Green's functions for anisotropic elastic plates containing polygonal holes

Abstract

Two sets of Green's functions for anisotropic elastic media containing a polygonal hole with rounded corners are presented. Analyzing non-elliptical holes poses a formidable mathematical challenge for anisotropic materials because existing mapping functions generally fail to meet certain criteria. The nonconformal mapping scheme and the perturbation method with conformal mapping are applied to derive two sets of solutions, respectively. They are solved using the method of analytical continuation. For the nonconformal approach, the hole is mapped onto a unit circle with nonconformal mapping functions, and the solution form is identical to elliptical holes only with a different mapping function. The perturbation solution is expressed in a series form based on the conformal mapping functions and solutions for elliptical holes. Proper adjustments of branch cuts of complex logarithm functions are introduced for both solutions. These adjustments are essential in numerical computation, as they ensure the continuity of an elastic body across branch cuts. Also, the two solutions presented in this paper are compared in-depth. Based upon their pros and cons, a combined use of these two solutions is suggested for application in boundary element method.