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 Γ subjected to a prescribed boundary condition, the problem can be solved by mapping Γ to a circle of unit radius. It is important that (i) each point on Γ is mapped to the same point for the three Stroh eigenvalues $p1,$$p2,$$p3$ and (ii) the mapping is one-to-one for the region outside Γ. In an earlier paper it was shown that conditions (i) and (ii) are satisfied when Γ is an ellipse. The paper did not address to the case when Γ is an open boundary, such as a parabola or hyperbola that was studied by Lekhnitskii. We examine the mappings employed by Lekhnitskii for a parabola and hyperbola, and show that while the mapping for a parabola satisfies conditions (i) and (ii), the mapping for a hyperbola does not satisfy condition (i). Nevertheless, a valid solution can be obtained for the problem with a hyperbolic boundary, although the prescription of the boundary condition is restricted. We generalize Lekhnitskii’s solutions for general anisotropic elastic materials and for more general boundary conditions. Using known identities and new identities presented here, real form expressions are given for the displacement and hoop stress vector at the parabolic and hyperbolic boundary.