# Mechanics Today

### Pergamon Mechanics Today Series, Volume 4

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Mechanics Today, Volume 4 focuses on solid mechanics and applied mathematics. This book is divided into six chapters. Chapter I provides a general description of the basic features and relevant concepts of mixed boundary-value problems in mechanics. The problem of crack extension in a solid under arbitrary loads is discussed in Chapter II, emphasizing the crack growth that leads from a planar to a nonplanar configuration. The third chapter reviews various methods of solving the scattering of elastic waves by inclusions. The interactions of electromagnetic field with deformable bodies in motion are elaborated in Chapter IV, while problems involving solids carrying high electric currents or being placed in high magnetic fields are deliberated in Chapter V. The last chapter concentrates on the implications of the second law of thermodynamics, and consequences of thermodynamic material stability and its corresponding restrictions on the evolutionary equations for internal variables. This publication is useful to specialists, but is also beneficial to non-experts with sufficient background in applied mechanics.

## Table of Contents

Contributors

Preface

Contents of Volume 1

Contents of Volume 2

Contents of Volume 3

Summary

I Mixed Boundary-value Problems in Mechanics

1. Introduction

2. Definitions: Multiple Series Equations, Multiple Integral Equations

2.1 Multiple Series Equations

2.2 Multiple Integral Equations

3. Application of Complex Potentials

3.1 A Problem in Potential Theory

3.2 The Case of Periodic Cuts

3.3 An Elasticity Problem for a Non-homogeneous Plane

4. Reduction to Singular Integral Equations

4.1 Reduction of Dual Series Equations to Singular Integral Equations

4.2 An Example on Triple Series Equations

4.3 Reduction of Multiple Integral Equations

4.4 Reduction of Multiple Series-Multiple Integral Equations

4.5 Remarks on the Selection of Auxiliary Functions

5. Numerical Solution of Singular Integral Equations of the First Kind

5.1 Solution by Gaussian Integration Formulas

5.2 Solution by Orthogonal Polynomials

6. Integral Equations with Generalized Cauchy Kernels

6.1 A Plane Elasticity Problem for Non-homogeneous Media

6.2 The Fundamental Functions

6.3 Numerical Method for Solving the Integral Equations with Generalized Cauchy Kernels

7. Singular Integral Equations of the Second Kind

7.1 The Fundamental Function

7.2 Solution by Orthogonal Polynomials

7.3 Solution by Gauss-Jacobi Integration Formulas

8. References

II On the Problem of Crack Extension in Brittle Solids Under General Loading

1. Introduction

2. Review of the Two-dimensional Non-planar Crack Problem

3. Statement and Formulation of the In-plane Problem

3.1 Boundary Value Problem for the Branched Crack

3.2 Modification of the Boundary-value Problem and Its Effect on the Strain Energy

3.3 The Strain Energy Change for the In-plane Problem

3.4 Solution of the Modified Boundary-value Problem

4. Determination of the Crack Branching Angle and of the Critical Load

5. Discussion of Results for the Two-dimensional Problem

5.1 The Crack Branching Angle

5.2 The Critical Stress

5.3 The Combination of Mode I and Mode II Stress Intensity Factors

5.4 Finite Crack Extension

5.5 Infinitesimal Deviation Angle

6. Experimental Work

6.1 Material Choice

6.2 Specimen Preparation and Test Procedure

6.3 Data Evaluation

6.4 Results for Mildly Ductile Solids

7. Crack Growth from a Crack Front Under a General, Three-dimensional State of Stress

8. An Experiment of Crack Extension in Anti-plane Deformation

8.1 Choice of Test Material

8.2 Specimen Geometry

8.3 Crack Propagation Observation

9. Related Work on Fracture Involving Mode III Deformations

10. An Approximate Analysis for Multimode Fracture in Brittle Solids

11. Extension to Mildly Ductile Solids

12. Appendix—Supplemental Definitions

13. References

III Scattering of Elastic Waves

1. Introduction

2. Scattering of P-waves by a Liquid Sphere or Cylinder

