Methods of the Classical Theory of Elastodynamics

328 Pages·1993·14.445 MB·English

by  Professor Dr. Vladimir B. Poruchikov (auth.)

Methods
of the Classical Theory
of Elastodynamics
Vladimir B. Poruchikov
Methods
of the Classical Theory
of Elastodynamics
With 74 Figures
Springer-Verlag
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Budapest
Professor Dr. Vladimir B. Poruchikov
Institute of Mechanics, Moscow State University
1, Michurinsky Prospect, 117192 Moscow, Russia
Translators:
Dr. V.A. Khokhryakov (Chaps. 1-4)
kv. 44, d. 1110, st. Malaya Ostroumovskaya, Moscow 107014 Russia
Dr. G. P. Groshev (Chaps. 5-7)
kv. 57, d. 4a, st. Serov, Zhukovsky town, Moscow region 140160, Russia
Title of the original Russian edition: Metody, dinamicheskol teorii uprugosti Nauka, Moscow 1986
ISBN-13:978-3-642-77101-9 e-JSBN-13:978-3-642-77099-9
DOl: 10.1007/978-3-642-77099-9
Library of Congress Cataloging-in-Publication Data. Poruchikov, V. B. (Vladimir Borisovich) [Metody di
namicheskol teorii uprugosti. English1 M ethods of the classical theory of elastodynamics / V. B. Poruchikov.
p. cm. Translation of: Metody dinamicheskol teorii uprugosti. Includes bibliographical references and index.
ISBN-13:978-3-642-77101-9 (US.) 1. Elasticity. l. Title. TA418.P6713 1992 531'.382--dc20 92-6511
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Preface
Numerous significant problems of present-day theoretical and applied science are
closely related to studies of wave propagation in continuous media, determination
of stress fields induced within bodies by impingement of waves or impulsive
loadings, as well as to calculations of the behavior of structures subjected to sudden
shocks. Primarily, this class of problems incorporates studying the effects of blast
and seismic waves, hypervelocity impacts of bodies upon obstacles, penetration of
solid bodies into compressible media, and crack propagation following impulsive
loadings.
To solve these problems, one has to employ methods of continuum mechanics
and, in particular, those of elastodynamics, which has recently been most in
tensively developed. However, until recently, there have been no books available
presenting a comprehensive and systematic analysis of rigorous analytical methods
of elastodynamics, which, although thoroughly developed during the last 25-30
years, are presented in a variety of hardly to obtain publications.
In 1986 a monograph by the present author, Methods o/the Dynamic Theory
0/ Elasticity (Nauka, Moscow) which attempted to fill the gap, was published in
Russian. It considered various methods of obtaining accurate analytical solutions
to dynamic problems of the linear theory of elasticity: both the classical ones
which were thoroughly developed during the last decades, and the ones proposed
quite recently.
The present book is an English translation of the revised Russian edition. It
should be noted that in preparing the English edition the list of references has been
thoroughly updated and extended and now adequately reflects recent progress in
analytical methods of elastodynamics. References difficult to obtain outside this
country were omitted while English language ones were retained. Although many
Soviet science journals are presently translated into English, in those cases where
the author had no reliable information on the availability of translated editions,
references to original Russian publications are given.
The monograph is addressed to research scientists, engineers and specialists
working in various branches of mechanical engineering and aerospace technology,
structural mechanics, seismology, as well as to students and postgraduates engaged
in studying transient processes in continuous media.
VI Preface
The author is sincerely grateful to Dr. V.A. Khokhryakov and Dr. G.P. Groshev
whose hard work in translating the book was of great help for publishing it in
English.
Moscow. August 1992 V. B. Poruchikov
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Formulation of Elastodynamic Problems.
