Fatigue crack growth in railway axle specimens – Transferability and model validation

Journal Pre-proofs Fatigue Crack Growth in Railway Axle Specimens – Transferability and Model Validation D. Simunek, M. Leitner, M. Rieger, R. Pippan, H.P. Gänser, F.J. Weber PII: DOI: Reference:

S0142-1123(19)30525-0 https://doi.org/10.1016/j.ijfatigue.2019.105421 JIJF 105421

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

31 October 2019 10 December 2019 11 December 2019

Please cite this article as: Simunek, D., Leitner, M., Rieger, M., Pippan, R., Gänser, H.P., Weber, F.J., Fatigue Crack Growth in Railway Axle Specimens – Transferability and Model Validation, International Journal of Fatigue (2019), doi: https://doi.org/10.1016/j.ijfatigue.2019.105421

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Fatigue Crack Growth in Railway Axle Specimens – Transferability and Model Validation D. Simunek 1,2, *, M. Leitner1, M. Rieger3, R. Pippan4,H.-P. Gänser5 and F.-J. Weber 2 1

Montanuniversität Leoben, Chair of Mechanical Engineering, Franz-Josef-Straße 18, 8700 Leoben, Austria

2 Siemens

Mobility GmbH, Eggenberger Straße 31, 8020 Graz, Austria

3 Graz

University of Technology, Institute of Thermal Turbomachinery and Machine Dynamics, Area of Structural Durability and Railway Vehicles, Inffeldgasse 25/D, 8010 Graz, Austria 4

Erich Schmid Institute of Materials Science, Jahnstraße 12, 8700 Leoben, Austria

5 Materials

Center Leoben Forschung GmbH, Roseggerstraße 12, 8700 Leoben, Austria

* Corresponding author. E-mail: [email protected]. Tel. +43 664 88557923

Keywords Crack growth testing; Residual lifetime assessment; Railway axle; Residual stresses; 1:3 scale specimens

Abstract The issue of transferability of crack propagation parameters from small-scale specimens up to realscale components is investigated. Crack growth material parameters are determined based on single edge notched bending SE(B) specimen tests and the crack growth model is validated by 1:3 scaled railway axle experiments as well as for 1:1 test axles focusing on the steel grade EA4T. Crack propagation assessment is performed with the aid of the software tool INARA demonstrating the importance of considering secondary mean stress states due to residual stresses and press fits. In addition, short crack behavior and the build-up of the fatigue crack growth threshold is considered by means of an enhanced NASGRO fatigue crack growth equation, which is of great importance at loads with stress intensity factors in the threshold region. A final comparison with results of a crack growth assessment model based on a previous research project shows the potential of the presented improved approach and highlights the importance to include secondary stress states as well as short crack behavior in order to ensure a proper transferability from small-scale specimens to real-scale components in practice.

Nomenclature a

crack length (crack depth for semi-elliptical cracks)

A5

percentage elongation after fracture

a0

initial crack depth

af

final crack depth

aN

notch depth

a/c

crack aspect ratio

a0/c0

initial crack aspect ratio

aN/cN

notch aspect ratio

a/D

crack depth normalized by the diameter

c

major semi axis of semi-elliptical surface cracks

C

crack growth material parameter

cN

half notch length on surface

COL

multiplicative constant

da/dN

crack growth rate in depth

Dmn

fitting coefficient for polynomial function

ds/dN

crack growth rate on surface

DS

diameter of specimen or axle

f

crack opening function

F

crack velocity factor

fmn

polynomial influence function

K

stress intensity factor

K0,OL

model parameter for overload induced retardation

KA

stress intensity factor in depth

KC

critical stress intensity factor

Keff

effective stress intensity factor

Kmax

maximum stress intensity factor in a loading cycle

Kmax,OL

maximum stress intensity factor inflicted by an overload

Kmin

minimum stress intensity factor in a loading cycle

Kres,OL

fictitious residual stress intensity factor due to the overload

KS

stress intensity factor at surface of the semi-elliptical crack

Ksec

stress intensity factor due to secondary stresses

2

li

length scale parameters (parameter for build-up of fatigue crack threshold)

LOL

material parameter to adapt the plastic zone size to actual boundary conditions

NCG

number of load-cycles at crack growth experiment

p

empirical constant describing the curvature that occur near the threshold

pOL

model parameter for overload induced retardation

q

empirical constant describing the curvature near the instability region of the crack growth curve

R

load ratio (stress ratio) (= Kmin/Kmax)

Reff

effective load ratio

Rm

tensile strength

Rp0,2

yield strength

RS

radius of specimen, axle

s

half surface crack length

s0

initial half surface crack length

sf

final half surface crack length

Sa

nominal stress amplitude

W

width of SE(B) specimens

Z

percentage reduction of area at fracture

ZOL

overload affected influence zone

γOL

fitting parameter

a

crack extension; change in crack depth

aOL

crack extension into the overload-affected influence zone

K

stress intensity factor range (= Kmax – Kmin)

Kth

fatigue crack propagation threshold

Kth,0

fatigue crack propagation threshold at a load ratio R = 0

Kth,eff

effective fatigue crack propagation threshold

Kth,lc

fatigue crack propagation threshold in the long crack regime

s

crack extension; change in surface crack length

νi

weighting factors

σ(x,y)

two-dimensional axial stress distribution

3

Abbreviations BL

block load

CA(L)

constant amplitude (loading)

eff

effective

lc

long crack

OL

overload

SIF

stress intensity factor

SOL

single overload

th

threshold

VA(L)

variable amplitude (loading)

4

1. Introduction The fatigue life assessment and estimation of inspection intervals to ensure a safe and reliable operation of railway axles [1 –4] in order to avoid failure or rupture is still a challenging task [5 ,6]. Especially the effect of variable amplitude loading due to service load spectra [7 –9] needs to be properly considered already in the design stage. One common approach to assess fatigue crack propagation is based on the model according to Forman and Mettu [10], which is known as the NASGRO equation. For an accurate crack growth assessment, the influence of the short crack behavior as well as crack closure [11 –14] needs to be incorporated, which is presented in detail in [15 ,16] and used for the crack growth assessment within this work (Equ. (1)). ⁠

