Theory of Elastic Waves

371 Pages·2022·8.391 MB·English

by  Peijun Wei

Peijun Wei
Theory of
Elastic Waves
Theory of Elastic Waves
Peijun Wei
Theory of Elastic Waves
Peijun Wei
Department of Applied Mechanics
School of Mathematics and Physics
University of Science and Technology
Beijing
Beijing, China
ISBN 978-981-19-5661-4 ISBN 978-981-19-5662-1 (eBook)
https://doi.org/10.1007/978-981-19-5662-1
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Preface
The theory of elastic wave propagation in complex media is of great importance. For
example, it is desired to understand the mechanism of propagation and attenuation
of seismic wave in the stratum for the seismic exploration. Based on the propagation
and attenuation law, the morphology and physical parameters can be obtained after
inverse analysis of the wave motion signals collected by seismic detector in the wild
field, and thus provide useful information about the distribution and the location
of underground oil and gas deposit. In the area of medical science, the ultrasound
inspection imaging also needs to understand the patterns of propagation and dissi-
pation of elastic wave in the bio-tissue. These laws can be used to design various
imaging algorithm to reconstruct the colored images of bio-tissue and organs based
on the inverse analysis of the reflection and attenuation information and thus help
clinical doctors to acquire the scientific evidence of disease diagnosis. In the area of
industrial manufacturing, the knowledge regarding elastic wave propagation char-
acteristics can be used in non-destructive testing (NDT) of materials and structures.
The information about the propagation constants in the interior, the reflection and
transmission at interface, the scattering at the inclusions and voids can be collected
by the sensor arranged at surface or interior of material. Further performing the
signal processing such as the de-noising and spectrum analysis by computer, the
location, distribution, geometry and size of the flaws inside the materials would be
available. This is crucial to maintain good quality of products and to guarantee the
security service of structures. No matter the seismic exploration, medical ultrasound
imaging and non-destructive testing of material, the basic scientific problem is the
law of elastic wave propagation in complex media. This book focuses on the basic
scientific problem. The contents are arranged by consideration of the completeness
of theory and research methods and the necessary involvement of latest research
results. Under such guideline, this book is finally finished and named Theory of
Elastic Waves.
Among all the books on elastic wave propagation theory, the Wave Propagation in
Elastic Solids (by J. D. Achenbach, Elsevier Science Publishers, 1973), Diffraction
of Elastic Waves and Dynamic Stress Concentrations (by Pao Y. H., Mao C. C. Grane,
Russak & Company Inc, 1973) and Ultrasonic Waves in Solid Media (by J. L. Rose,
v
vi Preface
Cambridge University Press, 1999) are of great influence. However, these books
were published early. Many new scientific problems and new analytic and numerical
methods have emerged during the past decades. It becomes necessary to recall and
summarize these analytic and numerical methods systematically which dispersed in
massive literatures. This is one motivation of writing this book. On the other hand,
the teaching experience of giving the lecture for graduate students for about ten
years makes me more and more feel that the advanced textbooks which are more
suited to the graduate students are rare and become the desideratum to accommodate
the research-type teaching. This is another motivation to write this book. Based on
my research efforts on the elastic wave propagation in complex media, I strive to
incorporate the latest research results into the contents of book. This book is finally
finished based on the lecture notes and after several addition and amendment.
