Numerical simulation model for the competition between short crack propagation and wear in the wheel tread

Journal Pre-proof Numerical simulation model for the competition between short crack propagation and wear in the wheel tread Makoto Akama, Takafumi Kimata PII:

S0043-1648(18)31696-X

DOI:

https://doi.org/10.1016/j.wear.2020.203205

Reference:

WEA 203205

To appear in:

Wear

Received Date: 26 December 2018 Revised Date:

22 January 2020

Accepted Date: 24 January 2020

Please cite this article as: M. Akama, T. Kimata, Numerical simulation model for the competition between short crack propagation and wear in the wheel tread, Wear (2020), doi: https://doi.org/10.1016/ j.wear.2020.203205. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

CRediT author statement

Makoto Akama: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing- Original draft preparation, Writing - Review & Editing, Visualization, Supervision. Takafumi Kimata: Software, Validation, Investigation, Writing - Review & Editing, Visualization.

Numerical simulation model for the competition between short crack propagation and wear in the wheel tread

Makoto Akama*, Takafumi Kimata

Department of Mechanical Engineering for Transportation, Osaka Sangyo University 3-1-1 Nakagaito, Daito, Osaka 574-8530, Japan

* Corresponding author. E-mail address: [email protected]

ABSTRACT A numerical model is developed to simulate the competition between RCF-initiated short crack propagation and wear in a wheel tread. The crack is assumed to initiate when the total accumulated plastic shear strain reaches the critical value. In the early crack growth simulations, the two-stage short crack growth model proposed by Hobson is applied. With regard to wear, the Archard model is adopted as the basis. The model is shown to operate properly when applied to the results of unidirectional twin disc tests. When an outward route comprising a 1000 km straight run is simulated based on the Shinkansen routes, it is found that the tread is hardly damaged. The results obtained from the rolling reversal fatigue tests show that if the rolling direction of the wheel is reversed, the delamination wear becomes dominant.

Keywords: Wheel tread; rolling contact fatigue; short crack; wear; Hobson model; Archard model.

1. Introduction Railway systems involve contact between rolling wheels and rail. The demand for higher train speed and greater wheel loads have led to increased contact pressure and tangential traction between the wheel and the rail. Under a cyclic wheel passage, these lead to the perpetual initiation of cracks known as “rolling contact fatigue” (RCF), and wear in both surfaces and these are major damages for wheels and rails. For example, the railway wheel turning data [1] indicated that the majority of wheels were turned due to RCF or tread/flange wear. In the initiation and early propagation stages of cracks, crack propagation competes with wear [2]. The mechanism that initiates cracks depends on factors such as the geometry of the rail and the wheel, properties of their steels, type of traffic, and lubrication. At the same time, wear changes the shape of both surfaces, thereby altering the wheel–rail contact region and associated contact stresses. Both crack propagation and wear are driven by stress, strain, and traction, which are all related to the forces in the contact region. When the wear rate is much greater than the crack growth rate, cracks cannot propagate and, in some cases, are eliminated. In contrast, very low wear rates have negligible influence on crack growth. Consequently, wear rate must be considered to estimate the net crack growth rate. The pioneering research on the competition between RCF crack propagation and wear in a wheel/rail system was carried out by Franklin et al. [3–5], who developed a “brick model” that employs a formulation based on the incremental plasticity by ratcheting within a number of sections (or bricks) within a cross section through a rail. If the bricks reach their critical shear strain, they are considered to fail and are marked as weak. Franklin et al. designed many patterns of weak and non-failed healthy bricks as scenarios in which a brick might be removed. Doing so can indicate situations in which bricks either are removed at the surface as wear debris or form crack-like defects inside the rail. Burstow [6–8] developed a “whole life rail model” that considers the competition between RCF crack propagation and wear in a rail. In this model, he proposed a “damage function” that describes the relationship between the wear number Tγ and the RCF crack initiation fatigue damage, where T is the shear force and γ is the creepage at the wheel–rail interface. The damage function comprises four regions: (i) a threshold below which RCF damage does not occur, (ii) a region where RCF damage linearly increases, (iii) the amount of damage falls as wear increases, and (iv) a region where damaged material is entirely removed by wear. Mazzù [9–11] proposed an integrated model that considers multiple interacting damage mechanisms in railway wheels. The procedure is based on models for wear, cyclic plasticity, and surface and subsurface fatigue cracks, all integrated in an algorithm that exchanges input and output data between each failure model. The wear is assessed by means of the Archard model of adhesive wear. The cracks are evaluated by means of linear elastic fracture mechanics (LEFM) terms such as stress intensity factors (SIFs), threshold SIFs, and a Paris-type law. Karttunen et al. [12,13] provided an engineering “meta-model” to predict gauge corner and flange root degradation concerning RCF and wear from measured rail, wheel, and track

geometries. The RCF impact is quantified by a shakedown based dimensionless “fatigue index” (FI) [14]. Deterioration due to wear is quantified using Tγ. Dirks and Enblom [15] introduced a model that can predict both wear and RCF of railway wheels and rails. Two existing RCF prediction models were analyzed and compared in a parametric study, the first being an FI model and the other being a “damage function” model. For the wear model, the Archard model was applied. Bevan et al. [16] developed a damage model to predict the deterioration rate of a wheel tread in terms of wear and RCF damage. This model uses a description of a fleet’s route diagram to characterize the duty cycle of the vehicle. Using this duty cycle and a combination of the Archard and Tγ damage models, many vehicle dynamics simulations were conducted to calculate the wheel and rail contact forces and predict the formation of wear and RCF damage. Brouzoulis [17] presented a two-dimensional (2D) finite element (FE) model to simulate the growth of a single RCF crack in a rail. This model accounts for wear and allows crack curving. The Archard wear model was adopted along with a Paris-type crack propagation law whose crack driving force is based on the concept of material forces. Jun et al. [18] studied the minimum size at which a crack would grow in a rail, which being defined as the size of the smallest crack that grew fast enough to stay ahead of removal by wear and periodic grinding. They used the Archard wear model and the “2.5D” fatigue crack growth model developed by Fletcher and Kapoor [19]. Trummer et al. [20] developed a predictive model for RCF crack initiation at the surface of rails and wheels, referred to as the “wedge model.” In this “wedge model”, both surface RCF crack initiation and delamination wear are deemed to be governed by the growth of the microscopic cracks in a severely shear deformed layer near the surface. Hiensch and Steenbergen [21] extended the concept of a “damage function” from a conventional rail to a premium pearlite rail. This was done both by simulating the dynamic train– track interaction and by using field observations. Values of the RCF damage index were established for the rails, describing the behavior of the associated “damage function.” Akama et al. [22] developed a numerical simulation model to study the competition between RCF-initiated short crack propagation and wear in a carbon steel having ferrite–pearlite structure under the RCF conditions. The crack is assumed to initiate when the total accumulated plastic shear strain reaches the critical value. In the early crack growth simulations, the two-stage short crack growth model proposed by Hobson is applied. With regard to wear, the Archard model is adopted as the basis. The model was applied to the railhead of the actual Shinkansen site, and the behavior of each function was confirmed. Herein, we use the developed model to study the competition between the RCF-initiated crack propagation and the wear in a railway wheel tread. Further, we have added to the model a new function to the model that accounts for the crack initiation caused because of the MnS present inside the steel. None of the aforementioned references considered the initiation of cracks from non-metallic inclusions even though several cracks are observed to be initiated from them.

