variational methods in nonlinear elasticity

VariationalMethodsin NonlinearElasticity

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VariationalMethodsin NonlinearElasticity

Pablo Pedregalliniversidad de Castitla-La ManchaCiudad Real, Spain

Society for Industrial and Applied MathematicsPhiladelphia

Copyright © 2000 by the Society for Industrial and Applied

1 0 9 8 7 6 5 4 3 2 1

All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, ortransmitted in any manner without the written permission of the publisher. For information, write to the Societyfor Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.

Library of Congress Cataloging-in-Publication DataPedregal, Pablo, 1963-

Variational 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-P83200053T.382--dc21

99-054264

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

is a registered trademark.

Copyright

ToDaniel,Silvia,

andJaime.

Contents

Preface ix

1 Elastic Materials and Variational Principles 11.1 Introduction 11.2 The Stress Principle and the Cauchy Theorem 21.3 Elastic and Hyperelastic Materials 41.4 Examples of Hyperelastic Materials 51.5 Appendix: Some Linear Algebra and Geometry 6

2 Quasi Convexity and Young Measures 92.1 Introduction 92.2 The Direct Method 102.3 Young Measures 122.4 Weak Lower Semicontinuity 162.5 Quasi Convexity and Gradient Young Measures 182.6 A General Existence Theorem 192.7 The Case p = +00 212.8 Appendix 22

3 Polyconvexity and Existence Theorems 253.1 Introduction 253.2 Polyconvexity 253.3 Coercivity and the Case p < oo 313.4 Existence Theorems 323.5 Hyperelastic Materials 34

4 Rank-one Convexity and Microstructure 414.1 Introduction 414.2 Elastic Crystals 414.3 Lack of Quasi Convexity 434.4 Generalized Variational Principles 464.5 Laminates 474.6 The Two-Well Problem 504.7 Continuously Distributed Laminates 59

vii

viii CONTENTS

5 Technical Remarks 635.1 Introduction 635.2 The Existence Theorem for Young Measures 635.3 Biting, Weak, and Strong Convergence 675.4 Homogenization and Localization 705.5 A Remarkable Lemma 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 grad-uate student and the newcomer to the subject may, however, have difficultyappreciating these statements and may feel disappointed to discover that thepath leading to a reasonable degree of understanding of the relevant issues, tothe point where they feel confident enough to pursue new directions by them-selves or under the guidance of a senior researcher, is not well trodden. Fillingconcisely this gap in the case of the mathematical theory of nonlinear elasticityis the main motivation for this text. It originated in the form of lecture notesfor a summer course in materials science and engineering held in Coimbra in1997 under the auspices of Centra Internacional de Matematica (CIM, Portu-gal) . Later, I completed that material with the idea of producing a referencethat might help readers get rapidly and efficiently to the heart of the matter ofvector variational methods in the context of nonlinear elasticity, as pointed outabove.

My purpose is in reality very modest: to focus on explaining the complex-ity of vector variational problems from the aspect of existence-nonexistence ofequilibrium configurations, with special emphasis on the relevance of structuralassumptions. In particular, my point is to communicate how the different no-tions of convexity arise in vector variational problems and to explain their sig-nificance with respect to the existence issue. The analysis does not go into anydeeper examination of polyconvexity, quasi convexity, or rank-one convexity. Ido not even attempt to differentiate among them or to define and study thecorresponding envelopes. I believe this must be an additional step for the inter-ested reader who, surprised to a certain degree at the need for different typesof genuine vector convexity concepts, finds the subject sufficiently appealing tomove on. I have also tried to motivate the main questions by unifying the treat-ment of weak lower semicontinuity, quasi convexity, polyconvexity, the failureof quasi convexity and rank-one convexity, oscillatory behavior, microstructure,etc., using gradient Young measures as the main tool. In this way, the readermay gain at the close of the book an overall picture of where the different tech-niques 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 convenientdevice to express and formalize concepts, ideas, and techniques. Almost nothing

ix

x PREFACE

is said about characterizing gradient Young measures. This would be anotherpossible direction for interested readers to investigate.

It is also important to point out that applications to real materials havenot been included, although they might have been. Two main reasons justi-fied this choice from my own perspective. In the first place, this field is orwill shortly be covered extensively and exhaustively by several references (seethe Bibliographical Comments) written by experts. I would do a poor job incomparison. Second, I would have had to sacrifice brevity for the sake of com-pleteness. Again, I think the readers must decide for themselves whether todeepen their knowledge of a particular area after understanding the basics. Forthe same reasons, a more formal and complete treatment of the foundations ofelasticity to establish how the variational nature of our problems arises has notbeen included.

The structure of the book is also oriented toward the main goal of helpingreaders comprehend the need, the reasons, and the motivation for the mainconcepts; the principal assumptions in results; the techniques in proofs; etc.Purely technical remarks have been deferred to the last chapter so that theydo not interfere with the main stream of reasoning. However, they have beencollected so that interested readers may find a guide to the proofs of such factsand results. Note that this choice has not compromised our insistence on brevity.Most of the technical proofs themselves, however, have not been written downbecause they can be taken almost entirely from other texts.

Chapter 1 is an introduction wherein the main problem examined in thebook, that of existence of equilibrium solutions, is precisely stated and motivatedfrom the general principles of continuum mechanics. Chapter 2 focuses on quasiconvexity as the main constitutive assumption for existence of solutions in thecontext of the direct method of the calculus of variations. Chapter 3 dealswith polyconvexity as the main source of quasi-convex functions and as thekey assumption for existence of solutions for some real materials. Failure ofquasi convexity, nonexistence, oscillatory behavior, and, ultimately, rank-oneconvexity and laminates, as the main example of microstructures within thecontext 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. Basicknowledge of real and functional analysis, measure theory, theory of distribu-tions, Sobolev spaces, geometry in R , and linear algebra is assumed, althoughappendices have been included in some chapters to remind readers of key factsin those areas.

Although implicit to some extent in the previous paragraphs, it may beworthwhile to point out that the book is particularly addressed to graduatestudents and researchers in applied analysis, applied mathematics, mechanics,materials science and engineering, etc. The book might also be an appropriatestarting point for those interested in the mathematical side of elasticity, or forthe physicist or engineer who could benefit from a better understanding of themathematical issues in nonlinear variational techniques. It could also serve asa 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 wouldlike to acknowledge them here. My thanks go first to CIM for giving me theopportunity to develop and implement the original draft of these notes. Inparticular, I appreciate the support of A. Ornelas. The feedback and commentsfrom many students who attended my lectures were most welcome and helped alot in convincing me of the need for this book. The instructors who generouslyagreed 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, andseveral anonymous referees for their very specific and detailed criticism that ledto the improvement of the original manuscript in various aspects.

Pablo PedregalCiudad Real, June 1999

Chapter 1

Elastic Materials andVariational Principles

1.1 Introduction

We describe in precise terms the mathematical formulation of the standardproblem in nonlinear elasticity, that of rinding equilibrium configurations ofelastic bodies under prescribed environmental conditions. Our aim is to justifyand clarify the main ingredients of the underlying mathematical problem and itsvariational nature and to show how this problem stems from the general theoryof continuum mechanics. This short chapter is purely descriptive and no proofof any kind has been included. It is intended for readers who are not familiarwith the mathematical theory of elasticity or continuum mechanics, stressingthe main points that must be covered in a more formal, rigorous derivation. Itcan be skipped by those who have previous exposure to such fields.

We focus from the outset on the situation of static equilibrium so that inertialeffects and conservation laws are neither considered nor referred to. We alsoconcentrate on the boundary value problem where the deformation is restrictedon all of the boundary of the reference configuration (global condition of place),which is the one that requires fewer prerequisites. Pure traction and mixedproblems can be handled as well using the same variational methods we willstudy in this text, although, for the sake of brevity, we will not cover thesesituations. Our emphasis will be on the variational methods themselves and wewill not be concerned with the different problems and situations that can betackled using them. At the end of the book, references for further reading aregiven for those interested readers who might wish to complete the expositionof this chapter or to deepen their understanding of continuum mechanics andelasticity. We have included an appendix at the end of this chapter to discussa few important facts in linear algebra that are needed in order to introduceexplicit examples of energy densities for elastic materials. Again, no proofs havebeen given, but they can be found in the references cited at the end of the book.

1

2 CHAPTER 1 ELASTIC MATERIALS AND VARIATIONAL PRINCIPLES

1.2 The Stress Principle and theCauchy Theorem

Assume fl c R3 is an open, bounded, connected subset with a sufficientlysmooth boundary. We will think of this domain as the part of space occupiedby a body before it is deformed. This is called the reference configuration. Oftenwe 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 offi), and orientation preserving. The matrix VM(X) is called the deformationgradient and it provides a measure of local strain. The orientation-preservingrequirement may be written det(Vu(x)) > 0 almost everywhere (a.e.) x € fi.

A deformed body associated with an arbitrary deformation u may be sub-jected 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 perunit volume in the deformed configuration. There might also exist surface forcesdenned 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 thedeformed configuration.

The following axiom is fundamental in continuum mechanics. It is known asthe 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 elementof 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 equilibriumof a deformed body is possible due to the overall effect of elementary surfaceelements t ( y , n). This tensor t ( y , n) is called the Cauchy stress tensor. The mainconsequence 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 continu-ous and the Cauchy stress tensor field t(y, n) is continuously differentiable withrespect to y and continuous with respect to n. There exists a continuously differ-entiable tensor field T: u(fi) ~> M such that the Cauchy stress tensor is givenby

Moreover,

where n is the unit outer normal to 71 and M denotes the space of 3 x 3mainces.

In order to transform the consequences of the Cauchy theorem to a partialdifferential boundary value problem, we need to express those conclusions inthe reference configuration O rather than in the deformed body u(fi). For thisreason we have to use the change of variables y — u(x), x e fJ. Note that alltensors and vectors have been expressed in terms of the Euler variable y. Wefirst 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 respectto the Euler variable y. For a matrix A, adj A indicates its adjugate or cofactormatrix (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.


