# Influences of Roughness Sampling Interval and Anisotropy on Shear Failure Mechanism of Rock Joint Surface

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## Abstract

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_{2}′.

## 1. Introduction

_{2}has become a widely used roughness evaluation parameter, although Z

_{2}only considers the effect of the local inclination angle and ignores the influence of the overall amplitude of the profiles on the roughness. Zhang et al. [3] introduced a new roughness index (λ) based on Z

_{2}, which considers the inclination angle, amplitude of asperities and their directions. Magsipoc et al. [11] conducted a comprehensive review of commonly used 2D and 3D surface roughness characterization methods. They suggested that 2D profiles may obscure or exaggerate the roughness features, and 3D roughness characterization methods should be considered for the development of surface measurement technology. A digital image processing (DIP) technique was used to reconstruct the surface morphology of joints [12], and a variogram function was adopted to characterize the anisotropy of the joint surface roughness and estimate the joint aperture. Moreover, the effects of the sampling interval and anisotropy on roughness parameters SF, Z

_{2}and R

_{p}were studied utilizing point cloud data obtained by 3D laser scanning [13]. By averaging the equivalent height difference in the shear direction to reflect the fluctuation and directivity of the joint surface morphology, Ban et al. [14] proposed a new roughness parameter called the average equivalent height difference (AHD), while the fractal dimension of AHD was used to characterize the scale effect of roughness. The effects of joint surface roughness and anisotropy on the size and shape of the excavation damage zone in jointed rocks were examined by numerical simulation [15]. Using a class ratio transform method, the geometric irregularities of the roughness parameters in the polar coordinate could be transformed to a regular roughness asperity pattern [16], which could be easily approximated by the ellipse function. However, the actual roughness in different directions may be ignored after smoothing processing.

_{s}) and (2A

_{0}θ*

_{max}/(C + 1)). It is noted that the sampling interval has a negative exponential relationship with the JRC value [19]. While the roughness of the joint surface AHD proposed by Ban et al. [20] has a positive size effect, there was a negative size effect and no size effect in a certain direction. The sampling interval has little influence when using volume and amplitude parameters (V

_{an}, Z

_{sa}, Z

_{rms}, and Z

_{range}) for JRC estimation, while it has some influence when other parameters (θ

_{s}, θ

_{g}, θ

_{2s}, S

_{sT}and S

_{sF}) are used [21]. Thus, equations with amplitude parameters are recommended to facilitate the rapid estimation of JRC in engineering practice for their easy calculation. Huang et al. [22] analyzed the variation in fitting parameters at different values of Δx based on the relationship between JRC and statistical parameter Z

_{2}, and they established empirical formulas for JRC, Z

_{2}and Δx. Due to the scale dependency of roughness, a changing scale will indirectly affect the surface contact mechanics, such as shear strength and dilation. Consequently, determining the optimal sampling interval is very important for accurately estimating the actual roughness and reducing the cost of data processing. Du et al. [23] proposed the concept of the effective length of the JRC scale effect and established a graphical method to identify the effective length of the JRC scale effect. Through a Fourier series reconstructed roughness profile [24], the maximum sampling interval is assigned as SI

_{max}= L/3n, where n is the order of the Fourier series and L is the length of the profile. A small sample size is generally used during roughness measurement in the laboratory, while the size of natural joint surfaces may be several meters or even longer. Yong et al. [25] developed a method for measuring the JRC values of large joints in the field. In order to overcome the deficiencies of existing approaches, Huan et al. [26] proposed a new JRC estimation method based on the back-calculation of shear strength.

_{p}, the angle standard deviation σ

_{i}, the root mean square method Z

_{2}and the modified root mean square Z

_{2}′, are used to analyze the scale effect and anisotropy of roughness. Moreover, the reasons for the scale effect of joint surface roughness are discussed. To compensate for the deficiencies of the regular interval sampling method that changes the geometrical morphology of the original profile, a non-equal interval sampling method and an equation for determining the sampling frequency on the roughness profile are proposed. Additionally, the influence of roughness anisotropy on the shear mechanism of the joint is further discussed. The distributions of shear stress and ranges of dilatancy and non-dilatancy on the joint with different shear directions are calculated based on the direct shear test. The results are helpful for understanding the influences of the roughness sampling interval and anisotropy on the shear failure mechanism of the rock joint surface.