We begin by applying the model to the results of twin disc fatigue tests to assess its capabilities and accuracy. Further, we apply it to obtain the realistic stress, strain, and slip-velocity states associated with the wheel tread. Finally, based on the twin disc fatigue tests, we study the manner in which reversing the wheel rolling direction affects the crack propagation and wear. This effect cannot be ignored because majority of the trains operate as a shuttle service; however, this important subject has not yet been addressed in the references.

2. Numerical simulation model Regarding the RCF-initiated cracks and wear, we have obtained some findings by observing the actual objects that have been used and the specimens after conducting the twin disc fatigue tests. Based on these observations, we developed a model to simulate the competition between the RCF-initiated crack propagation and wear of ferrite–pearlite steel in 2D rectangular regions (hereinafter referred to as the RCF-C&W model). The RCF-C&W model is implemented in bespoke software comprising a series of programs written in C++. The functions, such as establishing the microstructures and deformation, crack initiation, propagation, and wear, are all integrated into the software. Further, the results are visualized using ParaView, which is an opensource software. This section details the proposed model.

2.1. Short cracks An RCF crack begins life as a short crack. When the crack size becomes comparable to the scale of the characteristic microstructural dimension (i.e. grain size or less), obstacles such as grain boundaries, particle inclusions or phase changes hinder further growth. These obstacles are called “barriers” to crack propagation and retard or even arrest crack growth. In such cases, the crack is called a “microstructurally short crack” (MSC). After overcoming these microstructural barriers, the crack may be long enough to grow by continuum-mechanics mechanisms, whereby the crack length and applied stress or strain determine the crack growth rate. However, crack growth may occur within plastic regions, which invalidate LEFM analyses; the growth rate is greater than that predicted by LEFM. Such a crack is called a “physically small crack” (PSC) [23]. While treating the early stages of crack growth, such short-crack characteristics must be taken into account. First, we attempted to analyze the entire crack in a three-dimensional (3D) microstructure. However, the growth details for a short 3D crack are not yet well understood. Franklin et al. [24] used a model based not on the physical process, but only on how small 3D cracks depend on the geometry or morphology of the steel microstructure. Bomidi and Sadeghi [25] developed a 3D microstructural topology represented by randomly generated Voronoi tessellations to study RCF in bearings. Their material model was coupled with the continuum damage mechanics approach to capture the degradation due to fatigue. However, PSC propagation that can often occur in railhead or wheel tread was not simulated and the mechanism was not clarified. Therefore, this

study considers short crack propagation in a 2D microstructural area within the arbitrary cross sections of a material, separated to ensure that they do not affect each other. 2.1.1 Microstructure Generally, wheels and rails are manufactured using hypoeutectoid steel. The steel microstructure comprises pearlite grains bordered by regions of proeutectoid (PE) ferrite. A pearlite grain comprises pearlite colonies characterized by the joint arrangement of thin layers of ferrite and cementite. The lamellae orientation varies from colony to colony. When the steel is heated to keep all the grains austenite and subsequently cooled to room temperature, the PE ferrite thickens in some regions. In this model, this region can be referred to as a ferrite grain. The microstructure is modeled as a 2D area using equilateral hexagons with a side length of L and a diameter of d as an initial shape for the individual pearlite and ferrite grains (Fig. 1). The PE ferrite is divided into the ferrite grain and the grain boundary ferrite whose thickness is considered to be Tf for convenience. In addition, the PE ferrite contains square ductile inclusions with side lengths LD.

(i, j-1)

rc a

(i, j)

(i-1, j)

(i+1, j)

crack L

a

Tf (i, j+1)

d LD

x

z Figure 1 A hexagonal network of ferrite and pearlite grains bordered by grain boundary ferrite (the thickness; Tf). Crack length a and the origin (circles), slip line directions in the case of ferrite grains or lamellae directions in the case of pearlite grains (broken lines), side length L, diameter d of the grain, and the critical distance for the crack coalescence rc are shown. Black squares represent ductile inclusions.

The original shape of each hexagon is deformed randomly to represent a more realistic microstructure. All apices of the hexagon are moved from their initial positions within a limited circle by using a series of uniform random numbers so that the resultant hexagon shape remains convex [26]. The pearlite to ferrite grain ratio, Tf, and L values are obtained by microscopic observation of the actual microstructure. Multiple slip systems are active in each grain with a randomized crystallographic orientation.

2.1.2 Initiation Franklin and Kapoor [3–5] proposed a criterion based on ratcheting to predict crack initiation life. Here, we follow Franklin and Kapoor’s study and apply this criterion to predict crack initiation. The point at which a crack is initiated is randomly determined in the region of each ferrite grain and the ferrite of the lamellar structure in a pearlite grain. The grain boundary ferrite is divided by the six sides of the hexagonal grains that it surrounds, and a crack seed is generated on each side. The geometry of the given crack seed is a point, so the initial crack length is zero. The initial crack points are given at the beginning of the simulation. The crack is assumed to initiate when the total accumulated plastic shear strain γijn in the region reaches the critical shear strain γc. The strain increment ∆γij is calculated for each load cycle and for each grain (i, j) in row j and column i (Fig. 1) by using the maximum value τzxj(max) of the orthogonal shear stress at each depth and the effective shear yield stress keffij for each grain.

ij γ nij = γ n-1 + ∆γ ij

(1)

 τ j   zx(max) ∆γ = C ij  −1  keff     ij

(2)

where n is the number of load cycles and C the ratcheting rate parameter. The effective shear yield stress increases as the plastic shear strain accumulates. A modified Voce equation is used to calculate the new value:

(

ij keff = k0 max 1, βs 1 − e−γ n

ij

)

