Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

Abstract

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described. (2000): 74B05, 74E05, 74G05, 74K20. Mathematics Subject Classifications

A refined model for the buckling of film/substrate bilayers

“…The most attractive feature of this methodology is that all manipulations only involve the solutions of linear algebraic equations and the expansions can be carried out to any order in h on a symbolic manipulation platform. Also, our treatment of both the coating and half-space can be easily generalised to deal with anisotropic and/or inhomogeneous materials (Spencer, 1984(Spencer, , 2005, nonlinear effects (Erbay, 1997), liquid crystal elastomers (Goriely & Mihai, 2021;Liu et al, 2021), and the case when U and V depend on both of the in-plane coordinates.…”

A refined model for the buckling of film/substrate bilayers

“…The most attractive feature of this methodology is that all manipulations only involve the solutions of linear algebraic equations, and the expansions can be carried out to any order in h on a symbolic manipulation platform. Also, our treatment of both the coating and half-space can easily be generalised to deal with anisotropic and/or inhomogeneous materials [56,57], non-linear effects [58], liquid crystal elastomers [59,60], an infinite beam embedded in an infinite elastic solid [61], and the case when U and V depend on both of the in-plane coordinates.…”