Necas / Hlavácek Mathematical Theory of Elastic and Elasto-Plastic Bodies

An Introduction
1. Auflage 2017
ISBN: 978-1-4832-9191-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 343 Seiten, Web PDF

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.

Autoren/Hrsg.

Necas, J.

Hlavácek, I.

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;SUMMARY OF NOTATION;14
7;CHAPTER 1. STRESS TENSOR;16
7.1;1.1. Tensors. Green's Theorem;16
7.2;1.2. Stress Vector;21
7.3;1.3. Components of the Stress Tensor;22
7.4;1.4. Equations of Equilibrium;24
7.5;1.5. Tensor Character of Stress;25
7.6;1.6. Principal Stresses and the Quadric of Stress;25
8;CHAPTER 2. STRAIN TENSOR;30
8.1;2.1. Finite Strain Tensor;30
8.2;2.2. Small Strain Tensor;35
8.3;2.3. Equations of the Compatibility of Strain;37
9;CHAPTER 3. GENERALIZED HOOKE'S L AW;42
9.1;3.1. Tension Test;42
9.2;3.2. Generalized Hooke's Law;44
9.3;3.3. Elasto-Plastic Materials. Deformation Theory. (A Special Case of the Nonlinear Hooke's Law);51
9.4;3.4. Elasto-Inelastic Bodies. A Model with Internal State Variables;52
9.5;3.5. Hooke's Law with a Perfectly Plastic Domain;53
9.6;3.6. Flow Theory of Plasticity;55
10;CHAPTER 4. FORMULATION OF BOUNDARY VALUE PROBLEMS OF THE THEORY OF ELASTICITY;57
10.1;4.1. Lamé Equations. Beltrami-Michell Equations;57
10.2;4.2. The Classical Formulation of Basic Boundary Value Problems of Elasticity;59
11;CHAPTER 5. VARIATIONAL PRINCIPLES IN SMALL DISPLACEMENT THEORY;61
11.1;5.1. Principles of Virtual Work, Virtual Displacements and Virtual Stresses;61
11.2;5.2. Principle of Minimum Potential Energy in the Theory of Elasticity;63
11.3;5.3. Principle of Minimum Complementary Energy in the Theory of Elasticity;65
11.4;5.4. Hybrid Principles in the Theory of Elasticity. The Hellinger-Reissner Principle;67
12;CHAPTER 6. FUNCTIONS WITH FINITE ENERGY;72
12.1;6.1. The Space of Functions with Finite Energy;72
12.2;6.2. The Trace Theorem. Equivalent Norms, Rellich's Theorem;73
12.3;6.3. Coerciveness of Strains. Korn's Inequality;78
13;CHAPTER 7. VARIATIONAL FORMULATION AND SOLUTION OF BASIC BOUNDARY VALUE PROBLEMS OF ELASTICITY;87
13.1;7.1. Weak (Generalized) Solution;87
13.2;7.2. Solution of Basic Boundary Value Problems by the Variational Method;89
13.3;7.3. Solution of the First Basic Boundary Value Problem of Elasticity;96
13.4;7.4. Contact and Other Boundary Value Problems;101
13.5;7.5. Variational Formulation in Terms of Stresses. Method of Orthogonal Projections and Castigliano's Principle;104
13.6;7.6. Basic Boundary Value Problems of Elasticity in Orthogonal Curvilinear Coordinates;110
14;CHAPTER 8. SOLUTION OF BOUNDARY VALUE PROBLEMS FOR THE ELASTO-PLASTIC BODY. DEFORMATION THEORY;126
14.1;8.1. Formulation of the Weak Solution;126
14.2;8.2. Application of the Variational Method to the Solution of Basic Boundary Value Problems;129
15;CHAPTER 9. SOLUTION OF BOUNDARY VALUE PROBLEMS FOR THE ELASTO-INELASTIC BODY;132
15.1;9.1. Elasto-Inelastic Material;132
15.2;9.2. Solution of the First Boundary Value Problem for the Elasto-Inelastic Body;133
15.3;9.3. Solution of the Second Boundary Value Problem;139
16;CHAPTER 10. TWO- AND ONE-DIMENSIONAL PROBLEMS;142
16.1;10.1. Saint-Venant's Principle;142
16.2;10.2. Plane Elasticity;150
16.3;10.3. Axisymmetric Boundary Value Problems;179
16.4;10.4. Reduction of Dimension in the Theory of Elasticity;187
16.5;10.5. Torsion of a Bar;226
17;CHAPTER 11. RITZ-GALERKIN AND OTHER APPROXIMATE METHODS;234
17.1;11.1. Minimizing Sequence;234
17.2;11.2. The Ritz-Galerkin Method;235
17.3;11.3. Finite Element Method;236
17.4;11.4. A Posteriori Error Bounds. Two-Sided Energy Bounds. The Hypercircle Method;260
17.5;11.5. The Kacanov Method;263
17.6;11.6. Method of Steepest Descent;266
17.7;11.7. Method of Contraction;269
18;CHAPTER 12. LARGE DEFLECTIONS OF PLATES. THE EQUATIONS OF VON KÄRMÄN;274
18.1;12.1. Finite Elasticity;274
18.2;12.2. Large Deflections of Plates;278
18.3;12.3. Theory of Von Kärmän's Equations;282
19;CHAPTER 13. VARIATIONAL INEQUALITIES WITH APPLICATIONS TO PROBLEMS OF SIGNORINI'S TYPE AND TO THE THEORY OF PLASTICITY;295
19.1;13.1. Signorini's Problem;295
19.2;13.2. Elasto-Plastic Body with a Perfectly Plastic Domain;303
19.3;13.3. Approximate Solution of Variational Inequalities;310
19.4;13.4. Flow Theory of Plasticity. Elasto-Inelastic Body with a Perfectly Plastic Domain;323
19.5;13.5. Flow Theory. Elasto-Inelastic Body with Strain Hardening;330
20;BIBLIOGRAPHY;336
21;SUBJECT INDEX;341