# 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  
For the three-dimensional stress obtained under the action of contact stress, the main solution is the three-dimensional stress in the contact area. The structural stress of the lenticular sealing gasket is transformed into the coordinate system, and the structural stress and thermal stress caused by the internal pressure are unified under the same coordinate system. As shown in Figure 10a, the same as in Section 4.3.1, the Cartesian coordinate system \(O_{0}\)-\(x_{0}y_{0}z_{0}\) is established with any point \(O_{0}\) on the contact circle as the origin. In addition, the Cartesian coordinate system \(O\)-\(xyz\) is established with the center of the contact circle as the origin \(O\) and the axis of the sealing gasket as \(z\)-axis;\(\overrightarrow{OO_{0}}\) is the positive direction of the \(x\)-axis. Under the assumption of two-dimensional stress, the micro-body shown in Figure 10b is taken at point \(O_{0}\), and the stress components \(\sigma_{x0}\), \(\sigma_{z0}\) and \(\tau_{x0z0}\)have been calculated in Section 4.3.1 of Chapter 4. Figure 10c is the orthographic projection of the micro-body. Figure 10d is the orthographic projection obtained by intercepting the micro-body with a plane parallel to _Oxy_, and the balance equation can be obtained:  
\[\left\{ \begin{array}{l}\sum F_{x}=\sigma_{z_{0}}\cos\alpha\sin adS_{0}+\tau_{z_ {0}x_{0}}\sin^{2}\alpha dS_{0}-\sigma_{x_{0}}\sin\alpha\cos adS_{0}-\tau_{x_{0} z_{0}}\cos^{2}\alpha dS_{0}+\tau_{zx}dS_{0}=0\\ \sum F_{z}=\sigma_{z_{0}}\sin^{2}\alpha dS_{0}-\tau_{z_{0}x_{0}}\sin\alpha\cos adS _{0}+\sigma_{x_{0}}\cos^{2}\alpha dS_{0}-\tau_{x_{0}z_{0}}\cos\alpha\sin adS _{0}-\sigma_{z}dS_{0}=0\end{array} \right. \tag{66}\]  
Where \(\sigma_{z_{0}}\) is the normal stress on the section of the micro-body; \(\tau_{z_{0}x_{0}}\) is the shear stress on the section of the micro-body; \(dS_{0}\) is the section area of the micro-body. According to the reciprocal theorem of shear stress, \(\tau_{x_{0}z_{0}}\) and \(\tau_{z_{0}x_{0}}\) are equal in value. Equation (66) can be written as:  
\[\left\{ \begin{array}{l}\tau_{zx}=\frac{\sigma_{x_{0}}-\sigma_{z_{0}}}{\sin 2 \alpha}\sin 2\alpha+\tau_{x_{0}z_{0}}\cos 2\alpha\\ \sigma_{z}=\sigma_{x_{0}}\cos^{2}\alpha+\sigma_{z_{0}}\sin^{2}\alpha-\tau_{x _{0}z_{0}}\sin 2\alpha\end{array} \right. \tag{67}\]  
Figure 10e is the orthographic projection obtained by intercepting the micro-body with a plane parallel to _Oyz_. The same can be obtained:  
\[\sigma_{x}=\sigma_{x_{0}}\sin^{2}\alpha+\sigma_{z_{0}}\cos^{2}\alpha+\tau_{x_ {0}z_{0}}\sin 2\alpha \tag{68}\]  
Figure 10: Stress state analysis near the contact position. (**a**) The contact position; (**b**) The micro-body; (**c**) Orthographic projection of the micro-body; (**d**) The orthographic projection obtained by intercepting the micro-body with a plane parallel to _Oxy_. (**e**) The orthographic projection obtained by intercepting the micro-body with a plane parallel to _Oyz_. Substituting Equation (65) into Equation (67) and Equation (68), we can obtain:  
\[\left\{ \begin{array}{l}\sigma_{x}=\ -\frac{q_{\max}}{a}\frac{n_{0}\left(n_{0}^{2 }-z_{0}^{2}\right)\sin 2\alpha+\left(m_{0}-z_{0}\right)\left[n_{0}^{2}+n_{0}^{2 }+\left(m_{0}z_{0}-n_{0}^{2}\right)\cos 2\alpha\right]}{\left(m_{0}^{2}+n_{0} ^{2}\right)}\\ \sigma_{z}=\frac{q_{\max}}{a}\ \frac{n_{0}\left(n_{0}^{2}-z_{0}^{2}\right)\sin 2 \alpha-\left(m_{0}-z_{0}\right)\left[n_{0}^{2}+n_{0}^{2}+\left(n_{0}^{2}-m_{0} z_{0}\right)\cos 2\alpha\right]}{\left(m_{0}^{2}+n_{0}^{2}\right)}\\ \tau_{xz}=\frac{q_{\max}}{a}\frac{n_{0}\left(z_{0}^{2}-n_{0}^{2}\right)\cos 2 \alpha+\left(m_{0}-z_{0}\right)\left(m_{0}z_{0}-n_{0}^{2}\right)\sin 2 \alpha}{m_{0}^{2}+n_{0}^{2}}\end{array} \right.