Modelling the time-dependent rheological behaviour of heterogeneous brittle rocks

A 2-D numerical model for brittle creep and stress relaxation is proposed for the time-dependent brittle deformation of heterogeneous brittle rock under uniaxial loading conditions. The model accounts for material heterogeneity through a stochastic local failure stress field, and local material degradation using an exponential material softening law. Importantly, the model introduces the concept of a mesoscopic renormalization to capture the co-operative interaction between microcracks in the transition from distributed to localized damage. The model also describes the temporal and spatial evolution of acoustic emissions, including their size (energy released), in the medium during the progressive damage process. The model is first validated using previously published experimental data and is then used to simulate brittle creep and stress relaxation experiments. The model accurately reproduces the classic trimodal behaviour (primary, secondary and tertiary creep) seen in laboratory brittle creep (constant stress) experiments and the decelerating stress during laboratory stress relaxation (constant strain) experiments. Brittle creep simulations also show evidence of a critical level of damage before the onset of tertiary creep and the initial stages of localization can be seen as early as the start of the secondary creep phase, both of which have been previously observed in experiments. Stress relaxation simulations demonstrate that the total amount of stress relaxation increases when the level of constant axial strain increases, also corroborating with previously published experimental data. Our approach differs from previously adopted macroscopic approaches, based on constitutive laws, and microscopic approaches that focus on fracture propagation. The model shows that complex macroscopic time-dependent behaviour can be explained by the small-scale interaction of elements and material degradation. The fact that the simulations are able to capture a similar time-dependent response of heterogeneous brittle rocks to that seen in the laboratory implies that the model is appropriate to investigate the non-linear complicated time-dependent behaviour of heterogeneous brittle rocks.