Γ-convergene e for a geometrically exact Cosserat shell-model of defective elastic crystals

Abstract

I consider 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 functional 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 midsorfaee deformation m ∈ H ^1,2 (ω, ℝ³). For the true defective crystal case, however, μ_c=0 is appropriate. Without equi-coercivity, we obtain first an estimate of the Γ-lim in and Γ-lim sup 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.