Analytic solutions for effective elastic moduli of isotropic solids containing oblate spheroid pores with critical porosity

Abstract

Accurate characterization for effective elastic moduli of porous solids is crucial for better understanding their mechanical behaviour and wave propagation, which has found many applications in the fields of engineering, rock physics and exploration geophysics. We choose the spheroids with different aspect ratios to describe the various pore geometries in porous solids. The approximate equations for compressibility and shear compliance of spheroid pores and differential effective medium theory constrained by critical porosity are used to derive the asymptotic solutions for effective elastic moduli of the solids containing randomly oriented spheroids. The critical porosity in the new asymptotic solutions can be flexibly adjusted according to the elastic moduli – porosity relation of a real solid, thus extending the application of classic David-Zimmerman model because it simply assumes the critical porosity is one. The asymptotic solutions are valid for the solids containing crack-like oblate spheroids with aspect ratio $\alpha$< 0.3, nearly spherical pores (0.7 < $\alpha$< 1.3) and needle-like prolate pores with $\alpha$ > 3, instead of just valid in the limiting cases, for example perfectly spherical pores ($\alpha$= 1) and infinite thin cracks ($\alpha \to$0). The modelling results also show that the accuracies of asymptotic solutions are weakly affected by the critical porosity ${\varphi }_c$ and grain Poisson's ratio $v_0$, although the elastic moduli have appreciable dependency of ${\varphi }_c$ and $v_0$. We then use the approximate equations for pore compressibility and shear compliance as inputs into the Mori–Tanaka and Kuster–Toksoz theories and compare their calculations to our results from differential effective medium theory. By comparing the published laboratory measurements with modelled results, we validate our asymptotic solutions for effective elastic moduli.