finite element contact analysis: marc and dyna3d compared · finite element contact analysis: marc...

Finite Element Contact Analysis: MARC and DYNA3D compared

A.W. W. ha. Kuij pers

wfnr-report 94.033

Eindhoven University of Technology Faculty oflvíechanicai Engineering Division of Computational and Experimental Mechanics

Coaches: M.H.A. Claessens (TUE, wfk) A.A.H.J. Sauren (TUE, w h ) J.G.M. Thunnissen (TNO, iw)

March 1994

Abstract

This report deals about the research of contact modelling in the finite element c d e s MARC and DYNA3D. The research was performed to investigate which extra problems are introduced by modelling contact in dynamic finite element analyses.

In chapter 2, a description is given of the tasks that a finite element program has to carry out for contact modelling. The so-called contact-searching and contact interface algortihms that are designed for these tasks are discussed for both MARC and DYNA3D, in particular the solver contstraint method in MARC and the penalty method in DYNA3D.

In chapter 3, the contact impact problem that was used to test the finite element contact procedures in MARC and DYNA3D is presented. The problem configuration constist of a deformable cylindrical rod that impacts another object. The materid of the rod was modelled both as a linear elastic and linear visco- elastic solid. To investigate different types of contact modelling, the ‘object’ was modelled with boundary conditions, as a rigid body and as a deformable body. The resulting contact problems were analogous, in spite of the clear geometrical differences between the impacted objects. Some analytical solutions to the impact problem are also presented in this chapter. These solutions are used to validate the results of the finite element calculations.

Chapter 4 deals about the finite element calculations performed with MARC. First a description of the MARC system and its pre- and postprocessing facilities is given. Then the results for the finite element calculations with the three test configurations are presented. The results of these calculations are in good agreement with the analytical solutions of chapter 3. One discrepancy with the analyticially derived results was that stress peaks were created, immediately after contact is detected. These peaks seem to have a numerical nature and can be diminished by increasing the numerical damping in the model. In the problem of deformable to deformable body contact, some contact chattering occured. Some extra attention should be given to the problem of reducing this contact chattering.

The finite element calculations and results performed with DYNA3D are presented in chapter 5. In this chapter the DYNA3D system is shortly reviewed and then the results for the test problems are presented. There was a considerable difference between the analytical and MARC solutions on the one side and the results obtained with DYNA3D on the other side. The stresses and strains for the contact region of the rod were overestimated in DYNA3D. This discrepancy could be traced back to the contact algorithm used in DYNA3D. Apart from this problem, the main drawbacks from using the DYNA3D system are the analysis inflexibilities in the DYNA3D code and the impractical pre- and postprocessing facilities.

The conclusions that were drawn from this report are listed in chapter 6. The most important conclusion is that MARC offers the best options for finite element contact analysis and gives better results for the contact impact test problems. Also MARC is more flexible and easier extendable than DYNA3D. In Ou; opinion, chese dvantages iii MARC csiinrerbdance the advaneages of ehe DYNA3D program which comes with source code and a free licence.

.

1 Introduction 3

1.2 Finite element contact modellingwith MARC and DYNA3D . . . . . . . . . . . . . . 3 1.1 The need for finite element contact modelling . . . . . . . . . . . . . . . . . . . . . 3

2 Finite element contact-impact procedures 4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Contact-searching algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Contact interface algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Contact-searching algorithms in MARC and DYNA3D . . . . . . . . . . . . . 4

2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3.2 Contact interface algorithms in MARC and DYNA3D . . . . . . . . . . . . . 6

3 Test configurations for FEM contact impact tests 9 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Analytical solutions to the impact problem . . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 Wavepropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3.2 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Finite element contact calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4.1 Analytically derived solutions . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4.3 Rigid to deformable body contact problem . . . . . . . . . . . . . . . . . . 15 3.4.4 Deformable to deformable body contact problem . . . . . . . . . . . . . . . 16

3.4.2 Reference problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 FEM contact-impact tests using MARC 17 4.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.1 AnalysiswithMARC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.2 Pre- and postprocessing using MENTAT . . . . . . . . . . . . . . . . . . . . 19

4.3 FEM contact analysis using MARC . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3.1 Reference calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.2 Rigid to deformable body contact calculations . . . . . . . . . . . . . . . . . 24

4.2 UsingMARC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

I !

2 Contents .

4.3.3 Deformable to deformable body contact calculations . . . . . . . . . . . . . 27 4.3.4 Concluding remarks about finite element contact analysis with MARC . . . . . 29

5 E M contact-impact tests using DYNA3D 30 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 UsingDYNA3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2.1 Analysis with DYNA3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2.2 Preprocessingusing INGRID . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2.3 PostprocessingusingTAURUS . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 FEM contact analysis using DYNA3D . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3.1 Reference calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3.2 Rigid to deformable body contact calculations . . . . . . . . . . . . . . . . . 36 5.3.3 Deformable to deformable body contact calculations . . . . . . . . . . . . . 39 5.3.4 Concluding remarks about finite element contact analysis with DYNA3D . . . 40

6 Conclusions and recommendations 42

References 43

A Vector operations 44 Al Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 A 2 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Introduction

1.1 The need for finite element contact modelling

Finite element contact modelling is used in various engineering applications. Traditionally contact analysis is used for modelling processes like rolling iron, extrusion and other metal forming processes. These processes can all be described quasi-statically.

In the field of, for instance, car crashworthiness testing, finite element contact analysis is also used. However, inertial effects play an important role in this field of work and therefore accurate modelling results in dynamical contact analysis also known as contact-impact analysis.

Another use of finite element contact analysis is in the biomechanical research area. During the last two decades various finite element models to investigate the head dynamic response to impact have been proposed. Mostly, these models consist of isotropic structures with simple geometries based on ‘average’ human head dimensions. In these models the brain is often rigidly connected to the skull. This description of the head-brain interface is not correct considering the head’s anatomy. The use of dynamic contact analysis could improve the proper modelling of this interface.

1.2 Finite element contact modelling with MARC and DYNA3D

At the faculty of mechanical engineering at the Eindhoven University of Technology (TUE), the division computational and experimental mechanics has started a research program in 1992 to investigate the dynamical response of the human head under extreme loading conditions. In this scope the use of finite element contact procedures is investigated. The algorithms that impose the contact conditions have to be tested before finite element contact analysis can be used in head modelling. Otherwise, numerical effects arising from the contact algorithms cannot be distinguished from physical effects resulting from the head

In this report the use ofdynamic finite element contact analysis in two finite element codes; MARC and DYNA~E is Oiscwsed. The chûice fûr the -me ûf thac pïûgïams waj ïadï2ï aïbitïaïy. The MARC program has been used for modelling polymer and biological materials at the TUE mechanical engineering faculty. D Y N A 3 D is used by many researchers in the field of head injury biomechanics and has proved its value.

dYE2EiCS.

The simulations with MARC were performed at the Eindhoven calculations with D Y N A 3 D were carried out at the TNO Crash Safety

University of Technology and the Research Centre in Delft.

3

Chapter 2

Finite element contact-impact procedures

2.1 Introduction

In this chapter some basic concepts €or finite elemene coneace algorithms are presented Basically, the contact algorithms take care of two things. First, each node that is (or could come) into contact with its global remainder is identified. The algorithms performing this task are called the contact-searching algorithms. Then in the following stage the contact conditions are applied to the contacting nodes such that conservation of momentum and the non-penetration constraints are satisfied. The contact interface algorithms are designed for this stage (Wismans 1993, p. 193).

The finite element calculations in this report were executed with both MARC and DYNA3D. Since these programs have elementary different approaches to the contact problem, the contact algorithms in both codes will both be discussed briefly.

2.2 Contact-searching algorithms

2.2.1 Overview

The goal of contact searching algorithms is to find the intersection of a contact node with its global remainder, or equivalently, to find all segments in touch with (or near enough to) a given node. To search these contact nodes several different algorithms have been developed. It is never a problem to develop a contact-searching algorithm which can succesfully locate all contacting nodes or potential contacting nodes with the boundary surface defined mathematically. However the contact-searching algorithms must be sufficiently efficient while fulfilling the assigned tasks. Discussing all commonly used contact-searching algorithms is not within the scope of this report. For a survey of several contact-searching algorithms the reader is referred to for instance Zhong (1993, chap. 10).

Since the contact-searching algorithms are poorly described in both the MARC- and DYNA3D-manuds, only general remarks about these algorithms can be made. In both programs contact of a (possible) contact node with a contact surface is considered. In DYNA3D a node is considered to be in contact if its distance to the closest contact surface is less than zero, which means the node has penetrated the surface. In the

4

I

Finite element con tact-impact procedures 5 -

MARC code a node is in contact if its movement in normal direction to the closest surface brings it within a (user) specified distance tolerance from that surface. The two formulations are equivalent if we set the distance tolerance to zero in the MARC program. In DYNA3D the user cannot specify a contact distance tolerance.

Implementation in MARC

Contact in MARC between deformable bodies is such that each node on each body is mutually checked for contact with each surface of every body. There is no masterlslave relation; the default is that every body will be checked for contact versus every other body. Thus, a robust and reliable method for contact searching is implemented.

