Effect of plastic deformation and fretting wear on the fretting fatigue of scaled railway axles

Journal Pre-proofs Effect of plastic deformation and fretting wear on the fretting fatigue of scaled railway axles Lang Zou, Dongfang Zeng, Liantao Lu, Yuanbin Zhang, Yabo Li PII: DOI: Reference:

S0142-1123(19)30475-X https://doi.org/10.1016/j.ijfatigue.2019.105371 JIJF 105371

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

25 July 2019 6 November 2019 8 November 2019

Please cite this article as: Zou, L., Zeng, D., Lu, L., Zhang, Y., Li, Y., Effect of plastic deformation and fretting wear on the fretting fatigue of scaled railway axles, International Journal of Fatigue (2019), doi: https://doi.org/ 10.1016/j.ijfatigue.2019.105371

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.

© 2019 Published by Elsevier Ltd.

Effect of plastic deformation and fretting wear on the fretting fatigue of scaled railway axles Lang Zoua, Dongfang Zenga,*, Liantao Lua, Yuanbin Zhangb, Yabo Lib a State

Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China b CRRC

Qingdao Sifang CO., LTD., Qingdao 266111, China

Corresponding author. Tel.: +86 28 86466025; fax: +86 28 87600868. E-mail address: [email protected] Abstract ： In this paper, three fatigue methodologies based on the elastic material, elasto-plastic material and incremental wear models are developed for investigating the effect of plastic deformation and fretting wear on the fretting fatigue of scaled railway axles. By comparing with the experimental results, the most reasonable fatigue methodology is selected. Then, the influence of diameter ratio on the fretting behaviors is evaluated. The simulation results show that the profile evolution of fretting scar is mainly caused by fretting wear. The fatigue methodology based on either the elastic material model or the elasto-plastic material model could neither predict the fretting scar width nor the crack initiation site, while a relatively good prediction can be given using the wear-fatigue methodology. For the crack initiation life in the short-life region, a relatively accurate prediction could be given by using the elastoplastic material model based fatigue methodology and wear-fatigue methodology, while in the long-life region, the crack initiation life predicted using the wear-fatigue methodology is greater than that predicted by the fatigue methodology based on the elasto-plastic material model. The increment of diameter ratio can reduce the fretting wear and relative slip range through alleviating the stress concentration, and thereby improving the fatigue strength of the scaled railway axles. There exists a critical value of diameter ratio between 1.092 and 1.116. When the diameter ratio is less than the critical 1

value, the fretting wear and relative slip range are sensitive to the variation of diameter ratio and otherwise the opposite. Keywords: Fretting fatigue; Fretting wear; Plastic deformation; Railway axle; Diameter ratio 1 Introduction Axle is one of the most important mechanical components in railway vehicles, which is connected with the wheel by press-fitting [1, 2]. Press-fitted parts such as the wheel seats and gear seats are the critical part of the railway axle since the cracks are easily initiated due to the fretting behaviors [3, 4]. Numerous experimental studies have been conducted to investigate the fretting behaviors of the press-fitted axle [5-12]. Nishioka et al. carried out systematic research on the effect of geometric shapes on the fretting fatigue limit using 50 mm-dia scaled railway axles. They found that the fretting fatigue limit increased with a decrease in fillet radius r, and an increase in diameter ratio D/d. Meanwhile, the fretting damage was observed within the region 1-5 mm from the contact edge, and the fatigue fracture location was always inside the fretted zone [5]. Further studies showed that the fretting cracks could initiate even if the applied stress level far below the fatigue limit [6]. Juuma [7, 8] made a similar experimental study with a loading in alternated torsion, from which the similar fretting damage was observed. Alfredsson [12] studied the fretting damage of the shrink-pin, and found that the fretted zone can be clearly divided into one shallow but serve metallic scar with black oxides and the other deep scar but mild rust-red oxides. Recently, Zhang et al. [10] conducted interrupted rotating bending fatigue tests using 13 mm-dia press-fitted axles. According to the analysis of surface morphology, three characteristic fretted zones were identified, including zone I with severe abrasive wear, zone II with delamination and oxidative wear, and zone III with slight delamination, respectively. The width and depth of each zone varied with the increase of fretting cycles, and the fretting cracks initiated in zone II. The distribution of 2

three zones was generally uniform with that found in railway axles [11]. Although exhaustive experiments have been conducted, some important fretting variables that determine the fretting damage of the press-fitted axle, including contact pressure and local cyclic shear stress, et al. [2, 13, 14] cannot be measured from these tests due to the special structure of the press-fitted axle. Therefore, it is hard to reveal the mechanism of fretting fatigue in the press-fitted axle. With the advance in numerical method and computing power, the finite element method (FEM) was employed to investigate the fretting fatigue of the press-fitted axle. White et al. [15] studied the stress distribution in press-fitted axles using an elastic FE model, and found that the maximum axial stress appears at the contact edge. Then, an elastic FE analysis for the railway axle by Yang et al. [16] showed similar conclusions. It was found that the crack initiation at the inner side of the contact area cannot be explained by these FEM simulations. Almost all the experimental results found the profile change in the fretted zone, which is considered to be caused by the local plastic deformation and fretting wear. Kubota et al. [17] investigated the effect of stress relief groove on the fretting fatigue limit, and found that the cracks initiation at the inner side could be reasonably interpreted using an elasto-plastic FEM analysis. Zhu et al. [18] suggested that SWT model [19] could accurately predict the crack initiation site and life on bending fretting fatigue of LZ50 axle steel when considering ratcheting. Based on the works of Nishioka et al. [5], Lanoue et al. [20] established an elasto-plastic FE model and found that the Dang Van model [21] could predict the variation trend of fretting fatigue strength with the change of structure parameters. Meanwhile, the fretting wear simulation was used for complex configurations [22, 23] and operating conditions [24]. Recently, Zhang et al. [25, 26] proposed a wear-fatigue methodology for the press-fitted axle. The predicted results were validated with experimental data, but the effect of structure parameters has not been studied yet. Obviously, both the elasto-plastic material model based fatigue 3

methodology and wear-fatigue methodology could explain some phenomena observed in the test. However, there is no relevant literature reporting the difference between these two fatigue methodologies for studying the fretting behaviors of press-fitted axle. In addition, the influence of structure parameters on the fretting behaviors remains to be further studied. In this paper, the elastic material, elasto-plastic material and incremental wear FE models are used for predicting the fretting scar in scaled railway axles. Based on these FE models, the corresponding fatigue methodologies are proposed. Then, these fatigue methodologies are used to study the effect of plastic deformation and fretting wear on fretting fatigue of scaled railway axles. By comparing with the experimental data, the most reasonable fatigue methodology is selected. Finally, the influence of diameter ratio on the fretting behaviors of the scaled railway axles is evaluated. Based on these works, suggestions are given for selecting the fretting fatigue prediction methodology for the press-fitted axle. 2 Experimental details of Nishioka Systematic experiments were conducted by Nishioka et al. [5, 6] to study the effect of geometric shapes on the fretting fatigue of scaled railway axles. Lots of valuable information, including the material properties, fatigue limit, fretting scar width, crack initiation site and so on, was stated in the literatures in detail. Therefore, FE models are established based on these experiments, and some experimental information related to this study is briefly outlined in this section. 2.1 Specimens and materials The dimension of the scaled axle and hub specimen is shown in Fig. 1. The shoulder of the filleted axle coincides with the hub. Diameter of the axle at the press-fitted portion, D, is 50 mm, the outer diameter of the hub near the loading side is 75 mm, and a fillet with radius R of 7 mm is used to connect the press-fitted portion and the axle body. The effect of diameter ratio D/d on the fretting behaviors of 4

the press-fitted axle is studied by changing the axle body diameter d. Materials of the specimen are axle steel (SF55) for the axle and wheel steel (STY80) for the hub. The mechanical properties of both axle and hub are listed in Table 1.

