The effect of fluid pressure on wave speeds in a cracked solid

In a porous or cracked elastic solid, the effective stress (defined in term s of the loads applied to the solid part of the outer boundary) and effecti ve strain (defined in terms of the displacements at the solid part of the o uter boundary) occurring in small-amplitude deformations are connected by a linear relation along with the pressure within the fluid occupying the por es and cracks. We derive here a formula of this kind for a static system in which enough time is allowed for pressure to be equalized throughout the f luid (on the assumption that all pockets of fluid are connected in some way ). The formula depends on the overall stiffnesses relating stress to strain for the same material with the fluid removed (dry or empty cracks and pore s). For undrained conditions where no fluid is allowed to enter or leave th e body, the pressure is directly related to the effective stress and strain , and the Gassmann relations are obtained relating the stiffnesses for an i sotropic material in dry and undrained conditions. For an anisotropic mater ial, the Brown-Korringa relations are recovered. Externally imposed stresse s and fluid pressure distort the material structure and influence the wave speeds of elastic waves. The main way in which this occurs is in changing t he aspect ratios of flat cracks, the most compliant part of the microstruct ural geometry. This effect on the wave speeds is studied here both in terms of crack closure, with corresponding changes in crack number density, and in variations in crack aspect ratios. The principal way in which the latter influences the wave speeds is through the fluid incompressibility factor i n the formula for the properties of materials with connected cracks. An inc rease in aspect ratio of the cracks is equivalent to a reduction in the bul k modulus of the fluid. This effect is apparent in the limits of both high frequencies, when the material behaves as if the cracks were isolated, and low frequencies, when undrained conditions apply.