Displacement potential approach to solution of elasticity problems of orthotropic composite structures

Abstract

Due to increasing demands of composite materials in various applications, they have received wide attention of the researchers for their analyses with a view to using them safely, effectively, and efficiently. These materials are anisotropic and non-homogeneous from the macroscopic point of view which makes the analysis more difficult than that of isotropic homogeneous materials. Although experimental or numerical procedures can be used for the purpose of analysis, theatrics methods arc not always suitable. Experimental procedures arc usually costly and not practical for regular general purpose use. Therefore, a simple analytical method is always sought as an alternative. Besides, irrespective .of the accuracy of any numerical solution, analytical solution is always desirable. In this study, an analytical method is developed for the analysis of elastic field in structures of orthotropic composite materials under mixed mode of boundary conditions. Replacing the two displacement components by a single displacement potential function, the two dimensional elasticity problems are reduced to the solution of a single fourth order partial differential equation of a single unknown function. Also, all the boundary conditions are expressed in terms of the displacement potential function. The solution of the differential equation is obtained in the form of an infinite series, the coefficients of which are determined satisfying , the boundary conditions. The method is demonstrated by solving some practical problems of composite panels, columns, and beams under mixed boundary conditions. Further, a finite difference scheme based on the displacement potential function is formulated to cover wider category of boundary value problems which cannot be conveniently dealt with by the analytical method. In addition to the problems solved by the analytical method, some additional problems are also solved by the finite difference scheme. To verify the validity of the analytical method, a commercial code, ANSYS. is used to solve the same problems using rectangular plane e1cmcnts. The analytical results are compared with those obtained by the finite difference method and finite element method. It is noted that the results agree well within the acceptable limit. This establishes the soundness and reliability of the analytical method.