Variational Methods in Nonlinear Elasticity

3p/(4p - 3), and {det(Vu.,-)} is bounded in Lr($l), r > 1. Proof. The proof reduces to showing in each case that

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

3.4

Existence Theorems

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

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

3.4

EXISTENCE THEOREMS

33

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

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

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

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

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

34

CHAPTER 3

POLYCONVEXITY AND EXISTENCE THEOREMS

w is nonnegative and by Fatou's lemma,

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

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

By Fatou's lemma,

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

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

3.5

Hyperelastic Materials

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

3.5

HYPERELASTIC MATERIALS

35

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

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

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

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

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

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

is convex.

36

CHAPTER 3 POLYCONVEXITY AND EXISTENCE THEOREMS

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

Since g is convex, it suffices to show that

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

However, since g is symmetric we can actually put

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

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

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

3.5

HYPERELASTIC MATERIALS

37

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

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

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

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

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

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

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

38

CHAPTER 3

POLYCONVEXITY AND EXISTENCE THEOREMS

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

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

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

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

It is a matter of careful arithmetic to check that

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

and in this case,

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

3.5

HYPERELASTIC MATERIALS

39

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

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Chapter 4

Rank-one Convexity and Microstructure 4.1

Introduction

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

4.2

Elastic Crystals

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

42

CHAPTER 4 RANK-ONE CONVEXITY AND MICROSTRUCTURE

crystal given by matrix N. We assume that $ is frame indifferent as usual ($(JV) = $(QN) for any rotation Q), but it should also be invariant under any change of lattice basis: if N' is an equivalent choice of lattice basis (equivalent in the sense that the positions of the atoms are the same for TV and for N'), then <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 matrix N is fixed, we define the energy density per unit reference volume by putting

Altogether we have the invariance

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

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

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

4.3

LACK OF QUASI CONVEXITY

43

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

4.3

Lack of Quasi Convexity

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

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

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

44

CHAPTER 4

RANK-ONE CONVEXITY AND MICROSTRUCTURE

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

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

and

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

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

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

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

4.3

LACK OF QUASI CONVEXITY

45

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

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

46

4.4

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RANK-ONE CONVEXITY AND MICROSTRUCTURE

Generalized Variational Principles

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

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

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

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

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

4.5

LAMINATES

47

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

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

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

4.5

Laminates

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

48

CHAPTER 4

RANK-ONE CONVEXITY AND MICROSTRUCTURE

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

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

is

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

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

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

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

4.5

LAMINATES

49

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

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

the set of pairs {(si, Zl)}l• oc. In this case we consider infinite-order laminates. DEFINITION 4.5. Let v be a probability measure supported on M and let K = supp(z/) be a compact set. v is a laminate if there exists a sequence of sets of pairs {($ , Y f ) } ^ 2), verifying the (Hk) condition such that Si ^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 the matrices

50

CHAPTER 4

RANK-ONE CONVEXITY AND MICROSTRUCTURE

We claim that the probability measure

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

• R* is rank-one convex and v is a laminate, then

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

4.6

The Two-Well Problem

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

4.6

THE TWO-WELL PROBLEM

51

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

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

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

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

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

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

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

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

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

After some algebra,

4.6

THE TWO-WELL PROBLEM

53

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

with radius

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

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

As an intermediate step, we have used the fact

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

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or

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

If we consider again the decomposition

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

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

and hence

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

4.6

THE TWO-WELL PROBLEM

55

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

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

We have the inequalities

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

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

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

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

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Let v be the laminate supported in K:

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

and the corresponding Cauchy-Green tensor

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

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

In this case one obtains

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

is the reciprocal twin of

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

4.6

THE TWO-WELL PROBLEM

57

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

where

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

where

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

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where A is the average of A over fJ. Now

