Statistical analysis of acoustic emission avalanches generated during the compressive fracture process, and Mode I fracture process in cementitious composites

Abstract

This article reports a statistical analysis of acoustic emission (AE) avalanches in cementitious composites generated during unconfined uniaxial compression and flexural loading. The analysis emphasises on the probability distribution of absolute AE energies of the individual AE events and on the time correlations (aftershock rate for Omori’s law and waiting time between successive AE events for Universal Scaling Law). Under compression, the G-R exponent ($\epsilon$) remained constant at 1.15, 1.21, 1.21 and 1.32 for specimens having steel volume fraction (V_f) 0%, 0.8%, 1.6% and 2% respectively, for at least three decades (represents the powers of ten, i.e., $10^{1}$ is one decade, $10^{2}$ is two decades and so on) under uniaxial compression. A decrease in $\epsilon$ was observed as the damage in the material progressed for compressive fracture and Mode I fracture process. Distribution of other AE avalanche characteristics like inter-event time and waiting time ($\delta )$ distribution show almost similar characteristics for varying V_f.

Appendix A: Relation between seismic \(b\)-value and \(\epsilon\)

Appendix B: Implementation of Omori’s law

By analysing multiple aftershocks, the relation between average aftershock rate ($r_{AS}$) (i.e., number of aftershocks per unit time) as a function of the time difference from the instance of mainshock $\left( {\Delta t = t - t_{MS} } \right)$ can be studied (Utsu et al. 1995),

$$ r_{AS} \left( {\Delta t} \right) = K\left( {c + \Delta t} \right)^{p} $$

(B.1)

where K is a time-independent constant, p is the Omori’s exponent, c is a small shift used to avoid a divergence at Δt = 0. The shift (c) is due to the undercounting of aftershock (overlapped with the mainshock) very close to the MS.

1.1 B.1. Implementation of the Productivity law

The $r_{AS}$ would be constant $\left( {p = 0} \right)$ if the earthquakes were Poissonian in Eq. (B.1). However, p has a value close to $1.0$ for earthquakes. In Eq. (B.1), K determines the total number of aftershocks. K depends on the energy released during the mainshock (E_MS). A power-law function gives the relationship (Productivity law) between K and E_MS (Chen et al. 2019; Helmstetter 2003),

$$ K = K_{0} E_{MS}^{{\frac{2\alpha }{3}}} $$

(B.2)

where $K_{0}$ is a constant, and $\alpha$ is an exponent. A scaling plot (in log–log scale) can be drawn, using Eq. (B.3), by taking a combination of Eq. (B.1) and Eq. (B.2), if exponents $\alpha$ and $p$ are universal, and by plotting $E_{MS}^{{ - \frac{2\alpha }{3}}} r_{AS}$ versus ($\Delta t = t - t_{MS} ).$

$$ E_{MS}^{{ - \frac{2\alpha }{3}}} r_{AS} = K_{0} \left( {c + \Delta t} \right)^{ - p} $$

(B.3)