Fig. 1 Dimension of the scaled axle and hub specimen Table 1 Mechanical properties of materials Elastic modulus

Poisson’s

Yield strength

Tensile strength

Micro-hardness,

(GPa)

ratio

(MPa)

(MPa)

HV0.1

Axle (SF55)

206

0.3

346

572

170

Hub (STY80)

206

0.3

536

981

226

Materials

The nominal contact pressure that is determined by a grip diameter 2δr is 70 MPa. Therefore, the corresponding 2δr is calculated as 61 μm according to the Lame’s solution shown in Eq. (1).

P

2 E r   b   1     2b   a  

(1)

where E is the elastic modulus, b is the contact radius, a is the outer radius of the hub, and δr= bAxle-bHub is the grip radius. 2.2 Experimental results The fretting fatigue tests were conducted on a 4000 N·m cantilever rotating bending fatigue machine at a speed of 1250 rpm. Some valuable experimental results related to this investigation are summarized in Table 2. There are two types of fatigue limits listed in Table 2. The fatigue limit σw2 was determined under a 5

criterion whether the axle was completely broken or not after 2×107 fretting cycles. After stress cycles of 2×107, the unbroken specimen was saw-cut along the axial direction of the hub and the axle was pulled out of the hub with no damage. Then the abrasive oxide debris was removed, and the fretted zone was observed under a photomicroscope with a magnification of 400. The critical stress below which minute cracks (5μm in length) could not be found was defined as σw1. Some experimental findings related to the fretting damage were also measured. The fretting scar width Fb was measured at 100 MPa and 2.0×107 cycles. The fretting crack initiation site h means the distance between the fretting crack initiation site and the contact edge. In this paper, the fretting crack initiation site h of Test 1~4 were measured at 103, 136, 143 and 140 MPa, respectively, and the corresponding fretting cycles was 2.0×107. The relative slip range s between the hub face and the contact edge was measured at 100 MPa and 1.0×106 cycles. Table 2 Details of fatigue specimens and results of fatigue tests [6] Tes

Dimension of specimen

Fatigue limit (MPa)

Fb

h

s

t

D (mm)

d (mm)

D/d

σw1

σw2

(mm)

(mm)

(μm)

1

50

47.80

1.046

80

125

1.7

1.7

34

2

50

46.82

1.068

105

143

1.8

2.0

20

3

50

45.79

1.092

105

143

1.4

1.8

22

4

50

43.52

1.149

125

-

0.3

1.8

20

3 FEM methodology 3.1 Geometric shapes and stress levels The previous section outlines the experimental results under different conditions, and almost all test cycles reach 2×107. Obviously, the corresponding FEM simulation is very time-consuming. In order to save computational time, seven different cases are employed. Case 1 with nominal bending stress of 100 MPa is used to verify the validity of three FE models for the fretting scar simulation. Cases 2~7 are used to study the effect of diameter ratio on the fretting behaviors in scaled railway axles. Table 3 summarized 6

the geometric shapes and stress levels employed in this paper. Table 3 Geometric shapes and stress levels Case

Dimension of specimen D (mm)

d (mm)

Nominal bending

Loading

D/d

stress (MPa)

F0 (N)

1

50

47.80

1.046

100

2337

2

50

47.80

1.046

140

3272

3

50

46.82

1.068

140

3272

4

50

45.79

1.092

140

3272

5

50

44.80

1.116

140

3272

6

50

44.13

1.133

140

3272

7

50

43.52

1.149

140

3272

3.2 FE model The general-purpose non-linear commercial FE software ABAQUS was chosen for FE modeling. The FE model of the press-fitted axle with D/d=1.046 is shown in Fig. 2(a). As shown in Fig. 2(b), the mesh in the area of interest is refined to capture the complicated variations in stress-strain. Threedimensional eight-node linear brick elements (C3D8) are employed throughout. The contact surface interaction between the axle and hub is modeled via the master-slave contact algorithm. In this paper, the inner surface of the hub is chosen as master surface while the surface of the press-fitted portion as slave. Then the Coulomb friction model based on the penalty algorithm is employed to ensure the tangential behavior. The maximum elastic slip tolerance is set as 0.0001 and the friction coefficient is considered as constant 0.6 [3, 27]. The values of highly refined mesh size and the maximum elastic slip tolerance strongly affect the accuracy of simulation results, and the optimizations are described in Section 3.5. The loading and boundary conditions imposed on the FE model are shown in Fig. 2(a). During the simulation, x axis displacements on the left surface of the hub are not allowed, y and z axis displacements of the external surface of the hub are fastened. In step 1, the interference fit is introduced with a value of 61 μm. The concentrated force along the y axis (Fy=F0) is applied at point A in step 2. Then two 7

computational time-varying concentrated forces along y and z axis, Fy=F0cos(2πt) and Fz=F0sin(2πt), are applied at point A during the later steps. In this way, the stress state of the press-fitted portion under rotating bending is characterized. (a)

External surface_hub_Uy&Uz=0

Left surface_hub_Ux=0

Fy=F0cos(2πt)

Hub

Fz=F0sin(2πt)

Axle

A

Contact edge

Mesh refinement region (b)

Contact edge

Fig. 2 FE model: (a) overall geometry of the press-fitted specimen (D/d=1.046) and (b) mesh details in the region indicated by the rectangle in (a). 3.3 FEM simulation of the fretting scar 3.3.1 Elastic material model There is a simple assumption that only elastic deformation occurs in the fretted zone. Therefore, the axle and hub are considered as elastic material with an elastic modulus of 206 GPa and a Poisson’s ratio of 0.30. 3.3.2 Elasto-plastic material model Steep stress gradients are found near the contact edge when the press-fitted axle is subjected to rotating bending loads. Irreversible plastic deformation occurs once the yield stress σy is exceeded. Then 8

the materials near the contact edge are progressively rounded due to plastic deformation, which may release the stress concentration and cause the failure location shifting to the inner side of the contact area. In this study, the effect of plastic behavior was considered by using the linear kinematic hardening model [28]. In the kinematic hardening model, the translation of the center of the yield surface is dominated by a back-stress tensor χ, defined for linear hardening as follows:

