Evaluation method of local failure characteristics for joint based on white light scanning technology

Joint is an important factor affecting the stability of engineering rock mass. To quantitatively analyze the local failure characteristics, the point cloud data of joint before and after shear were obtained by 3D white light scanning technology. Firstly, the problem of cloud surface skew of point cloud data was solved by constructing rotation matrix and translation matrix, and the datum of joint was determined. Then, according to the marked points and the undamaged area, the joint point cloud data before and after shear are aligned, and the point cloud data with equal spacing were obtained by interpolation algorithm. Finally, a 3D description method of joint local failure characteristics is proposed. The proposed method can determine the scope of surface failure of joint, quantitatively analyze the locality of its failure characteristics, provide data support for distinguishing the types of shear failure and offer theoretical basis for further research on shear mechanism.


Introduction
Engineering rock mass generally contains a large number of crisscross joints with different scales, and these joints control the deformation and failure of the rock mass deformation and failure of rock mass (Pollard and Aydin.1988;Wang et al. 2020Wang et al. , 2022Song et al. 2022a). Engineering practices have proved that the instability of engineering rock mass is directly related to the failure of joint (Barton and Choubey.1977;Ladanyi and Archambault.1969;Han et al. 2022) Therefore, the exploration of the relationship between the surface morphological and failure characteristics of joint and its shear strength is of great significance for the analysis of rock mass stability.
There are two methods to describe the surface morphological characteristics of joint, namely 2D parameter description and 3D parameter description (Ge et al. 2012;Song et al. 2020;Du et al. 2022). 2D parameter description method usually selects multiple cross-sections on the surface of joint and uses the average values of 2D parameters of multiple cross-sections to represent the whole joint surface characteristics. The most common practice is using 2D roughness statistics parameters Z 2 (root mean square of the first deviation of the profile) (Thomas.1981), Rp (El-Soudani. 1978), SF (structure function of the profile) (Sayles and Thomas.1977)and Grasselli parameter * max ∕(C + 1) 2D (Tatone and Grasselli.2010) or fractal dimension D (Carr and Wardner.1987;Lee et al. 1990) to measure the 10 standard JRC (joint roughness coefficient) curves proposed by Barton (Barton.1973), establishing the fitting relationship between them and the JRC values, and then using this fitting Eq. to evaluate the JRC value of the joint profile line (Tatone and Grasselli. 2010;Yang et al. 2001;Yu and Vayssade.1991). However, due to the fact that the surface of real joint is three-dimensional, 2D parameter method is inadequate to evaluate the morphological characteristics of joint, leading to incomplete description of joint surface characteristics. To solve this problem, some scholars put forward 3D roughness algorithms considering the 3D morphological characteristics of joints (Ge et al. 2012;Song et al. 2020;Belem et al. 2000;Grasselli et al. 2002;Cai Y et al. 2018;Tatone and Grasselli. 2009;Chen et al. 2021). For example, Belem proposed to use five parameters, such as 3D average inclination θs, apparent anisotropy Ka, average gradient Z2s of joint surface and surface distortion parameter Ts, to describe the 3D morphological characteristics of joints (Belem et al.2000); Gresselli et al. considered the analysis directivity, dip angle characteristics and division characteristics of joints, and put forward the 3D roughness parameter * max (C + 1) (Grasselli et al. 2002;Tatone and Grasselli. 2009). These 3D parameters can describe the morphological characteristics of joint more comprehensively.
It is necessary to obtain the surface data of joint before describing its surface morphology using 3D parameters. Current measurement equipment for obtaining surface data of joint includes probe profiler (Barton and Choubey.1977;Weissbach.1978;Dong and Balasubramaniam. 2013;Du.1992), laser scanner (Ge et al. 2012;Belem et al. 2000;Grasselli et al. 2002) and white light scanner (Song et al.2017a(Song et al. , 2020(Song et al. , 2021). Probe profiler obtains the surface information of joint through the movement of probe. This method is cheap, but has slow measuring speed and may damage the surface morphology of joint. Laser scanner converts the concave-convex points of joint into optical signals through laser displacement meter, and then converts the optical signals into electrical signal, thus obtaining the point cloud data of joint surface. This method is a non-contact measurement method, which will not damage the structural surface. However, the measurement speed of this method is relatively low as it adopts a point-by-point scanning mode. In contrast, the white light scanning technology obtains the surface information of joint according to the variation of white light interference signals. This method can obtain tens of thousands of point cloud data in a short time, and the scanning accuracy is high. However, due to complex morphological characteristics and different sizes of natural joints, multi-stitching technology is usually needed in the scanning process, which leads to the skew of the point cloud data, so this technology is not suitable for being directly used to analyze the properties of joints (Song et al. 2021). Relevant research has shown that the analysis of joint morphology is related to the selection of its datum plane (Song et al. 2017a(Song et al. , 2022bWong et al. 2021).
The failure characteristics of joint can be analyzed by either direct comparison method or image description method. The direct comparison method mainly infers the shear mechanism by directly observing the shear morphology characteristics (Grasselli et al. 2002;Cai et al. 2018;Tatone and Grasselli. 2009;Brown et al. 1977;Riss et al. 1997) The digital image description method mainly describes the surface characteristics of joint by comparing the changes of gray scale and color depth before and after the shear test (Song et al. 2021;Liu et al. 2022;Shen et al. 2022;Babanouri and Nasab. 2015). These methods can quickly locate the scope and area of joint failure, which are intuitive, but they cannot meet the requirements for quantitative description of the failure characteristics of joint.
To this end, on the basis of obtaining point cloud data of joint by white light scanner, this paper solved the problems of point cloud data skew and inconsistent datum plane of point cloud data before and after shear test of joint by constructing specific rotation matrix and translation matrix, put forward a quantitative description method of local 3D parameters of joint based on calculus, and quantitatively analyzed the difference in failure characteristics of joint before and after shear test.

