# Tribological Analysis of Several Coatings under Flood and Cryogenic Cooling Conditions
## 2 Materials and Experimental Set-Up
### Determination of Contact Conditions in Machining of Ti-6 Al-4V Alloy
#### 2.3.2 Determination of the Normal Force Acting in the Pins
Due to the convergence issues, the application of a force to the pin was replaced with a displacement imposed on the pin in the direction normal to the workpiece surface until reaching the desired normal force. The materials of the workpiece and pin were considered as isotropic. The elastic properties of the Ti-6 Al-4V alloy were taken from [PERSON] et al. (2018), while the elastic properties of the pin were obtained from several studies found in the literature ([PERSON] et al., 2018; [PERSON] et al., 2019; [PERSON] et al., 2020). The Johnson-Cook constitutive model ([PERSON] et al., 2018) without the temperature term is used to represent the plastic behavior of the Ti-6 Al-4V alloy, as described by Equation (9):  
\[\widetilde{\sigma}=(A+B\cdot\varepsilon^{n})\bigg{[}1+C\cdot ln\bigg{(}\frac{ \dot{\varepsilon}}{\dot{\varepsilon}_{0}}\bigg{)}\bigg{]} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test No.**} & **Resultant Force** & **Contact Length,** & **Normal Force,** & **Contact Pressure, \(P\)** \\  & **R (N)** & \(l_{c}\) **(mm)** & \(F_{n\gamma}\) **(N)** & **(MPa)** \\ \hline Test 1 & 1299 & 0.276 & 1252 & 1136 \\ Test 2 & 1268 & 0.248 & 1233 & 1240 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Contact forces and pressures used in the pin-on-bar model ([PERSON] et al., 2018). Figure 7: Model of the pin-on-bar tribological test. where the first term represents the strain hardening effects, and the second term the strain-rate effect. \(\varepsilon_{p}\) and \(\dot{\varepsilon}\) (equal to 0.05 s\({}^{-1}\)) are the plastic strain and strain-rate, respectively. The coefficients \(A,B,C\) and \(n\) of this plasticity model were determined by [PERSON] et al. [52] via quasi-static and dynamic (different strain-rates) compression tests conducted over samples from the same bar used in the tribological tests. The mechanical properties of both the workpiece and pin are listed in Table 5.  
The friction coefficient in relation to the sliding speed was taken from the work of [PERSON] et al. [57], obtained for the same pin and workpiece materials using similar tribological tests. This friction coefficient can be represented by the following equation:  
\[\mu=0.359-0.042\cdot ln(V_{s}) \tag{10}\]  
where \(Vs\) is the sliding speed. The simulation was divided into two steps: indentation and sliding. In the indentation step, the pin penetrated to a given depth in the workpiece to establish the suitable average contact pressure. Then, in the sliding step, the pin slid 10 mm over the workpiece's top surface in the X-direction at a constant sliding speed. After the simulation, the contact pressure was extracted from the model and compared with that obtained by orthogonal cutting (see procedure in Figure 5). If they were different, the penetration depth was modified, and a new simulation was performed. This procedure was applied until both contact pressures were similar, and consequently the normal force acting in the pin was obtained. Both pin and workpiece were meshed with a four-node plane stress element type (CPS4R). A thin layer of 0.2 mm was placed on top of the workpiece to refine the mesh. Maximum element sizes of 1 mm and 0.5 mm were used for the pin and workpiece, respectively. To determine the minimum element size, the influence of element size on the results was studied. In addition, since the model is a simplification of the pin-on-bar geometry (initial bar diameter is equal to 80 mm), the influence of the equivalent radius used in the Hertz theory [58; 59] is also presented below. After this, the simulations were conducted for the five sliding speeds mentioned above. Furthermore, the minimum element size, the equivalent radius and the different sliding speeds were consulted to determine their influences on the determination of the normal force. The tested minimum element sizes are presented in Table 6, and the results are depicted in Figures 8 and 9.