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.
Highlights
We are interested of the asymptotic behavior of the linear elasticity system in a domain of R3 with a Tresca friction condition where the boundary of this domain has a fixed cross-section in dimension 2 and a small thickness
One of the objectives of this study is to obtain twodimensional equation that allows a reasonable description of the phenomenon occurring in the three-dimensional domain by passing the limit to 0 on the small thickness of the domain (3D)
Let us mention for example [1,2,3,4,5,6,7,8] in which the authors worked on the asymptotic behavior for the linearized elasticity system with different boundary conditions
Summary
We are interested of the asymptotic behavior of the linear elasticity system in a domain of R3 with a Tresca friction condition where the boundary of this domain has a fixed cross-section in dimension 2 and a small thickness. The static case with a nonlinear term for linear elastic materials has been considered in [3] See another situation in [4] where the paper concerns asymptotic derivation of frictionless contact models for elastic rods on a foundation with normal compliance. The authors in [5, 12] have proved the asymptotic behavior of a frictionless contact problem between two elastic bodies, when the vertical dimension of the two domain reaches zero. All these papers have been only restricted in a homogeneous and isotropic case of elastic materials.
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