Most of the existing 2D roughness parameters reflect only the morphological characteristics of 2D joint profiles and lack clear geometric meaning. To overcome this shortcoming, we construct asperities on rock joint profiles by the following process and give them clear geometric meaning.

(1) Extracting the joint profile at a certain sampling interval ( I s ) from the surface of a rock joint, as shown in Figure 1A. That is, the joint profile is formed by connecting a series of equally spaced discrete points.

Through the above process, the constructed asperities have clear geometric meaning. Thus, establishing new roughness parameters based on the joint profile containing various asperities will have clear geometric meaning.

Studying the influence of the morphological characteristics of joint surfaces on the shear mechanical behaviours of rock joints is necessary to obtain reasonable roughness quantitative characterization parameters. For the coarse and unfilled coupled joints, Grasselli (2001) indicated that in shear tests, only those zones facing the shear direction (i.e., where the quantity product of the unit outer normal vector and the shear direction unit vector is less than 0) could come into contact, while the others were detached gradually. Then, Grasselli et al. (2002) proposed the concept of potential contact zones to characterize those zones where contact may occur during shearing. This phenomenon that the resistance to shear can be provided only by the potential contact zones was also confirmed by the direct shear tests of several researchers (Zhang et al., 2016; Ban et al., 2020b). Therefore, taking the potential contact zones that control the shear mechanical behaviours of rock joints as major research objects has an experimental basis.

To further develop the new 2D roughness quantitative characterization parameters with clear physical and mechanical meanings, the extracted joint profile shown in Figure 1A is further treated as follows: (1) stretching the joint profile a certain thickness d along the direction perpendicular to the extension and fluctuation directions of the joint profile to form a joint surface, as shown in Figure 2A; and (2) constructing the rock joint shear mechanics analysis model shown in Figure 2B.

Grasselli et al. (2002) suggested that real contact is located only in the steepest part of all potential contact zones. That is, under normal and shear loads, all normal forces will act on the joint planes in real contact due to the climbing effect of the upper joint. In the model shown in Figure 2B, frictional sliding and shear-off are two possible failure modes of real contact asperities (i.e., those asperities whose joint planes are in contact during the shear test) with varying inclination angles, such as asperities i and j . Thus, based on the equilibrium conditions of the forces on asperity i , as shown in Figure 2C, the following two equations are easily obtained: N i J = T i ⋅ ⁡ sin ⁡ θ i + N i ⋅ ⁡ cos ⁡ θ i (2) T i J = T i ⋅ ⁡ cos ⁡ θ i − N i ⋅ ⁡ sin ⁡ θ i (3) where N i   J and T i   J are the local normal force and local shear force acting on the joint plane of asperity i , respectively, N i and T i are the local normal force and local shear resistance acting on asperity i , respectively, and θ i is the angle between the joint plane of asperity i and the horizontal plane, i.e., the inclination angle.

Similarly, the local forces acting on the joint plane of asperity i also satisfy the equilibrium condition: T i J = N i J ⋅ ⁡ tan ⁡ ϕ b (4)

Combining formulas 2–4, the following equation can be further obtained: T i = N i ⋅ ⁡ tan ϕ b + θ i (5)

However, the local normal force N j acting on asperity j exceeds the transition normal force N T , that is, the work required for the frictional sliding failure of this asperity exceeds the work required for the shear-off failure. The asperity j will experience shear-off failure (as shown in Figure 2D). At this time, the forces on asperity j satisfy Mohr‒Coulomb theory: T j = N j ⋅ ⁡ tan ϕ + C ⋅ I s ⋅ d 0 (6) N T = C ⋅ I s ⋅ d 0 tan ϕ b + θ j − ⁡ tan ⁡ ϕ (7) where T j is the local shear resistance acting on asperity j , ϕ and C are the internal friction angle and internal cohesion of the rock joint wall, respectively, θ j is the inclination angle of asperity j , and d 0 is the width of the rectangular joint plane of asperity, which can be taken as a constant.

Eq. 7 shows that the normal force corresponding to the transition of the failure mode of the asperity with a higher inclination angle is smaller. Therefore, when the normal load applied to the rock joint increases, more asperities will experience shear-off failure because of the local normal forces increasing and exceeding the respective transition normal forces.