⁠

⁠

⁠

⁠

da p  C  F  K m  p   K  K th  dN

(1)

with  a      F  1  (1  Flc )  1  e lF      

whereat

 1 f  Flc     1 R 

m

(2)

and

K th  K th,eff

a a      l1   K th,lc  K th,eff   1   1  e  2  e l2      

(3)

Herein, a is the crack depth, N is the number of load-cycles, F is the crack velocity factor, K is the stress intensity factor range, Kth is the threshold stress intensity factor range, and C, m and p are material constants. In equation (2), Flc is the crack velocity factor for the long crack, Δa is the crack extension and lf is a fictitious length giving the range of influence of short crack effects on the crack velocity factor. The crack velocity factor for the long crack Flc is determined with Newman’s crack opening function f [17] and the stress intensity factor ratio (load ratio) R. Equation (3) describes the build-up of the threshold in dependence of the crack extension Δa, where ΔKth,eff is the intrinsic (effective) threshold value, ΔKth,lc the threshold value of the long crack, l1 and l2 are fictitious length scales for the build-up of crack closure effects and ν1 and ν2 are weighting factors. Further details about the modified NASGRO equation are presented in [15]. It is known from literature, that the fatigue strength of materials decreases with increasing size, see [18 –22]. Different stress gradients and highly stressed volumes due to the geometries as well as effects from microstructure lead to differences of the fatigue limit from different specimens. In an assessment based on the notch root stress, these effects are typically accounted for by a correction ⁠

5

factor incorporating the stress gradient or the highly stressed volume of the loaded component, and a second correction factor based on the diameter of the pre-material for inhomogeneous microstructure (porosity, segregation etc.) and residual stresses from manufacturing (e.g., heat treatment), the socalled technological size effect. In a fracture mechanics assessment, different stress gradients are taken into account naturally by separate contributions from membrane and bending stress components in the stress intensity factor (SIF) solutions, or even more precisely by SIF solutions based on polynomial influence functions, where the stress field is approximated by a polynomial (typically of fourth or sixth order). Within this work, special attention is laid on the so-called technological sizeeffect. Thereby, residual stress states due to the manufacturing process may be significantly different for small- specimens in comparison to large-scaled components. Hence, the crack growth behavior of different types and size of specimens is investigated to study size-effects as well as the transferability from standardized small-scale specimens to real-scale railway axles. In the course of this analysis, the applicability of an improved crack growth model including material parameters from laboratory SE(B) specimens for 1:3-scaled as well as real railway axle specimens is validated. Besides the set-up of a proper crack propagation model, additionally the transferability from smallscale specimen tests to real-scale railway axles is another key task in the design stage [16 ,23 ,24]. Hence, this paper focuses on the transferability of crack growth parameters and model validation based on 1:3 and 1:1 scaled railway axle specimens. ⁠

⁠

This work was part of the research project “Eisenbahnfahrwerke 3 (EBFW 3)”, see [25 –27], and focuses on experimental analyses of fatigue crack growth in cylindrical specimens on a scale of 1:3 compared to the dimension of real wheelset axles. The base material investigated in this work is the commonly used railway axle steel EA4T, for details see [28]. As the fatigue testing of real scale components is commonly expensive as well as time-consuming, experimental analysis utilizing small-scale specimens and a further determination of crack growth parameters is mostly preferred. As described within [16], the material characterization and evaluation of crack growth model parameters is conducted based on SE(B) (single edge notched bending) specimens. If these parameters are applied in order to assess the residual lifetime of real-components, deviations due to additional effects, which result from the manufacturing routine, different geometry and size, may occur. In order to properly validate the applicability of the crack growth model parameters, 1:3 scaled cylindrical specimens and 1:1 scaled test axles are experimentally investigated to analyze additional effects on crack growth and enable the transferability from small-scale SE(B) specimens up to real size railway axles. ⁠

As presented in [16], the analytical assessment tool INARA (INtegrity Assessment for Railway Axles) for semi-elliptical crack growth in railway axles was developed within the project. The software provides the introduced crack growth model (see Equ. 1-3). Additionally, load sequence effects due to variable amplitude loading can be considered. A comprehensive description of the crack growth model including load sequence effects is provided by Maierhofer et al. in [16]. The stress 6

intensity factor solutions are based on numerical results from Varfolomeev [29 ,30]. Two dimensional axial stress distributions due to rotary bending in cylindrical and transition regions of axles can be considered as well as superimposed axial stresses due to press fits and residual stresses. The crack growth assessment procedure in INARA is presented in Figure 1 (a). Based on the two dimensional stress distribution from rotary bending, the load sequence (load amplitudes), interference (press) fits and residual stresses due to manufacturing a residual lifetime assessment can be conducted. Typically, the 2D bending stress profile is normalized and multiplied by the load amplitudes (absolute values). The improved crack growth model according to equation (1) is implemented in the software. Additionally, for variable amplitude loading, crack growth retardation due to load sequence effects can be considered. The model for load sequence effects is presented in section 4.2. In Figure 1 (b) the Graphical User Interface (GUI) of INARA with the geometry model and the two dimensional stress profile due to rotary bending is presented. ⁠

(a)

7

(b) Figure 1: Example for a typical crack growth assessment with the software INARA (a) crack growth assessment procedure and (b) GUI of INARA – Geometry model and normalized 2D stress profile due to rotary bending

This software is also used in this work; the calculation results are validated by experiments incorporating cylindrical 1:3 specimens as well as real-scale 1:1 railway test axles. Furthermore, the application of this tool to assess the crack growth of numerous test axles at variable amplitude loading is reported in [31]. Figure 2 summarizes the overall approach to investigate the transferability of the crack growth model and its parameters, starting from small-scale SE(B) specimens via 1:3 scaled cylindrical specimens up to real-scale experiments with 1:1 test axles.

