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Mechanics of Continuous Media and Analysis of Structures - 1st Edition - ISBN: 9780444861504, 9780080984612

Mechanics of Continuous Media and Analysis of Structures

1st Edition

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Author: R. Valid
eBook ISBN: 9780080984612
Imprint: North Holland
Published Date: 1st January 1981
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Mechanics of Continuous Media and Analysis of Structures is a six-chapter book that begins by elucidating the mechanics of solid continuous media. The text then describes the finite elements method, which undoubtedly dominates the methods used for structural analysis. Subsequent chapters explain the variational principles in linear elasticity, vibration of linear structure, non-linear deformations, and the shell theory. This book will be valuable to all those who need certain theoretical developments in mechanics, including mechanical engineers, economists, and mathematicians.

Table of Contents




Chapter I: Mechanics of Solid Continuous Media

1. General Hypotheses - Hyperelastic Media - Extension to Arbitrary Media

2. Stresses

3. Deformation

4. Equilibrium Equations

5. Homogeneous and Isotropic Linear Elasticity

5.1. Isotropy

5.2. Stress

5.3. Small Deformation-Linearization

5.4. Equilibrium Equations

5.5. Stability of Material

6. Variational Principle for a Linear Elastic Medium

7. Theorem of Reciprocity for a Linearized Hyperelastic Medium

8. Stress Functions

9. Polarized Media

Chapter II: The Finite Element Method

1. The Displacement Method in Static Problems

1.1. General

1.2. Remarks and Complements

2. Other Types of Elements and Static Problems

2.1. Axisymmetrical Three-dimensional Bodies with Axisymmetrical Loading

2.2. Improved Elements in the Bidimensional Case

2.3. Three-dimensional Elements and Problems

2.4. Plate Bending

2.5. Straight Beams in Bending

2.6. Shells as Assembly of Flat Elements

2.7. Axisymmetrical Shells

2.8. Structures under Elastoplastic Behaviour

2.9. Improvement of Elements

3. Other Types of Problems

Chapter III: Variational Principles in Linear Elasticity

1. Principle of Potential Energy

2. Hu-Washizu Principle, or Three Field Principle

3. Hellinger-Reissner Principle, or Two Field Principle

4. A Fraeijs de Veubeke Two Field Principle

5. Principle of Complementary Energy

6. Two Field Hybrid Principle of Pian

7. Principle of Virtual Displacements and Principle of Virtual Stresses

8. Application of Variational Principles

8.1. Stress Functions

8.2. Linearized Compatibility Condition

8.3. Application of the Finite Element Method

8.4. Upper Bound for the Error of an Approximate Solution

8.5. Non-conforming Elements - Patch Test

8.6. Hellinger-Reissner Principle and Saddle-Point

8.7. Legendre Transformation

Chapter IV: Vibration

1. Introduction

2. Discretized Structures

3. Riesz' Theorem

4. Variational Principles

5. Methods of Dynamic Reduction

6. Responses to Forced External Excitations

Chapter V: Non-Linear Deformations - Buckling

1. Formulation from the Natural State for a Hyperelastic Medium

2. Formulation from a Prestressed State for an Arbitrary Medium

3. Discretization

4. Methods of Solution

5. Static Buckling

5.1. General

5.2. The Criterion of Static Stability

5.3. Application of the Energy Criterion

5.4. Limit Point and Bifurcation Point

Chapter VI: Shell Theory

1. General

1.1. Introduction

1.2. Definitions

1.3. General Hypotheses

2. Equilibrium Equations

2.1. Equilibrium Equations in the Case of Hypothesis (w)

2.2. Equilibrium Equations in the Case of Hypothesis (wo)

2.3. Relation of Symmetry of Generalized Surface Stresses

3. Deformations

3.1. Deformation in the Case of Hypothesis (w), Linearized Case

3.2. Deformation in the Case of Hypothesis (wo), Linearized Case

3.3. Transverse Shear Deformation and Normal Deformation

3.4. Linearized Conditions of Compatibility

3.5. Non-linear Deformations

4. Stresses

4.1. Indeterminate Stresses in the Case of Hypothesis (wo)

4.2. Symmetrical Stresses in the Case of Hypothesis (wo)

4.3. Stress Functions in the Case of Hypothesis (wo)

4.4. Stress Functions in the Case of Hypothesis (w)

5. Variational Principles

6. Linear Constitutive Laws

7. Shells of Revolution

8. Discretization

Appendix : Notations and Formulae


Chapter I

Chapter II

Chapter III

Chapter IV

Chapter V

Chapter VI




© North Holland 1981
1st January 1981
North Holland
eBook ISBN:

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R. Valid

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