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Nonlinear static and dynamic finite element analysis of multilayer shell structures

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Nonlinear static and dynamic finite element analysis of multilayer shell structures
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Tan, Xiangguang, 1970-
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ix, 302 leaves : ill. ; 29 cm.

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Aspect ratio ( jstor )
Bending ( jstor )
Deformation ( jstor )
Eggshells ( jstor )
Engineering ( jstor )
Matrices ( jstor )
Patch tests ( jstor )
Sandwiches ( jstor )
Tangents ( jstor )
Tensors ( jstor )
Aerospace Engineering, Mechanics, and Engineering Science thesis, Ph.D ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics, and Engineering Science -- UF ( lcsh )
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Thesis:
Thesis (Ph. D.)--University of Florida, 2002.
Bibliography:
Includes bibliographical references (leaves 287-301).
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Also available online.
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Printout.
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Vita.
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by Xiangguang Tan.

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NONLINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS OF
MULTILAYER SHELL STRUCTURES














BY

XIANGGUANG TAN
















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2002






























To my parents.














ACKNOWLEDGMENTS

I wish to express my sincere gratitude to my advisor, Prof. Loc Vu-Quoc, for his pa-

tience, guidance, support and friendship throughout my Ph.D. education at the University

of Florida. I have greatly benefited from his stimulating approach to research and his re-

lentless pursuit of perfection in organization and documentation. Many thanks are extended

to him for his invaluable help in preparing this LINUX/LaTeX document.

I also wish to acknowledge the members of my examining committees, Professors

Martin A. Eisenberg, Raphael T. Haftka, Marc Hoit, Andrew J. Kurdila, and W. Gregory

Sawyer for their careful examination of the dissertation, and their invaluable comments

and insights, which made a deep impact on my research. I also benefited greatly from their

graduate courses and from their help in many other aspects.

I am indebted to several colleagues and mentors for their help in my present work:

in particular, Hui Deng for the use of finite element code, FEAP and many insightful dis-

cussions on the geometrically-exact shell theory; Fuller L. Brian for the installation of the

LLNL package; Paul Dionne, Andrzej Przekwas, Marek Turowski, and H.Q. Yang at the

CFDRC for discussion of the model reduction technique; Prof. Chen-Chi Hsu to work for

him as his teaching assistant; and to my friends, Joakim Andersson, Jonas Bjornstrom, Mat-

tias Horling, Stefan Jansson, Kil-Soo Mok, Mattias Quas, Simon. Sjogren, Xiang Zhang,

and Yuhu Zhai, and many others, who have made my stay at Gainesville one of the most

memorable periods of my life.

Last, but certainly not least, my heartfelt thanks go to my parents for their love, and

encouragement through my life. I am indebted to my girlfriend, Veronica Leung. Without

her love and care, I could not have accomplished so much.

This research is supported by a grant from the National Science Foundation, and also

by the CFDRC. This support is gratefully appreciated.

iii












TABLE OF CONTENTS

page

ACKNOWLEDGMENTS. ............... ............. 111

ABSTRACT ......................... ......... .. viii

CHAPTER

1 OVERVIEW ...................... ............ 1

1.1 Objectives and Motivation ................. ........... 1
1.1.1 Formulation and Kinematics ....................... 2
1.1.2 Computational Aspects .. ..................... 4
1.2 Chapter Overview ............... .............. 5

2 GEOMETRICALLY-EXACT SANDWICH SHELL FORMULATION ..... 7

2.1 Introduction .................. .......... ... 7
2.2 Virtual Powers ............... ..... .......... 8
2.2.1 Basic Kinematic Assumptions and Configurations ..... .. 9
2.2.2 Virtual Powers ........................... 11
2.2.2.1 Power of contact forces/couples and conjugate strain mea-
sures . . . 11
2.2.2.2 Power of assigned forces/couples . .... 13
2.2.3 Constitutive Relations ....................... 13
2.3 Weak Form and Linearization . . .. 14
2.3.1 Admissible Variations, Tangent Spaces . .... 14
2.3.2 Weak Form of Equations of Equilibrium . .... 15
2.3.3 Contact Weak Form ........................... 16
2.3.4 Assigned Weak Form ........................ .. 16
2.3.5 Linearization of Contact Weak Form . ... 17
2.3.5.1 Update of inextensible directors . ... 17
2.3.5.2 Perturbed configuration . ... 18
2.3.5.3 Linearized strain measures . .... 20
2.3.5.4 Linearized contact weak form . ... 22
2.3.6 Matrix-Operator Format of Contact Weak Form . 22
2.3.6.1 Material tangent operator . ... 31
2.3.6.2 Geometric tangent operator . ... 34
2.4 Numerical Examples for Statics of Sandwich Shells . .... 46
2.4.1 Roll-down Maneuver of a Sandwich Plate . .... 48


iv









2.4.2 Sandwich Plate with Ply Drop-offs . ... 49
2.4.2.1 Sandwich plate with ply drop-off . .. 49
2.4.2.2 Two-layer plate with ply drop-off: aspect ratio A = 5 : 1 :
(1,0.5) . . . .. 53
2.4.2.3 Two-layer plate with ply drop-off: aspect ratio A = 20: 1 :
(1,0.5) .. . .. .. .. 53
2.4.2.4 Two-layer plate with ply drop-off: aspect ratio A = 20 : 10 :
(1,0.5) ................... ........ 61

3 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER
COMPOSITES : STATICS ........... .. .... ...... 68

3.1 Introduction ... ... .. .. . .... 68
3.2 Kinematic Assumption and FHW Variational Formulation ... 74
3.2.1 Kinematics of Solid-Shell in Curvilinear Coordinates ... 74
3.2.2 Variational Formulation of EAS Method . ... 78
3.3 Finite-Element Discretization . . ... 83
3.3.1 The Weak Form of Modified Two-Field FHW Functional 83
3.3.2 Spatial Discretization . ... .. 84
3.3.3 Linearization of the Discrete Weak Form . ... 85
3.3.4 Material Law in Convected Basis . ... 89
3.3.5 The ANS Method .......................... .. .93
3.3.5.1 Transverse shear strains . ... 93
3.3.5.2 Transverse normal strain . ... 94
3.4 Interpolation of the Enhanced Strains . . ... 94
3.4.1 The Regular Enhanced Strains Treatment . 95
3.4.2 Proposed Efficient Enhancing Strains . . 100
3.4.3 Equivalence Between EAS Element and Incompatible Mode Element 102
3.4.3.1 Tensor form of enhancing strains . ... 103
3.4.3.2 Equivalence of condensed stiffness matrices ... 107
3.5 Numerical Examples ....... ............... ..... ..... 109
3.5.1 Patch Tests and Optimal Number of Parameters . 110
3.5.1.1 Membrane patch test . . Ill
3.5.1.2 Out-of-plane bending patch test ........ 111
3.5.2 Cantilever Plate ........................... 113
3.5.2.1 Cantilever beam: in-plane bending . ... 114
3.5.2.2 Cantilever plate: out-of-plane bending . ... 115
3.5.3 In-plane Bending Problem with Nearly Incompressibility 119
3.5.4 Snap-through of a Shallow, Cylindrical Roof under a Point Load 121
3.5.5 Pinched Hemispherical Shell . ..... 122
3.5.6 Multilayer Composite Plate . ... .. 125
3.5.6.1 Two-layer composite plate: linear solution ... 125
3.5.6.2 Multilayer composite plate with ply drop-offs ... 126
3.5.7 Multilayer Composite Hyperbolical Shell with Ply Drop-offs 129


v








4 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER
COMPOSITES: DYNAMICS .................... ... .... 132

4.1 Introduction .................... ............ 132
4.2 Dynamics of Solid Shells by an EM Conserving Algorithm ....... 133
4.2.1 Time Discretization on Dynamic Weak Form ... ...... 134
4.2.2 Linearization of Dynamic Weak Form ............... 136
4.3 Enhanced-Assumed-Strain Method Based on Deformation Gradient .. 143
4.3.1 Weak Form .............................. 143
4.3.2 Finite Element Discretization and Linearization . 147
4.3.3 Assumed Natural Strain (ANS) Treatment . ... 150
4.3.4 Simplified Formulation . . ... 151
4.4 Numerical Examples .... .... ............. .... ... .. 153
4.4.1 Double Cantilever Elastic Beam under Point Load . 154
4.4.2 Pinched Cylindrical Multilayer Shell. . 157
4.4.3 Free-Flying Single-Layer Plate . . 159
4.4.4 Free-Flying Multilayer Plate with Ply Drop-offs ... 161

5 EFFICIENT AND ACCURATE MULTILAYER SOLID-SHELL ELEMENT: NON-
LINEAR MATERIALS AT FINITE STRAIN . ... 175

5.1 Introduction ....... ....... .. ................ .. 175
5.2 Nonlinear Material Law .......................... .180
5.2.1 The Mooney-Rivlin Material Models . ... 180
5.2.2 The Hyperelastoplastic Model . . ... 182
5.2.2.1 Multiplicative decomposition of the deformation gradient F 183
5.2.2.2 Spectral form based on the right Cauchy-Green tensor C 185
5.3 Explicit Time Integration Method for Solid-Shell Elements ... 193
5.4 Numerical Examples ....... ..... ..... .......... .. 196
5.4.1 Large Deformation of Rubber Shells . 197
5.4.1.1 Stretch of a rubber sheet with a hole . ... 198
5.4.1.2 The snap-through of a conic shell . 198
5.4.1.3 Large motion of the pinched cylindrical shell .. 200
5.4.1.4 Rubber hemispherical shell . ... 203
5.4.2 Large Deformation of Elastoplastic Shells . ... 204
5.4.2.1 Bending of a cantilever beam . ... 206
5.4.2.2 Elastoplastic response of a channel beam . 208
5.4.2.3 Pinched hemisphere . . ... 211
5.4.2.4 Elastoplastic response of a simply supported plate 213
5.4.2.5 Elastoplastic response of a pinched cylinder . 215
5.4.2.6 Free-flying multilayer plate with ply drop-offs ... 218
5.4.2.7 The impact of a boxbeam . ..... ... 221
5.4.2.8 Pipe whip ......................... 223

6 SOLID SHELL ELEMENT FOR ACTIVE PIEZOELECTRIC SHELL STRUC-

vi








TURES AND ITS APPLICATIONS ....................... 228

6.1 Introduction .................... ............ 228
6.2 The Solid-Shell Formulation ................. ... ...... 231
6.2.1 The Kinematics of Piezoelectric Solid-Shell Formulation 231
6.2.2 Piezoelectric Solid-Shell Element ... .. 234
6.2.2.1 Functional and finite element formulation. ... 234
6.2.2.2 Linear piezoelectric material law in convected coordinate .239
6.2.3 Composite Solid-Shell Element . . ... 242
6.3 Simulation Control Design . . 244
6.3.1 Finite Element System Equation of Piezoelectric Structure 244
6.3.2 Reduced-Order Model of Piezoelectric Finite Element System 246
6.3.3 Controller Design ............. ............ .. 249
6.4 Numerical Examples . ..... . .. 252
6.4.1 Cantilever Plate: Out-of-Plane Bending . ... 253
6.4.2 Multilayer Composite Hyperbolical Shell . ... 255
6.4.3 Piezoelectric Bimorph Beam . .. .. 256
6.4.4 Cantilever Plate with PZT Actuators . ... 259
6.4.5 Cantilever Plate with PZT Actuator and Sensor . ... 263

7 CLOSURE ...................... ........... 268

7.1 Conclusion . ..... ........... .. 268
7.2 Directions for Future Research. . ..... ... 270

APPENDIX

A SOLID-SHELL FORMULATION . . ... 272

A.I Finite Element Approximation of Solid-Shell Element ... .. 272
A.2 Solution Procedure of Nonlinear Equations . ... 279
A.3 Explicit Integration Algorithm with EAS Method . ... 280
A.4 Return Mapping Algorithm for J2 Flow Theory with Isotropic Hardening 280
A.5 Elastoplastic Moduli Ej.P ................... ..... 281
A.6 Algorithmic Moduli for Return Mapping . .... 282

B PIEZOELECTRIC SOLID-SHELL FORMULATION . ... 285

B.1 Model Reduction Algorithm . ... 285
B.2 Solving Procedure on Control Design. ... . ...... 285

REFERENCES ................... ................. 287

BIOGRAPHICAL SKETCH ............................ 302




vii













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NONLINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS OF
MULTILAYER SHELL STRUCTURES

By

Xiangguang Tan

August, 2002

Chairman: Loc Vu-Quoc
Major Department: Aerospace Engineering, Mechanics, and Engineering Science

Firstly, the geometrically-exact sandwich shell formulation is developed to analyze

sandwich shells undergoing large deformation. Finite rotation of the director in each layer

is allowed, with shear deformation independently accounted for in each layer. The thick-

ness and the length of each layer can be arbitrary, thus allowing the modeling of multilayer

structures having ply drop-offs. The weak form of governing equations is constructed, and

the linearization and inextensible directors update are derived. Numerical examples on

elastic sandwich plates are presented to illustrate salient features of the formulation.

Furthermore, we present a low-order solid-shell element formulation-having only

displacement degrees of freedom (dofs) (i.e., without rotational dofs)-that has an opti-

mal number of parameters to pass the plate patch tests (both membrane and out-of-plane

bending), thus allowing for efficient and accurate analyses of large deformable multilayer

shell structures. The formulation is based on the mixed Fraeijs de Veubeke-Hu-Washizu

(FHW) variational principle leading to a novel enhancing assumed strain (EAS) tensor,

with improved in-plane and out-of-plane bending behaviors (Poisson thickness locking).

Shear locking and curvature thickness locking are treated using the Assumed Natural Strain

(ANS) method. We provide an optimal combination of the ANS method and the minimal

number of EAS parameters to pass the out-of-plane bending patch test and treat the locking


viii








associated with (nearly) incompressible materials. The energy-momentum (EM) conserv-

ing algorithm for the current element is presented. Two nonlinear 3-D material models are

applied directly without requiring the enforcement of the plane-stress assumption. More-

over, we present a low-order accurate piezoelectric solid-shell element formulation for

piezoelectric sensors and actuators used in active shell structures. Numerical examples in-

volving static analyses and implicit/explicit dynamic analyses of multilayer shell structures

having a large range of element aspect ratios for both material and geometric nonlinearities

are presented. Numerical examples involving static analyses and active vibration control of

piezoelectric shell structures are also presented. The developed element formulations are

accurate and efficient in modeling and analyzing general nonlinear multilayer composite

shell structures.
































ix














CHAPTER 1
OVERVIEW

Shells and shell structures are thin-walled, generally curved bodies in a three-dimens-

ional space. Their load-bearing behavior is dominated by stretching and bending. Shell

structures with different layers in the thickness direction are generally addressed as multi-

layer shells. For a comprehensive and valuable history and review of linear and nonlinear

shell theories, see Timoshenko and Woinowsky-Krieger [1959], Naghdi [1972] and Basar

and Kritzig [2000]. Below we describe the objectives and motivation for the current re-

search on multilayer shells. Some of the motivating factors behind the present work and

literature review are delineated in the following chapters.

1.1. Objectives and Motivation

Multilayer shell structures have widespread applications in engineering. Laminated

composite structures, initially developed for use in the aerospace industry, have played

an increasingly important role in robotics and machine systems that require high operat-

ing speed. The low weight and high stiffness offered by laminated composite structures

help reduce power consumption, increase the ratio of payload/self-weight, and improve

the accuracy of motion characteristics and reduce the level of acoustic emission of these

systems. It is shown from computer simulations with experimental corroboration that the

low weight/stiffness ratio of laminated composites is essential for obtaining high perfor-

mance in slider-crank and four-bar linkage systems (Sung, Thompson, Crowley and Cuccio

[1986], Thompson and Sung [1986]). More recently, considerable attention has been given

to a class of active structures with embedded piezoelectric layers as sensors and actuators

(Evseichik [1989], Tzou [1989], Saravanos, Heyliger and Hopkins [1997]) or interfero-

metric optical fiber sensors (Sirkis [1993]) for monitoring the strain level and for vibration

control. Large overall motion of multilayer structures can be found in robot arms or space


1





2


structures with embedded sensors/actuators. Another example of multilayer structures can

be found in the damping of structural vibration by using viscoelastic constrained layers (Al-

berts [1993], Dubbelday [1993], Rao [1993]) (Figure 1.1). The use of sandwich plates to

absorb energy in crashes (car, train, airplane) was investigated by Goldsmith and Sackman

[1991].












Figure 1.1. Multilayer shells with patches of constrained viscoelastic materials or of piezo-
electric materials.

The design and analysis of multilayer shell structure is a major challenge that in-

volves the proper modeling of composite materials with highly anisotropic properties, com-

plex geometric configuration, and strongly nonlinear material behavior. For example, only

a few studies so far have been performed on large deformation analysis of 3-D nonlin-

ear composite laminates. There have been no analytical studies involving 3-D analysis of

multilayer shells with nonlinear material behavior and large deformation.

1.1.1. Formulation and Kinematics

For nonlinear analysis of multilayer shell structures, we developed two different finite

element formulations: the geometrically-exact' multilayer shell formulation and multilayer

solid-shell formulation.

In the geometrically-exact multilayer shell formulation, the 3-D analysis is reduced

to a set of 2-D stress-resultant equations based on the kinematic assumptions. This model

accommodates large deformation and large overall motion. The layer directors at a point in
SThe term "geometrically-exact" reflects the fact that no additional kinematic assumptions are made
beyond the one-director assumption. In particular, approximations of the type sinO 0 03/6 are entirely
avoided.






3


the reference surface are connected to each other by universal joints, as in a chain of rigid

links. The thickness and length of each layer can be arbitrary, thus making it suitable to

model shell structures with ply drop-offs. The equations of motion of the multilayer shell

are derived based on the principle of virtual power, and expressed in terms of weighted

resultant forces and couples. The overall deformation of a sandwich shell can be described

by the deformation of a reference layer (which can be any layer; not necessarily the middle

layer). The unknown kinematic quantities are therefore the three displacement components

of the centroidal surface of the reference layer and two rotational components for each

layer director. No restriction is imposed on the magnitude of the displacement field, whose

continuity across the layer interfaces is exactly enforced. Finite rotations of the directors in

each layer are allowed, with shear deformation independently accounted for in each layer.

We have implemented the geometrically-exact sandwich shell element to illustrate the ver-

satility of formulation in the large deformable multilayer shell analysis involving linear

elastic material and small strain. Due to the kinematic assumptions, the present formula-

tion is more accurate than the equivalent single-layer shell models in the interlaminar stress

analysis, especially for thick and moderate thick shells.

In the solid-shell formulation, on the other hand, the shell kinematic descriptions

used are the displacement of the top and bottom surface of the shell. All kinematic quan-

tities such as displacements and the corresponding strains can be finite. For multilayer

shells, one solid-shell element in the thickness direction can be used for either one material

layer or several layers. In contrast to the shell formulation based on the degenerated shell

concept and the classical shell theory, the present element can incorporate the complex 3-D

material models without enforcing the zero transverse normal stress condition, can avoid

complex update algorithms for finite rotations, and can account for the transverse normal

stress. Based on the mixed Fraeijs de Veubeke-Hu-Washizu (FHW) variational principle,

the present low-order solid-shell element is designed to pass the plate patch tests and to

remedy volumetric locking, therefore allowing efficient and accurate nonlinear analyses

of multilayer shell structures. Moreover, the kinematic description provides a natural way





4


to connect solid-shell elements to regular solid elements without the need for transition

elements; such feature can also benefit the detailed modeling of shells with patches of

piezoelectric or viscoelastic materials. For the interlaminar stress analysis, with the refine-

ment through the thickness, the solid-shell element model can determine the localized 3-D

stress field (e.g., delamination, free-edge effect) accurately.

1.1.2. Computational Aspects

Several aspects can directly contribute to the success and generality of numerical

simulations: 1) element formulations; 2) time-integration schemes; and 3) equation solu-

tion strategies. The geometrically-exact sandwich shell formulation uses the resultant form

to avoid numerical integration in the thickness direction for elastic materials. The solid-

shell formulation uses the numerical integration for general nonlinear constitutive models.

All kinematic quantities such as displacements can be finite, and the update procedure is

proceeded in an exact manner, without approximations.

Engineering applications mandate the use of relatively coarse meshes for complex

geometries. The development of convergent elements, which are free of spurious numer-

ical locking, are variationally consistent, achieve good accuracy with coarse meshes, and

satisfy stability and completeness requirements, is essential. Flexural super-convergence

in membrane deformation is also important for applications involving in-plane bending.

Moreover, the use of low-order interpolations is extremely desirable for their simplicity,

efficiency and amenability to contact implementations. To this end, we use the methods

of enhanced assumed strains (EAS) and assumed natural strains (ANS) judiciously to con-

struct low-order elements possessing the above features for the analysis of multilayer shells.

In this work, we have implemented a number of dynamic time-stepping implicit/explicit

algorithms in the context of the present formulations for transient integration of the result-

ing semi-discrete finite element equations. The time step-size for the implicit integration

can be much larger than that for the explicit integration. The explicit method, on the other

hand, needs much less computational effort at each time step since the matrix factoriza-

tion is not needed. For elastodynamics, the introducing of numerical damping is essential





5


to increase the numerical stability of implicit integration methods, even for the energy-

momentum conserving algorithm.

The solution of discrete equations for problems involving large deformation and

long-term simulations can be accomplished with the Newton-Raphson scheme. The numer-

ical efficiency of this approach is a byproduct of the asymptotically quadratic convergence

of its iterations. To maintain this rate of convergence, the exact linearization of discrete

equations is explicitly obtained and implemented in the present work. For nonlinear mate-

rials, the consistent tangent moduli are crucial to be derived. For the quasi-static analysis of

unstable systems, arc-length method is used to find stability points and trace post-buckling

paths. Based on the above solution strategies, a large time or load increment is allowed to

use, while a good balance of accuracy and efficiency is maintained.

1.2. Chapter Overview

This dissertation is divided into six chapters. Two finite element models for multi-

layer shell structures, the geometrically-exact sandwich shell element and the solid-shell

element, are formulated and implemented.

Chapter 2 presents the static analysis of the geometrically-exact sandwich shell el-

ement formulation. The kinematic description and equilibrium equations of the sandwich

shell model are presented in Section 2.2. The corresponding weak form and linearization

are given in Section 2.3. Numerical examples for statics of sandwich shells are shown in

Section 2.4. This chapter has been published by Vu-Quoc, Deng and Tan [2000]. Readers

refer to Vu-Quoc, Deng and Tan [2001] for the corresponding dynamic analysis.

In Chapter 3, we carry out the static analysis of the optimal solid-shell element for-

mulation for multilayer composites. After a presentation of the kinematics assumption

and the formulation based on the Fraeijs de Veubeke-Hu-Washizu (FHW) variational prin-

ciple (Felippa [2000]) in Section 3.2, we discuss the finite-element discretization and its

implementation in Section 3.3. A review of the EAS method together with our proposed

modification is presented in Section 3.4. We present the numerical results in Section 3.5.

This chapter will be published by Vu-Quoc and Tan [2002a].





6


Chapter 4 addresses the dynamic analysis of the optimal solid-shell element formu-

lation for multilayer composites. We devoted Section 4.2 to the dynamic aspect and the use

of the energy-momentum algorithm for elastic materials. A variant of the EAS formulation

based on the deformation gradient (instead of Green-Lagrange strains) for solid shells is

the focus of Section 4.3. Numerical results are shown in Section 4.4. This chapter will be

published by Vu-Quoc and Tan [2002b].

In Chapter 5, we present static and dynamic analyses of the multilayer solid-shell

element formulation for nonlinear materials at finite strain. Two nonlinear material models

(i.e., Mooney-Rivlin material and hyperelastoplastic material), and their implementations

are discussed in Section 5.2. The explicit integration method for solid-shell elements is

addressed in Section 5.3. Numerical simulations, which illustrate the performance of the

proposed element formulation, and exhibit both material and geometric nonlinearities in the

large-scale implicit/explicit analyses, are given in Section 5.4. This chapter was submitted

for the publication by Tan and Vu-Quoc [2002a].

Chapter 6 discusses the solid shell element for active piezoelectric shell structures

and its applications. In Section 6.2, we introduce the kinematics and variational formu-

lation of the piezoelectric solid-shell element, and then present the composite solid-shell

element. The control design for structures with piezoelectric sensors and actuators is dis-

cussed in Section 6.3. Numerical simulations that illustrate the performance of the pro-

posed formulations, including comparisons with available experiment results and solutions

obtained from shell elements and solid elements, are given in Section 6.4. This chapter was

submitted for the publication by Tan and Vu-Quoc [2002b].

Chapter 7 gives the closure of our work. Conclusions are drawn in Section 7.1 and

directions for future investigation are suggested in Section 7.2.














CHAPTER 2
GEOMETRICALLY-EXACT SANDWICH SHELL FORMULATION

2.1. Introduction

Sandwich structures have played an important role in several areas of engineering.

Many background references were cited by Vu-Quoc and Ebcioglu [1995], and are not re-

peated in the present follow-up work, except for particularly relevant ones. We refer to

review papers such as Reddy [1989], Noor and Burton [1989], Noor [1990], Reddy and

Robbins [1994], and the references therein for various aspects on formulations for multi-

layer structures. The accuracy of layerwise theory, as compared to single-layer theory with

a shear correction factor, was demonstrated amply by Reddy [1989], where a comparison

of transverse shear stress with 3-D elasticity solution was provided (Reddy [1993]). We de-

scribe here a continuation of the results reported by Vu-Quoc, Ebcioglu and Deng [1997],

where the equations of motion for geometrically-exact sandwich shells are derived. Focus-

ing on the static case in the present work, we develop a Galerkin projection of the resulting

nonlinear governing equations of equilibrium.

In the present formulation, each layer in a sandwich shell structure can have different

thickness and side lengths. As such, the present formulation can be used to model an

important class of multilayer structures with ply drop-offs. Another important application

of the present formulation is the modeling of shell structures with patches of constrained

viscoelastic materials and/or patches of piezoelectric materials. No restriction is imposed

on the magnitude of the displacement field, whose continuity across the layer interfaces

is exactly enforced. Finite rotations of the directors in each layer are allowed, with shear

deformation independently accounted for in each layer. The layer directors at a point in the

reference surface are connected to each other by universal joints, and form a chain of rigid

links. The overall deformation of a sandwich shell can be described by the deformation of


7





8


a reference layer. The unknown kinematic quantities are therefore the three displacement

components of the deformed reference surface, and the unit directors associated with the

layers.

The starting point for the development of the Galerkin projection of the governing

equations of equilibrium is a nonlinear weak form based on the stress power of a sandwich

shell, from which the expressions of fully nonlinear strain measures are obtained (Vu-

Quoc et al. [1997]). A linearization of this nonlinear weak form is performed for use in the

solution for the kinematic quantities via the Newton-Raphson method. Together with the

update of the inextensible directors, the linearization leads to a symmetric tangent stiffness

operator, which is composed of a geometric part and a material part. The consistency

in the linearization leads to a quadratic rate of asymptotic convergence in the Newton-

Raphson iterative solution. Linear finite element functions are chosen to form a basis for

the Galerkin projection of the linearized equilibrium equations into a finite-dimensional

subspace of trial solutions. The tangent stiffness matrix is symmetric, and is evaluated

using selectively reduced integration in all layers to avoid shear locking.

Several numerical examples, including the bending and torsion of a sandwich plate,

are presented to illustrate the salient features of the present formulation. In particular, the

important case of sandwich shells with ply drop-offs under large deformation is presented.

Results are compared with those obtained using the commercial nonlinear finite element

code ABAQUS [1995].

2.2. Virtual Powers

In this section, we summarize the kinematic description of the sandwich shell model

developed in Vu-Quoc et al. [1997], and the equilibrium equations in weighted resultant

form.2 The component form of the stress power and of the constitutive relation are also

given at the end of this section.
2 The word "weighted" here is used to indicate that the resultants are in "weighted" tensor form and not
in true tensor form (the reader is referred to Vu-Quoc et al. [1997] for an explanation).





9

2.2.1. Basic Kinematic Assumptions and Configurations
1
Let A designate the material surface of the shell, and H := U (ce the total
thickness of the sandwich shell (Figure 2.1), where (e)- is the thickness of layer (e). Let
4o : x A 7-H Bo be the mapping from the material configuration to the initial (reference)
configuration, where A x H = B is the material configuration of the sandwich shell. The
material domain for layer (e) is denoted by (p)B such that
1
()B :=Ax ()H B:= U ()B (2.1)

e--




S ()h+ Layer (1)
(1)H
(I)h-

E (o) H (0) h+ Layer (0 2
ElI1- (o)h-



(-1)h+/

(-1)h- Layer (-1)


Figure 2.1. Sandwich shell: Profile and geometric quantities.

Let 4 : A x RH Bt be the deformation map from the material configuration to
the current deformed (spatial) configuration. We use the notation { := { (, 3 } E B to
denote the coordinates of a material point, where { := { Q1, 2 } E A is referred to as the
material surface coordinates, and 3 E the material through-the-thickness coordinate.
The deformation map for each of the three layers is written as follows

(1)4( (, t) = (e ( t) + (3 (e)Z) (e)t fore= -1,0,1, (2.2)





10

where (f)P : A R3 is the deformation map of the centroidal surface of layer (e), and
(e)t : A ri- S2 the unit director (represent transverse fiber vector) associated with layer
(f), (e)Z the distance from the centroidal surface of layer () to the centroidal surface of
the reference layer (0) (for which (o)Z = 0). The centroidal surface of each layer (e),
which does not necessarily correspond to the geometric center of the cross-section of layer
() are defined as follows: Let (e)Po and (t)Pt be the mass density in the initial and the
spatial configuration, Bo and Bt, respectively. We select the centroidal surface (e)(po of
layer (f) such that

/ (3 + (_)Z) (-J (1)pod3 = (3 + (_I)Z ( (-)j(-1)ptd3 = 0,
(-1)7 (_1i-

3 (o)jo (o)pod 3 = (ojt (o)Ptd3 = 0, (2.3)
(0)N (0)7

J (3 (IZ) (j (1)pod 3 = f (3 (1)Z) ( (1)pt dd3 = 0.
(1)7" (1)7-1
Using the assumption that the layer directors behave like a chain of rigid links connected to
each other by universal joints, the deformation maps of the centroidal surface of the outer
layers (1) and (-1) can be related to the deformation map of the centroidal surface of the
reference layer (0) as

(1)' ( t) := (o)O ( t", ) + (o)h+ (o)t + (i)h- ()t ,

(-1)o (p -, t) := (o)^ (-, t) (o)h- (o)t (-_lh+ (-i)t (2.4)

where (e)h+ and ()h- are the distances from the centroidal surface of layer (f) to its top
surface and to its bottom surface, respectively. The deformation of the sandwich shell is
therefore described by four vector-valued mappings collectively denoted by

{( := o)p, (e)t, for = -1,0,1} (2.5)

In terms of components, we have three components for (0)1, and two components for
each inextensible director (t)t, thus leading to a total of nine components, which are the
principal kinematic unknowns to be solved for.








The deformation gradient (e)F for layer (f) is

(t)F := [GRAD(,)-] o [GRAD-'(e)o] (2.6)

The Jacobian determinants of the mapping (e)4o, and (et) are given below

()jo := det [GRAD (),o ()] (2.7)

(e)jt := det [GRAD () (, t)] (2.8)

2.2.2. Virtual Powers
Here we summarize the expressions for the power Pc of contact forces/couples and
for the power P,, of assigned forces/couples. Together these powers play a crucial role in
the derivation of the equations of motion (see Vu-Quoc et al. [1997] for more details). The
balance of the power of contact forces/couples and the power of assigned forces/couples
as expressed by Pc = a,, leads to the equation of equilibrium for geometrically-exact
sandwich shells (see Eq. (84) and Eq. (85) of Vu-Quoc et al. [1997]).
2.2.2.1. Power of contact forces/couples and conjugate strain measures
The set of convected basis vectors on the spatial (current) centroidal surface are de-
noted by { (e)a }, where the underlined index i is to be expanded in the following sense

{()ai} :={()al, (g)a2, ()a3}: { ()o ,1, (1)P,2, ()t}3 =0, for t= -1,0,1.
(2.9)
The co-vectors { (a)j }, dual to the vectors {()a }, are defined by the standard orthogonal
relation
< (e)a3, (e)ai >= 6i (2.10)

where 63 is the Kronecker delta. The convected basis in the initial configuration is given
by specifying t = 0 in the spatial-centroidal-surface convected basis (P)ai to obtain

{(t)A } := ()Al, ()A2, (t)A3 := {(e)al, (e)a2, (e)a3 }lt fort = -1,0,1.
(2.11)





12


The basis { (t)Aj }, dual to the initial-centroidal-surface convected basis { (,)A }, is de-
fined similar to Eq.(2.10).
The membrane strain () the transverse shear strain (e 6, and the bending strain
measure (y)p, which (as we will see later) are conjugate to the effective resultant membrane
stress ()nf0, the transverse shear (e)q, and the resultant couple (t)~i~, respectively, can
be defined as follows

(e),E := (e)ea0 ()a"' 0 (e) a, (2.12)

(e)6 := (e)6a ()a', (2.13)

(e)P := (e)Pa,3 (t)a 0 (e)ao. (2.14)

The components of the membrane strain ()E, the transverse shear (t)6, and the
bending strain (e)P are given in the following relations, respectively,

(e)^Q5 = (()a ( )Aa l) (i)6 = ()l (g)7 ( )Paf = (0t)0a0 (0) ,

(2.15)
where

(t)aa := (e)a, (e)ap, (t)Aao := (i)Aa* (e)Aa (2.16)

are the components of the Riemannian metric tensors of layer (e), (t)ai (I)a}a @ (t)a
and (t)Aaf (e)A' 0 (e)AI, in the current configuration and in the initial configuration,
respectively (Naghdi [1972], Marsden and Hughes [1983]). The shear strain measures,
which are measures of how much the director (e)t and the director (,)to depart from the
normal to the centroidal surfaces in the current configuration and in the initial configuration,
respectively, are defined as

(t)7a = (t)a. ()t, () ()7^ = (t))A, (o)to (2.17)

Finally, the current and the initial (nonsymmetric) director metric (j)Kap and (e)~p for
layer () are defined as

(t)Ka3 := (t)aa' ()tl (t)3 := ()A ()to, .0 (2.18)





13

Using the above definitions, the stress power iPc of geometrically-exact sandwich shells
can now be written (Eq. (199) of Vu-Quoc et al. [1997]) as

j = E J [e (I )ai 4 + ()f ( + (f ()A l dA, (2.19)
e=-i A
where ()nP, (e,)mP and (e)^ are the components of the weighted effective membrane
force, the weighted resultant couple, and the weighted effective shear force, defined in
(169), (174), (175) of Vu-Quoc et al. [1997] respectively. See the footnote at the beginning
of this section for the meaning of the word "weighted."
2.2.2.2. Power of assigned forces/couples

Let n* denote the distributed assigned force on the centroidal surface of the refer-
ence layer (0), and (e)m* the distributed assigned moment on layer (e). On the boundary
9A, we assume that the normal to the lateral surface of the shell domain in the material
configuration is such that

(e) = (0) V = (o) V Ea (2.20)

The assigned force n*a and assigned couple (e)m*f on the boundary aA are then defined
such that
n* = n*Q (o)vc (e)ri* = (e)m*f (o)Va (2.21)

The power of the assigned forces and couples is written as follows (Vu-Quoc et al. [1997,

(56)])

Pa = + (e)r* (,)t dA
A ==-1

+ f n' u (o)va + (In *a)t (e)v d(OA). (2.22)
OA =- 1
2.2.3. Constitutive Relations

For layer (e), we employ the following constitutive relations between the above strain
measures and the mentioned effective resultant forces/couples

(-)jt (e)E (e)H (E (2.23)
1 ((e))V (e) .





14


a () (e )t (e)E ()H3 pa P (2.24)
121 ( (e) V(e)P

()rQ = (,)it (t) K (j)G (e)H ()AAP (e,)J (2.25)

where
(t)H := (e)h+ + ()h-, (e)Aa := (e,) A (e)A (2.26)

are the thickness of layer (t) of the sandwich shell, and the Riemannian metric tensor in
the initial configuration for layer (f) of the sandwich shell. In Eqs.(2.23) to (2.25), (y)E is
the Young's modulus, (e)G the shear modulus, (y)v the Poisson's ratio, and (I)K, the shear
5
correction coefficient, for all layer (e). With (e)t, = -, Eq.(2.25) is the same as that given
in Naghdi [1972, p.587]. The elastic constant ()E with its component form given as

( s = () () A a ()A + (1 () (()A(O' (t)A0` + (e)A' (,)A"') (2.27)

is a fourth-order elasticity tensor.
2.3. Weak Form and Linearization

In this section, we construct the weak form of the equations of equilibrium obtained
in Vu-Quoc et al. [1997], and linearize this nonlinear weak form, which plays an important
role in finite element implementation.
The shell layers are assumed to be inextensible in the thickness direction, and there
is no drill degree of freedom (dof) for the directors (i.e., the directors are not rotating about
themselves) considered. We have three translational dofs for displacement of the centroidal
surface of the reference layer (0), and two rotational dofs for the directors of each layer.
For a sandwich shell, the total number of dofs is nine (9). For the single-layer case, we
refer the readers to Simo and Fox [1989] and Simo, Fox and Rifai [1990] for the details.
2.3.1. Admissible Variations, Tangent Spaces
The admissible variations to the deformation map 4 = { (o), (-_)t, (o)t, ()t },
are denoted by

6 := {6 (o)W, 6 (-)t, 6(o)t, 6 (i)t} (2.28)





15


Let T S denote the tangent space formed by the Cartesian product of the tangent spaces

T )t S2 (i.e., the tangent spaces to the sphere S2 at (t)t S2, fore = -1,0,1). We write

1
TS := (I Tt S2 = T(_t S2 x T(t S2 X T()t S2. (2.29)

The space of admissible variations, denoted by T4 B3t (i.e., the tangent space to the current
configuration Bt ), at the current deformation ,, is then defined as


T4 Bt := {6 ( : A R3 x TS 6 (o)) = 0 on pA, (,)t = 0 on ,tA ,

(2.30)

where ayp A and 9)t A represent the portions of the boundary dA where the essential

boundary condition is imposed on W and on (e)t, respectively.
2.3.2. Weak Form of Equations of Equilibrium

The weak form of the equations of equilibrium for sandwich shells is readily provided
by the principle of virtual power, expressed by the following balance of power (Vu-Quoc

et al. [1997])

Tc = Ta (2.31)

where Tc and Ta are the power of the contact forces/couples and the power of the as-

signed forces/couples for the sandwich shell, respectively.
It suffices to replace the time rates in the expressions for the powers Tc and Ta by

the admissible variations S (P to obtain the weak form, which can now be written as

Find 4, such that

G, ( 6 ) = Ga(6 ) (2.32)

for all admissible variations 6 P<, where Gc ( 4, 64I) is the weak form of the contact

forces/couples (or contact weak form for short), and Ga (6 4 ) is the weak form of the

assigned forces/couples (or assigned weak form for short).





16


2.3.3. Contact Weak Form

The contact weak form of the contact power TP in component form is as follows:




1

e=-1A

where ()ia'~, (t) and (1)Q" are the components of the weighted effective membrane

force, the weighted resultant couple, and the weighted effective shear force, defined in Eqs.

(169), (174), (175) of Vu-Quoc et al. [1997] respectively, whereas 6 ()capo = (1/2)6 ()aag ,

6 (t)3ac~ = 6 (e)'cap, and 6 (e)6a = 6 (e) are respectively the variations of the strain mea-
sures conjugate to the above weighted resultant tensors (Eq.(2.15)). The component form

of the contact weak form is used in the computational formulation due to the constitu-

tive laws Eqs.(2.23) to (2.25) that relate the weighted resultant tensors to their respective

conjugate strains.

2.3.4. Assigned Weak Form

From the power 7Pa of the assigned forces/couples in Eq.(2.22), we obtain at once

the weak form of the assigned forces/couples


Ga(6 ) = n* *6(o) + (e in'* 6 (e)t dA
4A e=-1

+ 6 o(0)P (o)v, + ( ( ,)t (I)v d(MA). (2.34)
&A I=-1

Remark 2.1. For the dimension of the assigned forces/couples n* and (e)mi*, we refer

the readers to Vu-Quoc et al. [1997]. At the boundary 9 A of the sandwich shell, n* and

(e)m* are decomposed as follows

n* = n*a va on ,n A, (2.35)

(e)m* = (e)m*a (e)vc on (,)m A, (2.36)






17


where 9, A and 0,(m A are the portions of the boundary where the assigned forces and

the assigned couples are applied, respectively. I

2.3.5. Linearization of Contact Weak Form

To construct the linearization of the contact weak form Eq.(2.33) at a given configu-

ration ,P in the direction of an incremental tangent field:

A := (A(o)W, A (_l)t, A (o)t, A (1)t,) E TZ S, (2.37)

we consider a one-parameter family of perturbed configurations

E H e = ((0)( (-1)tE 1 (0) e (1)tE) (2.38)

such that

'p = = ',d =- A (2.39)
dE 16=0
The tangent contact weak form will be shown to be composed of a material tangent stiffness

operator and a geometric tangent stiffness operator. The linearization of the contact weak

form plays a central role in the computational procedure based on the Newton-Raphson

method.

2.3.5.1. Update of inextensible directors

The following steps are used in the update of the inextensible layer directors.

1. First, we must account for the assumption that the layer directors have no drilling

dofs in their increments. The removal of the drilling dof is realized by nullifying the com-

ponent A (e)T3 of the material incremental directors A ()T along the basis vector E3,

that is

A (e)T = A (f)T' Ea, (2.40)

where Greek indices take values in {1, 2}. Let the superscript k on a tensor quantity

denote the kth iteration in the Newton-Raphson procedure. The spatial incremental director

A (e)tk for layer () is related to its material counter part A () Tk through the orthogonal
tensor (e)Ak as follows

A ()tk= ()A A k ()T (2.41)





18

If we do not make the distinction between tensors and their matrices of components, and
the quantities in Eq.(2.41) in terms of matrices of component, then Eq.(2.41) can be written
as
A ()tk = ()A kA ()Tk (2.42)

where the matrix (t)X k E R3x2 is formed by the first two columns of the matrix ()Ak E
R]3x3
2. The spatial director is updated as follows


sin11 A (,)tk 11
()tk+1 :=exp^ [A(,)t] =cos| | )k I k S+^ I fI k (2.43)

3. The incremental director rotation matrix is obtained from the exponential map

A ()Ak := expso(3) [(I)O]

= cosll (e)0 I|1 + sin|| (e)0 II + [1 cos (110 eoI )] e e (2.44)

where 1 is the identity tensor, ()0 and e are skew-symmetric tensors with (t)0 and e
as their associated axial vectors respectively,

(e0 := ()t xA (e)tk e := ) (2.45)

4. Update the rotation matrix of layer ()

()Ak+1 A (e)A ()A (2.46)

The above procedure is very important for the linearization of the weak form.
2.3.5.2. Perturbed configuration
Let A 4 := A (o)p, A (-1)t, A (o)t, A (i)t } TT S be the incremental field
in the tangent space at the current configuration I,. The perturbed configuration along the
increments in the tangent space is defined as follows

= (O)~ (-l)tE ()te (I)te } (2.47)





19


with

(O) PE (O)O + A (o)OP, (2.48)

(e)t := exp ,t [EA (e)t], for = -1,0,1, (2.49)

where
ep [ (e]sinl e ( 't IE (f)t (2.50)
exp ( [EA (,)t := cosil eA (I)t (,) + EA (e)tf 1 t (2.50)

is the exponential map from T tt S2 to S2 (see Simo and Fox [1989] for details). We now
verify that Eqs.(2.48) and (2.49) satisfy (2.39). First, it is obvious from (2.48) and (2.49)
that

(0) E=o = (o)', (2.51)

()t = = (I)t for = -1,0,1. (2.52)

Second, by taking the directional derivative of (2.48) and (2.49), we obtain
d
dE (o) ~ =A (o) (2.53)


[ (e]e=0

[-1 t sn (1 t i) w at +11 A t lcos( I ) IA I)A(t)t II e

= A ()t for = -1, 0, 1. (2.54)

Next, to linearize the strain measures, the following formulas are useful

d d I
dE ( = A (O), d- ()t,a = A ()t,a (2.55)
Sc=O =0

With the above results, we can now proceed to the linearization of the strain measures
followed by the linearization of the contact weak form.

Remark 2.2. The directional derivative of 6 (I)t, along the direction of the increment

A (j)t can be expressed in terms of the variation 6 e)t and the director (t)t as follows. It






20


is clearer if one thinks of the symbol 6 in the variation 6 (e)te as a derivative with respect to
some variable, whereas the perturbation parameter e is a different variable. Since I t 11-
1, it follows that (e)t () = 0, and we can define ()w := ()t x (t)t such that

(e)t = (t)w x ()t or 6(e)t = 6(e) x (t)t (2.56)

where (I)w and 6 (0)O are vectors that play the same role, one for the time derivative, while
the other for the variation of (e)t. For the perturbed director (e)t,, we have

6(t)te = 6 ()0 x (e)te (2.57)

with the same rate (e)0 as for (e)t.
Hence

d d
d (t)te = 6 (t)0 X d (t)te = (t)O A ()t
e e=0 e=0
dE = -( t) d E tO =

-(6(e)tA(e)t) (e)t, for = -1,0,1. (2.58)

Since

S(t)O = (e)t x 6(e)t (2.59)

which is a result of (2.56)2 and 1 (e)t 11= 1, and since (t)t A (t)t = 0. I

2.3.5.3. Linearized strain measures

Let ((e)a' (e)' (e)~ ) be the strain measures corresponding to the perturbed
configuration (2.47), as defined in (2.16)-(2.17), and let the incremental strain measures be
defined as

(Ad ()a A 7,A(i)ap ) d+ ((y)a, (, ) c( =) o

for = -1, 0, 1. (2.60)

We obtain the following expressions

A(r)ao := (A(t) P, (), + (e0A( e)Pa), (2.61)





21

A()7a := (A ()t. ()o,a + (1)t.A(e)tp) (2.62)

A (e)Ka := (A(),a' ()tf 3 + (e)p *'A (e)t ,), for t= -1,0,1. (2.63)

For i = 1 and = -1 (i.e., the top layer and the bottom layer), we want to express the
incremental strains A (e)ap, A ()7, A()KQa in terms of the deformation map :=
{(o)), (-_)t, (o)t, ()t } and its increment A := {A(o) A (-)t, A (O)t, A ()t}.
This objective can be achieved by employing the constraints (2.4) in (2.63), and then we
obtain

A(,) a() = ((p (0) + (0) P,a (0) W)

+ (o)h ((o), ()t, + A (), (o))p + (O), + A (o) ,P (O)t,a + (O)WP,3 A (o)t,a)

+ h ( (io)( *O) (l0)t, + A ()t, (o)W,a + A (0)',p (l)t,a + (o))Wp *A (1)t,)

+(o)h+ (1)h (A ()t,a (1) t, + (o)t,a *A (1)t,, + A(1)t,a (o)t,

+(l)t,a *A (o)t,,) + ((o)h+)2 (A(o)t,a (O)t,3 + (0)t,a *A(o)t, )

+(()h- )2(A(1)t, (1)t, + (1)t, "A (1)t,O) (2.64)

A(1)7y = A(o)p,a (o)t + (o) W,a A (o)t + (o)h+ (A(o)t,. (1)t

+(o)t,a A(1)t) + (o)h- (A(1)t, (1)t + (1)t,a A ()t) (2.65)

A (1) / = A (o)Wa (l)t + (o)p,a AA (1)t,/ + (o)h+ (A (o)t, (1)t,p

+ (0)t,a *A (1)t) + (1)h-~ (A(1)t, (1)m3 + (1)t,a *A( )t,) (2.66)

For layer (-1), we obtain

A (-1)a, = A(o)),a (o)PW,p + (0) ,a *A(0) c,

(o)h- (o), (0)t,, + A (o0)t,3 (o0) ,a + A (o) ,p *) (o)t,a + (o),p A (o)t,a)

(-_)h+ (A (o), (-1)t,p + A (-1)t, (o) ,a + A (O)Y 8 (-1)t,a





22


+ (0) ,p *A(_)t,a) + (o)h- (-1)h+ (A(o)t,a (-1)t,/ + (o)t,a *A(-1)t,

+A(o)t,, (-l)t,a + (0)t,' A(-1)t,a) + ((o)h-)2 (A()t,a (O)t,

+(o)t,. A(o0)t,) + ((1)h+)2 (A (_)t,c. (J)t,o + (_i)t, AA(_)t,) (2.67)

A(-1)7,y = A(o)(j,c, (-l)t + (o)P,. o A(_l)t (O)h- (A(o)t, (-1)t

+(o)t,a *A()t) (-1)h+ (A(-l)t,a (-1)t + (-1)t,a *A(-)t) (2.68)

A (-I)r, = A (o)),, (-l)t,o + (o)cP, *A (-i)t,3 (o)h- (A (o)t, (-1)t,

+(0o)t,a *A(-1)t,3) (-)h+ (A(_-)t,a *(-1)t, + (-1)t,a *A(-1)t,#) (2.69)

Remark 2.3. The above (2.63)-(2.68) are for the incremental strain measures for sandwich
shell. To obtain the variation of these strain measures, we simply use the same relations
with A replaced by 6. I

2.3.5.4. Linearized contact weak form
We now derive the linearization of the contact weak form, which requires the lin-
earization of the resultant contact forces/couples. Substituting the one-parameter family
of the perturbed configuration (2.47) into the static weak form (2.32), and then taking the
directional derivative, we obtain

DGeC(@, ) *A := -G,(4,,5b ) .(2.70)
d=0
The complete linearization of the contact weak form G,( P 6 ) can be divided into two
parts, the material part and the geometric part. We will discuss these two parts in detail in
this section. To make the derivation simpler, we express the contact weak form in matrix-
operator format.
2.3.6. Matrix-Operator Format of Contact Weak Form
Let the material membrane force, shear force, and moment for layer (f) be defined as
follows

S (e) ll (In)22 ( et)fnl '





23

()Q = (() (e9 )t (2.71)
( := M o
(030
()M ( 1 (e)F m (t)i -22 -()12


where (e)Jo is the Jacobian determinant in the material configuration evaluated at the cen-
troidal surface of layer (f).
We also define the director rotation matrix for layer (f) as follows

(e)All (e)A12 (e)A13
(,)A I:= [()tt, (t)t2 ([)3 ()A21 (e)A22 (e)A23 (2.72)
(t)A31 (,)A32 (I)A33 x3
and let

S(e)A1 (e)Al2
(t)A := [(e)t, (e)t2 = (e)A21 (t)A22 (2.73)
S(e)A31 (e)A32 I 3x2
which simply represents the first two columns in (e)A From here on, we will not main-
tain a rigorous difference in notation between tensors and the matrices of their components.
Thus, bold-face symbols are also used to designate the matrices of components of tensors
with respect to the spatial basis { el, e2, e3 }. With this understanding in mind, the ma-
trices of components of the deformation map of the reference layer (0) and of the director
for layer () are written as follows

( (0)(P1 (t)t1
(0)O : (o) 2 (2 (2.74)
S(o0) 3X1 ( 3 3x1
while the variation and the increment of (o)W are written as

6 (o)p1 A (o)(p1
(o)W := 6(o)(2 (o) A(o) 2 (2.75)
3 (0)A x I / (0) 3x 1
'O)W 3x1 [ A -= 3x1
For the variation and the increment of the layer director (e)t, we need to account for the
no-drilling dofs condition. The matrices of components of 6 (e)t and A (t)t are

6 (e)t 1 (t)t1
6~)t ):= t2 A t := A(t2 J (2.76)
6 ()t3 3x A ()t3 3xl





24

The material counterparts of (e)t and A (e)t are respectively 6 (e)T and A (t)T, and are
related to 6 (e)t and A (e)t by

S(e)t = (e)A 6 )T, A(t)t = (t)AA ()T. (2.77)

The no-drilling-dof condition imposed on 6 (e)T and A (I)T is written as follows

6(e)T E3 =0, and A(e)T E3 =0, (2.78)

and thus if the matrices of components of 6 (e)T and of A (e)T (with respect to the material
basis { Ei }) are defined as


( = 6 ( 2x1 X A(TI 2 2x1
then

S(e)t = (A)X (e)T, A(t)t = ()A (e)T (2.80)

Also
J (o) A(o) ]
6 (:= 1 A(-l)T (2.81)
6c:=[ 6(_1)T [ := ~T
6 (o)T A (o)T
6 (1)T 9x1 A(l)T 9x1
We now will obtain the operator expression of the weak form for each of the three layers.
Since layer (0) is the reference layer, to which the two outer layers are referred to, we begin
with layer (0). From the membrane part of (2.33), the expression for 6 (o)aap similar to that
of A (o)ap in (2.63), and using the symmetry of the membrane forces (o)fI3P, we obtain

1 ~ 1 1 22
(o)n b( (o)aap = 2 (o)ll 6 (o)all + () 6 (o )a22

+ ((0) 126 (0)a12 + (0)i21 (0)a21)

S (0) 6) (o)1 1 + T O ()) + () (o) 22


+ (( ) OT2 6 (-o)P + (0),2 6) W (o)ii 2 (2.82)






25


Introducing the following operator for the membrane action in layer (0)

(O)tPi

()Bmm := (2.83)
0a 82
(0)P1 Bi 2 + (0) P,2 3x3

we can then rewrite (2.82) as

1 t
2 (o) p (o)aap = (0)o3 [() Bmm 6 (o)cP (0)N (2.84)

Similarly, we introduce the following operators related to the bending and shear actions in
layer (0):

(o)tl0




(0) t1 ( t
(O0B:= (0) t (2.85)



t a
(O)'Pj

(o)Bbb := (l 2 (2.86)
t t
(0) 2 + (0)',2 3x3



(O)Bsm = (0)Bsb := 1 (2.87)
(O) 2 (0) 'P,2 2x3
Ltt 2x3
Then following the similar procedure as described in (2.84), from (2.33), (2.63), (2.62), we
obtain the operator format for the shear part and for the bending part of the weak form as
follows

(o)4a(o)0ya (0)30 [(o)BsmJ(o)(P + (o)Bsb(o)t] (0)Q (2.88)


(o)n00 6o)na = (0) o [(o)Bbm6(o)(p + (o)Bb6(o)tt (o)M (2.89)





26

The contact weak form (2.33) for layer (0) can now be written as

(o)G, (',64)=

f [(0) Bmm 6(0)] (0) N + [(O)Bm 6 (0)3 + (o)Bsb (O) t (O)
A

+ [(0)Bbm((0) + (o)Bbb 6()t] (O)M V (o)Jo dA. (2.90)

Remark 2.4. We refer to Simo and Fox [1989, eq. (6.25)], which is an expression similar
to (2.90). I

To obtain a simple representation for all three layers, we define the following gener-
alized resultant force for layer (f)
) N
() R := ()Q (2.91)
(M 8xl1
Recalling the relationship between 6 (e)t and 6 (e)T as given in (2.77), we combine the
differential operators for membrane strain (2.83), for shear strain (2.86), and for curvatures
(2.86), all for layer (0), into
(O)Bmm 03x3 03x3 03x3
(o)B := (o)Bsm 02x3 (O)Bsb 02x3 A, (2.92)
(0)Bbm 03x3 (0) Bbb 03x3 8x12
where A is the director rotation matrix for all layers defined as
13
1:= ( )A (2.93)
(1)A 12x9
Then, the contact weak form (2.90) for layer (0) can be written concisely as

(o)Gc ( ) = (o) B 6 (o) R (o)Jo dA. (2.94)
A
For the membrane part of (2.33), the expression for 6 (-l)a, similar to that of A (-i)a,
in (2.67), and using the symmetry of (_1) P~, we obtain

1 1 1
(_l)na 6 ( -l1)ap =l (-) (a (-)i22 6 (-l)a22





27


+ (-1) 12 ()a12 + (_-1) 21 6(-1)a21

(-_i)n11 [65(o),i '(-1) ,1 (-)h+ 6 (-_)t,l (-1)?, (oh- (o)t,1 (-1)W,1]

+ (-,)f22 [6 (o),2 '(-1)P,2 (-1)h+ (-1)t,2 (-1),2 (o)h- (0)t,2 (-1)~,2]

+ (-1)12 [6 (0)(,1 (-1)',2 + 6 (0)P,2 (-1)(,1 (-l)h+ 6 (-l)t, (-1)',2

(-)h+ 6(-1)t,2 (-1)p,1 (o)h- (o)t,1 (-1) ,2 (o)h (0o)2 (-1),1 ]

(2.95)

Upon introducing the following operator (-1)Bmemb associated with the membrane action
in layer (-1)

(-1)Bmemb

(-) -1 (-1)h+ (-1)1 (0)h- (-1)WI1 01x3


(-1) W,2 2(-1) (-1),2 2 [-2(0) (-1) W2 2- 01x3

(-1)I1 )(- )1 (o)h- (-1)1 01Ox3

05x3 05x3 05x3 05x3

(2.96)

where the operator (_-)II is defined as
j a
(-1)1 := (_-)P2 (-1) 1 2 (2.97)

we can rewrite the membrane part for layer (-1) in (2.33) in a compact format as follows

2(-l)n (-)aa = (-1)Bmemb 6 (-1)R (2.98)
From (2.68), we define the differential operator associated with the shear action in layer
(-1) as follows

(-1) Bshear





28

03x3 03x3 03x3 03x3


(-1) (-1) 1 (- (-1) (o)h- (-1)tt 01x3
A .
(-1)o (-1)2 (-1)h+ (-,)t '-- (O)h- (-,)t- ] 01 x3


03x3 03x3 03x3 03x3

(2.99)

Similarly, from (2.69), we define the differential operator associated with the bending ac-

tion in layer (-1) as follows

(-1)Bbend :=

05x3 05x3 05x3 05x3

a a \- a 1
(-1) t (-1)H2 (o)h (-1)t 1 01x3


()1) 22-1 2 0x


(-1)14 (_i)n29 +(-1) [-(o)h (-1)114] O1x3

(2.100)

where the operators (_-1)I, for I = 2 ,3 ,4, are defined as

(-1)H2 := (-1)1 (-1)h+ (-1)t,1 ,

(-1)Hl3 := (-1)2 (-1)h+ (-1)*,2 (2.101)
,2 (2.101)
8 Q
(-1)H4 := (-1)tt2 a + (-1)t,1 2

We can easily verify that the shear part and the bending part for layer (-1) in the weak
form can be written in a compact format as follows

(-)e( -1a = (-1)Bshear5 (-1)R, (2.102)





29


(-1)a 6 t(-1)Kr, = (-1)Bbend 64 ((-1)R. (2.103)

Now let the combined differential operator for layer (-1) be

(-1)B := (-1)Bmemb + (-l)Bshear + (-1)Bbend (2.104)

From (2.96), (2.99), and (2.100), we obtain

(-1)B =

(-l)wl (-1) (-h )1 tI -(o) (-1),1 ( 01x
01x3

(_1) (P t+2 a i t 0




+ a9
(-1) ,2 (-I)h (-1)L 2 (o0) (-1),2 (pt 01X3

(-1)1 )+ (-) (o)h (- 01x3





(-1)t 2 (-1) 2 (- (- () (-1) Ox3


(-)t1 (-1)U2y (O)h- (-1)t 1 1 01X3


(-) t2 2 (-1) H3 2 -(o)h (-1)t2 2 Olx3


(-1)IH4 (-1)12 2- +(-1) I3- (0)h (-1)n4 01x3


(2.105)

We thus obtain the following compact expression

S1_) fl S (-_)ap + (-1)q (-)a 1) + (-1)ap 6 (-1)cap (-1)B 6(* (1)R (2.106)

and the contact weak form of layer (-1) as

(-)Gc (I, 65)= (_1)B5s R (-1) ()0 dA. (2.107)
A






30

For the top layer (1), similar to the definition of (-I)B for layer (-1), we define

(1)B :=
o a a
(1) W101 0x3 (O)h+ (1)W" T (I)h- (1)WPtl


(1) 2 o 01x3 (o) (1)h 2 (1)h- (1)2 W 2
ye-(1 W 2 9 2(9

(1) H1 01x3 ()h+ (1)111 ()h- (l)1I

S0
(1)tt Olx3 (oh+ ()tt 0 (1)2


2 a'2 M
8 8 A,
(I)t 2 01x3 (o)h+ (1)tt 2 (1)a3


(1) t, 01x3 (0)h (1)t g2 ( 1) 4

()t2 01x3 (o)h+ (1)tt2 2 (1)n5 2
W^ 0C 01X ])

(1)116 01x3 (o)h+ (1)6 (1) 5'+--(1) 04 -

(2.108)

where the operator (1)III, for I = 1,..., 6, are defined as follows

Q 8
(1)Hi := (1)W2 + (1) W1 2 '

(1)HI2 := (1) + ()h- (l)tt 1

(1)n3 := (1)t2 ( )h- 2 ,tt (2.109)

(1)114 := ()cpt + (i)h- ()t1 ,

(1)n5 := (1) t2 + (1)h- (1)t,2 ,
Q 3
(1)I6 := (1)tt2 1 + (1)tl 2 .
,2 1 a2






31

The contact weak form of layer (1) is then


(1)G(, ) = ()B 6 *(i)R (1)Jo dA (2.110)
A
2.3.6.1. Material tangent operator

The material part of the tangent stiffness operator, denoted by DMuG A 4I, arises as
a result of linearizing the resultant forces/couples at a fixed configuration. We now treat
each layer separately, as we did for sandwich beams in Vu-Quoc and Deng [1995]. Here,
we only consider hyperelastic materials. Let V be the energy function of the shell. We have
the following constitutive relation (Simo and Fox [1989])


riai -- = mPa- (j q. = (1)P q k' W a = W)p --- (2.111)
0 (t)E )q, a(i}0 (t) 6a WV) Pao

For each layer, we have

D(e)R A, = ()C (t)B A (2.112)

where the tensor (e)C of elastic moduli is given below


















ZlddeZ~d(?)e. Ud etd(l)ezde Ild(;)6etd(;)e zg(;),O'Zd(Jr~e Ty(;)ezld(l), ZTy)(0) er~rd(R~te U-90) Oztd(I)e 11.9(;) tzld(;)e
oze 'e ^e --e AeAz
Zrd(;)eZtd(If e ztd(J)egzd(;)e Tld(i)eztdzdIe tyl We d We lyQ;)ezzWd(I) zI (?) ezzd W~e ZZ3) ezzdWe Ii30W ottd(;)e
(AzeAe ^e (A z 'Ar -e- ---7--e

Zlrd(I)etrd(?)e 0? (d)eI dd(1 W tld(?) Il(;()g Z0(?)I3()0 19() lW(egTd l)g T30() gTTr(?)g ZZo3(d) grld(;)g 1t1(2) Oll(d(0)g
,Are eAzeAe (A cze A ze -Tz-(o 'Arelr

ZTd(?)gZp(1)0 ZZd( W) Zqg(1)g 0 d())e Zg(?)e Z0 ()op( )g lW(e)gp ()g Z13(0) 9g()e Z3(1) gZ9 (W) 11T3(M) Z9()We
tfoe Ae 'e Aze 'Aoe (A -- e -- -e I(d)
ZIC(g e 1(?)g ZZ (W)e g07)ge lrd(I)eglg(l)e Zq()gtg(?)g)e T9( We1gWe f) e^lg(ge Z Z.9(Wgo(T)g Il3(?)goIqWe
,Aze tg ^e c7zee (Ae --zg- Aze-i
zrd(I)e Z13(0)9 Ztd(l)g l3(;)ga C (ld)e Z 190)e zq(?)gIl( Zl?3(?e 190)e t130) e Z120) 6 zi~qaa 43_'? e .9) to 30) G Zl
(Ae( zee Ae fzee rA e-Aze-A- e

Zrdde ZZ.90) 0 zzd(I)e U90) 1e I Ida)e U30) e tg~le ZZ30) e T 9 We zz.? (1)e t 19 (1) LO l~ Z:Z () o zg?)e uq(?7 e I I.? We ztq(;) L
'Ilze 4zLe Aze Aze ---7e (Aze 'A t Ir q
Zrd(I)e 113(?)e zd We 11) 3(W)e T d()e tt13()9e zg()e 1.?3() 19 p(W)e 113(1)ge z()9 113(1)e E ZZ 3(?)e T13(1)g Il3(0)0 113(1)e
'A^e rAe o0 e ----Te (A --G ---ze- '

=: o(a)





33

For hyperelastic materials, the elastic moduli for the membrane, shear, and bending
actions for layer () are given as in (2.23)-(2.25). The matrix of the (tangent) elastic moduli
in this case takes the following form

(e)Cm 03x2 03x3

(e)C:= (e)Cs 02x3 (2.114)

sym. (,)Cb 8x8

with

(e)C11 (e)C 2 (t)C1 (e)C1 (e)C 2 ( 3)C1l3

(C) Cm := ( t) OC2 () 2 (e) Cb:= ( 1) CO2 () Cb23

Ssym. ()C33 I 3x3 sym. ()Cb33 3x3


( )C := 2 (2.115)
(MC 12 ()C"22 2x2
The coefficients of the above matrices are given below, for the membrane action:

(,C1 (eE ()H (A" (eA
W 2 (l)A (()A)


(em ()E (e)H2 [(e) ()A11 (,)A22 + (1 (e) ) (,)A12 (e)A12


()C3 = (e)E (e)H ( 2
1 (()) ()A (t)A
1-(m)v)
C'22 (t)E (j)H 22) H22
()C2 = (E ()H (e)A22 A (2.6)
1-((e))2

(e)C3 (e) E (2) H
1 ( (e)) (2 )A

( 3) ( )E )H 1 (e)/ ( 22 1 + ( A12
(t)m CM' ((v) 2 (A 2 V )A + 2T (t A ()A A
1-^ J 2 [





34

For the bending action:

I()C" = (t)E (y)H3
12 (1 (l2 ) (e)A (e)Al'


( 1)C (1)E (1)H3 [(,)v ()A1 (e)A22 + (- () Al2 A]

12 (1 ( l)2)

32 (t)E ()H3 11 12

12 (1- (e)v2) (e)A22 ()A22, (2.117)
( 3 (E ()H3 H A22 (t)A12
12 (1- (,) v2
[22 )E (l 22 d 22(




(C (e)E (e)H 1 ()v (-)Al (e)A22 1+ () e2 (.1172
12(1- (,)V2) 2 2

For the shearing action:

(M)C1 = () (e)s ()G ()H ()A11,

(e)C.2 = ()ts (e)G ()H (e)A12

((~22 = (e)K~ (e)G (e)H (t)Al (2.118)

The tangent material stiffness operator for layer (t) is thus

DM (eGc (, A b ) A a= [()B (c)C(t)BA ] dA. (2.119)
A
2.3.6.2. Geometric tangent operator
The geometric part of the tangent stiffness operator, denoted by DGG* A 4, arises
from the linearization of the geometric part of the contact weak form, while keeping the
material resultant forces/couples constant. We now treat each of the three layers separately,
as we did in dealing with the material stiffness operator.

