On the numerical study of fatigue process in rail heads by means of an isotropic damage based high-cycle fatigue constitutive law

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/356315814 On the numerical study of fatigue process in rail heads by means of an isotropic damage based high-cycle fatigue constitutive law ArticleinEngineering Failure Analysis · November 2021 DOI: 10.1016/j.engfailanal.2021.105915 CITATIONS 0 READS 69 4 authors: Some of the authors of this publication are also working on these related projects: Base Isolation and Energy Dissipation View project Pasarela peatonal sobre el río Besós. View project Sergio Jiménez Universitat Politècnica de Catalunya 5 PUBLICATIONS10 CITATIONS SEE PROFILE Lucia Gratiela Barbu Universitat Politècnica de Catalunya 16 PUBLICATIONS98 CITATIONS SEE PROFILE Sergio Oller Universitat Politècnica de Catalunya 268 PUBLICATIONS6,801 CITATIONS SEE PROFILE Alejandro Cornejo Universitat Politècnica de Catalunya 14 PUBLICATIONS49 CITATIONS SEE PROFILE All content following this page was uploaded by Alejandro Cornejo on 20 November 2021. The user has requested enhancement of the downloaded file. On the numerical study of fatigue process in rail heads by means of an isotropic damage based high-cycle fatigue constitutive law S. Jim´eneza,b,∗ , L. G. Barbua,b, S. Ollera,c, A. Cornejoa,b aCentre Internacional de M`etodes Num`erics en Enginyeria (CIMNE), Campus Norte UPC, 08034 Barcelona, Spain bUniversidad Polit´ecnica de Catalu˜na (UPC), Campus Norte UPC, 08034 Barcelona, Spain cConsejo Nacional y de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Facultad de Ingenier´ıa, Universidad Nacional de Salta, Av. Bolivia 5150, 4400, Salta, Argentina Abstract The fatigue phenomenon has been historically related to the railway industry. Nowadays, thanks to the high quality of the materials used, most of the recorded problems have been overcome although there are still degradation processes which are associated to fatigue and need to be considered for the definition of the maintenance campaigns and the general progress of the sector. In this paper, the latest improvements in the isotropic damage based high-cycle fatigue constitutive law proposed by Oller et al. [1] are presented and the approach is used for the study of two regions of the railway path where fatigue mechanisms are experimentally detected: a straight section and a crossing element. The analysis of the affected areas is performed through a finite element simulation identifying the critical regions liable to the fatigue degradation when the structure interacts with high speed vehicles and predicting the initiation of the degradation at the rail head while capturing the physics of the problem. The potential of the methodology is shown through the case studies and the current shortcomings and the future lines of research are clearly stated. Keywords: railway, finite element method, high-cycle fatigue simulation, isotropic damage, advance in time strategy, pearlitic steel, Hadfield steel ∗Corresponding author. Email address: [email protected] (S. Jim´enez) Preprint submitted to Engineering Failure Analysis November 20, 2021 1. Introduction Since the second half of the 19th century, the fatigue phenomenon has been deeply studied, originally associating it to the recurrent deterioration detected in the railway system which led to unexpected failures [2]. In this process the mechanical properties of the material are progressively degraded when cyclic5 loads below the ultimate material strength are applied [3, 4]. Nowadays, fatigue is determining in the failure of elements and structures in aerospace, naval and car industries as well as in some structures within the civil engineering field [4, 5]. The variety and complexity of the scenarios where these failures take place [6, 7, 8, 9, 10, 11, 12, 13, 14] make it necessary to10 develop numerical tools that allow for the correct study and prediction of the phenomenon. In this paper, a constitutive law based on the continuum damage mechanics and dedicated to the study of High-Cycle Fatigue (HCF) processes is presented and used for the study of one of these failure scenarios. HCF is a specific typology of fatigue characterized by the large number of cycles that15 lead to failure (105−107cycles) and by the conditions at which it takes place. This fatigue process occurs without the presence of plastic deformations and failure takes