# A Novel Finite Element Method Approach in the Modelling of Edge Trimming of CFRP Laminates
## 5 Results and Discussion
### Fibre Orientation of 45\({}^{\circ}\)  
Initially, the vertical forces \(F_{Y}\) experience a steady growth mainly motivated by the spring back phenomenon. This takes place because the range of machining relative angles given in the edge trimming of 45\({}^{\circ}\) laminates, from 45\({}^{\circ}\) to \(-15^{\circ}\), generate similar thrust forces in machining [55]. A parabolic decrease of the forces was observed because the spring back forces have a larger contribution to the X axis, which decreases notably the quantity measured in the vertical component. Horizontal forces \(F_{X}\) experience a parabolic increase due to the contribution of the spring back phenomenon to the horizontal axis. An analysis of this result reveals that cutting forces are not important in comparison to spring back forces, which are the leading cause of the \(F_{X}\) and \(F_{Y}\) force distribution. The global machining forces simulated for cutting tool edge radius of 10 \(\upmu\)m, 20 \(\upmu\)m, 40 \(\upmu\)m, and 50 \(\upmu\)m are presented in Figure 15.  
Note that the cutting forces recorded in the simulations do not follow a smooth distribution with the advancement of the tool, but rather an irregular distribution full of saw teeth. These fluctuations are explained due to the changes in the normal contact stiffness between the cutting tool and the laminate that occurs when the tool comes into contact with the different elements of the meshed laminate. When the cutting tool tip contacts with a new element, there is a slight rebound in the contact forces because of the fast increment of the contact rigidity that progressively diminishes as the tool advances its truck over the aforementioned element, as shown in Figure 16. The magnitude of these fluctuations increase notably with the implementation of higher contact stiffness and could create element distortion problems with the use of high values. In this research, a linear contact rigidity of 10 \(GPa\) was employed to avoid this problem. Regarding the forces \(F_{T}\) and \(F_{N}\), two steady rises with distinct slopes were observed. Therefore, it is concluded that the variation of the effect of the changes in machining relative angles is not crucial for this configuration. As discussed above, it was observed that spring back forces, which are responsible for the increment of \(F_{N}\) forces, are higher than the cutting forces that govern the \(F_{T}\) forces evolution. The representation of all graphs measuring these forces is showcased in Figure 17. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameters** & **Average \(F_{X}\)45\({}^{\circ}\)** & **Average \(F_{Y}\)45\({}^{\circ}\)** & **Average \(F_{X}\)90\({}^{\circ}\)** & **Average \(F_{Y}\)90\({}^{\circ}\)** \\ \hline \(f_{r}=800\) mm/min \(w=6000\) rpm & 1.1\% & 5.03\% & 5.19\% & 7.58\% \\ \(f_{r}=800\) mm/min \(w=8000\) rpm & 5.98\% & 0.98\% & 1.87\% & 8.18\% \\ \(f_{r}=1200\) mm/min \(w=6000\) rpm & 1.61\% & 1.31\% & 2.64\% & 2.34\% \\ \(f_{r}=1200\) mm/min \(w=8000\) rpm & 5.67\% & 2.30\% & 2.97\% & 18.23\% \\ \hline \end{tabular}
\end{table} 
Table 4: Numerical errors obtained with respect to the experimental average forces measured by [PERSON] et al. [44]. Figure 16: Horizontal force fluctuations simulated in a cutting tool edge of 30 \(\upmu\)m and a fibre orientation of 45\({}^{\circ}\). Figure 15: \(F_{X}\) and \(F_{Y}\) forces simulated in laminates with a fibre orientation of 90\({}^{\circ}\) and tool edge radius of (**a**) 10 \(\upmu\)m, (**b**) 20 \(\upmu\)m, (**c**) 40 \(\upmu\)m, and (**d**) 50 \(\upmu\)m.  
From the analysis of the results, it is inferred that tool wear does not increase significantly (less than 20%) the machining forces in the edge trimming of 45\({}^{\circ}\) laminates. The most significant increments are observed in forces \(F_{N}\) and \(F_{Y}\) with maximum rises of 19.90% and 13.76%, respectively. The force \(F_{X}\) keeps stable with small increments below 5%.