## 3 Automated Calibration of Chaboche Parameter
### Chaboche Model of Ratcheting  
The rate-independent form of the non-linear kinematic hardening model (NKH) recommended by [PERSON] [11; 40] is employed here. The model primarily consists of superposition of several (3, 4 or  
Figure 11: Experimental test data of (**a**) the biaxial cyclic stress-strain response of an AISI 316L SS under a strain-controlled condition; (**b**) the corresponding plastic strain to the peak stress showing the rate of ratcheting of an AISI 316L SS. Figure 10: Experimental test data of (**a**) the uniaxial cyclic stress-strain response of AISI 316L SS under a stress-controlled condition (**b**) The plastic strain corresponding to the peak stress showing the rate of ratcheting of AISI 316L SS. 5) Armstrong Frederick (A-F) kinematic hardening (KH) rules. The kinematic hardening (KH) rule encompasses a'recall term', which embeds the fading memory effect of the strain path. The constitutive equation is established on a von-Mises yield criterion, linear isotropic elasticity and the associated flow rule. The proposed equation for the backstress, \(d\alpha\), proposed by [PERSON] is given by [11]:  
\[d\alpha=\frac{2}{3}Cd\varepsilon^{p}-\gamma\alpha dp, \tag{3}\]  
\[\text{where }dp=\left|de^{p}\right|=\left[\frac{2}{3}d\varepsilon^{p}\cdot d \varepsilon^{p}\right]^{1/2} \tag{4}\]  
[PERSON] et al. [40] proposed a 'decomposed' nonlinear kinematic hardening rule in the form:  
\[d\alpha=\sum_{i=1}^{M}d\alpha_{i},d\alpha_{i}\ =\frac{2}{3}C_{i}\;de^{p}- \gamma_{i}\;\alpha_{i}\;dp,\]  
\[\text{where }dp=\left|de^{p}\right|.\]  
Both \(C_{i}\) and \(\gamma_{i}\) are constants in the Chaboche kinematic hardening model, which can be generated from the monotonic and uniaxial strain controlled stable hysteresis loop. As shown in Equation (5), the Chaboche model is an incorporation of several (A-F) hardening rules. Particular objectives are represented by each of these decomposed rules. For example, if M = 4, the first kinematic hardening rule \(\alpha_{1}\) predominantly signifies the first loop section by initiating with a quite large slope and following by quick stabilization. The fourth loop section is nearly linear along with a slight slope and is denoted by the third rule \(\alpha_{3}\). The transition knee part of the hysteresis curve is partitioned into two sections. The second section is represented principally by the fourth hardening rule \(\alpha_{4}\) and third section by the second hardening rule \(\alpha_{2}\). Hence, the initial slope of the first segment is a good estimation for \(C1\), the third segment for \(C2\), the fourth segment for \(C3\), and the second segment for \(C4\). The details on the physical description and calculation procedure of the Chaboche parameters are discussed in [11; 26]. The von-Mises yield criterion is represented as:  
\[f(\sigma-\alpha)=\left[\frac{2}{3}(s-a)\cdot(s-a)\right]^{1/2}=\ b_{0}, \tag{6}\]  
and the supplementary flow rule can be described by the following expression:  
\[d\varepsilon^{p}=\frac{1}{H}\left\{\frac{\partial f}{\partial\sigma}.d\sigma \right\}\frac{\partial f}{\partial\sigma}, \tag{7}\]  
where, \(\sigma\) is the stress tensor, \(s\) is the deviatoric stress tensor, \(\alpha\) is the current center of the yield surface, a \(s\) the current centre of the yield surface in deviatoric space, \(b_{0}\) is the size of the yield surface and is constant for a cyclically stable material, which defines the initial yield strength at the defined elastic limit of the material in an uniaxial tension test. \(H\) is the plastic modulus and in the above formulation, \(\left\{\mathbf{0}\right\}\) indicates the MacCauley bracket and the inner product is expressed by \(a\cdot b=\) aijbij. Therefore, for a one-dimensional representation of an NLK model, the \(\alpha\) backstress is achieved from Equation (8). \[\alpha=\sigma^{1}-\sigma_{0}^{1}, \tag{8}\]  
where, \(\sigma^{1}\) is the uniaxial stress of a tension specimen and \(\sigma_{0}^{1}\) is the elastic limit stress.