(3)

where k0 is the initial shear yield stress and βs the strain hardening ratio. In addition, we consider initiation at a ductile inclusion. Garnham et al. [27] used scanning electron microscopy (SEM) to examine the microstructural detail of the used service rails taken from the high rails of curves together with that of rail discs from the twin disc tests. They found that, near the gauge corner surface, MnS ductile inclusions became flattened both longitudinally and transversely due to the accumulated cyclic stressing. They concluded that these strainflattened MnS could be a source of surface and near-surface crack initiation with the wheel–rail contact. They also investigated spatial distribution in some rail steels of inclusions within certain size ranges. Nanohardness tests were also conducted to determine the difference in hardness between MnS and the pearlite and PE ferrite constituents of the matrix. In this study, square-shaped MnS inclusions are randomly located within the PE ferrite. A crack is assumed to initiate at the inclusion where the cumulated shear strain of the inclusion reaches a critical value. The stress concentration potentially caused by the MnS inclusions is not considered. Edwards [28] calculated the stress concentration factor, αs, when a uniform tensile stress was applied to an ellipsoidal nonmetallic inclusion in a base metal. When the base metal was ferrite and the nonmetallic inclusion was MnS, αs was approximately 1.2. An FE analysis performed in the present study confirmed that αs is similar for the square MnS. This level of stress concentration will produce no major effect on the analysis results. In addition, we do not account for the debonding that can occur at the MnS–ferrite interface. With regard to this phenomenon, Ervasti and Ståhlberg [29] studied the start of void formation close to an inclusion during steel slab hot rolling. The inclusion was chosen to be either three times harder or three times softer than the surrounding matrix. They found that voids were likely to be formed near the hard inclusion, while soft inclusions were only elongated, and no risk of void formation could be predicted. Because MnS is softer than ferrite significantly, debonding is unlikely to occur. To date, detailed behavior and mechanical properties of MnS have not been clarified; however, this simple treatment allows such inclusion “weak spots” to be represented in the microstructure.

2.1.3 Propagation In this study, crack propagation is modeled based on Hobson et al.’s [30] study because of the simplicity of the model. The Hobson model requires only a few material constants, which may be determined by standard experiments, and has been used extensively.

Pearlite colony Pearlite grain

Crack Pearlite block

Pearlite colony Figure 2 A schematic representation of the pearlite grain, pearlite block, pearlite colony, crack propagation through a boundary of pearlite colony and crack arrest at pearlite grain boundary.

The MSC crack growth equation describes the propagation of cracks within the first ferrite grain, the grain boundary ferrite and the ferrite of the lamellae structure in the pearlite colonies (stage I). We assume that such cracks are driven by cyclic shear stress range on the slip planes in the ferrite grain, in the grain boundary ferrite, or on the planes parallel to the lamellae structure in the pearlite grain, depending on the point of crack initiation. The pearlite grain is a lamellar structure in which ferrite and cementite are in formed in layers by the eutectoid reaction, originating from austenite. Subsequently, a hierarchical substructure is formed in the interior, in which a region with a uniform ferrite crystal orientation is defined as a block, and a region with a uniform lamellar orientation is called a colony. The orientations of the blocks and colonies differ both within one grain and among the grains. Urashima and Nishida [31] studied fatigue crack initiation and the propagation behaviors of rail steels using SEM observation. They observed that slip occurred parallel to the lamella of a pearlite colony in the very early stage of fatigue, and fatigue cracks were initiated. They also observed that the crack preferentially propagated along the lamella, and its direction was slightly deflected when it reached the perlite block boundary. The detailed structure of the pearlite grain is not considered in this study. Instead, we assume that the crack propagates along the lamellar direction of the colony. When a crack seed is generated in the ferrite of the lamellar structure located in a specific pearlite colony, only the lamellar direction of the pearlite colony existing at that position is considered. The lamella orientations in the remaining colonies are not considered because the boundary between the pearlite colonies cannot arrest a stage I crack. Also, the possible deflection of the crack propagation direction at the boundary of pearlite blocks is not considered. To ensure a random crack propagation direction, the direction of the slip line in the case of a ferrite grain and that of the lamella in the pearlite colony case are determined by connecting the randomly generated crack seed and the geometric center of the grain. The crack growth rate is influenced by the distance between the crack tip and the dominant microstructural barriers. The strength of the grain boundary makes it a major microstructural

obstacle to crack propagation. Therefore, the grain boundary is assumed to act as a barrier. Figure 2 shows a schematic representation for these explanations. In this study, the stage I crack propagation starts with the first load cycle just after the crack was initiated with a crack length of zero [32]. The crack growth rate in stage I is described by

da = A∆τ ωα ⋅ (d − a ) dn

(4)

where A and α are material constants, a the crack length, and ∆τω is the cyclic shear stress range on the slip planes. The unit is µm/cycle. The value of d is the crack length at the threshold of stage I growth. The stress considered in Eq. (4) is represented by that at the center of the individual grain in which each seed exists. This is also the case for the stress and strain used for performing the crack initiation calculations. Values of the stress and the strain are obtained by FE analysis. As a simplest approximation, the modeled microstructure is assumed to be subjected to the same stress and strain components as the bulk, because it is very difficult to calculate the generated stress and strain states in the microstructure based on its anisotropic stress and strain responses. It should be noted that Eq. (4) is applied to the interior of grains; therefore, the crack length a does not exceed the grain diameter d. When a crack is sufficiently long to permit a crack tip to open, PSC crack propagation occurs (stage II). In stage II, the influence of the microstructure is not considered, and crack growth is described by continuum mechanics in the form

da = B∆ε t β a − D dn (5)

where ∆εt is the total normal strain perpendicular to the crack faces, and B, β and D the experimentally determined material constants. The unit is µm/cycle.

2.1.4 Coalescence During the crack propagation stage, a sudden extension of crack length must be considered because cracks may link together. Crack coalescence is represented by assuming that linking occurs when crack length reaches a critical length lc and the tip-to-tip distance between the cracks is less than a critical distance rc (Fig. 1). After coalescence, crack length is defined by the length of the straight line between both crack tips. If a crack is formed by linking several cracks,

the overall crack length is taken to be the greatest separation between the tips of the cracks involved. In the transition from stage I to II, the crack length is introduced in terms of the number of grains, assumed to be three grain diameters in the present simulation [33]. Even if the crack has transitioned into a stage II crack, stage I cracks may grow at both ends after the transition, unless these cracks are arrested by a barrier. Therefore, the amount of crack growth after the transition is determined by comparing the amount of growth calculated by applying Eq. (4) to the stage I crack at both ends to that obtained by applying Eq. (5) to the entire crack, in which it is considered a stage II crack. The larger of the two values is adopted as the amount of crack growth. This concept is depicted schematically in Fig. 3. Figure 3(a) shows the case in which stage I crack growth occurs in each grain, and the growth of each crack is obtained by Eq.(4); (b) shows the situation when both ends of crack a1 are arrested; (c) demonstrates the case in which a1, a2, and a3 are coalesced, but the crack length is still less than 3d; and in (d), cracks a1, a2, a3, and a4 are coalesced, and the whole crack length exceeds 3d, so the coalesced crack is considered a stage II crack with total length a0. In this final case, the amount of crack growth, da0, is obtained using Eq. (5). Since a3 and a4 do not reach the barrier at one end, their growth amounts, da3 and da4, respectively, are obtained by Eq. (4). The total amount of growth, da34, is the sum of da3 and da4. The largest value among da0 and da34 is adopted as the amount of crack growth.

a3

a3 a1

a1 a2

a2 a4

(a)

a4

(b)

a3

a3

a1

a1

a0

a2

a2

a4

a4

(c)

(d)

Figure 3 Schematic representation for growth of stage I cracks, coalescence, and translation to stage II crack.