1.3 Elastic and Hyperelastic MaterialsA material is called elastic if the Cauchy stress tensor T(y] in each point ofa deformed configuration y e u(fi) is a function of x — u~l(y) and of thedeformation gradient Vu(x), exclusively. The constitutive equation can thus bewritten as

where T is called the response function of the material. Assuming that a bound-ary condition of place u = MO nas been given on the portion of the boundaryr0 = dtl \ FI , the equilibrium configuration u must satisfy the following bound-ary value problem:

The body and surface forces are assumed to have explicit dependence on u andVu, 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 —> Rdifferentiate with respect to F € M such that

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

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

In particular, minimizers of the total energy satisfying the global condition ofplace u = UQ on dfi will be (weak) solutions of the equilibrium equations. Thefirst contribution to the energy is called the strain energy.

Another way of stating the last assertion is by saying that the Euler-Lagrangesystem associated with the functional 7 is precisely the equilibrium equations.Under our structural assumptions this is a routine exercise if we are not con-cerned about technical details. Indeed, if u is a minimizer for / under thecondition 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 tojustify all the computations that follow, that

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

Making this derivation rigorous requires much more precision and work. Sincethis is not the goal of this chapter, we will not pursue this topic any furtherbut will be satisfied with the fact that minimizers of the total energy functionalcorrespond to equilibrium configurations of our body. We will concentrate onseeking those minimizers.

There are natural restrictions that a physically admissible energy density Wmust verify. The first one is the frame indifference

for all points x in Q, all matrices F, and all rotations Q. This invariancereflects 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 ideathat infinite stress is associated with extreme strains. This assumption leads usto 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 thetrace of a matrix. We shall no longer use the notation W for the stored-energyfunction but simply W

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

where r, s are positive integers, Oj > 0, bj > 0, a* > 1, /?,- > 1, and g is a convexfunction. 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 theeigenvalues of FTF. R is an orthogonal matrix. See the appendix for furthercomments. A material whose energy density is of the above type and whichsatisfies the additional property liniA-^o <?(A) = +00 is called an Ogden material.Particular examples are

1. Neo-Hookean materials:

2. Mooney-Rivlin materials:

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

1.5 Appendix: Some Linear Algebraand Geometry

We gather here a few important results from linear algebra that are relevant inelasticity. We also make some observations about the geometric significance ofthe adjugate matrix.

Given any 3 x 3 matrix F, both matrices FTF and FFT are symmetricpositive definite. Both have the same nonnegative eigenvalues Aj, whose squareroots Vi -= \f\i are called the singular values of F. The adjugate of F is definedthrough the formula

This identity uniquely determines the adjugate of nonsingular matrices and canbe extended by continuity to all matrices. It is often called the cofactor matrixand has the properties

Another important fact is the relationship between the eigenvalues of a matrixG and its adjugate: if Aj are the eigenvalues of G, then AiA2, AiAa, and A2Asare 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 isdefined 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 nonin-

creasing fashion, then

for any two matrices A, BTHEOREM 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 Cauchy-

Green tensor VuTVu, the adjugate adj Vu, and the determinant det Vu, where uis a deformation and Vw is the deformation gradient. If u is a smooth, invertiblemapping, then it is well known that det Vu gives a local measure of how thevolume changes under u. In the same way, it can be precisely shown that adjVumeasures the change in surface area under the deformation u. Finally, VuTVuis responsible for the change of length elements under the action of u. If F = Vwis interpreted as a deformation gradient, its singular values are called principalstretches, and the polar decomposition

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

Chapter 2

Quasi Convexity andYoung Measures

2.1 Introduction

The main point we tried to emphasize in the previous chapter is the variationalproblem, whose solutions correspond to equilibrium configurations of the mate-rial. It consists of finding minimizers (or more generally extremals) of the energyfunctional representing the internal energy associated with a deformation of thebody,

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

£1 C R3 is the reference configuration and u : fl —» R3 is an admissible de-formation of the body. R* stands for R U {+00}. Such minimizers describeequilibrium configurations of the material under the prescribed environmentalconditions. One of these important conditions is the global condition of placewhereby we impose

for some fixed deformation UQ representing the conditions we prescribe on <9fLSome other, more general conditions can also be handled by the methods de-scribed 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 gradientvariable, W = W(F). The important properties that the energy density Wmust 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 thatproperty 3 is stronger than the behavior for extreme deformations

The coercivity exponent in property 3 is important because it tells us thatminimizing sequences for I will be uniformly bounded in W1'p(£l) (using Poin-care's inequality to control the functions themselves). Thus, we are interestedin bounded sequences {uj} in Wl<p(fl) precisely for that exponent p. If thecoercivity hypothesis 3 fails for all such p > 1, the existence of equilibrium con-figurations may be compromised. For this reason, we may take the admissibledeformations u in (2.1) to belong to W l iJ)(fi). Typically, the deformation UQ giv-ing the boundary conditions on dfl will be a Lipschitz mapping UQ G W1>oc(fi)(we will not indicate the target space for deformations W^llp(f2;R3) since itis 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 bea 1-1 mapping. Other, more general possibilities may also be allowed.

Under these conditions, we would like to analyze the problem of the existenceof equilibrium configurations; that is, we are interested in understanding whenwe 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 weaklower semicontinuity of the functional /. This property in turn relies on thequasi convexity of W, which is the central issue of this chapter. In the appendixat the end of the chapter, some important points about Sobolev spaces andweak convergence are stated.

2.2 The Direct MethodIn order to appreciate the simplicity and elegance of the direct method, let uslook for a moment at the finite-dimensional situation. Let / : R™ -» R*. Wewould 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 theminimization problem: there cannot exist a minimizer. Let

and let {xj} be a minimizing sequence: T(XJ) \ m. If {xj} is relatively compactin Rra (this is the case if lim inf x^oc I(x) > rn) and I is continuous, then for someappropriate subsequence, not relabeled, x} -»• XQ and I(xj) -> m. ThereforeI(x0) = m and x0 is a minimizer. In fact, since we are interested in minimizersit 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 are1. 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 com-pactness properties are the weak topologies over these spaces. In particular, ifX is one of these spaces and is reflexive, it is well known that

possibly for a subsequence (Banach-Alaoglu-Bourbaki theorem). This propertyis extremely convenient and explains, from our perspective, why weak conver-gence is so important and why we are interested in deepening our understandingof it.

Finally, the most difficult step in applying the direct method is enforcing thesequential lower semicontinuity property with respect to these weak topologies:

We can summarize the previous considerations in the following abstract the-orem, 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:

(iii) / is sequentially lower semicontinuous with respect to the weak topology inX;

(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 inthe previous section, minimizing sequences {uj} will converge weakly to someu in W1>p(£l) (remember p > 1). According to the direct method, we mustconcern ourselves with the property of (sequential) weak lower semicontinuityof 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 lookat the appendix at the end of this chapter so that they can fully understandand appreciate the discussion that follows. From now on we will omit the term"sequential" when referring to weak lower semicontinuity, taking for grantedthat 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 concernourselves with the possible behavior of the integrals

when the sequence {uj} converges weakly in W1>p(fl). If this weak convergenceis in fact strong so that Uj ->• u in W1>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 isonly weak? What further hypotheses do we need to impose on W to ensurethe weak lower semicontinuity? Notice that in the case of weak convergence wecan also apply Fatou's lemma, but in this case liminfj^oo W(Vuj(x)) is not, ingeneral, W(Vu(x)).

2.3 YOUNG MEASURES 13

We study some simple examples to emphasize the behavior of weak conver-gence with respect to nonlinear functionals so as to gain some insight into thequestions raised.

1. Let n = (0,7r/2) and fj(x] = sin(jx), g^x) = ff(x), x 6 H. Promelementary 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 weaklimit of the squares. Nonetheless, we have the right inequality for weak lowersemicontinuity

where 0 is the weak limit of {/_>•}-Let us further take <p(x) = \Jx for x > 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 termsin the computations above converge to 0, and therefore

Notice that once again the weak limit of {<X<?j)} is not the composition of ipwith the weak limit of {gj}. Moreover,

In this case we do not have the weak lower semicontinuity inequality. We appar-ently need more conditions on W in order to have the weak lower semicontinuityproperty.

2. Let f(x) = 2x(0,i/2](x) - 1 for z € [0,1], and extend / to all of R byperiodicity. Take fj(x) = f ( j x ) . It is easy to see that the jumps from 1 to -1occur on a smaller and smaller scale as j tends to oo. On the other hand it isnot 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 thesquares. Moreover, if ip : R —» R is any continuous function, then

and

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

for any function (p in terms of the expression (1/2)(^(1) + tp(—l)). This is themain feature of the Young measure device. Through this property, weak lowersemicontinuity can be understood in a rather general way.

The examples examined above should have convinced us that there is some-thing special about sequences of a highly oscillatory nature when the conver-gence is only weak and not strong. In general terms, the problem we are tryingto understand can be formulated as follows. Suppose that fj -^ / in L°°(fi)so that ||/j||Loom) < C < co and (f is a continuous function. {<p(fj}} is alsoa uniformly bounded sequence in L°°(fi). Hence, some subsequence convergesweakly * in L°°(fl):

We have tried to explain through the examples that g is not </?(/). The Youngmeasure associated with {fj} furnishes the link among {/.,•}, /, g, and ip.

