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

It is common practice to refer to three independent stages of creep under s tatic 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 i nvoking two local mechanisms of positive and negative feedback applied to c onstitutive rules for time-dependent subcritical crack growth. In both phas es, the individual constitutive rule for measured strain epsilon takes the form epsilon(t) = epsilon(0) [1 + t/m tau](m), where tau is the ratio of in itial crack length to rupture velocity. For a local hardening mechanism (ne gative feedback), we find that transient creep dominates, with 0 < m < 1. C rack growth in this stage is stable and decelerating. For a local softening mechanism (positive feedback), m < 0, and crack growth is unstable and acc elerating. In this case a quasi-static instability criterion epsilon --> in finity can be defined at a finite failure time, resulting in the localizati on of damage and the formation of a throughgoing fracture. In the hybrid model, transient creep dominates in the early stages of damag e and accelerating creep in the latter stages. At intermediate times the li near superposition of the two mechanisms spontaneously produces an apparent steady-state phase of relatively constant strain rate, with a power-law rh eology, as observed in laboratory creep test data. The predicted acoustic e mission event rates in the transient and accelerating phases are identical to the modified Omori laws for aftershocks and foreshocks, respectively, an d provide a physical meaning for the empirical constants measured. At inter mediate times, the event rate tends to a relatively constant background rat e. The requirement for a finite event rate at the time of the main shock ca n be satisfied by modifying the instability criterion to having a finite cr ack 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 predict s the observed loading rate dependence of the breaking strength. The result ing scaling exponents imply systematically more non-linear behaviour for dy namic loading.