# Cyclic Elasto-Plastic Behaviour of 51 CrV4 Steel for Railway Parabolic Leaf Spring Design
## 1 Introduction  
Over the years, engineers and researchers have developed more efficient geometries and processes to maximise the strength of suspension elements. Although leaf springs are elements that have been used for many years, these components have evolved both in terms of their resistance and geometries. The evolution of this component is also dependent on the type of sectors where these elements are inserted, as for example, road vehicles [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and rail vehicles [11, 12, 13, 14, 15]. In the case of the railway freight sector, parabolic leaf springs (illustrated in Figure 1) that are standardised by the International Union of Railways (UIC) [16] have gained higher representation due to their optimised geometry, which allows an increase in the specific energy storage capacity and reduction in the friction effects, which consequently increases the safety and quality of wagon motion [17, 18]. This two-axle freight vehicle suspension component is produced from high-strength steel, designated as 51 CrV4 [19, 20, 21, 22], which might be applicable in various industry machinery components in addition to road and rail suspension vehicles [23, 24], such as steering knuckles, connecting rods, spindles, pumps, gear shafts, pinions, forged crankshafts, and gears [21, 25]. Despite the good mechanical resistance characteristics presented in several investigations [24; 26; 27; 28], due the presence of geometric notches [18; 29; 30; 31; 32] and high surface roughness of its leaves (average up to \(R_{a}=7\) um [33] and maximum roughness \(R_{z}=40\) um [34]), local stress levels can reach or exceed the yield limit of the material. Operating under these cyclic loading conditions, the cyclic elasto-plastic material behaviour model must be considered. Over the years, different constitutive models have been developed to represent cyclic plastic behaviour. Some examples are the linear kinematic hardening model proposed by [PERSON] [35], the multiple-yield surface model [36], two-surface plasticity model, yield and boundary surface model [37; 38], and others such as [PERSON] [39], [PERSON] [40], [PERSON] [41], and [PERSON] [42]. In fact, the development of the cyclic plasticity models helped in the comprehension of the different mechanical responses associated with the plastic domain. Transient effects associated with Bauschinger's effect, ratcheting, hardening or softening straining, and stress relaxation are some of the examples observed and investigated more frequently in cyclic plasticity [43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56]. Some of these cyclic plasticity models have been used in numerical models to perform fatigue prediction analysis in cyclic elasto-plastic conditions for both smooth and notched components [57; 58; 59; 60; 61; 62]. Even in components subjected to cyclic loadings below the yielding strength, due to the surface roughness, localised plastic deformation may occur [63; 64; 65]. Under these conditions, cyclic plasticity should be considered for fatigue crack initiation analysis. Moreover, in fatigue crack propagation analysis, there is localised plasticity at the crack tip. Thus, plasticity models should also be considered in determining the cyclic material behaviour at the crack tip [66; 67; 68; 69]. In this investigation the Chaboche model, widely used for designing structural components under cyclic loading, is considered. The Chaboche model considering the effect of combined kinematic and isotropic hardening was calibrated using experimental data on 51 CrV4 martensitic steel, obtained in strain-controlled fatigue tests. This combination of kinematic and isotropic hardening models has been widely used in different works, resulting in good prediction results when compared with experimental data [44; 70; 71; 72]. The parameters of the Chaboche kinematic hardening model were calibrated considering the stabilised cyclic elasto-plastic Ramberg-Osgood curve for 90% of the total life of the specimens.