IN DEGREE PROJECT MECHANICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2020

Fretting fatigue life analysis for a gas turbine compressor blade-disk material combination

ELHAM KHORAMZAD

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Royal Institute of TechnologyMaster’s Thesis

Fretting fatigue life analysis for a gas turbine compressorblade-disk material combination

Elham Khoramzad

August 12, 2020

Engineering Mechanics Department

Abstract

In order to analyse fretting fatigue life of dove-tail joints used in the compressor stage of a gasturbine, experimental study of different material combinations was conducted using rectangularfatigue specimen and bridge type pads. Specimen and pad interaction was simulated numericallyand obtained results was used as input for 2 different crack propagation life prediction methods.The combined effect of stress and fretting damage which was characterised by means of Ruizparameter was used in order to estimate the crack initiation and finally numerical results werecompared to experimental fretting fatigue life. The aim of this thesis was to study the contactfatigue behaviour of titanium alloy (Ti-6Al-4V) in combination with steel alloy (22NiCrMoV12-7).

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AcknowledgementI would like to express my deep gratitude to Professor Bo Alfredsson and Dr. Susanna Lungren,my master’s thesis supervisors, for their patient guidance and constitutive suggestions duringplanning and development of this project.

I would also like to extend my thanks to Mr. Martin Oberg and Mr. Magnus Boasen for theirhelp in conducting the experiments.

Finally, I wish to thank my parents for their support and encouragement throughout my study.

Elham KhoramzadStockholm, SwedenJuly, 2020

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Contents

List of Figures iv

List of Tables v

1 Introduction 1

2 Experiment setup and materials 2

3 Numerical analysis 33.1 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Numerical analysis of crack growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Results and Discussion 64.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Conclusion and Further work 12

References 13

Appendix A Edge through crack model 14

Appendix B Corner crack model 15

Appendix C Predicted crack propagation lives 16

Appendix D Additional pictures from experiments 17

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List of Figures

2.1 Schematic of the specimen and bridge type pad. . . . . . . . . . . . . . . . . . . . . . . . . 22.2 photograph of pads and specimen contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Photograph of test set up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1 Detailed mesh of contact region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 2D model of quarter of test setup and boundary conditions. . . . . . . . . . . . . . . . . . . 44.1 Fretting fatigue life of titanium alloy specimen vs steel pads. . . . . . . . . . . . . . . . . . 64.2 Cracks surface of titanium alloy specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.3 Contact Mark and crack initiation position in titanium alloy specimen. . . . . . . . . . . . 64.4 Fretting fatigue life of steel alloy specimen vs AM titanium alloy pads. . . . . . . . . . . . . 74.5 Cracks surface of steel alloy specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.6 Contact Mark and crack initiation position in steel alloy specimen. . . . . . . . . . . . . . . 74.7 Traction on titanium specimen (No bulk stress applied). . . . . . . . . . . . . . . . . . . . . 84.8 Traction on titanium specimen (Maximum bulk stress applied). . . . . . . . . . . . . . . . . 84.9 Traction on titanium specimen (Minimum bulk stress applied). . . . . . . . . . . . . . . . . 84.10 Bulk stress on contact region of titanium specimen. . . . . . . . . . . . . . . . . . . . . . . 84.11 Traction on steel specimen (No bulk stress). . . . . . . . . . . . . . . . . . . . . . . . . . . 94.12 Traction on steel specimen (Maximum bulk stress applied). . . . . . . . . . . . . . . . . . . 94.13 Traction on steel specimen (Minimum Bulk stress applied). . . . . . . . . . . . . . . . . . . 94.14 Bulk stress on contact region of steel specimen. . . . . . . . . . . . . . . . . . . . . . . . . 94.15 Titanium specimen vs steel pad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.16 Steel specimen vs AM titanium pad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.17 Predicted fretting fatigue life for titanium specimen vs experimental fretting fatigue life. . . 104.18 Predicted fretting fatigue life for steel specimen vs experimental fretting fatigue life. . . . . . 10A.1 configuration of trough crack at edge of plate. . . . . . . . . . . . . . . . . . . . . . . . . . 14B.1 Configuration of quarter elliptical corner crack at chamfer in plate. . . . . . . . . . . . . . . 15D.1 Cracks surface of titanium alloy specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17D.2 Contact Mark and crack initiation position in steel alloy specimen. . . . . . . . . . . . . . . 17D.3 Cracks surface of titanium alloy specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17D.4 Contact Mark and crack initiation titanium in steel alloy specimen. . . . . . . . . . . . . . 17

