Anisotropic elastic materials with a parabolic or hyperbolic boundary: A classical problem revisited

When an anisotropic elastic material is under a two-dimensional deformation that has a hole of given geometry Gamma subjected to a prescribed boundary condition, the problem can be solved by mapping Gamma to a circle of unit radius. It is important that (i) each point on Gamma is mapped to the same point for the three Stroh eigenvalues p(1), p(2), p(3) and (ii) the mapping is one-to-one for the region outside Gamma. In an earlier paper it was sho wn that conditions (i) and (ii) are satisfied when Gamma is an ellipse. The paper did not address to the case when Gamma is an open boundary, such as a parabola or hyperbola that was studied by Lekhnitskii. We examine the map pings employed by Lekhnitskii for a parabola and hyperbola, and show that w hile the mapping for a parabola satisfies conditions (i) and (ii), the mapp ing for a hyperbola does not satisfy condition (i). Nevertheless, a valid s olution can be obtained for the problem with a hyperbolic boundary, althoug h the prescription of the boundary condition is restricted. We generalize L ekhnitskii's solutions for general anisotropic elastic materials and for mo re general boundary conditions. Using known identities and new identities p resented here, real form expressions are given for the displacement and hoo p stress vector at the parabolic and hyperbolic boundary.