2.1.5 Surface roughness

Crack propagation near the surface can be strongly influenced by surface roughness, which locally modifies the pressure distribution of the contact. In this study, we use Seabra and Berthe’s [34] approach, in which surface roughness causes pressure peaks with amplitudes that depend mainly on the perturbation amplitude Amp and wavelength λ. The pressure concentration factor (PCF) is defined as the ratio of pressure peak Pmax to nominal maximum contact pressure P0:

αp

PCF =

β

 λ   b  p  λ γ p Pmax = C1   A   R   b  P0  mp   e 

(6)

where b is the contact area half width, Re the equivalent radius of the contacting bodies, C1 = 4.3884, αp = −0.4234, βp = −0.4204, and γp = −0.0155, respectively. The effect of surface roughness is taken into account in the RCF-C&W model by multiplying PCF to the contact pressure distribution obtained by FE analysis.

2.2. Wear

Franklin and Kapoor [4] considered that wear and RCF are two consequences of the same damage mechanism. Cracks and voids form near the contact surfaces and then join together by growth of voids, crack propagation and the plastic shear deformation of the material. Finally, the surface material spalls off as wear debris, a process well known as “delamination wear” [35]. The theory was first proposed to explain the adhesive wear against the Archard model, however considering the mechanism, this theory may be applicable to fatigue wear. In this study, wear is considered an independent phenomenon, and its interaction with crack formation and propagation is included. When modeling wear in the contact region, the Archard model for adhesive wear has been used and has provided results that are consistent with the measured data. Thus, the simulation in the present study is based on the Archard wear model, which is given by

V =K

Nds H

(7)

where V is the wear volume, K the wear coefficient, N the normal force, ds the sliding distance, and H the penetration hardness of the surface. In the wheel–rail contact region, we use nx points in the longitudinal direction (i.e. running direction) of the train, which is defined as the xdirection, and ny points in the transverse direction (y-direction). The intervals ∆x and ∆y between the points are constant. For a single point, the wear-depth increment ∆z is

∆z ( x, y ) =

Kp ( x, y ) H

s x ( x , y ) + s y ( x, y ) 2

2

∆x Vt

(8)

where p(x,y) is the normal pressure distribution, sx(x,y) and sy(x,y) are the slip velocities in the xand y-directions, respectively, and Vt is the train velocity. All contributions in the longitudinal array of the points sum to obtain

∆zl =

K ∆xl nx 2 2 p ( xkl , ykl ) sx ( xkl , ykl ) + sy ( xkl , ykl ) ∑ H Vt k =1

where (xkl, ykl) is the point inside a contact region, as shown in Fig. 4.

(9)

p (xkl-1, ykl-1) (xk-1l , yk-1l)

sy

sx

(xk+1l , yk+1l)

(xkl , ykl) (xkl+1 , ykl+1)

Contact region x y z Figure 4 Normal pressure p, sx and sy at point (xkl, ykl) inside a contact region.

In the RCF-C&W model, the contact surface can be specified to be the upper edge of the 2D simulation region prior to calculation. The wear-depth increment, calculated using Eq. (9), is eliminated from the edge. Within the removed region, the crack initiation and propagation calculations are no longer performed. If the wear eliminates the crack seeds, the crack initiation calculation from the seeds is not performed. If the wear eliminates the entire crack, the crack propagation calculation is not performed.

3. Application of the model to twin disc fatigue tests To investigate the capabilities and validity of the model described above, unidirectional rolling fatigue tests are performed on a Nishihara type twin disc wear testing machine. Then, the simulation is performed under the same conditions using the RCF-C&W model, and the results are compared.

3.1. Test procedure

Wheel discs are machined from the cross section of a standard wheel rim and run against rail discs from the railhead. Both discs are 12 mm thick and have a diameter of 30 mm. The chemical compositions and material properties of the constituent steels are given in Tables 1 and 2, respectively. Optical micrographs of microstructure for wheel and rail steels are presented in Fig.5(a) and (b), respectively.

Table 1 Chemical compositions (wt. %).

Material

C

Si

Mn

P

S

Wheel steel

0.65

0.26

0.73

0.016

0.01

Rail steel

0.68

0.26

0.93

0.016

0.01

Table 2 Mechanical properties.

Material

Ultimate tensile strength (MPa)

Proof stress (MPa)

Elongation (%)

Wheel steel

981 - 1030

470

8 - 12

Rail steel

934

508

≧10

0.1 mm

0.1 mm (a)

(b)

Figure 5 Optical micrographs for microstructure: (a) wheel steel. (b) rail steel.

All tests were conducted at 1% creepage with the wheel disc at faster speed. The maximum Hertzian contact pressure was 1100 MPa, and the rotational speed of the wheel disc was 1000 rpm. To prevent corrosion, the contact area was lubricated with water containing 5 % cutting oil (product name: MI cool X510) using a gravity drip system that supplied one drip per second approximately. As for the lubrication, in Japan, an oil lubricant is often applied to the gauge corners of the outer rail in a curved section as a countermeasure against rail wear. It is highly likely that this lubricating oil will stick to the wheel treads; therefore, lubrication is performed using a similar oil. After each predetermined number of cycles, the discs were removed and weighed. At the end of each test, all discs were sectioned and photographed for the optical metallography of their crack morphologies and microstructural changes.

3.2. Test results

The details of the tests were extensive, and should be reported in another paper; only those results that are relevant to model verification are shown herein. It was confirmed that a number of cracks initiated and propagated on the contact surface of the rail disc, whereas cracks were rarely found on the contact surface of the wheel disc even after around 1 × 106 cycles. Figure 6 shows an example of each section. Significant shearing plastic flow is evident near the surfaces of both discs. Wear particles are clearly present near the surface of the wheel disc. The crack was not seen as long as the entire circumference was observed for a certain cross section of the wheel disc. To observe the morphology of these wear particles, the discs were not cleaned up. Therefore, the mass of a disc sometimes increased because of wear particles being transferred from the mating disc.

Traction

0.1 mm

Traction

0.1 mm (a)

Traction

0.1 mm

Traction

0.1 mm (b) Figure 6 Circumferential sections of wheel and rail discs after 9.5×105 cycles: (a) wheel disc. (b) rail disc.

