Variational, Incremental and Energy Methods in Solid Mechanics and Shell Theory

Variational, Incremental and Energy Methods in Solid Mechanics and Shell Theory

1st Edition - January 1, 1980

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  • Author: J. Mason
  • eBook ISBN: 9781483289649

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Description

Studies in Applied Mechanics, 4: Variational, Incremental, and Energy Methods in Solid Mechanics and Shell Theory covers the subject of variational, incremental, and energy methods in Solid Mechanics and Shell Theory from a general standpoint, employing general coordinates and tensor notations. The publication first ponders on mathematical preliminaries, kinematics and stress in three-dimensional solid continua, and the first and second laws of thermodynamics. Discussions focus on the principles of virtual displacements and virtual forces, kinematics of rigid body motions, incremental stresses, kinematics of incremental deformation, description of motion, coordinates, reference and deformed states, tensor formulas for surfaces, and differentials and derivatives of operators. The text then elaborates on constitutive material laws, deformation and stress in shells, first law of thermodynamics applied to shells, and constitutive relations and material laws for shells. Concerns cover hyperelastic incremental material relations, material laws for thin elastic shells, incremental theory and stability, reduced and local forms of the first law of thermodynamics, and description of deformation and motion in shells. The book examines elastic stability, finite element models, variational and incremental principles, variational principles of elasticity and shell theory, and constitutive relations and material laws for shells. The publication is a valuable reference for researchers interested in the variational, incremental, and energy methods in solid mechanics and shell theory.

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

Product details

  • Language: English
  • Copyright: © North Holland 1980
  • Published: January 1, 1980
  • Imprint: North Holland
  • eBook ISBN: 9781483289649

About the Author

J. Mason

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