Mathematical Theory of Elastic and Elasto-Plastic Bodies (eBook)

An Introduction

I. Hlavacek, J. Necas (Autoren)

eBook Download: PDF

2017 | 1. Auflage
343 Seiten
Elsevier Science (Verlag)
978-1-4832-9191-8 (ISBN)

Infos

The book acquaints the reader with the basic concepts and relations of elasticity and plasticity, and also with the contemporary state of the theory, covering such aspects as the nonlinear models of elasto-plastic bodies and of large deflections of plates, unilateral boundary value problems, variational principles, the finite element method, and so on.

Inhalt

Front Cover 1
Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction 4
Copyright Page 5
Table of Contents 6
Preface 10
SUMMARY OF NOTATION 14
CHAPTER 1. STRESS TENSOR 16
1.1. Tensors. Green's Theorem 16
1.2. Stress Vector 21
1.3. Components of the Stress Tensor 22
1.4. Equations of Equilibrium 24
1.5. Tensor Character of Stress 25
1.6. Principal Stresses and the Quadric of Stress 25
CHAPTER 2. STRAIN TENSOR 30
2.1. Finite Strain Tensor 30
2.2. Small Strain Tensor 35
2.3. Equations of the Compatibility of Strain 37
CHAPTER 3. GENERALIZED HOOKE'S L AW 42
3.1. Tension Test 42
3.2. Generalized Hooke's Law 44
3.3. Elasto-Plastic Materials. Deformation Theory. (A Special Case of the Nonlinear Hooke's Law) 51
3.4. Elasto-Inelastic Bodies. A Model with Internal State Variables 52
3.5. Hooke's Law with a Perfectly Plastic Domain 53
3.6. Flow Theory of Plasticity 55
CHAPTER 4. FORMULATION OF BOUNDARY VALUE PROBLEMS OF THE THEORY OF ELASTICITY 57
4.1. Lamé Equations. Beltrami-Michell Equations 57
4.2. The Classical Formulation of Basic Boundary Value Problems of Elasticity 59
CHAPTER 5. VARIATIONAL PRINCIPLES IN SMALL DISPLACEMENT THEORY 61
5.1. Principles of Virtual Work, Virtual Displacements and Virtual Stresses 61
5.2. Principle of Minimum Potential Energy in the Theory of Elasticity 63
5.3. Principle of Minimum Complementary Energy in the Theory of Elasticity 65
5.4. Hybrid Principles in the Theory of Elasticity. The Hellinger-Reissner Principle 67
CHAPTER 6. FUNCTIONS WITH FINITE ENERGY 72
6.1. The Space of Functions with Finite Energy 72
6.2. The Trace Theorem. Equivalent Norms, Rellich's Theorem 73
6.3. Coerciveness of Strains. Korn's Inequality 78
CHAPTER 7. VARIATIONAL FORMULATION AND SOLUTION OF BASIC BOUNDARY VALUE PROBLEMS OF ELASTICITY 87
7.1. Weak (Generalized) Solution 87
7.2. Solution of Basic Boundary Value Problems by the Variational Method 89
7.3. Solution of the First Basic Boundary Value Problem of Elasticity 96
7.4. Contact and Other Boundary Value Problems 101
7.5. Variational Formulation in Terms of Stresses. Method of Orthogonal Projections and Castigliano's Principle 104
7.6. Basic Boundary Value Problems of Elasticity in Orthogonal Curvilinear Coordinates 110
CHAPTER 8. SOLUTION OF BOUNDARY VALUE PROBLEMS FOR THE ELASTO-PLASTIC BODY. DEFORMATION THEORY 126
8.1. Formulation of the Weak Solution 126
8.2. Application of the Variational Method to the Solution of Basic Boundary Value Problems 129
CHAPTER 9. SOLUTION OF BOUNDARY VALUE PROBLEMS FOR THE ELASTO-INELASTIC BODY 132
9.1. Elasto-Inelastic Material 132
9.2. Solution of the First Boundary Value Problem for the Elasto-Inelastic Body 133
9.3. Solution of the Second Boundary Value Problem 139
CHAPTER 10. TWO- AND ONE-DIMENSIONAL PROBLEMS 142
10.1. Saint-Venant's Principle 142
10.2. Plane Elasticity 150
10.3. Axisymmetric Boundary Value Problems 179
10.4. Reduction of Dimension in the Theory of Elasticity 187
10.5. Torsion of a Bar 226
CHAPTER 11. RITZ-GALERKIN AND OTHER APPROXIMATE METHODS 234
11.1. Minimizing Sequence 234
11.2. The Ritz-Galerkin Method 235
11.3. Finite Element Method 236
11.4. A Posteriori Error Bounds. Two-Sided Energy Bounds. The Hypercircle Method 260
11.5. The Kacanov Method 263
11.6. Method of Steepest Descent 266
11.7. Method of Contraction 269
CHAPTER 12. LARGE DEFLECTIONS OF PLATES. THE EQUATIONS OF VON KÄRMÄN 274
12.1. Finite Elasticity 274
12.2. Large Deflections of Plates 278
12.3. Theory of Von Kärmän's Equations 282
CHAPTER 13. VARIATIONAL INEQUALITIES WITH APPLICATIONS TO PROBLEMS OF SIGNORINI'S TYPE AND TO THE THEORY OF PLASTICITY 295
13.1. Signorini's Problem 295
13.2. Elasto-Plastic Body with a Perfectly Plastic Domain 303
13.3. Approximate Solution of Variational Inequalities 310
13.4. Flow Theory of Plasticity. Elasto-Inelastic Body with a Perfectly Plastic Domain 323
13.5. Flow Theory. Elasto-Inelastic Body with Strain Hardening 330
BIBLIOGRAPHY 336
SUBJECT INDEX 341

Produkt-Details

• Erscheint lt. Verlag: 1.2.2017
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 1-4832-9191-X / 148329191X
• ISBN-13: 978-1-4832-9191-8 / 9781483291918