## 3 Constitutive Model and Simulation Results with MATLAB  
### Several Nonlinear Constitutive Models  
Five advanced constitutive models were tentatively used to simulate the ratcheting strain of cryogenic temperatures for S30408 austenitic stainless steel.  
Figure 4: Experimental values of ratcheting.  
Ohno-Wang model  
To better describe ratcheting behavior, [PERSON] and [PERSON] [6] proposed an equation representing the critical state of the dynamic recovery term, and assumed that each component of backstress \(\alpha_{i}\) only works when its value reaches the critical value (OW-I). The OW-I model is proposed as follows  
\[\alpha=\sum_{i=1}^{M}\alpha_{i},d\alpha_{i}=\frac{2}{3}r_{i}\gamma_{i}de^{p}- \gamma_{i}H(f_{i})\bigg{/}de^{p}:\frac{\alpha_{i}}{\overline{\alpha_{i}}} \bigg{)}\alpha_{i}. \tag{2}\]  
where \(f_{i}=\overline{\alpha_{i}}^{2}-r_{i}^{2}\). However, the OW-I model produces closed hysteresis loops and hence cannot simulate uniaxial ratcheting. To eliminate this limitation, [PERSON] and [PERSON] [7] proposed a slight nonlinearity for each rule by introducing an exponential relation, and before reaching its critical state, the dynamic recovery term is partially operative (OW-II). The formula is proposed as follows  
\[\alpha=\sum_{i=1}^{M}\alpha_{i},d\alpha_{i}=\frac{2}{3}r_{i}\gamma_{i}de^{p}- \gamma_{i}\bigg{(}\frac{\overline{\alpha_{i}}}{r_{i}}\bigg{)}^{m_{i}}\bigg{/} de^{p}:\frac{\alpha_{i}}{\overline{\alpha_{i}}}\bigg{)}\alpha_{i} \tag{3}\]  
2. AKO model and its modification  
To improve the description of the uniaxial ratcheting behavior of the OW-I model, [PERSON] [8] combined the OW-I model with the A-F model (AKO). The AKO model is proposed in the following form  
\[\alpha=\sum_{i=1}^{M}\alpha_{i},d\alpha_{i}=\frac{2}{3}r_{i}\gamma_{i}de^{p}- \mu_{i}\gamma_{i}\alpha_{i}dp-\gamma_{i}H(f_{i})(d\lambda_{i})\alpha_{i}. \tag{4}\]  
where \(\mathrm{d}\lambda_{i}=d\varepsilon^{p}:\frac{\alpha_{i}}{r_{i}}-\mu_{i}dp\).  
[PERSON] [19] revised the \(\mu_{i}\) in the AKO model (AKO IV), where the \(\mu_{i}\) is expressed as follows  
\[\mu_{i}=\eta\bigg{/}\frac{\partial f}{\partial\sigma}:\frac{\alpha_{i}}{ \overline{\alpha_{i}}}\bigg{\rvert}^{\chi},d\eta=d\eta_{1}+d\eta_{2}. \tag{5}\]  
\[\mathrm{d}\eta_{1}=\omega_{1}(\eta_{\infty 1}-\eta_{1})dp,\mathrm{d}\eta_{2}= \omega_{2}(\eta_{\infty 2}-\eta_{2})dp. \tag{6}\]  
where \(\mu_{i}\) is the ratcheting parameter, \(\eta\) is a parameter affecting the ratchet deformation, \(\eta_{\infty 1}\), \(\eta_{\infty 2}\), \(\eta_{1}\), \(\eta_{2}\), \(\omega_{1}\) and \(\omega_{2}\) are material constants, \(\mathrm{d}p\) is the magnitude of the plastic strain increment tensor, and \(\chi\) is the multiaxial parameter. The uniaxial ratcheting studied in this study is \(\chi=0\).  
3. OW II-AF and its modified model  
[PERSON] [10] superimposed the A-F model and the OW-II model (OW II-AF) to improve the prediction of the uniaxial ratcheting effect of the OW-II model. The formula is proposed as follows  
\[\alpha=\sum_{i=1}^{M}\alpha_{i},d\alpha_{i}=\frac{2}{3}r_{i}\gamma_{i}de^{p}- \mu_{i}\gamma_{i}\alpha_{i}dp-\gamma_{i}\bigg{(}\frac{\overline{\alpha_{i}}}{ r_{i}}\bigg{)}^{m_{i}}\langle d\lambda_{i}\rangle\alpha_{i}. \tag{7}\]  
[PERSON] [20] proposed the modified OW II-AF model (M OW II-AF) by superimposing the OW-II model and the A-F model. The kinematic hardening is given by  
\[\alpha=\sum_{i=1}^{M}\alpha_{i},d\alpha_{i}=\gamma_{i}\bigg{(}\frac{2}{3}r_{i }de^{p}-\mu_{i}\phi(p)\alpha_{i}dp-\phi(p)\bigg{(}\frac{\overline{\alpha_{i} }}{r_{i}}\bigg{)}^{m_{i}}\bigg{/}de^{p}:\frac{\alpha_{i}}{r_{i}/\phi(p)}\bigg{)} \alpha_{i}\bigg{)}. \tag{8}\]  
where \(\phi(p)\) is cyclic hardening/softening function of materials \(\phi(p)=\phi_{\infty}+(1-\phi_{\infty})e^{-\omega_{\infty}p}\), and \(\phi_{\infty}\) is the stable value of Marquis.