# Wave Propagation in Elastic Solids

## 1st Edition

### North-Holland Series in Applied Mathematics and Mechanics

**Authors:**J. D. Achenbach

**Editors:**H. A. Lauwerier W. T. Koiter

**eBook ISBN:**9781483163734

**Imprint:**North Holland

**Published Date:**1st January 1973

**Page Count:**440

## Description

Wave Propagation in Elastic Solids focuses on linearized theory and perfectly elastic media.

This book discusses the one-dimensional motion of an elastic continuum; linearized theory of elasticity; elastodynamic theory; and elastic waves in an unbounded medium. The plane harmonic waves in elastic half-spaces; harmonic waves in waveguides; and forced motions of a half-space are also elaborated. This text likewise covers the transient waves in layers and rods; diffraction of waves by a slit; and thermal and viscoelastic effects, and effects of anisotropy and nonlinearity. Other topics include the summary of equations in rectangular coordinates, time-harmonic plane waves, approximate theories for rods, and transient in-plane motion of a layer.

This publication is a good source for students and researchers conducting work on the wave propagation in elastic solids.

## Table of Contents

Preface

Introduction

The Propagation of Mechanical Disturbances

Continuum Mechanics

Outline of Contents

Historical Sketch

Bibliography

1. One-Dimensional Motion of an Elastic Continuum

1.1. Introduction

1.2. Nonlinear Continuum Mechanics in One Dimension

1.2.1. Motion

1.2.2. Deformation

1.2.3. Time-Rates of Change

1.2.4. Conservation of Mass

1.2.5. Balance of Momentum

1.2.6. Balance of Energy

1.2.7. Linearized Theory

1.2.8. Notation for the Linearized Theory

1.3. Half-Space Subjected to Uniform Surface Tractions

1.4. Reflection and Transmission

1.5. Waves in One-Dimensional Longitudinal Stress

1.6. Harmonic Waves

1.6.1. Traveling Waves

1.6.2. Complex Notation

1.6.3. Standing Waves

1.6.4. Modes of Free Vibration

1.7. Flux of Energy in Time-Harmonic Waves

1.7.1. Time-Average Power Per Unit Area

1.7.2. Velocity of Energy Flux

1.7.3. Energy Transmission for Standing Waves

1.8. Fourier Series and Fourier Integrals

1.8.1. Fourier Series

1.8.2. Fourier Integrals

1.9. The Use of Fourier Integrals

1.10. Problems

2. The Linearized Theory of Elasticity

2.1. Introduction

2.2. Notation and Mathematical Preliminaries

2.2.1. Indicial Notation

2.2.2. Vector Operators

2.2.3. Gauss' Theorem

2.2.4. Notation

2.3. Kinematics and Dynamics

2.3.1. Deformation

2.3.2. Linear Momentum and the Stress Tensor

2.3.3. Balance of Moment of Momentum

2.4. The Homogeneous, Isotropic, Linearly Elastic Solid

2.4.1. Stress-Strain Relations

2.4.2. Stress and Strain Deviators

2.4.3. Strain Energy

2.5. Problem Statement in Dynamic Elasticity

2.6. One-Dimensional Problems

2.7. Two-Dimensional Problems

2.7.1. Antiplane Shear

2.7.2. In-Plane Motions

2.8. The Energy Identity

2.9. Hamilton's Principle

2.9.1. Statement of the Principle

2.9.2. Variational Equation of Motion

2.9.3. Derivation of Hamilton's Principle

2.10. Displacement Potentials

2.11. Summary of Equations in Rectangular Coordinates

2.12. Orthogonal Curvilinear Coordinates

2.13. Summary of Equations in Cylindrical Coordinates

2.14. Summary of Equations in Spherical Coordinates

2.15. The Ideal Fluid

3. Elastodynamic Theory

