3D structure–2D plate interaction model

Abstract

In this paper, we derive models for the interaction of a linearized three-dimensional elastic structure with a thin elastic layer of possibly different material attached to it. Rigorous derivation is performed by considering a thin three-dimensional layer and the asymptotics of the solution of the full remaining three-dimensional problem when the thickness $\epsilon$ of the thin layer tends to zero. Furthermore, the attached thin material is assumed to have the elasticity coefficients which are of order $1∕{\epsilon }^p$ , for $p\ge 0$ with respect to the coefficients of the three-dimensional body. In the limit, five different models are obtained with respect to different choices of p, namely $p\in [0,1⟩$ , $p=1$ , $p\in ⟨1,3⟩$ , $p=3$ , and $p\in ⟨3,\infty ⟩$ . Furthermore a three-dimensional–two-dimensional model is proposed that has the same asymptotics as the original three-dimensional problem. This is convenient for applications because one does not have to decide in advance which limit model to use.