# 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
\tag{28}\]  
[PERSON]'s thermal stress function \(\chi\) is written in the form of Fourier series  
\[\chi(r,\theta)=\sum_{j=0}^{\infty}f_{j}(r)\cos j\theta+\sum_{j=1}^{\infty}g_{j }(r)\sin j\theta \tag{29}\]  
where \(f_{j}(r)\) and \(g_{j}(r)\) are Fourier coefficients of [PERSON]'s thermal stress function. Equation (29) must satisfy Equations (19)-(25). Substituting \(f_{j}(r)\) of Equation (29) into Equation (19), Equation (19) can be expressed as  
\[f_{j}(^{4})+\frac{2}{r}f_{j}^{(3)}-\frac{1+2j^{2}}{r^{2}}f_{j}^{(2)}+\frac{1+2 j^{2}}{r^{3}}f_{j}^{(1)}+\frac{j^{2}(j^{2}-4)}{r^{4}}f_{j}=0 \tag{30}\]  
Substituting \(f_{j}(r)\) of Equation (29) into Equation (23), Equation (23) can be expressed as:  
\[\left\{ \begin{array}{ll}f_{j}(r_{o})=f_{j}^{(1)}(r_{o})=0\\ f_{j}(r_{i})\cos j\theta=a_{1}r_{i}\cos\theta+a_{2}r_{i}\sin\theta+a_{3}\end{array} \right. \tag{31}\]  
By comparing the corresponding coefficients, the following equation can be obtained:  
\[\left\{ \begin{array}{ll}f_{0}(r_{i})=a_{3},&f_{1}(r_{i})=a_{1}r_{i},f_{j}( r_{i})=0\ (j\geq 2)\\ f_{1}^{(1)}(r_{i})=a_{1},f_{j}^{(1)}(r_{i})=0\ (j\
eq 1)\end{array} \right. \tag{32}\]  
Therefore, for \(F_{j}\) in Equation (26), only the temperature distribution of \(j=0\) and \(j=1\) affects the stress function. Therefore, when \(j=1\), applying the Michell expression to \(F_{j}\) and \(f_{j}\), we obtain:  
\[r^{2}f_{1}(^{3})(r)-3f_{1}^{(1)}(r)+3r^{-1}f_{1}(r)=-k\Big{\{}r^{2}F_{1}^{(1)}( r)-rF_{1}(r)\Big{\}} \tag{33}\]  
When \(j=0\), it can be obtained from Equation (25):  
\[f_{0}^{(3)}(r)-r^{2}f_{0}^{(1)}(r)+r^{-1}f_{0}^{(2)}(r)=-kF_{0}^{(1)}(r) \tag{34}\]  
For \(g_{j}\), when \(j\
eq 1\), \(g_{j}=0\). When \(j=1\), the basic relations are:  
\[g_{1}^{(4)}+2r^{-1}g_{1}^{(3)}-\Big{(}1+2j^{2}\Big{)}r^{-2}g_{1}^{(2)}+\Big{(} 1+2j^{2}\Big{)}r^{-3}g_{1}^{(1)}+j^{2}\Big{(}j^{2}-4\Big{)}r^{-4}g_{1}=0 \tag{35}\]  
According to [PERSON]'s expression, we can obtain:  
\[\left(\frac{1}{r}-\frac{d}{dr}\right)\left[-\frac{g_{1}}{r^{2}}+\frac{1}{r} \frac{d}{dr}\left(r\frac{dg_{1}}{dr}\right)\right]=-k\bigg{(}\frac{1}{r}-\frac {d}{dr}\bigg{)}G_{1}(r) \tag{36}\]To sum up, the thermal stress is calculated for the following two cases:  
\[\tau^{(0)}(r,\theta)=F_{0}(r) \tag{37}\]  
\[\tau^{(1)}(r,\theta)=F_{1}(r)\cos\theta+G_{1}(r)\sin\theta \tag{38}\]  
First of all, for \(j=0\), the temperature distribution is only a function of \(r\), so it will not cause shear stress in the micro-body. The basic relations of the stress function can be obtained from Equation (30):  
\[f_{0}^{(4)}+2r^{-1}f_{0}^{(3)}-r^{-2}f_{0}^{(2)}+r^{-3}f_{0}^{(1)}=0 \tag{39}\]  
The solution of this differential equation is:  
\[f_{0}(r)=C_{1}+C_{2}\ln(r/\tau_{i})+C_{2}(r/\tau_{i})^{2}+C_{4}(r/\tau_{i})^{2} \ln(r/\tau_{i}) \tag{40}\]  
Since \(\tau^{(0)}\) in Equation (37) must satisfy Equation (28), we can obtain:  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dF_{0}}{dr}\right)=0 \tag{41}\]  
whereupon,  
\[F_{0}=K_{0}\ln(r)+K_{1} \tag{42}\]  
where \(K_{0}\), \(K_{1}\) are integral constants. Transformation of the above equation results in the following:  
\[f_{0}^{(3)}(r)-r^{2}f_{0}^{(1)}(r)+r^{-1}f_{0}^{(2)}(r)=-kF_{0}^{(1)}(r) \tag{34}\]0  
where \(K_{0}^{\prime}=K_{0},K_{1}=K_{1}+K_{0}\ln(r_{i})\). By substituting Equation (40) into Equation (22), the stress component of temperature variation along the radial direction is:  
\[\left\{ \begin{array}{l}\sigma_{rr}^{0}=\frac{1}{r}\frac{df_{0}}{dr}=\frac{C _{2}}{r^{2}}+\frac{2C_{3}}{r_{i}^{2}}+\frac{C_{4}}{r_{i}^{2}}\left(1+2\ln \frac{r}{r_{i}}\right)\\ \sigma_{\theta\theta}^{0}=\frac{d^{2}f_{0}}{dr^{2}}=-\frac{C_{2}}{r^{2}}+ \frac{2C_{3}}{r_{i}^{2}}+\frac{C_{4}}{r_{i}^{2}}\left(3+2\ln\frac{r}{r_{i}} \right)\\ \sigma_{\theta}^{0}=0\end{array} \right. \tag{44}\]  
According to the two boundary conditions where \(r=r_{i}\) and \(r=r_{o}\), \(\sigma_{rr}^{0}=0\) and Equation (34), the above constants are determined as:  
\[\left\{