Some General Results ...................................... 4
2. f Fundamental Equations of Elastodynamics ... . . . . . . . . . . . . . . 4
2.2 Initial and Boundary Conditions. Interfaces ................. 6
2.3 Constraints Imposed on the Solution Behavior
in the Neighborhood of Singular Points/Curves .............. 9
2.4 Continuous and Discontinuous Solutions ................... 12
2.5 Uniqueness Theorem for Solutions to Elastodynamic Problems
with Strong Discontinuities .............................. 15
2.6 The Green-Volterra Formula ............................. 21
2.7 Various Representations of Solutions to the Equations
of Motion of a Homogeneous Isotropic Medium ............. 26
2.7.1 Lame Representation ............................ 26
2.7.2 The Case of a Separable Solution
to the Vector Wave Equation ...................... 31
2.7.3 Iacovache's Representation ....................... 37
2.7.4 Representation Employing Papkovich-Neuber
Functions ...................................... 39
2.8 On the Relationships Between Solutions of Transient Dynamic
Problems and Those of Static, Steady-State and Stationary
Dynamic Problems ............................•........ 40
3. The Method of Functionally Invariant Solutions
(the Smirnov-Sobolev Method) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Functionally Invariant Solutions to the Wave Equation ........ 44
3.2 Plane and Complex Waves .............................. 46
3.2.1 Reflection of Plane Longitudinal
and Transverse Waves ........................... 47
3.2.2 The Case of Total Internal Reflection ............... 52
3.2.3 Rayleigh Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Homogeneous Solutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1 Construction of Homogeneous Solutions
to the Wave Equation ............................ 57
3.3.2 Diffraction of a Plane Shear Wave by a Wedge ....... 61
vm
Contents
3.4 The Case of an Elastic Half-Plane ........................ 64
3.4.1 Reduction of the Dynamic Problem to Superposition
of Problems for Symmetric and Antisymmetric
Components of the Displacement Vector ............ 65
3.4.2 Homogeneous Solutions .......................... 66
3.4.3 Solution of Some Problems ....................... 71
3.5 Mixed Boundary-Value Problems for an Elastic Half-Plane.
Crack Propagation ..................................... 77
3.5.1 Representations of Solutions
to Mixed Boundary-Value Problems ................ 77
3.5.2 Solution for the n ::::: I-Case. Some Examples ........ 79
3.5.3 Solution for the n :::; O-Case ....................... 87
3.6 Solution of Analogous Mixed Boundary-Value Problems.
Wedge-Shaped Punch ................................•. 91
3.6.1 Solution for the n ::::: I-Case ....................... 91
3.6.2 Indentation of a Wedge
with a Sub-Rayleigh Contact Speed ................ 94
3.6.3 Indentation of a Wedge
with Super-Rayleigh Contact Speed ................ 98
3.6.4 Solution Singularities at the Edges
of the Contact Region ........................... 104
3.7 Interrelation Between Three- and Two-Dimensional Problems .. 106
3.8 Application of the Smirnov-Sobolev Method to Solving
Axisymmetric Elastodynamic Problems .................... 112
3.8.1 Representation of Axisymmetric Solutions
by Employing Analytical Functions ... . . . . . . . . . . . .. 112
3.8.2 Solutions to Axisymmetric Problems . . . . . . . . . . . . . .. 117
3.9 Solutions to Some Axisymmetric Problems
with Mixed Boundary Conditions ......................... 120
3.9.1 Solution for the n ::::: I-Case ....................... 121
3.9.2 Circular Crack Expansion
and a Conical Punch Indentation ................... 123
3.9.3 Expansion of a Circular Crack Due
to a Concentrated Load .......................... 129
3.10 An Alternative Derivation
of the Smirnov-Sobolev Representations ................... 133
4. Integral Transforms in Elastodynamics ..................... .. 137
4.1 Application of Integral Transforms
to Solving Elastodynamic Problems ....................... 138
4.2 Lamb's Problem for a Half-Plane ......................... 139
4.2.1 Solution to the Problem .......................... 140
4.2.2 Cagniard-de Hoop Method ........................ 142
4.3 Diffraction of an Acoustic Wave by a Rigid Sphere .......... 149
Contents IX
4.4 Expansion of an Acoustic Wave Solution for a Sphere
Over a Time-Dependent Interval .......................... 155
4.5 Diffraction of Acoustic Waves by a Rigid Cone ............. 160
4.5.1 Diffraction of a Plane Acoustic Wave by a Rigid Cone 160
4.5.2 Diffraction of a Spherical Acoustic Wave by a Cone .. 168
4.6 Diffraction of Elastic Waves by a Smooth Rigid Cone ........ 173
4.6.1 Diffraction of a Plane Longitudinal Wave by a Cone .. 173
4.6.2 Diffraction of a Spherical Elastic Wave by a Cone .... 176
4.7 Impact of a Circular Cylinder on a Stationary Obstacle ....... 179
4.7.1 Formulation and Solution of the Problem ............ 179
4.7.2 Analysis of the Solution at the Points
of a Cylinder's Axis ............................. 183
5. Solution to Three-Dimensional Elastodynamic Problems
with Mixed Boundary Conditions
for Wedge-Shaped Domains ................................. 190
5.1 Combined Method of Integral Transforms .................. 191
5.1.1 Problem Formulation ............................ 191
5.1.2 Problem Solution ............................... 192
5.2 Diffraction of a Spherical Elastic Wave
