Elastic Wave Propagation in Structures and Materials

430 Pages·2022·23.629 MB·English

by  Srinivasan Gopalakrishnan

Elastic Wave
Propagation in
Structures and
Materials
Elastic Wave Propagation in Structures and Materials initiates with a brief in-
troduction to wave propagation, different wave equations, integral transforms
including fundamentals of Fourier Transform, and its numerical implementa-
tion. The concept of spectral analysis and procedure to compute the wave pa-
rameters, wave propagation in 1-D isotropic waveguides, and wave dispersion
in 2-D waveguides are explained. Wave propagation in different media such as
laminated composites, functionally graded structures, granular soils, and non-
local elasticity models is addressed. Separate chapters on signal processing and
wave propagation in viscoelastic waveguides are included in this book. The entire
book is written in modular form for easy understanding of the basic concepts.
Features:
• Brings out the idea of wave dispersion and its utility in the dynamic responses
• Introduces concepts such as Negative Group Speeds, Einstein’s Causality, and
escape frequencies using solid mathematical framework
• Discusses the propagation of waves in materials such as laminated composites
and functionally graded materials
• Proposes spectral finite element as analysis tool for wave propagation
• Supports each concept/chapter with homework problems and MATLAB® codes
This book is aimed at senior undergraduates and advanced graduates in all fields
of engineering, especially Mechanical and Aerospace Engineering.

Elastic Wave
Propagation in
Structures and
Materials
Srinivasan Gopalakrishnan
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks
does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of
MATLAB® software or related products does not constitute endorsement or sponsorship by The
MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Srinivasan Gopalakrishnan
Reasonable efforts have been made to publish reliable data and information, but the author and pub-
lisher cannot assume responsibility for the validity of all materials or the consequences of their use.
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Trademark notice: Product or corporate names may be trademarks or registered trademarks and are
used only for identification and explanation without intent to infringe.
ISBN: 9780367637576 (hbk)
ISBN: 9780367637637 (pbk)
ISBN: 9781003120568 (ebk)
DOI: 10.1201/9781003120568
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
This book is dedicated to my Late mother
Saraswathi, my father Srinivasan,
my wife Anu and
my Children
Karthik and Keerthana

5
Contents
Preface xv
Author Biography xvii
hapter
C 1(cid:4) Introduction to Wave Propagation 1
1.1 ESSENTIALCOMPONENTSOFAWAVE 2
1.2 INTERFERENCEOFWAVES 5
1.2.1 Interference of Two Similar Waves Propagating
in the Same Direction 6
1.2.2 Standing Waves 7
1.3 NEEDFORWAVEPROPAGATIONANALYSISIN
STRUCTURESANDMATERIALS 9
1.4 SOME WAVE PROPAGATION PROBLEMS IN SCIENCE
ANDENGINEERING 13
1.5 ORGANIZATIONANDSCOPEOFTHEBOOK 16
hapter
C 2(cid:4) Introduction to Fourier Transforms 21
2.1 FOURIERTRANSFORMS 22
2.1.1 Continuous Fourier Transforms (CFT) 22
2.1.2 Fourier Series 27
2.1.3 Discrete Fourier Transform 29
2.2 COMPARATIVEMERITSANDDEMERITSOFFFT 31
®
NOTEONMATLAB SCRIPTSPROVIDEDINTHIS
CHAPTER 33
SUMMARY 33
EXERCISES 33
vii
viii (cid:4) Contents
hapter
C 3(cid:4) Introduction to Wave Propagation in Structures 37
3.1 CONCEPT OF WAVENUMBER, GROUP SPEEDS AND
PHASESPEEDS 38
3.2 WAVEPROPAGATIONTERMINOLOGIES 41
3.3 SPECTRALANALYSISOFMOTION 43
3.3.1 Second-Order System 43
3.3.2 Fourth-Order System 46
3.4 GENERAL FORM OF WAVE EQUATION AND THEIR
CHARACTERISTICS 48
3.4.1 General Form of Wave Equations 49
3.4.2 Characteristics of Waves in Anisotropic Media 49
3.4.3 Characteristics of Waves in Inhomogeneous Media 50
3.4.4 Characteristics of Waves in Non-local Waveguides 51
3.5 DIFFERENTMETHODSOFCOMPUTING
WAVENUMBERSANDWAVEAMPLITUDES 52
3.5.1 Method - 1 : The Companion Matrix & the SVD
Technique 53
3.5.2 Method - 2 : Linearization of Polynomial
Eigenvalue Problem (PEP) 54
SUMMARY 55
EXERCISES 56
hapter
C 4(cid:4) WavePropagationinOne-DimensionalIsotropic
Structural Waveguides 59
4.1 WAVEPROPAGATIONIN1-DELEMENTARY
WAVEGUIDES 60
4.2 LONGITUDINALWAVEPROPAGATIONINRODS 61
4.2.1 D’Alemberts Solution 63
4.2.2 Spectral Analysis 63
4.2.3 PropagationofWavesinanInfiniteLongitudinal
Waveguide 64
4.2.4 Interaction of Waves with Fixed and Free
Boundaries 66
4.2.5 Reflection from an Elastic Boundary 67
4.2.6 ReflectionandTransmissionfromaJointHaving
Concentrated Mass and Stepped Rod 67
Contents (cid:4) ix
4.3 FLEXURALWAVEPROPAGATIONINBEAMS 73
4.3.1 Wave Propagation Analysis 74
4.3.2 Propagation of Waves in an Infinite Beam 76
4.3.3 Reflection from Boundaries 77
4.3.4 Reflection from Elastic Boundary 79
4.3.5 Reflection and Transmission from a Stepped
Beam and a Joint with Concentrated Mass 81
4.3.6 WavePropagationinBeamswithPre-Tensionor
Pre-Compression 85
4.3.7 Wave Propagation in a Beam on Elastic
Foundation 89
4.3.8 Wave Propagation in a Framed Structure 90
4.4 WAVEPROPAGATIONINHIGHER-ORDERWAVEGUIDES 97
4.4.1 Wave Propagation in Timoshenko Beam 97
4.4.2 Wave Propagation in Mindlin-Herrmann Rod 103
4.5 WAVEPROPAGATIONINROTATINGBEAMS 110
4.6 WAVEPROPAGATIONINTAPEREDWAVEGUIDES 113
4.6.1 Wave Propagation in Tapered Rod Having
Exponential Depth Variation 114
4.6.2 Wave Propagation in Tapered Rod Having
Polynomial Depth Variation 116
4.6.3 Wave Propagation in Tapered Beam 119
®
NOTEONMATLAB SCRIPTSPROVIDEDINTHIS
CHAPTER 121
SUMMARY 122
EXERCISES 123
hapter
C 5(cid:4) Wave Propagation in Viscoelastic Waveguides 129
5.1 CONSTITUTIVEMODELSFORVISCOELASTIC
WAVEGUIDES 131
5.1.1 Two-Parameter Models 132
5.1.2 Three-Parameter Models 133
5.2 WAVEPROPAGATIONINVISCOELASTICROD 137
5.2.1 Wave Propagation in Two-Parameter
Viscoelastic Rod 139
5.2.2 Wave Propagation in Three-Parameter
Viscoelastic Rod 141