1-s2.0-0045794b-main.pdf

Compufew 6: Slmc~ures Vol. 44, No. 3. pp. 499-514. 1992 004s7Y49/92 s5.w + 0.00 Printed in Great Britain. Pergamon Press Ltd

MECHANICS OF ANISOTROPIC PLATES AND SHELLS--A NEW LOOK AT AN OLD SUBJECT

A. K. NOOR

Center for Computational Structures Technology, University of Virginia, NASA Langley Research Center, Hampton, VA 23665, U.S.A.

(Received 29 April 1992)

Abstr&--A number of aspects of the mechanics of anisotropic plates and shells are discussed. The topics covered include computational models of anisotropic plates and shells, consequences of anisotropy on deformation couplings, symmetry types, stress ~on~ntrations and edge effects, and importance of transverse shear deformation; recent applications and recent advances in the modeling and analysis of anisotropic plates and shells; and new research directions.

INTRODUCTION

The. mechanics of anisotropic plates and shells has a long history. The first reported attempts to mode1 the response of orthogonally anisotropic (orthotropic) plates date back to the second half of the nineteenth century[l, 21, and the first known publication on orthotropic shells was in 1924 [3]. Prior to 1940 most of the work on anisotropic plates and shells was motivated by applications to wood, crystalline solids and reinforced concrete constructions (see, for example [4,5]).

Most of the early publications were limited to predicting the gross response characteristics (vibration frequencies, buckling loads, average through-the-thickness displacements and rotations) of thin anisotropic plates and shells. The classical Kirchhoff-Love plate and shell theory, based on neglecting transverse shear strains and transverse normal strains in the structure, is adequate for this purpose.

The advent and expanded use of high-performance composite materials in aerospace, automotive, ship building, medical and recreational industries, have stimulated interest in the development of refined two-dimensional theories and computational models for anisotropic plates and shells, as well as in advanced topics of mechanics (e.g., generalized continuum theories, damage mechanisms and criteria), such as required for the accurate prediction of the detailed response and failure characte~stics.

Significant advances in the mechanics of anisotropic plates and shells continue to take place on a broad front. The new advances are mani- fested by the advent of new application areas of anisotropic plates and shells (e.g., modeling of biological systems); the development of test techniques and methodologies for characterizing the stiffness, strength, and life of these structures; the development of ~mputational models to

simulate the mechanical, thermal, and electromagnetic behavior of anisotropic plates and shells; better understanding of the approximations involved in different computational models; more rational matching of the objectives of analysis with the computational models used; eRicient discretization techniques, computational strategies and numerical algorithms, as well as versatile and powerful software systems for the solution of complex mechanics problems.

Mechanics of anisotropic plates and shells is currently used to study various phenomena associated with the response, life, failure and performance of a variety of engineering systems. The phenomena involved cover a wide range of length scales, from microstructure to structural response. Within each scale has evolved a specialty mechanics discipline with several levels of sophistication. The literature on the mechanics of anisotropic plates and shells is nearly overwhelming. To the author’s knowledge, the first monographs totally devoted to anisotropic plates and shells were publish~ in 1947 and 1961, respectively (see the Appendix). Since then over 64 monographs and textbooks have been published on the subject. All the monographs and books that are known to the author are listed in the Appendix.

In the present paper a number of aspects of the mechanics of anisotropic plates and shells are described. The topics covered include computational models for anisotropic plates and shells; implications of anisotropy on deformation couplings, symmetry types, stress concentrations and edge effects, and importance of transverse shear deformation; recent applications and recent advances in the modeling and analysis of anisotropic plates and shells; and future directions of research. Both macroscopically homogeneous, piecewise homogeneous (e.g., laminated- see Fig. 1) and nonhomogeneous plates and shells are considered.

499

500 A. K. NOOR

er

Xl Nonhomogeneous

Laminated plate shell

Fig. 1. Laminated anisotropic plate and nonhomogeneous shell.

Computational models for anisotropic plates and shells

Since the kinematic relations and equations of motion are independent of the type of material used, it is the complex form of their constitutive relations that distinguishes the fundamental equations of an-

isotropic plates and shells from the corresponding equations of isotropic structures. Several computational models have been proposed for the prediction of the response of anisotropic plates and shells. Some of these models are extensions of similar

approaches used for homogeneous isotropic plates and shells. These models are discussed in [6, 71 and

are highlighted subsequently: 1. Three-dimensional continuum models, in which

the entire structure, or each of the individual layers of a laminated structure, is treated as a three-dimensional continuum.

2. Quasi-three-dimensional continuum models, in which simplifying assumptions are made regarding the stress (or strain) state in the structure (or in the

individual layers of a laminated structure), but no a priori assumptions are made about the distribution of the different response quantities in the thickness

direction. 3. Two-dimensional models. Four different

approaches are used for constructing these models,

namely: method of hypotheses; method of expansion; asymptotic integration technique; iterative methods

and methods of successive corrections (e.g., predictor<orrector procedures). Several classifications have been proposed for two-dimensional models (see [7]); however, for the purpose of the present paper it is convenient to divide the two-dimensional

models into the following three categories:

(a) classical Kirchhoff-Love models; (b) shear-deformable models based on global

through-the-thickness displacement, strain

y Undeformed t

and/or stress approximations, and for laminated anisotropic plates and shells; and

(c) shear deformable models based on piecewise, layer-by-layer, approximations of the response quantities in the thickness direction.

4. Microstructural and generalized continuum models. These models have been proposed for anisotropic fibrous composite plates and shells.

In Fig. 2 the deformations of a shell element

according to both the classical theory and the first- order shear deformation theory (based on linear displacement and/or strain variation through the

thickness) are contrasted with the actual deformation.