place in a brittle manner, driven by the coalescence of internal pores [1, 15]. Therefore, the problem has been historically approached from the damage mechanics point of view in several ways [4, 16, 17].20 The main objective of the undergone research has been to confront the presented HCF model towards real railway structures where fatigue degradation is detected, in particular it is devoted to study the degradation observed at the wheel-rail interaction region. The paper is organized in several sections, starting by introducing in Sections25 2 and 3 the problem that motivates the undergone research and the methodology applied, respectively. The degradation process, the effect over the rail components and the implications on the structure maintenance are presented in Section 2 while Section 3 is dedicated to review the numerical approach, pointing 2 out its main capabilities and introducing the latest developments with respect30 to the original constitutive model [1]. Section 4 focuses on the pre-processing operations that precede the simulations, i.e., the materials characterization is described paying special attention to their fatigue response and the built numerical models are presented including a description of the geometry, the finite element mesh used and how the train-induced loads are applied. The results35 obtained in the simulations are given in Section 5 and similarities are established between them and literature records. Finally, conclusions are drawn for the conducted research and the capabilities and the limitations of the applied methodology are stated in order to provide a full review and set the framework where reliable results can be obtained.40 2. Problem statement Fatigue problems associated to railway infrastructure have been numerous through the years and these actually control to a certain degree the maintenance operations at the rail level like grinding or ballast cleaning and replacement due to the unacceptable material degradation caused during the normal45 infrastructure operation [18]. This phenomenon is not exclusive of rails and it is convenient to distinguish between the regions where these fatigue problems are detected because, although the origin is similar in all the cases, the corresponding technical and economic consequences are different. Fatigue degradation is detected in vehicle components, mainly wheels but also engines, axles, body50 shells and suspensions, at subrail elements, including the fastening system, slippers and foundation, at surrounding infrastructure [19] and in rails, mostly on rail heads where contact with wheels takes place. There are many recent publications devoted to the study of these fatigue processes, mainly studying the degradation at the train axles and rails [20, 21, 22, 23, 24, 25, 26, 27, 28], but55 also on the bogies, the chassis and the engine [29, 30, 31], indicating the relevance of the topic for the railway community. The current research focuses on the HCF degradation processes at rail level, where the proposed constitutive law 3 has been used to reproduce the fatigue developed at two points of the railway path: a straight section and a crossing of a high-speed track.60 2.1. Straight section Fatigue degradation on these regions can develop at different points of the rail depending on the underlying trigger [18, 32, 33]. Due to the improvements in the steel manufacturing process, the degradation induced by internal inclusion or hydrogen shrinkage defects is not an issue anymore and failure is more related65 now with fatigue evolution on railheads, i.e. the rolling area. The predominant phenomenon is known as rolling contact fatigue (RCF) [32, 34, 35] which is produced by the normal wheel rolling along the rail. This means that the associated degradation can potentially appear at any point of the rail track although those areas with higher stress concentration, e.g. the inter-space between sleepers,70 sections in turns or welded sections, are the ones which concentrate the major part of the cases. This results in a superficial deterioration at the railhead level as shown in Figure 1, which can be solved through grinding operations if the problem is rapidly detected, otherwise the rail needs to be replaced. (a) (b) Figure 1: Rolling contact fatigue deterioration reported in bibliography: a) RCF microcracks reported by Kalousek [36] and b) fatigue degradation detected in the Medell´ın subway reported by Alarc´on et al. [34]. 