A Young measure is a family of probability measures v = {vx}x^fl associatedwith a sequence of functions fj : fi C HN —* Rm, such that supp(^) C

while

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

2.3 YOUNG MEASURES 15

Rm, 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 mea-sures 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 L1(fi). Intuitively, the Young measure can be thought of as givingthe limiting probability distribution of the values of {/_,-} when points are takenrandomly around each x e Q. If -B#(x) denotes the ball of radius R > 0 centeredat x e Q and E C Rm is any measurable set, then

where bars |-| denote the Lebesgue measure. This identification clearly showsthat the sequence {/_,-} is forced to oscillate near x among the different vectorsin the support of vx with relative frequency given by the weights correspondingto 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) extendedperiodically to all of R. Let u : R2 -¥ R2 be defined by u(x) — (g(x),g(x)),x = (xi,x-z) € R2, and Uj(x] ~ ( l / j ) u ( j x ) . If we compute the gradients ofthese 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

a weak lower semicontinuity result.

2.4 Weak Lower SemicontinuityAs pointed out, the fundamental property (2.3) of Young measures provides away of representing and manipulating the limits of the following integrals:

Prom this perspective, the Young measure is a convenient tool when dealingwith integral functionals in the calculus of variations.

The main issue is as follows: Under what type of conditions can we actuallyrepresent limits of integrals of the above type by Young measures? The answeris contained in the following basic existence theorem.

for all x 6 fi, where

For any continuous function ip : M2x2 —> R we have

4. Now take ip0(F) = \F\2 for F £ M2x2 so that

where Uj is the sequence of the previous example. We know that the sequenceof 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 <p$ is convex, byJensen's inequality,

which in this simple case is nothing but

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

2.4 WEAK LOWER SEMICONTINUITY 17

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

where g : [0, oo) —> [0, oo] is a continuous, nondecreasing function such thatlimôo g(t) = oc. There exist a subsequence, not relabeled, and a family ofprobability measures v = {^x}xâ ftê associated Young measure) dependingmeasurably on x with the property that whenever the sequence {ifr(x,Zj(x))} isweakly 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 xand 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 inLp(fl) contains a subsequence that generates a Young measure in the sense ofTheorem 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 onW. Since

must be the weak limit of {Vuj} by taking ij)(x,X) = A in Theorem 2.2, theweak lower semicontinuity property will hold true if

The assumption we have made that {W(Vuj}} is weakly convergent in L1(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 associatedYoung measure v = {^x}x^fi, then

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


In conclusion, if W verifies the coerciveness inequality

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

whenever Uj —l u in Wl'p(£l}. The proofs of Theorems 2.2 and 2.3 are sketchedin Chapter 5.

We have reduced the weak lower semicontinuity property to a certain in-equality involving probability measures, which in fact reminds us of Jensen'sinequality. What does this inequality mean in terms of weak convergence, andhow does Jensen's inequality relate to it? This is a profound question that wecan hardly answer here. We will merely provide an intuitive explanation ofwhy Jensen's inequality seems to be connected to weak lower semicontinuity.Roughly speaking, weak convergence of functions means convergence in the av-erage and therefore taking integrals plays a fundamental role in determining thelimit function from the members of the sequence. The only functionals thatcommute with the operation of taking integrals are the affine ones. However, ifwe understand weak convergence in terms of Young measures, Jensen's inequal-ity asserts that functionals associated with convex functions may not commutewith integration, but they do always respect an inequality. The direction of thisinequality is the appropriate one for lower semicontinuity. We will concentrateon condition (2.6) in the remaining sections of this chapter.

2.5 Quasi Convexity and GradientYoung Measures

The key constraint on the energy density W that ensures the weak lower semi-continuity property is (2.6). This inequality must be valid for all possible Youngmeasures generated by gradients of a bounded sequence in W1>p(fl). Since eachindividual vx is a probability measure, if W is convex then (2.6) will certainlybe true. This is the classical Jensen's inequality. In particular, weak lowersemicontinuity 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 tomore fully explore condition (2.6).

Let us call a family of probability measures v = {"zl^go. a W7l'p(fJ)-Youngmeasure (notice the explicit reference to the domain fi) if it can be generated asthe Young measure (according to Theorem 2.2) corresponding to a sequence ofgradients {Vuj}, where {uj} is a bounded sequence in W1>p(fl). The followinglocalization result asserts that the individual members that make up such afamily of probability measures are themselves Wl>p(Q}-Young measures for anydomain 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 fora.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 l ip(fi)-Ybim<7 measure. Fora.e. a e 0 and /or any domain Q, there exists a bounded sequence {vai]} inWl'p(Q) such that the homogeneous Young measure associated with {Vwaj-} isva. 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 Wllp-Young measure (withno reference to a particular domain) as a probability measure v supported on theset of matrices M such that for any domain Q c R3 there exists a sequence ofgradients {Vttj}, where {uj} is some bounded sequence in Wl'p(Q) generatingv according to Theorem 2.2 and such that Uj — up & W0

llp(Q), where F is thefirst moment of v. We immediately have that if v — {vx}x&n is a W1>p(fi)-Young measure, then a.e. vx is itself a homogeneous W1>p-Young measure. Wealso say that a function W : M -> R* is closed Wllp-quasi convex if

for all homogeneous Wl'p-Young measures. Notice that this definition requiresJensen's inequality for a (proper) subclass of probability measures supported inthe set of matrices M. The conclusion of the previous remarks and definitionsis that the closed W1|p-quasi convexity condition is sufficient (and necessary) inorder to have (2.6). In section 2.4, we showed that this inequality is sufficient forweak lower semicontinuity, but the necessity is a much more delicate point thatwill 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 followingterms:

for all (x, u) e fi x R3 and for all homogeneous W1>p-Young measures v. Someremarks on the proof of Proposition 2.4 can be found in Chapter 5.

2.6 A General Existence TheoremIt 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 thecoerciveness condition

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.


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

where UQ e W l ip(Q) is given such that I(uo) < oo. We are looking forminimizers

THEOREM 2.5. I f W is dosed W1<p-quasi convex, then there exists at leastone minimizer U.

To take care of the dependence of W on u it is interesting to keep in mind thefollowing fact about how strong convergence is reflected on the Young measure.

LEMMA 2.6. Let Zj = (uj,Vj) : Q -> Rd x Rm be a bounded sequence inLp(fl) such that {uj} converges strongly to u in Lp(£l). If i> — {^x}x€n is theYoung 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 convergentsequence in W l ip(fJ). In this situation we know that the functions themselvesconverge strongly to the weak limit by the compactness theorem of Sobolevspaces. If u € W1|p(fl) is the weak limit and {^x}x&^ is the Young measureassociated 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 andPoincare's inequality, it will be a bounded sequence in W1>p(fi) whose gradi-ents will generate a TV1'p(f2)-Young measure v = {VX}X£Q- For an appropriatesubsequence, Uj —v u and

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

2.7 THE CASE 21

2.7 The Case Once we have seen the relevant role of closed W1|p-quasi convexity for weaklower semicontinuity and its importance as a sufficient condition for the exis-tence of minimizers in vector variational problems, we are justified in making anattempt to better understand this condition so that we may gain some feelingfor and interpretation of it. Unfortunately, there is not much we can do in thisdirection, as quasi convexity still remains a largely unknown area. Since manytechnicalities are involved in the finite case p < +00, we consider in this briefsection the Lipschitz case

Let UF(X) be the affine Lipschitz function UF(X) = Fx for x € fi. Foru e W1>00(fi) such that u — up € W0>oc(fi) we can produce an admissiblefamily of probability measures in (2.6). This can be done through the Riemann-Lebesgue lemma, which is a particular case of a more general homogenizationfact (Theorem 5.12).

LEMMA 2.7. Let fi be a domain in HN and u £ W1>00(fi), u - up €W0'°°(f2). There exists a sequence {uj} bounded in WliOC(fi), Uj — up sW0

1>oc(fZ), such that the Young measure associated with {Vuj} is homogeneousand defined by

for any continuous <p.If we use v in (2.6), we get the inequality

for all F 6 M and all u £ W01>00(fi). A function that satisfies all these in-

equalities is simply called quasi convex. This property is important because itis in fact equivalent to (2.6) under weak * convergence in W1>00(fJ) provided Wtakes on finite values everywhere. In other words, quasi convexity is equivalentto closed WllOC-quasi convexity under the finiteness of W (why?). It is lessrestrictive than (2.6) under weak convergence in W1>p(f2) for finite p. Noticethat in (2.8) the choice of the domain O is irrelevant. From an energetic pointof view, (2.8) is much more transparent than (2.6), because it says that amongall deformations having affine boundary values determined by the matrix F, theleast energy is achieved by the affine deformation itself. This is a very importantinterpretation of quasi convexity that greatly aids our intuition.

In general, there is no way to obtain uniform bounds in W1'°°(fZ) for mini-mizing sequences from coerciveness, unless we assume that the energy densityW is +00 outside some compact set of matrices. However, this condition is notreally appealing. For this reason the case p — +00 is not usually considered inexistence theorems. More specific hypotheses might lead to that uniform boundin W1>00(fi), but these situations require very precise knowledge on W. Onthe other hand, closed W1>p-quasi convexity reduces to quasi convexity underupper bounds on W; this upper bound has been, however, rejected on physicalgrounds.

2.8 AppendixWe recall in this section a few general facts about the Lebesgue and Sobolevspaces Lp(O) and Wl>p(£l).

The space i1(fi) designates the set of all integrable functions over fJ, andwe write

The space Lp(f2) for p > 1 consists of all measurable functions / such thatI/I" el1 (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 L9(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 dualityas 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, continuousfunctions in O is dense in Lp(ft) for 1 < p < oo. An important inequality isHolder'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 itcan be generalized in the following way:

Let 1 < p < co. We say that {fj} converges weakly to / in I/P(Q) and writefj -* 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 measurablesubset 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 verypeculiar from the point of view of weak convergence and that peculiarity isprecisely condition (ii) above. Concentration effects are connected to the failureof (ii).

PROPOSITION 2.10. Let 1 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 generaltheorem.

THEOREM 2.11. Let X be a Banach space with dual X'. Every boundedsequence 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 € LP(O) such

that there exist functions jj^- £ Lp(fi) with the propertyt/iC-j

for any compactly supported, smooth function £. This definition can also bestated 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 ap-proximated by smooth functions in the norm of W1>p(fi).

