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.


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


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 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


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-ô
2. Mooney-Rivlin materials:

All these examples will be examined in some depth at the end of Chapter 3.

1.5

Appendix: Some Linear Algebra and Geometry

We gather here a few important results from linear algebra that are relevant in elasticity. We also make some observations about the geometric significance of the adjugate matrix. Given any 3 x 3 matrix F, both matrices FTF and FFT are symmetric positive definite. Both have the same nonnegative eigenvalues Aj, whose square roots Vi -= \f\i are called the singular values of F. The adjugate of F is defined through the formula This identity uniquely determines the adjugate of nonsingular matrices and can be extended by continuity to all matrices. It is often called the cofactor matrix and has the properties

Another important fact is the relationship between the eigenvalues of a matrix G and its adjugate: if Aj are the eigenvalues of G, then AiA2, AiAa, and A2As are the eigenvalues of adjG.

1.5

APPENDIX: SOME LINEAR ALGEBRA AND GEOMETRY

7

Given an invertible matrix F, its polar decomposition is

where R, S are orthogonal matrices, RRT = SST = 1, and U = (FTF)1/2, V = (FFT)ll2. Here the square root of a symmetric positive definite matrix is defined in the usual way: if P is an orthogonal matrix that diagonalizes G, i.e.,

then In the same manner, any positive power of G can be defined by

Finally we state two important results to be used in Chapter 3. THEOREM 1.3. If Vi(F) are the singular values of F ordered in a nonincreasing fashion, then

for any two matrices A, B THEOREM 1.4. Any matrix F can be decomposed as

where S and T are orthogonal matrices. We would like to point out the geometrical interpretation of the CauchyGreen tensor Vu T Vu, the adjugate adj Vu, and the determinant det Vu, where u is a deformation and Vw is the deformation gradient. If u is a smooth, invertible mapping, then it is well known that det Vu gives a local measure of how the volume changes under u. In the same way, it can be precisely shown that adjVu measures the change in surface area under the deformation u. Finally, Vu T Vu is responsible for the change of length elements under the action of u. If F = Vw is interpreted as a deformation gradient, its singular values are called principal stretches, and the polar decomposition

implies that any affine deformation can be decomposed and interpreted as a rotation followed by a stretching Vi(F)2 along three mutually orthogonal axes.


Chapter 2

Quasi Convexity and Young Measures 2.1

Introduction

The main point we tried to emphasize in the previous chapter is the variational problem, whose solutions correspond to equilibrium configurations of the material. It consists of finding minimizers (or more generally extremals) of the energy functional representing the internal energy associated with a deformation of the body,

where the continuous energy density W : O x R3 x M —> R* might also depend on additional physical parameters that are assumed to be held constant; W it ought to satisfy the appropriate physical requirements. We are taking here

£1 C R3 is the reference configuration and u : fl —» R3 is an admissible deformation of the body. R* stands for R U {+00}. Such minimizers describe equilibrium configurations of the material under the prescribed environmental conditions. One of these important conditions is the global condition of place whereby we impose for some fixed deformation UQ representing the conditions we prescribe on <9fL Some other, more general conditions can also be handled by the methods described in this chapter, but we will concentrate on the global condition of place. Let us assume, for the time being, that W depends only on the gradient variable, W = W(F). The important properties that the energy density W must verify are 9

10

CHAPTER 2

QUASI CONVEXITY AND YOUNG MEASURES

1. frame indifference: for all rotations Q and F £ M; 2. behavior for extreme deformations:

3. coerciveness:

Notice that upper bounds on W are incompatible with property 2 and that property 3 is stronger than the behavior for extreme deformations

The coercivity exponent in property 3 is important because it tells us that minimizing sequences for I will be uniformly bounded in W1'p(£l) (using Poincare's inequality to control the functions themselves). Thus, we are interested in bounded sequences {uj} in Wl 1, the existence of equilibrium configurations may be compromised. For this reason, we may take the admissible deformations u in (2.1) to belong to W liJ) (fi). Typically, the deformation UQ giving the boundary conditions on dfl will be a Lipschitz mapping UQ G W 1>oc (fi) (we will not indicate the target space for deformations W^ llp (f2;R 3 ) since it is always understood by the context) such that /(WQ) < oo. In particular, det VUQ(X) > 0 for a.e. x 6 fJ. It is also usual to assume UQ : fl —>• uo(O) to be a 1-1 mapping. Other, more general possibilities may also be allowed. Under these conditions, we would like to analyze the problem of the existence of equilibrium configurations; that is, we are interested in understanding when we can find an admissible deformation U so that for all admissible u. The so-called direct method will lead us to become concerned about the weak lower semicontinuity of the functional /. This property in turn relies on the quasi convexity of W, which is the central issue of this chapter. In the appendix at the end of the chapter, some important points about Sobolev spaces and weak convergence are stated.

2.2

The Direct Method

In order to appreciate the simplicity and elegance of the direct method, let us look for a moment at the finite-dimensional situation. Let / : R™ -» R*. We would like to find x0 e R™ such that I(x0) < I(x) for all x € Rn. The first

2.2

THE DIRECT METHOD

11

condition we need to ensure is that / be bounded from below, I ( x ) > c> —oo, for all x € R™. Otherwise, there is nothing we can do about the analysis of the minimization problem: there cannot exist a minimizer. Let

and let {xj} be a minimizing sequence: T(XJ) \ m. If {xj} is relatively compact in Rra (this is the case if lim inf xôc I(x) > rn) and I is continuous, then for some appropriate subsequence, not relabeled, x} -»• XQ and I(xj) -> m. Therefore I(x0) = m and x0 is a minimizer. In fact, since we are interested in minimizers it is enough to demand the lower semicontinuity of /:

whenever Xj —> x. The direct method consists of imitating the finite-dimensional case in the infinite-dimensional situation. The various important ingredients are 1. 7 is not identically +00; 2. 7 is bounded from below; 3. good compactness properties exist for the topology on the set of competing functions; 4. 7 should be lower semicontinuous with respect to the chosen topology. The function spaces of competing functions are usually Banach spaces with integral norms Lp(£l), Wl'p(£l), and the appropriate topologies with good compactness properties are the weak topologies over these spaces. In particular, if X is one of these spaces and is reflexive, it is well known that

possibly for a subsequence (Banach-Alaoglu-Bourbaki theorem). This property is extremely convenient and explains, from our perspective, why weak convergence is so important and why we are interested in deepening our understanding of it. Finally, the most difficult step in applying the direct method is enforcing the sequential lower semicontinuity property with respect to these weak topologies:

We can summarize the previous considerations in the following abstract theorem, whose proof has been sketched above. THEOREM 2.1. Let us consider the variational principle

where (i) A is a closed, convex subset of a reflexive Banach space X; (ii) 7 is coercive:

12

CHAPTER 2


(iii) / is sequentially lower semicontinuous with respect to the weak topology in X; (iv) there exists u £. A such that I(u] < oo. Then there exists UQ e A with I(UQ) < I(u] for all u e A. In our context, the functional I is given by (2.1), and because of property 3 in the previous section, minimizing sequences {uj} will converge weakly to some u in W 1>p (£l) (remember p > 1). According to the direct method, we must concern ourselves with the property of (sequential) weak lower semicontinuity of the functional /. The real issue is to decide when

This (sequential) weak lower semicontinuity property is the topic of this chapter. Those readers not familiar with weak convergence are advised to take a look at the appendix at the end of this chapter so that they can fully understand and appreciate the discussion that follows. From now on we will omit the term "sequential" when referring to weak lower semicontinuity, taking for granted that we always mean weak lower semicontinuity along sequences.

2.3

Young Measures

To study the weak lower semicontinuity property for a functional

where for the sake of simplicity we drop the dependence on x and u, is to concern ourselves with the possible behavior of the integrals

when the sequence {uj} converges weakly in W 1>p (fl). If this weak convergence is in fact strong so that Uj ->• u in W 1>p (f2) and, possibly for some subsequence, Uj(x) ->• u(x) and Vuj(x) —» Vu(x) for a.e. x € fi, then if W > 0 is continuous, by Fatou's lemma,

If W is continuous and bounded from below (by zero or some other constant), the associated functional enjoys the property of strong lower semicontinuity. Could one reasonably expect this inequality to hold even if the convergence is only weak? What further hypotheses do we need to impose on W to ensure the weak lower semicontinuity? Notice that in the case of weak convergence we can also apply Fatou's lemma, but in this case liminfjôo W(Vuj(x)) is not, in general, W(Vu(x)).

2.3

YOUNG MEASURES

13

We study some simple examples to emphasize the behavior of weak convergence with respect to nonlinear functionals so as to gain some insight into the questions raised. 1. Let n = (0,7r/2) and fj(x] = sin(jx), g^x) = ff(x), x 6 H. Prom elementary trigonometry we have

Hence

for any interval (a, b) C fl. This actually means that

Observe that the square of the weak limit does not coincide with the weak limit of the squares. Nonetheless, we have the right inequality for weak lower semicontinuity

where 0 is the weak limit of {/_>•}Let us further take 0, and let us examine the weak limit of the sequence {ip(gj)}. In this case, using the periodicity of sine, we get

where (a) stands for the integer part of a. We conclude that the last two terms in the computations above converge to 0, and therefore

Notice that once again the weak limit of {
14

CHAPTER 2


while

In this case we do not have the weak lower semicontinuity inequality. We apparently need more conditions on W in order to have the weak lower semicontinuity property. 2. Let f(x) = 2x(0,i/2](x) - 1 for z € [0,1], and extend / to all of R by periodicity. Take fj(x) = f ( j x ) . It is easy to see that the jumps from 1 to -1 occur on a smaller and smaller scale as j tends to oo. On the other hand it is not hard to see that fj ->• 0 for fi = (0,1). However, fj = 1 for all j so that /? —^ 1 and once again the square of the weak limit is not the weak limit of the squares. Moreover, if ip : R —» R is any continuous function, then

and

for any interval (a, b) C (0,1); hence for any such y>,

In this particular case we have been able to describe the limit behavior of the integrals

for any function (p in terms of the expression (1/2)(^(1) + tp(—l)). This is the main feature of the Young measure device. Through this property, weak lower semicontinuity can be understood in a rather general way. The examples examined above should have convinced us that there is something special about sequences of a highly oscillatory nature when the convergence is only weak and not strong. In general terms, the problem we are trying to understand can be formulated as follows. Suppose that fj -^ / in L°°(fi) so that ||/j|| L oom) < C < co and (f is a continuous function. {
2.3

YOUNG MEASURES

15

R m , depending measurably on x £ 0, which means that for any continuous (p : Rm -» R the function of x

is measurable. The fundamental property of this family of probability measures is that whenever {f(fj}} converges weakly * in L°°(fl) (or more generally, weakly in some £^(17)), the weak limit can be identified with the function Ip in (2.2): for all h G L 1 (fi). Intuitively, the Young measure can be thought of as giving the limiting probability distribution of the values of {/_,-} when points are taken randomly around each x e Q. If -B#(x) denotes the ball of radius R > 0 centered at x e Q and E C Rm is any measurable set, then

where bars |-| denote the Lebesgue measure. This identification clearly shows that the sequence {/_,-} is forced to oscillate near x among the different vectors in the support of vx with relative frequency given by the weights corresponding to such vectors. Let us look at some other examples. 3. Let g : R2 -»• R be denned by

where Xs/4 'ls the characteristic function of the interval (0, 3/4) C (0,1) extended periodically to all of R. Let u : R2 -¥ R2 be defined by u(x) — (g(x),g(x)), x = (xi,x-z) € R 2 , and Uj(x] ~ ( l / j ) u ( j x ) . If we compute the gradients of these functions we obtain or in matrix form

The tensor product a®n = a nT for a £ Rm and n e ILN is the rank-one matrix (diTij)^. We claim that the Young measure associated with {Vuj} is

CHAPTER 2

16


for all x 6 fi, where

For any continuous function ip : M 2 x 2 —> R we have

4. Now take ip0(F) = \F\2 for F £ M 2 x 2 so that

where Uj is the sequence of the previous example. We know that the sequence of gradients {Vu-j} converges weakly * in L°°(fl) to Vu = (3/4)A + (1/4)O = (3/4)A by simply taking (p to be the identity in (2.4). Since
which in this simple case is nothing but

By (2.4) applied to (po, this inequality is exactly

a weak lower semicontinuity result.

2.4

Weak Lower Semicontinuity

As pointed out, the fundamental property (2.3) of Young measures provides a way of representing and manipulating the limits of the following integrals:

Prom this perspective, the Young measure is a convenient tool when dealing with integral functionals in the calculus of variations. The main issue is as follows: Under what type of conditions can we actually represent limits of integrals of the above type by Young measures? The answer is contained in the following basic existence theorem.

