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

1555

4

3.2. Material and Mechanical Properties

In accordance with UIC517 (2007) standard, leaf springs are made of quenching and tempering spring steel. The quenching and tempering process are according to ISO683 (2004). Table 2 presents the typical values for chromium vanadium steels grades with designation 51CrV4.

Table 2. Chemical composition of spring steel and comparison with standard values (% weight). C Si Mn P S

Cr

V

DIN 51CrV4

0.47-0.55

≤ 0.40

0.70-1.10

≤ 0.025

≤ 0.025

0.90-1.20

≤ 0.10-0.25

The spring steel is typical a high-strength steel with a good strength to ductility ratio. The mechanical properties obtained from monotonic tests are presented in Table 3.

Table 3. Monotonic Properties for 51CrV4 steel grade applied in UIC parabolic leaf springs. E [GPa] ν σ 02 [MPa] σ uts [MPa] ε f [%] R A f [%]

H v [HV]

K [MPa]

n

202

0.29

1258

1414

7.970

48.07

424.3

1747

0.0519

4. Framework of Finite-element-based Model

4.1. Solution Scheme

The numerical procedure to compute the stress field on leaves is performed by finite element method. The numer ical problem is the di ff erential equation in static-equilibrium, which is solved by Newton-Raphson solution scheme assisted by line search (Bathe (1996)), such that K t i ∆ u n , i = F n − F nr n , i . The global tangent sti ff ness matrix, K t , is a symmetric matrix. and then it is used the DLDT procedure (George (1981)). With respect to the contact model, the augmented Lagrangian nested scheme without friction e ff ects is considered (Simo and Laursen (1992)). The contact search algorithm is the pinball algorithm (Belytschko and Neal (1991)). The numerical model is developed considering three type of solid elements, 20-node brick, 13-node pyramid , and 10-node tetrahedral. Uniform reduced integration is considered in solids elements constituting leaves, spring band, taper key and gid-headed key. Solid elements constituting the linings are integrated with a full integration scheme because only one element along thickness is considered. With respect to contact elements, 8-node and 6-node surface to-surface are used for deformable bodies and target bodies. A multiple constraint joint element is placed in the centre of shaft. A spring element is used to simulate the double link. Figure 3 illustrates a meshed geometry in analysis. Once that only vertical and lateral components of displacements are considered, a 1 / 2 of full numerical model is analysed. All components are considered to be linearly elastic, expect the master leaf. Master leaf is considered to have an elasticplastic behaviour. The time derivation of constitutive law is given by ˙ σ = C ˙ ε − ˙ ε p , where σ is the Cauchy stress tensor, ε is the total strain tensor, and ε p , the plastic strain tensor. The constitutive matrix, C , is defined by generalized Hooke’s law for an isotropic material. The total strain tensor is given by summing its elastic and plastic components, ε = ε e + ε p . In presence of large deflections, elastic strain tensor is defined by Biot strain tensor as ε e = U − I , with U the right-stretch tensor, and I the diagonal matrix. The limit elasticity surface is defined according the von Mises plasticity criterion, such that F = ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 = 2 σ 2 y . The associative flow rule follows the normality principle and it is defined as ˙ ε p = ˙ λ∂ F /∂ σ . The consistency condition, ˙ λ is computed using the radial-return mapping algorithm (Simo and Taylor (1985)), following an algorithm with an Euler backward scheme (Krieg and Krieg (1977)). 4.2. Constitutive law and Plasticity Criterion