Selection of datum plane before test
The measurement of joints of rock was completed by the 3D white light scanning system shown in Fig. 1. The system adopts international advanced microstructure white light projection technology and heterodyne multi-frequency phase-shift 3D optical measurement technology, with the advantages including high measurement accuracy (single measurement accuracy ± 0.005 mm), fast measurement speed (single scanning time < 3 S) and strong antiinterference ability. The 3D optical scanning system is mainly composed of 3D white light scanner, scanning control software and post-processing software. The measuring principle of the 3D white light scanner can be described: when measuring, the grating projection device projects a plurality of multi-frequency gratings onto the object to be measured, and two cameras with a certain angle synchronously collect the corresponding images; then, the image is decoded and phase calculated, and the 3D coordinates of pixels in the common viewing area of the two cameras are solved by using stereo matching technology based on triangle measurement principle.
The original point cloud data obtained by 3D white light scanner are established based on the coordinate system inside the equipment. However, due to the limitation of grating projection range, the range of single measurement is certain. To obtain the point cloud data of the whole joint surface, it is often necessary to scan it several times and splice the single point cloud data. Converting the point cloud data measured from different perspectives into a unified coordinate system often leads to a certain angle α between the spliced point cloud data and the X-O-Y plane of the unified coordinate system, as shown in Fig. 2a. In this case, the original point cloud data of joint cannot be directly applied to the subsequent research work. To solve this problem, a method to determine the datum plane of joint is proposed based on the principle of least square method after constructing the corresponding matrix. The specific steps are as follows.
(1) Determining the datum plane of original point cloud data: i. Let the coordinates of any point on the surface of rock joint be (x i , y i , z i ), and the plane equation of datum plane of original point cloud data is shown in Eq. (1); ii. According to the principle of least square method, the datum plane of joint should satisfy that the sum of distances from all points on the surface of joint to datum plane is the smallest. That is, the value of D in Eq. (2) is the smallest; iii. Find the first partial derivatives of Eq.
(2) about a, b and c, respectively, and make their derivatives zero, so that the values of coefficients a, b and c can be obtained by simplification. The calculation process is shown in Eqs. (3)-(8). Then the equation of the datum plane of the original point cloud data can be determined.
(1) z = a + bx + cy (2) Construct the translation matrix M and the rotation matrix R: according to the datum plane equation, construct the translation matrix M and the rotation matrix R, as shown in Eqs. (9), (10).
(3) Rotate and translate the point cloud data of the original joint: (1) construct an N * 4-order matrix B, where N is the number of point clouds. The first, second and third columns of the matrix are the x, y and z values of the point cloud data, respectively, and the fourth column elements are all numerical values "1"; (2) Perform the operation shown in Eq. (11) on matrix B to obtain matrix D. The data in the first three columns of matrix D are the point cloud data of joint under the datum plane.
By using software programming to realize the above steps, and processing the joint in Fig. 2a, the point cloud data of the joint in Fig. 2b under the datum plane can be obtained.
It should be pointed out that for a given rock joint surface, the datum plane determined by the above method is unique, and this method can effectively solve the problem that the original point cloud data obtained by white light scanning cannot be directly analyzed because of the dip angle.