2 d   cd p 3

(2)

where c is a hardening modulus, dεp is the increment of plastic strain and calculated via the follow rule: d

p

 d

f  3  dp  e 2

1 3 f   e   y  ( (    ) : (    )) 2   y 2 1 2 dp  ( d  p : d  p ) 2 3

(3) (4) (5)

where f is the von Mises yield function, dp is the increment of effective plastic strain, σe is the von Mises equivalent stress, dλ is the plastic multiplier, χ’ and σ’ are the back-stress tensor and deviator stress, respectively. In this paper, a yield stress σy of 346 MPa and a hardening modulus c of 20.6 GPa are employed for the SF55 axle steel, while a yield stress σy of 536 MPa and a hardening modulus c of 20.6 GPa are employed for the STY80 wheel steel. 3.3.3 Incremental wear model A modified version of the energy-based wear approach [29] is used to simulate the fretting wear. For completeness, it is briefly outlined here. For a given nodal point j on the contact surface of the fretted zone, the incremental wear depth Δhinc(j,m) at mth increment time can be expressed as:

hinc ( j , m)   q ( j , m)s ( j , m)

(6)

where α is the energy wear coefficient, q(j,m) and Δs(j,m) are the shear traction and the incremental slip, 9

respectively. A rotating bending load step is divided into n simulation increments. Then the total wear depth for one FE fretting cycle can be written as: n

hstep ( j , m)    q ( j , m)s ( j , m)

(7)

n 1

The wear simulation for the press-fitted axle in high-cycles cannot model individual wear cycle due to the limitation of computation capacity. Therefore, the cycle jump technique is employed, which assumes that the wear rate remains unchanged in ΔN fretting cycles. The nodal wear corresponding to ΔN cycles can be calculated in one FE fretting cycle by: n

hstep ( j , m)    q( j , m)s ( j , m)

(8)

n 1

Considering the balance between the computational precision and speed, the increment n and cycle jump ΔN are set as 40 and 4,000, respectively. The optimizations of these two wear parameters are described in Section 3.5. For the fretting wear simulation, the modified energy-based approach is implemented using an adaptive mesh technique via ABAQUS user subroutine UMESHMOTION. As described in the previous study [25], when the ratio of maximum bending stress to the normal contact stress exceeded a critical value, partial separation, namely open zone, occurs between the axle and the hub. Predictably, the debris would be removed easily when the open zone existed, and a fretting wear scar would occur near the contact edge. In this paper, the open zone appears near the contact edge when Case 1~Case 7 are employed. Therefore, the adaptive meshing algorithm of ABAQUS applies the local wear increment for nodes only in the open zone, and the local wear increment is implemented in two steps. First, it is implemented by moving the surface nodes in the local normal direction, and the geometry updates as a purely Eulerian analysis in this step. Then the second step is used to remap the material quantities 10

(variables) to the new positions. The linear kinematic hardening model is also employed for material behavior in this model. The energy wear coefficient is an inherent property of material, which is measured using a fretting wear test with ball-on-flat configuration [30]. In this paper, the energy wear coefficient of SF55 axle steel is 3.33×10-8 MPa-1 [31]. The energy wear coefficient of STY80 wheel steel cannot be found in the published literatures. However, the wear resistance is approximately proportional to the hardness, and 0.752 factor is used to relate the energy wear coefficient to that of the axle. Therefore, the energy wear coefficient of the hub is set as 2.50×10-8 MPa-1. 3.4 Damage modeling The contact surface and subsurface of the fretted zone is subjected to complex multi-axial stress as a result of the applied cyclic bending loads [2]. Several multi-axial fatigue approaches, such as SWT model and Dang Van model, have been used to evaluate the fretting fatigue. In this investigation, a mean stress correction multi-axial fatigue approach proposed by Ince et al. [32] is employed. This approach can give a good prediction for the material subjected to very large compressive mean stress, such as the materials in the fretted zone of the press-fitted axle. It is expressed as follows:

 eq ,a 2



 ae,eq 2



 ap  max  ae  ap  f (2 N i ) 2b   f (2 N i )c    2  f 2 2 E

(9)

For the FEM simulation with linear elastic material properties, the Eq. (9) can be written:

  ae,eq 2



 max  ae  f  (2 N i ) 2 b  f 2 E

where Δεeq,a/2 is a total equivalent strain amplitude,

∆𝜀𝑒𝑎,𝑒𝑞 2

=

𝜎max∆𝜀𝑒𝑎 𝜎′𝑓 2

(10)

is an equivalent elastic strain

amplitude corrected for the mean stress effect, σmax is the maximum normal stress, ∆𝜀𝑒𝑎/2 and ∆𝜀𝑝𝑎/2 are the elastic and plastic normal strain amplitudes, respectively. Ni is the number of cycles to initiate a crack 11

of a given length. 𝜎′𝑓 and b are the fatigue strength coefficient and exponent, 𝜀′𝑓 and c are the fatigue ductility coefficient and exponent. The strain-life constants for SF55 axle steel are given in Table 4. Table 4 Strain-life constants for the axle [33] 𝜎′𝑓 (MPa)

𝜀′𝑓

b

c

1553.8

0.4522

-0.13

-0.54

The FE-implementation of the mean correction model follows the method described by Sum at al. [34]. The total equivalent strain amplitude Δεeq,a /2 is calculated by using the commercial math software MATLAB. In this paper, there are 180×180 candidate planes for each element when the angle-interval of the candidate plane is set as 1°. During one FE fretting cycle, the σmax, ∆𝜀𝑒𝑎/2, ∆𝜀𝑝𝑎/2 and Δεeq,a/2 for each element in each candidate plane can be calculated according to Eq. (9) and Eq.(10) , and then the maximum value of Δεeq,a/2 and ∆𝜀𝑒𝑎/2 can be obtained. Three FE models are used for simulating the fretting scar, and therefore there are some differences in calculating the crack initiation life Ni. For the elastic material and elasto-plastic material models, the differences are detailed in Section 4.1.1. When the incremental wear model is adopted, the stress-strain corresponding to each FE fretting cycle l is used to calculate the crack initiation life Ni,l. However, the crack initiation life Ni,l predicted by Eq. (9) changes with the fretting cycles. Therefore, a cumulative damage approach is employed to capture this phenomenon. Madge et al. [35] found that the MinerPalmgren (M-P) linear fatigue damage accumulation rule can give a good prediction for the fretting fatigue when considering the fretting wear. Since the cycle jump technique is adopted in this study, a modified M-P rule can be given as:

D

N / N

D l 1

l



N / N

 l 1

N N i ,l

(11)

where D is the total damage in each element during FE simulation, Dl= ΔN/Ni,l is the calculated damage at FE fretting cycle l, N is the total number of experimental fretting cycles, Ni,l is the crack initiation life 12

predicted by Eq. (9) at the lth FE fretting cycle, N/ΔN is the total number of FE fretting cycles. The crack initiation is predicted when the accumulated damage D of surface material reached 1.0. As mentioned in Section 3.2.3, the local wear increment simulation is achieved by adaptive meshing algorithm in two steps: the mesh updating and remapping materials quantities (variables) to the new positions. In this method, the centroid position of each element changes from cycle to cycle, and specific element centroid is no longer linked uniquely to actual material during all fretting cycles. Obviously, the accumulated damage of an element centroid in the previous fretting cycle cannot be linearly accumulated to the new centroid position of the current cycle. To solve this problem, Madge et al. [35] proposed to create a material point mesh (MPM) as the global reference in the original element centroid. The nodes of the MPM have fixed coordinates during all simulation. Cyclic damage is calculated at the centroid of each element and linearly interpolated back to the MPM for accumulation. In this study, the MPM approach is employed to consider the effect of material removal due to fretting wear on fretting fatigue. The schematic diagram of the MPM approach is shown in Fig. 3. Material Point Mesh Axle

Material Point Mesh has fixed coordinates. Fretting wear

Red material points no longer accumulate.

