A Teaching Essay on Residual Stresses and Eigenstrains

Alexander M. Korsunsky

• Publisher: Butterworth-Heinemann
• Year: 2017
• ISBN:
  9780128109915
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

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

Chapter 1

Introduction and Outline

Abstract

This chapter introduces the fundamental principles and approaches that underlie the modeling and experimental approaches to residual stress analysis contained in the book. The first equation to appear in the text represents the decomposition of total strain into elastic and inelastic parts, and with its help the term eigenstrain is explained at the conceptual level, along with its significance as the source of residual stress. Experimental methods of residual stress evaluation are mentioned briefly, and the importance of diffraction techniques is highlighted. The content of the book is then outlined briefly, chapter by chapter.

Keywords

Diffraction; Eigenstrain; Residual stress; Strain decomposition

C.P. Cavafy: Ithaka. Collected Poems 1992

J. Haydn, Piano Concerto no. 11 in D major Igor Levit and Deutsche Kammerphilarmonie Bremen: 17 May 2015

Vassily Kandinsky, Black and Violet (oil on canvas, 78 × 100 cm) 1923 Private collection

When a solid is subjected to external loading, mechanical, thermal, or other, it changes its shape and dimensions. These geometrical changes, described quantitatively by displacements and strains, are the subject of kinematics. External loads are transmitted within the material via forces acting between elements across imaginary sections and interfaces. These interactions are quantitatively described by stresses (components of force per unit area) and form the subject of the disciplines of statics or dynamics. The kinematical and dynamical descriptions of solid deformation are linked by the so-called constitutive law that establishes the relationship between stresses and strains (and/or their derivatives and integrals) and captures the peculiarity of a particular material's deformation behavior.

If a solid recovers its original shape and dimensions fully after the removal of the external load, then the deformation (and material) is called elastic; otherwise, inelastic. If the relationship between strains and stresses in an elastic material is described by a simple linear proportion, then the material is called linear elastic; otherwise, nonlinear elastic. If a material does not recover its original shape, then it is said to have undergone inelastic deformation. In specific cases, the terms plastic strain, creep strain, misfit strain, transformation strain, and eigenstrain are used to refer to the permanent inelastic strain.

As this is the first time we encounter the term eigenstrain, it is worth introducing the associated notation, ε^∗, and recalling that the term was introduced by Toshio Mura (1987) in his book, together with “eigenstress” that he defined as “a generic name given to self-equilibrated internal stresses caused by one or several of these eigenstrains in bodies which are free from any other external force and surface constraint.” He points out that “eigenstress fields are created by the incompatibility of the eigenstrains,” and adds that “engineers have used the term ‘residual stresses’ for the self-equilibrated internal stresses when they remain in materials after fabrication or plastic deformation.” Below we flesh out the implications of the aforementioned definitions by discussing the relationships, form of equations, and problem formulations.

Total strain is additive: it is given by the sum of elastic and inelastic parts:

εij=eij+εij∗.

(1.1)

Total strain must satisfy the compatibility condition: it corresponds to a unique continuous deformation field.

After elastic loading and unloading of a stress-free body all strains return to zero, as do the stresses: no residual stress arises. Hence, inelastic deformation is a necessary (but not sufficient!) condition for the creation of residual stresses. Residual stresses arise as a body's response to permanent inelastic strains introduced during the loading–unloading cycle. Therefore, stresses exist only in association with elastic strains: there is no such thing as plastic stress. Stresses must satisfy the condition of static equilibrium.

Permanent inelastic strains (eigenstrains) are the source of residual stresses, not the other way round. If eigenstrain is introduced into a body, it responds by developing accommodating elastic strains, which in turn give rise to stresses. The direct problem of eigenstrain residual stress analysis is the determination of elastic strains and residual stresses from a given distribution of eigenstrains. The inverse problem of eigenstrain residual stress analysis is the problem of finding the eigenstrain distribution that gives rise to the (measured or partially known) distributions of residual elastic strains and/or residual stresses.

