_FLAG, STRESS(XYZ), STRAIN(XYZ),
ACCUMULATED_ETHERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE Plasticorthotropic (results saved in material system): PPLASTIC_STRAIN(ABC), HILL_EFFECTIVE_STRESS, YIELD_STRESS, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, THERMAL_
all strains: DruckerPrager: PLASTIC_FLAG,
PLASTIC_STRAIN(XYZ), CAP_LOCATION, YIELD_FUNCTION
Large strains: DruckerPrager: PLAPLASTI _FLAG2, STRESDEFORMATION_GRADIENT(XYYIELD_FUNCTION Small strains: Thermoplastic, creep,plasticcree creepvariable, plasticp, creepvariable, multilinearplasticcreepvariable: PLASTIC_FLAG,
STRAIN(XYZ), PLASTIC_STRAIN(XYZ), CREEP_STRAIN(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, ACCUMULATED_EFE_EFFECTIVE_SF
EFFECTIVE_CREEP_STRAIN Large strains: Thermoplastic, creep, plasticcreep, multilinear
lasticcpp sticcreepvariable: PLASTIC_FLAG, NUMBER_OF_SUBINCREMENTS, STRESS(XYZ), DEFORMATION_GRADIENT(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN,
Chapter 2: Elements
50 ADINA Structures ⎯ Theory and Modeling Guide
yce, hyperfoam (strains STRESS(XYZ), STRAIN(XYZ)
STRESS(XYZ), DEFORMATION_GRADIENT(XYZ) Small strains: Usersupplied: STRESS(XYZ), STRAIN(XYZ), USER_VARIABLE_I
RAIN(XYZ), ELEMENT_TEMPERATURE, OID_VOLUME_FRACTION
, RESS(XYZ , DEFORMATION_GRADIENT(XYZ),
E_STRE_EFFEC
THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, VOID_VOLUME_FRACTION Small strains: Camclay: PLASTIC_FLAG, STRESS(XYZ), STRA YZ), PLASTIC_STRAIN(XYZ),YIELD_SURFACE_DIAMETER_P, YIELD_FUNCTION, MEAN_STRESS, DISTORTIONAL_STRESS, VOLUMETRIC_STRAIN, VOID_RATIO, EFFECTIVE_STRESS_RATIO, SPECIFIC_VOLUME
FE_EFFECTIVE_STRESS, YIELD_STRESS, EFFECTIVE_CREEP_STRAIN MooneyRivlin , Ogden, ArrudaBoaved): s MooneyRivlin , Ogden, ArrudaBoyce, hyperfoam (deformation
radients saved): g
Large strains: Usersupplied: STRESS(XYZ), DEFORMATION_GRADIENT(XYZ), USER_VARIABLE_I
hermoorthotropic (results saved in material system): TSTRESS(ABC), STRAIN(ABC), THERMAL_STRAIN(ABC), FE_EFFECTIVE_STRESS, ELEMENT_TEMPERATURE Small strains: Gurson plastic: PLASTIC_FLAG, STRESS(XYZ), STRAIN(XYZ), PLASTIC_STRAIN(XYZ), FE_EFFECTIVE_STRESS, YIELD_STRESS, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, THERMAL_STV Large strains: Gurson plastic: PLASTIC_FLAGSTF
)VE_EFFECTI
ACCUMULATESS, YIELD_STRESS, TIVE_PLASTIC_STRAIN, D
IN(X
2.3: Threedimensional solid elements
ADINA R & D, Inc. 51
Large strains: Camclay: PLASTIC_FLAG, STRESS(XYZ), DEFORMATION_GRADIENT(XYZ), YIELD_SURFACE_DIAMETER_P, YIELD_FUNCTION, MEAN_STRESS, DISTORTIONAL_STRESS, VOLUMETRIC_STRAIN, VOID_RATIO, EFFECTIVE_STRESS_RATIO, SPECIFIC_VOLUME MohrCoulomb: PLASTIC_FLAG, STRESS(XYZ), STRAIN(XYZ), PLASTIC_STRAIN(XYZ), YIELD_FUNCTION
all strains: viscoelastic: STRESS(XYZ), STRAIN(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE Large strains: viscoelastic: STRESS(XYZ), DEFORMATION_GRADIENT(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE Small strains: creepirradiation: STRESS(XYZ), STRAIN(XYZ), PLASTIC_STRAIN(XYZ), CREEP_STRAIN(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, ACCUMULATED_EFFECTIVE_PLASTIC_STRAIN, EFFECTIVE_STRESS, YIELD_STRESS, EFFECTIVE_CREEP_STRAIN, IRRADIATION_STRAIN(XYZ). Large strains: creepirradiation: STRESS(XYZ), DEFORMATION_GRADIENT(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE, ACCUEFFECTIVE_STRESS, YIELD_STRESS,
STRESS(ABC) = STRESSAA, STRESSBB, STRESSCC, STRESSAB, STRESSAC, STRESSBC
Sm
MULATED_EFFECTIVE_PLASTIC_STRAIN,
EFFECTIVE_CREEP_STRAIN, IRRADIATION_STRAIN(XYZ).
In the above lists, STRESS(XYZ) = STRESSXX, STRESSYY, STRESSZZ, STRESSXY, STRESSXZ, STRESSYZ
Chapter 2: Elements
52 ADINA Structures ⎯ Theory and Modeling Guide
Chapter 2: Elements
52 ADINA Structures ⎯ Theory and Modeling Guide
with similar definitions for the other abbreviations used above. But the variable DEFORMATION_GRADIENT(XYZ) is interpreted as follows:
DEFORMATION_GRADIENT(XYZ) = DEFORMATION_GRADIENTXX, DEFORMATION_GRADIENTXY, DEFORMATION_GRADIENTXZ, DEFORMATION_GRADIENTYX,
TION_GRADIENTZX, DEFORMATION_GRADIENTZY, DEFORMATION_GRADIENTZZ,
Also note that you can request stretches instead of deformation gradients, and in this case STRETCH(XYZ) replaces DEFORMATION_GRADIENT(XYZ) in the above lists.
See Section 12.1.1 for the definitions of those variables that are not selfexplanatory.
! In ADINA, the stresses are calculated using the strains at the point of interest. Hence, they are not spatially extrapolated or smoothed. The AUI can be employed to calculate smoothed stresses from the results output by ADINA.
ies be output along ith the stresses.
ond to the element
tresses can also be requested in ADINA. This force vector is calculated in a linear analysis using
indisplacement matrix and is the
tress vector. The integration is performed over the volume of the
ref. KJBSection 4.2.1
DEFORMATION_GRADIENTYY, DEFORMATION_GRADIENTYZ, DEFORMA
! You can also request that strain energy densitw
Nodal forces: The nodal forces which corresps
T
V
where B is the element stra
dV= ∫F B τ
τselement; see Example 5.11 in ref. KJB, pp. 358359.
2.3: Threedimensional solid elements
ADINA R & D, Inc. 53
in the AUI using the variable ames NODAL_FORCEX, NODAL_FORCEY,
e calculation in a
2.3.6 Reco
lement can be costly.
ost effective (except in wave propagation analy
s; some examples are given in Fig. 2.37.
0node element for contact
a
is ressible analysis).
The nodal forces are accessible nNODAL_FORCEZ.
! The same relation is used for the element forcnonlinear analysis, except that updated quantities are used in the integration.
mmendations on use of elements
! The 27node element is the most accurate among all available elements. However, the use of this e
! The 20node element is usually the m
sis using the central difference method and a lumped mass idealization, see Chapter 7).
! The 20node element can be employed for analysis of thick structures and solid ! The 20node element is usually most effective if the element is rectangular (undistorted).
It is not recommended to use the 2!
an lysis because, in this case, the corresponding segment tractions will be very approximate. ! The 11node tetrahedral element should be used when tetrahedral meshing is used and a mixed interpolated formulationrequired (as in incomp
Chapter 2: Elements
54 ADINA Structures ⎯ Theory and Modeling Guide
Dam
Vessel
Tunnel
Figure 2.37: Some structures for which 3D solidelements can be used
! The 8node brick element and 4node tetrahedral element ot
bending effects are significant, consider the use of the incompatible m des element.
! When the structure to be modeled has a dimension which is
shell elements (see Sections 2.7 and
2.4 Beam
The beam element is a 2node Hermitian beam with a constant rosssection and 6 degrees of freedom at each node. The isplacements modeled by the beam element are (see Fig. 2.41):
< Cubic transverse displacements
should only be used in analyses when bending effects are nsignificant. If the 8node brick element must be used when
o
extremely small compared with the others, e.g., thin plates and shells, the use of the 3D solid element usually results in too stiff a model and a poor conditioning of the stiffness matrix. In this case, the use of the shell or the plate/
.6) is more effective. 2
elements
!
cd
v and w (s and tdirection displacements)
2.4: Beam elements
ADINA R & D, Inc. 55
< Linear longitudinal displacements u (rdirection
< Linear torsional displacements
displacement)
rθ and warping displacements
! The element is formulated based on the BernoulliEuler beam theory, corrected for shear deformation effects if requested.
! The behavior of the beam can be described using either a crosssection shape and a material model, or a momentcurvature input.
! The principal planes of inertia of the element coincide with the element local coordinate system (r,s,t). The (r,s,t) coordinate system is defined using the element end nodes (local nodes 1 and 2) and the auxiliary node K (see Figure 2.41). You must position node K appropriately with respect to the actual orientation of the principal planes of inertia of the element, especially in the case of composite crosssections.
Z
YX
K
1
2 w_
v_
u_
_ s
r
_ t
The element sdirection is defined as follows.
The sdirection lies in the plane defined by
nodes 1, 2, K, and the sdirection is perpendicular
to the rdirection.
(a) Geometry definition
Figure 2.41: Conventions used for 2node
Hermitian beam element
s
r
t
Chapter 2: Elements
56 ADINA Structures ⎯ Theory and Modeling Guide
Z
YX
S4 S1
1
S5
S2
S3S6
s
r
t
S11
S8
2
S7
S10
S12
S9
Neutral axis
(b) Element end forces/moments
Figure 2.41: Conventions used for 2node
Hermitian beam element
S = rdirection force at node 1 (axial force, positive in compression)1
S = rdirection force at node 2 (axial force, positive in tension)7
S = sdirection force at node 1 (shear force)2
S = sdirection force at node 2 (shear force)8
S = tdirection force at node 1 (shear force)3
S = tdirection force at node 2 (shear force)9
S = rdirection moment at node 1 (torsion)4
S = rdirection moment at node 2 (torsion)10
S = tdirection moment at node 1 (bending moment)6
S = tdirection moment at node 2 (bending moment)12
S = sdirection moment at node 1 (bending moment)5
S = sdirection moment at node 2 (bending moment)11
! ou can create offcentered beam elements using the rigid link (se
! moment and force release
tion can be used (see Fig. 2.43). Twisting moments and axial t
own in Fig. 2.43(a), a moment end release can be applied to local node 2 of element 1, or to local node 1 of element 2 (but no th
Ye Fig. 2.42 and Section 5.112).
To model beam internal hinges, aopforces can also be released. End releases are applied to the elemenlocal nodes (not to the global nodes). Therefore, to model the hinge sh
t to bo local nodes).
2.4: Beam elements
ADINA R & D, Inc. 57
IbeamIbeam
Rigid panel
Hollow square section
Beam element
Rigid links
Beam elements
Figure 2.42: Use of rigid link for modeling of offcentered beams
Physical problem: Finite element model:
Internal hinge
Element 1 Element 2
(a) Moment to be released at
internal hinge
Shear force
released
Element 1 Element 2
(b) Shear force to be released
Figure 2.43: Use of moment and shear force release options
r r rr
Specify moment release atelement 1, local node 2, orelement 2, local node 1
Specify shear force release atelement 1, local node 2, orelement 2, local node 1
Rigidend capability: For the modeling of assemblages of braces and struts, it is sometimes of interest that the joints of the structure be considered as infinitely rigid (see Figure 2.44). Hence, a rigidend capability exists for the beam element: with this capability, a fraction at each end of the element can be considered as rigid.
You can select either perfectly rigid ends (with infinite rigidity) or very stiff ends (with rigidity calculated by multiplying the rigidity of the element by a large number, such as 106).
e model.
The lengths of the beam rigid ends are specified in the units ofth
Chapter 2: Elements
58 ADINA Structures ⎯ Theory and Modeling Guide
Rigid end
Beam element
Finite element model:
Figure 2.44: Modeling of a beam intersection usingelements with rigid ends
Physical problem:
! The assumptions of the rigidend option are as follows:
< The elem cluding ts rig rm ated as one ent, in i id ends, is fo ulsingle element. Hence, as for all be ements, the internal deformations and element internal forces are all referred to a
of
< . Hence, gid ends may be very high (they are
not printed by the program), plasticity is not reached. This may be quite unrealistic in a model using the rigidend capability.
2.4.1 Line
ns are
! The beam element stiffness matrix is evaluated in closed form. The stiffness matrix used is discussed in detail in the following reference:
Ana rawHill Book Co., 1968.
am el
line connecting the endnodes, and using the assumptionssmall internal relative deformations. The rigid ends will never undergo plasticity
although the stress in the ri
< The mass of the rigid ends is taken into account in the mass matrix calculations.
ar beam element
! It is assumed that the displacements, rotations, and straiinfinitesimally small, and the elasticisotropic material is used.
ref. J.S. Przemieniecki, Theory of Matrix Structural lysis, McG
2.4: Beam elements
ADINA R & D, Inc. 59
Note that the element stiffness matrix is defined by the
following quantities:
E = Young's modulusν = Poisson's ratio
L = length of the beam , ,r s tI I I = moments of inertia about the local principal axes r,
s, and t A = crosssectional area
,
sh shs tA A = effective shear areas in s and t directions
Th
e
complete structure. ! You can enter either the section areas and moments of inertia, or the dimensions for various sections (see Fig. 2.45 and Table 2.41). ! The x' and y' axes are the principal axes of the crosssection, and have their origin at the centroid of the crosssection. Note that the x' and y' axes are oriented relative to the element local s and t coordinate axes according to Fig. 2.46, and that the element s and t coordinate axes directions are controlled by the location of the auxiliary node K (Fig. 2.41). The x' and y' axes can be offset from the element s and t coordinate axes using parameters SOFFSET, TOFFSET. ! When offsets are used they affect the moments of inertia Is and It of the crosssection as follows
Note that axial forces applied at Nodes 1 or 2 are assumed to be cting along th
shear forces applied at Nodes 1 or 2 are assumed to be acting through the beam
is stiffness matrix is transformed from the local degrees of freedom (in the r, s, t axes) to the global degrees of freedom (in thX, Y, Z axes) and is then assembled into the stiffness matrix of the
2' ' SOFFSETs x xI I A= + ×
2' ' TOFFSETt y yI I A= + ×
a e beams centroid and hence cause no bending. Also
s shear center and hence cause no twisting.
Chapter 2: Elements
60 ADINA Structures ⎯ Theory and Modeling Guide
n
d beam elements such as shown in Figure 2.42 without using rigid links
torsional moment of inertia in various cross sections (see Figure 2.45) Crosssection
Torsional moment of inertia Ir
In order to model the bending due to an offcentroidal axial force or a shear force applied away from the shear center, you caeither apply the resulting moments directly or apply the forces at anoffset location using rigid links (section 5.152). Beam offsets can be used to model multiple offcentere
(see Figure 2.47). In this case, there is only one node at the crosssection which should be located at the centroid of the combined crosssection. Table 2.41: Formulas used for
Rectangular
4
34
1 H1 H11 0.63 1 H1 H2, H1 H23 H2 12H2
⎛ ⎞⎛ ⎞− − ≤⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
Pipe 4 4(H1 H2 )32
− π
Box
2 2
2 2
2 H3 H4 (H1  H3) (H2  H4)H1 H3 + H2 H4  H3  H4
I
( 31 H4 H )3 31 + H6 H3 + H5 (H2  H4  H6)3
U
( )3 31 2H4 H1 + H3 (H2  2H4)) 3
L
( )3 31 H3 H2 + H4 (H1  H3)3
2.4: Beam elements
ADINA R & D, Inc. 61
y
xHEIGHT
(H2)
THICK 1
(H3)
THICK 2
(H4)
WIDTH
(H1)
x
y
HEIGHT
(H2)THICK 1
(H3)
THICK 2
(H4)
WIDTH
(H1)
Figure 2.45: Possible crosssection shapes for the linear elastic
2node Hermitian beam element
xHEIGHT
(H2)THICK 2
(H4)
THICK 1
(H3)
WIDTH
(H1)
H5=H2H4
y
C.G.C.G.
y
xC.G.HEIGHT
(H2)
WIDTH
(H1)
y
x
H2
DIAMETER
(H1)
C.G.
THICKNESS
a) Rectangular section b) Pipe section
c) Box section
xHEIGHT
(H2)
THICK 2
(H5)
THICK 3
(H6)
THICK 1
(H4)
WIDTH 2
(H3)
WIDTH
(H1)
y
d) I section
e) U section f) L section
Chapter 2: Elements
62 ADINA Structures ⎯ Theory and Modeling Guide
Figure 2.46: Relationship between principal crosssection axes
and element local coordinate system
C.G.C.G.
y
xC.G.
s
t
SOFFSET
TOFFSET
Auxiliarynode K
Node 1(or 2)
Beam #1
Offset #1 Offset #2
Offset #3
Beam #3
Beam #2
One node at centroid Figure 2.47: Using offsets to model offcentered beams
Note that war
! ping effects are only taken into account by a modification of the torsional rigidity, this modification to be spe
! Iben ed as follows:
cified in the input.
n the case of the L crosssection, the maximum and minimum ding moments of inertia (about the x' and y' axes) are obtain
22
2 2xx yy xx yy
x x xy
I I I II I′ ′
+ −⎛ ⎞= + +⎜ ⎟
⎝ ⎠
2
2
2 2xx yy xx yy
y y xy
I I I II I′ ′
+ −⎛ ⎞= − +⎜ ⎟
⎝ ⎠
2.4: Beam elements
ADINA R & D, Inc. 63
where the moments and products of inertia with respect to the axes x and y are:
22 335 3 34 1 1
1 4 5 312 2 12 2xx x xH H HH H HI H H C H H C⎛ ⎞⎛ ⎞= + − + + −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
22 333 5 51 4 4
1 4 5 3 412 2 12 2yy y yH H HH H HI H H C H H H C⎛ ⎞⎛ ⎞= + − + + + −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
3 55 3 42 2xy x y
H HI H H C H C⎛ ⎞⎛ ⎞= − + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
4 11 4 2 2y x
H HH H C C⎛ ⎞⎛ ⎞+ − + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
and where (Cx, Cy) are the centroid coordinates. In this case, the angles between the principal axes x and y and the horizontal (according to Figure 2.45) are
' ' ' '1' ' 2
' '
, arctan( )2
y y x xx y
x y
I II
θ θ−
=
These two angles are always 90o apart. For this Lshaped geometry, the angle defining the major principal axis θx will always be the angle between 0 o and 90o. ! You can specify the coefficient of thermal expansion as a material property. The coefficient of thermal expansion is constant (independent of the temperature). ! For the linear beam element are calculated either
< element end forces/moments or < section forces/moments at discrete locations along the
axial direction of the element.
The element end forces/moments are accessible in the AUI using the variable names NODAL_FORCER, NODAL_FORCES,
Chapter 2: Elements
64 ADINA Structures ⎯ Theory and Modeling Guide
NODAL_FORCET, NODAL_MOMENTR, NODAL_MOMENTS and NODAL_MOMENTT. The end forces/moments are computed at the element local nodes. In the AUI, element local nodes are defined as element points of type label. For example, to access the result computed at element 5, local node 2, define an element point of type label with element number 5, label number 2.
The section forces/moments along the element are accessible in
the AUI using the variable names AXIAL_FORCE, BENDING_MOMENTS, BENDING_MOMENTT, SHEAR_FORCES, SHEAR_FORCET and TORSIONAL_MOMENT. In the AUI, the section forces/moments along the axial direction of the element are reported starting at local node 1.
2.4.2 Large displacement elastic beam element
! The element is used with the elasticisotropic material model. ! It is assumed that large displacements/rotations can occur, but only small strains. ! The large displacement beam element assuming elastic conditions is used in the same way as the linear elastic beam element (see Section 2.4.1). The stiffness matrix and internal force vector are evaluated in closed form. ! The large displacements and rotations are taken into account through a corotational framework, in which the element rigid body motion (translations and rotations) is separated from the deformational part of the motion. For more information, see the following reference:
ref: B. NourOmid and C.C. Rankin, Finite rotation analysis and consistent linearization using projectors, Comput. Meth. Appl. Mech. Engng. (93) 353384, 1991.
! As for the linear elastic beam elements, the section flexural, shear and torsional properties can be directly input or the dimensions of the crosssection can be defined.
2.4: Beam elements
ADINA R & D, Inc. 65
! This beam element is primarily suited for elastic stability analysis, and in this case the option of a general beam section is important. ! Note that warping effects are only included very approximately by a modification of the torsional rigidity, as input. ! The calculated quantities for this beam element are only the element end forces/moments, and the output is as for the linear elastic beam element.
2.4.3 Nonlinear elastoplastic beam element
! The element is used with the plasticbilinear material model with isotropic hardening. ! The element can be used either with the small displacement formulation (in which case the formulation is materiallynonlinearonly) or with the large displacement formulation (in which case the displacements/rotations can be large and a UL formulation is employed). In all cases, the strains are assumed to be small.
! The nonlinear elastoplastic beam element can only be employed for circular and rectangular crosssections, see Fig. 2.48.
DI
DO
t
s
t
s
DI
DO
(WIDTH)
(HEIGHT)
Figure 2.48: Possible crosssections for nonlinear analysis
! The element can either be used using a 2D action option (in which case the element must be defined in the global XY, XZ or YZ planes) or a 3D action option (in which case the element can have an arbitrary orientation).
Chapter 2: Elements
66 ADINA Structures ⎯ Theory and Modeling Guide
! The beam element matrices are formulated using the Hermitian displacement functions, which give the displacement interpolation matrix summarized in Table 2.42.
Table 2.42: Beam interpolation functions not including shear effects
in which
Note: displacements correspond to the forces/moments Si in Fig. 2.41 after
condensation of the last two columns containing the shear effects in the s and tdirections.
1 1 2 2 1
4 5 6
4 5
1 3 3 1 1
1 7
6 1 7
1 6 6 0 6
0 0 1 0 0
0 0 1 0 0 0
6 0 (1 6 ) (1 6 )
0 0 0 ( )
0 ( ) 0
r
s
t
u
u
u
r s t r st sL L L L L
r t LL
r s LL
t t s t sL
r t r r LL
r s r r LL
Ψ Ψ Ψ Ψ Ψ
Ψ Ψ Ψ
Ψ Ψ
Ψ Ψ Ψ Ψ Ψ
Ψ Ψ
Ψ Ψ Ψ
⎡ − − −⎢⎢⎡ ⎤⎢⎢ ⎥ ⎛ ⎞= − −⎢ ⎜ ⎟⎢ ⎥ ⎝ ⎠⎢⎢ ⎥⎣ ⎦ ⎢ ⎛ ⎞− −⎢ ⎜ ⎟⎢ ⎝ ⎠⎣
⎤− − − − − ⎥⎥⎥− − − ⎥⎥⎥−⎥⎦
hhh
2 2 2
1 2 3
2 3 2 3
4 5
2 3 2 3
6 7
; 1 4 3 ; 2 3
1 3 2 ; 2
3 2 ; 3 2
r r r r r rL L L L L L
r r r r rL L L L L
r r r r rL L L L L
Ψ Ψ Ψ
Ψ Ψ
Ψ Ψ
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − = − + = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + = − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − = − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
2.4: Beam elements
ADINA R & D, Inc. 67
! In the formulation, shear deformations can be included in beams with a rectangular section in an approximate way; namely, if requested, constant shear distortions rsγ and rtγ along the length of the beam are assumed, as depicted in Fig. 2.49. In this case the displacement interpolation matrix of Table 2.42 is modified for the additional displacements corresponding to these shear deformations.
rs rt
r r
s t
Figure 2.49: Assumptions of shear deformations through element
thickness for nonlinear elastoplastic beam element
! The interpolation functions in Table 2.42 do not account for warping in torsional deformations (absent if twodimensional action is specified). The circular section does not warp, but for the rectangular section the displacement function for longitudinal displacements is corrected for warping as described in the following reference:
ref. K.J. Bathe and A. Chaudhary, "On the Displacement
Formulation of Torsion of Shafts with Rectangular CrossSections", Int. Num. Meth. in Eng., Vol. 18, pp. 15651568, 1982.
! Note that when using the small displacement formulation, the stiffness matrix is identical to the one used in linear elastic analysis if
< The crosssection of the beam is circular (or rectangular) with b a= or b a! , (because the exact warping functions are employed for b a= and b a! , and the appropriate torsional rigidity is used). < The numerical integration is of high enough order
Chapter 2: Elements
68 ADINA Structures ⎯ Theory and Modeling Guide
< The material is (still) elastic
! The derivation of the beam element matrices employed in the large displacement formulation is given in detail in the following paper:
ref. K.J. Bathe and S. Bolourchi, "Large Displacement
Analysis of ThreeDimensional Beam Structures," Int. J. Num. Meth. in Eng., Vol. 14, pp. 961986, 1979.
The derivations in the above reference demonstrated that the updated Lagrangian formulation is more effective than the total Lagrangian formulation, and hence the updated Lagrangian formulation is employed in ADINA.
! All element matrices in elastoplastic analysis are calculated using numerical integration. The default integration orders are given in Table 2.43. The locations and the labeling of the integration points are given in Fig. 2.410.
! The elastoplastic stressstrain relation is based on the classical flow theory with the von Mises yield condition and is derived from the threedimensional stressstrain law using the following assumptions:
 the stresses ssτ and ttτ are zero
 the strain stγ is zero
Hence, the elasticplastic stressstrain matrix for the normal stress
rrτ and the two shear stresses rsτ and rtτ is obtained using static condensation. ! In elastoplastic analysis, either end forces or element stresses can be requested. Note that if end forces are requested in an elasticplastic analysis, the state of stress in the element (whether elastic or plastic) is not indicated.
ref. KJBSection 6.6.3
2.4: Beam elements
ADINA R & D, Inc. 69
Table 2.43: Recommended integration orders in elastoplastic beam analysis
Integration orders Coordinate Section Analysis
Integration
eme Default
Maximumsch
r Any Any N.C.(**)
5
7
2D action in rs plane
3
7
Rect. 3D action
7(*)
7
s
7
Pipe
Any
N.C.
3
Rect.
2D actionin rs plane
3D
N.C.
1
7(*)
7
7
5
5
rs plane
t r θ
3D
omposite trapezoidal
rule
8
8
o C
Pipe
Str e following rmation is ccessible in the AUI using the variable names PLASTICFLAG,
ACCUM_EFF_PLASTIC_STRAIN
(*) Currently, seven point integration is used in the s and t directions for the rectangular section in 3D action, regardless of
ser input. u
(**) N.C.: NewtonCotes
esses: Each element outputs, at its integration points, thinformation to the porthole file. This info
aSTRESSRR, STRESSRS, STRESSRT, STRAINRR, STRAINRS, STRAINRT, PLASTIC_STRAINRR, PLASTIC_STRAINRS, PLASTIC_STRAINRT, IELD_STRESS, Y
See Section 12.1.1 for the definitions of those variables that are not selfexplanatory.
Chapter 2: Elements
70 ADINA Structures ⎯ Theory and Modeling Guide
s
sHEIGHT
Pipe section
WIDTH
t
Integration
points
equally
spaced
Rectangular section
r = 0r = 1 r = 1
r
s
t
13 2 4
Integration points
equally spaced
a) Integration point locations in rdirection
b) Integration point locations in sdirection
. .. .. .
3 1 2
213
F
2.4.4 Mom
y using these data without having to define a
ular.
igure 2.410: Integration point locations in elastoplastic beam analysis
Nodal forces: Same as for the elastic beam element.
entcurvature beam element
! In practical engineering analysis, the data available for the description of the behavior of beam members may be given only in the form of relationships between bending moment and curvature, and between torsional moment and angle of twist. ADINA offers the capability of directln "equivalent" stressstrain law and the exact beam crosssectional
shape. This element is suitable for modeling nonlinear elastic and
elastoplastic beam problems involving arbitrary crosssections, specially crosssections that are neither rectangular nor circe
2.4: Beam elements
ADINA R & D, Inc. 71
ss
Pipe section (3D action)Pipe section (2D action)
Rectangular section
c) Integration point locations in tdirection
1122
33
Integration points equally spaced
s
t
Integration
points
equally
spaced
......
213
Figure 2.410: (continued)
! In the momentcurvature input, it is assumed that both the centroid and the shear center of the beam crosssection lie on the raxis of the element: transverse forces applied to the element cannot generate twisting (as with a shear center offset) and axial forces applied to the element cannot generate bending (as with a centroid offset). However, you can create offcentered beam elements by using the rigid link (see Figure 2.42 and Section 5.11.2).
! The flexural and the torsional behavior of the beam element are respectively described by "bending moment vs. curvature" and "
torsional moment vs. angle of twist" relationships. These relationships are input in ADINA in the form of multilinear functions (see Figure 2.411).
Chapter 2: Elements
72 ADINA Structures ⎯ Theory and Modeling Guide
Curvature(or twist angle per unit length)
Fi
Fi+1
Fi1
Moment
Fi axial force
Typical beam data set:
F5
F3Curvature(or twist angle perunit length)
Moment
Axial force
F4
ADINA input:
(can be positiveor negative)
Figure 2.411: Input curves for the momentcurvature models
(curves are shown only for positive forces, moments and curvatures/twists)
! The "bending moment vs. curvature" and "torsional momeangle of twist" relationships are functions of the axial force. Note that the axial force is positive when the elem
nt vs.
ent is in tension and is neg wcon ion cal node 2 in F
l plane
of inertia. Curvature is defined as
ative hen the element is in compression. The sign vent s used for moments and torsion is that used for lo
igure 2.41(b).
! The flexural behavior of the beam element is defined by two "bending vs. curvature" relationships, one for each principa
1χρ
= where ρ is the radius
2.4: Beam elements
ADINA R & D, Inc. 73
of curvature. For a linear element (linear elastic material, small displacements/small strains), the relationship between bending
moment and curvature is M EIχ
= .
! The input for the torsional behavior of the element is similar to that for bending: A multilinear function is used to express the torsional moment in terms of the angle of twist per unit length. ! Note that the beam element does not include any shear deformation when the momentcurvature description is used. ! Note also that warping effects must be taken into account in the definition of the "torsional moment vs. twist" input curve if necessary. ! The rigidend and endrelease capabilities described in Section 2.4 can be used with this element. ! For computation of the beam element stiffness matrix and internal force vector, numerical integration (NewtonCotes) is only used for integration along the length of the element (raxis) and no integration is needed over the crosssection (saxis and taxis). ! Thermal effects can be included in momentcurvature relations.
Nonlinear elastic model: A nonlinear elastic model can be used (see Figure 2.412). In this case, the behavior for negative curvatures (resp. twist angles) may be different than the behavior for positive curvatures (resp. twist angles). The axial force versus axial deformation relationship is always linear elastic.
Note that the last segments of the momentcurvature curve are extrapolated if necessary, in order to calculate the moment when the curvature or twist angle per unit length is out of the input curve range. The end points of the curve do not represent rupture points.
Chapter 2: Elements
74 ADINA Structures ⎯ Theory and Modeling Guide
Last inputdata point
Moment
Curvature(or twist angle perunit length)
Extrapolation used in ADINA
First inputdata point
First and last data pointsare not rupture points
ADINA input for a given axial force:
Figure 2.412: Nonlinear elastic momentcurvature beam input
Elasticplastic model: For the analysis of beam members undergoing plastic deformations, ADINA offers bilinear and
ardening can be linear isotropic, linear kinematic or linear mixed, as shown in Figure 2.413. multilinear plasticity. H
1: kinematic hardening2: mixed hardening3: isotropic hardening
y
x
1
2
3
yield
yield0
y
yie
ld
y
yie
ld0
y
y
Figure 2.413: Hardening models for momentcurvature beams
2.4: Beam elements
ADINA R & D, Inc. 75
The momentcurvature plasticity model consists of uniaxial plasticity laws respectively applied to the axial strain, each bending curvature, and the twist angle per unit length:
Axial force/axial strain relationship: The relationship can either be symmetric or nonsymmetric with respect to the sign of the axial strain. Bending moment/curvature relationship: The relationship can either be symmetric or nonsymmetric with respect to the sign of the curvature. The relationship can depend on the axial force and can be different in axial tension and axial compression. Torsional moment/twist angle per unit length relationship: The relationship can either be symmetric or nonsymmetric with respect to the sign of the twist angle per unit length. The relationship can depend on the axial force and can be different in axial tension and axial compression.
In the symmetric case (Figure 2.414), you enter only the
positive section of the axial force/axial strain, bending moment/curvature or torsional moment/twist angle curve. The first data point always corresponds to yielding and the last data point always corresponds to rupture.
In the nonsymmetric case (Figure 2.415), you enter the entire axial force/axial strain, bending moment/curvature or torsional mo
o
ment/twist angle curve. The first data point always corresponds to negative rupture and the last data point always corresponds to positive rupture. One data point S the zero point S must be at the origin. The data point prior to the zero point corresponds to the negative yield and the data point after the zero point corresponds tthe positive yield. A different number of data points can be used for the positive and negative sections of the curve.
! To obtain bilinear plasticity, define a multilinear plasticity curvewith only two segments.
Chapter 2: Elements
76 ADINA Structures ⎯ Theory and Modeling Guide
xrupture
xx
yield
y
yrupture
yieldy
Yielding:First inputdata point
Rupture:last inputdata pointSlope must
decrease asx increases
(x, y): (axial strain, axial force), or
(curvature, bending moment) for a given axial force, or
(angle of twist/unit length, torsional moment) for a givenaxial force
Figure 2.414: Symmetric elastoplastic momentcurvature beam
input ! Bending moment and torsion relationships can depend on the axial force and this dependence can be different in tension and in compression. ! The same element section can be plastic with respect to the axial deformation, but still elastic with respect to bending or torsion. The same remark applies for rupture.
! Rupture depends on the value of the accumulated effective
plastic axial strain, accumulated effective plastic curvature or accumulated effective plastic angle of twist per unit length.
2.4: Beam elements
ADINA R & D, Inc. 77
x
x
positiverupture
negativerupture
xx
x
positiveyield
negativeyield
yy
y
positiverupture
negativerupture
positiveyield
negativeyield
y
y
One point mustbe at the origin
Curve cannotpass throughthis quadrant
Curve cannotpass throughthis quadrant
First point: negative rupture point
Last point: positive rupture point
Point before origin: negative yield pointPoint after origin: positive yield point
Slope mustdecrease asx increases
(x, y): (axial strain, axial force), or
(curvature, bending moment) for a given axial force, or
(angle of twist/unit length, torsional moment) for a givenaxial force
Figure t
rts al rigidities (see Figure 2.416). With this
option, the elastic rigidity
rigidity remains constant for the rest of the analy
2.415: Nonsymmetric elastoplastic momentcurvature beam inpu
! The elasticplastic model includes a special option for cyclic behavior, in which the elastic rigidities when unloading first stadiffer from the initi
(in bending, torsion or axial deformation) changes as soon as the yield stress is reached. The ratio of the newcyclic elastic rigidity to the initial elastic rigidity is defined by user input. Once updated, the elastic
sis, for loading and unloading.
Chapter 2: Elements
78 ADINA Structures ⎯ Theory and Modeling Guide
yu
yi
yi
yu
yuyi
x
yu
yi
y
yi : initial elastic rigidity
yu : cyclic elastic rigidity
Typical data curve:
y
x
x
y
Plastic strain
> 1
ADINA result (shown here for the bilinear case):
Figure 2.416: Typical material curve with a cyclic rigidity different than
ing or dynamic analysis). Theycomputation.
the initial rigidity
! In momentcurvature models, the crosssectional area and the crosssectional moments of inertia need to be input for the mass matrix computation (in case of mass proportional load
are used only for mass matrix
2.4: Beam elements
ADINA R & D, Inc. 79
! The results can be obtained in the formand moment en at the integration point locations on the element.
Stress resultants: Each element outputs, at the integration point locations on the element raxis, the following information to the p h s a
The integration point locations on the element raxis are shown in Figure 2.417. In the AUI, these integration point locations are
be section points, not element result points (see Section 12.1.1).
of element nodal forces s, or in the form of stress resultants giv
ort ole file, based on the material model. This information iccessible in the AUI using the given variable names.
considered to
Moments
Momentr
Forcer
Momentt
Int pt 1Int pt 2
Node 1
Node 2
s
r
t
Forcer is positive in tension.
Forces, Forcet are not computed. Figure 2.417: Stress resultant output option fo
curvature mr the moment
odels
ction point location points in the negative sdirection.
TWIST, URVATURES, CURVATURET
NDING_MOMENTS, ENDING_MOMENTT, AXIAL_STRAIN, TWIST,
The results are always given in the element local coordinate system (r,s,t). Note that a positive value of the Moments at a se
Nonlinear elastic, user supplied: AXIAL_FORCE, TORSIONAL_MOMENT, BENDING_MOMENTS, BENDING_MOMENTT, AXIAL_STRAIN, C Plasticmultilinear: PLASTIC_FLAG_AXIAL, PLASTIC_FLAG_TORSION, PLASTIC_FLAG_BENDINGS,PLASTIC_FLAG_BENDINGT, AXIAL_FORCE, TORSIONAL_MOMENT, BEB
Chapter 2: Elements
80 ADINA Structures ⎯ Theory and Modeling Guide
ORCE,
the element nodal
ess resultants at is e
s
Transverse forces are not computed at integration points since
gid ends, the element stress n point locations
CURVATURES, CURVATURET, PLASTIC_AXIAL_STRAIN, PLASTIC_TWIST, PLASTIC_CURVATURES, PLASTIC_CURVATURET, ACCUM_PLASTIC_AXIAL_STRAIN,
CUM_PLASTIC_TWIST, ACACCUM_PLASTIC_CURVATURES, ACCUM_PLASTIC_CURVATURET, YIELD_AXIAL_FYIELD_TORSIONAL_MOMENT, YIELD_BENDING_MOMENTS, YIELD_BENDING_MOMENTT
! Note that in a materially nonlinear analysis, moments are in general not equal to the moment strthe integration points located at the nodes. The reason for this
at in such cases the beam element does not exactly satisfy ththelement internal equilibrium on the differential level; that is, the element is a "true finite element". The differential equilibrium is, of course, satisfied more accurately as the finite element mesh irefined. !
the ADINA beam element formulation uses the equilibrium equations of the complete element to directly determine the transverse forces at the element end nodes. ! In the case of beam elements with riresultants are computed at the integratiocorresponding to the flexible part of the element, excluding the rigid ends. Nodal forces: Same as for the elastic beam element.
2.4.5 Beamtied element
! aft connecting shell surfaces. It consists of a beam ent with
torell surfaces. Specifically the torsional angle
at one end of the beam is computed from the inplane
beam end.
The beamtied element is used for the modeling of a sh elem
additional constraint equations. These constraints couple the sional angular displacements of the beam to the inplane
displacement of the sh
displacements of the nodes of the shell elements connected to the
2.4: Beam elements
ADINA R & D, Inc. 81
! A typical use of the beamtied element is shown in Figure 2.418. Note that the mesh around the boltshaft must be fine enough so that the first ring of shell elements around the boltshaft
an an :
modeling of a single isolated shaft. This feature is not m nt to be use
ends sp area about the size of the actual connection devicetoo coarse a mesh will result in an overconstrained model. Therefore, the use of the beamtied element is restricted to the
ead for the modeling of riveted, or spotwelded, or linewelded
plates.
ied
equations are not automatically generated at those nodes where the individual shell normals at the node differ than
the average shell normal at the node the bolt angle).
Figure 2.418: Example in which the beamtied element is used
! The automatically generated constraint equations can be applat either one or two ends of the beamtied element. If applied at only one end, the beamtied element is allowed to spin (please note that a mechanism is then introduced in the model). ! Constraint
more than a preset angle from(this preset angle is called
Chapter 2: Elements
82 ADINA Structures ⎯ Theory and Modeling Guide
! The beamtied element must have a finite length, i.e. the end
dimment should be used instead,
see Section 2.9.3.
2.4.6 Bolt element
! A bolt element is a prestressed Hermitian beam or pipe element with a given initial force. The bolt force is held constant during the
lution start, so that the initial force at the start of the zerosolution step is the input initial force,
! ng ution steps, the b e model. ! th the bolt option activated, e.g., EGROUP BEAM OPTION = BOLT. ThAUI comm irectly specified on the elements (e.g., AUI command EDATA).
displacementotropic elastic
a tion, only a circular crosssection is applicable. ! In the bolt force calculations we iterate as follows:
nodes must be further apart than a minimum distance. This minimum distance can be relative (proportional to the model
ensions) or absolute. For nodes that are closer than this minimum distance, the springtied ele
equilibrium iterations at the time of so
no matter what other external loads or initial strains/stresses are input.
Duri the solution for the first, second and successive sololt force can vary depending on the loads applied to th
Bolt elements are created in a beam or pipe element group wi
e initial force is entered either on the geometry (e.g., and LINEELEMENTDATA BEAM) or d
! The formulation used for the bolt elements can either be small
s or large displacements. Any crosssection available for the beam element can be used, but only the ism terial model can be used. For the pipebolt op
0 ( 1) ( ) ( 1) 0 ( 1)i i i i
I− − −K ∆ = −U R F
and 0 ( ) 0 ( 1) ( )i i i−= + ∆U U U
2.4: Beam elements
ADINA R & D, Inc. 83
here ( 1)Ii−Rw is the consistent nodal point force corresponding to the
forces in the bolt elements.
nt m:
! The bolt force convergence is checked for every bolt eleme
0
0 TOLLR
m m
m
< R R−
where s the current force in bolt element m. !
0Rm is the userinput bolt force for element m, and Rm i
The ( 1)Ii−R vector is calculated from
! mechanical deformations plus updated initial stresses:
0
T( 1) ( 1) ( 1)Ii i i
V
dV− − −= ∫R B τ
For every bolt element m, the stress vector corresponds to the
( ) (0) ( 1)i im m mτ τ α τ −= + ⋅
where 0
0 crosssectional area of bolt element R , m
m mm
mAA
τ = =
and α is a stresslevel based factor.
2.4.7 Element mass matrices
! The beam element can be used with a lumped or a consistent mass matrix, except for explicit dynamic analysis which always uses a lumped mass. ! The consistent mass matrix of the beam element is evaluated in closed form, and does not include the effect of shear deformations. This matrix is derived in the following reference.
ref. J.S. Przemieniecki, Theory of Matrix Structural
Chapter 2: Elements
84 ADINA Structures ⎯ Theory and Modeling Guide
Analysis, McGrawHill Book Co., 1968. • The beam element can be used with a lumped or a consistent mass matrix, except for explicit dynamic analysis which always uses a lumped mass. • The consistent mass matrix of the beam element is evaluated in closed form, and does not include the effect of shear deformations. • The lumped mass for translational degrees of freedom is M / 2 where M is the total mass of the element. The rotational lumped mass for all except explicit dynamic
analysis is 23 2
rrrr
M IMA
= ⋅ ⋅ , in which Irr = polar moment of
inertia of the beam crosssection and A = beam crosssectional area. This lumped mass is applied to all rotational degrees of freedom in order to obtain a diagonal mass matrix in any coordinate system. The rotational lumped mass for explicit dynamic analysis is
32
mrr
IMMA
= ⋅ ⋅ where ( )max ,m ss ttI I I= is the maximum
bending moment of inertia of the beam. This lumped mass is app g of do ! The rotational lumped masses can be multiplied by a userspecified multiplier ETA (except in explicit analysis).
2.5 Isobe
2.5.1 General considerations
! ele plane stress 2D beam with three degrees of freedom per node, plane strain 2D beamaxgen
lied to all rotational degrees of freedom. Note that this scalinrotational masses ensures that the rotational degrees of freedom not affect the critical stable time step of the element.
am elements; axisymmetric shell elements
The isobeam elements (which include the axisymmetric shell ments) can be employed in the following forms, see Fig. 2.51:
with three degrees of freedom per node, isymmetric shell with three degrees of freedom per node,
eral 3D beam with six degrees of freedom per node.
2.5: Isobeam elements; axisymmetric shell elements
ADINA R & D, Inc. 85
! he plane stress 2D beam element is identical to the 3D beam
Z plane (see Fig. 2.52).
tween the plane strain and the plane stress 2D
beam elements is that for the plane strain element, it is assumed that the outofplane strain εxx is equal to zero whereas the outofplane stress σxx is equal to zero for the plane stress element. ! Note that it can be significantly more effective to use the 2D beam option instead of the general 3D beam option, since then the numerical integration is only performed in two dimensions and the number of degrees of freedom is also reduced.
Telement but constrained to act only in the YZ plane. Hence, all motion of the 2D plane stress beam element must occur in the Y
! The difference be
Chapter 2: Elements
86 ADINA Structures ⎯ Theory and Modeling Guide
Examples of general 3D isobeam elements:
1 radian
Z
Y
Axisymmetric shell:
xx=0 e =0xx
X
X
X
ZZ
YY
Plane stress beam: Plane strain beam:
Figure 2.51: 2D beam, axisymmetric shell, and 3D beam elements
2.5: Isobeam elements; axisymmetric shell elements
ADINA R & D, Inc. 87
s
r
Axis ofsymmetry
Y
Element has varyingthickness crosssection
a) 2node plane stress or plane strain beam element
Z
Y
Element has constantrectangular crosssection
b) 4node axisymmetric shell element
Figure 2.52: Local node numbering; natural coordinate system
Z
sr
1
2
1
2
3
4
Element can have 2, 3or 4 nodes
Element can have 2, 3or 4 nodes
Auxiliary node is not used,set the auxiliary node to 0.
Auxiliary node is not used,set the auxiliary node to 0.
! The axisymmetric shell element formulation is anthe 2D beam element formulation in that the axstress/strain components are included in the model.The
extension of
isymmetric hoop
c shell element must be defined in the YZ plane, and half plane.
axisymmetrilie in the +Y
! The crosssectional areas of each of these elements are assumed to be rectangular. The 2D and 3D beam elements can only be assigned a constant area crosssection. The axisymmetric shell element can be assigned a varying thickness.
Chapter 2: Elements
88 ADINA Structures ⎯ Theory and Modeling Guide
Auxiliary node
s2node isobeam
Auxiliary node
Auxiliary node
rs plane
rs plane
3node isobeam
4node isobeam
c) General 3D isoparametric beam elements.
Figure 2.52: (continued)
12
rs plane
1
23
1
34 2
Elements have constantrectangular crosssection.
t
r
The elements c yed with 2, 3 or 4 nodes. Tode elements can be curved, but it should be not
! an be emplo he 3 and 4n ed that the element nodes must initially lie in one plane (which defines the rs pla ) ! element accurately, the warping behavior is represented as described in the following reference:
Sh ctions", Int. J. Num. Meth. in Eng., Vol. 18, pp.
ne .
To model the torsional stiffness of the 3D beam
ref. K.J. Bathe and A. Chaudhary, "On the Displacement
Formulation of Torsion of afts with RectangularCrossSe15651568, 1982.
2.5: Isobeam elements; axisymmetric shell elements
ADINA R & D, Inc. 89
! Note that the formulation directly models shear deformations in an approximate manner; namely, the shear deformations are
, Fig. 1.0.
sis3.
arametric beam element is available in ADINA
primarily to model < Curved beams < Stiffeners to shells, when the shell elem nt (described in Section 2.7) is used < Beams in large displacements < Axisy
2.5.2 Numerical integr
ulated using thCo s numerical l an yses. For the
2D beam and axisymmetric shell elements, numerical integration
ion
based upon a mixed interpolation of
displacements and stresses.
ref. KJBSections 5.4.1
and 6.5.1
assumed to be constant over the crosssection (see ref. KJB5.18, p. 398). This corresponds to using shear factors equal to The elements can be employed to model thin and thick beams and shells.
! Some applications using the element are published in the following reference.
ref. K.J. Bathe and P.M. Wiener, "On ElasticPlastic Analy of IBeams in Bending and Torsion", Computers and Structures, Vol. 17, pp. 711718, 198
! The isop
e
mmetric shells under axisymmetric loading
ation
! The element matrices and vectors are form e isoparametric interpolation, and Gauss or Newton teintegration is used to evaluate these matrices in al al
is only performed in the rs plane, hence these elements are considerably less expensive in terms of computer time than the general 3D element. The locations and labeling of the integratpoints are illustrated in Fig. 2.53.
! The elements are
Chapter 2: Elements
90 ADINA Structures ⎯ Theory and Modeling Guide
a) NewtonCotes integration (3 3)x b) Gauss integration (3 2)x
13 12 11
2133
32
31
s
r
label = 10(INR) + INS label = 10(INR) + INS
Figure 2.53: Examples of integration point numberingfor 2D beam, axisymmetric shell, and3D beam elements
2223
s
r32
12 11
21
31
22
the 2, 3 and 4node elements, i.e.:
not contain any spurious zero energy modes, do
sed
oint e r di ent
126
! Only the default integration order along the rdirection should be used for
< 2node isobeam: 1point integration
< 3node isobeam: 2point integration < 4node isobeam: 3point integration
These elements do not lock and are efficient in general nonlinear analysis.
! If, however, the default rintegration order is not u , theresults of an analysis can change drastically, for example, if 2pintegration along th rection is specified for the 2node elemmodel shown in Fig. 2.54, the resulting displacement (∆) is 0.0(i.e., locking is observed). ! For the 3D beam element, you can choose either 4×4 Gauss or 7×7 NewtonCotes integration over the beam crosssection (along sand t).
2.5: Isobeam elements; axisymmetric shell elements
ADINA Inc. 91
t
t
s
s
r
r
313
312
311
321
331
323
231
221
213
212
121
121
132
133
122
113
112
c) NewtonCotes integration (3 3 3)
d) Gauss integration (3 2 2)
Figure 2.53: (continued)
111
111
label = 100 (INR) + 10 (INS) + INT
label = 100(INR) + 10(INS) + INT
322
2.5.3 Linear isobeam elements
! The formulation of the element is presented in Chapter 5 of ref. KJB. ! It is assumed that the displacements, rotations, and strains are infinitesimally small and the elasticisotropic material is used.
ref. KJBSection 5.4.1
R & D,
Chapter 2: Elements
92 ADINA Structures ⎯ Theory and Modeling Guide
Figure 2.54: Results of the analysis of a cantilever
Element 1 Element 2
P
P
L
EI=10L=24P=20
5
th=PL
3EI= 0.9216
(theoretical solution,
no shear)
3
P
P
L
L/3 L/3L/3
L/2 L/2
= 0.9216
= 0.9224
= 0.8648
2node Hermitian beam (no shear):
Physical problem:
Element 1
2node isobeams (with shear):
Element 1
4node isobeam (with shear):
! Since the isobeam element stiffness matrix is calculated unumerical integration, it is clear that the 2node Hermitian beelement (see Section 2.4.1) is more effective when straight bmembers in linear elastic analysis are considered. Hence, for
sing am
eam the
linear analy
sis of many structures (such as frames, buildings, shafts) only the 2node Hermitian beam element should be used. ! Even when considering the linear analysis of a curved beam member, it is frequently more costeffective to model this beam as an assemblage of straight 2node Hermitian beam elements than to use the curved isobeam element.
2.5: Isobeam elements; axisymmetric shell elements
ADINA R & D, Inc. 93
! Note that the isobeam element can only be employed with a rectangular crosssection, whereas the linear 2node Hermitian bea
node
ell
etization of the stiffened shell structure (see Fig. 2.55).
m element can be used for any crosssection which can be defined by the principal moments of inertia, crosssectional area and the shear areas input to ADINA. ! In linear analysis, the isobeam element is primarily useful formodeling stiffeners to shells when the 4node or 8node or 16shell element (see Section 2.7) is employed. In this case the 2, 3or 4node isobeam elements, respectively, together with the shelements can provide an effective finite element discr
Shell nodeIsobeam node
Note: Select master nodes and use rigid links to tiethe shell nodes and the isobeam nodes together.
a) Stiffened plate
b) Stiffened shell
Figure 2.55: Models of stiffened plate and stiffened shell
Chapter 2: Elements
94 ADINA Structures ⎯ Theory and Modeling Guide
! The solution results obtained using the isoparametric beam elements are compared with those obtained using the 2node Hermitian beam elements in Fig. 2.54. Considering the tip loaded cantilever shown, a 2node Hermitian beam element representation gives the "exact" analytical value of displacements and section forces/moments (compared with BernoulliEuler beam theory). The 2node isobeam element yields approximate results, while the 4node isobeam element is very accurate. However, as is typical in finite element analyses, the discrepancy in the solution becomes negligibly small when a sufficiently fine mesh is used. ! The forces/moments at the element nodes are (in linear analysis)
=F KU
where Κ is the element stiffness matrix and U is the vector of nodal point displacements. Hence, at internal element nodes the forces/moments in F will be equal to the forces/moments, applied externally or by adjoining elements, and are not the internal section forces/moments.
2.5.4 Nonlinear isobeam elements
! The formulation of the element is presented in Chapter 6 of ref. KJB.
! The element can be used with the following material models: elastic, plasticbilinear, plasticmultilinear, thermoisotropic, thermoplastic, creep, plasticcreep, multilinearplasticcreep, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable, shapememory alloy.
! The element can be used either with a small displacement or a large displacement formulation. In the large displacement formulation, large displacements and rotations are allowed. In all cases, only small strains are assumed.
All of the material models in the above list can be used with either formulation. The use of a linear material with the small displacement formulation corresponds to a linear formulation, and the use of a nonlinear material with the small displacement
ref. KJBSection 6.5.1
2.5: Isobeam elements; axisymmetric shell elements
ADINA R
formulation corresponds to a materiallynonlinearonly formulation.
! nocha odeled accurately.
! The element matrices are evaluated using Gauss or NewtonCotes numerical integration. In elasticplastic analysis, the stressstr atrix is modified to include the effects of plasticity. This stressstrain matrix is based on the classical flow theory with the von Mises yield condition and is derived from the threedimensional stressstrain law with the appropriate stresses and strains set to zero.
! ma l mo he e ref ces elo
ion
2
ref. KJBSection 6.6.3
The isobeam element can be particularly effective in geometric
nlinear analysis of straight and curved members, because the nge in geometry due to the large displacements is m
ain m

For the elastoplastic and the thermoelasticplastic and creep teria dels, t ffectivestressfunction algorithm of theeren given b w is used.
ref. M. Kojiƒ and K.J. Bathe, "The EffectiveStressFunct
& D, Inc. 95
sticity and Creep", Int.
1987.
K.J. Bathe, "ThermoElasticPlastic and
987.
.5.5 Axisym
! casaxi case of the 2D solid element).
! strfor
! an
Algorithm for ThermoElastoPlaJ. Num. Meth. Engng., Vol. 24, No. 8, pp. 509532,
ref. M. Kojiƒ andCreep Analysis of Shell Structures", Computers & Structures, Vol. 26, No 1/2, pp. 135143, 1
metric shell element
The axisymmetric shell element can be thought of as a special e of the isobeam formulation (in the same way as an symmetric element is a special
For the axisymmetric shell element, the hoop (circumferential) ess and strain components are included in the isobeam mulation.
The use of this element can be significantly more effective th
Chapter 2: Elements
96 ADINA Structures ⎯ Theory and Modeling Guide
usibe ! ian of the axisymmetric shell is modeled when using this element (as with the 2D axisymmetric solid element).
2.5.6 Elemen
! The isobeam element can be used with a lumped or a consistent
. ! ametric for of ref
!
ng axisymmetric 2D solid elements when the shell structure to modeled is thin.
One rad
t mass matrices
mass matrix, except for explicit dynamic analysis which always uses a lumped mass
The consistent mass matrix is calculated using the isoparmulation with the displacement interpolations given on p. 408. KJB.
The lumped mass for degree of freedom i is iML
⎛ ⎞⋅⎜ ⎟⎝ ⎠!
, where M
ass, L is the elem
is the total m ent length and is a fraction of the length c
i! asso iated with node i. The rotational mass is
( )2 2i b d⎛ ⎞+ ⎟⎠
!. For a 2node isobeam element
1 13 6
ML
⋅ ⋅ ⎜⎝
1 2 2L
= = , while for a 4node isobeam element ! ! 1 2L
= =! ! and 6
3 4L3
= =! ! (see Fig. 2.56).
2.5.7 Elem
ref. KJBSection 5.4.1
! The rotational lumped mass can be multiplied by a userspecified scalar ETA (except in explicit dynamic analysis).
ent output
! You can request that ADINA either print or save stresses or forces.
Stresses: Each element outputs, at its integration points, the following information to the porthole file, based on the material
2.5: Isobeam elements; axisymmetric shell elements
ADINA R & D, Inc. 97
nformation is accessible in the AUI using the given variable names. model. This i
Figure 2.56: Construction of element lumped translational massmatrix for the beam option of the isobeam element
a) 2node element
b) 3node element
c) 4node element
Note: M = total mass
of element (2D or 3D
beam elements)
M2
M2
M2
M4
M4
M3
M3
M6
M6
General 3D isobeam, elasticisotropic material: STRESSRR, STRESSRS, STRESSRT, STRESSSS, STRAINRR, STRAINRS, STRAINRT, STRAINSS, FE_EFFECTIVE_STRESS Plane stress isobeam, elasticisotropic material: STRESSRR, STRESSRS, STRESSSS, STRAINRR, STRAINRS, STRAINSS, FE_EFFECTIVE_STRESS Plane strain isobeam or axisymmetric shell, elasticisotropic material: STRESSRR, STRESSRS, STRESSTT, STRESSSS, STRAINRR, STRAINRS, STRAINTT, STRAINSS, FE_EFFECTIVE_STRESS General 3D isobeam, elasticisotropic material, thermal effects: STRESSRR, STRESSRS, STRESSRT, STRESSSS, STRAINRR, STRAINRS, STRAINRT, STRAINSS,
Chapter 2: Elements
98 ADINA Structures ⎯ Theory and Modeling Guide
SRR, STRESSRS, STRESSSS, STRAINRR,
ST),
cmultilinear materials: PLASTIC_FLAG, STRESS(RST), STRAIN(RST), PLASTIC_STRAIN(RST), THERMAL_STRAIN(RST), FE_EFFECTIVE_STRESS, YIELD_STRESS, ACCUM_EFF_PLASTIC_STRAIN, ELEMENT_TEMPERATURE Thermopcreepvari ltilinearplasticcreepvariable materials: PLASTIC_FLAG,
NCREMENTS, STRESS(RST), FE_EFFECTIVE_STRESS,
EFFECTIVE_CREEP_STRAIN
SRS,
FE_EFFECTIVE_STRESS, THERMAL_STRAIN, ELEMENT_TEMPERATURE Plane stress isobeam, elasticisotropic material, thermal effects: STRESSTRAINRS, STRAINSS, FE_EFFECTIVE_STRESS, THERMAL_STRAIN, ELEMENT_TEMPERATURE Plane strain isobeam or axisymmetric shell, elasticisotropic material, thermal effects: STRESSRR, STRESSRS, STRESSTT, STRESSSS, STRAINRR, STRAINRS, STRAINTT, STRAINSS, FE_EFFECTIVE_STRESS, THERMAL_STRAIN, ELEMENT_TEMPERATURE Thermoisotropic material: STRESS(RFE_EFFECTIVE_STRESS, STRAIN(RST), THERMAL_STRAIN(RST), ELEMENT_TEMPERATURE Plasticbilinear, plasti
lastic, creep, plasticcreep, multilinearplasticcreep, able, plasticcreepvariable, mu
NUMBER_OF_SUBI
STRAIN(RST),YIELD_STRESS, PLASTIC_STRAIN(RST), ACCUM_EFF_PLASTIC_STRAIN, CREEP_STRAIN(RST), THERMAL_STRAIN(RST), ELEMENT_TEMPERATURE,
In the above lists,
STRESS(RST) = STRESSRR, STRESSTRESSRT, STRESSTT, STRESSSS
with similar definitions for the other abbreviations used above. See Section 12.1.1 for the definitions of those variables that are
not selfexplanatory.
2.5: Isobeam elements; axisymmetric shell elements
ADIN
2.5: Isobeam elements; axisymmetric shell elements
ADIN
Nodal forces: Nodal point forces are obtained using the relation
where is the straindisplacement matrix, the stresses are stored in represents the volume, and the superscript
t t
t t t t T t t
V
dV+∆
+∆ +∆ +∆= ∫F B τ
t t+∆ Bt t+∆ τ , t tV+∆
t t+ ∆ refers to the conditions at time (load step) . These forces are accessible in the AUI using the variable names
NODAL_FORCER, NODAL_FORCES, NODAL_FORCET, NODAL_MOMENTR, NODAL_MOMENTS,
ref. KJBSection 6.3
t t+ ∆
A R & D, Inc. 99 A R & D, Inc. 99
NODAL_MOMENTT. The end forces/moments are computed at the element local
nodes. In the AUI, element local nodes are defined as element points of type label. For example, to access the result computed at element 5, local node 2, define an element point of type label with element number 5, label number 2.
2.6 Plate/shell elements
! The plate/shell element is a 3node flat triangular element that can be employed to model thin plates and shells. ! The element can be used with the following material models: elasticisotropic, elasticorthotropic and Ilyushin (plastic). ! The element can be used with a small or large displacement formulation. In the large displacement formulation, very arge displacements and rotations are assumed. In both formulations, the strains are assumed small.
All of the material models in the above list can be used with either formulation. The use of a linear material with the small displacement formulation corresponds to a linear formulation, and
l
the use of a nonlinear material with the small displacement formulation corresponds to a materiallynonlinearonly formulation.
Chapter 2: Elements
100 ADINA Structures ⎯ Theory and Modeling Guide
2.6.1 Formulation of the element
! The element has 6 degrees of freedom per node.
! The stiffness matrix is constructed by superimposing the bending and membrane parts in the current local coordinate system.
The bending part is the discrete Kirchhoff bending element described in the following reference:
ref. J.L. Batoz, K.J. Bathe, and L.W. Ho, "A Study of
ThreeNode Triangular Plate Bending Elements", Int. J. Num. Meth. in Eng., Vol. 15, pp. 17711812, 1980.
The membrane part is the constant strain plane stress 3node
element (see ref. KJB, pp. 202205).