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List of Tables

1 Material parameters definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Fatigue life of titanium alloy specimen for normalised amplitude stresses. . . . . . . . . . . 63 Fatigue life of steel alloy specimen for normalised amplitude stresses. . . . . . . . . . . . . . 74 Predicted fatigue life of titanium alloy specimen for each normalised amplitude stress. . . . . 165 Predicted fatigue life of steel alloy specimen for each normalised amplitude stress. . . . . . . 16

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1 Introduction

Fretting fatigue is one of the major damage modes in compressor stage of gas turbines. The com-pressor blades are joined to rotating disk through an oblique contact where the mass forces fixesthe blade tail to the disk. The combined normal and cyclic tangential contact loads inevitably leadto fretting fatigue damage, where fatigue cracks may lead to loss of compressor blade and subse-quent complete engine failure. The aim of this thesis is to study the contact fatigue behaviour oftitanium alloy (Ti-6Al-4V) in combination with steel alloy (22NiCrMoV12-7).

Fretting is special wear process that occurs at the contact area between two materials due to cyclicshear stresses, generated by friction during small amplitude oscillatory motion or sliding betweentwo surfaces pressed together. During fretting fatigue, cracks can initiate at very low stress, wellbelow the fatigue limit of non-fretted specimens [4]. Under fretting conditions, fatigue strengthor endurance limits can be reduced by as much as 50 to 70% during fatigue testing [4]. As such,prevention of fretting fatigue is essential in the design process by eliminating or reducing slipbetween mated surfaces. However, prevention of fretting fatigue cracks is not always possible,hence reliable estimation of crack initiation and crack propagation is of great importance in orderto prevent catastrophic failure in structures [4].

The initiation of fatigue cracks in fretted regions depends mainly on the state of stress in thesurface, particularly stresses caused by high friction. The contributing factors to stress state atthe surface and hence crack initiation are contact stresses, slip amplitude and coefficient of fric-tion. Slip is relative displacement of contact surfaces and it covers a part of contact and the restexperiences no displacement and is usually called as stick region. The coefficient of friction hasthe greatest influence on fretting fatigue. It influences both slip amplitude and shear stress, the in-fluence of different material combinations on fretting fatigue life is due to coefficient of friction [6].

Contact stress includes both normal and shear stress imposed at the contact surface. The cyclicshear stresses at the surface are the causes of crack nucleation. The magnitude of the shear stressesdepends on the imposed forces, displacements, coefficient of friction, macroscopic stress concen-tration and local asperity geometry and distribution. Fretting fatigue strength based on cracknucleation decreases as the normal force increases [6].

Fretting fatigue life can be divided into 2 periods, crack initiation and crack propagation life untilfinal failure. Fretting reduces crack initiation life, however growth of long cracks does not differfrom plain fatigue. Fretting fatigue life or strength is determined by plotting the number of cyclesto failure on S-N curve where S is the alternating stress. A shift in cycles to failure at a given stresslevel is attributed to decrease in crack initiation life.

The purpose of this project is to obtain fretting fatigue life of 2 sets of material combinations,and numerically analyse the stress state in contact surface and through the specimen and use theobtained result in order to predict the fretting fatigue life of these 2 test series. Since the crackinitiates in contact region and is difficult to observe during inspections, it is of great importanceto predict the fretting fatigue life of material combinations used in engineering structures such asdovetail joints in compressor’s blades of a gas turbine.

