A Study of the Anisotropic Static Elasticity System in Thin Domain

Abstract

We study the asymptotic behavior of solutions of the anisotropic heterogeneous linearized elasticity system in thin domain of ${ℝ}^3$ which has a fixed cross-section in the ${ℝ}^2$ plane with Tresca friction condition. The novelty here is that stress tensor has given by the most general form of Hooke’s law for anisotropic materials. We prove the convergence theorems for the transition 3D-2D when one dimension of the domain tends to zero. The necessary mathematical framework and (2D) equation model with a specific weak form of the Reynolds equation are determined. Finally, the properties of solution of the limit problem are given, in which it is confirmed that the limit problem is well defined.