Hashin-Shtrikman bounds for multiphase composites and their attainability

Transcription

1 ARMA manuscript No. (will be inserted by the editor) Hashin-Shtrikman bounds for multiphase composites and their attainability Liping Liu October 16, 2007 Manuscript submitted to Archive for Rational Mechanics and Analysis Abstract This paper addresses the attainability of the Hashin-Shtrikman bounds for multiphase composites, including those of conductive materials and elastic materials. It presents a new derivation of these bounds that yield a necessary and sufficient condition for optimal microstructures. A key idea is a simple characterization of the gradient oung measures associated with optimal microstructures. Contents 1. Introduction Hashin-Shtrikman bounds for multiphase composites A necessary and sufficient condition for the attainment of Hashin-Shtrikman bounds Optimal microstructures: sequential E-inclusions Applications An outer bound for sequential E-inclusions An inner bound for sequential E-inclusions Composites of conductive materials Composites of elastic materials Summary and discussions Introduction Since the seminal works of Hashin & Shtrikman [16, 17], finding optimal bounds on the effective properties, with or without restriction on

2 2 Liping Liu the volume fraction, has become one of the central problems in the theory of composites (Milton [26]). The traditional approach of finding optimal bounds is first to derive a structure-independent bound, and then to study if this bound is attainable and if so, by what kind of structures/microstructures. The optimal bounds can in general be categorized into two types according to the methods of derivation: the Hashin-Shtrikman (HS) bounds and the translation bounds (Tartar [34]; Lurie & Cherkaev [23]). The known optimal microstructures are coated spheres and ellipsoids (Hashin [15]; Milton [25]), multi-coated spheres (Lurie & Cherkaev [24]) and multi-rank laminations. By this approach, the G-closure problem (Lurie & Cherkaev [23]; Tartar [36]) for two-phase well-ordered conductive materials has been resolved (Tartar [36]; Milton & Kohn [27]; Grabovsky [12]). However, for multi-phase composites of general materials, little is known about the attainability of the best bounds which, in many situations, are the HS bounds. The main purpose of this paper is to address the attainability of the HS bounds for general multi-phase composites. The approach of this paper is motivated by an observation that a given structure/microstructure is often optimal not only for the lower HS bound but also for the upper one, and is optimal for the lower or upper HS bounds of many different material systems. We therefore consider the problem of characterizing all the material systems for which a given structure/microstructure is optimal. This point of view appears new and is important for attacking the problem of optimal bounds. To explore the consequence of this alternate approach, we begin with a derivation of the HS bounds for multiphase composites. The derivation is based on the fact that a solution of (2.12) can be given by the gradient of a scalar function under some hypothesis on the polarization and material. We therefore avoid the difficulties associated with the concentration-factor in Walpole [38]. This derivation explains why the common choice of trial polarizations in the HS variational principle happens to be the right one. More importantly, it provides us a simple characterization of structures/microstructures that attain the bounds. In a periodic setting and in terms of a simple potential problem, the main condition is that the second gradient of the potential is constant in all but one phases. (The constants in different phases can be different.) At the first sight, this condition (cf., equation (3.10)) on the attaining structures seems too restrictive to be ever satisfied by any structure. This is however deceitful. In fact the familiar constructions of coated spheres and ellipsoids, multicoated spheres and multirank laminations satisfy this condition in the sense specified in Section 4 if they indeed attain the HS bounds. In a separate publication (Liu, James & Leo [22]) we present a method for constructing these special structures which we call periodic E-inclusions. A periodic structure attains the HS bounds if and only if the structure is a corresponding periodic E-inclusion. Therefore, from the attainability of the lower HS bound we can infer the attainability of the upper one, and the attainability of the bounds for many different material systems. We

3 Title Suppressed Due to Excessive Length 3 then adapt the arguments to sequences of structures or microstructures. It has proven to be useful to describe microstructures by gradient oung measures (Tartar [35]; Ball [6]). Based on the gradient field of a periodic E-inclusion, we define a particular form of gradient oung measures as sequential E-inclusions (cf., (4.2)). rom the estimates implied by the HS variational principles, we show that any microstructures that attain the HS bounds if, and only if their gradient fields generate a corresponding sequential E-inclusion. rom the basic relation between gradient oung measures and quasiconvex functions (Kinderlehrer & Pedregal [19, 20]), we can prescribe sequential E-inclusions, and therefore attainable HS bounds in terms of quasiconex functions. rom this viewpoint, the attainability of the HS bounds can be attacked by the standard approach in microstructure theory: we construct particular microstructures to find inner bounds of sequential E-inclusions, and use quasiconvex functions to find outer bounds of sequential E-inclusions. ollowing the above lines, we find an outer and inner bound for sequential E-inclusions. The outer bound is obtained using the null Lagrangian IR n n sym X m (Tr(X)X X 2 )m for any m IR n. The inner bound is based on a convex property of gradient oung measures (Kinderlehrer & Pedregal [19]) and the existence of periodic E-inclusions corresponding to a single matrix (Liu, James & Leo [22]). rom these bounds on sequential E-inclusions, we obtain an outer and inner bound on the attainable HS bounds for composites of any number of phases and in any dimensional space. We remark that the individual phases and the composites are not necessarily isotropic, though some symmetries on the softest (resp. stiffest) phase are required for the lower (resp. upper) HS bound. These attainability and non-attainability results appear new. When specialized to isotropic composites of isotropic phases, our attainability results recover what were first shown by Milton [25]. We remark that our result is mainly on the attainability of the HS bounds. The bounds (2.37) and the dual bounds (3.1) are known for various special cases, see Walpole [38]; Milton [25]; Allaire & Kohn [4, 5]; and Nesi [29]. Mentions should be made of the works of Grabovsky [13, 14] who, based on the translation method and for two-phase composites, derived a set of optimal conditions which are closely related to ours, see also Albin, Cherkaev & Nesi [1] for two dimensional three-phase composites and Silvestre [31] for the cross-property bounds. The paper is organized as follows. In Section 2 we derive the HS bounds for multiphase composites. In Section 3 we show the equivalence between the attainability of the HS bounds by a periodic structure and the existence of a corresponding periodic E-inclusion. Note that it is not the periodic structure per se that is needed in our derivation. or sequences of structures or microstructures, this result remains valid if one formulates it in terms of gradient oung measures. This issue is addressed in Section 4. In Section 5, we find an outer and inner bound for sequential E-inclusions, and hence an

4 4 Liping Liu outer and inner bound for the attainable HS bounds. inally, in Section 6 we summarize our results and discuss the directions of generalization. 2. Hashin-Shtrikman bounds for multiphase composites Let L (resp. L el ) be the collection of all positive definite self-adjoint tensors (resp. all positive semi-definite linear elasticity tensors), and Ω i (i = 0,, N) be a measurable disjoint subdivision of the unit cell = (0, 1) n. We consider a periodic (N + 1)-phase composite defined by L(x, O) = L i on Ω i (i = 0, 1,, N), (2.1) where the self-adjoint and positive semi-definite tensors L i : IR m n IR m n (i = 0,, N), either all in L or all in L el, describe the material properties of individual phases, O = (Ω 0,, Ω N ) is referred to as the structure of the composite, and θ i = Ω i / (i = 0,, N) are the volume fractions. Note that the tensors L i (i = 0,, N) include but are not limited to linear elasticity tensors. The benefit of this general setting is that many problems, including the conductivity problem and cross-property problem, can be treated simultaneously. or a given structure O = (Ω 0,, Ω N ) and applied average field IR m n, the effective tensor L e (O) of this composite is given by L e (O) = min u Wper(,IR 1,2 m ) ( u + ) L(x, O)( u + )dx, (2.2) where 1 V dx= volume(v ) dx is the average value of the integrand in V region V, and a minimizer of the right-hand side, which is unique within an additive constant and is denoted by u W 1,2 per(, IR m ), solves the following equation div [ L(x, O)( u + ) ] = 0 on periodic boundary conditions on. (2.3) Equivalently, the effective tensor is given by (Christensen [9]) L e (O) = L(x, O)( u + )dx IR m n. (2.4) In general the effective tensor L e (O) defined by equation (2.2) or (2.4) depends on many factors, such as the shape, topology and volume fractions (to mention a few), of the structure O. Therefore, it is often more useful to find the bounds on the effective tensors that depend only on some simple features of the structure, say, volume fractions, than to calculate the exact effective tensor.