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2 Experiment setup and materials

Twenty four experiment in two series were performed. Each series was composed of 12 specimenand 24 pads. Sharp edges of specimens which were in contact with pads had been rounded in or-der to reduce stress concentration of contact edge. Two bridge type pads were put in contact witheach specimen and the load P=200 MPa was applied on them through a ring. Load on the padwas chosen based on the real application, during operation blades are subjected to the specifiedload used in the experiments. Figure 2.1 presents schematic drawing of specimen and pad usedin these experiments.

Figure 2.1: Schematic of the specimen and bridge type pad.

The ring act as a spring to keep the load on the pads constant during the cycling loading. Ampli-tude stress σa was applied on the specimen by a servo-hydraulic test machine in the longitudinaldirection. Pictures of pads, specimen contact and the test set up can be seen in figures 2.2 and 2.3.

Figure 2.2: photograph of pads and specimen contact. Figure 2.3: Photograph of test set up.

In the first series of experiment twelve titanium alloy (Ti-6Al-4V) specimens were tested againststeel alloy (22NiCrMoV12-7) pads, see [13] and [12] for elastic properties of materials. Coefficientof friction for this material combination was considered to be µ = 0.78 [1], [3]. In the next series ofexperiment 12 specimens of same steel were tested against additive manufactured titanium alloyspecimens in which print direction was perpendicular to contact area, coefficient of friction wasconsidered to be same as first series.

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3 Numerical analysis

3.1 FEM

Finite element analyses of experiments were performed using Abaqus standard. Due to the sym-metry of test set up and constant pressure along the width of the specimen only a quarter of padsand specimen interaction were modelled in 2D. 41845 4-noded plain strain elements (elementcode CPE4R) were used to model the specimen and 6675 elements of same type as for the speci-men were used to model the pad. Finer element size was used to mesh the contact region in orderto obtain more accurate stress around the contact area, see figure 3.1, with element size of 15 µm .Elements in contact region and element distribution in the specimen and the pad is illustrated inFigure 3.1.

Figure 3.1: Detailed mesh of contact region.

The edges of the pads were rounded in order to prevent stress singularity. Elements on the padsclose to the contact region have the same size as elements on the specimen. A surface to surfacealgorithm was used in order to model the interaction between pad and specimen. Maximum bulkload applied on the specimen during the experiments was set to be less than yield strength, there-fore material was modelled as elastic. Hard contact has been chosen to define pressure-overclosurerelationship in contact region of both material combinations. In the first material combination,titanium specimen and steel pads, Lagrange multiplier method and in second material combina-tion, steel specimen and AM titanium pads, penalty method was applied for normal behaviour ofcontact interaction. In both material combinations, penalty method was used for enforcement oftangential constraint.

In order to model the applied bulk stress on the specimen and the pressure on pads, prescribeddisplacement have been applied on pad and specimen, see figure 3.2. Therefore, for keeping the

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force on the pad almost constant during the cyclic load, prescribed displacement is applied on thepad through a linear spring with stiffness of K=7965.5 N

mm .

Figure 3.2: 2D model of quarter of test setup and boundary conditions.

Figure 3.2 illustrates the boundary conditions used to model the specimen and pad interaction.Yellow circles represent the prescribed displacement applied on the pad and specimen. Red lineand yellow squares represent the interaction between the pad and specimen. Bottom boundary ofspecimen was fixed only in Y direction and was allowed to move freely in x direction, left bound-ary of pad and specimen was fixed in both X and Y direction in order to simulate the symmetry ofthe test setup.

3.2 Numerical analysis of crack growth

The fretting fatigue crack propagation lives were evaluated using two different crack growth mod-

els and crack growth ratesdadN

were estimated according to Paris’ equation 1 and NASGRO equa-tion 3, using material parameters as table 1 [12] ,[13].

dadN

= C(∆Keq)m (1)

∆Keq =√

∆KI + ∆KI I (2)

dadN

= C[(1− f1− R

)∆K]m (1− ∆Kth

∆K )q

(1− ∆Kma xKma x

Kc)

p (3)

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Table 1: Material parameters definition .