Implementation in DYNAQD

According to Hallquist and Whirley (1991), the contact in DYNA3D is formulated using a “node on surface” concept. One side of the contact-interface is calied the master side, the other side is cdled the slave side. Nodes that lie on the slave side, the so called slave nodes, are restricted from penetrating master surfaces. Conceptually, the mastedslave algorithms could be thought of as looping over all slave nodes, and for each slave node checking that there is no penetration through the any of the master surfaces defined for this contact interface. In the symmetric treatments, the designation of the two surfaces as slave or master are interchanged and the algorithm is applied a second time. Thus, each surface goes through the aigorithm once as a master surface and once as a slave surface. This symmetric approach has been found to increase the robustness and reliablility of the contact algorithm when both surfaces are deformable.

2.3 Contact interface algorithms

2.3.1 Overview

The contact interface algorithms apply the contact-impact conditions. In this stage ofthe contact algorithm, constraints are imposed on the contacting nodes that were found by the contact-searching algorithms, such that penetration ofthe node into its global remainder is prohibited. More exact, the contact constraints are imposed on the Contacting boundaries. This can be done by using several methods: Lagrange multipliers, penalty functions or solver constraints. Here the penalty functions and solver constraints are discussed. The Lagrange multiplier method (among others) is extensively reviewed in Zhong (1993, chapter 4).

Contact between bodies can be subdivided into two types of contact:

X rigid to deformable body contact,

X deformable to deformable body contact.

The rigid body in a contact analysis is assumed to have an infinite mass. Thus, the velocity of a hitting node dier ecat2ct shodd be equal te the velocity of the rigid body befoïe the cû!!isiûn. In c a e ofdefoïmable to deformable body contact extra considerations are involved because momentum has to be conserved at each node. In general, a collision of deformable bodies affects the velocities of the contact-boundary nodes in both bodies unlike in a collision of a deformable body with a rigid body for which the rigid body velocity stays unchanged. Apart from these considerations, the contact interface algorithms for rigid to deformable body contact and deformable to deformable contact are basically the same.

G Chap ter 2 -

2.3.2

Implementation in MARC

When contact is detected in MARC the contact interface conditions are applied with so-called solver constraints. In MARC the contact conditions that are imposed depend on the type of contact (deformable to rigid or deformable to deformable). Whenever contact between a deformable body and a rigid body is detected, imposed displacements are automatically created. Whenever contact between two deformable bodies is detected, multipoint constraints (called ties) are automatically created (see MARC user manual volume A., 1992a, p. 5.68).

The strategy implemented in MARC for deformable to rigid contact is based on the direct application of boundary conditions. When contact is detected a local coordinate transformation for the contacting node is defined such that one of the locai axes is normal to the rigid body surface (positive sign pointing into the body). Then a displacement increment utgid for the deformable body is imposed in that direction, in magnitude being equal to the projection of the rigid body motion along the normal. In this way a node follows the body surface. This is exemplified in figure 2.1.

Contact interface algorithms in MARC and DYNA3D

/

Figure 2.1: Rigid to deformable contact in MARC.

As a result of the local transformations made during contact, the reaction forces calculated after a solution of an increment, are available in the transformed system. The reaction force along the normal to the body surface is used to detect separation. If the reaction force is larger than the separation force tolerance (which can be specified by the user), the node separates and the contact transformations and boundary conditions associated with it are made inactive.

In case of deformable to deformable contact appropriate tying constraints will be imposed to maintain the contact. The essence of the method is that, once a node touches a surface, a new tie is activated that relates the displacement increments of the node to the displacement increments of the nodes of the closest boundary. This is illustrated in figure 2.2. The displacement increment of the tied node during

Finite element contact-impact procedures 7 -

Figure 2.2: Deformable to deformable contact in MARC.

contact T in normal direction to the plane which contains point of contact P is a weighted average of the displacement increments of the retained nodes during contact RI and R2, or:

Au: = aAuE1+ (1 - a)Au?,

where a is defined by

PR1 RlR2 *

a,=-

From the analysis it will follow that:

BV: = aBw,W.l + (1 - a)Av?, (2.2)

and

Au: = aAu: + (1 - ")Au?,

where Au,, AV, and Au, are the incremental nodal displacement, velocity and acceleration components normal to segment R1R2 respectively. The unit normal to the segment R1R2 is denoted by 9. When the tie is activated the contacting node follows the surface of the contacted surface; it may slide along it or be stbask assording t~ the genera! contact conditions (i.e. the friction m d e ! SSPO).

In each step the reaction forces of the nodes in contact are checked to determine whether these nodes stay in contact or have to be released. The modelling of contact friction is also supported in MARC but this feature will not be used in the impact tests presented in this report.

The main advantage of the contact option in MARC is that (undesired) penetration of nodes is preventedwhile no new unknowns are introduced (which is the main drawback of the Lagrange multipliers

8 Chapter 2 -

method). The non-penetration constraints are exactly satisfied at the extra expense of the evaluating impact, momentum and release conditions. A more detailed description of the contact interface algorithms used in MARC can be found in Konter (1993). For a more general review about dynamic contact analysis in MARC the reader is referred to de Graaf and Konter (1993).

Implementation in DYNAQD

The contact interface constraints in DYNA3D are imposed by means of a penalty function method. The penalty method consists of placing normal interface springs between all penetrating nodes and the contact surface. If a node has penetrated a contact surface we add an interface force vector f,, normal to the surface containing the contact node. This force vector depends on the amount of peneiration p and a stiffness factor kc which depends on a penalty parameter a, the bulk modulus Ki, volume V, and face area Ai of the element that contains the contact node i, or

with:

and as the unit normal vector to face area Ai. The normal force component is also projected to the nodes of the adjacent contact segment in the contact surface. For more information the reader is referred to Hallquist (1983).

In contrast with the Lagrange multiplier method and the solver constraints method used in MARC, the penalty method assumes that the impenetrability condition will be violated. Because too small penalty parameters may result in too large penetrations, the penalty parameter should be chosen very large. However, in explicit time integration schemes, large penalty parameters may cause numerical instability. The choice of penalty parameters is primarily determined by numerical experience. The modelling of Coulomb friction in the contact area is also supported in DYNA3D, but in this report this feature will not be looked at.

The main advantages of the penalty method are that no new unknowns are introduced, unlike with &e Lagrange multiplier method, and that the implementation is rather simple. Also momentum is exactly conserved with no extra considerations for impact and release.

Test configurations for FEM contact impact tests

3. P Introduction

In this chapter the test configurations which were used to test the finite element contact procedures in both MARC and DYNA3D are presented. The test configuration represents an axisymmetric rod which impacts another object with a prescribed initial velocity. The finite element solution will be compared with analytical one-dimensional impact solutions and the solutions to Navier‘s wave equations.

3.2 Geometry

The geometry that was used in the finite element tests is an axisymmetrical cylindrical rod with a length L = 0.05 [m] and a diameter D = 0.005 [m]. Due to the axisymmetrical nature ofthe problem movement in radial direction of the nodes that lie on the axis of symmetry was restrained. This geometry is referred to as the impacting body. Another geometry was modelled such that the front end of the rod would make contact with this geometry during the simulation. This body will be called the target body. A graphical representation ofthis configuration is given in figure 3.1.

The material of the rod was modelled both as isotropic linear elastic and isotropic linear visco-elastic. The shear relaxation modulus for the linear visco-elastic material behaviour is given by:

G(t) = G, + (Go - G,)e-$, (3.1)

where Go is the short term shear modulus, G, is the long term shear modulus and T is the relaxation time constant. The material properties for both material models are listed in table 3.1. The density for both materials was taken p = 7800 [kg/m3]. The material parameters for the elastic solid represent common stee!. The puzmeters f ~ r the visco-e!a9cic ate er ia! were cheser, such that the short term behaviouï of the material resembled that of the elastic material. The time constant was chosen such that noticeable visco-elastic effects occur during the time of the simulation. The long term shear modulus was chosen arbitrarily.

When contact between two bodies is considered, it is necessary to impose non-penetration constraints. In the finite element codes, definingso called ‘sliding surfaces’ in DYNA3D or the use of ‘contact elements’

9

10 Chap ter 3 -

v =-lS[ds] t--

L Y

n m

Target body L Y- 2

Impacbng Body

Figure 3.1 : Test configuration.

j Materiai modei

buik modulus K

Table 3.1 : Material properties for the impacting rod.

in MARC is sufficient to account for these contact constraints. For the target body both a rigid body and a deformable body were chosen. This was done because

most finite element codes implement a different approach for the contact problem between a rigid to deformable body and a deformable to deformable body.

3.3 3.3.1 Wave propagation

Generally speaking there are two types of waves that can occur in elastic media:

Analytical solutions to the impact problem

X pressure waves, also called longitudinal or dilationai waves,

X shear waves, also called transverse or distortional waves.

Here, only longitudinal waves are discussed.