For the rock joint, the local shear resistance of every real contact asperity can be calculated using formula 5 or 6 to calculate the shear strength of the rock joint. However, only Eq. 5 can reflect the influence of the morphological characteristics of varying asperities (i.e., their inclination angles) on the local shear resistance. Therefore, to adequately reflect the influence of the morphological characteristics of the joint surface on the shear resistance of the rock joint, this study assumes that the normal load applied to the rock joint is low (Barton and Choubey, 1977; Kulatilake et al., 2006; Cai et al., 2017). That is, in this condition, most real contact asperities experience frictional sliding failure. To further simplify the problem, all real contact asperities are assumed to experience frictional sliding failure. When contact occurs through surface roughness, the joint slips on the contact zones and forms angle α with the shear plane of the joint (see Figure 3). This is also the first failure mechanism (slippage) of rock joints proposed by Serrano et al. (2014).

If these arguments mentioned above are extended to the m contacts in n potential contact joint planes between the joint surfaces and a uniform distribution of force per unit of surface is assumed (Serrano et al., 2014), the following relationships are obtained: T = τ ⋅ L ⋅ d = ∑ i = 1 i = m T i (8) N = σ n ⋅ L ⋅ d = ∑ i = 1 i = m N i (9) N i = N ⋅ I s ⋅ d 0 ∑ i = 1 i = m I s ⋅ d 0 = N m (10) where T and N are the shear load and normal load applied to the rock joint, respectively, τ is the average shear stress applied to the rock joint, and L and d are the sizes of the nominal section of the rock joint.

Thus, the joint shear strength governing law may be obtained by combining formulas 5, 8–10. τ = 1 m ∑ i = 1 i = m σ n ⋅ ⁡ tan ϕ b + θ i (11)

Combining Figure 4 and Eq. 11, we find that parameters θ i   i = 1 , 2 , ⋯ , n can not only reflect geometric information of the potential contact asperities on the joint profile but also some of them (parameters obtained from those real contact asperities) have a direct mechanical relationship with the shear strength of the rock joint. Thus, the 2D roughness characterization parameter established by parameters θ i   i = 1 , 2 , ⋯ , n obtained from the potential contact asperities on the joint profile may better serve the shear strength estimation criterion. In addition, the greater contributions of higher inclination angles of joint line segments of potential contact asperities to roughness should be considered when establishing the roughness characterization parameter.

On the other hand, Yang and Chiang (2000) proved that the higher angle zones on the joint surface mainly controlled the mechanical behaviour. Considering this, we use the operator of root mean square to highlight the greater effects of steeper asperities on the rock joint shear strength instead of the arithmetic average when establishing the roughness characterization parameter based on all potential contact asperities on the joint profile.

Thus, a new average inclination angle parameter θ 2 D with clear geometric and physical meaning, based on a 2D rock joint profile containing n potential contact asperities (i.e., satisfying the relation of θ i   >   0 when the shear direction is taken as the positive direction), for quantitative characterization of 2D joint roughness is proposed. θ 2 D = 1 n ∑ i = 1 i = n θ i 2 (12)

Many studies found that those roughness parameters related only to the inclination angle could hardly reflect all the roughness information of rock joint profiles (Zhang et al., 2014; Liu et al., 2017; Wang et al., 2017). As shown in Figure 5A, when the horizontal projected length and inclination angle of the joint profile remain constant, the joint profile becomes flatter with the decrease in the average undulation height (i.e., h 1   >   h 2   >   h 3   >   h 4 ). Thus, the influence of the average undulation height on the roughness of the joint profile should not be ignored.

Barton and Bandis (1990) used the ratio of the maximum amplitude height h max to the horizontal projected length of the joint profile to characterize the undulation height characteristics of the joint profile (details in JRC straight-edge method). Considering that the maximum amplitude cannot fully represent the undulation height characteristics of the roughness profile, Zhang et al. (2014) proposed the average height parameter h a v e based on the mean line of a 2D joint profile. The mean line is a horizontal line that minimizes the area value of the region enclosed by it and the joint profile. However, the definition of the asperity in the calculation process of parameter h a v e has no clear geometric meaning (shown in Figure 6, M is the number of joint line segments). Moreover, it may be more reasonable to take the potential contact asperities on the joint profile rather than all asperities as calculation objects of the average undulation height parameter.

