Elastic moduli of solids containing spheroidal pores

We use asymptotic approximations for the elastic compliances (P, Q) of a spheroidal pore as input in the differential effective medium scheme to derive approximate analytical expressions for the effective moduli of an isotropic solid containing randomly oriented spheroids. The approximations are valid for crack-like pores having aspect ratios α as high as 0.3, needle-like pores having aspect ratios as low as 3, and nearly spherical pores (0.7 < α < 1.3). Analytical solutions for the differential scheme have previously only been available for the limiting cases of infinitely thin-cracks (α = 0) and spherical pores (α = 1). The relatively simple approximations found between the limiting cases can account for more realistic pore shapes, and are valid for a wide range of porosities. The behaviour of the effective Poisson’s ratio in the high concentration limit shows that ν is bounded between the Poisson’s ratio of the solid and a fixed point ν_c that only depends on the aspect ratio of the pores. The asymptotic expressions for P and Q can also successfully be used as input in any other effective medium theory, such as the Mori-Tanaka or Kuster-Toksoz schemes. The relatively simple expressions found for the various effective medium schemes, as well as the bounds found for the effective Poisson’s ratio, will be useful to simplify the process of inversion of elastic velocities in porous solids.