A Teaching Essay on Residual Stresses and Eigenstrains

By Alexander M. Korsunsky

Residual stresses are an important subject in materials science and engineering that has implications across disciplines, from quantum dots to human teeth, from aeroengines to automotive surface finishing. Although a number of monographs exist, no resource is available in the form of a book to serve as a good basis for teaching the fundamentals.

A Teaching Essay on Residual Stresses and Eigenstrains introduces eigenstrain methods as a powerful unified approach to residual stress modeling, measurement, and management. Starting with simple residual stress states, the key relationships are elucidated between deformation processes, inelastic strains (eigenstrains) these may introduce, and the resulting residual stress states. This book is written not only for the materials scientist, mechanical engineer, and student seeking to appreciate the origins of residual stress, but also for the more mature researcher and industrial engineer looking to improve their understanding of the eigenstrain approach to describing residual stress.

• Provides a unified basis for understanding the fundamentals of residual stress origins and consequences
• Introduces a classification of the most important residual stress states and their efficient description, as well as discussing measurement approaches, their limitations, and uses
• Approaches the nature and application of eigenstrain methods in a systematic way to describe residual stress fields

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 8, 2017
• ISBN: 9780128109915

Alexander M. Korsunsky

Professor Alexander Korsunsky is a world-leader in mechanical microscopy and rich tomography of materials systems and structures for the optimisation of design, durability and performance. He heads the Multi-Beam Laboratory for Engineering Microscopy (MBLEM) in the University of Oxford, and the Centre for In situ Processing Science (CIPS) in the Research Complex at Harwell Oxford. He consults Rolls-Royce plc on matters of residual stress and structural integrity, and is Editor-in-Chief of Materials & Design. In the last two decades, Alexander Korsunsky has been the most active proponent of eigenstrain theory for the analysis of inelastic deformation and residual stresses in materials and components. He teaches widely across the world, and each year gives several keynote and plenary lectures at major international conferences on engineering and materials. The broader context of Prof Korsunsky’s research interests concern improving the understanding of integrity and reliability of engineered and natural structures and systems, from high-performance metallic alloys to polycrystalline ceramics to natural hard tissues, such as human dentin and seashell nacre. He has co-authored books on fracture mechanics and elasticity and has published over 300 papers in scholarly periodicals on subjects ranging from multi-modal microscopy, neutron and synchrotron X-ray analysis, contact mechanics and structural integrity to micro-cantilever bio-sensors, size effects, and scaling transitions. Prof Korsunsky plays a leading role in the development of large-scale research facilities in the UK and Europe. He has chaired the Science Advisory Committee at Diamond Light Source, and is member of UK delegation to ESRF Council. His activities expand the range of applications of large-scale science to problems in real engineering practice. Prof Korsunsky’s research has received support from EPSRC and STFC (major UK Research Councils), the European Commission, the Royal Society, Royal Academy of Engineering (RAEng), CNRS (France), DFG (Germany), NRF (South Africa), and other international research foundations, as well as industrial partners, such as Rolls-Royce, Oxford Instruments, and Tescan-Orsay.

