Homogenization of weakly coercive integral functionals in three-dimensional linear elasticity

Marc Briane,Antonio Jesús Pallares Martín

doi:10.5802/jep.49

Abstract

This paper deals with the homogenization through Γ-convergence of weakly coercive integral energies with the oscillating density L(x/e)∇v : ∇v in three-dimensional elasticity. The energies are weakly coercive in the sense where the classical functional coercivity satisfied by the periodic tensor L (using smooth test functions v with compact support in R 3) which reads as Λ(L) > 0, is replaced by the relaxed condition Λ(L) ≥ 0. Surprisingly, we prove that contrary to the two-dimensional case of [2] which seems a priori more constrained, the homogenized tensor L 0 remains strongly elliptic, or equivalently Λ(L 0) > 0, for any tensor L = L(y1) satisfying L(y)M : M + D : Cof(M) ≥ 0, a.e. y ∈ R 3 , ∀ M ∈ R 3×3 , for some matrix D ∈ R 3×3 (which implies Λ(L) ≥ 0), and the periodic functional coercivity (using smooth test functions v with periodic gradients) which reads as Λper(L) > 0. Moreover, we derive the loss of strong ellipticity for the homogenized tensor using a rank-two lamination, which justifies by Γ-convergence the formal procedure of [8].

Highlights

R3 for any smooth function v with compact support in R3, with Λ(L) > 0, is replaced by the relaxed condition Λ(L) 0
On montre que le tenseur homogénéisé L0 reste fortement elliptique ou, de manière équivalente, Λ(L0) > 0, pour tout tenseur L = L(y1) vérifiant l’inégalité ponctuelle : L(y)M :M + D:Cof(M ) 0, p.p. y ∈ R3, ∀ M ∈ R3×3, par l’addition d’un lagrangien nul pour une matrice D ∈ R3×3 donnée, et en supposant la coercivité fonctionnelle périodique Λper(L) > 0
We focus on the case where the tensor L is weakly coercive, i.e., relaxing the condition Λ(L) > 0 by Λ(L) 0. In this case the homogenization of the elasticity system (1.3) associated with the energy (1.1) is badly posed in general, since one has no a priori L2-bound on the stress tensor ∇uε (assuming the existence of a solution uε to the elasticity system (1.3)) due to the loss of coercivity. It was shown by Geymonat et al [7] that the previous Γ-convergence result still holds when Λ(L) 0, under the extra condition of periodic functional coercivity, i.e., (1.8) Λper(L) := inf

Summary

Introduction

R3 for any smooth function v with compact support in R3, with Λ(L) > 0, is replaced by the relaxed condition Λ(L) 0. Contrary to the two-dimensional case of [2], we prove (see Theorem 3.3) that for any 1-periodic tensor L = L(y1), condition (1.11) combined with Λper(L) > 0 implies that αse(L0) > 0, making impossible the loss of ellipticity through homogenization.

Objectives

Results

Conclusion