Elastic waves

The general equation of motion for a linear isotropic elastic body written in terms ofdisplacement (Navier's equations), in the absence of a body force, can be written as:

4 - 4 4

p ü - pv2ü - (A + p)V(V * U) = o,

Test configurations for FEM contact impact tests 11 -

where X and p are the so-called Lamé constants, defined by:

Eu A =

( 1 + u)( l - 2 4 ’

A short review of the tensor operators used here, is given in appendix A.

here (Zukas et al. 1982). The components of the displacement vector U in Cartesian coordinates: Some solutions to equation (3.2) which correspond to longitudinal wave propagation are presented

27r 1 u, = uy = O, u, = Asin(-(z f c t ) ) ,

satisfy (3.2) where c assumes the special value C, with

(3.3)

(3.4)

The factor C, is also known as the elastic pressure wave propagation speed. Note that when the material becomes nearly incompressible and u + 0.5 then C, -+ 00.

visco-elastic waves

The wave propagation problem in visco-elastic media is more complex. Only analytical solutions for the one-dimensional wave propagation problem are presented here. The general equation of motion for a one-dimensional visco-elastic solid can be written as (Morrison 1956):

p a3u a2u d3u a2U -- E2 at3 + p q @ - - (1 + 2) + El’ldzz’

where Ei, E,, 7 and u are related to Go, G,, K and 7- by the following relationships:

GO - G,)K 3K+(Go-G,)’

E1 =

(3.5)

9GoK 3K + Go E2 = - El,

3K - 2Go 2(3K + Go) ’ E?

u =

- 7 - = U

Morrison found that the pressure wave propagation speed C, in the one-dimensional visco-elastic material satisfies:

El + E2 c; = P

12 Chap ter 3 -

The factor E1 + E2 is known as the impact modulus of the visco-elastic material (Fliigge 1975). This modulus represents the short term elastic behaviour of the visco-elastic material. From the one-dimensional elastic wave propagation theory it is known that the pressure wave propagation speed is defined by:

E P

cp = -. (3.7)

If we bear in mind that the short time behaviour of the visco-elastic solid is predominantly elastic, the resemblance between equations (3.6) and (3.7) is not surprising.

Extending the one-dimensional wave propagation theory of visco-elastic solids to three dimensions is not straightforward and not within the scope of this report. However, since the short time behaviour of a three-dimensional visco-elastic material resembles elastic behaviour, it is expected that the pressure wave propagation speed for this material will resemble the pressure wave propagation speed of the elastic material given by equation (3.4), where E assumes the value of the impact modulus El + E2 of the visco-elastic material.

Another way to look at the impact problem is by applying the laws of conservation of energy. The rod originally only has kinetic energy &in. Just afier the collision of the rod with another body, the kinetic energy in some point of the rod will be converted into elastic energy Eel in case of elastic rod material when the pressure wave arrives at this particular point. There the velocity vanishes instantaneously which results in a zero kinetic energy. This fact can be used to predict the occurring strains and stresses for the elastic material (Zukas et al. 1982).

The kinetic rod has an initial kinetic energy per unit volume of:

(3.8)

where p is the rod's density and V is the rod's initial velocity. For the sake of brevity we limit the following discussion to the one-dimensional wave propagation theory. Then the elastic energy per unit volume for a linear isotropic elastic material can be written as:

1 2 Ekin = p v 7

with u the stress component, E the strain component in z-direction and E the Young's modulus of the material. Equalizing these equations yields:

u = dpEV2, I

(3.10)

(3.11)

Thus, expressions for the produced stress and strain due to the impact have been derived. When the rod impacts another body, a stress (and strain) wave will propagate through the rod. Stresses and strains at a particular position will be zero before the wave front reaches this position and will be 'quasi-constant'

Test configurations for FEM contact impact tests 13 -

when the wave front has passed. The term ‘quasi-constant’ is used because in a rod, the stresses and strains fluctuate with a small (decreasing) amplitudes around constant levels. These levels are the solutions for the problem if the one-dimensionai wave theory applies.

The pressure wave will first be propagated through the rod in positive z-direction with a pressure wave propagation speed C, given by equation (3.4). When the pressure wave reaches the rear end of the rod the pressure wave will be reflected, resulting in e tensile w2ve whish will be propagzted with w2ve s p e d C, in negative z-direction. The strain that is produced by the tensile wave will cancel the strain caused by the pressure wave. Thus all elastic energy will be converted to kinetic energy. When the tensile wave reaches the front end of the rod, where the pressure wave was initiated, the rod will come free from the wall and will move in positive z-direction. The ‘rigid body’ velocity then equals -V = 15 [m/s] because all elastic energy has been converted back to kinetic energy and no energy has been dissipated. The collision is called perfectly elastic.

Visco-elastic materia

The visco-elastic and elastic rod have the same kinetic energy since their initial velocity, volume and density are identical. The short term behaviour of the visco-elastic material closely resembles the behaviour of an elastic material with a Young’s modulus of El + EZ which was presented in section 3.3.1. Therefore, it is expected that the initial strains and stresses at the front end of the bar initially are equal to the strains and stresses that were found for the elastic material.

Because of the time dependence of the constitutive relationships of visco-elastic material, the stresses and strains will not remain ‘quasi-constant’ when the wavefront has passed. Energy is lost due to viscous dissipation during wave propagation which is inherent in visco-elastic behaviour, so not ail of the kinetic energy can be transformed into elastic energy. Meanwhile, stress relaxation occurs in the part of the rod where the wavefront has already passed. This results in an uniform level ofstress in rod ‘&er‘ the wavefront.

The constitutive relationship of the visco-elastic material shows that the stiffness of the material decreases in time. This results in stress relaxation and creep which were already discussed in the previous paragraph. The time constant associated with these phenomena is the relaxation time constant T from equation (3.1).

3.4 Finite element contact calculations

In all finite element calculations, the same rod was used as the impacting body. For the target body two different geometries were chosen; a rigid and a deformable geometry. To be able to validate the results of the finite element calculations with proper modelling of contact phenomena, a reference finite element calculation was performed with rod a where a displacement constraint u, = O[m] was applied to the front rcd EG&§ of :he rûd.

This approach resulted in three different contact impact configurations to be analysed:

X ‘impact’ of the rod is accomplished by prescribing boundary conditions.

X the rod impacts a rigid wall structure.

8 the rod impacts an identical rod which has the same prescribed velocity but in the opposite direction.

14 Chapter 3 -

In the first case no real contact is considered, but the contact constraints are represented by the boundary conditions. In the second case contact between a rigid an a deformable body is considered, while in the third case contact benveen two deformable bodies occurs. The finite element approach for rigid to deformable body contact and deformable to deformable body contact is different. The results for these calculations should be exactly the same since all three contact impact configurations are essentially identical.

finite element calculations used for testhg the finite dement codes QYNAJD and ,MARC an initial velocity of V = -15 [m/s] was prescribed for the rod in axial direction. To investigate the contact phenomena that occur when the impacting body comes into contact with the target body, the displacements, velocities, strains and internal Cauchy stresses that are produced are mutuaíly compared for the impacting body These results are used to check whether the finite element representation of the collision is in good agreement with the the physical problem.

The results are presented for three nodes, which are all located on the axis of symmetry. Node nf is located at the front end of the rod and is a (possible) contact node, node nm is located in the middle of the rod and node n' is located at the rear end of the impacting body (see figure 3.1).

In

3.4.1 hdyticdly derived scoluaionns

It is expected that the compressive reaction forces, produced by the impact, will vanish when the pressure wave has travelled back and forth through the rod. The rod will then come free of the wall. The total time elapsed during the propagation of the pressure wave is also the total period of time that the contact occurs. Using the pressure wave speed from equation (3.4), this so-called contact timet, can be approximated by:

2 . L t, = -. CP

(3.12)

Applying the values oftable 3.1 for the elastic rod, the total contact timet, is about 1.7 - lo-' [SI. For the visco-elastic rod the contact time should be about the same since its short time behaviour is predominantly elastic.

To obtain values for the expected stress and strain levels we look at relations obtained in section 3.3.2. Using equations (3.10) and (3.1 1) we obtain the following values for the stress and strain in longitudinal direction:

U = dpEV2 = 6.1 * 108[Pa],

These values correspond to the elastic rod. In the visco-elastic rod the instantaneous stress and strain value of the front end nodes will be the same. However since stress relaxation and creep occur these values do not apply to the whole part of the rod where the wavefront has passed.

The vdüe for the pressure wave propagation speed san be astimated for the one 2nd thee-dimencicnal case for the elastic rod and in the one-dimensional case for the visco-elastic rod. Applying equation (3.7) for the one-dimensional elastic case the pressure wave speed is given by:

Test configurations for FEM contact impact tests

and applying equation (3.4) for the three-dimensional case by:

15 -

For the visco-elastic rod, equation (3.6) applies for the one-dimensional wave propagation theory. This results in:

which is the same result as for the one-dimensional elastic case. It is expected that the pressure wave propagation speed for the three-dimensional visco-elastic rod should also resemble the pressure wave speed for the three-dimensional elastic rod, because the short time behaviour of the visco-elastic rod is comparable to the behaviour of the elastic rod.

3.4.2 KeferenCe probiem

The finite element model for the reference problem is depicted in figure 3.2. We see that this configuration

V =-15 [mls]

Impdng body

Figure 3.2: Geometry used for reference calculation.

is an analogue of the contact impact configuration where the impacting and a target body are initially in contact and the target body prevents the (front end of the) impacting body from moving in the negative z-direction. The results for this calculation correctly describe ‘true’ contact as long as the reaction forces that are generated from the (nodal) constraints are positive (i.e. until the nodal constraints prevent the rod from moving in positive z-direction). The results can also be compared with analytical solutions for the impact problem.

