The evolution law of stress and acoustic emission ringing count of red sandstone under the uniaxial compression load is shown in Figure 2. It can be observed from Figure 2 that the damage stage of red sandstone can be divided into two parts: damage stable development stage (S) and damage unstable development stage (U). The two parts take the Y point as the dividing point. After the Y point, the stress appears as a V-shaped evolution phenomenon. This shows that the damage evolution process of red sandstone enters the unstable stage from the stable stage. When the red sandstone reaches its peak strength, the ringing count appears at the peak point. In addition, near the peak intensity, the ringing count is relatively low, and this response characteristic is called the relative quiet period phenomenon (Su G. S. et al., 2024). In the post-peak residual stage, the ringing count also showed a significant relative quiet period before the sample was completely destroyed. This is the AE precursor information of the instability failure of red sandstone. Before the Y point, the fluctuation range of the ringing count is small, the red sandstone is mainly elastic deformation, and the phenomenon of micro-crack development is not obvious. This shows that before the Y point, red sandstone will contain a significant amount of elastic energy. When the red sandstone reaches the ultimate bearing capacity, the elastic energy will be released sharply in a short time, and the brittle fracture failure characteristics will appear. The ratio of axial stress to peak stress at the Y point of red sandstone is equal to 92.9%, as calculated by the mean calculation method. Therefore, when the axial stress reaches 92.9% of the peak strength of red sandstone, the damage stage changes from the stable development stage (S) to the unstable development stage (U).

In Section 3.1, the damage stages of red sandstone are divided based on the evolution law of the stress–strain curve. In this section, we will conduct a more detailed exploration of the damage stage and failure type of red sandstone through acoustic emission characteristics. Figure 3 shows the macroscopic failure characteristics of red sandstone samples. It is very important to understand the evolution characteristics of failure types of red sandstone in the process of damage evolution. In this section, the damage type of red sandstone is explored by the acoustic emission parameter Ф (AF (ringing count/duration)/RA (rise time/amplitude) ). At present, there are two main methods to classify rock failure types using the RA-AF diagram (Du et al., 2020b), as shown in Table 1.

The abovementioned two methods define the area larger than Ф as tensile failure and the area smaller than Ф as shear failure. In this paper, the first method is adopted, and the 45° inclined line is taken as the dividing line. The region with Ф > 1 is defined as tensile failure, and the region with Ф < 1 is defined as shear failure.

It can be observed from Figure 5 that the damage stage of red sandstone can be divided into the following: the damage stable development stage (S), the damage unstable development stage (U), and the damage rapid development stage (R) according to the evolution process of data distribution degree of Ф value. By exploring D-1, it can be observed that when the axial stress reaches 83.1% of the peak strength, the damage stage changes from the damage stable development stage (S) to the damage unstable development stage (U). When the axial stress reaches the peak strength, the damage stage changes from the damage unstable development stage (U) to the damage rapid development stage (R). By exploring D-2, it can be observed that when the axial stress reaches 63.5% of the peak strength, the damage stage changes from the damage stable development stage (S) to the damage unstable development stage (U). When the axial stress reaches the peak strength, the damage stage changes from the damage unstable development stage (U) to the damage rapid development stage (R). According to the mean value method, the boundary point between the S and U stages is 73.3% of the peak stress of red sandstone, and the boundary point between the U and R stages is the peak stress of red sandstone. In the damage stable development stage (S), there is no high concentration of data in the Ф value, and the damage development stage is relatively stable. In the unstable damage development stage (U), there is a high concentration of data in the Ф < 1 region. This shows that the red sandstone is dominated by shear failure at this stage. In the rapid damage development stage (R), the data in the Ф < 1 region are still highly concentrated, but the degree of data concentration is generally lower than that in the U stage. This shows that the red sandstone is still dominated by shear failure at this stage, but the extent of shear failure is decreasing.

It can be observed from Figures 4, 5 that in the U and R stages, when the integrity of the data concentration phenomenon in the Ф < 1 region is good, the macroscopic fracture characteristics of the red sandstone sample show a single penetrating crack. When the integrity of the data concentration phenomenon in the Ф < 1 region is poor, the macroscopic fracture characteristics of the red sandstone sample show a serrated penetrating crack. It can be observed from Figure 5 that the samples are in a composite fracture state during the whole experiment. This shows that red sandstone is a highly heterogeneous material with micro-cracks. When red sandstone is subjected to compression, the propagation and interaction of micro-cracks result in a composite fracture pattern in the final fracture morphology (see Figure 4).