The use of both three-dimensional and quasi-three- dimensional models for predicting the response

characteristics of anisotropic plates and shells is computationally expensive, and therefore, not feas- ible for practical structures. On the other hand, the two-dimensional shear-deformable models are adequate for predicting the gross response character-

istics of medium-thick plates and shells, but they are not adequate for the accurate prediction of the transverse stresses and deformations. The predictor<orrector approaches described in a succeeding subsection appear to be very effective procedures for the accurate determination of the global as well as the detailed response characteristics of anisotropic plates and shells.

DEFINITION AND TYPES OF ANISOTROPY

Anisotropy of a property is the dependence of that property on direction in space. By contrast an

isotropic property is invariant with respect to coordinate rotation. Properties that can be anisotropic include mechanical, thermal, hygral, magnetic, elec- trical, optical and chemical. For the same plate (or shell) some properties can be isotropic, while others are anisotropic. A plate (or a shell) is usually called anisotropic if its properties of interest exhibit

anisotropy. Note that anisotropy depends on the length scale at which the property is observed. Com- binations of materials with isotropic properties can produce anisotropic properties at a higher length scale (e.g., case of fibrous composites). On the other hand, combinations of some materials with anisotropic properties can produce isotropic or

ClassIcal First-order Actual

Kirchhoff-Love theory shear deformation theory deformation

Fig. 2. Deformation of a shell element

Mechanics of anisotropic plates and shells

STIFFENED PANEL

S~FFENED SHELL Fig. 3. Stiffened panels and shells.

statistically isotropic properties at a different length scale (e.g., extensional properties of quasi-isotropic laminates, and mechanical properties of randomly oriented chopped fiber composites).

On a macroscopic level three types of anisotropy can be distinguished:

(1) intrinsic (or natural) anisotropy; (2) structural (or constructional) anisotropy; and (3) acquired (or induced) anisotropy.

Examples of the three types of anisotropy for mechanical and thermal properties are exhibited by: wood and fibrous composites; corrugated sheets and sheets stiffened with either corrugation or closely spaced stiffeners (see Fig. 3); and deformation-induced anisotropy in polycrystalline metals.

The presence of symmetries reduces the number of independent coefficients required in describing the property. All the possible symmetries are built up from three types of symmetry elements, namely, planes of reflection (or mirror) symmetry; pure rotation axes and axes of rotatory inversion. Speci& tally, the following four symmetries are encountered in anisotropic plates and shells:

1. ~o~oc~~n~~ s~~~~rr~~x~ibited by a property which has one plane of reflection symmetry.

2. Orthotropic symmetry-exhibited by a property having three mutually orthogonal planes of reflection symmetry.

3. Tr~nsverse~.v ~sotropjc symmerry exhibited by a property possessing a plane of isotropy on which the property is not dependent on direction.

4. Axial s~mmetry~xhibited by a property possessing an axis of rotational symmetry. This type of symmetry is limited to circular plates and shells of revolution.

Most of the anisotropic plates and shells considered in the literature have either orthotropic or transversely isotropic symmetries. Qualitatively, the response of these structures is not much different from that of isotropic structures. The present paper focuses primarily on anisotropic plates and shells

whose properties do not exhibit orthotropic or transversely isotropic symmetry (nonorthotropic structures).

CONSEQUENCE OF ANlSOTROPY

The differences between the isotropic and anisotropic elastic properties is reflected in the significant differences between the values of the elastic coefficients of the two materials. For example, Pois- son’s ratios for materials with isotropic elastic prop erties in excess of 0.5 are thermodynamically inadmissible, since they lead to negative strain energy under certain loads. By contrast, Poisson’s ratios as high as 1.97 have been reported for materials with anisotropic properties, Also, negative Poisson’s ratios have been observed for single crystals and pyrolytic graphite material used in thermal protection systems. Moreover, the dependence of the anisotropic elastic properties on loading, temperature, electromagnetic fields and other environmental effects is different from that of the corresponding isotropic properties (of traditional materials).

In addition to the differences in elastic coefficients, plates and shells with anisotropic (nonorthotropic) properties differ from those with isotropic (and orthotropic) properties in a number of ways, includ- ing deformation couplings; symmetry types exhibited by the response; stress concentrations and edge effects; and importance of transverse shear deformation. These are briefly discussed in succeeding subsections.

Plates and shells with only monoclinic symmet~ (nonorthotropic properties), exhibit coupling between the extensional and bending strains on the one hand and between shearing and twisting strains on the other hand. This coupling also occurs in plates and shells with orthotropic properties when the principal directions of orthotropy do not coincide with the directions of loading and/or the boundaries. In addition, nonhomogeneous, and

A. K, &bOR

Constitutive relations for first-order shear deformation theory

(Shells with monoclinic symmetry)

502

(4

"4

N2

N12 --_

Ml

M2

Ml2 ---

Q1

Q2

L ______ -i__

kt A55 ks A45 0 k2 A44

0 I Anisotrcpic (nono~hotra~c) terms

5 Cross elasticity effects

* E Zero slement

2E23

Fig. 4(a) and (b). Deformation couplings exhibited by the constitutive relations for nonhomogeneous anisotropic shells with monoclinic symmetry.

piecewise homogeneous (laminated) plates and shells exhibit coupling between extensional and bending strains (cross elasticity effects). This latter coupling distinguishes nonhomogeneous and laminated plates and shells from homogeneous ones.

The two types of couplings are depicted in Fig. 4 in which the relations between the stress resultants and the corresponding strain components, based on a first-order shear deformation theory are shown. The shell is assumed to be nonhomogeneous, anisotropic and has monoclinic symmetry (one plane of reflection symmetry parallel to the middle surface). The Ns, MS and Qs refer to the in-plane, bending and transverse shear stress resultants. The A,, Bij, D, (i, j = 1,2 and 6) and A, (Z, .I = 4,s) are the extensional, .coupling, bending and ~ansverse shear stiffnesses; k, and k2 are shear correction factors. Anisotropy is distin~ished by the presence of the A, B, D coefficients with subscripts 16 and 26; and the A coefficients with subscripts 45 (shown circled in Fig. 4).