Similarly, to propose the roughness parameter to characterize the average undulation height with clear physical meaning, understanding the influence of asperities with different amplitude heights on the rock joint shear strength is needed. Du et al. (1996) expounded the physical meaning of the JRC straight-edge method based on the mechanism study of the mechanical effect of the climbing angle. An indication can be obtained from their study that those asperities with larger amplitude heights can provide most of the shear resistance for a rock joint during shearing (see Figure 5B). Therefore, when establishing the undulation height parameter, it is a reasonable idea to take the root-mean-square operator to highlight the greater contributions of larger amplitude heights of asperities to roughness.

Figure 7 shows the calculation process for the amplitude height of potential contact asperity i on the joint profile. We further calculate the amplitude height of each potential contact asperity on the joint profile and take the root-mean-square amplitude height Eq. 13 as the average undulation height. Thus, a new parameter h 2 D independent of θ 2 D is used to quantify the average undulation height characteristics of the joint profile: h 2 D = 1 n ∑ i = 1 i = n y i c 2 (13) where the y-axis is the fluctuation direction of the joint profile, n is the number of potential contact asperities, and y i   c is the amplitude height of asperity i .

In this study, a sample population composed of 112 joint profiles (including 10 standard joint profiles) studied by Li and Zhang (2015) is used to establish the relationship between the new roughness parameters proposed in Section 2 and the JRC. They also provided the coordinates of points with a sampling interval of 0.4 mm on each profile, and the JRC value of each profile was determined by back-calculation of the rock joint direct shear test using the JRC-JCS model. These rock joint profiles come from a wide variety of rock types, such as sandstone and limestone, and range from 72 to 119.6 mm in horizontal projected length.

Studies have shown that the calculation and analysis of joint profile morphological characteristics are directly affected by the sampling interval (Jang et al., 2014; Wang et al., 2019). Thus, to verify the accuracy of different methods for JRC estimation of the same joint profile obtained at a particular sampling interval, only those established on the digitized results of joint profiles with the same sampling interval should be considered. Bearing in mind that most of the previously published JRC estimation methods were established on the digitized results of joint profiles with a sampling interval of 0.5 mm (Yu and Vayssade, 1991; Yang et al., 2001; Tatone and Grasselli, 2010; Jang et al., 2014; Liu et al., 2017; Yong et al., 2018b; Zhao et al., 2018; Ban et al., 2021), 112 digitized joint profiles are resampled at 0.5 mm sampling interval using linear interpolation in this study.

The results of digitized joint profiles used as the sample population directly affect the accuracy of JRC estimation values (Jang et al., 2014). Therefore, a reliability analysis of resampled digitized results should be conducted. Here, the resampled digitized results of the 10 standard joint profiles are selected for analysis.

The root mean square parameter closely related to the JRC, Z 2 in Eq. 14, is calculated to verify the reliability of the digitized results: Z 2 = 1 L ∑ i = 1 i = M y i + 1 − y i 2 x i + 1 − x i (14) where M is the number of joint line segments, and x i and y i are the coordinates of the discrete point.

We compare the Z 2 calculated for each standard profile with the data collected from previous studies (Tse and Cruden, 1979; Yu and Vayssade, 1991; Yang et al., 2001; Tatone, 2009; Jang et al., 2014; Sun et al., 2014; Liu et al., 2017). However, it should be noted that all collected data used for comparison must be at the same sampling interval of 0.5 mm because of the sampling interval effect of Z 2 (Yu and Vayssade, 1991). The results calculated in this study are consistent with those of previous studies, as shown in Figure 8.

The more accurate the values of JRC estimated using roughness parameters are, the better the estimation of the shear strength of rock joints. For this purpose, the functional relation between new roughness parameters and JRC should be established. According to formulas 12, 13, the roughness parameters of these 112 joint profiles have been calculated, and the results for the 10 standard joint profiles among them are shown in Table 1.

