## 4 Self-Organized Criticality During Damage and Fracture
### Self-Organized Process and Two Critical Points  
During uniaxial compression, damage evolution captured through real-time CT imaging and AE monitoring exhibits spontaneous localized accumulation with stochastic dispersion. This progression culminates in rapid fracture plane propagation along pre-existing damage trajectories post-critical threshold. These intrinsic features define the self-organized criticality governing sandstone fracture progression [32; 59]. The transition from self-organized damage accumulation to critical instability constitutes a non-equilibrium dynamic process with progressive self-organization. Self-organized critical progression during specimen R-3-1 damage evolution is documented in Figure 9a, and volumetric strain evolution specimen R-3-1 at critical transition points of Stage III in Figure 9b.  
Sandstone exhibiting self-organized criticality undergoes damage progression characterized by stage-dependent critical thresholds (Figure 9). The yield phase (Stage III)  
Figure 9: (**a**) Self-organized critical progression during sandstone (sandstone specimen R-3-1) damage evolution: Stage I (Compression): Initial structural compaction; Stage II (Elastic): Stable deformation regime; Stage III (Yield): Self-organization phase with meta-stability; Stage IV (Meta-instability): Precursor to critical failure; Stage V (Instability): Unstable collapse; and (**b**) Volumetric strain evolution at critical transition points of Stage III (sandstone specimen R-3-1).  
represents the self-organization regime. First critical threshold (volume expansion point) initiates microcrack nucleation from distributed microdamage. Second critical threshold (peak strength) drives microcrack coalescence into macroscale fractures. During pre-yield stages (I-II), stress accumulates through quasi-static deformation. Approaching Monitoring Point 3, strain rate escalation indicates damage localization. Volumetric strain (obtained from axial and radial strain gauges) transitions from compressive to dilative, accompanied by accelerated AE-microcrack coevolution. Crack propagation persists autonomously under constant-load conditions, demonstrating autocatalytic failure. Post Stage III, the system transitions to a meta-stable configuration. At Monitoring Point 4, stress intensification triggers macrocrack percolation. Localized damage zones evolve into stress singularities, enabling instantaneous energy release. Final fracture planes emerge through catastrophic coalescence of dominant cracks. Stage III serves as the instability precursor through self-organized damage patterning. Stage IV culminates in irreversible structural collapse. Final instability (Point 6) occurs substantially later than peak strength attainment (Point 4). Damage progression transitions from disordered low-energy states to organized high-energy configurations. Energy accumulates quasi-statically during early phases, whereas post-critical energy dissipation occurs catastrophically. Material heterogeneity induces nonlinear mechanical responses, driving self-organized damage evolution. Critical thresholds emerge at self-organization maturity. Self-organization patterns provide predictive signatures for impending instability. Thus, Stage III establishes the failure precursor framework.