The deformation couplings exhibited by anisotropic (nonorthotropic) plates and shells result in

more complicated governing equations than for the corresponding isotropic (and orthotropic) structures. fn general, exact analytic solutions based on the method of separation of variables (e.g., Navier-type and Levy-type solutions for rectangular plates) can- not be obtained. An exception to that is the case of rectangular antis~met~~lly laminated angle-ply composite piates with simply-supposed boundary conditions (see [8]). Aiso, closed-form solutions have been obtained, based on the classical Kirchhoff theory, for clamped anisotropic plates (see the monograph by Bazhanov et al. (1970~listed in the Appendix). Anisotropy was found to have a@ adverse effect on the accuracy and convergence of both numerical and analytic solutions (see, for example [9,10]).

Symmetry rypes for the response

The contrast between the more familiar reflection symmetries (and antisymmetries) exhibited by the response of plates and shells with isotropic and orthotropic properties, and those for plates having

Mechanics of anisotropic plates and shells 503

Anisotropic

Reflection symmetry with respect to x1 and x2 planes

inversion symmetry with respect to x3 axis

(a) Shallow shell

Web

Reflection antisymmetry with respect to x1 plane

(b) Zee stiffener

Inversion antisymmetfy with respect to origin (point c)

5. Normalized contour plots for the thermal buckling modes of a shallow shell and a Zee stiffener. Spacing of contour lines is 0.2 and dashed lines refer to negative contours.

anisotropic properties is shown in Fig. 5. Normalized contour plots for the thermal buckling modes of a shallow shell with rectangular planform and a Zee stiffener are shown in Fig. 5. Both structures are subjected to uniform temperature increase. The two cases of structures with isotropic and anisotropic thermoelastic properties are shown. The buckling modes for the isotropic structures exhibit reflection (mirror) symmetry (or antisymmetry). By contrast, the buckling modes for the anisotropic structures exhibit inversion symmetry (or antisymmetry) but no reflection symmetry. The inversion symmetries and

antisymmetries are characterized by the following relations:

shallow shell (inversion symmetry)

w(x,,d= w(-x,, -x2)

Zee stz@zer (inversion antisymmetry)

w(x*,xz,x3)= -w(-x,, -x2, -4

z&,x2, x3) = -u,(-x,, -x2, -xJ.

504 A. K. %OR

Detailed discussions of the different types of symmetry exhibited by the linear and nonlinear responses of anisotropic shells, plates, tires and stiffeners are given in [l l-141. Special procedures are presented in the cited references for exploiting the symmetries to significantly reduce the scope and cost of the finite element analysis.

Stress c5~ce~trati5~s and edge e_&ects

Stress concentrations occur in the vicinity of discontinuities in the material, geometry, loading and supports of plate and shell structures (e.g., defects, cracks, holes, and notches). For laminated structures stress concentrations also occur in the vicinity of through-the-thickness material discontinuities (such as ply drops and delamination), and free edges.

The presence of an edge zone (or boundary layer) in the vicinity of a free edge of a symmetrically laminated flat plate subjected to uniform axial loading is well established (see [15, 16]), and results in: (1) distortions of surface strains, due to the presence of interlaminar stresses and, (2) delamination induced failure, which initiate within the boundary layer.

Anisotropy results in a more complex stress distribution in the vicinity of discontinuities, and near free edges of laminated structures, than for the corresponding orthotropic structures (see [16], and the monograph of Pelekh and Siaskii (1975)-listed in the Appendix).

hportance of transuerse shear ~efor~zat~o~

Most of the advanced composites in use to date have a low ratio of the transverse shear modulus to the in-plane modulus, and therefore, the transverse shear deformation plays a much more important role

x3 t

hiL

in reducing the effective flexural stiffness of laminated plates and shells made of these composites than in the corresponding metallic structures. Also, it was found that anisotropy can amplify transverse shear deformation, i.e., transverse shear deformation effects can be more important in anisotropic plates and shells than in isotropic or even orthotropic structures of the same geometry (see [17, 181). This is depicted in Fig. 6 where the energy ratios iJ,/U, ~~i~~ U3/U, associated with the fundamental vibration frequencies, are plotted versus the thickness ratio h/L for ten-layer angle-ply square plates with different fiber orientation angles, 8. The energy components U, , U,, U, are associated with the three sets of stress components (cl,, Q, ot2); (f~,~. oZ3) and oX3, respectively, and U = U, + U, + U,. As 0 increases both the ratios of the anisotropic elastic coefficients, ~~~~~~,, and hB,,/D,, , and the transverse shear strain energy ratio G;/U increase (see [19]).

Despite its adverse effects, anisotropy has some positive aspects in design applications. For example, aeroelastic tailoring of composite wings, which is

used to achieve certain performance goals and to eliminate aeroelastic instabilities, relies on the deformation couplings resulting from anisotropy (see, for example [ZO]).

RECENT ADVANCES

In recent years considerable work has been devoted to various aspects of the mechanics of anisotropic plates and shells. Most of this activity was motivated by the interest in fibrous and laminated composites and covered a broad range of problems related to statics, dynamics, stability, fracture and experimental mechanics of composite plates and shells. A review of

“I -91~ “22,012

u2-013, O23

U3 - 033 033

4

0.4

0.3

0.2

0.1

Fig. 6. Energy ratios associated with the fundam~ntai vibration frequencies for ten-layer angle-ply composite plates (see [4]).

