Analytical simulation of the elastic moduli dispersion for an isotropic porous cylinder

Abstract

There are extensive laboratory measurements of elastic moduli dispersion of porous materials, yet the mechanisms for the dispersion are not fully understood. Related analytical solutions are missing in the literature, potentially due to complex mathematical derivations. In this paper, we use Biot's theory of poroelastodynamics (PED) and present the first analytical solution for an isotropic fluid-saturated porous cylinder subject to a forced deformation test. This is done by introducing Helmholtz decomposition, two scalar potentials, and two vector potentials to decouple the original governing partial differential equations. The methods of matrix diagonalization and separation of variables are then used to solve the decoupled equations. The elastodynamics (ED) solution is also presented. We demonstrate the coupled responses of displacement, pore pressure, and stress and interpret systematically the mechanisms for the elastic moduli dispersion. Furthermore, we investigate comprehensively the effects of loading frequency, material's poromechanical characteristics, sample size, and boundary conditions on the elastic moduli dispersion. Finally, excellent matches are found between the analytical solution and published measurements of the dynamic Young's moduli and Poisson's ratios of a water-saturated clastic sediment rock and a glycerin-saturated limestone. The analytical solution captures the materials’ elastic moduli dispersion at both seismic and sonic frequencies. This work should be of interest to scientists and engineers who work on dynamics, vibration, and poromechanics.