Table Of ContentElastic wave propagation in transversely isotropic media
Monographs and textbooks on mechanics of solids and fluids
editor-in-chief: G. .!E. Oravas
Mechanics of elastic and inelastic solids
editor: S. Nemat-Nasser
Books published under this series are:
1. G.M.L. Gladwell
Contact problems in the classical theory of elasticity
2. C. Wempner
Mechanics of solids with applications to thin bodies
3. T. Mura
Micromechanics of defects in solids
4. R.C. Payton
Elastic wave propagation in transversely isotropic media
5. S. Nemat-Nasser, H. Abe, S. Hirakawa (Eds.)
Hydraulic fracturing and geothermal energy
Elastic wave propagation in
transversely isotropic media
Robert G. Payton
Department of Mathematics and Computer Science
Adelphi University, Garden City, N. Y. US.A.
1983 MARTINUS NIJHOFF PUBLISHERS
a member of the KLUWER ACADEMIC PUBLISHERS GROUP
THE HAGUE / BOSTON / LANCASTER
Distributors
for the United States and Canada: Kluwer Boston, Inc., 190 Old Derby Street,
Hingham, MA 02043, USA
for all other countries: Kluwer Academic Publishers Group, Distribution Center,
P.O.Box 322,3300 AH Dordrecht, The Netherlands
Library of Congress Cataloging in Publication Data
Payton, Rocert G.
Elastic wave propagation in transversely isotropic
media.
(Mechanics of elastic and inelastic solids ; 4)
(Monographs and textbooks on mechanics of solids and
fluids)
Includes index.
1. Elastic waves. 2. Wave-motion, Theory of.
I. Title. II. Series. III. Series: Monographs and
textbooks on mechanics o~ solids and fluids.
C<A935.P34 1983 531' .3823 83-6289
ISBN-13: 978-94-009-6868-4
ISBN-13: 978-94-009-6868-4 e-ISBN-13: 978-94-009-6866-0
DOl: 10.1007/978-94-009-6866-0
Copyright
© 1983 by Martinus Nijhoff Publishers, The Hague.
Softcover reprint of the hardcover 1s t edition 1983
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means, mechanical,
photocopying, recording, or otherwise, without the prior written permission of
the publishers,
Martinus Nijhoff Publishers, P.O. Box 566, 2501 CN The Hague,
The Netherlands.
IV
Ter nagedachtenis aan mijn moeder, Ada Wright Payton (1896-1980),
voor mijn vader Paul Payton en voor mijn geliefde vrouw Marlene.
So teach us to number our days, that we may apply our hearts unto wisdom.
psalm 90, v. 12.
Table of contents
Preface Xl
Chapter 1. Basic equations
I. The Linearized Equations of Motion of an Anisotropic Elastic Solid
2. The Effect on the Equations of Motion of a Coordinate Rotation 2
3. The Elasticities for a Transversely Isotropic Solid 2
4. The Constraints on the ccxj3's of Positive Definiteness 4
5. The Constraints on the ccxj3's of Strong Ellipticity 4
6. The Uncoupled Equations of Motion in Two-Dimensions II
7. The Uncoupled Equations of Motion in Three-Dimensions 12
8. Some Other Transversely Isotropic Continuum Theories 14
Chapter 2. Wave front shape caused by a point source in unbounded media 15
Part I Two Space Dimensions IS
1. The Characteristic Partial Differential Equation for the Wave Front IS
2. The Normal Curve 17
3. Bitangents and the Existence of Inflection Points on the Normal
Curve 18
4. Bitangents Which Cross Both Coordinate Axes of the Normal Curve 20
5. Double Points of the Normal Curve 24
6. Classification of the Normal Curve Shape 26
7. Solution of the Characteristic Partial Differential Equation 31
8. Wave Front Construction as Envelope of Line Waves 34
9. Classification of the Wave Front Curve when Q is Strictly Hyperbolic 38
10. Normal and Wave Front Curves for some Hexagonal Materials 40
II. Wave Front Construction When the Normal Curve has Double Points
- Convex Hull 55
12. Necessary Condition for the Existence of Wave Front Lids 58
13. The Wave Front Shape When the Normal Curve has Double Points
- Conclusion 60
14. Remarks on the Wave Front Shape When the Line Load is Not Per-
pendicular to the Material Symmetry Axis 61
Part II Three Space Dimensions 62
15. The Characteristic Partial Differential Equation for the Wave Front 62
16. The Normal Surface-Rotational Symmetry 63
17. Multiple Points of the Normal Surface 64
18. Wave Front Construction as Envelope of Plane Waves 66
19. The Existence of Wave Front Lids 66
20. Normal and Wave Surfaces for some Hexagonal Materials 67
VII
Contents
Chapter 3. Green's tensor for the displacement field in unbounded media 68
Part I Two Space Dimensions 68
1. Integral Transform Representation of the Displacement Field 68
2. Transform Inversion and Reduction to a Residue Calculation 69
3. The Existence of Lacunas 74
4. Examination of the Complex p-Roots of Q(I, p, Y + zp) = 0 When
the Observation Point is on a Coordinate Axis 76
5. Displacement Components Along the Symmetry Axis for Some
Hexagonal Materials 78
6. The Solution Behaviour Near the Wave Front 81
7. The Stress Field Induced by a Line Load 91
8. Remarks on the Green's Tensor Representation When the Line Load
is Not Perpendicular to the Material Symmetry Axis 94
Part II Three Space Dimensions 95
9. Integral Transform Representation of the Displacement Field 95
10. Transform Inversion for the Special Case 'Y = IX + ~ 96
11. Transform Inversion for the General Case Along the Symmetry Axis 105
12. Application to Some Hexagonal Crystals 111
13. Herglotz-Petrowski Formula for the Displacement Field as an Integral
Over the Slowness Surface 114
14. Differential Geometry of the Slowness Surface N 119
15. Remarks Concerning the Displacement Behaviour Near the Wave
Front 123
Chapter 4. Surface motion of a two-dimensional half-space (Lamb's
problem) 130
1. Formulation of the Problem 130
2. Integral Transform Representation of the Solution When the Fourier
Inversion Path is Free of Branch Points 132
3. Transform Inversion for Materials Satisfying Condition (1) of Table 11 135
4. Transform Inversion for Materials Satisfying Condition (2) of Table 11 146
