# Scalability Analyses of Premium Connections Characterized by a Surface Fractal Function
## 2 Analysis of Fractal Surface Contact Behavior
### Analysis of the Contact Behavior of the Full-Size Area  
If the contact cutoff area of a single asperity is represented by \(a^{\prime}\), then the distribution density function of the asperity in the fractal contact region can be expressed as [21]  
\[n(a^{\prime})=\frac{D-1}{2a_{1}^{\prime}}\left(\frac{a_{1}^{\prime}}{a^{\prime} }\right)^{(D-2)/2} \tag{12}\]  
In Equation (12), \(a_{1}^{\prime}\) is the maximum cutoff area of a single asperity. The cutoff area of the contact region of the simplified model is  
\[A_{r}^{\prime}=\int_{0}^{a_{1}^{\prime}}n(a^{\prime})a^{\prime}da^{\prime} \tag{13}\]  
When \(a^{\prime}\leq a_{ncc}^{\prime}\), the contact area has perfect elastic deformation, and the real contact area is  
\[A_{e}=\int_{0}^{a}n_{e}(a)ada=\frac{D-1}{1.5-0.5D}a_{1}^{0.5D-0.5} \tag{14}\]  
The contact load is  
\[F_{te}=\int_{0}^{a_{ncc}}n_{e}(a)F_{e}(a)da=\frac{D-1}{3(2.5-D)}E^{\prime}\pi^{ 0.5D-2}2^{6.5-1.5D}(In\gamma)^{0.5}G^{D-2}a_{1}^{0.5D-0.5}(a_{1}^{2.5-D}-a_{c} ^{2.5-D}) \tag{15}\]  
When \(a_{ncc}^{\prime}\leq a^{\prime}\leq a_{nep}^{\prime}\), the contact area has elastoplastic deformation and the real contact area is  
\[A_{r}=A_{e}+A_{ep}=\int_{0}^{a_{ncc}}n_{e}(a)ada+\int_{a_{ncc}}^{a}n_{ep}(a)ada =\frac{D-1}{2(1.5-0.5D)}a_{1}^{1.5D-0.5} \tag{16}\]  
The contact load is  
\[F_{t}=F_{tr}+F_{trp}=\int_{0}^{a_{ncc}}n_{e}(a)F_{e}(a)da+\int_{a_{ncc}}^{a}n_ {ep}(a)F_{rp}(a)da=\frac{D-1}{3(2.5-D)}E^{\prime}\pi^{0.5D-2}2^{6.5-1.5D}(In \gamma)^{0.5}G^{D-2}a_{1}^{0.5D-0.5}(a_{1}^{2.5-D}-a_{c}^{2.5-D})\]  
\[+\frac{1.395\lambda cyga_{c}^{-n}(D-1)a_{1}^{0.5D-0.5}}{1.5-0.5D+n}(a_{c}^{1.5 -0.5D+n}-a_{p}^{1.5-0.5D+n}) \tag{17}\]  
When \(a^{\prime}\geq a_{nep}^{\prime}\), the contact area underwent complete plastic deformation, resulting in the real contact area:  
\[A_{r}=A_{e}+A_{ep}+A_{p}=\int_{0}^{a_{ncc}}n_{e}(a)ada+\int_{a_{ncc}}^{a_{ep}} n_{e}(a)ada+\int_{0}^{a_{ncc}}n_{e}(a)ada=\frac{D-1}{2(1.5-0.5D)}a_{1} \tag{18}\]  
The contact load is  
\[F_{t}=F_{tr}+F_{trp}+F_{tp}=\int_{0}^{a_{ncc}  
The contact pressure decreases exponentially with an increase in the fractal dimension \(D\), and when the fractal dimension \(D\) exceeds 2.7, the contact pressure is maintained at around 4 \(\times\) 10\({}^{8}\) Pa, and the change in the contact pressure becomes stable. When the fractal dimension \(D\) is less than 2.7, the volume of a single asperity is larger and the proportion of elastic deformation of asperities is higher, so both the contact area and the contact pressure change simultaneously with the fractal dimension \(D\). Based on Equation (20), for the same external load, the contact pressure is inversely proportional to the contact area, resulting in an increase in contact area and a decrease in contact pressure. However, when the fractal dimension \(D\) is greater than 2.7, the volume of a single asperity significantly decreases and leads to complete plastic deformation of the asperities, and as a result, the contact area continues to increase and the contact pressure remains constant. With an increase in the scale coefficient \(G\), the contact area decreases approximately linearly and the contact pressure increases linearly, as shown in Figure 5b. The scale coefficient \(G\) is the parameter that affects the amplitude of the fractal profile, but it does not affect the lateral spacing of the profile [22; 23]. As the scale coefficient \(G\) increases, the surface becomes rougher, which leads to an increase in stiffness of the asperities and a reduction in the ability of elastic deformations of the asperities, ultimately resulting in a decrease in the real contact area. Based on Equation (20), for the same external load, the contact pressure is inversely proportional to the contact area, resulting in an increase in contact area and a decrease in contact pressure. For the same reason, based on Equation (20), the contact pressure is inversely proportional to the contact area, resulting in a decrease in contact area and an increase in contact pressure.