# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 3 The Revised Model of a Single Fractal Asperity Contact  
In order to obtain the mechanical properties of contact between a cylinder-based or sphere-based fractal rough surface and a flat rigid surface, the contact load of single asperity at a microcontact of given interference must be determined first. In this chapter, a revised model of a single fractal asperity contact is presented based the MB model [16].  
Figure 4: Equivalent sphere-based fractal rough surface contacts with a rigid flat surface.  
Figure 3: Simulated three-dimensional sphere-based fractal rough surface (\(D=2.2\), \(G=2.4\times 10^{-3}\), \(U=S=1\)). (**a**) The three-dimensional sphere-based rough surface; (**b**) the section of the sphere-based rough surface.  
The cylinder-based or sphere-based fractal surface is assumed to be isotropic, asperities are far apart so there is no interaction between them, and no bulk deformation occurs during contact. Considering only one ridge in the cylinder-based or sphere-based fractal surface, the height functions of these asperities with a single variable in the cylinder-based and sphere-based fractal surface can be expressed, respectively, as follows [19].  
\[Z_{C}(x)=L_{C}\bigg{(}\frac{G}{L_{C}}\bigg{)}^{(D-2)}(\ln\gamma)^{1/2}\sum_{n=n _{\min}}^{n_{\max}}\gamma^{(D-3)n}\bigg{[}\cos\Psi_{1,n}-\cos\bigg{(}\frac{2 \pi\gamma^{n}x}{L_{C}}-\Psi_{1,n}\bigg{)}\bigg{]} \tag{4}\]  
\[Z_{S}(x)=L_{S}\bigg{(}\frac{G}{L_{S}}\bigg{)}^{(D-2)}(\ln\gamma)^{1/2}\sum_{n=n _{\min}}^{n_{\max}}\gamma^{(D-3)n}\bigg{[}\cos\Psi_{1,1,n}-\cos\bigg{(}\frac{2 \pi\gamma^{n}x}{L_{S}}-\Psi_{1,1,n}\bigg{)}\bigg{]} \tag{5}\]  
where \(L_{C}\) and \(L_{S}\) are sample lengths of cylinder-based and sphere-based fractal rough surfaces, respectively. The relation between the highest frequency index \(n_{\max}\) and lowest frequency index \(n_{\min}\) can be written as follows [20]:  
\[n_{\max}=n_{\min}+\text{int}\bigg{[}\frac{\log(L/L_{a})}{\log\gamma}\bigg{]}(L =L_{C},\,L_{a}=L_{aC}\text{ or }L=L_{S},\,L_{a}=L_{aS}\bigg{)} \tag{6}\]  
where \(\text{int}[\ ]\) is the integer part of the value in the parentheses, and \(L_{a}\) is the cut-off length of the order of approximately six lattice distances. In the present model, the frequency index \(n\) can be written as \(n=\ln(L/l_{n})/\ln\gamma\). Substituting this equation into Equations (4) and (5), the geometry of a single asperity for which the frequency index is \(n\) before deformation is given as follows:  
\[Z_{Cn}(x)=G^{(D-2)}(\ln\gamma)^{1/2}(l_{Cn})^{(3-D)}\bigg{[}\cos\Psi_{1,n}-\cos \bigg{(}\frac{2\pi x}{l_{Cn}}-\Psi_{1,n}\bigg{)}\bigg{]} \tag{7}\]  
\[Z_{Sn}(x)=G^{(D-2)}(\ln\gamma)^{1/2}(l_{Sn})^{(3-D)}\bigg{[}\cos\Psi_{1,1,n}- \cos\bigg{(}\frac{2\pi x}{l_{Sn}}-\Psi_{1,1,n}\bigg{)}\bigg{]} \tag{8}\]  
Figure 5 shows a single asperity for which the frequency index is \(n\) in the equivalent rough surface contacts with a rigid flat surface. The parameter \(l_{n}\) is asperity's base diameter, \(l^{\prime}_{n}\) is the true contact diameter, and \(l^{\prime}{}_{n}\) is the truncation diameter. The asperity height is \(\delta_{n}\), and contact interference is \(\omega_{n}\). Equations (7) and (8) have same form (mathematical structure, variables and parameters, physical meaning), and hence, the asperity height can be expressed as  
\[\delta_{n}=G^{(D-2)}(\ln\gamma)^{1/2}(l_{n})^{(3-D)},\,l_{n}=l_{Cn}\text{ or }l_{n}=l_{Sn} \tag{9}\]  
Figure 5: A single asperity contact model with a rigid flat surface.  
The radius of curvature \(R_{n}\), at the tip of the asperity, is given by  
\[R_{n}=\left|\frac{\left(1+{Z^{\prime}}^{2}\right)^{3/2}}{Z^{\prime}}\right|_{x=0} =\frac{l_{n}^{(D-1)}}{4\pi^{2}G^{(D-2)}(\ln\gamma)^{1/2}} \tag{10}\]