Anisotropic Materials with an Elliptic Boundary

The determination of stress distribution in a solid with the presence of a hole or an inclusion has been a mathematically interesting and challenging problem. It is also an important problem in applications. The simplest geometry of the hole is a circle. For isotropic materials a hole of arbitrary shape can be transformed, in theory, to a circle by a conformal mapping (Muskhelishvili, 1953; see also Section 3.12). Therefore a circular hole is all one needs to study for isotropic materials. For anisotropic materials there are three complex variables ɀα=x1+pαx2 (α=1,2,3). It is in general not possible to transform a hole of given shape to the same circle for all three complex variables. An exception is the ellipse (Lekhnitskii, 1950; Savin, 1961). In this chapter we study various problems involving an elliptic boundary. The ellipse can be a hole, a rigid body, or an inclusion of different anisotropic materials. We will also consider an elliptic body subjected to external forces. For anisotropic elastic materials even the circular hole needs a transformation. There is practically no difference in the analysis if the ellipse is replaced by a circle. We may employ dual coordinate systems. One coordinate system is chosen to coincide with a symmetry plane of the material when such a plane exists. The other coordinate system is to coincide with the principal axes of the ellipse. The analysis is no more complicated than when a single coordinate system is employed. For some problems employment of dual coordinate systems reveal that certain aspects of the solutions are invariant with the orientation of the ellipse in the material.