Remark 2.5. It is noted that while the principal kinematic unknowns are (o)P and
(e)t, for e = -1, 0, 1, the computational kinematic unknowns are (o)<, (e)0, for =





35


-1,0, 1, where (1)0 represent the rotation vectors that rotates E3 to (e)t at the current
state at time t. It is important to note that (e)0 does not represent the time history of the
motion of the director (p)t, but only relates the directors between the material configuration
and the current configuration.
In the linearization procedures, the primary variables (i.e., the variations to be held
constant in the linearization process) are 6 (e)0 instead of 6 (e)t, which we will explain in
Remark 2.6. We recall that

II (o)t II= 1 = 6(t)8 := (e)t x 6 ()t, or 6(e)t = 6() 0 x (e)t (2.120)

we thus obtain the increment of the 6 (e)t from (2.58) as

A (6l)t)= 6(t)0 xA ()t = (j) t X 6 (t) xA ()t =- (A)t (lt)t) ()t (2.121)

and also

[A (6 (Lt)],= -[(ACet'-Ceft) et],C

= (A()t,) *(t)t) (et (Am)t *6(m)t,a) (m)t (Ay)t *6(e)t) ()t,a (2.122)



Remark 2.6. From (4.19) and (4.12b) of Simo and Fox [1989], (84)3 and (156) of
Vu-Quoc et al. [1997], the equation of balance of angular momentum for a single-layer
shell is

t( m ) i -'+ Li* = IPt (2.123)
Jt
Alternatively, it also can be written in the following form

it m") + o,a x n" +m* = t x 7t (2.124)
it '
Since 11 t 11= 1, we differentiate it twice to obtain

tt = = 1= tt =0 = tt + t- t =0. (2.125)





36


Thus, t I t, let w be define as

w := t x t. (2.126)

We obtain

= tx + t x t x t. (2.127)

Since

( x t) St= [, t,6t] = (t x t)= O (2.128)


Sx t =(t x t')x t =(tt)'- (t'tt = t+ t 112t, (2.129)

and also since t t = 0, we obtain

( x t) *6t = t*t (2.130)

so that

7p' t 't = 7I, S0 (2.131)

In the above equation, 7p, corresponds to the inertia term on the right hand side of
(2.123) (i.e., the balance of angular momentum in terms of the weighted resultant moment
fmi ), whereas 7I w corresponds to the inertia term of (2.124) (as a result of (2.127)) (i.e.,
the balance of angular momentum in terms of the physical resultant moment m"). We
now point out that the reason to use 6 0 instead of 6 t, as primary variable: The use 6 0
is more convenient. Note that

t $ x t. (2.132)

Since t t = 0 and (2.126), we obtain

t= w xt (2.133)

Sx t=wx(w xt)=(w.t)W -(j.w)t =- II II 121.t, (2.134)





37

Making the derivative of t = x t, we obtain

t = xt+ xt = xt-II W 121 -t. (2.135)



For the reference layer (0), in (2.94), we hold the resultant forces/couples (o)R fixed,
and linearize the geometric part, that is, we are finding the expression

[D((o)B 6) A 4] (o)R (2.136)

in operator form, where the operator (o)B was given in (2.92). From (2.92) and (2.81),
(2.136) can be expressed as

D ((o)B 6) ) A (o)R = D[(o)Bmm 6(O(] A (0) 4} (O)N


+ {D [(o)Bm 6 (o) + (O)Bb 6 (O)T] A(O) (O)Q
[2]

+ D [(o)Bbm 6 (O) + (O)Bbb 6 (O)T] T A(o) (O)A, (2.137)
(3]
where the differential operator (o)Bm, (o)Bm,, (o)Bbm, (o)Bsb and (O)Bbb were given
in (2.83), (2.86), and (2.87), respectively. We now proceed to give a detailed expression for
part [1] in (2.137). From (2.83), we obtain

A (0) t 1 6 (0) ,1 (0)) 11
[1] = A (o)wPix 6 (0) ,2 (O)i22


= ([ 0)a 1 *(0)po, o + A (o), (o)( ) (o)i12.

+ (A(0) tl "6(0) W,2 +"A(o)cpil "6(o) W,1) (o) 12] (2.138)





38


We define the geometric differential operator for layer (0) as

13- 03x3 03x3 03x3


13-- 03x3 03x3 03x3

()T 03x3 03x3 13 03x3 (2.139)
(0) A 12x9, (2.139)

03x3 03x3 13@- 03x3
a8

03x3 03x3 13-9 03x3
15x12

and the tangent geometric moduli for the membrane action in layer (), for = -1, 0, 1, as

(e)1l 13 (e)j12 13 03x9
()K1 := ( ()n1213 (1) 22 3 03x9 (2.140)
09x3 09x3 09x9

It is easy to verify that

[1] = {D [(o)BmSo ] "A(o) } (o)i = (o)TY (o) K (O) TA (2.141)

Now, for Part [2] in (2.137), from (2.86), we obtain

[2] = D [(o)Bsm6 (O) + (o)Bsb 6 (o)t] A(O)} (O)

= [A(o)tt *5(0)o (0) 1 + A(o)tt 5(0)C,2 (o)O9 +A (o), 1t *6(o)t (0)q1

(0) 1 (o)t) ((o) A (0o)t) (,0) + A (o) W2 5(o0) (0)q

-((0)t2 (o)t) ( (o)t A(o)t) ()92] (2.142)

Remark 2.7. The matrix (o) T in (2.139) has five rows of submatrices and four columns
of submatrices. The four columns of submatrices correspond to (o) W, (-1)t, (o)t, (i)t ac-
cording to the ordering in 6 4 (See (2.81)), and 6 (e)t = (<)A (,)T (See (2.80)). I





39

Let the tangent geometric moduli for the shearing action in layer (t), for = -1,0, 1, be
defined as follows
03x3 03x3 (e)q1 13 03x6
1K2 03x3 03x3 ()Q 13 03x6 (2.143)
() (e)q1 13 (f)}q 13 (t)4 (t)71a 3 03x6
06x3 06x3 06x3 06x6
It can be verified that (2.142) can be written as

[2] = D [(o)Bsm,, (o)( + (O)Bsb 6 (0)] *A(O) (O)Q

= (o)T6 e (O)KG ()TA (2.144)

For part [3] in (2.137), from (2.87), we obtain

[3] = {D [(o)Bb 6 (O) + (o)Bbb6(O)t] *A(o)'}) (0)M

= [(AO)t1i (O)p,l +,A (O)ptl 0 (o)t,l + (O)Pi: 1AS (O)tj,) (0)rll
(A(O)tt2 + uA3( )t,2)
+ (A( 6(o) O) W, + A (,0, 6(0)t,2 (O) '2 -P ( ) t, )o2 22

+ (A (o)tt,1 (o),2 + A (o)tt *6 (0)oP, + A (O)W, *6 (o)t,2

+A(o) 2 (o)t, + (0)pl *A (0)t,2 +(0)2 oP A6 (O)t,l) (0)n12 (2.145)
Let the tangent geometric moduli for the bending action in layer (e), for i = -1, 0, 1, be
defined as follows
S13
(e)KG

03x3 03x3 03x3 (e)fill 13 (e)f112 13
03x3 03x3 03x3 (e)m12 13 ()m22 13
03x3 03x3 (E) f (e)rp3 13 (e)fla ()Ta l3 (t)Fn2 (e)7,t 13
()m11 13 (1)m12 13 (I)fla ()7ao 13 03x3 03x3
(e)f21 13 (t)f22 13 () m2a (I)a 13 03x3 03x3
(2.146)

Using Remark (2.5), we can verify that

[3] = { (o))Bbm (o) ( + (o) Bbb (O)t] A(O)} (0)

= (0)T6 (0)K* (O)TA0 (2.147)





40

Let the geometric stiffness moduli for each of the three layers be defined as

(e)K1G := (e)K + (e)KG + (t)KG =

(I)"11 13 (e)f12 13 (e)Ql 13 (e)rnl 13 (t)12 13

(t) 1n22 3 ( 13 13 ()ff12 13 (t)F22 13

(e)M' 13 ()q2 13 (e)C 13 ()fla (e)Ya 13 (f) R2a (1)7Ya 13

(e)lml 13 (e) 12 13 (e)rmlo (t)7a 13 03x3 03x3

(.)Fn21 13 (g)ra22 13 (e)n2a (I)7a 13 03x3 03x3

(2.148)

where

(t)C :=- ((e) ()7Y + (e) nP (C)e~a) (2.149)

Adding up (2.141), (2.144), and (2.147), we obtain the following expression for (2.137)

[ ((o)B 6) A ] (o) R = (o)T 6 (o)K1 (o)TA (2.150)

The tangent geometric stiffness operator for layer (0) can thus be written as

DoG(o)Gc (,, 6 ) 4 A = [(o)TP* (o)K (o )TAY ] (0)jodA. (2.151)
A
For the bottom layer (-1), in (2.107), we now hold the resultant forces/couples
(_-)R fixed, and linearize the geometric part, that is, we find the expression for

[P((-i)B 6 i)A1] *(-)R (2.152)





41


in operator form, where the operator (_1)B was given in (2.105). We define the two
differential operators (1)Tl, and (-1)T2 for the bottom layer (-1) as follows

a a a
13 1 (_)h+ 13 (o)h- 13 03x3

a a a
13 (-)h 13 2 ()h- 13a2 03x3

(-)Ti : 03x3 -13 03x3 03x3 12x9,

03x3 -13 03x3 03x3


03x3 -13 03x3 03x3
L x J 15x12
(2.153)


03x3 13 03x3 03x3

03x3 137i 03x3 03x3

a
03x3 13 19 03x3 03x3

(-1)T2 03x3 03X3 13 033 A12x9 (2.154)



a
03x3 03x3 13 -9 03x3

18x12

In addition to the tangent geometric moduli (_i)KI that corresponds to the bottom layer
(-1) as an independent single-layer shell, we have also the tangent geometric moduli

(-_)K that comes from the coupling between the bottom layer (-1) and the reference
layer (0). The tangent geometric moduli (_I)K2 for layer (-1) can be written as follows

(-1)KG := (_)K1 + (_-)Kc2 + (1)K2 =





42

(-_)KI 13 (-)K213 (-1)K3 13

(-l)K2 13 03x3 03x3 09x9

(-1)K3 13 03x3 03x3

(-1)K4 13 (-1)K5 13 (-)K6 13

09x9 (-1)K513 03x3 03x3

(-1)K613 03x3 03x3 18x18

(2.155)

The parameters in the above moduli matrix (_)KG are

(_I)K := (+ (-)h1_ +- (_1)2i12 (_1)2t) (-1)r,1

2t 12
+(-O)h ((-1)22 (-) ,2+ (-i)l2 (-1)1) (-1)t,2

+ (-)h+ ((-i)m11 (-1)t,, *(-1)t,i + (-1)m22 (-1)t,2 (-1)t,2

+2 (_l)h+ (_l)m2 (-1)t, (-)t2 ,






(_-)K4 := ()h (i)11 (_-1)3,i + (- 12 (-1)P2) (O)t,

-112
+(o0)h- (-1)22 (-1)t2 +(-1)12 (-1)1) (0),2

+ (o)h ((-1)q"1 (o(,i + (-1)q2 (O)t,2)" (-1)t

+ (0) (o (-I)1)m (-1)t(, (0)tl + (-1)22 (-l),2 (0)t,2)

+ (o)h- (-1)m12 ((-1)t,2 (0),1 + (-1)t,x (0)t,2) ,

(-1)K5 := (o)h- ((_1)i11 ("-1) + (_1i)n12 (_)-12) (o)W

+ (o)h- (_1)9 (0)t (-_)t





43


+ (o)h ((-1)i11 (-1)t,1 (0)t (-1) 12 (-1)t,2 (0" ))

(-1)K6: (0)h- ((1)22 (-1)2 + (-1)12 (-1) (0)t

+ (0)h (-1)Vh (-i)t 2 (0)t

+ ()h ((-)i)22 (-)t,2 (o)t + (-i)12 (-1)t,1 (o)t) (2.156)

The tangent geometric stiffness operator for layer (-1) can thus be written as

DG (-I)G, ( ,)A6 ) [ (i)T16 (-)KGP(-1) T1A
A

+(-1)T2^f (_1)K2 (-1)T2A6] (-1)JodA. (2.157)

For the top layer (1), in (2.110), we now try to obtain the expression

[D (d)B 6 -A P ]- () R (2.158)
in operator form, where the operator (I)B was given in (2.108). Similar to (2.157), we
obtain the following tangent geometric stiffness operator for layer (1)

DG(1)G, (c, 4)-A ) =

J [(1)Y5 (I) (1K i)T 6 (1)2+ (1)T(2 (1)KG (K1)T2A ] ()dA ,
A
(2.159)

where
13 03ax (0)h+13 1 (1)h- 3 9
S0 10
13i 03x3 (O)h+ 13 (l)h- 30




13
Y)Y := 1A2x9. (2.160)

09x9 13aI



15x12






44





03x3 03x3 13 03X3
33x3

03x3 03x3 13 03x3


(1)T2 := 03x3 O 1 A 12x9, (2.161)
03x3 03x3 03x3 13


03x3 03x3 03x3 13


03x3 03x3 03x3 13a2

18x12

(1)K

(1) K 13 (1)K2 13 (1)K3 13

(1)K2 13 03x3 03x3 09x9

(1)K3 13 03x3 03x3

(1)K4 13 (1)K5 13 (1)K6 13

09x9 (1)K513 03x3 03x3

(1)K613 03x3 03x3
18x18
(2.162)

The parameters in the matrix (1)K are defined as follows


(K := -(0) ()1 (1)W + ()12 () (0)t,1

(i)ii22 t + 12 t
-(o)h+ (()n22 (1)'2 2 (1) 12 (1) 1) (o)t,2

-(o)h+ ((1)1 (0) ,1 + (1) 2 (0)t,2) *(1)

(0)h+ ((1)ff1 (i)t, (o)t,1 + (,) m ()t,2 (O)t,2)





45


(O)h ()Fn12 ( ()t,2 ()t, + (1)t, (O)t,2

(l)K2 := -(0) h((I) (1)1 + (I)n 12 (1) t2 (0) (o)h+ (1) 1 (1) (0)t

(O) h ((l)n11 (I)t,l (0)t + (1)712 (1)t,2 (O)t)



122 12 t











(1)K6 :=- (o)h- ((1) 22 (1) o2 + ()12 (1) (1) () ()2 (2.163)

Remark 2.8. Even though (1)KG has the same form as (o)KG (i.e., the operator

of a stand-alone layer (1)), the operator (1) TI has the coupling terms in the submatrices
(1,3), (1,4), (2,3), (2,4). In (1)T2 of (2.160), there are five rows of submatrices and four
columns of submatrices. The four columns correspond to (0)t (1)t, (0)*, (1)t. The
coupling terms in (1)T, when multiplied with A (o)t and A ()t in (A A ), only affect
the submatrices (1,1), (1,2), (2,1), (2,2) related to the membrane forces ()i.e., the 12
tan21, 22. The reason is the offset of layer (1) with respect to reference surface, which
(1)ii2l (e)i2 The reason is the offset of layer () with respect to reference surface, which
is the centroidal surface of layer (0).
Another way to understand (I)TI (or the meaning of the coupling terms in (1)T1,
i.e., the difference between (o)T and (l)T ) is to think of layer (1) as a stand-alone layer,
initially at the same location as that of layer (0). Then (I)Ti would be similar to (o)T
(note the difference between the column corresponding to (i)t in (1)T1 and the column
corresponding to (o)t in (o) T). Now shift layer (1) to the top of layer (0); then the mem-
brane forces in layer (1) must generate some additional moments. The coupling terms in





46


(I)TY play the role of lever arms. On the other hand, in (1)K we have all the coupling
terms, but not in (1)T2. I

2.4. Numerical Examples for Statics of Sandwich Shells

The finite element formulation for the statics of geometrically-exact sandwich shells

presented in the previous sections has been implemented in the Finite Element Analysis

Program (FEAP), developed by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a

DEC ALPHA with the DEC UNIX V3.2D-1 operating system. Linear finite-element basis

functions are used in the examples in this section. To avoid shear locking, selective reduced

integration is used to evaluate the shear part of the tangent material stiffness matrix (I)KM

and the tangent geometric stiffness matrix (e)KG and also the shear part of the residual

force matrix, whereas the bending part and the membrane part of the tangent stiffness

matrix and of the tangent residual force matrix are evaluated using full integration.

To identify the correctness of the present theory and the related coding, we tested

several examples of sandwich plate with different aspect ratio. The aspect ratio is defined

as A := L : W : T, where (L, W, T) designate the length, width, and thickness of the

sandwich plate, respectively. For the bending stiffness and membrane stiffness, we use

the full 2 x 2 integration points, whereas for the shear stiffness, we use the reduced 1 x 1

integration point.

Firstly we tested the bending of a sandwich plate with three identical layers employ-

ing 90 four-node quadrilateral elements. The plate is clamped at the edge (' = 0, and is

free at all the other edges (See Figure 2.2). The vertical displacements and the tip rotations

are compared with the theoretical results and with the results obtained from the single-

layer theory. The results, with a maximum error less than 0.4%, also show the ability of

our sandwich shell elements to model the anticlastic curvature.

We also consider the Cook problem with only the core layer using 100 sandwich shell

elements. The results agree well with those of single-layer shells (Rifai [1993, p. 173]).

To demonstrate the ability of the formulation to capture large rotations and displace-





47

2
-*12
e =j w3 (l)i*12
E2 (o) )(m 1212

-el = E, 1
L

Figure 2.2. Sandwich shell with three identical layers

ments, we tested the torsion of a cantilever plate with three layers using 20 sandwich shell

elements (Figure 2.3). To justify the computed results, we use the theoretical rotation of

the torsion of an elastic bar given by 0 = TL/GJk as a basis for comparison,3 where L

is the length of the plate, G is the shear stiffness and Jk is the polar moment of inertia of

the plate cross section about the centroidal axis along the length of the plate. Since the

direction of the concentrated forces remains fixed along the (3 axis, the resulting couple T

generated by these forces decreases with the twisting of the bar (See Figure 2.4), as the dis-

tance d between these forces becomes smaller. Thus to make the comparison meaningful,

we use the final value of the couple at the last time step as the torque T used to compute

the theoretical rotation 0. The difference between the theoretical results and our FE results

on the twisted angle is 9%.












Figure 2.3. Torsion of a cantilever plate.


3 Roark and Young [1975, p.290] presented the formulation to calculate the angle of twist for beam with
solid rectangular section, which gives very close results when compared to the above formulation (< 0.2%).






48

'3





F\






d

S 0.2


Figure 2.4. Force couple, at the tip of the plate, generated by a couple of concentrated
forces.

2.4.1. Roll-down Maneuver of a Sandwich Plate

We now consider the roll-down maneuver of sandwich plates. First, we tested the

sandwich shell having only the core layer using 10 elements. Comparing to the theoretical

deformed shape (i.e., a full cylinder), the relative error in the tip displacements is 0.4%

in the 1 direction and 0.005% in the '3 direction. The displacements obtained with the

sandwich shell code are exactly the same as those with the single-layer shell code. We also

tested the roll-down of a sandwich plate with only one outer layer. We still obtained good

results even with the slower rate of convergence.

Next, we consider the sandwich plates with three identical layers. The material prop-

erties and geometric properties are chosen as follows:

(t)E = 1.2 x 107, ()v = 0.0, (t) s = 0.75, (i)h = 0.033333, for = -1,0,1.
(2.164)

where (e)E, (e)1, (t)ns are the Young's modulus, the Poisson ratio, and the shear correction

coefficient of layer (f), respectively. The geometrical dimensions of the plate are length

L = 10, width W = 0.1.

At first, we use 10 uniformly distributed sandwich shell elements to discretize the






49


sandwich plate. The computed tip displacements at the end of the last loading step are re-

ported in Table 2.1. The computed displacement u1 differs from the exact solution by 7.5%.

In the first loading step, convergence is achieved after 10 iterations; in the last loading step,

convergence is achieved after 11 iterations. Then we use 20 uniformly distributed sandwich

shell elements to discretize the sandwich plate. The computed tip displacements at the free

edge are reported in Table 2.1, where it can be seen that the displacement u1 is closer to the

exact solution of (-10) when the plate becomes a full cylinder. The relative error in ul is

now 1%. Convergence is obtained after 9 iterations in both the first loading step and in the

last loading step. Finally, the computed displacements at the free edge using 40 uniformly

distributed elements shown in Table 2.1 are clearly closer to the exact solution, in which

the value of u1 should be (-10), and the value of u3 should be zero. The relative error in

the displacement u1 is now 0.53%. Convergence is obtained after 10 iterations in the first

loading step, and 11 iterations in the last loading step.

Table 2.1. Roll-down of a sandwich plate with identical layers: Displacements of a corer
of the free edge.

Disp. 10 elements 20 elements 40 elements
n1 -9.25472 -9.81873 -9.94730
u2 -3.08237 x 10-11 -2.15727 x 10-10 -1.02421 x 10-9
u3 -1.98609 x 10-1 -1.05095 x 10-2 -8.77204 x 10-



Figure 2.5 shows the undeformed and deformed shapes of the sandwich plate using

40 sandwich elements.

2.4.2. Sandwich Plate with Ply Drop-offs

In this section, we present the computational results for sandwich plates that have

discontinuities due to disparities in the length of the layers, resulting in the so-called ply

drop-offs.

2.4.2.1. Sandwich plate with ply drop-off

We now consider a cantilever sandwich plate with three layers, and with a ply drop-

off in the top layer at mid length, see Figure 2.6. The free edge at the tip is subjected to






50





















Figure 2.5. Roll-down of a sandwich plate with three identical layers: Isometric view of
deformed shape.


a uniformly distributed force of n*13 = 100. The geometric and material properties are

listed below:

L =10, W = 0.1, Tb =0.3, Ta = 0.2, (2.165)

where L is the length, W the width, Tb the thickness before the ply drop-off, and Ta the

thickness after the ply drop-off, and


(y)E = 1.2 x 107, ()V = 0., (e)Ks = 0.75, for = -1,0,1. (2.166)

Before the ply drop-off, the layer thicknesses are


(e)h =0.1, for f=-1,0,1. (2.167)

After the ply drop-off, the layer thicknesses are

(_-)h = 0.1, (o)h = 0.1, ()h = 0. (2.168)

Since the plate has a large aspect ratio, we use the Euler-Bernoulli beam theory to

predict the deflection. The bending stiffness coefficients of the beam before and after the





51

A B C





Lb L/2 La = L/2 F



Figure 2.6. Sandwich plate with one ply drop-off

ply drop-off are

Elb = EW(Tb)3/12 = 2700, EIa = EW(Ta)3/12 = 800, (2.169)

respectively.
Let P = n*13 W and M = PLa be the resultant tip load, and the internal moment
at the ply drop-off. Let u3 be the transverse displacement at the ply drop-off B due to the
force P and the moment M. Let u2 be the transverse displacement at C of thin half of the
plate, with the section B at the ply drop-off clamped. The total transverse displacement u3

at the plate tip is the sum of u', the transverse displacement at C due to the rotation Ob of
the section at B (this rotation results from the bending of the portion AB), and u3:

u3 = u + LaOb + u

= (PLb3/3Eb + MLb2/2Eb) +

(PLb2/2EIb + M b/Elb)La + PL /3EIa

= 1.6010. (2.170)

Table 2.2 presents the computed results using 20 uniformly distributed linear sand-
wich shell elements and using 20 uniformly distributed equivalent linear single-layer shell
elements, respectively. The transverse displacement u3 obtained with sandwich shell ele-
ments has a relative error of 0.030% compared with the analytical result based on Euler-
Bernoulli beam theory, and a relative error of 0.033% compared with the computed result






52


using single-layer shell elements.4

Table 2.2. Sandwich plate with ply drop-off: Tip displacements
Disp. sandwich elements singer-layer elements
u -1.71076 x 10-1 -1.62236 x 10-1
~u2 -1.43540 x 10-1 3.58955 x 10-1
u3 1.55253 1.54842




Remark 2.9. Similar to the case of sandwich beams in Vu-Quoc and Deng [1995],

the result with single-layer shell elements is smaller than that with sandwich shell ele-

ments, since the equivalent single-layer plate has a symmetric ply drop-off, unlike the

non-symmetric ply drop-off in sandwich plate. Further, hinge is not allowed to form in

the cross section at the ply drop-off in the equivalent single-layer shell. Figure 2.7 depicts

the undeformed shape and the deformed shape; the effect of the ply drop-off is not easily

discernible. I


















Figure 2.7. Sandwich plate with ply drop-off subjected to tip force: Isometric view of
undeformed and deformed shapes.


4 For moderate thick plate and Poisson's ratio v = 0, Euler-Bernoulli beam theory gives accurate results
on displacements.






53


2.4.2.2. Two-layer plate with ply drop-off: aspect ratio A = 5 : 1: (1, 0.5)

Here the aspect ratio of the two-layer plate with ply drop-off is represented by A :=

L : W : (Tb, Ta). We now consider a cantilever sandwich plate with only two outer

layers (and with the core layer inexistent), subjected to the uniformly distributed force of
n*13 = 60 assigned at the free edge. The plate has a ply drop-off at mid length. The

geometric dimensions of the plate are (Figure 2.8)

L = 5, W= 1, Tb =, Ta =0.5, (2.171)

with the layer thickness before the ply drop-off being

(_-)h = 0.5, (o)h = 0.0, ()h = 0.5, (2.172)

and after the ply drop-off


(_i)h = 0.5, (o)h = 0.0, (i)h = 0.0. (2.173)

The material properties chosen are

(t)E = 29,000, (t)v = 0.294, (e);, = 1, for e -1,0,1. (2.174)

Ten uniformly distributed elements are used in the computation. The displacements

of a corer node at the tip are tabulated in Table 2.3. The undeformed and the deformed

shapes are shown in Figure 2.9, where a change in curvature at the ply drop-off is clearly

discernible.

Table 2.3. Two-layer plate with ply drop-off A = 5 : 1 : (1, 0.5): Tip displacements.
U1 2 U3
-5.18243 x 10-1 -4.22720 x 10-4 1.55017



2.4.2.3. Two-layer plate with ply drop-off: aspect ratio A = 20 : 1 : (1,0.5)

We now consider a cantilever sandwich plate with only two outer layers (and with the

core layer inexistent), subjected to a tip moment. The plate has a ply drop-off at mid length






54








I, Lb
L


Figure 2.8. Two-layer plate with ply drop-off: Geometry and assigned force.


















Figure 2.9. Two-layer plate with ply drop-off Aspect ratio A = 5 : 1 : (1,0.5): Isometric
view of undeformed and deformed shapes.


(see Figure 2.10). Since the connection between the thinner part La and the thicker part

Lb of the plate is flexible, the actual moment needed to bend the thinner part La into a full

circle is a little smaller than the theoretical result of M = 27rEIa/La, which is obtained for

an equivalent beam with a clamped end.

The geometric dimensions of the plate are as follows

L = 20, W = Lb =10, La =10, Tb= 1, Ta = 0.5. (2.175)

The layer thickness before the ply drop-off are

(-)h = 0.5, (o)h = 0.0, ()h = 0.5 (2.176)






55


and after the ply drop-off

(_-)h = 0.5, (o)h = 0.0, (i)h = 0.0 (2.177)

The material properties chosen are


(e)E = 29000, (e)v = 0.294, (t)s = 1. for = -1,0,1. (2.178)

Along the free edge at the tip of the cantilever plate, we assign a uniformly distributed
resultant couple (_l)m*12 = 189.8, which corresponds to the theoretical value of the tip

moment to bend a beam equivalent to the thinner part La of the plate into a full circle.




A B

Lb La


Figure 2.10. Two-layer plate with ply drop-off, Aspect ratio A = 20 : 1 : (1, 0.5): Geome-
try and assigned couple.

As mentioned above, since the connection between the thickness part and the thin-

ner part at the ply drop-off of the plate is flexible, the assigned resultant couple is higher
than what is needed to roll the thinner part of the plate into a full circle. Thus it is ex-

pected that the tip of the plate will be rolled past the ply drop-off location. The computed

displacements of the nodes at the ply drop-off and at the tip of the plate are reported in
Table 2.4.
In the case where a full circle is obtained, point B in Figure 2.10 should come back

to coincide with point A (in a projection onto the plane (Q1, c3)); in such case, the displace-
ment of point B should be u'(B) = -10 ul(A) = -10 0.969 = -10.969, where the
value of u1 (A) = -0.969 comes from Table 2.4. The computed displacements u1 for point
B is, however, equal to (-10.789), thus corresponds to a relative error of 1.6% compared






56

Table 2.4. Two-layer plate with ply drop-off Aspect ratio A = 20 : 1 : (1,0.5). Displace-
ments at the ply drop-off (point A in Figure 2.10) and at the tip (point B in Figure 2.10).

Node (1 J2 -3 u1 u2 3
101 10. 0. 0. -0.9689 7.875 x 10- -3.705
105 10. 1. 0. -0.9689 -7.875 x 10-3 -3.705
201 20. 0. 0. -10.789 8.768 x 10-3 -3.937
205 20. 1. 0. -10.789 -8.768 x 10- -3.937


















Figure 2.11. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll
down maneuver. Isometric view of deformed shapes (Peeling of a banana).

to the value of (-10.969) mentioned above. The deformed shape of the plate, shown in

Figure 2.11, evokes the action of peeling a banana.

To display the effects of anticlastic curvature that the sandwich shell elements can

capture, we refine the discretization to 160 sandwich shell elements. The three-dimensional

rendering of the deformed shapes are given in Figures 2.13, Figures 2.14, and Figures 2.15.

The anticlastic curvature can be seen clearly in Figure 2.13 and Figure 2.14. Since the

top surface of the plate is stretched in the (' direction, when the plate is roll down, by the

effect of the Poisson's ratio, this top surface experiences a contraction in the (2 direction.

The reverse is for the bottom surface of the plate (i.e., a contraction in the 1 direction

and a stretching (expansion) in the 2 direction). The combined effect of stretching and






57



















Figure 2.12. Ply drop-offproblem. Cantilever sandwich shell with drop-off subjected to tip
moment: Peeling of a banana.


contracting of the top and bottom surfaces of the plate in the (2 direction is the result
of bending in the (1 direction. To quantify the anticlastic curvature, and to compare the
result with a calculation employing 3-D solid elements using the nonlinear finite element
code ABAQUS, we look at the difference in the transverse displacement u3 of two points
located at (' = 13.5 (see Figure 2.15), one point at (2 = 0 (i.e., at the outer lateral edge of
the plate), and the other point at (2 = 0.5 (i.e., in the middle of the plate in (2 direction):

u3(13.5, 0.5) u3(13.5, 0) = -6.68096 (-6.70291) = 0.022. (2.179)

The quantity in (2.179) is to be compared to the quantity in (2.180) obtained from ABAQUS.

We note that the resultant couple needed to roll the thin part of the plate into a full
circle is (_l)m*12 = 177.5 which is 94% of the magnitude of the tip moment needed to roll
an equivalent clamped beam into a full circle. This lower magnitude is due to the flexibility
by the plate at the ply drop-off line, as discussed earlier.

To compare the results obtained with our sandwich shell element, we solve the same
problem using the solid elements in the nonlinear finite element code ABAQUS. In our






58




















Z



Figure 2.13. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1, 0.5): Roll
down maneuver. 3-D rendering of the deformed configuration. Isometric view.


















Figure 2.14. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5):
Roll down maneuver. 3-D rendering of the deformed configuration. Anticlastic curvature
viewed from (' direction.

ABAQUS model, we employ 960 C3D8I (8-nodes) linear brick elements, with 1453

nodes. These elements belong to the class of incompatible mode formulation. The final

value of the resultant couple assigned along the force edge at the tip of the plate is 189.8






59














1 = 13.5



Figure 2.15. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5):
Roll down maneuver. 3-D rendering of the deformed configuration. Anticlastic curvature
viewed from (2 direction.

(See Figure 2.19), which corresponds to that obtained from beam theory. Figure 2.16-2.18

provide various views of the final deformed configuration of the ABAQUS model.















Figure 2.16. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll
down maneuver. ABAQUS model using 960 solid elements with incompatible modes.
Undeformed and deformed configuration. Observer's viewpoint: (1,-1,1).



To quantify the anticlastic curvature, and to compare this quantification to the re-

sult obtained using the sandwich shell elements, we again consider the nodes having the

coordinates 61 = 13.5, 63 = 0, which lie on the top surface of the thinner part of the





60










L -


Figure 2.17. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1, 0.5): Roll
down maneuver. ABAQUS model deformed configuration. Observer's viewpoint: (1, 0,0).









L
Figure 2.18. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll
down maneuver. ABAQUS model. Undeformed and deformed configuration. Observer's
viewpoint: (0,-1,0).


two-layer plate. In the deformed configuration, these nodes are close to the points having
the lowest spatial coordinate x3 (or the z coordinate in Figure 2.15 and Figure 2.18). The
displacements of these nodes are given in Table 2.5.
Parallel to (2.179) for sandwich shell elements, the anticlastic curvature in the ABAQUS
model can be quantified using the results in Table 2.5 as follows

u3(13.5, 0.5) u3(13.5, 0) = -6.5948 (-6.6159) = 0.022. (2.180)

The above result agrees well with that obtained from the sandwich shell element.





61




Deformed plate








.0d
M=Fxd

Figure 2.19. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5):
ABAQUS solid model. Assigned forces at plate tip to create a resultant couple in the
roll-down maneuver.
Table 2.5. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1, 0.5): Anticlastic
curvature from ABAQUS solid model. Nodal displacements of nodes having coordinates
( = 13.5, (3 = 0.

T1 (2 1 u3
13.5 0.00 -6.007 -6.780
13.5 0.25 -6.007 -6.764
13.5 0.50 -6.007 -6.758
13.5 0.75 -6.007 -6.674
13.5 1.00 -6.007 -6.780



For the thinner part of the two-layer plate to roll into a complete circle, point B in

Figure 2.10 must roll back to coincide with point A which is itself moved by the deformed

plate. To compare the results obtained using sandwich elements and those obtained using

an ABAQUS model, we gather the coordinates of point A and B in the final deformed

configuration, corresponding to the resultant couple of M = 189.8 in Table 2.6 below.

2.4.2.4. Two-layer plate with ply drop-off: aspect ratio A = 20 : 10 : (1, 0.5)

We now consider the same plate as in the previous section, but with a width ten

times larger (i.e., W = 10.), instead of W = 1. as in the previous section. All other

geometric dimensions and material properties remain identical to those in the previous





62


Table 2.6. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5). Compari-
son between sandwich elements and ABAQUS solid model. Distance between point A and
point B in the final deformed configuration. ( A = (10, 0, 0), B = (20,0, 0)
A( T ( ((AA) 3(A)
Sandwich elements 9.03 0.00788 -3.71
ABAQUS model 8.64 0.0167 -3.56

( B) (2 B) (3 (B)
Sandwich elements 9.21 0.00877 -3.94
ABAQUS results 8.61 0.00930 -3.53

Distance between A and B =1| ) (, A) 4( B)II
Sandwich elements 0.293
ABAQUS results 0.0459


section, see Eqs. (2.175), (2.176), (2.177), and (2.178). The distributed couple (_l)m*12
assigned to the free edge of the plate tip is set as before to (_l)m*12 = 189.8, which is
the resultant couple that will roll an equivalent beam into a full circle. To discretize the
two-layer plate, we now employ 200 sandwich shell elements: 100 elements before the ply
drop-off, and 100 elements after the ply drop-off. The deformed shapes of the plate are
shown in Figure 2.20- 2.23. Figure 2.20 depicts the deformed shape in an isometric view.
One can clearly see the anticlastic curvature in the (2 direction in this figure, as well as
in Figure 2.22, which is the projection of the deformed shape on the (2, 63) plane. This
anticlastic curvature is the effect of the Poisson's ratio. The top surface of the undeformed
plate is extended in the 1 direction, this extension induces a contraction in the transverse

(2 direction, and thus the downward curvature is clearly seen at the bottom of the deformed
plate in Figure 2.22. Opposite to what take place at the top surface, the compression of the
bottom surface of the plate in the 1' direction induces an extension in the (2 direction, thus
resulting in the lateral bulging of the plate, as seen in Figure 2.22.
To quantify the anticlastic curvature similar to (2.179) and (2.180), we consider the
transverse displacement u3 of the nodes at F' = 14 and E3 = 0, which lie near the bottom






63





















Figure 2.20. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5): Roll-
down maneuver. Sandwich shell elements 3-D rendering of deformed shape. Isometric
view.








B







A

Figure 2.21. Two-layer plate with ply drop-off Aspect ratio A = 20 : 10 : (1, 0.5):
Roll-down maneuver. Zoom-in on the ply drop-off point.



of the deformed configuration of the two-layer plate (see Figure 2.23)


u3(14, 5) u3(14, 0) = -6.62581 (-6.64764) = 0.02183. (2.181)






64























Figure 2.22. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5):
Roll-down maneuver. Sandwich shell elements. 3-D rendering of deformed configuration.
Projection down the (1 axis.

To clearly depict the deformation at the ply drop-off and the tip of the deformed plate

around the area of the ply drop-off, we provide a zoom-in figure on this area in Figure 2.21.

The distance between point A and point B will be used to compare the results obtained

with sandwich shell elements and those obtained with an ABAQUS solid model, which

is composed of 2400 incompatible (solid) linear brick elements (type C3D8I). The final

moment at the plate tip assigned to the ABAQUS solid model has a magnitude of 1898,

which is obtained for the roll-down of an equivalent beam. Various views of the deformed

configuration obtained with the ABAQUS model are depicted in Figure 2.24, Figure 2.25,

Figure 2.26. The distance between point A and point B in the deformed configuration as

obtained both from sandwich shell elements and from the ABAQUS model are given in

Table 2.7.

From the above examples, we found that the present sandwich shell formulation gives

good results on displacements with very coarse mesh, when compared to the 3-D converged

results from ABAQUS. For the interlaminar stress analysis, due to the kinematic assump-






65


















1 =14, = 0




Figure 2.23. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5):
Roll-down maneuver. Sandwich shell elements. 3-D rendering of deformed configuration.
Projection down the (2 axis.
















Figure 2.24. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5):
Roll-down maneuver. ABAQUS solid model. Undeformed and deformed configurations.
Isometric viewpoint: (-4,-7,-3)


tion, the present formulation is expected to be more accurate than the single-layer shell

model, especially for thick and moderate thick shells.






66


























Figure 2.25. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5):
Roll-down maneuver. ABAQUS solid model. Projection down the 1' axis of deformed
configuration.















Figure 2.26. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5): Roll-
down maneuver. ABAQUS solid model. Projection of the deformed configuration along
the E2 axis.





67
















Table 2.7. Two-layer plate with ply drop-off. Aspect ratio A = 20: 10: (1,0.5). Compari-
son between sandwich elements and ABAQUS solid model. Distance between point A and
point B in the final deformed configuration. 4 A = (10, 0, 0), ( B = (20, 0, 0)
41 ( A) 2 A( A) 3 A)
Sandwich elements 9.16 0.0112 -3.5
ABAQUS model 9.16 0.0179 -3.53

a ( B) (D2 (_ B) (D3 ( B)
Sandwich elements 9.07 -0.0345 -3.41
ABAQUS results 8.23 -0.0294 -2.83

Distance between A and B =1l (( A) (C B)|I
Sandwich elements 0.142
ABAQUS results 1.17















CHAPTER 3
OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF
MULTILAYER COMPOSITES : STATICS

3.1. Introduction

The analysis of general shell structures have been of interest for several decades.

There is a continuing need to develop more reliable, accurate and efficient shell element,

especially for analyses of composite structures covering a wide range of physical scales (in-

cluding MEMS5) and material and geometric nonlinearities. Structures made of laminated

composites continue to be of great interest for engineering applications. For accurate anal-

yses of composites with a large number of layers, industry routinely employs FE meshes

with one solid element per ply in the thickness direction, and with element aspect ratio less

than 10 (Figure 3.1). It is therefore highly desirable to develop efficient finite elements that

are accurate at extreme aspect ratio to significantly reduce the computational effort.


SOne ply



S* 500 plies
*





Figure 3.1. Composite structure with 500 plies in the thickness direction; the ply thickness
is around 10-in.

Shell element formulations have been mainly developed within the context of the

so-called degenerated shell concept and the classical shell theory (Buechter and Ramm

[1992]). Both formulations are based on the common kinematic assumptions of inextensi-
5 MEMS stands for MicroElectroMechanical Systems.

68





69


bility in the thickness direction and the zero transverse normal-stress condition.6 Although

these approximations led to very good results in most cases, several difficulties could arise:

(i) Complex 3-D material models: the zero transverse normal stress condition must be im-

posed. (ii) Boundary conditions and finite rotations: use of rotational degrees of freedom

(dofs); especially those normal to the boundary, to describe soft support and hard support

(e.g., Zienkiewicz and Taylor [1991, p.92]); complex update algorithms for finite rota-

tions in geometrically-exact stress-resultant formulation (e.g., Vu-Quoc and Deng [1995],

Vu-Quoc and Ebcioglu [1996], Vu-Quoc and Ebcioglu [2000a], Vu-Quoc and Ebcioglu

[2000b], Vu-Quoc, Deng and Tan [2000]). (iii) Transverse normal stress: inconsistently a

posterior computation based on the computed in-plane stresses (see Reddy [1997, p.345],

and e.g. in the localized effects due to the concentrated surface loading and the delam-

ination of composite shells). (iv) Combination with regular solid elements: Transition

elements are needed to connect rotational dofs and displacement dofs (e.g., Kim, Varadan

and Varadan [1997] and the contact problem). (v) Through-the-thickness stress distribu-

tion in laminated composite with dissimilar materials: poor accuracy because of straight

director assumption (Bischoff and Ramm [2000]).

Accurate and robust low-order shell elements have always been in high demand for

development and for use in engineering analysis (e.g., DYNA3D [1993], NIKE3D [1995]),

particularly when complex nonlinear 3-D constitutive relations can be incorporated with-

out the added requirement to satisfy the constraint of zero transverse normal stress. Three

possible shell kinematic descriptions have been proposed: (i) The displacement of the refer-

ence surface together with the extensible transverse director (Simo, Rifai and Fox [1992],

Betsch, Gruttmann and Stein [1996]). (ii) The displacement of the reference surface to-

gether with the displacement vector of the tip of a director (Braun, Bischoff and Ramm

[1994], Roehl and Ramm [1996], Bischoff and Ramm [1997], Bischoff and Ramm [2000]).

(iii) The displacement of the top and bottom surface of the shell (e.g. Hauptmann and
6 Stress-resultant shell formulation can be generalized to account for thickness change, which relaxes
the zero transverse normal stress condition (e.g., Simo, Rifai and Fox [1992]).






70


Schweizerhof [1998], Klinkel, Gruttmann and Wagner [1999], Ramm [2000]). The kine-

matic descriptions (ii) and (iii) are attractive since they avoid the complex rotation updates

in stress-resultant elements. On the other hand, the kinematic description (iii) provides a

natural way to connect to regular solid elements without the need for transition elements;

such feature can benefit the detailed modeling of shells with patches of piezoelectric or

viscoelastic materials.

The present solid-shell element has the same displacement dofs as in regular lin-

ear (8-node) brick solid element. Displacement-based solid elements are known to have

poor performance in bending-dominated situation, such as in thin shells. To obtain the

same performance as stress-resultant shell formulations with plane stress assumption (e.g.,

Vu-Quoc, Deng and Tan [2000]), the Enhanced Assumed Strain (EAS) method and the

Assumed Natural Strain (ANS) method are employed here.

To improve the bending behavior of low-order elements, the EAS method based on

the Fraeijs de Veubeke-Hu-Washizu functional was proposed by Simo and Rifai [1990]. For

large deformation analyses, there are two ways to introduce the EAS method: (i) enhancing

the deformation gradient F (Simo and Armero [1992], Miehe [1998b]), and (ii) enhancing

the Green-Lagrangian strain tensor E (Bischoff and Ramm [1997], Klinkel and Wagner

[1997], Klinkel et al. [1999] etc.). From the computational standpoint, the latter is simpler

and more efficient, even though our numerical experience indicates that both approaches

lead to the same numerical results when the same EAS parameters are used.7 To incorporate

3-D constitutive laws in shell formulations, the transverse normal strain must have at least

a linear distribution over the shell thickness; otherwise, the so-called Poisson-thickness

locking would occur (Zienkiewicz and Taylor [1991, p.161], Bischoff and Ramm [1997]).

To relieve the Poisson-thickness locking, two methods were proposed in recent years: (i)

Assuming a quadratically distributed displacement field over the shell thickness (Parisch

[1995]), which then introduced an additional kinematic parameter, and (ii) using the EAS
7 Noted that the EAS method based on the displacement gradient as proposed in Miehe [1998b] and used
in Miehe and Schroeder [2001] does not pass the bending patch test, and there is no easy way to remedy this
problem, see the next paragraph for the details.





71

method to enhance the transverse normal strain (Buchter, Ramm and Roehl [1994]). In

our formulation, we enhance the transverse normal strain by the EAS method to include

bilinear terms 1~3 and 2 3 in terms of material coordinates. To improve the membrane

bending behavior, we also enhance the membrane strains in the similar manner as in Simo

and Rifai [1990].

On the other hand, to make the formulation more efficient, we propose a modified

EAS method while still keeping the same level of accuracy, where the inverse of element

Jacobian matrix and the Jacobian at the element center are no longer necessary. Further-

more, the present eight-node solid-shell element relies on a new optimal seven-parameter
EAS-expansion (for the transverse normal strain and for the membrane strains) together

with an ANS method (for the transverse shear strains); the present formulation is shown

to pass both the membrane patch test and the out-of-plane bending patch test. It should be

noted, however, that while the 30-parameter EAS expansion of Klinkel and Wagner [1997],

the five-parameter EAS expansion of Miehe [1998b] and of Klinkel et al. [1999] pass the

membrane patch test, all of them fail to pass the important out-of-plane bending patch test.

For the EAS approach using enhancing deformation gradient, we develop an EAS

expansion by superposing the enhancing converted basis to the compatible converted basis,

and then present a formulation that is much simpler than that employed in Miehe [1998b]

(see Section 4.3).

Two ANS modifications on the compatible covariant strains are employed to elimi-

nate the locking effects from the compatible low-order interpolations. ANS interpolation is

the most successful tool to overcome the shear-locking effect in the 4-node displacement-

based shell elements, even for initially distorted meshes (MacNeal [1978], Hughes and
Tezduyar [1981], Dvorkin and Bathe [1984], Parisch [1995]). We apply an ANS inter-

polation of the compatible transverse shear strains to treat shear locking. In the case of

curved structure with geometric nonlinearity, there is another locking effect: The so-called

curvature-thickness locking (Bischoff and Ramm [1997]), which is also known as the trape-
zoidal locking (Sze and Yao [2000]); this type of locking can be avoided by introducing the





72

ANS interpolation of the compatible transverse normal strain, as proposed by Betsch and

Stein [1995]. Such treatment can improve the performance of the formulation in Parisch

[1995] and Hauptmann and Schweizerhof [1998].

The features of this solid-shell formulation are summarized as below:

The kinematic description involves only displacement dofs that require no complex

finite rotation update and no transition elements to connect solid-shell elements to

regular solid elements (Hauptmann and Schweizerhof [1998]).

The use of covariant Green-Lagrange strain tensor, without neglecting any higher

order terms (e.g., as in Bischoff and Ramm [1997]). The stress and strain terms

quadratic in 3 become important in the analysis of relatively thick shells, for strong

curvatures or in the presence of large strains together with bending deformations

(Buchter et al. [1994]).

All stress and strain components are accounted for, thus allowing for an implementa-

tion of unmodified 3-D nonlinear constitutive laws, without the need for applying the

plane-stress constrain. The strain-driven character of the formulation also makes it
easier to implement nonlinear constitutive models, as compared to the hybrid finite-

element formulations (Simo, Kennedy and Taylor [1989]).

In contrast to EAS formulation based on the deformation gradient F (see Section 4.3),

EAS formulation based on enhancing the Green-Lagrange strain tensor (together

with the use of the second Piola-Kirchhoff stress tensor) are much simpler (Simo

and Armero [1992] and Andelfinger and Ramm [1993]).

An ANS method applied on the transverse shear strains is used to relieve the trans-

verse shear-locking problem (Dvorkin and Bathe [1984]), whereas an ANS method

applied on the normal strain components is used to remedy the curvature thickness
locking problem (Betsch and Stein [1995]).






73


In addition to the above features, our new contributions to the field are specifically listed as

below:


Optimal (minimum) number of EAS parameter to pass the patch tests for both the

membrane response, and the out-of-plane bending: (i) three EAS parameters on the

transverse normal strain to remedy the Poisson-thickness locking, and (ii) four EAS

parameters on the membrane strains to remedy the in-plane bending behavior.


Efficient EAS method that avoids the computation of the Jacobian at the element

center, and that no inverse of the Jacobian matrix at the element center is needed.

By using the tensor form, we prove the equivalence of the 2-D plane elasticity el-

ements of Simo and Rifai [1990], of Taylor, Beresford and Wilson [1976], and our

new enhancing formulation.


We justify through numerical experiments the relative importance of the separate

use of the EAS method and the ANS method, as compared to the pure displacement

formulation, and more importantly, the combined use of both the EAS method and

the ANS method in obtaining accurate results for plate bending problem over a large

range of aspect ratios.

The comparison among the above various solid-shell formulations is listed in Table 3.1.8

Table 3.1. Comparison of various solid-shell concepts.

Bending Locking-free Absence of rot. dofs/ Higher-order terms Absence of Model parameter-space Optimal
Patch test Disp. dofs only in thickness coord. pre-integration dimension EAS
Present emen yes yes yes yes yes 3-D yes
RameL l. 1997 yes yes yes no no 2-D no
Schweizerhofet. al.[I998 yes no yes yes yes 3-D no
Betsch eta.119961 yes yes no yes yes 2-D no
Miehe 1998) no yes yes yes no 3-D no


8 Parameter-space dimension is defined as: 1-D (beam), 2-D (stress-resultant plates and shells) and 3-D
(solids); Deformation-space dimension is defined as: 2-D (planar deformation), 3-D (general deformation)





74

The outline of the present chapter is as follows. After a presentation of the kinematics

assumption and the formulation based on the Fraeijs de Veubeke-Hu-Washizu (FHW) vari-
ational principle (Felippa [2000]) in Section 3.2, we discuss the finite-element discretiza-

tion and its implementation in Section 3.3. A review of the EAS method together with
our proposed modification is presented in Section 3.4. We present the numerical results in
Section 3.5.

3.2. Kinematic Assumption and FHW Variational Formulation

The extension of the EAS method to geometrically non-linear problems by Simo and
Armero [1992] employed an enhancement of the deformation gradient F and thus a multi-
plicative decomposition An alternative line of formulation for geometric and material non-
linearities based on the enhancement of the Green-Lagrange strain E leads to particularly
efficient computational effort (see, e.g. Bischoff and Ramm [1997]), with practically the
same results. We describe below the kinematics of a solid shell in curvilinear coordinates
and review the three field FHW variational principle and its role in the EAS method.
3.2.1. Kinematics of Solid-Shell in Curvilinear Coordinates

To overcome the known problems associated with the rotational degrees of freedom
in traditional shell elements, the shell kinematics of deformation is described by using the
position vectors of a pair of material points at the top and at the bottom of the shell sur-
face. In this kinematic description, a straight transverse fiber before deformation remains
straight after deformation. Such transverse fiber between two corresponding nodes at top
and bottom surfaces needs not be normal to the shell mid-surface before deformation, as
well as after deformation. We distinguish here three configurations of the shell: (i) the

material configuration, which is the biunit cube, (ii) the initial (or undeformed) configura-
tion, which could be curved, and (iii) the current (or deformed) configuration (see Fig.5 in

Vu-Quoc and Ebcioglu [2000b]). The initial (undeformed) three-dimensional continuum
of the shell geometry (Figure 3.2) is described by

( = [( + (3) x. (, 2) + (1- E3) X, (61,2)]





75


= ( E2,) E O := [-1,1] x [-1, 1] x [-1, 1], (3.1)

where X (*) is the mapping from the biunit cube 0, parameterized by the material coordi-
nation (1, 2, a), to the initial configuration. The image X (() of a point = (1, 2, 3) E
o is next represented by a linear combination of the position vectors X, (', *) and X, (*, *)
of a point in the upper material surface a = +1, and a point in the lower material surface
3 = -1, respectively.
3 2






X U ----
Xm G,
I1





el

Figure 3.2. Initial (undeformed) configuration of solid shell: convective coordinates ( and
position vectors X, and X1.

The kinematics of the present formulation is the same as in (Hauptmann and Schweiz-
erhof [1998]), and related to the director formulation (e.g. Bischoff and Ramm [1997]), if
one rewrites (3.1) as follows

X1 X, (1 ,2) + X, (1 2) X ( 2) X ( 2)

x= (e ) + 3h( 1 2)D (1, 2) (3.2)

where Xm is the position vector of the mid-surface in the initial configuration, D the unit
director, and h (1i, 22) the shell thickness.





76

Similarly, in the current (deformed) configuration, the geometry of the solid shell is
described by

) = z ( ,E2) +1 2 3 u E2) ( 2)]

= 1 [ (l+3) (C 2) + (1 _3)X( (1,2)] ,

= [, 2 3] E (3.3)

where x (') is the mapping from the biunit cube parameterized by ( 1, (2, 3) (material
configuration), to the current (deformed) configuration, x, (', -) and xa (', *) the position
vectors of the deformed upper and lower surface of the solid shell, respectively.
The initial configuration is related to the deformed configuration (Figure 3.3) by the
displacement field u as follows

x(W) = X(+) + u(W). (3.4)

The convected basis vectors Gi, i = 1, 2, 3, in the initial configuration Bo are related
to the position vector X and the converted coordinates i' by
SX (()
Gi( =- i=1,2,3, (3.5)

and satisfy the following relations

Gi*'G = 6i Gi'Gj= Gij i,j = 1,2,3, (3.6)

where Giy are the components of the metric tensor with respect to the basis G' 0 Gj
in the initial configuration Bo. configuration. To simplify the presentation, we will omit
the argument (() and simply write Gi and Gi. The covectors G' can be obtained by the
following relation

G" = GjGj with [Gj] = [Gi- R3x3. (3.7)

similarly, the convected basis vectors gi in the current configuration Bt are obtained by
using (3.4) and (3.5) as follows
Ox Ou
i = -G + i = 1, 2,3, (3.8)
aO~t





77

X












E3
E2 92
13
BBt








and satisfy the following relations
E3 E2 2







B

Figure 3.3. Solid-shell: material configuration B, initial configuration Bo, and deformed
configuration Bt.

and satisfy the following relations


g9i*9 = 68 gi'gj = ij i,j = 1,2,3, (3.9)

where gij are the components of the metric tensor with respect to the basis gi 0 gi in the
current configuration Bt. The covectors g' can be obtained as follows

gi = ggj with [g1] = [gi]-1 e R3x3 (3.10)

The deformation gradient expressed in converted basis vectors gi and G" takes the
form


F =- X = Gi (3.11)

Using (3.11), (3.6), and (3.9), we can then write the (compatible) Green-Lagrange strain





78

tensor with respect to the convective coordinates as follows

EC =1 (FTF I2) =' [(GI 0 og) (gj Gi) GijG' Gi]


= 2 (9ij Gij) G' 0 G = E G% G( (3.12)

where Efc is the covariant components of the strain tensor Ec, and if2 the identity tensor
expressed with respect to the convected basis G'i G7 using the relations ep = GiG' and
eq = GJGj as follows

A2 = 3,qe? 0 eq = 6pqGGjqG' Gj = GGk G -9 = G = G3G 0 G (3.13)

Using (3.8), the metric-tensor components gij in (3.9)2 can be expressed in terms of
the convected basis vector Gi and the displacement vector u as follows

9j = gig = (G + 9u G + a9

S_ u o u u u
= Gij + Gi* + Gj + (3.14)

Substituting (3.14) into (3.12), the covariant components E1 of the compatible Green-
Lagrange strain tensor Ec read as

(1 Ou u 9u Ou\
Ec=2 Gi + +Gu + (3.15)

The second Piola-Kirchhoff stress tensor S is conjugated with the Green-Lagrange
Strain Ec, and can be expressed with respect to the basis Gi 0 Gj as follows

S = S'Gi 9 Gj, (3.16)

with Sij being its contravariant components.
3.2.2. Variational Formulation of EAS Method
In this section, we provide a brief overview of the EAS method, which has the three-
field Fraeijs de Veubeke-Hu-Washizu (FHW) 9 variational principle with the following
9 We refer the readers to Felippa [2000] for a history on the contribution of Fraeijs de Veubeke to the
formulation of multifield variational principles.





79


functional as the point of departure

HI(u,E,S) = W, (E)dV + S: [E (u) -EdV
Bo Bo

fu (b' -iU)pdV+ (u* -u) tdS- u-t*dS,
Bo Su S,
(3.17)

where the displacement u, the Green-Lagrange strain E, and the second Piola-Kirchhoff

stress S are the independent variables. In (3.17), the expression of Ec (u) in terms of

u is given in (3.12) and (3.15); W, is the stored strain energy, t the traction vector, u*
and t* the prescribed displacement on the boundary S, and the prescribed traction on the

boundary S,, respectively. All variables here are expressed in the initial configuration Bo,
with S, U Su = 903o (i.e, the closure of the union of the boundary S, and the boundary S,

forms the complete boundary of B0).

The next step in the EAS method is to introduce an enhancement (or enrichment) to
the Green-Lagrange Strain EC and, as a result, of the variation 6Ec as follows

E=EC + E (3.18)

JE = JEc + JE, (3.19)
6E = 6E" + 6E (3.19)

where E and 6E represent the enhanced strain and its variation, whereas E and 6E repre-

sent the enhancing strain and its variation. Introduction of (3.18) into (3.17) yields

S(u,E,S) = W, (Ec (u) + E) dV S : Ed V
so Bo

fu (b*- i)pdV+ (u*-u).t(S)dS- ut*dS.
Bo Su S,
(3.20)

where it is noted that the traction t is actually a function of S(see also Malvern [1969,

p.69]), and not an independent variable.





80

Consider a perturbation of the displacement u as follows:

u, (X, t) = u (X, t) + e6u (X) (3.21)

since the variation 6u is only a function of space, and not of time, the variation of the
acceleration iu is then

6'U = 0, (3.22)

therefore the variation of n in (3.20) with respect to the displacement u is

d (U,E, S) : dEc(u) dV u(b* ) pdV
eo- I M
0 Bo 'E Bo

J utdSJ f 6ut*dS, Vu, (3.23)
s, So
8Ws.
Since the second-order tensor is symmetric, the first term in (3.23) can now be rewrit-
aE
ten as follows

8W, DEc (u) FWSW
S u = F- : GRAD (6u)
OE Bu aE

= Div( F 6u Div F ) J"6u (3.24)
( E ) E /

where the first equality is shown in Remark 3.1, and the second equality is obtained with
Leibniz rule.

Remark 3.1. With the deformation gradient F (u) expressed with respect to the basis
ei 0 Ej (see Figure 3.3), we obtain the following relations

F = Fjei 0 E FT = FEJe e (3.25)


FTF = Fg2 Fj3E' EJ, with j = ee.ej, (3.26)

OF OFJ
5F -- u = ei EJSu. (3.27)
5--1Ou





81

Note that, if {ei, e2, e3} were a system of orthonormal vectors, we would have gi = iy,
however, in general, it is not necessary to have {el, e2, e3} orthonormal. In this remark,
we retain the notation gij for generality. We define a symmetric second-order tensor S as

aw5
S := = SABEA 0 EB (3.28)
aE

By using (3.28), (3.12), and (3.26), we can then rewrite the first term in (3.23) as follows

OW, OEc 1 a(FT F)
-E : u = S: Su

!SAB (OFa g F3 +B
= 2 u gjF + Fi9j au )u. (3.29)

Since SAB = SBA from the symmetry of S, (3.29) becomes

awl F AB.. OF
:E J6E0= P FAS B u (3.30)
aE Aau

With the following component form of FS using (3.25)1 and (3.28), that is, we obtain

FS = FjSJBei 8 EB (3.31)

and then the following expression

aF jU = Fi SAB .FB 6U
FS : 6F = FS : u = FA ig (3.32)

Comparing (3.32) to (3.30) and using 6F = GRAD (Ju), we arrive at

E : 6EC = FS : 6F = FS : GRAD (6u) (3.33)




Using the divergence theorem, the integration of the first term in (3.24) becomes

Div ( "u) dV = Fa u) -ndS (3.34)
8E aE
Bo lBo
where n is the outward normal vector to the surface 3Bo.





82

With (3.24) and (3.34), we can write the variation of n in (3.23) with respect to u as

n(Ue ,S) -= f F ) udS Div (F aJudV
LE aE
E0 dBo=SUSE Bo

Ju (b* u) pdV Ju'tdS J6ut*dS
So SU S,


s, s.,
n \ / -[ + n F J


Div F---' +pb* p* -udV = O, V6u, (3.35)
Bo
thus the Euler-Lagrange equations resulting from the variation of II with respect to u in
above (3.35) are

t = no F- on S,, (3.36)
/ oW )
t*=n* F a on S, (3.37)


Div (F wa-)+ pb*=pi in Bo, (3.38)

Then make the variation of I of (3.20) with respect to E by the perturbation

Ee = E + e6E, (3.39)

it follows that

d n (u, ,, S) = i( S : 6EdV VSE (3.40)
dB0
the Euler-Lagrange equation associated with (3.40) is

Ow,
S = (3.41)
dE

Taking the variation of II of (3.20) with respect to S by the perturbation

S, = S + e6S (3.42)





83


we obtain

d
d-E1 (uE, S) = SS: EdV + (u* u) -tdS, V6S, (3.43)
e =0 o Su

where it is noted that t depends on S, then the Euler-Lagrange equations associated with

(3.43) are

E = 0, (3.44)

u = u* on S,, (3.45)

where in general E # 0 when the finite element approximation is introduced.

3.3. Finite-Element Discretization

In this section, we present the weak form and finite-element approximation of the

proposed solid-shell element. The orthotropic constitutive law of laminated composite is

then derived in convective coordinates. To avoid shear locking, we use the assumed natural

strain (ANS) method by Dvorkin and Bathe [1984] for the transverse shear strains. To

remedy the curvature-thickness locking (Bischoff and Ramm [1997]), we adopt the ANS

method by Betsch and Stein [1995] for the transverse normal strain.
3.3.1. The Weak Form of Modified Two-Field FHW Functional

By designing the approximation for the stress field S and the approximation for the
enhancing strain EC such that the following orthogonality condition holds:

S : EdV = 0. (3.46)
Bo
then the number of independent variables in the functional II in (3.20) is reduced to two,
that is

n(u, E) = W, (Ec(u) + )dV


u- (b* )pdV+ l(u*- u) -t(S)dS- fut*dS, (3.47)
Bo SU S,





84

leading to the following total variation

1 (u, E) = ,i mass (u) + lIsti (u, E+ l et (u) = 0, (3.48)

where from (3.35) and (3.40), we have

Imass = Ju-UpdV (3.49)


SI (u, ) = J(Ec (u) + S) W, (E (u) + E)dV, (3.50)
Bo

lext = Jub*pdV J6ut*dS. (3.51)
Bo S,
3.3.2. Spatial Discretization

Let the initial configuration B0 be discretized into nonoverlapping nel elements B e)
nel
with numnp nodes, such that Bo U J e). Let h denotes the characteristic size of the
e=l
finite element discretization.
In the element domain Boe), the displacement u, its variation 6u, and increment Au
are interpolated as follows

u N uh = N () d(e), (3.52)

u buh = N (() 3d(e), Au Auh = N (() Ad(e), (3.53)

where N is a matrix containing the basis functions restricted to element B(O), and d(e) E
R3xnu"np a matrix containing the nodal displacements. The readers are referred to the

Appendix A. 1 for the details.
The velocities u& and accelerations u are also interpolated by using the same shape
functions and the corresponding nodal values that is

in ilh = N (s) (fe) oU t eh = N (,) d(e) (3.54)

In what follows, for simplicity, we will omit the superscript h in uh, and simply write u.





85

Within a typical element (e), the variation and the increment of the (compatible)
Green-Lagrange Strain Ec is related to the variation and the increment of displacement,
respectively, based on (3.15), as follows

{6Ej} x = B (d(e) d(e) AE 61 = B (d(e)) Ad(e) (3.55)

where the components of Ef had been arranged into a 6 x 1 column matrix according to
the Voigt ordering (Brillouin [1946, p.221])

{ E } = {Efl, E2, 2E2, E3, 2E, 2E3} T (3.56)

and where B is the deformation-dependent displacement-to-strain operator, where detailed
expression is given in the Appendix A. I.
We denote the admissible variation of the element EAS parameter column matrix
a(e) E R"'IF associated with the enhancing strain {E~j} by 8a(e), where neas is the num-
ber of EAS parameters a(e). Interelement continuity is not required for the enhancing
strain E, where components can be approximated via an enhancing strain interpolation
matrix Q and the element EAS parameters ac(); an interpolation applies to the variation
and the increment of E, that is

{~j)x, = g (t) e), {iEJ,}x =)6 () Sa(e) {AE rl 1 (Aa(e). (3.57)

The number of internal parameters a(e) and the interpolation matrix 9 will be discussed in
Section 3.4.
3.3.3. Linearization of the Discrete Weak Form
The consistent tangent operator in the Newton solution procedure is constructed by
taking the directional derivative of the weak form at a configuration (k)u in the direction
of the increment A (k)u, where the left subscript k designates the iteration number. The
tangent operator can be viewed as the summation of the material and geometric tangent
operators. The geometric part results from taking the variation of the geometry while hold-
ing the material constant, whereas the material part results from taking the variation of the
material while holding the geometry constant.





86

Applying a standard finite-element procedure to discretize the weak form (3.48), we
obtain the following expression at the element level for the static case (ii = 0)
-I(e) = Si(e + H e) = 0, (3.58)


with the stiffness part stl.e from (3.50) and the external-force part 6S11 from (3.51)
written as follows


B e)0 0
stif = f {E T {SjI} dV+ f 8 j w {Sj} dV, (3.59)


sHt = f 6u.bpdV f ut*dS, (3.60)
5e") se)a
where we used the symbol definition, which corresponds to the second Piola-Kirchhoff
stress,

ow,
S W:- (3.61)

to alleviate the notation, and where the column matrix {Sij} has its coefficients arranged
in the same Voigt ordering as in (3.56)

{ } = [S11, S22, S12, S333, S23, 13] (3.62)
ow,
It should be noted that the symbol S in (3.61) is simply used in the place of and is
9E
not the independent variable in the Euler-Lagrange equation (3.41).

