Table Of ContentPLASTICITY
THEORY
Revised Edition (PDF)
Jacob Lubliner
University of California at Berkeley
Copyright 1990, 2006 by Jacob Lubliner
This book was previously published by Pearson Education, Inc.
Preface
WhenIfirstbegantoplanthisbook,IthoughtthatIwouldbeginthepreface
with the words “The purpose of this little book is...” While I never lost my
beliefthatsmallisbeautiful,Idiscoveredthatitisimpossibletoputtogether
a treatment of a field as vast as plasticity theory between the covers of a
truly “little” book and still hope that it will be reasonably comprehensive.
I have long felt that a modern book on the subject — one that would be
usefulasaprimaryreferenceand,moreimportantly,asatextbookinagrad-
uate course (such as the one that my colleague Jim Kelly and I have been
teaching) — should incorporate modern treatments of constitutive theory
(including thermodynamics and internal variables), large-deformation plas-
ticity, and dynamic plasticity. By no coincidence, it is precisely these topics
— rather than the traditional study of elastic-plastic boundary-value prob-
lems, slip-line theory and limit analysis — that have been the subject of
my own research in plasticity theory. I also feel that a basic treatment of
plasticity theory should contain at least introductions to the physical foun-
dations of plasticity (and not only that of metals) and to numerical methods
— subjects in which I am not an expert.
I found it quite frustrating that no book in print came even close to
adequately covering all these topics. Out of necessity, I began to prepare
class notes to supplement readings from various available sources. With
the aid of contemporary word-processing technology, the class notes came
to resemble book chapters, prompting some students and colleagues to ask,
“Why don’t you write a book?” It was these queries that gave me the
idea of composing a “little” book that would discuss both the topics that
are omitted from most extant books and, for the sake of completeness, the
conventional topics as well.
Almosttwoyearshavepassed,andsome1.2megabytesofdiskspacehave
been filled, resulting in over 400 pages of print. Naively perhaps, I still hope
that the reader approaches this overgrown volume as though it were a little
book: it must not be expected, despite my efforts to make it comprehensive,
to be exhaustive, especially in the sections dealing with applications; I have
preferred to discuss just enough problems to highlight various facets of any
topic. Some oft-treated topics, such as rotating disks, are not touched at
iii
iv Preface
all, nor are such general areas of current interest as micromechanics (except
on the elementary, qualitative level of dislocation theory), damage mechan-
ics (except for a presentation of the general framework of internal-variable
modeling), or fracture mechanics. I had to stop somewhere, didn’t I?
The book is organized in eight chapters, covering major subject areas;
the chapters are divided into sections, and the sections into topical subsec-
tions. Almost every section is followed by a number of exercises. The order
ofpresentationoftheareasissomewhatarbitrary. Itisbasedontheorderin
which I have chosen to teach the field, and may easily be criticized by those
partial to a different order. It may seem awkward, for example, that consti-
tutive theory, both elastic and inelastic, is introduced in Chapter 1 (which
is a general introduction to continuum thermomechanics), interrupted for a
surveyofthephysicsofplasticityasgiveninChapter2,andreturnedtowith
specific attention to viscoplasticity and (finally!) rate-independent plasticity
in Chapter 3; this chapter contains the theory of yield criteria, flow rules,
and hardening rules, as well as uniqueness theorems, extremum and varia-
tional principles, and limit-analysis and shakedown theorems. I believe that
the book’s structure and style are sufficiently loose to permit some juggling
of the material; to continue the example, the material of Chapter 2 may be
taken up at some other point, if at all.
The book may also be criticized for devoting too many pages to con-
cepts of physics and constitutive theory that are far more general than the
conventional constitutive models that are actually used in the chapters pre-
senting applications. My defense against such criticisms is this: I believe
that the physics of plasticity and constitutive modeling are in themselves
highly interesting topics on which a great deal of contemporary research is
done, and which deserve to be introduced for their own sake even if their
applicability to the solution of problems (except by means of high-powered
numerical methods) is limited by their complexity.
Another criticism that may, with some justification, be leveled is that
the general formulation of continuum mechanics, valid for large as well as
smalldeformationsandrotations,ispresentedasaseparatetopicinChapter
8, at the end of the book rather than at the beginning. It would indeed
be more elegant to begin with the most general presentation and then to
specialize. The choice I finally made was motivated by two factors. One is
that most of the theory and applications that form the bulk of the book can
beexpressedquiteadequatelywithinthesmall-deformationframework. The
otherfactorispedagogical: itappearstome, onthebasisoflongexperience,
that most students feel overwhelmed if the new concepts appearing in large-
deformation continuum mechanics were thrown at them too soon.
