Fretting Wear and Fretting Fatigue: Fundamental Principles and Applications 0128240962, 9780128240960

Table of contents :
Cover
Fretting Wear and Fretting Fatigue
Copyright
Contributors
Preface
Brief history of the subject
Early stages
Initial milestones in the understanding of the mechanics of fretting
Crucial steps toward a better understanding of fretting wear and fretting fatigue
State of the art at the beginning of the new millennium
Acknowledgments
References
Introduction to fretting fundamentals
Fretting within a wider context of tribology
Fretting wear
Fretting fatigue
Mitigating fretting damage
References
Contact geometry
Friction and fretting regimes
References
Transition criteria
Mapping approaches
References
Early developments
Basic test configurations
Fretting wear tests and analytical methods
Fretting fatigue tests and analytical methods
Combined fretting wear and fatigue approaches
References
Theoretical models
Wear mechanisms and phenomenological models of fretting wear
Fatigue crack initiation and propagation
Numerical models
Wear models using mechanistic approaches and advanced FEM and BEM simulations
Advanced numerical methods for crack initiation and propagation in fretting
Nano- and mesoscale models
References
The role of tribologically transformed structures and debris in fretting of metals
Overview
Wear in both sliding and fretting-Contrasts in the transport of species into and out of the contacts
The nature of oxide debris formed in fretting
Formation of oxide debris in fretting-The role of oxygen supply and demand
Oxygen supply-Contact size
Oxygen supply-Environmental oxygen concentration
Oxygen supply-Fretting in liquids
Oxygen demand-Fretting frequency
Tribo-sintering of oxide debris and glaze formation
Microstructural damage-Tribologically transformed structures in fretting
The critical role of debris in fretting: Godets third body approach
Godets third body approach revisited: Rate-determining processes in fretting wear
Wear in the steady state
The first potential RDP: Debris formation
The second potential RDP: Debris transport out of the contact
The third potential RDP: Oxygen transport into the contact
Implications of the RDP concept
Conclusion
References
Friction energy wear approach
Friction energy wear approach
Basics regarding friction energy wear approach
Fretting loop analysis and related friction energy parameters
Archard vs friction energy wear concept: The influence of the coefficient of friction
Friction energy wear concept
Third body theory (TBT)
Contact oxygenation concept (COC)
Influence of contact loadings regarding friction energy wear rate
Influence of the normal load
Influence of the sliding frequency
Influence of the contact size
Influence of the sliding amplitude
Extended wear coefficient approach: A power law formulation
Influence of ambient conditions
Influence of temperature
Influence of lubricated (grease) interface
Surface wear modeling using the friction energy density approach
Modeling the fretting worn profiles taking into account the dynamical evolution of debris layer
Multiphysics fretting wear modeling including friction energy density, third body, and contact oxygenation process
Predicting the coating durability using the friction energy density parameter
Conclusions
References
Lubrication approaches
Introduction
Parameter definition
Amplitude ratio
Damage ratio
Oil lubrication
Influence of viscosity
Influence of oscillation frequency
Mechanism for fretting wear reduction in oil lubrication
Grease lubrication
Influence of base oil viscosity
Influence of worked penetration
Influence of oscillation frequency
Mechanism for fretting wear reduction in grease lubrication
Conclusions
Acknowledgments
References
Impact of roughness
Introduction
Contact of rough surfaces
Stress distribution in rough contact
Effective contact area
Coefficient of friction
Bearing capacity
Surface anisotropy and orientation
Transition between partial and gross slip
Impact of surface roughness on fretting wear
Friction in lubricated contact conditions
Energy dissipated at the interfaces for smooth and rough surfaces
Impact of surface roughness on crack initiation
Dynamics of surface roughness evolution in fretting contact
Measurement of fretting wear using surface metrology
References
Materials aspects in fretting
Physical processes impacting materials in industrial fretting contacts
Factors affecting fretting behavior of different materials groups
Materials behavior vs fretting regimes
Materials behavior vs fretting contact load and geometry
Materials behavior vs fretting frequency
Materials behavior vs fretting temperature and environment
Materials hardness, stiffness, yield strength vs fretting behavior
Materials engineering approaches to the mitigation of fretting wear
Thermo-chemical surface treatments
Shot-peening treatment
Laser surface treatment
Application of coatings to mitigate fretting wear
Thermally sprayed coatings
Hard coatings
Adhesion of hard coatings
Soft metal coatings
Advanced coating designs and architectures
Multicomponent and composite coatings
Multilayered and superlattice coatings
Adaptive composite coatings
Duplex coatings
Concluding remarks
References
Contact size in fretting
Introduction
Experimental techniques for nano-/microscale fretting and reciprocating wear testing
Contact geometry effects
Pile-up
Contact size effects on deformation vs fracture
Indentation size effects
Lateral size effects
Size effects on yield and fracture in coatings
Contact size and friction
Case studies
MEMS-Silicon and thin hard carbon coatings on silicon
Coatings to protect silicon-Thin hard carbon films
Biomedical materials
DLC/steel
Conclusions
References
Partial slip problems in contact mechanics
Introduction
Global and pointwise friction
Global and local elasticity solutions
Half-plane contacts: Fundamentals
Conditions for full stick
Effects of tension and moments
Summary of full stick conditions
Sharp-edged (complete) contact: Fundamentals
Conditions for full stick
Partial slip of incomplete contacts
An introduction to corrective slip: The Cattaneo-Mindlin solution
Effect of bulk tension, cyclic loading, and change in normal load
Dislocation-based solutions
Introductory problem
Steady-state solution: Constant normal load
Steady-state solution: Varying normal load
Application to Hertzian contact
Asymptotic approaches
Summary
Eigenfunctions for the Williams wedge solution
Size of the permanent stick zone for a Hertz geometry with large remote tensions
References
Fundamental aspects and material characterization
Introduction
Mechanical models and metrics
The crack analogue approach
Modification of the crack analogue
Material testing and characterization
Looking ahead
References
Fretting fatigue design diagram
Equations for estimating fretting fatigue strength based on strength of materials approach
Fracture mechanics approach for fretting fatigue life prediction
Fretting fatigue crack path prediction
Fretting fatigue life prediction
Fretting-contact-induced crack closure
Fretting fatigue design diagram based on stresses on the contact surface
Summary
References
Further reading
Life estimation methods
Fretting fatigue features and fretting processes
Fretting fatigue features
Fretting fatigue processes
Fretting fatigue crack initiation limit
Crack initiation criteria using stress singularity parameter
Crack initiation criteria using critical distance stress theory
High-cycle fretting fatigue life estimations considering fretting wear
Process of fretting fatigue life analysis
Fretting wear analysis
Fracture mechanics analysis
Fretting fatigue life analysis
Low-cycle fretting fatigue life estimations without considering fretting wear
Application of failure analysis of several accidents and design analyses
Failure of bolted joint hubs in gear transmissions
Explanation of the accident situation
Failure analysis of accident
Failure of axle bolts in roller coasters
Explanation of the accident situation
Investigation of the cause of the accident
Fatigue life analysis of connecting rod bolts
Fretting fatigue strength improvements using stress release grooves
Conclusions
References
Further reading
Effect of surface roughness and residual stresses
Introduction
Effect of surface roughness on fretting fatigue
Numerical studies of the surface roughness effect on fretting fatigue
Experimental analyses of the surface roughness effect
Some general considerations
Residual stresses in fretting
Usual surface treatments
Stability of residual stresses
Modeling the effect of surface roughness on fretting fatigue
Analytical approaches
Numerical approaches
Residual stress modeling in fretting fatigue
References
Advanced numerical modeling techniques for crack nucleation and propagation
Introduction
Theoretical background
Crack nucleation
Mechanics of crack nucleation
Crack nucleation criteria
Critical plane approach
Findley parameter
Stress invariant approach
Crossland parameter
Fretting specific parameter
Ruiz parameter
Continuum damage mechanics approach
Lemaitre damage model
Crack propagation
Mechanisms of crack propagation
Methodologies to estimate propagation lives
Linear elastic fracture mechanics
Cyclic cohesive zone models (as an alternative for when LEFM assumptions are not satisfied)
Numerical modeling
Crack initiation models
FE models
Implementation of damage models
CP approach methodology
Crack propagation models
LEFM implementation
Implementation of cyclic cohesive zone models
Crack nucleation prediction
Crack nucleation location
Initial crack orientation
Nucleation life estimation
Effect of out-of-phase loading
Effect of stress gradient and stress averaging on life
Crack propagation lives estimation
LEFM and empirical laws (such as Paris Law)
Cyclic cohesive zone models (unifying initiation and propagation phases)
Summary and conclusions
Way forward
References
A thermodynamic framework for treatment of fretting fatigue
Introduction
Fretting fatigue models-background
Surface damage from irreversible thermodynamics framework perspective
Thermodynamically based CDM
CDM analysis of fretting fatigue crack nucleation with provision for size effect
Methodology and approach
Fretting contact stress formulation and analysis
Crack initiation parameter for CDM analysis
Averaging zone identification
Crack nucleation life by CDM
Fretting subsurface stresses with provision for surface roughness
Formulation of rough contact problem
Surface tractions and subsurface stress distribution
CDM-based prediction of fretting fatigue crack nucleation life considering surface roughness
Numerical procedure
Critical tangential force prediction
Crack nucleation life prediction
Conclusion and remarks
References
Aero engines
Introduction
Examples of engine events
Southwest Airlines flight 1380, April 17, 2018
RB211 Trent 892 turbofan engine Boeing 777, A6-EMM, January 31, 2001
Accident to the AIRBUS A380-861 with Engine Alliance GP7270 engines, September 30, 2017
Areas subject to fretting
Dovetail blade roots
Derivation of stresses
Design basis for a bladed disc
Fir-tree blade roots
Splines-Contact fatigue, notch fatigue, and wear
Flanges
Mitigation measures
Surface coatings
Surface treatment (residual stress)
Design criteria-Academic perspective
Short crack arrest
Contact asymptotics
Industrial applications perspective
Design and assessment approaches
Edge of contact stress prediction/fracture mechanics approaches
Bulk or net section stresses
Subcomponent test
Specimen test
Conclusions
References
Electrical connectors
Introduction
Effects of fretting on electrical contact resistance
Effects of materials on fretting in electrical contacts
Effects of contact load, frequency of motion, and slip amplitude on fretting in electrical contacts
Endurance of electrical contact resistance under fretting
Discussions
Fretting in industrial applications
Structure of connector
Connector and contact force
Connector materials
Fretting test with real-world products
Connectors with fretting occurrence
Case analysis of fretting damage
Alternative solutions for fretting in electrical contacts
Summary
Acknowledgments
References
Biomedical devices
Introduction
Common biomaterials
Metallic biomaterials
Cobalt-based alloys
Titanium and titanium-based alloys
Iron-based alloys
Ceramic biomaterials
Alumina and alumina composites
Silicon nitride
Zirconium and zirconium composites
The biological environment
Compound tribocorrosion degradation mechanisms of materials in the biological environment
Corrosion
Electrochemistry of corrosion
Passivity of metallic biomaterials
Mechanisms of fretting corrosion
In vitro assessment of fretting corrosion within the biological environment
The role of contact condition
The role of the material contact couple
The role of environment
In vivo fretting corrosion within the biological environment
Clinical implications of fretting corrosion
Orthopedics/trauma
Modular taper interfaces
Spinal instrumentation
Dental
Cardiovascular
Conclusions
References
Nuclear power systems
Introduction
Critical safety components of the nuclear reactor that are susceptible to fretting wear damage
Methodology for predicting fretting damage of nuclear structural components
Wear energy in impact-sliding fretting
Nonlinear response of the fretting tribo-system under random excitation
Fretting wear of nuclear steam generator tubes-Effects of process parameters
Effect of the tube-support radial clearance
Effect of temperature
Effect of water chemistry
Effect of materials and contact configuration/support geometry
Fretting Wear of nuclear fuel assembly-Effect of process parameters
Effect of the process parameters controlling the Wear energy
Effect of temperature
Effect of excitation mode
Effect of surface treatment
Concluding remarks and future outlook
Acknowledgments
References
Rolling bearings
Introduction
False brinelling vs true brinelling
Bearing applications at risk of false brinelling
Mechanisms of false brinelling in rolling bearings
Test methods for assessing lubricant protection against fretting wear in bearings
Progression of false brinelling damage
Influence of lubricant properties and contact conditions on false brinelling
Effect of lubricant properties
Effect of oscillating amplitude
Possible measures to mitigate false brinelling risk in rolling bearings
Fretting in nonworking surfaces of bearings
References
Overhead conductors
Introduction
Traditional approaches
Poffenberger-Swart formula
Endurance limit approach
Cumulative damage method
Recent results based on fatigue testing of conductors
Resonant fatigue test bench
Effect of the tensile load
Effect of the H/w parameter
Effect of elastomeric clamps
Effect of temperature
Recent progress toward a multiscale fatigue analysis
Motivation
Experiments
Methodology for fatigue life prediction
Evaluation of the methodology
References
Marine risers
Introduction
Design methodology for fretting in flexible marine riser
Experimental characterization of pressure armor material
Global riser loading conditions and analysis
Global riser analysis
Global-local loading conditions
Global riser axial tension
Global riser curvature
Local nub-groove fretting analysis
Fretting wear-fatigue predictions
Concluding remarks
Acknowledgments
References
Index

Citation preview

Fretting Wear and Fretting Fatigue

Elsevier Series on Tribology and Surface Engineering (ESTSE)

Fretting Wear and Fretting Fatigue Fundamental Principles and Applications

Edited by

Tomasz Liskiewicz Daniele Dini

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-12-824096-0 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Joshua Mearns Production Project Manager: Poulouse Joseph Cover Designer: Christian Bilbow Typeset by STRAIVE, India

Contributors

Magd Abdel Wahab Laboratory Soete, Faculty of Engineering and Architecture, Ghent University, Ghent, Belgium Jose Alexander Arau´jo Department of Mechanical Engineering—ENM, Faculty of Technology—FT, University of Brası´lia—UnB, Brası´lia, Brazil M. Helmi Attia Aerospace Manufacturing Technology Center, Institute for Aerospace Research, National Research Council of Canada, Ottawa, ON; Mechanical Engineering Department, McGill University, Montreal, QC, Canada Andrew R. Beadling Institute of Functional Surfaces, School of Mechanical Engineering, University of Leeds, Leeds, United Kingdom Ben D. Beake Applications Development, Micro Materials Ltd, Wrexham, Wales, United Kingdom Ali Beheshti Department of Mechanical Engineering, George Mason University, Fairfax, VA, United States Nadeem Ali Bhatti Laboratory Soete, Faculty of Engineering and Architecture, Ghent University, Ghent, Belgium Michael G. Bryant Institute of Functional Surfaces, School of Mechanical Engineering, University of Leeds, Leeds, United Kingdom Fa´bio Comes Castro Department of Mechanical Engineering—ENM, Faculty of Technology—FT, University of Brası´lia—UnB, Brası´lia, Brazil Adrian Connaire Wood plc, Parkmore, Galway, Ireland Pascale Corne Mines Saint-Etienne, Saint-Etienne, France Ian de Medeiros Matos Department of Mechanical Engineering—ENM, Faculty of Technology—FT, University of Brası´lia—UnB, Brası´lia, Brazil Daniele Dini Imperial College London, Faculty of Engineering, Department of Mechanical Engineering, London, United Kingdom

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Contributors

Jaime Domı´nguez Universidad de Sevilla, Sevilla, Spain Jorge Luiz De Almeida Ferreira Department of Mechanical Engineering—ENM, Faculty of Technology—FT, University of Brası´lia—UnB, Brası´lia, Brazil Siegfried Fouvry LTDS, Ecole Centrale de Lyon, Ecully, France Jean Geringer Mines Saint-Etienne, Saint-Etienne, France Antonios E. Giannakopoulos Mechanics Division, National Technical University of Athens, Athens, Greece Annette M. Harte Ryan Institute for Marine, Energy and Environment; Civil Engineering, School of Engineering, NUI Galway, H91 HX31, Ireland Toshio Hattori Gifu University, Gifu-City, Japan David A. Hills Department of Engineering Science, University of Oxford, Oxford, United Kingdom Ilkwang Jang School of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea Yong Hoon Jang School of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea Rachel Januszewski Department of Mechanical Engineering, Imperial College London, London, United Kingdom Murugesan Jayaprakash Department of Metallurgy and Materials Engineering, Indian Institute of Technology Indore, Indore, India Hyeonggeun Jo School of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea Amir Kadiric Department of Mechanical Engineering, Imperial College London, London, United Kingdom Remy Badibanga Kalombo Department of Mechanical Engineering—ENM, Faculty of Technology—FT, University of Brası´lia—UnB, Brası´lia, Brazil Chaosuan Kanchanomai Department of Mechanical Engineering, Faculty of Engineering, Thammasat University, Pathumthani, Thailand

Contributors

xv

Thawhid Khan Manchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom Michael M. Khonsari Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA, United States Krzysztof J. Kubiak School of Mechanical Engineering, University of Leeds, Leeds, United Kingdom Sea´n B. Leen Mechanical Engineering, School of Engineering; Ryan Institute for Marine, Energy and Environment, NUI Galway, H91 HX31; I-Form Centre for Advanced Manufacturing, Ireland Tomasz Liskiewicz Manchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom Yanfei Liu Beijing Institute of Technology, School of Mechanical Engineering, Beijing, China Taisuke Maruyama Core Technology R&D Center, NSK Ltd., Tokyo, Japan Thomas G. Mathia LTDS, Ecole Centrale de Lyon, Ecully, France Allan Matthews University of Manchester, Manchester, United Kingdom Matthew R. Moore Department of Physics & Mathematics, University of Hull, Kingston-Upon-Hull; Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, United Kingdom Yoshiharu Mutoh Department of System Safety, Nagaoka University of Technology, Nagaoka, Niigata, Japan Carlos Navarro Universidad de Sevilla, Sevilla, Spain David Nowell Department of Mechanical Engineering, Imperial College London, South Kensington Campus, London, United Kingdom Sinead M. O’Halloran Ryan Institute for Marine, Energy and Environment, NUI Galway, H91 HX31; SEAM Research Centre, School of Engineering, Waterford Institute of Technology, Waterford, X91 TX03, Ireland Abimbola Oladukon Institute of Functional Surfaces, School of Mechanical Engineering, University of Leeds, Leeds, United Kingdom

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Contributors

Youngwoo Park LG Electronics Vehicle Components USA, Troy, MI, United States Kyvia Pereira Laboratory Soete, Faculty of Engineering and Architecture, Ghent University, Ghent, Belgium John Schofield Consultant in Structural Integrity, Derby, United Kingdom Philip Howard Shipway Faculty of Engineering, University of Nottingham, Nottingham, United Kingdom Jesu´s Va´zquez Universidad de Sevilla, Sevilla, Spain Andrey Voevodin University of North Texas, Denton, TX, United States Aleksey Yerokhin University of Manchester, Manchester, United Kingdom Thanasis Zisis Mechanics Division, National Technical University of Athens, Athens, Greece

Preface

Applied research in fretting wear and fretting fatigue has gained significant momentum in recent years; however, there is no single reference book currently available that can educate researchers in a comprehensive way on both wear and fatigue aspects of the fretting phenomenon. Hence, this book takes a combined mechanics and materials approach, providing readers with fundamental understanding of fretting, related modeling and experimentation techniques, methods for design and mitigation against fretting, and robust examples of practical applications across an array of engineering disciplines. The aim of the book was to bring together, systematically in a single volume, the state-of-the-art knowledge on fretting. This was achieved through a collaborative approach between two editors specializing in different aspects of fretting. Often recognized as two distinct areas of research, fretting wear and fretting fatigue are addressed in one volume for the first time. This creates a unique synergy and opportunity to expand horizons and learn. We assembled the best experts in the field to contribute to this project, many of whom are members of the International Symposium on Fretting Fatigue (ISFF) community. We believe that this book can be a key reference for students and scholars from diverse science and engineering backgrounds such as those from mechanical engineering, materials science, structural design, and many more. We hope the book will also be of use to practitioners and engineers who frequently encounter frettingrelated problems, and who require timely, but adequate solutions. We tried to present the contents in a logical, easily assimilated manner. This book contains five sections written carefully to cover the fundamental aspects as well as the key theoretical developments and experimental methodologies associated with fretting wear and fretting fatigue. The first section of this book introduces the reader to the historical aspects of fretting, providing a useful starting point and background to the rest of the book. The second section discusses the underpinning basic theories, providing a good introduction to the subject to any learner new to fretting. The third and fourth sections of the book present contributions that cover different facets and provide a complete in-depth understanding of fretting wear and fretting fatigue, respectively. The subsequent seven chapters in section five of the book are dedicated to the studies of some of the main applications where understanding of fretting phenomena contributes to improved engineering design and practice. This book is meant to be an active source of reference, which a reader can flip through to find useful guidance on specific topics when there is a need for some perspective or advice. The fundamentals of fretting are covered in section two; however, when it is important for the context, some basic theories are also discussed in the subsequent chapters. There will be some degree of repetition between the chapters,

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Preface

which results from the fact that every chapter is a stand-alone contribution meant to provide a full understanding of a given topic. The reader is also advised to pay attention to the nomenclature used by the contributors, as this might differ between the chapters. We accepted this fact, as it spans from common practice in a field approached by scientists from different backgrounds, and did not try to manipulate the nomenclature, as any attempt to create a fretting unified glossary and list of symbols would be untrue to the field and could cause a degree of confusion. The editors are thankful to all contributing authors for agreeing to be a part of this project. This is especially appreciated as the book was edited during the COVID-19 global pandemic, when all of us were presented with unprecedented circumstances and challenges. We also thank colleagues and associates worldwide who have offered helpful comments and suggestions for improvement of the text. In particular, we acknowledge support and encouragement from our families in the process of completing this book. Tomasz Liskiewicz Daniele Dini Manchester and London June 2022

Brief history of the subject

1

Daniele Dinia and Tomasz Liskiewiczb a Imperial College London, Faculty of Engineering, Department of Mechanical Engineering, London, United Kingdom, bManchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom

1.1

Early stages

Eden et al. (1911) produced the first recorded report of fretting in the scientific literature. Fretting was revealed by the presence of oxide debris on the surface of the steel jaws of a testing machine where a fatigue specimen was gripped. Intensive research has been conducted to develop an understanding of fretting fatigue since this first observation of fretting in relation to fatigue tests. Although their findings may have been a curiosity of sorts, it started an activity that has culminated in significant progress being made in developing an understanding of fretting, including fretting corrosion and fretting wear. Fretting also acts conjointly with fatigue (cyclic loading) on components and specimens both in use and in laboratory evaluation programs. Since then, a large amount of work has been carried out with a substantial growth in the number of investigations with time. Recently, the number of publications on fretting fatigue has increased significantly and it has become a subject of active scientific debate. Shortly after the research of Eden and coworkers, Gillett and Mack (1924) performed research that showed a significant reduction in the fatigue of machine grips. Shortly thereafter, Tomlinson (1927) performed the first systematic investigation of fretting, and he clearly recognized that corrosion is a secondary factor and that surface damage could be caused by the movements of very small magnitude. He showed the dependence of fretting and the related damage phenomenon on tangential motion and addressed the tangential microdisplacements as a possible parameter to be involved in the understanding of fretting fatigue. Numerous investigators (Peterson and Wahl, 1935; Almen, 1937; Evans and Miley, 1937; Campbell and Thomas, 1938; Campbell, 1939; Tomlinson et al., 1939) have performed additional studies that aided the development of a phenomenological recognition and understanding of fretting. In 1941, Warlow-Davis conducted an extensive investigation into the effect of fretting on the fatigue “properties” of materials (Warlow-Davies, 1941). He was an early proponent of the idea that fretting accelerates fatigue degradation, and recognized a small decrease of fatigue performances (13%–17%) in previously fretted specimens (Warlow-Davies, 1941). Other investigators (Almen, 1948; Bowden and Tabor, 1950; Godfrey, 1950; Feng and Rightmire, 1952; Wright, 1952a,b) advanced the state of knowledge on fretting wear and fretting fatigue. Godfrey (1950) performed extensive

Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00001-9 Copyright © 2023 Elsevier Inc. All rights reserved.

4

Fretting Wear and Fretting Fatigue

microscopic evaluation of fretting corrosion to attempt to characterize the nature of fretting and clearly delineate fretting mechanisms. He concluded that adhesion resulted from contact, and extremely fine particles of debris were noted to have broken loose and oxidized.

1.2

Initial milestones in the understanding of the mechanics of fretting

Although it was not always recognized in these terms at the time, many of us now think of the fretting problem as being split into three parts, each capable of independent study. Firstly, the fretting process itself involves the reciprocating motion of one contacting surface (or part of one surface) over another. This has been the subject of much mathematical analysis since the pioneering work of Cattaneo in the 1930s (Cattaneo, 1938), rediscovered by Mindlin in the 1940s (Mindlin, 1949). Secondly, the presence of differential motion modifies the surfaces, causing wear, and may be instrumental in controlling the conditions under which cracks nucleate. Lastly, the presence of contact stresses will have an important effect on the early propagation of cracks, until the tip grows out of the influence of the contact stress field. It is easy, in hindsight, to see things in this way, but we must remember that fracture mechanics itself was still far from fully developed in the 1950s, so that even the propagation aspect of the problem was still a subject of study. The 1950s are therefore the first golden age for fretting research, which coincided with the booming of the studies in all three key aspects that characterize fretting: Bowden and Tabor (1950) published Part I of their classic work on friction and lubrication of solids, the fields of fatigue and fracture mechanics saw a plethora of developments (Irwin, 1948; Orowan, 1949; Bishop, 1955; Gordon, 1978) and Archard formulated its wear law (Archard, 1953; Archard and Hirst, 1956). In 1952, Feng and Rightmire proposed a theory on fretting mechanisms (Feng and Rightmire, 1952). As well, Wright and coworkers (Wright, 1952a,b; Fenner et al., 1956) performed extensive studies on the role of oxidation in fretting and recognized that ferric oxide was separating the two original surfaces. They observed that the formation of oxides accelerated the development of fretting damage. With most of the studies focused on the importance of oxidation and corrosion in fretting, by 1952 enough progress had been made to organize an ASTM symposium on fretting corrosion (ASTM, 1952), consisting of the effect of the interaction between fretting and corrosion in reducing the fatigue life of mechanical components. This was a landmark event because it brought together investigators in the field and provided focus and a stimulus for additional activity. Five papers were presented at this symposium. It is interesting to note that terms that were used to describe fretting at this meeting were: (i) friction oxidation, (ii) wear oxidation, (iii) false brinelling, (iv) bleeding, and (v) cocoa. Subsequently, Horger (1953), Uhlig et al. (1953a,b), Uhlig (1954), Waterhouse (1955), and Corten (1955) also made important contributions in this area.

Brief history of the subject

5

Studies by McDowell (1953) pointed out that the combined effects of fretting wear and fatigue were severe, with a quantification of the effect tentatively provided that showed a reduction in strength by a factor varying from 2 to 5 or even more. The absence of distinct fretting fatigue limits such as those found in plain fatigue tests (Ording and Ivanova, 1956) induced some researchers to state that it was possible to find some analogies between fretting fatigue and corrosion fatigue (Horger, 1956). Several papers on fretting fatigue were presented at the International Conference on Fatigue—reported in 1956 (Ording and Ivanova, 1956). Halliday and Hirst (1956) and Liu et al. (1957) also made important contributions. In the late 1950s, Fenner and Field (1958, 1960) carried out a basic investigation of fretting damage. Their work showed a series of innovations and concepts that still constitute the background of most of the research carried out in this field. Not only were they the first to demonstrate that fretting accelerated crack initiation, but they were also among the first to use a “bridge” type of fretting pad with two flat surfaces, which was to become popular with several researchers (e.g., Doeser, 1981; Edwards, 1981a). However, it has to be noted that up to the mid-to-late 1960s very little understanding of the importance of fretting and knowledge of the characteristic parameters governing the phenomenon had been achieved. In particular, further examinations of the influence of parameters such as contact pressure, contact size, relative slip displacement amplitude, environmental condition, fretting wear, and stress concentration on fretting strength were needed. By 1963, the field of fretting corrosion had reached such concern that the US Army issued a major literature review (Comyn and Furlani, 1963). Shortly thereafter, Bowden and Tabor (1964) published Part II of the Friction and Lubrication of Solids. Parts I and II of this classic have had a great influence on the evolution of knowledge on fretting wear and fretting fatigue. One of the pioneers of experimental work in the field was Robert Waterhouse, who started to make his mark in the early 1960s (Waterhouse, 1961; Waterhouse et al., 1962). Only in 1965, when Waterhouse and Allery (1965) tested the dependence of material strength on fretting corrosion, was it shown that there was no correlation between the rate of occurrence of corrosion and the reduction of fatigue strength. These researchers changed the rate of fretting corrosion by testing the same material under fretting conditions in atmosphere and in inert gas, but the reduction of corrosion did not give rise to a fatigue strength increase.

1.3

Crucial steps toward a better understanding of fretting wear and fretting fatigue

A turning point in the research field can be identified in the comprehensive series of tests and studies undertaken in the late 1960s by Nishioka et al. (1968) and Nishioka and Hirakawa (1969a,b,c,d, 1972) using cylindrical steel pads and steel specimens. They examined the influence of many of the factors reported above independently.

6

Fretting Wear and Fretting Fatigue

The most important conclusion of their first paper was the small influence of the frequency of cyclic loading on the life of fretted specimens. In the following series of papers, they reported the existence of nonpropagating fretting fatigue cracks [as also observed in Waterhouse and Allery (1965) and Fenner and Field (1958)], suggesting that, although fretting may assist the initiation and initial growth of cracks, there may be combinations of other parameters under which these cracks self-arrest. Another conclusion reached was that the amount of relative slip between the surfaces influences fatigue life. Below 5 μm of slip displacement, there appeared to be little reduction in life in the presence of fretting. Between 5 and 50 μm, the fatigue strength was reduced to as little as 1/8 of the value in the absence of fretting. Above 50 μm of slip displacement, significant wear took place and fatigue cracks were not observed, probably because they were worn away before they could start to propagate. Another interesting finding was the increase of the coefficient of friction during the first few cycles of the experiments. This was confirmed by other studies (Milestone and Janeczko, 1981; Endo et al., 1974) and highlighted as one of the essential features of the fretting fatigue mechanism in conjunction with an increase of the real contact area, as suggested in Wright and O’Connor (1972) and Bramhall (1973). Further investigations by Nishioka and Hirakawa (1969c), highlighted the initiation site and direction of initial propagation. Cracks were found to start in a region of high stress near to the edge of the contact and to propagate obliquely under the contact during the initial phase of growth. Furthermore, they found that the mean cyclic stress applied to the specimen had a little effect on the fretting fatigue life. Therefore, the fretting contribution to the total fatigue life of the components seems to assist the formation of embryo cracks whereas the remote bulk tension plays the main role in the later stages of propagation. In their last paper of the series of six, the two researchers investigated the influence of the contact pressure and discovered the small dependence of fretting fatigue life on material hardness. In 1970, the first part of a NATO/AGARD manual by Barrois (1970) emerged. This set the stage for his treatise on fretting corrosion (Barrois, 1975), which was truly a significant contribution to the literature. In 1970, Hurricks (1970) provided an extensive review of the mechanisms of fretting and noted that fretting wear mechanisms involved the following three stages: (i) initial adhesion and metal transfer, (ii) production of debris in a normally oxidized state, and (iii) steady-state wear condition. Additional research by Waterhouse et al. (1971), Waterhouse and Taylor (1971), and Hurricks (1972) provided additional insight into the knowledge related to the effect of the environment on fretting. In 1971, the first international conference on corrosion fatigue was held (Devereaux et al., 1972) and numerous papers on fretting were presented. Major fretting mechanisms of the day and the concept of a fretting fatigue damage threshold were presented. At about the same time that the conference proceedings were published, the first book of Waterhouse emerged (Waterhouse, 1972). This seminal book has become a classic in the field and has been used extensively in the past decades.

Brief history of the subject

7

During the 1970s, the number of investigations in fretting wear and fretting fatigue increased markedly. Noticeable works continued to emerge from Waterhouse and colleagues (see, e.g., Waterhouse and Taylor, 1972; Taylor and Waterhouse, 1972; Waterhouse and Dutta, 1973; Wharton et al., 1973a,b). Taylor’s work with Waterhouse (Taylor and Waterhouse, 1972) on surface treatments and the studies conducted at the microscopic scale clearly showed that the origin of fretting cracks was in the boundary between the slip and nonslip regions of the contact area. In the early 1970s, significant efforts were again focused on fretting corrosion and fretting wear. The new concept of fretting thresholds was proposed by Hoeppner and Goss (1972), who noted that a certain amount of fretting damage was necessary to give rise to any change in the fatigue strength of the material. By analyzing some tests conducted on Ti6Al4V and 7075-T6 aluminum, they showed that removal of the fretting source after fretting damage had occurred above the proposed threshold did not affect the fatigue life of the tested specimens. Later on, Endo and Goto (1976) confirmed the possibility of using fretting thresholds in order to predict the decrease of fatigue life due to fretting. At this stage, attempts were made by many researchers to find guidelines for designing mechanical components against fretting. The microdisplacement-based approach [firstly proposed by Tomlinson (1927) and later on used as a threshold by Nishioka and Hirakawa (1969b,c)] was the most convenient tool in orientating engineers facing this complex phenomenon. Prediction of the reduction of fatigue strength based on fretting microdisplacements is simple but it should only be used for qualitative ranking of material combinations due to the difficulties in applying results from simple specimen geometries to complex components (Lindley, 1997). A number of different approaches were developed in the 1970s and some of them still constitute the basis of more recent studies. A number of fretting models proposed during that decade have been developed in the last 50 years, during which many investigators have addressed the problem of fretting. A few researchers have focused on stress concentration effects (e.g., Wright and O’Connor, 1972). However, the objective difficulty of obtaining an accurate value for the elastic stress concentration factor, Kt, for frictional contact problems, and the fact that the very steep gradients of the stress field and the other complex aspects of the phenomenon require more complex modeling than the Kt-based approach, probably discouraged the early researchers. Another way to approach the problem was suggested by Collins and Tovey (1972). They proposed a microcrack-initiation mechanism based on fatigue stress concentration at the asperity-contact level. Similar methods have also been attempted more recently by Ballard et al. (1995) and Moobola (1998). These kinds of approach are based on the assumption that even though macroscopic stresses remain elastic, microscopic stresses on the asperity level can locally exceed yield. In particular, Ballard et al. (1995) analyzed the cyclic plastic response and predicted crack initiation when the elastic shakedown limit was exceeded. However, this asperity-based method, as is generally the case with the rough contact theories, still leaves unresolved the question of the resolution at which the roughness is measured.

8

Fretting Wear and Fretting Fatigue

What one sees as an asperity at one scale can be seen as a complex pattern of asperities at a scale just below. Additional research began to emerge from Japan during this period and provided significant contributions to the field. The research of Endo and Goto (1976) is extremely important as it, in conjunction with the work of Edwards et al. (1977) and Hoeppner (1977), supplied the basis for the application of fracture mechanics to fretting. They investigated crack growth in mild steel cylindrical fretting pads under full slip conditions. They were able to separate three main phases in the cracking process: (i) initiation: a first stage of crack initiation on shear planes, followed very shortly by a second stage in which the crack propagated in a direction normal to the surface, (ii) early propagation: crack growth strongly dependent on the frictional force arising in the contact; and (iii) propagation: crack propagation at a rate well reproduced by Paris law taking into account only the alternate remote stresses applied to the specimen. These results showed how fretting can be responsible not only for the initiation phase (as proposed by Hoeppner and Goss, 1974) but also for the early stage of crack propagation. This led to the conclusion that the fretting tangential force should have been included in the calculation of the range of stress intensity factors used in the Paris law. Another important “piece of the puzzle” was added in 1973 by Bramhall, who investigated variation in the contact size while keeping constant the magnitude of the peak contact pressure. Analysis of some series of these experiments showed that fretting substantially reduced fatigue life for contact size length above a certain critical value. In fact, shorter life was found for contact dimensions larger than the critical one while infinite life (>107) was found for smaller contact sizes. Therefore, he postulated that this was due to the high gradient of stresses induced in the case of small contact lengths that results in a stress field insufficiently extended throughout the specimen to cause a crack to propagate to a stage where it can grow under the influence of the remote bulk stress. Edwards (1981b) published an important, and extensive, paper on the application of fracture mechanics to fretting. This research, along with Endo and Goto’s earlier paper and the work of Hoeppner (1981) and other researchers, has formed the basis for extensive application of fracture mechanics up to the present time. More recently, Nowell (1988) confirmed the existence of a critical contact size by testing the same material (4% Cu Aluminum alloy) recording some important parameters, which were not fully taken into account by Bramhall. Similar results were obtained with Ti6Al4V alloy in a further investigation by Arau´jo (2000). The same group at Oxford, which has extensively contributed to the literature in this field in the last 40 years (e.g., Nowell and Dini, 2003; Nowell and Hills, 1987, 1990; Hills et al., 1988; Nowell, 1988; Hills, 1994; Hills and Nowell, 1994; Hills and Fellows, 1999; Mugadu, 2002; Mugadu and Hills, 2002; Sackfield et al., 2002; Navarro et al., 2003; Nowell et al., 2006; Hills and Andresen, 2021), has developed in the late 1980s and early 1990s a distributed dislocation method to calculate the stress intensity factor for cracks growing with different orientations with respect to the free surface (Hills and Nowell, 1994; Hills et al., 1996), allowing the investigation of the first stage of crack growth (oblique initial propagation had been shown by Nishioka and Hirakawa (1969c) and Endo and Goto (1976) as described above).

Brief history of the subject

9

In the late 1970s and early 1980s, there was a significant increase of activities and researchers published numerous works on fretting (Hoeppner, 1977, 1981; Reeves and Hoeppner, 1977, 1978a,b; Hoeppner and Salivar, 1977; Waterhouse, 1977, 1978, 1981; Sproles and Duquette, 1978; Czichos, 1978; Poon and Hoeppner, 1979; Wharton and Waterhouse, 1979; Alic et al., 1979; Alic and Hawley, 1979; Barrois, 1975; Waterhouse and Saunders, 1979; Sato et al., 1980; Hoeppner and Gates, 1981; NATO-AGARD, 1981; Edwards, 1981b). During this period, these researchers concluded that fretting is predominantly influenced by mechanical surface damage. The concept of the fretting damage envelope and damage threshold were introduced. Extensive research was done on aluminum and titanium alloys during this period. Also, principles and concepts that could be used in engineering design that could either prevent or alleviate fretting were presented (Hoeppner, 1981; Hoeppner and Gates, 1981). The relative role of the environment on fretting also was studied in a great deal. The systems view of fretting began to emerge during this period, which is extremely helpful in approaching both research in fretting fatigue and engineering design challenges in fretting. The book by Czichos (1978) is an extensive treatise on the systems approach to the science and technological challenges of tribology. Additional investigations in these two very important decades for fretting were conducted by numerous investigators and by 1981 a major book on fretting emerged Waterhouse (1981) and another ASTM symposium was held that was published in 1982 (ASTM, 1982). Various papers of importance on environmental effects in fretting fatigue emerged during this period (Bill, 1981a,b; Leadbeater et al., 1981; Brown and Merritt, 1981; SAE, 1982). In 1981, Bill published a noteworthy paper on a comparison between fretting wear and fretting fatigue (Bill, 1981b). In this period, significant developments on research on fretting began to emerge (Mann, 1982; ASTM, 1982; Kusner et al., 1982; Bill, 1982; Cook et al., 1983; Waterhouse, 1984; Colombie et al., 1984; Johnson, 1985; Lindley and Nix, 1985; Nix and Lindley, 1985a,b; Harris et al., 1985; Hoeppner, 1985; Tanaka et al., 1985; Iwabuchi, 1985; Waterhouse and Iwabuchi, 1985; Attia and D’Silva, 1985; Sato, 1985; Sato et al., 1985; Brown and Merritt, 1985). Some of these were more extensive studies related to orthopedic implants (Brown et al., 1988; Merritt and Brown, 1988). Even though fretting wear and fretting fatigue of orthopedic implants was recognized by 1980, the concern was accelerated around this time since orthopedic surgeons and implant companies were interested in modular implants. Because these implants possessed more mechanical joints, the concern for fretting damage increased markedly—especially in certain titanium alloy materials. This is due to the concern for infection in the body that may result from the debris. Turning our attention to wear, early wear models originated from empirical equations developed directly from tribological experiments. In the 1970s and 1980s, wear models started being elaborated on the basis of mechanical contact. Many of them considered the real contact area and mechanical properties of the materials such as Young’s Modulus and hardness. The most recognized wear model from that period was introduced by Archard (1953). This model was proposed well in advance to other

10

Fretting Wear and Fretting Fatigue

laws of contact mechanics and was derived by Archard from an equation previously given by Holm (1946), in which a dimensionless coefficient K was introduced to provide the conformity of the formula with experimental results. The K coefficient was interpreted by Archard as a probability to form a wear particle by the asperities of the interacting solid bodies; however, other authors propose different interpretations of this coefficient (Shaw, 1977; Hutchings, 1992; Johansson, 1993). A major book on contact mechanics ( Johnson, 1985) appeared and the concepts of a third body was reinforced (Colombie et al., 1984; Berthier et al., 1988a,b,c). This concept, and the analysis, was a significant contribution related to improving our understanding of fretting. Around the same time, Nix and Lindley (Lindley and Nix, 1985; Nix and Lindley, 1985a,b) performed extensive studies on fretting damage formation and fracture mechanics applied to fretting fatigue crack propagation. By the mid-1980s, Attia and colleagues were performing extensive research at Ontario Hydro in Canada (Attia and Ko, 1986; Attia, 1989). They were attempting to apply thermal evaluation and modeling techniques to the challenge of fretting. Attia also continued the work of many in ASTM (Horger, Grover, Hyler, Hoeppner, Niefert, Marble et al.) on fretting fatigue. This culminated in the ASTM symposium held in 1990 and planned by Attia and Waterhouse (ASTM, 1992). Sato and coworkers also studied fretting damage formation and crack propagation during this period (Sato et al., 1986a,b,c,d). Hattori also began his extensive studies on applying fracture mechanics to fretting fatigue (Hattori et al., 1988). The concept of fretting maps emerged during this period as well (Vingsbo and Soderberg, 1987, 1988; Berthier et al., 1988b). Vingsbo and Soderberg (1988) introduced the bi-dimensional diagrams (named fretting maps) in which the fretting regimes, i.e., stick, partial slip, and full slip, were plotted as a function of different combinations of normal load and tangential displacement. They also studied the effect of slip amplitude on fretting life confirming the existence of a critical amplitude (about 30 μm) corresponding to the minimum fretting life. Their work has become one of the major cornerstones in the field, and still today provides the basis for the identification of fretting regimes experienced by different components. The 1980s have also been the years in which research on fretting wear benefitted with interest in studying different mechanism tests of different materials. For example, experiments on steel/steel and chalk/glass were carried out by Colombie et al. (1984). Results revealed that the wear of the matrix material was governed by the competition of generation and maintenance of the debris layer with abrasion of the debris layer. Various studies (Blau, 1981; Quinn, 1984; Saka et al., 1984; Beard, 1988) demonstrated that wear mechanisms can be different according to different types of fretting couples, with experimental results showing that the abrasive mechanism was prevailing in the pair of steel/steel with accelerating the damage by debris, while for the combination of steel/bronze, the adhesive wear mechanism was predominant, and the debris acted as a kind of lubricant reducing the damage of fretting wear. Hardness of fretting couples may affect wear mechanisms. During the 1980s other books on wear and surfaces began to emerge (Budinski, 1988; Meguid, 1990). More recently,

Brief history of the subject

11

ASM published a new volume of the handbook which includes an important, and extensive, contribution by Waterhouse (1992). In 1994, Klaffke (1994) used a variant of Archard wear coefficient to compare different surface treatments as a function of relative humidity under fretting wear. A similar approach was used by Celis et al., who related the wear resistance of several hard coatings to the sliding distance, force normal, and number of fretting cycles (Blanpain et al., 1993). Considering that under fretting the wear volume results from the tangential effort induced by friction force, the dissipated energy approach was proposed (Mohrbacher et al., 1995; Fouvry et al., 1996; Fouvry and Kapsa, 2001; 2003). Here, the wear volume correlates with the quantity of cumulated dissipated energy in the contact area. This approach considers the normal load, evolution of the coefficient of friction, displacement amplitude, and the fretting test duration. It is possible to determine the energy wear coefficients to compare the wear resistance of different bulk materials and coatings. More recently, Farris’ group proposed quantifying fretting wear by local formulation of the Archard law Goryacheva et al. (1999, 2001). The evolution of wear was correlated with the dynamic modification of the contact geometry during fretting. This takes into account the evolution of contact geometry and the resulting redistribution of contact pressure. Those global analyses permit us to determine the scalar wear coefficients independently from the test conditions. However, the wear volume is expressed by the energy dissipated within the contact area, which does not consider the local aspects of degradation. Hence, a local energy approach was developed as a more reliable formulation to relate the local wear to the amount of energy dissipated (Fouvry et al., 1997). Another attempt to characterize the fretting damage was made by Ruiz et al. (1984) and Ruiz and Chen (1986). They were among the first to examine the design of dovetail joints of a turbine engine against fretting. Following the first attempts, using the slip amplitude δmax to characterize fretting (it was shown by Nishioka et al. (1968) that the most detrimental effect on life was the one obtained for δmax of approximately 15 μm), they proposed a second parameter viz., the frictional energy dissipation, D ¼ (τ δ)max, which, being an energy dissipation, may well be correlated to the damage mechanisms on the fretted surface, and indeed has the merit of being significant also in the gross slip regime. It has also been shown to give a measure of the accumulated shear strain according to the Bower-Johnson’s ratchetting mechanism (see Hills and Nowell, 1994). Furthermore, Ruiz et al. suggested a composite surface damage parameter, R ¼ (στ δ)max, which takes into account empirical evidence that cracks are more likely to develop in a region of tension rather than compression, therefore accounting also for the early stages of propagation. However, the last definition of damage parameter, R, although apparently dealing more completely with salient variables, does not have a pure physical interpretation. Also, it depends strongly on the surface stress, which in turn is the sum of three separate contributions (bulk stress, stress induced by tangential tractions, and stress due to the normal pressure). FEM and analytical and experimental results on dovetail joints indicated a qualitative good predictive capability of the Ruiz parameter, R, but the increased complexity of the

12

Fretting Wear and Fretting Fatigue

parameter does not necessarily correspond to an increased predictive capability as discussed in Hills and Nowell (1994).

1.4

State of the art at the beginning of the new millennium

In the early 1990s, some significant work was produced by the Oxford University group. In 1992, Hills and Nowell discussed the equipment and techniques needed to obtain a satisfactory standardization of fretting fatigue tests. In particular, they stated that the Hertzian geometry, i.e., cylindrical fretting pads pressed against a flat tensile specimen, was the most reliable in that it allows not only variation of the normal force/geometry ratio in order to preserve the peak contact pressure but also control of the external forces on the pads independently. They also studied the variation of the coefficient of friction within the contact area (Hills et al., 1988) and much of the work undertaken was published in a book (Hills and Nowell, 1994) that is only the second completely dedicated to fretting fatigue (the first was written by Waterhouse in 1981). Their subsequent work then focused on the improvement of analytical techniques and the extension of analytical solutions to different combinations of loading cycles and geometries. In 1995, Fellows et al. (1995) analyzed the differences between the stresses computed with the analytical half-plane solution and those induced in the specimen’s geometry. It was clearly pointed out that the stress component parallel to the surface in a typical specimen is less compressive than in a half plane, highlighting the care required in simulating the real stresses by analytical classical solutions. Later, Nowell and Dai (1998) developed a quadratic programming-based routine able to describe the variation of the shear tractions along the contact interface for a general loading cycle (for example in LCF/HCF interaction). This is a useful technique as it allows analysis of the case in which the bulk load has a mean value greater than zero (unsymmetrical) or is out of phase with the tangential load; in this case, the analytical “semistatic” solution cannot be used. Furthermore, an analytical solution for flat pads with rounded edges against a flat half-plane contact was found in 1998 (Ciavarella, 1998; J€ager, 1998). Such solutions facilitated the use of this specimen/pad geometry in the experimental and analytical work in order to achieve a more refined approximation of the dovetail joint geometry. Analytical and numerical solutions for partial slip, i.e., when localized slip occurs at the contact interface but there is no rigid body motion between the contacting bodies, have been also derived (see Navarro et al., 2003a). Arau´jo focused his D. Phil. work (2000) on short crack arrest methodologies (Arau´jo and Nowell, 1999) and multiaxial fatigue theories applied to fretting fatigue. Many tests were analyzed and a number of parameters applied in order to find a practical prediction criterion (more details will be provided later). Some of the alternatives look promising but further experimental work and investigations are needed to apply them to real component design. At the same time, asymptotic solutions for complete contacts have been applied in order to study fretting fatigue in turbine engine splines (Mugadu et al., 2002b). A more refined asymptotic analyses of flat and rounded contacts (in gross sliding,

Brief history of the subject

13

Sackfield et al., 2003) has been also carried out but the application of such methodology, although promising, needs further developments. With the approach of the new millennium, a large number of research groups and researchers, joined by an increased interest of aerospace companies, tried to predict fretting life with different techniques, most of which have been based on multiaxial fatigue models and fracture mechanics. In 1996, Szolwinski and Farris (1996) were the first to use the Smith, Watson, and Topper (SWT) (1970) multiaxial fatigue parameter to assess crack initiation life, the main crack site location, and the direction of the first stage of propagation. Their first analyses showed some overconservative predicted values and the inability of SWT to incorporate the size effect. Neu et al. (2000) used a number of critical plane approaches in order to correlate the location and the early crack growth with the ones resulting from tests on PH 13-8 Mo stainless steel. They showed that the Fatemi and Socie (1988) criterion was able to predict quite accurately both the location and the early-stage crack orientation while the SWT criterion was not able to predict the crack orientation. At the same time, Fouvry et al. (2000) carried out a series of analyses predicting fretting crack initiation (on Ti6Al4V) applying the macro/micromechanics approach developed by Dang Van et al. (1989). This may also be considered as a critical plane parameter as it predicts that crack initiation takes place on specific crystallographic planes but it assumes that the formation of cracks is a microscopic-dominated phenomenon. Elastic microscopic shakedown can be seen, then, as the cause of crack propagation because for a macroscopic elastic stress field only localized plasticity can be observed. They also obtained an overconservative life prediction avoidable only by averaging the parameter over a critical volume that was not clearly identified as a material constant. Further work by Arau´jo (2000) and Arau´jo and Nowell (2001, 2002) applied all these parameters to fretting fatigue experiments with both Al-4Cu and Ti6Al4V, finding that all of them needed to be averaged (along a line, over a certain area or volume) to take in account the “size effect” and that the averaging distance (area or volume) was similar to the grain size of the material even though it was impossible to define a unique dimension for the two materials. As the averaging of such parameters is computationally quite expensive, another technique was proposed by Murthy et al. (2001), who elaborated a stressinvariant multiaxial-fatigue approach consisting of the reduction of the multiaxial cyclic stress state to a single equivalent stress (see the Walker method, Doner et al., 1981) followed by the use of this equivalent stress history in conjunction with the modified Manson-McKnight multiaxial model (Slavik et al., 2001). In this case, the correction due to the high stress regime is based on a Weibull statistic approach and it involves only averaging on the surface. In parallel to these investigations, Giannakopoulos et al. (1998, 2000a,b) also suggested the existence of a strong analogy between the contact mechanics problem and fracture mechanics, whereby the contact edge could be treated as a crack front in order to assess the likelihood of crack initiation under various contact conditions. The early 2000s also saw a significant increase in efforts to study the effect of formation and accumulation of wear debris in the fretting process and the identification and classification of wear processes, as well as the advent of numerical simulations used to model wear. Elleuch and Fouvry (2002) studied the effects of different displacement amplitudes on the fretting behavior of aluminum alloy (A357)/52100 steel. They found

14

Fretting Wear and Fretting Fatigue

that a displacement amplitude threshold existed relating to the form and composition of the debris, which was independent of the sliding velocity and temperature. The work carried out by Fouvry and his group in Lyon (Fouvry et al., 2003) showed limitations of the Archard law for quantifying wear under fretting. The interfacial shear work was correlated with wear, and the approach was applied to study a variety of materials including different steels and hard coatings under reciprocating sliding conditions. By identifying wear energy coefficients for those materials, the wear resistance can be established. Shipway and coworkers looked at different aspects of fretting wear debris formation, including the development of predictive methods (Everitt et al., 2009), integrating a macrowear simulation tool and a microasperity model in a multiscale modeling (Ding et al., 2009), and the role of surface microtexturing in the acceleration of initial running-in during lubricated fretting ( Jibiki et al., 2010). By the mid-2000s, a few main approaches to fretting strength prediction had been identified. The recent research carried out by the “fretting community” was based on the following techniques: (i) linear elastic fracture mechanics (LEFM)-based approaches (see Arau´jo and Nowell, 1999; Nicholas et al., 2003; Conner et al., 2004); (ii) multiaxial initiation parameters (see Arau´jo and Nowell, 2001; Szolwinski and Farris, 1998; Neu et al., 2000); (iii) macro-micromechanics (see Fouvry et al., 2000; Arau´jo and Nowell, 2002); (iv) crystallographic plasticity (see Goh et al., 2003); and (v) asymptotics (see Mugadu et al., 2002a; Sackfield et al., 2003) A better understanding and improvement of the LEFM-based methods (i) (with particular application to the experimental data obtained during the research project) and the development of an asymptotic methodology (v) both to be applied to crack nucleation prediction in fretting fatigue have been pursued at the beginning of the millennium by Dini and coworkers (Dini, 2004; Dini and Nowell, 2004; Dini and Hills, 2004; Dini et al., 2005, 2006; Hills and Dini, 2006, 2016; Nowell et al., 2006; Hills et al., 2012, 2013; Flicek et al., 2013, 2015). While the above brief overview of the early developments in the field of fretting provides important historical information about this thriving research area, a plethora of developments have also been made in the past two decades, many of which will be reviewed in the following chapters and subchapters of the book.

Acknowledgments We gratefully acknowledge input to this chapter and pay our gratitude and our respects to our colleague David W. Hoeppner, who passed away on February 18, 2022. David was a long-time faculty member and former chair of the University of Utah’s Department of Mechanical Engineering. David worked on and made substantial contributions to experimental and analytical modeling of the behavior of structural materials in fatigue, corrosion, and corrosion fatigue, and was one of the key contributors to research on fretting fatigue. He founded FASIDE (Fatigue and Structural Integrity Design Engineering), the HOLSIP (Holistic Structural Integrity Process) Conference, and the Quality and Integrity Design Engineering Center at the University of Utah. David was also one of the main contributors and organizer of several NATO-AGARD and ASTM symposia on fretting. A successful scientist, whose academic endeavors led to the enhancement of industrial practice, he supervised hundreds of undergraduate, 130 masters, and 48 PhD students. David will be greatly missed as a colleague who contributed generously to the International Symposium on Fretting Fatigue (ISFF) community.

Brief history of the subject

15

References Alic, J.A., Hawley, A.L., 1979. On the early growth of fretting fatigue crack. Wear 56, 377–389. Alic, J.A., Hawley, A.L., Urey, J.M., 1979. Formation of fretting fatigue cracks in 7075-T7351 aluminum alloy. Wear 56, 351–361. Almen, J.O., 1937. Lubricants and false brinelling of ball and roller bearings. Mech. Eng. 59, 415. Almen, O.J., 1948. Fretting corrosion. In: Uhlig, H.H. (Ed.), Corrosion Handbook. John Wiley and Sons, Inc, pp. 590–597. Arau´jo, J.A., 2000. On the Initiation and Arrest of Fretting Fatigue Cracks. D. Phil. Thesis, University of Oxford. Arau´jo, J.A., Nowell, D., 1999. Analysis of pad size effects in fretting fatigue using short crack arrest methodologies. Int. J. Fatigue 21, 947–956. Arau´jo, J.A., Nowell, D., 2001. Prediction of fretting fatigue life using averaged multiaxial initiation parameters. In: Modern Practice in Stress and Vibration Analysis. IOP Publishing, pp. 223–234. Arau´jo, J.A., Nowell, D., 2002. The effect of rapidly varying contact stress fields on fretting fatigue. Int. J. Fatigue 24 (4), 763–775. Archard, J.F., 1953. Contact and rubbing of flat surfaces. J. Appl. Phys. 24 (8), 981–988. Archard, J.F., Hirst, W., 1956. The wear of metals under unlubricated conditions. Proc. R. Soc. A 236 (1206), 397–410. ASTM, 1952. In: ASTM (Ed.), Symposium on Fretting Corrosion. T. DeVilliers, Philadelphia. STP 144-EB. ASTM, 1982. Materials Evaluation under Fretting Conditions. ASTM, Philadelphia. STP 780. ASTM, 1992. In: Attia, M.H., Waterhouse, R.B. (Eds.), Standardization of Fretting Fatigue Test Methods and Equipment. ASTM, Philadelphia. STP 1159. Attia, M.H., 1989. Fretting Fatigue Testing: Current Practice and Future Prospects for Standardization. ASTM Standard News, pp. 26–31. 17, June 1989. Attia, M.H., D’Silva, N.S., 1985. Effect of mode of motion and process parameters on the prediction of temperature rise in fretting wear. Wear 106, 206–224. Attia, M.H., Ko, P.L., 1986. On the thermal aspects of fretting wear-temperature measurements in the subsurface layer. Wear 111, 363–376. Ballard, P., Dang Van, K., Deperrois, A., Papadopoulos, Y.V., 1995. High cycle fatigue and a finite element analysis. Fatigue Fract. Eng. Mater. Struct. 18 (3), 397–411. Barrois, W.G., 1970. Manual on the fatigue of structures—fundamental and physical aspects. NATO-AGARD Manual No 8, In: NATO, AGARD Symposium. Barrois, W.G., 1975. Manual on the fatigue of structures—II. Causes and prevention of structural damage 6. In: Fretting—Corrosion damage in Aluminum alloys, NATO, AGARD Symposium, Manual No 9. Beard, J., 1988. The avoidance of fretting. Mater. Des. 9 (4), 220–227. Berthier, Y., Flamand, L., Godet, M., Schmuck, J., Vincent, L., 1988a. Tribological behavior of titanium alloy Ti-6Al-4V. In: Sixth World Conference on Titanium, France, 6-9 June, 1988. Berthier, Y., Colombie, C., Vincent, L., Godet, M., 1988b. Fretting wear mechanisms and their effects on fretting fatigue. J. Tribol. 110, 517–524. Berthier, Y., Godet, M., Vincent, L., 1988c. Fretting wear and fatigue: initiations, mechanisms, and prevention. Mec. Mater. Electr., 20–26.

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Brief history of the subject

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Sato, K., Fujii, H., Kodama, S., 1986d. Effects of stress ratio fretting fatigue cycles on the accumulation of fretting fatigue damage to carbon steel S45C. Bull. JSME 29, 2759. Shaw, M.C., 1977. Dimensional analysis for wear systems. Wear 43, 263–266. Slavik, D.C., McClain, R.D., Farris, T.N., Murthy, H., 2001. Fatigue crack initiation modelling for applications with stress gradients in smooth and notched geometries. In: Proc. 6th Annual HCF Conference, Jacksonville, FL. Smith, R.N., Watson, P., Topper, T.H., 1970. A stress strain function for the fatigue of metals. J. Mater. 5 (4), 767–778. Sproles, E.S.J.R., Duquette, D.J., 1978. The mechanism of material removal in fretting. Wear 49, 339–352. Szolwinski, M.P., Farris, T.N., 1996. Mechanics of fretting fatigue crack formation. Wear 198, 93–107. Szolwinski, M.P., Farris, T.N., 1998. Observation, analysis and prediction of fretting fatigue in 2024-T351 aluminum alloy. Wear 221 (1), 24–36. Tanaka, K., Mutoh, Y., Sakoda, S., Leadbeater, G., 1985. Fretting fatigue in 0.55C spring steel and 0.45C carbon steel. Fatigue Fract. Eng. Mater. Struct. 8, 129–142. Taylor, D.E., Waterhouse, R.B., 1972. Sprayed molybdenum coatings as a protection against fretting fatigue. Wear 20, 401. Tomlinson, G.A., 1927. The rusting of steel surfaces in contact. Proc. R. Soc. Lond. Ser. A 115, 472–483. Tomlinson, G.A., Thorpe, P.L., Gough, H.J., 1939. An investigation of fretting corrosion of closely fitting surfaces. Proc. Inst. Mech. Eng. 141, 223. Uhlig, H.H., 1954. Mechanism of fretting corrosion. J. Appl. Mech. 21, 401–407. Uhlig, H.H., Tierney, W.D., McClellan, A., 1953a. Test equipment for evaluating fretting corrosion. In: Symposium on Fretting Corrosion, STP 144. American Society for Testing and Materials, Philadelphia, pp. 71–81. Uhlig, H.H., Feng, I.-M., et al., 1953b. Fundamental Investigation of Fretting Corrosion. NACA Technical Note 3029. Vingsbo, O., Soderberg, S., 1987. On fretting maps. In: Conference: Wear of Materials 1987, Texas, USA. ASME, NY, pp. 885–894. Publ. 1987. Vingsbo, O., Soderberg, S., 1988. On fretting maps. Wear 126, 131–147. Warlow-Davies, E.J., 1941. Fretting corrosion and fatigue strength: brief results of preliminary experiments. Proc. Inst. Mech. Eng. 146, 33–38. Waterhouse, R.B., 1955. Fretting corrosion. Proc. Inst. Mech. Eng. 169, 1159–1172. Waterhouse, R.B., 1961. Influence of local temperature increases on the fretting corrosion of mild steel. J. Iron Steel Inst. 197, 301–305. Waterhouse, R.B., 1972. Fretting Corrosion. Pergamon Press, New York. Waterhouse, R.B., 1977. The role of adhesion and delamination in the fretting wear of metallic materials. In: Wear of Materials 1977, The International Conference on Wear of Materials, Missouri. ASME, p. 55. Waterhouse, R.B., 1978. Effect of environment in wear processes and the mechanisms of fretting wear. In: Suh, N.P., Saka, N. (Eds.), Fundamentals of Tribology. MIT Press, London, pp. 567–584. Waterhouse, R.B., 1981. In: Waterhouse, R.B. (Ed.), Fretting Fatigue. Applied Science Publishers, Ltd., Essex, England. Waterhouse, R.B., 1984. Fretting wear. Wear 100, 107–118. Waterhouse, R.B., 1992. Fretting wear. In: ASM Handbook, Volume 18, Friction, Lubrication, and Wear Technology. ASM International, pp. 242–256.

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Fretting Wear and Fretting Fatigue

Waterhouse, R.B., Allery, M., 1965. The effect of non metallic coatings on the fretting corrosion of mild steel. Wear 8, 112–120. Waterhouse, R.B., Dutta, M.K., 1973. The fretting fatigue of titanium and some titanium alloys in a corrosive environment. Wear, 171–175. Waterhouse, R.B., Iwabuchi, A., 1985. The composition and properties of surface films formed during the high temperature fretting of titanium alloy. In: Proceedings of the International Tribology Conference, Tokyo, Japan, July 1985. Japan Society of Lubrication Engineers, Tokyo, Japan, pp. 53–58. Publ. 1985. Waterhouse, R.B., Saunders, 1979. The effect of shot peening on the fretting fatigue behavior of an austenitic stainless steel and a mild steel. Wear 53, 381–386. 1. Waterhouse, R.B., Taylor, D.E., 1971. The initiation of fatigue cracks in a 0.7% carbon steel by fretting. Wear 17, 139–147. Waterhouse, R.B., Taylor, D.E., 1972. Fretting debris and delamination theory of wear. Wear 29, 337–344. Waterhouse, R.B., Brook, P.A., Lee, G.M., 1962. The effect of electro deposited metals on the fatigue behavior of mild steel under conditions of fretting corrosion. Wear 5, 235. Waterhouse, R.B., Dutta, M.K., Swallow, P.J., 1971. Fretting fatigue in corrosion environments. In: Proceedings of the Conference on the Mechanical Behavior of Solids, Kyoto, Japan, pp. 294–300. Wharton, M.H., Waterhouse, R.B., 1979. Environmental effects in fretting fatigue of Ti-6Al4V. Wear, 287–297. Wharton, M.H., Waterhouse, R.B., Hirakawa, K., Nishioka, K., 1973a. The effect of different contact materials on the fretting fatigue strength of an aluminum alloy. Wear 26, 253–260. Wharton, M.H., Taylor, D.E., Waterhouse, R.B., 1973b. Metallurgical factors in the fretting fatigue behavior of 70/30 brass and 0.7 carbon steel. Wear 26, 251. Wright, K.H.R., 1952a. An investigation on fretting corrosion. Proc. Inst. Mech. Eng. 1B, 556– 571. Wright, K.H.R., 1952b. Discussion and communication on an investigation of fretting corrosion. Proc. Inst. Mech Eng. 1B, 571–574. Wright, T., O’Connor, J.J., 1972. The influence of fretting and geometric stress concentrations on the fatigue strength of clamped joints. Proc. Inst. Mech. Eng. 186, 827.

Introduction to fretting fundamentals

2.1

2

Fretting—complexities and synergies Tomasz Liskiewicza and Daniele Dinib a

Manchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom, bImperial College London, Faculty of Engineering, Department of Mechanical Engineering, London, United Kingdom

2.1.1

Fretting within a wider context of tribology

Fretting is considered as a specific type of surface damage resulting from cyclic stress between two nominally stationary components. It is important to situate fretting damage within a wider context of five basic wear mechanisms, namely (i) adhesive wear, (ii) abrasive wear, (iii) fatigue wear, (iv) corrosive wear, and (v) erosive wear. Tribological surface damage process generally involves more than one wear mechanism taking place at the same time, and these mechanisms can interact with synergistic effects. Hence, fretting is sometimes wrongly identified as a wear mechanism, rather than a phenomenon resulting from a synergy between existing basic wear mechanisms, i.e., adhesive wear, corrosive wear, and abrasive wear. Depending on the contact conditions, fretting causes damage by surface wear or surface fatigue. Fretting wear is a response of the tribo-system to the global overstraining of the surface, while fretting fatigue is a response of the tribo-system to the local overstressing. These two mechanisms are usually differentiated and considered separately by practicing engineers, however under certain conditions they can occur simultaneously. Cracking can be observed under fretting wear regime, while debris formation resulting in wear was reported in fretting fatigue configurations (Waterhouse, 1981; Zhou et al., 1992). Among these two main fretting regimes, fretting corrosion has been sometimes mentioned as a synonym of fretting. Nowadays, however, corrosion is typically recognized as a secondary process in relation to fretting resulting from synergistic effects between the mechanical and the oxidative damage. In such cases, surface degradation

Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00002-0 Copyright © 2023 Elsevier Inc. All rights reserved.

26

Fretting Wear and Fretting Fatigue

is accelerated by stresses generated by interacting solids, which increase the chemical reactivity of the surfaces. Fretting occurs in many assemblies where vibrations are likely to happen: rolling bearings, suspension cables, dovetail joints in turbine engines, rivet and screw joints, electrical connectors, heat exchangers, taper joints of hip implants, just to name a few examples. Fretting can be detrimental to all modes of transport, including the rail vehicles (e.g., contact between wheel and axis), aircrafts (e.g., clamping of the turbine blades) and automotive vehicles (e.g., loaded elements of immovable gear-box). Fretting is indeed a complex process and not all kinetics and mechanisms are well established. For example, in 1981, Collins defined more than 50 factors that influence fretting, as a result it is challenging to interpret complexities and synergies (Collins, 1981). As with other aspects of tribology and contact mechanics, fretting research benefits from improved capability of surface analysis and characterization techniques, novel numerical models, and sensor technologies.

2.1.2

Fretting wear

Fretting wear corresponds to the interfacial shear work resulting from the relative motion between interacting bodies, and is related to the debris formation and loss of matter. Under fretting wear, the interface firstly goes through plastic deformation followed by wear debris formation and detachment from the bulk material. In metallic materials, fretting wear process is usually accelerated by oxidation, resulting in synergy between the mechanical and chemical processes (Hurricks, 1970; Waterhouse, 1984; Aldham et al., 1985; Feng and Rightmire, 1956; Paulin et al., 2005). During tribological wear process, the crystallographic structure, physical properties, and chemical properties of the surface layer within the contact area are subjected to dynamic changes. Surface degradation is associated with the quantity of debris ejected outside the contact area, while debris remaining in the contact participate in the process of load transmission and protect indirectly the first-bodies against degradation (Godet, 1984). The third-body can be introduced to the interface voluntary (e.g., solid lubricant, grease, oil), or can result from the wear process of the firstbodies, leading to debris screen being maintained within the contact area. Under fretting wear regime, the extensive plastic shear leads to the contact surface deformation and formation of tribofilms, so that the modified top surface layer is usually described as a tribologically transformed structure (TTS) (Fouvry et al., 2001). The transformed layer corresponds to the intermediate stage between a material of the first-body and debris of the third-body. E. Sauger proposed a two-stage wear mechanism (Sauger et al., 2000): (i) accumulation of plastic deformation (without wear and TTS formation) and (ii) rapid formation of TTS leading to the generation of wear debris. Zhou et al. investigated the formation of the TTS layer during fretting wear under different conditions, including slip amplitude, normal load, frequency and number of cycles (Zhou et al., 1997). It was found that the TTS nucleated relatively quickly, and after 100 cycles a TTS layer of thickness 100 μm thickness was observed. To analyze the evolution of the TTS layer quantitatively, the dissipated energy

Introduction to fretting fundamentals

27

approach and fretting maps method were proposed (Sauger et al., 2000). It was also found that the increase of hardness during fretting can be related to the strainhardening in metals (Liskiewicz et al., 2017). Strain-hardening occurs during plastic deformation, which is induced by the dislocation and twinning of metal crystals (Liu et al., 2015; Gupta et al., 2016). The strain-hardening process is related to the stacking fault energy that varies for different materials (Peng et al., 2013), which can result in different TTS properties (Sauger et al., 2000). The through-thickness mechanical properties of the TTS layer have been investigated by nanoindentation of cross sections (Everitt et al., 2009). It was shown experimentally that TTS forms very quickly within the initial fretting cycles, and its mechanical properties remain almost constant during the entire test duration (Liu et al., 2019). Accurate prediction of fretting wear rates and its mechanism is challenging, due to a number of specific factors such as (Vingsbo and S€oderberg, 1988; Jin and Mall, 2004; McColl et al., 2004): (i) high-frequency modification of the interface and contact geometry during the wear process, (ii) occurrence of the third-body within the contact area, (iii) flow of debris, and (iv) mechanisms of material transport between bodies that are in contact. The classical Archard model is still probably the most used way for quickly assessing the wear performance of materials, including fretting wear (Archard, 1953; Archard and Hirst, 1956). In Archard’s model, wear is a function of hardness of the material, sliding distance and the applied normal load. Other approaches to quantify fretting wear include the use of dissipated energy (Fouvry et al., 2004), hardness to elastic modulus (H/E) ratio (Leyland and Matthews, 2000; Musil et al., 2002), and yield strength to elastic modulus ratio (Liskiewicz et al., 2013).

