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

TL;DR

This paper derives an explicit formula for the homogenized curvature energy in isotropic Cosserat shell models using Gamma-convergence, based on a 3D minimization problem involving dislocation density tensors.

Contribution

It provides an explicit Gamma-limit formula for the curvature energy in Cosserat shells, connecting 3D Cosserat models to 2D shell theories.

Findings

Explicit formula for homogenized curvature energy

Applicable to flat and curved Cosserat shells

Based on 3D minimization involving dislocation density tensor

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 $\alpha:=\overline{R} ^T {\rm Curl}\,\overline{R} \in \mathbb{R}^{3\times 3}$, $\overline{R}\in {\rm SO}(3)$ is used.