# Study on Mechanical Behavior of a New Three-Dimensional Sensor Under Rock Rheology
## 2 Three-Dimensional borehole Full Stress Monitoring Theoretical Analysis
After processing the above equations, we can obtain:  
\[S_{ij}=\hat{n}_{ij}^{T}.\sigma\cdot\hat{n}_{ij} \tag{11}\]  
\[\hat{n}_{ij}=\left(sin\theta_{i}cos\varphi_{j},\ sin\theta_{i}sin\varphi_{i}, cos\theta_{i}\right) \tag{12}\]  
\[S_{ij}=\sigma_{r}\left(sin^{2}\theta_{i}cos^{2}\varphi_{j}\right)+\sigma_{ \theta}\left(sin^{2}\theta_{i}sin^{2}\varphi_{i}\right)+\sigma_{\theta}\left( cos^{2}\theta_{i}\right)+2\tau_{\theta}\left(sin\theta_{i}cos\varphi_{j} \right)(\ sin\theta_{i}sin\varphi_{i}) \tag{13}\]  
Then, \(S_{ij}\) in the above equation can be expressed by the following equation:  
\[S_{ij}=f\left(\sigma_{r},\sigma_{\theta r},\sigma_{\varphi},\ \tau_{r\theta r}, \tau_{r\varphi},\tau_{\theta\varphi}\right) \tag{14}\]  
That is, the normal force \(S_{ij}\) measured at each measuring point can represent six stress components, which can be solved by the least square method:  
\[\left[\sigma\right]=\left(A^{T}A\right)^{-1}A^{T}S \tag{15}\]  
\(S\) is a column vector composed of normal forces obtained from eight measuring points. \(A\) is the coefficient matrix of \(8\times 6\), and the coefficient is calculated by the direction cosine. Finally, the stress component matrix \(\sigma\) is obtained. Therefore, from the generalized Hooke's law, the remaining three normal strain components (\(\epsilon_{r}\), \(\epsilon_{\theta r}\), and \(\epsilon_{\varphi}\)) and three tangential strain components (\(\gamma_{r\theta}\), \(\gamma_{r\varphi}\), and \(\gamma_{\theta\varphi}\)) can be obtained:  
\[ \begin{cases}\epsilon_{r}=\frac{1}{E}\left(\sigma_{r}-v\left(\sigma_{\theta}+ \sigma_{\varphi}\right)\right)\\ \epsilon_{\theta}=\frac{1}{E}\left(\sigma_{\theta}-v\left(\sigma_{r}+\sigma_{ \varphi}\right)\right)\\ \epsilon_{\varphi}=\frac{1}{E}\left(\sigma_{\varphi}-v\left(\sigma_{r}+ \sigma_{\theta}\right)\right)\end{cases}  \tag{16}\]  
\[\gamma_{r\theta}=\frac{\tau_{r\theta}}{G},\ \gamma_{r\varphi}=\frac{\tau_{r\varphi}}{G}, \ \gamma_{\theta\varphi}=\frac{\tau_{\theta\varphi}}{G} \tag{17}\]In the above formula, \(E\) is the elastic modulus of the hole wall rock; \(v\) is the Poisson's ratio of hole wall rock; and \(G\) is the shear modulus, \(G=\frac{E}{2(1+v)}\). The spherical co-ordinate system \(O_{xyz}\) where the stress meter is located, is converted to the spherical co-ordinate system \(O_{xyz}\), where the borehole is located. Assuming that the dip angle is \(V\), there is the following formula:  
\[R_{z}(V)= \begin{bmatrix}1&0&0\\ 0&cosV&-sinV\\ 0&sinV&cosV\end{bmatrix}  \tag{18}\]  
\[ \begin{cases}\sigma^{\prime}=R_{z}(V)\sigma R_{z}(V)^{\mathsf{T}}\\ \epsilon^{\prime}=R_{z}(V)\epsilon R_{z}(V)^{\mathsf{T}}\end{cases}  \tag{19}\]  
\(R_{z}\) is the transformation matrix of rotating \(V\) around the \(z(z^{\prime})\) axis, \(\sigma\prime\) and \(\epsilon^{\prime}\) are the stress tensor and strain tensor of the stress meter in the borehole co-ordinate system, respectively, and the tensor form is the same as that in the original co-ordinate system. By expanding the rotation transformation matrix operation, the specific formula of the stress component in the spherical co-ordinate system of the borehole can be obtained as follows:  
\[\left\{ \begin{aligned} \sigma^{\prime}{}_{r}=\sigma_{r}\\ \sigma^{\prime}{}_{\theta}=cos^{2}V\sigma_{\theta}+sin^{2}V\sigma_{ \varphi}+2 cosVsinV\tau_{\theta\varphi}\\ \sigma^{\prime}{}_{\varphi}=sin^{2}V\sigma_{\theta}+cos^{2}V\sigma_{ \varphi}-2 cosVsinV\tau_{\theta\varphi}\\ \tau^{\prime}{}_{\theta}=cosV\tau_{\theta\theta}-sinV\tau_{\tau\varphi}\\ \tau^{\prime}{}_{\varphi\theta}=sinV\tau_{\theta\theta}+cosV\tau_{\tau\varphi }\\ \tau^{\prime}{}_{\theta\varphi}=\left(cos^{2}V-sin^{2}V\right)\tau_{\theta \varphi}+cosVsinV\big{(}\sigma_{\varphi}-\sigma_{\theta}\big{)}\end{aligned} \right. \tag{20}\]  
The spherical co-ordinate system is converted into a cylindrical co-ordinate system (expressed in rectangular co-ordinates) that is consistent with the borehole co-ordinate system as follows:  
\[\left\{