Mechanics of anisotropic

some of this work is given in the survey papers listed in the Appendix. A state-of-the-art review of test methods for composite structures is given in [21]. The special phenomena associated with the microstructure of composite plates and shells are reviewed in [22-251. Fracture mechanics concepts for continuum, piecewise homogeneous and layered media are described in a recent monograph by Guz [26]. Herein, some of the recent work devoted to the development of effective computational models and strategies for the analysis (and reanalysis) of anisotropic plates and shells is described. Among the noteworthy contri- butions are development of: exact three-dimensional elasticity (and thermoelasticity) solutions for some anisotropic (nonorthotropic) plates; thermal lamination theory for multilayered composites; procedures for reducing the size of the analysis models of anisotropic plates, shells and stiffeners with symmetric geometry; effective analysis techniques for anisotropic shells of revolution; and predictor-corrector approaches for the accurate determination of the detailed response characteristics of anisotropic plates and shells. Each of these advances is highlighted subsequently.

Exact three-dimensional elasticity (and thermoelasticity) solutions for anisotropic plates

Although complete three-dimensional elasticity solutions for the free vibrations, bending and stability problems of simply-supported orthotropic plates were presented in the late 1960s and early 1970s (see, for example [27-291). Only in the last two years have exact solutions been reported for the steady-state heat conduction, free vibrations, bending and thermal buckling of antisymmetrically laminated anisotropic (nonorthotropic) plates. The solutions were based on decomposing each of the plate variables into symmetric and antisymmetric components in the thickness direction, and expressing each of the components in terms of double Fourier series in the Cartesian surface coordinates (see [ 18, 30-331). The three-dimensional solutions obtained were used as the basis for assessing the accuracy and range of validity of several two-dimensional theories reported in the literature

(see 16, 71).

Thermal lamination theory for mukilayered anisotropic plates and shells

Two-dimensional computational models have been presented in [34-361 for the linear and nonlinear heat transfer in multilayered anisotropic panels. The models are based on a preselected temperature variation through the thickness of the laminate. While [34, 351 considered only linear conduction and convection modes of heat transfer, [36] considered nonlinear conduction and convection and presented a computational procedure for generating the sensitivity derivatives of the response with respect to both material and lamination parameters.

plates and shells 505

Governing differential equation of transversely loaded

anisotropic plate (classical theory):

Lo (w) + h 1, (w) = p

where

h = perturbation parameter (identifying anisotropy)

-co = D,, #+2(Q+ 2 DG6) a:$+ Da2$

Lh.=4(D,6a~a,+D,,a, a:)

Perturbation series:

w = x Ft’ WI

j=O

Recursion formulas for WI:

j = 0, 1, (wo) = p

j > 1. L, (Wj) = -JI*(Wj_,)

Fig. 7. Application of the classical perturbation technique to transversely loaded slightly anisotropic plate.

Model-size reduction for symmetric anisotropic plates and shells

The response of slightly anisotropic plates and shells can be generated as a small perturbation from the response of the corresponding orthotropic structure [37, 381. The application of the classical perturbation technique to a transversely loaded plate is depicted in Fig. 7. The plate is modeled by using the classical Kirchhoff plate theory. For anisotropic structures with symmetric geometry, loading and boundary conditions, this approach results in reducing the size of the analysis model to that of the corresponding orthotropic structures. For highly anisotropic structures, a two-step computational procedure was developed by the author and his colleagues for reducing the size of the analysis model to that of the corresponding orthotropic structure. The basic idea of the procedure is to generate the response of the anisotropic structure using a large perturbation from that of the corresponding orthotropic structure. The two key elements of the procedure are: (a) operator splitting or additive decomposition of the stiffness matrix into the sum of an orthotropic part and a nonorthotropic (anisotropic) part; and either (b) successive application of the finite element method and the classical Ray- leigh-Ritz technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh-Ritz technique; or (c) selecting the orthotropic part of the stiffness matrix (with all the nonorthotropic terms set equal to zero) as the preconditioning matrix and applying the preconditioned conjugate gradient (PCG) technique to account for the nonorthotropic terms. Several successful applications of the procedure have been reported in the literature. These included linear stress analysis, free vibrations, and nonlinear problems (see [39-421). The similarities

506 A. K. NCKIR

between (b) and (c) above were identified in [43] and exploited to generate from the PCG technique, direct measures for the sensitivity of the different response quantities to the nonorthotropic material coefficients of the structure. The procedure was extended to structures with unsymmetric boundary conditions in [44] and to structures with unsymmetric geometry in [45,46].

Effective analysis technique for anisotropic shells oj revolution

An efficient technique has been presented in [47] for stress and free vibration analyses of anisotropic shells of revolution. The basic idea of the technique

is to express each of the shell variables in terms of Fourier series in the circumferential coordinate and

to approximate the response associated with a range of Fourier harmonics by a linear combination of a few global approximation vectors, which are generated at a particular value of the Fourier harmonic within that range. The full equations of the shell are solved for only a single Fourier harmonic, and the response corresponding to the other Fourier harmonics is generated using

a reduced system of equations with the un- knowns being the amplitudes of the global approximation vectors. The strategy can be applied to any linear shell of revolution model, regardless of the material (viz. isotropic, orthotropic or anisotropic). Its effectiveness for anisotropic shells of revolution can be enhanced by combining it with the model-size reduction technique described in the preceding subsection. The strategy has been applied in[47,48] to anisotropic toroidal shells (tires). In the latter study, a first-order shear deformation theory was used, and the tire was modeled as a laminated shell with variable thickness

(see Fig. 8).

Predictor-corrector approaches

Experience with two-dimensional plate and shell theories has shown them to be inadequate for the accurate prediction of transverse stresses and deformations. This is particularly true for medium-thick and thick multilayered anisotropic plates and shells (see [6,49]). A simple approach for the accurate evaluation of transverse stresses and strains in medium-thick and thick composite plates and shells is to use a two-dimensional shear deformation theory

for calculating the in-plane stresses, then three- dimensional equilibrium equations to determine the transverse stresses. Improvements on this approach

are the predictor<orrector procedures which are essentially iterational processes in which the infor- mation obtained in the first (predictor) phase of the analysis is used to correct key elements of the computational model and, hence, improve the response

predictions. Two predictor-corrector procedures have been developed by the author and his colleagues for solution of isothermal and thermoelastic stress analysis and free vibration problems (see [SO-531). Both procedures use first-order shear deformation theory in the predictor phase to calculate initial

estimates for the gross response characteristics of the structure (e.g., vibration frequency, buckling load, critical temperature, average through-the-thickness displacements and rotations), as well as the in-plane stresses; then three-dimensional equilibrium

equations and constitutive relations are used to calculate transverse shear and transverse normal stresses and strains.

