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# Mechanics Today

## Volume 3

- 1st Edition - January 1, 1976
- Editor: S. Nemat-Nasser
- Language: English
- Hardback ISBN:9 7 8 - 0 - 0 8 - 0 1 9 8 8 2 - 8
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 1 9 0 9 - 0
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 1 5 1 - 9

Mechanics Today, Volume 3 provides the advances in the fields of solid and fluid mechanics and applied mathematics. This volume is divided into six chapters that discuss the… Read more

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Request a sales quoteMechanics Today, Volume 3 provides the advances in the fields of solid and fluid mechanics and applied mathematics. This volume is divided into six chapters that discuss the fundamentals and analytical and experimental results of dynamic behavior of linear and nonlinear systems. Chapter I provides a formulation of the effective stiffness theory with equations of motion and boundary conditions presented for the case of plain strain motion. Chapter II summarizes some of the analytical results that have been obtained in an effort to improve understanding of elastodynamic fracture processes. Chapter III presents the matrix difference equations used to formulate problems related to random vibration of periodic and almost periodic structures, taking advantage of the identical construction of the interconnecting units. Chapter IV describes a basic approach to the Oseen problem through the use of integral representations of the velocity and pressure fields. Chapter V deals with an analysis of nonlinear gyroscopic systems and the motions of high-order nongyroscopic systems. Chapter VI focuses on the application of the WKB perturbation method in the study of static deformation, vibration, wave propagation, and instability of elastic bodies. This volume is of great value to solid and fluid mechanics specialists and also to non-specialists with sufficient background of the field.

Contributors

Preface

Contents of Volume 1

Contents of Volume 2

Summary

I On Modeling the Dynamics of Composite Materials

1. Introduction

2. Theoretical and Experimental Background

2.1 Steady-State Waves

2.2 Transient Waves

3. Structured Continuum Theory

3.1 Coordinate Systems

3.2 Displacement Expansions

3.3 Interface Continuity Conditions

3.4 Equations of Motion

3.5 Boundary Conditions

4. Results of the Theory

4.1 Dispersion Results

4.2 Modal Results

4.3 Transient Results

5. Experimental Methods

5.1 Ultrasonic Waves

5.2 Low-Amplitude Transient Waves

5.3 High-Amplitude Transient Waves

6. Potential Future Developments

7. References

II The Analysis of Elastodynamic Crack Tip Stress Fields

1. Introduction

2. Equations of Linear Elastodynamics

2.1 Equations of Motion

2.2 Work and Energy Relations

2.3 The Laplace Transform

3. Crack Tip Stress Fields

4. Energy Considerations

4.1 Conservation Laws

4.2 A Path-Independent Integral for Dynamic Loading

4.3 Energy Flux into a Moving Crack Tip

4.4 Uniqueness of Solution for Running Cracks

5. Stationary Crack under Dynamic Loading

5.1 The Weight Function Method

5.2 An Example

6. Crack Extension at Nonuniform Rates

6.1 The Stress Intensity Factor

6.2 The Crack Tip Equation of Motion

7. Continuous Distribution of Moving Dislocations

7.1 The Superposition Method

7.2 An Example

8. Discussion

9. References

III Random Vibration of Periodic and Almost Periodic Structures

1. Introduction

2. Discrete Periodic Systems

3. Continuous Periodic System—Point Loading

4. Continuous Periodic System—Convected Loading

5. Concluding Remarks

6. References

Appendix A

IV Integral Representations and the Oseen Flow Problem

1. Introduction

2. Integral Formulation

2.1 The Fundamental Solution

2.2 Integral Representation of the Oseen Solution

2.3 Integral Equations for the Stress Components

3. Structure of the Problem

3.1 The Inhomogeneous Oseen Problem

3.2 Force Relationships and Drag Invariance

3.3 Asymptotic Behavior at Infinity

3.4 Uniqueness of the Oseen Solution

3.5 Existence of the Oseen Solution

3.6 The Navier-Stokes Solution

3.7 The Oseen Problem in Two Dimensions

4. Results on Integral Equations

4.1 Equations on a Semi-infinite Interval

4.2 Equations on a Finite Interval

4.3 Equations on Two Disjoint Intervals

4.4 Equations on Circular Arcs

4.5 Singularities and Positivity of the Solutions

4.6 Variational Problems

5. Solutions of the Oseen Problem for Planar Flows

5.1 Semi-infinite Plate Problems

5.2 Green's Tensor for the Semi-infinite Plate

5.3 Finite Plate Problems

5.4 Circular Arc Problems

5.5 Application of Variational Methods

5.6 Injection-Suction Problems

5.7 Free Surface Problems

6. Concluding Remarks

7. References

V On Nonlinear Gyroscopic Systems

1. Introduction

2. Derivation of Equations of Motion

2.1 Equations of Motion for a Holonomic Dynamical System with Imposed Motions

2.2 Equations of Motion for a Holonomic Dynamical System with Cyclic Coordinates

3. Weakly Nonlinear Systems and Linearization

4. Weakly Nonlinear Systems—Hamiltonian Formulation

4.1 An Example

5. High Spin Gyroscopic Systems

5.1 Initial Motions

5.2 Analysis in the Case When Initial Velocities are Small

5.3 Example: Single Axis Stable Platform

6. References

VI Application of the WKB Method in Solid Mechanics

1. Introduction

2. First Approximation

2.1 Green-Liouville Transformation

2.2 Boundary-Layer Problems

2.3 Eigenvalue Problems

2.4 Steady-State Wave Propagation

2.5 Transient Pulse Propagation

2.6 Conservative Systems

3. Successive Corrections

3.1 Formal Asymptotic Expansion

3.2 Convergent Expansion

3.3 Corrections to Eigenvalues

3.4 Reflections in Pulse Propagation

4. Formal Expansions for Transition Points

4.1 Coalescence of Two Roots at Zero

4.2 More General Coalescence of Two Roots

4.3 Coalescence of Four Roots at Zero

5. Vibration with Transition Point

5.1 Torsion of Elastically Constrained Rod

5.2 Taut Beam on Elastic Foundation

5.3 Beam on Elastic Foundation

5.4 Shell of Revolution

6. References

Author Index

Subject Index

- No. of pages: 318
- Language: English
- Edition: 1
- Published: January 1, 1976
- Imprint: Pergamon
- Hardback ISBN: 9780080198828
- Paperback ISBN: 9781483119090
- eBook ISBN: 9781483151519

SN

### S. Nemat-Nasser

Affiliations and expertise

La Jolla, CA, USARead

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