Parameter Value (Titanium)Value (Steel) Description

C Material parameterm Material parameterR Stress ratioq empirical coefficientsP empirical coefficientsKc Critical stress intensity factor

∆Keq is equivalent stress intensity range, f is the Newman crack closure function and ∆Kth isThreshold stress intensity factor range as R → 1.0. The crack grow into material perpendicular tothe contact surface. The stress intensity was calculated by loading crack with σx from first cyclein the uncracked configuration. Initial crack size was a0= 15 µm. Stress intensity factors used inParis’ equation were computed according to equations 4 and 5 [8]. See appedndices A and B forcrack models used in NASGRO equation.

KI =2√πa

σx(1√

a− ( ba )

2)F(

ba) (4)

KI I =2√πa

τ(1√

a− ( ba )

2)F(

ba) (5)

where a is crack length, b is and F is a weight function and can be calculated through equation 6.

F(ba) = 1.3− 0.3(

ab)

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(6)

Fatigue failure consist of crack initiation, crack propagation and final failure. To begin with, cracksgrow at stress raisers or material defects. Here, due to frictional contact, crack initiation positiondepend on cyclic slip rate δ between pads and specimen, shear stress τ and bulk stress σx alongthe contact patch . The combined effect of stress and fretting damage could be characterised bymeans of a parameter

k = σx × τ × δ (7)

was calculated in order to predict the nucleation position of cracks for different bulk stresses.

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4 Results and Discussion

4.1 Experimental Results

In the first series, see table 2, the normalised amplitude stress σa applied on the specimen rangedfrom 0.073 to 0.19, two or three test for each amplitude stress was performed, applied amplitudestress in the experiments is normalised by yield stress of specimen material. Figure 4.1 illustratesthe fatigue life of titanium specimens obtained from experiments, fitted to Basquin’s equation 8.

σa = σ0Nb (8)

Figure 4.1: Fretting fatigue life of titanium alloy specimenvs steel pads.

Table 2: Fatigue life of titanium alloy specimen for nor-malised amplitude stresses.

σa Pad pressure/MPa Fatigue life0.19 250 130000.19 200 120700.19 200 159100.14 200 574100.14 200 746800.12 200 1802000.12 200 2156000.09 200 6199000.09 200 883700

0.087 200 8369000.073 200 58750000.073 200 5162000

In this series, fatigue cracks started from the edge of the pad to specimen contact surface andpropagated through the specimen. Cracks initiated at the damaged edge of the contact, figure 4.3.A small part of the specimen in contact with the pads was damaged due to slip-stick conditionhappening during the test.

Figure 4.2: Cracks surface of titanium alloy specimen. Figure 4.3: Contact Mark and crack initiation position intitanium alloy specimen.

Figure 4.3 shows wider damaged area close to the edge of specimen and figure 4.2 shows the crack

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surface of the titanium alloy specimen tested with σa = 0.19, main crack started at the bottomedge of the specimen and continued growing into the specimen, several other cracks started onthe upper edge of the specimen and continued growing into the specimen however their length issignificantly smaller than the main crack. when the cracks grow enough throu specimen, remain-ing material cannot hold the applied load on the specimen and final rupture happens.

In the second material combination, the load on the pads was set to be same as first test series.Normalised amplitude stress σa ranged from 0.17 to 0.48 (amplitude stresses have been normalisedby yield stress of specimen material). Figure 4.4 illustrates the fatigue life of specimens obtainedfrom experiments, fitted to equation 8. Each amplitude stress was tested 1 or 2 times in order tocover larger stress range on the S-N curve.

Figure 4.4: Fretting fatigue life of steel alloy specimen vsAM titanium alloy pads.