The two procedures differ in the elements of the

computational model being adjusted in the corrector phase. The first procedure calculates, a posteriori, estimates of the composite correction factors, k, and k, (see Fig. 9) and uses them to adjust the transverse shear stiffnesses of the plate (or shell). By contrast,

_ \A,:-- L-^,.^l^

NL = number of layers

Fig. 8. Laminated anisotropic shell model used for the space shuttle nose-gear tire (see [48]).

Mechanics of anisotropic plates and shells

Flnt-ordor shear dsformatlon theory Three-dlmonslonal quatlons

I l Calculate shell stlffnesses, A’s, B’s,

and D’s using lamination theory

l Select Initial values of composite correction factors k:, kg

I Predict the gross response characteristics (reference surface displacements,

strains, and stress resultants) I

Calculate through-the-thickness In-plane stresses o&

I 4

Calculate transverse stresses and stralns

(a) Predictor phase

I I I

I I Calculate corrected composite shear factors k,, k,

i

I I I

I + I

I Use a reanalysis procedure to correct I

gross response characteristics

I I

I JI I

I I

Correct through-the-thickness I dlsolacements and In-alane stresses 1 I

------a-----------------A

(b) Corrector phase in first procedure

-a------------mm-- _____T

I l Calculate through-the-thickness

’ I I displacements O,,S I

I l Use the thickness dlstributlons as I I basis functions

- I

I + I I Apply Rayleigh-Ritz technique In

conjunction with minimum potential I I energy prfnclple to determine I I amplitudes of basis functions

- I I J* I Calculate corrected displacements &, 6,

‘I I I

I ”

I in-plane stresses ui8 I d

and transverse stresses a,s, oa3 I I

/ __-__----___----____--__ I

(c) Corrector phase in second procedure

Fig. 9. Schematic representation of the steps involved in the two phases of the predictor-corrector procedures. Superscript 0 refers to predictions of first-order shear deformation theory; a bar over a symbol refers to response quantities obtained by three-dimensional equations; a caret refers to corrected quantities

(see 171).

508 A. K. NOOR

the second procedure calculates, a poster~ori, the functional dependence of the displacement components on the thickness coordinate. The corrected quantities are then used in conjunction with three- dimensional equations to obtain better estimates for the different response quantities. A schematic representation of the steps involved in the two phases of the predictor-corrector procedures is given in Fig. 8. The details of the first procedure are given in 149, SO], and the application of the second procedure to isothermal stress and free vibration analysis of multilayered composite plates and cylindrical shells is discussed in [52]. The appIi~ation of the second procedure to thermal buckling problems is described in [53]. Numerical experiments presented in the cited references have shown that only one iteration is needed (in the corrector phase) to obtain highly accurate response predictions.

Damage mechanics

Considerable progress has been made in recent years in identifying damage mechanisms as well as in predicting damage initiation and propagation in anisotropic plates and shells. This is particularly true for fibrous composite structures, for which classical fracture mechanics and traditional continuum mechanics models are inadequate. Damage mechanisms in continuous fiber reinforced structures include delamination and delamination buckling, fiber-matrix- interface debonding, matrix micro- and macro-cracking, fiber micro-buckling and fiber breakage. For regions of high stress concentrations deterministic damage criteria have been proposed, and for regions of low stress concentrations probabilistic damage criteria appear to be more appropriate (see [54]). An integrated multiscale modeling approach has been proposed in [55,56] to relate local effects to globa behavior of composite structures. The local effects accounted for include fiber-matrix-interface debonding, constituent material behavior which exhibits cyclic nonlinear history dependence, progressive damage and failure processes.

RECENT APPLICATtONS

Anisotropic plates and shells are currently used in several modern engineering systems (see, for example [57-591). In addition, anisotropic plates and shells are important components of a number of new aeronautics and space technology programs such as National Aerospace Plane (NASP), High-Speed Civil Transport (HSCT), Advanced Tactical Fighter (ATF), Integrated High-Performance Turbine Engine Technology (IHPTET), and Boost-Glide Vehicle (BGV). Among the recent important applications are:

(1) skin structure for wings, tails, control surfaces and engines of commercial aircraft {e.g., Boe- ing 777, Airbus A340 and McDonnell Douglas

various components of new cars and other transportation vehicles (see, for example [61]); and coastal and marine structures, liquid tanks, ceiling and panel units in indust~al buildings, bridge decks, highway embankments, and other infrastructure components ([62-t%]).

Anisotropic plates, sheils and membranes have also been used in modeling:

(1) airbags in new cars (1661); (2) directional, densely distributed cracks in

reinforced concrete slabs and shells ([67]); (3) biological systems such as the incised human

cornea [68]; cochlea of the ear [69], and the left ventricular wall of the heart [70] (see Fig. 11).

FUTURE D[RECTIONS FOR RESEARCH ON MODELING OF ANISOTROPW PLATES AND SHELLS

The expanded use of anisotropic plates and shells in high-tech and other industries is likely to accelerate in the next decade. Therefore, there is a need for exploiting the significant advances in both the com- puting technology and test methods to improve the current predictive ~pability of the response, life, performance and failure of anisotropic plates and shells. This is particularly true for structures made of new materials and/or subjected to harsh environments.

