# A Fractal Prediction Model for the Friction Coefficient of Wet Clutch Friction Plates
## 2 Friction Plate Surface Roughness Contact Characteristics Research
### Rough Surface Fractal Prediction Model
The expression of the deformation, \(\delta\), of the micro-convex body in the M-B fractal model is:  
\[\delta=G^{D-1}l^{2-D}=G^{D-1}a^{\frac{(2-D)}{2}} \tag{12}\]  
The magnitude of the critical deformation, \(\delta_{c}\), can be compared when analyzing the deformation state of the micro-convex, and the current state of the micro-convex can be described by establishing the ratio of the current micro-convex deformation to the critical deformation:  
\[\frac{\delta_{c}}{\delta}=\left(\frac{K\sigma_{y}}{2E}\right)^{2}\frac{a^{D-1 }}{G^{2(D-1)}} \tag{13}\]  
In order to facilitate calculation, the deformation area of a micro-convex body in a critical deformation state is defined as the current critical contact area, ac, that is, when \(\delta=\delta_{c}\), the critical contact area ac can be expressed as:  
\[a_{c}=\frac{G^{2}}{\left(\frac{K\sigma_{y}}{2E}\right)^{\frac{2}{(D-1)}}} \tag{14}\]Substituting Equation (13), we obtain:  
\[S(\tau)=\Big{\langle}[Z(x+\tau)-Z(x)]^{2}\Big{\rangle}=G\tau^{4-2D}\]0  
As can be seen from Equation (15), the deformation area of the micro-convex can also describe the deformation state of the micro-convex, when the deformation amount of the micro-convex \(\delta>\delta_{c}\), the deformation area of the micro-convex \(a<a_{c}\), then micro-convex deformation occurs. When the deformation of the micro-convex \(\delta<\delta_{c}\), the contact area of the micro-convex \(a>a_{c}\), then micro-convex elastic deformation occurs. It can be seen that the micro-convex with a deformation area smaller than the critical contact area produces plastic deformation, and the contact point with deformation area larger than the critical contact area produces elastic deformation, which is due to the small curvature radius of the micro-convex with less than the critical contact area, resulting in its critical deformation, \(\delta_{c}\), also being small, and vice versa. Therefore, the M-B model based on fractal theory is more reasonable to some extent. Assuming that the section of a single micro-convex body cut along the horizontal plane is circular, according to the geometric shape of the micro-convex body before deformation:  
\[S(\tau)=\Big{\langle}[Z(x+\tau)-Z(x)]^{2}\Big{\rangle}=G\tau^{4-2D}\]1  
When \(a>a_{c}\), according to the above conclusions, the micro-convex body will undergo elastic deformation. According to the Hertz contact theory, the elastic load generated by the micro-convex body is as follows:  
\[S(\tau)=\Big{\langle}[Z(x+\tau)-Z(x)]^{2}\Big{\rangle}=G\tau^{4-2D}\]2  
By substituting Equations (10) and (16) into Equation (17), the elastic load generated by the micro-convex deformation area can be characterized by:  
\[S(\tau)=\Big{\langle}[Z(x+\tau)-Z(x)]^{2}\Big{\rangle}=G\tau^{4-2D}\]3  
When \(a<a_{c}\), that is, the deformation area of the micro-convex is less than the critical deformation area, micro-convex deformation occurs, and its plastic load is:  
\[S(\tau)=\Big{\langle}[Z(x+\tau)-Z(x)]^{2}\Big{\rangle}=G\tau^{4-2D}\]4  
Assuming that the size distribution of the deformed micro-convex satisfies the function \(n(a)\), if the micro-convex in contact at this time contains not only elastic deformation but also plastic deformation, then there must be a case where the maximum deformation area of the micro-convex is greater than the critical deformation area, that is, \(a_{1}>a_{c}\). In this state, the axial load borne by the micro-convex can be expressed as the sum of all the stress of elastic deformation to the stress of plastic deformation. That is:  
\[S(\tau)=\Big{\langle}[Z(x+\tau)-Z(x)]^{2}\Big{\rangle}=G\tau^{4-2D}\]5  
where the former term is the sum of the bearing capacity of all the micro-convex bodies undergoing elastic deformation, and the latter term is the sum of the stress of all the micro-convex bodies undergoing plastic deformation. [PERSON] [15] revised the contact point distribution function:  
\[n(a)=\left\{ \begin{array}{ll}\frac{D}{2}\psi^{(2-D)/2}a_{l}^{D/2}a^{-(D+2)/ 2},&0<a<a_{l}\\ 0,&a_{l}<a<+\infty\end{array} \right.