# Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities
## 2 A New Fractal Characterization Approach for a Rough Surface Texture
### Contact Mechanics Model of Single Elliptical Asperity
#### 3.2.2 Elastic-Plastic Contact of Elliptical Asperity  
As shown in Figure 5, [PERSON] and [PERSON] [7] investigated the mechanism of elastic-plastic deformation occurring in asperities on rough surfaces by performing a finite element contact analysis of a single elastic sphere with a smooth rigid plane; the results of the study showed that the asperities are in elastic-plastic deformation when the actual deformation \(\omega_{n}\) of the asperities on the rough surface is greater than the critical elastic deformation \(\omega_{nec}\) and less than or equal to 110 \(\omega_{nec}\). In addition, a large number of simulation results have proven that the elastic-plastic deformation is divided into two stages: the asperities are in the first elastic-plastic deformation stage when the actual deformation \(\omega_{n}\) is greater than the critical elastic deformation \(\omega_{nec}\) and less than or equal to 6 \(\omega_{nec}\), and the asperities are in the second elastic-plastic deformation stage when the actual deformation \(\omega_{n}\) is greater than the critical elastic deformation 6 \(\omega_{nec}\) and less than or equal to 110 \(\omega_{nec}\). 6 \(\omega_{nec}\) and 110 \(\omega_{nec}\) are defined as the critical elastic-plastic deformation and critical plastic deformation, respectively, and their expressions can be expressed as follows, respectively:  
\[\omega_{nec}=6\omega_{nec}=6K(e)E(e)R_{m}\bigg{(}\frac{KH}{E^{\prime}}\bigg{)}^ {2} \tag{29}\]\[\omega_{npc}=110\omega_{ncc}=110K(e)E(e)R_{m}\bigg{(}\frac{KH}{E^{\prime}}\bigg{)}^ {2} \tag{30}\]  
Contact mechanics demands that the contact process of the asperity should be continuous. Therefore, the elliptical asperity is in the first elastic-plastic deformation when the real contact area satisfies \(a_{ncc}\geq a_{n}\geq a_{npc}\), and the elliptical asperity is in the second elastic-plastic deformation when the real contact area satisfies \(a_{npc}\geq a_{n}\geq a_{npc}\). The critical elastic-plastic contact area \(a_{npc}\) and the critical plastic contact area \(a_{npc}\) can be expressed as:  
\[a_{npc}\left(\frac{2^{(3-D_{1})}\pi^{\frac{D_{1}}{2}}G_{1}{}^{(D_{1}-1)}(\ln \gamma_{1})^{\frac{1}{4}}}{(1-e^{2})^{-\frac{D_{1}}{4}}}a_{ncc}{}^{-\frac{D_{1 }}{2}}+\frac{2^{(3-D_{2})}\pi^{\frac{D_{2}}{2}}G_{2}{}^{(D_{2}-1)}(\ln\gamma_ {2})^{\frac{1}{4}}}{(1-e^{2})^{\frac{D_{2}}{2}}}a_{ncc}{}^{-\frac{D_{2}}{2}} \right)^{2}=4f_{1}(e)\pi K(e)E(e)\bigg{(}\frac{KH}{E^{\prime}}\bigg{)}^{2} \tag{31}\]  
\[a_{ncc}\left(\frac{2^{(3-D_{1})}\pi^{\frac{D_{1}}{2}}G_{1}{}^{(D_{1}-1)}(\ln \gamma_{1})^{\frac{1}{4}}}{(1-e^{2})^{-\frac{D_{1}}{4}}}a_{ncc}{}^{-\frac{D_{1 }}{2}}+\frac{2^{(3-D_{2})}\pi^{\frac{D_{2}}{2}}G_{2}{}^{(D_{2}-1)}(\ln\gamma_ {2})^{\frac{1}{4}}}{(1-e^{2})^{\frac{D_{2}}{2}}}a_{ncc}{}^{-\frac{D_{2}}{2}} \right)^{2}=4f_{1}(e)\pi K(e)E(e)\bigg{(}\frac{KH}{E^{\prime}}\bigg{)}^{2} \tag{32}\]  
The expressions of the contact area and contact load when the asperity is in the first elastic-plastic deformation can be expressed as follows, respectively:  
\[\frac{a_{nep1}}{a_{ncc}}=m_{nep1}\bigg{(}\frac{\omega_{n}}{\omega_{ncc}}\bigg{)} ^{n_{nep1}} \tag{33}\]  
\[\frac{F_{nep1}}{F_{ncc}}=c_{nep1}\bigg{(}\frac{\omega_{n}}{\omega_{ncc}}\bigg{)} ^{d_{nep1}} \tag{34}\]  
where \(m_{nep1}\) = 0.93, \(n_{ep1}\) = 1.136, \(c_{ep1}\) = 1.03, \(d_{ep1}\) = 1.425.  
The expressions of the contact area and contact load when the asperity is in the second elastic-plastic deformation can be expressed as follows, respectively:  
\[\frac{a_{nep2}}{a_{ncc}}=m_{nep2}\bigg{(}\frac{\omega_{n}}{\omega_{ncc}}\bigg{)} ^{n_{nep2}} \tag{35}\]  
\[\frac{F_{nep2}}{F_{ncc}}=c_{nep2}\bigg{(}\frac{\omega_{n}}{\omega_{ncc}}\bigg{)} ^{d_{nep2}} \tag{36}\]  
where \(m_{nep2}\) = 0.94, \(n_{ep2}\) = 1.146, \(c_{ep2}\) = 1.40, \(d_{ep2}\) = 1.263.