Remark 3.2. The linearization of the weak form 6I (u, E) can be accomplished by
the truncated Taylor series about the kth iterate ((k)U, (k)E):

0m((k+) U, (+,) )((k)U, (kE)
+ (811) ,)
+ ( ()) (A( (E)
a u (Uk)U,U E= E)

= 611( (k)U (k)E ) + D (6H) ((k)U, (k)E) (Au, AE) (3.63)




87

where Au = (k+1)U (k)U, AE = (k+1)E (k)E. To compute the increment
(Au, AE) in the Newton's solution process, we simply set the expression in (3.63) to
zero. I

To alleviate the notation, we will omit the left subscript k designating the iterative index.
Using the approximation (3.52), (3.53), (3.55) and (3.57) in (3.63), the increments
Ad(e) and Aa(e) can be computed in the Newton's solution process, as mentioned above,
by using the following equation
f (sn(e) + ,H(e)
,D ((6s(e)) (d(e), a)(e)) (Ad(e), Aa(e))= stiff ( de) A(e))
a (d(e), a(e)) (Ad Aa)
e())
a (Wn) (
=fd(e), (Ad(e), Aa(e)) (6,1e) + 611e) V6d(e), V6a(e) (3.64)
a (dne) a(e))

in which the variation 61ge, in (3.59) and 6I e in (3.60) now take the form

8I) (d(e) a(e)) = 6d(e)Tf(e + (e)Tf (es (3.65)

with f = B BT Sj dV, f, = AST (Si}dV, (3.66)
1,'>) B
le (d(e) = -d(e)Tf (3.67)

with fe = NT b*pdV + NTt*dS. (3.68)
03(e) st'e
Thus the left hand side of (3.64) becomes
8b(e) O 8 ( e)~) Aoz(e)
V (6sti) Ad(e) a, Aa()) Ad() + a A (ee
= [d(e)Tk(e + 6a(e)Tk()] Ad(e) + [Jd(e)Tk (e + 6a(e)Tk(e] Aa(e)

= 6d(e)T [kF(Ad(e) + k a(e)] + 6aC(e)T [k Ad(e) + keAa(e (3.69)

Let the matrix of tangent elastic moduli C be defined as

C [Cijkl] := Ea6 (3.70)
1 -' LOEk' I





88

where Cijkl are the components of constitutive tensor C in the convected basis, and are
subsequently arranged in the matrix C according to the ordering of the strain components
in (3.56) and of the stress components in (3.62).
Using (3.69), (3.66)1, (3.70), and (3.55), we obtain the following expressions

k e) == / (GTS + BTCB) dV (3.71)
S BC dV (3.72)
B(e)

af (e)
(e)a = -J stiff = I BTc r dV
u ~ -9 J BCgdv (3.72)
0B(e )

where the matrix
B (d(e))
G .- e (3.73)

(which is a function of the coordinates 0), and the stress matrix $ (which is related to the
matrix {S'j} in (3.62)) have their detailed expressions given in the Appendix A.1. It is
noted that the dimension of $ and G are 144 x 24 for the present element with six stress
components, eight nodes per element, and three dofs per node. From (3.69), (3.66)2, (3.70),
(3.73), and (3.55), we obtain the remaining parts of the stiffness matrix

e Od(=e) i- I= TCBdV, (3.74)
SB(oe)

k( = 9 TC G dV (3.75)
B(e)
(3 0
It follows from (3.64), (3.65), (3.67), and (3.69) that the discrete linearized system
of equations to solve for the increments Ad() and Aa(e) is given by
ke Ad(e) + k(e)Aa) )= f (e)-f (3.76)
,ua J ext s stiff (3.76)
k(e)Ad(e) + keAot(e) = _f (e) (3.77)


or in matrix form as
[ (e) k$e). Ad(e) (e) (e)
k k (e) A ( ) tff (3.78)
k(e) k(e) Ac(e=) (e)3.78)
oI oau EAS





89

Since the enhancing strain E is chosen to be discontinuous across the element bound-
aries, it is possible to eliminate the EAS parameter increment Aa(e) at the element level,
before proceeding to assemble the element matrices into global matrices. Solving for the
increment Aa(e) using (3.77)

Az(e) [k 1 (f~ ) s + k, Ad(e) (3.79)


then substituting (3.79) into (3.76), we obtain the following condensed symmetric element
stiffness matrix k) and the element residual force vector r(e)

k() = k [k ] T[k ke ,) (3.80)
T uu aaj au

r(e) f(e) f(e) + Tu] T (e) f (e)
r -e =f sff + kau [k ]a1 EAS (3.81)

An assembly of the element matrices k) and r(e) leads to the global system

KTAd = R, (3.82)
nel nel
with KT = Ake) R = Ar(e) (3.83)
e=l e=l
where A denotes the finite-element assembly operator.
The incremental displacement Ad can be solved by using (3.82), and the displace-
ment d and d(e) updated. With (3.79), the incremental displacement Ad(e) is used to com-
pute the increment Aa(e), which is in turn used to update the EAS parameter a(e). The
details of this iterative procedure are provided in the Appendix A.I.
3.3.4. Material Law in Convected Basis

For the Saint-Venant-Kirchhoff material, the fourth-order material tensor C is de-
fined as the second derivative of the stored energy function W, with respect to the Green-
Lagrange strain tensor E,
092WS
C = EE (3.84)

and the second Piola-Kirchhoff stress tensor S is then expressed as

S=- C : E. (3.85)





90


The constitutive relation of laminated composites can be described by using an or-

thotropic material law. For that purpose, we express the components ijk' of tangent elastic

moduli tensor C relative to the fiber reference axis {al, a2, a3} of a lamina, and arrange

these components in a matrix [Cijkl] (see, e.g., Reddy [1997, p.41] and Figure 3.4), using

the same ordering of the strain components in column matrix form as in (3.56) (see also

(3.62)).
b2




a2
al











0 b
b3 = a3

Figure 3.4. A fiber-reinforced lamina and fiber reference axes {al, a2, a3}.


C1111 C1122 0 01133 0 0
01122 02222 0 02233 0 0
"ijkl] 0 0 01212 0 0 0
0 J C1133 2233 0 '3333 0 0 (3.86)
0 0 0 0 02323 0
0 0 0 0 0 01313
6x6
where the components ,ijkl take the following expressions

011111 El (1 v23V32) 02222 E2 (1 1331) 3333 E3 (1 12V21)


A A A
^ ~ ,^13 ^ ___^ ___





91


A = 1 v12V21 -- V2332 V21v13 2V12V32V13

01212 = G12, 2323 = G2 1313 = G13 ,


vijEj = vjiEi, for (i,j = 1,2,3 ,andi j),

and El, E2, E3 are the Young's moduli in the principal material directions {al, a2, a3},

respectively, and vi and Gyj the Poisson's ratio and the shear modulus in the (es, ej) plane,

respectively. Note that, for the special case of isotropy, only two material parameters E
and v are needed:

E = E2 =E3 = E, v12 =23 =13 =v,

E
G12 = G23 = G13 = (3.87)
2 (1+ v)

Since matrix [Ci3kl] of elastic moduli is associated with the principal material direc-

tions, we need to transform it from the lamina coordinate axes {a,, a2, a3} to the global

Cartesian coordinate axes {bl, b2, b3}. With 0 being the fiber direction angle relative to the

global Cartesian system (see Figure 3.4), the relationship between the lamina coordinate

system and the global Cartesian coordinate system is given by

al = cos Obl + sin Ob2 a2 = sin Obl + cos Ob2, a3 = b3 (3.88)

Since we are developing the formulation in the convective coordinates associated

with the basis {Gi}, we have to express the tensor C of elastic moduli in the same convec-

tive coordinates. Thus,

C = Cab6,d a 0 b a c ad = CjklGi ( Gj Gk ( G (3.89)

where the components Cabcd are given in (3.86), and the components Cijkl are to be com-
puted for use with the present solid-shell formulation.

From the following component forms of the second Piola-Kirchhoff stress tensor S

S = Sabaab ab = S3Gi Gj (3.90)




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/' QL WLJ


NONLINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS OF
MULTILAYER SHELL STRUCTURES
BY
XIANGGUANG TAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2002

To my parents.

ACKNOWLEDGMENTS
I wish to express my sincere gratitude to my advisor, Prof. Loc Vu-Quoc, for his pa¬
tience, guidance, support and friendship throughout my Ph.D. education at the University
of Florida. I have greatly benefited from his stimulating approach to research and his re¬
lentless pursuit of perfection in organization and documentation. Many thanks are extended
to him for his invaluable help in preparing this LINUX/LaTeX document.
I also wish to acknowledge the members of my examining committees, Professors
Martin A. Eisenberg, Raphael T. Haftka, Marc Hoit, Andrew J. Kurdila, and W. Gregory
Sawyer for their careful examination of the dissertation, and their invaluable comments
and insights, which made a deep impact on my research. I also benefited greatly from their
graduate courses and from their help in many other aspects.
I am indebted to several colleagues and mentors for their help in my present work:
in particular, Hui Deng for the use of finite element code, FEAP and many insightful dis¬
cussions on the geometrically-exact shell theory; Fuller L. Brian for the installation of the
LLNL package; Paul Dionne, Andrzej Przekwas, Marek Turowski, and H.Q. Yang at the
CFDRC for discussion of the model reduction technique; Prof. Chen-Chi Hsu to work for
him as his teaching assistant; and to my friends, Joakim Andersson, Jonas Bjomstrom, Mat-
tias Horling, Stefan Jansson, Kil-Soo Mok, Mattias Quas, Simon. Sjogren, Xiang Zhang,
and Yuhu Zhai, and many others, who have made my stay at Gainesville one of the most
memorable periods of my life.
Last, but certainly not least, my heartfelt thanks go to my parents for their love, and
encouragement through my life. I am indebted to my girlfriend, Veronica Leung. Without
her love and care, I could not have accomplished so much.
This research is supported by a grant from the National Science Foundation, and also
by the CFDRC. This support is gratefully appreciated.
iii

TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
ABSTRACT viii
CHAPTER
1 OVERVIEW 1
1.1 Objectives and Motivation 1
1.1.1 Formulation and Kinematics 2
1.1.2 Computational Aspects 4
1.2 Chapter Overview 5
2 GEOMETRICALLY-EXACT SANDWICH SHELL FORMULATION 7
2.1 Introduction 7
2.2 Virtual Powers 8
2.2.1 Basic Kinematic Assumptions and Configurations 9
2.2.2 Virtual Powers 11
2.2.2.1 Power of contact forces/couples and conjugate strain mea¬
sures 11
2.2.2.2 Power of assigned forces/couples 13
2.2.3 Constitutive Relations 13
2.3 Weak Form and Linearization 14
2.3.1 Admissible Variations, Tangent Spaces 14
2.3.2 Weak Form of Equations of Equilibrium 15
2.3.3 Contact Weak Form 16
2.3.4 Assigned Weak Form 16
2.3.5 Linearization of Contact Weak Form 17
2.3.5.1 Update of inextensible directors 17
2.3.5.2 Perturbed configuration 18
2.3.5.3 Linearized strain measures 20
2.3.5.4 Linearized contact weak form 22
2.3.6 Matrix-Operator Format of Contact Weak Form 22
2.3.6.1 Material tangent operator 31
2.3.6.2 Geometric tangent operator 34
2.4 Numerical Examples for Statics of Sandwich Shells 46
2.4.1 Roll-down Maneuver of a Sandwich Plate 48
IV

2.4.2Sandwich Plate with Ply Drop-offs 49
2.4.2.1 Sandwich plate with ply drop-off 49
2.4.2.2 Two-layer plate with ply drop-off: aspect ratio A = 5 : 1 :
(1,0.5) 53
2.4.2.3 Two-layer plate with ply drop-off: aspect ratio A = 20 : 1 :
(1,0.5) 53
2.4.2.4 Two-layer plate with ply drop-off: aspect ratio A = 20 : 10 :
(1,0.5) 61
3 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER
COMPOSITES : STATICS 68
3.1 Introduction 68
3.2 Kinematic Assumption and FHW Variational Formulation 74
3.2.1 Kinematics of Solid-Shell in Curvilinear Coordinates 74
3.2.2 Variational Formulation of EAS Method 78
3.3 Finite-Element Discretization 83
3.3.1 The Weak Form of Modified Two-Field FHW Functional 83
3.3.2 Spatial Discretization 84
3.3.3 Linearization of the Discrete Weak Form 85
3.3.4 Material Law in Convected Basis 89
3.3.5 The ANS Method 93
3.3.5.1 Transverse shear strains 93
3.3.5.2 Transverse normal strain 94
3.4 Interpolation of the Enhanced Strains 94
3.4.1 The Regular Enhanced Strains Treatment 95
3.4.2 Proposed Efficient Enhancing Strains 100
3.4.3 Equivalence Between EAS Element and Incompatible Mode Element 102
3.4.3.1 Tensor form of enhancing strains 103
3.4.3.2 Equivalence of condensed stiffness matrices 107
3.5 Numerical Examples 109
3.5.1 Patch Tests and Optimal Number of Parameters 110
3.5.1.1 Membrane patch test Ill
3.5.1.2 Out-of-plane bending patch test Ill
3.5.2 Cantilever Plate 113
3.5.2.1 Cantilever beam: in-plane bending 114
3.5.2.2 Cantilever plate: out-of-plane bending 115
3.5.3 In-plane Bending Problem with Nearly Incompressibility 119
3.5.4 Snap-through of a Shallow, Cylindrical Roof under a Point Load . . 121
3.5.5 Pinched Hemispherical Shell 122
3.5.6 Multilayer Composite Plate 125
3.5.6.1 Two-layer composite plate: linear solution 125
3.5.6.2 Multilayer composite plate with ply drop-offs 126
3.5.7 Multilayer Composite Hyperbolical Shell with Ply Drop-offs .... 129
v

4 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER
COMPOSITES : DYNAMICS 132
4.1 Introduction 132
4.2 Dynamics of Solid Shells by an EM Conserving Algorithm 133
4.2.1 Time Discretization on Dynamic Weak Form 134
4.2.2 Linearization of Dynamic Weak Form 136
4.3 Enhanced-Assumed-Strain Method Based on Deformation Gradient . . . 143
4.3.1 Weak Form 143
4.3.2 Finite Element Discretization and Linearization 147
4.3.3 Assumed Natural Strain (ANS) Treatment 150
4.3.4 Simplified Formulation 151
4.4 Numerical Examples 153
4.4.1 Double Cantilever Elastic Beam under Point Load 154
4.4.2 Pinched Cylindrical Multilayer Shell 157
4.4.3 Free-Flying Single-Layer Plate 159
4.4.4 Free-Flying Multilayer Plate with Ply Drop-offs 161
5 EFFICIENT AND ACCURATE MULTILAYER SOLID-SHELL ELEMENT: NON¬
LINEAR MATERIALS AT FINITE STRAIN 175
5.1 Introduction 175
5.2 Nonlinear Material Law 180
5.2.1 The Mooney-Rivlin Material Models 180
5.2.2 The Hyperelastoplastic Model 182
5.2.2.1 Multiplicative decomposition of the deformation gradient F 183
5.2.2.2 Spectral form based on the right Cauchy-Green tensor C . 185
5.3 Explicit Time Integration Method for Solid-Shell Elements 193
5.4 Numerical Examples 196
5.4.1 Large Deformation of Rubber Shells 197
5.4.1.1 Stretch of a rubber sheet with a hole 198
5.4.1.2 The snap-through of a conic shell 198
5.4.1.3 Large motion of the pinched cylindrical shell 200
5.4.1.4 Rubber hemispherical shell 203
5.4.2 Large Deformation of Elastoplastic Shells 204
5.4.2.1 Bending of a cantilever beam 206
5.4.2.2 Elastoplastic response of a channel beam 208
5.4.2.3 Pinched hemisphere 211
5.4.2.4 Elastoplastic response of a simply supported plate 213
5.4.2.5 Elastoplastic response of a pinched cylinder 215
5.4.2.6 Free-flying multilayer plate with ply drop-offs 218
5.4.2.7 The impact of a boxbeam 221
5.4.2.8 Pipe whip 223
6 SOLID SHELL ELEMENT FOR ACTIVE PIEZOELECTRIC SHELL STRUC-
vi

TURES AND ITS APPLICATIONS
228
6.1 Introduction 228
6.2 The Solid-Shell Formulation 231
6.2.1 The Kinematics of Piezoelectric Solid-Shell Formulation 231
6.2.2 Piezoelectric Solid-Shell Element 234
6.2.2.1 Functional and finite element formulation 234
6.2.2.2 Linear piezoelectric material law in convected coordinate . 239
6.2.3 Composite Solid-Shell Element 242
6.3 Simulation Control Design 244
6.3.1 Finite Element System Equation of Piezoelectric Structure 244
6.3.2 Reduced-Order Model of Piezoelectric Finite Element System . . . 246
6.3.3 Controller Design 249
6.4 Numerical Examples 252
6.4.1 Cantilever Plate: Out-of-Plane Bending 253
6.4.2 Multilayer Composite Hyperbolical Shell 255
6.4.3 Piezoelectric Bimorph Beam 256
6.4.4 Cantilever Plate with PZT Actuators 259
6.4.5 Cantilever Plate with PZT Actuator and Sensor 263
7 CLOSURE 268
7.1 Conclusion 268
7.2 Directions for Future Research 270
APPENDIX
A SOLID-SHELL FORMULATION 272
A.l Finite Element Approximation of Solid-Shell Element 272
A.2 Solution Procedure of Nonlinear Equations 279
A.3 Explicit Integration Algorithm with EAS Method 280
A.4 Return Mapping Algorithm for J2 Flow Theory with Isotropic Hardening . 280
A.5 Elastoplastic Moduli £f? 281
A.6 Algorithmic Moduli for Return Mapping 282
B PIEZOELECTRIC SOLID-SHELL FORMULATION 285
B.l Model Reduction Algorithm 285
B.2 Solving Procedure on Control Design 285
REFERENCES 287
BIOGRAPHICAL SKETCH 302
vii

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NONLINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS OF
MULTILAYER SHELL STRUCTURES
By
Xiangguang Tan
August, 2002
Chairman: Loc Vu-Quoc
Major Department: Aerospace Engineering, Mechanics, and Engineering Science
Firstly, the geometrically-exact sandwich shell formulation is developed to analyze
sandwich shells undergoing large deformation. Finite rotation of the director in each layer
is allowed, with shear deformation independently accounted for in each layer. The thick¬
ness and the length of each layer can be arbitrary, thus allowing the modeling of multilayer
structures having ply drop-offs. The weak form of governing equations is constructed, and
the linearization and inextensible directors update are derived. Numerical examples on
elastic sandwich plates are presented to illustrate salient features of the formulation.
Furthermore, we present a low-order solid-shell element formulation—having only
displacement degrees of freedom (dofs) (i.e., without rotational dofs)—that has an opti¬
mal number of parameters to pass the plate patch tests (both membrane and out-of-plane
bending), thus allowing for efficient and accurate analyses of large deformable multilayer
shell structures. The formulation is based on the mixed Fraeijs de Veubeke-Hu-Washizu
(FHW) variational principle leading to a novel enhancing assumed strain (EAS) tensor,
with improved in-plane and out-of-plane bending behaviors (Poisson thickness locking).
Shear locking and curvature thickness locking are treated using the Assumed Natural Strain
(ANS) method. We provide an optimal combination of the ANS method and the minimal
number of EAS parameters to pass the out-of-plane bending patch test and treat the locking

associated with (nearly) incompressible materials. The energy-momentum (EM) conserv¬
ing algorithm for the current element is presented. Two nonlinear 3-D material models are
applied directly without requiring the enforcement of the plane-stress assumption. More¬
over, we present a low-order accurate piezoelectric solid-shell element formulation for
piezoelectric sensors and actuators used in active shell structures. Numerical examples in¬
volving static analyses and implicit/explicit dynamic analyses of multilayer shell structures
having a large range of element aspect ratios for both material and geometric nonlinearities
are presented. Numerical examples involving static analyses and active vibration control of
piezoelectric shell structures are also presented. The developed element formulations are
accurate and efficient in modeling and analyzing general nonlinear multilayer composite
shell structures.
IX

CHAPTER 1
OVERVIEW
Shells and shell structures are thin-walled, generally curved bodies in a three-dimens¬
ional space. Their load-bearing behavior is dominated by stretching and bending. Shell
structures with different layers in the thickness direction are generally addressed as multi¬
layer shells. For a comprehensive and valuable history and review of linear and nonlinear
shell theories, see Timoshenko and Woinowsky-Krieger [1959], Naghdi [1972] and Basar
and Kratzig [2000]. Below we describe the objectives and motivation for the current re¬
search on multilayer shells. Some of the motivating factors behind the present work and
literature review are delineated in the following chapters.
1.1. Objectives and Motivation
Multilayer shell structures have widespread applications in engineering. Laminated
composite structures, initially developed for use in the aerospace industry, have played
an increasingly important role in robotics and machine systems that require high operat¬
ing speed. The low weight and high stiffness offered by laminated composite structures
help reduce power consumption, increase the ratio of payload/self-weight, and improve
the accuracy of motion characteristics and reduce the level of acoustic emission of these
systems. It is shown from computer simulations with experimental corroboration that the
low weight/stiffness ratio of laminated composites is essential for obtaining high perfor¬
mance in slider-crank and four-bar linkage systems (Sung, Thompson, Crowley and Cuccio
[1986], Thompson and Sung [1986]). More recently, considerable attention has been given
to a class of active structures with embedded piezoelectric layers as sensors and actuators
(Evseichik [1989], Tzou [1989], Saravanos, Heyliger and Hopkins [1997]) or interfero¬
metric optical fiber sensors (Sirkis [1993]) for monitoring the strain level and for vibration
control. Large overall motion of multilayer structures can be found in robot arms or space
1

2
structures with embedded sensors/actuators. Another example of multilayer structures can
be found in the damping of structural vibration by using viscoelastic constrained layers (Al¬
berts [1993], Dubbelday [1993], Rao [1993]) (Figure 1.1). The use of sandwich plates to
absorb energy in crashes (car, train, airplane) was investigated by Goldsmith and Sackman
[1991].
Figure 1.1. Multilayer shells with patches of constrained viscoelastic materials or of piezo¬
electric materials.
The design and analysis of multilayer shell structure is a major challenge that in¬
volves the proper modeling of composite materials with highly anisotropic properties, com¬
plex geometric configuration, and strongly nonlinear material behavior. For example, only
a few studies so far have been performed on large deformation analysis of 3-D nonlin¬
ear composite laminates. There have been no analytical studies involving 3-D analysis of
multilayer shells with nonlinear material behavior and large deformation.
1.1.1. Formulation and Kinematics
For nonlinear analysis of multilayer shell structures, we developed two different finite
element formulations: the geometrically-exact1 multilayer shell formulation and multilayer
solid-shell formulation.
In the geometrically-exact multilayer shell formulation, the 3-D analysis is reduced
to a set of 2-D stress-resultant equations based on the kinematic assumptions. This model
accommodates large deformation and large overall motion. The layer directors at a point in
1 The term “geometrically-exact” reflects the fact that no additional kinematic assumptions are made
beyond the one-director assumption. In particular, approximations of the type sinO «Í - 03/6 are entirely
avoided.

3
the reference surface are connected to each other by universal joints, as in a chain of rigid
links. The thickness and length of each layer can be arbitrary, thus making it suitable to
model shell structures with ply drop-offs. The equations of motion of the multilayer shell
are derived based on the principle of virtual power, and expressed in terms of weighted
resultant forces and couples. The overall deformation of a sandwich shell can be described
by the deformation of a reference layer (which can be any layer; not necessarily the middle
layer). The unknown kinematic quantities are therefore the three displacement components
of the centroidal surface of the reference layer and two rotational components for each
layer director. No restriction is imposed on the magnitude of the displacement field, whose
continuity across the layer interfaces is exactly enforced. Finite rotations of the directors in
each layer are allowed, with shear deformation independently accounted for in each layer.
We have implemented the geometrically-exact sandwich shell element to illustrate the ver¬
satility of formulation in the large deformable multilayer shell analysis involving linear
elastic material and small strain. Due to the kinematic assumptions, the present formula¬
tion is more accurate than the equivalent single-layer shell models in the interlaminar stress
analysis, especially for thick and moderate thick shells.
In the solid-shell formulation, on the other hand, the shell kinematic descriptions
used are the displacement of the top and bottom surface of the shell. All kinematic quan¬
tities such as displacements and the corresponding strains can be finite. For multilayer
shells, one solid-shell element in the thickness direction can be used for either one material
layer or several layers. In contrast to the shell formulation based on the degenerated shell
concept and the classical shell theory, the present element can incorporate the complex 3-D
material models without enforcing the zero transverse normal stress condition, can avoid
complex update algorithms for finite rotations, and can account for the transverse normal
stress. Based on the mixed Fraeijs de Veubeke-Hu-Washizu (FHW) variational principle,
the present low-order solid-shell element is designed to pass the plate patch tests and to
remedy volumetric locking, therefore allowing efficient and accurate nonlinear analyses
of multilayer shell structures. Moreover, the kinematic description provides a natural way

4
to connect solid-shell elements to regular solid elements without the need for transition
elements; such feature can also benefit the detailed modeling of shells with patches of
piezoelectric or viscoelastic materials. For the interlaminar stress analysis, with the refine¬
ment through the thickness, the solid-shell element model can determine the localized 3-D
stress field (e.g., delamination, free-edge effect) accurately.
1.1.2. Computational Aspects
Several aspects can directly contribute to the success and generality of numerical
simulations: 1) element formulations; 2) time-integration schemes; and 3) equation solu¬
tion strategies. The geometrically-exact sandwich shell formulation uses the resultant form
to avoid numerical integration in the thickness direction for elastic materials. The solid-
shell formulation uses the numerical integration for general nonlinear constitutive models.
All kinematic quantities such as displacements can be finite, and the update procedure is
proceeded in an exact manner, without approximations.
Engineering applications mandate the use of relatively coarse meshes for complex
geometries. The development of convergent elements, which are free of spurious numer¬
ical locking, are variationally consistent, achieve good accuracy with coarse meshes, and
satisfy stability and completeness requirements, is essential. Flexural super-convergence
in membrane deformation is also important for applications involving in-plane bending.
Moreover, the use of low-order interpolations is extremely desirable for their simplicity,
efficiency and amenability to contact implementations. To this end, we use the methods
of enhanced assumed strains (EAS) and assumed natural strains (ANS) judiciously to con¬
struct low-order elements possessing the above features for the analysis of multilayer shells.
In this work, we have implemented a number of dynamic time-stepping implicit/explicit
algorithms in the context of the present formulations for transient integration of the result¬
ing semi-discrete finite element equations. The time step-size for the implicit integration
can be much larger than that for the explicit integration. The explicit method, on the other
hand, needs much less computational effort at each time step since the matrix factoriza¬
tion is not needed. For elastodynamics, the introducing of numerical damping is essential

5
to increase the numerical stability of implicit integration methods, even for the energy-
momentum conserving algorithm.
The solution of discrete equations for problems involving large deformation and
long-term simulations can be accomplished with the Newton-Raphson scheme. The numer¬
ical efficiency of this approach is a byproduct of the asymptotically quadratic convergence
of its iterations. To maintain this rate of convergence, the exact linearization of discrete
equations is explicitly obtained and implemented in the present work. For nonlinear mate¬
rials, the consistent tangent moduli are crucial to be derived. For the quasi-static analysis of
unstable systems, arc-length method is used to find stability points and trace post-buckling
paths. Based on the above solution strategies, a large time or load increment is allowed to
use, while a good balance of accuracy and efficiency is maintained.
1.2. Chapter Overview
This dissertation is divided into six chapters. Two finite element models for multi¬
layer shell structures, the geometrically-exact sandwich shell element and the solid-shell
element, are formulated and implemented.
Chapter 2 presents the static analysis of the geometrically-exact sandwich shell el¬
ement formulation. The kinematic description and equilibrium equations of the sandwich
shell model are presented in Section 2.2. The corresponding weak form and linearization
are given in Section 2.3. Numerical examples for statics of sandwich shells are shown in
Section 2.4. This chapter has been published by Vu-Quoc, Deng and Tan [2000], Readers
refer to Vu-Quoc, Deng and Tan [2001] for the corresponding dynamic analysis.
In Chapter 3, we carry out the static analysis of the optimal solid-shell element for¬
mulation for multilayer composites. After a presentation of the kinematics assumption
and the formulation based on the Fraeijs de Veubeke-Hu-Washizu (FHW) variational prin¬
ciple (Felippa [2000]) in Section 3.2, we discuss the finite-element discretization and its
implementation in Section 3.3. A review of the EAS method together with our proposed
modification is presented in Section 3.4. We present the numerical results in Section 3.5.
This chapter will be published by Vu-Quoc and Tan [2002a],

6
Chapter 4 addresses the dynamic analysis of the optimal solid-shell element formu¬
lation for multilayer composites. We devoted Section 4.2 to the dynamic aspect and the use
of the energy-momentum algorithm for elastic materials. A variant of the EAS formulation
based on the deformation gradient (instead of Green-Lagrange strains) for solid shells is
the focus of Section 4.3. Numerical results are shown in Section 4.4. This chapter will be
published by Vu-Quoc and Tan [2002¿].
In Chapter 5, we present static and dynamic analyses of the multilayer solid-shell
element formulation for nonlinear materials at finite strain. Two nonlinear material models
(i.e., Mooney-Rivlin material and hyperelastoplastic material), and their implementations
are discussed in Section 5.2. The explicit integration method for solid-shell elements is
addressed in Section 5.3. Numerical simulations, which illustrate the performance of the
proposed element formulation, and exhibit both material and geometric nonlinearities in the
large-scale implicit/explicit analyses, are given in Section 5.4. This chapter was submitted
for the publication by Tan and Vu-Quoc [2002a],
Chapter 6 discusses the solid shell element for active piezoelectric shell structures
and its applications. In Section 6.2, we introduce the kinematics and variational formu¬
lation of the piezoelectric solid-shell element, and then present the composite solid-shell
element. The control design for structures with piezoelectric sensors and actuators is dis¬
cussed in Section 6.3. Numerical simulations that illustrate the performance of the pro¬
posed formulations, including comparisons with available experiment results and solutions
obtained from shell elements and solid elements, are given in Section 6.4. This chapter was
submitted for the publication by Tan and Vu-Quoc [2002¿].
Chapter 7 gives the closure of our work. Conclusions are drawn in Section 7.1 and
directions for future investigation are suggested in Section 7.2.

CHAPTER 2
GEOMETRICALLY-EXACT SANDWICH SHELL FORMULATION
2.1. Introduction
Sandwich structures have played an important role in several areas of engineering.
Many background references were cited by Vu-Quoc and Ebcioglu [1995], and are not re¬
peated in the present follow-up work, except for particularly relevant ones. We refer to
review papers such as Reddy [1989], Noor and Burton [1989], Noor [1990], Reddy and
Robbins [1994], and the references therein for various aspects on formulations for multi¬
layer structures. The accuracy of layerwise theory, as compared to single-layer theory with
a shear correction factor, was demonstrated amply by Reddy [1989], where a comparison
of transverse shear stress with 3-D elasticity solution was provided (Reddy [1993]). We de¬
scribe here a continuation of the results reported by Vu-Quoc, Ebcioglu and Deng [1997],
where the equations of motion for geometrically-exact sandwich shells are derived. Focus¬
ing on the static case in the present work, we develop a Galerkin projection of the resulting
nonlinear governing equations of equilibrium.
In the present formulation, each layer in a sandwich shell structure can have different
thickness and side lengths. As such, the present formulation can be used to model an
important class of multilayer structures with ply drop-offs. Another important application
of the present formulation is the modeling of shell structures with patches of constrained
viscoelastic materials and/or patches of piezoelectric materials. No restriction is imposed
on the magnitude of the displacement field, whose continuity across the layer interfaces
is exactly enforced. Finite rotations of the directors in each layer are allowed, with shear
deformation independently accounted for in each layer. The layer directors at a point in the
reference surface are connected to each other by universal joints, and form a chain of rigid
links. The overall deformation of a sandwich shell can be described by the deformation of
7

8
a reference layer. The unknown kinematic quantities are therefore the three displacement
components of the deformed reference surface, and the unit directors associated with the
layers.
The starting point for the development of the Galerkin projection of the governing
equations of equilibrium is a nonlinear weak form based on the stress power of a sandwich
shell, from which the expressions of fully nonlinear strain measures are obtained (Vu-
Quoc et al. [1997]). A linearization of this nonlinear weak form is performed for use in the
solution for the kinematic quantities via the Newton-Raphson method. Together with the
update of the inextensible directors, the linearization leads to a symmetric tangent stiffness
operator, which is composed of a geometric part and a material part. The consistency
in the linearization leads to a quadratic rate of asymptotic convergence in the Newton-
Raphson iterative solution. Linear finite element functions are chosen to form a basis for
the Galerkin projection of the linearized equilibrium equations into a finite-dimensional
subspace of trial solutions. The tangent stiffness matrix is symmetric, and is evaluated
using selectively reduced integration in all layers to avoid shear locking.
Several numerical examples, including the bending and torsion of a sandwich plate,
are presented to illustrate the salient features of the present formulation. In particular, the
important case of sandwich shells with ply drop-offs under large deformation is presented.
Results are compared with those obtained using the commercial nonlinear finite element
code ABAQUS [1995],
2.2. Virtual Powers
In this section, we summarize the kinematic description of the sandwich shell model
developed in Vu-Quoc et al. [1997], and the equilibrium equations in weighted resultant
form.2 The component form of the stress power and of the constitutive relation are also
given at the end of this section.
2 The word “weighted” here is used to indicate that the resultants are in “weighted” tensor form and not
in true tensor form (the reader is referred to Vu-Quoc et al. [1997] for an explanation).

9
2.2.1. Basic Kinematic Assumptions and Configurations
i
Let A designate the material surface of the shell, and H := U (ejH the total
f=-i
thickness of the sandwich shell (Figure 2.1), where is the thickness of layer (£). Let
: AxTi h-> be the mapping from the material configuration to the initial (reference)
configuration, where A x Tí = B is the material configuration of the sandwich shell. The
material domain for layer (£) is denoted by ^B such that
{t)B :=Ax {t)H , B:=\J{e)B. (2.1)
i=-i
e
Figure 2.1. Sandwich shell: Profile and geometric quantities.
Let $ : A x 7i >-*• Bt be the deformation map from the material configuration to
the current deformed (spatial) configuration. We use the notation £ := { £-, £3 } e B to
denote the coordinates of a material point, where £- := { £* , £2 } e A is referred to as the
material surface coordinates, and £3 G H the material through-the-thickness coordinate.
The deformation map for each of the three layers is written as follows
(£, t) := {e)ip (£-, t) + (£3 - (i)Z) (e)t , ior £ = -1,0,1,
(2.2)

10
where (¿)

• 1R3 is the deformation map of the centroidal surface of layer (£), and
(¿)t : A S2 the unit director (represent transverse fiber vector) associated with layer
(£), (i)Z the distance from the centroidal surface of layer (£) to the centroidal surface of
the reference layer (0) (for which (0)Z = 0). The centroidal surface of each layer {£),
which does not necessarily correspond to the geometric center of the cross-section of layer
{£) are defined as follows: Let (¿)p0 and y)pt be the mass density in the initial and the
spatial configuration, B0 and Bt, respectively. We select the centroidal surface ^) layer (£) such that
/ (í3 + (-i)Z) (-i)j0 (-i)P0d£?
(-i)W
/
£3 (o)jo (o)Pod^3
J (£3 + (-i)jt (-i)Ptd^3 = 0,
<-i>â„¢
/ ? (o)Jt(o)Ptde =0, (2.3)
(O)W (0)W
J (^3 - (i)^) (l)jo (i)pod^ = J (£3 - (i)Z) (i)jt (i)ptd^ = 0.
Wn 0)n
Using the assumption that the layer directors behave like a chain of rigid links connected to
each other by universal joints, the deformation maps of the centroidal surface of the outer
layers (1) and (—1) can be related to the deformation map of the centroidal surface of the
reference layer (0) as
(i)<£ (£“) 0 (o)<£ (£a, t) + (o)h+ (o)t + (i)h (i)t ,
(-i)¥> (£“, t) ■= (o)<¿> (£“, t) - (o)h~ (o)t - (_i)h+ (_i)t , (2.4)
where y)h+ and ^h~ are the distances from the centroidal surface of layer (£) to its top
surface and to its bottom surface, respectively. The deformation of the sandwich shell is
therefore described by four vector-valued mappings collectively denoted by
$ := { (o), (<)t, for £ = -1,0, l} . (2.5)
In terms of components, we have three components for (o)<£, and two components for
each inextensible director (*)t, thus leading to a total of nine components, which are the
principal kinematic unknowns to be solved for.

11
The deformation gradient mF for layer {€) is
F :=
GRAD
GRAD"1 w*0
(2.6)
The Jacobian determinants of the mapping (£)$0, and (¿)4> are given below
(e)jo ■= det [GRAD (¿) (t)jt •— det GRAD(¿)<£ (£, t)
(2.7)
(2.8)
2.2.2. Virtual Powers
Here we summarize the expressions for the power 7C of contact forces/couples and
for the power 7a of assigned forces/couples. Together these powers play a crucial role in
the derivation of the equations of motion (see Vu-Quoc et al. [1997] for more details). The
balance of the power of contact forces/couples and the power of assigned forces/couples
as expressed by 7C = CPa leads to the equation of equilibrium for geometrically-exact
sandwich shells (see Eq. (84) and Eq. (85) of Yu-Quoc et al. [1997]).
2.2.2.1. Power of contact forces/couples and conjugate strain measures
The set of convected basis vectors on the spatial (current) centroidal surface are de¬
noted by | (¿)a¿ j, where the underlined index i is to be expanded in the following sense
\W°i} :== {w°i» (r)a2 i (i)0>3 } := { Wí}|í3=0 » for ¿=-1,0,1.
(2.9)
The co-vectors j ^cJ- j, dual to the vectors j j, are defined by the standard orthogonal
relation
< (<)“J, (e)ai >= , (2.10)
where 6¡ is the Kronecker delta. The convected basis in the initial configuration is given
by specifying t = 0 in the spatial-centroidal-surface convected basis to obtain
{ mAf } := { > m A2, (t)Az } := { (f)ai, wa2, wa3 }|f=o ,
for i = -1,0,1.
(2.11)

12
The basis {(*)■A¿ }> dual to the initial-centroidal-surface convected basis | (e)A¿ j, is de¬
fined similar to Eq.(2.10).
The membrane strain ^e, the transverse shear strain , and the bending strain
measure , which (as we will see later) are conjugate to the effective resultant membrane
stress , the transverse shear ^qa , and the resultant couple (e)^a, respectively, can
be defined as follows
M€
:= {i)£ap (<)«“ ®
(2.12)
(*)S
:= {t)Sa (¿)“Q ,
(2.13)
(t)P
:= {e)Pap (¿)a° {t)a3.
(2.14)
The components of the membrane strain ^e, the transverse shear
(1)6a , and the
bending strain are given in the following relations, respectively,
(£)¿>a = {1)1 a - {1)1 a >
[í)Pa¡3 = {t)Ka0 — (/)««/? ,
(2.15)
where
(i)°a/3 := (¿)aa * (i)a/3 > {t)Aap '■= (t)Aa • (£)Ap , (2.16)
are the components of the Riemannian metric tensors of layer (£), (e)aQp ^aa (t)ap
and {t)Aap (e)Aa ® (i)A0, in the current configuration and in the initial configuration,
respectively (Naghdi [1972], Marsden and Hughes [1983]). The shear strain measures,
which are measures of how much the director and the director (¿)£0 depart from the
normal to the centroidal surfaces in the current configuration and in the initial configuration,
respectively, are defined as
{t)la = (<)«a * {i)t, (t)la = WAa * (i)*0 • (2.17)
Finally, the current and the initial (nonsymmetric) director metric (¿)Kap and (e)^°ap for
layer (£) are defined as
{¿)Ka0 (0°a * {H)t,0> (t)Ka0 {t)Aa * (e)to,P .
(2.18)

13
Using the above definitions, the stress power Tc of geometrically-exact sandwich shells
can now be written (Eq. (199) of Vu-Quoc et al. [1997]) as
e=-i
E / \\wnl3a 3 + W'W + «)?“ (07a
where {¿)ña^, (¿)raQ/3 and ^q01 are the components of the weighted effective membrane
force, the weighted resultant couple, and the weighted effective shear force, defined in
(169), (174), (175) of Vu-Quoc et al. [1997] respectively. See the footnote at the beginning
of this section for the meaning of the word “weighted.”
2.2.22. Power of assigned forces/couples
Let n* denote the distributed assigned force on the centroidal surface of the refer¬
ence layer (0), and (¿)m* the distributed assigned moment on layer ((). On the boundary
dA, we assume that the normal to the lateral surface of the shell domain in the material
configuration is such that
(/)*' = (o)*' = (o)^ Ea . (2.20)
The assigned force n*a and assigned couple [¿)fh*a on the boundary dA are then defined
such that
n* = n*a (0)ua , {l)m* = {i)m*a {0)uQ . (2.21)
The power of the assigned forces and couples is written as follows (Vu-Quoc et al. [1997,
(56)])
wm* • (/)£ | dA
ya = Í l n* • ú + ]T
a \ *=-'
+ I n*a • ú (0)i/Q + J2 {i)™a * (t)i {t)Va ] d(dA). (2.22)
dA \ <=-i /
2.2.3. Constitutive Relations
For layer (£), we employ the following constitutive relations between the above strain
measures and the mentioned effective resultant forces/couples
.1 (() E (i) H ^fforfS
Wn ~ 7 We7<5>
1 ~ (wu)
(2.23)

14
where
(¿)77r =
0a {f)Jt {t)E (l)H ^00^5
(l)P-f6
12 [l-((o^)'
«)?“ = (i)Jt (t)K, (flG «)ÍÍ (íj-4'"! (Í)í/J ,
(2.24)
(2.25)
(¿)íf := (f)^+ + (£)/i , (e)Aai3 := (¿)Aa • (¿jA^ , (2.26)
are the thickness of layer (£) of the sandwich shell, and the Riemannian metric tensor in
the initial configuration for layer (£) of the sandwich shell. In Eqs.(2.23) to (2.25), is
the Young’s modulus, the shear modulus, y)V the Poisson’s ratio, and (¿)Ks the shear
5
correction coefficient, for all layer (£). With ^k3 = -, Eq.(2.25) is the same as that given
in Naghdi [1972, p.587]. The elastic constant (¿)3 , with its component form given as
(oS** = mv (,A#" (i)/Ps + Í (l - wv) mA“ + (OA* WA<”) , (2.27)
is a fourth-order elasticity tensor.
2.3. Weak Form and Linearization
In this section, we construct the weak form of the equations of equilibrium obtained
in Vu-Quoc et al. [1997], and linearize this nonlinear weak form, which plays an important
role in finite element implementation.
The shell layers are assumed to be inextensible in the thickness direction, and there
is no drill degree of freedom (dof) for the directors (i.e., the directors are not rotating about
themselves) considered. We have three translational dofs for displacement of the centroidal
surface of the reference layer (0), and two rotational dofs for the directors of each layer.
For a sandwich shell, the total number of dofs is nine (9). For the single-layer case, we
refer the readers to Simo and Fox [1989] and Simo, Fox and Rifai [1990] for the details.
2.3.1. Admissible Variations, Tangent Spaces
The admissible variations to the deformation map = j (0)<¿?, (_i)t, (0)t, j,
are denoted by
6 := {, ó(-i)t, S(0)t, .
(2.28)

15
Let TS denote the tangent space formed by the Cartesian product of the tangent spaces
T t S2 (i.e., the tangent spaces to the sphere S2 at y)t € S2, for l = — 1,0,1). We write
i
TS := FT T f S2 = T f S2 x T t S2 x T * S2. (2.29)
11 (t)l (_1)I (0)*' (l)1 v '
t=-1
The space of admissible variations, denoted by Bt (i.e., the tangent space to the current
configuration Bt ), at the current deformation $, is then defined as
7$ Bt := i t-4 M3 x T S \ ó (0)<¿> = 0 on dtp A, S — 0 on d^ A| ,
(2.30)
where dip A and d(()tA represent the portions of the boundary dA where the essential
boundary condition is imposed on ip and on , respectively.
2.3.2. Weak Form of Equations of Equilibrium
The weak form of the equations of equilibrium for sandwich shells is readily provided
by the principle of virtual power, expressed by the following balance of power (Vu-Quoc
et al. [1997])
= ya, (2.3i)
where yc and Ta are the power of the contact forces/couples and the power of the as¬
signed forces/couples for the sandwich shell, respectively.
It suffices to replace the time rates in the expressions for the powers Tc and Ta by
the admissible variations ó $ to obtain the weak form, which can now be written as
Find $, such that
Gc() , (2.32)
for all admissible variations 5 $, where Gc ( $ , 8 ) is the weak form of the contact
forces/couples (or contact weak form for short), and Ga $) is the weak form of the
assigned forces/couples (or assigned weak form for short).

16
2.3.3. Contact Weak Form
The contact weak form of the contact power 7C in component form is as follows:
i ,rl
Gc( & ,5&) = Y J - (t)npa 6 {e)áQ0 + {l)mPa ti {e)ka/} + <5 (e)7a
f=-r
dA
= Y i [{t)ña0 ti {t)eap + {i)ma0 S wpaf) + S{e)Sa] dA , (2.33)
i=-l A
where {¿)ña0, (i)ma0 and y)0° are the components of the weighted effective membrane
force, the weighted resultant couple, and the weighted effective shear force, defined in Eqs.
(169), (174), (175) of Vu-Quoc et al. [1997] respectively, whereas 6 (e)€ap = (l/2)<5 (t)áap,
<5 (e)Pa/3 = <5 [i)kap, and 5 ^Sa = 6 (e)7a are respectively the variations of the strain mea¬
sures conjugate to the above weighted resultant tensors (Eq.(2.15)). The component form
of the contact weak form is used in the computational formulation due to the constitu¬
tive laws Eqs.(2.23) to (2.25) that relate the weighted resultant tensors to their respective
conjugate strains.
2.3.4. Assigned Weak Form
From the power iPa of the assigned forces/couples in Eq.(2.22), we obtain at once
the weak form of the assigned forces/couples
A V
n* ‘<5(0)93 + Y (e)m* •ó (¿)£ J dA
A V «—1
+
/
dA L
1
n*Q *<5(o)

e=-i
d(dA). (2.34)
Remark 2.1. For the dimension of the assigned forces/couples n* and ^rh*, we refer
the readers to Vu-Quoc et al. [1997]. At the boundary d A of the sandwich shell, n* and
(t)m* are decomposed as follows
n* = n*a va on dn A,
{e)m* = wro*a {e)ua on d{t)m A,
(2.35)
(2.36)

17
where dn A and d(e)m A are the portions of the boundary where the assigned forces and
the assigned couples are applied, respectively. I
2.3.5. Linearization of Contact Weak Form
To construct the linearization of the contact weak form Eq.(2.33) at a given configu¬
ration in the direction of an incremental tangent field:
A := (A (o) we consider a one-parameter family of perturbed configurations
£■ * * = ^(o)^£) (—i)^£) (o)^e ) (i)^e) ) (2.38)
such that
= A 3» . (2.39)
e=0
The tangent contact weak form will be shown to be composed of a material tangent stiffness
operator and a geometric tangent stiffness operator. The linearization of the contact weak
form plays a central role in the computational procedure based on the Newton-Raphson
method.
2.3.5.1. Update of inextensible directors
The following steps are used in the update of the inextensible layer directors.
1. First, we must account for the assumption that the layer directors have no drilling
dofs in their increments. The removal of the drilling dof is realized by nullifying the com¬
ponent A (¿)T3 of the material incremental directors A ^)T along the basis vector E 3,
that is
A {l)T = A wTa Ea, (2.40)
where Greek indices take values in {1, 2}. Let the superscript A; on a tensor quantity
denote the /cth iteration in the Newton-Raphson procedure. The spatial incremental director
A (,i)tk for layer (£) is related to its material counter part A ^Tk through the orthogonal
tensor ^ Ak as follows
A {e)tk = {e)Ak-A {t)Tk .
(2.41)

18
If we do not make the distinction between tensors and their matrices of components, and
the quantities in Eq.(2.41) in terms of matrices of component, then Eq.(2.41) can be written
as
A {l)tk = wAfcA {t)Tk ,
(2.42)
where the matrix mAk G R3*2 is formed by the first two columns of the matrix mAk G
p3x3
2. The spatial director is updated as follows
(i)tk+1 := exp tk A {i)tk — cos11 A {e)tk || (t)tk +
sinll A
hpA^t*. (2.43)
«*' || Awtk ||
3. The incremental director rotation matrix is obtained from the exponential map
A(*)Afc :=exp50(3)
W
e
= cos11 (t)0 || 1 + sin11 (¿)0 II é + [l - cos (|| w0 ||) e ® e , (2.44)
where 1 is the identity tensor, ^)0 and é are skew-symmetric tensors with ^0 and e
as their associated axial vectors respectively,
,0
(¿)0 := (t)tk x A , e :=
4, Update the rotation matrix of layer (£)
(*)
0
(2.45)
Al+1 =AWA*WA*
(2.46)
The above procedure is very important for the linearization of the weak form.
2.3.5.2. Perturbed configuration
Let A := {A(o)<£, A (_i)t, A(0)t, A^jij 6 T S be the incremental field
in the tangent space at the current configuration . The perturbed configuration along the
increments in the tangent space is defined as follows
| (0)iPe > (—1)^£ > (0)^e i (l)^e } )
(2.47)

19
with
(O)V’e := (O)V5 +£A(0)¥>,
:= expw*
eA(£)t
, for ¿ = —1,0,1 ,
(2.48)
(2.49)
where
exiV
eA {l)t
. ,, sin 11 eAmt || .
:=cos|| eAwt ll«>*+ || eA(()t || (')* •
(2.50)
is the exponential map from T ^ S2 to S2 (see Simo and Fox [1989] for details). We now
verify that Eqs.(2.48) and (2.49) satisfy (2.39). First, it is obvious from (2.48) and (2.49)
that
(0)Â¥>e
e=0
= (o)f
=o = for i = —1, 0,1 .
(2.51)
(2.52)
Second, by taking the directional derivative of (2.48) and (2.49), we obtain
d
de
(0)Vs
= A (o)i^ ,
(2.53)
d
£=0
£=0
-II A(t)t l|sin(|| eA {t)t ||) {e)t + || A {e)t ||cos (|| eA (l)t ||)
= A {t)t for t = —1,0,1.
£=0
(2.54)
Next, to linearize the strain measures, the following formulas are useful
= A (f)i a
d
de (0)V?e’°
— A (0)^,0 >
£=0
d_
de
(2.55)
£=0
With the above results, we can now proceed to the linearization of the strain measures
followed by the linearization of the contact weak form.
Remark 2.2. The directional derivative of 6 ^ te along the direction of the increment
A (¿)f can be expressed in terms of the variation S (¿)t and the director y)t as follows. It

20
is clearer if one thinks of the symbol 6 in the variation S ^tE as a derivative with respect to
some variable, whereas the perturbation parameter £ is a different variable. Since || y)t ||=
1, it follows that (¿)£ • y)t = 0, and we can define := y)t x such that
(e)t = (i)w x (i)t ) or 6 (e)t = <5 (£)& x (e)t , (2.56)
where wu; and 5 ^6 are vectors that play the same role, one for the time derivative, while
the other for the variation of ^)t. For the perturbed director y)tE, we have
5 (¿)tE =6^6 x y)tE ,
(2.57)
with the same rate 5 ^)G as for ^t.
Hence
e=0
5we x ¿“W4*
= 5(£)0 x A (f)t
(ey
e=0
= - (5(¿)t *A(¿)t) (/)t, for £=-1,0,1.
(2.58)
Since
5 (i)& — (i)t x 6(t)t ,
which is a result of (2.56)2 and || ||= 1, and since ^)t • A (/)£ = 0.
(2.59)
I
2.3.5.3. Linearized strain measures
Let ((t)a£ap , (e)lap > (i)Kap ) be the strain measures corresponding to the perturbed
configuration (2.47), as defined in (2.16)—(2.17), and let the incremental strain measures be
defined as
(A waap , A (f)7Q/3, A WKa/3) := — ((e)a£a0, (¿)7«/3. (i)Kap )
£=0
for £=-1,0,1. (2.60)
We obtain the following expressions
A (t)aap : = * (l)V,p + (f)V5a *A ,
(2.61)

21
A (O 7a := (a {()t • (*)¥>>Q + wt • A (¿)v>,a ) , (2.62)
A (t)K,ap := (A(f)(9 + (^./A^), for ¿=-1,0,1. (2.63)
For ¿ = 1 and ¿ = -1 (i.e., the top layer and the bottom layer), we want to express the
incremental strains A A(¿)7a, A(¿)KQj3 in terms of the deformation map $ :=
{(o)^, (-1 )t, (o)t, (ijt } and its increment A $ :={A(0)V?, A(_i)t, A(0)t, A(1)t}.
This objective can be achieved by employing the constraints (2.4) in (2.63), and then we
obtain
A (1)Gq/J = (A (0)^,0 * (())Â¥>,0 + (0) + (0)/l+ (a (0),0 * A (0)i,Q )
+ (1)^~ (A(0)¥>iQ • + A (!)t ^ • (0)V?,a + A(0)(^ijS* (I)t,a + (0) V3,/? * A (i)t.Q )
+ (o)^+ (i)h (A(o)t,Q • + (o)i,a *A(i)tj + A(i)tiQ. • (o)t,p
+ (l)¿,a • A (0)i,j9 ) + ((0)^+) (A(0)t,a * (0)*,/? + (0)¿,a * A (o)f ,0 )
+ ((1)^ ) (A(i)t,a • (i)i,/3 + (i)t,Q •A(1)ti/3 j , (2.64)
A(1)7q = A(0)V5iQ • (0)t + (0)^,0 *A(0)i + (0)^+ (A(0)t,a * (1)*
+ (o)£,a *A(i)t) + (o)h- (A(i)ta • (1)4 + (1)4,0, *A(1)i) , (2.65)
A(1 )Ka0 = A(0)V?,a * (l)t,0 + (0)¥>,a *A(i)í,0 + (0)^+ (A (0)t a * (i)í,0
+ (0)í,a *A(i)ti(gj + (1 )h (A(1)£q • (i)£,0 + (l)í,a *A (i)í,^) . (2.66)
For layer (—1), we obtain
A(_i)aQ/3 = A(o)V?iQ * (o)V>,/8 + (o)V>,a *A(o)Â¥>,0
- (o)h~ (A (0)¥>iÉt • (0)t,p + A (0)t, 0 • (0)V>,a + A (O)<¿>,0 • (0)*, a + (O)<£,0 • A (0)t,a )
- (-i)/r+ (a (0)V>,a * (-1)^ + A (-1 )tji * (0)V>,a + A (o)Â¥>,0 * (_i)t,a

22
Remark 2.3. The above (2.63)-(2.68) are for the incremental strain measures for sandwich
shell. To obtain the variation of these strain measures, we simply use the same relations
with A replaced by <5.
I
2.3.5.4. Linearized contact weak form
We now derive the linearization of the contact weak form, which requires the lin¬
earization of the resultant contact forces/couples. Substituting the one-parameter family
of the perturbed configuration (2.47) into the static weak form (2.32), and then taking the
directional derivative, we obtain
X>GC($,<$$)• A$ := ^-Gc{$e,0$)
ae
(2.70)
The complete linearization of the contact weak form Gc( , 6 ) can be divided into two
parts, the material part and the geometric part. We will discuss these two parts in detail in
this section. To make the derivation simpler, we express the contact weak form in matrix-
operator format.
2.3.6. Matrix-Operator Format of Contact Weak Form
Let the material membrane force, shear force, and moment for layer (£) be defined as
follows

23
(â– f)Q := ((VO1 > WQ2) >
WJo v '
(2.71)
(t)
M
:= (wm11, wm22, wm12)t ,
(¿)7o
where (¿)j0 is the Jacobian determinant in the material configuration evaluated at the cen-
troidal surface of layer {£).
We also define the director rotation matrix for layer (£) as follows
(A
A :== [(£)*! i [l)t2 , (¿fa
and let
(A
A :=
(/)^i> (e)h
(¿)Aii (f)A12 (<)Ai3
(£)A2i (¿) A22 (^)A23
(í) A31 (¿)A32 (£)A33
(f) An (í)Ai2
(¿)A2i (f)A22
(flAsi (€)A32 j 3x2
(2.72)
3x3
(2.73)
which simply represents the first two columns in . From here on, we will not main¬
tain a rigorous difference in notation between tensors and the matrices of their components.
Thus, bold-face symbols are also used to designate the matrices of components of tensors
with respect to the spatial basis { e\, e2 , e3 }. With this understanding in mind, the ma¬
trices of components of the deformation map of the reference layer (0) and of the director
for layer (£) are written as follows
(2.74)
3x1
(o)^1
' (A*1 '
(o)Â¥> :=
(o)2
» (A4 :=
00*2
. (o)^3 .
3x1
L w*3 J
while the variation and the increment of are written as
’ A (o)^1 ‘
<5(o)<¿> :=
, A(0)ip :=
A (o ) . 5(o)V?3 .
3x1
. A(o)^3 .
(2.75)
3x1
For the variation and the increment of the layer director , we need to account for the
no-drilling dofs condition. The matrices of components of 6 and A are
1
T-H
-+o
1
A {i)tl
II
-to
II
<
A (¿)f2
.5 w¿3 .
3x1
.A w¿3 .
(2.76)
3x1

24
The material counterparts of 8 ^)t and A y)t are respectively 8 and A (¿)T, and are
related to 8 and A by
6(t)t = (g)A5(e)T, A(¿)í = (ijAA^r . (2.77)
The no-drilling-dof condition imposed on 6{e)T and A is written as follows
8 {t)T -E3 = 0, and A {l)T • E3 = 0 , (2.78)
and thus if the matrices of components of 8 and of A ^T (with respect to the material
basis { Ei }) are defined as
ÓWT =
V1 '
S«)T2 J
AWT :=
A {i)Tl '
A WT2 \
(2.79)
then
S{g)t = (*)A 8(t)T, A (f)t = (<)AA(i)T .
(2.80)
Also
<5$ :=
’ A (o)¥>
1
9-
§
<1
1
, A$ :=
A(-i)T
5(0)T
A(0)T
LA(1)T J
9x1
[A(1)r J
(2.81)
We now will obtain the operator expression of the weak form for each of the three layers
Since layer (0) is the reference layer, to which the two outer layers are referred to, we begin
with layer (0). From the membrane part of (2.33), the expression for 8 (0)aap similar to that
of A (o)Oq/j in (2.63), and using the symmetry of the membrane
1 ~a/3 r 1 -11 X I 1 —22 r
2(0)^ ^(0)Oq/3 — ^ (°)n ®(0)all +2^n ®(°)a22
+ 2 ((°)^12<^(°)ai2 + (°)^21 A (0)021 )
forces (0)ñQ^, we obtain
(0)ñU +
T ^ c \ —22
'.2-q^6(0)^> I (0)U
^-<5(0)^ + (0)V>5 ^r¿(0)V> J (0)ñ12 . (2.82)
+
T d
(o)V».i

25
Introducing the following operator for the membrane action in layer (0)
t d
(o)V’i
7^ ^ n rn • ”
(0 )-Dmm
we can then rewrite (2.82) as
1
(0)^,2
ae
d
t d t d
(0)V>,1 ^2 + (0)^,2
(2.83)
3x3
2 (0)« 3 (0)Qa/3 (0)7 0 (O)-^mm^(O)1^
(0)
AT
(2.84)
Similarly, we introduce the following operators related to the bending and shear actions in
layer (0):
d
B lnn - —
(0 )-°6m
{0)t’1 af1
*
it,
(0)C,2 ac2
W*.! + (0)É,2 ^T
(2.85)
3x3
Bbh :=
(0)^>66
(0)^,1
5
ae1
a
(°)^,2 ^£2
t a t a
(0)^.1 ^ + (0)^,2 g^T
a
(2.86)
3x3
(0)-L* sm
(0)tl
(0 )**
ae1
a
ae2
:=
(0)-°s6
(o)v>;i
. (0)V>f2 .
(2.87)
2x3
2x3
Then following the similar procedure as described in (2.84), from (2.33), (2.63), (2.62), we
obtain the operator format for the shear part and for the bending part of the weak form as
follows
(0)9° 6 (o)7a = (o).7o (o)Bsmd(o)1? + (,o)Bsb5{o)t (o)Q , (2.88)
(o)ñaf3 6 {o)Ka¡) = (o)70 + (o)-Bfc6^(o)i] (o)M . (2.89)

26
The contact weak form (2.33) for layer (0) can now be written as
(0)GC( J | [(0)-®mm ^(O)1^ (0)N + (0)Bsm d (Q'jip + (0)Bsb & (0)t (0)Q
+ + {o)Bbb5(o)t (o )M 1 (o)jodA.
(2.90)
Remark 2.4, We refer to Simo and Fox [1989, eq. (6.25)], which is an expression similar
to (2.90). I
To obtain a simple representation for all three layers, we define the following gener¬
alized resultant force for layer (£)
R :=
WN
wSL
{e)M
(2.91)
8x1
Recalling the relationship between 5 y)t and S (e)T as given in (2.77), we combine the
differential operators for membrane strain (2.83), for shear strain (2.86), and for curvatures
(2.86), all for layer (0), into
(o)B :=
where A is the director rotation matrix for all layers defined as
I3 _
(-i)A _
(o)A
0)A J 12x9
(0)^mm
(0 )Bsm
03x3
02x3
03x3
(0)Bsb
03x3
02x3
A ,
(2.92)
(0 )Bbm
03x3
(0 )Bbb
03x3
8x12
A :=
(2.93)
Then, the contact weak form (2.90) for layer (0) can be written concisely as
(0)GC(&,6&):=J (ojBSQ • (o)i? (o)j0dA . (2.94)
.4
For the membrane part of (2.33), the expression for 8 (_i)aQ/J similar to that of A (-i)aap
in (2.67), and using the symmetry of (-i)ña(}, we obtain
1 —• OlQ r 1 —. 11 ^ 1 22 C
2 (-!)n ° [-l)aa/3 — 2 (-i)n O (_!)On + - (_1)W <5 (-1)022

+ 2 ( (—l)^12 3 (-1)^12 + (-l)rc21 ¿ (-1)^21 )
27
= (-i)ñ11 <5(o)V>.i * (-i)V,i ~ (~i)^+ <5(-i)*,i * (-1)^,1 “ (o)h S(o)t,i * (-i)V’.i
+ (_l)n22 ¿(0)^,2 * (-!)¥>,2 “ (-l)^+ ¿(-1)*,2 * (-!)¥>,2 - (0)/l ^ (0)^,2 * (-1) + (-l)n12 5 (0)^,1 * (-1)^,2 + ^ (0)^,2 * (-1)^,1 ~ (-l)^+ $ (-1)^,1 * (-1)^,2
-(_i)h+ ¿(_i)t2 • (-!)¥>,! - (0)fr ¿(0)*, 1 * (-1)^,2 “ (O)/* ¿(0)*,2 * (-1)¥>,1
(2.95)
Upon introducing the following operator (_i)Smemb associated with the membrane action
in layer (—1)
(—l)-®memb :=
t 9
dt1
,+ t 9 '
(~1)« (-1)^,1
, _ t 9
(o)h (-DV’.i d£i
0ix3
t 9
(-*)¥>,2 d£2
,+ t 9
(-1 )k (-1)^,2
t 9
(0)« (-1)^,2 q £2
0lx3
A ,
(—!)ni
~(-i)h+ (_i)n
L
“(o)h~ (-i)n
0iX3
05x3
05x3
05x3
05x3
(2.96)
where the operator (-1)11! is defined as
d d
(-1)11! := (_i)V?*2 + (-i)V>fi -Qp , (2.97)
we can rewrite the membrane part for layer (-1) in (2.33) in a compact format as follows
2 (—1)^ ^ (—1)®q/3 (—l)-®memb^^ * (—1)-R • (2.98)
From (2.68), we define the differential operator associated with the shear action in layer
(—1) as follows
(— l)-®shear • —

28
O
3x3
(-l)t
(-1 )tl
t d
de
d
de
o
3x3
03x3
(-d^£i - (-i)^+ (-i)£t ¿^r
(-1)V>?2 “ (~Vh+ (-i)*
t d
de
o
3x3
03x3 03x3
“(O )h (-1)£
9e
(O )h (-1)£Í
d
de
O
1x3
O
1x3
03x3 03x3
(2.99)
Similarly, from (2.69), we define the differential operator associated with the bending ac¬
tion in layer (—1) as follows
(—1 )^bend -=
05x3
t 9
(_l)í’1 Je
tt 9
(—1)£,2
0
5x3
(-i)n2
d
â– de
(-1)Ü3
d
de
(0 )h (-i)iji
05x3 05x3
d
“(0 )h (-1)^2
9e
d
0
1x3
0
1x3
9 tt 9
(—1) I14
de
— (o)h (_i)n4] olx3
A ,
(2.100)
where the operators (_i)Il/, for I = 2,3,4, are defined as
(-i)n2 := (-i)¥>£i - (-i)^+ (-1)^,1 .
(_i)II3 := (-1)^2 — (-i)^+ (—1)^,2 > (2.101)
n - j.t 9 t d
(-t)n4 — (-l)t 2 gp + (-1)* i ^5* •
We can easily verify that the shear part and the bending part for layer (—1) in the weak
form can be written in a compact format as follows
(-1)9 ^(—l)Ta = ( — l)-^shear & ^ * (—1)R > (2.102)

29
(_l)ma/3¿ (-ij/Co/j = (_i)-Bbend<5$ * (-l)-R
(2.103)
Now let the combined differential operator for layer (—1) be
(—1)B (—l)^memb ~b (—l)-®shear "h ( — l)-^bend
(2.104)
From (2.96), (2.99), and (2.100), we obtain
(~i)B =
d
(_1)V>'1 d?
t d
d£2
, + t d
a
(-i)í
(-i)ni
* ^
(-i )t
ae1
ae
ae
- (_X)/i+(-l)IIi
a
(-1)^1 - (-l)h+ Qg¡
3
- (o)h~ (-1)^1 0ix3
3
- (0)h~ (-1)^*2 ^7 °1X3
— (0)^ (-l)Hl 0ix3
3
- (0)h~ (-i)t1 0ix3
(-1)^2 - (~\)h+ (-1)*
* ^
(_1)t’1 d£}
d?
ae
a
(0)h (-i)t
ae
t d
ae
(_1)Il2a^
a
(-i)n3
(-1)114
a
a
- (0)h (_ijt;2
i ^
ae
(-i)n2ae ^3ae
0
1x3
3
{0)h~ (-ijtfj 0ix3
0
1x3
— (o)^ (-i)H4 0jx3
(2.105)
We thus obtain the following compact expression
1
2 (-i)nQ/3^(-i)aQ/3 + (-i)e^(-i)7a + (_i)mQ/3(5(_1)«:ct/3 = (_i)J3 and the contact weak form of layer (—1) as
(-i)Gc ($, 6 $) = J (_i)BS $ • (_x)R (_x)J0 dA .
(2.107)

30
For the top layer (1), similar to the definition of (-i)-B for layer (-1), we define
(i)B :=
f\
(D^i -qJT °1x3 (°)^+ ^7T
d
d
t 5
(i)^ («^l ^r
(1)^2 ^2" °lx3 (0)/l+ (1)^2 (1)A (1)^2
a
d£2
(l)nl
0lx3
(o )h+ (i)IIi
Wh
t d
{1)t d?
0lx3
A
mh+ o )*'aii
ft 9
{1)t d?
0ix3
A
(0)/i+ (1)7
9
(1)í)1 dt1
0lx3
a
(o)" (0*1 ¿)¿i
(l)n
tt 9
(1)í’2 di2
0ix3
.+ ,i 9
(^,2^2
(l)11
d)n6
0ix3
(0)^+ (1)116
3
(i)i-*-5 q +(i) n4
(l)n2
(l)n3
a
a
‘d£2
where the operator (1)11/, for 7=1,6, are defined as follows
TT — 9 , L,i 5
(l)11! — (1)^,2 + (1)^.1 >
A
(i)n2 := (i)V»fi + (i)^- (0^ »
d
(l)n3 := (l)V*f2 + (l)^- (l)4* .
(l)n4 := (!)<£*! + (i)/l" (i)t,i ,
0)n5 := (1)^*2 + (1)^ (1)7,2 ,
rr — ft 9 , *t ^
{1)II6 (1)t 2 — + (1)4,! ■
(2.108)
(2.109)

31
The contact weak form of layer (1) is then
(i )GC(*&,6&) = J (i)B<5$ • (i)J2 (i)j0dA. (2.110)
A
2.3.6.1. Material tangent operator
The material part of the tangent stiffness operator, denoted by DmG‘ A 3>, arises as
a result of linearizing the resultant forces/couples at a fixed configuration. We now treat
each layer separately, as we did for sandwich beams in Vu-Quoc and Deng [1995], Here,
we only consider hyperelastic materials. Let ip be the energy function of the shell. We have
the following constitutive relation (Simo and Fox [1989])
~ _ dip _ dip __ _ dtp
= = m~pa^ â–  ai10
For each layer, we have
VWR-A& =wCwBAf , (2.112)
where the tensor of elastic moduli is given below

