# 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  
In the working state, the subsea connector is subjected to multiple loads at the same time. Since the sealing structure is regarded as a thick-walled cylinder, the three-dimensional stress caused by the internal pressure load can be obtained from the Lame formula [24]:  
\[\left\{\begin{array}{l}\sigma_{r}^{p}=\frac{{p{r_{o}}^{2}}}{{{r_{i}}^{2}-{r_{ f}}^{2}}}-\frac{{p{r_{i}}^{2}{r_{o}}^{2}}}{{{r_{i}}^{2}-{r_{i}}^{2}}}\frac{1}{{ \tau^{2}}}\\ \sigma_{\theta}^{p}=\frac{{p{r_{o}}^{2}}}{{{r_{i}}^{2}-{r_{i}}^{2}}+\frac{{p{r_{i} }^{2}{r_{o}}^{2}}}{{{r_{i}}^{2}-{r_{o}}^{2}}+\frac{1}{{r_{i}^{2}}-{r_{o}}^{2}}{ {r_{i}}^{2}-{r_{i}}^{2}}}\frac{1}{{\tau^belt is also called the sealing width. In order to correctly describe the sealing performance of the lenticular sealing structure, the sealing width, contact stress mean value, and the contact stress distribution are essential key parameters. Therefore, the sealing mechanics of the lenticular sealing structure should be studied.  
The contact surface between the spherical surface of the lenticular sealing gasket and the conical surface of the hub is expanded around the axis, which can be regarded as the contact between a cylinder and a plane, and it is expressed as a non-conforming surface [27] contacting the \(O_{0}\) point. As shown in Figure 9, the semi-sealing width is \(a\), the radius of the sealing sphere is \(r\), and the elastic modulus and Poisson's ratio of the hub and sealing gasket are \(E_{1}\), \(E_{2}\), \(\mu_{1}\) and \(\mu_{2}\) respectively. Therefore, the equivalent elastic modulus, \(E^{*}\), can be expressed as:  
\[E^{*}=\left(\frac{1-\mu_{1}^{2}}{E_{1}}+\frac{1-\mu_{2}^{2}}{E_{2}}\right)^{-1} \tag{56}\]  
According to Hertz contact [28], for this type of contact stress, it can be expressed as:  
\[q(x_{0})=\left\{ \begin{array}{l}\frac{2P}{\pi a^{2}}(a^{2}-{x_{0}}^{2})^{1/2 },-a\leq x_{0}\leq a\\ 0\,,x_{0}<-a\ \&\ x_{0}>a\end{array} \right. \tag{57}\]  
where \(P\) is the line load, \(P=(\pi E^{*}a^{2})/4r\). Then, the maximum contact stress, \(q_{\max}\), is:  
\[q_{\max}=q(0)=\frac{E^{*}a}{2r} \tag{58}\]  
The mean value of contact stress, \(\overline{q}\), is:  
\[\overline{q}=\frac{\int_{-a}^{a}q(x_{0})dx_{0}}{2a}=\frac{\pi E^{*}a}{8r} \tag{59}\]  
Substituting Equation (59) into Equation (58), it can be seen that:  
\[q_{\max}=\frac{4\overline{q}}{\pi} \tag{60}\]  
Meanwhile, the semi-sealing width, \(a\), is:  
\[a=\frac{8\overline{q}r}{\pi E^{*}} \tag{61}\]  
Figure 9: Contact principle diagram of the seal gasket. In the sealing width, any segment of the micro-unit is marked as \(ds\). For the distributed force of normal contact subjected to \(q(s)\), the stress at any point (\(x_{0}\),\(z_{0}\)) inside the sealing gasket is:  
\[\left\{ \begin{array}{l}\sigma_{x_{0}}=-\frac{2z_{0}}{\pi}\int_{-a}^{+a}\frac{ q(s)(x_{0}-s)^{2}ds}{\left(\left(x_{0}-s\right)^{2}+z_{0}\right)^{2}}\\ \sigma_{y_{0}}=\mu_{2}(\sigma_{x_{0}}+\sigma_{z_{0}})\\ \sigma_{z_{0}}=-\frac{2z_{0}^{2}}{\pi}\int_{-a}^{+a}\frac{q(s)ds}{\left(\left(x _{0}-s\right)^{2}+z_{0}\right)^{2}}\\ \tau_{x_{0}z_{0}}=-\frac{2z_{0}z^{2}}{\pi}\int_{-a}^{+a}\frac{q(s)(x_{0}-s)ds}{ \left(\left(x_{0}-s\right)^{2}+z_{0}\right)^{2}}\end{array} \right. \tag{62}\]  
For the contact stress distribution shown in Equation (57), on the contact interface, \(\sigma_{x0}=\sigma_{z0}=q(x_{0})\); outside the contact area, the stress components on the surfaces of the two objects are all zero. Substituting \(q(s)=\frac{2P}{\pi^{2}}\left(a^{2}-s^{2}\right)^{1/2}\) into Equation (62) and integrating along the \(z_{0}\)-axis, the principal stress inside the sealing gasket in the contact area can be obtained:  
\[\left\{ \begin{array}{l}\sigma_{x_{0}}=-\frac{q_{\max}}{a}[(a^{2}+2z_{0}{}^{ 2})(a^{2}+z_{0}{}^{2})^{-1/2}-2z_{0}]\\ \sigma_{y_{0}}=-\frac{2z_{0}q_{\max}}{2}[(a^{2}+z_{0}{}^{2})^{1/2}-z_{0}]\\ \sigma_{z_{0}}=-q_{\max}a(a^{2}+z_{0}{}^{2})^{-1/2}\\ \tau_{x_{0}z_{0}}=q_{\max}a[z_{0}-z_{0}{}^{2}(a^{2}-z_{0}{}^{2})^{-1/2}]\end{array} \right.