Theory of laminated elastic plates I. isotropic laminae

Abstract

In this paper (part I) we establish a theory for stretching and bending of laminated elastic plates in which the laminae are different isotropic linearly elastic materials. The theory gives exact solutions of the three-dimensional elasticity equations that satisfy all the interface traction and displacement continuity conditions, with no traction on the lateral surfaces; the only restriction is that edge boundary conditions can be satisfied only in an average manner, rather than point by point. The method, which is based on a generalization of Michell’s exact plane stress theory, yields exact solutions for each lamina. These solutions are generated in a very straightforward manner by solutions of the approximate two-dimensional classical equations of laminate theory and contain sufficient arbitrary constants to enable all the continuity and lateral surface boundary conditions to be satisfied. The values of the constants depend only on the lamina thicknesses and the elastic constants. Thus, for a given laminate and for any boundary-value problem , it is necessary only to solve the appropriate two-dimensional plane problem, and the corresponding exact three-dimensional laminate solution follows by straightforward substitutions. The two-dimensional solution may be derived by any of the available methods, including numerical methods. An important feature of the theory is that it determines the interfacial shearing tractions, as well as the in-plane stress components. The procedure is illustrated by applying the theory to three problems involving stretching and bending of laminated plates containing circular holes.