# Characterization and Finite Element Modeling of Microperforated Titanium Grade 2
## 2 Materials and Methods
### Finite Element Modeling  
The authors also propose studying the material using a Nakajima test finite element model (FE model). Strain limits can be obtained for each mechanical behavior (equivaxial, plane strain, uniaxial tension, etc.) by deforming sheet metal blanks of different geometries with a hemispherical punch. The proposed model is a dynamic/explicit analysis which uses an explicit solution technique that integrates the equations of motion through time. This procedure is conditionally stable and has a robust contact functionality that easily solves even the most complex contact simulations. The FE model of the Nakajima includes the main components such as the punch, blank holder, die, and sheet. The analysis was performed by modeling just a quarter of the model, taking advantage of the double symmetry of all Nakajima geometries (see Figure 2), which reduces the number of elements and saves time consumption. The following image shows the FE model for geometry G7. The punch, blank, and holder have been modeled as discrete rigid components. The sheet is a 0.8 mm thick deformable shell (S4R elements) with variable geometry according to the shapes shown in Figure 2. The approximate size of the sheet elements is 2 mm. The material titanium grade 2 is anisotropic, so it is necessary to use a model that faithfully reproduces the mechanical properties. Considering that the material exhibits different yield in different directions, the Hill anisotropic yield criterion has been used [8]. This behavior is introduced in Abaqus through user-defined stress ratios (\(R_{ij}\)) that are applied in the quadratic Hill's potential function. Among the multiple yield criteria mentioned in the introduction, Hill48 is particularly well suited for modeling materials that exhibit anisotropic plasticity. This criterion was an extension of the isotropic [PERSON] yield criterion and can be presented as follows:  
\[f(\sigma)=\sqrt{F(\sigma_{22}-\sigma_{33})^{2}+G(\sigma_{33}-\sigma_{11})^{2}+ H(\sigma_{11}-\sigma_{22})^{2}+2L\sigma_{23}{}^{2}+2M\sigma_{31}{}^{2}+2N \sigma_{12}{}^{2}} \tag{1}\]  
where \(F\), \(G\), \(H\), \(L\), \(M\), and \(N\) are constants characterizing the anisotropy behavior of the material. Considering the model being studied in this work, it can be approximated as a plane strain case, and the Hill yield criterion is as follows:  
\[f(\sigma)=\sqrt{(\text{G}+H)\sigma_{11}{}^{2}+(F+H)\sigma_{22}{}^{2}-2H\sigma_ {11}\sigma_{22}+2N\sigma_{12}{}^{2}} \tag{2}\]  
The material constants (\(G\), \(H\), \(F\), and \(N\)) can be expressed using stress ratios \(R_{ij}\). The yield ratios are expressed relative to a reference yield stress (\(\sigma_{0}\)), so if \(\sigma_{ij}\) is applied as the only non-zero stress, the corresponding yield stress will be \(R_{ij}\sigma_{0}\). \[F=\frac{1}{2}\bigg{(}\frac{2}{R_{22}{}^{2}}+\frac{2}{R_{33}{}^{2}}+\frac{2}{R_ {11}{}^{2}}\bigg{)} \tag{3}\]  
\[G=\frac{1}{2}\bigg{(}\frac{2}{R_{33}{}^{2}}+\frac{2}{R_{11}{}^{2}}+\frac{2}{R_ {22}{}^{2}}\bigg{)} \tag{4}\]  
\[H=\frac{1}{2}\bigg{(}\frac{2}{R_{11}{}^{2}}+\frac{2}{R_{22}{}^{2}}+\frac{2}{R_ {33}{}^{2}}\bigg{)} \tag{5}\]  
\[N=\frac{3}{R_{12}{}^{2}} \tag{6}\]  
In the case of planar anisotropy, the unknown ratios will be \(R_{22}\), \(R_{33}\), and \(R_{12}\) since the rest has a common value of 1 (\(R_{11}=R_{23}=R_{32}=1\)). These yield ratios are the study variables which will be obtained by means of an RSM optimization process.