Shear compliance of two-dimensional pores possessing N-fold axis of rotational symmetry

We use the complex variable method and conformal mapping to derive a closed-form expression for the shear compliance parameters of some two-dimensional pores in an elastic material. The pores have an N-fold axis of rotational symmetry and can be represented by at most three terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. We validate our results against the solutions of some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from two and three terms of the Schwarz-Christoffel mapping function for regular polygons, we find the shear compliance of a triangle, square, pentagon and hexagon. We explicitly verify the fact that the shear compliance of a symmetric pore is independent of the orientation of the pore relative to the applied shear, for all cases except pores of fourfold symmetry. We also show that pores having fourfold symmetry, or no symmetry, will have shear compliances that vary with cos 4 theta. An approximate scaling law for the shear compliance parameter, in terms of the ratio of perimeter squared to area, is proposed and tested.