Book preview

A Teaching Essay on Residual Stresses and Eigenstrains

Alexander M. Korsunsky

University of Oxford

Table of Contents

Cover image

Title page

Copyright

Biography

Preface

Chapter 1. Introduction and Outline

Chapter 2. Elastic and Inelastic Deformation and Residual Stress

2.1. Deformation and Strain

2.2. Stress

2.3. Equations of Equilibrium

2.4. Formulation and Solution of Problems in Continuum Mechanics

2.5. Strain Energy Density

2.6. Contracted Notation

2.7. Elastic Isotropy

2.8. Elastic Constants

2.9. Uniform Deformation

2.10. Thermoelasticity

2.11. Plane Stress and Plane Strain

2.12. Fundamental Residual Stress Solutions and the Nuclei of Strain

2.13. The Relationship Between Residual Stresses and Eigenstrains

Chapter 3. Simple Residual Stress Systems

3.1. Additivity of Total Strain

3.2. Constrained Elastic–Plastic Bar Loaded at a Point Along Its Length

3.3. Elastoplastic Composites: Uniform Stress (Reuss) and Strain (Voigt)

3.4. On the Composite Mechanics of Polycrystals

3.5. The Ramberg–Osgood Stress–Strain Relationship

3.6. Continuum Plasticity

Chapter 4. Inelastic Bending of Beams

4.1. Slender Rods: Columns, Beams, and Shafts. Saint-Venant's Principle

4.2. Inelastic Beam Bending

4.3. Direct Problem: Residual Stress in a Plastically Bent Beam

4.4. Case: Residual Stresses due to Surface Treatment

4.5. Case: Residual Stresses in Coatings and Thin Layers

Chapter 5. Plastic Yielding of Cylinders

5.1. Inelastic Expansion of a Thick-Walled Tube

5.2. Case: Autofrettaged Tubes and Cold-Expanded Holes

5.3. Case: Quenching of a Solid Cylinder

Chapter 6. The Eigenstrain Theory of Residual Stress

6.1. Generalization

6.2. The Eigenstrain Cylinder

6.3. The Eigenstrain Sphere

6.4. Eshelby Ellipsoidal Inclusions

6.5. Nuclei of Strain

Chapter 7. Dislocations

7.1. Dislocations

7.2. Screw Dislocation: Antiplane Shear Solution

7.3. Edge Dislocation: Plane Strain Solution

7.4. Dislocation Phenomenology: Forces, Dipoles, Initiation, and Annihilation

7.5. The Dynamics of Dislocation Motion

7.6. Dislocation Dynamics Examples

Chapter 8. Residual Stress Measurement

8.1. Classification

8.2. Layer Removal and Curvature Measurement

8.3. Hole Drilling

8.4. The Contour Method

8.5. Physical Methods

8.6. Method Overview and Selection

Chapter 9. Microscale Methods of Residual Stress Evaluation

9.1. Peculiarities of Residual Stress Evaluation at the Microscale

9.2. Microfocus X-ray Diffraction Methods

9.3. Electron Diffraction Methods

9.4. Spectroscopic Methods

9.5. Introduction to FIB-DIC Microscale Residual Stress Analysis and Error Estimation

9.6. FIB-DIC Milling Geometries

9.7. Error Estimation and Propagation in FIB-DIC Residual Stress Analysis

9.8. Case: Sequential Milling FIB-DIC Micro-Ring-Core Residual Stress Analysis in a Shot-Peened Ni-Superalloy Aeroengine Turbine Blade

9.9. Parallel Milling FIB-DIC Residual Stress Analysis

9.10. Case: Stress Analysis in a Carbon Core of a SiC Fiber

Chapter 10. The Inverse Eigenstrain Method of Residual Stress Reconstruction

10.1. Fundamentals of Inverse Eigenstrain Analysis

10.2. Inverse Eigenstrain Analysis of an Inelastically Bent Beam

10.3. Inverse Eigenstrain Analysis of Welds

10.4. Inverse Eigenstrain Analysis of Laser Shock-Peened Samples

10.5. Strain Tomography and Related Problems

Chapter 11. Eigenstrain Methods in Structural Integrity Analysis

11.1. Eigenstrain Analysis in Tribology

11.2. Eigenstrain Analysis in Fracture Mechanics

Chapter 12. Conclusions and Outlook

Appendix A. The Eshelby Solution

Appendix B

Appendix C

Bibliography

Index

Copyright

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Biography

Professor Alexander Korsunsky is a world leader in mechanical microscopy and rich tomography of materials systems and structures for optimization of design, durability, and performance. He heads the Multi-Beam Laboratory for Engineering Microscopy in the University of Oxford and the Centre for In Situ Processing Science in the Research Complex at Harwell Oxford. He consults Rolls-Royce plc on matters of residual stress and structural integrity and is Editor-in-Chief of Materials & Design.

In the last two decades, Prof. Korsunsky has been the most active proponent of eigenstrain theory for the analysis of inelastic deformation and residual stresses in materials and components. He teaches widely across the world, and each year he gives several keynote and plenary lectures at major international conferences on engineering and materials. The broader context of Prof. Korsunsky's research interests concern improving the understanding of integrity and reliability of engineered and natural structures and systems, from high-performance metallic alloys to polycrystalline ceramics to natural hard tissue, such as human dentin and seashell nacre. He has coauthored books on fracture mechanics and elasticity and has published ∼300 papers in scholarly periodicals on subjects ranging from multimodal microscopy, neutron and synchrotron X-ray analysis, contact mechanics and structural integrity to microcantilever biosensors, size effects, and scaling transitions. Prof. Korsunsky plays a leading role in the development of large-scale research facilities in the United Kingdom and Europe. He has chaired the Science Advisory Committee at Diamond Light Source and is member of UK delegation to ESRF Council. His activities expand the range of applications of large-scale science to problems in real engineering practice.

