Variational Methods in Nonlinear Elasticity

Variational Methods in Nonlinear Elasticity

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Variational Methods in Nonlinear Elasticity Pablo Pedregal

liniversidad de Castitla-La Mancha Ciudad Real, Spain

Society for Industrial and Applied Mathematics Philadelphia

Copyright Copyright © 2000 by the Society for Industrial and Applied 10987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Pedregal, Pablo, 1963Variational methods in nonlinear elasticity/Pablo Pedregal. p. cm. Includes bibliographical references and index. ISBN 0-89871-45 2-4 (pbk) 1. Elasticity. 2. Nonlinear theories. 3. Calculus of variations. I. Title. QA931-P832000 53T.382--dc21 99-054264

Portions of this book were adapted with permission from material presented in Parametrized Measures and Variational Principles, Progress in Nonlinear Differential Equations and Their Applications, Volume 30, Birkhauser Verlag AG, Basel, Switzerland, 1997.

is a registered trademark.

To Daniel, Silvia, and Jaime.

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Contents Preface

ix

1 Elastic Materials and Variational Principles 1.1 Introduction 1.2 The Stress Principle and the Cauchy Theorem 1.3 Elastic and Hyperelastic Materials 1.4 Examples of Hyperelastic Materials 1.5 Appendix: Some Linear Algebra and Geometry

1 1 2 4 5 6

2 Quasi Convexity and Young Measures 2.1 Introduction 2.2 The Direct Method 2.3 Young Measures 2.4 Weak Lower Semicontinuity 2.5 Quasi Convexity and Gradient Young Measures 2.6 A General Existence Theorem 2.7 The Case p = +00 2.8 Appendix

9 9 10 12 16 18 19 21 22

3 Polyconvexity and Existence Theorems 3.1 Introduction 3.2 Polyconvexity 3.3 Coercivity and the Case p < oo 3.4 Existence Theorems 3.5 Hyperelastic Materials

25 25 25 31 32 34

4 Rank-one Convexity and Microstructure 4.1 Introduction 4.2 Elastic Crystals 4.3 Lack of Quasi Convexity 4.4 Generalized Variational Principles 4.5 Laminates 4.6 The Two-Well Problem 4.7 Continuously Distributed Laminates

41 41 41 43 46 47 50 59

vii

viii

CONTENTS

5 Technical Remarks 5.1 Introduction 5.2 The Existence Theorem for Young Measures 5.3 Biting, Weak, and Strong Convergence 5.4 Homogenization and Localization 5.5 A Remarkable Lemma

63 63 63 67 70 74

Bibliographical Comments

77

Bibliography

81

Index

97

Preface The underlying mathematical problems in nonlinear elasticity are fascinating. The variational methods developed to tackle them are even more so. The graduate student and the newcomer to the subject may, however, have difficulty appreciating these statements and may feel disappointed to discover that the path leading to a reasonable degree of understanding of the relevant issues, to the point where they feel confident enough to pursue new directions by themselves or under the guidance of a senior researcher, is not well trodden. Filling concisely this gap in the case of the mathematical theory of nonlinear elasticity is the main motivation for this text. It originated in the form of lecture notes for a summer course in materials science and engineering held in Coimbra in 1997 under the auspices of Centra Internacional de Matematica (CIM, Portugal) . Later, I completed that material with the idea of producing a reference that might help readers get rapidly and efficiently to the heart of the matter of vector variational methods in the context of nonlinear elasticity, as pointed out above. My purpose is in reality very modest: to focus on explaining the complexity of vector variational problems from the aspect of existence-nonexistence of equilibrium configurations, with special emphasis on the relevance of structural assumptions. In particular, my point is to communicate how the different notions of convexity arise in vector variational problems and to explain their significance with respect to the existence issue. The analysis does not go into any deeper examination of polyconvexity, quasi convexity, or rank-one convexity. I do not even attempt to differentiate among them or to define and study the corresponding envelopes. I believe this must be an additional step for the interested reader who, surprised to a certain degree at the need for different types of genuine vector convexity concepts, finds the subject sufficiently appealing to move on. I have also tried to motivate the main questions by unifying the treatment of weak lower semicontinuity, quasi convexity, polyconvexity, the failure of quasi convexity and rank-one convexity, oscillatory behavior, microstructure, etc., using gradient Young measures as the main tool. In this way, the reader may gain at the close of the book an overall picture of where the different techniques and basic ideas have their place and of the connection between them. You will assess to what extent this has been accomplished. On the other hand, this is not a book about Young measures. They are used merely as a convenient device to express and formalize concepts, ideas, and techniques. Almost nothing ix