3.3. FE analysis of the twin disc fatigue test

In order to simulate the competition between RCF-initiated crack propagation and wear using the RCF-C&W model, stress and strain inside the material, and contact pressure distribution and sliding velocities in the contact region are required. FE analysis of the twin disc fatigue test was conducted using the general-purpose FE code MARC to obtain these values. The discs were modeled with first-order plane strain elements, and the numbers of elements and nodes were 24,600 and 25,622, respectively. This study used a model that combines a nonlinear kinematic hardening rule with the isotropic hardening rule developed by Chaboche and Lemaitre [36] for the rail disc material, namely

ti +∆ti

{

}

σ y = 0σ y + Q 1 − exp ( −Bct +∆t e p )

2 dα = hde p −ζαde p 3

i

i

(10)

(11)

where

ti + ∆ ti

σ y is the updated yield stress at time ti+∆ti, 0σ y is the initial yield stress (namely 508

MPa), Q and Bc are material constants, ti + ∆ti e p is the accumulated effective plastic strain at ti+∆ti, α is the shift of the center of the yield surface, ep is the plastic strain, h and ζ are material constants, and d implies an increment in the variable following it. Q, Bc, h and ζ were determined by means of strain controlled uniaxial fatigue experiments using normal rail steel. Meanwhile, bilinear plasticity is used for the wheel steel because of the lack of detailed data. Data for the plasticity parameters as well as Young’s modulus E and Poisson’s ratio ν are presented in Table 3.

Table 3 Material properties of the rail and wheel steels. Material

E (MPa)

ν

Rail

183,008

Wheel

197,300

0

σ y (MPa)

Q

b

h (MPa)

ζ

E 0 (MPa)

0.3

508

-208

24.2

85,248

193

-

0.3

470

-

-

-

-

1973

A Coulomb type friction law is presumed, and the friction coefficient between the discs is set to 0.1[37]. At each disc center, a pilot node is connected to the nodes at the inner circumference of the disc using a multipoint constraint. All the external loading and boundary conditions are applied on the pilot nodes. As an example of the analytical results, Fig. 7 shows the in-plane shear stress distribution after 10 cycles. In the ratcheting state, the incremental strain becomes stable after receiving some load repeatedly. The stress distribution including the residual stress became almost saturated at 10 cycles. Therefore, stress, strain, contact pressure, and slip velocity were evaluated based on the results at the cycle. Strain hardening by increasing the number of cycles to millions of revolutions is performed in the RCF-C&W model using Eq. (3).

Figure 7 In-plane shear stress distribution near the contact region.

3.4. Competition between short crack propagation and wear in rail disc

The simulation using the RCF-C&W model can be performed in any rectangular region on any area, as long as the area is within the FE analysis area. Here, the region is decided to be the rectangular region of 0.9 mm (x-direction) × 0.3 mm (z-direction) in the rail disc whose center of gravity is 0.15 mm below the surface and an upper edge coincides with the rail disc surface. A schematic representation showing the relationships between these dimensional values, as well as that between the region and the FE analysis area is depicted in Fig. 8. FE analysis was used to obtain the values of τjzx (max) used in Eq. (2) for crack initiation; ∆τω used in Eq. (4) for stage I crack growth; ∆εt used in Eq. (5) for stage II crack growth; and p (xkl, ykl), sx (xkl, ykl), and sy (xkl, ykl) for wear depths at the corresponding positions.

Figure 8 Schematic representation showing the dimensions of the simulation region using the RCF-C&W model (gray area) and the relation between the region and the FE analysis area (yellow area). The black point in the gray region is the center of gravity.

Various parameters are determined as follows. The volume fraction of ferrite is 0.02 as estimated from the carbon content and Lineal analysis using the photographs of microstructure of rail steel. Although only the area fraction of ferrite was obtained using this method, it was considered to be equal to the volume fraction for simplicity. Note that this value of 0.02 does not include the volume of ferrite inside the pearlite grain. The thickness of grain boundary ferrite Tf was also set to 0.2 µm based on the observations. The material constants in Eqs. (2) and (3) for crack initiation are indicated in Table 4. k0 for pearlite and ferrite were obtained from nanohardness measurements. Each measurement was performed three and four times for the pearlite and ferrite regions, respectively, and then averaged. Figure 9 shows photographs of the microstructure and indentations. Each βs was estimated from values of k0 before and after the experiments. The k0 of MnS, taken from [38], neglects strain hardening, i.e., βs = 0.

Table 4 Material constants for crack initiation.

k 0 (MPa)

βs

C

γc

Pearlite

320

1.41

0.00237

11

Ferrite

190

1.41

0.00237

11

MnS

19

0

-

11

(a)

(b)

Figure 9 SEM images displaying the microstructure and nanoindentation. (a) pearlite with indentation depth of 1000 nm and (b) ferrite with indentation depth of 200 nm.

The angle of the deformed microstructure θ at 0.03 mm below the contact surface was used to determine γc (i.e.γc = tanθ) for the ferrite and the pearlite, while that of MnS was assumed to be identical. Figure 10 shows the illustration of the method [39]. It is difficult to obtain C, and therefore, the value of BS 11 rail steel determined by Tyfour et al. [40] is used. The volume fraction of MnS is set to 0.002. The initial shape is assumed to be a square and the average length of a side LD is set to 2.5 µm. The distribution density is 5 mm−2 on average, as per Garnham et al. [27].

Figure 10 Schematic representation of the method used to find the γc.

Regarding crack propagation, data for the material constants in Eqs. (4) and (5) are summarized in Table 5. The value for d was obtained from the microscopic observations, while those for A and α are from Hobson et al. [30] and those for B, β, and D are from independent

fatigue tests using the same rail steel. In the present simulation, crack coalescence occurs when the cracks reach 75% of d (lc = 0.75 × d) while rc is less than 50 % of d.

Table 5 Material constants for crack propagation.

d (µm) 31.8

A -31

1.85×10

α

B

β

D (µm)

11.14

0.504

1.466

3.57×10

-3

The data for the PCF parameters Amp and λ in Eq. (6) presented in Table 6 were estimated by surface roughness measurements, while b was obtained by FE results.

Table 6 Parameters for PCF.

A mp (µm)

λ (µm)

b (mm)

R e (mm)

0.5

150

0.28

15

Regarding wear, data for the parameters in Eq. (9) are presented in Table 7. The value for K is set by considering the test condition as the boundary lubrication [41], while Vt is calculated using the diameter of the disc (30 mm) and the rotational speed (1000 rpm), and ∆xl is the distance between nodes in the FE analyses. Table 7 Constants for wear.

