Homogenization of nonconvex unbounded singular integrals

Anza Hafsa, Omar; Clozeau, Nicolas; Mandallena, Jean-Philippe

Annales mathématiques Blaise Pascal, Tome 24 (2017) no. 2, pp. 135-193.

We study periodic homogenization by $\Gamma$-convergence of integral functionals with integrands $W(x,\xi )$ 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., $W(x,\xi )={\sum }_{j\in J}{\mathbf{1}}_{V_j}\left(x\right)H_j\left(\xi \right)$ 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 $W(x,\xi )=a\left(x\right)H\left(\xi \right)$ when $H\left(\xi \right)\to \infty$ as $det\xi \to 0$ and $a\in L^{\infty }({ℝ}^N;[0,\infty \left[\right)$ 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.

Mots clés : Homogenization, $\Gamma $-convergence, Unbounded integrand, Singular growth, Determinant constraint type, hyperelasticity

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