Estimation of Joint Roughness Coefficient from Three-Dimensional Discontinuity Surface

Appendix

See Figs. 7, 8 and 9.

Fig. 7

3D digital surface model (note: the positive x-direction is set to the zero angle and contra-rotates with the positive z-direction)

Fig. 8

Space relationship of triangle of the unit with the mean plane (notes ABC—part of roughness surface, A’B’C’—the projection of roughness surface in mean plane)

Fig. 9

Calculation of the equivalent length

1.1 The Mean Plane

As shown in Fig. 7a, a 3D surface can be digitized by 3D laser technology. According to the surface size and scanning precision, it can be divided into N + 1 in x-direction and M + 1 in y-direction. Thus, the 3D surface is discretized into N×M units and made up of (N + 1)×(M + 1) points, and each unit contains four points. Any unit made up of points A (i, j), B (i + 1, j), C (i, j + 1) and D (i + 1, j + 1) will have coordinates of (x _{i, j} , y _{i, j} , z _{i, j} ), (x _{i+1, j} , y _{i+1, j} , z _{i+1, j} ), (x _{i, j+1}, y _{i, j+1}, z _{i, j+1}) and (x _{i+1, j+1}, y _{i+1, j+1}, z _{i+1, j+1}), respectively. The unit can be thought of as two triangles of ABC and BCD, as shown in Fig. 7b. The whole 3D surface can be divided into 2(N × M) triangles. Their relationship with the horizontal plane A’B’C’D’ is shown in Fig. 7b, where A’, B’, C’ and D’ are the projection of A, B, C and D in the horizontal plane, respectively. Then, for the whole 3D roughness surface, the mean asperity height can be obtained by:

$$h = \frac{{V_{\text{total}} }}{{L_{x} L_{y} }} = \frac{1}{{L_{x} L_{y} }}\int_{0}^{{L_{x} }} {\int_{0}^{{L_{y} }} {\left| {z - z^{\prime}} \right|{\text{d}}x} {\text{d}}y} = \sum\limits_{i = 1}^{N + 1} {\sum\limits_{j = 1}^{M + 1} {\frac{{V_{(i,j)(i + 1,j)(i,j + 1)} + V_{(i + 1,j)(i,j + 1)(i + 1,j + 1)} }}{{L_{x} L_{y} }}} }$$

(9)

$$V_{(i,j)(i + 1,j)(i,j + 1)} = V_{ABCC'B'A'} ,V_{(i + 1,j)(i,j + 1)(i + 1,j + 1)} = V_{BCDD'C'B'}$$

(10)

where z’ represents the position of the horizontal plane, L _x and L _y are the length in the x-direction and y-direction of roughness surface sample, respectively. The value of h varies by changing the position of the horizontal plane, which is called the mean plane, where the value of h is the minimum. In fact, according to the relative relationship between each triangle and the mean plane, the volume in Fig. 7b can be divided into four conditions as shown in Fig. 8.

As shown in Fig. 8, the volume of each case should be calculated by different formulae because of the different shapes. However, no matter how complex the shape is, it always can be divided into several tetrahedrons, for which the volume can be calculated easily according to the coordinates of four vertexes. Assume there is a tetrahedron with known vertexes, such as (x ₁, y ₁, z ₁), (x ₂, y ₂, z ₂), (x ₃, y ₃, z ₃) and (x ₄, y ₄, z ₄). Its volume can be calculated by

$$V = \frac{1}{6}(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{a} \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{b} ) \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{c}$$

(11)

where $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{a}$, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{b}$ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{c}$ are the vectors of the tetrahedron sides, and they can be calculated by

$$\left\{ \begin{aligned} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{a} = (x_{2} - x_{1} ,y_{2} - y_{1} ,z_{2} - z_{1} ) \hfill \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{b} = (x_{3} - x_{1} ,y_{3} - y_{1} ,z_{3} - z_{1} ) \hfill \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{c} = (x_{4} - x_{1} ,y_{4} - y_{1} ,z_{4} - z_{1} ) \hfill \\ \end{aligned} \right.$$

(12)

In Fig. 8, the first case can be divided into AA’B’C’, ABCC’ and ABB’C’, the second case is made up of AA’B’D, ABB’D, BB’OD and CC’OD, the third is made up of CC’B’D, CBB’D, BB’OD and AA’OD, and the last is made up of AA’C’D, ACC’D, CC’OD and BB’OD. Therefore, the volume of each can be calculated by Eq. (11). Additionally, it is substituted into Eqs. (10) and (9), such that the mean asperity height can be obtained.

In addition, the length of the rock sample in the shear direction should be obtained in order to calculate the parameter, λ, in Eq. (2). However, it is difficult to determine this shear length in reality. Instead, an equivalent value can be used, as shown in Fig. 9, where the ellipse is made up of the end-point of the equivalent length. Then, the formula for the equivalent shear length is given as:

$$L_{\text{eq}} = \frac{{L_{x} L_{y} }}{{\sqrt {(L_{x} \sin \alpha )^{2} + (L_{y} \cos \alpha )^{2} } }}$$

(13)

where L _eq is the equivalent length in the shear direction, and α is the angle between the shear direction and the x-coordinate.