Weak convergence in W1'p(fl) means weak convergence in LP(Q) for thefunctions together with their gradients.

A fundamental fact about Sobolev spaces is the compactness theorem.THEOREM 2.12. Iffl is an open, regular, bounded domain, then1. p < N: W1>p(fi) is compactly embedded in Lq(fl) for any I < q < p*,

where p* — Np/(N — p);2. p = N: W1>p(fl) is compactly embedded in L9(fi) for any 1 < q < oo;3. p > N: W l ip(f2) is compactly embedded in the space of continuous func-

tions up to the boundary under the sup norm.An important corollary is that a weak convergence sequence in W1>P(Q) will

converge strongly in Lp(fl) to the same limit.The space WQ'P(&.) is the closure in W1>P(Q) of the set of smooth func-

tions with compact support in O. It is important to keep in mind Poincare'sinequality, 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<p(tl), 1 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 ) £ Kfor n a.e. x 6 f2, where K c Rm is a convex set. If (p is a convex functiondefined in K, then

Chapter 3

Polyconvexity andExistence Theorems

3.1 Introduction

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

3.2 Polyconvexity

We showed in Chapter 2 that if the energy density of a hyperelastic materialis convex with respect to the gradient variable, then we have equilibrium con-figurations under prescribed boundary conditions. Under the assumption ofconvexity of the energy density W, our existence theorem, Theorem 2.5, ap-plies almost immediately. We would like to argue, however, that this hypothesiscontradicts the physical requirements imposed on W. Indeed, for a convexfunction 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 indifferenceimplies some inequalities that the eigenvalues of the Cauchy stress tensor in anypoint of any deformed configuration must verify. Nonetheless, these inequalitiescannot always be expected to hold. There are counterexamples, although theyare beyond the scope of this book. Hence, we have to rule out convexity as abasic assumption.

If we cannot reasonably rely on the convexity assumption for our free energydensities 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 thosemeasures. If g : R -» R* is convex, the composition W(A) = g(M(A)} will ver-ify (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 functionof all its arguments, the composition W(A) = g(M(A)} will verify (2.6) aswell. What does (3.1) mean in terms of a sequence {Vwj} that generates vlIf the two sequences {Vu,-} and {M(Vuj)} converge weakly in L1(Q) to Vwand M, respectively, then (3.1) means precisely that M = M(Vu), which isagain elementary to check using the fundamental property of the Young mea-sure and (3.1). Conversely, if we would like (3.1) to be true, we would have toshow that v can be generated by a sequence {Vw-,-} such that Vuj —*• Vw andAf(Vwj) ->> M(Vw) in Ll(£l), where u e W1>P(JT). In practice, since if p > 1the convergence VMJ —*• VM is always guaranteed, the fact M(Vuj) —*• M(Vu)in L1(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 uniformbounds 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 con-

vergence M(Vuj) —*• Af(Vu) in W1|30(fi) implies that both M and —M haveto be quasi convex, we are thus led to determine continuous functions M, de-fined on matrices, such that both M and — M are quasi convex. An initial ideais that since quasi convexity implies rank-one convexity (this will be provedin Chapter 4), M and — M will have to be rank-one convex, i.e., the functionM(A + tB) must be affine in t whenever B has rank one. The functions thatenjoy this property are called quasi-afRne functions and they are identified inthe following theorem.

THEOREM 3.1. Affine functions of the minors of a matrix are the onlyquasi-affine functions.

In the case of 2 x 2 matrices, there are five minors: the four entries (the 1x1minors) and the determinant. For 3 x 3 matrices, we have 19 minors: the nineentries, nine 2 x 2 minors, and the unique 3 x 3 minor, the determinant. Andso on it goes.

The fact that these minors are quasi affine is easy to check. As a matterof fact, it is not hard to convince oneself that for one of these minors, M, thefunction of t, M(A + tB), is a polynomial of degree rank(B). To show thatthese are the only quasi-affine functions requires more work and a little bit ofalgebraic manipulation.

For the complete proof of Theorem 3.1, let M(F) denote a quasi-affine func-tion 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 sothat €i <g) €j is a basis for the space of matrices. Let M^(F) denote the linearfunctional defined on matrices and determined by the numbers M(I)(CJ ® Cj):

and let

We claim that this function vanishes on rank-one matrices. In order to provethis claim we state the following elementary property of quasi-affine functionsthat 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 ofrank one, we can write

Certainly, M(2) is quasi affine (difference of two quasi-amne functions) and van-ishes 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 corre-

sponding to rows i, j and columns k, I. Define

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 elementaryproperty 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* identicallyvanishes on M and therefore

Our main task consists now of showing that the minors indeed generateweak continuous functionals. For a matrix A 6 M, let A' denote some squaresubmatrix 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<p($l) andp > 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 theproof in the particular case of 2 x 2 matrices and N = r = 2 so that there areno 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 ofdistributions. If v is smooth, then it is very easy to check that

due to the equality of the mixed partial derivatives. For a nonsmooth functionin W l t p ( f l ) , take a sequence of smooth functions {vj} converging strongly to vin W1|p(fi). Obviously, adj(Vwj) will converge strongly to adj(Vu) so that forany 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 convergeweakly in Lp(^l) precisely to adj(Vu) (notice that adj(Vw) are just the singlederivatives 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 Sobolevfunctions, {uj} converges strongly to u in Lq(£l) for any finite q. The secondterm 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 de-formation u, let (Vw)' denote the corresponding submatrix of order r of Vu.There are r independent variables and r components of the deformation u as-sociated with the choice of this submatrix indicated by '. Let i correspond tosome component of u chosen in the submatrix. div will indicate the divergenceoperator 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 ofthe mixed partial derivatives, we find indeed that div'(adj(Vu)') = 0. For a gen-eral v 6 Wl>p(fl), take a sequence of smooth functions {vj} converging stronglyto v in VF l ip(ft). adj(Vuj)' converges strongly to adj(Vu)' in Lp/(-T-1\tl) (byHolder'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, weobtain that

as distributions. By induction, let us assume that adj(Vuj)' —*• adj(Vw)' in thesense of distributions. Since adj(Vuj)' involves terms with r — 1 products, itwill be bounded in £,p^r~^(0), p/(r — 1) > 1, and therefore that sequence ofadjugates will converge weakly (in Lp/(r~^(£l)) precisely to adj(Vu)' due to theuniqueness 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 Sobolevfunctions, {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. Thesecond 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 wheneverUj —>• u in W l ip(fi) and {M(Vuj)} converges weakly in Ll(£l), M being anyminor of order r and p > r, then we have in fact M(Vw^) —^ M(Vu) inand 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 theweak limit of {«?}.

PROPOSITION 3.3. // M is any minor and v is any homogeneous jy1'00-Young measure, then (3.1) holds.

Let M(A) denote the vector of all possible minors of a matrix A consideredin some fixed order. A function W : M —> R* is called polyconvex if it can berewritten 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 discussionis that every polyconvex function W verifies (2.6) for any homogeneous W1'00-Young 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 sequencesis usually enforced through coercivity assumptions on the integrand W. Unfor-tunately, this weak convergence can very rarely be obtained in W1'°°(17) unlessW(A) = +00 whenever |.4| > R for some R > 0, but this hypothesis is notinteresting. Much more natural are coercivity assumptions assuming a certainrate 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 theoremproved in section 3.4 is compromised. This is the reason why the technicaldetails that follow, although tedious to some extent, are important. The growthexponent p determines the Sobolev space in which we have to work, and inparticular we need hypotheses to ensure (3.1) for homogeneous Wrl'p-Youngmeasures precisely for that value of p.

The following result lists several situations in which we can achieve the weakconvergence 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 thefollowing 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 Wlip(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 equi-

integrability for the case p = 3. In case 3, notice that Theorem 3.2 applies forr = 2 so that the adjugates converge in the sense of distributions. Since theexponent of boundedness is greater than 1. the convergence takes place weakly inL3/2(Q). For the determinant we cannot directly apply Theorem 3.2. However,under these assumptions on the adjugate, (3.2) shows that the determinantsconverge in the sense of distributions by using Holder's inequality with theexponents 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|adjJ4|3/2 shows that {detVwj} converges weakly in L1^). Case 4is left to the reader.

3.4 Existence Theorems

Our main existence theorem for polyconvex integrands yields minimizers for theenergy

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 ip(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 assumption1 in Theorem 3.6.

Proof. After Theorem 3.5, the only delicate point in this proof is the casep = 3. The following remarkable lemma yields the clue to the proof. Its completeproof can be found in Chapter 5.

LEMMA 3.7. Let {vj} be a bounded sequence in W l > p ( f y , p > 1. Therealways exists another sequence {uj} of Lipschitz functions (uj € Wl'°°(£l) forall 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 areallowed in the variational principle. In certain cases some of them might bethe optimal configurations (cavitation and fracture). This issue is, however, farbeyond the scope of this text.

We finally provide some results concerning the orientation-preserving and in-jectivity requirements. Concerning the former we have the following elementarylemma.

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 ofA imply that z > 0 a.e. x e SI. Let E = {z = 0}. Define w in E by

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 beFor our purposes, it is enough to apply Lemma 3.8 in the context of Theo-

rem 3.6 to Zj = det Vttj, z = det Vu, and

Note that this function h is continuous due to the uniform coercivity of W withrespect to (x,u).

The injectivity issue is much more delicate. We simply state without proofthe following fact that is sufficient in many cases.

THEOREM 3.9. Assume thatuo is a continuous-up-to-the-boundary, injectivedeformation of a domain fl and let u be another deformation such that u GWltp(£i) for some p > 3, det Vu > 0, a.e. x € J7; and u = UQ on <9fi. Then u isa.e. injective.

