The assumptions in the Shakedown Map and its previous applications

The Shakedown Map and ratchetting limit are an efficient method to define the material’s response to the applied forces and provides an indication of the propensity to generate surface and sub-surface damage. But, the model was developed based on certain theoretical assumptions:

1) Hertzian contact theory: Although it has certain assumptions which were detailed in the Chapter 2.1, the Shakedown Map extended the Hertzian contact theory by including the inelastic material response and rolling friction (Ringsberg, 2001). 2) The sliding contact case was in full slip and the effect of partial-slip condition was

mainly neglected. Nevertheless, while partial slip condition showed a higher influence on line contacts, it showed a lesser effect on point contacts (Dirks & Enblom, 2011).

3) It was mainly applicable for particularly point contacts which have larger lateral width than its longitudinal semi axis (b≥a) and circular contact (a=b).

In one of the earlier studies, Beagley (1976) linked the transition between mild and severe wear by the help of the Shakedown Map. It was stated that the traction coefficient of 0.32

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played a crucial role as it was indicated as the transition point from subsurface to surface deformation. On lubricated surfaces, the T/N was less than 0.32 and hence the maximum shear stress occurred below the surface. Additionally, the wear was experienced particularly under high contact stresses and it became more critical than cracks as it was often encountered at the rail flange contact in rail traffic operations.

Another study used the Shakedown Map to determine the failure mechanism and the related RCF prediction model for different type of material responses. When the material is below the elastic shakedown limit, the failure would occur eventually by high cycle fatigue (HCF) mechanism. As it was expected, no cracks were initiated for the simulated magnitudes of contact pressure and friction coefficient. However, if it was above this limit and inside the plastic shakedown limit, the material was defined to be failed by Low Cycle Fatigue (LCF). In this regime, lower number of cycles were generated to initiate crack development than HCF. The model results for the ratchetting mechanism showed the severity of this regime since the cracks were generated at the lowest number of cycles (Ringsberg, Loo-Morrey, Josefson, Kapoor, & Beynon, 2000). In the subsequent study, the site observations revealed that the large shear deformations of the material microstructure in the zone of head checks occurred as a consequence of both ratchetting and LCS mechanisms. The HCF produced less visible damage on the surface of damaged rails (Ringsberg, 2001).

Figure 2.38: Shakedown Map for dry and FM conditions (Eadie et al., 2008)

A more recent study demonstrated the effect of friction modifiers (FM) on the rail damage by using the Shakedown Map. As it is presented in Figure 2.38, the dry conditions led to more severe deformations (ratchetting) than the FM conditions. While the ratchetting mechanism gave rise to wear and head check crack formation, the FM applications delayed the onset of head checking (Eadie et al., 2008).

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One of the well-known applications of the shakedown theory is the Surface Fatigue Index (FIsurf). It was calculated from the horizontal projection of the shortest distance between

contact points and the ratchetting limit. (Ekberg, Kabo, & Andersson, 2001). As the equation of the boundary curve for surface flow (ratchetting) in the Shakedown Map is

𝑣 =1

𝑓= 1

𝑇/𝑁 (2.48)

Then, the surface fatigue index becomes; 𝐹𝐼𝑠𝑢𝑟𝑓= 𝑓 −

1 𝑣= 𝑓 −

2𝜋𝑎𝑏𝑘

3𝐹𝑁 (2.49)

Previous studies were conducted to validate this model using twin-disc and full-scale testing. It can be seen in Figure 2.39 that the RCF was predicted for all the test conditions with the exception of wet twin-disc cases in which the (FIsurf) value was given as negative.

During the experiments, the higher FIsurf values led to early crack initiation. However, it

was stated that the further detection of cracks was not reliable as the wear removed the initiated cracks. It should be noted that the FIsurf was applicable for surface initiated

damage, another parameter the surface fatigue index FIsub was also developed in which

the cracks were assumed to initiate at depths approximately 3 mm from the surface (Ekberg, Åkesson, & Kabo, 2014).

Figure 2.39: The surface fatigue index (FIsurf) results for the studied test conditions (Innotrack, 2009b) FIsurf and WLRM predictions were compared in a number of studies. One of the studies

stated that the damage index provided a better correlation to the reality than the FIsurf,as

it underestimated some of the damage (Stichel, Mohr, Ågren, & Enblom, 2008). In tighter curves, the damage index became negative showing the larger wear rate over crack initiation but, in these areas, the FIsurf, had the highest values (more distant to ratchetting

limit). It was later stated that although the FIsurf might underestimate the RCF prediction

for high creepages due to limitation of traction coefficient values by the maximum friction coefficient, the damage index might overestimate as it was previously observed in the laboratory experiments that for high creepages (> 5%), the RCF life was unaffected by creepage.

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Recently, new models: Stress Index (SI) and Energy Index (EI) have been developed by incorporating the longitudinal and lateral shear stresses to these models (Dirks, Enblom, Ekberg, & Berg, 2015).

𝑆𝐼 = √𝜏𝑧𝑥(𝑥, 𝑦)2+ 𝜏𝑧𝑦(𝑥, 𝑦)2− 𝑘 (2.50)

𝐸𝐼 (𝑥, 𝑦) = 𝜏𝑧𝑥(𝑥, 𝑦) ∗ (𝛾𝑥− (𝜑 ∗ 𝑦)) + 𝜏𝑧𝑦(𝑥, 𝑦) ∗ (𝛾𝑦+ (𝜑 ∗ 𝑥)) (2.51)

where 𝜏𝑧𝑥(𝑥, 𝑦) and 𝜏𝑧𝑦(𝑥, 𝑦) are respectively the longitudinal and lateral shear stresses in

the cell element of 𝑥 and 𝑦 directions. 𝛾𝑥, 𝛾𝑦 are the creepages and 𝜑 is the spin moment.

These new models were also extended by the crack propagation model in order to predict crack depth for small crack sizes. In the study, the model results were correlated with measured crack depths and surface crack lengths on a curve of the Dutch railways. It was found that while EI provided better predictions regarding crack depth estimations, the SI was superior in surface length. Nonetheless, it was mentioned that further validation was required considering a range of operational conditions as the models were only applied on a single curve.