2.1.3

Fretting fatigue

Fretting fatigue occurs as a result of cyclic stresses and is related to the crack nucleation and propagation within or at the edge of the contact area, which can lead to a catastrophic damage if the crack propagates. As reported by Waterhouse in 1981, the critical relative displacement amplitude for fretting crack propagation is from 10 to 25 μm (Waterhouse, 1981). With further increase of displacement, degradation process moves toward debris formation. Particles’ detachment eliminates the microcracks before they begin to propagate. Fretting fatigue is often observed in tribological contacts like splines, shaft keys, or suspension ropes. Fretting fatigue cracks are typically classified into two categories, i.e., short cracks with a length of some tens of microns, and the long ones that propagated further. For high contact pressures and small displacement amplitudes crack is expected to nucleate at the contact boundary, and with increasing displacement amplitude multicracks are observed within the contact area. In 1961, Forsyth proposed two stages of cracking (Forsyth, 1961): l

Stage I: Crack initiates with the orientation of 45° to the surface with the direction corresponding to the maximum shearing plane. The length of the crack varies and depends on the tribo-system, however the material properties are a dominating factor.

28 l

Fretting Wear and Fretting Fatigue

Stage II: The crack propagation is controlled by the principal stresses traction and point of cracking. Crack initially oriented at 45°, propagates perpendicularly to the surface.

Crack nucleation can be predicted by one of the multiaxial fatigue criteria: l

l

l

l

Dang Van fatigue approach: the crack-nucleation process is a function of two macroscopic stresses: shear and hydrostatic pressure. If these two variables reach the limit value, there is a high risk of cracking (Dang Van, 1993). Crossland fatigue description: the macroscopic loading path is considered and cracknucleation risk is expressed as a combination of stress and hydrostatic pressure (Crossland, 1956). McDiarmid fatigue formulation: identifies the maximal shear stress amplitude and indicates the critical plane where the crack is expected (McDiarmid, 1991). Smith-Watson-Topper fatigue criterion: crack will initiate in shear and grow on a certain plane perpendicular to the principal stress (Smith et al., 1970).

Fretting fatigue process is controlled by cyclic stresses, which leads to surface degradation by cracking. In pure fatigue, the lifetime of uniaxially tensioned sample is often described by W€ ohler fatigue curves, representing relation between maximum amplitude of cyclic stresses and the number of cycles to sample failure. This approach allows to establish the endurance limit of materials, i.e., the maximal tension value before the sample failure. The approach can be also applied to fretting fatigue studies by adding the loaded pads to the classical fatigue test configuration. This has an effect on W€ohler curve position and the influence of fretting can be expressed by the decrease of endurance limit, which was reported by, e.g., McDowell (McDowell, 1953), Endo (Endo, 1976), Lindley and Nix (Lindley and Nix, 1985), Hoeppner (Hoeppner, 1994), Del Puglia et al. (Del Puglia et al., 1994), and (Sato and Kodoma, 1994).

2.1.4

Mitigating fretting damage

Although tribological damage is a complex interaction of different mechanisms, a number of common factors have emerged as being very influential in determining the rate of wear. These include load, speed, temperature, oxides and contaminant surface films, compatibility of surface materials, surface treatments and coatings, and lubrication. Practical strategies can be implemented to reduce wear, e.g., select materials for the surfaces with low adhesion tendency; promote effective lubrication, control relative hardness of the two surfaces; make harder surface smooth, thereby minimizing the number of hard asperities capable of ploughing; apply filtration to remove hard particles; select materials with high fatigue resistance; reduce stress levels and cycling; increase lubricant film thickness; modify the corrosive environment; and/or seal the surfaces from the corrosive environment. When possible, protection against fretting damage should be considered at the design stage. By adjusting the contact geometry, the concentration of stresses and consequently fatigue cracking and wear debris generation can be normally avoided. Nevertheless, due to design constrains it is difficult to exclude fretting completely as fretting is observed even under amplitude as low as 1 μm (Kennedy et al., 1983).

Introduction to fretting fundamentals

29

The following methods have been identified as adequate for increasing fretting resistance at the design stage of the tribosystem (Harris et al., 1985; Labedz, 1988; Waterhouse, 1988; Fu et al., 2000): l

l

l

l

Induce residual compressive stresses; one of the most effective mechanisms for the mitigation of fretting fatigue (shot-peening, nitriding, carburizing, ion implantation, etc.). Decrease the coefficient of friction; higher friction forces lead to higher shear stresses which can promote fatigue failure and delamination (PVD and CVD techniques, solid lubricated coatings, nitriding, carburizing). Increase the surface hardness: increase in surface hardness prevents adhesion and abrasive wear in fretting (nitriding, carburizing, shot-peening, PVD and CVD techniques, ion implantation, laser cladding). Control of surface chemistry: tribo-film formation (oxide, nitride, or carbide layers).

The interim method to decrease fretting damage is employing, where possible, the lubricating techniques. Among promising approaches, application of solid lubricants on susceptible contact areas is considered to avoid catastrophic fatigue failure under pure stick and stick-slip contact regimes (Lin et al., 2021). Several studies have reported that the DLC films can effectively improve the fretting wear and fracture resistance (Shi et al., 2019). To counter both fretting regimes, contradicting requirements of a low shear strength surface (to prevent stick-slip and associated fatigue) and a high shear strength surface (to resistant abrasion) could be provided by duplex surface treatments and coatings to provide a low shear and soft top surface and high shear strength and hard underlying layer (Liu et al., 2018).

References Aldham, D., Warburton, J., Pendlebury, R.E., 1985. The unlubricated fretting wear of mild steel in air. Wear 106, 177–201. Archard, J.F., 1953. Contact and rubbing of flat surfaces. J. Appl. Phys. 24, 981–988. Archard, J.F., Hirst, W., 1956. The wear of metals under unlubricated conditions. Proc. Roy. Soc. Lond. Math. Phys. Sci. 236, 397–410. Collins, J.A., 1981. Failure of Materials in Mechanical Design. J. Wiley, New York. Crossland, B., 1956. Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. In: Proc. of Intr. Conf. on Fatigue of Metals, London, pp. 138–149. Dang Van, K., 1993. Macro-micro approach in high-cycle multiaxial fatigue. In: Advances in Multiaxial Fatigue. ASTM STP 1191, pp. 120–130. Del Puglia, A., Pratesi, F., Zonfrillo, G., 1994. In: Waterhouse, R.B., Lindley, T.C. (Eds.), Experimental Procedure and Involved Parameters in Fretting-Fatigue Test, ESIS 18. Mechanical Engineering Publication, London, pp. 219–238. Endo, K., 1976. Initiation and propagation of fretting-fatigue cracks. Wear 38, 311–324. Everitt, N.M., Ding, J., Bandak, G., Shipway, P.H., Leen, S.B., Williams, E.J., 2009. Characterisation of fretting-induced wear debris for Ti-6Al-4V. Wear 267, 283–291. Feng, I.M., Rightmire, B.G., 1956. An experimental study of fretting. Proc. Inst. Mech. Eng. 170, 1055–1064. Forsyth, P.J.E., 1961. A two stage fatigue fracture mechanisms. In: Proc. Cranfield Symposium on Fatigue, p. 76.

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Fretting Wear and Fretting Fatigue

Fouvry, S., Kapsa, P., Vincent, L., 2001. An elastic-plastic shakedown analysis of fretting wear. Wear 247, 41–54. Fouvry, S., Duo, P., Perruchaut, P., 2004. A quantitative approach of Ti-6Al-4V fretting damage: friction, wear and crack nucleation. Wear 257, 916–929. Fu, Y., Wei, J., Batchelor, A.W., 2000. Some considerations on the mitigation of fretting damage by the application of surface-modification technologies. J. Mater. Process. Technol. 99, 231–245. Godet, M., 1984. The third-body approach: a mechanical view of wear. Wear 100, 437–452. Gupta, R.K., Kumar, V.A., Mathew, C., Rao, G.S., 2016. Strain hardening of titanium alloy Ti6Al-4V sheets with prior heat treatment and cold working. Mater. Sci. Eng. A 662, 537–550. Harris, S.J., Overs, M.P., Gould, A.J., 1985. The use of coatings to control fretting wear at ambient and elevated temperatures. Wear 106, 35–52. Hoeppner, D.W., 1994. In: Waterhouse, R.B., Lindley, T.C. (Eds.), Mechanisms of Fretting Fatigue, ESIS 18. Mechanical Engineering Publication, London, pp. 3–19. Hurricks, P.L., 1970. The mechanism of fretting—a review. Wear 15, 389–409. Jin, O., Mall, S., 2004. Effects of slip on fretting behavior: experiments and analyses. Wear 256 (7–8), 671–684. Kennedy, P.J., Stallings, L., Petersen, M.B., 1983. A study of surface damage at low amplitude slip. In: Proc. of Conf. “ASLE-ASME Lubrication Conf.”, Hartford. Labedz, J., 1988. Metal Treatments Against Wear, Corrosion, Fretting Fatigue. Pergamon Press, Oxford, pp. 87–98. Leyland, A., Matthews, A., 2000. On the significance of the H/E ratio in wear control: a nanocomposite coating approach to optimised tribological behaviour. Wear 246, 1–11. Lin, M., Nemcova, A., Voevodin, A.A., Korenyi-Both, A., Liskiewicz, T.W., Laugel, N., Matthews, A., Yerokhin, A., 2021. Surface characteristics underpinning fretting wear performance of heavily loaded duplex chameleon/PEO coatings on Al. Tribol. Int. 154, 106723. Lindley, T.C., Nix, K.J., 1985. The role of fretting in the initiation and early growth of fatigue cracks in turbo-generator materials. In: Conference: Multiaxial Fatigue, ASTM, Philadelphia, USA, pp. 340–360. Liskiewicz, T., Kubiak, K., Comyn, T., 2017. Nanoindentation mapping of fretting-induced surface layers. Tribol. Int. 108, 186–193. Liskiewicz, T.W., Beake, B.D., Schwarzer, N., Davies, M.I., 2013. Short note on improved integration of mechanical testing in predictive wear models. Surf. Coating. Technol. 237, 212–218. Liu, F., Dan, W.J., Zhang, W.G., 2015. Strain hardening model of twinning induced plasticity steel at different temperatures. Mater. Des. 65, 737–742. Liu, Y., Liskiewicz, T., Beake, B.D., 2019. Dynamic changes of mechanical properties induced by friction in the Archard wear model. Wear 428–429, 366–375. Liu, Y.-F., Liskiewicz, T., Yerokhin, A., Korenyi-Both, A., Zabinski, J., Lin, M., Matthews, A., Voevodin, A.A., 2018. Fretting wear behavior of duplex PEO/chameleon coating on Al alloy. Surf. Coat. Technol. 352, 238–246. McColl, I.R., Ding, J., Leen, S.B., 2004. Finite element simulation and experimental validation of fretting wear. Wear 256 (11–12), 1114–1127. McDiarmid, D.L., 1991. A general criterion for high-cycle multiaxial fatigue failure. Fatigue Fract. Eng. Mater. Struct. 14, 429–453. McDowell, O.J., 1953. Fretting corrosion tendencies of several combinations of materials. In: Symposium on Fretting Corrosion, ASTM, Philadelphia, USA, pp. 40–53.

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Musil, J., Kunc, F., Zeman, H., Polakova, H., 2002. Relationships between hardness, Young’s modulus and elastic recovery in hard nanocomposite coatings. Surf. Coat. Technol. 154, 304–313. Paulin, C., Fouvry, S., Deyber, S., 2005. Wear kinetics of Ti-6Al-4V under constant and variable fretting sliding conditions. Wear 259, 292–299. Peng, X., Zhu, D., Hu, Z., Yi, W., Liu, H., Wang, M., 2013. Stacking fault energy and tensile deformation behavior of high-carbon twinning-induced plasticity steels: effect of Cu addition. Mater. Des. 45, 518–523. Sato, K., Kodoma, S., 1994. In: Waterhouse, R.B., Lindley, T.C. (Eds.), Effect of TiN Coating by the CVD and PVD Processes on Fretting Fatigue Characteristics in Steel, ESIS 18. Mechanical Engineering Publication, London, pp. 513–526. Sauger, E., Fouvry, S., Ponsonnet, L., Kapsa, P., Martin, J.M., Vincent, L., 2000. Tribologically transformed structure in fretting. Wear 245, 39–52. Shi, X., Liskiewicz, T.W., Beake, B.D., Sun, Z., Chen, J., 2019. Fretting wear behavior of graphite-like carbon films with bias-graded deposition. Appl. Surf. Sci. 494, 929–940. Smith, K.N., Watson, P., Topper, T.H., 1970. A stress-strain function for the fatigue of metals. J. Mater. 5 (4), 767–778. Vingsbo, O., S€oderberg, S., 1988. On fretting maps. Wear 126 (2), 131–147. Waterhouse, R.B., 1981. Fretting Fatigue. Applied Science, London. Waterhouse, R.B., 1984. Fretting wear. Wear 100, 107–118. Waterhouse, R.B., 1988. Metal Treatments Against Wear, Corrosion, Fretting Fatigue. Pergamon Press, Oxford, pp. 31–40. Zhou, Z.R., Fayeulle, S., Vincent, L., 1992. Cracking behaviour of various aluminium alloys during fretting wear. Wear 155, 317–330. Zhou, Z.R., Sauger, E., Liu, J.J., Vincent, L., 1997. Nucleation and early growth of tribologically transformed structure (TTS) induced by fretting. Wear 212, 50–58.

2.2

Contact mechanics in fretting Daniele Dinia and Tomasz Liskiewiczb a

Imperial College London, Faculty of Engineering, Department of Mechanical Engineering, London, United Kingdom, bManchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom

Many real fretting problems, including those encountered in bolted joints, splined shafts, biomedical implants, etc., involve contact of components of varying geometric complexity. These problems usually require advanced numerical methods (with the finite element method being often the favored technique) to evaluate the state of stress and the relative displacement experienced by the interacting bodies. However, it is very difficult to address fundamental questions and perform analyses and tests on actual components as this will require laborious efforts in obtaining accurate

32

Fretting Wear and Fretting Fatigue

information about local stick-slip zone boundaries and stress and strain fields in the areas subjected to fretting damage. For this reason, researchers prefer to use test and conduct analyses and theoretical investigations on defined idealized geometries, so that the nature of the contact, the induced stress, and displacement fields are well defined, easily controllable, repeatable, and insensitive to manufacturing imperfections. This has led to many investigations, with a number of authors and researchers studying simplified geometries in greater depth (Gladwell, 1980; Hills and Nowell, 1994; Hills and Andresen, 2021). In this chapter, we briefly review the fundamental aspects needed to study the contact mechanics of such simplified geometries and other more advanced techniques before turning our attention to the classification of contact regimes for fretting and hysteresis loops defining the energy dissipation and state of the contact interface.

2.2.1

Contact geometry

In order to simplify the real complex tribosystems that are present in mechanical devices and engineering systems of interest, geometries that are easier to treat and control have been proposed to perform fundamental investigations. There are three main contact representations studied in the laboratories: l l l

plane/plane, cylinder/plane, sphere/plane.

The plane/plane configuration allows investigating the wear process on the widespread surface and is easy to perform, there is a discontinuity in the pressure distribution on the border of the contact. That influence makes the mechanical analysis of the interacting bodies difficult. To solve this problem linear and point contact geometries were proposed where the border effect is eliminated and the maximum pressure values are present along the axis of the cylinder and at the center of the contact with sphere. State of stresses and elastic deformations for the cylinder/plane and sphere/plane configurations can be determined analytically by theory of Hertz (1881). More complex and deeper investigations of contact mechanics were carried out by Johnson (1985) and Hills et al. (1993). It is very useful also to introduce a comprehensive classification of contacts (Hills and Andresen, 2021), which in itself is capable of providing some useful information about the nature of the contact. Cases in which the size of the contact is determined by an abrupt change in the front-face profile of the contact-defining body are called complete contacts and are exemplified by the fundamental problem of an elastic block pressed into an elastically similar half-space shown also in Fig. 2.2.1 (plane/plane). These are not the kinds of contact usually met with first in textbooks on contact mechanics, e.g., Johnson (1985), Hills et al. (1993), and Barber (2018), because they are not capable of analysis, at least completely, in a closed form. A class of contacts that has features both complete and incomplete contacts, called almost complete, has also emerged recently as a very interesting problem to be solved in the context of applications whereby complete contacts are characterized by finite radii at their edges,

Introduction to fretting fundamentals

plane / plane

33

cylinder / plane

sphere / plane

Fig. 2.2.1 Three fretting contact configurations (Liskiewicz, 2004).

which makes the problem somewhat easier to treat (Ciavarella, 1998; J€ager, 1998; Dini, 2004; Fleury et al., 2016). By contrast, looking at the cylinder/plane and sphere/plane configurations shown in Fig. 2.2.1, when the two bodies are brought together the contact is made along a line (cylinder) or point (sphere) and, as the load is increased, it will broaden to form a strip (cylinder) or a circle (sphere). These are example of incomplete contacts, whose extents are not fixed by the geometry but depend on the applied load. A second property of such contacts is that, at least locally near the contact edge, the contacting bodies behave like half-spaces, and mathematical treatments can be used to determine the state of stress and displacements in the neighborhoods of the contact, where fretting damage is likely to occur, and fretting wear and fatigue analyses need to be performed. A second kind of classification often used is one which concerns conformality. Looking at the cylinder/plane configuration shown in Fig. 2.2.1, under a relatively light load the contact extension will be much less than the characteristic radius of the cylinder. This means that we can approximate the cylinder as a half-space when evaluating the deformation and stresses as the contact is nonconformal. Other classes of incomplete contacts, for example, are formed by pressing a cylinder within a hole (e.g., journal bearings), see portion of contacts whose characteristic dimension is a significant fraction of the radius of the bodies being pressed together; these are known as conformal contacts. There are also cases, such as a strip pressed by a localized load on a surface, in which the application of the load will cause the outer edges to lift and the contact area to decrease; these are called receding contacts. Each of the above problems in their simplified form can be treated using analytical, semianalytical, or numerical methods; solutions exist that allow to determine contact pressure and the local state of stress induced by the application of normal and tangential load. For a more in-depth analyses of the above simplified set of contact problems in the context of fretting, the reader should refer to Hills and Nowell (1994) and Hills and Andresen (2021).

2.2.2

Friction and fretting regimes

The law of friction most commonly used to study fretting contacts is that associated with the names of Amontons and Coulomb, which remains quite acceptable for most engineering purposes, but probably not on the micro- or nanoscales. The macroscopically determined ratio between the shear force, Q, and the normal force, P, on a sliding block is given the symbol f (or μ), the coefficient of friction. It is apparently independent of the shape of the block, and the value of f is substantially independent of the normal load. Particles on the surfaces of two components pressed together may be in one of the two states: they may be stuck together or they may be

34

Fretting Wear and Fretting Fatigue

slipping. A contact where at least some of the interface is stuck cannot be sliding. Most fretting problems are concerned with the study of contacts where at least part of the contact is slipping for some of the time, see, e.g., Figs. 2.2.2 and 2.2.3. By defining point-wise in a contact the shear tractions, q(x,y), and contact pressure, p(x,y), at a point characterized by planar coordinates x,y in the plane in which the contact is established, in a region of contact that is slipping, the ratio of shear traction to normal pressure jq(x,y)j/p(x,y) is equal to the coefficient of friction, f, i.e., the point-wise value and the global value are assumed to be the same. When a contact is in partial slip, there are two sets of conditions which hold in the slip and stick regions of contact: l

Stick region—here, the local shear traction must be lower in magnitude than the contact pressure multiplied by the coefficient of friction, i.e., jq(x,y)j < f p(x,y), and the relative slip displacement of pairs of contacting points is preserved at its current value.

force normal : P

unexposed surface

2δg

fretted surface

e=1 fretted surface

a fretting regime e1

0 0

displacement amplitude : δ

Fig. 2.2.2 Definition of sliding ratio “e” and identification of transition between fretting regime and reciprocating sliding (contact sphere/plane) (Liskiewicz, 2004, after Fouvry et al., 1996).

2a 2c

P

Q

sliding zone

stick zone

Fig. 2.2.3 Sphere on plane contact with a constant normal force (P) and a variable tangential force (Q) (Liskiewicz, 2004).

Introduction to fretting fundamentals l

35

Slip region—the magnitude of the shear traction is limited to the product of the coefficient of friction and the contact pressure, so that q(x,y) ¼  fp(x,y) and the choice of sign must be consistent with the change in slip of surface points, so that it opposes the motion.

The presence of friction in a solid mechanics problem, rather like plasticity, makes the problem inherently nonlinear, even if it was linear in the first place (doubling the load on the contacting cylinder shown in Fig. 2.2.1, for example, does not double the contact pressure and, hence, does not double the magnitude of the state of stress). The state of the contact interface, that is, the distribution of regions of stick and slip depend not just on where the problem lies in, say, PQ space, but also how that point was reached. One of the most important distinction between fretting processes is defined by the fretting “regime,” which characterize a contact problem in fretting based on whether part of the contacts are still adhering (partial slip) or the entire contact is sliding—this is an extremely important distinction as it generally identified if fretting fatigue (more likely damage mechanism in partial slip) could dominate as a damage mechanism as compared to wear (more likely damage mechanism in full sliding). Bibliography sources quote different values of displacement amplitudes as a border between fretting process and reciprocating sliding motion. This value is variously interpreted and comprises lies between 50 and 300 μm (Halliday and Hirst, 1956; Vaessen et al., 19681969; Ohmae and Tzukizoe, 1974; Gordelier and Chivers, 1979). Although, the engineering activities can lead to the strict limitation of relative displacement between particular elements of mechanical system, it is nearly not possible to eliminate fretting as this kind of matter degradation was reported even with displacement amplitude δ < 1 μm (Gordelieen and Stallings, 1982; Kennedy et al., 1983). Hence, knowing the lower limit of the fretting displacement amplitude, to establish the fretting regime field the top limit of displacement should be determined. It can be clarified by introducing the coefficient e (Fouvry et al., 1996), which is defined as a sliding ratio and is expressed as e ¼ δg =a where δg is the sliding amplitude, which is different from the displacement amplitude due to the contact and testing device compliance, and a is the Hertzian contact radius. The tribosystem remains in the fretting regime (e < 1) when the unexposed surface is maintained at the center of the fretted surface and transits toward reciprocating sliding condition (e > 1) when all the plane surface is exposed to the atmosphere. When considering the fretting process the state of stresses and elastic deformations as well as friction forces should be taken into account. For a small value of tangential force, the microdisplacements between two bodies are observed. Nevertheless, when tangential stresses are higher than the unit-friction forces, stick and sliding zones in the Hertzian contact area can be distinguished as introduced by Cattaneo (1938) and Mindlin (1949) (see, e.g., Hills et al., 1993). For the describing geometry of ball-on-flat surface the circular stick zone at the center of the contact and surrounding annular slip zone can be separated. Increase of the relative tangential force involves change of the amplitude toward higher

36

Fretting Wear and Fretting Fatigue

(a)

(b)

Q

δ

(c)

Q

δ

Q

δ

Fig. 2.2.4 Characteristic experimental fretting cycles: (A) stick regime, (B) stick-slip regime, and (C) gross-slip regime.

displacements, which influences the fretting contact conditions. Dynamic recording of the tangential force and displacements amplitudes during the fretting tests as well as some wear scar observations allowed Vingsbo and Soderberg (1988) to map the following fretting regimes: l

l

l

Stick regime: The interfacial sliding between two bodies is not observable as displacement is accommodated by elastic deformations. Stick domain is maintained by locked asperities that can be plastically shared in the direction of micromovement. Low fretting fatigue damage is observed mainly by crack nucleation and propagation. For this regime evolution of Q ¼ f(δ), called fretting loop, takes form of closed linear relation (Fig. 2.2.4). Stick-slip regime: Even though sliding is present, the stick zone is a dominating area in the contact. Surface degradation is characterized by cracking as a result of contact fatigue particularly close to the stick-slip boundary. If rough surfaces are subjected to contact the stick zone can be spread for a number of single contacts. The Q–δ curve assumes shape of characteristic elliptic hysteresis (Fig. 2.2.4). In the case of sphere/plane contact geometry the  1=3 Q ratio between stick zone radius and contact radius is expressed by ac ¼ 1  μP . Gross-slip regime: With higher displacement amplitude values, the stick zone is not present any more and entire contact area is subjected to sliding. Dominating wear mechanism is debris formation by creating and breaking the adhesive junctions. Wear debris can remain within the contact area as abrasive particles. The maximal tangential force is not dependent on the sliding amplitude and can be described by the classic Amonton’s friction law: Q ¼ fP. The evolution of fretting loop takes more quadratic shape (Fig. 2.2.4).

With successive increasing of sliding amplitude the surface degradation process moves toward reciprocating sliding regime where fretting wear principles are no longer applicable.

References Barber, J.R., 2018. Contact Mechanics, SMIA 250. Springer Cham, Switzerland. Cattaneo, C., 1938. Sul contatto di due corpi elastici: distribuzion local degli sforzi. Reconditi dell’Accademia nazionale dei Lincei 27, 342–348. 434–436, 474–478. Ciavarella, M., 1998. The generalised Cattaneo partial slip plane contact problem, Part I Theory, Part II Examples. Int. J. Solids Structs 35, 2349–2378. Dini, D., 2004. Studies in Fretting Fatigue With Particular Application to Almost Complete Contacts. PhD Thesis, University of Oxford. Fleury, R.M.N., Hills, D.A., Barber, J.R., 2016. A corrective solution for finding the effects of edge-rounding on complete contact between elastically similar bodies. Part I. Contact law and normal contact considerations. Int. J. Solids Struct. 85, 89–96.

Introduction to fretting fundamentals

37

Fouvry, S., Kapsa, P., Vincent, L., 1996. Quantification of fretting damage. Wear 200, 186–205. Gladwell, G.M.L., 1980. Contact Problems in the Classical Theory of Elasticity. Sijthoff and Noordhoff, Dordrecht. Gordelieen, S.C., Stallings, L., 1982. An evaluation of fretting at small slip amplitudes. ASTM Spec. Tech. Publ. 780, 30–48. Gordelier, S.C., Chivers, T.C., 1979. A literature review of palliatives for fretting fatigue. Wear 56, 177–190. Halliday, J.S., Hirst, W., 1956. The fretting corrosion of mild steel. Proc. Roy. Soc. A 236, 411–425. € Hertz, H., 1881. Uber die Ber€uhrung fester elastischerK€ orper. J. Reine Angew. Math. 92, 156–171. Hills, D.A., Andresen, H.N., 2021. Mechanics of Fretting and Fretting Fatigue, SMIA 266. Springer Cham, Switzerland. Hills, D.A., Nowell, D., 1994. Mechanics of Fretting Fatigue. Kluwer Academic Publishers, Dordrecht, The Netherlands. Hills, D.A., Nowell, D., Sackfield, A., 1993. Mechanics of Elastic Contact. ButterworthHeinemann, Oxford, UK. J€ager, J., 1998. A new principle in contact mechanics. ASME J. Tribol. 120 (4), 677–684. Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge. Kennedy, P.J., Stallings, L., Petersen, M.B., 1983. A study of surface damage at low amplitude slip. In: Proc. of Conf. “ASLE-ASME Lubrication Conf.”, Hartford. Liskiewicz, T., 2004. Hard Coatings Durability Under Variable Fretting Wear Conditions. PhD  Thesis, Ecole Centrale de Lyon. Mindlin, R.D., 1949. Compliance of elastic bodies in contact. J. App. Mech. 16, 259–268. Ohmae, N., Tzukizoe, T., 1974. The effect of slip amplitude on fretting. Wear 27, 281–293. Vaessen, G.H.G., Commissaris, C.P.L., De Gee, A.W.J., 1968-1969. Proc. Inst. Mech. Eng. 183, 125. London. Vingsbo, O., Soderberg, S., 1988. On fretting maps. Wear 126, 131–147.

2.3

Transition criteria and mapping approaches Tomasz Liskiewicza, Daniele Dinib, and Yanfei Liuc a

Manchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom, bImperial College London, Faculty of Engineering, Department of Mechanical Engineering, London, United Kingdom, cBeijing Institute of Technology, School of Mechanical Engineering, Beijing, China

2.3.1

Transition criteria

Depending on the working conditions, different fretting regimes associated with wearor fatigue-dominated mechanisms can be achieved. A correct identification of the regime is important in understanding the dominant degradation phenomenon. The correct theory can be applied with robust regime identification for a better understanding

38

Fretting Wear and Fretting Fatigue

of elastic and plastic compliance. The use of parameters to differentiate between regimes allows correlation to other data sets, enabling the innovation of asset monitoring devices. The accurate understanding of transition between fretting regimes based on robust data sets can allow development of online asset monitoring devices, which can lead to prolong service and increase efficiency, supporting the concept of Industry 4.0. This can also allow the prediction of the onset of transition with possible application as early warning systems highlighting the risk of a catastrophic failure. Mindlin’s ratio is a purely theoretical value and based on assumptions that are difficult to qualify. The progression of friction coefficient used to identify regime transition requires observation throughout experiments or using predetermined values of coefficient of friction (Fouvry et al., 1995). However, differentiation between regimes can be achieved using criteria directly derived from the characteristic parameters of fretting loops (Vingsbo and S€ oderberg, 1988; Zhou et al., 2006), as described above. The energy ratio A between the dissipated energy (area bound by the fretting loop) and the total energy (area bound by the smallest possible rectangle containing a fretting loop) was introduced by Mohrbacher et al. (Mohrbacher et al., 1993) as a constant independent of the material properties, the contact geometry, and the coefficient of friction. A ¼ 0.2 was demonstrated to be the transition between partial-slip and gross-slip conditions. The evolution of the aperture of fretting loops expressed as the sliding amplitude to displacement amplitude ratio (D) is an alternative method of identifying regime transition (Sandstrom et al., 1993). This parameter is independent of the contact dimension and the mechanical properties of the tribopair. The transition between the partial-slip and gross-slip conditions takes place at D ¼ 0.26. Fouvry et al. postulated that the determination of the transition between the partialslip and gross-slip regimes (GSRs) would be easier if a parameter independent of the system compliance was introduced (Fouvry et al., 1995). By coupling the energy ratio (A) and sliding ratio (D) criteria, they introduced the system-free (B) criterion independent of the testing device with the transition occurring at B ¼ 0.77. The criterion is not affected by the tangential accommodation of the testing apparatus; however, it is highly dependent on the aperture of the fretting loop variation, and it is relatively difficult to obtain an accurate value for the transition. It was observed further by Hereida and Fouvry that there is a conflicting argument in the literature about whether cracking in the partial-slip regime (PSR) is amplified by local wear mechanisms (Heredia and Fouvry, 2010). To investigate that discrepancy, they introduced a new fretting regime sliding criterion defined as the proportion of gross-slip cycles during the test. It is an expansion of the energy ratio ‘A,’ and it allows to record the history of the fretting sliding imposed at the interface. It was confirmed that wear processes present in mixed fretting are mainly controlled by the debris ejection process and cannot be approximated by gross-slip wear rates. The authors concluded that much simulation research based on the surface profile evolution under partial-slip condition using Archard or energy wear laws should be treated with caution. Measuring and differentiating between slip amplitude and displacement amplitude in practice can be challenging and introduces a certain level of error. Hence,

Introduction to fretting fundamentals

39

Varenberg et al. used the Buckingham-π theorem to define slip ratio as a function of contact stiffness, displacement amplitude, and normal load otherwise known as slip index (Varenberg et al., 2004). Regime transition using the slip index was established by experimentally observing the relationship with the coefficient of friction, providing a transition criterion independent of the system and fretting rig mechanical response. The PSR is characterized by the slip index 0.8. By computing the fretting loop hysteresis, Suciu and Uchida developed a model where ellipse, super-ellipse, and parallelogram can simulate the fretting loops for different regimes (Suciu and Uchida, 2010). They hypothesized that “ellipticity” and rotation of the ellipse can allow regime identification by drawing the relationship between the ellipticity, angle of rotation, and the power parameter of the super-ellipse. Fretting contacts were also investigated using acoustic emission (AE) techniques. Fretting fatigue crack formation was investigated using AE with an applied tensile force to the flat specimen and a fretting motion using a Hertzian contact in the PSR (Meriaux et al., 2010; Wang et al., 2012; Cadario and Alfredsson, 2006). An increased in accumulative energy and number of hits associated with the rapid release of strain upon crack initiation and propagation was observed. However, it was clear that a correlation between mechanical data, wear morphology, and acoustic data was difficult to establish, often only comparing general trends as opposed to specific events in data sets. Fretting wear was investigated by Ito et al. (2009) using a metal-on-metal contact who found that peaks in AE amplitude did not occur in partial slip but occurred in the gross slip when pure slip was achieved, which was associated with the sudden fracture of contacting asperities. This was also seen in another fretting wear study that used a ductile metal-on-metal contact, which found increased hits during the sliding portion of the gross-slip fretting loop at the start of the test (Benı´tez et al., 2016). However, as the experiment progressed ploughing action of the contact resulted in the increased amplitude of the hits at the loop extremities. Effects of the ploughing action was also seen in fretting experiments using ceramic-on-metal contacts, identifying collision with the ends of the wear tracks as a possible source of AE (Merhej et al., 2009). More recently, a study investigated the transition between fretting regimes using the relationship between AE and the mechanical response of a dry, steel-on-steel, ball-on-flat contact under different tangential displacements (Wade et al., 2020a), see Fig. 2.3.1. Increased AE response occurred during gross-slip events and there was strong positive correlation between AE and fretting energy ratio. The relationship was strongest when gross sliding was experienced allowing identification of transition from the partial-slip to the GSR. This makes AE a good candidate to detect regime transition in situ due to ease of integration and its nondestructive nature. From the tribological contact characterization point of view, it is also important to understand when the transition between fretting gross slip and reciprocating sliding takes place. This can be achieved by using the slip index explained above (Varenberg et al., 2004). Varenberg et al. established that reciprocal sliding occurs for slip index > 11, and that fretting takes place for slip index < 10. Other method for determining transition between fretting gross slip and reciprocating sliding includes the sliding ratio e, which captures a relationship between the relative sliding amplitude and the Hertzian contact radius (Fouvry et al., 1996), see Section 2.2.2.

40

Fretting Wear and Fretting Fatigue

Fig. 2.3.1 (A) The experiment fretting loops, (B) energy ratio and AE hit amplitude, and (C) subloop AE hit amplitude over the whole experiment with regions of stick and slip indicated (Wade et al., 2020a).

Transition between fretting and reciprocating sliding can be also associated with changes in wear coefficient, wear volume, coefficient of friction, profile of the scar, and wear debris character before and after the transition point. For steel/steel contact wear coefficient and wear volume were reported as the most appropriate for identifying the transition (Chen and Zhou, 2001).

2.3.2

Mapping approaches

A wear map is a two-dimensional, less often three-dimensional, representation of wear data. With carefully selected parameters such as axes, maps capture the wear behavior of a tribological system. A wear map can provide means to systematically compare

Introduction to fretting fundamentals

41

and select materials on a common basis and can serve as a quick guide to a designer in the selection of suitable operating conditions for tribological components. A wellconstructed wear map defines dominant wear mechanisms and reveals the transition criteria based on contact conditions (mechanical approach) or the wear mechanisms (structural approach) (Rapoport, 1995). The mapping approach has been also used to investigate fretting contacts. It allows to establish the transition between the broad fatigue and wear regimes, and also to predict a general evolution of fretting response under combined experimental conditions. When discussing fretting maps, one should recognize the importance of contact mechanics theory introduced by Mindlin in 1949, see Chapter 2.2. Based on Mindlin’s theory, the concept of fretting map was initially proposed by Vingsbo and S€ oderberg in 1988 (Vingsbo and S€oderberg, 1988). The authors distinguished three fretting regimes and correlated them with varied displacement amplitude, normal and tangential force, and frequency. The three regimes were the PSR, GSR, and mixed fretting regime (MFR), and the corresponding modes of surface damage were identified from posttest metallographic examination (Fig. 2.3.2). Building on Vingsbo and S€ oderberg’s theory, Zhou and Vincent proposed in 1992 two fretting map approaches: running condition fretting map (RCFM) and material response fretting map (MRFM) (Zhou et al., 1992), see Fig. 2.3.3. The RCFM is plotted from the direct analysis of the friction logs, while the MRFM requires more elaborate metallographic analyses including the cross-sectional specimen analysis and microscopic observations. The RCFM shows the fretting condition as a function of normal force and displacement amplitude and captures the fretting regime characteristics of the test throughout its whole duration. The running condition relates here to the local slip behavior of a contact. Other running condition maps like the RCFM can be created for other parameters such as frequency, test length, temperature, etc. The MRFM expands on the running condition mapping approach by showing the failure modes as a function of the normal force and displacement amplitude. The material response modes fall into four categories: (i) no surface degradation—damage too small to be observed, (ii) fatigue cracking—occurs in the PSR under relatively high normal force and low displacement amplitude, (iii) cracking and wearing— Fig. 2.3.2 Fretting map in terms of normal force N vs. displacement amplitude D (Vingsbo and S€ oderberg, 1988).

42

Fretting Wear and Fretting Fatigue

Fig. 2.3.3 Fretting maps, running condition fretting map (RCFM), and material response fretting map (MRFM) (Fouvry and Kapsa, 2001).

corresponds to mixed fretting running condition where the contact experiences competition between the two damage mechanisms, and (iv) wear regime—corresponds to gross slip under high displacement amplitudes where the progressing wear process damages fatigue cracks before they can propagate. Compiling a MRFM can be a time-consuming process as it requires performing a large number of experiments and subsequent metallographic analysis. Nevertheless, the fretting regimes can be often estimated based on analysis of the RCFM. Specific mapping approaches have been developed for fretting fatigue. The displacement amplitude used in most fretting wear maps is typically replaced here by the maximum fatigue stress. In 1994, Nakazawa et al. studied the effect of relative slip amplitude on fretting fatigue in high strength steel using varied fretting pad lengths (Nakazawa et al., 1994). A map representing fretting fatigue life vs contact pressure and relative slip amplitude captured a minimum life in relation to slip and contact pressure, and the contact durability was interpreted in terms of local stress concentration within the fretted area. Based on running condition and MRFMs, in 1995 Petiot et al. developed fretting-fatigue maps by varying normal load and maximum cyclic stress under fretting fatigue conditions (Petiot et al., 1995). The authors used

Introduction to fretting fundamentals

43

an elastoplastic finite element method to simulate the mechanical contact conditions based on the Dang Van criterion to predict the specimen failure. Fouvry and Kubiak introduced fretting fatigue mapping concept based on dual crack-nucleation and crack-propagation approach (Fouvry and Kubiak, 2009). The crack-nucleation boundary was formalized by applying the Crossland high cycle fatigue criterion, and the crack-propagation condition was described using short crack arrest description. Other fretting mapping approaches include determination of the influence of contact geometry, as proposed by Warmuth et al. (2015). The fretting corrosion map was developed by Narayanan et al. to identify the impact of chemical processes under different fretting test conditions (Narayanan et al., 2007). More recently, a dynamic fretting transition map was developed using a direct AE measurement during the experiment (Wade et al., 2020b). The “dynamic” context of the approach relates to the fact that the transition between the fretting regimes can be identified online during the test. This method enables smart asset monitoring of engineering components that are subject to fretting.

References Benı´tez, A., Denape, J., Paris, J.-Y., 2016. Interaction between systems and materials in fretting. Wear 368–369, 183–195. Suciu, C.V., Uchida, T., 2010. Modeling and simulation of the fretting hysteresis loop. In: 2010 International Conference on P2P, Parallel, Grid, Cloud and Internet Computing, pp. 560–564. Cadario, A., Alfredsson, B., 2006. Fretting fatigue experiments and analyses with a spherical contact in combination with constant bulk stress. Tribol. Int. 39 (10), 1248–1254. Chen, G.X., Zhou, Z.R., 2001. Study on transition between fretting and reciprocating sliding wear. Wear 250 (1–12), 665–672. Fouvry, S., Kapsa, P., 2001. An energy description of hard coating wear mechanisms. Surf. Coat. Technol. 138 (2–3), 141–148. Fouvry, S., Kubiak, K., 2009. Introduction of a fretting-fatigue mapping concept: development of a dual crack nucleation—crack propagation approach to formalize fretting-fatigue damage. Int. J. Fatigue 31 (2), 250–262. Fouvry, S., Kapsa, P., Vincent, L., 1995. Analysis of sliding behaviour for fretting loadings: determination of transition criteria. Wear 185 (1–2), 35–46. Fouvry, S., Kapsa, P., Vincent, L., 1996. Quantification of fretting damage. Wear 200 (1–2), 186–205. Heredia, S., Fouvry, S., 2010. Introduction of a new sliding regime criterion to quantify partial, mixed and gross slip fretting regimes: Correlation with wear and cracking processes. Wear 269 (7–8), 515–524. Ito, S., Shima, M., Jibiki, T., Akita, H., 2009. The relationship between AE and dissipation energy for fretting wear. Tribol. Int. 42 (2), 236–242. Merhej, R., Beguin, J.D., Paris, J.-Y., Denape, J., 2009. Acoustic emission for investigations on fretting wear of ceramic-metal contacts. Eur. Conf. Tribol. 1 (June), 1–6. Meriaux, J., Boinet, M., Fouvry, S., Lenain, J.C., 2010. Identification of fretting fatigue crack propagation mechanisms using acoustic emission. Tribol. Int. 43 (11), 2166–2174. Mohrbacher, H., Blanpain, B., Celis, J.P., Roos, J.R., 1993. Low amplitude oscillating sliding wear on chemical vapour deposited diamond coatings. Diam. Relat. Mater. 2, 879.

44

Fretting Wear and Fretting Fatigue

Nakazawa, K., Sumita, M., Maruyama, N., 1994. Effect of relative slip amplitude on fretting fatigue of high strength steel. Fatigue Fract. Eng. Mater. Struct. 17, 751–759. Narayanan, T.S.N.S., Park, Y.W., Lee, K.Y., 2007. Fretting-corrosion mapping of tin-plated copper alloy contacts. Wear 262 (1), 228–233. Petiot, C., Vincent, L., Dang Van, K., Maouche, N., Foulquier, J., Journet, B., 1995. An analysis of fretting-fatigue failure combined with numerical calculations to predict crack nucleation. Wear 181–183 (Part 1), 101–111. Rapoport, L., 1995. The competing wear mechanisms and wear maps for steels. Wear 181–183 (Part 1), 280–289. Sandstrom, P.W., Sridharanan, K., Conrad, J.R., 1993. A machine for fretting wear testing of plasma surface modified materials. Wear 166, 163–168. Varenberg, M., Etsion, I., Halperin, G., 2004. Slip index: a new unified approach to fretting. Tribol. Lett. 17, 569–573. Vingsbo, O., S€oderberg, S., 1988. On fretting maps. Wear 126 (2), 131–147. Wade, A., Copley, R., Clarke, B., Alsheikh Omar, A., Beadling, A.R., Liskiewicz, T., Bryant, M.G., 2020a. Real-time fretting loop regime transition identification using acoustic emissions. Tribol. Int. 145, 106149. Wade, A., Copley, R., Alsheikh Omar, A., Clarke, B., Liskiewicz, T., Bryant, M., 2020b. Novel numerical method for parameterising fretting contacts. Tribol. Int. 149, 105826. Wang, D., Zhang, D., Ge, S., 2012. Effect of displacement amplitude on fretting fatigue behavior of hoisting rope wires in low cycle fatigue. Tribol. Int. 52, 178–189. Warmuth, A.R., Shipway, P.H., Sun, W., 2015. Fretting wear mapping: the influence of contact geometry and frequency on debris formation and ejection for a steel-on-steel pair. Proc. R. Soc. A 471 (2178), 20140291. Zhou, Z.R., Fayeulle, S., Vincent, L., 1992. Cracking behaviour of various aluminium alloys during fretting wear. Wear 155 (2), 317–330. Zhou, Z.R., Nakazawa, K., Zhu, M.H., Maruyama, N., Kapsa, P., Vincent, L., 2006. Progress in fretting maps. Tribol. Int. 39 (10), 1068–1073.

2.4

Experimental methods Tomasz Liskiewicza, Daniele Dinib, and Thawhid Khana a

Manchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom, bImperial College London, Faculty of Engineering, Department of Mechanical Engineering, London, United Kingdom

2.4.1

Early developments

In the early 1970s, theoretical work advances have coincided with important developments of experimental equipment and techniques. Pioneering experiments were conducted by Waterhouse to determine the effect of fretting on fatigue life (Fretting Corrosion, 1972). He employed a rotary four-point bend fatigue test, very similar to that

Introduction to fretting fundamentals

45

of W€ oehler, but with “bridges” clamped on the dog-bone specimen. The bridges exerted the contact load and, because of the natural tendency of pairs of particles on the specimen surface to move harmonically toward and away from each other during each cycle, a shearing force on the pads was also developed. In these early experiments, there were few attempts to quantify the mechanics of the problem, beyond noting the average bearing pressure at each pad, and the reduction in fatigue life, when compared with a plain fatigue specimen. The advantages of such tests are that they are relatively cheap to carry out, and they are fast because the rotary speed of the dog-bone specimen may be set fairly high, assuming balance of the specimen and pad assembly is maintained. The shortcomings are considerable; it is not possible to add a mean stress, it is virtually impossible to determine the stick/slip regime existing at one of the pad feet, and it is impossible to vary any of the independent variables in a test (the shearing traction, tension in the specimen surface, and slip displacement). The first real improvement to fretting testing was made by Bramhall, who devised an ingenious experiment in which a tensile test specimen in a conventional uniaxial test machine had cylindrical pads clamped onto its surface, restrained by stiff springs that impose controlled fretting damage (Bramhall, 1973). It was the use of cylinders, giving rise to a Hertzian contact, which provided the big advance; it was now possible to use the Cattaneo-Mindlin analysis to understand the nature of the stick/slip regimes present and, hence, to quantify the salient variables that cause an acceleration of fatigue crack development. The same basic configuration was redeveloped later by Hills and Nowell, and further sets of tests were conducted (Hills and Nowell, 1992). Two further developments of the test apparatus have been recently made by the Oxford University group. The first is the use of a second actuator in place of the springs to apply the shearing force (Hills and Nowell, 1994), and in this form the apparatus has been widely used by other groups, such as that at MIT (Wittkowsky et al., 1999). The second is the use of a self-aligning pad fixture, to facilitate the use of complete or almost complete pad geometries (Mugadu and Hills, 2002). An alternative approach to studying fretting is to use an apparatus designed to reproduce the prototypical contact conditions, in a test laboratory, with as much fidelity as possible. A few attempts to design fretting rigs for prototypical testing have been carried out in the last decade. For example, Nottingham University has used one-third scale models to study fretting and plain fatigue in spline joints (Leen et al., 2001). Dovetail failures of gas turbine fan blades are another very important practical problem, and these demand the representation of both high-cycle and low-cycle fatigue loads. Tests at Oxford using a combination of hydraulic and shaker loads have succeeded in achieving highly reproducible test results that correlate well with service experience (Nowell, 2000).

2.4.2

Basic test configurations

Fretting damage by surface fatigue or wear formation is dependent on loading conditions such as displacement amplitudes and normal loading. Two common test configurations utilized for fretting experiments are as follows (Beake et al., 2015; Ma et al., 2019): l

Fretting wear tests involve generation of a contact load by the relative motion between two bodies. A reciprocating wear test configuration is used to analyze the wear process due to

46

l

Fretting Wear and Fretting Fatigue

microdisplacements. These reciprocal microdisplacements induce gross slip at the contact surface between a counterpart and a specimen. Fretting fatigue tests are derived from classical fatigue tests where two pads are pressed on a fatigue specimen. Two counterparts are needed for a single test, and it is necessary to induce the same contact load to the contact surfaces between counterparts and the specimen. Cyclic tangential load and relative displacement are induced by the strain generated through the fatigue sample which is subjected to external loading. This applied external loading initiates a crack that nucleates and then propagates. This configuration allows for high contact loads to be applied to replicate conditions found in, e.g., aerospace components.

The two types of fretting tribometers described serve different experimental objectives (see Fig. 2.4.1). The clamped surface-oscillating load-type tribometer is mostly used in fretting fatigue studies, i.e., studies of the effect of fretting on material fracture, while a small amplitude reciprocating sliding tribometer is often used in the studies of friction and wear in the gross-slip fretting regime (Stachowiak and Batchelor, 2004). Fretting fatigue tests are carried out in the partial-slip regime as this increases the likelihood of the initiation of fretting fatigue cracks, whereas the use of the gross-slip regime (larger displacements) leads to the formation of wear debris and replicates fretting wear (Kong et al., 2020).

2.4.3

Fretting wear tests and analytical methods

Fretting wear tests using a tribometer involve the application of an oscillatory tangential displacement and microsliding contact, which can be achieved using an electrodynamic shaker (Stachowiak and Batchelor, 2004; Wade et al., 2020a). The electrodynamic shaker provides a precisely controlled high-frequency reciprocating movement at small amplitudes, typically from 10 μm up to several millimetres (Stachowiak and Batchelor, 2004). Fretting is induced by relative displacement through external vibrations without applying a bulk stressing of the system (Fouvry et al., 1998). At the contact interface a normal load can be applied by a cantilever

Fig. 2.4.1 Basic fretting test configurations (Liskiewicz, 2004).

Introduction to fretting fundamentals

47

system. Outputs from the fretting tests include tangential displacement (δ) and tangential load (Q), which can be measured using an optical displacement sensor and axially mounted load cell, respectively. Both sensors are calibrated (externally and in situ) in static conditions (Warmuth et al., 2015). The contact mechanics of industrial applications involving fretting wear are relatively complex. Simplistic studies involve the analysis of conforming geometries such as cylinder/plane and sphere/plane contacts (Fouvry et al., 1998). The most common fretting configuration consists of a ball-on-flat contact, with the device controlled and data recorded using an acquisition program. With these apparatuses a ball specimen is mounted to the upper specimen mounting block using a ball holder which is connected to an actuator. A normal load between the specimens can be produced by using a dead weight configuration. The flat/plate specimen is mounted on a lower specimen mounting block, which is stationary (Warmuth et al., 2015; Chen and Zhou, 2001). The ball-on-flat surface test geometry is the most widely used in laboratory testing environments. In such configuration, the ball is often treated as an approximation of a single asperity contact. Three different fretting slip modes can be distinguished for a ball-on-flat configuration (Mohrbacher et al., 1995), see Fig. 2.4.2: l l l

linear displacement (mode I), radial displacement (mode II), circumferential displacement (mode III).

Most fretting wear literature refers to mode I experiments, where similarities are drawn with reciprocating sliding friction phenomena. Fretting mode II occurs in

P S

δ

mode I P = constant S=0 δ = variable

mode II

mode III

P = variable S=0 δ=0

P = constant S = variable δ=0

Fig. 2.4.2 Three basic fretting modes with representative displacement tracks (Liskiewicz, 2004).

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Fretting Wear and Fretting Fatigue

stationary ball bearings or electrical connectors, while fretting mode III can be observed in the heat exchanger or steam generator components. Fretting loops that are a plot of a cycle of the measured lateral force as a function of displacement amplitude give good indication of the behavior of the contact, see Section 2.2.2. Fretting wear experiments are carried out in the gross sliding regime, where slip occurs across the entire contact so the fretting loop demonstrates a quadrilateral shape (Warmuth et al., 2015). Recent studies using novel numerical method for parameterizing fretting contacts showed that fretting loop regimes can be identified using acoustic emissions (AEs) in real time (Wade et al., 2020a; Wade et al., 2020b). This method provides additional insights into the mechanical response of the contact, and the AE signal was strongly correlated with the fretting dissipated energy. Two typical methods are used to evaluate damage due to fretting wear. One involves using surface analysis equipment such as a confocal microscope or a 3D mechanical profilometer to measure the wear volume on a worn surface after a fretting test. The second method involves the measurement of the kinetic friction coefficient. A load-cell attached on the test rig allows the friction coefficient, which is the ratio of the maximum tangential force to normal force at a contact surface, to be measured (Ma et al., 2019). Weighing test samples before and after fretting tests has infrequently been used for attaining wear volume. However, typically wear scars of worn ball and flat samples from the fretting tests can be assessed using a white-light interferometer. Some interferometers have a vertical resolution of 0.15nm and lateral resolution of 0.4 μm. Scans on the worn surfaces extend outside the wear scar to create a reference surface on the unworn surface. Volume below the reference surface is regarded as wear volume and volume of material above the reference surface is regarded as transferred volume. The total wear and transfer volumes for the couple are typically defined as the sum of respective volumes for the ball and flat samples (Warmuth et al., 2015). Wear assessment can be also conducted using a 3D confocal laser scanning microscope. The 3D microscopes can image in three dimensions, allowing for precise measurement of wear of the surface area and volume due to fretting. The 3D microscopes have the capability to stitch together several images to create a high-resolution wide-angle image allowing 3D measurements to be carried out (Sidun and Da˛browski, 2017). Scanning electron microscopy (SEM) can be used to characterize the nature of the wear scars. Energy dispersive X-ray (EDX) analysis is used to identify the chemical and elemental composition of an area of interest. The specimen surface is bombarded with electrons, the energy of the X-ray signal released is measured which is then used to identify the atom which released it. Cross-sectional imaging and EDX scans through the specimens are often used to investigate subsurface damage. Microstructural features can be revealed after polishing and etched with Nital (Warmuth et al., 2015).

2.4.4

Fretting fatigue tests and analytical methods

When setting up fretting fatigue tests the immediate question to ask is: “What is the most appropriate test configuration to employ?” In order to answer this question two categories of fretting fatigue tests will be considered here (Hills and Nowell, 1994):

Introduction to fretting fundamentals l l

49

Simulation of real engineering fretting problems Idealized fretting fatigue tests

It is recognized that the simpler the contact model the grosser the idealization, and hence the sooner the results obtained from the tests carried out using that model will fail to reproduce the characteristics of the real component. Therefore, it seems appealing to seek to simulate the fretting fatigue conditions as accurately as possible so that the results found from the test rig can readily be extended to the real configuration. On the other hand, the reader should be reminded that the purpose of the experimental stage of some of the investigation is to complement the theoretical findings. In order to do this, a relatively large number of tests must be conducted and a sufficiently simple model is needed to perform experiments that are easy to analyze and can be performed in a limited amount of time. Therefore, it seems more sensible to carry out such tests employing an idealized geometry. For example, two levels of approximation have been therefore employed in order to simulate the fretting phenomenon in dovetail joints and other configurations: l

l

A simplified multiaxial model of the real joint to be tested (Rajasekaran and Nowell, 2006; Jutte, 2004; Abbasi et al., 2020). A coupon model employing different pads and specimens (De Pauw et al., 2011; Limmer et al., 2001).

In the multiaxial rig approximations of the coupling usually three-dimensional effects arising at the prototype contact interface when it is subjected to in-service loading conditions are ignored. Very good fidelity is, however, obtained by matching the geometries of the plane model and the prototype (or at least a two-dimensional “slice” of it) and the loads applied remotely from the contact interface. The limitations of these experiments are the cost of the specimens to be tested, the amount of time needed to run a single test, and the complexity of the analysis to be performed in order to obtain refined simulation of the contact pairs. Although they constitute the best possible means of simulating the fretting conditions (aside from the real structures) as accurately as possible, these test configurations may be employed to verify the accuracy of the developed design criteria after they have been fully validated using classical geometries. This will speed up the process and turn the analysis more accurate. Therefore, a coupon fretting fixture will be used to test simple couplings consisting of plane specimens indented by ideally shaped fretting pads. This will undoubtedly increase the level of approximation and also offer a number of advantages. The experiments performed using these geometries will be (i) cheaper to perform, (ii) easier to set up and rapid to execute (and hence easier to check in terms of reproducibility), (iii) comparable with a great deal of data already available in the literature, and (iv) easier to model by analytical and numerical techniques, thus allowing a number of parameters to be found in closed form. Coupon scale testing is therefore a common practice for material characterization and is used by academics to perform research. The idealized geometries available are the configurations employing a fretting pad (whose geometry is chosen so as to match that of the prototype, and where the degree of approximation depends on the nature of

50

Fretting Wear and Fretting Fatigue

idealization required) and a flat specimen. In particular, the geometries frequently adopted are as follows: l

l

l

l

Flat pads loaded against a plane specimen (for a two-dimensional analysis) (Lindley, 1997; Sato, 1992; Mugadu et al., 2002). Flat and rounded pads loaded against a plane specimen (for a two-dimensional analysis) (Hutson et al., 2001, 2003; Farris et al., 2000; Dini, 2004). Cylindrical pads loaded against a plane specimen (two-dimensional Hertzian contact) (Nowell, 1988; Szolwinski and Farris, 1996; Lykins et al., 2000). Spherical pads loaded against a plane specimen (three-dimensional Hertzian contact) (Navarro et al., 2003).

The most common type of fretting fatigue test consists of a clamped contact where microsliding is induced by an oscillating load on one of the specimens which cyclically stretches and contracts in response to the load. Tribometers of this type can simulate very small amplitude (1–100 μm) microsliding (Stachowiak and Batchelor, 2004). In this configuration the loading of the contact is generated by the relative displacement between interacting surfaces. The displacement is induced by external stresses applied causing bulk deformations of the system’s components (Fouvry et al., 1998). The fretting fatigue experiment is intended for use as standardized fatigue test setup configurations. For symmetry reasons two pads are used and installed on the fretting fixture (see Fig. 2.4.3). A hydraulic actuator is fitted to the fretting fixture which is moved to control the slip between the pad and coupon specimen. The displacement of the fretting fixture is controlled by the displacement of the hydraulic actuator and the

Fig. 2.4.3 Example of fretting fatigue test apparatus (schematic) developed at the University of Oxford (left), and schematic diagram of loading and geometry configuration used in the flat and rounded experiments (right) (Dini, 2004).

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compliance of the structural frame (De Pauw et al., 2011). This rig allows the pads to move freely laterally until a pressure load is applied, allowing for its fine alignment with the specimen. The specimen is held vertically in the chucks of the test rig, with an oscillating axial load applied periodically. The loading frequency corresponds to the natural frequency of the system, which corresponds to the natural frequency of the system relating to the mass of the machine and stiffness of the specimen. A normal contact force is applied horizontally, alongside a perpendicular axial cyclic load (Nesla´dek et al., 2012). Different tools can be used to apply the normal force such as a clamp with spring system, screw tightening, or dead load. The forces are monitored by a dynamometer and the initiation and growth of fatigue cracks are sometimes monitored using a microscopic camera (Nesla´dek et al., 2012; Cheikh and Fermy, 2014). Variations of this fretting fatigue test setup involve the use of a single hydraulic shaker that controls the applied fatigue load rather than two hydraulic shakers controlling the applied load and sample displacement. To realistically replicate fretting behavior a slip between the specimen and a loading pad can be simulated. Relative displacement between the fatigue coupon and pad can be achieved by the compliance of the specimens and the test rig (De Pauw et al., 2011). Traditionally destructive techniques are utilized to carry out analysis on fretting fatigue samples. This involves optical analysis of a specimen after failure or analyzing an unbroken specimen from an experiment that is interrupted and the sample is sectioned and analyzed for cracks/defects. With the latter samples the fretting test is paused at a certain time and the specimen is sectioned in the middle of its fretting scar, polished and multiple repeat images are taken to obtain the maximum crack length. The crack-nucleation fretting load threshold can be calculated from the crack length. Using SEM, crack initiation, location, growth paths, length, and propagation modes can be identified. The key limitation of this technique is that it is destructive and can only be carried out when the test has failed or been interrupted. Also, it is not possible to monitor the contact zone during fretting fatigue tests (Gandiolle and Fouvry, 2015; Fouvry et al., 2014). Recently nondestructive testing (NDT) techniques are used to monitor test specimens without causing damage and can be used in situ. These techniques are widely used for fretting fatigue tests, which allows the evaluation of cracks and obtain data on crack initiation, monitor crack growth, identify crack-propagation modes, and detect crack locations. The in situ techniques typically used include advanced optical microscopy, ultrasound, and AE (Kong et al., 2020). Classical optical microscopy can be used to image the contact area of the test samples, where tests are interrupted allowing images to capture crack development alongside measuring the crack length. Another method involves filming the contact area during testing, where the tests can be paused to capture images and measure crack lengths. Key advantages of this visual method are the visualization of the crack location, crack shape, and evolution (Kong et al., 2020; Arora et al., 2007; Abbasi and Majzoobi, 2018). During ultrasound testing, ultrasound waves enter the sample and the receiving wave response is recorded. Ultrasound focuses on the area of interest where a crack is expected to occur and changes in the receiving wave response allows the

52

Fretting Wear and Fretting Fatigue

identification of detects within the material. This technique has allowed fatigue cracks to be detected during fretting fatigue tests without direct access to the region of contact. The disadvantage of using ultrasound includes low sensitivity for detecting crack initiation (Wagle and Kato, 2009; Hutson and Stubbs, 2005). AE involves the detection of sound produced from mechanical vibrations created by energy release from defect or cracks within a specimen when a load is applied. A sensor is utilized to convert the acoustic waves into electrical AE signals which are processed. This technique has successfully detected crack nucleation and propagation during fretting fatigue tests and the transition between slip regimes during fretting wear tests. AE can detect internal cracks and defects, but the main limitation is the production of noise in the system making the signal difficult to interpret (Meriaux et al., 2010; Cadario and Alfredsson, 2006).

2.4.5

Combined fretting wear and fatigue approaches

Using modified fretting fatigue rig, combined fretting wear-fatigue experiments can be carried out to study the effect of wear and wear debris in fretting fatigue performance (Madge et al., 2007; Done et al., 2017), and can also feed in semianalytical and analytical model for the prediction of wear-fatigue interactions (Cwiekala and Hills, 2021; Ding et al., 2008). This can be obtained, as an example, by converting a uniaxial fatigue machine into a biaxial fatigue machine, through the addition of a transversal axis. This enables a normal force to be applied at contacting surface when the specimen and pins are interacting. As in a conventional fatigue test, the specimens reciprocating movement is controlled by longitudinal actuator. This actuator will also apply the same pressure to the two pins. Friction and wear are observed at the contact between the specimen and pins. The ability of the pins to apply a symmetrical load on the specimen is an advantage of this set-up. A load cell is applied directly to each pin allowing for the force applied to be directly measured.

References Abbasi, F., Majzoobi, G.H., 2018. An investigation into the effect of elevated temperatures on fretting fatigue response under cyclic normal contact loading. Theor. Appl. Fract. Mech. 93, 144–154. Abbasi, F., Majzoobi, G., Mendiguren, J., 2020. A review of the effects of cyclic contact loading on fretting fatigue behavior. Adv. Mech. Eng. https://doi.org/10.1177/1687814020957175. Arora, P.R., Jacob, M.S.D., Salit, M.S., Ahmed, E.M., Saleem, M., Edi, P., 2007. Experimental evaluation of fretting fatigue test apparatus. Int. J. Fatigue 29 (5), 941–952. Beake, B., Harris, A.J., Liskiewicz, T., 2015. Advanced Nanomechanical Test Techniques. Bramhall, R., 1973. Studies in Fretting Fatigue. D. Phil. Thesis, University of Oxford, UK. Cadario, A., Alfredsson, B., 2006. Fretting fatigue experiments and analyses with a spherical contact in combination with constant bulk stress. Tribol. Int. 39 (10), 1248–1254. Cheikh, M., Fermy, A., 2014. Prototype fretting device and some experimental results. Proc. Inst. Mech. Engineers J: J. Eng. Tribol. 228 (3), 266–275. Chen, G., Zhou, Z., 2001. Study on transition between fretting and reciprocating sliding wear. Wear 250 (1–12), 665–672.

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Cwiekala, N., Hills, D.A., 2021. Asymptotic description of fretting wear and fretting fatigue for incomplete contacts. Eng. Fract. Mech. 256, 107986. De Pauw, J., De Baets, P., De Waele, W. (Eds.), 2011. Review and classification of fretting fatigue test rigs. In: Sustainable Construction and Design 2011 (SCAD). Ghent University, Laboratory Soete. Ding, J., Leen, S.B., Williams, E.J., Shipway, P.H., 2008. Finite element simulation of fretting wear-fatigue interaction in spline couplings. Tribol. Mater. Surf. Interfaces 2, 10–24. Dini, D., 2004. Studies in Fretting Fatigue With Particular Application to Almost Complete Contacts. PhD Thesis, University of Oxford. Done, V., Kesavan, D., Krishna, R.M., Chaise, T., Nelias, D., 2017. Semi analytical fretting wear simulation including wear debris. Tribol. Int. 109, 1–9. Farris, T.N., Harish, G., McVeigh, P.A., Murthy, H., 2000. Prediction and observation of fretting fatigue of Ti6Al4V subjected to blade/disk type contacts. In: 5th National Turbine Engine High Cycle Fatigue (HCF) Conference, Chandler, Arizona. Fouvry, S., Kapsa, P., Vincent, L., 1998. Developments of fretting sliding criteria to quantify the local friction coefficient evolution under partial slip condition. Tribol. Ser. 34, 161–172 (Elsevier). Fouvry, S., Gallien, H., Berthel, B., 2014. From uni- to multi-axial fretting-fatigue crack nucleation: development of a stress-gradient-dependent critical distance approach. Int. J. Fatigue 62, 194–209. Fretting Corrosion, R.B., 1972. Waterhouse, International Series of Monographs on Materials Science and Technology. vol. 10 Pergamon Press Ltd, Oxford. 253 pp. Gandiolle, C., Fouvry, S., 2015. FEM modeling of crack nucleation and crack propagation fretting fatigue maps: plasticity effect. Wear 330–331, 136–144. Hills, D.A., Nowell, D., 1992. The development of a fretting fatigue experiment with welldefined characteristics. In: Attia, M.H., Waterhouse, R.B. (Eds.), Standardization of Fretting Fatigue Test Methods and Equipment, ASTM STP 1159. ASTM, West Conshohocken, PA. Hills, D.A., Nowell, D., 1994. Mechanics of Fretting Fatigue. Kluwer Academic Publishers, Dordrecht, The Netherlands. Hutson, A., Stubbs, D., 2005. A fretting fatigue crack detection feasibility study using shear wave non-destructive inspection. Exp. Mech. 45 (2), 160–166. Hutson, A.L., Nicholas, T., Olson, S.E., Ashbaugh, N.E., 2001. Effect of sample thickness on local contact behavior in a flat-on-flat fretting fatigue apparatus. Int. J. Fatigue 23 (1), 445–453. Hutson, A.L., Neslen, C., Nicholas, T., 2003. Characterization of fretting fatigue crack initiation processes in Ti-6Al-4V. Tribol. Int. 36, 133–143. Jutte, A.J., 2004. Effect of a variable contact load on fretting fatigue behavior of Ti-6Al-4V. MSc Thesis, Air Force Institute of Technology, USA. Kong, Y., Bennett, C.J., Hyde, C.J., 2020. A review of non-destructive testing techniques for the in-situ investigation of fretting fatigue cracks. Mater. Des. 196, 109093. Leen, S.B., Richardson, I.J., McColl, I.R., Williams, E.J., Hyde, T., R., 2001. Macroscopic fretting variables in a splined coupling under combined torque. J. Strain Anal. 36 (5), 481–498. Limmer, L., Nowell, D., Hills, D.A., 2001. A combined testing and modelling approach to the prediction of fretting fatigue performance of splined shafts. Proc. I. Mech. E., Part G. J. Aero Eng. 215, 105–112. Lindley, T., 1997. Fretting fatigue in engineering alloys. Int. J. Fatigue 19 (1), S39–S49. Liskiewicz, T., 2004. Hard Coatings Durability Under Variable Fretting Wear Conditions. PhD  thesis, Ecole Centrale de Lyon.

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Lykins, C.D., Mall, S., Jain, V.K., 2000. A shear stress based criterion for fretting fatigue crack initiation. In: 5th National Turbine Engine High Cycle Fatigue (HCF) Conference, Chandler, Arizona. Ma, L., Eom, K., Geringer, J., Jun, T.-S., Kim, K., 2019. Literature review on fretting wear and contact mechanics of tribological coatings. Coatings 9 (8), 501. Madge, J.J., Leen, S.B., Shipway, P.H., 2007. The critical role of fretting wear in the analysis of fretting fatigue. Wear 263 (1–6), 542–551. Meriaux, J., Boinet, M., Fouvry, S., Lenain, J.C., 2010. Identification of fretting fatigue crack propagation mechanisms using acoustic emission. Tribol. Int. 43 (11), 2166–2174. Mohrbacher, H., Celis, J.-P., Roos, J.R., 1995. Laboratory testing of displacement and load induced fretting. Tribol. Int. 28, 269–278. Mugadu, A., Hills, D.A., 2002. A generalised stress intensity approach to characterising the process zone in complete fretting contacts. Int. J. Solids Struct. 39, 1327–1335. Mugadu, A., Hills, D.A., Nowell, D., 2002. Modifications to a fretting fatigue testing apparatus based upon an analysis of contact stresses at complete and nearly complete contacts. Wear 252, 475–483. Navarro, C., Garcia, M., Dominguez, J., 2003. A procedure for estimating the total life in fretting fatigue. Fatigue Fract. Eng. Mater. Struct. 26 (5), 459–468. Nesla´dek, M., Sˇpaniel, M., Jurenka, J., Ru˚zˇicka, J., Kuzˇelka, J., 2012. Fretting fatigue— experimental and numerical approaches. Int. J. Fatigue 44, 61–73. Nowell, D., 1988. An Analysis of Fretting Fatigue. D. Phil. Thesis, Oxford University. Nowell, D., 2000. Advances in the understanding of fretting fatigue with reference to gas turbine engines. In: Morton, J., Paris, F. (Eds.), Progress in Structural Mechanics. University of Seville, pp. 61–72. Rajasekaran, R., Nowell, D., 2006. Fretting fatigue in dovetail blade roots: experiment and analysis. J. Tribol. Int. 39, 1277–1285. Sato, K., 1992. Determination and control of contact pressure distribution in fretting fatigue. In: Attia, M.H., Waterhouse, R.B. (Eds.), ASTM STP 1159: Standardisation of Fretting Fatigue Methods and Equipment. ASTM, Philadelphia, pp. 85–100. Sidun, J., Da˛browski, J.R. (Eds.), 2017. The method of fretting wear assessment with the application of 3D laser measuring microscope. Conference on Innovations in Biomedical Engineering. Springer. Stachowiak, G., Batchelor, A.W., 2004. Experimental Methods in Tribology. Elsevier. Szolwinski, M.P., Farris, T.N., 1996. Mechanics of fretting fatigue crack formation. Wear 198, 93–107. Wade, A., Copley, R., Clarke, B., Alsheikh Omar, A., Beadling, A.R., Liskiewicz, T., et al., 2020a. Real-time fretting loop regime transition identification using acoustic emissions. Tribol. Int. 145, 106149. Wade, A., Copley, R., Alsheikh Omar, A., Clarke, B., Liskiewicz, T., Bryant, M., 2020b. Novel numerical method for parameterising fretting contacts. Tribol. Int. 149, 105826. Wagle, S., Kato, H., 2009. Ultrasonic wave intensity reflected from fretting fatigue cracks at bolt joints of aluminum alloy plates. NDT & E Int. 42 (8), 690–695. Warmuth, A., Shipway, P., Sun, W., 2015. Fretting wear mapping: the influence of contact geometry and frequency on debris formation and ejection for a steel-on-steel pair. Proc. R. Soc. A: Math. Phys. Eng. Sci. 471 (2178), 20140291. Wittkowsky, B.U., Birch, P.R., Dominguez, J., Suresh, S., 1999. An apparatus for quantitative fretting-fatigue testing. Fatigue Fract. Eng. Mater. Struct. 22, 307–320.

