## 4 Self-Organized Criticality During Damage and Fracture
### Power-Law Scaling Behavior Analysis  
Probability distributions characterize the occurrence likelihood of discrete and continuous random variables. This framework effectively quantifies both discrete stochastic events and continuous stochastic processes. Such distributions capture the stochastic nature inherent in AE signal patterns and crack network evolution. Power-law exponents derived from AE-crack probability distributions reveal load/time-dependent damage progression in sandstone. These exponents quantify nonlinear damage dynamics, including power-law scaling transitions during structural degradation (Figures 10 and 11). Elevated power-law indices correlate with enhanced nonlinearity, localized damage intensification, and spatial damage focusing. The power-law exponent functions as both damage quantification metric and evolution mechanism indicator for AE-crack systems. The sandstone fracture process is bifurcated into pre-critical (I-II) and post-critical (III-V) regimes using the first critical threshold (Monitoring Point 3) as the demarcation. The post-critical regime initiates with Stage III, characterized by autonomous damage reorganization. Probability density functions are computed for both AE energy release and pore volume evolution. The AE energy distribution is modeled through Probability Density Function \(P(E)\), defined as follows:  
\[P(E_{AE})=\frac{E^{-r_{AE}}}{E_{\min}^{1-r_{AE}}}E>E_{\min} \tag{2}\]  
where \(E_{\min}\) is the lower intercept of the AE energy distribution, and \(r_{AE}\) is the critical exponent of the AE energy distribution. The probability density distribution function of the pore volume is described by Equation (3):  
\[P(V)=\frac{V^{-r_{V}}}{V_{\min}^{1-r_{V}}}V>V_{\min} \tag{3}\]  
where \(V_{\min}\) is the lower intercept of the pore volume distribution and \(r_{V}\) is the critical exponent of the pore volume distribution. Figures 10 and 11 present power-law distributions characterizing acoustic emission (AE) energy and pore volume evolution across three distinct loading phases: initial (I-II), terminal (III-V), and complete (I-V) regimes. Table 4 quantifies these regimes through their respective power-law exponents. Distinct probability density profiles and critical scaling exponents define each phase's mechanical signature. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & \multicolumn{3}{c}{_r_ **of AE Signal**} & \multicolumn{3}{c}{_r_ **of Pore Volume**} \\ \cline{2-7}  & **I–II** & **III–V** & **I–V** & **I–II** & **III–V** & **I–V** \\ \hline R-3-1 & 1.41 & 1.68 & 1.68 & 1.68 & 1.67 & 1.66 \\ R-3-2 & 1.61 & 1.71 & 1.71 & 1.71 & 1.65 & 1.67 \\ R-3-3 & 1.57 & 1.66 & 1.66 & 1.66 & 1.67 & 1.68 \\ R-3-4 & 1.60 & 1.61 & 1.61 & 1.61 & 1.64 & 1.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Power–law exponents \(r\) characterizing AE energy and pore volume distributions across three loading regimes: initial (I–II), terminal (III–V), and complete (I–V). Figure 10: AE energy probability density functions \(P(E_{AE})\) for sandstone specimens: (**a**) R-3-1, (**b**) R-3-2, (**c**) R-3-3, (**d**) R-3-4 across three loading regimes: initial (I–II), terminal (III–V), and complete (I–V). The linearity of AE energy and pore volume probability densities in double-logarithmic coordinates (Figures 10 and 11) demonstrates power-law scaling governing damage evolution during uniaxial compression. Absolute magnitudes of distribution slopes (\(|r_{E}|\), \(|r_{V}|\)) serve as power-law exponents (\(r\)). Exponent values derived from histogram analysis show strong consistency with maximum likelihood estimations. This methodological agreement confirms the statistical robustness of power-law exponents (\(r\)). Post-critical phase (III-V) distributions exhibit spatiotemporal continuity with complete loading history (I-V), showing negligible statistical divergence. Within equivalent measurement ranges, exponents for III-V (1.66-1.83) and I-V (1.64-1.80) regimes demonstrate quantitative equivalence, exceeding I-II values (1.41-1.71).