Many rock mechanics tests have shown that the stress-axial strain curve of rock has an approximately linear change stage. Therefore, the crack closure stress can be determined directly from the rock stress-axial strain curve. As shown in Figure 2, the crack closure stress intensity corresponds to the turning point of the stress-axial strain curve from nonlinearity to linearity; the crack initiation stress corresponds to the turning point of axial stress-axial strain curve from nonlinear to linear. The advantage of this method is that the idea is simple, and the physical meaning is clear, but the deficiency is that it is greatly affected by human factors. For example, the accurate determination of the turning point strongly depends on the subjective judgment of engineers.

According to the stress-lateral strain and stress-axial strain test data obtained from the rock compression test, the volumetric strain under a certain level of stress can be calculated: ε v = ε 1 + 2 ε 3 (1) Where, the axial strain ( ε 1 ) and lateral strain ( ε 3 ) are under the same level of stress.

Rock volume strain ( ε v ) includes elastic volume strain ( ε e v ) and crack volume strain ( ε c v ). According to the elastic theory, the elastic volumetric strain ( ε e v ) of the specimen can be calculated as follows (Cieślik, 2014): ε e v = 1 − 2 ν E ( σ 1 + σ 2 + σ 3 ) (2) Where, σ ₁, σ _2, and σ ₃ are the axial stress and confining pressure; v and E are the Poisson^’s ratio and elastic modulus of the rock. Then the crack volume strain can be calculated by the following equation: ε c v = ε v − 1 − 2 ν E ( σ 1 + σ 2 + σ 3 ) (3)

Under uniaxial conditions, Eq. 3 can be simplified as: ε c v = ε v − 1 − 2 ν E σ 1 (4)

The crack volume strain method is based on the relationship between the crack volume strain and the loading stress σ ₁, σ ₂ and σ ₃ established according to Eq. 3. The strain value that the crack volume strain first tends to zero as the closing stress intensity, and the axial strain value at the point where it deviates from zero again is taken as the crack initiation stress intensity. The solution diagram of this method is shown in Figure 3. Compared with the axial strain method, this method reduces the subjectivity of artificial estimation to a certain extent, and the solution results are relatively objective. However, the method has the disadvantage of over-reliance on the elastic parameters v and E, and the insufficient accuracy of the parameters may easily lead to errors in the solution results.

The strain response method is a method proposed by Nicksiar and Martin, 2012 to identify crack initiation stress by transverse strain parameters. The idea of this method is as follows: First, determine the damage stress point A according to the stress-volume strain curve (Figure 4), the corresponding point B in the lateral strain curve is determined, and OB is connected as the reference line. Then, under the same axial stress, the strain value corresponding to the transverse strain curve and the reference line is taken, and the difference between the two is the transverse strain difference. Finally, polynomial fitting is performed on the data points, and the peak point C of the fitting curve is the crack initiation stress intensity. To further simplify the solution process, it is not necessary to fit the transverse strain difference curve, and the stress corresponding to the maximum value of the transverse strain difference is directly taken as the initiation stress intensity. The advantage of this method is that the value standard is objective and unique, which is convenient for programming. However, the deficiency is the lack of theoretical basis, so it can consider that the method is an empirical method, which is based on the trend of the stress-lateral strain curve trend.

Similar to the transverse strain method proposed by Nicksiar and Martin, 2012, the method of determining the closure stress based on the axial strain response was proposed by Peng et al. (2015). The idea is as follows: Firstly, the damage stress point A is determined according to the stress-volume strain curve, and its projection point on the axial strain curve is determined as B, and OB is connected as the reference line. Then, under the same axial stress condition, the strain values corresponding to the axial strain curve and the reference curve are solved. The difference between the two is used as the axial strain difference. Finally, polynomial fitting is performed on the data points, and the peak point C of the fitted curve is the closure stress, or the maximum value of the axial strain difference is directly calculated. The solution principle is shown in Figure 5. The advantages and disadvantages of this method are the same as the transverse strain response method. Similarly, the method is an empirical method of taking based on of rock stress-axial strain curve trend.