x

PREFACE

is said about characterizing gradient Young measures. This would be another possible direction for interested readers to investigate. It is also important to point out that applications to real materials have not been included, although they might have been. Two main reasons justified this choice from my own perspective. In the first place, this field is or will shortly be covered extensively and exhaustively by several references (see the Bibliographical Comments) written by experts. I would do a poor job in comparison. Second, I would have had to sacrifice brevity for the sake of completeness. Again, I think the readers must decide for themselves whether to deepen their knowledge of a particular area after understanding the basics. For the same reasons, a more formal and complete treatment of the foundations of elasticity to establish how the variational nature of our problems arises has not been included. The structure of the book is also oriented toward the main goal of helping readers comprehend the need, the reasons, and the motivation for the main concepts; the principal assumptions in results; the techniques in proofs; etc. Purely technical remarks have been deferred to the last chapter so that they do not interfere with the main stream of reasoning. However, they have been collected so that interested readers may find a guide to the proofs of such facts and results. Note that this choice has not compromised our insistence on brevity. Most of the technical proofs themselves, however, have not been written down because they can be taken almost entirely from other texts. Chapter 1 is an introduction wherein the main problem examined in the book, that of existence of equilibrium solutions, is precisely stated and motivated from the general principles of continuum mechanics. Chapter 2 focuses on quasi convexity as the main constitutive assumption for existence of solutions in the context of the direct method of the calculus of variations. Chapter 3 deals with polyconvexity as the main source of quasi-convex functions and as the key assumption for existence of solutions for some real materials. Failure of quasi convexity, nonexistence, oscillatory behavior, and, ultimately, rank-one convexity and laminates, as the main example of microstructures within the context of phase transitions in crystalline solids, is addressed in Chapter 4. Comments on the bibliography are given at the end of the book. I have also made a serious attempt to keep prerequisites to a minimum. Basic knowledge of real and functional analysis, measure theory, theory of distributions, Sobolev spaces, geometry in R , and linear algebra is assumed, although appendices have been included in some chapters to remind readers of key facts in those areas. Although implicit to some extent in the previous paragraphs, it may be worthwhile to point out that the book is particularly addressed to graduate students and researchers in applied analysis, applied mathematics, mechanics, materials science and engineering, etc. The book might also be an appropriate starting point for those interested in the mathematical side of elasticity, or for the physicist or engineer who could benefit from a better understanding of the mathematical issues in nonlinear variational techniques. It could also serve as a basic reference for a graduate course in the mathematical theory of elasticity.

PREFACE

xi

Many people have contributed to the realization of this project and I would like to acknowledge them here. My thanks go first to CIM for giving me the opportunity to develop and implement the original draft of these notes. In particular, I appreciate the support of A. Ornelas. The feedback and comments from many students who attended my lectures were most welcome and helped a lot in convincing me of the need for this book. The instructors who generously agreed to conduct several problem-oriented sessions were also very supportive, especially A. Barroso, J. Matias, and J. Matos. Finally, I want to thank R. Kohn for very encouraging comments on an earlier version of the book, and several anonymous referees for their very specific and detailed criticism that led to the improvement of the original manuscript in various aspects. Pablo Pedregal Ciudad Real, June 1999

