Physical Basis of Plasticity in Solids

275 Pages·3.478 MB·English

by  Toledano J.C.

Physical Basis of
Plasticity in Solids
8295.9789814374057-tp.indd 1 11/22/11 3:09 PM
Physical Basis of
Plasticity in Solids
Jean-Claude Tolédano
Ecole Polytechnique, France
World Scientific
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PHYSICAL BASIS OF PLASTICITY IN SOLIDS
Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-4374-05-7
ISBN-10 981-4374-05-9
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October14,2011 13:37 WorldScientificBook-9inx6in master
Preface
This book is based on a physics course first launched in 2002 for the stu-
dentsatEcolePolytechnique,aFrenchscientificinstitutionaccessibleafter
two years of university-level studies. The course was attended by students
majoring in a second year program(corresponding to a fourth year in uni-
versity) entitled “Structure Mechanics”, and intending to specialize, after
their graduation from Ecole Polytechnique, in various fields such as civil
engineering, aeronautics, mechanical engineering, nuclear engineering, or
research in the field of materials science. The program included a set of
courses onthe mechanicalproperties of solidstate materials,among which
was an introduction to the methods used in the mechanical sciences to
analyze the plastic deformation of solids or their fracture.
The physics course introduces the physical mechanism of the plastic
deformation, that relies essentially on the occurence and motion of dislo-
cations. Three of the eight chapters of this book are devoted to the study
of these linear defects of crystalline solids. Two other chapters provide the
elements of crystallography and mechanics required to discuss their prop-
erties. One chapter describes other types of crystal defects with which the
dislocations interact. The main chapter follows, devotedto the description
of the mechanism of plastic deformation, based on the preceding elements.
The appendices contain over thirty exercises and their solutions, which
were used during classes or written examinations. Many of these exercises
are adapted from texts found in various standard textbooks, as quoted in
Chapter 1.
On a topic involving many different aspects and a great variety of ma-
terials behaviour, we had to elaborate to eight lectures, each of duration
of an hour and a half, and as many two-hour exercise classes. One had to
thereforerestricttothemainideas,andshowtheirrelevanceininterpreting
phenomena well known to everyone (Why certain metals are harder than
v
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vi Preface
others? Why heating metals makes them more compliant, etc...). It was
also necessary to describe the methods used in calculating the properties
of dislocations, and train students to practice these calculations.
It appeared that a good understanding of the subject, by the students,
could be obtained by the following organization: Chapters 1 and 2 (one
course), Chapters 3 and 4 (one course), Chapter 5 (two courses), Chapter
6 (one course), Chapter 7 (one course), Chapter 8 (two courses).
Itisworthmentioningthattheinitialinspirationofthiscourseisrooted
in earlierones on the same matter givento the students of Ecole Polytech-
nique. Yves Qu´er´e, had first taught the subject in 1989. His course was
based ontwo chapters of his book “Physiquedes Mat´eriaux”( Ellipses ed-
itor 1988, English translation “Physics of Materials” Gordon and Breach
1998). TheteachingofthesubjectwascontinuedbyDenisGratias,between
1994and2002,relyingonhis lecturenotes“Introduction`alaPhysiquedes
Mat´eriaux”(unpublished). Thesetwotextscoveralargersetoftopicsthan
the present book. In particular, they devote substantial attention to the
metallurgical phase changes and to the mechanism of fracture.
At the end of Chapter 1, we have listed a few reference books, writ-
ten in French or in English, and discussing more thoroughly the subjects
considered in this book. Most of these references were written during the
years 1960 1970 (with, sometimes, revised versions published later), that
−
is,towardstheendoftheperiodofclarificationofthecentralideasrelative
to dislocations and to their role in the mechanism of plasticity. With the
exceptionoftheabovementionedworks,byYvesQu´er´eandDenisGratias,
andofthetreatisebyD.HullindicatedattheendofChapter1,mostlisted
books are rather aimed at researchers than at undergraduate students.
CollaboratingwithDenisGratiasduringhisteachingofthissubjectbe-
tween 2000 and 2002 was invaluable for me to outline the composition of
this course. I am indebted, to severalcolleagues with whom I have taught,
for their useful remarks, namely Alain Barbu, Fran¸cois Beuneu, Yann Le
Bouar, and Jean-Eric Wegrowe. I am grateful to my colleagues from the
Department of Mechanics, Dominique Barthes-Biesel, Pierre Suquet, and
Andr´eZaouifor their friendly welcome to their “structuremechanics”pro-
gram. Finally, I would like to thank Jean-Paul Coard for encouraging me
to write the French edition of this course.
