# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 3 The Revised Model of a Single Fractal Asperity Contact
### The Existing Conditions for Asperity Elastic Deformation  
During the contact between a single asperity and a rigid flat surface, elastic, elasto-plastic, or full plastic deformation should occur in the asperity. The single asperity for which the frequency index is \(n\) can be regarded as a hemisphere body with radius \(R_{n}\), so the elastic critical interference of asperity \(\omega_{nec}\) at yielding inception has been obtained as Reference [1].  
\[\omega_{nec}=\left(\frac{\pi KH}{2E}\right)^{2}R_{n} \tag{11}\]  
where \(K=0.454+0.41\
u\), \(\
u\) is [PERSON]'s ratio, \(E\) is [PERSON]'s modulus, and \(H\) is the hardness of the softer material.  
To frequency index \(n\), \(R_{n}\gg\omega_{n}\), the asperity contact area in the elastic deformation is given by the Hertzian theory to be the following [20]:  
\[a_{nec}=\pi R_{n}\omega_{n} \tag{12}\]  
In the elastic deformation, the relation between the asperity contact area and the asperity truncated area can be given as  
\[{a^{\prime}}_{nec}=2a_{nec} \tag{13}\]  
For interference \(\omega_{n}=\omega_{nec}\), we obtain the elastic critical contact area  
\[a_{nec}=\pi R_{n}\omega_{nec} \tag{14}\]  
Then, the elastic critical truncated area is  
\[{a^{\prime}}_{nec}=2\pi R_{n}\omega_{nec} \tag{15}\]  
According to Hertzian theory, the asperity is in elastic deformation with \(\omega_{n}\leq\omega_{nec}\). Using Equations (12)-(15), the relations among the elastic critical contact area, elastic critical truncated area and contact area, and truncated area for the asperity elastic deformation are obtained as  
\[a_{nec}\leq a_{nec} \tag{16}\]  
\[{a^{\prime}}_{nec}\leq{a^{\prime}}_{nec} \tag{17}\]  
Substituting Equations (10) and (11) into Equation (14), the elastic critical contact area is written as follows:  
\[a_{nec}=\pi\left(\frac{\pi KH}{2E}\right)^{2}R_{n}^{2}=\left(\frac{KH}{2E} \right)^{2}\frac{l_{n}^{2(D-1)}}{16\pi G^{2(D-2)}\ln\gamma} \tag{18}\]  
Therefore, the elastic critical contact area is not a constant value for given fractal parameters and is proportional to the radius of curvature of asperity. The condition accords with classical contact mechanics.  
In the elastic deformation, the elastic contact load of the asperity is  
\[F_{ne}=\frac{4}{3}ER_{n}^{1/2}\omega_{n}^{3/2} \tag{19}\]  
Substituting Equation (12) into Equation (19), the relation between the contact load and contact area is developed to  
\[F_{ne}=\frac{16E\pi^{1/2}G^{(D-2)}(\ln\gamma)^{1/2}}{3l_{n}^{(D-1)}}a_{ne}^{3/2} \tag{20}\]  
Substituting Equation (13) into Equation (20), the relation between the contact load and truncated area is given as  
\[F_{ne}=\frac{8E\pi^{1/2}G^{(D-2)}(\ln\gamma)^{1/2}}{3\sqrt{2}l_{n}^{(D-1)}} \big{(}a^{\prime}{}_{ne}\big{)}^{3/2} \tag{21}\]  
For interference \(\omega_{n}\leq\omega_{nec}\), contact area \(a_{ne}\leq a_{nec}\), the asperity of frequency index \(n\) is in elastic deformation. When the asperity height is less than or equal to its own elastic critical interference, namely \(\delta_{n}\leq\omega_{nec}\), the asperity is only in elastic deformation under contact load. According to Equations (6), (9) and (11), the elastic critical frequency index can be expressed as  
\[n_{ec}=\text{int}\left[\frac{\log\!\left(\left(\frac{KH}{2E}\right)^{2}\! \frac{L^{2(D-2)}}{4G^{2(D-2)}\ln\gamma}\right)}{2(D-2)\log\gamma}\right] \tag{22}\]