Plasticity Theory







PLASTICITY

THEORY

Revised Edition (PDF)

Jacob Lubliner

University of California at Berkeley



Preface

When I first began to plan this book, I thought that I would begin the preface

with the words “The purpose of this little book is...” 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 “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

useful as a primary reference and, more importantly, as a textbook in a graduate

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 plasticity,

and dynamic plasticity. By no coincidence, it is precisely these topics

— rather than the traditional study of elastic-plastic boundary-value problems,

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 foundations

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.

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



all, nor are such general areas of current interest as micromechanics (except

on the elementary, qualitative level of dislocation theory), damage mechanics

(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 subsections.

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 constitutive

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 variational

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 concepts

of physics and constitutive theory that are far more general than the

conventional constitutive models that are actually used in the chapters presenting

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 largedeformation

continuum mechanics were thrown at them too soon.

Much of the material of Chapter 1 — including the mathematical fundamentals,

in particular tensor algebra and analysis — would normally be

covered in a basic course in continuum mechanics at the advanced under-



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 consistent

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 elasticplastic

problems) for which analytical solutions have been found: some elementary

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 numerical

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. Chapter

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 contains

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

problems in the dynamic loading of elastic–perfectly plastic structures treated

by an extension of limit analysis, and with wave-propagation problems, onedimensional

(with the significance of rate dependence explicitly discussed)

and three-dimensional. The content of Chapter 8 has already been mentioned.

As the knowledgeable reader may see from the foregoing survey, a coherent

course may be built in various ways by putting together selected portions

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

those whose orientation and interests are different from mine will nonetheless

find this would-be “little book” useful.

In shaping the book I was greatly helped by comments from some out-



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 Vassilis

Panoskaltsis, who was especially helpful in the writing of the sections

on numerical methods (including some sample computations) and who suggested

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 professional

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.



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