PSI - Issue 39

Davide Leonetti et al. / Procedia Structural Integrity 39 (2022) 9–19 Author name / Structural Integrity Procedia 00 (2019) 000–000

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1. Introduction Rolling-contact fatigue, RCF, is caused by the fluctuating stresses resulting from bodies in rolling contact and it is characterized by the presence of cracks nucleating either at the surface of slightly below it, see Magel (2001) and Sadeghi (2009). Many examples of these cracks can be found in railway rails, such as squat, head checks, and in rolling bearings. The main danger associated with RCF defects is that the direction of growth the subsurface leading crack may suddenly change towards the transverse plane of the section, determining complete failure when not detected on time. Planning maintenance according to damage tolerant philosophy allows the determination of inspection interval for the safe use of the structure, on the basis of the predicted fatigue crack growth rate, and the critical crack size, both requiring stress intensity factors, SIFs, to be determined. A squat is one of the most frequent RCF crack in rails, and it usually initiates at the running surface of the head of the rail and it can be visually identifiable, as it appears as a dark spot, Zerbst (2009). The crack typically propagates 3–5 mm below the rail surface with an angle of about 20 degrees with the rail surface. RCF cracks in rails were studied from a phenomenological point of view from many authors collecting relevant information concerning the influencing factors of crack initiation and growth, see Li (2008), Li (2011), and Pal (2012), other studies concerned the prediction of crack growth using fracture mechanics. For example, Olzak (1991) presented one of the first 2D models for the analysis of stress states in surface breaking cracks under contact load, considering and neglecting the friction between the crack faces. Bogdanski (1996) proposed one of the first models in which the squat is modelled in 3D, concluding that the contribution of the second mode of loading of the crack is dominant. Moreover, as found by Wong (1996) the mode I effective stress intensity range and the degree of overlap between mode I and mode II are the main parameters for rolling contact fatigue that govern the choice of crack growth direction. Ringsberg (2003) gave an extensive overview of the early crack propagation problem in railheads. A two-dimensional finite element (FE) model was used to question the applicability of linear elastic fracture mechanics (LEFM), concluding that rolling contact fatigue analysis of short crack growth behaviour of surface‐breaking cracks can be studied within LEFM only if the material has shaken‐down. Farjoo (2012) found that both the elastic foundation supporting the track and the water entrapped in the crack are key factors in crack propagation since the obtained results related to mode mixity differ as compared with previous models. A more recent numerical study to model the propagation of squat defects is published by Trollé (2012), based on the extended finite element method, X-FEM, stabilization technique of a three-field weak formulation of frictional crack problems. The following study of Trollé (2013) concentrated on the prediction of fatigue crack growth and branch conditions under rolling contact loading by making use of a 2D model, including frictional contact at the crack faces and residual stresses. Amongst the other conclusions, the results produced highlight the role of residual stresses in the crack growth rate. The accurate determination of the stress intensity factors of RCF crack is particularly difficult due to the complex stress field induced by the rolling contact characterised by high stress gradients. The finite element models developed with this scope require a large computational effort as compared to other crack problems studied with LEFM in civil infrastructures, often involving cracks loaded in the far field, as done for example by Maljaars (2012) or Carpinteri (2005), see Figure 1. Most frequently, the SIFs are estimated through the use of geometric functions, F I and F II through the following: K I =F I σ ( π a ) 0.5 (1) K II =F II σ ( π a ) 0.5 (2) where is the nominal stress applied in the far field, and a is the crack depth. The geometric functions are often determined using the finite element method and depend on the crack depth, loading condition and the geometry, and are grouped in handbooks, e.g. Tada (2000). An alternative to this approach is to use the weight functions, introduced by Bueckner (1970). The weight functions solely depend on the geometry and the crack depth, and not on the type of applied load and boundary conditions, see Wu (1991) and Fett (1997) for a collection of weight functions for the most common geometries. Moreover, using the weight functions, the determination of the SIFs requires the stress state along the crack line determined in the uncracked body. Weight functions are often used to estimate the SIFs when a