PSI - Issue 42

Vítor M. G. Gomes et al. / Procedia Structural Integrity 42 (2022) 1552–1559 V.M.G. Gomes et al. / Structural Integrity Procedia 00 (2019) 000–000

1557

6

cyclic curve, given by Ramberg-Osgood’s curve and the the equation: σ a = σ ′ y + n = 3 a (Chaboche (2008)). The objective function for minimization is given by the Least-squares method, and the unconstrained numer ical optimization is performed by using the iterative method, Broyden–Fletcher–Goldfarb–Shanno (BFGS) (Martins and Ning (2021); Nocedal and Wright (2006)). The computed, estimated and assigned values for parameters regarding isotropic and kinematic hardening laws are presented in the Table 5. i = 1 C i /γ i tanh γ i ε p

Table 5. Isotropic and Kinematic Hardening Parameters σ y [MPa] R ∞ [MPa]] b ∞

C 1 [MPa]

C 2 [MPa]

C 3 [MPa]

γ 1

γ 2

γ 3

1258

-386

2

33628

427.72

19548

95.84

9268.4

17.56

4.4. Definition of the Loading Scenarios

Two numeric experiments were performed with distinct vertical and lateral loadings. The definition of the random amplitude loading is needed to determine the initial stress-strain state in the parabolic leaf spring. The initial stress strain state represents the loading of the freight wagon and hence the initial deformation of the leaf springs. Once the leaf spring is deformed, the loading from the circulation of the train is implemented. This latter is defined as a random amplitude loading during the time and its definition is given as δ j ( t ) = δ mean j + N E δ amp j ( t ) , var δ amp j ( t ) , where index j denotes the direction of the applied displacement (vertical or lateral). mean and amp denote respectively the mean value for initial condition and the amplitude value. The amplitude value is assigned following a Gaussian distri bution as shown in the rainflow matrices for vertical and lateral displacements, respectively illustrated in figure 3 - B. The amplitude value is generated by using a Twisted GFSR Random Number Generator of 32 bits MT19937 denomi nated as Mersenne Twister (Matsumoto and Nishimur (1998)). Notice that in the figure 3 - C, the two experiments are defined with an initial vertical displacement of 75 mm. After this value is reached, two scenarios are defined. In the first scenario, is assumed that the initial displacement is null, existing only variation of the amplitude of the lateral and vertical displacements. However, in the second scenario is assumed that the leaf spring is initially deformed laterally, with a lateral displacement around 20 mm. This value is almost in the limit of the gap existing in the suspension play. Besides these di ff erences, these two scenarios are distinguished by its variance. The variance in scenario 1 is greater than in scenario 2. According to the relationship between the maximum variance of the resolved shear stress and the equivalent shear stress amplitude found in (Susmel (2010)) and which is given by τ a = 2 Var τ q max , a possible assumption for the prediction of the spots with the highest crack initiation potential can be formulated. That is since the component of stress amplitude has a greater impact than the component of the mean stress, and since a greater amplitude of shear strain is associated with a greater variance of the resolved shear stress, then, from the visualization of the maximum variance at each of the nodes it is possible to analyse the most critical spots for fatigue initiation on the surface of the master leaf spring. Figure 4 shows the maximum variance of the resolved shear stress for loading scenarios 1 and 2 along the coordinates associated with the longitudinal direction and width of the master leaf spring surface. Notice that only the master leaf is analysed as fatigue cracks always tend to initialize in this component. With respect to the case of scenario 1, there is a tendency towards a symmetrical profile in the width along the entire length of the leaf. However, from the analysis of the maximum variance values for each of the nodes, the nodes with the greatest potential for fatigue initiation are those located at the edge of the leaf width with coordinates in length between 100 and 350 mm. Nodes with z coordinate greater than 350 mm tend to have a lower value of the variance of resolved shear stress. On the other hand, evaluating the overlap zone by the spring buckle, the nodes that have the 5. Results and Discussion 5.1. The Highest Potential Spots for Failure