5 Title Suppressed Due to Excessive Length 5 To proceed, we introduce a few notations. Denote by N ( ) and R( ) the null space and the range space of a self-adjoint linear mapping ( ), respectively. or two self-adjoint linear mappings T 1, T 2 : IR m n IR m n, we write T 1 (resp. )T 2 if T 1 T 2 is positive (resp. negative) semidefinite. We follow the conventions 1/ = 0, 1/0 =, and interpret the self-adjoint inverse T 1 of a self-adjoint positive semi-definite linear mapping T : IR m n IR m n as Z T 1 Z = Clearly, (T 1 ) 1 = T and sup 2Z X X TX} Z IR m n. (2.5) X IR m n Z T 1 Z c Z 2 Z IR m n, (2.6) where c > 0 is independent of Z. Clearly, Z / R(T) if and only if inequality (2.6) holds for arbitrary c. In this case we write Z T 1 Z =. It is useful to notice Lemma 2.1. Let L c, L 0,, L N be all in L or all in L el. Consider (N +1)- phase composites (2.1) with the effective tensor L e (O) given by (2.2). If 0 L c L(x, O) (resp. L c L(x, O) 0), and θ i = Ω i / = 0 for all i = 0,, N, then which is equivalent to N (L e (O) L c ) = N i=0n (L i L c ), (2.7) R(L e (O) L c ) = N i=0r(l i L c ). (2.8) Proof. In the case 0 L c L(x, O), from equations (2.2) and (2.3) we have (L e (O) L c ) = ( u + ) (L(x, O) L c )( u + )dx + u L c u dx 0. (2.9) Thus, L e (O) L c. If N (L e (O)L c ), u = const. is clearly a minimizer of the right-hand side of (2.2). Thus, equation (2.9) implies 0 = θ i (L i L c ). Therefore, N (L i L c ) for all i = 0,, N. Conversely, if N i=0 N (L i L c ), then L i = L c for all i = 0,, N. Thus, u = const. solves (2.3). Therefore, L e (O) = L c, i.e., N (L e (O) L c ). We thus have proved (2.7). or equation (2.8), we only need to show that [ N i=0 N (L i L c )] = N i=0 R(L i L c ). This is standard. The proof for

6 6 Liping Liu the case 0 L c L(x, O) is now completed. The case L c L(x, O) 0 can be handled similarly and will not be repeated here. We now reformulate the effective tensor through the well-known HS variational principles. Let L c L L el be a comparison material. We recall the classic Wiener bounds H Θ L e (O) L Θ, (2.10) where L Θ = N i=0 θ il i and H Θ = [ N i=0 θ il 1 i ] 1 are the arithmetic mean and harmonic mean, respectively. rom (2.10), it is clear that if 0 L c L(x, O) (resp. L c L(x, O) 0), then L e (O) L c (resp. L e (O) L c ). Let E L (P) = [ vp L c v P + P (L(x, O) L c ) 1 P ] dx E U (P) = [ vp L c v P + P (L c L(x, O)) 1 P ] dx, (2.11) where P L 2 per(, IR m n ) and v P Wper(, 1,2 IR m ) satisfies div[l c v P + P] = 0 on periodic boundary conditions on. (2.12) By the divergence theorem, we have 0 v P L c v P dx = P v P dx P L 1 c P dx. (2.13) In particular, the second inequality follows from (P L 1 c P v P L c v P ) dx = (P L c v P ) L 1 c (P L c v P ) dx 0. With the inverse of a positive semi-definite linear mapping interpreted as (2.5), we have Lemma 2.2. (HS variational principles) Let L c, L 0,, L N be all in L or all in L el, L e (O) be given by (2.2), P 0 R(L e (O) L c ), P(P 0 ) := P L 2 per(, IR m n ) : P(x)dx = P 0 }, (2.14) u be given by (2.3) with = (L e (O) L c ) 1 P 0, and P (x) = (L(x, O) L c )( u + ) P(P 0 ). (2.15) or a structure O = (Ω 0,, Ω N ),

7 Title Suppressed Due to Excessive Length 7 (i) if L(x, O) L c and P P(P 0 ) (cf., (2.11)), E L (P) P 0 (L e (O) L c ) 1 P 0 c where c > 0 is independent of P, and (ii) if L(x, O) L c and P P(P 0 ), E U (P) P 0 (L c L e (O)) 1 P 0 c where c > 0 is independent of P. P P 2 dx, (2.16) P P 2 dx, (2.17) be a solu- Proof. We follow Milton & Kohn [27] for the proof. Let v P tion of (2.12) for P. Plugging (2.15) into (2.12), we verify v P = u. (2.18) rom u dx = 0 and (2.4), we verify P (x) dx = L(x, O)( u + ) dx L c = P 0. (2.19) or any P P(P 0 ), by (2.18) we have v P L c u P dx = P u P dx = v P P dx. (2.20) We now show equation (2.16). Since L(x, O)L c 0, by equations (2.5)- (2.6) we have c P P 2 dx E L (P P ) = E L (P) + (P 2P) [(L(x, O) L c ) 1 P u P ] dx = E L (P) + [ (P 2P) dx] = E L (P) P 0 (L e (O) L c ) 1 P 0, (2.21) where c > 0 is independent of P, and in the first equality we have used equations (2.18), (2.20), and the fact u L c u dx = P u dx, see (2.13). To show equation (2.17), we notice, from L c L(x, O) 0 and equation (2.5), that for any X R(L c ), X (L c L(x, O)) 1 X X L 1 c X = sup 2Z X Z (L c L(x, O))Z} X L 1 c X Z IR m n X L 1 c L(x, O)L 1 c X, (2.22)