Support for Prof. Korsunsky's research comes from EPSRC and STFC (major UK Research Councils) and the European Union, Royal Society, Royal Academy of Engineering, Rolls-Royce, Oxford Instruments, Tescan, NRF (South Africa), DFG (Germany), and other international research foundations.

Preface

A.S.Pushkin: On the hills of Georgia. 15 May 1829 (AMK translation in Appendix B)

V. Gavrilin, Tarantella (Ballet version: V. Vasiliev and E. Maksimova in the film Anyuta, 1982)

K. Petrov-Vodkin, Bathing of the red stallion 1912 (oil on canvas, 160 × 186 cm) Tretyakov Gallery, Moscow

I began working on residual stress analysis in the context of geomechanics and the exploration for coal, gas, and oil. This was back in the late 1980s, during my study and research practice at the Moscow Physico-Technical Institute (PhysTech) under Sergei A. Christianovich, an outstanding academician who headed the Laboratory for the Mechanics of Non-linear Media at the Institute for Problems in Mechanics, my research base (placement) at the time. He became interested in the gas contained in coal seams and sudden hazardous outbursts during mining, a fascinating problem that is both multidisciplinary and multiscale in nature. At the atomic scale, it is related to the molecules of natural gas physically adsorbed at the pore surfaces within coal seams. The strength of this attachment is highly sensitive to the confining pressure: when it is reduced as a result of mining, desorption leads to the release of gas. At the next scale-up, further course of the process is governed by the permeability and strength of the rock mass. In coal seams of low permeability, the buildup of pressure may lead to catastrophic blowout of matter and equipment. This example is an instance of a physicochemically nonlinear system that undergoes significant structural change as a function of residual stress. The change may be sudden and catastrophic, as in the case of coal seam gas outburst, or directed and intended, as in the case of shale gas extraction.

The influence of residual stresses on structural integrity is often similarly drastic and can make the difference between failure and safety. Interestingly, the effect may be particularly strong in cases of short-duration or long-term loading, i.e., under extremes of loading time dependence, such as impact or creep-fatigue. This, in turn, can be understood in terms of the particular significance of tensile stress for damage accumulation and fracture within solid materials. The importance of residual stress control during manufacture has been recognized in engineering design practice and manifests itself in a wide range of ingenious technology solutions employed across the spectrum of applications and material systems: metallic alloys, polymers and composites, ceramics and construction materials, thin films and coatings, semiconductors used in electronics applications, and systems such as armor and spacecraft.

In common with many aspects of engineering, development of ideas and methods in the field often proceeded by trial and error, through identifying treatment conditions that have deleterious or positive effects, and optimizing them by means of trial and error. Although the advent of computers meant a ramp change in the access to numerical modeling tools and versatile characterization equipment, the conventional approach seems to have persisted until now. At least two reasons may be identified why this is the case: (1) the challenge of measuring the complex residual stress tensor at the right location, with the required spatial resolution, preferably in a nondestructive fashion, and (2) the difficulty of interpreting the results in way that is compatible with modern simulation tools used for the determination of safe design life of components and assemblies.

I think this book is timely, because the early decades of the 21st century saw the accumulation of insights, theoretical and experimental approaches, and techniques that led to a step change in our understanding of residual stress states, their origins, nature, and evolution under thermal, mechanical, and other loading conditions. It is timely to introduce some pivotal terms that arose in the context of this subject, such as eigenstrain theory, mechanical microscopy, and residual stress engineering, to describe specific branches in the system of approaches aimed at systematic description, classification, measurement, modeling, and control of residual stresses.