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Modelling approaches Daniele Dinia and Tomasz Liskiewiczb a

Imperial College London, Faculty of Engineering, Department of Mechanical Engineering, London, United Kingdom, bManchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom A plethora of theoretical methods and numerical simulations and modeling techniques have been developed to study the fretting phenomenon in terms of wear, fatigue, and fracture. Here we will first briefly summarize the different methodologies developed in the last few decades for wear and fretting fatigue modeling and simulations. Theoretical advances will be first described and discussed, with particular attention being paid to fundamental understanding of wear processes and fatigue, and fatigue crack initiation and propagation approaches being developed. Our attention will then turn to numerical model used to predict wear and study fretting fatigue and crack initiation and propagation methods. To that end, multiaxial fatigue parameters are introduced putting an emphasis on the physical basis of the fretting phenomena and the suitability of each model. On the other hand, the propagation phase based on linear elastic fracture mechanics (LEFM) via the finite element method (FEM) and the extended finite element method (X-FEM) analysis methods is presented and compared. Finally, different approaches and latest developments for fretting fatigue lifetime prediction will be discussed.

2.5.1

Theoretical models

2.5.1.1 Wear mechanisms and phenomenological models of fretting wear Modeling fretting wear requires capturing contact geometry adaptation, which can be formulated as a spatial-temporal contact problem concerned with the evaluation of the surface profiles of the contacting bodies due to wear. Given the complexity of the problem and its dependence on several material, geometric, loading, and displacements parameters and their time variation, theoretical analyses are often limited to the understanding of the relevant wear mechanisms and the construction of wear laws and approaches that can be used to study local surface alterations and can be implemented in numerical schemes that capture the evolution of the contacting bodies. It is important and necessary to simulate the fretting wear behavior of different materials under various work environments. Therefore, a reliable wear model connecting working parameters to the wear damage of a given fretting couple is needed. Over the past decades two main approaches have emerged to model fretting wear from a phenomenological point of view (i.e., capable of capturing the main aspects of the phenomenon as observed macroscopically but not of delving into the complex physical

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and chemical mechanisms explicitly): the Archard model and Energy model, both based on contact mechanics (Meng and Ludema, 1995). The Archard wear model and its generalizations are traditionally the most adopted (Argatov and Chai, 2020; Yue and Wahab, 2019) methods, while the energy-based models, pioneered by Matveevsky (1965) (see also Jahangiri et al., 2012), have been promoted and further improved for applications to fretting wear since the 1990s by Fouvry and his coworkers (Fouvry et al., 1996). These methods and their subsequent improvements (see, e.g., Fouvry et al., 2003; Paulin et al., 2008), which include considerations about material transformation and the presence of wear debris, form today the basis of most numerical studies of fretting wear (see also Chapter 3). Analytical and semianalytical methods are infrequently used to describe the modification of the surfaces and the response of the tribosystems of interest. These are often limited to the study of (i) the so-called wearing-in period in gross-slip fretting wear, when the initial contact state progresses into a steady state, in which the applied contact load is redistributed along the contact area in accordance with the wear equation and (ii) the evolution of the contact toward a limiting state, characterized by transferring the contact load primarily through the stick zone, under partial-slip fretting wear conditions. For a recent review of approaches dealing with these problems, see Argatov and Chai (2020). While a number of solutions can be obtained for these states for simple geometries under specific loading conditions (e.g., Popov, 2014; Dini et al., 2008; Goryacheva et al., 2001; Goryacheva and Goryachev, 2006; Hills et al., 2009; Argatov and Chai, 2018), the main limitation of the current state-of-the-art analytical techniques is their inability to effectively deal with varying geometry due to wear loss for arbitrary geometries and the presence of nonlinear wear equations. The semianalytical methods proposed, for example, by Johansson (1993), Gallego et al. (2006), and Hegadekatte et al. (2008) are also limited in terms of applicability but provide alternative methods of the computationally expensive FEM wear depth models discussed below.

2.5.1.2 Fatigue crack initiation and propagation Looking at the other face of the fretting medal (i.e., fatigue), as explained in Chapter 1, many theoretical studies have been performed since the pioneering work performed in the 1970s (e.g., Endo and Goto, 1976; Hoeppner, 1977) to establish criteria to predict the first stage of crack initiation and the early propagation of fretting fatigue cracks. The most noticeable early efforts were made in the 1980s and 1990s, when a number of crack initiation parameters based on continuum mechanics (some of which have already been successfully applied to low- and high-cycle uniaxial fatigue) were evaluated; these include the strain-life parameter, the maximum strain corrected for strain ratio effects, the maximum principal strain corrected for principal strain ratio effects, the Smith-Watson-Topper (SWT) parameter, the critical plane SWT parameter, the Fatemi and Socie (F-S) parameter, and the Ruiz parameters (which were specifically developed to fretting fatigue condition). The evaluation has been based on the parameter’s ability to predict the number of cycles to initiation and location of crack and has

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covered a number of applications and has been often coupled to numerical studies aimed at determining the detailed state of stress and strain fields required to derive such parameters and their evolution in time during a loading cycle (Nowell and Hills, 1990; Szolwinski and Farris, 1996, 1998; Lykins et al., 2000; Alfredson and Cadario, 2004; Nowell et al., 2006). Microstructurally based methods have also emerged in the past decades, where strain-gradient methodologies are often used for the prediction of microstructure-sensitive length-scale effects in crack initiation (initially developed for standard fatigue) in fretting fatigue conditions (e.g., Ashton et al., 2017, 2018; Minaii et al., 2019). These often require complex numerical implementations, which are covered in the next section. Alternative approaches consider the inherent length scale associated with the microstructural features of the materials undergoing fretting considering line and volume averages and/or damage criteria that capture the intrinsic length scale governing the problem (Arau´jo et al., 2006, 2017; Nowell et al., 2006). An alternative approach to study the origin of fretting fatigue failures using asymptotic approaches has also been recently developed (Hills and Dini, 2006, 2016; Hills et al., 2012; Flicek et al., 2013; Hills and Andresen, 2021). Theoretical approaches are often used to predict crack propagation from a given crack size (reached after initiation). These are often based on Paris law and rely on the definition of an initial crack size as well as the potential to generalize the use of simplified crack-propagation laws based on LEFM to multiaxial states of stress (Hattori and Nakamura, 1997; Szolwinski and Farris, 1996), which are common in fretting. A study of crack-propagation models in fretting fatigue was carried out by Navarro et al. (2006), who looked at fatigue crack growth models and concluded that the combined use of Paris law and the modified stress intensity factors (SIFs) model (Hattori et al., 2003) was suitable to capture the salient features of crack propagation in fretting for aluminum alloys conventionally used in aerospace and other applications. Concerning crack orientation criteria, they are generally based on the analysis of the stress and strain fields. The suitability of each criterion mainly depends on the evolution of the stresses and strains along a loading cycle. It should be noted that fretting fatigue usually induces friction between crack faces prone to slip motion during the loading cycle. The problem is therefore nonproportional, and the classical orientation criteria for proportional loading are not applicable. Giner et al. (2014) investigated the suitability of the nonproportional loading criteria available in the literature for fretting fatigue problem and concluded that the prediction of the crack path observed in the complete contact experiments did not present a good agreement with the models available. More recently, advanced methodologies that consider the presence of microstructural features and the interplay between wear (and wear debris) and fatigue have also been considered (e.g., McCarthy et al., 2014; Madge, 2009; Llavori et al., 2019; Wang et al., 2022). A recent review of the most common methods adopted to model various aspects of fretting is provided by Llavori et al. (2017). A significant amount of these approaches relies on complex modeling, which requires advanced numerical implementations, which are covered in Section 2.5.2 and, in much greater depth, in Chapters 3 and 4.

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2.5.2

Fretting Wear and Fretting Fatigue

Numerical models

2.5.2.1 Wear models using mechanistic approaches and advanced FEM and BEM simulations Fretting wear is a complicated problem involving material properties of tribosystem and their working conditions. While phenomenological models based on simplified theories allow to study fretting wear in first approximation and often require experimental inputs for calibration, mechanistic models are developed to capture underlying mechanisms such as plasticity, third-body interactions, dislocation formations, interactions between wear and crack initiation, adhesion, and chemistry ( Jacq et al., 2002; Iordanoff et al., 2002; Suh, 1973; Maegawa et al., 2010; Johnson et al., 1971; Quinn, 1971). Although a mechanistic model can capture one or more of the underlying mechanisms, real engineering surfaces can undergo multiple mechanisms, thus these mechanistic models also come with limitations and also these models require calibration with experimental data (Bastola et al., 2022). Application of models beyond the test conditions may result in erroneous results. So, it is very unlikely that there will be a single theory that applies to every situation of wear (Lim et al., 2017). Therefore, often fretting wear experiments are still the most reliable method to understand tribological behaviors for specific components and applications. However, tribological experiments are inherently time-consuming and costly experiments as these tests must mimic operating conditions. Also, the different parameters involved during the sliding test require a high number of tests to capture the results within the operating conditions. For these reasons, advanced numerical methods (which overcome the limitations of the semianalytical models discussed above), such as the FEM and the boundary element method (BEM), have recently emerged as an alternative to tests, especially at design stage, where a better understanding of the most influential parameters affecting wear is needed. The study of wear by solving nonlinear contact problems using FEM has been ongoing since the mid-1990s. Johansson (1994) presented an algorithm for applying Archard’s wear law locally to evaluate the contact pressure and the resultant change in geometry. A wear model is applied at an elemental level. Johansson’s algorithm was modified by McColl et al. € (2004), Po˜dra and Andersson (1999), and Oqvist (2001) for the simulation of fretting wear using an incremental wear approach. These modified versions are faster and useful to predict a high number of cycles ranging from tens of thousands to millions for 2D models. However, these versions assume wear rates are constant or vary at a predefined rate for a given number of cycles. It was noted by the authors that the effect of incremental number of fretting cycles per solution step is dependent on element height and wear rate. Coworkers of McColl further developed the method by McColl et al. (2004) for simulating fretting wear on 3D models. Ding et al. (2007) calculated wear on realistic helical spline coupling teeth of aeroengines and then (Ding et al., 2008) further developed this model considering the interaction of wear with fretting fatigue on spline couplings. The 3D fretting wear models were also developed by Cruzado et al. (2012) to capture wear on both contacting bodies.

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An alternative to FEM is the BEM, which is often more efficient than FEM. Only the surfaces are discretized in the BEM method instead of the whole volume. This reduces the dimension of the problem by one thus reducing computational time. However, for nonlinear contact problems, the volume integrals present in the boundary integral formulation need to be evaluated first. Various methods are developed over the years to evaluate the volume integrals without volume discretization for different contact problems. These methods generally involve approximating volume integrals with linear equations (Katsikadelis, 2002; Coleman et al., 1991; Sfantos and Aliabadi, 2007; Ghanbarzadeh et al., 2016); however, they often require simplifying assumptions in terms of boundary conditions and are not efficient when dealing with finite geometries, especially in 3D. Zografos (2011) and Ghanbarzadeh et al. (2016) reported interesting examples of the application of BEM to capture surface evolution. The BEM is used to model boundary lubrication to study the effect of tribofilms in reducing friction and wear. The BEM can be a powerful alternative to FEM where only surface field outputs are required.

2.5.2.2 Advanced numerical methods for crack initiation and propagation in fretting There are many numerical approaches that have been used by researchers in order to model fretting fatigue crack initiation lifetime based on multiaxial criteria such as critical plane approaches (Szolwinski and Farris, 1998; Navarro et al., 2008), which have also been recently reviewed by Wahab and coworkers (see, e.g., Bhatti and Wahab, 2016; Hojjati-Talemi et al., 2014). More recently, continuum damage mechanics (CDM) has been introduced for modeling fretting fatigue damage (Hojjati-Talemi and Wahab, 2013a, b; Zhang et al., 2012; Aghdam et al., 2012). In terms of crack propagation, the initial crack was inserted in the contact model and advanced up to the final rupture of specimen. For fretting fatigue crack propagation, there are numerous studies (Rooke and Jones, 1979; Kondo and Mutoh, 2000; Nicholas et al., 2003; Munoz et al., 2007; Giner et al., 2011; Sabsabi et al., 2011; Hojjati-Talemi et al., 2012) that have used a fracture mechanics approach to calculate fretting fatigue crack-propagation lifetime. In this study, the crack-propagation part was modeled by means of the LEFM approach under mixed-mode loading conditions. From an initial crack length, whose size is related to the microstructure of material, the advance of crack was modeled using the FE method. Recently, a new FE meshindependent formulation, i.e., extended finite element method (XFEM) (e.g., Sabsabi et al., 2011), became more popular due to its capability to model the crack inside the mesh without any remeshing technique. Nonetheless, the conventional FEM is still a promising approach and has been implemented by Mohd Tobi et al. (2013) to study fretting fatigue behavior, where the conventional FE method including the remeshing technique was chosen to model fretting fatigue crack propagation. Since fretting fatigue is highly nonlinear and subjected to the nonproportional loading condition, finding the right crack-propagation trajectory is an issue, when dealing with fretting fatigue crack propagation. Giner et al. (2014) have proposed a new approach

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for fretting fatigue problem under the complete contact condition. The model they proposed was based on the minimum value of shear stress range evaluated ahead of a crack tip during a full cyclic loading. In the last two decades, increased attention has been paid to microstructurally based models to be used to study the origin of fretting damage (Dick and Cailletaud, 2006). In fretting contacts, small material volumes at the microstructural scale typically experience high stresses and high stress gradients. Polycrystal plasticity models are more adequate than continuum plasticity models to explicitly capture the influence of microstructure and slip systems, as has been shown by Morrissey et al. (1999, 2001) and Schoenfeld and Kad (2002). The model of Morrissey and coworkers, which showed that the microstructural plastic response depends on the morphology of the microstructure and on the crystallographic texture, has been applied to a FE fretting model by Goh and coworkers (Goh et al., 2001, 2003), in the first example of application of the crystal plasticity finite element (CPFE) method to fretting. This has been followed more recently by several attempts to use CPFE to predict fretting fatigue crack initiation; these include studies by McCarthy et al. (2013, 2014), who reasonably predicted partial-slip cracking and wear scar by using microstructure-sensitive CPFE approach and the investigations by Sun et al. (2020), who studied the fretting fatigue behavior of a Ni-based superalloy via CPFE calculations, and claimed that the crack initiation sites coincide with the location of maximum equivalent plastic strain. More recently, Wang et al. (2021) have explored the fretting fatigue behavior of an aluminum alloy via a CPFE method considering the effect of grain orientation and morphology, and developed an energy criterion to predict the fretting fatigue crack initiation (FFCI) life in metallic materials.

2.5.2.3 Nano- and mesoscale models The models presented can be limited to certain length scales, whereas the real engineering surfaces span multiple length scales. For instance, the adhesive wear model proposed by Pham-Ba et al. (2020) predicts the formation of wear particles in microns and its validity diminishes below a critical microcontact. Likewise, the adhesive wear model proposed by Brink and Molinari (2019) is at the nanoscale. While, given the extraordinary complexity of the problem, systematic attempts have been made to study fretting fatigue and wear with approaches spanning all the scales, from atomistic to component levels, several attempts have been made to perform and bridge simulations, as well as theoretical and experimental investigations, across multiple scales. These include studies incorporating molecular dynamics simulations to assess the influence of nanoscale interactions on fretting (Zhang et al., 2021), the use of discrete dislocation dynamics to study surface deformation and plasticity at the submicron scales and nanofretting (Xu et al., 2021), the use of hybrid simulations and multiscale homogenization techniques to capture the effect of microstructural features at test sample level (Wang et al., 2022), and the use of multiscale frameworks to focus on fretting damage from the study of component-level problems (Zografos et al., 2009).

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References Aghdam, A., Beheshti, A., Khonsari, M., 2012. On the fretting crack nucleation with provision for size effect. Tribol. Int. 47, 32–43. Alfredson, B., Cadario, A., 2004. A study on fretting friction evolution and fretting fatigue crack initiation for a spherical contact. Int. J. Fatigue 26, 1037–1052. Arau´jo, J.A., Vivacqua, R.C., da Silva Bernardo, A.T., Mamiya, E.N., 2006. A crack initiation threshold methodology in fretting fatigue. J. Strain Anal. Eng. Des. 41 (5), 363–368. Arau´jo, J.A., Susmel, L., Pires, M.S.T., Castro, F.C., 2017. A multiaxial stress-based critical distance methodology to estimate fretting fatigue life. Tribol. Int. 108, 2–6. Argatov, I., Chai, Y.S., 2020. Contact geometry adaptation in fretting wear: a constructive review. Front. Mech. Eng. 6, 51. Argatov, I.I., Chai, Y.S., 2018. Limiting shape of profiles in fretting wear. Tribol. Int. 125, 95–99. Ashton, P.J., Harte, A.M., Leen, S.B., 2017. Statistical grain size effects in fretting crack initiation. Tribol. Int. 108, 75–86. Ashton, P.J., Harte, A.M., Leen, S.B., 2018. A strain-gradient, crystal plasticity model for microstructure-sensitive fretting crack initiation in ferritic-pearlitic steel for flexible marine risers. Int. J. Fatigue 111, 81–92. Bastola, A., Stewart, D., Dini, D., 2022. Three-dimensional finite element simulation and experimental validation of sliding wear. Wear 204402. Bhatti, N.A., Wahab, M.A., 2016. A review on fretting fatigue crack initiation criteria. In: Proceedings of the 5th International Conference on Fracture Fatigue and Wear, pp. 78–85. Brink, T., Molinari, J.-F., 2019. Adhesive wear mechanisms in the presence of weak interfaces: insights from an amorphous model system. Phys. Rev. Mater. 3, 053604. Coleman, C.J., Tullock, D.L., Phan-Thien, N., 1991. An effective boundary element method for inhomogeneous partial differential equations. Z. Angew. Math. Phys. ZAMP 42, 730–745. Cruzado, A., Urchegui, M.A., Go´mez, X., 2012. Finite element modeling and experimental validation of fretting wear scars in thin steel wires. Wear 289, 26–38. Dick, T., Cailletaud, G., 2006. Fretting modelling with a crystal plasticity model of Ti6Al4V. Comput. Mater. Sci. 38, 113–125. Ding, J., McColl, I.R., Leen, S.B., 2007. The application of fretting wear modelling to a spline coupling. Wear 262, 1205–1216. Ding, J., Leen, S.B., Williams, E.J., Shipway, P.H., 2008. Finite element simulation of fretting wear-fatigue interaction in spline couplings. Tribol. Mater. Surf. Interfaces 2, 10–24. Dini, D., Sackfield, A., Hills, D.A., 2008. An axi-symmetric Hertzian contact subject to cyclic shear and severe wear. Wear 265, 1918–1922. Endo, K., Goto, H., 1976. Initiation and propagation of fretting fatigue cracks. Wear 38, 311–324. Flicek, R., Hills, D.A., Dini, D., 2013. Progress in the application of notch asymptotics to the understanding of complete contacts subject to fretting fatigue. Fatigue Fract. Eng. Mater. Struct. 36, 56–64. Fouvry, S., Kapsa, P., Vincent, L., 1996. Quantification of fretting damage. Wear 200, 186–205. Fouvry, S., Liskiewicz, T., Kapsa, P., Hannel, S., Sauger, E., 2003. An energy description of wear mechanisms and its applications to oscillating sliding contacts. Wear 255, 287–298. Gallego, L., Nelias, D., Jacq, C., 2006. A comprehensive method to predict wear and to define the optimum geometry of fretting surfaces. J. Tribol. 128, 476. Ghanbarzadeh, A., Wilson, M., Morina, A., Dowson, D., Neville, A., 2016. Development of a new mechano-chemical model in boundary lubrication. Tribol. Int. 93, 573–582.

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Giner, E., Navarro, C., Sabsabi, M., Tur, M., Dominguez, J., Fuenmayor, F., 2011. Fretting fatigue life prediction using the extended finite element method. Int. J. Mech. Sci. 53, 217–225. Giner, E., Sabsabi, M., Ro´denas, J.J., Fuenmayor, F.J., 2014. Direction of crack propagation in a complete contact fretting-fatigue problem. Int. J. Fatigue 58, 172–180. Goh, C.H., Wallace, J.M., Neu, R.W., McDowell, D.L., 2001. Polycrystal plasticity simulations of fretting fatigue. Int. J. Fatigue 23, S423–S435. Goh, C.H., Neu, R.W., McDowell, D.L., 2003. Crystallographic plasticity in fretting of Ti-6AL4V. Int. J. Plast. 19, 1627–1650. Goryacheva, I.G., Goryachev, A.P., 2006. The wear contact problem with partial slippage. J. Appl. Math. Mech. 70, 934–944. Goryacheva, I.G., Rajeev, R.T., Farris, T.N., 2001. Wear in partial slip contact. J. Tribol. 123, 848–856. Hattori, T., Nakamura, M., 1997. Initiation and propagation behavior of fretting fatigue cracks. Trans. Eng. Sci. 14, 183–192. Hattori, T., Nakamura, M., Watanabe, T., 2003. Simulation of fretting-fatigue life using stresssingularity parameters and fracture mechanics. Tribol. Int. 36, 87–97. Hegadekatte, V., Kurzenh€auser, S., Huber, N., Kraft, O., 2008. A predictive modeling scheme for wear in tribometers. Tribol. Int. 41, 1020–1031. Hills, D.A., Andresen, H.N., 2021. Mechanics of Fretting and Fretting Fatigue, SMIA 266. Springer Cham, Switzerland. Hills, D.A., Dini, D., 2006. A new method for the quantification of nucleation of fretting fatigue cracks using asymptotic contact solutions. Tribol. Int. 39, 1114–1122. Hills, D.A., Dini, D., 2016. A review of the use of the asymptotic framework for quantification of fretting fatigue. J. Strain Anal. Eng. Des. 51, 240–246. Hills, D.A., Sackfield, A., Paynter, R.J.H., 2009. Simulation of fretting wear in halfplane geometries. Part I. The solution for long term wear. J. Tribol. 131, 031401. Hills, D.A., Thaitirarot, A., Barber, J.R., Dini, D., 2012. Correlation of fretting fatigue experimental results using an asymptotic approach. Int. J. Fatigue 43, 62–75. Hoeppner, D.W., 1977. Comments on initiation and propagation of fretting fatigue cracks (letter to the editor). Wear 43, 267–270. Hojjati-Talemi, R., Wahab, M.A., 2013a. Fretting fatigue crack initiation lifetime predictor tool: using damage mechanics approach. Tribol. Int. 60, 176–186. Hojjati-Talemi, R., Wahab, M.A., 2013b. Numerical estimation of fretting fatigue lifetime using damage and fracture mechanics. Tribol. Lett. 52, 11–25. Hojjati-Talemi, R., Wahab, M.A., De Baets, P., 2012. Numerical investigation into effect of contact geometry on fretting fatigue crack propagation lifetime. Tribol. Trans. 55, 365–375. Hojjati-Talemi, R., Wahab, M.A., De Pauw, J., De Baets, P., 2014. Prediction of fretting fatigue crack initiation and propagation lifetime for cylindrical contact configuration. Tribol. Int. 76, 73–91. Iordanoff, I., Berthier, Y., Descartes, S., Heshmat, H., 2002. A review of recent approaches for modeling solid third bodies. J. Tribol. 124, 725–735. Jacq, C., Nelias, D., Lormand, G., Girodin, D., 2002. Development of a three-dimensional semianalytical elastic-plastic contact code. J. Tribol. 124, 653–667. Jahangiri, M., Hashempour, M., Razavizadeh, H., Rezaie, H.R., 2012. Application and conceptual explanation of an energy-based approach for the modelling and prediction of sliding wear. Wear 274–275, 168–174. Johansson, L., 1993. Model and numerical algorithm for sliding contact between two elastic half-planes with frictional heat generation and wear. Wear 160, 77–93.

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Johansson, L., 1994. Numerical simulation of contact pressure evolution in fretting. J. Tribol. 116, 247–254. Johnson, K.L., Kendall, K., Roberts, A., 1971. Surface energy and the contact of elastic solids. Proc. R. Soc. Lond. A Math. Phys. Sci. 324, 301–313. Katsikadelis, J.T., 2002. Boundary Elements: Theory and Applications. Elsevier Science. Kondo, K., Mutoh, Y., 2000. Crack behavior in the early stage of fretting fatigue fracture. ASTM Spec. Tech. Publ. 1367, 282–294. Lim, S., Batchelor, A.W., Lim, C., 2017. Introduction and basic theory of wear. In: Friction, Lubrication, and Wear Technology. ASM International, pp. 1–2. Llavori, I., Esnaola, J.A., Zabala, A., Gomez, M.L.X., 2017. Fretting: review on the numerical simulation and modeling of wear, fatigue and fracture. In: Darji, P.H., Darji, V.P. (Eds.), Contact and Fracture Mechanics. IntechOpen, London. Llavori, I., Zabala, A., Urchegui, M.A., Tato, W., Go´mez, X., 2019. A coupled crack initiation and propagation numerical procedure for combined fretting wear and fretting fatigue lifetime assessment. Theor. Appl. Fract. Mech. 101, 294–305. Lykins, C.D., Mall, S., Jain, V., 2000. An evaluation of parameters for predicting fretting fatigue crack initiation. Int. J. Fatigue 22, 703–716. Madge, J.J., 2009. Numerical modelling of the effect of fretting wear on fretting fatigue. PhD Thesis, University of Nottingham. Maegawa, S., Suzuki, A., Nakano, K., 2010. Precursors of global slip in a longitudinal line contact under non-uniform normal loading. Tribol. Lett. 38, 313–323. Matveevsky, R.M., 1965. The critical temperature of oil with point and line contact machines. Trans. ASTM 87, 754. McCarthy, O.J., McGarry, J.P., Leen, S.B., 2013. Microstructure-sensitive prediction and experimental validation of fretting fatigue. Wear 305, 100–114. McCarthy, O.J., McGarry, J.P., Leen, S.B., 2014. Micro-mechanical modelling of fretting fatigue crack initiation and wear in Ti-6Al-4V. Int. J. Fatigue 62, 180–193. McColl, I., Ding, J., Leen, S., 2004. Finite element simulation and experimental validation of fretting wear. Wear 256, 1114–1127. Meng, H.C., Ludema, K.C., 1995. Wear models and predictive equations: their form and content. Wear 181, 443–457. Minaii, K., Farrahi, G.H., Karimpour, M., Bahai, H., Majzoobi, G.H., 2019. Investigation of microstructure effect on fretting fatigue crack initiation using crystal plasticity. Fatigue Fract. Eng. Mater. Struct. 42, 640–650. Mohd Tobi, A., Shipway, P., Leen, S., 2013. Finite element modelling of brittle fracture of thick coatings under normal and tangential loading. Tribol. Int. 58, 29–39. Morrissey, R.J., McDowell, D.L., Nicholas, T., 1999. Frequency and stress ratio effects in high cycle fatigue of Ti-6Al-4V. Int. J. Fatigue 21, 679–685. Morrissey, R.J., McDowell, D.L., Nicholas, T., 2001. Microplasticity in HCF of Ti-6Al-4V. Int. J. Fatigue 23, 55–64. Munoz, S., Navarro, C., Dominguez, J., 2007. Application of fracture mechanics to estimate fretting fatigue endurance curves. Eng. Fract. Mech. 74, 2168–2186. Navarro, C., Mun˜oz, S., Domı´nguez, J., 2006. Propagation in fretting fatigue from a surface defect. Tribol. Int. 39, 1149–1157. Navarro, C., Mun˜oz, S., Domı´nguez, J., 2008. On the use of multiaxial fatigue criteria for fretting fatigue life assessment. Int. J. Fract. 30, 32–44. Nicholas, T., Hutson, A., John, R., Olson, S., 2003. A fracture mechanics methodology assessment for fretting fatigue. Int. J. Fract. 25, 1069–1077. Nowell, D., Hills, D.A., 1990. Crack initiation criteria in fretting fatigue. Wear 136, 329–343.

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Nowell, D., Dini, D., Hills, D.A., 2006. Recent developments in the understanding of fretting fatigue. Eng. Fract. Mech. 73, 207–222. € Oqvist, M., 2001. Numerical simulations of mild wear using updated geometry with different step size approaches. Wear 249, 6–11. Paulin, C., Fouvry, S., Meunier, C., 2008. Finite element modelling of fretting wear surface evolution application to a Ti-6A1-4V contact. Wear 264 (1-2), 26–36. Pham-Ba, S., Brink, T., Molinari, J.F., 2020. Adhesive wear and interaction of tangentially loaded micro-contacts. Int. J. Solids Struct. 188–189, 261–268. Po˜dra, P., Andersson, S., 1999. Simulating sliding wear with finite element method. Tribol. Int. 32, 71–81. Popov, V.L., 2014. Analytic solution for the limiting shape of profiles due to fretting wear. Sci. Rep. 4, 3749. Quinn, T.F., 1971. Oxidational wear. Wear 18, 413–419. Rooke, D., Jones, D., 1979. Stress intensity factors in fretting fatigue. J. Strain Anal. Eng. Des. 14, 1–6. Sabsabi, M., Giner, E., Fuenmayor, F., F., 2011. Experimental fatigue testing of a fretting complete contact and numerical life correlation using X-FEM. Int. J. Fract. 33, 811–822. Schoenfeld, S.E., Kad, B., 2002. Texture effects on shear response in Ti-6Al-4V plates. Int. J. Plast. 18, 461–486. Sfantos, G.K., Aliabadi, M.H., 2007. A boundary element formulation for three-dimensional sliding wear simulation. Wear 262 (5–6), 672–683. Suh, P.N., 1973. The delamination theory of wear. Wear 25, 111–124. Sun, S., Li, L., Yue, Z., Yang, W., Zhao, Z., Cao, R., et al., 2020. Experimental and numerical investigation on fretting fatigue behavior of Nickel-based single crystal superalloy at high temperature. Mech. Mater. 150, 103595. Szolwinski, M.P., Farris, T.N., 1996. Mechanics of fretting fatigue crack formation. Wear 198, 93–107. Szolwinski, M.P., Farris, T.N., 1998. Observation, analysis and prediction of fretting fatigue in 2024-T351 aluminum alloy. Wear 221, 24–36. Wang, J., Chen, T., Zhou, C., 2021. Crystal plasticity modeling of fretting fatigue behavior of an aluminum alloy. Tribol. Int. 156, 106841. Wang, S., Yue, T., Wang, D., et al., 2022. Effect of wear debris on fretting fatigue crack initiation. Friction 10, 927–943. Xu, Y., Balint, D.S., Dini, D., 2021. On the origin of plastic deformation and surface evolution in nano-fretting: a discrete dislocation plasticity analysis. Materials 14 (21), 6511. Yue, T., Wahab, M.A., 2019. A review on fretting wear mechanisms, models and numerical analyses. Comput. Mater. Contin. 59, 405–432. Zhang, T., McHugh, P., Leen, S., 2012. Finite element implementation of multiaxial continuum damage mechanics for plain and fretting fatigue. Int. J. Fract. 44, 260–272. Zhang, Z., Pan, S., Yin, N., et al., 2021. Multiscale analysis of friction behavior at fretting interfaces. Friction 9, 119–131. Zografos, A., 2011. Numerical and Experimental Investigation of Fretting in Wheel-Hub Type Bolted Joints. PhD Thesis, Imperial College London. Zografos, A., Dini, D., Olver, A.V., 2009. Fretting fatigue and wear in bolted connections: a multi-level formulation for the computation of local contact stresses. Tribol. Int. 42, 1663–1675.

The role of tribologically transformed structures and debris in fretting of metals

3.1

Philip Howard Shipway Faculty of Engineering, University of Nottingham, Nottingham, United Kingdom

3.1.1

Overview

In fretting, the scale of the oscillatory displacements between the bodies in contact is typically much less than the size of the contact patch between the bodies themselves, meaning that the contact remains largely closed throughout the process lifetime. It is well understood that wear behavior in fretting is different from that observed in sliding, with this being a result of the closed nature of the contact. As outlined by Godet almost 40 years ago (Godet, 1984), fretting wear of metals normally involves (i) the formation of oxide debris within the fretting contact and (ii) the expulsion of that debris from the contact. If oxide debris is not expelled from the contact, it can form a protective bed within the contact which serves to restrict further wear, with this indicating that fretting wear is a phenomenon that requires a thorough understanding of debris transport. To form oxide debris in the contact requires the transport of oxygen into the contact to the point where the debris is formed (a process which consumes oxygen), and again it has long been recognized that this second transport process may also control the mechanisms and rates of damage and wear in fretting. In particular, if the rate of oxygenation of the contact is insufficient, then the damage is driven by adhesion and microwelding of the metallic surfaces with subsurface deformation being observed (these are often termed tribologically transformed structures) and the rate of wear being reduced (Fouvry et al., 2017). While there has been general progress in understanding these key concepts over the years, there are some recent and significant developments that provide a useful framework for the development of understanding here. 1. Progress has recently been made in modeling the transport of oxygen into fretting contacts (Baydoun et al., 2020), facilitating an understanding of adhesive and abrasive behavior in fretting, including the spatial distribution of these zones in scars as a function of various fretting parameters. Although various mechanisms of oxygen transport were considered, it was concluded that oxygen transport into fretting contacts is mainly driven by diffusion (Baydoun et al., 2021). 2. While the role of debris transport has been more widely acknowledged since the work of Godet, subsequent developments in this area have been largely phenomenological. However, progress has recently been made in beginning to model debris transport within the Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00005-6 Copyright © 2023 Elsevier Inc. All rights reserved.

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contact, where it has been shown that the flow rate of debris is inversely proportional to some measure of the size of the contact (Zhu et al., 2019). It should be noted that (i) since the wear rate is dependent upon the contact size, the rate of fretting wear (with either conforming or non-conforming contact geometries) is no longer in accord with the Archard wear equation; (ii) specifically for fretting with nonconforming contact geometries, the wear rate is expected to vary with exposure to wear (i.e., the wear volume is not proportional to the exposure as predicted by the Archard wear equation) (Zhu and Shipway, 2021).

Together, these insights regarding the key role of debris formation and ejection in fretting contacts pave the way for the development of an all-encompassing physically based model of fretting wear, although it is recognized that this is “practically not easy” (Baydoun et al., 2019). The interrelationships between the key processes need to be acknowledged in seeking to assimilate the information presented in the following sections. The identification of the rate-determining process is critical in the development of an understanding of wear in fretting (Shipway, 2021; Zhu et al., 2019) with a clear understanding of debris formation and ejection being central to this.

3.1.2

Wear in both sliding and fretting—Contrasts in the transport of species into and out of the contacts

One of the foundational insights in understanding wear in a sliding contact was presented in the work of Archard (1953) where he proposed that contact of two bodies occurs at asperities, with the true contact area summed across the asperities in a contact being closely proportional to the normal load. Via a number of assumptions, Archard’s law was proposed which indicates that the wear volume is proportional to the total distance slid and to the load carried by the contact; however, in the context of fretting, it is notable that Archard’s law indicates that the wear volume should be independent of the size of the contact (i.e., wear is proportional to load and not to contact pressure). The diversity of mechanisms that have been proposed to describe the manner in which material is removed from surfaces in sliding wear indicates the complexity of the process. A broad distinction is made between mild wear (where the debris is typically made up of small particles of metal oxide) and severe wear (where the debris emanating from the contact is made up of larger flakes of metallic material). In mild wear, worn surfaces are often covered with a debris bed, and under some circumstances, such debris beds can build to form a smooth, coherent glaze layer that protects the underlying material from further wear (Quinn, 1992). Wear in sliding or fretting requires consideration of both mechanical and chemical aspects of the process as follows: l

mechanical damage to the materials, by plastic deformation (from asperity level to macroscopic), fracture or fatigue;

The role of tribologically transformed structures and debris in fretting of metals l

l

69

chemical reactions of the surface materials, normally involving environmental species such as oxygen, and the influence of the resulting reaction products on subsequent rates of reaction or on the ensuing mechanical damage to the materials in the contact; increases in temperature associated with the frictional energy dissipated as heat, and the effect of this on the dominant mechanisms and rates of any mechanical or chemical degradation processes.

These influences cannot be considered independently, since they each influence the others. Some of the earliest work on fretting wear termed the process "fretting corrosion" on account of the nature of the debris as being metal oxide. However, it was also recognized that fretting wear could still take place in environments that do not support oxidation (Godfrey, 1951; Feng and Uhlig, 1954). In a naı¨ve sense, fretting wear can be considered simply as reciprocating sliding with small amplitudes, but there are in fact very significant differences between them in terms of key processes. Sliding wear and fretting wear are often distinguished from each other by reference to the amplitude of the sliding motion; sliding wear is said to occur with larger sliding amplitudes, although a very wide range of values have been suggested at which this transition occurs (Waterhouse, 1972; Ohmae and Tsukizoe, 1974; Vingsbo and S€ oderberg, 1988; Chen and Zhou, 2001). In fretting, the wear rate (volume lost per unit sliding distance) increases with increasing slip amplitude as illustrated in Fig. 3.1.1 (Vingsbo and S€ oderberg, 1988). As can be seen, it was proposed that across the gross-slip fretting regime (i.e., with displacement amplitudes increasing from
20 to 300 μm), the wear coefficient increases by almost two orders of magnitude. In contrast, in reciprocating sliding (i.e., at slip amplitudes >300 μm), it is proposed that the wear rate is independent of the slip amplitude, as predicted by Archard’s equation. It is suggested that rather than attempting to distinguish sliding wear and fretting wear from each other in terms of slip amplitude, it may be more fruitful to make the

Fig. 3.1.1 Influence of the displacement amplitude on the specific wear rate in fretting, after (Vingsbo and S€oderberg, 1988).

70

Fretting Wear and Fretting Fatigue

distinction based on the key processes associated with degradation and wear in the two cases. In both cases, the following processes must occur: l

l

l

mechanical deformation at asperities; formation of debris at those regions of deformation which may be dependent on chemical reactions with active species (such as oxygen) from the environment; if so, these active species need to be transported to the sites where these reactions take place; transport of debris away from those sites so that further wear can occur (i.e., the process of wear can continue).

In fretting, the slip amplitude is generally small compared to the smallest dimension of the contact patch; as such, the transport of active species into the contact from the environment and the transport of debris out of the contact into the environment will be much more difficult than it is in sliding wear. Within the regime of fretting, there are further complexities associated with partial slip and gross-sliding fretting; however, wherever slip occurs, there is a need for the transport of species. In the case of partial slip fretting, wear will occur so that the contact pressure falls in the region which is slipping, with more of the load being carried by the region which is stuck; this will lead to an overall reduction in the rate of wear as wear proceeds. In contrast, in gross-sliding fretting, wear will continue as long as the transport processes allow. It has been shown that when a nonconforming contact geometry is being employed, the rate of wear (in terms of the rate of recession of the surfaces) is the same at any point within the contact at any point in time, although it may vary with time as the contact patch changes in size (Zhu et al., 2019).

3.1.3

The nature of oxide debris formed in fretting

The debris associated with fretting of steel contacts in normal atmospheric conditions is generally seen to be dominated by very finely divided oxide with a hematite (Fe2O3) structure. Within this debris, small amounts of metallic material may also be observed (Halliday and Hirst, 1956; Feng and Rightmire, 1956; Kirk et al., 2019). Similarly, the debris produced for other metallic alloys is generally found to be in the form of finely divided oxides. The influence of the oxidational aspect is highlighted by the significant change in damage observed during fretting in a nonoxidizing environment, either under an inert atmosphere or in vacuum (Uhlig, 1954; Feng and Uhlig, 1954; Cai et al., 2009). Where oxidation cannot occur, fretting results in a continuous process of metal transfer between the surfaces resulting in severe roughening and plastic deformation; any fretting debris is metallic in character but is generally produced in a very much smaller quantity than when oxidation occurs. While oxide debris is generally described as “fine,” the size range reported tends to cover a wide range. The particle size of fretting debris has been reported over a range of values, and even within single tests, debris particle distributions have been seen to range from submicron up to tens of microns (Halliday and Hirst, 1956; Waterhouse,

The role of tribologically transformed structures and debris in fretting of metals

71

1972); it was proposed that these larger debris particles may be the result of the tribosintering of originally smaller debris particles (Kirk et al., 2019).

3.1.4

Formation of oxide debris in fretting—The role of oxygen supply and demand

The supply of oxygen into a fretting contact is a requirement for the formation of oxide debris, with the key issue here being the balance between the rate of supply of oxygen into the contact and the rate of consumption of oxygen in the contact in forming the debris. Depletion of oxygen may occur under certain fretting conditions when the rate of oxygen transport into the interface is not sufficient to replenish oxygen consumed in the fretting process itself (Mary et al., 2009, 2011). This thinking has been utilized in the development of a model to explain the presence of what are termed adhesive zones toward the center of fretting contacts in certain conditions (Fouvry et al., 2017; Baydoun et al., 2020); in these zones, oxide debris is absent on the surface following fretting and it is argued that this occurs due to the consumption of oxygen in the outer regions of the scar resulting in the oxygen pressure in the center being too low to support debris formation. When considering supply and demand of oxygen, care is required to ensure that rates are considered in comparable terms; the supply of oxygen into the contact is normally considered in amount per unit time, and therefore the rate of consumption (associated with the formation of wear debris) must also be considered in amount per unit time as opposed to the more normally considered amount per unit distance of sliding or amount per unit energy dissipated in the contact (Shipway, 2021). It has been suggested that the primary transport mechanism of oxygen into a fretting contact is diffusion (Fouvry et al., 2017; Baydoun et al., 2021). A number of issues influence oxygen supply into the contact, key among these being: l

l

l

Contact size: the oxygen supply is more difficult as the size of the fretting contact increases; Oxygen availability in the environment: a lower concentration of oxygen in the environment will make oxygen supply into the contact more difficult; Other media in the fretting contact: fretting in a liquid environment will reduce the access of oxygen into the contact.

The oxygen demand will be increased by any factor which results in an increase in the time-based rate of debris production (amount per unit time); for a particular contact pair with a fixed contact size, this is equivalent to saying that the oxygen demand increases as the frictional power dissipated in the contact increases. The frictional power dissipated is proportional to the slip amplitude, to the applied load and to the fretting frequency, and so changes in any of these parameters will affect the oxygen demand. However, changes in the frictional power dissipated will also result in a change in contact temperature and may also have other more subtle influences (e.g., on debris transport out of the contact). It is noted that, of the three parameters highlighted,

72

Fretting Wear and Fretting Fatigue

the fretting frequency is the one where the widest range is generally examined in fretting research and where other conflating effects are minimized, and therefore the effect of oxygen demand is most clearly seen in test programmes where the fretting frequency is considered.

3.1.4.1 Oxygen supply—Contact size It has been noted that oxygen transport into contacts will be less readily achieved as the contact size increases, with this resulting in oxygen-starved regions toward the center of larger contacts (Fouvry and Merhej, 2013; Warmuth et al., 2013, 2015; Fouvry et al., 2017) with this issue recently being both clearly demonstrated experimentally and addressed in model developments (Baydoun and Fouvry, 2020; Baydoun et al., 2020). In this latter work, some fretting tests were conducted with flat-on-flat pairs with grooves cut in one of the surfaces (parallel to the fretting direction) to facilitate oxygen transport into the contact, and it was shown the presence of grooves significantly reduced the adhesive wear zones associated with oxygen starvation.

3.1.4.2 Oxygen supply—Environmental oxygen concentration It has been observed that environmental oxygen concentration exerts a significant influence on the mechanism of fretting (Feng and Uhlig, 1954; Feng and Rightmire, 1956; Iwabuchi et al., 1983, 1986; Pendlebury, 1988). For example, in fretting of steel pairs, a critical pressure exists below which a transition in wear mechanism occurs from an oxidative-abrasive mechanism to an adhesive mechanism; when the adhesive mechanism was operative, the wear rate was observed to fall with decreasing oxygen pressure, with metallic debris being formed which was transferred between surfaces while being mostly retained in the contact (as opposed to being expelled from it). Interestingly, in torsional fretting (where oxide debris expulsion from the contact is more limited), the depth of damage was seen to increase in nonoxygen containing environments and was associated with adhesive damage (Cai et al., 2009).

3.1.4.3 Oxygen supply—Fretting in liquids It has been long recognized that in liquid-lubricated fretting contacts, the lubricant will both reduce the rate of damage in the contact (and thus reduce the demand for oxygen in debris formation) but that will also restrict the flow of oxygen into the contact (McDowell, 1958; McColl et al., 1995; Shima et al., 1997; Warmuth et al., 2016), with the restriction of oxygen supply increasing with increasing lubricant viscosity (Liu and Zhou, 2000), and being even more effective for semisolid greases (McColl et al., 1995; O’Halloran et al., 2018; Wang et al., 2011). When the oxygen supply is effectively restricted, microwelding of surfaces is generally observed if the lubricant itself is not sufficient to prevent damage.

The role of tribologically transformed structures and debris in fretting of metals

73

3.1.4.4 Oxygen demand—Fretting frequency As previously stated, many parameters affect oxygen demand in a fretting contact, but due to its relative simplicity, only the effect of fretting frequency will be discussed in this regard. It has generally been reported in the literature that increasing the frequency of oscillation results in a reduction in wear (Feng and Uhlig, 1954; Toth, 1972; Warmuth et al., 2015; Fouvry et al., 2017), and it has generally been argued that this is primarily linked to its effect on debris behavior, since the effect of frequency on wear has been found to be negligible when fretting occurs in a vacuum or an inert atmosphere (Feng and Uhlig, 1954; Vaessen et al., 1968). Increases in fretting frequency promote an increase in the time-based rate of debris formation, and thus result in an increase in the time-based demand for oxygen in the contact; in contrast, it is proposed that the effect of frequency on the supply of oxygen into the contact is small. This change in supply and demand is thought to be a key issue in the observed effects of frequency on fretting rates and mechanisms (Baydoun et al., 2021).

3.1.5

Tribo-sintering of oxide debris and glaze formation

In both fretting and sliding wear, it has been observed that above a certain critical temperature (itself dependent on the particular circumstances), the oxide-based debris which is produced in contact can form what is termed a glaze (Kayaba and Iwabuchi, 1981; Stott, 2002; Rybiak et al., 2010; Dreano et al., 2019). This layer is often described as consisting of a smoothly burnished layer on top of the compacted oxide bed below. The formation of a glaze results in a significant reduction of the rate of wear and in many cases, the wear rate essentially falls to zero (Hurricks, 1972; Kayaba and Iwabuchi, 1981; Pearson et al., 2013; Dreano et al., 2020). Glaze formation on a high strength steel has been observed at temperatures as low as 85°C (Pearson et al., 2013); however, while temperature is the key parameter in glaze formation, it has also been shown that the temperature at which glaze formation first occurred was elevated by an increase in the fretting displacement (Hayes and Shipway, 2017). The significant reduction in wear rate as glaze formation starts to occur is associated with the effect of temperature on the rate of expulsion of the debris from the contact (Dreano et al., 2020). It has been suggested (Pearson et al., 2013) that the increase in temperature first results in stronger agglomeration of individual debris particles and then (at higher temperatures) leads to a process known as tribo-sintering to form a glaze (Kato and Komai, 2007). Tribo-sintering occurs at much lower temperatures than normal sintering (i.e., sintering without the mechanical shearing action encountered in fretting); for example, neck growth has been observed to take place between 300 nm Fe2O3 particles at temperatures as low as room temperature when these particles were supplied to a sliding interface (Kato and Komai, 2007; Kato, 2008). More recently Viat et al. (2017a) proposed that the formation of a stable debris layer was promoted by high diffusion rates within the oxide, indicating the key role of sintering in the process.

74

Fretting Wear and Fretting Fatigue

In work on glaze formation in a cobalt-based alloy, the glaze material formed was shown to be complex in structure; the bulk of it was made up of metallic (unoxidized) nanocrystalline ( PO2,th) for low sliding frequency until the typical “W” fretting scar structure when adhesive wear phenomenon is operating within the inner part of the contact under high sliding frequency conditions (Fig. 3.2.19B). Despite its simplicity and limitations, rather correlations with experiments were observed underlying the potential interest of such tribological approach to predict complex fretting wear processes and seizure phenomena as well.

Mechanical analysis of freng contact (FEM)

-3

-2

200 150 100 50 0 -1 0

X (mm) 1

2

3

Wear influence on mechanical loadings

(Pa)

100000 1000 10 -3

Computaon of wear increment: modificaon of surfaces & TB

-2

0.1 -1 0

1

2

-3

-2

0 -1 0 -0.02

1

2

-0.04

0.025

-1 0 -0.025

0 -1 -0.02

3

-0.04 -0.06 -0.08 -0.1

-0.06

End

0.025

1

2

plane

3

-3

-2

(b)

0 -1 0 -0.025 -0.05

-0.075

h (mm)

(a)

0.05

cylinder

1

2

3

plane

-0.075

3

ADR influence on wear

-2

0.05

-0.05

-3

-3

0.1 0.075

0

X (mm)

X (mm) Cycle number reached?

1st 300th 2500th 5000th

0.075

cylinder

f= 5 Hz

cycles

0.1

Pressure effect on ADR ADR computaon of freng contact (DFM)

f= 1 Hz

(J/mm²)

wear profiles (mm)

Start

plane worn profile (mm)

WTO modeling

(c)

-0.12

1

3

WTO (th.)

0 -1 -0.02

-3

-0.04

WTO (th.) -0.06

exp. abrasive wear zone adhesive wear zone

1

3

exp.

-0.08 -0.1 -0.12

Fig. 3.2.19 (A) Global algorithm of the WTO multiphysics wear modeling combining friction energy (i.e., mechanical), third body simulation and ADR simulation of the contact oxygenation process (Arnaud et al., 2021), (B) Multiphysics simulation (WTO) of plane and cylinder worn profiles and debris layer thickness profiles as a function of the applied sliding frequency, (C) Comparison between WTO and the experimental wear profiles as a function of the applied sliding frequency (B and C: Ti-6AL-4 V cylinder-on-flat interface, R ¼ 80 mm, Fn,L ¼ 1066 N/mm, δs ¼ 75 μm, N ¼ 5000 cycles) (Arnaud et al., 2021).

112

Fretting Wear and Fretting Fatigue

3.2.5.3 Predicting the coating durability using the friction energy density parameter Many technological developments have taken place during the past decades to improve wear resistance by applying soft, hard, and solid lubricant coatings. These surface treatments reduce the coefficient of friction and limit seizure phenomena (Liskiewicz and Fouvry, 2005; Cai et al., 2020; Toboła et al., 2021; Langlade et al., 2001; Fridrici et al., 2003; Fouvry and Paulin, 2014). Their performances are usually established by comparing the wear volume extension although in practice the coating durability needs to be related to the critical loading cycles (Nc) when the sliding interface reaches the substrate interface. For solid lubricant coatings, Nc can be easily detected by measuring the discontinuous increase of the coefficient of friction (Fig. 3.2.20A). When the coating does not lead to significant fluctuation of the coefficient of friction, Nc can be estimated by performing interrupted tests and measuring the maximum wear depth. The coating durability Nc can be expressed as a function of the applied displacement amplitude for a given normal load as detailed by Langlade et al. (2001) (Fig. 3.2.20B). It can be also formalized as a function of the maximum friction energy density (or Archard density parameter) inputted in the

Fig. 3.2.20 Quantification of the coating durability under fretting wear (analysis of a MoS2 solid lubricant) (Fouvry and Paulin, 2014): (A) coating failure Nc when the substrate is reached (friction discontinuity), (B) evolution as a function of δs; (C) quantification of Nc as a function of the maximum local friction energy density for various contact geometries (CP: cylinder/ plane, PP: punch/plane), sliding amplitude (δs) and maximum contact pressure (pmax).

Friction energy wear approach

113

fretted interface (Fig. 3.2.20C). Investigating thin coating layers (i.e., less than few micron thicknesses), the worn surface extension can be then neglected so that the initial maximum friction energy density appears as a reliable parameter to establish the fretting wear durability of coatings. Using this approach, contact pressure and sliding amplitude can be combined through a single energy loading variable. A power law formulation can be defined so that the coating endurance can be related to a given friction energy capacity (χ) (Liskiewicz and Fouvry, 2005; Fridrici et al., 2003; Fouvry and Paulin, 2014; Liskiewicz et al., 2005) characterizing the fretting performance of given surface treatment (i.e., coating). Nc ¼

χ φ max

(3.2.24)

with φmax being the maximum friction energy density inputted in the contact during a fretting cycle given by φ max ¼ 4  δs  μe  p max

(3.2.25)

where pmax is the maximum pressure value in the contact. For hard coating layers (i.e., TiC PVD coating, (Liskiewicz and Fouvry, 2005)), the friction energy capacity could be related to the energy wear coefficient (α) assuming a linear extension of the maximum wear depth vs the maximum accumulated friction energy density so that (Fig. 3.2.21) χ¼

tn α

(3.2.26)

However, in many situations, a brutal spalling phenomenon inducing a coating failure before the fretted interface reaches the substrate can be observed. This significantly reduces the expected fretting wear endurance. This aspect can be formalized by considering the effective coating thickness (te) related to the worn thickness before the coating layer is definitively removed from the interface due to a cracking cohesion process so that (Liskiewicz and Fouvry, 2005) (Fig. 3.2.21) Nc ¼

χe φ max

(3.2.27)

with χe ¼

te tn  tr ¼ α α

(3.2.28)

with tr being the residual coating thickness just before failure. Hence, using this basic approach it is possible to formalize the fretting wear endurance of hard coatings such as TiC.

114

Fretting Wear and Fretting Fatigue

Fig. 3.2.21 Application of the friction energy density approach to formalize the fretting wear durability of a thin TiC hard coating (Liskiewicz and Fouvry, 2005): (A) evolution of the maximum wear depth as a function of the accumulated friction energy density; (B) coating failure occurring at h ¼ te due to brutal decohesion phenomenon so that an effective coating thickness (te) smaller than the nominal one (tn) must be considered to rationalize the coating durability; and (C) formalization of the coating durability Nc as a function of the maximum friction energy density inputted in the interface φmax through the effective friction energy capacity of the studied TiC coating (χe).

3.2.6

Conclusions

This chapter illustrates some aspects regarding the quantification of the fretting wear degradation using the friction energy approach. Because friction energy takes into account the friction coefficient, it appears more reliable than the usual Archard work theory in predicting the fretting wear volume extension at least under fluctuating friction coefficient conditions. However, many investigations underline that the energy wear coefficient (α) is not constant but depends on the wear mechanics involved in the fretted interface which itself depends on the loading conditions. To interpret such fluctuations both Third Body Theory (TBT) and Contact Oxygenation Concept (COC) must be considered. TBT suggests that wear kinetics is a function of the balance

Friction energy wear approach

115

between the debris formation and the debris ejection flows. The COC allows formalizing the transition from abrasive to adhesive wear response when the dioxygen partial pressure within the debris layer becomes lower than a threshold value (i.e., PO2(x,y) < PO2,th). Hence, the fresh metal exposed by the friction work cannot be oxidized so metal transfers are activated. Using TBT and COC, it is possible to interpret and rationalize the effect of contact pressure, sliding frequency, and sliding amplitude but also contact size effects regarding the fluctuation of the energy wear rate parameter. Multiphysics models combining friction energy density, third body conversion parameter but also ADR approach were recently developed to formalize, at the local scale, the surface damage evolution observed in the fretting scars. Using this multiphysics strategy, pure abrasive “U-shape” but also abrasive-adhesive “W-shape” fretting scar morphologies can be approximated. Besides, by better predicting the wear depth extension, it is also possible to better predict the durability of fretting wear palliatives such as solid lubricants and also hard coatings.

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Mary, C., Fouvry, S., Martin, J.M., Bonnet, B., 2011. Pressure and temperature effects on Fretting Wear damage of a Cu–Ni–In plasma coating versus Ti17 titanium alloy contact. Wear 272 (1,3), 18–37. McColl, I.R., Ding, J., Leen, S.B., 2004. Finite element simulation and experimental validation of fretting wear. Wear 256, 1114–1127. Merhej, Fouvry, S., 2012. Contact size effect on fretting wear behaviour: application to an AISI 52100/AISI 52100 interface. Lubr. Sci. 24, 273–290. Mohrbacker, H., Blanpain, B., Celis, J.P., Roos, J.R., Stals, L., Van Stappen, M., 1995. Oxidational wear of TiN coatings on tool steel and nitride tool steel in unlubricated fretting. Wear 188, 130–137. Nurmi, V., Hintikka, J., Juoksukangas, J., Honkanen, M., Vippola, M., Lehtovaara, A., et al., 2019. The formation and characterization of fretting-induced degradation layers using quenched and tempered steel. Tribol. Int., 258–267. Sauger, E., Fouvry, S., Ponsonnet, L., Kapsa, P., Martin, J.M., et al., 2000. Tribologically transformed structure in fretting. Wear 245 (1), 39–52. Shima, M., Suetake, H., McColl, I.R., Waterhouse, R.B., Takeuchi, M., 1997. On the behaviour of an oil lubricated fretting contact. Wear 210, 304–310. Shipway, P.H., Kirk, A.M., Bennett, C.J., Zhu, T., 2021. Understanding and modeling wear rates and mechanisms in fretting via the concept of rate-determining processes—contact oxygenation, debris formation and debris ejection. Wear 486–487, 20406. Stachowiak, G.W., 2005. Wear, Materials, Mechanics and Practice, Tribology in Practice Series. Wiley (ISBN: 0470016280, 458 pages). Stott, F.H., Wood, G.C., 1978. The influence of oxides on the friction and wear of alloys. Tribol. Int. 11, 211–218. Toboła, D., Liskiewicz, T., Yang, L., Khan, T., Boron, Ł., 2021. Effect of mechanical and thermochemical tool steel substrate pre-treatment on diamond-like carbon (DLC) coating durability. Surf. Coat. Technol. 422, 127483. Tumbajoy-Spinel, D., Descartes, S., Bergheau, J.-M.l., Lacaille, V., Guillonneau, G., Michler, J., Kermouche, G., 2016. Assessment of mechanical property gradients after impact-based surface treatment: application to pure α-iron. Mater. Sci. Eng. A 667, 189–198. Warmuth, A.R., Pearson, S.R., Shipway, P.H., Sun, W., 2013. The effect of contact geometry on fretting wear rates and mechanisms for a high strength steel. Wear 301 (1–2), 491–500. Waterhouse, R.B., 1978. Fretting. In: Scott, D. (Ed.), Treatise on Materials Science and Technology. Academic Press, pp. 259–286. Waterhouse, R.B., 1984. Fretting wear. Wear 100 (1–3), 107–118. Yue, T., Wahab, A.l., 2017. Finite element analysis of fretting wear under variable coefficient of friction and different contact regimes. Tribol. Int. 107, 274–282. Zhou, Z.R., Vincent, L., 1999. Lubrification in fretting a review. Wear 225 (9), 962–968. Zhou, Z.R., Fayeulle, S., Vincent, L., 1992. Cracking behaviour of various aluminium alloys during fretting wear. Wear 155, 317–330. Zhu, T., Shipway, P.H., 2021. Contact size and debris ejection in fretting: the inappropriate use of Archard-type analysis of wear data and the development of alternative wear equations for commonly employed non-conforming specimen pair geometries. Wear 474–475, 203710. Zhu, T., Shipway, P.H., Sun, W., 2019. The dependence of wear rate on wear scar size in fretting; the role of debris (third body) expulsion from the contact. Wear 440–441, 203081.

Lubrication approaches Taisuke Maruyama Core Technology R&D Center, NSK Ltd., Tokyo, Japan

3.3

Nomenclature aY h hc hmin k p qD qP t u x y z A C D E’ F M N QC QP R Rx Ry Vmax α β γ_ η η0 λ μ0 ρ ρ0 ν

constant in the Carreau-Yasuda equation (
) film thickness (m) central oil film thickness calculated using Hamrock-Dowson equation (m) minimum oil film thickness calculated using Hamrock-Dowson equation (m) permeability (m2) contact pressure (Pa) Darcy flow rate of base oil per unit width (m2/s) Poiseuille flow rate of the thickener network per unit width (m2/s) time (s) entrainment speed (m/s) coordinate in the rolling direction (m) coordinate in the transverse direction (m) coordinate in the film thickness direction (m) amplitude of oscillation (m) thickener concentration (
) diameter of Hertzian contact area (m) reduced modulus of elasticity (Pa) normal applied load (N) constant in Eq. (3.3.12) (
) constant in Eq. (3.3.12) (
) Couette flow rate per unit width (m2/s) Poiseuille flow rate of grease per unit width (m2/s) ratio of the Darcy flow rate of base oil to the Poiseuille flow rate of the thickener network (
) effective radius in rotation direction of rolling element (m) effective radius perpendicular to rotation direction of rolling element (m) maximum oscillating velocity (m/s) pressure-viscosity coefficient (Pa
1) constant in the modified Kozeny-Carman equation (m2) shear rate (s
1) equivalent viscosity (Pas) equivalent viscosity at atmospheric pressure (Pas) relaxation time in the Carreau-Yasuda equation (s) base oil viscosity at atmospheric pressure (Pas) 9

+ 1:34p density of lubricant expressed as ρ ¼ 0:5910 ρ0 (kg/m3) 0:59109 + p

density of lubricant at atmospheric pressure (kg/m3) kinematic viscosity at atmospheric pressure (mm2/s)

Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00007-X Copyright © 2023 Elsevier Inc. All rights reserved.

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3.3.1

Fretting Wear and Fretting Fatigue

Introduction

Fretting wear occurs in the presence of minute oscillation between two objects in contact (Zhu and Zhou, 2011). This wear leads to the damage of various mechanical parts, such as deterioration of acoustic characteristics of the bearing, increase in bearing torque, and peeling, which starts from wear tracks. This in turn deteriorates the function, performance, and reliability of the entire machine besides the affected parts and results in major economic losses in terms of maintenance, repair, and replacement of parts. Therefore, identifying appropriate lubrication approaches for preventing fretting wear that occurs in the contact area of mechanical parts is very important. As fretting wear is caused by minute oscillations, it can be easily imagined that even if a lubricant is used, the oil film will break down and result in metallic contact when the amplitude decreases. Therefore, understanding the lubrication mechanism in detail during this minute oscillation leads to the prevention of fretting wear by the lubricant, and research on fretting wear in various cases of oil and grease lubrication have already been conducted. First, the research conducted on fretting wear in oil lubrication is introduced. Kalin and Vizˇintin (2006) conducted fretting wear tests under oil lubrication in pure sliding point contacts and investigated the effectiveness of DLC (diamond-like carbon) films. They found that, at a low amplitude, fretting wear could be reduced by applying a DLC film; however, at a high amplitude, no significant differences from the untreated test specimen were observed, indicating the presence of an oil film. Maruyama and Saitoh (2010) measured the oil film thickness under minute oscillation in point contacts by using optical interferometry and found that the oil film collapse occurred at a critical amplitude, and with the amplitudes above this level, resulted in the oil film becoming thicker with an increase in the viscosity of the lubricating oil. Furthermore, they conducted fretting wear tests using thrust ball bearings and confirmed that, at a high amplitude, fretting wear could be reduced by increasing the viscosity of oil (Maruyama and Saitoh, 2011a; Maruyama et al., 2017). Han et al. (2019) also observed oil film behavior under minute oscillation and found that the oil film broke at a critical amplitude, and below this level, wear was found to increase. Next, the research conducted in grease lubrication is described. Kita and Yamamoto (1997) conducted fretting wear tests under minute oscillation by using lithium soap grease with thrust ball bearings and indicated that fretting wear decreased with decrease in the base oil viscosity of the grease. Yano et al. (2010) showed that fretting wear decreased with decrease in the base oil viscosity under minute oscillation, but decreased with an increase in the base oil viscosity in case of variable load. Maruyama and Saitoh (2011a), Maruyama et al. (2017) conducted minute oscillation tests using urea grease and showed that fretting wear decreased with decrease in the base oil viscosity in the case of high oscillations, whereas viscosity had almost no effect in the case of minute oscillations. Schwack et al. (2020) conducted fretting wear tests using angular contact ball bearings and confirmed that fretting wear could be reduced by using grease with a low-viscosity base oil. Thus, it has been reported that the design guidelines for lubricants for preventing fretting are exactly the opposite in the case of oil and grease lubrication. In this

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chapter, the mechanism for reducing fretting wear that occurs during minute oscillation is explained for each case of oil and grease lubrication, and the optimal lubrication approaches for each case are presented.

3.3.2

Parameter definition

3.3.2.1 Amplitude ratio Before explaining the lubrication approach applied to prevent fretting wear, the degree of minute oscillation is defined, which differs according to the size of the contact area. In this chapter, the amplitude ratio (Maruyama and Saitoh, 2010, 2011a; Maruyama et al., 2017; Schwack et al., 2020) is defined as a parameter for quantifying the degree of amplitude. If the amplitude is set as A and the diameter of the Hertzian contact diameter is set as D, then the amplitude ratio is expressed as A/D. As indicated in Fig. 3.3.1, when A/D < 1, the contact areas undergo minute oscillation in the overlapping ranges. For example, in a ball bearing with a groove on the transfer surface, when the rolling element swings in the direction of rolling in the bearing, the variable D in Fig. 3.3.1 becomes the minor axis of the contact ellipse (Schwack et al., 2020).

3.3.2.2 Damage ratio The damage ratio (Maruyama and Saitoh, 2011a; Maruyama et al., 2017; Schwack et al., 2020) is the ratio between the surface roughness (i.e., maximum height roughness) obtained before and after the fretting wear tests, expressing the degree of adhesion between the two rubbing surfaces.

3.3.3

Oil lubrication

3.3.3.1 Influence of viscosity First, the effect of viscosity of the lubricating oil on fretting wear in point contact under oil lubrication is described (Maruyama and Saitoh, 2011a; Maruyama et al., 2017). Fig. 3.3.2A shows the fretting wear tracks when the amplitude ratio A/D is changed using additive-free poly-α-olefin oil (PAO) with two types of viscosities. Here, the maximum oscillating velocity Vmax remains constant (Vmax ¼ 20 mm/s) even if A/D is changed. Annular fretting wear is observed when A/D < 1, which is Fig. 3.3.1 Definition of amplitude ratio. Credit: ©2016 Taylor & Francis.

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Fig. 3.3.2 Influence of oil viscosity on fretting wear for varying amplitude ratio at Vmax ¼ 20 mm/s: (A) photographs of fretting wear track and (B) relationship between amplitude ratio and damage ratio; oil B: ν ¼ 30 mm2/s at 40°C, oil E: ν ¼ 396 mm2/s at 40°C. Credit: ©2016 Taylor & Francis.

attributed to the Mindlin slip (Mindlin, 1949). Schwack et al. (2020), Schwack et al. (2018) and Shima and Jibiki (2008) also found that an annular wear track similar to that shown in Fig. 3.3.2A occurred when the oscillation angle of the rolling bearing was low. Furthermore, when A/D > 1.5, there was almost no wear with an increase in the viscosity of PAO. Zhu and Zhou (2011) also conducted fretting wear tests in oil lubrication and indicated that an oil film can be formed in the contact area by increasing the amplitude. Fig. 3.3.2B shows the relationship between the amplitude ratio and damage ratio. From Fig. 3.3.2B, at A/D < 1.5, it is suggested that the oil film collapses regardless of the viscosity because the damage ratio is not affected by the viscosity. Furthermore, at A/D  1, the damage ratio is the maximum. This is because the wear

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particles increase with the amplitude in the range of A/D < 1, and repeatedly adhere in the contact area; however, the particles are removed from the contact area if A/D > 1. Besides, when A/D > 1.5, the damage ratio can be reduced by increasing the viscosity of oil; hence, an oil film is presumed to have formed in the contact area.

3.3.3.2 Influence of oscillation frequency Fig. 3.3.3 shows the effect of oscillation frequency (i.e., Vmax) on fretting wear at A/D ¼ 1.9 (Maruyama et al., 2017). At low frequencies, a similar degree of fretting occurs regardless of the viscosity. Meanwhile, at high frequencies, fretting wear can be reduced by using a high-viscosity oil. As fretting wear can be reduced with higher oscillation frequency and viscosity, it is suggested that the oil film thickness is thicker than the surface roughness of the contacting areas under this condition. Details of the oil film behavior under minute oscillation are presented in the next section.

3.3.3.3 Mechanism for fretting wear reduction in oil lubrication Dowson and Higginson (1996) theoretically predicted the oil film distribution in the contact ellipse, which considered not only the elastic deformation of the contact area but also the viscosity and density of the lubricant under the high contact pressure. Such a lubrication state is called elastohydrodynamic lubrication (EHL, (Dowson and Higginson, 1996)), and its contact state is called elastohydrodynamic contact (EHD contact, (Gohar and Cameron, 1967)). Hamrock and Dowson (1977a) used numerical analysis to obtain approximate expressions for the minimum oil thickness hmin [m] and central oil film thickness hc [m], respectively. Eqs. (3.3.1) and (3.3.2) are the approximate equations for the contact ellipse, generally called Hamrock-Dowson equations (Zolper et al., 2012; Maruyama et al., 2019). Fig. 3.3.3 Relationship between maximum oscillating velocity and damage ratio with varying viscosity at A/D ¼ 1.9; Oil A: ν ¼ 19 mm2/s at 40°C; Oil E: ν ¼ 396 mm2/s at 40°C. Credit: ©2016 Taylor & Francis.

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0:073
0:64 !! Ry hmin η0 u 0:68 F 0:49 ¼ 3:63 0 ðαE0 Þ 1
exp
0:70 E Rx Rx Rx E 0 Rx 2


0:067 hc η0 u 0:67 F 0 0:53 ¼ 2:69 0 ðαE Þ E Rx Rx E0 Rx 2

(3.3.1)
0:64 !! Ry 1
0:61exp
0:75 Rx (3.3.2)

The above equations can be applied in a steady state under the fully flooded lubrication (Hamrock and Dowson, 1977a), where the effects of shear thinning (Liu et al., 2007) are ignored. The mechanism of oil film formation is extremely complicated in nonsteady states that can occur in actual bearings, such as for starved lubrication (Wedeven et al., 1971; Hamrock and Dowson, 1977b; Cann et al., 2004; Maruyama and Saitoh, 2015; Nogi, 2015), thermal EHL (Murch and Wilson, 1975; Goksem and Hargreaves, 1978a,b; Zhou and Hoeprich, 1991; Kumar et al., 2010; Bair, 2005), acceleration/deceleration motion (Sugimura et al., 1998), and minute oscillatory motion (Maruyama and Saitoh, 2010; Han et al., 2019; Wang et al., 2005). Here, the following Eq. (3.3.3), which is a simplification of the Reynolds equation (Reynolds, 1886), is used as the transport equation for the lubricating oil flowing through the contact area when deriving Hamrock-Dowson equations as shown above:

 ∂ ρh3 ∂p ∂ ρh3 ∂p ∂ðρhÞ + ¼u ∂x 12η ∂x ∂y 12η ∂y ∂x

(3.3.3)

However, as the entrainment speed u ¼0 m/s at the stroke end of minute oscillation, the wedge action on the right side of Eq. (3.3.3) disappears, which means the oil film would collapse. In reality, the oil flow resistance accompanying the outflow of oil outside the contact area (i.e., squeeze action) occurs when the wedge action becomes lower and the oil film decreases near the stroke end, and a squeeze film (Dowson and Jones, 1967; Herrebrugh, 1970; Kaneta et al., 2007, 2011) is formed in the contact area to contribute to the maintenance of the oil film. The transport equation of oil flowing through the contact area under minute oscillation is expressed as Eq. (3.3.4), which considers the squeeze action:

 ∂ ρh3 ∂p ∂ ρh3 ∂p ∂ðρhÞ ∂ðρhÞ + + ¼u ∂x 12η ∂x ∂y 12η ∂y ∂x ∂t

(3.3.4)

That is, the time when u ¼ 0 m/s at the end of the stroke is too short to break down the oil film immediately. Fig. 3.3.4 compares the experimental results obtained by optical interferometry and the analysis results that consider the squeeze action for the oil film profile in the oscillatory EHD contact (Wang et al., 2005). The experimental results are highly consistent with the analysis results, and the oil film can be retained even at

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Fig. 3.3.4 Comparison of experimental and theoretical results under the oscillation. Credit: ©2005 Elsevier.

the stroke end by considering the squeeze action. Wang et al. (2005) indicated that the decreased amount of oil supplied to the contact area at high frequencies required the consideration of starved lubrication, but that its effect on the minimum oil film thickness was low.

