## 1 Introduction
### Overview of Material Models  
A material model included in software for finite element analysis is essentially a mathematical representation of the behavior of a material under a given load [3]. The nonlinear relationship between stress and strain can be simplified and represented with material models of different complexity. Good examples include the practical and widely used Ramberg-Osgood model [4] or one of its variations [5; 6; 7], and more complex models like the Chaboche model [8]. A reason for the wide acceptance of the Ramberg-Osgood model is its ability to effectively describe the elasto-plastic stress-strain behavior of most metallic materials, particularly steels. The monotonic stress-strain curve obtained from tensile tests can then be approximated using the following Ramberg-Osgood Equation (4):  
\[\varepsilon=\varepsilon_{\text{e}}+\varepsilon_{\text{P}}=\frac{\sigma}{E}+ \left(\frac{\sigma}{K}\right)^{\frac{1}{2}}. \tag{1}\]  
This equation establishes the relationship between true stress, \(\sigma\), and true strain, \(\varepsilon\), under monotonic loadings. True strain can be further divided into elastic and plastic components of strain, denoted as \(\varepsilon_{\text{e}}\) and \(\varepsilon_{\text{P}}\), respectively. Material properties include Young's modulus, \(E\), the strength coefficient, \(K\), and the strain-hardening exponent, \(n\)[4]. When the relationship between \(\sigma\) and \(\varepsilon_{\text{P}}\) is plotted on a double logarithmic scale, it forms a straight line. In this plot, the slope of the line is represented by the strain-hardening exponent \(n\), while the strain-hardening coefficient \(K\) corresponds to the stress value at plastic deformation \(\varepsilon_{\text{P}}=1\)[4]. The cyclic stress-strain curve, obtained from cyclic tests, defines the relationship between stress amplitude, \(\sigma_{\text{a}}\), and strain amplitude, \(\varepsilon_{\text{a}}\), and can be represented with the following cyclic Ramberg-Osgood equation [4]:  
\[\varepsilon_{\text{a}}=\varepsilon_{\text{a,e}}+\varepsilon_{\text{a,p}}= \frac{\sigma_{\text{a}}}{E}+\left(\frac{\sigma_{\text{a}}}{K\prime}\right)^{ \frac{1}{2}}, \tag{2}\]  
where \(K^{\prime}\) is the cyclic strength coefficient and \(n^{\prime}\) is the cyclic strain-hardening exponent. When plotting the relationship between the stress amplitude \(\sigma_{\text{a}}\) and the plastic strain amplitude \(\varepsilon_{\text{a,p}}\) on a double logarithmic scale, the result is a straight line. Thus, the cyclic Ramberg-Osgood parameters can be interpreted in a similar way as in the case of monotonic loading: The cyclic strain-hardening exponent, \(n^{\prime}\), is the slope of the line, and the cyclic strength coefficient, \(K^{\prime}\), is the stress amplitude corresponding to a plastic deformation amplitude of \(\Delta\varepsilon_{\text{P}/2}=1\)[4]. Employing the Ramberg-Osgood model to characterize the stress-strain relationship of specific materials has shown some inaccuracies in the range of larger strains [5; 7]. To address this limitation, numerous alternative models based on the Ramberg-Osgood expression have been introduced in the literature. [PERSON] et al. [5] introduced a novel model describing the mechanical behavior of advanced high-strength steels by refining the Ramberg-Osgood expression. They developed an updated two-stage, plus linear stress-strain model, addressing the post-yield inaccuracies observed in conventional models [5]. The proposed formulation for two stages is as follows:  
\[\varepsilon=\left\{ \begin{array}{ll}\frac{\sigma}{E}+p\left(\frac{\sigma}{ \varepsilon_{\text{p}}}\right)^{n}&\text{ for }0\leq\sigma\leq\sigma_{\text{P}}\\ \frac{\sigma-\varphi_{\text{p}}}{E_{\text{P}}}+\left(\varepsilon_{\text{eu}} -\varepsilon_{\text{p}}-\frac{\sigma_{\text{eu}}-\varepsilon_{\text{p}}}{E_{ \text{P}}}\right)\left(\frac{\sigma-\sigma_{\text{p}}}{\varepsilon_{\text{eu}} -\varepsilon_{\text{p}}}\right)^{m}+\varepsilon_{\text{p}}&\text{ for }\sigma_{\text{p}}<\sigma\leq\sigma_{\text{eu}}\end{array}