8 8 Liping Liu where the inequality follows by choosing Z = L 1 c X. Therefore, from (2.22) and the second inequality in (2.13) we have E U (P P ) c P P 2 dx, (2.23) where c > 0 is some constant independent of P. Then, we obtain (2.17) by the same calculations as in (2.21). The proof of the lemma is now completed. The classical HS bounds are obtained by choosing a comparison material L c and a trial polarization P. Then equations (2.16) and (2.17) yield bounds on the effective tensor L e (O). If the right trial polarization P are chosen, the bounds depend only on the volume fractions of the structures, and in many but not all situations, are attainable. We now present a new derivation of the HS bounds for (N + 1)-phase composites. This derivation will provide us a necessary and sufficient condition for the bounds to be attainable. We first consider a scalar function u Wper( 2,2 ) defined as a solution of u = f(x) on periodic boundary conditions on, (2.24) where f L 2 per( ). The solvability of (2.24) requires f(x)dx = 0. (2.25) By ourier analysis, we can represent the second gradient of a solution of (2.24) as u(x) = k K\0} where K = (2π)ZZ n is the reciprocal lattice of ZZ n and ˆf(k) = f(x) exp(ik x) dx. k k ˆf(k) k 2 exp(ik x), (2.26) A key observation of our derivation is that for some special L c and P, a solution of (2.12) is given by the gradient of the scalar potential (2.24). To see this, let δ ij (i, j = 1,, n) be the components of the identity matrix I, m = n 2, 0 a IR, and (L c ) piqj = µ c 1δ ij δ pq + µ c 2δ pj δ iq + λ c δ ip δ jq, P (x) = k c I[a + f(x)], (2.27) where k c = µ c 1 + µ c 2 + λ c > 0. Note that L c L (resp. L c L el ) implies the constants µ c 1, µ c 2, λ c satisfy µ c 1 > µ c 2 (resp. µ c 1 = µ c 2), µ c 1 + µ c 2 > 0 and λ c > µc 1 + µ c 2. (2.28) n

9 Title Suppressed Due to Excessive Length 9 Again by ourier analysis (Khachaturyan [18]), we can represent the gradient of a solution of (2.12) with P = P in (2.27) as v P (x) = k K\0} By comparing (2.26) with (2.29), it is clear that k k ˆf(k) k 2 exp(ik x). (2.29) v P (x) = u(x). (2.30) Assuming I R(L e (O) L c ) and choosing P (x) as a trial polarization of the right-hand sides of equations (2.16) and (2.17), we obtain I (L e (O) L c ) 1 I E L (P ) = I 1 (f) + I 2 (f) if L(x, O) L c, I (L c L e (O)) 1 I E U (P ) = I 1 (f) I 2 (f) if L(x, O) L c. (2.31) Here the integrals I 1 (f) and I 2 (f) are as follows: I 1 (f) := 1 a 2 2 v P L c v P dx = 1 k c a 2 f(x) 2 dx, (2.32) k c where the equality follows from (2.30), (2.24), and the fact that u 2 dx = u 2 dx. Also, I 2 (f) := 1 a 2 (a + f(x)) 2 I (L(x, O) L c ) 1 Idx = 1 N a 2 θ i c i (a + f(x)) Ω 2 dx, (2.33) i where θ i = Ω i i=0 and c i = I (L i L c ) 1 I if L i L c I (L c L i ) 1. (2.34) I if L c L i Note that only the volume fractions of the structure O appear in I 1 (f) and I 2 (f). To achieve the best bounds, we minimize the right-hand sides of (2.31) over all admissible f(x). That is, mini 1 (f) + I 2 (f) : f(x)dx = 0} mini 1 (f) I 2 (f) : if L(x, O) L c. (2.35) f(x)dx = 0} if L(x, O) L c Using the method of Lagrangian multiplier, we obtain the minimizers, which have the same form for both minimization problems, as follows: f (x) = 1 N p i χ Ωi, p i = a[ γ(1/k c + c i ) 1], γ = θ i. (2.36) 1/k c + c i i=0 i=0

10 10 Liping Liu where χ Ωi is the characteristic function of region Ω i. With f (x) chosen, equation (2.31) reads I (L e (O) L c ) 1 I c := 1 γ 1 kc I (L c L e (O)) 1 I c := 1 γ + 1 k c if L(x, O) L c. if L(x, O) L c (2.37) Note that the two bounds in (2.37) do not contradict each other since the comparison materials L c for these two bounds are different for the given L 0,, L N, and hence that numbers c, γ, k c are not the same in these two bounds. Moreover, the bounds are structure-independent in the sense that the numbers c depend only on the material properties L i and the volume fractions θ i of the structures O. In the next section we study if the equality part of in (2.37) can hold, and if so, for what kind of periodic structures O. 3. A necessary and sufficient condition for the attainment of Hashin-Shtrikman bounds The bounds in (2.37) are inconvenient since they are restrictions on I (L e (O) L c ) 1 I. In this section we first convert the bounds (2.37) into direct restrictions on L e (O) using the duality of convex functions. We then show a necessary and sufficient condition for the attainment of these bounds restricted to periodic structures. As far as the bounds are concerned, the restriction to periodic structures has no loss of generality, as is well-known that any effective tensor can be approximated arbitrarily well by those of periodic composites. At the same time, the condition being necessary and sufficient implies a set of material-independent parameters that relate the attainability of these bounds for different materials. Therefore, it suffices to study the restrictions on this set of material-independent parameters to describe the attainability of these bounds for all materials of the type specified below. We now rephrase the bounds (2.37) as direct restrictions on L e (O). Theorem 3.1. Consider a periodic (N +1)-phase composite defined by (2.1) with volume fractions θ i = Ω i / (i = 0,, N). If L c, L 0,, L N are all in L or all in L el, L c satisfies (2.27) and (2.28), and I R(L e (O)L c ) (see Lemma 2.1 for a sufficient condition), then the effective tensor L e (O) given by (2.2) satisfies (2.10) and (2.37). More, the bounds (2.37) are equivalent to (L e (O) L c ) Tr() 2 / c if L(x, O) L c (L e (O) L c ) Tr() 2 (3.1) / c if L(x, O) L c for any IR n n. urther, one of the inequalities in (3.1) holds as an equality for R(L e (O) L c ) with Tr() 0 if, and only if the corre-

11 Title Suppressed Due to Excessive Length 11 sponding inequality in (2.37) holds as an equality. In this case, we have Tr() = 1 (L e (O) L c ) 1 I. (3.2) c Proof. We first consider the case L(x, O) L c. To show (2.37) implies (3.1), for any structure O, by the fact L e (O) L c, (2.5) and (2.37) we have sup 2 I (L e (O) L c )} = I (L e (O) L c ) 1 I c. (3.3) IR n n Choosing with Tr() = c, we see that (L e (O) L c ) c = Tr() 2 / c, which, by multiplying by a such that atr() = c, in fact holds for any with Tr() 0. If Tr() = 0, the first bound in (3.1) is obvious. We therefore conclude the first bound in (3.1) for all IR n n. urther, = (L e (O) L c ) 1 I is a maximizer of the left-hand side of (3.3). Therefore, if the first bound in (2.37) holds as an equality, we have Tr( ) = I (L e (O) L c ) 1 I = c 0, and (L e (O) L c ) = I (L e (O) L c ) 1 I = Tr( ) 2 / c. Thus, the first inequality in (3.1) holds as an equality for a with any a 0, i.e., all that satisfy (3.2). Conversely, from the first bound in (3.1), choosing = (L e (O) L c ) 1 I we obtain the first bound in (2.37). urther, if the first bound in (3.1) holds as an equality for R(L e (O) L c ) with Tr() 0, we have sup 2P 0 P 0 (L e (O) L c ) 1 P 0 } (3.4) P 0 IR n n Choosing P 0 = Tr()I/ c we have and hence 2Tr() 2 c = (L e (O) L c ) = Tr() 2 / c. Tr()2 c 2 I (L e (O) L c ) 1 I Tr() 2 / c, I (L e (O) L c ) 1 I c, which, together with the first bound in (2.37), implies that I (L e (O) L c ) 1 I = c, and that P 0 = Tr()I/ c R(L e (O) L c ) is in fact a maximizer of the left-hand side of (3.4). On the other hand, the maximization problem in (3.4) admits the unique maximizer (L e (O) L c ) in R(L e (O) L c ), which then implies equation (3.2). Thus, we complete the proof of Theorem 3.1 for the case L(O, x) L c. The case L c L(O, x) can be handled similarly and will not be repeated here. Below we refer to the first and the second inequalities in (2.37), (3.1) as the lower and upper HS bounds, respectively. To achieve the best bounds and

