On the Solution of Plane, Orthotropic Elasticity Problems by an Integral Method

Abstract

The present paper is concerned with the application of integral equation techniques to problems in plane orthotropic elasticity. Two approaches for solving such problems are outlined, both of which are characterized by embedding the real body in a “fictitious” body for which the appropriate influence functions are known. Fictitious tractions are then introduced such that the boundary conditions on the real body are satisfied. This results in a coupled set of integral equations in the fictitious traction components. Once these are found the unknowns, i.e., stresses, etc., are found in a straightforward manner. The difficulty is in introducing the fictitious traction field such that the resulting integral equations are useful computationally, i.e., are Fredholm equations of the second rather than the first kind. A sufficient condition for this is that the fictitious traction field is applied to the boundary of the real body. The two approaches just mentioned differ in the choice of influence function used, in one case the influence function being singular in the field and the other singular on the boundary. A solution method already exists in the isotropic case using the boundary influence function [3]. An alternate formulation is presented using an internal influence function which is shown to have computational advantages in the anisotropic (orthotropic) case. To illustrate the methods, the stress field is found in a “truncated” orthotropic quarter space, under the condition of a given traction on the truncated surface, traction-free elsewhere. This problem is of interest in certain Rock Mechanics calculations, e.g., to a first approximation the stress field is that due to a rigid wedge penetrating a brittle, orthotropic elastic solid (prior to chip formation).