3.5 Hyperelastic Materials

In this section we seek to analyze the examples of hyperelastic materials includedin 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 Ogdenmaterials 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 convexfunction.

We must demonstrate the poly convexity and coerciveness for this W. Wetreat these two issues successively.

Concerning the polyconvexity, it is convenient to express W(F) in terms ofthe singular values of F and adj.F. We know that if Uj are the singular valuesof F then t^fs, ui^s, and v\v^ are the singular values for adjF. We can rewriteW(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 > r2 > 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 fixedQ and R, the function ti(FQDR) is linear in F, and the maximum of linearfunctions 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 theequality.

LEMMA 3.11. Let g : [0, +oc)3 —>• R be a convex, symmetric function,nondecreasing with respect to each variable. The composition

is convex.

Proof. The symmetry requirement on g means that it is invariant under anypermutation of its variables. Let a, 6, and u denote the vectors of singular valuesof the matrices A, B, and XA+(1— A)J5, respectively, ordered in a nondecreasingfashion, 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 thoseobtained by permutations of the last three vectors. V contains at most 13vectors. We would like to show that u belongs to the convex hull of V. If thisis 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 anyof 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 coordi-

nates 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 possibilitiesmust hold:

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 coordinatesof u are ordered in a nondecreasing fashion. The third inequality is a directconsequence of Lemma 3.10 for an appropriate choice of TJ. The last inequalityis true because the vector with coordinates va^ for i < j and 0 for i > j belongsto V.

Since the function

verifies all the requirements of Lemma 3.11, keeping in mind the observationmade earlier about the singular values of adj(F), we conclude that the polycon-vexity 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-dimensionalspaces, there exists some positive constant c7 such that

Because |.F|2 = tr FTF = Vi(F)2 + v2(F)2 + 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 existencetheorem provided we have the proper relationship between the exponents a, j3and the one for the coercivity of g.

The particular case of Mooney-Rivlin materials for which

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. Nev-ertheless, in dimension N = 2 the existence theorem is recovered because weregain 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 energydensities are not polyconvex. In this regard, the failure of the existence theoremis 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 musthold:

for any e > 0. This is clearly impossible, and hence W cannot be poly-convex.

The energy density of a St. Venant-Kirchhoff material can be easily rewrittenin the form

where a < 0 and b,c,d> 0. By the previous proposition such a function cannotbe polyconvex. Nonetheless, such energy densities can be approximated nearthe identity by integrands corresponding to Ogden materials.

Chapter 4

Rank-one Convexity andMicrostructure

4.1 Introduction

We describe a class of materials whose stored-energy densities lack the propertyof quasi convexity. This situation leads us to consider generalized variationalprinciples where Wl'p-Young measures are allowed to enter the minimizationproblem. These objects are physically interpreted as microstructures and rep-resent highly oscillatory minimizing sequences on smaller and smaller spatialscales. 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 convexityand laminates. Although the issue is beyond the scope of the book, it turnsout that this kind of microstructure can be characterized in terms of Jensen'sinequality for rank-one convex functions and so rank-one convexity is central tothis chapter.

The reader will not find applications to real materials. Our objective is tostress the ideas in order to understand, from an energetic point of view, theappearance 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 periodicfashion. Probably the simplest way of describing this array is to place the originat one of the atoms and then refer the position of the remaining atoms to thechosen origin using three independent lattice vectors {HI, n2,713}. We let N € Mbe 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 anychange of lattice basis: if N' is an equivalent choice of lattice basis (equivalentin the sense that the positions of the atoms are the same for TV and for N'),then <E>(./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 matrixN 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 groupof GL(Z3). Furthermore, we also impose the conditions

In practice, however, the invariance above is assumed only when H e P, whereP is the point group of the reference crystal lattice consisting of all the matricesH e N GL(Z3) N"1 that are rotations. This is a finite group. For example, ifthe atoms in the reference configuration aligned themselves on cubic cells, thenP would include the 24 rotations that leave a cube invariant.

As mentioned, W and $ depend on temperature 6. Above a certain criticaltemperature 90 there is a stable phase, taken as reference. By "stable" wesimply mean that it minimizes W, so that Wg(\) = 0, where 1 is the identitymatrix and B > 90. At the transition temperature OQ, there is a change ofstability or of crystal structure so that below 6*0 the stable phase is no longerrepresented by 1 but by some other nonsingular matrix f/o describing the changein crystal structure that has taken place. Thus Wg(U0) = 0 but Wg(l) > 0for Q < 00. At the transition temperature 9 — OQ both phases may coexist:WgQ(Uo) — Wg0(l) — 0. Because of the invariance that the energy densityW — Wg0 must satisfy, we should have at the critical temperature

for any H £ P and any rotation R. We have found many matrices for whichthe 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 thesets

a potential well associated with U% = HiUQH:[ and make the further assumptionthat the free energy density W is positive outside the set of the wells: the zeroset for W, {W = 0}, is exactly the set of the wells.

Under these circumstances, we are looking for minimizers of the energy func-tional

among all deformations u of the reference configuration ft c R3 satisfyingappropriate boundary conditions. We are explicitly assuming that W is thesame 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 energydensity for an elastic crystal cannot be closed Wlip-quasi convex. Recall thatW : M —> R* enjoys this property when Jensen's inequality holds for all homo-geneous Wl'p-Young measures. There is a necessary condition for a function tobe closed W1>p-quasi convex (or quasi convex for that matter) that is also veryimportant for the description of equilibrium configurations of elastic crystals.This condition is called rank-one convexity and it requires the usual convexitycondition 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 necessityof this condition we rely on the following convenient reformulation of the quasiconvexity condition.

LEMMA 4.1. // a function W : M —>• R* is closed W1'1'-quasi convex forsome p > 1, then

for all T'-periodic Lipschitz deformations u and every matrix F, where T is anyunit cube in R3.

The converse of this result requires suitable upper bounds on W for finite por the finiteness of W for p = +oc. Because the format of the statement of thelemma is not the usual one in which this result is presented, we give the proofhere.

Proof. We may take F = 0 without loss of generality. Given a T-periodicLipschitz deformation u, consider the sequence

Trivially, Vu}(x) = Vu(jx] for x e T. We would like to identify the Youngmeasure associated with this sequence of gradients. This is in fact a homog-enization result in the spirit of Theorem 5.12. Indeed, let £ be a continuousfunction in T and <p a continuous function defined on matrices. By the meanvalue theorem for integrals we have

where we have used the periodicity of u and certain pointsyf € 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-quasiconvexity 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 thecharacteristic function of the interval (0,1/2) in (0,1), extended by periodicityto 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, andtherefore is eligible in Lemma 4.1. Let us examine the gradient

and

4.3 LACK OF QUASI CONVEXITY 45

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

for any matrix F and vectors a and n. This inequality is the rank-one convexitycondition. 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 intervalbetween 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 concludethat

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 wholesegment 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 vectorit 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 haveb • 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 es-

tablishing the existence of equilibrium configurations for elastic crystals. Eventhough there are results on existence despite lack of convexity, this lack of quasiconvexity makes it impossible to apply the direct method to show existence ofequilibrium states in this case and indeed leads us to think about nonexistencein many interesting cases.

4.4 Generalized Variational PrinciplesThe lack of quasi convexity shown in the previous section is typically a precursorof the oscillatory behavior of minimizing sequences for nonconvex (nonquasi-convex) integrands. Indeed, fine-phase mixtures provide minimizing sequenceswhose weak limits are not minimizers. These highly oscillatory minimizingsequences represent the behavior of elastic crystals. The configurations withlow energy resemble the construction used in the above analysis of the rank-oneconvex condition where very fine layers of different phases (different potentialwells) alternate. In this sense, we would like to understand the possible behaviorof minimizing sequences. The Young measures associated with the gradients ofminimizing sequences may serve as a device to account for this behavior.

As an illustration of the type of phenomenon we are trying to understand, wecan study the following academic, one-dimensional example. Assume we havean elastic bar with unit length and clamped endpoints. If we take fi = (0,1)as the reference configuration, we postulate that the internal energy associatedwith 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 ismade of is, for instance,

where the preferred states are given by a and /3 and 0 < a < 1 < /3. Weclaim that there is no equilibrium configuration for this bar. Indeed, if we letX 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 wouldhave 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 thisfunctional oscillate more and more between a and /3.

We can set up a new generalized variational principle where we let Wl>p-Young measures compete in the energy-minimization process. Let A denote theset of admissible deformations for the old variational principle:

4.5 LAMINATES 47

where u0 e Wl>p(Q) is given with J(u0) < oo. Let ^4 be the set of Youngmeasures 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 ) sothat inf_4 I < inf^ /. The equality of the two infima holds under upper boundson 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 minimizingsequences such that I(uj) \ 0. In this case, if v = (vx}x&u ^s t^ie Youngmeasure associated with {Vwj}, by Theorem 2.3 we have

Hence, I(v) = 0 and because of the nonnegativity of W this can only happenif supp(i^x) C {W = 0} for a.e. x e fL v is called a stress-free microstructureand in real problems these are the ones in which we are interested. As wehave argued, they are the families of probability measures that, satisfying allthe restrictions of the problem, have their support contained in the zero set ofthe 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 tofinding all gradient Young measures supported in the set of the wells. Given aminimizing sequence such that I(uj] \ 0, there exists an associated gradientYoung measure that describes the behavior of {Vi/j}. Conversely, if we find agradient Young measure supported in the set of wells, the sequence of functionswhose gradients generate such a Young measure will be a stress-free minimizingsequence, and hence will describe a possible behavior of the material. What iscrucial is the gradient requirement. We can find many families of probabilitymeasures supported on the set of wells. But only those that are gradient Youngmeasures are associated with stress-free minimizing sequences and these are theones relevant to our original variational problem. In other words, only gradientYoung measures are physically meaningful for our problem. The others do nothave any physical significance.