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Next, the effect of amplitude ratio A/D on the oil film behavior in EHD point contacts is shown (Maruyama and Saitoh, 2010). Fig. 3.3.5 shows the mid-plane thickness profiles along the entraining direction at the stroke end of oscillation when the amplitude ratio is changed by using PAO, which indicates that the oil film is entrained from the left to right side of figures. These figures show that the oil film can be retained even at the stroke end if A/D > 1.5. Besides, the position of the maximum oil film thickness at the stroke end moves toward the center of the contact area and the area where the oil film is thick becomes wider as the amplitude ratio is increased. In other words, the amount of oil entrained into the contact area increases with the amplitude ratio. Meanwhile, Fig. 3.3.5B shows oil film profiles when Vmax is changed at A/D ¼ 1.5, where the oil film breaks down. The maximum oil film thickness in the contact area increases with Vmax; however, the oil film remains collapsed. Fig. 3.3.6A–C show the relationship between amplitude ratio and minimum oil film thickness during one stroke under varying viscosity, oscillation frequency, and maximum contact pressure, respectively. Even if the viscosity, frequency, and contact pressure are changed, the critical amplitude ratio at which the oil film breaks down is approximately 1.5. However, when A/D > 1.5, the minimum oil film thickness increases with increasing viscosity and frequency, whereas it is hardly affected by the contact pressure. A similar tendency can be observed with the load dependence of the oil film thickness in steady state shown in Eqs. (3.3.1) and (3.3.2). In other words, this is attributed to the fact that increases in viscosity and velocity improve oil film formation due to wedge action; however, the increase in contact pressure (i.e., normal load) leads to the increase in the contact area, which offsets the decrease in oil film thickness due to the increase in load. Fig. 3.3.6 also shows that the minimum oil film thickness is not affected by the amplitude ratio if A/D > 3.0 even when viscosity, oscillation frequency, and contact pressure are changed. Fig. 3.3.5A shows that the thicknesses of the thin part of the oil film formed at the inlet and outlet of the contact area gradually becomes equal as the amplitude ratio increases, and it is believed that sufficient lubricating oil is entrained to the contact area. Therefore, if A/D > 3.0, increasing the amplitude ratio is not expected to reduce fretting wear. From Figs. 3.3.2B and 3.3.6A, the fretting wear can be decreased if A/D > 1.5 by using a high-viscosity oil because the oil film thickness in the contact area would increase. Han et al. (2019) also observed the oil film behavior under minute oscillation by optical interferometry and confirmed that the critical amplitude ratios of pure rolling and pure sliding are virtually identical. However, they reported that the critical amplitude ratio of the EHD point contact is 1.0. The Reynolds equation shown in Eq. (3.3.4) assumes that wall slip of lubricant does not occur at the contact surface (Kaneta et al., 1990; Guo et al., 2009); hence, the theoretical critical amplitude ratio should be 1.0 when the effects of surface roughness are ignored. However, as the viscosity of oil is affected by the pressure distribution in the contact area according to the Barus equation (Barus, 1893) shown below, the viscosity at the contact edge becomes considerably lower than that within the contact area. η ¼ η0 exp ðαpÞ

(3.3.5)

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Fig. 3.3.5 Mid-plane film thickness profiles at the stroke end of oscillation; oil: PAO (viscosity at 40°C: ν ¼ 411 mm2/s); (A) influence of amplitude ratio on oil film behavior at Vmax ¼ 20 mm/s and (B) influence of Vmax on oil film behavior at A/D ¼ 1.5. Credit: ©2010 Elsevier.

128

Fig. 3.3.6 Relationship between amplitude ratio and minimum oil film thickness under minute oscillation variation: (A) kinematic viscosity at 40°C, (B) maximum oscillating velocity, and (C) maximum contact pressure. Credit: ©2010 Elsevier.

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As the viscous resistance associated with the leakage of oil to the outside of the EHD contact area decreases, the minimum thickness of the squeeze film formed at the contact edge may be lower than the surface roughness. Furthermore, Sugimura et al. (1998) experimentally confirmed that the oil film thickness during acceleration was thinner than that in the steady state, and pointed out the effect of the squeeze action. Therefore, the theoretical critical amplitude ratio is 1.0 when it is assumed that the fluid does not generate wall slip; however, the minimum oil film thickness around the squeeze film may not be sufficiently thicker than the surface roughness because of the low high-pressure viscosity at the contact edge. Besides, it is difficult for an oil film to form during acceleration when the contact area begins to move from the stroke end. As shown in Fig. 3.3.5A, a higher amplitude ratio increases the flow rate of the lubricating oil to the contact area, expands the thick area of the oil film at the stroke end, and gradually increases the minimum oil film thickness around the squeeze film; hence, the minimum oil film thickness can be sufficiently increased beyond the surface roughness if A/D > 1.5. The fretting wear test results shown in Fig. 3.3.2 are also considered to be the results obtained by reflecting the influence of the surface roughness. Note that, in order to reduce the critical amplitude ratio as shown in this section to less than 1.5, the minimum oil film thickness of the squeeze film at the stroke end must be sufficiently thicker than the surface roughness. This can be achieved using two approaches. One is to use a lubricating oil with a high pressure-viscosity coefficient (i.e., traction oil). Ohno and Hirano (2001), Ohno and Yamada (2007) indicated that a lubricating oil behaves as a viscoelastic solid when its high-pressure viscosity in EHD contact exceeds 107 Pas, and that there is no leakage from the squeeze film when using an oil with a high pressure-viscosity coefficient. Fig. 3.3.7A shows the relationship between amplitude ratio and minimum oil film thickness during one stroke when using traction oil (ν ¼ 31 mm2/s at 40°C) (Maruyama and Saitoh, 2008). When PAO (α  15 GPa
1 at 25°C) is used, the critical amplitude ratio is constant at 1.5 irrespective of how much the viscosity is increased; however, when using traction oil (α  30 GPa
1 at 25°C), the critical amplitude ratio can be reduced. Fig. 3.3.7B shows the mid-plane film thickness profiles at the stroke end of A/D ¼ 1.0 comparing various oils. When traction oil is used, an oil film with a sufficient thickness is formed in the vicinity of the outlet of the contact area. Meanwhile, when PAO is used, the oil film is not formed near the outlet. This is true even when the viscosity of PAO is increased. For reference, Fig. 3.3.7B also shows the mid-plane film thickness profiles when using polyol ester (POE, ν ¼ 245 mm2/s at 40°C). The pressure-viscosity coefficient of POE is similar to that of PAO, and the critical amplitude ratio is found to be similar to that of PAO. Therefore, it is indicated that the critical amplitude ratio is more significantly affected by the high-pressure viscosity rather than the viscosity of the atmospheric pressure. Fig. 3.3.7C shows a graph in which the traction oil results are added to the fretting wear test results shown in Fig. 3.3.2B, which also indicates that the critical amplitude ratio is reduced by increasing the pressure-viscosity coefficient. Therefore, traction oil has a very high pressure-viscosity in the contact area and acts as a viscoelastic solid, thereby suppressing any leakage of oil from the contact area. As shown in Fig. 3.3.7B, the thick region of the squeeze film is expanded, thus

130

Fig. 3.3.7 Experimental results obtained using traction oil under EHD point contact: (A) relationship between amplitude ratio and minimum oil film thickness during minute oscillation, (B) mid-plane film thickness profiles at the stroke end of A/D ¼ 1.0, and (C) relationship between amplitude ratio and damage ratio. Credit: ©2008 Elsevier (Fig. 3.3.7B, C are original.)

Fretting Wear and Fretting Fatigue

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resulting in a critical amplitude ratio becoming 1.0. Actually, using traction oil as a bearing lubricant increases the bearing torque, which is not favorable as a means of lowering the critical amplitude ratio. Ohno and Yamada (2007) experimentally confirmed that even with PAO, the leakage from the squeeze film can be reduced by increasing the contact pressure. Therefore, it is predicted that the critical amplitude ratio of PAO can be lowered to 1.0 by increasing the contact pressure. The second approach of reducing the critical amplitude ratio is to oscillate minutely under line contact instead of point contact (Maruyama and Saitoh, 2011b). In the case of line contact, the second term on the left-hand side of Eq. (3.3.4) can be ignored, resulting in the following equation:
 ∂ ρh3 ∂p ∂ðρhÞ ∂ðρhÞ + ¼u ∂x 12η ∂x ∂x ∂t

(3.3.6)

Therefore, the Poiseuille flow that occurs at the inlet and outlet of the line contact in the direction perpendicular to the oscillation direction can be ignored (Vichard, 1971), and thus the leakage of oil from the contact area can be suppressed, compared to point contact. Fig. 3.3.8A shows the relationship between amplitude ratio and minimum oil film thickness in EHD line contacts during one stroke when using various oils. The critical amplitude ratio can be reduced to 1.0 in case of line contact even if PAO is used, similar to the case of using traction oil. Fig. 3.3.8B shows the mid-plane film thickness profiles at the stroke end of A/D ¼ 1.0. Oil films of sufficient thickness can be formed near the outlet of the contact area regardless of the viscosity and the pressure-viscosity coefficient. Since the contact area of the ball bearing is elliptical, side leakage is less likely to occur than point contact, and the critical amplitude ratio might be less than 1.5.

3.3.4

Grease lubrication

3.3.4.1 Influence of base oil viscosity Next, an approach for reducing fretting wear in grease lubrication will be described (Maruyama and Saitoh, 2011a; Maruyama et al., 2017). Fig. 3.3.9 shows the effect of varying A/D using two types of additive-free grease with different base oil viscosities on fretting wear. For both types of grease, the thickener is urea, the base oil is PAO, and the worked penetration is approximately 210 (Maruyama et al., 2017). Fig. 3.3.9 indicates that the grease base oil viscosity does not affect the fretting wear if A/D < 1.0, which is similar to the test results shown in Fig. 3.3.2 for oil lubrication. In contrast to the results obtained for oil lubrication, the fretting wear can be decreased by decreasing the base oil viscosity if A/D > 1.0. Fig. 3.3.10 shows photographs of the rubbing part of the test piece after ultrasonic cleaning with petroleum benzene (Maruyama et al., 2017). Fig. 3.3.10A indicates that white deposits formed by the thickener (i.e., thickener layer) are confirmed not only in the rubbing part but also in its surrounding areas when using low-viscosity grease. Meanwhile, only fretting wear tracks are observed when high-viscosity grease is used,

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Fig. 3.3.8 Experimental results obtained using traction oil under EHD line contacts: (A) relationship between amplitude ratio and minimum oil film thickness during minute oscillation at Vmax ¼ 10 mm/s and (B) mid-plane film thickness profiles at the stroke end of A/D ¼ 1.0. Credit: ©2011 Japanese Society of Tribologists (Fig. 3.3.8B is original.)

as shown in Fig. 3.3.10B. Maruyama and Saitoh (2011a) used an EHL test rig to perform in situ observations of the EHD contact area during minute oscillation under grease lubrication. Fig. 3.3.11 shows that the interference fringes occasionally disappear when low-viscosity base oil grease is used (e.g., 10, 15, 20, and 30 min after the start of the tests). These results indicate that a thickener layer that is thick enough that interference fringes cannot be observed is formed in the contact area, and is repeatedly

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Fig. 3.3.9 Relationship between amplitude ratio and damage ratio at Vmax ¼ 20 mm/s for varying base oil viscosity; base oil: PAO, thickener: urea, worked penetration: approximately 210, Grease C: ν ¼ 19 mm2/s at 40°C, Grease H: ν ¼ 396 mm2/s at 40°C. Credit: ©2016 Taylor & Francis.

Fig. 3.3.10 Photographs of washed specimens after fretting wear tests at A/D ¼ 1.9 and Vmax ¼ 20 mm/s using (A) Grease C: ν ¼ 19 mm2/s at 40°C and (B) Grease H: ν ¼ 396 mm2/s at 40°C. Credit: ©2016 Taylor & Francis.

formed and collapsed. Cen et al. (2014) also noted the presence of a thickener layer under grease lubrication. Meanwhile, as interference fringes are always observed when high-viscosity base oil grease is used, the thickener layer is predicted to be thinner than when low-viscosity grease is used. Therefore, in the case of grease lubrication, lowering the viscosity of the base oil forms a thickener layer in and around the contact area, leading to a reduction in fretting wear. As the viscosity effects are confirmed if A/D > 1.0 (as indicated in Fig. 3.3.9), the critical amplitude ratio of the EHD point contact in grease lubrication is approximately 1.0. The fact that the critical

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Fig. 3.3.11 left adjust hspace Time evolution of interference fringe images of the contact area and its vicinity at A/D ¼ 2.0 and Vmax ¼ 20 mm/s; (top) Grease C: ν ¼ 19 mm2/s at 40°C and (bottom) Grease H: ν ¼ 396 mm2/s at 40°C. Credit: ©2011 Japanese Society of Tribologists.

amplitude ratio of grease lubrication is lower than that of oil lubrication regardless of the fact that PAO is used in the base oil of grease is attributed to the effects of the thickener. In other words, it is supposed that the thickener layer formed in the contact area behaves as a viscoelastic solid in a manner similar to the test results obtained using traction oil (Fig. 3.3.7). In the next section, the effects of the thickener concentration of grease (i.e., worked penetration) on fretting wear are described.

3.3.4.2 Influence of worked penetration Fig. 3.3.12A shows the relationship between the worked penetration and damage ratio at A/D ¼ 1.9, and Fig. 3.3.12B shows the relationship between the worked penetration and oil separation of the grease (Maruyama et al., 2017). Fig. 3.3.12A shows that fretting wear can be decreased with higher worked penetration and lower viscosity of the base oil. Besides, Fig. 3.3.12B shows that the oil separation increases with higher worked penetration and lower viscosity of the base oil. Kita and Yamamoto (1997) and Yano et al. (2010) also found that a higher worked penetration of the grease increases oil separation and decreases fretting wear. Therefore, in the case of grease lubrication, reducing base oil viscosity and increasing worked penetration increase oil separation, which is believed to reduce fretting wear because of the formation of a thickener layer in the contact area, as shown in Figs. 3.3.10 and 3.3.11.

3.3.4.3 Influence of oscillation frequency Fig. 3.3.13 shows the relationship between oscillation frequency and damage ratio at A/D ¼ 1.9 under varying grease base oil viscosity and worked penetration (Maruyama et al., 2017). When using low-viscosity grease, the damage ratio increases as oscillation frequency decreases regardless of worked penetration. In contrast, when using high-viscosity grease, the damage ratio increases with increase in oscillation frequency. It means that the relationship between oscillation frequency and damage ratio is reversed depending on the base oil viscosity. Furthermore, fretting wear also increases with lower worked penetration (i.e., increased thickener concentration) when high-viscosity base oil grease is used. Fig. 3.3.14 shows the results of observing

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Fig. 3.3.12 Influence of worked penetration on fretting wear and oil separation: (A) relationship between worked penetration and damage ratio at A/D ¼ 1.9, Vmax ¼ 20 mm/s and (B) relationship between worked penetration and oil separation; base oil: PAO, thickener: urea; closed circles: base oil viscosity ν ¼ 19 mm2/s at 40°C; open squares: ν ¼ 396 mm2/s at 40°C. Credit: ©2016 Taylor & Francis.

fretting wear tracks after ultrasonic cleaning of the test piece with petroleum benzene (Maruyama et al., 2017). When both base oil viscosity and oscillation frequency are high, only wear tracks are observed, as shown in Fig. 3.3.14D. In contrast, when viscosity and oscillation frequency are both low, the thickener layer is observed around the contact area, as shown in Fig. 3.3.14A. However, as shown in Fig. 3.3.13, the damage ratio increases when viscosity and oscillation frequency are both low. That is, the grease with a low-viscosity base oil grease (i.e., high oil separation) forms a thickener layer around the contact area; however, it does not necessarily form within the contact

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Fig. 3.3.13 Relationship between maximum oscillating velocity and damage ratio at A/D ¼ 1.9 for varying viscosity and worked penetration; base oil: PAO, thickener: urea; Grease C: base oil viscosity ν ¼ 19 mm2/s at 40°C and worked penetration ¼ 200; Grease E: ν ¼ 19 mm2/s at 40°C and worked penetration ¼ 320; Grease H: ν ¼ 396 mm2/s at 40°C and worked penetration ¼ 215; and Grease K: ν ¼ 396 mm2/s at 40°C and worked penetration ¼ 339. Credit: ©2016 Taylor & Francis.

area. Furthermore, as shown in Fig. 3.3.14C, a thickener layer is confirmed by lowering oscillation frequency even though high-viscosity grease is used, and fretting wear decreases as shown in Fig. 3.3.13. Therefore, the formation of the thickener layer is affected by not only oil separation but also oscillation frequency. The mechanism of thickener layer formation in the EHD contact area for grease lubrication is described in the following section.

3.3.5

Mechanism for fretting wear reduction in grease lubrication

To explain the mechanism of reducing fretting wear during minute oscillation in grease lubrication, the experimental results of oil and grease lubrication are compared (Maruyama et al., 2017). Fig. 3.3.15A shows a graph comparing the results of oil and grease lubrication in terms of the relationship between viscosity and damage ratio at A/ D ¼ 1.9. Besides, Fig. 3.3.15B shows the relationship between the base oil viscosity and oil separation of grease. Here, the worked penetration is constant at approximately 210 even if the base oil viscosity is changed. Fig. 3.3.15A shows that fretting wear decreases with increased viscosity in the case of oil lubrication. This is because the oil film becomes thicker with increased viscosity if A/D > 1.5, as shown in

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Fig. 3.3.14 Photographs of washed specimens after fretting wear tests at A/D ¼ 1.9; base oil: PAO, thickener: urea; Grease E: ν ¼ 19 mm2/s at 40°C and worked penetration ¼ 320, and Grease K: ν ¼ 396 mm2/s at 40°C and worked penetration ¼ 339; (A) Vmax ¼ 2.3 mm/s using Grease E (low viscosity); (B) Vmax ¼ 91.6 mm/s using Grease E (low viscosity); (C) Vmax ¼ 2.3 mm/s using Grease K (high viscosity); and (D) Vmax ¼ 91.6 mm/s using Grease K (high viscosity). Credit: ©2016 Taylor & Francis.

Fig. 3.3.6A. In contrast to the oil lubrication results, the fretting wear can be decreased with lower base oil viscosity in the case of grease lubrication. Fig. 3.3.15B shows that oil separation increases with lower base oil viscosity, which is the opposite of the relationship between viscosity and damage ratio, as shown in Fig. 3.3.15A. It has been conventionally considered that grease with a higher oil separation allows for more supply of bled oil to the contact area; thus, fretting wear can be reduced (Kita and Yamamoto, 1997; Yano et al., 2010). However, Fig. 3.3.15A shows that the damage ratio is lower for grease lubrication than that for oil lubrication in case of low viscosity; hence, this cannot be explained with conventional thinking. That is, the thickener clearly contributes to lubrication in case of grease lubrication. Furthermore, Figs. 3.3.10 and 3.3.11 show that a thickener layer is more likely to form with grease, which has higher oil separation (i.e., lower base oil viscosity). (Hurley and Cann, 2000a,b) confirmed that an oil-rich material existed near the contact area and a deposited film of thickener was formed within the track when tetraurea grease was used.

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Fretting Wear and Fretting Fatigue

Fig. 3.3.15 Influence of kinematic viscosity on fretting wear and oil separation: (A) relationship between kinematic viscosity and damage ratio at A/D ¼ 1.9 comparing oil and grease lubrication, Vmax ¼ 20 mm/s and (B) relationship between kinematic viscosity and oil separation of grease; base oil: PAO, thickener: urea, worked penetration 210. Credit: ©2016 Taylor & Francis.

Therefore, it is not the case that higher oil separation of the grease allows for an easier supply of the bled oil in the contact area. Rather, as shown in Fig. 3.3.16A, the bled oil is more easily removed from the contact area and its surroundings, and the thickener concentration near the contact area increases (Maruyama et al., 2017). Then, the grease with increased thickener concentration is entrained into the contact area to form a thickener layer, thereby reducing the fretting wear. In contrast, when the oil separation is low, the grease is removed from the contact area and its surroundings, as shown in Fig. 3.3.16B, which results in starved lubrication, thereby increasing the fretting wear (Maruyama et al., 2017). Fig. 3.3.17 shows the relationship between oil separation and damage ratio (Maruyama et al., 2017). As this graph shows that the damage ratio with lower oil separation approaches the result of no lubrication, it means that a lower oil separation

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Fig. 3.3.16 Schematic representation of EHD contact area and its vicinity in the case of (A) high oil separation and (B) low oil separation. Credit: ©2016 Taylor & Francis.

Fig. 3.3.17 Relationship between oil separation and damage ratio at A/D ¼ 1.9 and Vmax ¼ 20 mm/s using various greases; base oil: PAO, thickener: urea. Credit: ©2016 Taylor & Francis.

139

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Fretting Wear and Fretting Fatigue

results in the replenishment of grease being less likely to occur. According to Fig. 3.3.9, the fact that the damage ratio of the high-viscosity grease always has a high value without being affected much by the amplitude ratio can be simply explained by the starved lubrication. (Hurley and Cann, 2000a,b) confirmed that tetraurea grease had more thickener within the track than lithium grease. They indicated that replenishment did not occur and starved lubrication was more likely to occur based on the result that the released base oil was not confirmed in the vicinity of the contact area when lithium grease was used. Yano et al. (2010) also confirmed that the fretting wear during minute oscillation was greater in lithium grease than in urea grease. Therefore, as grease with low oil separation is more likely to result in starved lubrication, fretting wear is considered to have increased. Next, we compare the results of oil and grease lubrication in terms of the effect of oscillation frequency on fretting wear. Fig. 3.3.18A shows the results of low viscosity Fig. 3.3.18 Relationship between maximum oscillating velocity and damage ratio at A/D ¼ 1.9; (A) comparison of lowviscosity oil and grease (ν ¼ 19 mm2/s at 40°C) and (B) comparison of highviscosity oil and grease (ν ¼ 396 mm2/s at 40°C); base oil: PAO, thickener: urea; low worked penetration ( 210): Grease C and H; high worked penetration ( 330): Grease E and K. Credit: ©2016 Taylor & Francis.

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and Fig. 3.3.18B shows those of high viscosity (Maruyama et al., 2017). These are experimental results at A/D¼ 1.9, and the effect of worked penetration on the fretting wear is also shown, particularly for grease lubrication. Fig. 3.3.18A shows that in the case of low-viscosity grease, fretting wear occurs to the same extent as that of the oil lubrication case regardless of the worked penetration in low frequency. In the case of oil lubrication, as shown in Fig. 3.3.6B, as the oil film in the EHD point contact breaks down by reducing the oscillation frequency even if oscillations occur at an amplitude higher than the critical amplitude ratio, the thickener layer is similarly broken in the case of grease lubrication. In other words, when a grease with a high oil separation (i.e., low-viscosity base oil) is used, a thickener layer that is sufficiently thicker than the surface roughness cannot be formed within the contact area if both the oscillation frequency and viscosity are too low, although a thickener layer is formed around the contact area, as shown in Fig. 3.3.14A. Meanwhile, the damage ratio of grease lubrication is lower than that of oil lubrication when the oscillation frequency is high. Therefore, grease with increased thickener concentration around the contact area is considered to behave like a viscous fluid as the thickener layer thickness might increase with the oscillation frequency. However, as indicated by in situ observations in Fig. 3.3.11, the thickener layer repeatedly forms and collapses, and thus, the thickener layer within the contact area might not act as a viscous fluid but as a solid. Fig. 3.3.9 shows that the critical amplitude ratio of the EHD point contact for grease lubrication is approximately 1.0; hence, it is suggested that the thickener layer acts like a viscoelastic solid (Ohno and Hirano, 2001; Ohno and Yamada, 2007) in the contact area in a manner similar to that for traction oil. Meanwhile, as shown in Fig. 3.3.18B, in the case of high viscosity, the damage ratio at high frequency is greater for grease lubrication than that for oil lubrication. As the thickener layer cannot be observed from Fig. 3.3.14D, these results indicate that the grease around the contact area is removed and starved lubrication occurs. However, at low frequency, fretting wear can be reduced further for grease lubrication than that for oil lubrication. These results clearly indicate that the thickener contributes to lubrication in case of grease lubrication. In other words, although the oscillation frequency is so low that an oil film is not even formed in oil lubrication, the thickener layer in the grease lubrication maintains the thickener layer thickness that is sufficient for preventing fretting wear. Furthermore, despite the grease having a high-viscosity base oil [i.e., a low oil separation, as shown in Fig. 3.3.15B], the thickener layer is formed around the contact area, as shown in Fig. 3.3.14C. Hence, both oscillation frequency and oil separation affect the formation of a thickener layer. Cen et al. (2014), Dong et al. (2009), and Laurentis et al. (2016) reported that the thickness of a grease film increases with decreased velocity in the low-velocity domain, and this tendency agrees with the results shown in Fig. 3.3.18B. Here, the central oil film thickness hc is virtually calculated using the HamrockDowson equation to investigate the effects of both viscosity and oscillation frequency on fretting wear. Specifically, hc is calculated by substituting the maximum oscillating velocity Vmax and the base oil viscosity μ0 (¼ ρ0ν) into u and η in Eq. (3.3.2), respectively. Fig. 3.3.19A shows the relationship between the calculated hc and the damage ratio at A/D ¼ 1.9 (Maruyama et al., 2017). It appears as though the results obtained

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Fig. 3.3.19 Influence of kinematic viscosity and maximum oscillating velocity on fretting wear at A/D ¼ 1.9 comparing oil and grease lubrication: (A) relationship between calculated oil film thickness and damage ratio and (B) relationship between calculated oil film thickness and damage ratio under the assumption that the viscosity of the thickener layer is approximately 30 times higher than that of the base oil; base oil: PAO, thickener: urea; low viscosity (ν ¼ 19 mm2/s at 40°C): Oil A, Grease C and E; high viscosity (ν ¼ 396 mm2/s at 40°C): Oil E, Grease H and K; low worked penetration ( 210): Grease C and H; high worked penetration ( 330): Grease E and K. Credit: ©2016 Taylor & Francis.

for different viscosities are connected by a single line for the results of not only oil lubrication but also grease lubrication. In the case of grease lubrication, in particular, different types of grease with similarly worked penetration appear to be connected to each other. That is, the virtually calculated oil film thickness can serve as an index that indicates the lubrication state during minute oscillation, which considers the effects of viscosity and oscillation frequency. Specifically, in the case of oil lubrication, the damage

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ratio increases under the test conditions where hc < 100 nm is satisfied. This is because the oil film thickness decreases if the viscosity or oscillation frequency is too low. Meanwhile, for grease lubrication, the damage ratio achieves the minimum value at hc  10 nm. The increased damage ratio at hc < 10 nm signifies that the thickener layer thickness in the contact area decreases below the surface roughness if the base oil viscosity or oscillation frequency decreases considerably, as in the case of oil lubrication. As the effect of worked penetration is also not confirmed at hc < 10 nm, the thickener layer breaks down regardless of the fluidity of the grease. Meanwhile, the fact that the damage ratio increases as the worked penetration decreases at hc > 10 nm indicates the occurrence of starved lubrication. Furthermore, Fig. 3.3.19A shows that the critical hc value at which the damage ratio reaches a minimum value is significantly different between oil and grease lubrication. If we assume that the viscosity of the thickener layer is approximately 30 times that of the base oil viscosity, then as shown in Fig. 3.3.19B, the oil and grease lubrication results agree in the range where hc < 100 nm (Maruyama et al., 2017). Therefore, it is suggested that the thickener concentration increases as the base oil is released in front of the contact area, and as a result, the equivalent viscosity of the grease becomes much higher than the base oil viscosity. Jonkisz and Krzeminski-Freda (1979) and Kaneta et al. (2000) reported that the film thickness in the grease increases beyond that in the base oil under the fully flooded lubrication. Nogi et al. (2020) cited an increase in the thickener concentration in the contact area and its vicinity as the reason why the film thickness in the lowvelocity domain of grease lubrication is thicker than that in the domain of oil lubrication. Specifically, the flow of grease in the steady state is regarded as a two-phase flow of the base oil and the thickener network, and the thickener concentration in the contact area and its vicinity is theoretically obtained from the continuity equation for the thickener. The mechanism by which the thickener concentration increases around the EHD contact area is presented below. As shown in Fig. 3.3.20A, the flow rate qD of the base oil through the thickener network under a pressure gradient can be expressed using Darcy’s law, as follows: qD ¼


kh rp μ

(3.3.7)

Here, the variablek in the above equation is the permeability that represents the degree of oil separation. From the Kozeny-Carman equation, assuming that the densities of the base oil and thickener are equal, k is obtained from the following equation by using the thickener concentration C: k¼β

ð1
C Þ3 C2

(3.3.8)

Furthermore, as shown in Fig. 3.3.20B, the Poiseuille flow rate QP of grease is given by the superposition of the Darcy flow rate qD of the base oil and the Poiseuille flow rate qP of the thickener network: Q P ¼ qD + qP

(3.3.9)

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Fig. 3.3.20 Grease flow as a two-phase flow of base oil and thickener: (A) flow of base oil through the thickener network and (B) Poiseuille and Darcy flow velocities. Credit: ©2020 Taylor & Francis.

Here, assuming that the thickener is entirely contained in the thickener network, the continuity equation of the thickener is expressed as follows:
 ∂ðρCQC Þ 1 + r∙ ρCQP ¼ 0 ∂x 1+R

(3.3.10)

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where Qc indicates the Couette flow rate of the grease and R is the ratio of qD to qP, as shown in the following equation:
 kqD k 12k η0 R¼
1 ¼ kqP k h 2 μ 0

(3.3.11)

Here, μ0 is the base oil viscosity, and η0 is the equivalent viscosity of the grease given by the following equation in reference to the Carreau-Yasuda model:
1

η0 ¼ MCN ð1 + ðλ_γ ÞaY Þ aY + μ0

(3.3.12)

where γ_ is the shear rate, which means that decreases in γ_ result in increases in η0. Fig. 3.3.21 shows the analysis results of the thickener concentration C at low and high velocities. Fig. 3.3.21B shows that the thickener concentration in the grease increases in the EHD contact area and its vicinity at lower velocities. Nogi et al. (2020) explained the mechanism by which these analysis results are obtained as follows. The flow rate ratio R increases from Eq. (3.3.11) as the film thickness h of the contact area inlet decreases due to the wedge shape. Furthermore, as the equivalent viscosity η0 increases at low velocities from Eq. (3.3.12), R similarly increases. Therefore, the second term on the left-hand side of Eq. (3.3.10) can be ignored when R ≫ 1; hence, the following equation is established: ∂ðρCQC Þ ¼0 ∂x

(3.3.13)

As the grease flows toward the contact, QC decreases in proportion to the film thickness, thereby increasing C. Fig. 3.3.22A shows the results of measuring the thickener concentration at the center of the contact using Fourier transform infrared spectroscopy analysis, where the results of the experiments and analysis are mostly in agreement (Nogi et al., 2020). Besides, Fig. 3.3.22B presents the analysis result of the oil film thickness obtained by calculating the equivalent viscosity of the grease from the thickener concentration as shown in Fig. 3.3.22A, where the values obtained using optical interferometry are found to be approximately equal in all velocity domains (Nogi et al., 2020). That is, at low velocities, the base oil is more easily squeezed out of the inlet of the contact than the thickener network, which leads to an increase in the thickener concentration. This can increase the equivalent viscosity of the grease in the inlet of the contact and consequently the grease film thickness by orders of magnitude compared with those of base oil. Meanwhile, as the flow rate ratio R is low at high velocities, the grease behaves as a single-phase flow and the film thickness is consistent with the analytical result of the base oil, as shown by the dashed line in Fig. 3.3.22B. According to Fig. 3.3.12B, the reason why the oil separation increases as the base oil viscosity decreases and the worked penetration increases (i.e., the thickener

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Fig. 3.3.21 Thickener concentration at different entrainment speeds: (A) 1600 mm/s and (B) 2 mm/s. Credit: ©2020 Taylor & Francis.

concentration decreases) is that R increases according to Eqs. (3.3.8) and (3.3.11). As a result, the thickener concentration around the contact area increases according to Eq. (3.3.13) and the equivalent viscosity of the grease increases according to Eq. (3.3.12); hence, the fretting wear decreases, as shown in Fig. 3.3.12A. However, when the thickener concentration is excessively reduced (i.e., C  0), the damage ratio can eventually approach that of the oil lubrication. Besides, the thickener concentration increases around the contact area at low speed, as shown in Fig. 3.3.21B. That is why the thickener layer is observed around the contact area at the low oscillation frequency despite the high-viscosity base oil, as shown

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Fig. 3.3.22 Comparison of experimental and theoretical results: (A) thickener concentration at the center of the contact as a function of entrainment speed and (B) central film thickness as a function of entrainment speed. Credit: ©2020 Taylor & Francis.

in Fig. 3.3.14C. This means that the equivalent viscosity of the grease increases as a result, the fretting wear can be reduced by lowering the oscillation frequency, as shown in Fig. 3.3.18B. On the other hand, as shown in Fig. 3.3.18A, in the case of low viscosity, the damage ratios at low frequencies are similar for oil and grease lubrication although the thickener layer is observed around the contact area shown in Fig. 3.3.14A. It is considered that the thickener concentration increases with a decrease in the frequency, and the release of the bled oil is suppressed according to Eq. (3.3.8), thereby gradually slowing down the increase in the equivalent viscosity of the grease. That is, the wedge action that occurs in front of the contact area decreases and the thickener layer within the contact area breaks down under the

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conditions where the amount of decrease in the oscillation frequency exceeds that of increase in the equivalent viscosity of the grease. Furthermore, Fig. 3.3.22B shows that the grease film thickness in the high-velocity range assuming the fully flooded lubrication is almost the same as the oil film thickness under the oil lubrication. However, by using high-viscosity grease as shown in Fig. 3.3.18B, the starved lubrication is likely to occur with minute oscillatory motion as fretting wear is greater than oil lubrication when the oscillation frequency is increased. The fact that the damage ratio is affected by worked penetration in this high-frequency domain also suggests the occurrence of starved lubrication.

3.3.6

Conclusions

This chapter shows that the approach for preventing fretting wear in minute oscillation is completely different between oil and grease lubrication. Oscillations occurring at an amplitude ratio above the critical amplitude ratio are the most important factor in preventing fretting wear, regardless of the oil or grease lubrication. It can be said that not only increasing the amplitude of the oscillation, but that decreasing the contact area (i.e., normal load) is also an appropriate approach. Alternatively, increasing the high-pressure viscosity or changing to line contact in order to lower the critical amplitude ratio is effective. Moreover, in the case of oil lubrication, after oscillating at an amplitude ratio above the critical value, fretting wear can be prevented by increasing viscosity and oscillation frequency in order to increase the oil film thickness in the EHD contact. Meanwhile, in the case of grease lubrication, lower viscosity of base oil and higher worked penetration (i.e., lower thickener concentration) would increase oil separation, which forms a thickener layer due to the entrainment of the grease which thickener concentration increases as a result, and this can reduce fretting wear. Furthermore, according to Darcy’s law, decreasing oscillation frequency can also promote the release of bled oil from the grease, which contributes to an increased film thickness of the thickener layer formed within the contact area. However, as the wedge action in front of the contact area will be low if the base oil viscosity or oscillation frequency is too low, the thickener layer in the contact area will break down and cause fretting wear. In contrast, starved lubrication occurs if the base oil viscosity or the oscillation frequency is too high and the replenishment of grease is not occurred, similarly causing wear. This signifies the existence of optimal values of base oil viscosity, worked penetration, and oscillation frequency for minimizing the fretting wear in grease lubrication. Although so far, there is no theory for calculating the grease film thickness under minute oscillation in starved EHD contacts, it has been experimentally suggested that the equivalent viscosity of the thickener layer is approximately 30 times higher than the base oil viscosity.

Acknowledgments In writing this chapter, I received much advice through discussions conducted with Professor Fumihiro Itoigawa and Associate Professor Satoru Maegawa of Nagoya Institute of

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Technology, Assistant Professor Satoshi Momozono of Tokyo Institute of Technology, and Dr. Takashi Nogi of Kyodo Yushi Co. Ltd. Furthermore, Atsushi Oda, Fumiaki Aikawa, Faidhi Radzi, Michita Hokao, and Nobuaki Mitamura of NSK Ltd. provided much support in writing this chapter. I would like to express my gratitude to these people.

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Kaneta, M., Ogata, T., Takubo, Y., Naka, M., 2000. Effects of thickener structure on grease elastohydrodynamic lubricant films. Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol. 214, 327–336. Kaneta, M., Ozaki, S., Nishikawa, H., Guo, F., 2007. Effects of impact loads on point contact elastohydrodynamic lubrication films. Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol. 221 (3), 271–278. Kaneta, M., Nishikawa, H., Mizui, M., Guo, F., 2011. Impact elastohydrodynamics in point contacts. Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol. 225 (1), 1–12. Kita, T., Yamamoto, Y., 1997. Fretting wear performance of lithium 12-hydroxystearate greases for thrust ball bearing in reciprocating motion. Jpn. J. Tribol. 42 (6), 492–499. Kumar, P., Anuradha, P., Khonsari, M.M., 2010. Some important aspects of thermal elastohydrodynamic lubrication. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 224 (12), 2588–2598. Laurentis, N.D., Kadiric, A., Lugt, P., Cann, P., 2016. The influence of bearing grease composition on friction in rolling/sliding concentrated contacts. Tribol. Int. 94, 624–632. Liu, Y., Wang, Q.J., Bair, S., Vergne, P., 2007. A quantitative solution for the full shear-thinning EHL point contact problem including traction. Tribol. Lett. 28 (2), 171–181. Maruyama, T., Saitoh, T., 2008. Oil film behavior under minute vibrating conditions in EHL point contacts. In: Proceedings of 35th Leeds-Lyon Symposium on Tribology. Elsevier, Netherlands. Maruyama, T., Saitoh, T., 2010. Oil film behavior under minute vibrating conditions in EHL point contacts. Tribol. Int. 43 (8), 1279–1286. Maruyama, T., Saitoh, T., 2011a. Difference in preventive mechanism for fretting wear between oil and grease lubrication. Jpn. J. Tribol. 56 (12), 788–796. Maruyama, T., Saitoh, T., 2011b. Oil film behavior under minute oscillatory conditions in EHL line contacts. In: Proceedings of International Tribology Conference, October 30–November 3. Hiroshima, Japan. Maruyama, T., Saitoh, T., 2015. Relationship between supplied oil flow rates and oil film thicknesses under starved elastohydrodynamic lubrication. Lubricants 3, 365–380. Maruyama, T., Saitoh, T., Yokouchi, A., 2017. Differences in mechanisms for fretting wear reduction between oil and grease lubrication. Tribol. Trans. 60, 497–505. Maruyama, T., Maeda, M., Nakano, K., 2019. Lubrication condition monitoring of practical ball bearings by electrical impedance method. Tribol. Online 14 (5), 327–338. Mindlin, R.D., 1949. Compliance of elastic bodies in contact. ASME Trans. J. Appl. Mech. 16, 259–268. Murch, L.E., Wilson, W.R.D., 1975. A thermal elastohydrodynamic inlet zone analysis. ASME J. Lubr. Technol. 97 (2), 212–216. Nogi, T., 2015. An analysis of starved EHL point contacts with reflow. Tribol. Online 10 (1), 64–75. Nogi, T., Soma, M., Dong, D., 2020. Numerical analysis of grease film thickness and thickener concentration in elastohydrodynamic lubrication of point contacts. Tribol. Trans. 63 (5), 924–934. Ohno, N., Hirano, F., 2001. High pressure Rheology analysis of traction oils based on free volume measurements. Lubr. Eng. J. STLE 57 (7), 16–22. Ohno, N., Yamada, S., 2007. Effect of high-pressure rheology of lubricants upon entrapped oil film behaviour at halting elastohydrodynamic lubrication. Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol. 221 (3), 279–285. Reynolds, O., 1886. On the theory of lubrication and its application to Mr Beauchamp Tower’s experiments including an experimental determination of the viscosity of olive oil. Philos. Trans. R. Soc. Lond. 177 (i), 157–234.

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Schwack, F., Prigge, F., Poll, G., 2018. Finite element simulation and experimental analysis of false brinelling and fretting corrosion. Tribol. Int. 126, 352–362. Schwack, F., Bader, N., Leckner, J., Demaille, C., Poll, G., 2020. A study of grease lubricants under wind turbine pitch bearing conditions. Wear 454–455, 203335. Shima, M., Jibiki, T., 2008. Fretting wear. Jpn. J. Tribol. 53 (7), 462–468. Sugimura, J., Jones, W.R., Spikes, H.A., 1998. EHD film thickness in non-steady state contacts. ASME J. Tribol. 120, 442–452. Vichard, J.P., 1971. Transient effects in the lubrication of Hertzian contacts. J. Mech. Eng. Sci. 13 (3), 173–189. Wang, J., Hashimoto, T., Nishikawa, H., Kaneta, M., 2005. Pure rolling elastohydrodynamic lubrication of short stroke reciprocating motion. Tribol. Int. 38 (11–12), 1013–1021. Wedeven, L.D., Evans, D., Cameron, A., 1971. Optical analysis of ball bearing starvation. ASME J. Lubr. Technol. 93 (3), 349–361. Yano, A., Noda, Y., Akiyama, Y., Watanabe, N., Fujitsuka, T., 2010. Evaluation of fretting protection property of lubricating grease applied to thrust ball bearing. Tribol. Online 5 (1), 52–59. Zhou, R.S., Hoeprich, M.R., 1991. Torque of tapered roller bearings. ASME J. Tribol. 113 (3), 590–597. Zhu, M.H., Zhou, Z.R., 2011. On the mechanisms of various fretting wear modes. Tribol. Int. 44 (11), 1378–1388. Zolper, T., Li, Z., Chen, C., Jungk, M., Marks, T., Chung, Y.W., Wang, Q., 2012. Lubrication properties of polyalphaolefin and polysiloxane lubricants: molecular structure–tribology relationships. Tribol. Lett. 48 (3), 355–365.

Impact of roughness Krzysztof J. Kubiaka and Thomas G. Mathiab a School of Mechanical Engineering, University of Leeds, Leeds, United Kingdom, bLTDS, Ecole Centrale de Lyon, Ecully, France

3.4.1

3.4

Introduction

This chapter provides the didactical concept of surface roughness influence and its evolution at the fretting interface. The surface roughness in a fretting contact is constantly evolving and changing, steady-state conditions do not exist. The interface is changing from initial roughness through a transitional stage and into the final state of a much rougher surface as illustrated in Fig. 3.4.1. In almost all cases, the roughness will increase as the fretting progresses. There are many additional aspects that can influence this evolution: material, microstructure, contact size, loading conditions, relative motion amplitude, residual stress in the material, deformation, heat treatment, coatings, lubrication, humidity, etc. They all contribute to the transitional nature of the roughness evolution in tribological contact. The speed at which the evolution of the surface roughness is happening seems to follow the rate at which the energy dissipation is happening in the fretting contact. To change the surface roughness, the removal of material must happen in certain parts of the contact. To transform the surface and remove some material or create grooves, therefore, deform the surface plastically, a significant amount of energy must be delivered and dissipated at the interface. Therefore, the higher the dissipated energy density is, the higher the wear and surface roughness will be. In terms of the dynamics of the roughness evolution, the longer the surface is exposed to the fretting conditions, the more energy will be dissipated in the contact due to the rougher surface and the higher number of asperities in the contact that will have to be deformed, displaced, or cutoff from the surface.

3.4.2

Contact of rough surfaces

Smooth surfaces do not exist in nature, and every surface will have some roughness associated with it. Surface roughness can be considered at different scales, from molecular through microroughness, waviness, and form to the roughness of the Earth’s surface in the form of mountains. Therefore, when talking about roughness, one must distinguish the scale associated with it. In the case of fretting wear and fretting fatigue, the scales will usually be in the micrometer range and the contact area is usually no larger than a few millimeters and often in the millimeter range. Considering the contact of rough surfaces, it is clear that not all of the surface area will be in contact but only the tips of the asperities will be touching each other. Another common misconception is the ratio of the height to the width of the asperities. When Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00008-1 Copyright © 2023 Elsevier Inc. All rights reserved.

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Initial state

Transitional

Final state

Fretting

Morphology

Rheology

Physico-chemistry

Smooth surface

Material

Transformed structure

Wear

Wear dynamics

Roughness

Intermediate roughness

Rough surface Time

Fig. 3.4.1 Transitional nature of surface roughness evolution in fretting contact.

representing roughness on a graph, there is usually about a 10-times difference in scale between the length and height of the asperities. It can be seen in Fig. 3.4.2 that the height of the asperities is about 10 μm and the length of each peak is about 100 μm. The peaks may appear much larger than they actually are. When we consider the contact of such surfaces, it may appear that there will be even more deformation at the peaks of asperities than we anticipated. In practice, if the peak experiences very high contact pressure, it most likely will be above the plastic deformation threshold and it will be deformed permanently. In this case, this will allow other asperities to come into contact until the surface is able to accommodate the load and support local pressure and overall load along the interface. Peak asperities coming into contact are illustrated in Fig. 3.4.3. When the roughness is plotted using the same scale in both directions, it can be seen that it looks much different now. It can be seen in Fig. 3.4.4 that the surface is overall very flat and able to accommodate a significant load in the contact between two surfaces. It has to be noted that pressure distribution along the contact is changing and the maximum value of the contact pressure will be in the center. It will, of course, depend on the contact geometry. However, most of the surfaces exposed to fretting wear and fatigue damage are sphere on plane, sphere on sphere, or cylinder on plane configurations.

3.4.3

Stress distribution in rough contact

In a rough contact, stress distribution is much higher at pick asperities and much lower everywhere else. Therefore, there will be a very uneven distribution of the pressure along the contact line, and local stress can very easily exceed the elastic deformation limit. However, according to the St. Venant principle, the influence of those high contact stress regions will be limited in most cases to the size of the asperities, and the subsurface contact stress distribution will be similar to the theoretical value of a smooth surface contact. Considering fretting wear where the thickness of the

Length = 1.00 mm Pt = 44.7 µm Scale = 100 µm

µm 50 40

Scale ratio z/x = 100µm /1000µm = 1/10

30 20 10 0 -10 -20 -30 -40 -50 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 3.4.2 The ratio of the length to the height of the peaks can make the surface appear much rougher than it really is.

0.9

1 mm

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Fretting Wear and Fretting Fatigue

Fig. 3.4.3 Contact between two real surfaces is not continuous.

No min al f orm

Surface 1 s

nes

ugh

ro eal

R

Contact

Contact

Z X

Surface 2

tribologically transformed structure is of the same order as the roughness of the surface, it can be clearly seen that the surface roughness may have a significant influence on the formation and removal of the asperities. As it can be seen in Fig. 3.4.5, small surface roughness can influence contact pressure locally. However, the overall pressure distribution remains very similar to the nominal pressure distribution that can be observed in the case of a smooth surface. If we consider two spherical balls, one indenting a smooth surface and the other indenting a rough surface with an identical load, there could be a slightly different contact area. However, both will support the applied load in a similar manner.

3.4.4

Effective contact area

From the previous analysis, it can be concluded that effective contact area will be reduced when surface roughness is increased. In the case of high peak asperities, it can be noted that at the peaks where a level of stress will be higher than the plastic deformation limit of a material, the surface will flatten to accommodate the high load at the peak asperities. Subsequent peaks coming into contact can also be deformed plastically until the bearing ratio will be able to support the applied load. Once the surface has regained the balance, an effective surface area can be estimated considering that local contact pressure cannot be higher than the plastic deformation limit. σ¼

P P in case of a rough surface σ ¼ σ p therefore A ¼ A σp

where σ is a contact pressure, σ p is a contact pressure corresponding to plastic deformation, P is a normal load, and A is a surface area.

3.4.5

Coefficient of friction

The friction mechanism of a rough surface is significantly different from the mechanism of a smooth surface, especially under very severe contact conditions in fretting. The experimental results presented in Fig. 3.4.6 compare the friction behavior of

Length = 1.00 mm Pt = 44.7 µm Scale = 1000 µm

µm 500 400

Scale ratio z/x = 1000µm /1000µm = 1/1

300 200 100 0 -100 -200 -300 -400 -500 0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 3.4.4 Real surface roughness with the correct aspect ratio that appears very smooth.

0.7

0.8

0.9

1 mm

Fig. 3.4.5 An example of contact pressure distribution between a rough flat surface and a smooth spherical surface.

Fretting Wear and Fretting Fatigue

Normalised Contact pressure

158

1

0.5

0 -1.5

-1

-0.5 0 0.5 Contact width

1

1.5

smooth and rough surfaces under fretting loading with a gradually increasing amplitude of displacement after each 1000 cycles. The coefficient of friction is increasing until it reaches its maximum value and a transition from partial slip to full sliding will occur. In this case, we can notice that a much higher tangential load was required to provoke full sliding and the coefficient of friction reached a value of μ ¼ 1. This was in contrast with a rough surface, where the coefficient reached only 0.7 before transitioning to full sliding. Therefore, it can be concluded that a smooth initial surface with many small asperities requires more energy to wear off the peaks and create lateral sliding along the full length of the contact interface.

Fig. 3.4.6 Comparison of the coefficient of friction at the transition between partial/full slip for smooth Surf7 (Sz ¼ 1.02 μm) and rough surface Surf1 (Sz ¼ 35 μm) (Kubiak et al., 2010).

Impact of roughness

159

Fig. 3.4.7 The effect of initial surface roughness on the transition of a sliding regime (Kubiak et al., 2010).

1200

Normal force, N

1000 Partial Slip

800 600

Full Slip

400

Surf.1 Surf.3 Surf.4 Surf.7

200 0 0

2

4

6

amplitude of sliding distance δ*, µm

It is a very interesting fact that in this experiment, the final value of the coefficient of friction seems to be converging to a similar level for smooth and rough surfaces. This may indicate that other processes related to material and physico-chemistry are becoming dominant at higher sliding values. However, in the fretting regime, especially at the transition from partial to full sliding surface roughness, the effect is very apparent. Initial surface morphology can influence the transition between partial and full sliding conditions, especially in the case of larger values of normal load in the contact. In Fig. 3.4.7, it can be noted that in the case of low contact load, the transition from partial to full slip is taking place at an amplitude of 1 μm of sliding distance inside the contact. And at low normal force, surface roughness is less significant. However, at much higher normal loads, the transition is moved to much higher values of sliding distance amplitude and, for a smooth surface with a roughness of Sa ¼ 0.09 μm it is about 5 times bigger, reaching 5 μm. To explain this phenomenon, one can look at the peak densities and peak amplitudes. In the case of a smooth surface, there will be many small peaks in contact that will have to be cut off to allow the transition and full sliding in the contact. In the case of a rough surface with a smaller amount but much larger peaks, more material will have to be cutoff to remove those large peaks, and this will require more energy and, therefore, higher sliding amplitude to transition from partial to full sliding. A similar trend can be observed in Fig. 3.4.8, where the coefficient of friction at the transition is plotted as a function of Hertzian pressure in the contact. At higher values of contact pressure, a smooth surface will experience a higher coefficient of friction at the transition. It is clear that initial surface roughness will influence contact pressure, local forces, and friction behavior. This knowledge can be used to our advantage, and by designing surface morphology or surface texture, we can control the contact condition. For example, by decreasing surface roughness or by introducing fine surface texture,

160

Fretting Wear and Fretting Fatigue

Fig. 3.4.8 Influence of pressure and surface roughness on the coefficient of friction at the transition between partial/full slip regimes (Kubiak et al., 2010).

1.1 Roughness influence 1.0 0.9

μt

0.8 0.7 Surf.1 Surf.3 Surf.4 Surf.7

0.6 0.5 0.4 400

600

800

1000

1200

maximum Hertzian's contact pressure, MPa

we can favor partial slip conditions and we can prevent fretting wear. However, this approach will have to be carefully considered because, by changing the sliding mode from full sliding to partial slip, we may unintentionally expose parts or components to much more sinister mode of fretting cracking or fatigue damage.

3.4.6

Bearing capacity

As mentioned previously, higher surface roughness will lead to a lower number of asperities in contact. Therefore, for the same overall load, local contact pressure will be much higher and will expose the surface asperities to plastic deformation. Therefore, the bearing ratio curve will be much different, and the capacity of the surface to withstand normal load can be lower as well. Fig. 3.4.9 illustrates how the bearing ratio is changing for smooth and rough surfaces.

3.4.7

Surface anisotropy and orientation

Different machining processes will create surfaces with different roughnesses. However, they may also create surface anisotropy and unidirectional surface features. For example, abrasive polishing will create groves aligned with the direction of individual µm 20

Surf.1

Surf.2

Surf.3 15 10 5 0

Fig. 3.4.9 Illustration of bearing capacity in the case of a rough surface, 20% of peaks top has been removed to illustrate possible surface contact.

Impact of roughness

161

Fig. 3.4.10 Comparison of the coefficients of friction at the transition and in the stable friction regime in the full slip regime for various machining processes (Kubiak et al., 2010).

1.0 ut

ut

0.8

ustab

0.6

0.4

abrasion

cutting

polishing

milling

0.2

0.0

0

10 20 30 roughness parameter Sz , µm

40

(surface peak-to-valley av. max. height)

cutting grains’ movement and, therefore, can also influence contact interface and friction. In Fig. 3.4.10, we can notice different friction behaviors of abrasion polished and milled surfaces. The difference is much more pronounced in the case of the coefficient of friction at the transition between partial slip and full sliding. In the case of full sliding conditions and stable wear taking place, the initial surface roughness was removed and it does not play a significant role anymore.

3.4.8

Transition between partial and gross slip

Smooth initial surface roughness will delay the transition to a full sliding regime. Therefore, it will favor the partial slip conditions and associated cracking damage mode. Whereas the initial rough surface will tend to transition to a full sliding regime, and therefore, it will favor wear damage mode and may prevent cracking. This behavior can be observed in Fig. 3.4.11, where friction coefficient μt at the transition between partial and full sliding is plotted as a function of initial surface roughness and Hertzian contact pressure. Also, higher contact pressure or higher normal contact load generally increases this transition but is not dominant.

3.4.9

Impact of surface roughness on fretting wear

The initial surface roughness of a fretting contact can also influence wear behavior. In this example, the rough milled surface can be seen after the fretting wear test. In Fig. 3.4.12, the transfer of material can be clearly observed. However, adhesion and transfer will be highly dependent on the material type, hardness, and surface

162

Fretting Wear and Fretting Fatigue

Fig. 3.4.11 The influence of initial surface roughness and contact pressure on the coefficient of friction at the transition between partial/ full slip regimes (Kubiak and Mathia, 2009). µt

1.2

0.6

0 0

10

20 30 Roughness Sz (µm)

40

1000 900 re su 800 s e 700 Pr 600 ’s n 500 ia rtz e H

treatment or coating type. In this case, the relatively soft material of low-carbon steel is in contact with the hard chromium-based high alloy ball-bearing material. Transfer of the material under fretting conditions is inevitable and can be observed for both smooth and rough initial morphologies. However, this transfer is predominantly visible in ZDDTP lubricated contact, therefore, physiochemical processes are occurring at the interface (Fig. 3.4.13), and they are not occurring in dry contact conditions. In terms of fretting wear, the process will have a few different stages. The initial activation, where a small amount of wear can be observed, is followed by a steady state and usually linear wear rate regime. In cases where high adhesion and material transfer can be observed, the wear rate can have a more exponential character. Initial activation of wear is related to the tribologically transformed structure (TTS) of hardened and much more brittle material that is created in close proximity to the contact interface. This transformed structure will gradually fracture and will be removed from the interface. However, to create TTS the initial surface will have to be transformed, which requires a certain amount of energy to be dissipated in the contact. It can be seen in Fig. 3.4.14A that the wear rate is linear, but it does not cross the origin of the graph. This shift corresponds to the initial TTS activation in the material. Now considering initial surface roughness, it can be noted that a rough surface has slightly lower activation energy due to more severe conditions created by a smaller number of asperities in contact bearing the same load, therefore, higher local pressure. This will accelerate TTS formation in the early stages of the wear process. Later dynamics of the wear rate are also dependent on the initial surface roughness, as confirmed by the slightly different slopes of the curves in Fig. 3.4.14A. This behavior and the influence of initial surface roughness on wear rate are illustrated in Fig. 3.4.14B, and it can be seen that it is independent of the material, where AISI 1034 and TI-6AL-4 V have been tested. The wear rate is slightly higher for low-carbon steel than for titanium alloy, but the initial roughness influence is identical (Fig. 3.4.14C).

4.45mm 5.1mm

a)

mm

mm

0

150

30

130

50

110

70

90

90

70

110

3.97mm

50

3.47mm 130

30

150

10

b)

0

Fig. 3.4.12 Transfer of the material takes place from a soft, rough surface to a hard sphere during lubricated contact in fretting condition with ZDDTP additive (Kubiak and Mathia, 2009).

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Fretting Wear and Fretting Fatigue

Wear of Material Dry contact 0 10

Plane

Sphere

mm 150 130 110

20

90

30

70

40

50 50 30 60 10 0

70 mm

Transfer of Material ZDDTP lubricated contact 0

Plane

mm 150

10 20

Sphere

30

130 110 90

40

70

50

50

60

30

70 mm

10 0

Fig. 3.4.13 Wear of material in dry conditions and transfer of material in lubricated conditions due to tribochemical interface interaction.

3.4.10

Friction in lubricated contact conditions

In the case of contact lubricated with base mineral oil with ZDDTP additive, we can observe an increased coefficient of friction in the lubricated contact under a partial slip regime. These counterintuitive results that can be seen in Fig. 3.4.15 are related to the lack of sliding and, therefore, low shear rate that is required to activate tribofilm formation. Such conditions create more adhesion in the contact and material transfer from the soft surface to the hard ball-bearing material. This type of condition can be observed in the ball bearings of wind turbine generators subjected to fretting vibration and occasional movement of the generator head to align with the wind direction. However, in the full sliding regime with more energy dissipated in the contact, a tribofilm can be formed and it will result in a lower value of the coefficient of friction μ ¼ 0.35. Wear debris will remain in contact and provide conditions for a “colloidal lubrication regime” (Mathia and Georges, 1978), whereas in dry contact, a “particulate lubrication regime” known as third body agglomeration (Mathia, 1978; Louis, 1983; Mathia and Louis, 1987) will be dominant.

Impact of roughness

165

a)

b)

Low Carbon Alloy (AISI1034)

Titanium Alloy (Ti-6Al-4V)

3.0 Ti6Al4V (Ra=1.51 µm)

AISI1034 (Ra=1.52 µm)

Ti6Al4V (Ra=0.45 µm)

AISI1034 (Ra=0.15 µm)

Ti6Al4V (Ra=0.23 µm)

Wear volume V, µm3 x 106

2.0 Ed thP3

ΔV α= ΔΣ E d

1.5

1.0

Ed thP2

0.5

Ed thP1

6 Ed thP3

α=

4

ΔV ΔΣ E d

Ed thP2 2 Ed thP1

0

0

0

5

10

15

20

25

0

cumulated dissipated energy ΣEd , J

C)

5

15

20

25

cumulated dissipated energy ΣEd , J

initial roughness vs wear rate 5.6

1.5

wear rate , µm 3/J x 105 (AISI 1034)

10

α= 1.4

ΔV ΔΣ E d

5.2

4.8

1.3

1.2

4.4 AISI 1034 Ti-6Al-4V

1.1 0

0.2 0.4 0.6 0.8

wear rate , µm3/J x 105 (Ti-6Al-4V)

Wear volume V, µm3 x 106

2.5

8

AISI1034 (Ra=0.34 µm)

4.0 1.0 1.2 1.4 1.6

initial surface roughness Ra, μm

Fig. 3.4.14 Wear volume (A), (B) and wear rate (C) for low-carbon alloy and titanium alloy (Kubiak et al., 2011).

3.4.11

Energy dissipated at the interfaces for smooth and rough surfaces

Once the process of fretting wear is activated and TTS is formed, the wear volume is proportional to the amount of energy dissipated in the contact at each cycle. This is, of course, assuming steady-state operating conditions. If the conditions are changing, for example, during the variable displacement test, that is, after each 1000 cycles, the displacement amplitude will increase, and the amount of energy dissipated per cycle will also increase. In the full sliding regime, this increase is quite linear, as can be seen in Fig. 3.4.16 for dry conditions. In the case of lubricated contact, it can be seen that after tribofilm formation, friction dynamic is changing and dissipated energy per cycle is

166

Fretting Wear and Fretting Fatigue

1.0 0.9

µt (lubr.)

0.8

Plane Surf.7 Ref. 7 dry contact

µ=Q*/P

0.7 0.6

µt (dry)

0.5

sliding transitions

0.4

lubricated contact Plane Surf.7 Ref. 7

Partial Slip

0.3 0.2 0.1

Full Sliding

0.0 0

4

d *t

6

8

10

12

14

sliding amplitude (µm)

Fig. 3.4.15 Coefficient of friction comparison at the transition between dry and lubricated contact (Kubiak and Mathia, 2009).

Dissipated energy per cycle, Ed (J)

0.06

Q

0.05

Ed

d

Smooth surface Smooth surface (ZDDTP) (dry, no tribofilm)

0.04 Ed (smooth) 0.03 Energy of tribofilm activation

Ed (rough)

0.02

0.01

Rough surface (ZDDTP)

0.00 0

100k 50k Number of fretting cycles

150k

Fig. 3.4.16 The effect of surface roughness on energy dissipation and the formation of tribofilms in fretting.

Impact of roughness

167

much lower. Also, there is a very visible influence of initial surface roughness on this tribofilm formation. In the case of a rough surface, tribofilm is formed at Ed ¼ 0.02 J/cycle and in the case of a smooth surface, the energy required for full sliding and tribofilm formation is Ed ¼ 0.38 J/cycle almost twice the threshold for a rough surface. Of course, the effective contact area in smooth and rough surface contact will be different, as well as local pressure distribution and shear rate. Therefore, a certain level of roughness can be beneficial for tribofilm formation under fretting wear condition.

3.4.12

Impact of surface roughness on crack initiation

The influence of surface roughness under a partial slip sliding regime has been shown above to be very limited and it is not influencing friction behavior in any significant way. Therefore, its influence on crack nucleation is limited. The only exception could possibly be a fretting fatigue situation with very high roughness, where such roughness can create stress concertation and potential crack nucleation sites. However, the contact stress and shear stresses at the contact edges will dominate. Therefore, the initial surface roughness influence will be limited.

3.4.13

Dynamics of surface roughness evolution in fretting contact

Once the fretting wear process is initiated, there is continuous evolution of the interface, and therefore, surface roughness is constantly evolving. In Table 3.4.1, we can see the results of the surface roughness parameter evolution under full sliding conditions with 130 μm displacement. Such conditions will create full sliding and very rapid wear of the surface. Just after the 2000 cycles, we can observe a significant increase in all surface parameters. Sq increases from 0.045 to 2.09 μm, which is even higher than later values where it stabilizes at around 1.5 μm after 7000 and 10,000 cycles. Table 3.4.1 shows surface roughness parameters with a significant variation. The surface profiles extracted through the centerline of the fretting wear scar are shown in Fig. 3.4.17. It can be noted how quickly the wear is progressing and how the roughness and size of the wear scar are evolving. It has been shown before in Fig. 3.4.14 that wear rate depends on initial surface roughness and since it is a continuous process, it depends on an interface history throughout the lifetime. However, the evolution of the roughness parameters itself seems to stabilize and remain at a certain level. It is more likely to be dependent on the material, microstructure, and dynamics of tribologically transformed structure formation and removal. As shown in Fig. 3.4.17, after 5000 cycles, almost all the initial isotropic surface features have been removed. Fig. 3.4.18A presents a very isotropic initial surface, and Fig. 3.4.18B presents a significant drop in isotropy and a change in first direction to align with the direction of fretting movement during the test. A small amount of isotropy in the initial direction of 90° remains, but the peak is quite small and is now a second direction. There are also many more peaks on the direction graph, scattered in all directions.

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Table 3.4.1 Dynamic evolution of surface morphology during wear process, measured at different number of cycles in a wear trace on surface with removed spherical form and filtered with Gaussian filter 0.08 mm (plane material Ti-6Al-4 V, P ¼ 20 N, δ∗ ¼ 130 μm, F ¼ 15 Hz) (Kubiak et al., 2014). Cycles

Initial

2000

5000

7000

10,000

0.045 0.229 3.23 0.197 0.272 0.47 0.035

2.09 0.249 3.77 15.3 15.3 30.6 1.61

1.82 0.214 5.05 19.9 23.1 43 1.43

1.5 0.228 3.27 12.4 16.5 28.9 1.22

1.56 0.802 5.16 12.9 17.2 30.2 1.23

0.058 0.083

2.66 4.72

2.17 4.05

1.97 3.04

1.87 3.52

277 7.81 0.32 0.12 0.20

184 319 23.5 10.3 13.2

94.7 547 31.4 16.1 15.3

137 324 18.7 9.14 9.56

113 422 25 9.51 15.5

Height parameters Sq (μm) Ssk Sku Sp (μm) Sv (μm) Sz (μm) Sa (μm)

Functional parameters Smc (μm) Sxp (μm)

Feature parameters Spd (1/mm2) Spc (1/mm) S10z (μm) S5p (μm) S5v (μm)

3.4.14

Measurement of fretting wear using surface metrology

Recent developments in optical surface metrology have made techniques such as white light interferometry, focus variation, and confocal chromatography very popular for wear analysis. However, the methods’ limitations to capturing the surface at high slopes and the generally very low reflectance of the wear scar surface, which is usually very dark compared to the surrounding surface, make such measurement very challenging and require a highly experienced operator. This makes the optical measurements highly unreliable. The most reliable technique is still a stylus-type contact profiler, preferably measuring the surface in three-dimensional (3D). It is also possible to estimate the wear using a few two-dimensional (2D) profiles looking for the deepest wear track that can be used to estimate the overall wear volume. However, such a technique carries an error of about 10% of the wear volume and we assume that the wear track is elliptical, cylindrical, or of other simple and known shape. The drawback of full 3D stylus profilometry is the much longer measurement time, but it is still the most accurate technique. Therefore, if using optical metrology

Impact of roughness

169

L ength = 1.94 mm P t = 0.196 µm S cale = 3.00 µm

µm

N=0 cycles , Initial s urface

1 0 -1 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 mm

L ength = 1.40 mm P t = 20.4 µm Sc ale = 30.0 µm

µm 10

N=2000 c yc les

0 -10 0

0.2

0.4

0.6

0.8

1

1.2

1.4 mm

L ength = 1.51 mm P t = 23.0 µm Sc ale = 30.0 µm

µm

N=5000 c yc les

10 0 -10 0

0.2

0.4

0.6

0.8

1

1.2

1.4 mm

L ength = 1.41 mm P t = 20.2 µm Sc ale = 30.0 µm

µm

N=7500 cycles

10 0 -10 0

0.2

0.4

0.6

0.8

1

1.2

1.4 mm

L ength = 1.43 mm P t = 18.4 µm Sc ale = 30.0 µm

µm

N=10 000 c yc les

10 0 -10 0

0.2

0.4

0.6

0.8

1

1.2

1.4 mm

Fig. 3.4.17 Dynamics of surface roughness evolution during fretting wear of Ti-6AL-4 V (Kubiak et al., 2014).

110° 120° 130°

a)

100°

90°

80°

b)

70°

100° 110° 120° 130°

60° 50°

140°

20°

170° 180° 0

0.4

0.8

1.2

10°

170°

Second Direction: 83.2° Third Direction: 97.0°

0.4 0.6 0.8 1 1.2 1.4

50° 40° 30°

0°

180°

20° 10° 0°

1.6 mm

0

0

First Direction: 90.0°

60°

160°

0.2

0.4

0 0.1

0.2

Isotropy: 2.70 %

70°

150°

30°

160°

80°

140°

40°

150°

90°

Isotropy: 64.5 % First Direction: 0.214°

0.2 0.3 0.4 0.5 0.6

Second Direction: 90.0°

0.7

Third Direction: 42.4°

0.9

0.8

1 1.1

1.6 mm

mm

Fig. 3.4.18 Surface roughness isotropy before and anisotropy after the fretting wear test on AISI 1045 (Kubiak et al., 2014).

0.6

0.8

1

1.2 mm

Impact of roughness

171

instruments, one should validate the methodology using reference stylus profilometry. Another validation method could be to create an imprint with a sphere or any other indenter, where expected volume can be calculated and measured using a profilometer and preferred software. For example, MountainsMap to calculate wear volume. Several steps could be required to arrive at the correct volume, for example, nonmeasured point restoration, surface filtration, shape removal, reference level restoration, etc. There could also be several other factors contributing to systematic error, such as device, magnification, operator, profilometer calibration, data storage format, etc. To summarize, it is a highly skilled job to measure the war volume correctly.

References Kubiak, K.J., Mathia, T.G., 2009. Influence of roughness on contact interface in fretting under dry and boundary lubricated sliding regimes. Wear 267 (1–4), 315–321. Kubiak, K.J., Mathia, T.G., Fouvry, S., 2010. Interface roughness effect on friction map under fretting contact conditions. Tribol. Int. 43 (8), 1500–1507. Kubiak, K.J., Liskiewicz, T.W., Mathia, T.G., 2011. Surface morphology in engineering applications: influence of roughness on sliding and wear in dry fretting. Tribol. Int. 44, 1427–1432. Kubiak, K.J., Bigerelle, M., Mathia, T.G., Dubois, A., Dubar, L., 2014. Dynamic evolution of interface roughness during friction and wear processes. Scanning 36, 30–38. Louis, J.F., 1983. Comportement rheologique des produits interfaciaux et phenome`ne d’usure.  Ph.D. Thesis, 25/04/1983, Ecole Centrale de Lyon, France.  Mathia, T.G., 1978. Etude d’une interface de glissement en lubrification monomoleculaire. D. Sc. Thesis, Universite Claude Bernard & Ecole Centrale de Lyon. Mathia, T.G., Georges, J.M., 1978. Wear initiation of boundary lubricated surfaces. Wear 50 (1), 191–194. Mathia, T.G., Louis, F., 1987. Tribological consequences of compaction and interface shear of micronised powders. In: Briscoe, B.J., Adams, M.J., Hilger, A. (Eds.), Tribology in Particulate Technology. Institute of Physics Publishing and Taylor & Francis Ltd (ISBN-10: 0852744250 & ISBN-13: 978-0852744253).

