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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