Discussion on the conformal mapping of a half-plane onto a unit disk in anisotropic elasticity and related applications

Abstract

The conformal mapping, which transforms a half-plane into a unit disk, has been used widely in studies involving an isotropic elastic half-plane under anti-plane shear or plane deformation. However, very little attention has been paid to the possibility of utilizing this mapping in the study of an anisotropic elastic half-plane under the same deformation. In this paper, we discuss a general case of an arbitrarily located anisotropic elastic half-plane that corresponds to several affine counterparts (resulting from corresponding complex variable formalism). We show that this mapping is indeed applicable to each of the affine half-planes if and only if the key parameters in the mapping satisfy simple geometrical conditions. In addition, we introduce the application of this mapping with the corresponding geometrical conditions to the related study of anisotropic thin films under two-dimensional deformation.