As mentioned above, the results of the axial stress-transverse strain and the axial stress-axial strain obtained from the rock compression test are all discrete point data. According to many rocks’ mechanical compression tests, the rock axial stress-strain curve presents an approximately linear characteristic in the elastic stage. The slope of the stress-strain curve is approximately unchanged at this stage. Based on this, the elastic stage can be determined by changing the first derivative of the stress-axial strain curve (i.e., the slope, which can be interpreted as stiffness in physical terms).

According to the knowledge of computational mathematics: The first-order derivative of the discrete points can be approximated by the difference method (Xue and Chen, 2004; Behforooz, 2006). The commonly used difference methods include forward difference equation, center difference equation, five-point difference equation, etc.

The forward difference equation (Xue and Chen, 2004) is defined as: k 1 = σ i + 1 − σ i ε i + 1 − ε i (5) Where, σ _i and ε i are the axial stress and strain of the discrete point at position i. σ _i+1, and ε i + 1 are the axial stress and strain of the discrete point at position i+1.

The center difference equation (Xue and Chen, 2004) is defined as: k 2 = σ i + 1 − σ i − 1 ε i + 1 − ε i − 1 (6) Where, σ _i-1 and ε i − 1 are the axial stress and strain of the discrete point at position i-1.

The five-point difference equation (Xue and Chen, 2004) is defined as: k 3 = − σ i + 2 + 8 σ i + 1 − 8 σ i − 1 + σ i − 2 12 ( ( ε i + 2 − ε i − 2 ) / 4 ) (7) Where, σ _i+2 and ε i + 2 are the axial stress and strain of the discrete point at position i+2. σ _i-2 and ε i − 2 is the axial stress and strain of the discrete point at position i-2.

Theoretically, the accuracy of the derivative calculated by Eq. 7 is relatively higher than that of Eq. 6–5. However, due to the higher degree of nonlinearity of the stress-strain curve, the derivative obtained by several methods may have a larger deviation. To intuitively show the response of several methods to the slope, take the same set of discrete curve data and calculate the slope characteristics determined by various methods, as shown in Figure 6. The coordinates of the x-axis and y-axis in Figure 6 are all dimensionless.

It can be seen from Figure 6 that the slopes calculated by the three equations are quite different. Using volatility to measure the slope obtained by the appeal method, the volatility of Eq. 7 is smaller than that of Eq. 6–5. But the derivatives calculated by the three equations all reflect the characteristics of the true slope.

The determination of rock closure stress and crack initiation stress is based on the linear and nonlinear changes of rock stress–axial strain, but the stress–strain data obtained by uniaxial compression test or triaxial compression test are discrete. The over-subjective determination of closure stress and crack initiation stress obtained directly from rock stress–axial strain curve may lead to the deviation of these data from the actual value. Therefore, using the difference formula to process the original data and combining it with statistical knowledge (Aslam et al., 2021) to determine the rock closure stress and crack initiation stress can reduce the influence of human subjective factors so that the results are more realistic. The specific ideas are as follows:

1) The closure stress and crack initiation stress of rock determined by n (n > 2) existing methods (such as axial strain method, crack volume strain method, strain response method, etc.);

The calculation of closure and initiation stress intensity of fractured rock is transformed into an optimization problem by the new method proposed in this paper. Determining the values of the two through the optimal idea is a fine value method (or called the optimal value method), which has good universality.

Through the similar ideas of Nicksiar and Martin, 2012 and Peng et al. (2015), an empirical method to determine the closure stress and crack initiation stress through the volumetric strain curve and the stress-axial strain curve was proposed in this paper. The idea and principles of this method are as follows (Figures 7, 8):

It should be noted that the proposed empirical method is obtained from the intuitive observation of the trend of the rock stress-axial strain standard curve (Figure 1). Since it is an empirical value method, it may have a greater impact in some cases-error cases: such as rocks with insignificant deformation characteristics at various stages. Compared with the refine value method, the empirical value method has the advantage of being simple to implement. Compared with the axial strain method, the empirical value method has more objective results. However, due to the complex deformation laws of rock and many factors affecting its mechanical properties, the characteristics of its stress-strain curve are not as evident as the standard strain curve, which limits the application of this empirical value method.

In this paper, the the uniaxial compressive stress-strain data of five rock samples are used as examples to verify the proposed methods (Eberhardt et al., 1998; Zhao et al., 2013; Zhao et al., 2015).

