Explicit formula for the Gamma-convergence homogenized quadratic curvature energy in isotropic Cosserat shell models

Abstract

We show how to explicitly compute the homogenized curvature energy appearing in the isotropic $\Gamma$-limit for flat and for curved initial configuration Cosserat shell models, when a parental three-dimensional minimization problem on $\Omega \subset {\mathbb{R}}^3$ for a Cosserat energy based on the second order dislocation density tensor $\alpha :={\overline{R}}^T\mathrm{Curl}\overline{R}\in {\mathbb{R}}^{3\times 3}$, $\overline{R}\in \mathrm{SO}(3)$ is used.