! The complete element formulation is summarized and discussed in the following references:
ref. K.J. Bathe and L.W. Ho, "A Simple and Effective
Element for Analysis of General Shell Structures", J. Computers and Structures, Vol. 13, pp. 673681, 1981.
ref. K.J. Bathe, E. Dvorkin, and L.W. Ho, "Our Discrete
Kirchhoff and Isoparametric Shell Elements for Nonlinear Analysis An Assessment", J. Computers and Structures, Vol. 16, pp. 8998, 1983.
tes how the element stiffness matrix is constructed. Note that in the local coordinate system the matrices
! Figure 2.61 illustra
BK and MK do not contain stiffness corresponding to the rotatio z
n θ In the program, this stiffness is set equal to
( ) 4a value oz
Kθ = ×f , 10x x
k kθ θ
in sis res
! of flat plates that lie parallel to the global coordinate planes, the rotational degrees of freedom normal to the
order to remove the zero stiffness but yet not affect the analyults significantly.
In the linear analysis
2.6: Plate/shell elements
ADINA R & D, Inc. 101
pla elefre
ne of the plates would best be deleted, since the plate/shell ments do not provide actual stiffness to these degrees of edom. Fig. 2.62 illustrates these considerations.
zy
x
zw
y
x
u
v1
2
3Z
YX
z z z
x x x
y y y
vu
+ =
x
x
y y
w w
u v
z
KM KB K
KM
KB
K Z
K =
(18 18)
KM is the local stiffness matrix of the constantstrain plane stress element (6 6)
KB is the local stiffness matrix of the discreteKirchhoff plate bending element (9 9)
K Z
Z
is the artificial stiffness matrix for therotation (3 3)
Figure 2.61: Local node numbering; natural coordinate system;
construction of total element stiffness matrix
! Considering the formulation of the element, the following observations should also be noted:
Chapter 2: Elements
102 ADINA Structures ⎯ Theory and Modeling Guide
< The element does not model shear deformations and is only applicable to the analysis of thin plates and shells.
< The element does not contain any spurious zero energy
e to the analysis of very thin plates ock, see Section 2.7.10).
lar, a mesh orientation effect can esh is used that is not spatially
isotropic. Figs. 2.63 and 2.64 illustrate how meshes that are more appropriate can be constructed.
mode. < The element is applicabl
and shells (it does not ever l
< Since the element is triangube noted in analyses when a m
Z
Z
X
X
Y
Y
z DOF can be deleted
on top plate
x DOF can be deleted on front plate
y DOF can be deleted
on side plate
Beam
Note: Torque is taken by the beam only (except for the very
small value resisted by the stiffness of the plate).K z
Figure 2.62: Considerations for the zero stiffness
corresponding to the rotational degreeof freedom normal to the plate
x
y
z
< In the element formulation, stress resultants (membrane forces, bending moments) are emppoints are only located on the mid
loyed, so that the integration surface of the element.
2.6: Plate/shell elements
ADINA R & D, Inc. 103
P
P
P
P
P
P
1
1
1
2
2
2
(a) Cantilever plate model, 1 2
(b) Cantilever plate model, = 1 2
(c) Cantilever plate model, 1 2
Figure 2.63: Models of a cantilever indicating effect of
discretization on the solution results
Z
Z
Z
Y
Y
Y
X
X
X
! Numerical integration is performed using the Hammer's formulas for triangles (see Fig. 2.65):
< In elastic analysis, the membra
ne stiffness is evaluated in
ss is evaluated using a threepoint (interior) scheme over which you have no control. < In elastoplastic analyses, the membrane and bending stiffnesses are in general coupled. In this case both stiffnesses are integrated using numerical integration.
closedform and the uncoupled bending stiffne
Chapter 2: Elements
104 ADINA Structures ⎯ Theory and Modeling Guide
! onepoint integration shown in Fig. 2.65 corresponds to a reduced integration scheme and is therefore not recommended.
! Note that, in elastic analysis, the mwithin the element.
The
embrane forces are constant
Figure 2.64: Models of a plate indicating effect of discretization
on the solution results
Z
Z
ZY
Y
Y
X
X
X
(a) Model of clamped plate. Corner
element shows zero response.
(b) Model of clamped plate. Corner
region shows nonzero response.
(c) Spatially isotropic mesh for
analysis of clamped plate.
2.6.2 Elast
! The elastoplastic model for the plate/shell element in ADINA is b oncondition.resultants, nd hence num is only performed over the midsurface
ent. The Ilyushin yield condition and its use with the described in Section 3.4.4.
rete
mputers and Structures, Vol. 16, pp.
oplastic analysis
ased the flow theory and the use of the Ilyushin yield This yield condition operates directly on the stress i.e., the membrane forces and bending moments, aerical integration
of the elemelement are
ref. K.J. Bathe, E. Dvorkin, and L.W. Ho, "Our Disc
Kirchhoff and Isoparametric Shell Elements An Assessment", J. Co8998, 1983.
2.6: Plate/shell elements
ADINA R & D, Inc. 105
1
1
1
61
3
3
4
3
2
2
7 52
r
r
r
r
s
s
s
s
1
1
1
1
2
2
2
3
3 0.
0.
34567
0.33333 0.33333
0.10129 0.101290.101290.797630.797630.10129
0.47014 0.470140.47014
0.47014
0.05972
0.05972
0.33333 0.33333
0.500000.50000 0.50000
0.50000
0.16667
0.16667
0.166670.16667
0.66667
0.66667
Figure 2.65: Hammer's integration point locations
s
s
s
s
r
r
r
r
Seven integration points (interior):
One integration point (centroid):
Three integration points (interior):
Three integration points (midside):
! In practice the threepoint integration schemes (see Fig. 2.65are most effective for elastoplastic analysis. The sevenpoint scheme can also be used to provide more output lo
)
cations for the stress resultants.
2.6.3 Elemen
! e employed with a lumped or a
t mass matrices
The plate/shell element can b
Chapter 2: Elements
106 ADINA Structures ⎯ Theory and Modeling Guide
consistent mass matrix (which is actually only an approximate consistent mass matrix).
! The translational lumped mass assigned to each node is 3M
and
the rotational lumped mass is 2
3 6M t
⋅ , in which t is the thickness of
all translational degrees of
freedom of the element. Since the actual bending displacements KJB,
nal lumped mass can be multiplied by a userspecified scalar ETA (except in explicit analysis).
2.6.4 Element output
! You can request that ADINA either print or save stress resulta n
Stress a oints, the following ile, based on the material model h invariable name
Elastic p , MEMB FBENDING_MOMENTXL, BENDING_MOMENTYL,
NE_STRAINXLYL, CURVATUREXL, CURVATUREYL, CURVATUREXLYL Ilyushin: PLASTIC_FLAG, MEMBRANE_FORCEXL, MEMBRANE_FORCEYL, MEMBRANE_FORCEXLYL, BENDING_MOMENTXL, BENDING_MOMENTYL, BENDING_MOMENTXLYL, MEMBRANE_STRAINXL,
the plate element. ! The element consistent mass matrix is calculated using the samelinear displacement functions for
are of cubic order, the mass matrix is approximate (see ref. Section 4.2.4). ! The rotatio
nts or odal point forces.
result nts: Each element outputs, at its integration pinformation to the porthole f
. T is formation is accessible in the AUI using the given s.
isotro ic, elasticorthotropic: MEMBRANE_FORCEXLRANE_ ORCEYL, MEMBRANE_FORCEXLYL,
BENDING_MOMENTXLYL, MEMBRANE_STRAINXL, MEMBRANE_STRAINYL, MEMBRA
2.6: Plate/shell elements
ADINA R & D, Inc. 107
MEMBRANE_STRAINYL, MEMBRANE_STRAINXLYL, CURVATUREXL, CURVATUREYL, CURVATUREXLYL
The indices XL and YL indicate that the results are output with
respect to the element local coordinate sy em. See Section 12.1.1 for the definitions of those variables that are
not selfexplanatory.
Nodal forces: Nodal point force vectors which correspond to the
NODAL_MOMENTY, NODAL_MOMENTZ.
2.7 Shell e
! etric ll
and Section 2.7.10).
2.7.1 Basic assum s
! The basic equations used in the formulation of the element are given ! Th edimensional co llowing two assumptions used in the Ti e theory:
ref. KJBSections 5.4.
st
element membrane forces and bending moments can also be requested in ADINA.
The nodal forces are accessible in the AUI using the variable names NODAL_FORCEX, NODALFORCEY, NODAL_FORCEZ, NODAL_MOMENTX,
lements
The shell element is a 4 to 32node (degenerate) isoparamelement that can be employed to model thick and thin general shestructures (see Fig. 2.71). However, depending on the application,the appropriate number of nodes on the element must be employed (see Fig. 2.72
ption in element formulation
in ref. KJB. 2 and 6.5.2
e shell lement is formulated treating the shell as a threentinuum with the fo
moshenko beam theory and the Reissner/Mindlin plat
Chapter 2: Elements
108 ADINA Structures ⎯ Theory and Modeling Guide
Analysis of a turbine blade:
Figure 2.71: Some possible applications of shell elements
Analysis of a shell roof:
that straight line during the deformations.
o.
, and
! In the calculations of the shell element matrices the following geometric quantities are used:
The coordinates of the node k that lies on the shell element
midsurface at k
Assumption 1: Material particles that originally lie on a straight line "normal" to the midsurface of the structure remain on Assumption 2: The stress in the direction normal to the midsurface of the structure is zer
For the Timoshenko beam theory, the structure is the beam
for the Reissner/Mindlin plate theory, the structure is the plate under consideration. In shell analysis, these assumptions correspond to a very general shell theory.
< , ,t t t
k kx y z (see Fig. 2.73); (the left superscript denotes the configuration at time t) < The director vectors pointing in the direction "normal" to the shell midsurface
t knV
2.7: Shell elements
ADINA R & D, Inc. 109
Figure 2.72: Examples of shell elements
a) Midsurface description
c) Topbottom description
b) Triangle
e) Multilayered shell
d) Transition shell to solid
< The shell thickness, ka , at the nodal points measured in the
direction of the director vectors (see Fig. 2.74 and Section 2.7.2).
ref. KJBFig. 5.33 t k
nVpage 437
Chapter 2: Elements
110 ADINA Structures ⎯ Theory and Modeling Guide
r
r
s
s
t
t
1
17
1
5
21
5
9
25
9
2
18
2
6
22
6
10
26
10
3
19
3
7
23
7
11
27
11
4
20
4
8
24
8
12
28
12
15
31
15
16
32
16
13
29
13
14
30
14
a) Midsurface nodes
b) Topbottom nodes
Figure 2.73: Some conventions for the shell element; local
x node numbering; natural coordinate system
For the 9node shell, themidsurface center nodeis local node number 13.
For the 9node shell, thetopbottom center nodesare local node numbers13 and 29.
! Fig smidsurface nodal points and the nodal point director vectors. Using rface is inte d
ref. KJB. Simat any using
. 2.74 hows a 4node shell element with the shell
the midsurface nodal point coordinates, the shell midsurpolate using the interpolation functions ( ),kh r s given in
ilarly, using the vectors nV , the director vector point P on the midsurface is obtained by interpolation
t tnV
2.7: Shell elements
ADINA R & D, Inc. 111
the functions ( ),k r s . Hence, with the shell thickness at nodal hpoint k
the direction o etry of the shell at any time t is defined by
equal to ka and the isoparametric coordinate t measured in
f t V , the geomn
1 1
1 1
1 1
2
2
q qt t
2
q qt tt k
k k k k nz
t t kk k k k nx
ky k k k ny
k k
k k
k kq q
t tt
x h x a h V
h y a h V= =
= =
+
+
∑
∑
shell m dsurface interpolation for material points not on the midsurface
where q is the number of element nodes and are
rection vectors at nodes can be either directly input or
automatically generated by the program. When they are generated
rs
lution in that the particles along the director ector (interpolated from the nodal point director vectors
) remain on a straight line during defor
demonstrates this observation for a vthe shell initial configuration. Furthermore, even if is originally normal to the shell midsurface, after deformations have
ty
= =
= ∑
= ∑
tz h z a h V= +∑ ∑
i interpolation
, ,t k t k t knx ny nzV V V
the direction cosines of the shell director vector t knV .
! The di
by the program, they can be created as the normal vectors of a geometrical surface, or as the averaged vectors of the surrounding elements, which may not be exactly normal to the corresponding geometric surface.
! The assumption 1 on the kinematic behavior of the shell entethe finite element sov t
nVt k
nV mation.
Note that in the finite element solution, the vector nV is not necessarily exactly normal to the shell midsurface. Figure 2.75(a)
t
ery simple case, considering t
nV
Chapter 2: Elements
112 ADINA Structures ⎯ Theory and Modeling Guide
taken place this vector will in general not be exactly perpendicular to the midsurface because of shear deformations (see Fig. 2.75(b)).
(a) Programcalculated normal vectors (two such vectors)
at node k. These vectors are used as director vectors
for the respective elements. Node k has 6 DOFs.
0 knV
elt.1
0 knV
elt.2
element 1
element 1
element 2
element 2
Programcalculated midsurface
normal vector at node k using
the geometry of element 2
Programcalculated midsurface
normal vector at node k using
the geometry of element 1
k
k k
Figure 2.74: Convention for shell element thickness
The configuration showncorresponds to time t=0.
formulation employed:
element: The stress in the tdirection (i.e., in the direction of ) is imposed to be zero. This is achieved by using
! The assumption 2 on the stress situation enters the finite element solution in a manner that is dependent on the
All formulations except for the large displacement/large strain ULH shell
tnV
the stressstrain relationship in the , ,r s t coordinate system, shown in Fig. 2.76(a), with the condition that the stress in the direction t is zero.
Large displacement/large strain ULH shell element: The stress in the direction (not necessarily in the direction of is
ref. KJBSection 5.4.2
page 440
t tnV
2.7: Shell elements
ADINA R & D, Inc. 113
is zero.
ent (the coordinates of this
ption 2 is only approximately ful ns of the
imposed to be zero. This is achieved by using the stressstrain relationship in the , ,r s t coordinate system, shown in Fig. 2.76(b), with the condition that the stress in the direction t ! Note that if the nodal point midsurface director vectors are inputor generated (see Section 2.7.3) to be initially "exactly" normal to the midsurface of the shell elemmidsurface are obtained by interpolation from the nodal points on the midsurface), then assumption 1 is also exactly satisfied in the finite element solution, but assum
filled in geometric nonlinear analysis, because the directio initially normal vectors are updated.
b) Programcalculated director vector (single vector)
at node k. Node k has 5 DOFs.
0 knV
0 knV
element 1
element 1
element 2
element 2
Program calculates director
vector by taking average of
all normal vectors at node k.
k
k k
Figure 2.74: (continued)
be hickne tor
ref. KJB ! The transverse shear deformations are assumed by default toconstant across the shell ss. The use of the correction facof 5/6 can be specified to improve the prediction of the displacement response.
tpp. 399, 440
Chapter 2: Elements
114 ADINA Structures ⎯ Theory and Modeling Guide
! The interpolation of the geometry of the shell element is always
rrent the current
of the ons at the
ysis).
ode for
tion 2.2.1 and ref KJB, Section 4.4.1. ent
node urface
as described above, but for a specific solution time the cucoordinates of the midsurface nodal points are used, and irector vectors are employed. The midsurface nodal point d
coordinates are updated by the translational displacementsnodes and the director vectors are updated using the rotatinodes (rotation increments in large displacement anal
! Incompatible modes can be used in conjunction with the 4nshell element. The theory used is analogous to the theory used the 2D solid element, see Sec The incompatible modes are added to the midsurface displaceminterpolations; only the membrane action of the 4node shell element is affected. A typical use of the incompatible modes feature would be to
prove the inplane bending response of the 4node element. im The incompatible modes feature is only available for 4single layer shell elements in which all nodes are on the midsof the element. The incompatible modes feature is available in linear and nonlinear analysis, for all formulations except for theULJ formulation.
2.7: Shell elements
ADINA R & D, Inc. 115
c) User input director vector (single vector)
at node k. Node k has 5 DOFs.
0 knV
0 knV
element 1
element 1
element 2
element 2
Userdefined midsurface director vector at node k
k
k k
Figure 2.74: (continued)
2.7.2 Mate
!
lasticcreepvariable, viscoelastic.
mall displacement/small e displacement/large ent/large strain ULH
ll strain and the large be used with any of the
icultilinear or the plasticorthotropic material models. The large
sed with the ultilinear material models.
rial models and formulations
The shell element can be used with the following material models: elasticisotropic, elasticorthotropic, plasticbilinear, plasticmultilinear, plasticorthotropic, thermoisotropic, thermoorthotropic, thermoplastic, creep, plasticcreep, multilinearplasticcreep, creepvariable, pvariable, multilinearplasticcreep
! The shell element can be used with a sstrain, large displacement/small strain, largstrain ULJ formulation or a large displacemformulation. The small displacement/smadisplacement/small strain formulations canabove material models. The large displacement/large strain ULJ formulation can be used with the plasticbilinear, plastmdisplacement/large strain ULH formulation can be uplasticbilinear or plasticm
Chapter 2: Elements
116 ADINA Structures ⎯ Theory and Modeling Guide
Angle 90
Angle = 90
Initial configuration
Final configurationZ
XY
Figure 2.75: Example of change in direction of director vector
due to deformations. At time t the vector
is not normal to the shell midsurface.
tnV
( )L2
L1 + L2
k k + 1
L1 L2
0 knV
0 k+1nV
0nV
0nV
tVn
a) Interpolation of director vector inside element
b) Change in direction of director vector due todisplacements and deformations (with shear)
In the small displacement formulation/sma ormulation, the displacements and rotations are assumed to be infinitesimallysmall. In
ll strain f
the large displacement formulation/small strain
formulation, the displacements and rotations can be large, but the strains are assumed to be small. In the large displacement/large strain ULJ formulation, the total strains can be large, but the incremental strain for each time step should be small (< 1%). In the large displacement/large strain ULH formulation, the total strains and also the incremental strains can be large.
2.7: Shell elements
ADINA R & D, Inc. 117
r
r
s
s
s
t
t
r
rs t
=
s ts
t r=
t r2 2
Figure 2.76a: Definition of the local Cartesian system ( , , )
at an integration point
r s t
The large displacement/large strain formulation can only be used in conjunction with 3node, 4node, 9node or 16node single layer shell elements, in which the shell geometry is described in terms of midsurface nodes.
r
r
s
s
s
t
t
r
t =sr
r sr =
s t
s t
Figure 2.76b: Definition of the midsurface Cartesian system ( , , )
at an integration point
r s t
, , s t r= ( = 0)t
t
Chapter 2: Elements
118 ADINA Structures ⎯ Theory and Modeling Guide
nt/small nd the use
loading are ation.
atrix is optionally updated
can improve the obtained frequencies and mode shapes when the shell structure is relatively flexible and
2.7.3 Shel
!
5 degrees of freedom node: A node "k" that is assigned 5 degrees of freedom incorporates the following assumptions:
Cartesian
system (or to the skew system if such a system is defined at the
< O e director imassociated with the node. You can directly enter the director
the prog late it (see F . In the latte he progra ates the direc by taking the of all nor rs (one norm r is gen shell e ched to nod node.
If two (or more) e tached to thopp rected n DINA reverdirected normals, so ch (nea same dir
The use of a linear material with the small displacemestrain formulation corresponds to a linear formulation, aof a nonlinear material with the small displacement/small strain formulation corresponds to a materiallynonlinearonly formulation.
! The effects of displacementdependent pressure taken into account in the large displacement formul
In frequency analysis, the stiffness mby deformationdependent pressure loads acting onto shell elements. The update
deformationdependent pressure loads are applied.
l nodal point degrees of freedom
Either 5 or 6 degrees of freedom can be assigned at a shell element midsurface node.
< The translations uk, vk, wk are referred to the global
node)
e = 0 as 0 knV ) is nly on vector (denoted at t
vector or let ram calcu ig. 2.74)r case, t m calcul tor vectoraverage mal vecto al vecto
e k) at thee node ha
erated per lement attalements at ve
ses the oppositely ositely di ormals, Athat all normals atta ed to the node have
rly) the ection.
< The rotations ,k kα β are referred to the local midsurface system (see Fig. 2.77) defined at time = 0 by
2.7: Shell elements
ADINA R & D, Inc. 119
0
01 0
2
kk n
kn
×=
×Y VVY V
k
For the special case when the vector is parallel to the Y axis, the program uses the following conventions:
0 0 02 1k k
n= ×V V V
0 knV
0 0 01 2 when k k k
n≡ ≡ ≡V Z V X V Y +
and
0 0 01 2 when k k k
n≡ − ≡V Z V X V Y ≡ −
vk
wk
uk k
Vkn
k
No rotational
stiffness for
this DOF
Z
YX
Figure 2.77: Shell degrees of freedom at node k
When using the large displacement formulation, the definitions of and are only used at time=0 (in the initial
and are d
is The is automatically
01kV 0
2kV
configuration) after which the vectors t knV 1
t kVupdated using incremental rotations at the nodal points, an
calculated by the crossproduct 2 1t k t k t k
n= ×V V V .
rotational degree of freedom along
2t kV
0 knV<
deleted by ADINA.
Chapter 2: Elements
120 ADINA Structures ⎯ Theory and Modeling Guide
6 degrees es of freedom
The translations uk, vk, wk are referred to the global Cartesian
m < A director vector would in general not be input for this node.
Note that any director vector input is ignored for a shell node
< The ny normal vectors at node k as
ere are shell elements attached to the node (see Fig. 2.74(a)).
vector elemen nding to the rotational degrees of freedom at this node are first formulated in the local midsurface
he
< The three rotational degrees of freedom at node k referred to at
of
of freedom node: A node "k" that is assigned 6 degre incorporates the following assumptions:
<syste or any skew system assigned to node k.
(with six degrees of freedom.)
program generates as math Hence each individual shell element establishes at node k a
normal to its midsurface. The components of the shell t matrices correspo
system defined by the normal vector and then rotated to the global Cartesian system (or to the skew system, if defined at tnode).
the global Cartesian system (or to the skew system, if defined the node) can be free or be deleted.
! Some modeling recommendations are given below for the use the shell elements:
< Always let all director vectors be established by the program; director vectors should only be defined via input if specific director vectors are required in the modeling.
< Always specify 5 degrees of freedom at all shell midsurface nodes except for the following cases in which 6 degrees of freedom should be used:
(i) shell elements intersecting at an angle, (ii) coupling of shell elements with other types of
structural elements such as isoparametric beams (e.g., in the modeling of stiffened shells),
(iii) coupling of rigid links (see Section 5.11.2) to the shell midsurface nodes
2.7: Shell elements
ADINA R & D, Inc. 121
of together with the deletion of the X
tation degree of freedom is only applicable in small displacement
r uk, al
(iv) imposing specific boundary conditions.
All of the above considerations are illustrated in the schematic example shown in Fig. 2.78. Note that in the example, the use 6 degrees of freedom at node 1 roanalysis.
If a large displacement formulation is used in this example, node 1 must be assigned 5 degrees of freedom and the applied concentrated moments must then refer to the local midsurface system at that node, which consists of the global directions fovk, wk and the local directions 1
t kV and 2t kV for the increment
rotations kα and kβ . (Alternatively, 6 degrees of freedom could be used if a beam of very small stiffness is defined along the edge of the plate.)
Chapter 2: Elements
122 ADINA Structures ⎯ Theory and Modeling Guide
X
YZ
9 8 7 65
4 3 2 1
No X,Y,Z translations
No X,Y rotations
2node isobeam
elements
Rigid link
(Node 4 as
master node)
Concentrated
loads at node 1
MY FX
MZ FZ
4node shell elements
Figure 2.78: Example on the recommended use of shell elements
NodeNumberof DOF
1
2
3
54
6
7
89
6
5
5
6
5
6
6
6
6
Degree of freedom
1 2 3 4 5 6
*
*
*
Referencesystem
Global
Global
Global
Global
Global
Global
Midsurface
Midsurface
Midsurface
= Deleted because of no stiffness at this DOF= Free DOF
* = The AUI automatically deletes the 6th DOFwhen a midsurface system is specified for the node
2.7.4 Transition elem
! The sh lso be employed as transition elements. A transition element is obtained by using, instead of a shell
transition element.
ents
ell elements can a
midsurface node, one node on the top surface and one node on the bottom surface of the shell element. Each of these nodes has thenonly 3 degrees of freedom corresponding to translations in the global X, Y, Z coordinate directions. Fig. 2.79(a) shows a cubic
2.7: Shell elements
ADINA R & D, Inc. 123
vkukwk
Transition nodes
Z
XY
L
L
t
s
r
L uniform
over element
b) Zero energy mode for flat element
with only transition nodes
a) Example of a transition element
Figure 2.79: Transition elements
! Note that although the degrees of freedom at a shell transition node are those of a threedimensional isoparametric element, the assumption that the stress "normal" to the midsurface of the shell (in the tdirection) is zero distinguishes a transition element from a 3D solid element even when a shell element has only transition nodes.
ent,
i.e.,
! In particular, if only transition nodes are used on the elemthen the element has a zero energy mode corresponding to auniform strain in the tdirection. Fig. 2.79(b) shows this zeroenergy mode. For a 3D solid element, this mode corresponds, of course, to a uniform strain in the tdirection.
If not all nodes on a transition element are transition nodes,
Chapter 2: Elements
124 ADINA Structures ⎯ Theory and Modeling Guide
there is at least one midsurface node, then the zeroenergy mode
o to solid intersections, see Fig. 2.710 and the
following reference:
shown in Fig. 2.79(b) is not present. ! The transition element is particularly useful in modeling shell tshell and shell
Shell to shell intersection
Shell to solid intersection
Figure 2.710: Examples of use of transition element
ref. K.J. Bathe and L.W. Ho, "Some Results in the Analysis of Thin Shell Structures," in Nonlinear Finite Element Analysis in Structural Mechanics, W. Wunderlich et al
2.7.5 Numerical integration
! Numerical integration is used for the evaluation of the element matrices, and the default integration or a higher order should always be used; then no spurious zero energy modes are present.
! Gauss numerical integration is used in the rs plane. For the regular 4node element, the default order the regular 8node and 16node elements, the default integrations are respectively 3×3 and 4×4 point integration.
ref. KJBSection 5.
(eds.), SpringerVerlag, 1981.
is 2×2 integration. For
5
2.7: Shell elements
ADINA R & D, Inc. 125
! Either Gauss or NewtonCotes numerical integration is used through the shell thickness. Usually, 2point Gauss or 3point Ne a an
wtonCotes integration is appropriate for an elastic material, buthigher integration order may be more effective for elasticplastic alysis (see Fig. 2.711).
ref. KJBSection 6.8.4
t
tt
ttt
tt
2
1
Order = 2 Order= 3
3
2
1
43
21
5
5
5
4
43
3
2
22
1
11
654321
Order = 4
Order = 5
Order = 3 Order = 5 Order = 7
Order = 6
a) Gauss integration in the thickness direction
3
2
1
6
7
b) NewtonCotes integration in the thickness direction
Figure 2.711: Numerical integration through the shellelement thickness
3
4
Chapter 2: Elements
126 ADINA Structures ⎯ Theory and Modeling Guide
tion points for quadrilateral and triangular elements is given in Fig. 2.712. The locations of the
solid ee Section 2.2.3).
! The labeling of the integra
integration points in the rs plane are the same as for the 2D elements (s
s
s
r
r
INR=1,INS=1,INT=2
INR=1,INS=2,INT=3
INR=1,INS=1,
INT=1
INR=3,INS=1,INT=1
INR=2,INS=1,INT=1
3 3 2 Gauss integration:
2 2 3 NewtonCotes integration:
Figure 2.712: Examples of integration point labeling
a) Examples of integration point labeling for quadrilateral shell elements
t
t
2.7: Shell elements
ADINA R & D, Inc. 127
t
t
s
s
r
r
INS=4,INT=1
INS=7,INT=1
INS=7,INT=2
INS=1,INT=1
INS=1,INT=2
INS=4,INT=1
4 3 integration:
7 2 integration:
b) Examples of integration point labeling for triangular shell elements
Figure 2.712: (continued)
2.7.6 Composite shell elements
! The composite shell elements are kinematically formulated in the same way as the single layer shell elements, but
total thickness of the shell.
< Each layer can be assigned one of the different material m dels available. The element is nonlinear if any of the
he
< An arbitrary number of layers can be used to make up the
omaterial models is nonlinear, or if the large displacement formulation is used.
< The computation of accurate transverse shear stresses based on a threedimensional theory can be requested. This way t
Chapter 2: Elements
128 ADINA Structures ⎯ Theory and Modeling Guide
! Layers are numbered in sequential order starting from 1 at the bottom of the shell.
! The layer thicknesses can be assigned using one of two general approaches:
< Specify the total element thickness and the percentage of thickness for each layer.
< Specify the data of each layer (or ply) in terms of the weight per unit surface of the fiber Wf, the density of the fiber
condition of zero transverse shear stresses at the top and bottom surfaces of the shell are satisfied, and the transverse shear stresses are continuous at layer interfaces.
fρ and
the fiber volume fraction of the fibermatrix compound fφ . The AUI then computes the thickness of each layer
automatically using the formula f
f f
Wh
ρ φ= , in which it is
assumed that the ply is made of uniaxial fibers, so that the fiber thickness fraction perpendicular to the plate midsurface is proportional to the fiber volume fraction. The total thickness of the multilayered shell is then the sum of the ply thicknesses.
This approach is especially useful when the layers are fibermatrix composites.
! In order to take into account the change of material properties from one layer to another, numerical integration of the mass and stiffness matrices is performed layer by layer using reduced natural coordinates through the thickness of the element (see Fig. 2.713 nd 2.714). The relation between the element natural coordinate t d
aan the reduced natural coordinate tn of layer n is:
( )1
11 2 1n
i n n
it t
a =
⎡ ⎤⎛ ⎞= − + − −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦∑! ! (2.71)
2.7: Shell elements
ADINA R & D, Inc. 129
with
layer inate through the thickness = thickness of layer i a = total element thickness
are ions
t = element natural coordinate through the thickness tn = n natural coord
i!
a and ! funct of r and s. i
1
lV1
n
lV
11
lV
12
1
1
2
3
4a4
t
r
s
Z
Y
X
x,u
e1
e2
e3
l
l
l
l l l
Ve V
e V
V V V
kknkn
k kn
k
12
2 2
2 1
= 
=
Figure 2.713: 8node composite shell element
y,v
z,w
ln
aLayer n
Layer 1
E Gnxx
nxz n
xy
Figure 2.714: Multilayered plate
Chapter 2: Elements
130 ADINA Structures ⎯ Theory and Modeling Guide
The geometry of layer n is given by:
1 1 2 2 2
n nN N nk i n kk k k
k i k k nik k
a1
ii
x h x t h V= =
⎡ ⎤⎛ ⎞+ − + − +⎢ ⎥⎜ ⎟
⎝⎣∑ ∑ ∑ !! ! !! (2.7
=
=⎠ ⎦
! 2)
with
ix! = coordinates of a point inside layer n N = number of nodes = interpolation functions
khkix! = Cartesian coordinates of node k
= components of normal vector = total element thickness at node k
= thickness of layer i at node k
= 0 in the initial configuration, 1 in the deformed
configuration
In the above formula, the expression
kniV! k
nV!
ka ik!
[ ]!
12 2
nnn ik kk k
i
am=
= − + −∑ !! (2.73)
corresponds to the distance at node k between the element midsurface and the midsurface of layer n.
Accordingly, the displacements in layer n are:
( )0 02 1
1 1 2
n nN Nk n k kk
i k i k i k ik k
tu h u m V V kα β= =
⎛ ⎞= + + − +⎜ ⎟
⎝ ⎠∑ ∑ !
(2.74) with
= components of nodal displacements at node k
kiu
,k kα β = rotations of about and (see Fig. 2.713)
Using the expressions of the coordinates and displacements
0 knV 0
1kV 0
2kV
2.7: Shell elements
ADINA R & D, Inc. 131
defined in Eqs. 2.72 and 2.74, the contribution of each layer to
! In the analysis of thick plates and shells or in the analysis of mu w she be negligible compared to the flexural energy and high transverse shear stresses may be encountered. In these cases, the shell kinemexpressions of the shear deformation energy and transverse shear stresses.
the plate equilibrium equations and the 3dimensional equilibrium equations,
ty at at the top and bottom
surfaces of the shell can be obtained. Simultaneously, a shear red t the shear deformation energy approximation.
the element stiffness and mass matrices can be evaluated.
ltilayered structures such as composite sandwiches with loar rigidity, the transverse shear deformation energy may not
atic formulation must be amended to obtain more accurate
By combining for a plate in cylindrical bending,
an expression of the shear stress which satisfies stress continuithe layer interfaces and zero boundary values
uc ion factor is obtained which can be used to improve
The expression of the shear factor is given by
2 22a a
a a
xxxz
xzxz
kgG dz dzG− −
=
∫ ∫ (2.75
2 2
2
xz
I)
and in value the transverse shear stress in terms of the shear stra
xzγ is
( ) 2
2
a
a
xz xzxz
xz
( )xx xzI g zzσ γ= (2.76)
where
g dzG−∫
( )2
2
20
a
axx xxI D z z dz−
= −∫
1
nn xxxx n n
xy yx
EDν ν
=−
xxD = nD in layer n
= Young's modulus xx
xxE
Chapter 2: Elements
132 ADINA Structures ⎯ Theory and Modeling Guide
= shear modulus
xxG
,n nxy yxν ν = Poisson's ratios
a = total thickness
( ) ( )
2
0a
z
xx xxg z D z dξ ξ−
= − −∫
on point. The shear correction factor calculation is only available for the
edings, STRUCOME 90, DATAID AS & I, Paris, France,
2.7.7 Mas
can be employed with a lumped or a
consistent mass matrix, except for explicit dynamic analysis which
ss matrix is calculated using the isoparametric
formulation with the shell element interpolation functions.
n
r to ccount for element distortion.
The rotational lum sis
is,
Equations 2.75 and 2.76 are used in ADINA to calculate shear correction factors for each element and shear stress profiles at each integrati
linear elastic and linear orthotropic material models.
ref. O. Guillermin, M. Kojiƒ, K.J. Bathe, "Linear and Nonlinear Analysis of Composite Shells," Proce
November 1990.
s matrices
! The shell element
always uses a lumped mass.
• The consistent ma
• The lumped mass for translational degrees of freedom of midsurface nodes is M / n where M is the total element mass and is the number of nodes. No special distributory concepts are employed to distinguish between corner and midside nodes, oa
ped mass for all except explicit dynamic analy
( )2av
112n
rotational m ss matrix is assumed for 5 and 6degree of freedom nodes, and is applied to all m.
M t⋅ where is the average shell thickness. The same
a rotational degrees of freedo
The rotational lumped mass for explicit dynamic analysis is
avt
2.7: Shell elements
ADINA R & D, Inc. 133
( )2av12
An
+ , where avt is the average shell thickness and A is
the crosssectional area. The rotational mensure that the rotational degrees of
21M t⋅
asses are scaled up to freedom do not reduce the
critical time step for shell elements. The same rotational mass atrix is assumed for 5 and 6degree of freedom nodes and is
! The rotational lumped mass can be multiplied by a userlar ETA (except in explicit analysis).
2.7.8 Anisotropic failure criteria
n
e provided in ADINA for the analysis of shell tructures using the elasticisotropic, elasticorthotropic, thermo
ated during the analysis to determine
material of this
e in linear elastic or ator in nonlinear
oint of
es (and A
mapplied to all rotational degrees of freedom.
specified sca
! The study of material failure in composite structures has showthat various criteria can be used to assess failure. The following failure criteria arsisotropic or thermoorthotropic material models:
< Maximum stress failure criterion < Maximum strain failure criterion < TsaiHill failure criterion < Tensor polynomial failure criterion < Hashin failure criterion < Usersupplied failure criterion
These criteria can be evalu!
whether the material has failed. However, the stresses and properties of the model are not changed as a consequence calculation. It is therefore intended only for usthermoelastic analysis, or as a stress state indicanalysis.
! The material failure is investigated at each integration peach element during the analysis. The failure criteria valupossibly modes) are printed and/or saved according to the ADINstress printing and saving flags.
! The maximum stress failure criterion compares each component of the stress tensor (referred to the material principal
Chapter 2: Elements
134 ADINA Structures ⎯ Theory and Modeling Guide
irections) to maximum stress values. The input constants for this
t
second
the material second
ress in the material third cipal direction (cdir.)
c imum compressive stress in the material third
tress in the ac plane
Th
fied anymore:
t
t
dcriterion are:
Xt = maximum tensile stress in the material first principal direction (adir.)
Xc = maximum compressive stress in the material firsprincipal direction
Yt = maximum tensile stress in the materialprincipal direction (bdir.)
Yc = maximum compressive stress in principal direction
Zt = maximum tensile stprin
Z = maxprincipal direction
Sab = maximum absolute shear stress in the ab plane Sac = maximum absolute shear sSbc = maximum absolute shear stress in the bc plane
e failure of the material occurs when any of the following inequalities is not satis
c a t
c b
c c
X XY Y
τ< <
Z Zττ
ab ab ab
ac ac ac
bc bc bc
S SS SS S
τττ
< << <
wh al
t c ac bc
f
− < <− < <− < <
ere (τa,τb,τc,τab,τac,τbc) is the stress vector referred to the principmaterial directions.
The above conditions are applicable to 3D analysis. In plane stress analysis, the parameters Z , Z , S and S are not used.
! Note that the maximum stress criterion indicates the mode ofailure. Note that this criterion does not take into account any interaction between failure modes.
2.7: Shell elements
ADINA R & D, Inc. 135
iterion compares each component of the strain tensor (referred to the material principal
is
Xεt = maximum tensile strain in the material first principal
st
second principal direction (bdir.)
al ab
c
The failure of the material occurs when any one of the following ine tie
t
! The maximum strain failure cr
directions) to maximum strain values. The input constants for thcriterion are:
direction (adir.) Xεc = maximum compressive strain in the material fir
principal direction Yεt = maximum tensile strain in the material
Yεc = maximum compressive strain in the material secondprincipal direction
Zεt = maximum tensile strain in the material third principal direction (cdir.)
Zεc = maximum compressive strain in the material third principal direction
Sεab = maximum absolute shear strain in the materiplane
Sεac = maximum absolute shear strain in the material ac plane
Sεbc = maximum absolute shear strain in the material bplane
quali s is not satisfied anymore:
c a t
c b
X XY
ε εε< <Yε εε
c c t
ab ab ab
ac ac ac
bc bc bc
Z ZS SS SS S
ε ε
ε ε
ε ε
ε ε
εγγγ
< <
rred to the principal material directions.
sis. In plane however, the parameters Zεt, Zεc, Sεac and Sεbc are not
used.
< <− < <− < <− < <
where (εa, εb, εc, γab, γac, γbc) is the strain vector refe
The above conditions are applicable to 3D analystress analysis,
Chapter 2: Elements
136 ADINA Structures ⎯ Theory and Modeling Guide
count any interaction between failure modes.
s is an ailure of the material
occurs when the following inequality is not satisfied anymore:
b c
bc ac ab
! Note that the maximum strain criterion indicates the mode of failure. Note that this criterion does not take into ac
! The TsaiHill failure criterion provides a yield criterion applicable to materials with anisotropic failure strengths. Thiextension of the von Mises yield criterion. F
2 2 2
2 2 2
) ( ) ( ) 2 2 2
2 2 2 1a b c a b a cG H F H F G H G F
L M N
( τ τ τ τ τ τ τ
τ τ τ
+ + + + + − − −
+ + + <
τ τ
with
2 2 2
2 2 2
2 2 2
1 1 1 121 1 1 121 1 1 1
FX Y Z
G
2
1
2 ab
12 bc
2 ac
1
X Y Z
HX Y Z
S
⎞
⎝ ⎠
h
principal direction (adir.) Y
LS
=
M =S
N =
⎛ ⎞= − + +⎜ ⎟⎝ ⎠⎛= − +⎜ ⎟⎝ ⎠⎛ ⎞= + −⎜ ⎟
w ere
X = maximum absolute stress in the material first
= maximum absolute stress in the material second principal direction (bdir.)
Z = maximum absolute stress in the material third principal direction (cdir.)
Sab = maximum absolute shear stress in the ab plane Sac = maximum absolute shear stress in the ac plane Sbc = maximum absolute shear stress in the bc plane
2.7: Shell elements
ADINA R & D, Inc. 137
In the case when plane stress conditions are used, the material
versely isotropic (Y=Z), and the failure criterion reduces to behavior with respect to failure is considered as trans
2 2 2τ τ τ τ τ2 2 2 2 1a a b b ab
X X Y Sab
− + + <
In this case, the maximum allowable stresses Z, Sbc and Sac are not used. ! Note that this criterion includes the effect of interactions between failure modes, but does not indicate a specific mfailure.
ode of
! The tensor polynomial failure criterion assumes the existence of a failure surface in the stress space of the form:
1 with , 1,...,6i i ij i jF F i jτ τ τ+ = =
wi
th the convention
1 2 3
4 5 6
; ;; ;
a b c
ab ac bc
τ τ τ τ τ ττ τ τ τ τ τ
= = == = =
and with
1 111 1 1;F F
2 221 1 1;
bt bc
3 33;ct cc ct cc
1 1 1
at ac at ac
bt bc
X X X X
F FX X X X
F FX X X X
= + = −
0F F F
= + = −
= + = −
4 5 6, , =
44 55 661 1 1; ;F F F
S S S= = =
ab ac bc
Chapter 2: Elements
138 ADINA Structures ⎯ Theory and Modeling Guide
where the following strength components need to be defined:
rial first principal direction (adir.)
Xbt = maximum tensile stress in the material second
principal direction Xct = maximum tensile stress in the material third
S = maximum absolute shear stress in the material ab
e material ac plane
Sbc = maximum absolute shear stress in the material bc plane
For plane stress conditions, the parameters Zt, Zc, Sbc, Sac, Fbc
and Fac are not used.
! The interaction strengths Fij with i…j can be input, or calculated by ADINA according to Hoffman's convention:
Xat = maximum tensile stress in the mate
Xac = maximum compressive stress in the material first principal direction
principal direction (bdir.) Xbc = maximum compressive stress in the material second
principal direction (cdir.) Xcc = maximum compressive stress in the material third
principal direction ab
plane Sac = maximum absolute shear stress in th
1 1
with and , , ,
ijit ic jt jc
FX X X X
i j i j a b c
⎛ ⎞= − +⎜ ⎟⎜ ⎟
⎝ ⎠≠ =
! Note that this failure criterion includes interactions between the failure modes, but that the mode of failure is not given by the analysis.
! The Hashin failure criterion applies especially to fibrous composites where the fiber and the matrix failure mechanisms are
ers are predominantly distinct. In ADINA, we assume that the fib
2.7: Shell elements
ADINA R & D, Inc. 139
s
aligned with the first material principal direction (adirection), thumaking the material transversely isotropic about the adirection. Failure will occur when any of the following inequalities is not satisfied anymore:
( )2 22
2 2 1 (if 0)ab acaa
t abX Sτ ττ τ
++ < > (2.77
)
1 (if 0)aτ τ acX
< < (2.78)
( ) ( ) ( ) ( )( )
2 2 2
2 2 2 1 if 0bc b ab acb c
t tr abY S Sτ τ τ τ
τ τ− +2
cb cττ τ+
+ + < + >
(2.79)
( )( ) ( ) ( )
2⎡ ⎤⎛ ⎞
( )( )2 2 24
if 0c tr tr ab
b c
Y S S S
τ τ+ <
2 2 2 21
21
cb c
tr b b c ab acb c
YS
τ ττ τ τ τ ττ τ
+ −⎢ ⎥⎜ ⎟− +⎢ ⎥⎝ ⎠ +⎣ ⎦ + + + <
(2.710)
ction Yt = maximum tensile stress in the material second
principal direction (bdir.)
te
ane
are usually referred to as: tensfailure (2.77), compressive fiber failure (2.78), tensile matrix
with
Xt = maximum tensile stress in the material first principaldirection (adir.)
Xc = maximum compressive stress in the material first principal dire
Yc = maximum compressive stress in the material second principal direction
Sab = maximum absolute shear stress in the ab plane (nothat Sab = Sac)
Str = maximum absolute shear stress in the bc pl
The above inequalities ile fiber
failure (2.79), compressive matrix failure (2.710).
Chapter 2: Elements
140 ADINA Structures ⎯ Theory and Modeling Guide
n allows for the description of a failure envelope using up to four quadratic failure relations of
i=
! The usersupplied failure criterio
the form:
6
1if1 0i i ij i j i iF Fτ τ τ α τ+ = >∑
1 2 3
4 5 6
with; ;; ;
a b c
bc ac ab
τ τ τ τ τ ττ τ τ τ τ τ
= = == = =
The material failure occurs when any one of the failure surfaces is reached and the corresponding stress condition is satisfied at the same time.
! This flexible and very general usersupplied failure m
odel
allows you to input any quadratic failure criteria with failure mode
ref. Jones R.M., Mechanics of Composite Materials,
w Hill, 1975.
iteria site Materials," Journal of Reinforced Plastics
and Composites, Vol.3, pp.4062, January 1984.
l of Applied Mechanics, Transactions of the ASME, Vol. 50, pp. 481505, September 1983.
2.7.9 Elem t
r forces.
fol
interactions.
! The following references contain additional information about composite failure criteria:
McGra
ref. Tsai S.W., "A Survey of Macroscopic Failure Crfor Compo
ref. Hashin Z., "Analysis of Composite Materials A
Survey," Journa
ent outpu
You can request that ADINA either print and save stresses o
Stresses: Each element outputs, at its integration points, the
lowing information to the porthole file, based on the material
2.7: Shell elements
ADINA R & D, Inc. 141
be referred to the global (X,Y,Z) al
model. This information is accessible in the AUI using the given variable names.
The results can be requested to
axes, or to the loc ( ), ,r s t axes (see Fig. 2.76a), or to the
midsurface ( ) , ,r s t axes (see Fig. 2.76b). For the large
displacement/large strain ULH formulation, both the local and midsurface axes are the ( ) , ,r s t axes, so the results in both the
local and midsurface systems are identical. In addition, for orthotropic material models, the results referred to the material axecan be requested.
s
s: STRESS(XYZ), STRAIN(XYZ), FE_EFFECTIVE_STRESS,
ropic: STRESS(XYZ), STRAIN(XYZ), THERMAL_STRAIN(XYZ), FE_EFFECTIVE_STRESS,
Plasticbilinear, plasticmultilinear, without thermal effects: STRESS(XYZ), STRAIN(XYZ),
PLASTIC_STRAIN(XYZ), FE_EFFECTIVE_STRESS,
YIELD_STRESS, ACCUM_EFF_PLASTIC_STRAIN, MPERATURE
Thermcreepvariable, plasticcreepvariable, multilinearplasticcreepvar STRESS(XYZ), STRAIN(XYZ), PLTHERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE,
Elasticisotropic, elasticorthotropic: STRESS(XYZ), STRAIN(XYZ), FE_EFFECTIVE_STRESS Elasticisotropic with thermal effect
THERMAL_STRAIN, ELEMENT_TEMPERATURE Thermoisot
ELEMENT_TEMPERATURE
PLASTIC_FLAG,
YIELD_STRESS, ACCUM_EFF_PLASTIC_STRAIN Plasticbilinear, plasticmultilinear with thermal effects: PLASTIC_FLAG, STRESS(XYZ), STRAIN(XYZ), PLASTIC_STRAIN(XYZ), FE_EFFECTIVE_STRESS,
THERMAL_STRAIN(XYZ), ELEMENT_TE
oplastic, creep, plasticcreep, multilinearplasticcreep,
iable: PLASTIC_FLAG, NUMBER_OF_SUBINCREMENTS,
ASTIC_STRAIN(XYZ), CREEP_STRAIN(XYZ),
Chapter 2: Elements
142 ADINA Structures ⎯ Theory and Modeling Guide
ACFE_EFFECTIVE_STRESS, YIELD_STRESS, EFFECTIVE_CREEP_STRAIN Plasticorthotropic without therm PLASTIC_FLAG, STPLASTIC_STRAIN(XYZ), HILL_EFFECTIVE_STRESS, YIPlaSTRESS(XYZ), STRAIN(XYZ), PL
CUM_EFF_PLASTIC_STRAIN,
al effects: RESS(XYZ), STRAIN(XYZ),
ELD_STRESS, ACCUM_EFF_PLASTIC_STRAIN sticorthotropic with thermal effects: PLASTIC_FLAG,
ASTIC_STRAIN(XYZ), HILL_EFFECTIVE_STRESS, YIELD_STRESS, ACCUM_EFF_PLASTIC_STRAIN, THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE Viscoelastic: STRESS(XYZ), STRAIN(XYZ), THERMAL_STRAIN(XYZ), ELEMENT_TEMPERATURE
In the above lists,
when results are referred to the global (X,Y,Z) axes, STRESS(XYZ) = STRESSXX, STRESSYY, STRESSZZ, STRESSXY, STRESSXZ, STRESSYZ
when results are referred to the ( ), ,r s t or ( ) , ,r s t axes,
STRESS(XYZ) = STRESSRR, STRESSSS,
.
re
, each element outputs,
at e in
STRESSTT, STRESSRS, STRESSRT, STRESSST
when results are referred to the material axes, STRESS(XYZ) = STRESSAA, STRESSBB, STRESSCC, STRESSAB, STRESSAC, STRESSBC
with similar definitions for the other abbreviations used aboveSee Section 12.1.1 for the definitions of those variables that a
not selfexplanatory.
! In addition, when a failure model is usedach integration point, the following information to the porthole
file, based on the failure criterion. This information is accessible the AUI using the following variable names:
2.7: Shell elements
ADINA R & D, Inc. 143
Maximum stress, maximum strain: FAFAILURE_FLAG_COMPRESSIONA, FAFAFAFAFAFA TsaiHill, tensor polyFA HaFAFAILURE_FLAG_TENSMATRIX, FAFAILURE_CRITERION_TENSFIBER, FAILURE_CRITERION_COMPFIBER,
UsFAFAFAFAFAFAFA
no
Semo thelat
promi
ILURE_FLAG_TENSIONA,
ILURE_FLAG_TENSIONB, ILURE_FLAG_COMPRESSIONB, ILURE_FLAG_TENSIONC, LURE_FLAG_COMPRESSIONC, I
ILURE_FLAG_SHEARAB, FAILURE_FLAG_SHEARAC, ILURE_FLAG_SHEARBC, FAILURE_FLAG
nomial: FAILURE_FLAG, ILURE_CRITERION
shin: FAILURE_FLAG_TENSFIBER, ILURE_FLAG_COMPFIBER,
ILURE_FLAG_COMPMATRIX,
FAILURE_CRITERION_TENSMATRIX, FAILURE_CRITERION_COMPMATRIX
ersupplied: FAILURE_FLAG_SURFACE1, ILURE_FLAG_SURFACE2, ILURE_FLAG_SURFACE3, ILURE_FLAG_SURFACE4, FAILURE_FLAG, ILURE_CRITERION_SURFACE1, ILURE_CRITERION_SURFACE2, ILURE_CRITERION_SURFACE3, ILURE_CRITERION_SURFACE4
See Section 12.1.1 for the definitions of those variables that are t selfexplanatory.
ction results: The computation of stress resultants, forces and ments, the computation of membrane strains and curvatures, and position of the neutral axes can be requested in ADINA. The ter is especially useful for multilayered shell elements. These results are given at the locations corresponding to the jections of the integration points on the shell element
dsurface (see Fig.2.715).
Chapter 2: Elements
144 ADINA Structures ⎯ Theory and Modeling Guide
mo ate the stress e alw
Fig. 2.716 shows the convention used for positive forces and ments. Table 2.71 contains the formulas used to evalu
and strain resultants and the neutral axes positions. They arays referred to the local Cartesian system ( ), ,r s t (see Fig. 15). 2.7
A
B
C
D
r
s
+ h_2
 h_2
Z0r
b) Definition of the neutral axes
Figure 2.715: Conventions and definitions for section
results output
E zdzr
E zdzs
E dzr
E dzs
Z0s =
Z0r =
Point labels INR INS
A
B
C
D
1
1
2
2
1
2
1
2
Layer number
5
432
1
h/2
h/2
h/2
h/2
h/2
h/2
h/2
h/2
a) Point labeling for section results(2 2 2 Gauss integration)
2.7: Shell elements
ADINA R & D, Inc. 145
sults output for th Stress resultants
Table 2.71: Section re e shell element
(forces and moments) / 2 / 2
/ 2 / 2
/ 2
/ 2
/ 2
h h
r r r r r r s sh hh
s s s sh h
t t
r s r s
r t r th
R dz M z dz/ 2
/ 2
h
s s r r
/ 2h
/ 2 t th
/ 2 / 2
/ 2
h h
r s r s/ 2
/ 2
h h
h
/ 2
/ 2
h
s t s th
R dz
R dz
M z dz
R dz
R dz
R
σ σ
σ σ
σ
− −
− −
−=
∫∫ ∫
∫ ∫∫
= = ∫= = −
M z dzσ σ= =
σ−
= ∫
− −
σ−
= ∫ dzMembrane strains and curvatures
( ) ( )12 1
2r r 1r r 2r r 1r r1r r z
z zε ε ε ε
ε χ
( ) ( )
( ) ( )
2 1mr r s s
m s s
2r s 1r s 2r s 1r smr s
z zε
12 1 2 1
2s s 1s s 2s s 1s s1s s r rz
z z z z
12 1 2 1
1r s r szz z z z
ε ε ε εε χε
ε ε ε ε
− −− =
− −
− −
ε ε
=−
=
= −
Indices 1 and 2 refer e lowest and highest integration points on a normal to the elemen
the shell elements.
− −− =
−
χ =− −
espectively to th rt midsurface.
results are given with respect to the local Cartesian system of Note that the section
Chapter 2: Elements
146 ADINA Structures ⎯ Theory and Modeling Guide
r
r
s
s
t
t
Rrr
Mrs
Rrs
Rrt
Rrs
Rst
Rss
MrsMss
Mrr
Figure 2.716: Nomenclature for shell stress resultants
(positive)
The corresponding AUI variables are (for all material models): MEMBRANE_FORCERB, MEMBRANE_FORCESB, MESHEAR_FORCESB, BENDING_MOMENTRB, BENDING_MOMENTSB, BENDING_MOMENTRBSB, MEMBRANE_STRAINRB, MEMBRANE_STRAINSB, MEMBRANE_STRAINRBSB, CURVATURERB, CURVATURESB, CURVATURERBSB,
to
uested in ADINA. The pro dKJ
namNODAL_FORCEZ, NODAL_MOMENTX, NODAL_MOMENTY,
ref. KJBExample 5.11
pp. 358359
MBRANE_FORCERBSB, SHEAR_FORCERB,
SHEAR_STRAINRB, SHEAR_STRAINSB, NEUTRAL_AXIS_POSITIONRB, NEUTRAL_AXIS_POSITIONSB.
See Section 12.1.1 for the definitions of those variables that are not selfexplanatory.
Nodal forces: The nodal point force vector which corresponds the element internal stresses can also be req
ce ure for the calculation of the force vector is given in ref. B. The nodal forces are accessible in the AUI using the variable es NODAL_FORCEX, NODAL_FORCEY,
NODAL_MOMENTZ.
2.7: Shell elements
ADINA R & D, Inc. 147
2.7.10 Selecti
! The following types of shell elements are available:
on of elements for analysis of thin and thick shells
Number of nodes
Single layer with midsurface nodes only
Multilayer or top & bottom
4node MITC4 MITC4 8node MITC8 MITC8 9node MITC9 DISP9
Quadrilaelement
teral s
16node MITC16 DISP16 3node MITC3 MITC4 Collapsed 6node DISP7 Collapsed DISP7 Collapsed
Trianguelement
lar s
9node DISP10 Collapsed DISP10 Collapsed The types of triangular elements are illustrated in Fig. 2.717.
Figure 2.717. Types of triangular elements
! mousually th
The st effective element for analysis of general shells is e 4node element, shown in Fig. 2.718(a). This element
Chapter 2: Elements
148 ADINA Structures ⎯ Theory and Modeling Guide
does not lo e can be used for e :
anics
ar 88,
ck and has a high predictive capability and hencffective analysis of thin and thick shells. See references
ref. Dvorkin, E. and Bathe, K.J., "A Continuum MechBased FourNode Shell Element for General NonlineAnalysis," Engineering Computations, Vol. 1, pp. 771984.
r
r
r
s
ss
t
t t
Recommended element: MITC4
MITC16 MITC9
(a) 4node shell element for thick and thin shells.
Sometimes also useful elements: MITC16 and MITC9
Figure 2.718: Recommended elements for analysis of any shell
(b) 16node cubic shell element for thick and thin shells.9node shell element for thick and thin shells.
ref. Bathe, K.J. and Dvorkin, E., "A FourNode Plate Bending Element Based on Reissner/Mindlin Plate Theory and a Mixed Interpolation," Int. J. Num. Meth. in Eng., Vol. 21, pp. 367383, 1985.
2.7: Shell elements
ADINA R & D, Inc. 149
ref. Bathe, K.J. and Dvorkin, E., "A Formulation of General e of Mixed Interpolation of Int. J. Num. Meth. in Eng., Vol.
22 pp. 697722, 1986.
alem, M.L and Bathe, K.J., "HigherOrder MITC General Shell Elements," Int. J. for Num. Meth in Engng,
An Evaluation of the MITC Shell Elements," Comp. Struct,
p. Struct,
elemestrains, but the physm ring strains, then the element becomes very stiff as its thi
ima 4 lateral elechoMITC9 and MITC16 elem
! she
ref. KJBpp. 40340
Shell Elements The UsTensorial Components,"
ref. Buc
Vol. 36, pp. 37293754, 1993. ref. Bathe, K.J., Iosilevich, A. and Chapelle, D., "
Vol. 75, pp.130, 2000. ref. Lee, P.H. and Bathe, K.J., Development of MITC
isotropic triangular shell finite elements, ComVol.82, 945962.
In linear analysis it can be effective to use the 9node shell
element (the MITC9 element) or the 16node shell element (the MITC16 element) (see Fig. 2.718(b)). However, the use of these elements can be costly.
! The phenomenon of an element being much too stiff is in the literature referred to as element locking. In essence, the phenomenon arises because the interpolation functions used for an
ment are not able to represent zero (or very small) shearing or mbrane strains. If the element cannot represent zero shearing
sical situation corresponds to zero (or very l) sheaal
ckness over length ratio decreases. Our MITC3, MITC4, MITC9 and MITC16 elements are
ented to overcome the locking problem. When you choose plemnode shell element, the MITC4 element is used for quadriments and MITC3 is used for triangular elements. When you ose a 3, 9 or 16node single layered shell element, the MITC3,
ents are used. In order to arrive at an appropriate element idealization of a thinll, it may be effective to consider the behavior of a single
element in modeling a typical part of the shell. As an example, if a shell of thickness h and principal radii of curvatures R1 and R2 is to
8
Chapter 2: Elements
150 ADINA Structures ⎯ Theory and Modeling Guide
d
tes (e.g., constant bending moments) tells what ize of element, and hence element idealization, can be used to
e mployed together with the elements of Fig. 2.718 in the analysis
ll enough in size.
be analyzed, a single element of this thickness and these radii ansupported as a cantilever could be subjected to different simple stress states. The behavior of the single element when subjected to the simple stress stassolve the actual shell problem. ! For the analysis of thick shells, the elements depicted in Fig. 2.719 can be used as well. If necessary, these elements can also beof a thin shell, but then only a few elements should be employed to model a special region, where necessary, such as a transition region, a cutout, and a triangular region. Also, the elements should be sma
Nodes 1 to 4 (corner nodes) must be input
Optional nodes
a) Quadrilateral elements b) Triangular elements
1
2 23
34
Figure 2.719: Elements for analysis of thick shells
1 and 4
node,
s
.9.3) to model plates connected by bolts.
! It is recommended that, as often as possible, only the 34node, 9node and 16node elements be employed. ! Note that the elements in Fig. 2.719 can also be used atransition elements by assigning top and bottom surface nodes instead of a midsurface node.
! The shell elements can be used in conjunction with the boltshaft elements (Section 2.4.5) and boltspring elements (Section 2
2.8: Pipe elements
ADINA R & D, Inc. 151
2.8 Pipe elem
Two types of pipe element are available: pipebeam element (no or 4 no lement has 4 nodes). Figure 2.81 shows the conventions used.
! ressure, see Section 2.8.2.
KJB.
ref. KJBSections 5.4.1,
6.5.1
ents
!
ovalization/warping of the crosssection, the element has 2des); pipeshell element (with ovalization/warping, the e
Assemblages of pipe elements can be subjected to internal p
! The pipebeam element has 6 displacement degrees of freedomper node (3 translations and 3 rotations, see Fig. 2.81).
The basic equations used in the formulation of the pipebeam element are given in ref.
Chapter 2: Elements
152 ADINA Structures ⎯ Theory and Modeling Guide
R
ac
aa
ab
cc
Convention forpipe stresses
a
b
cs
t r
Integration point
4
2
3
1
Auxiliary node KK
Nodes 1 to 4and KK mustlie in one plane(rs plane).
XY
Z
Element coordinate system convention:
longitudinal direction = adirection (or )
radial direction = bdirection (or )
circumferential direction = cdirection (or )
Figure 2.81: Pipe element configuration and displacementdegrees of freedom and stress convention
Pipe global displacementdegrees of freedom
! The pipeshell element has, in addition to the 6 pipebeam degrees of freedom, also 3 or 6 ovalization and 3 or 6 warping deg
The formulation of the pipeshell elemen (ovalization/w
included) is described in the following references:
aly. 1, pp. 93100,
1980.
rees of freedom per node allowing the crosssection to ovalize and warp.
t arping
ref. K.J. Bathe and C.A. Almeida, "A Simple and Effective Pipe Elbow Element Linear An sis," J. Appl. Mech., Transactions of the ASME, Vol. 47, No
2.8: Pipe elements
ADINA R & D, Inc. 153
ref. K.J. Bathe, C.A. Almeida and L.W. Ho, "A SimpEffective Pipe Elbow Element Some Nonlinear Capabilities," Comp. & Structures, Vol. 17, No. 5/6, pp.
h pi
et i
1,
.
916, 1982.
le and
659669, 1983.
