# Thermal-Structural Coupling Analysis of Subsea Connector Sealing Contact
## 4 Thermal-Structural Coupling Mathematical Model of the Subsea Connector
### Three-Dimensional Stress Caused by Internal Pressure Load
#### 4.3.2 Conversion of Contact Stress
\tag{69}\]  
As shown in Figure 11a, the coordinate system \(O_{0}\)-\(x_{0}y_{0}z_{0}\) is transformed by the rotation matrix \({}_{o}^{0}\)R and the position vector \({}^{\alpha}\)P\({}_{o}\) to obtain the coordinate system \(O\)-\(xyz\), where:  
\[{}_{o}^{\alpha_{0}}\text{R}=\text{R}(y_{0},\frac{\pi}{2}+\alpha)=\left[ \begin{array} []{ccc}\cos(\frac{\pi}{2}+\alpha)&0&\sin(\frac{\pi}{2}+\alpha)\\ 0&1&0\\ -\sin(\frac{\pi}{2}+\alpha)&0&\cos(\frac{\pi}{2}+\alpha)\end{array} \right]= \left[ \begin{array}{ccc}-\sin\alpha&0&\cos\alpha\\ 0&1&0\\ -\cos\alpha&0&-\sin\alpha\end{array} \right] \tag{70}\]  
\[{}^{\alpha_{0}}\text{P}_{o}=\left[ \begin{array}{ccc}\frac{D_{k}}{2}\sin \alpha&0&\frac{D_{k}}{2}\cos\alpha\end{array} \right]^{T} \tag{71}\]  
Then, the transformation relationship is:  
\[{}^{o}\text{P}={}_{o_{1}}^{o}\text{R}\,{}^{\alpha_{1}}\text{P}+{}^{\alpha} \text{P}_{o_{1}} \tag{72}\]  
where \({}^{o}\text{P}=\left[ \begin{array}{ccc}x_{0}&y_{0}&z_{0}\end{array} \right]^ {T}\) is the coordinate description in \(O_{0}\)-\(x_{0}y_{0}z_{0}\); \({}_{o_{1}}^{o_{1}}\text{P}=\left[ \begin{array}{ccc}x&y&z\end{array} \right]^ {T}\) is the coordinate description in \(O\)-\(xyz\). Considering Equations (70)-(72) comprehensively, we can obtain:  
\[\left[ \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\end{array} \right]=\left[ \begin{array}{c}\frac{D_{k}}{2}\sin\alpha-x \sin\alpha+z\cos\alpha\\ y\\ \frac{D_{k}}{2}\cos\alpha-x\cos\alpha-z\sin\alpha\end{array} \right] \tag{73}\]  
The cylindrical coordinate system \(O\)-\(r\theta\)z, as shown in Figure 11b, is established at the origin of the coordinate system \(O\)-\(xyz\), where the \(r\)-axis coincides with the \(x\)-axis. Then, according to Equation (73), it can be obtained that:  
\[\left\{ \begin{array}{c}x_{0}=\frac{D_{k}}{2}\sin\alpha-r\sin\alpha+z\cos \alpha\\ z_{0}=\frac{D_{k}}{2}\cos\alpha-r\cos\alpha-z\sin\alpha\end{array} \right. \tag{74}\]  
In the Cartesian coordinate system \(O\)-\(xyz\) and the cylindrical coordinate system \(O\)-\(r\theta\)z, the \(O_{0}\) point has the same plane stress state, namely:  
\[\left[ \begin{array}{c}\sigma_{r}\\ \sigma_{z}\end{array} \right]=\left[ \begin{array}{c}\sigma_{x}\\ \sigma_{z}\end{array} \right] \tag{75}\]  
and  
\[\sigma_{\theta}=\mu_{2}(\sigma_{r}+\sigma_{z}) \tag{76}\]  
Figure 11: Coordinate transformation. (**a**) Coordinate system transformed; (**b**) The cylindrical coordinate system. From Equations (69) and (74)-(76), it can be obtained:  
\[\left\{ \begin{array}{l}\sigma_{r}^{0}=-aq_{\max}\cos^{2}\alpha\frac{1}{\sqrt{a^{ 2}+z_{0}^{2}}}-aq_{\max}\sin 2\alpha\left[z_{0}-\frac{z_{0}^{2}}{\sqrt{a^{2}-z_{0} ^{2}}}\right]\\ \quad-a^{-1}q_{\max}\sin^{2}\alpha\left[-2z_{0}+\frac{a^{2}+2z_{0}^{2}}{\sqrt {a^{2}+z_{0}^{2}}}\right]\\ \sigma_{\theta}^{0}=\frac{-2\mu_{2}q_{\max}}{a}\left[\sqrt{a^{2}+z_{0}^{2}}-z_ {0}\right]\\ \sigma_{z}^{0}=-aq_{\max}\sin^{2}\alpha\frac{1}{\sqrt{a^{2}+z_{0}^{2}}}+aq_{ \max}\sin 2\alpha\left[z_{0}-\frac{z_{0}^{2}}{\sqrt{a^{2}-z_{0}^{2}}}\right]\\ \quad-a^{-1}q_{\max}\cos^{2}\alpha\left[-2z_{0}+\frac{a^{2}+2z_{0}^{2}}{\sqrt {a^{2}+z_{0}^{2}}}\right]\end{array} \right. \tag{77}\]  
where \(z_{0}=\frac{D_{k}}{2}\cos\alpha-r\cos\alpha-z\sin\alpha\).