## 4 Numerical Simulation
### Plasticity and Fracture Models  
An isotropic hardening rule was employed, since no cyclic plasticity was involved. The true stress-true strain data were obtained from the corresponding coupon test results, and post-necking \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Yield Force \(F_{\mathrm{y}}\)(Na)**} & \multirow{2}{*}{\(F_{\mathrm{y,\textsc{FE}}}/F_{\mathrm{y,\textsc{test}}}\)} & \multirow{2}{*}{**Peak Load (0N)**} & \multirow{2}{*}{\(F_{\mathrm{x,\textsc{FE}}}/F_{\mathrm{y,\textsc{test}}}\)} & \multirow{2}{*}{\(F_{\mathrm{x,\textsc{FE}}}/F_{\mathrm{y,\textsc{test}}}\)} & \multirow{2}{*}{\(d_{\mathrm{x,\textsc{FE}}}/d_{\mathrm{x,\textsc{test}}}\)} & \multirow{2}{*}{\(d_{\mathrm{x,\textsc{FE}}}/d_{\mathrm{x,\textsc{test}}}\)} \\ \cline{5-10}  & \multicolumn{2}{c}{**Test**} & & & & & & **Test** & **FE** \\ \hline H40A85R0.5 & 106.91 & 115.29 & 1.08 & 187.06 & 197.41 & 1.06 & 15.24 & 17.04 & 1.12 \\ H50A85R0.5 & 109.93 & 116.89 & 1.07 & 197.94 & 197.70 & 1.00 & 17.00 & 16.13 & 0.95 \\ H40A85R0.5 & 110.06 & 119.65 & 1.09 & 210.55 & 216.04 & 1.03 & 17.26 & 17.38 & 1.01 \\ H40A85R0 & 107.07 & 115.80 & 1.08 & 187.19 & 197.54 & 1.06 & 16.98 & 15.78 & 0.93 \\ H40A85R1 & 110.32 & 115.37 & 1.05 & 190.76 & 199.94 & 1.04 & 14.83 & 15.45 & 1.04 \\ CoV & & & & & & \(4.96\times 10^{-4}\) & & & \(4.60\times 10^{-3}\) \\ Correlation & 0.50 & & & & & & & & \(4.60\times 10^{-3}\) \\ \hline \multicolumn{10}{c}{0.90} \\ \hline \hline \end{tabular} Notes: \(F_{\mathrm{y,\textsc{FE}}}\) and \(F_{\mathrm{y,\textsc{test}}}\) denote the yield force of the FE analysis result and the experimental result, respectively; \(F_{\mathrm{x,\textsc{FE}}}\) and \(F_{\mathrm{y,\textsc{test}}}\) denote the ultimate force of the FE analysis result and the experimental results, respectively; \(d_{\mathrm{x,\textsc{FE}}}\) and \(d_{\mathrm{x,\textsc{test}}}\) denote the ultimate displacement of the FE analysis result and the experimental result, respectively; the ultimate displacement refers to the instant when the force decreased to 85% of the peak load.  
\end{table} 
Table 2: Experimental and numerical results of specimens. Figure 6: FE model of Specim H40A85R1. modification of the data was also conducted using a method proposed by one of the authors [42]. Based on the void growth model proposed by [PERSON] and [PERSON] and the conclusion found in the literature that the fracture strain approaches infinity when \(T\) is below \(-1/3\), the relationship between fracture strain, \(\varepsilon_{f}\), (equivalent strain), and the stress triaxiality, \(T\), can be given as follows for an ideal loading case with a constant \(T\)[42]. \[\varepsilon_{f}=\left\{ \begin{array}{cc}\chi_{\mathrm{cr}}\mathbf{e}^{-1.5T}& T\geq-\frac{1}{3}\\ \infty&T<-\frac{1}{3}\end{array} \right. \tag{4}\]  
where \(\chi_{\mathrm{cr}}\) is a material constant determining the fracture strain of a material. In reality, the stress triaxiality, \(T\), commonly varies throughout the whole loading process until rupture, and a damage index in incremental form shown in the following equation was obtained according to Equation (4) [42], which postulated that crack would initiate when \(D_{ini}\) reached 1.0. \[\mathrm{d}D_{\mathrm{ini}}=\left\{ \begin{array}{cc}\frac{\mathrm{d}\varepsilon _{\mathrm{eq}}^{\mathrm{P}}}{\chi_{\mathrm{cr}}\mathbf{e}^{-1.5T}}&T\geq- \frac{1}{3}\\ 0&T<-\frac{1}{3}\end{array} \right. \tag{5}\]  
where \(D_{\mathrm{ini}}\) is the damage index for crack initiation and \(\varepsilon_{\mathrm{eq}}^{\mathrm{P}}\) is the equivalent plastic strain. This ductile fracture model was found to be effective for crack initiation prediction of monotonic loading and cyclic incremental loading, and was thus employed in this study. The value of \(\chi_{\mathrm{cr}}\) obtained from the coupon test results was 2.4. A simple crack propagation rule was employed, which postulated instant crack propagation after crack initiation.