# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 4 The Total Real Contact Area and Total Contact Load
### The Cylinder-Based Fractal Surface Is in Contact with a Rigid Flat Surface
#### 4.1.3 Total Real Contact Area and Total Contact Load  
The right contact zone with the \(z\) axis is uniformly divided into \(N\) subdivisions, and each subdivision can be regarded as a smaller flat fractal surface contacting a flat rigid surface. These asperities for which the frequency indexes range from \(n_{\rm min}\) (\(n_{\rm min}<n_{ec}\)) to \(n_{\rm max}\) (\(n_{\rm max}>n_{epc}\)) in the right contact zone are in elastic, elasto-plastic, and full plastic deformation, respectively. The total real truncated area of the right contact zone can be expressed as  
\[A^{\prime}{}_{r}=A^{\prime}{}_{re}+A^{\prime}{}_{rep}+A^{\prime}{}_{rp} \tag{54}\]  
where \(A^{\prime}{}_{re}\), \(A^{\prime}{}_{rep}\), and \(A^{\prime}{}_{rp}\) are the real truncated areas in the elastic, elasto-plastic, and full plastic deformation, respectively. The three portions are expressed as  
\[A^{\prime}{}_{re}=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{\rm min}}^{n_{ec}} \int_{0}^{a^{\prime}{}_{i,nl}}n_{i,n}^{C}(a^{\prime})d^{\prime}da^{\prime}+ \sum\limits_{i=1}^{N}\sum\limits_{n=n_{ec}+1}^{n_{\rm max}}\int_{0}^{a^{\prime }{}_{i,nrec}}n_{i,n}^{C}\big{(}a^{\prime}\big{)}a^{\prime}da^{\prime} \tag{55}\]  
\[A^{\prime}{}_{rep}=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{ec}+1}^{n_{rpc}} \int_{a^{\prime}{}_{i,nrec}}^{a^{\prime}{}_{i,nl}}n_{i,n}^{C}(a^{\prime})a^{ \prime}da^{\prime}+\sum\limits_{i=1}^{N}\sum\limits_{n=n_{epc}+1}^{n_{\rm max }}\int_{a^{\prime}{}_{i,nrec}}^{a^{\prime}{}_{i,nrec}}n_{i,n}^{C}\big{(}a^{ \prime}\big{)}a^{\prime}da^{\prime} \tag{56}\]  
\[A^{\prime}{}_{rp}=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{epc}+1}^{n_{\rm max}} \int_{a^{\prime}{}_{i,nrec}}^{a^{\prime}{}_{i,nl}}n_{i,n}^{C}\big{(}a^{\prime }\big{)}a^{\prime}da^{\prime} \tag{57}\]  
Substituting Equations (13) and (53) into Equation (55), the real contact area in the elastic deformation can be expressed as  
\[ \begin{array}{ll}A{}_{re}&=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{\rm min}}^ {n_{ec}}\int_{0}^{a^{\prime}{}_{i,nl}}a^{\prime}n_{i,n}^{C}(a^{\prime})da^{ \prime}+\sum\limits_{i=1}^{N}\sum\limits_{n=n_{ec}+1}^{n_{\rm max}}\int_{0}^{a ^{\prime}{}_{i,nrec}}\frac{a^{\prime}}{2}n_{i,n}^{C}(a^{\prime})da^{\prime}\\ &=\sum\limits_{i=1}^{N}\lambda_{i}(N)\lambda_{i}\sum\limits_{n=n_{\rm min}}^ {n_{ec}}\int_{0}^{a^{\prime}{}_{i,nl}}\frac{a^{\prime}}{2}n(a^{\prime})da^{ \prime}+\sum\limits_{i=1}^{N}\lambda_{i}(N)\Lambda_{i}\sum\limits_{n=n_{ec}+1 }^{n_{\rm max}}\int_{0}^{a^{\prime}{}_{i,nrec}}\frac{a^{\prime}}{2}n(a^{\prime })da^{\prime}\\ &=\lambda(N)\frac{D-1}{6-2D}\sum\limits_{i=1}^{N}\Lambda_{i}\left(\sum \limits_{n=n_{\rm min}}^{n_{ec}}a^{\prime}{}_{i,nl}+\sum\limits_{n=n_{ec}+1}^ {n_{\rm max}}\big{(}a^{\prime}{}_{i,nl}\big{)}\frac{D-1}{2}\big{(}a^{\prime}{ }_{i,nrec}\big{)}\frac{3-D}{2}\right)\end{array}  \tag{58}\]  
The total elastic deformation contact load can be written as  
\[F{}_{re}=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{\rm min}}^{n_{ec}}\int_{0}^{a ^{\prime}{}_{i,nl}}F{}_{i,nre}\big{(}a^{\prime}\big{)}n_{i,n}^{C}\big{(}a^{ \prime}\big{)}da^{\prime}+\sum\limits_{i=1}^{N}\sum\limits_{n=n_{ec}+1}^{n_{ \rm max}}\int_{0}^{a^{\prime}{}_{i,nrec}}F{}_{i,nre}\big{(}a^{\prime}\big{)}n_{ i,n}^{C}(a^{\prime})da^{\prime} \tag{59}\]  
Substituting Equation (21) into Equation (59), we can obtain  
\[F{}_{re}=T_{0}\lambda(N)\frac{D-1}{4-D}\sum\limits_{i=1}^{N}\Lambda_{i}\left( \sum\limits_{n=n_{\rm min}}^{n_{ec}}l_{i,n}^{(1-D)}\big{(}a^{\prime}{}_{i,nl} \big{)}\frac{3}{2}+\sum\limits_{n=n_{ec}+1}^{n_{\rm max}}l_{i,n}^{(1-D)}\big{(} a^{\prime}{}_{i,nl}\big{)}\frac{D-1}{2}\big{(}a^{\prime}{}_{i,nrec}\big{)}\frac{4-D}{2}\right) \tag{60}\]  
where \(T_{0}=8E\pi^{1/2}G^{(D-2)}(\ln\gamma)^{1/2}/3\sqrt{2}\). Substituting Equations (26) and (53) into Equation (56), the real contact area in the elasto-plastic deformation can be expressed as  
\[