The calculated results in Table 1 show that both the average inclination angle ( θ 2 D ) and the average undulation height ( h 2 D ) are generally positively correlated with the JRC. Previous studies (Zhang et al., 2014; Liu et al., 2017; Wang et al., 2019; Ban et al., 2021) have also pointed out that there are positive correlations between the quantization parameters of inclination angle and undulation height and the JRC. Additionally, for a flat rock joint, the values of its average undulation height and average inclination angle are obviously 0, so its JRC value should also be 0. That is, when the roughness parameters values in the JRC estimation equation are 0, the JRC estimation value should be 0.

Based on these findings and two functional relations between roughness parameters and JRC developed in earlier studies (Liu et al., 2017; Ban et al., 2021), three function models (including two modified models and a new proposed model) are applied to fit the relationship between the variables in this study (details are shown in Table 2). A total of 102 profiles were selected from these 112 joint profiles to determine the fitting parameters in these function models. Table 2 indicates that the new function model proposed in this study can better characterize the functional relationship between the new roughness parameters and JRC. Thus, the JRC value of a target rock joint profile is suggested to be estimated by the nonlinear regression equation as following: J R C s u g g e s t e d = 31.82 × 1 − 1 1 + θ 2 D 1.01 ⋅ h 2 D 0.44 / 37.23 (15)

To visualize the fitting effect of the new proposed model, a 2D roughness index, P, is introduced as: P = θ 2 D b ⋅ h 2 D c (16)

Figure 9 shows the correlations between P and JRC at sampling intervals of 0.5 mm and 1.0 mm. Fitting parameters b and c in Eq. 16 can be determined by Model 3 in Table 2. In addition, the data points of the remaining 10 joint profiles suggested as standard profiles by Barton and Choubey (1977) are also plotted in Figure 9A as a comparison (Zhang et al., 2014; Wang et al., 2019) to further verify the accuracy of Eq. 15 for JRC estimation. Figure 9A shows that the region formed by the predicted boundaries at 90% covers the majority of their data points. Hence, Eq. 15 can be used to accurately estimate the JRC values of the 10 standard rock joint profiles.

In this section, the results of direct shear tests on a total of 10 tensile joint replicas with two different types of real 3D surface morphologies under different constant normal load conditions collected from the studies of Xia et al. (2014) and Liu et al. (2017) are used. Eight peak shear strength estimation models, named Barton Models 1 to 6, Liu’s model and Tatone’s model, are introduced for comparative validation. Table 3 shows the mechanical properties of the materials used for the rock joint replicas. The two joint surface morphologies were represented by nine and fourteen joint profiles parallel to the chosen shear direction in their studies, respectively. We used GetData software to acquire the coordinate data of each joint profile at 0.5 mm sampling interval.

The JRC in Barton Models 1 to 6 are determined by Eq. 15, the visual comparison method (Barton and Choubey, 1977), the root mean square method ( Z 2 ) (Yang et al., 2001), the fractal dimension method (D) (Sanei et al., 2015), the roughness parameter method ( θ max * / C + 1 2 D ) (Tatone and Grasselli, 2010), and the method proposed by Ban et al. (2021), respectively. Liu’s model proposed by Liu et al. (2017) is an improved JRC-JCS model, and the corresponding JRC estimation method was provided in their study. Furthermore, Tatone’s model (Tatone et al., 2010) was established based on 3D roughness parameters of rock joints. To visualize the comparison results, a bar chart (see Figure 10) of the estimation errors calculated by Eqs 17 and 18 is further plotted. Compared with the peak shear strength estimated results from the seven other models, the maximum estimation error and the average estimation error of Barton Model 1 are the smallest, i.e., 0.1324 and 0.0493, respectively. This is an indication that Eq. 15 can give relatively more accurate JRC estimation values for rock joints with real 3D joint surfaces.

Rock joints with triangular asperities are usually used for direct shear tests to obtain a better understanding of the shear mechanical behaviour of rock joints with real 3D joint surfaces (Ghazvinian et al., 2010). Li et al. (2016), Liu et al. (2018) and Wu et al. (2018) performed direct shear tests on a series of artificial joint specimens with triangular asperities under different normal stresses. In their tests, the mechanical properties of the casting materials of joint samples (i.e., material No. Ⅲ to Ⅴ) are shown in Table 3.