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Chapter 1

Elastic Materials and Variational Principles 1.1

Introduction

We describe in precise terms the mathematical formulation of the standard problem in nonlinear elasticity, that of rinding equilibrium configurations of elastic bodies under prescribed environmental conditions. Our aim is to justify and clarify the main ingredients of the underlying mathematical problem and its variational nature and to show how this problem stems from the general theory of continuum mechanics. This short chapter is purely descriptive and no proof of any kind has been included. It is intended for readers who are not familiar with the mathematical theory of elasticity or continuum mechanics, stressing the main points that must be covered in a more formal, rigorous derivation. It can be skipped by those who have previous exposure to such fields. We focus from the outset on the situation of static equilibrium so that inertial effects and conservation laws are neither considered nor referred to. We also concentrate on the boundary value problem where the deformation is restricted on all of the boundary of the reference configuration (global condition of place), which is the one that requires fewer prerequisites. Pure traction and mixed problems can be handled as well using the same variational methods we will study in this text, although, for the sake of brevity, we will not cover these situations. Our emphasis will be on the variational methods themselves and we will not be concerned with the different problems and situations that can be tackled using them. At the end of the book, references for further reading are given for those interested readers who might wish to complete the exposition of this chapter or to deepen their understanding of continuum mechanics and elasticity. We have included an appendix at the end of this chapter to discuss a few important facts in linear algebra that are needed in order to introduce explicit examples of energy densities for elastic materials. Again, no proofs have been given, but they can be found in the references cited at the end of the book. 1

2

CHAPTER 1

1.2

ELASTIC MATERIALS AND VARIATIONAL PRINCIPLES

The Stress Principle and the Cauchy Theorem

Assume fl c R3 is an open, bounded, connected subset with a sufficiently smooth boundary. We will think of this domain as the part of space occupied by a body before it is deformed. This is called the reference configuration. Often we refer to fi as the body. A deformation of the body is a mapping u : Q —>• R3 assumed to be smooth enough, injective (except possibly on the boundary of fi), and orientation preserving. The matrix VM(X) is called the deformation gradient and it provides a measure of local strain. The orientation-preserving requirement may be written det(Vu(x)) > 0 almost everywhere (a.e.) x € fi. A deformed body associated with an arbitrary deformation u may be subjected to body forces represented by a vector field / : u(Jl) —> R3. The mapping / must obviously depend on u and represents the density of applied forces per unit volume in the deformed configuration. There might also exist surface forces denned as a vector field on a part of the boundary 7! C du(fl), g : 71 -» R3, referred to as the density of the applied surface forces per unit area in the deformed configuration. The following axiom is fundamental in continuum mechanics. It is known as the stress principle of Euler and Cauchy. AXIOM 1.1. There exists a vector field

where S is the unit sphere o/R3, such that the following apply: 1. Axiom of force balance: For any subdomain E c u(fi),

where n is the unit outer normal along dE and dS represents the element of area,. 2. Axiom of moment balance: For any subdomain E c w(O),

where a A b is the cross product in R3. 3. For any subdomain E C u(fl) and any y 6 71 n dE where the unit outer normal vector n to 71 n dE is well defined,

This stress principle expresses intuitively the idea that static equilibrium of a deformed body is possible due to the overall effect of elementary surface elements t ( y , n). This tensor t(y, n) is called the Cauchy stress tensor. The main consequence of this stress principle is the Cauchy theorem, which establishes, among other things, the linear dependence of t(y, n) on n.

1.2

THE STRESS PRINCIPLE AND THE CAUCHY THEOREM

3

THEOREM 1.2. Assume that the applied body-force density f is continuous and the Cauchy stress tensor field t(y, n) is continuously differentiable with respect to y and continuous with respect to n. There exists a continuously differentiable tensor field T: u(fi) ~> M such that the Cauchy stress tensor is given by Moreover,

where n is the unit outer normal to 71 and M denotes the space of 3 x 3 mainces. In order to transform the consequences of the Cauchy theorem to a partial differential boundary value problem, we need to express those conclusions in the reference configuration O rather than in the deformed body u(fi). For this reason we have to use the change of variables y — u(x), x e fJ. Note that all tensors and vectors have been expressed in terms of the Euler variable y. We first let