Fig.3 The schematic diagram of MPM approach. 3.5 Optimization of computational accuracy and speed In this study, a 3D FE model is used to study the press-fitted axle subjected to rotating bending loads. The computational accuracy and speed are governed by the mesh size, maximum elastic slip 13

tolerance, increments per FE fretting cycle and cycle jump ΔN. So this section presents the analysis of these principal parameters involved in the FEM simulation. All the results are taken from the incremental wear FE model. The frictional shear traction and axial slip range are obtained from the axle without fretting wear, and all the fretting scars are predicted in the axle by 120,000 fretting cycles. 3.5.1 Mesh size in fretted zone Fig. 4 shows the effect of mesh size on the frictional shear traction distribution and predicted fretting scar. It can be seen that the smaller mesh size can capture the stress concentration near the contact edge more accurate, and give a better prediction for the fretting scar. When the mesh size is less than 20 μm, both of the frictional shear traction and predicted fretting scar almost remain unchanged, while the computational time increases rapidly. Therefore, the mesh size is set as 20 μm in this study.

(b)

(a)

Fig.4 Effect of mesh size on the (a) frictional shear traction distribution of the compression side and (b) predicted fretting scar profile. The maximum elastic slip tolerance, increments per FE fretting cycle and cycle jump ΔN are set as 0.0001, 40 and 4,000, respectively. 3.5.2 Maximum elastic slip tolerance for penalty approach Different contact algorithms within ABAQUS can describe the stick-slip behavior in the contact surface of the press-fitted axle. Rajasekaran et al. [36] found that the penalty approach with an appropriate maximum elastic slip tolerance can give a result as good as Lagrange multipliers approach, but with less contact convergence problem. Therefore, the penalty formulation is used in this study and 14

the distribution of axial slip range with different slip tolerance is shown in Fig. 5. It can be seen that the smaller slip tolerance can give a better description for the stick-slip, and the axial slip range gradually reaches a stable state after the slip tolerance is less than 0.0001. Since the smaller slip tolerance requires more iteration times, it is selected as 0.0001 in this study.

Fig.5 Effect of the maximum elastic slip tolerance on axial slip range distribution of the tension side. The mesh size is set as 20 μm. 3.5.3 The increments per FE fretting cycle and the cycle jump ΔN As can be seen from the Eq. (8), the computational accuracy of the predicted fretting scar is directly associated with the increments per FE fretting cycle and cycle jump ΔN. The impact of increments per FE fretting cycle on the predicted fretting scar is shown in Fig. 6. It has been found that the smaller increments per cycle produce convergence problems, while the predicted fretting scar almost keeps unchanged when the increments per cycle is greater than 40. Fig. 7 shows the effect of cycle jump ΔN on the predicted fretting scar. The wear depth decreases with a decrease of ΔN, and the predicted fretting scar finally reaches a stable state as the cycle jump ΔN is less than 4,000. Therefore, the increments per FE fretting cycle and cycle jump ΔN are set as 40 and 4,000, respectively.

15

Fig. 6 Effect of the increments per FE fretting cycle on predicted fretting scar profile. The mesh size, maximum elastic slip tolerance and cycle jump ΔN are set as 20 μm, 0.0001 and 4,000, respectively.

Fig. 7 Effect of the cycle jump ΔN on predicted fretting scar profile. The mesh size, maximum elastic slip tolerance and increments per FE fretting cycle are set as 20 μm, 0.0001 and 40, respectively. 4 Results and discussions 4.1 Fretting fatigue prediction based on three models In this section, three FE models described above are combined with the corresponding multi-axial fatigue approach to study the fretting behaviors of the press-fitted axle, including the fretting scar width, crack initiation site and life. Then, the most reasonable fatigue methodology is selected to explain some phenomena observed in the fatigue test. The results presented in this section are taken from Case 1, as listed in Table 3. 4.1.1 Fatigue methodologies based on the elastic material and elasto-plastic material models 16

When these two material models based fatigue methodologies are used to study the press-fitted axle under rotating bending loads, the stress-strain near the contact edge may oscillate in the first few revolutions, especially for the elasto-plastic material model. Therefore, the prediction of crack initiation life according to Eq. (9) and Eq. (10) should be carried out when the steady-state is reached. Fig. 8(a) shows the predicted evolution of axial stress S11 with fretting cycles in elastic material model based fatigue methodology. As indicated by the rectangle in Fig. 8(a), it can be seen that the axial stress has a slight oscillation in the first fretting cycle, and achieves stability since the second fretting cycle. The axial stress in the steady-state is a standard sinusoid, which indicates that the contact area is subjected to rotating bending stress. Fig. 8(b) displays the predicted evolution of equivalent elastic strain amplitude ∆𝜀𝑒𝑎,𝑒𝑞/2 distribution with fretting cycles. Due to the elastic material model is adopted, ∆𝜀𝑒𝑎,𝑒𝑞 /2 almost remains unchanged during fretting cycles 1~8. Therefore, the crack initiation life is calculated based on the ∆𝜀𝑒𝑎,𝑒𝑞/2 obtained in the 8th fretting cycle. The predicted crack initiation life is 4.58×105 cycles, and the corresponding fretting crack is predicted to initiate at the contact edge.

(a)

(b)

Fig. 8 Predicted evolution of (a) the axial stress S11 of each revolution and (b) total equivalent elastic strain amplitude ∆𝜀𝑒𝑎,𝑒𝑞/2 with fretting cycles in elastic material model based fatigue methodology. Fig. 9(a) shows the predicted evolution of the equivalent plastic strain PEEQ with fretting cycles in elasto-plastic material model based fatigue methodology. It can be seen that the value of PEEQ at the contact edge gradually increase with the increase of fretting cycles, but the increase rate gradually slows 17

down. The increment of PEEQ at the inner side is very small and almost remains unchanged. Assuming that the fretting scar is mainly caused by the plastic deformation, the width of the plastic zone shown in Fig. 9 is considered as the fretting scar width, and its value is 0.16 mm. The predicted evolution of the total equivalent strain amplitude Δεeq,a/2 with fretting cycles is shown in Fig. 9(b). The value of Δεeq,a/2 in the plastic zone decreases quickly in the first few fretting cycles, and it reaches to a nearly stable state since the 10th fretting cycle. Due to the high non-linearity cyclic plastic deformation, existed cycle jump technique cannot be applied in here. Therefore, the crack initiation life is calculated based on the Δεeq,a/2 obtained in the 100th fretting cycle. The predicted crack initiation life is 6.67×106 cycles, and the corresponding fretting crack is predicted to initiate at the contact edge.