! T e peshell formulation also takes into account the effects of stiffening due to internal pressure, and the effects of interactions b ween pipes and flanges, and p pes with different radii, see Fig. 2.82. The formulations are described in the following references:
ref. K.J. Bathe and C.A. Almeida, "A Simple and Effective Pipe Elbow Element Interaction Effects," J. Appl. Mech., Transactions of The ASME, Vol. 49, pp. 165171982.
ref. K.J. Bathe and C.A. Almeida, "A Simple and Effective Pipe Elbow Element Pressure Stiffening Effects," JAppl. Mech., Transaction of the ASME, Vol. 49, pp. 914
Chapter 2: Elements
154 ADINA Structures ⎯ Theory and Modeling Guide
Pipe elements
Pipe centroidal axis
Pipe centroidal axis
Rigid flange
a) Pipe connected to a rigid flange
b) Pipes of different radii joined together
Junction
R1
R2
Figure 2.82: Interaction effects with the pipe element
! In the description of the pipeshell element kinematics, reference is made to the inplane and outofplane bending deformation modes. otN e that the inplane bending corresponds to
e bending in the rs plane and the outofplane bending orresponds to the bending in th rt plane (see Fig. 2.81 and .83).
be element models the usual beam strains
odels in addition ovalization and warping effects by including the following strains (see Fig. 2.81 for notation):
he on
thc e2
! Whereas the pipe amonly, the pipeshell element m
< T ovalization is included by the von Karman ovalizatimodes with
2.8: Pipe elements
ADINA R & D, Inc. 155
w
w ξζ φ
∂=
∂
and
( )
2sin cos 1ov w w wξ ζ ζφ φ2 2cos cos
aaeR a R a
ζφ θφ⎜ ⎟− ∂−⎝ ⎠
⎛ ⎞− ∂= − ⎜ ⎟
0ovabγ = (2.81)
1ov
ac
wcosR a
ζγφ θ
=∂
− ∂
r
r
s
s
t
t
Mt(inplane moment)
M
(outofplane moment)s
Mt
Ms
Plane of pipe element
Plane of pipe element
Auxiliary node
Auxiliary node
a) Inplaneonly bending conditions
b) Outofplane bending conditions
Pipe element
Pipe element
Figure 2.83: Loading conditions for the pipe element
Chapter 2: Elements
156 ADINA Structures ⎯ Theory and Modeling Guide
21ov wξ2 2cce w
a ζ ζφ
⎜ ⎟⎜ ⎟∂
⎛ ⎞∂= − +
⎝ ⎠
< The warping is included using shell theory as
cos 11cos cosaa R a R a
w we ηφ ζ
φ φ θ⎜ ⎟⎜ ⎟∂⎛ ⎞
= −⎛ ⎞
− − ∂⎝ ⎠⎝ ⎠
0wabγ
=
sin1cosac a R a
w w wη η φγ
φ φ∂ −
0wcce
∂= − (2.82)
=
ere, a is the pipe
mean radius, R is the bend radius and θ and Hφ are the angles defined in Fig. 2.81.
placements wξIn the formulation of the element the dis for the
ov in Eq. 2.8alization 1 and wη for the warping in Eq. 2.82 are erpolated by sine and int cosine functions associated with the
ovalization and warping degrees of freedom.
! elem
pipe bends.
degrees of freedom is recommended:
Note that the above equations for the pipebeam and pipeshell
ents give the same stress response if there is no ovalization. Hence, the ovalization and warping of the pipeshell element canbe thought of as an additional kinematic mode to the pipebeam response which results in an appropriate reduction of stiffness of
! For the pipeshell element, the following use of the ovalization/warping
< Activate the inplane ovalization/warping degrees of freedom if the pipe elements undergo only inplane deformations, see Fig. 2.83(a).
2.8: Pipe elements
ADINA R & D, Inc. 157
< Activate the outofplane ovalization/warping degrees of
< Activate all ovalization/warping degrees of freedom if the
!
deletion of all nodal
ovalization/warping degrees of freedom
ow, a nondimensional geometric factor λ is
defined as
freedom if the pipe elements undergo only outofplane deformations, see Fig. 2.83(b).
pipe elements undergo general threedimensional deformations.
In order to model the presence of a flange at a pipe node (of a pipeshell element), all ovalization/warping degrees of freedomshould be deleted at the node. The
implies that the pipe section at that node remains circular. In addition, the option of enforcing the zeroslopeofpipeskin must be employed to model the flange (see Fig. 2.84).
! Note that a symmetry condition implies zero warping but not necessarily zero ovalization.
! For a pipe elb
Rδλ
2 21a ν=
−
pipe bend radius, δ = pipe wall thickness, a = pipe cross
section mean radius, v = Poisson's ratio.
where R =
Chapter 2: Elements
158 ADINA Structures ⎯ Theory and Modeling Guide
W
W
W
W
W
Wa
a
a a
aa
a
b
b
b
b
b
b
= 0
= 0
= 0
Node j
Node j
Node i
Element i
ii+1
Symmetry plane,
Nonzero ovalization andzero warping at node jfor element i.
Figure 2.84: Zeroslopeofpipeskin conditions
(b) Symmetry condition
(a) Flange conditions
Ovalization and warping
equal to zero at node j
for elements i and i + 1.
Ovalization and warping
equal to zero at node j
for element i.
Element i + 1 Element i
Element i
W valu on/warping effects of the crosssection become mpipe
ith a decreasing e of λ, the ovalizatiore pronounced. Note that for a straight
element, R → ∞ and thus the crosssection does not ovalize/warp except if connected to another pipe element with a finite bend radius.
2.8.1 Material models and formulations
! T e used material mod plastic ultilinear, thermoisotropic, thermoplastic, creep, plasticcreep, multilinearplasticcreep, creepvariable, plasticcreepvariable, multilinearplasticcreepvariable.
he pipe elements can b with the following els: elasticisotropic, bilinear, plasticm
2.8: Pipe elements
ADINA R & D, Inc. 159
! or a
cement In all cases,
only small strains are assumed.
ulation
material with the small displacement formfor
In the leffects arenot for the displacement effects are not accounted for in the element cross
!
2.8.2 Pipe internal pressures
ent end nodes, as
s
pressure. The transversal force acting on the elementary axial length dL of the pipe axis is
The pipe elements can be used with a small displacementlarge displacement formulation. In the small displacement formulation, the displacements and rotations are assumed to be infinitesimally small. In the large displa formulation, the displacements and rotations are assumed to be large.
All of the material models in the above list can be used with either form . The use of a linear material with the small displacement formulation corresponds to a linear formulation, and the use of a nonlinear
ulation corresponds to a materiallynonlinearonly mulation.
arge displacement formulation, large displacement only included for the overall beam displacements and ovalization degrees of freedom. Hence, large
sectional deformations.
An element is to be considered a nonlinear pipe element if a large displacement analysis is performed and/or a nonlinear material is used and/or pipe internal pressure is present.
! If the element is loaded by internal pressure the load vector corresponding to that pressure is calculated as follows: First, two xial forces ( )pF and ( )pF are acting at the elema a1 a2
shown in Fig. 2.85. Their intensities are
( ) 2 ( ) 21 2π ; πp p
a1 i a2 iF a p F a p= =
where p1 and p2 are the internal pressures at nodes 1 and 2 (input anodal point pressures to the program), and ai is the internal radius of the crosssection. Also, if the element is curved, transversal forces are generated by the internal
2π i
Tad p dLR
= −F s
Chapter 2: Elements
160 ADINA Structures ⎯ Theory and Modeling Guide
where hk is the interpolation function c rresponding to no
and the nodal transversal forces are obtained as
k k TL
h d= ∫F F
o de k.
2
Fa(p)
2
Fa(p)
1
1R
dF(p) dL
Auxiliary node
3
4
Figure 2.85: Forces due to pipe internal pressure
T
s
! The nodal forces due to the internal pressure for an element present a selfbalanced system. Consequently, if a structure is
, as in the case of the can ilever in Fig. 2.86support reactions are equal to zero. Ho ever, in the case restrained system, the reactions at a support balance the forces and
( .
refree to deform t (a), the
w of a
moments which are a result of the internal stresses and the nodal forces that are equivalent to the internal pressure, see Fig. 2.86 b)
2.8: Pipe elements
ADINA R & D, Inc. 161
a) Cantilever pipe free to deform (zero reactions)
A
MA
RT
Ra
Internalpressure p
R = R = M = 0a T A
p
2a+RA
A B
RB
2
b) Fully restrained pipe
Figure 2.86: Support reactions for pipe loaded by internal pressure
In elastic deformations:
R =R = a p(a 2 a)A B i i
a = a  /2= internal radius
i
a = mean radius
= Poisson's ratio
! In the stress calculation the following assumptions are made regarding the internal pressure effects.
< For the pipebeam element it is assumed that the hoop stress t t
ccσ+∆ is determined by the given pressure, ( )t t t t pcc ccσ σ+∆ +∆= ,
where ( )t t pccσ+∆ is the hoop stress equivalent to the pressure
t t p+∆
norma (using thinwalled cylinder theory). Then, using that the l stress through the pipe skin is e al to zero, the
stress qu axial
t taaσ+∆ is obtained as
Chapter 2: Elements
162 ADINA Structures ⎯ Theory and Modeling Guide
( ) ( )t t t t t t t t IN t t TH t t t t p aa aa aa ccE e e eσ ν σ= − − +
where t t E+∆ and t t
+∆ +∆ +∆ +∆ +∆ +∆ +∆
ν+∆ are the Young's modulus and Poisson's
tio, e+∆ is the thermal strain corresponding to the mperature
and t t THrate t tθ+∆ ; and ainelastic strains, respectively.
o he ip s the
deformation of the pipe skin is that the material is free to
t taae+∆ t t IN
aae+∆ re the total mechanical and
< F r t p e hell element the basic condition regarding
deform in the circumferential direction. Hence, the total hoop strain is
( )
( )2
12
1 + 1
t t t t t t t t IN t t IN t t OVaa aa aa cc
t t TH
e e e e
e
ν
νν
+∆ +∆ +∆ +∆ +∆ +∆
+∆
− − +
⎛ ⎞−+ + ⎜ ⎟
( )
cc
pcc
e
Eσ
= −
⎝ ⎠
. ptio that th
etal plasticity
! Pressures at all nodal points of a piping model using the pipebeam or pipeshell elements (and other elements) must be defined
he pipebeam and pipeshell elements. See Section 5.7 for information about specifying int pr
2.8.3 Numerical inte
rs are in l
where t t OVcce+∆ is the strain due to ovalization of the crosssection
This relation is based on the assum n e inelastic deformation is incompressible (as it is in m and creep).
if internal pressure effects are to be included in some of the elements. The pressures are only used for t
ernal essures.
gration
! The element stiffness and mass matrices and force vectoall formulations evaluated using Gauss or NewtonCotes numericaintegration in the axial and the thickness directions and the composite trapezoidal rule along the circumference of the pipe section. The locations and labeling of the integration points are given in Fig. 2.87.
2.8: Pipe elements
ADINA R & D, Inc. 163
2
2
2
3
3
3
1 4
4
5
5 6 71
1
Node 1
Node 1
Node 1
Node 2
Node 2
Node 2
Order 3
Order 5
Order 7
Node 1
Node 1
Node 1
Node 1
Node 2
Node 2
Node 2
Node 2
1
1
2
321
Order 2
Order 1
Order 3
Order 44321
NewtonCotes formulas:
Gauss formulas:
a) Integration point locations on pipe centroidal axis (label INR)
Figure 2.87: Locations and labeling of integration points
Chapter 2: Elements
164 ADINA Structures ⎯ Theory and Modeling Guide
s
s s
ss
s
t
t t
tt
t
b
b
1
1 1
11
22
22
22
33
3
3
3
3
4
4
4
5
5
Gauss: order 3 NewtonCotes: order 5
Order 4 Order 8
Order 24Order 12
6
7
8
8
8
4
455 6
6
7
7
9
9
10
10
11
11
12
12
131415
1617 18 19 20
2122
2324
b) Integration point locations along thickness direction (label INB)
c) Integration point locations along circumferential direction (label INC)
Figure 2.87: (continued)
1
! For the pipebeam elements the following integration orders should be used (and are default in ADINA):
Ga th xial < 1point and 3point uss integration along e a
direction for the 2node and 4node elements, respectively
2.8: Pipe elements
ADINA R & D, Inc. 165
< 2point Gauss integration through the thickness o tskin
tegration alo he circumference
! sho
< 3point Gauss integration along the a < 2point Gauss integration through the thickness of the pipe
! ar
no inelastic strain effects are considered, use the same
ielding or creep is to be represented. For the integration along the length of the elements and the circumference of the crosssection, an integration order higher than that used in linear analysis may for the same reasappropriate.
2.8.4 Bolt
art, so that the initial force at the r
f he pipe
< 8point in ng t
For the pipeshell element the following integration ordersuld be used (and are default in ADINA):
xial direction
skin < 12point integration along the circumference if only the inplane ovalization degrees of freedom are activated < 24point integration along the circumference if the outofplane ovalization degrees of freedom are activated
The recommended numerical integration orders for nonlineanalysis are:
< Whenintegration orders as in linear analysis.
< When material nonlinear conditions are considered, use a higher integration order through the pipe skin if the pipe skin progressive y
on also be
element
! A bolt element is a prestressed pipe element with a given initialforce. The initial force is held constant during the equilibrium iterations at the time of solution ststart of the first solution step is the input initial force, no mattewhat other external loads or initial strains/stresses are input.
Chapter 2: Elements
166 ADINA Structures ⎯ Theory and Modeling Guide
d succes n
! lt
r be sm ll tropic e stic
2.8.5 Elem
! stent
matrix associates mass only with the pipes
translational and rotational degrees of freedom and is calculated using the isoparametric formulation with the displacement interpolations given on p. 408 of ref. KJB. Henceassociated with the ovalization/warping degrees of freedom.
slational degree of freedom i is
! During the solution for the first, second an sive solutiosteps, the bolt force can vary depending on the loads applied to themodel.
Bolt elements are created in a pipe element group with the booption activated. The initial force is entered either on the geometry (e.g., AUI command LINEELEMENTDATA PIPE) or directly specified on the elements (e.g., AUI command EDATA). ! The formulation used for the bolt elements can eithe adisplacements or large displacements. But only the iso lamaterial model can be used.
ent mass matrices
The pipe element can be used with a lumped or a consimass matrix, except for explicit dynamic analysis which always uses a lumped mass.
! The consistent mass
, no masses are
! The lumped mass for traniM
L⎛ ⎞⋅⎜ ⎟⎝ ⎠!
where M is the total elem
length and
ent mass, L is the element
i! is a fraction of the length associated with node i. The
rotational lumped mass is ( )2 21 13 4
io iM d d
L⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅ +⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠!
where
tlum
and warping degrees of freedom is
and d are the outside and inside diame ers of the pipe. The ped ovalization and warping mass for each of the ovalization
o id
iML⎝ ⎠
l lumped mass can be multiplied by a usercified scalar ETA (except in explicit analysis).
⎛ ⎞⋅⎜ ⎟!
.
! The rotationaspe
ref. KJBSection
5.4.1
2.8: Pipe elements
ADINA R & D, Inc. 167
2.8.6 Elemen
You can request that ADINA either print or save stresses or forces.
Strfolmovar ElaFE_EFFECTIVE_STRESS, EQEQ ElasticisST BC), FEQEQUIV_INTERNAL_HOOP_PRESSURE, THERMAL_STRAIN, ELEMENT_TEMPERATURE ThTHERMAL_STRAIN(ABC), EQUIV_INTERNAL_AXIAL_PRESSURE,
P_PRESSURE, ELEMENT_TEMPERATURE
TRAIN(ABC), THERMAL_STRAIN(ABC),
EQUIV_INTERNAL_HOOP_PRESSURE, FE_EFFECTIVE_STRESS, YIELD_STRESS, ACCUM_EFF_PLASTIC_STRAIN, ELEMENT_TEMPERATURE
creepvariable, plasticcreepvariable, multilinearplasticcreep: PLASTIC_FLAG, NUMBER_OF_SUBINCREMENTS,
FE
t output
esses: Each element outputs, at its integration points, the lowing information to the porthole file, based on the material del. This information is accessible in the AUI using the given iable names.
sticisotropic: STRESS(ABC), STRAIN(ABC),
UIV_INTERNAL_AXIAL_PRESSURE, UIV_INTERNAL_HOOP_PRESSURE
otropic with thermal effects: STRESS(ABC), RAIN(A E_EFFECTIVE_STRESS, UIV_INTERNAL_AXIAL_PRESSURE,
ermoisotropic: STRESS(ABC), STRAIN(ABC),
EQUIV_INTERNAL_HOO
Plasticbilinear, plasticmultilinear: PLASTIC_FLAG, STRESS(ABC), STRAIN(ABC), PLASTIC_SEQUIV_INTERNAL_AXIAL_PRESSURE,
Thermoplastic, creep, plasticcreep, multilinearplasticcreep,
variableSTRESS(ABC), STRAIN(ABC), PLASTIC_STRAIN(ABC), CREEP_STRAIN(ABC), THERMAL_STRAIN(ABC), ELEMENT_TEMPERATURE, ACCUM_EFF_PLASTIC_STRAIN,
_EFFECTIVE_STRESS, YIELD_STRESS,
Chapter 2: Elements
168 ADINA Structures ⎯ Theory and Modeling Guide
EF
STRESS(ABC) = STRESSAA, STRESSAB, STRESSAC, STRESSCC, STRESSBB
Fig. 2.81 for the pipe stress coordinate system convention. Se are
Nodal forces: In linear analysis, the forces/moments at the ele
EQUIV_INTERNAL_AXIAL_PRESSURE, EQUIV_INTERNAL_HOOP_PRESSURE,
FECTIVE_CREEP_STRAIN In the above lists,
with similar definitions for the other abbreviations used above. See
See ction 12.1.1 for the definitions of those variables that not selfexplanatory.
ment nodes are =F K U
where K is the element stiffness matrix and U is the vector of nodal point displacements which may include ovalizations and warpings. Hence, the forces/moments in F will be equal to the forces/moments applied externally or by
adjoining elements, and are
ces are accessible in the AUI using the variable names NODAL_FORCER, NODALFORCES, NODAL_FORCET, NODAL_MOMENTR,S, NODAL_MOMENTT.
The end forces/moments are computed at the element local nodes. In the AUI, element local nodes are defined as element
ype d at
num
2.8.7 Recommendations on use of elements
ailable in ADINA to model general pipe networks subjected to general boundary and loading
not necessarily the internal section forces/moments. The nodal for
NODAL_MOMENT
points of t label. For example, to access the result computeelement 5, local node 2, define an element point of type label with element ber 5, label number 2.
! The pipe elements are av
conditions including internal pressure and thermal loading. ! When ovalization effects are insignificant (straight pipes or pipes of large thickness) the 2 or 4node pipebeam element
2.8: Pipe elements
ADINA R & D, Inc. 169
sed. In this case, linear or nonlinear behavior can be modeled with large displacements but small strains. ! When o de pipeshell element should be used. In this case, linear or nonlinear behavior can
nd that the strains are small. The displacements are always relatively small because in the calculation of the flexibility due to ovalization/warping the pipe radius is constant (and equal to the initial pipe radius) and no large displacement strain terms are included in the ovalization/warping calculations. The only large displacement strain terms accounted for are those corresponding to the beam behavior.
However, if the large displacement option is employed, the change of the pipe radius is taken into account in the calculation of
pl
,
! Since the pipe element stiffness matrix is calculated using numerical integration, it is clear that the linear 2node Hermitian beam element (see Section 2.4.1) for which thevaluated in closedform can be much more costeffective in analyzing linear piping systems with negligible ovalization effects.
2.9 General and spring/damper/mass elements
2.9.1 General elements
eral elements are linear elements which can have from one node to as many nodes as there are in the structure.
! General elements are useful in the definwhose formulations are not directly available in ADINA. General
ul in providing flexibility in obtaining desirable stiffness/mass/damping matrices.
should be u
valization effects are important, the 4no
be modeled with the assumptions that the displacements are always relatively small a
the element nodal forces due to the pipe internal pressure.
ref. K.J. Bathe and C.A. Almeida, "A Sim e and Effective Pipe Elbow Element Linear Analysis," J. Appl. Mech., Transactions of the ASME, Vol. 47, No. 1, pp. 931001980.
e stiffness matrix is
! Gen
ition of special elements
elements are also usef
Chapter 2: Elements
170 ADINA Structures ⎯ Theory and Modeling Guide
! Note that in a dynamic analysis, the only other alternative besides the use of general elemdamping matrix is to specify R
tri ysis, the mass and damping matrices) of the general element.
The dimension of the stiffness/mass/dampexceed ND, defined as NDOF×IELD, where
eneral element and NDOF is the number of active egrees of freedom. The components for the
stiffness/mass/damping matrices must be defined for all ND degrees of freedom. Note that only the upper triangular half of the matrices needs to be input (the stiffness/mass/damping matrices
For example, for a 4node 3D solid element with one active degree of freedom, ND = 1 x 4 = 4; the stiffness matrix is 4 x 4 symmetric, and the matrix is entered as follows:
1 0.5E8 0.5E7 0.1E8 0.5E7 0.1E8
ment, see Se
ADINA if there are skew systems at the element group nodes, or all matrices are input in the coordinate systems used at the nodes and
in YES
mand).
ents in introducing a consistent ayleigh damping.
! You directly enter the stiffness ma x (and in dynamic anal
ing matrices cannot IELD is the number of
nodes of the gd
must be symmetric).
matrix stiffness 1 4
2 0.5E8 0.5E7 0.1E8 3 0.5E8 0.5E7 4 0.5E8 dataend
! An alternative is to define a usersupplied general ele
ction 2.9.4.
Skew system input option: Either all matrices are input in the global coordinate system and proper transformations are made by
no transformations between global and skew systems are madeADINA (SKEWSYSTEMS = in the EGROUP GENERAL or EGROUP SPRING com
! Forces are calculated for general elements using
=F K U
where U stores the displacements fo
r the ND degrees of freedom. The forces are accessible in the AUI using the variable names
2.9: General and spring/damper/mass elements
ADINA R & D, Inc. 171
NODAL_FORCEX, NODAL_FORCEY, NODAL_FORCEZ, OMENT
Z.
! Stresses can also be calculated for general elements if you define a stressdisplacement matrix obtained using
NODAL_MOMENTX, NODAL_MOMENTY, NODAL_M
S. The stress vector σ is
=σ S U
Note that the matrix S must be input as a full matrdimension NS×ND, where NS equals the number components in (NS must be less than or equal to 600).
to the porthole file. This e
GENERAL_ELEMENT_STRESS. selectin the AUI by setting the label point number component number in .
2.9.2 Linear and nonlinear spring/damper/mass elements
Linear spring singledegreeoffreedom elemen
atrix has only ponent and corresponds to a ting (us f
! The damping matrix corresponds to a grounded damper acting in the single degree of freedom, see Fig
ix and is of of stress
σEach element outputs the vector σ
vector is accessible in the AUI using the variable nam the component of σ equal to the
You
σ
t
! The stiffness m 1 comgrounded spring ac in the single erspecified) degree ofreedom, see Fig. 2.91(a).
! The mass matrix corresponds to a concentrated mass acting in the single degree of freedom, see Fig. 2.91(b).
. 2.91(c).
Chapter 2: Elements
172 ADINA Structures ⎯ Theory and Modeling Guide
k u
Single
translational
DOF spring
k
Single
rotational
DOF spring
ü
m
Single
translational
DOF mass
Single
rotational
DOF mass
(b) Mass = [m]M
c c
Single
translational
DOF damper
Single
rotational
DOF damper
(c) Damping = [c]C
Figure 2.91: Singledegreeoffreedom element
(a) Stiffness = [k]K
.u
t
asses acting in the two degrees
consistent mass matrix is obtained using a linear int
Linear spring twodegreeoffreedom elemen
! The stiffness matrix is shown in Fig. 2.92(a) and corresponds to a spring coupling the two (userspecified) degrees of freedom. ! The lumped mass matrix is diagonal, see Fig. 2.92(b), and corresponds to two concentrated mof freedom. ! The
erpolation of the accelerations in the two degrees of freedom, see Fig. 2.92(b) and ref. KJB, Example 4.5, pp. 166170. ! The damping matrix is shown in Fig. 2.92(c) and correspondsto a damper coupling the two degrees of freedom.
2.9: General and spring/damper/mass elements
ADINA R & D, Inc. 173
K =
C =
Mlumped =
Mconsistent =
k
c
c
m/2
m/2
m/6
m/6
m/3
m/3
k
c
c
0
0k k
k = spring constant
c = damping constant
m = total massof element
a) Stiffness matrix
c) Damping matrix
b) Mass matrix
Figure 2.92: Twodegreesoffreedom element matrices
Nonlinear spring/damper/mass element
! A nonlinear "spring" element is available in ADINA for static and dynamic analysis. This element can be used to connect two nodes or to attach a node to the ground. It can be a translational or a rotational "spring" (see Figure 2.93). The stiffness, damping and mass of the element can each be either nonlinear or linear. ! Large displacements can be applied to nonlinear spring elements.
! In the case of a translational spring, the nonlinear stiffness is defined by a multilinear elastic force versus relativedisplacement relationship. The corresponding curve can be different for negative relativedisplacement than for positive relativedisplacement. The input curve need not pass through the origin (0,0).
The nonlinear damping is given by a function of the form NDF C U= ! where DF is the damping force in the element, U!
is the relative velocity between the element end nodes, and C and N are real constants.
The mass of the element can vary with time (see Figure 2.94).
Chapter 2: Elements
174 ADINA Structures ⎯ Theory and Modeling Guide
U2
U1
Nonlinear translational damping: Nonlinear torsional damping:
U = U  U2 1
U = U1
Nonlinear translational stiffness:
Nonlinear translational stiffness,grounded spring:
1
ME
ME
U2
U 1
2
FE
2
1
1
FE
FE
U 1
U2
U1
U = U  U2 1
Nonlinear torsional stiffness:
FD
1
U1
2
FD
U2
MD
1
2
MD
Figure 2.93: Description of the nonlinear spring/damper/mass element
n
The relative displacement U is defined as follows: Let U1 andU
d
U U− wh
about o use
2 be the displacement vectors at local nodes 1 and 2, and 0x1 an 0x2 be the respective initial positions of these nodes.
For initially coincident nodes (0x1 = 0x2), U = 2 1dxid id
ere id specifies a global direction in the global Cartesian coordinate system, and is specified in the input. id = 1,2,3 for the X, Y, Z translational directions and id = 4,5,6 for the rotationthe X, Y, Z directions. There is also an option (using id = 0) tanother arbitrary direction.
2.9: General and spring/damper/mass elements
ADINA R & D, Inc. 175
12
FE
u
21
Case 1: no ruptureCase 2: rupture
Force versus displacement ormoment versus rotation:
F D
F D
U
CUN
Force versus velocity or momentversus torsional velocity:
Totalmass
Time
Mass versus time:
Figure 2.94: Nonlinear stiffness, damping and mass
=
ent can be used to define a translational spring along the direction given by thetra
dis t is given by( )
If the nodes are not initially coincident, then the elem
vector from node 1 to node 2. Let U1 store only the nslational displacements, then the relative translational
lacemen( )0 2 0 1−x x
p 2 10 2 0 1
( )U = − ⋅U U , i.e., the
reldir
! loc
!
! d on th value of the relative displacement may or ma not be included in the nonlinear spring element (see Figure
−x xative displacement is the difference in displacements in the ection of the line (spring) element.
In a similar manner, a torsional spring is defined between the al nodes 1 and 2.
Similar definitions are used for the relative velocity.
Rupture base e y
2.94). If rupture is not desired, the last segments of the elasticforce versus relative displacement curve are extrapolated when necessary. If rupture is desired, then the elastic force in the element is set to zero whenever the relative displacement is larger
Chapter 2: Elements
176 ADINA Structures ⎯ Theory and Modeling Guide
t (resp. first) point on the curve. Note
into the curve displacement range.
g
differently: the linear spring element connects ment e input
lement output
r print or save stress
ment: GENERAL_ELEMENT_STRESS (NS values per
element)
Nonlinear element: DAMPING_FORCE, ELASTIC_FORCE
ORCEZ, NO
2.9.3 Spri
spring elements and additional constraint equations. Springtied elements can also be used to model a connection between a shell surface and ground.
he constraint equations are the same as are employed for the bea
(resp. smaller) than the lasthat this effect is reversible, as the force will be computed again according to the curve if the displacement value comes back
! Note that the linear spring element and the nonlinear sprinelement act quitespecific degrees of freedom, whereas the nonlinear spring eleconnects nodes and acts in a specified direction. Likewise thfor these elements is also different.
E
! You can request that ADINA eitheresultants or forces.
Stress resultants: Each element outputs the following information to the porthole file. This information is accessible in the AUI usingthe given variable names.
Linear ele
Nodal forces: These are the elastic nodal forces (in the global Cartesian reference system) equivalent to the axial elastic stress.
The forces are accessible in the AUI using the variable names NODAL_FORCEX, NODAL_FORCEY, NODAL_F
DAL_MOMENTX, NODAL_MOMENTY, NODAL_MOMENTZ.
ngtied element
! The springtied element is used for the modeling of a connection between touching, or closelyspaced, shell surfaces. The springtied element consists of six
Tmtied element, see Section 2.4.5.
2.9: General and spring/damper/mass elements
ADINA R & D, Inc. 177
! ote that the distance between the nodes of the springtied ele m
element must be used when the beamtied
ele se to
bout modeling given for the beamtied
ele
2.9.4 Usersupplied element
! With this option you code a nonlinear general element in the ADINA subroutine CUSERG. Subroutine CUSERG calculates the element stiffness matrix and nodal force vector, and also prints the results and outputs the results into the porthole file.
! The following element types are allowed:
< Triangular 3node, 6node, or 7node 2D solid element Quadrilateral 4node, 8node, or 9node 2D solid element
Variable number of nodes up to maximum 32 nodes
ser be generated by the AUI mesh generation
Nment must be smaller than a maximum distance. This maximu
distance can be a relative value (proportional to the model dimensions) or an absolute value. For longer connections, the beamtied element should be used.
! The springtied ment cannot be used due to element nodes being too clo
each other.
! Also see the remarks ament, see Section 2.4.5.
< < Tetrahedral 4node, 10node, or 11node 3D solid element < Hexahedral 8node, 20node, 21node, or 27node 3D solid element < 2node, 3node, or 4node beam element < 4node, 8node, 9node, or 16node shell element <transition shell element
The nodal connectivity of the usersupplied elements must be compatible with the ordinary ADINA elements so that the usupplied elements can
Chapter 2: Elements
178 ADINA Structures ⎯ Theory and Modeling Guide
com
de is always assumed for the shell usersupplied element.
!
neral element
con
t
vided (of course, the mass and damping ma
set group
general elements, subroutine CUSERG calculates the stif
UI, atrix from the
Ma
e element subtype (2D solid, 3D solid, beam or shell)
< The material (the material number from the usersupplied material definition). The material properties can be temperatureindependent or temperaturedependent. < The total number of userprovided integration points NUIPT. The default value of NUIPT is NUIT1*NUIT2*NUIT3. < The integration order in the first local direction NUIT1. NUIT1 is ignored if NUIPT is entered.
mands and so that the usersupplied elements can be loaded into the AUI for postprocessing.
Six degrees of freedom per no
It is necessary to include the usersupplied general element coding into subroutine CUSERG. A template for CUSERG is distributed in file ovl160u.f. The Makefile includes instructions for compiling and linking this file.
! The input to the AUI for the usersupplied gesists of four parts: element group definition, matrix set
definition, matrix definition and material definition. The usersupplied general elements belong to a general elemen
group, which is nonlinear. As with any general element, the stiffness, mass and damping
matrices must be protrices are necessary only for certain analysis types). The
stiffness, mass and damping matrices are grouped into a matrixand the matrix set number is included as part of the elementdefinition. In ordinary general elements (Section 2.9.1), you directly specify the stiffness matrix and other matrices. In usersupplied
fness matrix, and the mass matrix and damping matrix are calculated as in ordinary general elements.
The element subtype, material label, integration schemes, etc. are provided to ADINA through a matrix definition (in the Athe matrix definition is accessed as a usersupplied m
trix Set dialog box). This matrix definition includes
< Th
2.9: General and spring/damper/mass elements
ADINA R & D, Inc. 179
< The integration order in the second local direction NUIT2. NUIT2 is ignored if NUIPT is entered. < The integration order in the third local direction NUIT3. NUIT3 is ignored if NUIPT is entered.
! The calls from ADINA to CUSERG are divided into several phases, as controlled by integer variable KEY.
EY=1: Integration point locations are to be provided, these
oints are used to store stress and strain results. The array is ugh
3. The are the same as the
ration point rders. The array XYZ gives the element nodal coordinates and
ns of
d on
cements DISP2 and the displacements DISP1 at e last time step. Also, the material properties and parameters
an be used if necessary. RE has the dimension size of ND, the tal number of element degrees of freedom. The nodal forces
o an additional array REBM which can store the beam
odal forces in the local system for display purposes.
a
ment coordinates, latest
rties
KpXYZIPT provides these integration location coordinates andarranged through the sequence of 1 to NUIPT points or throthe local integration orders by NUIT1, NUIT2 and NUITdefinitions of NUIT1, NUIT2 and NUIT3 ADINA 2D solid, 3D solid and shell element integois passed from ADINA, then can be used for the calculatiointegration point coordinates.
KEY=2: Element nodal forces RE are to be calculated basethe latest displathctoare the equilibrating forces during the equilibrium iterations andthe resultant nodal forces in the converged solution. RE must be expressed in terms of the global coordinate system. We alsproviden
If variables are history dependent, they must be calculated and stored in the ARRAY and IARRAY arrays. Each column of these arrays stores the variables for one integration point.
KEY=3: The element stiffness m trix AS(ND,ND) is to be calculated, based on the eledisplacements DISP2, last converged displacements DISP1, and the constants and parameters defined in the usersupplied material. Some material properties are temperaturedependent and are interpolated by ADINA before they are passed to CUSERG. Temperature dependent and independent prope
Chapter 2: Elements
180 ADINA Structures ⎯ Theory and Modeling Guide
re used to calculate the material matrix. Note that the RE vector and AS matrix must be expressed in
rms of the global coordinate system.
e d se
NG Element group number NEL Element number IELD Number of nodes in the element ND Number of degrees of freedom in the
element NOD(IELD) Global node numbers of the element TEMP1(IELD) Element temperature at time t TEMP2(IELD) Element temperature at time TREF Reference temperature
CP(99) Solution control parameters TD(98,IELD) Temperaturedependent constants at time t
a
te
KEY=4: Some element results are to be written to the porthole file, e.g. stress and strain components. These quantities must bassigned to the indicated addresses of the arrays RUPLOT anIUPLOT in order to be displayed during postprocessing. Pleasee the source code comments for RUPLOT and IUPLOT formore information.
! Some useful arguments passed to CUSERG are described below:
t t+ ∆
SC
for each node CTDD(98,IELD) Temperaturedependent constants at time
t t+ ∆ for each node CTI(99) Temperatureindependent constants ALFA(IELD) Coefficient of thermal expansion for each
node at time t TIME Solution time at the current time step. DT Time step increment. IDEATH Element birth/death options, =0; option not used
=1; elements become active at time of birth
irth,
Birth time of the current element.
=2; elements become inactive at time of
death =3; elements become active at time of b
then inactive at time of death ETIME
2.9: General and spring/damper/mass elements
ADINA R & D, Inc. 181
TIMV2 Death time of the current element. GTH1 Length of real working array ARRAY at
each integration point GTH2 Length of integer working array IARRAY at
each integration point
s
fine
SUPPLIED command to define the
the MATRIX MASS and MATRIX DAMPING commands to define the mass and the damping matrices if
element matrixset.
ES
l
8) Define the rest of the input (i.e. boundary conditions, enerate a data file. Run ADINA as
usual and display the results as usual.
EL
L
Steps for using the usersupplied element option. These stepare given in terms of the AUI command language.
1) Compile ovl160u.f and link as described in the Installation Notes for your computer.
2) Use the MATERIAL USERSUPPLIED command to dethe material properties, control parameters and working array sizes, i.e. CTI, CTD, SCP, LGHT1 and LGTH2. 3) Use the MATRIX USERstiffness matrix. 4) Use
necessary.
5) Use the MATRIXSET command to define the general
6) Enter the EGROUP GENERAL USERSUPPLIED=YMATRIXSET=(defined in the MATRIXSET command) command. 7) You can generate the usersupplied elements using the usuamesh generation commands, as long as the element subtype is restricted to 2D solid, 3D solid, beam and shell elements.
loadings) as usual and g
Chapter 2: Elements
182 ADINA Structures ⎯ Theory and Modeling Guide
2.10 Disp
! ing ass
er
ost incompressible medium < Relatively small displacements
! (see the ADINAF Theory and Modeling Guide for details of the formulations employed). In the remainder of this section, only the ADINA fluid elements are discussed.
! esbas
f
s
eledy 1) is m ed for use for solving this class of problems.
! Simply stated, the displacementbased fluid elemthought of as derived from the solid two and threedimensional
lacementbased fluid elements
The elements discussed in this section incorporate the followumptions:
< Inviscid, irrotational medium with no heat transf < Compressible or alm
< No actual fluid flow
For other types of fluid flow, use ADINAF and ADINAFSI
The typ of problems for which the ADINA displacemented fluid elements can be employed are:
< Static analyses, where the pressure distribution in the fluid and the displacement and stress distribution in the structure is ointerest < Frequency analyses, where natural frequencies and mode shapes of a structure/fluid medium are to be determined < Transient analyses, where a pressure wave propagates rapidly through the fluid, which does not undergo large motion(transient acoustic problems)
However, in practice, the use of the displacementbased ments is rather restricted to special applications in static and namic analyses. The potential based element (see Section 2.1
uch more general and is recommend
ents can be
2.10: Displacementbased fluid elements
ADINA R & D, Inc. 183
elements (see Sections 2.2 and 2.3) by using an elastic stressstrain relation with a bulk modulus K and a zero shear modulus.
! ed in 2D and 3D analyses. Twodim
e ie in the
ust have positive y
rmulation, the allowed fluid motion is relatively se the elements must not become
annot be analyzed using the elem
ces
ref. K.J. Bathe and W. Hahn, "On Transient Analysis of
F l. 10, pp. 383391, 1981.
of FluidStructure Interaction Problems," Computers &
Engineering and Design, Vol. 76, pp. 137151, 1983.
it is noted that problems v
The elements can be employensional fluid elements can be employed in planar and
axisymmetric analyses. Twodimensional fluid elements must bdefined in the YZ plane, and axisymmetric elements must l+Y half plane (all nodal point coordinates mvalues).
! Although the fluid elements can be employed using a large displacement fosmall in a practical analysis, becaudistorted. Actual flow of a fluid c
ents. Use ADINAF or ADINAFSI if the allowed fluid motion is large.
! Difficulties in the use of the elements and various experienin solutions obtained are discussed in the following references:
luidStructure Systems," Computers & Structures, Vo
ref. J. Sundqvist, "An Application of ADINA to the Solution
Structures, Vol. 17, pp. 793808, 1983. ref. L. Olson and K.J. Bathe, "A Study of Displacement
Based Fluid Finite Elements for Calculating Frequencies of Fluid and FluidStructure Systems," Nuclear
or example, in Olson and Bathe,F
in olving structures moving through fluids that behave almost incompressibly (e.g., an ellipse vibrating on a spring in water) cannot be solved satisfactorily with the displacementbased fluid elements.
! In linear analysis, you can impose irrotational conditions in the element formulation. This is achieved using a penalty constraint.
Chapter 2: Elements
184 ADINA Structures ⎯ Theory and Modeling Guide
ote that if the penalty constraint is imposed, rigid body rotations of
I
ements, see Sections 2.2.3 and 2.3.3.
Nthe elements are no longer possible.
! These elements are defined within 2D and 3D fluid element groups (in ADINA). Set the formulation of the element group to either displacementbased without rotation penalty or displacementbased with rotation penalty (if you are using the AUuser interfaces, set the formulation with the Ainterpolation type@ field).
! Pressures are output at the integration points. The integration point numbering is the same as the numbering convention used for the solid el
The pressure is evaluated using the following relation:
vp K e= −
where p is the pre
ssure, K is the bulk modulus and ev is the vo
XZ, AINXY and STRAINXZ are applicable only
for 3D elem
2.11 Potentialbased fluid elements
g
lumetric strain (∆ volume/volume). The pressures and strain components are accessible in the AUI
using the variable names FE_PRESSURE, STRAINXX, STRAINYY, STRAINZZ, STRAINXY, STRAINSTRAINYZ (STR
ents). See Section 12.1.1 for the definitions of those variables that are not selfexplanatory.
! You can also request nodal force output. The nodal forces are accessible in the AUI using the variable names NODAL_FORCEX, NODAL_FORCEY, NODAL_FORCEZ (NODAL_FORCEX is applicable only for 3D elements).
! The elements discussed in this section incorporate the followinassumptions:
< Inviscid, irrotational medium with no heat transfer < Compressible or almost incompressible medium
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 185
r types of fluid flow, use ADINAF and ADINAFSI
(se he
! he types of problems for which the ADINA potentialbased flu
Static analyses, where the pressure distribution in the fluid
otions
small motions.
and
ts must
nodal point coordinates must have nonnegative y values).
A d
< Relatively small displacements of the fluid boundary < Actual fluid flow with velocities below the speed of sound(subsonic formulation) or no actual fluid flow (linear infinitesimal velocity formulation)
! For othee the ADINAF Theory and Modeling Guide for details of t
formulations employed). In the remainder of this section, only the ADINA fluid elements are discussed.
Tid elements can be employed are:
<and the displacement and stress distribution in the structure is of interest < Frequency analyses, where natural frequencies and mode shapes of a structure/fluid medium are to be determined < Transient analyses, where a pressure wave propagates rapidly through the fluid, which does not undergo large m(transient acoustic problems) < Transient analyses, where fluid flows through the domain, and the boundaries of the domain undergo only
! The potentialbased fluid elements can be employed in 2D3D analyses. Twodimensional elements can be employed in planar and axisymmetric analyses. Twodimensional elemenmust be defined in the YZ plane, and axisymmetric elements lie in the +Y half plane (all
! The potentialbased fluid elements can be coupled with ADINstructural elements, as described in detail below. The structuralmotions cause fluid flows normal to the structural boundary, anthe fluid pressures cause additional forces to act on the structure.
Chapter 2: Elements
186 ADINA Structures ⎯ Theory and Modeling Guide
! e l
lement boundary). This feature can be used to model free surfaces.
onal on the ADINAF boundary.
potentialbased elements can model (approximately) an
infinite domain through the use of special infinite elements.
n, which is nonlinear, and an infinitesimal velocity for e
!
the potentialbased fluid elements and can model a much wider range of flow con
are
less
requires a minimum of 4 degrees of freedom, whereas each node in the interior of a potentialbased mesh
al
The potentialbased fluid elements can be coupled to a pressurboundary condition (i.e., no structure adjacent to the potentiabased fluid e
! The potentialbased fluid elements can be coupled directly to ADINAF fluid elements. The ADINAF fluid element motions cause potentialbased fluid flows normal to the ADINAF boundary, and the potentialbased fluid pressures cause additiforces to act
! The
! Two formulations are available, a subsonic velocity formulatio
mulation, which is linear. These are described in detail in thfollowing sections.
In some analyses, either ADINAF/ADINAFSI or the potentialbased fluid elements can be used in the modeling. ADINAF/ADINAFSI is far more general than
ditions. In addition, in ADINAFSI, the fluid mesh need not becompatible with the structural mesh.
However, for the class of problems in which the assumptions given above are acceptable, the potentialbased fluid elements more efficient. This is because
< The number of degrees of freedom in the fluid region isfor the potentialbased formulation. In 3D analysis, each ADINAF node
requires only one degree of freedom.
< If the velocities are small, then the linear infinitesimvelocity formulation can be used. ADINAF is always nonlinear.
< In addition, frequency analysis is not possible in ADINAF.
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ADINA R & D, Inc. 187
! ely rlier versions of
DINA are superseded by new features of ADINA 8.0. Features of earlier versions of ADINA that are superseded by new features of ADINA 8.0 include:
so that models developed for earlier versions of ADINA still work in ADINA 8.0. However these features are not described here, and are not recommended for new models.
2.11.1 Theory: Subsonic velocity formulation
! Figure 2.111 shows a fluid region with volume and bounding surface. In the fluid, we use the basic equations of continuity and energy/momentum, as written in terms of the velocity potential: 0
The potentialbased element formulation has been extensivrevised in version 8.0. Some of the features of eaA
The 0P degree of freedom (superseded in ADINA 7.4) Volume infinite elements (superseded in ADINA 8.0) These features are retained in ADINA 8.0
( )ρ ρ φ+ =# $i $ (2.111) and ( ) 1
2h φ φ φ= Ω − − ∇ ∇x # i (2.112)
h w ere ρ is the density, φ is the velocity potential ( φ=v $ wv is the fluid velocity), h is the specific enthalpy (defined as
here
dphρ
= ∫ ), p is the pressure and ( )Ω x is the potential of the
(conservative) body force accelerations at position x . For example, when the body forces are due to gravity, Ω = g$ , where g is the acceleration due to gravity.
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188 ADINA Structures ⎯ Theory and Modeling Guide
Fluid region V
Bounding surface S
Unknowns in V:
Unknowns on S:
= fluid potential
= fluid potentialu = displacements
Essential boundary condition: prescribed
Figure 2.111: Fluid region
n
Body force acceleration
body force acceleration potential
g,
Natural boundary condition: prescribed u n
Equations (2.111) and (2.112) are valid for an inviscid irrotational fluid with no heat transfer. In particular, (2.112) is valid only when the pressure is a function of the density (and not of, for example, the density and temperature).
For the pressuredensity relationship, we use the slightly compressible relationship
0
1 pρρ κ
= + (2.113)
where κ is the bulk modulus and 0ρ is the nominal density. (2.113) then directly gives the densityenthalpy relationship and
e
th pressureenthalpy relationships:
00 exp hρρ ρ
κ⎛ ⎞= ⎜ ⎟ ,⎝ ⎠
0exp 1hp ρκκ
⎡ ⎤⎛ ⎞= −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
he continuity equation (2.111) is then approximated using the sta
(2.114a,b)
Tndard Galerkin approach to obtain
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( )( ) 0V
F dVφδ ρ ρ φ δφ= + =∫ # $i $ (2.115)
Fφδwhere δ is the variation of symbol and is the variation in
the
mass flux rate. (2.115) can be rewritten as
( )V S
F dV dSφδ ρ δφ ρ φ δφ ρ φ δφ= − ∇ ∇ − ∇∫ ∫ n# i i (2.11
where S i
6)
s the boundary of and is the inwards normal on .
From this equation, we observe that the natural boundary condition is
V n S
ρ φ n$ i prescribed, in other words, prescribed mass flux rate.
At this point, we notice that ρ is a function of the fluid veloand position through the densityenthalpy relationship (2.114a)because the enthalpy is a function of fluid
city ,
velocity and position. In , we compute the enthalpy using (211.2). Hence, in , the
density is a function only of the fluid potential and position. But on , we anticipate that part or all of might be a moving boundary with velocity . We adopt the convention that the fluid velocity used in the enthalpy calculation on is computed in terms of both the boundary and the internal fluid velocity
V V
S S ( )u x#
v S velocity u#
φ$ using ( )n =v u n n , ( )t# i φ φ= −v n n , n t= +v v v $ $ i (2.117a,b,c)
In other words, the fluid velocity normal to the surface is taken to be the velocity of the moving boundary and the remainder of the fluid velocity (which is tangential to the surface) is taken from the
fluid potential. We also include the motions of in the body force potential calculations in the enthalpy. Combining the above considerations, we take
S
( ) 1 12 2n n t th φ= Ω + − − −x u v v v v# i i (2.118)
on S .
(2.116) can be written as
Chapter 2: Elements
190 ADINA Structures ⎯ Theory and Modeling Guide
V S
h dV dShφF ρδ δφ ρ φ δφ ρ δφ= − ∇ ∇ + −⎜ ⎟∂⎝ ⎠∫ ∫ u n# #i i
The surface integral is obtained using (2.117).
On S , we assume that the boundary motion ( )u x is small enough so that the volume V and the normal n can be assumed constant. In other w
∂⎛ ⎞ (2.119)
ords, we perform all integrations on the un
itions are applied to S, the boundary condition
deformed fluid boundary. We discuss the implications of this assumption later. ! From (2.119), if no boundary cond
0ρ ⋅ =u n# is implied. This boundary condition corresponds to no fluid flow through the boundary.
From (2.119), the natural boundary condition is prescribedρ ⋅ =u n# . We use this boundary condition to apply the
structural motions to the fluid dom
ain. We also use this boundary condition to model infinite fluid regions. Both of these cases are discussed in more detail below.
Notice that there is no boundary condition corresponding to flow tangential to the boundary. In other words, fluid can slip tangential to the boundary without restriction.
Also from (2.119), the essential boundary condition prescribedφ = is possible. Through (2.112) this boundary
condition corresponds to a partially prescribed enthalpy and hence prescribed pressure. The enthalpy and pressure are not a partially
fully prescribed since φ$ is not prescribed when φ is prescribed only on the boundary. Modifications to equations of motion for the structure
! We assume that part of the boundary is adjacent to the structure (Figure 2.112). The part of the boundary adjacent to the structure is denoted .
S
1S
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Fluid
Pressure p
Structure
Traction pn
Figure 2.112: Forces on structure from fluid
n
Surface S1 (Fluid and structureare separated forclarity)
The fluid pressure on 1S provides additional forces on the structure adjacent to 1S :
1
1uS
F p dSδ δ− = −∫ n ui (2.1110)
where F
uδ is the variation in the applied force vector and the mi
d
nus signs are used in anticipation of uF being considered an internal force vector in (211.12) below. The pressure is evaluateusing
( )( )1 12 2( ) n n t tp p h p φ= = Ω + − − −x u v v v v# i i (2.1111)
e emphasize that points into the fluid (and out of the structure).
Finite element equations of motion
! Equations (2.116) and (2.119) are linearized using standard procedures to obtain
nW
Chapter 2: Elements
192 ADINA Structures ⎯ Theory and Modeling Guide
( )( )UU UF
FU FF FFFF S
⎡ ⎤∆ ∆⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ +⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− +− ∆ ∆⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎢ ⎥⎣ ⎦
C C0 0 u uC C C0 M
## ### #φ φ
(2.1112) where the increment in the vector of unknown potentials
( )( ) ( )UU UF U
F FFU FF FF SS+− +
⎡ ⎤ ⎡ ⎤∆⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥∆⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦
K K F
F FK K K
u 00φ
φ is written ∆φ and the increment in the vector of unknown displacements is written u ∆u . (We drop the left superscript t t+ ∆ here and below for ease of writing.) In the linearization process, increments in both the nodal displacements and increments in the nodal potentials (and their time derivatives) are considered. In (211.12), = vector from (2.1110) = vector from volume integration term in (2.119)
UF
FF( )F S
and the matrices are obtained by linearization. Note that vectors and matrices with the subscript S are integrated over the surface. Also the matrices with displacement degrees of freedom areinte
F = vector from surface integration term in (2.119)
grated over the surface.
The sum of FF and ( )F S
F can be interpreted as an outofbalance mass flux vector. (211.12) is satisfied only if this sum is zer ss fluxes
(2.1112), we do not include any of the structural system matrices. (2.1112) only gives the contribution of the potentialbased fluid elements to the system matrices. Of course, the rest of the of
o at every node in the fluid, that is, if the consistent maat each of the nodes from all of the attached fluid elements sum tozero.
In
structure will make additional contributions to the first rowthe above equations.
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ADINA R & D, Inc. 193
e everal sources:
! Equation (2.1112) is a nonlinear system of equations. Thnonlinearities come from s
1) The 12 φ φ− ∇ ∇i term in h . This term is sometimes called
the Bernoulli effect.
) The nonlinear relationship between the density, pressure and the enthalpy (2.114).
Because the system is nonlinear, equilibrium iterations must be employed for an accurate solution, just as in nonlinear structural analysis.
! There is no explicit restriction on the magnitude of the fluid velocities in equation (2.1112). However, in practice, there are several restrictions
1) The velocity must be smaller than the speed of sound, otherwise, the equations (2.119) become hyperbolic and cannot be solved using ordinary finite element techniques.
2) The density change should not be too great, otherwise the change in mass flux may be quite different than the change in volume flux when the boundary velocities are perturbed.
3) Also, the pressuredensity relationship in (2.113) only holds for relatively small density changes.
it
tion
rs
lower as the velocities become larger.
2
With these restrictions, the Mach number of the fluid flow should not exceed about 0.3 or so.
! Equation (2.1112) is in general nonsymmetric. But in the limof very small velocities, equation (2.1112) becomes symmetric (see equation (2.1119) below). Even for finite velocities, equa(2.1112) is nearly symmetric. This means that we can symmetrize (2.1112) and use the usual symmetric equation solveof ADINA for the solution of (2.1112). The convergence rate isver fast for small velocities and becomes sy
Chapter 2: Elements
194 ADINA Structures ⎯ Theory and Modeling Guide
2.11.2 The
ory: Infinitesimal velocity formulation
! If we assume that the velocities and the density changes are infinitesimally small, then the continuity equation (2.111) becomes
2 200 0( ) 0pρρ ρ φ ρ ρ φ ρ φ
κ+ ∇ ∇ ≈ + ∇ ≈ + ∇ =
## #i (2.1113
The momentum/equilibrium equation (2.112) becomes
)
( )h p φρ
≈ ≈ Ω −x # (2.1114)
fro
m which we see
( )( ) ( )( )0p ρ φ ρ φ≈ Ω − ≈ Ω −x x# # (2.
Substituting (2.1115) into (2.1113) gives
20 0
1115)
ρ φ κ ρφ− + ∇ = − Ω## # (2.1116)
Equation (2.1116) is a special form of the wave equation. It is
linear in the solution variableφ . (2.1116) can be written in variational form using standard techniques. The result is
0
V V S
dV dV dSρ φ δφ κ φ δ φ κ δφ− − ∇ ⋅ ∇ − ⋅∫ ∫ ∫ u n## #
0V
(2.1117) dVρ δφ= − Ω∫ #
The fluid pressure onto the structure becomes
1 1
1 0 0 0 1uS S
p dS dSδ ρ ρ ρ φ δ∂Ω⎛ ⎞= ≈ Ω + −⎜ ⎟∂⎝ ⎠∫ ∫n u u n ux
#i i i (2.1118) Fδ
The finite element contributions to the system matrices corresponding to (2.1117) and (2.1118) are
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 195
( )TUUFU S
⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤+ ⎢ ⎥⎢ ⎥⎥−
K 00 0 U0 CU UM C 0
## ###φ φ FFFF FU
( )UB S =⎡ ⎤ ⎡ ⎤
+⎢ ⎥ ⎢ ⎥−⎣
0RR0 #
FB
+ ⎢ ⎥⎢ ⎥⎢ ⎢ ⎥− ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
⎣ ⎦⎦
0 K0 # φ(2.1119)
where
FFM = matrix from φ δφ## term in (2.1117) K φ δ φ∇ ⋅ ∇FF = matrix from term in (2.1117)
FUC = ma rix from t δφ⋅u n# term in (2.1117)
( )UU SK = matrix from 0ρ δ∂Ω⎛ ⎞
⎜ ⎟∂⎝ ⎠u n u
xi i term in (2.1118)
( )UB SR = loads vector from ( )0ρ δΩ n ui term in (2.1118)
FBR# = loads vector from 0ρ δφΩ# term in (2.1117) = vector containing unknown nodal displacements
nown nodal fluid potentials. Uφ = vector containing unk
We note that the term ( )UU S
K is numerically very smallcompared with the rest of the structural stiffness matrix, when theis a structure adjacent to the fluid. But
re
( )K is important in thcase when there is no structure adjacent to the fluid.
! The lefthandside of equation (2.1119), with the exception of the term
UU Se
( )UU SK , is identical to the fo
following reference: rmulation presented in the
and
2.11.3 The
! ulations
ref. L.G. Olson and K.J. Bathe, AAnalysis of fluidstructure interactions. A direct symmetric coupled formulation based on the fluid velocity potential@, J. ComputersStructures, Vol 21, No. 1/2, pp 2132, 1985.
ory: Infinite fluid regions
Both the subsonic and infinitesmal velocity form
Chapter 2: Elements
196 ADINA Structures ⎯ Theory and Modeling Guide
include special boundary conditions for the modeling of infinite fluid regions. ! he basic approach is to replace the infinite fluid region with a boundary condition that simulates the infinite fluid region. The boundary condition allows outwardsgoing waves to be propagated into the infinite fluid region without reflection. ! here are many approaches for setting up infinite elements. One approach that is physically intuitive is to use the results from acoustic analysis regarding outwardsgoing waves. Planar waves: To illustrate the procedure, we consider a planar boundary, that is, a boundary on which plane acoustic waves propagate outwards through the boundary (Figure 2.113). If the waves have very small amplitude,
T
T
vp cρ∆ = ∆ (2.1120)
where p∆ is the change in pressure applied to the boundary from the fluid on the outside of the boundaryin press e fluid that is not explicitly modeled),
(in other words, the change ure applied to the boundary from th
v∆ is the change in the outwards velocity of the fluid and is the speed of sound in the fluid. Hence the pressure on the boundary is
c
( )v vp p cρ∞ ∞= + − (211.21)
where is the outwards velocity at the boundary, and v p∞ and v∞ are the pressure and outwards velocity at infinity ( p∞ and v∞ must be specified as part of the model definition).
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 197
Fluid
p=p + c(v  v )
Figure 2.113: Planar infinite boundary
n
The pressure andvelocity on the boundaryare related by
At infinity
v=v
p=p
a) Physical problem
b) Modeling with planar infinite boundary
Direction ofwave propagation
e variation in mass flow through the boundary is
Th
vB B
BS S
F dS dSφδ ρ φ δφ ρ δφ= − ∇ =∫ ∫ni (2.1122)
We need to express the mass flow through the boundary in terms of
as follows. First, we assume that there is no fluid flow tangential to the boundary the potential on the boundary. This can be done
and that the boundary does not move. Under those conditions,
21 2
Given a value of
( ) ( v )p h p φ= Ω − −# (2.1123)
φ# , v can be numerically evaluated by combining en a value of φ#equations (2.1121) and (2.1123) and hence, giv ,
(2.1122) can be numerically evaluated. We use this procedure in ic
ere are no cities are small, and the velocities and pressure
the implementation of the planar infinite element for the subsonvelocity formulation.
As an illustrative example, in the special case when thbody forces, the velo
Chapter 2: Elements
198 ADINA Structures ⎯ Theory and Modeling Guide
at infinity are zero
v= pc c
ρφρ −=
# (2.1124)
o in this special case,
s
that the variation in mass flow rate is,
BSB c
F dSφρφδ δφ= − ∫#
(2.1125)
1125) in the implementation of the planar infinite
warel
We use (2.element for the infinitesimal velocity formulation. Spherical waves: The same technique can be used for spherical
ves, provided that the appropriate relationship is used for the ationship between p and vρ . The relationship that we use for erical waves is based on the relationship between pressure and ss flux in the frequency domain
sphma
p pv=c i r
ρω
+ (2.1126)
where 1i = − , ω is the frequency of the outwardsgoing
is the radius of the boundary. Here we assume that bwave
and oth the pre avelocity itdomain is
rssure nd velocity at infinity are equal to zero and that the
self is small. The equivalent of (2.1126) in the time
v=p dtpρ +
c r∫ (2.1127)
We now neglect body force effects, so that p ρφ= − # on the
. Then boundary
v=c rρφ ρφρ −
−#
(2.1128)
so that the variation in mass flow rate is
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B
BF dSc rφ
ρφ ρφδ δφ⎛ ⎞
= − +⎜ ⎟∫#
(2.1129S ⎝ ⎠
)
ω Note that this derivation holds regardless of the wave frequency
domain is
Cylindrical waves: The same technique and derivation can be usedfor cylindrical waves. For cylindrical waves, the relationship between pressure and mass flux in the frequency
1 1
0 0
( ) ( )v=( ) ( )
p J kr iY kric J kr iY kr
ρ⎡ ⎤−⎢ ⎥−⎣ ⎦
(
ere 0J , 1J , 0Y , 1Y are the Bessels functions of the first and
2.1130)
wh
seconr
d kinds of orders 0 and 1, and krc
ω= . (2.1130) cannot be
ansform e domain, but for frequencies higher than
abo
tr ed into the tim
ut 1rc
ω= , it turns out that (2.1130) is well approximated by
(2.1126), so that the spherical wave derivation can be used, provided that the frequency is high enough.
! For planar, spherical and cylindrical infinite boundaries, only surface integrations are required. Also, no additional degrees of freedo dary. For planar waves, the infinit
m are required on the boune boundaries contribute to ( )F S
F ( )FF SC and for spherical
and cylindrical waves, the infinite boundaries contribute to ( )F SF ,
( )C and ( )K . FF S FF S
! It i above approach with the doublyasymptotic approach in Olson and Bathe.
ref. L.G. Olson and K.J. Bathe, AAn infinite element for
teractions@, Engineering Computations, Vol 2, pp 319329, 1985.
s instructive to compare the
analysis of transient fluidstructure in
Chapter 2: Elements
200 ADINA Structures ⎯ Theory and Modeling Guide
n. The added mass approximation do encies and the pla wave approximation do uencies. The plan wave approximation gives exactly the results given above for planar waves (equation 2.1 ss approximation in the DAA requires a volume integral and hence volume infinite elements are required for modeling nonplanar waves. Provided that the nodes in the vo ents at infinity are placed at the correct locations, the volume infinite elements give exactly the same results as the spherical infinite elem
iven in equations (2 120) to (2.1130) gives the same results as the DAA approach, and since the surface integral formulation does not require volume ele feel that the surface integ al formulation is preferable.
2.11.4 The
,
(2.1131)
nd
is the vector of nodal point displacements relative to the ground motion. The ground motions are expressed as
In the DAA, the infinite fluid region is modeled as the superposition of two effects, an added mass approximation and a plane wave approximatio
minates for low frequ neminates for high freq e
125). But the added ma
lume infinite elem
ents (equation (2.1129). Since the surface integral formulation g .1
ments, we r
ory: Ground motion loadings
! Ground motion loadings require special treatment. In this case
r gU = U + U
where gU is the vector of nodal point ground displacements a
rU
g gk k
ku= ∑U d (2.1132)
hw ere gku are the ground displacements in direction k and is
e ve of nodal point values in which if equation i
e.
otions are known, the increment in displacem
kd
ctor ( ) 1k i =dth
corresponds to a translation in direction k and ( ) 0k i =d otherwis
Because the ground m∆Uents is equal to the increm
placements r
ent in the relative dis ∆U He ndnce the left ha sides of equations (2.1112) and (2.1119) are unchanged and the righthandsides of these
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 201
equations are updated only by additional internal forces. For exam the flu
ple, the additional internal forces added to (2.1119) fromid elements are
( )UU S k
FUgk gk
k
uu⎡ ⎤ ⎡ ⎤
−− ⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦
0C d
K d0
#
Note that φ still c luontains the nodal point va es of φ and φ is the potential corresponding to the absolute (not relative) velocities.
lso note that ground motions are handled differently than ordinary
tructural analysis, in which physical body forces and ground .
