Cumulative Benioff strain-release, modified Omori's law and transient behaviour of rocks

Irreversible thermodynamic theories with internal state variables can be used to derive a general constitutive law for both transient and steady-state behaviours of rocks. This constitutive law can represent the concepts of damage and damage evolution in either the fibre-bundle model or continuum damage mechanics. We have previously proposed an empirically based constitutive law for both the transient and steady-state behaviours of rocks ultimately derived from laboratory experimental data. We show here that this law is concordant with the general constitutive law derived from irreversible thermodynamic theories, and that the relaxation modulus has a temporal power–law that depends on a structural fractal property of rocks. Our constitutive law predicts forms for the cumulative Benioff strain-release for precursory seismic activations and the modified Omori's laws of aftershocks, both aspects of the temporal fractal properties of seismicity. These seismic properties can also be derived by the fibre-bundle model or continuum damage mechanics. Our model suggests that these time-scale invariant processes of seismicity may be regulated by the fractal structures of crustal rocks.