Table 3: Fatigue life of steel alloy specimen for normalisedamplitude stresses.

σa Pad pressure/MPa Fatigue life0.48 200 395400.47 200 405500.45 200 532000.4 200 1440000.34 200 3274000.31 200 4954000.28 200 11570000.28 200 19080000.27 200 Run out0.25 200 Run out0.22 200 Run out0.17 200 Run out

Specimens subjected to amplitude stresses less than σa = 0.28 didn’t break after 10 million cy-cles, therefore they were considered to be run out, while final rupture happened for 2 specimenssubjected to σa = 0.27, hence it can be said that fretting fatigue limit for this material is betweennormalised amplitude stress of σa = 0.27 and σa = 0.28 .

Figure 4.5: Cracks surface of steel alloy specimen. Figure 4.6: Contact Mark and crack initiation position insteel alloy specimen.

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Here, cracks were initiated from corner of specimen, figure 4.5 shows the profile of crack afterspecimen failure, this specimen was subjected to amplitude stress of σa = 0.28. The crack startedfrom bottom left corner of the specimen and final rupture was out of plane, see figure 4.6, (Seeappendix D for more pictures). Fatigue specimens with rectangular cross-sections are prone tocorner crack, small miss alignment while applying the load can results in the corner cracks. Hav-ing wider pads than specimen width can also lead to stress concentration on the corner of thepad and subsequently causing corner cracks, in steel specimen vs AM titanium pads test series,pads were wider than the specimen width which can be reason for observing corner cracks in thespecimens.

4.2 Numerical Results

Contact pressure and tangential traction along the contact surface obtained from FE analysis forfirst material combination in which titanium specimen is loaded by normalised amplitude stressσa = 0.19 is plotted in figures 4.7 to 4.10.

Figure 4.7: Traction on titanium specimen(No bulk stress applied).

Figure 4.8: Traction on titanium specimen(Maximum bulk stress applied).

Figure 4.9: Traction on titanium specimen(Minimum bulk stress applied).

Figure 4.10: Bulk stress on contact region of titaniumspecimen.

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Contact pressure and tangential traction along the contact surface of AM titanium pads and steelspecimen subjected to normalised amplitude stress σa = 0.22 can be found in figures 4.11 to 4.14.The peak of contact pressure in the titanium specimen subjected to same amplitude stress wasslightly lower than the steel specimen showed in figures 4.7 to 4.10.

Figure 4.11: Traction on steel specimen(No bulk stress).

Figure 4.12: Traction on steel specimen(Maximum bulk stress applied).

Figure 4.13: Traction on steel specimen(Minimum Bulk stress applied).

Figure 4.14: Bulk stress on contact region of steel speci-men.

The combined effect of stress and fretting damage can be characterised by the Ruiz parameteralong contact surface for titanium specimen subjected to amplitude stress σa = 0.19 and steelspecimen subjected to normalised amplitude stress σa = 0.22 is illustrated in figures 4.16 and 4.15[10]. In titanium specimen maximum ruiz parameter is calculated at x = 4.9293 mm and in steelspecimen it belongs to x = 4.9543 mm, these are considered to be crack initiation position in these

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specimens.

Figure 4.15: Titanium specimen vs steel pad. Figure 4.16: Steel specimen vs AM titanium pad.

Comparing the coordinate of the node on which Ruiz parameter was maximum and position ofmaximum traction on the contact surface for both material combinations, it is clear that the fret-ting fatigue life of specimen does not only depend on the peak stress alone but fretting damagehas significant effect.

Knowing the position of crack initiation and using fatigue life prediction models explained insection 3.2, fatigue lives obtained from Paris equation and NASGRO equation using both crackmodels are compared to experimental results in Figure 4.17 and 4.18. For more detailed resultssee Appendix C.

Figure 4.17: Predicted fretting fatigue life for titaniumspecimen vs experimental fretting fatigue life.

