A damage mechanics model for power-law creep and earthquake aftershock and foreshock sequences

Abstract

It is common practice to refer to three independent stages of creep under static loading conditions in the laboratory: namely transient, steady-state, and accelerating. Here we suggest a simple damage mechanics model for the apparently trimodal behaviour of the strain and event rate dependence, by invoking two local mechanisms of positive and negative feedback applied to constitutive rules for time-dependent subcritical crack growth. In both phases, the individual constitutive rule for measured strain ε takes the form ε(t) = ε0[1 + t/mτ](m), where τ is the ratio of initial crack length to rupture velocity. For a local hardening mechanism (negative feedback), we find that transient creep dominates, with 0 < m < 1. Crack growth in this stage is stable and decelerating. For a local softening mechanism (positive feedback), m < 0, and crack growth is unstable and accelerating. In this case a quasi-static instability criterion ε → ∞ can be defined at a finite failure time, resulting in the localization of damage and the formation of a throughgoing fracture. In the hybrid model, transient creep dominates in the early stages of damage and accelerating creep in the latter stages. At intermediate times the linear superposition of the two mechanisms spontaneously produces an apparent steady-state phase of relatively constant strain rate, with a power-law rheology, as observed in laboratory creep test data. The predicted acoustic emission event rates in the transient and accelerating phases are identical to the modified Omori laws for aftershocks and foreshocks, respectively, and provide a physical meaning for the empirical constants measured. At intermediate times, the event rate tends to a relatively constant background rate. The requirement for a finite event rate at the time of the main shock can be satisfied by modifying the instability criterion to having a finite crack velocity at the dynamic failure time, dx/dt → V(R), where V(R) is the dynamic rupture velocity. The same hybrid model can be modified to account for dynamic loading (constant stress rate) boundary conditions, and predicts the observed loading rate dependence of the breaking strength. The resulting scaling exponents imply systematically more non-linear behaviour for dynamic loading.