## 4 Novel Method of Crack Stress Threshold Calculation
### Modified Strain Decomposition Method  
The volumetric strain of a rock can be decomposed into elastic strain \(\varepsilon_{\mathrm{v}}^{\mathrm{e}}\) and plastic strain \(\varepsilon_{\mathrm{v}}^{\mathrm{p}}\). During cyclic loading and unloading, the volumetric strain after unloading is the plastic volumetric strain at the upper limit stress. The difference between the total strain at the upper limit stress and the plastic body strain is the elastic volumetric strain.  
The elastic strain of a rock comprises three parts (Figure 9): (a) The linear elastic strain of the solid-phase is dominated by rock particles (\(\varepsilon_{\mathrm{v}}^{\mathrm{s}}\)) [46]. (b) The recoverable strain of pores and cracks is the elastic compaction strain, where the growth rate is inversely proportional to the stress level [20]. The original pores and cracks gradually close in the axial direction as stress gradually increases. (c) Elastic expansion strain along the transverse direction produced by the reversible deformation in the form of secondary tensile cracks in the rock is caused by the relative displacement between shear cracks. The strain gradually increases with increasing stress [15]. The latter two strains can be regarded as gas-phase nonlinear elastic volumetric strains (\(\varepsilon_{\mathrm{v}}^{\mathrm{v}}\)).  
During the crack closure stage, there are a few cracks in the rock, and the volumetric strain is mainly generated by solid-phase deformation and crack closure. Therefore, the stress-axial strain curve exhibits a concave curve with a positive slope. As the load increases, the elastic expansion along the transverse direction increases, thereby increasing the slope of the stress-volumetric strain curve during the unloading stage.  
The solid phase of rock can be modeled as linear elastomers, and the deformation of the solid phase of rock under load follows the generalized [PERSON]'s law:  
\[\varepsilon_{i}^{\mathrm{s}}=\frac{1}{E}\left[\sigma_{i}-\
u(\sigma_{j}+\sigma _{k})\right], \tag{1}\]  
where \(\varepsilon_{i}^{\mathrm{s}}\) is the solid-phase strain; \(i=1\), \(2\), \(3\); \(\sigma_{i}\), \(\sigma_{j}\) and \(\sigma_{k}\) are the stresses on the rock with \(i,j,k\in\{1,2,3\}\); \(E\) is the elastic modulus; \(\
u\) is the Poisson's ratio; and \(K\) is the bulk modulus.  
According to Equation (1), the volumetric strain of the solid phase can be expressed as follows:  
Figure 9: Components of elastic volumetric strain: (a) solid-phase strain, (b) axial compaction strain, and (c) elastic expansion of a tension crack.  
\[\varepsilon_{\rm v}^{\rm s}=\frac{1-2\
u}{E}\big{(}\sigma_{\rm i}+\sigma_{\rm j}+ \sigma_{\rm k}\big{)}, \tag{2}\]  
where \(\varepsilon_{\rm v}^{\rm s}\) is the solid-phase volumetric strain.  
Equation (2) can be expressed in the form of the hydrostatic pressure and bulk modulus. The bulk modulus is measured by the unloading tangent modulus of the stress-volumetric strain curve at the maximum volumetric strain. This approach minimizes the influence of gas-phase deformation (Figure 10).  
\[\varepsilon_{\rm v}^{\rm s}=\frac{p}{K}, \tag{3}\]  
\[K=\frac{E}{3(1-2\
u)}, \tag{4}\]  
\[p=\frac{\sigma_{\rm i}+\sigma_{\rm j}+\sigma_{\rm k}}{3}, \tag{5}\]  
where \(K\) is the bulk modulus, and \(p\) is the hydrostatic pressure.  
The gas-phase volumetric strain is calculated by subtracting the plastic- and solid-phase volumetric strains from the total volumetric strain.  
\[\varepsilon_{\rm v}^{\rm v}=\varepsilon_{\rm v}-\varepsilon_{\rm v}^{\rm P}- \varepsilon_{\rm v}^{\rm s}, \tag{6}\]  
where \(\varepsilon_{\rm v}^{\rm v}\) is gas-phase volumetric strain, \(\varepsilon_{\rm v}\) is the total strain at the upper limit, and \(\varepsilon_{\rm v}^{\rm P}\) is the plastic volumetric strain.  
Figure 10: Schematic of volumetric strain decomposition (\(\sigma_{\rm 3}=0\) MPa) for different loading and unloading cycles: (**a**) 2 nd, (**b**) 4 th, (**c**) 6 th, and (**d**) 8 th cycle.