4.5 Laminates

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


by gradients but we can restrict attention to a particular class of such familiesof probability measures called laminates.

Let us briefly recall (in a slightly different form) how the rank-one convexitycondition was introduced in section 4.3. Let Fi & M, i = 1,2, a € R3, and aunit 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 byperiodicity, the Young measure associated with the sequence of gradients

is

where O is any bounded domain in R3. Therefore the probability measure vin (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. Throughoutthe rest of this chapter we will use the term "gradient Young measure" to meanW1'°°-Young measure.

By Lemma 5.13, applied to (4.2), we can assume without loss of generalitythat Uj - UF 6 Wo1'00(ft), F = tF\ + (1 - i)F2. Moreover, Vuj takes the valuesFI and Fy except in small sets E j , Ej —> 0. We would like to go one stepfurther 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 ofQ where Vu., — Fi. For j and i fixed, based on the compatibility conditionbetween F2 and F2 > we can construct a sequence of gradients {Vv£1}, v£ —up? e Wo'00 (ft?), whose values essentially alternate between Fy,2 itn

preassigned frequency to £ (0,1) and normal e to the layers. Let E% be theset where Vv^ does not take either of the two values F2 -^2 - Choosek — 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 l i00(n) and satisfies w(j' — UF £W'r0

1'oc(O). The homogeneous Young measure associated with {Vu^} is

The probability measure in (4.4) is a gradient Young measure provided we havethe compatibility conditions (4.1) and (4.3).

It is not difficult to generalize this construction when a finite number ofmatrices is involved if we have the rank-one condition in a recursive way. Thisbasic 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<i<lr where U > 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<i<l_l satisfies the (#1-1) condition.An immediate consequence of our previous discussion is the following.PROPOSITION 4.4. If {(ti,Yi)}l<i<l satisfies the (Hi) condition, the proba-

bility measure v = ^Ji ij£y; is a gradient Young measure.We can even take weak * limits in the sense of measures for sequences of fi-

nite convex combinations of Dirac masses verifying (Hi) conditions. These weaklimits will also be gradient Young measures: the argument is elementary andinvolves taking diagonal sequences. Note that in fact the set of homogeneousgradient Young measures is weak * closed. This remark motivates the definitionof laminates. For those readers not familiar with the notion of weak * conver-gence of measures, these weak * limits can be interpreted as (Hi) conditionswhen I —>• oc. In this case we consider infinite-order laminates.

DEFINITION 4.5. Let v be a probability measure supported on M and letK = supp(z/) be a compact set. v is a laminate if there exists a sequence ofsets of pairs {($ , Y f ) } ^<i<k, (k > 2), verifying the (Hk) condition such that

Si ^i &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 thematrices

We claim that the probability measure

is a laminate despite the fact that none of the pairs Ai — Aj has rank one (seethe Bibliographical Comments to find several places in the literature where youcan decipher why this probability measure is a laminate).

In order to fully understand condition (2.6), we need to analyze the defin-ing properties of Young measures associated with the gradients of boundedsequences in Wl<p(ft). These have been called Wl'p-Young measures in pre-vious sections. This is a deep issue, not completely understood except in anabstract way. Indeed, an important result emphasizes that (2.6) represents aduality between quasi-convex functions and gradient Young measures. In thestatement that follows, £p is essentially (not quite) the space of functions withgrowth of order at most p:

THEOREM 4.7. Let v — {vx}xe.£i be a family of probability measures sup-ported in the space of matrices M. v is a Wl'p-Young measure if and only

if(i) there exists u e W1>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 boundedfrom below and a.e. x £ O;

(iii) fafM\A\pdvx(A)dx<™.For the case p = oo, the condition <p € £p drops out (Jensen's inequality

ought to be true for all quasi-convex functions regardless of their growth) andthe third requirement must be replaced by the condition of uniform compactsupport of vx.

A necessary condition for a probability measure to be a laminate is Jensen'sinequality for rank-one convex functions: if W : M ->• R* is rank-one convexand v is a laminate, then

This is easy to derive because of the recursive way in which (Hi) conditions aredefined. It turns out that this condition is also sufficient so that laminates areto 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 foundin Chapter 5. We refer readers to the Bibliographical Comments at the end ofthe 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 previoussections, 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 weassume that the zero set of the energy density W7, {W = 0}, is the union oftwo disjoint wells. We restrict our attention to dimension 2 so that M denotesthroughout this section the set of 2 x 2 matrices. The case of dimension 3 isconsiderably more complex.

The problem we want to address is the search for minimizers of the varia-tional 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 findingstress-free microstructures I ( v ) = 0 and would like to draw some conclusionson the following issues:

1. conditions on the wells to ensure the existence of nontrivial, stress-freemicrostructures;

2. affine boundary values UF(X) = Fx that may support such microstruc-tures;

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 theset 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 isalso true for i'p(A) = det(A — F) for a fixed matrix F because t/j is also a weakcontinuous function. This fact can also be proved by using the formula thatfollows, 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.


We would like to consider whether it is possible to have nontrivial gradientYoung measures supported in a single well SO(1)F. This is impossible becauseif 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 thisimplies that v has to be trivial, i.e., a delta measure.

A second step is to consider gradient Young measures supported in just twomatrices, F\ and F%, rather than two wells. In this case any probability measurecan 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 aDirac mass.

We now get into the two-well problem fully and treat the three issues indi-cated above sucessively.

Our main result concerning restrictions on the set of two wells is the follow-ing. 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 Diracmass.

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,


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

By continuity, this last circle does not meet the solid unit circle either. This isthe 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

with radius


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 lettingA — 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 havingdet(RFl - PF2) > 0 for all such P. In particular, det(fif\ - P2F2) > 0 for allrotations 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 sumof three nonnegative terms vanishes only if each one vanishes individually. Theconclusion is that if the wells are incompatible, then either A = 0 or A = 1 andin 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 assumeaccordingly that the two wells are compatible. After an appropriate change ofcoordinates we can take

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

and hence

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


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

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 comefrom the Cauchy-Green tensor corresponding to a gradient Young measure vsupported in Kl In order to answer this question, we need to review brieflyhow laminates supported in four matrices can be easily constructed.

Given four matrices, A, B, (7, D, a set of compatibility conditions that allowsus to build a laminate consists of

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

so that

and consequently


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 andthen 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-onerelated to F^1. Namely, after some computations,

The matrix QgF0 is called the reciprocal twin of F^1. Thus we may considerthe 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

we eventually reach every point in D as (cr, A) e [0,1] x [0,1], for A lets us moveup 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 deter-mined.

The next step is to study, for each possible F whose Cauchy-Green tensor liesin D, the set of gradient Young measures supported in the two wells with suchan underlying deformation or at least to say something about the structure orthe complexity of that set. As we will see shortly, this is a much harder problemthat cannot be solved completely except for some special matrices.

Suppose that v = {VX}X€Q is a nonhomogeneous gradient Young measuresupported 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,

where A is the average of A over fJ. Now

are probability measures, and hence reduce to Dirac masses if the Mt are ro-tations. Furthermore, the M, are averages of rotations, and hence lie in theconvex hull of 5O(2).

We now have the equation

Multiplying to the right by

where H - (F^1)2 = I + tei <g> e2, 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 themixed 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 thatmay support nontrivial microstructures such uniqueness fails drastically: thereeven exist continuously distributed gradient Young measures supported in thetwo wells. This is the aim of the final section of this chapter.

4.7 CONTINUOUSLY DISTRIBUT

4.7 Continuously Distributed Laminates

We provide here some insight into how complicated a laminate can be. Wewill show that for a given affine deformation F supporting nontrivial stress-freemicrostructures there is a whole continuum of laminates. As a consequence,by taking convex combinations of these we can build microstructures in whicha continuum of matrices participates. These are continuously distributed lam-inates. We stick to the notation of the previous section. In particular, wedesignate 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 thereciprocal twin. Let the underlying affine deformation F be given that supportsnontrivial, stress-free microstructures such that

The additional constraint (a2 + c2)(62 + d2) > I is a direct consequence of thefact that detF = 1.

THEOREM 4.11. Let A e K. A sufficient condition for the existence ofa stress-free laminate with first moment F whose support contains A and iscontained in a set of at most four matrices from K is the following, accordingto whether

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

By (4.7), we get

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

59


If we put

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

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

is a laminate with barycenter F and whose support contains A — RF0 and iscontained in K.

We would like to solve the inequalities expressed in the statement of theprevious result and see how often those conditions can be met. For the sake ofsimplicity 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, andtherefore the above two families of inequalities will hold when R is a rotation ofangle 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 isorthogonal to rz- This final situation is similar to the one previously discussedbut 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 sameunderlying deformation F. By combining more and more laminates of thesefamilies through convex combinations, we can eventually have continuously dis-tributed laminates supported on the set of the two wells. This makes it clearthat 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 resultsand facts about Young measures used in the previous chapters. We have triedto keep these remarks to a minimum, including only the necessary results andfacts to facilitate understanding of the structure of the complete proofs. Promthis perspective, this chapter should be referred to when the reader might beinterested in a particular technical proof. The complete proofs themselves havenot been included here because they can all be found in [176]. The statementsof 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 andeven some selected proofs, so as to avoid this chapter becoming a mere list ofresults. Nevertheless, we warn the reader that it is not intended to be read inthe same spirit as the preceding chapters.

5.2 The Existence Theorem forYoung Measures

The starting point for the use of Young measures in variational principles is thefollowing existence theorem.

THEOREM 2.2. Let fi C Rw be a measurable set and let Zj : £1 -> Rm bemeasurable functions such that

where g : [0, 00} —» [0, oo] is a continuous, nondecreasing function such thatlimt^oo g(t) = oo. There exist a subsequence, not relabeled, and a family ofprobability measures v = {vx}X£fi (the associated Young measure) dependingmeasurably 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 i1(fJ) ofuniformly bounded sequences in the i1-norm because it is this condition thatenables us to represent weak limits through Young measures. Remember thata 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 veryuseful.

