## 2 Materials and Methods
### Finite Element Model Validation  
The FE model validation has been performed theoretically comparing the simulations results for the extrusion force with the results using [PERSON] semi-empirical model [24; 25; 26], typically used in this kind of analysis. In this model the deformation is defined in Equation (2):  
\[\varepsilon_{x}=a+b\cdot ln(r_{x}) \tag{2}\]  
where \(r_{x}\) is the extrusion ratio (\(A_{0}\)/\(A\)) and \(a\) and \(b\) are constants obtained by semi-empirical methods whose values are 0.8 and 1.2, respectively.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{**Strain**} & \multicolumn{5}{c}{**Stress (MPa)**} \\ \cline{3-6}  & & **T = 150 °C** & **T = 200 °C** & **T = 250 °C** & **T = 300 °C** \\ \hline \multirow{8}{*}{**Strain rate = 0.001**} & 0.02 & 112.69 & 91.54 & 74.35 & 60.40 \\  & 0.05 & 133.12 & 107.86 & 87.40 & 70.82 \\  & 0.10 & 168.45 & 136.07 & 109.91 & 88.78 \\  & 0.15 & 202.14 & 163.00 & 131.44 & 105.99 \\  & 0.20 & 230.06 & 185.44 & 149.47 & 120.49 \\  & 0.25 & 248.31 & 200.35 & 161.65 & 130.42 \\  & 0.30 & 254.18 & 205.56 & 166.23 & 134.43 \\ \hline \multirow{8}{*}{**Strain rate = 0.01**} & 0.02 & 127.46 & 105.42 & 87.19 & 72.11 \\  & 0.05 & 149.13 & 122.70 & 100.95 & 83.06 \\  & 0.10 & 186.21 & 152.12 & 124.27 & 101.52 \\  & 0.15 & 221.25 & 179.81 & 146.14 & 118.77 \\  & 0.20 & 250.18 & 202.67 & 164.18 & 133.01 \\  & 0.25 & 269.21 & 217.81 & 176.23 & 142.58 \\  & 0.30 & 275.67 & 223.20 & 180.72 & 146.32 \\ \hline \multirow{8}{*}{**Strain rate = 0.1**} & 0.02 & 144.17 & 121.41 & 102.25 & 86.11 \\  & 0.05 & 167.06 & 139.58 & 116.61 & 97.43 \\ \cline{1-1}  & 0.10 & 205.84 & 170.06 & 140.50 & 116.08 \\ \cline{1-1}  & 0.15 & 242.17 & 198.36 & 162.48 & 133.08 \\ \cline{1-1}  & 0.20 & 272.07 & 221.51 & 180.34 & 146.83 \\ \cline{1-1}  & 0.25 & 291.87 & 236.80 & 192.13 & 155.88 \\ \cline{1-1}  & 0.30 & 298.99 & 242.36 & 196.46 & 159.25 \\ \hline \hline \end{tabular}
\end{table}
Table 4: AZ31B flow curves values based on Wen-Juan exponential model.  
The extrusion force is defined in Equation (3):  
\[F=A_{0}\sigma_{f}\bigg{(}\varepsilon_{x}+\frac{2L}{D_{0}}\bigg{)} \tag{3}\]  
where \(A_{0}\) is the initial area, \(L\) is the contact length of the container, \(D_{0}\) is the initial diameter of the billet and \(\sigma_{f}\) is the average flow stress.  
Three simulations have been performed to validate the model. In all the simulations the cylinder had 12 mm of external diameter and 20 mm of height and the final external diameter was 9 mm, therefore the extrusion ratio (\(r_{x}\)) is 1.77. The first simulation is performed using a Ti6 Al4V cylinder, the second one using a AZ31B cylinder and the last one using a bimetallic billet with a core of 6 mm diameter.  
The extrusion parameters were a die semi-angle of 45\({}^{\circ}\), a friction coefficient of 0.1, a ram speed of 2 mm/s and a temperature of 200 \({}^{\circ}\)C.  
Equation (4) [27] was used to obtain the flow stress for the multi-material cylinder:  
\[\sigma_{m}=\sigma_{c}\cdot\frac{V_{c}}{V_{c}+V_{s}}+\sigma_{s}\cdot\frac{V_{s} }{V_{c}+V_{s}} \tag{4}\]  
where \(\sigma_{m}\) is the yield stress for the multi-material part, \(\sigma_{c}\) is the yield stress for the core, \(\sigma_{s}\) is the yield stress for the sleeve, \(V_{c}\) is the volume of the core and \(V_{s}\) is the volume of the sleeve.  
The comparison results are shown in Figures 4-6.  
After comparing the analytical results with the simulation ones, it can be concluded that the FE model is robust enough to be validated.  
Figure 4: Comparison chart between Johnson and FEM extrusion force for a billet of Ti6 Al4V.  
Figure 5: Comparison chart between Johnson and FEM extrusion force for a billet of AZ31B.