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

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

4.7 CONTINUOUSLY DISTRIBUT

4.7

59

Continuously Distributed Laminates

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

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

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

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

By (4.7), we get

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

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If we put

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

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

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

and we set

with

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

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

or

4.7

CONTINUOUSLY DISTRIBUTED LAMINATES

61

The two lines

intersect at the unique point

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

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

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

or

If we consider the line

then r\ and r3 meet in the point

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

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Chapter 5

Technical Remarks 5.1

Introduction

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

5.2

The Existence Theorem for Young Measures

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

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

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

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

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

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

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

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

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

5.2

THE EXISTENCE THEOREM FOR YOUNG MEASURES

65

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

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

under the duality

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

In this case we have the duality

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

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

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

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CHAPTER 5 TECHNICAL REMARKS

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

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

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

It is even enough to have

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

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

5.3

BITING. WEAK. AND STRONG CONVERGENCE

67

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

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

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

if

5.3

Biting, Weak, and Strong Convergence

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

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

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

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

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

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

5.3

BITING, WEAK, AND STRONG CONVERGENCE

69

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

Moreover, the whole sequence {fj} if

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

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

then

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

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

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

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then by Theorem 5.5

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

and this proves the statement. Strict inequality in (5.7) occurs when the sequence {ip(x,Zj(x))} develops concentrations. In this sense we say that Young measures do not capture concentration effects. Finally, we would like to understand how strong convergence gets translated into the Young measure. Since Young measures are used to keep track of oscillations and strong convergence rules out this phenomenon, one can expect that Young measures associated with strong convergent sequences are trivial. We restrict our attention to the case in which g(i) = tp. LEMMA 5.8. Let {zj} be a sequence in Lp(Sl] such that {\ZJ\P} is weakly convergent in L 1 (O) for p < oo and v — {^x}zen is the associated Young measure. Then Zj —> z strongly in Lp(£i] if and only if vx — &z(x) f°r a-ex e ft. It is also helpful to consider Young measures coming from sequences for which we have strong convergence only for some components of the sequence. In this case strong convergence reflects the triviality of the Young measure for the corresponding components. LEMMA 2.6. Let Zj — (uj,Vj) : O —>• Rd x Rm be a bounded sequence in P L (Q) such that {uj} converges strongly to u in Lp($l). Ijv — {vx}x<E£i is the Young 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 Young measures. We first define in general terms homogenization and localization in the next two lemmas and then apply them to the situation of gradients. LEMMA 5.9. Let 0 and D be two regular, bounded domains in TtN with 1901 = \dD — 0. Let {zj} be a sequence of measurable functions over O such that for g a continuous, nondecreasing, nonnegative function with lim^oo g(t) = oo. Let v = {vx}X£n be the Young measure associated with some subsequence, still denoted {zj}. There exists a sequence {wj} of measurable functions defined over D such that

5.4

HOMOGENIZATION AND LOCALIZATION

71

and its homogeneous Young measure v is given by

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

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

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

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

and its homogeneous Young measure is va.

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

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

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

Condition (5.8) can be fulfilled by simply putting

for x G dij + €{j£l. It may not be true, however, that such a Wj belongs to W 1>p (fi). Indeed, we know that Wj should be much more regular than simply belonging to Lp(£l). For p sufficiently large, Wj even ought to be continuous, and this might not be the case if we are not more careful about our definition of Wj. The only known way of ensuring the continuity property for Wj is to enforce affine boundary values for Uj. Let Y £ M mxAr and let UY be the affine function UY(X) = Yx. If Uj <E W 1)P (Q) and Uj - UY £ W01>P(O), the function

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

Lemma 2.7 is a direct corollary of this theorem.

5.4

HOMOGENIZATION AND LOCALIZATION

73

For the localization property, we should have

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

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

Take

where the constant M0(^p is chosen so that

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

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

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CHAPTER 5 TECHNICAL REMARKS

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

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

5.5

A Remarkable Lemma

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

where

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

and, in particular, for any A > 0,

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

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

5.5

A REMARKABLE LEMMA

75

Proof.

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

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

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

Let

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

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

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

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

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

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

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

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

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

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

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79

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

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

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

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

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

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

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

Compatibility condition, 48, 55 Compensated compactness, 80 97

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

INDEX

98

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

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

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

99

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