Mathematical Theory of Elastic and Elasto-Plastic Bodies

Mathematical Theory of Elastic and Elasto-Plastic Bodies

An Introduction

1st Edition - January 1, 1981

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  • Authors: J. Necas, I. Hlavácek
  • eBook ISBN: 9781483291918

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Description

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.

Table of Contents


  • Preface

    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

    Bibliography

    Subject Index

Product details

  • Language: English
  • Copyright: © North Holland 1981
  • Published: January 1, 1981
  • Imprint: North Holland
  • eBook ISBN: 9781483291918

About the Authors

J. Necas

I. Hlavácek

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