## 2. Materials and Methods

#### 2.1. Collection of Rock Joint Specimen Point Clouds

#### 2.2. Direct Shear Tests

_{n}, was applied on top of the mobile block before the shear tests. This force F generates a normal stress σ

_{n}= F/A

_{s}, where A

_{s}denotes the area of the horizontal section of the contact specimen. Then, the superior block was displaced horizontally by applying a horizontal force, thereby shearing the sample. Displacement sensors were used to record the horizontal and vertical displacement during the test. The accuracy of the displacement sensor was 0.01 mm. A YY-8 Type geotechnical mechanics data acquisition instrument was applied to acquire the experimental data, such as normal stress, shear stress and shear displacement, and the data collection interval was 10 ms.

## 3. Influence of Non-Stationary Joint on Roughness

#### 3.1. Reasons for Non-Stationary Point Clouds

#### 3.2. Removing the Non-Stationary Feature of Point Clouds

_{0}, y

_{0}, z

_{0}) is a source point cloud with non-stationary features and that Q(x, y, z) is the target point cloud without non-stationary features. The following steps are required for removing the influence of non-stationarity.

_{0}, y

_{0}, z

_{0}) in the grid data to the fitting plane $d=\frac{A{x}_{0}+B{y}_{0}+C{z}_{0}}{\sqrt{{A}^{2}+{B}^{2}+{C}^{2}}}$ (Figure 3c).

_{0}, y

_{0}, z

_{0}) as the X and Y coordinates of the target point, and letting the calculated distance d be the Z coordinate of the target point, namely Q(x

_{0}, y

_{0}, d) (Figure 3d).

## 4. Estimation of Joint Surface Roughness

#### 4.1. Methods Used to Evaluate JRC

_{i}and the root mean square of the first deviation of the profile Z

_{2}are the best. In this paper, 14 parallel profile lines along the shear direction with an interval of 10 mm were intercepted from the study region after the pretreatment process of original point clouds obtained by a 3D laser scanner (Figure 4). Digitized information about the intercepted lines was extracted, and the coordinate values for each point on the lines were obtained. The roughness ratio R

_{p}, σ

_{i}, Z

_{2}and modified Z

_{2}′ were taken to estimate the roughness of each profile. Finally, the average roughness of each profile was adopted to characterize the roughness of the joint surface.

_{p}

_{p}is the ratio of the actual length of a profile to the nominal length of the profile [31]:

_{i}and y

_{i}are points along the profile of the x and y coordinates, respectively.

_{i}

_{2}

_{2}, which has become one of the most commonly used parameters.

_{2}is the root mean square of the first deviation of the profile.

_{2}′

#### 4.2. Size Effect of Regular Sampling Interval

_{p}, σ

_{i}, Z

_{2}and Z

_{2}′ decreased with the increase in sampling interval, and eventually stabilized, showing a negative size effect. As the sampling interval increased, the roughness parameters reduced rapidly and finally stabilized gradually. As the existing roughness estimation methods are sensitive to the sampling interval (Δx), it is extremely important to study the relationship between the sampling interval and roughness in order to determine the maximum sampling interval. Huang et al. [22] established an empirical formula of JRC, Z

_{2}and Δx.

#### 4.3. Non-Equal Interval Sampling Method

_{0}were applied. The coordinate data were recorded and saved if the calculated curvature radius R < R

_{0}. Otherwise, the data of this point were not recorded. The values of the four roughness parameters mentioned above under different curvature radius thresholds were calculated using this method (Figure 9). In the process of reducing the curvature radius threshold R

_{0}, the values of the parameters showed different degrees of reduction. However, the reduction in each parameter was trivial when R

_{0}was between 150 mm and 30 mm, indicating that the roughness parameters are not sensitive to the variation in the curvature radius in this range. Contrarily, the roughness parameters decreased rapidly with the decrease in the curvature radius when R

_{0}< 30 mm.

