Experimental study of the shear strength criterion of rock structural plane based on three dimensional surface description

: The failure of rock mass is mainly due to the failure of the structural plane, which is an important factor to reduce the mechanical properties and stability of rock mass. The shear strength of rock mass is one of the parameters for the stability calculation of large - scale rock mass engineering. The shear strength of a rock structural plane is strongly in ﬂ uenced by surface morphology. Considerable research has been conducted regarding the correlation between two - dimensional structural plane morphology and shear strength. However, quantitative research on three - dimensional ( 3D ) morphology is rela tively limited. In this study, 3D printing technology was used to create molds. Using cement and sand as the main materials, additives such as early strength and water reducing agents were added, and test samples of irregular surface topography were created. The 3D roughness was quanti ﬁ ed by formula calculation. Using a ZScanner ® 800 hand - held 3D laser scanner to perform scanning on the structural surface, the parameter curve was analysed by generating 3D coordinate information and a 3D image of the fracture surface, and the quantitative parameter M p3D describing the 3D morphology of the structural surface was constructed. The change rule of R p3D and joint roughness coe ﬃ cient ( JRC ) were analysed under di ﬀ erent scanning resolutions, Δ ( r ) , the scanning precision was suggested, and the functional relationship between JRC and M p3D was established. Finally, a formula for shear strength para meters considering the 3D characteristics of a structural plane surface was established. The model validation results show that the experimental data were within the 95% con ﬁ dence band of the model curve, the average error of the shear strength was 10.4%, the errors of friction angle and cohesion, C , were 3.4 and 9.4%, and the reliability was ﬁ ne.

