Plasticity Teory
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Plasticity Teory


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PLASTICITY
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
When I first began to plan this book, I thought that I would begin the preface
with the words \u201cThe purpose of this little book is...\u201d While I never lost my
belief that small is beautiful, I discovered that it is impossible to put together
a treatment of a field as vast as plasticity theory between the covers of a
truly \u201clittle\u201d book and still hope that it will be reasonably comprehensive.
I have long felt that a modern book on the subject \u2014 one that would be
useful as a primary reference and, more importantly, as a textbook in a grad-
uate course (such as the one that my colleague Jim Kelly and I have been
teaching) \u2014 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
\u2014 rather than the traditional study of elastic-plastic boundary-value prob-
lems, slip-line theory and limit analysis \u2014 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
\u2014 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,
\u201cWhy don\u2019t you write a book?\u201d It was these queries that gave me the
idea of composing a \u201clittle\u201d 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.
Almost two years have passed, and some 1.2 megabytes of disk space have
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\u2019t 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
of presentation of the areas is somewhat arbitrary. It is based on the order in
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
survey of the physics of plasticity as given in Chapter 2, and returned to with
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\u2019s 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
small deformations and rotations, is presented as a separate topic in Chapter
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
be expressed quite adequately within the small-deformation framework. The
other factor is pedagogical: it appears to me, on the basis of long experience,
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 \u2014 including the mathematical fun-
damentals, in particular tensor algebra and analysis \u2014 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
classical quasi-static problems of rate-independent plasticity theory. Chapter
4 contains a selection of problems in contained plastic deformation (or elastic-
plastic problems) for which analytical solutions have been found: some ele-
mentary problems, and those of torsion, the thick-walled sphere and cylinder,
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 \u201cexact\u201d 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