Mechanics Today

Contributors

Preface

Contents of Volume 1

Contents of Volume 2

Contents of Volume 3

Contents of Volume 4

Contents of Volume 5

Summary

I Macroscopic Response of Continua with Random Microstructures

1. Introduction

2. Formal Solution

2.1 The Governing Equations

2.2 The Method of Smoothing

2.3 Renormalization

2.4 The Case of a Homogeneous £(0) Field

2.5 A Few References

3. The Local Approximation (Effective Modulus Theory)

3.1 The Weakly Inhomogeneous Limit and Corrections

3.2 The Dilute Suspension and Corrections

3.3 Bounds on the Effective Elastic Moduli Tensor

3.4 Self-Consistent Calculations

3.5 Miscellaneous Calculations

4. Nonlocal Theory

4.1 Form of the Theory and a Uniqueness Theorem

4.2 The Kernel Function Cijkl(x-x')

4.3 Solutions of the Nonlocal Theory

5. Wave Propagation Experiments

5.1 Extension of General Formalism

5.2 Other Statistical Treatments

5.3 Nonstatistical and Phenomenological Treatments

6. References

II Degenerate Duality, Catastrophes and Saddle Functional

1. Morse Functions

1.1 Introduction

1.2 Morse Functions

1.3 Two-Variable Morse Functions

1.4 Legendre Transformations

1.5 Saddle Functions

1.6 Example: Plastic Constitutive Equations at a Yield Vertex

1.7 Sets of Equivalent Problems

1.8 Examples

2. Non-Morse Functions

2.1 Co-rank

2.2 Structural Instability of Non-Morse Functions

2.3 Legendre Transformations

2.4 One-Parameter Family of Legendre Transformations

2.5 Two-Parameter Family of Legendre Transformations

2.6 Pause

2.7 Two-Parameter Functions in a Minimum Norm Problem

2.8 A Pure Catastrophe Machine

2.9 Swallowtail Free Enthalpy

3. Saddle Functionals

3.1 Two Inner Product Spaces

3.2 Generating Functional

3.3 Saddle Functional

3.4 Uniqueness Criteria

3.5 Bifurcation Set

3.6 Dual Extremum Principles

3.7 Problems with Non-Linear Programming Structure

3.8 Estimates for Linear Functionals

4. Examples of Generating Functionals

4.1 Non-Morse Functionals

4.2 Solid Mechanics

4.3 Classical Elasticity

4.4 Rigid/Plastic Yield Point Problem

4.5 Small-Deflection Cantilever Beam or Strut

4.6 Potential Energy of Strut

4.7 Finite Elasticity

4.8 Incipient Plastic Distortion from a Stressed State

4.9 Theoretical Physics

4.10 Legendre Transformation for Compressible Fluids

4.11 Magnetohydrodynamic Pipe Flow

4.12 Minimum Norm Extremum Principles

5. References

III Elastic-Plastic Response of a Structure to Cyclic Loading: Practical Rules

1. Introduction

2. The Symmetric Three-bar System

2.1 Hypothesis

2.2 Elastic-Perfectly Plastic Material

2.3 Kinematically Hardening Material

3. Bree's Tube

3.1 Hypothesis

3.2 Perfectly Plastic Material

3.3 Kinematically Hardening Material

4. Plane Rectangular Plate

4.1 Hypothesis

4.2 Perfectly Plastic Material

4.3 Kinematically Hardening Material

5. A Non-symmetric Three-bar System

5.1 Hypothesis

5.2 Perfectly Plastic Material

5.3 Kinematically Hardening Material

6. General Analysis of Periodic Limiting Responses of Structures. Practical Rules

6.1 Initial-boundary-value Problem for an Elastic-plastic Structure of Kinematically Hardening Material

6.2 Fundamental Properties for Kinematically Hardening Material

6.3 Elastic Shakedown for the Kinematically Hardening Material

6.4 Plastic Shakedown for the Kinematically Hardening Material

6.5 Application Examples

6.6 Extension to Other Materials

7. References

IV Mixed Boundary-value Problems in Mechanics: Addendum

Author Index