## 3 Finite Element Analysis Coupled with Damage Evolution  
The CFBE process has some advantages, since it forms a cup-rod shaped part effectively. However, CFBE processes induce complicated material flows due to the opposing directions of extruded material flows, as shown in Figure 1. Such material flows in a CFBE are a mutually restraining and unsteady state. Thus, it is not easy to understand deformation behaviors during a CFBE process, as well as defining the process conditions for a successful CFBE process. Thermomechanical finite element analysis may aid in understanding the deformation characteristics of AZ31 Mg alloys during the forming process at an elevated temperature, especially the deformation behaviors inside a work piece that could not be observed in experiments. In this work, the elastic behavior of AZ 31 Mg ally at warm temperatures was so small that it was neglected. Its inelastic behavior at warm temperatures was assumed as hardening and viscoplastic. Since the finite element formulation for the hardening, rigid viscoplastic deformation has been reported previously in References [19; 20], its detailed procedure was not given in this study. Here, a common finite element \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Forward Extrusion**} & \multicolumn{2}{c}{**Backward Extrusion**} & \multirow{2}{*}{\begin{tabular}{c} **Thickness t** \\ **mm** \\ \end{tabular} } \\ \hline \begin{tabular}{c} **Lower Die Dia., D\({}_{\text{F}}\)**,** \\ **mm** \\ \end{tabular} & \begin{tabular}{c} **Forward Extrusion Ratio** \\ **(FER)** \\ \end{tabular} & \begin{tabular}{c} **Punch Dia., D\({}_{\text{P}}\)**,** \\ **mm** \\ \end{tabular} & \begin{tabular}{c} **Backward Extrusion Ratio** \\ **(BER)** \\ \end{tabular} & \begin{tabular}{c} **mm** \\ \end{tabular} \\ \hline \begin{tabular}{c} 15 \\ 10 \\ 15 \\ 20 \\ \end{tabular} & \begin{tabular}{c} 15 \\ 2.25 \\ \end{tabular} & \begin{tabular}{c} 1.33 \\ 22 \\ \end{tabular} & \begin{tabular}{c} 7.5 \\ 2.16 \\ \end{tabular} & \
\begin{tabular}{c} 4 \\ 4.02 \\ \end{tabular} &
\begin{tabular}{c} 2 \\ 1 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental conditions for CFBE of AZ31B Mg alloy. Figure 2: Geometrical parameters in a Combined Forward–Backward Extrusion (CFBE) process. package DEFORM 2D [19] was used for the thermomechanical finite element analysis of AZ31 Mg alloy at warm temperatures. Damage in this work was defined as a void volume fraction or porosity, since AZ31 Mg alloy has moderate ductility at warm temperatures. [PERSON] et al. [21] reported the comparisons of four damage evolution models by [PERSON] et al. [22], [PERSON] et al. [23], [PERSON] [24], and [PERSON] [25], for the analysis of damage evolution in pure shear and backward extrusion of AZ31 Mg alloy. In the damage evolution model by [PERSON] and [PERSON], the mean stress was scaled by a hardness (\(\sigma^{*}\)), which represents the rate-independent limit of the hardening state, whilst the mean stress was scaled by flow stress in all others. [PERSON] et al. [21] concluded that the model proposed by [PERSON] and [PERSON] could predict damage evolution in extrusion better than the other models, especially for complicated deformation, including severe shear deformation. The first reason is that the exponential function with a ratio of mean stress, with respect to a flow stress, would overestimate for the damage evolution when the flow stress is low. The second reason is that the damage models, except for one by [PERSON] and [PERSON], were unable to capture accurately the effects of shear deformation on the damage evolution. The detailed explanation on the comparison is given in the Reference [21]. The damage evolution model by [PERSON] and [PERSON] is coupled with DEFORM 2D through a user subroutine, which is given by the following equation. \[\dot{\phi}=C_{1}\frac{\phi}{1-\phi}\;Exp\Big{[}C_{2}\frac{\sigma_{m}}{\sigma^{ *}}\Big{]}\dot{\epsilon} \tag{1}\]  
where \(\phi\) is a damage variable, defined as a porosity in this work.