4 2.2. Crossing element75 Crossings are the elements of the railway system that allow switching the trains from one rail to another when required. These are composed by two parts: the first one controls the transition between tracks through switch units that redirect the train movement and the second part takes charge of accommodating the train on its transition from one rail to the other. Ensuring the80 smoothness along this transition is not always possible and this can lead to the fatigue degradation of the metal piece that interacts with the train wheel, the so called frog nose. The deterioration here is similar to the one observed on the straight elements, i.e., a spalled region is formed along the frog nose being more important on those points where the first contact with the wheel takes85 place. Figure 2 shows the degradation experienced by a frog nose placed in the Puertollano high-speed railtrack in Spain. (a) (b) Figure 2: Fatigue degradation detected at the frog nose element placed at the Puertollano line. a) General and b) detail views. Source: CIMNE. The study undergone for the straight section and the crossing element in this paper focuses on the behaviour of the rail and the frog nose, respectively, while trains go through. Therefore, the corresponding numerical simulations intend90 to reproduce the superficial degradation patterns observed along the rail and exemplified on Figures 1 and 2. The spalling observed on the studied elements appears on these simulations as a progressive loss of the material strength and stiffness at those regions where fatigue degradation is predicted, driving the 5 material to a fully fatigued-damaged state with fully dissipated fracture energy.95 The studied scenarios have been selected for their simplicity compared with other areas of the railtrack where fatigue could appear as a second order effect. Therefore, the study here can be focused on the fatigue phenomenon without major interactions of other wearing processes. Finally, although at this stage some simplifications have been taken into account, e.g., nor thermal effects, nor100 plasticity, nor residual stresses, etc. have been taken into account, the present research can be considered as a starting point in the direction of building a tool for the advanced study of fatigue processes at rail level and it can be useful in the planning of the maintenance campaigns. 3. HCF constitutive law105 The HCF constitutive law presented [1] is an extension of the isotropic damage model proposed by Oliver et al. [37]. The underlying model predicts the progressive degradation of the material strength and stiffness once a certain stress threshold is reached. The isotropic damage model is based on the definition of the damage scalar110 variable d, which accounts for the stiffness degradation in the material with absence of plastic strains. Therefore, the constitutive response of a material is σ=m0∂Ψ ∂ε= (1 −d)C0:ε=C0:ε |{z} predictive stress, σ0 −d·C0:ε | {z } damage correction (1) where σand εare the stress and strain tensors, respectively, C0is the undamaged constitutive tensor, m0is the material density and Ψ is the free Helmholtz energy which can be formulated in the reference configuration as a function of115 strains and the damage internal variable [38]. Ψ (ε, d) = (1 −d)1 2m0(ε:C0:ε) (2) This is a direct constitutive model where dis explicitly defined for each stress level and no iterative process is required during the calculation at the integration point level. The damage variable takes values [0 −1] from intact to fully 6 damaged, remaining constant, ˙ d= 0, while the yielding criterion is satisfied,120 i.e., F(σ0) = f(σ0)− K ≤ 0 (3) where f(σ0) is the equivalent predictive stress and Kthe material threshold function which is a historical variable of the maximum stress level reached on the material. The equivalent stress is a uniaxial measure of the tensorial stress state which depends on the yield surface chosen. For the purpose of this work, the125 Von Mises yield surface has been adopted for the steel elements and equivalent stresses are computed as f(σ0) = p3J2(4) where J2is the second invariant of the stress deviator tensor [38, 39]. Therefore, the equivalent predictive stress is computed at each integration point of the structure from the predictive stress σ0and governs the response of the material130 along the simulation. When Eq. (3) is not satisfied, the evolution of the damage internal variable is controlled through an exponential softening function [37, 40] d= 1 −f0(σ0) f(σ0)·exp A·1−f(σ0) f0(σ0) (5) where f0(σ0) = K0is the initial equivalent stress threshold and A, A= gf·E (f0(σ0))2−1 2!