A model for non-linear rock deformation under compression due to sub-critical crack growth

Time dependency in rock deformation under compression is modelled by considering an elastic body containing cracks that grow under compressive stresses due to sub-critical crack growth. This is considered the prime mechanism for the time-dependent deformation of brittle rocks at low temperatures. The growth of cracks under compressive stresses is formulated using the "sliding crack" model, which considers extensile crack growth due to stress concentrations around pre-existing flaws. Subcritical crack growt is included into the sliding crack model by utilizing the empirical Charles power law relation between crack velocity and the crack tip stress intensity factor. The model is able to predict the dependence of the stress-strain curve on the applied strain rate, and agrees extremely well with experimental data. Also, the model is able to predict the occurrence of both transient and tertiary creep. The transient creep behavior is derived in closed-form, and is found to give creep that depends on the logarithm of time, which is similar to many empirical formulae for creep in brittle rocks. Tertiary creep in the model is due to crack interaction, and is found to occur at a critical value of crack density. This allows time-to-failure predictions to be made, which could be useful for underground structures required to remain open for long periods of time.