Figure 2: Investigation of transferability of crack growth parameters within the research project EBFW 3

To sum up, this paper, dealing with fatigue crack growth of railway axles, scientifically contributes to the following research topics: 

Crack growth tests with 1:3 scaled round specimens incorporating both constant and variable amplitude loading. 8



Evaluation and incorporation of residual stress states as one major influencing factor for fatigue crack propagation [32 ,33]. Validation of the improved crack growth model and the software tool INARA based on 1:3 scaled round specimens Application of the fatigue crack growth model [16] and final comparison with experimental results of constant amplitude tested real scale 1:1 test axles to validate the transferability of the approach and the involved model parameters from small-scale SE(B) specimens to real railway axles. ⁠

 

2. Experimental procedure and material Fatigue crack growth testing of cylindrical 1:3 scale specimens was conducted on a four-point rotary bending test rig, which was built-up on the university of Leoben (Montanuniverität Leoben, Chair of mechanical engineering). The design and functionality of the test rig is depicted in Figure 3. The bending moment is applied by a pneumatic cylinder, which is connected to coupling rods. The load is split and a constant bending moment is applied between these rods. Surface crack growth was measured by an optical measurement system. Details and calibration of the optical measurement system are reported in detail in [34]. After testing, all specimens were cooled down in liquid nitrogen and ruptured. Subsequently afterwards, microscopical investigations of the fracture surfaces were performed to analyze crack extension in depth and crack shape evolution via inspection of the beach marks and the final fatigue crack front.

Figure 3: Four-point rotary bending test rig for 1:3 scale specimen experiments

The 1:3 specimens were manufactured out of railway axle blanks, which originally exhibited a diameter of 190 mm. The axle blanks were cut into segments with a length of approximately 320 mm. Each segment was divided into four parts and specimens with a diameter of DS = 55 mm in the test cross section were manufactured by turning. In the middle of each specimen a semi-elliptical notch with a depth of aN = 1 mm (minor semi-axis) and a surface length of 2cN = 2.5 mm (major axis) was 9

manufactured by electrical discharge machining. The spark eroded notch acts as the crack initiation point and the location at which the optical system is focused for further crack growth measurement. The position of the semi-elliptical surface notch was approximately 20 mm below the surface of the primary axle blank, see Figure 4. As depicted in [16], the notch root of the SE(B) specimens is at the same depth below the original axle blank surface to guarantee identical material properties for the different specimen types.

Figure 4: Extraction of cylindrical 1:3 scale specimens from a railway axle blank

The constant amplitude 1:1 test axle experiments were performed on a resonance test rig by the university of technology in Graz (Graz University of Technology, Institute of Thermal Turbomachinery and Machine Dynamics, Area of Structural Durability and Railway Vehicles). The axles exhibited a diameter of 180 mm in the cylindrical region and included a mounted wheel on one end, whereas on the other end an unbalance motor unit was mounted. The axles were positioned vertically and fixed at the wheel on the fundament. The unbalance motor unit induced a rotating bending moment in the axle. The functionality and components of the test rig are demonstrated schematically in Figure 5. Crack growth testing of 1:1 test axles was performed on axles with identical semi-elliptical initial notches similar to 1:3 specimens (aN = 1 mm and 2cN = 2.5 mm) in the cylindrical region and axles with starting notches in the transition region. The notches in the transition region were positioned 32 mm away from the clamping wheel seat because prior finite element calculations [35] showed the highly stressed zone at this position.

10

Figure 5: Functionality of the resonance test rig for 1:1 test axles

Similar to 1:3 experiments, the tested axles were cooled down in liquid nitrogen after tesing and ruptured for microscopical investigations on the Erich Schmid Institute of Materials Science. The chemical analysis and the mechanical properties of the base material (axle blanks) are given in Table 1 and Table 2.

Table 1: Chemical analysis of base material (all values in [wt %]).

EA4T

C 0.28

Si 0.31

Mn 0.74

P 0.012

S 0.006

Cr 1.12

Mo 0.20

V 0.04

Ni 0.24

Al 0.016

Cu 0.04

Sn 0.002

Table 2: Mechanical properties of base material

EA4T

Rp0,2 [MPa] 541

Rm [MPa] 690

A5 [%] 20.5

Z [%] 65

After manufacturing, Vickers hardness measurements of each type of specimen were conducted 5 mm below the surface of the initial notch, see Table 3. All measurements were comparable and highlight no major differences, which again validates the similar material properties within the testing region for all three tested geometries. Table 3: Mean values of hardness for different types of specimens

HV10

Base Material 232

SE(B) specimen 226

1:3 specimen 234

1:1 test axle 228

In Table 4 the different types of specimens are summarized and an overview about testing conditions, the diameter in the test cross section and the nominal stress amplitude are provided.

11

Table 4: Investigated specimens in this work

Specimen Type 1:3 Specimen 1:3 Specimen 1:3 Specimen 1:1 Test axle 1:1 Test axle

Evaluation Region

Test Condition

Nominal load [MPa]

Cylindrical Cylindrical Cylindrical Cylindrical Transition

CA CA BL CA CA

100 150 150/ 100 100 100

Diameter in test cross section [mm] 55 55 55 180 180.654

3. Experimental Results As mentioned, within the research project experimental SE(B) specimen crack growth testing was performed to determine crack growth material parameters, see [16] for detailed information. To investigate crack propagation on components with different crack shape and size, experimental investigations on cylindrical 1:3 scale railway axle specimens and 1:1 test axles were performed (type of specimens and test rigs are shown in section 2). In this section the experimental investigations are shown, which are reference results for the validation of the latter crack growth approach in section 4. 3.1 Cylindrical 1:3 scale specimens Constant amplitude (CA) and block-load (BL) fatigue crack testing was performed on 1:3 specimens. In case of the CA tests, two different load levels were tested at nominal stress amplitudes of Sa = 150 MPa and Sa = 100 MPa, respectively. Block load testing was conducted with two alternating block loads. Each block was NBL = 50.000 load-cycles with 150 and 100 MPa nominal stress amplitude respectively. Before testing, a crack initiation procedure was performed to initiate a fatigue crack with its origin at the spark eroded semi-elliptical notch. For the CA test with Sa = 150 MPa and the BL test, both experiments were started with Sa = 150 MPa, because a crack was initiating at this load level. For the CA experiment with Sa = 100 MPa a higher load was necessary for crack initiation. Generally, crack initiation at 1:3 specimens with the described notch is achieved at nominal stress amplitudes Sa = 110 -130 MPa. This variation can be explained by minor deviations at manufacturing of the initial notch (tip) and hence variation of the local notch intensity. Anyway, after a crack was initiated and a surface crack extension Δs = 0.5 mm (absolute surface crack length 2s = 3.5 mm) was obtained, a load reduction procedure was performed to prevent overload effects on the further crack growth tests. In that procedure the load was reduced in steps of 10 MPa to the load level the crack growth test was started and a minimum crack extension per load reduction step of Δs = 0.25 mm was obtained.