−1 ≥0 (6) is a parameter that guarantees that the available energy for the non-linear pro-135 cess at each Gauss point is equal to the given volumetric fracture energy of the material, gf=Gf lc=R∞ t=0 Ψ0˙ d dt, where Eis the Young’s modulus, Gf is the fracture energy, lcis the characteristic length of the finite element and Ψ0=f0(σ0) 2Eis the free energy. Figure 3 schematically reflects the main features of this constitutive law in a uniaxial stress-strain chart.140 7 Figure 3: Uniaxial strain-stress scheme of the isotropic damage model. The absence of plastic strains in HCF processes [1, 15] is the key point in deciding to use the isotropic damage model as the underlying model in the proposed HCF constitutive law. The main modification introduced by this routine affects the isotropic damage yielding criterion definition (Eq. (3)), which is now145 F(σ0) = f(σ0) fred (Nc, R, Smax,)− K ≤ 0 (7) where fred is the so called fatigue reduction factor and ranges from 1 to 0. This parameter takes into account the effect of the acting cyclic load and, consequently, amplifies the stress state depending on the number of cycles applied, Nc, the reversion factor, R=Smin Smax , and the maximum stress generated by the applied load, Smax. The number of cycles, Nc, along the simulation is updated150 based on the evolution of the equivalent stresses at the integration point level, i.e., when maximum and minimum values of the stress are detected this indicates that a new cycle has overcome. Therefore, Smax,Smin and Rvariables are updated at each new cycle, if necessary. Figure 4 compares the performance of the HCF constitutive law assuming that155 a value of fred = 0.8 has been reached in the material with the isotropic damage prediction. 8 of UIC 60 profiles which are used in those tracks with speed limits >160km/h, including high-speed rail systems [46]. On the other hand, Hadfield steels are alloys with a high content of manganese. This component provides them with255 high impact strength [47, 48] and thus these steels have been traditionally used in railway crossings. 4. Computational model and materials characterization The analysis of the two regions described in Section 2 has been done through the Finite Element Method (FEM). Two models have been created in order260 to study the HCF effects that the continuous traffic circulation has over the rails. The geometries and the finite element meshes have been created using the pre and post-processor tool GiD [49] and the calculations have been performed using the open-source code Kratos Multi-physics [50] where the HCF algorithm presented in Section 3 has been implemented.265 4.1. Straight section The straight section study has been undergone building the 3.1mlong railway straight section shown in Figure 6. The model includes the main elements of a standard high-speed railway track, i.e., the rail, the elastomeric pads, the ties, the ballast and the main sub-structure ground layers. The rail profile cor-270 responds to a UIC 60 defined in the EN 13674-1:2002 [51] and a detailed view of the modelled rail is shown in Figure 7. The model has been constrained according to the confining conditions of the surrounding soil, i.e., limiting the displacements on the bottom of the subsoil material and on the four sides of the ballast and ground layers, and the same275 hypothesis has been considered to the rail, where the longitudinal displacement has been blocked on both ends. These boundary conditions are sufficient for the undergone analysis considering the localized effect that the applied loads introduce in the model and because of the assumed simplifications (see Section 2).280 15 Ballast Gravel Subsoil Ties Rail 2.3m 3.1m 1.8m 1.0m 1.3m 0.3m Elastomeric pads Figure 6: Numerical model of the straight railway section. Main elements and dimensions. (a) 16.5 150 72 172 (b) Figure 7: UIC 60 profile view, a) EN standard scheme [51] and b) model detail. Units in [mm]. 