Abstract: The failure of rock mass is mainly due to the failure of the structural plane, which is an important factor to reduce the mechanical properties and stability of rock mass. The shear strength of rock mass is one of the parameters for the stability calculation of large-scale rock mass engineering. The shear strength of a rock structural plane is strongly influenced by surface morphology. Considerable research has been conducted regarding the correlation between two-dimensional structural plane morphology and shear strength. However, quantitative research on three-dimensional (3D) morphology is relatively limited. In this study, 3D printing technology was used to create molds. Using cement and sand as the main materials, additives such as early strength and waterreducing agents were added, and test samples of irregular surface topography were created. The 3D roughness was quantified by formula calculation. Using a ZScanner ® 800 hand-held 3D laser scanner to perform scanning on the structural surface, the parameter curve was analysed by generating 3D coordinate information and a 3D image of the fracture surface, and the quantitative parameter M p 3D describing the 3D morphology of the structural surface was constructed. The change rule of R p 3D and joint roughness coefficient (JRC) were analysed under different scanning resolutions, Δ(r), the scanning precision was suggested, and the functional relationship between JRC and M p 3D was established. Finally, a formula for shear strength parameters considering the 3D characteristics of a structural plane surface was established. The model validation results show that the experimental data were within the 95% confidence band of the model curve, the average error of the shear strength was 10.4%, the errors of friction angle and cohesion, C, were 3.4 and 9.4%, and the reliability was fine.
Keywords: structural plane, three-dimensional, topography, laser scanning, shear strength, roughness coefficient List of symbols 3D fluctuation coefficient of the structural plane. S Q area of the structural surface slope S T area of the horizontal projection Δr scanning resolution σ 1 axial stress σ 3 confining pressure σ n normal stress τ p peak shear strength φ internal friction angle of the material φ b basic friction angle of the structural plane complex system with geometric space characteristics. The instability of rock mass engineering and the occurrence of geological disasters are mostly related to the weakening of the mechanical properties of rock mass [4]. By controlling the shear strength of the jointed rock mass, the roughness of the joint plane has become one of the important factors affecting the mechanical properties of the rock mass structural plane [5]. The influence of joint surface morphology on the rock's mechanical properties is notable. To eliminate or reduce the rock morphological effects, model tests have been widely used. In the study by Patton, by introducing a structural plane physical model, the effect of serrated structures on the shear strength is studied [6]. By analysis of a total of 136 natural rock joints, the index of joint roughness coefficient (JRC) was proposed by Barton and Choubey [7] to describe the roughness of rock surfaces. Since the 1970s, joint morphology and its effects on mechanics have become a hot topic [7][8][9][10][11][12][13][14].
To evaluate the quantitative characteristics of the structural surface, Li et al. [15] introduced a meshing method of Schmidt projection to calculate the fractal dimension for joint orientation for the first time. By introducing the minimum spacing/trace length value and fractal dimension, the corresponding fractal probability density function was derived. Wei et al. [16] used cumulative relative fluctuation amplitude and weighted average gradient to characterize the morphological characteristics of the joint profile, and the JRC calculation formula was put forward. The quantitative characterization of ten standard JRC curves has been analysed by many scholars [17][18][19][20][21][22][23], and a set of statistical formulas have been developed. Some roughness parameters, such as root mean square of the deviation [17][18][19][20][21][22], structural function [17,18,21], and roughness profile index [18][19][20][21]24], were often mentioned. By using a total station, Feng et al. [25] put forward a new way to describe non-contact fracture roughness. In the study by Morelli [26], a new index for describing the JRC was put forward. In addition, some scholars have found that self-affine fractal and multifractal natures exist in joint morphology [27][28][29]. By analysing ten standard JRC curves, the effect of fractal dimension on joint strength has been studied. It was noted by Grasselli [9] that the initial contact area was correlated with the damaged zone and the peak shear strength. There are many methods to study and determine the roughness of a structural plane, but how to apply it to the deformation and failure of rock mass and quantitatively combine the roughness with the failure strength of rock mass has become an important research direction.
In the study by Barton and Choubey [7], a formula for the peak shear strength that combines the JRC, normal stress, and mechanical properties was put forward. Zheng et al. [30] proposed a generalized three-dimensional (3D) fractal dimension, which can comprehensively characterize the roughness, undulation, and anisotropy of structural surface morphology, and established the mathematical expression of a generalized 3D fractal dimension. Taking the effect of the slippage resistance of convex teeth in the shear direction into account, Grasselli [9] used the morphological quantification of 3D joint surfaces and proposed a peak shear strength prediction model. Furthermore, considerable work in this field has also been done by other researchers [5,[31][32][33][34][35]. The research on structural surfaces mentioned in the above literature mainly focuses on 2D morphology, and it is necessary to further develop the research on 3D models. Combining 3D scanning and direct shear technology, it is feasible to explore and establish the criterion of peak strength of a structural plane [36]. The computation of JRC is cumbersome and requires a lot of study. The 3D morphological characteristics or the anisotropy of joint surfaces are not reflected. As a quantitative model to describe the 3D surface morphology, the parameters in the Grasselli model are too complex.
In this study, via material preparation and the 3D printing (3DP) method, structural test samples of different topographies were developed. The ratio of slope area to horizontal projection area was used to reflect the fluctuation in the 3D structure. Then, the change rule of the fluctuation state of the structural surface under different scanning resolutions was analysed using the 3D laser scanning method. A function model of the 3D structural surface and shear strength was established, then the shear test was used for validation ( Figure 1).

Material preparation
To prepare a rock model, a mix proportion procedure was used. The ratio of cement to sand is 1:1.5, and the ratio of cement to water is 1:0.3. In addition, 10% silica powder, 15% silicon carbide, 1.5% water reducer, and 2% accelerator were used in the whole mixture [37][38][39][40][41][42][43][44][45][46][47]. In the test, 52.5R composite silicate cement and sand were the main ingredients, silica powder and silicon carbide were additives, and a small amount of early strength admixture. To test the compressive strength and shear strength of the material, the dimensions and test standards of the modulus materials are shown in Table 1 ( Figure 2).

3DP
3DP is a computer-controlled manufacturing technique that can fabricate objects with complicated geometry and internal structures repeatedly and accurately. Various printing techniques have been used to fabricate polymer composites. In this study, 3DP equipment and corresponding printing materials have been studied to produce molds of structural surface topography. Among the equipment, 3D printers using powder ink binders (PIB) are widely used. PIB printers are based on powder and liquid binders. Because PIB printers are made from sand powder, the resulting product may be more visually like rock material.