Figure 4.18: Predicted fretting fatigue life for steel speci-men vs experimental fretting fatigue life.

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As it can be seen in figure 4.17, prediction of crack propagation shows longer life time than experi-ments, it can be due to the fact that in experiments several cracks happens on both side of the spec-imen, while in the prediction models, fatigue life of specimen subjected to one crack on one sideof specimen was predicted. Decreasing the load can prevent multiple cracks initiation, thereforethe predicted crack propagation life gets lower than the experiment. Prediction of crack growthin steel specimen done by using two crack models, see figure 4.18. However, in experiments onlycorner cracks were observed and results from corner crack model deviated significantly from theedge through crack which was used for both Paris’ equation and NASGRO equation. It signifiesthe importance of choosing correct crack model. The corner crack model was only used in NAS-GRO equation and for highest amplitude stress it predicts the crack initiation at first couple ofcycles. As the load on the specimen is decreased, crack initiation takes longer time to happen, seefigure 4.18.

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5 Conclusion and Further work

Comparing Numerical and experimental results and crack initiation position in both cases, it canbe concluded that FEM model is capable of simulation the specimen and pad interaction. Also byconsidering experimental and numerical results, following conclusions can be drawn:

• Fretting wear mostly effect the crack initiation life and the crack propagation can be simu-lated as plain fatigue.

• Considering the fact that blades are made out of titanium alloy and rotating disk of com-pressors are made of steel alloy, it is more probable to get the fretting fatigue crack in theblades. This will make the operations more economically efficient since it is easier to replacethe blades rather than changing the disks.

• Edge through crack model cannot be used to predict the crack propagation life in steel spec-imen in which corner crack happens during the experiment.

• for the loads higher than 0.12, results from the prediction of crack growth are not valid andfor predicting the crack growth life in titanium specimen results from Paris’ equation aremore valid.

• The fretting damage is predicated by a parameter equal to shear stress times the relative slipat the contact surface, hence the combined stress and fretting damage parameter should beused to estimate the fatigue life and predict the location of the crack.

In order to fully complete the project, a third series of experiments composed of 3 types of AMmanufactured specimens with printing directions of: 0, 45 and 90 degrees should be tested againstconventionally manufactured steel pads and the result can be compared with the experimentalstudy results presented in this thesis. Finally, it can be concluded if it is possible to change con-ventionally manufactured titanium blades with additive manufactured blades.

One can also analyse the effect of printing direction in more details by conducting a three dimen-sional FEM analyses and consider the printing direction will the simulation.

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References

[1] B. Alfredsson and A. Cadario, A study of fretting friction evolution and fretting fatigue crackinitiation for spherical contact, International Journal of Fatigue, 26(10), 1037-1052, 2004.

[2] D. Hannes and B. Alfredsson, A fracture mechanical life prediction method for rolling contactfatigue based on asperity point load mechanism, Engineering Fracture Mechanics, 83:62-74, 2012.

[3] D. A. Hills and D. Nowell, Mechanics of Fretting Fatigue,

[4] ASTM E647-00 (2005) Standard test method for measurement of fatigue crack growth rates.In:Annual book of ASTM standards,West Conshohocken, PA.

[5] Lindley, T. C. (1997) Fretting fatigue in engineering alloys. Int. J. Fatigue 19, S39–S49.

[6] K. Nishiko and K. Hirakawa, Fundamental Investigation of Fretting Fatigue, Part 4, The Effect ofMean Stress, Bull. Jpn. Soc. Mech. Eng. Vol 12(No.51), 1969, p408-414.

[7] John M. Barsom, Fracture and fatigue control in structures, Application of fracture mechanics.c

[8] H. Tada, Paul C. Paris, George R. Irwin, The stress analysis of cracks handbook, 2nd edition.

[9] NASGRO 9.1, 2019, Reference Manual. Southwest Research Institute.

[10] C. Ruiz, P.H.B Boddington and K.C. Cohen, An Investigation of Fatigue and Fretting in a DovetailJoint, 1984.