3.4.3 Next the rigid to deformable contact is investigated, again both for elastic and visco-elastic materials. The rigid body is modelled as a wall without velocity. This configuration is depicted in figure 3.3. In this figure we see diat ai iiiiiial gap between the impacting body and target body was introduced to allow some rigid body motion of the impacting body before contact should be detected. For the rigid to deformable body contact an initial gap G = 1.5 * [m] was chosen. Considering the initial velocity, contact should occur after i - 1 0 - ~ [SI simulation time.

The analytical solutions for pressure wave speed, initial stress and strain, derived in the previous section also apply for this configuration since the contact configuration is essentially the same.

Rigid to deformable body contact problem

16 Chapter 3 -

Contact elements f.-----

Inpadiiig My

Y- 2

Figure 3.3: The rod impacting a rigid wall.

3.4.4 Deformable to deformable body contact problem A third series of finite element calculations was carried out with an impact configuration modelling the contact of two deformable bodies. The target body was chosen identical to the impacting body (see figure 3.4). Thus the contact configuration was identical to the formerly described probiems.

Contact elements

V =lS[m/sj V =-15 [mls] - -.

Target body 2G

Figure 3.4: The rod impacts an identical rod.

For the deformable to deformable body contact a gap of 2G = 3.0 * [m] between impacting and target body was used. Since the initial velocity V equals 15 [m/s], contact should again occur after i - [SI simulation time.

As in the rigid to deformable body contact configuration, the analytical solutions derived for the reference problem can be used here to predict the occurring stress, strain etc.. The contact situation is identical to that of the former problems but since deformable to deformable body contact is considered, the finite e!ement pgrzms ' s e 2 digerex apprmch ,V: chis yrob!em :hm foï the rigid io deformable body contact problem.

Chapter 4

FEM contact-impact tests using MARC

4.1 Introduction

In this chapter an overview is given about the MARC system. Then, the finite element contact-impact calculations performed with MARC are presented and discussed. The testing configurations used in the finite element calculations are described in chapter 3.

4.2 U s i n g M ~ ~ c The MARC system contains a series of integrated programs that facilitate analysis of engineering problems in the fields of structural mechanics, heat transfer and electromagnets. The MARC system consists of the following programs:

X MARC

X MARC-PLOT

X MARC-PIPE

X MENTAT

These programs work together to:

X Generate geometric information that defines the structure of the problem (MARC, MESH3D, PIPE and MENTAT)

X Analyse the structure (MARC)

X Graphically depict the results (MARC, MARC-PLOT and MENTAT)

Figure 4.1 shows the interrelationships among these programs. A full description of these programs can be found in the MARC user manual volume A (1992a). Here only the pre- and postprocessor MENTAT and analysis grogram MARC will be discussed.

17

18 Chapter 4 -

MESH3D MENTAT MARC-PIPE

MARC-PLOT

Figure 4.1 : The MARC system.

4.2.1 Analysis with MARC

MARC can be used to perform linear or nonlinear stress analysis in the static and dynamic regimes and to perform heat transfer analysis. The nonlinearities may be due to either material behaviour, large deformations or boundary conditions (e.g. contact modelling).

Physical problems in one, two or three dimensions may be modelled using a variety of elements. These elements include trusses, beams, shells and solids. Mesh generators, graphics and post-processing capabilities, which assist in the preparation of input and the generation of results, are all available in the MARC system. The equations governing mechanics and implementation of these equations in the finite element method are discussed in the MARC user manual volume A (1992a, Appendix).

For dynamic analysis with MARC, the user can choose between explicit or implicit (time) integration schemes. The schemes can be used successively. The user can choose between the implicit Newmark-@ or Houbolt method or the explicit central difference method. The modal superposition method is also implemented. A lumped or consistent mass matrix can be chosen for the dynamic analysis. The damping in the model can be expressed as modal, Rayleigh or numerical damping.

Awidevariety ofoptions such as non-linear loading, user-defined material models, non-linear boundary conditions etc. can be added to the program Using user sübroütines. The structure of these subroutines is documented in MARC user manual volume D (1992b).

The input-file for MARC consists of three parts. These parts are called decks and one can distinguish the problem definition, model definition and history definition deck. In the problem definition deck general parameters concerning the type of problem and the solver are set. In the model definition deck the finite element mesh is established, geometric and material properties are set and the boundary conditions

FEM contact-impact tests using MARC 19 -

are defined (including non-linear boundaries such as contact elements). Finally, in the history definition deck the time steps are defined and the variation of quantities in the MARC model deck can be defined.

4.2.2 Pre- and postprocessing using MENTAT I

Overview

Pre- and postprocessing for MARC is most conveniently done with MENTAT. It is an interactive, mouse and command-line driven computer program that prepares and processes data for use with the finite element method. Graphical presentation of data is supported to review the large quantity of data typically associated with finite element analysis. Different types of graphical output including colour postscript can be produced.

The MENTAT program can process both two- and three-dimensional meshes to do the following:

8 generate and display a mesh,

8 generate and display boundary conditions and loadings,

8 perform post-processing to generate contour, deformed shape and time history plots.

The data that is processed includes nodal coordinates, element connectivity, nodal boundary conditions, nodal coordinate systems, element material and geometric properties, element loads, nodal loads/nonzero boundary conditions and elemendnodal sets.

MENTAT in practice

The use of MENTAT for pre- and postprocessing proved to be very useh1 in the contact-impact tests. MENTAT does not require to learn several command sequences to generate meshes and input problem parameters such as material and geometrical properties, element types, etc., because all options can be accessed using mouse sequences. However, this sometimes involves using many different (pull-down) menus. This disadvantage can be overcome with the use of command sequences or editing the MARC input file directly.

However some remarks must be made. In the preprocessing stage, MENTAT I! vl.2 is not capable of utilising all features of the MARC program. Sometimes not all values of some options can be set. Thus, it is often necessary to inspect the MARC-input file that MENTAT writes. When experience with the format of the MARC-input files has been obtained, it is often faster to edit the MARC-input file directly in stead of using MENTAT to alter the input.

Postprocessing can be done very accurately with MENTAT. The user can produce plots of deformed shapes with different postprocessingvariables such as strain and stras levels in coloured contours, as vectors etc. For an arbitrary number of nodes, time history or path plots can be pïodüced. MENTAT a h suppcrts making eigenmode or transient analysis movies from the produced data.

4.3 FEM contact analysis using MARC

The contact problem that was investigated with MARC is an axisymmetrical impact problem. Hence, the rod and the target bodies were also modelled as axisymmetric bodies since MARC supports the use of

20 Chap ter 4

axisymmetric elements. The arbitrary linear quadrilateral ring element (type “10”) was used. This way, a great reduction in degrees of freedom could be accomplished. The spatial discretization of the rod was done with 50 elements along the longitudinal side and 5 elements in radial direction. We chose physical linear material models and a finite element representation which accounted for possible large displacements.

In ail dynamic finite element calculations with MARC the Newmark-B time integration method vas wed. Furthermore9 numerical damping was introduced to prevent exciting non-physical modes associated with the time step value. The value of the numerical damping constant y was initially chosen y = 0.1 (de Graafand Konter 1993). The use ofa consistent mass matrix was chosen. Convergence testing for stress analysis in MARC can be done on the residual forces, the displacements and the strain energy. Since rigid body motion occurs in the problems we choose to check on the relative displacements. With this method convergence is satisfied if the maximum displacement of the last iteration is small compared to the actual displacement change of the increment.

In MARC, the user can choose to use a fixed time step, or automatic time stepping. In the latter case MARC determines the ideal time step for every increment, which can have certain advantages in the contact algorithms (i.e. the increment can be split if too many nodes come into contact at once). The user can specify between which bounds MARC should choose the time steps. Because of these advantages, we choose the automatic time stepping option. It is also possible to use both time stepping methods successively. In ail simulations MARC chose about 50 time steps per 1 . [SI which was the minimum number of time steps that we prescribed for the simulations.

To impose (possible) contact constraints the user defines the elements that could possibly come into contact with the global remainder as so-called ‘contact elements’. The MARC program then automatically uses the nodes on the boundary of these elements as the (possible) contact nodes. The user can specify a contact distance tolerance. If a contact node is within this distance from a contact element then the node is considered to be in contact. In all simulations this tolerance was set to 1 [m]. In MARC the user can also specify a separation force tolerance. If a tensile force occurs at a contacting node, the node separates fiom the contact body if this tensile force is larger than the separation force tolerance. In all simulations with MARC this tolerance was set to zero.

4.3.1 Reference calculations

The purpose of the finite element calculations was to investigate whether the MARC program produced correct and accurate results for contact-impact problems. In order to make a fair judgement on the exact influence of the impact algorithms in MARC, we first simulated an impacting body without contact elements (as described in section 3.4).

[SI. Thus, the simulation time was longer than the time of contact t , to be expected. At the end of the simulation time, tensile reaction forces were expected. The results for stress, strain, displacement and veielocicy iiì longitUdina! direction for nodes nf, E“‘ and n‘ a6 the front, middle and rear end of the rod, respectively, can be found in figures 4.2, 4.3, 4.4 and 4.5 for the elastic rod and in figures 4.6, 4.7, 4.8 and 4.9 for the visco-elastic rod. The results for strains and stresses can only be accessed per element, not per node, but in the figures, the values for the elements that contain the nodes nf, nm and n‘ is presented. The solutions for the nodes nf are given by the solid line (-), for nm by a dashed-dotted line (a - .) and for n‘ by a dashed line (- -).