by a Smooth Rigid Wedge ............... . . . . . . . . . . . . . .. 204
5.2.1 Solution to the Problem .......................... 205
5.2.2 Analysis of the Obtained Solution .................. 210
5.3 Diffraction of an Arbitrary Incident Plane Elastic Wave
by a Rigid Smooth Wedge .............................. 212
6. Wiener-Hopf Method in Elastodynamics ...................... 220
6.1 Problems with a Stationary Boundary ..................... 220
6.1.1 A Semi-Infinite Punch ........................... 221
6.1.2 Analysis of the Punch Solution .................... 229
6.2 A Finite-Width Punch .................................. 231
6.2.1 Solution to the Problem .......................... 231
6.2.2 The Acoustic Case .............................. 233
6.3 Problems with Moving Boundary Edges ................... 236
6.3.1 Problem Formulation and Application
of Integral Transforms ........................... 236
6.3.2 Splitting of Fundamental Solutions ................. 240
6.3.3 Solution of the Two-Dimensional Dynamic Problem ... 243
6.4 Some Crack and Punch Problems ......................... 246
6.4.1 The Plane-Strain Problem
of a Semi-Infinite Crack Propagation ............... 246
6.4.2 The Antiplane Problem
of a Semi-Infinite Crack Propagation ............... 248
6.4.3 The Acoustic Problem for a Punch ................. 254
X Contents
7. Homogeneous Solutions to Dynamic Problems
for Anisotropic Elastic Media (Willis' Method) ................. 256
7.1 Studies in Elastodynamics for Anisotropic Media ............ 256
7.2 Solution to the First Boundary Value Problem . . . . . . . . . . . . .. 257
7.2.1 Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . .. 258
7.2.2 Two-Dimensional Case ........................... 269
7.2.3 Radon Transform ..•............................ 271
7.3 Solution to the Second Boundary-Value Problem ............ 272
7.3.1 Three-Dimensional Case ......•.................. 272
7.3.2 Two-Dimensional Case ........................... 274
7.4 Lamb's Problem ....................................... 275
7.4.1 Three-Dimensional Case ....... . . . . . . . . . . . . . . . . .. 276
7.4.2 Two-Dimensional Case ........................... 277
7.4.3 Isotropic Half-Space ............................. 277
7.5 The Wedge-Shaped Punch Problem ....................... 281
7.6 Representing the Solutions for an Anisotropic Space
in Terms of Displacement/Stress Discontinuities
Across a Plane ......................................... 288
7.7 Expansion of an Elliptic Crack ........................... 294
7.7.1 Solution to the Problem .......................... 295
7.7.2 Isotropic Media ................................. 298
7.8 Two-Dimensional Problems .............................. 300
7.8.1 A Strip-Shaped Crack ............................ 301
7.8.2 The Axisymmetric Case .......................... 304
References ................................................... 309
Subject Index ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 317
1. Introduction
Many scientific and technological problems are closely related to studying wave
processes in continuous media, the behavior of structures subjected to impulsive
loadings, and the phenomena of penetration and destruction. These problems can
often be solved by applying the methods of elastodynamics.
Since disturbances arising in both solid and dense liquid media, such as metals,
rock materials, and water, may remain small even at large absolute values of ap
plied loadings or pressure drops at the wavefronts, such media may be treated with
the linear approach within a wide range of mechanical parameters [1.1]. There
fore, from the mathematical point of view, for such physical problems one has to
solve a system of linear equations coupled through linear boundary conditions and
satisfying given initial conditions.
Thus, many applied problems may be solved by employing the linear theory
of dynamics of a solid deformable body. This has stimulated rapid development
of this approach during the last few decades, and has called to life numerous
monographs, textbooks and reviews [1.2-26].
It should be noted that analytical methods play an important role in the study
of dynamic processes. They are needed for solving model and test problems, car
rying out qualitative studies of properties and particularities of solutions, and for
obtaining numerical solutions to complicated three-dimensional dynamic problems
which require the use of combined numerical and analytical methods. Asymptot
ic methods have been thoroughly treated by Babich and Buldyrev [1.27] in their
book on short-wavelength asymptotics of diffraction problems of acoustics, and
by Achenbach et al. [1.20] in the monograph on applying ray methods to elasto
dynamics.
Until recently there have been no books presenting a systematic discussion of
rigorous analytical methods of elastodynamics, and this circumstance has hindered
their study.
The present monograph is intended to fill this need. A variety of practical
methods of obtaining accurate analytical solutions to dynamic problems in the
linear theory of elasticity are considered. The scope of the book is limited to
transient problems which play the most important role in dynamic processes. A
long with the the classical methods thoroughly developed in recent years, such
as the Smirnov-Sobolev method, the integral transform, and the Wiener-Hopf ap
proach, this monograph treats new and highly efficient methods of solving two and