2.11.5 Theory: Static a
ecial attention. It is
btaining a static solution as a
(in
Aphysical body forces. This is quite different than in smotions are both modeled using massproportional loads
nalysis
The case of static analysis also requires sp!
not correct simply to set the velocities to zero in (2.111), because (2.111) is identically satisfied for zero velocities.
We envision the process of o!
quasistatic process. In this process, all fluid velocities are assumed negligible and therefore the fluid potential is constant space). Under these conditions,
h φ= Ω − # (2.1132)
)
and the variational statement of continuity is
V S
F dV dSφδ ρ δφ ρ δφ= −∫ ∫ u n# # i (2.1133
Equation (2.1133) is simply the integral equation of conservation of mass because δφ is constant in space.
! We now numerically integrate (2.1133) in time. We use the Euler backwards approximations
( ) /t t t t t tρ ρ ρ+∆ +∆= − ∆# , ( ) /t t t t t t+∆ +∆= − ∆u u u# (2.1134a,b)
Chapter 2: Elements
202 ADINA Structures ⎯ Theory and Modeling Guide
and then obtain, at time t t+ ∆ ,
( ) ( )t t t t t t t t
V S
F dV dSφ ρ ρ δφ ρ δφ+∆ +∆ +∆= − − −∫ ∫ u u n# #i (2.1δ 135)
where we write / tδφ δφ= ∆# .
The modification to the structural equations of motio
n is
u ∫
1t tF p dSδ δ+∆= n ui (2.1136)
1S
which is the same as equation (2.1110), except that the t t+ ∆ iexplicitly written.
We prefer the Euler backwards approximations because then stiffness matrix remains
s
the symmetric when the displacement
crements are infinitesimally small. Of course, the steps must be
pressible. Note that we assume that the domain of integration and its
Also note that the solution in the fluid is the single value
insmall enough so that the Euler backwards approximations apply, but in many problems, these approximations are quite good, especially when the fluid is almost incom
boundary remain (nearly) unchanged during the time integration. All integrations are performed over the original domain of integration.
φ# .
on for the subsonic formulation
)FU FF FF SS
⎡⎢⎢ ⎥ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
&
are
gion in the proble φ
This value represents the Bernoulli constant. The linearized equations of moti
become
)( )( ) (
UU UF US
F F
⎤ ⎡ ⎤∆⎡ ⎤ ⎡ ⎤⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥∆ +⎢⎣ ⎦ ⎣ ⎦− +
K K Fu 00 F FK K
& & &
# && & & φ (2.1137)
(K
where we use the ~ to emphasize that the vectors and matrices different than the corresponding dynamic vectors and matrices, along with the constraints constant# =φ (if there is more than one fluid re constant= in each fluid m, then #
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 203
gion). As in the dynamic case (equation 2.1112), equation
So, as in the dynami mmetrize and use the usual symmetric equation solver.
.1137) the sum of
re(2.1137) is in general nonsymmetric, but (2.1137) becomes symmetric when the displacement increment t t t+∆ −u u becomes infinitesimally small. c case, we sythe system
In (2 ( )F F S+F F& & can be interpreted as an
outofbalance mass vector. (2.1137) is only satisfied if the totamass of each fluid regionhat do not change the to
l is conserved. Motions of the boundary tal mass of any fluid region are associated stem matrices, unless there is structural
tiffness associated with these motions. Note that the K&
ma s,
the infinitesimal velocity
formulation can be obtained either from the above derivation, or
119) and applying the final value eorem. The result is
twith zero pivots in the sy
( )UUsS
trix also provides structural stiffness, so if there are body forcemotions of the boundary that do not change the total mass of any fluid region are not associated with zero pivots.
The equations of motion for
can be formally derived from (2.1119) by applying the Laplace transform to both sides of (2.1th
( ) ( )TUU FU UBS S=FU FF FB
⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥− −⎢ ⎥ ⎣ ⎦ ⎣ ⎦
K C U R (2.1138)
⎣ ⎦C M R#φ
together with the condition
FF =K 0φ (2.1139)
There are a number of unusual characteristics of (2.1137),
!
(2.1138) and (2.1139):
< The solution involves #φ instead of φ . This makes sense as (2.1132) then implies that p is constant (in time) in a static solution. < The condition (2.1139) m e satisfied. When there are ust bno infinite boundaries, this condition is satisfied whenever
Chapter 2: Elements
204 ADINA Structures ⎯ Theory and Modeling Guide
of unknow potential degrees of freedom in static analysis is equal to the number of separate fluid regions in the
< The condition within each separate fluid on implies that
constant# =φ within each separate fluid region. Hence the number n
analysis.
constant# =φ
0p Cρ= Ω + wher
d within and any
e C is a constant regidetermined from the solution. Hence the variation of pressure within each separate fluid region is containe C
Ωchoice of constant of integration within 0ρ Ω (recall that is a potential and therefore includes an arbitrary constant of integration) is balanced by an opposite change in C . < If the φ degree of freedom is fixed at a node, or if infinite boundaries are included in the fluid region, then you must set
0=φ for all the nodes in the fluid region. This condition implies 0p ρ= Ω within the fluid region. Any choice of constant of integration within Ω then affects the solution.
2.11.6 The
s used, and
< It is necessary to enter the density of the fluid in static analysis, even when the solution does not depend on the density.
ory: Frequency analysis
! Frequency analysis is possible when there is no structural damping, when the infinitesimal velocity formulation iwhen there are no infinite boundaries. The eigenvalue problem to be solved is
( )T ( j)UU2 FU S
( j)FFFF
=FU
j jω ω⎛ ⎞+⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤
− − +⎜ ⎟⎢⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎣ ⎦⎣ ⎦K 0F
⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠
K K 0M 0 00 C U00 M C 0
)
(2.1140
where (j) (j)i= −F φ , 1i = − and in which we also include the structural stiffness matrix K and structural mass matrix M .
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 205
sform
ll of al s are real and nonnegative. It is solved using either inant search method or the Lanczos iteration method.
(2.1140) is derived from (2.1119) by taking the Fourier tranof the lefthandside.
(2.1140) is a nonstandard eigenvalue problem in which athe eigenv ue
e determthThe eigenvectors are scaled according to the following
orthogonality condition:
( )( )(i) T (j) (i) (j) (i) (j) (i) (j)FU FU UU FF
=2
i j
i j ij
ω ω
ω ω δ
+ + + +u C F F C u u M u F M F
(2.1141)
where ijδ is the Kronecker delta. Notice that when there is no fluid, ( onality condition
(i)2.1141) reduces to the usual orthog
(j)UU = ijδu M u .
(2.1140) has one rigid body mode for each separate fluid region in which all of the φ degrees of freedom are free. Each rigid body mode is of the form in each
e fluid region. Fo ture reference, wody modes
(j) (j)0, constant= =U Fseparat e term these rigid r fu
φb rigid body modes. The determinant search method can determine all φ rigid body modes.
The modal pressures and modal fluid particle displacements acomputed using
re
( ) ( ) ( ) ( )1,j j j jj
j
p f fρ ωω
= = ∇u (2.1142)
which ( )jpin is the modal pressure at the point of interest for
mode j, ( )jf is the fluid potential eigenvector ( )jF interpolated to the point of interest and ( )ju is the odal displacement vector athe point of interest for mode j.
m t
2.11.7 The
Mode superposition is possible when there is no structural
ory: Mode superposition
!
Chapter 2: Elements
206 ADINA Structures ⎯ Theory and Modeling Guide
theory used to determine the odal equations of motion is given in Chapter 6 of the following
tural
Dyna
form
damping, the infinitesimal velocity formulation is used and when there are no infinite boundaries. Themreference:
ref. L. Meirovitch, Computational Methods in Strucmics, Sijthoff & Noordhoff, 1980.
The derivation starts with the equations of motion (2.1119) being put into the
( )TUU rFUr r S
FFFF FU
( ) ( )UB UUS S
FUFB
=
+⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤⎤⎡ ⎡ ⎤+ + ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎥⎢ ⎢ ⎥− ⎣ ⎦⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
K K 0 U0 CU U0 KC 0
## ### # φφ φ
gk gku u
⎣
M 00 M
(2.1143)
ffness matrix and structural otion loading terms
in (2.1143) in anticipation of the response spectrum.
The m
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤+ + − +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦k
k
00R R K dC dR0 0 0
##
Kwhere we include the structural stimass matrix M . We also include the ground m
derivation below. (2.1143) corresponds to an undamped gyroscopic system
odal expansion used is
(j) (j)r , =
n nj
j1 1j jj
ξξ= −∑ ∑U U F#φ
ω= =
(2.1144a,b)
where jξ is the generalized coordinate for mode j. The modal equation is
2 ( )j
j j jξ ω ξ Γ+ =## (2.114 5)
wh
ere
( ) ( )T T( ) (j) (j)φ u
jjΓ ω= − +F R U R (2.1146)
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ADINA R & D, Inc. 207
and, from (2.1143),
( ) ( )u UB UUS S= + −R R R K k gkud
FU k gkuφ =R C d # (2.1 147a,b)
With modal damping (damping ratio jζ ), (2.1145) becomes
2 ( )2 jj j j j j jξ ω ζ ξ ω ξ+ + =## # (2.1148)Γ
Then we use
(j) (j) (j)r r r
1 1 1, ,
n n nj j j
j j jj j j
ξ ξ ξω ω ω= = =
= = =∑ ∑ ∑U U U U U U# ##
#
(2.1149a,b,c)
= , = , =n n
dtξ ξ ξ− − −∑ ∑ ∑∫ F F F# ## #φ φ φ
(2.1150a,b,c)
to obtain the structural and fluid response. The initial modal coordinates are computed from the initial
e al a ial
second time derivatives of fluid potentials are not used.
(j) (j) (j)tn
1 1 10j j j
j j j= = =
conditions (displacements, velocities, fluid potentials and timderivatives of fluid potentials). Initi ccelerations and init
In the first solution step, we use the Newmark method with 1/ 2, 1α δ= = because this choice of Newmark paramete
not require initial accelerations. In the successive solution steps, we use the Newmark method with the usual New
rs does
mark parameters 1/ 4, 1/ 2α δ= = .
ory: Response spectrum, harmonic and random vibration alysis
2.11.8 The
an
are calculated for response spectrum nalysis and random vibration
a ar
! Modal participation factorsanalysis, harmonic vibration an lysis. There are two cases: modal participation factors
corresponding to applied forces (in which case we do not conside
Chapter 2: Elements
208 ADINA Structures ⎯ Theory and Modeling Guide
response spectrum analysis) and modal participation factors
n factor calculation, we only consider the dynamic solution (due to dynamically applied loads).
corresponding to ground motions. In the modal participatio
To begin the derivation, we introduce
jjx
j
ξω
= (2.1151)
as the new generalized coordinate. The modal expansion is then
(j) (j), =n n
x xω= −∑ ∑U U F#φ (2.1152a,b)
1j j j
j=
d
r1j=
an the modal equation of motion is
2 (x x x )2 jj j j j j jω ζ ω Γ+ + = ## # (2.1153)
where
( ) ( )T T( ) (j) (j)u
1jΓω φ= − +F R U R (2.115
j
4)
We will compute
( )jΓ rather than ( )jΓ because the modal expansion for Ur is then the commonly used one.
Applied forces: As it is assumed that the physical body forces are ent
tor is simply
irely static and that there are no ground motions, the modal participation fac
( )T( ) (j)jΓ = U R (2.1155)
As a consequence, the modal participation factor for any φ rigid body mode is zero.
Ground motions: The loads vectors are
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 209
( )u UU S k gk k gku u= − −R K d Md ## ,
FU k gkuφ =R C d # (2.1156a,b)
Hence
( ) ( )
( ) ( )
T T( ) (j) (j)FU
T(j)UU S
1
jk k
j
k gk
u
u
Γω
⎛ ⎞= − +⎜ ⎟⎜ ⎟
⎝ ⎠
−
F C d U Md
U K d
##gk (2.1157)
e term
( ) ( )T(j)UU S k gkuU K dWe neglect th (which is probably
small anyway) and observe that the ground motion modal participation factor is then
( ) ( )T T( ) (j) (j)FU
1
jk k
j
Γω
= +F C d U Md (2.1158)
The modal participation factor for a φ rigid body mode can be zero or nonzero. It can be shown that the modal participation factor for a φ rigid body mode is zero if each fluid region is completely surrounded by interface elements.
Note: static corrections (residual calculations) are not implemented for the potentialbased fluid element because residual
culations are based upon nonconstant (in space)
potentialbased fluid elements.
2.11.9 Modeling: Formulation choice and potential master degree of freedom
The potentialbased formulation discussed in this section is introduced in ADINA 7.4 and extensively modified in ADINA 8.0.
You can select the potentialbased formulation in the AUI commandline input using the parameter FLUIDPOTENTIAL in the MASTER command:
displacement calbody force loading. Such loading is not possible in general for the
Chapter 2: Elements
210 ADINA Structures ⎯ Theory and Modeling Guide
FLUIDPOTENTIAL=AUTOMATIC: The potentialbased for
FLUIDPOTENTIAL=YES: The potentialbased formulation of ADINA 7.3 and lower is employed. FLUIDPOTENTIAL=NO: The potentialbased formulation is not employed. Note, it is not possible to select the potentialbased formulation using the AUI user interfaces. The potentialbased formulation of ADINA 8.0 is employed when using the AUI user interfaces.
When FLUIDPOTENTIAL=AUTOMATIC, the AUI automatically detects the presence of potentialbased elements, and, if there are any potentialbased elements, activates the potential
2.11.10 M
).
Infinitesimal velocity formulation: Deleting the potential degree of freedom along part of the bounding surface has the effect of setting the pressure equal to
mulation of ADINA 7.4 and 8.0 is employed. This is the default.
master degree of freedom.
odeling: Potential degree of freedom fixities Subsonic formulation: Deleting the potential degree of freedom along part of the bounding surface has the effect of partially specifying the enthalpy along that surface, see equation (2.112This has no physical meaning.
0ρ Ω along that part of the bounding surface, forces, then see equation (2.1115). If there are no body 0Ω =
and the pressure is therefore set to zero along that part of the bounding surface. This boundary condition can be used if the displacements of the boundary are not of interest.
2.11.11 Modeling: Elements
The volume V of the fluid domain is modeled using twodimensional or threedimensional fluid elements. These elements
It is recommended that the potential degrees of freedom not bedeleted anywhere in the fluid.
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 211
dimensional solid elements and the nodal point numbering of the fluid elements is the sam
The tw either planar (unit thickness of fluid assumed) or axisymmetric (1 radian of fluid assumed).
or
e S of the fluid domain is modeled with potentialinterfaces and/or interface elements, as discussed in detail below.
It is required that each separate fluid domain be modeled with separate fluid element groups. This is because the AUI constrains the potential degrees of freedom of each element group together in static analysis during phi model co pletion, step 7, see Section 2.1
is not permitted to have fluid regions of different densities sharing the same potential degrees of freedom. This is because the nodal pressure would be different as computed from the fluid regions connected to the node.
is recommended that the fluid and adjacent structure not share appropriate
constraint equations between the fluid and structural degrees of freedom that are most appropriate during phi model completion, see Section 2.11.15.
2.11.12 Modeling: Potentialinterfaces
ne potentialinterfaces of
various types along the surface of the fluid domain. When you generate a data file, the AUI places interface elements along the
by the potentialinterfaces, except as noted
pes of potentialinterface: Fluidstructure: Place a fluidstructure potential interface on the
boundary between a potentialbased fluid and the adjacent
atically
are analogous to the twodimensional or three
e as the nodal point numbering of the solid elements. odimensional elements are
These elements are defined within 2D and 3D fluid element groups (in ADINA). Set the formulation of the element group to either Linear PotentialBased Element (for the infinitesimal velocity formulation) or Subsonic PotentialBased Element (fthe subsonic velocity formulation).
he bounding surfacT
m1.15. It
Itthe coincident nodes. This allows the AUI to construct
For ease of modeling, you can defi
boundaries specified below. ADINA itself does not use the potentialinterfaces.
There are several ty
structure. In many cases, the AUI can autom generate fluid
Chapter 2: Elements
212 ADINA Structures ⎯ Theory and Modeling Guide
re interface elements along the boundary between the fluid nd structure during phi model completion, step 1, see Section
2.11.15. So you typically do not need to define fluidstructure potentialinterfaces.
Free surface: Place a free surface potential interface on the
a
t of the waves is assumed to e small.
an terface on the boundary adjacent to an ADINAF mesh.
Usually the AUI automatically generates ADINAF interface elements along the boundary adjacent to an ADINAF mesh during phi model completion, step 1. So you typically do not need to
DINAF potentialinterfaces.
Infinite: Place an infinite potentialinterface wherever infinite boundary conditions are desired.
There are three types of infinite potentialinterface. Planar: In the subsonic formulation, the pressure and velocity at infinity must be specified. In the infinitesimal
formulation, the pressure and velocity zero.
and
Cylindrical: The radius of the boundary must be specified. ro, and
the velocities at the boundary are assumed to be small. This element cannot accurately model lowfrequency waves, that is,
waves with
structua
boundary where the pressures are to be prescribed and the displacements are desired, for example, on the free surface offluid. Surface waves can be approximately modeled in this manner, but note that the displacemenb
ADINAF: Place ADINAF potential in
define A
velocity are assumed to be Spherical: The radius of the boundary must be specified. The pressure and velocity at infinity are assumed to be zero,the velocities at the boundary are assumed to be small.
The pressure and velocity at infinity are assumed to be ze
1rc
ω< .
Inletoutlet: Place an inletoutlet potentialinterface wherever the pressure of the boundary is specified, and where the displacements
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 213
of the boundary are not of interest. For example, the outlet of a pipe on which the pressure is known can be modeled using an inletoutlet potentialinterface. If there is more than one inlet or outlet in the model, each inlet and outlet must have its own inletoutlet potentialinterface.
Fluidfluid: Place a fluidfluid potentialinterface on the boundary between two potentialbased fluid elements of two different element groups.
nts that share a common
y condition is modeled in ADINA by the absence of any interface element. Therefore a rigidwall potentialinterface suppresses any automatic generation of interface elements along the boundary of the rigidwall potentialinterface. However the AUI uses the rigidwall potential interface during phi model completion, step 2, in constructing structural normals, see Section 2.11.15.
2.11.13 M
ents are used on the surface of the fluid domain to specify a boundary condition. In many cases, you do not have to define interface elements. Rather you define potentialinterfaces and
ame element group as the fluid elements themselves. In 2D analysis, the fluidstructure interface elements are 2 or 3 node line segments, in 3D analysis, the fluidstructure interface elements are 3 to 9 node area segments. The nodal point numbering conventions are shown in Figure
The AUI automatically reorders all interface elements enerating the ADINA input data file so that the interf e
element normals point into the fluid.
Note that only one fluidfluid potentialinterface need be defined for each boundary. The AUI generates a fluidfluid interface element for each of the two elemeboundary during phi model completion, step 1, see Section 2.11.15. Rigidwall: Place a rigidwall potential interface wherever the fluidis not to flow through the boundary. Note that this boundar
odeling: Interface elements
Interface elem
the AUI then generates the interface elements. However we describe the interface elements here for completeness.
Interface elements are defined within the s
2.114.when g ac
Chapter 2: Elements
214 ADINA Structures ⎯ Theory and Modeling Guide
b) 3D elements
Fluid
n
1Solid
Fluid
1
n
2 Solid 32
a) 2D elements
111
9
5
6Solid
Fluid
n1
2
3
4
Solid
Fluid
n1
2
3
4
Solid
Fluid
n
72
3
4
Solid
Fluid
n 8
2
3
4
Figure 2.114: Fluidstructure interface elements, showingthe local node numbering convention
ace elements: There are severa types of interf F re interfa e
potentialbased fluid element with an adjacent structural element (Figure ). Each nod ial degree of freedom and displacement degrees of freedom. It is assumed that the displacements of the nodes of the interface element are small.
It is assumed that the structure provides stiffness to all translational degrees of freedom, because the fluidstructure interface element does not provide stiffness, mass or damping to
structure interface element and adjacent structural element must all
l
luidstructu ce element: This element connects th
2.115 e of the element contains the potent
the tangential directions. We emphasize that the potentialbased fluid element, fluid
be compatible.
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 215
Fluid element
Fluidstructureinterface element
Structural element
Nodal unknowns:
Nodal unknown:
Nodal unknown:
u
u
Assumptions: is small.u
Nodes circled byshare degrees offreedom
Figure 2.115: Fluidstructure interface element Free surface interface element: This element is placed onto the
boundary of a potentialbased fluid element wherever the pressure is to be prescribed, and wherever the displacements of the fluid are required (Figure 2.116). For example, the free surface of a fluid in a basin can be modeled using free surface interface elements. Each node of the element contains the potential degree of freedom and displacement degrees of freedom. It is assumed that displacements and velocities of the nodes of the interface element are small.
Fluid elementNodal unknowns: u
Nodal unknown:
Assumptions: is small.unun is small.
Free surfaceinterface element
Applied pressureor no applied pressure
.
Figure 2.116: Free surface interface element It is necessary to fix all displacements that are tangential to the
free surface interface ecause the free surface interface element does not provide stiffness, mass or
element, b damping to the
tangential directions. In many cases, the AUI can generate skew systems and fixities
corresponding to the tangential directions during phi model generation, see Section 2.11.15.
Chapter 2: Elements
216 ADINA Structures ⎯ Theory and Modeling Guide
boundary is adjacent e 2.117). Each node of the element contains the potential degree of freedom and displacement deg of s of the nod
Note ththrough the displacements of the shared boundary. Therefore, bec
luid flow between the ADINA and AD
ts that are tangential to the ADINAF interface element, because the ADINAF interface
an generate these fixities automatically during phi model completion, see Section 2.11.15.
The ADINAF interface element need not be compatible with the e
DINAF fluidstructure boundary in the ADINA model to connect the ADINA and ADINAF mo l
NAF interface elements during phi model completion, step 1, see Section 2.1
ADINAF interface element: This element is placed onto the of a potentialbased fluid element wherever the boundary to an ADINAF mesh (Figur
rees freedom. It is assumed that displacements and velocitiees of the interface element are small. at the coupling between ADINA and ADINAF is
ause the displacements of the ADINAF interface element are assumed to be small, the ADINA/ADINAF coupling cannot be used for modeling actual f
INAF models. It is necessary to fix all displacemen
element does not provide stiffness, mass or damping to the tangential directions. In many cases, the AUI c
el ments from the adjacent ADINAF mesh . It is necessary to define an A
de s, just as in ordinary ADINAF FSI analysis. In most cases, the AUI automatically generates the ADI
1.15.
ADINAFinterface element
Applied tractionsfrom ADINAF
Figure 2.117: ADINAF interface element
Assumptions: is small.unun is small..
Fluid element
Nodal unknown:
Nodal unknowns: u
Infinite interface elements: This element is placed onto the
boundary of a potentialbased fluid element wherever infinite boundary conditions are desired. Each node of the element
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 217
contains only a potential degree of freedom. There are three types of infinite interface element: Planar infinite element: In the subsonic formulation, the pressure and velocity at infinity must be specified. In the infinitesimal velocity formulation, the pressure and velocity are assumed to be zero. A planar infinite element is shown in Figure 2.118.
Planar infiniteinterface element
specifiedp , v
Assumptions: constant along boundary.
Flow velocity is close to v
Fluid element
Nodal unknown:
Nodal unknown:
Figure 2.118: Planar infinite interface element Spherical infinite element: The radius of the boundary must be specified. The pressure and velocity at infinity are assumed to be zero, and the velocities at the boundary are assumed to be small.
of the boundary must be specified. The pressure and velocity at infinity are assumed to
small. This element cannot accurately model lowfrequency
Cylindrical infinite element: The radius
be zero, and the velocities at the boundary are assumed to be
waves, that is, waves with 1rc
ω< .
When considering where to place the infinite interface elember that the infinite elements are derived under the
ements, remassbono oundary (Figure 2.119).
umption that waves travel normal to the boundary. Hence the undary must be placed so that any anticipated waves travel rmal to the b
Chapter 2: Elements
218 ADINA Structures ⎯ Theory and Modeling Guide
Figure 2.119: Direction of wave propagation must be normal to theinfinite interface element
a) Correct modeling
Direction ofwave propagationindicated by
b) Incorrect modeling
tlet interface element: This element is placed onto the ent wherever the pressure
is specified, and where the displacem
Inletouboundary of a potentialbased fluid elemof the boundary ents of the
ents. Each node of the element ent degrees
used to m displacements themselves
boundary are not of interest (Figure 2.1110). For example, theoutlet of a pipe on which the pressure is known can be modeled using inletoutlet interface elemcontains the potential degree of freedom and displacemof freedom. The displacement degrees of freedom are onlyco pute velocities and accelerations; theare not used.
Assumptions: constant along boundary.
constant along boundary.
Figure 2.1110: Inletoutlet interface element
Inletoutletinterface element
un.
Nodal unknowns: u
Fluid element
Nodal unknown:
Applied pressureor no applied pressure
It is necessary to fix all displacements that are tangential to the
inletoutlet interface element, belement does not provide stiffness, mass or damping to the
f the potential
ecause the inletoutlet interface
tangential directions. In addition, it is necessary to set the tangential fluid velocity to zero by constraining all o
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 219
y to set the
Fluidfluid interface element: This element is placed along the
interface between two potentialbased fluid elements of two different element groups (Figure 2.1111). For example, the interface between air and water can be modeled using fluidfluid interface elements.
degrees of freedom to be equal. Finally it is necessarnormal velocity to be uniform along the element, by constrainingthe normal velocities to be equal. In many cases, the AUI can generate these fixities and constraints automatically during phi model completion, see Section 2.11.15.
Fluid element ofelement group 2
Fluid element ofelement group 1
Fluidfluid interfaceelement of elementgroup 2
Fluidfluid interfaceelement of elementgroup 1
Nodal unknowns:
Nodal unknowns:
Nodal unknown:
Nodal unknown:
(1), un
(2), un
(2)
(1)
Assumptions: is small.unun is small..
Figure 2.1111: Fluidfluid interface element Each node of the interface element contains the potential degree
idlement
mon boundary requires a fluidfluid interface element. The two fluids
responding normal displacements of the other interface element.
of freedom and displacement degrees of freedom. The normal velocities and displacements are assumed to be small. It is necessary to fix all displacements that are tangential to the flufluid interface element, because the fluidfluid interface edoes not provide stiffness, mass or damping to the tangential directions.
Each of the potentialbased elements that share a com
are connected by constraining the normal displacements of one of the interface elements to the cor
Chapter 2: Elements
220 ADINA Structures ⎯ Theory and Modeling Guide
In many cases, the AUI can generate these fixities and constraints automatically during phi model completion, see Section 2.11.15.
2.11.14 M
Co
Concentrated forces, pressure loads and/or prescribed displacements can be applied directly to any part of the fluid boundary on which there are fluidstructure, free surface, inletoutlet or fluidfluid interface elements. However, when applying
tld make sure
tha
Massproportional loads
Massproportional loads can be applied. However the AUI and
ADINA make a distinction between those massproportional loads use del physical body forces and those massproportional loads used to enter ground accelerations. For each massproportional load, you must specify its interpretation: body force or ground acceleration.
s ds interpreted as physical
ody forces must be constant in time when there are finitesimal velocity potentialbased fluid elements in the odel. These loads are used in the construction of and
odeling: Loads
ncentrated forces, pressure loads, prescribed displacements
concentrated forces, remember that the AUI can apply skew systems o certain nodes on the fluid boundary during phi model completion, see Section 2.11.15. Therefore you shou
t the nodes on which you apply concentrated forces have the degree of freedom directions that you anticipate.
d to mo
Body force: Mas proportional loabinm kg Ω as follows:
MAGNITUDE A(k) TFkg × ×=
here MAGNITUDE is the magnitude of the massproportional ading, A(k) is the vector giving the direction of the mass
wloproportional loading and TF is the value of the time function. Then
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 221
(2.1159) ( )3
1k k k0
kg x x
=
Ω = −∑where k0x is a datum value (entered as part of the fluid material escription). Notice that the choice d
thof the datum value affects
solu
region.Bod only
be k rt dy .
an be timevarying, and are used in the construction of ugk as follows: Suppose that you specify a massproportional load interpreted as ground motions of magnitude gk(t), where gk(t) is entered as for body force loads (but the time function need not be constant). Then ADINA computes the ground motions using
Note that it is assumed that 0 .
t affect the structura ent
cluded in the potentialbased
fluid elements.
Mass flux loads (also referred to as phiflux loads) can be
e tion only when infinite interface elements are present, or if at least one potential degree of freedom is fixed in the fluid
y forces can be applied in a static analysis as the
loads in the analysis. Then, if other timevarying loads are present, a restart to dynamic analysis can be performed. The body forces should ept in the resta namic analysis Ground acceleration: Massproportional loads interpreted as ground motions c
, ,t t
gk k gk gk gk gktstart tstart
u g u u dt u u dt= − = =∫ ∫## # ## # (2.1160a,b,c)
( ) 0, ( )gk gku tstart u tstart= =#
The choice of interpretation for the massproportional loads does no l elem s used in the model.
Centrifugal loads
Centrifugal load effects are not in
Mass flux loads (phiflux loads)
prescribed directly onto potentialbased fluid elements. The mass
Chapter 2: Elements
222 ADINA Structures ⎯ Theory and Modeling Guide
fluxes can be distributed or concentrated. The dimensions of
distributed mass flux are [mass]
[time] [area]× in dynamic analysis and
are [mass][area]
in static analysis). The dimensions of concentrated
ma[mass]
ss flux are [time]
in dynamic analysis and are[mass] in static
analysis. Positive mass flux is assumed to represent mass flowing into the
fluid domain. No interface elements or potentialinterfaces should be defined
on boundaries with distributed or concentrated mass fluxes.
odeling: Phi model completion
2.11.15 M
s
performs phi model completion whenever generating a data file
ary of each fluid element region. If the fluid element side already has
att
with no
ele his step is to cover as much of the fluid boundary as possible with interface elements.
Note that phi model completion relies on determining the nodes that are close together. The tolerance used for this determination is the same tolerance as is used in coincident node checking
As can be seen above, there are many restrictions and condition
that must be considered when specifying boundary conditions on potentialbased fluid elements. Many of these conditions have been automated in the AUI in the following way. The AUI
in which potentialbased fluid elements are used. The steps in phimodel completion are:
1) The AUI loops over all fluid element sides on the bound
an interface element, the side is skipped. If the fluid element side has a potentialinterface, an interface element of the appropriate type is generated. Otherwise, the side is checked to see if it is
ached to a structure (shares structural degrees of freedom with structural elements), close to a structure (nodes coincident
des of a structural element) or on a ADINAF fluidstructure boundary; and, if any of the above conditions are met, an interface
ment of the appropriate type is generated. The intent of t
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 223
eter COINCIDENCE of comm
ts. d.
a) If the node is attached only to a free surface interface, AD Fthen the nand zero sThe free n ro stiffness directions are identified, and if they are not aligned with the global directions, a skew system is
ess
tion
3D, another direction orthogonal to the free normal and structural normal directions, which may be a zero stiffness direction or another structural normal direction. The free
to be orthogonal to the structural normal directions. The free normal, structural normal and zero
f d, if they are not aligned with the o generated that is aligned wi h normal and zero stiffness dir i
det the structure is smooth. This tolerance is parameter PHIANGLE in command TOLERANCES GEOMETRIC. See
The intent of step 2 is to identify the zero stiffness and free
(param and TOLERANCES GEOMETRIC).
2) The AUI loops over all nodes attached to interface elemen
If the node is attached to structural elements, the node is skippeOtherwise the types of the attached interface elements are determined. Then
INA interface, inletoutlet interface or fluidfluid interface, ode has a free normal direction (normal to the interface) tiffness directions that are tangential to the free normal. ormal and ze
generated that is aligned with the free normal and zero stiffndirections.
b) If the node is attached to a free surface interface, ADINAF
interface, inletoutlet interface or fluidfluid interface, and is also attached to a fluidstructure interface or rigidwall interface, the AUI proceeds as follows. The node has a free normal direc(determined from the free surface interface, ADINAF interface, inletoutlet interface or fluidfluid interface), a structural normal direction (determined from the fluidstructure interface or rigidwall interface), and, in
normal direction is modified
sti fness directions are identified, an gl bal directions, a skew system isth t e free normal, structural ect ons. This process relies on an angle tolerance in 3D analysis to ermine if
Example 2 below.
normal directions of the nodes.
Chapter 2: Elements
224 ADINA Structures ⎯ Theory and Modeling Guide
nterface elements. If t skipped.
or node de
g for differences in skew systems between A and B, accounting for B possibly being a slave node in a constraint
constrained if the degree of eedom is a free normal direction (see 2 above).
fluid mesh with the str
vots in the system matrices tha
. If the node is attached to a structural element, the node is skipped. Otherwise constraint equations are defined for all pairs of nodes, so
e
s
m a
3) The AUI loops over all nodes attached to ihe node is attached to a structural element, the node is
If the node (node A) is attached to a fluidstructure interface element and is close to a structural node B, node A is constrained to node B as follows. Each displacement degree of freedom fA is constrained to the corresponding degrees of freedom for noB, accountin
equation (but not accounting for B possibly being a slave in a rigid link), accounting for B possibly being fixed. But a displacement degree of freedom for node A is not fr
The intent of step 3) is to connect the
uctural mesh, when different nodes are used for the fluid andstructure. The connection still allows the fluid nodes to slip relative to the structural nodes on intersections between free surfaces and the structure.
4) In static analysis, when there are no body force loads, the
AUI loops over all nodes on a free surface, ADINAF interface, inletoutlet or fluidfluid interface. If the node is attached to astructural element, the node is skipped. Otherwise, constraint equations are defined for all nodes so that the displacements in the direction of the free normal are equal.
The intent is to remove the zero pit are otherwise present (see the discussion after equation
(2.1137)). 5) In dynamic analysis, or in static analysis when there are body
force loads, the AUI loops over all nodes on a fluidfluid interface
that the displacements in the direction of the free normal are compatible.
The intent is to enforce displacement compatibility between thfluids.
6) The AUI then loops over all nodes with zero stiffness degreeof freedo nd defines fixities for each zero stiffness degree of freedom.
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 225
al
on an inletoutlet. Note that the model that results from the phi model completion
rocess is not stored in the AUI, but is immediately written to the ADINA data file. Therefore the model that results from phi model completion cannot be displayed during model definition. Phi model completion does write some messages to the log file or user interface, which you may find helpful. However, we recommend that you also check the model by running the model through ADINA, for example, for one load step; then use ADINAPLOT to display the model.
Example 1: We now present a detailed example for a 2D fluid filled basin with flexible walls.
Figure 2.1112(a) shows the model before phi model completion. The model is defined with separate nodes for the fluid and the structure. One way to guarantee that the nodes on the fluid and structure are separate is to generate the structural elements as usual, but to set coincidence checking to group when generating the fluid elements.
A potentialinterface of type free surface is defined on the geometry line corresponding to the free surface.
7) In static analysis, the AUI constrains all of the potential degrees of freedom for an element group together.
8) In dynamic analysis, the AUI constrains all of the potenti
degrees of freedom on an inletoutlet together. The intent is to remove the tangential flows
p
Chapter 2: Elements
226 ADINA Structures ⎯ Theory and Modeling Guide
Potentialinterface of typefree surface is defined onthis line
Fluid nodes are at the samecoordinates as structural nodes.The fluid nodes and structuralnodes are separated in this figurefor clarity.
a) Finite element model before phi model completion
Figure 2.1112: Example 1 of phi model completion
Potentialbasedfluid elements
Beam elements
1 10
11
12131430o
15
16 23
456
79 8
str nt ts where the fluid is adjacent to the structure, and free surface interface elements corresponding to the potentialinterface (Figure 2.1112(b)).
In step 1 of phi model completion, the AUI generates fluid
ucture i erface elemen
Interface elementsof type free surface
Interface elements oftype fluidstructure interface
b) Step 1 of phi model completion;interface elements are created
Figure 2.1112: (continued)
123
46
79 8
In step 2 of phi model completion, the AUI classifies the placement directions on the free surface (Figure 2.1112(c)). tice that the free normal for node 2 is taken from the free
disNo
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 227
sur of becthe global coordinate directions. node 2 will be fixed in step 6 below.
face, but the free normal for node 3 is modified by the presencethe adjacent structure. A skew system is defined for node 3 ause the free normal and structural normal are not aligned with
The zero stiffness direction of
Node 1: y = structural normal direction,z = free normal direction
Node 2: y = zero stiffness direction,z = free normal direction
Node 3: b = free normal direction,c = structural normal direction
c) Step 2 of phi model completion: classificationof displacement directions on free surface
Figure 2.1112: (continued)
12y
b
cy
zz
3
In step 3 of phi model completion, the AUI constrains the fluid
displacement directions to the structure (Figure 2.1112(d)). At node 1, only the y displacement direction is constrained; the z displacement is left free so th the free surface c slip along the wall. At node 3, only the c displacement direction is constrained; the b displacement is left free so that the free surface can
at an
slip along the
ice that at node 4, both the y and z displacements are constrained to the structure. The fluid still slips in the z direction
ment (the y displacement in this case) is used by the fluid equations. Similar statements hold for
Nodes 7 and 9 are fixed because corresponding nodes 12 and 14 are
wall. Not
because only the normal displace
nodes 6 and 8.
fixed.
Chapter 2: Elements
228 ADINA Structures ⎯ Theory and Modeling Guide
d) Step 3: Creation of constraint equations and fixities
Figure 2.1112: (continued)
1 10
11
121314
15
16 3
46
79 8
Node 6: = , =u u u uy y z z15 15
Node 8: = , =u u u uy y z z13 13Node 7: = =fixedu uy z
Node 9: = =fixedu uy z
Node 4: = , =u u u uy y z z11 11
Node 1: =u uy y10
Node 3: =cos30  sin30u u uc y zo 16 o 16
orces, then the AUI performs step 4 of phi model completion (Figure 2.1112(e)). The free surface can only translate vertically as a rigid body.
If the analysis is static without body f
e) Step 4 of phi model completion: defining constraintequations to set normal displacements equal on free surface
Step 4 is only performed in static analysis whenthere are no body forces.
Figure 2.1112: (continued)
12y
b
cy
zz
3
Node 2: =u uz z1
Node 3: =u ub z1 / cos30
o
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 229
This motion affects the total mass of the fluid region, so that
e all i .
Step 5 of phi model completion is skipped because there are no fluidfluid interfaces.
s fixed (Figure 2.1112(f)). Vertical motions of the nodes
attached only to free surface interface elements are allowed, but hoflu ss to horizontal mo ons).
there is no zero pivot in the system matrices. If there are body forces, then step 4 is not necessary becaus
boundary motions are given stiffness by the matr x ( )UU SK
In step 6 of phi model completion, the zero stiffness direction atnode 2 i
rizontal motions of these nodes are not allowed (because the id does not provide stiffness, damping or mati
f) Step 6 of phi model completion: defining fixities toeliminate zero stiffness degrees of freedom
Figure 2.1112: (continued)
2y
z
Node 2: =fixeduy
. If the analysis is static, then the AUI performs step 7 of phi
mo y constant (in space) po
del completion (Figure 2.1112g). Onltentials are allowed in static analysis.
Chapter 2: Elements
230 ADINA Structures ⎯ Theory and Modeling Guide
g) Step 7: defining constraint equations to set all potentialdegrees of freedom equal
Step 7 is performed only in static analysis
Figure 2.1112: (continued)
123
456
79 8
2 9 1 == , ...,1
Exam An alternative way to model the problem of Example 1 is shown in Figure 2.1113. Here the free surface is shifted downwards slightly so that the nodal coincidence checking feature of the AUI mesh generation algorithms sting nodes on the free surface boundary. Depending on how much you shift the fluid free surface, yuse
ple 2:
will not reuse any exi
ou may need to adjust the tolerance d in the coincidence checking algorithms.
Potentialinterface of typefree surface is defined onthis line
Figure 2.1113: Alternative modeling of problem in Figure 2.1112
Potentialbasedfluid elements
Beam elements
1
10
30o
11 2
3
456
79 8
During phi model completion, it is necessary to set the
coincidence tolerance to be loose enough so that node 1 is
on ocesses nodes 1 and 3.
T
considered close to node 10 and node 3 is considered to be closeto node 11. Then steps 1, 2, 4, 6, 7 of phi model completiproceed as in Example 1, and step 3 only pr
he solution results will be almost exactly the same as in Example
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 231
is
edge of the nodal coincidence checking features of the AUI mesh generation commands.
ider the model shown in Figure ral
1 (the solutions will be slightly different because the geometryslightly different).
We think that the user input for Example 2 is more difficult thanthe user input for Example 1 because Example 2 requires a good working knowl
Example 3: In 3D analysis of a fluidfilled basin, there is one additional consideration. Cons2.1114, in which only the free surface and the adjacent structunodes are shown.
Fluid nodes are at the samecoordinates as structural nodes.The fluid nodes and structuralnodes are separated in this figurefor clarity.
a) Top view of finite element model
Figure 2.1114: Example 3 of phi model completion
Sides of 3D potentialbasedfluid elements on free surface
Sides of shell elements
1
10 11 12
13 14 15
16
17
1820 19
21
22
2 3
4 5 6
7 98
y
x
ines the
ice that node 3 has two structural normals, but node 6 has only one structural normal. That is because the angle between
NGLE, 6 is less
an PHIANGLE. because the
During step 2 of phi model completion, the AUI determstructural normal direction(s), zero stiffness direction(s) and free normal direction for the nodes of the free surface (Figure 2.1114(b)). Not
the two structural normals for node 3 is greater than PHIAbut the angle between the two structural normals for nodeth
Nodes 3, 6, 9 and 12 are assigned skew systems
Chapter 2: Elements
232 ADINA Structures ⎯ Theory and Modeling Guide
structural normal directions are not aligned with the global system.
b) Classification of structural normals and zero stiffness directionsfor some nodes on the free surface
n14
n12
n23
n36
n69
Node 1: Structural normal 1 =Structural normal 2 =Free normal =
nn
z
14
12
Node 3: Structural normal 1 =Structural normal 2 =Free normal =
nn
z
23
36
Node 2: Structural normal = =Zero stiffness direction =Free normal =
n nx
z
12 23
Node 5: Zero stiffness direction 1 =Zero stiffness direction 2 =Free normal =
xy
z
Node 6: Structural normal = average of andFree normal =Zero
n nz
36 69
stiffness direction = remaining orthogonal direction
Figure 2.1114: (continued)
1
10 11 12
2 3
4 65
87 9
y
x
During step 3 of phi model completion, the AUI creates
r example, node 1 is constrained in both the x and y to node 13, because both directions are structural normal
constraint equations for the fluid nodes adjacent to the structural nodes. Fodirections directions. Node 2 is also constrained in both the x and y
tion.)
ause these directions are zero stiffness directions, and there is no adjacent structure.
directions to node 14, here because the y direction is a structural normal direction and the x direction is a zero stiffness direction. (It is assumed that the structure provides stiffness in the x direc
During step 6 of phi model completion, the AUI fixes the x and
y directions for nodes 5 and 8, bec
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 233
present some of the steps for phi model completion of an enclosure with two distinct fluid regions. Figure Example 4: We now
2.1115(a) shows the model before phi model completion.
Fluid nodes are at the same coordinates as structural nodes.The fluid nodes and structural nodes are separated in this figurefor clarity.
Enclosed nodes all lie onthe same geometry line,potentialinterface of typefluidfluid is defined onthis line
a) Finite element model before phi model completion
Figure 2.1115: Example 4 of phi model completion
2D potentialbasedfluid elements, group 1
2D potentialbasedfluid elements, group 2
Beam elements
1
101112
131415
1617
1820 19
23
456
79 8
In step 1 of phi model completion, the AUI generates fluide
e
he rf ree normal is always in the z direction and the zero stiffness directions for nodes 5 and 8 are in the y direction.
In step 3 of phi model completion, the fluid nodes are constrained to the adjacent structural nodes (Figure 2.1115(b)). Notice that the nodes on the fluidfluid interface are allowed to slip relative to the structure.
structure interface elements where the fluid is adjacent to thstructure, and fluidfluid interface elements corresponding to thpotentialinterface. Four fluidfluid interface elements are generated, two for each shared element side.
In step 2 of phi model completion, the AUI classifies the displacement directions on t fluidfluid inte ace. Here the f
Chapter 2: Elements
234 ADINA Structures ⎯ Theory and Modeling Guide
b) Step 3: constraining fluid nodes to adjacent structural nodes
Figure 2.1115: (continued)
1
101112
131415
1617
1820 19
23
456
79 8
Node 4: =u uy y16
Node 6: =u uy y17
Node 9: =u uy y17
Node 7: =u uy y16
Node 10: = , =u u u uy y z z18 18
Node 12: = , =u u u uy y z z20 20Node 11: = , =u u u uy y z z
19 19
Node 1: = , =u u u uy y z z13 13
Node 3: = , =u u u uy y z z15 15
Node 2: = , =u u u uy y z z14 14
If the analysis is static without body forces, then the AUI
performs step 4 of phi model completion. In this case, the z displacements of nodes 5 to 9 are constrained to be equal to the z displacement of node 4. The free surface can only translate
ly
fluid interface are constrained to each other (Figure 2.1115(c)). e that the
potential degrees of freedom are not constrained.
vertical as a rigid body. In step 5 of phi model completion, the fluid nodes on the fluid
Step 5 is not performed if step 4 was performed. Notic
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 235
c) Step 5: constraining adjacent nodes of the two fluid groups together
Figure 2.1115: (continued)
1
101112
131415
1617
1820 19
23
456
79 8
Node 4: =u uz z7
Node 6: =u uz z9
Node 5: =u uz z8
In step 6 of phi model completion, the zero stiffness direction of
id element groups.
nodes 5 and 8 are fixed. Vertical motions of the nodes attached only to fluidfluid interface elements are allowed, but horizontal motions are not allowed (because the fluid does not provide stiffness, damping or mass to horizontal motions).
If the analysis is static, then the AUI performs step 7 of phi model completion (Figure 2.1115(d)). Only constant (in space) potentials are allowed in static analysis, but the potential can be different for the two flu
Chapter 2: Elements
236 ADINA Structures ⎯ Theory and Modeling Guide
d) Step 7: defining constraint equations to set the potentialdegrees of freedom of each element group together
Figure 2.1115: (continued)
1
101112
131415
1617
1820 19
23
456
79 8
Step 7 is performed only in static analysis
2 6 1 == , ...,1 8 12 7 == , ...,7
Exam
ple 5: We demonstrate how to model an initial pressure in confined fluid region. This ca be done in a static analysis as
fluid element, physically unattached
d apply the specified initial pressure to the inletutlet.
The constraint equations that the AUI defines in static analysis fluid element to the confined fluid region.
a nfollows: Define an auxiliaryto the fluid region but belonging to the same fluid element group. On the auxiliary fluid element, place a potentialinterface of type inletoutlet, ano
connects the auxiliaryThe result is that the value of φ# in the confined fluid region is thsame as the value of
e φ# in the auxiliary fluid element; the value of
φ# in the auxiliary ent is determined by the pressure that you apply to the inletoutlet.
If you restart to a dynamic or frequency analysis, fix all of the egrees of freedom of the auxiliary fluid element. Since, in
onstraint equations connecting the potential degrees of freedom, the aux
fluid elem
ddynamic or frequency analysis, the AUI does not define c
iliary fluid element is no longer connected to the confined fluid
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 237
region. The initial values of φ# in the confined fluid region are thafro
t m the static analysis, so the confined fluid region has the same
initial pressure as in static analysis. The modeling process is schematically illustrated in Figure
2.1116.
Figure 2.1116: Modeling an internal pressure in a confined fluid region
Confinedfluid regionwith nonzerofluid pressure
a) Physical problem b) Modeling in static analysis
Finite fluidelements
Auxiliaryfluid element,in same fluidelement group
Applied pressure
Constraint equationsautomatically generatedby the AUI during phimodel completion
Potentialinterfaceof type inletoutlet
c) Restart to dynamic or frequency analysis
Fix all degrees of freedomattached to the auxiliary fluidelement.
Pressure atbeginning ofrestart analysis= pressure atend of staticanalysis.
2.11.16 M
odeling: Considerations for static analysis
In static analysis, the potential degree of freedom output should
Chapter 2: Elements
238 ADINA Structures ⎯ Theory and Modeling Guide
be interpreted as φ# , not φ . 2.11.17 Modeling: Considerations for dynamic analysis
t time
2.11.18 Modeling: Considerations for frequency analysis and m
participation factor analysis
frequency data) cannot be performed if infinite interface elements
ns, it is y
Any of the direct time integration methods available in ADINA,
except for the central difference method, can be used in direcintegration. Rayleigh damping can be specified in the structure.
Please remember that there is no damping within the fluid. Therefore the choice of initial conditions is extremely important, because poorly chosen initial conditions will cause transient responses that are not damped out.
odal
Frequency analysis (and all analyses that depend upon
are present in the analysis. For ground motion modal participation factor calculatio
recommended that each fluid region be completely surrounded binterface elements. This sets the modal participation factors for the φ rigid body modes to zero.
Figure 2.1117 shows a onedimensional example in whichthere is a φ rigid body mode and the ground motion modal participation factor is nonzero. The motion excited by the nonzero gro rtic ation factor corresponds to a constant ground acceleration. Notice that this ground motion causes the unbounded expansion of the fluid region.
und motion modal pa ip
2.11: Potentialbased fluid elements
ADINA R & D, Inc. 239
Figure 2.1117: Problem in which the ground motion modal
participation factor for the rigid body modeis nonzero
u = constant ground motionz
Fluid
u = 0z
b) Interpretation of rigid body mode
Fluid
No boundarycondition
a) Finite element model
u is the absolutedisplacement
z
k k
Fixed toground
2.11.19 Modeling: Elem
ation to the porthole file (the integration point
TY is applicable only for 3D elements). is the value of p and
ent output
Each fluid element outputs, at its integration points, the following informnumbering is the same as the numbering convention used for the solid elements): FE_PRESSURE, FLUID_REFERENCE_PRESSURE, ELEMENT_XVELOCITY, ELEMENT_YVELOCITY, ELEMENT_ZVELOCITY (ELEMENT_XVELOCIFE_PRESSUREFLUID_REFERENCE_PRESSURE is the value of ρΩ (see Equation (2.1159)).
In the output of modal quantities, ELEMENT_XVELOCITYELEMENT_YVELOCITY, ELEMENT_ZVELOCITY shouldinte
, be
s, in
. ot have any output.
rpreted as modal particle displacements, not velocitieaccordance with equation (2.1142). However in all other types of analysis (static, direct time integration, modal superposition, response spectrum, harmonic and random), these quantities are velocities. The velocities are absolute velocities, even when ground motions are entered using massproportional loads
Interface elements do n
Chapter 2: Elements
240 ADINA Structures ⎯ Theory and Modeling Guide
Th
is page intentionally left blank.
3.1: Stress and strain measures in ADINA
ADINA R & D, Inc. 241
3. Materiar is to summarize the theoretical basis
models and formulations available i
3.1 Stress
3.1.1 Summ
f the input data in which the material with respect to these stress and strain
A
the stress and strain measures used for the input data and a
e
utput: All elements a m put Cauchy stresses
and engineering strains
Input of material parameters: 2nd PiolaKirchhoff stresses and GreenLagrange strains. Note that under small strain conditions, 2nd PiolaKirchhoff stresses are nearly equal to engineering stresses, and GreenLagrange strains are nearly equal to
l models and formulations
The objective of this chapteand practical use of the material n ADINA.
and strain measures in ADINA
ary
! It is important to recognize which stress and strain measures are employed in the use of a material model:
< In the preparation oparameters are definedmeasures < In the interpretation of the analysis results in which the type of stresses and strains output must be considered
! The practical use of the material models available in ADINgardingre
an lysis results is summarized in the following. These stress/strain measures are described in detail in ref. KJB, Section 6.2.
Small displacement/small strain formulation
Input of material parameters: All elements and material models usthe engineering stressengineering strain relationship.
O nd aterial models out.
Large displacement/small strain formulation
engineering strains.
Chapter 3: Material models and formulations
242 ADINA Structures ⎯ Theory and Modeling Guide
Output:
odel
GreenLagrange strains.
For the two and three
creep, multilinearplasticcreep, plasticcreepvariable, multilinearplasticcreepvariable, viscoelastic, usersupplied. In these cases the updated Lagrangian Hencky formulation is used.
Input of material parameters: Cauchy (true) stresses and logarithmic (true) strains
Output: Cauchy stresses and deformation gradients.
Kin atic Hardening Using the Logarithmic Stress and Strain Measures", Int. J.
ation is used.
(1) 2D, 3D elements: all material m s output Cauchy stresses and GreenLagrange strains.
(2) Beam, isobeam, pipe, shell elements: all material models output 2nd PiolaKirchhoff stresses and
Large displacement/large strain formulation, 2D and 3D elements
dimensional solid elements, the following material models can be used:
(1) Plasticbilinear, plasticmultilinear, DruckerPrager, Mroz bilinear, plasticorthotropic, thermoplastic, creep, plastic
ref. A.L. Eterovic and K.J. Bathe, "A HyperelasticBased Large Strain ElastoPlastic Constitutive Formulation with Combined Isotropic em
Numerical Methods in Engineering, Vol. 30, pp. 10991114, 1990.
(2) MooneyRivlin, Ogden, ArrudaBoyce, hyperfoam. In this case the total Lagrangian formul
Input of material parameters: MooneyRivlin, Ogden, ArrudaBoyce or hyperfoam constants.
Output: Cauchy stresses and deformation gradients.
3.1: Stress and strain measures in ADINA
ADINA R & D, Inc. 243
ment Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis," J. Computers and Structures, Vol. 26, 1987.
Large displacement/large strain formulation, shell elements For the shell elements, the following material models can be used, provided that the elements are single layer elements described using midsurface nodes:
(1) Plasticbilinear, plasticmultilinear. Either the updated Lagrangian Jaumann (ULJ) formulation or the updated Lagrangian Hencky (ULH formulation) , can be used if the shell elements are 3node, 4node, 9node or 16node shell elements. The default is the ULH formulation, except in the following cases:
a) the rigidtarget contact algorithm is employed,or b) explicit time integration is employed
(2) Plasticorthotropic. The shell elements must be 3node, 4node, 9node or 16node elements. The updated Lagrangian Jaumann (ULJ) formulation is employed.
When the ULJ formulation is employed for shell elements:
Input of material parameters: Cauchy (true) stresses and logarithmic (true) strains.
Output: Cauchy stresses and logarithmic strains.
When the ULH formulation is employed for shell elements:
Input of material parameters: Kirchhoff stresses and logarithmic (true) strains.
Output: Kirchhoff stresses and left Hencky strains.
! Small strains are strains less than about 2%.
ref. T. Sussman and K.J. Bathe, "A Finite Ele
Chapter 3: Material models and formulations
244 ADINA Structures ⎯ Theory and Modeling Guide
Strain measures: The strain measures used in ADINA are illustrated hereafter in the simplified case of a rod under uniaxial tension (see Fig. 3.11).
Engineering strain: 00
0
e −=! !!
2 20
20
12
−ε =
! !!
GreenLagrange strain:
2 20
2
12a
−ε =
! !!
Almansi strain:
00
ln de⎡ ⎤⎛ ⎞
= =⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦
∫!
!
! !! !
Logarithmic strain and Hencky strains:
0
λ =!!
Stretch:
! GreenLagrange strains are used in the large displacement/small strain formulations. This is because large rotations do not affect the GreenLagrange strains (GreenLagrange strains are invariant with respect to rigidbody rotations), and for small strains, small rotations, GreenLagrange strains and engineering strains are equivalent.
! In the small strain formulations, the current area is always assumed to be equal to the initial undeformed area.
0
F
F
A
A0
Figure 3.11: Rod under uniaxial tension
A = initial section area0
A = final section areaF = applied force
= +0
ref. KJBSec. 6.2.2
3.1: Stress and strain measures in ADINA
ADINA R & D, Inc. 245
! Engineering strains are also called nominal strains in the
Stress measures: The stress measures used in ADINA are illu case of a rod under uniaxial tension (see Fig. 3.11).
ngineering stress:
literature.
! Logarithmic strains are also known as true strains.
strated hereafter in the simplified
E0
FA
σ =
0AF
A Aστ = =Cauchy stress:
olaKirchhoff stress: 0 0
0
FSA
σ= =! !! !
2nd Pi
Kirchhoff stress: 0 0 0
FJA
στ = =! !! !
! resses. !
Cauchy stresses are also known as true st
For the case in which the material is incompressible,
0
J στ τ= =!!
can be used to compute the Cauchy stress and the
Kirchhoff stress from the engineering stress.
When the strains are small, the 2nd PiolaKirchhoff stresses are nearly equal to the Cauchy stresses from which the rigid body rotations have been removed. ! 2nd PiolaKirchhoff stresses are input only for the large dis rmass e str ! 2nd PiolaKirchhoff stresses are output only for the large
!
placement/small strain fo ulation. Because the strains are umed to be small, the engineering stresses can be entered as thess input quantities.
Chapter 3: Material models and formulations
246 ADINA Structures ⎯ Theory and Modeling Guide
diswh te system of the dedostr uchy stresses, because the strains are assumed to be small. If the element also undergoes rigid body rotatio inate sys change (because the element coordinate system also rotates). Since the 2nd PiolaKirchhoff stresses, as expressed in the undeformed element coordinate system, also do not change during a rigid body rotation, these 2nd PiolaKirchhoff stresses remain nearly equal to the Cauchy stresses. The 2nd PiolaKirchhoff stresses here provide a convenient way of removing the rigid body rotations from the output stresses. ! When the material is nearly incompressible, the Kirchhoff stresses are nearly equal to the Cauchy stresses. ! Since Kirchhoff stresses are input/output only for materials that are nearly incompressible, practically speaking, the differences bet irchhoff and Cauch stresses are negligib
3.1.2 Large strain thermoplast naly
! wi llowimu r, Mroz bilinear, plasticorthotropic, the mucreep e, ltilinearplasticcreepvar ! rge strain ine Section 6.6 ref e strpla
refStr lag, 2003.
placement/small strain formulation and for element types in ich the stresses expressed in the element coordina
formed element are of physical significance. If the element es not undergo rigid body rotations, the 2nd PiolaKirchhoff esses are nearly equal to the Ca
ns, the Cauchy stresses expressed in the element coordtem do not
ween K y le.
icity and creep a sis
This section discusses large strain analysis (ULH formulation) th the fo ng material models: plasticbilinear, plasticltilinear, DruckerPragermoplastic, creep, plasticcreep, ltilinearplasticcreep,
variable, plasticcreepvariabl muiable, usersupplied.
The following is a quick summary of the theory of lalastic analysis. For further information, see ref KJB,.4 and also the following references:
. F.J. Montáns and K.J. Bathe, "Computational issues in largain elastoplasticity: an algorithm for mixed hardening and stic spin", Int. J. Numer. Meth. Engng, 2005; 63;159196.
. M. Kojić and K.J. Bathe, Inelastic Analysis of Solids and uctures, SpringerVer
3.1: Stress and strain measures in ADINA
ADINA R & D, Inc. 247
To efodeformatioconfiguratinclude the . Polar decomposition into rotation and right stretch tensor: The tot tensor X can be decomposed into a ma rial rigidbody rotation tensor R and a symmetric positivedefinite (right) stretch tensor U (polar decomposition)
tal d rmation gradient tensor: Let X be the total n gradient tensor at time t with respect to an initial
ion taken at time 0. For ease of writing, we do not usual left superscripts and subscripts
al deformation gradientte
:
=X R U (3.11) Principal directions of right stretch tensor: The right stretch tensor U can be represented in its principal directions by a diagonal tensor , such that (3.12) where is a is a rotation tensor with respect to the fixed global axes (see Figure 3.12).
Λ
TL L=U R Λ R
LR
Chapter 3: Material models and formulations
248 ADINA Structures ⎯ Theory and Modeling Guide
RL
R
(t)
(0)x
y
R
(0)
(t)
RL
Initial configuration at time 0
Configuration at time t not including rigid body rotation
Configuration at time t
Directions of maximum/minimum total stretchesand strains
Material rigidbody rotation between time 0 and time t
Directions of initial configuration fibers withmaximum/minimum total stretches and strains
Figure 3.12: Directions of maximum/minimumtotal stretches and strains
(Note that the rotation does not correspond to a material
rdinate system: U nd are two representations of the same deformed state,
in the global coordinate system and in the U principal irections coordinate system.)
ight Hencky strain tensor: The Hencky strain tensor (computed the right basis) is given by
LRrigidbody rotation, but to a rotation of the coo
Λarespectivelyd
Rin ln lnR T
L L= =E U R Λ R (3.1 The superscript R symbolizes the right basis. Polar decomposition into rotation and left stretch tensor: The total deformation gradient tensor X can also be decomposed into a materia
3)
l rigidbody rotation R and a symmetric positivedefinite (left) stretch tensor V (polar decomposition):
3.1: Stress and strain measures in ADINA
ADINA R & D, Inc. 249
=X V R (3.14)
i
tensor: The left stretch tensor V can be represented in its principal directions by a diagonal tensor
R n (3.13) is the same as R in (3.11). Principal directions of left stretch
Λ , such that
TE E=V R Λ R (3.15)
where ER is a rotation tensor with respect to the fixed global axes. Note that E L=R R R . Left Hencky strain tensor: The Hencky strain tensor (computed in the left basis) is given by ln lnL T
E E= =E V R Λ R (3.16) The superscript L symbolizes the left basis.
Comparison of left and right Hencky strain tensors: Theprincipal values of the left and right identical, and equal to the logarithm
Hencky strain tensors are s of the principal stretches.
Hence both of these strain tensors can be considered to be logarithmleft and right Hencky strain tensors are different. The principal directions of the right Hencky stra tensor do not contain the rigid body rotations of the material, but the principal directions of the left Hencky strain tensor contain the rigid body rotations of the ma Therefore, for a material undergoing rigid body rotations, the principal directions of the right Hencky strain tensor do not rotate, however the principal directions of the left Hencky strain tensor rotate with the material. Hence, the left Hencky strain tensor is preferred for output and visualization of the strain state.
Multiplicative decomposition of deformation gradient in inelastic analysis: In inelastic analysis, the following multiplicative decomposition of the total deformation gradient into
ic strain tensors. However, the principal directions of the
in
terial.