12

All experiments were performed at a nominal load stress ratio R = -1. In practice additional stresses due to interference fits (press fits) of attachments and manufacturing-induced residual stresses may lead to a shift of the effective stress ratio and hence to different crack growth rates. In case of the 1:3 specimens, no attachments were applied and the clamping area did not influence the cross section of the semi-elliptical crack, which was evaluated by strain gauge measurements during the clamping process. In Table 5 the number of load-cycles of the crack growth experiments are presented. Table 5: Experimental results of the 1:3 experiments

Test condition

Stress amplitude [MPa]

CAL #1 CAL #2 BL

150 100 150/100

Start surface crack length 2s0 [mm] 4,0 4,0 4,0

Final surface crack length 2sf [mm] 18,0 18,0 18,0

Number of Load-Cycles [-] 965,328 3,531,934 1,740,876

After testing the fracture surfaces of the specimens were analyzed and the crack aspect ratio (a/cratio) was determined. For this purpose the initial notch, the detectable beach marks and the final fatigue fracture surface were evaluated by non-linear least-square fitting and the axes of the semielliptical crack extension were determined. In Figure 6 the evaluation of a beach mark is demonstrated.

Figure 6: Crack shape evaluation of the block load experiment by fracture surface analysis

Minor longitudinal (axial) residual stresses due to manufacturing were expected and also reported for 1:3 specimens of EA1N in [32]. Hence, X-Ray diffraction (XRD) residual stress measurements were conducted on four specimens up to depths of 2 mm and 4 mm respectively. Although the investigations show only minor residual stresses, crack propagation may be significantly influenced, especially when testing at comparably low stress amplitudes. The results showed a compressive residual stress peak on the surface due to machining, but immediately drop to zero in a depth of approximately 100 μm. Below the effect of machining vanishes and minor tensile residual stresses are measured. Hence, as the influence of the compressive residual stress peak at the surface is small, the measurement point on the surface can be neglected, see also [32 ,36] to reduce convergence problems of numerical and analytical calculations. Based on the measurements below the surface, a 13 ⁠

polynomial fit and an extrapolation in depth were conducted for later consideration in the crack growth assessment. The extrapolation is based on the fact, that the sum of the residual stresses must lead to an equilibrium of forces, hence tensile and compressive residual stresses must be in an equilibrium state. For simplicity, a radial symmetric residual stress distribution was adopted. The measurements and the fitted distribution excluding the surface measurement point are depicted in Figure 7. Although the diagram shows minor tensile residual stresses up to approximately +30 MPa, crack propagation is significantly influenced because testing is performed at low stress amplitudes of Sa = 100 and Sa = 150 MPa.

Figure 7: XRD-Measurements of residual stresses in depth and fitted distribution of 1:3 scaled specimens

14

3.2 Test axles (1:1 experiments) Within the research project, constant amplitude tests at a load ratio R = -1 with SE(B), cylindrical 1:3-specimens (see section 3.1) as well as 1:1 test axle experiments were performed. Similar to 1:3 specimens, CA testing of 1:1 test axles with a nominal stress amplitude Sa = 100 MPa in the crack plane was performed at an axle with semi-elliptical initial notches in the transition region next to the wheel (32 mm next to the wheel seat) as well as for an axle with initial notches in the cylindrical region of the axle. After testing, similar to the 1:3 experiments, microscopical investigations of the fracture surfaces are conducted and the crack aspect ratio (a/c-ratio) is analyzed. Similar to 1:3 tests, a crack initiation procedure at 1:1 test axles was performed before the crack growth experiment. For comparison, both 1:1 experiments are evaluated starting from a surface crack length 2s0 = 7mm to a final crack length 2sf = 35mm. Due to the press fit of the wheel and the stress concentration in the transition region, crack growth is expected faster compared to a crack in the cylindrical region at the same nominal stress amplitude. Contrary to expectations, crack growth at the axle with a crack in the cylindrical region (Test axle #1) showed higher crack propagation compared to the axle with a fatigue crack in the transition region (Test axle #2), see Table 6. Table 6: Experimental results of the 1:1 experiments

Test condition CAL Test axle #1 CAL Test axle #2

Notch/ Crack position Cylindrical region Transition region

Stress amplitude [MPa]

Start surface crack length 2s0 [mm]

Final surface crack length 2sf [mm]

Number of Load-Cycles [-]

100

7,0

35,0

1,731,120

100

7,0

35,0

3,923,120

XRD residual stress measurements at test axles showed different results. Whereat one stress profile exhibited tensile residual stresses in depth with a maximum of +40 MPa, compressive residual stresses were measured for other axles. The measured tensile residual stress profile is depicted in Figure 8. Similar to the measurements of the 1:3 residual stress profiles, an inter- and extrapolation by a polynomial fit was conducted to generate a residual stress profile for the latter crack growth calculations.

15

Figure 8: XRD residual stress measurements and fitted profile for test axle #1

The residual stress measurements and the interpolated profile for test axle #2 are shown in Figure 9 with compressive residual stresses of -40 MPa. Additionally, the axial stress distribution of the press fit of the clamping plate, which is superimposed with the residual stress profile is depicted. The axial stresses due to the press fit depend on the interference of the axle and the clamping plate. The surface stresses of the press fits were measured by strain gauge measurements at the assembling process and a stress profile in depth was determined by numerical analysis, see [35 ,37]. As shown in Figure 9, the press fit stress on the surface is approximately +30 MPa and decreases in depth. The superimposed stress distribution of both, the residual stress profile due to manufacturing and the press fit are used for the latter crack growth assessment of test axle #2. As depicted, the superimposed stresses are almost neutralized in the near surface region. ⁠

Figure 9: XRD residual stress measurements and fitted profile for test axle #2 and superimposed stresses from the press fit

16

4. Study of transferability and model validation Within the study of transferability and crack assessment validation, analytical stress intensity factor solutions have been analyzed based on 1:3 scale specimens and compared to numerical investigations. Afterwards, crack growth assessment of experimental 1:3 constant amplitude (CA) and block-load (BL) tests was conducted to validate the crack growth model for semi-elliptical surface cracks in round bars. Finally, experimental constant amplitude (CA) tests of SE(B), 1:3 scaled specimens and 1:1 test axles are compared and the issue of transferability is evaluated and discussed. 4.1 Evaluation of stress intensity factor solutions Within the research project, the software INARA (INtegrity Assessment for Railway Axles) [38] was developed, which is capable to analyze semi-elliptical crack growth in railway axles analytically. The nomenclature of a semi-elliptical fatigue crack in a round bar is depicted in Figure 10. The surface crack length is characterized by the arc length 2s or the major axis 2c of the semi-elliptical crack.

