# Contact Analysis and Friction Prediction of Non-Gaussian Random Surfaces
## 2 Methodology
### Friction Model  
Friction is a mechanical resistance for transmitting tangential forces when one surface moves over another. At the micrometer level, the engaging surfaces always exhibit protrusions and depressions, and mechanical contact occurs primarily on the asperity summits. At present, the topography of the contact surfaces is considered to influence the friction behavior at the macroscopic scale. In this section, a friction coefficient prediction model is established to investigate the effect of micro-rough interactions on the friction behavior. When rough surfaces come into contact, the applied load is shared by protrusions. Therefore, the overall friction coefficient determined by the ratio of the sum of all asperity frictions to the normal force can be expressed as  
\[\mu=\frac{\sum_{i=1}^{n}f_{i}}{W}=\frac{\sum_{i=1}^{n}\mu_{i}w_{i}}{W}, \tag{11}\]  
where \(W\) indicates the applied normal load; and \(\mu_{i}\), \(f_{i}\), and \(w_{i}\) represent the friction coefficient, friction force, and normal load of individual asperities, respectively. The macroscale friction behavior of the engaging surfaces can be characterized by investigating the friction mechanics of single asperities. To address the behaviors of a single asperity, they were modeled as perfect protuberances of conical shapes for convenience. During the contact of two asperities at the microscale, adhesion and deformation (plowing) at asperity summits collectively contribute to friction. Figure 2 depicts a schematic of the single-asperity friction model. Figure 1: The flowchart of the generation of the non-Gaussian surface. In the case of elastic contact, the contribution of asperity elastic deformation to friction can be negligible. Therefore, the overall friction force is equivalent to the interfacial adhesion force, which can be expressed as  
\[f_{a}=\tau A_{a} \tag{12}\]  
where \(A_{a}\) denotes the interfacial contact width, and \(\tau\) indicates the interfacial shear strength, which varies with the contact pressure. [PERSON] et al. [41] conducted boundary-layer shearing experiments under different lubricant conditions. They determined the shear strength at different nominal pressures, which can be calculated using \(\tau=c_{p}P_{m}^{m}\), where \(c_{p}\) and \(n\) are model parameters, and \(P_{m}\) denotes the maximum contact pressure in the contact width \(A_{a}\). Furthermore, \(c_{p}=2.05\) and \(n=0.88\) were used for the unlubricated condition. Consequently, the friction coefficient of the elastic contact can be expressed as  
\[\mu=\frac{f_{a}}{w_{i}}=\frac{c_{p}A_{a}P_{m}^{m}}{w_{i}} \tag{13}\]  
Plastic deformation occurs when the contact pressure exceeds 1.1 \(Y\); in other words, \(P\geq 1.1Y\), where \(Y\) denotes the yield stress that can be estimated as \(Y=\frac{H}{3}\). Herein, \(H\) denotes the hardness of the material. Owing to the appearance of plastic deformation, part of the deformed asperity was assumed to penetrate the substrate. Under this condition, the following effect of asperity combined with interfacial adhesion contributed to the friction force during sliding. The friction coefficient formula proposed by [PERSON] et al. [42] is presented in Equation (14). This was obtained from the conical model to estimate the contributions of both adhesion and flowing mechanisms to the total friction force:  
\[\mu=\frac{2}{\pi}\Big{[}tan\theta+\frac{1}{6}\Big{(}\frac{\tau}{\tau_{m}} \Big{)}sec\theta\Big{]}, \tag{14}\]  
where \(\theta\) denotes the attack angle, which is determined by the penetration depth \(h\) and contact width \(A_{p}\). \(\tau\) and \(\tau_{m}\) indicate the interfacial and metal shear strengths, respectively. Therefore, the ratio \(\frac{\tau}{\tau_{m}}\) can assume values between zero and unity depending on the interfacial conditions. In this study, an intermediate value of 0.6 was adopted in the elastic-plastic interaction. Complete plastic deformation occurred when the contact pressure approached material hardness. The contribution of interfacial adhesion to friction was negligible.