# Scalability Analyses of Premium Connections Characterized by a Surface Fractal Function
## 2 Analysis of Fractal Surface Contact Behavior
### Contact Behavior Analysis of Single Asperities  
At the microscale, the contact on the fractal rough surface may be regarded as contact pairs between the plane composed of asperity and the rigid plane. Assuming that the rough surface is isotropic, it can be concluded that the asperities on the surface which have a hemispherical geometric shape, do not interact with each other. Then, the undeformed profile equation of the asperity in the rectangular plane coordinate system is [20]  
\[z(x)=G^{D-1}\sum_{n=n_{1}}^{\infty}\frac{\cos(2\pi\gamma^{n}\pi+\varphi_{n})}{ \gamma^{(2-D)n}} \tag{2}\]  
The process of an asperity contacting the rigid plane and deforming is shown in Figure 4. As the external load increases, the single asperity will transform from elastic to elastoplastic to perfect plastic in succession. The corresponding radius of curvature can be obtained using the curvature calculation equation [20]:  
\[R_{j,n}=\left|\frac{\left[1+\left(dz(x)/dx^{2}\right)^{3/2}\right]}{d^{2}z(x) /dx^{2}}\right|_{x=L\psi_{j,n}/2\pi\gamma^{n}}=\frac{\left(L/\gamma^{n}\right) ^{(D-1)}}{4\pi^{2}G^{(D-2)}\left(ln\gamma\right)^{1/2}} \tag{3}\]  
When \(\omega\leq\omega_{cc}\), part of the asperity will only undergo elastic deformation [21]. The equation for critical elastic compression is  
\[\omega_{cc}=\left(\pi KH/2E\right)^{2}R \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \(G\) **(m)** & \(M\) & \(\gamma\) & \(L\) **(m)** \\ \hline \(1\times 10^{-13}\) & 10 & 1.5 & \(1\times 10^{-5}\) \\ \hline \end{tabular}
\end{table}
Table 1: Fractal parameters.  
Figure 3: 3D fractal rough surface. (a) D = 2.1; (b) D = 2.5; (c) D = 2.9.  
where \(K\) is the hardness coefficient that is related to [PERSON]'s ratio \(\mu\), \(K=0.454+0.41\mu\); \(E\) is the elastic modulus, in Pa; and \(H\) is the material's hardness, in Pa.  
The corresponding critical elastic area is  
\[a_{ec}=\pi R\omega_{ec} \tag{5}\]  
In the elastic deformation state only, the relationship between the contact area \(a_{e}\), the cutoff area \(a_{e}^{\prime}\) and contact load \(F_{e}\) of the asperities is  
\[a_{e}=\pi R\omega=\frac{a_{e}^{\prime}}{2} \tag{6}\]  
\[F_{e}=\frac{8E\pi^{1/2}G^{(D-2)}(In\gamma)^{1/2}}{3\sqrt{2}(L/\gamma^{n})^{(D -1)}}(a_{e}^{\prime})^{3/2} \tag{7}\]  
When \(\omega_{ec}\leq\omega\leq 76.8\omega_{ec}\), some of the asperities will undergo elastoplastic deformation. The relationship between the contact area and contact load of the asperities is  
\[a_{ep}=0.5(a_{ec}^{\prime})^{-0.1597}(a_{ep}^{\prime})^{1.1597} \tag{8}\]  
\[F_{ep}=\frac{1}{3}KH(a_{ec}^{\prime})^{-0.3801}(a_{ep}^{\prime})^{1.3801} \tag{9}\]  
where \(a_{ec}^{\prime}\) is the critical cutoff area of the elastoplastic deformation stage, \(a_{ec}^{\prime}=\pi R\omega\); \(a_{ep}^{\prime}\) is the critical cutoff area of the plastic deformation stage, \(a_{ep}^{\prime}=153.66a_{ec}^{\prime}\).  
When \(\omega\geq\omega_{\text{epc}}\), perfect plastic deformation will occur. \(\omega_{\text{epc}}\) is the maximum deflection at the stage of elastoplastic deformation. The following relationships can be obtained:  
\[a_{p}=a_{p}^{\prime}=2\pi R\omega \tag{10}\]  
\[F_{p}=Ha_{p}^{\prime} \tag{11}\]  
where \(a_{p}\) and \(a_{p}^{\prime}\) are the contact and cutoff areas of the perfect plastic deformation stage, respectively, and \(F_{p}\) is contact load of the perfect plastic deformation stage.  
Figure 4: Deformation diagram of a single asperity.