2.4

WEAK LOWER SEMICONTINUITY

17

THEOREM 2.2. Let 0 c R A be a measurable set and let Zj : fi -> Rm be measurable functions such that

where g : [0, oo) —> [0, oo] is a continuous, nondecreasing function such that limôo g(t) = oc. There exist a subsequence, not relabeled, and a family of probability measures v = {^x}xâ ftê associated Young measure) depending measurably on x with the property that whenever the sequence {ifr(x,Zj(x))} is weakly convergent in Ll (fi) for any Caratheodory function if?(x, A) : O x Rm —> R*, the weak limit is the function

By a Caratheodory function, we simply mean a function measurable in x and continuous in A. A particularly important example is obtained by taking g(t) = tp for p > 1 (we can even allow 0 < p < 1). In this case, every bounded sequence in Lp(fl) contains a subsequence that generates a Young measure in the sense of Theorem 2.2. Assume now that our sequence {^(Vuj)} is weakly convergent in Ll(£i), where {uj} is minimizing for the functional /, so that

Herei/ = {^x}xen is the Young measure associated with the sequence {^- = Vu,-}, which is bounded in 1^(0.) because of the coerciveness hypothesis assumed on W. Since must be the weak limit of {Vuj} by taking ij)(x,X) = A in Theorem 2.2, the weak lower semicontinuity property will hold true if

The assumption we have made that {W(Vuj}} is weakly convergent in L 1 (fi) is not always true. The following result establishes that even if equality (2.5) might not hold, inequality (2.6) is, however, always valid. THEOREM 2.3. // {zj} is a sequence of measurable functions with associated Young measure v = {^x}x^fi, then

for every Caratheodory function ip, bounded from below, and every measurable subset E C f!.

18

CHAPTER 2


In conclusion, if W verifies the coerciveness inequality

and (2.6) holds for any v coming from the gradients of a bounded sequence in Wl>p(ty, then

whenever Uj —l u in Wl'p(£l}. The proofs of Theorems 2.2 and 2.3 are sketched in Chapter 5. We have reduced the weak lower semicontinuity property to a certain inequality involving probability measures, which in fact reminds us of Jensen's inequality. What does this inequality mean in terms of weak convergence, and how does Jensen's inequality relate to it? This is a profound question that we can hardly answer here. We will merely provide an intuitive explanation of why Jensen's inequality seems to be connected to weak lower semicontinuity. Roughly speaking, weak convergence of functions means convergence in the average and therefore taking integrals plays a fundamental role in determining the limit function from the members of the sequence. The only functionals that commute with the operation of taking integrals are the affine ones. However, if we understand weak convergence in terms of Young measures, Jensen's inequality asserts that functionals associated with convex functions may not commute with integration, but they do always respect an inequality. The direction of this inequality is the appropriate one for lower semicontinuity. We will concentrate on condition (2.6) in the remaining sections of this chapter.

2.5

Quasi Convexity and Gradient Young Measures

The key constraint on the energy density W that ensures the weak lower semicontinuity property is (2.6). This inequality must be valid for all possible Young measures generated by gradients of a bounded sequence in W1>p(fl). Since each individual vx is a probability measure, if W is convex then (2.6) will certainly be true. This is the classical Jensen's inequality. In particular, weak lower semicontinuity will hold for any convex integrand. The convexity of W is not, however, a necessary condition in our situation. Our aim in this section is to more fully explore condition (2.6). Let us call a family of probability measures v = {"zl^go. a W7l'p(fJ)-Young measure (notice the explicit reference to the domain fi) if it can be generated as the Young measure (according to Theorem 2.2) corresponding to a sequence of gradients {Vuj}, where {uj} is a bounded sequence in W1>p(fl). The following localization result asserts that the individual members that make up such a family of probability measures are themselves Wl>p(Q}-Young measures for any domain Q. By a domain we mean an open, bounded, regular set so that, in

2.6

A GENERAL EXISTENCE THEOREM

19

particular, \dQ\ — 0. If the Young measure v = {^x}xfcn is such that i/x = VQ for a.e. x e fi, where t/o is a fixed probability measure, we say that v is homogeneous. PROPOSITION 2.4. Let v = {z^l^n &e a W lip (fi)-Ybim<7 measure. For a.e. a e 0 and /or any domain Q, there exists a bounded sequence {vai]} in Wl'p(Q) such that the homogeneous Young measure associated with {Vwaj-} is va. Moreover, each function vaj can be chosen in such a way that va,j —Up(a) € W0'P(Q), where UF(X) is the linear function Fx and

Due to this result, we can define a homogeneous W llp -Young measure (with no reference to a particular domain) as a probability measure v supported on the set of matrices M such that for any domain Q c R3 there exists a sequence of gradients {Vttj}, where {uj} is some bounded sequence in Wl'p(Q) generating v according to Theorem 2.2 and such that Uj — up & W 0 llp (Q), where F is the first moment of v. We immediately have that if v — {vx}x&n is a W 1>p (fi)Young measure, then a.e. vx is itself a homogeneous W1>p-Young measure. We also say that a function W : M -> R* is closed W llp -quasi convex if

for all homogeneous Wl'p-Young measures. Notice that this definition requires Jensen's inequality for a (proper) subclass of probability measures supported in the set of matrices M. The conclusion of the previous remarks and definitions is that the closed W1|p-quasi convexity condition is sufficient (and necessary) in order to have (2.6). In section 2.4, we showed that this inequality is sufficient for weak lower semicontinuity, but the necessity is a much more delicate point that will not be treated here. For a general Caratheodory function W : Q xR3 x M —> R* the closed WliJ)-quasi convexity condition can be formulated in the following terms: for all (x, u) e fi x R3 and for all homogeneous W1>p-Young measures v. Some remarks on the proof of Proposition 2.4 can be found in Chapter 5.

2.6

A General Existence Theorem

It is now easy to prove a general existence theorem of minimizers under (2.7) by using Theorem 2.1. Assume is a Caratheodory function representing an energy density that satisfies the coerciveness condition

20

CHAPTER 2


for all pairs ( x , u ) . Consider the variational principle

where UQ e W lip (Q) is given such that I(uo) < oo. We are looking for minimizers THEOREM 2.5. I f W is dosed W1 Rd x Rm be a bounded sequence in p L (fl) such that {uj} converges strongly to u in Lp(£l). If i> — {^x}x€n is the Young measure associated with {zj}, then vx = 6U(X) <&{j,x for a.e. x 6 fi, where {fJ-x}xeQ is the Young measure corresponding to {vj}. This lemma is applied to sequences {uj,Vuj} for {uj} a weakly convergent sequence in W lip (fJ). In this situation we know that the functions themselves converge strongly to the weak limit by the compactness theorem of Sobolev spaces. If u € W1|p(fl) is the weak limit and {^x}x&^ is the Young measure associated with the gradients {Vu./}, then vx — 6U(X) <8> fj,x for a.e. x G f2. Lemma 2.6 is discussed further in Chapter 5. The proof of Theorem 2.5 is simple. Let be a real number. If {u^} is a minimizing sequence, by coerciveness and Poincare's inequality, it will be a bounded sequence in W1>p(fi) whose gradients will generate a TV1'p(f2)-Young measure v = {VX}X£Q- For an appropriate subsequence, Uj —v u and

By Lemma 2.6, Theorem 2.3, and the closed W1>p-quasi convexity.

u is then a minimizer in W1'p(fl). The uniqueness of minimizers does not hold in many situations. We will not study this issue here.

2.7

2.7

THE CASE

21

The Case

Once we have seen the relevant role of closed W1|p-quasi convexity for weak lower semicontinuity and its importance as a sufficient condition for the existence of minimizers in vector variational problems, we are justified in making an attempt to better understand this condition so that we may gain some feeling for and interpretation of it. Unfortunately, there is not much we can do in this direction, as quasi convexity still remains a largely unknown area. Since many technicalities are involved in the finite case p < +00, we consider in this brief section the Lipschitz case Let UF(X) be the affine Lipschitz function UF(X) = Fx for x € fi. For u e W 1>00 (fi) such that u — up € W0 >oc (fi) we can produce an admissible family of probability measures in (2.6). This can be done through the RiemannLebesgue lemma, which is a particular case of a more general homogenization fact (Theorem 5.12). LEMMA 2.7. Let fi be a domain in HN and u £ W 1>00 (fi), u - up € W0'°°(f2). There exists a sequence {uj} bounded in W liOC (fi), Uj — up s W 0 1>oc (fZ), such that the Young measure associated with {Vuj} is homogeneous and defined by for any continuous
for all F 6 M and all u £ W 0 1>00 (fi). A function that satisfies all these inequalities is simply called quasi convex. This property is important because it is in fact equivalent to (2.6) under weak * convergence in W 1>00 (fJ) provided W takes on finite values everywhere. In other words, quasi convexity is equivalent to closed WllOC-quasi convexity under the finiteness of W (why?). It is less restrictive than (2.6) under weak convergence in W 1>p (f2) for finite p. Notice that in (2.8) the choice of the domain O is irrelevant. From an energetic point of view, (2.8) is much more transparent than (2.6), because it says that among all deformations having affine boundary values determined by the matrix F, the least energy is achieved by the affine deformation itself. This is a very important interpretation of quasi convexity that greatly aids our intuition. In general, there is no way to obtain uniform bounds in W 1 '°°(fZ) for minimizing sequences from coerciveness, unless we assume that the energy density W is +00 outside some compact set of matrices. However, this condition is not really appealing. For this reason the case p — +00 is not usually considered in existence theorems. More specific hypotheses might lead to that uniform bound in W 1>00 (fi), but these situations require very precise knowledge on W. On the other hand, closed W1>p-quasi convexity reduces to quasi convexity under upper bounds on W; this upper bound has been, however, rejected on physical grounds.

22

2.8

CHAPTER 2


Appendix

We recall in this section a few general facts about the Lebesgue and Sobolev spaces L p (O) and Wl>p(£l). The space i 1 (fi) designates the set of all integrable functions over fJ, and we write The space L p (f2) for p > 1 consists of all measurable functions / such that I/I" el 1 (ft)- We put

L°°(fi) is the space of all (essentially) bounded, measurable functions over ft; set Under these norms the spaces V($l), 1 f(x) for a.e, x e Q. Z/^fi) is a reflexive space if 1 < p < oo with dual L 9 (fi) for the conjugate exponent q = p/(p - 1). The duality is given by

The dual space of Ll (Q) can be identified with L°°(£l) through the same duality as before. L : (ft) and L°°(O) are not reflexive. Lp(£l} is separable for 1 < p < oo. L°°(S1) is not separable. The space of compactly supported, continuous functions in O is dense in L p (ft) for 1 < p < oo. An important inequality is Holder's inequality: if / e Lp(ty and g £ L«(fi) with q = p/(p - 1), then

The inequality is valid for the limit cases p = 1, oc as well. For several terms it can be generalized in the following way:

Let 1 < p < co. We say that {fj} converges weakly to / in I/P(Q) and write fj -* f in L"(n) if

2.8

APPENDIX

23

for every g e Lq(£t), l/p + l/q — 1. For p = oo, {/,•} converges weakly * to / in L°°(n) (in compact form

for every 5 6 L1(i7). The criterion for weak compactness is contained in the following proposition. The case p = 1 is very special. PROPOSITION 2.9. Let 1 0 such that \\fj\\LPim < K uniformly for all j. Let p = 1. The sequence {/_,-} is relatively compact in Ll(£l) if and only if (i) there exists a constant K > 0 such that \\fj\\Lim) < K for all j; (ii) for every e > 0 there exists a 6 = 6(e) > 0 such that for every measurable subset E with \E\ < S, we have

uniformly for all j. Condition (ii) is the equi-integrability property or Dunford-Pettis criterion of weak compactness in Ll(Q). It is a well-established fact that L1^) is very peculiar from the point of view of weak convergence and that peculiarity is precisely condition (ii) above. Concentration effects are connected to the failure of (ii). PROPOSITION 2.10. Let 1 < p < oo. fj -* f in L P (O) (or weakly * in L°°(Q); if and only if (i) \\fJ\\LP(n}0; (ii) lim^-nx, JD(fj(x) — f(x)) dx = 0 for all cubes Let p = 1. fj^finL1 (fj) if and only if (i) \ \ f j \ \ L 1 ( f l ] < K , K > 0 ; (ii) the equi-integrability property holds; (iii) linij-i.oo fD(fj(x) - f(x)} dx = 0 for all cubes D C 17. These results on weak compactness are a particular case of a more general theorem. THEOREM 2.11. Let X be a Banach space with dual X'. Every bounded sequence in X' is sequentially weakly * compact. Sequential weak * convergence in X', fj —^ f , means that for every x € X. The Sobolev space W1>p(17) is defined as the set of functions u € L P (O) such that there exist functions jj^£ L p (fi) with the property t/iC-j

24

CHAPTER 2


for any compactly supported, smooth function £. This definition can also be stated by saying that the distributional partial derivatives of u belong to Lf(fl). We write Wl'p(fl)

is a Banach space under the norm

It is reflexive for 1 p (fi) can be approximated by smooth functions in the norm of W 1>p (fi). Weak convergence in W1'p(fl) means weak convergence in L P (Q) for the functions together with their gradients. A fundamental fact about Sobolev spaces is the compactness theorem. THEOREM 2.12. Iffl is an open, regular, bounded domain, then 1. p < N: W 1>p (fi) is compactly embedded in Lq(fl) for any I < q < p*, where p* — Np/(N — p); 2. p = N: W 1>p (fl) is compactly embedded in L 9 (fi) for any 1 < q < oo; 3. p > N: W l i p (f2) is compactly embedded in the space of continuous functions up to the boundary under the sup norm. An important corollary is that a weak convergence sequence in W 1>P (Q) will converge strongly in Lp(fl) to the same limit. The space WQ'P(&.) is the closure in W 1>P (Q) of the set of smooth functions with compact support in O. It is important to keep in mind Poincare's inequality, which is given below. THEOREM 2.13. Under the same assumptions on f2 as in Theorem 2.12, and for 1 0 such that for every u £ Wo' p (f2). Another version of Poincare's inequality is the following. THEOREM 2.14. If £1 is a bounded domain and u e Wl 0 such that where We finally recall the classical Jensen's inequality. THEOREM 2.15. Let p be a positive measure over a a-algebra in a set fl such that fj,(fl) — 1. Let f be a vector-valued function in Ll(^) such that f ( x ) £ K for n a.e. x 6 f2, where K c Rm is a convex set. If (p is a convex function defined in K, then

Chapter 3

Polyconvexity and Existence Theorems 3.1

Introduction

We would like to specify situations in nonlinear elasticity where the general techniques described in the previous chapter lead to explicit existence results for equilibrium configurations. The central contribution of this chapter is to provide the main source of nontrivial (nonconvex) examples of functions that satisfy condition (2.6). Many of the typical energy densities for various classes of materials fall into this category and hence we will show that equilibrium configurations for these materials exist.

