PSI - Issue 57

Olivier Vo Van et al. / Procedia Structural Integrity 57 (2024) 104–111 O. VoVan / Fatigue Design 2023 00 (2023) 000–000

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extreme values. When too low, fragile points of the rail, for example where a crack has already initiated and is in its propagation phase, can experience brittle fracture. When too high, risk is of rail buckling, which is one major concern regarding climate change as [20] shows. The neutral temperature T N at which the average stress along the rail profile is null, is fixed to tackle both risks. One study of extreme temperature impact on crack initiation has been led by [9], considering the indirect impact temperature has on track geometry, which changes rolling stock dynamic behaviour and then induces change in contact force and thus in RCF. As regular cyclic thermal load is very low, as detailed in Section 2.1, its impact on fatigue (crack initiation) has generally been neglected. In order to decide whether to consider temperature cycles or not in further work, experimental observations have to be studied. A deep database is available concerning RCF but it appears that too many correlated parameters impact RCF to make a simple sensitivity analysis. Using available data at that time, the french railway network started to build a machine learning model detailed by [10], which will be the base of the present work. The goal here is to consider and quantify the contribution of temperature in the damage process of the rail. For this purpose, temperatures are aggregated using the method proposed in Section 2 and the results from the model are discussed in Section 3. The available dataset consists of daily minimum, maximum, and mean air ambient temperatures for a period of 6 years across the entire French railway network, with measurements taken at every 5 kilometers. These temperatures have to be converted into mechanical stresses. Previous studies such as [4] showed that maximal temperatures can reach T rail = 1 , 5 ∗ T ambient if weather conditions are fulfilled. The approximation made in this context is to assume that the rail temperature is equal to the ambient temperature ( T rail = T ambient ) at all times and locations. It mitigates the level of damage, the calculation of which will be presented later, without altering the rail ordering in relation to this temperature-induced damage, which is an important aspect for machine learning. It means that the temperature variation of the rail in one location over a day will be lower compared to another location, regardless of whether the measured ambient temperature T ambient is multiplied by a factor or not. To convert the temperature variation into rail stresses, we assume that the rail is at the same temperature in all its vertical profile. Knowing the linear coe ffi cient of thermal expansion, α th and using the Hooks law with E Young Modulus of the steel and ϵ the strain, one gets the stress σ , σ [ MPa ] = E ∗ ϵ = E ∗ α th ∗ ∆ T ≈ 2 , 5 ∗ ( T ambient − T N ) , (1) with T N the neutral temperature of the rail, corresponding to the temperature at which the cumulated stresses over the rail section are null. In France, this temperature is generally fixed at 25°C. We thus obtain time series of stresses that have to be converted into equivalent set of constant amplitude stress reversals. The Rainflow-counting algorithm, as standardised by [7] helps to do so. Figure 2 shows the distribution of the obtained cycles. Once cumulated, amplitude and average of cycles could reach 140 MPa , which is below the fatigue limit of rail steel, which is evaluated to half of the ultimate tensil strength and thus sets between 340 MPa and 440 MPa respectively for steel R200 and R260 [6]. The Miner’s Rule cannot be applied in this case, as it would result in 0 damages. This observation explains why temperature has never been considered as a fatigue loading for the rail. Few cumulative damage models allow low amplitude cycles to contribute. As developed by [18], only continuum damage mechanics model can handle these low amplitudes and it’s general thermodynamic framework was developed by Lemaitre and Chaboche [3]. The equation linking the damage at step i − 1 to the one at step i is given by 1 − (1 − D i ) β + 1 =  1 − (1 − D i − 1 ) β + 1  . exp    ( β + 1)    σ maxi − σ m M 0 (1 − σ m R u )    β    (2) with σ m the mean stress of the considered cycle, ( σ maxi − σ m ) defined by half of the range displayed in Figure 2, the coe ffi cients β = 5 . 5, M 0 = 2537 MPa and R u = 1689 MPa fixed using results available in [18] for steel SAE 4130, which was the steel with the closest mechanical properties to rail steel for which these coe ffi cients could be found. Since damage remains zero for low amplitude loading in the absence of any initial damage, D 0 has been fixed to 0 . 1. 2. Counting method and sensitivity analysis 2.1. Temperature data aggregation