12 12 Liping Liu to simplify algebraic calculations, we assume the materials (L c, L 0,, L N ) satisfy R(L 0 L c ) xi : x IR} and (3.5) R(L i L c ) = R(L c ) IR n n sym for i = 1, 2,, N. Now we are ready to state the necessary and sufficient condition for the bounds (2.37) or (3.1) being attained by periodic structures. Theorem 3.2. Consider a periodic (N + 1)-phase composite (2.1) of L i (i = 0,, N) with structure O = (Ω 0,, Ω N ) and volume fractions (θ 0,, θ N ). Assume that L c, L 0,, L N, all in L or all in L el, satisfy equation (3.5), and that (L c ) piqj = µ c 1δ ij δ pq + µ c 2δ pj δ iq + λ c δ ip δ jq, k c = µ c 1 + µ c 2 + λ c (3.6) satisfy (2.28). Let I (L i L c ) 1 I if L i L c c i = I (L c L i ) 1, I if L c L i γ = i=0 θ i 1/k c + c i, and c = 1 γ 1 k c. (3.7) Also, for a given R(L c ) with Tr() 0, let (1 + k c c ) Q i = Tr() c (1 + k c c i ) (L i L c ) 1 I (i = 1,, N), (3.8) p i = Tr(Q i ) = Tr()( c c i ) c (1 + k c c i ) and θ 0 p 0 + θ i p i = 0. (3.9) Then the effective tensor L e (O), given by (2.2), satisfies (2.10) and (2.37) or (3.1). urther, the lower or upper HS bound in (3.1) holds as an equality for if, and only if the following overdetermined problem u = N i=0 p iχ Ωi on u = Q i on Ω i, i = 1,, N (3.10) periodic boundary conditions on admits a solution u W 2,2 per( ). Proof. Note that the Theorem is trivial if θ 0 = 1. Below we assume θ 0 < 1. rom Lemma 2.1 and (3.5), we have R(L c ) = R(L e (O) L c ) I. Let us first consider the lower HS bound in (3.1). rom Lemma 2.2 and Theorem 3.1, equation (3.2), for a given structure O, we infer the following statements are equivalent:

13 Title Suppressed Due to Excessive Length 13 (i) The first inequality in (3.1) holds as an equality for R(L c ) = R(L e (O) L c ) with Tr() 0. (ii) The first inequality in (2.37) holds as an equality. (iii) Let a = Tr()/k c c (cf., (3.2)), u and satisfy (2.3), and P = (L(x, O) L c )( u + ). (3.11) Then = k c a(l e (O)L c ) 1 I, P dx = k c ai (cf., (2.4)), and on, P = k c I(a + f (x)) = P = (L(x, O) L c )( u + ), (3.12) where (cf., (2.36) and (3.9)) f 1 (x) = a [ γ(1/k c + c i ) 1]χ Ω i (x) = p i χ Ωi (x). (3.13) i=0 In particular, (i) (ii) follows from Theorem 3.1. (ii) (iii) follows from (3.2) and Lemma 2.2, (2.16). Since P = P, we have v P = v P, where v P and P ( v P and P ) satisfy equation (2.12). rom equations (3.12), (2.18) and (2.30), we obtain P = (L(x, O) L c )( u + ) = (L(x, O) L c )( v P + ) = (L(x, O) L c )( u + ) = P = k c I[a + f (x)] on, (3.14) where u W 2,2 per( ) satisfies equation (2.24) with f(x) replaced by the f (x) in (3.13). rom equations (2.1) and (3.13), equation (3.14) is equivalent to that on each Ω i (i = 0, 1,, N), (L i L c )( u + ) = k c I[a + f (x)] = I Tr()(1 + k c c ) c (1 + k c c i ) i=0. (3.15) The proof for the case of lower HS bound is complete if we show equation (3.15) with u Wper( 2,2 ) satisfying equation (2.24) with f (x) as in (3.13) is equivalent to (3.10) with Q i given by (3.8). rom equation (3.5) and the fact R(L c ) = R(L i L c ) for i = 1,, N, it follows the equivalence between equation (3.15) and the second equation in (3.10) with Q i given by (3.8) on Ω 1,, Ω N. Since R(L 0 L c ) xi : x IR}, direct calculations reveal that On Ω 0, equations (3.15) is equivalent to u = Tr() Tr() c 0(1 + k c c ) c (1 + k c c 0 ) = p 0 on Ω 0, where θ 0 p 0 + N θ ip i = 0, see equations (3.8) and (3.7). We have thus completed the proof for the lower HS bound in (3.1). The proof for the upper HS bound in (3.1) is similar and will not be repeated here.

14 14 Liping Liu Remark 1. or a sequence of periodic structures, Theorem 3.2 also holds in certain sense. More specifically, consider the lower HS bound in (3.1) and let O (k) = (Ω (k) 0,, Ω(k) ) be a sequence of structures such that N lim k (Le (O (k) ) L c ) = Tr() 2 / c, (3.16) where R(L c ) with Tr() 0, and lim k θ (k) i = lim k Ω (k) i / = θ i. By a similar argument as in the proof of Theorem 3.1, we can show that and lim I k (Le (O (k) ) L c ) 1 I = c (3.17) lim k (Le (O (k) ) L c ) = Tr() I. (3.18) c Let a = Tr()/k c c, u (k) and satisfy (2.3) with O replaced by O(k), and (cf., (3.12)) P (k) = (L(O(k), x) L c )( u (k) + ), (3.19) P (k) (x) = k c I[a + f (k) (x)], (3.20) where (cf., (3.13)) f (k) 1 (x) = a [ γ(1/k c + c i ) 1]χ (x) = p Ω (k) i χ (k) i Ω (x). (3.21) i i=0 Direct calculation shows that lim Let k P (k) dx = Tr() I and lim c P (k) 1 = i=0 E L (P (k) k P (k) dx P (k) (x)dx R(L c ) for all k = 1, 2,. rom (3.18), (2.4), and (3.19), we have lim P (k) Tr() dx = I, k c ) = Tr()2 c. (3.22)

