Homogenization of weakly coercive integral functionals in three-dimensional elasticity

This paper deals with the homogenization through $\mathrm{\Gamma }$-convergence of weakly coercive integral energies with the oscillating density $\mathbb{L}(x\mathrm{/}ϵ)\mathrm{\nabla }v:\mathrm{\nabla }v$ 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$,isreplacedbytherelaxedcondition$\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 $\mathrm{\Lambda }({\mathbb{L}}^0)>0$, for any tensor $\mathbb{L}=\mathbb{L}(y_1)$ satisfying $$\mathbb{L}$(y)M : M + D : {\rm Cof}(M) \ge 0$,a\mathrm{.}e\mathrm{.}$y \in $\mathbb{R}$^3$,$\forall M \in $\mathbb{R}$^{3\times 3}$,forsomematrix$D \in $\mathbb{R}$^{3\times3}$(whichimplies$\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 $\mathrm{\Gamma }$-convergence the formal procedure of [8].