3.2

Polyconvexity

We showed in Chapter 2 that if the energy density of a hyperelastic material is convex with respect to the gradient variable, then we have equilibrium configurations under prescribed boundary conditions. Under the assumption of convexity of the energy density W, our existence theorem, Theorem 2.5, applies almost immediately. We would like to argue, however, that this hypothesis contradicts the physical requirements imposed on W. Indeed, for a convex function W : M -> R*, the set where it is finite has to be convex, but the set {F e M : det F < 0} is not. On the other hand, the axiom of frame indifference implies some inequalities that the eigenvalues of the Cauchy stress tensor in any point of any deformed configuration must verify. Nonetheless, these inequalities cannot always be expected to hold. There are counterexamples, although they are beyond the scope of this book. Hence, we have to rule out convexity as a basic assumption. If we cannot reasonably rely on the convexity assumption for our free energy densities W, how can we find some other condition on W to guarantee (2.6)? 25

26

CHAPTER 3

POLYCONVEXITY AND EXISTENCE THEOREMS

One fundamental idea is the following. Suppose we have found a function M : M —> R that has the property

for any homogeneous Wljp-Young measure v or for a selected subclass of those measures. If g : R -» R* is convex, the composition W(A) = g(M(A)} will verify (2.6), which is elementary to check by the usual Jensen's inequality. Likewise, if M(A) now represents the vector, in some order, of all (essentially different, i.e., linearly independent) functions verifying (3.1) and g is a convex function of all its arguments, the composition W(A) = g(M(A)} will verify (2.6) as well. What does (3.1) mean in terms of a sequence {Vwj} that generates vl If the two sequences {Vu,-} and {M(Vuj)} converge weakly in L1(Q) to Vw and M, respectively, then (3.1) means precisely that M = M(Vu), which is again elementary to check using the fundamental property of the Young measure and (3.1). Conversely, if we would like (3.1) to be true, we would have to show that v can be generated by a sequence {Vw-,-} such that Vuj —*• Vw and Af(Vwj) ->> M(Vw) in Ll(£l), where u e W1>P(JT). In practice, since if p > 1 the convergence VMJ —*• VM is always guaranteed, the fact M(Vuj) —*• M(Vu) in L 1 (n) is directly achieved as a result of the following two main ingredients: 1. {M(Vuj)} converges weakly in L1(J7) (which is a consequence of uniform bounds in some Lr(fl) with r > 1); 2. M(Vuj) —*• M(Vu) in the sense of distributions. Where can we look for functions M satisfying (3.1)? Since the weak convergence M(Vuj) —*• Af(Vu) in W 1|30 (fi) implies that both M and —M have to be quasi convex, we are thus led to determine continuous functions M, defined on matrices, such that both M and — M are quasi convex. An initial idea is that since quasi convexity implies rank-one convexity (this will be proved in Chapter 4), M and — M will have to be rank-one convex, i.e., the function M(A + tB) must be affine in t whenever B has rank one. The functions that enjoy this property are called quasi-afRne functions and they are identified in the following theorem. THEOREM 3.1. Affine functions of the minors of a matrix are the only quasi-affine functions. In the case of 2 x 2 matrices, there are five minors: the four entries (the 1x1 minors) and the determinant. For 3 x 3 matrices, we have 19 minors: the nine entries, nine 2 x 2 minors, and the unique 3 x 3 minor, the determinant. And so on it goes. The fact that these minors are quasi affine is easy to check. As a matter of fact, it is not hard to convince oneself that for one of these minors, M, the function of t, M(A + tB), is a polynomial of degree rank(B). To show that these are the only quasi-affine functions requires more work and a little bit of algebraic manipulation. For the complete proof of Theorem 3.1, let M(F) denote a quasi-affine function and let, for notational convenience, M^(F) denote the constant M(0).

3.2

POLYCONVEXITY

27

The difference M^)(F) = M(F) - M^(F) vanishes on rank-zero matrices, that is to say, on the zero matrix. Let et denote the standard basis in R3 so that €i
and let

We claim that this function vanishes on rank-one matrices. In order to prove this claim we state the following elementary property of quasi-affine functions that we are going to use several times: if M is a quasi-affine function such that

then M(A + B) = 0. The proof of this fact is almost trivial. Since A — B is of rank one, we can write

Certainly, M(2) is quasi affine (difference of two quasi-amne functions) and vanishes on gj ® ej by definition of M^. It is also true, bearing in mind that M^ is quasi affine, that for any scalar s,

By using the above-mentioned property, we can write

Proceeding in this way, one can easily show that

for any vector a. By using the same technique with respect to the second factor, we prove that for any two vectors a, n. For given indices i, j, fc, and I, let A.l^(F) be the 2 x 2 minor of F corresponding to rows i, j and columns k, I. Define

28

CHAPTER 3


We know that M^ is quasi affine so that the difference

is also quasi affine. We claim that M(3) vanishes on matrices of rank two or less. This is clearly the case for matrices of rank one or less and also, by definition, for all the matrices ti ® ek + €j ®ei. By again using repeatedly the elementary property mentioned earlier, one can show successively

for arbitrary vectors a, n, 6, m. This proves the claim. Finally, set

Once again M* is quasi affine, and just as before it can be shown that M* vanishes on matrices of rank three or less. This means that M* identically vanishes on M and therefore

Our main task consists now of showing that the minors indeed generate weak continuous functionals. For a matrix A 6 M, let A' denote some square submatrix of order r of A. The adjugate of A' is denned by the identity

where 1 is the identity matrix (of order r). THEOREM 3.2. Ifuj -± u in Wl r, then det(Vuj)' ->• det(Vu)' in the sense of distributions. Proof. Because of the importance of this result and in order to make the argument for the general case more transparent, we are going to first give the proof in the particular case of 2 x 2 matrices and N = r = 2 so that there are no primes in the proof and

We divide the proof into two steps.

3.2

POLYCONVEXITY

29

Step 1. Let v 6 W^> P (Q). We claim that div(adj(Vu)) = 0 in the sense of distributions. If v is smooth, then it is very easy to check that

due to the equality of the mixed partial derivatives. For a nonsmooth function in W l t p ( f l ) , take a sequence of smooth functions {vj} converging strongly to v in W 1|p (fi). Obviously, adj(Vwj) will converge strongly to adj(Vu) so that for any test function,

as desired. Step 2. If uW is the ith component of u and adj^(Vw) is the ith row of adj(Vu), then, as a consequence of Step 1, we obtain that

as distributions. This identity is rather easy to check in this situation. Since (adj(Vuj)} is bounded in Lp(£l), some subsequence (not relabeled) will converge weakly in Lp(^l) precisely to adj(Vu) (notice that adj(Vw) are just the single derivatives of u). If ^ is a smooth test function, then

The first term is bounded by

which tends to 0 as j —>• oc because, by the compactness theorem for Sobolev functions, {uj} converges strongly to u in Lq(£l) for any finite q. The second term converges to 0 due to the weak convergence of the adjugates in LP(Q). This is the conclusion of the theorem in this particular situation. In the general setting, we require a little bit of notation. For a given deformation u, let (Vw)' denote the corresponding submatrix of order r of Vu. There are r independent variables and r components of the deformation u associated with the choice of this submatrix indicated by '. Let i correspond to some component of u chosen in the submatrix. div will indicate the divergence operator with respect to the independent variables involved in the submatrix, and the same will be understood for the vector operator V.

30

CHAPTER 3

POLYCONVEXITY AND EXISTENCE THKOREMS

We divide the proof again into two steps. Step 1. Let v e W l ' p ( S l ) . We claim that div'(adj(Vv)') = 0 in the sense of distributions. Assume first that v is actually smooth. Based on the equality of the mixed partial derivatives, we find indeed that div'(adj(Vu)') = 0. For a general v 6 Wl>p(fl), take a sequence of smooth functions {vj} converging strongly to v in VF lip (ft). adj(Vuj)' converges strongly to adj(Vu)' in Lp/(-T-1\tl) (by Holder's inequality and pointwise convergence) because the terms in adj(Vuj)' are products of at most r -1 factors. For a smooth test function ip,

for all j. By the strong convergence just pointed out,

and this is our claim. Step 2. If u^ is the tth component of u (one of the chosen ones in A') and adj^(Vu)' is the ith row of adj(Vu)', then, as a consequence of Step 1, we obtain that

as distributions. By induction, let us assume that adj(Vuj)' —*• adj(Vw)' in the sense of distributions. Since adj(Vuj)' involves terms with r — 1 products, it will be bounded in £,p^r~^(0), p/(r — 1) > 1, and therefore that sequence of adjugates will converge weakly (in Lp/(r~^(£l)) precisely to adj(Vu)' due to the uniqueness of both weak limits. If tp is a smooth test function, then

The first term is bounded by

which tends to 0 as j -» oo because, by the compactness theorem for Sobolev functions, {u.j} converges strongly to u in £*(fi) for q < 3p/(3 — p) if p < 3

3.3

COERCIVITY AND THE CASE p < CO

31

and in L°°($7) if p > 3. Notice that p / ( p - r + 1) < 3p/(3 - p) if p > r. The second term converges to 0 due to the weak convergence of the adjugates in £p/(r-i)(Q)_ The theorem is proved. This result is important because it allows us to conclude that whenever Uj —>• u in W l i p (fi) and {M(Vuj)} converges weakly in Ll(£l), M being any minor of order r and p > r, then we have in fact M(Vw^) —^ M(Vu) in and consequently (3.1) will hold for the Young measure associated with {Vuj}. For instance, in the case p = oo we always have the weak convergence in L°°(Q) of {M(Vitj)} for any minor M so that M(Vuj) ->• M(Vu) in L°°(n) if u is the weak limit of {«?}. PROPOSITION 3.3. // M is any minor and v is any homogeneous jy1'00Young measure, then (3.1) holds. Let M(A) denote the vector of all possible minors of a matrix A considered in some fixed order. A function W : M —> R* is called polyconvex if it can be rewritten in the form W(A) = g ( M ( A ) ) , where g is a convex (in the usual sense) function of all its arguments. A direct consequence of our previous discussion is that every polyconvex function W verifies (2.6) for any homogeneous W1'00Young measure, and consequently we have the following result. PROPOSITION 3.4. IfW : M -> R* is polyconvex and Uj -^ u in W1'00^), then

It is very easy to give examples of polyconvex integrands that are not convex. For instance, g(det A) is always polyconvex for any convex function g.

3.3

Coercivity and the Case

In variational principles, the weak convergence Uj —^ u of minimizing sequences is usually enforced through coercivity assumptions on the integrand W. Unfortunately, this weak convergence can very rarely be obtained in W1'°°(17) unless W(A) = +00 whenever |.4| > R for some R > 0, but this hypothesis is not interesting. Much more natural are coercivity assumptions assuming a certain rate of power growth for W:

The exponent p is crucial, and in particular its size relative to the dimension 3. Note that if no such coercivity assumption holds (for instance, if W( A) = det A), then the direct method cannot be implemented and the existence theorem proved in section 3.4 is compromised. This is the reason why the technical details that follow, although tedious to some extent, are important. The growth exponent p determines the Sobolev space in which we have to work, and in particular we need hypotheses to ensure (3.1) for homogeneous Wrl'p-Young measures precisely for that value of p.

32

CHAPTER 3 POLYCONVEXITY AND EXISTENCE THEOREMS

The following result lists several situations in which we can achieve the weak convergence of (M(Vuj)} in Lp(fl) for finite p. THEOREM 3.5. Let v = {vx}xeft be a Wl'p(£l)-Young measure with p > 2. If v can be generated by the gradients of a sequence {uj} such that one of the following conditions holds, then (3.1) holds for a.e. vx: 1. {uj} is bounded in Wl>p(£l) and p > 3; 2. {uj} is bounded in Wl's(£l) and {|V«j|3} is equi-integrable; 3. {uj} is bounded in W lip (Q), 2 3/2; 4. {uj} is bounded in W1'"(&.), 2
3p/(4p - 3), and {det(Vu.,-)} is bounded in Lr($l), r > 1. Proof. The proof reduces to showing in each case that

and then concluding that F = adj(Vw) and / = det(Vu). Cases 1 and 2 are taken almost directly from Theorem 3.2 and the equiintegrability for the case p = 3. In case 3, notice that Theorem 3.2 applies for r = 2 so that the adjugates converge in the sense of distributions. Since the exponent of boundedness is greater than 1. the convergence takes place weakly in L 3 / 2 (Q). For the determinant we cannot directly apply Theorem 3.2. However, under these assumptions on the adjugate, (3.2) shows that the determinants converge in the sense of distributions by using Holder's inequality with the exponents 3p/(4p - 3) and 3p/(3 - p) rather than p/(r - 1) and p/(p - r + I). Notice that 3/2 > 3p/(4p - 3) if 2 < p < 3. On the other hand, the inequality |detv4| < <7|adj J 4| 3/2 shows that {detVwj} converges weakly in L1^). Case 4 is left to the reader.