Materials aspects in fretting

3.5

Thawhid Khana, Andrey Voevodinb, Aleksey Yerokhinc, and Allan Matthewsc a Manchester Metropolitan University, Faculty of Science and Engineering, Department of Engineering, Manchester, United Kingdom, bUniversity of North Texas, Denton, TX, United States, cUniversity of Manchester, Manchester, United Kingdom

3.5.1

Physical processes impacting materials in industrial fretting contacts

Fretting wear can be observed in several industrial applications when either oscillating loads or vibrations are applied to static contacts. It can be found in biomedical, aerospace, and nuclear components such as the blade-disc interface in aeroengines, stem-cement interface in hip replacement prosthesis, and fuel rod-supporter interface in a nuclear reactor. In-depth analysis of the role of fretting within industrial applications can be found in Chapter 5; however, the material aspects of these applications and how they influence fretting behavior will be discussed here. With aeroengines, engine vibration and centrifugal force lead to small amplitude and high frequency oscillations at the dovetail blade-disc interface, leading to both fretting wear and fretting fatigue (Korsunsky et al., 2008). With hip implants, contact degradation at interfaces such as the stem-cement is due to the low frequency gait movement (Kim et al., 2013), whereas a nuclear power generator fuel rod-supporter interface has medium frequency and variable amplitude oscillations in water leading to fretting wear (Kim et al., 2006; Ferrari et al., 2020). Fretting wear can be observed with various mechanical components in a range of working environments and temperatures (Ma et al., 2019; Pearson et al., 2013). With gears of wind-powered electrical turbines, especially for the ones positioned off-shore or near the sea coast, fretting corrosion is one of the most critical maintenance needs (Papaelias et al., 2019). Dobromirski (1992) proposed that there are over 50 parameters that influence fretting behavior. The fretting process is comprised of a number of complex physical processes, whose balance depends on fretting contact stress, amplitude, frequency, environment, temperature, and material characteristics. The following are important considerations (Warmuth et al., 2015; Hurricks, 1970): l

l

l

Interaction of asperities leading to either adhesion and plastic deformation (in the case of metal alloys), or fracture of asperities promoting abrasion (in the case of ceramics), resulting in material transfer and/or debris entrapment in the contact. Accumulation of dislocation density and subsurface fatigue crack initiation. For metal alloys in oxidative environments, oxidation of nascent metal surfaces opened by asperity plastic deformation or fracture, friction-induced heating, and increased contact temperature.

Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00009-3 Copyright © 2023 Elsevier Inc. All rights reserved.

174 l

l

Fretting Wear and Fretting Fatigue

Materials transfer into the contact interface, compaction, oxidation/reduction, sintering, and formation of the so-called “third body” and its degradation by wear in the fretting zone. Removal of contact debris, which is influenced by the contact geometry, oscillation amplitude, surface topography, and the agglomeration or sintering of the debris particles.

3.5.2

Factors affecting fretting behavior of different materials groups

3.5.2.1 Materials behavior vs fretting regimes The mechanics of fretting contacts distinguish three regimes determined by the amplitude of relative motion of contact interfaces in fretting: stick, stick-slip, and gross-slip regimes. These are defined in Chapter 2.2 and are used here as a guide for materials behavior discussions. While there are no commonly accepted numerical amplitude values for transition boundaries between these regimes, the specific regime dictates the dominating mode of failure, which is of prime concern and hence leads to various mitigation strategies in materials designs for such dominating failure modes. In the stick regime, there is a negligible amount of sliding and most of the damage is due to accumulation of dislocation density and fatigue crack initiation, commonly resulting in a catastrophic failure of the part. Since this is less predictable and the practically most feared failure mode, material strategies are designed to avoid such a regime altogether, where one example is liquid or solid lubricant application to contact interfaces to avoid 100% stick even at very small oscillation amplitudes. One common example is the use of a solid lubricant for bolts and other fasteners in joining parts that experience vibrations in operation. While the stick regime is the most difficult to predict material failure in subsurface volumes, the mixed stick-slip regime is one of the most difficult in terms of impact on the materials surface, since it combines both subsurface fatigue and surface wear, where amplitude is also small and does not allow efficient debris and heat rejection, leading to debris trapping, accumulation, sintering, and continuing modification of the material in the contact. To counteract this regime both surface hardening and low shear interface strength are needed, which are contradictory requirements to each other and are difficult to achieve with one material. Most common surface modifications, coatings, and lubricants are used as discussed later in this chapter. One common example would be the dovetail joint in the blade-disc assembly of jet engines, where metal alloy solid lubricants are applied on the hardened joint surfaces. Gross sliding wear with higher amplitudes is closer to a reciprocating sliding contact, and failure of such interfaces can be reasonably predicted and monitored using wear rates. The materials design strategies are similar to the mixed regimes with a reduced concern for subsurface fatigue failure. Discussions in this chapter are mainly focused on the mixed stick/slip regime, which commonly distinguishes fretting wear from contact fatigue and reciprocated sliding contact wear. Thus, from the interface material response behavior, fretting wear occupies the region between elastic contact fatigue and reciprocated sliding wear failure, incorporating elements of both to various degrees, depending on contact mechanics, environment and materials as outlined

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175

earlier, and also bringing additional concerns due to debris trapping and reduced possibility for dissipation of frictional heat.

3.5.2.2 Materials behavior vs fretting contact load and geometry The contact geometry plays an important role in controlling the flow of debris and therefore the wear rate in fretting (Warmuth et al., 2013; Godet, 1984). To reduce fretting wear, the debris must either be ejected or should promote the formation of a protective layer which can influence mechanisms and correspondingly the rates of wear. Studies have suggested that due to geometrical changes to the contact in the process of wear, the contact area can increase, and debris can become entrapped more easily in the contact, leading to a decrease in wear rates if such entrapping leads to sintering of a stable protective layer. Ding et al. (2005, 2007) proposed that the pressure distribution in the contact can be impacted by the retention of debris, which would change the development of the wear scar shape and affect the debris flow within the contact. Fouvry et al. (2009) and Warmuth et al. (2013) have shown that with increasing contact conformity the wear rate decreases toward an asymptote. Merhej and Fouvry (2009) found that for simplified contact experiments (ball-on-flat), wear rates reduced with the increase of the radius of curvature of the nonplane body. They believed this was due to a reduction in debris removal from the contact, which in turn formed a protective layer against wear. Warmuth et al. (2013, 2015) argued that these trends were due to a change in wear mechanism with increasing ball radius. They proposed that oxygen was able to penetrate less-conforming contacts easily, helping with the formation of oxides which then flowed out of the contact. In comparison, as the contact is more conforming, there is a lack of oxygen at the center of the contact leading to metal-metal adhesion and transfer between the contacting surface, at high displacement amplitudes, resulting in severe pits and peaks within the contact.

3.5.2.3 Materials behavior vs fretting frequency For all other equal conditions, the frequency of oscillation is another known factor to influence material damage in terms of fretting wear and fatigue. In testing, running experiments at high frequencies and small amplitudes includes several complexities that make it challenging due to the control of mechanical motion and the accurate reading of actual slip and contact forces; hence, many test experiments utilize lower frequencies for experimental test programs (Warmuth et al., 2015; Ramalho and Celis, 2003), extrapolating results to higher frequencies. Early major studies investigating the role of frequency have been carried out by Feng and Uhlig (1954) and Uhlig (1954), who demonstrated that for mild steel the wear rate generally decreased with increasing fretting frequency. While possible mechanisms can be suggested for such effects, high frequency normally leads to increased heating in the contact zone influencing oxidation reactions, and in the case of steels the formation and softening of oxides may play one of the dominant roles. Correspondingly, it was also determined that at higher displacement amplitudes (higher velocities for the same frequency) the influence of frequency change was stronger (Warmuth et al., 2015). Additionally, the

176

Fretting Wear and Fretting Fatigue

impact of environment on steels and other metal alloys can be significant. Some studies suggest corrosion-induced cracking in fretting contact to be very sensitive to the frequency of sliding (Bill, 1983; Stachowiak and Batchelor, 2013). Feng and Uhlig’s (Feng and Uhlig, 1954) study also demonstrated that the influence of frequency on wear was not evident when fretting tests were conducted in a nonoxidizing environment. From this early study of steels, both chemical and mechanical factors were determined to influence fretting wear. The chemical factor refers to the formation and evolution of oxides in the contact when a fresh surface is exposed to an oxidizing environment. With surface rubbing/interaction the oxide could be removed, and a nascent metal surface exposed allowing the new oxide growth and removal cycle to repeat. Alternatively, the developed oxide can stay in the contact, compact and sinter, forming a “glaze”—a stable compact oxide layer. In fretting wear, the mechanical aspect is defined as the removal or adherence of debris when two surfaces and their asperities interact. The role of oscillation frequency in fretting wear is thus explained by a combined influence on the rate of oxide growth, heat dissipation, and subsurface fatigue accumulation. The significance of the frequency factor is, therefore, material dependent in addition to contact geometry, load, and environment (Warmuth et al., 2015; Uhlig, 1954).

3.5.2.4 Materials behavior vs fretting temperature and environment Temperature is shown to be a significant fretting wear factor due to its influence on kinetics, thermodynamics, and material properties in the contacts (Pearson et al., 2013). Temperature has an impact on fretting by several factors (Hirsch and Neu, 2013, 2016): (1) changing the mechanical properties of the bodies undergoing fretting; (2) controlling the rate of oxide formation and oxide phase formed on the interacting surfaces; (3) and influencing if oxide particles in the contact are retained or removed. For metal contacts, the coefficient of friction and wear rate usually decrease with an increase in temperature over a certain range defined as the transition temperature (Pearson et al., 2013). It is generally understood that with rising temperatures the yield and shear strengths of metals fall, and it is expected to lead to significant shear plastic deformation during fretting (Nikulin et al., 2010; Jin et al., 2017). Similar behavior has been widely observed with reciprocating sliding wear experiments (Pearson et al., 2013). With mild steels, a temperature of 200°C in argon resulted in a reduction in fretting wear, whereas above 300°C a sharp increase in fretting wear was observed. The reduction in fretting wear at 200°C was attributed to the steel being strain hardened, whereas above 300°C such an effect disappeared due to annealing (Stachowiak and Batchelor, 2013; Waterhouse, 1955). Studies have also demonstrated that during fretting at low temperatures a smooth surface finish increases surface damage, whereas rougher surfaces are preferred to minimize wear. However, at elevated temperatures, smoother surface finishes suffer less damage than rough surfaces (Hurricks and Ashford, 1970). Such behavior is most likely to be linked to the temperaturecontrolled surface oxidation in fretting contacts discussed later.

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177

Both atmospheric oxygen and water have significant impacts on fretting wear, especially with metals (Waterhouse, 1955; Bill, 1980). Water and oxygen generally increase surface wear and corrosion damage, but for fretting regimes the oxidation may decrease fretting wear by preventing asperities to adhere and lock together (Stachowiak and Batchelor, 2013). Temperature-actuated oxidation rates and formation of protective oxides strongly depend on metal alloy composition. A well-adhered oxide layer acts as a solid lubricant helping to reduce friction and fretting wear by preventing metal-on-metal contact. The effectiveness of protective oxide films is determined by the metal alloy composition, mechanical properties, and fretting contact severity (Bill, 1983; Stachowiak and Batchelor, 2013). At high temperatures, the formation of a protective layer has been observed with carbon steel (Hurricks, 1974), titanium alloys (Waterhouse and Iwabuchi, 1985), stainless steel (Kayaba and Iwabuchi, 1981), and nickel alloys (Hamdy and Waterhouse, 1981). With mild steel after a transition temperature between 200°C and 500°C, fretting wear in air falls to a very low rate (Hurricks, 1974). With the different materials and their different temperature ranges different thicknesses and morphology oxide films are formed. A glaze type of oxide film was found to form on nickel alloys and steels (Kayaba and Iwabuchi, 1981; Hamdy and Waterhouse, 1981), whereas with titanium (Waterhouse and Iwabuchi, 1985) a thin, protective oxide film was observed. With titanium alloys, fretting has also led to the formation of nitrides alongside oxides (Stachowiak and Batchelor, 2013). Debris build-up within a fretting contact may be significantly influenced by temperature in the following ways ( Jin et al., 2017): 1. Oxide formation rate: The oxidation rate of metals generally increases exponentially with temperature. 2. Oxide nature: The temperature at which oxidation develops influences the dominant oxide present or a mixture of oxides. Hurricks (1972) found that the oxide debris formed in fretting would contain more Fe3O4 rather than α-Fe2O3 under higher temperatures. Kayaba and Iwabuchi (1981) argued that with an increasing presence of Fe3O4 in the oxide layer wear rates would fall. 3. Oxide debris retention: With rising temperatures, debris particles forming in the fretting contact have higher surface energy, which leads to their mutual adhesion, agglomeration, sintering, retention within the contact and the formation of a stable debris bed. For steels, studies have suggested that Fe3O4-containing debris adhere strongly to worn surfaces and encourage stable debris beds ( Jiang et al., 1998; Clark et al., 1967).

Consolidation of oxide debris layer by sintering results in the growth of a protective glaze layer. Pearson et al. (2013) proposed that debris sintering in the fretting of high-strength steel occurred at temperatures as low as 85°C. For titanium alloys, fretting tests at 450°C in air led to the formation of such protective surface films composed of a mixture of titanium nitride and titanium oxide, where oxygen starvation within wear contacts at small oscillation amplitudes leads to the reaction of atmospheric nitrogen with exposed nascent titanium on the worn surface. It is believed that nitride formation only occurs with extremely depleted supplies of oxygen in the fretting contact of titanium alloys (Stachowiak and Batchelor, 2013; Mary et al., 2009).

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Fretting Wear and Fretting Fatigue

The role of temperature on fretting wear is then directly related to the nature, adherence, and mechanical (shear strength and hardness) properties of oxides formed during fretting, where these range from hard and abrasive to low-shear-strength and friction reducing, from loose and weakly adhered to compacted (glazed) stable third bodies. If the formed oxides are hard, then the temperature leads to an increase of abrasive oxide debris and fretting wear accelerations. However, if oxides have a low shear strength, then both friction and wear rates are reduced. Oxide chemistry, structure, and properties (easy to shear, protective, adhered, etc.) also depend on the temperature. One widely studied example is the formation of a low friction glazing oxide layer in fretting wear of nickel-base superalloys, e.g., Incoloy 901, Nimonic 75, and Nimonic C263. The glaze layer formed on the surface of these alloys in fretting contains a mix of NiO, CoO, FeO, and Fe2O3, as for example described by Stott et al. (1973). The formation of nickel oxide glaze layer is highly temperature sensitive, where studies by Jiang et al. (1994) with Nimonic 80A alloy had demonstrated that it takes about 250°C to help form a continuous oxide glaze later, leading to the reduction of both friction and the fretting wear. In recent works by Hager Jr et al. (2009, 2010), the fretting wear was clearly correlated to the temperature-driven evolution of nickel oxide from a more abrasive Ni2O3 at lower temperatures to a low shear-strength NiO oxide at 300–400°C. Such temperature-dependent behavior and oxide evolution in the fretting contact can be effectively used to reduce friction, wear, and subsurface fatigue by shifting the stick regime to smaller oscillation amplitudes.

3.5.2.5 Materials hardness, stiffness, yield strength vs fretting behavior Materials hardness, stiffness (Young’s and shear moduli) and yield strength all affect the mechanics of the fretting contact (Warmuth et al., 2015). Toth (1972) proposed frequency dependence of fretting wear, which was observed with softer materials due to metal-on-metal adhesion being encouraged at lower frequencies. Fretting wear is generally decreased with the use of hardened surfaces. However, modifying only material hardness is not necessarily a reliable indicator of improved fretting resistance, with several studies demonstrating that the hardness of alloyed steel surfaces has no direct relation to fretting wear rates, where microstructural factors, such as the presence, for example, of martensitic or austenitic structures, have a strong influence on fretting wear rate (Batchelor et al., 1992). Most commonly for hardened surfaces there are complex property modifications, which help to reduce fretting wear. For example, surface nitriding and carburizing discussed later in this chapter increase surface hardness, stiffness and yield strength, and at the same time also improve resilience to surface oxidation and reduce adhesive interactions. Fretting wear rates can be negatively impacted by the presence of hard phases, if such can be dislodged and trapped inside the fretting contact during the wear process. Aluminum-silicon alloy matrix composites with silica and graphite particles combined with alumina fibers demonstrated moderate coefficients of friction during fretting against hardened steel. With a high concentration of hard alumina fibers or silica particles significant wear is

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179

observed (Stachowiak and Batchelor, 2013; Leech et al., 1992; Goto and Uchijo, 2004). Some studies have demonstrated that the harder material in a fretting pair can be worn alongside the softer material as a result of transfer and oxidation processes (Stachowiak and Batchelor, 2013). When aluminum alloy (A357) is fretted against a harder material (bearing steel 52,100), initial adhesion of the aluminum alloy to the steel surface is observed. If the materials during fretting are exposed to air, the aluminum debris formed can oxidize to form aluminum oxide. These alumina particles are harder than the steel, leading to the steel sample being abraded. The steel sample begins to wear after initially being protected by a transfer film. It was also observed that by increasing the amplitude of fretting, fewer fretting cycles were required to initiate wear (Stachowiak and Batchelor, 2013; Elleuch and Fouvry, 2002).

3.5.3

Materials engineering approaches to the mitigation of fretting wear

Several approaches to mitigate or reduce the impact of fretting wear are available, generally split into three preventive measures (Fu et al., 2000). l

l

l

Design changes: Geometry modification of components and changing contact materials can reduce fretting. However, this option can be costly and time-consuming and can lead to other problems such as overloading and decrease in fatigue strength. Use of surface engineering: Fretting is strongly tied to wear, corrosion, and fatigue, which can be influenced by surface modification treatments (thermo-chemical or mechanical) and application of coatings. Some coatings have been shown to have the ability to reduce fretting wear volume by a factor of 100 or improve fretting fatigue life by a factor of 10. Use of lubricants: The use of appropriate lubricants (grease, liquids, or solid lubricants) to reduce fretting by reducing friction and shear stress of the surfaces in contacts. Not all applications allow for grease and oil lubrication of fretting contacts due to constraints associated with the operating environment, overall system weight, small amplitude of displacements, and other factors that favor solid lubricant applications as soft coatings for fretting contacts.

The following sections of this chapter focus on fretting contact surface material changes, using different surface engineering approaches: thermo-chemical surface modification, mechanical surface modification, hard coatings, soft solid lubricating coatings, as well as their hybrids and duplex surface engineering methods. For liquid/grease lubricants, the reader is referred to Chapter 3.3.

3.5.3.1 Thermo-chemical surface treatments Nitriding is usually carried out within the temperature range of 400–590°C and, if the nitrogen concentration is greater than 2.5 wt%, a single or multiphase nitride layer is created, which can improve the wear resistance of the component alongside other dynamic characteristics of the ferrous material. The process is mostly used as a thermochemical case-hardening process where the surface of a component is enriched with nitrogen interstitially incorporated into the ferrite matrix through diffusion into

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Fretting Wear and Fretting Fatigue

the surface zone. One of the advantages of nitriding (as well as nitrocarburizing) is the low temperature at which the process is carried out, which prevents phase transformation to austenite and corresponding physical distortion of the component that would essentially reduce its service life (Brinke et al., 2006; ASM Handbook: Steel Heat Treating, 2013; Holm and Sproge, 1954). Nitriding refers to the process where only nitrogen is introduced to the surface, but if carbon is also diffused in simultaneously this treatment is referred to as nitrocarburizing. Post nitriding/nitrocarburizing, oxidation can be carried out producing a layer of magnetite on top of the nitrided zone. This layer acts as a running-in coat and it can further improve the treated component’s friction and wear properties (Brinke et al., 2006). The key properties produced by nitriding can be summarized as follows (Prabhudev, 1988): l

l

l

Higher fatigue strength. Higher surface hardness and wear resistance alongside reduced risk of scuffing and galling. Improved corrosion resistance.

An attractive way to improve the tribological and corrosion properties of high strength steels and other nitriding alloys is plasma nitriding, which utilizes active nitrogen species generated in a low-pressure glow-discharge plasma. At elevated temperatures, this facilitates interaction with steel surfaces to produce compound and diffusion layers. A postnitriding oxidation treatment forms a superficial oxide layer over the compound layer on the surface of the metallic material. Plasma nitriding and postnitriding oxidation treatments find increasing application on low-alloy and stainless steels, cast irons and tool steels due to the possibility of using a lower process temperature and shorter treatment duration, thereby minimizing distortion to the surface and lower use of energy compared to conventional surface modifications (Ramesh and Gnanamoorthy, 2006; Prakash and Bennett, 2017). The process of oxy-nitriding is a method of passivating the surface and increasing fretting wear resistance. The high surface hardness achieved by nitriding has a higher wear loss compared to the oxy-nitrided layers, which indicates hardness has less influence under fretting conditions than chemical stability. With structural steels, build-up of hard debris in the contact area and oxidation-assisted wear leads to severe wear damage. On surfaces of hardened and tempered steels thin oxide films are formed due to repeated oxidation of the freshly worn surface followed by scraping/removal of the surface oxide layer and generating abrasive debris trapped in the fretting contact. With the presence of thicker initial oxide layers, much looser wear particles are produced, and the oxide layer provides an effective oxygen diffusion barrier to protect the underlying alloy from further oxidation. The fretting wear behavior is therefore directly influenced by the thickness of the oxide layer, with thicker layers producing superior fretting resistance (Kayaba and Iwabuchi, 1981; Ramesh and Gnanamoorthy, 2006). Carburizing involves the dissolving of carbon in the surface layers of the material at a temperature sufficient to form an interstitially carbon supersaturated modified layer known as S-phase or expanded austenite. Carburizing allows also obtaining a large thickness of the diffusion later with high hardness and stiffness as compared with nitriding,

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and thus provides a better load support, wear resistance, and corrosion resistance. Nickelfree austenitic stainless steel has gained interest in the medical field due to its biocompatibility and corrosion resistance; however; the material demonstrates relatively low hardness and poor resistance to fretting wear. The development of S-phase has helped to pave the way toward improved wear- and corrosion-resistant austenitic stainless steel for medical device application (Pan et al., 2000; Davis, 1994; Saklako
glu et al., 2006). Buhagiar et al. (2010) demonstrated that carburizing can effectively improve the fretting-wear resistance of Ni-free austenitic stainless steel. The hardened surface after the carburizing treatment was shown to support an oxide film due to improved stability and wear resistance. The oxide film helps to reduce metal-on-metal contact (which can cause detrimental adhesive wear) and provides protection against corrosion. The carburizing treatment was shown to reduce the coefficient of friction and increase fretting-wear resistance. Carburizing normally involves considerably higher temperatures compared to nitriding (order of 900–1000°C), and hence is restricted to a certain class of steels, which allow quenching after the carburizing process for the required core mechanical properties. Carbonitriding is becoming one of the common methods used for strengthening fretting contact surfaces, as the method utilizes a balance of temperature (higher than nitriding and less then carburizing), which allows for a direct oil quenching of steel parts, provides a flexibility and more precise control of the diffusion-modified carbonitrided surface layer and also helps in generating surface compressive stress after the quench treatment. This, combined with relatively low costs of using hydrocarbon and ammonia gas mixtures, makes the method suitable for treatment of gears, bearings, valves, and other components, which may experience fretting wear (Davies and Smith, 1978; Herring, 2002; Mittemeijer, 2013).

3.5.3.2 Shot-peening treatment Fretting damage can be successfully suppressed by shot-peening, which work-hardens and roughens the surface as well as generating compressive stresses that improve fretting fatigue and wear resistance. However, excessive shot-peening can cause surface damage, which will lead to fretting fatigue strength deterioration. Innovative surface engineering approaches include incorporating multilayers combined with shotpeening, e.g., CrN film deposition on shot-peened Ti6Al4V substrate (Waterhouse, 1982; Waterhouse and Saunders, 1979; Bell et al., 1998). This duplex treatment provides the highest wear resistance, combined with excellent fretting fatigue resistance. Depositing the CrN on a shot-peened hardened surface can improve abrasive wear resistance and the presence of CrN can improve anticorrosion and antioxidative resistance (Fu et al., 2000).

3.5.3.3 Laser surface treatment Laser beams can be used to deposit coating material or create a surface layer of altered microstructure. Lasers are used in tribological surface engineering for surface alloying, hardening, and cladding. Studies have demonstrated that laser surface

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hardening and alloying increased the fretting wear resistance of 2Cr13 stainless steel by up four times in comparison to untreated material (Dehua et al., 1994). Alternatively, the laser alloying of zirconium and chromium films on steel substrates helps to reduce fretting wear damage (Batchelor et al., 1992). Laser alloying of Ni and Cr to mitigate the fretting wear of Al 061 alloy has also been investigated, with fretting wear reduced by a factor of three (Fu and Batchelor, 1998). It was also reported that a reduction in friction is observed, most likely due to the hardening effect of the laser, which prevents adhesive and abrasive wear during fretting. An increase in surface hardness helps to reduce adhesion, plastic deformation, and abrasion during fretting alongside improving corrosive and oxidation resistance (Fu et al., 1998a). Laser surface treatment is usually considered for the mitigation of fretting wear rather than fretting fatigue. This may primarily be due to the formation of surface cracks and other defects during the treatment (Stachowiak et al., 1994). The roughened and hardened surface after the treatment may lead to rapid failure in fretting fatigue critical applications due to stress and wear in small parts of the coating surface. By optimizing the laser process parameters, fretting resistance can be improved (Dehua et al., 1994; Yongqing et al., 1997).

3.5.4

Application of coatings to mitigate fretting wear

3.5.4.1 Thermally sprayed coatings Thermal spraying involves the heating of coating materials in a hot gaseous medium and then spraying on to a surface at high velocity to create a coating. The advantage is a thick coating layer, the composition of which can include metals (as a main component or a binder) and ceramic phases helping to develop lubricating oxides in the contact while achieving a high hardness and load-supporting characteristics. For example, the formation of thin layers of molybdenum oxide during wearing contributes to the enhanced fretting resistance of sprayed Mo coatings. MoO2 and MoO3 oxides form and spread across the surface acting as a lubricant, which leads to a low friction coefficient and prevents metal contact. At high temperatures, the oxides behave as effective lubricants leading to a low coefficient of friction and reduced fretting wear. The dispersion and volume of these oxides as well as the temperature of fretting determines their effectiveness (Fu et al., 2000). The use of certain oxides in the coating makes it a good option for use in fretting applications. The coating platelet structure allows it to delay crack propagation while the presence of oxides reduces adhesion between two contacting surfaces (Fu et al., 2000). Fretting fatigue strength of steel can be increased by 120% and can prevent material loss at a range of temperatures (Taylor and Waterhouse, 1972; Overs et al., 1981). The spray deposition of hard ceramic coatings such as tungsten carbide (WC) can help to inhibit fretting wear. The limitations of these sprayed coatings during fretting can be attributed to defects within the coatings such as tiny cracks, high porosity, poor adhesion, and cohesion. Postspray treatment methods such as annealing and laser treatments can help to reduce defects and improve adhesion properties (Fu et al., 1997).

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Plasma-sprayed hydroxyapatite (HA) coatings can be utilized in fretting regions within orthopedic applications under lubricated and unlubricated conditions (Fu et al., 1998b, 1999). The coatings demonstrated inferior fretting wear properties to the Ti substrate due to defects and a lamella structure, which generated debris particulates. Hot isostatic pressing (HIP) treatment can be utilized to minimize the debris formation during fretting, as it improves wear resistance and the cohesive bonding of the coating due to the decrease of defects and the densification of the microstructure. The HIP treatment, which applies high temperatures and gas pressures, can lead to interlamellar diffusion and an improved bonding between the lamellae and with the substrate, which can improve fretting wear resistance (Fu et al., 1998c; Khor and Loh, 1995).

3.5.4.2 Hard coatings Hard coatings are commonly applied to reduce erosive, abrasive, or adhesive wear. A hard coating layer on a softer substrate provides protection against abrasion by hard debris and counterface asperities. There is extensive literature on ceramic-based hard coatings based on nitrides, carbides, borides, oxides, and their combinations applied to metal alloys and cermet composites, providing protection against wear and oxidation at high contact loads, speeds, and temperatures (Musil et al., 2014). Here, we briefly highlight the major families of hard coatings applied for fretting wear contacts before proceeding to describe more complex coating designs utilized by industry. TiN coatings have been widely implemented and successful for mitigating fretting damage (Blanpain et al., 1995). Typical friction behavior of TiN coatings during fretting tests involves a running-in period where a low friction coefficient is observed followed by a period of high friction after a several thousand cycles. In this period, the coefficient of friction reaches a maximum level, until reaching a lower steady state (Fu et al., 2000). de Wit et al. (1998) investigated the tribo-chemical reaction of TiN coatings in fretting tests. Results demonstrated that, at the beginning of the tests, the debris consisted of both amorphous and nanocrystalline materials with a nanometer crystal size. During the friction rise period, the debris was composed of amorphous material, said to be related to the presence of nitrogen in the contact. When the COF reached a low steady-state value, debris with a nanocrystalline structure was found, indicating the change in friction behavior was due to the transformation of new phases (Fu et al., 2000). Some studies have shown Ti2N-containing coatings are more effective than TiN coatings in mitigating fretting wear due to enhanced ability to form lubricating oxides. However, these coatings are softer, which leads to a reduced endurance over long-term fretting tests (De Bruyn et al., 1995). The performance of TiN coatings in fretting is influenced by factors such as the normal load, frequency, and slip amplitude. By increasing the normal load a reduction in friction is observed, most likely due to an increase in oxide debris. Large fretting displacement may greatly damage TiN coatings and wear is observed to increase with higher amplitudes of slip. With low displacement fretting, the central wear section is covered by a layer of debris, which separates the two interacting surfaces and reduces wear. With a large amplitude of slip, an increase in volume of wear is observed due to

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the escape of abrasive debris from the contact. By increasing the frequency of fretting, the oxidation time for debris is reduced and debris can be removed from the contact more easily; hence, an increase in the coefficient of friction is observed. At lower frequencies, debris are exposed to the atmosphere for longer periods and hence oxide debris are formed and the coefficient of friction is reduced. Humidity can also impact on the performance of TiN coatings through the formation of lubricious layers such as TiO2 x within the fretting contact and hence a decrease in the coefficient of friction can be observed (Wei et al., 1997a; Mohrbacher et al., 1995a,b). There are many derivatives of TiN coatings, including TiCN, TiAlN, TiCrN, and others, which maintain the fcc lattice of TiN and tailor hardness, toughness, oxidation, and corrosion resistance to specific applications (Sundgren and Hentzell, 1986; Chauhan and Rawal, 2014; Schalk et al., 2022; Mitterer, 2014; Mayrhofer et al., 2014). Chemical vapor deposition (CVD) and physical vapor deposition (PVD) are predominant methods used by the coating industry to produce these hard coatings, where PVD methods using magnetron sputtering and cathodic arc evaporation (and their combination) are currently dominating. Other hard coatings currently used in industry include oxides such as Al2O3, Cr2O3, carbides such as TiC, WC, SiC, and B4C, and borides such as TiB2 and HfB2. Multicomponent composition and multilayer coating architectures from these materials (in addition to the nitride-based coatings) with a broad spectrum of properties have been adapted by the coating industry and the current research focuses on microstructure designs in composite coatings to improve hardness, wear resistance, and tribological properties (Mitterer, 2014). A key limitation of hard coatings deposited on soft substances relates to high stresses induced in the coatings and at the interface with the substrate by the applied load. Additionally, at high deposition temperatures, internal stresses are generated, which are compressive in the plane of the surface and can benefit wear and fatigue failure behaviors (Holmberg and Matthews, 2009). Diamond-like carbon (DLC) is another important family of materials for commercial wear protective coatings, which form a hard film on the surface, which offers good load-carrying capabilities and reduces scratching, alongside having the ability to produce low shear strength transfer layers on interacting surfaces, which results in weak shear planes and low friction. The coating offers excellent wear resistance combined with coefficients of friction levels, which are a magnitude lower compared to many other coatings (Holmberg and Matthews, 2009). Depending on the synthesis method (CVD and PVD) and precursor (hydrocarbon gas and graphite), the material ranges from relatively soft hydrogenated DLC to a hard hydrogen-free tetrahedral coordinated polycrystalline diamond with a range of compositions in between, where hydrogen incorporation, sp2/sp3 ratio of near atom arrangements, and doping with various metals (W, Ti, and Si) are used to tune from soft and lubricious to harder and abrasionresistant DLC variants. For all of these DLC coating compositions, the fretting process leads to a formation of lubricating graphitic film in the contact by stress and temperature-induced phase transformation (Tambe and Bhushan, 2005), which adds to improved fretting wear resistance and self-lubricating properties. Correspondingly, in high-frequency fretting tests a decrease in friction coefficient is observed due to the thermal degradation of the coating and its partial graphitization. Similarly, with a high

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humidity a reduction of friction is observed due to chemisorbed moisture on the DLC coating and water molecular intercalation in the graphitic film formed in the contact (Schouterden et al., 1995; Blanpain et al., 1993; Wei et al., 1997b).

3.5.4.3 Adhesion of hard coatings A common problem associated with hard coatings can be the poor adhesion to soft substrates due to several issues such as a low degree of prime (metallic, covalent, or ionic) bonding, high interface shear stresses originated from a mismatch in elastic moduli and, coefficients of thermal expansion, resulting in residual stresses in both the coating and the underlying substrate. The presence of brittle and low shear strength interfacial phases, or poor mechanical interlocking at the interface are also issues. The selected coating process influences these factors; hence, one of the prime development focuses in coating technology is in optimizing the deposition process to achieve effective adhesion (Holmberg and Matthews, 2009). While there are a number of coating design approaches directed on the adhesion enhancement depending on a selected deposition process (Sundgren and Hentzell, 1986; Chauhan and Rawal, 2014; Schalk et al., 2022; Mitterer, 2014; Mayrhofer et al., 2014), those most commonly accepted by the coating industry are: (i) introduction of metallic bond layers, where Ti and Cr are most typical for modern PVD hard coatings; (ii) substrate surface plasma etching prior to coating deposition to minimize oxide and hydrate phase on the interface for prime bonding enhancement; (iii) metallurgical modification of the surface by subimplanting of bonding metal into the underlying substrate alloy, as for example in PVD growth with combined cathodic arc for bonding and follow-up magnetron sputtering for the main coating layer; and (iv) controlled substrate surface roughening and texturing for mechanical interlocking, as, for example, in plasma sprayed coating technologies. For example, improved tribological properties and adhesion were observed with the application of a 100-nm titanium interlayer between TiN coatings and steel substrate. On cast-iron substrates, increases in wear resistance are achieved through the deposition of a 1-μm thick pure titanium layer between a TiN coating and the substrate, which helps to reduce the influence of graphite on the growth of the hard coating (Matthes et al., 1990). The crack resistance of these coatings can be improved by also improving the interface roughness (Matthews, 1980; Matthews, 1985; Matthews et al., 1993; Valli et al., 1985; Cheng et al., 1989; Rickerby et al., 1990; Zoestbergen and De Hosson, 2002). Gerth and Wiklund (2008) compared the adhesion of several different interlayers to increase the adhesion strength of TiN coatings on steel substrates. W, Mo, Nb, Cr, Ti, Ag, and Al interlayers ranging from 100 to 150 nm were assessed, with Mo and Nb providing the best adhesion results. The hardness of these interlayers had a greater influence on adhesion than the elastic modulus of the interlayer. Over the years, designs with graded interfaces, allowing for a controlled transition of thermal expansion coefficient, stiffness, and metallic-to-covalent bond transition have been developed and are being commonly used nowadays by the coating industry to avoid peak shear stresses at the bonding interface and enhance adhesion (Voevodin et al., 1995a, 1997a). For fretting contacts at high contact loads, the attention to the coating

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adhesion is one of the most critical parameters dictating the overall effectiveness of the coating application, where methodologies outlined above are all being successfully used together with enhancing high load supporting capacity by underlying substrate strengthening. As an example, for DLC coatings used in fretting contacts on plasma-nitrided steel surfaces, a Si bond interlayer contributes to a significant improvement of the coating performance (Dalibon et al., 2019).

3.5.4.4 Soft metal coatings Using soft metals such as silver, lead, copper, gold, indium, and nickel provides the advantage of low shear conditions, which can provide low friction properties. However, other parameters such as adhesion to the substrate, film thickness, load, and surface roughness would need to be optimized. Studies with fretting tests, where displacement amplitudes ranged between 50 and 100 μm, have shown that chromium and zirconium metal coatings provided durable films (Ohmae et al., 1978). Other metal coatings such as cadmium and gold also act as effective films; however, their toxicity or cost is disadvantages (Cobb and Waterhouse, 1987). Where electroplating is impractical, vacuum deposition methods can help to deposit homogeneous films with greater adhesion. Bowden and Tabor (2001) demonstrated that soft metal coatings with a thickness ranging from 0.1 to 10μm help in achieving low values of friction, even less than 0.1, when sliding against a metal counterface. It was assumed that at high normal loads and sliding speeds, the metal coating softens due to high localized flash temperatures and behaves as a lubricant. However, key issues were the wear and lifetime of these coatings. With the application of thin films, factors such as surface roughness, film thickness, and environmental influence on the coating oxidation play an important role. Studies have shown for a 0.5 μm-thick lead film on a steel substrate rubbing against a steel counterface, friction reaches a minimum value; however, with thicker or thinner films friction increases. Thicker films usually have a larger contact area, hence higher friction whereas very thin films may suffer from asperity breakthrough (Holmberg and Matthews, 2009). Thinner films can produce lower wear rates, since with thick coatings the wear mechanism is the microcutting of the coating by the interacting surface asperities. The thin film failure mechanism is shifted toward fatigue where the interacting surface asperities can break through the film and contact the substrate material (Sherbiney and Halling, 1977). Jahanmir et al. (1976) demonstrated that optimum film thicknesses less than 1 μm were effective for a number of soft metal coatings such as Cd, Ag, Au, and Ni. The explanation was based on an onset of wear particle generation, where the coated material must be softer than the substrate to minimize the wear rate and easy shear continuously to avoid debris. Thin soft coatings deposited on a hard coating can retard the wear of the harder substrate through preventing plastic deformation in the substrate by sliding shear in the top soft layer. If the soft layer is thicker than a critical value, the plastic deformation results in forming loose wear particles. These particles can further oxidize and lead to surface plowing and accelerating plastic deformation and wear of the top layer. With thick coatings on very rough surfaces there is also an increase in microcutting wear, while for thinner coatings on such rough counterfaces they may

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interact with the substrate through the soft coating. Overall, soft coatings applied to smoother surfaces last longer in sliding contacts ( Jahanmir et al., 1976). In aerospace turbine engine applications, the blade/disk contact surfaces are susceptible to high-frequency fretting and may experience stick-slip regime wear damage leading to blade root fatigue cracking and catastrophic failure. To protect the components from fretting wear the blade dovetail is often plasma sprayed with a thick soft copper-nickel-indium (CuNiIn) coating followed by a solid lubricant for compressor blades made of titanium alloys. The CuNiIn coating and solid lubricant acts as a sacrificial layer which: (i) avoids stick and minimizes the stick-slip regime by shifting contact toward the gross slip mode of operation to reduce fatigue crack risk; and (ii) protects the disk and blade components from wear by accommodating sliding shear in the contact. The CuNiIn coating enhances protection against plastic deformation while the solid lubricant reduces the friction coefficient and promotes sliding. The surface roughness of the CuNiIn coatings helps with the bonding to the solid lubricant. This strategy leads to an increase in component life; however, recent studies have demonstrated that once the solid lubricant layer was worn out, the CuNiIn-coated dovetails caused significant damage to the uncoated Ti-alloy disk due to galling at the interface. There are tremendous costs associated with the maintenance of these coated systems (Hager Jr et al., 2008; Mary et al., 2011). Plasma-sprayed Ni coatings have been suggested as a good choice for fretting wear contacts at high temperatures (in excess of 800°C) due to the formation of a glaze oxide layer (Hager Jr et al., 2008). Freimanis et al. (2002) observed similar trends with the application of cobalt coatings for applications with temperatures above 450°C. These coatings are suitable for jet engine blades in hot sections and other components, which are exposed to extremely high temperatures in oxidizing environments.

3.5.5

Advanced coating designs and architectures

New deposition techniques and their combination provide the ability to produce coated surfaces with a combination of complex designs and architectures such as multicomponent coatings, gradient coatings, multilayer coatings, superlattice coatings, and duplex coatings (Subramanian and Strafford, 1993; Hogmark et al., 2000). Fig. 3.5.2 shows some of these design variants.

3.5.5.1 Multicomponent and composite coatings Multicomponent (single phase) and composite (multiphase) coatings, which have a mixed composition of several materials, offer a possibility to improve the tribological properties of single-layer surface coatings (Fig. 3.5.1). The mechanical and physical characteristics of a coating can be modified by changing the relative concentration of nonmetallic elements, which changes the valence electron concentration (Subramanian and Strafford, 1993; Hogmark et al., 2000; Freller and Haessler, 1988; Holleck, 1991; Knotek et al., 1992). While some of these coating remains single phase, as for example a class of TIN-based coatings with addition of carbon and other metals and maintaining fcc single phase, most often phase separations in these coatings are

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Fig. 3.5.1 Architectures of advanced tribological coatings.

used to obtain enhanced wear-resistant performance. Approaches with a controlled formation of an amorphous phase tissues around nanosized crystalline grains made of nitrides, borides, and carbides create hard and superhard (above 40 GPa) coatings (Veprek, 1999; Voevodin et al., 1997b; Musil, 2012; Mayrhofer et al., 2006a). Also, multicomponent coating design approaches with a controlled spinodal decomposition of fcc to wurtzite and hcp phases in TiAlN coating (Mayrhofer et al., 2003) or TiN and TiB2 nanocrystalline phase formations in TiBN coatings (Mayrhofer et al., 2006b) help maintain and sometimes increase hardness in high-temperature operations. Hard composite coatings obtained by PVD methods which incorporate an excess of carbon in the hard ceramic matrix coatings result in segregation of amorphous carbon around hard crystalline grains (Voevodin et al., 1997b; Rebholz et al., 1997) and are especially beneficial for fretting wear situations to provide a hard wear-resistant surface and form a continuous lubricious film in the contact by carbon phase graphitization under repeated cycling. Examples with WC:C and TiC:C are most commonly reported for this class of multicomponent coatings in available reviews (Musil, 2000; Voevodin and Zabinski, 2005; Pogrebnyak et al., 2009) and widely adapted by modern PVD coating industry.

3.5.5.2 Multilayered and superlattice coatings One coating does not always have all the properties needed for a certain application; however, this can be achieved through using layered coatings. Each layer

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contributes with one or a set of specific properties (Holmberg and Matthews, 2009) and algorithms for the coating layer sections and stacking for improved wear resistance designs were proposed (Voevodin et al., 1994). There are a number of review articles (Holleck and Schier, 1995) devoted specifically to the optimization of the design of multilayered wear protective coating architectures, where considerations of balancing of hardness and toughness for optimum mechanical performance, surface crack deflection at high and cycled contact loads, diffusion barrier, and oxidation resilience for high-temperature applications and enhancing lubrication of the coating surfaces for tribological endurance are all considered toward choosing individual layer material, thickness, stacking sequencing, and layer-to-layer transitions. For example, multilayer coatings alternating superhard, hydrogen-free DLC and less hard TiC or more ductile metallic Ti layers has been reported to provide a considerable endurance under high-contact load sliding (Voevodin et al., 1995b, 1997c). One branch of multilayer coatings design considers specifically the selection of layers with matched crystal lattices and stacking hundreds of these layers in a nanoscale-level-controlled thickness, where induced modulation of a lattice parameter through the stack had been aimed at creating a “superlattice” effect, which aims to block dislocation mobility and provide a superhard surface (Helmersson et al., 1987; Sproul, 1996; Hovsepian and M€ unz, 2002). Although early endeavors in this direction were found to be challenging (due to the difficulty in creating completely heteroepitaxial growth on real components), now it is possible to obtain coatings that can exhibit some of the benefits of “superlattice” films and such coatings have been adapted and adopted by the PVD protective coating industry (M€unz et al., 2001). Currently, their application use is reduced in favor of composite superhard coating designs discussed earlier, which are more manufacturing versatile as they avoid technological constraints associated with maintaining the superlattice lattice parameter over complex substrate shapes.

3.5.5.3 Adaptive composite coatings The term “adaptive” generally refers to smart or intelligent multicomponent composite coatings, which can respond to the changing operating conditions and environment. This behavior is of great appeal in tribology applications and an example is the use of these coatings in gas turbine engines to resist high-temperature corrosion as well as in aerospace applications where pressure, temperature, humidity, and surrounding atmosphere composition fluctuate significantly (Voevodin and Zabinski, 2005). In this application, the coating develops a tribofilm of varying composition, depending on the operating environment, and such a tribofilm self-adjusts depending on the environment temperature, humidity, and chemistry. Such coatings are sometimes referred as “chameleon” in the adaptive solid lubrication literature (Voevodin et al., 2002; Muratore and Voevodin, 2009). Zabinski et al. (1995a,b) developed solid lubricant coatings with the capability of adapting to temperatures from ambient up to 800°C. At such high temperatures, graphite or metal dichalcogenides (MoS2 or WS2) are used because conventional oils and greases are ineffective. The lubricious properties of these coatings are due to their low

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shear strength. However, at high temperatures these materials may oxidize; this may also occur at a lower temperature of 350°C; hence, research has involved the addition of graphite fluoride (CFx) additives to the coating. This extends the operating condition of the coating to 450°C and combined with a pulsed laser deposition method, this modified coating was less sensitive to moisture and achieved low friction (0.01–0.04) values against a stainless steel counterface. These coatings are effective over a wide operating range. Other material additions can be considered to provide lubricity at higher temperatures (500–800°C). Oxides (ZnO) and fluorides (CaF2 and BaF2) offer effective lubricious properties in this regime due to their low shear strength and high ductility properties. However, at ambient temperatures these materials demonstrate brittle behavior and are subject to high wear rates. To produce a coating that can operate over a wide range of temperatures requires the uniform dispersion of a range of low- and high-temperature materials throughout the coating in either nanolayered or nanocomposite structures (Holmberg and Matthews, 2009). Studies have highlighted that to achieve reactive coatings the aim is to develop chameleon-like coatings through using diffusion barrier coatings with layered structures, allowing them to operate reversibly through multiple temperature cycles. Ideally, these chameleon coatings will include the nanodispersion of nonpercolated active constituents. Nanocomposite structured coatings are favored as they provide a degree of control over the availability of active constituents, meaning the release rate can be adapted to suit any operating life requirements (Holmberg and Matthews, 2009).

3.5.5.4 Duplex coatings Duplex wear-resistant coating technologies are normally referred to as processes in which a combination of significantly different processing routes in sequence to each other is used to engineer surfaces for improved combination of load-bearing support, endurance, and tribological performance. A common example adapted by industry is the combination of nitriding, carburizing, and carbonitriding outlined earlier in this chapter with described coating applications. Of these, plasma nitriding followed by a PVD hard coating application is probably one of the most widely accepted and practically used duplex technology, stemming back to early days of single-phase TiN hard-coating application on steels (Korhonen et al., 1983) to the current application of multicomponent TiBN-TiB2 and CrAlTiBN coatings to plasma nitrided surfaces (Weinhold et al., 2021; Klimek et al., 2003), as well as plasma nitriding process enhancement by high current pulsed plasmas, which take their origin from the high ionization rate PVD process technologies (Sugumaran et al., 2021). PVD methods are also combined with other methods for generating a thick load supporting layer before applying hard and low friction coatings, which include both using a powder spray (Bemporad et al., 2006) and electro-spark alloying (Kiryukhantsev-Korneev et al., 2020) followed by sputtering and cathodic arc-evaporation PVD methods.

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One nonvacuum-based duplex method, which appeared more recently in the literature, is a combination of plasma electrolytic oxidation (PEO) and chameleon solid lubricant coating applied by burnishing, which was especially effective for mitigating fretting wear contacts on aluminum alloys (Lin et al., 2021; Liu et al., 2018; Shirani et al., 2020). PEO is an advanced electrochemical surface treatment used to generate oxide ceramic coatings on light alloys, including magnesium, aluminum, titanium, and zirconium alloys. Produced coatings have a relatively high hardness and adhere well to the substrate. Morphologically, PEO coatings typically comprise an outer porous region and a dense inner layer followed by a thin interfacial barrier layer, which provides a suitable load-bearing sublayer for the application of chameleon coatings (Fig. 3.5.2). The outer porous region of the PEO coating can be beneficial in combination with chameleon coatings as it can provide lubricant reservoirs for continuous release into the fretting contact. A chameleon coating (MoS2/Sb2O3/C) and PEO combination has been applied to AA6082 Al series alloy substrates. The friction and wear performances were investigated under fretting conditions in humid air and dry nitrogen environments. This combination provided significant fretting wear reduction; however, adaption mechanisms and surface evolution under different fretting wear conditions are still not fully understood and need further investigation (Lin et al., 2021; Liu et al., 2018).

3.5.6

Concluding remarks

In this chapter, we have endeavored to scrutinize the available literature on materialsrelated aspects in fretting. That literature is sometimes contradictory and does not lend itself to a simple set of guidelines, since, as indicated, the processes involved are complex and are strongly influenced by many different factors. Nevertheless, we hope that the chapter will help to provide some insight into and guidance on the range of materials-related effects occurring, which need to be considered when dealing with fretting. The mainstream strategies to ensure performance of engineering components subjected to fretting rely upon shifting materials failure from fretting fatigue to more predictable fretting wear. Therefore, aspects of materials tribology related to reciprocating sliding under high contact loads at small displacement magnitudes become relevant. Under such conditions, the efficiency of approaches, based on bulk materials engineering, involving simple surface hardening and single layer coatings, is rather limited, and more complex surface engineering strategies including combinations of surface hardening (especially using thermal-chemical treatments or thick hard coatings) followed by deposition of soft lubricious topcoats become of increasing interest. These functionally graded and composite surface structures are generally more robust and show a better performance under laboratory conditions. However, changing the modality of mechanical loading, temperature, and composition of the environment characteristics to demanding real-life applications represent further challenges for such advanced surface architectures. Associated problems are currently being addressed by more intricate approaches, which include interfacial engineering to

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Fig. 3.5.2 Example of nonvacuum duplex coating system comprising a Gr-MoS2-Sb2O3 “chameleon” topcoat burnished on a load-bearing PEO sublayer formed on Al 6082 alloy substrate and its fretting performance against the steel counterpart in humid air. Adopted from Lin, M., Nemcova, A., Voevodin, A.A., Korenyi-Both, A., Liskiewicz, T.W., Laugel, N., et al., 2021. Surface characteristics underpinning fretting wear performance of heavily loaded duplex chameleon/PEO coatings on Al. Tribol. Int. 154, 106723 with permission from Elsevier©.

optimize adhesion of the hard load-bearing underlayer with both a metal substrate and a soft lubricious topcoat as well as providing the latter with smart materials’ functionalities to allow it to respond to changing environmental conditions. Research in this direction is currently in its infancy and we can expect it to be developed further in the future.

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Contact size in fretting Ben D. Beake Applications Development, Micro Materials Ltd, Wrexham, Wales, United Kingdom

3.6.1

3.6

Introduction

In 2007, Matthews and coauthors identified five important “scales” of tribology—unitribology, decitribology, macrotribology, microtribology, and nanotribology (Matthews et al., 2007). They noted that a key trend in tribology research was the continuing move down the length scale in terms of the fundamental understanding of tribological contacts, which was particularly useful in aiding the development of new coatings by identifying their property requirements at different scale levels. Deformation and wear begin at the asperities between contacting surfaces (Greenwood and Williamson, 1966; Bhushan, 1996; Sawyer and Wahl, 2008; Stoyanov and Chromik, 2017), but the contact pressures acting on these are not generally accurately known in a macroscale tribological test with multiasperity contact. The contact zone is hidden and the real area of contact is initially a small fraction of the apparent contact area and the contact pressure larger than the nominal pressure (Bhushan, 1996; Sawyer and Wahl, 2008; Dieterich and Kilgore, 1994). The simplified contact conditions in single-asperity tribological tests with much lower forces and sharper probes provide an alternative approach to study the onset of wear, its correlation with friction, and the influence of surface topography and mechanical properties (Sawyer and Wahl, 2008; Stoyanov and Chromik, 2017; Jacobs et al., 2019; Szlufarska et al., 2008). This chapter focuses on micro-/nanoscale tribological testing involving singleasperity contacts in sliding (scratch) and reciprocating contacts including fretting, covering gross slip and partial slip regimes. The most common experimental configuration is the sphere-on-flat test geometry which combines experimental convenience with being amenable to indentation modeling analysis. In single-asperity microtribology, the length scale (contact size) plays an important role and contact mechanics is key. Contact mechanics is critical in tribology, as it provides quantitative descriptions of the contact area, elastic indentation, contact stiffness, and the stress and strain fields of a mechanically loaded asperity (Szlufarska et al., 2008). Experimental techniques for studying small-scale mechanical contact are introduced. The micro-/nanoscale experiments are contrasted with those in single-asperity nanotribological tests with AFMs. Industrially relevant material systems from the MEMS, biomedical and coating sectors are selected as example case studies. In each case, the contact size plays a Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00010-X Copyright © 2023 Elsevier Inc. All rights reserved.

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crucial role in the observed behavior. For example, the surface scratches observed in retrieved protheses are typically only a few microns wide and less than a micron deep, so in order to develop a fuller mechanistic understanding for reliable artificial joint design, it is helpful to study mechanical and tribological properties of biomedical materials at the relevant contact scale as these properties can be highly size dependent.

3.6.2

Experimental techniques for nano-/microscale fretting and reciprocating wear testing

Different experimental scanning probe techniques have been employed using, for example, TEMs ( Jacobs et al., 2019; Liao et al., 2015; Liao and Marks, 2015, 2016; Milne et al., 2019), AFMs (Szlufarska et al., 2008; Beake et al., 2001a, 2004; Prioli et al., 2003; Schiffmann, 1998; Yu et al., 2009a,b,c, 2012, 2010; Chen et al., 2011; Varenberg et al., 2005; Peng et al., 2009; Colac¸o, 2009; Mitchell and Shrotriya, 2007; Degiampietro and Colac¸o, 2007; Garabedian et al., 2019; Liu et al., 2020), nanoindenters (Qian et al., 2007; Schiffmann and Hieke, 2003; Schiffmann, 2004, 2008; Wilson et al., 2008, 2009; Wilson and Sullivan, 2009; Wilson, 2008; Beake et al., 2012, 2011, 2010; Liskiewicz et al., 2010, 2013; Stoyanov et al., 2010, 2012), or microtribometers (Drees et al., 2004; Achanta et al., 2005, 2008; Gee and Gee, 2007; Gee et al., 2011) as test platforms. Transitions between different wear regimes can be conveniently studied by changing the experimental conditions (e.g., applied load, probe geometry, sliding distance) that combine to alter the length scale of the contact. To date, many but not all nano
/microscale fretting tests have been performed under gross slip conditions. Potentially one of the reasons is that in many cases the wear rates are higher under gross slip conditions, as mechanisms where lower wear due to stabilized third body or restricting tribooxidation as occurs in partial slip are less common (Fouvry et al., 2009). Information about the stress distributions in small-scale fretting contacts can be supported by performing tests and analysis of simplified contact situations that occur in indentation or scratch (sliding) contact. For coated systems, modeling of these provide simulated initial stress distributions which can reveal, e.g., whether the coating or substrate is overloaded. TEM single-asperity and MD approaches have been used ( Jacobs et al., 2019; Liao et al., 2015; Liao and Marks, 2015, 2016; Milne et al., 2019). Liao and Marks have reviewed TEM sliding experiments. Liao, Hoffman and Marks showed that in nanoscale abrasive dry sliding of the fcc phase of CoCrMo against an Si tip in vacuum in a TEM, the attack angle was critical in controlling the deformation behavior (Liao et al., 2015). Milne and coworkers reported that adhesion between nanoscale Si contacts increased after sliding, hypothesizing that sliding temporarily removed passivating terminal species such as hydrogen or hydroxyl groups (Milne et al., 2019). Combining small radius probe tips, higher resolution and low noise floor atomic force microscopes (AFMs) have been a popular choice for single-asperity tribological tests (Szlufarska et al., 2008; Degiampietro and Colac¸o, 2007). Degiampietro

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and Colac¸o used an AFM with a 250 nm end radius diamond tip to wear 2 μm  2 μm regions of 316 L stainless steel at 2–20 μN (Degiampietro and Colac¸o, 2007). They reported that the worn volume increased linearly with the number of interactions between the surface and the abrasive tip and the applied load but decreased with increasing scan velocity. Mitchell and Shrotriya in 70 pass wear experiments with an Si3N4 AFM probe of 50 nm end radius found that while 2–3 nN (0.9 GPa) was insufficient to produce plastic deformation of CoCr, 25 nN (2.3 GPa) resulted in breaking of the thin passive oxide and alloy surface wear (Mitchell and Shrotriya, 2007). At this scale, it can prove difficult to detect the onset of wear and in cases where wear is dominated by plastic deformation rather than material removal, measures of surface roughness, such as the Ra surface roughness, may be a more useful indicator of the progression of damage than the wear depth (Beake et al., 2001a, 2004; Achanta et al., 2005). Varenberg et al. studied the partial and gross slip fretting behavior of 1.55 μm radius SiO2 probes tested against Si wafers. They reported a substantial increase in friction at the transition between the partial and gross slip regimes and differences in wear scar appearance between the two fretting regimes (Varenberg et al., 2005). For potential application in MEMS devices operating in vacuum conditions, Yu et al. studied the nano-fretting behavior of monocrystalline silicon by using SiO2 AFM tips with end radii of 0.15–0.9 μm (Yu et al., 2009a,b,c, 2012, 2010) over displacement amplitudes 0.5–250 nm. The energy ratio related to the transition from the partial to gross slip regime was measured at 0.32–0.64 and compared to the same energy ratio observed in classic macroscale fretting (0.2). The role of adhesion in enhancing stiction was considered. Depending on the applied load and test environment, either “hillocks” or a typical wear trench was observed (Yu et al., 2009a, 2012). At lower contact pressure, hillocks formed, but as the Hertzian contact pressure is close to the hardness of silicon, grooves were observed (Yu et al., 2009a). Despite its popularity, there are certain limitations in using AFM technology for nano-wear tests. These include: (i) short track distances, (ii) low sliding velocity, (iii) high contact pressures, (iv) susceptibility to probe wear due to the small tip radius, and (v) as a piezo-based technology, it does not have the necessary stability for long duration tests (Mitchell and Shrotriya, 2007). A complementary approach to the AFM nano-wear test is to perform nano-/ microscale tribological tests with a nanomechanical test instrument (nanoindenter). These tests typically have (i) larger probe radii (often in the micron range, minimizing indenter wear), (ii) a larger available force range, (iii) a longer sliding distance, (iv) a higher sliding velocity, and (v) better instrumental stability. Nanomechanical testing has proved to be an important technique in improving our fundamental understanding of the basis of mechanical properties of materials and the importance of nanoscale behavior on their performance. Nanomechanical and nano-/microtribological test techniques can be broadly divided into two main categories: (i) those primarily designed for characterization of mechanical properties—i.e., nanoindentation and (ii) those primarily involved with simulating contact conditions—including nanoand microscratch testing, nano-/microimpact and reciprocating nano-wear, nano-fretting tests.

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When sliding at low sliding speeds under nearly elastic conditions with low friction, the indentation and scratch stress fields are similar, but they increasingly diverge for higher friction contacts as occur at a higher load and/or on more ductile materials. In ramped load nano-scratch tests that cause coating failure in a single cycle, the contact pressures are often necessarily high and the ploughing component of the friction force can be relatively large. In a repetitive subcritical load scratch test, where the conditions are chosen so that the peak stresses can be placed in the coating or in the vicinity of the interface, the contact pressure remains high, but the test shows enhanced sensitivity to subtle differences in interfacial bonding strength (Shi et al., 2008). Nanoindenters and microtribological instruments have been used to study solid lubricant coating behavior in reciprocating friction and wear tests at the mN force range. In these microscale tests, the residual wear depth combines contributions from real material loss and plastic deformation. Schiffmann investigated the correlation between friction and the evolution of elastic, plastic deformation and wear in reciprocating testing of DLC and Si-doped DLC on glass (Schiffmann and Hieke, 2003). By performing additional nanoindentation tests on the same loads, it was possible to separate the individual contributions of plastic deformation and material loss from the residual depth (since in the low load single-cycle indentation tests the contribution of wear is minimal and the residual depth is due to plastic deformation). Stoyanov and coworkers studied the microtribological properties of Au, Au-MoS2 composite, and Au/MoS2 bilayer coatings (Stoyanov et al., 2010, 2012). The composite and bilayer coatings showed more improved behavior in the reciprocating tests than the monolayer Au. Tribofilm formation was revealed by ex situ characterization of the wear track. Achanta and coworkers performed reciprocating sliding tests at different length scales, showing that the influence of surface topography on friction was more significant in smaller scale contact (Drees et al., 2004; Achanta et al., 2005, 2008). Gee and coworkers developed a microtribometer that could be used as a benchtop system or in an SEM (Gee and Gee, 2007; Gee et al., 2011). Wilson and coworkers used a modular addition to a commercial nanoindenter (NanoTest system) with stability to run reciprocating (nano-fretting) tests to over 200,000 cycles (Wilson et al., 2008, 2009; Wilson and Sullivan, 2009; Wilson, 2008). The configuration for small-scale fretting includes an additional oscillating stage with a multilayer piezo-stack to generate sample motion. The piezo movement is magnified by means of a lever arrangement to achieve larger amplitudes. The much larger number of wear cycles that can be conveniently run with this module enabled nano-fretting tests at lower contact stresses, via probes with larger radii and/or using smaller contact loads. Table 3.6.1 compares the typical conditions in nano-fretting and multipass sliding contact with the same instrumentation to AFM nano-wear. Using a 150 μm ruby tip under a 10–200 mN applied load and 2–14 μm displacement amplitude, the transition from the fretting/partial slip to gross slip was studied in wear of Cr doped carbon and amorphous carbon films (Wilson et al., 2008, 2009; Wilson and Sullivan, 2009; Wilson, 2008). The authors identified two distinct fretting wear regimes, with a classic W-shaped wear scar under low oscillation amplitude and a full U-shaped wear scar at larger amplitudes. Zhu and Zhou noted that in addition to

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Table 3.6.1 Typical test conditions in nano- and microtribological tests. AFM wear Type of motion Sliding speed (mm/s) Track length (mm) Number of cycles Total sliding distance (m) Probe radius (μm)

Reciprocating 0.001–0.25 0.001–0.1 1–50 0.000001– 0.001 0.02–1

Repetitive nanoscratch

Nanofretting

Unidirectional 0.001–0.1 0.01–1 1–20 0.00001–0.01

Reciprocating 0.01 0.02 1000–200,000 0.01–0.1

5–25

5–200

classic tangential fretting, three other modes can be studied in ball-on-flat contacts including torsional, rotational, and radial fretting (Zhu and Zhou, 2011). Nanoindenters have been used to perform the so-called radial nano-fretting experiments involving multicycling load-partial unload-reload contacts (Qian et al., 2009; Yu et al., 2011) and true nano-impact experiments involving high strain rate repetitive impacts (Beake et al., 2001b; Beake and Smith, 2004).

3.6.2.1 Contact geometry effects As mentioned above, the probe radius, track length (if in a partial slip regime), and applied load influence the friction and wear behavior, particularly for tests on thin coating systems. In general, in a nano- or microscale sphere-on-flat tribological test, the wear does not proceed at a constant rate. There are several reasons behind this. As wear progresses, there are opposing effects from the changes in contact geometry. Although the initial contact pressure is high, this can rapidly decrease through wear increasing the contact depth and contact area. This increase in depth increases the contact strain. The influence of contact strain on the predominant wear mechanism has been studied across several length scales. Using a pin-on-disc tribometer inside an SEM, Kato and coworkers investigated wear mechanisms during low-pass (1–10 cycles) scratching of steels with a 30 μm WC-Co indenter at 0.07–1.5 N (Kitsunai et al., 1990; Hokkirigawa et al., 1998). They observed that increasing the load and contact strain resulted in a transition in wear mode from ploughing to wedge formation and then cutting at >40°. They introduced a degree of penetration parameter, Dp, defined as the on-load contact depth/contact radius to characterize the severity of the abrasive contact. Increased steel hardness promoted the transition to cutting wear which then occurred at a lower attack angle. In microtribological tests with a 1.8 μm radius diamond probe, Schiffmann (Achanta et al., 2005) reported a critical attack angle of 19  2° for the transition between ploughing and cutting, commenting that this was in good agreement with models from the literature that predict 21°. In TEM sliding, Liao et al. observed ploughing at 24° and cutting at 64° (Liao and Marks, 2016).

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3.6.2.2 Pile-up To understand what is behind the wedge formation of wedges in sliding on ductile materials, insights can be gained from indentation where more closely controlled experiments can be performed. When spherical indenters are used, the presence of pileup or sink in in nonwork hardening materials can change during an indentation as the indentation depth increases. FEA by Taljat and Pharr has shown that increasing h/R is qualitatively similar to increasing E/σ y (Taljat and Pharr, 2004). Suresh and coworkers have shown that in sliding contact there can be 2 3 more pileup in ductile materials at the edge of the scratch track than occurs in comparable indentation contact (Bellemare et al., 2008; Prasad et al., 2009).

3.6.2.3 Contact size effects on deformation vs fracture Rhee et al. noted that the deformation in an indentation test is a function of the radius of the indenter, with larger radii indenters (i.e., larger contact size) producing more brittle deformation and smaller radii indenters (i.e., small contact size) more plasticity (Rhee et al., 2001). These authors proposed a brittleness index Py/Pc reflecting the competition between plastic yield (deformation) and fracture in an indentation test. The relative proportion of deformation and fracture has a large influence on the subsequent wear in a tribological test and explains why hardness alone can be insufficient to predict the wear rate.

3.6.2.4 Indentation size effects Metallic materials can show a “smaller is harder” indentation size effect (ISE) where hardness and yield stress increase as the contact size is reduced (Spary et al., 2006; Hou et al., 2008). Jennett and coworkers have shown that the yield stress increases with the inverse cube root of the indenter radius in relatively large-grained fcc metals (Spary et al., 2006; Hou et al., 2008). Shim et al. (2008) noted that the increase in strength as the size of the contact decreases can be considered to be a different type of indentation size effect to that commonly seen in hardness, since the latter depends on the yielding and work-hardening behavior of the material and the former on the stress to initiate dislocation plasticity. An estimate for the size of the volume of material influenced in an indentation contact is given by 2.4a  πa2, where a is the contact radius and 2.4a is the depth of the primary indentation zone (Pathak and Kalidindi, 2015).

3.6.2.5 Lateral size effects Kareer and coworkers have reported that similar size effects occur in sliding contacts, during nano-scratch testing of single crystal copper scratched by a diamond Berkovich (pyramidal) indenter, whether the indenter was sliding face-forward or edge-forward (Kareer et al., 2016a,b). The higher scratch hardness was ascribed to a lateral size effect (LSE). Subsequent crystal plasticity finite element simulations confirmed that

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the deformation state in sliding was different from that in indentation (Kareer et al., 2020). The lateral size effect was smaller for small-grained materials than for larger grained or single crystal materials. The authors note that this provides evidence that the length scales combine rather than superimpose as they do in indentation, and it is this combined length scale that is responsible for the observed lateral size effect.

3.6.2.6 Size effects on yield and fracture in coatings In addition to these plasticity size effects in seen in bulk materials for coating systems, the interfaces and layers add further complexity. Chudoba and coworkers studied the dependence of the critical load for plastic deformation in an indentation contact on the yield stress of 2–3 μm TiN layers on a steel substrate (Chudoba et al., 2002). Critical load vs probe radius curves showed a sharp discontinuity at the transition between plastic yield in coating (at a low radius) and substrate (at a higher radius). For this coating-substrate system when indented by very large radii the coating no longer had a protective effect. This was due to the much higher stiffness of the coating than the substrate resulting in a stress concentration at the interface. This led to yield at a lower load than the uncoated steel, showing that although a hard stiff coating on a softer substrate is often assumed to mechanically protect the substrate, it is not always the case. Substrate properties can also play an important role in scratch tests, where higher critical loads are found for harder substrates. Sensitivity to coating and interfacial properties can be increased by performing tests at a smaller contact size. This can be achieved by reducing the test probe radius and using instrumentation with greater sensitivity at a lower load to perform nano- or microscale scratch tests. “Dimensioning” the test in this way reduces the influence of substrate deformation on the overall coating system response (Schwarzer et al., 2011). With a suitable choice of indenter radius and load, potential deficiencies in coating adhesion can be investigated by positioning the maximum von Mises stress at the interface, or at interfaces between different layers in a multilayer coating system. In comparison with progressive load nano-scratch testing, repetitive (subcritical load) constant load nano- and micro-scratch tests can be performed. This has the advantage that the constant load can be varied to locate the maximum von Mises stress near the coating-substrate interface and minimize (and potentially even eliminate in some cases) substrate deformation as a precursor of coating debonding. The experiments can be more informative regarding the influence of thin film stress leading to poor adhesion than single scratch tests. Shi and coworkers performed repetitive constant load nano-scratch testing with a 4 μm diamond indenter on a set of 1 μm a-C films deposited at different substrate bias (Shi et al., 2008). The carbon coatings failed in the nano-scratch test at a critical load of 200 mN, so repetitive tests were performed at subcritical loads of 50 and 150 mN. At 50 mN, the maximum stresses were within the carbon coatings, so their mechanical properties dominated and interfacial strength was less important. At 50 mN, the contact was almost completely elastic with residual wear depths under 100 nm, being lower on the harder films. At 150 mN, Hertzian analysis showed that the maximum von Mises stress was very close to the interface.

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Under more severe loading conditions, plasticity and microfracture dominated and the harder films deposited under high substrate bias performed poorly with extensive delamination outside of the scratch track within a few repetitive passes.

3.6.2.7 Contact size and friction The friction coefficient in fretting and full sliding reciprocating tests can be determined from the frictional energy dissipation as shown by Fouvry and Liskiewicz (Liskiewicz and Fouvry, 2005; Liskiewicz et al., 2005; Rybiak et al., 2008; Laporte et al., 2015). μ ¼ frictional energy dissipation=ð2  actual track length  applied loadÞ (3.6.1) The appearance of the friction loop in a fretting test is dependent on the lateral contact size through the sliding ratio. Fouvry et al. (1996) defined a sliding ratio e ¼ δg/a, where δg ¼ sliding amplitude (which can be different from the displacement amplitude due to the contact and testing rig compliance); a ¼ contact radius. The tribosystem remains in the true fretting regime when the unexposed surface is maintained at the center of the fretted surface (e < 1). The system moves into the reciprocating sliding regime when the center of the contact area becomes exposed to the atmosphere (e > 1). In studies of small-scale fretting, several authors (Martı´nez-Nogu
es, 2016; Raeymaekers et al., 2010; Beake et al., 2013) have reported that a marked reduction in track length can occur at a higher load, which may be connected to changing the contact stiffness of the tribosystem. In sliding tests, the observed friction is the sum of the interfacial and ploughing components. The friction coefficient can be considered to be the sum of interfacial and ploughing components (Eq. 3.6.2). μðtotalÞ ¼ μðploughingÞ + μðinterfacialÞ

(3.6.2)

Lafaye and Troyon showed how the ploughing friction component is influenced by contact depth and indenter geometry (Lafaye and Troyon, 2006).