Jean-Claude Tol´edano
Paris, April 2011
October14,2011 13:37 WorldScientificBook-9inx6in master
Contents
Preface v
1. Introduction 1
1.1 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Mechanical properties of solids . . . . . . . . . . . 2
1.1.2 Microscopic mechanisms . . . . . . . . . . . . . . 4
1.2 Organizationand contents of the chapters . . . . . . . . . 9
1.3 General References . . . . . . . . . . . . . . . . . . . . . . 11
2. The structure of crystalline solids 13
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Crystal geometry . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Ideal crystal . . . . . . . . . . . . . . . . . . . . . 14
2.3 Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Unit cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Primitive unit cells . . . . . . . . . . . . . . . . . 22
2.4.2 Conventional unit cells . . . . . . . . . . . . . . . 22
2.4.3 Classification of the Bravais lattices.
Cubic lattices . . . . . . . . . . . . . . . . . . . . 24
2.5 Examples of crystal structures . . . . . . . . . . . . . . . 26
2.5.1 Simple monoatomic structure packings . . . . . . 26
2.5.2 Physical realizations in metals . . . . . . . . . . . 31
2.5.3 Simple covalent structures . . . . . . . . . . . . . 34
2.6 Non-crystalline solids. . . . . . . . . . . . . . . . . . . . . 35
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3. Mechanics of deformable solids 39
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Fundamental tensors . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Strain and stress . . . . . . . . . . . . . . . . . . . 39
3.2.2 Stiffness . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Coordinate changes. . . . . . . . . . . . . . . . . . . . . . 42
3.4 Stiffness tensor and crystal symmetry . . . . . . . . . . . 44
3.4.1 General constraints . . . . . . . . . . . . . . . . . 44
3.4.2 Crystal symmetry . . . . . . . . . . . . . . . . . . 45
3.4.3 Mathematical transformation of tensors . . . . . . 45
3.5 Isotropic solids . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 Stiffness tensor . . . . . . . . . . . . . . . . . . . . 47
3.5.2 Basic equations . . . . . . . . . . . . . . . . . . . 48
4. Vacancies, an example of point defects in crystals 51
4.1 Classification of defects in crystals . . . . . . . . . . . . . 51
4.2 Stability of point-defects in solids . . . . . . . . . . . . . . 55
4.2.1 Statistical equilibrium . . . . . . . . . . . . . . . . 55
4.2.2 Concentration of defects at thermal equilibrium . 57
4.3 Formation of vacancies . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Formation energy . . . . . . . . . . . . . . . . . . 59
4.3.2 Random displacement of vacancies, diffusion . . . 64
5. The geometry of dislocations 73
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Straight edge dislocation . . . . . . . . . . . . . . . . . . . 75
5.2.1 Hypothetical procedures of formation . . . . . . . 75
5.2.2 Burgers circuit and Burgers vector . . . . . . . . . 78
5.2.3 Edge dislocation loops . . . . . . . . . . . . . . . 82
5.3 Other types of dislocations . . . . . . . . . . . . . . . . . 84
5.3.1 Screw dislocation . . . . . . . . . . . . . . . . . . 84
5.3.2 Mixed dislocation-loops . . . . . . . . . . . . . . . 86
5.3.3 General properties of the Burgers vector . . . . . 88
5.4 Volterra process of formation . . . . . . . . . . . . . . . . 89
5.4.1 Edge and screw dislocations . . . . . . . . . . . . 90
5.4.2 General case . . . . . . . . . . . . . . . . . . . . . 93
5.5 Observation of dislocations . . . . . . . . . . . . . . . . . 94
5.5.1 Reflexion of electrons by a crystal . . . . . . . . . 95
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Contents ix
5.5.2 How can a dislocation be observed? . . . . . . . . 97
5.5.3 Lattice planes and reciprocal lattice . . . . . . . . 99
5.5.4 Diffraction of electron beams . . . . . . . . . . . . 102
