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

Patrizio Neff¹  ; Krzysztof Chelminski²

¹Technische Hochschule Darmstadt, Germany
²Technical University, Warszawa, Poland

Interfaces and free boundaries, Tome 9 (2007) no. 4, pp. 455-492

Résumé

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 μc​≥0, equi-coercivity needs a strictly positive μc​>0. Then the Γ-limit model determines the midsurface deformation m∈H1,2(ω,R3). For the true defective crystal case, however, μc​=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 μc​=0.

Détail

DOI : 10.4171/ifb/173

Classification : 35-XX, 65-XX, 76-XX, 92-XX
Mots-clés : Shells, plates, membranes, thin films, polar materials, non-simple materials, <em>Γ</em>-convergence, homogenization, transverse shear, shear correction factor, defective elastic crystals, lattice rotations <em>p</em>-harmonic map, finite elements, full discretization, discrete energy law