Much of the material of Chapter 1 — including the mathematical fun-
damentals, in particular tensor algebra and analysis — would normally be
covered in a basic course in continuum mechanics at the advanced under-
Preface v
graduate or first-year graduate level of a North American university. I have
included it in order to make the book more or less self-contained, and while
I might have relegated this material to an appendix (as many authors have
done), I chose to put it at the beginning, if only in order to establish a con-
sistent set of notations at the outset. For more sophisticated students, this
material may serve the purpose of review, and they may well study Section
8.1 along with Sections 1.2 and 1.3, and Section 8.2 along with Sections 1.4
and 1.5.
The core of the book, consisting of Chapters 4, 5, and 6, is devoted to
classicalquasi-staticproblemsofrate-independentplasticitytheory. Chapter
4containsaselectionofproblemsincontainedplasticdeformation(orelastic-
plastic problems) for which analytical solutions have been found: some ele-
mentaryproblems,andthoseoftorsion,thethick-walledsphereandcylinder,
and bending. The last section, 4.5, is an introduction to numerical methods
(although the underlying concepts of discretization are already introduced
in Chapter 1). For the sake of completeness, numerical methods for both
viscoplastic and (rate-independent) plastic solids are discussed, since nu-
merical schemes based on viscoplasticity have been found effective in solving
elastic-plastic problems. Those who are already familiar with the material
of Sections 8.1 and 8.2 may study Section 8.3, which deals with numerical
methods in large-deformation plasticity, immediately following Section 4.5.
Chapters 5 and 6 deal with problems in plastic flow and collapse. Chap-
ter 5 contains some theory and some “exact” solutions: Section 5.1 covers
the general theory of plane plastic flow and some of its applications, and
Section 5.2 the general theory of plates and the collapse of axisymmetrically
loaded circular plates. Section 5.3 deals with plastic buckling; its placement
in this chapter may well be considered arbitrary, but it seems appropriate,
since buckling may be regarded as another form of collapse. Chapter 6 con-
tains applications of limit analysis to plane problems (including those of soil
mechanics), beams and framed structures, and plates and shells.
Chapter 7 is an introduction to dynamic plasticity. It deals both with
problemsinthedynamicloadingofelastic–perfectlyplasticstructurestreated
by an extension of limit analysis, and with wave-propagation problems, one-
dimensional (with the significance of rate dependence explicitly discussed)
and three-dimensional. The content of Chapter 8 has already been men-
tioned.
As the knowledgeable reader may see from the foregoing survey, a coher-
entcoursemaybebuiltinvariouswaysbyputtingtogetherselectedportions
of the book. Any recommendation on my part would only betray my own
prejudices, and therefore I will refrain from making one. My hope is that
thosewhoseorientationandinterestsaredifferentfromminewillnonetheless
find this would-be “little book” useful.
In shaping the book I was greatly helped by comments from some out-
vi Preface
standing mechanicians who took the trouble to read the book in draft form,
and to whom I owe a debt of thanks: Lallit Anand (M. I. T.), Satya Atluri
(Georgia Tech), Maciej Bieniek (Columbia), Michael Ortiz (Brown), and
Gerald Wempner (Georgia Tech).
An immeasurable amount of help, as well as most of the inspiration to
write the book, came from my students, current and past. There are too
many to cite by name — may they forgive me — but I cannot leave out Vas-
silis Panoskaltsis, who was especially helpful in the writing of the sections
on numerical methods (including some sample computations) and who sug-
gested useful improvements throughout the book, even the correct spelling
of the classical Greek verb from which the word “plasticity” is derived.
Finally, I wish to acknowledge Barbara Zeiders, whose thoroughly pro-
fessional copy editing helped unify the book’s style, and Rachel Lerner
and Harry Sices, whose meticulous proofreading found some needles in the
haystack that might have stung the unwary. Needless to say, the ultimate
responsibility for any remaining lapses is no one’s but mine.
A note on cross-referencing: any reference to a number such as 3.2.1,
without parentheses, is to a subsection; with parentheses, such as (4.3.4), it
is to an equation.
Addendum: Revised Edition
Despite the proofreaders’ efforts and mine, the printed edition remained
plagued with numerous errors. In the fifteen years that have passed I have
managed to find lots of them, perhaps most if not all. I have also found it
necessary to redo all the figures. The result is this revised edition.