In my opinion, in this context the theory of eigenstrains and their relationship with residual stresses must play a central role. Over the years I have used the opportunities to teach this subject to different cohorts of undergraduate and postgraduate students: at Oxford, at the National University of Singapore, at Ecole Nationale Supérieure d'Ingénieurs de Caen in Normandy, France. Frequently the students approached me for advice on the choice of a good book that would introduce the concepts and methods that I presented to them in my lectures. Although a number of good volumes exist on the subject, including some collections of contributed articles from prominent researchers in the field, I felt that there was not a text available that would combine an exposition of fundamental ideas with some projections onto particular application fields. After a while, my answer came to be that such book is still being written—so eventually I had to make an effort and express my perception of the subject in the form of this script, which I refer to as a teaching essay. By that I want to indicate that it does not presume to be complete but nevertheless seeks to encapsulate in one narration what I consider to be the key elements of knowledge that someone entering the subject would wish to have at their disposal. In the modern day, information is becoming ever more readily available through online resources. The ease of searching for keywords means that, on the one hand, a plethora of articles can be found in seconds, but on the other, one can easily get lost in this sea of detailed studies that may obscure the overall picture. I think that the purpose of an essay like this is to pull together into a coherent narrative various pointers to specific results, so that the reader can follow the threads that they find important and relevant to their particular purpose.

Even with the various references I include, I do not consider this essay exhaustive or even complete in any way—it is a snapshot of a work in progress. For example, many miniature modeling tools that I have written in Matlab or Mathematica over the years (and that were the source of some of the illustrations provided) always remain in a state of flux—because new ideas emerge and also because software providers keep changing the underlying functions, which makes old versions stop working. My choice in preparing this report has been to make reference to these tools and give an indication of their functionality: for some readers this will be enough to enable them to write their own code, others may choose to get in touch to ask for help or advice, and then may be able to complete writing something that I never got round to finishing yet! Alternatively, questions at the end of some sections may ask for analysis that is best accomplished by putting together some code. I do not presume to aspire to the level of Landau and Lifschitz physics textbook, some exercises in which, if done properly, to this day may give a diligent student enough material to publish an article.

It is logical and inevitable that the narrative in this essay combines and links the theoretical and numerical modeling frameworks with the experimental approaches for residual stress evaluation. It may be argued that the history of technology is a story of man finding ways to augment physical and mental strengths with ever more complex handy tools, from crowbar to computer.¹ For my research group, throughout the years the tight link between modeling and experimentation has become modus operandi, perhaps even raison d'être! As will become clear to the reader, the cause of this lies in the very definition of stress (including residual stress), which relies on a mind experiment involving severing internal material bonds and replacing them with equivalent distributed forces, something we can never hope to achieve practically but may only continue to come up with new, ever more ingenious ways of approximating.² For this reason I am adamantly against using the phrase stress measurement, and much prefer stress evaluation: some sort of a model is always required to convert observable parameters into stress, the mental concept that we find so convenient as a vehicle for thinking about the limits of structural integrity and the conditions of failure.

Throughout these years, teaching has been an integral and important part of my activity. My teaching style has evolved, and I found myself progressively devoting more time to trying to educate by drawing similarities, pointing out connections with something already familiar to the listeners, often from other walks of life that may lie as far afield as philosophy, literature, art, music. I also found that during periods of working on a particular aspect of the subject of residual stresses I was accompanied by some particular image or tune, or even a poem. Pointers to these snippets appear at the start of each chapter. I hope the reader will be lenient with me in cases when the connection appears tenuous—often it is circumstantial or coincidental, but I hope that those references might prove amusing for those who wish to follow them and wonder about the link and relevance.

I must conclude this Preface with an expression of gratitude to the wonderful people who were by my side on this journey and without whom I would not know how to go on. My wife Tanya is by far the cleverest person I know—she not only taught me a lot but also always supported me with her inexhaustible reserve of patience, strength, wisdom, and generosity. I am indebted to the members of my research group over the years—too many to mention, truly, but all of whom I remember, and care for each and every one, who in their unique and different ways contributed to the development of my understanding and vision, challenged my views, and focused my mind on the connections that I might have overlooked otherwise.

Alexander M. Korsunsky

Oxford, January 2017

---

¹ Put shortly, If experiment is an extension of our hands, then modeling must be an extension of our mind.

² Put shortly, Experiment is the art of the possible.

Chapter 1

Introduction and Outline

Abstract

This