Figure 10: Nomenclature of a semi elliptical crack front in a round bar

Crack propagation is affected by a multitude of factors such as the crack shape geometry, superimposed secondary stresses from press fits and manufacturing induced residual stresses as well as by short crack behavior and load sequence effects. Hence, fatigue crack growth has to be considered by means of a fatigue crack growth model taking all these effects into account. In INARA, a modified NASGRO equation according to Maierhofer et al. [15] is implemented to consider short crack effects and build-up of the fatigue crack growth threshold with crack extension. Variable amplitude loading may lead to load-sequence effects, which can significantly affect crack growth. Crack growth retardation due to overload effects and hence, an increase of plasticity induced crack closure are considered as well as crack growth retardation due to oxide-induced crack closure when the stress intensity factor range is below the crack growth threshold. The implemented crack growth parameters are summarized in Table 7. Table 7: Crack growth parameters for EA4T [16]

17

EA4T

C [mm/MPa√m] 1.92.10-8

m [-]

p [-]

ν1 [-]

ν2 [-]

l1 [mm]

l2 [mm]

lF [mm]

2.64

0.319

0.439

0.561

0.00265

1.83

0.01

ΔKth,eff [MPa√m] 2.0

ΔKth,0 [MPa√m] 7.35

As the stress intensity factor is a main parameter for crack growth assessment, its exact determination is of great importance. Investigations of stress intensity factor solutions according to INARA are performed for different semi-elliptical crack fronts with a surface crack length 2s = 5 mm and varying aspect ratios between 0.5 ≤ a/c ≤ 1.0. Comparison of fatigue crack growth assessment with INARA and numerical calculations are also reported for EA1N in [32] and show sound accordance. In [39] a collection of different stress intensity factor solutions for cracks in railway axles are presented, including the solutions from Varfolomeev. Based on the stress intensity factor solutions according to Varfolomeev surface cracks in different regions of a railway axle can be determined and stress concentration due to notches as well as stresses due to press fits of attachments (wheels, gear wheels, bearings) and residual stresses due to manufacturing can be considered within residual lifetime assessments. The stress intensity factor solutions implemented in INARA are based on numerical calculations according to Varfolomeev et al. [29 ,30 ,40]. The two-dimensional axial stress distribution in axles under rotary bending with semi-elliptical surface cracks is defined by the polynomial function ⁠

⁠

m n m n  y y  (1)  x  (2)  x    ( x, y )    Dmn    Dmn   sgn  y  RS  RS RS  RS  m0 n 0      4

4

(4)

Dmn(1) and Dmn(2) are fitting coefficents of the polynomial functions, which represent the even and odd terms of the stress field about the symmetry plane of the semi-elliptical crack (Figure 10) and RS is the corresponding radius of the axle in the cross section of the fatigue crack. Varfolomeev determined the stress intensity factor solutions along the crack front for different crack aspect ratios a/c and crack depths a. Based on the numerical results, polynomial influence functions were generated. The stress intensity factors can be determined according to equation (5). 4 4  a a     a  1 1  a a  2  2  a a K I  , ,      a   Dmn  f mn  , ,    Dmn  f mn  , ,       m0 n 0   c RS   c RS   c RS    RS  (5)

mn

a   c

n

Here, fmn(1) and fmn(2) represent the normalized stress intensity factors which were determined numerically for the crack depth point A and the surface points S1 and S2, the angle ϕ describes the position on the crack front. The two-dimensional stress profiles can be determined f.e. by finite element calculations and the results can be imported in INARA, hence stress concentration in transition regions can also be considered.

18

Additionally, numerical investigations are performed using the software tools Abaqus (Version 6.143) [41] and Franc3D (Version 7.1.0.2) [42] and compared to the INARA solutions. The geometry and boundary conditions as well as the meshed un-cracked model were set-up with Abaqus. Subsequently, this model was imported in the software Franc3D and a semi-elliptical crack was inserted. A rotating bending moment was defined, whereat the nominal bending stresses on the surface of the un-cracked specimen is identical to the definition of the stress amplitude Sa of INARA (Sa = 100 MPa). Semielliptical cracks with a constant surface crack length 2s = 5 mm and different a/c-ratios were implemented to analyze the stress intensity factors for the surface points S1 and S2, as well as the crack depth point A. Hexahedral elements (denoted as C3D20R elements in Abaqus) were chosen for the basic mesh. The sub-model technique was used to reduce numerical effort and calculation time. In Franc3D the crack growth material parameters identical with INARA were imported and the initial crack front geometry was defined. The semi-elliptical crack was inserted and re-meshed automatically by Franc3D considering the singularity at the crack tip by 3D quarter point singular elements; for detailed information see [43]. The comparison of the results is shown in Figure 11. Minor deviations between the different assessment methods arise with a maximum of +2% at a/c = 0.5. These results show that the software tool INARA provides an accurate assessment of the stress intensity factors at the crack front.

Figure 11: Comparison of stress intensity factor solutions for semi-elliptical cracks as a function of the a/c - ratio

Numerical assessments of crack propagation are mesh sensitive and time consuming depending on the size of the model and number of elements. Special experiences on numerical methods combined with fracture mechanics are needed for an accurate crack growth assessment The advantages of INARA are a time efficient and accurate assessment of stress intensity factors for semi-elliptical cracks in axles by polynomial influence function method. Additionally, the software provides solutions to consider secondary stresses due to press fits and manufacturing combined with rotary bending.

