Analogy of scaling law for heterogeneous interfacial fracture to explain the Gutenberg–Richter formula for earthquakes

Abstract

We propose a new approach for the statistical law due to the fracture of a heterogeneous interface involving spatial correlation of disorders. The dynamic process of interfacial fracture is governed by three coupled integral equations, which further become a system of linear algebraic equations after discretizing the interface to a set of prismatic elements. By tuning parameters, this model covers the whole cases of interfacial fracture from local-load-sharing to almost equal-load-sharing, extending the classical fiber bundle models to a general form. Numerical simulations present that in all cases, the statistical frequency distribution of bursts follows a power law with the exponent in the range (1.5, 2.5), the corresponding b-value in (0.75, 2.25), which well explains the empirical Gutenberg–Richter scaling. The exponent depends on stiffness of elastic spaces, heterogeneous properties of interface, and the distribution of displacements induced by loading. Furthermore, the exponent drops temporally with the evolution of fracture, to its final value before rupture of interface, a phenomenon that may be treated as a precursor for imminent catastrophic failure.