3.6.3

Case studies

Industrially relevant material systems from the MEMS, biomedical and coating sectors are selected as example case studies. Since a multifunctional nanomechanical platform (NanoTest system) has been used to perform these tests, further information could be gained by performing different contact tests—such as nanoindentation or a repetitive nano-scratch—on these materials using a test probe of the same geometry. In each case, the contact size played a crucial role in the observed behavior and transitions between the different deformation regimes. This was through (i) whether pressures are > phase transformations in silicon, and thin hard coating systems on silicon,

Contact size in fretting

209

(ii) where the system is overloaded in DLC/steel, and (iii) whether the thin passive oxide layer on the titanium alloy remains intact.

3.6.3.1 MEMS—Silicon and thin hard carbon coatings on silicon The high surface-to-volume ratio makes interfacial interactions a dominant factor in the wear and lifetime of MEMS (or MST) devices. While MEMS can be fabricated from several different materials, by far the most popular is silicon due to the large infrastructure for Si-based devices of the microelectronics industry. At room temperature, silicon is a brittle material with little or no conventional plasticity and low fracture toughness. It exhibits highly complex mechanical and tribological behavior with phase transformations and lateral cracking being observed in indentation and brittle fracture in a wide range of mechanical contacts (Cook, 2006; Bhowmick et al., 2009; Domnich and Gogotsi, 2002). Consequently, wear and stiction forces have limited the reliability of Silicon-based MEMS when/if mechanical contact occurs (Williams and Le, 2006; Tanner et al., 1998; Ku et al., 2011). Commercial applications have therefore limited device motion, but if friction issues could be solved, many more applications could be developed (Kim et al., 2007). There has been considerable interest in the mechanical characterization of silicon, with its behavior under the idealized quasi-static loading conditions of a nanoindentation experiment being the subject of several rigorous studies (Oliver et al., 2008; Bradby et al., 2001; Juliano et al., 2003, 2004; Chang and Zhang, 2009a,b). It has been established that a phase transformation to metallic behavior occurs beneath the indentation contact site and the “popout” during unloading is a consequence of phase transformation and its accompanying volumetric expansion. However, much less is understood about its behavior under more complex loading geometries that occur in practical tribological situations. Beake and coworkers have studied the behavior of Si(100) using the same 4.6 μm spheroconical indenter as a test probe for nanoindentation, nano-scratch, and nano-fretting tests (Beake et al., 2011). The influences of tangential oscillation, applied load, and loading rate were investigated and the interrelationships between these examined in detail. In the nano-fretting tests, the track length was set at 2 μm and the oscillation frequency at 10 Hz. Fretting experiments were performed over a range of applied loads from 30 to 300 mN, resulting in 12–13 GPa contact pressure. Hertz equations estimate a contact radius of 0.9 μm under a 30 mN normal load and 2.7 μm under 300 mN. This suggests that in most cases the dominant fretting wear regime is gross slip, as the track length (2 μm) is smaller than 2 the Hertzian contact radius, in agreement with SEM images showing an elongated wear scar geometry. Two types of fretting tests were performed that differed in the rate of application of the applied load. The load was applied either abruptly (< 0.3 s) or slowly (10 s) and held for 1000s at this level in both cases before removal at the same rate. In contrast to its behavior in the nanoscratch and nanoindentation tests, where it showed relatively little rate sensitivity, in the nano-fretting test silicon wear was more strongly dependent on the rate of initial loading. Fig. 3.6.1 shows this. When the load was applied abruptly, radial and lateral cracking and material removal was observed and large displacement jumps (pop-ins)

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Fig. 3.6.1 Load and rate effects on wear scar volume in nano-fretting and nanoindentation tests. Key to symbols: Open circles ¼ slow loading without oscillation. Closed circles ¼ slow loading with oscillation and hold. Open diamonds ¼ abrupt loading with oscillation.

were observed during the subsequent constant load fretting test. The crack morphology was very similar to that in repetitive nano-impact tests and conventional nanoindentation at a higher applied load with the same probe. In contrast, when the load was applied more slowly, radial cracking was not observed and there was a distinct threshold load at around 100 mN, marking the transition to a more severe wear mode with extensive lateral cracking and material removal. In Ref. (Beake et al., 2011), a novel method was introduced allowing quantitative comparison of deformation during loading in the nanoindentation, nano-scratch, and nano-fretting tests. The loading curves in all three tests were almost identical at a very low load. Fig. 3.6.2 shows loading curves from indentation, scratch, and fretting tests to 200 mN. Tangential loading in the nano-scratch and nano-fretting tests promoted yield resulting in greater penetration depths at a higher load than in nanoindentation. This is in agreement with analytical results which have shown that in comparison with indentation, tangential loading with friction facilitates yielding. Zok and Miserez proposed that the magnitude of the decrease in critical load is related to the friction coefficient by (Eq. 3.6.3). Py =Py 0 ¼ 1
2:9μ2

(3.6.3)

where P0y is the yield load in the absence of friction, Py is the yield load with friction, and μ is the friction coefficient (Zok and Miserez, 2007). Eq. (3.6.3) is valid for friction forces up to 0.3, which Hamilton and Goodman showed is the point at which the maximum stress occurs on the surface at the edge of the contact and surface yield occurs (Hamilton and Goodman, 1966). Beake and coworkers reported Ly ¼ (40  5) mN in indentation, (37  5) mN in scratch, and  30 mN in fretting (Beake et al., 2011). These experimental data are in reasonable agreement with Eq. (3.6.3).

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Fig. 3.6.2 Comparison of loading curves on Si(100) from indentation, scratch, and fretting tests to 200 mN.

3.6.3.1.1 Coatings to protect silicon—Thin hard carbon films Various coating strategies have been considered for protective low friction overcoats for Si-based MEMS such as (i) liquid lubrication, (ii) solid lubrication with selfassembled monolayers, (iii) atomic layer deposition (ALD), and (iv) hard carbon films (Ku et al., 2012; Maboudian et al., 2000; Nistorica et al., 2005; Scharf et al., 2006; Smallwood et al., 2006; Akita et al., 2001; Chan et al., 2001; Xu et al., 1996; Beake and Lau, 2005; Beake et al., 2009). Promising results on actual MEMS devices have been obtained for conformal deposition strategies (WS2 by ALD, DLC by PECVD). Tetrahedral amorphous carbon (ta-C) films deposited by filtered cathodic vacuum arc (FCVA) have been developed for MEMS applications including capacitive sensors and protective coatings for micromachined components. The mechanical and interfacial behavior of the contacting silicon surfaces is modified by these thin, low surface energy films. Fretting, nano-scratch, and nanoindentation of 5, 20, and 80 nm ta-C films deposited on Si(100) have been performed using spherical indenters to investigate the role of film thickness, tangential loading, contact pressure, and the deformation mechanism in the different contact situations (Beake et al., 2013). The influence of the mechanical properties and phase transformation behavior of the silicon substrate in determining the tribological performance (critical loads, damage mechanism) of the ta-C coated samples was investigated. Nanoindentation showed that the films are hard (e.g., 80 nm ta-C has H  22 GPa) and elastic (Beake et al., 2013; Beake and Lau, 2005). This high hardness and H/E is due to >70% sp3, but the films can be highly stressed if they are too thick. Nano-scratch testing of ta-C films has shown that they are sufficiently thin to not show large area delamination and scratch resistance increases with H/E (Beake and Lau, 2005; Beake et al., 2009). Increasing the ta-C thickness from 5 to 80 nm was

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found to increase the critical load for film delamination by a factor of two. The ta-C film alters the phase transformation behavior of Si by providing load support, reducing the effective load reaching the substrate and spreading the deformation out over a wider area. Critical loads for pop-ins and pop-outs were modified by the presence of the ta-C overlayer, but their rate dependence was not influenced by the presence or absence of any film, confirming that they are due to transitions in the underlying Si rather than the film. The fretting tests were performed with a 10 μm track length and oscillation frequency of 5 Hz using 5 and 37 μm end radii diamond probes (Beake et al., 2013). The mean initial Hertzian contact pressures were 6 GPa using the 5 μm probe and ranged from 3–4 GPa at 10 mN to 10 GPa at 200 mN using the larger probe. The fretting wear occurred at a significantly lower contact pressure than was required for plastic deformation and phase transformation in nanoindentation and nano-scratch testing. Fig. 3.6.3 shows the friction-depth correlation from a 6000 cycle test on the 80 nm ta-C film with the 37 μm probe at 200 mN. Close examination shows that the slope in the on-load fretting depth shows some slight correlation with the measured friction, as changes in the depth wear rate were observed when friction changes, most noticeably after 3200 cycles. Deformation proceeded by a fatigue mechanism with a gradual wearing away of the film, as shown by the absence of any abrupt changes in depth or friction. This contrasts with the behavior in the nano-scratch test where abrupt changes in depth and friction occurred at film failure. There was a correlation in coating performance between the fretting and nano-scratch test results, despite the differences in contact pressure and failure mechanism in the two tests. Increasing the film thickness provided more load support and protection for the Si substrate. Thinner films offered significantly less protection, failing at lower load in the scratch test and more rapidly

Fig. 3.6.3 Friction-depth correlation from a 6000 cycle nano-fretting test on the 80 nm ta-C film. Probe end radius ¼ 37 μm. Applied load ¼ 200 mN.

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and/or at lower load in the fretting test. Chen and coworkers studied AFM nano-wear of 2 and 5 nm DLC films deposited by a filtered cathodic arc on Si(100) against R ¼ 1 μm SiO2 and R ¼ 0.5 μm diamond tips (Chen et al., 2011). Although the thicker DLC showed the best wear resistance, the thinner film also was more wear resistant than the Si(100). Wilson and coworkers studied the influence of film thickness on specific wear rate in high cycle (to 216,000 cycles) nano-fretting tests of 10–2000 nm sputtered Cr doped amorphous C films (Wilson et al., 2008, 2009; Wilson and Sullivan, 2009; Wilson, 2008). The counterbody was a ruby sphere of radius 150 μm and the displacement amplitude was 14 μm and oscillation frequency 10 Hz. Under these conditions, the specific wear rate (defined as worn volume of the coating per unit load per unit slid distance) reduced exponentially with decreasing film thickness. The lowest wear rates were in the range of 0.1–6.1  10
17 m3 N
1 m
1. The specific wear rate reduced as the applied load increased from 100 μN to 10 mN. A rapid reduction of the specific wear rate was observed during the first 3000 oscillation cycles. Greater wear for the thicker films may relate to their being more stressed than the thinner films.

3.6.3.2 Biomedical materials In hip implants, wear occurs mainly on bearing surfaces leading to adverse metallic ions and particulate debris generation. The rate at which an implant experiences wear depends on the bearing surfaces selected for the ball-and-socket components which typically include metal-on-metal, metal-on-polyethylene, ceramic-on-polyethylene, and ceramic-on-ceramic implant designs, each combination having unique advantages and distinct drawbacks (Buford and Goswami, 2004). Although artificial joints are designed as fully lubricated systems, asperity-to-asperity contact occurs leading to partially dry contact at the microscopic level (i.e., potential areas with boundary lubrication). In order to understand the failure mechanism of artificial joints, it is necessary to understand the mechanism that governs roughening of the metallic surface and subsequent damage nucleation and wear at the bearing interface. Scratching of the metallic surface by entrapped wear debris leading to increased UHMWPE wear rates has been recognized as one of the main causes of early failure of TJRs (Najjer et al., 2000). The hard particles responsible for the scratches can be zirconia or complex carbides present in the as-cast femoral head. In reciprocating sliding tests, Co- and Mocarbides are fractured and torn off the surfaces, resulting in additional surface fatigue and abrasion. From the presence of these scratches, it has been suggested that the current biomedical materials do not provide adequate load support (Li et al., 2004). In order to develop a full mechanistic understanding of reliable artificial joint design, it is necessary to investigate the mechanical and tribological properties of biomedical materials at the relevant contact scale as these properties are size dependent. Nano-scratch and nano-fretting tests have been performed on highly polished biomedical grade Ti6Al4V, 316 L stainless steel, and wrought high carbon CoCrMo alloy samples using a 3.7 μm end radius spheroconical diamond indenter in a commercial nanomechanical test system (NanoTest system) (Beake and Liskiewicz, 2013). By performing repetitive constant load unidirectional and bidirectional scratches in

214

Fretting Wear and Fretting Fatigue

Table 3.6.2 Mechanical properties of biomedical alloys.

316 L stainless steel CoCrMo Ti6Al4V

H (GPa)

E (GPa)

H/E

H3/E2 (GPa)

3.1 6 4.4

222 277 146

0.014 0.022 0.030

0.0006 0.0028 0.0040

From nanoindentation to 100 mN.

addition to the single scratches, it was attempted to simulate the repetitive and complex contact that can occur in the body when abrasion occurs by third body wear. The relationship between the mechanical properties of biomaterials and their friction and nano
/microscale wear behavior was investigated. Their mechanical properties from nanoindentation tests to 100 mN are summarized in Table 3.6.2. Although it was not as hard as the CoCrMo alloy, the Ti alloy had the highest H/E and H3/E2 due to its relatively low elastic modulus. The alloys tested showed a predominantly ductile response to scratching with ploughing and pileup and debris at the sides of the scratch track, with the onset of chipping events at increasing applied load. In ramped load scratch tests to 500 mN, the hardest alloy, CoCrMo, showed the best scratch resistance and the softest alloy, while 316 L stainless steel generally showed the lowest scratch resistance. By performing the nano-scratch tests as 3-pass (topography-scratch-topography) experiments, it was possible to estimate the contact pressure at low contact force where the friction coefficient is sufficiently low for the Hertzian analysis to be applied. Mean pressures for measurable plastic deformation (as defined by the first point in the final topographic scan showing nonzero depth) were 2.3 GPa for Ti6Al4V, 2 GPa for CoCr, and 1 GPa for the 316 L stainless steel. The mean contact pressure required to produce plastic deformation is slightly less than 1.1 times the yield stress due to shear stress. Macroscopic yield stresses in these biomedical materials have been reported to be in the range 0.3–1 GPa, with stainless steel being the lowest and Ti6Al4V being the highest. The ranking found in the nano-scratch test correlates with the parameter, H3/E2, a measure of the resistance to plastic deformation, in Table 3.6.2. There is a lateral size effect on plastic yielding. The nano-scratch data are consistent with higher yield stress at the nanoscale, emphasizing the importance of testing at the relevant scale rather than using bulk values. The scratch recovery ([on-load depth
residual depth]/on-load depth) at 30 mN correlated with H/E. At the end of the 10 cycle test at 30 mN, scratch recovery decreased significantly due to plastic deformation and wear. The change in on-load scratch depth and friction with scratch cycles are shown in Fig. 3.6.4A and B. Although initially the scratch depth was similar on the Ti alloy and steel differences grew with cycling, wear resistance did not directly correlate to hardness. For a wide range of hard metals and ceramics, Gee has shown that scratch resistance in low load multiple pass scratch tests was not correlated with hardness, as it was in single scratches, due to differences in the contribution of fracture to the damage development

Contact size in fretting

215

Fig. 3.6.4 Evolution of (A) on-load scratch depth and (B) friction in repetitive nano-scratch testing of biomedical materials. Probe end radius ¼ 3.7 μm. Applied load ¼ 30 mN.

(Gee, 2001). The observed decreases in friction during the initial wear cycles of this low-cycle test were consistent with a reduction in the ploughing component on CoCrMo and 316 L stainless steel as asperities are ploughed out. The decrease in friction with increasing scratch cycles was much less on Ti6Al4V than on either CoCr or 316 L stainless steel. In repetitive scratch testing with larger probes, titanium (98 cycles at 1.25 N, 25 μm/s, 500 μm track length, R ¼ 800 μm sapphire indenter, unidirectional sliding (Hanlon et al., 2005)) showed little (Drees et al., 2004) decrease in the friction coefficient. This is explained by a lower scan-on-scan decrease in the ploughing contribution on pure Ti and Ti6Al4V in the nano- and macrotests. The same probe was used for nano-fretting experiments from 1,2,3…30 mN with track length ¼ 10 μm; oscillation frequency ¼ 10 Hz. In each experiment, the load was linearly increased to the test load in 10 s and held at this level for 290 s before unloading in 10 s, so that the total number of fretting cycles during the loading and hold segments ¼ 3000. CoCrMo exhibited the highest wear resistance and smallest increase in depth during the constant load period. The wear resistance of the 316 L

216

Fretting Wear and Fretting Fatigue

steel and Ti6Al4V markedly deteriorates as the fretting load increases. SEM images of fretting scars after 2900 cycles at 1 mN applied load revealed an appreciable plastic fretting track on 316 L stainless steel though the fretting tracks were barely visible on CoCrMo and Ti6Al4V. At 2 mN, the fretting scars are still not distinct on CoCr and Ti6Al4V, but there was a transition to a more severe fretting wear on the 316 L stainless steel. At 3 mN, the fretting scars are more developed on CoCrMo and on Ti6Al4V and the 316 L stainless steel exhibits a more pronounced delamination wear. The probe depth under load was used to track the fretting wear. At the beginning of the test, the stainless steel performed better than did the Ti6Al4V alloy. A marked increase in probe depth occurred after typically 200 repeat wear cycles on the stainless steel. Abrupt increases in probe depth were commonly observed on both Ti6Al4V and the 316 L stainless steel but were absent on the CoCr alloy. The thickness and structure of the passive oxide films formed in air on the alloys on their nanoscale tribological behavior. Several factors appear to be responsible for the tribological behavior on Ti6Al4V with a marked transition to more severe wear as the load increases. It has a harder surface oxide (Moharrami et al., 2013) which effectively protects the underlying alloy at low load. With fretting cycles at a higher load, the hard oxide layers break exposing the Ti6Al4V surface causing an abrupt increase in wear rate. A range of wear mechanisms were observed in the nano-fretting tests as the load was increased from 1 to 30 mN, from ironing at very low load to ploughing and then more extreme cutting and delamination wear as the load increased. This transition was observed during the tests in the probe depth vs time plots and in the posttest SEM imaging. Fretting scars at 1–15 mN on Ti6Al4V are shown in Fig. 3.6.5. Between 3 and 4 mN, there was an abrupt transition to a more severe wear mode. The variability above this is another contact size effect, i.e., the complex surface/near-surface microstructure in Ti6Al4V is variable at the scale of the tests. Similar load-dependent transitions were observed on 316 L stainless steel. SEM images of wear scars at 1 and 4 mN are shown in Fig. 3.6.6A and B, respectively.

Fig. 3.6.5 SEM image of fretting scars at 1–15 mN on Ti6Al4V.

Contact size in fretting

217

Fig. 3.6.6 SEM image of fretting scars on 316 L stainless steel at (A) 1 mN and (B) 4 mN.

As the on-load probe depth increases as wear progresses in the higher load tests, there is a transition from the spherical part of the probe to its conical part. This sphere-cone transition increases the attack angle promoting more severe wear modes. Martı´nez-Nogu
es and coworkers investigated the nano-scratch and nano-fretting behavior of different CoCrMo alloys (Martı´nez-Nogu
es, 2016; Martı´nez-Nogu
es et al., 2016). They reported that in nano-scratch tests on CoCrMo alloys with 5 and

218

Fretting Wear and Fretting Fatigue

200 μm radii diamonds, the friction coefficient was independent of load for the blunter probe (Martı´nez-Nogu
es, 2016). It was strongly load dependent for the sharper probe, with higher friction coefficients as the load increased. They showed that the increased friction was correlated with a higher degree of penetration (on-load depth/contact radius). Martı´nez-Nogu
es reported that an increase in applied load from 50 to 500 mN in nano-fretting of the as-cast CoCrMo alloy resulted in less sliding with a reduction in track length from 10 to 2 μm and accompanied changes to the shape of the friction loops (Martı´nez-Nogu
es, 2016).

3.6.3.3 DLC/steel Friction reduction in automotive engines is necessary to reduce fuel and thus meet environmental and legislative requirements (Holmberg et al., 2012; Holmberg and Erdemir, 2017; Woydt et al., 2019). Diamond-like carbon (DLC) coatings are applied to many components in the power train such as tappets, pistons, piston rings, and fuel injectors (Erdemir and Donnet, 2006; Lawes et al., 2010; Ga˚hlin et al., 2001). Under fully lubricated conditions, DLC coatings have been shown to perform well but with the current power train trends to downsizing, turbocharging, low lubricant viscosity and start-stop, the number of components operating in the boundary/mixed regime will increase, so it will be necessary to design for components operating under mixed or boundary lubrication. Under severe loading, the performance of DLC coatings is limited by their resistance to contact damage (Bernoulli et al., 2015a,b). Typically, they behave poorly at higher load despite being hard and elastic. In pin-on-disk tests, DLC films that exhibited very low rates of wear under low contact pressure sliding were susceptible to abrupt increases in wear rate as the pressure increases beyond a critical threshold at higher load ( Jiang and Arnell, 1998). Low friction Si-doped DLC coatings have been explored for a range of applications and WC/C (a-C:H:W) coatings have been noted to perform well in a wide range of tribological conditions (W€anstrand et al., 1999; Ramı´rez et al., 2015; Mutafov et al., 2014; Lanigan et al., 2016). The mechanical and microtribological behavior of Si-doped DLC and WC/C coatings on hardened tool steel have been compared to a typical hard hydrogenated DLC (Liskiewicz et al., 2013; Beake et al., 2015). The coatings were 3–4 μm thick with multilayered (Cr/W-C:H/a-C:H, Cr/W-C:H/ Si-a-C:H, and CrN/a-C:H:W) architectures to improve their adhesion to the steel substrate. In the a-C:H and Si-a-C:H PECVD coating systems, the adhesion layer is a thin Cr and then gradient layers are applied to adapt the elastic modulus of the substrate to the elastic modulus of the hard top coating, thus giving the coating both abrasive wear resistance and impact fatigue wear resistance. In a-C:H:W (a commercial coating, Balinit C Star), the hard CrN sublayer provides load support and improved adhesion. Coating-only hardness and elastic modulus were determined from nanoindentation data according to the approach of ISO 14577-4. Nano- and microscratch tests and nano-fretting tests were performed using spheroconical diamond probes with end radii of 5 μm for the nano-scratch and nano-fretting tests and 25 μm for the microscratch tests. The scratch tests were carried out as three-scan procedures involving a prescan surface profile, ramped load scratch, and postscan surface profile. The critical loads in

Contact size in fretting

219

Table 3.6.3 Critical loads for yield, cracking and failure in nano- and microscratch tests. Coating

a-C:H Si:a-C:H a-C:H:W a

R 5 5 μm—nano-scratch test

R 5 25 μm—microscratch test

Ly (mN)

Lc1 (mN)

Lc2 (mN)

Ly (mN)

Lc1 (mN)

Lc2 (mN)

206  5 110  10 68  4

422  4 445  12 >500a

> 500a > 500a > 500a

356  9 383  52 375  49

2179  120 1827  111 2256  116

2612  127 2830  367 3695  132

The peak load of the instrument used for the nano-scratch tests (500 mN) was reached before Lc1 failure on a-C:H:W and Lc2 on all of the coatings.

the nano-scratch tests with R ¼ 5 μm and microscratch tests with R ¼ 25 μm are shown in Table 3.6.3. The table shows that the probe radius plays an important role in the yield behavior. With the sharper probe, there is a very clear correlation between the coating properties and critical load for yield, but this is absent for the larger radius probe. Simulations of the initial stresses in sliding contact at 100 mN with the R ¼ 5 μm probe provide some insight into the differences in wear resistance reported in 4500 cycle nano-fretting tests on these coatings where the best wear resistance was found for a-C:H and the worst for a-C:H:W. The simulated stress distributions were obtained with the Surface Stress Analyzer (from SIO). Input parameters for the simulations were (i) mechanical properties of the coating (taken as monolayered) and substrate, i.e., H, E, and H/Y (H/Y ¼ 1.2 for coatings, H/Y ¼ 2.5 for the steel substrate), (ii) coating thickness, (iii) Poisson’s ratios, and (iv) probe radius, applied load, and measured friction coefficient in the nano-scratch test. The initial stresses in the first sliding cycle in the nano-fretting tests are equated to those at the same load in the nano-scratch test with a probe of the same nominal geometry. 2D projections (through the centerline of scratch) of the simulated von Mises stress distributions at 100 mN in the nano-scratch test are shown in Fig. 3.6.7. In this figure, the sliding direction is left to right and the overstressed areas—where the magnitude of the von Mises stress is greater than the yield stress at that point—are shown by hashed regions. The simulations show at the 100 mN contact is elastic for aC:H; there is a small isolated zone of plasticity for Si:a-C:H and a more extensive yielded region for a-C:H:W. There was good agreement between the simulated stresses and the experimental data. In practice, contact is not completely wear-free even on a-C:H due to higher contact pressure at asperities as the probe and sample are not perfectly smooth and flat, respectively. The simulations were also able to explain probe geometry on yield location The location of initial yield in a scratch test is important as it has been shown to influence how the interface is weakened and the extent of subsequent coating delamination and wear in studies of other hard coatings on silicon (e.g., TiSiN, TiFeN), as well as in studies of the influence of temperature on coating and substrate mechanical properties in hard coatings on cemented carbide for metal cutting applications.

Fig. 3.6.7 2D projections of the simulated von Mises stress distributions on hard carbon coatings at 100 mN. Diamond probe end radius ¼ 5 μm. Sliding direction is left to right. The hashed regions mark the overstressed areas where the magnitude of the von Mises stress is greater than the yield stress at that point.

Contact size in fretting

3.6.4

221

Conclusions

Studying single-asperity contacts can be very informative in explaining deformation and failure mechanisms under a highly loaded mechanical contact. Nano-wear testing is complementary to short duration, low cycle, and high contact pressure scratch testing. The longer, higher cycle and lower contact pressure nano-wear tests can be an effective method to study damage tolerance. In this chapter, several examples where the contact size influences the deformation mechanism are introduced as case studies and the role of contact size in effecting these transitions is explained.

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Partial slip problems in contact mechanics

4.1

David A. Hillsa and Matthew R. Mooreb,c a Department of Engineering Science, University of Oxford, Oxford, United Kingdom, b Department of Physics & Mathematics, University of Hull, Kingston-Upon-Hull, United Kingdom, cMathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, United Kingdom

4.1.1

Introduction

Fretting is a process that involves small relative displacements between the surfaces of contacting components, which are believed to exacerbate the phenomenon of fatigue crack nucleation. Often, absolute slip displacements in the range of about 10–100 μm are cited as those within the range of definition of fretting (Vingsbo and Søderberg, 1988). In physics, meaningful quantities are invariably dimensionless, and the range of slip displacements cited may be comparable with length scales such as the topography of the surface finish or the characteristic grain size of very fine-grained alloys. However, in this chapter, we shall concern ourselves with the mechanics of fretting contacts, and so it is natural to think of the ratio of the maximum slip displacement, Δ, to the contact half-width, a, as the key dimensionless parameter. Moreover, before we can use this information effectively, we must remind ourselves about the different classes of contact (Fig. 4.1.1). The two most fundamental classes of contact are those shown in Fig. 4.1.1A and B. Fig. 4.1.1A shows an incomplete or convex (or Barber, 2018 calls them nonconformal) contact characterized by increasing contact size with increasing normal load and by a contact pressure that falls smoothly to zero in a square-root bounded manner as the contact edge is approached. This property suggests that, even far from the sliding (rigid body motion) condition, local slip may be prevalent. Fig. 4.1.1B shows a complete (or, in Barber’s taxonomy, conformal) contact where the contacting bodies have a common surface profile and, therefore, touch over a large region under an infinitesimal load, and where the contact size is defined by a discontinuity in surface profile in one body. The contact size is, therefore, independent of the normal load and the contact pressure adjacent to the contact edges will be power-order singular in form, that is, p(s)
sλ1, where p(s) is the contact pressure, s is a coordinate measured in from the contact edge, and λ is the dominant root (0 < λ < 1) of an eigenvalue problem defining the contact edge geometry. Since the edge contact pressure is singular it seems, subjectively, that slip is less likely; anticipating what we shall discover later. Significant slip will occur only if a corner lifts off, rendering the contact incomplete. Broadly, the contact edge may exhibit, if the contacting components are elastically pffiffi similar, two classes of behavior; it may be square-root bounded (pðsÞ
s), or it may Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00012-3 Copyright © 2023 Elsevier Inc. All rights reserved.

232

(A)

(C)

Fretting Wear and Fretting Fatigue

(B)

(D)

Fig. 4.1.1 Different classes of contacts. (A) An incomplete contact, in which the contact pressure vanishes at the contact edge. (B) A complete contact, in which one of the bodies has sharp corners, leading to a singular contact pressure. (C) A receding contact, in which the application of a point force on the thin plate causes the contact region to reduce in size. (D) A limiting case of a complete contact, in which both bodies simultaneously define the contact edge. The contact pressure for this case is finite everywhere.

be power-order singular. Other kinds of contact are found: in Fig. 4.1.1C, we show a plate lying on an elastically similar substrate and subject to a point (or line) force. When this is first lightly applied, the contact will jump to a size comparable with the plate thickness and the contact edges will not subsequently move. The contact has therefore receded, but the important thing to note is that when any further normal loads are applied the contact edge will remain stationary and exhibit behavior very similar to the incomplete class. Lastly, as we shall see shortly in Section 4.1.5, we state without proof that in the case of complete contacts the value of λ increases with decreasing internal angle in the contact-defining body, and approaches unity when the angle becomes vanishingly small. This state of affairs is very similar to that when the contact edge is simultaneously defined by both components (Fig. 4.1.1D), when p(s)
s0, and this is best viewed as a limiting case of the complete contact class—it is the only kind of finite contact where the pressure is finite everywhere, but it is hard to deduce general properties.

4.1.2

Global and pointwise friction

Here, we shall assume a classical concept of friction, based on the historical ideas of Amontons and Coulomb, so that we denote by μ the ratio jQj/P where P is the normal force supported by the contact and Q is the shear force, when the contacting pair is

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233

exhibiting rigid body motion—or sliding; this ratio is independent of the sliding velocity. The assumption is made that the same ratio applies in a pointwise sense, between the shear traction, q(x), and the contact pressure, p(x), at points which slip (i.e., jq(x)j/ p(x) ¼ μ). These statements are self-consistent because, in sliding, Z

Z μ pðxÞ dx Q μ ¼ ¼ Zcontact ¼ Z contact : P pðxÞ dx pðxÞ dx qðxÞ dx

contact

contact

Frictional contacts, like components exhibiting plastic flow, have internal states of stress which depend not only on the current load state (e.g., (P, Q)), but also how that state was reached. It is perfectly possible to have either a complete or incomplete contact at a general point (P1, Q1) (where jQ1j/P1 < μ and hence there is no rigid body movement) where one load path leads to the contact being fully adhered (i.e., stuck at all points), while a different load path leads to there being significant regions where there is interfacial slip—a so-called state of partial slip. Contacts may also be subject to a moment, M, and a tension parallel with the free surface. The former simply redistributes the contact pressure, while the latter excites shearing traction but has no resultant; neither, therefore, affects the condition for sliding.

4.1.3

Global and local elasticity solutions

By a global solution to a contact problem, we mean a formulation in which we have a full description of contact pressure, shear traction, and slip regime. In order to do this, we must have an elasticity solution for the whole of the contacting body, which may be challenging for prototypical problems, which may have complicated geometries. However, if the contact is incomplete in character, it will normally be perfectly permissible to exploit the difference in scales of the prototype geometry and the size of the contact region to model local behavior using so-called half-plane theory, in which each body is assumed to have the form of an elastic half-plane. When the half-plane idealization holds, it will be possible, for a wide range of contacting profiles and load histories, to describe in detail the traction and slip distribution over the entire contact length. When the contact is sharp edged (complete), no such dramatic simplification of the contacting domains is possible, and it will not normally be possible to obtain closedform solutions for the contact pressure or other details. In these cases, the best that can be done is to consider just the contact edges, and to model these using wedge theory. This enables a moderately full description of contact edge slip and stick to be revealed, but in a more limited way than the incomplete contact solutions permit. On the other hand, it is perfectly possible to use wedge theory to model the behavior of the edges of incomplete contacts, although there are some fundamental differences from the complete contact solutions.

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Fretting Wear and Fretting Fatigue

4.1.4

Half-plane contacts: Fundamentals

Throughout this chapter, we will assume that the contacting bodies are made from the same materials. As a consequence, contacts which may be represented using the halfplane idealization—including Hertzian contacts, wedges, and approximately, at least, the “flat-and-rounded” geometry—exhibit no form of coupling so that the presence of any interfacial shear tractions does not affect the normal contact solution. Thus, the normal contact problem may be solved independently, and regardless of loading path. We state without proof—for details, the reader is directed to, for example, Barber (2018) and Hills and Andresen (2021)—that the following integral represents the relationship between the contact pressure, p(x), and the relative surface profile, g(x), of the bodies being pressed together ∗

E dg 1 ¼ 2 dx π

Z

a a

pðξÞdξ  a  x  a, xξ

(4.1.1)

where a is the contact half-width and E* is the composite plane strain modulus of the material from which the bodies are made, that is, 1 2ð1  ν2 Þ , ¼ E∗ E

(4.1.2)

where E is Young’s modulus and ν is Poisson’s ratio. Note in Eq. (4.1.1), we have assumed that the contact region is symmetric about x ¼ 0 for simplicity. This is natural for symmetric contacts, but also, after a suitable shift of the coordinate origin, covers nonsymmetric contacts where the contacting region may not be symmetric with respect to the minimum of the contacting body. The relationship (4.1.1) is an integral equation with a Cauchy kernel, which can be inverted to find the contact pressure distribution for a given profile. In order to determine the size of the contact overall, normal equilibrium must be imposed by setting Z P¼

a

pðxÞdx:

(4.1.3)

a

There is an analogous relationship between the interfacial shear traction, q(x), and the relative surface strain, E(x), present in the bodies ∗

E 1 EðxÞ ¼ π 2

Z

a

qðξÞdξ  a  x  a, a x  ξ

(4.1.4)

and use may be made of this correspondence to establish very useful general relationships between the contact pressure and developing shear traction distributions. In order to do this effectively, we must first derive the so-called Mossakovskii-Barber relationships. The basic idea is to consider the contact as a set of stacked punches so that an infinitesimal increase in normal load increases the surface displacement

Partial slip problems in contact mechanics

235

of the corresponding body by a constant amount, except near the edges where the effect of the profile comes into play and it is related to the incremental contact law, a(P). While we will touch on these ideas when we consider the conditions for full stick in incomplete contacts in Section 4.1.4.1, the procedure is not fully developed in this chapter and the interested reader is referred to Hills et al. (2011), Moore and Hills (2020), and Hills and Andresen (2021) for further details.

4.1.4.1 Conditions for full stick We can deduce some results without making use of the contact law. Fig. 4.1.2 introduces the load space (P, Q). Lines of gradient μ represent the applied forces needed to achieve sliding. When the contact is first formed under a normal load, P0, we move from the origin along the P > 0 axis to (P0, 0). As subsequent loads are applied, we shall move from (P0, 0) to a general point in the load space (P, Q) and providing we always stay within the bounding lines the contact will suffer no rigid body motion. However, the immediate question which arises is: “Will the contact be fully stuck, or will it be in a state of partial slip?” To answer this, we make use of Mossakovskii-Barber the ideas introduced at the end of the previous section, and consider the incremental change in contact pressure due to an infinitesimal increase in load due to a stacked punch. If the current halfwidth of the contact is a, an increase in load, ΔP, will cause an increase in contact pressure over the length of the contact corresponding to a constant displacement, which corresponds to a change of ΔP ΔpðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : π a2  x 2

(4.1.5)

A quick substitution of this into Eq. (4.1.1) confirms that this is, indeed, the expression we seek. For the benefit of the wary reader, we illustrate this for the common example of the pressure in a Hertzian contact, in which the contacting bodies are cylindrical with Q Sli

μ μ

din

g

P

(P0, 0) Sli

Fig. 4.1.2 Load paths in (P, Q)-space showing regions of full stick (green) and regions of partial slip (red).

Partial slip Stick din

g

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Fretting Wear and Fretting Fatigue

radius R. As shown by, for example, Barber (2010), the contact pressure and contact law for this geometry are given by pðxÞ ¼

ffi 2P pffiffiffiffiffiffiffiffiffiffiffiffiffi 4PR a2  x 2 , a2 ¼ ∗ : 2 πa πE

(4.1.6)

Then, taking the total derivative with respect to P, we see  ∗ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∗ 0 dp d E E aa ðPÞ 1 ¼ aðPÞ2  x2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 2 dP dP 2R 2R aðPÞ2  x2 π a  x

(4.1.7)

where a prime denotes differentiation with respect to argument. Clearly, this retrieves Eq. (4.1.5). Similarly, let us now consider the tangential loading for two elastically similar bodies in glancing contact aside from the contact interval where they are bonded together. Due to the similarity of the normal and tangential problems, an identical argument to that above allows us to show that the appropriate change in shear traction distribution caused by a change in the applied shear force, ΔQ, leading to zero relative tangential surface strain over a < x < a is given by ΔQ ΔqðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : π a2  x 2

(4.1.8)

Therefore, we see that the resultant traction ratio, (q(x) + Δq(x))/(p(x) + Δp(x)) necessarily falls below the coefficient of friction at all points within the contact provided only that jΔQj=ΔP < μ:

(4.1.9)

We may then make the following deductions: 1. The condition for full stick with an increasing normal load (and, therefore, increasing size of contact) depends only on the ratio of the increments of normal load and shear force, and not the absolute load state. Load contours, such as those depicted in Fig. 4.1.2, therefore exhibit regions where there will be full stick over the whole contact, and regions where there will be partial slip. 2. When the normal load is reduced and the contact contracts, unless the unloading trajectory precisely matches the loading trajectory the contact will be in a state of partial slip. 3. It follows directly from inequality (4.1.9) that whenever the shear force is changed but the normal load remains constant, there will always be regions of slip present.

4.1.4.2 Effects of tension and moments Often contacts will be subject not only to normal and shear loads, but also tensions parallel with the free surface. If these are exerted before the contact is formed, and

Partial slip problems in contact mechanics

237

P

(A)

Q y σ0

x

⇐

Q

⇒ σ0

P

(B)

M P Q y x

σ0

Q

σ0

P M

Fig. 4.1.3 (A) A symmetric incomplete contact problem subject to a normal load, P, an applied shear, Q, and a bulk tension σ 0. (B) A nonsymmetric incomplete contact problem, including an applied moment, M, in addition to the other loads.

held constant, they will have no influence on the slip state. On the other hand, if they are exerted afterwards, they will cause interfacial shear tractions to arise and, therefore, modify the partial slip problem (Fig. 4.1.3A). It may be shown that, if all interfacial slip is inhibited, and with the axis set as shown, the application of a remote tension σ 0 to the lower body will induce the following interfacial shear traction (Hills et al., 2011): σ 0 x qðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 4 a2  x 2

(4.1.10)

Note that if, simultaneously, a tension σ 1 arises in the upper body, it is the difference in bulk tensions that is the important quantity, so that σ 0 in Eq. (4.1.10) and the following equations should be replaced by σ 0  σ 1. If the contact is now subject to three small simultaneous changes in load, ΔP (>0), ΔQ, Δσ, the change in the ratio of tractions along the interface is given by

238

Fretting Wear and Fretting Fatigue

ΔqðxÞ 4ΔQ  πΔσ 0 x ¼ ΔpðxÞ 4ΔP

(4.1.11)

so that the largest value of this ratio will occur at x ¼ a (depending on the relative sizes and signs of ΔQ and Δσ), and the condition for full stick becomes j4ΔQ  πΔσ 0 aj < μ: 4ΔP

(4.1.12)

We note that, if we had an extended form of Fig. 4.1.2 in which there was a third axis representing the tension present, the condition for sliding would be independent of the bulk tension, and that remarks similar to those made at the end of Section 4.1.4.1 for the (P, Q)-problem continue to apply; in particular that the condition for full stick is an incremental one. Lastly, the application of a small moment, ΔM, positive in the sense shown in Fig. 4.1.3B will cause a small change in the contact pressure (which must nevertheless remain positive—compressive—at all points) given by Sackfield et al. (2001) as ΔpðxÞ ¼

2ΔMx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , a2  x 2

πa2

(4.1.13)

where we note that here it is necessary to choose our coordinate axes appropriately so that the contact region may continue to be written symmetrically. Thus, it follows that, if the contact now suffers simultaneous changes in ΔP (>0), ΔQ, Δσ, ΔM, and the magnitude of M is small so that both edges of the contact advance, combining Eqs. (4.1.5), (4.1.8), (4.1.11), (4.1.13) yields ΔqðxÞ 4a2 ΔQ  πa2 Δσ 0 x ¼ ΔpðxÞ 4a2 ΔP + 8ΔMx

(4.1.14)

and we must ensure that ΔqðaÞ π, the dominant roots are shown in Fig. 4.1.5. From this, it is clear that the order of the singularity increases with angle until, when 2α ¼ 2π and we have a crack tip (or, with some qualifications, an incomplete contact edge), the dominant roots are λI ¼ λII ¼ 1/2. Back substitution of the roots into Eqs. (4.1.21), (4.1.22) reveals the ratios KI0 =A, KII0 =B—the eigenvectors—and hence the spatial distribution of stresses. We can then write down the state of stress near the wedge apex (or contact edge) in the form σ ij ðr, θÞ ¼ KI r λI 1 fijI ðθÞ + KII r λII 1 fijII ðθÞ,

(4.1.24)

where we rescale the solution (which uncouples along the line θ ¼ 0) so that I ð0Þ ¼ 1, frθI ð0Þ ¼ 0 fθθ

(4.1.25)

II frθII ð0Þ ¼ 1 fθθ ð0Þ ¼ 0,

(4.1.26)

Fig. 4.1.5 The dominant symmetric and antisymmetric eigenvalues λ  1 as a function of the wedge angle, 2α found by solving Eq. (4.1.23).

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Fretting Wear and Fretting Fatigue

pffiffiffiffiffi and the factor of 2π customarily included in fracture mechanics for historical reasons is omitted. The eigenfunctions fijk ðθÞ are explicit and given in Appendix 4.1.1.

4.1.5.1 Conditions for full stick We now apply the solutions just found to the particular case of the square-edged contact mentioned in the introduction to this section. When 2α ¼ 3π/2 radians, λI ¼ 0.5445, λII ¼ 0.9085 so that the symmetric terms are even more strongly singular than the antisymmetric terms. The contact pressure is (with a sign change), σ θθ(r, π/4) and the shear traction is given by σ rθ(r, π/4) so that the traction ratio near the contact edge is given by qðsÞ KI r λI 1 f I ðπ=4Þ + KII r λII 1 frθII ðπ=4Þ frθI ðπ=4Þ I ¼  λ 1 rθ ’  I ðπ=4Þ + K r λII 1 f II ðπ=4Þ I ðπ=4Þ ≡  grθ pðsÞ KI r I fθθ fθθ II θθ

(4.1.27)

as r ! 0, where s ¼ r cos π=4 is a coordinate measured inwards from the contact edge along the interface. The quantity gIrθ ¼ 0:543, and the negative sign indicates that the contact-defining body tends to spread relative to the half-plane. It follows that, provided only that the coefficient of friction is at least this big, the original assumption— stick—holds. It is perhaps surprising that this result holds independently of the nature of the applied load, as long as intimate contact persists right up to the edge of the block and there is no separation. The geometric requirement is that the straight edge defining the contact edge must extend upwards far enough—meaning a big enough fraction of the contact face—for the eigensolution derived to be valid. This result is in stark contrast to that derived for incomplete contacts, and is completely different in character. Strictly speaking what we have derived is a sufficient condition for full stick at the edges of the contact. It is of course feasible that there may be a so-called “detached” slip zone wholly within the contact, although, from the fretting fatigue point of view it should be stated that the nucleation of cracks from within contacts is very rare. Moreover, the slip displacement within any detached slip region must necessarily be small as the integral of its value across the slip region must vanish. If the inequality f > gIrθ is not satisfied, what happens? This question is harder to answer. Initially, there must be slip, but under cyclic loading we state without proof (see Churchman and Hills, 2006a, b) that frictional shakedown to an almost fully stuck state usually ensues. This is again in stark contrast to the case of incomplete contacts.

4.1.6

Partial slip of incomplete contacts

We now return to our consideration of contacts which are capable of representation using half-plane theory. This book is concerned with fretting, that is, surface damage when the applied loading is oscillatory in character (Andresen and Hills, 2020). Some problems encountered involve loads, which act in a varying direction: a good example

Partial slip problems in contact mechanics

243

(A)

F

(B)

T

M

(C) P Q σ0

σ0

Fig. 4.1.6 (A) An out-of-balance rotor with eccentric force rotating with the same speed as the rotor will experience oscillatory applied normal and shear loads 90 degrees out of phase. (B) The dovetail root between a blade and the central rotating disk in a gas turbine. The centrifugal force, F, varies slowly with time, while the vibrational force, T, oscillates even more rapidly. (C) The flat-and-rounded punch modeling the contact in figure (B) within the limit of half-plane theory.

is an out-of-balance rotor, where the eccentric force is of constant magnitude but rotates at the same speed as the rotor. So, for example, it could lead to normal and shear forces, which vary with a 90 degrees phase shift (Fig. 4.1.6A). Problems of this kind have been widely studied, for example, in Barber et al. (2011) and Davies et al. (2012). On the other hand, there are many problems involving loads that change in magnitude rather than direction, such as those developed within a gas turbine rotor (Fig. 4.1.6B). This problem has another typical property, namely that there is one force which varies only very slowly with time—in this case the centrifugal force, F, which increases from zero at engine start up and varies with throttle setting—while a second force—here the vibration force, T—has a typical frequency content measured in hundreds of hertz or kilohertz, and hence there are hundreds of thousands of cycles of this component of load for every cycle of throttle setting. The dovetail flank, where the blade root is the contact-defining body for one contact edge, and the rotor is the contact-defining body for the other edge, may be replaced by a substantially equivalent contact problem in the form of a flat-and-rounded elastic punch pressed onto a half-plane (Fig. 4.1.6C). Each component of load at the contact—P, Q, M, σ—is (approximately) linearly related to the remote loads, by a full matrix of the form

244

Fretting Wear and Fretting Fatigue

0

1 0 P ? BQ C B? B C¼B @MA @? σ ?

1 ?   ?C C F , ?A T ?

(4.1.28)

and the elements of the matrix connecting these loads may be found by a commercial finite element program. The point is made that a practical problem which occurs frequently is one where one set of contact loads (here stemming from the first column of the matrix) has a constant value while there is, in addition an oscillatory set, and these change at high frequency and are usually fully reversing in character. So, although transient problems may be profitably studied (Hills and Andresen, 2021), it is the steady-state solution to a partial slip problem that is of most practical use. It is the steady state that we analyze here, and with no phase shift between any of the components of load. Partial slip contact problems where the bodies may be represented using half-plane theory normally have a central region which is permanently stuck and edge regions which suffer oscillatory slip, and the core of the problem is to find the size of the permanent stick region. Once this is known, other aspects of the solution, such as the slip displacement, fall into place. We will begin our studies with cases where the contact is not subject to an applied moment, and where the contact-defining body is symmetrical. Hence, the contact pressure distribution p(x) will also be symmetrical and the contact will span the interval ½a a. The contact law, a(P), is defined by the indenter shape, and we will use the notation from Barber (2018) that the contact pressure distribution may be written in the compact notation p(x;P). So, for example, returning to the Hertzian contact between cylinders of relative radius of curvature R, the contact law and pressure (4.1.6) can be rewritten as rffiffiffiffiffiffiffiffi∗ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∗ PE πE x2 4PR pðx;PÞ ¼ 1 for jxj < aðPÞ, a2 ðPÞ ¼ ∗ : πE πR 4PR

(4.1.29)

This relationship is completely independent of the effects of shearing traction, and hence of (Q, σ). In the stick regions, two conditions must hold. One is that any interfacial relative slip displacement arising before stick took hold—i.e., (uB(x)  uA(x)) or equivalently, (EB(x)  EA(x))—is preserved, and second jqðxÞj < μpðxÞ 8x within the stick region:

(4.1.30)

Meanwhile, in the slip regions, the magnitude of the shear traction is precisely the coefficient of friction multiplied by the contact pressure jqðxÞj ¼ μpðxÞ,

(4.1.31)

Partial slip problems in contact mechanics

245

(A)

(B)

Q (and/or σ)

x a

2

g

din

1

oa ll

ia nit

I

1

2

Time

−a P

Permanent stick zone

Fig. 4.1.7 (A) The loading trajectory in (P, Q, σ)-space. There is an initial transient before the steady-state loading cycle between states 1 and 2 begins. (B) The stick-slip zone pattern as the steady-state loading cycle is traversed. There is a permanent stick zone in the center of the contact that is reached at each end of the cycle.

while its sign is such that it opposes relative motion or changes in relative slip displacement, that is, sgn ðqðxÞÞ ¼ sgn ðu_B ðxÞ  u_A ðxÞÞ,

(4.1.32)

where sgn () is the signum function and the overdot denotes derivatives with respect to time. Before looking at possible formulations, we can make some general observations about the expected behavior, and in reading these, reference should be made to Fig. 4.1.7. We are going to study problems where, generally, the quantities (P, Q, σ) all vary in time between end points 1 and 2 (see Fig. 4.1.7A). First, we note that, if P2 > P1, the contact size will vary cyclically, as sketched in Fig. 4.1.7B. Second, it is clear that a shear force induces shear of the same sign at each edge of the contact, whereas a bulk tension induces shear of opposite sign. So, if the tension applied is zero or small, the slip zones at each edge will be of the same sign (moderate tension), whereas if the tension exerted is large they will be of opposite sign. The sketch in Fig. 4.1.7B shows a case with moderate tension. Third, we see that as we move from one end of loading to the other (both 1 ! 2 and 2 ! 1) the slip zones increase in size. As we “move around” either end of the loading trajectory, condition (4.1.31) does not hold in either slip zone, so that full stick ensues. The sign of slip at both ends of the contact changes when we go from 1 ! 2 compared with going from 2 ! 1. In fact, we can make a stronger statement than that because, bearing in mind that we are in a steady state, and material must be preserved, when we go through the cycle 1 ! 2 ! 1 every surface particle (and indeed all material) must go back to where it came from. So, ðu_B ðxÞ  u_A ðxÞÞj1!2 ¼ ðu_B ðxÞ  u_A ðxÞÞj2!1 :

(4.1.33)

This statement applies everywhere, not just within the permanent contact region. There are two basic philosophies that we can adopt when solving problems of this kind; either we can start off with a sliding shear traction distribution over the whole

246

Fretting Wear and Fretting Fatigue

contact, and superpose an initially unknown shear traction distribution over the stick region to achieve preservation of slip displacement there (subject to inequality (4.1.30) being satisfied), or we can assume that the contact is stuck everywhere, and add distributions of glide dislocations in the slip regions to enforce Eqs. (4.1.31), (4.1.32) within them. Both approaches have their respective merits, as we shall outline in the following sections.

4.1.6.1 An introduction to corrective slip: The Cattaneo-Mindlin solution The idea of using a corrective shear traction on the sliding distribution goes back to the origins of partial slip solutions for constant normal load problems. The first solution, limited to a Hertzian contact, was found by Cattaneo (1938), and this solution was rediscovered and extended by Mindlin (1949) a decade or so later. It was noted that the form of the corrective shear was a scaled form of the sliding traction, and therefore the contact pressure, and many years later ( J€ager, 1997) and independently Ciavarella (1998a, b) showed that this was not coincidental, but generally true. The basic idea is very straightforward, but in this original form is restricted to the constant normal load form of the problem—and an applied shear force only—but it is instructive to follow the ideas, and so we present it here. Eq. (4.1.4) relates the relative surface strains created in the contact by a shear traction distribution, and we will now write the latter as the sum of a sliding distribution together with a corrective shear, q*(x), applied over a potential stick region b  x  b, so that it becomes ∗

E μ EðxÞ ¼ π 2

Z

a

pðξ,PÞ dξ 1 + xξ π a

Z

∗

q ðξÞ dξ for  a  x  a: b x  ξ b

(4.1.34)

We now impose the requirement within the stick region that the strain difference must remain zero, that is, EðxÞ ¼ 0 for  b  x  b, and making use of Eq. (4.1.1) we see that ∗

μE dg 1  ¼ 2 dx π

Z

∗

q ðξÞ dξ for  b  x  b: b x  ξ b

(4.1.35)

Now, comparison of Eqs. (4.1.1), (4.1.35) shows that the corrective traction is simply a ∗ ^ where the ficscaled form of the contact pressure, and in particular q ðxÞ ¼ μpðx, PÞ, ^ titious normal force, P, can be determined from tangential equilibrium, namely Z Q¼

a

a

Z μpðx, PÞ dx +

b

Q ∗ q ðxÞ dx ) P^ ¼ P  : μ b

(4.1.36)

Partial slip problems in contact mechanics

247

Thus, the size of the stick zone, b, can then be retrieved from the contact law. In particular, in terms of the indenter profile, this can be expressed as (Ciavarella, 1998a) Q JðbÞ ¼1 , where JðxÞ ¼ x μP JðaÞ

Z

0

sg ðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ds: x x2  s2 x

(4.1.37)

For the particular example of a Hertzian (cylindrical) geometry for which the contact law is given by Eq. (4.1.6), the size of the stick zone is given by b2 ¼

4ðμP  QÞR , μπE∗

(4.1.38)

so that we are able to derive the simple relationship  2 b Q ¼1 : a μP

(4.1.39)

4.1.6.2 Effect of bulk tension, cyclic loading, and change in normal load In this section, several steps are taken beyond the previous solution. First, the normal load is allowed to vary during the loading cycle, while at the same time a moment may be present so that symmetry is no longer demanded, and the contact zone thus spans the interval ½a c. Second, we now jump straight to a steady-state solution, rather than looking just at the transient loading problem. Lastly, we add in the effects of differential bulk tension, all under the assumption of proportional loading conditions (Eq. 4.1.16). Note, however, that the slip zones must be of the same sign, that is, slip must be of the same sense at each edge of the contact. The formulation is, therefore, appropriate for cases that we defined earlier as “moderate tension.” As previously, we write down the shear traction as the sum of the sliding term plus a correction, except now with the additional contribution from the bulk tension. We see that, at load point 2, the difference between the surface strains is given by ∗

πE πσ 2 E2 ðxÞ ¼ +μ 2 4

Z

p2 ðξÞ dξ + a2 x  ξ c2

Z

q∗2 ðξÞ dξ for  a2  x  c2 , m x  ξ n

(4.1.40)

where the steady-state stick zone region is taken to be ½m n. At the other end of the loading range, point 1, the sign of shear is reversed but the steady-state stick zone (the permanent stick zone) is the same, and so ∗

πE πσ 1 E1 ðxÞ ¼ μ 2 4

Z

p1 ðξÞ dξ + a1 x  ξ c1

Z

q∗1 ðξÞ dξ for  a1  x  c1 , m x  ξ n

(4.1.41)

248

Fretting Wear and Fretting Fatigue

and we note that, within the permanent stick region the differential strain locked in must be constant, that is, E1 ðxÞ ¼ E2 ðxÞ for  m  x  n:

(4.1.42)

Eqs. (4.1.40)–(4.1.42) may be combined to give π  ðσ 2  σ 1 Þ  μ 4

Z

 Z n ∗ p1 ðξÞ dξ q2 ðξÞ  q∗1 ðξÞ dξ ¼ xξ a2 x  ξ m

Z

p2 ðξÞ dξ + a1 x  ξ c1

c2

(4.1.43) for m  x  n. It is probably best if, at this point, we develop the solution in stages. Suppose, for the time being, that the normal load is kept constant and the contact symmetrical, so that a1 ¼ a2 ¼ c1 ¼ c2 ¼ a. Then, Eq. (4.1.43) simplifies to π  ðσ 2  σ 1 Þ  2μ 4

Z

a

pðξÞ dξ ¼ a x  ξ

Z

q∗2 ðξÞ  q∗1 ðξÞ dξ xξ m n

(4.1.44)

for m  x  n, so that after making use of Eq. (4.1.1) again (as with Eq. 4.1.35), we have Z 1 1 n q∗2 ðξÞ  q∗1 ðξÞ ∗ dg (4.1.45)  ðσ 2  σ 1 Þ  μE ¼ dξ 4 dx π m xξ for m  x  n. If there were no differential bulk tension this would, as before, imply that the corrective shear traction were a scaled form of the contact pressure. However, now there is an additional term, which is a constant, so that this means that we have an integral equation for the change in corrective traction which is a scaled form of the normal contact problem, but with a rotation, say α, where 1 ∗ (4.1.46) ðσ 2  σ 1 Þ ¼ μE α: 4 Therefore, a precursor calculation would need to be carried out for this class of problem, in which, instead of solving for just the symmetrical profile, an additional solution is required where the indenter is tilted, and the associated contact patch is not symmetrically positioned. For more details of this solution procedure, the reader is directed to see Andresen et al. (2019, 2020a). We now return to the more general problem, where the normal load varies, and, therefore, Eq. (4.1.43) describes the solution, which will provide us with the permanent stick zone and, if required, the corrective shear traction in the stick region. Guided by our treatment of the constant normal load problem, above, we see that the left-hand side may be simplified considerably in appearance by making use of the solution for normal load, and write   Z dg 1 1 n q∗2 ðξÞ  q∗1 ðξÞ ∗ μE α0 + dξ  m  x  n,  ðσ 2  σ 1 Þ ¼ dx 4 π m xξ

(4.1.47)

Partial slip problems in contact mechanics

249

where 1 α0 ¼ ðα1 + α2 Þ, 2

(4.1.48)

which represents the average angle of tilt. Scrutiny of this equation and comparison with Eq. (4.1.45) shows that, in each case, the left-hand side incorporates just two terms; one describes the profile of the punch (dg/dx) and the other is simply a constant. There is therefore a very close parallel with the formulation for a normal contact problem, where in Eqs. (4.1.45), (4.1.47), the unknown function in the right-hand integral is the difference between the corrective shear tractions at each end of the cycle, q∗2 ðxÞ  q∗1 ðxÞ, rather than the contact pressure, p(x). Once we have solved the normal problem, therefore, we can deduce immediately the solution for the partial slip, steady-state shear-loading problem, simply by dint of the following mapping: ½a

c ! ½ m

n

1 ∗ ½q ðxÞ  q∗1 ðxÞ 2μ 2 ΔQ P ! P0  2μ 1 α ! α0 + ðσ 1  σ 2 Þ: 4μE∗

pðxÞ ! 

4.1.7

(4.1.49)

Dislocation-based solutions

We are very familiar with solving elasticity problems starting from the effect of a force. This is how this chapter began, by looking at the effect of a line force (generally) at the apex of a wedge, determining the displacement field generated, and going on to match the depression of the surface to the form of the contacting body, thereby solving a contact problem. But this is not the only possible starting point, and we may instead use a number of possible strain nuclei. The most common and the easiest one to interpret is the dislocation, and the edge dislocation is appropriate for plane elasticity. A cut is made from the core of the dislocation to a free surface, and then a constant small displacement, whose magnitude we call the Burgers vector, we impose everywhere along the cut before we glue it back together. The path cut is made contiguous with whatever line of displacement discontinuity is present in the component or assembly to be modeled. So, for example, it is perfectly possible to solve the contact problem itself with this as a starting point, and the solutions found are compact and elegant (Moore and Hills, 2018). In these cases, the Burgers vector of the dislocation is made perpendicular to the path cut so that layers of material are added by a dislocation array (climb dislocations), and then the material is trimmed off again by adding dislocations of opposite sign. Here, we shall not make use of this principle, but we will use glide dislocations, that is, where the Burgers vector is parallel with the path cut, and these

250

Fretting Wear and Fretting Fatigue

will be used to introduce slip displacement. The reason that this is attractive is that it averts the limitation implicit in the strategy for solving partial slip problems developed in the previous section where, it will be recalled, the slip zones at each edge of the contact must be of the same sign. The basic idea behind the approach is to make the reverse assumption from that above, where we assumed a sliding traction, and therefore satisfied the slipping conditions, and added a corrective shear traction in the stick region so as to enforce stick there. Here, we will start off assuming that the whole contact sticks, and then distribute glide dislocations in the slip regions so as to correct the solution there. The piece of analysis which made this possible was the discovery by Barber (private communication) of the traction arising along the interface line when two half-planes are glued together over the given interval ½a a , and a glide dislocation, of magnitude bx(c) is installed at point c. No direct traction is developed but the shear traction, q(x), is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∗ a2  c 2 E bx ðcÞ for jxj < a: qðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2π a2  x2 c  x

(4.1.50)

Note, to begin with, we have returned to the case of a symmetric contact with no moment. A distribution of corrective glide dislocations in the slip zones of density Bx(ξ) ¼ dbx/dξ, therefore, generates a shear traction given by ∗

E qðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2π a2  x2

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2  ξ2 Bx ðξÞdξ for jxj < a: ξx slip zones

(4.1.51)

4.1.7.1 Introductory problem We will start by resolving a partial slip problem under a constant normal load, so that the size of the contact is fixed. If, simultaneously, a shear force, Q, and a differential bulk tension, σ, are developed, from Eqs. (4.1.4), (4.1.10) we see that, when full stick is enforced over the contact, the shear traction distribution, qst(x), is given by Q σx qst ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi + pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 2 π a  x 4 a2  x 2

(4.1.52)

Suppose that the applied tension is large, so that the slip zones are of opposite sign at each end of the contact patch, and that the stick interval spans the distance ½m n. The integral equations defining the dislocation densities in the slip zones are thus given by ∗

E qðxÞ¼ qst ðxÞ + pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2π a2  x2 ¼ μ sgn ðxÞpðxÞ

Z

m a

ffi Z a  pffiffiffiffiffiffiffiffiffiffiffiffiffi a2  ξ2 Bx ðξÞ dξ + ξx n

(4.1.53)

Partial slip problems in contact mechanics

251

for a  x m and n  x  a, which can be concisely rewritten as ∗

E q0 ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2π a2  x2

Z

m a

ffi Z a  pffiffiffiffiffiffiffiffiffiffiffiffiffi a2  ξ2 Bx ðξÞ dξ + ξx n

(4.1.54)

for a  x m and n  x  a, where Q σx q0 ðxÞ ¼ μ sgn ðxÞpðxÞ  qst ðxÞ ¼ μ sgn ðxÞpðxÞ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : π a2  x 2 4 a2  x 2 (4.1.55) An inversion of Eq. (4.1.54) enables the dislocation density to be found, alongside two side conditions which must be satisfied. They are Z

m

a

Z

m

a



Z a n



Z a n

q0 ðξÞ dξ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0, ðξ  nÞðξ + mÞ

(4.1.56)

ξq0 ðξÞ dξ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0: ðξ  nÞðξ + mÞ

(4.1.57)

These are extremely useful in themselves as they enable the locations of the stick-slip points to be found. We note that, although we included the signum function sgn x in Eqs. (4.1.53), (4.1.55) to illustrate the solution procedure using dislocations and its applicability to large tension cases, there is nothing preventing us from tackling moderate tension cases using dislocation as well. In this case, the solution procedure would be exactly the same, but with the signum functions omitted everywhere.

4.1.7.2 Steady-state solution: Constant normal load Suppose that we have a problem similar to the problem just set out, but where the quantities exciting shear stress (Q, σ) vary between two load states 1 and 2. So, in moving to state 2 from state 1, the following describes the change in shearing traction: Q2  Q1 ðσ 2  σ 1 Þx q2 ðxÞ¼ q1 ðxÞ + pffiffiffiffiffiffiffiffiffiffiffiffiffiffi + pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 a2  x 2 π a x ffi Z m Z a  pffiffiffiffiffiffiffiffiffiffiffiffiffi ∗ a2  ξ2 B12 E x ðξÞ dξ + pffiffiffiffiffiffiffiffiffiffiffiffiffiffi + ξx 2π a2  x2 a n

(4.1.58)

for a  x  a, whereas, in moving from state 2 back to state 1 we have Q2  Q1 ðσ 2  σ 1 Þx q1 ðxÞ¼ q2 ðxÞ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi π a2  x 2 4 a2  x 2 ffi Z m Z a  pffiffiffiffiffiffiffiffiffiffiffiffiffi ∗ a2  ξ2 B21 E x ðξÞ dξ + pffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 2 2 ξ  x 2π a  x a n

(4.1.59)

252

Fretting Wear and Fretting Fatigue

for a  x  a. We may now impose the following conditions in the slip zones: q2 ðxÞ ¼ q1 ðxÞ ¼ μpðxÞ sgn ðxÞ for  a  x  m and n  x  a,

(4.1.60)

where the signum function is omitted where bulk tension is moderate, and included when bulk tension is large. In addition, we impose the material continuity condition implied by Eq. (4.1.33), that is, particles which displace laterally return to their original positions at the end of the loading cycle. Here, this means that we must have 21 B12 x ðxÞ ¼ Bx ðxÞ for  a  x  m and n  x  a:

(4.1.61)

Eqs. (4.1.58)–(4.1.61) together give Q2  Q1 ðσ 2  σ 1 Þx q0 ðxÞ ≡ 2μpðxÞ sgn ðxÞ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 π a  x 4 a2  x 2 ffi Z m Z a  pffiffiffiffiffiffiffiffiffiffiffiffiffi ∗ a2  ξ2 B12 E x ðξÞdξ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi + ξx 2π a2  x2 a n

(4.1.62)

for a  x m and n  x  a. The side conditions (4.1.56), (4.1.57) continue to be apposite, with the new definitions of the function q0(x), and, in connection with this, we note the following standard integral results: Z

m

a

where Ik ¼

ξk dξ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ða2  ξ2 Þðξ  nÞðξ + mÞ

8 >
: ðm  nÞ 2

Z

a n

ξk dξ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ik , ða2  ξ2 Þðξ  nÞðξ + mÞ

(4.1.63)

k ¼ 0, k ¼ 1, k ¼ 2,

which may well be useful for tackling specific problems.

4.1.7.3 Steady-state solution: Varying normal load The ideas set out in the previous section may be extended to the case where the normal load changes. We shall, however, retain the assumption that the contact remains symmetrical and that no moment is exerted, using a combination of the ideas set out earlier. We now have a load set (P, Q, σ) varying between states 1 and 2 (cf. Fig. 4.1.7A). The first step in the solution is to understand the change in shear tractions, under conditions of full stick, when we move between the end points of the cycle, and this is given by q2 ðxÞ  q1 ðxÞ ¼ ½pðx;P2 Þ  pðx;P1 Þ½λ + ηx,

(4.1.64)

Partial slip problems in contact mechanics

253

where λ¼

Q2  Q1 , P2  P 1

(4.1.65)

η¼

π σ2  σ1 , 4 P2  P 1

(4.1.66)

and the notation p(x;Pi) means the pressure distribution at load Pi, for whatever contact geometry is being considered. With this as a starting point, we can easily set up corrections for the slip regions in a manner analogous to the procedure adopted in the previous section. So, for example, in place of Eq. (4.1.58) we have q2 ðxÞ ¼ q1 ðxÞ + ½pðx;P2 Þ  pðx;P1 Þ½λ + ηx ffi Z m Z a  pffiffiffiffiffiffiffiffiffiffiffiffiffi ∗ a2  ξ2 B12 E x ðξÞdξ + + pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ξ  x 2π a  x a n

(4.1.67)

for a  x  a and the following steps are very similar, leading to the following integral equation in terms of the dislocation density (cf. Eq. 4.1.62): q0 ðxÞ≡ pðx;P2 Þ½μ sgn ðxÞ + λ + ηx + pðx;P1 Þ½μ sgn ðxÞ  λ  ηx ffi Z m Z a  pffiffiffiffiffiffiffiffiffiffiffiffiffi ∗ a2  ξ2 B12 E x ðξÞdξ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi + ξx 2π a2  x2 a n

(4.1.68)

for a  x m and n  x  a. Again, with the new definition of q0(x), side conditions (4.1.56), (4.1.57) apply, and the standard integrals (4.1.63) may be useful.

4.1.7.4 Application to Hertzian contact We return to the example of a Hertzian geometry to illustrate how this methodology works in more detail. Recall that the contact pressure and contact law for the Hertz problem are given by Eq. (4.1.6). We shall consider a steady-state cycle from state 1 to 2 such that ΔQ ¼ Q2  Q1 and Δσ ¼ σ 2  σ 1, and we shall, for simplicity, take P2 ¼ P1 ¼ P so that the contact is over ½a a. As shown by Andresen et al. (2020b), the solution via corrective slip—that is, the moderate tension regime with slip zones of the same sign that we considered in Section 4.1.6—is valid provided that the inequality ∗

4E μ a Δσ  R

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2RΔQ a2  πμE∗

(4.1.69)

254

Fretting Wear and Fretting Fatigue

is satisfied. When this is the case, the edges of the permanent stick zone can be found from the side conditions (4.1.56), (4.1.57) and are given by m ¼ d + e and n ¼ d + e where   4R P ΔQ RΔσ d ¼ ∗  : , e¼ πE 2 2μ 4μE∗ 2

(4.1.70)

It is interesting to note that in the moderate tension case, the change in bulk tension only effects the eccentricity, e, of the permanent stick zone, not its size, d, and vice versa for the change in applied tangential load. When the inequality (4.1.69) is violated, the slip zones are of opposite sign and we must deploy our dislocation solution to find the size of the permanent stick zone. The signum function makes the evaluation of Eqs. (4.1.56), (4.1.57) more complicated, with the solution being given in terms of elliptic integrals, which may be found in Appendix 4.1.2. It is worth noting here, however, that when the remote tensions are large, the size and eccentricity of the permanent stick zone do not decouple as readily as we saw for the moderate tension regime, with the solution being highly nonlinear.