3.4

Existence Theorems

Our main existence theorem for polyconvex integrands yields minimizers for the energy

under Dirichlet boundary conditions u = UQ on <9fi. THEOREM 3.6. Assume the following assumptions on the energy density W: 1. W : tt x R3 x M ->• R* is a Caratheodory function; 2. W(x,u, •) is polyconvex for a.e. x e fl and every u 6 R3;

3.4

EXISTENCE THEOREMS

33

3, one of the following coerciveness conditions holds for some constant c > 0:

If I(UQ) < oo for some UQ 6 W l i p (f2), the variational principle

admits minimizers. Note that our situation in nonlinear elasticity, where W(x, u, A) = +00 when det^4 < 0 and W(x,u,A) ~> +00 when det/1 -> 0 + , is covered by assumption 1 in Theorem 3.6. Proof. After Theorem 3.5, the only delicate point in this proof is the case p = 3. The following remarkable lemma yields the clue to the proof. Its complete proof can be found in Chapter 5. LEMMA 3.7. Let {vj} be a bounded sequence in W l > p ( f y , p > 1. There always exists another sequence {uj} of Lipschitz functions (uj € Wl'°°(£l) for all j) such that {|Vttj p} is equi-integrable and the two sequences of gradients, {V?Xj} and {Vuj}, have the, same underlying Wl>p(ty-Young measure. Note that in the cases when 2 < p < 3, discontinuous deformations are allowed in the variational principle. In certain cases some of them might be the optimal configurations (cavitation and fracture). This issue is, however, far beyond the scope of this text. We finally provide some results concerning the orientation-preserving and injectivity requirements. Concerning the former we have the following elementary lemma. LEMMA 3.8. Assume z3 is such that Zj(x) > 0 a.e. x S fi and

where h > 0 is continuous and h(t) = +00 when t < 0. // Zj —>• z in Lp($l), p > 1, then z(x) > 0 a.e. x e fi. Proof. Since Zj -*• z in I/ p (fi), for any measurable subset A C 0 we have

The nonnegativity of the integrals on the left-hand side and the arbitrariness of A imply that z > 0 a.e. x e SI. Let E = {z = 0}. Define w in E by

34

CHAPTER 3


w is nonnegative and by Fatou's lemma,

Due to the weak convergence Zj -*• 0 in E where z = 0, we conclude that

This implies w(x) = 0 a.e. in E, and hence

By Fatou's lemma,

Since h(0) = +00, \E\ must be For our purposes, it is enough to apply Lemma 3.8 in the context of Theorem 3.6 to Zj = det Vttj, z = det Vu, and

Note that this function h is continuous due to the uniform coercivity of W with respect to (x,u). The injectivity issue is much more delicate. We simply state without proof the following fact that is sufficient in many cases. THEOREM 3.9. Assume thatuo is a continuous-up-to-the-boundary, injective deformation of a domain fl and let u be another deformation such that u G Wltp(£i) for some p > 3, det Vu > 0, a.e. x € J7; and u = UQ on <9fi. Then u is a.e. injective.

3.5

Hyperelastic Materials

In this section we seek to analyze the examples of hyperelastic materials included in the last section of Chapter 1, in the context of Theorem 3.6. The main example to which Theorem 3.6 can be applied is the case of Ogden materials whose energy densities are of the form

3.5

HYPERELASTIC MATERIALS

35

where r, s are positive integers, a« > 0, bj > 0, a^ > 1, /3j > 1, and 5 is a convex function. We must demonstrate the poly convexity and coerciveness for this W. We treat these two issues successively. Concerning the polyconvexity, it is convenient to express W(F) in terms of the singular values of F and adj.F. We know that if Uj are the singular values of F then t^fs, ui^s, and v\v^ are the singular values for adjF. We can rewrite W(F) as

In this case, it suffices to check the convexity of the function

when 7 > 1, in order to show the polyconvexity of W. The convexity of such a (p is a consequence of the next two lemmas. LEMMA 3.10. Let n > r 2 > r3 > 0 be given. The function

where Vi > V2 > v$ > 0, is convex. Proof. The conclusion of the lemma will follow from the identity

where the maximum is taken among orthogonal matrices. Note that for fixed Q and R, the function ti(FQDR) is linear in F, and the maximum of linear functions is convex. In order to establish the equality (3.4), by Theorem 1.3 we have

By Theorem 1.4, the particular choices Q = T^1 and R — S'1 yield the equality. LEMMA 3.11. Let g : [0, +oc)3 —>• R be a convex, symmetric function, nondecreasing with respect to each variable. The composition

is convex.

36

CHAPTER 3 POLYCONVEXITY AND EXISTENCE THEOREMS

Proof. The symmetry requirement on g means that it is invariant under any permutation of its variables. Let a, 6, and u denote the vectors of singular values of the matrices A, B, and XA+(1— A)J5, respectively, ordered in a nondecreasing fashion, A e (0,1). We have to check that

Since g is convex, it suffices to show that

Let V consist of the vectors (0,0,0), (ui,0,0), (vi,V2,Q), («i, ^2,^3), and all those obtained by permutations of the last three vectors. V contains at most 13 vectors. We would like to show that u belongs to the convex hull of V. If this is so, since g is convex, we can write

However, since g is symmetric we can actually put

where s« > 0 and J]isi = 1- Moreover, g is nondecreasing with respect to any of its variables so that

and hence The lemma will then have been proved. It remains to show then that u G co(V) (the convex hull of V). To this end, assume we have a vector d e R3 and e e R such that

We would like to conclude that in this case u • d < e as well. First, observe that because (0,0,0) 6 V, e > 0. Consequently, if all coordinates di < 0 we would trivially have u • d < e because all coordinates Ui > 0. For a suitable permutation of indices a, one of the three following possibilities must hold:

3.5


37

Whatever the situation, let j denote the greatest index such that da-\^ > 0. The following chain of inequalities proves our claim:

The first inequality is an easy exercise if you bear in mind that the coordinates of u are ordered in a nondecreasing fashion. The third inequality is a direct consequence of Lemma 3.10 for an appropriate choice of TJ. The last inequality is true because the vector with coordinates va^ for i < j and 0 for i > j belongs to V. Since the function

verifies all the requirements of Lemma 3.11, keeping in mind the observation made earlier about the singular values of adj(F), we conclude that the polyconvexity property holds for the energy density of Ogden materials. Concerning the coerciveness, simply notice that the map

is a norm in R3 if 7 > 1. Since all norms are equivalent in finite-dimensional spaces, there exists some positive constant c7 such that

Because |.F|2 = tr FTF = Vi(F)2 +2(F)2 v + vz(F)2, this inequality means that

Applying this fact to all terms in (3.3), we get the coercivity

where a, b > 0, a = maxQi, j3 = max/?i. We can apply our main existence theorem provided we have the proper relationship between the exponents a, j3 and the one for the coercivity of g. The particular case of Mooney-Rivlin materials for which

38

CHAPTER 3


is also covered by Theorem 3.6. This is not so for neo-Hookean materials

due to the absence of the term involving the adjugate: coercivity fails. Nevertheless, in dimension N = 2 the existence theorem is recovered because we regain the necessary coercivity. The St. Venant-Kirchhoff materials with energy densities of the form

where a, (3 are constants, exhibit a real difficulty in the sense that these energy densities are not polyconvex. In this regard, the failure of the existence theorem is deeper. PROPOSITION 3.12. An energy density of the form

is not polyconvex if a < 0 and b, c > 0. Proof. Consider the matrices

It is a matter of careful arithmetic to check that

where M is the vector of all minors for 3 x 3 matrices, in other words, M(F) = (F,adjF,detF). If W(F) were polyconvex, there would exist a convex function g such that

and in this case,

However, if one performs these computations, the following inequality must hold:

3.5


39

for any e > 0. This is clearly impossible, and hence W cannot be polyconvex. The energy density of a St. Venant-Kirchhoff material can be easily rewritten in the form where a < 0 and b,c,d> 0. By the previous proposition such a function cannot be polyconvex. Nonetheless, such energy densities can be approximated near the identity by integrands corresponding to Ogden materials.


Chapter 4

Rank-one Convexity and Microstructure 4.1

Introduction

We describe a class of materials whose stored-energy densities lack the property of quasi convexity. This situation leads us to consider generalized variational principles where Wl'p-Young measures are allowed to enter the minimization problem. These objects are physically interpreted as microstructures and represent highly oscillatory minimizing sequences on smaller and smaller spatial scales. Here we study a particularly important class of such microstructures: the so-called laminates. The title of this chapter refers to the relationship between rank-one convexity and laminates. Although the issue is beyond the scope of the book, it turns out that this kind of microstructure can be characterized in terms of Jensen's inequality for rank-one convex functions and so rank-one convexity is central to this chapter. The reader will not find applications to real materials. Our objective is to stress the ideas in order to understand, from an energetic point of view, the appearance of microstructure and fine-phase mixtures in elastic crystals.

4.2

Elastic Crystals

We will consider a crystal as a countable set of atoms arranged in a periodic fashion. Probably the simplest way of describing this array is to place the origin at one of the atoms and then refer the position of the remaining atoms to the chosen origin using three independent lattice vectors {HI, n2,713}. We let N € M be the matrix with columns n;. We postulate the existence of a nonnegative, free energy $ that depends on the change of shape and on temperature as well. As before, it is a function of each particular periodic array of the atoms of the 41

42

CHAPTER 4 RANK-ONE CONVEXITY AND MICROSTRUCTURE

crystal given by matrix N. We assume that $ is frame indifferent as usual ($(JV) = $(QN) for any rotation Q), but it should also be invariant under any change of lattice basis: if N' is an equivalent choice of lattice basis (equivalent in the sense that the positions of the atoms are the same for TV and for N'), then (./V) — $(JV'). Since N and N' must be related by a matrix of the set

where Z is the set of integers, we can write

Once a basis of lattice vectors has been chosen and the corresponding matrix N is fixed, we define the energy density per unit reference volume by putting

Altogether we have the invariance

where Q is any rotation and H £ NGL(Z>3) N~l, which is a conjugate group of GL(Z3). Furthermore, we also impose the conditions

In practice, however, the invariance above is assumed only when H e P, where P is the point group of the reference crystal lattice consisting of all the matrices H e N GL(Z3) N"1 that are rotations. This is a finite group. For example, if the atoms in the reference configuration aligned themselves on cubic cells, then P would include the 24 rotations that leave a cube invariant. As mentioned, W and $ depend on temperature 6. Above a certain critical temperature 90 there is a stable phase, taken as reference. By "stable" we simply mean that it minimizes W, so that Wg(\) = 0, where 1 is the identity matrix and B > 90. At the transition temperature OQ, there is a change of stability or of crystal structure so that below 6*0 the stable phase is no longer represented by 1 but by some other nonsingular matrix f/o describing the change in crystal structure that has taken place. Thus Wg(U0) = 0 but Wg(l) > 0 for Q < 00. At the transition temperature 9 — OQ both phases may coexist: WgQ(Uo) — Wg0(l) — 0. Because of the invariance that the energy density W — Wg0 must satisfy, we should have at the critical temperature

for any H £ P and any rotation R. We have found many matrices for which the free energy density W vanishes:

4.3

LACK OF QUASI CONVEXITY

43

where P = {l,Hi,Hi,...,Hn} and R is any rotation. We call each one of the sets a potential well associated with U% = HiUQH:[ and make the further assumption that the free energy density W is positive outside the set of the wells: the zero set for W, {W = 0}, is exactly the set of the wells. Under these circumstances, we are looking for minimizers of the energy functional among all deformations u of the reference configuration ft c R3 satisfying appropriate boundary conditions. We are explicitly assuming that W is the same for all points in the reference domain (i.e., no dependence on or).

