# Approximation Method for Stress-Strain Using Metamodel Parameter Updating
## 2 Pre-Processing of the Constitutive Model
### Initialization of Constitutive Model Parameters for True Stress-Strain Curve  
The load-displacement data, which are the result of the tensile test, can be expressed by the constitutive model for engineering stress-strain. The engineering stress-strain relationship is linear. The engineering strain can be expressed by the relationship between the initial gauge length and the strain amount of the gauge length, and is as shown in Equation (1). The engineering stress can be expressed by the relationship between the elastic modulus and the engineering strain, and can be expressed by the relationship between the load and the central cross-sectional area, as shown in Equation (2). When a material is subjected to yield stress, the stress-strain relation is transformed into a non-linear property, which is shown in Equations (3) and (4). The constitutive model for true stress-strain is an  
Figure 3: Results of the tensile test (ASTM E8/E8M). Figure 2: Schematic of the ASTM E8/E8M tensile test. equation to briefly explain this non-linear behavior. Constitutive models for stress-strain are equations that can easily describe such nonlinear behaviors. \[ \begin{array}{ll}\sigma_{e}&=E\,\varepsilon_{e}\\ &=\frac{F}{\Lambda}\\ \\ \sigma_{e}&=E\,\varepsilon_{e}\\ &=\frac{F}{\Lambda}\\ \\ \varepsilon_{t}=\ln(1+\varepsilon_{e})\\ \\ \sigma_{t}=\sigma_{y}+K\left(\varepsilon_{y}+\varepsilon_{t}\right)^{n}\\ \end{array}  \tag{1}\]  
In these equations, \(E\) denotes Young's modulus, \(\delta\) is the displacement, \(L\) is the gauge length, \(\sigma_{e}\) is the engineering stress, \(\varepsilon_{e}\) is the engineering strain, \(F\) is the external force, \(\sigma_{e}\) is the central cross-sectional area of the specimen, \(\varepsilon_{t}\) is the true strain, \(\sigma_{y}\) is the yield stress, \(K\) is the plastic strength coefficient, \(n\) is the work-hardening parameter, and \(\varepsilon_{y}\) is the yield strain. Equations (1) and (2) represent the material's elasticity (\(0\leq\varepsilon\leq\varepsilon_{y}\)). Equations (3) and (4) represent the material's plastic area as a polynomial function and Gosh models, respectively, where (\(\varepsilon_{y}\leq\ \varepsilon\)). However, various constitutive models must be established because each material has a different curve characteristic for the plastic area. Table 2 lists the names, equations, and the number of parameters of the true stress constitutive models for the plastic area. Among the presented constitutive models, only the engineering dataset is an engineering stress-strain model, and the rest (Models 1-3) are true stress-strain models. In these models, \(\sigma_{s}\) is the saturation stress, \(\sigma_{y}\) is the yield stress, \(\varepsilon_{y}\) is the yield strain, \(b\) and \(p\) are material properties, \(n\) is the work-hardening parameter, \(K\) is the plastic strength coefficient, and \(\alpha\) is the combination factor. The true stress-strain constitutive model for metals is distinguished by the curved shape of the polynomial (Model 1) [44; 45; 46] and exponential models (Model 2) [47; 48; 49] listed in Table 2. If the constitutive model cannot sufficiently express the nonlinear behaviors of the specimen, combination functions (Model 3) [29; 42] can be considered. The combination function uses \(\alpha\) to combine the properties of the power and exponential functions. In addition, various component equations were used to express the nonlinear properties. The parameters of the constitutive models listed in Table 2 were initialized using the numerical linear algebra method proposed by the ASTM E646 standard, and the results are shown in Figure 4. The X-axis and Y-axis in Figure 4 represent the engineering strain and true stress values, respectively. Model 1 presents the engineering stress-strain values of the experimental data. The \(\sigma_{y}\), \(\varepsilon_{y}\), and \(\sigma_{s}\) were initialized using \(E\) and slope values between the data.