# Study on Mechanical Behavior of a New Three-Dimensional Sensor Under Rock Rheology
## 2 Three-Dimensional borehole Full Stress Monitoring Theoretical Analysis
The proposed stress measurement method is shown in Figure 1.  
According to rock mechanics [32], the distribution stress of borehole surrounding rock is:  
\[\sigma_{\rho}=\frac{\sigma_{x}+\sigma_{y}}{2}\bigg{(}1-\frac{a^{2}}{\rho^{2}} \bigg{)}+\frac{\sigma_{x}-\sigma_{y}}{2}\bigg{(}1-4\frac{a^{2}}{\rho^{2}}+3 \frac{a^{4}}{\rho^{4}}\bigg{)}cos2\theta+\tau_{xy}\bigg{(}1-4\frac{a^{2}}{\rho ^{2}}+3\frac{a^{4}}{\rho^{4}}\bigg{)}sin2\theta \tag{1}\]  
\[\sigma_{\theta}=\frac{\sigma_{x}+\sigma_{y}}{2}\bigg{(}1+\frac{a^{2}}{\rho^{2} }\bigg{)}-\frac{\sigma_{x}-\sigma_{y}}{2}\bigg{(}1+3\frac{a^{4}}{\rho^{4}} \bigg{)}cos2\theta-\tau_{xy}\bigg{(}1+3\frac{a^{4}}{\rho^{4}}\bigg{)}sin2\theta \tag{2}\]  
\[\sigma_{z}^{\prime}=-v\bigg{[}2\big{(}\sigma_{x}-\sigma_{y}\big{)}\frac{a^{2}} {\rho^{2}}cos2\theta+4\tau_{xy}\frac{a^{2}}{\rho^{2}}sin2\theta\bigg{]}+\sigma _{z} \tag{3}\]  
\[\tau_{\rho\theta}=\frac{\sigma_{x}-\sigma_{y}}{2}\bigg{(}1+2\frac{a^{2}}{\rho ^{2}}-3\frac{a^{4}}{\rho^{4}}\bigg{)}sin2\theta+\tau_{xy}\bigg{(}1+2\frac{a^{2 }}{\rho^{2}}-3\frac{a^{4}}{\rho^{4}}\bigg{)}cos2\theta \tag{4}\]  
\[\tau_{\theta Z}=\big{(}-\tau_{zx}sin\theta+\tau_{yz}cos\theta\big{)}\bigg{(}1+ \frac{a^{2}}{\rho^{2}}\bigg{)} \tag{5}\]  
\[\tau_{\theta z}=\big{(}\tau_{zx}cos\theta+\tau_{yz}sin\theta\big{)}\bigg{(}1- \frac{a^{2}}{\rho^{2}}\bigg{)} \tag{6}\]  
The stress of borehole surrounding rock is studied by a cylindrical co-ordinate system, where \(a\) is the radius of the borehole; \(\rho\) is the distance from point to polar axis; \(\theta\) is the angle between the point and the polar axis, and the \(x\)-axis rotates counterclockwise to be positive; \(z\) is the co-ordinate (height) on the polar axis; \(\sigma_{z}^{\prime}\) is the stress component in the polar axis direction of any point in the surrounding rock of the hole affected by excavation. Figure 1: Stress measurement method: (**a**) stress distribution of borehole surrounding rock; (**b**) force direction of total stress meter. In this paper, the stress monitoring meter has eight measuring points, which are located on four mutually symmetrical straight lines, that is, four directions with the same angle as the co-ordinate axis \((x,y,z)\). The measured results of each measuring point are all normal forces, represented by \(S_{ij}\); then, the co-ordinates of the measuring points in the spherical co-ordinate system are:  
\[\left(r,\theta t_{i},\varphi_{j}\right),\ i=1,2,3,4,\ j=1,2 \tag{7}\]  
Here, \(r\) is the radius of the measuring point in the spherical co-ordinates; \(\theta t_{i}\) is the azimuth angle; \(\varphi_{j}\) is the elevation angle, and the stress components at the center of the sphere are \(\sigma_{r}\), \(\sigma_{\theta}\), \(\sigma_{\varphi}\), \(\tau_{r\theta}\), \(\tau_{r\varphi}\) and \(\tau_{\theta\varphi}\). Because the measuring points are arranged in the four directions with the same angle of the spherical co-ordinate axis, the symmetry of the co-ordinate system and the stress component conversion in the spherical co-ordinate system are considered, and the following relationship can be obtained:  
\[\hat{n}_{ij}=\left(\hat{r}_{ij},\hat{\theta}t_{ij},\hat{\varphi}_{ij}\right) \tag{8}\]  
\[\sigma=\left[ \begin{array}{ccc}\sigma_{r}&\tau_{r\theta},&\tau_{r\varphi}\\ \tau_{r\theta},&\sigma_{\theta},&\tau_{\theta\varphi}\\ \tau_{r\varphi}&\tau_{\theta\varphi}&\sigma_{\varphi}\end{array} \right] \tag{9}\]  
\[S_{ij}=\sigma_{r}\left(\hat{r}_{ij}\right)^{2}+\sigma_{\theta r}\left(\hat{ \theta}t_{ij}\right)^{2}+\sigma_{\varphi}\left(\hat{\phi}_{ij}\right)^{2}+2 \tau_{r\theta}\hat{r}_{ij}\hat{\theta}t_{ij}+2\tau_{r\theta}\hat{r}_{ij}\hat{ \phi}_{ij}+2\tau_{\theta\varphi}\hat{\theta}t_{ij}\hat{\phi}_{ij} \tag{10}\]  
\(\hat{n}_{ij}\) is the unit vector of the normal vector on the sphere, where \(\hat{r}_{ij}=\left[ \begin{matrix}1&0&0\end{matrix} \right]\), \(\hat{\theta}t_{ij}=\left[ \begin{matrix}0&1&0\end{matrix} \right]\), and \(\hat{\phi}_{ij}=\left[ \begin{matrix}0&0&1\end{matrix} \right]\).