## 2 Deformation Characteristics of Rock Joint  
### Basic Equations  
As shown in Figure 2, the bolted rock joint at the compression-shear side was taken as the research object, and its dimensions in \(x\) and \(y\) directions are much larger than those in the \(z\) direction. Thus, rock joints at the compression-shear side can be regarded as an infinite plane, and the bolt can be considered as a rigid disk in the infinite plane. According to the stress analysis of the bolted rock joint, in addition to the impact of the shear force \(F\) on the bolt on plane \(x=0\) in Figure 2, shear stress also exists on the surface of the rock block. Therefore, the basic equation of the bolted rock joint in polar coordinates (excluding body force) is  
\[\left\{ \begin{array}{l}\frac{\partial\sigma_{r}}{\partial r}+\frac{1}{r} \frac{\partial\tau_{\theta}}{\partial\theta}+\frac{\sigma_{r}-\sigma_{\theta} }{r}=0\\ \frac{1}{r}\frac{\partial\sigma_{\theta}}{\partial\theta}+\frac{\partial\tau _{r\theta}}{\partial r}+\frac{2\tau_{r\theta}}{r}=0\end{array} \right. \tag{1}\]  
In the formula, \(r\) is the radial distance between the point and bolt center, mm; \(\sigma_{r},\sigma_{\theta}\), and \(\tau_{r\theta}\) are, respectively, the radial normal stress, tangential normal stress, and shear stress of rock joints in polar coordinates, MPa. They are all unknowns, and the conditions of geometry and physics must be applied to solve them. The geometric relationship between the displacement and strain in polar coordinates can be expressed as  
\[\left\{ \begin{array}{l}\varepsilon_{r}=\frac{\partial u_{r}}{\partial r}\\ \varepsilon_{\theta}=\frac{u_{r}}{r}+\frac{1}{r}\frac{\partial u_{\theta}}{ \partial\theta}\\ \gamma_{r\theta}=\frac{1}{r}\frac{\partial u_{r}}{\partial\theta}+\frac{ \partial u_{\theta}}{\partial r}-\frac{u_{\theta}}{r}\end{array} \right. \tag{2}\]  
where \(u_{r}\) and \(u_{\theta}\) are, respectively, the radial and circumferential displacements of rock joints, mm. \(\varepsilon_{r},\varepsilon_{\theta}\), and \(\gamma_{r\theta}\) are the radial strain, circumferential strain, and shear strain of rock joints, respectively. Considering that the size of the microunits on the bolted rock joint in \(x\) and \(y\) directions is much larger than that in the \(z\) direction, and the external forces received are parallel to  
Figure 2: Calculation model of bolted rock joints. the rock joint and uniformly distributed around the joint, it can be regarded as a plane stress problem. The stress-strain relationship in polar coordinates can be expressed as  
\[\left\{ \begin{array}{l}\varepsilon_{r}=\dfrac{(\sigma_{r}-\mu\sigma_{\theta})}{E }\\ \varepsilon_{\theta}=\dfrac{(\sigma_{\theta}-\mu\sigma_{r})}{E}\\ \gamma_{r\theta}=\dfrac{\tau_{r\theta}}{G}=\dfrac{2(1+\mu)\tau_{r\theta}}{E} \end{array} \right. \tag{3}\]  
where \(\mu\) is the Poisson's ratio of rock mass, \(E\) is the elastic modulus, and \(G\) is the shear modulus of rock. The basic solution of the radial stress of rock joints at the compression-shear side can be deduced according to the Lame solution [32] in elastic mechanics, as shown in Figure 3 and Equation (4). \[\sigma_{r}=-\dfrac{R^{2}}{r^{2}}q_{1},\sigma_{\theta}=\dfrac{R^{2}}{r^{2}}q_{1} \tag{4}\]  
In Equation (4), \(R\) is the inner radius of the  
Substituting the stress \(\sigma_{r}\) and \(\tau_{r\theta}\) expressions assumed by Equation (5) into the left side of Equation (6) and letting \(r=R\) in Equation (5) after integration, Equation (7) can be obtained. \[-\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{R^{2}}{\pi r^{3}}F\cos^{2}\theta rd \theta-\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{R^{2}}{\pi r^{3}}F\sin^{2} \theta rd\theta\]  
\[=-\frac{F}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\sin^{2}\theta+\cos^{2} \theta)d\theta\]  
\[=-\frac{F}{\pi}(\frac{\pi}{2}+\frac{\pi}{2})=-F\]  
The value on the right side of Equation (7) is equal to the value on the right side of Equation (6). Therefore, Equation (5) satisfies the condition that the stress of the contact part between the bolt and joint is equal.