6. Strain field of dislocations 105
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Strain and stress fields . . . . . . . . . . . . . . . . . . . . 106
6.2.1 Screw dislocation . . . . . . . . . . . . . . . . . . 107
6.2.2 Straight edge dislocation . . . . . . . . . . . . . . 110
6.2.3 Mixed straight dislocation . . . . . . . . . . . . . 113
6.2.4 Elastic energy of a dislocation loop . . . . . . . . 113
6.3 Action of a stress on a dislocation . . . . . . . . . . . . . 114
6.3.1 Effective force applied to a dislocation. . . . . . . 114
6.3.2 Peach and Koehler formula . . . . . . . . . . . . . 115
6.4 Line tension of a dislocation . . . . . . . . . . . . . . . . . 117
6.5 Interaction between dislocations . . . . . . . . . . . . . . 120
6.5.1 Qualitative study . . . . . . . . . . . . . . . . . . 120
6.5.2 Quantitative study . . . . . . . . . . . . . . . . . 123
6.6 Interaction between a dislocation and a vacancy. . . . . . 125
7. Interactions with the lattice 127
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Core structure of an edge dislocation . . . . . . . . . . . . 128
7.2.1 Lattice planes coordinates . . . . . . . . . . . . . 129
7.2.2 Components of the core energy. . . . . . . . . . . 131
7.2.3 Determination of the core width . . . . . . . . . 134
7.3 Peierls-Nabarrostress . . . . . . . . . . . . . . . . . . . . 136
7.3.1 Gliding of a dislocation . . . . . . . . . . . . . . . 136
7.3.2 Principle of calculation of the
Peierls-Nabarrostress . . . . . . . . . . . . . . . . 138
7.4 Dissociation of the dislocation core . . . . . . . . . . . . . 139
8. Microscopic mechanism of plasticity 143
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 Plastic deformation and local shear . . . . . . . . . . . . . 145
8.2.1 Amplitude of the slips . . . . . . . . . . . . . . . . 145
8.2.2 Resolved shear stress . . . . . . . . . . . . . . . . 145
8.2.3 Crystallographic characteristics of the slips . . . . 147
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8.3 Elementary slip and yield strength . . . . . . . . . . . . . 149
8.3.1 Edge dislocations . . . . . . . . . . . . . . . . . . 149
8.3.2 Screw dislocations . . . . . . . . . . . . . . . . . . 150
8.3.3 Upper limit of the yield strength . . . . . . . . . . 152
8.3.4 Slip assisted by dislocation motion . . . . . . . . . 152
8.3.5 Number of dislocations . . . . . . . . . . . . . . . 153
8.4 Dislocation sources . . . . . . . . . . . . . . . . . . . . . . 154
8.4.1 Change of configuration of dislocations . . . . . . 154
8.4.2 Frank-Read source. . . . . . . . . . . . . . . . . . 155
8.4.3 Spiral source . . . . . . . . . . . . . . . . . . . . . 158
8.5 Dislocation glide and climb . . . . . . . . . . . . . . . . . 158
8.6 Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.6.1 Pinning dislocations . . . . . . . . . . . . . . . . . 160
8.6.2 Hardening mechanism . . . . . . . . . . . . . . . . 166
8.6.3 Unpinning of the motion of dislocations . . . . . . 167
8.7 Effect of temperature. . . . . . . . . . . . . . . . . . . . . 169
8.8 Deformation rate . . . . . . . . . . . . . . . . . . . . . . . 172
8.8.1 Orowan formula . . . . . . . . . . . . . . . . . . . 172
8.8.2 Dislocation velocity . . . . . . . . . . . . . . . . . 173
8.9 Origin of the diversity of plastic behaviours . . . . . . . . 174
8.9.1 Easy-glide planes; glide systems . . . . . . . . . . 174
8.9.2 Metals with an FCC or HC structure . . . . . . . 174
8.9.3 Other solids . . . . . . . . . . . . . . . . . . . . . 177
Appendix A Exercises 181
A.1 Structure of crystalline solids (chap. 2). . . . . . . . . . . 181
A.2 Dislocations, Burgers vector (chap. 5) . . . . . . . . . . . 187
A.3 Deformations and forces (chap. 6) . . . . . . . . . . . . . 193
A.4 Core of a dislocation (chap. 7) . . . . . . . . . . . . . . . 199
A.5 Mechanism of plasticity (chap. 8). . . . . . . . . . . . . . 204
Appendix B Solutions to Exercises 217
Index 267