Chapter 3: Material models and formulations
250 ADINA Structures ⎯ Theory and Modeling Guide
an elastic deformation gradient EX and an inelastic deformation gradient is assumed:
PX
E P=X X X (3.17) To understand (3.17), consider a small region of material under a given stress state with deformation gradient X. If this region of material is separated from the rest of the model and subjected to the same stress state, the deformation gradient is still X. Now if the stress state is removed, (3.17) implies that the deformation gradient of the unloaded material is . The stresses are due ent s associated with the elastic deformation gradient
PXirely to the strain
EX . It can be shown (see Montáns and Bathe), that (3.17) is equivalent to the additive decomposition of the displacements into elastic displacements and plastic displacements. Fo dered here, det 1P =X . r the materials consi Polar decomposition of elastic deform tion gradient: The elastic deformation gradient can be decomposed into an elastic rotation ten
a
sor ER and elastic right and left stretch tensors EU , EV : E E E E E= =X R U V R .18a,b Elastic Hencky strain tensors: The elastic Hencky strain tensorsin the right and left bases are giv
(3 )
en by
ln , lnER E EL E= =E U E V (3.19a,b) Stressstrain relationships: The stresses are computed from the elastic Hencky strain tensors using the usual stressstrain law of isotropic elasticity. However, the stress measures used depend upon the strain measures used. When the right Hencky strain measure is used, the stress measure used is the rotated Kirchhoff stress
3.1: Stress and strain measures in ADINA
ADINA R & D, Inc. 251
( )TE EJ=τ R τ R (3.110)
and when the left Hencky strain measure is used, the stress measure is the (unrotated) Kirchhoff stress Jτ . detJ = X ume change of the material, and, using det 1,
is the vol.
With theses choices of stress and strain measures, the stresses and t ins are workconjugate.
e num strain
Imdifference between the Cauchy and Kirchhoff stresses is neglected. The stress measure used with the right Hencky strains is
detP EJ= =X X
s
ra
The choice of right Hencky strain and rotated Kirchoff stresses gives the sam erical results as the choice of left Henckyand (unrotated) Kirchhoff stresses).
plementation notes: For 2D and 3D solid elements, the
( )TE Eτ R τ R . The input of material properties is assumed to be
in terms of Cauchy stresses, and the output of stresses is in terms of uchy stresses.
For shell elements, Kirchhoff stresses a
=
Ca
re used throughout. The rchhoff
tresses, and the output of stresses is in terms of Kirchhoff stresses. These assumptions are justified because they are used with material models in which the plastic deformations are incompressible and
tic deformations are genera ic ations.
3.2 Linea
l models are discussed in this section:
Elasticorthotropic: orthotropic linear elastic
input of material properties is assumed to be in terms of Kis
the plas lly much larger than the elastdeform
r elastic material models
! The following materia
Elasticisotropic: isotropic linear elastic
Chapter 3: Material models and formulations
252 ADINA Structures ⎯ Theory and Modeling Guide
In each mostrain.
s. In all cases, the strains are assumed small.
When the elasticisotropic and elasticorthotropic materiaused with the small displacement formulation, the formulation is
merically more effective.
! In the small displacement formulation, the stressstrain relationship is
del, the total stress is uniquely determined by the total
! These models can be employed using the small displacement or large displacement formulation
ls are
linear.
! If the material models are employed in a large displacement analysis, the total or the updated Lagrangian formulation is automatically selected by the program depending on which formulation is nu
0 0t tσ = C e
in which 0t σ = engineering stresses and 0
t e = engineering strains.
In the total Lagrangian formulation, the stressstrain relationship is !
0 0t t= εS C
in which = second PiolaKirchhoff stresses and = Green
ge strains.
is
0t S 0
Lagran
! In the updated Lagrangian formulation, the stressstrain relationship
t t at
t ε
τ = εC
in which t τ = Cauchy stresses and t at ε = Almansi strains.
on
ref. KJBSection 6.6.1
! The same matrix C is employed in all of these formulati s. As long as the strains are small (with large displacements), the difference in the response predictions obtained with the total and
3.2: Linear elastic material models
ADINA R & D, Inc. 253
updated Lagrangian formulations is negligible.
e, the difference in the response ns is very significant (see ref. KJB, pp 589590). If the
strains are large, it is recommended that the linear elastic material model not be used.
3.2.1 Elas
s ma el is available for the truss, 2D solid, 3D solid, beam, isobeam, plate, shell and pipe elements.
! The two material constants used to define the constitutive relation (the matrix C) are
E = Young's modulus, v = Poisson's ratio
! The same constants are employed in the small and large displacement formulations, and hence the matrices C are identically
nt rm f
3.2.2 Elas ial model
! The elasticorthotropic material model is available for the 2D solid, 3D solid, plate and shell elements.
2D solid elements: Figure 3.21 shows a typical twodimensional ele "a"
ref. KJBTable 4.3,
p. 194
! However, if the strains are largpredictio
ticisotropic material model
! Thi terial mod
the same in all formulations.
! You can specify the coefficie of the al expansion as part othe elasticisotropic material description. The coefficient of thermal expansion is assumed to be temperatureindependent.
ticorthotropic mater
ment, for which the inplane orthogonal material axes areand "b". The third orthogonal material direction is "c" and is perpendicular to the plane defined by "a" and "b". The material constants are defined in the principal material directions (a,b,c).
Chapter 3: Material models and formulations
254 ADINA Structures ⎯ Theory and Modeling Guide
Z
Y
3
24
1
a
b
Principal inplane material axes orientation for the
orthotropic material model for 2D solid elements
Figure 3.21:
5
67
8
The stressstrain matrices are as follows for the various 2D lement types:
Plane stress:
e
1 0
1 0b ba bc
a b
eE
1 0 0
10 0 0
ca cb
a b c
ab
E E E
a aab ac
a b c
bc
c c
abab
eE E E
E E
e
G
σ⎡ ⎤ ⎡ ⎤ν ν⎡ ⎤− −⎢ ⎥ ⎢⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ν ν⎢ ⎥
− − σ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ν ν
− −⎢ ⎥ ⎢ ⎥σ =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥γ σ
⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
Plane strain:
1ae0
1 0
1 00
aab ac
a b c
ba bc
ca cbc c
ab
E E E
e
G
σ⎡ ⎤ ⎡ ⎡ ⎤ν ν ⎤− −⎢ ⎥ ⎢ ⎥⎢ ⎥
be⎢ ⎥ ⎢⎢ ⎥⎥⎢ ⎥ ⎢
ba b cE E E
⎢ ⎥ν ν ⎥− − σ⎢ ⎥ ⎢ ⎥⎢ ⎥
10 0 0
a b c
ab
E E E
⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢− −
⎢ ⎥⎥ν ν⎢ ⎥ ⎢ ⎥= σ⎢ ⎥
ab
⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥γ σ⎢ ⎥
⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
3.2: Linear elastic material models
ADINA R & D, Inc. 255
Axisymmetric: 1 0
1 0
a aab ac
a b c
b ba bc
E E Ee
1 0ca cb
ce
1a b c
0 0 0
ba b c
c
e
E E E
E E E
σ⎡ ⎤ ⎡ ⎤ν ν⎡ ⎤− −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ν ν⎢ ⎥
− − σ⎢ ⎥ ⎢ ⎢ ⎥⎥⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ν ν
− −⎢ ⎥ ⎢ ⎥σ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥
ab ababG
⎢ ⎥ ⎢ ⎥γ σ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
The seven material constants (Ea, Eb, Ec, vab, vac, vbc and Gab)def 1−ine the symmetric compliance matrix C! .
1
0
1 0
1 0
ab ac
a b c
bc
b c
ab
E E E
E E
G
−
1
1symmetric
ν ν⎡ ⎤− −⎢ ⎥⎢ ⎥
ν⎢ ⎥−⎢ ⎥
⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢
C!
cE⎥
⎢ ⎥⎢ ⎥⎣ ⎦
where the subscript " ! " in 1−C! indicates that the material law given in
is the local sy aterial axes. Note that the
determstem of the m
1−inant of C! must not be zero in order to be able to calcula. This imposes the following restrictions on th
te the inverse e constants e stressstrain co
C!. Th material constants must be defined so that the
nstitutive matrix is positivedefinite; i.e.:
Chapter 3: Material models and formulations
256 ADINA Structures ⎯ Theory and Modeling Guide
12
iEν⎛ ⎞
< =, , , ,ji i j a b c⎜ ⎟⎜ ⎟
jE⎝ ⎠
20.5 1 a bE 2 2 0.5cab bc ca ab bc ca
E E
b c aE E Eν ν ν ν ν
⎞− − ≤⎜ ⎟
Based on the input values for
ν⎛
< −⎝ ⎠
ijν , ADINA calculates jiν so as to have a sy x; i.e.,
mmetric constitutive matri
ji ij
E Eν
i j
ν
See also x wing reference:
ref. f Composite Materials,
G 1975.
In AD ,
=
for e ample the follo
Jones, R.M., Mechanics oMc rawHill p. 42,
INA the Poissons ratio ijν is defined t differen ly from Jones, alth g igure 3.22, a uniaxial tensile t under test is loaded in the defined in ADINA by
ou h it is consistent with Jones nomenclature. In Fest is illustrated in which the material
adirection. The Poissons ratio is
bba
ε
a
νε
= −
b
a
Definition of baFigure 3.22:
F F
a
b
In general,
3.2: Linear elastic material models
ADINA R & D, Inc. 257
( , , , )i i j a b cijj
ενε
= − =
In Jones, however, the same Poissons ratio is notated
jJ
iji
εν
ε= − .
Using the correspondence benomenclature for the Poissons ratio,
J
tween the Jones and ADINA
ji ijν ν relati
= and the onship
ji ij
Ei jEν ν
=
it is possible to calculate the value of jiν given a value for J
ijν from
j Jij ij
i
EE
ν ν⎛ ⎞
= ⎜ ⎟⎝ ⎠
When material data is available from sources that follow Jones notation of Poissons ratio, the above conversion allows the equivalent ADINA Poissons ratio to be calculated from the Jones Poissons ratio for input to ADINA.
To obtain the stressstrain matrix C corresponding to the global es, first is calculated and then
where
C!material ax
TC = Q C Q!
Chapter 3: Material models and formulations
258 ADINA Structures ⎯ Theory and Modeling Guide
2 21 1 1 22 22 2 1 2
1 2 1 2 1 2 2 1
00
2 20 0 0
mm m m
m m m m=
+Q
! ! !!
! ! ! !
01
! are the direction cosines of the a, b material
axes to the global axes. Note that in a large displacement analysis it is assumed that the
angle β remains constant throughout the incremental solution. Hence, the use of the total Lagrangian formulation is usually most appropriate.
3D solid elements: In threedimensional analysis we have
1/
a ab b ac c
b bc c
c
ab
ac
bc
E E EE E
EG
GG
−
−ν −ν
and 1 2! 1 2, and ,m m
1
1/ / / 0 0 01/ / 0 0 0
1/ 0 0 01/ 0 0
1/ 0
⎡ ⎤⎢ ⎥−ν⎢ ⎥⎢ ⎥
= ⎢ ⎥ ⎢ ⎥⎢
symmetric
⎥⎢ ⎥⎢ ⎥⎣ ⎦
C!
aterial directions (a,b), for which we have
a a b bc c b
e E E E
e
−ν −ν σ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎥ ⎢ ⎥
and, as in twodimensional analysis, this matrix is inverted and thentransformed to the global coordinate axes. Plate elements: Figure 3.22 shows a typical plate element for which the inplane orthogonal material axes are "a" and "b". The material constants are defined in the principal m
1/ / / 0/ 1/ / 0
a a ab b ac c a
b be E E E⎢ ⎥ ⎢ ⎥ ⎢ ⎥−ν −ν σ⎢ ⎥ ⎢=⎢ / / 1/ 0 0
0 0 0 1/c ca a cb b c c
ab ab ab
E E EG
⎥ ⎢ ⎥ ⎢ ⎥−ν −ν σ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥γ σ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
3.2: Linear elastic material models
ADINA R & D, Inc. 259
Note that the transverse normal strain ec is not calculated for the plate elements.
2
3
1
a
y
b
z,c
Z
X
Y
x
Figure 3.23: Principal inplane material axes orientation for the
orthotropic material model for plate elements
The ab and xy planes
are coplanar. Nodes
1, 2 and 3 belong to
the xy plane.
g to the
whsi sin cos sin
sin cos sin
In order to obtain the stressstrain matrix C correspondinlocal x y z directions of the element, we use
T T=C Q C Q!
cos cos nβ β β β β β⎡ ⎤ere sin sin cos cos cos sin
cos sin cos
T
2 2
⎢ ⎥= β β β β − β β⎢ ⎥⎢ ⎥−2 β β 2 β β β − β⎣ ⎦
β is the angle between the a and x axes, see Fig. 3.22.
Shell elements: Figure 3.23 shows typical shell elements for which the orthogonal material axes are "a", "b" and "c".
The constitutive relation defined in the a,b,c system is
Q
Chapter 3: Material models and formulations
260 ADINA Structures ⎯ Theory and Modeling Guide
0
a
! Table 3.21 summarizes which material constants are necessary, depending on the type of element used in the analysis.
Table 3.21: Required material constants for the element types available in ADINA
Element type (these constants must be nonzero except for
Poisson's ratios)
1/ / / 0 0 0/ 1/ / 0 0 0/ / 1/ 0 0 0
0 0 0 1/ 0 00 0 0 0 1/ 00 0 0 0 0 1/
a a ab b ac c
b ba a b bc c b
c ca a cb b c c
ab ab ab
ac ac ac
bc bc bc
e E E Ee E E Ee E E E
GG
G
−ν −ν σ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−ν −ν σ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−ν −ν σ =
=⎢ ⎥ ⎢ ⎥ ⎢ ⎥γ σ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥γ σ⎢ ⎥ ⎢ ⎥ ⎢ ⎥γ σ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
Required material constants
2D solid 3D solid
Plate Shell
Ea, Eb, Ec, νab, ν
Ea, Eb, Ec, νab, νac, νEa, Eb, νab, Gab
ac, νbc, Gab
bc, Gab, Gac, Gbc
Ea, Eb, Ec, νab, νac, νbc, Gab, Gac, Gbc
3.2: Linear elastic material models
ADINA R & D, Inc. 261
(a) 4node element
a
d
b
t
s
r
,c
1
2
r12
3
a
d
b
t, c
s
r
1
2
r12
(b) 16node element
Figure 3.24: Definition of axes of orthotropy for shell elements
3
Note:
is the input material angle.
(r, s, t) is the local Cartesian system.
The ab and rs planes are coplanar.
d ris the unit projection of onto the rs plane.12
Chapter 3: Material models and formulations
262 ADINA Structures ⎯ Theory and Modeling Guide
3.3 Nonlinear elastic material model (for truss elements)
! For the truss element, an elastic material model is available for which the stressstrain relationship is defined as piecewise linear. Fig. 3.31 illustrates the definition of the stressstrain law.
Stress
6
5
4
3
2
1
e1 e2 e3
e4 e5 e6 Strain
Figure 3.31: Nonlinear elastic material for truss element
Note that the stress is uniquely defined as a function of the strain only; hence for a specific strain te, reached in loading or unloading, a unique stress is obtained from the curve in Fig. 3.31.
! A typical example of the truss nonlinear elastic model is shown in Fig. 3.32. Note that the stressstrain relation shown corresponds to a cablelike behavior in which the truss supports tensile but no compressive loading.
3.3: Nonlinear elastic material model (for truss elements)
ADINA R & D, Inc. 263
Stress
1.0 1.0 Strain
Point 1 Point 2
Point 3
Figure 3.32: Example of nonlinear elastic stressstrainmaterial model for truss
4.32 10 4
A sufficient range (in terms of the strain) must be used in the definition of the stressstrain relation so that the element strain evaluated in the solution lies within that range; i.e., referring to Fig. 3.31, we must have 1 6
te e e≤ ≤ for all t.
! The truss element with this material model is particularly useful in modeling gaps between structures. This modeling feature is illustrated in Fig. 3.33. Note that to use the gap element, it is necessary to know which node of one body will come into contact with which node of the other body.
ref. S.M. Ma and K.J. Bathe, "On Finite Element Analysis of
Pipe Whip Problems," Nuclear Engineering and Design, Vol. 37, pp. 413430, 1976.
ref. K.J. Bathe and S. Gracewski, "On Nonlinear Dynamic
Analysis using Substructuring and Mode Superposition," Computers and Structures, Vol. 13, pp. 699707, 1981.
Note that an alternate gap element can be obtained by the use of
the 2node truss element employing the plasticbilinear material model, see Section 3.4.1.
A more general way of modeling contact between bodies is the use of contact surfaces, see Chapter 4.
! Other modeling features available with the truss element and this material model are shown in Figure 3.34.
Chapter 3: Material models and formulations
264 ADINA Structures ⎯ Theory and Modeling Guide
Pipe (beam)
m
n
Gap
Gap element
_L
Strain
Stress
Figure 3.33: Modeling of gaps (pipe whip problem)
L
< The model shown in Fig. 3.34(a) corresponds to a compressiononly behavior. < The model shown in Fig. 3.34(b) corresponds to a tension cutoff behavior, the snapping of a cable, for example. < The model shown in Fig. 3.34(c) corresponds to a behavior exhibiting both tension and compression cutoffs.
3.3: Nonlinear elastic material model (for truss elements)
ADINA
(a) Compression only
Stress
Strain
e4e3
e2
e1
(b) Tension cutoff
Stress
Strain
e6
e3
e4
e5
e2e1
e7 e8
Compression
cutoff limit
Tension
cutoff limit
(c) Tension and compression cutoffs
Figure 3.34: Various modeling features available with the
nonlinear elastic truss model
Stress
Strain
e1 e2 e3 e4
3.4 Isotherm y at
! l m
har
model for plates Gurson: Gurson plastic model for fracture/damage analysis
ref. KJBSection 6.6.3
al plasticit m erial models
This section describes the following materia odels:
Plasticbilinear, plasticmultilinear: von Mises model with isotropic, kinematic hardening or mixed hardening Mrozbilinear: von Mises model with Mroz hardening Plasticorthotropic: Hill yielding with bilinear proportional
dening Ilyushin: Ilyushin
R & D, Inc. 265
Chapter 3: Material models and formulations
266 ADINA Structures ⎯ Theory and Modeling Guide
! Allelamo z mo
Formalism of Cyclic Plasticity," J. Appl. Mech., Trans. ASME, Vol. 43, pp. 645651, 1976.
Th ref
ref. K.J. Bathe, E. Dvorkin and L.W. Ho, "Our DiscreteKirchhoff and Isoparametric Shell Elements An Assessment," J. Computers and Structures, Vol. 16, pp. 8998, 1983.
! The DruckerPrager material model is described in Section 3.9.2.
3.4.1 Plasticbilinear and plasticmultilinear material models
! These material models are based on
< The von Mises yield condition (see p. 597, ref. KJB) < An associated flow rule using the von Mises yield function < An isotropic or kinematic, bilinear or multilinear, hardening rule
Figs. 3.41 to 3.43 summarize some important features of these material models.
! These models can be used with the truss, 2D solibeam (plasticbilinear only), isobeam, shell and pipe elements.
elasticplasticity models use the flow theory to describe the sticplastic response; the basic formulations for the von Mises dels are summarized on pp. 596604, ref. KJB, and for the Mrodel in:
ref. Y.F. Dafalias and E.P. Popov, "Plastic Internal Variables
e formulation for the Ilyushin model is given in the followingerence:
d, 3D solid,
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 267
0y
Stress
Strain
Figure 3.41: von Mises model
Bilinear stressstrain curve
Multilinear stressstrain curve
! These models can be used with the small disstrain
placement/small
dis be used with
2D solid, 3 shell elements must be 3node, 4node, 9node or 16node single layer shell elements des
, large displacement/small strain and large placement/large strain formulations. The large
displacement/large strain formulation can only the D solid and shell elements (the
cribed entirely by midsurface nodes).
3
s3
ss
a) Principal stress space
23
0y
23
0y
Elastic region
b) Deviatoric stress space
Figure 3.42: von Mises yield surface
(1,1,1)
Chapter 3: Material models and formulations
268 ADINA Structures ⎯ Theory and Modeling Guide
Figure 3.43: Isotropic and kinematic hardening
b) Bilinear kinematic hardening
E
E
E
E
E
T
T
0y
a) Bilinear isotropic hardening
Stress
Straint
y
ty
E
E
E
E
E
T
T
0y
Stress
Strain
20y
0y 0
yt
y
ty
0y
Strain Strain
Stress Stress
c) Multilinear isotropic hardening
2
d) Multilinear kinematic hardening
When used with the small displacement/small strain
tion
truss element, multilinear plasticity, you can only enter
up
formulation, a materiallynonlinearonly formulation is employed, when used with the large displacement/small strain formulation, either a TL or a UL formulation is employed, and when used with the large displacement/large strain formulation, a ULH formula(solid elements) or either a ULH or ULJ formulation (shell element) is employed.
! For the to 7 stressstrain points in the stressstrain curve. For other
elements, multilinear plasticity, there is no restriction on the number of stressstrain points in the stressstrain curve.
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 269
t
ing
! Plane strain, axisymmetric or 3D solid elements that referencethese material models should also employ the mixed displacemenpressure (u/p) element formulation.
! In the von Mises model with isotropic hardening, the followyield surface equation is used:
( ) 2 02 3
t t tσ1 1ty yf = ⋅ − =s s
where ts is the deviatoric stress tensor and 0 2
yσ the updated yield
stress at time t. In the von Mises model with kinematic hardening, the following
yield surface equation is used:
( ) ( ) 0 21 1 02 3
t t t t ty yf σ= − ⋅ − − =s α s α
where tα
is the shift of the center of the yield surface (back stress tensor) and 0 2
yσ is the virgin, or initial, yield stress. the von Mises model with mixed hardening, the following
yieInld surface equation is used:
( ) ( ) 21 1 02 3
t t t t t ty yf σ= − ⋅ − − =s α s α
where 0t p
y y pMEσ σ= + e The back stress tα is evolved by ( )1pd C M d= −α e
p
Cp is Pragers hardening parameter, related to the plastic mo ldu us Ep by
Chapter 3: Material models and formulations
270 ADINA Structures ⎯ Theory and Modeling Guide
2C E= 3p p
and M is the factor used in general mixed hardening (0 < M < 1)
which can be a variable, expressed as ( ) ( )0 exp eM M M M∞ ∞ −= + − pη
e v n ises model with mixed hardening
tures, Vol. 82, No. 6, pp. 535  539, 2004.
o MThe formulation for th
is g iven in the following reference: ref K.J. Bathe and F.J. Montáns, On Modeling Mixed
Hardening in Computational Plasticity, Computers and Struc
Note that the convergence might not be good when the mixed
hardening parameter 0η ≠ . Therefore, 0η = is preferred. The yield stress i a function of the effective plastic strain,
which defines the h ens
str
ard ing of the material. The effective plastic ain is defined as
0
2tPt p p
3e d d= ⋅∫ e e
in which is the tensor of differential plastic strain increments.
In finite element analysis, we approximate
pdePt e as the sum of all of
the plastic strain increments up to the current solution time:
Pt p
all solution steps
e = ∆∑ e
23
P p pe d d∆ = ⋅e e and p∆e iswhere the tensor of plastic strain
increments in a solution step. Because of the summation over the
solution steps, we refer to the calculated value of Pt e as the
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 271
ted effective plastic strain.
t but the material characteristics are considered to be temperature independent.
.
ear stressstrain curve, a maximum allowable effective lastic strain
accumula
! If a thermal load is applied to the structure, the thermal strains are taken into accoun
Modeling of rupture: Rupture conditions can also be modeledFor the bilin
Pp A
ultilinear stressstrain curve, the rupture plastic strain ce can be specified for the rupture condition. For the
orresponds t e r the stress
str on point,
the (see Section 11.5).
code the p
RUP4 ents, CURUP5 for isobeam elements, CURUP7 for
hell e ents. These subroutines are in file ovlusr.f. ADINA provides the calculated latest stresses,
tal strains, thermal strains, plastic strains, creep strains, yield str and th
Ratedependent plasticity for the truss element: The elasticpla c strainrate ffects in dynamic analysis. In this case, the yield stress including
mto he ffective plastic strain at the last point input fo
ain curve. When rupture is reached at a given element integrati corresponding element is removed from the model
There is also the option of usersupplied rupture. Youru ture condition into one of the CURUP subroutines: CURUP2 for 2D solid elements, CURUP3 for 3D solid elements, CUfor beam elems lements and CURUP8 for pipe elem
toess and accumulated effective plastic strain to these subroutines
ese subroutines return the current rupture state.
sti material models in the truss element can includeestrainrate effects ( )D
yσ is interpolated based on the current plastic
e, from the strain ratDyσ⎛ ⎞
yσ⎜ ⎟⎜ ⎟⎝ ⎠
versus plastic strain rate relation that
ou define ( is the yield styσy re t c
Ratedependent plasticity for truss, 2D, 3D and shell e rate dent mo ADINA sed to sim te the y tress with an increase in strain rate.
ependent model only applies to the isotropic plasticity sotrop rdening ( ar or m ear).
ss withou strain rate effe ts).
elements: Ththe increase in
T ted
depenield s
del in is u ula
he ramodels with i ic ha biline ultilin
Chapter 3: Material models and formulations
272 ADINA Structures ⎯ Theory and Modeling Guide
ependent model is implemented for truss, 2D solid, 3D solid and shell (either singlelayered or multilayered) elements.
ive yield stress including strain rate effects is
The rated
The effect
0 1 ln 1y y b εσ σε0
P⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
##
where 0yσ is the static yield stress, 0ε# is the transition strain ra
and b is the strain rate hardening parameter. te
0ε#The material parameter must be directly specified. The aterial parameter b can either be directly specified or can be
he .
For more information, see the following reference:
Drysdale and A.R. Zak, AMechanics of Materials l Theories C A Theory for Rate Dependent
, Computers and Structures, Vol. 20, pp. 259264, 1985.
When used in
o h uss elements; see Section 3.3). In this case the gap elements can n
hen the
3.4.2 Mro material model
This material model is based on < The von Mises yield condition
An associated flow rule il t
mcalculated using an additional userinput stressstrain curve for tstressstrain behavior at a given strain rate
ref. W.H.and StructuraPlasticity@
Modeling of gaps using the truss element: conjunction with the 2node truss element the elasticplastic material models can be used to model gaps in the elements (the nonlinear elastic m del can also be used to model gaps wit thetro ly resist compressive loads, i.e., gap elements have no tensile stiffness. The gap width input for each element is used to determine a strain egap (compressive strain is defined as negative). The compressive stiffness of a gap element is zero if the strain in the element is greater than or equal to egap and is nonzero wstrain is less than egap.
zbilinear
!
< < B inear Mroz hardening, including a rule for displacemen
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 273
displacement of the bounding surface
The Mrozbilinear model can be used with the D solid elements.
odel can be used with the small
isplacement/ all strain, ge disprge displacement/large strain formulations.
ll
lation is employed and when used with the large ement/large strain formulation, the ULH formulation ed.
rain, axisymmetric or 3D solid elements that reference
is material model should also employ the mixed displacemente (u/p) element formulation.
ilinear a bounding line.
A representation of the stressstrain curve is shown in Fig. s at
of the yield surface and a rule for
! 2D solid and 3
! The Mrozbilinear md sm lar lacement/small strain and la
When used with the sma displacement/small strain formulation, a materiallynonlinearonly formulation is employed, when used with the large displacement/small strain formulation, the TL formudisplac is
ployem
!
th Plane st
pressur ! The material behavior is characterized by a uniaxial bstressstrain curve plus
3.44. Under uniaxial loading the plastic deformation startstress 0
yσ σ= and continues along the line AB until the bounding
stress
yBσ is reached. With further stress increase, plastic eformation continues and the stressstrain relation follows the d
bounding line (segment BC in the figure). The material response corresponding to deformation ABC is the same as the plasticbilinear model with kinematic or isotropic hardening (Section 3.4.1). If the tangent modulus ETB of the bounding surface is equal
zero, the deformation on the segment BC corresponds to a perfectly plastic von Mises material.
The uniaxial plastic deformation is repstress space in Fig. 3.45(a). The stress path ANBNCN corresponds to
in the stress space with the common stress point moving along the line BNCN.
to
resented in the deviatoric
the stressstrain path ABC. After the yield surface touches the bounding surface at point BN, the two surfaces translate
Chapter 3: Material models and formulations
274 ADINA Structures ⎯ Theory and Modeling Guide
! When the load is reversed, plastic deformation starts at point D and follows the segments DF and FG, see Fig. 3.44.
In the reverse loading, the material response differs from the response corresponding to the plasticbilinear model with isotropic or kinematic hardening. For example, in the case of kinematic hardening, the stressstrain relation in the reverse loading follows the line D F with the slope ETB while in the Mroz hardening, the reverse plastic deformation starts with the slope ET and continues with that slope until the bounding line at point F is reached. Also, in all subsequent reverse loadings, yielding starts with the tangent modulus ET (as at point H in Fig. 3.44). The stress path FNGN corresponding to segment FG, and the yield and bounding surfaces are shown in Fig. 3.45(b).
Figure 3.44: Mroz model; bilinear stress strain curve
Stress
Strain
Bounding line
Bounding line
ETB
ETB
0y
20y
0y
F
A
BC
E
E
E
ET
ET
F
G
HO
DET
2
yB
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 275
Yield surface
at stress state G
Bounding surface
at stress state G
(a) Loading path OABC (b) Reverse loading CDFG
Figure 3.45: Mroz model behavior in deviatoric stress space
Yield surface
at stress state C
Bounding surface
at stress state C
yB
0y A
G
B
F
C C
D
23
23
s3 s3
s1s1s2
s2
! nal assum tiodescribed
< The y
In the case of a general nonradial loading, some additiop ns are used. The model stressstrain behavior is
as follows:
ield surface
( ) ( ) 0 21 1 02 3y yf σt t t t t= − ⋅ − − =s α s α
and the bounding surface
( ) ( ) 0 21 1 02 3
t t t t tyB B B yBf σ= − ⋅ − − =s β s β
t
are given in Fig. 3.46 where is the back stress and defines the position of the bounding surface.
α tβ
< The plastic flow rule is
( )t P t t t= Λ −e s α##
Chapter 3: Material models and formulations
276 ADINA Structures ⎯ Theory and Modeling Guide
where tΛ# is the plastic multiplier (positive scalar).
tyBf = 0
tyf = 0
0y
yB
2
2
3
3
t
t
Rt Pe
tn
tn
tm
tBs
ts
t
T
0
Figure 3.46: Mroz model in case of general loading conditions
s3
s1
s2
the change of the back stress is given by
< The constitutive relation governingtα
23
t Pt t t
pd eC
dt=α m#
with
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 277
2 t
pt E3p t tC =
⋅m n
t
t E Tp
E E=
Here
TE E−
t Pd edt
represents the rate of the effective
is the plastic modulus, E is the elastic modulus and is re unit
normals shown in Fig. 3.46. The above definition of the
plastic strain,
tpE t
TEthe hardening modulus. The variables tm and tn a
modulus tpC is based on the assumption that the work p
time of the back stress onding to the plastic straining general loading conditions is equal to the same quaradial loading conditions. The direction of tα# is such that the
er unit corresp in
ntity in the
ontact
e and the l
yield and bounding surfaces have the same normal at the cpoint.
< The bounding surface moves in the stress space only when the yield surface is in contact with the bounding surfacloading continues in the direction of the then common norma(see Figs. 3.45(b) and 3.46, i.e. t t=β α# # ).
< Note that when the
tangent modulus of the bounding surface ETB is equal to zero, the bounding surface does not move in the stress space. In this case, the effective stress tσ remains smaller than yBσ for any loading conditions.
! If a thermal loading is applied to the structure, thermal strains are taken into account, but the material characteristconsidered to be temperature independent.
ue
ics are
PAe ! Rupture conditions can be modeled. When a positive val
is input for the maximum allowable effective plastic strain, the rogram compares the accum t Pe p ulated effective plastic strain
with PAe for each material point used in the analysis and over the
Chapter 3: Material models and formulations
278 ADINA Structures ⎯ Theory and Modeling Guide
whole history of deformation. It is considered that the rupture at a material point occurs when t Pe is greater than P
Ae . When ruptuis r ached at
re a given element integration point, the corresponding
3.4.3 Plast
< The Hill yield condition < An associated flow rule < A bilinear proportional hardening rule
! This model can be used with the 2D solid, 3D solid and shell elements.
e ith the
e
the TL formulation is employed. When used with the large displacement/large strain formulation, the ULJ formulation is
!
)2bb cc cc aa aa bb
ab bc ac
F G H
L M N
σ σ σ σ σ σ
σ σ σ
− + − + −
+ + + −
eelement is removed from the model. (see Section 11.5).
The usersupplied rupture option is also available, see Section 3.4.1.
icorthotropic material model
! The plasticorthotropic model is based on:
! The plasticorthotropic model can be used with the small displacement/small strain, large displacement/small strain and large displacement/large strain formulations. Small strains are always assumed with the 2D solid and 3D solid elements. Thlarge displacement/large strain formulation can be used wsh ll elements (which must be 3node, 4node, 9node or 16node single layer elements described entirely by midsurface nodes).
When used with the small displacement/small strain formulation, a materiallynonlinearonly formulation is employed and when used with the large displacement/small strain formulation,
employed.
The Hill yield condition is given by:
0=
( ) ( ) (2 2
2 2 2 2 2 2 1
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 279
b, c) are the material principal axes, and F, G H, L, M, N al cons nts. The constants a given by
where (a, , are materi ta se re
2 2 2
1 1 1 12
FY Z X
⎛ ⎞= + −⎝ ⎠
⎜ ⎟
2 2 2
1 1 1 1G2 Z X Y⎝ ⎠
⎛ ⎞= + −⎜ ⎟
2 2 2
1 1 1 12
HX Y Z
⎛ ⎞= + −⎜ ⎟⎝ ⎠
2 2 2
1;2 2 2
NY Y Y
=
ld stres , b,
in the planes (a,b), (b,c), and (a,c) (see Figure 3.47).
1 1;L M= =ab bc ac
where X, Y, Z, are the yie ses in the material directions ac, and Yab, Ybc, Yac are the yield stresses for pure shear
Yield surface
at time t
Figure 3.47: Orthotropic yield surface
Y
A B
0
X
cc
bb
C
aa
Z
Initial yield surface
a, b, c material axes initial yield stressesX, Y, Z
Chapter 3: Material models and formulations
280 ADINA Structures ⎯ Theory and Modeling Guide
here are three ways to specify the material constants F, G, H, L, M, N:
< Enter the yield stresses, then ADINA computes the material constants using the equations given above.
T
< Enter the Lankford coefficients r0, r45, r90; then
( )( ) ( )
( )
( )
( )
0 90 450 2 12
r r rrF L90 0 90 0
0
0
1 11 1.5
1
1.5
r r r r
G Mr
rH N0 1r
+ ⋅ += =
g direction and r90 is the Lankford coefficient for the direction 90° to the rolling direction (the transverse direction).
< Directly specify F, G, H, L, M, N .
! Please note that F, G, H, L, M, N must all be positive. In addition, for 2D axisymmetric elements, the coefficiesatisfy the conditions 2
⋅ + ⋅ +
= =+
= =
+
in which r0 is the Lankford coefficient for the (rolling) direction, r45 is the Lankford coefficient for the direction 45° to the rollin
nts must , ,F G M N L F H= = − = .
! Hardening can be specified in three wa
< Enter the universal plasti
ys:
c modulus . In this case the )
u and
puE
plastic moduli for directions a, b, c and planes (a,b), (a,c), (b,c
are given by p p p p= = =a b cE E E E3
pp p p u
ab ac bcEE E E= = = .
nt
irectio equal to
< E er the independent moduli , ,T T Ta b cE E E for the a, b, c
ns and for the (a,b), (a,c), (b,c) planes. In this case, puE d
is
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 281
( )2 2 2 2 2 21 12 3
p p p p p pa b c ab ac bcE E E E E E⎛ ⎞+ + + + +⎜ ⎟
⎝ ⎠
where
T
p i ii T
i i
E EEE E
=−
i = a,b,c
andT
ij ijpij T
ij ijE E−
< Enter the parameters of a stressstrain curve which is used to calculate the universal plastic modulus. The stressstrain curve is of the form
E EE = ij = ab,ac,bc
( )n
0Cσ ε ε= ⋅ +
in which σ is the yield stress, ε is the plastic strain, 0ε is the al plastic strain, n is the strain hardening coefficient and C is
a constant.
used, ma rial axis a must be aligned with the Y axis and material axis c
into account but the material characteristics are
initi
! When the 2D solid element (plane stress or plane strain) or shell element is used, material axes a and b must lie in the plane of the element. When the 2D solid element (axisymmetric) is
temust be aligned with the Z axis.
! If a thermal load is applied to the structure, the thermal strains are takenconsidered to be temperature independent.
! Rupture conditions can also be modeled: a maximum allowable effective plastic strain P
Ae can be specified for the rupture condition. When rupture is reached at a given element integration point, the corresponding element is removed from the model (see Section 11.5).
Chapter 3: Material models and formulations
282 ADINA Structures ⎯ Theory and Modeling Guide
also be used (see Sect3.4
is used to calculate
stresses and plastic strains when plasticity occurs. ! Note that orthotropic proportional hardening reduces to isotropic hardening for the appropriate values of the elastic and tangent moduli.
3.4.4 Ilyushin material model
yield condition (see Fig. 3.48)
le he Ilyushin function < Bilinear isotropic hardening
he Ilyushin model is only available for the plate element.
! large displacement rmulatio . In all ca s the stra
When used with the small displacement formulation, a ed
formulation, the UL formulation is m
resultants, the membrane forces, and bending moments.
The Ilyushin yield surface equation is given by
The usersupplied rupture option can ion .1).
! The effectivestressfunction algorithm
! The Ilyushin model is based on
< The Ilyushin < An associated flow ru using t
T
The Ilyushin model can be used with the small displacement or fo ns se ins are
assumed to be small.
materiallynonlinearonly formulation is employed, and when uswith the large displacemente ployed.
! The Ilyushin model is formulated in terms of the stress
!
0t t t ty M B MB yf Q Q Q γ σ 2= + + − =
where γ is an input parameter, t
yσ is the yield stress, and
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 283
Figure 3.48: Ilyushin model for use with the plate element
(a) Element section forces and moments
_x3 _
x1
_x2
t12M
t2M
t1M
t1N
t2N
t12N
(b) Uniaxial yield condition
0
1.0
1.0
= 0
= 1
3
__
t
ty
_N
( h)
t
ty
2
M
h
4( )
Node 1 Node 2
Node 3
( ) ( ) ( ) 2 2
1 2 1t t t tN N N N+ − ⋅
2
2 122
1 3t tMQ N
h= +
( ) ( ) ( ) 2 2 2
14
16 3t t t t t tBQ M M M M M
h= + − ⋅ + 2 1 2 12
Chapter 3: Material models and formulations
284 ADINA Structures ⎯ Theory and Modeling Guide
( ) ( )1 1 2 2 12 12 1 2 2 13
12 1 13 6
t t t t t t t t t tMBQ N M N M N M N M N
h⎧ ⎫= ⋅ + ⋅ + ⋅ − ⋅ + ⋅⎨ ⎬⎩ ⎭
t M
! Since the Ilyushin model is formulated using stress resultants, no integration through the element thickness is performed in the calculation of element matrices.
Thus, considering the yielding of a plate element at an integration point, the element is elastic until the entire section becomes plastic. This delay in the change of state from elastic to plastic can be observed when comparing the response predicted by the plate element and the shell element; see Fig. 6 of the following reference:
! In an elastoplastic plate/shell analysis uthe membrane forces and bending moments are, in general, coupled. Hence, even though the element constant strain
t stress
3.4.5 Gurs
! The Gurson model is intended for use in fraanalysis. The purpose of the Gurson plastic mductile crack growth by void growth and coalescence. This is a
.
! The Gurson model can be used with the small displacement/small strain, large displacement/small strain and large displacement/large strain formulations.
When used with the small displacement/small strain ed, , the
ref. K.J. Bathe, E. Dvorkin and L.W. Ho, "Our DiscreteKirchhoff and Isoparametric Shell Elements An Assessment," J. Computers and Structures, Vol. 16, pp. 8998, 1983.
sing the Ilyushin model,
conditions still hold in the membrane representation, constanconditions may not be predicted.
on material model
cture/damage odel is to predict
micromechanical model based approach. ! The Gurson model is available for the 2D solid and 3D solid elements
formulation, a materiallynonlinearonly formulation is employwhen used with the large displacement/small strain formulation
3.4: Isothermal plasticity material models
ADINA R & D, Inc. 285
large ulation, the ULH formulation is
TL formulation is employed and when used with the displacement/large strain formemployed.
! In the Gurson model, the yield function has the form
( )2
2* *21 3
0 0
32 cosh 1 02
q q pq f q fφσ σ
⎛ ⎞ ⎛ ⎞= + − − + =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
in which φ is the yield function, q is the von Mises stress, 0σ is
flow stress, is the pressure (positive in the equivalent tensile pcompression), *f is based on the void volume fraction and q 1,q2, q3 are the Tvergaard constants. Note that this reduces to the von Mises yield function when *f equals zero. *f is defined as
( )
* *
for
for
c
u cc c
F c
f f ff f ff f f f f
f f
≤⎪= −⎨ + − >⎪ −⎩
in which f is the void volume fraction, c
⎧
f is the critical void
volume fraction, Ff is the void volume fraction at final failure and*
11/u
f q . =
! Initially, the void volume fraction is 0f , which must be input. The increase of void volume fraction is controlled by
( )1 p p
kkdf f d Adε ε= − +
in which pkkd
ε is the sum of the normal plastic strain components,
pε is the equivalent pensity. A is defined
lastic strain and A is the void nucleation int as
Chapter 3: Material models and formulations
286 ADINA Structures ⎯ Theory and Modeling Guide
2
1exp22
pN N
NN
fASS
ε ε⎡ ⎤⎛ ⎞−⎢ ⎥= − ⎜ ⎟
π ⎢ ⎥⎝ ⎠⎣ ⎦
in which Nf is the volume fraction of void nucleating particles,
Nε is the mean void nucleation burst strain and is thcorresponding standard deviation.
NS e
! The uniaxial stressstrain curve is
0 0 3N
p
y y y⎝ ⎠
where y
Gσ σ εσ σ σ
⎛ ⎞= +⎜ ⎟⎜ ⎟
σ is the tensile yield stress of the matrix, G is the shear
modulus and N is a material constant (typically
! For more fo
on the order of 0.1).
in rmation, see the following reference:
ref. N. Aravas, AOn the numerical integration of a class of pressuredependent plasticity models@, Int. J. Num. Meth. in Engng., Vol. 24, 13951416 (1987).
3.5 Thermoelastic material models
! ls are latem For thermal stress analysis, the material moduli can e temperaturedependent.
odel is available for the truss, 2D solid,
D solid, isobeam
ic model is available for the 2D solid, ents.
Both models can be used with the small displacement and
When used with the small displacement formulation, a
The thermoisotropic and thermoorthotropic material mode used to calcu te stress distributions due to imposed peratures.
constant orb
! The thermoisotropic m3 , shell and pipe elements. ! The thermoorthotrop3D solid and shell elem
!
large displacement formulations. In all cases the strains are assumed to be small.
3.5: Thermoelastic material models
ADINA R & D, Inc. 287
materiallynonlinearonly formulation is employed, and when used
rmulation is employed. ! In the data input formust be defined for all tim a file created by ADINAT, reading these temperatures from a ma ping file or defining them by time functions). If the model inc
! For these models, the elastic moduli, thPoisson's ratios and the coefficients of thermal expansion are input as piecewise linear functions of the temperature, as illustrated in
properties
with the large displacement formulation, either the TL or UL fo
the analysis, the nodal point temperatures e steps (reading these temperatures from
pludes axisymmetric shell or shell elements, the nodal point
temperature gradients must also be defined for all time steps. See Section 5.6 for more information on prescribing temperatures andtemperature gradients.
e shear moduli, the
Fig. 3.51. Linear interpolation is used to calculate the material between input points.
E
tEt
t
t t
t
4 4
4
1 1
1
2 2
2
3 3
3
Temperature Temperature
Temperature
Figure 3.51: Variation of material properties for thermoelastic
model ! re calculated
The thermal strains at a given element integration point a at time t using
Chapter 3: Material models and formulations
288 ADINA Structures ⎯ Theory and Modeling Guide
( )0t TH t tij ije α θ θ δ= − (3.51)
where
( ) ( ( )( ) ( )( ))0 0t t
(3.52)
0 REF REFtθ θ θ α θ θ θ
θ θ− − −
−
and where the temperatures t
1tα α=
θ and 0θ are obtained by interpolation from the nodal point temperatures (and temperature gradients), tα is obtained as a function of tθ or 0θ from the inpdefinition of the material parameters, 0
ut θ is the initial temperatu
corresponding to zre
ero thermal strains at the integration point, REFθ is the material reference temperature, and ijδ is the Kronecker de
( ij
lta
δ = 1 for i = j and ijδ = 0 for for i ≠ j).
! Equations (3.51) and (3.52) are derived as follows: Suppose that, from experimental data, the dependence of the length of a bar as a function of temperature is obtained, as shown in Figure (3.52).
Temperature, REF
Len
gth
,L Secant to curve
L
LREF
Figure 3.52: Length of bar vs. temperature The thermal strain with respect to the reference length may be calculated as
REFth
REF
L LL
ε −=
3.5: Thermoelastic material models
ADINA R & D, Inc. 289
Th dgiven tem
en we efine the mean coefficient of thermal expansion for a perature as follows:
( ) ( )thε θα θ =
REFθ θ− With this definition, the secant slope in Figure (3.52) is
( )REFL α θ . Now, in ADINA, we assume that the thermal strains are initially
zero. To do this, we subtract off the thermal strain corresponding to 0θ to obtain
( )( ) ( )( )0 0t t t
th REF REFε α θ θ θ α θ θ θ= − − −
and
this may be rewritten as
( )0t t tthε α θ θ= −
in which
( )( ) ( )( ) ( )0 00 REF REFt θ θ α θ θ θ− −
1t t tα α θ= −θ θ−
Notice that if the mean coefficient of thermal expansion is constant, then REFθ no longer enters into the definition of tα . In general, when the mean coefficient of thermal expansion is not constant, REFθ must be chosen based on knowledge of the experiment used to determine ( )α θ (for the same material curve,
dif tferen oices of REF ch θ yield different values of ( )α θ ). terials that accept mean coefficients of thermal
perature as an input .
The maexpansion all accept the reference temarameter
! For the
p
evaluation of the temperatures tθ and 0θ at the point considered, the isoparametric inintegration terpolation
functions hi are used; e.g., in twodimensional analysis we have
Chapter 3: Material models and formulations
290 ADINA Structures ⎯ Theory and Modeling Guide
1
q
i ii
hθ θ=
= ∑
where iθ is the temperature at element nodal point i (see Fig. 3.53). Note that when higherorder elements are used the
mperatures at the integration point can be significantly different
tefrom the values at the nodal point (for example negative although all nodal point temperatures are greater than or equal to zero).
Figure 3.53: Interpolation of temperature at integration points
Temperature t1
Integration points
s
r
1
ic shell and shell elements, the temperature at an tegration point is established using the temperatures at the
face nodes and the temperature g
midsurface node is defined in the direction of the mid director v tion 2.7.3.
ly the tem at m nodes are oints.
3.6 Therm stoplasticity and creep m
ep, sticcre
multilinearplasticcreepvariable and irradiation reep material models.
! For axisymmetrinmidsur radients if they are specified through input data. Note that the temperature gradient at a shell surface ector at that node, see Sec
For transition shell elements, no temperature gradients are associated with the top/bottom nodes and on peraturesthe top/botto used to evaluate the temperatures at theintegration p
oela aterial models
! This section describes the thermoplastic, creep, plasticcremultilinearpla ep, creepvariable, plasticcreepariable,v
c
3.6: Thermoelastoplasticity and creep material models
ADIN
The thermoelastoplasticity and creep models include the of
< Timeindependent plastic strains,
< Timedependent creep strains,
!
effects
< Thermal strains, rse
t TH
t prse
t C
rse
and the constitutive relation used is
( )t C t THt t E t t Pij ijrs rs rs rs rsC e e e eσ = − − −
ref. KJBSection 6.6.3
A R & D, Inc. 291
herew tijσ is the stress tensor at time t and is the elasticity
nsor at the temperature corresponding to time t. The tensor can be expressed in terms of Young's modulus tE and Poisson's
tio tv which are both temperaturedependent.
The irradiation creep model includes the effects of
< Thermal strains, eφ
< Irradiation strains, < Neutron fluencedependent creep strains,
and the constitutive relation used is
t EijrsC
te t EijrsC
ra
!
THrs
Wrseφ
Crseφ
( )E C Wij ijrs rs rs rs rsC e e e eφ φ φ φ φ φσ = − − − TH
where ijφσ is the stress tensor corresponding to the neutron fluence
φ at time t and is the elasticity tensor corresponding to the
neutron fluence
EijrsCφ
φ at time t. The tensor can be expressed in
terms of Young' odulus which also corresponds to the eutron fluence
EijrsCφ
s m Eφ
n φ at time t.
Chapter 3: Material models and formulations
292 ADINA Structures ⎯ Theory and Modeling Guide
aterial models can be used with the trussid, isobeam, shell and pipe elements, w
odel whicn can only be used with 2D solid nd 3D solid elements.
! These models can be used with the small displacement/small strain, large displacement/small strain and large displacement/ large strain formulations. The large displacement/large strain
rmulation can only be used with the 2D solid and 3D solid elements.
When used with the small displacement/small strain formulation, a materiallynonlinearonly formulation is employed,
hen used with the large displacement/small strain formulation, either a TL or a UL formulation is employed and when used with
e large displacement/large strain formulation, the ULH
nt
ese material models should also employ the mixed displacementpressure (u/p) element formulation. ! Note that the constitutive relations for the thecreep strains are independent of each other; hence the onlyinteraction between the strains comes from the fact that all strains ffect the stresses. Figure 3.61 summarizes the constitutive
n m ch
r . Snyder and K.J. Bathe, "A Solution Procedure for
ThermoElasticPlastic and Creep Problems," J. Nuclear Eng. and Design, Vol. 64, pp. 4980, 1981.
ref. M. Kojiƒ and K.J. Bathe, "The EffectiveStressFunction
Algorithm for ThermoElastoPlasticity and Creep," Int. J. Numer. Meth. Engng., Vol. 24, No. 8, pp. 15091532, 1987.
! These m , 2D solid, 3D sol ith the exception of the creepirradiaion ma
fo
w
thformulation is employed.
! Plane strain, axisymmetric or 3D solid eleme s that referenceth
rmal, plastic and
adescription for a onedimensional stress situation and a bilinear stressstrain curve.
Since there is no direct coupling in the evaluation of the
differe t strain co ponents, we can discuss the calculation of ea strain component independently.
ef. M.D
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 293
f t material based on the following threeinvariant
! Rupture o hecriterion can be specified:
( ) ( ) ( ) ( )3 1 21t t tuJ J Jα β α β σσ + σ + − − σ ≤
in which t σ is the stress tensor, are the first, second nd third stress tensor invariants,
1J , 2J , 3J α , βa are material properties
(input data) and uσ is tupture criteria can be used in combination with other
pture criteria described later in this section. en rupture is reached at a given element i
ent is removed from the m
he ultimate effective stress (input data). This r
ruWh ntegration point,
the corresponding elem odel (see Section 11.5).
Chapter 3: Material models and formulations
294 ADINA Structures ⎯ Theory and Modeling Guide
(b) Strains considered in the model
Figure 3.61: Thermoelastoplasticity and creep constitutive
description in onedimensional analysis
tP
Area A
(a) Model problem of truss element under constant load
Stress
t
tT
tE ( )
t tE( )
tyv
t Pe (time independent)
t Ee (time independent)
Strain
Creep
strain
t Time
te (time dependent)C
Temperature t
( )t
temperature
! The nodal point temperatures (and possiblygradients) are input to ADINA as described in Section 5.6.
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 295
3.6.1 Eval
l on being
. ! The thermal strains of the irradiation creep model are dependent on temperature and neutron fluence. The value of thermal strain is etermined using the Shifting rule described in Section 3.6.5.
3.6.2 Evaluation of plastic strains
Plasticity effects are included in the thermoplastic, plastic
uultilinearplasticcreepvariable material models. For the material odels that have the word "multilinear" in their names, the stress
strain curve is
The plastic strains are calculated using the von Mises plasticity model (see Section 3.4.1) with aterial parameters (Young' us, strain hardenin lus, Poisson's ratio, yield stress, etc...). Figure 3.62 shows the assumptions used for the dependence of the material parameters on temp re (in
a sss law). (Note the ic
t
s in isotropic hardening
uation of thermal strains
! The thermal strains are calculated from the prescribed nodapoint temperatures with the coefficient of thermal expansitemperature dependent. The procedure used is the same as for the thermoisotropic material model, see Section 3.5
d
!
creep, m ltilinearplasticcreep, plasticcreepvariable and mmstrain curve is multilinear, otherwise the stressilinear. b
!
temperaturedependent ms modul g modu
eratuthe case of a biline r stre train that plastmodulus
EP = E · ET /(E ET)
is interpolated as a function of temperature, not the tangenodulus ET .) Hence, the yield function im
2v
1 12 3
t t t ty yf σ= ⋅ −s s
nd in kinematic hardening
a
( ) ( ) 2v
1 12 3
t t t t t ty yf σ= − ⋅ − −s α s α
Chapter 3: Material models and formulations
296 ADINA Structures ⎯ Theory and Modeling Guide
where is the deviatoric stress tensor, ts vt
yσ is the virgin yield
tress corresponding to temperature tθ (see Fig. 3.62) and is the shift of the stress tensor due to kinematic hardening. In the case of
ultilinear stressstrain curves, the yield curves are interpolated during the solution as shown in Fig. 3.63.
The expressions for plastic strain increments resulting from the
are
s
m
!
flow theory P tij ijde d sλ= for isotropic hardening and
( )P t tij ij ijde d sλ α= − for kinematic hardening, in which dλ is the
multiplier (positive scalar) which can btfy = 0. In the case of kinem
express the change of the yield surface position in the form
odulus
plastic e determined from the yield condition atic hardening, we
t Pd Cdeα =
where
ij ij
tC is the m
23
t Tt t
E ECE
=−
t t
TE
In the case of multilinear yield curves, tET represents the tangent s of the segment on the yield curve c
ccumulated effective plastic strain
modulu orresponding to the t Pea .
odel in kinematic ardening conditions. Namely, nonphysical effects can result when
iation in ET as a function of temperatuis recommended to use this model only when the variation in ET as function of temperature is small.
rial
effective plastic strain
Care should be exercised in the use of this m!
hthe var re is large. Hence, it
a ! Rupture due to plasticity can be modeled using this matemodel. When using a bilinear stressstrain curve, a maximumallowable P
A
m ltilinear stressstrain curve, the maximum allowable effective e can be input. When using a
ulastic strain at a given temperature corresponds to the last point on
the yield curve at that temperature (see Fig. 3.63).
p
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 297
1
1
2
2
.
.
.
.
.
.
.
.
.
.
.
.
i
i
ET1
ET2
ETi
E1
E2
Ei
Strain
Stress
Stress
yv1
yv1
yv2
yv2
yvi
yvi
E E
E  E
1 T1
1 T1
E E
E  E
2 T2
2 T2
E E
E  E
i Ti
i Ti
(a) Uniaxial relation of stress versus total strain
(b) Uniaxial relation of stress versus plastic strain
Figure 3.62: Variation of material properties with temperature
for the thermoelastoplasticity and creep models;
bilinear stressstrain curves
Plastic strain
Chapter 3: Material models and formulations
298 ADINA Structures ⎯ Theory and Modeling Guide
a) Stressstrain curves input data
b) Yield curves
Stress
Stress
1
1
2
2
3
3
4
4
ei1
e
ei2
e
ei3
e
ei4
e
Strain
i+1
i+1 i+1 i+1 i+11 2 3 4
i
Temperature
Figure 3.63: Interpolation of multilinear yield curves
with temperature
eeTemperature
Interpolated
yield curve
i+1 i+1P P2 3
Plastic
strain
1
1
2
2
3
3
4
4
t
i
i+1
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 299
3.6.3 Eval
plasticcreep, multilinearplasticcreep material models
effective creep strain is ca ulated using one of the
uation of creep strains
Creep,
! The lcfollowing creep laws:
Power creep law (creep law 1) :
1 20
a at C te a tσ= in which a0, a1, a2 are material constants. Exponential creep law (creep law 2) :
( )1t C Rte F e G− t= − +
with
4
610 2
3
; ;tt
ataaF a e R a G a e
a 5σσ σ ⋅⋅ ⎛ ⎞
= = =⎜ ⎟⎝ ⎠
in which a0 to a6 are material constants. Eightparameter creep law (creep law 3) ) :
t C He S T e−= ⋅ ⋅ w
ith
61 2 4 70 3 5; ,
273.16tS a =θ +
aa a at aT t a t a t Hσ= = + +
in which a to a are material constants.
In the above equations,
0 7
t Ce , tσ and tθ denote the effective p strain, stress and temperature at timeperature can either be EK or EC.
cree t. The unit of the tem
Chapter 3: Material models and formulations
300 ADINA Structures ⎯ Theory and Modeling Guide
B ployed
Th p d and 3D s t can be used in conjunction with strainhardening or timehardening.
the followingequations:
LUBBY2 creep law: The LU BY2 creep law can be emto predict longterm behavior of rock salt mass.
e law is implemented for 2D solie LUBBY2 creolid elements. I
The LUBBY2 creep law is specified by
( )( )( ) ( )
1 1exptt C
K tGd e t
σt t t
K K Mdtσ
η σ η σ η σ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟= − +
⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
or
( )( )( ) ( ) ( )
1 1expt
t C tt t t t
K K M K
e tG G
σKG t σσ η σ η σ σ
⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟= − + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
ere
wh ( ) *tM e
t
M
m , ( ) 1* t
K
ktKG G e σσ = and ση σ η=
( ) 2* t
K
ktK e ση σ η= . Also, as usual, t Ce is the effective creep
strain, t Cd edt
is the effective creep strain rate and tσ is the
ions, m, , k2, effective creep stress.
*M
η *K
GIn these equat k1 , and *K
η are six , kmaterial parameters. m or the stress effect on 1, k2 account f
* * *K
G , parameters M
η , K
η . The s eters a
vely. Unloadi
ix material param re identified in ADINA as a0, a1, acrp2, a3, a4 and a5 respecti
ng and cyclic loading effects are considered in the .
ry
the
LUBBY2 creep lawAnalytical solutions with the LUBBY2 creep law are ve
sensitive to the three parameters m, k1, k2, so these parameters must be very carefully specified.
For more information about the LUBBY2 creep law, see following reference:
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 301
re eep Tests on Rock Salt
with Changing Load as a Basis for the Verification of Theoretical Material Laws@, Proc. 6th Symp. on Salt, Toronto, Vol. 1, pp. 417435, 1983.
Blackburn creep law:
f. K.H. Lux and S. Heusermann, ACr
1 1 2 21 exp( ) 1 exp( )t Ce C rt C r t Gt= − − + − − + with
310 2
1C 21 2
,AAA G A GC
r r= =
8AR
5 7
1 4 2 6, ,A AR Rr A t r A t G Ft− − −= = =
( )2log log t Aσ10 0 1 10 2 10 9( ) log t
Rt B B Bσ= + +
+
1110 exp
273.16t
AF Aθ
⎛ ⎞= ⎜ ⎟+⎝ ⎠
1512 ,AB A B= + = 140 13 1 2,
273.16 273.16 273.16t t t
AA Bθ θ θ
=+ + +
5 are material constants.
tic
fective stress. aluated within user
supplied subroutines. In the usersupplied subroutines, these
0 1 2 1, , ,...,A A A A
Creepvariable, plasticcreepvariable, multilinearplascreepvariable material models
The following creep laws can be employed:
Eightparameter creep law (creep law 3) with variable coefficients: The same functional form as in creep law 3 described above is employed, but coefficients a0 to a7 are considered to be functions of temperature and ef
The values of a0 to a7 can also be ev
Chapter 3: Material models and formulations
302 ADINA Structures ⎯ Theory and Modeling Guide
parameters can be functions of the temperature and the effective stress.
The usersupplied subroutines are:
UCOEF2 for 2D solid elements (file ovl30u.f) UCOEF3 for 3D solid elements (file ovl40u.f) UCOEFB for isobeam elements (file ovl60u.f)
EFS for shell elements (file ovl100u.f) UCOEFP for pipe elements (file ovl110u.f)
n
bove nsidered to be functions
f the temperature (but not the stress, because the stress effects on ters
UCO
These subroutines are called by ADINA every time the stressstrailaw is updated.
Sixparameter LUBBY2 creep law with variable coefficients: The same functional form as in the LUBBY2 creep law described ais employed, but the six parameters are coo
Mη , KG , Kηparame have already been taken into account sing the three parameters m, k1, k2). The six parameters can also e evaluated using the usersupplied subroutines mentioned above.
Irradiation creep model
ub
The irradiation creep strain is calculated from
( ) 21
0
aE
1 exp ,Ce a Mφ
φ φσφ σ φ= − − +⎡ ⎤⎣ ⎦
( )3 4exp 5M a a= aθ +
here φ isw the neutron fluence at time t, a1 ,, a5 are material onstants, θ is the Kelvin temperature at time t, and 0E is
tial Youngs modulus. The subsequent Youngs modulus canendent on neutron fluence and temperature. In this case, its ue is determined
c the ini be depval by using the Shifting rule described in Section
.6.5.
3
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 303
l s
rd
lic ad nd
W ,
n
Hence, the incremental creep strains are calculated using
where the creep multiplier
A l creep material model
! The creep strains are evaluated using the strain ha eningprocedure for load and temperature variations, and the O.R.N.L. rules for cyc lo ing co itions.
ref. C.E. Pugh, J.M. Corum, K.C. Liu and .L. Greenstreet"Currently Recommended Constitutive Equations for Inelastic Design of FFTF Compo ents," Report No. TM3602, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 1972.
C t tij ije t sγ= ∆
tγ varies with stresses and creep strains, and the tsij are the deviatoric stresses (see Section 3.6.2 and ref. KJB, p. 607).
The procedure used to evaluate the incremental creep strains is summarized in the following: Given the total creep strains and
the deviatoric stresses tsij at time t,
1) Calculate the effective stress
t Cije
123
2t t t
ij ijs sσ ⎡ ⎤= ⎢ ⎥⎣ ⎦
Calculate the pseudoeffective creep strain
2)
( )( )1
23
t C te2C orig t C orig
ij ij ij ije e e e⎡ ⎤− −= ⎢ ⎥⎣ ⎦
3) Substitute t t Cσ , e and tθ into the generalized uniaxial creep law
Chapter 3: Material models and formulations
304 ADINA Structures ⎯ Theory and Modeling Guide
( )t C t tce F tσ θ= , ,
4) Solve for the pseudotime t .
