## 2 Materials and Methods
### Material Characterization
#### 2.2.2 Strain Rate-Dependent Hardening  
Contrary to the traditional assumption that strain rate effects are negligible in predicting cold-work metal flow, recent studies have demonstrated that strain rate can play a significant role, particularly with respect to springback phenomena. Increased strain rate sensitivity enhances the limiting draw ratio [40], thereby altering the stress distribution  
\begin{table}
\begin{tabular}{l l l l l} \hline
**Thickness** & \(r_{0}\) & \(r_{45}\) & \(r_{90}\) & \(r_{n}\) \\ \hline
0.8 mm & 0.93 & 1.29 & 0.80 & 1.077 \\ \hline \end{tabular}
\end{table}
Table 2: The \(r\) values at 20% elongation and normal anisotropy.  
Figure 3: Miyauchi test schematic.  
Figure 2: The \(r\) vs. \(\varepsilon\) values for 0.8 mm blank thickness.  
during forming and implying a corresponding influence on springback behavior. Similarly, high strain rates delay the onset of plastic instability, promoting more homogeneous deformation and reducing localized stress concentrations that typically exacerbate springback upon unloading [41]. Furthermore, investigations by [42] observed that high-speed processes, such as impact hydroforming, yield enhanced formability and a more favorable springback response, as the rapid deformation minimizes the time available for stress relaxation and residual stress accumulation. A methodology that correlates higher strain rates with reduced springback in sheet metals further substantiated these findings [43], thereby emphasizing the importance of incorporating strain rate effects into predictive models for improved accuracy in industrial applications.  
To determine the sensitivity to strain rate of the material, further uniaxial tensile tests are conducted. A strain rate bracketing is defined with a factor of 10 relative to the quasi-static tensile reference strain rate 0.004 s\({}^{-1}\). In Figure 5, experimental results are shown.  
The evaluation of the strain hardening in the uniaxial tension test is limited to the uniform elongation. Significantly higher equivalent strains can be reached in a deep drawing process. It is not known which numerical strain-hardening law can be applied for the extrapolation of the strain hardening up to high strains. In order to describe the flow behavior for high equivalent strains correctly, testing techniques other than uniaxial tension, that allow the measurement of the flow stress up to high levels of plastic strain, must be used. A bulge test with a controlled strain rate is performed. The hydraulic bulge test gathers the strain-hardening curve in a regime beyond the uniform elongation of the tensile tests and is used to identify the best analytical strain-hardening model in addition to the tensile test data.  
[PERSON] [44], [PERSON] [45] and Hockett-Sherby [46] have been compared in this study. A fitting procedure is applied to identify the strain-hardening law that describes the hardening best. Since Hockett-Sherby produces the optimal fit to the experimental curves, work hardening is approximated by its stress-strain relation from reference strain rate data [46].  
Figure 4: Experimental/modeled yield locus comparison.  
The initial yield stress (\(\sigma_{i}\)), saturation yield (\(\sigma_{sat}\)), exponent in the Hockett-Sherby approach (n) and plastic strain (c) parameters are shown in Table 3.  
Determination of the yield locus parameters involves normalizing experimental yield stresses and r-values using the rolling direction as a reference. Data from uniaxial, biaxial, and shear tests are integrated into a gradient-based optimization that minimizes squared differences between predicted and measured values. Weighting factors are applied to balance the contributions from each test. The yield locus exponent is constrained to remain within a realistic range (typically capped at 8) to avoid unphysical sharp corners. Iterative plane strain simulations fine-tune the exponent to accurately capture the material's anisotropic behavior.