## 3 Identification of Feature Intensity  
The whole process of the deformation of rock specimens under uniaxial loading can be roughly divided into five stages: pore compacting stage, linear elastic deformation stage, crack stable expansion stage, crack unstable expansion stage, and post-peak deformation stage, of which, the pore compacting stage and linear elastic deformation stage rocks do not produce damage. It is usually considered that the beginning of the crack stable expansion stage is the starting point of damage produced within the rock. Therefore, accurate identification of the characteristic strength is the focus when analyzing the degree of damage in sandstone. The crack volumetric strain method [21] can accurately determine the characteristic strength of rocks. For uniaxial rock mechanics tests, volumetric strain \(\varepsilon_{\mathrm{v}}\) can be calculated from \(\varepsilon_{\mathrm{axial}}\) and \(\varepsilon_{\mathrm{lateral}}\) measured in the test as follows:  
\[\varepsilon_{v}=\varepsilon_{axial}+2\varepsilon_{lateral} \tag{1}\]  
The crack volume strain \(\varepsilon_{\mathrm{v}}\)c is the volume strain \(\varepsilon_{\mathrm{v}}\) minus the elastic volume strain \(\varepsilon_{\mathrm{v}}\)e, which can be expressed as:  
\[\varepsilon_{v}^{\mathrm{c}}=\varepsilon_{v}-\varepsilon_{v}^{\mathrm{e}} \tag{2}\]where the elastic volume strain \(\varepsilon_{v}\)e is the strain caused by the elastic deformation of the rock mass, which can be solved by the elastic modulus \(E\) and Poisson's ratio \(\mu\) through elastic mechanics as follows:  
\[\varepsilon_{v}^{c}=\frac{1-2\mu}{E}\sigma_{1} \tag{3}\]  
Bringing Equation (3) into Equation (2), the solution of the crack volume strain \(\varepsilon_{v}\)c is obtained as:  
\[\varepsilon_{v}^{c}=\varepsilon_{v}-\frac{1-2\mu}{E}\sigma_{1} \tag{4}\]  
The evolution curves of volume strain and fracture volume strain during the uniaxial loading stress-strain of intact sandstone were plotted based on the calculated results of Equations (1)-(4), as shown in Figure 2.  
The principle of dividing the stress-strain curve by the fracture volume strain method is as follows:  
(I) Pore compression and density stage (OA section): The primary crack closes slowly under axial load, with the axial strain dominating and the radial strain almost not generated; the compression rate of the crack volume strain decreases, and the point where the reduced rate tends to level off for the first time is point A. The stress level corresponding to point A is the crack closure strength \(\sigma_{\rm cc}\). (II) Linear elastic deformation stage (AB section): The axial strain and the radial strain grow in an approximately linear pattern with the increase in stress; no nascent cracks are generated in this stage, and the crack volume strain is almost unchanged. (III) Crack stable extension stage (BC section). After the loading exceeds point B, the crack inside the rock starts to sprout, and the point where the crack volume strain starts to tilt in the negative direction is point B. The stress level corresponding to point B is the crack initiation strength \(\sigma_{\rm ci}\). Namely:  
\[\sigma_{ci}=\sigma_{\rm ci,max} \tag{5}\]  
(IV) Crack instability expansion stage (CD section): After the loading exceeds point C, the crack inside the rock expands rapidly, the dominant strain changes from the axial strain  
Figure 2: Identification of stress–strain curves of intact sandstone specimens and their characteristic strengths. to the radial strain, leading to an inflection point in the volume strain curve, and the rock starts to enter the expansion stage, which is the C point and also the expansion point. The stress level corresponding to the C point is the damage strength (also called the expansion strength) \(\sigma_{\text{cd}}\).