4.3

Lack of Quasi Convexity

The most striking consequence of the previous description is that the energy density for an elastic crystal cannot be closed Wlip-quasi convex. Recall that W : M —> R* enjoys this property when Jensen's inequality holds for all homogeneous Wl'p-Young measures. There is a necessary condition for a function to be closed W1>p-quasi convex (or quasi convex for that matter) that is also very important for the description of equilibrium configurations of elastic crystals. This condition is called rank-one convexity and it requires the usual convexity condition along directions of rank one: W is said to be rank-one convex if

whenever FI — F% is a matrix of rank one. In order to establish the necessity of this condition we rely on the following convenient reformulation of the quasi convexity condition. LEMMA 4.1. // a function W : M —>• R* is closed W1'1'-quasi convex for some p > 1, then

for all T'-periodic Lipschitz deformations u and every matrix F, where T is any unit cube in R 3 . The converse of this result requires suitable upper bounds on W for finite p or the finiteness of W for p = +oc. Because the format of the statement of the lemma is not the usual one in which this result is presented, we give the proof here. Proof. We may take F = 0 without loss of generality. Given a T-periodic Lipschitz deformation u, consider the sequence

44

CHAPTER 4

RANK-ONE CONVEXITY AND MICROSTRUCTURE

Trivially, Vu}(x) = Vu(jx] for x e T. We would like to identify the Young measure associated with this sequence of gradients. This is in fact a homogenization result in the spirit of Theorem 5.12. Indeed, let £ be a continuous function in T and

where we have used the periodicity of u and certain points yf

and

€ T. If we take limits as j —¥ oo we get

This equality holds because we have obtained a Riemann sum for the integral of £ in T. By the remark after the sketch of the proof of Theorem 2.2 in Chapter 5, we conclude that the Young measure is homogeneous (why?) and given by

Since {uj} is a bounded sequence in Wl'p(T) for allp > 1, the closed Wlip-quasi convexity of W applied to this i/ yields the desired result. The construction that follows is a generalization of example 3 in section 2.3. Let F 6 M, a € R3, and a unit vector n £ R3 be given. Let Xi/z be the characteristic function of the interval (0,1/2) in (0,1), extended by periodicity to all of R, and x — 2^1/2 ~ 1- Consider the vector function

This function is T-periodic if n is one of the three orthogonal axes of T, and therefore is eligible in Lemma 4.1. Let us examine the gradient

4.3

LACK OF QUASI CONVEXITY

45

(recall that the tensor product a ® n is another way of writing the rank-one matrix anT and (r) denotes the integer part of r). If W is quasi convex, by Lemma 4.1 we must have

for any matrix F and vectors a and n. This inequality is the rank-one convexity condition. Rank-one convexity is thus a necessary condition for quasi convexity. We can now prove the main conclusion of this section. PROPOSITION 4.2. Let W : M -> R* be nonnegative and If there exist matrices R, H and non-vanishing vectors a, n as before such that the function W cannot be quasi convex. Proof. The proof reduces to the observation that a nonnegative, convex function of one variable that vanishes at two points must vanish in the interval between them, too. If we apply this argument to the function that vanishes for t = 0 and t = 1 and is convex if W is quasi convex, we conclude that However, this is not possible. Due to the fact that each well is a compact set, and we have a finite number of them, the only possibility is that the whole segment be contained in the same well. In this case, for some rotation Q. This will clearly imply that 1 — Q is a rank-one matrix: This equation forces b to be an eigenvector of Q so that if b is not the zero vector it must be the axis of rotation of Q. Then and if v is not the zero vector, then b • v = 0. In this case we must also have b • Qv = 0, but Therefore either b or v must vanish, but this contradicts our assumption. This fact is a clear indication that there might be real difficulties in establishing the existence of equilibrium configurations for elastic crystals. Even though there are results on existence despite lack of convexity, this lack of quasi convexity makes it impossible to apply the direct method to show existence of equilibrium states in this case and indeed leads us to think about nonexistence in many interesting cases.

46

4.4

CHAPTER 4


Generalized Variational Principles

The lack of quasi convexity shown in the previous section is typically a precursor of the oscillatory behavior of minimizing sequences for nonconvex (nonquasiconvex) integrands. Indeed, fine-phase mixtures provide minimizing sequences whose weak limits are not minimizers. These highly oscillatory minimizing sequences represent the behavior of elastic crystals. The configurations with low energy resemble the construction used in the above analysis of the rank-one convex condition where very fine layers of different phases (different potential wells) alternate. In this sense, we would like to understand the possible behavior of minimizing sequences. The Young measures associated with the gradients of minimizing sequences may serve as a device to account for this behavior. As an illustration of the type of phenomenon we are trying to understand, we can study the following academic, one-dimensional example. Assume we have an elastic bar with unit length and clamped endpoints. If we take fi = (0,1) as the reference configuration, we postulate that the internal energy associated with a particular deformation of the bar u : fi —t R, u(0) = 0, w(l) = !,?/> 0, is given by the integral

where the energy density W corresponding to the material which the bar is made of is, for instance,

where the preferred states are given by a and /3 and 0 < a < 1 < /3. We claim that there is no equilibrium configuration for this bar. Indeed, if we let X be the function defined as a in the interval (0, (/J - !)/(/? - a)) and /? in ((/? - !)/(/? - a), 1), extended by periodicity to all of R, then for

it is elementary to check that /(%) \ 0. Therefore

There is, however, no admissible u such that I(u) — 0. In this case we would have W(u') = 0, which implies u' = a or f3 and at the same time u(x) = x, which is clearly impossible. The derivatives of the minimizing sequence for this functional oscillate more and more between a and /3. We can set up a new generalized variational principle where we let Wl>pYoung measures compete in the energy-minimization process. Let A denote the set of admissible deformations for the old variational principle:

4.5

LAMINATES

47

where u0 e Wl>p(Q) is given with J(u 0 ) < oo. Let ^4 be the set of Young measures generated by the gradients of bounded sequences in A. Define

for v G A. Trivially, the choice vx = <5vu(z) for u e A takes us back to I ( u ) so that inf_4 I < inf^ /. The equality of the two infima holds under upper bounds on W but, as pointed out, this condition violates our basic hypothesis on W, which takes the value +oc when the determinant is nonpositive. Nevertheless, oftentimes we seek stress-free microstructures. By this we refer to minimizing sequences such that I(uj) \ 0. In this case, if v = (vx}x&u ^s tîe Young measure associated with {Vwj}, by Theorem 2.3 we have

Hence, I(v) = 0 and because of the nonnegativity of W this can only happen if supp(i^x) C {W = 0} for a.e. x e fL v is called a stress-free microstructure and in real problems these are the ones in which we are interested. As we have argued, they are the families of probability measures that, satisfying all the restrictions of the problem, have their support contained in the zero set of the energy density. For this reason the structure of that set is very important. We know that for an elastic crystal it is a finite union of wells. In this way, we have reduced the problem of understanding the behavior of the material to finding all gradient Young measures supported in the set of the wells. Given a minimizing sequence such that I(uj] \ 0, there exists an associated gradient Young measure that describes the behavior of {Vi/j}. Conversely, if we find a gradient Young measure supported in the set of wells, the sequence of functions whose gradients generate such a Young measure will be a stress-free minimizing sequence, and hence will describe a possible behavior of the material. What is crucial is the gradient requirement. We can find many families of probability measures supported on the set of wells. But only those that are gradient Young measures are associated with stress-free minimizing sequences and these are the ones relevant to our original variational problem. In other words, only gradient Young measures are physically meaningful for our problem. The others do not have any physical significance.

4.5

Laminates

We would like to focus on generalized minimizers with zero energy. We know that this requirement is equivalent to asking for the support of each individual member to be contained in the zero set of the energy density W. However, the restrictions placed on such generalized minimizers must be met. The most fundamental of all is the fact that they must be generated by sequences of gradients. We cannot understand in full generality Young measures generated

48

CHAPTER 4


by gradients but we can restrict attention to a particular class of such families of probability measures called laminates. Let us briefly recall (in a slightly different form) how the rank-one convexity condition was introduced in section 4.3. Let Fi & M, i = 1,2, a € R3, and a unit vector n 6 R3 be given in such a way that

If Xt is the characteristic function of the interval (0, t) in (0,1) extended by periodicity, the Young measure associated with the sequence of gradients

is

where O is any bounded domain in R 3 . Therefore the probability measure v in (4.2) is a gradient Young measure (more precisely a Wli00-Young measure) for any t e [0,1] provided the compatibility condition (4.1) holds. Throughout the rest of this chapter we will use the term "gradient Young measure" to mean W1'°°-Young measure. By Lemma 5.13, applied to (4.2), we can assume without loss of generality that Uj - UF 6 Wo 1 ' 00 (ft), F = tF\ + (1 - i)F2. Moreover, Vuj takes the values FI and Fy except in small sets E j , Ej —> 0. We would like to go one step further in this construction. Assume, in addition to (4.1), that

where b e R3 and e g R3 is another unit vector. Let ft^ be the part of Q where Vu., — Fi. For j and i fixed, based on the compatibility condition between F2 and F2 > we can construct a sequence of gradients {Vv£1}, v£ — tn up? e Wo'00 (ft?), whose values essentially alternate between iFy , 2 preassigned frequency to £ (0,1) and normal e to the layers. Let E% be the

set where Vv^ does not take either of the two2 values -^2 F- Choose k — k(j, i) in such a way that

as j' -» oo uniformly in i = 1,2. Define

4.5

LAMINATES

49

This sequence {u^)} is uniformly bounded in W li00 (n) and satisfies w (j ' — UF £ W'r01'oc(O). The homogeneous Young measure associated with {Vu^} is

The probability measure in (4.4) is a gradient Young measure provided we have the compatibility conditions (4.1) and (4.3). It is not difficult to generalize this construction when a finite number of matrices is involved if we have the rank-one condition in a recursive way. This basic construction has been referred to in the literature as "layers within layers" and accurately reflects the situation. It motivates the following definition. DEFINITION 4.3. A set of pairs {(ti,Yi)}l 0, £]»*» = !» Yj € M, is said to satisfy the (Hi) condition if (i) for I = 1, rank(Y1 -YZ)<1; (ii) if I > 2 and possibly after a permutation of indices, rank(Yi — Y%) < 1, and if we set

the set of pairs {(si, Zl)}l• oc. In this case we consider infinite-order laminates. DEFINITION 4.5. Let v be a probability measure supported on M and let K = supp(z/) be a compact set. v is a laminate if there exists a sequence of sets of pairs {($ , Y f ) } ^ 2), verifying the (Hk) condition such that Si î &Yk —^ v in the sense of measures. PROPOSITION 4.6. Every laminate is a homogeneous gradient Young measure. It may be extremely hard to decide whether a given probability measure is a laminate even for innocent-looking examples. For instance, consider the matrices

50

CHAPTER 4


We claim that the probability measure

is a laminate despite the fact that none of the pairs Ai — Aj has rank one (see the Bibliographical Comments to find several places in the literature where you can decipher why this probability measure is a laminate). In order to fully understand condition (2.6), we need to analyze the defining properties of Young measures associated with the gradients of bounded sequences in Wl
THEOREM 4.7. Let v — {vx}xe.£i be a family of probability measures supported in the space of matrices M. v is a Wl'p-Young measure if and only if (i) there exists u e W 1>p (fi) such that Vw(x) — fMAdi>x(A) for a.e. x € fi; (ii) JM tp(A)dvx(A) > 9j(Vu(x)) for every ip e £p quasi convex and bounded from below and a.e. x £ O; p ( i ai fM\A\ i dv ) f x(A)dx<™. For the case p = oo, the condition
• R* is rank-one convex and v is a laminate, then

This is easy to derive because of the recursive way in which (Hi) conditions are defined. It turns out that this condition is also sufficient so that laminates are to rank-one convexity what gradient Young measures are to quasi convexity. The proof of Theorem 4.7 or its rank-one convex counterpart cannot be found in Chapter 5. We refer readers to the Bibliographical Comments at the end of the book for further reading on these issues.

4.6

The Two-Well Problem

An interesting way of making precise some of the ideas described in the previous sections, and of demonstrating how complicated some issues might be concerning

4.6

THE TWO-WELL PROBLEM

51

the description of microstructures, is to study the two-well problem, where we assume that the zero set of the energy density W7, {W = 0}, is the union of two disjoint wells. We restrict our attention to dimension 2 so that M denotes throughout this section the set of 2 x 2 matrices. The case of dimension 3 is considerably more complex. The problem we want to address is the search for minimizers of the variational problem

where UQ is some prescribed Lipschitz deformation with finite energy I(UQ) < oo. Our main assumption here is that the free energy density W is nonnegative, W(F) = +00 if det F < 0, and

SO(2) designates the space of rotations in the plane. We are interested in finding stress-free microstructures I ( v ) = 0 and would like to draw some conclusions on the following issues: 1. conditions on the wells to ensure the existence of nontrivial, stress-free microstructures; 2. affine boundary values UF(X) = Fx that may support such microstructures; 3. examples of stress-free microstructures. Before analyzing these topics, and as a preparation for the discussion that will follow, we first focus on several elementary facts. The convex hull of the set SO(2) consists of all matrices P of the form

Furthermore, if /j, is a probability measure on SO(2),

is in the convex hull of SO(2). If det P = 1, then P e SO(2) and /j, = SP. The main tool in deriving necessary conditions in this context is the minor relation

which should be valid whenever v is a gradient Young measure. This equality is also true for i'p(A) = det(A — F) for a fixed matrix F because t/j is also a weak continuous function. This fact can also be proved by using the formula that follows, which will play a role in some proofs. It is only valid for 2 x 2 matrices:

Note that (adj A)T • B is a linear function on the entries of A.