_{i}is the curvature at the i-th point on the profile line; Δx

_{0}relates to the accuracy of the 3D laser scanner.

_{p}, Z

_{2}and Z

_{2}′ are, respectively, 0.01%, −1.15% and 0.60%, while the angle standard deviation σ

_{i}has an error of 50.10%. The roughness values calculated from the original profile and the non-equidistant profile are in good agreement. Numerous methods, including UAV aerial surveys and three-dimensional laser scanning, can obtain massive point cloud data. The processing of these point cloud data is a huge challenge in terms of computer storage and computing performance. However, the amount of data required by the proposed non-equal interval sampling method is only 54.7% of the original data. In other words, the non-equidistant roughness evaluation method can greatly reduce the amount of data to improve the efficiency of data processing, but has little effect on the overall roughness.

#### 4.4. Roughness Anisotropy

_{AHD}considering the roughness characteristics of different shear directions was proposed by Ban et al. [20]. Using a method similar to that of Ban et al. [20], the anisotropy of the roughness parameter (ARP) is defined as:

_{i}is the roughness for the i-th analysis direction of a specific evaluating parameter, and four parameters R

_{p}, σ

_{i}, Z

_{2}and Z

_{2}′ are used in this paper; RP

_{ave}is the average value of RP

_{i}of all analysis directions, and n is the total number of analysis directions. The roughness anisotropy parameter ARP is composed of RP

_{i}, which can reflect the shear mechanism and takes into consideration the roughness characteristics of the joint surface in all shear directions; thus, it is more reasonable and comprehensive. When ARP = 0, the roughness of the joint surface is isotropic, and when ARP > 0, the roughness of the joint surface is anisotropic. The larger the value is, the more significant the anisotropic character.

_{p}, σ

_{i}, Z

_{2}and Z

_{2}′ are 0.004, 0.078, 0.187 and 0.137, respectively. The calculation results show that Z

_{2}and Z

_{2}′ can accurately reflect the anisotropy of the joint surface, while the anisotropy of R

_{p}is the least obvious, followed by σ

_{i}. It can be seen from Figure 11a that there are several obvious ridges on the joint surface with an attitude of around 50°~70°. The calculated roughness of σ

_{i}, Z

_{2}and Z

_{2}′ in this direction is the smallest, indicating that the resistance is minimal when the joint surface is sheared along the direction of 50°~70°. However, the direction of 80°~140° intersects the direction of the ridge with a large angle, and the calculated roughness is relatively large, indicating that the joint surface receives the greatest resistance when shearing in this direction.

## 5. Shear Failure Mechanism of Rock Joints

_{c}:

_{f}is the internal friction angle of an intact rock.

_{f}is the cohesion of an intact rock and σ

_{n}is the normal stress. m = h/l is the aspect ratio of the local convex area (Figure 12).

_{B}− y

_{A}> 0) and a negative area (y

_{D}− y

_{C}< 0) along the shear direction (Figure 5). The shear stress was 0 on the negative region during the shearing process, i.e., τ = 0. Two failure modes existed on the positive region in the process of shearing (Figure 12). When m is less than m

_{c}, the shear strength increases linearly with m. When m is greater than m

_{c}, the shear strength remains unchanged.

_{f}= 48.81 kPa and φ

_{f}= 38.26°. Figure 13a illustrates that, under any normal stress conditions, an increase in shear stress occurs twice in the process of shearing. The variation in shear stress with shear displacement can be divided into several stages. The shear stresses increased rapidly at the initial stage of the test (OA stage), and they were almost unchanged or only slightly increased at stage AB; the shear stress increased again at stage BC and finally decreased (CD stage).