This book focuses on the elastic wave propagation in isotropic media; no contents
of special subjects are included. If the basic theory and methods about elastic wave
propagation in the isotropic solid are grasped, then a firm and strong foundation
is established to further study the elastic wave propagation in anisotropic solid,
viscoelastic solid, porous solid, etc. With the consideration of systematicness and
completeness, the bulk wave, the surface wave and the guided wave are all involved in
this book. Firstly, the dispersion and attenuation features of elastic wave propagating
in elastic media of infinite extension are addressed. Many important conceptions
about elastic wave propagation are provided in this chapter. Then, the reflection and
transmission problems at interfaces are addressed. The reflection and transmission
of single interfaces is first discussed. Apart from flat interfaces, the periodic undula-
tion interfaces are also involved. Moreover, the various imperfect interfaces are also
involved beside the perfect interface. The layered media are often met in actual engi-
neering fields as a common type. Then, the elastic wave propagation through layered
structure is addressed based on the reflection and transmission investigation at single
interface. Consider the fact that the reflecting and transmitting features of elastic wave
in layered media can be embodied in the sandwiched structure with two interfaces; we
addressed the reflection and transmission of sandwiched structure with two interfaces
with great interest. The simultaneous interface condition method, the transfer matrix
method, the stiffness matrix method, the multiple reflection/transmission method, the
super-interface method and the state transition matrix method, basically all main-
stream research methods, are all included. These methods can easily be extended to
the laminated structure with arbitrary N layers with appropriate modification. In the
chapter of surface wave, not only the classic Rayleigh wave, Love wave and Stanley
wave, but also the rotating surface wave is also addressed. The rotation surface wave
is a natural result in the cylindrical coordinate system of the surface waves studied in
the rectangular coordinates. In the chapter of guided wave, the guided wave propa-
gation in bar, pipe, beam, plate, cylindrical shell and spherical shell is all involved. In
particular, the elastic wave propagation in spherical shells is rarely mentioned in the
existing published books and literatures. The guided waves propagation in spherical
shell is addressed in this chapter, and the comparison with the vibration mode is also
made. Moreover, the leaky waves due to the liquid loads are also addressed in this
chapter. These contents make a distinguishing feature of this book.
Preface vii
The main focus of this book is the basic theory and analytic methods of elastic
wave propagation problem. This book is suitable for those who work in the field of
seismic survey, the material characterizing and non-destructive testing, the medical
ultrasound imaging, the phononic crystal/metamaterial and the structure health moni-
toring, especially for high-grade undergraduate and postgraduate students as textbook
for the systematic study of elastic wave propagation theory.
During the process of writing this book, my Ph.D. students, including Guo Xiao,
Zhang Peng, Li Yueqiu, Li Li, Wei Zibo, Xu Chunyu, Ma Zhanchun, Xu Yuqian,
Zhao Lingkang, Wang Ziwei, Zhao Lina, etc., did a great deal of work, including
but not limited to the typing and checking of manuscript, text layout and drawing
illustrations. Here, I’ll express my loyal thanks for their helpful works. Moreover,
I’ll also express my thanks for the financial support from “National Natural Science
Foundation of China (No. 11872105, No. 12072022)” and “Project of Graduate
Textbook Construction of University of Science and Technology Beijing.”
There are inevitably some mistakes existing in the book due to the author’s
limited level and careless omissions, and I’m pleased and encourage whoever to
give comments and correct mistakes.
Beijing, China Peijun Wei
September 2020
Introduction
The theory of elastic wave propagation in complex media is widely used in many
fields, such as geophysical exploration, seismic survey, medical ultrasound imaging
and non-destructive testing of material and structure. However, the books which
systematically introduce the theory of elastic wave propagation are rare. This book
systematically introduced the basic theory of elastic wave propagation in isotropic
solid media, including elastic wave propagation in infinite media, reflection and
transmission of elastic wave at interfaces, reflection and transmission of elastic wave
through layered structure with finite thickness, Rayleigh wave and Love wave prop-
agating along the surface of semi-infinite solid and covering layer, the guided waves
and leaky waves in flat plates and in cylindrical rods. The propagation patterns and
features of guided waves in cylindrical shells and spherical shells are also intro-
duced. The single scattering and multiple scattering of elastic waves, although very
important also, but are not included due to the limitation of length. The author has
been teaching the course of Theory of Elastic Wave for graduate students for over ten
years. At the same time, the author also has been conducting the research works on
the elastic wave propagation in complex media and the actual applications for over
two decades. Hence, this book is written based on the lecture notes of “elastic wave
theory” and has combined with the related research results. The entire book is divided
into six chapters and is mainly focused on the basic theory and the systematicness
of analytic methods.