(a)

(b)

Fig. 9 Predicted evolution of (a) the equivalent plastic strain PEEQ and (b) total equivalent strain amplitude Δεeq,a/2 with fretting cycles in elasto-plastic material model based fatigue methodology. 4.1.2 Wear-fatigue methodology It is known that the fretting wear contributes to material removal, which induces the evolution of contact surface profile and the stress-strain redistribution. In this section, the evolution of the fretting scar is predicted using the incremental wear model, and the fretting fatigue prediction is performed based on the predicted fretting scar. Fig. 10 shows the predicted evolution of fretting scar profile with fretting cycles. The depth of predicted fretting scar reaches the maximum at the contact edge for any given fretting cycles, and it 18

gradually decreases when moving toward the inner side of the contact area. Besides, both the depth and width of the fretting scar increase with an increase of fretting cycles.

Fig. 10 Predicted evolution of fretting scar profile with fretting cycles. Based on the predicted fretting scar profile of the axle, the total equivalent strain amplitude Δεeq,a/2 distribution with fretting cycles is calculated, as shown in Fig. 11. It can be seen that the maximum value of Δεeq,a/2 appears at the contact edge on the unworn surface. Under the action of fretting wear, the material near the contact edge is progressively removed, and therefore the severe stress concentration is relieved. Thereby the value of Δεeq,a/2 at the contact edge is gradually decreased and a new peak value appears at the worn-unworn boundary. With an increase of fretting cycles, the peak value of Δεeq,a/2 gradually increases and its location moves toward the inner side. Fig. 12 shows the predicted evolution of fatigue accumulated damage D distribution with fretting cycles. During the fretting cycles 1~7.68×106, as shown in Fig. 13(a), the predicted accumulated damage DE at the contact edge sharply increases in the early stage, and a peak value appears there. The fatigue damage also accumulates at the inner side of the contact area, where a new peak value DI appears. When the number of fretting cycles reaches 2.16×106, the accumulated damage DI in the inner side almost equals to the value of DE. With further increase of fretting cycles, the increase rate of accumulated damage DE gradually slows down because of material removal, while the accumulated damage DI 19

dramatically increases. Therefore, the maximum value of D shifts from the contact edge to the inner side. During the fretting cycles 7.68×106~1.24×107, as shown in Fig. 13(b), the evolution trend of the accumulated damage D is somewhat different from the previous stage. It can be seen that the accumulated damage DE decreases throughout this stage. According to the simulation of the cylinder on flat configuration by Madge et al. [35], the material removal induced by severe fretting wear would remove the old outermost surface material, but it also introduces fatigue damage to the new outermost surface material at the same time. Obviously, there is a competition between the fretting wear and fatigue damage. When the number of fretting cycles exceeds 7.68×106, fatigue damage near the contact edge is slight in each fretting cycle while the fretting wear almost remains unchanged. Therefore, the material with serious fatigue damage is removed by fretting wear and replaced by the new surface material with less damage. With an increase of fretting cycles, the accumulated damage DI increases and moves toward the inner side, while the accumulated damage DE decreases. Finally, the accumulated damage DI reaches 1.0 and the fretting crack is predicted to initiate at the inner side.

Fig. 11 Predicted evolution of the total equivalent strain amplitude Δεeq,a/2 distribution with fretting cycles in wear-fatigue methodology.

(a) 20

(b)

( c )

Fig. 12 Predicted evolution of fatigue accumulated damage D distribution with fretting cycles: (a) 1~7.68×106 fretting cycles, (b) 7.68×106~1.24×107 fretting cycles and (c) 1.24×107~2.0×107 fretting cycles. During the fretting cycles 1.24×107~2.0×107, the accumulated damage D distribution is shown in Fig. 13(c). The accumulated damage DI is greater than 1.0, and further increases with an increase of

21

fretting cycles. It can be predicted that the fretting crack could initiate within the region 1.42 ~2.06 mm away from the contact edge in this stage. 4.1.3 Comparison of experimental and predicted results The comparisons of the experimental and predicted results are summarized in Table 5. As described above, for both the fatigue test and FEM simulation, the fretting scar width and fretting crack initiation site are obtained at stress near 100 MPa. The fretting scar widths obtained using the material models alone are much smaller than that measured in the test. On the contrary, the scar width predicted by the incremental wear model is slightly larger than that obtained from the fatigue test. For the crack initiation site, statistical analysis of the experimental results found that there is a correlation between the fretting scar width and the crack initiation site, as shown in Fig. 13. It can be seen that the fretting cracks initiate within the region of fretting scar naturally, and the boundary of the fretting scar coincides completely with the fretting crack initiation site in some cases. As shown in Table 3, the predicted result based on the wear-fatigue methodology shows that the fretting crack initiates within the region near the wornunworn boundary, which agrees with the phenomenon found in the test. The experimental results did not give the crack initiation life at 100 MPa. However, they found that the σw1 corresponding to 2×107 fretting cycles is 80 MPa. Therefore, it can be speculated that the experimental crack initiate at the cycles less than 2×107 fretting cycles. Compared with the experimental data, the crack initiation life predicted using the elastic material model based fatigue methodology is obviously shorter, while it predicted using the wear-fatigue methodology is greater than that predicted by the fatigue methodology based on the elastoplastic material model.

Table 5 Comparison of the predicted results and experimental data Fretting scar 22

Initiation

Initiation

Experimental

width Fb (mm)

life, Ni

Site h (mm)

1.7

<2.0×107

1.7

0

4.58×105

0

0.16

6.67×106

0

2.02

1.24×107

1.42 ~2.06

Elastic material model based fatigue methodology Elasto-plastic material model based fatigue methodology Wear-fatigue methodology

Fig. 13 The correlation between the fretting scar Fb and the fretting crack initiation site h [6] In this study, the phenomenon that the predicted fretting scar width is slightly larger than the experimental result can mainly be attributed to two reasons. (a) The wear simulation employed an average energy wear coefficient throughout the simulation, while the local energy wear coefficient in the fretted zone changes from cycle to cycle. (b) The fretting wear debris trapped in the fretted zone which can relieve the fretting wear is not considered in the simulation [37]. Ding et al. [38] conducted a simulation based on the cylinder on flat configuration, and found that the predicted fretting scar width when considering the effect of debris is more consistent with the experimental result. The wear debris is also observed in the fretted zone, and it may have a similar influence. Through the FEM simulation considering the effect of plastic deformation, Zhu et al. [18] found that the crack initiation site for the pad-on-flat configuration could be accurately predicted, while it cannot be predicted for the press-fitted axle in this study. For the bending fatigue test conducted on the 23

