## 3 Cyclic Elasto-Plasticity in Fatigue
### Determination Hardening Parameters
#### 3.2.2 Cyclic Elasto-Plastic Chaboche Hardening Model  
The cyclic elasto-plastic model provided by [PERSON] (RO) is widely applicable to many different fatigue approaches. However, since this model is an empirical model, it has no hardening formulation; as a result, the RO model cannot predict transient effects such as cyclic hardening or softening, ratcheting, mean stress relaxation, etc. Instead, the Chaboche hardening model may be applied to deal with this concern. The Chaboche hardening model has been widely used in various studies to characterize cyclic hardening behavior, mainly in steels. Chaboche kinematic hardening was developed considering the [PERSON] plasticity criterion (Equation (7)).  
For strain-controlled stable cyclic conditions, the kinematic hardening is written using Equation (16) as [29]:  
\[\sigma_{a}=\sigma^{\prime}_{y}+\frac{\Delta\alpha}{2}=\left(\sigma_{y}+R^{ \prime}_{y}\right)+\sum_{i=1}^{n_{a}}\frac{C_{i}}{\gamma_{i}}\tanh\left( \gamma_{i}\varepsilon^{p}_{a}\right), \tag{16}\]  
where the parameters \(C_{i}\) and \(\gamma_{i}\) are determined by fitting the Equation (16), and \(R^{\prime}_{y}\) denotes the variation of the elasticity limit surface for the stabilized-state hysteresis loop.  
A useful feature of the Chaboche hardening model is the possibility to connect the isotropic hardening to the kinematic hardening behavior.The isotropic hardening permits a change in the size of the elasticity limit surface and is given by  
\[\dot{R}=b_{\infty}(R_{\infty}-R)\dot{p}, \tag{17}\]  
where \(R_{\infty}\) denotes the saturation of isotropic hardening and \(b_{\infty}\) denotes the rate of isotropic hardening. Both \(R_{\infty}\) and \(b_{\infty}\) are parameters depending on the material and temperature.  
Higher values of \(b_{\infty}\) indicate a faster saturation of the model, otherwise the yield surface tends to evolve more slowly. Integrating the Equation (17) results that  
\[R=R_{\infty}[1-\exp(-b_{\infty}p)]. \tag{18}\]  
Equation (18) gives the size of the elasticity limit surface, and it may be used both in cyclic and monotonic loading conditions. Under strain-controlled cyclic conditions, the evolution of the elasticity limit surface (Equation (18)) may be represented in terms of the normalized maximum stress, as a function of the accumulated plastic strain, such that  
\[R^{*}=\frac{R}{R_{\infty}}=\frac{\sigma_{max}(N_{i})-\sigma_{max}(N_{i=1})}{ \sigma_{max}(N_{i=s})-\sigma_{max}(N_{i=1})}, \tag{19}\]  
with \(\sigma_{max}(N_{1})\), \(\sigma_{max}(N_{i=s})\), and \(\sigma_{max}(N_{i})\) denoting the maximum stress corresponding to the first cycle, the stabilized cycle, and the \(i\)-th cycle, respectively. \(R_{\infty}\) denotes the difference between the value of yielding stress in the stabilized cycle and the value of yielding stress in the first cycle.  
The value of \(R_{\infty}\) is dependent on the imposed deformation amplitude. Increasing the total strain amplitude, the quantity of hardening/softening, \(R_{\infty}\), tends to increase with the maximum value, \(R_{\infty,max}\). \(R_{\infty,max}\) indicates that, from a certain value of applied strain, which is around 2 or 3% for medium and high strength steels, the quantity of variation in the elasticity limit surface tends to remain unchanged [40; 41; 42]. The variation of \(R_{\infty}\) with the applied strain amplitude, \(\varepsilon_{a}\), is given by the decreasing power law, written as [43]  
\[R_{\infty}=R_{\infty,max}[1-\exp(k_{0}(\varepsilon_{a}-\varepsilon_{0}))], \tag{20}\]  
where parameters \(k_{0}\) and \(\varepsilon_{0}\) are estimators for the amount of cyclic softening.  
With respect to the hardening parameter that governs the saturation rate, \(b_{\infty}\), this parameter may also be computed explicitly as  
\[b_{\infty}=-\frac{\ln[1-R^{*}]}{2N_{f}\Delta e^{p}}. \tag{21}\]