This book is suitable to those who work in the fields of geophysical exploration,
non-destructive testing, medical ultrasound imaging, phononic crystal, metamate-
rial, structure health monitoring and so on. Especially, it is suited to the high-grade
undergraduate and graduate students to study the elastic wave theory systematically
as textbook.
ix
Contents
1 Fundamentals of Elastodynamics ................................ 1
1.1 Basic Hypothesis of Elastodynamics .......................... 1
1.1.1 Continuity Hypothesis ................................ 1
1.1.2 Elasticity Hypothesis ................................. 1
1.1.3 Small Deformation Hypothesis ......................... 2
1.1.4 Homogeneous Hypothesis ............................. 2
1.1.5 Isotropic Hypothesis .................................. 2
1.1.6 Zero Initial Stress Hypothesis .......................... 3
1.2 Basic Conservation Laws of Elastodynamics ................... 3
1.2.1 Law of Mass Conservation ............................ 3
1.2.2 Law of Conservation of Momentum .................... 5
1.2.3 The Law of Conservation of Energy .................... 6
1.3 Variational Principle of Elastodynamics ........................ 7
1.4 The Initial Boundary Value Problem of Elastodynamics .......... 10
1.5 Transient and Steady-State Problems .......................... 12
2 Elastic Waves in an Infinite Medium ............................. 15
2.1 Scalar Potential and Vector Potential .......................... 15
2.2 Solution of Wave Equation ................................... 19
2.3 Properties of Plane Waves ................................... 31
2.3.1 Propagation Mode of Plane Waves ...................... 31
2.3.2 The Stress Distribution on the Wavefront ................ 34
2.3.3 The Energy Flow Density of a Plane Wave ............... 35
2.4 Inhomogeneous Plane Wave ................................. 51
2.5 Spectrum Analysis of Plane Wave ............................. 59
3 Reflection and Transmission of Elastic Waves at Interfaces ......... 63
3.1 Classification of Interfaces and Plane Waves .................... 64
3.1.1 Perfect Interface and Imperfect Interface ................ 64
3.1.2 P Wave, S Wave and SH Wave ......................... 70
3.2 Reflection of Elastic Waves on Free Surface .................... 73
3.2.1 Reflection of P Wave on Free Surface ................... 73
xi
xii Contents
3.2.2 Reflection of SH Waves on Free Surface ................. 83
3.2.3 Reflection of SV Waves on Free Surface ................. 84
3.2.4 Incident P Wave and SV Wave Simultaneously ........... 90
3.3 Reflection and Transmission of Elastic Waves at the Interface ..... 95
3.3.1 Reflection and Transmission of P Waves at the Interface ... 95
3.3.2 Reflection and Transmission of SH Waves
at the Interface ....................................... 103
3.3.3 Reflection and Transmission of SV Waves
at the Interface ....................................... 106
3.3.4 P Wave and SV Wave Incidence Simultaneously .......... 112
3.4 Reflection and Transmission of Waves at the Periodic
Corrugated Interface ........................................ 128
4 Reflection and Transmission of Elastic Waves in Multilayer
Media ......................................................... 151
4.1 Simultaneous Interface Conditions Method ..................... 151
4.2 Transfer Matrix Method ..................................... 161
4.3 Stiffness Matrix Method ..................................... 167
4.4 Multiple Reflection/Transmission Method ...................... 175
4.5 Super-Interface Method ..................................... 179
4.6 The State Transfer Equation Method .......................... 193
4.7 Bloch Waves in Periodic Layered Structures .................... 207
5 Surface Wave and Interface Wave ............................... 225
5.1 P-type Surface Waves and SV-Type Surface Waves .............. 225
5.2 Rayleigh Wave ............................................. 228
5.2.1 Rayleigh Wave’s Wave Function ....................... 228
5.2.2 Rayleigh Equation ................................... 231
5.2.3 The Displacement Field of the Ryleigh Wave ............ 233
5.2.4 Rayleigh Wave in Elastic Half-Space with Cover Layer .... 236
5.3 Love Wave ................................................ 251
5.3.1 The Displacement Distribution of Love Wave ............ 252
5.3.2 The Dispersion Equation of Love Wave ................. 255
5.4 Stoneley Wave ............................................. 258
5.4.1 Wave Function of Stoneley Wave ....................... 259
5.4.2 Stoneley Equation .................................... 262
5.5 Torsional Surface Wave ..................................... 264
6 Guided Waves ................................................. 283
6.1 Flexural Waves in Beams .................................... 283
6.2 Flexural Waves in Plate ...................................... 299
6.3 Guided Waves in Plate (Lamb Wave) .......................... 309
6.3.1 Mixed Boundary Condition ............................ 312
6.3.2 Free Boundary Conditions ............................. 316
6.3.3 Fixed Boundary Condition ............................ 318
6.3.4 Liquid Load on Both Sides ............................ 320