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.

Further results concerning the refined theory of anisotropic laminated composite plates

“…As in the case of MPE principle (see also [24]), it could be shown that (53) assumes its absolute minimum for the actual solution and in addition that ()min = -(WC)mi We mention in passing that these energetic principles constitute as basis, among others, of many approximate numerical solution schemes, the most popular one being the finite-element method.…”

“…In addition to this shortcoming, the requirement of continuity of transverse shear stresses at the layer interfaces is violated. Within the discrete-layer model, the continuity requirements of transverse shear stresses at the interlaminar interfaces have been fulfilled in [16][17][18][19][20][21][22][23][24] for plates and shells and in [25] for beam-type structures.…”

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

“…The origins of the method can be found in the classical solutions by Michell [9] for plane stress of moderately thick elastic plates, and a reformulation of Michell's equations by Kaprielian, Rogers and Spencer [10]. The principal result obtained and applied in [1Y8] is that any solution of the classical thin plate or classical laminate theory equations (which describe a two-dimensional theory) can be applied by straightforward substitutions, to generate an exact solution of the three-dimensional linear elasticity equations for a material with arbitrary inhomogeneity in a specified direction, which is here taken to be the direction normal to the surface of a thick flat plate.…”

Concentrated force solutions for an inhomogeneous thick elastic plate

“…The origins of the method reside in classical solutions by Michell [7] for plane stress of moderately thick elastic plates, and a reformulation of Michell's equations by Kaprielian, Rogers and Spencer [8]. The principal result obtained in [1][2][3][4][5] is that any solution of the classical thin plate or classical laminate theory equations (which describe a two-dimensional theory) can be applied by straightforward substitutions, to generate an exact solution of the three-dimensional linear elasticity equations for a material with arbitrary inhomogeneity in a specified direction, which is here taken to be the direction normal to the surface of a thick flat plate.…”