PLASTICITY THEORY pdf

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 “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” b ook 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 grad- uate course (such as the one that my colleague Jim Kelly and I have been teaching) — s hould 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 — s ubjects in which I am not an expert. I found it quite frustrating that no bo ok 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 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 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 bo ok’s structure and style are s ufficiently 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 — 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 bo ok 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 b oth 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 problems in the dynamic loading of elastic–perfectly plastic structures treated 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- ent course m ay 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 help e d 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 s pelling 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 Cartes ian 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 Displacem ent 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-Indep endent Plasticity 116 3.1.3 Visc oplasticity 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 Rec tangular 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 [...]... 434 438 446 448 xii Contents Section 7.3 Three-Dimensional Waves 7.3.1 Theory of Acceleration Waves 7.3.2 Special Cases 452 453 459 Chapter 8: Large-Deformation Plasticity Section 8.1 Large-Deformation Continuum Mechanics 8.1.1 Continuum Deformation 8.1.2 Continuum Mechanics and Objectivity Section 8.2 Large-Deformation Constitutive Theory 8.2.1 8.2.2 8.2.3 8.2.4 Thermoelasticity Inelasticity: Kinematics... Limit Analysis of Shells: Theory 6.4.3 Limit Analysis of Shells: Examples 338 338 341 347 355 355 358 364 369 374 374 380 385 390 398 398 404 407 Chapter 7: Dynamic Problems Section 7.1 Dynamic Loading of Structures 7.1.1 Introduction 7.1.2 Dynamic Loading of Beams 7.1.3 Dynamic Loading of Plates and Shells Section 7.2 One-Dimensional Plastic Waves 7.2.1 7.2.2 7.2.3 7.2.4 Theory of One-Dimensional... 8.3 Numerical Methods in Large-Deformation Plasticity 8.3.1 Rate-Based Formulations 8.3.2 “Hyperelastic” Numerical Methods 465 465 473 478 478 480 485 487 491 492 496 References 501 Index 517 Chapter 1 Introduction to Continuum Thermomechanics Section 1.1 1.1.1 Mathematical Fundamentals Notation Solid mechanics, which includes the theories of elasticity and plasticity, is a broad discipline, with experimental,... solid bodies that arise in civil and mechanical engineering, geophysics, physiology, and other applied disciplines These aims are not in conflict, but complementary: some important results in the general theory have been obtained in the course of solving specific problems, and practical solution methods have resulted from fundamental theoretical work There are, however, differences in approach between workers... cumbersome, requiring several lines of long equations where other notations permit one short line, and it sometimes obscures the mathematical nature of the objects and processes involved Workers in constitutive theory tend to use either one of several systems of “direct” notation that in general use no indices (subscripts and superscripts), such as Gibbs’ dyadic notation, matrix notation, and a combination of... the basis vectors themselves vary A central role is played by the metric tensor with components gij = gi · gj , having the property that dx · dx = gij dξ i dξ j An alternative approach is based on the theory of differentiable manifolds (see, e.g., Marsden and Hughes [1983]) Curvilinear tensor analysis is especially useful for studying the mechanics of curved surfaces, such as shells; when this topic . 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. 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. 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