It should be noted that the JRC estimation methods adopted in Barton Model 5, Barton Model 6 and Liu’s model cannot be used to estimate the JRC values of this type of rock joint. For example, the roughness parameter, θ max * / C + 1 2 D , proposed by Tatone et al. (2010) is based on the cumulative distribution of the inclination angles of all potential contact line segments on a profile. A continuous function was proposed in their study to describe the cumulative distribution: L θ * = L 0 θ max * − θ * θ max * C (19) where L 0 is the normalized length of the joint profile consisting of all potential contact line segments (i.e., the ratio of the joint profile consisting of all potential contact line segments to the total length of joint profile), L θ * is the normalized length of the joint profile consisting of those potential contact line segments with inclination angle of >   θ * (i.e., the ratio of the joint profile consisting of those potential contact line segments with inclination angle of >   θ * to the total length of joint profile), θ * is the angular threshold, θ max * is the maximum value among the inclination angles of all potential contact line segments, and C is the fitting parameter, which reflects the distribution characteristics of inclination angles of potential contact line segments.

However, the cumulative distribution of the inclination angles of the potential contact line segments on a triangular joint profile, such as a rock joint with regular triangular asperities used in the research by Liu et al. (2018) is piecewise and cannot be described by a continuous function (see Figure 11). That is, Eq. 19 cannot be used to fit the curve shown in Figure 11. Furthermore, Table 4 shows the results of estimated peak shear strength obtained by the three models mentioned above and Tatone’s model for the rock joint shown in Figure 11 under a normal stress of 0.5 MPa. The results show that it is a wrong viewpoint of using C = 0 to represent a regular triangular joint profile or a regular triangular joint surface. Moreover, the JRC value in Liu’s model is not even available. Obviously, these models are not applicable for estimating the peak shear strength of rock joints with triangular asperities.

Thus, Barton Model 1 and Barton Model 3 selected from Section 4.1 are used to estimate the peak shear strength of rock joints with triangular asperities. In addition, Barton Model 7 is further introduced for comparative analysis. The fractal dimension method proposed by Xie and Pariseau (1994), which is commonly used to estimate the JRC values of rock joints with triangular asperities (Indraratna et al., 1998; Ghazvinian et al., 2010; Mirzaghorbanali et al., 2014; Wu et al., 2018), is used to determine the JRC in Barton Model 7. Corresponding estimated values of peak shear strength for 63 rock joint samples with triangular asperities by these three estimation models are obtained. Then, a box diagram (see Figure 12) is further plotted for the statistical analysis of estimation errors. According to Figure 12, the box diagram for Barton Model 1 is flatter and positioned lower, indicating that the estimation errors of this model are more concentrated and closer to 0. Furthermore, the average estimation error and the estimation error of abnormal data (represented by blue rhombuses) of Barton Model 1 are lower than those of other models, which implies that the estimated results of Barton Model 1 are more accurate. Thus, Eq. 15 can also be used for accurate estimation of the roughness of rock joints with triangular asperities.

Studies have shown that rock joint roughness has anisotropy, i.e., the JRC varies with the shear direction (Tatone and Grasselli, 2010; Zhang et al., 2014; Wang et al., 2019). Thus, reasonable quantitative characterization parameters of 2D roughness should reflect the difference in roughness in different shear directions. The 10 standard joint profiles with a sampling interval of 0.5 mm are taken as examples to investigate whether the JRC estimation method proposed in this study can reflect the anisotropy of joint roughness. As shown in Table 5, the values of θ 2 D , h 2 D and J R C s u g g e s t e d of each standard joint profile are calculated via forward direction analysis and reverse direction analysis. Forward and reverse directions represent the analysis direction from left to right and from right to left with respect to each standard joint profile, respectively.

The difference in the values of J R C s u g g e s t e d for the same joint profile indicates that parameters θ 2 D and h 2 D can reflect the directional characteristics of the JRC. The fundamental reason is that the roughness parameters proposed in this paper are established by the potential contact parts on a joint profile, which are directly related to the shear direction. Thus, the analysis direction of the rock joint roughness should be consistent with the shear direction of direct shear tests considering the important effect of shear direction on the JRC.