The total simulation timet, for the calculations for both the elastic and visco-elastic rodwas 2.0

21

E;igure 4.2: Srress Q,, for e!asric rod in reference pro bíem . problem.

Figure 4.3: Strain eZz for elastic rod in reference

Discussion of the results for the elastic rod

The resuits seem to describe the contact correctly until t , = 1.9 [s] as can be seen from figure 4.4 because at that time the nodes nm and n' move past their initial position U = O [m] into the positive z-direction. This results in a tensile stress for node nf in figure 4.2. The value of the contact time is somewhat higher (about 10%) than was estimated in chapter 3.

An estimation for the pressure wave propagation speed C, can be found for instance when looking at figure 4.2. There we see that the pressure wave reaches the rear end node n' after about 0.8 loe5 [SI. The distance travelled by the wavefront equals L = 0.05 [m] so an estimation for the pressure wave speed is:

= 6.25 103[m/s]. L

0.8 - 10-5 Cp,estï,ei =

In section 3.4.1 of chapter 3 we found a pressure wave speed for the three-dimensional case of Cp,3d,el = 6.02 lo3 [m/s]. Thus the value for Cp,estl,el seems tenable. Another way to estimate the pressure wave speed is by looking at the contact time t,. This yields:

0.1 = 5.26 103[m/s]. 2L

Cp,est2,ei = - = t, 1.9 -10-5

This value is more close to the estimated pressure wave speed for the one-dimensional rod: Cp,ld,el =

The estimated stress and strain values from chapter 3 accurately describe the values resulting from the 5.19 ' IO3 [m/s].

finik? de,?.,enC 2r?dï§k (SPP fig'lrC5 4.2 2nd 4.3):

u = UFEM N dpEV2 = 6.1 * 108[Pa],

E = EFEM x --=2.9.10-~[-]. JE'

22

o ,< .- .............. , ;

i :

Chapter 4 I -

-14

’\ i I \ i ‘ _.. .......... <.._i ................ *.+ ................. j ............. ...i .......

Figure 4.4: Displacement u, for elastic rod in reférence problem.

Stress s-zz vs. time I

I as I 1 5 2 25

time [SI x 10s

Figure 4.6: Stress uzz for visco-elastic rod in reference problem.

Velocity v-z vs. time I

I 0.5 1.5 2 5 1 2

time [SI *io”

Figure 4.5: Velocity ‘u, for elastic rod in reference problem.

Strain e-zz vs. time osxl~’ I I

5 time [s] * 16’

Figure 4.7: Strain cZz for visco-elastic rod in refi eren Ce problem.

Finally, when looking at figure 4.5 we see that the velocities of the nodes before and &er the ‘collision’ are equal in amplitude. Hence, the contact has been perfectly elastic because all kinetic energy that had been converted to elastic energy is reconverted into kinetic energy resulting in the rod to come free of the waf! (had the f~on: nodes fist heen rescrzined).

Discussion of the results for the visco-dasde ïad

Looking at figure 4.6 we see that a tensile stress is produced after t , = 1.9 . [SI. This time can be interpreted as the total contact time for the visco-elastic rod. Comparing this value to the contact time of the elastic rod, we can conclude that the wave propagation phenomena in the visco-elastic rod are predominated by the elastic terms in the constitutive relations for the visco-elastic material. Since these


Hgsre 4.8: Displacement u, for visco-elastic rod in reference problem. reference problem.

Figure 4.9: Velocity v, for visco-elastic rod in

elastic terms were chosen to represent the behaviour of the elastic rod, it is not surprising that the values for contact time (and consequently for the pressure wave propagation speed) of the visco-elastic rod so closely resemble the values for the elastic rod.

It is more difficult to predict the stress and strain levels throughout the visco-elastic rod. As was expected, the initial stress and strain level for the front end node nf in figures 4.6 and 4.7 respectively, resemble the stress and strain levels ofthe elastic rod. Because of the exponential decay in the shear relaxation modulus for the visco-elastic rod (equation (3.1)), the elasticity modulus of the material decreases. The occurring stress relaxation and creep can be explained by this shear relaxation effect.

Looking at the displacements for the nodes in figure 4.8 we see that because of the creep in the visco-elastic rod, the middle and rear end node(s) have not reached their initial position after reciding hom contact. In figure 4.7 we see that this results in a non-zero strain when the contact is released. It is expected however that the strain level in the rod will eventually become zero because the rod is no longer loaded after it has come loose of the wall.

In figure 4.9 we see that some energy has been dissipated due to the viscous terms in the visco-elastic constitutive relations. The magnitude of the velocity of the rod before and after the ‘collision’ is not equal. Thus the kinetic energy that was transformed during the collision was partly converted to elastic energy and partly lost due to viscous dissipation.

24 Chapter 4 -

4.3.2 The rigid to deformable body contact calculations in MARC were carried out to investigate whether the contact searching and contact interface algorithms work properly and give accurate results. A simulation time t, = 5 - [SI was chosen to allow rigid body motion of the rod before and after the collision. The configuration is described in section 3.4.3

The time history plots for stress, strain, displacement and veiocity in longitudinal direction can be found in figures 4.10,4.11,4.12 and 4.13 for the elastic rod and in figures 4.14,4.15,4.16 and 4.17 for the visco-elastic rod. Again the results for three nodes are presented; the solution for the front end node nf is given by a solid line (-), for the middle node nm by a dashed-dotted line (. - e) and for the rear end node n' by a dashed line (- -).

Rigid to deformable body contact calculations

4 î l o ~ , Stress- VS. timq , ,

time [SI x1o.s time [SI x 10-1

Figure 4.10: Stress u,, for elastic rod in rigid to deformable body contact problem.

Figure 4.1 1: Strain E=, for elastic rod in rigid to deformable body con tact problem.


In figure 4.12 we see that the rod initially has a rigid body movement towards the wall. At timet = 1 [s] contact is detected and the front nodes of the rod are restrained. The total time of contact is the time that node nf is held at a displacement of 1.5 * [SI which is slightly larger than the value oft, = 1.9 -

[SI the rod separates from the wall and moves as a 'rigid' body away from the rigid wall. In figure 4.13 we see that the velocities of the nodes are not constant but fluctuate around a constant value of TI, = -15 [mls]. This is probably caused by the fact that pressure and tensile longitudinal (and trasverra!) w2ves stil! propagate throligh the rod. These waves 2re reflectior?s of the iriitiz! pressure wave that was initiated in the contact area.

The stress and strain levels for the nodes depicted in figures 4.10 and 4.11 are, during contact, comparable to the stress and strain levels in the reference problem. The main difference between the reference problem and the rigid to deformable body contact problem is that a stress peak is created when the contact is detected first. This peak can be diminished by increasing the numerical damping in the

[m]. This results in a contact time t , = 2 . [SI obtained in the reference calculation.

After t = 3 -


Velocity v-z vs. time 30 1

time [SI I10"

Figure 4.12: Di~placerne~~t 3, fcr dastic rsd in rigid to deformable body contact problem.

Figure 4.13: Ve!~~itg.~, f ~ r elastic deformable body contact problem.

in rigié te

model. The conclusion that these peaks are numerical effects rather than physical effects, seems justifiable. [s] can also be explained by

the fact that there are still (reflected) tensile and pressure waves travelling through the rod. These waves pass the nodees at certain times, producing stress peaks. Thus these peaks have a physical nature.

The peaks in the stress and strain graphs after the collision t > 3.0 -

Discussion of the results for the visco-elastic rod

The phenomena occurring during the contact of the visco-elastic rod with the rigid wail are essentially the same as described in the reference problem with the visco-elastic rod, so only the main difference will be discussed here.

Looking at figure 4.14 we see that a stress peak is produced when the contact is first detected. Like in the case of the elastic rod, this peak can be diminished by increasing the numerical damping. Another resemblance between the visco-elastic and elastic contact between a rigid and a deformable body is that after the rod has come loose of the wall, there still exist: some stress waves in the rod.

Ifwe look at the strain levels in the rod after the contact in figure 4.15 we see that the strains that are still present in the rod gradually decrease to zero, when the impacting body separates fiom the target body. This effect was already expected in section 4.3.1 in the discussion of the results for the visco-elastic rod.

2 6 Chapter 4 -

Stress s-zz vs. time

I I 1 2 3 4 5 6

time [SI I

Figure 4.14: Stress u,, for visco-elastic rod in rigid to deformable body contact problem.

timé [SI x IOJ

Figure 4.16: Displacement u, for visco-elastic rod in rigid to deformable body contact problem.

Strain e-zz vs. time I

4t ,' 50 I 2 3 4 5 6

time [s] .IO*

Figure 4.15: Strain E=, for visco-elastic rod in rigid to deformable body contact problem.

Velocity v-z vs. time

time [SI x Ibs

Figure 4.17: Velocity u, for visco-elastic rod in rigid to deformable body contact problem.