16 The mechanical properties used in the simulation to characterize the behaviour of the straight section model are included in Table 1. These basic properties characterize the elastic response of the structure. This has been considered sufficient for all the elements of the structure except for the rail, where considerably higher stresses are concentrated. Table 1: Mechanical properties of the materials included in the straight section model. Element Density [kg/m3]Young modulus [MPa] Poisson Rail UIC60 7850 210000 0.32 Elastomeric pad 1200 100 0.45 Concrete tie 2200 25000 0.32 Ballast 1390 300 0.27 Gravel 1390 280 0.27 Subsoil 1390 2750 0.27 285 Therefore, these properties have been complemented with the fatigue characterization done for the pearlitic Grade 900A steel used in the rail. This calibration of the non-linear response has been done by adjusting the control variables of the model to experimental results, as explained in Section 3. For this material, the R= 0.1 tensile fatigue tests undergone by Christodoulou290 et al. [43] have been used. The samples tested there were obtained from rail heads in unused conditions and thus the properties exhibited by the material are suitable for the study of the rail fatigue behaviour. Despite this, the effect that the maintenance operations introduce in the rail along its life [52, 53] are not reflected in these original properties and have not been considered in the295 undergone study. Results of the calibration process are summarized in Table 2 and Figure 8 where the values of the HCF constitutive law control variables and the SN curves for the reversion factor spectra R∈[0 −1] are shown. Considering that the fitting has been done using only values corresponding to R= 0.1,300 the use of this calibrated material should be limited to fatigue cases where the reversion factor of the applied cyclic loads is close to the experimental one. This condition is satisfied in the undergone analysis as the oscillating load applied in 17 the straight section model has R= 0.0. Table 2: Pearlitic Grade 900A steel fatigue characterization. HCF model variables for the spectra R∈[−1; 1]. Variable Value Su 924 MPa Sth(R=−1) 369.1 MPa αf0.0068 βf3.20 Sth,R12.00 AUX R1 0.01 400 500 600 700 800 900 1000 1E+04 1E+05 1E+06 1E+07 Maximum stress [MPa] Number of cycles, Nc R 1 . 0 R 0 . 0 Experimental values, R=0.1 Numerical adjustment Figure 8: Fatigue calibration of the pearlitic Grade 900A steel according to the experimental results obtained by Christodoulou et al. [43] for R= 0.1. Run-outs points are highlighted with arrows. The mechanical rolling over effect of the train is considered by imposing305 a cyclic vertical displacement at the mid-span of the space between ties where maximum stresses are generated. This displacement has been introduced through the model of a train wheel as shown in Figure 9 in order to guarantee the correct force distribution over the rail during the simulation. Other actions like thermal or chemical effects are out of the scope of this research and only a displacement310 which generates a stress state compatible with fatigue is being considered. 18 (a) (b) Figure 9: Load application at the straight section model: a) general view and b) section view. Finally, the finite element mesh of the model has been built using 8-nodded hexahedral elements. Higher mesh density has been provided to the rail element, particularly to the region between ties where the imposed displacement is applied as this is the region where fatigue deterioration is expected. Figure 10315 shows this refinement, where 71,400 elements of the total of 142,309 hexahedra are concentrated. (a) (b) Figure 10: Finite element mesh detail of the rail element, a) elevation view of the refined region between ties where the displacement is imposed and b) section view. 4.2. Crossing element The crossing element study has been undergone building a 2.1mx 2.0m model which focuses on the frog nose where fatigue degradation is studied. The320 model includes the main elements of the crossing system, i.e., elastomeric pads, concrete ties and a 0.3mballast layer as shown in Figure 11. No sub-ballast 19 soil layers have been considered for this simulation due to the high localization of the phenomenon as will be seen on the results section. The dimensions of all the elements have been adjusted to the Puertollano line crossing element shown325 in Figure 2. The corresponding constrictions have been set according to the confining conditions as in the straight section model however, only the downstream end of the frog nose has its longitudinal displacement blocked, while the tip is free. These boundary conditions are again sufficient for the crossing element model330 due to the localization of the applied loads and because of the assumed simplifications (see Section 2). 