## 1 Introduction  
An accurate description of the effect of plastic strain on the mechanical behaviour of materials is necessary for a reliable prediction of the mechanical response beyond the elastic limit. There are many industrial applications in which plasticity plays a fundamental role, such as contact mechanics [1], fatigue analysis [2] and residual stresses' analysis [3]. In our previous paper, Ref. [4], it is evident that, without considering plasticity around the notch, the stress predictions could reach very high values, which are approximately similar to the ultimate stress of the material and not meaningful for cyclic loading analysis. The Chaboche kinematic hardening (CKH) model is a powerful constitutive law to model the plasticity of metals. It was introduced by [PERSON] [5], extending the Armstrong and Frederick model [6] by adding multiple backstress components with different properties. It can also be efficiently combined with the isotropic hardening rule proposed by [PERSON] [7], as in [8; 9; 10], and both hardening rules are implemented in Ansys commercial finite element software [11]. The combination of a CKH model and Voce isotropic hardening model results in the Chaboche isotropic kinematic hardening (CIKH) model. The CKH model has often been involved in fatigue analyses, as in [12; 13; 14]. In these papers, the theory of critical distances and the strain energy density criterion, which are efficient tools for predicting the fatigue life of components, were used, combined with the CKH model showing better predictions rather than maintaining a simple, linear, elastic behaviour. This model was also employed in contact mechanics problems, as in [15]. In recent years, researchers made some modifications to the original formulation of the CIKH model, as in [16; 17]. In this latter paper, a refinement was proposed to properly model the ratcheting rate, then elaborated in [18]. A backstress with a threshold was introduced. Below this threshold the backstress behaves according to linear Prager rule, while it behaves like Chaboche nonlinear hardening model above the threshold. The concept of the backstress with threshold was extended in [19] to model both the ratcheting rate and the mean stress relaxation. Alternatively, to properly model the ratcheting rate, in Ref. [20] a multiplicative hardening rule was introduced, in which one of the two coefficients of some backstress components is a function of another dimensionless, second-order, internal variable that nonlinearly evolves as the backstress components of the CIKH model. This approach was applied to model the cyclic plastic behaviour of Steel 316 L and Carbon Steel 1026 in [20], and was then extended to aluminium alloy 7075-T6 in [21; 22], and to Ti-6 Al-4V in [23]. The CKH model is generally combined with the Von Mises yielding law but, in Ref. [24], it was employed together with the Hill yielding law for anisotropic metals [25]; for example, to model the cyclic plastic behaviour of Managing 300, obtained through additive manufacturing technologies. In principle, the stabilized cycles of strain-controlled tests alone can be engaged to determine the parameters. However, if the transient of the stress-controlled test plays a fundamental role in further analyses, the ratcheting rate must also be considered. In Refs. [26; 27], the ratcheting rate was employed to calculate the parameters, whereas in Refs. [28; 29] both uniaxial stress-controlled tests and strain-controlled tests were exploited. The algorithms used to calculate these parameters can follow different approaches, such as genetic algorithms, as in [30; 31], particle swarm optimization in [32], and the differential evolution algorithm in [33]. All of these techniques require a great computational effort. Alternatively, the Bouc-Wen (B-W) model [34; 35] is widely used to replicate the hysteretic behaviours of systems such as wire rope isolators [36], piezoelectric actuators [37] and seismic isolation in [38].