3.1. Introduction

3.2. Uniqueness of Solution

3.3. The Dynamic Reciprocal Identity

3.4. Scalar and Vector Potentials for the Displacement Field

3.4.1. Displacement Representation

3.4.2. Completeness Theorem

3.5. The Helmholtz Decomposition of a Vector

3.6. Wave Motion Generated by Body Forces

3.6.1. Radiation

3.6.2. Elastodynamic Solution

3.7. Radiation in Two Dimensions

3.8. The Basic Singular Solution of Elastodynamics

3.8.1. Point Load

3.8.2. Center of Compression

3.9. Three-Dimensional Integral Representation

3.9.1. Kirchhoff's Formula

3.9.2. Elastodynamic Representation Theorem

3.10. Two-Dimensional Integral Representations

3.10.1. Basic Singular Solutions

3.10.2. Antiplane Line Load

3.10.3. In-Plane Line Load

3.10.4. Integral Representations

3.11. Boundary-Value Problems

3.12. Steady-State Time-Harmonic Response

3.12.1. Time-Harmonic Source

3.12.2. Helmholtz's Equation

3.12.3. Helmholtz's First (Interior) Formula

3.12.4. Helmholtz's Second (Exterior) Formula

3.12.5. Steady-State Solutions in Two Dimensions

3.13. Problems

4. Elastic Waves in an Unbounded Medium

4.1. Plane Waves

4.2. Time-Harmonic Plane Waves

4.2.1. Inhomogeneous Plane Waves

4.2.2. Slowness Diagrams

4.3. Wave Motions with Polar Symmetry

4.3.1. Governing Equations

4.3.2. Pressurization of a Spherical Cavity

4.3.3. Superposition of Harmonic Waves

4.4. Two-Dimensional Wave Motions with Axial Symmetry

4.4.1. Governing Equations

4.4.2. Harmonic Waves

4.5. Propagation of Wavefronts

4.5.1. Propagating Discontinuities

4.5.2. Dynamical Conditions at the Wavefront

4.5.3. Kinematical Conditions at the Wavefront

4.5.4. Wavefronts and Rays

4.6. Expansions Behind the Wavefront

4.7. Axial Shear Waves by the Method of Characteristics

4.8. Radial Motions

4.9. Homogeneous Solutions of the Wave Equation

4.9.1. Chaplygin's Transformation

4.9.2. Line Load

4.9.3. Shear Waves in an Elastic Wedge

4.10. Problems

5. Plane Harmonic Waves in Elastic Half-Spaces

5.1. Reflection and Refraction at a Plane Interface

5.2. Plane Harmonic Waves

5.3. Flux of Energy in Time-Harmonic Waves

5.4. Joined Half-Spaces

5.5. Reflection of SH-Waves

5.6. Reflection of P-Waves

5.7. Reflection of SV-Waves

5.8. Reflection and Partition of Energy at a Free Surface

5.9. Reflection and Refraction of SH-Waves

5.10. Reflection and Refraction of P-Waves

5.11. Rayleigh Surface Waves

5.12. Stoneley Waves

5.13. Slowness Diagrams

5.14. Problems

6. Harmonic Waves in Waveguides

6.1. Introduction

6.2. Horizontally Polarized Shear Waves in an Elastic Layer

6.3. The Frequency Spectrum of SH-Modes

6.4. Energy Transport by SH-Waves in a Layer

6.5. Energy Propagation Velocity and Group Velocity

6.6. Love Waves

6.7. Waves in Plane Strain in an Elastic Layer

6.8. The Rayleigh-Lamb Frequency Spectrum

6.9. Waves in a Rod of Circular Cross Section

6.10. The Frequency Spectrum of the Circular Rod of Solid Cross Section

6.10.1. Torsional Waves

6.10.2. Longitudinal Waves

6.10.3. Flexural Waves

6.11. Approximate Theories for Rods

6.11.1. Extensional Motions

6.11.2. Torsional Motions

6.11.3. Flexural Motions - Bernoulli-Euler Model

6.11.4. Flexural Motions - Timoshenko Model

6.12. Approximate Theories for Plates

6.12.1. Flexural Motions - Classical Theory

6.12.2. Effects of Transverse Shear and Rotary Inertia

6.12.3. Extensional Motions