2.1 Diffraction by a Liquid Sphere

2.2 Diffraction by a Liquid Circular Cylinder

3. Wave Propagation in a Half-space Containing a Cylindrical Cavity

3.1 Method of Line Source Potentials

3.2 Method of MAE

4. Scattering of Elastic Waves by Rigid Spheroids

4.1 Scattering by a Single Rigid Spheroid

4.2 Wave Propagation in the Presence of a Random Distribution of Rigid Spheroids

5. Scattering by a Rigid Circular Disc

5.1 Equations for Pn, Qn

5.2 Equations for An, Bn and Cn

5.3 Far-field Scattering Amplitudes

6. References

Appendix A

Appendix B

Appendix C

Appendix D

IV Electromagnetic Forces in Deformable Continua

1. Introduction

2. Balance Equations of Continuum Mechanics

3. Maxwell Equations for Media at Rest

3.1 The Maxwell Equations

3.2 Forces on Free Charges and Free Currents

4. Maxwell Equations for Moving Media

4.1 The Minkowski Formulation (EBDH)

4.2 The Lorentz Formulation (EBPMv)

4.3 The Statistical Formulation (EBPM)

4.4 The Chu Formulation (EHPM)

4.5 Global Laws for Electrodynamics

5. Maxwell Stress Tensor and Minkowski Energy-momentum Tensor

5.1 The Maxwell Stress Tensor

5.2 Balance Laws of Electromagnetic Momentum and Energy

5.3 The Minkowski Energy-momentum Tensor

5.4 Interaction of Fields with Matter

6. Total Energy-momentum Tensors

6.1 Closed Systems and Open Systems

6.2 Total Energy-momentum Tensor

6.3 The Principle of Virtual Power

6.4 Discussion

7. The Theory of Electrons and Statistical Mechanics

7.1 Microscopic and Macroscopic Field Equations

7.2 Momentum Equation for Composite of Particles

7.3 Equations of Statistical Mechanics

7.4 Discussion

8. Macroscopic Maxwell-Lorentz Forces

8.1 The EBPMv and EBPM Formulations

8.2 The Chu Formulation

8.3 Discussion

9. Magnetostatic Forces on a Whole Body

9.1 Pole, Dipole, and Current-circuit Models of Magnetizations

9.2 Body Forces and Surface Forces

9.3 Various Stress Tensors and Momentum Equations

9.4 Discussion

10. Models for Field-matter Interactions

10.1 Electric and Magnetic Dipoles and Current-circuits

10.2 Force, Couple, and Energy Supply of the Two-dipole Model

10.3 The Dipole-current Circuit Model

10.4 Discussion

11. Summary of Electromagnetic Forces and Energy

12. Constitutive Equations and Boundary Conditions

12.1 Constitutive Equations for the Two-dipole Formulation

12.2 Constitutive Equations for the Dipole-current Circuit Formulation

12.3 Boundary Conditions

12.4 Summary and Conclusion

13. References

V Problems in Magneto-solid Mechanics

1. Introduction

2. Methods

2.1 Magnetic Forces—Field Method

2.2 Magnetic Forces—Energy Method

3. Stability of Ferroelastic Structures in Magnetic Fields

3.1 Magnetoelastic Buckling of Beam-plates

3.2 Comparison of Buckling Theory and Experiment

3.3 Magnetoelastic Stability of Circular Rods

3.4 Plate Vibrations in a Magnetic Field

4. Mechanics of Elastic Conductors

4.1 Introduction

4.2 Continuum Models

4.3 Stresses in High Current Magnets and Coils

4.4 Virial Theorem and Force-free Magnets

4.5 Superconducting Magnets

4.6 Stability of Current-carrying Rods

4.7 Conducting Rods in Magnetic Fields

4.8 Effect of Currents on Plate Vibrations

4.9 Elastic Stability of Superconducting Magnets

4.10 Mechanical Properties of Superconductors

5. Dynamic Magnetic Forces in Solids

5.1 Introduction

5.2 Magnetic Generation of Stress Waves

5.3 Magnetic Forming in Metals

5.4 Magnetic Impulse Testing of Solids and Structures

5.5 Magnetic Forming Forces in Ferromagnetic Conductors

6. Epilogue

7. References

VI On Non-equilibrium Thermodynamics of Continua: Addendum

1. Introduction

2. Preliminaries

3. Second Law

3.1 On the Principle of Nondecreasing Entropy

3.2 Internal Forces

4. On the Nature of Evolutionary Equations

5. Normality Rules

6. References

Author Index

Subject Index

## Product details

- No. of pages: 446
- Language: English
- Copyright: © Pergamon 1978
- Published: January 1, 1978
- Imprint: Pergamon
- eBook ISBN: 9781483146317

## About the Editor

### S. Nemat-Nasser

#### Affiliations and Expertise

La Jolla, CA, USA