FEM contact-impact tests using MARC

5

__ 27

. . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . ; . . . . . : ..... ; . .: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.3

The deformable to deformable body contact calculations were performed in MARC to find out whether the deformable to deformable contact algorithms and the rigid to deformable body contact algortithms procduce similar results. The configuration to investigate the contact between two deformable bodies is described in section 3.4.4.

In these sirnuiations the numerical damping was doubled to 7 = 0.2 because contact Chattering occurred. According to Konter (1993) this chattering can also be diminished by increasing the separation force tolerance or by decreasing the time step. As can be seen from the figures with the nodal velocity vs. time, for these simulations, doubling the numerid damping was not enough to entirely prevent the chattering.

The levels for stress, strain, displacement and velocity in longitudinal direction can be found in figures 4.18,4.19,4.20 and 4.21, respectively for the elastic rod and in figures 4.22,4.23,4.24 and 4.25, respectively for the visco-elastic rod. The solution for the front end node nf is given by a solid line (-), for the middle node nm by a dashed-dotted line (a - -) and for the rear end node n‘ by a dashed line (- -).

Deformable to deformable body contact calculations

Figure 4.18: Stress u,, for elastic rod in deformable to deformable body contact problem.

Figure 4.19: Strain cZz for elastic rod in deformable to deformable body contact problem.

Because most physical effects are already discussed in the previous sections, we limit the discussion of the results only to considerable differences in these graphs with respect to the graphs of the rigid to deformable body contact problem.

DkXZSSiQn Q f f h EESdkS for the €!hk rod

The m i n dlRerence Setween this analysis and the rigid to deformable body contact analysis for the elastic rod is illustrated in figure 4.21. There we see that the velocity of the front end node(s) of the rod does not become zero when contact is first detected but fluctuates severely around a constant level of zero. This is caused by contact chattering. This results in a large stress peak in figure 4.1 8. This contact chattering occurred in spite of having doubled the numerical damping as was suggested in one of the MARC-manuals. The other possibilities that exist for diminishing the contact chattering were not investigated.

28

Figwe 4.20: Disp12cemenc uz ,cjr d2scir rod In deformable to deformable body contact problem.

Figure 4.22: Stress uzz for visco-elastic rod in deformable to deformable body contact problem.

Chapter 4

Velocity v-z vs. time I

time [SI x 10-5

time [SI x 10.1

Figure 4.23: Strain for visco-elastic rod in deformable to deformable body contact problem.

Apart from the effects caused by the contact chattering, the results for the elastic deformable to deformable contact globally resemble the results obtained for the elastic rigid to deformable body contact problem.

Disc*~slo.n. efthe resdt, fsr the visci-elastic red

In the analysis of the visco-elastic deformable to deformable body contact, contact chattering occurs despite of doubling the numerical damping. This can be seen in figure 4.25 where the velocity of the front end node nf fluctuates heavily around a constant level of zero. The other resuits resemble the solutions of the rigid to deformable body contact problem.

FEM con tact-impact tests using MARC 29 -

. , . . . . . .

. . . . , . . .

-30 o.^ l . ~ 2 25 3 3.5 4 4.5 5 time [SI x IOJ

Velocity v-z vs. time

time [SI I IO-*

Figure 4.24: Disp!asement u, for visco-elastic rod in deformable to deformable body contact problem.

Figure 4.25: Velocity u, for visco-elastic rod in deformable to deformable body con tact problem.

4.3.4

The analysis of the reference problem with MARC showed that the program is capable of describing ‘contact’ by prescribing boundary conditions. The resulting stress, strain, displacement and velocity levels can be validated by the analytical solutions to the wave propagation problem.

Having investigated the dynamic contact modelling in MARC we would like to conclude that the contact algorithms in the MARC program seem accurate and robust enough to describe the various phenomena. The rigid to deformable body contact option can be used when both mass and stiffness of a certain body dominate the problem. Designating that body as a rigid body, the time step taken by MARC will not be based on that body’s (significantly larger) stiffness and thus larger time steps can be taken without loss of accuracy.

The modelling of contact between two deformable bodies is not more complex than modelling the contact between a rigid and a deformable body. However, in the problem presented here, measures against the introduced contact chattering should be taken.

The contact algorithms implemented in MARC do not negatively influence the results for the calculations, as can be concluded by comparing the results from the analyses with and without the use ofcontact elements. The main difference between the former and the latter was that in the results from the latter a stress peak was introduced, which can be diminished by increasing the numerical damping.

Concluding remarks about finite element contact analysis with MARC

FEM contact-impact tests using DY NA3 D

In this chapter, contact modelling using the finite element package DYNA3D is presented. First an overview is given about the DYNA3D system including its pre- and postprocessor. Then the finite element calculations with the contact-impact problems described in chapter 3 are presented. The chapter is ended with a discussion about the results obtained for the contact-impact problems.

5.2 using DYNA3D

As a free licence code, DYNA3D has seen wide application to a variety of problems. The DYNA3D system consists of the following programs:

% preprocessor INGRID

% analysis program (solver) DYNA3D

% postprocessor TAURUS

For a íÛll description of DYNA3D the reader is referred to Hallquist and Whirley (1991). A description of INGRID is given in Hallquist, ñainsberger and Stillman (1985) and a description of TAURUS is given in Brown, Hallquist and Rainsberger (1984).

5.2.1 Analysis With DYNA3D

DYNA3D is an explicit finite element code for analysing the transient dynamic response of three- dimensional solids and structures. The element formulations available include one-dimensional truss anid beam eiements, two-dimensional quadiilareïal and rïiaïìgdar shel! elements and tkree-dimensiona! continuum elements. Many material models are available to represent a wide range of material behaviour including thermal effects. In addition, DYNA3 D has a contact interface capability, including frictional sliding and single surface contact, to handle arbitrary mechanical interactions between independent bodies. As an explicit code, DYNA3D is appropriate for problems where high rate dynamics or stress wave propagation effects are important (Hallquist and Whirley 1991, p. 15).

30

FEM contact-impact rests using DY NA3 D 31 -

In dynamic analysis with DYNA3D the user has to use the build-in time integration scheme. DYNA3D uses the explicit central difference time integration with a lumped mass matrix. DYNA3D automatically calculates the maximum time step size at each step in a solution for this conditionally stable method. This feature minimizes the cost of the analysis while assuring that stability is maintained.

Damping in DYNA3D can be introduced, based on the concepts of Rayleigh damping. The Rayleigh damping concept is extended to apply to nonlinear analysis with large displacements and nonlinear material behaviour.

The finite element formulation used in DYNA3D is based on one-point Gauss quadrature for the element integration. This approach gives rise to spurious zero energy deformation modes, or “hourglass modes,” within the element. The element must bestabilized to eliminate the spurious modes while retaining legitimate deformation modes. This stabilization is effectively accomplished in DYNA3 D (Hallquist 1983, pp. 14-20).

In DYNA3D initial conditions are specified as initial velocities. All initial velocities may be set to zero or the initial velocity of every node or a subset of nodes may be explicitly defined. The time variation of quantities in DYNA3D is specified by “load curves”. An arbitrary number of load curves may be defined and any number of boundary conditions or loads may reference one load curve. Each load curve may have an arbitrary number of points.

DYNA3D contains a number of options for modelling the wide range of boundary conditions en- countered in engineering analysis. Nodes may be constrained from translation or rotation in the global coordinate system. Alternatively, any number of local coordinate systems may be defined and nodes can be constrained in these local systems using single point constraints. Nodes may be given prescribed velocities as a function of time in any global coordinate direction, or in

an arbitrary direction specified by a given vector. In case where prescribed boundary velocities at t = O are not equal to defined initial velocities, the user is not warned about the incompatibility of these boundary and initial conditions.

5.2.2 Preprocessing using INGRID

Preprocessing for DYNA3D is done with INGRID. This preprocessor is a (semi-)interactive preprocessor using a command-line interface. The term semi-interactive is used because most data is defined in input fles for INGRID. This reduces the chance on mistakes when generating a mesh. The program writes a DYNA3D input file and can produce some graphical output both on paper and for the postprocessor TAURUS. The display of boundary conditions and element and node numbers in INGRID is possible but these facilities are not well elaborated.

The mesh generation in INGRID is based on the use of a so-called index space. The mesh is built in the index space and this index space is then mapped into the true geometry of the problem. The main advantage of this method is that complex geometries can be built with only few commands. The main disadvantage of this method is that generating a mesh becomes far from straightforward and requires good . . spatial ins!ght=

5.2.3 Postprocessing using TAURUS

The postprocessor for DYNA3D is called TAURUS. Like INGRID this program is an interactive program with a command-line interface. The program supports contour plots of the deformed shapes, simple transient

analysis ‘movies’ and a limited facility to make history plots for selected nodes. The program accepts input hom MARC but also from INGRID. This allows the meshes produced by I N G R I D to be printed easier (see for instance figure 5.1 in the next section).

The use of TAURUS is rather straightforward once the user knows the commands to manipulate the data produced by DYNA3D. However having used MENTAT for postprocessing MARC-data the large disadvantages of a command-line driven user-interface as in TAURUS became evident.

