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

### Patrizio Neff

Technische Hochschule Darmstadt, Germany### Krzysztof Chelminski

Technical University, Warszawa, Poland

## Abstract

We derive the $Γ$-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 $Γ$-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 $μ_{c}≥0$, equi-coercivity needs a strictly positive $μ_{c}>0$. Then the $Γ$-limit model determines the midsurface deformation $m∈H_{1,2}(ω,R_{3})$. For the true defective crystal case, however, $μ_{c}=0$ is appropriate. Without equi-coercivity, we obtain first an estimate of the $Γ−liminf$ and $Γ−limsup$ which can be strengthened to the $Γ$-convergence result. The Reissner-Mindlin model is "almost" the linearization of the $Γ$-limit for $μ_{c}=0$.

## Cite this article

Patrizio Neff, Krzysztof Chelminski, A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via $Γ$-convergence. Interfaces Free Bound. 9 (2007), no. 4, pp. 455–492

DOI 10.4171/IFB/173