15 Title Suppressed Due to Excessive Length 15 which, by (3.22), implies lim k P (k) 1 = 0. rom the first equation in (3.17) and (2.16), we have lim k P (k) = lim k P(k) P (k) 1 2 dx = lim k (L(O (k), x) L c )( u (k) P (k) P(k) 2 dx (3.23) + ) P(k) 2 dx = 0, Also, let u (k) Wper( 2,2 ) satisfy equation (2.24) with f(x) replaced by f (k) (x) + f (k) 1 in (3.21), where f (k) 1 IR is such that (f (k) + f (k) 1 )dx = 0. Clearly, lim k f (k) 1 = 0. Let v P (k) satisfy (2.12). By equation (2.30), v P (k) and P (k) = u (k). Since u (k) (2.18), we have [u (k) v P (k) equations (2.13) and (3.23) we obtain L c [ u (k) and P(k) ] satisfies (2.12) with P = P (k) satisfy (2.12) by P(k). rom v ] = L P (k) c [ u (k) + u(k) ] 0 (3.24) strongly in L 2 per(, IR n n ). By equations (3.23), (3.24), and similar calculations as in (3.14)-(3.15), we have u (k) = N i=0 p iχ (k) Ω + f (k) 1 on, i lim k u Ω (k) Q (k) i 2 dx = 0 on Ω (k) i, i = 1,, N, (3.25) i periodic boundary conditions on. Conversely, if a sequence of structures O (k) is such that equation (3.25) holds, then it follows equation (3.16). The proof of this statement is presented in the proof of Theorem 4.1. The overdetermined problem (3.10) does not have a solution unless the structure O = (Ω 0,, Ω N ) is very special. or their analogy with ellipsoids and their extremal properties as shown in the above theorem, we call (Ω 1,, Ω N ) a periodic E-inclusions corresponding to symmetric matrices K = (Q 1,, Q N ) and volume fractions Θ = (θ 1,, θ N ) if the overdetermined problem (3.10) admits a solution u Wper( 2,2 ), see Liu, James & Leo [22]. It is well-known that some special microstructures, say, confocal ellipsoids and multi-rank laminations, attain the optimal bounds for many different material systems. rom the viewpoint of equations (3.10) and (3.8), this corresponds to the fact that equation (3.8) has many different solutions of (L c, L 0, L N ) and for a given periodic E-inclusion corresponding to K and Θ. Therefore, it is useful to know all the material systems (L c, L 0, L N ) and for which the composites of this periodic E-inclusion attains the lower or upper HS bound in (3.1). Corollary 3.1. Let tensors L c, L 0,, L N, constants k c, c, c 0,, c N, be as in Theorem 3.2, and (Ω 1,, Ω N ) be a periodic E-inclusion corre-

16 16 Liping Liu sponding to symmetric matrices K = (Q 1,, Q N ) and volume fractions Θ = (θ 1,, θ N ). If L(x, O) L c (resp. L(x, O) L c ), then the (N + 1)- phase periodic composite (2.1) of this periodic E-inclusion attains the lower (resp. upper) HS bounds (3.1) for R(L c ) with Tr() 0 if, and only if = Q i + Tr(Q i ) 1 + k c c c c i (L i L c ) 1 I i = 1,, N. (3.26) Proof. This is a restatement of Theorem 3.2. We only need show that (3.26) is equivalent to (3.8), which is obvious by (3.5). rom Theorem 3.2, we can relate the attainability of the HS bounds of different material systems. Corollary 3.2. Consider a periodic (N + 1)-phase composite L(x, O) = L i on Ω i (i = 0, 1,, N). (3.27) Let (θ 0,, θ N ) be the volume fractions. If (L c, L 0,, L N ), all in L or all in L el, satisfy L(x, O) L c (resp. L(x, O) L c ), (3.5) and (3.6) for some comparison material L c, and if the effective tensor L e (O) attains the lower (resp. upper) HS bound in (3.1) for R(L c ) with Tr() 0, then the periodic composite (cf., (3.27)) L (x, O) = L i on Ω i (i = 0,, N), (3.28) attains the corresponding lower (resp. upper) HS bounds (3.1) for R(L c) with Tr( ) = 0 if (L c, L 0,, L N ), all in L or all in L el, satisfy L (x, O) L c (resp. L (x, O) L c), (3.5) and (3.6) for some L c, and if (1 + kc c ) = Tr() c (1 + k c c i ) (L i L c ) 1 I (3.29) + ( c c i ) 1 + k c c } c (1 + k c c i ) c c (L i L c) 1 I i for all i = 1,, N. Here k c, c 0,, c N, c are as in (3.6)-(3.7) with L i replaced by L i for all i = c, 0,, N. Proof. rom Theorem 3.2, the attainment of the lower or upper HS bound in (3.1) is equivalent to the existence of the periodic E-inclusion corresponding to (1 + k c c ) Q i = Tr() c (1 + k c c i ) (L i L c ) 1 I (i = 1,, N). (3.30) rom Corollary 3.1 and equation (3.26), it follows that the attainment of the lower or upper HS bound in (3.1) for materials (L c, L 0,, L N ) and if = Q i + Tr(Q i ) 1 + k c c c c (L i L c) 1 I (i = 1,, N). (3.31) i

17 Title Suppressed Due to Excessive Length 17 Plugging (3.30) into (3.31) yields (3.29). 4. Optimal microstructures: sequential E-inclusions The overdetermined problem (3.10) places non-obvious restrictions on matrices K and volume fractions Θ for which we can find a periodic E- inclusion. or instance, it has been shown in Liu, James & Leo [22] that K and Θ necessarily satisfy [θ 0 Tr(Q i ) + θ j Tr(Q j )]θ i Q i θ 0 j=1 N θ i Q 2 i + [ θ i Q i ] 2. (4.1) A natural question arises: for what K and Θ, can we find a corresponding periodic E-inclusion? rom Theorem 3.2, an answer to this question would gives us all the attainable HS bounds (3.1) by periodic structures. urther, it is desirable to allow sequences of structures or microstructures to attain the HS bounds. In fact, the familiar construction of coated spheres and multi-rank laminations are sequences of structures that attain the bounds in a periodic setting. To find the restrictions on K and Θ and to include sequences of structures, it is useful to introduce the concept of sequential E-inclusions. Before the formal definition, let us observe the following common feature of the sequence u (k) satisfying (3.25) and a periodic E-inclusion specified by (3.10). In the case of a periodic E-inclusion, we let u (k) = u with u satisfying (3.10) for all k. Using L p estimates for the Laplace operator we see that the sequence u (k) is in fact bounded in Wper( 2,p ) for any 1 p < since u (k) is bounded in L per( ) (Gilbarg & Trudinger [11], page 235). Then for an open bounded domain D, the gradient sequence v (k) of v (k) (x) = u (k) (kx)/k D generates a homogeneous oung measure ν that satisfies ν = θ i δ Qi + θ 0 µ and supp µ X IR n n sym : Tr(X) = p 0 }, (4.2) where (Q 1,, Q N ) [IR n n sym ] N, (θ 0, θ 1,, θ N ) [0, 1] N+1, N i=0 θ i = 1, are the Dirac masses at Q i, µ is a probability measure, and p 0 IR δ Qi is such that θ 0 p 0 + N θ itr(q i ) = 0. Note that the Dirac masses at Q i (i = 1,, N) arise from region Ω (k) i in (3.25) or Ω i in (3.10), and the condition Tr(X) = p 0 arises from u (k) = p 0 f (k) 1 on Ω (k) 0 in (3.25) or u = p 0 on Ω 0 in (3.10). Motivated by (4.2), we now define sequential E-inclusions, see Liu, James & Leo [22]. Definition 1. A sequential E-inclusion is a homogeneous gradient oung measure that satisfies (4.2), has zero center of mass, and is generated by a