1.2 The Modified Root Mean Square Method

The Root Mean Square is the second step in calculating the roughness index. We will now transfer Eq. (4) to a new form in order to meet the three-dimensional requirement as follows:

$$Z_{2}^{'} = \sqrt {\int {\frac{1}{{L_{x} L_{y} }}\left( {\hbox{max} (0,\frac{{{\text{d}}S_{\tau } }}{{{\text{d}}S_{xy} }})} \right)^{2} {\text{d}}S_{xy} } } = \sqrt {\frac{1}{{L_{x} L_{y} }}\sum\limits_{i = 1}^{2(N \times M)} {\frac{{\left( {\hbox{max} (0,S_{\tau ,i} )} \right)^{2} }}{{S_{xy,i} }}} }$$

(14)

where dS _τ and dS _xy are the projections of a triangle in the shear plane whose normal vector is the shear direction and the horizontal plane, respectively. They can be calculated by the coordinates of three vertexes in the triangle. Taking ABC as an example in Fig. 7b, the inner normal vector of ABC plane can be expressed by (x _n , y _n , z _n ) (here z _n should be a negative), which are calculated by

$$\left\{ \begin{aligned} x_{n} = (y_{i + 1,j} - y_{i,j} )(z_{i,j + 1} - z_{i,j} ) - (z_{i + 1,j} - z_{i,j} )(y_{i,j + 1} - y_{i,j} ) \hfill \\ y_{n} = (z_{i + 1,j} - z_{i,j} )(x_{i,j + 1} - x_{i,j} ) - (x_{i + 1,j} - x_{i,j} )(z_{i,j + 1} - z_{i,j} ) \hfill \\ z_{n} = (x_{i + 1,j} - x_{i,j} )(y_{i,j + 1} - y_{i,j} ) - (y_{i + 1,j} - y_{i,j} )(x_{i,j + 1} - x_{i,j} ) \hfill \\ \end{aligned} \right.$$

(15)

Assuming the shear direction is horizontal, its unit vector might be written as (x _τ , y _τ , 0), where we can calculate x _τ and y _τ as:

$$x_{\tau } = \cos \alpha ,y_{\tau } = \sin \alpha$$

(16)

Therefore, according to the formula for calculating the included angles of two vectors, the angles of the plane ABC (as shown in Fig. 7b) with the shear direction and ABC with the horizontal plane are given by, respectively

$$\left\{ \begin{array}{l} \cos \theta_{\tau } = \frac{{x_{n} \cos \alpha + y_{n} \sin \alpha }}{{\sqrt {x_{n}^{2} + y_{n}^{2} + z_{n}^{2} } }} \hfill \\ \cos \theta_{xy} = \frac{{z_{n} }}{{\sqrt {x_{n}^{2} + y_{n}^{2} + z_{n}^{2} } }} \hfill \\ \end{array} \right.$$

(17)

where θ _τ and θ _xy are the included angles of ABC with the shear plane and the horizontal plane, respectively. Thus, the projected areas of ABC in the shear plane and the horizontal plane can be given by,

$$S_{\tau ,ABC} = S_{ABC} \cos \theta_{\tau } ,S_{xy,ABC} = S_{ABC} \cos \theta_{xy}$$

(18)

where S _ABC is the area of ABC, S _τ,ABC is the projected area of ABC in the shear plane, and S _xy,ABC is the projected area of ABC in the horizontal plane. Similarly, S _τ,i and S _xy,i can be calculated in the same manner.

It should be pointed out that the revised RMS method proposed by Zhang et al. (2014) considers the shear direction. Firstly, the plane does not affect the shear strength calculated by the method proposed in this paper if it reverses to the shear direction. Therefore, those planes are not to be considered in Eq. (14) if the angles of the inner normal vector of those planes with the shear direction are larger than 90°. Furthermore, the projected area of any ABC in the shear plane is considered if the plane faces to the shear direction as given in Eq. (18). The detailed steps for calculating the 3D JRC are as follows:

• 1. Obtain the real joint surface profile, 3D geometry (x _i , y _i , z _i ), by scanning the joint surface shown in Fig. 7a. Then, triangulate the roughness surface.

• 2. According to the shear direction, divide the roughness surface into a rectangle for which the long edge is L _eq and parallel to the shear direction. The short edge is perpendicular to the shear direction as shown in Fig. 7. Then, L _eq is equal to the quadrature length of the inner ellipse of the roughness surface in the shear direction given in Eq. (13).

• 3. Assess every triangle to determine whether it is in reverse or in the shear direction. If the triangles are in the reverse direction, then they are to be disregarded in the analysis. Only the projected areas of the triangle that are in the shear direction and the horizontal plane are to be calculated in Eq. (18).

• 4. Lastly, obtain the modified RMS, Z ₂′, using Eq. (14).

According to the above equations and steps, we implemented the theoretical approach using Compaq Visual FORTRAN 6 to calculate 3D JRC. Furthermore, the 2D JRCs of every roughness surface in different shear direction were calculated using the method proposed by Zhang et al. (2014).