A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via $\Gamma$-convergence

Abstract

We derive the $\Gamma$-limit to a three-dimensional Cosserat model as the aspect ratio $h>0$ of a flat domain tends to zero. The bulk model involves already exact rotations as a second independent field intended to describe the rotations of the lattice in defective elastic crystals. The $\Gamma$-limit based on the natural scaling consists of a membrane like energy and a transverse shear energy both scaling with $h$, augmented by a curvature energy due to the Cosserat bulk, also scaling with $h$. A technical difficulty is to establish equi-coercivity of the sequence of functionals as the aspect ratio $h$ tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption. While the three-dimensional problem is well-posed for the Cosserat couple modulus ${\mu }_c\ge 0$, equi-coercivity needs a strictly positive ${\mu }_c>0$. Then the $\Gamma$-limit model determines the midsurface deformation $m\in H^{1,2}(\omega ,{\mathbb{R}}^3)$. For the true defective crystal case, however, ${\mu }_c=0$ is appropriate. Without equi-coercivity, we obtain first an estimate of the $\Gamma -\mathrm{liminf}$ and $\Gamma -\mathrm{limsup}$ which can be strengthened to the $\Gamma$-convergence result. The Reissner-Mindlin model is "almost" the linearization of the $\Gamma$-limit for ${\mu }_c=0$.