32
CO
(N
CO
CN
CO
¡6
CO
"$â–  co
% 2
05
9-
CN
CO
-$-
CN
CO
i^co"
CN X,
CO S
a
co"
-$-
CN
CO
a
C'
co"
% 2
co"
co
CN _
CO -
â– $- co
CO CN
lO
coT
CN
CO
CO
N
CO
CO
CN
CN
CN
CN
a
CN
a
S
co"
CN
c6"
1^-
CN
co"
•-©
CO
CN
K5
CO
a
'W
c6"
coT
co"
â– 5H
CN
CO
a
C“
co"
CN
CN
co"
-S>-
CN
CO
co
CN
a
&
CN
CO
CO
•$- co
CN r\i
•5-
CN
CO
CN
CO
CN
CO
CO
CO
'S'co'
CN Z.
CO 2
a
«o
c6"
-S>-
CN
CO
CO
CN
CN
«V
co"
CN
CO
co"
iH'
CN
CO
CN CN
co
CN CN
cr
â– 5-
CN
CO
CN
CO
â– ^jco
CN
CO
co"
co
CN
CO
a
«w
co"
co
CN
40 D,
«V
co"
^.co
CN r->
<*> £
%>
CO
.^.co
CN CN
•3-
CN
CO
co
CN
CO
I*-
CN
CO
^-co
CN
CO
a
co"
^|co
CN
CO
co
co
CN
tO
V»
CN
to
«V
CN
<0
CN
to
CN
(0
CN
tO
CN
to
CO
9
CN
CO
•9-
CN
co"
CN
CO
CN
w
CO
1*-
CN
co"
CN
CO
-9
CN
(0
CO
CO
CN
CN
<0
co"
CO
CN
U)
«V»
co"
CO
co"
CO
CN
to
co"
CO
'V
c6"
CO
CN
CN
co"
CO
co
CN
a
co"
-3-
CN
CO
i*-
CN
CO
CN
CO
CO
-3-
CN
CO
CN
CO
CO
iH
CN
CO
CO
S
co"
iN
CN
CO
CO
CN
CN
a
<6"
CN
CO
CO
CN
co"
u
-
-
£
1-H
to
to
to
to
to
9
CN
ciT ^
CN
'W
co" ^
X, CN
co
CN
to
2
CO CN
to
s ^
CO CN
« «?
CO
£ <0
CN (TN
CN VJ
2 CO
9; co
CN CO
z CO
CN ^
to
to
to
«-o
*0
co"
co"
CO
c6"
co"
co"
co"
o
IQ.
CO

33
For hyperelastic materials, the elastic moduli for the membrane, shear, and bending
actions for layer (£) are given as in (2.23)—(2.25). The matrix of the (tangent) elastic moduli
in this case takes the following form
(¿)Cm 03x2 03x3
(£)CS 02x3
(0
C :=
sym.
WCb J
(2.114)
8x8
with
Cm :=
(€)W
1
(flC“
&
1
«A12
mQ3'
(1)C
(0 , (£)Cfc :=
(flCf
(flCf
sym.
o
3x3
sym.
wCf.
3x3
C< :=
(t)^s
(flC,11 mcp
L«)C,12 (<)Cf
(2.115)
2x2
The coefficients of the above matrices are given below, for the membrane action:
nil __ WE WH All All
(0°m - 7 72 (t)A (t)A >
W(
yOl
(t)Cn
r> 22
WGm =
(0
C" =
H
wu)
(t)H
W*')2
(Off
>-(
(0^
l-(
W*)2
A11 mA12
2 W
4 22 4 22
\A (f\A
(2.116)
1 — (wI/)
,„r'33 - w-51 (i)#
[Wu)
2 (O'1 (02
1 - (O*'
4 11 4 22 .
(i)A {l)A +
22 1 + W" 412 412
a i2 41
(024 (024

34
For the bending action:
SI
II
(t)E (t)H3
all all
â–  (Va (Va >
12(1- K)"2)
SI
to
it
(VE (VH3
■ [(I)” (l)A (l)
12(1- mA
r> 13
(e)Gb -
(t)E (i)H3
. ,^An ,^A12
12(1- m"2)
(t)H (Va >
r< 22
(t)cb =
(t)E {t)Hd
12(1- (fl*2)
(ty1 (Va >
r< 23
(i)Gb =
(i)E (t)H3
. ... a22 a 12
12 (l - ^ )
(ty1 (vA )
r< 33
mOi =
(i)E (t)H3
. i1-.1'’" «4
112
(2.117)
11 A22-h
(Va w
1 + [l)V
A12 4 12
(Va (Va
For the shearing action:
(e)G¡1 = (VKs (t)G (t)H (i)^u ,
(i)C]2 = (£)ks (t)G (i)H {t)A12,
)C22 = (f\G (t\H (f\A22 .
(t)^a = (t)Ks (l)^ (t)” (ty
The tangent material stiffness operator for layer (£) is thus
E>m (i)Gc (, S ) • A 4» = J (t)B 5 *(¿)C (i)B A $ j dA
(2.118)
(2.119)
2.3.6.2. Geometric tangent operator
The geometric part of the tangent stiffness operator, denoted by DgG* A $, arises
from the linearization of the geometric part of the contact weak form, while keeping the
material resultant forces/couples constant. We now treat each of the three layers separately,
as we did in dealing with the material stiffness operator.
Remark 2.5. It is noted that while the principal kinematic unknowns are and
(t)t, for £ = —1,0,1, the computational kinematic unknowns are (0)
35
-1,0,1, where ^6 represent the rotation vectors that rotates E3 to y)t at the current
state at time t. It is important to note that ^6 does not represent the time history of the
motion of the director (/)t, but only relates the directors between the material configuration
and the current configuration.
In the linearization procedures, the primary variables (i.e., the variations to be held
constant in the linearization process) are S y)G instead of á ^t, which we will explain in
Remark 2.6. We recall that
|| (¿)t ||= 1 6(i)0 x á(¿)£, or 5y)t —5^)6 x ^t . (2.120)
we thus obtain the increment of the ó ^)t from (2.58) as
A (£)í ) = S (¿jé? xA(f)í = ( (¿)£ x 5 (£)i ) xA^jt = — (A mS y)t) ^t , (2.121)
and also
[A (<5(*)*)] q = - [(A(e)t •S(e)t) (i)t]>a
= - '8(/)£) (f)í - (A(¿)£ •6(t)ti0i') (e)t - (A(¿)£,a • (2.122)
I
Remark 2.6. From (4.19) and (4.12b) of Simo and Fox [1989], (84)3 and (156) of
Vu-Quoc et al. [1997], the equation of balance of angular momentum for a single-layer
shell is
Jt
(2.123)
Alternatively, it also can be written in the following form
(2.124)
Since || t ||= 1, we differentiate it twice to obtain
t • t = 1 =>■ t • t = 0 => t • t + t • t = 0.
(2.125)

36
Thus, í 1 i,let w be define as
u> := t x t . (2.126)
We obtain
w = í x í + i x t = t x t . (2.127)
Since
(w x t) ’St = Ú), t, 61 = ¿o • (t x St) — ¿J •S 9 ,
(2.128)
u) x t = (t x x t = (t • t) i — • t^j t = t + || t ||2 t , (2.129)
and also since t m6t = 0, we obtain
x t^j’6t = t *6t , (2.130)
so that
7pt'5t = 7pú'60 . (2.131)
In the above equation, Ipuj corresponds to the inertia term on the right hand side of
(2.123) (i.e., the balance of angular momentum in terms of the weighted resultant moment
m"), whereas I pCj corresponds to the inertia term of (2.124) (as a result of (2.127)) (i.e.,
the balance of angular momentum in terms of the physical resultant moment ma). We
now point out that the reason to use 5 9 , instead of 51, as primary variable: The use 5 9
is more convenient. Note that
V ^ ¿J x t . (2.132)
Since t • t = 0 and (2.126), we obtain
t — U) x t
(2.133)
U) x t = U) x(co x t) = (u> • t) OJ — (u) • uf) t = -|| u> II2 1 -t , (2.134)

37
Making the derivative of t = u x t, we obtain
•• • •
t = U X t + ui X t
¿J X £ — || LJ ||2 1 * t .
(2.135)
I
For the reference layer (0), in (2.94), we hold the resultant forces/couples (0)B fixed,
and linearize the geometric part, that is, we are finding the expression
[P((0)B<5$) • A
(o)
R
(2.136)
in operator form, where the operator (0)B was given in (2.92). From (2.92) and (2.81),
(2.136) can be expressed as
V *A$ • (o)R = [(o)Bmm£(o)V> * A (0) $ } • (0)N
[i]
+ {p + (o)Bsb5(o)T *A(0)^}*(o)Q
[2]
+ [ (o)Bbm <5 (0)V> + (o)Bbb <5 (o)T ] * A (o)$ } * (o)M, (2.137)
[3]
where the differential operator (0)Bm , (0)Bsm , (0)Bbm , (0)Bsb , and (0)Bbb were given
in (2.83), (2.86), and (2.87), respectively. We now proceed to give a detailed expression for
part [1] in (2.137). From (2.83), we obtain
1
i-H
•
V
o'
<1
i
t
— 11
(op
[1] =
A(0)^‘i * <5(0)95,2
• <
(0 p22
. A(0)^1 *<5(0)95,2 + A (0)9^4 *<5(0)95,1 .
. (op12 .
= [A(0)^!i *<5(o)¥>,i (op11 +A(0)^1 •S(0) + (A(o)V>4 (0)V>,2 + A(o)<^i •<5(o)V>,i) (op12. .
(2.138)

38
We define the geometric differential operator for layer (0) as
I3"
(0)
T :=
lde
03x3
03x3
03x3
d
1d£2
03x3
03x3
03x3
03x3
03x3
I3
03x3
03x3
03x3
1, 9
03x3
03x3
03x3
l3ae2
03x3
A 12x9 ,
(2.139)
15x12
and the tangent geometric moduli for the membrane action in layer (£), for £ = — 1,0,1, as
(2.140)
Kln :=
(O^G
(^n11 13 (¿)n1213 03x9
[í)ñu 13 (e)ñ22 1 3 03 x9
O9X3 09x3 09x9
It is easy to verify that
[1] = [(o)Bm5(0)

Now, for Part [2] in (2.137), from (2.86), we obtain
[2] = [(o)-Bsm <5 (o)

— [A(o)^í •Ó(0) ~ ((o)V>fi (o)t) (<5(o)t 'A(o)i) (o)Q1 + A(o)V?*2 '¿(o)t (o)<72
~ ((0)^2 (0)*) (¿(0)i *A(0)t) (0)72] • (2.142)
Remark 2.1. The matrix (0) T in (2.139) has five rows of submatrices and four columns
of submatrices. The four columns of submatrices correspond to (o)V5 > (-i)¿ > (o)t, (i)t ac¬
cording to the ordering in ó 3> (See (2.81)), and 5 wt = (e)A 8 {l)T (See (2.80)). I

39
Let the tangent geometric moduli for the shearing action in layer (£), for t — — 1,0,1, be
defined as follows
K12 :=
03x3
03x3
(e)Q} I3
O3X6
03x3
03x3
(í)T2 I3
O3X6
ml is
(erf213
- (e)q° (¿)7a Is
O3X6
06x3
06x3
06x3
06x6
It can be verified that (2.142) can be written as
[2] = {P [(o)Bsm5(0)ip + (o)Bsb6(0)t • A(0)4> } • (o)Q
= (o)T <5$ • {o)KXq {o)T .
For part [3] in (2.137), from (2.87), we obtain
[3] — jx> [ (0)^6rn ¿ (O)V3 +
(o)Bbb6 (o)t
' A (0)$ | • (o)M
(2.143)
(2.144)
= [(A(o)*íi *¿(0)^,1 + A(o)<^i (o)*,i + (0)¥>¡i *A¿(o)í i) (o)m11
+ (A (0)^*2 (0)^,2 + A (0)V^2 (0)^,2 + (0)^2 *A^ (0)^,2) (0)â„¢22
+ (A (0)^1 *^(0)^,2 + A(0)^2 *^(0)^,1 + A (o)^*! *¿(o)í,2
+a(0)93*2 *<5(0)t,i + (0)V»fi *A<^ (0)^,2 + (0)V»f2 *A^(o)A 1) (o)™12] • (2.145)
Let the tangent geometric moduli for the bending action in layer (¿), for Í = —1,0,1, be
defined as follows
03x3
03x3
03x3
(t)rñn 13
{i)rnu 13
03x3
03x3
03x3
â– {e)mu 13
(()m22 13
03x3
03x3
- (i)rna0 (t)Ka0 I3
- [t)rñla {i)7q I3
- {e)m2a (t)'Ya I3
(fjm1113
wmn 13
- (l)Tñla (¿)7Q 13
03x3
03x3
{e)mn 13
wm2213
- {i)â„¢?a (<)7a I3
03x3
03x3
(2.146)
Using Remark (2.5), we can verify that
[3] = |l> [(o)-06m¿(O)<¿> + (0)#66<5(0)t] *A(0)3> I • (0)M
(0)T^*(0)ü:^3 (0)TA$ .
(2.147)

40
Let the geometric stiffness moduli for each of the three layers be defined as
II
-O
X
[í)Kg +
Ü)K% +
tv-13
(()kg
=
(£)ñn is
(£)ñ1213
(í)Ql I3
(i)rñn I3
wm1213 '
(í)ñ1213
(e)ñ2213
(i)Q I3
{e)m1213
(/)íñ2213
(i)Ql I3
(1)0* I3
(i)C I3
-(t)ñí
la(07al3
- (e)™-2a (í)7q 13
wmn 13
( - (e)mla
(1)1 a I3
03x3
03x3
. (e)m2113
(i)â„¢?2 13
~ «)™2a
(t)la 13
03x3
03x3 .
(2.148)
where
(i)C := - ((f) Adding up (2.141), (2.144), and (2.147), we obtain the following expression for (2.137)
[2>((0)B<$$) -A#
• (o)R = (o) Y ¿ <3? • en)Kn mi Y A $ .
(o)^G (o)
(2.150)
The tangent geometric stiffness operator for layer (0) can thus be written as
{o)Gc ($, ó ) • A = J [(0)T Ó& • (o )KlG (0)T A«I>] (o)3odA . (2.151)
A
For the bottom layer (—1), in (2.107), we now hold the resultant forces/couples
(_i)R fixed, and linearize the geometric part, that is, we find the expression for
[z> ((_!)# <5 ) *A3>J • (_i)jR
(2.152)

41
in operator form, where the operator (-i)-B was given in (2.105). We define the two
differential operators (-i)Ti, and (_i)T2 for the bottom layer (-1) as follows
(-!)*! :=
1, 9
3ds1
~(-l)
h ude
1
§
1
1,9
3oe
03x3
d
“(-I)
k Ud£2
~(o)h
l3o?
03x3
03x3
I3
03x3
03x3
03x3
-*w
03x3
03x3
03x3
13 o e
03x3
03x3
03x3
13
03x3
03x3
03x3
'w
03x3
03x3
03x3
13 a e
03x3
03x3
II
CN
rH
03x3
03x3
I3
03x3
A
03x3
03x3
13 9
3ae
03x3
03x3
03x3
1,a
3ae
03x3
_
18x12
12x9
15x12
(2.153)
12x9
(2.154)
In addition to the tangent geometric moduli (-\)KlG that corresponds to the bottom layer
(—1) as an independent single-layer shell, we have also the tangent geometric moduli
(-i)Kq that comes from the coupling between the bottom layer (-1) and the reference
layer (0). The tangent geometric moduli (-i)Kq for layer (-1) can be written as follows
(-i)K2g â– â– = (-i)K% + (-y*? + (-i)K% =

42
1
T
%
M
CO
(-1)^2 I3
(-1)^3 13
(-1)#2 13
03x3
03x3
0gx9
(-1)^3 I3
03x3
03x3
(-1)^4 I3
(-1)^5 13
(-1)^6 I3
09x9
(-i)-Ks I3
03x3
03x3
(-1)^6 I3
03x3
03x3 .
18x18
(2.155)
The parameters in the above moduli matrix (_i)K2G are
<-D*i
•= (-D^+ (
(-i)n11 (-i)V»a + (-D^12 (-1)^2)
(-!)*,!
+ (-l )h+
((-1)7722 (-1)^*2 + (-1)”12 (-1)^1
) (-1)^2
+ (-1 )h+
((-1 )mn * (-i)t,i + (-i)m22 (—1)^,2 * (-1)^,2 )
+2 (_i )h
f (-i)m12 (_i)t,i • (_i)t,2 ,
(-1)^2
â– - (-i)^+ (
-11 ,A , —12 ,A )
(-1 )n (-1)^,1 + (~l)n (-1)^,2 )
(-1 )t + (-i)^+ (-1 )Ql
(-1)^3
~ (-i)*+ (
—22 . .t , -12 \
(-l)n (-1)^,2 + (-l)n (-1)V>,1 )
(_i)t + (-1 )h+ (_1 )92
(-1)^4
:= (o)h~ ((
Jill _i_ —12 ,„t \ +
-I)77 (-l)V5,! + (-l)n (-1)^,2 J (0)t,l
+ (0)/»“ (
'(_!)ñ22 (_d^2 + (_!)ñ12 (-1)^)
(0)*,2
+ (fi)h~ ((-I)?’ (0)^,1 + (-1)(? (0)^,2 ) * (-1)*
+ (0)h ((-l)mU (—1)^,1 * (0)*,1 + (-l)m22 (-1)^,2 * (0)^,2 )
+ (0)^ (-l)771-12 ( (—1)^,2 * (0)^,1 + (—1)^,1 * (0)^,2 ) ,
(-1)^5 := (0)h~ ((-i)ñ11 (-1)^! + (_i)ñ12 (-1)^2) (0)t
>
+ (o)h (-i)Q1 (o)t * (-i)t

43
+ {o)h ((-i)mu (_i)t,i • (o)t + (-i)^12 (-1)^,2 ’ (o)t) ,
(-i)K& := (o)h~ ((-i)ñ22 (-1)^2 + (-i)^12 (-i)V’a) (O)*
+ (o)^ (-1)?2 (-i)i * (o)t
+ (o)h~ ((-iyrn22 (_i)t,2 * (0)t + (-l)7™12 (-i)*,i * (o)*) •
(2.156)
The tangent geometric stiffness operator for layer (—1) can thus be written as
T^g (-1 )GC = J K1g (.uTiA*
A
+ (.1)T2¿Í-(.1)4(_1)T2A$] (-i)JodA. (2.157)
For the top layer (1), in (2.110), we now try to obtain the expression
j *A$ • (i)-R
(2.158)
in operator form, where the operator (i)B was given in (2.108). Similar to (2.157), we
obtain the following tangent geometric stiffness operator for layer (1)
T>g{\)Gc ($,5$) *A$ =
J[mr1s*-wK1a (d^a# + mr26$-mK2a (1)t2a$] (i)7o (2.159)
where
(i)Ti
d
'dp
o
sh+ 1
d
3x3 (0)« (i)
a
o
3¿^7 U3x3 (o)« X3^72' (0
h+ 1?
dp
d
h 1;
dp
\h~ 1
0
9x9
d
dp
d
'dp
I3
d
'dp
d
'dp
A12x9. (2.160)
15x12

44
03x3
03x3
I3
03x3
03x3
03x3
'4
03x3
03x3
03x3
03x3
(1)^2
:=
03x3
03x3
03x3
I3
03x3
03x3
03x3
'4
03x3
03x3
03x3
1, 9
Kl :=
(i)Ki I3
(1)^2 I3
(1)^3 I3
(1)^2 I3
03x3
03x3
09xc
(1)K3 13
03x3
03x3
12x9 >
18x12
(1)^4 I3
(1)^5 I3
(1 )K& I3
09x9
(l)-^5 I3
03x3
03x3
(1)^6 I3
03x3
03x3
18x18
The parameters in the matrix (\)K2G are defined as follows
(i)^i := ~(0)^+ ((i)”11 (i)¥>‘i + (i)ñ12 (1)^*2) (o)*,i
-(0)h+ ((1)«22 (1)^2 + (l)^12 (l)V’Jl) (0)i,2
~(0)h+ ((l)?1 (0)*,1 + (l) - (0)h+ ((l)Tn11 (l)t,l • (0)i,l + (1)77T,22 (l)t,2 * (0)^,2 )
(2.161)
(2.162)

45
(i)
(i)
(i)
(i)
(i)
“(o)h+ (i)m12 ((i)t,2 * (o)t,i + (i)t i • (0)t,2) ,
K2 := -(o)/?+ ((i)^U (1)^1 + (i)”12 (1)^2) (o)* - (o)h+ (i)?1 (i)* * (o)t
— (0)h+ ((l)^U (1)£,1 * (0)£ + (l)m12 (1)^,2 * (0)í) >
K3 := -(o)^+ ((i)”22 (1)^2 + (i)^12 (1)^1) (o)* “ (0)h+ (i)q* (1 )t • (o)t
~(o)h+ ((i)^22 (i)*,2 * (o)t + (1 )m12 (i)ti • (0)t) ,
K4 := -(i)h~ ((í)ñ11 (1)^4 + (i)ñ12 (1)^2) (i)*,i “ (í)^” ((i)^22 (1)^2
+ (i)«12 (1)^1) (i)*,2 - (i)h~ ((i)íñu (i)*,i * (i)*,i
+ (l)^22 (1)* 2 * (1)^,2 - 2 (1 )m12 (l)t, 1 * (l)í,2 ) )
#5 := ((i)”11 (1)^1 + (i)”12 (1)^2) (i)¿ ~ (i)^_ (i)^1 >
K& ■— ~ (o)h~ ((i)ñ22 (1)^2 + (i)”12 (í)V’íi) (i)¿ ~ (i)^- (í)í2 • (2.163)
Remark 2.8. Even though (i)KG has the same form as (0)KG (i.e., the operator
of a stand-alone layer (1)), the operator (i)Yi has the coupling terms in the submatrices
(1,3), (1,4), (2,3), (2,4). In (1)Yi of (2.160), there are five rows of submatrices and four
columns of submatrices. The four columns correspond to (0)V?, (_i)t, (0)t, (1 )t. The
coupling terms in (1) Yj, when multiplied with A (0)£ and A (1 )t in (A A ), only affect
the submatrices (1,1), (1,2), (2,1), (2,2) related to the membrane forces (¿)ñu , (t)ñu,
{e)ñ21, (¿)ñ22 . The reason is the offset of layer (1) with respect to reference surface, which
is the centroidal surface of layer (0).
Another way to understand (i)Yi (or the meaning of the coupling terms in (i)Yj,
i.e., the difference between (0)Y and (ijYx ) is to think of layer (1) as a stand-alone layer,
initially at the same location as that of layer (0). Then (i)Yj would be similar to (0)Y
(note the difference between the column corresponding to (!)£ in (^Yj. and the column
corresponding to (0)f in (0) Y). Now shift layer (1) to the top of layer (0); then the mem¬
brane forces in layer (1) must generate some additional moments. The coupling terms in

46
(i)Yi play the role of lever arms. On the other hand, in (\)KG, we have all the coupling
terms, but not in (i)Y2. 1
2.4. Numerical Examples for Statics of Sandwich Shells
The finite element formulation for the statics of geometrically-exact sandwich shells
presented in the previous sections has been implemented in the Finite Element Analysis
Program (FEAP), developed by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a
DEC ALPF1A with the DEC UNIX V3.2D-1 operating system. Linear finite-element basis
functions are used in the examples in this section. To avoid shear locking, selective reduced
integration is used to evaluate the shear part of the tangent material stiffness matrix (t)KM
and the tangent geometric stiffness matrix {(.)Kg and also the shear part of the residual
force matrix, whereas the bending part and the membrane part of the tangent stiffness
matrix and of the tangent residual force matrix are evaluated using full integration.
To identify the correctness of the present theory and the related coding, we tested
several examples of sandwich plate with different aspect ratio. The aspect ratio is defined
as A := L : W : T, where (L, W, T) designate the length, width, and thickness of the
sandwich plate, respectively. For the bending stiffness and membrane stiffness, we use
the full 2x2 integration points, whereas for the shear stiffness, we use the reduced lxl
integration point.
Firstly we tested the bending of a sandwich plate with three identical layers employ¬
ing 90 four-node quadrilateral elements. The plate is clamped at the edge £* = 0, and is
free at all the other edges (See Figure 2.2). The vertical displacements and the tip rotations
are compared with the theoretical results and with the results obtained from the single¬
layer theory. The results, with a maximum error less than 0.4%, also show the ability of
our sandwich shell elements to model the anticlastic curvature.
We also consider the Cook problem with only the core layer using 100 sandwich shell
elements. The results agree well with those of single-layer shells (Rifai [1993, p. 173]).
To demonstrate the ability of the formulation to capture large rotations and displace-

47
r
h c
s
h <
2
h a
« (D™
*12
*12
i-um
Figure 2.2. Sandwich shell with three identical layers
merits, we tested the torsion of a cantilever plate with three layers using 20 sandwich shell
elements (Figure 2.3). To justify the computed results, we use the theoretical rotation of
the torsion of an elastic bar given by 6 = TL/GJk as a basis for comparison,3 where L
is the length of the plate, G is the shear stiffness and <4 is the polar moment of inertia of
the plate cross section about the centroidal axis along the length of the plate. Since the
direction of the concentrated forces remains fixed along the £3 axis, the resulting couple T
generated by these forces decreases with the twisting of the bar (See Figure 2.4), as the dis¬
tance d between these forces becomes smaller. Thus to make the comparison meaningful,
we use the final value of the couple at the last time step as the torque T used to compute
the theoretical rotation 9. The difference between the theoretical results and our FE results
on the twisted angle is 9%.
Figure 2.3. Torsion of a cantilever plate.
3 Roark and Young [1975, p.290] presented the formulation to calculate the angle of twist for beam with
solid rectangular section, which gives very close results when compared to the above formulation (< 0.2%).

48
Figure 2.4. Force couple, at the tip of the plate, generated by a couple of concentrated
forces.
2.4.1. Roll-down Maneuver of a Sandwich Plate
We now consider the roll-down maneuver of sandwich plates. First, we tested the
sandwich shell having only the core layer using 10 elements. Comparing to the theoretical
deformed shape (i.e., a full cylinder), the relative error in the tip displacements is 0.4%
in the direction and 0.005% in the £3 direction. The displacements obtained with the
sandwich shell code are exactly the same as those with the single-layer shell code. We also
tested the roll-down of a sandwich plate with only one outer layer. We still obtained good
results even with the slower rate of convergence.
Next, we consider the sandwich plates with three identical layers. The material prop¬
erties and geometric properties are chosen as follows:
{i)E = 1.2 x 107, (l)v = 0.0, {í)ks = 0.75, {l)h = 0.033333, for l = -1,0,1.
(2.164)
where , ^ks are the Young’s modulus, the Poisson ratio, and the shear correction
coefficient of layer (£), respectively. The geometrical dimensions of the plate are length
L = 10, width W — 0.1.
At first, we use 10 uniformly distributed sandwich shell elements to discretize the

49
sandwich plate. The computed tip displacements at the end of the last loading step are re¬
ported in Table 2.1. The computed displacement ul differs from the exact solution by 7.5%.
In the first loading step, convergence is achieved after 10 iterations; in the last loading step,
convergence is achieved after 11 iterations. Then we use 20 uniformly distributed sandwich
shell elements to discretize the sandwich plate. The computed tip displacements at the free
edge are reported in Table 2.1, where it can be seen that the displacement u1 is closer to the
exact solution of (-10) when the plate becomes a full cylinder. The relative error in ul is
now 1%. Convergence is obtained after 9 iterations in both the first loading step and in the
last loading step. Finally, the computed displacements at the free edge using 40 uniformly
distributed elements shown in Table 2.1 are clearly closer to the exact solution, in which
the value of u1 should be (—10), and the value of u3 should be zero. The relative error in
the displacement u1 is now 0.53%. Convergence is obtained after 10 iterations in the first
loading step, and 11 iterations in the last loading step.
Table 2.1. Roll-down of a sandwich plate with identical layers: Displacements of a comer
of the free edge.
Disp.
10 elements
20 elements
40 elements
ul
-9.25472
-9.81873
-9.94730
u2
-3.08237 x 10-11
-2.15727 x 10-10
-1.02421 x 10~9
u¿
-1.98609 x 10-1
-1.05095 x 10~2
-8.77204 x 10"4
Figure 2.5 shows the undeformed and deformed shapes of the sandwich plate using
40 sandwich elements.
2.4.2. Sandwich Plate with Ply Drop-offs
In this section, we present the computational results for sandwich plates that have
discontinuities due to disparities in the length of the layers, resulting in the so-called ply
drop-offs.
2.4.2.1. Sandwich plate with ply drop-off
We now consider a cantilever sandwich plate with three layers, and with a ply drop¬
off in the top layer at mid length, see Figure 2.6. The free edge at the tip is subjected to

50
Figure 2.5. Roll-down of a sandwich plate with three identical layers: Isometric view of
deformed shape.
a uniformly distributed force of n*13 = 100. The geometric and material properties are
listed below:
L = 10, W = 0.1, T6 = 0.3, Ta = 0.2, (2.165)
where L is the length, W the width, Tb the thickness before the ply drop-off, and Ta the
thickness after the ply drop-off, and
(t)E = 1.2 x 107, = 0., {i)Ks = 0-75, for £=—1,0,1. (2.166)
Before the ply drop-off, the layer thicknesses are
{t)h= 0.1, for £ = -1,0,1. (2.167)
After the ply drop-off, the layer thicknesses are
(_x)/i =0.1, (o )h =0.1, (i )h =0. (2.168)
Since the plate has a large aspect ratio, we use the Euler-Bernoulli beam theory to
predict the deflection. The bending stiffness coefficients of the beam before and after the

51
Figure 2.6. Sandwich plate with one ply drop-off.
ply drop-off are
EIb = EW(Tb)3/12 = 2700, EIa = EW{Ta)3/12 = 800, (2.169)
respectively.
Let P = n*13 W and M = PLa be the resultant tip load, and the internal moment
at the ply drop-off. Let u\ be the transverse displacement at the ply drop-off B due to the
force P and the moment M. Let u\ be the transverse displacement at C of thin half of the
plate, with the section B at the ply drop-off clamped. The total transverse displacement u3
at the plate tip is the sum of uf, the transverse displacement at C due to the rotation 6b of
the section at B (this rotation results from the bending of the portion AB), and u3 :
u3 = u3 -f- La9b -(- l¿2
= (PLb3/3EIb + MLb2/2EIb) +
(.PLb2/2EIb + MLb/EIb)La + PL3a/3EIa
= 1.6010. (2.170)
Table 2.2 presents the computed results using 20 uniformly distributed linear sand¬
wich shell elements and using 20 uniformly distributed equivalent linear single-layer shell
elements, respectively. The transverse displacement u3 obtained with sandwich shell ele¬
ments has a relative error of 0.030% compared with the analytical result based on Euler-
Bemoulli beam theory, and a relative error of 0.033% compared with the computed result

52
using single-layer shell elements.4
Table 2.2. Sandwich plate with ply drop-off: Tip displacements
Disp.
sandwich elements
singer-layer elements
ul
-1.71076 x 10-1
-1.62236 x 10"1
u¿
-1.43540 x 10-13
3.58955 x 10"13
V?
1.55253
1.54842
Remark 2.9. Similar to the case of sandwich beams in Vu-Quoc and Deng [1995],
the result with single-layer shell elements is smaller than that with sandwich shell ele¬
ments, since the equivalent single-layer plate has a symmetric ply drop-off, unlike the
non-symmetric ply drop-off in sandwich plate. Further, hinge is not allowed to form in
the cross section at the ply drop-off in the equivalent single-layer shell. Figure 2.7 depicts
the undeformed shape and the deformed shape; the effect of the ply drop-off is not easily
discernible. I
Figure 2.7. Sandwich plate with ply drop-off subjected to tip force: Isometric view of
undeformed and deformed shapes.
4 For moderate thick plate and Poisson’s ratio v — 0, Euler-Bernoulli beam theory gives accurate results
on displacements.

53
2.4.2.2. Two-layer plate with ply drop-off: aspect ratio A = 5 : 1 : (1,0.5)
Here the aspect ratio of the two-layer plate with ply drop-off is represented by A :=
L : W : (Tb,Ta). We now consider a cantilever sandwich plate with only two outer
layers (and with the core layer inexistent), subjected to the uniformly distributed force of
n*13 = 60 assigned at the free edge. The plate has a ply drop-off at mid length. The
geometric dimensions of the plate are (Figure 2.8)
L = 5, W = l, Tb= 1, Ta = 0.5, (2.171)
with the layer thickness before the ply drop-off being
(_i )h =0.5, (o )h =0.0, (\)h =0.5, (2.172)
and after the ply drop-off
= 0.5, [Q)h = 0.0, (\)h = 0.0. (2.173)
The material properties chosen are
WE = 29,000, wu = 0.294, = 1, for ¿ = -1,0,1. (2.174)
Ten uniformly distributed elements are used in the computation. The displacements
of a corner node at the tip are tabulated in Table 2.3. The undeformed and the deformed
shapes are shown in Figure 2.9, where a change in curvature at the ply drop-off is clearly
discernible.
Table 2.3. Two-layer plate with ply drop-off, A = 5 : 1 : (1,0.5): Tip displacements.
u1
u2
tt3
-5.18243 x 10"1
-4.22720 x 10-4
1.55017
2.4.2.3. Two-layer plate with ply drop-off: aspect ratio A — 20 : 1 : (1,0.5)
We now consider a cantilever sandwich plate with only two outer layers (and with the
core layer inexistent), subjected to a tip moment. The plate has a ply drop-off at mid length

54
Figure 2.8. Two-layer plate with ply drop-off: Geometry and assigned force.
Figure 2.9. Two-layer plate with ply drop-off. Aspect ratio >1 = 5:1: (1, 0.5): Isometric
view of undeformed and deformed shapes.
(see Figure 2.10). Since the connection between the thinner part La and the thicker part
Lb of the plate is flexible, the actual moment needed to bend the thinner part La into a full
circle is a little smaller than the theoretical result of M — 2nEIa/La, which is obtained for
an equivalent beam with a clamped end.
The geometric dimensions of the plate are as follows
L = 20, W = 1, Lb = 10, La = 10, Tb = 1, Ta = 0.5. (2.175)
The layer thickness before the ply drop-off are
(_i )h = 0.5, (o )h = 0.0, (!)/i = 0.5
(2.176)

55
and after the ply drop-off
(_!)&= 0.5, (o)h = 0.0, (1)/i = 0.0 (2.177)
The material properties chosen are
(t)E = 29000, {l)v = 0.294, wk, = 1. for ¿ = -1,0,1. (2.178)
Along the free edge at the tip of the cantilever plate, we assign a uniformly distributed
resultant couple (_!)m*12 = 189.8, which corresponds to the theoretical value of the tip
moment to bend a beam equivalent to the thinner part La of the plate into a full circle.
Figure 2.10. Two-layer plate with ply drop-off, Aspect ratio A = 20 : 1 : (1,0.5): Geome¬
try and assigned couple.
As mentioned above, since the connection between the thickness part and the thin¬
ner part at the ply drop-off of the plate is flexible, the assigned resultant couple is higher
than what is needed to roll the thinner part of the plate into a full circle. Thus it is ex¬
pected that the tip of the plate will be rolled past the ply drop-off location. The computed
displacements of the nodes at the ply drop-off and at the tip of the plate are reported in
Table 2.4.
In the case where a full circle is obtained, point B in Figure 2.10 should come back
to coincide with point A (in a projection onto the plane (£\ £3)); in such case, the displace¬
ment of point B should be ux(B) = —10 — w*(A) = —10 — 0.969 = —10.969, where the
value of ufA) = —0.969 comes from Table 2.4. The computed displacements u1 for point
B is, however, equal to (—10.789), thus corresponds to a relative error of 1.6% compared

56
Table 2.4. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5). Displace¬
ments at the ply drop-off (point A in Figure 2.10) and at the tip (point B in Figure 2.10).
Node
e1
£2
£3
u1
u2
u3
101
10.
0.
0.
-0.9689
7.875 x 10~3
-3.705
105
10.
1.
0.
-0.9689
-7.875 x 10"3
-3.705
201
20.
0.
0.
-10.789
8.768 x 10“3
-3.937
205
20.
1.
0.
-10.789
-8.768 x 10~3
-3.937
Figure 2.11. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll
down maneuver . Isometric view of deformed shapes (Peeling of a banana).
to the value of (—10.969) mentioned above. The deformed shape of the plate, shown in
Figure 2.11, evokes the action of peeling a banana.
To display the effects of anticlastic curvature that the sandwich shell elements can
capture, we refine the discretization to 160 sandwich shell elements. The three-dimensional
rendering of the deformed shapes are given in Figures 2.13, Figures 2.14, and Figures 2.15.
The anticlastic curvature can be seen clearly in Figure 2.13 and Figure 2.14. Since the
top surface of the plate is stretched in the £* direction, when the plate is roll down, by the
effect of the Poisson’s ratio, this top surface experiences a contraction in the £2 direction.
The reverse is for the bottom surface of the plate (i.e., a contraction in the f1 direction
and a stretching (expansion) in the £2 direction). The combined effect of stretching and

57
Figure 2.12. Ply drop-off problem. Cantilever sandwich shell with drop-off subjected to tip
moment: Peeling of a banana.
contracting of the top and bottom surfaces of the plate in the £2 direction is the result
of bending in the f1 direction. To quantify the anticlastic curvature, and to compare the
result with a calculation employing 3-D solid elements using the nonlinear finite element
code ABAQUS, we look at the difference in the transverse displacement u3 of two points
located at f1 = 13.5 (see Figure 2.15), one point at £2 = 0 (i.e., at the outer lateral edge of
the plate), and the other point at £2 = 0.5 (i.e., in the middle of the plate in £2 direction):
u3(13.5,0.5) - u3(13.5,0) = -6.68096 - (-6.70291) = 0.022 . (2.179)
The quantity in (2.179) is to be compared to the quantity in (2.180) obtained from ABAQUS.
We note that the resultant couple needed to roll the thin part of the plate into a full
circle is (_!)m*12 = 177.5 which is 94% of the magnitude of the tip moment needed to roll
an equivalent clamped beam into a full circle. This lower magnitude is due to the flexibility
by the plate at the ply drop-off line, as discussed earlier.
To compare the results obtained with our sandwich shell element, we solve the same
problem using the solid elements in the nonlinear finite element code ABAQUS. In our

58
Figure 2.13. Two-layer plate with ply drop-off. Aspect ratio A — 20 : 1 : (1, 0.5): Roll
down maneuver. 3-D rendering of the deformed configuration. Isometric view.
L.
Figure 2.14. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5);
Roll down maneuver. 3-D rendering of the deformed configuration. Anticlastic curvature
viewed from direction.
ABAQUS model, we employ 960 C3D8I (8—nodes) linear brick elements, with 1453
nodes. These elements belong to the class of incompatible mode formulation. The final
value of the resultant couple assigned along the force edge at the tip of the plate is 189.8

59
= 13.5
Figure 2.15. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5):
Roll down maneuver. 3-D rendering of the deformed configuration. Anticlastic curvature
viewed from £2 direction.
(See Figure 2.19), which corresponds to that obtained from beam theory. Figure 2.16-2.18
provide various views of the final deformed configuration of the ABAQUS model.
Figure 2.16. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll
down maneuver. ABAQUS model using 960 solid elements with incompatible modes.
Undeformed and deformed configuration. Observer’s viewpoint: (1,-1,1).
To quantify the anticlastic curvature, and to compare this quantification to the re¬
sult obtained using the sandwich shell elements, we again consider the nodes having the
coordinates f1 = 13.5, £3 = 0, which lie on the top surface of the thinner part of the

60
Figure 2.17. Two-layer plate with ply drop-off. Aspect ratio = 20 : 1 : (1, 0.5): Roll
down maneuver. ABAQUS model deformed configuration. Observer’s viewpoint: (1, 0,0).
3
.2 1
Figure 2.18. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll
down maneuver. ABAQUS model. Undeformed and deformed configuration. Observer’s
viewpoint: (0,-l,0).
two-layer plate. In the deformed configuration, these nodes are close to the points having
the lowest spatial coordinate x3 (or the z coordinate in Figure 2.15 and Figure 2.18). The
displacements of these nodes are given in Table 2.5.
Parallel to (2.179) for sandwich shell elements, the anticlastic curvature in the ABAQUS
model can be quantified using the results in Table 2.5 as follows
it3(13.5,0.5) — it3(13.5,0) = -6.5948 - (-6.6159) = 0.022. (2.180)
The above result agrees well with that obtained from the sandwich shell element.

61
Figure 2.19. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5):
ABAQUS solid model. Assigned forces at plate tip to create a resultant couple in the
roll-down maneuver.
Table 2.5. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Anticlastic
curvature from ABAQUS solid model. Nodal displacements of nodes having coordinates
f1 = 13.5, £3 = 0.
I1
£2
u1
1?
13.5
0.00
-6.007
-6.780
13.5
0.25
-6.007
-6.764
13.5
0.50
-6.007
-6.758
13.5
0.75
-6.007
-6.674
13.5
1.00
-6.007
-6.780
For the thinner part of the two-layer plate to roll into a complete circle, point B in
Figure 2.10 must roll back to coincide with point A which is itself moved by the deformed
plate. To compare the results obtained using sandwich elements and those obtained using
an ABAQUS model, we gather the coordinates of point A and B in the final deformed
configuration, corresponding to the resultant couple of M = 189.8 in Table 2.6 below.
2.4.2.4. Two-layer plate with ply drop-off: aspect ratio A = 20 : 10 : (1,0.5)
We now consider the same plate as in the previous section, but with a width ten
times larger (i.e., W = 10.), instead of W = 1. as in the previous section. All other
geometric dimensions and material properties remain identical to those in the previous

62
Table 2.6. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5). Compari¬
son between sandwich elements and ABAQUS solid model. Distance between point A and
point B in the final deformed configuration. £ a = (10, 0,0), £ b = (20,0,0)
4>2(í¿)
Sandwich elements
9.03
0.00788
-3.71
ABAQUS model
8.64
0.0167
-3.56
$‘(Íb)
4>2(Íb)
*3(ís)
Sandwich elements
9.21
0.00877
-3.94
ABAQUS results
8.61
0.00930
-3.53
Distance between A and B= (£ a) ~ *£ (£ b)
Sandwich elements
0.293
ABAQUS results
0.0459
section, see Eqs. (2.175), (2.176), (2.177), and (2.178). The distributed couple
assigned to the free edge of the plate tip is set as before to = 189.8, which is
the resultant couple that will roll an equivalent beam into a full circle. To discretize the
two-layer plate, we now employ 200 sandwich shell elements: 100 elements before the ply
drop-off, and 100 elements after the ply drop-off. The deformed shapes of the plate are
shown in Figure 2.20- 2.23. Figure 2.20 depicts the deformed shape in an isometric view.
One can clearly see the anticlastic curvature in the £2 direction in this figure, as well as
in Figure 2.22, which is the projection of the deformed shape on the (£2, £3) plane. This
anticlastic curvature is the effect of the Poisson’s ratio. The top surface of the undeformed
plate is extended in the £* direction, this extension induces a contraction in the transverse
£2 direction, and thus the downward curvature is clearly seen at the bottom of the deformed
plate in Figure 2.22. Opposite to what take place at the top surface, the compression of the
bottom surface of the plate in the £* direction induces an extension in the £2 direction, thus
resulting in the lateral bulging of the plate, as seen in Figure 2.22.
To quantify the anticlastic curvature similar to (2.179) and (2.180), we consider the
transverse displacement u3 of the nodes at £* = 14 and £3 = 0, which lie near the bottom

63
Figure 2.20. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5): Roll-
down maneuver. Sandwich shell elements 3-D rendering of deformed shape. Isometric
view.
Figure 2.21. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5):
Roll-down maneuver. Zoom-in on the ply drop-off point.
of the deformed configuration of the two-layer plate (see Figure 2.23)
u3(14, 5) - u3(14, 0) = -6.62581 - (-6.64764) = 0.02183.
(2.181)

64
Figure 2.22. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5):
Roll-down maneuver. Sandwich shell elements. 3-D rendering of deformed configuration.
Projection down the axis.
To clearly depict the deformation at the ply drop-off and the tip of the deformed plate
around the area of the ply drop-off, we provide a zoom-in figure on this area in Figure 2.21.
The distance between point A and point B will be used to compare the results obtained
with sandwich shell elements and those obtained with an ABAQUS solid model, which
is composed of 2400 incompatible (solid) linear brick elements (type C3D8I). The final
moment at the plate tip assigned to the ABAQUS solid model has a magnitude of 1898,
which is obtained for the roll-down of an equivalent beam. Various views of the deformed
configuration obtained with the ABAQUS model are depicted in Figure 2.24, Figure 2.25,
Figure 2.26. The distance between point A and point B in the deformed configuration as
obtained both from sandwich shell elements and from the ABAQUS model are given in
Table 2.7.
From the above examples, we found that the present sandwich shell formulation gives
good results on displacements with very coarse mesh, when compared to the 3-D converged
results from ABAQUS. For the interlaminar stress analysis, due to the kinematic assump-

65
É1 = 14, £3 = O
L.»
Figure 2.23. Two-layer plate with ply drop-off. Aspect ratio A — 20 : 10 : (1,0.5):
Roll-down maneuver. Sandwich shell elements. 3-D rendering of deformed configuration.
Projection down the £2 axis.
Figure 2.24. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5):
Roll-down maneuver. ABAQUS solid model. Undeformed and deformed configurations.
Isometric viewpoint: (-4,-7,-3)
tion, the present formulation is expected to be more accurate than the single-layer shell
model, especially for thick and moderate thick shells.

66
Figure 2.25. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5):
Roll-down maneuver. ABAQUS solid model. Projection down the f1 axis of deformed
configuration.
Figure 2.26. Two-layer plate with ply drop-off. Aspect ratio A - 20 : 10 : (1, 0.5): Roll-
down maneuver. ABAQUS solid model. Projection of the deformed configuration along
the if axis.

67
Table 2.7. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5). Compari¬
son between sandwich elements and ABAQUS solid model. Distance between point A and
point B in the final deformed configuration. £ a = (10, 0, 0), £ b = (20,0,0)
&Ua)
Sandwich elements
9.16
0.0112
-3.5
ABAQUS model
9.16
0.0179
-3.53
*1Mb)
Sandwich elements
9.07
-0.0345
-3.41
ABAQUS results
8.23
-0.0294
-2.83
Distance between A and B= <& (£ a) - (£ b)
Sandwich elements
0.142
ABAQUS results
1.17

CHAPTER 3
OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OP
MULTILAYER COMPOSITES : STATICS
3.1. Introduction
The analysis of general shell structures have been of interest for several decades.
There is a continuing need to develop more reliable, accurate and efficient shell element,
especially for analyses of composite structures covering a wide range of physical scales (in¬
cluding MEMS5) and material and geometric nonlinearities. Structures made of laminated
composites continue to be of great interest for engineering applications. Lor accurate anal¬
yses of composites with a large number of layers, industry routinely employs FE meshes
with one solid element per ply in the thickness direction, and with element aspect ratio less
than 10 (Figure 3.1). It is therefore highly desirable to develop efficient finite elements that
are accurate at extreme aspect ratio to significantly reduce the computational effort.
Figure 3.1. Composite structure with 500 plies in the thickness direction; the ply thickness
is around 10-3m.
Shell element formulations have been mainly developed within the context of the
so-called degenerated shell concept and the classical shell theory (Buechter and Ramm
[1992]). Both formulations are based on the common kinematic assumptions of inextensi-
5 MEMS stands for MicroElectroMechanical Systems.
68

69
bility in the thickness direction and the zero transverse normal-stress condition.6 Although
these approximations led to very good results in most cases, several difficulties could arise:
(i) Complex 3-D material models: the zero transverse normal stress condition must be im¬
posed. (ii) Boundary conditions and finite rotations: use of rotational degrees of freedom
(dofs); especially those normal to the boundary, to describe soft support and hard support
(e.g., Zienkiewicz and Taylor [1991, p.92]); complex update algorithms for finite rota¬
tions in geometrically-exact stress-resultant formulation (e.g., Vu-Quoc and Deng [1995],
Vu-Quoc and Ebcioglu [1996], Vu-Quoc and Ebcioglu [2000a], Vu-Quoc and Ebcioglu
[2000b], Vu-Quoc, Deng and Tan [2000]). (iii) Transverse normal stress: inconsistently a
posterior computation based on the computed in-plane stresses (see Reddy [1997, p.345],
and e.g. in the localized effects due to the concentrated surface loading and the delam¬
ination of composite shells), (iv) Combination with regular solid elements: Transition
elements are needed to connect rotational dofs and displacement dofs (e.g., Kim, Varadan
and Varadan [1997] and the contact problem), (v) Through-the-thickness stress distribu¬
tion in laminated composite with dissimilar materials: poor accuracy because of straight
director assumption (Bischoff and Ramm [2000]).
Accurate and robust low-order shell elements have always been in high demand for
development and for use in engineering analysis (e.g., DYNA3D [1993], NIKE3D [1995]),
particularly when complex nonlinear 3-D constitutive relations can be incorporated with¬
out the added requirement to satisfy the constraint of zero transverse normal stress. Three
possible shell kinematic descriptions have been proposed: (i) The displacement of the refer¬
ence surface together with the extensible transverse director (Simo, Rifai and Fox [1992],
Betsch, Gruttmann and Stein [1996]). (ii) The displacement of the reference surface to¬
gether with the displacement vector of the tip of a director (Braun, Bischoff and Ramm
[1994], Roehl and Ramm [1996], Bischoff and Ramm [1997], Bischoff and Ramm [2000]).
(iii) The displacement of the top and bottom surface of the shell (e.g. Hauptmann and
6 Stress-resultant shell formulation can be generalized to account for thickness change, which relaxes
the zero transverse normal stress condition (e.g., Simo, Rifai and Fox [1992]).

70
Schweizerhof [1998], Klinkel, Gruttmann and Wagner [1999], Ramm [2000]). The kine¬
matic descriptions (ii) and (iii) are attractive since they avoid the complex rotation updates
in stress-resultant elements. On the other hand, the kinematic description (iii) provides a
natural way to connect to regular solid elements without the need for transition elements;
such feature can benefit the detailed modeling of shells with patches of piezoelectric or
viscoelastic materials.
The present solid-shell element has the same displacement dofs as in regular lin¬
ear (8-node) brick solid element. Displacement-based solid elements are known to have
poor performance in bending-dominated situation, such as in thin shells. To obtain the
same performance as stress-resultant shell formulations with plane stress assumption (e.g.,
Vu-Quoc, Deng and Tan [2000]), the Enhanced Assumed Strain (EAS) method and the
Assumed Natural Strain (ANS) method are employed here.
To improve the bending behavior of low-order elements, the EAS method based on
the Fraeijs de Veubeke-Hu-Washizu functional was proposed by Simo and Rifai [1990]. For
large deformation analyses, there are two ways to introduce the EAS method: (i) enhancing
the deformation gradient F (Simo and Armero [1992], Miehe [ 1998¿>]), and (ii) enhancing
the Green-Lagrangian strain tensor E (Bischoff and Ramm [1997], Klinkel and Wagner
[1997], Klinkel et al. [1999] etc.). From the computational standpoint, the latter is simpler
and more efficient, even though our numerical experience indicates that both approaches
lead to the same numerical results when the same EAS parameters are used.7 To incorporate
3-D constitutive laws in shell formulations, the transverse normal strain must have at least
a linear distribution over the shell thickness; otherwise, the so-called Poisson-thickness
locking would occur (Zienkiewicz and Taylor [1991, p. 161], Bischoff and Ramm [1997]).
To relieve the Poisson-thickness locking, two methods were proposed in recent years: (i)
Assuming a quadratically distributed displacement field over the shell thickness (Parisch
[1995]), which then introduced an additional kinematic parameter, and (ii) using the EAS
7 Noted that the EAS method based on the displacement gradient as proposed in Miehe [1998¿>] and used
in Miehe and Schroeder [2001] does not pass the bending patch test, and there is no easy way to remedy this
problem, see the next paragraph for the details.

71
method to enhance the transverse normal strain (Buchter, Ramm and Roehl [1994]). In
our formulation, we enhance the transverse normal strain by the EAS method to include
bilinear terms £!£3 and £2£3 in terms of material coordinates. To improve the membrane
bending behavior, we also enhance the membrane strains in the similar manner as in Simo
and Rifai [1990],
On the other hand, to make the formulation more efficient, we propose a modified
EAS method while still keeping the same level of accuracy, where the inverse of element
Jacobian matrix and the Jacobian at the element center are no longer necessary. Further¬
more, the present eight-node solid-shell element relies on a new optimal seven-parameter
EAS-expansion (for the transverse normal strain and for the membrane strains) together
with an ANS method (for the transverse shear strains); the present formulation is shown
to pass both the membrane patch test and the out-of-plane bending patch test. It should be
noted, however, that while the 30-parameter EAS expansion of Klinkel and Wagner [1997],
the five-parameter EAS expansion of Miehe [1998/?] and of Klinkel et al. [1999] pass the
membrane patch test, all of them fail to pass the important out-of-plane bending patch test.
For the EAS approach using enhancing deformation gradient, we develop an EAS
expansion by superposing the enhancing converted basis to the compatible converted basis,
and then present a formulation that is much simpler than that employed in Miehe [1998b]
(see Section 4.3).
Two ANS modifications on the compatible covariant strains are employed to elimi¬
nate the locking effects from the compatible low-order interpolations. ANS interpolation is
the most successful tool to overcome the shear-locking effect in the 4-node displacement-
based shell elements, even for initially distorted meshes (MacNeal [1978], Hughes and
Tezduyar [1981], Dvorkin and Bathe [1984], Parisch [1995]). We apply an ANS inter¬
polation of the compatible transverse shear strains to treat shear locking. In the case of
curved structure with geometric nonlinearity, there is another locking effect: The so-called
curvature-thickness locking (Bischoff and Ramm [1997]), which is also known as the trape¬
zoidal locking (Sze and Yao [2000]); this type of locking can be avoided by introducing the

72
ANS interpolation of the compatible transverse normal strain, as proposed by Betsch and
Stein [1995]. Such treatment can improve the performance of the formulation in Parisch
[1995] and Hauptmann and Schweizerhof [1998].
The features of this solid-shell formulation are summarized as below:
• The kinematic description involves only displacement dofs that require no complex
finite rotation update and no transition elements to connect solid-shell elements to
regular solid elements (Hauptmann and Schweizerhof [1998]).
• The use of covariant Green-Lagrange strain tensor, without neglecting any higher
order terms (e.g., as in Bischoff and Ramm [1997]). The stress and strain terms
quadratic in £3 become important in the analysis of relatively thick shells, for strong
curvatures or in the presence of large strains together with bending deformations
(Buchter et al. [1994]).
• All stress and strain components are accounted for, thus allowing for an implementa¬
tion of unmodified 3-D nonlinear constitutive laws, without the need for applying the
plane-stress constrain. The strain-driven character of the formulation also makes it
easier to implement nonlinear constitutive models, as compared to the hybrid finite-
element formulations (Simo, Kennedy and Taylor [1989]).
• In contrast to EAS formulation based on the deformation gradient F (see Section 4.3),
EAS formulation based on enhancing the Green-Lagrange strain tensor (together
with the use of the second Piola-Kirchhoff stress tensor) are much simpler (Simo
and Armero [1992] and Andelfinger and Ramm [1993]).
• An ANS method applied on the transverse shear strains is used to relieve the trans¬
verse shear-locking problem (Dvorkin and Bathe [1984]), whereas an ANS method
applied on the normal strain components is used to remedy the curvature thickness
locking problem (Betsch and Stein [1995]).

73
In addition to the above features, our new contributions to the field are specifically listed as
below:
• Optimal (minimum) number of EAS parameter to pass the patch tests for both the
membrane response, and the out-of-plane bending: (i) three EAS parameters on the
transverse normal strain to remedy the Poisson-thickness locking, and (ii) four EAS
parameters on the membrane strains to remedy the in-plane bending behavior.
• Efficient EAS method that avoids the computation of the Jacobian at the element
center, and that no inverse of the Jacobian matrix at the element center is needed.
• By using the tensor form, we prove the equivalence of the 2-D plane elasticity el¬
ements of Simo and Rifai [1990], of Taylor, Beresford and Wilson [1976], and our
new enhancing formulation.
• We justify through numerical experiments the relative importance of the separate
use of the EAS method and the ANS method, as compared to the pure displacement
formulation, and more importantly, the combined use of both the EAS method and
the ANS method in obtaining accurate results for plate bending problem over a large
range of aspect ratios.
The comparison among the above various solid-shell formulations is listed in Table 3.1.8
Table 3.1. Comparison of various solid-shell concepts.
Bending
Locking-frec
Absence of rot. dofs/
Higher-order terms
Absence of
Model parameter-space
Optimal
Patch lest
Disp. dofs only
in thickness coord.
pre-integration
dimension
EAS
Present clement
yes
yes
yes
yes
yes
3-D
yes
Ramm cL al.[1997]
yes
yes
yes
no
no
2-D
no
Schweizcrhof ct. al.[1998]
yes
no
yes
yes
yes
3-D
no
Betsch eL al.[1996]
yes
yes
no
yes
yes
2-D
no
Miehe [1998]
no
yes
yes
yes
no
3-D
no
8 Parameter-space dimension is defined as: 1-D (beam), 2-D (stress-resultant plates and shells) and 3-D
(solids); Deformation-space dimension is defined as: 2-D (planar deformation), 3-D (general deformation)

74
The outline of the present chapter is as follows. After a presentation of the kinematics
assumption and the formulation based on the Fraeijs de Veubeke-Hu-Washizu (FHW) vari¬
ational principle (Felippa [2000]) in Section 3.2, we discuss the finite-element discretiza¬
tion and its implementation in Section 3.3. A review of the EAS method together with
our proposed modification is presented in Section 3.4. We present the numerical results in
Section 3.5.
3.2. Kinematic Assumption and FHW Variational Formulation
The extension of the EAS method to geometrically non-linear problems by Simo and
Armero [1992] employed an enhancement of the deformation gradient F and thus a multi¬
plicative decomposition An alternative line of formulation for geometric and material non-
linearities based on the enhancement of the Green-Lagrange strain E leads to particularly
efficient computational effort (see, e.g. Bischoff and Ramm [1997]), with practically the
same results. We describe below the kinematics of a solid shell in curvilinear coordinates
and review the three field FHW variational principle and its role in the EAS method.
3.2.1. Kinematics of Solid-Shell in Curvilinear Coordinates
To overcome the known problems associated with the rotational degrees of freedom
in traditional shell elements, the shell kinematics of deformation is described by using the
position vectors of a pair of material points at the top and at the bottom of the shell sur¬
face. In this kinematic description, a straight transverse fiber before deformation remains
straight after deformation. Such transverse fiber between two corresponding nodes at top
and bottom surfaces needs not be normal to the shell mid-surface before deformation, as
well as after deformation. We distinguish here three configurations of the shell: (i) the
material configuration, which is the biunit cube, (ii) the initial (or undeformed) configura¬
tion, which could be curved, and (iii) the current (or deformed) configuration (see Fig.5 in
Vu-Quoc and Ebcioglu [2000/?]). The initial (undeformed) three-dimensional continuum
of the shell geometry (Figure 3.2) is described by
x«) = 5[(i+f3)^»(e1,ís) + (i-í3)-x'(í1>«2)] ■

75
£ = (¿\ <£2, e) e □ := [-1,1] X [-1,1] x [-1,1], (3.1)
where X (•) is the mapping from the biunit cube □, parameterized by the material coordi¬
nation (f1, £2,£3), to the initial configuration. The image X (£) of a point £ = (f1,^2,^3) G
□ is next represented by a linear combination of the position vectors Xu (*, •) andX; (*, •)
of a point in the upper material surface £3 = +1, and a point in the lower material surface
£3 = —1, respectively.
Figure 3.2. Initial (undeformed) configuration of solid shell: convective coordinates and
position vectors Xu and X¡.
The kinematics of the present formulation is the same as in (Hauptmann and Schweiz-
erhof [1998]), and related to the director formulation (e.g. Bischoff and Ramm [1997]), if
one rewrites (3.1) as follows
xa) = i[x„(ei,52)+x,(íi,í2)] + iís[xu(íi.í2)--x'df‘,í2)]
= Xm({',í2) + ^3ft(í‘,í2)D(e‘,í2) , (3.2)
where Xm is the position vector of the mid-surface in the initial configuration, D the unit
director, and h (f£2) the shell thickness.

76
Similarly, in the current (deformed) configuration, the geometry of the solid shell is
described by
= í[(i+e8)*.(e1,0 + (i-^3)aí*(^0] •
í=[^,e2,-e3]ea,
(3.3)
where x(') is the mapping from the biunit cube parameterized by (£\£2,£3) (material
configuration), to the current (deformed) configuration, xu (•, •) and x¡ (*, •) the position
vectors of the deformed upper and lower surface of the solid shell, respectively.
The initial configuration is related to the deformed configuration (Figure 3.3) by the
displacement field u as follows
x (o = x({)+ «({).
(3.4)
The convected basis vectors G¿, ¿ = 1,2,3, in the initial configuration Bo are related
to the position vector X and the converted coordinates £* by
0,(0 - ^ . ¿ = 1,2,3, (3.5)
and satisfy the following relations
Gi• Gj = S{ , Gi • Gj = Gij , i,j = 1,2,3 , (3.6)
where are the components of the metric tensor with respect to the basis G1 (8)
in the initial configuration B0. configuration. To simplify the presentation, we will omit
the argument (£) and simply write and G1. The covectors G1 can be obtained by the
following relation
& = GijGj , with [Gij] = [GijY1 € R3x3 . (3.7)
similarly, the convected basis vectors g¡ in the current configuration Bt are obtained by
using (3.4) and (3.5) as follows
dx
du
Q£i Gi + pta »
i = 1,2,3,
(3.8)

77
B
Figure 3.3. Solid-shell: material configuration B, initial configuration Bo, and deformed
configuration Bt.
and satisfy the following relations
9i '9* = % , 9i’9j = 9ij , i,j = 1,2,3 , (3.9)
where are the components of the metric tensor with respect to the basis gl ® gi in the
current configuration Bt. The covectors gl can be obtained as follows
9X = 91J9j , with gl] = foy] G
1 — 1 r- ]n)3x3
(3.10)
The deformation gradient expressed in converted basis vectors g{ and Gl takes the
form
3x
F=-=g,®G'.
(3.11)
Using (3.11), (3.6), and (3.9), we can then write the (compatible) Green-Lagrange strain

78
tensor with respect to the convective coordinates as follows
£? = \ (FtF - I2) = Í [(G‘ ® Si) (s, ® &) - GyCr ® G>]
= l (s« - Gij) G‘ ® G' = £“ G' ® G’, (3.12)
where Efj is the covariant components of the strain tensor Ec, and I2 the identity tensor
expressed with respect to the convected basis Gl ® Gj using the relations ep = GpGl and
eq = GqjGj as follows
I2 = 6Pqep ® e9 = 5pqGviG]Gi ® Gj = GfGjG* ® Gj = GZJGl ® Gj, (3.13)
Using (3.8), the metric-tensor components gtj in (3.9)2 can be expressed in terms of
the convected basis vector and the displacement vector u as follows
/ du\ / du\
9a - - (Gi + —J • (G, + ^-J
_ _ du du du du
= Gy + G,.^- + —-Gi + —
(3.14)
Substituting (3.14) into (3.12), the covariant components E'?• of the compatible Green-
Lagrange strain tensor Ec read as
_c 1 (n 9u du du du\
E» = 2[G''d(i + W , + d?"d(i) '
(3.15)
The second Piola-Kirchhoff stress tensor S is conjugated with the Green-Lagrange
Strain Ec, and can be expressed with respect to the basis Gj as follows
S = SijGi ® Gj , (3.16)
with Slj being its contravariant components.
3.2.2. Variational Formulation of EAS Method
In this section, we provide a brief overview of the EAS method, which has the three-
field Fraeijs de Veubeke-Hu-Washizu (FHW) 9 variational principle with the following
9 We refer the readers to Felippa [2000] for a history on the contribution of Fraeijs de Veubeke to the
formulation of multifield variational principles.

79
functional as the point of departure
II (it, E, S)
I Ws (E) dV + J SI [Ec (it) - E\ dV
B0 B0
j u* (b* — u) pdV + J (it*
Bq Su
it) mtdS — J u’t*dS ,
sa
(3.17)
where the displacement it, the Green-Lagrange strain E, and the second Piola-Kirchhoff
stress S are the independent variables. In (3.17), the expression of Ec (it) in terms of
it is given in (3.12) and (3.15); Ws is the stored strain energy, t the traction vector, it*
and t* the prescribed displacement on the boundary Su and the prescribed traction on the
boundary Sa, respectively. All variables here are expressed in the initial configuration B0,
with Sa U Su — dBo (i.e, the closure of the union of the boundary Su and the boundary S„
forms the complete boundary of B0).
The next step in the EAS method is to introduce an enhancement (or enrichment) to
the Green-Lagrange Strain Ec and, as a result, of the variation 6EC as follows
E = Ec + E ,
(3.18)
6E = 6EC + SE ,
(3.19)
where E and SE represent the enhanced strain and its variation, whereas E and 6E repre¬
sent the enhancing strain and its variation. Introduction of (3.18) into (3.17) yields
n (it, E, s) = Í W8 (Ec (it) + E)dV - f S : EdV
Bo Bo
- Ju-(b*-u)pdV + j (it* — it) ’t (5) dS — J u-t*dS .
Bo Su S„
(3.20)
where it is noted that the traction t is actually a function of Sisee also Malvern [1969,
p.69]), and not an independent variable.

80
Consider a perturbation of the displacement u as follows:
ue (X, t) = u (X, t) + eóu (X) , (3.21)
since the variation 6u is only a function of space, and not of time, the variation of the
acceleration ü is then
¿ü = 0,
(3.22)
therefore the variation of II in (3.20) with respect to the displacement u is
dWs . (dEc(u)
Bo
|nM'S)| „ = /W: (^■6u)dV-fSu-(b--ü)PéV
e=0 Bo V 7 Bo
— j Su'tdS — J 6u’t*dS , \/6u ,
(3.23)
Su
dWs
Since the second-order tensor —— is symmetric, the first term in (3.23) can now be rewrit-
oE
ten as follows
dws. (dEc{u) _dws
dE
a. *•) - rW =
â– Su , (3.24)
where the first equality is shown in Remark 3.1, and the second equality is obtained with
Leibniz rule.
Remark 3.1. With the deformation gradient F (u) expressed with respect to the basis
e¿ Ej (see Figure 3.3), we obtain the following relations
F = Fje¿ 0 EJ , Ft = FjEJ 0 e¿, (3.25)
FTF = F}gijFjE1 0 EJ , with glj = e^e,- , (3.26)
SF = —5u = ^ei®EJ5u.
ou ou
(3.27)

81
Note that, if {e^ e2, e3} were a system of orthonormal vectors, we would have glj = Sij,
however, in general, it is not necessary to have {ej, e2,e3} orthonormal. In this remark,
we retain the notation g'i for generality. We define a symmetric second-order tensor S as
S:=^-=SabEa®Eb.
dE
(3.28)
By using (3.28), (3.12), and (3.26), we can then rewrite the first term in (3.23) as follows
■ dECsu - s: iaJ£nSu
dE du
2 du
1 ~AB fan _ _ dF%
= -S
2 \ du
â– 9ijF]B + j Su .
(3.29)
Since SAB = SBA from the symmetry of S, (3.29) becomes
W . r. c pi qAB r
ae-5e=fas
(3.30)
With the following component form of FS using (3.25)i and (3.28), that is, we obtain
FS = F)SJBei ® EB ,
(3.31)
and then the following expression
dF
li &Fb
FS : ÓF = FS : — = F\sABgij 0
du 71 J du
<5u .
(3.32)
Comparing (3.32) to (3.30) and using SF = GRAD (Su), we arrive at
dW,
S .
dE
: 6EC = FS : 6F = FS: GRAD (<5u)
(3.33)
Using the divergence theorem , the integration of the first term in (3.24) becomes
,dWs
“v r ~dEouiav = j [r-dE
Bo
/Div(
Su mndS,
(3.34)
where n is the outward normal vector to the surface dBo.

