## 2 Mathematical Formulations for Cyclic Plasticity Constitutive Model  
A realistic description of the cyclic behavior of materials in the framework of infinitesimal strain theory requires the use of proper mathematical models in obtaining evolutionary equations of plastic deformations. In this study, the constitutive plasticity model is developed based on the Chaboche nonlinear kinematic hardening model alongside von Mises yield criteria and the normality flow rule. Furthermore, the total strain increment \(\dot{\epsilon}\) is expressed by the sum of the elastic (or reversible) \(\dot{\epsilon}^{e}\), thermal \(\dot{\epsilon}^{T}\), and plastic (or permanent) \(\dot{\epsilon}^{p}\) increments:  
\[\dot{\epsilon}\ =\ \dot{\epsilon}^{e}+\dot{\epsilon}^{p}+\dot{\epsilon}^{T} \tag{1}\]  
Henceforth, through the current work, tensors are identified by the bold symbols. Under non-isothermal conditions, the constitutive relation between the strain components and the stress \(\sigma\) through generalized linear elastic Hooke's law is defined as:  
\[\sigma\ =\ \textbf{W}\Big{(}\epsilon-\epsilon^{p}-\epsilon^{T}\Big{)} \tag{2}\]  
where **W** is the fourth-order material stiffness tensor. It is noteworthy that for isothermal conditions, \(\epsilon^{T}\ =\ 0\). The [PERSON] yield criterion assumes that plastic yielding will occur when the second invariant of the deviatoric stress attains some critical value. Thus, the yield function \(F\) is defined as:  
\[F(\sigma,\textbf{X})\ =\ \sqrt{\frac{3}{2}}\|(\textbf{o}^{{}^{\prime}}-\textbf{X}^ {{}^{\prime}})\|-\sigma_{y} \tag{3}\]  
where  
\[\|(\textbf{o}^{\prime}-\textbf{X}^{{}^{\prime}})\|\ =\ \sqrt{(\textbf{o}^{ \prime}-\textbf{X}^{{}^{\prime}})\cdot(\textbf{o}^{\prime}-\textbf{X}^{{}^{ \prime}})} \tag{4}\]  
where \(\textbf{o}^{\prime}\ =\ \textbf{o}-\frac{1}{3}tr(\sigma)\textbf{I}\) signifies the stress deviator in which **I** represents second-order unit tensor, \(\textbf{X}^{{}^{\prime}}\) represents the deviatoric back stress and \(\sigma_{y}\) is the yield stress. In addition, (.) identifies the inner product between \(\textbf{o}^{{}^{\prime}}\) and \(\textbf{X}^{{}^{\prime}}\). Moreover, the increment of plastic strain is evaluated based on the classical normality rule. Thus, taking into account the von Mises yield function as the flow potential, the plastic strain rate is defined as [25]:  
\[\dot{\epsilon}^{p}\ =\ \dot{\lambda}(\partial F/\partial\sigma)\ =\ \sqrt{\frac{3}{2}}\dot{ \lambda}\Big{(}\Big{(}\textbf{o}^{{}^{\prime}}-\textbf{X}^{{}^{\prime}}\Big{)} /\|\Big{(}\textbf{o}^{{}^{\prime}}-\textbf{X}^{{}^{\prime}}\Big{)}\|\Big{)}\ =\ \sqrt{\frac{3}{2}}\dot{ \lambda}\dot{\textbf{N}} \tag{5}\]  
Keeping in mind:  
\[\dot{\textbf{N}}\ =\ \frac{\Big{(}\textbf{o}^{{}^{\prime}}-\textbf{X}^{{}^{ \prime}}\Big{)}}{\|\Big{(}\textbf{o}^{{}^{\prime}}-\textbf{X}^{{}^{\prime}} \Big{)}\|} \tag{6}\]  
Here, \(\dot{\lambda}\) is the rate of plastic multiplier. The flow rule in Equation (5) ensures that the plastic strain rate develops along the normal to the yield surface having convex shape at the current loading point. The kinematic hardening model, which assumes that the yield surface undergoes a rigid translation in the stress space, is defined by the three-decomposition rule of the A-F model [1] proposed by [PERSON] [26; 27]. According to this model, the increment of back stress is defined by inserting a dynamic recovery term to linear kinematic hardening model of Prager as:  
\[\dot{\textbf{X}}\ =\ \sum_{I\ =\ 1}^{k}\dot{\textbf{X}}^{I} \tag{7}\]  
\[\dot{\textbf{X}}^{I}\ =\ \frac{2}{3}C^{l}\dot{\epsilon}^{p}-\gamma^{l}\dot{ \textbf{X}}^{I}\dot{\epsilon}^{p}_{e} \tag{8}\]  
where \(\dot{\textbf{X}}\) designates the total back stress rate, \(\dot{\textbf{X}}^{I}\) represents a part of the total back stress increment and \(k\) signifies the number of pairs (in the present research \(k\) = 3).