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Observational studies from rock fractures to earthquakes indicate that fractures and many large earthquakes are preceded by accelerating seismic release rates (accelerated seismic deformation). This is characterized by cumulative Benioff strain that follows a power law time-to-failure relation of the form C(t) = K + A(Tf – t)m, where Tf is the failure time of the large event, and m is of the order of 0.2-0.4. More recent theoretical studies have been related to the behavior of seismicity prior to large earthquakes, to the excitation in proximity of a spinodal instability. These have show that the power-law activation associated with the spinodal instability is essentially identical to the power-law acceleration of Benioff strain observed prior to earthquakes with m = 0.25-0.3. In the present study, we provide an estimate of the generic local distribution of cracks, following the Wackentrapp-Hergarten-Neugebauer model for mode I propagation and concentration of microcracks in brittle solids due to remote stress. This is a coupled system that combines the equilibrium equation for the stress tensor with an evolution equation for the crack density integral. This inverse type result is obtained through the equilibrium equations for a solid body. We test models for the local distribution of cracks, with estimation of the stress tensor in terms of the crack density integral, through the Nash-Moser iterative method. Here, via the evolution equation, these estimates imply that the crack density integral grows according to a (Tf – t)0.3-law, in agreement with observations.
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