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CHAPTER 4


We would like to consider whether it is possible to have nontrivial gradient Young measures supported in a single well SO(1)F. This is impossible because if v is a gradient Young measure supported in SO(2)F, we must have

The right-hand side is detF and the left-hand side can be written as det(PF), where P G co(SO(2)). Hence, detP — I and by the observation above this implies that v has to be trivial, i.e., a delta measure. A second step is to consider gradient Young measures supported in just two matrices, F\ and F%, rather than two wells. In this case any probability measure can be decomposed as Again by the weak continuity of the determinant, The left-hand side can be factored out as This formula is also valid only for 2 x 2 matrices. It clearly implies that det(Pi — F2) — 0 and thus FI — F2 must be a rank-one matrix. Otherwise, v must be a Dirac mass. We now get into the two-well problem fully and treat the three issues indicated above sucessively. Our main result concerning restrictions on the set of two wells is the following. We say that the wells 5O(2)F: and SO(2)F2 are incompatible if FI - QF2 is never a rank-one matrix for all rotations Q. THEOREM 4.8. Let v be a homogeneous gradient Young measure with If the wells SO(2)Fi and SO(2)F2 are not compatible, v = &QHi is a Dirac mass. A crucial technical fact in the proof is the next lemma. LEMMA 4.9. Let A be a matrix such that det(A — Q) > 0 for all rotations Q e 50(2). Then det(A - P) > 0 for every P € co(5O(2)). Proof. Write

After some algebra,

4.6


53

If det(A - Q) > 0 for all (a, 0) in the unit circle, this means that the unit circle does not meet the circle centered at

with radius

By continuity, this last circle does not meet the solid unit circle either. This is the conclusion of the lemma Proof of Theorem 4.8. Set

Consider the weak continuous function On the one hand, by direct substitution. On the other hand, and due to the weak continuity of det and by (4.7),

As an intermediate step, we have used the fact

By putting together these two ways of computing ift(F), we obtain the equality

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CHAPTER 4


or

Assume that A 6 (0,1) and the wells are incompatible, so that det(RFi — QF2) > 0 for all rotations Q and R. Multiplying by F^1 to the right and letting A — RFiF^1 we have det(A - Q) > 0 for all rotations Q. By Lemma 4.9, det(^4 — P) > 0 for all P in the convex hull. This is equivalent to having det(RFl - PF2) > 0 for all such P. In particular, det(fif\ - P2F2) > 0 for all rotations R. Therefore

If we consider again the decomposition

we observe that the first term on the right-hand side is positive by (4.11), and the second one is nonnegative by (4.9). Hence tl>(F) > 0, and by (4.8), det(Pi.Fi — P2F2) > 0. This is a clear contradiction to (4.10) because the sum of three nonnegative terms vanishes only if each one vanishes individually. The conclusion is that if the wells are incompatible, then either A = 0 or A = 1 and in this case the probability measure is trivial (the case of one well). We would like to characterize the affine boundary conditions Uo(x) = Fx, F e M, that may support nontrivial, stress-free microstructures. We assume accordingly that the two wells are compatible. After an appropriate change of coordinates we can take

where 6 > 0 is a fixed parameter and a is the canonical basis for R2. If v is a homogeneous gradient Young measure, we write

and hence

where P; e co(SO(2)) and

4.6


55

We have kept the notation -F\ = -Fo, -Fj = F^1 for convenience. Placing these expressions into F,

and for C — FTF, the Cauchy-Green tensor, write

We have the inequalities

On the other hand, by the weak continuity of det,

so that and consequently In the Cn-c22 plane we have found the constraints

These determine a region D that is easy to draw. Does every point in D come from the Cauchy-Green tensor corresponding to a gradient Young measure v supported in Kl In order to answer this question, we need to review briefly how laminates supported in four matrices can be easily constructed. Given four matrices, A, B, (7, D, a set of compatibility conditions that allows us to build a laminate consists of

for some X, a G (0,1). In this case, any convex combination of X$A + (1 — A)£B and crSc + (1 — cr)So will be a gradient Young measure (a laminate), using again the idea of layers within layers to find the corresponding sequence of gradients.

56

CHAPTER 4


Let v be the laminate supported in K:

(FQ and Fg"1 are rank-one related). For this v,

and the corresponding Cauchy-Green tensor

As A moves from 0 to 1, c22 = 1 + 52(1 - 2A) 2 decreases from 1 + S2 to 1 and then increases back to 1 + 52, while en stays constant at 1. There is another matrix Qg € 5O(2) with the property that Q$F0 is rank-one related to F^1. Namely, after some computations,

The matrix QgF0 is called the reciprocal twin of F^1. Thus we may consider the laminate and find

In this case one obtains

so as A runs through [0,1], 032 is fixed at 1 + 52 but en goes from 1 to 1/(1 + ^ 2 ) and back to 1. The same computations show that for a given A S [0,1] and Q(\) — Qs(i-"2,\)^ given by (4.12) with 6(1 — 2A) replacing S, the matrix

is the reciprocal twin of

because (1 — A)_Fo + AFg"1 is a matrix of the same type as FQ. For

4.6


57

we eventually reach every point in D as (cr, A) e [0,1] x [0,1], for A lets us move up and down and a from left to right. This F corresponds to the measure

where

This probability measure is a laminate because the rotation Q(\) was so determined. The next step is to study, for each possible F whose Cauchy-Green tensor lies in D, the set of gradient Young measures supported in the two wells with such an underlying deformation or at least to say something about the structure or the complexity of that set. As we will see shortly, this is a much harder problem that cannot be solved completely except for some special matrices. Suppose that v = {VX}X€Q is a nonhomogeneous gradient Young measure supported in K where we take again FI = FQ, F% = F^1, vx = (1 — A(or)) v]. + \(x)v^,. Denote by y(x] the deformation underlying is, that is,

where

belong to the convex hull of SO(2). We have the following uniqueness result. THEOREM 4.10. Suppose thaty(x] satisfies for some 8, 0 < 9 < 1. Then Proof. Assuming that |fi| — 1, by the divergence theorem,

58

CHAPTER 4


where A is the average of A over fJ. Now

are probability measures, and hence reduce to Dirac masses if the Mt are rotations. Furthermore, the M, are averages of rotations, and hence lie in the convex hull of 5O(2). We now have the equation Multiplying to the right by where H - (F^1)2 = I + tei e 2 , t = -26. Suppose that

Then Ictil < 1 and imply QJ = 1. Moreover, forces j3z — 0. Finally, can only happen if j3\ = 0, and likewise implies 9 = A. Consequently, the matrices Mi = 1, v\. — Si, i = 1,2, and We now need to show that \(x) is actually a constant function. First, using the mixed second partial derivatives in (4.14) with P, = 1, we conclude that A(or) is a function of x<2 alone. Then Applying the boundary condition, we see that f(xz) = 9tx2, and going back to (4.14), we obtain A(x) = 0 This uniqueness result is very special. Indeed, for most of the matrices that may support nontrivial microstructures such uniqueness fails drastically: there even exist continuously distributed gradient Young measures supported in the two wells. This is the aim of the final section of this chapter.

4.7 CONTINUOUSLY DISTRIBUT

4.7

59

Continuously Distributed Laminates

We provide here some insight into how complicated a laminate can be. We will show that for a given affine deformation F supporting nontrivial stress-free microstructures there is a whole continuum of laminates. As a consequence, by taking convex combinations of these we can build microstructures in which a continuum of matrices participates. These are continuously distributed laminates. We stick to the notation of the previous section. In particular, we designate by Qi, i = 1,2, the rotations for which

One of the Qj's is precisely the identity matrix 1 and the other one yields the reciprocal twin. Let the underlying affine deformation F be given that supports nontrivial, stress-free microstructures such that

The additional constraint (a2 + c 2 )(6 2 + d2) > I is a direct consequence of the fact that detF = 1. THEOREM 4.11. Let A e K. A sufficient condition for the existence of a stress-free laminate with first moment F whose support contains A and is contained in a set of at most four matrices from K is the following, according to whether

Proof. We will concentrate on the first case. The second one is similar. Let us set

By (4.7), we get

which says that F is rank-one related to the matrix

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CHAPTER 4


If we put

consider the half-line F + sBi, s > 0. It is easy to check that for small values of s such matrices may support nontrivial laminates (i.e., verify the three requirements on the Cauchy-Green tensor of the last section) and moreover, if so > 0 is the infimum of the values of s such that F + sBi does not admit stress-free laminates, then either the square of its first column is 1 or the square of the norm of its second column is 1 + 62. According to our discussion in the preceding section, F + s$Bi must be a convex combination of two rank-one related matrices, one on each well,

for some rotation Q, ii £ [0,1], j = 1 or 2. By construction, it must be clear that for some A € [0,1],

is a laminate with barycenter F and whose support contains A — RF0 and is contained in K. We would like to solve the inequalities expressed in the statement of the previous result and see how often those conditions can be met. For the sake of simplicity in the computations, we will take 5 = 1. In this case,

and we set

with

Let us first take Q, — 1. If we solve explicitly for t in

we obtain The condition t G [0,1] translates, after some manipulation, into either

or

4.7

CONTINUOUSLY DISTRIBUTED LAMINATES

61

The two lines

intersect at the unique point

Under the condition a2 + c2 < 1, the point P lies inside the unit circle, and therefore the above two families of inequalities will hold when R is a rotation of angle B with 0i < 9 < <92 or 03 < 0 < 04, where 6»i < 6>2 < 6»3 < 6>4. If

the computations are similar. The value of t is given by

Again the restriction t £ [0,1] holds true if

or

If we consider the line

then r\ and r3 meet in the point

This point lies inside the unit circle provided that b2 + d2 < 2. Notice that r3 is orthogonal to rz- This final situation is similar to the one previously discussed but for different values of the angles 9r. The important conclusion is, anyway, that we have a whole continuum of laminates supported in K for the same underlying deformation F. By combining more and more laminates of these families through convex combinations, we can eventually have continuously distributed laminates supported on the set of the two wells. This makes it clear that there exist very complex stress-free microstruetures.


Chapter 5

Technical Remarks 5.1

Introduction

We gather in this chapter several remarks on the proofs of the main results and facts about Young measures used in the previous chapters. We have tried to keep these remarks to a minimum, including only the necessary results and facts to facilitate understanding of the structure of the complete proofs. Prom this perspective, this chapter should be referred to when the reader might be interested in a particular technical proof. The complete proofs themselves have not been included here because they can all be found in [176]. The statements of the main results to be discussed are taken from this reference. We have, however, given some hints on some of the nontechnical parts of some proofs and even some selected proofs, so as to avoid this chapter becoming a mere list of results. Nevertheless, we warn the reader that it is not intended to be read in the same spirit as the preceding chapters.

5.2

The Existence Theorem for Young Measures

The starting point for the use of Young measures in variational principles is the following existence theorem. THEOREM 2.2. Let fi C R w be a measurable set and let Zj : £1 -> Rm be measurable functions such that

where g : [0, 00} —» [0, oo] is a continuous, nondecreasing function such that limtôo g(t) = oo. There exist a subsequence, not relabeled, and a family of probability measures v = {vx}X£fi (the associated Young measure) depending measurably on x with the property that whenever the sequence {i/j(x,Zj(x})} is 63

64

CHAPTER 5

TECHNICAL REMARKS

weakly convergent in Ll(fl) for any Caratheodory function ifr(x, A) : Q x Rm —>• R*, the weak limit is the function

It is therefore important to understand the weak convergence in i 1 (fJ) of uniformly bounded sequences in the i1-norm because it is this condition that enables us to represent weak limits through Young measures. Remember that a sequence of ^-functions {fj} is said to be equi-integrable if, for given e > 0, one can find 5 > 0 (depending only on e) such that

for all j, if \E\ < 6. The following version of this property turns out to be very useful. LEMMA 5.1. Let {fj} be a bounded sequence in Ll(Q),

The sequence is weakly relatively compact in i1(r2) if and only if

The limit (5.2) prevents the existence of concentration effects. For the proof of the existence theorem of Young measures, we need a few basic notions of Lp-spaces when the target space for functions is some general Banach space X with dual X'. For £7 c RN we write f is strongly measurable and

Such a function / is said to be strongly measurable if there exists a sequence of simple (i.e., taking a finite number of values) measurable functions {fj} such that fj(x) —>• f ( x ) a.e. x e 0 and

We write Lpw($l;X} = I f : fi -> X : f is weakly measurable, ||/(o;)||x is a measurable function of x, and

5.2

THE EXISTENCE THEOREM FOR YOUNG MEASURES

65

A function / is weakly measurable if for every T € X' the function of a;, x i—> (/(x),T), is measurable. In the same way,

measurable function of x, and Lp(tt;X), L£, (fl;X), and L ^ , t ( f l ; X ' ) are Banach spaces under the Lp-norm. THEOREM 5.2. Let X be a separable Banach space with dual X'. Then

under the duality

where / 6 U>(Sl; X) and g e Ll*(Sl;X'). The particular case we are interested in is

In this case we have the duality

Sketch of the proof of Theorem 2.2. The proof proceeds in several steps. The first step consists of showing the existence of the Young measure. This is the nontechnical part of the proof, because it relates to where the Young measure comes from. The vector space

is a Banach space under the supremum norm. Its dual space is the space of Radon measures supported in R m , denoted M(Hm), with the dual norm of the bounded variation. Since C 0 (R m ) is separable, we have, according to the above discussion, that under the duality

for t/j e L 1 (Q;C 0(Rm)) and p. £ L~ (ft; M(Rm)). The norm in L%f(fl;M(Rm)) is

66

CHAPTER 5 TECHNICAL REMARKS

For each j, we define Vj e L^x(£l\M(Rm)) through the identification Vj = &zj(x)-> where Sa is the usual Dirac mass centered at a € R m . For ip € Ll(Cl;C0(R.m)),

It is easy to check that By the Banach-Alaoglu-Bourbaki theorem there exist some subsequence, not relabeled, and v e L^(tt; M(Rm)) such that Vj -^ v:

for every $ € L 1 (fi;C 0 (R m )). The rest of the proof is an extension of (5.3) for an arbitrary Caratheodory function ijj such that {^(x, Zj(x})} converges weakly in L1^). It is of a highly technical nature. An important remark to bear in mind when working with Young measures is that in order to identify the one associated with a particular sequence of functions {zj} (obtained perhaps in some constructive way or using some scheme), it is enough to check for every

It is even enough to have

for £ and (p belonging to dense, countable subsets of Ll(fl) and C 0 (R TO ), respectively. If this is so for a given family of probability measures v = {vx}x&n and a sequence of functions {zj} satisfying (5.1), then v must be the Young measure associated with {zj} and therefore

for every Caratheodory function i/> such that {i{>(x, Zj(x))} is weakly convergent in L 1 (ri). The reason for this is that probability measures v are identified by their action on Co(R m ). Equation (5.4) identifies each vx for a.e. x 6 Q. There are two interesting situations for which this remark can have some relevance. For reference, we include them in the following lemma. LEMMA 5.3. Assume that we have two sequences, {zj} and {wj}, both bounded in Lp(fl).