K -7

1×10

H (MPa)

V t (m/s)

∆x l (mm)

3494

1.57

0.1

Because of the high number of cycles, simulation of every loading cycle is not efficient. Therefore, stresses and strains on the crack surfaces are assumed to be constant over a finite number of cycles dn in Eqs. (4) and (5). In this case, dn is set to 1000. Henceforth, these values are referred to as a “standard case” of the twin disc fatigue test that is used for comparison. The simulation was conducted on the rail disc for a maximum of 1 × 106 cycles and the result was investigated in detail by comparing the crack morphologies of the cracks propagated near the surface of the rail disc surface.

When the lamellae direction of the pearlite colonies, along which the stage I cracks had already propagated, were approximately that of the cracks, which had grown by coalescence and fulfilled the coalescence conditions, it was found that these cracks coalesced. The resulting stage II crack propagated in a significantly different direction from the plastic flow near the surface. As shown in Fig. 6 (b), the actual cracks propagated approximately along the plastic flow of the surface. Therefore, we added a coalescence condition that the cracks could coalescence when the difference between the directions of the cracks was between −10° and +10°, tentatively. The simulation results for the standard case after the above-mentioned modification are shown in Fig. 11. The plastic flow near the surface was already remarkable and many stage I cracks were propagating at 5 × 105 cycles. At 1 × 106 cycles, these cracks coalesced to form several stage II cracks. The wear from the surface can also be seen clearly. The simulation was run on a personal computer and the computational time was approximately 3 minutes in this case. A comparison of the result of the test for the rail disc shown in Fig. 5(b) with that of the simulation shown in Fig. 10(c) under similar conditions indicates that the plastic flow near the surface, the crack length, and the inclination angle are all very similar. Specifically, of the 100 simulations performed using the random numbers generated, the average crack length was 0.36 mm, with an average inclination angle of 85° from the horizontal. Conversely, the measurements from the images in Fig. 5 (b) were 0.32 mm and 86°. Although the verified example is insufficient, the capabilities and behavior of the RCF-C&W model are considered to be reasonable.

Figure 11 Simulation results for the interaction between short fatigue crack propagation and wear in the rail disc for the standard case: (a) initial (n = 0), (b) after 5 × 105 cycles, (c) after 1 × 106 cycles.

3.5. Influence of various factors To assess how the various factors influence the results, a series of simulations were performed.

3.5.1 Thickness of grain boundary ferrite Figure 12(a) shows the result at 1 × 106 cycles, when Tf was changed to 0.1 µm, which is half of that in the standard case. In this case, the cracks did not progress to stage II eventually, and the plastic flow was low compared to that in the standard case. This is because fewer cracks were initiated from the grain boundary ferrite as the thickness decreased. As the hardness increased because of the decreased amount of low hardness ferrite, the plastic flow also decreased.

3.5.2 Volume fraction of non-metallic inclusion The result obtained by changing the volume fraction of MnS from 0.002 to 0.001 is shown in Fig. 12(b). When the fraction is specified, the model automatically calculates the MnS amount and the position. Therefore, cracks are initiated in the ferrite region at identical points as in the standard case. Compared to the standard case, no cracks progressed to stage II. This is because the initiation of cracks is low when the MnS inclusion amount is reduced, since these are crack initiation points.

3.5.3 Wear coefficient The result of changing K (wear coefficient) from 1 × 10-7 to 2.5 × 10-7 is shown in Fig. 12(c). Compared with the standard case, the initiated cracks and plastic flow near the surface are completely scraped off, i.e., wear had overcome crack propagation. The above analysis results seem reasonable and, therefore, confirm that the developed simulation model can operate correctly.

Figure 12 Influence of various factors on the results at 1 × 106 cycles: (a) thickness of grain boundary ferrite (0.2µm → 0.1µm), (b) volume fraction of nonmetallic inclusion (0.002 → 0.001), and (c) wear coefficient (1 × 10-7 → 2.5 × 10-7).

4. Application of the model to the wheel tread

Next, the RCF-C&W model is applied to realistic stress, strain, and slip velocity situations associated with the wheel tread using the FE analyses presented in the next section and the data for the microstructure, crack initiation, propagation, and wear.

4.1. FE analysis of wheel/rail contact A 3D FE model of a wheel and a rail is shown in Fig. 13(a). The wheel shape is for newly manufactured Shinkansen vehicles and the diameter is 860 mm. The shape of the tread surface is modified arc. The shape of the railhead is that of a new normal 60 kg rail. The rail was developed as a rail for the Shinkansen, with a mass of 60 kg per meter. The lateral symmetries (y-direction in Fig. 13(a)) of the wheel and the rail are utilized, implying that the rolling of the wheel on the center line of the rail is straight. The modeled parts are large enough to neglect boundary effects and do not require much computational resources. Regions close to the contact area are meshed with the first-order brick elements. The smallest element size close to the contact area is roughly 0.2 mm, thanks to the adaptive remeshing, and the final numbers of elements and nodes are 536,488 and 632,784, respectively. The constitutive models for the rail and wheel materials are the same as those used in Section 3.3. To rotate the wheel on the rail surface, an axle is modeled by elastic straight beam elements. The diameter and elastic modulus of the beam elements are identical to those of the actual axle

and are connected to the wheel by rigid elements [42]. The wheel rotates on the rail by 5º in a pure rolling condition macroscopically. The wheel load is set to 98 kN as a dynamic force considering the traveling train through an additional dynamic factor. Thermal loads due to tread braking are not considered because in this study, disc braked vehicles are considered. The friction coefficient between the wheel and the rail is set to 0.3, considering the contact between rusty steels. Only one wheel passage is analyzed because of the large computational time due to the high mesh density and the demand of small time steps. Figure 13(b) shows part of the results, namely the radial stress distribution on the wheel tread, almost at the center of the contact between the wheel and the rail model.

(a)

(b) Figure 13 (a) FE mesh for the wheel-rail contact problem. (b) Radial stress distribution on the wheel tread. The arrow indicates the rolling direction of the wheel.