3D laser scanning
Hand-held 3D laser scanners are used to measure structural surface topography as they have high scanning speed and high precision. In this study, a ZScanner ® 800 handheld 3D laser scanner was used, in which the maximum scanning accuracy can reach 0.04 mm. The instrument is shown in Figure 3, and the process of obtaining the surface morphology of the structural plane   is shown in Figure 4, which included instrument installation, scanning specimen preparation, pasting the sensitive plate, open scanning of software, adjusting the laser power and other configurations, scanning the specimen, saving the file, data processing, model reconstruction, and the analysis result. After the scan is completed, a coordinate transformation of the data is required. Geomagic Qualifier & Studio can quickly complete the transformation of coordinates. The principle of transformation can be expressed as Formula 1 as follows: where R(∂, β, γ) represents the rotating vector and [x 0 , y 0 , z 0 ] T represents the transformation vector between different coordinate systems.

Model creation
In this study, a 3D zigzag structural plane was processed. The model size was 10 cm × 10 cm × 5 cm. The geometric features of the convex teeth were controlled by the fluctuation angle (i) and the fluctuation height (h). Six serrated models were created, and their dimensions were , h = 0.5 cm; and i = 30°, h = 0.5 cm, as shown in Figure 5.
3 Roughness quantization of the3D surface of the structural surface 3.1 3D roughness quantization formula The profile roughness coefficient R pl is used to represent the fluctuation of the structural plane. R pl = EF/OF, and its coordinate expression is as follows: Triangle ΔABE and triangle ΔAOB have a public side AB, EF⊥AB, and OF⊥AB, and ΔABE is a projection of the ABCD in plane ΔAOB, S ΔABE = ½(AB × EF), S ΔAOB = ½(AB × OF).    Thus, we obtain formula (3) as follows: where S Q represents the slope area of the structural plane and S T represents the projected area in the horizontal plane.
The profile roughness coefficient R p can reflect the undulating state of the structure. However, to avoid the surface area of the structural surface becoming 0, the situation of R p 3D tends to infinity. The reciprocal M P of the profile roughness coefficient R p is adopted, and the expression of M p is as follows:

Effect of scanning resolution on 3D roughness
In general, the higher the resolution, the more detailed the model is, and the closer it is to the real shape of the  structural plane. To study the effect of scanning accuracy on the 3D roughness coefficient R p 3D , different scanning resolutions were used to scan the structural surface topography to explore the relationship between Δr and R p 3D .
The resolution of the scanning instrument was 0.4, 0.6, 0.8, 1.0, 2.0, 3.0, and 4.0 mm, respectively, and 3D data of the structural surface were obtained. Then, the data were imported into the ploy-works 8.0 reconstruction software to process, generate 3D images of the structural surface, and develop statistics on the related geometric parameters. The 3D scanning model and grid are shown in Figure 7. The geometric parameters of the structural plane are shown in Table 2. Table 2 shows, when the resolution of Δr = 0.4 mm, the scanning model is most similar to that of the original rock surface, and the detailed characteristics are very clear. When the resolution is gradually reduced, the details of the structural surface become blurred, and the model surface becomes smooth. When the resolution reaches 4 mm, the detailed morphology can hardly be observed, and only the areas with prominent fluctuations can be reflected; the whole plane gradually tends to a smooth plane. The relationship between scanning resolution and R p 3D is shown in Figure 8. With a decrease in scanning resolution (Δr increasing), R p 3D gradually decreases, representing a power index relationship. The JRC value is the parameter reflecting the roughness of the structural surface. At different resolutions, the roughness coefficient JRC of the structural surface image changes. To study the change rule of JRC under different scanning accuracies, six profile lines were taken of the different scanning resolution models. Based on the Barton model, which links the roughness of the structural plane with the shear strength and compressive strength, the roughness coefficient JRC value of the profile line was calculated. The relationship between Δr and JRC was studied. Similar to Figure 9, the JRC values of the profile lines are shown in Figure 10.
The relationship curve of JRC to Δr is shown in Figure 11. With a decrease in resolution (Δr increasing), the roughness of the structural surface gradually decreases, and they are related exponentially. Its change trend is basically consistent with the change trend of M p 3D , and a relationship between M p 3D and JRC is established as follows: We can see from the aforementioned figures that with a decrease in the scanning resolution (an increase in Δr), R p 3D and JRC gradually decrease. When the scanning resolution is reduced to 10.6 mm, the R p 3D value is 1.061.
Substituting 10.6 into the formula in Figure 11, the JRC value is 0.01, which is near 0, indicating that the structural surface is near the plane. When the scanning resolution is enhanced, R p 3D and JRC also increase, but the JRC value does not increase when the scanning resolution reaches 0.2 mm. Therefore, we suggest that during the scanning process of the structural surface topography, the scanning resolution should be adjusted to be from 0.4 mm. At the same time, the function relationship between JRC and M p 3D was established, as shown in the following equation: indicates that the surface is near an ideal horizontal plane). S Q represents the slope area of the structural surface. S T represents the area of the horizontal projection.

