Loss of polyconvexity by homogenization: a new example

Abstract

Abstract. This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that polyconvex energies are not closed with respect to periodic homogenization. The counterexample is obtained through a rank-one laminated structure assembled by mixing two polyconvex functions with p-growth, where p ≥ 2 can be fixed arbitrarily.

New Counterexamples to the Cell Formula in Nonconvex Homogenization

“…The previous lemma shows that in the case of a mixture of the type W = χW 1 + (1 − χ)W 2 , the zero levelset of W cell depends only on the zero levelsets of W 1 , W 2 and not on their global shapes or growths. The same property can be proved for the zero levelset of W hom (see [6,Theorem 1.3]). This fact is one of the keys of our counterexamples: we have to introduce suitable zero levelsets first, and only afterwards construct suitable functions.…”

Characterization of Two-Scale Gradient Young Measures and Application to Homogenization

“…Let ν ∈ M u , by Remark 2.6 there exists a sequence {u n } ⊂ W 1,p (Ω; R d ) such that {( ·/ε n , ∇u n )} generates the Young measure {ν (x,y) ⊗ dy} x∈Ω and u n ⇀ u in and that {|∇u n k | p } is equi-integrable, which is always possible by the Decomposition Lemma (see Lemma 1.2 in Fonseca, Müller & Pedregal [20]). In particular, due to the p-growth condition (1.8), the sequence {f (·, ·/ε n k , ∇u n k )} is equi-integrable as well and applying Theorem 2.8 (ii) in Barchiesi [10] we get that…”

“…Arguing exactly as before we can assume that (a subsequence of) {∇ū n } generates a two-scale gradient Young measure {ν (x,y) } (x,y)∈Ω×Q , that ν ∈ M u and {f (·, ·/ε n , ∇ū n )} is equi-integrable. According to Theorem 2.8 (ii) in Barchiesi [10] and using the fact that {ū n } is a recovering sequence,…”

“…Extract a subsequence {ε n k } ⊂ {ε n } such that lim sup εn k (u n k )and that {|∇u n k | p } is equi-integrable, which is always possible by the Decomposition Lemma (see Lemma 1.2 in Fonseca, Müller & Pedregal[20]). In particular, due to the p-growth condition (1.8), the sequence {f (•, •/ε n k , ∇u n k )} is equi-integrable as well and applying Theorem 2.8 (ii) in Barchiesi[10] we get that Taking the infimum over all ν ∈ M u in the right hand side of the (4.2) yields to (4.1).Let η > 0 and {un } ⊂ W 1,p (Ω; R d ) such that u n ⇀ u in W 1,p (Ω; R d ) and lim inf n→+∞ F εn (u n ) Γ-lim inf n→+∞ F εn (u) + η. (4.5)For a subsequence {n k }, we can assume that there exists ν∈ L ∞ w (Ω × Q; M(R d×N )) such that {( •/ε n k , ∇u n k )} generates a Young measure {ν (x,y) ⊗ dy} x∈Ω and lim k→+∞ F εn k (u n k ) = lim inf n→+∞ F εn (u n ).…”

A variational approach to the local character of G-closure: the convex case

“…, W (N ) . We refer to [9] for a similar characterization of the zero level set of (W χ ) hom . The result obtained has a lot of interesting consequences because it will enable us to build new counter-examples to the validity of the cell formula even in the quasiconvex case (Example 6.1) and to the continuity of the determinant with respect to the two-scale convergence (Example 6.5).…”