19

4.2 Model validation and transferability based on 1:3 specimens In a next step, the fitted residual stress distribution, presented in section 3.1, was considered within the crack growth assessments. First, crack growth calculations at constant amplitude loading were performed. The comparison of the CA fatigue crack growth experiments and the analytical assessments are shown in Figure 12. Crack propagation was evaluated from a surface crack length 2s = 4mm to a final surface crack length 2s = 18 mm. Good agreement is observed between prediction and experiment.

Figure 12: Comparison of CA fatigue crack growth tests and analytical assessments

Additionally, crack growth calculations neglecting residual stresses were performed to show the importance of considering the residual stress state within the crack growth assessments. The results of the crack growth calculations are summarized in Table 8. While all assessments considering residual stresses show conservative and sound results, neglecting the residual stress state leads to nonconservative results with more than 30% deviation from the experimental results. Table 8: Comparison of residual lifetime: Experimental study and analytical assessment with INARA

Experiment CAL 150 MPa CAL 100 MPa

965,328 3,531,934

INARA w. o. Residual stresses 1,199,590 4,683,700

Deviation from Experiment +24.3% +32.6%

INARA considering Res. stresses 920,620 3,254,562

Deviation from Experiment -4.6% -7.8%

In practical application, components are subjected to variable amplitude (VA) loading and load interaction effects may influence crack growth behavior. Increased plasticity induced crack closure, when loading is reduced from a higher to a lower level and build-up of an oxide layer when loading is below the crack growth threshold value lead to crack growth retardation during operation. Within the research project, appropriate load-interaction models were investigated. Finally, the generalized Willenborg-Gallagher [44 ,45] model was used and modified by implementing additional parameters to consider the overload effect. It should be pointed out, that within this work an overload is any load, which is higher than the following load. The overload affected zone is described by equation (6). LOL ⁠

20

is a parameter considering the boundary conditions, which are between the plane strain and plane stress state. K0,OL and pO are parameters which are originally K0,OL = 0 and pOL = 2 in the Willenborg model [44] and are implemented for more accurate approximation of experiments. The maximum stress intensity factor due to the overload is considered by Kmax,OL .

Z OL  LOL  ( K max,OL  K 0,OL ) poL

Kmax,OL  K0,OL  0

if

(6)

The overload is considered by means of a fictitious compressive residual stress intensity factor Kres,OL according to equation (7).

K res,OL

 a   COL  K max,OL  1  OL  Z OL  

 OL

 K max

(7)

Herein, COL is a constant factor, ΔaOL denotes the crack extension within the overload-affected zone, Kmax is the maximum stress intensity factor of the load following the overload and γOL is a fitting parameter. The crack growth retardation effect is considered by an effective stress intensity factor ratio Reff according to equation (8), which is incorporated with the crack growth model, see equation (1). The minimum and maximum stress intensity factor Kmin and Kmax are due to the primary load, Ksec are secondary loads (due to press fits and residual stresses).

R eff 

K min  K sec  K res,OL

(8)

K max  K sec  K res,OL

Based on laboratory SE(B) specimens [15 ,16], load-sequence parameters have been determined and implemented in INARA. The verification of overload parameters is shown in [33] for single overloads at constant amplitude loaded 1:3 specimens and show sound accordance. The parameters for the overload retardation model (equations (6)-(8)) for the material EA4T are represented in Table 9. ⁠

Table 9: Crack growth retardation parameters for EA4T

EA4T

COL [-] 1.0

LOL [mm/(MPa√m)] 7.62.10-4

pOL [-] 2.72

γOL [-] 0.369

K0,OL [MPa√m] 0

Based on SE(B) specimen experiments, it was found that the retardation model may overestimate the crack growth retardation leading to non-conservative results in some cases. Hence the model was limited by a minimum effective stress intensity factor due to the overload Reff,min = -5 and a retardation factor RF = 0.1. The factor RF = 0.1 limits the retardation by a maximum decrease of the crack growth rate down to 10% of the crack growth rate before the overload. Within this study a block load test was performed with two different alternating block loads and a comparable crack growth assessment considering load-sequence effects was conducted. The results 21

are depicted in Figure 13 and show sound accordance with the analytical calculation. Especially up to a surface crack length of 2s = 6 mm, crack propagation was almost identical. At higher crack lengths the retardation of crack growth at the low load level (Sa = 100 MPa) is more pronounced in the experiment. The results within this work as well as calculation of single overload experiments in [33] confirm the potential of the enhanced load-sequence model.

Figure 13: Comparison of a simple block load test and calculation

The block-load results of the crack growth assessment with the software tool INARA are compared to the experimental investigations, summarized in Table 10. Within this comparison results of crack growth assessments neglecting the residual stress state are shown and establish again the importance of secondary stresses. Additionally, results without consideration of crack growth retardation effects are presented. The comparison between the experiment and the calculation considering the residual stress distribution shows an improved assessment and a reduction of the deviation down to -11.2%. Table 10: Residual lifetime of experimental studies and analytical assessments with INARA

Experiment Without Retardation Effect Including Retardation Effect

1,740,876

INARA w. o. Residual stresses 1,856,550

Deviation from Experiment +6.6%

INARA considering Res. stresses 1,343,690

Deviation from Experiment -22.8%

2,118,800

+21.7%

1,546,533

-11.2%

Crack aspect ratio evaluation was performed starting from the spark eroded initial notch to the end of each experiment and the data points were fitted by an asymmetric double sigmoidal function. Crack growth assessment was performed starting from an initial crack with a surface length 2s0 = 4 mm, which was the initial value of the experimental crack growth tests. Crack initiation and crack propagation are different periods of fatigue life. Different mechanisms are affecting these two different phases. Whereas the stress concentration factor is the main parameter for crack initiation, the stress intensity factor is used to predict crack growth [46]. Nevertheless, for simplification the associated start value of the aspect ratio a0/c0 = 0.9 was determined by simulating the crack initiation 22

procedure based on fracture mechanics with INARA starting from the initial notch with aN = 1 mm and aN/cN = 0.8 to the surface crack length 2s0 = 4 mm. The microscopical evaluation and the fitted data curve of the crack aspect ratios, based on ten different experiments are compared to the analytical assessments by INARA, see Figure 14. The diagram shows an increase of the fitted data curve after crack initiation up to approximately a/c ≈ 1.0 (half-circular crack front) but also lower aspect ratios (see sample #4) of about a/c ≈ 0.92. Hence, the assumption of the crack growth start dimensions a/c = 0.9 at 2s = 4 mm is legitimate. At a/D ≈ 0.075 the crack growth assessments and experiments show sound accordance and only a minor deviation.

