# 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  
According to the characteristics of the sealing structure of the subsea connector, the sealing structure can be simplified to a thick-walled cylinder when the temperature field achieves a steady-state distribution, and the stress state at any point inside can be determined by three-dimensional stress, which includes the circumferential stress \(\sigma_{\theta}\), axial stress \(\sigma_{z}\), and radial stress \(\sigma_{r}\) in the cylindrical coordinate system [22]. In order to use the generalized Hooke law and the thermal stress function method to calculate thermal stress, the following assumptions need to be made for the structure. Suppose that (1) the material selected for the sealing structure of the connector is isotropic; (2) the connector is only subjected to the internal oil and gas pressure, and there is no other external load; (3) there is steady-state thermal conduction among the components of the connector sealing structure; (4) the length of the top hub and the bottom hub is infinite and the constraints on the hub, such as a jumper, are ignored; (5) the connector sealing structure satisfies the generalized Hooke law and small-deformation theory. Usually, when the temperature of an object rises, it expands. Assuming that the change in temperature is \(\tau\) (\(\tau=T-T_{0}\): \(T_{0}\) is the initial temperature; \(T\) is the final temperature. ), the expansion of any segment of the micro-body is not limited, and the strain of the isotropic body under free expansion is [23]  
\[\left\{ \begin{array}{l}\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon_{zz}= \alpha^{*}\tau\\ \varepsilon_{xy}=\varepsilon_{yx}=\varepsilon_{zx}=0\end{array} \right. \tag{12}\]  
where \(\alpha^{*}\) is the thermal expansion coefficient. However, each element is not free. There must be mutual restraint between them, so thermal stress must be generated. The strain of each element is caused by the combined action of the change of temperature and stress. For the plane strain problem, the relationship between strain and stress under the action of temperature can be expressed by [PERSON]'s law [23] as  
\[\left\{ \begin{array}{l}\varepsilon_{xx}=\frac{1+\
u}{\zeta_{xx}}\left\{ \sigma_{xx}-\frac{\
u}{1+\
u}\left(\sigma_{xx}+\sigma_{yy}+\sigma_{zz}\right) \right\}+\alpha^{*}\tau\\ \varepsilon_{yy}=\frac{1+\
u}{\zeta_{yy}-\frac{\
u}{1+\
u}\left(\sigma_{xx}+ \sigma_{yy}+\sigma_{zz}\right)\right\}+\alpha^{*}\tau\\ \varepsilon_{zz}=\frac{1+\
u}{\varepsilon_{zz}}\left\{\sigma_{zz}-\frac{\
u}{ 1+\
u}\left(\sigma_{xx}+\sigma_{yy}+\sigma_{zz}\right)\right\}+\alpha^{*} \tau\\ \varepsilon_{xy}=\frac{\left(1+\
u\right)\sigma_{yy}}{E},\varepsilon_{yz}= \frac{\left(1+\
u\right)\sigma_{yy}}{E},\varepsilon_{zx}=\frac{\left(1+\
u \right)\sigma_{yz}}{E}\end{array} \right. \tag{13}\]  
where \(\
u\) is Poisson's ratio; \(E\) is the elastic modulus. For a plane problem, if the coordinates are \(x\) and \(y\), which are independent variables and do not consider the volume force, then the balance equation can be written as:  
\[\left\{ \begin{array}{l}\frac{\partial\sigma_{xx}}{\partial x}+\frac{\partial \sigma_{yy}}{\partial y}=0\\ \frac{\partial\sigma_{yy}}{\partial x}+\frac{\partial\sigma_{yy}}{\partial y}=0 \end{array} \right. \tag{14}\]  
Then, the compatibility equation of the strain can be expressed as:  
\[\frac{\partial^{2}\varepsilon_{xx}}{\partial x^{2}}+\frac{\partial^{2}\varepsilon _{yy}}{\partial y^{2}}=2\frac{\partial^{2}\varepsilon_{xy}}{\partial x \partial y} \tag{15}\]  
In the plane strain state, it can be obtained by \(\varepsilon_{zz}=0\) in Equation (13)  
\[\sigma_{zz}=\
u(\sigma_{xx}+\sigma_{yy})-\alpha^{*}E\tau \tag{16}\]By plugging Equation (16) into Equation (13), the relationship between strain and stress in the plane strain state can be expressed as:  
\[\left\{