6.13. Problems

7. Forced Motions of a Half-Space

7.1. Integral Transform Techniques

7.2. Exponential Transforms

7.2.1. Exponential Fourier Transform

7.2.2. Two-Sided Laplace Transform

7.2.3. One-Sided Laplace Transform

7.3. Other Integral Transforms

7.3.1. Fourier Sine Transform

7.3.2. Fourier Cosine Transform

7.3.3. Hankel Transform

7.3.4. Mellin Transform

7.4. Asymptotic Expansions of Integrals

7.4.1. General Considerations

7.4.2. Watson's Lemma

7.4.3. Fourier Integrals

7.4.4. The Saddle Point Method

7.5. The Methods of Stationary Phase and Steepest Descent

7.5.1. Stationary-Phase Approximation

7.5.2. Steepest-Descent Approximation

7.6. Half-Space Subjected to Antiplane Surface Disturbances

7.6.1. Exact Solution

7.6.2. Asymptotic Representation

7.6.3. Steepest-Descent Approximation

7.7. Lamb's Problem for a Time-Harmonic Line Load

7.7.1. Equations Governing a State of Plane Strain

7.7.2. Steady-State Solution

7.8. Suddenly Applied Line Load in an Unbounded Medium

7.9. The Cagniard-de Hoop Method

7.10. Some Observations on the Solution for the Line Load

7.11. Transient Waves in a Half-Space

7.12. Normal Point Load on a Half-Space

7.12.1. Method of Solution

7.12.2. Normal Displacement at z = 0

7.12.3. Special case λ = μ

7.13. Surface Waves Generated by a Normal Point Load

7.14. Problems

8. Transient Waves in Layers and Rods

8.1. General Considerations

8.2. Forced Shear Motions of a Layer

8.2.1. Steady-State Harmonic Motions

8.2.2. Transient Motions

8.3. Transient In-Plane Motion of a Layer

8.3.1. Method of Solution

8.3.2. Inversion of the Transforms

8.3.3. Application of the Method of Stationary Phase

8.4. The Point Load on a Layer

8.5. Impact of a Rod

8.5.1. Exact Formulation

8.5.2. Inversion of the Transforms

8.5.3. Evaluation of the Particle Velocity for Large Time

8.6. Problems

9. Diffraction of Waves by a Slit

9.1. Mixed Boundary-Value Problems

9.2. Antiplane Shear Motions

9.2.1. Green's Function

9.2.2. The Mixed Boundary-Value Problem

9.3. The Wiener-Hopf Technique

9.4. The Decomposition of a Function

9.4.1. General Procedure

9.4.2. Example: The Rayleigh Function

9.5. Diffraction of a Horizontally Polarized Shear Wave

9.6. Diffraction of a Longitudinal Wave

9.6.1. Formulation

9.6.2. Application of the Wiener-Hopf Technique

9.6.3. Inversion of Transforms

9.7. Problems

10. Thermal and Viscoelastic Effects, and Effects of Anisotropy and Non-Linearity

10.1. Thermal Effects

10.2. Coupled Thermoelastic Theory

10.2.1. Time-Harmonic Plane Waves

10.2.2. Transverse Waves

10.2.3. Longitudinal Waves

10.2.4. Transient Waves

10.2.5. Second Sound

10.3. Uncoupled Thermoelastic Theory

10.4. The Linearly Viscoelastic Solid

10.4.1. Viscoelastic Behavior

10.4.2. Constitutive Equations in Three Dimensions

10.4.3. Complex Modulus

10.5. Waves in Viscoelastic Solids

10.5.1. Time-Harmonic Waves

10.5.2. Longitudinal Waves

10.5.3. Transverse Waves

10.5.4. Transient Waves

10.5.5. Propagation of Discontinuities

10.6. Waves in Anisotropic Materials

10.7. A Problem of Transient Nonlinear Wave Propagation

10.8. Problems

Author Index

Subject Index

## Details

- No. of pages:
- 440

- Language:
- English

- Copyright:
- © North Holland 1973

- Published:
- 1st January 1973

- Imprint:
- North Holland

- eBook ISBN:
- 9781483163734