5.3 FEM contact analysis using DYNA3D

The contact problem geometry for DYNA3D was generated with INGRID. Since D Y N A 3 0 does not support axisymmetric elements, the rod was modelled with three-dimensional elements. Because of the symmetry of the problem only one quarter of the rod was modelled. The mesh that was created for the contact-impact problem using DYNA3D is depicted in figure 5.1. Along the longitudinal side (in a-direction), 50 elements were used. The elements used are valid for large displacements and strains. Spurious hourglass modes are ‘stabilized‘ using an “hourglass viscosity”. The constitutive equations are evaluated once based on the state at the centre of the element.

Figure 5.1 : Spatial discretization of the rod in D Y N A 3 D.

In DYNA3D the user cannot specify the way time integration is carried out. The time integration is based on the explicit central difference scheme with a lumped mass matrix. The user control on time stepping is also pre-empted by DYNA3D so only the material parameters and boundary conditions have to be specified in :he DYNA3C)-inputfi!e. In d! simu!atiûns, mwement of the d e s that !ie on :he symmetry planes was restrained in tangential direction. As a result of using explicit time integration, no convergence tolerance has to be specified. To give an idea on the number of time steps needed for integration: DYNA3D used more than 1300 time steps internally for the rigid to deformable and deformable to deformable contact problems.

The (possible) contact conditions are imposed in D Y N A 3 D by means of defining master and slave

FEM contact-impact tests using DY N A 3 D 33 __

segments in so called “sliding surfaces”. The user must specify the boundaries of the bodies that could come into contact. The traditional meaning of master and slave side is not used internally by DYNA3D because the definitions for master and slave side are interchanged during contact searching analysis (see section 2.2.2).

5.3.1 Reference cdcdations

The reference calculations in DYNA3D are carried out to serve as a comparison to the contact-impact calculations. The problem definition for the reference dculations is described in section 3.4). The analytical solutions to the impact problem are described in section 3.4.1 and can be compared with the solutions obtained for the reference problem calculations. The results of the reference problem calculation with DYNA3D can also be compared to the solutions for the same problem calculated with MARC.

The total simulation timet, for the calculations for both the elastic and visco-elastic rod was 2.0. [SI, as in the reference calculations with MARC.The stress, strain, displacement and velocity results in longitudinal direction for nodes nfy nm and n‘ at the fiont, middle and rear end of the rod, respectively, can be found in figures 5.2,5.3,5.4 and 5.5 for the elastic rod and in figures 5.6,5.7, 5.8 and 5.9 for the visco-elastic rod. For all graphs, the solutions for the node nf are given by the solid line (-), for n”’ by a dashed-dotted line (. - .) and for n‘ by a dashed line (- -).

0.5 1.5 2 25 time [SI x 10-5

21 10.’ Strain e z z vs. time I I

Figure 5.2: Stress bZz for elastic rod in reference problem. problem,

Figure 5.3: Strain for elastic rod in reference


The total contact time t , for the reference calculations with DYNA3D can be approximated as the time eiapsed before a tensiie stress is produced for the front end noáe n’. Looking at figure 5.2, the totai contact time is estimated as t , = 1.9 loq5 [SI. The same result was obtained with MARC.

The pressure wave speed can be estimated by:

= 6.25 103[m/s], L

0.8.10-5 Cp,estl,el =

34

i . . .. . .. . . , . , . . . . -

, : I \ jl

0.5 1 1.5 2 25 -14

time [SI x 10.’

Figure 5.4: Displacement u, for elastic rod in referen Ce problem .

or:

Chapter 5 __

Velocity v-z vs. time ,

time [SI =lo-’

Figure 5.5: Velocity o, for elastic rod in reference problem.

0.1 - = 5.26 103[m/s].

2L Cp,est2,e1 = - - t , 1.9.10-5

These are exactly the same results as obtained with the reference calculations in MARC. The main difference between the results obtained by MARC and DYNA3D is that the results of the latter give a smoother plot. This is probably caused by the lumping of the mass-matrix in DYNA3D. This measure has the side effect that the higher frequency terms are filtered out of the solutions. Note that lumping the mass matrix gives a less accurate description of the spatial mass distribution of the ‘real’ geometry.

The stresses and strains obtained from the DYNA3D finite element calculations are in good agreement with the analytically derived solutions and the values calculated by MARC. The small deviations can again be explained with the fact that the mass matrix in the DYNA3D calculations was lumped.

Discussion ofthe results for the visco-elastic rod

The resulting graphs for the visco-elastic rod reference problem obtained by DYNA3D are globally identical to the results obtained by MARC (Compare figures 5.6,5.7,5.8 and 5.9 with figures 4.6,4.7,4.8 and 4.9 respectively). The difference in smoothness can again be explained by mass matrix lumping in DYNA3D.

The effects of stress relaxation and creep that are found in figures 5.6 and 5.7 were described when discussing the results obtained by MARC for the same problem. The remarks made there also apply to the results presented here, i.e., it is expected that the non-zero strain level in the rod will eventually become zero, when the rod recedes from the wall

FEM contact-impact tests using DYNA3D 35 -

Figure 5.6: Stress u,, for visco-elastic rod in reference problem.

Strain e-zz vs. time o.5x 10.' I

I a 5 I 1 5 2 2 5

time [SI x 10"

I

Figure 5.7: Strain erence problem.

for visco-elastic rod in re&

Velocity v z vs. time

Figure 5.8: Displacement u, for visco-elastic rod in reference problem.

Figure 5.9: Velocity u, for visco-elastic rod in reference problem.

36 Chapter 5 __

5.3.2

The calculations with DYNA3D for the rigid to deformable body contact problem were carried out to investigate the influence of the contact algorithms on the contact problem and to compare the contact algorithms of MARC and DYNA3D. A total time period oft, = 5 [SI was simulated to allow rigid body motion of the rod before and after the impact. The problem is described in section 3.4.3

The time history piots for the important quantities can be found in figures 5.10,5.11, 5.12 arid 5.13 for the elastic rod and in figures 5.14, 5.15,5.16 and 5.17 for the visco-elastic rod. The solution for the fiont end node nf is given by a solid line (-), for the middle node nm by a dashed-dotted line (a - e) and for the rear end node n' by a dashed line (- -).

Rigid to deformable body contact calculations

1 1 2 3 4 5

time [SI *los

Figure 5.10: Stress c,, for elastic rod in rigid to deformable body contact problem.

time [SI li IO*

Figure 5.12: Displacement u, for elastic rod in rigid to deformable body contact problem.

-4' 1 2 3 4 I time [SI >i 11

Figure 5.1 i : Strain E=, for elastic rod in rigid to deformable body contact problem.

Veiocitv v z vs. time

time [SI x 16'

Figure 5.13: Velocity w, for elastic rod in rigid to deformable body contact problem.

FEM contact-impact tests using DYNA3D 37

-10-

Figuïe 5.14: Stress o,, for visco-eliâsric rod in rigid to deformable body contact problem.

Figure 5-15: Scrain cZz for visco-elascic rod in rigid to deformable body con tact problem.

............... .... ............... ........ ............... ......... ! ' ! 1:

! ;. . . . . . . . ; .... .I. . . . . . . . . .: . . . . . . . . . . . . . . . . . .

, 1: i I :

1 :

l.lo' Displacement u-z vs. time

-25 _....... ................. .! ,"": i ....., I ..... ;- ............ ............ .. \ : , \ : I \:I

- 3 ~ I 2 3 4 5 6 time [SI x IOJ

2 3 4 S 6

Figure 5.16: Displacement u, for visco-elastic rod in rigid to deformable body contact problem.

Figure 5.17: Velocity u, for visco-elastic rod in rigid to deformable body con tact problem.


The displacements and velocities presented in figures 5.12 and 5.13 are in good agreement with the results obtained by MARC in figures 4.12 arid 4.13. In the DYNA3D plot for the velocity we see a little pakwher. contact is first detected. This indicates that there might be some contact chattering or may be caused by the penalty method, which is used by DYNA3D to impose the contact conditions.

Ifwe compare the results for stress and strain for D Y N A 3 D and MARC we see a clear difference (compare figures 5.10 and 5.1 1 with figures 4.10 and 4.1 1 respectively). We see that the graphs for nodes nm and n' are in good agreement but that DYNA3D finds a higher stress and strain level for the front end node nf,

38 . Chapter 5 __

According to the analytically derived solutions for the elastic rod, the occurring stress and strain can be approximated by:

CI = d s ' = 6.1 108[Pa],

Furthermore, the stress and strain levels in the part where the pressure wave front has passed should be equal throughout the rod. Clearly, the results obtained by DYNA3D are not accurate.

The discrepancy between the results obtained by DYNA3D and the values obtained both by MARC and analytically, is in all probability caused by the contact interface algorithms used in DYNA3D. The contact interface conditions are imposed by a penalty method which penalizes any penetration by applying a normal interface force to a penetrating node (see chapter 2). This causes incorrectly calculated stress and strain levels for front end node nf which is a penetrating node.

Inspecting the stress and strain level of the node next to node nf on the axis of symmetry we found that the stress and strain level for this node were physically correct. This confirmed the assumption that the higher stress and strain of the front end node were caused by the contact interface algorithm used in DYNA3D.

The incorrect stress and strain levels might be brought into closer range with the MARC and analytical solutions by changing the penalty parameters. However, this has not been tested, since the choice for these penalty parameters is (or at least should be) based on numerical experience and we lack that experience with DYNA3D. Using the penalty method requires a compromise between accuracy and simulation time.