5.3

BITING. WEAK. AND STRONG CONVERGENCE

67

(i) If \{zj ^ Wj}\ —> 0, the Young measure for both sequences is the same. (ii) // {zj} and {wj} share the Young measure. Sketch of the proof. Let 99 e C 0 (R m ) and £, e L l ( S l ) . Then

The integrand on the right-hand side is an L 1 (Q)-function and it is integrated over a sequence of sets of vanishing measure. Hence the limit vanishes as j ->• oc, and this in turn implies that the weak limits for {
f o r ^ e l , 1 ^ ) and (,2 e C 0 (R m ). A helpful example of this situation is the following. Assume {zj} is uniformly bounded in Lp($l) and let v — {vx}x€fl be its associated Young measure. Consider the truncation operators

We claim that for any subsequence k(j) —> oc as j —>• oc the Young measure corresponding to { T k ^ ) ( z j ) } is also v. To this end, we simply notice that

if

5.3

Biting, Weak, and Strong Convergence

Whenever a bounded sequence in L 1 (fi) is not equi-integrable, one can "bite" (remove) the set where concentrations occur and be left with a well-behaved sequence. This is essentially Chacon's biting lemma. The proof can be made in a very general and abstract setting. We restrict our attention, however, to the framework in which we will be using this fact. THEOREM 5.4. Let {/j} be a uniformly bounded sequence in L l ( f l ) , where

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TECHNICAL REMARKS

There exist a subsequence, not relabeled, a nonincreasing sequence of measurable sets fin C fi, | fin \ 0, and f e Ll(£l) such that

for all n. The exceptional sets fira may be taken to be

where the subsequence {j'TO} is appropriately chosen (again we have to discard the places where concentrations may occur). Lemma 5.1 is used to show the weak convergence outside the exceptional sets. In order to formalize this important fact, we say that the sequence {fj} C i 1 (fi) converges in the biting sense to / e i 1 (fi) and is denoted

if there exists a nonincreasing sequence of measurable sets {fin} such that |fi n | \ 0 a n d for all n. We may restate Chacon's biting lemma by saying that a uniformly bounded sequence in L 1 (fi) contains a subsequence converging in the biting sense to a function in L 1 (J7). The relationship between biting convergence and Young measures is given in the following theorem. THEOREM 5.5. Let {zj} be a sequence of functions with associated Young measure v = {v?\x^. If (p '. SI x. Rm —>• R* is a Caratheodory function such that the sequence {
Sketch of the proof. By Chacon's biting lemma, there exists a collection of subsets {fln} and (p e Ll(fl) such that |fin \ 0 and

for all n. By Theorem 2.2, whenever weak convergence in Ll(E) holds for any subset E C fi, the weak limit has to be !p in (5.5). Since |fin| \ 0, we conclude that <£ = ip a.e. x e fi. In some circumstances, biting convergence may be improved to weak convergence so that Young measures will provide weak limits. This amounts to discarding the possibility of concentrations. The following lemma gives a necessary and sufficient condition for such an improvement.

5.3

BITING, WEAK, AND STRONG CONVERGENCE

69

LEMMA 5.6. Let fj : Q —> R+ (fj > Oj be a sequence of measurable functions in Ll(Q,), converging in the biting sense to f 6 Ll(Q}. A subsequence converges weakly in Ll(Q) if and only if

Moreover, the whole sequence {fj} if

converges weakly in Ll(£l) to f if and only

The proof is not difficult. Assuming biting convergence and the failure of the Dunford-Pettis criterion leads to the failure of (5.6). The following corollary is a straightforward fact. COROLLARY 5.7. Let {zj} be a sequence of functions with associated Young measure v = {^x}x^^- If, for po, a nonnegative Caratheodory function, we have

then

for any measurable subset E C £1 and for any tp in the space

If, in spite of all efforts, Corollary 5.7 cannot be applied to a particular situation so that concentrations may arise, we can still deduce some information that might be helpful in some circumstances. This is Theorem 2.3. Notice the direction in which the inequality holds and how it is exactly what we need for weak lower semicontinuity. THEOREM 2.3. If{zj} is a sequence of measurable functions with associated Young measure v = {fz}xeQ, then

for every Caratheodory function ifr, bounded from below, and every measurable subset E C fi. Sketch of the proof. If the left-hand side of (5.7) is infinite, there is nothing to be proved. If it is finite, the sequence {i/)(x, Zj(x}}} is a bounded sequence in Ll(E). If we set

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TECHNICAL REMARKS

then by Theorem 5.5

By Lemma 5.6, it is not possible to have the strict inequality

and this proves the statement. Strict inequality in (5.7) occurs when the sequence {ip(x,Zj(x))} develops concentrations. In this sense we say that Young measures do not capture concentration effects. Finally, we would like to understand how strong convergence gets translated into the Young measure. Since Young measures are used to keep track of oscillations and strong convergence rules out this phenomenon, one can expect that Young measures associated with strong convergent sequences are trivial. We restrict our attention to the case in which g(i) = tp. LEMMA 5.8. Let {zj} be a sequence in Lp(Sl] such that {\ZJ\P} is weakly convergent in L 1 (O) for p < oo and v — {^x}zen is the associated Young measure. Then Zj —> z strongly in Lp(£i] if and only if vx — &z(x) f°r a-ex e ft. It is also helpful to consider Young measures coming from sequences for which we have strong convergence only for some components of the sequence. In this case strong convergence reflects the triviality of the Young measure for the corresponding components. LEMMA 2.6. Let Zj — (uj,Vj) : O —>• Rd x Rm be a bounded sequence in P L (Q) such that {uj} converges strongly to u in Lp($l). Ijv — {vx}x
5.4

Homogenization and Localization

Homogenization and localization are two basic operations used to analyze Young measures. We first define in general terms homogenization and localization in the next two lemmas and then apply them to the situation of gradients. LEMMA 5.9. Let 0 and D be two regular, bounded domains in TtN with 1901 = \dD — 0. Let {zj} be a sequence of measurable functions over O such that for g a continuous, nondecreasing, nonnegative function with limôo g(t) = oo. Let v = {vx}X£n be the Young measure associated with some subsequence, still denoted {zj}. There exists a sequence {wj} of measurable functions defined over D such that

5.4

HOMOGENIZATION AND LOCALIZATION

71

and its homogeneous Young measure v is given by

The main technique for the proof of this lemma is the Vitali covering lemma (Theorem 5.10 below). It requires the notion of a Vitali covering of a set. For a given point x & R m , a sequence of sets {Ei} shrinks suitably to x if there is a > 0 such that each Ei c 5(x, r^), where B(x,Ti} is a ball centered at x and radius TI > 0 and where r^ —»• 0 as i —> oo. A family of open subsets {ÂJAÂ ^s called a Vitali covering of f2 C Rm if for every x 6 fJ there exists a sequence {î} of subsets of the given family that shrinks suitably to x. THEOREM 5.10. Let A = {A\}XEA be a Vitali covering of Ft. There is a sequence \i e A such that

and the subsets A\i are pairwise disjoint. Sketch of the proof of Lemma 5.9. The sequence {wj} is defined in terms of {zj} as follows:

is obtained by means of Theorem 5.10. It is not hard to see that the sequence {wj} (or a suitable subsequence) gives rise to the homogeneous Young measure. The rest of the proof is essentially technical. LEMMA 5.11. Let $1 and D be defined as in Lemma 5.9. Let {zj} be such that

where, as usual, g is a continuous, nonnegative, nondecreasing function with limt-j.oo g(t) = oo. Let v = \yx}x^$i be its Young measure. For a.e. a G £1 there exists a sequence {z^ defined on D such that

and its homogeneous Young measure is va.

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TECHNICAL REMARKS

Sketch of the proof. The sequence generating the localized Young measure is obtained through a typical blow-up argument around each point a £ fi:

{z"} is then defined as a suitable subsequence of the above functions. Technicalities involve several tools from measure theory. The case to which we would like to apply Lemmas 5.9 and 5.11 is Zj•, = Vuj, where {uj} is a bounded sequence in W 1)P (J7). Going back to Lemma 5.9, we realize that all we need is a sequence of functions Wj e W l i p (ft) such that

for x € dij + 6jjJ7, where once again by Vitali's covering lemma

Condition (5.8) can be fulfilled by simply putting

for x G dij + €{j£l. It may not be true, however, that such a Wj belongs to W 1>p (fi). Indeed, we know that Wj should be much more regular than simply belonging to Lp(£l). For p sufficiently large, Wj even ought to be continuous, and this might not be the case if we are not more careful about our definition of Wj. The only known way of ensuring the continuity property for Wj is to enforce affine boundary values for Uj. Let Y £ M mxAr and let UY be the affine function UY(X) = Yx. If Uj P(O), the function

otherwise is well defined as a function in W1)P(Q) since it is continuous (easy to check), Wj — My e WQ >p (fi), and its gradient satisfies (5.8). Hence we have the following theorem. THEOREM 5.12. Let {uj} be a bounded sequence of functions in W1)P(O) with affine boundary values given by UY- Let v — \yx\xî be the Young measure associated with {Vuj}. There exists a sequence {wj}, bounded in W1>P(J1) with the same boundary values, such that the homogeneous Young measure 17 associated with {Vwj} is given by

Lemma 2.7 is a direct corollary of this theorem.

5.4

HOMOGENIZATION AND LOCALIZATION

73

For the localization property, we should have

The sequence {Vu£p} is uniformly bounded in LP(J7) in j and p for a.e. a e O, and if z/ = {^j^g^ is the Young measure associated with {Vwj}, then {Vw°>p} generates the homogeneous va for an appropriate subsequence. In order to have a bounded sequence of W 1>p (fi)-functions {UJP} as p —> 0, define

the linear function ua(x) = F(a)x for a e O, and its average over f2,

Take

where the constant M0(^p is chosen so that

By Poincare's inequality, the sequence [u'j p} is truly bounded in W 1>P (J7) independently of j and p and for almost every a 6 J7. We would also like to incorporate the affine boundary values for the new sequence defining the localized, homogeneous Young measure at almost every point in Ct. In order to do this we have the following lemma. LEMMA 5.13. Let {vj} be a bounded sequence in W 1>p (fi) such that the sequence {Vvj} generates the Young measure v = {^x}x&^- Let

so that Vj —^ u in W 1>p (f2). There exists a new sequence {uk}, bounded in W lip (Q), such that {Vwfc} generates the same Young measure v and u^ — u G W 0 ' P (O) for allk. If for p < oo {|V^j|p} is equi-integrable, then so is {|Vwfc| P }The proof of this lemma makes use of a standard sequence of cut-off functions in order to enforce the appropriate boundary values. If we apply this lemma to the subsequence of {W")P} given in (5.9), we obtain our version of the localization property for gradients. PROPOSITION 2.4. Let v = {vx}x&^ be a Wl'p(0,}-Young measure. For a.e. a 6 il and for any domain Q, there exists a bounded sequence in Wl>p(Q), {va,j}, such that the homogeneous Young measure associated with {Vvaj} is va.

74


Moreover, each function vaj can be chosen such that where up (x) is the linear function Fx and

Notice that in all these facts about homogenization and localization, Lemma 5.3 is used to identify the Young measure determined by the particular sequence of functions tailored for a specific purpose.

5.5

A Remarkable Lemma

This section contains the proof of Lemma 3.7. We have decided to include this proof so that the reader may get a feeling for the type of technicalities involved, and because it is a good example of Young measure techniques. We remind the reader of some facts about maximal operators. For any v G (^(R^) we set

where

is the maximal function of /. It is well known that if v 6 (^(R^), -M*^ € C(RN) and

and, in particular, for any A > 0,

This last inequality is also valid for p — I even though the previous one is not. LEMMA 5.14. Let v e C^(RN) and A > 0. Set Hx = [M*v < A}. Then

where C(N) depends only on N. It is also interesting to remember that any Lipschitz function denned on a subset of R^ may be extended to all of R^ without increasing its Lipschitz constant. LEMMA 3.7. Let {vj} be a bounded sequence in Wl>p(£l), p > I. There always exists another sequence {uj} of Lipschitz functions (uj e W 1>00 (Q) for all j) such that {\Vuj p} is equi-integrable and the two sequences of gradients, {Vuj} and {Vvj}, have the same underlying Wl'p(£l)-Young measure.

5.5

A REMARKABLE LEMMA

75

Proof.

Step 1. Assume, in addition to all the hypotheses of the statement, that Vj € (^(R^) and replace 0 by R^. Consider the sequence { M * ( v j ) } , where M* is the maximal operator of a function and its gradient. This sequence is bounded in LP(HN). Let \i = {A*a;}x6R>jv be its corresponding Young measure (possibly for an appropriate subsequence). Consider the truncation operators Tfe defined by

Since for fixed fe, {TkM*(vj}} is bounded in L°°(RN),

We have used the monotone convergence theorem for the second limit. Notice that is an L 1 (R Ar )-function. We can find a subsequence k(j) —> oo as j —> oo such that On the other hand, by the observation made about these truncation operators after Lemma 5.3, the Young measure associated with the sequence {Tk(j)M*(Vvj}} is also IJL. By Corollary 5.7, we conclude that

Let

\Aj\ -)• 0 because {M*(vj}} is bounded in LP(RN) and k(j) -)• oo. By Lemma 5.14, there exist Lipschitz functions Uj such that Uj = Vj (and therefore Vwj = Vfj) outside Aj and, moreover,

The fact that \Aj\ —>• 0 implies that the Young measure for both sequences is the same (Lemma 5.3). It follows easily (because M*(VJ) > \Vvj\} that

Since the right-hand side is equi-integrable in I/ 1 (R JV ), the conclusion of the lemma follows.