Examples of the new material systems are smart/intelligent materials (see, for example [7 1,721) and composites with three-dimensional reinforce- ments (as occurs with automated textile processes such as three-dimensional weaving, stitching, knit- ting, and braiding, which place reinforcing fibers through the thickness of the material to lock the composite plies into place and suppress delamination-see [73]). Harsh environment includes high temperatures, severe heat fluxes, electromagnetic and radiation effects. The key tasks for the accurate modeling of anisotropic plates and shells are:

1. E~~~orat~an of fundamental rne~~~ani~s theories to accurately represent the physics governing the plate for shelf) behavior through the entire range of temperature, strain and stress of interest.

2. Formuiation and implementation of effective computational strategies and numerical techniques for the efficient generation of the response, and for the simulation of failure of the structure. This includes, among other things, development of:

(a) hierarchical modeling and analysis strategies for anisotropic plates and shells with complicated geometry (in which different computational models are used for different regions of the structure, and are adaptively refined as needed);

(b) practical models and numerical techniques for predicting, in measurable and controllable parameters, the damage initiation and propagation

MD-IZX[60], see Fig. IO); in anisotropic plates and shells subjected to


Fig. IO. Boeing 777 aircraft. Top anisotropic composite panels and shells (shown shaded in the sketch). Bottom a deve~opmen~l version of the carbon/epoxy horizontal stabilizer with a span of 13.4 m (courtesy

of Boeing, Seattle, WA.

510 A. K. Noon

left ventricle

Model of heart’s ventricles

(Y.C. Pao, et al.) (P. Pinsky, et al.) Model of inner ear

anisotropic membrane (M. H. Holmes)

Fig. 11. Biological systems modeled as anisotropic plates and shells.

different loading conditions. For fibrous composite structures these models must incorporate local physics (e.g., microstructural effects).

3. Quality assessment, adaptive improvement, and

validation of numerical simulations. In addition to developing practical error estimates and selecting a benchmark set of anisotropic plate and shell problems for assessing new computational strategies and numerical solutions, a high degree of interaction and communication is needed between computational modelers and experimentalists. Carefully conducted experiments are required not only to vali- date the computational models, but also to under- stand the physical phenomena associated with the shell behavior.

4. Probabilistic analysis and stochastic modeling.

This is needed to account for uncertainties in geometry, material properties, loads and boundary conditions, as well as to quantify inherent uncertainties in the response of anisotropic plates and shells. However, the principal benefit of using any stochastic method consists of the insights into engineering, safety and economics that are gained in the process of arriving at those quantitative results and carrying out reliability analyses. As future anisotropic plate and shell structures become more complicated, failure mechanisms will be probabilistically modeled from the beginning of the design process, and potential design improvements will be evaluated to assess their effects on reducing overall risk. The results, combined with economic considerations, will be used in systematic cost-benefit analyses (perhaps also done on a probabilistic basis) to determine the structural design with the most acceptable balance of cost and risk.

5. Multidisciplinary analysis and design optimiz-

ation. The realization of new complex anisotropic plates and shells (e.g., those made of piezoelectric materials) requires integration between the structures discipline and other traditionally separate disciplines

such as heat transfer and electromagnetics. This is mandated by significant interdisciplinary interactions and couplings which need to be accounted for in predicting response, as well as in optimal design of these structures.

Acknowledgemenrs-The present work is partially supported by a NASA Cooperative Agreement NCCW-0011 and by an Air Force Office of Scientific Research Grant No. AFOSR-90-0369. The author acknowledges useful discussions with Samuel L. Venneri of NASA Headquarters and Spencer T. Wu of AFOSR.

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APPENDIX: LIST OF MONOGRAPHS, BOOKS AND SURVEY PAPERS ON ANISOTROPIC, RIBBED

AND LAYERED PLATES AND SHELLS

To the author’s knowledge the first monographs on anisotropic plates and shells were published in 1947 and 1961, respectively [Lekhnitskii, S. G., Anisotropic Plutes, Fizmatgiz, Moscow, 1947 (in Russian)] and [Ambartsumian, S. A., Theory of Anisotropic Shells, Fizmatgiz, Moscow, 1961 (in Russian)]. Since then, over 67 books and monographs have been published on the subject. For the benefit of the readers, all the monographs, books and survey papers on anisotropic, ribbed and layered plates and shells are listed subsequently (in chronological order).

I. Books and monographs

S. G. Lekhnitskii, Anisotropic Plates. Gostekhizdat, Moscow (in Russian) (1947).

2. A. V. Alexandrov, L. E. Bryuker, L. M. Kurshin and A. P. Rasskazov, Analysis of Three-Layered Panels. Oborongiz, Moscow (in Russian) (1960). S. A. Ambartsumian, Theory of Anisotropic Shells. Fizmatgiz, Moscow (in Russian) (1961); English translation NASA TTF-118 (1964).

4.

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9.

10.

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12.

13.

14.

15.

16.

17.

18.

