## 4 Self-Organized Criticality During Damage and Fracture
### Power-Law Scaling Behavior Analysis
Stage III manifests power-law acceleration through abrupt exponent escalation. Abrupt power-law modifications during self-organization constitute quantifiable precursors for impending structural failure. Figure 11: Pore volume 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). Compared with stages I-II prior to the first critical point, the directionality of damage and fracture evolution in III-V stages is enhanced, and the degree of disorder is lower. In self-organized Stage III, most AE signals and pores begin to accumulate in large numbers, and the accumulated AE energy and total pore volume are comparable to those in the final failure state (Figures 6 and 7). This is because with the accumulation of sandstone damage, when approaching the first critical point (volume expansion point), the cohesion force used to resist the compression deformation of the sandstone sample decreases. Friction between fracture surfaces acts to resist sandstone compression and deformation, and the weakening of friction increases the likelihood of the eventual failure of sandstone. Thus, the evolution of AE energy signal and pore structure of sandstone has self-organized critical characteristics, and it is reasonable to regard self-organized Stage III as the precursor stage of sandstone failure. This study establishes a coupled AE-pore framework to probe self-organized criticality during sandstone fracture, where acoustic emission signatures and pore network evolution jointly quantify damage progression. The analysis highlights autonomous state transitions from mechanical equilibrium to critical instability, accompanied by power-law scaling acceleration (Figures 12 and 13). Spatiotemporal expansion of damage zones exhibits progressive escalation during failure evolution. Integrating these findings with [PERSON]'s continuity-damage theory [60], the dimensionless AE damage parameter \(D_{AE}\) is formulated as follows:  
\[D_{AE}=\frac{A_{i}}{A_{f}} \tag{4}\]  
where \(A_{i}\) is the effective cross-sectional area of the damaged region during sandstone loading and \(A_{f}\) is the effective cross-sectional area of the final failure region. In addition, \(N_{i}\) and \(E_{AE_{i}}\) are the number of AE events and the cumulative AE energy during sandstone loading, respectively, and \(N_{f}\) and \(E_{AE_{f}}\) are number of AE events and the cumulative AE energy of the final failure region, respectively. The relationship between \(N_{AEi}\) and \(E_{AE_{i}}\) is described by the following:  
\[N_{AEi}=\frac{E_{AE_{i}}}{E_{AE_{f}}}N_{AEf} \tag{5}\]  
The dimensionless damage variable \(D_{AE}\) during sandstone loading is formulated through cumulative AE energy integration, defined as follows:  
\[D_{AE}=\frac{E_{AE_{i}}}{E_{AE_{f}}} \tag{6}\]  
Notably, \(t_{i}\) represents the time of each stress stage of sandstone, the initial loading is 0, and \(t_{f}\) represents the time of final failure and instability. \(t_{i}/t_{f}\) is the normalized loading time of the sandstone sample, and \(\Delta=1-t_{i}/t_{f}\) is the change rate of the loading time, that is, the control parameter. When parameter \(\Delta\) is treated as a temporal scaling factor, the loading-path-independent nature of sandstone's self-organized critical behavior during damage progression is demonstrated [58]. Figure 12 presents power-law distributions of AE parameters across all loading phases (I-V), while Tables 5 and 6 quantify their scaling exponents for three distinct regimes: initial (I-II), terminal (III-V), and complete (I-V) stages. Figure 12: Power-law scaling of acoustic emission signals during progressive loading (I–V): (**a**) \(D_{AE}\)-\(t_{i}/t_{f}\), (**b**) \(D_{AE}\)-\(1-t_{i}/t_{f}\), (**c**) \(N_{AE}\)-\(D_{AE}\) and (**d**) \(N_{AE}\)-\(1-t_{i}/t_{f}\).