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This study aims at proposing a reasonable roughness parameter that can reflect the peak shear strength (PSS) of rock joints. Firstly, the contribution of the asperities with different apparent dip angles to shear strength is studied. Then the shear strength of the entire joint asperities is derived. The results showed that the PSS of the entire joint asperities is proportional to a key parameter θτ, which is related to the geometric character of the joint surface and the joint material properties. The parameter θτ is taken as the new roughness parameter, and it is reasonable to associate the PSS with the geometric characteristics of the joint surface. Based on the new roughness parameter and shear test results of 20 sets of joint specimens, a new PSS model for rock joints is proposed. The new model is validated with the artificial joints in this paper and real rock joints in published studies. Results showed that it is suitable for different types of rock joints except for gneiss joints. The new model has the form of the Mohr-Coulomb model, which can directly reflect the relationship between the 3D roughness parameters and the peak dilation angle.

The area ratio content of the asperities with different apparent dip angles is studied based on Grasselli’s apparent dip angle distribution function. An approximate expression of the minimum dip angle of the asperities in contact is further obtained according to the contact theory. For regular joints, the peak dilation angle is the average dip angle of all asperities on the contact part. Extending the above idea to the peak dilation angle of rough joints, the peak dilation angle model is derived based on the apparent dip angle distribution of the real contact asperities. In the derivation process, the peak dilation angle of an arbitrary stress state is directly obtained, instead of obtaining the initial dilation angle first and then defining the relation between the peak dilation angle and the initial dilation angle. The more important innovation is that all the parameters in the new model are of physical significance and easy to obtain; furthermore, they are not obtained by fitting test results. Based on 89 sets of test data, the predicted values of the new model are compared with those of the other 4 existing models. The results show that the prediction accuracy of the new model is the best. Besides, the mesh scales of rock joint are discussed, and the size range of the triangle mesh is obtained. It is proposed to remove the isolated points with a large apparent dip angle when processing test data. The shear mechanism of rough joints is further clarified in this study: the dilation is regarded as the average apparent dip angle of the asperities in contact at a certain stress level. This view is easy to understand and follow. It is as simple and beautiful as many natural principles.

The peak shear displacement (PSD) of artificial joints was studied by the shear test of cement mortar duplicate joints under constant normal load. First, the splitting rock joints were produced by the Brazilian splitting method, and the high-precision point cloud of joint surface morphology was acquired by three-dimensional (3D) scanning method. The polylactic acid (PLA) base mold was produced by 3D printing technology. The duplicate joints with the same joint surface as the splitting rock joints were obtained by pouring cement mortar on the PLA base mold. Then, shear tests of 20 cement mortar specimens with 5 groups of joint surface morphologies under 4 different normal loads were carried out, and the joint shear stress-displacement curves were obtained. The influencing factors of the PSD were studied. The results indicate that the PSD is inversely related to the joint roughness and positively related to the normal stress (NS). The limitation of Barton’s empirical formula is that it cannot predict (fit) the PSD of the joints with JRC = 0 and reflect the effect of NS on the PSD. To quantitatively study the PSD of joints, an empirical formula that can comprehensively consider the influence of surface roughness and NS is introduced based on the test results. The modified model overcomes the limitation of Barton’s empirical formula which ignored the NS and can well reflect the sensitivity of joint roughness coefficient (JRC) to PSD. The acquisition of PSD can provide a basis for the numerical simulation of rock engineering and provide quantitative research methods for the analysis of stick-slip events.

Rock is a material that is affected by wear, and the curvature of the asperities on a rock joint surface increases with the degree of wear after shearing. Based on the Greenwood and Williamson (GW) model, a new model considering the change of asperity curvature is proposed to explain the wear behaviour of rock joints. First, the shear stiffness formula for a joint surface is derived when the asperity curvature is constant, which shows that the shear stiffness increases with increase of asperity curvature. According to the Mohr–Coulomb criterion, the yield position of a single asperity under normal force and tangential friction force is discussed. Then, the critical normal force for a single asperity at a specific friction coefficient is obtained, which shows that the normal force corresponds to the curvature radius of the asperity. A rough surface model with multi-level curvature radius is proposed. With increase of normal force, the higher-order asperities gradually fail and the curvature radius become larger. A specific pressure value excites a specific radius of curvature, and the larger the pressure, the larger the radius of curvature. The relation between the normal force and the curvature radius is proposed and a shear stiffness formula considering the change of curvature radius of the asperity is derived. The proposed model is verified on the basis of the published experimental results. The calculation results of the proposed model can reflect the test results well: for a given joint surface, with increase in normal force the joint surface gradually becomes smooth; for different joint surfaces, with increase in roughness, the joint surface is more easily smoothed.