The explanation for this transformation of the tensor T is that we have

Notice the use of divy to emphasize that the divergence is taken with respect to the Euler variable y. For a matrix A, adj A indicates its adjugate or cofactor matrix (see the appendix at the end of this chapter). On the other hand, we let

and These definitions are motivated by the following identities:

where dS represents the element of surface area in the appropriate variables. The conclusions of the Cauchy theorem can now be written in the reference configuration as follows:

where w(Fi) = 71 and n is the unit outer normal to FI.

4

1.3

CHAPTER 1

ELASTIC MATERIALS AND VARIATIONAL PRINCIPLES

Elastic and Hyperelastic Materials

A material is called elastic if the Cauchy stress tensor T(y] in each point of a deformed configuration y e u(fi) is a function of x — u~l(y) and of the deformation gradient Vu(x), exclusively. The constitutive equation can thus be written as where T is called the response function of the material. Assuming that a boundary condition of place u = MOnas been given on the portion of the boundary r0 = dtl \ FI , the equilibrium configuration u must satisfy the following boundary value problem:

The body and surface forces are assumed to have explicit dependence on u and Vu, respectively, because this is the case in most of the interesting situations. An elastic material is hyperelastic if there exists a function W : f2 x M —> R differentiate with respect to F € M such that

The function W is called the stored-energy function of the material. If in addition there exists a function f ( x , u) such that

and for simplicity we assume TO — dSl, then equilibrium configurations are extremals of the total energy functional

In particular, minimizers of the total energy satisfying the global condition of place u = UQ on dfi will be (weak) solutions of the equilibrium equations. The first contribution to the energy is called the strain energy. Another way of stating the last assertion is by saying that the Euler-Lagrange system associated with the functional 7 is precisely the equilibrium equations. Under our structural assumptions this is a routine exercise if we are not concerned about technical details. Indeed, if u is a minimizer for / under the condition u = UQ on 90, then for any smooth, compactly supported test field v, we have that u + tv is admissible for any real t £ R. Therefore the function of t

1.4

EXAMPLES OF HYPERELASTIC MATERIALS

5

has its global minimum at t = 0. This implies, under smoothness conditions to justify all the computations that follow, that

We have used the divergence theorem and the fact that the boundary term vanishes due to the compact support of v in J7. Since the above equality is valid for arbitrary v, we can conclude that

Making this derivation rigorous requires much more precision and work. Since this is not the goal of this chapter, we will not pursue this topic any further but will be satisfied with the fact that minimizers of the total energy functional correspond to equilibrium configurations of our body. We will concentrate on seeking those minimizers. There are natural restrictions that a physically admissible energy density W must verify. The first one is the frame indifference

for all points x in Q, all matrices F, and all rotations Q. This invariance reflects the fact that no change of energy is associated with rotations of bodies. Another restriction has to do with extreme strains and it should reflect the idea that infinite stress is associated with extreme strains. This assumption leads us to postulate the behavior of W for large strains:

1.4

Examples of Hyperelastic Materials

A St. Venant-Kirchhoff material has a stored-energy function of the form

where a, /3 are constants, 1 stands for the identity matrix, and tr indicates the trace of a matrix. We shall no longer use the notation W for the stored-energy function but simply W

6

CHAPTER 1

ELASTIC MATERIALS AND VARIATIONAL PRINCIPLES

An important class of functions that appear as energy densities in nonlinear elasticity is

where r, s are positive integers, Oj > 0, bj > 0, a* > 1, /?,- > 1, and g is a convex function. The matrix (FTF)~*/2 is denned by the identity

where the singular values of F, fi(-F), are given by Vi = \/A7 and A* are the eigenvalues of FTF. R is an orthogonal matrix. See the appendix for further comments. A material whose energy density is of the above type and which satisfies the additional property liniA-^o