# Scalability Analyses of Premium Connections Characterized by a Surface Fractal Function
## 5 Conclusions  
On the basis of fractal contact theory, the impact of fractal parameters on the contact behavior of a sealing surface was assessed. The rough surface profile and the deformation of asperities on the sealing surface of a premium connection were taken into account. Through the use of ABAQUS software and contact strength theory, the fractal FEM of a full-size premium connection was established, and the sealability of the premium connection under axial tension and internal pressure was analyzed. The following conclusions can be drawn from this study. Compared with the scale coefficient \(G\), the influence of fractal dimension \(D\) on contact area and contact pressure is more significant. With the increase in fractal dimension, the number of asperities on the fractal surface increases, and the contact area exhibits exponential growth, while the contact pressure decreases exponentially. When the fractal dimension \(D\) is less than 2.5, the maximum Von Mises stress is 8.81 \(\times\) 10\({}^{8}\) Pa and the maximum contact pressure on the sealing surface is 1.20 \(\times\) 10\({}^{9}\) Pa, making it prone to gluing and ultimately leading to sealing failure. Conversely, when the fractal dimension \(D\) is greater than 2.7, the contact pressure distribution on the sealing surface is more uniform, which improves sealability. As the axial tension increases, stress concentration in the area gradually shifts from the sealing surface to the threaded portion, resulting in a reduction in the contact pressure and the effective contact length of the sealing surface. When the axial tension reaches 1.2 \(\times\) 10\({}^{6}\) N, the sealing surface experiences significant displacement along the axial direction, the effective contact length is reduced from 2.72 \(\times\) 10\({}^{-3}\) m to 2.24 \(\times\) 10\({}^{-3}\) m, and the maximum contact pressure is reduced from 8.10 \(\times\) 10\({}^{8}\) Pa to 6.39 \(\times\) 10\({}^{8}\) Pa, which leads to a 30% decrease in sealing strength and therefore poses a high risk of sealing failure. As the internal pressure increases, the plastic deformation ratio of the fractal surface asperities significantly increases, leading to a proportional increase in the contact pressure and effective contact length of the sealing surface. When the internal pressure reaches 1 \(\times\) 10\({}^{8}\) Pa, the effective contact length is increased from 9.08 \(\times\) 10\({}^{-3}\) m to 1.06 \(\times\) 10\({}^{-2}\) m, the maximum contact pressure is increased from 8.67 \(\times\) 10\({}^{8}\) Pa to 1.37 \(\times\) 10\({}^{9}\) Pa, and the sealing strength is increased by 23%. At the same time, the maximum Von Mises stress on the sealing surface reached 9 \(\times\) 10\({}^{8}\) Pa, resulting in a significant stress concentration on the sealing surface. In summary, the sealability of the premium connection is primarily affected by the axial tension. Within the range in which the structural integrity is not compromised by internal pressure, increasing the internal pressure and ensuring smoother sealing surfaces can both contribute to maintaining sealability. Therefore, to ensure the sealability of the premium connections, axial tension should be kept below 1.2 \(\times\) 10\({}^{6}\) N, and internal pressure should be limited to below 1 \(\times\) 10\({}^{8}\) Pa. Moreover, to avoid the sealing surfaces gluing during assembly, surface roughness should be minimized during manufacture. Compared to the traditional method of assuming a smooth sealing surface for premium connection, utilizing fractal functions to describe the rough morphology on the sealing surface of premium connections can reveal the contact behavior of the sealing surface more realistically. The premium connections' fractal FEM established by this method is closer to the actual situation, considering the effect of the asperities' elastic-plastic deformation on the sealability.