82
With (3.24) and (3.34), we can write the variation of II in (3.23) with respect to u as
d„, ~ ^ r {^dWA , _ f„dWs
Bo
- / »(^)-^5-/Div
e=0
5F
d!3o=SaUSu
— J 5u* (b* — ü) pdV — J Su’tdS — J Su‘t*dS
Bo Su Sa
-I
Su
-/
Bo
.aw.
•6udS + J
n. (j,dWs\ ..
Div F —— + pb — pu
dE
(„dWa\
SudV = 0 , Viw ,
Scr L
•SudS
(3.35)
thus the Euler-Lagrange equations resulting from the variation of II with respect to u in
above (3.35) are
.dWs
t = n• F
dE
on Su ,
= onS-
DÍV (F^w) + pb* = P“ in B° ’
(3.36)
(3.37)
(3.38)
Then make the variation of IT of (3.20) with respect to E by the perturbation
Ee — E T eSE ,
(3.39)
it follows that
én (-■*■•s) 10=/(
e=0 \
'aw,
dE
- S : 6EdV , VSE ,
(3.40)
E=u Bo
the Euler-Lagrange equation associated with (3.40) is
dWa
S =
dE ’
(3.41)
Taking the variation of II of (3.20) with respect to S by the perturbation
S£ = S + e6S ,
(3.42)

83
we obtain
±n(uE,St)
= J SS 1 EdV + J (u* - u) •StdS , VÓS ,
6=0 Bo Su
(3.43)
where it is noted that t depends on S, then the Euler-Lagrange equations associated with
(3.43) are
E = 0,
(3.44)
it = it* on Su,
(3.45)
where in general E 0 when the finite element approximation is introduced.
3.3. Finite-Element Discretization
In this section, we present the weak form and finite-element approximation of the
proposed solid-shell element. The orthotropic constitutive law of laminated composite is
then derived in convective coordinates. To avoid shear locking, we use the assumed natural
strain (ANS) method by Dvorkin and Bathe [1984] for the transverse shear strains. To
remedy the curvature-thickness locking (Bischoff and Ramm [1997]), we adopt the ANS
method by Betsch and Stein [1995] for the transverse normal strain.
3.3.1. The Weak Form of Modified Two-Field FHW Functional
By designing the approximation for the stress field S and the approximation for the
enhancing strain Ec such that the following orthogonality condition holds:
J S I EdV = 0 . (3.46)
Bo
then the number of independent variables in the functional II in (3.20) is reduced to two,
that is
n (u, E) = J Ws (.Ec (u) + E) dV
Bo
— j it* (b* — it) pdV + J (it* — u)*t (S) dS — J wt*dS , (3.47)
Bo Su Sa

84
leading to the following total variation
(u, jg?) = inmass (u) + ¿nsíijflr (u, e) + snext («) = o, (3.48)
where from (3.35) and (3.40), we have
SUmass = Jóu-üpdV, (3.49)
Bo
SUstiff (u, E) = I (SEC (u) + 6E) : ^Ws (ec («) + E)dV , (3.50)
So
¿lie* = - J 6u-b*pdV - f 5wt*dS . (3.51)
So
3.3.2. Spatial Discretization
Let the initial configuration B0 be discretized into nonoverlapping nel elements ]
nel / ,
with numnp nodes, such that B0 ~ (J Bq . Let h denotes the characteristic size of the
e=l
finite element discretization.
In the element domain Bq\ the displacement u, its variation 6u, and increment Au
are interpolated as follows
u^uh = N(£) d{e) , (3.52)
Su « Suh = N (£) 6d{e) , Am w Auh = N {£) Ad(e), (3.53)
where AT is a matrix containing the basis functions restricted to element Bq\ and d^ E
j^3xnumnP a matrix containing the nodal displacements. The readers are referred to the
Appendix A. 1 for the details.
The velocities ú and accelerations ü are also interpolated by using the same shape
functions and the corresponding nodal values that is
ú ~ uh = N (£) d^ , ü w üh = N (f) d^ . (3.54)
In what follows, for simplicity, we will omit the superscript h in uh, and simply write u.

85
Within a typical element (e), the variation and the increment of the (compatible)
Green-Lagrange Strain Ec is related to the variation and the increment of displacement,
respectively, based on (3.15), as follows
{"$}.*! =B(dW)W<*)- {AS«}Sxl = ^ (dW) Ad ' (3'55)
where the components of had been arranged into a 6 x 1 column matrix according to
the Voigt ordering (Brillouin [1946, p.221 ])
= ni E'22i‘2El2, E33,2E23,2E13} , (3.56)
and where B is the deformation-dependent displacement-to-strain operator, where detailed
expression is given in the Appendix A.l.
We denote the admissible variation of the element EAS parameter column matrix
a(e) g Rnaw associated with the enhancing strain by 5a^e\ where neas is the num¬
ber of EAS parameters oSe\ Interelement continuity is not required for the enhancing
strain E, where components can be approximated via an enhancing strain interpolation
matrix Q and the element EAS parameters an interpolation applies to the variation
and the increment of E, that is
{4'L, = s (i) a<‘>, {íÉ¡,}6xl = S ({) , {A4}6X1 = s «) Aa<"). (3.57)
The number of internal parameters a^ and the interpolation matrix Q will be discussed in
Section 3.4.
3.3.3. Linearization of the Discrete Weak Form
The consistent tangent operator in the Newton solution procedure is constructed by
taking the directional derivative of the weak form at a configuration in the direction
of the increment A (k)U, where the left subscript k designates the iteration number. The
tangent operator can be viewed as the summation of the material and geometric tangent
operators. The geometric part results from taking the variation of the geometry while hold¬
ing the material constant, whereas the material part results from taking the variation of the
material while holding the geometry constant.

86
Applying a standard finite-element procedure to discretize the weak form (3.48), we
obtain the following expression at the element level for the static case (ü = 0)
= ¿n^+¿n2 = o, (3.58)
with the stiffness part from (3.50) and the external-force part SU[eJ, from (3.51)
written as follows
¿ng, = j 6 {Etf {&} dV + J S {Enf {&} dV , (3.59)
B(oe) B(0C)
6n{£t = - [ Su-b*pdv- f Su-t'dS, (3.60)
where we used the symbol definition, which corresponds to the second Piola-Kirchhoff
stress,
S :=
dWs
dE ’
(3.61)
to alleviate the notation, and where the column matrix {tS1-7} has its coefficients arranged
in the same Voigt ordering as in (3.56)
{sij} = [sn,s22,sl2,s33,s23,sn]r.
(3.62)
It should be noted that the symbol S in (3.61) is simply used in the place of
not the independent variable in the Euler-Lagrange equation (3.41).
dWa
dE
, and is
Remark 3.2. The linearization of the weak form (it, E^j can be accomplished by
the truncated Taylor series about the kth iterate ((*)«, (^¿2
SU ({fc+1)u, {k+1)E ) « SU (lk)u, ik)E )
+
d(sn)
d (u, E)
(u=wu, E=(k)Ej
(Au, AE)
— ((k)U, (k)E) + v (¿n) ( (fc)U, (jk)E) • aeJ ,
(3.63)

87
where Au = (fc+1 )U — (fc)U, AE = ^+i)E - (k)E. To compute the increment
(Au, AE^j in the Newton’s solution process, we simply set the expression in (3.63) to
zero.
To alleviate the notation, we will omit the left subscript k designating the iterative index.
Using the approximation (3.52), (3.53), (3.55) and (3.57) in (3.63), the increments
Ad^ and Aq^ can be computed in the Newton’s solution process, as mentioned above,
by using the following equation
P («!<•>) (dM.oM) • (Ad<'\Aa<‘>) = • (**■>, A««)
= ‘ (Adw, AaW) = - (SH% + «S) , , (3.64)
in which the variation ¿II ^ in (3.59) and ¿fl^ in (3.60) now take the form
(d(e) , a(e))
= + S<*{e)Tf'&s .
(3.65)
with f%
= J bt {s«} dv, /gu = / eT (siJ) (3.66)
»(e) o(e)
(d(e))
(3.67)
with
= J NTb'pdV + J NTfiS.
(3.68)
?(*)
j(e)
Thus the left hand side of (3.64) becomes
V(6U%) • (Ad^, Aa(e)) = "Ad(e) + .Aa(«)
= [5d(e)Tfcg + Sa^Tk^] • Ad^ + [5d^Tk[e¡ + Sa^k^] • A«W
= ¿<¿(e)r [fc¡2Ad(e> + Aa] + <$a Aa] , (3.69)
Let the matrix of tangent elastic moduli C be defined as
C = [Cijkl] :=
dS*’
dEkl
d6x6
(3.70)

88
where are the components of constitutive tensor C in the convected basis, and are
subsequently arranged in the matrix C according to the ordering of the strain components
in (3.56) and of the stress components in (3.62).
Using (3.69), (3.66)i, (3.70), and (3.55), we obtain the following expressions
*£ = d-£jk= / (gt$ + btcb) dv,
k(e)
= l3) = / BT°^dV ■
«(<=)
(3.71)
(3.72)
where the matrix
G :=
dB (d(e))
dd[e)
(3.73)
(which is a function of the coordinates £), and the stress matrix $ (which is related to the
matrix {5lJ} in (3.62)) have their detailed expressions given in the Appendix A.l. It is
noted that the dimension of $ and G are 144 x 24 for the present element with six stress
components, eight nodes per element, and three dofs per node. From (3.69), (3.66)2, (3.70),
(3.73), and (3.55), we obtain the remaining parts of the stiffness matrix
U(fi)
dfie)
EAS
dd(e)
k(e) df{EAS
dot*)
= = / QTCBdV ,
= J gTcgdv.
(3.74)
(3.75)
It follows from (3.64), (3.65), (3.67), and (3.69) that the discrete linearized system
of equations to solve for the increments Aand Aa^ is given by
é'>
ffeíjAaW =
Ae) _ Ae)
J ext J stiff j
(3.76)
fcSAd<‘>
+ fcgAaW =
Ae)
J EAS >
(3.77)
or in matrix form as
U(e) uifi)
"'UU "'ua
*.(«) *.«
. "'au "'act _
II
'w' s
^ a
< <
Ae) _ /.(e) -j
J ext J stiff [
_Ae) f '
J EAS )
(3.78)

89
Since the enhancing strain E is chosen to be discontinuous across the element bound¬
aries, it is possible to eliminate the EAS parameter increment Aa(e' at the element level,
before proceeding to assemble the element matrices into global matrices. Solving for the
increment Aa^ using (3.77)
Aa<«> = - [*«]'' (ffAS + fcAd<'») , (3.79)
then substituting (3.79) into (3.76), we obtain the following condensed symmetric element
stiffness matrix kft and the element residual force vector r^
fc(®) = ¿.(e) — r*wi
kT kuu ['cauJ [^araj Kau >
r(e) _ f(e) _ Ae) , \u(,e)]T ~1 f ^
T — J ext Jstiff + [^auj [KaaJ JEAS
An assembly of the element matrices kj) and r^ leads to the global system
(3.80)
(3.81)
Kt Ad = R , (3.82)
n el net
with Kt = Akij) , R = Ar(e* , (3.83)
e=l e=l
where A denotes the finite-element assembly operator.
The incremental displacement Ad can be solved by using (3.82), and the displace¬
ment d and updated. With (3.79), the incremental displacement Ad(e^ is used to com¬
pute the increment Aa*e), which is in turn used to update the EAS parameter a^. The
details of this iterative procedure are provided in the Appendix A.l.
3.3.4. Material Law in Convected Basis
For the Saint-Venant-Kirchhoff material, the fourth-order material tensor C is de¬
fined as the second derivative of the stored energy function Ws with respect to the Green-
Lagrange strain tensor E,
„ d2Ws
~ dEdE ’
and the second Piola-Kirchhoff stress tensor S is then expressed as
(3.84)
S =
dWs
dE
= c: e
(3.85)

90
The constitutive relation of laminated composites can be described by using an or¬
thotropic material law. For that purpose, we express the components Cljkl of tangent elastic
moduli tensor C relative to the fiber reference axis {di, a2, a3} of a lamina, and arrange
these components in a matrix ^Ctjklj (see, e.g., Reddy [1997, p.41 ] and Figure 3.4), using
the same ordering of the strain components in column matrix form as in (3.56) (see also
(3.62)).
Figure 3.4. A fiber-reinforced lamina and fiber reference axes {di, a2, a3}.
CU11
£1122
0
£1133
0
0
£1122
£2222
0
£2233
0
0
0
0
£1212
0
0
0
£1133
£2233
0
£3333
0
0
0
0
0
0
£2323
0
0
0
0
0
0
£1313
(3.86)
where the components Clikl take the following expressions
— i
"A A
^31^23) ^1133 = £3^
A
*1111 _ (1 — ^23^32) *2222 ^2(1-^13^31) *3333 ^3 (1 ~ ^I2v2l)
O - T ,c - x ,C = ,
* 1122 _ ^1 (^21 + ^31^23) *1133 ^3(^13 + ^12^23) *2233 E2 (u32 + Ui2iy3i)
° “ A -C A 'C = A ■

91
A = 1 — 1^12^21 — ^23^32 — ^21 ^13 — 2l^i2^32l/13 >
/Ó1212 /~< /S2323 /~i /S1313 /o
O = (ji2 , O — Ct23 j — ^13 i
VijEj = VjiEi, for (z, J = 1,2,3 , and i ± j) ,
and E\,E2,E3 are the Young’s moduli in the principal material directions {ai,a2,a3},
respectively, and 14, and G¿j the Poisson’s ratio and the shear modulus in the (e¿, e¿) plane,
respectively. Note that, for the special case of isotropy, only two material parameters E
and v are needed:
E\ — E2 = E3 — E , V\2 — ^23 — ^13 — v >
(?12 — G 23 — G13 —
E
2 (1 +1/)
(3.87)
Since matrix of elastic moduli is associated with the principal material direc¬
tions, we need to transform it from the lamina coordinate axes {ai, a2, a3} to the global
Cartesian coordinate axes {bi, b2, b3}. With 9 being the fiber direction angle relative to the
global Cartesian system (see Figure 3.4), the relationship between the lamina coordinate
system and the global Cartesian coordinate system is given by
ai = cos 6bi + sin 0b2 , a2 = — sin 6bi + cos 6b2 , a3 = b3. (3.88)
Since we are developing the formulation in the convective coordinates associated
with the basis {G,}, we have to express the tensor C of elastic moduli in the same convec¬
tive coordinates. Thus,
C = Cabcdaa ® ab ® ac ® ad = CijklG{ ® Gj ®Gk®Gt, (3.89)
where the components Cabcd are given in (3.86), and the components Cljkl are to be com¬
puted for use with the present solid-shell formulation.
From the following component forms of the second Piola-Kirchhoff stress tensor S
S = Sabaa ®ab = SijGi ® Gj ,
(3.90)

92
we obtain the relation between the components and Sab as
S**= (G^o») (GJ-at) S'*6,
where G2,Gj = <5j, and a¿ = a1.
Similarly for the Green-Lagrange strain tensor E,
E = .É^a0 ® ad = EijG1 ® GJ ,
(3.91)
(3.92)
we obtain the following relation
Ecd= (Gk-ac) (iGl-ad)Ekl. (3.93)
Using (3.91) and (3.93) in the following component form of the stress-strain relation (3.85)
with respect to the basis {oj}
gab = cabcdEcd , (3.94)
we obtain
S* = (Gl-aa) (Gj-ab) (Gk-ac) (Gl-ad) CabcdEkl,
(3.95)
which when compared to the component form of (3.85) with respect to the basis {G¿}
Sij = CijklEkl,
(3.96)
leads to
= (G‘-aa) (&-ab) (Gfc*ac) («Gl-ad) C"
abed
(3.97)
The above relation can also be obtained directly by using (3.89).
If we expressed (3.97) in matrix form by using the same ordering of strain and stress
components described in (3.56) and (3.62), which resulted in (3.86), the constitutive matrix
[Cljkl] in the convective coordinates associated with the basis {G¿} is given by
\cijkl] = Tq [cabcd
(3.98)

93
with
Tg =
(¿I)2
(tí)2
t\t\
(t\)2
t\t\
(t\f
(ti?
t\t\
(tlf
t\t\
t\t\
2 t\t\
2t\t\
t\t\ + t\t\
2 t\t\
t\t\ + t22t{
t\t\+t\t
(t¡)2
(t¡)2
t\t\
(tV)
44
t\t\
2t^t\
2t\t\
t\t\ + t\t\
2t\t\
t\t\ + tjtl
t\t3 + t\i,
2t\t\
2 t\t\
t\t¡ + t\t\
2 t\t\
t2A + t\t\
At3 4. At:
lxl3 -r i3t
and t{ = Gj 'cii.
3.3.5. The ANS Method
(3.99)
The assumed natural strain (ANS) method was originally prepared to relieve the shear
locking problem that typically arises as the thickness of the shell goes to zero (MacNeal
[1978] Hughes and Tezduyar [1981], Dvorkin and Bathe [1984]), and was later given a
mixed variational foundation (Simo and Hughes [1986]). Here we use the ANS method to
treat shear locking caused by the transverse shear strains and curvature thickness locking
caused by the transverse normal strain in the present solid-shell element.
3.3.5.1. Transverse shear strains
To avoid shear locking, we adopted the ANS method as applied to the four-noded
shell element in Dvorkin and Bathe [1984]. Here, a linear interpolation of the compatible
transverse shear strains E±3 and E%3 in (3.12), evaluated at the four midpoints A, B, C, D
of the element edges, at £3 = 0 (see Figure 3.5), is applied
E&NS El3 {iA) + (1 + £2) El3 (ic)
i a -e)E^uD)+(i+e)Ec23^B)
eír
(3.100)
where the coordinates of points A,B,C,D are £A = (0,-1,0), = (1,0,0), £c =
(0,1,0), = (—1,0,0), respectively.
The above interpolation on the transverse shear strains eliminates the shear-locking
problem, and allows for pure bending deformation without parasitic transverse shear strains.

94
3.3.5.2. Transverse normal strain
In the case of curved thin shell structures or in the nonlinear analysis, to circum¬
vent the locking effect from parasitic transverse normal strain, we employ an assumed-
strain approximation for the covariant component E%3 of the compatible Green-Lagrangian
strain tensor, as done in Betsch and Stein [1995] for a stress-resultant shell formulation.
Here, a bilinear interpolation of the transverse normal strains sampled at the four comers
E, F, G, H of the element midsurface (Figure 3.5) is imposed, that is
£$*» = ¿JV¡ (£*,{») £3,(£), (3.101)
t=l
with Ni = | (1 + £/ £*) (1 + £,? £2), and the coordinates of the corner points E, F, G, H
being Cr = ZE = (-1,-1,0), £2 = = (1,-1,0), £3 = SG = (1,1,0), £4 =
=(-1,1,0).
Figure 3.5. Eight-node solid shell element in isoparametric coordinates: Sampling points
for ANS interpolations for transverse shear strains (A, B, C, D) and for transverse normal
strain (E, F, G, H).
3.4. Interpolation of the Enhanced Strains
In this section, we first review the regular enhanced-strain method (Klinkel et al.
[1999]) and establish the optimal number of internal parameters for the enhancing strains

95
in the present solid-shell element to pass the membrane patch test and the out-of-plane
bending patch test. We then propose a new efficient way to enhance the strains, and prove
the equivalence of the 2-D plane elasticity elements of Simo and Rifai [1990], Taylor et al.
[1976] and our new enhancing formulation.
3.4.1. The Regular Enhanced Strains Treatment
To include the constant stress in the element (e), the orthogonal condition of EAS
must hold in (3.20), that is
f S I EdV = 0. (3.102)
We define the following component forms of the enhancing strain tensor E as
E = (£) (8) Gj (£) = EklGk (0) ® Gl (0) , (3.103)
where the enhancing strain components with respect to the covectors Gl (£) at any arbitrary
point £ are denoted by Eij, while those with respect to the covectors Gk (0) at the element
center £ = 0 by Ek¡. From (3.103), E^ can be expressed in terms of Ek¡ as follows
4 = %i [G, (Í) ■Gt (0)] \G‘ (0) -Gj ({)] , (3.104)
where the covector Gk (0) can be computed from the vector Gi (0) by
Gk (0) = GqGi (0) ,
with [G0fc/] = [Goki] 1 and Gold = Gk (0) *G¡ (0) , (3.105)
The matrix form of (3.104) is
{A*} = r0|Ey} , (3.106)
where the components of the enhancing strain are arranged in the same order as in (3.56),
and T0 is the matrix that transforms the strain components relative to the basis {G, (0)} to
those components relative to the basis {G¿ (£)}.

96
Using the Column-matrix form {S,J} for the stress components, as in (3.62), and
using (3.106), we can rewrite (3.102) as
I {S”}T {En} dV = I {5^}TT0 dV — 0 . (3.107)
b<•>
For constant stresses, we have the following condition on
I {fy} dV = £ £ jfy} Jdedfde = 0 , (3.108)
where J is the determinant of element Jacobian matrix of the mapping from the isopara¬
metric space â–¡ to the initial configuration B^ of element (e). Let ji?y j be defined by
using the interpolation matrix M and the element parameter as follows
= jM(í)a('>. (3.109)
Substituting (3.109) into (3.106), the enhancing strain {.Ey} can be written as
{Éy} = Q (O a , with Q = jT0M . (3.110)
Remark 3.3. In Simo and Rifai [1990] and other papers such as Klinkel et al. [1999],
the matrix Q involves the calculation of the determinant J0 of the Jacobian matrix evaluated
at eh element center, that is
g = jT0M. (3.111)
From our numerical experiments, both expressions for Q (without J0 as in (3.110) and
with J0 as in (3.111)) led to exactly the same results. We therefore use only (3.110) for
computational efficiency.
If we only enhance the membrane strains [En, E22, 2En], and the transverse normal

97
strain £33, the transformation matrix T0 in (3.106) should be presented as follows
' (al)2
(a?)2
a\al
(alf '
(al)2
(a\f
a\a\
(al)2
2a\a\
2 a\a\
a\al + a\a\
2a\al
. (<4)2
(a!)2
a\aj
(a!)2 .
where the coefficients a? are evaluated by
aj = Gi (£) *Gj (0) , *,¿ = 1,2,3. (3.113)
The interpolation matrix M should be constructed to satisfy (3.108) for arbitrary
matrix The selection of M is not unique. In the present solid-shell element, the
matrix M with the minimum internal parameters of five is in the form of
M =
r f1 0 0 0 0
0 £2 0 0 0
0 0 f1 f2 0
_ 0 0 0 0 £3 J
(3.114)
which is the same as that used in Klinkel et al. [1999]. Our numerical experiments showed
that the selected M as in (3.114) cannot pass the out-of-plane bending patch test, while
passing the membrane patch test.
Remark 3.4. The concept of patch test was first introduced by Bazeley, Cheung,
Irons and Zienkiewicz [1965] and has since demonstrated to give a sufficient condition
for convergence (e.g., Irons and Loikkanen [1983], Taylor, Simo, Zienkiewicz and Chan
[1986], Zienkiewicz and Taylor [1997]). A reviewer pointed out that there is no consensus
about the necessity of passing the out-of-plane bending patch test for convergence, while
passing the membrane patch test is necessary for convergence. On the other hand, we show
in Section 5.4 that the solid shell formulation with Five EAS parameters, which does not
pass the out-of-plane bending patch test, cannot provide accurate results for problems in¬
volving nonlinear material behavior (in addition to large deformation), whereas the present

98
formulation with seven EAS parameters, which does pass the out-of-plane bending patch
test, provides accurate results. I
To pass the membrane patch test and out-of-plane bending patch test, the bilinear
polynomials for the transverse normal strain E33 are necessary (i.e., the minimum number
of EAS parameters for E33 should be three, instead of just one as in (3.114)). Therefore,
the optimal number of EAS parameters should be seven, as shown in the matrix M below
M =
£* 0 0 0 0 0 0
0 £2 0 0 0 0 0
o o £* £2 o o o
0 0 0 0 £3 £*£3 £2£3 J
(3.115)
A computationally more expensive choice for passing both patch tests is to include
the trilinear polynomials for E33 and bilinear polynomials for 7712 (Bischoff and Ramm
[1997] and Betsch and Stein [1996]). In this case, the number of EAS parameters is nine,
with the matrix M as shown below
M =
£*000 0
0 £2 0 0 0
0 0 £* £2 £*£2
0 0 0 0 0
0
0
0
£3 ^e3 £2e
0
0
0
rlc3
0
0
0
2r 3
0
0
0
m3 j
(3.116)
The results of our numerical experiments showed that there is little advantage in using
(3.116), since improvements compared to the use of (3.115) were insignificant.
If we enhance all the six strain components [En, 2£22, Ei2, E33,2E23,2Ei3], the
interpolation matrix M contains complete sets of polynomials up to the trilinear one, and
thus corresponds to a set of 30 EAS parameters (Andelfinger and Ramm [1993] and Klinkel
and Wagner [1997]). In this case, the matrix M is as shown below
M = [m(1),M(21),M(22),M{3)] ,
(3.117)

99
where the submatrices M(21\ Ml'22\ and M(3) are
M(1) =
^ 00000000
0^2 0000000
00 £:£2 00000
0000 £3 0000
00000 f2f3 00
0000000 f1 £3
m(21)
M<22) =
m(3) =
0 0
0 0
£2<£3
0 0
0
0
0
0
0
0
0
0
0
0
e^3
0
0
0
0
0
eee
0
0
0
0
0
0 ^3
0
0
0
0
0
0
0
0
0
0
0
0
0 ^£2 ^
0 0 0 0 0
£^2 £2£3 0 0 0
0 0 £x£2 0 0
0 0 0 f1^3 £2£3
0
0
0
tlee
0
0
0
0
0
0
0
0
0
0
0
0
¿^2
0
0
0
0
0
0
0
0
0
£2£3
0
0
0
0
0
0
£^3 J
0 0
0 0
C1^3 o
0 ^2£3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
e1^3 J
The corresponding transformation matrix T0 previously discussed in (3.106) is now ex¬
pressed as follows
T o =
(a\)2
(a?)2
a\aj
(a?)2
a\a\
aja3
(a2)2
a\a\
(a3)2
a\a\
a^al
2a}a2
2a2a2
a\a\ + a\a\
2a3a3
a2a3 + a2a3
a}a3 + ala'
(<4)2
(a|)2
a3a3
(a!)2
2 T
«3a3
a3a3
2a\a\
2a2a2
a\al + a\a\
2a\a\
a\a\ + a2 a3
a\a| + a\a',
2a\a\
2 a\a\
a\a\ + a\a\
2a3a3
a2a3 +
ajag -f- a^a'
. (3.118)
where the coefficients aj are the same as (3.113).
Without a combination with the ANS method to remedy the shear-locking problem,
the above 30-parameter EAS element (Klinkel and Wagner [1997]) cannot pass the out-of¬
plane bending patch test by itself.

100
3.4.2. Proposed Efficient Enhancing Strains
In the traditional EAS method, as presented above, the 3 x 3 matrix [Go¿j] has to be
inverted in each element so as to obtain the covector G1 (0). For models of composite struc¬
tures that involve a lot of elements in the thickness direction, such inversion clearly adds
to the computational cost. Here we propose a new method (for calculating the enhancing
strain) that avoids the inversion of [Goij], while still passing the required membrane patch
test and the bending patch test. Moreover, this new method yields the same performance in
terms of accuracy when compared to the traditional method that needs the covector G1 (0)
in (3.106), (3.112) and (3.113).
Similar to (3.92), the enhancing strain tensor E (£) can be expressed with respect to
either the convected basis {GJ or the Cartesian orthonormal basis {ea} as follows
E = E{j G' (O Gj (0 = ekl ck e', (3.119)
where ek = ek, and the components eki have the same structure as that of Eij as expressed
in (3.109).
With the use of the orthonormal basis {e^}, we thus avoid the computation of the
covectors Gk (0) and the inversion of a 3 x 3 matrix, as mentioned above.
To compute the tangent stiffness matrix, we need to compute the enhancing strain
components Ey (£) at the Gauss points. Since our formulation is based on convective
coordinates, we need to express the compatible strain tensor Ec and the enhancing strain
tensor E in the same convected basis, so to add these components together to form the total
strain tensor E as expressed in (3.18).
Once the components eki in (3.119) are known, the components E^, can be computed
from eki and (G, (£)} as follows
Éi¡ (() = hi ({)] [t'-Gj («)] , (3.120)
where, unlike the use of the covectors |G’ (0)J in (3.104), we do not need to invert any
matrices, since the basis {e1} is orthonormal and thus c¿ = e*.

101
The next question is how to select the orthonormal basis |efc| for use in (3.119) and
for each element. For the case of flat plates, the convected basis {GJ can be chosen to be
colinear with the global Cartesian basis {e¿}. In this case, we simply choose the basis {ej
to be the same for all elements such that
tí j % 1,2,3 .
(3.121)
For the case of curved shell, in each element (e), we select the orthonormal basis {e¿} such
that e3 is colinear with the convected basis G3 evaluated at the first Gauss point of the
element (e), that is
(e) (£i)
C3 "HcM^ir
(3.122)
and ei and e2 are obtained by rotating the basis vectors ex and e2 through same rotation
operation that rotates e3 to coincide with c3 as defined above. The computation of the
rotation matrix for the above operation is given in Remark 3.5. It is noted that this rotation
matrix yields directly the components of the basis vectors ej and e2.
Remark 3.5. Given any two vectors e, e with e / e there exists a unique rotation
tensor A such that
e = Ae ,
(3.123)
where
A = (e*e) IT2 + e x c + -—-— (e x e) 1 + e*e
where the symbol ” over a vector designates the skew symmetric tensor associated with
the vector (i.e. having the vector as its axial vector, for more details, see e.g., Vu-Quoc,
Deng and Tan [2000]).
Let e be the global Cartesian basis vector e3, and e be the basis vector e3 = e^e, =:
fle¿ as defined in (3.122). The rotation tensor A can then be expressed in terms of the
components t* = el, as follows
A = ti e¿ such that A • ej = tj ,
(3.125)

102
Let ct = U for easy recognition of the symbols. We have
ti = t-ej , (3.126)
With (3.126), (3.125) becomes
A = c, e¿ = t{e{ ej,
(3.127)
with
With the superscript i in A' designating the row index, and the subscript j designating the
column index. We have the expression for
t\ei = Vei as follows
Aj
in terms of the components of e3 = =
t3 +
(*2)2
1 + t3
t1 t2
1 + t3
f1 t2
1+ t3
t3
(tM
i +
t2
-11
(3.128)
Thus the first column in |^A*j contains the components t\ of fj = ei = t\ei. The second
column of [A}j contains the components of f2 =
I
3.4.3. Equivalence Between EAS Element and Incompatible Mode Element
In this section, we will show the equivalence of an EAS four-node element in plane
elasticity as presented in Simo and Rifai [1990] and the incompatible-node four-node ele¬
ment of Taylor et al. [1976] by using the tensor form. In addition, we will also derive a new
element formulation and prove that it too leads to a condensed stiffness matrix identical
to that of the incompatible-mode element of Taylor et al. [1976], The tensor form of the
enhancing strain does indeed allow one to see the connection between various formulations
in an elegant and simple manner. This approach has not been exploited in the literature.
Note that even though Simo and Rifai [1990] stated that their element is in fact identical

103
to the incompatible-mode element of Taylor et al. [1976], they provided no proof, which
is not immediately obvious, even though numerical experiments did confirm that the two
elements are identical. Moreover, we have not seen any such proof of equivalence in the
literature, to the best of our knowledge. In this section, we are mainly concerned with the
small strain case, and thus the small strain notation e is used throughout the section.
3.4.3.1, Tensor form of enhancing strains
The second-order enhancing strain tensor e can be expressed as
e = El a, (3.129)
where IE is a fourth-order interpolation tensor containing the polynomial basis functions,
and a a second-order tensor containing the EAS parameters of an element.
The enhancing strain e can be expressed in either the basis {Ga (0)} or the basis
{ea} as follows
e (0 = £ap (0 Ga (0) G0 (0) = eab (0 ea 0 e6, a, (3, a, b = 1,2 , (3.130)
where £ap are the components of £ with respect to the convected basis {Ga (0)} evaluated
at the element center, and eab the components of £ with respect to the global Cartesian basis
{e“ = ea}. Also, note that indices a and a in (3.130) take values in {1,2}, since we are
dealing with 2-D elements here.
Using (3.130), we obtain the following relationship between components £ap and
components eab of £
= £ab [ea * Gq (0)] [e6 • Gp (0)] , (3.131)
or in matrix form
(3.132)

104
with the strain matrix
{•Sq/j} = £ll> ^22) 2(£i2
" (°i)2 (<*?)2
{a\f (ai)2
2a\a\ 2a\a\ a\a\ + ala\
where the coefficients are obtained by
and the matrix Fl defined below
Fo =
a\al
a\a\
(3.133)
af = Ga (0) ‘ep , a,/? =1,2.
(3.134)
Similarly, the EAS parameter tensor a can also be expressed with respect to different
bases. Here, we choose to express e* with respect to the basis Gm (0) en and to the basis
es <8> e9 as follows
a = (0) en = ® eff, (3.135)
where the superscript (s) in the components a£l of a with respect to the basis Gm (0)en
represents the EAS parameters in the Simo and Rifai [1990] formulation, and the super¬
script (t) in the components of a with respect to the basis ef e9 represents the EAS
parameters in the Taylor et al. [1976] formulation.
From (3.135), the transformation between the components a$n and a¡l is then
«S = a/n(e/,G!m(0)) , (3.136)
or in matrix form
(3.137)
where the indices (ran) and (/n) are arranged in the order {11, 22,12, 21}, and the matrix
M-1 take the expression
M_1
a\ 0 0 a\ "
0 al a\ 0
0 a\ a} 0 ’
a\ 0 0 af _
(3.138)

105
where the coefficients were given in (3.134).
For the relation between the enhancing strain components ¿y and the EAS parame¬
ters a$n, it follows from (3.129), (3.130)i and (3.135)i that the fourth-order interpolation
tensor IE can be expressed in component form as follows
E = (0) 0 Gj (0) 0 Gk (0) 0 ei, (3.139)
leading to the following component matrix equation
(3.140)
Similarly, for the relation between the enhancing strain components eab and the EAS
parameters a^, it follows from (3.129), (3.130)2, and (3.135)2 that the interpolation tensor
IE can be expressed in component form as follows
IE = E(t)Cadbea 0 eb 0 ec 0 ed ,
(3.141)
leading to the following component matrix equation
IE(i)
{“2}
(3.142)
Substituting (3.137) into (3.140), then using the result in the left hand side of (3.132), and
next substituting (3.142) into the right hand side of (3.132), we obtain the relation between
E<'>“
V
and
E(t)
as follows
E(s)
= Fn
E{t)
M .
(3.143)
The interpolation matrix
E{t)
is chosen to be the same as in Taylor et al. [1976],
and the derivatives evaluated at the element center, that is
1
E(t)
¿(0
-0y^(o) 0 £2y,$>(0) o
o 0 a* (0) o -e*,e(0)
0*4* (°) (°) -0 x,(i (o) 0y,ci (o) _
(3.144)

106
where the rows correspond to the strain components {en> £22, and the columns cor¬
respond to the EAS parameters arranged following the order cd = {11,22,21,22}.
By substituting (3.144) into (3.143), and by rearranging the columns in the resulting
component matrix
IBM"
, we obtain 10
1
r^1
0
0
01
1E^‘
0
e
0
0
V
7(0
_ 0
0
e1
(3.145)
where the rows correspond to the strain components jin, €22, 2i12}, and the columns cor¬
respond to the EAS parameters ak¡ arranged following the order kl = {11,22,12,21}. It
can be seen that (3.145) is the same strain-enhancing interpolation matrix for 2-D elasticity
elements as suggested in Simo and Rifai [1990].
Using the tensor formalism, we can derive an EAS formulation that is different from,
but equivalent to, the EAS formulation in Simo and Rifai [1990] and the incompatible
element of Taylor et al. [1976]. To this end, let’s express the interpolation tensor E with
respect to the basis vectors {G¿ (0)} and jG1 (0) j at the element center as follows
E = E(0) (8) Gj (0) (8) Gk (0) ® G, (0) . (3.146)
In parallel to the above, let’s express the EAS parameter tensor a in the same basis:
a = a$Gp (0) <8> Gq (0) . (3.147)
Similar to (3.135), we find the relation between cS*¡j and in matrix form to be
(3.148)
where the indices (pq) and (kl) follow the order {11,22,12,21}, and the matrix T 1 takes
the form
’ a\ 0 af 0
0 a\ 0 a\
a\ 0 a\ 0
0 a\ 0 a}
(3.149)
10 Symbolic computation was used to carry out the computation in (3.143) to obtain the result shown in
(3.145).

107
when the coefficient are computed as in (3.134).
Since the enhancing strain jí¿j| with respect to the convected basis {Gi (0)} can be
expressed as
{«»}=r>”] «>}={4?},
it follows from (3.148) and (3.150) that the relation between the interpolation matrix
and the interpolation matrix is given by
(3.150)
E
(3.151)
There are thus infinitely many ways to define the interpolation matrix E^- that
are equivalent to each other from the tensor viewpoint. The enhancing-strain component
matrix {gab} relative to the global Cartesian basis {ea} can then be expressed in several
ways
\E^
V J
{ffab}
= g® -
(«&} ,
with g(i)
oT {íB*'1"} M-*
, (3.152)
{¿aft}
= g^.
Ri} ,
, with g(s)
-*T\
â–º = G[t)M,
(3.153)
= gw
Ri} ,
, with g^
f = Gl,)T,
(3.154)
where the matrices £(i) and are the interpolation matrices relating to the EAS parame¬
ters Qf^n and a^n, respectively, to the Cartesian strain components eab, and is another
choice of interpolation matrix.
3A3.2. Equivalence of condensed stiffness matrices
The matrix form of the strain tensors is used in finite-element formulation. In a
typical element (e), the compatible strain matrix and the enhancing strain matrix
{¿ab} are interpolated respectively as follows
(3.155)
{<4} = ,
{?<4 = 6a1” ,
(3.156)

108
where B is the strain-displacement matrix, and Q one of the interpolation matrices given
in (3.152)—(3.154), depending on the formulation used.
The enhanced strain matrix {£ab} is obtained by adding the enhancing strain {ea&} to
the compatible strain {e£6}
{£-} = fó} + {54 = B | ^ ^ | . (3.157)
The element stiffness matrix k^e\ which is similar to (3.78), can be obtained by
fc(e) = J BTCBdV = j \*l]c[BQ)dV
h<«) »»(«) ^ *
f BrCBdV f BTCQdV
(«>
j gTCBdv / gTcgdv
*(e)
L 0
B
(«)
U(e) u(e)
^uu A'i¿a
JU(«) ju(e)
(3.158)
where C is matrix of linear elastic moduli.
The condensed element stiffness matrix has the same form as in the formulation by
Taylor et al. [1976], Simo and Rifai [1990], and in the formulation presented in (3.146)-
(3.149)
(3.159)
For the formulation of Simo and Rifai [1990], the second term in (3.159) leads to
t.(e) [/u(e)] 1 ju(e) _
'"íí/v '•'aiA
J BTcg{s)dv
T -l
f g[s)Tcg[s)dv
,(<)
I g{s)TCBdV . (3.160)
,<«>
Substituting in (3.160) the following relation as given in (3.153)2
g{3) (S) = G{t) (0 M,
(3.161)

109
where M is the inverse of M 1 in (3.138), and is a constant nonsingular matrix.
Then substituting (3.161) into (3.160), we obtain
ju(e) fju(e)] 1 u(e) _
^uq ["'aaj "'em
l n'rcg">dv
MM~X
Í g[t)Tcg[t)dv
-i
m~tmt
Í gMTCBdv
&
&
-1
= J BTcgwdV
I guTcg(t)dv
I gWTCBdV,
(3.162)
which is exactly the same as the second term of the condensed element stiffness matrix k^
in (3.159) obtained from the formulation of Taylor et al. [1976]. Since the compatible part
*¡2 of fc(I) are the same, the condensed element stiffness matrix kin Simo and Rifai
[1990] is exactly the same as in Taylor et al. [1976].
From (3.154), we have the following relation between g^ and
g(s) (£) = gW (i)r1,
(3.163)
where T is the inverse of T-1 in (3.149), and is a constant non-singular matrix. Following
the same procedure as in (3.159) and (3.162), one can easily establish that the new enhanc¬
ing strain as given in (3.146)—(3.149) yields a condensed stiffness matrix identical to that
obtained by Taylor et al. [1976] and Simo and Rifai [1990].
It should be noted that a formulation similar to (3.146)—(3.149) but evaluated at an
arbitrary point £ inside the biunit cube will also produce an element that passes the patch
test. In the previous section, we have indeed selected £ = £¡ (i.e. the first Gauss point).
3.5. Numerical Examples
The finite element formulation of the present low-order solid-shell element for static
analyses of multilayer composite shell structures, presented in the previous sections, has
been implemented in both Matlab and the Finite Element Analysis Program (FEAP), devel¬
oped by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a Compaq Alpha work¬
station with UNIX OSF1 V5.0 910 operating system. The tangent stiffness matrix and the

110
residual force vector are evaluated using full 2x2x2 Gauss integration in each element.
A tolerance of 10“18 on the energy norm is employed in the Newton iteration scheme for
the convergence. Below we present numerical examples involving geometrically nonlinear
static analysis, with isotropic and orthotropic elastic materials.
3.5.1. Patch Tests and Optimal Number of Parameters
The patch tests for the membrane behavior and the transverse out-of-plane bending
behavior of plate and shell elements were suggested by MacNeal and Harder [1985]. In
these tests, a patch of five plate/shell elements with four external nodes and four internal
nodes at the (X, Y) coordinates in Figure 3.6. (see also MacNeal and Harder [1985]):
Since we are interested in submitting the solid-shell element formulation described in the
previous sections to the above patch test, the number of nodes is actually doubled, with
one series of nodes at the top surface of the plate (Z = h/2), and one series at the bottom
surface (Z = —h/2). The aspect ratio is defined by the triplet A L : W : h, where
(L, W, h) designates the length, width, and thickness of the plate, respectively. We will
subject the following elements to the above mentioned patch tests: The proposed solid-
shell element with seven EAS parameters, the solid element with thirty EAS parameters in
Klinkel and Wagner [1997], the solid-shell elements with five EAS parameters in Miehe
[1998b] and Klinkel et al. [1999], It is noted that we will discuss the EAS formulation
based on the displacement gradient in details in Section 4.3, which is more efficient than
Miehe [1998b],
Coord. (X,Y):
1 : (0.04,0.02)
2 : (0.18,0.03)
3 : (0.16,0.08)
4 : (0.08,0.08)
Figure 3.6. The five-element patch of a plate: Geometric dimension (L : W : h = 0.24 :
0.12 : 0.001); E = 106, v = 0.25; 1,2, 3, 4 are node numbers on top surface.

Ill
3.5.1.1. Membrane patch test
The following displacements u, v, w along the X, Y, Z, axes, respectively, are pre¬
scribed at the top and bottom exterior nodes of the plate:
u = 1(T3 {X + Y/2) , v = 1(T3 (Y + X/2) , w = 0 . (3.164)
The theoretical solution is a constant in-plane membrane stress field in all five ele¬
ments of the patch as shown below
Our numerical results show that the proposed solid-shell element, together with the
element formulations by Klinkel and Wagner [1997], Miehe [1998b], and Klinkel et al.
[1999] all pass the membrane patch test, that is, the computed displacements at the interior
nodes agree exactly with (3.164), and the computed stresses in all elements agree exactly
with (3.165).
3.5.1.2. Out-of-plane bending patch test
To construct a constant stress state, the displacements (u0, v0, w0) and rotations
(Ox , 0y) at the midsurface of the plate (i.e., Z = 0) should be
u0 = v0 = 0 , wo = 10-3 (X2 + XY + Y2) /2 ,
9X = 10-3 {Y + X/2) , dY = -10"3 {X + Y/2) . (3.166)
With the deformation as prescribed by (3.166), the displacements of the external nodes at
the top surface and at the bottom surface of the plate can then be prescribed by
[u,v,w\T = [uq, vo, wo]r± ^[9y ,~9x ,0f , (3.167)
which is the prescribed boundary conditions for the out-of-plane bending patch test.
The exact nodal displacements at the interior nodes are calculated by using (3.167)
(see, e.g., Table 3.2 for the finite-element displacement of the interior nodes at the top

112
Table 3.2. Displacements at internal nodes in bending patch test. Present element with 7
EAS parameters, and with ANS.
node number
u
V
w
1
2.500 x 10~8
2.000 x 10~8
1.400 x 10“6
2
9.750 x 10~8
6.000 x 10'8
1.935 x 10~5
3
1.000 x 10~7
8.000 x 10“8
2.240 x 10~5
4
6.000 x 10"8
6.000 x 10-*
9.600 x 10~6
surface). The theoretical solutions of the stresses at the top and bottom surfaces of the plate
are
(3.168)
Table 3.3. Displacements at internal nodes in bending patch test. Solid element with 30
EAS parameters, and without ANS.
node number
u
V
w
1
1.15827 x 10"7
2.34981 x 10~8
5.44389 x 10~6
2
1.40031 x 10"s
8.58585 x 10~8
2.51957 x 10-5
3
1.81490 x 10~8
6.05335 x 10“8
2.89471 x 10~5
4
1.37490 x 10~7
4.70933 x 10"8
1.65964 x 10-5
Without the use of the ANS method on the transverse shear strains, the solid element
even with the full 30 EAS parameters (Klinkel and Wagner [1997]) cannot pass the out-
of-plane bending patch test, that is, it cannot obtain the same nodal displacements shown
in Table 3.2, and the stress state shown in (3.168). The calculated displacements at the
interior nodes on the top surface for the solid element with 30 EAS parameters are listed in
Table 3.3.
Some researchers used the ANS method in combination with the EAS method with
one parameter for transverse normal strain E33 (e.g., Miehe [1998/?], Klinkel et al. [1999]).
Here, we show, however, that this formulation cannot pass the out-of-plane bending patch
test (Table 3.4), while the computed results are much better than 30-parameter EAS solid
element (Klinkel and Wagner [1997]). These results indicate that the ANS method plays
an effective role for remedying shearing locking in thin-shell problem.

113
In the present work, we propose an optimal formulation for EAS treatment, that is,
the minimal number of parameters that is required for the element to pass both the mem¬
brane patch test (easy) and the out-of-plane bending patch test (more difficult), in which
we use both the ANS method and a seven-parameter EAS method (four parameters for
the membrane strains (£n, £22, 2£12), and three parameters for the for transverse normal
strain £33). It is noted that the three EAS parameters for the transverse normal strain £33
correspond to a polynomial with one linear term £3 and two bilinear terms i;3^1 and £3£2.
Numerical results show that this solid-shell element formulation passes the out-of-plane
bending patch test, namely, obtains exactly the same displacements at the interior nodes as
shown in Table 3.2, and the stresses at the top surface and at the bottom surface as shown
in (3.168).
Bischoff and Ramm [1997] used a nine-parameter EAS method to treat the incom¬
pressibility problem (five parameters for the membrane strains, and the full four parameters
for the transverse normal strain). It turns out that this choice also passes the membrane and
the out-of-plane bending patch tests, but with higher computational effort, compared to the
seven EAS parameters that we are proposing. In this sense, the combination of the ANS
method and the proposed seven-parameter EAS method (four for membrane strains, and
three for transverse normal strain) is computationally optimal.
Table 3.4. Displacements at internal nodes in bending patch test. Solid-shell element with
5 EAS parameters, and with ANS.
node number
u
V
w
1
2.49406 x 10~8
2.00207 x 10"8
1.39803 x 10“6
2
9.74938 x 10~8
5.94341 x 10"8
1.93334 x 10~5
3
1.00572 x 10~7
8.01161 x HT8
2.23496 x 10~5
4
5.93407 x IQ"8
5.97675 x 10"8
9.55657 x 10“6
3.5.2. Cantilever Plate
We use a single-layer cantilever plate to establish the correctness of the present for¬
mulation, by comparing the computed results to those published in the literature (e.g., those

114
from geometrically-exact shell theory (Vu-Quoc, Deng and Tan [2000])).
3.5.2.1. Cantilever beam: in-plane bending
This problem has been previously investigated by Simo et al. [1990] to show the
superior performance of their mixed finite element shell formulation for the membrane
behavior based on the Hellinger-Reissner functional. A cantilever beam subjected to an
end load is discretized with ten elements. The first mesh contains elements with uniform
and regular geometry (Figure 3.7), whereas the second contains highly distorted elements
(Figure 3.8). The beam has length L — 1.0, width W = 0.1, and the thickness h = 0.1.
The material properties are
£=1.0x 107, i/ = 0.3 , (3.169)
where E and v are the Young’s modulus and the Poisson’s ratio, respectively.
The load deflection curves for both meshes are shown in Figure 3.9. The present
solid-shell element based on the Fraeijs de Veubeke-Hu-Washizu functional shows the
same accurate results and insensitivity to mesh distortion as for the element reported in
Simo et al. [1990],
Figure 3.7. In-plane bending: Deformed and undeformed regular mesh with 10 solid-shell
elements.

115
Figure 3.8. In-plane bending: Deformed and undeformed distorted mesh with 10 solid-
shell elements.
Figure 3.9. In-plane bending: Load deflection curve for both regular mesh and distorted
mesh with 10 load steps.
3.5.22. Cantilever plate: out-of-plane bending
A cantilever plate of length L = 10 and width W = 1 is subjected to the transverse
shear loading F at the free end. We consider three different values of the plate thickness

116
h(l, 0.1,0.01), which correspond to three different aspect ratios L/h( 10,100,1000), re¬
spectively. Ten elements are used to model this problem (Figure 3.10 and Figure 3.12). We
use various different aspect ratios to test the performance of the present solid-shell element.
To have the same level of deflection magnitude regardless of the thickness h, the applied
loading F is set to be proportional to the thickness raised to power three (i.e., h3) in the
numerical examples in this section.
The material properties are prescribed to be
£=1.0xl07, i/ = 0.4, (3.170)
where E and v are the Young’s modulus and the Poisson’s ratio, respectively.
Figure 3.10. Out-of-plane bending: Geometry and mesh of cantilever plate.
First, we point out the importance of using a combination of EAS and ANS methods
in the present element, by solving the linear problem with the transverse loading F =
104/i3 at the free end. A comparison of the tip deflection of the plate for different options
in the present solid-shell element and beam theory is shown in Figure 3.11, where all
results are normalized to the current solid-shell element. There are differences in the results
obtained from elements with EAS and from elements without EAS (purely displacement
formulation): the results show that the EAS method clearly improves the bending behavior,
particularly for low aspect-ratio structures; its influence diminishes dramatically, however,
as the aspect ratio increases. The ANS method, on the other hand, remains important
throughout the large range of aspect ratio, from low to high. For aspect ratio larger than 100,
the ANS method plays a more important role than the EAS method. But the ANS method
alone cannot provide accurate results, when compared to the exact solutions for beams,

117
i
0.8
v
3 o-6
ÍT
0.4
0.2
10° 10’ L/h 1°2 1°3
Figure 3.11. Out-of-plane bending: Relative importance of EAS and ANS method in re¬
lieving shear locking.
as shown in Figure 3.11. A combination of both EAS and ANS methods is important to
achieve accurate results, regardless of the aspect ratio. In Figure 3.11, the small discrepancy
between the exact solution for the Timoshenko beam and the present sold-shell formulation
(with EAS and ANS) is due to the effect of the Poisson’s ratio. If the Poisson’s ratio is set
to zero, the present solid-shell formulation produces results that agree with the Timoshenko
beam theory.
For flat plates undergoing small deformation, numerical results show that it is suffi¬
cient to consider the ANS treatment for only the transverse shear strain E13 and the
additional ANS treatment for the transverse normal strain E33 does not change the numer¬
ical results.
For the geometrically nonlinear problems, we applied the tip loading in five load
steps to reach the total force F = 5 x 104/i3. The free-tip transverse displacement along
the force direction at the comer of the midsurface of the plate agrees well with the tip
displacement obtained from the geometrically-exact shell element (e.g., Vu-Quoc and Tan
~ -fc,--- 1 fr fr 1
* ~ ~ - -I
N \
\ \
w
V'
\'
\ \
\\
\\
\\
\\
-O- disp
EAS only
—l— ANS only
-0- EAS&ANS
-0- Timoshenko beam
— E-B beam

118
[2002a]) in which the selectively reduced integration was employed; see Table 3.5. Even in
the extremely thin plate case (aspect ratio = 6667 or h = 1.5 x 10-3), the present solid-shell
element yields excellent results, without any sign of shear locking.
Table 3.5. Out-of-plane bending: Tip deflection of cantilever plate for wide range of aspect
ratios. Comparison between proposed solid-shell and geometrically-exact shell.
Aspect ratio L/h
Present element
Geometrically-exact shell element
10
7.5083
7.4897
100
7.4146
7.4144
1000
7.4137
7.4137
5000
7.4140
-
6667
7.4139
-
To verify the coarse-mesh accuracy of the proposed solid-shell formulation, we use
the numerical solution obtained from using 640 geometrically-exact shell elements (80
elements along the length of the plate, and 8 elements along its width) as the reference
solution. The results tabulated in Table 3.6 show that even the coarsest mesh of two solid-
shell elements can capture the geometrically nonlinear response with good accuracy. In
general, the results obtained from the present solid-shell formulation agree well with those
from the geometrically-exact shell formulation.
Table 3.6. Out-of-plane bending: Convergence of computed solution, for h = 0.01.
Mesh (elem. aspect ratio)
geometrically-exact shell
present (relative error (%))
2 x 1(500)
7.61149
7.61355 (5.6)
10 x 1(100)
7.41366
7.41366 (2.8)
20 x 2(50)
7.15862
7.16073 (0.7)
40 x 4(25)
7.19335
7.19744 (0.2)
80 x 8(12.5)
7.20918
7.21180 (0.0)
To check the conditioning of the tangent stiffness matrix of the proposed solid-shell
element, we present in Table 3.7 the number of Newton iterations for each load step and
the total number iterations for five load steps for three different shell formulation: The
present solid-shell formulation, the four-node solid-shell of Bischoff and Ramm [1997]

119
Figure 3.12. Out-of-plane bending: Undeformed and deformed mesh with 10 solid-shell
elements.
(see its implementation in the Appendix A.l), and the geometrically-exact shell element
(Vu-Quoc, Deng and Tan [2000]). For thick and moderately thin plates (10 < aspect ratio
< 200), the three shell element formulations have a similar performance. For thin plates
(aspect ratio > 500), the geometrically-exact shell element formulation provides the best
performance. The result indicates that the kinematic description of the geometrically-exact
formulation, in which the displacement of a reference surface and the finite rotation of a
transverse fiber are unknown kinematic quantities, leads to a better-conditioned tangent
stiffness matrix than that of the solid-shell formulation, but with more complex operation
(Vu-Quoc, Deng and Tan [2000], Vu-Quoc et al. [2001]).
3.5.3. In-plane Bending Problem with Nearly Incompressibility
As noted before, the displacement-based formulation exhibits severe locking when
using the full 3-D constitutive models in the incompressible limit. Here we demonstrate
that the proposed solid-shell element is able to alleviate the parasitic phenomenon.
The beam is clamped at one end and subjected to an in-plane bending moment at the

120
Table 3.7. Out-of-plane bending: Number of iterations in each load step and total number
of iterations in five load step.
aspect ratio
present solid-shell
4-node solid-shell
geometrically-exact shell
10
9,10, 8, 7, 7(41)
9,10, 9, 8, 7(43)
8, 9, 8, 8, 7(40)
100
12,13,10, 9, 9(53)
12,13,11,10,10(56)
11,15,10,10, 9(55)
200
13,13,12, 9, 9(56)
13,14,13,10,10(60)
12,14,11,10,11(58)
500
20,15,16,10,10(71)
20,17,20,12,10(79)
13,14,12,10,10(59)
1000
25,16,15,10,10(76)
26,21,17,13,11(88)
15,16,15,11,10(67)
other (Figure 3.13). The material properties are
£ = 4.0xl03, v- 0., 0.25,0.49999, (3.171)
where E and u are the Young’s modulus and the Poisson’s ratio respectively.
The beam is modeled with 10 x 4 x 1 finite element mesh with the plane strain
constraint (constraining the thickness change). The results of this linear problem are listed
in Table 3.8. The results from the standard displacement formulation, and the classical
B-bar element (Nagtegaal, Parks and Rice [1974], Simo and Hughes [1986]), and the exact
solution from the Euler-Bernoulli beam theory for v = 0 are given for the comparison.
Figure 3.13. In-plane bending problem with nearly incompressibility: Geometry and mesh.
Table 3.8. In-plane bending problem with nearly incompressibility: Deflection at center of
free end
u
Displacement
B-bar
Present
Exact
0
0.1834
0.1868
0.2063
0.20625
0.25
0.17434
0.1810
0.1921
0.49999
1.591E-04
0.1453
0.1456
It can be seen that the present solid-shell element performs very well for all cases,

121
including the incompressible limit, while the displacement formulation and B-bar element
lags behind for small values of v. The displacement formulation locks severely for the
nearly incompressible limit in plain strain case. This indicates that for problems where
the thickness stretch may be constrained (contact or external surface loading) and the in¬
compressible constraints (e.g., plasticity) be involved, the present element will produce a
reliable results.
3.5.4. Snap-through of a Shallow, Cylindrical Roof under a Point Load
This example illustrates the use of the arc-length method (e.g., Simo, Wriggers,
Schweizerhof and Taylor [1986] and Schweizerhof and Wriggers [1986] for the implemen¬
tation in FEAP) to obtain the unstable static equilibrium response of an elastic shell struc¬
ture that exhibits snap-through behavior. The shell in this case is a shallow, cylindrical roof,
pinned along its straight edges and loaded by a point load at its midpoint. The dimensions
of roof and material properties are shown in Figure 3.14. The roof is assumed to deform
in a symmetric manner, so that one quadrant is discretized, as shown in Figure 3.14. A
regular 4x4 in-plane mesh of solid-shell elements are used, and two elements are used in
the thickness direction since the hinged boundary need to be prescribed at the middle line
of the straight edge of roof. In the R/h — 200 case, the load-deflection path contains two
Figure 3.14. Snap-through of a shallow, cylindrical roof under a point load. A 4 x 4 x 2
mesh is used for one quarter of the panel, with symmetric boundary conditions. Two cases
are considered by varying the thickness: R/h = 200 and R/h = 400.
limit points, and the displacement control can be used to solve the equations successfully
(left of Figure 3.15, with 30 load steps). In the R/h = 400 case, the path, however, “kicks

122
in” after the first limit point, the arc-length control has to be used (right of Figure 3.15, with
36 load steps). The results shown in Figure 3.15 are in close agreement with those reported
in Rifai [1993, p.245], and Example 4.2.6 of ABAQUS [1995].
deflection path for the R/h = 200 case, where displacement control is employed (left),
and load-deflection path for the R/h — 400 case, where arc-length control is employed
(right), both compared to the geometrically-exact shell element.
3.5.5. Pinched Hemispherical Shell
The pinched hemispherical shell can be considered as one of the most severe (and
meaningful) benchmark problem for nonlinear analysis of shells (Stanley [1985]). The
undeformed configuration of the hemispherical shell has an 18° hole at the top (North
Pole), and is subjected to two inward forces at 0° and at 180° longitude on the equator, and
two outward forces at 90° and 270° longitude on the equator, respectively (see Figure 3.16).
The material and geometric properties are
E = 6.825 x 107, v = 0.3 ,
R= 10, h = 0.04, (3.172)
where E and u are the Young’s modulus and the Poisson’s ratio, R the radius, and h the
thickness of the shell, respectively.
Because of symmetry, only one quadrant of the shell is modeled (Figure 3.16). The
computed displacements for the small-deformation case along the direction of unit loads

123
are listed in Table 3.9. For four different meshes with increasing number of elements,
the values of displacements are normalized with respect to the converged value of 0.094
(MacNeal and Harder [1985]). The performance of the present solid-shell element is quite
remarkable, compared to the geometrically-exact shell element.
Table 3.9. Pinched hemispherical shell: Normalized displacement for linear small defor¬
mation.
Node per side
present element
geometrically-exact element
3
1.083
0.909
5
1.040
0.993
9
1.003
0.987
17
0.995
0.988
For the large-deformation case, we choose a smaller thickness of h = 0.01 and the
radius-over-thickness ratio R/h = 1000. The same problem was considered by Parisch
[1995] for investigating the behavior of several types of elements in thin-shell applications.
The mesh is composed of 16 x 16 x 1 solid-shell elements. The total load is applied in
fifteen equal steps. The final deformed mesh configuration is shown in Figure 3.18 without
any magnification of the deformation. A plot of the pinching loads versus the deflections
at the corresponding pinching points is shown in Figure 3.17, by comparing with the 4-
node degenerated shell element in Parisch [1995]. From Table 3.10, it is observed that
the present solid-shell element is somewhat better than the four-node degenerated shell
element in Parisch [1995], when both are compared to the converged results of eight-node
degenerated shell element in Parisch [1995],
Table 3.10. Pinched hemispherical shell: large-deformation displacements due to pinched
force F = 5.
Element type
u at B
u at C
4-node shell elem. (Parisch [1995])
3.24803
5.43434
present solid-shell elem.
3.26055
5.48331
8-node shell elem. (Parisch [1995])
3.32798
5.84238

124
Figure 3.16. Pinched hemispherical shell: One quadrant of hemisphere with 18° hole.
Figure 3.17. Pinched hemispherical shell: Load deflection curve of the nonlinear calcula¬
tion.

125
C
B
Figure 3.18. Pinched hemispherical shell: Deformed hemisphere at F/2 = 2.5, viewing
through hole.
3.5.6. Multilayer Composite Plate
While the results in the previous section are restricted to single-layer shell, we now
provide numerical examples related to multilayer composite shells in both liner and non¬
linear deformation regimes.
3.5.6.1. Two-layer composite plate: linear solution
Consider a two-layer laminated plate with angle ply (±0) construction (Figure 3.19,
0° along axis X), are clamped on all sides, and subjected to an uniformly distributed trans¬
verse downward load on the top surface. The side length of the square plate is a = 20.0,
the layer thickness ^)h = 0.01, and the total thickness h = 0.02. The magnitude of the
uniformly distributed load is q = 1.0. The layer material properties are
Eu = 40 x 106 , E22 = E33 = 106 ,
^12 = ^13 = ^23 ~ 0.25 ,
G\2 -- G\3 — G23 — 0.5 x 106,
(3.173)

126
Because of fiber-orientation induced stretching/bending coupling, which eliminates the
symmetry condition found in single-layer homogeneous plates, the entire plate has to be
modeled. A mesh of 6 x 6 x 2 element, with one element per layer in the thickness direction,
is used.
In Table 3.11, the transverse displacement of the plate center is compared to both
the series solution given by Whitney [1969] and the computational results obtained with
a high-order hybrid multilayer shell element with the same in-plane mesh in Spilker and
Jakobs [1986]. The present solutions are more accurate than those in Spilker and Jakobs
[1986] , when taking the series solution of Whitney [1969] as reference. Both sets of nu¬
merical results (Spilker and Jakobs [1986] and Whitney [1969]) show an increased relative
error when compared to the series solution as the ply angle 9 decreases, with our solution
being always closer to the series solution of Whitney [1969]. The magnified deformed
configuration of the two-layer composite plate is shown in Figure 3.20.
Table 3.11. Two-layer composite plate: linear transverse displacement at plate center.
Angle ±9
series solution
Spilker et al.
present element
relative error (%)
±45
57.80
58.58
58.92
1.95
±35
55.26
56.88
56.75
2.71
±25
47.10
51.44
50.22
6.62
±15
33.82
40.18
38.15
12.8
To test the performance of the present solid-shell element for high aspect ratios by
decreasing the plate thickness, we decrease the magnitude of the loading as the cube of
the plate thickness, so that the transverse displacement at the plate center remains the same
in the series solution. The results compiled in Table 3.12 show that the computed linear
solutions are accurate for a large range of aspect ratios, for the ply angle 9 = ±45°.
3.5.6.2. Multilayer composite plate with ply drop-offs
This example demonstrates the applicability of the present solid-shell formulation to
analyze composite structure with ply drop-offs; an example of such structures would be a
(composite) plate with piezoelectric patches at the top or bottom surface.

127
Figure 3.19. Two-layer composite plate: Undeformed mesh with solid-shell elements.
Figure 3.20. Two-layer composite plate: Deformed shape with solid-shell elements.
Table 3.12. Two-layer composite plate: Transverse displacement at plate center for large
plate aspect ratios, with ply angle ±45.
Layer aspect ratio (a/ y)h)
series solution
present element
relative error (%)
1000
57.80
58.92
1.95
10000
57.80
58.91
1.93
20000
57.80
57.54
0.44
In this example, each layer is made of unidirectional fiber-reinforced material, with
the fiber directions aligned at 45/-45/45/-45/45/-45 degrees with respect to the length di¬
rection (Figure 3.21, 0° along axis X). The plate, with length L = 12 and width W = 6,
has a total of six layers at the thick end, which is clamped; the free thinner end is sub¬
jected to a transverse normal load distribution uniformly along the free edge. The location
of the ply drop-offs are at X = 4 and X = 8 with the top two layer removed after each
drop-off. The layer material properties are Eu = 25 x 109 , E22 = E33 = 109 , i/12 =
^13 = ^23 — 0.2 , C712 = Gi3 = G23 = 0.5 x 109 . Three different values of the

128
thickness of any given layer (all six layers have the same thickness) are considered, that is,
y)h = 0.1,0.01, and 0.004, for £ = 1,..., 6, where (£) represents the layer number.
The FE mesh is composed of 288 elements with 12 elements along the length, six
elements along the width, and one element for each layer through the thickness direction.
The applied load on the free tip is increased in five load steps up to the total force F —
6 x 109 (¿)h3, which is proportional to the cube of the layer thickness. The computed free-
tip transverse displacement along the force direction at the corner of the bottom surface
of the plate are presented in Table 3.13. It is observed that unlike the isotropic plate in
Subsection 3.5.2.2, the level of deflection magnitude is not proportional to the cubic of
the thickness in this nonlinear composite plate problem. The deformed plate is shown in
Figure 3.22.
Figure 3.21. Multilayer composite cantilever plate with ply drop-offs: Undeformed mesh
with (t)h = 0.1.
Table 3.13. Multilayer composite cantilever plate with ply drop-offs: Nonlinear transverse
displacement.
Layer aspect ratio L/ (¿)h {(t)h)
Transverse disp.
120(0.1)
6.72325
1200(0.01)
6.11453
3000(0.004)
6.01374
To test coarse-mesh accuracy, we consider the plate with layer thickness = 0.01,
and use the computed solution obtained from the FE mesh with 24 elements along the
length and 12 elements along the width as the reference solution. From Table 3.14, it can
be seen that the coarse mesh with six elements along the length and three elements along

129
the width already captures the geometrically nonlinear response with a great degree of
accuracy.
Table 3.14. Multilayer composite cantilever plate with ply drop-offs: Performance at coarse
mesh, (¿)h = 0.01.
Mesh (elem. aspect ratio)
present element
relative error (%)
6 x 3(200)
6.03514
0.02
12 x 6(100)
6.11453
1.50
24 x 12(50)
6.02229
0.00
Figure 3.22. Multilayer composite cantilever plate with ply drop-offs: Deformed mesh with
(.t)h = 0.1.
3.5.7. Multilayer Composite Hyperbolical Shell with Ply Drop-offs
This examples was considered to test the current solid shell element formulation in
shell structures having discontinuous geometry and strong geometric nonlinearity. The
shell structure consists of three layers with the same thickness y)h = h/3 placed sym¬
metrically with respect to the middle surface and two ply drop-offs at Z = 9 and Z = 15,

130
respectively (Figure 3.23). For the shell without ply drop-offs, we have compared with
Basar, Ding and Schultz [1993] and the results agree each other, while a layerwise shell el¬
ement with complex rotation update was employed in Basar et al. [1993]. Only one eighth
of the shell structure is modeled with a mesh of 14 x 14 solid-shell elements for inner layer,
14 x 10 middle layer, 14 x 6 outer layer by assuming the symmetry (Figure 3.23, 0° along
circumferential direction). The layer material properties are En — 40 x 109 , Eyi =
•£33 = 109 , 1^12 = ^13 =: ^23 = 0.25 , O12 = G13 = ^23 = 0.6 x 109 . The anal¬
ysis was carried out for three different stacking sequences: [0°/90o/0°], [90°/0°/90°] and
[—45°/0°/45°]. The deformed shapes shown in Figure 3.24 exhibit a considerable influence
from the stacking sequence. The shell with ply drop-offs with the [—45°/0°/45°] stacking
sequence has larger deformation, and is less resistant to the loading than the shell with ply
drop-offs with the [0°/90°/0°] and [90°/0°/90°] stacking sequences. The deformed shapes
in Figure 3.25 for the final load P = \AQKN demonstrate clearly that large rotations and
displacements are involved in this example.
0 15
0°/90°/0°,90o/0°/90°,—45°/0°/45°
Stacking sequences:
= °-04
7*1 = 7.5
7*2 = 15.0
h'p = 21
r = ^(b2 + Z2)1/2
b
Figure 3.23. Pinched multilayer composite hyperbolical shell with ply drop-offs: Unde¬
formed mesh.

