# Scalability Analyses of Premium Connections Characterized by a Surface Fractal Function
## 3 Contact Analysis of Premium Connections
### Establishment of the Fractal Model of a Premium Connection  
A P110 cone-cone premium connection (\(\Phi\)8.89 \(\times\) 10\({}^{-2}\) m \(\times\) 6.45 \(\times\) 10\({}^{-3}\) m) is taken as the example for analysis. As shown in Figure 6, the sealing surface taper is 1:2, the bearing surface angle of the thread is \(-\)3\({}^{\circ}\), the guide surface angle of the thread and the torque shoulder angle are 10\({}^{\circ}\), and the thread cone angle is 1:16. SolidWorks software was utilized to established a 3D fractal rough surface for the sealing surfaces.  
Through use of ABAQUS software, the fractal profile of the sealing surface was imported into the full-size FEM of the premium connection for preprocessing, as shown in Figure 7. The elastic modulus \(E\) of the premium connection was 2.06 \(\times\) 10\({}^{11}\) Pa, the Poisson's ratio \(\mu\) was 0.3, the yield strength was 7.58 \(\times\) 10\({}^{8}\) Pa, and the ultimate strength was 8.35 \(\times\) 10\({}^{8}\) Pa [24; 25]. Due to the thread compound, the friction coefficient between the contact pairs is 0.02 [26].  
In order to improve the calculation and the accuracy of calculation while controlling the convergence rate, the CAX4R element, which is the axisymmetric subtraction of an integral element, was used when meshing. Because the sealing surface and thread are involved in contact analysis, refinement of the local grid was carried out when meshing to improve the calculation accuracy. Meanwhile, because the sealing surface is a rough surface generated by the fractal function, its profile is composed of a large number of asperities. In order to avoid the calculation failing due to mesh distortion, the fractal surface of the sealing surface is again refined [27; 28].  
Figure 6: Construction of the cone–cone premium connection.  
Two contact pairs were established in the FEM of the premium connection, namely the sealing surface-torque shoulder contact pair and the thread contact pair. According to the criteria of the contact pair in ABAQUS software, the box tip with the greater hardness was set as the main contact surface, and the pin tip was set as the lesser contact surface. According to the existing research, the sealing surface will have slight plastic deformation when the premium connections make up [29]. Therefore, the FEM needs to define the plastic parameters of the material. When setting the material parameters, ABAQUS software needs to convert the nominal stresses and strains obtained from tests into real stresses and strains. Therefore, the following transformation relationships should be used:  
\[\varepsilon=In(1+\varepsilon_{nom}) \tag{21}\]  
\[\sigma=\sigma_{nom}(1+\varepsilon_{nom}) \tag{22}\]  
where \(\varepsilon\) is the real strain, \(\sigma\) is the real stress (in Pa), \(\varepsilon_{nom}\) is the nominal strain and \(\sigma_{nom}\) is the nominal stress (in Pa).  
The boundary conditions and load settings of the FEM are shown in Figure 8. Because the premium connection has an axisymmetric structure, the position of the radial axis at the box tip is constrained at UX, UY, RX and RY. To simulate the downhole loads, axial tension is applied to the pin tip, and internal pressure to its inner wall.