# The Static Frictional Behaviors of Rubber for Pipe-Laying Operation
## 4 Discussion
### Normal Loading Condition  
It is confirmed that the friction is directly proportion to the real contact area or the adhesive zone, the classical Hertz theory gives a relationship between the real contact area and normal load [17]:  
\[A_{real}\ =\ \pi(\frac{3}{4}\frac{(1-\
u^{2})F_{N}R}{E})^{2/3} \tag{1}\]  
where \(F_{N}\), \(\
u\), \(R\), and \(E\) are the normal load, [PERSON]'s ratio, curvature radius and Young's modulus of the rubber, respectively.  
Apparently, the real contact area increases with increasing normal load within a certain range. Furthermore, according to [PERSON] et al. [8], the real contact area is larger than the value which is calculated by the Hertz theory. They deduced a new approach- the JKR approach- from the Hertz theory by taking adhesion into consideration:  
\[A_{real}\ =\ \pi(\frac{3}{4}\frac{F_{n1}R}{E})^{2/3} \tag{2}\]  
\[F_{n1}\ =\ F_{N}\ +3\cdot W_{12}\cdot\pi\cdot R+\sqrt{6\cdot W_{12}\cdot\pi \cdot R\cdot F_{N}\ +(3\cdot W_{12}\cdot\pi\cdot R)^{2}} \tag{3}\]  
where \(F_{n1}\) is the apparent Hertzian load which would cause the same contact area, \(W_{12}\) is the work of adhesion [26].  
The calculating values of the contact area by the JKR theory is compared with the measured values from the video frames in Figure 8. Even difference between the two groups of values of the contact area exists, they increase as normal load increases with a same trend. It indicates that the JKR theory is appropriate to estimate the contact in this study. In addition, the static frictions of rubber-steel and PMMA contacts are plotted in Figure 8, either. As being proportion to the real contact area, the increasing trends of the static frictions show good agreement with that of the real contact areas [7,27].  
Figure 7: Evolution of the rubber A-PMMA contact area. Normal load: 30 N; increasing rate \(F_{T}\): 1.5 N/s; (**a**) Stable contact area; (**b**) The contact area deforms due to the tangential load; (**c**) The folds appear as the tangential load increases; (**d**) The propagation of the folds; (**e**) The folds is stopped adjacent the boundary of the central adhesive zone; (**f**) The instant of the macroscopic slip. Six black lines are marked from 1–6.  
There are two major soft contact models to predict the level of friction. Model 1 is appropriate for the condition that the roughness of the rubber surface is extremely high. The other one for when the rubber surface is smooth in a certain range. Apparently, the second model is suitable for this research. Model 2 assumes that the real contact area will increase with normal load due to the viscoelastic deformation of the entire rubber surface, whereas the deformations of asperities on the rubber surface contribute little to the increment of the real contact area. In addition, the friction is proportional to the real contact area. Thus, according to Equations (1) and (2), the real contact area can be seen as proportional to \({\rm F_{N}}^{2/3}\), the COF \(\mu\) which is acquired from \(F_{T}/F_{N}\), is proportion to \({F_{N}}^{-1/3}\). The tendencies of real contact area and \(\mu\) with increasing normal load are plotted in Figure 9. At the range of normal load which is used in this research, the testing results of COF in Figure 5 are shown in good agreement with \(\mu\) which comes from the model 2.  
Figure 8: Calculating values by the JKR theory and measured values from the video frames versus the normal load. The maximum static friction undergoes the same trend with the adhesive contact area.  
Figure 9: The soft contact model for relative smooth interfaces; \(\mu\) is coefficient of friction; \(A_{real}\) is real contact area; \(\psi\) is constant factor for the rubber sample. The \(\mu/\psi\) and \(A_{real}/\psi\) indicate that even for different rubber samples, coefficient of friction and real contact area behave in their same tendencies, respectively.