18 18 Liping Liu bounded sequence in W 1,p (D) for any 1 p <. Let K = (Q 1,, Q N ) and Θ = (θ 1,, θ N ) be as in (4.2). We say such a sequential E-inclusion corresponds to matrices K and volume fractions Θ. In connection with the familiar microstructures such as coated spheres and multi-rank laminations, a sequential E-inclusion singles out the gradient fields associated with optimal microstructures, which geometrically not necessarily resemble coated spheres or multi-rank laminations. This is convenient since it is the special gradient field, not the geometric or topological properties that makes a microstructure optimal. Algebraically, the matrices (Q 1,, Q N ) associated with a sequential E-inclusion are a better set of parameters to describe the optimal microstructures since they directly relate the optimal bounds and material properties, see (3.30). urther, direct connections between optimal microstructures with gradient oung measures and quasiconvex functions make possible the use of many tools for constructing and restricting new microstructures, see Theorem 5.2 and Theorem 5.5 for such examples. In terms of sequential E-inclusions, from Theorem 3.2 and Remark 1 we have Theorem 4.1. Let tensors L c, L 0,, L N, volume fractions Θ = (θ 1,, θ N ), average field R(L c ) with Tr() 0, matrices K = (Q 1,, Q N ), and constant c be as in Theorem 3.2. Consider (N + 1)-phase periodic composites (2.1) of L 0,, L N with the effective tensors L e (O) given by (2.2). Then, for any structure O with the prescribed volume fractions, the effective tensor L e (O) satisfies (2.10) and (2.37) or (3.1). urther, or inf (L e (O) L c ) : Ω i = θ i } = Tr() 2 / c if L(x, O) L c (4.3) sup (L e (O) L c ) : Ω i = θ i } = Tr() 2 / c if L(x, O) L c (4.4) if, and only if there exists a sequential E-inclusion corresponding to K and Θ. Proof. Let O (k) be a minimizing (resp. maximizing) sequence of (4.3) (resp. (4.4)). rom Remark 1, (3.25) and the above calculations on the gradient oung measure generated by v (k) (x) = u (k) (kx)/k D, it is clearly that equation (4.3) (resp. (4.4)) implies the existence of the corresponding sequential E-inclusion. Conversely, let ν be a sequential E-inclusion corresponding to K and Θ. Since ν has zero center of mass and is supported on IR n n sym, by Lemma 1 in Šverák [32], we can assume that ν is generated by ξ (k) with ξ (k) being a bounded sequence in Wper( 2,p ) for any 1 p <. Let Ω (k) i x : ξ (k) (x) Q i < 1/k} for i = 1,, N (4.5)

19 Title Suppressed Due to Excessive Length 19 and Ω (k) 0 = \ ( N Ω(k) i ). By (4.2) we can assume θ (k) i as k. Let (cf., (3.21)) f (k) = i=0 p i χ Ω (k) i = Ω (k) i / θ i + f (k) 1 on, (4.6) where p 0, p N are as in (3.9) and f (k) 1 IR are such that N i=0 p iθ (k) i + f (k) 1 = 0. Again, we have lim k f (k) 1 = 0. rom (4.2), (4.6) and (4.5), the oung measure associated with ξ (k) f (k) is the Dirac mass supported at 0 IR a.e. on, and hence up to a subsequence and without relabelling, we have for any 1 p < (see Tartar [35], Proposition 2), ξ (k) f (k) 0 strongly in L p per( ). (4.7) Now consider the composites (2.1) with structures given by O (k) = (Ω (k) 0,, Ω(k) N ). Direct calculations reveal that for any comparison material L c : IR n n IR n n, ( ξ (k) + ) L(x, O (k) )( ξ (k) + ) dx = ( ξ (k) + ) (L(x, O) L c )( ξ (k) + )dx (4.8) + ξ (k) L c ξ (k) dx + L c Let L c and c be as in (4.3) or (4.4). Sending k, from equations (4.5), (3.8) or (3.15), we have lim ( ξ (k) + ) L(x, O (k) )( ξ (k) + ) dx (4.9) k = Tr() 2 / c + L c. In particular, equation (4.5) and (3.15) are used to calculate the first term on the right-hand side of (4.8), and the divergence theorem is used to calculate the second term on the right-hand side of (4.8). The rest of the proof has two steps. Step 1. We first claim lim k w L(x, O (k) )( ξ (k) + ) dx = 0 w W 1,2 per( ). (4.10) To show this, we let u (k) W 2,2 per( ) satisfy u (k) = f (k) on. rom equation (4.7) and the L 2 estimate of the Laplace operator, we have ξ (k) u (k) 0 strongly in L 2 per( ). (4.11)

20 20 Liping Liu urther, for any w Wper( 1,2 ), w L(x, O (k) )( u (k) + ) dx = w [L c u (k) + k c I(a + f (k) )] dx (4.12) + w [(L(x, O (k) ) L c )( u (k) + )] k c I(a + f (k) )} dx, where a = Tr()/(k c c ). The first term on the right-hand side vanishes by equations (2.12), (2.24), (2.27) and (2.30). or the second term on the right-hand side, we notice from the proof of Theorem 3.2, (3.15) that (L i L c )(Q i + ) = k c I(a + Tr(Q i )) i = 1,, N. By equations (4.5) and (4.11), we have (L(x, O (k) ) L c )( u (k) + ) k c I(a + f (k) ) 0 strongly in L 2 per(, IR n n ), which, together with (4.11) and (4.12), completes the proof of equation (4.10). Step 2. Let u (k) W per( 1,2 ) be a solution of (2.3) for L(x, O (k) ) and. The weak form of (2.3) reads w L(x, O (k) )( u (k) 1,2 + ) dx = 0 w Wper( ). (4.13) Subtracting (4.13) from (4.10), we obtain lim k w L(x, O (k) )( ξ (k) + u (k) 1,2 ) dx = 0 w Wper( ). (4.14) Since ξ (k) + u (k) and L(x, O(k) )( ξ (k) + u (k) ) are bounded in L 2 per(, IR n n ), up to a subsequence and without relabelling, we have ξ (k) + u (k) u L(x, O (k) )( ξ (k) + u (k) ) S weakly in L 2 per(, IR n n ), (4.15) where u Wper( 1,2 ; IR n ), and S L 2 per(, IR n n ) satisfies w S dx = 0 w Wper( 1,2 ).

21 Title Suppressed Due to Excessive Length 21 rom (4.14), (4.15), and the Div-Curl Lemma (Murat [28]; Tartar [34]), we have lim ( ξ (k) + u (k) ) L(x, k O(k) )( ξ (k) + u (k) ) dx = u S = 0. Since (L 0,, L N ) are either all in L or all in L el, we have L(x, O (k) )[ ξ (k) + u (k) ] 0 strongly in L2 per(, IR n n ). Therefore, the effective tensors of L(x, O (k) ) satisfy lim k Le (O (k) ) = lim ( u (k) + ) L(x, k O(k) )( u (k) + ) dx = lim ( ξ (k) ) L(x, O (k) )( ξ (k) ) dx k = L c + Tr() 2 / c, where equation (4.9) has been used in the last equality. Thus, the bounds (4.3) or (4.4) are attained by the sequence of structures O (k). The proof of the theorem is now completed. 5. Applications 5.1. An outer bound for sequential E-inclusions rom the basic relation between gradient oung measures and quasiconvex functions (Kinderlehrer & Pedregal [19, 20]), we have Theorem 5.1. Let (Q 1,, Q N ) [IR n n sym ] N and (θ 0,, θ N ) [0, 1] N+1. Let ν be a probability measure with zero center of mass satisfying ν = θ i δ Qi + θ 0 µ, where µ is a probability measure with supp µ X IR n n sym : Tr(X) = p 0 }. Then ν is a sequential E-inclusion if, and only if IR n n ψ(x) dν(x) = θ i ψ(q i ) + θ 0 ψ(x) dµ(x) ψ(0) (5.1) IR n n for all quasiconvex functions ψ : IR n n IR satisfying ψ(x) C( X p + 1) for some C > 0 and 1 p <. As a first application of Theorem 5.1, we show equation (4.1) is also a restriction on sequential E-inclusions. Since a sequential E-inclusion ν is

