Complex Variable Solutions for Inhomogeneous and Laminated Elastic Plates

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 thin plate or classical laminate theory equations (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 formulate this theory in terms of complex potentials. It is shown that the displacement and stress in the inhomogeneous material can be expressed in terms of four complex potentials that are analytic functions of ζ = x + i y. Expressions for stress and moment resultants are derived in terms of these complex potentials. As examples we solve the problems of biaxial stress in an infinite plate and of an in-plane point force in an infinite plate and at the boundary of a semiinfinite plate. We also formulate the general boundary-value problem for a half-plane and develop a procedure to determine the complex potentials for an inhomogeneous plate in terms of the complex potentials for the corresponding plane-strain problem.