## 4 Novel Method of Crack Stress Threshold Calculation
### Determination of Crack Stress Threshold Based on Gas-Phase Volumetric Strain  
The gas-phase volumetric strain directly reflects the extent of crack development. Therefore, gas-phase volume strain is suitable for determining the crack stress threshold. The gas-phase volumetric strain tends to be constant near the maximum point, indicating minimal crack propagation during this stage. The change from positive to negative in the rock gas-phase strain indicates that the volume of cracks has exceeded their initial state. Thereafter, the bearing area of the rock becomes smaller than its original value and gradually decreases with increasing stress and crack propagation. Therefore, the initiation and termination points of the stationary stage of the gas-phase volumetric strain curve can be considered as the crack closure and initiation stresses, respectively. The stress at which the gas-phase volumetric strain changes from positive to negative is considered the crack damage stress (Figure 13).  
To quantitatively determine the stationary stage of the gas-phase volumetric strain curve, the gas-phase volumetric strain before the damage stress is generalized into three segments: nonlinear segment \(OB\), linear segment \(BC\), and nonlinear segment \(CD\). The nonlinear segment could be quadratically generalized into bilinear segments (\(OA\)-\(AB\) and \(CD\)-\(DE\)). Connecting \(OB\), \(AC\), \(BD\), and \(CE\) as reference lines, points \(A\), \(B\), \(C\), and \(D\) correspond to the maximum difference between the gas-phase volumetric strain and the reference line (Figure 14a).  
Figure 12: Volumetric strain decomposition of granite for different values of \(\sigma_{3}\): (**a**) 0 MPa, (**b**) 4 MPa, (**c**) 8 MPa, (**d**) 12 MPa, (**e**) 16 MPa, and (**f**) 20 MPa.  
Figure 14: Determination of the stationary stage: (**a**) three stages of gas-phase volumetric strain curve, (**b**) initialization, (**c**) updated points \(B\) and \(C\), and (**d**) updated points \(A\) and \(D\).  
Figure 13: Novel calculation method of crack stress thresholds: (**a**) stress–strain curve, (**b**) bulk modulus calculation, and (**c**) crack stress thresholds calculation.  
Owing to the interaction between the decomposition points, it is necessary to calculate them using an iterative method (Figure 15). The steps are as follows:  
1. Initialization: Point \(A\) is initialized to origin \(O\), point \(D\) is initialized to point \(E\) where the volume strain changes from positive to negative, and points \(B\) and \(C\) are initialized to the peak of gas-phase volumetric strain (Figure 14b);
2. Update points \(B\) and \(C\): Draw reference lines \(AC\) and \(BD\) and move points \(B\) and \(C\) to the maximum deviation between the gas-phase volumetric strain and the reference line, respectively (Figure 14c);
3. Update points \(A\) and \(D\): Draw reference lines \(OB\) and \(CE\) and move points \(A\) and \(D\) to the maximum deviation between the gas-phase volumetric strain and the reference line, respectively (Figure 14d);
4. Termination judgment: The calculation is terminated if the stress corresponding to points A, B, C, and D changes by less than 0.1 MPa compared to its previous value. Otherwise, steps (b) and (c) are repeated. The final positions of points \(B\) and \(C\) correspond to crack initiation and damage stresses (Figure 14a).