22 22 Liping Liu supported on symmetric matrices, inequality (5.1) is valid for quasiconvexity functions restricted to IR n n sym, see Corollary 1 in Šverák [32]. It is easy to show that the quadratic function ψ(x) = m (Tr(X)X X 2 )m is convex on symmetric rank-one matrices and therefore quasiconvex restricted to IR n n sym for any m IR n. (In fact, ψ is a null Lagrangian in this second gradient context.) An application of (5.1) to ±ψ yields equation (4.1), see details in Liu, James & Leo [22]. To see the implication of equation (4.1) on the attainable HS bounds, we define the following hyper-surfaces S L = ˆL L : I ( ˆL L c ) 1 I = c, ˆL Lc } S U = ˆL L : I (L c ˆL) 1 I = c, ˆL, (5.2) L c } where constants c are as in (2.37). Similarly, Sel L and SU el can be defined by requiring ˆL L el. Recall that G Θ -closure of composites of material (L 0,, L N ), denoted by G Θ (L 0,, L N ), is the closure of all effective tensors that can be achieved by a composite of (L 0,, L N ) with volume fractions (θ 0,, θ N ). rom (4.1) and Theorem 4.1, we have Theorem 5.2. Let tensors L c, L 0,, L N, volume fractions Θ = (θ 1,, θ N ), and constant k c, c, c 0,, c N be as in Theorem 3.2. or a given ˆL L L el, we define 1 Q i = [ ( c ˆL L c ) 1 (1 + k c c ) c (1 + k c c i ) (L i L c ) 1 ]I (5.3) for i = 1,, N. If L i L (resp. L el ) and L i L c for all i = 0,, N, and if ˆL S L (resp. Sel L) is such that K = (Q 1,, Q N ) defined by (5.3) and Θ do not satisfy (4.1), then ˆL is not attainable in the sense that there is no sequence of structures O (k) = (Ω (k) 0,, Ω(k) N ) satisfying lim k Ω (k) i / = θ i (i = 1,, N) such that In another word, ˆL / G Θ (L 0,, L N ). lim k Le (O (k) ) = ˆL. (5.4) It is clear that the analogous result holds for the upper bound, which will not be repeated here. Proof. Assume there exists a sequence of structure O (k) with the prescribed volume fractions such that (5.4) holds. Since ˆL S L (or Sel L), = ( ˆL L c ) 1 I with Tr() 0 is well-defined if c <. Direct calculations reveal that O (k) is a minimizing sequence such that (4.3) holds for this, and hence by Theorem 4.1 (see also Remark 1), we have the existence of the corresponding sequential E-inclusion with matrices K = (Q 1,, Q N ) given by (5.3) and volume fractions Θ. This however contradicts (4.1) by assumption.

23 Title Suppressed Due to Excessive Length 23 We remark that Theorem 5.2 gives an outer bound on the attainable HS bounds (2.37) or (3.1). Although it is algebraically tedious to verify (4.1), we emphasize that this non-attainability result is nontrivial if N 2, see discussions in Section 5.3. Of course, there are many other quasiconvex functions that could be used in equation (5.1) that would evidently give further restrictions on the K and Θ, and therefore by Theorem 4.1, give further restrictions on the attainable HS bounds An inner bound for sequential E-inclusions In this section we derive an inner bound for sequential E-inclusions, which then implies an inner bound on the attainable HS bounds. or brevity, we refer to a gradient oung measure that is generated by a bounded sequence in W 1,p (D) as a W 1,p gradient oung measure, see Kinderlehrer & Pedregal [20]. Similarly, a sequential E-inclusion is a W 1, sequential E-inclusion if it can be generated by a bounded sequence in W 1, (D). Recall the following two theorems: Theorem 5.3. (Theorem 3.1, Kinderlehrer & Pedregal [19]) Let ν 1 and ν 2 be two homogeneous W 1, gradient oung measures with zero center of mass. Then for each λ [0, 1], the measure (1 λ)ν 1 + λν 2 is also a homogeneous W 1, gradient oung measure with zero center of mass. Theorem 5.4. (Theorem 3, Liu, James & Leo [22] ) Let Q IR sym n n be either negative semi-definite or positive semi-definite. Then for each θ [0, 1], corresponding to Q and θ, there exists a W 1, sequential E-inclusion ν = θδ Q + (1 θ)µ, (5.5) where µ is a probability measure and supp µ X IR n n sym : θtr(q) + (1 θ)tr(x) = 0}. rom Theorem 5.4 and Theorem 5.3, we have Theorem 5.5. Let K = (Q 1,, Q N ) be either all negative semi-definite or all positive semi-definite, and Θ = (θ 1,, θ N ) be any array satisfying θ 1,, θ N [0, 1], θ 0 = 1 θ i [0, 1]. (5.6) Then corresponding to K and Θ, there exists a W 1, sequential E-inclusion ν = θ i N δ Qi + θ 0 µ, (5.7) where µ is a probability measure and supp µ X IR n n sym : N θ itr(q i )+ θ 0 Tr(X) = 0}.

24 24 Liping Liu Proof. We prove the theorem by induction. If N = 1, the theorem holds by Theorem 5.4. Assume the theorem holds for 1 N k, below we show the theorem holds for N = k + 1. Let Θ = (θ 1,, θ k+1 ) satisfy (5.6) for N = k + 1. By multiplying the generating sequence v (k) by any constant a IR, we see that there exists a W 1, sequential E-inclusions corresponding to (aq 1,, aq k+1 ) and Θ if there exists a W 1, sequential E-inclusions corresponding to (Q 1,, Q k+1 ) and Θ. Therefore, without loss of generality we may assume K = (Q 1,, Q k+1 ) are all negative semi-definite. Let p 0 IR be such that θ 0 p 0 + k+1 θ itr(q i ) = 0 and α [0, 1] be such that αtr(q k+1 ) + (1 α)p 0 = 0. If any θ i = 0 or p 0 = 0, the theorem for N = k +1 follows trivially from the inductive assumption. In particular, if θ 0 = 0 or p 0 = 0, from the negative semi-definiteness of (Q 1,, Q k+1 ), the only possible form of ν in (5.7) is the Dirac measure supported at the zero matrix. We subsequently assume θ i (0, 1) for all i = 0,, k + 1 and p 0 > 0. It is easy to verify that α = and that p 0 p 0 Tr(Q k+1 ) > p 0 p 0 k θ i + k θ itr(q i ) = θ k+1, (5.8) p 0 Tr(Q k+1 ) θ k+1 Tr(Q k+1 ) + θ 0 p 0 0 = θ k+1 α (1 α) = Tr(Q k+1)θ k+1 θ 0. (5.9) p 0 Define λ and θ i (i = 1,, k) by λ = θ k+1 α, (1 λ)θ 0 + λ(1 α) = θ 0 and (1 λ)θ i = θ i. (5.10) rom equations (5.8) and (5.9), we see that λ (0, 1) and θ 0,, θ k 0. In particular, θ 0 0 follows from (5.9) and (1 λ)θ 0 = θ 0 λ(1 α) = θ 0 θ k+1 α (1 α). urther, k θ i = 1 k λ(1 α) θ i 1 λ 1 λ = λ + k+1 i=0 θ i = 1. 1 λ i=0 i=0 Thus, (θ 1,, θ k ) satisfy (5.6) for N = k. By the inductive assumption, for N = k we have the existence of a W 1, sequential E-inclusion ν 1 = k θ iδ Qi + θ 0µ 1, (5.11)

