The complex variable method and conformal mapping are used to derive a closed-form expression for the compressibility of an isolated pore in an infinite two-dimensional, isotropic elastic body. The pore is assumed to have an n -fold axis of symmetry, and be represented by at most four terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. The results are validated against some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from three and four terms of the Schwarz–Christoffel mapping function for regular polygons, the compressibilities of a triangle, square, pentagon and hexagon are found (to at least three digits). Specific results for some other pore shapes, more general than the quasi-polygons obtained from the Schwarz–Christoffel mapping, are also presented. An approximate scaling law for the compressibility, in terms of the ratio of perimeter-squared to area, is also tested. This expression gives a reasonable approximation to the pore compressibility, but may overestimate it by as much as 20%.

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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 4q. An approximate scaling law for the shear compliance parameter, in terms of the ratio of perimeter squared to area, is proposed and tested.

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The boundary perturbation method is used to solve the problem of a nearly circular rigid inclusion in a two-dimensional elastic medium subjected to hydrostatic stress at infinity. The solution is taken to the fourth order in the small parameter epsilon that quantifies the magnitude of the variation of the radius of the inclusion. This result is then used to find the effective bulk modulus of a body that contains a dilute concentration of such inclusions. The corresponding results for a cavity are obtained by setting the Muskhelishvili coefficient κ equal to −1, as specified by the Dundurs correspondence principle. The results for nearly circular pores can be expressed in terms of the pore compressibility. The pore compressibilities given by the perturbation solution are tested against numerical values obtained using the boundary element method, and are shown to have good accuracy over a substantial range of roughness values.