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 generalBanach 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 sequenceof simple (i.e., taking a finite number of values) measurable functions {fj} suchthat 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. Thefirst step consists of showing the existence of the Young measure. This is thenontechnical part of the proof, because it relates to where the Young measurecomes from.

The vector space

is a Banach space under the supremum norm. Its dual space is the space ofRadon measures supported in Rm, denoted M(Hm), with the dual norm of thebounded variation. Since C0(Rm) is separable, we have, according to the abovediscussion, that

under the duality

for t/j e L1(Q;C0(Rm)) and p. £ L~ (ft; M(Rm)). The norm in L%f(fl;M(Rm))is

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 € Rm. For ip € Ll(Cl;C0(R.m)),

It is easy to check that

By the Banach-Alaoglu-Bourbaki theorem there exist some subsequence, notrelabeled, and v e L^(tt; M(Rm)) such that Vj -^ v:

for every $ € L1(fi;C0(Rm)).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 highlytechnical nature.

An important remark to bear in mind when working with Young measuresis that in order to identify the one associated with a particular sequence of func-tions {zj} (obtained perhaps in some constructive way or using some scheme),it is enough to check

for every <p € Co(Rm), where as usual

It is even enough to have

for £ and (p belonging to dense, countable subsets of Ll(fl) and C0(RTO), respec-tively. If this is so for a given family of probability measures v = {vx}x&n and asequence of functions {zj} satisfying (5.1), then v must be the Young measureassociated with {zj} and therefore

for every Caratheodory function i/> such that {i{>(x, Zj(x))} is weakly convergentin L1(ri). The reason for this is that probability measures v are identified bytheir action on Co(Rm). Equation (5.4) identifies each vx for a.e. x 6 Q.

There are two interesting situations for which this remark can have somerelevance. For reference, we include them in the following lemma.

LEMMA 5.3. Assume that we have two sequences, {zj} and {wj}, bothbounded 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 C0(Rm) and £, e L l ( S l ) . Then

The integrand on the right-hand side is an L1(Q)-function and it is integratedover a sequence of sets of vanishing measure. Hence the limit vanishes as j ->• oc,and this in turn implies that the weak limits for {<p(zj}} and {<p(wj)} are thesame. By the above remark, both sequences share the Young measure.

For (ii), use the dominated convergence theorem to examine the difference

for^el,1^) and (,2 e C0(Rm).A helpful example of this situation is the following. Assume {zj} is uni-

formly 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 measurecorresponding 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 L1(fi) is not equi-integrable, one can "bite"(remove) the set where concentrations occur and be left with a well-behavedsequence. This is essentially Chacon's biting lemma. The proof can be made ina very general and abstract setting. We restrict our attention, however, to theframework in which we will be using this fact.

THEOREM 5.4. Let {/j} be a uniformly bounded sequence in L l ( f l ) , where

There exist a subsequence, not relabeled, a nonincreasing sequence of measurablesets 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 discardthe places where concentrations may occur). Lemma 5.1 is used to show theweak convergence outside the exceptional sets.

In order to formalize this important fact, we say that the sequence {fj} Ci1(fi) converges in the biting sense to / e i1(fi) and is denoted

if there exists a nonincreasing sequence of measurable sets {fin} such that|fin| \0and

for all n. We may restate Chacon's biting lemma by saying that a uniformlybounded sequence in L1(fi) contains a subsequence converging in the bitingsense to a function in L1(J7).

The relationship between biting convergence and Young measures is givenin the following theorem.

THEOREM 5.5. Let {zj} be a sequence of functions with associated Youngmeasure v = {v?\x^. If (p '. SI x. Rm —>• R* is a Caratheodory function suchthat the sequence {<p(x,Zj(x))} is uniformly bounded in Lx(fi) then, possibly fora subsequence,

Sketch of the proof. By Chacon's biting lemma, there exists a collection ofsubsets {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 anysubset E C fi, the weak limit has to be !p in (5.5). Since |fin| \ 0, we concludethat <£ = ip a.e. x e fi.

In some circumstances, biting convergence may be improved to weak con-vergence so that Young measures will provide weak limits. This amounts todiscarding the possibility of concentrations. The following lemma gives a nec-essary 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 functionsin 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} converges weakly in Ll(£l) to f if and onlyif

The proof is not difficult. Assuming biting convergence and the failure ofthe 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 particularsituation so that concentrations may arise, we can still deduce some informationthat might be helpful in some circumstances. This is Theorem 2.3. Notice thedirection in which the inequality holds and how it is exactly what we need forweak lower semicontinuity.

THEOREM 2.3. If{zj} is a sequence of measurable functions with associatedYoung measure v = {fz}xeQ, then

for every Caratheodory function ifr, bounded from below, and every measurablesubset E C fi.

Sketch of the proof. If the left-hand side of (5.7) is infinite, there is nothingto be proved. If it is finite, the sequence {i/)(x, Zj(x}}} is a bounded sequence inLl(E). If we set

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 con-centration effects.

Finally, we would like to understand how strong convergence gets translatedinto the Young measure. Since Young measures are used to keep track of oscil-lations and strong convergence rules out this phenomenon, one can expect thatYoung measures associated with strong convergent sequences are trivial. Werestrict 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 weaklyconvergent in L1(O) for p < oo and v — {^x}zen is the associated Youngmeasure. Then Zj —> z strongly in Lp(£i] if and only if vx — &z(x) f°r a-e-x e ft.

It is also helpful to consider Young measures coming from sequences forwhich we have strong convergence only for some components of the sequence.In this case strong convergence reflects the triviality of the Young measure forthe corresponding components.

LEMMA 2.6. Let Zj — (uj,Vj) : O —>• Rd x Rm be a bounded sequence inLP(Q) such that {uj} converges strongly to u in Lp($l). Ijv — {vx}x<E£i is theYoung measure associated with {zj}, then vx = 6U(X) ® ̂ x for a.e. x € Q, where{^x}x€.fi is the Young measure corresponding to {vj}.

5.4 Homogenization and Localization

Homogenization and localization are two basic operations used to analyze Youngmeasures. We first define in general terms homogenization and localization inthe 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 with1901 = \dD — 0. Let {zj} be a sequence of measurable functions over O suchthat

for g a continuous, nondecreasing, nonnegative function with lim^oo g(t) = oo.Let v = {vx}X£n be the Young measure associated with some subsequence, stilldenoted {zj}. There exists a sequence {wj} of measurable functions defined overD 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. Fora given point x & Rm, a sequence of sets {Ei} shrinks suitably to x if there isa > 0 such that each Ei c 5(x, r^), where B(x,Ti} is a ball centered at x andradius TI > 0 and

where r^ —»• 0 as i —> oo. A family of open subsets {ÂJAÂ ^s called a Vitalicovering of f2 C Rm if for every x 6 fJ there exists a sequence {î} of subsetsof the given family that shrinks suitably to x.

THEOREM 5.10. Let A = {A\}XEA be a Vitali covering of Ft. There is asequence \i e A such that

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 suchthat

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

and its homogeneous Young measure is va.

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:

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

{z"} is then defined as a suitable subsequence of the above functions. Techni-calities 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 W1)P(J7). Going back to Lemma 5.9, werealize that all we need is a sequence of functions Wj e W l ip(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 toW1>p(fi). Indeed, we know that Wj should be much more regular than simplybelonging 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 definitionof Wj. The only known way of ensuring the continuity property for Wj is toenforce affine boundary values for Uj. Let Y £ MmxAr and let UY be the affinefunction UY(X) = Yx. If Uj <E W1)P(Q) and Uj - UY £ W0

1>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 followingtheorem.

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\x^i be the Young mea-sure associated with {Vuj}. There exists a sequence {wj}, bounded in W1>P(J1)with the same boundary values, such that the homogeneous Young measure 17associated 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 havea bounded sequence of W1>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 W1>P(J7) in-dependently of j and p and for almost every a 6 J7.

We would also like to incorporate the affine boundary values for the newsequence defining the localized, homogeneous Young measure at almost everypoint in Ct. In order to do this we have the following lemma.

LEMMA 5.13. Let {vj} be a bounded sequence in W1>p(fi) such that thesequence {Vvj} generates the Young measure v = {^x}x&^- Let

so that Vj —^ u in W1>p(f2). There exists a new sequence {uk}, bounded inW l ip(Q), such that {Vwfc} generates the same Young measure v and u^ — u GW0'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 functionsin order to enforce the appropriate boundary values.

If we apply this lemma to the subsequence of {W")P} given in (5.9), we obtainour version of the localization property for gradients.

PROPOSITION 2.4. Let v = {vx}x&^ be a Wl'p(0,}-Young measure. Fora.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.

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 5.3is used to identify the Young measure determined by the particular sequence offunctions tailored for a specific purpose.

5.5 A Remarkable Lemma

This section contains the proof of Lemma 3.7. We have decided to include thisproof 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 thereader of some facts about maximal operators. For any v G (^(R^) we set

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 Lipschitzconstant.

LEMMA 3.7. Let {vj} be a bounded sequence in Wl>p(£l), p > I. Therealways exists another sequence {uj} of Lipschitz functions (uj e W1>00(Q) forall 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.

where

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 ) } , whereM* is the maximal operator of a function and its gradient. This sequence isbounded in LP(HN). Let \i = {A*a;}x6R>jv be its corresponding Young measure(possibly for an appropriate subsequence). Consider the truncation operatorsTfe defined by

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

We have used the monotone convergence theorem for the second limit. Noticethat

is an L1(RAr)-function. We can find a subsequence k(j) —> oo as j —> oo suchthat

On the other hand, by the observation made about these truncation operators af-ter 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. ByLemma 5.14, there exist Lipschitz functions Uj such that Uj = Vj (and thereforeVwj = Vfj) outside Aj and, moreover,

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

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


Step 2 (Approximation). We can assume that Vj —^ u in Wlip(J7) for someu G Wl'p(£l). Moreover, by Lemma 5.13, we can assume that v3•, - u G W0

1>P(Q).Let Wj = Vj — u be extended by 0 to all of R^. By density, we can findZj € 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 consideredrestricted 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 onthe topics covered in this book and new directions which follow after goingthrough the material of this book. The comments also serve to complete ourexposition 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 mainobjectives of this book. In this way we seek to point out new directions wheremore 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 arounded, closed, well-established, well-understood field, and there is room formuch improvement. The final section of this chapter is devoted to the reviewof possible research directions that can be pursued by interested readers.

