## 3 Parameter Optimization of the Constitutive Model
### Optimal Selection of Thermal Stress Parameters  
The parameters for the determination of thermal stress include \(\overline{\
abla}\), \(\hat{\sigma}_{sh}\), \(n_{2}\), \(\alpha\), \(\beta\), \(\hat{\varepsilon}_{0}\), \(\hat{\varepsilon}_{0}\), \(p\), and \(q\). Since \(\hat{\varepsilon}_{0}\) and \(\hat{\varepsilon}_{0}\) are not independent in parameter fitting, and both are included  
Figure 6: True stress–strain curve at a strain rate of \(10^{4}\) s\({}^{-1}\). Figure 7: Curve-fitting results of non-thermal stress at a strain rate of \(10^{4}\) s\({}^{-1}\). in the logarithmic function, the influence of changes in value on thermal stress is much smaller than that of the previous \(\alpha\) and \(\beta\). These values need to be predicted to better determine the parameters. S32760 duplex stainless steel has both BCC and FCC structures, and the constitutive equation of HCP metal can be regarded as a linear combination of BCC and FCC. Therefore, the reference strain rate of HCP metal can be used to determine the reference strain rate of S32760. According to [PERSON] (1960), \(\dot{\epsilon}_{0}=\dot{\gamma}_{0}/f(\gamma,T)\), where \(\dot{\gamma}_{0}=bl_{0}p_{w}w_{0}\), \(b=3\times 10^{-10}\) m, \(l_{0}=500\) b, \(p_{w}=10^{11}/\text{cm}^{2}\), \(T=598\) k, and \(\gamma=0.25\), and using these values, \(\dot{\epsilon}_{0}=3\times 10^{9}\) s\({}^{-1}\). Since the value of \(\dot{\epsilon}_{s0}\) is generally two orders of magnitude larger than that of \(\dot{\epsilon}_{0}\), \(\dot{\epsilon}_{s0}=1\times 10^{11}\) s\({}^{-1}\). After determining \(\dot{\epsilon}_{0}\) and \(\dot{\epsilon}_{s0}\), the remaining parameters can be solved using the method of reverse identification. The purpose of solving parameters is to find a set of material parameters that correlate with the physical meaning so that the curve solved using the constitutive equation is as consistent as possible with the curve obtained through experimentation. The objective function is defined using the principle of the least-square method. The sum of the squares of the difference between the stress value calculated using the constitutive equation and the stress value measured through experimentation is taken as the objective function so that the objective function is as small as possible. The objective function can be expressed as follows:  
\[F(x)=\sum_{i=1}^{N}(Y_{i}(x)-Y_{i}^{*})^{2} \tag{41}\]  
where \(x=[x_{1},x_{2},\ldots,x_{m}]\) is the parameter to be optimized, \(N\) is the number of points taken, \(\gamma_{i}\) is the value calculated at the \(i\)-th point, and \(Y_{i}^{*}\) is the value measured using the \(i\)-th point experiment. In this study, the MATLAB program was used to optimize the computation of the objective function. The flowchart of the constitutive model parameter optimization program is shown in Figure 8. In addition, in order to ensure the accuracy of the calculation results, it was necessary to determine the theoretical range of each parameter. The remaining unknown parameters were grouped as follows:  
1. Determination of \(\overline{Y}\) and \(\hat{\sigma}_{sh}\): Based on the work carried out by [PERSON] et al. [PERSON] (1960), the terms \(1560\times f(\varepsilon,T)\) and \(\hat{\sigma}_{sh}+\overline{Y}\varepsilon^{n_{2}}\) in Equation (8) belong to the same category, and thereby, this equation could be effectively employed as a basis for determining the ranges of \(\overline{Y}\) and \(\hat{\sigma}_{sh}\). The median value of the range of \(\hat{\sigma}_{sh}\) was 1560, and it was extended to both ends to determine \(\hat{\sigma}_{sh}\in[1000,2000]\). The range of temperature \(T\) was set as 77-998 K, while strain was maintained in the range of 0-0.6. Thus, we estimated \(\overline{Y}\in[1500,3500]\). 2. Determination of \(\alpha\) and \(\beta\): The Boltzmann constant was \(k=1.3806505\times 10^{-23}\) J/K and for S32760 stainless steel, \(b=3\times 10^{-10}\) m, shear modulus \(E=159.2\) GPa, \(a_{0}\in(0.2,2)\) [PERSON] (1960); [PERSON] (1960), and \(\alpha=k/(b^{3}Ea_{0})\).