Contents
Chapter 1: Introduction to Continuum Thermome-
chanics
Section 1.1 Mathematical Fundamentals
1
1.1.1 Notation 1
1.1.2 Cartesian Tensors 3
1.1.3 Vector and Tensor Calculus 6
1.1.4 Curvilinear Coordinates 9
Section 1.2 Continuum Deformation
14
1.2.1 Displacement 14
1.2.2 Strain 15
1.2.3 Principal Strains 21
1.2.4 Compatibility Conditions 23
Section 1.3 Mechanics of Continuous Bodies
26
1.3.1 Introduction 26
1.3.2 Stress 29
1.3.3 Mohr’s Circle 33
1.3.4 Plane Stress 35
1.3.5 Boundary-Value Problems 36
Section 1.4 Constitutive Relations: Elastic
44
1.4.1 Energy and Thermoelasticity 44
1.4.2 Linear Elasticity 49
1.4.3 Energy Principles 54
vii
viii Contents
Section 1.5 Constitutive Relations: Inelastic
59
1.5.1 Inelasticity 59
1.5.2 Linear Viscoelasticity 61
1.5.3 Internal Variables: General Theory 65
1.5.4 Flow Law and Flow Potential 69
Chapter 2: The Physics of Plasticity
Section 2.1 Phenomenology of Plastic Deformation
75
2.1.1 Experimental Stress-Strain Relations 76
2.1.2 Plastic Deformation 80
2.1.3 Temperature and Rate Dependence 85
Section 2.2 Crystal Plasticity
89
2.2.1 Crystals and Slip 89
2.2.2 Dislocations and Crystal Plasticity 94
2.2.3 Dislocation Models of Plastic Phenomena 100
Section 2.3 Plasticity of Soils, Rocks and Concrete
103
2.3.1 Plasticity of Soil 104
2.3.2 “Plasticity” of Rock and Concrete 108
Chapter 3: Constitutive Theory
Section 3.1 Viscoplasticity
111
3.1.1 Internal-Variable Theory of Viscoplasticity 111
3.1.2 Transition to Rate-Independent Plasticity 116
3.1.3 Viscoplasticity Without a Yield Surface 118
Section 3.2 Rate-Independent Plasticity
122
3.2.1 Flow Rule and Work-Hardening 122
3.2.2 Maximum-Dissipation Postulate and Normality 127
3.2.3 Strain-Space Plasticity 130
Section 3.3 Yield Criteria, Flow Rules and Hardening Rules
135
3.3.1 Introduction 135
Contents ix
3.3.2 Yield Criteria Independent of the Mean Stress 137
3.3.3 Yield Criteria Dependent on the Mean Stress 141
3.3.4 Yield Criteria Under Special States of Stress or Deformation 144
3.3.5 Hardening Rules 146
Section 3.4 Uniqueness and Extremum Theorems
152
3.4.1 Uniqueness Theorems 152
3.4.2 Extremum and Variational Principles 154
3.4.3 Rigid–Plastic Materials 159
Section 3.5 Limit-Analysis and Shakedown Theorems
162
3.5.1 Standard Limit-Analysis Theorems 162
3.5.2 Nonstandard Limit-Analysis Theorems 168
3.5.3 Shakedown Theorems 170
Chapter 4: Problems in Contained Plastic Defor-
mation
Section 4.1 Elementary Problems
177
4.1.1 Introduction: Statically Determinate Problems 177
4.1.2 Thin-Walled Circular Tube in Torsion and Extension 178
4.1.3 Thin-Walled Cylinder Under Pressure and Axial Force 181
4.1.4 Statically Indeterminate Problems 184
Section 4.2 Elastic–Plastic Torsion
189
4.2.1 The Torsion Problem 189
4.2.2 Elastic Torsion 191
4.2.3 Plastic Torsion 194
Section 4.3 The Thick-Walled Hollow Sphere and Cylinder
205
4.3.1 Elastic Hollow Sphere Under Internal and External Pressure 206
4.3.2 Elastic–Plastic Hollow Sphere Under Internal Pressure 208
4.3.3 Thermal Stresses in an Elastic–Plastic Hollow Sphere 213
4.3.4 Hollow Cylinder: Elastic Solution and Initial Yield Pressure 216
4.3.5 Elastic–Plastic Hollow Cylinder 220
x Contents
Section 4.4 Elastic–Plastic Bending
229
4.4.1 Pure Bending of Prismatic Beams 229
4.4.2 Rectangular Beams Under Transverse Loads 239
4.4.3 Plane-Strain Pure Bending of Wide Beams or Plates 245
Section 4.5 Numerical Methods
250
4.5.1 Integration of Rate Equations 251
4.5.2 The Finite-Element Method 256
4.5.3 Finite-Element Methods for Nonlinear Continua 262
Chapter 5: Problems in Plastic Flow and Collapse
I: Theories and “Exact” Solutions
Introduction 275
Section 5.1 Plane Problems
276
5.1.1 Slip-Line Theory 277
5.1.2 Simple Slip-Line Fields 287
5.1.3 Metal-Forming Problems 291
Section 5.2 Collapse of Circular Plates
298
5.2.1 Introduction to Plate Theory 299
5.2.2 Elastic Plates 303
5.2.3 Yielding of Plates 308
Section 5.3 Plastic Buckling
313
5.3.1 Introduction to Stability Theory 314
5.3.2 Theories of the Effective Modulus 319
5.3.3 Plastic Buckling of Plates and Shells 326
Chapter 6: Problems in Plastic Flow and Collapse
II: Applications of Limit Analysis
Introduction 337
