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# Variational, Incremental and Energy Methods in Solid Mechanics and Shell Theory, Volume 4

## 1st Edition

**Authors:**J. Mason

**Hardcover ISBN:**9780444418999

**eBook ISBN:**9781483289649

**Imprint:**North Holland

**Published Date:**1st January 1980

**View all volumes in this series:**Studies in Applied Mechanics

## Table of Contents

Preface

Introduction

Chapter 1. Mathematical Preliminaries

1.1 Tensor Formulas in Three-Dimensional Euclidean Spaces

1.2 Tensor Fomulas For Surfaces

1.3 Basic Concepts and Definitions in the Theory of Abstract Spaces and Operators

1.4 Fundamentals of the Classical Calculus of Variations

1.5 Differentials and Derivatives of Operators — Variations

1.6 Variational Boundary Value Problems

1.7 Variational Methods of Approximation

Part I — Three-Dimensional Theory

Chapter 2. Kinematics and Stress in Three-Dimensional Solid Continua

2.1 Description of Motion, Coordinates, Reference and Deformed States

2.2 Kinematics of Deformation

2.3 Kinematics of Incremental Deformation

2.4 Definitions of Stress — Stress Rates

2.5 Incremental Stresses

2.6 External Loadings — Boundary Conditions

2.7 Illustrative Example of Physical Components of Displacements and Stress

Chapter 3. General Principles - The First and the Second Laws of Thermodynamics

3.1 Kinematics of Rigid Body Motions — Invariance Requirements

3.2 the First Law of Thermodynamics

3.3 the Second Law of Thermodynamics

3.4 the Principles of Virtual Displacements and Virtual Forces

3.5 Equations of Equilibrium in Physical Components For Cylindrical and Spherical Coordinates

Chapter 4. Constitutive Material Laws

4.1 Elastic, Hypoelastic and Hyperelastic Materials

4.2 Elastic-Plastic Materials

4.3 Visco-Elastic Materials

Part II - Shell Theory

Chapter 5. Deformation and Stress in Shells

5.1 Shell Middle Surface: Coordinates, Base Vectors

5.2 Description of Deformation and Motion in Shells

5.3 Kinematics of Incremental Deformation in Shells

5.4 Stress Resultants and Stress Couples for Shells - Three-Dimensional Approach

5.5 Load Vectors and Incremental Load Vectors

Chapter 6. the First Law of Thermodynamics Applied to Shells — General Principles

6.1 Formulation of the First Law - Invariance Requirements

6.2 Reduced and Local Forms of the First Law

6.3 The First Law in the Reference State

6.4 Some Consequences of the First Law — Discussion — Boundary Condition

6.5 Incremental Forms of the First Law of Thermodynamics

6.6 Incremental Theory and Stability

6.7 the Principles of Virtual Displacements and Virtual Forces For Shells

Chapter 7. Constitutive Relations and Material Laws For Shells

7.1 Material Laws For Thin Elastic Shells

7.2 Hyperelastic Incremental Material Relations

7.3 A Note on the Constitutive Relations for Inelastic Shells

Part III. Variational Principles, Finite Element Models and Applications, Elastic Stability

Chapter 8. Variational Principles of Elasticity and Shell Theory — Classical Approach