131
Figure 3.24. Pinched multilayer composite hyperbolical shell with ply drop-offs: Load-
displacement diagrams, v(B) is the displacement along axis Y at point B, u(A) the dis¬
placement along axis X at point A.
Figure 3.25. Pinched multilayer composite hyperbolical shell with ply drop-offs: Deformed
shape with stacking sequence [0°/90°/0°] (left) and [-45°/0°/45°] (right).

CHAPTER 4
OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF
MULTILAYER COMPOSITES : DYNAMICS
4.1. Introduction
The formulation and implementation of the solid shell element for the dynamic anal¬
ysis of flexible multilayer shell structures undergoing large deformation and large overall
motion is addressed here. In the present formulation, the dynamics of the motion of multi¬
layer shells is referred directly to an inertial frame, thus simplifies considerably the inertia
operator. The starting point is the nonlinear dynamic weak form based on the internal en¬
ergy and kinetic energy of shells. A linearization of this nonlinear dynamic weak form is
performed for use in the solution for the kinematic quantities via Newton’s method. Time
discretization is introduced to obtain the time-discrete weak form. A time-stepping al¬
gorithm based on the energy-momentum (EM) conserving algorithm (Simo, Tamow and
Wong [1992], Kuhl and Crisfield [1999]) is employed in the time-discrete weak form. This
algorithm preserves the total momentum and the total energy exactly. Compared to the
classical Newmark algorithm (Newmark [1959]), the energy-momentum conserving algo¬
rithm gains a more robust stability behavior. On the other hand, a high-frequency algo¬
rithmic dissipation is still desirable to incorporate in the EM algorithm for the stability. A
number of the modifications of the Newmark family of algorithm have been proposed, typ¬
ically in the form of linear multi-step methods, which introduce high-frequency dissipation
and preserve second order accuracy (see e.g., Hilber, Hughes and Taylor [1977], Wood,
Bossak and Zienkiewicz [1981], Bazzi and Anderheggen [1982]). A small modification
to the present energy-momentum conserving algorithm, in a form similar to generalized-
a method (Chung and Hulbert [1993]), results in high-frequency dissipation. Numerical
simulations show that the effect on the transient response is minor for small amount of
132

133
numerical dissipation.
Since the kinematic description is the displacement of the top and bottom surface
of the shell in the present formulation, the special treatment on the rotational degrees of
freedom (Simo and Vu-Quoc [1988], Vu-Quoc et al. [2001]) is not needed. Unlike the
stress-resultant shell formulation, the resulting consistent mass matrix of the present ele¬
ment is symmetric and independent from the configuration.
The present eight-node solid-shell element relies on a new optimal (minimum) seven-
parameter EAS-expansion together with the Assumed Natural Strain (ANS) method that
passes both the membrane patch test and the out-of-plane bending patch test. In the for¬
mulation, the transverse normal strain and the membrane strains are enhanced by the EAS
method; and ANS modifications on both the compatible transverse shear strains and the
compatible transverse normal strain are employed to eliminate the locking effects from the
compatible low-order interpolations.
For the EAS approach using enhancing deformation gradient, we develop an EAS
expansion by superposing the enhancing converted basis to the compatible converted basis,
and then present a formulation that is much simpler than that employed in Miehe [1998¿].
The presentation of the present chapter is as follows: we devoted Section 4.2 to
the dynamic aspect and the use of the energy-momentum algorithm for elastic materials.
A variant of the EAS formulation based on the deformation gradient (instead of Green-
Lagrange strains) for solid shells is the focus of Section 4.3. In Section 4.4, several implicit
direct integration methods with/without numerical dissipation are then used and compared
in terms of the accuracy, stability and cost in analyses of multilayer shell structures.
4.2. Dynamics of Solid Shells by an EM Conserving Algorithm
To obtain a fully discrete formulation, the temporal and spatial discretization of the
weak form need to be applied subsequently. The readers are referred to Section 3.3 for
the finite-element discretization of the static weak form. The classical Newmark family of
implicit integration scheme (Bathe [1996]) is widely used for the temporal discretization.
The Newmark algorithm is unconditionally stable for linear problems, but only condition-

134
ally stable for nonlinear problems (e.g., Simo, Tamow and Wong [1992]). Since the stabil¬
ity of the time integration scheme in nonlinear problems is related to the conservation of
energy, Simo, Tamow and Wong [1992] proposed the energy-momentum conserving algo¬
rithm in an attempt to obtain numerically stable and accurate long-term response. On the
other hand, the Newmark family are symplectic and momentum preserving, and often have
excellent global energy behavior with smaller time step (Kane, Marsden, Ortiz and West
[2000]). In this section, we will present some results of dynamic analyses of solid shells
using both energy-momentum conserving algorithm (Simo, Tarnow and Wong [1992] and
Kuhl and Ramm [1999]) and classical Newmark method.
4.2.1. Time Discretization on Dynamic Weak Form
v
The time interval of interest [0, T\ = (J [in> ¿n+i] is subdivided in N time steps. The
n=l
state variables such as displacement un, velocity iin, and acceleration un at the beginning
of the time interval [tn, in+1] are assumed to be known. The state variables un+1, ún+1, iin+1
at the end of the time step [fn, fn+1] are calculated by a time-discretization algorithm.
A class of time-discretization algorithms can be presented altogether as a general¬
ization of the Newmark method as follows. Let the inertial (mass) term be evaluated, not
at the time station tn+1, but at the time tn+am = amtn+i + (1 — am)tn, which is a point
inside the interval [tn, £n+i], with the internal force and external force evaluated at another
time point tn+af = a/fn+1 + (1 — a¡)tn inside the interval [tn, i„+i]- The time-integration
parameters tn+Qm and tn+af have their subscript “m” and “f” being mnemonic for “mass”
and “force,” and are positive numbers between zero and one. The classic Newmark algo¬
rithm corresponds to = am = 1 (Hughes [1987]); the Bossak-a algorithm corresponds
to a¡ — 1 (Wood et al. [1981]); the energy-momentum conserving algorithm corresponds
to Oi¡ = am — 4 (Kuhl and Ramm [1999]).
The dynamic weak form at time tn+a/ is
¿n (un+ctf, En+atf') = J (^-É'n+Q/ * ^n+a/ + • Sn+aj') dV
Bo

135
- í 6un+aj • (b*n+af - ün+Qm) pdV - í 5un+af-t*n+afdS = O . (4.1)
B0 S„
The displacement un+a/ at the time point tn+af and the acceleration ün+Qm at time
point tn+arn are generated by a convex combination of the corresponding quantities at time
tn and time fn+1 as follows
Un+af — ^/^n+l T" (1 ^/)^n > (4-2)
Ün+am = «mÜn+l + (1 ~ Oím)Ün . (4.3)
The variation óun+Q/ from (4.2)
6un+af = ctfSu . (4.4)
The velocity iin+1 and the acceleration ün+1 at time tn+\ can be expressed as a function
of the displacement un+1 at tn+\ and the computed state variables at time tn by using the
Newmark method, that is
un+l
7
pAt
(tin+l
At un ,
un+i =
pAV
pAt
u,
(4.5)
(4.6)
where P and 7 are the parameters in the Newmark method, and At — tn+1 — tn the time
step size. The compatible-strain tensor Ec and the enhancing-strain tensor E at time tn+a/
are
1 c
n+aj
= Ec
(un+a/J >
(4.7)
n-\-a¡
=
(4.8)
where the EAS parameter ocn+af is a convex combination of an and a„+i in the same
manner as that for the displacement un+af as given in (4.2).
For linear elastic materials, the above convex combination for the compatible strain
Ecn+OCf and the enhancing strain En+a/ naturally leads to the identical convex combination

136
in the algorithmic stress Sn+Clf as follows
&n+oif ~~ ®fBn+l "b (1 &f)Bn
— Cl [«/ {Ec (wn+i) + E (an+i) j + (1 — otf) ^Ec (un) + E (c*n)jj ,
(4.9)
which is an essential point in energy-momentum conservation algorithm, see Remark 4.2
on the conservation of energy and momentum where the need for the definition of Sn+af
becomes clearer.
Substituting (4.3) into (4.1), it becomes
n+aji
'n+aj
+ ÓE
n+otf
+
OLm
/3At2
(^71 + 1 Wn)
Qfri
PA t
Ur
~^P
\
pdV
)
~b I I dun-\-a¡ 'bn+afPdV / Sun+afmtn+afdS
V Bo L
— ¿nst¿jgr + 6Umass + 5Y[exl = 0 , (4.10)
where SUstiJj is the virtual work generated by the internal forces, ¿nmass the virtual work
by the inertia forces, and 5Uext the virtual work by the external forces.
4.2.2. Linearization of Dynamic Weak Form
Within a typical element (e), the variation of the internal energy Ustiff in (4.10) at
time tn+a/ is
j HE^, {S“} dV, (4.11)
where the variation of the strain components Eij in matrix form is given by
«iU = KU+KL.,
= Bn+ocfSd^la/ + &óa(nela/
(4.12)

137
where the strain-displacement matrix Bn+aj is a function of the discrete displacement
dn+of > whereas the interpolation matrix Q given in Section 3.4 is time independent. Note
that the variations
«&«, = af5d^ ,
¿o$a/ = afSa^ . (4.13)
With (4.12), the variation of the internal energy II^ in (4.11) in matrix form be¬
comes
XTr(e)
011 stiff
Sd{e)T
oan+a¡
, / B^,{s>il+a,dV + s°‘™, J ST {s“l+a/dv
bP B = Sd
(e)T
n+otf
f
, ¿r*(e)T
stiff + ocxn+af
Ae)
J EAS )
(4.14)
where and f^As are respectively the internal force related to the displacement d^
and the internal force related to the EAS parameter a(e\
The linearization of (4.14) with respect to the nodal displacement d„h and internal
parameter at time tn+1, with increment Ad^ = d„h - and Aa^e) = ol^+1 -
ct£\ respectively, is then
V6n%-(Aé‘\Ao,M) = Sd¡,% (fcSAd + 6<£Z, (*£>Ad(,) + fcgAa<'>) , (4.15)
where the stiffness submatrices fc^|, k^, and k^l are derived from internal forces
f^stiff ar>d Íeas > whose expressions were given in (4.14).
Using (4.2) and differentiate Bn+Ctf with respect to d^h, we obtain
dBr
’n+af
ftfj(e)
oan+\
dBn+a dd[nla
7~\ T\ — Gat
ddn+aj dd£h
(4.16)
Also, by using the definition of Sn+a/ as stated in (4.9) for elastic material, it follows that
d{S%
dd
n+a/ n ^ {Slj}n+i
- af
(e)
n+1
dd.
(e)
n+1
= a/CBn+1 , and
(4.17)

138
Hsij}n+af dm^
00ín+1
af p (e)
d = afcg .
(4.18)
Remark 4.1. It is noted that in (4.18), the stress {Sij}n+a/ should not be computed
from {Eij}n+a¡ by using the constitutive relation S = C l E, that is
1 n-Ya¡ ' " n+Q/ ’
instead, {SIJ}n+Q/ should be computed from (4.9).
(4.19)
I
By using (4.16) and (4.17), the tangent stiffness submatrix k^¿ in (4.15) at time tn+Qf is
r(e)
c)f( ’ r
f (GTS„+„, + BT CBm) dV , (4.20)
ddnl 1 J
,(=)
where Sn+Q/ is the same stress matrix as explained in Section 3.3, but evaluated at time
Ln+otf
Next, by using (4.18), the tangent stiffness submatrix in (4.15) at time tn+a takes the
form
*£ = pk- = al J BL«,C9dV .
d(Xn+l „(.)
(4.21)
Similarly, using (4.17), we obtain the following expression for the submatrix k^¿ in (4.15)
as
k(e) _ dfsAS
„f(e)
n+l
dd
= af j QTCBn+1dV ,
(4.22)
and using (4.18), the submatrix kpl in (4.15) at time tn+af as
*2 = pér = a, [ grcgdv.
3“n+l J.,
(4.23)
It can be observed in (4.20) that the two matrices Bn+aj and Bn+1 are evaluated
at two different configurations d|1+Q/ and d^jl5 respectively, it follows that the submatrix

139
is non-symmetric. Similarly, we have fcjfj ^ • The consequence is that the
condensed tangent stiffness matrix kj) (see (4.31)) will be also non-symmetric.
From (4.10), the discrete weak form of the inertia force is
am (,{e) _ ,(e)\ _ _ Qm ~ 2/?» (e
pAt2 V1n+1 n ) 3 At. 23
XTT(e) _ f)(lWr m(e)
011mass — oan+aj'n
PAt
= Sd('e')T f^
u u,n+a¡ J mass »
(4.24)
where the element mass matrix m^ is obtained by using the same spatial discretization as
in Section 3.3 for 6un+af, u, it, and ii:
m(e) = [ NTNpdV. (4.25)
The linearization on the weak form of inertia force (4.24) is
V6tf±,• Ad<'> = SdUZ, Ad"’ • <4-26>
Likewise, the contribution to the weak form (4.10) from the external forces takes the form
ext
-sd
{e)T Ae)
n+a / J ext >
(4.27)
where the element external force f^t at time tn+aj is
/2 = [ Nb*n+afpdV+ f Nt*n+afdS. (4.28)
<> s<‘>
It follows from the above expressions that the linearization of the dynamic weak form
(4.10) can be written as
vm%-(Ad">, Aa(e|) + VSTl^-Ad^ = -OT Substituting (4.14), (4.15), (4.24), (4.26) and (4.27) into (4.29), for any admissible ód„la/
and So¡.n\ar we obtain the algorithmic tangent stiffness matrix and the force vector as
follows
L,(e)
A d[e)
Aa^
f(e) _ f(e) _ f{e)
J ext J stiff J mass
(4.30)

140
The condensation of the incremental EAS parameters Aa*e) in (4.30) is similar to the static
case, as presented in Section 3.3. On the other hand, the non-symmetry of the tangential
stiffness operator as explained above in (4.20)-(4.22), together with the addition of the
inertia force f^ass given in (4.24), have to be taken into account. Consequently, the effec¬
tive element dynamic tangent stiffness kj) and the effective element residual force are
computed from (4.30) as follows:
I» _ u(e) , am (e) _ r.(e) hu(e)l 1 ju(e)
— ^uu ' /JAt2 " ua L aQJ au ’
r(e) — f(e) _ f(e) _ f(e) i ju(e) ir.(e)l_1 f(e)
' J ext Jstiff J mass ' la [^aQ] J EAS
(4.31)
(4.32)
After an assembly process as in Section 3.3, a global matrix equation for the incremental
displacement Ad is obtained. Once solved, the incremental displacement Ad is used to
compute the incremental EAS parameter at the element level , used for the update
of the EAS parameter a^.
With the converged solution d^h and ol^\x obtained at time tn+i, the acceleration
and velocity d^h are updated in the classical manner as shown in (4.6).
Recall that the above general form of time stepping algorithm encompasses the clas¬
sical Newmark algorithm (a/ = am = 1), the Bossak-a algorithm («/ = 1), and the
energy-momentum conserving algorithm (a/ = am — |).
To check the conservation property of the above algorithms, the total energy and
the linear and angular momenta have to be calculated. The total energy is the sum of the
internal energy £mt and the kinetic energy K.
£t°t =
¡C + £int,
(4.33)
K =
^ J pú • údV ,
(4.34)
So
£int =
\Je: Sdv.
(4.35)
So
where the velocity ú, the strain E, and the stress S are calculated at each time step tn, and

141
so are the linear momentum £ and angular momentum J
C = f púdV , (4.36)
Bo
J = J px XX dV , (4.37)
Bo
where x is the position vector in the current configuration, and is related to the position
vector X in the initial configuration and the displacement vector u, as given by
x(Z,t) = X(S) + u(t,t) , (4.38)
similar to (3.4).
Remark 4.2. In the case where a.¡ = am — | and b* = t* = 0 (no external
loading), energy and momentum in the system is conserved. The total energy £tot, defined
in (4.33), varies within the time interval [tn, tn+\] according to
C+l - C = Kn+1 -Kn + SStl ~ C* â–  (4.39)
From (4.2)-(4.3), it follows that
Un+¿ ~ 7y {un+1 T Un) ,
For elastic material, that is
(4.40)
S = C
E = c:
Ec (u) + E (a) ,
(4.41)
the internal energy £int can be rewritten as follows
£im = }_JE: SdV = 1 J E . c ; EdV , (4.42)
B0 Bo
leading to the following expression for the increment of the internal energy within the time
interval [tn, tn+i]:
C+1 -£nnt = \l (En+i: C : En+1 -En:c : En) dV
Bo
= \j {En+1 - En) : C : (En+1 + En) dV .
Bo
(4.43)

142
Let the algorithmic stress 5n+i be defined as follows:
Sn+i := 2 (^n+l + Sn) = ^ * (-®n+l + £?„) »
From Taylor expansion, we expand the strains En+X and En at time tn+i to get
En+1 — En = AtÉn+x + O (Af3) ,
Using (4.44) and (4.45), (4.43) becomes
Cl - C = Af / ¿„+J -S^dv + O (At3) ,
Bo
Likewise, it follows that
itn+1 -un = Atun+i + O (At3) ,
and hence the change of the kinetic energy 1C in the interval [tn, tn+x ] is
ICn+l ^n ~ 2 JP (^n+1 * ^n+1 — * ^n) dV
Bo
= \ J P (Wn+1 + «n) * (Wn+X - U„) dV
Bo
= Af / pún+i-ün+idU + 0(Af3) ,
Bo
Substituting (4.48) and (4.46) into (4.39), it shows
Cl - C = / K+i •«„+! + ¿„+i: s„+ij AWK + o (Ai3)
Bo
= n„+iAf + o(Af3) ,
(4.44)
(4.45)
(4.46)
(4.47)
(4.48)
(4.49)
If nn+i = 0, it means that the stationary condition of functional II at time fn+i, namely
¿n (un+x,En+1) = 0 ,
therefore the total energy is conserved in the time interval [in, tn+1]
Cl - C = o (Ai3) .
(4.50)
(4.51)
I

143
4.3. Enhanced-Assumed-Strain Method Based on Deformation Gradient
In this section, we will present the weak form and the finite-element discretiza¬
tion for the proposed solid-shell element by using the enhanced-assumed-strain (EAS)
method based on the deformation gradient, instead of the reparametrization of strains in
Section 3.2."
4.3.1. Weak Form
The non-linear version of EAS method by Simo and Armero [1992] is based on the
decomposition of the deformation gradient F into the compatible part Fc and the enhanc¬
ing part F as follows
F = Fc (it) + F . (4.52)
In this case, the three-field Fraeijs de Veubeke-Hu-Washizu functional, depending on the
displacement field u, the enhancing deformation gradient F, and the nominal stress tensor
P = FS is given by
n (u,f,p)
where U.stiff corresponds to the internal energy, IImass the work by the inertia force, and
next the work by the external forces.
Remark 4.3. The naming of the stress P in (4.53) is not unique, and can be confus¬
ing. In component form, P is written as
p = PaAea Ea , such that t = P» N , (4.54)
11 The resulting element cannot pass the out-of-plane bending patch test while pass the membrane patch
test, see the numerical examples of Section 3.5 for details.

144
where ea is the basis tangent vector in the current configuration, Ed4 the basis tangent vector
in the initial configuration, t = ta ea the traction force acting on a facet in the current
configuration, and N = EA the normal of the same facet in the initial configuration.
Truesdell and Noll [1992, p.100], Chadwick [1976, p. 124] andMarsden and Hughes [1983,
p. 135] refer to P as the first Piola-Kirchhoff stress tensor, whereas Malvern [1969, p.222]
refers to T0 = Pr, namely, the transpose of P, as the first Piola-Kirchhoff stress:
T0 = Tq0, Ea <8> ea , such that t = N • T0 . (4.55)
Chadwick [1976, p.99], on the other hand, refers to T0 = PT as the nominal stress. The
reason for the difference in the naming of this stress tensor lies in the convention adopted
for the indices in the component of the (Cauchy) stress tensors (Malvern [ 1969, p.224]), and
thus the way these tensors operate on the normal to yield the traction vector. Even though
Chadwick [1976] adopted the same convention for the indices as in Malvern [1969]), he
defined the traction vector as:
t = P- N = Tl • N = N -T0 , (4.56)
and called T0 the nominal stress, and P — Tq the first Piola-Kirchhoff stress. Here we
follow the convention and naming as in Malvern [1969], that is, T0 = PT being called
the first Piola-Kirchhoff stress, and thus call P the nominal stress.
The stress power, which is related to the term f P • FdV in (4.53) is written as
Bo
follows (Malvern [1969, p.224])
IP *. FdV = I T0 • •FdV , (4.57)
Bo Bo
since
P : F = PaA (p)“ = TAa (f)“ = To • -p . (4.58)
The nominal stress P and the first Piola-Kirchhoff stress T0 = PT is related to the second
Piola-Kirchhoff stress as follows
P=Tl =FS .
(4.59)

145
Remark 4.4. Regarding the functional nmass related to the inertia force, we recall
that the variation of the displacement u is taken to be independent of time (see (3.21)), and
thus 'úe — ti, since 5u = 0. I
Using (3.12) and (4.52), the variation of the strain E is given by
SE = Ft6F = Ft (¿Fc + 6F) ,
(4.60)
we have
dWs _ _ „dWs
dE
*. SE = F
dE
: SF,
d\Vs
where —— is a symmetric second order tensor (see Section 3.2).
oE
The weak form of (4.53) is expressed as
sn F, P^ ^flstiff + T SYlexi ,
(4.61)
(4.62)
where the weak form 6Ustiff of the internal energy Y[stiS is obtained by using (4.52) and
(4.61)
dW„ . f _dW„
8U
stiff
I 1: SEdV - / ¿ (p : f) dv = / : sfw
B0 Bo B0
+
r QW ~ r ~ r ~
J f-^- : SFdv - J sp: Fdv - j p: SFdV,
Bo Bo Bo
whereas the weak form ¿nmass for the inertia force is simply
snr
= J Su • up dV ,
Bo
and the weak form óTI^ for the externally applied forces
sn
ext
= - j 8u-b*pdV - j 5u-t*dS
Bo S„
(4.63)
(4.64)
(4.65)

146
Again the nominal stress P can be eliminated from the above weak form (4.63) by en¬
forcing the orthogonality condition between P and the enhancing deformation gradient F,
namely
16P : FdV + J PI SFdV = 0 . (4.66)
Bo B0
Using the definition of the notation S as in (3.28), the weak form <ünsí¿^ becomes
sn stiff = j SI SEdV = j (.FS : SFC + FS : SF) dV . (4.67)
B0 B0
If we develop the formulation in the convected basis, the deformation gradient F is ex¬
pressed by
F = Fc + F = g, <8> G1 , (4.68)
where the additive decomposition of the total deformation gradient F directly leads to an
additive decomposition of the convected base vectors g{ in the current configuration into
the compatible part g¡ and the enhancing part gt, namely
9r = 9i +9i,
(4.69)
Fc = g\ ® Gi, F = gi®&.
(4.70)
It follows that the Green-Lagrange strain tensor E, which takes the form
E = ±(FTF-I2) = i(gt-gj-G,-Gj)Gi®Gi
= l ! 9' •;/; + 'J, ■ = Ec + E. (4.71)
where
E' = 5 (g‘-g'j-Gi-G])Gi®G’ , (4.72)
E = 1 (S¡% + 9(*9| + g,-g,) G‘G>~ 1 jJG* ® G>. (4.73)

147
The variation of components of the Green-Lagrange strain E is given by
SEi:i = - {Sgi»gj + g{*5g^ = sym (Sg^g^ . (4.74)
Substituting (4.69) and (4.74) into (4.67), the weak form ÓUstiff now reads as
SUstiff = JsijSEijdV = jsijsym(5gi-gj)dV
B0 B0
= J5"Jsym (dg'i'gj + dg^g^j dV . (4.75)
Bo
4.3.2. Finite Element Discretization and Linearization
Following the same spatial discretization procedure as in Section 3.3, the compatible
convected basis vectors gc at a point inside an element (e) in the current configuration are
functions of the displacement d^ of element (e).
To pass the patch test, Simo and Armero [1992] suggested the use of the enhancing
deformation gradient F tensor in the following form
F = ^j-FoJqFJq1 , (4.76)
where J is the Jacobian determinant at a point £ inside an element, and J0 the Jacobian
determinant evaluated at the element center £ = 0. In convected coordinates, the compat¬
ible deformation gradient F0, and the element Jacobian J0, both evaluated at the element
center, are expressed as
■P’o = 9oí ® CT0 , Jo — Gqí ® el, J01 = e¿ ® Gl0 , (4.77)
where g0i and G0i are given by
_a^o) ax (o)
9oi ' d£ ’ 01 ' d£
(4.78)
and Gq are obtained from G0i, the vectors e¿ = é form the global orthonormal basis. The
tensor T in (4.76) is the same as in Simo and Armero [1992], namely
3 __
F = £ r; ® GRADcN1 = F]ei ® ej , 7 = 1,2,3,
1=1
(4.79)

148
where T/ 6 M3 is the vector of the internal EAS parameters o¿e\ and N1 the Wilson’s
incompatible shape function for the tri-linear brick element
Ñ1 =^((V)2-l) , 7 = 1,2,3. (4.80)
The components of are listed in Appendix A.l.
Substituting (4.77) and (4.79) in (4.76), and comparing it with (4.70), the enhancing
convected base gt, depending on d^ and can be expressed as
9i = J-jgl¡Pk (G*-G.) = glj (d“>) P¡ (<*<">) ,
with j? («<*>)-, a* := J-j (g$-G,) . (4.81)
The variation and the linearization of the convected basis vector g{ and its variation Sg,
with respect to the displacement d^ and the EAS parameter are given by
Sg, (<¡w,aw)
= $gci + Sg,, with Sg, = Sgc0kEf + gcokSE* ,
(4.82)
= ^9Ci + AgcokEt + gcokAEt,
(4.83)
A (í9¡ (d('\a))
= SgcokAE? + AgcokSE? .
(4.84)
Using (4.82), the variation of components of the strain tensor E takes the form
6Ei:i = sym (Sg^g^ = sym (Sg^gj + SgcokT^g^ + sym (g^SE^g^) , (4.85)
which can be recast in matrix form as
{<5En) = B (d(e), a where B and B are the strain-displacement matrices associated with Sd^ and Scx^, re¬
spectively.
The linearization of SEij in (4.85) is
A (SEi:i) = sym [Sg^Agj + A (Sg,) »g3] ,
(4.87)

149
where 6g{, Agx, and A [bg¿] are obtained in (4.82)-(4.84).
Using (4.86) and (4.75), the discrete weak form <511^^ of the stiffness operator for
element (e) is then
fllgy = J {SEtjf {Sij} dV = 5d[e)T I BT {Sij}dV + 5a{e)T J BT{Sij}dV
B
= + Sa^Tfg\s ,
where /^ and fg¿s are the internal forces associated with the nodal displacement d
and the EAS parameter a^, respectively.
The linearization of SU^ in matrix form then be written as
V (tfn^) • (ASe\ AqW) = 5d^T (fegAd^ + fe^Aa^)
+ 8o¿e)T (fcWAd(e) + k^Aa(e)) . (4.89)
With the constitutive relation {S1J} = C {Eij} for the elastic material, the stiffness
r(e)
(4.88)
W
submatrices k$, k(J¿, and k^l in (4.89) are
Af(e) ,
*2 = -£$■= j {GT„$„ + BTCB)dV,
B (4.90)
t.(e) _
Kua
a/Sr
<9a(e)
= / + BtCB) dV ,
4->
(4.91)
"'em
dfEAS
dd[e)
= [<=2]T = / (gLS. + Brcs) (¡1/ ,
«¡•>
(4.92)
ii
Já
df^EAS
da (e)
= / (cL$« + btcb) dU ,
(4.93)
»(*)
where the matrices associated with the geometrical stiffness parts in (4.90-4.93) are
Guu
dB(d[e\a.^) dB(d{e\a^)
5d(e) ’ Gua :=
Gau .
dB (Se\ a^)
dd&
, Gc
da^
dB (d(e),aM)
da(e)
(4.94)

150
and S>u and $a are the stress matrices, whose expressions are listed in Appendix A.l.
The linearized stationarity condition for the mixed functional n in (4.53) is obtained
by using (4.89), (4.88), and (4.65), and can be expressed in the following matrix relation
t.(e) ju(e
â„¢uu ^UOL
u(e) u(e)
A'/voi
A d(e)
Aa^
/.(e) _ p(e)
J ext J stiff
Ae)
~J EAS
(4.95)
recalling that the discretization of SUext in (4.65), when restricted to element (e) can be
written as
án2 = -Mw-/2, (4.96)
and with /(¡L. /e)s in (4.88), we follow the same procedure for the nonlinear solution as
in Section 3.3.
Remark 4.5. It should be noted that in the EAS formulation based on the Green-
Lagrange Strain E, while the tangent stiffness in (3.71) contains both the geometric
part and the material parts, the other three tangent stiffness submatrices k^, k^¿ and k^l
contain only the material part, because while the strain-displacement matrix B (see (3.55)
and (3.73)) depends on the displacement Se\ the interpolation matrix Q does not. In the
EAS formulation based on the displacement gradient F, all tangent stiffness submatrices
as given in (4.90)-(4.93) contain both the geometric part and the metrical part, because
both strain-displacement matrices B and B (see (4.86) and (4.94)) depend on both the
displacement d^ and the EAS parameter I
4.3.3. Assumed Natural Strain (ANS) Treatment
To treat the locking problems due to the transverse shear strains and the transverse
normal strain, we consider the same approach as in Section 3.3 (i.e., the use of the ANS
method). To avoid destroying the enhancement of the transverse and normal strains as
already introduced in the previous section, we need to replace the compatible strain in
the enhanced strain Eij by the assumed strain E(-NS, for ij = 13,23,33 (i.e., the transverse
and normal strains only) as follows
= E^ - + E*ns , for ij = 13,23,33.
(4.97)

151
where Éij is the modified strain, the compatible strain Efj is given in (4.73), and the as¬
sumed strain EfiNS is given in (3.100) and (3.101).
Since the variation and the increment of the components of the compatible strain Ec
are
(4.98)
(4.99)
respectively, and the linearization of the variation of the compatible strain is
A [SEf^ = A [sym (óg'»gfj\ = sym (óg^Ag^ , (4.100)
we use the same procedure in Section 3.3 to obtain the variation SE(-NS of the assumed
natural strain and its linearization A (6E^Nsy The expressions in (4.85) and (4.87) are
then replaced by the following
6Éi:i = SEij — SEfj + 6E£ns , (4.101)
A (SÉu) = AOSEyJ-A^J+A^"») , (4.102)
for the computation of the residual in (4.88) and for the tangent stiffness submatrices in
(4.90M4.93).
4.3.4. Simplified Formulation
In the present EAS formulation based on the deformation gradient F, both the strain-
displacement matrices B and B are functions of the nodal displacement d^ and element
EAS parameter c*(e). Such strong coupling makes this EAS formulation more complex than
the EAS formation based on the Green-Lagrange Strain E, where the strain-displacement
matrix B (see (3.55)) depends only on and the interpolation matrix Q depends on
neither d^ nor o:^^. Recall that the tangent vector gt can be written as
9i = 9i (d[e)) + <7¿ (<¿(e), c*(e)) , with g{ = gcok (d(e)) F- (a(e)) .
(4.103)

152
With the Green-Lagrange strain E expressed as
E=(E¡j + Éij) Gi Gj, (4.104)
where compatible Green-Lagrange strain component Efj and the enhancing strain com¬
ponent Eij are given in (4.71). Note that g*j and thus E^ are the functions of d(e) and
ot^.
Recall from (3.57) that the enhancing strain in the EAS formulation based on the
Green-Lagrange Strain E actually depends linearly on a^ only.
To simplify the formulation, it is possible to omit the high order term in the
expression for 5* in (4.73), that is, consider the approximation
Sv&ffi’ffj+ffi'rf, (4.105)
which, together with (4.82), lead to the following expression for 6Etj
SEij = ^5gij » (g^dj + gl'9j)
= sym (óg\ • + Sg^ff • gfj + sym {gc0k^i ^f) » (4.106)
where the two above terms in turn lead to simplified expressions for the strain-displacement
matrices B and B, respectively.
We then obtain the following linearization of the variation SE of the strain
A(SEtj) =
= sym [Sgcr (agcj + Agcok^j) + s9ok^i'A9j]
+sym (Sg^g^Aff + SgcokAT^gf)
+sym(Agcok6Etk>gcj+ gc0k5Ek-Agfj , (4.107)
where the above three terms correspond to the simplified geometric stiffness submatri¬
ces Guu, Gua, and Gau associated with {5Se\ Ad(e^, (Sd^, Aa(e)j, Ad(e^,

153
respectively, while the submatrix Gaa associated with (Sct^e\ Aq^J (similar to the ex¬
pression in 4.93) vanishes completely as a result of the approximation in (4.105). Readers
are referred to Appendix A. 1 for details of the above matrices.
We follow the same procedures described in (4.86), (4.88), and (4.89) to obtain the
tangent stiffness submatrices that are similar to (4.90)-(4.93) and the internal forces similar
to (4.88), and then solve the assembled nonlinear problem. Our numerical tests show that
it is virtually identical for both the full formulation and the simplified formulation.
4.4. Numerical Examples
The finite element formulation of the present low-order solid-shell element for dy¬
namic analyses of multilayer composite shell structures, presented in the previous sections,
has been implemented in both Matlab and the Finite Element Analysis Program (FEAP),
developed by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a Compaq Alpha
workstation with UNIX OSF1 V5.0 910 operating system. The tangent stiffness matrix,
the dynamic residual force vector, and the consistent mass matrix are evaluated using full
2x2x2 Gauss integration in each element. A tolerance of 10~18 on the energy norm
is employed in the Newton iteration scheme for the convergence. Below we present nu¬
merical examples involving geometrically nonlinear dynamic analyses, with isotropic and
orthotropic elastic materials.
To assess the performance of the present solid shell formulation in dynamic analyses
of multilayer composite shells, we implemented four different second-order implicit inte¬
gration schemes, and compared their performance in relation to our solid-shell formulation.
For the classical Newmark (trapezoidal) algorithm, the time-integration parameters
in (4.2)-(4.6) are chosen as follows
a/ = am = 1, 7 = ^- (4.108)
For the energy-momentum conserving algorithm, the time-integration parameters in
(4.2)-(4.6) are chosen as follows
a' = ftTTT ’ “™ = , /? = j (1 + Qm - a,f , 7 = j - Q/ . (41°9)

154
where p^ is the user-specified spectral radius or high-frequency dissipation coefficient
(Chung and Hulbert [1993] and Kuhl and Ramm [1999]). The value Pt» = 1 corresponds
to the case of non numerical dissipation, while a smaller spectral radius (p00 < 1) corre¬
sponds to a greater numerical dissipation for the generalized energy-momentum algorithm
by Kuhl and Ramm [1999]. It is noted that the numerical dissipation can also be introduced
by shifting the algorithmic stress in (4.9) with a small damping parameter, as done in the
modified energy-momentum algorithm (Armero and Petocz [1998]). On the other hand,
both the modified energy-momentum algorithm by Armero and Petocz [1998] and the gen¬
eralized energy-momentum algorithm by Kuhl and Ramm [1999] lose the second-order
accuracy (Kuhl and Crisfield [1999]).
By simply replacing the algorithmic stress (4.9) of the average between the config¬
urations at the beginning and at the end of the time step with the stress at the mid-point
configuration, we recover the mid-point rule from the energy-momentum conserving algo¬
rithm without numerical dissipation.
The Bossak-a algorithm (Wood et al. [1981]) can be contained as a special case
of generalized-a method (Chung and Hulbert [1993]), which is second-order accurate,
and has the controllable numerical dissipation in the higher-frequency modes. The time-
integration parameters of Bossak-a algorithm are
2 1 2 1
o./ = 1 , am = —— , (3 — — (am) , y o.m — . (4.110)
Poo i ¿
Although the general ized-a method possesses an optimal combination of low numerical
dissipation in the low-frequency range and high numerical dissipation in the high-frequency
range (Chung and Hulbert [1993]), the numerical results in Kuhl and Ramm [1999] showed
that larger numerical dissipation was necessary to obtain a stable integration for nonlinear
elastodynamics, when compared to the Bossak-a algorithm.
4.4.1. Double Cantilever Elastic Beam under Point Load
This example concerns the dynamic responses of an linear elastic beam with rect¬
angular cross-section, built-in at both ends, subject to a suddenly applied step load at its

155
midspan (see Figure 4.1). This central part of the beam undergoes displacement several
times its thickness, so that the solution quickly becomes dominated by membrane effects
which significantly stiffen its response. The purpose of this example is to verify the present
numerical implementation of implicit dynamic analysis. Five solid-shell elements are used
0-
Z
Y
P = 640
f:
20.0
I
E = 3 x 107
v — 0.3
p = 2.54 x 10“4
L
1.0
T
0.125
Figure 4.1. Double cantilever elastic beam under point load. Geometry and material prop¬
erty.
to model one half of the beam, with symmetric conditions (u = 0) applied at the midspan.
The fixed time step-size is At = 50 x 10_6sec, and the Newmark method without numer¬
ical damping is used. The displacement in axis Z at the midspan along time are shown in
Figure 4.2, in which the results with lumped mass matrix (obtained via (5.82) or (5.83)) is
close to that from the beam element with cubic interpolation reported in Example 5.2.1 of
ABAQUS [1995]. For flexural problems such as beams and shells, it is noted that the use
of consistent mass matrix leads to more accurate results (Cook, Malthus and Plesha [1989,
p.375]). The linear and angular momenta along with time are demonstrated in Figure 4.3.
Table 4.1, which depicts the values of the Euclidean norm of both the residual and the en¬
ergy norm at each iteration, clearly exhibits the quadratic rate of asymptotic convergence
in the Newton’s solution procedure.
In nonlinear analyses, it is informative to print the energy balance, which allows us
to assess how much energy has been lost. To this end, the kinetic energy, the strain energy,
the total energy (kinetic + strain energies), and the work of the external forces are plotted as
a function of the integration time. Figure 4.4 shows the energy values from both Newmark
algorithm and the EM algorithm, where the very small energy balance error (total energy
versus external work) indicates the Newmark algorithm gives the satisfactory solutions.

156
Figure 4.2. Double cantilever elastic beam under point load. Dynamic response at midspan
of beam.
Table 4.1. Double cantilever elastic beam under point load: Convergence results (residual
norm, energy norm).
Iter.
Time step 1 (t=5E-5 sec)
step 40 (t=2E-3 sec)
step 100 (t=5E-3 sec)
0
4.525£+02,8.236E+00
2.009£+03,1.804£+02
2.442£+03, 2.476£+02
1
4.479£+03,3.851£-02
2.555^+04,1.327^+00
2.419£'+04,1.179EJ+00
2
3.558£—01,4.407£—10
1.551£+01,3.053£—05
1.029£’+01,1.515^—06
3
1.125E—07,1.731£—22
3.917£—02,2.737£—12
4.757£—04, 5.898£-16
4
4.833£-09,5.385£-26
2.218£-08,1.536E1—24
momentum (left) and of angular momentum (right), using Newmark algorithm.

157
Figure 4.4. Double cantilever elastic beam under point load. Energy conservation using
the Newmark algorithm (left) and the EM algorithm (right).
4.4.2. Pinched Cylindrical Multilayer Shell
Here we simulate the large deformation of a pinched cylindrical shell with ply drop¬
off by using the classical Newmark algorithm. The geometry properties, material parame¬
ters, finite-element mesh and loading conditions for one-eighth are given in Figure 4.5. The
cylindrical shell is subjected to two opposite forces acting at the mid-section and on the
outer surface of the shell. The initial conditions for displacements and velocities at t = 0
are set to zero. One-eighth of the cylinder is discretized with a FE mesh 32 x (32 + 8)
solid-shell elements, with appropriate boundary conditions at the plane of symmetry. The
cylinder is pinched with three different rates of loading as shown in Figure 4.5. Smaller
time-step size is used for larger rate of loading, that is, 0.04 sec for Rate 1, 0.02 sec for
Rate 2, and 0.01 sec for Rate 3. The deformed shape for static analysis is shown on the
left of Figure 4.7, while the deformed shapes for dynamic analyses with the above men¬
tioned rates of loading are shown in the right of Figure 4.7 and Figure 4.8. On average,
it took roughly five iterations to convergence for each time step in the dynamic analyses.
Figure 4.6 displays the relationship between the magnitude of the pinching forces and the
displacement at the point of application of a pinching force. The dynamic analyses clearly
lead to patterns of deformation that are more complex than that obtained with static anal¬
ysis. Both the inertia (mass) and the loading rates have important influence on the final
deformed shapes, in which buckling modes along both the circumferential direction and

158
the longitudinal direction can be clearly observed. At a higher rate of loading, larger mag¬
nitude of the pinching force is required for the same displacement; the overall resulting
deformed shape is also more severe.
Figure 4.5. Pinched cylindrical shell with ply drop-off: Geometry, material, loadings, 0°
along circumferential direction.
Figure 4.6. Pinched cylindrical shell with ply drop-off: Pinching force amplitude versus
displacement under pinching force.

159
Figure 4.7. Pinched cylindrical shell with ply drop-off: Deformed shape for static case
(left), deformed shape at t = 2.64 for Rate 1, time-step size 0.04 sec (right).
Figure 4.8. Pinched cylindrical shell with ply drop-off: Deformed shape at t = 1.98 for Rate
2, time-step size 0.02 sec (left), deformed shape at t = 1.5 for Rate 3, time-step size 0.01
sec (right).
4.4.3. Free-Flying Single-Layer Plate
The same example as in Kuhl and Ramm [1999] is used here to compare the perfor¬
mance of the present solid-shell formulation to that of the eight-node shell element with
reduced integration. The geometry, the loading configuration and the time history loading
amplitude are described in Figure 4.9. In this example, we use thirty solid-shell elements,
with a time-step size of At = 50 x 10-6sec and a total simulation time of t — 0.1 sec. The
effect of gravity force is not considered.

160
The material properties are
E = 206.GPa, v = 0., p = 7800.Kg/m3 , (4.111)
where E, u, and p are the Young’s modulus, the Poisson’s ratio, and the mass density,
respectively.
Snap shots of the plate undergoing large overall motion and large deformation are
taken every 4 x 10-3sec from the simulation by using the energy-momentum (EM) con¬
serving algorithm, and are displayed in Figure 4.11. Furthermore, the conservation of
linear and angular momenta is demonstrated in Figure 4.12. On the left of Figure 4.16, the
kinetic energy, the strain energy, the total energy (kinetic + strain energies), and the work
of the external forces are plotted as a function of the integration time. Unlike the slightly
increasing total energy caused by reduced integration technique in Kuhl and Ramm [1999],
the total energy in the present solid-shell formulation is exactly conserved. The classical
Newmark method and the mid-point rule lead to a loss of stability at an early time stage of
the integration, as indicated by the dramatic increase of the total energy (Figure 4.17). The
mid-point rule has a stability that is somewhat worse than that of the Newmark algorithm
in this example. On the other hand, the mid-point rule conserves the angular momentum,
while the Newmark algorithm conserves only the linear momentum (Figure 4.13 and Fig¬
ure 4.14). With a numerical dissipation set at p0Q = 0.975, the Bossak-a method yields a
stable integration in the whole time range (the right of Figure 4.16), and conserves both the
linear momentum and the angular momentum. The variation of the number of iterations per
time-step as the integration progressed indicates a trend of increasing number of iterations
and a potential for lack of convergence at some future station in the EM algorithm (left of
Figure 4.18). On the other hand, the Bossak-a algorithm possesses a stable number of iter¬
ations and thus a better rate of convergence at each time step (right of Figure 4.18). While
the difference in the displacements obtained from the EM algorithm and from the Bossak-
a algorithm is negligible (left of Figure 4.19), the difference in the total energy is about
18% at t — 0.1 sec (Figure 4.16). In parallel, the difference in the velocity (right hand side

161
of Figure 4.19) is small, but the difference in the acceleration (Figure 4.20) is large. The
magnitude of the acceleration obtained from the EM algorithm increases rapidly, while the
one obtained from the Bossak-a algorithm hovers at around the level of 106. The relative
higher magnitude of acceleration suggests a higher level of “noise” in the high-frequency
range of the response obtained from the EM algorithm, compared to the Bossak-a: algo¬
rithm, whose parameter am in (4.110) has a direct influence on the acceleration update as
expressed in (4.3).
0.04m
Figure 4.9. Free-flying single-layer plate: Geometry, loading distribution, and loading
amplitude.
Figure 4.10. Free flying single-layer plate: Initial undeformed configuration at t = 0.
4.4.4, Free-Flying Multilayer Plate with Ply Drop-offs
This example establishes the capability and performance of the present solid-shell
in modeling multilayer plates/shells with ply drop-offs. Comparison of the stability and
accuracy of the four different time-integration algorithms is provided for this example. The
geometry and the mesh of a three-layer plate with ply drop-offs is shown in Figure 4.21.

162
Figure 4.11. Free-flying single-layer plate: Perspective view.
Figure 4.12. Free-flying single-layer plate: Three components of linear momentum (left)
and of angular momentum (right), using EM algorithm.
The length, width, and layer thickness of the plate are the same as those of the single¬
layer plate in Subsection 4.4.3 (i.e., L = 0.3, W = 0.06, and wh — 0.001). The plate
is divided into three equal parts and two ply drop-offs along its length, with each part
having a length of 0.1. We use the same material properties as listed in (4.111) so that the
three-layer plate with ply drop-offs here has the same weight as the the single-layer plate

163
Figure 4.13. Free-flying single-layer plate: Linear momentum (left) and angular momen¬
tum (right) using the Newmark algorithm.
Figure 4.14. Free-flying single-layer plate: Linear momentum (left) and angular momen¬
tum (right) using the mid-point rule.
Figure 4.15. Free-flying single-layer plate: Linear momentum (left) and angular momen¬
tum (right) using the Bossak-a algorithm.
in Subsection 4.4.3. The load distribution and the time history of the loading amplitude
are also the same as Figure 4.9. A total of sixty solid-shell elements are used to model the

164
Figure 4.16. Free-flying single-layer plate: Energy conservation using the EM algorithm
(left) and the Bossak-a algorithm (right).
Figure 4.17. Free-flying single-layer plate: Energy conservation using the Newmark algo¬
rithm (left) and the mid-point rule (right).
Time (sec) Time (sec)
Figure 4.18. Free-flying single-layer plate: Number of iterations to convergence at each
time step using the EM algorithm (left) and Bossak-a algorithm (right).
three-layer plate with ply drop-offs. The same time-step size and time range are used as in
Subsection 4.4.3.

165
Figure 4.19. Free-flying single-layer plate: Difference in displacements (left) and in veloc¬
ities (right) as obtained from the EM algorithm and the Bossak-a algorithm.
Figure 4.20. Free-flying single-layer plate: Difference in accelerations as obtained from
EM algorithm and Bossak-a algorithm.
Snap shots of the deformed shapes taken at every 4 x 10_3sec time interval from
the computations using the Bossak-a algorithm are presented in Figure 4.22. The energy
distribution for this example is given in Figure 4.28, which when compared to Figure 4.16
reveals a smaller level of total energy in the present example of the three-layer plate in

166
relative to the single-layer plate in Subsection 4.4.3. The deformed shapes of the single¬
layer plate and of the three-layer plates are shown in Figure 4.23, where it can be seen
that the three-layer plate has a more flexible (thinner) end and a more rigid (thicker) end
when compared to the single-layer plate. Further, when comparing the initial undeformed
configuration in Figure 4.21 to the deformed configuration on the right of Figure 4.23, we
can see that the thick end of the plate has moved from left to right as a result of the large
overall rotation of the plate. Both Figure 4.21 and Figure 4.23 were plotted using the same
perspective point of view.
In this example, the energy-momentum conserving algorithm could not carry on the
integration beyond the time t = 56 x 10-3sec due to lack of convergence (right of Fig¬
ure 4.28) . On the other hand, the Bossak-a algorithm with the numerical dissipation
Poo = 0.975 provides a stable integration in the whole time range, and conserves the lin¬
ear and angular momenta (Figure 4.27). With the EM algorithm, the linear and angular
momenta (Figure 4.24) together with the energy (left of Figure 4.28) are conserved for
the present solid-shell element up to the time t = 56 x 10_3sec. The difference between
the displacements obtained with the EM algorithm and with the Bossak-ct algorithm up
to t = 56 x 10~3sec is negligible (left of Figure 4.31) ; the difference in the velocity is
negligible up to about f = 20 x 10_3sec, and then increases to a noticeable level until
t — 56 x 10_3sec (right of Figure 4.31). The total energy with the Bossak-a algorithm
exhibits a loss of about 24%. Again the difference in the acceleration is large (Figure 4.32).
The increasing number of iterations to convergence in each time step when using the EM al¬
gorithm (Figure 4.30) eventually leads to a numerical failure at time t = 56 x 10~3sec. The
reason behind this increase in the number of iterations and the eventual numerical failure
is the increasingly active high-frequency response in the numerical solution as manifested
in the rapid increase of the acceleration (Figure 4.32). Similar to the single-layer plate in
Subsection 4.4.3. The Newmark algorithm and the mid-point rule lead to a loss of stability
at an early stage in the solution process, as shown in Figure 4.29. On the other hand, the
loss of stability of the mid-point rule occurred earlier compared to the Newmark algorithm,

167
even though the mid-point rule conserves both the linear and angular momenta, while the
Newmark algorithm conserves only the linear momentum (Figure 4.25 and Figure 4.26).
Among the recent energy-momentum conserving algorithms with numerical dissi¬
pation, the algorithm by Kuhl and Ramm [1999] is shown to be more robust than the
algorithm by Armero and Petocz [1998]. The level of acceleration in the Kuhl and Ramm
[1999] algorithm plateaued out after an initial stage of increase in magnitude, and stayed
that way until the end of the simulation time range (left of Figure 4.36). The acceleration
in the Armero and Petocz [1998] algorithm kept increasing until a lack of convergence that
prematurely halted the solution process in the same manner as encountered with the orig¬
inal EM algorithm by Simo, Tarnow and Wong [1992] (right of Figure 4.36). Figure 4.37
displays the number of iterations to converge in the Kuhl and Ramm [1999] algorithm and
in the Armero and Petocz [1998] algorithm. The difference in displacements obtained from
both EM algorithms with numerical dissipation and the Bossak-a algorithm is negligible
(Figure 4.34), and the difference in the velocity (Figure 4.35) is small. Nevertheless, unlike
the Bossak-a algorithm, both EM algorithms with numerical dissipation lose the desired
second-order accuracy (Kuhl and Crisfield [1999]).
Figure 4.21. Free-flying three-layer plate with ply drop-offs: Initial undeformed configura¬
tion at t = 0.
From the results of above numerical examples, it is clear that the Newmark (trape¬
zoidal) algorithm and the mid-point rule do not guarantee a robust time integration in non-

168
Figure 4.22. Free-flying three-layer plate with ply drop-offs using the Bossak-a algorithm:
Perspective view.
Figure 4.23. Free-flying single-layer plate (left) and three-layer plate (right) with ply drop-
offs: Deformed shapes at t = 16 x 10-3sec. .
linear elastic dynamics due to their rapidly increasing energy in the integration process. By
algorithmically enforcing the conservation of the total energy within each time step, the
energy-momentum algorithm gains a more robust stability behavior compared to the New-
mark algorithm and the mid-point rule. Yet, the EM algorithm remains unstable due to a
continuous growth in the acceleration; such accelerated growth would eventually terminate

169
Figure 4.24. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu¬
lar momentum using the EM algorithm.
Figure 4.25. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu¬
lar momentum using the Newmark algorithm.
Figure 4.26. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu¬
lar momentum using the mid-point rule.
the integration process due to a lack of convergence within a finite time length. Quenching
the acceleration growth in the EM algorithm by introducing numerical damping in the high-

170
Figure 4.27. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu¬
lar momentum using the Bossak-a algorithm with p^ = 0.975.
Figure 4.28. Free-flying three-layer plate with ply drop-offs: Energy conservation and
divergence using the EM algorithm (left); energy loss and continued integration using the
Bossak-a algorithm with p0„ = 0.975 (right).
Figure 4.29. Free-flying three-layer plate with ply drop-offs: Energy balance and diver¬
gence using the Newmark algorithm (left) and the mid-point rule (right).

171
Figure 4.30. Free-flying three-layer plate with ply drop-offs: Number of iterations till
convergence in each time step for the EM algorithm (left) and for the Bossak-a algorithm
with poo = 0.975 (right).
Figure 4.31. Free-flying three-layer plate with ply drop-offs: Difference in displacements
(left) and in velocities (right) as obtained from the EM algorithm and the Bossak-a algo¬
rithm with poo = 0.975.
frequency range may or may not prolong the termination of the integration process, while
paying the price of losing the desired second-order accuracy. All EM algorithms, with or
without numerical damping, lead to non-symmetric tangent stiffness matrices. By contrast,
the Bossak-a algorithm with an appropriate amount of numerical dissipation provides a
stable, and second-order accurate integration process that yields practically the same dis¬
placements as obtained in all other algorithms (Newmark, mid-point rule, EM algorithm
with or without numerical damping) before the failure. We note on the other hand that
there is a smaller loss of total energy in the algorithm of Kuhl and Ramm [1999] (left of
Figure 4.33) when compared to the Bossak-a algorithm (right of Figure 4.28).

172
Figure 4.32. Free-flying three-layer plate with ply drop-offs: Difference in accelerations
obtained from the EM algorithm and Bossak-a algorithm with p^ = 0.975.
algorithm with p^ = 0.985 as in Kuhl and Ramm [1999] (left) and with f = 0.0001 as in
Armero and Petocz [1998] (right).
I

173
Figure 4.34. Free-flying three-layer plate with ply drop-offs: Difference in displacements
using the EM algorithm with p^, — 0.985 as in Kuhl and Ramm [1999] (left) and with
f = 0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-o: algorithm
with poo = 0.975.
1 1 1 1 1 1 1 1
600
' i.l
500 -
/i ft
500
ft l\
400 â–  .
iyv pV(M/V\aaVv.
400 •
/vVi
> 300-1
I~£L*|
5 300 -
I —EM I
200 j
200
100
100
â– 
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0
Tim* (sec)
1 0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.
Tim® (sec)
Figure 4.35. Free-flying three-layer plate with ply drop-offs: Difference in velocities using
the EM algorithm with p^ = 0.985 as in Kuhl and Ramm [1999] (left) and with f =
0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-a algorithm with
Poo = 0.975.

174
Figure 4.36. Free-flying three-layer plate with ply drop-offs: Difference in accelerations
using the EM algorithm with p00 = 0.985 as in Kuhl and Ramm [1999] (left) and with
£ = 0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-a algorithm
with poo = 0.975.
Figure 4.37. Free-flying three-layer plate with ply drop-offs: Number of iterations till
convergence in each time step using the EM algorithm with p^ = 0.985 as in Kuhl and
Ramm [1999] (left) and with £ = 0.0001 as in Armero and Petocz [1998] (right).

CHAPTER 5
EFFICIENT AND ACCURATE MULTILAYER SOLID-SHELL
ELEMENT: NONLINEAR MATERIALS AT FINITE STRAIN
5.1. Introduction
The analysis of general shell structures have been of interest for several decades.
There is a continuing challenge to develop reliable, accurate and efficient low-order shell
elements, especially for analyses of shell structures with arbitrary geometries, loadings,
boundary conditions and nonlinear materials.
Because of the high cost of 3-D continuum elements, shell structures are mainly mod¬
eled by shell elements based on either the degenerated shell concept or the classical stress-
resultant shell theory. Both formulations are based on the common kinematic assumptions
of inextensibility in the thickness direction and the zero-transverse-normal-stress condition.
Although these approximations led to good results in most cases, several difficulties and
appreciable errors could arise. Since the zero-transverse-normal-stress condition must be
imposed, the implementation of 3-D material models proves to be a difficult task, and the
complexity of the algorithmic treatment is increased. For example, even for the simplest
von-Mises elastoplastic model, the stress-resultant constitutive models is rather complex
(Simo and Kennedy [1992]). Moreover, a proper description of 2-D constitutive equations
at finite strain remains a question (Schieck, Pietraszkiewicz and Stumpf [1992]). On the
other hand, in many applications involving (i) the localized effects due to surface load¬
ings, (ii) the contact interaction of different shell structures, or (iii) the delamination of
multilayer shells, it is important to include the transverse normal-stress and the associated
thickness change to obtain a better accuracy (Cho, Yang and Chung [2002], Fox [2000]).
Furthermore, when both shell elements and solid elements are used in one FE model such
175

176
as folded shell structures,12 additional transition elements (e.g., Liao, Reddy and Engelstad
[1988], Cofer and Will [1992]) or multipoint constraints (e.g., MPC in ABAQUS [2001])
are needed to connect rotational dofs and displacement dofs.
Figure 5.1. Discretization at shell junction: Combination of solid elements and solid-shell
elements.
The proposed solid-shell element formulation overcomes the above mentioned diffi¬
culties, and improves the computational accuracy for wider shell applications. The kine¬
matic description of the present element consists of only displacement dofs at the top and
bottom surfaces of the shell. Complex finite rotation updates such as those found in stress-
resultant shell elements (Vu-Quoc, Deng and Tan [2000]). The present formulation also
provides a natural way to connect to regular solid elements (see Figure 5.1) without the
need for transition elements or submodeling technique as in ABAQUS [2001], in which
the modeling processes are laborious and error prone. For bending-dominated problem of
homogeneous shells, in contrast to the use of 3-D solid elements where a large number of
layers of elements must be used in the thickness direction together with a dense mesh on
the shell surface (thus leading to ill-conditioned stiffness matrices), only a single solid shell
elements across the shell thickness, together with a much coarser mesh of solid shell ele¬
ments on the shell surface, are sufficient to provide accurate results. Unlike stress-resultant
shell formulations, 3-D nonlinear complex material models can be used directly in a solid-
shell, without further treatments which add complexity of the model, such as accounting for
12 It is not possible for stress-resultant shell elements to describe the detailed strain and stress distributions
at shell intersections (Chroscielewski, Makowski and Stumpf [1997]).

177
the plane-stress constraint condition. Moreover, the strain-driven character of the formu¬
lation further simplifies the implementation of nonlinear constitutive models, in contrast
to the hybrid finite-element formulations derived from the Hellinger-Reissner functional
(Simo et al. [1989]). Because of the use of the covariant Green-Lagrange strain tensor
without ignoring the higher order terms, all kinematic quantities such as displacements and
the corresponding strains can be finite, and the update procedure can be proceeded in an
exact manner, without approximations. It is noted that the quadratic terms in the strains
become important in the analysis of relatively thick shells, involving strong curvatures or
presenting large strains together with bending deformations (Buchter et al. [1994]).
Displacement-based solid elements are known to have poor performance in bending-
dominated situation and/or with incompressible materials, such as in thin shell analysis
with elastoplastic material. To obtain the same performance in bending as with stress-
resultant shell formulations with plane-stress assumption (e.g., Vu-Quoc, Deng and Tan
[2000]), the Enhanced Assumed Strain (EAS) method and the Assumed Natural Strain
(ANS) method are employed in the present solid-shell formulation.
As demonstrated in Chapter 3, many solid-shell formulations unfortunately do not
pass the out-of-plane bending patch test (e.g., Miehe [ 1998¿>], Klinkel et al. [1999]). To
avoid the Poisson-thickness locking problem (Zienkiewicz and Taylor [ 1991, p. 161 ], Bischoff
and Ramm [1997]), the transverse normal strain must have at least a linear distribution
across the shell thickness. In the present formulation, an optimal number of EAS parame¬
ters is established to enhance both the transverse normal strain (linear distribution in thick¬
ness) and the in-plane strains. Moreover, the present solid shell formulation passes both
the membrane patch test and the out-of-plane bending patch test, and thus with minimum
computational effort.
ANS interpolation has been a most successful tool to tackle the shear-locking effect
in the 4-node displacement-based shell elements, even for initially distorted meshes (Mac-
Neal [1978], Hughes and Tezduyar [1981], Dvorkin and Bathe [1984], Parisch [1995]), as
compared to the (selectively) reduced integration. To treat transverse shear locking, we ap-

178
ply an ANS interpolation on the compatible transverse shear strains. In the case of curved
structure with geometric nonlinearity, to treat the curvature-thickness locking (Bischoff and
Ramm [1997]), which is also called the trapezoidal locking (Sze and Yao [2000]), we apply
an ANS interpolation on the compatible transverse normal strain, as proposed by Betsch
and Stein [1995].
In addition to the above features, our new contributions in this chapter are specifically
listed below:
1) Demonstrate that the proposed optimal seven EAS parameters (three for the transverse
normal strain to treat the Poisson-thickness locking, and four for the membrane strains to
treat the in-plane bending locking) are sufficient to avoid locking problems with incom¬
pressible materials, in addition to passing the membrane and out-of-plane bending patch
tests.
2) Justify the use of the present element with various nonlinear materials in problems in¬
volving multilayer composite shells, including junctions with regular solid elements and
contact/impact.
3) Show that dynamic analyses can be carried out using either the consistent mass matrix
or lumped mass matrix (in explicit integration13) without spurious modes (Belytschko, Lin
and Tsay [1984] , Zeng and Combescure [1998]). Recall that the consistent mass matrix
for multilayer stress resultant shells is complex and configuration dependent (Vu-Quoc et
al. [2001]).
Two nonlinear 3-D material models at finite strain have been implemented in our
solid-shell formulation, and the simulation results reported in the present chapter. For
the (compressible) Mooney-Rivlin Model, the approach of the incompressibility limit is
tackled by the use of the penalty method or the augmented Lagrangian method. Other
forms of hyperelastic constitutive models such as the Ogden-type model (Ogden [1984])
are amenable to be implemented in the present element formulation. On the hyperelasto-
13 It is noted that the rotational dofs in traditional shell formulations correspond to high frequency modes,
which will drive the stable time-increments to very small size, thus requiring an artificial scaling of the
rotational masses (Hughes [1987, p.564]).

179
plastic model, the current implementation possesses the following advantages: (1) the re¬
turn mapping algorithm of infinitesimal plasticity can be carried over to the present finite
deformation context without any modification, and with a simplification of the compu¬
tational procedure: The closest-point projection algorithm is now formulated in princi¬
pal stretches. In particular, the algorithmic elastoplastic moduli tensor is symmetric, and
the incompressibility is automatically ensured; (2) With the elastic response emanating
from the hyperelastic form of the free-energy function, the elastic predictor in the return¬
mapping algorithm is exact, and computed without resorting to the use of incrementally
objective algorithms (Simo and Hughes [1998, (p.276)]) as for hypoelastic models; The
present implementation (3) employs the consistent tangent moduli tensor, instead of the
continuum elastoplastic moduli tensor, thus achieving quadratic rate of convergence in the
Newton iterative procedure (Simo [1988Z?]), and (4) finite-strain elastoplasticity based on
the Cauchy-Green strain tensor, thus avoiding the computation of the deformation gradient
F via the costly polar decomposition.
Even though there is no consensus on the necessity of passing the out-of-plane bend¬
ing patch test for convergence, we demonstrate that elements (with insufficient EAS pa¬
rameters) that did not pass the out-of-plane bending patch test perform poorly in problems
involving nonlinear material behavior and large deformation, as opposed to the present
formulation, which provides accurate results.14
The outline of the present chapter is as follows. We discuss the implementation of the
Mooney-Rivlin material model and the hyperelastoplastic material model, in Section 5.2.
The explicit integration method for solid-shell elements is present in Section 5.3. Several
numerical examples that illustrate the performance of the present formulation involving
large deformation, implicit and explicit dynamic analyses, together with a comparison of
the computed results to those obtained from other shell formulations and from a meshless
method, are presented in Section 5.4.
14 There is on the other hand a consensus that passing the membrane patch test is necessary for
convergence.

180
5.2. Nonlinear Material Law
A major advantage of the present solid-shell element is that all algorithms concerning
the 3-D nonlinear material models can be implemented without any modification. For non¬
linear materials, the second Piola-Kirchhoff stress tensor S in (3.16) and the fourth-order
consistent tangent moduli tensor C in (3.70) are crucial for the numerical solution of initial-
boundary-value problems with the quadratically convergent Newton-Raphson scheme.
5.2.1. The Mooney-Rivlin Material Models
There are two possible ways to define the stored energy function Ws for the modified
Mooney-Rivlin material, in which both satisfy the zero stress condition S — 0 at the initial
configuration (i.e., E = 0).
According to Fried and Johnson [1988], the stored energy function Ws of a modified
Mooney-Rivlin material is expressed as follows
Wa = Cx (h - 3/31/3) + C2 (J2 - 3732/3) + ^ (ln/3)2 , (5.1)
where the invariants Iu /2, and /3 are expressed in terms of the right Cauchy-Green tensor
C
c = FtF = 2E + 1,
(5.2)
as follows
Ix := trace (C) , /2 := ^ Ix2 — trace (C2)j , I3 := det (C) , (5.3)
where Cx, C2 are material constants, and A the penalty parameter for incompressibility.
From (5.3), the following derivatives are obtained
dh _ dl2 dh _ 2/ c_i dC _
8É-21’ dÉ~2hl~2C' dÉ~2hC ’ dE~2/)
(5.4)
where 1 represents the second-order identity tensor, and I the fourth-order identity tensor,
both expressed in the convected basis {G¿} as follows
1 = GijGi 0 Gj = GijGi 0 Gj , (5.5)
I=\ (GikGjl + GilGjk) Gi 0 Gj 0 Gk 0 Gl , (5.6)

181
The second Piola-Kirchhoff stress tensor S is the derivative of the stored energy function
Ws with respect to the strain tensor E. Using (5.1) and (5.4), it follows that
S = ^ = 2 [(Cl + Call) 1 - C2C - (C,I1/3 + 2C2I¡13 - Ain/,) CT1] .
where from (3.11) and (5.2) we obtain the following expressions for C and its inverse
as
(5.7)
C~x
c = FtF = gijGi ® Gj , C1 = gijGi ® Gj , (5.8)
respectively, with and gtj computed as shown in (3.9) and an equivalent equation.
The fourth-order material moduli tensor C is the second derivative of the stored en¬
ergy function Ws with respect to the strain tensor E, that is
d2Wa _ dS
dEdE ~ dE
= 46'21 ® 1 - 4C,I + 4 (Ain/, - I¡/3C, - 2I¡/3C2) —-
+ 4 (A - - i/|/3Ca) O"1 ® C-' , (5.9)
dc-x 1S
where the term -â–  can be shown to take the form as follows 3
oC
dC~x 1
-qq- = ~2 (ffV + 9il9jk) Gi ® Gj ®Gk®Gt. (5.10)
Another choice of the energy function Ws for the Mooney-Rivlin material is given
below (Bathe [1996, p.593]):
Ws — C\ (I\ — 3) + C2 (I2 — 3) — (Ci + 2C2) ln/3 + — (ln/3)2 , (5.11)
where the parameters C\, C2, A have the same meaning as in (5.1).16
15 By differentiating the identity g'^gjq = <5* with respect to gki, and by recognizing that =
: 9gkt
2 Í9jrgqa + gjsgqr), one arrives at (5.10) after postmultiplying the resulting equation with gqr> and reusing
the identity gjqgqp = 5P once more.
16 Both (5.1) and (5.11) are modified forms of the original Mooney-Rivlin model (Truesdell and Noll
[1992, p.350]), which corresponds to the first two terms in (5.11).

