Magnitude and variation of the critical power law exponent and its physical controls

We study the physical controls on the scatter of exponents in the critical power law relation that describes an acceleration in precursory signals of deformation (displacements) or seismicity (damage) in the vicinity of failure time. Based on the time-dependent fiber bundle model and equal load share (ELS) rule, we find that the critical power law exponents range from −0.5 to −1.0. And values of the critical power law exponents depends on a parameter ρ, which defines the sensitivity of damage growth in a fiber to the local stress. Both the simulation results and theoretical analysis demonstrate that the critical power law precursor exponent −β has a relationship −β=−(1−1∕ρ) with ρ. Thus, our results illustrate a physical mechanism of variation of the critical power law exponent that is determined by the degree of the local stress controlling the damage evolution of a fiber.