JournalsifbVol. 9, No. 4pp. 455–492

A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via <em>Γ</em>-convergence

  • Patrizio Neff

    Technische Hochschule Darmstadt, Germany
  • Krzysztof Chelminski

    Technical University, Warszawa, Poland
A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via <em>Γ</em>-convergence cover
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Abstract

We derive the Γ\Gamma-limit to a three-dimensional Cosserat model as the aspect ratio h>0h>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\Gamma-limit based on the natural scaling consists of a membrane like energy and a transverse shear energy both scaling with hh, augmented by a curvature energy due to the Cosserat bulk, also scaling with hh. A technical difficulty is to establish equi-coercivity of the sequence of functionals as the aspect ratio hh 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 μc0\mu_c\ge 0, equi-coercivity needs a strictly positive μc>0\mu_c>0. Then the Γ\Gamma-limit model determines the midsurface deformation mH1,2(ω,R3)m\in H^{1,2}(\omega,\R^3). For the true defective crystal case, however, μc=0\mu_c=0 is appropriate. Without equi-coercivity, we obtain first an estimate of the Γlim inf\Gamma-\liminf and Γlim sup\Gamma-\limsup which can be strengthened to the Γ\Gamma-convergence result. The Reissner-Mindlin model is "almost" the linearization of the Γ\Gamma-limit for μc=0\mu_c=0.

Cite this article

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

DOI 10.4171/IFB/173