English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

This paper deals with the homogenization through $\Gamma$-convergence ofweakly 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 functionalcoercivity satisfied by the periodic tensor L (using smooth test functions vwith 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] whichseems a priori more constrained, the homogenized tensor $\mathbb{L}^0$ remainsstrongly elliptic, or equivalently $\Lambda(\mathbb{L}^0) >0$, for any tensor$\mathbb{L} = \mathbb{L}(y_1)$ satisfying $\mathbb{L}(y)M : M + D : {\rmCof}(M) \ge 0$, a.e. $y \in \mathbb{R}^3$, $\forall M \in \mathbb{R}^{3\times3}$, for some matrix $D \in \mathbb{R}^{3\times3}$ (which implies$\Lambda(\mathbb{L}) \ge 0$), and the periodic functional coercivity (usingsmooth test functions $v$ with periodic gradients) which reads as $\Lambda_{\rmper}(\mathbb{L})>0$. Moreover, we derive the loss of strong ellipticity for thehomogenized tensor using a rank-two lamination, which justifies by$\Gamma$-convergence the formal procedure of [8].

Homogenization of weakly coercive integral functionals in three-dimensional elasticity