5. Transform Inversion for Materials Satisfying Condition (3) of Table 11 151
6. Graph of the Surface Displacements for Some Hexagonal Crystals 154
Chapter 5. Epicenter and epicentral-axis motion of a three-dimensional
balf-space 157
1. Problem Formulation for the Epicenter Motion of a Half-Space Due
to the Sudden Application of a Buried Point Source 157
2. Transform Inversion at the Epicenter for Materials Satisfying Con-
dition (1) of Table 11 162
3. Discussion of the <.v-Plane Branch Points and the Cagniard Path When
the Real <.v-Axis is Not Free of Branch Points 166
4. Transform Inversion at the Epicenter for Materials Satisfying Con-
ditions (2) and (3) of Table 11 169
5. Discussion of the Epicenter Vertical Displacement for Some Hexag-
onal Crystals 175
VIII
Contents
6. Vertical Displacement Along the Epicentral-Axis Due to a Surface
Point Load Acting Normal to the Surface 179
7. Body Forces Equivalent to Internal Discontinuities in an Aniso-
tropic Elastic Solid 183
References 186
Author index 189
Subject index 191
IX
Preface
In this monograph I record those parts of the theory of transverse isotropic elastic
wave propagation which lend themselves to an exact treatment, within the framework
of linear theory. Emphasis is placed on transient wave motion problems in two- and
three-dimensional unbounded and semibounded solids for which explicit results can be
obtained, without resort to approximate methods of integration. The mathematical
techniques used, many of which appear here in book form for the first time, will be of
interest to applied mathematicians, engeneers and scientists whose specialty includes
crystal acoustics, crystal optics, magnetogasdynamics, dislocation theory, seismology
and fibre wound composites.
My interest in the subject of anisotropic wave motion had its origin in the study of
small deformations superposed on large deformations of elastic solids. By varying the
initial stretch in a homogeneously deformed solid, it is possible to synthesize aniso
tropic materials whose elastic parameters vary continuously. The range of the parameter
variation is limited by stability considerations in the case of small deformations super
posed on large deformation problems and (what is essentially the same thing) by the
notion of hyperbolicity (solids whose parameters allow wave motion) for anisotropic
solids. The full implication of hyperbolicity for anisotropic elastic solids has never
been previously examined, and even now the constraints which it imposes on the
elasticity constants have only been examined for the class of transversely isotropic
(hexagonal crystals) materials. In chapter one these constraints are discussed along
with their relation to the more familiar constraints of positive definiteness.
Chapter two treats the shape and geometric features of the wave front caused by a
point source. This is a beautiful subject on which significant work has been done by
Musgrave (86). The wave front lends itself to an elegant solution in the form of an
envelope to a plane wave solution of the nonlinear first order characteristic partial
differential equation of the system. One of the high points of this chapter is an explicit
set of conditions which, when imposed on the elasticity parameters, insure the
existence of inflection points on the slowness curve. These conditions are the key to
the classification of the wave front shape and to other propagational phenomena.
It is not generally known that an explicit solution may be found for the two
dimensional anisotropic elastic wave Green's tensor. But such is indeed the case and a
detailed presentation of this remarkable result is presented in chapter three. This
elegant solution is made possible by employing an integration technique due to
Burridge (31). In this connection a full discussion of lacunas (gaps in the wave field
behind the leading front) is given. Further explicit results are obtained along the
symmetry axis for the three-dimensional solid.
The study of transient elastic waves in an elastic half-space began in 1904 when
XI
Preface
Lamb analyzed the surface motion of a two-dimensional isotropic solid. Later Kraut
(58), using the method of Cagniard, extended Lamb's results to transversely isotropic
materials, in particular to the crystal Beryl. In chapter four the two-dimensional
surface motion problem of a point loaded half-space is completely solved for the full
range of hexagonal crystals. Finally the monograph ends in chapter five with some
very recent research concerned with the epicentral-axis motion of a (surface) point
loaded, three-dimensional half-space. Some intriguing results (for Zinc) caused by the
head wave are herein given.
Musgrave's 1970 book 'Crystal Acoustics' [20] was the first' ... to describe in
detail the conservative mechanics of waves in crystals whether regarded as continua
or as structures of discrete particles.' This book went far toward exploring the under
lying physics associated with elastic waves in crystals. During the past decade several
excellent books on the subject of linear, isotropic elastic wave motion have appeared
in the English language. These books, by Achenbach [87], Eringen and Suhubi [88] ,
Miklowitz [89] and Hudson [90] are written from the viewpoint of the applied
mathematician. It has been my intention to write a monograph on transversely
isotropic elastic waves, very much in the spirit of the above mentioned books.
Emphasis has been placed on closed form solutions (no asymptotics!) to an interesting
and important class of problems in continuum mechanics.
My thanks are due to Professor S. Nemat-Nasser for encouragement during the
preparation of this monograph.
Garden City, New York R.G. Payton
1983
Oh that my words were now written! oh that they were printed in a book!
Job 19, v. 23.
XII
