# Interfaces and Free Boundaries

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**Volume 9, Issue 4, 2007, pp. 455–492**

**DOI: 10.4171/IFB/173**

Published online: 2007-12-31

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

^{[1]}and Krzysztof Chelminski

^{[2]}(1) Technische Hochschule Darmstadt, Germany

(2) Technical University, Warszawa, Poland

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 $m\in H^{1,2}(\omega,\R^3)$. For the true defective crystal case, however, $\mu_c=0$ is appropriate. Without equi-coercivity, we obtain first an estimate of the $\Gamma-\liminf$ and $\Gamma-\limsup$ which can be strengthened to the $\Gamma$-convergence result. The Reissner-Mindlin model is "almost" the linearization of the $\Gamma$-limit for $\mu_c=0$.

*Keywords: *Shells, plates, membranes, thin films, polar materials, non-simple materials, *Γ*-convergence, homogenization, transverse shear, shear correction factor, defective elastic crystals, lattice rotations
*p*-harmonic map, finite elements, full discretization, discrete energy law

Neff Patrizio, Chelminski Krzysztof: A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via *Γ*-convergence. *Interfaces Free Bound.* 9 (2007), 455-492. doi: 10.4171/IFB/173