4.1.8

Asymptotic approaches

We have already seen how asymptotic approaches may be used to explain the stick/slip/ separation behavior at the edges of complete contacts in Section 4.1.5. These approaches apply equally well to incomplete contacts and may be used to tackle regimes that we have thus far been unable to reach. In the case of incomplete contacts, a moderately full range of solutions has been found in an exact form, depending on the complexity of the contact geometry, and therefore on the tractability of the resulting integrals. If we classify the problems into moderate and large tension cases, then the full set of solutions is available for the moderate tension cases up to and including the case where there is a varying moment present, as this class of problem can be tackled using a superposition of the sliding traction distribution. However, in the case of large tension there is no choice but to use the glide dislocation-based formulation, and this demands a knowledge of the change in shearing traction under full stick conditions between the extremes of the loading cycle, and we have been unable to discover this for the case when there is a changing applied moment. It is here that asymptotic approaches shine. Asymptotic methods, generally, have relevance to the matching of laboratory experiments with complicated prototypes, but that is outside the scope of this chapter (see, e.g., Andresen et al., 2021b; Hills and Andresen, 2021). Suppose that the coefficient of friction is initially so high that all slip is inhibited. A change in the shearexciting quantities (Q, σ) will induce a square-root singular shear traction at the contact edge. So, if we consider material near the right-hand contact edge x ! a, the shear traction near there will be given by KII qðxÞ ¼ pffiffiffiffiffiffiffiffiffiffi as x ! a, ax

(4.1.71)

Partial slip problems in contact mechanics

255

where (cf. Eq. 4.1.10) Q σ KII ¼ pffiffiffiffiffi  π 2a 4

rffiffiffi a : 2

(4.1.72)

The contact pressure at the edge of an incomplete contact is always squareroot bounded, so that as we approach the same edge we may write the pressure down as pffiffiffiffiffiffiffiffiffiffi pðxÞ ¼ LI a  x as x ! a:

(4.1.73)

The calibration for LI depends on the contact geometry, but for a Hertzian contact, for example, rffiffiffi 2 LI ¼ p0 , a

(4.1.74)

where the peak Hertzian contact pressure, p0, is given by 2P/πa. Since q(x) is singular and p(x) is finite as x ! a, if the coefficient of friction is now reduced to a finite value, μ, a slip zone is necessarily present. In the case of sequential loading, if the range of shear stress intensity is ΔKII the size of the slip zone, measured from the contact edge, d, in the steady state is given by (Dini et al., 2005; Hills et al., 2016) d¼

ΔKII : μLI

(4.1.75)

We turn, now, to the study of a contact oscillating between two points in general load space (P, Q, M, σ). First, the normal contact problem should be solved to find, in particular, two things: the distance to the edge of contact of interest at each end of the loading cycle (a1, a2) and the value of the asymptotic multiplier at the same points (LI1, LI2). We then initially assume full stick and find, for the ends of the loading cycle, the multipliers on the shear traction (KII1, KII2). These six quantities are sufficient to determine, together with the coefficient of friction, an estimate of slip zone size, and hence the extent of the permanent stick zone (Fig. 4.1.8). The method will work best when the movement of the contact edge is small, because the solution hinges on a single-term description of the contact pressure (Eq. 4.1.73) being valid over a distance from the outermost position of the contact edge to a point within (preferably well within) the permanent stick zone. As shown in Fig. 4.1.8, the slip extent at each end of the cycle is given by d1 and d2, where d1 ¼

ΔKII a2  a1 ΔKII a2  a1 and d2 ¼ :  + μLI 2 μLI 2

For further details of this analysis, see Andresen et al. (2021a).

(4.1.76)

256

Fretting Wear and Fretting Fatigue

d1 a1 Δa

Contact lies here

a2 d2

Fig. 4.1.8 The principles of the asymptotic approach in which we focus on the contact problem local to one of the edges of contact: here the right-hand edge is depicted without loss of generality. At each end of the load cycle, we use an approximate solution for the traction profiles to get an estimate of the size of the slip zones, denoted by ai  di where Δa ¼ a2  a1 is the change in the size of the contact region as the normal load increases.

4.1.9

Summary

In this chapter, we have looked at conditions for sliding of a contact and for full stick. We have seen how these considerations manifest themselves for both complete and incomplete contacts. Complete contacts can be expected normally to remain fully stuck or almost fully stuck for a wide range of conditions, so our attention has been focused mostly on incomplete contacts. For incomplete contacts, we have developed two methods for solving partial slip problems under monotonic loading and when the applied loads vary harmonically but without phase shift. The first method relies on assuming a fully sliding solution and superposing a correcting shear traction to maintain stick in the center of the contact. This method works well for a wide range of problems, provided that the differential bulk tensions are moderate and the slip zones are of the same sign. When this is not the case, we instead approach the problem in the opposite manner, assuming a fully stuck solution and distributing dislocations in the stick zones to correct the shear there. Dislocation-based methods allow for a full set of solutions when the slip zones are of the opposite (or, indeed, the same) sign, except in the case that a varying moment is present. In this complicated regime, there are, as of the time of writing, no closedform solutions for the half-plane problem, and we thus need to resort to asymptotic methods to tackle them.

Appendix 4.1.1

Eigenfunctions for the Williams’ wedge solution

We recall from Eq. (4.1.24) that the state of stress near the wedge apex in the Williams’ solution can be written as σ ij ðr, θÞ ¼ KI r λI 1 fijI ðθÞ + KII r λII 1 fijII ðθÞ:

(4.1.77)

Partial slip problems in contact mechanics

257

The eigenvalues λI and λII are given by solutions to Eq. (4.1.23), while the symmetric eigenfunctions are given by ðλI  3Þ cos ððλI + 1ÞαÞ cos ððλI  1ÞθÞ ðλI + 1Þ , cos ððλI + 1ÞαÞ  cos ððλI  1ÞαÞ

cos ððλI  1ÞαÞ cos ððλI + 1ÞθÞ  frrI ðθÞ ¼ frθI ðθÞ¼

sin ððλI  1ÞαÞsin ððλI + 1ÞθÞ  sin ððλI + 1ÞαÞsin ððλI  1ÞθÞ , ðλI + 1Þ sin ððλI + 1ÞαÞ sin ððλI  1ÞαÞ  ðλI  1Þ (4.1.78)

I ðθÞ¼ fθθ

cos ððλI  1ÞαÞcos ððλI + 1ÞθÞ  cos ððλI + 1ÞαÞ cos ððλI  1ÞθÞ , cos ððλI  1ÞαÞ  cos ððλI + 1ÞαÞ (4.1.79)

and the antisymmetric eigenfunctions are ðλII  3Þ sin ððλII + 1ÞαÞ sin ððλII  1ÞθÞ ðλII + 1Þ , ðλII  1Þ sin ððλII + 1ÞαÞ sin ððλII  1ÞαÞ  ðλII + 1Þ

sin ððλII  1ÞαÞ sin ððλII + 1ÞθÞ  frrII ðθÞ¼

frθII ðθÞ¼ II fθθ ðθÞ ¼

cos ððλII  1ÞαÞcos ððλII + 1ÞθÞ  cos ððλII + 1ÞαÞcos ððλII  1ÞθÞ , cos ððλII  1ÞαÞ  cos ððλII + 1ÞαÞ sin ððλII  1ÞαÞ sin ððλII + 1ÞθÞ  sin ððλII + 1ÞαÞsin ððλII  1ÞθÞ : ðλII  1Þ sin ððλII + 1ÞαÞ  sin ððλII  1ÞαÞ + ðλII + 1Þ

Appendix 4.1.2

Size of the permanent stick zone for a Hertz geometry with large remote tensions

For the Hertzian problem considered in Section 4.1.7.4, the permanent stick zone ½m n can be found from the following pair of simultaneous equations: ∗

Δσπ μE ¼ ½α1 EðχÞ + β1 KðχÞ + γ 1 Πðϕ, χÞ + δ1 Πðω, χÞ, 8 2R

(4.1.80)

∗

ΔQ ðn  mÞΔσπ μE + ¼ ½α2 EðχÞ + β2 KðχÞ + γ 2 Πðϕ,χÞ + δ2 Πðω,χÞ, 2 16 2R

(4.1.81)

258

Fretting Wear and Fretting Fatigue

where ma an , ω¼ , m+a a+m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðm  nÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða + mÞða + nÞ, α1 ¼ 2 ða + mÞða + nÞ, α2 ¼ 2 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi   a+m 3 a + m n 1 2a β1 ¼ ðn  m + 4aÞ , β2 ¼  + 2a m  ðm2 + n2 Þ  ða + nÞ , a+n 2 a+n 3 2 3   2 2aðn  mÞ 3a 2mn 4a γ 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , , γ 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2  + m2  3 3 2 ða + mÞða + nÞ ða + mÞða + nÞ   m2  n 2 3ðm + nÞ 2mn 4a2 2 2 δ2 ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  , +m  δ1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 3 3 4 ða + mÞða + nÞ ða + mÞða + nÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2  m2 Þða2  n2 Þ , χ¼ ða + mÞða + nÞ (4.1.82) ϕ¼

and the elliptic integrals of the first, second, and third kinds are given by Z

1

1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi dt, 2 2 1  k t 1  t2 0 p Z 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  k 2 t2 pffiffiffiffiffiffiffiffiffiffiffi dt, EðkÞ ¼ 1  t2 0 Z 1 1 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi dt: Πðν,kÞ ¼ 2 1  k 2 t2 1  t2 0 1  νt KðkÞ ¼

References Andresen, H., Hills, D.A., 2020. A review of partial slip solutions for contacts represented by half-planes including bulk tension and moments. Tribol. Int. 143, 106050. Andresen, H., Hills, D.A., Barber, J.R., Va´zquez, J., 2019. Frictional half-plane contact problems subject to alternating normal and shear loads and tension in the steady state. Int. J. Solids Struct. 168, 166–171. Andresen, H., Hills, D.A., Barber, J.R., Va´zquez, J., 2020a. Steady state cyclic behaviour of a half-plane contact in partial slip subject to varying normal load, moment, shear load, and moderate differential bulk tension. Int. J. Solids Struct. 182, 156–161. Andresen, H., Hills, D.A., Moore, M.R., 2020b. The steady state partial slip problem for half plane contacts subject to a constant normal load using glide dislocations. Eur. J. Mech. A. Solids 79, 103868. Andresen, H., Fleury, R.M.N., Moore, M.R., Hills, D.A., 2021a. Explicit and asymptotic solutions for frictional incomplete half-plane contacts subject to general oscillatory loading in the steady-state. J. Mech. Phys. Solids 146, 104214.

Partial slip problems in contact mechanics

259

Andresen, H., Hills, D.A., Moore, M.R., 2021b. Representation of incomplete contact problems by half-planes. Eur. J. Mech. A. Solids 85, 104138. Barber, J.R., 2010. Elasticity, third ed. Springer. Barber, J.R., 2018. Contact Mechanics. Springer. Barber, J.R., Davies, M., Hills, D.A., 2011. Frictional elastic contact with periodic loading. Int. J. Solids Struct. 48 (13), 2041–2047. Cattaneo, C., 1938. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Rend. Accad. Naz. dei Lincei 27. 342–348, 434–436, 474–478. Churchman, C.M., Hills, D.A., 2006a. General results for complete contacts subject to oscillatory shear. J. Mech. Phys. Solids 54 (6), 1186–1205. Churchman, C.M., Hills, D.A., 2006b. Slip zone length at the edge of a complete contact. Int. J. Solids Struct. 43 (7–8), 2037–2049. Ciavarella, M., 1998a. The generalized Cattaneo partial slip plane contact problem. I. Theory. Int. J. Solids Struct. 35 (18), 2349–2362. Ciavarella, M., 1998b. The generalized Cattaneo partial slip plane contact problem. II. Examples. Int. J. Solids Struct. 35 (18), 2363–2378. Davies, M., Barber, J.R., Hills, D.A., 2012. Energy dissipation in a frictional incomplete contact with varying normal load. Int. J. Mech. Sci. 55 (1), 13–21. Dini, D., Sackfield, A., Hills, D.A., 2005. Comprehensive bounded asymptotic solutions for incomplete contacts in partial slip. J. Mech. Phys. Solids 53 (2), 437–454. Hills, D.A., Andresen, H., 2021. Mechanics of Fretting and Fretting Fatigue. Springer. Hills, D.A., Davies, M., Barber, J.R., 2011. An incremental formulation for half-plane contact problems subject to varying normal load, shear, and tension. J. Strain Anal. Eng. Des. 46 (6), 436–443. Hills, D.A., Fleury, R.M.N., Dini, D., 2016. Partial slip incomplete contacts under constant normal load and subject to periodic loading. Int. J. Mech. Sci. 108, 115–121. J€ager, J., 1997. Half-planes without coupling under contact loading. Arch. Appl. Mech. 67 (4), 247–259. Mindlin, R.D., 1949. Compliance of elastic bodies in contact. J. Appl. Mech., ASME 16, 259–268. Moore, M.R., Hills, D.A., 2018. Solution of half-plane contact problems by distributing climb dislocations. Int. J. Solids Struct. 147, 61–66. Moore, M.R., Hills, D.A., 2020. Extending the Mossakovskii method to contacts supporting a moment. J. Mech. Phys. Solids 141, 103989. Sackfield, A., Truman, C.E., Hills, D.A., 2001. The tilted punch under normal and shear load (with application to fretting tests). Int. J. Mech. Sci. 43 (8), 1881–1892. Vingsbo, O., Søderberg, S., 1988. On fretting maps. Wear 126, 131–147. Williams, M.L., 1952. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. Appl. Mech. 19 (4), 526–528.

Fundamental aspects and material characterization

4.2

Antonios E. Giannakopoulos and Thanasis Zisis Mechanics Division, National Technical University of Athens, Athens, Greece

4.2.1

Introduction

The mechanics of fretting fatigue involve a two (or more)-body interaction and its pylons are contact mechanics and fracture mechanics (Hills and Nowell, 1994; Nowell et al., 2006). The related modeling can be broadly categorized as full finite element analysis (FEA), partial FEA, and theoretical analysis. The emphasis here will be on the theoretical analysis that we believe must always be in the core of the FEA developments. Let us consider the example shown in Fig. 4.2.1, which often describes actual tests. Depending on the thickness W, part of the body is close to plane stress condition (with zero out-of-plane stress), whereas the interior is close to plane strain condition (with zero out-of-plane strain). The influence of the elastic mismatch enters through the constants α and β (the celebrated Dundurs’ constants (Comninou, 1976). Denoting the elastic modulus by E, the Poisson’s ratio by ν, and the shear modulus by G ¼ E/(2 (1 + ν)), we have Dundur’s parameters ðplane strainÞ, 0  ν1, 2  0:5 α¼

G1 ð1  ν2 Þ  G2 ð1  ν1 Þ G 1 ð1  ν 2 Þ + G 2 ð1  ν 1 Þ

β¼

G1 ð1  2ν2 Þ  G2 ð1  2ν1 Þ G1 ð2  2ν2 Þ + G2 ð2  2ν1 Þ

(4.2.1)

with suffixes 1 and 2 indicating the pad and the substrate, respectively. In many contact mechanics analyses, elastic dissimilarity is more conveniently described by Dundur’s parameter β and the parameter γ:
 E2 1  ν22  γ¼1+
E1 1  ν21

(4.2.2)

For elastically similar materials γ ¼ 2 and for rigid pads γ ¼ 1. Assume in what follows that the punch is rigid (α ¼ 1) and the substrate is incompressible (β ¼ 0). If slip is allowed and the punch is rigid, Dundurs and Lee (1972) Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00013-5 Copyright © 2023 Elsevier Inc. All rights reserved.

262

Fretting Wear and Fretting Fatigue

M= + -QL

W

y

B

xx

(1) L

D B

E

P

x F

+ -Q A

(2) C z

H

a a

B

xx

S

Fig. 4.2.1 A generalized plane contact punch problem used in fretting fatigue analysis. Note that S > > H > > 2α and W > > 2α. Loading is applied by normal line load P, tangential line load Q, and line moment M (per unit width).

predicted a square root stress singularity at point E. If the two materials are elastically similar (α ¼ β ¼ 0) and the interface fully adheres, the singularity at point E becomes 0.543 (Williams, 1952). However, the singularity at point F for the fully adhering contact would be 0.65 (Bogy, 1971). Now suppose that the friction is Coulomb type (isotropic and homogeneous) with a friction coefficient f ¼ + 0.2 (meaning that the punch moves in the positive ydirection). In this case the results are given (Gdoutos and Theocaris, 1975) as: for a rigid punch the singularity at E is 0.8, and for elastically similar materials it is 0.68. We cannot find singularity at F because the direction of the friction needs to be assessed. Suppose that the friction coefficient readjusts from +0.2 to +0.6. Interestingly, the stress singularity at point E drops to 0.32 following the increase in the friction coefficient f. So friction may not be independent of the normal load, as suggested by the Coulomb law. It is important to emphasize the influence of the far field boundary conditions for two-dimensional problems (plain stress or plain strain). The particular boundary conditions have a strong influence on displacements and stresses. One such example regarding normal contact loading has been investigated by Dahlberg and Alfredsson (2005), who found analytically, experimentally, and numerically that the surface axial stress σ xx depends on the geometry as !
2     2 P 2 + 3 Hx x x arctan 3 σ xx ðxÞ ¼ π H 1 +
x 2 H H H

(4.2.3)

Fundamental aspects and material characterization

263

Note that the analytical solution (Flamant’s problem) for infinite boundaries would have given σ xx ¼ 0. The theoretical analysis is primarily based on contact mechanics of incomplete contact between idealized contacting pad geometries and semiinfinite substrates. The analysis is performed in the context of isotropic linear elasticity, Coulomb friction or adhesion, and Archard’s wear law. Typical pads are encountered as 2D cylinders (Smith and Liu, 1953) and 3D spheres (Hamilton, 1983; Mindlin and Deresiewicz, 1953). Regarding the complete contact problems, several contact solutions exist and stress singularities exist at the contacting edges. Coulomb friction does not guarantee uniqueness of solution in all cases (except under some particular incremental loading conditions). Nevertheless, the stress singularities predicted by linear elasticity offer a great possibility to apply many direct methodologies and results from the mechanics of notches, as well as from linear elastic fracture mechanics (LEFM) regarding the yet uncracked surfaces. Vast information has been accumulated historically on fatigue of cracks from LEFM, and several books exist on fatigue and fracture, notably Suresh (1998). Such approaches have the advantage of utilizing only the stress field. The fretting fatigue process can be grossly separated into three different regimes according to the relative slip between the contacting bodies, and Vingsbo and S€ oderberg (1988) classified the wear and fatigue into their famous fretting maps. At very low slip the contact region almost sticks the contacting bodies together and as a result there is little wear of the sticking part of the contacting surfaces. At higher slip, but not so high for the contacting bodies to slide, part of the region slips and part of it sticks. The slip region is the region where small surface (edge) cracks nucleate. At gross slip, wear is the dominant phenomenon. In Fig. 4.2.2 we show a complete contact with flat cylindrical pad under a normal load P (applied first) and a tangential load Q (applied sequentially). The contacting body is a semiinfinite isotropic and homogeneous linear elastic space. Under Mindlin-Cattaneo assumptions, the contact is either sticking (Q < f P) or completely slipping (Q ¼ f P). The normal and tangential tractions at the contacting region (circular area of radius a) are square root singular and are given by E∗ h σ zz ðr Þ ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , π α2  r 2

r  a, r 2 ¼ x2 + y2

G∗ Stick solution : σ zx ðr Þ ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ, π α2  r 2

δ  δc ,

G∗ Slip solution : σ zx ðr Þ ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δc , π α2  r 2

δ > δc ,

(4.2.4)

(4.2.5)

The equivalent elastic modulus and equivalent shear modulus are given by 1 1  ν1 1  ν2 ¼ + , G1 G2 E∗ 1 2  ν1 2  ν2 ¼ + : 4G1 4G2 G∗

(4.2.6)

264

Fretting Wear and Fretting Fatigue

P Q (1)

z y

x

h

(2)

Fig. 4.2.2 A cylindrical punch under normal and tangential loading.

where suffixes 1 and 2 denote the elastic constants of the contacting bodies (1) and (2), respectively. Note that the pad (together with the surface that sticks with the pad) can be displaced up to a critical value δc corresponding to Q ¼ f P. A subtle point in all contact problems is that all forces and displacements under consideration refer to the contact surface and the issue regarding the application of displacements and forces at other points of the pad will not be discussed herein. Denote by h the vertical displacement of the rigid punch contacting the elastic surface. It can be found that full stick condition turns to full slip condition at δc: δc ¼

f E∗ h G∗

(4.2.7)

Since this problem is 3D, the vertical displacement h and the tangential displacement δ of the punch are linearly related to the applied forces P and Q, respectively, with zero level of displacement far away from the contact region: P ¼ 2E∗ h α, (4.2.8) ∗

Q ¼ 2 G δ α: For this specific problem we cannot construct a friction-related energy dissipation loop because friction does not participate in the solution, except under gross sliding of the pad (which we cannot classify as a true fretting condition).

Fundamental aspects and material characterization

265

Consider now the 2D (plane strain) rigid, square-ended punch resting on an incompressible half-plane, just about to slide. Sackfield et al. (2002) showed that the corner asymptotic solution holds for a certain range of loads that depends on the friction coefficient and the roundness of the corner. They also constructed from linear elasticity the isobars of the von Mises equivalent stress (in the context of small-scale yielding), around the region of the trailing edge of the punch. It is interesting to note that these von Mises stress contours resemble the ones at the tip of an infinite crack with a ratio of stress intensity factors equal to the friction coefficient KII/KI ¼ f (Maccagno and Knott, 1989). Friction and punch angle change the order of singularity, as expected. The particular case of a 2D, 90o rigid punch (α ¼ 1) on an elastic substrate with Coulomb friction was investigated by Mugadu et al. (2002). Their results revealed that for low friction coefficient  0.1 < f < 0.1, the stress singularity is in the range of 0.525 and 0.475, regardless of the value of β.

4.2.2

Mechanical models and metrics

The mechanical models start with an assessment of the normal and shear tractions exerted on the body destined to be fatigue fretted by the contacting pad. The pad transmits initially a normal load P (often remaining constant), followed by a tangential load Q that varies cyclically (often in the range of + Q and  Q). Nevertheless, the essential aspects of the contact mechanics related to fretting fatigue can be preserved by considering several simplifications (rigorously satisfied if β/f < 0.1). A good part of the simplifications relies on the approximation β ¼ 0 which strictly holds under certain combinations of the elastic properties of the contacting materials, e.g., elastically similar materials, incompressible materials, and rigidincompressible materials. It is very instructive for the discussions that follow to present the plain strain problem of a cylindrical incomplete contact under oscillating loads + Q and  Q. Taking the cylinder to be of radius R (the only length introduced a priori in the analysis), the contact width 2α can be estimated from the loading P and the radius R. The wellknown estimates for the contact tractions at the phase +Q (in the x-direction) and Q ¼ 0 with phase  Q opposite to phase + Q can be found in the studies by Mindlin and Deresiewicz (1953) and Cattaneo (1938). The normal surface pressure (compression) due to the normal line force P is pð x Þ ¼

1=2 2P
2 α  x2 2 πα

2P πα   1  ν21 1  ν22 α2 ¼ 4 + RP π E1 π E2 p max ¼

(4.2.9) (4.2.10) (4.2.11)

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Fretting Wear and Fretting Fatigue

In the presence of Coulomb friction (with friction coefficient f), a monotonically applied tangential line force + Q will create a shear surface traction: 1=2 2 fP
2 α  x2 , c  jxj  α, 2 πα h 1=2
2 1=2 i 2f P
stick zone : qðxÞ ¼  2 α2  x2  c  x2 , πα

slip zone : qðxÞ ¼ 

(4.2.12) jxj  c:

The solution distinguishes an adherence zone of width 2c:  2 c Q : ¼1 α fP

(4.2.13)

Monotonic decrease of the tangential load back to zero produces reverse slipping in the region c0  jx j  α:  0 2 c Q : ¼1 2f P α

(4.2.14)

Reloading the cylindrical pad with a tangential force in the opposite direction ( Q), a reverse slip appears in the region c  j xj  c0 : 1=2 2f P
2 α  x2 , c0  jxj  α, πα2  1=2

 2f P
2 2 1=2 02 2 q¼ α  x  2 c  x , c  jxj  c0 , πα2

 1=2
  2f P
2 2 1=2 02 2 2 2 1=2 α x 2 c x + c x q¼ , πα2

q¼

(4.2.15) jxj  c:

The stick region may be offset due to the presence of a bulk tensile stress σ Βxx (acting in phase with the tangential load Q) close to the surface of the semiinfinite body. The eccentricity for the cylindrical contact is approximately σ Bxx e c ¼ 1 , α 2 γ P max α

(4.2.16)

and, if the bulk stress is tensile, the eccentricity shifts the sticking zone closer to the leading contact edge. Moreover, the elastic mismatch between the contacting bodies can also offset the stick zone as found in a study by Nowell et al. (1988). However, the mathematical structure of the stress asymptotes do not change much around two particular regions in the contact area: Close to the contact perimeter there is a square root variation with

Fundamental aspects and material characterization

267

distance from the contact edge, r½, and inside the stick zone there is a square root singular variation from the contact edge, r1/2. This has been recognized first in the slightly rounded flat punch model of Giannakopoulos et al. (2000b) and was later generalized by Dini and Hills (2004). For a very high friction coefficient, the stick boundary approaches the contact perimeter and almost the whole contact sticks. Then the shear traction becomes Q qQ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : π α2  x2

(4.2.17)

If an underlying axial stress σ Bxx acts (in phase with the tangential load Q) in the substrate (Fig. 4.2.3), additional shear tractions at the sticking contact emerge (see Hills et al., 2012): qB ¼

σ Bxx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: 2γ α2  x2

(4.2.18)

The above equation indicates that a “crack-like” stress field may be envisaged at the trailing contact edge (Fig. 4.2.3) with stress intensity factors: pffiffiffiffiffiffi σ Bxx πα Q K I ¼ 0 and K II ¼ pffiffiffiffiffiffi + : 2γ πα

(4.2.19)

Then, at the edges of the sticking zone, the crack initiation problem can be thought of as a problem of crack kinking from the preexisting major cracks that extend at the free surface outside the contact. The inclination of the crack kink with the surface φ can be obtained by maximizing the local mode I stress intensity factor according to wellknown kinking criteria or tables (Melin, 1994). Fig. 4.2.3 A sticking incomplete contact with underlying axial stress σ Bxx acting simultaneously with Q.

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Fretting Wear and Fretting Fatigue

Following Melin’s tables, the predicted inclination angle is φ ¼ 76.58o and the maximum local stress intensity is kI ¼ 1.2316KII. Subsequently, we can establish a crack arrest criterion and essentially a fretting fatigue endurance limit of the form:

th  th  ΔkI ¼ 1:23ΔK II < ΔK th I ¼ max ΔK I R¼0 ð1  RÞΔK I jR¼1 ,

φ ¼ 77o , (4.2.20)

where Δ denotes range and ΔKth I is the mode I fatigue threshold for long cracks, for a max particular load ratio (R ¼ kmin /k ) of the problem. The fatigue thresholds depend on I I the material and the environmental conditions around the contact edge. In terms of the fretting problem, the threshold criterion can be stated as pffiffiffiffiffi Δσ Bxx πa ΔQ ΔK II ¼ pffiffiffiffiffi + < 0:81ΔK th I : 2γ πa

(4.2.21)

This last result seems to be supported by the experiments reviewed by Hills et al. (2012). We can observe that Eq. (4.2.21) can be restated as a crack nucleation type pffiffiffiffi Δσ Bxx πa th πa + < 0:81ΔK of fretting fatigue map, ΔQ I fP . If the above inequality is violated 2γfP fP a small crack may develop, and the next question is if this small crack will be arrested due to crack closure from the normal contact load. The last problem depends strongly on the pad geometry and the friction coefficient (Moobola et al., 1998). For cases other than the full stick (adherence) contact conditions, we may approach the crack initiation problem in the spirit of Kitagawa (1976) and El Haddad et al. (1979). A characteristic small crack length b0 can be established from the long crack limit stress intensity factor threshold ΔKth : I and the plain fatigue stress limit σ f ΔK th 1 I b0 ¼ π 1:12 σ limit f

!2 :

(4.2.22)

El Haddad et al. (1979) proposed a modification of the Kitagawa-Takahashi stress intensity factor threshold that holds for all crack lengths b and is given by ΔK short th

¼

ΔK th I

rffiffiffiffiffiffiffiffiffiffiffiffiffi b : b + b0

(4.2.23)

More generally, de Pannemaecker et al. (2015) suggested an experimental-based stress intensity factor threshold of the El Haddat type in the form  ΔK short th

¼

ΔK th I

b b + b0

n :

(4.2.24)

Fundamental aspects and material characterization

269

Experimental fretting fatigue results imply that n ¼ 0.4 and other investigators give n ¼ 1/3 (Asai, 2014). Note that the “short crack” characteristic length b0 is an uncertain parameter, probably not a true material constant. Fretting fatigue falls basically into the category of high cycle fatigue (HCF) and as such it was early on argued that methodologies based on endurance and CoffinManson type of crack initiation models available from plain fatigue experiments could be applicable, at least to incomplete contacts. In this approach, the stress amplitudes have to be computed at different points and on all planes that pass through such points (mainly at the contact surface). Then this information passes on to particular HCF models in order to predict: (a) the location of initiation of the fretting crack, (b) the direction of the initial crack, and (c) the number of cycles needed to create the initial crack. Typically, the location of the crack initiation in incomplete contacts with friction is at the boundary of the stick-slip region and it is expected that metrics will be able to predict them. The basic models that have been used extensively in fretting fatigue are: (a) stress-based (combined with Basquin type of life prediction) and (b) strain-based (combined with Coffin-Manson type of life prediction) models. Stress-based criteria (with search for the critical plane) include Findley, Dang Van, and McDiarmid criteria, see, for example, Alfredsson and Olsson (2001) for an overall description of these models. Strain-based criteria (with search for the critical plane) include Fatemi-Socie and Smith-Watson-Topper criteria. Stress criteria based on stress invariants include Crossland criterion (Crossland, 1956), however, these criteria do not predict the initial crack direction. None of these models is really adequate (usually fail to predict the crack initiation point), with the Findley and Fatemi-Socie models proving to the most successful. Findley (with Basquin-type crack initiation law) (Findley, 1959).


b max τa  kF σ nmax ¼ τIF 2N f ,

(4.2.25)

where max is the maximum value from all planes around a point of crack nucleation, τa the amplitude of shear stress on a plane through a point, σ max the maximum of norn mal stress on a plane through a point, Nf the number of cycles to initiate crack, τIF, b the material fatigue strength in shear and the corresponding exponent, and kF is the material constant (0.2 for soft metals). Fatemi-Socie (with Coffin-Marson-type crack initiation law) (Fatemi and Socie, 1988). Δ γ max 2

1 + kFS

τ0f
b
c σ nmax 2 N f + γ 0F 2N f , ¼ σy G

(4.2.26)

where max is the maximum value from all planes around a point of crack nucleation, Δ γ max the maximum shear strain range on a plane through a point, σ y the uniaxial yield stress, G the shear modulus, and γF0 , c are material fatigue shear strain and exponent. Stress gradients contribute to the enhancement of the fatigue endurance. Indeed, the first attempt to include such effects in ordinary HCF methodology can be found in the work of Papadopoulos and Panoskaltsis (1996), who amended the Crossland formulation with the gradient of the maximum pressure. Such amended stress criterion

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Fretting Wear and Fretting Fatigue

(modified Crossland’s criterion) was provided by Zepeng et al. (2015) and is stated as follows: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi   J 2,α + α0 P max  log g0 r J 2,α + g0 rP max   b0

(4.2.27)

pffiffiffiffiffiffiffi where ‘g is the material length, J 2,a the amplitude of variation of the second invariant of the deviatoric stress, Pmax the maximum hydrostatic stress during a loading cycle, a0, b0 material constants related to the endurance limit in alternate torsion and tension compression, and r the gradient operator. Note that a material-related length is introduced into the amended model, which probably includes the microstructural effects in an average sense (e.g., the geometrically necessary dislocation patterns) and is the only additional parameter required for this metric. Observing the surface tractions of incomplete contact problems, Eq. (4.2.15), a problem appears: the stress gradients have a r 1/2 singularity at the contact perimeter and at the stick-slip perimeter. Therefore this physically motivated stress gradient metric becomes infinite at these locations, as noted by Amargier et al. (2010) among others, and the criterion needs further amendment, losing its local significance. Focussing on the fretting fatigue endurance limit, and for fairly convex pads, a “knockdown” factor Kff was established by Ciavarella (2003) and later refined by Ciavarella and Dini (2005). They used a “notch analogue” to describe both the stress like singularity r 1/2 and the stress concentration r 1/2 encountered in a fretting incomplete contact problem, inspired from the notch analogue initially presented by Giannakopoulos et al. (2000b). Ciavarella (2003) provides the knockdown factor for a cylindrical pad in contact with an elastically similar substrate as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 α K ff ¼ min 1 + Y ff , K ft , b0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 fP 4 Q 1 Y ff ¼ B pffiffiffiffiffi 1  1  + pffiffiffiffiffi : fP π σ xx 2π 2 2π

(4.2.28)

In this case the fretting stress concentration factor for large contacts is

4f P K ft ¼ 1 + π aσ Bxx

rffiffiffiffiffiffi Q : fP

(4.2.29)

Once the fretting crack starts developing at the contact surface, the Paris’ law for crack advance with loading cycles N can be utilized (Paris, 1961): db ¼ CΔkm I ðbÞ: dN

(4.2.30)

Fundamental aspects and material characterization

271

where b is the crack length, N the loading cycles, C and m material constants, and ΔkI the local mode I stress intensity difference per loading that includes the crack geometry change. Several modifications of the Paris’ law have been investigated for fretting fatigue by Navarro et al. (2006) and the best (regarding experimental results) was the one that included the fatigue threshold and the El Haddad’s characteristic crack length b0: " # rffiffiffiffiffiffiffiffiffiffiffiffiffi!n
th n db b + b0 ¼ C ΔkI  ΔK I : dN b

(4.2.31)

Crack advance also needs a crack direction criterion. A frequently utilized orientation criterion for the crack advance direction is the maximum mode I local stress intensity factor range ΔkI criterion, based on the maximum circumferential stress criterion of Erdogan and Sih (1963) for proportional loading. Another criterion is that of minimizing the local shear stress range, based on the kinking criterion of Cotterell and Rice (1980), which zeroes the initial local mode II stress intensity factor, kII ¼ 0. Various investigators (e.g., Giner et al., 2014; Baietto et al., 2013; Llavori et al., 2019) have utilized and compared these criteria in actual FEA simulations and it seems that the best criterion is the second one, while keeping the crack tip open: ΔkII minimum with ΔkI > 0:

(4.2.32)

An important issue appears in case that fretting fatigue life needs to be assessed and has to do with the crack’s initial length and its initial direction. These problems do not appear in the crack analogue approach that will be discussed next.

4.2.3

The crack analogue approach

A model that points to a strong analogy between the state of stress adjacent to the contact region and a double edge semiinfinite crack was proposed by Giannakopoulos et al. (1998, 2000a). The “crack analogue” was amended by introducing a small roundness at the sharp edges by Giannakopoulos et al. (2000b), producing the “notch analogue.” As already mentioned, “friction” in its Coulomb formulation is a vague concept whereas adhesion is not. This brought the concept of adhesive contact to naturally complement the “crack analogue” by Giannakopoulos et al. (1999). As a starting point, we will revisit the crack analogue method by examining the problem of a rigid punch that adheres fully with a semiinfinite elastic body (not necessarily incompressible as in the original formulation) under plane strain conditions as shown in Fig. 4.2.4. If the stress-strain behaviors of both materials are characterized by piecewise linear elastic/power law plastic relations, the stress and strain fields in the lower hardening material will approach asymptotically those of a material with identical plastic properties bonded to a rigid surface (Shih and Asaro, 1990). Thus, with a pad that is plastically harder than the substrate, plasticity is confined only to the substrate.

272

Fretting Wear and Fretting Fatigue y

M

Q

P B

B

xx

xx

a

a

x

Fig. 4.2.4 A cylindrical punch under normal and tangential loading.

The substrate is an isotropic, homogeneous linear elastic solid with Poisson’s ratio ν and elastic modulus E. The punch is chosen to be rigid but this can be relaxed, provided that the punch is stiffer (elastically) and harder (plastically) than the substrate. The punch is loaded with normal, tangential, and moment line loads, P, Q, and M, respectively. The original crack analogue model dealt with the incompressible case (ν ¼ 1/2). The line moment M is required to keep the punch horizontal due to the action of a tangential load Q. Finally, a horizontal bulk stress σ Bxx is applied in the substrate to include possible prestressing, in-service loading (e.g., substrate bending), or other experimental conditions. The problem was solved by Adams (2016) regarding the P, Q, and M loading. The elastic mismatch of the two contacting bodies brings about a square root stress singularity and localized oscillations of the surface displacements outside the contact width ( a < x < a), which do not affect the solution at all since the surface outside contact is free to deform. The contact traction inside the contact area due to the normal load P is h   i jCj a+x pðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos ε ln +φ : ax a2  x2

(4.2.33)

The tangential contact traction inside the contact area due to the shear force Q is h   i jCj α+s qðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin ε ln +φ : αs α 2  s2 where.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jCj ¼ coshπ ðπεÞ P2 + Q2 , 0:175, θ ¼ 212ν ð1νÞ :

φ ¼ tan 1


Q P

,  π2 < φ < π2 ,

(4.2.34)

ε ¼ π1 tanh 1 θ, 0  ε 

Fundamental aspects and material characterization

273

If ν ¼ 0.5 then ε ¼ 0, as in the original version of the crack analogue. The balancing moment to keep the punch horizontal depends on Poisson’s ratio and is zero for incompressible substrate (ν ¼ 1/2). For other Poisson’s ratios we use the numerical results of Adams and provide the approximation: M  0:5ν2  0:45ν + 0:35: Qα

(4.2.35)

The stress field is exactly the same with a double edge interfacial crack of two semiinfinite cracks with their crack-tip ligament equal to the contact width 2α. The crack analogue has an asymptotic field along the interface that can be written according to Sun and Jih (1987): h      i 1 s s σ yy ðsÞ ¼ pffiffiffiffiffiffiffi K CI cos ε ln  K CII sin ε ln , 2α 2α 2πs h      i 1 s s σ xy ðsÞ ¼ pffiffiffiffiffiffiffi K CI sin ε ln + K CII cos ε ln : 2α 2α 2πs

(4.2.36)

s is the surface length from the punch cornrer. Comparing these forms with the contact tractions, we obtain two stress intensity factors that refer to the length scale of the problem (the contact width 2α), at the trailing edge of the contact pad: P K CI ¼  pffiffiffiffiffiffi cosh ðπεÞ, πα K CII

(4.2.37)

Q ¼ pffiffiffiffiffiffi cosh ðπεÞ: πα

Such stress intensity factors were also established in the original crack analogue problem for an incompressible substrate (for ε ¼ 0). The mode I stress intensity factor is negative due to the compression of the normal load. Turning our attention to the bulk stress, we observe that this loading brings out another crack analogue result (Mulville et al., 1976). The bulk stress σ Bxx creates a square root singularity with stress intensity factors: K BII

pffiffiffiffiffiffi σ xx πα ¼ , 2

K BI ln ð3  4νÞ : B ¼ 2ε ¼ π K II

(4.2.38)

The mode I stress intensity factor is tensile, if the bulk stress is tensile. An interesting scenario may emerge when the bulk load dominates: the stick condition breaks at the contact edges and the pad lifts from the surface. The adhesive bonding between the rigid pad and the substrate will hold, if the energy release rate (the J-integral in this case) stays below the critical adhesive energy

274

Fretting Wear and Fretting Fatigue

(enhanced by localized plasticity at the edges) giving a combined critical energy release rate Jc which depends on the material and environmental conditions. The energy release rate for a bi-material crack can be found in a work by Shih and Asaro (1988) and other related works and expressed as J¼

 1 1  ν2
2 K I + K 2II and max J < J c , 2 2 cosh ðπεÞ E

(4.2.39)

with, KI ¼ KCI + KBI , KII ¼ KCII + KBII. If the adhesion condition fails, then sliding (and possibly fretting wear) occurs, until the adhesion stabilizes to a new sticking zone 2c < 2α. There are at least two mechanisms that have been investigated and can help in readjusting the crack analogue model. The first mechanism is to assume that Coulomb friction (with friction coefficient f) holds and so utilize the slip-stick maps as those of Churchman and Hills (2006). We can also use particular frictional slip scenarios as in the study by Ciavarella and Macina (2003) with the approximation of incompressible contacting bodies (but otherwise elastically dissimilar). In this case the normal surface tractions show a mode I stress intensity factor as P K I ¼ K CI ¼  pffiffiffiffiffi : πa

(4.2.40)

Ciavarella and Macina (2003) found that the full stick conditions apply for σ Bxx 

  2γ f P Q 1 , aπ fP

(4.2.41)

Then the shear surface tractions show a mode II stress intensity factor as pffiffiffiffiffi πa Q Trailing edge : K II ¼ K CII + K BII ¼ pffiffiffiffiffi + σ Bxx , 2γ πa

(4.2.42)

Leading edge : K II ¼ K CII  K BII : For Q ¼ 0, the mode II stress intensity factor becomes Trailing edge : K II ¼ K BII ¼ σ Bxx Leading edge : K II ¼

pffiffiffiffiffi πa , 2γ

(4.2.43)

K BII :

If the full stick inequality condition does not hold, then slip is expected. For Q ¼ 0, the slip region appears symmetrically at the contact edges and the stick zone 2c can be approximated from the results of Ciavarella and Macina (2003) as

Fundamental aspects and material characterization

 1

1


c 2  2 a

4

 π σ Bxx

a : 2γ f P

275

(4.2.44)

2 γ f Pð1Q=ðf PÞÞ 2γf P B For intermediate bulk loads a π ð1Q= , slip appears at the ðf PÞÞ  σ xx  aπ trailing edge only and the stick zone 2c is approximately

2c 2γ ðf P  QÞ ¼  1: a π aσ Bxx

(4.2.45)

In this case, the mode II stress intensity factor differentiates between the leading (+ Q direction) and the trailing edge: fP Trailing edge : K II ¼ pffiffiffiffiffi , πa fP Leading edge : K II ¼ pffiffiffiffiffi  πa

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðf P  QÞσ Bxx : γ

(4.2.46)

2γ f P For σ Bxx  aπ ð 1Q= ðf PÞÞ , the mode II stress intensity factor becomes approximately

fP Trailing edge : K II ¼ pffiffiffiffiffi , πa

(4.2.47)

fP Leading edge : K II ¼  pffiffiffiffiffi : πa At the contact boundary, however, the shear stresses at the leading edge (+ f) drop to zero, whereas at the trailing edge ( f) the stress singularity λ rises from 0.5 to 1 with increase in friction coefficient from 0 to infinity (σ ij > Fij(φ)r λ), according to the smallest positive root of tan ðπλÞ ¼

2ð1  νÞ : f ð1  2νÞ

(4.2.48)

For incompressible bodies ν ¼ 1/2, so λ ¼ 1/2, regardless of the friction coefficient. Note that an elastic stress intensity coefficient higher than 0.5 does not give finite strain energy density around the singularity and this has not been looked carefully so far. The second mechanism to deal with the slip region is to accept that wear at the pad corners will change the contact conditions. The model of Archard (1953) is often utilized and (as for the Coulomb friction) is considered a bi-material constant, influenced by the environment, which can be applied pointwise along the slipping contact region.

276

Fretting Wear and Fretting Fatigue

Wear models assume that the rate of wear w_ (as a change of the surface position) depends linearly on the wear coefficient kw, and possibly nonlinearly on the local normal pressure p and the local slip velocity u_ according to _ w_ ¼ kw pn u,

(4.2.49)

(n ¼ 1 for Archard’s law). An advanced wear model can be found in the work of Fouvry et al. (2003). TheFouvry et al. wear model is based on the work of the shear traction q(x,t) with relative slip u(x,t), which are both local quantities (along the contact position x and time t). We can interpret the wear coefficient as a characteristic stress σ w and present the depth reduction due to wear as a function of the local energy dissipation due to friction according to wðxtÞ ¼

1 σw

ðu

qðxtÞdu:

(4.2.50)

0

The problem, however, is more complex because wear is affected by the kinetics of the wear debris. It will be shown in the next section that if wear debris is constantly removed from the contact region at the rate it is produced, the fretting fatigue reshaping of the contacting bodies may form a sticking-only contact region, so stress singularities in the reshaped incomplete contacts appear and can be very detrimental. All the previously mentioned stress intensity factors have to be considered as timevarying, since all loading factors can vary with time (P(t), Q(t), M(t), σ Bxx(t)). Therefore, regarding fracture-related fatigue, we need to establish the ranges of these values (max-min) at each loading step. We can then treat the problem as two long “cracks” (the surfaces outside the contact region) that will try to kink inside the material, if the maximum local mode I stress intensity range is above a stress intensity threshold: th max Δkth I  ΔK I :

(4.2.51)

This threshold value depends on the substrate material, the closure effects due to stress intensity factor ratios (min/max), and the environment (see, for example, Nix and Lindley, 1985, de Pannemaecker et al., 2016). Regarding the kinking formation condition and the angle of kinking (at the trailing edge of the contact area), we can utilize the results of He and Hutchinson (1989), who based their results on the maximum energy release criterion. These results indicate that kinking is allowed only toward the less stiff material and the angle of kinking is not very different from the kinking of a crack in the homogeneous case. This approach precludes the possibility of cracking of the pad. Practically, the maximum energy release rate is similar to the criterion in which the crack kinks with an initial angle φ that makes the maximum local mode II stress intensity factor zero:

        1 K II φ 3φ φ 3φ   max + sin + 3 cos : ¼ sin  cos  2 2 2 2 KI (4.2.52)

Fundamental aspects and material characterization

277

A plain strain, lattice-based numerical method (something close to an atomistic formulation) with brittle erosion algorithm was developed by Mohammadipour and Willam (2018), who verified the above crack kinking relation (4.2.52) (for adhesive contact conditions) and confirmed many other high cycle fatigue aspects in the context of linear elasticity. This reaffirms indirectly that small-scale plasticity needs to hold for these theories to be valid. Once the kinked crack starts to advance according to Paris’ law, the local surface stresses diminish inversely with the surface depth and bulk stresses dominate, rendering the remaining fretting problem a classic fatigue crack propagation problem. If the dominant stress conditions favor a db/dN higher in the new stress environment, the crack turns to a new direction (following the kink criterion at the new location) and Paris’ law can be applied again. Otherwise, the kinked crack arrests at a length where the local mode I stress intensity range in the new direction is less than the corresponding threshold value.

4.2.4

Modification of the crack analogue

Regarding the crack-like notch analogue, the asymptotic results of Ciavarella and Dini (2005) can be utilized to obtain an estimate of the slip length d for a flat punch that has a very small roundness of radius R and a contact width that is approximately equal to the flat width. The result can be cast in an approximate closed form as d ¼ a

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi  σ Bxx πa Q 1+  1

> <  ∂Ψ ; σ > σ e > 0

∂D ¼ ∂ε > > :

(4.7.14)

∂D 0; σ < σ e

where σ e denotes the material endurance limit. The efficacy of the above CDM model was verified experimentally (Bhattacharya and Ellingwood, 1998) and shown to be able of predicting the number of cycles to failure, Nn (here referring to crack nucleation). However, it is worth mentioning that the fretting contact loading generally generates a multiaxial stress state where finding the solution to Eq. (4.7.13) becomes challenging. Nevertheless, the multiaxial analysis is considered unnecessary in fretting because cracks normally initiate in a region with a highly dominant quasi-uniaxial stress state (Proudhon et al., 2005; Szolwinski and Farris, 1996, 1998) (see Section 4.7.3.1.2 for more discussion). Employing the Ramberg-Osgood-type equation for the hysteresis loop provides the value of Di, indicating damage in ith cycle (Bhattacharya and Ellingwood, 1998): Di ¼ 1  ð1  Di1 ÞFi ; σ max > σ e where  Fi ¼ 

1+

1+

 1 1 1 Δεoi 1+M M  1 1 1 Δεmi 1+M M

1

 Δεli M Δεoi + Ci  Δεli M Δεmi + Ci 1

otherwise Di ¼ Di1 where 1

3σ f 1 Δε 1+M Ci ¼  1   oi 1 + Δεli M Δεoi 1M 1 + 4 2 G M

(4.7.15)

416

Fretting Wear and Fretting Fatigue

Fig. 4.7.3 Stress and strain coordinates in a typical loading cycle. Adapted from Bhattacharya, B., Ellingwood, B., 1998. Continuum damage mechanics analysis of fatigue crack initiation. Int. J. Fatigue 20(9), 631–639.

in which σ max is the maximum cyclic stress and σ f symbolizes the true failure stress. M denotes the cyclic hardening exponent and G represents the cyclic hardening modulus. The specific strain range parameters in Eq. (4.7.15) are depicted in Fig. 4.7.3. A detailed explanation of these parameters can also be found elsewhere (Bhattcharya and Ellingwood, 1998). Here, the material is assumed to be initially defect-free, i.e., D0 ¼ 0, where the elastoplastic material behavior is assumed. In the case of high-cycle fatigue, the material is generally subjected to stresses smaller than the bulk yield stress (σ yield) and may remain in the elastic range. In such a case, comparing to the elastic strain, the plastic strain remains very small and, therefore, negligible at the macrolevel. Nonetheless, it should be emphasized that from the CDM theory standpoint (Lemaitre and Desmorat, 2005), the material exhibits elastoplastic behavior in the microscale and, hence, is treated as an elastoplastic damageable volume.

4.7.3

CDM analysis of fretting fatigue crack nucleation with provision for size effect

In this section, an overview of a method for the prediction of the nucleation life in fretting fatigue contact conditions is presented using the thermodynamically based CDM approach. The CDM method is suitable for crack nucleation analyses, as it defines damage to be independent of the crack size. However, incorporating a local approach and the wide scatter in available experimentally obtained fatigue lives

A thermodynamic framework for treatment of fretting fatigue

417

can impede the application of the CDM approach in fretting crack nucleation study. To counter this issue, an improved technique that incorporates a nonlocal analysis is presented.

4.7.3.1 Methodology and approach 4.7.3.1.1 Fretting contact stress formulation and analysis Referring to Fig. 4.7.4A, consider a cylinder-on-a-flat configuration with an oscillatory sliding motion. The contact pressure distribution associated with a normal load, Fn, is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð X Þ ¼

1  X2

(4.7.16)

Fig. 4.7.4 Partial slip fretting contact; (A) cylinder-on-flat and (B) hystersis loop due to loading and unloading.

418

Fretting Wear and Fretting Fatigue

where pðXÞ is the dimensionless pressure distribution with respect to the maximum Hertzian contact pressure (p ¼ p/po) and X is the dimensionless coordinate (X ¼ x/a), where a denotes the contact half-width. The maximum Hertzian pressure (po) is obtained using the normal load as well as the contact half-width and contact length (l), po ¼ 2Fn/(πal). Fig. 4.7.4B shows a typical hysteresis loop illustrating the oscillation of the tangential force, Ft, with an amplitude Fmax associated with the maximum relative displacet ment, δmax. Following the experimental observations (Fouvry et al., 2004; Proudhon et al., 2005, 2006), it is reasonable to assume that the partial slip condition dominates while the bulk of the material under contact remains elastic. The following expression for the shear traction distribution is provided by Hills and Nowell (Nowell and Hills, 1994) for the cylindrical line contact. It follows a general solution for the shear traction, q(x), in Hertzian partial slip contact derived by Mindlin and Deresiewicz (1953) and Mindlin et al. (1989). pffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðxÞ ¼ μ 1  X2 ; b=a  jXj  1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 2 qðxÞ ¼ μ 1X 2  X2 ; c=a  jXj  b=a a2 (4.7.17) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 c2 qðxÞ ¼ μ 1  X2  2  X2 +  X2 ; jXj  c=a 2 a a2 where μ represents the coefficient of friction, qðxÞ is the dimensionless surface traction, c is the stick zone’s half-width, and b is the instantaneous stick zone’s half-width associated with the instantaneous tangential force. The positive sign in Eq. (4.7.17) represents loading, where the negative sign signifies the unloading half cycle. The resultant stresses generated by the normal and tangential tractions at an arbitrary point of A below the surface and under elastic plane strain conditions are ( Johnson, 1985): Z Z 2 3 2Z 1 pðSÞðX  SÞ dS 2 1 qðSÞðX  SÞ dS σx ðX, ZÞ ¼      2 2 π 1 π 1 ð X  SÞ 2 + Z 2 ðX  SÞ2 + Z2 Z Z pðSÞdS 2Z 3 1 2z2 1 qðSÞðX  SÞdS σz ðX, Z Þ ¼  2  π  2 π 1  1 ðX  SÞ2 + Z 2 ð X  SÞ 2 + Z 2 σy ¼ vðσx + σz Þ Z Z 2 2Z2 a pðSÞðX  SÞdS 2z a qðSÞðX  SÞ dS τxz ðX, ZÞ ¼      2 2 π a π a ð X  SÞ 2 + Z 2 ð X  SÞ 2 + Z 2 (4.7.18) where Z is the dimensionless coordinate along contact depth and ν denotes the Poisson’s ratio.

A thermodynamic framework for treatment of fretting fatigue

419

4.7.3.1.2 Crack initiation parameter for CDM analysis As discussed before, the fretting crack nucleation generally occurs on the surface and is restricted to close vicinity of the contact edges. Since a “quasi-uniaxial state” of stress exists at the contact edges, a uniaxial analysis approach can be applied to the case of fretting-induced fatigue. This approach has been adopted in other studies. For example, research reported by Lykins et al. (2000) reveals that there is no significant difference in the results of the uniaxial analysis for fretting fatigue as compared to the multiaxial approach using the Smith-Watson-Topper (SWT) criterion. It should be noted that σ x at the edges of contact is the only in-plane stress component that can keep fluctuating above the endurance limit (σ e) and can be a potential candidate for the nucleation parameter. Nevertheless, it is necessary to implement an averaging technique that goes beyond a point at the contact edge, making the selection of σ x less favorable since the averaging techniques use regions at which other in-plane stress components are not necessarily zero. Accordingly, here, the more inclusive principal stress components are considered. Moreover, in regions close to the contact edges, σ 2 principal stress component is always negative and, hence, σ 2 (compressive in nature) is assumed to have a minimal impact on crack formation (Raje et al., 2008; Jalalahmadi and Sadeghi, 2010). Thus, the first principal stress component is selected for the current analysis. Although σ 1 is the largest at the contact edge, the fretting cracks nucleate mostly close to the vicinity, but not always, at the contact edges (Fouvry et al., 2002; Arau´jo and Nowell, 2002; Hills and Nowell, 2009). In fact, σ 1 distributions show intense stress gradients forming near the surface in the proximity of the contact edges (Aghdam et al., 2012). This region is often treated as the averaging zone, where several studies have pointed that identifying such a zone and applying some sort of averaging technique over that region is necessary for predicting fretting behavior (Fouvry et al., 2004; Namjoshi et al., 2002; Proudhon et al., 2005; Mun˜oz et al., 2006; Rudas et al., 2017; Arau´jo and Nowell, 2002; Lykins et al., 2001b; Naboulsi and Mall, 2003).

4.7.3.1.3 Averaging zone identification Research shows that the size of contact significantly affects the fretting fatigue life (Bramhall, 1973; Nowell et al., 2000; Nowell and Hills, 1990), particularly in small contacts at which relatively large stresses exist over a length scale comparable to that of the microstructure of the material. While the need for the including size effect in fretting fatigue has been stated in many studies (Fridrici et al., 2005; Fouvry et al., 2004, 2002; Proudhon et al., 2005; Arau´jo and Nowell, 2002; Arau´jo et al., 2004), there exists no universally accepted standard for the averaging zone size and shape (e.g., see Proudhon et al. 2006; Mun˜oz et al., 2006; Arau´jo and Nowell, 2002; Swalla and Neu, 2002). Some studies used experimental results for its identification; others implemented variable zone sizes correlated with the slip width (ls ¼ a-c) (Proudhon et al., 2006). Regarding the fretting contact, we assume that crack initiation depends on the stress field coupled with the severity of its gradient. This hypothesis

420

Fretting Wear and Fretting Fatigue

Fig. 4.7.5 Distribuation of the first principal stress gradient for (A) c/a ¼ 0.48, μ ¼ 1.05; (B) c/a ¼ 0.83, μ ¼ 0.85. Adapted from Aghdam, A.B., Beheshti, A., Khonsari, M.M., 2012. On the fretting crack nucleation with provision for size effect. Tribol. Int. 47, 32–43.

has been adopted by others (Proudhon et al., 2006; Naboulsi and Mall, 2003). Thus, we seek to determine a correlation between the stress gradients and the averaging zone. Given that predominant crack nucleation sites occur at or close to the contact edges, they should be the core of the averaging zone. Fig. 4.7.5 demonstrates the first principal stress gradient near the contact edges for two different loading conditions calculated as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 ∂σ 1 ∂σ 1 + krσ 1 k ¼ ∂x ∂z

(4.7.19)

As seen, the highest stress gradients (HSG) are created at the slip zone borders since surface traction derivatives are not continuous at those locations. Therefore, a plausible assumption is that the center of the HSG zone is on the contact edge with a length equal to the slip width (ls). The essence of the averaging zone concept is to account for high-stress gradients over a microstructural length scale. Thus, for a target averaging zone to be determined, the microstructure of material should also be considered and compared against the region of HSG. It follows, therefore, that averaging may not be essential for large contact sizes wherein the material microstructure is much smaller than the HSG region. Referring to Fig. 4.7.5, the stress gradient drops rapidly at higher depths, rending the stress gradients effect in the x-direction is quite significant compared to the z-direction. Therefore, it can be hypothesized that the fretting crack initiation is chiefly controlled by the surface stresses and their gradients. The average value is, thus, given as σ 1avg

1 ¼ X2  X1

Z

X2

σ 1 dX

(4.7.20)

X1

where X1 and X2 specify the boundaries of the averaging zone with X1 ¼ 1  ls/2a and X1 ¼ 1 + ls/2a.

A thermodynamic framework for treatment of fretting fatigue

421

Here, the boundary limits for the averaging zone can only be defined qualitatively because the exact link between the material’s microstructure and the averaging zone is yet to be fully comprehended. A reasonable guideline for the use of the HSG region, as the averaging zone, is to check if the size of the HSG region is on the same scale as the material microstructure.

4.7.3.2 Crack nucleation life by CDM Fouvry et al. (Proudhon et al., 2005, 2006) studied the fretting crack nucleation between an Al 7075-T6 cylinder and a flat specimen made of Al 2024-T351. The mechanical properties of Al 2024-T351 are listed in Table 4.7.1. Referring to Eq. (4.7.15) and Fig. 4.7.3, this information was also used along with the relations between the Ramberg-Osgood and Coffin-Manson parameters to obtain the M and G parameters (Boller and Seeger, 1987). The critical damage parameters for the Al 2024-T351 (Lemaitre, 1992) are also shown in Table 4.7.1. It is necessary to point out that it is generally not easy to find values of Dc for specific materials. In the absence of available information, the concepts of Elastic Strain Equivalence or Elastic Energy Equivalence may be used to find a reasonable range, including theoretical lowest and highest limits for the critical damage parameter (Beheshti and Khonsari, 2011). We selected two types of experiments to evaluate and establish the validity of the current approach. The first type of experiment (Proudhon et al., 2005) was targeted at finding the crack initiation boundary, i.e., the tangential force critical amplitude, Fmax , t resulting in cracking for a given number of cycles (Nn) and normal force. According to Proudhon and coworkers (Proudhon et al., 2005), tests were terminated at the specific number of cycles, Nn, to examine the specimen for initiated cracks. Then, by trial and error, the value for tangential force critical amplitude was determined. The second type of experiment (Proudhon et al., 2006) was conducted similar to the first type except to find the number of cycles for crack initiation at given normal and tangential forces. For the simulations of the first type of experiments, an initial value equal to μFn/2 is assigned to the program for the tangential force amplitude. The number of cycles to crack nucleation is input here. The program computes the variation of the first principal stress (and its minimum/maximum) during a full fretting cycle for either the contact edge (local analysis) or the averaging zone (nonlocal analysis). It is then used to calculate the stress and strain range parameters associated with the Ramberg-Osgood hysteresis curve (Bhattcharya and Ellingwood, 1998) as shown in Fig. 4.7.3. Next, Table 4.7.1 Mechanical properties used in CDM analysis (Proudhon et al., 2005, 2006; Boller and Seeger, 1987). Material type Al 2024T351

E (GPa)

ν

σe (MPa)

σ yield(MPa)

G (MPa)

σf (MPa)

M

Dc

72.4

0.33

140

325

851

684

10.2

0.1

422

Fretting Wear and Fretting Fatigue

based on Eq. (4.7.15), the damage evolution per fretting cycle is attained until the final cycle of Nn. The critical tangential force is identified using trial and error that results in damage equal to Dc for given Nn cycles. For the simulations pertaining to the second type of experiment, the input parameter is defined as the tangential force amplitude. An analogous computational procedure is used, except that the solution for these simulations is straightforward and lasts until the damage reaches Dc where the associated number of cycles is recorded, representing the number of cycles for crack initiation. Table 4.7.2 shows a summary of the other input parameters for each category of tests. The coefficient of friction is assumed to be constant, which is a valid assumption for aluminum alloys with a low dependency of friction coefficient on contact pressure (Proudhon et al., 2005). In addition, for each class of tests, two different sets of simulations, considering the local and nonlocal approaches, are performed. Fig. 4.7.6A shows the boundaries of crack nucleation predicted using the CDM approach (shown by solid lines) along with the published experiments (shown by points) reported in Proudhon et al. (2005). The predictions using the nonlocal CDM analysis (curve a) and the experimental results are in reasonable agreement, whereas the local analysis predictions (curve b) differ significantly from the measured results. This shows the necessity of including the stress gradients effect in the fretting analysis. It is worth noting that the predicted averaging zone’s size varies with the normal load from 150 to 418 μm for the aluminum alloy having an average grain size of 150 μm (Proudhon et al., 2005, 2006). The nonlocal approach considers the contact field stress; hence, the cracking risk slightly increases with a higher normal load at a constant coefficient of friction. However, as curve b illustrates, the local analysis results in a cracking boundary entirely independent of the normal load. In contrast to the nonlocal approach, the local analysis only uses contact edge stress, which is only dependent on the tangential force and the coefficient of friction. Hence, the predicted straight vertical cracking boundary is justified. For materials such as titanium alloy, however, the constant friction coefficient assumption should be used with cautions since a relatively high dependency on pressure is reported for the friction coefficient (Fouvry et al., 2004). In fact, the opposite trend for titanium alloy Ti-6Al-4V is reported (Fouvry et al., 2004), where it is shown that the friction coefficient decreases with the normal pressure. Table 4.7.2 Input parameters for the fretting simulations. Simulation objective

Material type

Fn (N/mm)

Fmax t (N/mm)

Nn

μ

R (mm)

Contact length (mm)

Critical tangential force (Type 1) Number of cycles to nucleation (Type 2)

Al 2024T351 Al 2024T351

110–700

N/A

5  104

1.1

49

4.4

400

90–250

N/A

1.1

49

4.4

A thermodynamic framework for treatment of fretting fatigue

423

Fig. 4.7.6 (A) Crack nucleation boundary for Al 2024-T351 and at Nn ¼ 5  104 where curves a and b represent nonlocal analysis and local analysis, respectively; (B) Number of cycles to crack initiation versus the amplitude of tangential force, curves a and b represent nonlocal approach with normal averaging zone and confined averaging zone, respectively and curve c is related to local approach. Results shown are adapted from Aghdam, A.B., Beheshti, A., Khonsari, M.M., 2012. On the fretting crack nucleation with provision for size effect. Tribol. Int. 47, 32–43.

Fig. 4.7.6B shows the evolution of the number of cycles to crack initiation and the corresponding experimental results reported in Proudhon et al., 2006). As shown, both the local and nonlocal prediction results yield similar trends to what is typically seen in high-cycle fatigue of the SdN curves in plain fatigue problems. These are often interpreted as possessing an infinite nucleation life, i.e., Nn  1  106. For the results shown in Fig. 4.7.6B, the averaging zone size varies between 118 and 418 μm, which is fairly close to the reported average grain size of aluminum (150 μm (Proudhon et al., 2005; Proudhon et al., 2006)). Nonetheless, for a few loading conditions, the averaging zone’s predicted size gets smaller than the value of average grain size. In Fig. 4.7.6B, curve “a” includes loading conditions that have this range of averaging zone size. To further verify if the grain size is an appropriate lower threshold for the size of the averaging zone, an additional set of simulations is performed where the size of the averaging zone is kept more than 150 μm

424

Fretting Wear and Fretting Fatigue

(curve “b”). It is evident that applying the new composite criterion and confining the lower limit to the material grain size enhances the accuracy of predictions. It should be added that again here local approach (curve “c”) results in considerable under prediction of life. While the critical value for the damage parameter is reported in the literature for Al 2024-T351, the predictions are not appreciably sensitive to the exact value of Dc. For instance, evaluating Ti-6Al-4 V, we have applied two values of 0.1 and 0.4 for Dc, and yet, observed quite close outcomes (see Aghdam et al., 2012 for more details). This is anticipated from the perspective of damage growth rate evolution as demonstrated by (Bhattcharya and Ellingwood, 1998) where after sufficiently large values of damage (e.g., D  0.1), the damage increases exponentially.

4.7.4

Fretting subsurface stresses with provision for surface roughness

This section presents a simple approach for deterministic prediction of pressure and tangential traction distributions in a rough contact that is needed for treating the fretting fatigue crack initiation risk and life as described in Section 4.7.5.

4.7.4.1 Formulation of rough contact problem Fig. 4.7.7 illustrates a contact of a “smooth cylinder” with a rough flat surface (line contact configuration). The equivalent radius is R and the equivalent modulus of elasticity is denoted by (E∗) where the flat surface has a roughness profile, λ(x). The nondimensioned separation between the two surfaces, hðXÞ, can be then expressed by     Z+ ∞ X2 4Fn 1 4Fn hð X Þ ¼ h0 + pðSÞ ln ðjX  SjÞdS + λðXÞ  π π 2 π ∞

hðXÞ ¼ 0, pðXÞ > 0 if X  Ωc hðXÞ > 0, pðXÞ ¼ 0 if X⊄Ωc

Fig. 4.7.7 Line contact configuration for a smooth cylinder on a rough flat surface.

(4.7.21)

A thermodynamic framework for treatment of fretting fatigue

425

where h ¼ h=R and λ ¼ λ=R and Fn ¼ Fn =lE∗ R, h0 is the integration constant and Ωc shows the contact domain. The surface profile, λ(x) can be generated numerically where the asperity heights are randomly generated with a given standard deviation, Rq, and considering Gaussian distribution (see Kasarekar et al., 2007; Akbarzadeh and Khonsari, 2010 for details). The load balance equation in dimensionless form is Z pðSÞdS ¼

π 2

(4.7.22)

Ωc

If local pressure reaches or exceeds the hardness, the surface asperities deform plastically, and hence, it is assumed that the asperity tips deformation is elastic up to a limiting pressure (material hardness) and experience perfectly plastic behavior beyond that limit. To account for this, an elastic-perfectly plastic behavior is considered as follows: pðXÞ ¼ pðXÞ if pðXÞ < H=p0 pðXÞ ¼ H=p0 if pðXÞ  H=p0

(4.7.23)

where H is the surface hardness. In a fretting application, the contact region experiences a partial slip with an inner stick zone surrounded by a slip region. An expression for the distribution of tangential traction, qðXÞ, is provided by J€ager (1998) and Ciavarella (1998):

qðXÞ ¼ μ pðXÞ  p∗ ðXÞ

(4.7.24)

where p∗ ðXÞ symbolizes the corrective normal pressure. This parameter is similar in form to the normal pressure distribution with an addition of a corrective (smaller) nor∗

mal force value, Fn ( J€ager, 1998; Ciavarella, 1998). If the total tangential force, Ft, is ∗

known, then the value of Fn can be determined by integrating Eq. (4.7.24) over the entire contact domain. That is: ∗

Fn ¼ Fn  Ft =μ

(4.7.25)

Eqs. (4.7.24) and (4.7.25) apply to monotonic tangential loading. In the case of the max partial-slip condition, the tangential load oscillates between Ft , causing cyclic variation in the stick zone size. To determine the tangential traction pertaining to the instantaneous force, Ft, considering unloading, we first assume that the tangential max force drops from + Ft to instantaneous force Ft, and as a result, the contact edges experience a reverse slip where the tangential traction then becomes μpðXÞ. This reduction in traction can be assumed to happen due to the application of a fictitious max opposite tangential force, ΔFt ¼ Ft  Ft, with the associated tangential traction of

426

Fretting Wear and Fretting Fatigue

2μpðXÞ in the new reverse slip zone (Nowell and Hills, 1994). This results in the creation of a new stick zone, and the associated traction distribution can be expressed analogous to Eq. (4.7.24) as

ΔqðXÞ ¼ 2μ pðXÞ  p∗∗ ðXÞ

(4.7.26)

where 2μp∗∗ ðXÞ is the corrective tangential traction pertaining to the new stick zone. In a similar fashion to the monotonic loading, the value of the normal force corresponding to p∗∗ ðXÞ can be then obtained: Fn

∗∗

¼ Fn  ΔFt =2μ

(4.7.27)

Similarly, for the loading half-cycle, the fictitious opposite tangential force is max + Ft . The tangential traction solution during cyclic partial slip is then ΔFt ¼ Ft (Beheshti et al., 2013):

qcl ðXÞ ¼ ½qðXÞ  ΔqðXÞ ¼  pðXÞ  p∗ ðXÞ + 2p∗∗ ðXÞ

(4.7.28)

The numerical solution procedure begins by generating the surface profile λ(x) and evaluating the surface tractions by solving first Eqs. (4.7.21)–(4.7.23) simultaneously to obtain stable pressure distribution and then surface tangential tractions using Eqs. (4.7.24)–(4.7.28). Further, traction distributions are employed to obtain the surface and subsurface stress fields using Eq. (4.7.18).

4.7.4.2 Surface tractions and subsurface stress distribution Illustrated in Fig. 4.7.8 are pressure and tangential traction distributions at the maximum tangential force for four different dimensionless values of surface roughness, Rq ¼ Rq =R. Also plotted in Fig. 4.7.8, is the subsurface dimensionless von Mises stress contour (σ vM ¼ σ vM =p0). Analytical Hertzian pressure and tangential traction distributions are illustrated by discrete marks, whereas solid lines demonstrate the numerical solution based on the current model. The discrepancy between Hertzian (Fig. 4.7.8A) and deterministic approaches becomes more noticeable as the surface becomes rougher. Note also that results are illustrated at the maximum tangential load corresponding to the minimum stick zone in a complete loading cycle. However, the stick zone size fluctuates between its smallest value (shown) and the entire contact boundaries with the oscillation of the tangential load (Nowell and Hills, 1994). Again, it should be noted that, here, for all numerical simulations, the traction distributions and their evolution in the slip and stick zones are taken into account for the full loading cycle calculations but are not shown here for brevity. Also, the maximum pressure is restricted to a certain limit (hardness) following the elastic-perfectly plastic assumption (see, e.g., Fig. 4.7.8D). As anticipated, the stress distribution follows that of the Hertzian type for the smooth surface, whereas, the maximum stress is located

A thermodynamic framework for treatment of fretting fatigue

(a) 3 – Rq =2.5u10–8(smooth) 2.5

– – p(x), q(x)

1.5

2

1

2c–

0.5

1.5 1 0.5

0

0

0.5

Z

Z

0.5

1 1.2 –1.5

(c) 3

1

–1

–0.5

0 X

0.5

1

1.2 –1.5

1.5

(d)

– Rq= 6u10–6

–1

–0.5

(s–vM)max= 1.36

0

0.5

–

2

2

1.5

1.5

1 0.5

1.5

(s–vM)max= 1.92

Rq = 1.5u10–5 2.5

1

X

3

2.5

– – p(x), q(x)

– – p(x), q(x)

(s–vM)max= 0.80

– Rq = 1.25u10–6

2.5

–p(X)-numerical –p(X)-analytical –q(X)-numerical –q(X)-analytical

– – p(x), q(x)

2

(b) 3

(s–vM)max= 0.70

427

1 0.5

0

0

0.5

Z

Z

0.5

1 1.2 –1.5

1 –1

–0.5

0 X

0.5

1

1.5

1.2 –1.5

–1

–0.5

0 X

0.5

1

1.5

Fig. 4.7.8 Surface traction as well as subsurface von Mises stress distributions for different   max values of surface roughness Fn ¼ 1:7  104 , Ft ¼ 5  105 , H=p0 ¼ 0:016, μ ¼ 0:4 . From Beheshti, A., Aghdam, A.B., Khonsari, M.M., 2013. Deterministic surface tractions in rough contact under stick-slip condition: application to fretting fatigue crack initiation. Int. J. Fatigue 56, 75–85.

randomly at scattered spots for the case of a rough surface. It should be mentioned that, generally, when the total load increases, the rough surface behavior becomes more Hertzian due to the increased dominance of the bulk deformation. The same behavior has been observed using the statistical approaches (Beheshti and Khonsari, 2014, 2012). A review of the literature pertaining to the surface roughness effect on the fretting crack initiation indicates experimental data scarcity. Indeed, to the best of the authors’ knowledge, the only quantitative experimental study, considering surface roughness effect, was done by Proudhon et al. (Proudhon et al., 2005) where, the findings, to a large extent, corroborate what is discussed in the following sections.