semi-cylindrical fretting pad-on-flat configuration, the flat is bounded by two narrow semi-cylindrical fretting pads. The bending load is imposed on the flat, which causes a narrow fretted zone. Severe stress concentration occurs in this zone, causing severe plastic deformation. Besides, fretting wear also occurs in the fretted zone, but it is weak. The fretting scar is mainly caused by plastic deformation, and therefore the FEM simulation considering the effect of plastic deformation could accurately predict the crack initiation site. For the press-fitted axle in Case 1, a relatively small nominal bending load was applied at the axle, which causes a slight plastic deformation near the contact edge. On the contrary, the fretting wear occurs in the fretted zone and continuously changes the contact surface profile throughout the test. Under the action of slight plastic deformation and sustained fretting wear, the fretting scar is mainly caused by the fretting wear, especially for the press-fitted axle with long service life. In summary, as compared with the fatigue methodologies based on these two material models, the wear-fatigue methodology can give a relatively good prediction for the fretting scar width and fretting crack initiation site of the press-fitted axle. 4.1.4 Effect of fretting scar profile variation on the stress-strain Fig. 14 shows the predicted evolution of contact pressure and frictional shear traction distribution in the tension side when considering the fretting wear. The evolution trend of contact pressure is similar to that of the frictional shear traction since these two contact variables can be converted by the Coulomb friction law. The von Mises stress contours at different stages are shown in Fig. 15. It can be seen that the open zone appears near the contact edge and a serious stress concentration locates at the edge of the open zone, as shown in Fig. 15(a). Under the action of continuous fretting wear, the material is removed and therefore the stress concentration near the contact edge is significantly relieved, as shown in Fig. 15(b). Fretting wear causes the change of contact stiffness and the open zone moves toward the inner 24

side of the contact area. Then, the fretting wear occurs in the new open zone. A new stress concentration appears at the worn-unworn boundary due to the serious geometric discontinuity. With an increase of fretting cycles, the peak contact variables and von Mises stress gradually increase and move toward the inner side, as shown in Fig.14 and Fig.15. This leads to a similar evolution trend of the peak total equivalent strain amplitude Δεeq,a/2, as shown in Fig. 11. (b)

(a)

Fig. 14 Predicted evolution of (a) contact pressure and (b) frictional shear traction distribution with fretting cycles. Contact edge

Contact edge

(b)

(a) Contact edge

(c) Fig. 15 Von Mises stress contours at different stages: (a) 0 fretting cycles, (b) 1.0×107 fretting cycles and (c) 2.0×107 fretting cycles. It can be seen from Fig. 11~Fig. 15 that the stress-strain varies with the change of fretting scar. 25

Severe fretting wear near the contact edge greatly reduces the stress concentration and grinds off the surface material with great accumulated fatigue damage. Meanwhile, the new fatigue damage at the inner side is introduced by the continuous fretting wear. Consequently, the stress redistribution induced by the fretting wear should be considered in the fretting fatigue prediction of the press-fitted axle. 4.2 Effect of diameter ratio on the fretting behaviors As shown in Table 2, both the fretting scar width and the fretting fatigue strength of the press-fitted axle are strongly influenced by the diameter ratio D/d. In this section, Cases 2~7 with different D/d listed in Table 3 are employed to study the evolution of the fretting scar width and relative slip range with diameter ratio. Then, the effect of diameter ratio on the fretting fatigue in the press-fitted axle is evaluated. 4.2.1 Fretting scar and relative slip range According to the results of the previous section, the fretting scar widths predicted using the material models based fatigue methodologies are far less than that obtained by the test. Therefore, the effect of the diameter ratio on the fretting scar profile and relative slip range is predicted using the wear-fatigue methodology, and the predicted fretting scar profiles of the axle are shown in Fig. 16. All the fretting scar profiles are obtained at 140 MPa and 5.0×106 fretting cycles. It can be seen that the maximum depth and width of fretting scar decrease with an increase in diameter ratio. The predicted fretting scar almost coincides when the diameter ratio is greater than 1.133. Nishioka et al. found that there exists a close correlation between the σw1 and fretting scar width [6]. Therefore, it is necessary to study the evolution of fretting scar width with diameter ratio, and the predicted evolution is shown in Fig. 17(a). As can be seen, all the fretting scar widths decrease with an increase in diameter ratio, and there exists a critical value between the diameter ratio of 1.092 and 1.116. When the diameter ratio is less than the critical value, the fretting scar width is sensitive to the variation 26

of diameter ratio and otherwise the opposite. Besides, the fretting scar widths increase with an increase of fretting cycles. The experimental results listed in Table 2 show that the fretting scar width can be reduced by increasing the diameter ratio, and the fretting scar width obtained in D/d=1.149 is much smaller than that of the others. Therefore, it can be seen that the predicted evolution trend of fretting scar width agrees with that found in the fatigue test. Fig. 17(b) shows the predicted evolution of relative slip range with diameter ratio. It can be seen that the predicted evolution of the relative slip range decreases with an increase in diameter ratio, and the evolution trend is almost consistent with the experimental data. This is attributed to the factor that the bending stress near the contact edge is effectively alleviated with an increase in diameter ratio. With an increase of fretting cycles, the relative slip range gradually increases. This evolution trend is similar to that measured in the practical railway axle tests conducted by Zhang and Cao [39]. When the fretting wear appears at the fretted zone, the grip diameter is reduced. Then the constraint between the axle and hub is alleviated, which causes the increase of relative slip range with fretting cycles.

Fig. 16 Predicted evolution of fretting scar profiles with diameter ratio.

27

(a)

(b)

Fig. 17 Predicted evolution of the (a) fretting scar width and (b) relative slip range with diameter ratio. 4.2.2 Fretting crack initiation behaviors Fig. 18 shows the predicted evolution of the total equivalent elastic strain amplitude ∆𝜀𝑒𝑎,𝑒𝑞/2 with diameter ratio in elastic material model based fatigue methodology. It can be seen that all the ∆𝜀𝑒𝑎,𝑒𝑞/2 reach its maximum value near the contact edge. With an increase in diameter ratio, the maximum value of ∆𝜀𝑒𝑎,𝑒𝑞/2 decreases since the stress concentration is effectively relieved. When the diameter ratio reaches 1.149, severe stress concentration shifts from the contact edge to the inner side of the contact area. The predicted evolution of total equivalent strain amplitude Δεeq,a/2 obtained based on the elastoplastic material fatigue methodology is shown in Fig. 19. As can be seen, the evolution of Δεeq,a/2 is nearly the same as ∆𝜀𝑒𝑎,𝑒𝑞/2, but its values are different. Compared with Fig. 17, the predicted value of Δεeq,a/2 near the contact edge is smaller than that of ∆𝜀𝑒𝑎,𝑒𝑞/2 since the materials were progressively rounded with the plastic deformation. Based on the predicted results of ∆𝜀𝑒𝑎,𝑒𝑞/2 and Δεeq,a/2, the fretting crack initiation life is calculated and added in Table 6. Based on the fretting scar profiles predicted in wear-fatigue methodology, the total equivalent strain amplitude Δεeq,a/2 and fatigue accumulated damage D are calculated. Fig. 20 shows the predicted evolution of the total equivalent strain amplitude Δεeq,a/2 with diameter ratio, and all the values are calculated at 5.0×106 fretting cycles. It can be seen that all the Δεeq,a/2 reach its maximum value near the worn-unworn boundary, and the maximum value decreases with an increase in diameter ratio.

28

Fig. 18 Predicted evolution of the total equivalent elastic strain amplitude ∆εea,eq/2 with the diameter ratio in elastic material model based fatigue methodology.