Figure 14: Crack aspect ratio distribution of 1:3 experiments and calculations

4.3 Model validation and transferability based on 1:1 test axles The material parameters were determined by SE(B) crack growth tests, as reported in [16]. Within this section the crack growth model according to equation (1) and the material parameters are investigated based on constant amplitude test results and the application for 1:1 test axles is analysed. The validation of the overload model according to equation (6)-(8) as well as for an oxide-induced crack closure model based on 1:1 test results is reported within [31]. In this section, the transferability of model parameters from laboratory specimens to real-scale components with different shape and size is shown. First, constant amplitude experiments of SE(B), 1:3 specimens and 1:1 test axles were conducted and the crack growth curves were compared. The different crack shapes and dimensions of the specimens were considered by the presented stress intensity factor solutions, which incorporate the geometry factor K = f(a/W) for SE(B) exhibiting a straight crack front and K = f(a/RS; a/c) for semi-elliptical cracks in shafts. The crack growth curves of the test axles with different locations of the fatigue crack are shown in Figure 15. Additionally, this diagram includes crack growth curves of SE(B) specimens and 1:3 specimens for comparison. Besides the differences in the stress intensity factor between the investigated types, the technological size-effect in terms of different residual stress conditions is also incorporated. 23

As described in section 3.2, analyses of the residual stresses revealed tensile residual stresses (with a maximum of about +40 MPa) at the test axle #1 with the crack in cylindrical region whereas the test axle #2 with the fatigue crack in the transition region exhibits compressive residual stresses (minimum value approximately -40 MPa). Strain gauge measurements at assembling of the wheel showed a maximum press fit value on surface of approximately +30 MPa. Hence, the superimposed mean stresses due to the press fit and the residual stresses due manufacturing are almost neutralized for axle #2 near the surface region. This fact is shown within the comparison of test axle #2 and the results of the SE(B) specimens. Due to this neutralization of the secondary stresses, the curves of the 1:1 test axle #2 (crack in transition region) and the SE(B) specimens, which exhibit negligible residual stresses (see also [11]), show sound accordance, whereas the crack propagation curve of test axle #1 is comparable with the investigated 1:3 specimens. As the effect of the crack shape and specimen dimension is considered within the calculation of the stress intensity factor, differences in the resulting crack growth curves can be drawn to different effective stress intensity factor ratios, which are influenced by secondary stresses. Therefore, the results highlight that technological size-effects, in terms of manufacturing-induced residual stress states, need to be considered to ensure a sound transferability of small-scale crack growth results to real-scale railway axle applications.

Figure 15: Comparison of crack growth rate da/dN vs. ΔK of different specimens

Due to the shift of the effective load ratio (equation (9)) for the different experiments, a comparison depending on the stress intensity factor range ΔK is not feasible. To that purpose, the crack growth rate depending on the maximum effective stress intensity factor Kmax,eff according to equation (10) is illustrated in Figure 16 and shows an improved accordance of the different types of specimens. This again highlights the importance of considering secondary stress states in the crack growth assessment.

Reff 

K min  K sec K max  K sec

Kmax,eff  Kmax  Ksec

(9)

(10) 24

Figure 16: Comparison of crack growth rate depending on Kmax,eff

To sum up, different types of specimens are well comparable if secondary stresses due to residual stresses and press fits are considered. Beside the well-known effect from the crack geometry, the residual stress state due to manufacturing, as technological size-effect, is one major issue for a proper transferability. To validate the assessment method implemented in INARA, crack growth calculations of 1:1 test axles are performed. For this purpose, the same approach as for 1:3 specimens (see section 4.2) was used. The dimension of the axle is considered within the calculation by the nominal two-dimensional stress profile in the crack plane. Compared to test axle #1 with the crack in the cylindrical part of the axle, the crack position of test axle #2 is in the transition region and additionally influenced by the stress concentration. Based on prior finite element calculations, the stress profile at the crack position was determined exhibiting a stress concentration at the surface of approximately 1.18. The stress profiles are important for the calculation of the stress intensity factors according to equation 5. Calculations with both considering and neglecting the residual stresses are performed. Crack growth assessment results are compared from an initial surface crack length 2s0 = 7 mm to a final surface crack length of 2sf = 35 mm, see Table 11. The results again emphasize the importance of residual stresses for crack growth assessment. Whereas calculations neglecting residual stresses exhibit comparably high deviations, the maximum difference of the assessment considering residual stresses was only 8%. Table 11: Residual lifetime of experimental studies and analytical assessments of 1:1 test axles with INARA

Deviation Crack in Cylindrical Region Crack in Transition Region

Experiment

INARA w. o. Residual stresses

Deviation from Experiment

INARA considering Res. stresses

Deviation from Experiment

1,731,120

3,243,000

+87.3%

1,631,800

-5.7%

3,923,120

1,518,000

-61.3%

3,610,000

-8,0%

25

The comparison of the experiments and the INARA assessments including residual stresses are shown in Figure 17. In [31] the application of the improved crack growth model considering overload effects as well as oxide-induced crack closure and secondary stresses for crack growth assessment with INARA for test axles under VA loading is reported.

Figure 17: Comparison of crack propagation assessment by INARA and experiments of 1:1 test axles

The crack growth rate of 1:3 specimens and 1:1 test axles were determined during testing with optical measurement systems on the surface, for details see [34] and [47]. Crack growth in depth can only be determined after testing by fracture surface analyses. Hence, fracture surface evaluations of 1:1 test axles were conducted to investigate the evolution of the semi-elliptical crack front. In Figure 18 the distribution of crack aspect ratios (a/c-ratio) of test axles are compared. The results show the differences of cracks in the cylindrical and the transition region of the axles. Test axles with cracks in the shafts reveal a pronounced increase of a/c-ratio compared to cracks in the transition region. Additionally, the crack aspect ratio evolution of the crack growth assessments with INARA is depicted. For simplification, equal to assessments of 1:3 specimens (see section 4.2), the crack initiation stage is considered with INARA and fracture mechanical approaches to determine the initial a/c-ratio of the crack growth period (denoted as “CG” in Figure 18). Afterwards, crack growth assessment of the constant amplitude experiment with the nominal stress amplitude of Sa = 100 MPa is conducted.

