The Changeable Power Law Singularity and its Application to Prediction of Catastrophic Rupture in Uniaxial Compressive Tests of Geomedia

Abstract

The acceleration precursor of catastrophic rupture in rock‐like materials is usually characterized by a power law relationship, but the exponent exhibits a considerable scatter in practice. In this paper, based on experiments of granites and marbles under quasi‐static uniaxial and unconfined compression, it is shown that the power law exponent varies between −1 and −1/2. Such a changeable power law singularity can be justified by the energy criterion and a power function approximation. As the power law exponent is close to the lowest value of −1, rocks are prone to a perfect catastrophic rupture. Furthermore, it is found that the fitted reduced power law exponent decreases monotonically in the vicinity of a rupture point and converges to its lower limit. Therefore, the upper bound of catastrophic rupture time is constrained by the lowest value of the exponents and can be estimated in real time. This implies that, with the increase of real‐time sampling data, the predicted upper bound of catastrophic rupture time can be unceasingly improved.

Localization of deformation and its effects on power-law singularity preceding catastrophic rupture in rocks

“…The dot represents the first and second derivatives of the response quantity X with respect to time. The validation of predicting rupture time based on equations (2) and (3) has been widely verified in many retrospective predictions (Boue´et al, 2015;Hao et al, 2016;Smith and Kilburn, 2010;Voight and Cornelius, 1991;Xue et al, 2018) of laboratory experiments (Hao et al, 2013;Kilburn, 2012), landslides (Helmstetter et al, 2004), earthquakes (Ben-Zion and Lyakhovsky, 2002;Sornette and Sammis, 1995), and volcanic eruptions (Kilburn, 2003;Main, 1999). Especially when b F ¼ À1 (i.e., ¼ 2), the rupture time can be determined by extrapolating the inverse rate R À1 to the abscissa (Boue´et al, 2015;Voight, 1988).…”

“…It is a self-sustaining process that does not need any supplying of external work. In order to characterize the catastrophic rupture, Hao et al (2013) and Xue et al (2018) introduced a response function…”

“…Especially when b F ¼ À1 (i.e., ¼ 2), the rupture time can be determined by extrapolating the inverse rate R À1 to the abscissa (Boue´et al, 2015;Voight, 1988). However, the power-law singularity exponent b F is not always a constant in reported measurements but exhibits a large dispersion (Boue´et al, 2015;Cornelius and Scott, 1993;Hao et al, 2017;Jin et al, 2012;Kilburn, 2003;Voight and Cornelius, 1991;Xue et al, 2018). The uncertainty resulting from the scatter of the exponent makes it difficult to use such methods for prediction of the rupture time (Boue´et al, 2015;Hao et al, 2017;Xue et al, 2018).…”

Forecasting Catastrophic Rupture in Brittle Rocks Using Precursory AE Time Series

“…As illustrated in Figure 1 (see details in section 2), an evident acceleration in cumulative AE events is typically registered in the vicinity of catastrophic rupture, which is quite similar to “accelerating creep” phenomenon (Amitrano et al, 2005; Heap et al, 2009; Koivisto et al, 2016; Lennartz‐Sassinek et al, 2014; Nechad et al, 2005b; Pál et al, 2016; Voight, 1988, 1989; Vu et al, 2019). The acceleration behavior has been analyzed as a critical phenomenon with a power‐law divergence of geophysical precursors at failure (Abaimov & Cusumano, 2014; Bak & Tang, 1989; Ben‐Zion & Lyakhovsky, 2002; Renard et al, 2018; Rundle et al, 2003; Toussaint & Pride, 2002; Xue et al, 2018). Such physical process underlies the failure forecast (Poli, 2017; Xue et al, 2018).…”

“…The acceleration behavior has been analyzed as a critical phenomenon with a power‐law divergence of geophysical precursors at failure (Abaimov & Cusumano, 2014; Bak & Tang, 1989; Ben‐Zion & Lyakhovsky, 2002; Renard et al, 2018; Rundle et al, 2003; Toussaint & Pride, 2002; Xue et al, 2018). Such physical process underlies the failure forecast (Poli, 2017; Xue et al, 2018). To quantitatively describe the acceleration process toward failure, an empirical relation (Voight's relation) between a response quantity Ω (e.g., strain and AE) and its rate was established as follows (Voight, 1988, 1989): where α is an exponent measuring the degree of nonlinearity and A is a constant.…”