Compressibility and shear compliance of spheroidal pores: Exact derivation via the Eshelby tensor, and asymptotic expressions in limiting cases

We explicitly calculate the elastic compliance of a spheroidal pore in an isotropic solid, starting from Eshelby’s tensor. The exact expressions found for the pore compressibility, P, and the shear compliance, Q, are valid for any value of the aspect ratio α, from zero (cracks) to infinity (needles). This derivation clarifies previous work on this problem, in which different methods were used in different ranges of α, or typographical errors were present. The exact expressions obtained for P and Q are quite complex and unwieldy. Simple expressions for both P and Q have previously been available for the limiting cases of infinitely thin-cracks (α = 0), infinitely long-needles (α = ∞), and spherical pores (α = 1). We have now calculated additional terms in the asymptotic expansions, yielding relatively simple approximations for P and Q that are valid for crack-like pores having aspect ratios as high as 0.3, needle-like pores having aspect ratios as low as 3, and nearly spherical pores. Their relatively simple forms will be useful for incorporation into various schemes to estimate the effective elastic moduli.