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

Abstract

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. Then the Γ -limit model determines the midsurface deformation m ∈ H 1,2(ω,R3). For the true defective crystal case, however, μc = 0 is appropriate. Without equi-coercivity, we first obtain an estimate of the Γ -lim inf 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 μc = 0.