Calcu ne
5) late the effective creep strain rate from the ge ralizeduniaxial creep law
( )t C
t tFe ∂∂ c tσ θt t
= , ,
∂ ∂
6) Calculate tγ
t C
tt
et⎜ ⎟∂3 ⎝ ⎠γ
σ
⎛ ⎞∂
=
d
2
an the increment
t C t tije t γ∆ = ∆ s
Steps 1) through 4) correspond to a strain hardening procedure.
There is also the option of using a time hardening procedure, in which steps 1) through 4) are replaced by the formula t t= . See ref. KJB, pp 607608 for a discussion of the two hardening procedures. The strain hardening procedure is recommended for general use (and is the default).
! Rupture of the material due to excessive creep strain can be ase, a uniaxial creep strain failure envelope must
be defined as a function of temperature and effective stress (Fig.
point lysis
specified. In this c
3.64). The following criterion is then checked at each integration
during the ana :
,Cu rt C e
eTF
≤
h is the uniaxial creep strain at failure, ,Cu re t Ce is the in whic
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 305
ffective creep strain, TF is the triaxiality factor, and e
11 22 33t t t t
mσ σ σ σ= + + is the volumet tress.
You can choose to use the triaxiality factor
ric st
TF mt
σ
σ= (the
default) or to use no triaxiality factor (that is, 1TF = ).
Figure 3.64: Multilinear interpolation of uniaxial creep strain
at rupture
Temperature
Effective
stress
!
!
"!
#!
#!
"! !
!
cu,rc (1)
u,r 1
c (2)u,r 1
c (3)u,r 1
c (4)u,r 1Uniaxial creep strain
at rupture
You can choose to apply the rupture model only under tensile compressiv
only in tension, based on the sign of
conditions ( 0tmσ > ) or under both tensile and e
conditions. The default is to apply the rupture model only under tensile conditions.
Rupture can be allowedt
mσ .
! You can apply a time offset to any of the creep material models (for example, creep law 1 becomes ( ) 21
0 0aat C te a t tσ= − , where
0t is the time offset, which can be positive or negative). The time offset can be modified in a restart run.
Chapter 3: Material models and formulations
306
! The accumulated effective creep strain is calculated as
all solution steps
t C Ce e= ∆∑
where 23
C Ce C∆ = ∆ ⋅ ∆e e and C∆e is the tensor of creep strain
increments in a solution step. Because of the summation over the solution steps, we refer to the calculated value of t Ce as the accumulated effective creep strain.
3.6.4 Com
! esses at the integration points are evaluated using the
ns.
putational procedures
The streffectivestressfunction algorithm.
ref. M. Kojiƒ and K.J. Bathe, "The EffectiveStressFunction Algorithm for ThermoElastoPlasticity and Creep," Int. J. Numer. Meth. Engng., Vol. 24, No. 8, pp. 15091532,1987.
Briefly, the procedure used consists of the following calculatioThe general constitutive equation
( )( ) ( ) ( ) ( )t t i t t E t t i t t P i t t C i t t THσ+∆ +∆ +∆ +∆ +∆ +∆= − − −C e e e e (3.61)
ref. KJBSection 6.6.3
ADINA Structures ⎯ Theory and Modeling Guide
is solved separately for the mean stress and for the deviatoric tresses. In this equation the index (i) denotes the iteration counter
quilibrium. For easier writing this cussion to follow. The mean stress
sin the iteration for nodal point eindex will be dropped in the disis calculated as
( )1 2
t tt t t t t t TH
m mt t
E e eσ+∆
+∆ +∆ +∆+∆= −
− ν (3.62)
The deviatoric stresses t t+∆ s depend on the inelastic strains and they can be expressed as
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 307
( )1 1t t t t tt t
E
ta t
ττ α γ
α γ λ+∆ +∆
+∆′′⎡ ⎤= − − ∆ ⎣ ⎦+ ∆ + ∆
s e s (3.63)
where 1
t tt t
E t t
Ea+∆
+∆+∆=
+ ν, ts = deviatoric stress at the start of the
me step and α is the integration parameter used for stress tievaluation ( )0 1α≤ < The creep and plastic multipliers τγ and
λ∆ are functions of the effective stress t tσ+∆ only, and they account for creep and plasticity; also
t t t t t P t C+∆ +∆′′ ′= − −e e e e is known since the deviatoric strains t t+∆ ′e , plastic strains t e
ep strains t Ce are known from
P and cre the current displacements and the str /
alar function
ess strain state at the start of the current time step.
he following sc ( )t tf σ+∆T is obtained from Eq.
(3.63) ( ) 2 2 0t t t tf a b c dτ τσ σ γ γ+∆ +∆ 2 2 2= + − − = (3.64)
The zero of equation (3.64) provides the solution for the effective stress t tσ+∆ , where
t tEa a t τ
α γ λ+∆= + ∆ + ∆
( )3 1 t t tij ijb t e sα +∆ ′= − ∆
( )1 tc tα σ= − ∆
2 3
2 ij ijd e et t t t+∆ +∆′′ ′′=
with summation on the indices i, j. Once the solution for has been determined from Eq. (3
simultaneously with the scalars
.64), τγ and λ∆ from the creep and
Chapter 3: Material models and formulations
308 ADINA Structures ⎯ Theory and Modeling Guide
t t C t C t t
plasticity conditions, the deviatoric stress t t+∆ s is calculated fromEq. (3.63), and the plastic and creep strains at the end of the time step are obtained as
t t P t P t tλ
(1 t t τα α γ+∆ +∆⎡ ⎤= + − ) + ∆⎣ ⎦e e s s
The above equations correspond to isotropic hardening
conditions and a general 3D analysis. The solution details for kinematic hardening conditions and for special problems (for the
eam elements) are given in the
the, "ThermoElasticPlastic and
3.6.5 Shifting rule for the irradiation creep model
e have been carried out generally at a constant temperature. They cannot be determined directly from the test data due to variation of temperature.
Therefore, the irradiation strain at temperature
+∆ +∆= + ∆e e s
plane stress, shell, isoparametric babove cited reference and also in the following reference.
ref. M. Kojiƒ and K.J. BaCreep Analysis of Shell Structures", Computers & Structures, Vol. 26, No 1/2, pp. 135143, 1987.
! Irradiation tests obtaining irradiation strain data and material property changes of graphit
θ and under fast neutron fluence φ is calculated as follows
)( ) ( 1 11
, , ,n
W Wi i i i
iε θ φ ε θ φ φ φ− −
=
= ∆ −∑
where ( ) ( ) (, , , ,W W W
i i )iε θ φ φ ε θ φ φ ε θ φ∆ ∆ = + ∆ − This is illustrated in Figure 3.61, and documented in the
followin ref. nd M. Ishihara, "Development of
Thermal/Irradiation Stress Analytical Code VIENUS for HTTR Graphite Block", Journal of Nuclear Science and
g reference.
T. Iyoku a
3.6: Thermoelastoplasticity and creep material models
ADINA R & D, Inc. 309
Technology, 28[10], pp. 921931, October 1991.
(a) irradiated from zero to φunder temperature θ 1
2
1(b) irradiated from φ to φunder temperature θ(c) irradiated from φ to φunder temperature θ
1 2
2 33
(a)
(c)(b) θ
θ
θ
1
2
3
φ φφφ1 2 30
εW
Figure 3.61: Shifting rule for irradiation creep model
3.7 Concrete material model
3.7.1 Gene
model can be employed with the 2D solid and D solid elements.
! The concrete model can be used with the small displacement and large displacement formulations. In all cases, small strains are assumed.
When used with the small displacement formulation, a materiallynonlinearonly formulation is employed, and when used with the large displacement formulation, a TL formulation is employed.
! Although the model is entitled "concrete model", the basic constitutive characteristics are such that the model can also be useful when representing other materials. The basic material characteristics are:
ral considerations
The concrete!
3
Chapter 3: Material models and formulations
310 ADINA Structures ⎯ Theory and Modeling Guide
< Tensile failure at a maximum, relatively small principal tensile stress < Compression crushing failure at high compression < Strain softening from compression crushing failure to an ultimate strain, at which the material totally fails.
The tensile and compression crushing failures are governed by tensile failure and compression crushing failure envelopes.
hese material characteristics pertain, for example, to a variety cks.
! It is well accepted that concrete is a very complex material. The model provided in ADINA may not contain all the detailed material c a
to model most of the commonly aviors.
3.7.2 Formulation of the concrete material model in ADINA
ation used in the formulation of the model is given
below.
tE = equivalent multiaxial tangent Young's modulus at time t (the left superscript "t" refers to time t)
= uniaxial initial tangent modulus (all uniaxial
Tof ro
h racteristics that you may look for. However, considering the variability of concrete materials that need to be described in practice, and recognizing that the model may also be useful in themodeling of rock materials, the objective is to provide an effectivemodel with sufficient flexibilityused material beh
! The not
0E& qualities are identified with a curl "~" placed over them)
sE& = uniaxial secant modulus corresponding to uniaxial
maximum stress, cs
c
Eeσ
=&&&
uE = uniaxial secant modulus corresponding to uniaxial
ultimate stress,
&
uu
u
Eeσ
=&&&
3.7: Concrete material model
ADINA R & D, Inc. 311
tpiE& = uniaxial tangent modulus in the direction of t
piσ t
ije = total strains
incremental strains
ce& = uniaxial strain corresponding to ( )0c ceσ <& &
ue& = ultimate uniaxial compressive strain ( )0ue <& t
ij
ije =
= uniaxial strain te&
σ = total stresses
ijσ = incremental stresses tσ& = uniaxial stress
tσ& = uniaxial cutoff tensile strength ( )0tσ >&
tpσ& = postcracking uniaxial cutoff tensile strength
( )0tpσ >& . Note that if 0tpσ =& , ADINA sets
tp tσ σ=& & .
cσ& = ( )0cσ <& maximum uniaxial compressive stress
uσ& = ultimate uniaxial compressive stress ( )0uσ <& t
piσ = principal stress in direction i ( )1 2 3t t t
p p pσ σ σ≥ ≥
Three basic features are used in the concrete model:
!
< A nonlinear stressstrain relation to allow for the weakening of the material under increasing compressive stresses < Failure envelopes that define failure in tension and crushing in compression
Strrel
< A strategy to model the postcracking and crushing behavior of the material
essstrain relations: The general multiaxial stressstrain ations are derived from a uniaxial stressstrain relation tσ& sus te& . ver
Chapter 3: Material models and formulations
312 ADINA Structures ⎯ Theory and Modeling Guide
A typical uniaxial stress tσ& to uniaxial strain relation is sho e t
te&wn in Fig. 3.71. This stressstrain relation shows that there ar
hree strain phases; namely, and0, 0 , t t tc c ue e e e e e≥ > ≥ > ≥& & & & & &
where ce& is the strain corresponding to the minimum (crushing) stress c
σ& that can be reached, and is the ultimate compressive strain.
tion
ue&
If 0 0E >& , the material is in tension and the stressstrain relais linear until tensile failure at the stress tσ& . A constant Young's
modulus 0E& is employed, i.e., & & (3.71)
For e
tσ =& 0tE e
t ≤ 0& , we assume the following relation
0t E eσ ⎜ ⎟⎜ ⎟
=
&& &&
2 3
t
s c
t t t
c c c
E e
e e eσ
⎛ ⎞⎛ ⎞
⎝ ⎠⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞
&
& & & &
1c A B Ce e e
+ + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠& & &
3.7: Concrete material model
ADINA R & D, Inc. 313
~
~ t
~eu~ec
~u~c
Figure 3.71: Uniaxial stressstrain relation used in the concrete
model
~et~e
t p~
and hence
2 3
0
22 3
1 2
1
t t
c ct
t t t
c c c
e eE B Ce e
Ee e eA B C
e e e
⎡ ⎤⎛ ⎞ ⎛ ⎞− −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦=⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞
+ + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎠ ⎝ ⎠ ⎝ ⎠⎝⎣ ⎦
& &&& &
&& & &
& &
(3.72)
here
&
w
Chapter 3: Material models and formulations
314 ADINA Structures ⎯ Theory and Modeling Guide
( ) ( )( )
3 2 3 20 02 2 3 1u s
p p p pE E
2
0
0
2 1
2 3 2
2s
E E
Ap p
EB A
EE
⎡ ⎤+ − − − +⎢ ⎥
⎣ ⎦& &
p
sE
C A
=⎡ ⎤− +⎣ ⎦
⎡ ⎤⎛ ⎞= − −⎢ ⎥⎜ ⎟
⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
& &
&
&&
and the parameters
⎝ ⎠⎣ ⎦
= −
&
0 , , , , , , ,c uc c s u u u
c c
eE e E e p Ee e
u
ueσ σσ σ= =& && & && & & && &
=&&
are obtained from uniaxial tests.
t o ze he
following modulus:
The stressstrain relation in Eq. (3.72) assumes monotonic loading conditions. For unloading conditions and loading back to the stress state from which unloading occurred, the initial Young's modulus 0E& is used. For strain states beyond ue& in compression, iis assumed that stresses are linearly released t ro, using t
u cu
u c
Ee e
σ σ−=
−& &
Note that confined concrete can values for
&& &
therefore be modeled using close andu cσ σ& & .
evaor unloading. Poisson'ten
loa tg = tσe (3.73)
wh thedet
Under multiaxial stress conditions, the stressstrain relations areluated differently depending on whether the material is loading
s ratio is assumed to be constant under sile stress conditions and can vary in the compressive region. To characterize loading and unloading conditions we define a ding scalar tg for each integration point,
ere tσe is the effective stress at time t. The material is loading at integration point except when the unloading conditions are ermined,
3.7: Concrete material model
ADINA R & D, Inc. 315
tg < gmax (3.74)
where gmax is the maximum value of the loading scalar that has been reached during the complete solution.
During unloading, the material is assumed to be isotropic and the initial Young's modulus, , is used to form the incremental stressstrain matrix, both for stiffness and stress calculations.
In loading conditions, the principal stresses
0E&
tpiσ (with
3p1 2t t t
p pσ σ σ≥ ≥
dire
) are calculated. For each principal stress
giv
In ith the
parameters
ction, a tangent Young's modulus tpiE& corresponding to the
en strain state tpie is evaluated using Eqs. (3.71) and (3.72).
Eq. (3.72), the strains tpie are used together w
, ,c u ceσ σ′ ′ ′& & and ue′ defined in Eq. (3.78) which accounts for the multiaxial stress conditions.
ortcrapo
The material is considered as orthotropic with the directions of hotropy being defined by the principal stress directions. Once cking occurs in any direction i, that direction is fixed from that int onward in calculating t
piσ . The stressstrain matrix corresponding to these directions is, for
threedimensional stress conditions,
( )( )( )
( )( )
(0.5 )
( )
1 1 1
23
3
12
23
11 1 2
11 0 0 0
0 01 2 0 0
symme0.5 1 2
t tp
t t
tp
t
E E E
EE
2 3
2
0 0 0
1 0
t
pE E
( ) 13tric 0.5 1 2 0t
t
EE
= ×+ ν + ν
− ν ν
− ν
− ν
− ν
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥⎣ ⎦
C
&&
&
5)
ν
ν
− ν
− ν⎢ ⎥⎢ ⎥
(3.7
Chapter 3: Material models and formulations
316 ADINA Structures ⎯ Theory and Modeling Guide
where v is the constant Poisson's ratio and the tEij with i … j are evaluated using
t t t t
pi pi pj pjtij t t
pi pj
E EE
σ σ
σ σ
+=
+
& & (3.76)
e stress increment from t to time t + ∆t, we have
(3.77)
where the stressstrain matrix is the one defined in Eq. (3.75) but using the Young's moduli . These moduli are evaluated
using three point Gaussian integration of between and
, with and the strain components measured in
Note that the above stressstrain relations for material loading
conditions are only employed in the calculation of the stiffness matrix at time t. Considering the evaluation of th
=σ CeC
piEτ &
piE& tpie
t tpie+∆ t
pie t tpie+∆
the directions of the respective principal stresses tpiσ and pi
t tσ+∆
Material failure envelopes3.7 tlawwhether tensile or crushing failure of the material has occurred.
Unestablished the principal stresses
.
: The failure envelopes shown in Figs. 2 o 3.75 are employed to establish the uniaxial stressstrain accounting for multiaxial stress conditions, and to identify
iaxial stress law under multiaxial stress conditions: Having t
piσ with 3p p p1 2t t tσ σ σ≥ ≥ the
stresses t1pσ and 2p
tσ are held constant and the minimum stress that would have to be reached in the third principal direction to cause crushing of the material is calculated using the failure envelope, see Fig. 3.75(a). Let this stress be cσ ′& , we have
1c
c
σγσ
′=&&
and
3.7: Concrete material model
ADINA R & D, Inc. 317
( )
( )
21 1 1
1 1 2 1
;u u
u ue C C e
σ γ σ γ
γ γ
′ ′ +
= +
& &
where C
2 1
2
;c ce C C eγ= =
′ (3.78)
1 and C2 are input parameters. Normally, C1 = 1.4 and C2 =
0.4. The constants , , ec u cσ σ′ ′ ′& & and ue′ are employed instead of the ish, using Eq. (3.72), the
uniaxial stressstrain law under multiaxial conditions (see Fig. 3.75(b)).
unprimed variables in order to establ
Figure 3.72: Threedimensional tensile failure envelope of
concrete model
( , 0, 0)c~
tp2
tp3
tp1
t t' =~
c
t t
tp3
c
= (10.75 ~ )
t t
tp2
c
tp3
c)= (10.75 ~ ) (10.75 ~
~
~
~
~
t t
tp2
c= (10.75 ~ )
(0, ,0)c~
~ = Uniaxial cutoff tensile stress under multiaxial conditions~ = Uniaxial compressive failure stress under multiaxial conditions
t
c
Chapter 3: Material models and formulations
318 ADINA Structures ⎯ Theory and Modeling Guide
Figure 3.73: Biaxial concrete compressive failure envelope
ADINA
Experimental
tp2
c
~
tp2
c
~
1
Figure 3.74: Triaxial compressive failure envelope
Input to ADINA
c
pt
3
t
t
t
tp2
p2
p3
p3
t tp2 p1
Experimental
tp2
c
~
~
Curve 1
p1t i
t ~c
= constantCurve i
3.7: Concrete material model
ADINA R & D, Inc. 319
~eu~eu ~ec
~ec
te~
t~
tu
~
tu
~
tc
~
tc
~
(b) Determine uniaxial behavior for multiaxial stress conditions
tp2
tp3 =
tp2
tp3 =
(a) Determination of~
c from given ( , ) t tp1 p2
Figure 3.75: Definitions used for evaluation of onedimensional
stressstrain law under multiaxial stress conditions
tp3
c
~
1
c
c~
~t
p1
c
~
tp2
c
~
tp2
c
~
= constant
= constant
1
tp1
tp2
is determined from
failure envelope by
holding andconstant.
~ /~ =
c c 1
~ /~ =u u 1
e~ / e~ = C + Cu u 1 12
2 1 e~ / e~ = C + Cc c 1 1
22 1
Chapter 3: Material models and formulations
320 ADINA Structures ⎯ Theory and Modeling Guide
pe used in the concrete model is shown in Fig. 3.72. To identify whether the material has failed, the principal stresses are used to locate the current stress state. Note that the tensile strength of the material ina principal direction does not depend on tensile stresses in the other principal stress directions, but depends on compressive stresses in the other directions.
pal stress direction exceeds the tensile failure stress. In this case it is assumed that a plane of failure develops perpendicular to the corresponding principal stress direction. The effect of this material failure is that the normal and shear stiffnesses and stresses across the plane of failure are reduced and plane stress conditions are assumed to exist at the plane of tensile failure, as discussed in more
elow. Prior to tensile failure the stressstrain material law is given by
Eqs. (3.75) to (3.77). Assuming that tσp1 is larger than the tensile failure stress, the new material stressstrain matrix used in the st
Tensile failure envelope: The tensile failure envelo
Tensile failure occurs if the tensile stress in a princi
detail b
iffness matrix calculation is given by
( )
( )
( )
( )
0
2 23
31 tpE2− ν ⎢ ⎥⎣ ⎦
0
0
23
0 0 0
0 0 0
0 0
symmetric 0
2 1
2 1
2 1
tp
s
s
t
E
E
E
E
E
η
η
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
⎤ν⎢ ⎥⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
+ ν⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+ ν⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
0 0 0 0 0
1
n
t E
η
⎡
+ ν⎢ ⎥⎣ ⎦
C
&
&
&
(3.79)
&&
3.7: Concrete material model
ADINA R & D, Inc. 321
where the are the uniaxial Young's moduli evaluated in the principal stress directions using Eqs. (3.71) or (3.72), and the
are evaluated using Eq. (3.76).
The constants ηn and ηs are the stiffness and shear reduction factors, respectively. Typically, ηn = 0.0001 and ηs = 0.5. The factor ηn is not exactly equal to zero in order to avoid the possibility of a singular stiffness matrix. The factor ηs depends on a number of physical factors and you must use judgement to choose its value. For the concrete model in ADINA, ηn and ηs are both input parameters.
For stress calculation, the following stressstrain matrices are used:
< For the tensile stress normal to the tensile failure plane and the shear stresses in this plane, we use the total strains to calculate the total stresses with
tpiE&
tijE
12
13
0 0 0
sym.
ff
f
EG
G
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
C (3.710a)
where Ef and are evaluated as shown in Fig. 3.76. In this figure,
12fG , 13
fGξ is a user input variable that defines the amount of
tension stiffening. Fig. 3.76(b) shows that is evaluated from the initial shear modulus. Also, Fig. 3.76(a) assumes loading from zero stress directly into the tensile region. If the tensile stress is reached by unloading from a compressive stress, the strain normal to the tensile failure plane is measured from the specific strain value at which the stress is zero (see Fig. 3.77 point 11).
Special care in the analysis must be taken if
fijG
ξ is chosen to be greater than 1.0, because strain softening may yield nonunique solutions.
Chapter 3: Material models and formulations
322 ADINA Structures ⎯ Theory and Modeling Guide
be provided instead of
To obtain a mesh independent solution, the fracture energyGf can ξ . In doing so, ξ is evaluated at each integration point, based on the size of the finite elem
ents (see Fig. 3.76(c)). In this case, ADINA also calculates ηs internally, overwriting the user input.
~t
~
t
1
1
Ef
~et~ete ~e (= ~e )m t
Strain normal
to tensile
failure plane
a) Calculation of Young's modulus Ef, normal to tensile failure plane.~ is the postcracking uniaxial cutoff tensile stress.tp
b) Calculation of shear modulus in tensile failure plane
Figure 3.76: Material moduli for stress calculation
after tensile failure
~et~em
~e
Strain normal
to tensile
failure plane
f
1.0
s
G =
~E
2(1+ )fij f
0
0
~E0
~tp
3.7: Concrete material model
ADINA R & D, Inc. 323
are
< For the remaining stress components, the increments evaluated from the strain increments using
h2
Integration point
Figure 3.76: (continued)
Element
Physical state with
multiple cracks
Model with
unique crack
with G energy released per unit area in
direction perpendicular to h directionf
2
2
~E G~ h
0 f2t 2
c) Smeared crack approach for the calculation of the parameter
( ) ( )
2 23
3
23
01 0
1sym. 1
p
p
E EE
G
τ τ
τ2
2 τ
⎡ ⎤ν⎢ ⎥
= ⎢ ⎥− ν ⎢ ⎥
− ν⎢ ⎥⎣ ⎦
C
& && (3.710b)
where the are the uniaxial Young's moduli used in Eq.
(3.77) and and τG23 are evaluated using Eq. (3.76) but
lope
ific envelopes by the input explained below, and this flexibility makes it possible to model various concrete and rock materials.
piEτ &τE23
with piEτ & instead of tpiE& .
Compressive failure envelope: The triaxial failure enve used inthe concrete model is shown in Fig. 3.74. Note that the biaxial failure envelope shown in Fig. 3.73 is curve 1 of the triaxial failure envelope. Also, note that the biaxial and triaxial envelope curves can be used to represent a large number of spec
Chapter 3: Material models and formulations
324 ADINA Structures ⎯ Theory and Modeling Guide
The triaxial compression failure envelope is shown in Fig.
3.74. First, the values ip1
c
σσ&
are input. These values define for
which levels of the tp1σ the twodimensional failure envelopes
functions of stresses tp2σ and t
p3σ are input. Six failure envelopes must be defined, each with three points
corresponding to the locations , ,t t t tp2 p1 p2 p3σ σ σ β σ= = and
t tp2 p3σ σ= where β is an input parameter. To identify compression failure, the largest principal stress,
tp1σ , is employed to establish the biaxial failure envelope function
of tp2σ and t
p3σ , using the interpolation shown in Fig. 3.75(a). If
the stress state corresponding to tp2σ and t
p3σ lies on or outside this biaxial failure envelope, then the material has crushed.
It should be noted that you must choose appropriate values for and other parameters. These may aterials and structures and must be
o
rial
is otherwise active. Therefore, a tensile failure plane c
nsile failure plane becomes active. This tensile failure plane remains active while the strain continues to increase to point 2 and then decreases to point 3 and then increases to point 4 and then decreases to point 5. The failure plane becomes inactive at point 5 and remains so while the
the input of the failure surfaces vary significantly for different m
btained from experimental data.
Material behavior after failure: The post failure matebehaviors considered in the concrete model in ADINA include the post tensile cracking, post compression crushing, and strainsoftening behaviors.
Post tensile cracking behavior: Once a tensile plane of failure has formed, it is checked in each subsequent solution step to see whether the failure is still active. The failure is considered to be inactive provided the normal strain across the plane becomes negative and less than the strain at which the "last" failure occurred. Itan repeatedly be active and inactive. Consider the uniaxial cyclic
loading case shown in Fig. 3.77 in which the strain is prescribed. As the strain reaches the tensile cutoff limit, a te
3.7: Concrete material model
ADINA R & D, Inc. 325
strain decreases to point 8 and increases again to point 11. At point 11, the tensile failure plane again becomes active, and it remains active until the strain reaches point 13 and beyond.
Figure 3.77: Uniaxial cyclic loading stress/strain experiment
with the concrete model
Stress
Stress
Strain
Strain
1
A
A
35
2
4
8
1112
20
Point numbers:
25,1112 crack open
610,1320 crack closed
>20 material crushed
If a tensile failure plane has developed, which mbe active, the material stressstrain relations are alwayas described above, corresponding to the directions along and perpendicular to the plane of failure. Hence, instead of using the princ the
l, th y the material tensile failure plane are used to evaluate the stressstrain matrix, corresponding to those directions. Once a failure plane has been initiated, a subsequent failure plane is assumed
ay or may not s established,
ipal stresses and corresponding directions as done forunfailed materia e stresses in the directions defined b
to form
Chapter 3: Material models and formulations
326 ADINA Structures ⎯ Theory and Modeling Guide
perpendicular to the direction of the first failure plane whenever a no sile
y ind in
.
n, it is assumed that the material strainsoftens in all directions.
Consider first uniaxial stress conditions. As shown in Fig. 3.71, when the uniaxial strain is smaller than , the material has
crushed and softens with increasing compressive strain, i.e., is negative.
Under multiaxial stress conditions the compression crushing is identified using the multiaxial failure envelope, and once the material has crushed, isotropic conditions are assumed. As in uniaxial conditions, in the subsequent solution steps the Young's modulus is assumed to be very small but positive in the stiffness matrix calculations, but the stress increments are computed from the uniaxial stressstrain law with the const
rmal stress along the original failure plane has reached the tenfailure stress. It follows that at an tegration point, the direction of the third tensile failure plane is fixed once failure has occurretwo directions
Postcompression crushing behavior: If the material has crushed in compressio
ce&t E&
ants , c ceσ ′ ′& & and so on, see Eq. (3.78), corresponding to the multiaxial conditions at crushing. The Young's modulus τE corresponding to the current strain increment ep3 is evaluated using the uniaxial stressstrain relationship in Fig. 3.71
t t
p3 p3 p3e e e
p3
Ee
σ σ+τ
⎡ ⎤−⎣ ⎦=& &
where
the nt,
solus . Note that when p3
es equal to or less than
tep3 and ep3 are the strain component at time t measured indirection of the principal stress tσp3. To obtain the stress incremeEq. (3.77) is used where the matrix C corresponds to i tropic material conditions with Young's modu τE tebecom ue′& , the stresses are linearly released to zero.
3.7: Concrete material model
ADINA R & D, Inc. 327
If unloading of the crushed material in the strainsoftening region occurs, characterized by ep3 ≥ 0, the initial Young's modulus
is used. At any time during postcrushing calculations, if any idually reaches a positive
thermal incremental strain and is calculated from the temperature
g material properties can be defined as temperaturedependent:
< Uniaxial initial tangent modulus
0E&one of the principal stresses checked indivvalue, this stress is set to zero.
Temperature effects: In some analyses, temperature strains t THeneed to be included. These strains are taken into account by replacing the total incremental strain e in the governing incremental equations with the strain e  e
TH, where eTH is the
conditions. The followin
0
< Poisson's ratio E&
ν fficient of thermal expansion α < Coe
Uniaxial cutoff tensile stress tσ& <
< Postcracking uniaxial cutoff tensile stress tpσ&
axial maximum compressive stress c< Uni σ& < Uniaxial compressive strain at cσ& , ce& < Ultimate uniaxial compressive stress uσ& < Uniaxial strain at uσ& , ue& < Fracture energy gfi
< Constant for tensile strain failure xsii
The nodal point temperatures are input to ADINA as discussed
in Section 5.6.
Poisson's ratio in the compressive region: It has been observed in experiments that the ratio of lateral strain to principal compressive strain remains constant until approximately 80% of the maximum compressive stress cσ& .
Usually, we assume that the Poisson ratio is constant throughout the analysis; however, as an option, the value vs can be used when
Chapter 3: Material models and formulations
328 ADINA Structures ⎯ Theory and Modeling Guide
pression. The value of vs is given by
the material dilates under com
( )
2
2
when
1
p3
22when
1
t
s s ac
s f f
σ
aa
a
ν ν ν γ γσ
γ γν ν ν ν γ γγ
⎧= = = ≤⎪ ′⎪
⎨⎛ ⎞−⎪
= − −⎪
&
− >⎜ ⎟−
f
failure = 0.42, a = 0.7.
3.8 Rubbe
! material models in ADINA are the o l
lid and
! displacement/large strainem !
⎝ ⎠⎩
with v = initial Poisson's ratio, v = maximum Poisson's ratio at γ
r and foam material models
The rubber and foamM oneyRivlin, Ogden, ArrudaBoyce and hyperfoam materiamodels.
! These material models can be employed with the 2D so3D solid elements.
These material models can be used with the large formulation. A TL formulation is
ployed.
These material models include the following effects:
< isotropic effects (required), see Section 3.8.1 < viscoelastic effects (optional), see Section 3.8.2 < Mullins effects (optional), see Section 3.8.3 < orthotropic effects (optional), see Section 3.8.4 < thermal strain effects (optional), see Section 3.8.5
! These material models allow three types of temperature dependence:
< no temperature dependence
< timetemperature superposition for viscoelastic effects, see
3.8: Rubber and foam material models
ADINA R & D, Inc. 329
Section 3.8.6. This type of material is referred to as a thermorheologically simple material (TRS material).
< full temperature dependence, see Section 3.8.7.
ws a summary of the features. Tamo
ain
Table 3.81 sho
ble 3.81: Summary of features for rubber and foam material dels
Viscoelastic effects1
Mullins effects1
Orthotropic effects
Thermal streffects
No perature yes yes yes no tem
dependence TRS
temde
perature pendence
yes yes yes yes
Full perature yes yes yes tem
dependence yes
1) Viscoelastic effects and Mullins effects cannot be used together. ! When the material temperature dependence
is TRS or full, heat eneration from the rubberlike materials is included in TMC
3.8.1 Isotropic hyperelastic effects
! and foam material models include the following models for isotropic hyperelastic effects:
< MooneyRivlin
< hyperfoam ! Only the MooneyRivlin and Ogden material models are supported in explicit dynamic analysis.
g(thermomechanicalcoupling) analysis.
The rubber
< Ogden < ArrudaBoyce
Chapter 3: Material models and formulations
330 ADINA Structures ⎯ Theory and Modeling Guide
! Thby specifyorigin
e isotropic hyperelastic effects are mathematically described ing the dependence of the strain energy density (per unit
al volume) W on the GreenLagrange strain tensor ijε . ! use , refer to ref KJB, section 6.6.2. Here and below, we omit the usual left superscripts and subscripts for ease of writing. Unless otherwise stated, all quevaluated at time and referred to reference ti Cgiv
We now give a brief summary of the quantities and conceptsd. For more information
antities are me 0 . t
Useful quantities are the CauchyGreen deformation tensor ij , en by
2ij ij ijC
ε δ= + (3.81) where ijδ is the Kronecker delta; the principal invariants of the
CauchyGreen deformation tensor,
( )22 2 1
1ij ijI I C C− , = 3 detI = C1 kkI C= , (3.82a,b,c)
the reduced invariants
131 1 3J I I −= ,
232 2 3 J I I −= ,
123 3J I= , (3.83a,b,c)
the stretches iλ where the iλ s are the square roots of the principal stretches of the CauchyGreen deformation tensor; and the reduced stretches
( )1
*iλ 3
i 1 2 3λ λ λ λ −= (3.84) Note that 33 1 2J λ λ λ= (3.85) is the volume ratio (ratio of the deformed volume to the undeformed volume).
3.8: Rubber and foam material models
ADINA R & D, Inc. 331
The strain energy density is written in terms of the invariants or stretches. In m cases, the strain energy density is conveniently written as the sum of the deviatoric strain energy density
Wany
DW and the volumetric strain energy density . With knowledge of how the strain energy density depends on the GreenLagrange strain tensor (through the invariants or stretches), the 2nd PiolaKirchhoff stress tensor is evaluated using
VW
W
12ij
ij ji
W WS⎛ ⎞∂ ∂
= +⎜ ⎟⎜ ⎟∂ε ∂ε⎝ ⎠ (3.86)
and the incremental material tensor is evaluated using
12
ij ijijrs
rs sr
S SC
∂ ∂⎛ ⎞= +⎜ ⎟∂ε ∂ε⎝ ⎠
(3.87)
3.8.1.1 MooneyRivlin model
! The MooneyRivlin material model is based on the following expression:
3 +
(3.88)
where C1 to C9 and D1 to D2 are material constants.
! incompressible material
( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( ) ( )( )( )
21 1 2 2 3 1 4 1 2
2 3 25 2 6 1 7 1 2
2 38 1 2 9 2 1 2 1
3 3 3 3
3 3 3 3
3 3 3 exp 3 1
DW C I C I C I C I I
C I C I C I I
C I I C I D D I
= − + − + − + − −
− + − + − − +
− − + − + − −
This strain energy density expression assumes a totally
( )3 1I = and is modified as explained below for plane strain, axisymmetric or 3D analysis. Plane stress analysis In plane stress analysis, the material is assumed to be totally incompressible. Therefore is zero and VW
Chapter 3: Material models and formulations
332 ADINA Structures ⎯ Theory and Modeling Guide
DW W= . A displacementbased finite element formulation is used, in which the incompressibility condition of the material is
calculating the appropriate thickness of the material.
Plane strain, axisymmetric and 3D analysis: In plane strain, axisymmetric and 3D analysis, the material is modeled as
set the bulk modulus high so that the material is almost incompressible. by
imposed by
compressible (that is, the bulk modulus is not infinite), but you can
The MooneyRivlin strain energy density equation is modified: 1) substituting for the invariants 1 2,I I the reduced invariants , J , 2) removing the co1 2J ndition I 3 1= , and 3) adding the
volumetric strain energy density
( )23
1 12VW Jκ= − (3.89)
where κ is the bulk modulus. This expression for the volumetric
strain energy density yields the following relationship between the pressure and the volume ratio:
( )3 1p Jκ= − − (3.810) The MooneyRivlin material model is intended for use when the bulk to shear modulus ratio is large. The large bulktoshear modulus ratio causes numerical difficulties for axisymmetric, plane strain and 3D elements. The mixed u/p formulation pressure formulation can be used (and is the default) for these elements. This formulation avoids the numerical difficulties by use of a separately interpolated pressure, as described in the following reference:
ref. T. Sussman and K.J. Bathe, "A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis," J. Computers and Structures, Vol. 26, No. 1/2, pp. 357409, 1987.
The displacementbased formulation can also be used, but is not recommended.
ref. KJBSection 6.6.2
3.8: Rubber and foam material models
ADINA R & D, Inc. 333
Se tdes s D1 to D2 and the higChchoMocho
ed for modeling certain biological materials and need not be used otherwise.
mo
lec ion of material constants: The MooneyRivlin material cription used here has constants C1 to C9, constant bulk modulus κ. Strictly speaking, this material law is termed aherorder or generalized MooneyRivlin material law. oosing only C1 … 0 yields the neoHookean material law, and osing only C1 … 0, C2 … 0 yields the standard twoterm oneyRivlin material law. In ADINA, constants C3 to C9 can be sen to more closely fit experimental data. Constants D1 and D2 are primarily intend
The small strain shear modulus and small strain Young=s dulus can be written in terms of these constants as (assuming
κ = ∞ ) ( )1 2 1 22G C C D D= + +⎡ ⎤⎣ ⎦ (3.811)
( )1 2 1 26E C C D D= + +⎡ ⎤⎣ ⎦ (3.812)
The ! ility of the
! n small strain near
se moduli must be greater than zero.
The bulk modulus κ is used to model the compressibmaterial for plane strain, axisymmetric and 3D analysis.
ADINA assumes a default for the bulk modulus based oincompressibility, i.e.,
( ) with
3 1 2Eκ ν
ν= = 0.499
− (3.813)
where E is the small strain Young's modulus or, in terms of the
small strain shear modulus G,
( )( )
2 1500 for
3 1 2G
Gν
κ νν
+= = = 0.499
− (3.814)
This rule of thumb can be used to estimate the bulk modulus in the
sab ence of experimental data. However, you can use lower values of the bulk modulus to model compressible materials.
Chapter 3: Material models and formulations
334 ADINA Structures ⎯ Theory and Modeling Guide
s
s
equently an acceptable assumption since the material is almost incompressible.
io may
3.8.1.2
ssion:
• In explicit analysis, the program assumes the same bulk modulus based on small strain nearincompressibility. However, this can significantly reduce the stable time step. In such cases, ibetter to use one that results in ν = 0.49.
• When explicit analysis with automatic time step calculation iused for a MooneyRivlin material, the critical time step is governed by the dilatational wave speed. This is most fr
! As the material deforms, the bulk to shear modulus ratchange, because the instantaneous shear modulus is dependent on the amount of deformation. A value of the bulk modulus that corresponds to near incompressibility for small strains may not belarge enough to correspond to near incompressibility for large strains.
Ogden material model
! The Ogden material model is based on the following expre
9
1 2 31
3n n nnD
n n
W α α αµ λ λ λα=
⎛ ⎞⎡ ⎤= + + −⎜ ⎟⎣ ⎦
⎝ ⎠∑ (3.815)
herew nµ and nα are the Ogden material constants.
! This strain energy density expression assumes a totally incompressible material ( )3 1I = and is modified as explained below for plane strain, axisymmetric or 3D analysis. Plane stress analysis: The material is assumed to be totally incompressible. Therefore is zero and VW DW W= . A displacementbased finite element formulation is used, exactly as for the MooneyRivlin material model described above.
Plane strain, axisymmetric and 3D analysis: The material is modeled as compressible (that is, the bulk modulus is not infinite), but you can set the bulk modulus high so that the material is almost incompressible.
ref. KJBSection 6.6.2
3.8: Rubber and foam material models
ADINA R & D, Inc. 335
The Ogden strain energy density equation is modified by 1) substituting the reduced stretches for the corresponding stretches, 2) removing the condition 11 2 3λ λ λ = , and 3) adding the volumetric strain energy density
( ) ( )2 21 11 12 2V 1 2 3 3W Jκ λ λ λ κ= − = − (3.816)
where κ is the bulk modulus. The relationship between the pressure and the volumetric ratio is the same as for the MooneyRivlin material descripition. Either the u/p formulation (default) or the displacementbased formulation can be used. For comments about the u/p formulation, see the corresponding comments in the MooneyRivlin material description.
Selection of material constants: The Ogden material description used here has 19 constants: , , 1,...,9n n nµ α = and the bulk
o Chodulus. osing only , 0, 1, 2,3n n nµ α ≠ = the standard three Ogden material description is recovered.
mterm
mo n bThe small strain shear modulus and small strain Young=s dulus ca e written as (assuming κ = ∞ )
9
1
12 n n
nG µ α
=
= ∑ (3.817)
9
1
32 n n
nE µ α
=
= ∑ (3.818)
Th
analysis with automatic time step calculation is used for an Ogden material, the critical time step is governed by the
! The bulk modulus κ is used to model the compressibility of the
ese moduli must be greater than zero.
• When explicit
dilatational wave speed. This is most frequently an acceptable assumption since the material is almost incompressible.
Chapter 3: Material models and formulations
336 ADINA Structures ⎯ Theory and Modeling Guide
material for plane strain, axisymmetric and 3D analysis. For comments about the bulk modulus, see the corresponding comments about the bulk modulus in the MooneyRivlin material description.
3.8.1.3 ArrudaBoyce material model
! The ArrudaBoyce model is based on the following expression:
( )5
12 21
3i iiD i
i m
CW Iµλ −
=
⎡ ⎤= −⎢ ⎥
⎣ ⎦∑ (3.819)
in which 112
C = , 2120
C = , 311
1050C = , 4
197050
C = ,
5 673750519
= and in which there are two material constants: C µ is
the initial shear modulus and mλ is the locking stre ! The Arruda
tch.
Boyce material model is described in the following
l
reference:
ref. E.M. Arruda and M. C. Boyce, A threedimensionaconstitutive model for the large stretch behavior of rubber elastic materials, J. Mech. Phys. Solids, Vol,. 41 (2), pp 389412 (1993).
! This strain energy density expression assumes a totally incompressible material ( )3 1I = and is modified as explained below for plane strain, axisymmetric or 3D analysis.
Plane stress analysis: The material is assumed to be totally incompressible. Therefore VW is zero and DW W= . A displacementbased finite element formulation is used,for the MooneyRivlin material model described above Plane strain, axisymmetric and 3D analysis: The mmodeled as compressible (that is, the bu
exactly as .
aterial is lk modulus is not infinite),
ref. KJBSection 6.6.2
3.8: Rubber and foam material models
ADINA R & D, Inc. 337
o that the material is
sity equation is modified
but you can set the bulk modulus high salmost incompressible. The ArrudaBoyce strain energy denby: 1) substituting for the strain invariant 1I the reduced strain invariants 1J , 2) removing the condition 1I = , and 3) adding the 3
volum
etric energy term
( )2 1Jκ 33ln
2 2VW J⎡ ⎤−⎢ ⎥= −⎢ ⎥⎣ ⎦
(3.820)
here κ is the smallstrain bulk modulus. The relationship between w
the pressure and the volume ratio is
31p Jκ ⎛ ⎞
= −⎜ ⎟ (3.82132 J⎝ ⎠
)
ion,
default values of the material constants are
Either the u/p formulation (default) or the displacementbased formulation can be used. For comments about the u/p formulatsee the corresponding comments in the MooneyRivlin material description. When the u/p formulation is used, there should be at least one solution unknown. This is because the constraint equation used in the u/p formulation is nonlinear in the unknown pressures. Therefore equilibrium iterations are required for convergence, even when all of the displacements in the model are prescribed. Selection of material constants: The
12.6008µ = , 1mλ = , 63000κ = . It is likely that you will need to change these values in your analysis.
3.8.1.4 Hyperfoam material model
! The hyperfoam model is based on the following expression:
Chapter 3: Material models and formulations
338 ADINA Structures ⎯ Theory and Modeling Guide
( )1 2 3 31
13 1n n n n n
Nn
n n n
W Jα α α α βµ λ λ λα β
−
=
⎡ ⎤= + + − + −⎢ ⎥
⎣ ⎦∑ (3.822)
in which there are the material constants , , , 1,...,n n n n Nµ α β = .
he maximum value of N is 9.
A material model similar to the hyperfoam material model is described in the following reference:
ref. B. Storåkers, On material representation and constitutive branching in finite compressible elasticity, J. Mech. Phys. Solids, Vol,. 34(2), pp 125145 (1986).
this reference,
T
!
nβIn is the same for all values of n.
When it is required to compute the deviatoric and volumetric parts of the strain energy density, we use
!
/31 2 3 3
1
3n n n n
Nn
Dn n
W Jα α α αµ λ λ λα=
⎡ ⎤= + + −⎣ ⎦∑ (3.823)
( ) ( )/ 33 3
1
13 1n n
Nn
Vn n n
W J Jα αµα β
−
=
1nβ⎡ ⎤= − + −⎢ ⎥
⎣ ⎦∑ (3.824)
Notice that D VW W W= + . This decomposition of the strain energy density has the advantage that the stresses obtained from the
no deformations: deviatoric and volumetric parts separately are zero when there are
0
1 02ij
D D Dij
W WSε =
∂ ∂
0ijij ji ε =
⎛ ⎞= + =⎜ ⎟⎜ ⎟∂ε ∂ε
, ⎝ ⎠
0
0
1V V Vij
W WS⎛ ∂ ∂ 0
2ij
ijij ji
ε =ε =
⎞= +⎜⎜ ⎟ =⎟
∂ε ∂ε⎝ ⎠
3.8: Rubber and foam material models
ADINA R & D, Inc. 339
Notice that DW contains the volumetric part of the motion through the nJ α . Therefore /3
DWterm 33 is not entirely deviatoric.
Plane stress analysis: The material is not assumed to be totally inco lement formulation is used.
Plane strain, axisym not assu otally incompressible.
Because bo
mpressible. A displacementbased finite e
metric and 3D analysis: The material is med to be t
th DW a ntain the volumetric part of the motion, the mixed u/p formulation cannot be used with the hyperfoam rial A displacementbased formulation is used.
Selection of material constants: The hy erfoam material description used here has 27 constants:
nd VW co
mate .
p, , , 1,...,9n n n nµ α β = .
T e s train bulk modulus can w
h mall strain shear modulus and small s be ritten as
91G n n12 nµ α= ∑ (3.825)
=
9
1
13n n
nnκ β µ
=
⎛ ⎞= +⎜ ⎟⎝ ⎠
∑ α (3.826)
These moduli must be greater than zero, hence we note that nβ should be greater t When all of the
han 1/3. nβ are equal to each other β= , then the
Poissons ratio is related to β using
ν
1 2β
ν=
− (3.827)
aterial constants areThe default values of the m 1 1.85µ = ,
1 4.5α = , 1 9.2β = , 2 9.2µ = , 2 4.5α = − , 2 9.2β = . It is likely that you will need to change these values in your analysis.
Chapter 3: Material models and formulations
340 ADINA Structures ⎯ Theory and Modeling Guide
! The hyperfoam material model is generally used for highly compressible elastomers. the ratio of the bulk modulus to shear modulus is high (greater than about 10), the material is almost incompressible and we recommend that one of the other hyperelastic materials be used.
3.8.1.5 Curve fitting
! ou can have the AUI determine the material constants for the Moa curvefit to experimental stressstrain data.
! he AUI allows for data from ental cases as shown in Figure 3.81: simple tension, pure shear and equibiaxial tena combination of data from any two, or all three of these experimental cases is allowed.
squares sense is calculated. You must select the chosen order of the rmodel). Note: the total number of data points must be greater than the number of material constants to be determined.
ental case, you can enter either the stretch or engineering strain in the strain column and you enter the eng er see the definitions of stretch, engineering strain and engineering stress for the experimental cases.
If
YoneyRivlin, Ogden, ArrudaBoyce or hyperfoam materials by
T three experim
sion. Input from a single experiment from one of these cases or
A single set of constants representing a best fit in the least
app oximation (giving the number of constants used in the
For each experim
ine ing stress in the stress column. Refer to Figure 3.81 to
3.8: Rubber and foam material models
ADINA R & D, Inc. 341
X
X
Y
Y
Z
Z
L
L
L
L
Original area A
Original area A
Deformed area A/
Deformed area A/
= (L+ )/L
= (L+ )/L
1 2 31/2= ; = =
1 2 3= ; = 1; = 1/
0 p1 = F/A
0 p1 = F/A
F
F
a) Simple tension test (incompressible material)
b) Pure shear test (incompressible material)
e= /L
e= /L
Figure 3.81: Experimental test cases for rubber and foam materials
Chapter 3: Material models and formulations
342 ADINA Structures ⎯ Theory and Modeling Guide
X
Y
Z
Original area A
Deformed area A/
1 2 32= ; = ; =
F
F
c) Equibiaxial tension test (incompressible material)
L
L +
= (L+ )/L e= /L
0 p1 = F/A
Figure 3.81: (continued)
e strain energy density for total incompressibility is given in Eq. 3.88 with the material constants D1 and to zero.
Given aterial constants, Ci, then M is equal to 2, 5 and 9 for first, second and third order approximations, respectively.
The principal stress in the direction of principal stretch
MooneyRivlin model: Th
D2 set M nonzero m
1λ is given by
( )0 0 , ( 1,..., )p1 p1 iW C i Mσ σ λλ
∂= = =
∂ (3.828)
Notice that this is the engineering principal stress, that is, the force divided by the original area of the specimen. Thus, we have an expression for the principal stress that is linear in terms of the material constants Ci, and dependent on the principal stretch determined from the allowed experimental cases. Hence, given a set of data points ( ),k kλ σ from one or more of the test cases, we can establish the least squares fit by a curve of the form given by
q. (3.828). Consider the sum of discrete error terms at the given data points
of one of the experimental test cases (simple tension, pure shear or
E
3.8: Rubber and foam material models
ADINA R & D, Inc. 343
quibiaxial tension)
(3.829)
where
e
( )( )2
1
N
k p1 kk
S σ σ λ=
= −∑
p1σ is the approximating function dependent on the constants C (i = 1,...,M) given by Eq. (3.828). We can expand thi
11 1
k k
j j j
j kS σ σ λ
= =
= −⎜ ⎟⎝ ⎠
∑ ∑ (3.830)
j denotes the experimental test case (simple ar j = 2, equibiaxial tension j = 3). Note: the
f
is sum to include data from all three test cases, i.e.:
3 2jN⎛ ⎞( )( )
where the superscript tension j = 1, pure sheorm of 1 jσ differs for j = 1, 2, 3 due to the different principal
stretch terms λ2 and λ3 . The least squares fit is determined by minimizing S in Eq. (3.8
0) with respect to the constants C ,, which gives M algebraic
incompressibility, given by Eq. (3.815). Thus, for an Mterm )
3 iequations for the constants Ci (i = 1,...,M).
Ogden model: The strain energy density function is, for total
Ogden model, there are 2 M constants, i.e., the pairs ( ,i iµ α for i
hown in the principal stress
= 1,...,M. Assuming the principal stretches for the three test cases s
Figure 3.81, the following form for 0 p1σ can be derived:
i
( )1 10
i i
Mc
p1 iσ µ λ λ1
α − α −= −∑ =
(3.831)
wi simple tension, 1.0 for pure shear, 2.0 for equibiaxial tension.
th c = 0.5 for
Equation (3.831) is linear in terms of the coefficients iµ , but nonlinear in terms of the exponents iα . A nonlinear least squares
Chapter 3: Material models and formulations
344 ADINA Structures ⎯ Theory and Modeling Guide
ion for the constants algorithm could be employed to iterate to a solut
( ),i iµ α . However, in our curvefitting implementation
in the AUI, you specify the desired exponent terms iα (with tdefault i
he α = i), and the least squares fit of the resulting function
yields the corresponding coefficients iµ . You can select the ger number of terms M desired (1 # M # 9). The assumption of inte
values for iα has in practice given excellent fits to experimental data for rubber materials. Nevertheless, you can use othefor i
r values α to possibly give a better fit for specific materials.
ArrudaBoyce model: The strain energy density function is, for total incompressibility, given by Eq. (3.819). There are only two material constants, µ and mλ .
Assuming the principal stretches for the three test cases shoin Figure 3.81, the following forms for the principal stress 0 p1
wn σ
can be derived:
Uniaxial tension:
( )5
2 1 20 1 12 2
1
2 , 2iip1 i
i m
iC I I 1σ µ λ λ λ λλ
− −−
=
= − = +∑ (3.
Pure shear:
− 832)
( )5
3 1 20 1 12 2
1
2 ,ii 2 1ii m
iC I Iλ λ λ λλ
− − −−
=p1σ µ= − = + +∑ (3.833)
Equibiaxial tension:
( )5
5 1 20 12 2
1
2 ,iip1 i
i m
iC I I 41 2σ µ λ λ λ λ
λ− − −
−=
= − = +∑ (3.834)
The AUI uses a nonlinear curvefitting algorithm to determine µ
mλ . and
Hyperfoam model: The strain energy density function is given by Eq. (3.822). However, we seek a curvefit for which all of the nβ terms are equal. Thus, for an Mterm hyperfoam model, there are
3.8: Rubber and foam material models
ADINA R & D, Inc. 345
2 1M + material constants, i.e., the constants ( ),n nµ α for n =
1,...,M, and β . First β is determined using information about the lateral stretch
Tλ . There are two options for entering the lateral stretch information:
1) Specify the Poissons ratio. β is then calculated using (3.827).
2) Enter the lateral stretches as part of the stressstrain data input. Then β is determined with the use of the following formulas:
Uniaxial tension:
1 2T
ββλ λ
−+= (3.835)
Pure shear:
1T
ββλ λ
−+= (3.836)
tension:
Equibiaxial
21
T
ββλ λ
−+= (3.837)
Since each of these formulas can be also written in the form ln ( ) lnT fλ β λ= − , a linear curvefit is performed to obtain β . Next, the constants ( ),n nµ α
principal stretches for the three test cases shown in Figure 3.81, for the principal stress
are determined. Assuming the
the following form can be derived:
0 p1σ
( )0 31
1 nn
M
p1 nn
J α βσ µ λλ
−α
=
= −∑ (3.838)
where 2
3 TJ λλ= for uniaxial tension, 3 TJ λλ= for pure shear and
Chapter 3: Material models and formulations
346 ADINA Structures ⎯ Theory and Modeling Guide
T2
3J λ λ for e 38) iconstants
= quibiaxial tension. (3.8 s then used as the basis of a nonlinear curvefitting algorithm to obtain the ( ),n nµ α .
nd Ogden Notes for curvefitting for the MooneyRivlin a
material models: The system of equations used to solve for the
function
least squares fitted constants is likely to be illconditioned. Direct use of Gaussian elimination may yield constants with large magnitudes, which are balanced to cancel out when the fitted
σ is evaluated. Furthermore, and perhaps more
. stem
f
mao
gular
.E., Malcom, M.A. and Moler, C.B.,
Computer Methods for Mathematical Computations, ,
The essence of the SVD algorithm is that combinations of the basis
near zero eigenvalues of the matrix system are eliminated.
ngular terms always increases the error (in the least squares sense). However, the inc
allest eigenvalues until a fitted function is obtained which gives a monotonic increase in stress with strain.
Depending on data, the number of terms eliminated from a higher order approximation may be large, giving a poor fit, even though the monotonicity criterion has been achieved.
importantly, there is no guarantee that the fitted function has physically feasible behavior outside the range of the data supplied
The reason for the illconditioning of the least squares symatrix is that the data may be fitted equally well by two or more othe "basis" functions or combinations thereof, thus rendering the
trix nearly singular. The AUI provides an alternative matrix solution algorithm t
the GaussJordan elimination technique, namely that of SinValue Decomposition (SVD) described in
ref. Forsythe, G
Chapter 9, Prentice Hall, Inc. Englewood Cliffs, NJ1977.
functions which correspond to
Successive elimination of the near si
rease in error is often small compared to the overall improvement in the fitted solution the constants obtained are usually smaller in magnitude and physically reasonable.
The SVD algorithm is controlled by successively eliminating combinations corresponding to the sm
3.8: Rubber and foam material models
ADINA R & D, Inc. 347
As an example, consider the curve fitting results illustrated in Fig. 3.82. Fig. 3.82(a) shows the fitted curves for a 9term ( iα = i, i = 1, . . ., 9) Ogden model approximating a set of data points for a simple tension test. The sum of error squares S is given in the key of the graphs.
STRETCH
STRETCH
STRETCH
0.0
0.0
0.0
0.5
0.5
0.5
1.0
1.0
1.0
1.5
1.5
1.5
2.0
2.0
2.0
2.5
2.5
2.5
3.0
3.0
3.0
3.5
3.5
3.5
ST
RE
SS
ST
RE
SS
ST
RE
SS
2.0
2.0
2.0
2.0
2.0
2.0
Input data
Input data
Input data
Ogden (9term; Gauss) S=1.56E03
Ogden (9term; SVD(1)) S=1.61E03
Ogden (9term; SVD(2)) S=1.75E03
Ogden (6term; Gauss) S=1.94E03
Ogden (6term; SVD(1)) S=2.53E03
Ogden (3term; Gauss) S=2.76E03
b)
a)
c)
Figure 3.82: Curve fitting results for Ogden material model
While the Gaussian elimination curve has the minimum S, its
solution is asymptotically unacceptable. Two terms were removed to give a which still gives monotonic increasing solution (SVD(2))
Chapter 3: Material models and formulations
348 ADINA Structures ⎯ Theory and Modeling Guide
good approximation to the input data. Fig. 3.82(b) shows the tted curves for a 6term Ogden model for the same input data. A
monotonic increasing solution was obtained after removal of one term (SVD(1)), while the Gauss elimination solution was not asymptotically feasible. For the 3term Ogden fit to the data, the Gauss elimination solution was accepted since it provided a monotonic increasing curve, as shown in Fig. 3.82(c).
Notes for curvefitting for the ArrudaBoyce and hyperfoam material models: The MarquardtLevenberg algorithm from the following reference is used in the curve fitting:
ref. W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes the Art of Scientific Computing (Fortran Version), Cambridge University Press, pp 521528 (1990).
Ge nique
a) Use a lower order approximation. b) Use a different material model
ion (λ < 1).
Spcur e cur ch of the req
3.8.2 Viscoela
! Viscoelastic effects can be included in the MooneyRivlin, Og
afi
neral comments: It can be seen that the curve fitting techprovided gives a valuable tool, but one which must be used with caution. Always check the resulting fitted function. If a physicallyreasonable fit is not achieved, we recommend the following strategy
c) Provide more data points, covering a greater range of anticipated strains, including values for compress
ecification of input: All of the information required for the ve fitting is contained in a data set of type curvefitting. Thvefitting data contains the numbers of the data sets for ea three test cases, the order of the curve fit and other informationuired for the curve fit.
stic effects
den, ArrudaBoyce and hyperfoam material models.
3.8: Rubber and foam material models
ADINA R & D, Inc. 349
The viscoelastic model used is due to Holzapfel, see the following
to elastomeric structures, Int. J. Num. Meth. Engng., Vol. 39, pp 39033926, 1996.
ref. G. A. Holzapfel, Nonlinear solid mechanics. A
continuum approach for engineering. John Wiley &
Lemaitre (ed.), Handbook of Materials Behavior
1, pp 10571071. In for finite strain viscoelasticity. Eq n in Fig 3.8chains. A generic chain is denoted with superscript
references:
ref. G. A. Holzapfel, On large strain viscoelasticity: continuum formulation and finite element applications
Sons, Chichester, pp 278295, 2000.
ref. G. A. Holzapfel, Biomechanics of soft tissue, in
Models: Nonlinear Models and Properties, Academic Press, 200
the following, we give a brief discussion of the Holzapfel model
uivalent 1D model: The equivalent 1D model is show3. It is the same as a generalized Maxwell model with many
α , as shown in the figure.
E
E1
1
E
Strain
g
Strainstress
e,
Strain
...
The spr
Figure 3.83: Equivalent 1D model for viscoelastic effects
ing E∞ is equivalent to the elastic stiffness of the del. Each chain contains a spring with stiffness Emo α and dashpot
Chapter 3: Material models and formulations
350 ADINA Structures ⎯ Theory and Modeling Guide
with viscosity αη . (Note that the superscripts and ∞ α do not denote exponentiation.) The strain in each chain is the sum of the strain in the spring gα and the strain in the dashpot αΓ . The observed stress is qα
α
σ σ ∞= + ∑ (3.839)
where E eσ ∞ ∞ q E gα α α α αη= = Γ# is = is the elastic stress and
the stress in chain α . Using the definition E
αα
α
ητ = and the
assumption E Eα αβ ∞= , the following expression is obtained:
1q qα α αα β σ
τ∞+ =# # (3.840)
Assuming that q0 0α = , (3.840) can be written in convolution form as
0
expt
tt tq dα αα β σ
τ′ ∞′−⎛ ⎞ ′= −⎜ ⎟
⎝ ⎠∫ # t (3.841)
from which the total stress is
0
1 expt
tt tE e dtαα
α
σ βτ
′∞ ′⎡ − ⎤⎛ ⎞ ′= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∑∫ # (3.84
Evidently the relaxation modulus is
2)
( ) 1 exp tE t E αα
α
βτ
∞ ⎡ ⎤⎛ ⎞= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∑ which is a Prony series
expression. The dissipation in dashpot α is
3.8: Rubber and foam material models
ADINA R & D, Inc. 351
1 q( )D q q e g q e
Eα α α α α α
αβ ∞= Γ = − = −⎜ ⎟⎝ ⎠
## # # # (3α⎛ ⎞
.843)
and the total dissipation is D Dα
α
= ∑ . In the above, the
for each chain are and .α ατ β viscoelastic material constants Potentialbased 1D model: The 1D model can be written in terms of
a potential as follows:
( ) ( )e gα α
α
∞Ψ = Ψ + Ψ∑ (3.844
ere
)
wh 21( )2
e E e∞ ∞Ψ = is the strain energy of the elastic chain
and ( )21( )2
g E gα α α αΨ = is the strain energy in the spring of
chain α . In terms of αβ , ( ) ( )g gα α α αβ ∞Ψ = Ψ . The 1D
del is recovered by defining mo
fixede α
σΓ
∂Ψ=
∂,
fixede
qαα
∂Ψ= −
∂Γ (3.845a,b)
45a) imply Notice that (3.839) and (3.8
fixede gα
qα α
α ∂Ψ ∂Ψα= = .