428

4.7.5

Fretting Wear and Fretting Fatigue

CDM-based prediction of fretting fatigue crack nucleation life considering surface roughness

In this section and through combinations of methodologies described in Sections 4.7.3 and 4.7.4, we further present a model to estimate the number of cycles for crack onset under fretting loading conditions with provision for surface roughness.

4.7.5.1 Numerical procedure Following what is established in Section 4.7.3.1.2, the first principal stress, σ 1 is selected as a nucleation parameter in CDM analysis with the same averaging technique. Fig. 4.7.9 shows that the maximum first principal stress located (at Xm) exactly at the contact edge for an ideally smooth surface is similar to Fig. 4.7.5. For the case of a rough surface, however, while the maximum value of σ 1 always occurs at locations very close to the contact edge, it is not necessarily located at the contact edge. Also, as anticipated, for an ideally smooth surface and if the remote stress is zero, the fretting surface traction distribution is symmetrical, and so either of the edges of the contact can be used as the center of the averaging zone. Nevertheless, in the case of rough surfaces where the roughness profile is generated randomly, the situation is quite different. To properly identify the location as well as the maximum amount of first principal stress, the stresses should be rigorously analyzed throughout a complete fretting cycle.

4.7.5.2 Critical tangential force prediction Fig. 4.7.10 presents the CDM simulation results for the predicted critical tangential force amplitude for Al 2024-T351 pertaining to three different values of surface roughness. Also demonstrated are the corresponding experimental results of Proudhon et al., who identified the crack initiation boundaries in their laboratory tests (Proudhon et al., 2005). They performed several fretting tests with different tangential force amplitudes in the range of 135–454 N/mm. The experiments were terminated at a given number of cycles (5  104), and the test samples were subsequently examined for the existence of an initiated crack (Proudhon et al., 2005). Here, to take into account the effect of the randomly generated surface profile, the simulations are carried out at least 10 times for each roughness value in successive batches to get the range of possible initiation lives corresponding to a given tangential force amplitude. Therefore, a “prediction band” exists due to the randomly generated profiles (hatched area), which are different for the same value of surface roughness. As seen, the prediction band is relatively larger for greater values of surface roughness owing to the more pronounced impact of randomness at greater values of surface roughness. The band reduces into a line when the value of roughness, and hence the associated randomness of the profile, become effectively zero, similar to what is depicted in Fig. 4.7.6. The detrimental surface roughness effect on the crack initiation, can be

A thermodynamic framework for treatment of fretting fatigue

429

Fig. 4.7.9 Averaging regions identification for (A) ideally smooth, and (B) rough surface. From Aghdam, A.B., Beheshti, A., Khonsari, M.M., 2014. Prediction of crack nucleation in rough line-contact fretting via continuum damage mechanics approach. Tribol. Lett. 53(3), 631–643.

distinctly gleaned by the comparison of crack initiation boundary for the rough surface with those estimated for the relatively smoother surfaces. The simulations generally predict a lower crack initiation boundary for greater roughness values, consistent with the experimental observations. Minor discrepancies can be attributed to the differences between the randomly generated surface profiles and the actual surfaces. Despite these differences, the good overall accordance between the experiments and simulations demonstrates the CDM technique authenticity for the analysis of crack initiation in rough surfaces fretting. In addition, the results show that the adopted averaging technique can appropriately take the effect of high-stress gradients into account even for rough surfaces.

430

3

2 1.5

Smooth Predictions

Sliding Boundary

1 0.5

Rq = 2.245 u10–6 0

0.5

–

Ft (10–4)

(c)

3

–

1

Smooth Predictions

2 1.5

Sliding Boundary

1

0

1.5

–

Rq = 1.22 u10–5 0

Experiments Predictions (Rough)

2.5

Fn (10–4)

Experiments Predictions (Rough)

0.5

–

0

3 2.5

–

–

Fn (10–4)

(b)

Experiments Predictions (Rough)

2.5

Fn (10–4)

(a)

Fretting Wear and Fretting Fatigue

0.5

–

Ft (10–4)

1

1.5

Smooth Predictions

2 1.5

Sliding Boundary

1 0.5

–

Rq = 1.53 u10–5

0 0

0.5

–

Ft (10–4)

1

1.5

Fig. 4.7.10 The CDM-based as well as experimental determination of the tangential force critical amplitude (at Nn ¼ 5  104) for different normal loads and dimensionless roughness values. The dimensional roughness values considered are (A) 0.11 μm, (B) 0.60 μm, and (C) 0.75 μm. Adapted from Aghdam, A.B., Beheshti, A., Khonsari, M. M., 2014. Prediction of crack nucleation in rough line-contact fretting via continuum damage mechanics approach. Tribol. Lett. 53(3), 631–643.

4.7.5.3 Crack nucleation life prediction It is often desirable to predict the nucleation life of a component under a variety of loading conditions. To show the efficacy of the approach, several numerical simulation results are illustrated in Fig. 4.7.11 pertaining to the dimensionless normal load of Fn ¼ 1:36  104 . As explained earlier, due to the randomness in generated surface profiles, at each value of tangential force amplitude, the model predicts a range of possible numbers for the initiation life. The span of the range increases monotonically with a decrease in the tangential force amplitude. Yet, for each roughness value, a further evaluation of the results uncovers the variation of the cracking lives remains relatively constant if one considers their mean values, irrespective of tangential force amplitudes. This trend can be attributed to the influence of randomly generated surface profiles and not to be confused with scatters typically observed in fatigue life. Higher fatigue lives are noticable for lower tangential force amplitude despite the considerable effect of randomness. Also depicted are power-law curve-fits, encompassing the minimums and maximums of the number of cycles. The curve-fits are similar to the well-known fatigue life SdN curves. It should also be noted that the maximum amount of the initiation life considered here is restricted to N ¼ 1  106 where infinite initiation life is assumed beyond this point. The prediction bands notably indicate that greater values of surface roughness lead to faster crack initiation incidents along with the experimental findings reported in Proudhon et al. (2005).

A thermodynamic framework for treatment of fretting fatigue

431

Fig. 4.7.11 Crack initiation lives for different tangential force amplitude depicted for three values of surface roughness at Fn ¼ 1:36  104 . Adapted from Aghdam, A.B., Beheshti, A., Khonsari, M.M., 2014. Prediction of crack nucleation in rough line-contact fretting via continuum damage mechanics approach. Tribol. Lett. 53(3), 631–643.

4.7.6

Conclusion and remarks

A numerical approach is utilized here for the analysis of the crack initiation behavior in rough line-contact fretting fatigue. The approach combines a thermodynamically based continuum damage mechanics (CDM) along with the deterministic prediction

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of surface and subsurface stress field for rough surface contact to model the accumulation of microdamage under fretting fatigue loading conditions. In contrast to classical fracture mechanics, the current approach is not hampered by the ambiguous critical crack length definition. Instead, it requires a simple critical damage value for the identification of material failure. This parameter can be determined through simple tension or experimental fatigue tests, and more importantly, the exact determination of the critical damage value is not necessarily required as predictions seem to be insensitive to its value beyond a certain limit. A combination of the thermodynamically based CDM methodology, nonlocal averaging technique, and rough surface contact model, presented here, resulted in the development of a robust predictive fretting crack initiation methodology. Collating the general conformity observed between the predictions and the experimental observations shows the capabilities of the current technique to predict fretting fatigue crack initiation behavior for rough surface contact. Owing to its inherent advantages, the thermodynamically based framework adopted here is believed to be amenable to be extended for the crack nucleation analysis in a wide range of contact problems along with different material types.

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Aero engines

5.1

John Schofielda and David Nowellb a Consultant in Structural Integrity, Derby, United Kingdom, bDepartment of Mechanical Engineering, Imperial College London, South Kensington Campus, London, United Kingdom

5.1.1

Introduction

Fretting fatigue occurs in regions of contact between two bodies under compressive load where small relative tangential motions and cyclic forces occur. High localized tensile stress fields are subsequently generated at and adjacent to the trailing edge of the contact zone. In the presence of bulk load, cracks can initiate and propagate at or close to the so-called edge of contact and threaten component integrity. Fretting fatigue is often associated with wear. In the context of this chapter, wear is defined as surface damage due to the relative motion of the two contacting surfaces. Relative motion is often denoted in this context as slip. Slip can occur across the whole contact surface (bulk slip, or sliding) or just at the edge of the contact surface (partial slip). In partial slip, some of the contacting surfaces remain stuck and this area is often defined as the stick zone. Partial slip occurs at the edge of the contact with magnitudes of typically a few tens of microns or less. The slip zone size will increase as the shear forces across the interface increase. Bulk or gross slip is predicted when the total shear force (Q) exceeds the total normal force (P) multiplied by the coefficient of friction. This apparently simple relationship is complicated by the fact that the coefficient of friction (μ) is affected by surface damage caused by previous bulk slip or partial slip. In many cases, μ is expected to vary across the contact surface. During bulk slip, the evidence of wear is often clear and its effects are more easily characterized. Wear in partial slip zones is more difficult to characterize and how it contributes and combines with the local stress field to potentially initiate cracks is much less understood. Fig. 5.1.1 provides a commonly used diagram to describe the effect of slip amplitude on fatigue life, originally proposed by Vingsbo and S€oderberg (1988). The y-axis provides a qualitative view of fretting fatigue lives and wear lives, the latter being very much component-specific. Minimum fretting fatigue lives are evident at slip amplitudes of 10–50 μm. As sliding amplitude increases beyond these values the wear rate increases and, in some cases, it can become the life-limiting feature. In engine components wear is most associated with reciprocating or cyclic motion, for example, a bolt located in a hole, a spline, or a dovetail joint, and slip zone sizes are limited by feature design. Most practical fretting fatigue issues in aircraft engines are generally dominated by wear or by partial slip, which may lead to fatigue Fretting Wear and Fretting Fatigue. https://doi.org/10.1016/B978-0-12-824096-0.00020-2 Copyright © 2023 Elsevier Inc. All rights reserved.

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Gross Slip

Reciprocat. sliding

10–14

Wear [m3/Nm]

Stick

107

106

10–15

10–16 1

3

10

30

100

300

Fatigue Life [Number of Cycles]

Mixed stick and Slip

105 1000

Δ [μm]

Fig. 5.1.1 Variation of fatigue life and wear volume with amplitude of slip (Vingsbo and S€ oderberg, 1988).

crack initiation. There are some real issues, for example, dovetail slot cracking, where the combination of global sliding and partial slip that threaten crack initiation is a more complex combination. This issue is covered in some detail later in this chapter. The description of fretting fatigue in this chapter is based on predicting the potential for crack initiation and propagation at the edges of a contact area. The focus is on understanding and predicting integrity robustness based on the singular and combined effects of slip amplitude, partial slip zone size, mean and alternating local and bulk stresses, and surface damage. In real applications understanding and predicting surface deterioration or wear is, in the authors’ experience, an empirical science. Many attempts have been made to derive a parameter that can be used to transfer material property data from a laboratory specimen to an actual component. Such approaches can be useful to provide a qualitative view of contributing parameters and options for mitigating crack initiation. However, very few have been successful in providing useful quantitative data. Archard and Hirst (1956) provided an equation that has been used by many to fit laboratory wear data. It recognizes the importance of contact force and sliding distance on wear levels in providing the best fit to individual data sets. In most components, there are other parameters that significantly influence wear rate. These include corrosion, the generation and ejection or ingestion of wear debris, and the environment (humidity, temperature, chemistry). Wear rates are therefore geometry and applicationdependent, and unless near-exact conditions are replicated in the laboratory attempts at useful quantitative data generation are often futile. Most studies of wear have focused on the effects of bulk slip. The contribution of partial slip to wear is less clear, but components in partial slip often see relatively little wear.

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This chapter focuses on fretting fatigue within gas turbine engines, although the principles can be applied more widely. A gas turbine is a rotary engine that extracts energy from an axial flow of combustion gas. The main features are a compressor, combustion chamber, and a turbine. The compressor sucks air into the engine and compresses it, the hot air is mixed with fuel in the combustor and ignited, and the hot expanding gas then drives the turbine. The compressor and turbine sections contain aerofoils mounted on a rotating disc to impart energy to and extract energy from the air, respectively. In an aerospace gas turbine, the momentum change of the air across the engine provides thrust. Gas turbines can have one, two, or three shafts all mounted on separate bearings. The purpose of the shafts is to couple together each stage of the turbine and the corresponding compressor stage. The inlet or cold end is designed to tolerate some foreign body ingestion, including birds and ice, while at the combustor exit, gas temperatures can exceed the melting point of the metals. A typical engine requires a secondary air system for cooling, bearing load balancing, and for providing cabin air. An oil system is also required to provide lubrication and cooling. In such a complex product there are many contact interfaces between the multiple structures and materials and hence there are multiple opportunities for fretting fatigue to challenge the mechanical integrity of the engine. Fig. 5.1.2 illustrates an aerospace gas turbine engine, in this instance a Rolls-Royce Trent 1000. Listed in the figure are some of the potential areas where contact fatigue can occur. An understanding of fretting fatigue in gas turbine engines requires an appreciation of the types of loading that engine components can see. For design and analysis purposes the loads are generally defined as being either steady or vibration. Steady loads are those associated with engine cycles (e.g., flights) and include centripetal, thermal, pressure, assembly, and residual stresses. Acceptable cyclic lives of engine parts are generally derived based on a knowledge of stresses associated with steady loads and comparison with a curve showing stress range (S) versus cycles (N) curve, generally

Fan and Compressor–dovetail wear and fretting, tip-rub,variable stator vane mechanisms.

Engine structures–Joints: pinned, bolted and keyed, interference fits, end stops.

Combustor–pipe contact and supports, flanges and interfaces.

Hot end–firtree roots, dynamic and static seals, lock-plates, cover-plates

Shaft and power transmissions: splines, gears, bearings

Fig. 5.1.2 Example areas of contact fatigue in a Rolls-Royce Trent 1000 Engine.

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termed an S-N curve. Many S-N curves (in the case of safety-critical parts, it is all curves) have a datum point (or a series of datum points) generated from component tests. These component tests attempt to mimic the steady stresses as accurately as possible and achieve a stress state at least as high as that seen by the component. Vibration stresses are generally not reproduced, although any increases in the total stress range due to combined steady and vibration stresses are considered. Vibration loads are, as the name suggests, associated with the vibration of components or subcomponents. Vibrations can be induced by multiple mechanisms, with the two main drivers being periodic aerodynamic or mechanical forces. The frequency of load cycling under vibration can be from a few tens of Hertz to tens of kilohertz. These frequencies are generally sufficiently high to accumulate many cycles in a short period of time. Furthermore, the S-N curve is generally close to being horizontal at the high cycle end, which makes a quantitative assessment of the cycles to crack initiation problematic. Should a crack initiate, then it can propagate rapidly to failure in short timescales. Given the points above and the difficulties in quantifying vibration amplitudes, the adopted industry assessment approach is to show that any vibration stresses remain below a level that could cause any fatigue damage. A common industry approach is to demonstrate significant margins of vibratory stress against a so-called fatigue endurance limit (EASA, 2018; Nicholas, 2006). This margin is against crack initiation. On safety-critical parts, engine manufacturers are also required to show some level of damage tolerance to components. Manufacturers are required to recognize the potential for component imperfections through demonstration of defect tolerance. A minimum specified cyclic life must be tolerated for a specific defect type and size. In this case, the maximum tolerable defect size is defined as a size where the crack driving force reaches the material toughness or, as is most likely at the edge of contact when the vibration-driven fatigue crack propagation threshold is reached. This latter condition requires a much smaller stress range than the fatigue threshold for crack initiation. Defect tolerance calculations at the edge of contact zones are subject to all the same analysis challenges discussed in this chapter. Vibration is the most significant source of gas turbine component failures (Davenport, 2006). As stated above, the industry-wide principle adopted is to keep vibration stresses below a threshold value such that crack initiation is prevented. The gas turbine industry uses widely the so-called Goodman Diagram approach (Goodman, 1930) to determine acceptable levels of vibrational stress. The Goodman Diagram defines the fatigue endurance limit for all applied levels of mean and vibration stress. The appropriateness of the name the Goodman Diagram, although in widespread use, is disputed by some who argue that it should be called a Haigh Diagram (Haigh, 1917). In this chapter, the general industry adopted name of Goodman Diagram is used. The diagram in its simplest form plots vibration stress on the y-axis versus steady stress on the x-axis. A line is then drawn from the UTS/2 (ultimate tensile stress
2) on the y-axis to the UTS on the x-axis. If a point representing the stress state in a component lies above the line, then fatigue cracking is predicted, whereas if the point is below the line, then cracking should not occur. The simple straight-line approximation, although having proven valuable, cheap to generate, and reliable for many materials, is not appropriate for all gas turbine materials. Consequently, it is now common to draw

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a line based on the results of material component and/or specimen tests and for the line to be geometry specific. The above description of a Goodman Diagram is applicable to fatigue thresholds in a plain geometry, i.e., in the absence of any stress concentration. Its use has been adopted in several ways to describe fatigue thresholds for stress concentrations. Arguably the simplest approach is to move the point on the x-axis to a point that corresponds to the stress concentration value “Kt” multiplied by the UTS. A more accurate approach is to plot actual specimen data that represent a similar stress gradient and a similar stressed volume. The combined approach using a life prediction based on a calibrated S-N curve and a below endurance level Goodman Diagram assessment approach allows a safe cyclic life to be predicted for most components. It also ensures that there is no steady and vibration stress interaction, or a sometimes-called “combined cycle fatigue” effect on a structure. This approach cannot be easily adopted when analyzing the edge of contact. Deriving an S-N curve for fretting fatigue initiation is difficult for many reasons and its relevance is questionable. Firstly, providing an accurate recreation of the local stress state is difficult as is the contributing effect of recreating the actual contact geometry and surface finish due to the differing influences of geometric tolerancing and contact surface wear. The stress state in and particularly at the edge of the contact region involves very high peak stresses, extremely steep stress gradients, multiaxial stress states, and differing mean stresses. Obtaining convergence in a finite element model is challenging and the stress state is load path-dependent. Secondly, deriving a “Goodman Diagram” for vibration stresses is problematic for all the same reasons. Thirdly, the assumption that there will be no interaction of steady stress-induced damage and vibration stresses is questionable. Due to the high, but very localized stress field, steady loads can potentially introduce micro-cracking that can be picked up and grown by vibration stresses that could be below any threshold derived assuming that steady stresses do not cause any prior damage. Deterioration plays a significant role in determining the mechanical integrity of contacting parts. Mechanisms include corrosion (including oxidation and the formation of corrosion products), wear (including the removal of wear debris), chemical reactions, and the introduction of other debris. Wear and corrosion, as the main contributors to deterioration, can have a major effect on mechanical integrity and can in some cases be beneficial. Wear remains largely an empirically based science, but largely governed by the magnitude of the contact loads, the amount of sliding, the local environment, and the ability or otherwise to eject wear debris. In gas turbine engines relative movements can occur due to steady loads and/or vibration-induced loads. Sliding can occur across the whole contact surface or locally at the edges of the contact surface (partial slip). It is this latter phenomenon that is generally associated with fretting fatigue. It is noted that many engineers refer loosely to fretting fatigue when they are describing issues of wear/contact fatigue/galling. In this chapter fretting fatigue refers to the mechanism that can induce cracking at the edge of contact due to stress fields associated with partial slip. As stated in the introduction to this chapter, wear refers to the deterioration of contact surfaces due to relative motions. Galling is used to describe the transfer of material between metallic contact surfaces.

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5.1.2

Examples of engine events

The aerospace industry safety culture, the mandatory requirements for early warning of any potential failures, and the additional requirements of the regulatory authorities restrict engine events to very small numbers. In an extreme case, the worst consequence of any mechanical integrity issue is an in-flight shutdown (IFSD) of a single-engine. Any high-energy debris resulting from component failure should be contained and not hazard the airframe. In addition, any engine damage and loss of thrust should not prevent the aircraft from continuing to fly. Consequently, fretting fatigue issues are generally an additional maintenance and cost burden and suitable actions are taken to prevent any engine malfunction. If specific fretting-related maintenance is essential, then there are significant drivers to restrict it to planned shop visits whose frequency is dictated by the life of more sensitive components. A small number of potential fretting fatigue or wear-related crack initiation events have led to single-engine shutdowns on large civil aircraft. Two examples of singleengine events are provided below. Both events had major cost implications and one resulted in cabin depressurization. A further example is provided, showing evidence acquired during the investigation of an unrelated failure, of a critical part that highlights the additional maintenance burden that can be required to prevent possible fretting fatigue. All the three examples provided relate to the dovetail joint in the low-pressure compressor. These examples are used as they are perhaps most visible, most likely to cause significant damage, and details are readily available in the open literature.

5.1.2.1 Southwest Airlines flight 1380, April 17, 2018 An engine event, most likely due to fretting fatigue occurred in a CFM56-7B24 turbofan engine on Southwest Airlines flight 1380 on April 17, 2018. A fan blade cracked and fractured the dovetail part of the blade root. The released part of the blade hit the fan casing, ultimately causing the release of the fan cowl and some of the engine inlet. Some of the engine debris hit the fuselage of the aircraft causing a depressurization event and tragically one casualty. The aircraft landed safely 17 min after the event. Further details can be found on the following websites: https://www.flightglobal.com/ntsb-finds-fatigue-cracking-on-southwest-cfm56-7b-failedblade/128006.article https://www.ntsb.gov/investigations/AccidentReports/Pages/AAR1903.aspx

5.1.2.2 RB211 Trent 892 turbofan engine Boeing 777, A6-EMM, January 31, 2001 An aborted take-off occurred of a Boeing 777, due to engine failure associated with a fan blade release. Although not at the edge of contact, the stress field in the cracking location was influenced by the conditions of the contact surfaces in the dovetail root.

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The released fan blade was contained, and the aircraft returned safely to its stand (https://www.atsb.gov.au/media/33974/tr200100445_001.pdf).

5.1.2.3 Accident to the AIRBUS A380-861 with Engine Alliance GP7270 engines, September 30, 2017 An engine failure on an Airbus A380-861, although not associated with fretting fatigue provides an insight into the maintenance burden on some engines that are required to manage potential fretting fatigue and wear issues. The event report referenced below describes how the root is shot-peened, the contact surfaces are coated with a copper-aluminum plasma spray, and finally, a layer of dry film lubricant (DFL) is applied. A reapplication of a molybdenum disulfide lubricant to these fan dovetail roots is required every 500 cycles. This repeated application ensures that a low coefficient of friction is maintained throughout the operation (https://www.bea. aero/uploads/tx_elydbrapports/BEA2017-0568.en.pdf).

5.1.3

Areas subject to fretting

5.1.3.1 Dovetail blade roots A dovetail is a single lobe contact joint used to fix blades into a disc, Fig. 5.1.3. The design is widely used due to ease of assembly and disassembly and allows the replacement of individual blades. Earlier large civil gas turbine engines, pre the mid-1980s, used multiple lobes on the low pressure and intermediate pressure compressors, but more modern engines use the more cost and space optimized dovetail design. These can be axially or circumferentially oriented. Multiple lobe joints are still widely used in the hot end of the engine, where design space and load path redundancy requirements still dictate their use. Such multiple lobe joints are commonly referred to as “fir trees.”

Fig. 5.1.3 An example of a dovetail joint attachment on a fan blade/disc assembly.

Fretting Wear and Fretting Fatigue

σ

A

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Fc

σ

Q

FV

σ

B

σ

A

P

B

M

T

blade

T

Fig. 5.1.4 Schematic illustration of a dovetail root showing load transfer.

Fig. 5.1.4 provides a schematic two-dimensional illustration of the load transfer across a dovetail joint. As the engine starts to rotate, centrifugal loads (Fx) pull the blade away from the disc and the blade will try to move radially outwards. The disc will also expand; thus the dovetail slot will dilate and allow the blade to move further from the engine centerline. The fan blade will be subject to steady lateral or tangential forces due to pressure loads. Despite the design intent to minimize vibration forces, some vibration sources are present, and some level of vibration of the fan blade is inevitable. Any vibration will manifest itself as cyclic forces, ΔFx and ΔFy, on the fan blade. The fan blade loads are transferred into the blade disc at the contact flanks. At the contact flanks the resulting load transfer can be idealized as a combination of a direct load P, a shear load Q, and a moment M. There will also be an underlying or bulk stress in both the fan blade and disc. These bulk stresses will be three-dimensional in nature and will vary both in and out of the plane of the figure. During vibration, all the loads at the joint and the underlying stresses will see a cyclic variation about a mean value. The joint will slide outwards when: Q ¼ μP, and will slide inwards when Q ¼  μP. These two lines, as plotted in Fig. 5.1.5 and denoted as “slide out” or “slide in” lines, respectively, bound all combinations of P and Q that can exist. If the loads correspond to either line, then the bodies can slide with respect to one another. All other points within the bounded region represent partial slip. Where any contact surface sits on the diagram is dependent on the rotational speed and its history. Initially, during acceleration, the joint will lie on the “slide out” line until a maximum speed is achieved. The relationship between sliding distance and speed is nonlinear. Firstly, the centrifugal force is proportional to the rotational speed squared and, secondly, the wedge effect of the joint provides increasing resistance with increasing radial movement. When the speed is reduced, P will increase, and Q decreases as the effective disc poststiffness

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Fig. 5.1.5 Q versus P for two bodies in contact. slide out line A

Q

incr rpm

D 0

C decr rpm slide in line B 0

P

changes with rotational speed. Once point B is reached on the graph the blade can slide inwards. Vibration forces can occur at any engine speed and will cause a cyclic variation in P, Q, M, and bulk stresses. Based on the above, the contact joint will see bulk slip during initial engine acceleration to an operating speed. Partial slip will then often occur during subsequent changes in engine speed (e.g., reduction from take-off to climb and cruise conditions). Vibration forces will also result in a partial slip. As a result, the contact face will see wear due to bulk slip, some wear in partial slip zones during vibration, and fretting fatigue. This last phenomenon will be due to high local steady stresses, low-frequency cyclic stresses due to speed changes, and high-frequency cyclic stresses due to vibration. The points of maximum tensile and cyclic stresses in the dovetail joint geometry will be where the convex geometry presses on the concave geometry. The two positions that fretting fatigue cracks could occur in a dovetail joint are illustrated by the red lines in Fig. 5.1.6.

5.1.3.1.1 Derivation of stresses The modern approach to stress analysis very much favors the use of finite element models. For a dovetail joint, this generally takes the form of a single sector blade and disc model and the assumption of cyclic symmetry. At a dovetail joint the principal limitations of this approach include: l

l

l

l

l

l

an exact geometry (often nominal) is assumed, the blade is assumed to take the same position in the disc slot every time the centripetal load is applied, during the steady stress analysis, a coefficient of friction (CoF) is assumed between the contact surfaces, a common assumption being that this is uniform, converged stresses are achieved at the edge of contact, the effects of load history change the predicted stresses, during the vibration or dynamic stress analysis, the contact surfaces are effectively locked together.

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Fretting Wear and Fretting Fatigue

Fig. 5.1.6 Positions of crack initiation in a dovetail joint due to fretting fatigue.

Taking each of the above points in turn: l

l

l

l

Contact geometries are manufactured to very tight tolerances. Despite this, small withintolerance variations can and do result in significant differences in load distribution within the slot and hence in the contact stresses. The tight tolerances reduce the effect of the variation in blade position, but blades can take up slightly different positions in the slot each time the engine is operated. There is undoubtedly some “wear to fit” that takes place with the cyclic operation that reduces the effect of the two phenomena described above. However, this only reduces the effects of the mechanisms described and does not remove them. The assumed coefficient of friction affects the local and bulk stresses in the dovetail root. Fig. 5.1.7 shows the dependence of bulk loads and hence bulk stresses on the coefficient of friction. The figure shows how the bulk stress in the dovetail correlates directly with the CoF and how bulk stresses increase as the COF increases.

Looking at Fig. 5.1.7 and resolving forces vertically, F ¼ 2(P + Q) cos θ. Q ¼ μP, where μ is the coefficient of friction F ¼ 2Pð1 + μÞ cos θ

F P a X Q b X

Fig. 5.1.7 Forces acting on and net section bending and membrane stresses in a dovetail root.

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F 2ð1 + μÞcos θ

The net section bending stress (σ b) on section X-X beneath the edge of contact is σb ¼

My I

where the bending moment, M ¼ (P  a) + (Q  b) I ¼ second moment of area of section X-X Furthermore, looking perpendicular to section X  X : σm ¼

Q A

where σ m is the membrane stress and A is the cross-sectional area of section X-X. As b > a in most metal root designs, M is a maximum, hence σ b is a maximum when Q is a maximum. Q is a maximum when μ is a maximum. Although generally much lower, σ m is also a maximum when Q is a maximum. Given the above, the higher the value of the coefficient of friction (μ) the higher the net section bending and membrane stresses. The above assumes that μ is uniform over all the contact surfaces. This is unlikely to be the case. Localized wear or galling, due to localized slip can be expected to provide an area of increased friction coefficient and therefore attract load and increase local stresses. Wear patterns can also be expected to show significant variability for the reasons stated previously regarding manufacturing variability and the “wear to fit” concept. The correct practice with finite element models is always to derive a converged stress solution. This means that the mesh should be continued to be refined until the prediction of localized stress no longer changes. To achieve this, a blade dovetail would require element sizes of  5% of the end corner radius (Rajasekaran, 2005). Despite the massive advances in computing capability, this is not yet realistic. Computing times remain long and the achievement of model convergence is problematic. In any case, once the element size starts to approach that of the material grain size, it is questionable whether an elastic isotropic continuum analysis is appropriate. Stresses at the edge of contact are load history-dependent. The variations within an engine cycle due to differing thrust requirements and external temperature will change local stress predictions. Exact stress solutions often involve more than one surface stress peak, when plotting stress versus distance in the slip zone, and the positions of maximum steady stress and cyclic stress range are often different. One of the most significant issues with fretting fatigue capability prediction is the handling of vibration or dynamic stresses. Dynamic stress assessment generally involves the prediction of a vibration mode and shape and then the prediction or measurement of vibration amplitude. The former relies on a finite element (FE) model. The FE software carries out a “modal analysis” to determine possible vibration modes. Stresses are derived for a unit displacement amplitude. These stresses are then scaled

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to a specific displacement that is either determined by detailed analysis or test. The FE model can simulate the sliding that occurs during the application of centripetal load. However, for modal analysis, the contact areas of the dovetail are by necessity “locked together.” In the analysis, a spring is placed between nodes to provide a contact stiffness. This approach removes the possibility of a stick/slip zone and hence forces the edge of the contact stress field to be singular. It also does not allow the position or values of P and Q, as illustrated in Fig. 5.1.5, to be derived accurately. One means of overcoming the issue of locked contact surfaces is to extract vibration displacements from the FE solution, to reapply them to a subset of the finite element model, and to derive a quasistatic solution. This approach requires the choice of which nodes local to the contact zone are given a prescribed displacement. Care has also to be taken to ensure the disc contribution to the mode shape is modeled correctly. When using this approach, the nodes on and close to the contact surfaces must be left free to develop the stick and partial slip zones. If displacements are applied to all nodes, then the singular stress field at the edge of contact will remain. However, if too few nodes are constrained then the joint will not conform correctly to the shape imposed by the vibration mode. The choice of which nodes to constrain requires some judgment and sensitivity studies. One possible approach is to prescribe displacement for all blade root nodes and for the centerline only of the dovetail disc post. An alternative approach to determining local contact stresses is to use contact loads and bulk stresses (P, Q, M, σ m+ b) as an input to a semianalytical solution (Rajasekaran, 2005). This is in effect a submodeling approach. The best current models are only able to represent a half-space for the punch and the plane that is being impacted. These approaches require significant assumptions as to how the underlying or bulk stress is modeled. In a real component, the bulk stress varies both radially and axially in the blade and in the disc. In a half-space solution, a constant bulk stress in the punch and half-space is assumed. There are several other assumptions that require addressing. These include the presence of a local notch stress field in the real component and not in the half-space, the three-dimensional effects, particularly of an axially curved root, that prevent local slip, and the varying bulk stress throughout the blade and disc dovetail.

5.1.3.1.2 Design basis for a bladed disc All integrity analysis of aerospace gas turbine components is preceded by an assessment of the consequences of “failure.” Consequences can range from high-energy debris release (and hence a threat to the airframe), an engine fire (contained or not contained), multiple engine failure, a controlled single engine shutdown to simply an increased cost due to additional maintenance requirements or inspections. The first three consequences listed are described as hazardous events. If failure of the component leads to a hazardous event, then the component is classified as being a “critical part” and additional steps are necessary to ensure adequate safety. These steps include the adoption and fixing of engineering, manufacture, and service management plans (EASA, 2018). In the following text, a blade is defined as having a root (or dovetail) section and an aerofoil (the part that contacts the gas stream). A blade failure is generally not classified as a critical part failure. Credible blade failures can be “contained”

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and hence demonstrated not to be hazardous. However, if a disc should break up the energy involved precludes containment within the engine then the part is classified as critical. An additional requirement for critical parts is the forewarning of failure. In a bladed disc assembly, a forewarning of a mechanism that could result in hazardous consequences is the cracking of a blade. All components are designed not to fail. However, a design requirement is that if a previously unidentified mechanism, in particular a vibration mode, should come to light then forewarning of failure of the critical disc component is provided by cracking of the noncritical blade. This can be provided by demonstrating higher endurance ratios (maximum predicted stress
allowable stress) for the blade than the disc. Further assurance is provided by demonstrating that aerofoil cracking (an event that is much easier to contain) precedes blade root cracking. The required failure sequence for a bladed disc assembly if subject to higher than anticipated stress levels is aerofoil, blade root, or dovetail then disc dovetail slot. It is worthy of note that cracking in a disc dovetail could be viewed as “nonhazardous” as cracking will not result in a hazardous event (i.e., failure of the entire disc). This can add value to a forewarning of failure argument, but nevertheless, a disc remains a critical component. In smaller compressor blades, due to the slender nature of the aerofoil, most vibration modes are more likely to cause aerofoil cracking than root cracking. This ensures that any released fragments can be contained and any consequences “nonhazardous.” Avoiding resonances that are more likely to crack the blade root than the aerofoil, therefore, provides a good design basis. The above requirements pose a challenge to the analyst. Given the limitations discussed earlier of finite element analysis and the interpretation of the results, the obvious challenge is to quantify an endurance ratio for fretting fatigue that can be compared with more conventional fatigue analysis as expressed on a Goodman type diagram. Means of doing this are discussed later in Sections 5.1.5 and 5.1.6.

5.1.3.1.3 Fir-tree blade roots A fir tree is a multilobed feature for fitting a blade into a disc (Fig. 5.1.8). The use of a fir tree is more common in the hotter sections of the compressor and behind the combustor in the turbine.

Fig. 5.1.8 A fir-tree blade root.

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Fretting Wear and Fretting Fatigue

The use of multiple lobes provides a larger shear area for carrying load. It also provides some redundancy if cracking initiates first on the inner lobes. The principles of analyzing contact stresses in a fir tree are the same as those used to analyze a dovetail. A complication with a fir tree is that the multiple lobes of fir-tree blade make the load path determination, and hence the load assigned to each lobe is difficult. A common first approach is to use the nominal dimensions in an FE model. If an exact fir-tree profile and geometry match between the blade and the disc at cold-fit is adopted as the design basis then there is a risk that the outer lobe will carry a high proportion of the steady load. Cracking of the outer lobe could lead to blade release. Temperature differences between the aerofoil, the blade root, and the disc, due to the aerofoil seeing the hot gas path, increase the likelihood of increasing the proportion of the load carried by the outer lobe. Allowances are therefore made in the design to accommodate potential thermal gradients by progressively offsetting the nominal or “cold-fit” pitch as the distance from the center of the engine changes. Such pitch changes are often small and of a similar order of magnitude to the geometric tolerances. Offsetting the pitch in this way has the benefit of increasing the steady load on the inner lobes. Cracking of any of the inner lobes would not result in blade release and cracking generally only comes to light when the blades are removed during routine maintenance. Within tolerance variation of pitch, dimensions can potentially increase the amount of steady load carried by the radially outboard lobes. Realistic analysis assumptions and some sensitivity analysis on lobe pitch are recommended in carrying out integrity calculations to ensure a robust design, in that the inner lobe cracking will occur first. However, ensuring that inner lobe cracking occurs first under vibration loads is more problematic. Most vibration modes involve aerofoil displacement and hence the bulk of the vibration loads are likely to react on the radially outer-most lobes. This observation increases the importance of a robust design approach to ensure vibration stresses are below any level that might cause crack initiation. Due to their higher operating temperatures, many blades with fir-tree roots tend to be manufactured from nickel-based alloys or single crystals. At high operating temperatures, it is common for oxide or glaze to form. This glaze can fundamentally change the coefficient of friction and in many cases reduce contact stresses. Due to the smaller size, the edge of the contact position can be very close to a notch fillet. This complicates the stress field somewhat and limits the value of extracting contact forces and transferring them to a half-space model.

5.1.3.1.4 Splines—Contact fatigue, notch fatigue, and wear As illustrated in Fig. 5.1.9, aircraft engines typically have two or three concentric shafts on which the discs and blades are mounted. These rotate independently, and in each case, power is transferred from the turbine components through the shaft to the compressor components, therefore requiring a torque as well as an angular velocity. Given the power involved, these torques can be quite significant. In order to allow for engine assembly/disassembly, the shafts are each manufactured in two parts with a joint that allows a male part to be slid into a matching female part. Transmission of

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Fig. 5.1.9 Schematic representation of a three-shaft engine.

Fig. 5.1.10 (A) Spline coupling of two shafts and (B) cross section of male and female teeth.

torque requires that a spline joint be used here. For ease of manufacture, the tooth shape on the male spline is generally an involute profile, and the female part has a matching female involute. Fig. 5.1.9 shows a diagram of the two shafts and Fig. 5.1.10 shows a cross-section of the two shafts when mated together. Although the splines can be straight, frequently they are formed on a helix. Shaft failures are not particularly common in practice, and the splines tend to be designed on rather empirical rules. Increasing requirements for greater efficiency leads to larger, slower fans and both these features will push up the level of torque required from the shaft. Hence there is a need to reduce the levels of conservatism in existing shaft design methods in order to keep the weight of the shaft as low as reasonably possible. In general, the shaft will see two main forms of variable load: i. Variations in torque due to changes in load and speed. These will cause changes in pressure on the flanks of the spline together with small amounts of radial motion, which can lead to relative radial movement of the surfaces. ii. Rotating bending. The shaft will carry a bending moment due to the weight of the rotor, and hence the upper part of the shaft will be in compression, whereas the bottom will be in tension. As the shaft rotates, any point will see a cyclic tension/compression loading cycle. This will cause changes in axial strain and may lead to partial slip towards the ends of both the male and female parts of the spline.

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Fretting Wear and Fretting Fatigue

A number of potential failure mechanisms need to be considered. Firstly, there is fretting fatigue, associated with the edge of contact in both radial and axial directions (i.e., at the end of the tooth and at the end of the spline). Secondly, there may be conventional notch fatigue at the spline root, caused by bending of the spline tooth. This is essentially similar to tooth root fatigue in an involute gear. Finally, one must consider that wear may take place between the surfaces, leading to loss of the original spline geometry. Eventually, this may become a source of engine vibration. Testing of shaft splines under the complex loads experienced is difficult to do without carrying out an expensive engine test. However, there have been attempts at developing smaller-scale test rigs. These include an experiment carried out at the University of Nottingham (Leen et al., 2002) using a multiactuator testing machine. Researchers have also attempted to characterize the loading at the edge of the teeth, so that a simpler experimental configuration may be used. A particular feature is that the edge of contact has an extremely sharp radius so that the pressure is almost singular. Leen et al. (2000) have carried out a finite element analysis and proposed a single actuator test geometry. Mugadu et al. (2002) have used the asymptotic approach similar to that described in Chapter 5.1 to characterize conditions at the edge of contact and therefore to allow simple experiments to be performed at conditions that are representative of those in real spline couplings. Some practical issues need to be taken into account when designing spline couplings. First, there may be the same tolerancing issues that occur in the case of fir-tree roots (see the earlier discussion). Hence, the assumption that an applied torque is shared equally between all teeth of a spline may not be true in practice. This can mean that one or more pairs of mating teeth may see significantly higher stresses. However, such teeth will also tend to wear more than those which are less heavily loaded, so to some extent the situation will be self-correcting. There is often a degree of lubrication present in splines, even if only residual lubrication from the assembly process. The closely fitting nature of the male/female spline contact means that often such lubrication is retained. Hence, shear stresses are likely to be lower than might be the case for the blade root contacts described above and this will have a significant effect in lower stresses at the edge of contact. In order to reduce wear, spline joints are often hardened, using processes such as nitriding or carburizing (Kalpakjian, 1989). This means that, in addition to the increase in hardness and a reduction in wear, there is frequently compressive residual stress in the surface layers of the material. Of course, since this is residual stress (i.e., one present with no applied load on the component) there must be balancing tensile stress further beneath the surface so that the net force on any section is zero. Nevertheless, most fatigue cracks start at or very close to the surface, so that compressive residual stress with a depth of tens to hundreds of microns will have a significant effect in inhibiting the initiation of fatigue cracks. In practice, spline couplings are designed using very simple rules, for example, limits on the average contact pressure and/or conventional tooth bending stresses at the root of the teeth. There is clearly room for more sophisticated analysis in optimizing design. This will become increasingly important as shaft torques increase with the move to large, slower rotating fans. In some designs, this trend may be mitigated by the introduction of a geared fan, but the gearbox itself will introduce significant design challenges in the area of contact fatigue.

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5.1.3.1.5 Flanges Aircraft engines contain a number of flange joints between components, for example between sections of the outer casing. Such joints provide a relatively straightforward means of joining different components and allowing for assembly disassembly. They also play an extremely useful role in introducing frictional damping which will control vibration amplitudes. A useful demonstration of this may be obtained by comparing the difference in response to a gentle tap on a cracked and on an uncracked wine glass. In this case friction between the surfaces of the crack plays a similar role to friction between surfaces of a flange joint. A number of factors should be taken into account when designing a flange. These include: l

l

l

l

The necessity to control overall stress levels in the flange, particularly at the radius at the root of the flange. The necessity to prevent leakage under service pressure and axial loads. This requires a sufficient preloading of the flange, in turn requiring a given number of bolts and a prescribed preload for each bolt, usually specified in practice by a bolt torque, applied during assembly. The requirement for the bolts themselves not to fail by fatigue, particularly when considering the preload that has been applied. The necessity to consider fretting may take place at the contact surfaces between the two flanges and/or at the contact between the flanges and the bolts/washers which apply the preload.

Hoeppner (2006) reports a case of a fretting fatigue failure at a flange joint (in this case on a propeller shaft). Anderson (1994) gives a further example, which is discussed by Majzoobi and Jaleh (2007). In general, though, there are relatively few reports of failures in the literature, flanges are probably less highly stressed than the blade roots and splines as discussed above. Whilst the careful design of flange joints is necessary, fretting is not usually the principal design consideration.

5.1.4

Mitigation measures

A number of mitigation measures have been successfully used in practice and these can be particularly useful when a problem manifests itself during engine operation. Two main methods are available: surface coatings and surface treatments.

5.1.4.1 Surface coatings Modification of the coefficient of friction can be a useful means of addressing a fretting fatigue problem. In particular, reducing the coefficient will generally increase the amplitude of partial slip, but reduce the stress range. On the other hand, increasing the coefficient will reduce slip, and therefore surface damage, at the expense of raised stress levels. Hence, one might conclude that from a fretting fatigue perspective the “best” friction coefficient is either very low or very high. However, we should note that frictional slip plays an important role in reducing vibration amplitudes by providing damping. Hence, there is an optimum value of μ from a vibration perspective. Thus, if we reduce the level of stress for a given level of vibration amplitude by

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Fretting Wear and Fretting Fatigue

reducing friction, thereby also reduce damping and increase the level of vibration response, then the result may not be what we intended. Nevertheless, surface coatings are frequently used, generally to reduce the friction coefficient. In low-temperature areas of the engine, such as the dovetail fan blade roots, DFLs with a high molybdenum disulfide content are frequently used. These can produce a friction coefficient of around 0.1. However, they do not adhere very well to the base material of the blade root, so that an intermediate layer of copper-nickel-indium is sometimes specified in order to improve the adherence of the DFL and to accommodate any unevenness of the surface, thus ensuring a more even pressure distribution. The Cu-Ni-In is not a particularly low friction material in itself. One of the issues with coatings, however, is that they can wear relatively quickly, particularly if there is gross slip, which there is likely to be in a dovetail blade root during the engine run-up. One can neither guarantee equal wear rates, nor necessarily a uniform initial coating thickness. This means that the coating may wear through to base metal in some areas more quickly than in others. The result of this is rather unfortunate since there will now be worn patches with rather high friction (e.g., 0.5–0.6). These patches now have a tendency to stick and have increased tangential stiffness. Hence, they will attract load, and finish up more highly loaded than the remaining coated area. Some analyses have concluded that this part-worn condition (sometimes called “patchy friction”) leads to a worse stress state than the original uncoated contact. Hence, coatings can be useful, but they must be employed carefully and replaced or renewed before significant wear takes place. It should be noted that it is more difficult to employ low friction coatings of the type described above in the hot parts of the engine. However, as discussed earlier, an oxide glaze can form under a combination of temperature and sliding which has the fortuitously beneficial effect of reducing the coefficient of friction significantly.

5.1.4.2 Surface treatment (residual stress) As described above in the context of splines, it can be useful to introduce compressive residual stress. For steel components, this can be achieved at the same time as surface hardening by nitriding or carburizing. For other materials, this can best be achieved by mechanically inducing residual stress at the surface. A very common process is shotpeening, where small spherical particles are fired at the surface to plastically deform the surface layer and induce compressive residual stress up to a depth of approximately 100–200 μm (Soady, 2013). Shot-peening generally provides a useful improvement in fatigue performance and the compressive layer can assist in preventing small flaws, initiated at the edge of contact, from propagating. The treatment is relatively inexpensive to apply and is therefore used in a number of areas that are potential initiation sites for cracks. These include dovetail blade roots. However, although there is a benefit in terms of life, this is rarely taken into account explicitly in fatigue life calculations. Rather, it is typically considered to constitute an additional “safety factor.” Where fretting conditions are more extreme there may be a benefit in achieving a deeper layer of residual compression. A useful process for this is laser shock peening

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(LSP). Here, a laser is used to vaporize an aluminum tape applied to the surface (frequently called an ablative layer). This creates a shock wave in the material and can induce compression up to 1 mm deep. A useful review of the process is given by Montross et al. (2002). Zou et al. (2018) described an application of LSP to a compressor blade aerofoil, with particular emphasis on the leading edge, where foreign object damage (FOD) might be anticipated. Rolls-Royce plc has used LSP on the fan blade roots of some Trent engines in order to extend inspection intervals (King et al., 2005). Whilst LSP can prove extremely effective in resisting fatigue crack initiation, it is a relatively expensive process, involving the use of a high-powered laser and robotic manipulation of the components in addition to the manual process of applying the ablative tape. Further, there are limitations caused by the requirement to access the area with the laser beam. It is difficult, for example, to treat the female dovetail slots in fan and compressor discs. For these reasons, LSP is sometimes thought of as a method of last resort, i.e., its use should only be contemplated where other less expensive methods have not proved sufficient to eliminate fretting crack initiation and propagation.

5.1.5

Design criteria—Academic perspective

Many designers are used to designing based on constraints on the stress state at any point in a component. This is particularly true of fatigue design based on the “safe life” approach. Here it is sufficient to ensure that all surface points are below the fatigue limit, which is established through specimen testing of samples with a similar surface finish. For the case of fretting fatigue, this approach will lead to an extremely conservative design. The reason for this is that there are very high stresses at the edge of the contact, but these decay extremely rapidly. Hence there are two reasons why a fatigue limit derived from smooth specimen tests will not be appropriate: i. In contrast to the plain specimen used to determine the fatigue limit for the material, only a very small surface area and associated volume will be subjected to the high-stress levels. In a plain specimen, there is a much larger surface area, so the probability of finding a “poor” microstructure is significant. In the case of the contact stress field, it is much more likely that we will subject a “typical” microstructure to the high-stress levels. Hence, the performance of the material will be better than we might expect from the plain specimen results. ii. The stress levels will decrease rapidly as we move away from the edge of contact so that a growing crack will experience rapidly reducing stress. Although this does not generally lead to a decreasing crack driving force, there are situations where the contact stress is applied in a region of relatively low bulk stress where a crack that initiates from the edge of the contact will arrest as it grows into the reducing stress field. This situation has been explicitly investigated for fretting by Fouvry et al. (2008).

Both of these features are of course present with notch fatigue and the well-known size effect for notches is also present in fretting (Nowell, 1988). Various means have been proposed for dealing with this, including averaging the stress over a representative volume (Arau´jo and Nowell, 2002). In essence, the method is very similar to the “Theory of Critical Distances,” proposed by Taylor (2008). However, it should be noted

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that the stress gradients associated with contacts are very much higher than might be the case with normal geometric features such as notches and holes. Therefore, calibration of the method using specific fretting experiments is likely to be required. Two approaches do appear to have some potential for application in aerospace components:

5.1.5.1 Short crack arrest

Fig. 5.1.11 Example KitagawaTakahashi diagram.

Normalized Stress

The method of short crack arrest was first proposed by Arau´jo and Nowell (1999). Essentially the method relies on the observation of Kitagawa and Takahashi (1976) that the threshold for the propagation of short cracks is lower than the long crack threshold stress intensity. In many cases of fretting fatigue, the number of cycles of loading is very large. This is certainly true for problems driven by vibration. Hence one is primarily interested in the threshold condition where the component life is, in effect, infinite. There are two ways to assure this. Firstly, one may seek conditions where a crack is never nucleated. The problem with this is that it requires precise knowledge of the conditions at the location of nucleation. These might include local geometry, loading, interfacial friction, and material microstructure as well as macroscopic loading. Even if they could be determined, they might vary significantly between different examples of the same component, and it would be extremely difficult to guarantee no failures during service. An alternative approach is to make the conservative assumption that the conditions at the point of nucleation are always sufficiently severe to nucleate a crack. Hence the problem that needs to be investigated is whether the crack can escape from the small zone of very high stresses (in this case close to the edge of the contact), which caused the crack to nucleate. This leads to the concept of short crack arrest, proposed for fretting fatigue by Arau´jo and Nowell (1999). In order to do this, one needs to be familiar with the concepts expressed in the Kitagawa and Takahashi (K-T) diagram (Kitagawa and Takahashi, 1976). An example of such a diagram is shown in Fig. 5.1.11. In this diagram, the threshold stress which produces crack growth is plotted against crack length. For small cracks, it is found that the threshold is approximately constant (consistent with the fatigue limit for the material, Δσ o). Beyond a certain crack length, however, the threshold stress falls in proportion to √ a, where a is the crack length. This is, of course, consistent with the concept of a long crack threshold ΔK, denoted

1

0.1 0.1

1 10 Normalized Crack Length

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ΔK0. The crack length, a0, at which the two regimes meet may be found from the requirement that pffiffiffiffiffiffiffi ΔK0 ¼ YΔσ 0 πa0

(5.1.1)

where Y is the usual geometry factor in the expression for stress intensity factor. When Y is set to 1, then the resulting value for a0 is often called the El-Haddad length. This may be thought of as a material property. However, its use in this manner must be treated with caution, since both ΔK0 and △ σ 0 vary with load ratio and other parameters and they are both notoriously difficult to measure. Further, it should be noted that a0 is found (Eq. 5.1.1) by taking the ratio of ΔK0 and △ σ 0 and squaring the result. Hence, even a small uncertainty in ΔK0 and △ σ 0 will result in significant uncertainty in a0. Hence, a0 might be better thought of as a physically based fitting parameter rather than a true material property. Rather than plotting stress against crack length as in the original K-T diagram, an alternative approach is to plot the results as a stress intensity factor range. This presentation is sometimes called the modified K-T diagram. An example is shown in Fig. 5.1.12. In any situation, a growing crack may be thought of as tracing a locus on this diagram, and the area beneath the two thresholds may be considered as a region of nonpropagation. Let us now refer to Fig. 5.1.13. A crack of type 1 (green) is always above the threshold and will always propagate. In contrast, a crack of type 2 (red) will propagate initially, but then falls below the threshold and will arrest. A crack of type 3 (blue) represents the boundary between the safe (self-arrest) and unsafe (propagation) conditions, where the crack growth curve just touches the threshold. Hence, in order to investigate self-arrest, one merely has to postulate the nucleation of a crack, examine the increase of its stress intensity as it grows, and compare this trajectory to the K-T threshold. In practice, there is a shortcut to this process, since with the K-T threshold as described here, the critical point is almost always at the kink in the threshold (a ¼ a0). Hence, one simply has to calculate the stress intensity factor range for a ¼ a0 and check whether it is greater than or less than the long crack threshold Fig. 5.1.12 Example of the modified K-T diagram.

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Fretting Wear and Fretting Fatigue

1.4 1.2

ΔKth/ΔK0

1

0.8 0.6 K-T Thres hold

Type Curve 1

Type Curve 2

Type Curve 3

0.4

0.2 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Crack size, a/a 0

Fig. 5.1.13 Crack growth curves in the modified K-T diagram.

(ΔK0). That is, if ΔK(a0) > ΔK0, there will be propagation, whereas if ΔK(a0) < ΔK0 there will be arrest. Rajasekaran (2005) has used this approach to predict the failure or otherwise of dovetail blade root specimens in fretting fatigue. It may also be used in other rapidly varying stress fields, such as those around sharp notches, and has been used by Duo to predict failure from FOD on aeroengine blades (Nowell et al., 2003). Further, Oakley and Nowell (2007) have extended the approach to address the situation where there is a combination of LCF (engine speed) and HCF (vibration) loading. An advantage of the short crack arrest approach is that it is relatively straightforward to incorporate residual stress into the calculation, as these can be included in the evaluation of stress intensity factors. Hence, the effects of surface treatments such as shot-peening and LSP may readily be taken into account.

5.1.5.2 Contact asymptotics As noted by Hills and Murray in Chapter 5.1, a convenient means of describing conditions at the edge of contact is by using contact asymptotics. Most of the contacts encountered in aero-engines will be of the “incomplete” type (i.e., where the contact area increases with increasing load). This means (see Chapter 1.3) that the pressure close to the edge of contact is of the form pffiffi pð x Þ ¼ L 1 r

(5.1.2)

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where r is the distance from the edge of contact and L1 is a parameter that describes the magnitude of the pressure.a Similarly, it can be shown that if the stick zone size is small, then the shear traction close to the edge of contact, but remote from the stick zone itself may be described by KII qðxÞ ¼ pffiffi r

(5.1.3)

where KII describes the intensity of shear loading. Fig. 5.1.14 shows these two asymptotic solutions, together with the exact shear traction for example, the problem of a Hertzian contact subject to a shear force less than that necessary to cause sliding. It will be immediately apparent that the dimensions of L1 and KII are different, and it may be shown (Chapter 5.1) that the slip zone size d may be approximated by d¼

ΔKII μLI

(5.1.4)

where ΔKII is now interpreted as the change in KII since the last load reversal and μ is the friction coefficient. These observations permit correlation between an experiment and a full-scale component. If an experiment is run at the same values of L1 and KII, as are present in a component, both the stress levels and the slip zone size (and hence the amplitude of slip) will be the same. Hence, one would expect the initiation and short growth phases of life to be the same in each case. Andresen (2000) has recently suggested that the fretting fatigue life may be plotted as a series of ΔKII versus N curves, one for each value of L1. If the life is dominated by the initiation and short crack phases then this holds out the prospect of creating universal fretting fatigue life

Fig. 5.1.14 (A) Comparison of asymptotic and full solutions for the pressure close to the edge of a Hertzian contact and (B) similar comparison for shear tractions in a Hertz/Mindlin contact.

a

Different notations are currently used for L1 and KII. To avoid confusion, we will use here the notation adopted in Chapter 1.3.

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data for each material pair using data from specimen tests. This could then be used directly in the design process by evaluating the L1 and KII parameters for the design conditions, using conventional finite element analysis and looking up the corresponding life from specimen tests. The authors are not, however, aware that this approach has yet been applied in any practical aeroengine application.

5.1.6

Industrial applications perspective

It is useful at this stage to summarize the three main phases in the cracking process. (1) Initiation (or nucleation): the first stage on shear planes and quickly followed by propagation normal to the surface. (2) Early (or short-crack) propagation: crack propagation is strongly dependent on local frictional force and local microstructure. (3) Propagation: at a rate defined by the “long crack” Paris Equation and driven by bulk stresses.

Traditional fatigue criteria tend to focus on crack “initiation.” This is usually defined as the cyclic life to initiate a crack and grow it to an engineering crack size, stages (1) and (2) above. The definition of engineering crack size varies, but common definitions are in the range of 0.15–0.25 mm. As stated in the introduction to this chapter, the prediction of safe cyclic lives due to fatigue crack initiation in gas turbine engines is generally based on steady loads only. The definition of a small engineering crack size in plain surface or notch fatigue precludes crack propagation due to vibration stress. The validity of the application of this simple interpretation in the case of the edge of contact stress fields is less clear. The high localized stress due to steady loads could feasibly initiate short cracks that could propagate at vibration stress ranges that are below conventional fatigue threshold limits. This phenomenon is made more complex by the continually degrading surface due to global and partial slip. Many academic studies have focused upon the application of a single stress range. A much smaller number recognize and address the fatigue interaction question. Quantification of fatigue lives remains essentially an empirical activity. The task is to find a parameter that provides “transferability” from measured lives on specimen, subcomponent or even component tests to predicted lives in actual in-service engines for all geometries and stress fields. Predicted safe lives can then be provided based on tests carried out using representative material coupons. Mean stress corrected strain range is commonly used as a fatigue initiation assessment parameter, but this cannot be universally applied across all geometries. Research continues to make steps forward in providing a more mechanistic basis for fatigue life prediction for all stress fields and geometries. The issues of nonlinear material behavior, stressed volume, multiaxiality, out-of-phase or nonproportional loading have all been addressed to some degree. Similarly, steps have been made to provide more mechanistic-based arguments that use genuine material properties and avoid the need to use fitting parameters. The holy grail of an all-encompassing single mechanistic-based framework to explain quantitatively fatigue lives for all geometries remains a noble vision.

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The current best practice is to use the developed understanding to define the most appropriate component tests that can be carried out to derive experimentally fatigue lives and then use this understanding to infer component lives. The better the understanding is, the smaller the amount of experimental data required and the smaller the margins or factors that can be applied to account for any ignorance. Section 5.1.5 describes the possible criteria that allow correlation of specimen data to fretting fatigue cyclic life component predictions. These approaches include stress at a point, stress averaged over an area or volume, stress resolved on to a critical plane, linear-elastic fracture mechanics (LEFM), and asymptotic approaches. All these approaches rely on the accurate prediction of local stresses and/or slip amplitudes and correlation with specimen or coupon tests. Section 5.1.4 describes why accurate prediction of local stresses is credible in most specimen tests, but not so in actual service components. Local stresses will vary significantly due to variation within geometric tolerances, global and local friction, and load history effects. If local stresses cannot be predicted to the required accuracy, then the transferability of specimen data to actual must be questionable. A second issue is how to handle the potential individual contributions of steady and vibration stresses and potential combined effects. Steady stress load cycles can be expected to contribute to first stage initiation and early propagation. They can also be expected to contribute to bulk slip and hence contact surface wear. Vibration stress thresholds can reasonably be expected to be dependent on prior steady stress cycles and conditions, both in terms of the condition of the contact surfaces following wear and possible progression of first stage initiation. Fracture mechanics approach, including the short crack arrest approach, do lend themselves to separate and combined assessment of steady and vibration stresses. However, crack driving force predictions rely on accurate prediction of local stress fields, which as described above cannot be viewed as accurate in an actual component. Given the above, it is advised that any assessment includes a more holistic approach that looks at more predictable bulk parameters along with service experience as well as local stress predictions.

5.1.6.1 Design and assessment approaches We will take here the example of a dovetail blade root, in order to illustrate the decisions that need to be made in a practical design or assessment problem. At the initial design stage, a good first step is to establish and apply limits for average and local “crushing stresses” and for net section bending plus membrane stresses for the steady loads. The average crushing stress can be defined as the centripetal load normal to the contact patch divided by the total contact area. This normal load is dependent on the coefficient of friction, Fig. 5.1.15. Any finite element model generated must recognize and use a CoF that represents a level as high as any that can be achieved in service. The thickness of the blade root and disc post, Sections “B-B” and “D-D” in Fig. 5.1.15, can be defined based on prescribed maximum net section stresses. Fig. 5.1.15 shows the simple formulae that can be applied to use the finite element model defined stresses to determine membrane (σ membrane) and bending (σ bending)

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D B

D

x t B

Fig. 5.1.15 Derivation of net section stresses.

stresses. The two sections “B-B” and “D-D” should represent the planes of maximum normal stress and are close to the planes of maximum principal stress. Good advice for assurance of the mechanical integrity of the dovetail is to try and spread contact loads out axially along the contact flank as much as possible. Axial skewing of dovetails to maximize aerofoil performance and three-dimensional computational fluid dynamics (CFD)-driven blade design makes this very difficult and often necessitates a need to analyze local effects on mechanical integrity. In this case, the net section stresses should be limited to the prescribed value at the maximum net section stress location. A further check on steady stress levels is to ensure that stress levels in the adjoining notch fillet remain at acceptable levels. The finite element model should produce a converged solution in this region and the derived numbers assessed against notch material specimen or component data to give acceptable fatigue lives. The finite element model can be used to define maximum local steady stress levels at the edge of contact. However, as previously stated a very fine mesh would be needed to get a converged solution. Alternatively, submodeling approaches can be used to determine maximum local stress levels. This is possible but computationally expensive. Furthermore, if converged stresses are achieved, there is no readily available criterion to assess them against. Maintaining maximum net section steady stress levels is considered an acceptable alternative approach. The next stage is to assess the acceptability or otherwise of potential vibration stresses. Definition of potential vibration modes and vibration amplitude levels is a fundamental input to this analysis. The definition of mode and amplitude is assumed to be known in this instance as their definition is beyond the scope of this chapter. Vibration modes are readily identified through a combination of analysis, experience, and test. Vibration amplitudes are similarly defined, and it should be recognized that the complex physics involved in their definition creates a reasonable level of uncertainty that must be catered for in the design process. Given the difficulties of determining and assessing the local stress field at the edge of contact, several approaches are listed below that offer a good level of insight into

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the acceptability or otherwise of vibration stress levels. It is suggested that at least two approaches are chosen, and a consistent interpretation of their results is sought. The choice should include wherever possible a comparison of predictions for good and bad service experiences.

5.1.6.2 Edge of contact stress prediction/fracture mechanics approaches The best prediction of local steady and vibration stresses under idealized loading conditions can be carried out and used in an integrity assessment. Through wall maximum and minimum stress distributions can be derived from finite element models that assume a nominal component geometry. The assessment by necessity should consider an idealized (simplified) flight profile. Vibration stresses for each potential mode should be derived and applied at the appropriate steady stress level and a “shakedown” stress range established using the finite element “prescribed displacement” approach is described in Section 5.1.3. Stresses can be derived by submodeling or using contact forces and bulk stresses as input to a half-space model. The assessment is required for all locations along the length of the contact zone for both the blade and the disc. Any assessment requires a prediction of the coefficient of friction on the contact surfaces. It is suggested that sensitivity analysis is carried out using assumptions that bound the range of potential friction coefficient values. Anything other than assuming uniform friction levels across the joint will complicate the analysis significantly. Once through-wall stresses have been derived, an assessment criterion from those listed in Section 5.1.5 should be chosen. The currently favored approach by the authors is the use of the proposed crack arrest method. This approach defines crack lengths at which the vibration crack propagation threshold is reached. A life to propagate to that size due to steady stresses can also be estimated. Values of what are acceptable minimum crack lengths before the vibration threshold is reached can be derived from analysis of service experience or from subcomponent test results. The vibration amplitude required to allow a crack of a prescribed size to propagate due to the applied vibration stress range can also be estimated. The asymptotic or fretting fatigue intensity approach is considered to offer some potential, but the approach requires further development before use.

5.1.6.3 Bulk or net section stresses A simpler and more robust approach, in terms of analytical procedure, is the use of net section stresses and net section stress ranges. This approach requires only a modest finite element mesh refinement and does not need a converged stress solution at the edge of contact. It uses the fact that the load balance across the section must satisfy equilibrium and the assumption that any spurious perturbations in stress near the edge of contact will only have a small effect on bulk stresses. The bulk or net section stresses for the steady loads, the bulk stress levels for “steady + vibration” and for “steady  vibration” should be derived using the approach described in Fig. 5.1.15.

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Fretting Wear and Fretting Fatigue

These values can then be plotted on a Goodman type diagram. Using good and bad service experience an envelope of good experience can be defined and used in the assessment of similar designs.

5.1.6.4 Subcomponent test Some researchers have had some success in replicating service behavior in subcomponent tests, an example being the biaxial dovetail fretting fatigue rig at the University of Oxford (Rajasekaran and Nowell, 2006). Successful tests have used near-identical dovetail geometries in a two-dimensional representation of the highest stressed section of a three-dimensional fan blade root. The tests allow steady and vibration loads to be applied simultaneously and thus can attempt to replicate wear due to global slip associated with steady loads and partial slip associated with both steady and vibration loads. The major challenges with such tests represent and maintain a three-dimensional stress field in a two-dimensional geometry, deriving an acceptable means of applying representative loads to the test apparatus and to maintain the correct load levels on each side of the dovetail throughout the test. The tests are easier to carry out on fan blade roots but become more challenging with the smaller root sections found in the compressor. The comments about the ability to establish representative wear levels made in Section 5.1.1 remain relevant to this test. Nevertheless, such tests can provide an invaluable contribution to understanding and predicting root integrity. The tests can also provide valuable data to inform decisions on the benefits of fretting fatigue palliatives such as service coatings or residual stress-inducing techniques such as shot or laser peening.

5.1.6.5 Specimen test Current specimen test databases do not lend themselves to the prediction of component fretting fatigue lives as it is very difficult to mimic real component conditions to any degree of accuracy. However, simple specimen or coupon tests do provide a good means of establishing the importance of individual and collective parameters on fretting fatigue. Representative contact and shear loads (P and Q) can be readily applied. However, applying and measuring the bending moment across the contact face is more problematic. The choice of bulk stress field is also problematic. In a specimen, the underlying bulk stress is constant through the specimen, whereas in the component the stress field varies in all dimensions. In a component, the detection of microcracking can be made much easier by applying a sufficiently large stress range such that small cracks propagate. However, this makes the interpretation of crack initiation and propagation phases more problematic. Progress continues to be made on proposals for matching component analysis and test. At the time of writing this chapter, most if not all specimen data have been used to inform analysis choice and fatigue data generation approaches on both subcomponent tests and real components. Specimen tests have provided useful qualitative data on

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fretting fatigue palliatives and for understanding the individual effect of parameters in a much-simplified loading regime. Further specimen tests are considered of academic value. Suggestions for improved analysis and specimen design include: l

l

l

l

Improving the understanding of how to match specimen and components stress fields. Investigating and understanding the combined effects of steady and vibration loads. Understanding and quantifying the effects of the bending moment across the contact surface. Understanding the combined effect of a notch stress field with an edge of contact stress field.

5.1.7

Conclusions

Significant progress has been made in recent years in the understanding and prediction of mechanical integrity in areas subject to fretting fatigue. A simple definition of fretting fatigue and associated wear mechanisms is provided in this chapter along with a summary of the progress made in mechanical integrity assessment. Areas of potential cracking due to fretting fatigue in an aerospace gas turbine engine have been highlighted. An insight into the loads and mechanisms that can result in high stresses is provided along with the challenges in providing accurate quantification and assessment. This judgment of areas that are potentially vulnerable to fretting fatigue is supported by examples of cracking in engines in service. The chapter provides a significant focus on the issue of fretting fatigue in dovetail joints. This is partially in response to service events, partially to highlight the challenges in carrying out integrity assessments of this component and the fact that this example highlights many of the areas that must be addressed in a generic fretting fatigue assessment. An insight into fretting fatigue in fir-tree joints, shaft splines, and flanges is also provided. The best method to prevent fretting fatigue is generally the reduction of stress raising mechanisms through design changes. The two main means of fretting fatigue mitigation without any fundamental design changes are coatings and surface treatments. The benefits and typical applications of these approaches are described. Section 5.1.5 provides, from an academic perspective, an assessment of the possible criteria that can be used to assess the potential for fretting fatigue. The challenges of predicting the localized local stress to a satisfactory degree of accuracy are highlighted. Two approaches are proposed as having potential for application to aerospace components. These are short crack arrest and fretting fatigue intensity or asymptotic approaches. Details are provided on the basis of these approaches and their application. Section 5.1.6 provides a view on fretting design criteria from an industrial perspective. Using the dovetail joint as a design example, several of the additional complexities of assessing real components are introduced. Advice is also provided on how to address the uncertainties that are present in the design phase. This section proposes that by combining the use of bulk and local stress assessment approaches and criteria a robust design space can be established.

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Fretting Wear and Fretting Fatigue

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Electrical connectors

5.2

Yong Hoon Janga, Ilkwang Janga, Youngwoo Parkb, and Hyeonggeun Joa a School of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea, bLG Electronics Vehicle Components USA, Troy, MI, United States

5.2.1

Introduction

The use of connectors in electrical devices has increased significantly across a wide range of industrial sectors over the past few decades. Therefore, the performance and reliability of electrical connectors in various electrical systems, including electric vehicles in the limelight, have become increasingly important. In particular, the electrical contact resistance (ECR) of electrical connectors must remain stable to ensure the stable transmission of electrical signals. However, vibration is inevitably induced by the working environment, causing displacements at the microscale and resulting in wear. The oxides accumulate as a result of fretting wear damage, eventually leading to electrical failure. In previous studies, it was reported that at least 15% of connector failures are due to wear damage (Laporte et al., 2017). In addition, connectors are subjected to macroscale reciprocating sliding owing to repetitive locking actions. Such large sliding movements may cause damage to the contact interface, which, combined with fretting sliding, affects the electrical responses. The basic condition for the occurrence of fretting is the relative interfacial movement between the objects in contact. When these movements exceed a certain amplitude, they cause slip and damage. In a previous study, it was reported that an amplitude of approximately 10
8 cm (