# Thermal-Structural Coupling Analysis of Subsea Connector Sealing Contact
## 4 Thermal-Structural Coupling Mathematical Model of the Subsea Connector
### Three-Dimensional Stress Caused by the Steady-State Temperature Field
\begin{array}{l}\varepsilon_{xx}=\frac{1-\
u^{2}}{E}\big{(}\sigma_{xx}- \frac{\
u}{1-\
u}\sigma_{yy}\big{)}+(1+\
u)\alpha^{*}\tau\\ \varepsilon_{yy}=\frac{1-\
u^{2}}{E}\sigma_{yy}-\frac{\
u}{1-\
u}\sigma_{xx} \big{)}+(1+\
u)\alpha^{*}\tau\\ \varepsilon_{xy}=\frac{(1+\
u)\sigma_{xy}}{E}\end{array} \right. \tag{17}\]  
By introducing [PERSON]'s thermal stress function \(\chi\), Equation (14) can satisfy  
\[\sigma_{xx}=\frac{\partial^{2}\chi}{\partial y^{2}},\sigma_{yy}=\frac{ \partial^{2}\chi}{\partial x^{2}},\sigma_{xy}=-\frac{\partial^{2}\chi}{ \partial x\partial y} \tag{18}\]  
Because the stress of the connector on the \(z\)-axis cannot be ignored, the plane strain model can be selected as its analysis model. Consideration of Equations (15), (17), and (18) in the plane strain state:  
\[\Delta\Delta\chi=-\frac{\alpha^{*}E}{1-\
u}\Delta\tau=-k\Delta\tau \tag{19}\]  
where \(k=\alpha^{*}E/(1-\
u)\), and  
\[\Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} \tag{20}\]  
According to the simplified sealing structure, \(\Delta\) can be converted into the expression in a cylindrical coordinate system by Equation (20)  
\[\Delta=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r }+\frac{1}{r^{2}}\frac{\partial}{\partial\theta^{2}} \tag{21}\]  
Then, the thermal stress component is:  
\[\left\{ \begin{array}{l}\sigma_{rr}=\frac{1}{r^{2}}\frac{\partial^{2}\chi}{ \partial\theta^{2}}+\frac{1}{r}\frac{\partial\chi}{\partial r}\\ \sigma_{\theta\theta}=\frac{\partial^{2}\chi}{\partial r^{2}}\\ \sigma_{r\theta}=-\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial \chi}{\partial\theta}\right)\end{array} \right. \tag{22}\]  
Since the sealing structure is a multi-connected domain, according to the heating condition of the thick-walled hollow cylinder with an inner diameter \(r_{i}\) and an outer diameter \(r_{o}\), the boundary conditions are as follows:  
\[\left\{ \begin{array}{l}\chi=\frac{\partial\chi}{\partial r}=0\ \ \ \ \ (r=r_{o})\\ \chi=a_{1}x+a_{2}x+a_{3}\\ \frac{\partial\chi}{\partial r}=a_{1}\cos\theta+a_{2}\sin\theta\end{array}  \right\}(r=r_{i}) \tag{23}\]  
where \(a_{1},a_{2},a_{3}\) are unknown constants. In order to determine the three unknown constants, the Michell expression [23] is added and expressed in polar coordinates. When \(r=r_{i}\):  
\[\left\{ \begin{array}{l}x=r\cos\theta\\ y=r\sin\theta\end{array} \right. \tag{24}\]  
Then, the Michell expression can be described as:  
\[\left\{ \begin{array}{l}\int_{0}^{2\pi}\!\left(y\frac{\partial\Delta x}{ \partial\tau}-x\frac{\partial\Delta x}{\partial\theta}\right)rd\theta=-k\int_ {0}^{2\pi}\!\left(y\frac{\partial\tau}{\partial\tau}-x\frac{\partial\tau}{r \partial\theta}\right)rd\theta\\ \int_{0}^{2\pi}\!\left(x\frac{\partial\Delta x}{\partial\tau}-y\frac{\partial \Delta x}{\partial\theta}\right)rd\theta=-k\int_{0}^{2\pi}\!\left(x\frac{ \partial\tau}{\partial\tau}+y\frac{\partial\tau}{r\partial\theta}\right)rd \theta\\ \int_{0}^{2\pi}\!\frac{\partial\Delta x}{\partial\tau}rd\theta=-k\int_{0}^{2 \pi}\!\frac{\partial\tau}{\partial\tau}rd\theta\end{array} \right. \tag{25}\]The general temperature field distribution, \(\tau\), is calculated by Fourier series  
\[\tau(r,\theta)=\sum_{j=0}^{\infty}F_{j}(r)\cos j\theta+\sum_{j=1}^{\infty}G_{j}(r )\sin j\theta \tag{26}\]  
where \(F_{j}(r)\) and \(G_{j}(r)\) are the Fourier coefficients of \(\tau;j=0\), 1, 2.... If \(\tau\) is the harmonic function in the plane, then  
\[\Delta\tau(r,\theta)=0 \tag{27}\]  
Substituting Equation (26) into Equation (27), \(F_{j}\) and \(G_{j}\) must satisfy the following relations:  
\[\left\{ \begin{array}{ll}\frac{1}{r}\frac{d}{dr}\left(r\frac{dF_{j}}{dr} \right)-\frac{j^{2}}{r^{2}}F_{j}=0&(j=0,1,2,\cdots\cdots)\\ \frac{1}{r}\frac{d}{dr}\left(r\frac{dF_{j}}{dr}\right)-\frac{j^{2}}{r^{2}}G_{j }=0&(j=1,2,\cdots\cdots)\end{array} \right.