182
The second Piola-Kirchhoff stress tensor S corresponding to Ws in (5.11) is then
given by
5 = ^ = 2 [(Cx + C2h) 1 -C2C- (Ci + 2C2 - Aln/s) C~\
(5.12)
while the material moduli tensor C takes the form
+ 4AC-1 0 C~l .(5.13)
Our numerical experiments show that both models in (5.1) and (5.11) lead to the
same results if the same value of penalty parameter A is used.
Remark 5.1. To satisfy the incompressibility constraint (i.e., /3 = 1), the penalty
parameter A must be large enough so that the error on incompressibility is negligible (i.e.,
/3 is approximately equal to 1). On the other hand, the penalty parameter A cannot be
so large that numerical ill-conditioning occurs, and makes Newton’s method difficult to
converge. Alternatively, the augmented Lagrangian method is designed to avoid the char¬
acteristic ill-conditioning of the above penalty method, and enforces the incompressibility
accurately (Simo and Taylor [1991]), but destroys the desired quadratic rate of asymptotic
convergence of Newton’s method (See Section 5.4 for some examples). I
5.2.2. The Hyperelastoplastic Model
Up to the beginning of the 1980s, computational methods for finite-strain elastoplas-
ticity typically relied on hypoelastic extensions of the classical infinitesimal model (Simo
and Hughes [1998]), and hence not suitable for applications involving large elastic strains
(e.g., metal forging). In the last 15 years, computational approaches based on the multi¬
plicative decomposition have received considerable attention in the literature. Simo and
Ortiz [1985] and Simo [1988a] proposed a computational approach entirely based on the
multiplicative decomposition, and pointed out the role of the intermediate configuration
in a definition of the trial state via the hyperelastic stress-strain relations. Subsequently,

183
Eterovich and Bathe [1990] and Weber and Anand [1990] used the multiplicative decompo¬
sition in conjunction with a logarithmic stored energy function and an exponential approx¬
imation to the flow rule cast in terms of the full plastic deformation gradient. Based on the
above multiplicative decomposition, Simo [1992] showed that the closest-point-projection
algorithm of infinitesimal plasticity could be carried over to the finite-deformation context
without modification. A computational treatment of plasticity is to interpret the local evo¬
lution equations describing the plastic flow in the framework of the principle of maximum
dissipation, which lead to the return mapping algorithm with an operator split procedure.
Within a typical time step, an elastic trial state is first computed for prescribed strain in¬
crements and converged internal variables. Then, in the corrector stage the actual stress
is obtained by the closest-point projection of the trial stress state onto the elastic domain.
This projection is computed locally at each quadrature point of a typical finite element, and
depends exclusively on the functional form adopted by the yield criterion in stress space.
For the J2 flow theory, the closest-point projection reduces to the classical radial return
mapping.
There are two methods to implement the hyperelastoplastic model at finite strains:
(i) multiplicative decomposition of the deformation gradient F (Simo [1988a] and Simo
[19886]), and (ii) the spectral form of the right Cauchy-Green tensor C (Ibrahimbegovic
[1994], Miehe [1998a], Betsch and Stein [1999]). Our numerical experiments show that
while both methods lead to the same computed results, the second method does not require
the expensive polar decomposition. For method (ii), we provide a more convenient way to
decompose the right Cauchy-Green tensor C.
5.2.2.1, Multiplicative decomposition of the deformation gradient F
As already discussed in previous section, the present locking-free solid-shell element
formulations are based on the modified Green-Lagrangian strain E. Therefore, a modified
deformation gradient F, which is consistent to the modified strain E, is required for the
algorithm of the finite strain elastoplasticity (e.g., Simo [19886]). The evaluation of the
modified deformation gradient F consistent with the modified strain E, however, is rather

184
time-consuming.
The compatible deformation gradient Fc from the displacement field is split into an
orthogonal rotation tensor R and a right-stretch tensor Uc as follows
Fc = RUC, (5.14)
substituting (5.14) into the compatible Green-Lagrangian strain Ec of (3.12)
Ec = - (FcTFc - l) ,
(5.15)
the orthogonal rotation tensor R drops out and the compatible stain Ec will depend solely
on the right-stretch tensor Uc as
Ec
UcT (.RtR) Uc - 1
\ {UcTUc - l) ,
(5.16)
where Uc can be obtained by using a polar decomposition on (5.16), and then used in
(5.14) for the rotation tensor R by the following
R = FC [t/c]_1 .
(5.17)
The right-stretch tensor U, consistent with the current strain E including the EAS
and ANS treatments, can be computed by employing another polar decomposition on the
following
E = i (UTU - l) . (5.18)
With the unmodified orthogonal rotation tensor R calculated from (5.17) and the
current right-stretch tensor U obtained from (5.18), the deformation gradient F consistent
with the current strain E is computed through the following
F = RU. (5.19)
Thus, the above procedure to evaluate the deformation gradient F, which is consis¬
tent to the current strains E, needs twice a time-consuming polar decomposition at each

185
integration point. We refer to Simo and Hughes [1998, p.244] for the closed-form of the
polar decomposition.
With the deformation gradient F, it is straightforward to implement the return map¬
ping algorithm for the J2 finite strain plasticity model. For more details, readers refer to
Simo and Armero [1992].
5.2.3.1. Spectral form based on the right Cauchy-Green tensor C
Recently, the more advantageous algorithm of isotropic hyperelastoplasticity is based
on the principal stretch (Simo [1992], Miehe [1998c]), in which all the algorithms previ¬
ously developed for the small-strain plasticity can be used directly.
The phenomenological description of large-strain elasto-plasticity relies on the local
decomposition of the deformation gradient F,
F — FeFp
(5.20)
where Fe is the deformation caused by the stretching and rotation, Fp the deformation
associated with the plastic flow (see e.g., Maugin [1992, p. 167]).
To construct the nonsymmetric tensor Ze associated with elastic stretches, we de¬
compose the right Cauchy-Green tensor C similar as in (5.20), namely
C = ZeCp with Ze := C (Cp)_1 , ZeT := (Cp)_1 C
(5.21)
where Cv := FpTFp is the plastic Cauchy-Green tensor.
To represent the constitutive relation in terms of the eigenvectors jV¿ associated with
Ze, the standard eigenvalue problem needs to be solved
(5.22)
where eigenvalues {A,2}i=1 23 represent the elastic principal stretches, and eigenvector iV¿
and covector Nj satisfies
NrNj = S¡ , i,j = 1,2,3,
(5.23)

186
and Nl is normalized with respect to the plastic Cauchy-Green tensor Cp
Ni‘Cp»Ni = 1, ¿ = 1,2,3,
which leads to simple expressions of Cp and Cp 1 as follows
(5.24)
c” = ÉíVí®jví,
t=l
CP-1 = ¿ AT 0 N1,
i=l
respectively, where the summation convention is not used in the section.
From (5.22), we can express Ze in the following form
3
Ze = '£\2iNi®Ni,
2=1
therefore, with (5.25) and (5.27), the spectral form of C becomes
3
C = ZeCp = Y, A?JVi 0 iV¿.
1=1
Assuming the isotropic free energy ip takes the form
ip = ip(eitt2,e3,h) ,
(5.25)
(5.26)
(5.27)
(5.28)
(5.29)
where the logarithmic elastic principal strain e, is defined as the function of elastic principal
stretches A¿, namely
1
e¿ := InAj = ^lnA¿ , i = 1,2,3 ,
and h is the equivalent plastic strain.
Define the corresponding principal stress r, as follows
dip
Ti jr~ , * = 1,2,3.
(5.30)
(5.31)
According to (5.27) and (5.28), the derivatives of A2 with respect to Ze and C are
respectively
f) a2 «\2
=JVi®lVi, ZZ- = Ni®Nt.
dZ
dC
(5.32)

187
With (5.30), (5.31) and (5.32)2, the stresses S in the spectral representation are
c _ fty _ o^t
8E dC
= 2E
50 de.
= 2 E
i, de, d\?
U de, dC i=1
‘5A?ac
The Kuhn-Tucker loading/unloading condition is
k = E-¿¡NÍ®NÍ-
(5.33)
7 > 0 , <¡> < 0,70 (ri, r2, r3, y) = 0 ,
(5.34)
50
where 0 is the yield function, 7 plastic parameter, and y = —. The flow rule for the
oh
internal variables h and Cp in spectral form can be expressed as the ordinary differential
equations with respect to the time
• 50
h = ’
dy
Cf 1 = -27Cp"1 ¿ 0 N,
<=1 5r0
(5.35)
(5.36)
Remark 5.2, As mentioned in Simo [1988a], the Kuhn-Tucker optimality conditions
and constraints on the evolution equations for the internal variables can only be obtained
naturally through the principal of maximum plastic dissipation.
Consider the following local dissipation inequality (Simo and Hughes [1998, p.229])
dt
= ifs-2^'
2 l dC
50 *p-l 50.
c~dc^c ~dhh-°■
(5.37)
Since
o _ 0^0
5£ ac7
(5.38)
50
and denote y := —f, (5.37) is rewritten as
oh
50 Ap-i f . n
3T : C ~yh> 0 ,
acp
27 :=
(5.39)

188
taking the derivatives of eigenvalue problem (5.22) while keeping C fixed, along with
(5.21)2, it becomes
(cdCp~l - Ni + (CCP-1 - At2l) dNi = 0, (5.40)
by multiplying iV, at both sides of (5.40), using the orthogonality condition iV¿ • dNi = 0,
and with (5.28), we have
<9A2
dCp
t = A
(5.41)
it follows that
dx2i -yWX2N&N
dCp~l ~ h9>ZdCp-' hi9Kl 1 *'
(5.42)
With (5.32)2, we have
tty 3 00 0A? * ity , ,
ac ¿í 9A? ac ® •
(5.43)
Therefore, with the above (5.42) and (5.43), and combining the expression of C in (5.28),
Cp in (5.25), and S in (5.33), we obtain the relation
J* = C—C” = l-CSC” .
dcP_1 dC 2
With (5.33) and (5.28), CS in (5.44) becomes
3
(5.44)
cs = y.t'ní® iyi-
(5.45)
i=l
thus the dissipation inequality in (5.39) can be rewritten as follows
1 3
V = --Y, tíní <8> N*CP : Cf~l -yh> 0 .
Z •
(5.46)
i=i
The classical rate-independent plasticity model in the spectral form is obtained by
postulating the following local principle of maximum dissipation:
For admissible (r¿,/i), which satisfies the yield criterion <¿>(r¿,/i), the actual state
(tí, h) maximizes the dissipation function V. Using the Lagrange multiplier method, we
define the constrained minimization problem in the following
£ (r»> y, 7) := -V fa, y) +1<\> 0Ti, y) ,
(5.47)

189
where 7 is the plastic consistency parameter. By finding the extremal point of £, the con-
dr
strained minimization problem of (5.47) leads to the flow rule in (5.35) (i.e., — = 0) and
oy
(5.36) (i.e., — = 0 ), and the Kuhn-Tucker loading/unloading condition of (5.34). I
on
For the evolution equation, the plastic hardening variable h is obtained by a standard
implicit backward Euler method with the time increment At = fn+i — tn on (5.35)
d(j)
hn+i hn 4" (3
dy
, (3 = 7n+iAf.
(5.48)
71+1
As originally proposed by Weber and Anand [1990], we integrate the flow rule of
Cp_1 in (5.36) by means of the exponential map. The advantage of this method is two
folds: i) preserve the plastic incompressibility condition of the pressure insensitive yield
function 0; ii) return mapping algorithm of small strain plasticity can be used without any
modification.
By using the backward Euler integration with an exponential map on (5.36) (simi¬
larly, for y — c(t)y, it follows that yn+1 = [exp (cn+i At)] yn), it yields
c;;i = exp('-2,s¿|hvi«jv.') CP.^
\ i=1 OTi 1 n+l
(5.49)
where Cvn 1 is associated with the eigenvalue problem (5.22) for the trial state, that is
(c^.cr1 - Af 1) JV| = 0,
(5.50)
which gives
i=l
In parallel, C„+\ is associated with the eigenvalue problem at tn+\
(Cn+1Cpn-+\ - At2l) Ni = 0 ,
(5.51)
(5.52)
which gives
C.wC'-tbE^SJV1,
J=1
(5.53)

190
where AT¿ is the eigenvectors associated with the eigenvalue A¿ for time tn+1.
Similar to (5.25) and (5.26), the plastic Cauchy-Green tensor Cvn and its inverse at
the trial state are expressed as follows
3 3
cpn = £ JVj ® N\, Cv~l = ENti® N* ■
(5.54)
t=l
i=l
Substituting (5.54)2 into (5.51), and recalling (5.28), we have
3
l
i= 1
C„+1 = t X?N\ ® N\ = ¿ X?Ni ® N,.
3
i=l
(5.55)
Therefore, the eigenvectors Nt at fn+1 and N\ at the trial state are proportional
\t
AT. = -IN*
1 Xi "
(5.56)
by multiplying Nl and Ntl at both sides of (5.56), and with (5.23), the eigenvectors Nl
associated with A¿ at tn+\ and Nu associated with A- at the trial state are also proportional
AT = Nu .
A,
Considering the above (5.57) in (5.54)2, we have the following
3 3 \ i2
Cv~l = Y,Nti® N* = E ~\2N* ® .
t=l i=l
substituting (5.58) into (5.49) and using (5.26), it becomes
(5.57)
(5.58)
A,2 - exp ( -2/?Sr ) A,“,
(5.59)
with the definition of (5.30), the update of the principal strain is by simply taking the
logarithm on both side of (5.59), namely
*-¡-C
(5.60)
Denote
6 ^ ’et {e01T ~{dt}’ 1,2,3’

(5.61)
T ^ d2(j> = dn _ d2(j) _ dnh
dr dr dr ’ h <9y<9y
the solution of the set of nonlinear algebraic equations (5.48), (5.60) along with the yield
criterion 0 = 0 is typically obtained by a Newton procedure. A Newton procedure based
on the systematic linearization of these equations gives rise of a plastic-corrector return
to the yield surface based on the concept of closest-point projection. We write the plastic
updates in (5.48), (5.60) and yield condition 0 = 0 at time fn+1 in the form of
rt = e — el + (3n = 0 , (5.62)
rh =h-ht + (3nh = 0 , (5.63)
0 (r, y) = 0 , (5.64)
linearizing (5.62)-(5.64) gives the incremental form at iteration (k) of time in+1,
rt + Ae + A/3n + /3An = 0 , (5.65)
77, + A/i + A/3n/, + (3Anh = 0 , (5.66)
0 + nAr + rihAy = 0 . (5.67)
By making use of the following from (5.61)
At = £Ae , Ay = ShAh , (5.68)
An = TAt , Anh = J^Aj/, (5.69)
and substituting (5.68) into (5.67), substituting (5.69) into (5.65) and (5.66), it yields

192
rh + £h 1ShAh + A/3nh = 0 , (5.71)
(f> + n£Ae 4- n^ShAh = 0 , (5.72)
with S =(£-'+pry1 , £h= (£¿l+(3Thyl , (5.73)
then we can solve for the increments A/?, Ae, Ah from (5.70)—(5.72) as follows
Ap = (p - nT£ re - nh£h rh ) , (5.74)
Ac = -£-x£ (r£ + Apn) , (5.75)
Ah = -£hl£h(rh + Apnh) , (5.76)
with D = nT£n + £hn2h . (5.77)
With the above obtained increments, we update the strains, internal variables and plasticity
parameter for the iteration (k + 1) at time tn+\. The local Newton iteration procedure is
continued until the convergence to the yield surface within a sufficient tolerance.
The Newton’s method relies on the fourth-order consistent tangent moduli C (also
called the algorithmic tangent moduli), which is crucial in the development of the material
tangent stiffness matrix (see Section 3.3). In Simo and Taylor [1985], it was first shown
that the disappointing rates of convergence exhibited by Newton-type iterative methods
arise from lack of consistency between the continuum elastoplastic moduli and the return
mapping algorithm. The development of the consistent tangent moduli is based on a sys¬
tematic linearization of the stress update algorithm, which is defined as
dS__2dS_
~ dE - 2dc ’
(5.78)
where S at ¿n+1 is obtained from (5.33).
Since the principal stress r¿ at time tn+1 is the function of e‘, with (5.32)2 and (5.33),
the derivation of (5.78) leads to the algorithmic tangent moduli C (see also Ogden [1984],
Zienkiewicz and Taylor [20006, p.343] for the details) in the following
C = ¿ 2--fe^2) Ntl ®Ntl + Y, 2Ti Ntl ® NU
dC ^
h ac

193
3 3 ,reP_2r ¿ -
1 ^-Nu ® Nu ® ATtJ ® NtJ
= EE —
\t2\t2
i=lj=l Ái Áj
3 3
+ E E áj-N* ® JVy ® {nu <8> ivtj + ATÍJ (8) TV4') ,
¿=1 j=l j'/i
where the elastoplastic moduli £¡p is defined in the eigenspace as follows
(5.79)
cep =
ij • de) ’
(5.80)
which can be derived based on the linearization of (5.60), (5.48) along with = 0. The
elastoplastic moduli Stef is derived in Appendix A.5, and the term gij listed in (A.48).
Since we develop the present solid shell formulation in the convected basis, it is nec¬
essary to transform the second Piola-Kirchhoff stress tensor S and the consistent tangent
moduli tensor C from the basis of eigenvector Nu to the basis G¿, namely
S = SfjN*® Ntj = SijGi®Gj
C = cyjklNu ®Ntj ®Ntk ®Ntl =CijklGi®Gj®Gk®Gl, (5.81)
which involve the transformation procedure similar to that used between the Cartesian basis
el and the convected basis in Section 3.3.
In summary, we identify {C, Cp, h} as the state variables. Once the state variables
are known, the stress tensor S and the consistent tangent moduli C will be determined
through (5.33) and (5.79) respectively. The detailed implementation on the stress update
and consistent tangent moduli are listed in Appendix A.4 and Appendix A.6, respectively.
5.3. Explicit Time Integration Method for Solid-Shell Elements
Here we discuss the conditional-stable explicit method associated with the present
solid-shell element, in which the reliable full numerical integration is employed. Since
the maximum stable time-step size At decreases with the mesh refinement (see, e.g., Be-
lytschko, Liu and Moran [2000, p.314], LS-DYNA [1998, Chap. 19]), it is beneficial to use
the present element, which has coarse mesh accuracy, and needs only one element through
the thickness, in explicit analyses of shell structures. Reduced-integration (RI) elements

194
have been widely used together with explicit time integration (DYNA3D [1993], ABAQUS
[2001]). Due to the use of ad hoc assumptions on the kinematics and the material properties,
the proper stabilization (hourglass control) techniques for the spurious zero-energy modes
caused by the RI scheme are, however, still an active research area, especially for physi¬
cally nonlinear problems such as crashworthiness problems (Zhu and Cescotto [1996], Zhu
and Zacharia [1996], Zeng and Combescure [1998]). Furthermore, reduced-integration
elements are highly sensitive to mesh distortion (see, e.g., Stanley [1985]).
Although there is a disadvantage with the need for using small time increments for
stability, the advantages of explicit methods are significant in that the construction, stor¬
age, decomposition and back substitution of the effective tangent stiffness matrix, which
is required in implicit methods, are completely avoided. There are many situations where
explicit methods are preferable. For example, in high-speed events such as car crash simu¬
lations, a small time step is required due to the noise introduced by the contact and impact
between different structural parts.
Two types of mass matrices can be considered: The non-diagonal consistent mass
matrix and the diagonal lumped mass matrix. To increase computational efficiency, lumped
mass matrix is often chosen to avoid the decomposition and back-substitution. Although
the procedures for diagonalizing the mass matrix are quite ad hoc and questionable, espe¬
cially for high-order elements and stress-resultant shell elements (Zienkiewicz and Taylor
[1989, p.605], Hughes [1987, p.565]), the lumped element mass matrix for the present
eight-node solid-shell element can be obtained easily without problem. One common ap¬
proach is the row-sum technique (Zienkiewicz and Taylor [1989, p.474]), in which the
diagonal entries mlu of the lumped element mass matrix mlü are obtained by
ndof
m
a = 5Z mij >
(5.82)
3=1
where the sum is over the entire row of the consistent element mass matrix
jg>ndof xndof^ where ncj0f js the number 0f degrees of freedom in an element (see Vu-Quoc
and Tan [20026]).

195
Alternatively, the lumped element mass matrix
from the following expression
m\Á can also be evaluated directly
(5.83)
where in the element domain Bq \ M is the matrix containing the shape functions corre¬
sponding to degrees of freedom of the element nodes, and p the mass density. Both (5.82)
and (5.83) lead to the same lumped element mass matrix for the present element.
The global semi-discretized nonlinear ordinary differential equation (ODE) is in the
form
Mil - Fext - Fint ,
(5.84)
where M is the assembled mass matrix, and the internal force Fint
is computed as follows
F
int
(5.85)
with the matrices f^EAS being the terms associated with the EAS method in
(3.78), and EAS parameters o¿e) are updated through (3.79).
To solve the nonlinear ODE (5.84), the effective and most widely used explicit
method is the central difference method. In a typical time step [£n,£n+i] with time-step
size tn+i — tn = At, the displacement un+1 at £n+1 is updated by using the displacement
un, the velocity un, and the acceleration ün at tn as follows
Wn+i T¿ri “b UjiAt -f- Uyi
At2
~Y'
(5.86)
By using (5.84) and (5.86), the acceleration ün+1 at time £n+1 is obtained via
^n+1 = M-1 [Fext (un+1) - Fint (un+1)] ,
(5.87)
where only divisions are required for the solution of ün+j due to the diagonal form of
M, in contrast to the expensive decomposition and back-substitution needed in implicit

196
methods such as the Newmark and energy-momentum conserving algorithms (Vu-Quoc
and Tan [20026]).
With un+l from (5.87), the velocity un+x at tn+x is approximated by
Ú„+, = + y (“»+ K"«) ■ <5-88)
Updating for the displacement un+x by (5.86) does not require the solution of any
algebraic equations. Thus, in this sense, explicit integration is simpler than implicit inte¬
gration. As shown in Appendix A.3, the explicit program is a straightforward evaluation of
the governing equations and the time integration formulas. It can be seen in (5.85) that the
computation of the internal nodal forces involves the calculation of strains, stresses, and
the constitutive matrix. When the element nodal forces are calculated, they are assembled
to the global array according to the node connectivity. By prescribing the nodal velocities
at prescribed velocity boundaries (see, e.g., in Appendix A.3), the correct nodal displace¬
ments result from (5.86). The reaction forces at prescribed velocity nodes can be obtained
from the total nodal forces
Rn+i = Fen% - . (5.89)
Since the time step in explicit integration must be below a critical value (otherwise
the numerical solution will blow up), it is not recommended to use explicit methods in
quasi-static or low-speed events such as the springback effect in metal forming process.
On the other hand, it is appropriate to combine explicit methods and implicit methods to
maximize computational efficiency, that is, for example, in metal forming or crash analysis,
explicit time integration can be used for the initial time stepping and then a static or implicit
dynamic solution be used for the rest (see, e.g., ADINA [2002]).
5.4. Numerical Examples
The finite element formulation of the present low-order solid-shell element for non¬
linear analysis of shell structure presented in the previous sections has been implemented in
both the Finite Element Analysis Program (FEAP), developed by R.L. Taylor (citef:zie.89a),

197
and the NIKE3D by the Lawrence Livermore National Laboratory (NIKE3D [1995]), and
run on a Compaq Alpha workstation with UNIX OSF1 V5.0 910 operating system. In each
element, the mass matrix is evaluated using the Gauss integration 2x2x2, the tangent
stiffness matrix, the dynamic residual force vector are evaluated using the Gauss integration
2 x 2 in the in-plane direction, three Gauss points in the thickness direction for Mooney-
Rivlin material, and five Gauss points in the thickness direction for elastoplastic material.
Below we present the examples involving the static and dynamic large deformation analysis
with nonlinear materials. In the nonlinear dynamics analysis, we use the trapezoidal rule
for implicit time integration (with the consistent mass matrix) and the central difference
method for explicit time integration (with the lumped mass matrix), without introducing
numerical dissipation.
5.4.1. Large Deformation of Rubber Shells
Here we analyze the large deformation of rubber shells by using the present solid-
shell element and Mooney-Rivlin material model. For the augmented Lagrangian method,
a function of 7ln/3 is appended to the strain energy function Ws in either (5.1) or (5.11),
where 7 is augmented Lagrangian multiplier. The augmented Lagrangian procedure can be
accomplished by a nested iteration algorithm (Zienkiewicz and Taylor [2000a, (p.323)]).
Within a typical iteration of one step, we solve the linear system with fixed penalty pa¬
rameter A. Then the Lagrangian multiplier 7 is updated by 7 = 7 + Ain/3 at the end of
nested iteration . The nested iteration can be repeated with the increased penalty parameter
to reach the desired accuracy for the incompressibility constraint. Here we use an initial
value of A = 250 for the penalty parameter. Our numerical experience on the uniform
extension/compression test of one element (Bathe [1996, p.593]) showed that with four
nested iterations the incompressibility can be enforced to the tolerance of 10-8 in the vol¬
ume change between the initial configuration and the deformed configuration, compared to
10~5 by using the penalty method with a large value of penalty parameter A = 5000.

198
5.4.1.1. Stretch of a rubber sheet with a hole
This problem has been analyzed by Gruttmann and Taylor [1992]. The material con¬
stants for Mooney-Rivlin model are C\ = 25 and C2 = 7. The length of the square is
L = 20, the radius of the circle is R — 3, and the thickness h = 1 (left of Figure 5.2). Due
to the symmetry, only one quarter of the sheet has been modeled with 64 solid-shell ele¬
ments. The augmented Lagrangian method is used here, with five nested iterations for each
load step. Figure 5.2 depicts the initial geometry and the corresponding final deformed
mesh configuration for q = q0 = 90. A full agreement with the membrane element devel¬
oped by Gruttmann and Taylor [1992] is shown in left of Figure 5.3. It is clear that the large
strains and the thickness stretching are involved in the present problem. For instance, the
sheet thickness at point D becomes one half of the initial thickness. It is interesting to note
that the sheet thickness at point B is increased rather than decreased (right of Figure 5.3).
Figure 5.2. Stretch of a rubber sheet with a hole: Initial configuration (left) and deformed
shape at q0 = 90 (right).
5.4.1.2. The snap-through of a conic shell
Next we show the robustness of present solid-shell elements with large elastic strains.
This problem was appeared in Li, Hao and Liu [2000] for the application of their meshless
method, in which 12,300 particles with three particles in the thickness are used. A total
of 1,800 solid-shell elements (3,720 nodes) are used in the discretization, with only one
element in the thickness direction. It is noted that the ongoing intensive research is directed

199
thickness stretching (right).
to make meshless methods more computationally efficient, which includes the interpolation
scheme, numerical integration procedures and techniques of imposing boundary conditions
(De and Bathe [2001 b], De and Bathe [2001a], Atluri and Shen [2002]).
The material and geometric properties are
Ci = 18.35, C2 = 1.468 ,
A = 1.468 x 103 , p = 1.4089 x 10“4 ,
Rtop = 1, Rbot = 2, H = 1, h = 0.05 , (5.90)
where C\, C2 are the material constants, A the penalty for incompressibility, p the initial
density of the rubber shell, and Rtop, Rb0t the radius at the top and the bottom of conic shell,
respectively, H the height, and h the thickness of the conic shell.
The time increment is chosen as At = 25 x 10-6sec, and total 400 steps are used
in the implicit Newmark method. In the computation, we fix the bottom edge of the conic
shell and do not allow the horizontal movement of the upper inner edge, and prescribe
the vertical displacement of upper inner edge such that it drags the whole shell structure
down. At the end of the computation, the conic shell turns inside out. In Figure 5.4-5.5,
several snap-shots are taken to form a deformation sequence. The deformation is large and
involves both material nonlinearity and geometry nonlinearity. Such deformation process

200
belongs to a so-called snap-through instability problem because the reaction force at top
edge is alternating along with the advance of vertical displacement of the top edge (see
Figure 5.6). In the static case, the magnitude of the reaction force is much smaller than in
the dynamic case, which is true when the inertial effect is not considered (left of Figure 5.6).
Disp. control
• •7777777777. •
Figure 5.4. Snap-through of the conic shell: Initial configuration (left) and deformed shape
at t — 2.5 x 10-3sec (right).
Figure 5.5. Snap-through of the conic shell: Deformed shape at time t — 5.0 x 10 3sec
(left) and t = 10.0 x 10_3sec (right).
Finally, we point out that, both the penalty method and augmented Lagrangian method
leads to similar behavior (right of Figure 5.6) in this problem, while the latter needs more
iterations at each time step (Table 5.1).
5.4.1.3. Large motion of the pinched cylindrical shell
This problem was also appeared in Li et al. [2000] for the application of their mesh¬
less method. We prescribe the inward radial displacement for two opposite nodes of the
inner-surface at the middle section of the cylinder (Figure 5.7).

201
Figure 5.6. Snap-through of the conic shell: Deflection versus reaction force curve by
penalty method (left) and augmented Lagrangian method (right).
Table 5.1. Snap-through of the conic shell: Convergence results for both penalty method
and augmented Lagrangian method at a typical time step (energy norm)
Iter.
Penalty
1 st augm.
2nd augm.
3rd augm.
4th augm.
5th augm.
0
3.20£-02
1.44£-03
1.32)5—05
2.64)5—06
4.7477—07
8.30£-08
1
7.1977—03
7.35E-03
2.34£—08
1.2077—10
0°
CO
o
1
i—1
co
8.1877—15
2
4.86£—05
2.8077—06
2.4877—16
1.9777—20
2.8677—24
5.4177—28
3
1.16£-08
2.10£-12
4
9.02£-16
3.0277—23
The material properties are the same as in (5.90) and geometric properties are
R = 1, H = 1, h = 0.02 , (5.91)
where R is the radius, H the height, and h the thickness of the cylinder.
Because of the symmetry, only one-eighth of the cylinder needs to be modeled by
32 x 32 x 1 solid-shell elements with 2,178 nodes (Figure 5.7). For the temporal integration
of explicit central difference algorithm, the time increment is At = 0.5 x 10-6sec and total
20,000 time step has been taken to finish the run. For the integration by implicit Newmark
algorithm, the time increment is At = 10 x 10“6sec and total 1,000 time step has been
taken. In Li et al. [2000], total 30,300 particles with three particles in the thickness are used
and 21,000 time steps are taken in the explicit analysis.
Both the implicit method and the explicit method lead to the similar deformation
sequence shown in Figure 5.8- 5.9, The deformation of cylinder under pinched loading
is drastic. At the end of our computation, the two opposite points of inner surface of the

202
cylindrical shell come together. Compared to Li et al. [2000], the deformation from the
current calculation is more severe and appears more buckling modes.
Figure 5.7. Large motion of pinched cylindrical shell: Geometry and loading.
Figure 5.8. Pinched cylinder: The deformation at time t — 2.0 x 10 3sec (left) and
t — 4.0 x 10-3sec (right).
The relation between the reaction force and the deflection both at the pinch point
is presented in Figure 5.10. It is observed that for both the implicit method and the ex¬
plicit method, the results of reaction forces are very close, which justify the use of explicit
method in the present solid-shell element for the high-speed dynamics.

203
Figure 5.9. Pinched cylinder: The deformation at time t — 8.0 x 10 3sec (left) and
t — 10.0 x 10_3sec (right).
5.4.1.4. Rubber hemispherical shell
The clamped rubber hemispherical shell is subjected to a point load at the pole. This
problem is used to demonstrate the localized effects of thickness change and importance
of applying the external surface loading for moderately thick shell problem. Without any
modification on the element formulation, the surface loading can be considered naturally
in the present solid-shell element.
The material and geometric properties are
Ci = 25, C2 = 7 ,
A = 1. x 103,
R = 26.3, h = 4.4 , (5.92)
where C\, C2 are the material constants, A the penalty for incompressibility, respectively,
and R the radius, and h the thickness of the spherical shell. Due to the symmetry, one
quadrant is modeled by 192 solid-shell elements (Figure 5.11). Displacement control is
used to drive the top of the hemisphere down. Two different loading cases, top surface
loading at point A and both surfaces loading at point A and B to approximate the midsurface
loading in classic shell element (with inextensible director), are considered.
The both-surface loading keeps the thickness unchanged between point A and B, and

204
0.035
— explicit
— implicit
0.03
0.025
0.02
0.015
0.01
0.005
-0.005
0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
U
Figure 5.10. Large motion of the pinched cylindrical shell: Deflection versus reaction force
curve.
produces a ’’kink” at point B, which is true only for thin shells (left of Figure 5.12). The top-
surface loading exhibits the anticipated localized effects and the thickness change by about
70% between point A and B, which produces the physically more reasonable deformation
(right of Figure 5.12).
The relation between the deflection at loading location and the corresponding total
force to produce such deflection for both cases is presented in Figure 5.13. The difference
on two loading cases shows the importance of localized thickness change effects.
5.4.2. Large Deformation of Elastoplastic Shells
For finite deformation J2 plasticity, the free energy function is assumed to take the
uncoupled form as follows
(5.93)
where A and p are the Lame parameters, H is a parameter which characterizes the isotropic
hardening in the material, and h the equivalent plastic strain. The Von-Mises yield criterion

205
Figure 5.11. Rubber hemispherical shell: Mesh, boundary conditions and loading for the
one quarter of shell.
Figure 5.12. Rubber hemispherical shell: Displacement loading at both surface shows
”kink” at point B (left), and displacement loading at top surface has the smoothness at
point B. The thickness between point A and B is changed by approximately 70% (right).
is expressed in terms of principal stresses as
(,Tl,T2,T3,y)
(5.94)
where ry is the initial flow stress, y = Hh.

206
Figure 5.13. Rubber hemispherical shell: Load-deflection curve showing the difference of
the two loading cases.
5.4.2.1. Bending of a cantilever beam
This example investigates an elastoplastic cantilever beam illustrated in Figure 5.14
for several aspect ratios L/h, and confirms the correctness of the current solid-shell element
for elastoplastic material. The mesh for the plate is 20 x 1 x 1.
The material properties are
E= 1.2 x 107, p = 0.3, Ty = 2.4 x 104, H = 1.2 x 105, (5.95)
where E, p, ry, H are the Young’s modulus, the Poisson’s ratio, the initial yield stress, and
the isotropic hardening parameter, respectively.
Numerical results are compared to Eberlein and Wriggers [1999] where the solutions
are computed from the classical 5-parameter shell element (volumetric locking does not
appear in elements with the plane-stress condition (Hughes [1987, p. 193])). The good
agreement in all range of aspect ratios can be observed. For the aspect ratio of L/h = 100,
the final deformed configuration with the contour of the equivalent plastic strains at the

207
plate surface is shown in right of Figure 5.14.
plastic stra n
5 0.0047
0.0039
0.0026
0.0013
0.0003
clamped F
jS UUUUJ
â–  0.0000
Figure 5.14. Out-of-plane bending: Geometry and mesh of cantilever plate (left), and
deformed shape and upper-surface contour plot of equivalent plastic strains (right).
A load deflection curves for different aspect ratio are shown in Figure 5.15-5.16.
Compared to our seven-parameter EAS formulation, the solid-shell element with five-
parameter EAS in Miehe [1998¿>] and Klinkel et al. [1999] behave stiffer and cannot obtain
the collapse load for the thick and moderate thin plate (Figure 5.15), which may be caused
by the inability of the element to deal with the volumetric locking in the plastic deforma¬
tion stage. It is observed for the thin case (aspect ratio = 1000) the plastic stage doesn’t
appear till the end of loading, so two Gaussian points through the thickness are enough
for this elastic problem, and the results from Miehe [1998¿>] and Klinkel et al. [1999] is
close to the present though their element formulations cannot pass the out-of-plane bend¬
ing patch test (Subsection 3.5.1). The plasticization appears at load F = 475 for aspect
ratio L/h = 10, and F — 5 for aspect ratio L/h= 100.
w
w
Figure 5.15. Out-of-plane bending: Load deflection curve for aspect ratio L/h— 10 (left)
and for aspect ratio L/h= 100 (right).

208
Cantilever beam loaded by shear force, L/h = 1000
Figure 5.16. Out-of-plane bending: Load deflection curve for aspect ratio L/h — 1000.
Since the exact tangent (Jacobian) matrix is employed in the Newton’s solution pro¬
cedure, the quadratic rate of asymptotic convergence was actually observed in all problems
we examined. Table 5.2, which depicts the values of the Euclidean norm of both the resid¬
ual and the energy norm at each iteration, clearly exhibits the quadratic rate of asymptotic
convergence, which is in the sharp contrast to the extremely slow convergence (hundreds
of iterations in one step is not uncommon) with the hypoelastoplastic rate-form material
model (Choudhry and Wertheimer [1997]).
5A2.2. Elastoplastic response of a channel beam
A far larger and equally important class of structures are the nonsmooth shell which
consists of folds, kinks or branches. Here we present the warping of an angle iron to prove
the validity and applicability of the proposed solid-shell element formulation, in which the
simulation of such structures on both the overall behavior and local stress concentration ef¬
fect at one time becomes straightforward, and does not introduce any assumption on the in-

209
Table 5.2. Out-of-plane bending: Convergence results for plate with L/h - 100 (residual
norm, energy norm)
Iter.
Step 1 (F=0.5)
Step 10 (F=5.0)
Step 13 (F=6.5)
0
2.505-01,8.275-02
6.835+00, 7.735-02
1.765+01,7.275—02
1
1.005+04,1.795+00
9.025+03,1.455+00
8.215+03,1.205+00
2
3.965+00,3.295-07
5.275+00,3.625-04
4.585+01,1.535-02
3
3.465-05,3.925-14
3.835+01,2.595-05
2.125+03,8.335-02
4
5.515-09,7.105-25
4.145-03,9.195-10
8.225+01,1.115-02
5
1.305—04,3.015—16
3.835+03,2.575-01
6
1.705-08, 7.015-24
2.515+00,3.415-03
7
1.995+03,6.915-02
8
1.345-01,5.705-06
9
3.405+00,2.025-07
10
6.835-06,6.165-14
11
4.575-08,3.995-23
terconnection or penalty parameters for the shell intersection. A cantilever beam subjected
to a point load is considered as given in Figure 5.17. Because the centroid (inside of the
open cross section) and the shear center (outside of the open cross section) of the beam do
not coincide, if the concentrated forces are not applied at the shear center of the cross sec¬
tion, the beam section will twist significantly. Originally, the purely elastic material behav¬
ior of this example was presented by Chroscielewski, Makowski and Stumpf [1992], where
the numerical results show a strong dependence on a penalty multiplier which accounts for
the drilling stiffness within their shell formulation. The same example for elastoplasticity
was also investigated by Eberlein and Wriggers [1999], where the converged results are
presented for their 6/7-parameter shell concept along with a penalty method for the shell
intersection. Since the present element uses the upper and lower surface to describe the
deformation, the intersection can be simply modeled by solid elements. For the detailed
analysis of the intersection, a refined solid element mesh can be used, which can be easily
connected to other shell parts modeled by solid-shell elements. It is noted that it is not
possible for stress-resultant shell element to describe the detailed strain and stress distribu¬
tion at the intersection (Chroscielewski et al. [1997]), while the solid-shell element offers

210
a more realistic and convenient representation for the physical structure.
The ideal elastoplastic material properties are used as follows
E = 107, [i = 0.333, Ty = 5.0 x 103, H = 0 , (5.96)
where E, /i, ry, H are the Young’s modulus, the Poisson’s ratio, the initial yield stress, and
the isotropic hardening parameter, respectively.
For the computations, a discretization of 1,584 elements is used as shown in Fig¬
ure 5.17. In Figure 5.20, the load-deflection curves for both the elastic material and elasto¬
plastic material obtained from the present element coincide very well with solutions by
Betsch et al. [1996] and Eberlein and Wriggers [1999], respectively. The deformed con¬
figurations for u = 4 are illustrated in left of Figure 5.18 for the elastic solution and right
of Figure 5.18 for the elastoplastic solution. For the elastic solution, the buckling of the
upper flange can be observed in the vicinity of the clamped edge, whereas the free end is
twisted, since the external load does not act in the shear center. The elastoplastic solution
also shows a buckling phenomenon in the upper flange, but it is more concentrated than
in the purely elastic case. Since the plastic strains occur very early during the deformation
process as shown in Figure 5.20, the equivalent plastic strains for the deformed configura¬
tion u = 0.2 are reported in left of Figure 5.19. As expected from the beam theory, there
is a maximum of plastic deformation in the lower and upper flange at the clamped edge of
the steel channel. The equivalent plastic strains for the deformed configuration u = 4.0 are
shown in right of Figure 5.19.
It is noted that for the current shell with intersection, there are two ways to model
the intersection as in Mesh 1 and Mesh 2 of Figure 5.21. From our numerical experience,
the results from the Mesh 2 is stiffer than Mesh 1, which is more pronounced for the linear
elastic material (Figure 5.20). The possible reason for it lies on the sensitivity of assumed
strain method to the mesh distortion through the thickness. For folded shells with multi¬
intersections, Mesh 3 in Figure 5.21 is more advantageous for the modeling, compared to
the more involved shell elements with drilling couples (Chroscielewski et al. [1997]).

211
Figure 5.17. Channel beam: Initial configuration with mesh.
Figure 5.18. Channel beam: Deformed mesh for elastic material at u = 4.0 (left) and
deformed shape for elastoplastic material at u = 4.0 (right).
5.4.2.3. Pinched hemisphere
The present example investigates an elastoplastic hemisphere, which has been calcu¬
lated previously by Simo and Kennedy [1992] by means of a 5-parameter shell formulation
in combination with a plasticity model formulated in stress resultants, based on an addi-

212
Figure 5.19. Channel beam: Deformed mesh and contour plot of equivalent plastic strains
at u = 0.2 (left) and at u = 4.0. (right).
u .u .
Figure 5.20. Channel beam: Load deflection curve by model with solid elements at comer
(left), and model without solid elements at comer (right).
tive decomposition of the strains into elastic and plastic parts. The radius and thickness of
hemisphere are R = 10 and h — 0.5, respectively. Figure 5.22 illustrated the mesh and
the boundary condition for the present calculation, where total 432 solid-shell elements
are used to model the one quadrant of the hemisphere due to the symmetry. The material
properties are
E = 10, [i = 0.2, Ty = 0.2, H = 9 , (5.97)
where E, /i, ry, H are the Young’s modulus, the Poisson ratio, the initial yield stress, and
the isotropic hardening parameter, respectively.
Figure 5.23 shows the deformed hemisphere including an outer-surface contour plot
of the equivalent plastic strains at a load level of F — 30 x 10~3. The load-displacement
curves are depicted in Figure 5.24, where the results reported in Eberlein and Wriggers
[1999] are included. Our results agree with Eberlein and Wriggers [1999] very well.

213
1—1—1—
1 1 1 7
1 1
1
1
Mesh 1 Mesh 2 Mesh 3
Figure 5.21. Mesh at the cross section of shells with intersection.
12 -12
Figure 5.22. Pinched hemisphere: Geometry and mesh of one quadrant of hemisphere.
5.4.2.4. Elastoplastic response of a simply supported plate
We consider the elastoplastic deformation of a simply supported square plate. The
square plate has a length of L = 0.508 and thickness h = 2.54 x 10-3. Due to the symmetry
of geometry and the boundary conditions, only a quarter of the plate has been discretized
with 16x16x1 solid-shell elements (Figure 5.25). It is simply supported so that horizontal
displacements and rotations may occur. Only the transversal displacements are set to zero

214
plastic stra
n
0.0486
0.0347
0.0208
0.0069
0.0041
0.0020
0.0000
Figure 5.23. Pinched hemisphere: Deformed shape and outer-surface contour plot of equiv¬
alent plastic strains.
Figure 5.24. Pinched hemisphere: Load F versus the deflection under load F.
at the boundaries of the upper surface of plate.
The ideal elastoplastic material properties are
E = 6.9 x 1010, p = 0.3, Ty = 2.48 x 108, H = 0 , (5.98)

215
Figure 5.25. Inflation of a plate: Mesh, boundary conditions and loadings for the one
quadrant of square plate.
where E, p, ry, H are the Young’s modulus, the Poisson ratio, the initial yield stress, and
the isotropic hardening parameter, respectively.
A uniform transverse dead-load of po = 104 has been applied at the upper surface of
plate. Figure 5.27 depicts the load deflection curve where the load factor / in p = fp0 has
been plotted as a function of the vertical displacement w of the center point of the square
plate. The curve of the present formulation is in agreement with the converged results from
the high-order shell element in Schieck, Smolenski and Stumpf [1999], which is available
up to f = 27.
Figure 5.26 displays the deflection of the plate at the load level / = 70, along with
the upper-surface contour plot of the equivalent plastic strains. The remarkably distorted
element in the comer zones of the plate is observed. And the localization of the plastic
strains in the four comer zones of the square plate is appeared.
5.4.2.5. Elastoplastic response of a pinched cylinder
The example is concerned with the elastoplastic deformation of a thin-walled cylin¬
der with large equivalent plastic strain (about 100%). This example has already been inves-

216
plastic stra n
0.2567
0.2007
0.1345
0.0701
0.0141
0.0005
Figure 5.26. Inflation of a plate: Deformed shape and upper-surface contour plot of equiv¬
alent plastic strains.
Figure 5.27. Inflation of a plate: Load factor / versus vertical displacement w of center
point of square plate.
tigated by Simo and Kennedy [1992], Wriggers, Eberlein and Reese [1996], and Eberlein
and Wriggers [1999]. The geometry and the boundary conditions of the cylinder are dis¬
played in Figure 5.28, where the mesh of 32 x 32 x 1 solid-shell elements are used for one

217
eighth of the cylinder due to the symmetry. The cylinder is loaded with two radial pinched
displacements in the middle of the structure. At both ends the boundary conditions have
been prescribed such that the circular shape of the end cross section is preserved and free
deformation in axial direction is allowed. The von Mises-type elastoplastic material with
isotropic hardening response is governed by the material parameters
(5.99)
£ = 3000, n = 0.3, tv = 24.3, £ = 300
where E, [i, tv, H are the Young’s modulus, the Poisson’s ratio, the initial yield stress, and
the isotropic hardening parameter, respectively.
Rigid diaphragm
Figure 5.28. Pinched cylinder: Geometry and loading.
Figure 5.29-5.30 display the deformation shapes of half of the cylinder at w = 150,
w = 200, w = 250, and w = 280, along with the the contour plot of the equivalent plastic
strains on the outer surface of the shell, respectively. The maximum equivalent plastic
strains happened at the loading areas. Figure 5.31 depicts the load deflection curve of the
problem which has been achieved in an incremental deformation controlled process. The
curve is in agreement with the Wriggers et al. [1996] within a wide region.

218
Figure 5.29. Pinched cylinder: Deformed shape and outer-surface contour plot of equiva¬
lent plastic strains at w = 150 (left) and at w = 200 (right).
Figure 5.30. Pinched cylinder: Deformed shape and outer-surface contour plot of equiva¬
lent plastic strains at w = 250 (left) and at w = 280 (right).
5.4.2.Ó. Free-flying multilayer plate with ply drop-offs
This example establishes the capability and performance of the present solid-shell
in modeling elastoplastic multilayer plates/shells with ply drop-offs. The same model for
linear elastic material was demonstrated in Subsection 4.4.4. The geometry, the loading
configuration and the time history loading amplitude are described in Figure 4.9. The mesh
of a three-layer plate with ply drop-offs is shown in Figure 4.21 (i.e., L = 0.3m, W =
0.06m, and ^h = 0.001m). The plate is divided into three equal parts and two ply drop-
offs along its length, with each part having a length of 0.1. We use sixty solid-shell elements
to model the three-layer plate with ply drop-offs, with a time-step size of At = 25 x 10-6sec
and a total simulation time of t — 0.08sec. The effect of gravity force is not considered.
The material properties are
E = 206. x 109Pa, v = 0., p = 7800.Kg/m3, ry = 245 x 106Pa, H = 1.x 109Pa ,
(5.100)

219
Figure 5.31. Pinched cylinder: Load deflection curve.
where E, ¿i, p, ry, H are the Young’s modulus, the Poisson’s ratio, the mass density, the
initial yield stress, and the isotropic hardening parameter, respectively.
Snap shots of the multilayer plate undergoing large overall motion and large defor¬
mation are taken in every 4 x 10~3sec, from the simulation using the implicit Newmark
algorithm without numerical dissipation, and are displayed in Figure 5.32. Furthermore,
the variation of linear and angular momenta along the time are shown in Figure 5.34. Both
the linear momentum and the angular momentum are conserved. On the left of Figure 5.35,
the kinetic energy, the strain energy (computed via (5.93)), the total energy (kinetic + strain
energies), and the work of the external forces are plotted as a function of the integration
time. Due to the physical plastic dissipation, the total energy is much less than the work
of the external forces, without any numerical damping. Unlike in the elastodynamics, the
strain energy remains on a very low level, which indicates that the free-flying of plate is
essentially a rigid-body movement (comparing Figure 5.33 to Figure 5.32). For the elasto-
plastic dynamic analysis, the Newmark algorithm possesses a stable number of iterations

220
and thus a good rate of convergence at each time step (right of Figure 5.35). The smooth¬
ness of both kinetic energy and strain energy along the time suggests a very low level of
high-frequency “noises” in the response, compared to the elastic case where the introduc¬
tion of numerical dissipation is crucial to suppress the high-frequency modes.
Figure 5.32. Free-flying three-layer plate with ply drop-offs using the Newmark algorithm:
Perspective view.
Figure 5.33. Free-flying three-layer plate with ply drop-offs: Deformed shapes for linear
material material (left) and elastoplastic material (right) at t = 16 x 10~3sec.

221
Figure 5.34. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu¬
lar momentum using the Newmark algorithm.
10
9
3
2
O' 1 ' 1 1 1 ' 1
0 0.01 0.02 0.03 0.04 0.06 0.06 0.07 0.08
Tim« (sac)
Figure 5.35. Free-flying three-layer plate with ply drop-offs: Energy balance (left) and
Number of iterations (right) using the Newmark algorithm.
Overall, the Newmark method yields a stable integration for the elastoplastic shells.
It is interesting to note that a more expensive mid-point rule may conserve the angular
momentum exactly via a product formula algorithm, while the accuracy of the dynamic
response remains about the same as in using the Newmark method (Simo [1992, Fig. 14]).
5.4,2.7. The impact of a boxbeam
In this example, we simulate a boxbeam being impacted at one end while the other
end being fixed. The rigid impactor is assumed having an infinite mass with a fixed velocity
of v = 20m/s (Figure 5.36). For this high-speed problem with rough response, the explicit
method is suitable.

222
The material properties are
E = 21. x 109Pa, p = 0.3, p = 7800.Kg/m3, Ty = 1.06 x 109Pa, H = 40.9 x 106Pa ,
(5.101)
where E, p, p, ry, H are the Young’s modulus, the Poisson’s ratio, the mass density, the
initial yield stress, and the isotropic hardening parameter, respectively.
Neglecting frictions between the impactor and the boxbeam, and without considering
the self-contact on the surface of the boxbeam, we assumed that once the impact occurs,
the rigid impactor stays with the impacted end of the boxbeam, which moves down with
the same constant velocity v as the impactor, while its displacements in both Y-direction,
and Y-direction are constrained.
Due to the symmetry, only one quarter of the boxbeam is modeled by 960 solid-shell
elements (2,080 nodes), as compared to 7,952 particles used in meshless method of Li et al.
[2000]. With time-step size At = 0.4 x 10_6sec, a sequence of deformations are displayed
in Figure 5.37-5.38, where half of the structure is displayed for a better visualization of
the buckling modes. The experiment results show that the first few buckling modes should
appear immediately at the impact location (Zeng and Combescure [1998]). Our results
give the same prediction on the locations where the buckling modes should appear. The
final deformed shape and the contour of equivalent plastic strains of outer surface of the
boxbeam at t — 1.6 x 10_3sec is shown in left of Figure 5.39. It is observed that the large
equivalent plastic strains take place at the comer of the boxbeam. The relation between the
reaction forces and the deflection at the collision end is shown in right of Figure 5.39. For
the comparison, the calculation by using the implicit Newmark method was also carried
out with time-step size At = 4 x 10~6sec, and similar results on deformations confirm
the validity of explicit method with the present solid-shell element in elastoplastic large
deformation shell analysis.

223
Rigid Impactor
v = 20 m/s
L = 203mm
a = 50.8 mm
b — 38.1 mm
h — 0.914 mm
a
Y
h
Cross section
X
Figure 5.36. The impact of a boxbeam: geometry and loading.
Figure 5.37. The impact of a boxbeam: The deformation at time t = 0.4 x 10~3sec (left)
and at time t = 1.0 x 10_3sec (right).
5.4.2.8. Pipe whip
This example is to show the modeling capability of the present solid-shell element
in contact problems. The element provides a natural and efficient way for shell contact
problem since double-side surfaces of shell are available and the transverse normal stress
is included. To authors’ knowledge, the shell contact problem using solid-shell elements

224
Figure 5.38. The impact of a boxbeam: The deformation at time t — 1.3 x 10 3sec (left)
and at time i = 1.6x 10~3sec (right).
0 0.005 0.01 0.015 , 0.02 0.026 0.03
w{m)
Figure 5.39. The impact of a boxbeam: Deformed shape and outer-surface contour plot of
equivalent plastic strains at time t — 1.6 x 10_3sec (left), and deflection versus reaction
force curve (right).
x 10
has not been discussed in the literature before.
This transient dynamic analysis simulates the impact of two steel pipes without con¬
sideration of friction. The pipes have the outside diameter of 3.3125, the thickness of 0.432,
and the length of 50. The target pipe is supported with a fixed boundary at each end. The
impacting pipe swings freely about a point at one end with an angular velocity of 75 radians
per second (Figure 5.40). The material properties for both pipes are the same
£ = 3.0 x 107, p = 0.3 , p = 7.324 x 10-4, ry = 7.0 x 104, H = 0, (5.102)
where E, p, p, ry, H are the Young’s modulus, the Poisson’s ratio, the mass density, the
initial yield stress, and the isotropic hardening parameter, respectively.

225
Because of the symmetry, only one half of the geometry was modeled, in which
total 840 solid-shell elements are used with one element in the thickness direction (1,830
nodes). Total 300 time steps are used with an equal time-step size At = 5.0 x 10_5sec
in the implicit Newmark time integration without numerical dissipation. Slide surfaces are
defined in potential contact outer surfaces of both pipes. This prevents nodes on one pipe
from penetrating element surfaces of the other pipe, and allows the contact areas to evolve
as the pipes deform. Here the slave surface is defined over the region of refined mesh at the
outer surface of the impacting pipe, and master surface is defined over the region of refined
mesh at the outer surface of the impacted pipe.
Figures 5.41-5.42 shows deformed configurations at various time stages of the calcu¬
lation. The contact begins at a single point, and then evolves to two separate regions. Late
in the analysis, a gap opens at the initial contact point. At the final state of t = 15 x 10-3sec,
the upper pipe has obviously rebounded. Figure 5.43 shows the contour of equivalent plas¬
tic strain at outer surface of both pipes at the final state of deformation, in which the cross
section of the upper pipe is severely collapsed along with a local yielded region around the
contact area.
Initial angular vel.
Fix p
Fi
Fix end
Figure 5.40. Pipe whip: Geometry and loading.
Since this problem involves dynamics and contact between deformable bodies with

226
Figure 5.42. Pipe whip: Deformed shape at t = 10 x 10 3sec (left) and t = 15 x 10_3sec
(right).
nonlinear material, it provides a very good illustration of the applicability of the proposed
solid-shell element to a wide range of problems.

227
Figure 5.43. Pipe whip: Deformed shape and outer-surface contour plot of equivalent
plastic strains, t = 15 x 10-3sec.

CHAPTER 6
SOLID SHELL ELEMENT FOR ACTIVE PIEZOELECTRIC SHELL
STRUCTURES AND ITS APPLICATIONS
6.1. Introduction
An active shell structure has distributed sensors and actuators, along with the control
algorithm to control the response of the host structure. The applications range from the
vibration and buckling control (Berger, Gabbert, Koppe and Seeger [2000], Balamurugan
and Narayanan [2001a]), to the shape control (Gabbert, Koppe and Seeger [2001]) and
the noise suppression (Lim, Gopinathan, Varadan and Varadan [1999]). The active struc¬
tures have great potential in the design of light-weight and high-strength structures that are
widely used in areas such as aerospace (Thirupathi, Seshu and Naganathan [1997], Loewy
[1997]) and automotive industries (Chopra [1996]).
In recent years, considerable effort has been devoted to the modeling and control¬
ling of active piezoelectric shell structure, see, e.g., Chee, Tong and Steven [1998], Sunar
and Rao [1999], Benjeddou [2000] and references therein. The coupled electromechani¬
cal properties of piezoelectric ceramics and their availability in the thin shell form make
them increasingly popular for the use as distributed sensors and actuators (Niezrecki, Brei,
Balakrishnan and Moskalik [2001]). The direct and converse piezoelectric effects govern
the electromechanical interaction in these materials. The direct piezoelectric effect states
that a strain applied to the material is converted to an electric charge, while the converse
piezoelectric effect states that an electric potential applied to the material is converted to a
strain.
The finite element modeling based on the classical laminate plate theory (Hwang and
Park [1993]) and first-order shear deformation theories (e.g., Detwiler, Shen and Venkayya
[1995], Balamurugan and Narayanan [2001/?]) has certain limitations due to their improper
228

229
modeling of the piezoelectric shell structure (Gopinathan, Varadan and Varadan [2000]).
Improvements are made by using high-order shear deformation theories (Correia, Gomes,
Suleman, Soares and Soares [2000]) and layerwise shell element formulations (Tzou and
Ye [1996], Saravanos [1997]), while some shortcomings still remain in that 1) they do not
consider the transverse normal stress in the element formulation, which may affect the be¬
havior of multilayer structure (Fox [2000]); 2) for the large deformation analysis, the finite
rotation update associated with rotational degrees of freedom (dofs) in shell formulations
is very complex to handle (Vu-Quoc, Deng and Tan [2000]); and 3) it is troublesome, if not
impossible, to incorporate nonlinear piezoelectric material models associated with large
input signals (e.g., Ghandi and Hagood [1997], Kamlah and Tsakmakis [1999], Kamlah
and Bohle [2001]) into shell elements based on the plane-stress assumption.
On the other hand, due to i) the complex geometry, ii) the material anisotropy, iii)
the coupling of electric field and mechanical field, and iv) the need to satisfy boundary
conditions of both electric field and mechanical field, 3-D detailed modeling for piezoelec¬
tric shell structures is used extensively. For example, twenty-node solid element in Koko,
Smith and Orisamolu [1999] and eight-node solid element with incompatible modes in Ha,
Keilers and Chang [1992] and Panahandeh, Cui and Kasper [1999] were used to model such
structures. As mentioned in Chapter 3 and demonstrated here, the eight-node solid element
with incompatible modes and the twenty-node solid element (with reduced integration) are
not as accurate as shell elements for thin plate/shell problems. To achieve better accuracy,
excessive number of solid elements are needed (see, e.g., Figure 2.24). To be computation¬
ally economical, Kim et al. [1997] has proposed the use of 3-D solid elements in modeling
the piezoelectric devices, shell elements for the host structure, and transition elements to
connect 3-D solid elements in the piezoelectric region with shell elements used for the
structure. The modeling is complex to handle because of the mixing of several types of el¬
ements and the need for tuning of the aspect ratio of the transition elements. Moreover, the
use of 3-D solid elements still lead to unnatural stiffening of piezoelectric devices, and as
a result, artificially high natural frequencies. Alternatively, an assumed-stress piezoelectric

230
solid-shell element is proposed in Sze, Yao and Yi [2000], which had coarse mesh accuracy
and could give accurate results for linear piezoelectric shell analysis; By construction, how¬
ever, the assumed stress method, obtained from the Hellinger-Reissner variational principle
(Zienkiewicz and Taylor [2000a, p.284]), encounters the difficulty when incorporating the
classical strain-driven nonlinear material models (Simo et al. [1989]).
Therefore, the objective of our investigation is to develop efficient and accurate fi¬
nite elements which have the ability to model the nonlinear coupled mechanical-electrical
response of multilayer composite shell structure which contains distributed piezoceramics.
In the present work, a piezoelectric solid-shell element is formulated, and active vibration
control of multilayer plate/shell with distributed piezoelectric sensors and actuators is re¬
alized by the control algorithm of the linear-quadratic regulator (LQR). The simplicity and
efficiency of solid-shell elements for general nonlinear shell applications were proved in
our previous work (Chapters 3-5). By including the electric dofs in the solid-shell ele¬
ment, the piezoelectric solid-shell element is developed within the context of the Fraeijs
de Veubeke-Hu-Washizu (FHW) variational principle. Furthermore, a composite solid-
shell element is proposed based on the solid-shell element previously developed, which
can reduce considerably the number of equations, therefore, the amount of memory and
the running-time, while at the same time keeping the same accuracy for thin multilayer
composite shells and having the flexibility for the refinement through the thickness. The
combination of the present piezoelectric solid-shell element and the solid-shell element en¬
ables the analysis of general curved active shell structures in the nonlinear regime, which
in practice, becomes increasingly important such as in aerospace, MEMS applications. By
combining the control algorithm, the sensor outputs predicted by the FE analysis could be
used to determine the amount of input to the actuators for controlling the response of the
integrated structures in a closed loop.
The outline of this chapter is as follows: Section 6.2 reviews the kinematics and
presents variational formulation of the piezoelectric solid-shell element, and discusses
the composite solid-shell element. Section 6.3 presents the control design for structures

231
with piezoelectric sensors and actuators. Numerical simulations which illustrate the per¬
formance of the proposed formulations, including comparisons with available experiment
results and solutions obtained from shell elements and solid elements, are given in Sec¬
tion 6.4.
6.2. The Solid-Shell Formulation
We have developed the solid-shell formulation for geometric and material nonlinear¬
ities based on the enhancement of the Green-Lagrange strain E (see Chapter 3, Chapter 5),
which leads to particularly efficient computational effort. We briefly describe below the
kinematics of a piezoelectric solid shell in curvilinear coordinates, and present the piezo¬
electric solid-shell element formulation based on the three field FHW variational principle
and the EAS method, and then discuss the composite solid-shell element. Readers are
referred to Chapter 3 for more details.
6.2.1. The Kinematics of Piezoelectric Solid-Shell Formulation
To overcome the known problems associated with the rotational degrees of freedom
in the shell elements, we describe the shell kinematics by linear combination of a pair
of material points at the top and bottom surfaces of the shell. Therefore, the assumption
of the shell element that the normals to the element mid-surface remain straight but not
necessarily normal during deformation, is still adopted. Thus, the initial (undeformed)
three-dimensional continuum of the shell geometry (Figure 3.2)
X(& = i[(l + {,)x.({1,{a) + (l-i,)x, ({■,{“)] ,
Í = 53] e □ := [-1,1] X [-1.1] X [-1,1] , (6.1)
where Xu and X¡ are the position vectors on the upper surface and lower surface of solid-
shell element in initial configuration, respectively, (f £2) the convective coordinates in the
in-plane direction, £3 the convective coordinate in the thickness direction, and □ represents
the bi-unit cube.
Similarly, in the deformed configuration, the current three-dimensional continuum is

described by
232
*(«) = i[(i+í3)*,(íI,{1) + (i-{,)*i(í1>ía)]-
í = [?\í2,?3] e □ , (6.2)
where xu and x¡ are the current position vectors on the upper surface and the lower surface
of the solid-shell element in the deformed configuration, respectively.
The initial configuration is related to the deformed configuration (see Figure 3.3) by
the displacement field u as
x(() = X(S) + tt(t).
(6.3)
The convected basis vectors in the initial configuration are obtained by partial deriva¬
tives of the position vector X with respected to the convective coordinates f * as
= ¿ = 1,2,3, (6.4)
which satisfies
Gi-Gj=6i, Gij = Gi• Gj , i,j = 1,2,3, (6.5)
where Gij are the metric coefficients of the initial configuration. To simplify the presen¬
tation, we will omit (£) in vectors like G¿(£)- The covectors G1 can be obtained by the
following
Gl = G'JGj
with
Gij
= [Gy]
-1
d3x3
(6.6)
and the convected basis vectors in the deformed configuration are defined in the similar
way, by using (6.3) and (6.4),
dx du
9i = dli=Gi + d^'
i = 1,2,3,
(6.7)
similar to (6.5) and (6.6), the covector gl, the metric coefficients g^, and gli in the deformed
configuration are obtained.

