Tension and shear cracks in the Dugdail-Bahrenblatt model
© Khazan Y.M., Fialko Y.A.
We derive exact solutions for two-dimensional equilibrium tensile and shear Dugdale-Barenblatt (DB) cracks in an infinite elastic body. We show that in the DB approximation the solutions depend on critical length l* that equals 3-10 cm for limestone and granite under zero confining pressure. For long cracks, i.e. in the limit, where l, σT and p stand for the halflength of the developed part of a crack, cohesive stress acting at a crack tip and equilibrium internal pressure, correspondingly, a crack propagates preserving the length of the cohesive zone, and structure of the stress field in the vicinity of the tip. In this limit fracture energy, and apparent stress intensity factor do not depend on a crack length and could be considered as material properties. Microcracks, i.e. cracks having length, differ in many aspects from macrocracks. For mode I microcracks under atmospheric pressure and ambient extension, an equilibrium halflength of an open crack is much smaller than the critical length, l<<l*, but the total crack halflength (open crack halflength plus a cohesive zone) may greatly exceed l*. Also, for such microcracks the DB model predicts a scaling, in agreement with experimental data. Such scaling implies that for small tensional cracks the fracture energy is not a material property. The total length of fluid driven microcracks filled up to the cohesive zone base is of the order of where ρ°° stands for confining (lithostatic) pressure, and an equilibrium internal pressure greatly exceeds cohesive stresses and confining pressure. Microcracks become interconnected when the total crack length approches the characteristic spacing between the cracks. In the regions of mantle upwellings, the permeability of partially molten rocks may increase as a result of an increase in the total crack length due to decompression. For long cracks under high confining pressure the DB model predicts an increase of KQ with confining pressure and the length of region at the crack tip not penetrated by fluid. An increase in the pressure induced fracture resistance for fluid driven cracks results in domination of the confining pressure effects over intrinsic rock strength starting from quite shallow depth (hundred of meters).