## 2 Analysis of Forming Process
### Finite Element Process Simulation and Analysis
#### 2.3.2 Material Setting  
The metal plastic deformation process is accompanied by the generation of energy and the conduction of heat, which in turn causes a change in the temperature field inside  
Figure 2: Three-dimensional finite element model. the blank and the mold. Therefore, it is necessary to use thermal-mechanical coupling technology to analyze the interaction relationship of the internal physical field. Since plastic deformation mainly occurs in the metal forming process, and the elastic deformation is relatively small, the elastic deformation can be ignored, and the physical model of the material is simplified to a rigid-plastic finite element model. The rigid-plastic model should satisfy the following equations in the deformation process [14]. 1. The equilibrium differential equation is as follows: \[\sigma_{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{ \bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar}}}}}}}}}}}}}}}=0\] (1) where \(\sigma_{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{ \bar{\bar{\bar{   }}}}}}}}}}}}}}\) is the stress tensor on any particle \(j\). 2. The constitutive equation is as follows: \[\dot{\epsilon}_{ij}=\frac{3\bar{\bar{\bar{\epsilon}}}}{2\overline{\sigma}^{ \prime}}\sigma_{ij}\] (2) where \(\bar{\bar{\bar{\epsilon}}}=\sqrt{\frac{2}{\bar{\bar{\bar{\bar{\bar{\bar{ \epsilon}{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{ \bar{    \bar{         }}}}}}}}}}}}}}\) is the equivalent strain rate, \(\overline{\sigma}=\sqrt{\frac{3}{2}\sigma_{ij}^{\prime}\sigma_{ij}^{\prime}\) is the equivalent stress, and \(\sigma_{ij}^{\prime}\) is the deviatoric stress. 3. The geometric equation is as follows: \[\dot{\epsilon}_{ij}=\frac{1}{2}\big{(}V_{i,j}+V_{j,i}\big{)}\] (3) where \(\dot{\epsilon}_{ij}\) is the strain rate tensor, and \(V_{i,j}\) and \(V_{j,i}\) are the velocity components of each node. 4. The Mises yield criterium is as follows: \[\overline{\sigma}=\sqrt{3{J^{\prime}}_{2}}=Y\] (4) where \(Y\) is the yield stress of the material. \({J^{\prime}}_{2}\) is the second invariant of the stress deviator tensor. 5. The principle of volume incompressibility is expressed as follows: \[\dot{\epsilon}_{V}=\dot{\epsilon}_{ij}\dot{\epsilon}_{ij}=0\] (5) where \(\dot{\epsilon}_{V}\) represents the volume strain rate and D is the Kronecker symbol. 6. The boundary conditions are as follows: \[\sigma_{ij}n_{j}=\overline{p}_{i}(S\in S_{p})\] (6) \[v_{i}=\overline{v}_{i}(S\in S_{V})\] (7) where \(n_{j}\) denotes the cosine in the tensor direction at any point on \(S_{p}\), \(\overline{p}_{i}\) denotes the force vector given by the stress boundary \(S_{p}\), and \(\overline{v}_{i}\) denotes the velocity vector given by the velocity boundary \(S_{v}\). The blank was designated as a plastic material, and we selected 16 Mn, the composition of which is shown in Table 1[15]. According to the Johnson-Cook model [16] in JMatpro 14.0, the flow stress curve at 1100-1250 \({}^{\circ}\)C was generated, as shown in Figure 3. Its constitutive equation is as follows:  
\[\overline{\sigma}=\overline{\sigma}\Big{(}\bar{\bar{\bar{\epsilon}}},\bar{ \bar{\bar{\bar{\epsilon}}}},T\Big{)} \tag{8}\]  
where \(\overline{\sigma}\) represents the stress of flow, \(\bar{\bar{\bar{\epsilon}}}\) is material strain, \(\dot{\bar{\bar{\bar{\epsilon}}}}\) denotes the rate of material strain, and \(T\) denotes temperature. The initial forging temperature of 16 Mn should be lower than 1280 \({}^{\circ}\)C, and the final forging temperature should be controlled between 600 and 750 \({}^{\circ}\)C. Combined with the initial temperature of the actual forging of CITIC Heavy Industries Co., Ltd., this study selected 1100-1250 \({}^{\circ}\)C as the forging temperature range [17]. This temperature range not only avoids the performance degradation caused by the over-burning or over-heating of the material but also ensures the recrystallization temperature of the material and ensures that the material has a good plastic deformation ability.