## 2 Materials and Methods
### Finite Element Modeling  
Finite element simulations were performed by the commercial finite element program DEFORM 3D. Taking advantage of the axial symmetry of the problem, only a quarter of the process has been modeled in order to reduce the computation time and other computational resources, such as storage needs. The ram and the container were modeled as rigid objects and the bimetallic cylinders were modeled as an assembly between two plastic objects (sleeve and core).  
The plastic parts have been meshed using tetrahedral elements. A mesh study has been realized in order to determine the best mesh configuration. Meshes with 5000, 7000, and 13,000 elements have been considered. In Figure 3, the extrusion force results versus the die stroke are shown for different mesh sizes.  
Taking into account that the results for the extrusion force using several mesh sizes are quite similar, 7000 elements mesh size is considered the optimum one due to the compromise between the precision reached and the computational time to perform each simulation.  
The contact conditions among the different objects have to be defined as well. Rigid objects are considered \"master\" (the ones who deform) and the plastic objects are considered \"slave\" (the ones who are deformed). In the case of sleeve and core interaction, where both objects are plastic, titanium alloy is defined as \"master\" and magnesium alloy as \"slave\", according to the properties presented in Table 3. The friction model used in all the simulations is the Coulomb model that, along with the shear friction model, is one of the classical friction models in metal forming analysis [20].  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Ti** & **Al** & **V** & **Fe** & **C** & **O** & **N** & **H** \\  & **(wt.\%)** & **(wt.\%)** & **(wt.\%)** & **(wt.\%)** & **(wt.\%)** & **(wt.\%)** & **(wt.\%)** & **(wt.\%)** \\ \hline Ti6 Al4V & Bal. & 5.5–6.5 & 3.5–4.5 & 0.25 & 0.08 & 0.13 & 0.040 & 0.012 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Chemical composition of titanium alloy Ti6 Al4V.  
Figure 3: Extrusion force for different mesh sizes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **AZ31B** & **Ti6 Al4V** \\ \hline Density (g/cm\({}^{3}\)) & 1.74 & 4.46 \\ Tensile strength (MPa) & 260 & \(\geq\)895 \\ Yield strength (MPa) & 200 & \(\geq\)828 \\ Elastic modulus (GPa) & 44.80 & 105–120 \\ Poisson’s ratio & 0.35 & 0.31 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Physical and mechanical properties of the magnesium alloy AZ31B and titanium alloy Ti6 Al4V.  
Materials AZ31B and Ti6 Al4V were assumed to be isotropic during all the process. The flow curves used for Ti6 Al4V are from DEFORM library.  
In the case of AZ31B, as there was no reference in DEFORM, an exponential model defined by [PERSON] [21] was used to define the flow curves, obtaining the results shown in Table 4.  
Damage factor has been also used in this work as a parameter to evaluate the quality of the extrudate. Damage factor is evaluated in DEFORM using normalized Cockcroft and Latham criterion [22], typically used for predicting damage in metal forming operations [23], due to its simplicity and the accessibility of material data required for its calculation. This criterion considers that the fracture is mainly governed by the maximum principal stress, as it is shown in Equation (1):  
\[\int_{0}^{\varepsilon_{f}}(\sigma_{max}/\sigma_{H})d\varepsilon=C \tag{1}\]  
where \(\varepsilon\) is the equivalent plastic strain, \(\varepsilon_{f}\) is the equivalent strain to fracture, \(\sigma_{max}\) is the maximum principal stress, \(\sigma_{H}\) is the stress according to the Huber-Mises hypothesis and \(C\) is a constant depending on the material and experimental determined.