We derive the -limit to a three-dimensional Cosserat model as the aspect ratio 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 -limit based on the natural scaling consists of a membrane like energy and a transverse shear energy both scaling with , augmented by a curvature energy due to the Cosserat bulk, also scaling with . A technical difficulty is to establish equi-coercivity of the sequence of functionals as the aspect ratio 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 , equi-coercivity needs a strictly positive . Then the -limit model determines the midsurface deformation . For the true defective crystal case, however, is appropriate. Without equi-coercivity, we obtain first an estimate of the and which can be strengthened to the -convergence result. The Reissner-Mindlin model is "almost" the linearization of the -limit for .
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. 455–492DOI 10.4171/IFB/173