Selection of datum plane after test
Taking joint A shown in Fig. 3a as the research object, several model samples were made by using 3D scanning technology and 3D printing technology. The model production method can prepare joint samples with the same morphological characteristics in batches, and the reliability of this method was verified (Jiang and Song 2018;Song et al. 2022b). The process of sample preparation is shown in Fig. 3a. First, establish 3D digital model based on the joint point cloud data obtained by 3D scanning technology and then import the 3D digital model into the 3D printer to fabricate the 3D printing model of the joint. Finally, take the 3D printing model as the template, and pour the model sample containing the morphological characteristics of joint a using cement and quartz sand as the main materials. The mass ratio of cement, quartz sand and water is 1:1:0.25. In order to understand the mechanical properties of the pouring material, several cylindrical samples with a  Table 1.
Shear tests were carried out on joint model samples under normal stresses of 1.0 MPa and 5.0 mpa. The test results are shown in Fig. 3b, c. As can be seen from Fig. 3c, the local area of the joint surface fails after the shear test. According to the principle of least square method, the datum plane of point cloud data determined above is the minimum distance from all data points to this plane. However, the surface morphology of the joint will change after shear test, which leads to inconsistency of datum plane of the joint before and after the shear test, bringing great challenges to the quantification of the failure characteristics of the joint.
To solve the above problems, a feasible method is to adopt the datum plane before the shear test for the joint after shear test. The specific practices are as follows: (1) Compare the morphological characteristics of joint before and after shear under normal stress of 1.0 MPa, find the area where the joint surface is not damaged during shearing, and record the data points before and after shear in this area as A 11 -A nn and B 11 -B nn (as shown in Fig. 4), respectively, and then use these two groups of data points for least square fitting to align the point cloud data before and after shear.
(2) According to the method proposed in 2.1, the rotation matrix and translation matrix of the joint before the test are determined, based on which the point cloud data of the joint before and after the shear is adjusted and the point cloud data of the joint under the same datum plane can be guaranteed before and after the test.
For the joint under normal stress of 5 MPa as shown in Fig. 5a, the whole surface of joint is almost completely damaged after shear, and it is difficult to visually determine the unfailed area of the surface, so it is impossible to keep the datum plane before and after shear consistent by the above method. To this end, locating points can be introduced to address the above problems. The specific method is as follows (Fig. 5b): Firstly, the joints before and after the shear test are placed on the positioning plate, and four identical positioning marks A, B, C and D are set on the positioning plate; then, the sample is scanned, and the point cloud data of joint including four positioning marks before and after shear are obtained, respectively. After that, select the point cloud data of the four positioning marks before and after shear as the alignment basis, and continue the operations according to the same operation method as above, so that the consistent datum plane of the point cloud data of the joint before and after shear can be guaranteed. Figure 6 shows the point cloud data measured by the 3D white light scanner. It can be found that the data are not equidistant, which is caused by the complex morphology of joint surface and the grating deformation projected by the instrument and the data splicing. However, when analyzing or measuring the morphological characteristics of joints,  the point cloud data are often required to be equidistant (Yang et al. 2001;Yu and Vayssade.1991;Ge et al. 2012;Grasselli et al. 2002;Song et al. 2017b;Cai et al. 2018;Tatone and Grasselli. 2009), because the non-equidistant data will bring certain challenges to the subsequent analysis of joints. To solve this problem, the interpolation method is used to interpolate the irregular point cloud data so as to obtain equidistant point cloud data. At present, there are two main methods to convert irregular point cloud data into equidistant point cloud data. One is one-dimensional interpolation, and the other is two-dimensional interpolation.

Equal-distance processing of point cloud data of joint
Before carrying out one-dimensional interpolation for the point cloud data of joint, rasterization of the joint is required. Specifically, the point cloud data along the X and Y axes are divided according to the required spacing (Fig. 7). Then the interpolation position is selected according to the requirements, and equidistant processing is carried out according to a certain interpolation algorithm. However, due to the irregularity of the point cloud data of the joint, there may be fewer data points on a straight line at a certain interpolation position. If using only the points on the straight line for interpolation, the error between the obtained equidistant data and the original data will be large. To obtain more reliable interpolation results, small data interval measured by the white light scanner is adopted, and the point within a certain range close to the straight line is translated to the straight line as the coordinate of the point on the straight line, thus increasing the number of points on the straight line (Fig. 7). After completion of translation, the local interpolation algorithm is used to interpolate the points on each straight-line segment by segment until the error is acceptable. Generally, the error value is not more than 5%.
Although the equidistant data of obtained by the above-mentioned one-dimensional equidistant interpolation method can keep the authenticity of joint surface information to a certain extent, it has a certain impact on the surface morphology of the joint as it changes the coordinates of the original data of the joint. To over the shortcomings of this method, two-dimensional interpolation can be adopted. Firstly, Delaunay triangulation of the joint is carried out based on the split-merge algorithm of adaptive mesh division (Su et al. 2020;Kumbhar et al. 2012), and then the point cloud data of the joint after triangulation can be obtained, as shown in Fig. 8a. Then the coordinates of any point m in the triangle can be obtained by interpolation algorithm. For instance, as shown in Fig. 8, the coordinate of point E can be obtained by interpolation algorithm using the coordinates of points A and B. Similarly, the coordinate of point F can be obtained by using the coordinates of points Fig. 7 One-dimensional equidistant interpolation of point cloud data A and C, and then the coordinate of point m can be obtained by using the coordinates of points E and F, so that the coordinate of any point in the joint can be obtained.