Constructing a mathematical model of rock damage evolution law is an important method for quantitatively evaluating rock instability and failure. The core of constructing the mathematical model is how to define the damage variable. At present, the definition of damage variable mainly includes dissipation energy/total energy, damage area/apparent area, and elastic modulus/initial elastic modulus. Regardless of the method used to define the damage variable, it is necessary to follow the following assumptions: the rock is a homogeneous, isotropic continuum; the damage deformation of rock is a continuous failure, and the assumption is suitable for all mesoscopic units. The distribution of rock structural units conforms to the distribution law of the Weibull function. The Weibull distribution function is as follows: F x ; k , m = m k x k m − 1 exp − x / k m   x > 0 0   x ≤ 0 . (1) In Formula 1, x is the red sandstone strain, k is the proportional parameter, and m is the Weibull modulus.

It is assumed that the rock is composed of several identical mesoscopic units. A mesoscopic unit rupture corresponds to a ringing count. Therefore, when the rock strain reaches x , the number of ruptures of the mesoscopic unit can be expressed by the cumulative ringing count ( N A E x ). The number of meso-unit ruptures during rock failure can be expressed by the cumulative ringing count ( N A E ). Therefore, based on continuous damage mechanics, the damage variable can be expressed as follows: D = N A E x N A E . (2)

The continuity factor ( φ ) can be expressed as follows: φ = 1 − D . (3)

When the strain of rocks changes from 0 to x , the fracture number of rock meso-units can be expressed as follows: N A E x = N A E ∫ 0 x F x ; k , m d x = N A E 1 − exp − x / k m . (4)

Combining Formulas 2–4, the expression of damage variable D is obtained as follows: D = 1 − exp − x / k m . (5)

It is assumed that the mechanical properties of rock materials satisfy Hooke‘s law under the uniaxial compression load. The axial stress–strain relationship of rock materials can be expressed as follows: σ = E 1 − D ε . (6) In Formula 6, σ is the axial stress, ε is the axial strain, and E is the initial elastic modulus.

Combining Formulas 5, 6, the following expression of axial stress–strain of the rock material under the uniaxial compression load is obtained: σ = E exp − x / k m ε . (7)

Based on the ringing counting parameters, the damage evolution characteristics of red sandstone are obtained. The practical application results of Formula 7 are obtained (see Figure 8). In the mathematical calculation, the initial elastic modulus calculated by the secant method is 6.3 GPa. Based on Figure 6, the damage variable value of the red sandstone damage boundary point is calculated by the mean value calculation method. When D < 0.18, the red sandstone is in the damage stable development stage (S). When 0.18 < D < 0.75, the red sandstone is in the damage unstable development stage (U). When 0.75 < D < 1.0, the red sandstone is in the damage rapid development stage (R). Therefore, the point D = 0.18 is defined as the damage acceleration point (YY). The point D = 0.75 is defined as the damage critical point (P). At this time, the average value of the axial stress ratio corresponding to the YY point is 73.3%, and the axial stress corresponding to the P point is the peak stress. The abovementioned results are the same as those obtained in Section 3.2. This shows that the abovementioned damage variables can well-characterize the damage evolution characteristics of red sandstone. Compared with the damage stage described in Section 3.1, the damage stage divided by acoustic emission characteristic parameters is more detailed. Moreover, the axial stress value of the YY point is lower than that of the Y point, which provides more decision-making time for predicting the instability and failure of red sandstone in advance. Furthermore, it can be observed from Figure 7 that the abovementioned damage model can better characterize the damage evolution process of red sandstone.

It can be observed from Figure 8 that when the value of E is the initial elastic modulus of red sandstone, the linear evolution law appears in the initial stage of the mathematical model calculation results. This is obviously inconsistent with the actual evolution law of red sandstone. This is mainly due to the fact that the E value of red sandstone is constantly changing during the stress process. However, in the calculation, E is usually set to a fixed value, which is the main factor leading to a large deviation in the calculation results. In order to explore the variation law of the E value, in this paper, we use the piecewise calculation method to calculate the E value of red sandstone (see Figure 9). The calculation method is shown in Formula 8. Using Formula 8, the evolution law of the E value of D-1 and D-2 is obtained, as shown in Figure 10. E i j = σ i − σ j ε i − ε j . (8) In Formula 8, E i j is the elastic modulus between i and j, σ i is the axial stress at i, σ j is the axial stress at j, ε i is the axial strain at i, and ε j is the axial strain at j.

It can be observed from Figure 10 that the evolution law of the E value of red sandstone samples conforms to the distribution law of the Gauss function. The expression is as follows: E = A W π / 2 exp − 2 ε − χ W 2 . (9)

Substituting Formula 9 into Formula 7, the expression of axial stress–strain of red sandstone can be obtained. σ = A W π / 2 exp − 2 ε − χ W 2 exp − x / k m ε . (10) In this formula, A , W , and χ are the fitting parameters.

Formula 10 is applied in the actual application, and the modified stress–strain evolution process is obtained (see Figure 11). It can be observed from Figure 11 that the calculation results of the modified mathematical model change from a linear evolution law to a nonlinear evolution law, aligning more closely with the deformation behavior of red sandstone. This shows that Gauss function combined with the Weibull distribution function and Hooke‘s law can better represent the stress–strain behavior of red sandstone.