Homogenization of nonconvex unbounded singular integrals
Annales mathématiques Blaise Pascal, Volume 24 (2017) no. 2, pp. 135-193.

We study periodic homogenization by $\Gamma$-convergence of integral functionals with integrands $W\left(x,\xi \right)$ having no polynomial growth and which are both not necessarily continuous with respect to the space variable and not necessarily convex with respect to the matrix variable. This allows to deal with homogenization of composite hyperelastic materials consisting of two or more periodic components whose the energy densities tend to infinity as the volume of matter tends to zero, i.e., $W\left(x,\xi \right)={\sum }_{j\in J}{\mathbf{1}}_{{V}_{j}}\left(x\right){H}_{j}\left(\xi \right)$ where ${\left\{{V}_{j}\right\}}_{j\in J}$ is a finite family of open disjoint subsets of ${ℝ}^{N}$, with $|\partial {V}_{j}|=0$ for all $j\in J$ and $|{ℝ}^{N}\setminus {\bigcup }_{j\in J}{V}_{j}|=0$, and, for each $j\in J$, ${H}_{j}\left(\xi \right)\to \infty$ as $det\xi \to 0$. In fact, our results apply to integrands of type $W\left(x,\xi \right)=a\left(x\right)H\left(\xi \right)$ when $H\left(\xi \right)\to \infty$ as $det\xi \to 0$ and $a\in {L}^{\infty }\left({ℝ}^{N};\left[0,\infty \left[\right)$ is $1$-periodic and is either continuous almost everywhere or not continuous. When $a$ is not continuous, we obtain a density homogenization formula which is a priori different from the classical one by Braides–Müller. Although applications to hyperelasticity are limited due to the fact that our framework is not consistent with the constraint of noninterpenetration of the matter, our results can be of technical interest to analysis of homogenization of integral functionals.

Published online:
DOI: 10.5802/ambp.367
Keywords: Homogenization, $\Gamma$-convergence, Unbounded integrand, Singular growth, Determinant constraint type, hyperelasticity
Omar Anza Hafsa 1, 2; Nicolas Clozeau 3; Jean-Philippe Mandallena 2

1 LMGC, UMR-CNRS 5508 Place Eugène Bataillon 34095 Montpellier, France
2 Université de Nîmes Laboratoire MIPA, Site des Carmes Place Gabriel Péri 30021 Nîmes, France
3 École Normale Supérieure de Cachan 61 avenue du président Wilson 94230 Cachan, France
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Omar Anza Hafsa; Nicolas Clozeau; Jean-Philippe Mandallena. Homogenization of nonconvex unbounded singular integrals. Annales mathématiques Blaise Pascal, Volume 24 (2017) no. 2, pp. 135-193. doi : 10.5802/ambp.367. https://ambp.centre-mersenne.org/articles/10.5802/ambp.367/

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