At the same time, we caution the reader that our references may be incom-plete 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 coveredin the preceding chapters. In this sense, the same principle of brevity has led usin the selection of references, which we have tried to keep specific to the topicsof the book.

We divide our remarks on the bibliography according to the previous fivechapters of the book.

Chapter 1

The expository nature of the first chapter forces us to include several referencesin which the reader may find proofs, further reading, more discussions, etc.,about the foundations of continuum mechanics and elasticity. Since this is notthe aim of this work, our list of references for this chapter is rather short. Thereader interested in these topics will have to perform a more in-depth analysisof the literature.

Some elementary texts which could help someone not yet exposed to contin-uum mechanics or whose main interest is the underlying mathematical problemsbut 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 mathematicalproblems in elasticity more closely and hence are more advanced. The inter-

77

78 BIBLIOGRAPHICAL COMMENTS

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

Chapter 2

Variational problems have always been an important part of applied mathemat-ics. Reference [209] is a nice way to be exposed to such problems for the firsttime. Some interesting, more classical approaches are found in [80] and [183],while for more complete treatments based on the direct method and weak lowersemicontinuity where vector problems are considered, the reader may look at[41], [65], and [67]. Reference [110] is encyclopedic in character but deals withthe indirect method through optimality conditions associated with minimizers.A full discussion of frame indifference, behavior for extreme deformations, andappropriate physical requirements is contained in [52].

Young measures (also called parametrized measures) were introduced inanalysis many years ago in [211], [212], and [213]. Since then, many appli-cations in applied mathematics and analysis have been explored. More recentbooks 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 lowersemicontinuity has systematically been considered in [28], [88], and [171]. Seealso [129] and [201].

The notion of quasi convexity and its pivotal role in weak lower semiconti-nuity for vector problems was identified in [150]. See [151] for a full analysis ofthis issue. Recently, the quasi convexity condition has been examined in moregeneral settings in [4], [25], and [140]. This condition was introduced in termsof gradient Young measures in [171]. General existence results similar to theone included here can be found in many places, in particular in almost all theabove references dealing with vector problems. See also [133], [143], and [144].Reference [112] contains a different approach to existence theorems in nonlinearelasticity.

If the reader would like to study more deeply the facts contained in theappendix, 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 andproved in [181] and [182]. Since then, polyconvexity has been reexamined fromthe point of view of weak continuity in different settings and under various sets ofassumptions in an attempt to improve previous and known results. In particular,the effort to push weak continuity of determinants to the limit, and consequently

BIBLIOGRAPHICAL COMMENTS 79

obtain new weak lower semicontinuity results and existence theorems, is takenas 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 orderare analyzed in [19].

The existence theorem presented in section 3.4 is classical, except perhapsfor the case p = 3, which is based on a quite remarkable lemma (see [176],[214]). Concerning the issue of discontinuous deformations and cavitation, seesome comments in the final section of this chapter. The injectivity and inter-penetration 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 par-ticular, 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 effortsin many distinct areas in the past 15 years. The incredible number of worksdealing with this topic almost certainly makes our list appear incomplete in thisregard. In the context of phase transitions in crystals, nonconvexity was raisedin 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 anumber of important mathematical developments: [94], [117], [118], [123], and[155]. In particular, the analysis of Young measures generated by gradientswas 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 ideasand the description and interaction of energy minimization related to materialsmodels are [22], [157], and [179].

Some papers dealing specifically with laminates and the two-well problemare [147], [148], [160], [169], [170], [200], and [216]. The notion of the (Hn)condition was first recognized and studied in [66]. The remarkable example ofthe laminate supported on four matrices not pairwise rank-one related, as wellas similar examples, can be explored in [32], [149], [206], and in other referencesnot listed here.

Chapter 5

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

80 BIBLIOGRAPHICAL COMMENTS

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

Additional Areas of Research

It is probably worthwhile to point out several areas that can be pursued inmore depth once the main ideas we have tried to convey in this book are un-derstood. In most cases, challenging and fascinating problems are waiting tobe answered. We have selected a number of key references for each area, againwithout attempting to be exhaustive either in the list of different areas or in thereferences within each area. A survey with very interesting comments on someof 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, 6Affine functionals, 18Affine Lipschitz function, 21

Banach spaces, 11, 22Banach-Alaoglu-Bourbaki theorem,

11, 66Blow-up argument, 72Boundary conditions, 10, 25

affine boundary values, 21, 73Dirichlet boundary conditions,

32environmental conditions, 9global condition of place, 1, 4,

9Bounded variation, 65

Calculus of variations, 16Caratheodory function, 17, 32Cauchy theorem, 2Cauchy-Green tensor, 55Cavitation, 79Chacon's biting lemma, 67

biting convergence, 68exceptional sets, 68

Characteristic function, 15Coerciveness, 10, 17, 19, 21, 33, 37

coercivity, 31, 34coercivity exponent, 10growth exponent, 31

Cofactor matrix, 3, 6Compactness, 11Compactness theorem, 24, 30Compactness theorem of Sobolev spaces,

20Compatibility condition, 48, 55Compensated compactness, 80

Computations, 80Concentration effects, 23, 64Concentrations, 68Constitutive equation, 4Continuum mechanics, 1, 77Convexity

convex envelopes, 80convex functions, 18convex hull of 50(2), 51nonconvexity, 79

Crystal structure, 42change of lattice basis, 42change of shape, 41conjugate group of GL(Z3), 42lattice basis, 42lattice vectors, 41point group, 42reciprocal twin, 56

Deformation, 2, 9deformation gradient, 2, 4deformed configuration, 2discontinuous deformations, 33,

79extreme deformations, 10extreme strains, 5periodic Lipschitz deformation,

43underlying deformation, 57

Direct method, 10, 78Duality, 65Dynamics, 80

Elastic bar, 46Energy

energy densities, 1, 9, 25free energy, 41

97

INDEX

Energy (cont'd.)stored-energy functions, 4strain energy, 4surface energy, 80total energy functional, 4

Equilibriumequilibrium configuration, 1, 4,

9, 10, 25equilibrium equation, 4static equilibrium, 1, 2

Euler variable, 3Euler-Lagrange system, 4Existence

existence results, 78existence theorem, 19, 32existence without convexity, 80nonexistence, 45

Fatou's lemma, 12, 34Forces

body forces, 2, 4surface forces, 2, 4

Fracture and cavitation, 80Frame indifference, 5, 10

Generalized variational principles, 41Gradient requirement, 47

Holder's inequality, 22, 30Homogenization, 21

Indirect method, 78Injectivity, 33, 79

Jensen's inequality, 16, 18, 24, 26,50

Laminates, 41(Hi) conditions, 49(Hn) conditions, 79continuously distributed lami-

nates, 59continuum of laminates, 59fine layers, 46layers within layers, 49

Localization result, 18

Maximal operators, 74, 80Microstructure, 41, 79

fine-phase mixtures, 41, 46stress-free microstructure, 47, 51

Minimizersminimizing sequences, 10, 12of the total energy, 4, 9uniqueness of, 20

Minors, 26, 31minor relation, 51

Mixed problems, 1Mooney-Rivlin materials, 6, 37

Neo-Hookean materials, 6, 38Nonlinear functionals, 13

Ogden material, 6, 34Optimality conditions, 78Orientation-preserving requirement,

2, 33Oscillatory minimizing sequences, 41

Phase transitions, 79, 80change of stability, 42critical temperature, 42stable phase, 42transition temperature, 42

Poincare's inequality, 10, 20, 24, 73Polar decomposition, 7Potential wells, 43

single well, 52two-well problem, 51, 79

Probability measures, 14Pure traction problems, 1

Quasi convexityclosed W^1'°°-quasi convexity, 21closed Vt^'P-quasi convexity, 19-

21lack of, 45

Quasi-affine functions, 26

Radon measures, 65Rank-one matrix, 15Reference configuration, 2, 9Regularity, 80Relaxation, 80

98

INDEX 99

Response function, 4Riemann-Lebesgue lemma, 21

Singular values, 6, 35Sobolev spaces, 10, 22St. Venant-Kirchhoff materials, 5,

38Stress

Cauchy stress tensor, 2, 4, 7principal stretches, 7stress principle of Euler and Cauchy,

2Strong convergence, 20, 70

strong lower semicontinuity, 12Strongly measurable, 64

Tensor product, 15Thermodynamics, 78Truncation operators, 67, 75

Uniform bounds, 21Uniqueness, 58

Vector problems, 78Vitali covering lemma, 71

Weak continuity, 55weak continuity of determinants,

78

weak continuous functionals,28

weak continuous functionals ofhigher order, 79

Weak convergence, 10weak * convergence of measures,

49weak compactness, 23weak limit, 13, 15, 17, 20weak topologies, 11

Weak convergence in L1(O), 64Dunford-Pettis criterion, 23equi-integrability property, 23

Weak lower semicontinuity, 10, 12Weakly measurable, 65

Young measurescharacterization of, 80homogeneous gradient Young mea-

sures, 52homogeneous W1'00-Young mea-

sures, 31homogeneous Wl'p-Young mea-

sures, 19, 26, 31parametrized measures, 78W1>P(Q)-Young measures, 18, 41

Zero set, 43, 47

variational methods in nonlinear elasticity

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