Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach

Abstract

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.