Stress Relaxation Near the Tip of a Stationary Mode I Crack in a Poroelastic Solid

We study the short time transient stress and pore pressure fields near the tip of a stationary crack when a sudden load is applied to a poroelastic solid. These fields are determined using a small scale “yielding” (SSY) analysis where the stress relaxation due to fluid flow is confined to a small region near the crack tip. They are found to exhibit the usual inverse square root singularity characteristic of cracks in linear elastic solids. Analysis shows that these fields are self-similar; the region of stress relaxation that propagates outward from the crack tip is proportional to $Dct$⁠, where $Dc$ is the cooperative diffusion coefficient and t is time. The pore pressure at the crack tip vanishes immediately after loading. The stress intensity factor at the crack tip is found to be reduced by a factor of $1/[2(1-v)]$⁠, where $v$ is the Poisson's ratio of the drained solid. Closed form approximations are found for the pore pressure and the trace of the effective stress. These approximate analytical solutions compare well with finite element results.