Analysis of an orthotropic elastic plane problem of a half plane with an edge crack using a mapping function

Abstract

It is a challenging problem to derive a solution for an orthotropic elastic plane problem using a mapping function. The complex variable method is employed to introduce a mapping function. Then, comparatively arbitrary configurations can be analyzed. The boundary condition equation is represented by two complex variables. This fact makes solving the present problem difficult. This difficulty is conquered by applying Cauchy integrals. Also, mapping functions represented by an infinite terms of fractional expressions are introduced to conquer the mathematical difficulties of the Cauchy integral in the process of the derivation of the stress function. One of two stress functions is derived by analytical continuation. The final exact stress functions are represented by an irrational mapping function in a closed form. Stress components are represented by one complex variable. Therefore, the calculation of the stress components is easier than that of an isotropic problem. Half planes with a notch or a mound expressed by an irrational mapping function can be solved using their mapping functions. Such a polygonal mapping function can be derived by Schwarz–Christoffel’s transformation. As a demonstration, a half plane with a vertical edge crack subjected to uniform tension is analyzed and the stress distributions are shown for two examples of two characteristic roots of the characteristic equation for the orthotropic elastic plane.