Fig. 19 Predicted evolution of the total equivalent strain amplitude Δεeq,a/2 with the diameter ratio in elasto-plastic material model based fatigue methodology. Fig. 21 shows the predicted evolution of the fatigue accumulated damage D distribution with diameter ratio when the maximum value of D equals 1.0. The predicted results are listed in Table 6. For D/d=1.149 (Test 4 in Table 2), the experimental crack initiation site is 1.8 mm away from the contact edge, which is measured at 140 MPa and 2.0×107 fretting cycles. The crack is predicted to initiate at the cycles of 1.72×107, and the initiation site is 1.4 mm away from the contact edge. It can be seen that the predicted initiation site is slightly closer to the contact edge than that measured in the test. For the D/d of 1.068 and 1.092 (Test 2 and 3 in Table 2), the experimental crack initiation sites are measured at stress near 140 MPa, and the distances between the crack initiation site and the contact edge are 2.0 mm and 1.8 mm, while the corresponding simulation distances are 0.64 mm and 0.76 mm. These differences can 29

be attributed to that the fretting cycles used in the FE simulations are less than 2.0×107. As discussed earlier, the crack initiation is predicted when the fatigue accumulated damage D of surface material reaches 1.0. For the above listed cases, the fretting cycles used in the simulation are less than 2.0×107. As the number of fretting cycles continues to increase and finally reaches 2.0×107, the boundary of fretting scar would further move toward the inner side of the contact area. Thereby the peak value of accumulated damage D further increases and its location moves toward the inner side, and the fretting crack is predicted to initiate within the region between the predicted initiation site and the boundary of the fretting scar corresponding to 2.0×107 fretting cycles. That is to say, the fretting crack could initiate within the region near the worn-unworn boundary. This phenomenon agrees with those found in scaled railway axles, as shown in Fig. 13.

Fig. 20 Predicted evolution of the total equivalent strain amplitude Δεeq,a/2 with the diameter ratio in wear-fatigue methodology. As shown in Table 6, the crack initiation life predicted using the elastic material model based fatigue methodology is obviously shorter. When the other two fatigue methodologies are used, the predicted crack initiation life is almost the same in the short-life region, while there is a significant difference in the long-life region.

30

Fig. 21 Predicted evolution of the accumulated damage D distribution with diameter ratio when the maximum value of D equals 1.0. Table 6 Comparisons of the predicted results of three fatigue methodologies

Case

D/d

Elastic material model

Elasto-plastic material model

Wear-fatigue

based fatigue methodology

based fatigue methodology

methodology

Initiation

Initiation site

Initiation

Initiation site

Initiation

Initiation

life, Ni

(mm)

life, Ni

(mm)

life, Ni

site (mm)

2

1.046

3.49×103

0

3.42×105

0

4.44×105

0.84

3

1.068

6.73×103

0

6.76×105

0

7.60×105

0.64

4

1.092

1.33×104

0

1.03×106

0

1.68×106

0.76

5

1.116

7.88×104

0

2.62×106

0

9.77×106

0.80

6

1.133

2.85×105

0

4.25×106

0

1.36×107

0.72

7

1.149

1.14×106

0.04

5.74×106

0.04

1.72×107

1.40

According to the S-N curve obtained by Nishioka et al. [5], the total fatigue life is about 2.5×106 fatigue cycles when the diameter ratio and bending stress are set as 1.046 and 140 MPa, respectively. It has been reported that the crack initiation life is often approximately 10%~30% of the total fatigue life in press-fitted axle [10, 40], and therefore the calculated fretting crack initiation life is approximately 2.5×105~7.5×105 fretting cycles. By comparison, the elastic material model based fatigue methodology underestimates the crack initiation life, while a relatively accurate prediction for the short-life region could be given by using the other two fatigue methodologies. When the diameter ratio is increased from 1.116 to 1.149, the predicted crack initiation life is in the

31

long-life region, and there is a difference among the life predicted by these fatigue methodologies. For D/d=1.149 (Test 4 in Table 2), the fatigue limit σw1 corresponding to 2×107 fretting cycles is 125 MPa, and therefore the experimental crack initiates at the cycles less than 2×107. As shown in Table 6, the crack initiation life predicted by these three fatigue methodologies are less than 2×107, and the life predicted using the elastic material model based fatigue methodology is obviously shorter. It is hard to determine the prediction accuracy of the elasto-plastic material model based fatigue methodology and the wear-fatigue methodology. However, there are significant differences in the calculation of the crack initiation life. As shown in Fig. 16, although the fretting wear becomes slighter as the diameter ratio increases, the experimental fretting crack still initiates at the inner side. Obviously, the stress redistribution induced by the evolution of the fretting scar greatly affects the fretting fatigue. The elastic material and elasto-plastic material models based fatigue methodologies cannot fully consider the evolution of the fretting scar, and the crack initiation life is predicted based on the stress-strain obtained near the contact edge where severe stress concentration appears. This may underestimate the crack initiation life. When the wear-fatigue methodology is selected, the role of fretting wear can be taken into account, and the fretting crack is predicted to initiate at the inner side. In addition, the fretting scar width, fretting crack initiation site and relative slip range predicted using the wear-fatigue methodology agree with the findings from the fatigue tests. Therefore, as compared with the fatigue methodologies based on these two material models, it can be inferred that the crack initiation life in the long-life region could be reasonably predicted using the wear-fatigue methodology. 5 Conclusions In this paper, three fatigue methodologies based on the elastic material, elasto-plastic material and incremental wear models are developed for investigating the fretting behaviors of scaled railway axles. 32

Some parameters related to these fatigue methodologies are optimized, and these methodologies are used to study the effect of plastic deformation and fretting wear on fretting fatigue. Then, the influence of diameter ratio on the fretting behaviors of scaled railway axles is evaluated. The main conclusions obtained in this work are summarized as follow: (1) The profile evolution of the fretting scar in the press-fitted axle is mainly caused by fretting wear. The fatigue methodology based on either the elastic material model or the elasto-plastic material model could neither predict the fretting scar width nor the crack initiation site, while a relatively good prediction can be given using the wear-fatigue methodology. (2) Under the action of fretting wear, serious stress concentration near the contact edge is alleviated and therefore the fretting crack initiation is suppressed, while a new stress concentration appears at the inner side, which promotes the fretting crack initiation within the region near the worn-unworn boundary. (3) For the crack initiation life in the short-life region, a relatively accurate prediction could be given using the elasto-plastic material model based fatigue methodology and wear-fatigue methodology. While in the long-life region, the fretting crack initiation life predicted using the wear-fatigue methodology is greater than that predicted by the fatigue methodology based on the elasto-plastic material model. (4) The increment of diameter ratio can reduce the fretting wear and relative slip range through alleviating the stress concentration, and thereby improving the fatigue strength of press-fitted axle. There exists a critical value of diameter ratio between 1.092 and 1.116. When the diameter ratio is less than the critical value, the fretting wear and relative slip range are sensitive to the variation of diameter ratio and otherwise the opposite. 33

Acknowledgments This work was supported by the Independent Research Project of State Key Laboratory of Traction Power (2018TPL_Z01) and the National Natural Science Foundation of China (51375406).

34

References [1]

Maedler K, Geburtig T, Ullrich D. An experimental approach to determining the residual lifetimes of wheelset axles on a full-scale wheel-rail roller test rig. Int J Fatigue 2016;86:58-63.

[2]

Gürer G, Gür CH. Failure analysis of fretting fatigue initiation and growth on railway axle pressfits. Eng Failure Anal 2018;84:151-66.