With the help of 3D printing and 3D laser scanning techniques, cement mortar joint samples with a certain surface morphology were prepared. Shear tests of 20 sets of matching joint samples and 8 sets of joint samples with different percentages of cavity area were performed under constant normal load and uniform shear displacement. The results show that the distribution characteristics of the equivalent height difference based on the new roughness description were in accordance with the distribution of the wearing area after shearing, and the effect of cavities on the peak shear strength is essentially due to the influence of the cavities on the roughness of the joint surface. The relationships between the peak shear strength and the two roughness parameters were discussed, and a new criterion for predicting the peak shear strength of rock joints was proposed. It was noted that the roughness parameter system adopted in this paper, which can describe the peak shear strength, was reasonable. The roughness anisotropy of the five joint surfaces was discussed, and a corresponding quantification parameter AAHD accounting for the roughness anisotropy was proposed. The roughness of the joint surface has a positive size effect, a negative size effect and no size effect in a certain direction. However, no matter the direction, the roughness parameters will gradually stabilize as the research scale increases. A similar relationship between the roughness anisotropy parameters and research scales was also observed. To check the applicability of the proposed criterion for estimating the peak shear strength of natural rock joints, 12 sets of rock joints with the same surface morphology were produced based on a numerically controlled engraving technique. Under the same loading conditions as those in the shear tests of the cement mortar replication joints, the peak shear strengths of the rock joints were tested, and the results indicated that the new criterion was also applicable to predict the peak shear strength of natural rock joints.

To clearly portray the mechanical behavior of the shear process and to relate three-dimensional surface morphology to the shear strength of rock joints, we propose a new method, the equivalent height difference system, which can reflect that different geometric parameters of the asperities contribute differently to shear strength. A roughness parameter, the average equivalent height difference (AHD), is obtained by averaging the equivalent height difference in the shear direction, which can reflect the fluctuation and directivity of the joint surface morphology. The values of AHD at different sampling intervals were studied. The relationship between AHD values and sampling intervals is power law. Another roughness parameter, fractal dimension (DAHD), was proposed based on the fractal component, which characterizes the relationship of roughness at different scales and describes the roughness comprehensively from the point of view of scale. We used these two newly proposed roughness parameters to characterize the roughness of five joint surfaces. The results show that the new method can well reflect the three-dimensional morphology of the joint and describe the anisotropy of roughness, and it can also overcome the influence of the sampling interval.

To overcome the drawbacks of the existing two-dimensional (2D) roughness parameters, this paper proposes two parameters to characterise the roughness of rock joints. One is used to represent the local features of the joint profiles, and the other is used to characterise the overall features. The proposed local 2D parameter θ2d can provide an alternative approach and facilitate the characterisation of the roughness of the joint rock surface when a three-dimensional measuring apparatus is unavailable. This parameter is calculated using 10 standard joint roughness coefficient (JRC) profiles, and the value of θ2d does not increase monotonically with an increase in the JRC. The results suggest that θ2d can reflect the local waviness but fails to describe the overall features of the profiles. Another parameter, h1, is introduced to consider the overall features of the profiles. A new equation for the JRC is derived by combining the small- and large-scale roughness parameters. Predictions based on the new equation are compared with the experimental data, and the results indicate the applicability of the new equation.

Based on the index, the root mean square of the first derivative of the profile (Z2), the contribution of the asperities with different geometric characteristics to the peak shear strength (PSS) is considered, and an improved roughness index (Z2)ds is proposed. (Z2)ds is reasonably related to the PSS and can reflect the directionality of the PSS. The direct shear tests of 60 joints under different normal stress are carried out. Test results show that the PSS is related to normal stress, surface morphology and shear direction. The joint equivalent dip angles and the abrasion zone of joints after shearing share a similar distribution. There is a clear linear relationship between (Z2)ds and the joint PSS. Accordingly, a new relationship between the roughness index (Z2)ds and PSS is proposed. The new relationship can eliminate the influence of sampling intervals. The proposed index (Z2)ds establishes a reasonable relationship between the surface roughness and shear mechanical properties of rock joints. This is helpful to study the shear behavior of rock masses in a specific direction in engineering geology.