Prediction model of shear strength based on 3D roughness
On the basis of the quantitative description of the 3D structural surface and the Barton model [48], we established a 3D shear strength estimation formula by adopting the M p 3D of the structural surface undulating coefficient a follows:    indicates that the surface is near an ideal horizontal plane). S Q represents the slope area of the structural surface. S T represents the area of the horizontal projection. Δr is the scanning resolution. The joint compression strength (JCS) represents the compressive strength of the rock. φ b is the basic friction angle of the structural plane.

Discussion
A shear test under different stress levels was conducted to verify the reliability of the shear strength model. The reliability of the theoretical formula was evaluated by comparing the error of the test value to the predicted value. During the experiment, the normal stress was 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 MPa. The structural surface sample is shown in Figure 14. The compressive strength was 52.5 MPa and the basic friction angle was 45.48°after 28 days of standard curing. The resolution of the scanning instrument was 0.4 mm, via scanning and calculation, M p 3D = 0.7642. Figure 15 shows the laser scanning image following the shear test under different stresses. With an increase in the normal stress, the loss degree of the structural surface gradually increases. When the normal stress is relatively low, the surface of the structural plane is significantly damaged, and this results in a sliding type after climbing over the convex body.
When the normal stress was increased gradually, the convex was cut and slippage shear failure occurred. According to the experimental data of the shearing results, the relationship curve between the shear stress and the positive stress was plotted. The shear stress was the vertical coordinate and the normal stress was the horizontal coordinate, and C and φ were obtained according to formula (9). At the same time, the theoretical prediction model was applied. C and φ were obtained after accurate determination of M p 3D , JCS, and φ b as shown in Figure 16. Experimental values and theoretical prediction values were calculated, and the results are shown in Table 3. The friction angle was 41.5°and the cohesion coefficient C was 0.32 MPa by experiment. The friction angle was 42.9°and the cohesion C was 0.29 MPa using the theoretical prediction model. Through the linear fitting of the 95% confidence interval of the model formula, the experimental values are within the confidence band. The average error of the shear strength was 10.4%. The errors of friction angle and the cohesion C were 3.4% and 9.4%. The results show that the formula is reliable.
The test assumes that the structural plane is hard and does not consider the anisotropy of the structural plane. The established theoretical prediction model can well predict the friction angle and the cohesion C, but the shear strength is not very ideal when the normal stress is small. We believe that the cohesion C still has some internal relationships with M p 3D . Through the shear test of   samples with different proportions, we may be able to find the relationship between them and then modify the theoretical model.

Conclusions
The model of an irregular structure was prepared via material preparation and 3DP. Laser scanning was used to obtain the surface topography of the rock structural surface and produce a quantitative model of the 3D topography of the irregular structure. Based on the fluctuation   (1) Using the composite silicate cement and sand as main ingredients, silica powder and silicon carbide were added as additives, and with a small amount of early strength admixture, rock model materials were prepared. The ratio of cement to sand is 1:1.5, cement to water is 1:0.3. In addition, 10% silica powder, 15% silicon carbide, 1.5% water reducer, and 2% accelerator were used. (2) The undulation coefficient M p 3D was used to describe the 3D fluctuation of the structural surface, and its value was the ratio of the slope area to the horizontal projection area for the structural surface.    Figure 15: Laser scanning image of the structural plane after the shear test.