Description method of local failure characteristics of joint and application
At present, the values of 3D parameters of the joint are determined by the overall morphological characteristics of the joint surface (Belem et al. 2000;Grasselli et al. 2002;Ge et al. 2012;Cai et al. 2018;Tatone and Grasselli. 2009;Song et al. 2020;Chen et al. 2021). Therefore, these parameters can only reflect the overall difference of the 3D surface characteristics of the joint before and after shear, rather than characterizing the local failure characteristics of the joint nor the difference between the failed area and the unfailed area after the test.
To solve this problem, the joint is divided into square micro-analysis units of the same size along the X-axis and Y-axis directions by using the idea of calculus, and the 3D parameters of the analysis unit can be used as the measurement results of the morphological characteristics in this area. Taking the joint shown in Fig. 9 as an example, the interval of the point cloud data of this joint is 0.1 mm, and the size of the divided analysis unit is 1 mm 2 . Therefore, each analysis unit will contain 121 data points, and the information of these data points can be used to evaluate the morphological characteristics of the joint. When analyzing the local morphological characteristics of the joint using the average height of the center line, the average height of the center line of the unit body H Ave can be defined as the average of the distances from all points in the unit to the datum plane of the joint, which can be expressed as Eq. (12): where: Z i,j is the height of the (i,j) th sampling point relative to the datum plane; m and n are the number of points in the x and y directions, respectively.
The above measurement method of joint local morphology can effectively analyze the local failure characteristics of joint after shear. Figure 10a shows the shear failure characteristics of the joint under the normal stress of 0.5 MPa, which shows obvious locality. The area from y = 75 mm to y = 76 mm along the direction parallel to the X-axis of the joint surface is selected as the research area, and the size of each analysis unit is 1 mm 2 , with a total of 150 micro units (Fig. 9a). To visually observe the change of failure morphology of joint, the layered coloring method in map representation is used to show the 3D morphology of joint surface morphology before and after shear, as shown in Fig. 10b, c. By comparison, it can be found that the joint surface of the study area is obviously damaged in the range of 45-55 mm along the shear direction. Using the local 3D parameter description method, the average height of the center line of the study area of the joint before and after shear is calculated, and the calculation results are shown in Fig. 10d. It can be found that the average height of the center line of the joint before and after shear changed dramatically in the area of x = 45-55 mm along the shearing direction, which indicates that the shear failure occurred in this area. There are certain differences in the average height of the center line of the joint in the area of x = 86-105 mm and x = 128-145 mm, which indicates that sliding wear occurred in this area. In addition, in the area of x = 145-150 mm, the average height of the center line of the joint also changed dramatically, which is mainly due to the boundary effect. The above rules are consistent with the failure characteristics shown in Fig. 10a, which indicates that the local 3D parameter description method can well describe the local failure characteristics of joints, determine the scope and degree of failure, and provide data support for identifying the failure types, which is of great significance for analyzing the shear mechanism of joint.

Conclusion
The point cloud data of joint were obtained by white light scanning technology, and the local failure characteristics of joint surface after shear test were described by 3D parameters. The main research results are as follows: (1) Based on the principle of least square method, the problem of point cloud data skew was solved by constructing rotation matrix and translation matrix, and the datum plane of joint was determined.
(2) Using the point cloud data of the unfailed area of the joint or positioning mark, the datum planes of the joint before and after shear were adjusted to be consistent, which lays a solid foundation for quantitative description of the failure characteristics of the joint.
(3) A description method using 3D local parameters of joint was proposed. This method can describe the local failure characteristics of joint, quantitatively analyze the difference of morphological characteristics of joint before and after shear, determine the scope and degree of surface failure of joint and provide data support for distinguishing the types of joint surface failure, laying an important basis for further research on joint shear mechanism from the perspective of shear wear characteristics.