76


Step 2 (Approximation). We can assume that Vj —^ u in W lip (J7) for some u G Wl'p(£l). Moreover, by Lemma 5.13, we can assume that v3•, - u G W01>P(Q). Let Wj = Vj — u be extended by 0 to all of R^. By density, we can find Zj € Cg°(RN) such that

Apply Step 1 to {zj} and find a sequence of Lipschitz functions {%} such that {|V%|P} is equi-integrable in Ll(RN) and {V^-^Vuj}| -»• 0. Therefore, again by Lemma 5.3, the Young measures for the sequences (now considered restricted to Q) {Vuj}, {Vzj}, and {Vwj} are the same. Take Uj = uj n + u. The sequence {wj} verifies the conclusion of the theorem.

Bibliographical Comments Our comments on the bibliography are intended to suggest further reading on the topics covered in this book and new directions which follow after going through the material of this book. The comments also serve to complete our exposition or to fill gaps which have purposely been left for the sake of brevity, or because according to our judgment the material does not focus on the main objectives of this book. In this way we seek to point out new directions where more work is needed, where open problems are still waiting to be understood, or where new situations need to be clearly stated. The subject is far from a rounded, closed, well-established, well-understood field, and there is room for much improvement. The final section of this chapter is devoted to the review of possible research directions that can be pursued by interested readers. At the same time, we caution the reader that our references may be incomplete in some regards. We do not pretend to have exhausted all related works, and many people could probably enrich our comments and list of references. We have included only the items most closely connected to the topics covered in the preceding chapters. In this sense, the same principle of brevity has led us in the selection of references, which we have tried to keep specific to the topics of the book. We divide our remarks on the bibliography according to the previous five chapters of the book.

Chapter 1 The expository nature of the first chapter forces us to include several references in which the reader may find proofs, further reading, more discussions, etc., about the foundations of continuum mechanics and elasticity. Since this is not the aim of this work, our list of references for this chapter is rather short. The reader interested in these topics will have to perform a more in-depth analysis of the literature. Some elementary texts which could help someone not yet exposed to continuum mechanics or whose main interest is the underlying mathematical problems but who would like to better understand the general theory of continua are [78] and [114]. References [9], [135], and [145] treat the theory and the mathematical problems in elasticity more closely and hence are more advanced. The inter77

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BIBLIOGRAPHICAL COMMENTS

play between mechanics and thermodynamics can be explored in [192], where some material related to our main interests in variational problems can also be found. Reference [52] is also a good book that shares some of our goals but offers information on many more topics.

Chapter 2 Variational problems have always been an important part of applied mathematics. Reference [209] is a nice way to be exposed to such problems for the first time. Some interesting, more classical approaches are found in [80] and [183], while for more complete treatments based on the direct method and weak lower semicontinuity where vector problems are considered, the reader may look at [41], [65], and [67]. Reference [110] is encyclopedic in character but deals with the indirect method through optimality conditions associated with minimizers. A full discussion of frame indifference, behavior for extreme deformations, and appropriate physical requirements is contained in [52]. Young measures (also called parametrized measures) were introduced in analysis many years ago in [211], [212], and [213]. Since then, many applications in applied mathematics and analysis have been explored. More recent books such as [13], [176], and [210] treat them from a more modern perspective. See also [36] and [188]. The use of Young measures in examining weak lower semicontinuity has systematically been considered in [28], [88], and [171]. See also [129] and [201]. The notion of quasi convexity and its pivotal role in weak lower semicontinuity for vector problems was identified in [150]. See [151] for a full analysis of this issue. Recently, the quasi convexity condition has been examined in more general settings in [4], [25], and [140]. This condition was introduced in terms of gradient Young measures in [171]. General existence results similar to the one included here can be found in many places, in particular in almost all the above references dealing with vector problems. See also [133], [143], and [144]. Reference [112] contains a different approach to existence theorems in nonlinear elasticity. If the reader would like to study more deeply the facts contained in the appendix, some general references have been included to cover them: [6], [38], [77], [113], [115], and [189].

Chapter 3 The fundamental ideas in this chapter were fully explored for the first time in [14]. The key fact for the weak continuity of determinants was also noted and proved in [181] and [182]. Since then, polyconvexity has been reexamined from the point of view of weak continuity in different settings and under various sets of assumptions in an attempt to improve previous and known results. In particular, the effort to push weak continuity of determinants to the limit, and consequently


79

obtain new weak lower semicontinuity results and existence theorems, is taken as the main motivation for a series of papers: [3], [45], [71], [72], [107], [138], [139], [152], [153], [154], and [214]. Weak continuous functionals of higher order are analyzed in [19]. The existence theorem presented in section 3.4 is classical, except perhaps for the case p = 3, which is based on a quite remarkable lemma (see [176], [214]). Concerning the issue of discontinuous deformations and cavitation, see some comments in the final section of this chapter. The injectivity and interpenetration of matter is a complex issue. Some references in this direction are [15], [53], [98], and [120]. The topics covered in this chapter are essentially contained in [52] (in particular, the treatment in section 3.5) where many other references can be found. Some of those topics can also be studied in [192].

Chapter 4 Nonconvexity has played an important role in leading important research efforts in many distinct areas in the past 15 years. The incredible number of works dealing with this topic almost certainly makes our list appear incomplete in this regard. In the context of phase transitions in crystals, nonconvexity was raised in a number of important papers. We would like to mention some of them: [20], [21], [50], [81], and [84]. The interest in such variational problems motivated a number of important mathematical developments: [94], [117], [118], [123], and [155]. In particular, the analysis of Young measures generated by gradients was recognized as central to the understanding of microstructure in [124], [125], [129], [147], [148], and [186]. See [131] for nonconvexity in a different context. New and fundamental applications to materials science have been made in [1], [30], [31], [32], and many others. Some texts dealing with all these ideas and the description and interaction of energy minimization related to materials models are [22], [157], and [179]. Some papers dealing specifically with laminates and the two-well problem are [147], [148], [160], [169], [170], [200], and [216]. The notion of the (Hn) condition was first recognized and studied in [66]. The remarkable example of the laminate supported on four matrices not pairwise rank-one related, as well as similar examples, can be explored in [32], [149], [206], and in other references not listed here.

Chapter 5 Our main source for Chapter 5 is [176]. Further reading related to the technical facts involved is suggested in this book. Some relevant references we would like to mention here are [17], [26], and [40]. The technical tools from measure theory used in this chapter can be studied in [190]. Reference [77] contains material about Lp-spaces when the target space is another Banach space. The main

80


results concerning maximal operators that have been invoked can be discovered in [194].

Additional Areas of Research It is probably worthwhile to point out several areas that can be pursued in more depth once the main ideas we have tried to convey in this book are understood. In most cases, challenging and fascinating problems are waiting to be answered. We have selected a number of key references for each area, again without attempting to be exhaustive either in the list of different areas or in the references within each area. A survey with very interesting comments on some of these areas can be found in [156]. 1. Characterization of Young measures: [99], [124], [125], [129], [158], [172], [186], [201]. 2. Polyconvexity, quasi convexity and rank-one convexity: [7], [8], [10], [18], [24], [27], [44], [68], [69], [105], [116], [130], [168], [174], [177], [178], [191], [196], [197], [198], [199], [208], [215]. 3. Relaxation and convex envelopes: [2], [12], [37], [42], [64], [66], [92], [93], [95], [100], [102], [106], [126], [142]. 4. Regularity: [5], [89], [90], [91], [111], [141]. 5. Materials science and phase transitions: [34], [35], [46], [73], [74], [76], [82], [83], [84], [85], [86], [87], [119]. 6. Existence without convexity: [11], [70], [79], [109], [146], [180], [187], [195]. 7. Compensated compactness: [54], [55], [108], [162], [163], [164], [184], [202], [203], [204], [205], [207]. 8. Fracture and cavitation: [16], [159], [161]. 9. Surface energy: [96], [97], [127], [128]. 10. Dynamics: [23], [29], [33], [75], [103], [104], [121], [122], [193]. 11. Computations: [39], [43], [47], [48], [49], [51], [56], [57], [58], [59], [60], [61], [62], [63], [101], [132], [134], [137], [149], [165], [166], [167], [173], [175], [185].

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Index Adjugate matrix, 6 Affine functionals, 18 Affine Lipschitz function, 21 Banach spaces, 11, 22 Banach-Alaoglu-Bourbaki theorem, 11, 66 Blow-up argument, 72 Boundary conditions, 10, 25 affine boundary values, 21, 73 Dirichlet boundary conditions, 32

environmental conditions, 9 global condition of place, 1, 4, 9 Bounded variation, 65 Calculus of variations, 16 Caratheodory function, 17, 32 Cauchy theorem, 2 Cauchy-Green tensor, 55 Cavitation, 79 Chacon's biting lemma, 67 biting convergence, 68 exceptional sets, 68 Characteristic function, 15 Coerciveness, 10, 17, 19, 21, 33, 37 coercivity, 31, 34 coercivity exponent, 10 growth exponent, 31 Cofactor matrix, 3, 6 Compactness, 11 Compactness theorem, 24, 30 Compactness theorem of Sobolev spaces, 20

Compatibility condition, 48, 55 Compensated compactness, 80 97

Computations, 80 Concentration effects, 23, 64 Concentrations, 68 Constitutive equation, 4 Continuum mechanics, 1, 77 Convexity convex envelopes, 80 convex functions, 18 convex hull of 50(2), 51 nonconvexity, 79 Crystal structure, 42 change of lattice basis, 42 change of shape, 41 conjugate group of GL(Z3), 42 lattice basis, 42 lattice vectors, 41 point group, 42 reciprocal twin, 56 Deformation, 2, 9 deformation gradient, 2, 4 deformed configuration, 2 discontinuous deformations, 33, 79 extreme deformations, 10 extreme strains, 5 periodic Lipschitz deformation, 43 underlying deformation, 57 Direct method, 10, 78 Duality, 65 Dynamics, 80 Elastic bar, 46 Energy energy densities, 1, 9, 25 free energy, 41

INDEX

98

Energy (cont'd.) stored-energy functions, 4 strain energy, 4 surface energy, 80 total energy functional, 4 Equilibrium equilibrium configuration, 1, 4, 9, 10, 25 equilibrium equation, 4 static equilibrium, 1, 2 Euler variable, 3 Euler-Lagrange system, 4 Existence existence results, 78 existence theorem, 19, 32 existence without convexity, 80 nonexistence, 45 Fatou's lemma, 12, 34 Forces body forces, 2, 4 surface forces, 2, 4 Fracture and cavitation, 80 Frame indifference, 5, 10 Generalized variational principles, 41 Gradient requirement, 47 Holder's inequality, 22, 30 Homogenization, 21 Indirect method, 78 Injectivity, 33, 79 Jensen's inequality, 16, 18, 24, 26, 50 Laminates, 41 (Hi) conditions, 49 (Hn) conditions, 79 continuously distributed laminates, 59 continuum of laminates, 59 fine layers, 46 layers within layers, 49 Localization result, 18

Maximal operators, 74, 80 Microstructure, 41, 79 fine-phase mixtures, 41, 46 stress-free microstructure, 47, 51 Minimizers minimizing sequences, 10, 12 of the total energy, 4, 9 uniqueness of, 20 Minors, 26, 31 minor relation, 51 Mixed problems, 1 Mooney-Rivlin materials, 6, 37 Neo-Hookean materials, 6, 38 Nonlinear functionals, 13 Ogden material, 6, 34 Optimality conditions, 78 Orientation-preserving requirement, 2, 33 Oscillatory minimizing sequences, 41 Phase transitions, 79, 80 change of stability, 42 critical temperature, 42 stable phase, 42 transition temperature, 42 Poincare's inequality, 10, 20, 24, 73 Polar decomposition, 7 Potential wells, 43 single well, 52 two-well problem, 51, 79 Probability measures, 14 Pure traction problems, 1 Quasi convexity closed W^1'°°-quasi convexity, 21 closed Vt^'P-quasi convexity, 1921 lack of, 45 Quasi-affine functions, 26 Radon measures, 65 Rank-one matrix, 15 Reference configuration, 2, 9 Regularity, 80 Relaxation, 80

INDEX Response function, 4 Riemann-Lebesgue lemma, 21 Singular values, 6, 35 Sobolev spaces, 10, 22 St. Venant-Kirchhoff materials, 5, 38 Stress Cauchy stress tensor, 2, 4, 7 principal stretches, 7 stress principle of Euler and Cauchy, 2 Strong convergence, 20, 70 strong lower semicontinuity, 12 Strongly measurable, 64 Tensor product, 15 Thermodynamics, 78 Truncation operators, 67, 75 Uniform bounds, 21 Uniqueness, 58 Vector problems, 78 Vitali covering lemma, 71 Weak continuity, 55 weak continuity of determinants, 78

99

weak continuous functionals, 28 weak continuous functionals of higher order, 79 Weak convergence, 10 weak * convergence of measures, 49 weak compactness, 23 weak limit, 13, 15, 17, 20 weak topologies, 11 Weak convergence in L 1 (O), 64 Dunford-Pettis criterion, 23 equi-integrability property, 23 Weak lower semicontinuity, 10, 12 Weakly measurable, 65 Young measures characterization of, 80 homogeneous gradient Young measures, 52 homogeneous W1'00-Young measures, 31 homogeneous Wl'p-Young measures, 19, 26, 31 parametrized measures, 78 W1>P(Q)-Young measures, 18, 41 Zero set, 43, 47

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