G. 1. Pshenichnov, Analysis of Ribbed Cylindrical Shells. Izdatel’stvo Akademi Nauk SSSR, Moscow (in Russian) (1961). M. Sh. Mikeladze, Statics of Anisoiropic Plastic Shells. Izdatel’stvo Akademi Nauk Gruzinskoi SSR, Tbilisi (in Russian) (1963). G. Savin and N. P. Fleishman, Rib-Reinforced Plates and Shells. Naukova Dumka, Kiev (in Russian) (1964); English translation NASA TTF-427 (1967). V. I. Korolev, Laminated Anisotropic Reinforced Plastic Plates and Shells. Mashinostroenie, Moscow (in Russian) (1965); English translation NASA TM-76585 (1981). E. I. Grigolyuk and P. P. Chulkov, Critical Loads for Three-Layered Cylindrical and Conical Shells. Zapadno- Sibiriskoe Knizhnoe Izdatel’stvo, Novosibirisk (in Russian) (1966). F. S. Plantema. Sandwich Construction-The Bending and Buckling of Sandwich Beams. Plates and Shells. John Wiley, New York (1966). 0. G. Tsyplakov, Fundamentals of Fabrication of Glass Reinforced Plasric Shells. Mashinostroenie, Leningrad (in Russian) (1968). N. P. Abovskii, (Ed.), Computer Program for the Calculation of Ribbed Shallow Shells. Krasnoyarsk (in Russian) (1969). L. Librescu, Sratica si Dinamica Structurilor Elasrice Anizotrope si Eterogene (Statics and Dynamics of Elastic Anisotropic and Heterogeneous Type Structures). Editura Academiei Republicii Socialiste Romania, Bucharest (in Romanian) (1969). J. E. Ashton and J. M. Whitney, Theory of Laminated Plates. Technomic, Westport, CT (1970). V. K. Kabulov and K. Sh. Babamuradov, Computer Calculation of Three-Layer Shells. Izdatel’stvo Fan, Tashkent (in Russian) (1970). V. L. Bazhanov, I. I. Goldenblat, V. A. Kopnov, A. D. Pospelov and A. M. Siniukov, Plates and Shells Constructed From Glass-Reinforced Plastics. Vishaya Shkola, Moscow (in Russian) (1970). A. N. Elpatevskii and V. V. Vasilev, Strength of Cylindrical Shells Made of Reinforced Materials. Mashinostroenie, Moscow (in Russian) (1972). I. Ya. Amiro, V. A. Zarutskii and P. C. Poliakov. Ribbed Cylindrical Shells. Naukova Dumka, Kiev (in Russian). E. I. Grigolyuk and P. P. Chulkov, Stability and Vibrations of Three-Layered Shells. Mashinostroenie, Moscow (in Russian) (1973).

Mechanics of anisotropic platers and shells 513

19. Ya. M. Grigorenko, Isotropic and Anisotropic tayered 43. A. V. Alexandrov, R. E. Lamper and V. G. Suvernev, Shells of Revolution of Variable Stiffness. Naukova l&mka,*Kiev (in Russian) (1973). “-

Calculation of Structural Elements of Aircraft: Sandwich Plates and Shells. Mashinostroenie, Moscow (in

20. S. A. Ambartsumian, General Theory of Anisotropic Russian) (1985). Shells. Nauka, Moscow (in Russian) i1974). 44.

2 I. R. B. Rikards and G. A. Teters. Sfabilitv of Shells Made From Composite Materials. Zinatne, Riga (in Russian) (1974). 45.

22. S. A. Timashev, Stability of St@ned Shells. Stroiizdat, Moscow (in Russian) (1974).

23. Ya. M. Grigorenko, A. T. Vasilenko, E. I. Bespalova, 46. N. D. Pankratova, ‘I’. I. Polishchuk, and I. F. Latsinnik, Numerical Solution of Statics Problems for Orthotropic Shells with Variable Parameters. Naukova Dumka, Kiev (in Russian) (1975). 47.

24.

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26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

i. Librescu,, l&to;fotics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures. Noordhoff, Leyden (1975). 8. L. Pelekh and A. A. Siaskii, Stress Distribution Near Holes in Shear-Compliant An~sotropic Shells. Naukova Dumka, Kiev (in Russian) (1975). I. F. Obraztsov, V. V. Vasilev and V. A. Bunakov, Optimal Reinforcement of Shells of Revolution Con- structed From Composite Materials. Mashinostroenie, Moscow (in Russian) (1977). 0. M. Palii and V. E. Spiro, Anisotropic Shells in Ship Construction-Theory and Analysis. Sudostroenie, Leningrad (in Russian) (1977). I. A. Tsuroal and N. G. Tamurov. Ana&sis of Muiticon- netted Gyered and Nonlinear El&c Plates and Shells. Izdatel’stvo Vishcha Shkola, Kiev (in Russian) (1977). G, A. Teters, R. B. Rikarz, and V. L. Narusberg, Optimization of Shells Constructed From Layered &mposites. Zinatne, Riga (in Russian) (1978). G. A. Vanin. N. P. Semeniuk and R. F. Emelianov, Stability of’ Shells Constructed From Reinforced Materials. Naukova Dumka, Kiev (in Russian) (1978). C. W. Bert, Recent research in composite and sandwich plate dynamics. Shock Vibr. Digest ll(lO), 13-27 (1979). I. Ya. Amiro and V. A. Zarutskii, The Theory of Rib-Stiffened Shells-Methods for the Calculation of Shells, Vol. 2. Naukova Dumka, Kiev (in Russian) (1980). V. V. Bolotin and Yu. N. Novichkov, Mechanics of Multilavered Structures. Mashinostroenie, Moscow (in Russian) (1980). B. L. Pelekh and M. A. Sukhorolski. Contact Problems of the Theory of Elastic Anisotropic Shells. Naukova Dumka, Kiev (in Russian) (1980). V. A. Eltyshev, Stress-Deformation States of She& with Fillers. Nauka, Moscow (in Russian) (1981). V. I. Miachenkov and 1. V. Grigorev, Analysis of Composite Shell Structures on Electronic Computers. Mashinostroenie, Moscow (in Russian) (1981). I. Ya. Amiro, Mechodr of Vibration Analysis of Ribbed Shells on Electronic Computers. Naukova Dumka, Kiev (in Russian) (1982). B. Ya. Kantor, S. I. Katarzhnov and V. V. Arii, On the Theory of Sti~ned Shells. Akademii Nauk Ukrainskoi SSR, Kharkov (in Russian) (1982). B. L. Pelekh and V. A. Lazko, Layered An&otropic Plates and Shells with Stress Concentration. Naukova Dumka, Kiev (in Russian) (1982). G. I. Pshenichnov, Theory of Thin Elastic Ribbed Shells and Plates. Nauka, Moscow (in Russian) (1982). I. Ya. Amiro, V. A. Zarutskii and V. G. Palamarchuk, Dynamics of Ribbed Shells. Naukova Dumka, Kiev (in Russian) (1983). N. A. Alfutov, P. A. Zinovev and B. G. Popov, AnaIysis of Mu~filayer Plates and Shells of composite Materials. Mashinostroenie, Moscow (in Russian) (1984).