[3]

Foletti S, Beretta S, Gurer G. Defect acceptability under full-scale fretting fatigue tests for railway axles. Int J Fatigue 2016;86:34-43.

[4]

Hirakawa K, Toyama K, Kubota M. The analysis and prevention of failure in railway axles. Int J Fatigue 1998;20:135-44.

[5]

Nishioka K, Komatsu H. Researches on Increasing the Fatigue Strength of Press-Fitted Shaft Assembly. Bull JSME 1967;10:880-9.

[6]

Nishioka K, Komatsu H. Researches on Increasing the Fatigue Strength of Press-Fit Shaft : 4th Report, Geometry of the End of Press-Fitting and σw1. Bull JSME 1972;15:1019-28.

[7]

Juuma T. Torsional fretting fatigue strength of a shrink-fitted shaft. Wear 1999;231:310-8.

[8]

Juuma T. Torsional fretting fatigue strength of a shrink-fitted shaft with a grooved hub. Tribol Int 2000;33:537-43.

[9]

Cervello S. Fatigue properties of railway axles: New results of full-scale specimens from Euraxles project. Int J Fatigue 2016;86:2-12.

[10] Zhang Y, Lu L, Gong Y, Zhang J, Zeng D. Fretting wear-induced evolution of surface damage in press-fitted shaft. Wear 2017;384:131-41. [11] Song C, Shen M, Lin X, Liu D, Zhu M. An investigation on rotatory bending fretting fatigue damage of railway axles. Fatigue Fract Eng Mater Struct 2014;37:72-84. 35

[12] Alfredsson B. Fretting fatigue of a shrink-fit pin subjected to rotating bending: Experiments and simulations. Int J Fatigue 2009;31:1559-70. [13] Faanes S. Distribution of contact stresses along a worn fretting surface. Int J Solids Struct 1996;33:3477-89. [14] Iyer K. Peak contact pressure, cyclic stress amplitudes, contact semi-width and slip amplitude: relative effects on fretting fatigue life. Int J Fatigue 2001;23:193-206. [15] White DJ, Humpherson J. Finite-element analysis of stresses in shafts due to interference-fit hubs. J Strain Anal Eng Des 1969;4:105-14. [16] Yang G, Xie J, Zhou S, Xiao N, Xie Y. Research on the Influence of Axle Design Parameters

on

Contact Pressure between Axle and Hub. J China Railw Soc 2009;31:31-5. [17] Kubota M, Kataoka S, Kondo Y. Effect of stress relief groove on fretting fatigue strength and index for the selection of optimal groove shape. Int J Fatigue 2009;31:439-46. [18] Zhu Y, Kang G, Ding J, Zhu M. Study on bending fretting fatigue of LZ50 axle steel considering ratchetting by finite element method. Fatigue Fract Eng Mater Struct 2013;36:127-38. [19] Smith KN, Watson P, Topper TH. A stress-strain function for the fatigue of metals. Journal of Materials 1970;5:767-78. [20] Lanoue F, Vadean A, Sanschagrin B. Fretting fatigue strength reduction factor for interference fits. Simul Model Pract Th 2011;19:1811-23. [21] Dang Van K, Griveau B, Message O. On a new multiaxial fatigue limit criterion:theory and applications. In: Brown MW, Miller KJ, editors. EGF 3. London: Mechanical Engineering Publications; 1989. p. 479–96. [22] Zhang T, Harrison NM, Mcdonnell PF, Mchugh PE, Leen SB. A finite element methodology for 36

wear–fatigue analysis for modular hip implants. Tribol Int 2013;65:113-27. [23] Cruzado A, Leen SB, Urchegui MA, Gómez X. Finite element simulation of fretting wear and fatigue in thin steel wires. Int J Fatigue 2013;55:7-21. [24] Hu Z, Wang H, Thouless MD, Lu W. An approach of adaptive effective cycles to couple fretting wear and creep in finite-element modeling. Int J Solids Struct 2018;139-140:302-11. [25] Zhang Y, Lu L, Gong Y, Zeng D, Zhang J. Finite Element Modeling and Experimental Validation of Fretting Wear Scars in Press-Fitted Shaft with Open Zone. Tribol Trans 2017;61:585-95. [26] Zhang Y, Lu L, Zou L, Zeng D, Zhang J. Finite element simulation of the influence of fretting wear on fretting crack initiation in press-fitted shaft under rotating bending. Wear 2018;400401:177-83. [27] Makino T, Kato T, Hirakawa K. Review of the fatigue damage tolerance of high-speed railway axles in Japan. Eng Fract Mech 2011;78:810-25. [28] Mohd Tobi AL, Ding J, Bandak G, Leen SB, Shipway PH. A study on the interaction between fretting wear and cyclic plasticity for Ti–6Al–4V. Wear 2009;267:270-82. [29] Fouvry S, Liskiewicz T, Kapsa P, Hannel S, Sauger E. An energy description of wear mechanisms and its applications to oscillating sliding contacts. Wear 2003;255:287-98. [30] McColl IR, Ding J, Leen SB. Finite element simulation and experimental validation of fretting wear. Wear 2004;256:1114-27. [31] Lee DH, Kwon SJ, You WH. Characteristics of Fretting Wear in a Press-Fitted Shaft Subjected to Bending Load. Adv Mater Res 2010;97-101:1269-72. [32] Ince A, Glinka G. A modification of Morrow and Smith–Watson–Topper mean stress correction models. Fatigue Fract Eng Mater Struct 2011;34:854-67. 37

[33] Lee DH, Goo BC, Lee CW, Choi JB, Kim YJ. Fatigue Life Evaluation of Press-Fitted Specimens by Using Multiaxial Fatigue Theory at Contact Edge. Key Eng Mater 2005;297-300:108-14. [34] Sum WS, Williams EJ, Leen SB. Finite element, critical-plane, fatigue life prediction of simple and complex contact configurations. Int J Fatigue 2005;27:403-16. [35] Madge JJ, Leen SB, Mccoll IR, Shipway PH. Contact-evolution based prediction of fretting fatigue life: Effect of slip amplitude. Wear 2007;262:1159-70. [36] Rajasekaran R, Nowell D. On the finite element analysis of contacting bodies using submodelling. J Strain Anal Eng Des 2005;40:95-106. [37] Ghosh A, Wang W, Sadeghi F. An elastic–plastic investigation of third body effects on fretting contact in partial slip. Int J Solids Struct 2016;81:95-109. [38] Ding J, McColl IR, Leen SB, Shipway PH. A finite element based approach to simulating the effects of debris on fretting wear. Wear 2007;263:481-91. [39] Zhang J, Cao Z. Report on the full-scale fatigue of RD_2 railway axle. Roll Stock 1987;5:3-11. [40] Lee CW, Kwon SJ, Choi JB, Kim YJ. Observation of fatigue damage in the pressfitted shaft under bending loads. Key Eng Mater 2006;326-328:1071-4.

38

Highlights 1. Three fatigue methodologies are used to investigate the fretting fatigue of scaled railway axles. 2. The effect of plastic deformation and fretting wear on fretting fatigue is studied. 3. The influence of the diameter ratio on fretting behavior is studied. 4. The predicted results agree with the findings from the fatigue tests.

39

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:

40