Discussion of the results for the visco-elastic rod

Comparison of the results from DYNA3D and MARC for the visco-elastic rod revealed two global differences between the simulations. Both effects causing these differences were previously discussed but will be summarized here shortly. The graphs for the DYNA3D results are smoother than the MARC graphs because of a different approach in building the mass matrix for the dynamic finite element representation of the problem; MARC uses a consistent mass matrix, DYNA3D uses a lumped mass matrix. The discrepancy between the stress and strain levels for both programs is caused by the difference in contact interface algorithms; MARC uses the solver constraints while DYNA3D uses the penalty method (see section 2.3).

FEM con tact-impact tests using D Y N A 3 D 39 -

5.3.3 The contact problem between two identical deformable bodies is described in section 3.4.4. The calculations in D Y N A 3 D were carried out to test if the contact searching and contact interface algorithms implemented in D Y N A 3 D produce appropriate results.

The levels for stress, strain, displacement and velocity in longitudinal direction can be found in figures 5.18,5.19,5.20 and 5.21, respectively for the elastic rod and in figures 5.22,5.23,5.24 and 5.25, respectively for the visco-elastic rod. The solution for the front end node nf is given by a solid line (-), for the middle node n" by a dashed-dotted line (- - a) and for the rear end node n' by a dashed line (- -).

Deformable to deformable body contact calculations

- t d Stress s-zz vs. time

9 I time [SI x 10s

1 2 3 4 5 6

Figure 5.18: Stress u,, for elastic rod in deformable to deformable body contact problem.

time [SI x IO1

Figure 5.20: Displacement u, for elastic rod in deformable to deformable body con tact problem.

Strain e-zz vs. time

time [SI x16'

Figure 5.19: Strain E,, for elastic rod in deformable to deformable body con tact problem.


Figure 5.21: Velocity v, for elastic rod in deformable to deformable body con tact problem.

40 Chapter 5

Stress s-zz vs. time io* 1

time [SI x 10.’

Figure 5.22: Stress B,, for visco-elaseic rod in deformable to deformable body con tact problem.

Figure 5.23: §train for visco-elastic rod ifi deformable to deformable body contact problem.

Discussion of the results for the elastic and visco-elastic rod

The results obtained with DYNA3D for the deformable to deformable body contact problem are qualitat- ively equivalent to the results for the rigid to deformable body contact problem. The remarks made for the latter problem also apply here. However, some minor differences have been found.

Looking at the results for longitudinal displacement in figures 5.20 and 5.24 we see that the displacement for the front end node nf in both graphs exceeds -1.5 - [m]. This means that the front end node has penetrated the other rod.

When examining at the velocities for the front nodes in figures 5.21 and 5.25 we see that the velocities fluctuate around a constant level of zero. This indicates contact chattering. There is no user control on the time step in DYNA3D nor can the user specify numerical damping or a separation tolerance. Hence no means ofpreventing contact chattering are present in DYNA3D.

Finally when looking at the stress and strain levels for the front node, we see that these are again incorrectly calculated compared to the analytical solution. In fact the difference between analytical values for stress and strain and the values obtained by DYNA3D is about 20 %.

5.3.4

The ability of DYNA3D in describing ‘contact’ by prescribing boundary conditions at the ‘contact’ nodes has been shown by the analysis of the reference problem. The calculated values for stress, strain, displacement and velocity for both the elastic and the visco-elastic rod were in agreement with the analytically derived sohions in section 3.4.1, in spke of the model sirnp!ifications c a w d by mass rnat:ix !urnping.

Contact analysis has traditionally been one of the main features of the DYNA3D code. However the simulations with contact modelling, between a deformable body on the one side and both a rigid and deformable body on the other side, learnt that not in all cases accurate results can be obtained. The incorrect stress and strain levels are, in all probability, caused by the contact interface algorithms since the calculations without the contact interface did produce the accurate stress and strain results. Another

Concluding remarks about finite element contact analysis with DYNA3D

__ FEM contact-impact tests using DYNA3D 41


time [SI time [SI

Figure 5.24: Displeasemet? t u, for ~isco-e_las~!'c rod in deformable to deformable body contact problem.

Figure 5.25: Yekxjy w, -f8h viso-elastic rod in deformable to deformable body con tact problem.

disadvantage of the DYNA3D code is that no measures can be taken to prevent contact chattering between two deformable bodies.

The main disadvantage of the DYNA3D program in general is that the user is limited in choice of mass discretization, time integration method, element types etc. The use of implicit time integration is not supported by DYNA3D but with little alteration, the DYNA3D input file can be used in NlKE3D the implicit compeer of DYNA3D. When two-dimensional analysis is required the user has to switch to DYNA2D which involves learning to work with a new pre- and postprocessor since both INGRID and TAURUS can only be used for three-dimensional analysis data.

Chapter 6

Conclusions and recommendations

Conclusions

The main conclusions that can be drawn from the finite element contact analyses are:

X Both MARC and DYNA3D are capable of accurately describing ‘contact’ by means of prescribing boundary conditions at the area where contact occurs.

Both MARC and DYNA3D have reliable and robust contact searching algorithms as far as may be concluded from the performed calculations.

The solver constraints method used in MARC to impose the impenetrability constraints in the contact area, gave accurate results for both the rigid to deformable body problem and the contact problem with two deformable bodies.

X The penalty method used in DYNA3D to account for the contact conditions produces accurate results for displacement and velocity fields but incorrect values for the stresses and strains of the elements in the contact area.

I( In DYNA3D the user has no means for diminishing the contact chattering.

Recommendations

In view of the conclusions that have been drawn from the finite element contact analyses in both MARC and DYNA3D we would like to recommended MARC for modelling contact interfaces in the research project for dynamic head impact modelling. Apart from the conclusions this recommendation is based on the greater flexibility MARC offers the user in choice of analysis and element type, material model, contact parameters, integration meehocls etc.

Pre- and postprocessing for MARC, using MENTAT is preferred to the old-fashioned pre- and postprocessing for ûYNA3D, using iNGRiD and TAÜRÜS. Besides that the documentation for ÎuikRC is more complete and better. This greatly decreases the amount of time needed for learning to use the packages.

In our opinion the only reason why D Y N A 3 D should be preferred to MARC is because the former is a free licence code while the latter is a commercial package.

42

References

Brown, B. E., Hallquist, J. O. and ñainsberger, R (1984), TAURUS: An Interactive Post Processor for the Ana& Codes NIECE3D, DIIvA3D, TACO3D, and GEMINI Lawrence Livermore National Laboratory, rep. nr. UCID-19392, Revised.

de Graaf, A. I? and Konter, A. W. A. (1993), Dynamic analyss with MARC. MARC Analysis Research Corporration - Europe, Zoetermeer.

Flügge, W. (1975), ViscoeLasticity, chapter 6, pp. 121-128. Springer-Verlag, 2nd rev. edition.

Hallquist, J. O. (1 983), TbeoreticalManualfor D W B D . Lawrence Livermore National Laboratory.

Hallquist, J. O., Rainsberger, R and Stillman, D. W. (1985), INGRID: A Three-DimensionalMesh Generatorfor Modeling Nonlinear Systems. Lawrence Livermore National Laboratory, rep. nr. UCID- 20506, Revised.

Hallquist, J. O. and Whirley, R. G. (1991), D W B D A Nonlinear, Eqlicit, Three-Dimensionalfi- nite Element Code For Solid and Structural Mechanics- User Manual. Lawrence Livermore National Laboratory, rep. nr. UCRL-MA- 107254.

MARC Analysis Research Corporration - Europe, Zoetermeer. Konter, A. W. A. (1993), FEM anabsis of contact problems in metal forming and rubber applications.

Marc Analysis Research Corporation (1992a), Marc Reference Library, Volume A: User Information.

Marc Analysis Research Corporation (1992b), Marc Reference Library, Volume D: User Subroutines and

Morrison, J. A. (1956), Wave propagation in rods of Voight material and visco-elastic materiais with

Wismans, J. S. H. M. (1993), Injury Biomechanics. Course notes nr. 4721, Eindhoven University of

Zhong, Z.-H. (1 993), Enite elementproceduresfor contact-impact problem. Oxford University Press.

Zukas, J. A., Nicholas, T., Swift, H. E, Greszczuk, L. IB. and Curran, D. K (1 9823, Impact Dynamics,

Special Routines.

three-parameter models. Quarter4 $Applied Mathematics xuL(2): 1 53-1 69.

Technolog, NL.

chapter 1, pp. 9-15, John W k y gb Sons.

43

Appendix A

Vector operations

In this report some vector operations are used. These operations will be represented in Cartesian and in cylindircai coordinates.

Al Cartesian coordinates

The gradient operator is defined as:

+ a a a ax a y ûz

V = grad = ëx- + ëy- + ëz-.

The divergence of a vector ä = a,ëx + ayey + a,& is defined as:

and the Laplace operator is defined as:

A2 Cylindrical coordinates

The gradient, divergence and Laplace operators in cylindrical coordinates are defined as:

+ a i a a V = grad = ër- + ëe-- + ëz-.

ar r a 6 ûz (A.4)

with ä = arër + aeëe + a,&

44

finite element contact analysis: marc and dyna3d compared · finite element contact analysis: marc...

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