Γ∂ ∂
Finpodef
ite strain model: The finite strain model is derived from the tentialbased 1D model as follows. The elastic potential is ined as
( ) ( )W∞Ψ =ε ε (3.846) where ( )ijW ε is the strain energy density from the elastic part of
Chapter 3: Material models and formulations
352 ADINA Structures ⎯ Theory and Modeling Guide
the material model. The potential of each chain α is defined as
( ) ( )( )( )
, usage=deviatoric
, usage=volumetric
D ij
V ij
W G
W G
α α
α α
β
β
=
=
(3.847a,
which the usage flag (which is a userinput flag) determines
, , usage=combinedij ij ijW Gα α α αε βΨ Γ =
b,c)
in whether the entire elastic strain energy density, deviatoric strain energy density or volumetric strain energy density is taken for chain α . Here ijGα is analogous to the strain in the 1D spring gα ,
and we assume ij ij ijGα αε= − Γ . Note that with this definition of
ijGα , we have
, fixedij
αΓ
ijij
Sε
∂Ψ=
∂fixed fixedij ij
ijij ij ij
QGα
αα
α αεεΓ
∂Ψ ∂Ψ ∂Ψ= = − =
∂ ∂Γ ∂
wh
(3.848a,b)
ere ijS are the 2nd PiolaKirchhoff stresses and ijQα is analogou
to the stress q
s α .
Following exactly the same arguments as in the 1D case, we obtain
1
ij ijQ Q Sα α αα β ijτ
∞+ =# # (3.849)
Assuming that 0 0ijQα = , (3.850) can be written in convolution form as
0
expt
tij ij
t tQ Sα αα β
τ′ ∞′−⎛ ⎞ ′= −⎜ ⎟
⎝ ⎠∫ # dt (3.850)
and (3.850) can be numerically approximated using
3.8: Rubber and foam material models
ADINA R & D, Inc. 353
( )1 exp
expt t t
ttQ Q t t t
ij ij ij ijS St
αα α αβ+∆
∆⎛ ⎞− −⎜∆⎛ ⎞ ⎝= − + α αα
α
ττ
τ
+∆⎟⎠ −⎜ ⎟ ∆⎝ ⎠
.
(3.851) (3.851) is exact when ijSα does not change during the time step, and is more accurate than the formula given by Holzapfel:
( )exp exp2
t t t t t tijQ Q ij ijS S ij
t tα α α α αα αβ
τ τ+∆ +∆∆ ∆⎛ ⎞ ⎛ ⎞= − + − −⎜ ⎜
⎝ ⎝
expecially when
⎟⎠
⎟⎠
(3.852)
tατ
∆ is large.
alculations: If the dissipation is required, it is sing
Dissipation ccalculated u
( ) ij ij ij ijQ Q ijD Gα α α ε= Γ = −# #α α# (3.853)
where
2
rs ijij rs
G QG G
αα α
α α
∂ Ψ=
∂ ∂# # (3.854)
is used to compute the unknown rsGα# from the known ijQα# . If usage=combined,
2 2
ijrsij rs ij rs
W CG G
αα α
α α β βε ε
∂ Ψ ∂= =
∂ ∂ ∂ ∂. (3.855)
t the strain state where the tensor C is evaluated aijrs ijGα . The dis sipation calculation requires the solution of a set of simultaneouslinear equations of at most order 6 (in the 3D case) at each integration point.
Chapter 3: Material models and formulations
354 ADINA Structures ⎯ Theory and Modeling Guide
If usage=deviatoric,
( )2 2
DD ijrs
ij rs ij rs
W CG G
αα α
α α β βε ε
∂ Ψ ∂= =
∂ ∂ ∂ ∂ (3.856
)
where the tensor ( )D ijrsC is evaluated at the strain state ijGα . Here,
the dissipation calculation requires a singular value decomposition of ( )D ijrs
C , since ( )D ijrsC has a zero eigenvalue. A similar
situation applies when usage=volumetric, except that the corresponding material tensor has only one nonzero eigenvalue. The procedure given in (3.853) to (3.856) is only approximate,
ental assumption ij ij ijGα αε= − Γsince the fundam strictly speaking only holds for small strain analysis. Restrictions and recommendations: The allowed values of the usage flag depend upon the material model and finite element type, s shown in Table 3.82.
In view of the restrictions, we recommend that usage=combined
e used in conjunction with the hyperfoam material model, and Mooney
imetemperature superposition: The preceding derivation
nt. One method of including the effects of temperature is e method of timetemperature superposition.
In timetemperature superposition, the actual tim is replaced
a
bthat usage=deviatoric be used in conjunction with theRivlin, Ogden and ArrudaBoyce material models. Tassumes that the viscoelastic response is not temperaturedependeth
e tby the reduced time ζ . The relationship between the actual time
dan reduced time is given by
1( )t
T
ddt aζ
θ= (3.857)
where tθ is the temperature and ( )t
Ta θ is the shift function
.
3.8: Rubber and foam material models
ADINA R & D, Inc. 355
MooneyRivlin, Ogden, ArrudaBoyce
hyperfoam
Table 3.82: Allowed values of the usage flag
Plane stress1
Plane strain, axisymmetric, 3D2
Plane stress3
Plane strain, axisymmetric, 3D
usage=combined yes no yes yes usage=deviatoric yes yes no yes usage=volumetric no no no yes 1) Usage cannot be equal to volumetric. This is because the material is assumed to be fully incompressible, hence the volumetric strain energy density is zero. 2) When the u/p formulation is used, the usage flag cannot be com
formulation. 3) The only bined. This
must be zero, and in the Holzapfel finite strain viscoelastic model, the only way that this can happen is if
bined or volumetric. This is because the modification to the volumetric stresses caused when the usage flag is combined or volumetric is not taken into account in the u/p
allowable value of the usage flag is comis because the outofplane stress component xxS
xxS ∞ is zero.
Evidently
1tt
0 ( )Ta θtdtζ ′
′= ∫ (3.858)
The shift function used here is the WLF shift function,
110
2
log ( )TaC
θ( )t
reftt
ref
C θ θθ θ−
(3.859) = −+ −
where refθ is the reference temperature and are material 1C , 2C
Chapter 3: Material models and formulations
356 ADINA Structures ⎯ Theory and Modeling Guide
constants. Notice that as ddtζ
tθ increases, ( )tTa θ decreases and
increases.
40) becomes
For the viscoelastic model used here, the differential equation of the 1D model (3.8
1dq dq
αα α
d dσβαζ τ ζ
∞
+ = (3.860)
and using (3.857), (3.860) can be written as
1( )T
q qa
α α αα β σ
θ τ∞+ =# # (3.8 61)
It is seen that the effect of temperature is to modify the time constants. As the temperature increases, the modified time constants become smaller, that is, the material relaxes more
. The convolution equation of the finite strain model becomes
quickly
0
expt t
ijij
d SQ d
d
ζα α
α
ζ ζ β ζτ ζ
′ ∞′−⎛ ⎞ ′= −⎜ ⎟⎝ ⎠∫ (3.862)
and (3.862) is numerically approximated by
( )1 exp
expt t t t t tij ij ij ijQ Q S
α
Sα α α α αα
α
ζζ τβ ζτ
τ
+∆ +∆
∆⎛ ⎞− −⎜ ⎟∆⎛ ⎞ ⎝ ⎠= − + −⎜ ⎟ ∆⎝ ⎠
(3.863) The only additional consideration is to calculate ζ∆ , and this is done using
3.8: Rubber and foam material models
ADINA R & D, Inc. 357
1
( )tTt
dta
ζθ′
t t+∆
′∆ = ∫ (3.864)
This integration is performed numericall ln ( )Ta θ y assuming that varies linearly over the time step. Heat generation: A userspecified fraction of the energy dissipated by the viscoelastic model can be considered as heat generation. This heat generation can cause heating in a TMC (thermomechanicalcoupling) analysis. Specification of input: To add viscoelastic effects to the rubberlike material model, you need to define the viscoelastic data using a data set of type rubberviscoelastic, then specify the number of the rubberviscoelastic data set in the rubberlike material model. The rubberviscoelastic data set includes:
ed.
< A table with one row for each chain. Each
< A flag indicating whether the WLF shift function is us < The shift function material constants 1C , C 2
row contains αβ , ατ , the heat generation factor (fraction of dissipation
e is 0.0), and the
on the number of chains permitted. The usage flag can be different for each chain.
In order to include timetemperature superposition, the material mu ! It is seen that the dissipation calculation can be quite expensive. Furthermore the dissipation is not rTherefore it is the default in ADINA to not perform the dissipation
considered as heat generation, default valuusage flag (default value is deviatoric). There is no restriction
st have a TRS temperature dependence.
equired for the stress solution.
calculation. The dissipation is only calculated for the chain α when the heat generation factor is nonzero.
Chapter 3: Material models and formulations
358 ADINA Structures ⎯ Theory and Modeling Guide
3.8.3 Mullins effect
When rubber is loaded to a given strain state, unloaded, then reloaded to the same strain state, the stress required for the reloading is less than the stress required for the initial loading. This phenomenon is referred to as the Mullins effect. The Mullins effect can be included in the rubberlike materials. The material model used is the one described in the following reference:
We briefly summarize the main concepts below. Fig 3.84 shows the Mullins effect in simple tension. On initial loading to force , the specimen follows curve abc. When the load is removed, the specimen follows the
ng curve cda. On reloading to force , the specimen follows the reloading curve adc, and on further loading to force
, the specimen follows the loading curve cef. When the load oved, the specimen follows the unload
on reloading to force
ref. R.W. Ogden and D. G. Roxburgh, A pseudoelastic model for the Mullins effect in filled rubber, Proc. R. Soc. Lond. A (1999) 455, 28612877.
cF the forcedeflection
unloadi cF
fFis rem ing curve fga, and,
gF , the specimen follows the reloading curve agf.
Force
a
b
c
d
ef
g
h
Deflection Figure 3.84: Mullins effect loadingunloadingreloading curve
s
Note that any permanent set associated with the Mullins effect
3.8: Rubber and foam material models
ADINA R & D, Inc. 359
is not included in the OgdenRoxburgh model used here. The OgdenRoxburgh model, as implemented in ADINA, uses the following strain energy density exp
(3.865a,b)
where
ression:
( ) ( ), MooneyRivlin,Ogden,ArrudaBoyce= ( ) ( ), hyperfoam
D ij
ij
W WW
η ε φ η
η ε φ η
= +
+
&
( )ijW ε is the total elastic strain energy density, ( )D ijW ε is
the deviatoric elastic strain energy density, η is an additional solution variable describing the amount of unloading and ( )φ η is the damage function. is referred to as the pseudofunction. In our implementation, the deviatoric strain energy density is used for the (almost) incompressible materials and the total strain energy density is used for compressible materials. For ease of writing, we discuss only the case of compressible materials; for incompressible materials, replace by
W& energy
W DW in the equations below. η is computed as
( )1 11 erf mW Wr m
η ⎡ ⎤= − −⎢ ⎥⎣ ⎦ (3.866)
where erf( )x is the error function
( )2
0
2erf( ) expx
x u duπ
= −∫ (3.867)
is the maximum value of encountered during the deformation history and and are material constants.
mW Wm r
( )φ η is defined by
( )d W
dφ η
η= − (3.868)
and is computed by numerical integration of
( ) Wφ η η= −# # . For a
Chapter 3: Material models and formulations
360 ADINA Structures ⎯ Theory and Modeling Guide
given value of , there is a minimum value of mW η computed as
11 erf m
mmW
rη ⎡ ⎤= − ⎢ ⎥⎣ ⎦
(3.869)
The value of ( )φ η at mη η= is written ( )mφ η . (Note: the subscript m in the term means maximum, but the subscript m in the term
mW mη means minimum.) The time rate of change of
( )mφ η can be written as
( ) 1( ) 1 erf mm m m mWWW
r mφ η η ⎡ ⎤= − = ⎢ ⎥⎣ ⎦
# (3.870) # #
Physically, ( )mφ η is interpreted as the dissipation.
During loading, mW W= , 0mW ># and 1η = . Therefore
( ) 0φ η =# and during loading.
During unloading or reloading, and
( ) 0mφ η >#
mW W> , 0mW =#
1mη η≤ < . Therefore ( ) 0φ η ≠# anor reloading.
rs. It is seen that, for an unloadingreloading loop in which , the initial slope of the reloading curve is reduced by
d ( ) 0mφ η =# during unloading
Material constants m and r do not have any direct physical significance. However Fig 3.85 shows the dependence of an unloadingreloading curve in simple tension on these paramete
mW m>> the factor
11r
− . must therefore be greater than 1.
r
3.8: Rubber and foam material models
ADINA R & D, Inc. 361
Force
Slope ka
Slope kb
Wm = strain energy density at b
Fb
Slope
Deflection
11 erf m
a
Wk
r m
! "
Slope22 1
b bk Frm#
Figure 3.85: Dependence of reloading curve on Mullins effect
material constants It can also be shown that the dissipation of a loadingunloading cy le, as shown in Fig 3.86, can be written as
c
2
01E
m mW W Wmφ σ erf 1 exp mdeA r m m mπ
⎡ ⎤⎞⎞⎞⎛ ⎛⎛ ⎞ ⎛⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟= = − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
∫
(3.871)
mA
W deσ= ∫ (3.872)
where
C
0
0σ is the engineering stress, e is the engineering strain. Therefore, given φ and mW from two loadingunloading cycles of different amplitude, m and r can be computed.
Chapter 3: Material models and formulations
362 ADINA Structures ⎯ Theory and Modeling Guide
0=Force/original area
Dissipation = shaded area
e=displacement/original length
A
E
B
C
D
Figure 3.86: Dissipation in Mullins effect in a loadingreloading
cycle
in a TMC (thermomechanicalcoupling) analysis. Specification of input: To add Mullins effects to the rubberlike material model, you need to define the Mullins data using a data set of type rubberMullins, then specify the number of the rubberMullins data set in the rubberlike material model. The rubberMullins data set includes: The .
tion o red
3.8.4 Orthotropic effect
A 2D network of orthotropic fibers can be included in the rubber
Heat generation: A userspecified fraction of the energy dissipated by the Mullins effect model can be considered as heat generation. This heat generation can cause heating
material constants r and m
The heat generation factor (frac f dissipation consideas heat generation). The default value is 0.
like material descriptions, see Figure 3.87. The directions of the fibers are described by the normal vectors an , bn , with
omponents ( )n and ( )n respectivelyb i. These normal vectors c a i
are defined in the undeformed configuration, and can change directions due to material deformations.
3.8: Rubber and foam material models
ADINA R & D, Inc. 363
nanb
Figure 3.87: Network of orthotropic fibers The orthotropic fibers are included by appending to the strain energy density the following additional term
( ) ( ) 2 212 4 2 6
22kexp 1 exp 1 2O
kW k J k J⎡ ⎤ ⎡ ⎤= − + − − (3.873)
where
⎣ ⎦ ⎣ ⎦
1/3 1/3
4 4 3 6 6 3,J I I J I I− −= = , ( ) ( )4 ij a ai jI C n n= ,
( ) ( )6 b bi jijI C n n= C is the CauchyGreen deformation tensor,
Gasser, R. W. Ogden, A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models, Journal of Elasticity, 61: 148, 2000.
We note: The normal directions and are defined as shown in
igure 3.88. In 2D analysis, the figures show how and are defin
, ij
and 1 2,k k are material constants. This material description is described in the following
reference:
G. A. Holzapfel, T. C.
an bn
an bnFed in terms of the input material axis angle β and the offset
angles aβ , bβ . and lie in the plane of the element. In x)
a b
axisymmetric analysis, it is also allowed to set bn to the hoop (direction. In 3D analysis, the figure shows how an and bn are
n n
Chapter 3: Material models and formulations
364 ADINA Structures ⎯ Theory and Modeling Guide
erms of the material directions and and lie
a b . an bndefined in tin the plane defined by a and b . an and bn need not be perpendicular to each other.
a) 2D element, hoop option not used
na
a
b
nb1
a
b2
3
4
Figure 3.88: Orientation of orthotropic axes
b) 2D element, hoop option used,
= xnb
na
a
b
y
z1
a
2
3
4
Figure 3.88: (continued)
3.8: Rubber and foam material models
ADINA R & D, Inc. 365
c) 3D element, , lie in the  planen n a ba b
na
nb
a
b
c
1
5
87
a
b
2
3 4
Figure 3.88: (continued) 4I is interpreted as the square of the stretch of the fiber initiallorientated in direction n ;
y
a 6I is interpreted as the square of the stretch of the fiber initially orientated in direction ; see Ref KJB, Example 6.7. is interpreted as the deviatoric part of
bn
4J 4I ; 6J is interpreted as the deviatoric part of 6I . It is assumed that the material is nearly incompressible. When 4 1J < , the material is in compression in direction ; wh
anen 6 1J < , the material is in comp ion bn .
ove the orthotropic effect when the m terial is in ession in either of the fiber direc .
Specification of input: To add orthotropic effects to the rubberlike material model, you need to define the orthotropic data usindata set of ty
ression in direct It is possible to rem acompr tions
g a pe rubberorthotropic, then include the number of the
rubberorthotropic data set in the rubberlike material model
Chapter 3: Material models and formulations
366 ADINA Structures ⎯ Theory and Modeling Guide
The rubberorthotropic data set includes: description.
< The material offset angles aβ , bβ (default values are 0). < The material constants 1k , 2k . < A flag indicating whether the orthotropic effect is removed
iber directions (the default is to remove the orthotropic effect when the material is in compression).
he default is that x (hoop) direction).
3.8.5 Thermal strain effect
terial is temperaturedependent, you can include a th rmal expansion. The coefficient of thermal
expansion is
when the material is in compression in either of the f
< A flag indicating whether direction bn is the x (hoop)
direction (t bn is not the
When the macoefficient of e
( )α θ , where θ is the temperature. Notice that the of thermal expansion is temperature dependent.
The thermal strain is calculated as
coefficient
( ) ( )0 0( ) ( )th REe θ − F REFα θ θ α θ θ θ= − − (3.874)
where
0θ is the initial temperature and REFθ is the material perature. This is the same formula as is used for the
oelastic materials, see Section 3.5. rmal strain is assumed to be isotropic.
al strain is nonzero, the deformation gradient ed to be decomposed into a thermal deformation and a mechanical d or ation gradient , using
reference temother therm The the When the thermX is assumgradient ef m
thX mX
m th=X X X (3.875) The thermal deformation gradient is
3.8: Rubber and foam material models
ADINA R & D, Inc. 367
(1 )th the= +X I (3.87 therefore th
6)
e mechanical deformation gradient is
the mechanical CauchyGreen deformation tensor is
e −= +C C (3.878)
1(1 )m the −= +X X (3.877)
2(1 )m th and the mechanical GreenLagrange strain tensor is
( )2 21(1 ) 1 (1 )m e e− −= + − − +ε ε I (3.82th th 79)
d thermal strains, s in small strain analysis. However, we do not assume that the
thermal strains are small. The strain energy densities are computed using the mechanical
eformations. This is done by computing all invariants and al
hhoff stresses are obtained by differentiating density with respect to the total strains. Since the
strain energy density is a function of the mechanical strains, we obtain
For small thermal strains, (3.879) reduces to m the≈ −ε ε I , so that he strains are nearly the sum of the mechanical ant
a
dstretches using the mechanical deformations, e.g. the mechanicCauchyGreen deformation tensor.
The 2 the strain energy
nd PiolaKirc
( )( )2
1ij
W WS⎛ ⎞∂ ∂
2 ) ( )
2 ) ) ( ) )
112 ) ( )
ij ji
m ab ij m ba ji
thm ij m ji
W We
ε
ε ε ε
ε ε−
= +⎜ ⎟⎜ ⎟∂( ∂ ε⎝ ⎠
= +⎜ ⎟⎜ ⎟∂( ∂( ∂ ε ∂(⎝ ⎠⎛ ⎞∂ ∂
= + +⎜ ⎟⎜ ⎟∂( ∂⎝ ⎠
(3.880)
ith this definition, the 2nd PiolaKirchhoff stresses are conjugate
to the GreenLagrange strains.
) )1 m ab m baW Wε ε⎛ ⎞∂( ∂(∂ ∂
W
Chapter 3: Material models and formulations
368 ADINA Structures ⎯ Theory and Modeling Guide
3.8.6 TRS material
The rubberlike materials can include the TRS (thermorheologically simple) material assump n. In this assumption, the material properties are independent of temperature, bu ture as described in the viscoelastic msuperposition). TRS material requires a rubbertable of type TRS. The number of the rubbertable data set is specified as part of the rubberlike material description. The rubbertable of type TRS data set is a table with a different temperature for each row of the table. Each row of the table contains the coefficient of thermal expansion (enter 0 if there are no thermal effects).
3.8.7 Temperaturedependent material
lly temperaturedependent material is done using a rubbertable. The rubbertable has different forms depending upon whether the MooneyRivlin, Ogden, ArrudaBoyce or hyperfoam relationships are used, hence there is a rubbertable of type MooneyRivlin, a rubbertable of type Ogden, etc. For brevity, we discuss only the case of a MooneyRivlin material. A MooneyRivlin material that is fully temperaturedependent requires a rubbertable of type MooneyRivlin. The number of the rubbertable data set is specified as part of the MooneyRivlin material description. The rubbertable of type MooneyRivlin data set is a table with a different temperature for each row of the table. Each row of the table contains
< The coefficient of thermal expansion. < The material properties (C1 to C9, constants D1 to D2 and the bulk modulus κ).
tio
t the viscoelastic time constants are functions of temperaaterial effect (timetemperature
A
The rubberlike materials can be fully temperaturedependent. Specification of input: The input of the material properties for a fu
3.8: Rubber and foam material models
ADINA R & D, Inc. 369
< The curvefitting number (0 if curvefitting is not used). < The rubberviscoelastic data set number, if the model inclu < The rubberMullins data set number, if the model includes Mullins effects. < The rubberorthotropic data set number, if the model includes orthotropic effects.
You can specify a different curvefitting number for each temperature. You can specifiy a different rubberviscoelastic data set number, rubberMullins data set number or rubberorthotropic data set number for each temperature. In this way, the material constants for the viscoelastic, Mullins or orthotropic effects can be temperaturedependent. However, if one row of the rubbertable has viscoelastic data, all rows must have viscoelastic data. Similarly, if one row of the rubbertable has Mullins or orthotropic data, all rows must have this data. Solution process: The strain energy density is computed by interpolation from the strain energy densities of the bounding temperatures as follows: Let
des viscoelastic effects.
θ be the current temperature, 1θ be the temperature in the rubbertable just below the current temperature and 2θ be the temperature in the rubbertable just above the current temperature. Then the program computes the strain energy and , and the derivatives of the strain energy densities, using the material properties for
densities 1W 2W
1θ and 2θ respectively. Finally the program computes the strain energy density and its derivatives using linear interpolation, for example
2 11 2
2 1 2 1
W W Wθ θ θ θθ θ θ θ
− −= +
− − (3.881)
Chapter 3: Material models and formulations
370 ADINA Structures ⎯ Theory and Modeling Guide
ial models
3.9.1 Curv
d axisymmetric) and 3D solid elements.
! small dis ses, sm s
mawith the large displacemem
! strain law used to represent the response of geological mThe m oduli as pie own in Fig. 3.91. An explicit yma ined by the history of the volume strain only. ! o present the governing constitutive relations, let
3.9 Geotechnical mater
e description material model
! The curve description model can be employed with the 2D olid (plane strain ans
The curve description model can be used with the
placement and large displacement formulations. In all cal trains are assumed. al
When used with the small displacement formulation, a teriallynonlinearonly formulation is employed, and when used
ent formulation, a TL formulation is ployed.
The curve description model is a simple incremental stressaterials.
odel describes the instantaneous bulk and shear mcewise linear functions of the current volume strain, as sh
ield condition is not used and whether the terial is loading or unloading is determ
Tt
ije = total strains (the left superscript "t" always refers to time t)
ije = incremental strains
= tme
3
tii
i
e∑ = mean strain (negative in compression)
me = 3ii
i
e∑ = incremental mean strain
= t tij m ije et
ijg δ− = deviatoric strains
ijg = incremental deviatoric strains t
ijσ = total stresses (negative in compression)
ijσ = incremental stresses
3.9: Geotechnical material models
ADINA R & D, Inc. 371
tmσ =
3
tii
i
σ∑ = mean stress
mσ = 3
ii
i
σ∑ = incremental mean stress
pmin = minimum mean stress ever reached t = t t
ijs ij m ijσ σ δ− = deviatoric stresses
ijs = incremental deviatoric stresses
,t tG K = shear and bulk moduli
The incremental stressstrain relations using the curve description model are then
2
3
tij ij
tm m
s G g
K eσ
=
=
Chapter 3: Material models and formulations
372 ADINA Structures ⎯ Theory and Modeling Guide
Lo
adin
gsh
ear
mo
du
lus
Lo
adin
gb
ulk
mo
du
lus
e1v
e1v
e1v
K
G
G
K
K
KLD
LD
LD
LD
t
t
t
UN
UN
UN
Compressive volume strain
is assumed positive
Note: G =K
GK
tUN t
LDtLD
Un
load
ing
bu
lkm
od
ulu
st
e2v
e2v
e2v
e3v
e3v
e3v
e4v
e4v
e4v
e5v
e5v
e5v
e6v
e6v
e6v
ev
ev
ev
et v
et v
et v
Figure 3.91: Input of the curve description model
Note:
3.9: Geotechnical material models
ADINA R & D, Inc. 373
moduli, tK and tG, are nctions of the loading condition, and the volumetric strain te is
! The instantaneous bulk and shear fu vdefined as
( )3t tgrave e eν = + − m
where egrav is the volumetric strain (three times the mean strain andtaken positive in compression) due to t
the gravity pressure and em
the mean strain at time t. Definin emin as the minimum mean strain ever reached during the solution, we have that the material is loading if
Note that the loading conditions for both the bulk and the shear
of te only. The values of L 1,
g is
mine≤ and the material is unloading if tme >t
me min
i.e., e
min
min
when when
t LD mt t
UN m
K e eK
K e e⎧ ≤
= ⎨>⎩
and
min
min
when when
t tt LD m
t tUN m
G e eG
G e e⎧ ≤
= ⎨>⎩
t t
!
moduli are determined by the history mtK , tK , and tG are obtained using the curves in Fig. 3.9D UN LD
t ⎞and the modulus t t
t
KG G⎛
= UN
LDK ⎟ .
! The incremental solution at time t is obtained using the
UN LD ⎜⎝ ⎠
equations
( )2 t t t t tij ij ij ijs G g g s+∆t t+∆ = − +
and ( )3t t t t t t t
ij ij ij ijK e eσ σ+∆ +∆= − +
mparing the current volumetric strains nd previous volumetric strains. To start the procedure at time 0,
To obtain tG and tK, we check whether the loading or unloading conditions are active by coa
Chapter 3: Material models and formulations
374 ADINA Structures ⎯ Theory and Modeling Guide
loading conditions are assumed. It should be noted that the stresses at time t + ∆t are calculated using the material moduli pertaining to time t. ! An important additional analysis option is that the material can weaken under loading conditions if tensile stresses exceed preassigned values. Since the curve description model has been developed primarily for the analysis of geological materials, the material weakening is assumed to occur once the principal tensile stresses due to the applied loading exceed the insitu gravity pressure p (taken positive). The gravity pressure p is calculated as
1
N
i ii
p h p=
= ∑ where hi are the element interpolation functions and
is the pressure at the element nodes. The nodal pressure pi is pi calculated as i ip Zγ= ⋅ where γ is the material density and Zi the nodal Z coordinate, which coincides with the vertical direction. The material weakening can be included using a tension cutoff model or a cracking model.
< In both modes of behavior (tension cutoff and cracking) the
al
sses
e stress are reduced, but the stresses are fully retained. The model therefore simulates elasticplastic flow of the material.
On the other hand, in the cracking mode the stiffnesses are reduced, and in addition the normal and shear stresses due to external loads and corresponding to the direction of stiffness reduction are released. The reduction of stiffness together with the stress release models a crack, i.e., a tensile failure plane, that forms at right angles to the direction of the maximum principal tensile stress.
principal stresses due to the external loading are calculated and compared with the insitu gravity pressure. Once the principal tensile stress is equal to the insitu gravity pressure, the materiis treated as being orthotropic, with the modulus corresponding to the direction of the principal tensile stress being multiplied by a stiffness reduction factor (an input parameter). Another factor, also an input parameter, is applied to reduce the shear stiffness.
< In the tension cutoff mode, the normal and shear stiffnecorresponding to the direction of the maximum principal tensil
3.9: Geotechnical material models
ADINA R & D, Inc. 375
< In subsequent load steps, tension cutoff or cracking only in the direction already determined and/or other(s) perpendicular to it may be active, i.e., no change in the tension cutoff or cracking direction is considered. The tension cutoff or cracking mode is considered to be inactive provided the strain normal to the crack (in the direction of tension cutoff) becomes both negative and less than the strain at which the failure occurred initially; otherwise it remains active.
Fig. 3.92 illustrates the tension cutoff and cracking options.
! Note that the Z direction should be the vertical direction and that the ground level is assumed to be at Z = 0 in the calculation of the insitu gravity pressure.
ADINA checks for tension cutoff or cracking at each integration point.
(b) Option of cracking
ab
ab
x2
x2
a
a
x1
x1
Maximum =
insitu gravity pressure
a
a
a
ea
ea
(a) Option of tension cutoff
a failure stress =
insitu gravity pressure
Figure 3.92: Tension cut off and cracking options
! The insitu gravity pressure is evaluated using the input for material density which corresponds to the weight per unit volume.
Chapter 3: Material models and formulations
376 ADINA Structures ⎯ Theory and Modeling Guide
This is different than the mass density which is used in the element mass matrix calculation.
3.9.2 DruckerPrager material model
! The DruckerPrager model is based on
< The DruckerPrager yield condition (see p. 604, ref. KJB) < A nonassociated flow rule using the DruckerPrager and cap yield functions
< A perfectl vior < Tension cutoff < Cap hardening
!
,
yplastic DruckerPrager yield beha
! The DruckerPrager model can be used with the 2D solid and 3D solid elements.
The DruckerPrager model can be used with the small displacement/small strain, large displacement/small strain and large displacement/large strain formulations.
When used with the small displacement/small strain formulation, a materiallynonlinearonly formulation is employedwhen used with the large displacement/small strain formulation, a
L formulation is employed and when used with the large Tdisplacement/large strain formulation, the ULH formulation is employed.
! Fig. 3.93 summarizes some important features of the DruckerPrager model (with tension cutoff and cap hardening). The DruckerPrager yield function is given by:
1 2t t t
DP Df J J kα= + −
α and k are material where parameters and are functions of frictional angle φ and cohesion coefficient . The corresponding potential function is given by
c
3.9: Geotechnical material models
ADINA R & D, Inc. 377
1 2DP Dg J J uβt t t= + −
where
β is a function of dilatation angle ψ , u is not requiinput parameter. The parameters
red as
α , k and β can be determined by matching the ru
or triaxial compression test:
D ckerPrager criterion with the MohrCoulomb criterion. F
( ) ( ) ( )2sin 6 cos 2sinc
3 3 sin 3 3 sin 3 3 sinkφ φ ψα β
φ φ ψ= = =
− − −
For triaxial tensile test:
( ) ( ) ( )2sin 6 cos 2sin
3 3 sin 3 3ck
sin 3 3 sinφ φ ψα β
φ φ ψ=
+ = =
+ + For the plane strain test:
2 2
tan 3ck9 12 tan 9 12 tan
φαφ φ+ +
The cap yield function depends on the shape of th
plane cap: t a
= =
e cap. For a
11t t
Cf J= − J+
where 1
t aJ is a function of the volumetric plastic strain t PVe :
1 1
01 ln 1t P
t a aVeJ J⎛ ⎞
= − − + D W⎝ ⎠
0 a
⎜ ⎟
here J is the cap initial position and W and D are material w
1
constants.
Chapter 3: Material models and formulations
378 ADINA Structures ⎯ Theory and Modeling Guide
Tension
cutoff
Yield surface
Elliptical cap
Plane cap
Elastic region
J2D$
TJ1
H
 J (elliptical cap)a1
 J (plane cap)a1
Figure 3.93: DruckerPrager model
b) DruckerPrager model in J versus J space1 2D$
%
&
'
a) DruckerPrager yield surface in principal stress space
&&&
O A C
For an elliptical cap:
( ) ( )2 2 21 2
t t t t tC Df J L R J B= + + −
L is equal to OA. The dependence of on the volumetric plastic strain is as for the plane cap. In the case of tension cutoff, T is the maximum value that can take.
where tB is the vertical semiaxis of the ellipse (AH), R is the cap tio (AC/AH) and t
1
0 aJ
1
t aJ
ra
3.9: Geotechnical material models
ADINA R & D, Inc. 379
! We note that:
< Unlike for the von Mises model, the DruckerPrager yield function cannot be used with hardening (the material is always assumed to be elastic perfectlyplastic, except for cap hardening). < If α(f) approaches zero (minimum value of α(f) is taken to be 105), the initial position of the cap is moved far to the right in Figure 3.93 and is not reached in the analysis, and the position of the tension cutoff is moved far to the left and is also not reached in the analysis, then the DruckerPrager yield
ic
ises yie y) = 0
< Cap yielding leads to an increase of compressive plastic volumetric strain. If the plane cap is used, there is no change of deviatoric plastic strains, while if the elliptical cap is used, the deviatoric components of the plastic strain change during the
that is, the constraint that exists between these two quantities. rresponds to a hardening and
aller
1
0 aJ
T
condition approaches the von Mises elasticperfectlyplastyield condition. < In the case of DruckerPrager yielding, the dilatency (volume expansion of the material in shear) is governed by the magnitude of α(y) (for the von M ld condition α(and there is no dilatency).
cap yielding. Fig. 3.94 shows the relation between the cap position t aJ and the volumetric strain
1
11 22 33t P t P t P t P
Ve e e e= + +
We note that the cap movement cothat the volumetric plastic strain is constrained to be smthan the input parameter W in absolute value.
Chapter 3: Material models and formulations
380 ADINA Structures ⎯ Theory and Modeling Guide
Figure 3.94: Relation between cap position J and volumetric
plastic strain e
t a1
t Pv
t1aJ
tvPe
W
 1_D
0 a1J
Possible volumetric
plastic strain
Material parametersW and D are negative.
(negative)
this limit is
< In the case of vertex yielding (the stress state is representedby point H in Fig. 3.93), the plastic deformation corresponds to the DruckerPrager and cap yielding. The vertex yielding leads to changes in volumetric and deviatoric plastic strains with the cap hardening behavior. < Stress states beyond the tension cutoff value T (input to ADINA as a positive value) are not possible. Whenreached or exceeded, the program sets all the shear stress components equal to zero and the normal stress components all
equal to 3T
. The elastic constitutive law is used for the stiffness
matrix.
3.9.3 Cam aterial model
y
clay m
! The Camclay material model is a pressuredependent plasticitmodel. It is based on
3.9: Geotechnical material models
ADINA R & D, Inc. 381
< An associated flow rule using an elliptical yield function
ehavior of clayey materials
The Camclay model can be used with the small
large displacement/small strain and large displacement/large strain formulations.
sed L
rmulation is employed and when used with the large is
The Camclay model is able to simulate the following
y
< Strain hardening and softening under normal consolidation
stic volumetric strain on the hydrostatic pressure
ity at which plastic shearing can continue indefinitely without changes in volume or
The Camclay yield function is given by
< The critical state line, which controls the failure of the material < The consolidation b
! The Camclay model can be used with the 2D solid and 3Dsolid elements.
!
displacement/small strain,
When used with the small displacement formulation, a materiallynonlinearonly formulation is employed and when uwith the large displacement/small strain formulation, a Tfodisplacement/large strain formulation, a ULH formulationemployed.
!
mechanical behaviors for clayey materials, which are confirmed blab tests and insitu tests:
states or overconsolidation states < Nonlinear dependence of the ela
< An ultimate condition of perfect plastic
effective stress
!
( )2tq
0 0t t tp p p2t F
M= + − =
Chapter 3: Material models and formulations
382 ADINA Structures ⎯ Theory and Modeling Guide
where the mean stress t p and the distortional stress tq at timre related t
e t o the first s invariantstres 1
t I and the second deviatoric a
stress invariant 2t J by 1
3t
t Ip = and 23t tq J= . M, a ma
constant, is the slope of the critical state line and
terial
0t p , called the
is the diameter of the ellipsoid at time t along the axis p. All of the variables are shown in Fig. 3.95(a).
The hardening rule is written as
preconsolidation pressure,
!
Figure 3.95: Camclay model
Critical state line
Elliptical yieldsurface
( p, q)t t
1M
q
p
p= pt cs
p= pt cs
p= pt 0
p= pt 0
a) Camclay yield surface
v
ln p
N
p=1
( p, v)t t
1
1k
1
Isotropic consolidation line
Critical state line
Overconsolidation line
b) Compression behavior
00v ln ln t
tt t pN p k
pλ= − +
3.9: Geotechnical material models
ADINA R & D, Inc. 383
or, in rate form,
( ) 0v lnt
t Pt
pkp
λ= − −#
where is the specific volume at time t and is the plastic
evt vt P#
sp cific volume rate at time t. λ , k and N are material constants, in which λ is the slope of the isotropic consolidation line, k is thslope of the
e overconsolidation e and N is the specific volume at
the isotropic consolidation state when lin
t p is 1.0. N is related to , the specific volume at the critical state when Γ t p is 1.0, by
( ) ln 2N kΓ λ= + −
! The effective bulk modulus at time t can be expressed as
v=t
t t pKk
The corres t
ponding shear modulus at time is obtained using ( )( )
3 1 2ν−2 1
G Kν
=+
, in which t t ν is the Poisson=s ratio (constan
throughout the analysis).
In an analy
t
! sis using the Camclay model, ADINA requires tial stresses or an initial stiffness to be defined.
induced stresses), or can be the full gravityinduced stresses.
When the analysis starts with initial stresses, the initial size
e
either the ini
Initial stresses: The initial stresses are either directly input or elastically calculated before or at the first load step for any specified element group. They can be tiny isotropic compressive stresses (such as 5% of the gravity
of the yield surface can either be directly input or computed from the initial stresses. If the initial stresses are induced by thfull gravity load, the initial size of the yield surface (pre
Chapter 3: Material models and formulations
384 ADINA Structures ⎯ Theory and Modeling Guide
the
definition. If the initial stresses are not directly input, ADINA calculates
Then, input
elastic material properties E and
consolidation pressure) would be estimated by employing input parameters OCR and KNULL, unless the initial size of the yield surface is specified in the Camclay model
the initial stresses as the stresses computed at load step 1.(a) The results for load step 1 are calculated using the
ν . (b) The results for the remaining load steps depend upon the
solution for load step1. stresses, ADINA takes
ust be directly specified.
When the analysis starts with initialthe true initial size of the yield surface as the greater of the specified initial size and the computed initial size.
Initial stiffness: The analysis can be performed with an initial stiffness, which is calculated using the initial Young=s modulus and Poisson=s ratio. In this case, the initial size of the yield surface m
! Parameter KNULL is defined as the coefficient of earth pressureat the normal consolidation state. Usually it is different than the coefficient of current earth pressure. KNULL is approximated by
0 1 sinK φ= − where φ is the internal friction angle of the soil. ! The material constant M is computed based on φ . If triaxial
6sin
3 sinM φ
φ=compression tests are performed, M is given by
−. If
triaxial extension tests are performed, M is given
by6sin
3 sinM φ
φ=
+.
! For more information, see references:
ref. D.M. Wood, Soil behavior and critical state soil mechanics, Cambridg
e University Press, pp. 84225,
s 184,
1990. ref. A.M. Britto and M.J. Gunn, Critical state soil mechanic
via finite elements, Ellis Horwood Limited, pp. 1611987.
3.9: Geotechnical material models
ADINA R & D, Inc. 385
< Tension cutoff
ain and ment/large strain formulations.
When used with the small displacement formulation, a
all strain formulation, a TL rmulation is employed and when used with the large
The MohrCoulomb yield function is given by
3.9.4 MohrCoulomb material model
! The MohrCoulomb model is based on
< A nonassociated flow rule < A perfectlyplastic MohrCoulomb yield behavior
! The MohrCoulomb model can be used with the 2D solid and 3D solid elements. ! The MohrCoulomb model can be used with the small displacement/small strain, large displacement/small strlarge displace
materiallynonlinearonly formulation is employed and when used with the large displacement/smfodisplacement/large strain formulation, a ULH formulation is employed. !
( ) ( )( )
1
2
MC sint tf I φ=
21 3 1 sin sin 3 3 sin cos 3 cost t t J Cφ θ φ θ φ+ − + + −
and the corresponding potential function is written as
( ) ( )( )1
2
sin1 3 1 sin sin 3 3 sin cos 3 cos2
t
t t t
I
J C
ψtMCg
ψ θ ψ θ ψ+ − + + −
where
=
1 33/ 22
1 3 3cos3 2
tt
t
JJ
θ − ⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠
Chapter 3: Material models and formulations
386 ADINA Structures ⎯ Theory and Modeling Guide
φ is the friction angle (a material constant), C is the cohesionmaterial constant),
(a ψ is the dilatation angle (a material constant),
t1I is the first stress invariant at time t, t J is the second deviatoric 2
stress invariant at tim t and is the third deviatoric stress invariant at time t.
emonstrated in Figures 3.96 to 3.99.
e 3t J
! Some important features of the MohrCoulomb model are d ! It is noted that:
< A hardening rule does not apply to the MohrCoulomb model; the MohrCoulomb model is only used in conjunction with the elasticperfectlyplastic yield condition.
'
'
'
3
Figure 3.96: The yield surfaces of the MohrCoulomb modeland the DruckerPrager model
DruckerPragermodel
modelMohrCoulomb
2
1
3.9: Geotechnical material models
ADINA R & D, Inc. 387
T
2J
1I
The yield surface
Figure 3.97: The MohrCoulomb model inI vs J space1 2
'
1
'
2'
3
DruckerPrager yield
surface with and (for a tensile meridian
DruckerPrager yield
surface with and ( for a
compressive meridian
MohrCoulomb
yield surface
Figure 3.98: Shapes of the MohrCoulomb and the
DruckerPrager yield criteria on the plane#
Chapter 3: Material models and formulations
388 ADINA Structures ⎯ Theory and Modeling Guide
CA
B
p
ij)
p
ij)
p
ij)
DruckerPrager
potential surface
MohrCoulomb
potential surface
Figure 3.99: Corner treatment of theMohrCoulomb model
< When using the MohrCoulomb model, the volume dilatency due to shearing is only governed by the dilatency angle ψ . In order to reduce the pathologically large material dilatation, one usually specifies the dilatation angle smaller than the friction angle. < In the case of vertex yielding (the stress states represented by points A, B and C in Figure 3.99), the plastic strain increments are obtained by applying the flow rule to the DruckerPrager equation. In other words, the flow rule is taken to be identical with the projection of the axis of either the tensile or the
pressive
< Stress states beyond the tension cutoff value T are not pMaxim
mponent to the hydrostatic ion cut
f the
com
meridian on the deviatoric plane.
ossible. When this limit is reached or exceeded, the umTensileStress Criterion (Rankine) is applied by
shifting the hydrostatic stress copressure corresponding to the tension cutoff. The tensoff value T is three times as large as the tensile strength omaterial. < The MohrCoulomb yield equation can be written as
1 2t t t t t
MCf I J kα= + −
3.9: Geotechnical material models
ADINA R & D, Inc. 389
eHer tα and are stress dependent:
tk
( ) ( )2sint
t3 1 sin sin 3 3 sin cos t
φαφ θ φ θ− + +
=
( ) ( )6 sin
3 1 sin sin 3 3 sin cost
t t
Ck φφ θ φ θ
=− + +
)
If stress states fall on a compressive meridian ( π / 3tθ = ,
then
( ) ( )2sin 6 cos,
3 3 sin 3 3 sint t Ckφ φα
φ φ= =
− −
If stress states fall on a tensile meridian ( )0tθ = , then
( ) ( )2sin 6 cos,
3 3 sinCkt t
3 3 sinφ φα
φ φ= =
+
3.10 Fabric material model with wrinkling
or of fabric structures.
lly very thin and flexible. They can
+
! For more information, see the following reference:
ref. W.F. Chen, Nonlinear analysis in soil mechanics, Elsevier, pp. 139370, 1990.
! The elasticorthotropic material model (discussed in Section 3.2) can be used with the 2D solid element (3D plane stress option) and the large displacement/small strain formulation to model the behavi ! Fabric materials are typicasupport only inplane tensile stresses, and will wrinkle under compressive stresses.
Chapter 3: Material models and formulations
390 ADINA Structures ⎯ Theory and Modeling Guide
f fabric structures with ADINA is performed in two steps: first, initial strains must be specified, Young's moduli
to a fraction (usually 104) of their actual values, and the wrinkling of the material must not be taken into account. Then,
the same orthotropic material
! The analysis o
must be set
in a second step (restart analysis), model is used except that the Young's moduli are now set to their actual values, and wrinkling is taken into account. ! For each integration point, wrinkling occurs if the smallest of the two inplane principal stresses ( ),t
p1σ tp2σ becomes negative.
If wrinkling occurs, then this smallest principal stress tp2σ is set
o zero. If the other inplane principal stress tp1σ is gret ater than
zero, the program calculates a new "wrinkling stress" tp1σ ′ using
t tp1 p1A
σ ε′ ′=
here t
tp1σw p1ε is the strain in the direction of and A is calculated
, using the condition 0tp2σ =from the orthotropic material law . If
tp1σ is smaller than zero, ADINA sets t
p1σ to zero; in this case,
ling behavior and becomes
wrinkling occurs in two directions. The element constitutive matrix corresponding to the principal
directions is also modified for the wrink
0 00 0A
Aα
η⎡ ⎤
0 0 Aη
⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
or where η is a reduction factor equal to 106, and α = 1 if 0tp1σ ≥
α = η if 0tp1σ < .
3.11 Viscoelastic material model
solid, ! The viscoelastic model can be used with the truss, 2D3D solid and shell elements.
3.11: Viscoelastic material model
ADINA R & D, Inc. 391
the small n and
D solid and 3D
ly formulation is employed, hen used with the large displacement/small strain formulation, a
ULH formulation is mployed.
n isotropic and linear viscoelastic
tation as
! The viscoelastic model can be used withdisplacement/small strain, large displacement/small strailarge displacement/large strain formulations (2solid elements only).
When used with the small displacement/small strain formulation, a materiallynonlinearonwTL formulation is employed and when used with the large displacement/large strain formulation, thee
! The mechanical behavior for amaterial may be expressed in tensor no
0
( )( ) 2 (0) ( ) 2 ( )t
ij ij ijdGs t G e t e t dτ
dτ τ= + −∫ (3.111)
τ+
0
( ) 3 (0) ( ) 3 ( )kk kk kkt K e t t dd
( )t dK τσ ε τ ττ+
= + −∫ (3.112)
where t is the time,
13ij ij ij kks σ δ σ= − is the deviatoric stress, ijδ
is the Kronecker delta, ijσ is the stress, 13ij ij ij kke ε δ ε= − is the
atoric strain, ijε is the strain, devi G(t) is the shear modulus and (t) is the bulk modulus.
r
KIn the presence of a temperature variation T(t), the stresses fo
an isotropic and thermorheologically linear viscoelastic material may be written as
0
( )( ) 2 (0) ( ) 2 ( )ij ijdGe t e d
dζξ
ijs t G ξ ζ ζζ+
−∫ (3.113)
= +
[ ] [ ]0 0( )0) ( ) 3 ( ) ( ) 3 ( ) 3 ( )kk kk kk
dKe t t e t dξ ζ
0
( ) 3 (kk t Kd
σ α θ ε ξ ζ α θ ζ− + − −∫ ζζ+
=
(3.114)
Chapter 3: Material models and formulations
392 ADINA Structures ⎯ Theory and Modeling Guide
,T dτ
where
( ) ( )0 0
t
T dζ ψ η η ξ= ψ τ τ+ +
=⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦∫ ∫ (3.115)
the ( )t
pseudotemperature θ is given by
( ) ( )0
00 T
T (3.116)
( )t
1
01( ) ,
T
t T dTθ α α αα
′ ′= =∫
α is the temperatureexpansion and ( )t
dependent coefficient of thermal ψ is the shift function, which obeys
0( ) 1, ( ) 0, 0dT TdTψψ ψ= > > (3.117)
Please note that the definition of the temperaturedependent
coefficient of thermal expansion ( )tα is slightly different than the definition of tα used in other material models of ADINA; see Section 3.5 for the definition of .tα If the coefficient of thermal expansion is independent of temperature, then both definitions are the same.
In equations (3.113) and (3.114) it is assumed that the mechanical and thermal responses are uncoupled. Furthermore if the temperature is constant, equations (3.113) and (3.114) reduce to equations (3.111) and (3.112).
! We assume the following thermomaterial properties:
1
( )G
iti
iG t G G e
ηβ−
∞=
= + ∑ (3.118)
1
( )K
iti
iK t K K e
ηγ−
∞=
= + ∑ (3.119)
3.11: Viscoelastic material model
ADINA R & D, Inc. 393
( ) 0T t ≠ (3.1110) ( )Tα α= (3.1111)
where G∞ and K∞ are the longtime shear modulus and bulk modulus respectively, iβ and iγ are the decay constants for the shear modulus and bulk modulus respectively and Gη and Kη are the number of timedependent terms for the shear modulus and bulk modulus respectively. Equations (3.118) and (3.119) are referred to in the literature as Prony or Dirichlet series. Gη and Kη are limited to a maximum value of 15.
The shift function used is the WilliamsLandellFerry (WLF) equation, written as follows
( )( )
1 010
2 0
log ( )C T T
TC T T
ψ−
=+ −
(3.1112)
in which C1 and C2are material constants. ! You can either enter the Prony series constants in the AUI, or you can enter the measured data for the shear and bulk modulus and let the AUI construct the Prony series. In the latter case, the AUI constructs the Prony series using a leastsquares technique, see the reference by Frutiger and Woo given below.
! The noda cussed in Section 5.6.
! For more information, see the following references:
ref. W.N. Findley, J.S. Lai and K. Onaran, Creep and
relaxation of nonlinear viscoelastic materials, Dover Publications, 1976.
ref. R.L. Frutiger and T.C. Woo, AA thermoviscoelastic
analysis for circular plates of thermorheologically simple material@, Journal of Thermal Stresses, 2:4560, 1979.
l point temperatures are input to ADINA as dis
Chapter 3: Material models and formulations
394 ADINA Structures ⎯ Theory and Modeling Guide
3.12 Porous media formulation
! The porous media formulation is applicable to porous structures subject to static or dynamic loading. It deals with the interaction between the porous solids and pore fluids, which flow through the porous solid skeleton. ! The porous media formulation only applies to 2D and 3D solid elements. These elements have pore pressure variables at their corner node points.
! The pore pressures are considered to be independent variables, along with the displacements. The pore pressures are output by ADINA along with the displacements, velocities, stresses and other solution results. ! The porous media model is implemented with a fully coupled formulation, as described in detail below. ! The ADINA porous media formulation is based on the following assumptions:
< The pore fluids are incompressible or slightly compressible. < The porous solid skeleton is fully saturated with pore fluid. < The pore fluid flows through the porous solid according to Darcy=s law.
With the porous media formulation, undrained analysis (static
large displacement/small strain formulation and rge displacement/large strain
ulation.
!
pore fluid) and consolidation or swelling analysis (pore fluid owing through the porous solid skeleton) can be performed. fl
! In the analyses of porous structures, all the nonlinear formulations in ADINA (small displacement/small strain, large displacement/small strain and large displacement/large strain) canbe used. A materiallynonlinearonly formulation is employed for the small displacement/small strain formulation; the TL formulation for the
e ULH formulation for the lathform
3.12: Porous media formulation
ADINA R & D, Inc. 395
be employed in the analyses of porous structures. However the incompatible modes feature cannot
e
ion): The pore pressure distribution, the displacement and stress distribution, and their
ssure, e
d small time steps.
dia analysis. You can use whichever material model is
! The mixed (u/p) formulation can
be employed in the analyses of porous structures. ! The types of problems for which the porous media model can bemployed are:
Transient static analysis (consolidat
changes with time are of great interest.
Transient dynamic analysis: In addition to the pore predisplacement and stress results, the potential failure of thporous structure will draw much more attention. Liquefaction, which is an extremely important failure mode for infrastructuresin an earthquake disaster, can be studied with the porous media model.
Undrained analysis: No specific formulation is given to relate the pore pressure to total stresses under undrained conditions. The undrained analysis can be performed by specifying undrained boundary conditions an
! All the material models available for the solid elements (including elasticplastic material models) can be employed in the orous mep
most physically meaningful for your problem. ! The general equations governing the porous media model are as follows:
< The macroscopic stresses must satisfy the equilibrium condition:
, 0ij j ifσ + = (3.121)
,3C P i j k lσ ε= + = (3.122)
where the total (Cauchy) stress is
, , , , 1, 2ijkl
e eij kl f ijδ
Chapter 3: Material models and formulations
396 ADINA Structures ⎯ Theory and Modeling Guide
tensor, fP is the pore pressure, ijkl
eC is the fourth order elasticityeklε is the elastic strain tensor and ijδ is the Kronecker delta.
< The continuity condition for slightly compressible fluid
flows must be satisfied:
( ) fiif
f
PnPt t
εκ
∂∂∇ ⋅ ⋅∇ = +
∂ ∂k (3.123)
where k is the permeability matrix of the porous material, iiε is
n is the porosity of the volumetric strain of the porous skeleton, the material and fκ is the bulk modulus of the fluid. If the fluid is assumed incompressible, the last term in (3.123) will be zero.
! After a finite element discretization, equations (3.121) and
ent statics:
t t T t t t tt t t t+∆ +∆ +∆+∆ +∆
⎡
(3.123) yield the following equations.
Transi
f
f f f f f
uu u p(1) (2)
u p p p p pf f
t t t tt t t t+∆ +∆+∆ +∆⎡ ⎤⎤ ⎡ ⎤ ⎡ ⎤ + ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
0K K 0 KP P#
+∆
0 K KU U#
f
u
p
t t
t t+∆
⎡ ⎤= ⎢ ⎥
⎢ ⎥⎣ ⎦(3.124)
Transient dynamics:
u p p pf f
uu u p(2)p p pf
t t t tt t
t t
t t t t t tt t
t t t tt t
+∆ +∆+∆
+∆
+∆ +∆ +∆+∆
+∆ +∆+∆
RR
(1)t t T t tt t +∆ +∆+∆
⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤+ +⎥⎢⎢ ⎥⎢ ⎥
f f f
f
f f f
u
⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦
⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
C 0
K K RU0 K RP
## ##
(3.125)
U UM 0K KP P0 0 ##
3.12: Porous media formulation
ADINA R & D, Inc. 397
For a finite element, these matrices are defined as follows (omitting the left superscript for brevity):
(T dV= +∫ ∫R H f H )
( )
f f
f
uu u u
u p u p
p p p pf f f
(2)p p p pf f f f
u u u
p pf f
f
q
S
S
T
V
T
V
f V
T
VT
fV S
T
qS
dV
dV
n dV
dV
dS
dS
κ
=
=
=
=
=
∫
∫
∫
∫
∫
K B DB
K B I H
K H H
K B k B
f
R H q
In the above expressions, D is the constitutive matrix (possibly elastoplastic), f and q are load vectors (force, traction and flux),
and
(1) 1 T
uB pfB are gradient matrices for the displacements U and
pore pressures fP , uH and Hpf are gradient matrices for the
displacements and pore pressures , and is the identity tensor.
ting equations (3.124) and (3.125) with respect to time gives the final equations used for the porous media mo
U fP I
! Numerically integra
del:
Transient statics: the Euler backward method is used:
Chapter 3: Material models and formulations
398 ADINA Structures ⎯ Theory and Modeling Guide
u p(2)
u p p p p p f
u
t t t
t t t t
t t
t
∆ +∆
+∆ +∆
+∆
⎤ ⎡ ⎤⎥ ⎢ ⎥− ∆⎢ ⎥ ⎣ ⎦⎣ ⎦
f
f f f f f
uu (1)
t t t
t t T t t
+∆ +
+∆ +∆
⎡⎢
K
⎡
f f f f
(1)p u p p p f
t t t t T t t t tt +∆ +∆ +∆
⎤= ⎢ ⎥−∆ + +⎢ ⎥⎣ ⎦R K U K P
K UK K K P
R
(3.126)
e
t t
t t t tt
+∆
+∆ +∆
⎡ ⎤
Transient dynamics: the Euler backward method is used for the continuity equation and the Newmark method is used for thequilibrium equation:
fuu 0 1 u p(1)
t t t t t t t t
t t T t t
a a+∆ +∆ +∆ +∆
+∆ +∆
+ +
f f f f f
f f f f
u p p p p p f
u(1)
p u p p p f
t t d
t t t t T t t t tt
+∆
+∆ +∆ +∆
(2)
⎡ ⎤⎢ ⎥ ⎢ ⎥− ∆⎢ ⎥ ⎣ ⎦
⎡ ⎤= ⎢ ⎥−∆ + +⎢ ⎥⎣ ⎦
RR K U K P
in which
⎣ ⎦K P
(3.127)
K M C K UK K
( )( )
u u 0 2 3
1 4 5
t t d t t t t t t t
t t t t t
a a a
a a a
+∆ +∆ +∆
+∆
= + + + +
+ +
R R M U U U
C U U U
# ##
# ##
(3.128)
. KJB, Section 9.2.4).
ote the following:
and where a0, a1, ... , a5 are the integration constants for the ewmark method (see RefN
! Please n
< Usually, the global stiffness matrix is no longer positive definite in porous media analysis. This is because the diagonal components of
f f f f
(1) (2)p p p p
t t t tt+∆ +∆− ∆K K in equations (3.126)and (3.127) are usually negative. Therefore, you shourequest that ADINA continue execution after ADINA detects anonpositive definite stiffness matrix (see Section 7.1.1).
ld
3.12: Porous media formulation
ADINA R & D, Inc. 399
ocess is unconditionally stable for any time step size because of the use of the Euler backward method. However, this does not necessarily imply that any time step size is permissible. A small time step size and large mesh size will result in the zigzag pore pressure distribution at the initial stage of pore fluid dissipation (Britto & Gunn, 1987). It is logical to use progressively larger time steps in the porous media analysis.
< The pore pressures are only computed by ADINA at the corner nodes of solid elements. During postprocessing, the AUI interpolates the pore pressures from the corner nodes to the other nodes (e.g. midside nodes) in higherorder elements. < Once the porous solid material properties are specified, the corresponding pore fluid properties are automatically computed. Certainly, they can be redefined if necessary.
< Porous media cannot be used in explicit dynamic analysis.
3.13 Gask
! Gaskets are relatively thin components placed between two bodies/surfaces to create a sealing effect and prevent fluid leakage (see Figure 3.131). While most gaskets are flat, any arbitrary gasket geometry can be modeled in ADINA.
< It is important to have a reasonable scheme for choosing time steps. Theoretically, the solution pr
! For more information, see the following reference:
ref. Britto, A.M. and Gunn, M.J., Critical state soil mechanics via finite elements, Ellis Horwood Limited, 1987.
et material model
Chapter 3: Material models and formulations
400 ADINA Structures ⎯ Theory and Modeling Guide
Gasket thickness direction(orthotropic axis A)
Only one elementthrough the thickness
Figure 3.131: Schematic of gasket
Gasket inplanedirections
! The sealing effect is created when the compressive load, applied in the direction of the gasket thickness, exceeds the initial yield stress of the gasket. The sealing effect is maintained as long as the compressive stre threshold
alue. The gaske xceeds the askets ultimate stress. Unlike rupture, if a gasket leaks it still
n characteristics.
ed with 2D solid and 3D solid ith small displacement/small
en
represented by pressurelo ed to be constant or
e closure strain is lw ness divided by the
tional plasticity o flexibility in
sket, elastic material properties are
ss does not drop beyond a specifiedt ruptures if the compressive stress ev
gmaintains its loaddeflectio ! The gasket model can be uslements. It can also be used we
strain, large displacement/small strain formulations.
! The gasket behaves as a nonlinear elasticplastic material whompressed in the thickness or gasket direction. Its loadc
deformation characteristics are typicallysure curves. Tensile stiffness can be assumc
zero. Hardening is assumed to be isotropic. Thays measured as the change in gasket thicka
original gasket thickness. The gaskets unidirecdel speeds up computations, and allows morem
defining the shape of the loading and unloading curves. ! Two gasket types are implemented in ADINA, the full or regular gasket, and the simple gasket. In the simple gasket all inplane deformations are neglected. In this case, the gasket only possesses thickness stiffness and the inplane degrees of freedom can be eliminated. In the full ga
3.13: Gasket material model
ADINA R & D, Inc. 401
lastic segments and any number of plastic loading segments.
.
assumed for the gasket plane. ! Figure 3.132 shows a typical pressureclosure relationship. It consists of a main loading curve consisting of any number ofeLoading/unloading curves can be provided for different points on the loading curve. However, it is not necessary to define a loading/unloading curve for each point on the main loading curve
Loading curve
Leakagepressure
Rupturepoint
Closure strain
Gas
ket
pre
ssu
re
Multiple loading/unloading curves
Initial yieldpoint
Figure 3.132: Pressureclosure relationship for a gasket material
an the leakage pressure.
igher than the leakage pressure but has not yet caused plasticity.
There has been plastic gasket deformation and the current pressure is above the gasket leakage
deformation, the gasket pressure has dropped below gasket leakage pressure.
! Each gasket can have one of the following five states:
Open: The gasket pressure is less th
Closed: The gasket pressure is h
Sealed:
pressure. Leaked: After plastic
Chapter 3: Material models and formulations
402 ADINA Structures ⎯ Theory and Modeling Guide
.
es
! The gasket should be modeled as a single layer of 2D or 3D elements. Only linear elements are possible (3 and 4noded elements in 2D and 4,6 and 8noded elements in 3D). ! Since the gasket has different properties in different directions, it is considered by ADINA to be an orthotropic material. Material axis A is set to the gasket normal direction. The AUI attempts to automatically define the gaskets material axes if they are not xplicitly defined by the user.
et mal
automatically define the gaskets skew directions if they are not set by the user. ! he top and bottom surfaces of a gasket can be separate from tho of e nected via contact. The gasket can also share a common surface with the intended mating surface. In this case, contact is not needed, however, the gasket cannot separate from its target. ! The number of points in all loading/unloading curves must be identical for efficiency. Also the last point in each loading/unloading curve must be one of the input point on the main loading curve.
The input: In addition to the pressureclosure relationships the following material properties are needed for a gasket: density,
The following properties are only needed for a full gasket: in
Crushed: Gasket closure strain has exceeded the rupture value
Modeling issu
e
! If a simple gasket is selected, the local Xaxis at the gasknodes (or Yaxis in 2D) must coincide with the gaskets nordirection, using skew coordinate systems if necessary. The simple gasket is disconnected from the remaining inplane degrees of freedom. The AUI attempts to
Tse th mating surfaces. In this case, they should be con
transverse shear modulus, tensile Youngs modulus, thermal expansion coefficient in thickness direction, reference temperature, and leakage pressure.
plane Youngs modulus, inplane Poissons ratio and inplane thermal expansion coefficient.
