# 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  
#### 3.2.1 Elastic Contact of Elliptical Asperity  
According to [PERSON]'s theory, when an elliptical asperity is deformed significantly less than the critical elastic deformation, namely, \(\omega_{n}\leq\omega_{nec}\), it merely experiences elastic contact deformation. The maximum contact pressure \(P_{m}\), the semimajor radius of the contact ellipse \(r_{x}\), and the elastic deformation \(\omega\) can be expressed, respectively, as:  
\[P_{m}=\frac{3F}{2a} \tag{16}\]  
\[r_{x}={\left[\frac{3E(e)FR_{m}}{2\pi E^{\prime}(1-e^{2})}\right]}^{1/3} \tag{17}\]  
\[\omega=\frac{2K(e)}{\pi}{\left[\frac{\pi{\left(1-e^{2}\right)}}{2E(e)R_{m}} \right]}^{1/3}{\left(\frac{3F}{4E^{\prime}}\right)}^{2/3} \tag{18}\]  
where \(K(e)\) and \(E(e)\) are the complete elliptic integrals of the first and second kind, respectively, whose expressions are:  
\[K(e)=\int_{0}^{\pi/2}{\frac{d\varphi}{\sqrt{1-e^{2}\sin^{2}\varphi}}} \tag{19}\]  
\[E(e)=\int_{0}^{\pi/2}{\sqrt{1-e^{2}\sin^{2}\varphi}d\varphi} \tag{20}\]  
where \(E^{\prime}\) is the effective elastic modulus, which can be calculated by the following equation:  
\[\frac{1}{E^{\prime}}=\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}} \tag{21}\]where \(\
u_{1}\), \(\
u_{2}\) and \(E_{1}\), \(E_{2}\) are Poisson's ratio and the elastic modulus of the two rough surface materials in contact with each other, respectively.  
Combining Equations (16)-(18), the real contact area and real contact load of elastic deformation for elliptical asperity can be presented as:  
\[a_{ne}(\omega)=\left[\frac{E(e)}{K(e)(1-e^{2})^{1/2}}\right]\pi R_{m}\omega=f_{ 1}(e)\pi R_{m}\omega \tag{22}\]  
\[F_{ne}(\omega)=\left[\frac{\pi E(e)^{-1/2}}{2K(e)^{3/2}(1-e^{2})^{1/2}}\right] \frac{4}{3}E^{\prime}R_{m}{}^{1/2}\omega^{3/2}=f_{2}(e)\frac{4}{3}E^{\prime}R_ {m}{}^{1/2}\omega^{3/2} \tag{23}\]  
Therefore, the average contact pressure \(P_{e}\) can be expressed as:  
\[P_{e}(\omega)=\frac{F_{e}(\omega)}{a_{e}(\omega)}=\frac{4f_{2}(e)E^{\prime}} {3f_{1}(e)\pi}\bigg{(}\frac{\omega}{R_{m}}\bigg{)}^{1/2} \tag{24}\]  
The asperity entirely enters a plastic deformation state when the maximal contact pressure \(P_{m}\) is equal to \(KH\), and the equation is:  
\[H=2.8\sigma_{y} \tag{25}\]  
where, \(H\) denotes the material hardness, \(K\) denotes the hardness coefficient, and the relationship between the material Poisson's ratio is:  
\[K=0.454+0.41\
u \tag{26}\]  
The critical normal deformation of a single elliptical asperity between elastic and inelastic deformation is:  
\[\omega_{nec}=K(e)E(e)R_{m}\bigg{(}\frac{KH}{E^{\prime}}\bigg{)}^{2} \tag{27}\]  
By combining Equations (15), (22) and (27), the expression for the critical contact area \(a_{nec}\) between elastic and inelastic deformation can be deduced as:  
\[a_{nec}\left(\frac{2^{(3-D_{1})}\pi^{\frac{D_{1}}{2}}G_{1}{}^{(D_{1}-1)}(\ln \gamma_{1})^{\frac{1}{2}}}{(1-e^{2})^{-\frac{D_{1}}{4}}}a_{nec}{}^{-\frac{D_{1 }}{2}}+\frac{2^{(3-D_{2})}\pi^{\frac{D_{2}}{2}}G_{2}{}^{(D_{2}-1)}(\ln\gamma_{ 2})^{\frac{1}{2}}}{(1-e^{2})^{\frac{D_{2}}{4}}}a_{nec}{}^{-\frac{D_{2}}{2}} \right)^{2}=4f_{1}(e)\pi K(e)E(e)\bigg{(}\frac{KH}{E^{\prime}}\bigg{)}^{2} \tag{28}\]