## 3 Constitutive Model and Simulation Results with MATLAB
### The Constitutive Model with Martensite Transformation  
The Garion model [30] used the linear mixed hardening model to calculate the backstress. The content of deformed martensite affects the linear kinematic hardening and dynamic recovery term, which can better simulate the secondary hardening phenomenon under uniaxial loading. However, it is difficult to describe the primary hardening phenomenon of materials after they enter plasticity. The accurate simulation of the ratcheting strain depends on the simulation effect of the model on thehysteresis loop. As the Garion model cannot describe hysteresis well, it cannot accurately describe the ratcheting strain. However, the description of the content of deformed martensite and its influence on the hardening of the material by this model is worth referencing. The volume fraction of martensite is the most popular macroscopic internal variable specifying the growth of the martensitic phase [35; 36]. Bain strain is added to the strain calculation due to the influence of the volume strain caused by the martensitic transformation. The martensite content and cumulative plastic strain values are obtained by experiments, and the evolution law of martensitic content is obtained by fitting the curve. In this paper, the martensite content dependent variable \(\xi\) is correlated with the Ohno Wang II model and isotropic hardening model, which can simulate the ratcheting of S30408. We propose the following constitutive model. 1. According to the study of [PERSON], we assumed that strain \(\epsilon\) is divided into elastic part \(\epsilon^{\rm e}\) inelastic part \(\epsilon^{\rm p}\) and phase transformation part \(\epsilon^{\rm bs}\) \[\epsilon=\epsilon^{\rm e}+\epsilon^{\rm p}+\epsilon^{\rm bs}.\] (11)
2. The elastic part obeys [PERSON]'s law \[\mathbf{\sigma}=\mathbf{\mathrm{E}}:\left(\epsilon-\epsilon^{\rm p}-\epsilon^{\rm bs }\right).\] (12)
3. The plastic part can be stated as \[\mathrm{d}\epsilon^{p}=\frac{3}{2}\frac{\langle ds\mathbf{\cdot}\mathbf{\mathrm{n}} \rangle}{h_{s}}\mathbf{\mathrm{n}}.\] (13)
4. The martensitic transformation part can be expressed in terms of relative volume change \(\Delta v\), due to the phase transformation [29], as \[\epsilon^{\rm bs}=\frac{1}{3}\Delta v\xi\mathbf{\mathrm{I}}=\frac{1}{3}\frac{(V_{ m}-V_{a})}{V_{a}}\xi\mathbf{\mathrm{I}}.\] (14) where \(V_{a}\) and \(V_{m}\) represent the unstresses specific volumes occupied by the austenite and martensite, respectively. The value of \(\Delta v\) is approximately 0.02-0.05, depending on the alloy composition [24]. 5. We assumed the material follow von [PERSON] yield criterion, which can be given by \[f(\mathbf{\sigma},\mathbf{\alpha},R)=J_{2}(\sigma-\mathbf{\alpha})-\sigma_{y}-R.\] (15)
6. The backstress \(\mathbf{\alpha}\) is altered by the induced martensite, and the evolution law is postulated in the following form: \[\mathbf{\alpha}=\sum_{i=1}^{M}\mathbf{\alpha}_{i}.\] (16) \[\mathrm{d}\mathbf{\alpha}_{i}=\mathrm{d}\mathbf{\alpha}_{i}^{a}+\mathrm{d}\mathbf{ \alpha}_{i}^{a\xi}=\gamma_{i}\left(\frac{2}{3}\tau_{i}\delta e^{p}-\left( \frac{\mathbf{\overline{\alpha}}_{i}}{r_{i}}\right)^{m_{i}}\left(d\epsilon^{p}: \frac{\mathbf{\alpha}}{\mathbf{\overline{\alpha}}_{According to [28], austenitic stainless steel S30408 is characterized by rapid hardening due to martensitic transformation. The martensite content of the material increases rapidly with the increase of cumulative plastic strain, and then tends to be stable. The occurrence of martensite transformation has a certain influence on the cyclic hardening of the material, which can be attributed to isotropic hardening.  
\[\mathrm{d}R=\frac{2}{3}(1-\xi)b(Q\xi-R)dp. \tag{19}\]  
where \(R\) is the isotropic hardening parameter, and \(p\) is the cumulative plastic strain. The term \((1-\xi)\) is added to compensate for the assumption that the martensite is elastic [29]. In reality, the martensite shall rather be considered elasto-plastic. Therefore, the contribution to the hardening of the material is slightly smaller.