Mathematical Theory of Elastic and Elasto-Plastic Bodies - 1st Edition - ISBN: 9780444997548, 9781483291918

Mathematical Theory of Elastic and Elasto-Plastic Bodies, Volume 3

1st Edition

An Introduction

Authors: J. Necas I. Hlavácek
eBook ISBN: 9781483291918
Imprint: North Holland
Published Date: 1st January 1981
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Table of Contents


Summary of Notation

Chapter 1. Stress Tensor

1.1. Tensors. Green's Theorem

1.2. Stress Vector

1.3. Components of the Stress Tensor

1.4. Equations of Equilibrium

1.5. Tensor Character of Stress

1.6. Principal Stresses and the Quadric of Stress

Chapter 2. Strain Tensor

2.1. Finite Strain Tensor

2.2. Small Strain Tensor

2.3. Equations of the Compatibility of Strain

Chapter 3. Generalized Hooke's Law

3.1. Tension Test

3.2. Generalized Hooke's Law

3.3. Elasto-Plastic Materials. Deformation Theory. (A Special Case of the Nonlinear Hooke's Law)

3.4. Elasto-Inelastic Bodies. A Model with Internal State Variables

3.5. Hooke's Law with a Perfectly Plastic Domain

3.6. Flow Theory of Plasticity

Chapter 4. Formulation of Boundary Value Problems of the Theory of Elasticity

4.1. Lamé Equations. Beltrami-Michell Equations

4.2. The Classical Formulation of Basic Boundary Value Problems of Elasticity

Chapter 5. Variational Principles in Small Displacement Theory

5.1. Principles of Virtual Wo;k, Virtual Displacements and Virtual Stresses

5.2. Principle of Minimum Potential Energy in the Theory of Elasticity

5.3. Principle of Minimum Complementary Energy in the Theory of Elasticity

5.4. Hybrid Principles in the Theory of Elasticity. The Hellinger-Reissner Principle

Chapter 6. Functions with Finite Energy

6.1. The Space of Functions with Finite Energy

6.2. The Trace Theorem. Equivalent Norms, Rellich's Theorem

6.3. Coerciveness of Strains. Korn's Inequality

Chapter 7. Variational Formulation and Solution of Basic Boundary Value Problems of Elasticity

7.1. Weak (Generalized) Solution

7.2. Solution of Basic Boundary Value Problems by the Variational Method

7.2.1. Solution of the Abstract Variational Problem

7.2.2. Application to Basic Problems of the Theory of Elasticity

7.3. Solution of the First Basic Boundaiy Value Problem of Elasticity

7.4. Contact and Other Boundary Value Problems

7.5. Variational Formulation in Terms of Stresses. Method of Orthogonal Projections and Castigliano's Principle

7.6. Basic Boundary Value Problems of Elasticity in Orthogonal Curvilinear Coordinates

7.6.1. Tensors in Curvilinear Coordinates

7.6.2. Physical Components of Strain and Stress Tensors

7.6.3. Formulation of Variational Principles in Curvilinear Coordinates

7.6.4. Weak Solution of Basic Boundary Value Problems Formulated in Terms of Displacements or Stresses

Chapter 8. Solution of Boundary Value Problems for the Elasto-Plastic Body. Deformation Theory

8.1. Formulation of the Weak Solution

8.2. Application of the Variational Method to the Solution of Basic Boundary Value Problems

Chapter 9. Solution of Boundary Value Problems for the Elasto-Inelastic Body

9.1. Elasto-Inelastic Material

9.2. Solution of the First Boundary Value Problem for the Elasto-Inelastic Body

9.3. Solution of the Second Boundary Value Problem

Chapter 10. Two- and One-Dimensional Problems

10.1. Saint-Venant's Principle

10.2. Plane Elasticity

10.2.1. Basic Cases of Plane Elasticity

10.2.2. Solution of Problems of Plane Elasticity in Terms of Displacements

10.2.3. Solution of Problems of Plane Elasticity in Terms of Stresses

10.3. Axisymmetric Boundary Value Problems

10.4. Reduction of Dimension in the Theory of Elasticity

10.4.1. Kantorovičs Method

10.4.2. Bending of a Beam

10.4.3. Bending of a Plate

10.4.4. Shells

10.4.5. Solution of a Boundary Value Problem for a Cylindrical Shell

10.5. Torsion of a Bar

Chapter 11. Ritz-Galerkin and Other Approximate Methods

11.1. Minimizing Sequence

11.2. The Ritz-Galerkin Method

11.3. Finite Element Method

11.3.1. Compatible Models

11.3.2. Equilibrium Models

11.3.3. Mixed Models

11.4. A Posteriori Error Bounds. Two-Sided Energy Bounds. The Hypercircle Method

11.5. The Kacanov Method

11.6. Method of Steepest Descent

11.7. Method of Contraction

Chapter 12. Large Deflections of Plates. The Equations of von Kármán

12.1. Finite Elasticity

12.2. Large Deflections of Plates

12.3. Theory of Von Kármán's Equations

Chapter 13. Variational Inequalities with Applications to Problems of Signorini's Type and to the Theory of Plasticity

13.1. Signorini's Problem

13.2. Elasto-Plastic Body with a Perfectly Plastic Domain

13.3. Approximate Solution of Variational Inequalities

13.4. Flow Theory of Plasticity. Elasto-Inelastic Body with a Perfectly Plastic Domain

13.5. Flow Theory. Elasto-Inelastic Body with Strain Hardening


Subject Index


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.


© North Holland 1981
North Holland
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About the Authors

J. Necas Author

I. Hlavácek Author