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57.

58.

59.

60.

61.

62.

63.

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65.

66.

I. V. Andrianov, V. A. Lesnichaia and L. I. Manevich, Method of Averaging of Statics and Dynamics of Ribbed Shells. Nauka, Moscow (in Russian) (1985). C. W. Bert, Research on dynamic behavior of composite and sandwich plates-IV. Shock Vibr. Digest 17(11), 3-15 (1985). Ya. M. Grigorenko, A. T. Vasilenko and N. D. Pankratova, Statics of Anisotropic Thick Walled Shells. Izdatel’stvo Vishcha Shkola, Kiev (in Russian) (1985). V. I. Klimanov and S. A. Timashev, Nonlinear Problems of Stiffened Shells. Akademi Nauk SSSR, Uralskii Nauchnii Tsenter, Sverdlovsk (in Russian) (1985). V. N. Kobelev and V. A. Potonakhin, Dynamics of Multilayered Shells. Izdatel’stvo Universitieta, Rostovna-donu (in Russian) (1985). V. Kovarik, Stresses in Luyered Shells of R~olution. Prague (1985); English translation by Elsevier, New York (1989). I. V. Andrianov, V. A. Lesnichaia, V. V. Loboda and L. I. Manevich, Strength Analysis of Ribbed Shells in Engineering Structures. Vishcha Shkola, Kiev (in Russian) (1986). N. 3. Hoff. Monocoque, Sandwich and Composite Aerospace Structures. Technomic, Lancaster, PA (1986). V. P. Ilin and V. V. Karpov, Strength and Stability of Rib~d Shells with Large Detections. Stroiizdat, Leningrad (in Russian) (1986). Yu. I. Khoma, Generalized Theory of Anisotropic Shells. Naukova Dumka, Kiev (in Russian) (1986). A. 0. Rasskazov, I. I. Sokolovskaya and N. A. Shul’ga, Theory and Analysis of Layered Orthotropic Plates and Shells. Izdatel’stvo Vishcha Shkola, Kiev (in Russian) (1986). S. A. Ambartsumian, Theory I$ Anisotropic Plates. Nauka, Moscow (in Russianj (1987). 1. Ya. Amiro. 0. A. Grachev, V. A. Zarutskii, A. S. Pal’Chevskii and I. A. Sannikov, Stability of Ribbed Shells of Revolution. Naukova Dumka (in Russian) (1987). A. E. Bogdanovich, Nonlinear Problems in the Dynamics of Cvlindrical Composite Shells. Izdatel’stvo Zinatne, Riga- (in Russian) (1987). V. N. Filatov. Elastic Textile Shells. Lecrmombvtizdat, Moscow (in Russian) (1987).

_. _

Ya. M. Grigorenko, A. T. Vasilenko and G. P. Golub, Statics of Anisotropic Shells with Finite Shear Rigidity. Naukova, Dumka, Kiev (in Russian) (1987). J. N. Reddy and K. Chandrasekhara, Recent advances in the nonlinear analysis of laminated composite plates and shells. Shock Vibr. Digest 19(4), 3-9 (1987). G. A. Vanin and N. P. Semeniuk, Stability of Shells of Composite Muterials with Imperfections. Naukova Dumka, Kiev (in Russian) (1987). J. M. Whitney, Structural Analysis of Laminated An- isotropic Plat&. Technomic, Lancastdr, PA (1987). E. I. Grieolvuk and G. M. Kulikov. Mult~lavered Reinforced- ihells. AnaQsis of Pneumatic Fires. Mashinostroenie, Moscow (in Russian) (1988). Ya. M. Grigorenko and N. N. Kriukov, Numerical Solution of Static Problems for Elastic Luyered Shells with Variable Parameters. Naukova Dumka, Kiev (in Russian) (1988). I. Ya. Amiro, V. A. Zarutskii and V. W. Revutskii, Vibration of Ribbed Shells of Revolution. Naukova Dumka, Kiev (in Russian) (1989). G. D. Gavrilenko, Sfabiiity of Ribbed Cyiindric~~ Shells Under Nonhomogeneous Stress Reformation States. Naukova Dumka, Kiev (in Russian) (1989).

CAS 4413-B

514 A. K. Noort

67. R. K. Kapania and S. Raciti, Recent advances in analysis and laminated beams and plates, Part I: shear effects and buckling, Part II: vibrations and wave propagation. AIAA Jnl 27(7), 923-946 (1989).

II. Survey papers

1. L. M. Habip, A review of recent work on multilayered structures. Int. J. Mech. Sci. 7, 589-593 (1965).

2. S. A. Ambartsumian, Some current aspects of the theory of anisotropic layered shells. In Applied Mechanics Surveys (Edited by H. N. Abramson, H. Liebowitz, J. M. Crowley and S. Juhasz), pp. 301-314. Spartan Books, Washington, DC (1966).

3. S. A. Ambartsumian, Specific features of the theory of shells made of currently available materials. Izvestia Akad. Artnianskoi SSR, Mekhanika 21(4), (in Russian), 3-19 (1968).

4. C. W. Bert and D. M. Egle, Dynamics of composite, sandwich and stiffened shell-type structures. J. Spacecraft and Rockets 6( 12), 1345-1361 (1969).

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7. C. W. Bert, Analysis of shells. In Structural Design and Analysis (Edited by C. C. Chamis), Part 1, Vol. 7, pp. 207-258. Academic Press, New York (1975).

8. C. W. Bert, Analysis of plates. In Composite Materials-Structural Analysis and Design (Edited by C. C. Chamis), Vol. 7, Part I, pp. 149-206. Academic Press, New York (1975).

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