# Approximation of Non-Linear Stress-Strain Curve for GFRP Tensile Specimens by Inverse Method
## 3 Optimization of Inverse Methods
#### Parameter Optimization  
D-optimal DOE was applied to minimize the number of iterations in the optimization process [31]. Here, the process of deriving the error range between the experimental and FEA results and adjusting the parameters was repeated. The same optimization method and number of iterations were applied to all inverse methods. Figure 9 shows the analysis results. As the engineering stress and true stress models based on ASTM D638-14 were excluded from the optimization targets, the errors indicated by Figure 7 could be observed. When the optimized inverse methods were used in the FEA, the results were very similar to the engineering S-S curve obtained from the experiment. The error of the optimized inverse method (\(\mathrm{e}_{\mathrm{q}}\)) can be seen in Figure 10. The error patterns of the power law and that of the exponential law models were reversed, and it can be seen that the Hockett-Sherby formula was very accurately optimized.  
The average of cumulative errors (\(\overline{e}_{\mathrm{q}}\)) is listed in Table 3. Considerable errors were observed in the three data points excluded from the optimization process. Among the power law models, the [PERSON] model obtained the most accurate results, followed by the [PERSON], [PERSON], and Hollomon models. Among the exponential law models, the [PERSON] model obtained the most accurate results, followed by the Hockett-Sherby model.  
Figure 9: _Cont._  
Figure 10: _Cont._  
Figure 9: Engineering S–S curve of the optimized models: (**a**) inverse models of GFRP wt 20%GF; (**b**) inverse models of GFRP wt 30% GF; (**c**) inverse models of GFRP wt 40% GF.  
Figure 10: Error values of the optimized inverse method models: (**a**) optimized inverse models of GFRP wt 20% GF; (**b**) optimized inverse models of GFRP wt 30% GF; (**c**) optimized inverse models of GFRP wt 40% GF.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Eq.**} & \multirow{2}{*}{**Applied S-S**} & \multicolumn{3}{c}{\(\hat{e_{\textbf{q}}}\) [MPa]} & - \\ \cline{5-6}  & & **Curve Model** & **wt 20\% GF** & **wt 30\% GF** & **wt 40\% GF** & **\(\mathbf{c_{r}}\) [MPa]** \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{1} & \multirow{2}{*}{ASTM Engineering model} & \multirow{2}{*}{13.54} & \multirow{2}{*}{17.00} & \multirow{2}{*}{15.75} & \multirow{2}{*}{15.43} \\  & & & & & & \\
2 & 3 & ASTM True model & 12.62 & 15.97 & 14.67 & 14.42 \\
3 & 4 & Hollomon & 0.96 & 1.20 & 1.18 & 1.11 \\
4 & 5 & Ludwik & 0.68 & 1.09 & 0.82 & 0.86 \\
5 & 6 & Swift & 0.57 & 0.77 & 0.71 & 0.69 \\
6 & 7 & Gosh & 0.73 & 0.95 & 0.91 & 0.86 \\
7 & 8 & Voce & 0.85 & 0.97 & 0.87 & 0.90 \\
8 & 9 & Hockett-Sherby & 0.17 & 0.06 & 0.29 & 0.17 \\
9 & 19 & Kim–Tuan & 0.12 & 0.16 & 0.32 & 0.20 \\
10 & 20 & Swift–Voce & 0.11 & 0.15 & 0.29 & 0.18 \\
11 & 21 & Ludwik–Voce & 0.38 & 0.26 & 0.45 & 0.36 \\  & & Swift & 1.25 & 0.18 & 0.39 & 0.61 \\
12 & 22 & Hockett–Sherby & & & & \\
13 & 23 & [PERSON] & & & & \\  & & Hockett–Sherby & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Optimized values of the advanced inverse method.