_{n}= 400 kPa. Shear stress distribution for shearing in the positive direction of the x axis is shown in Figure 14, where white indicates that no shear stress was exerted in this area during the shearing process. The calculation result reveals that the protruding parts on the roughness surface contributed to the shear strength. As mentioned previously, two different failure modes were exhibited on the joint surface during the shearing process. The dilatancy failure area and fracture failure area for shearing in the positive direction of the x axis are shown in Figure 15. Accompanying the appearance of dilatancy, a large area on the rough surface expressed a certain degree of slip (Figure 15a). Stress concentration occurred when the upper and lower blocks of the roughness surface slipped to a certain extent, resulting in shear fracture failure of the protruding parts on the roughness surface due to shear stress greater than their shear strength (Figure 15b).

_{2}′ among the four roughness evaluation parameters selected in this paper can consider the failure mechanism in the case of changing the shearing direction of the rock joints.

## 6. Conclusions

_{p}, σ

_{i}, Z

_{2}and Z

_{2}′, exhibiting an obvious negative size effect. The reason for the scale effect of joint surface roughness is that the regular sampling interval method changes the geometrical morphology of the original profile, thereby ignoring the contribution of secondary fluctuations to roughness.

_{2}′.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Joint surfaces in rock mass. (

**a**) Inclined state of natural rock; (

**b**) Schematic diagram of forces exerted on joint surface.

**Figure 2.**Reasons for non-stationary point clouds. (

**a**) Joint surface with an inclination angle; (

**b**) Non-perpendicular application of light from a 3D laser scanner to the joint surface.

**Figure 3.**Method for removing the inclination features of rock joints. (

**a**) Grid data processed by denoising and minimum curvature interpolation; (

**b**) Fitting plane; (

**c**) Calculating the distance from original point cloud to the fitting plane; (

**d**) Joint without inclined features.

**Figure 6.**Relationships between roughness parameters with sampling interval. (

**a**) R

_{p}; (

**b**) σ

_{i}; (

**c**) Z

_{2}; (

**d**) Z

_{2}′.

**Figure 11.**The anisotropy of roughness. (

**a**) Profiles along circumferential directions; (

**b**) Roughness in radar plot.

**Figure 12.**Failure modes of rock joints during direct shear test. (

**a**) Joint matching; (

**b**) Dilatancy failure; (

**c**) Fracture failure. The aspect ratio of the local convex area is m = h/l, and the dilatancy is δ.

**Figure 13.**Results of direct shear test. (

**a**) The variation in shear stress with shear displacement; (

**b**) The relationship between peak shear strength and normal stress.

**Figure 15.**Two types of failure areas for shearing in the positive direction of the x axis. (

**a**) Dilatancy failure area; (

**b**) Fracture failure area.

**Figure 17.**Two types of failure areas for shearing in the negative direction of the x axis. (

**a**) Dilatancy failure area; (

**b**) Fracture failure area.

Roughness Parameters | R_{p} | Z_{2} | Z_{2}′ | σ_{i} |
---|---|---|---|---|

Original profile | 1.0061 | 0.8934 | 0.4783 | 1.0017 |

Non-equidistant profile | 1.0060 | 0.9036 | 0.4755 | 0.5000 |

Relative errors (%) | 0.01 | −1.15 | 0.60 | 50.10 |

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## Share and Cite

**MDPI and ACS Style**

Chen, F.; Yu, H.; Yang, Y.; Wu, D.
Influences of Roughness Sampling Interval and Anisotropy on Shear Failure Mechanism of Rock Joint Surface. *Energies* **2021**, *14*, 6902.
https://doi.org/10.3390/en14216902

**AMA Style**

Chen F, Yu H, Yang Y, Wu D.
Influences of Roughness Sampling Interval and Anisotropy on Shear Failure Mechanism of Rock Joint Surface. *Energies*. 2021; 14(21):6902.
https://doi.org/10.3390/en14216902

**Chicago/Turabian Style**

Chen, Fan, Hongming Yu, Yilin Yang, and Daoyong Wu.
2021. "Influences of Roughness Sampling Interval and Anisotropy on Shear Failure Mechanism of Rock Joint Surface" *Energies* 14, no. 21: 6902.
https://doi.org/10.3390/en14216902