# 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
\begin{array}{ll}A{}_{rep}&=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{ec}+1}^{n_ {epc}}\int_{a^{\prime}{}_{i,nrec}}^{a^{\prime}{}_{i,nl}}\frac{1}{2}\big{(}a^{ \prime}{}_{i,nrec}\big{)}^{-0.1597}(a^{\prime})^{1.1597}n_{i,n}^{C}(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}}\frac{1}{2}\big{(}a^{\prime}{}_{i,nrec} \big{)}^{-0.1597}(a^{\prime})^{1.1597}n_{i,n}^{C}(a^{\prime})da^{\prime}\\ &=\lambda(N)\frac{D-1}{6.6388-2D}\sum\limits_{i=1}^{N}\Lambda_{i}\left[\sum \limits_{n=n_{ec}+1}^{n_{\rm{spec}}}\left(\big{(}a^{\prime}{}_{i,nl}\big{)} \frac{D-1}{2}\big{(}a^{\prime}{}_{i,nrec}\big{)}^{-0.1597}\big{(}a^{\prime}{}_{ i,nrec}\big{)}^{-0.1597}-(a^{\prime}{}_{i,nl}\big{)}\frac{D-1}{2}\big{(}a^{ \prime}{}_{i,nrec}\big{)}\frac{3-D}{2}\right)\\ &+\sum\limits_{n=n_{epc}+1}^{n_{\rm max}}\left(\big{(}a^{\prime}{}_{i,nl}\big{)} \frac{D-1}{2}\big{(}a^{\prime}{}_{i,nrec}\big{)}^{-0.1597}\big{(}a^{\prime}{}_{ i,nrec}\big{)}^{-}\frac{D}{2}+^{1.6597}-(a^{\prime}{}_{i,nl}\big{)}\frac{D-1}{2}\big{(}a^{ \prime}{}_{i,nrec}\big{)}\frac{D-1}{2}\right)\end{array}  \tag{61}\]The total elasto-plastic deformation contact load can be written as  
\[F_{rep}=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{qc}+1}^{n_{rep}}\int_{a^{\prime} _{i,inc}}^{a^{\prime}_{i,incl}}F_{i,nep}\big{(}a^{\prime}\big{)}n_{i,n}^{C} \big{(}a^{\prime}\big{)}da^{\prime}+\sum\limits_{i=1}^{N}\sum\limits_{n=n_{qc} +1}^{n_{\max}}\int_{a^{\prime}_{i,inc}}^{a^{\prime}_{i,spec}}F_{i,nep}\big{(}a^ {\prime}\big{)}n_{i,n}^{C}\big{(}a^{\prime}\big{)}da^{\prime} \tag{62}\]  
Substituting Equation (27) into Equation (62), we can obtain  
\[ \begin{array}{ll}F_{rep}&=\lambda(N)\frac{KH(D-1)}{11.2806-3D}\sum\limits_{ i=1}^{N}\Lambda_{i}\left[\sum\limits_{n=n_{qc}+1}^{n_{rep}}\left((a^{\prime} _{i,incl})^{1.3801}(a^{\prime}_{i,inc})^{-0.3801}-(a^{\prime}_{i,incl}) \frac{D-1}{2}\left(a^{\prime}_{i,inc}\right)\frac{3-D}{2}\right)\right.\\ &+\sum\limits_{n=n_{qc}+1}^{n_{\max}}\left((a^{\prime}_{i,incl}) \frac{D-1}{2}\left(a^{\prime}_{i,inc}\right)^{-0.3801}\left(a^{\prime}_{i, nepc}\right)^{-\frac{D}{2}+1.8801}-(a^{\prime}_{i,incl})\frac{D-1}{2}\left(a^{ \prime}_{i,inc}\right)\frac{3-D}{2}\right)\right]\end{array}  \tag{63}\]  
In the full plastic deformation, the asperity contact area equals the asperity truncated area, so the real contact area in the full plastic deformation can be expressed as  
\[ \begin{array}{ll}A_{rp}&=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{qc}+1}^{n_ {\max}}\int_{a^{\prime}_{i,inc}}^{a^{\prime}_{i,incl}}n_{i,n}^{C}(a^{\prime })a^{\prime}da^{\prime}\\ &=\lambda(N)\frac{D-1}{3-D}\sum\limits_{i=1}^{N}\Lambda_{i}\sum\limits_{n=n_{qc }+1}^{n_{\max}}\left(a^{\prime}_{i,incl}-(a^{\prime}_{i,incl})\frac{D-1}{2 }\left(a^{\prime}_{i,nepc}\right)\frac{3-D}{2}\right)\end{array}  \tag{64}\]  
The total contact load in the full plastic deformation can be written as  
\[F_{rp}=\sum\limits_{i=1}^{N}\sum\limits_{n=n_{qc}+1}^{n_{\max}}\int_{a^{ \prime}_{i,inc}}^{a^{\prime}_{i,incl}}F_{i,nep}\big{(}a^{\prime}\big{)}n_{i, n}^{C}\big{(}a^{\prime}\big{)}da^{\prime} \tag{65}\]  
Substituting Equation (31) into Equation (65), we can obtain  
\[F_{rp}=\lambda(N)H\frac{D-1}{3-D}\sum\limits_{i=1}^{N}\Lambda_{i}\sum\limits _{n=n_{qc}+1}^{n_{\max}}\left(a^{\prime}_{i,incl}-\big{(}a^{\prime}_{i,incl} \big{)}\frac{D-1}{2}\left(a^{\prime}_{i,nepc}\right)\frac{3-D}{2}\right) \tag{66}\]  
The total real contact area and total contact load for the right contact zone can be obtained as  
\[A_{r}=A_{re}+A_{rep}+A_{rp} \tag{67}\]  
\[F_{r}=F_{re}+F_{rep}+F_{rp} \tag{68}\]  
We define the nominal area \(A_{a}\) as [19]  
\[A_{a}=L_{c}^{2} \tag{69}\]  
where the sample length \(L_{c}\) equals \(\sqrt{2\pi\rho L_{h}}\), where \(L_{h}\) is the thickness of cylinder-based fractal surface. In this paper \(L_{h}=\rho\).