A comparison of finite-difference and finite-element methods for calculating free edge stresses in composites

Abstract

This study considers the accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber-reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points. Moreover, forward and backward finite difference formulas are used to enforced continuity of interlaminar stress components for the third problem. The study shows that the finite difference method employed in this investigation provides solutions to the three elasticity problems considered that are as accurate as the corresponding finite element solutions. Furthermore, the finite difference method appears to give a solution for the laminate problem that characterizes the stress distributions near an interface corner in a more realistic manner than the finite element method.