# Approximation of Non-Linear Stress-Strain Curve for GFRP Tensile Specimens by Inverse Method
## 2 Experiment and FEA Setup
### Inverse Methods  
Inverse methods are useful when the change in the thickness of the necking zone cannot be accurately measured after the yield stress is reached. They can be used as correction measures for the stress value when the engineering S-S curve obtained experimentally is different from the engineering S-S curve obtained using the FEA.  
\[\sigma_{hollomon}=c_{1}\epsilon_{e}^{c_{2}} \tag{4}\]  
\[\sigma_{huduxi}=c_{1}+c_{2}\epsilon_{e}^{c_{3}} \tag{5}\]  
\[\sigma_{swift}=c_{1}(c_{2}+\epsilon_{e})^{c_{3}} \tag{6}\]  
Figure 5: Results of the finite element analysis: (**a**) result of the finite element analysis; (**b**) total CPU time by element type and solver type.  
\[\sigma_{gosh}=c_{1}+c_{2}(c_{3}+\varepsilon_{e})^{c_{4}} \tag{7}\]  
\[\sigma_{vocz}=c_{1}-(c_{1}-c_{2})\times\exp(-c_{3}\varepsilon_{e}) \tag{8}\]  
\[\sigma_{h.s}=c_{1}-(c_{1}-c_{2})\times\exp(-c_{3}\varepsilon_{e})^{c_{4}} \tag{9}\]  
Inverse methods to obtain the true stress are largely divided into power law models and exponential law models. The models of [PERSON] [24], [PERSON] [25], [PERSON] [26], and [PERSON] [27] shown in Equations (4)-(7) can be considered as representative power law models. The models of [PERSON] [28] and [PERSON] [29] shown in Equations (8) and (9) can be considered as representative exponential law models. In each model, \(c\) is a material parameter, and its initial value is determined using the logarithmic curve method given in ASTM E646-16 [30].  
In this theoretical model, parameters of the test environments such as the ambient temperature and test speed can be included. However, in this study, the parameters of the test environments were not considered, as the aim of the study was to propose an inverse method that could approximate the plastic deformation behavior of GFRP.  
Figure 6 shows the true S-S curves derived using Equations (4)-(9). In the figure, test data for the curves were included for a comparison between the true stress and the engineering stress.  
It was found that the true S-S data obtained using the ASTM D638-14, Hollomon, Ludwik, and Swift equations were not significantly different from the experimental data. In contrast, the Gosh model and the exponential law model exhibited significant differences from the experimental data. The physically reliable and accurate data of the engineering S-S curve and all true stress curves were estimated values.  
The true S-S curve data were used as the mechanical properties of the FEA software, and were entered as the point data of 100 divisions based on the total strain. The engineering S-S curve data shown in Figure 7 were derived using the simulation shown in Figure 4f.  
To quantitatively demonstrate the expression accuracy of the analysis results for an actual physical phenomenon, the engineering S-S curve obtained experimentally was compared with the engineering S  
Figure 6: Theoretical true S–S curves: (**a**) GFRP wt 20% GF; (**b**) GFRP wt 30% GF; (**c**) GFRP wt 40% GF.  
Figure 7: _Cont._  
Figure 7 shows the results of the FEA that was conducted using the estimation equations. For the engineering S-S curves, the experimental data were significantly different from the analysis data, and the \(c\) value of ASTM E646-16 resulted in inaccurate results. This can also be observed in Figure 8, which shows the graphs obtained using Equation (10) for a comparison. The arithmetic average values for \(\text{e}_{\text{q}}\) are listed in Table 2. Among the power law models, the Gosh model exhibited the most accurate approximations of the deformation when compared to the actual values, and among the exponential law models, the Voc model exhibited the best approximation.  
Figure 8: \(Cont\).  
Figure 7: Experimental curves and finite element analysis (FEA) results: (**a**) inverse models of GFRP wt 20% GF; (**b**) inverse models of GFRP wt 30% GF; (**c**) inverse models of GFRP wt 40% GF.