25 Title Suppressed Due to Excessive Length 25 where µ 1 is a probability measure with supp µ 1 X IR n n sym : k θ itr(q i ) + θ 0Tr(X) = 0}. By Theorem 5.4, we also have the existence of a W 1, sequential E-inclusion where µ 2 is a probability measure with ν 2 = αδ Qk+1 + (1 α)µ 2, (5.12) supp µ 2 X IR n n sym : Tr(X) = p 0 }. Let p 0 be such that k θ i Tr(Q i) + θ 0p 0 = 0. rom equation (5.10), we have k θ itr(q i) 1λ + p 0 1λ [θ 0 λ(1 α)] = 0, which, by equation (5.9) and the definition of p 0, implies 0 = [θ 0 + θ k+1tr(q k+1 ) p 0 ]p 0 θ k+1 Tr(Q k+1 ) p 0 θ 0 (5.13) = 1 p 0 [θ 0 p 0 + θ k+1 Tr(Q k+1 )](p 0 p 0 ). If θ 0 p 0 + θ k+1 Tr(Q k+1 ) = 0, by equations (5.9) and (5.10) we have θ 0 = 0. If θ 0 p 0 + θ k+1 Tr(Q k+1 ) 0, by equation (5.13) we have p 0 = p 0. or either case, define k+1 ν := λν 2 + (1 λ)ν 1 = θ i δ Qi + θ 0 µ, (5.14) where µ = λ(1α) θ 0 µ 2 + (1λ)θ 0 θ 0 µ 1 is a probability measure with supp µ X IR n n sym : Tr(X) = p 0 }. rom Theorem 5.3 and Definition 1, we see that ν in (5.14) is a W 1, sequential E-inclusion corresponding to K and Θ. The proof of the theorem is completed. rom Theorem 4.1 and Theorem 5.5, we have the following sufficient conditions for the HS bounds to be attainable. Theorem 5.6. Let tensors L c, L 0,, L N, volume fractions (θ 0,, θ N ), and constants k c, c 1, c N, c be as in Theorem 3.2. Consider (N + 1)-phase periodic composites (2.1) of L 0,, L N with the effective tensors L e (O) given by (2.2). Then the effective tensors satisfy the HS bounds (2.10) and (2.37) or (3.1). The bounds in (3.1) are attainable for R(L c ) with Tr() 0 if the matrices (1 + k c c ) Q i = Tr() c (1 + k c c i ) (L i L c ) 1 I (i = 1,, N) (5.15) are all symmetric and either all negative semi-definite or all positive semidefinite.

26 26 Liping Liu Below we illustrate the applications of Theorem 5.2 and Theorem 5.6 by two examples Composites of conductive materials or the first example, let us consider conductive composites of (N + 1)- phases with conductivity tensors 0 < A 0,, A N IR sym n n and volume fractions θ 0 [0, 1], Θ = (θ 1,, θ N ) [0, 1] N. The effective conductivity tensor of a composite is denoted by A e. In accordance with (3.5), we assume and A 0 = k 0 I, A N = k N I, (5.16) A 0 < A 1, A 2,, A N1 < A N. (5.17) To use Theorem 3.1, we set (L i ) pjqk = δ pq (A i ) jk for i = 0,, N and choose (L c ) pjqk = k 0 δ pq δ jk for the lower bound and (L c ) pjqk = k N δ pq δ jk for the upper bound. With the effective tensors L e defined as in (2.2), one can easily verify that they can be written as (L e ) piqj = δ pq (A e ) ij. By (2.10) and (5.17), we have A 0 H Θ A e A Θ A N, (5.18) where A Θ = N i=0 θ ia i and H Θ = [ N i=0 θ ia 1 i ] 1 are the arithmetic mean and harmonic mean, respectively. rom equations (2.37) and (3.1), we have Tr[(A 1/2 0 A e A 1/2 0 I) 1 ] c L Tr[(I A 1/2, (5.19) )1 ] c U N Ae A 1/2 N and Tr( T ) + Tr() 2 / c L A 1/2 0 A e A 1/2 0 T Tr( T ) + Tr() 2 / c U A 1/2 N Ae A 1/2 N T IRn n, (5.20) where, by (3.7), c L = N i=0 θ i c L i /(1 + cl i ) N i=0 θ i/(1 + c L i ), (5.21) c U = N i=0 θ i c U i /(1 + cu i ) N i=0 θ i/(1 + c U i ), (5.22) and c L i = Tr[(A 1/2 0 A i A 1/2 0 I) 1 ] > 0 c U i = Tr[(I A 1/2 N A ia 1/2 N )1 ] < 0 (i = 0,, N). (5.23)

27 Title Suppressed Due to Excessive Length 27 We remark that through a linear transformation, one can show that the bounds (5.18), (5.19) and (5.20) are valid without assuming (5.16). urther, according to Theorem 3.1, the bounds (5.18) and (5.19) are equivalent to the bounds (5.18) and (5.20). Let us denote by G out Θ (A 0,, A N ) the set of symmetric matrices A e that satisfy (5.18) and (5.19) or (5.20). Let G Θ (A 0,, A N ) be the G Θ -closure of (A 0,, A N ) with volume fraction (θ 0,, θ N ). If no confusion arises, both G out Θ (A 0,, A N ) and G Θ (A 0,, A N ) are sometimes shortly written as G out Θ is a closed and convex set in IR n n Sco L = A e G out Θ : Tr[(A1/2 0 A e A 1/2 S U co = A e G out Θ and G Θ, respectively. Clearly, G out Θ sym and contains G Θ. Let (cf. (5.2)) 0 I) 1 ] = c L } : Tr[(A1/2 N Ae A 1/2 N I) 1 ] = c U } (5.24) be two hypersurfaces in IR n n sym. It is worthwhile noticing that Grabovsky [12] has shown that G out Θ = G Θ for two-phase well-ordered conductive composites. We now discuss the attainable and non-attainable points on Sco L and Sco. U Let A Sco L and define Q L i = (A 1/2 0 AA 1/2 0 I) cl 1 + c L (A 1/2 0 A i A 1/2 0 I) 1 (5.25) i for i = 1,, N. By Theorem 4.1, equations (3.2) and (3.8), we see that A is attainable if and only if there exists a sequential E-inclusion corresponding to K L = (Q L 1,, Q L N ) and ΘL = (θ 1,, θ N ). Taking the traces of both sides of (5.25), we have rom (5.21), it is clear that Tr(Q L i ) = cl c L i (1 + c L i ). (5.26) c L min c L i : i = 1,, N} = c L N > 0. (5.27) Thus, Q L i cannot be all negative semi-definite for i = 1,, N unless c L = c L N. Similarly, let A SU co and define Q U i = (I A 1/2 N AA1/2 N ) cu 1 + c U (I A 1/2 N A ia 1/2 N )1 (5.28) i for i = 0,, N 1. Then A is attainable if, and only if there exists a sequential E-inclusion corresponding to K U = (Q U 0,, Q U N1 ) and ΘU = (θ 0,, θ N1 ). Also, Tr(Q U i ) = cu cu i 1+ c U i, c U max c U i : i = 0,, N 1} = c U 0 < 1. (5.29) Thus, Q U i cannot be all negative semi-definite for i = 0,, N 1 unless c U = c U 0.