Taking sample BS06 as an example, the closing stress and crack initiation stress calculated by the crack volume strain method are 30 MPa and 70 MPa, respectively; the closing stress and crack initiation stress determined by the axial strain method are 40 MPa and 90 MPa, respectively. The search interval for closing stress is determined to be [30, 40] MPa, and the search interval for crack initiation stress is [70, 90] MPa. Through the fine value method, the optimization is carried out, and the final two characteristic stress values are 34.67 MPa and 86.07 MPa, respectively. Similarly, the closing stress and crack initiation stress via the empirical value method are about 35.90 MPa and 85.54 MPa. It can be seen that the empirical value method can also obtain relatively accurate value results, which can be used as a supplement to the refine value method.

The calculation results of all cases are summarized in Table 1, and the reliability of the method proposed in this paper is verified by the variance (<10 MPa) indicating the dispersion.

The determination of rock closure stress and initiation stress intensity is attributed to determining the linear change of rock stress-axial strain, which is based on many rock mechanical deformation tests. However, these data may deviate from the true value because it is too subjective to obtain the closure stress and crack initiation stress directly from the rock stress-axial strain curve. Therefore, how to deal with these discrete data is the key to determining the characteristic stress intensity of the rock. Nowadays, some scholars (Brace et al., 1966; Bieniawski, 1967; Brown, 1981; Eberhardt et al., 1998; Cai et al., 2004; Everitt and Lajtai, 2004; Cieślik, 2014; Peng et al., 2015; Yang, 2016) have determined the closure stress and crack initiation stress by the slope of the stress vs. strain curve (calculated by the forward difference equation). However, the existing methods still depends on subjective observation in determining the stable stage of the slope curve. The optimization value method proposed in this paper can avoid the subjectivity of artificially determining the stationary phase of the strain slope curve and can effectively improve the accuracy of the value. The following improvements can be made to the optimization method:

1) In addition to variance function, other functions such as information entropy can be tried to select objective function (Shi et al., 2012; Aslam et al., 2021).

In this paper, the first-order derivative of the original curve is selected for optimization. Compared with the original curve, this method refines the variation characteristics of the original curve and simplifies the calculation amount compared with the second-order derivative. According to the given optimization method framework (Figure 9), more methods for determining rock closure and crack initiation stress can be obtained, which will not be repeated here.

The empirical value method proposed in this paper is based on the law summarized by many rock compression deformation tests. However, due to the different compositions and structures of different rocks, not all rocks can be divided into four deformation stages before the peak. Therefore, the curve characteristics should be observed before determining the characteristic stress intensity of rock. For rocks with no obvious characteristics in the deformation stage, the fine value method is still applicable; For the rock with obvious deformation characteristics, the empirical value method and the fine value method have obtained reasonable results.

In this paper, the study on the determination methods of closure stress and crack initiation stress of fractured rock focuses on the mathematical value method or geometric meaning. However, the detailed physical meaning of the value method is not introduced in-depth, which is a deficiency of this paper. On the one hand, the author believes that rock mechanics have a great experience, and it is difficult to quickly obtain many physical rates in physical. However, most empirical rates are summarized, which can be widely accepted. The research method of this paper is to solve the closure stress and crack initiation stress problem of fractured rock employing a little mathematical value method based on empirical rate; On the other hand, not all rocks can be divided into four deformation stages before the peak. The author believes that all the deformation stages before the peak still need to be considered (Zhong et al., 2017). The initiation stress of fractured rock mass is closely related to the fracture toughness of rock considered by some scholars (Zhao et al., 2020a; Zhao et al., 2020b; Zhao et al., 2016; Lin et al., 2019). For example, Zhao et al. (2020a) found that rock fracture toughness is related to rock stress by rock crack rheology, fracture toughness, and subcritical crack propagation test. Hang Lin et al. (2019) studied the tip stress field and crack initiation angle of open defects under uniaxial compression. The above research shows that rock fracture mechanics can determine the crack initiation stress of fractured rock. Therefore, when the four deformation stages before the peak cannot be divided in the following study, the method of rock fracture mechanics is considered to determine the initiation stress.