The surfaces of natural rock joints are continuous. However, the digitized joint profiles extracted from a rock joint surface can only be obtained when a certain sampling interval is applied. The morphologies of digitized joint profiles extracted from the same location on a rock joint surface at different sampling intervals are apparently different, which will lead to the difference in roughness estimation. Thus, four of the ten standard joint profiles (the actual JRC values of the four profiles are 0.4, 6.7, 12.8, and 18.7) are selected to study the sampling interval effect of roughness.

Figure 13A shows four different morphologies of the same standard joint profile whose JRC value is 12.8 at four different sampling intervals (i.e., 0.5 mm, 1.5 mm, 4.0 mm and 10.0 mm). According to Figure 13A, some features on the profile with a base length less than a certain interval have been lost with increasing sampling interval, leading to smoothing of the joint profile. The parameters Z 2 , θ 2 D and h 2 D for the four joint profiles sampled at intervals ranging from 0.5 mm to 10 mm are calculated to further quantitatively study the sampling interval effect. The analysis direction of each standard profile is from left to right. Figures 13B, C intuitively show the general decrease in parameters Z 2 and θ 2 D with increasing sampling interval, respectively. However, as the sampling interval increases, the parameter h 2 D remains almost constant, as shown in Figure 13D. The reasons for these phenomena shown in Figures 13B–D are that: (1) the features on a joint profile with a small base length generally have greater inclination angles but lower amplitude heights. (2) when a joint profile is digitized with a larger sampling interval, more roughness features with base lengths smaller than the sampling interval will be lost. Therefore, the larger the sampling interval, the less or even no information can be acquired from those high-inclination-angle or low-amplitude-height roughness features.

The rock joint profiles at the laboratory scale (approximately 100 mm) have been digitized at sampling intervals of 0.25 mm (Yu and Vayssade, 1991), 0.4 mm (Li and Zhang, 2015; Wang et al., 2019), 0.5 mm (Yu and Vayssade, 1991; Yang et al., 2001; Tatone and Grasselli, 2010; Liu et al., 2017; Ban et al., 2021), 1.0 mm (Yu and Vayssade, 1991; Tatone and Grasselli, 2010) and 1.27 mm (Tse and Cruden, 1979) by previous researchers to characterize their roughness. Figure 13 also indicates that when the sampling interval adopted is greater than 1.5 mm, a significant portion of the roughness information (especially the inclination angle information) of these joint profiles have been lost. Hence, when digitizing laboratory-scale joint profiles, a sampling interval of less than 1.5 mm is recommended to obtain adequate roughness information. We further use function Model 3 mentioned in Section 3.3 to fit the relationship between the roughness parameters θ 2 D and h 2 D at 1.0 mm sampling interval and the JRC (see Figure 9B). From Figure 9, we know that the relationships between roughness parameters and JRC vary considerably for different sampling intervals. This phenomenon has also been observed by previous researchers (Yu and Vayssade, 1991; Tatone and Grasselli, 2010; Wang et al., 2019). Accordingly, when using relationships between roughness parameters and JRC, we should focus attention on the sampling intervals to which those relationships apply.

The innovation of this paper is to propose two parameters with clear geometric and physical meanings to characterize the 2D roughness of rock joints. One represents the average inclination angle of all the potential asperities on the whole joint profile, and the other describes their average undulation height. Both parameters are easily obtained and closely related to the shear strength of rock joints. The roughness information of rock joint profiles can be more fully reflected by combining these two parameters. Additionally, it is significant that the new proposed nonlinear regression equation of these two parameters with JRC can accurately estimate the JRC values for realistic and triangular joint surfaces. The improved JRC estimation equation further facilitates the application of the JRC-JCS model in practice.

The limitation is that only the peak shear strength of laboratory-scale rock joints under perfectly matched conditions can be estimated by the modified model proposed in this paper (i.e., Barton Model 1). However, the joint scale and the joint matching degree significantly affect the peak shear strength of rock joints (Zhao, 1997; Johansson, 2016; Ríos-Bayona et al., 2021). Therefore, investigations on the estimation models for the peak shear strength of rock joints on the site scale when considering the joint matching degree are urgent.