Poisson's ratio of porous and microcracked solids: Theory and application to oxide superconductors

We present a theoretical study of the effective Poisson's ratio of elastic solids weakened by porosity and microcracks. Explicit expressions of the effective Poisson's ratio are obtained using the Mori-Tanaka mean-field approach as applied to macroscopically isotropic solids containing randomly distributed and randomly oriented spheroidal pores. We focus on the influence of pore shape and concentration and devote special attention to the limiting cases of spherical, penny-shape, and needle-shape pores. A key result of this study is that the effective Poisson's ratio depends only on pore concentration, pore shape, and Poisson's ratio of the bulk solid. In other words, it is independent of any other elastic constants of the bulk solid. Also, theratioof the shear and bulk moduli behaves similarly. Unlike other elastic constants which monotonically decrease with pore concentration, Poisson's ratio may increase, decrease, or remain unchanged as a function of pore concentration, depending on the pore shape and Poisson's ratio of the bulk solid. We discuss ramifications of these findings with regard to the elastic constants of oxide superconductors, especially the bismuth cuprates, which show unusually low Poisson's ratios. We also discuss these low Poisson's ratios, including the possibility of negative Poisson's ratios.