# Localized Necking Model for Punching Fracture Simulation in Unstiffened and Stiffened Panels
## 3 Fracture Model Formulation
### Fracture Model Parameters Identifications  
The loading path to fracture is defined as the evaluation of the stress state parameters \(\{\eta,\ \overline{\theta}\}\) according to the increase in the equivalent plastic strain when fracture initiates. The commercial software package Abaqus/Explicit was used to extract the stress state at the element, which showed the highest equivalent plastic strain at the fracture initiation obtained experimentally. Figure 8 shows a comparison of the force-stroke displacement curves between the test and simulation. The numerical results show a higher accuracy in the prediction of the force levels of the tests. Therefore, the plasticity model worked well. The determination of fracture initiation was defined as a sudden drop in force.  
Figure 6: Comparison of experimental and simulated force-displacement curves for NT.  
Figure 7: Flow stress beyond the onset of necking of JIS G3131 SPHC.  
Figure 9 shows the variation in the stress state parameters with increasing equivalent plastic strain. The loading path of the CH specimen was almost constant at fracture, which corresponds to the uniaxial tension stress state \(\left\{\eta=1/3,\ \overline{\theta}=1.0\right\}\). The stress state of the PST specimen initially showed plane strain tension \(\left\{\eta=2/3,\ \overline{\theta}=0.0\right\}\), but the Lode angle parameter changed linearly with the increasing equivalent plastic strain. In the case of the shear specimen, two stress state parameters changed to the mixed shear/tension state from pure shear \(\left\{\eta=0.0,\ \overline{\theta}=0.0\right\}\) because the symmetric plane of the gauge part in the shear specimen rotated with increasing the loading step.  
An optimization was conducted to identify the HC model parameters \(\left\{a,b,c\right\}\). The objective function \(f(\chi)\) is used to minimize the sum of the square of residues of the numerical error, and the design variables were \(\chi=\left\{a,b,c\right\}\). The optimization algorithm with the linear damage accumulation law is as follows:  
\[\chi=\text{argmin}\{\left\langle\left.\left|\chi\right|\right\}\text{ with }f(\chi)=\sum_{i=1}^{3}\left(\left(\int_{0}^{\overline{s}_{f}}\frac{d\overline{ \varepsilon}_{p}}{\overline{\varepsilon}_{HC}^{pr}[\eta,\overline{\theta}]} \right)_{i}-1\right)^{2}, \tag{27}\]  
where index \(i\) corresponds to the number of calibration tests, which is three for CH, PST, and SH specimens. The MATLAB toolbox with a derivative-free algorithm was used to solve the optimization problem. The calibrated HC fracture model parameters are \(a=1.5979,\ b=1.3599,\ c=0.0012\).  
Figure 8: The comparisons of force-displacement between experimental and numerical results.  
The DSSE parameter \(d\) is identified using the isotropic hardening function \(k\big{[}\bar{\varepsilon}_{p}\big{]}\). The equivalent plastic strain at the onset of localized necking under the plane strain tension condition, \(\bar{\varepsilon}_{DSSE}^{PST}\), is obtained based on the Considence instability criterion:  
\[k\Big{[}\bar{\varepsilon}_{DSSE}^{PST}\Big{]}=\frac{2}{\sqrt{3}}\frac{dk\Big{[} \bar{\varepsilon}_{DSSE}^{PST}\Big{]}}{d\bar{\varepsilon}_{p}}. \tag{28}\]  
Then, parameter \(d\) is solved using an implicit equation:  
\[\Big{(}1+2^{d-1}\Big{)}^{\frac{1}{d}}=\sqrt{3}\Bigg{(}\frac{\bar{\varepsilon }_{DSSE}^{PST}}{b}\Bigg{)}^{-p}. \tag{29}\]  
For the JJS G3131 SPHC steel, \(d\) was 1.7027. Figure 10 presents a 3D HC fracture locus, where the red line corresponds to the plane stress condition. For the HC model, the JIS G3131 SPHC steels tend to be more sensitive to the Lode angle parameter than the stress triaxiality. Figure 11 shows the calibrated HC ductile fracture with the DSSE-localized necking envelope under the plane stress condition. The DSSE model is valid only between uniaxial and equibiaxial tension (\(1/3\leq\eta\leq 2/3\)).  
Figure 9: Variation of stress state parameters as function of equivalent plastic strain.