1.3m 2.1m 2.0m Ballast Ties Frog nose Elastomeric pads 0.3m Figure 11: Crossing element numerical model. Main parts and dimensions The same basic mechanical properties used for the straight section model and shown in Table 1 are used to characterize the materials of the crossing element model and a new calibration of the HCF constitutive law has been done for the335 manganese steel present in the frog nose. The R= 0.1 bending fatigue results obtained by Kang et al. [48] have been used to do this fatigue characterization, where the differences between tension and bending fatigue [54, 55] have been taken into account by defining a R= 0.1 S-N curve which predicts earlier failure than the ones obtained through the bending experiments. Results of340 this calibration process are summarized in Table 3 and Figure 12 where the 20 values of the HCF constitutive law control variables and the S-N curves for the reversion factor spectra R∈[0 −1] are shown. As for the straight section case, the use of this calibrated material should be limited to fatigue cases where the applied cyclic loads have reversion factors close to the experimental one, i.e.,345 R= 0.1. This condition is satisfied in the undergone analysis as the oscillating load considered in the frog nose model has R= 0.0. Table 3: Hadfield steel fatigue characterization. HCF model variables for the spectra R∈ [−1; 1]. Variable Value Density 7850 kg/m3 Young modulus 210000 MPa Poisson 0.30 Su 1100 MPa Sth(R=−1) 300 MPa αf0.0008 βf4.46 Sth,R11.58 AUX R1 -0.00032 500 600 700 800 900 1000 1100 1200 1E+04 1E+05 1E+06 1E+07 Maximum stress [MPa] Number of cycles, Nc R 1 . 0 R 0 . 0 Experimental values, R=0.1 Numerical adjustment Figure 12: Fatigue calibration of the Hadfield steel used for the frog nose according to the experimental results obtained by Kang et al. [48] for R= 0.1. Run-outs points are highlighted with arrows. The interaction between the wheel and the frog nose has proven to take 21 place at a distance of 0.47mfrom the tip and it is at this point where the cyclic action has been applied. The application of this external displacement350 has been performed analogously to the straight section, as shown in Figure 13, and neither thermal nor chemical effects have been considered in this case. (a) (b) Figure 13: Load application at the crossing element model: a) general view and b) section view. Finally, the finite element mesh of the model has been build using 8-nodded hexahedral elements. Higher mesh density has been provided to the frog nose element where fatigue study is intended, particularly to the section located at355 0.47mfrom the tip where the wheel-rail interaction takes place. Figure 10 shows this refined region, where 47,648 elements of the total of 82,686 hexahedra are concentrated. 22 (a) (b) (c) Figure 14: Finite element mesh detail of the frog nose, a) general of the frog nose refined region where the displacement is imposed, b) section view and c) frontal view of the frog nose finite element mesh. 5. Results In this section the results obtained for the previous scenarios are presented.360 A preliminary control case has been included where one of the tensile experiments used on the fatigue characterization of the pearlitic steel is reproduced. The main features of the HCF constitutive law are reviewed through this ini23 tial case where result are already known. Figure 15 shows the model built and an overall view of the finite element mesh used. Displacements in one end of365 the sample have been blocked while on the opposite end a cyclic displacement has been imposed to reproduce the experiment conditions of a normalized axial fatigue test according to ASTM E466 [56]. (a) (b) Figure 15: Model of the samples used in the tensile fatigue experiments undergone by Christodoulou et al. [43]: a) geometry with main dimensions and b) finite element mesh made of 29,051 8-nodded hexahedra. On this simulation stresses oscillate between 55.9−559MPa. For this stress rate, damage initiates after 318,663 cycles and rapidly evolves and propagates.370 Damage is triggered by the decrease of the fred variable on those areas where stress exceeds the fatigue limit. This susceptible to fatigue region extends throughout the entire narrowed section as shown in Figure 16, however, two rings showing higher degradation level are recognizable and set the starting point for damage propagation.375 24 the physics of the problem have been captured. 