3.13: Gasket material model
ADINA R & D, Inc. 403
available for both gasket types: GASKET_PRESSURE, ASKET_CLOSURE_STRAIN, GASKET_THERMAL_STRAIN,
GASKET_STRESSBB, GASKET_STRESSCC, GASKET_STRESSAB, GASKET_STRESSAC, GASKET_STRESSBC, GASKET_STRAINBB, GASKET_STRAINCC, GASKET_STRAINAB, GASKET_STRAINAC, GASKET_STRAINBC, GASKET_THERMAL_STRAINBB, GASKET_THERMAL_STRAINCC, GASKET_THERMAL_STRAINAB, GASKET_THERMAL_STRAINAC, GASKET_THERMAL_STRAINBC, GASKET_YIELD_STRESS, GASKET_PLASTIC_CLOSURE_STRAIN, GASKET_STATUS, GASKET_DEFORMATION_MODE
See Section 11.1.1 for the definitions of these variables. Note
that in 2D analysis, only the AA, BC, CC components are output.
3.14 Shape Memory Alloy (SMA) material model ! The Shape Memory Alloy (SMA) material model is intended to model the superelastic effect (SE) and the shape memory effect (SME) of the shapememory alloys using the finite element method.
clinic
Output variables: The following gaskets output variables are
G
! One representative of SMA is Nickel Titanium (NiTi), can undergo solidtosolid phase transformations induced by stress or temperature. The high temperature phase is called austenite (A) with a bodycentered cubic structure and the lowtemperature phase is called martensite with a monocrystal structure in several variants. Fig.3.141 shows a schematic SMA stresstemperature diagram. The martensite(M) phase in two generalized variants and the austenite phaseare shown, as well as stress and temperaturedependent transformation conditions. The martensite phase is favored at low temperatures or high stresses.
Chapter 3: Material models and formulations
404 ADINA Structures ⎯ Theory and Modeling Guide
Detwinnedmartensite
T
Twinnedmartensite
Austenite
MfMs
AsAf
Figure 3.141: SMA stresstemperature phase diagram
! The typical uniaxial isothermal stressstrain curve is shown in Fig. 3.142. The SE effect is defined at temperature T>A
.
residual transformation strain remains after unloading; however heating the material to temperature above Af leads to thermally induced M→A transformation and the recovery of transformation strain.
f and is displayed in Fig.3.142(a). The stress cycle application induces transformations from A→M and then from M→A to exhibit the hysteresis loop. The SME effect isdefined at temperature T<As and is displayed in Fig.3.142(b)A
3.14: Shape Memory Alloy (SMA) material model
ADINA R & D, Inc. 405
T > Af
)
(a)
T < As
)
(b)
Figure oys
ity, and (b) shape memory effect
! The equation of total strain,
3.142: Schematic of stressstrain curves for shapememory all(a) pseudoelastic
! The SMA material model in ADINA is based on the following equations:
e t θε ε ε ε= + + where ε e= elastic strain εθ = thermal strain εt = transformation strain; to be evaluated ! The equations of onedimensional macroscale modeling,
s tξ ξ ξ= + ; 0 ≤ ξ ≤1 ξ + ξΑ = 1 εt = εt
max ξs
smax((1 ) )( )tA ME Eσ ξ ξ ε ε ξ= − + −
Chapter 3: Material models and formulations
406 ADINA Structures ⎯ Theory and Modeling Guide
ξt = twinned martensite volume fraction ξs = detwinned martensite volume fraction ξA = austenite volume fraction εt
max = maximum residual strain; a material property ! The flow rule of threedimensional constitutive model,
t
max
t tij s ijnε ξ ε∆ = ∆
3 ijtij
sn
2 σ⎛ ⎞
= ⎜ ⎟ ; for the martensitic transformation ⎝ ⎠
32
tijt
ij tnεε
⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠; for the reverse transformation
sij = deviatoric stresses
32 ij ijs sσ =
32
t tij ij
tε ε ε=
! The phase transformation conditions, 1. for the martensitic transformation as starting condition
23 ( )SM M Sf J C Mθ= − −
2. for the martensitic transformation as ending condition
23 (f
)M M ff J C Mθ= − −
3.14: Shape Memory Alloy (SMA) material model
ADINA R & D, Inc. 407
3. for the reverse transformation as starting condition
23 (SA A )Sf J C Aθ= − −
4. for the reverse transformation as ending condition
23 ( )fA A ff J C Aθ= − −
! The phase transformation rate using linear kinetic (Auricchio and E. Sacc
rule o, 1999),
f
Rf
fξξ∆ = ∆
2 23 ( )
2
t t tt t t
t t
J Jf c θ θσ
+∆+∆
+∆
−∆ = − −
where c = the specific heat per unit volume
 ff M Mf f c C= = and Rξ = 1  ξ (for the martensitic
transformation)
 ff A Af f c C= = and Rξ = ξ (for the reverse transfo
rmation)
• Computational steps for the stressintegration of the SMA model are as follows (Kojic and Bathe, 2005):
1. Calculate trial stresses, '( ) ( )1 ( )
tt t t t t t
ij ij ijt
ES ξ ε εν ξ
+∆ +∆= −+
Chapter 3: Material models and formulations
408 ADINA Structures ⎯ Theory and Modeling Guide
2. Check for martensitic transformation, 0
f sM Mf f < and 1tξ < and 0f∆ > Check for reverse transformation,
0f sA Af f < and and 0tξ > 0f∆ <
3. Solve for martensitic transformation with the governing equations as follows,
2 2
2
"max
3 ( )( ) ( )2
1( )2
( ) ( ( ) )1 ( )
t t t t t tt t t
t t t tf
t t t t t tij ij
t tt t t t t t t t
ij ij s ijt t
R J Jg cf
J s s
Es n
ξ ξξ ξ θσ
ξ
ξ ε ξ ξ εν ξ
+∆ +∆ +∆+∆
+∆ +∆
+∆ +∆ +∆
+∆+∆ +∆ +∆
+∆
θ⎡ ⎤∆ −
∆ = ∆ − − −⎢ ⎥⎣ ⎦
∆ =
∆= − ∆ ∆ ⋅
+ ∆
(3.141)
1Rξ ξ= − and ff Mf f= −
4. Solve for reverse transformation using the governing equations (3.141) with Rξ ξ= and
ff Af f= 5. Solve for martensitic reorientation with the governing equations as follows,
3.14: Shape Memory Alloy (SMA) material model
ADINA R & D, Inc. 409
2
"
3( )2
( ) (1 ( )
t t t t t ts ij ij R R
t tt t t t t t t t
ij t t
g s s C
Es
ξ θ σ
ξν ξ
+∆ +∆ +∆
+∆+∆ +∆ +∆
+∆
∆ = − −
∆=
+ ∆
(3.142
max( ) )ij s ijnε ξ ξ ε− ∆ ∆ ⋅
)
. Update historydependent variables for this time step/iteration step. 7. Calculate the consistent tangent constitutive matrices.
ref. M. Kojic and K.J. Bathe, Inelastic Analysis of Solids and Structures, Springer, 2005
ref. F. Auricchio and E. Sacco, A TemperatureDependent
The martensite reorientation calculation step is optional; it is activated when σR > 0 is input. 6
Beam for ShapeMemory Alloys: Constitutive Modelling, FiniteElement Implementation and
Numerical Simulation, Computer Methods in Applied Mechanics and Engineering, 174:pp. 171190 (1999)
3.15 Usersupplied material model
3.15.1 General considerations
! The usersupplied model is provided in ADINA to allow you to construct any type of material model for use with 2D or 3D solid lements.
! The constitutive relation must be arranged in the form of an algorithm and inserted into subroutine CUSER2 for 2D solid elements or into subroutine CUSER3 for 3D solid elements.
! The material model can be used with the following formulations:
e
Chapter 3: Material models and formulations
410 ADINA Structures ⎯ Theory and Modeling Guide
< Materiallynonlinearonly, for infinitesimally small displacements and strains (MNO) < total Lagrangian, in which the displacements and rotations can be very large, but the strains may have to be small or can be large depending on the material characteristics (TL) < updated Lagrangian Hencky, in which case the strains can be large (ULH)
! Either the displacement based elements or the mixedinterpolated displacement/pressure elements can be used with the usersupplied material model. The mixedinterpolated elements are preferred for (nearly) incompressible materials. ! The use of the usersupplied model is described in the following, first for 2D solid elements and then for 3D solid elements. Then the usersupplied material models for which the source code is available in ADINA are described. ! Note that even if the material properties are not temperature dependent, or the temperature loading is zero, both the temperature table and the temperature loading input must be entered accordingly in order that the .dat file be generated for computation.
3.15.2 2D solid elements
The total solution process for a
ad (time) step involves updating the stresses from time t to time t endent quantities. ADINA
performs this updating by subdividing the total strain increment
ER2 (see Figs. 3.151 and .152).
e viscoplastic model (see below).
Basic concept of implementation:lo+ ∆t and updating any history dep
from time t to time t + ∆t into subincrements and, for each subincrement, calling the subroutine CUS3
! A sample version of CUSER2 may be found in file ovl30u_vp1.f. The coding within this sample version is fullydiscussed in the discussion of th
3.15: Usersupplied material model
ADINA R & D, Inc. 411
Time t
Time * Time **
Time +t t
Subincrements
Strain
Time
teij eij
eij(k)
+ eij t+teij
*
* **
t**
t+tDirection ofsubincrementation
Values provided and passed back through
the calling arguments of subroutine CUSER2
At entry, the arguments
correspond to time:
At exit, the arguments
correspond to time:
Variable or
Array name
EPS(4)
STRAIN(4)
THSTR1(4)
THSTR2(4)
TMP1
TMP2
ALFA
ALFAA
CTD(*)
CTDD(*)
ARRAY(*)
IARRAY(*)
STRESS(4)
t
t+t
t
t t
**
* ** *
* *
* *
* *
**
* *
**
* *
**
*
*
*
* *
* *
* *
* *
No change
in value
Figure 3.151: Subincremenation process for forward stress integration
Convention used in subroutine CUSER3
! The stress integration can be performed by employing forward integration or backward integration. Details regarding these two stress integration schemes are given below.
Chapter 3: Material models and formulations
412 ADINA Structures ⎯ Theory and Modeling Guide
Z
Y
X
STRESS(1)=*yy
STRESS(2)=*zz
STRESS(3)=*yz
STRESS(4)=*xx
*zz
*yz
*xx
*yy
Figure 3.152: Stress convention for 2D elements
Forward stress integration
In this case, the stress at the end of a subincrement k is
calculated as
k (3.151)
( )kσ
( ) ( 1) ( )k k −= + ∆σ σ σ
where (0) t=σ σ . The stress increment ( )k∆σ corresponds to the t ( )k∆e and temperature increment ∆θ. strain incremen
For example
( )∆ = ∆ − ∆ − ∆σ C e e e (3.152)
where C
( ) ( ) ( ) ( )k E k th k IN k
and ∆e are
e increments of thermal and inelastic strains in the subincrement.
LFAA, CTD(*), CTDD(*) and the temperatureindependent properties CTI(*) in the calculation of the stresses and the constitutive matrix. The temperaturedependent properties at time τ are calculated by ADINA at each integration point prior to calling subroutine CUSER2.
E is the elastic constitutive matrix and ∆eth(k) IN(k)
th It is assumed that the total strain varies linearly in the time step.
! The evaluation of the strain subincrements is performed by ADINA prior to calling subroutine CUSER2. Your coding must update the stresses (in array STRESS(4)) and other history dependent variables for each subincrement, and print the stressesonce the final stresses for the load (time) step have been evaluated.
Your coding uses the temperaturedependent properties ALFA, A
3.15: Usersupplied material model
ADINA R & D, Inc. 413
ess integration in the subincrement can be
explicit or implicit. If the integration is implicit, your coding must include the appropriate iterative scheme for the stress calculation.
Backward stress integration
! The stress calculation is as follows:
First, the subroutine CUSER2 is called for the evaluation of the trial elastic stress corresponding to the total and thermal strains at time t + ∆t, t + e and t + ∆teth, respectively, and to the inelastic trains at time t, teIN. For example
! Note that the str
*Eσ
∆t
s
( )*t t E t t E t t t t th t IN+∆ +∆ +∆ +∆= − −σ C e e e (3.153)
where t + ∆tCE is an elastic constitutive matrix corresponding to temperature t + ∆tθ. Calculation of the trial elastic stress must be provided by your coding.
Next, the subroutine CUSER2 is called within the subincrementation loop, as schematically shown in Fig. 3.153. The stress correction ∆σ(k) in subincrement k (Eq. (3.151)) now corresponds to the increment of inelastic strain ∆eIN(k). For example
k
where is the elastic constitutive matrix corresponding to temp
( ) ( ) ( )k E k IN∆ = − ∆σ C e
( )E kCerature τ τ θ−∆ . As indicated in Fig. 3.153, the direction of
subincrementation is now from t + ∆t toward t.
Chapter 3: Material models and formulations
414 ADINA Structures ⎯ Theory and Modeling Guide
E EEETime t
Time *Time**
Time +t t
Subincrements
Elasticstrain
Time
teij eij
eijE(k)
 eij t+teij
*
***
t**
t+tDirection ofsubincrementation
E INt t te e eij ij ij= 
Values provided and passed back through
the calling arguments of subroutine CUSER2
At entry, the arguments
correspond to time:
At exit, the arguments
correspond to time:
Variable or
Array name
EPS(4)
STRAIN(4)
THSTR1(4)
THSTR2(4)
TMP1
TMP2
ALFA
ALFAA
CTD(*)
CTDD(*)
ARRAY(*)
IARRAY(*)
STRESS(4)
No change
in value
No change in value(small/large
strains)
(ULH large strains)DPSP(4)
t
t t
*
* *
*
* *
*
* *
*
* *
**
**
* *
* *
**
t
t t
*
* *
*
* *
*
* *
*
* *
*
*
*
*
*
**
**
**
**
Figure 3.153: Subincrementation process for backward stress integration Convention used in subroutine CUSER3
! The concept of evaluation of variables for the current subincrement is the same as above for the forward integration. The
3.15: Usersupplied material model
ADINA R & D, Inc. 415
uation of the trial elastic solution, and should not be changed during the subincrementation until the criteria for discontinuing the
entation are satisfied (in case of viscoplasticity, for he stopping criterion corresponds to the time when the
stress state falls inside the yield surface). When the criteria for sto
ntegration point. The stresses and other history parameters evaluated at the condition
. ing
to ADINA for the calculation of the inelastic eformation gradient in ADINA (array DPSP(4)).
be
ust include the appropriate iterative procedure for the stress calculation.
istory dependent variables are stored at each integration point
to characterize the history of the material behavior:
STRAIN(4): Strain components at time t + ∆t. For the MNO and TL formulations, STRAIN(4) are total strains; that is, STRAIN(4) includes all strain contributions (e.g, elastic, inelastic, thermal). For the ULH formulation, STRAIN(4) are elastic strains (total minus inelastic).
RRAY(LGTH2): Variable length working array of material
istory parameters (integer variables).
Material properties, algorithm control parameters: You can specify the material properties and algorithm control parameters to the usersupplied material model using the AUI, material type user
additional parameter in the backward (the return mapping) cedure is ISUBM. Namely, the flag ISUBM is set to zero at pro
eval
subincremexample, t
pping the subincrementation are satisfied, your coding must reset ISUBM to 1 and this condition signals ADINA to stop the subincrementation loop for the current i
ISUBM = 1 are considered by ADINA to be the solutions for the current time step.
Note that in the case of the updated Lagrangian Hencky formulation only the backward stress integration can be employed Also, the increments of inelastic strains evaluated by your codmust be passedd
Note that the integration of stresses in the subincrement can explicit or implicit. In case of an implicit scheme, your coding m
H
ARRAY(LGTH1): Variable length working array of material history parameters (floating point variables).
IAh
Chapter 3: Material models and formulations
416 ADINA Structures ⎯ Theory and Modeling Guide
sup
NSCP: Number of solution control parameters (maximum 99).
CTD(NCTD): Temperaturedependent material constants for
plied. This information is passed to subroutine CUSER2 as follows:
LGTH1: Length of array ARRAY. LGTH2: Length of array IARRAY. NCTI: Number of material property constants (maximum 99).
NCTD: Number of temperaturedependent material constants (maximum 98). CTI(NCTI): Temperatureindependent material constants.
the given temperature at time τ . ADINA calculates CTD(NCTD) from the table of CTD(NCTD) vs. temperature entered into the AUI.
or CTDD(NCTD): Temperaturedependent material constants fthe given temperature at time τ τ+ ∆ (forward stress integration) or τ τ− ∆ (backward stress integration). Acalculates CTDD(NCTD) fro
DINA m the table of CTD(NCTD) vs.
temperature entered into the AUI. SCP(NSCP): Solution control parameters. ALFA: Coefficient of thermal expansion for the given temperature at time τ . ADINA calculates ALFA from the tablof
e α vs. temperature entered into the AUI.
ALFAA: Coefficient of thermal expansion for the given temperature at time τ τ+ ∆ (forward stress integration) or τ τ− ∆ (backward stress integration). ADINA calculates ALFAA from the table of α vs. temperature entered into the AUI. mperatures: The temperatures are passed to subroutine SER2 as follows:
TeCU
3.15: Usersupplied material model
ADINA R & D, Inc. 417
TMP1: Temperature at time τ . TMP2: Temperature at time τ τ+ ∆ (forward stress integration) or τ τ− ∆ (backward stress integration).
The variable KEY: Subroutine CUSER2 is called for all integration points of each 2D solid element and is used to perform the following four types of operations during the various phases. The different operations to be performed are controlled by the integer variable KEY.
KEY = 1: called once at the beginning of the analysis.
This indicates the initialization of the variables stored at the
integration points, and is performed only once during the input phase. The arrays STRAIN, STRESS, ARRAY and IARRAY are automatically initialized by ADINA to zero. Your coding must set ARRAY and IARRAY to their proper initial values, if different from zero.
KEY = 2: called for each subincrement within a load (time) step.
the nodal displacements, and in the stress calculation phase after the nodal disstrsupplied coding inserted in subroutine CUSER2. The variables ava are giv In addition to the stresses, the arrays ARRAY and IARRAY (if used to store history dependent variables) must be updated when KE =
of int
ubroutine CUSER2 is in the form
This indicates the calculation of the element stresses in the
solution of the governing equilibrium equations for
placements have been evaluated. The calculation of the element esses at each integration point is performed using the user
ilable in the subroutine and the notations/dimensions used en in the listing of the sample subroutine (in file ovl30u_vp1.f).
Y 2. Two stress integration schemes are available for the calculation stresses from strains: forward integration and backward egration.
Forward stress integration: In this case, the stresses arecalculated according to Eq. (3.131) and the usersupplied coding in the s
Chapter 3: Material models and formulations
418 ADINA Structures ⎯ Theory and Modeling Guide
DO 100 I = 1, N 0. 1 J)
where N is the number of stress or strain components, D(I,J) is the constitutive matrix which is to be evaluated also using the usersupplied coding, and DEPS stores the components of the elastic and inelastic strain subincrements. Note that DEPS is evaluated by dividing the total strain increment for the step into the appropriate number of equal subincrements and the thermal strain subincrement is subtracted from DEPS prior to calling CUSER2. Note also that the array STRESS contains the stresses calculated at the end of the previous strain subincrement when the array STRESS is passed to CUSER2.
If an elastic constitutive matrix is used, then all inelastic rain subincrements must be subtracted from DEPS before the
above DO loop is executed. If the material model employs a total strain theory, then the
algorithm used to perform the stress calculation can be of the (assuming that in this case there are no thermal
ffects): DO 100 J = 1, N 100 + D(I,J)*STRAIN(J)
here N and D(I,J) are defined above and STRAIN stores the total strain components for the current step. Using the total strain procedure, the number of subdivisions used for the strain subincrements should be set to 1 for solution effectiveness.
Backward stress integration: In this case, your coding must
3);
formation variables, e.g., the inelastic strains eIN, etc. and use them in the calculation of stresses.
STRESS(I) = D0 DO 100 J = 1, N 00 STRESS(I) = STRESS(I) + D(I,J)*DEPS(
st
following forme
DO 100 I = 1, N STRESS(I) = 0.D0
STRESS(I) = STRESS(I)
w
first calculate the trial elastic stress according to Eq. (3.15arrays STRAIN and THSTR2 contain the total strains andthermal strains as described in the listing of subroutine CUSER2. Your coding must store into the working arrays ARRAY and/or IARRAY the history of de
3.15: Usersupplied material model
ADINA R & D, Inc. 419
meter ISUBM must be
ents of inelastic strains DEPS(4) for each subdivision. Stresses and strains passed to subroutine CUSER2 correspond to deformation without rigid body rotations. The stresses STRESS(4) which are true (Cauchy) stresses are passed to subroutine CUSER2 only for printout (KEY = 4). The transformation that takes into account the material rotation is performed in ADINA prior to calling subroutine CUSER2.
An example of coding for the backward stress integration is given in the listing of the viscoplastic material example.
Note that when using the TL formulation, the stresses in array
STRESS(4) correspond to the second PiolaKirchhoff stress tensor. The Cauchy stresses, obtained by the transformation of the second PiolaKirchhoff stress tensor in ADINA, are passed to the subroutine CUSER2 only for printout (KEY=4).
If you require the stress invariants, your coding can call subroutine XJ1232. The call statement
CALL XJ1232(STRESS,XI1,XI2,XI3,XJ1,XJ2,XJ3)
returns the three invariants XI1, XI2 and XI3 of the stress tensor, and the three invariants XJ1, XJ2 and XJ3 of the deviatoric stress tensor. These are calculated as follows by XJ1232:
In the calls to CUSER2 following calculation of the trial elastic stress, you must provide the coding for the stress increments due to the inelastic deformation within the subincrements. It is also important to note that the stoppingcriteria and the resetting of the paraproperly coded.
Note that when using the ULH formulation, the array STRAIN(4) contains the trial elastic strains for KTR = 0; the array is further updated in ADINA by subtracting increm
XI1 , XI2 1/ 2 , XI3 1/ 3
XJ1 0, XJ2 1/ 2 , XJ3 1/ 3ii ij ij ij jk ki
ij ij ij jk kis s s s s
τ τ τ τ τ τ= = =
= = =
where ijτ is the stress tensor passed in STRESS(4) and is the
deviatoric stress tensor corresponding to ijs
ijτ . The summation convention is used in the above expressions.
Note that if the large displacement/large strain options are used,
Chapter 3: Material models and formulations
420 ADINA Structures ⎯ Theory and Modeling Guide
stresses energetically conjugated with the corresponding strains (see Table 3.151) are employed in these calculations, and you cannot control this.
Table 3.151: Stress and strain measures used in the usersupplied material model. Note that the stresses are transformed internally by ADINA into true (Cauchy) stresses for printout (KEY=4)
Kinematic formulation
Strain measure
Stress measure
MNO (materially nonlinearonly): small displacements, small strains
Infinitesimal strains
Cauchy stresses
TL (total Lagrangian): large displacements, small or large strains (depending upon the material model)
GreenLagrange strains
2nd PiolaKirchhoff stresses
ULH (updated Lagrangian Hencky): large displacements, large strains
Logarithmic Hencky (elastic) strain and plastic velocity strains
Cauchy stresses
KEY=3: called when the new tangent stiffness matrix is to be calculated.
This indicates the calculation of the constitutive matrix D which
is to be employed in the evaluation of the tangent stiffness matrix. atrix calculation
ed or calculated in the last subdivision of the stress calculation when KEY=2, as described above.
ent response in the stress
The variables available for the constitutive mare the values defin
KEY=4: called for stress printout.
This indicates the printing of the elemcalculation phase. No stress calculations need to be performed when KEY=4; your coding must only include the required format statements for the printing of the desired element results.
Note that if the TL formulation is employed, the ADINA program automatically transforms, just for printout, the stresses
3.15: Usersupplied material model
ADINA R & D, Inc. 421
used in the constitutive relations (see Table 3.151) to the Cauchy stresses in the global coordinate system. Then subroutine CUSER2 is called for stresses and strains printout.
If the ULH formulation is employed, total stretches PRST(3) are calculated and passed to subroutine CUSER2. PRST(1) stores the maximum inplane stretch (maximum deviation from 1.0), PRST(2) is the smaller inplane stretch and PRST(3) is the stretch in the Xdirection. Also, the angles of the first and the second principal stretches with respect to the Yaxis, stored in the array ANGLE(2), are passed to CUSER2.
3.15.3 3D solid elements
! The usersupplied material model for 3D solid elements is entirely analogous to that for 2D solid elements. Hence, refer to the above section for detailed descriptions. ! The usersupplied algorithm for the usersupplied model for 3D solid elements is to be inserted in subroutine CUSER3. ! A sample version of CUSER3 may be found in file ovl40u_vp1.f. The coding within this sample version is fully discussed in the discussion of the viscoplastic model (see below). ! The convention used for stresses and strains is shown in Fig. 3.134.
STRESS(1)=*xx
STRESS(2)=*yy
STRESS(3)=*zz
STRESS(4)=*xy
STRESS(5)=*xz
STRESS(6)=*yz
*zz
*yz
*yy
*xx
*xz
*xy
Z
Y
X
ure 3.154: Stress/strain convention for 3D elements Fig
! hen the ULH formulation is employed, the principal stretches and the direction cosines of the maximum principal stretch (with
W
Chapter 3: Material models and formulations
422 ADINA Structures ⎯ Theory and Modeling Guide
maCU ! If you require the stress invariants, your coding can call subroutine XJ123. The call statement
CALL XJ123(STRESS,XI1,XI2,XI3,XJ1,XJ2,XJ3) returns the three invariants XI1, XI2 and XI3 of the stress tensor, and the three invariants XJ1, XJ2 and XJ3 of the deviatoric stress tensor. These invariants are defined as for the 2D elements described above.
3.15.4 Usersupplied material models supplied by ADINA R&D, Inc.
3.15.4.1 Viscoplastic material model
! A viscoplastic material model for axisymmetric or plane strain conditions is given as an example of the use of the CUSER2 subroutine. A viscoplastic model for fully 3D conditions is given as an example of the CUSER3 subroutine. The coding in subroutine CUSER2 is based on the equations given hereafter.
! The CUSER2 and CUSER3 subroutine coding can be found in files ovl30u_vp1.f and ovl40u_vp1.f respectively.
! The rate of viscoplastic strain is given by
ximum deviation from 1.0) are passed to the subroutine SER3 for the printout (KEY=4).
32
tijt VP
ij t
se γ φ
σ= < ># (3.154)
where
= component of deviatoric stress tijs
32
t tij ijs sσ = t = effective stress
γ = fluidity parameter
φ< > = φ if 0φ >
3.15: Usersupplied material model
ADINA R & D, Inc. 423
In
= 0 otherwise
the above,
1tσφ
yσ= − (3.155)
wh per
! ing and the material constants are considered to be temperature ind e ! int
ere σy is the yield stress (above which the material is consideredfectly plastic).
The material is subjected to mechanical and thermal load
ep ndent.
Stress integration is performed using forward or backwardegration.
Forward integration: In forward integration, the increment of stresses in the subincrement (Eq. (3.152)) are
( )( ) ( ) ( ) ( )k E k th k VP ki ij j j jσ C e e e∆ = ∆ − ∆ − ∆ (3.15
wh
6)
,
ere
1 2 3 4, , ,yy zz yz xxσ σ σ σ σ σ σ σ∆ = ∆ ∆ = ∆ ∆ = ∆ ∆ = ∆
( )( )
t t tj jk
j
e ee
+∆ −∆ =
NSUBD(3.157)
NSUBD is the number of subincrements,
( )( ) for 1, 2, 4th kje jτ τ τα θ θ+∆∆ = − =
( )( )( ) 3VP kτ
1 jj
se
τ
τ
στ γ 2 yσ σ⎝ ⎠
⎛ ⎞∆ = ∆ −⎜ ⎟⎜ ⎟ (3.158)
Chapter 3: Material models and formulations
424 ADINA Structures ⎯ Theory and Modeling Guide
Here α is the mean coefficient of thermal expansion, τ and ∆τ aredefined in Fig. 3.151, and the strains ej are defined as
1 2 3 4, , ,yy zz yz xxe e e e e e eγ= = = =
for time t and t+∆t. The same convention is used for increments of viscoplastic strains and deviatoric stresses. The coefficients represent the elastic constitutive matrix
EijC
( )( )( )
( ) ( )
( )( )
( )
11 1
1 01 11 1 2
1 20
2 11
E E
SYMM
ν ν⎡ ⎤0⎢ ⎥− ν − ν⎢ ⎥⎢ ⎥ν
− ν ⎢ ⎥− ν= ⎢ ⎥+ ν − ν ⎢ ⎥− ν⎢ ⎥
− ν⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
C
(3.159)
where E and ν are the Young's modulus and Poisson's ratio.
Backward integration: In backward integration, the trial elastic state is determined according to Eq. (3.153),
( )*
t t E E t t t t th t VPij j j jC e e eσ+∆ +∆ +∆= − − (3.1510)
The increments of stresses are obtained from
kC eσ∆ = − ∆ (3.1511)
( ) ( )k E VPi ij j
and the ( )VP kje∆ are given by Eq. (3.158).
If during subincrementation the effective stress ( )kσ is ( )k
yσ σ< (3.1512)
3.15: Usersupplied material model
ADINA R & D, Inc. 425
the subdivision loop is stopped by setting the parameter ISUBM to 1. Also the stresses are corrected as follows:
t t k kx+∆ ( ) ( )= − ∆ +σ σ σ (3.1513)
where
( )
( )
ky
kxσ σ
σ−
=∆
and the t t+∆ σ are the final stresses at time
t+∆t. Similarly, the increments of the viscoplastic strains are corrected using
t t VP VP k VP kx+∆ ( )= − ∆ +e e e ( ) (3.1514)
where are the final viscoplastic strains at time t+∆t.
Constitutive matrix: For the calculation of the constitutive matrix, the constitutive relations (3.156) can be written as
t t VP+∆ e
( )thm m mc e eσ∆ = ∆ − ∆ (3.1515)
( )1 VPi i
E
s e ea
′∆ = ∆ − ∆ i (3.1516)
where
( )1 2 4
3 3
1312
me e e
e e
e∆ = ∆ + ∆ + ∆
′∆ = ∆ (3.1517)
are increments of the mean and deviatoric strains, and cm and aE are elastic constants
1 21
mEc =
EaE
− ν+ ν
(3.1518=
)
Chapter 3: Material models and formulations
426 ADINA Structures ⎯ Theory and Modeling Guide
The derivatives of the mean stress and of the deviatoric stress follow from Eqs. (3.1515) and (3.1516):
( )
1 1,2,43
mm
j
c je
1 ii i
j E j
s ee a e e
VP
j
e
σ∂= =
∂
⎛ ⎞∂ ∆
⎠
(3.1519)
The derivatives
′∂ ∂⎜ ⎟= −⎜ ⎟∂ ∂ ∂⎝
ie′∂
je∂ can be written in matrix form as
2 1 1
1 1 2 1 , 1, 2, 4ie i j− −⎡ ⎤
′∂ ⎢ ⎥= − − =3
1 1 2je∂⎢ ⎥− −⎣ ⎦⎢ ⎥ (3.1520a)
3
3
1e2e
′∂ =
∂ (3.1520b)
0 for 3 or 3 and i
j
e i j i je
′∂= = = ≠
∂ (3.1520c)
In the case of viscoplastic flow in the time step, an approximate
expression for ( )VP
ie∆
je∂
∂ can be obtained from Eq. (3.157) as
( )VP
i VP i
j j
e sce e
∂ ∆ ∂=
∂ ∂ (3.1521)
where
3 12
t tVP
t tc τ σγ+∆
+∆
⎛ ⎞∆
yσ σ= −⎜ ⎟⎜ ⎟ (3.1522)
⎝ ⎠
3.15: Usersupplied material model
ADINA R & D, Inc. 427
It follows from Eqs. (3.1521) and (3.1519) that the derivatives i
j
se
∂∂
are
i i j j
s ece e
′∂ ∂′=∂ ∂
(3.1523)
(3.1524)
The constitutive matrix CVP can be obtained using the relation
where
VPEc a c′ = +
3 3
1, 2, 4i i ms is
σ σσ
= + ==
(3.1525)
from which it follows that
1, 2, 4VP i mij
j j
sC ie e
σ∂ ∂= + =
∂ ∂ (3.1526a)
33VP
jj
sCe
∂=
∂ (3.1526b)
By substituting Eqs. (3.1519) and (3.1523) into (3.1526) and using Eq. (3.1520), the constitutive matrix CVP is obtained as
Chapter 3: Material models and formulations
428 ADINA Structures ⎯ Theory and Modeling Guide
( ) ( ) ( )
( )
( )
1 1 12 03 3 3
1 2 0 03
1SYMM 02
1 23
m m m
mVP
m
c c c c c c
c c
c
c c
⎡ ⎤′ ′ ′+ − −⎢ ⎥⎢ ⎥⎢ ⎥′+⎢ ⎥
= ⎢ ⎥⎢ ⎥′⎢ ⎥⎢ ⎥
′+⎢ ⎥⎢ ⎥⎣ ⎦
C
3.15.4.2 Plasticity model
! An isothermal von Mises plasticity model with isotropic hardening is given as an example of the use of the CUSER2 and CUSER3 subroutines. ! The implementation provided includes the MNO, TL and ULH formulations. ! The CUSER2 and CUSER3 subroutine coding can be found in files ovl30u_pl1.f and ovl40u_pl1.f respectively. ! The implementation is based on the equations given in Section 6.6.3 of Ref. KJB. The effectivestressfunction algorithm is used for
3.15.4.3
ample odel
inc
! the equations given in Section 6.6.3 of ref. KJB.
The creep strain increments are defined using the PrandtlReuss equations:
stress integration.
Thermoplasticity and creep material model
! A model for thermoplasticity and creep is given as an exof the use of the CUSER2 and CUSER3 subroutines. The m
ludes coding for the MNO, TL and ULH formulations. ! The CUSER2 and CUSER3 subroutine coding can be found in files ovl30u_cn1.f and ovl40u_cn1.f respectively.
The implementation is based on
3.15: Usersupplied material model
ADINA R & D, Inc. 429
Cij ijd sε γ=
where C
ijε are the creep strain components, are the deviatoric stress components and
ijsγ is the creep multiplier. Consequently,
note that the effect of the hydrostatic pressure on the creep deformations is considered to be negligible.
The creep law is the eightparameter creep law, i.e: ( ) ( ) ( )1 2 3
C f f fε σ θ= t
where Cε is the effective creep strain, σ is the effective stress, θ is the temperature and t is the time, and
( ) ( ) ( )( )
( ) 62 4
3 3 5aa af t t a t a t= + +
The same creep law also is used in the standard ADINA creepmaterial model (see Section 3.6).
For multiaxial creep deformation, the calculation of the stress
11 0 2 7; exp / 273.16af a f aσ σ θ θ= = − +
es
same as the uniaxial stressstrain curve of the material.
tensor
ed
ffective stress.
The tangent constitutive matrix
is based on the use of an effective stress versus effective (creep) strain curve. The curve is assumed to be the
In case of cyclic loading, the origin of the creep straindepends on the stress tensor reversal and is determined using the ORNL rule (subroutines XCRCY2 and XCRCY3).
The effectivestressfunction algorithm is used for stress integration (subroutines XEFSF2 and XEFSF3). Either the timehardening procedure or the strain hardening procedure can be usin the calculation of the creep multiplier (subroutines XGAMA2 and XGAMA3). A rootfinding algorithm without acceleration (subroutine UBSECT) is used for computation of both the pseudo time (strain hardening) and the e
t dtt dt EC i
ij t dtj
C σε
++
+= , where
t dtiσ+ are the stress components and t dt
jε+ are the total strain components, is calculated using an approximation based on the effectivestressfunction algorithm (subroutines XELMA2 and
Chapter 3: Material models and formulations
430 ADINA Structures ⎯ Theory and Modeling Guide
XELMA3).
3.15.4.4 Concrete material model, including creep effects
! A model for concrete is given as an example of the use of the CUSER2 and CUSER3 subroutines. The model includes coding for the MNO, TL and ULH formulations. ! The CUSER2 and CUSER3 subroutine coding can be found in files ovl30u_cn2.f and ovl40u_cn2.f respectively. ! This usersupplied material model has been developed for the analysis of concrete structures, in which creep deformations depend on the loading, on material ageing, on the temperature, and possibly on other material or load related parameters. ! The creep strain increments are defined by the PrandtlReuss equations
C
ij ijd sε γ=
where Cijε are the creep strain components, are the deviatoric
stress components and ijs
γ is the creep multiplier. Consequently, ote that the effect of the hydrostatic pressure on the creep
ndeformations is considered to be negligible. ! A generalized creep law of the following form is assumed:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 3 4 5 6 7 8f f f t f x f f f t f xε σ θ σ θ= +
where
ε is the effective creep strain, σ is the effective stress, θ is the temperature, t is the time, and x is a userdefined variable.
Appropriate coding for the creep member functions f1 to f8 is provided (subroutines XFNC2 and XFNC3). In addition, coding for the time derivatives of f3 and f7 is also provided (subroutines XDFNC2 and XDFNC3).
! Two different time variables are used (see Figure 3.155): t = time since concrete was cast; t0 = time since ADINA analysis (and
3.15: Usersupplied material model
ADINA R & D, Inc. 431
presumably loading) started. The quantity (t  t0) represents the age of the concrete at the start of the finite element analysis. The input data array SCP is used to store (t  t0) (in SCP1 = SCP(1)). Note that t = t0 + SCP1.
Time whenconcrete wascast
Current time
Time
t
t t 0
t0
Time whenfinite elementanalysis started
Figure 3.155: Time variables used for creep in concrete
! The variable SCP2 (= SCP(2)) is used to store the flag for choosing between the strain hardening (SCP2 = 1) or the time hardening (SCP2 = 0) procedure in the effectivestressfunction algorithm. ! The elastic modulus is assumed to be time dependent and given by (see Figure 3.156):
Elasticmodulus
Time
Asymptotic limit
Figure 3.156: Time dependent elastic modulus for creep in concrete
Chapter 3: Material models and formulations
432 ADINA Structures ⎯ Theory and Modeling Guide
( ) ( )( )4
1 2 3exp 1 / aE t a a a t⎡ ⎤= −⎣ ⎦
! The creep law member functions are:
( )
( ) ( )
( ) ( )( )( )
( )
6 8
2523 24 201612
221814
1
2 5 7 9 10
19 015 0113 0
13 17 0 21 0
4
with =exp
1,
1
a a
aa a aaa
aaa
f
f a a a a
a t ta ta tf t tE t a t a t a t t
f x
σ σ
θ φ φ φ θ
=
= + +
⎡ ⎤+ −⎡ ⎤ ⎡ ⎤++= ⎢ ⎥⎢ ⎥ ⎢ ⎥+ + + −⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
=
and
( )( ) ( )
( ) ( )( )
28
5
6 26 27
7
8
1
1
a
f
f a a
f tE t
f x
σ σ
θ θ
=
= +
=
=
Note that the term of f3 which depends only on (t  t0) could also be programmed as the member function f4 (t  t0).
! The creep constants are taken from array CTI as follows: a1 = CTI(1), a2 = CTI(2), ... . One possible set of values is a1 = 2.0 H 1010, a2 = 0.5, a3 = 28, a4 = 0.5, a5 = 1.0, a12 = 0.6, a13 = 10, a14 = 0.6, a23 = 1.0.
! In the case of multiaxial creep deformation, the calculation of the stresses is based on the use of an effective stress versus effective (creep) strain curve. The curve is assumed to be the same as the uniaxial stressstrain curve of the material.
In case of cyclic loading, the origin of the creep strain tensor
depends on the stress tensor reversal and is determined using the ORNL rule (subroutines XCRCY2 and XCRCY3).
3.15: Usersupplied material model
ADINA R & D, Inc. 433
The effectivestressfunction algorithm is used for stress integration (subroutines XEFSF2 and XEFSF3). The time hardening procedure or the strain hardening procedure can be used in the calculation of the creep multiplier, in case of variable stress and/or variable temperature conditions (subroutines XGAMA2 and XGAMA3). A rootfinding algorithm without acceleration (subroutine UBSECT) is used for computation of both the pseudo time (strain hardening) and the effective stress.
The tangent constitutive matrix t dt
t dt EC iij t dt
j
C σε
++
+= , where
t dtiσ+ are the stress components and t dt
jε+ are the total strain components, is calculated using an approximation based on the effectivestressfunction algorithm (subroutines XELMA2 and XELMA3).
3.15.4.5 Viscoelastic material model
! A model for linear viscoelasticity is given as an example of the CUSER2 and CUSER3 subroutines. The model includes coding for the MNO, TL and ULH formulations. ! The CUSER2 and CUSER3 subroutine coding can be found in files ovl30u_vel.f and ovl40u_vel.f respectively. ! The implementation is fully described in Section 3.11. ! The material constants are specified in the Constant Material Properties area of the UserSupplied Material dialog box of the AUI as follows:
Constant, 1: Gη Constant, 2: G∞ Constant, 3: 1
Constant, 4: 1
G β
... Constant, ( )2 1 1Gη + +⎡ ⎤⎣ ⎦ : Kη
Chapter 3: Material models and formulations
434 ADINA Structures ⎯ Theory and Modeling Guide
Constant, ( )2 1 2Gη + +⎡ ⎤⎣ ⎦ : K∞
( )2 1 3Gη + +⎡ ⎤Constant, ⎣ ⎦ : 1K
Constant, ( )2 1 4⎡ Gη + + ⎤⎣ ⎦ : 1γ
... Constant, ( )2 2 1G Kη η+ + +⎡ ⎤⎣ ⎦ : 0T
Constant, ( )2 2 2η ηG K+ + +⎡ ⎤⎣ ⎦ 0: α
Constant, ( )2 2 3G Kη η+ + +⎡ ⎤⎣ ⎦ : 1C
Constant, ( )2 2 4η η 2C G K+ + +⎡ ⎤ : ⎣ ⎦ and the total number of constants is ( )2 2 4G Kη η+ + + . In ad ition, you must set the Length of Real Working Array to 70 or larger for materials used in 2D solid elements and the Length of Real Working Array to 90 or larger for materials used in 3D solid elements.
Using the AUI com
d
mandline interface command MATERIAL eter
TI2, ... .
3.15.4.6
the CUSER2 and CUSER3 subroutines.
30u_pl2.f and ovl40u_pl2.f respectively.
! Additional files ovl30u_pl3.f and ovl40u_pl3.f are shown to use a more effective bisection algorithm, calling the ADINA subroutine BISECT.
The RambergOsgood stressstrain description of the yield curve is in the form
USERSUPPLIED, enter the number of constants using paramNCTD and the constants using parameters CTI1, C
RambergOsgood material model with mixed hardening
! The RambergOsgood material model with mixed hardening isrovided as an example in p
The model includes coding for the axisymmetric, plane strain, plane stress and 3D elements and for the MNO, TL and ULH formulations. ! The CUSER2 and CUSER3 subroutine coding can be found in
les ovlfi
!
3.15: Usersupplied material model
ADINA R & D, Inc. 435
( )0 nt py yCσ σ= + e
where Cy and n are material constants and 0yσ is the initial yiel
stress. d
ng behaviour is described by
matic f the mixed hardening model
is
! The mixed hardeni
( )1i kp pp p pd d d Md M d= + = + −e e e e e
where ipde and kpde are, respectively, the isotropic and kinecontributions, and the yield function o
( ) ( ) 21 1 02 3
t t t t t ty yf σ= − ⋅ − − =s α s α
where the back stress tα is evolved by
( )1 ppd C M d= −α e
Cp is Pragers hardening parameter and is calculated as
23 1
Mp p
p
E MEC
M−
=−
where MpE
p
is the derivative of the yield curve at the plastic
strain Me , pE is the plastic modulus corresponding to , and M is the factor used in general mixed hardening (0 < M < 1) which can be a variable, expressed as
pe
( ) ( )0 exp peM M M M η∞ ∞ −= + −
! The effectivestressfunction algorithm is used to calculate the plastic multiplier, see subroutines XEFSA and XEFSB in 2D (subroutine CUSER2) and XEFSF3 in 3D (subroutine CUSER3).
Chapter 3: Material models and formulations
436 ADINA Structures ⎯ Theory and Modeling Guide
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4. Contact conditions
ADINA R & D, Inc. 437
4. Contact
INA to model contact
• Very general contact conditions are assumed:
! The points of contact are assumed not known a priori. ! Friction can be modeled according to various friction laws (only standard Coulomb friction for explicit dynamic analysis). ! Both sticking and sliding can be modeled. ! Repeated contact and separation between multiple bodies is
permitted in any sequence. ! Selfcontact and doublesided contact are permitted.
! Nodetosegment or nodetonode contact can be assumed.
lems," Int. J.
es, Vol.
Pantuso, D., Bathe, K.J. and Bouzinov, P.A."A Finite
ref. KJBSection 6.
conditions Contact conditions can be specified in AD•
involving solid elements (2D and 3D solids) and/or structural elements (truss, beam, isobeam or axisymmetric shell, plate, shelland pipe elements).
! Tied contact can be modeled.
Some of the contact algorithms used in ADINA are described in thefollowing references:
ref. Bathe, K.J. and Chaudhary, A., "A Solution Method for
Planar and Axisymmetric Contact ProbNum. Meth. in Eng., Vol. 21, pp. 6588, 1985.
ref. Eterovic, A. and Bathe, K.J., "On the Treatment of
Inequality Constraints Arising From Contact Conditions in Finite Element Analysis," J. Computers & Structur
40, No. 2, pp. 203209, July 1991. ref.
Element Procedure for the Analysis of Thermomechanical Solids in Contact," J. Computers & Structures, Vol. 75, No. 6, pp. 551573, May 2000.
7
Chapter 4: Contact conditions
438 ADINA Structures Theory and Modeling Guide
ntact defined between contact surfaces. Contact segments
are the building blocks for contact surfaces. • Note that the presence of contact grounonlinear even when no nonlinear element groups are defined.
2 or ly to all contact
TCONTRO commands.
4.1 Overv
faces N e
metric or planar and
must lie in the global YZ plane, with all X coordinates equal to zero. Typical twodimensional contact surfaces are shown in Fig. 4.11(b).
es of higher order elements are automatically divided into 2 or 3 linear segments. The new contact surface representation supports both linear and quadratic contact segments.
A 2D contact surface is a closed surface if all the contact segments on the surface taken together form a closed path (see Fig. 4.11, contact surface 3). If the segments do not form a closed path, then the contact surface is open (see Fig. 4.11, contact surfaces 1 and 2).
• Contact in ADINA is modeled using contact groups. Each contact group is composed of one or more contact surfaces. Copairs are then
ps renders the analysis
• Most of the settings and tolerances needed for contact are provided in the contact group definition command (CGROUPCGROUP3). Some contact settings however appgroups (such as contact convergence tolerances, suppression ofcontact oscillations). These settings are provided in the MASTER or CONTAC L
iew • Contact groups (and their contact sur ) in ADI A can beither 2D or 3D. The contact surfaces should be defined as regions that are initially in contact or that are anticipated to come into contact during the solution.
! 2D contact surfaces are either axisym
The default contact surface representation only supports linear contact segments. Hence, contact segments on the fac
4.1: Overview
ADINA R & D, Inc. 439
Body 2
Body 3
Contact surface 3
Contact surface 2
A
Body 1
Contact surface 1
D
B
C
Figure 4.11: Typical 2D contact surfaces
! A 3D contact surface is made up of a group of 3D contact ate
ports
o
segments (faces) either on solid elements, shell elements, plelements, or attached to rigid nodes. See Fig. 4.12 for an illustration. The default contact surface representation suponly linear (and bilinear) segments. Hence, segments on thefaces of higher order elements are automatically divided up intlinear 3node or 4node faces. The new contact surface representation supports higher order (quadratic) contact segments.
Body 2
Body 1
Contact surface 2
(top surface of
body 2)
Contact surface 1
(surface of cylinder)
Figure 4.12: Typical contact surfaces and contact pair
Chapter 4: Contact conditions
440 ADINA Structures Theory and Modeling Guide
• A contact pair consists of the two contact surfaces that may come into contact during the solution. One of the contact surfaces in the pair is selected to be the contactor surface and the other contact surface to be the target surface. In the case of selfcontact, the same surface is selected to be both contactor and target. • Within a contact pair, the nodes of the contactor surface are prevented from penetrating the segments of the target surface, and not vice versa. • Target surfaces may be rigid, while contactor surfaces cannot be rigid. A contactor node should not also have all of its degrees of freedom dependent. • In explicit dynamic analysis, both contactor and target surfaces can be rigid if the penalty algorithm is used. Otherwise, the same restriction mentioned above applies.
• Rigid surfaces have no underlying elements and therefore no f
• Fig. 4.13 shows the effect of contactor and target selection on the different contact configurations.
lexibility apart from rigid body motions. All their nodal degrees of freedom must be either fixed, have enforced displacement, or berigidly linked to a master node.
Contactorsurface
No penetration
No penetration
Targetsurface
Targetsurface
Contactorsurface
Figure 4.13: Contactor and target selection
4.1: Overview
ADINA R & D, Inc. 441
lable
command). The new representation involves more accurate traction calculation, more detailed contact search and more accurate constraint enforcement for quadratic contact segments. The new contact surfaces are currently restricted to work with only the following options:  Constraintfunction algorithm  All 3D and shell element types except for 16node shells
All 2D element types except for 4node beams  No consistent contact stiffness  Contact surface offsets = 0  Singlesided contact only  Frictionless and regular Coulomb friction (no usersupplied
friction)  Eliminate or ignore initial penetrations (no discarding of initial
penetrations)  Regular contact surface definitions (no contact body or contact
point definitions, no analytical rigid targets)  Static, implicit dynamic and frequency analysis The direction of a contact surface is marked by a normal vector
n pointing towards the interior of the contact surface as shown in Fig. 4.14 and Fig. 4.15.
• Selfcontact is when a contact surface is expected to come intocontact with itself during the solution. • A default and a new contact surface representation are avaiin ADINA (set via the CSTYPE parameter in the MASTER

•
Chapter 4: Contact conditions
442 ADINA Structures Theory and Modeling Guide
k1
k k+1
k+2
Contactor
surface
Continuum
Contactor segment
n
nr
Z
Ynn
=normal vector, acting into the continuum
r=tangent vector
igure 4.14: NF ormal vector of a contact segment in 2D analysis
N4
N33D contact surface
segment (area = A)
n
Normal to the segment
at (r = 0, s = 0)
r
s
N2
N1
Segment contact force =(integration of segment tractions)
F
Average normal traction = [( . ) ]/AF n n
Average tangential traction = [  ( . ) ]/AF F n n
Figure 4.15: Normal vector of a contact segment in 3D analysis • he contact
ternal and the other side to be external. Any contactor node within the internal side of a target surface is
For singlesided contact (see Fig. 4.16), one side of tsurface is assumed to be in
4.1: Overview
ADINA R & D, Inc. 443
assumed to be p face. This singlesided option is ideal for contact surfaces on the faces of
elements since in that case it is clear that one side is internal e solid while the other is external. In this case, the external
side can usually be predicted from the geometry. This option is also useful for shells when it is known that contact will definitely occur from one direction. In this case, however, the program cannot intuitively predict the internal side of the contact surface.
enetrating and will be moved back to the sur
solidto th
Target surface
External side
Internal side(no contactor nodes allowed)
Figure 4.16: Singlesided contact surface
• In doublesided contact (see Fig. 4.17), there are no internal or external sides. The contactor surface nodes in this case are
rface
et
prevented from crossing from one side of the target contact suto the other during solution. This option is more common for shellbased contact surfaces. If a contactor node is one side of the targsurface at time t, it will remain on the same side at time t + ∆t.
Contactor node cannotpenetrate upper side
Contactor node cannotpenetrate lower side
Targetsurface
Figure 4.17: Doublesided contact
Chapter 4: Contact conditions
444 ADINA Structures Theory and Modeling Guide
• When the tied contact feature is selected for a contact group, ADINA performs an initial contact check at the start of the analysis. All contactor nodes that are found to be in contact are permanently attached to their respective target segments. This is conceptually similar to using rigid links or constraints to attach the node to the target surface. The main difference is that the coefficients for the rigid elements are automatically determined by the program and they are only applied for the nodes that are initially in contact. The basic idea is illustrated in Fig. 4.18.
Contactor surface
Target surface
Gap exaggeratedbetween
contact surfaces
Permanent rigidconnections
Figure 4.18: Tied contact option
Tied contact is not "real" contact because there can be tensiobetwee
n
n tied contact surfaces. Also no sliding can occur between tied contact surfaces.
e
If the contact surfaces initially overlap, they are not pushed
puted based on the initial nodal positions only
e of the contactor surface is paired with a node on the target contact surface (a node
The tied contact option can be used to glue two surfaces together if they have incompatible meshes. However, the surfaceglueing feature described in Section 5.16 produces more accuratglue equations.
back to eliminate the overlap.The tied contact constraint equations are com
. The constraints generated in tied contact are not updated during the analysis. Hence, they may be less accurate if the bodies experience large rotations. • When nodenode contact is used, each nod
4.1: Overview
ADINA R & D, Inc. 445
ode contact pair). A node on the contactor surface can come into contact with the infinite plane passing through the paired node on the target surface; the infinite plane is perpendicular to the normal
irection indicating the direction within the target body (Fig. 4.1t
cting each ontactor and target nodes.
n
d9). This normal direction, and hence the infinite plane, is constanduring the response. The normal directions can be based on the geometry of the target segments or on the vector connec
Contactor nodein nodenodecontact pair
Target nodein nodenodecontact pair
Infinite plane passingthrough target node
Target node normalContinuum
Nodenode contact pair
n
ct tatus information.
• The normal contact conditions can ideally be expressed as
Figure 4.19: Nodetonode contact
• The user can request output of nodal contact forces and/or contact tractions. Using the default contact surface representatiogives output of segment tractions. The new contact surface epresentation outputs nodal tractions as well as gap and contar
s
0; 0; 0g gλ λ≥ ≥ = (4.11) where g is a gap, and λ is the normal contact force. Different algorithms may vary in the way they impose this condition. • For friction, a nondimensional friction variable τ can be definedas
TFτµλ
)
ref. KJBSection 6.7.2
= (4.12
Chapter 4: Contact conditions
446 ADINA Structures Theory and Modeling Guide
• The standard Coulomb friction condition can be expressed as
where FT is the tangential force and λ is the normal contact force.
( ) ( )
1and 1 implies 0 while 1 implies sign sign
uu
ττ
τ τ
≤< =
= =
##
(4.13)
where is the sliding velocity. • In static analysis, the sliding velocity is calculated by dividing the incremental sliding displacement by the time increment. Hence, time is not a dummy variable in static frictional contact problems. • When (Coulomb) friction is used, the friction coefficient can be constant or calculated from one of several predefined friction laws.
ble states of the contactor nodes and/or segments are
n the contactor node and the target
rce less than the limit Coulomb force.
as long as the tangential force on the contactor node (equal to
the normal force times the Coulomb friction coefficient), the contactor node sticks to the target segment.
4.2 Contact algorithms for implicit analysis
ADINA offers three contact solution algorithms (set via the ALGORITHM flag): ! Constraintfunction method,
u#
• The possi
No contact: the gap between the contactor node and target segment is open. Sliding: the gap betweesegment is closed; a compression force is acting onto the contactor node and the node kinematically slides along the target segments (either due to frictionless contact to a frictional restrictive fo
Sticking:that initiates sliding is less than the frictional capacity
•
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e multiplier (segment) method, or
ethod
• Each contact group must belong to one of these three contact
4.2.1 Cons
! Lagrang
! Rigid target m
algorithms. However, different contact groups can use different algorithms. • All 3 contact algorithms can be used with or without friction.
traintfunction method
• In this algorithm, constraint functions are used to enforce the nopenetration and the frictional contact conditions. The inequality constraints are replaced by the following normal constraint function:
( )2
,2 2 N
g gw g λ λλ ε+ −⎛ ⎞= − +⎜ ⎟⎝ ⎠
where εN is a small userdefined parameter. The function is shown in Fig. 4.21. It involves no inequalities, and is smooth and differentiable.
g
w(g, )
Figure 4.21 Constraint function for normal contact
Chapter 4: Contact conditions
448 ADINA Structures Theory and Modeling Guide
• The constraint function method regularizes the stickslip transition of Eq. (4.13). This results in a smooth transition from tick to slip and vice versa, and it also results in a smooth
rgence sdifferentiable friction law that is less likely to cause convedifficulties.
The frictional constraint function ( )v ,u τ# is defined by
vv arctan
T
uτε
⎛ ⎞2 − 0+ − =⎜ ⎟π ⎝ ⎠
#
Here εT is a small parameter which can provide some "elasticity" to the Coulomb friction law as shown in Fig. 4.22.
u
1
1
.
*
Figure 4.22: Constraint functions for tangential contact ( )v ,u τ#
ange multiplier (segment) method
• In this method, Lagrange multipliers are used to enforce the contact conditions of Eq. (4.11). The kinematic conditions areenforced at the contactor nodes, and the frictional conditions are enforced over the c
4.2.2 Lagr
ontact segments.
s. It also
hould not be used with a force or energy/force convergence criterion.
• This method involves distinct sticking and sliding statecalculates this state for each contactor node based on the contact forces on the target segment. • This method s
4.2: Contact algorithms for implicit analysis
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• This is a significantly simplified contact algorithm intended
et surface must be rigid, and contact is enforced using a enaltybased approach. The normal modulus is set via the
NORMALSTIFFNESS parameter. In addition, two major simplifying assumptions are used which facilitate convergence, but at the same time reduce the acceptable time step size. These simplifications are: ! New contactor nodes can only come in contact at the first iteration of a load step. Contact is not enforced at any contactor node that comes into contact after the first iteration. These
re penetrate the target.
! Contactor nodes currently in contact can only be released
h
st be checked to ensure that the solution does the original contact conditions.
• If a tensile force (from the second simplification) is higher than a userspecified tolerance (RESIDUALFORCE parameter), the
me step is reduced and the load step is repeated. Similarly, if the um of all tensile contact forces is greater than a userspecified
tolerance (LIMITFORCE parameter) the time step is reduced and the load step is repeated.
he manner in which the time step is reduced due to tensile forces governed by the RTSUBDIVIDE parameter in the MASTER
command. It can be based on the magnitude of the tensile contact forces, or on the general subdivision parameters set for Automatic Time Stepping (ATS) in Section 7.2. •
4.2.3 Rigid target method
primarily for metal forming applications. • The targp
nodes may therefo
also in the first iteration of a load step. Nodes trying to be released at intermediate iterations will be kept in tensile contact.
• These simplifications require that a small time step be used witthe Rigid Target contact algorithm. Also the resulting penetrationsand contact forces munot excessively violate
tis
Tis
Rigid target penetration checking can be selected (via the RTPCHECK parameter) to ensure that the penetrations (due to the first
Chapter 4: Contact conditions
450 ADINA Structures Theory and Modeling Guide
simplification) do not exceed a preset level. If the penetration exceeds the tolerance, the time step is reduced by the program and the load step is repeated. • Friction in the Rigid Target algorithm is also treated using a penaltybased approach. The tangential modulus is set via the TANGENTIALSTIFFNESS parameter.
4.2.4 Selection of contact algorithm
• Our experience is that in most frictionless contact problems the constraint function method is more effective than the segment method. The constraint function method is the default. • For friction contact dominated contact problems, the performance of the constraint function and segment methods are comparable. The default is still the constraint function method. • The rigid target method requires much smaller time steps and has several input parameters which need to be properly adjusted. It should only be used for metal forming type problems. • Note that the target surface can be rigid in all three contact algorithms. The presence of a rigid target does not mean that the rigid target algorithm should be used.
4.3 Contact algorithms for explicit analysis • ADINA offers three contact solution algorithms for explicit analysis: ! Kinematic constraint method, ! Penalty method, or ! Rigid target method
• Each contact set must belong to one of these three contact algorithms. However, different contact sets can use different algorithms.
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• fric
4.3.1 Kine
e default explicit contact algorithm.
• conpendis al system of equations. Instead, a good approximation is done. In this case, for eac
All these contact algorithms can be used with or without tion.
matic constraint method
• This is th
A predictor step is first done without applying contact straints or forces. Then displacements are evaluated and etration is detected and corrected. The exact correction of
placements requires the solution of a nondiagon
h penetrating contactor node, a penetration force
*N N N2C C C CM M
tδ
= =∆
F a N
alculated. This is the force required to remove the penetration at contactor node. However, not all the penetration will be oved by moving the contactor. The target will get
is ctherem some motion depending on its mass relative to the contactor and how many contactor nodes are touching it. So, the force above is projected to the target segment nodes:
wh nt to that of each target node. Similarly, the mass of the contactor node is projected to the target in the sam way:
NCF
i
N NT i CN=F F
ere Ni is the shape function relating the contactor displaceme
e
iT i CM N M= and th d to that of the target node itself. Then the accele
is mass is adderation of the target node is determined as
( )i i
N NT T T TM M+ =∑ ∑a F
This correction is then used to update the target displacements. The contactor acceleration is
Chapter 4: Contact conditions
452 ADINA Structures Theory and Modeling Guide
i
*N N NC C T N= − ∑a a a
• For friction, a similar approach is used. A correction force is calculated
*T TC CM
t=
∆vF
where vT is the tangential sliding velocity. However, this force cannot exceed the limit force based on the normal force and the coefficient of friction
( )*min ,T NC Cµ=F F T
CF
Th
ping,
• A modification is also required for rigid targets, which are common in contact. The form of the equations in this case depends on whether the rigid target has natural or essential boundary conditions.
.3.2 Penalty method
• In this algorithm, contact conditions are imposed by penalizing the interpenetration between contacting surfaces. When a penetration is detected, a normal force of
e rest of the procedure is very similar to the case of normal contact. The form of the equations is different if there is damand is also different if the previous and current time steps are not the same.
4
( )N N D NA A K Kδ δ= = +F P N # is applied to the contactor node, where KN is the normal stiffness, KD is a normal rate stiffness, Nδ is the penetration, Nδ# is the penetration rate, N is the normal vector pointing towards the contactor, A is the contact area and P is the normal contact traction. An opposing force is distributed to the target nodes. • Similarly, in the presence of friction, the relative sliding
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ADINA R & D, Inc. 453
velocity between the two bodies is penalized as follows:
T TA K=F v where vT is the tangential sliding velocity. • The normal and tangential penalty stiffnesses KN and KT can be
ram. te ss
y
the
lty stiffnesses will lead to
exc
4.3.3 Rigid
uld only be used when the analysis might involve restarts to or from an implicit analysis involving the im rig mal contact coalgorithm onstraint method. Friction conditions are imposed using the same penalty approach as in the
parameter. H
4.3.4 Selec
the simplest and fastest of the explicit
selected by the user, or determined automatic