233
By using (6.4) and (6.7), the deformation gradient F relates the basis vectors G, of
the initial configuration to the basis vectors gi of the deformed configuration as follows
F=j£ = 9i®G\ (6-8)
and with (6.5) and (6.8), the compatible Green-Lagrange strain tensor Ec is obtained as
E‘ = \ (.FtF - I2) = i (Slj - Gij) G1® G1 = Erfi' ® GJ, (6.9)
where I2 is a second-order metric tensor, are the covariant components of the strain
tensor Ec.
The corresponding second Piola-Kirchhoff stress tensor S is expressed in the same
convected basis G¿, i.e.
5 = SijGi ® Gj . (6.10)
where Stj are the covariant components of the stress tensor S.
Similar to (6.1), the electric potential 0 is described as follows
m = i [(i+e3) ^ (í'.e2) +11 - s3) * («’.'i2)] >
í=[í\f2.?3]e°, (6.11)
where 4>u and 0/ are the electric potentials on the upper surface and lower surface of solid-
shell element, respectively.
The electric field £ is derived from the gradient of the electric potential with respect
to the position vector X as
£«.) = -GRADM =-JjsG* ■ (6-12)
The corresponding electric displacement vector D is expressed in the same convected
basis Gj
D = DlGi.
(6.13)

234
6.2.2. Piezoelectric Solid-Shell Element
Here we present the piezoelectric solid-shell element formulation used for piezoelec¬
tric sensors and actuators in active shell structures. The variational formulation of the EAS
method, and the finite element approximation of the developed solid-shell elements for the
multilayer composite shells are described in Chapter 3.
6.2.2.1. Functional and finite element formulation
The electric flux conservation for the piezoelectrics is described as follows
nE = -J D-£dV
Bo
- J Qv dV ~ J Qs dS ,
B0 sq
(6.14)
where D is the electric displacement vector, £ the electric field defined in (6.12), the
electric potential, qs the electric flux per unit area on the surface, and qy the electric flux
per unit volume.
The variation of functional (6.14) with respect to the electric potential 4> is
5Ue = - J D’6£dV - J qv 5(j)dV - J qs StpdS ,
Bo
(6.15)
B0 Sg
and the variation of the functional of mechanical energy IIm was presented in Section 3.3
and rewritten as follows
6 n
= J S I SEdV - j 6u-b*pdV - J 5u‘t*dS .
(6.16)
Bo Bo Sa
where the enhanced Green-Lagrange strain E = Ec (u) + E (a).
The total variation is the combination of (6.15) and (6.16)
in which
SUstiff = J SI 5EdV - J D'6£dV , (6.18)
B0 Bo
<5rlext = — J óu^b*pdV — J 6u‘t*dS — J6(f)qv dV — J StfiqsdS , (6.19)
Bo Sa Bo Sq

235
where the second Piola-Kirchhoff stress S and the electric displacement D are determined
by the constitutive law with the knowns E and £.
Following the standard finite element discretization as explained in Hughes [1987]
or Bathe [1996], we discretize the reference configuration B0 with numnp nodes and nel
elements, i.e., the discretization of the reference configuration Bo into a collection of finite
( ) nel (e)
element subdomains Bq , such that Bo ~ U &o •
e=l
Besides the same interpolations used on displacements u and Green-Lagrange strains
Ec, E in Section 3.3, additional linear interpolation is used on the electric potential , and
the variation and increment of electric field £ by denoting
p
(j) » NVe) , {¿£} = -B^(e) , {ASt} = -B+A<£(e), (6.20)
where N* and B0 take similar forms as in the Appendix A. 1.
We apply a standard finite element procedure to the discrete weak form (6.17) on the
element domain Bq'1
= ail? + ¿nff = + 6i¡(¿ = o, (6.21)
where the discrete weak forms of the stiffness operator ¿II^L and the external forces 511^
become
to
stiff
I S{e^}T {sij}dV+ I 5{Éij}T {sij}dV- J 6{£if{Dl}dV ,
b? <
= - J Su’b*pdV - J 6u‘t*dS — j S^qydV- j Scj)qsdS ,
»(c) e*(e) »(e) o(e)
where {Stji} is the vector of the second Piola-Kirchhoff stresses in the order of
(6.22)
(6.23)
{S#} = [5n,522,512,533 , 523,
and the vector of electric field {£¿} is as follows
(6.24)
{Si} — [£i, £2, SzY >
(6.25)

236
and the vector of electric displacement {D1} is given as
{£>*} = [.D\D2,D2
(6.26)
To simplify the presentation, we will omit the iteration index k, in the following lineariza¬
tion procedure.
The linearization of (6.21) with respect to the primary unknowns (cfie\ (f¿e\ a^) is
V (¿n) • (aS‘\ A«W) = - (íiig, + ¿n'¿) , (6.27)
where the variations <511 in (6.22) and in (6.23) are
«nSj (d('>. 0(”>, a<«)) = W(')T/£>w + <50(')T/SW + 6<*(‘)TffAS , (6.28)
with /Stiff = / BT {S«} dV , = J B*T{D‘}dV, (6.29)
o(e)
,(«)
/Ss= / eT{sy}<¿v',
ing (<*•>, «)<•>) = -<5d wjth/SLi = J NTb'pdV+ J NTfdS,
?(e)
f%L= j N*TqvpdV+ j N*TqsdS,
o(e) o(e)
(6.30)
(6.31)
(6.32)
(6.33)
by linearizing (6.28) and (6.31), the left hand side of (6.27) becomes
D (¿n(e^) • (ASe\ A^e\ AqW) = Sd^T (k^Ad^ + k%A(j>M + AqW)
+¿0(e)r (*$Ad + A0^ + fcgAaW)
We)T (fc^Ad(e) + fcgA0(e) + jfeWAaW) .
(6.34)

237
Defining the matrices of tangent material moduli C, e, eT, e as
C = [Cijkl\ :=
e =
0kij
eT =
0ijk
e =
cO
as#'
dEki_
'dDk'
dEij
'dSij'
[d£k\
1
'dDr
â– 
dE,
d6x6
1)3x6
d6x3
d3x3
(6.35)
(6.36)
(6.37)
(6.38)
where material moduli C, e, eT, e are expressed in the convected basis and subsequently
arranged in matrices according to the ordering of the stress components in (6.24) and the
electric field components in (6.25).
From (6.29)i, (6.35) and (6.37), it follows that
u(e)
^uu
Mstiff
dd{e)
*.(«)
Ku —
UJ Mstiff
d^e)
k[e) -
Qi
II
ua
<9cre)
fL= f (GTS + BTCB)dV ,
«(=)
(6.39)
(6.40)
(6.41)
where the matrix G, the stress matrix $, and the enhancing strain interpolation matrix Q
are given in Section 3.4.
From (6.30), (6.35), and (6.37), it follows that
= [fcS]r = / 9TCBdV , (6.42)
f- = f BTeTBdV,
= f BTCQdV,
«<«)
k(e) _ dfEAS
aU ~ dd^
,(=)
*S = = / STeTB*dV
*£ = = / 5l'CedV' •
R (6.43)
(6.44)

238
From (6.29)2, (6.36), and (6.38), it follows that
K df^Estiff _
dd{e)
k
T
= f B*TeBdV ,
(6.45)
K a —
dfEstiff _
da(e)
k
â– \T
= [ B*TeGdV,
*(«>
(6.46)
/if (e) .
fc(e) = = - / B^eB+dV. (6.47)
U Q,(e) J V '
B(=)
°0
By substituting (6.30), (6.32), (6.33) and (6.34) into (6.27), the linearized system of
equations on the element level is obtained for arbitrary variations of the element nodal dis¬
placement d(e) , the element nodal potential a^ as follows
fc£Ad<*> + ki$A*<*> + fc«Aa<'> = (6.48)
+ fc^A^'1 + fcW Aa<«> = -/g5, Via<*>, (6.49)
fe£Ad<«> + fcgA0W + fcgAaW = 9„, -. (6.50)
Since the enhancing strains i? are chosen discontinuously across the element bound¬
aries, the elimination of the local internal parameters at the element level is possible.
From (6.49), the increment of internal variable vector a^ can be expressed as
Aa<«> = - [fc£]" ‘ (ffAS + k<2Ad(e> + fcSA0<*>) , (6.51)
then substituting (6.51) to (6.48) and (6.50), the condensed system of equations in element
(e) is as follows
fc(e) k[e)
Kuu KWt>
k(e) k[e)
Ku K J
A d{e)
A 0(e)
where the condensed tangent stiffness matrices
(6.52)
(6.53)

239
and the condensed element residual force vectors
~ fl£m + k£ [*£>] ■’ f%\s ■ (6.54)
After assembling the element matrices in (6.53) and element residual force vectors in
(6.54), we obtain the incremental displacement-potential problem as follows
K UU ^ U<¡>
(f)U
net / \ ne^ ~(e)
with Kuu = Akuu , KU(j> = [K+u] = Afcu0 , Kh = Afc^ , (6.56)
e—1 c—1 c—-*■
nel nel
Rm = Arg , Re = Arg> , (6.57)
where the action of the assembly algorithm is denoted by the assembly operator A. For
the nonlinear dynamic response calculation, the implicit time integration schemes can be
employed. The incremental dynamic equations are obtained by including the weak form
of inertia] forces and its linearization in (6.21) and (6.27) respectively, we refer readers to
Section 4.2 for the details.
6.2.2.2. Linear piezoelectric material law in convected coordinate
The constitutive relation for linear piezoelectrics is expressed as
S = C : E-eT‘£ ,
(6.58)
D = el E + f£ ,
(6.59)
where the stress tensor S are expressed in the Cartesian basis
a¿ and convected basis as
S = Sabaa 0 ab = SijGl 0 Gj ,
(6.60)
we obtained the relation between components Sand Sab
Slj = (G'-aa) (.Gj-ab) Sab,
(6.61)
(6.55)

240
where Gl,Gj = ¿j, a¿ = a1.
The strain tensor E is expressed in different basis a, and G¿ as
E = Éc¿ac ad = EkiGk ® G* , (6.62)
we obtained the relation between the components Ecd and Eki
Écd^(Gk-ac)(Gl-ad)Ekl. (6.63)
The electric displacement vector D is expressed as
D = PGi = Daaa , (6.64)
thus the relation between the components Dl and Da is
D{ =(Gi-aa)ba. (6.65)
The electric potential gradient vector £ is expressed as
£ = £i& = £aaa , (6.66)
thus the relation between the components Si and Sa is
Sa=(aa-G1)Si. (6.67)
The fourth-order elastic constitutive tensor C is as follows
C = Cabcdaa ® ab® ac® ad = CijklGi Gj ®Gk®G¡, (6.68)
and the component form is
Ciikl = (G? • o0) ( The third-order piezoelectric tensor e determined at constant strain takes the form
e = eabcaa ab (8> ac = ey'feG* ® Gj ® ,
(6.70)

241
and the component form is the following
eijk = (G1 • aa) ( and the second-order dielectric tensor e determined at constant strain is expressed as
e = iabaa ab = e^Gi Gj , (6.72)
thus the component form is
eij = (G¿-aa) (G^ab)éab.
(6.73)
From (6.58) and (6.59), the relationship between the stress components and the strain com¬
ponents, the electric displacement components and the electric field components, with re¬
spect to different basis a¿ and are, respectively
oab fiabcd ip ~mab c
O — U Ihc(i — e Cm ,
(6.74)
Da = iabcEbc + eabEb,
(6.75)
and
Sij = CijklEki - enij£n ,
(6.76)
Dl = eijkEjk + eijSj .
(6.77)
If we express (6.61), (6.63), (6.65), and (6.67) in the matrix form by the same com¬
ponent ordering as in (6.24) and (6.25), we obtain
{s«} = Tl {S“k} ,
{O*} = Tl {£>“} ,
{^cd} = To {£*(} ,
= T, {£i},
where the matrix Tq is in Section 3.3, and the matrix Te is
T
-1- e
t\ t\ tf
t\ t\ t\
t\ t\ t\
(6.78)
(6.79)
(6.80)

242
with the coefficients t\ = a¿ • , «, j = 1,2,3.
Since in the Cartesian coordinate, the constitutive relation of (6.74) and (6.75) in the
vector form are expressed as follows
[C°“] {E^} - [«”“*] {£„} ,
(6.81)
(6.82)
and in the convected coordinate, the constitutive relation of (6.76) and (6.77) in the vector
form is as follows
{^1
{°‘}
{Ejk} + [e«] {£ji
>
(6.83)
(6.84)
substituting (6.78)2 and (6.79)2 into (6.81) and (6.82), and then substituting (6.81) and
(6.82) in (6.78)j and (6.79)i. By comparing the resulting equations with (6.83) and (6.84),
the constitutive matrix ^Cljklj, [eljfc , and [eu] in the convective coordinates associated with
basis Gi are transformed from the ones in the Cartesian coordinate
(6.85)
(6.86)
-ab
(6.87)
6.2.3. Composite Solid-Shell Element
To improve the modeling efficiency of laminated composite shells, a composite solid-
shell element is developed here. As shown in Figure 6.1, the material layer can be stacked
in parallel to the upper surface and lower surface of the eight-node solid-shell element. The
element matrices are obtained by using the numerical integration, in which 2x2 Gauss
quadrature is used in the plane of the lamina, and two Gaussian points are used for each
material layer in the thickness direction.

243
Figure 6.1. Composite solid-shell element: Eight-node composite solid-shell element in the
isoparametric space, two Gaussian points (x) for each layer, and eight collocation points
for assumed natural strain methods.
Assuming that the thickness ^)h of layer (£) within an element remains constant
n
and total thickness H = ^ of n layers is much smaller than other dimensions of the
i=i
element, it is straightforward to find isoparametric coordinates £3 in the thickness direction
of Gaussian points at each layer. For a typical layer (i), the isoparametric coordinates (¿)£f
and (j)£3 in the thickness direction at its lower surface and upper surface are respectively
¿-i
(i
>)$ = JjYl (t)h ~ 1 > (0« = 7/S Wh ~ 1 •
e=i
H
(6.88)
e=i
By using (¿)^3 and (¿)£3, the isoparametric coordinates of layer Gaussian points through
the thickness direction are obtained. Accordingly, the weight factor for each integra-
(i)h
tion point at layer (i) is scaled by
H
To avoid shear-locking of the displacement formulation, we employ the assumed
natural strain method, as applied to the four-node shell element in Dvorkin and Bathe
[1984]. The assumed transverse shear strains are based on the the constant-linear interpo¬
lations of compatible transverse shear strains E$3, E%3 in (6.9), evaluated at the midpoints
Q = A, B,C, D of the element boundaries with £3 = 0 (Figure 6.1).
In the case of curved thin shell structures or the nonlinear analysis, to circumvent
the locking effect from parasitic transverse normal strain, we employ an assumed strain
approximation for the covariant component E33 of the compatible Green-Lagrangian strain

244
tensor, refer to Betsch and Stein [1995], We assume bilinear interpolations of the transverse
normal strain field, where the points Q = E,F,G,H at the comers of element midsurface
(Figure 6.1) serve as sampling points of the compatible transverse normal strain. The
readers are referred to Section 3.3 for more details.
6.3. Simulation Control Design
In this section, we present the procedure in integrating the finite element analysis
with the control algorithm to simulate and control the response of an active structure with
piezoelectric sensors and actuators (Figure 6.2). The static condensation is employed to
eliminate the zero-mass degrees of freedom (dofs) associated with the electric field. A
modal analysis is then performed to transform the coupled finite element equations of mo¬
tion into the reduced-order model in the modal coordinates. The linear quadratic regulator
(LQR) is then employed to emulate the optimal controller by solving the Riccati equations
from the modal state space model.
Feedback <\>a
Actuator
Host structure
F7\V71 (
Controller
k\\N
¿sensor
Sensor
Figure 6.2. The typical active structure configuration.
6.3.1. Finite Element System Equation of Piezoelectric Structure
The finite element equation for the linear piezoelectric structure, without considering
damping, is as follows
M
0
I d \
0
0
UM
UU u K
H
(6.89)
where stiffness matrices Kuu, Ku and are in (6.56), and external forces Fext and
Qext are from (6.57) without considering the internal forces, M the mass matrix. Note
that the dofs with the electric field are “massless.”

245
We partition the piezoelectric dofs into two parts, that is, sensor dofs and actuator
dofs, as follows
0
*' \ Q = I ^xt
0“ / ’ 1 Qext
(6.90)
where the actuator voltage dofs 0“ are known from the input, and the external electric flux
Qext associated with the sensor dofs, in general, is zero; the voltage dofs 0s and electric
flux Q“xt are unknowns.
The matrices and Ku become
—
Tf83 TSsa
^(ficfi ^(fi(fi
jy~CLS Tfaa
^U
I/- 1 T/'SS jy-aal ry- r ry lT1 Í ry-ss ry-aa 1'
j\.u where K" = [jf“]T, = [*r“]T, and K% =
Accordingly, the system equations of (6.89) becomes
' M
0
0 ’
• • 'x
d
uu
Tfss
^ U(fi
Ty^aa
^ U(fi
r d '
' Fex t '
0
0
0
<
V
> +
Tfss
(fiu
lfss
Ku
Tfsa
Ku
<
0s
> = <
Qext
0
0
0
[PI
Tyaa
. (fiu
TF'Cls
Kcfi(fi
Ty^aa
K l P J
. Qext .
From (6.93)2, the sensor voltage increments 0s are obtained by
‘) ,
substituting (6.94) into (6.93)i, we obtain
M d + ( Kuu - K" K%
Fcxt - K
u
ryss
KH>
u
-1
1-1
Ks;u)d =
Qext + ( K%
K
H>
-1
jy'sa jv-aa
^(fiffi ***u (6.91)
(6.92)
(6.93)
(6.94)
) 0“ • (6.95)
During the solving procedure, we calculate the displacements d by (6.95), then solve the
sensor nodal voltage 0s by (6.94).

246
There are two particular cases in (6.93), that is, only sensors existing or only actuators
existing in the structure. If only sensors exist (i.e, (p — (f>s), the system equation (6.93) is
reduced to
M 0
0 0
+
Tf J^ss
TfSS TfSS
d
ext
Q
(6.96)
ext
we can solve for d by setting cpa = 0 in (6.95), that is
Md + (if„, - AT” [ifj;]"1 K¿) d = Fext - K
then solve for (ps by setting (pa = 0 in (6.94), that is
Ulj)
K

Qext ,
(6.97)
TfSS
Ku
-1
(%. -K£d) .
(6.98)
On the other hand, if there are only actuators in the structure (i.e, (p = (pa), the
system equation (6.93) becomes
M
0
Í d
i*
7CUU
jyaa
^ U(p
0
0
m
rydd
lj>U
T^aa
Ku \
d
r
we use (6.99)i to solve for d with the known (pa, that is
F ext
Ql
(6.99)
ext
M d+Kuud= Fext -Kau“(Pa,
(6.100)
then solve for the electric flux Q“xt with (6.99)2, that is
Qext =K‘£ld + K™cpa. (6.101)
6.3.2. Reduced-Order Model of Piezoelectric Finite Element System
Consider a typical structure bonded with piezoelectric actuators and sensors. The
goal of the design is to suppress unwanted vibrations and increase damping of the structure,
which can be achieved by proper controlling of voltage signals (pa to the actuators.
With the external electric flux Q*xt = 0, (6.95) and (6.94) become respectively
Md+ (kuu - AT" [AX]'" AX) d = F^t +
{k"* [-KX]"1 kZ ~ KZ) • (6-102)
4>’={K‘¿Y'(-Kgd-K^r) .
(6.103)

247
To compute the signal outputs (f)s from piezoelectric sensors, we assume that there is
no coupling between sensors and actuators such that in (6.102) and (6.103)
K% = 0 ,
(6.104)
and denote
K =
(6.105)
At the right hand side of (6.102), the actuation force vector Fa , is related to the coupling
stiffness matrix K°^ and the vector of applied voltages “ by
Fa =-Klir . (6.106)
Using (6.104), (6.105), and (6.106) in (6.102) and (6.103), the governing dynamic equa¬
tions of the structure under both mechanical excitations and actuation forces can be ex¬
pressed as follows
Md + Kd= Fext - Fa , (6.107)
and the sensor output is
SS
00.
Kud â– 
(6.108)
It is noted that the response of the system in (6.107) is regulated by the control voltage
vector 4>a, which depends on the information of the states of the system measured through
sensors. To construct a control law much more efficiently, all matrices in (6.107) should be
transformed into diagonal forms. For the linear system, (6.107) can be decoupled by means
of the modal transformation, which is based on the solution of the generalized eigenvalue
problem
KV = M^n.2 € Rnxn , (6.109)
where Í22 = diag [u2, • • •, u>2] G MnX71 contains the eigenvalues, and = [tpl, ■ ■ •, tpn] £
Rnxn contains the eigenvectors.

248
The cost of solving the generalized eigenvalue problem (6.109), however, can be pro¬
hibitively high for the large size n. For structural dynamics problems, the typical modal
analysis studies on the frequency content and spatial distribution of the excitation have
shown that the response is controlled by a relatively small number of low frequency modes.
On the other hand, the finite element analysis approximates the lowest frequencies and
the associated mode shapes best, and has worse accuracy in higher frequencies and mode
shapes. Therefore, in practice only the mode shapes with low frequencies are used for
the dynamic response of the structure. Two iteration methods, WYD Ritz vector approach
(Wilson, Yuan and Dickens [1982], Leger [1986]) and Lanczos approach (Leger [1986],
Parlett [1998]), can generate a few eigenpairs in low frequencies much more effectively,
compared to the traditional subspace iteration method (e.g., Hughes [1987, p.576]). Both
algorithms were implemented by authors in the code of CFD-ACE+ and listed in the Ap¬
pendix B.l. For the details of the performance of the above two algorithms, please see the
series reports by Vu-Quoc, Tan and Zhai [1998-2000].
The transformation is given by
d = tyq ñ¡ *I>rqr , d = ^q « 'S'/dr > (6.110)
where ^ is the modal matrix and q the vector of modal coordinates for the full-order
model (6.107), \Pr and qT contain the first nr( coordinates. The eigenvectors d>r are orthogonal with respect to both the stiffness matrix
K and the mass matrix M, and normalized to M as follows
^jK^r=n2r, T = I,
(6.111)
where ÍÍJ: = diag uf, ■ ■ • , o£r] contains the first nT eigenvalues uf of the structure. By
substituting (6.110) in (6.107), and premultiplying 'SfJ at both sides of (6.107), and then
using (6.111), we obtained the uncoupled reduced-order model
qT + tt2rqr = tyj Fext + fr ,
(6.112)

249
where fT is the modal control force vector as follows
f, (6.113)
In general, it is not necessary to control all the first nr modes in the reduced-order
uncoupled model of (6.112). Instead, only the first few modes are used for the control
design. For this purpose, the nr modal equations in (6.112) are separated into the first nc
equations of controlled modes and the rest nu (= nr—nc) equations of uncontrolled modes.
Accordingly, the modal matrix \I>r, the modal coordinate vector qr, the modal control force
vector fr, and the modal excitation force vector \I>rr Fext are partitioned into two parts
as follows
= [Vc tfu] , qr
*?f,
ext
t
Fext
therefore, the dynamic equation for the first nc controlled modes are
(6.114)
(6.115)
Qc +
tfcqc
= fc+*l F,
ext
(6.116)
where f2c2 = diag
nu modes are
and the dynamic equation for the rest of the uncontrolled
Qu + Qu — f u + *1Fext , (6.1 17)
where fl2u = diag [w2c+1, • • •, w2r .
6.3.3. Controller Design
The controller or control law describes the algorithm or signal processing used by
the control process to generate the actuator signals from the sensor and command signals
it receives. Since the 1960s, modern state-space controller design methods such as the
linear quadratic regulator (LQR) and the linear quadratic Gaussian (LQG) (e.g., Anderson
and Moore [1990]) have been developed in the linear time-invariant (LTI) system. In the

250
uncoupled system (6.116), the linear-quadratic regulator (LQR) is used to determine the
modal control force for any given mode, which depends on only the modal coordinate and
modal velocity of that mode. As a result, the independence of the open-loop equations for
each mode is preserved for the closed-loop system. For the details, readers refer to e.g.
Boyd and Barratt [1991].
Letting
Qc
he
12ncx 1
(6.118)
one can transfer the dynamic equations (6.116) for controlled modes into state space equa¬
tions that are suitable for control design. The state space equations from (6.116) can be
written in the matrix form as
Xc — A Xc + B f c + D F ext ,
(6.119)
where
I
0
0
0
, B =
I
, D =
1
hi
*
i
(6.120)
For the system described by (6.119), the linear feedback control law is defined as follows
fc=-KcXc, (6.121)
where matrix Kc £ RncX2ric is partitioned into the displacement gain matrix Gd £
KncXnc corresponding to modal coordinates qc and the velocity gain matrix Gv £ RncX”c
corresponding to modal velocities qc
Kc=[GdGv], (6.122)
substituting (6.121) into (6.119), it yields the closed-loop state-space equation
Xc = (A - B Kc) Xc + D Fext , (6.123)
for the response of the active control system. In terms of modal coordinates of controlled
modes, the control force vector fc in (6.115)i and (6.121) takes the form
fc = = -KcXc = -(Gdqc+ G„ qc) .
(6.124)

251
The optimal regulator state feedback gain matrix Kc may be obtained by minimizing
a linear-quadratic cost function J, which is defined as follows
J = Jâ„¢ (XCTQXC+ fcTRfc)dt, (6.125)
where fc is the input force in (6.124), Q the weight for the effectiveness, and R the weight
for forces fc.
The weighting matrix Q in the performance function J of (6.125) is usually assumed
to be diagonal (Meirovitch [1990]) as follows
Q = diag (lü¡, •••,<, 1, • • •, l) G , (6.126)
however, there are no general guidelines for the choice of R, the weighting matrix for
control forces fc. A diagonal R, as assumed in many applications, may be used as
R = diag (Ru ■ • •, Rnc) € Rn‘xne . (6.127)
Define the optimal gain matrix Kc
Kc = R~lBTPc, (6.128)
which is obtained once Pc is solved from the following Riccati equation
AtPc + PcA +Q - PcBR1BtPc = 0 , (6.129)
where the above matrices A and B are from (6.120).
With matrix Kc in hand, the closed-loop response of the system is obtained by
integrating (6.123). Subsequently, the control voltages “ for the actuators are calculated
through (6.124)
r =[*/*■£]KcX0=[*!cTK%Y1(Giqc+ G„qc) ,
(6.130)
where the number of controlled modes nc and the size of vector 4>a of actuators may not
be the same, hence the inverse, 41 in (6.130) is usually operated by a pseudo¬
inverse process. Therefore, the modal forces fc are only approximately independent,
depending on how close 'TcTi
252
The response of the structure is calculated by combining of the contribution from the
controlled modes \I/C and uncontrolled modes \FU
d ^ (6.131)
the sensor voltages 0 s are then approximated by using (6.108) and (6.131)
tcqc + *„ (6.132)
and the rate of the sensor voltages 0 are obtained by differentiating (6.132) with respect
to time
0S = -
k:
Ks;u (M>cqc + *uqu)
(6.133)
Under a given structural excitation, the structure would vibrate accordingly, and the
distributed sensor outputs could be calculated from (6.132). Then the voltage supplied to
actuators could be determined from the control law by using the calculated sensor outputs
as input. Then the new state of the structure could be calculated under both the external
excitations and the actuation voltages applied to the structure through actuators. The pa¬
rameters in (6.127) of the controller is then modified to optimize (or tune) the performance
for the desired closed-loop response. From the closed-loop system of (6.123), it is clear
that the displacement feedback - Gd qc modifies the stiffness, and the rate feedback of
— Gv qc modifies the damping of the open-loop system (6.120), that is
A -BKr =
0 I
nl-Gd -gv
(6.134)
The solving procedure on the above control design is given in the Appendix B.2.
6.4. Numerical Examples
The finite element formulations of the present low-order solid-shell element for anal¬
ysis of piezoelectric shell structure presented in previous sections have been implemented
in the Matlab, and run on a Compaq Alpha workstation with UNIX OSF1 V5.0 910 oper¬
ating system. In each element, the mass matrix is evaluated by using the Gauss integration

253
2 x 2 x 2, the tangent stiffness matrix and the dynamic residual force vector are evaluated
by using the Gauss integration 2 x 2 in the in-plane direction, and two Gauss points in the
thickness direction for each material layer. Below we present the examples involving the
static, dynamic analyses and active vibration control of piezoelectric shell structures.
6.4.1. Cantilever Plate: Out-of-Plane Bending
We present this example to show that the eight-node solid element with incompatible
modes and the high-order twenty-node solid element (with reduced integration) suffer from
the shear-locking in thin shell applications.
A cantilever plate of length L — 10 and width W = 1 is subjected to the transverse
shear loading F at the free end (Figure 3.10 and Figure 3.12). We consider various models
with different aspect ratios L/h to compare the performances of several elements. To have
the same level of deflection magnitude regardless of the thickness h, the applied loading
F is set to be proportional to the thickness raised to power three (i.e., h3) in the numerical
calculation.
The material properties are prescribed to be
E = 1.0 x 107, i/ = 0.4 , (6.135)
where E and u are the Young’s modulus and the Poisson’s ratio, respectively.
A comparison of the tip deflection of the plate for different elements is shown in
Figure 6.3. All results are normalized to the solution obtained from the geometrically-
exact shell element (Vu-Quoc, Deng and Tan [2000]). The FE models involved are made
of ten eight-node current solid-shell elements, ten eight-node solid elements with full in¬
tegration, ten eight-node solid elements with incompatible modes (Taylor et al. [1976]),
five twenty-node solid elements with full integration, and five twenty-node solid elements
with 14-point reduced integration scheme (Hoit and Krishnamurthy [1995]), all with one
element in the thickness. It is noted that the direct use of twenty-node solid element with
2x2x2 reduced integration scheme encounters the singularity in this problem. In the
linear problem, the transverse loading F = 104h3 is applied at the free end. The free-tip

254
transverse displacement along the force direction at the comer of the midsurface of the
plate obtained from the geometrically-exact shell element, approaches w = 3.9236 at the
thin limit (the solution from Euler-Bernoulli beam theory w — 4). For the large deforma¬
tion case, the loading F = 5 x 104/i3 is applied in five equal load steps. The tip deflection
in the transverse direction obtained from the geometrically-exact shell element approaches
w = 7.41366 at the thin limit. In both linear and nonlinear situations, the present solid-
shell element yields excellent results even for the extremely thin plate (aspect ratio = 6667
or h — 1.5 x 10-3). The eight-node solid element with full integration suffers from the se¬
vere locking. The twenty-node element (with reduced integration) and the eight-node solid
element with incompatible modes cannot obtain correct results, especially for the large de¬
formation analysis, while the twenty-node element with 14 integration points had better
performance than the eight-node solid element with incompatible modes and the twenty-
node element with full integration. It is interesting to note that a twenty-node solid element
with another 14 integration points rule by Irons [1971] gives the same unsatisfactory results
as that with the full Gaussian integration (3x3x3 integration points), but taking almost
half of the computational effort.
For flat plates undergoing small deformation, numerical results show that it is suffi¬
cient to consider the ANS treatment of only the transverse shear strain £’13 and 3; the
additional ANS treatment of the transverse normal strain E33 does not change the numerical
results.

255
6.4.2. Multilayer Composite Hyperbolical Shell
This examples is used here to verify the correctness of the proposed composite solid-
shell element in thin shell applications. The same example was considered in Basar et al.
[1993] to test their shell element formulation. The shell structure consists of three layers
with the same layer thickness ^)h — h/3, which are placed symmetrically with respect to
the middle surface. Due to symmetry, only one-eighth of the shell structure is modeled with
FE meshes of 14 x 14 x 3 solid-shell elements (one element per material layer) and 14 x
14 x 1 (one element through the thickness h) composite solid-shell elements respectively
(Figure 6.4, 0° along circumferential direction). The layer material properties are En =
40 X 109 , Ell = ^33 = 109 , 1^12 = v 13 — 1^23 = 0.25 , G\2 = G\z = G23 =
0.6 x 109 . The analysis was carried out for two different stacking sequences: [0°/90°/0°]
and [90°/0°/90°]. The load-displacement diagram Figure 6.5 shows that results obtained
from the model with composite solid-shell elements (1,260 equations totally) agree with the
refined model by having one solid-shell element for each material layer (2,520 equations
totally). Therefore, the composite solid-shell element is accurate and more efficient to
capture the overall global response such as the deflection for thin shells. The shell with the
[90°/0°/90°] stacking sequence has larger deformation, and is less resistant to the loading
than the shell with the [0°/90°/0°] stacking sequence. The computed results agree with
those in Basar et al. [1993], where a layerwise shell element with complex rotation update
was employed. It is noted that for the shell with the [90°/0°/90°] stacking sequence, a
more refined mesh is needed to achieve the converged results (see also Basar et al. [1993]).
The deformed shapes in Figure 6.6 for the final load P = 160 x 103 demonstrate clearly
that large rotations and displacements are involved in this example. It is noted that for
moderately thick shells or with nonlinear materials, the model with one element through
thickness is incapable of accurately determining the structure response such as in-plane
displacements, transverse shear stresses (e.g., Vu-Quoc, Tan and Mok [2002]).

256
n = 7.5
r2 = 15.0
r = r-±(b2 + Z2)1'2
b
h = 0.04
X J Stacking sequences:
[0°/90°/0°], [907079O0]
Figure 6.4. Pinched multilayer composite hyperbolical shell: Undeformed mesh.
Figure 6.5. Pinched multilayer composite hyperbolical shell: Load-displacement diagrams
from both solid-shell elements and composite solid-shell elements, v(B) is the displace¬
ment along axis Y at point B, u(A) the displacement along axis X at point A.
6.4.3. Piezoelectric Bimorph Beam
This numerical application is used to validate the developed piezoelectric solid-shell
element in both an actuating and a sensing mechanism. The experiment consists of a can-

257
Figure 6.6. Pinched multilayer composite hyperbolical shell: Deformed shape with stack¬
ing sequence [0°/90°/0°] (left) and [90°/0°/90°] (right).
tilevered piezoelectric bimorph beam with two equal polyvinylidene fluoride (PVDF) lay¬
ers bonded together, and polarized in parallel or anti-parallel directions, with the dimen¬
sions indicated in Figure 6.7. The beam is discretized into 10 equal solid-shell elements.
The mechanical and piezoelectric properties of the PVDF are
E = 2. x 109Pa , v = 0.,
e3J = e32 = 0.0460C/m2 , (6.136)
Pn = P22 - P33 = 0.1062 x 10 9F/m ,
where E and u are the Young’s modulus and the Poisson’s ratio, e31 and e32 are the piezo¬
electric stress coefficients, and pn,P22, and p33 the electric permittivity coefficients,
clamped
Figure 6.7. Piezoelectric bimorph beam: geometry and mesh (left), and electric loading
for anti-parallel polarization type (a) and parallel polarization type (b) (right).

258
For the anti-parallel polarization type (Figure 6.7(a)) and parallel polarization type (Fig¬
ure 6.7(b)), the theoretical results on the transverse deflection of the free-tip (Andersson
and Sjogren [2001]) are calculated by
e31^ y2
2 Eh2
wb
3
e^V
Eh2
(6.137)
respectively, where h is the beam thickness, V the voltage applied on the surfaces of the
PVDF layers.
The present results obtained from both cases agree exactly with the theoretical re¬
sults. The deflections along the beam length for the anti-parallel polarization case are
given in Table 6.1, in which the results obtained from the present piezoelectric solid-shell
element are compared with that of a four-node shell element (Detwiler et al. [1995]), a
nine-node shell element (Balamurugan and Narayanan [200\b]), and the experiment (Ha
et al. [1992]).
Table 6.1. Piezoelectric bimorph beam: Deflections (xlO 7ra) for anti-parallel polariza¬
tion type.
Location (m)
4-node shell
9-node shell
present and theory
experiment
0.02
0.139
0.144
0.138
0.04
0.547
0.557
0.552
0.06
1.135
1.240
1.242
0.08
2.198
2.192
2.208
0.1
3.416
3.415
3.450
3.15
The sensing voltage distribution of the bimorph beam with anti-parallel polarization
under the prescribed free-tip deflection is also analyzed. The voltage distribution for a
prescribed free-tip deflection of 0.01 (or equivalently F = 0.0254371 at the free tip) is
given in Figure 6.8, which agrees well with the results from a laminated triangle shell
element by Tzou and Ye [1996]. The highest sensor voltage at X = 0 indicates that the
largest induced-strain at the clamped end take places under the free-tip loading.

259
-©- top surface
midsurface
o I i i i i i i i i i
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
X(m)
Figure 6.8. Piezoelectric bimorph beam: Sensor voltages along the length of bimorph
beam.
6.4.4. Cantilever Plate with PZT Actuators
This problem has been studied experimentally by Crawley and Lazarus [1991], and
used extensively for the verification of various finite element formulations (Ha et al. [ 1992],
Detwiler et al. [1995] and Saravanos et al. [1997]).
The cantilever actuator/plate system, consists of a plate with piezoelectric actuators
symmetrically bonded to both upper and lower surfaces (Figure 6.9). For the plate of
aluminum, a constant voltage of 252. V was applied to the outer surfaces of actuators; for
the composite plate with lay up [0/ ± 45]s, a constant voltage of 157.61/ was applied to the
outer surfaces of actuators; and for the composite plate with lay up [+302/0]s, a constant
voltage of 188.81/ was applied. The finite element model is shown in Figure 6.10, where
144 solid-shell elements for the host plate and 98 piezoelectric solid-shell elements for the
actuators are used.

260
The material properties of the aluminum are
E = 63. x 109Pa, v = 0.4 , p = 2800Kg/m3 , (6.138)
where E, u and p are the Young’s modulus, the Poisson’s ratio, and the mass density,
respectively. The layer material properties for graphite-epoxy composites are prescribed to
be
En = 143 x 109Pa , E22 = E33 = 9.7 x 109Pa ,
^12 = ui3 = v23 = 0.3 , (6.139)
Gn = Gi3 = 6. x 109Pa , G23 = 2. x 109Pa , p = mOKg/rn3 ,
and material properties for PZT (lead zirconate titanate) actuators are
E = 63 x 109Pa , v = 0.3 ,
dn = d12 = 254 x I6~um/V , (6.140)
d15 = d2A = 584 x 10"12m/V , d33 = 374 x 10~um/V ,
Pn = P22 = 15.3 x 10 9F/m , p33 = 15.0 x 10 9F/m , p = 7600Kg/m3,
where E and v are the Young’s modulus and the Poisson’s ratio, dn, di2, d15, d24, and
d33 are the piezoelectric strain coefficients, and pn, P22, and p33 the electric permittivity
coefficients.
For the comparison, the dimensionless displacements are used as follows
M2 M2 — (Mi + M3)/2 Mi — M3
WL = W,WT= ¡y , WR = pjT ’
(6.141)
where W is the width of the plate, and Mi, M2, and M3 are the transverse deflections along
axis Z at locations shown in Figure 6.9.
The longitudinal bending wi, transverse bending wT and lateral twisting wr of alu¬
minum plate, [0/ ± 45]s and [+302/0]s composite plates are shown in Figureó.l 1-6.13,

261
51
0.83
H H-
/
/
/
/
/
/
/
/
/
/
/
/
/
/
clamped
L = 292
Figure 6.9. Cantilever plate with PZT actuator: Geometry of cantilever actuator/plate
system (Unit:mm).
Figure 6.10. Cantilever plate with PZT actuator: Mesh of cantilever actuator/plate system.
respectively. In spite of the scattering of the test data, the numerical results provide ex¬
cellent accuracy as compared with the experiments. It is noted that the results from solid
elements with incompatible mode (e.g., Ha et al. [1992]) are overstiff (locking). The re¬
sults from QUAD4 elements are too flexible (Detwiler et al. [1995]), which may be caused
by the inability to model the 3-D structure with ply drop-offs by using single-layer shell
elements.
For the large deformation analysis with linear material relation, we apply the total

262
*io-*
*10"*
Figure 6.12. Cantilever plate with PZT actuator: Longitudinal bending wl (left), trans¬
verse bending wt and lateral twist wr (right) of graphite/epoxy [0/ ± 45]s plate.
Figure 6.13. Cantilever plate with PZT actuator: Longitudinal bending wl (left), trans¬
verse bending wt and lateral twist wr (right) of graphite/epoxy [+302/0]s plate.
voltage 3152F in 20 equal steps. The final deformation of the structure, and the relation
between the electric loading and the free-tip deflection M2 are shown in Figure 6.14. Ta¬
ble 6.2, which depicts the values of the Euclidean norm of both the residual and the energy

263
norm in each iteration of one load step, clearly exhibits the quadratic rate of asymptotic
convergence, which confirms the correctness of the present implementation.
Figure 6.14. Cantilever plate with PZT actuator: deformed shape (left) at voltage 31521/,
and nonlinear load deflection curve (right).
Table 6.2. Cantilever plate with PZT actuator: Convergence results for large deformation
(residual norm, energy norm).
Iter.
Step 10 (V=1576.)
Step 20 (V=3152.)
0
2.082e T 05, 2.306e + 02
2.0825 + 05, 2.3205 + 02
1
9.418e + 05,1.029e + 04
6.4315 + 05,5.1475 + 03
2
3.937e T 03,3.636e -|- 00
2.5845 + 03,3.5305 + 00
3
1.530e + 04, 2.678e + 00
5.6165 + 04,4.0275 + 00
4
9.056e + 02,1.189e + 00
1.3155 + 02,2.0255 + 00
5
2.829e + 04,9.100e + 00
7.709E + 04,7.789757 + 01
6
3.502e + 01,4.893e — 01
2.0655 + 01,5.6675 — 03
7
2.383e + 04, 6.261e + 00
1.9855 + 02,5.0215-04
8
3.320e + 00,3.523e - 03
6.2435 - 02, 6.3815 - 08
9
2.052e + 02,4.557e - 04
2.5795 - 03,8.8285- 14
10
1.788e — 02,2.098e — 08
2.0875 - 06,4.8575 - 20
11
1.282e - 03,1.796e - 14
12
1.430e — 06,2.009e — 20
6.4.5. Cantilever Plate with PZT Actuator and Sensor
Next, a thin cantilever plate with PZT actuator and sensor patches as shown in Fig¬
ure 6.15, is studied to show the efficiency of the present modeling and validity of the
implemented control algorithm. The cantilever aluminum plate has the thickness h =
0.965 x 10~3m, the width W = 0.025m, and the length L = 0.226m (Figure 6.15).

264
Unit voltage is applied to the upper surface of actuator. The finite element model shown
in Figure 6.16 uses 34 solid-shell elements for the host plate, six piezoelectric solid-shell
elements for the actuator, and four piezoelectric solid-shell elements for the sensor. The
material of both actuator and sensor is PZT-5H, and the thickness of actuator and sensor
are 0.5 x 10_3m and 0.25 x 10~3m, respectively.
Figure 6.15. Cantilever plate with PZT actuator and sensor: Geometry of cantilever system
(Unit:mm).
The material properties of the aluminum are the same as in (6.138). The material properties
of PZT-5H are
Eu = E22 = 62.05 x 109Pa , E33 = 57.17 x 109Pa ,
u12 = 0.334 , i/13 = t/23 = 0.444 , (6.142)
Gu = 23.3 x 109Pa , Gi3 = G23 = 23.0 x 109Pa ,
^31 = £32 = -0.5C/m2, e33 — 23.3C/m2, = e24 = l7C/m2,
P11 = P22 = 0.1503 x 10_9F/m, p33 = 0.13 x 10~9F/m, p = 7500Kg/m3 .
It is assumed that the PZT sensor and actuator are bonded perfectly to the plate. In Kim et

265
al. [1997], a complex and expensive modeling was used to calculate the static response, in
which twenty-eight twenty-node solid elements are used to model the PZT regions includ¬
ing a part of the plate underneath the PZT patches, and twenty nine-node shell elements
used for the remaining part of the plate structure, and two thirteen-node transition elements
for the transition region between solid and flat-shell elements. Moreover, the element as¬
pect ratio has to be tuned for good accuracy. The comparison of the maximum deflection at
the tip of the cantilever plate is present in Table 6.3. The present results agree well with the
theoretical results (Hong [1992]), which were computed by applying an equivalent force
corresponding to the actuator voltage on the structure. This example again verifies that the
actuator performance is being simulated correctly and economically by using the present
combination of solid-shell elements and piezoelectric solid-shell elements.
Table 6.3. Cantilever plate with PZT actuator and sensor: Deflections (xlO 6m) at free-
tip.
Kimet. al.[1997]
44 present elems
128 present elems
theory
w
3.94
3.3049
3.4980
3.53
relative error (%)
11.6
6.64
1.19
0.
For the vibration control, an step force of 0.17V is applied at the free tip of the plate.
The dynamic response of the FE model of forty-four elements is calculated by using mode
superposition technique with the first nine eigenmodes. Among the first nine modes, the
first four modes are controlled by LQR optimal control described in Section 6.3, by setting
Ri = 0.01, i ~ 1,4 in (6.127). The uncontrolled response and controlled dynamic response
are shown in Figure 6.16. Both the corresponding output voltage from the sensor and input
voltage for the actuator at certain locations along with time are presented in Figure 6.17.
Consider the same cantilever plate in Figure 6.15 made by [0/90/ ± 45]s composite
with the same material properties as in (6.140). Without changing anything else, we use
thirty-four composite solid-shell elements for the plate. The uncontrolled response and
controlled response of the free-tip are shown in Figure 6.18. Both the output voltage from
the sensor and input voltage for the actuator at certain locations along with the time are

266
«io-
Figure 6.16. Cantilever plate with PZT actuator and sensor: The mesh of cantilever system
(left), closed-loop step response and open-loop step response of aluminum plate (right).
Figure 6.17. Cantilever plate with PZT actuator and sensor: Output voltage at point 2 of
sensor (left), and input voltage at point 1 of actuator (right).
shown in Figure 6.19. Again, the vibration of the plate is suppressed successfully. Com¬
pared with the previous aluminum plate, the vibrating magnitude of the composite plate is
a little smaller, and the time period is shorter, which indicate the fact that the composite
plate is stiffer and lighter than the aluminum plate.
Finally, we want to point out in these two examples, high values of voltage are applied
on the actuator (~ lO4)/) to suppress the relatively large magnitude of vibration, thus
the assumption of the linear piezoelectric material may not be appropriate and the use of
nonlinear piezoelectric models is necessary.

267
Figure 6.18. Cantilever plate with PZT actuator and sensor: Closed-loop step response
and open-loop step response of [0/90/ ± 45]s composite plate.
x io4
Figure 6.19. Cantilever plate with P7J actuator and sensor: Output voltage at point 2 of
sensor (left), and input voltage at point 1 of actuator (right) with [0/90/ ± 45]s composite
plate.

CHAPTER 7
CLOSURE
7.1. Conclusion
This dissertation has addressed many computational aspects of multilayer shell struc¬
tures based on two finite element models.
Firstly, we have developed the finite element formulation for analyzing the large de¬
formation of geometrically-exact sandwich shell model, whose governing equations were
developed in Vu-Quoc et al. [1997]. In our formulation, the layer directors form a chain of
rigid links connected to each other by universal joints. Finite rotations of the directors in
every layer are allowed, with shear deformation independently accounted for in each layer.
The thickness and the length of each layer can be arbitrary. We have derived the weak form
of the equations of equilibrium of our sandwich shell model. The tangent stiffness matrix
is thus obtained from the linearization of the weak form and the update of the inextensi-
ble directors, which results in an asymptotically quadratic rate of convergence in numerical
analysis. We have illustrated the essential features and generality of the present formulation
by presenting several examples, including the sandwich plates with ply drop-offs. We refer
to Vu-Quoc et al. [2001] for the dynamic computational formulation for the geometrically-
exact sandwich shell, and to Vu-Quoc and Ebcioglu [2000¿] for a generalization of the
dynamic formulation to the multilayer case. For geometrically-exact multilayer beams
with through-the-thickness deformation, we refer to Vu-Quoc and Ebcioglu [2000a],
Then, an efficient eight-node solid-shell element for the analysis of multilayer shells
with a large range of aspect ratios and with nonlinear materials has been presented. We
have proposed a new optimal number of EAS parameters in the formulation, together with
the ANS method, to pass the membrane and out-of-plane bending plate patch tests and to
remedy the volumetric locking. Furthermore, a modification for the efficient EAS proce-
268

269
dure, which avoids the inverse operation at each element, was presented. We also proved
the equivalence between various choices of the enhancing strains in tensor form, and com¬
pared them in terms of the relative efficiency. In contrast to Miehe [1998¿>], the alternative
EAS approach using enhancing deformation gradient was reformulated in a much simpler
manner. The nonlinear dynamic weak form and linearization have been derived based on
the energy-momentum conserving algorithm for the current solid-shell element. Numer¬
ical damping was introduced to the time-integration algorithms for smoothing of high-
frequency modes common in structural analysis. Although this destroyed the conservation
properties of the algorithm, numerical simulations demonstrated that only minor deriva¬
tions resulted from small amounts of dissipation. With the enhancement on the compatible
transverse normal strain, the full three-dimensional nonlinear constitutive models can be
incorporated without resorting to the plane-stress assumption. Due to the parametrization
of the displacements on the top and bottom surface of the present solid-shell element, the
complicated finite rotation update in the stress-resultant shell models is no longer neces¬
sary. Moreover, it is convenient to model the shell contact problems and multilayer shell
structure with geometry discontinuity such as piezoelectric patches.
For the extension, a new eight-node piezoelectric solid-shell element for the analy¬
sis of active composite shells with a large range of aspect ratios has also been presented.
The combination of ANS method and EAS method was used to deal with various locking
mechanisms. The composite solid-shell element was proposed to make the 3-D analysis of
thin composite shells even more efficient, while keeping the good accuracy. The active vi¬
bration control of multilayer plate/shell with distributed piezoelectric sensors and actuators
was realized effectively by the linear-quadratic regulator design.
Numerical examples confirmed that the present solid-shell element performance is
competitive against more elaborate shell formulations. The combined use of both the EAS
method and the ANS method in obtaining accurate results was justified. The present solid-
shell element was proven for the following applications: 1) thick or extremely thin aspect
ratio (=6667) in the linear and nonlinear regime; 2) isotropic material, composite lami-

270
nates with dissimilar material layers, and incompressible nonlinear materials; 3) the im¬
plicit/explicit dynamic analysis with/without numerical dissipation; 4) 3-D modeling of
linear piezoelectric shell structure.
7.2. Directions for Future Research
Several directions for improvement or addition can be further investigated:
Element technology. The development of new and improved elements is always in
high demands. In particular, low-order triangle-type solid-shell element free of element
deficiency (membrane locking, volumetric locking, shear locking, and thickness locking)
and possessing good in-plane bending behavior is vital in bringing together shell analysis
and automatic meshing techniques that rely on triangulation to fill arbitrarily shaped regions
(Newsletter Vol.2( 1) of ADINA [2002]). On the other hand, for stress analysis of laminated
composite thick plates, a new Hybrid-EAS solid element is under development (Vu-Quoc et
al. [2002]), which can predict the interlaminar stresses accurately and satisfy the transverse
shear stress continuity at layer interface and the vanishing transverse shear stress at free
surfaces of the laminates.
Constitutive models. To better characterize the actual response of structures, the
nonlinear material models including anisotropy, hysteresis, and multi-field coupling (me¬
chanical, electrical, thermal, and magnetic) are needed to be developed and incorporated
into the present element formulation.
Adaptive mesh. In events involving extremely large deformation, such as metal
forming, extrusion, and rolling, the element mesh is severely distorted so that the Jaco¬
bian determinants may become negative at quadrature points to abort the calculations. In
addition, the conditioning of implicit analysis deteriorates and explicit stable time steps
decrease rapidly. Therefore, the incorporation of remeshing with the present element for¬
mulation becomes necessary.
Control and Optimization of active structure. For general nonlinear analysis of
active shell structure, the nonlinear controller may be designed based on the current LTI
controller (Boyd and Barratt [1991, p.45]). On the other hand, the optimal design for the

271
weight, size and location of piezoelectric sensors and actuators subjected to certain con¬
straints (e.g., stress failure criterion, maximum deflection) can be developed (e.g., Correiaa,
Soares and Soares [2001], Han and Lee [1999]), based on the general-purpose sensitivity
analysis and structural optimization theory (e.g., Giirdal, Haftka and Hajela [1999]).

APPENDIX A
SOLID-SHELL FORMULATION
A.l. Finite Element Approximation of Solid-Shell Element
In the following section, we provide the detailed derivation of the finite element
approximation in the solid-shell elements.
The geometry in the initial configuration is
xa = •
7=1 ¿
£ = (£\£2,£3)eü, (A.l)
where the position vector X = [X, Y, Z]T, the nodal position vector at upper surface
Xui = [Xuj, YuI, ZuI]T, the nodal position vector at lower surface Xu = [Xu, Yu, Zu]T-
For eight-node solid-shell element, the two-dimensional shape functions N¡ in the in-plane
direction
m*i’*2M(i+&i) (i+^2) • (a-2)
where £] and are the coordinates of node /. The convected basis vector Gl in the initial
configuration is computed asG¿ =
The displacement can be interpolated in the same way (A.l)
*»<€)=¿ N, (e,e) 5 f(i+f) u i (i - e3) h] 4° = Ndp>. (a.3)
/=1 z
where N = [N¡ ,N2,N3 , N4], with N¡ = N¡\ [(1 + £3) J3 (1 - £3) J3] and I3 being
a 3x3 identity matrix, d^ =
,(e)T j(e)TlT , .
dj . du being
d[e)T, 4e)T, 4)T, 4e)T]r with d{je) =
the displacements of the upper surface and of lower surface, respectively, at node /.
The partial derivatives of the displacement field u with respect to natural coordinate
(f1,^3) are
— =JV ,.<*•> — =JVrtd'e> —-1
¿)£l -Í ° > Q£2 -'v,i2a . ^3 iV,i3a •
(A.4)
272

273
To use a general finite element notation, the components of the second-order Green-Lagrange
strain tensor, the second Piola-Kirchoff stress tensor and the fourth-order constitutive ten¬
sor are contained in the related matrices E, S, andC of the dimension 6 x 1 or 6 x 6,
respectively.
Table A. 1. Transformation of indices from tensor to matrix form.
Tensor Index
11
22
12(21)
33
23(32)
13(31)
Matrix Index
1
2
3
4
5
6
With (A.4) and the expression of the compatible Green-Lagrange strain tensor (3.15), we
can write the compatible Green-Lagrange strain Ec in the vector form:
ipc
' GjNÁid(e) + ld{e]TNTeN¿id(e)
&22
GT2N¿2d[e) + \d(e)T NTeN ¿2díe)
2-^12
> = ,
G[lVií2d(e) + GlN¿idie) + d(e)TlYj,A/>d(e)
E33
G3N¿3d^ + ¡d{e)TNTeN^3d{e)
2Ec23
G%N¿3d{e) + G£jV>d(e) + d(e) T AT^2 N ¿3 d(e)
. 2Ef3 á
G^N^d^ + G3N¿id^ + dUTNTeN¿3dU
The strain-displacement matrix B is the derivatives of (A.5) with respect to nodal displace¬
ment d(e)
’ GfjV¿i+d(e)TiVjiiV)fi
G^A/> + d{e)TNTpN¿2
GjN¿2 + G%N¿ i + dUTNTeN¿2 + d^TNTeNte
G*N¿3 + d{e)TNT^N^3
G%N¿ 3 + G¡N¿2 + d^TNTpN¿3 + d{e)T NT¡:3N ¿2
GfN¿3 + <%N¿i + d^N^N^ + d{e)TNTeN¡e _
(A.6)
6x24

274
The derivatives of (A.6) with respect to nodal displacement d^ are
NTCN¿2 + NTeN¿i
NT?N¿ 3
Gâ„¢=<
dd{e)
(A.7)
where the corresponding stress matrix S> should be
144x24
Sfcllr c22 r ol2 r o33 r o23 r cl3 T ^ cl
— p -*24)>-> -*24>*-> -* 24) *-> -*24>*J -*24)<-> 24j £
»144x24
(A.8)
where I2a is the identity matrix with the dimension 24 x 24.
In Bischoff and Ramm [1997], they use different kinematics for the solid-shell el¬
ement, namely, the midsurface of the shell Xm and the director Xr which defines the
thickness direction of the shell (see Figure 3.2). The geometry in the initial configuration
is
4
Jf(i) = £IV¡ (A «’)[*<”■ (A ,
t=l
^ = (c1,^2,^3) e n, (A.9)
where the position vector X = [X, Y, Z]T, the nodal position vector at midsurface Xim =
P^im> ^¿m? ) the nodal director vector -X^ — Z. For four-node solid-
shell element, the two-dimensional shape functions N{ in the in-plane direction is the same
dX
as (A.2). and the convected basis Gt in the initial configuration is G{ = , and note that
G3 = Xr.
The displacement can be interpolated in the same way as (A.9)
4
£
i=i
u -¿m (^,^2) [h, eh] (A. 10)

275
where the nodal displacement vector d-e) includes the nodal displacement at the midsurface
and the nodal difference of the director between the initial configuration and deformed
configuration. d,(e) = d[^T , d$T , the interpolation matrix
N = [AT1 , AT2, AT3 , AT4] , with AT = N{ [j3 £3/3] ,
and J3 is the identity matrix with dimension 3x3. The nodal displacement vector d^ =
[d?Ttd?TAe)TA)T]T.
For the Green-Lagrange strain Ec, the strain-displacement matrix B and the matrix
G, we follows the same procedure as in (A.4)-(A.8), by using the different interpolation
matrix in (A. 10).
In the formulation based on the deformation gradient F, the compatible convected
basis is interpolated as
= % = + • 9l> = ^ + JVj, (0) d<*) , (A.11)
If we choose the five parameters for EAS method,
rx = [4e),4e),o]r , r2 = [4e),aie),o]r , r3 = [o,o,4e)]T , (A.12)
for element (e), the EAS parameter vector is = a[e\ af, aje), a^e)]T. Then the
matrix form of in (4.79) is
Kl =
?24e>
0
e4e) o
o o £34e)
(A. 13)
The matrix form of Fj in (4.81) is
a}^1»
n-
+o?í24e) «JíM*1 + “lf24e)
a¡í‘4*» + a\e<*S° a'2('4'> + a?f2Q + a|f2a‘e)
aU34‘'
a5íS«í,)
ak34e>
, (A.14)
where indices i and j in F{ and T{ are the row index and column index, respectively.

276
Define the components T{ — H^a^, where ¿Ty are
Hn = (ajC1,0, a\i2,0, o) , ií21 = (o, aje\ 0, ao) ,
H31 = (0,0,0,0, a\e) , Hn = (a&\0, a^2,0, o) ,
H22 = (0, alt1,0, a2e2,0) , H32 = (o, 0,0,0, a^3) ,
#13= (<&\0>alÉ2,O,0) , H23= (0,^,0, alf2,0) ,
H33= (o, 0,0,0, a¡e) ,
and denote
(■f^i)3x5 = 9okHki, ®> & = 1,2,3,
where the index k uses the summation convention.
The spatial convected basis g{ is
9i = 9i+9i,
with = Gi + N¿id(e), g{ = iT,a(e) , ¿ = 1,2,3,
we define the following operators ¿y, ¿3)°, Q¿
(¿«),4x5 = (°) ■ (5°)3x24 = ^ (°) ■
{Qi)3>i2t = N.(‘ + D°,i,k=l,2,3,
The strain vector E is
9h
922
E = EC +
9*2 + 92i
933
923 + 932
. 513 + g31 .
(A. 15)
(A. 16)
(A. 17)
(A. 18)
(A. 19)
(A.20)
(A.21)
6x1

277
where Ec is the same as (A.5), and The strain-displacement matrix B is
' g\Qi
92Q2
9ÍQ1 + 9IQ2
9IQ 3
92Q3 "T 93Q2
, 9ÍQ3 + 9ÍQ1
6x24
(A.22)
and the strain-displacement matrix 1? is
9¡H2
gTxH2 + gT2Hl
B = <
9ÍH3
g\H3 + g\H2
gjH3 + g3Hi
(A.23)
Similarly, the geometrical stiffness matrices Guu, Gua, GQU, and C?QQ in (4.94) are the
derivatives of the strain-displacement matrices B in (A.22) and B in (A.23) with respect
to nodal displacements d^ and internal parameters respectively, which are
G
UU
dB
deft
QjQx
Q2 Q2
Qi Q2 Q2 Qi
Q3 Q3
Q2 Q3 + Q3 Q2
Qi Q3 T Q3 Qi
(A.24)
144x24

278
G
UCt
dB
da
Q^H.+Ln
Q2h 2 ■H -£22
Qi H2 + Q2 Hi + ¿12 + ¿21
Qs H3 + ¿33
Q2 H3 + Q3 H? + ¿23 + ¿32
Qj i/3 + Q3-fíl + ¿13 + ¿31 ,
(A.25)
5d(e)
h[Qi
+
tT
^11
J/2 Q2
+
fT
-^22
HjQ,
+
h[q2
+
r T
â– ^12
+
¿51
// j Q3
+
tT
-^33
H3Q2
+
H2 Q3
+
fT
â– Li23
+
a
H:i Qi
+
H1 Q3
+
fT
-^13
+
¿51
(A.26)
30x24
G
aa
(973
5a
7/ i H1
Il'llz
h[h2 + H^Hi
—r—
i/3 #3
i/^i/3 + H^z
(A.27)
i/[i/3 + //[//! J30><5
where the corresponding stress matrices S>u in (4.90) and (4.92) is the same as (A.8) and
$a in (4.91) and (4.93) should be
Sa = [Sn/5, S22I5, S12/5, 533/5,S23/5, S13/5]T € M30x5 (A.28)
where /5 is the identity matrix with the dimension 5x5.
For the simplified formulation, without the high-order term gt • the term in the
strain of (A. 18) is
9ij = 9i*9j + 9i'9j + 9i’9j > (A.29)
subsequently, in (A.22), we replace by 7Vi?i; in (A.23), replace g{ by gf; in (A.24),
replace Qr Qj by + 7Vj¿>° + D°T7V A in (A.25), replace Qz by 7Vifi, and
replace g{ in ¿¿J by g¡\ in (A.27), all terms are zero (i.e., Gaa = 0).

279
A.2. Solution Procedure of Nonlinear Equations
The iterative algorithm for solid-shell elements is as follows:
1.Update on element level for iteration (k + 1)
- nodal displacements:
(t+1)dw = wd<‘> +A wd('> ,
- EAS parameters
(fc+l)<*(e) = (k)Oc(e)
(k)
*=S]_1(w*SAwd<') + (t)/Ws) ,
2.At each gauss point of each element
- enhancing strain
E = Q (fc+1)a(e) ,
ANS on components of compatible strain
PC
^33 >
E¡3,
T?c
iZ/23 J
- element tangent stiffness
(k+l)^T ~ (fc+l)^iu ~ [(fc+1)
- element residual vector
(fc+1)^ = (fc+l)/Lrt — (fc+1) fluff +
- save EAS arrays
f[ <*+.)*£ ]"'<*+>)*&! ,
k(e) ]T \ t.(e)
(fc+l)"'^
(fc+l)<
-1 ito
(fc+1)/
£>15
f(e)
(fc+1) J £MS
(fc+1)
a
(e)
3. Assembling from each element for [k+\)K, {fc+1)i£.
4. Solving and global convergence control
(A.30)
(A.31)
(A.32)
(A.33)
(A.34)
(A.35)
(A.36)
A(fc+i)W — (fc+i)-?f 1 (k+i)R ,
(A.37)

280
if | (k+i)R | < Tol or |A {fe+1)u • (k+i)R I goto next time step
else
k = k + 1
go to 1
endif
A.3. Explicit Integration Algorithm with EAS Method
The dynamic system can be integrated over a typical time step [fn,fn+1] using an
explicit central difference scheme.
1. Given un, iin, iln at time tn,
2. Enforce essential moving b.c. on ún, un,
3. Update un+\ at time tn+\
A t2
un+i = un + itnAt + ün— , (A.38)
4. Update un+1 at time £n+1
Sn+1 = M-1 , (A.39)
where Rn+i = Fe^x — and EAS parameters are condensed inside each element.
5. Update itn+1 at time fn+i
un+i = iin + y («« + «n+l) , (A.40)
6. n = n + 1 and go to next time step.
A.4. Return Mapping Algorithm for J2 Flow Theory with Isotropic Hardening
The model can be integrated over a typical time step [tn, ¿n+1] using a backward Euler
difference scheme leading to the closed-form return mapping algorithm.
Given Cpn~\ hn at tn, current Cn+1, solve for Cp~\, hn+1, S, C at in+1.
1. Trial calculation:
Nu (Cn^CP-1 - At2l) = 0 , with Nti-Cp-Nti = 1,
(A.41)

281
e\ = InA- , h* = K , i = 1,2,3 . (A.42)
2. Constitutive relation: compute the elastic strain principal stresses r¿, and moduli £ff
in the eigenspace form, hardening parameter hn+j.
3. Update for fn+1
Ai = exp (e¿) , N1 = ^Nu ,
3
(A.43)
clA = £¿v®n\
i=1
(A.44)
CO
II
%
¿
%
(A.45)
3 3 pep _ 2t S â– 
C = J;°tjNu ® ATÍ¿ ® ATtji (8
i=l »=1 Aj
(A.46)
3
+ 9ijNu ® 0 (iV* ® iViJ + ATtj ® JV£l) ,
(A.47)
where
Ti/\f - Tj/Xf
9ij = J whenA^Aj,
cep _ pep _ 2r.
ft; = -when Aj = A*-, (A.48)
where in the numerical calculation, the equal eigenvalues form is used when the difference
of A â–  is less than a small tolerance (e.g., 10-7).
A.5. Elastoplastic Moduli Elf
To find out the elastoplastic moduli Elf, we can write the incremental form from
(5.80)
{Ar¿} = [£®/] {Aej} or At = £ep A e£ , (A.49)
From (5.48) and (5.60), the corresponding incremental form are
Ae = Ae! — /3An — A¡3n ,
(A.50)

282
Ah = /3/O.rih + Afirih , (A.51)
substituting (5.69)i in (A.50), then in (5.68)i, and substituting (5.69)2 in (A.51), then in
(5.68)2, solving for At and Ay respectively, we obtain
At = £ (A ef — Afin) , Ay = —ShAfirih , (A.52)
where £ and £h are defined in (5.73).
The consistency condition requires
Aí0(r,y) = O, (A.53)
with definition of n and in (5.61), the incremental consistency condition of (A.53) gives
A(f> = w At + nhAy = 0 , (A.54)
substituting (A.52)i and (A.52)2 in (A.54), we obtain Afi in terms of A el
A/? = -^nT£A e‘ , (A.55)
where D is listed in (5.77).
Then substituting (A.55) back into (A.52)i, we have
Ar = £ep A et , (A.56)
where the elastoplastic moduli £ep is
£ep = £- £n ® £n , (A.57)
A.6. Algorithmic Moduli for Return Mapping
The constitutive relation for elastoplastic material in step 2 of A.4 is as follows:
1. Initialize the trial status:
e=et,h = ht, ¡3 = 0,
(A.58)
2. Loop on local iteration k :

283
3. obtain derivatives of free energy ip, flow rule Define
r
dip dip
de ' ^ dh
d2ip f _ d2ip d2(p d2(p
dede ’ h dhdh ’ drdr ' h dydy
4. If cp < tol then
dip d2ip
T=&’£ =siai
exit
Else
5. Compute the residual
rf = e — el + /3n ,
rh = h - hl + pnh , r = [re, rh]T ,
6. Define
£ = (s-' + pry' , & = (fi-' + zWv.)'
D = nrBn + ,
7. If ( rT r + r = — , £ep = £ - 1-£n £n ,
oe D
exit
(A.59)
(A.60)
(A.61)
(A.62)
(A.63)
(A.64)
(A.65)
(A.66)
(A.67)
endif

284
8. compute the incremental plastic parameters
A(3 = ( (A.68)
Ae = —£~l£ (rt + A(3n) ,
(A.69)
Ah = -£hl£h (rh + A (3nh) ,
(A.70)
9. update strain and internal variables
e = e + Ae ,
(A.71)
h — h + Ah ,
(A.72)
P = P + A p.
(A.73)
endif
ENDLoop

APPENDIX B
PIEZOELECTRIC SOLID-SHELL FORMULATION
B.l. Model Reduction Algorithm
Lanczos Method for Generalized Eigenproblem:
Given Data :
AT n x n Mass Matrix
K n x n Stiffness Matrix
Triangularized Stiffness Matrix :
K = Lt D L n x n
Choose an Arbitrary Starting Vector X :
b — (XT AT X )1/2 M — Normalization
X i = X /b Vector one
Solve for Additional Vectors with bi = 0 and i = 2, ...r :
(а) K X i = M X ¿-i solve for X ¿
(б) a¿_i = ~Xi M X j_x
(c) X i = X i — a¿_i X ¡_i — X j_2 M — Orthog.
(d) bi = (X ¿ AT X j)1/2 AT — Normalization
Xi^lCi/bi
Construct Symm. Tridiagonal Matrix T (optional) :
Calculate Eigenvalues and Eigenvectors of Tr :
TrZ = Z [A]
u2 = 1/A
Expand Eigenvectors to Full System Size :
$ = X Z
B.2. Solving Procedure on Control Design
The procedure for the vibration control of a linear time-invariant (LTI) system
285

286
1. Solve eigenproblem (6.109) by using Lanczos method, and then form the reduced-order
model (6.112), and partition the reduced system into controlled and uncontrolled parts.
2. Form state-space system of reduced-order model with controlled modes from (6.116)
and form state-space system of uncontrolled reduced-order model from (6.117);
3. Obtain the optimal gain matrix Kc in (6.128), where R may be adjusted for better
performance;
4. The closed-loop response of the reduced-order model by including the solution of
(6.123) and (6.117);
5. Feedback of actuator voltages in (6.130) and Sensor voltage outputs in (6.132).

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