Homogenization of weakly coercive integral functionals in three-dimensional elasticity

Abstract

This paper deals with the homogenization through $\Gamma$-convergence of weakly coercive integral energies with the oscillating density $\mathbb{L}(x/\epsilon)\nabla v : \nabla v$ 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 $\mathbb{R}^3$) which reads as $\Lambda(\mathbb{L}) >0$, is replaced by the relaxed condition $\Lambda(\mathbb{L}) \ge 0$. Surprisingly, we prove that contrary to the two-dimensional case of [2] which seems a priori more constrained, the homogenized tensor $\mathbb{L}^0$ remains strongly elliptic, or equivalently $\Lambda(\mathbb{L}^0) >0$, for any tensor $\mathbb{L} = \mathbb{L}(y_1)$ satisfying $\mathbb{L}(y)M : M + D : {\rm Cof}(M) \ge 0$, a.e. $y \in \mathbb{R}^3$, $\forall M \in \mathbb{R}^{3\times 3}$, for some matrix $D \in \mathbb{R}^{3\times3}$ (which implies $\Lambda(\mathbb{L}) \ge 0$), and the periodic functional coercivity (using smooth test functions $v$ with periodic gradients) which reads as $\Lambda_{\rm per}(\mathbb{L})>0$. Moreover, we derive the loss of strong ellipticity for the homogenized tensor using a rank-two lamination, which justifies by $\Gamma$-convergence the formal procedure of [8].