4.2. Competition between short crack propagation and wear in the wheel tread Similarly to the methods described in Section 3.4, the simulation can be performed for any rectangular region in any area, as long as the area is within the FE analysis area. As mentioned in Section 2.1, we consider crack propagation and wear in 2D microstructural areas within arbitrary cross sections in the wheel tread, separated to ensure that they do not affect each other. As an example, simulations were conducted for the region of a square represented by A with sides of 2 mm and an upper edge coinciding with the tread surface in a plane parallel to the circumferential direction of the wheel, and the initial state is depicted in Fig. 14(a). In the transverse direction, the center of gravity of the region is at the center of the wheel, which is 65 mm away from the outer surface of the flange. In addition, we consider two regions B and C, which are located roughly 2.0 mm away from region A in the transverse direction. Region B is closer to the flange side, whereas region C is closer to the anti-flange side. All regions are under the wheel–rail contact patch. Unfortunately, certain parameters of the wheel steel that are necessary for simulating the competition between short crack and wear, such as k0, βs and γc for crack initiation and A, α, B, β and D for crack propagation are not available. However, the chemical compositions of wheel steel and rail steel are similar, as shown in Table 1, and both are ferrit-pearlite hypoeutectoid steels. Although the grain boundary ferrite in wheel steel appears to be slightly thicker than that in rail steel as shown in Fig.5, the missing wheel steel parameters are taken as those of rail steel as given in Sections 3.3 and 3.4. The value of K is set to 1 × 10-4, which is the typical value for the contact between the wheel tread and the rail top surface in dry conditions [43]. The parameters for PCF are as follows: Amp = 1.0 µm, λ = 1 mm, b = 5.0 mm, Re = 430 mm. τjzx (max), etc. were obtained from the FE analysis as discussed in Section 4.1. Vt is set to 55.6 m/s (200km/h). ∆xl is 0.2 mm that is the distance between nodes in FE analysis. In this case, dn is set to 10, which corresponds to the running distance of 27 m. Hereafter, these values are referred to as a “standard case” of the wheel tread. With reference to the Shinkansen routes, the train travels a distance of 1000 km at an average speed of 200 km/h. Shinkansen routes are mostly straight sections; any curved section has a sufficiently large radius that it does not require consideration. Moreover, the accelerating and braking section lengths are proportionally extremely low compared to the coasting section length during normal operation. Consequently, the simulations were performed under macroscopically pure rolling conditions that were considered to be those of coasting approximately. Only the outward route run was simulated. In this case, the computational time was about 2000 seconds.

One set of simulation results is shown in Fig. 14(b). Under these conditions, the stress and strain generated near the tread surface were so low that almost no macroscopic plastic flow occurred. The conditions were straight running with pure rolling macroscopically, therefore, the maximum shear stress related to crack initiation occurred approximately 2.5 mm beneath the surface. Many stage I cracks, initiated at MnS, developed below 0.5 mm depth. However, no cracks progressed to stage II. Because there was almost no slip, the wear of the tread surface was insignificant.

4.3. Influence of various factors To assess how various factors influenced the results, a series of simulations were performed. The factors whose values were changed were the macroscopic slip ratio (0% to 1%), the wheel load (P = 98 kN to 118 kN) and the wear coefficient (K = 1 × 10-4 to 1 × 10-3). Only when the slip ratio was changed from 0% to 1% did the crack initiation positions clearly move toward the surface, and stage I crack propagation occurred in these regions. Results for the increasing slip ratio are shown in Fig. 15. Slip on the tread surface increases the shear strain and shear stress near the surface due to tangential traction. In all cases, stage II crack propagation was not seen and there was very little wear, namely almost the same as in the standard case.

Traction

Initial tread surface

C 2 mm A

B

2 mm

2 mm

2 mm

(a)

(b)

Figure 14 Simulation results for the interaction between short fatigue crack propagation and wear near the surface of the wheel tread for the standard case at different travel distances, Td. (a) Td = 0 km; (b) after Td = 1000 km. (blue, ferrite; yellowish-green, pearlite; orange, stage I crack).

Traction

Figure 15 Simulation results near the wheel tread’s surface after the macroscopic slip ratio increased to 1% after Td = 1000km.

5. Tests involving reversal of rolling direction

Generally, trains operate as a shuttle service. When a train arrives at the terminal station, its direction of travel is reversed, and therefore so is the rolling direction of the wheels. It is therefore important to investigate how the crack morphology, propagation, and wear are changed by reversing the rolling direction. Tyfour and Beynon [44,45] studied how the rolling direction reversal affected the fatigue crack morphology, propagation, RCF life, and wear rate of pearlitic rail steel. Considering a single line supporting a two-directional traffic, they used a twin disc fatigue machine to simulate the conditions, reversing the rail discs every predetermined number of cycles. They reported a beneficial effect on both RCF life and wear rate. At present, the model for the competition between crack propagation and wear does not allow the wheel rolling direction to be reversed. This is because it is yet to be clarified how the crack propagation and wear near the wheel tread behave under such circumstances. The results obtained by Tyfour and Beynon cannot be referred to because the direction of relative slip and the inclinations of the generated propagating cracks differ completely between the wheel and the rail. This is why we did not simulate the return route in Section 4.2. To elucidate how reversing the rolling direction of the wheel affects crack initiation and wear, a series of tests were conducted using a Nishihara type twin disc wear testing machine.

5.1. Test procedure The rolling direction was reversed by inverting the wheel disc after a certain number of rolling cycles. That number was determined as that after which cracks were definitely initiated in the wheel discs as obtained in the previously performed unidirectional tests. Other items such as

the machine and materials used and testing conditions were the same as those presented in Section 3.1.

5.2. Test results Test details will be reported in the near future; only those results that are relevant to improving the model are presented here. The unidirectional tests confirmed that cracks were initiated and propagated near the contact surface of the wheel disc after 1.2 × 106 cycles, so the disc rotation was reversed after that number of cycles. Figure 16 (a) shows a crack in the wheel disc when it was run for 1.2 × 106 cycles, and (b) shows the crack in the wheel disc when the disc rotation was reversed after 1.2 × 106 cycles and run for 3 × 105 reversed cycles. To capture the entire image of the cracks, low magnification micrographs of the same disc are presented in Fig. 17. The cracks clearly tended to propagate parallel to the surface when the disc was reversed, and the resulting long and thin wear sheets delaminated. There was no plastic flow near the surface where the wear sheets were peeled off.

Figure 16 Circumferential section through the detected crack zone of a wheel disc. (a) After 1.2 × 106 cycles run unidirectionally. (b) After 3.0 × 105 cycles reversed at 1.2 × 106 cycles.

Figure 17 Circumferential sections through the detected crack zone of a wheel disc at low magnification after 3.0 × 105 cycles reversed after 1.2 × 106 cycles. (a) Long and thin sheet delaminated and (b) delaminated sheet broken into pieces.

6. Discussion

A numerical model named RCF-C&W model was developed to simulate the competition between RCF-initiated crack propagation and wear in the ferrite-pearlite steel. First, unidirectional rolling fatigue tests were performed, and the model was validated using the experimental results of crack initiation and propagation in a rail disc. A series of sensitivity analyses indicated that the present simulation model can operate properly at least qualitatively. However, the values used for the variables concerning microstructure, crack initiation, propagation, and wear were often uncertain. For example, it was assumed that the crack coalescence occurs when lc is greater than 0.75d and while rc is smaller than 0.5d. At present, the crack coalescence process is difficult to observe; thus, sufficient information cannot be obtained. Considering that coalescence occurs on the order of the grain size, the critical distances can at least be estimated as less than or equal to the grain size. A comparison of the results obtained by simulation using these values and the crack observation results in twin disc tests under the same conditions as in Section 3 reveals similar crack sizes and inclined angles. Therefore, the assumed values are not considered to be outliers. Socie and Furman [33] set rc to 0.25d for a ferritepearlite steel microstructure under the multiaxial loading, whereas Hoshide et al. [46] set rc to 0.85d for a pure copper microstructure under biaxial stresses. To achieve quantitative accuracy, these values should be made certain by situation-specific experimental results.