7. Acknowledgments This work has been done within the framework of the RESILTRACK (IDI-475 20171003) project: resilencia de infraestructuras ferroviarias frente a cambio clim´atico. This project has received funding from the Spanish Government. The work has been also supported by the Spanish Government program FPU17/04196. The authors gratefully acknowledge all the received support. Appendix A. Fracture energy dissipation in the HCF process480 Here it is derived the expression for the variation of the available fracture energy when the HCF constitutive law is used. The expression for the total dissipated energy in the isotropic damage constitutive law [38] is wmax t=Z∞ t=0 Ξdt =Z∞ t=0 Ψ0˙ d dt =Z¯σmax ¯σ0 [¯σ]2 2·E |{z} free energy ∂ˆ G(¯σ) ∂¯σd¯σ=gf(A.1) where ¯σ≡f(σ0) is the predictive equivalent stress, ¯σ0≡f0(σ0) is the yield stress, ¯σmax ≡fmax (σ0) is the predictive stress level when the fracture energy485 has been fully dissipated, Eis the Young modulus and ˆ G≡dis the damage internal variable function. The result, wmax t=gfis independent of the definition chosen for ˆ G. Two cases are studied here: the linear and the exponential descriptions of the damage. Linear case: d=1−¯σ0 ¯σ·1 1−A∧A=¯σ02 2gfE Exponential case: d= 1 −¯σ0 ¯σ·exp hA·1−¯σ ¯σ0i ∧A=1 gfE (¯σ0)2−1 2 (A.2) The ¯σmax value reached in each scenario is ¯σmax lin. =¯σ0 Afor the linear description490 and ¯σmax exp. = +∞for the exponential case. These stress values are obtained from Eq. A.2 when d= 1.0. 31 The extrapolation to the HCF case is done by affecting Eqs. A.1 and A.2 with fred. This function results in a reduction of the normal yield surface of the underlying damage model as a consequence of the fatigue process. Finally,495 Linear: wmax, HCF t=Z∞ t=0 Ξdt =Z∞ t=0 Ψ0˙ d dt =Z¯σ0·fred/A ¯σ0·fred [¯σ]2 2E ∂ˆ G(¯σ) ∂¯σd¯σ= =Z¯σ0·fred/A ¯σ0·fred [¯σ]2 2E ¯σ0·fred [¯σ]2(1 −A)d¯σ=¯σ·¯σ0·fred 2E·(1 −A) ¯σ0·fred/A ¯σ0·fred = =¯σ02·f2 red 2EA =gf·f2 red (A.3) Exponential: wmax, HCF t=Z∞ t=0 Ξdt =Z∞ t=0 Ψ0˙ d dt =Z+∞ ¯σ0·fred [¯σ]2 2E ∂ˆ G(¯σ) ∂¯σd¯σ= =Z+∞ ¯σ0·fred [¯σ]2 2E A ¯σ+fred ¯σ0 [¯σ]2·exp A·1−¯σ fred ¯σ0d¯σ= =− fred ¯σ02·fred ¯σ0+A¯σ·exp A·(1 −¯σ) fred ¯σ0 2AE  +∞ ¯σ0·fred = =f2 red ¯σ02(A+ 2) 2AE =gf·f2 red (A.4) Appendix B. Computing the advance in time (AIT) once damage has initiated In order to smartly advance once the internal damage variable has initiated, i.e., Nfhas been reached at some GP, the user defines the ∆d∗-parameter which500 controls the AIT. The time increment that analytically induces a variation of the damage internal variable ∆d= ∆d∗is computed through this input. The relation between the time increment and the variation of the damage internal variable is obtained using dand fred definitions (Eq. (5), (9)) and that the stress state in the HCF regime is f(σ0) fred .505 32 The damage: Linear case d=1−fyield f(σ)·1 1−A⇒fred = [1 −d·(1 + A)] f(σ0) f0(σ0)(B.1) Exponential case d= 1 −f0(σ0)·fred f(σ0)·exp A·1−f(σ0) f0(σ0)·fred  ⇒(1 −d)·f(σ0) f0(σ0)·fred = exp A·1−f(σ0) f0(σ0)·fred  ⇒ln (1 −d)·f(σ0) f0(σ0)+ ln 1 fred =A·1−f(σ0) f0(σ0)·fred  ⇒A·f(σ0) f0(σ0)·fred + ln 1 fred =A−ln (1 −d)·f(σ0) f0(σ0) ⇒exp A·f(σ0) f0(σ0)·fred ·1 fred =exp (A)·f0(σ0) (1 −d)·f(σ0) ⇒fred =A·f(σ0) f0(σ0)·1 ωA·exp (A) 1−d (B.2) Where ω(·) = P∞ n=1 (−n)n−1 n!(·)nis the Lambert function. And the fatigue reduction factor:510 fred = exp n−B0·(log10 Nc)β2 fo⇒Nc= 10  − ln (fred) B0   1/β2 f (B.3) Therefore, Linear case: ∆Nc= 10         − ln [1 −∆d∗·(1 + A)] ·f(σ0) f0(σ0) B0         1/β2 f Exponential case: ∆Nc= 10                     − ln     A·f(σ0) f0(σ0)·1 ωA·exp (A) 1−∆d∗    B0                     1/β2 f (B.4) 33 References [1] S. Oller, O. Salom´on, and E. O˜nate. A continuum mechanics model for mechanical fatigue analysis. Computational Materials Science, 32(2):175– 195, 2005. ISSN 0927-0256. doi: https://DOI.org/10.1016/j.commatsci.515 2004.08.001. [2] H. Zenner and K. Hinkelmann. August W¨ohler – founder of fatigue strength research. Steel Construction, 12(2):156–162, 2019. doi: https://DOI.org/ 10.1002/stco.201900011. [3] ASTM E1823-13. Standard terminology relating to fatigue and fracture520 testing. Technical report, ASTM International, West Conshohocken, PA, 2013. URL www.astm.org. 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