8.1 Variational Principles of Three-Dimensional Elasticity

8.2 Variational Principles of Elastic Shell Theory

8.3 Special Cases — The Principles of Reciprocity, Clapeyron, Castigliano and Menabrea

8.4 General Remarks — The Role of Legendre Transformation

8.5 Illustrative Examples

8.6 Variational Principles — Weak Solutions

Chapter 9. Variational and Incremental Principles — General Abstract Approach

9.1 Basic Equations and Operators

9.2 Inner Products-Formal Adjoints — Special Notations and Admissible Variations

9.3 the Principles of Virtual Displacements and Virtual Forces

9.4 General Conditions for Potentialness of the Operators of Dynamics and Kinematics

9.5 Strain and Complementary Energy — Loading Potentials

9.6 General Variational Principles Derived from the Principles of Virtual Displacements and Virtual Forces

9.7 the Variational Principles of Non-Linear Elasticity and Elastic Shell Theory

9.8 Incremental Variational Principles

9.9 Incremental Principles of Non-Linear Elasticity and Shallow Shell Theory

Chapter 10. Finite Element Models — Selected Applications

10.1 Matrix Representation of Important Results

10.2 Matrix Formulas for Physical Components

10.3 Finite Element Models — Stiffness, Flexibility and Tangent Matrices

10.4 the Numerical Solution of Finite Element Equations in Statics and Dynamics

10.5 Illustrative Examples

10.6 the Problem of Initial Stresses and Temperature Effects

10.7 Variational Problems with Discontinuous Fields

10.8 Concluding Remark

Chapter 11. Elastic Stability

11.1 Incremental Expansions for the Potential Energy

11.2 Neutral Equilibrium — Critical State

11.3 Illustrative Examples

11.4 General Theory of Elastic Stability — Postbuckling Behaviour

11.5 Stability of an Equilibrium State

11.6 Critical Behaviour

11.7 Limit Points and Branching Points

11.8 Imperfection Sensitivity, Coincident Branching Points and Multiparameter Loadings

11.9 Critical and Limit Points — Illustrative Examples

Selected References

Index

## Description

Preface

Introduction

Chapter 1. Mathematical Preliminaries

1.1 Tensor Formulas in Three-Dimensional Euclidean Spaces

1.2 Tensor Fomulas For Surfaces

1.3 Basic Concepts and Definitions in the Theory of Abstract Spaces and Operators

1.4 Fundamentals of the Classical Calculus of Variations

1.5 Differentials and Derivatives of Operators — Variations

1.6 Variational Boundary Value Problems

1.7 Variational Methods of Approximation

Part I — Three-Dimensional Theory

Chapter 2. Kinematics and Stress in Three-Dimensional Solid Continua

2.1 Description of Motion, Coordinates, Reference and Deformed States

2.2 Kinematics of Deformation

2.3 Kinematics of Incremental Deformation

2.4 Definitions of Stress — Stress Rates

2.5 Incremental Stresses

2.6 External Loadings — Boundary Conditions

2.7 Illustrative Example of Physical Components of Displacements and Stress

Chapter 3. General Principles - The First and the Second Laws of Thermodynamics

3.1 Kinematics of Rigid Body Motions — Invariance Requirements

3.2 the First Law of Thermodynamics

3.3 the Second Law of Thermodynamics

3.4 the Principles of Virtual Displacements and Virtual Forces

3.5 Equations of Equilibrium in Physical Components For Cylindrical and Spherical Coordinates

Chapter 4. Constitutive Material Laws

4.1 Elastic, Hypoelastic and Hyperelastic Materials

4.2 Elastic-Plastic Materials

4.3 Visco-Elastic Materials

Part II - Shell Theory

Chapter 5. Deformation and Stress in Shells

5.1 Shell Middle Surface: Coordinates, Base Vectors

5.2 Description of Deformation and Motion in Shells

5.3 Kinematics of Incremental Deformation in Shells

5.4 Stress Resultants and Stress Couples for Shells - Three-Dimensional Approach

5.5 Load Vectors and Incremental Load Vectors

Chapter 6. the First Law of Thermodynamics Applied to Shells — General Principles

6.1 Formulation of the First Law - Invariance Requirements

6.2 Reduced and Local Forms of the First Law

6.3 The First Law in the Reference State

6.4 Some Consequences of the First Law — Discussion — Boundary Condition

6.5 Incremental Forms of the First Law of Thermodynamics

6.6 Incremental Theory and Stability

6.7 the Principles of Virtual Displacements and Virtual Forces For Shells

Chapter 7. Constitutive Relations and Material Laws For Shells

7.1 Material Laws For Thin Elastic Shells

7.2 Hyperelastic Incremental Material Relations

7.3 A Note on the Constitutive Relations for Inelastic Shells

Part III. Variational Principles, Finite Element Models and Applications, Elastic Stability

Chapter 8. Variational Principles of Elasticity and Shell Theory — Classical Approach

8.1 Variational Principles of Three-Dimensional Elasticity

8.2 Variational Principles of Elastic Shell Theory

8.3 Special Cases — The Principles of Reciprocity, Clapeyron, Castigliano and Menabrea

8.4 General Remarks — The Role of Legendre Transformation

8.5 Illustrative Examples

8.6 Variational Principles — Weak Solutions

Chapter 9. Variational and Incremental Principles — General Abstract Approach

9.1 Basic Equations and Operators

9.2 Inner Products-Formal Adjoints — Special Notations and Admissible Variations

9.3 the Principles of Virtual Displacements and Virtual Forces

9.4 General Conditions for Potentialness of the Operators of Dynamics and Kinematics

9.5 Strain and Complementary Energy — Loading Potentials

9.6 General Variational Principles Derived from the Principles of Virtual Displacements and Virtual Forces

9.7 the Variational Principles of Non-Linear Elasticity and Elastic Shell Theory

9.8 Incremental Variational Principles

9.9 Incremental Principles of Non-Linear Elasticity and Shallow Shell Theory

Chapter 10. Finite Element Models — Selected Applications

10.1 Matrix Representation of Important Results

10.2 Matrix Formulas for Physical Components

10.3 Finite Element Models — Stiffness, Flexibility and Tangent Matrices

10.4 the Numerical Solution of Finite Element Equations in Statics and Dynamics

10.5 Illustrative Examples

10.6 the Problem of Initial Stresses and Temperature Effects

10.7 Variational Problems with Discontinuous Fields

10.8 Concluding Remark

Chapter 11. Elastic Stability

11.1 Incremental Expansions for the Potential Energy

11.2 Neutral Equilibrium — Critical State

11.3 Illustrative Examples

11.4 General Theory of Elastic Stability — Postbuckling Behaviour

11.5 Stability of an Equilibrium State

11.6 Critical Behaviour

11.7 Limit Points and Branching Points

11.8 Imperfection Sensitivity, Coincident Branching Points and Multiparameter Loadings

11.9 Critical and Limit Points — Illustrative Examples

Selected References

Index

## Details

- Language:
- English

- Copyright:
- © North Holland 1980

- Published:
- 1st January 1980

- Imprint:
- North Holland

- eBook ISBN:
- 9781483289649

## Reviews

@qu:This book is a significant contribution to the field of solid mechanics, especially to shell theory. It serves as an excellent textbook, and as a reference... Mathematical rigor, and comprehensive examples make this book a valuable tool for research and education. @source:Mathematics Abstracts