Damage Rheology as an Emergent Property of Sub-Critical Crack Growth

Griffith's nucleation theory can be combined with chemical kinetics at an atomic level to predict an exponential relationship between crack growth velocity V and the energy release rate G under sub-critical, quasi-static conditions. In turn this predicts an inverse logarithmic acceleration of crack length to a singularity at the failure time. However, laboratory experiments commonly report a power law relationship between V and G, implying a strong statistical mechanical overprint or geometrical constraint on the way in which a crack population can grow in a composite medium. This additional constraint results in an inverse power-law acceleration of crack growth to a singularity, equivalent to that found in the evolution of correlation length towards the critical point. A mean field damage model based on these observations predicts the observed form of the modified Omori law. By allowing for a combination of positive and negative feedback processes between crack length and G (e.g. chemically-assisted compaction followed by sub-critical crack growth), a power-law creep rheology emerges with a finite yield strength. We summarise a suite of laboratory data carried out at different strain rates that validates the predictions of the theory, and highlight the utility of simultaneous measurement of rheological (stress, strain) seismological (or acoustic emission) and chemical processes. The results also show that deformation at slow strain rate results in a sharper singularity at the failure time, implying a lower degree of predictability at low strain rate. Further work is required, particularly in the strain softening regime, to account for crack-crack interactions not included explicitly in the mean field theory presented, and to discriminate between competing damage hypotheses given the relative narrow bandwidth in space and time of much of the available data.