[11] X. R. Wu and A. J. Carlsson, Weight Functions and Stress Intensity Factor Solutions, NY: Perga-mon Press, 1991.

[12] Siemens material data, document number: K-190-670902

[13] Siemens material data, document number: K-190-259655

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Appendix A Edge through crack model

Crack case TC12 is a weight function solution for an edge through crack in a finite plate subjectedto loadings resulting in a general nonlinear stress distribution.The crack geometry is defined bythree parameters, plate full widthW, plate thicknesst and flaw size c. The crack size range for TC12is as follows:

0 ≤ cW≤ 0.9 (9)

Figure A.1: configuration of trough crack at edge of plate.

The weight function for TC12 is analytically derived. the weight function is based on five termsexpansion around the crack tip location. The stress intensity factor from crack tip is thus given by

KI = f√

πc (10)

where

f =1√2πc

∫ c

0σ(x)[

5

∑i=1

βi(1−xc)i− 3

2 ]dx (11)

Where c and x are normalised crack length and coordinate wit respect to the plate width W, andthe coefficient of the weight functions (βi) are determined through interpolation from tabulateddata for given c

v ratios. For stress variations that can be represented by the pre-defined polyno-mials, the solution has been pre-integrated analytically and hard-coded into the routine for fastcomputing. For stress variations defined by stress array, the integration is performed numericallyusing the Gauss-Chebyshev quadrature with convergence check [11].

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Appendix B Corner crack model

Crack model CC12 is a quarter-elliptical corner crack model at a chamfered corner of a finite plate.the two chamfer legs must be equal. The corner crack must completely span the chamfer. Loadingis specified by general bivariant stress distributions on a crack plane in an uncracked plate. Thecrack model is illustrate in figure 7.1 and geometric inputs are:

• t: plate thickness

• W: plate width

• d: chamfer depth

• a: initial flaw size in the thickness

• ac : aspect ratio of the corner crack

Figure B.1: Configuration of quarter elliptical corner crack at chamfer in plate.

The stress intensity factor for a corner crack at a chamfered corner is based on the weight functionsolution implemented for the crack case of corner crack model without chamfer[9]. In order to usethis model the corner crack must completely span the chamfer as shown in the figure above. Inorder to account for the effect of reduction in corner crack area due to chamfering a multiplicationfactor was applied to determined corner crack with out chamfer solution. This correction factor isbased on the ratio of two quarterly infinite domain solutions, as illustrated in figure B.1.

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Appendix C Predicted crack propagation lives

• Amplitude stresses have been normalised by specimen’s material yield stress.

Table 4: Predicted fatigue life of titanium alloy specimen for each normalised amplitude stress.

σaParis’ equation NASGRO equation

(edge through crack) (edge through crack model)0.073 1600000 74545340.09 350170 5116690.12 182780 1972230.14 93409 962880.19 37038 37233

Table 5: Predicted fatigue life of steel alloy specimen for each normalised amplitude stress.

σaParis’ equation NASGRO equation NASGRO equation

(edge through crack) (edge through crack) (corner crack model)0.48 18016 10349 399030.45 20825 15400 471520.4 32011 27883 65424

0.34 46688 42600 904010.31 60483 56743 1144690.28 81090 78685 1493790.27 96125 94665 1741380.25 119750 119372 2141390.22 175770 200116 2716280.17 707860 757507 817329

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Appendix D Additional pictures from experiments

• Cracks surface and cracks initiation position in steel alloy specimen subjected to amplitudestress σa = 0.4 .

Figure D.1: Cracks surface of titanium alloy specimen. Figure D.2: Contact Mark and crack initiation position insteel alloy specimen.

• Cracks surface and cracks initiation position in titanium alloy specimen subjected to ampli-tude stress σa = 0.9.

Figure D.3: Cracks surface of titanium alloy specimen. Figure D.4: Contact Mark and crack initiation titaniumin steel alloy specimen.

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