Residual stresses cannot be measured directly, nor can, indeed, any stress: stress is an imaginary tool used for convenience of description. Instead, other quantities related to residual stresses are measured, usually some displacements or strains, strain increments, or some indirectly related physical quantities, e.g., magnetization. Experimental methods differ in the level of accuracy and detail with which the residual stresses can be assessed: some are only qualitative, whereas others provide highly detailed information that is resolved in terms of spatial location and orientation.

Diffraction provides a uniquely powerful method of evaluating lattice parameters, from which crystal strains can be readily deduced. Diffraction utilizes the simple relationship provided by Bragg's law between the interplanar lattice spacing parameter d, the wavelength λ (which can be expressed through photon energy E and universal constants h and c), and the scattering angle 2θ:

2dsinθ=λ=hcE.

(1.2)

The lattice spacing d is found experimentally by fixing one of the two parameters E or θ, and varying the other one to find a peak. Consequently, two principal experimental diffraction methods are in use, the angle dispersive (monochromatic, λ = const) or energy dispersive (white beam, 2θ = const). Highly detailed information can be extracted by simultaneous interpretation of multiple diffraction peaks, e.g., the strain average across all peaks provides information about the equivalent macroscopic strain, whereas the differences reflect the inhomogeneity of deformation in terms of the interaction, and load and strain transfer between crystallites of different orientation.

Solid objects are very rarely (in fact, almost never) the idealized homogeneous and isotropic continua studied in the classical mechanics of deformable solids. Real materials consist of distinct grains and phases, e.g., polycrystals that are made up of constituent grains whose mechanical properties depend on orientation. Loading applied to such “composites” induces inhomogeneous deformation: some regions experience greater strain or stress than others; some may yield, whereas others remain elastic. As a consequence of this inhomogeneity, after unloading, some microscopic intergranular residual stresses arise.

Within individual grains inelastic deformation is often mediated by crystallographic defects, such as dislocations. Dislocations appear and move under the applied shear stress; they exert forces upon each other and can organize themselves into more or less stable arrangements, such as persistent slip bands, pile-ups, ladders, and cell-wall structures. Dislocations can escape through the material free surface, leaving behind steps, and pairs of dislocations of opposite sign can annihilate. Defects such as dislocations give rise to residual stresses at the intragranular level.

This book contains an introductory course devoted to the analysis of residual stresses and their interpretation and understanding with the help of the concept and modeling framework of eigenstrain. The following order of presentation is adopted. After a brief introduction to the subject in this chapter, the continuum mechanics fundamentals for the description of deformation and stresses are given in Chapter 2. Some basic types of inelastic deformation and residual stress states are defined and analyzed: the consideration of constraint and stress balance conditions allows the introduction of the simple (i.e., 0-D, or “point”) residual stress states in Chapter 3. In Chapters 4 and 5, the important case of one-dimensional residual stress states is addressed, using the examples of inelastic beam bending and inelastic expansion of a hollow tube under internal pressure.

Chapter 6 is devoted to the eigenstrain theory of residual stress. First, the procedure for the solution of the direct problem of eigenstrain is illustrated using the examples of “shrink-fit” eigenstrain cylinder and eigenstrain sphere. The important Eshelby solution for the eigenstrain ellipsoid is introduced next, followed by the concept of nuclei of strain as the sources of residual stress concentrated to a point.

Chapter 7 is devo...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Biography
• Preface
• Chapter 1. Introduction and Outline
• Chapter 2. Elastic and Inelastic Deformation and Residual Stress
• Chapter 3. Simple Residual Stress Systems
• Chapter 4. Inelastic Bending of Beams
• Chapter 5. Plastic Yielding of Cylinders
• Chapter 6. The Eigenstrain Theory of Residual Stress
• Chapter 7. Dislocations
• Chapter 8. Residual Stress “Measurement”
• Chapter 9. Microscale Methods of Residual Stress Evaluation
• Chapter 10. The Inverse Eigenstrain Method of Residual Stress Reconstruction
• Chapter 11. Eigenstrain Methods in Structural Integrity Analysis
• Chapter 12. Conclusions and Outlook
• Appendix A. The Eshelby Solution
• Appendix B
• Appendix C
• Bibliography
• Index