PSI - Issue 54

Sjoerd T. Hengeveld et al. / Procedia Structural Integrity 54 (2024) 34–43 S.T. Hengeveld et al. / Structural Integrity Procedia 00 (2023) 000–000

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Fig. 2: Overview geometry

maximum stress P 0 = 845 MPa, which is an 2D approximation of a wheel load of 80 kN on a rail, assuming a 10 mm transversal contact width. Hertzian contact theory predicts a semi-elliptical normal stress distribution with a contact half width of b = 6 . 75 mm. Within one FE simulation, the load is translated from a begin position indicated with x b and t b to and end position indicated with x e and t e , see Figure 2. The stress distribution σ n and shear stress distribution τ xy K at every increment and location are defined by Equation 1. σ n ( x , x c ) =   P 0  1 −  x − x c b  2 if | x − x c | ≤ b 0 otherwise (1) Assuming full sliding conditions, the tangential stress τ ( x , x c ) is described by Equation 2. τ ( x , x c ) = µ r − w · σ n ( x , x c ) (2) inwhich µ r − w is the constant traction coe ffi cient. Braking results in µ r − w > 0 and acceleration is denoted with µ r − w < 0. Friction is assumed on the contact between the crack faces and is modelled using a Mohr-Coulomb friction model with friction coe ffi cient µ c . A structured FE mesh is used, with quadratic elements (type CPE8) using a full integration scheme. A refined mesh is used near the crack tip, see Figure 3. For a sharp crack the theoretical strain field at the crack tip in a FE model becomes singular. Using quadratic elements this singularity can be approximated by translating the midside nodes to the quarter point position close to the crack tip. Contact formulations using the penalty method including augmented Lagrange iterations are added between the crack faces to limit penetration.

Fig. 3: Example of mesh refinement in FE model

The SIF in this study is computed using the virtual crack closure technique method. In VCCT the SIFs are cal culated by assuming that the energy required to advance the crack with a small increment is the same as the energy required to close the crack for the same increment, see Rybicki and Kanninen (1977) and Krueger (2004). The equa tions derived by Raju (1987) are used within this study, see Equation 3 and Equation 4. G 1 = − 1 2 ∆ a F c y ′  t 11 ∆ U m y ′ + t 12 ∆ U n y ′  + F j y ′  t 21 ∆ U m y ′ + t 22 ∆ U n y ′  G 2 = − 1 2 ∆ a F c x ′ � t 11 ∆ U m x ′ + t 12 ∆ U n x ′  + F j x ′ � t 21 ∆ U m x ′ + t 22 ∆ U n x ′  (3)