Effects of permeability conditions on time-dependent fracture of poroelastic media

The time-dependent fracture of fluid-infiltrated porous materials is known to be affected by fluid diffusion, which is strongly influenced by the initial pore pressure field attributed to loading and the permeability conditions on the boundaries. In this research, the concurrent solid deformation and fluid migration for a traction-free crack of finite length 2a in a poroelastic medium are analyzed. The remote boundaries of the medium are subjected to constant stress or constant stain in the direction perpendicular to crack faces. Without confining the range of the fluid diffusion to the neighborhood of the crack, four types of permeability conditions are considered on the boundaries: (I) the remote boundaries and the crack faces are both impermeable, (II) the remote boundaries are permeable to constant pore pressure, but the crack faces are impermeable, (III) the remote boundaries are impermeable, but the crack faces are permeable to constant pore pressure, and (IV) the remote boundaries and the crack faces are both permeable to constant pore pressure. The singularities of the stress and the pore pressure gradient at small flow times are obtained from finite element simulations and compared to asymptotic predictions. The instantaneous fracture energy J is evaluated using a method that uses the J-integral around the cohesive zone embedded ahead of the crack tip in the simulations. The numerical results of J are validated with the asymptotes predicted from stress intensities at small flow times. When there is a difference between the imposed pore pressure and the initial pore pressure caused by applying loading on the boundaries, the variation in J(t) is affected by short and long ranges of fluid flow, and J(t) reaches its drained limit at a time on the order of W²/c, where W is the dimensional size and c is the consolidation coefficient of porous materials. The drained limit may not be the maximum value of J, especially when the difference on the remote boundaries is positive under strain control. For stress-control cases, whether the maximum J occurs before the drained state depends not only on the difference in pore pressure but also on the location of the permeable boundaries.