## 4 Self-Organized Criticality During Damage and Fracture
### Power-Law Scaling Behavior Analysis
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{**\(\boldsymbol{r}\) of AE Signal**} \\ \cline{2-10}
**Specimen** & \multicolumn{3}{c}{\(\boldsymbol{D_{AE}}\)**–\(\boldsymbol{1}\)**–\(\boldsymbol{T_{i}}\)/\(\boldsymbol{T_{f}}(\boldsymbol{\alpha})\)**} & \multicolumn{3}{c}{\(\boldsymbol{N_{AE}}\)**–\(\boldsymbol{D_{AE}}(\boldsymbol{\beta})\)**} & \multicolumn{3}{c}{\(\boldsymbol{N_{AE}}\)**–\(\boldsymbol{1}\)**–\(\boldsymbol{T_{i}}\)/\(\boldsymbol{T_{f}}(\boldsymbol{\gamma})\)**} \\ \cline{2-10}  & I–II & III–V & I–V & I–II & III–V & I–V & I–II & III–V & I–V \\ \hline R-3-1 & 0.82 & 1.09 & 1.11 & 0.62 & 0.74 & 0.75 & 0.65 & 0.77 & 0.79 \\ R-3-2 & 0.88 & 1.02 & 1.08 & 0.66 & 0.74 & 0.76 & 0.72 & 0.80 & 0.82 \\ R-3-3 & 0.75 & 1.13 & 1.12 & 0.68 & 0.76 & 0.78 & 0.75 & 0.89 & 0.87 \\ R-3-4 & 0.78 & 1.04 & 1.06 & 0.59 & 0.63 & 0.62 & 0.73 & 0.85 & 0.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Power-law scaling exponents (\(r\)) characterize AE signals evolution across three loading regimes: initial (I–II), terminal (III–V), and complete (I–V). Figure 13: Power-law scaling of pore structure during progressive loading (I–V): **(a)**\(D_{CT}\)-\(t_{i}/t_{f}\), **(b)**\(D_{CT}\)-\(1-t_{i}/t_{f}\), **(c)**\(L_{CT}\)-\(D_{CT}\) and **(d)**\(L_{CT}\)-\(1-t_{i}/t_{f}\). Equation (12) and Figure 13b demonstrate that parameters \(D_{CT}\) and \(\Delta\) exhibit \(D_{CT}\sim\Delta^{-\delta+1}\)-type power-law scaling throughout the sandstone damage-to-failure continuum. Correspondingly, variables \(L_{CT}\) and \(D_{CT}\) conform to \(L_{CT}\sim D_{CT}\delta\)-type correlation as shown in Figure 13c. Spatial variation analysis reveals: Parameter \(L_{CT}\) displays fault-aligned anisotropy, Parameters \(L_{CT}\) and \(\Delta\) follow \(L_{CT}\sim\Delta^{-(\delta-1)\epsilon}/L_{CT}\sim\Delta^{-\zeta}\)-type power-law distributions, Scaling exponents \(\zeta=(\delta-1)\epsilon\) yield absolute magnitudes: 0.89, 1.07, 1.12, 0.95 (Figure 13d). Throughout the entire damage-fracture continuum, structural parameters (\(N_{i}\), \(E_{i}\), \(D_{AE}\), \(L_{CT}\), \(\phi_{i}\), \(D_{CT}\)) exhibit positive correlations with control parameters (\(t_{i}/t_{f}\), \(1-t_{i}/t_{f}\)). Fault-aligned damage divergence provides empirical validation for critical phenomena in sandstone failure mechanics. This observation confirms scaling criticality in complex rock systems approaching mechanical instability. These findings align with the scaling framework established by [PERSON] et al. and further validate [PERSON] critical threshold model [61]. The AE-porosity coevolution demonstrates power-law scaling during damage progression. This universal scaling behavior remains invariant to sample size and structural heterogeneity. Power-law variations in fault evolution originate from AE source mechanism diversity [62]. Parameter \(r\) demonstrates strain-rate sensitivity [63], with additional dependence on: Strain energy density, Microcrack network evolution, structural reorganization. The sandstone is in a steady state in stages I-II, and the internal parameters of the sandstone internal structure increase slowly. In stages III-V, starting from self-organized process (III), the sandstone enters the meta-unstable state from the meta-stable state through the critical instability point and then fails to reach the unstable state. In stages III-V, the very small stress disturbance is sufficient to cause the internal crack to expand to the macroscopic magnitude, which also plays a decisive role in the final failure. The damage theory in fracture mechanics states that damage in the early stage can promote fracture to form a \"chain reaction\" in the later stage. The power-law exponents \(r^{\prime}\)III-V (0.84-1.15) of the sandstone AE signal and pore parameters in stages III-V are approximately equal to those of \(r^{\prime}\)I-V (0.87-1.20) in stages I-V, and both are greater than those of \(r^{\prime}\)I-II (0.71-0.85) in stages I-III.