26

Figure 18: Aspect ratio of cracks in cylindrical region vs. transition region of 1:1 test axles

Comparison of normalized crack depth evolution depending on the normalized surface crack length between 1:3 specimens and 1:1 test axles is depicted in Figure 19. The diagram clearly demonstrates the differences between semi-elliptical crack front evolutions in cylindrical region and transition region. Crack growth testing of 1:3 specimens was only conducted with cracks in cylindrical region without any influence of press fits. Hence, the results are comparable with crack front evolution in the cylindrical region of 1:1 test axles as demonstrated in Figure 19. Due to the press fit and stress concentration in the transition region, crack growth on the surface is enhanced compared to cracks in cylindrical regions. Surface cracks of identical surface lengths in the cylindrical and transition region of 1:1 test axles show different crack depths as depicted in Figure 19.

Figure 19: Comparison of crack depth evolution (experiments): Cylindrical region vs. Transition region

5. Summary 27

Crack growth parameters were determined by SE(B) specimen testing. The crack growth model for constant amplitude loading was validated based on 1:3 and 1:1 experiments. The crack growth assessment for CA tests with the software tool INARA and the implemented improved crack growth model show sound accordance for 1:3 specimens and 1:1 test axles considering residual stresses. Although, minor residual stresses were measured, the influence on residual lifetime is significant. Hence, neglecting residual stresses within crack growth assessment lead to uncertainties of the residual lifetime. Compared to the crack growth model of the former project Eisenbahnfahrwerke 2 (EBFW 2) [30], the results of the improved model exhibit minor deviations. Additionally, an assessment with INARA considering overload effects of a block load 1:3 experiment was conducted. The comparison of the approaches for the 1:3 specimens is depicted in Table 12. Whereas the crack growth assessment according EBFW 2 of the constant amplitude test at Sa = 150 MPa and the block load show very conservative results, the assessment of the CA assessment at Sa = 100 MPa showed no crack growth. The results of the assessment based on the improved crack growth model show sound accordance and minor deviation from experimental investigations. Table 12: Residual lifetime of 1:3 experiments and analytical assessment with INARA a) EBFW2 and b) EBFW 3 model

Experiment

a) INARA EBFW 2

965,328 3,531,934 1,740,876

637,300 No Crack Growth 1,040,800

CAL 150 MPa CAL 100 MPa Block Load

Deviation from Experiment

b) INARA EBFW 3

-34.0% Not evaluable -40.2%

920,620 3,254,562 1,546,533

Deviation from Experiment -4.6% -7.8% -11,2%

Crack growth assessment results of constant amplitude loaded 1:1 test axles again show the importance of secondary stresses within the calculations. Considering the residual stress distribution and the modified NASGRO crack growth model according to Maierhofer [15 ,16] improves the results with deviations of 5.7 and 8%. The results are summarized in Table 13 and compared to the calculations of the former research project EBFW 2. ⁠

Table 13: Residual lifetime of 1:1 experiments and analytical assessment with INARA a) EBFW2 and b) EBFW 3 model

Experiment Crack in Cylindrical Region Crack in Transition Region

a) INARA EBFW 2

Deviation from Experiment

b) INARA EBFW 3

Deviation from Experiment

1,731,120

961,000

-44.5%

1,631,800

-5.7%

3,923,120

5,698,000

+45.2%

3,610,000

-8,0%

Crack growth curves (ds/dN-K curves) at identical load ratio of different specimens should be comparable because the stress intensity factor includes the influence of geometry and size. Although, SE(B), 1:3 scaled railway axle specimens and 1:1 test axles have been tested at same conditions, deviations of crack growth curves was observed. Residual stress measurements highlighted, that different residual stress distributions lead to a shift of the effective load ratio at the crack tip and hence to differences of crack propagation and residual lifetime. Considering residual stresses within the

28

crack growth assessment showed sound accordance with experimental investigations and confirms the importance of the residual stress state on residual lifetime.

6. Conclusion Based on the conducted research and presented results within this paper, the following main scientific conclusions can be drawn: 

Consideration of secondary stresses due press fits and manufacturing-induced residual stresses is of great importance. Minor uncertainties at stress profile determination may lead to high deviations in residual lifetime assessment.



X-Ray residual stress measurements of SE(B) specimens (see also [11]) showed negligibly small residual stresses (< 10 MPa), whereat 1:3 specimens exhibited minor tensile residual stresses of 20 – 30 MPa below the surface, which lead to significant reduction of residual lifetime.



Residual stress measurements of 1:1 test axles are reported in [31] and highlight mean values of approximately -40 up to +40 MPa, which was also observed for the investigated CA 1:1 experiments within this article.



Due to the cutting the SE(B) and 1:3 specimens out of railway axle blanks, residual stresses may be reduced, which was also confirmed by residual stress measurements in this work. Hence, the influence of residual stresses is more pronounced in 1:1 test axles.



The improved crack growth model according to Maierhofer et al. [15 ,16] additionally considers short crack behaviour and showed sound accordance to experiments.



Transferability of the crack growth model and the involved parameters from small-scale specimens to real-scale components is ensured based on the presented results considering secondary stresses and short crack behaviour.

⁠

Acknowledgments The authors gratefully acknowledge the financial support under the scope of the COMET program within the K2 Center “Integrated Computational Material, Process and Product Engineering (ICMPPE)” (Project No 859480). This program is supported by the Austrian Federal Ministries for Transport, Innovation and Technology (BMVIT) and for Digital and Economic Affairs (BMDW), represented by the Austrian research funding association (FFG), and the federal states of Styria, Upper Austria and Tyrol.

29

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Highlights 

Experimental and analytical crack growth analysis of railway axle specimens

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Consideration of short crack behaviour and residual stresses for assessment

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Fracture surface analysis and aspect ratio evaluation of semi-elliptical cracks

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Validation and verification of crack growth model based on experiments

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Evaluation of transferability based on experimental results of different specimens

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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:

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