In these tests, infinitesimal wear particles as shown in Fig. 6(a) were frequently observed near the surfaces of the discs. These are inferred to be either wear particles that have grown as a result of combining the transfer particles that were previously proposed by Sasada [47] in the adhesive wear theory or delaminated sheets broken into small pieces by entrapment between the contacting surfaces. In the former case, it is reasonable to use the Archard model while changing the wear coefficient to approximate values. If these particles are fragments of delaminated sheets, applying the model is questionable. These particles must be investigated in more detail. Next, the model was applied to the situations in the wheel tread considering the actual operation of Shinkansen vehicles. Simulations were conducted for three square regions under the contact patch with sides of length 2 mm and with an upper edge that coincides with the tread surface. Only a straight outward route of 1000 km was simulated, changing the various factors within their practical ranges. In these cases, no stage II crack propagation was seen, and very little wear occurred. This is because the shape of the wheel tread is a modified arc, and the contact between wheel and rail is almost conforming contact. The generated stress near the surface does not exceed the yield stress of ferrite and pearlite. In this situation, it is necessary to simulate a return route in which the rolling direction of the wheel is reversed, considering also the conditions during acceleration, deceleration, and curve negotiation, although those portions in the overall section are extremely small. Rolling direction reversal tests were performed using a twin disc machine and it was found that when the rolling direction of the wheel disc was reversed, long and thin wear sheets were apt to delaminate. The phenomenon was never observed in the unidirectional tests. The mechanism behind the formation of such sheets is considered as follows. Figure 18 (a) is a schematic of a state in which a crack initiated and propagated slightly along the direction of shear plastic flow of the wheel disc in the unidirectional test. In this case, the crack tip enters the contact zone between the two discs before the mouth. Therefore, no fluid is expected to be trapped because the crack is closed starting from the tip, while the mouth remains open. Because the presence of fluid in the crack is known to be essential for RCF cracking, it is difficult for the crack to continue to propagate. By contrast, Fig. 18(b) shows a state in which the disc is reversed after a crack has initiated on its surface. The crack is filled with fluid just before its mouth reaches the contact area, whereupon the mouth is sealed. In this case, the fluid is trapped in the crack and acts as a lubricant to either reduce the friction coefficient or pressurize the crack faces. With either mechanism, cracks tend to propagate easily. During unidirectional rolling–sliding contact, the microstructure near the surface of the wheel disc deforms in such a way that the carbide plate in the pearlite lamellae and the grain boundaries are aligned almost parallel to the surface as shown in Fig. 16(a). These become obstacles to the crack propagation. In this case, once a crack initiates between these obstacles, it propagates easily along them because there are no obstacles in that direction, as shown schematically in Fig. 19. Therefore, the thin and long wear sheets shown in Fig. 17 are likely to form and then delaminate.

In these ways, when short cracks are present on the wheel tread, reversing the rolling direction seems to make delamination wear dominant. Therefore in future work, we will add to the present model a function that can simulate delamination wear by devising the mechanisms properly, considering the detailed running conditions. Furthermore, even though there is very little information, we would like to extend the RCF-C&W model to include the exact 3D microstructure regions. Also, rolling direction of wheel reversal tests should be continued by changing the load, slip ratio, and ambient conditions.

(a)

(b)

Figure 18 Schematic representation of the directions of rolling and traction for a wheel disc in rolling– sliding contact: (a) during a unidirectional test; (b) after the rolling direction was reversed.

Figure 19 Schematic representation of a lamellar structure formed near the contact surface of the wheel disc.

7. Conclusions A numerical model was developed to simulate the competition between RCF-initiated short crack propagation and wear in a wheel tread. The simulation model assumes that the wheel materials are polycrystalline ferrite and pearlite and that RCF crack initiation is determined by the total accumulated plastic shear strain. The growth of RCF cracks is calculated using the Hobson model and the Archard model is used to calculate wear.

First, the model was applied to the results of the twin disc fatigue tests. We used parametric studies to investigate how the various factors influence the competition between cracks and wear. It was shown that the present simulation model can operate properly. Next, the model was applied to the situations in the wheel tread by using the stresses, strains and relative sliding velocities obtained from FE analyses together with the variables for microstructure, crack initiation, propagation and wear. When a straight outward route of 1000 km was simulated based on the Shinkansen routes, it was found that the tread was hardly damaged. Finally, how reversing the rolling direction of the wheel affected crack propagation and wear was studied using the twin disc fatigue tests. It was shown that reversing the rolling direction makes delamination wear dominant.

Acknowledgments This study was part of the research and development program for the future of railways entitled “Creation of rail damage/ballast track deterioration models and evaluation of maintenance work saving technologies” at the Railway Technical Research Institute. Part of this study was supported by MEXT KAKENHI Grant No. 26390137. The authors are grateful for the financial support. The authors would like to thank Enago for the English language review.

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Table 1 Chemical compositions (wt. %). Material

C

Si

Mn

P

S

Wheel steel

0.65

0.26

0.73

0.016

0.01

Rail steel

0.68

0.26

0.93

0.016

0.01

Table 2 Mechanical properties. Ultimate tensile strength

Proof stress

Elongation

(MPa)

(MPa)

(%)

Wheel steel

981 - 1030

470

8 - 12

Rail steel

934

508

≥ 10

Material

Table 3 Material properties of the rail and wheel steels. Material

E (MPa)

ν

Rail

183,008

Wheel

197,300

0

σy (MPa)

Q

b

h (MPa)

ζ

E0 (MPa)

0.3

508

-208

24.2

85,248

193

-

0.3

470

-

-

-

-

1973

Table 4 Material constants for crack initiation. k0 (MPa)

βs

C

γc

Pearlite

320

1.41

0.00237

11

Ferrite

190

1.41

0.00237

11

MnS

19

0

-

11

Table 5 Material constants for crack propagation. d (µm)

A

α

B

β

D (µm)

31.8

1.85×10-31

11.14

0.504

1.466

3.57×10-3

Table 6 Parameters for PCF. Amp (µm)

λ (µm)

b (mm)

Re (mm)

0.5

150

0.28

15

Table 7 Constants for wear. K

H (MPa)

Vt (m/s)

∆xl (mm)

1×10-7

3494

1.57

0.1

Highlights A numerical model is developed to simulate the competition between crack propagation and wear. The model is shown to operate properly when applied to the results of twin disc tests. When a 1000 km straight run is simulated, the wheel tread is hardly damaged. If the rolling direction of the wheel is reversed, the delamination wear becomes dominant.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: