Wave Propagation in Layered Anisotropic Media with Applications to Composites (North-Holland Series in Applied Mathematics and Mechanics, Volume 39)

WAVE PROPACIAI'ION IN LAYERED ANISOTROPIC MEDIA with Applications to Composites

N O R T H - H O L L A N D SERIES IN

APPLIED MATHEMATICS AND MECHANICS EDITORS:

J.D. ACHENBACH Northwestern University

B. BUDIANSKY Harvard University

H.A. LAUWERIER University of Amsterdam

EG. SAFFMAN California Institute of Technology

L. VAN WIJNGAARDEN Twente University of Technology

J.R. WILLIS University of Bath

VOLUME 39

ELSEVIER AMSTERDAM

9L A U S A N N E

9N E W Y O R K

9O X F O R D

9S H A N N O N

9T O K Y O

WAVE PROPAGATION IN LAYERED ANISOTROPIC MEDIA with Applications to Composites

ADNAN

H. NAYFEH

Aerospace Engineering and Engineering Mechanics University of Cincinnati Cincinnati, OH, U.S.A.

1995 ELSEVIER AMSTERDAM

9L A U S A N N E ~ N E W Y O R K 9O X F O R D ~ S H A N N O N ~ T O K Y O

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN: 0-444-89018-1

9 1995 ELSEVIER SCIENCE B.V. 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, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B. V., Copyright& Permissions Department, P.O. Box 521,1000 AMAmsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisherfor any injury and~or damage to persons orproperty as a matter of products liability, negligence or otherwise, orfrom any use or operation of any methods, products, instructions or ideas contained in the material herein.

This book is printed on acid-flee paper. PRINTED IN THE NETHERLANDS

Dedicated to

my wife Sana and to our children

vi

Preface This book has been motivated by the recent advances in the study of the dynamic behavior of layered materials in general, and laminated fibrous composites in particular. The need to understand the microstructural behavior of such classes of materials brought a new challenge to existing analytical tools. These classes of materials differ from isotropic homogeneous materials in that they are both anisotropic and inhomogeneous, the combination of which leads in many cases to dispersive effects. These important effects are due to the presence of material interfaces (between fibers and matrix for composites and between layers). The degrees of anisotropy and dispersivity depend upon the specific materials under consideration, the interfacial conditions, and upon the scale lengths involved, however. Layered media could exhibit anisotropy on micro- as well as on macroscales. In this book we shall refer to these as micro- and macro-anisotropy. Micro-anisotropy arises when one or more of the individual layers exhibits point anisotropy whereas macro-anisotropy arises from combinations of different layers. Similarly, we can speak of micro- and macro-dispersion. Micro-dispersion is produced by the presence of microstructural interfaces, between fibers and matrix, for example whereas macro-dispersion is produced in bounded media due to the restrictions imposed on their outer boundaries. Thus, for unbounded media only micro-dispersion can exist, whereas both micro- and macro-anisotropy could be present in bounded structures. It is the intent of this book to touch upon these effects. The fundamental question we wish to ultimately answer is how mechanical waves propagate and interact with layered anisotropic media. In order to reach there, we organize the material in this book in accordance with a building block type approach, which follows a logical sequence depending upon the complexity of the physical model and its mathematical treatment. After the introduction of chapter 1, we present, in chapter 2, a complete description of the relevant field equations together with their tensorial properties for general anisotropic media. Here classification of the various material symmetries of such materials, using linear transformation properties, are discussed. Chapter 3 is devoted to the propagation of bulk waves in infinite homogeneous anisotropic media. In chapter 4, we discuss the generalized Snell's law and relate it to interfaces and then proceed to define the critical angle phenomenon. In chapter 5, we present formal solutions for a bounded medium in the form of an infinite layer bounded by two

vii parallel faces. This constitutes the backbone of the building block approach and will be applied in the remainder of the book in a variety of applications. Chapter 6 is devoted to the study of reflections and refractions from interfaces separating two half-spaces; these include a combination of two solids, a solid and a liquid or a solid bounded by a vacuum (a free halfspace). Chapter 7 is devoted to the study of interface waves which include Rayleigh surface, pseudo-surface, Scholte and Stoneley waves. The study of the propagation of free waves on anisotropic plates in vacuum and in contact with fluid is covered in chapter 8. In chapter 9, we present solutions for the interaction of elastic waves with multilayered anisotropic media. Specialization to cases involving propagation along axes of symmetry is covered in chapter 10. Chapter 11 is devoted in its entirety to fluid-loaded solids. Here semi-spaces and single and multilayered systems in contact with fluids are analyzed. Chapter 12 extends the results of the previous chapters to include piezoelectric coupling. In chapter 13, the techniques introduced for harmonic wave motions are modified to study transient motions in unbounded and in semi-space media. In chapter 14, we present an example of wave interaction with layered coaxial systems. Specifically, we discuss scattering of horizontally polarized shear waves from multilayered anisotropic cylinders. The book concludes by presenting, in chapter 15, some model calculations for the effective elastic properties of fibrous composite materials needed in applications covered in earlier chapters. In writing this book, I have attempted to strike a balance between the way I presented the theory and its simple adaptation to numerical computations. I am a strong fan of computers and their experimental-like power. I have best understood the material when aided on the spot with computer programs. The significant checking power on the accuracy of the analytical models brought about by the meticulous experiments of Dale Chimenti is acknowledged. The invaluable help over the years of my graduate students in developing this material is acknowledged. Of these I single out Drs. T. Taylor, H .T. Chien, M. Hawwa, Y. Y. Kim and my current graduate students H. Hu and N. Al-huniti. I also extend my appreciation to my colleagues J. Wade and G. Bahr who carefully read through several versions of this book. The technical suggestions resulting from the critical reading of the entire manuscript by my colleague P. Nagy constituted an invaluable asset.

Cincinnati, Ohio June 10, 1995

This Page Intentionally Left Blank

Contents 1

INTRODUCTION 1.1

2

Historical b a c k g r o u n d

. . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

M o s t l y isotropic m e d i a . . . . . . . . . . . . . . . . . .

2

1.1.2

Mostly anisotropic media

6

1.1.3

F l u i d - l o a d e d solids . . . . . . . . . . . . . . . . . . . .

9

1.1.4

P i e z o e l e c t r i c effects . . . . . . . . . . . . . . . . . . . .

11

1.1.5

S c a t t e r i n g f r o m layered cylinders . . .. . . . . . . . . .

12

1.1.6

E l a s t i c p r o p e r t i e s of c o m p o s i t e s . . . . . . . . . . . . .

13

. . . . . . . . . . . . . . . .

FIELD EQUATIONS AND TENSOR ANALYSIS 2.1 T h e stiffness t e n s o r . . . . . . . . . . . . . . . . . . . . . . . . 2.2 M a t e r i a l s y m m e t r y . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1

15 16 17

The transformation . . . . . . . . . . . . . . . . . . . .

17

2.3 2.4

M a t r i x forms of stiffness . . . . . . . . . . . . . . . . . . . . . Engineering constants . . . . . . . . . . . . . . . . . . . . . .

21 23

2.5

Transformed equations . . . . . . . . . . . . . . . . . . . . . .

24

2.6

2.5.1 A d v a n t a g e s of o r t h o g o n a l t r a n s f o r m a t i o n s . . . . . . . E x p a n d e d field e q u a t i o n s . . . . . . . . . . . . . . . . . . . .

25 26

2.7

2.6.1 Monoclinic . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Orthotropic . . . . . . . . . . . . . . . . . . . . . . . . P l a n e s of s y m m e t r y . . . . . . . . . . . . . . . . . . . . . . .

27 28 29

BULK

31

WAVES

3.1

A n overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.2

T h e Christoffel e q u a t i o n . . . . . . . . . . . . . . . . . . . . .

33

3.2.1

G e n e r a l f e a t u r e s of t h e Christoffel e q u a t i o n

34

3.2.2

L i m i t a t i o n s of a n a l y t i c s o l u t i o n s . . . . . . . . . . . .

37

Material symmetry . . . . . . . . . . . . . . . . . . . . . . . .

38

3.3.1

Analytical solutions

. . . . . . . . . . . . . . . . . . .

38

3.3.2

Higher s y m m e t r y . . . . . . . . . . . . . . . . . . . . .

41

3.3

ix

......

x

CONTENTS

. . . . . . . . . . . . . . . . . . . . .

42

. . . . . . . . . . . . . . . . . . . .

46

C o m p u t e r aided analysis . . . . . . . . . . . . . . . . . . . . .

48

3.5

G r o u p velocity

. . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.6

E n e r g y flux

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.4

3.3.3

Cubic s y m m e t r y

3.3.4

T h e isotropic case

GENERALIZED 4.1

SNELL'S LAW AND INTERFACES

61

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . .

62

4.1.1

62

T y p e s of interface c o n d i t i o n s

..............

4.2

C h a r a c t e r i z a t i o n of incident waves

4.3

Critical angles . . . . . . . . . . . . . . . . . . . . . . . . . . .

................

64 66

4.4

T w o fluid m e d i a

68

4.5

T w o isotropic m e d i a

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.1

C o m m o n f o r m of s o l u t i o n s . . . . . . . . . . . . . . . . . . . .

71 72

5.2

Triclinic layer . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.3 5.4

T h e m o n o c l i n i c case

74

FORMAL

SOLUTIONS

. . . . . . . . . . . . . . . . . . . . . . .

Higher s y m m e t r y m a t e r i a l s

. . . . . . . . . . . . . . . . . . .

5.4.1

P r o p a g a t i o n a l o n g off-principal-axes

..........

5.4.2

P r o p a g a t i o n a l o n g an axis of s y m m e t r y

5.4.3

Isotropic media . . . . . . . . . . . . . . . . . . . . . .

5.5

F o r m a l s o l u t i o n s in fluid m e d i a

5.6

T h e c ~ - c r e l a t i o n a n d t h e Christoffel e q u a t i o n

SCATTERED

........

. . . . . . . . . . . . . . . . . ........

WAVE AMPLITUDES

75 77 77 79 80 80

83

6.1 6.2

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reflection f r o m a free surface . . . . . . . . . . . . . . . . . .

84 85

6.3

S c a t t e r i n g f r o m fluid-solid interfaces

88

6.4

S c a t t e r i n g f r o m solid-solid interface . . . . . . . . . . . . . . .

INTERFACE

..............

WAVES

90

93

7.1

Surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

7.2

P s e u d o - s u r f a c e waves . . . . . . . . . . . . . . . . . . . . . . .

95

7.3

Scholte waves

. . . . . . . . . . . . . . . . . . . . . . . . . .

99

FREE WAVE IN PLATES

103

8.1

Free waves in triclinic p l a t e s . . . . . . . . . . . . . . . . . . .

105

8.2

Free waves in m o n o c l i n i c p l a t e s . . . . . . . . . . . . . . . . .

106

8.2.1

T h e d r y case

106

8.2.2

M o n o c l i n i c p l a t e s i m m e r s e d in fluids . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

108

CONTENTS 8.2.3

xi

Fluid-monoclinic plate-vacuum system .........

109

8.3

Higher s y m m e t r y material plates

. . . . . . . . . . . . . . . .

110

8.4

Numerical computation strategy

. . . . . . . . . . . . . . . .

112

9.1

G e o m e t r i c d e s c r i p t i o n of u n i t cell . . . . . . . . . . . . . . . .

117 118

9.2

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 T h e local t r a n s f e r m a t r i x . . . . . . . . . . . . . . . .

118 122

9.3 9.4

P r o p e r t i e s of t h e t r a n s f e r m a t r i x

Free waves on t h e l a y e r e d cell . . . . . . . . . . . . . . . . . .

126

9.5

W a v e s in a p e r i o d i c m e d i u m . . . . . . . . . . . . . . . . . . . 9.5.1 D i s p e r s i o n curves . . . . . . . . . . . . . . . . . . . . .

127 128

9.5.2

. . . . . . . . . . . . . . . . . .

129

9.5.3 S p e c i a l i z a t i o n to a single m a t e r i a l . . . . . . . . . . . B o t t o m b o u n d i n g solid s u b s t r a t e . . . . . . . . . . . . . . . .

130 131

GENERAL

9.2.2

9.6

LAYERED

T h e global t r a n s f e r m a t r i x . . . . . . . . . . . . . . . .

D i s p e r s i v e slownesses

10 P R O P A G A T I O N 10.1 G e o m e t r y 10.2

MEDIA

. . . . . . . . . . . . . . .

ALONG AXES OF SYMMETRY

123 . 124

135

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

10.2.1 Free waves . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 P e r i o d i c m e d i a . . . . . . . . . . . . . . . . . . . . . .

138 139

S H waves

10.2.3 Effective elastic p r o p e r t i e s . . . . . . . . . . . . . . . .

143

10.3 M o t i o n in t h e s a g i t t a l p l a n e . . . . . . . . . . . . . . . . . . .

145

10.4 Free waves on t h e l a y e r e d cell . . . . . . . . . . . . . . . . . . 10.5 Waves in a p e r i o d i c m e d i u m . . . . . . . . . . . . . . . . . . . 10.6 B o t t o m b o u n d i n g solid s u b s t r a t e . . . . . . . . . . . . . . . .

147 148 149

11 F L U I D - L O A D E D

SOLIDS

153

11.1 R e f l e c t i o n f r o m a s u b s t r a t e . . . . . . . . . . . . . . . . . . . 11.1.1 Q u a l i t a t i v e d i s c u s s i o n . . . . . . . . . . . . . . . . . . 11.2 P l a t e s c o m p l e t e l y i m m e r s e d in fluids . . . . . . . . . . . . . . 11.2.1

Cremer's correspondence principle . . . . . . . . . . .

155 157 160 162

11.2.2

Fluid-plate-vacuum system

. . . . . . . . . . . . . . .

170

11.2.3

T h e g e n e r a l layered m e d i a . . . . . . . . . . . . . . . .

173

11.2.4 B o t t o m s u b s t r a t e . . . . . . . . . . . . . . . . . . . . . 11.3 H i g h e r s y m m e t r y cases . . . . . . . . . . . . . . . . . . . . . .

178 181

11.4 L e a k y waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

11.4.1

F i e l d of t h e incident finite b e a m

11.4.2 F i e l d of t h e reflected b e a m

............

. . . . . . . . . . . . . . .

11.4.3 A n overview of t h e reflection coefficient

........

184 187 188

CONTENTS

xii 11.4.4 11.4.5

R a y l e i g h pole . . . . . . . . . . . . . . . . . . . . . . . R e f l e c t e d b e a m profile . . . . . . . . . . . . . . . . . .

11.5 E x p e r i m e n t a l t e c h n i q u e

12 P I E Z O E L E C T R I C

. . . . . . . . . . . . . . . . . . . . .

EFFECTS

190 193 197

201

12.1 Basic r e l a t i o n s of piezoelectric m a t e r i a l s

............

202

12.2 Simplified field e q u a t i o n s . . . . . . . . . . . . . . . . . . . 12.3 A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 F o r m a l s o l u t i o n s . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Surface waves . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Free p l a t e m o d e s . . . . . . . . . . . . . . . . . . . .

. . . . .

203 204 205 207 209

12.5 H i g h e r s y m m e t r i c m a t e r i a l s . . . . . . . . . . . . . . . . . . . 12.5.1 O r t h o t r o p i c - 2 2 2 . . . . . . . . . . . . . . . . . . . . . 12.5.2 B.-G. waves . . . . . . . . . . . . . . . . . . . . . . .

209 210 211

12.6 R e m a r k s on t h e m o n o c l i n i c - m case

213

...............

12.7 R e f l e c t i o n a n d t r a n s m i s s i o n coefficients . . . . . . . . . . . . . 12.7.1 R e f l e c t i o n a n d t r a n s m i s s i o n f r o m a s u b s t r a t e

.....

12.7.2 R e f l e c t i o n a n d t r a n s m i s s i o n f r o m a p l a t e . . . . . . . . 12.8 S a m p l e i l l u s t r a t i o n s . . . . . . . . . . . . . . . . . . . . . . . 12.9 R e m a r k s on layered piezoelectric m e d i a . . . . . . . . . . . .

13 T R A N S I E N T

WAVES . . . . . .

. . .

. . .

13.6.1 D i s p l a c e m e n t s o l u t i o n s . . . . . . . . . . . . . . . . . . 13.7 S e m i - s p a c e m e d i a . . . . . . . . . . . . . . . . . . . . . . . . .

FROM LAYERED

14.1 F i e l d e q u a t i o n s

215 215 218

221

13.1 T h e o r e t i c a l d e v e l o p m e n t . . . . . . . . . . . . . . . . . . 13.2 S o u r c e c h a r a c t e r i z a t i o n . . . . . . . . . . . . . . . . . . 13.3 I n t e g r a l t r a n s f o r m s of f o r m a l s o l u t i o n s . . . . . . . . . . . . . 13.3.1 M e t h o d s of i n v e r t i n g t h e t r a n s f o r m s . . . . . . . . . . 13.4 I s o t r o p i c m e d i a . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 T h e C a g n i a r d - d e H o o p t r a n s f o r m a t i o n . . . . . . . . . 13.4.2 D i s p l a c e m e n t d i s t r i b u t i o n . . . . . . . . . . . . . . . . 13.5 A n i s o t r o p i c m e d i a . . . . . . . . . . . . . . . . . . . . . 13.6 C a g n i a r d - d e H o o p t r a n s f o r m a t i o n . . . . . . . . . . . . . . . .

14 S C A T T E R I N G

213 214

CYLINDERS

. . . . . . . . . . . . . . . . . . . . . . . . . .

14.2 F o r m a l s o l u t i o n s in isotropic cylinders

.............

14.3 C h a r a c t e r i z a t i o n of i n c i d e n t waves . . . . . . . . . . . . . . . 14.4 F o r m a l s o l u t i o n s for a layer . . . . . . . . . . . . . . . . . . .

221 223 225 229 230 234 237 238 239 242 245

253 255 256 258 260

14.4.1

Local transfer matrix

. . . . . . . . . . . . . . . . . .

262

14.4.2

Global transfer matrix . . . . . . . . . . . . . . . . . .

263

CONTENTS 14.4.3

xiii

P r o p e r t i e s of t h e t r a n s f e r m a t r i c e s

14.5 S c a t t e r i n g a m p l i t u d e s

...........

263

. . . . . . . . . . . . . . . . . . . . . .

264

S c a t t e r i n g f r o m a solid core . . . . . . . . . . . . . . .

265

14.5.2 S c a t t e r i n g f r o m a n i n n e r c a v i t y . . . . . . . . . . . . . 14.5.3 S t r e s s e s in t h e h o s t m e d i u m . . . . . . . . . . . . . . . 14.5.4 S c a t t e r i n g cross s e c t i o n . . . . . . . . . . . . . . . . .

265 265 266

14.5.1

15 E L A S T I C P R O P E R T I E S

OF COMPOSITES

267

15.1 G e n e r a l d e s c r i p t i o n of f i b r o u s c o m p o s i t e s ........... 15.2 T h e m o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 269

15.3 T h e l a y e r e d m o d e l

. . . . . . . . . . . . . . . . . . . . . . . .

269

15.3.1

Averaging . . . . . . . . . . . . . . . . . . . . . . . . .

271

15.3.2

Strain and stress compatibilities

............

272

15.3.3 A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 T h e s q u a r e f i b r o u s case . . . . . . . . . . . . . . . . . . . . . 15.4.1

. . . . . . . . . . . . . . . . . . . . . .

276

15.4.2 A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 A n i s o t r o p i c fiber a n d m a t r i x . . . . . . . . . . . . . . . . . .

277 279

15.5.1 15.5.2

Compatibilities

272 275

The layered model . . . . . . . . . . . . . . . . . . . . T h e f i b r o u s case . . . . . . . . . . . . . . . . . . . . .

280 281

15.6 S t r a i n e n e r g y a p p r o a c h . . . . . . . . . . . . . . . . . . . . . .

282

15.6.1 15.7

The layered model

. . . . . . . . . . . . . . . . . . . .

282

15.6.2 T h e f i b r o u s m o d e l

. . . . . . . . . . . . . . . . . . . .

284

. . . . . . . . . . . . . . . . . . . . . . . . .

284

U n d u l a t e d fiber

15.7.1 D i s c r e t i z a t i o n . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

INTRODUCTION Studies of the propagation of elastic waves in layered media have long been of interest to researchers in the fields of geophysics, acoustics and nondestructive evaluation. Common to all of these studies is the investigation of the degrees of interaction among the layers, which manifest themselves in the forms of reflection and transmission agents and give rise to geometric dispersion. These interactions depend, among other factors, upon the mechanical properties, geometric arrangements, number and nature of the interfacial conditions and on the loading conditions. The variability in the mechanical properties of the individual layers ranges from that of the simple case of isotropic materials to that of the most general anisotropic ones, namely the " triclinic materials". Often, the layer's properties exhibit other effects such as dissipative, piezoelectric and thermal. These effects will undoubtedly result in further complications in the behavior of the system. Typically, a layered medium consists of two or more material components attached at their interface in some fashion. A plate made up of an arbitrary number of different material components and whose outer boundaries are either free or supported by semi-infinite media constitutes a general layered medium. Often the above definition is relaxed to include semi-infinite solids, single-layer plates and two semi-infinite solids in contact as degenerate cases of layered media. Layered media are, of course, not limited to fiat layers and can be curved in the form of coaxial cylinders, for example. Rigid (perfect) bondings which insure continuity of all field variables (stress and displacement components) are mostly encountered in applications. Smooth, as well as rough, interfacial conditions which require modifications of the continuity conditions are, to a lesser extent, encountered. These constitute classes of imperfect interfaces. Loading conditions include either plane harmonic or transient waves.

2

CHAPTER1.

INTRODUCTION

Plane harmonic waves are single frequency waves which propagate indefinitely into the system. The transient waves, on the other hand, are time arbitrary dependent processes, that are usually caused by concentrated sources within the system. Common types of transient loads are the point and line sources with prescribed time-history. The simple statement of the problem stated above belies the formidable effort needed to study even the simplest cases. Up to quite recently, this is why, in this author's opinion, most of the available work on the subject has been focused on phenomena that strictly deal with isotropic materials of simple geometric arrangements such as the ones just mentioned. Studies of elastic waves in such "simple" and mostly isotropic systems are widely available in the books by Rayleigh [208], Achenbach [3], Graft [97], Miklowitz [147], Officer [193], Auld [12], Pilant [200], Wasley [249], Kolsky [125], Viktorov [248], Beltzer [17], Aki and Richards [9] and Morse and Ingard [149]. The previously published fewer works on multilayered isotropic media can be found in the books by Ewing, Jardetzky and Press [77], Brekhovskikh [28], and Kennett [120]. Whatever is available on studies of responses caused by transient loadifigs of these simple systems can be found, in one way or another, in the above references.

1.1

Historical background

1.1.1

Mostly isotropic media

To give a chronological account of the available literature on the propagation of elastic waves in layered isotropic media, we start with the original work presented by Lord Rayleigh [2071, who in 1885 studied the propagation of elastic waves on the free surface of a semi-infinite solid. His work was motivated by the interest in understanding near surface motions caused by earth tremors. This classical work was, and still is, a foundation of all works on layered media. Due to its originality, it has carried his name and has been widely known as the Rayleigh or surface wave problem. In 1911 Love [138] added another interface in order to simulate a finite thickness layer and attempted to solve the simplest case of wave interaction with it, namely that of a horizontally polarized S H wave. He found out that such waves can exist in the layer only when it is attached to a solid semi-space of a different material. The problem has since been widely known as the Love wave problem. Lamb [129] in 1917 studied the more complicated problem of the propagation of free waves in the layer. The wave corresponds to the coupled

1.1. HISTORICAL BACKGROUND

3

longitudinal and transverse (normal to the layer) motion, namely the one that propagates in the sagittal plane. He presented exact solutions and was able to identify the two possible types of wave modes that exist in the layer, namely the symmetric and the antisymmetric ones. Whenever and wherever the phrase " Lamb waves" is mentioned, it precisely refers to the above problem. In 1924 Stoneley [230] generalized Rayleigh's surface wave problem by studying the propagation of free waves at the interface between two solid semi-spaces. Such waves have since then been referred to as " Stoneley waves". Scholte [221] in 1947, while attempting to specialize Stoneley's problem to the case where one of the semi-spaces is water, discovered a new and unusual wave, the "Scholte wave" that propagates at a speed which is lower than any of the bulk wave speeds belonging to the solid and the water (see more recent works on Scholte waves in deBilly and Quentin [58, 59] and Chamuel and Brooke [41]). We should mention that most of the above pioneering works were conducted for applications in geophysical systems. The first significant work on the study of elastic wave interaction with multilayered media is attributed to Thomson [237] who introduced the matrix transfer method in order to facilitate his analysis. A small error in his work was subsequently corrected by Haskell [105]. According to this technique we construct the transfer matrix for a stack of an arbitrary number of layers by extending the solution from one layer to the next while satisfying the appropriate interfacial continuity conditions. In its original form, the specific steps taken can be summarized as follows: formal solutions are first obtained for the individual layer in terms of its wave potential amplitudes. By specializing these solutions to the top and bottom surfaces of the layer, followed by eliminating its common wave amplitudes, we can directly relate the field variables (the stress and displacement components) of one of its surfaces to the other. This ensuing relation is conveniently written in a matrix form which defines the individual (local) transfer matrix. Such a matrix relation can be used, in conjunction with satisfying appropriate interface conditions across neighboring layers, to directly relate the stresses and displacements at the top of individual layer to the bottom of its neighbor. If this procedure is carried out consequently for all layers, a global transfer matrix, the product of the individual transfer matrices, will result, which relates the field variables at the top of the layered system to those at its bottom or visa-versa. The global matrix can then be used to present results for a wide variety of problems. All what we need to do is to invoke the appropriate values of the field variables at the outer surfaces of the layered system. Because the method is used to transfer (propagate) boundary conditions from one outer surface of the layered system to the other via matrix

4

CHAPTER1.

INTRODUCTION

multiplications, it is also known as "the propagator matrix method". The emergence of computational capabilities, in the sixties and the early seventies, extensively expanded the use of the matrix transfer method by applying it to a wide range of wave propagation investigations in layered media; the bulk of which was directed towards seismological applications. We cannot be all encompassing in referring to the vast literature, but we have been eclectic in citing some of the foundation landmark works of contributors such as Press et al. [204], Randall [206], Watson [250], Phinney [197], Gilbert [93], Gilbert and Backus [94], iayfeh and Taylor [176, 178], Fahmy and Adler [78] and Adler [7]. In situations involving inhomogeneous (evanescent) waves, the matrix transfer method is found to suffer from an important problem concerning the instability of the solution when the propagation sensitive parameter f d (the product of the frequency and the layered plate thickness) becomes large. The cause of this problem is the poor conditioning of the transfer matrix due to the combinations of both decaying and growing coefficients when evanescent waves are present. Undoubtedly, the problem is application dependent. Several attempts by Dunkin [72], Thrower [238], Kind [122, 123], Abo-Zena [1], Menke [144], Evans [75], L~vesque and Pich(~ [131] and Castaings and Hosten [36, 37] have been carried out to fix this problem. Limited success has been established, and even this is at the expense of the simple form inherent in the original formulation of the method. Alternative to the matrix transfer method is the use of the so-called direct approach. Here, once again, solutions for each layer are obtained by expressing its displacements and stresses in terms of its wave potentim amplitudes. By directly satisfying appropriate interfacial conditions at all layer's interfaces, a large single matrix is assembled which involves the amplitudes of all layers. This technique has been used by Knopoff [124], Schwab [222], Schmidt and Jensen [217, 218], Mal [140] and Lowe [139]. The degree of complication in the algebraic manipulation of such a matrix will thus depend upon the number of components. For relatively few components, the direct approach is appropriate ( Achenbach and Keshava [4], Nayfeh et al. [173], Chimenti et al. [50], and Bogy and Gracewski [22]). Although robust and can be simply and directly implemented, as the number of components increases, the direct approach becomes cumbersome and may lead to substantial increase in computational times. For systems consisting of layered periodic media, we mention the work of Rytov [215], who studied a bilayered periodic medium for the purpose of obtaining effective mechanical properties for "thinly layered systems". This has been known as the zero frequency limit. Hegemier and Nayfeh [108]

1.1. H I S T O R I C A L B A C K G R O U N D

5

studied similar model and derived exact and asymptotic results for the propagation of waves having arbitrary frequencies in such system. Subsequently, i a y f e h [164] generalized the results of [108] to the case of multilayered media. i e m a t - N a s s e r [179] and i e m a t - N a s s e r et al. [180, 183, 184] studied harmonic waves in a variety of composites consisting of isotropic layers. Further studies of harmonic waves on similar systems have been conducted by many authors such as Delph et al. [65, 66, 67] and others [23, 24, 95, 112,214]. Approximate solutions have also been widely available on wave propagation in the bilayered periodic composite (see, for example, Bedford and Stern [16], Sun et al. [232], and Murakami et al. [154]).

Transient waves and layered isotropic m e d i a In comparison to the extensive literature on the interaction of plane harmonic waves with isotropic media, much less work is available on the response of such media to concentrated source loadings. Here concentrated sources include point as well as line loads of which harmonically pulsating and transient sources are common types of such loadings. Understanding the response of elastic solids to internal mechanical sources has long been of interest to researchers in classical fields such as acoustics, seismology, as well as in modern fields of application like ultrasonic and acoustic emission. It is known that whenever a material undergoes a local failure, elastic waves are generated due to the rapid release of localized strain energy. Such radiation, for example, is known as acoustic emission in the field of nondestructive testing of materials. In seismology it is of course known as earthquake. An account of the literature dealing with this problem through 1957 can be found in Ewing, Jardetsky and Press [77]. Most of the earlier work [73, 130, 196] followed Lamb [128], who apparently was the first to consider the motion of a semi-infinite space caused by a vertically applied line load on the free surface or within the medium. It was shown that displacements at large distances consist of series of events which correspond to the arrival of longitudinal, shear, and Rayleigh surface waves. The analytical approach used in the above mentioned investigations can be summarized as follows: the steady state problem for a harmonically pulsating source in an infinite isotropic media is solved at first and then generalized to the case of a half-space using superposition techniques. For transient source loading, results are obtained from those corresponding to the harmonic ones by a Fourier integral approach. The resulting double integral is evaluated only by considering large distances. A modern alternative approach has been suggested by Cagniard [35]. He

6

CHAPTER1.

INTRODUCTION

showed that a suitable deformation of the integral contour not only resulted in considerable analytical simplification but led to exact, closed form, algebraic expressions for the displacements as functions of time. Subsequently, Garvin [90] applied the Cagniard method and obtained the disturbances due to sources in isotropic media. Since then, de Hoop modified the Cagniard method [61] and used it in a variety of applications [64, 63, 60, 62].

1.1.2

Mostly anisotropic

media

In recent years, continued efforts have been expended upon modeling wave propagation interaction with layered anisotropic media mostly for applications in fields other than seismology, such as nondestructive evaluation. This interest has been prompted by the recent expansion of the use of composite materials in a wide variety of applications. Typically, composite structural components are made up of a stack of layers (called plies or lamina) to form a laminate. The individual lamina in structural composites is composed of brittle, stiff fibers embedded in a matrix of more ductile material which bonds the fibers together and acts as a load-transfer medium. However, the morphology of these classes of materials, as compared with that of homogeneous isotopic media, can seriously complicate their mechanical response. For example, these classes of materials differ from isotropic homogeneous materials in that they are both anisotropic and inhomogeneous. These two important effects are due to the presence of material interfaces (between fibers and matrix and between plies). The degree of anisotropy depend upon the specific material under consideration, the interfacial conditions, and upon the scale lengths involved, however. Fiber reinforced composites exhibit anisotropy on micro- as well as on macro-scales, referred respectively to as micro- and macro-anisotropy. Micro-anisotropy arises from two effects: the first results if one or more of the composite's basic constituents (fiber or matrix) are anisotropic whereas the second is produced as a consequence of constituent combinations (microstructure) and is highly dependent upon their specific geometric ordering. Macro-anisotropy, on the other hand, arises from combinations of different composite laminates to make a new composite structure. A vivid example of this is a plate made up of several arbitrarily oriented layers of the same basic composite medium. Now that we have shown, due to the anisotropy of the individual lamina, that composite components constitute a more complicated class of layered media, we move to review available works on cases involving anisotropy. We first start with the literature on dry anisotropic media.

1.1. H I S T O R I C A L B A C K G R O U N D

7

D r y solids As compared with isotropic cases, solutions to the anisotropic problem are much more difficult to obtain. This is because of the added algebraic complications in handling of the pertinent equations even in simple cases involving propagation in unbounded media (the bulk waves). Based on the widely used bulk wave slowness surface techniques (see, for example, the books by Musgrave [156], Synge [233] and Fedorov [82]), several differences between wave propagation characteristics of isotropic and anisotropic media can be identified. In the isotropic case, the slowness surface consists of two concentric spherical sheets, the inner one represents longitudinal wave and the outer represents two coincident shear waves. For the anisotropic case however, there are three general surfaces, one for a quasi-longitudinal wave and two for quasi-shear waves. This means that incident and reflected waves in anisotropic media can no longer be thought of as purely longitudinal or shear with appropriate directionally independent wave speeds. This also implies that the direction of energy flow (i.e., group velocity) does not, in general, coincide with the normal to the wave front. The uncoupling of the longitudinal and shear waves in the isotropic case simplifies the algebraic treatment of their wave propagation characteristics when compared to the corresponding anisotropic case. Generally speaking, in the case of anisotropic media, the three waves are coupled, and in order to identify them one needs to solve a sixth order polynomial characteristic equation. For special material symmetry directions, one of the shear waves uncouples, leading to fourth order polynomial equations which are relatively easier to handle. Taken together, these complications perhaps explain why many of the available treatments of wave propagation in anisotropic media emphasize wave motions along material symmetry directions. While the propagation of bulk waves in anisotropic materials has received attention and is well understood, comparatively fewer quantitative results have been reported on solutions of problems containing either internal or external boundaries, which may classify them as layered media. Here, we mention that several authors including Merkulov [146], Musgrave [155], Gates [91], Joel [115], Henneke [111], Lim and Musgrave [135], Johnson [116], Chadwick and Currie [38] and Rokhlin et al. [209] have discussed, in varying degrees of depth, reflection and refraction of sound waves from interfaces of anisotropic media. Several new insights in the behavior of Stoneley waves in anisotropic media are well covered in these works. The related problem of Rayleigh waves on anisotropic surfaces has received the vast majority of attention from a large number of investigators. It appears that Stoneley [231] was the first to calculate the secular equation

8

CHAPTER1.

INTRODUCTION

of Rayleigh surface wave on cubic semi-spaces. Buchwald and Davis [30, 31], Gazis et al. [92], Rollins et al. [211] and Lim and Farnell [134] also studied surface waves on anisotropic media having cubic symmetry. Apparently, Lim and Farnell [134] also discovered the pseudo-surface wave, a wave that is unique to anisotropic media and has many of the characteristics of the normal surface wave. We defer further description of this type of wave until later on where it will be extensively studied in chapter 7. Other authors studied surface waves in various details on materials possessing various degrees of anisotropy. Of these we mention the works of Chadwick and Smith [39], Barnett and Lothe [13], Gavazza et al. [14], Lothe and Barnett [136], Rose et al. [212], Crean and Waintal [54], Shaikh et al. [226], Neubauer [187], Velasco and Garcis [246], Engan et al. [74], Verevkina et al. [247], Diachok et al. [68], Farnell [80], Every [76], Burridge [32] and Woodhouse [255]. For Lamb waves, limited theoretical analyses have been reported in plates of orthotropic or higher symmetry. Of these we mention the works of Markus et al. [143], Nikiforenko et al. [192], Abubakar [2], Kosevich and Syrkin [126], Baylis and Green [15] and Solie and Auld [229]. Kaul and Mindlin [119], Nayfeh and Chimenti [168] and Li and Thomson [133] have given solutions for free waves on plates consisting of the less symmetric monoclinic materials. On problems relating to multilayered anisotropic media, Nayfeh [161, 162] recently used the matrix transfer method and presented solutions for horizontally and generally polarized waves on multilayered anisotropic media, respectively. Fryer and Frazer [88, 87] studied wave interaction with stratified geophysical media. Harmonic wave propagation in a variety of anisotropic layered systems have been forwarded by many researchers. Of these we mention Nemat-Nasser et al. [185, 186, 182], Yamada and NematNasser [256], Helbig [109], Braga and Herrmann [25], Ting and Chadwick [240], Yang and Lee [257] and Shah and Datta [225]. Approximate results on wave interactions with layered anisotropic media are also available ( Nayfeh [158], Murakami [153, 151,152] and Toledano and Murakami [241, 242]). T r a n s i e n t waves a n d a n i s o t r o p y In comparison with the reasonably rich literature on the interaction of plane harmonic waves with anisotropic media, very little work is available on the response of such media to concentrated source loadings. Following the classical work of Lamb [128] who obtained the exact solution for the disturbances

1.1. H I S T O R I C A L B A C K G R O U N D

9

that are generated by an impulsive, concentrated load applied along a line on the free surface of a homogeneous isotropic elastic half-space, Kraut [127] examined the influence of transverse isotropy on such a problem. In [33] Burridge used the Cagniard-de Hoop method to obtain displacements in an anisotropic semi-space where the surface has no particularly symmetrical orientation with respect to the crystallographic axes of the medium. Burridge conducted his study on cubic symmetry materials. Van der Hijden in [245] also used the Cagniard-de Hoop method to study unbounded anisotropic media using an eigenvalue problem approach. Taylor [236] studied Lamb's problem for semi-spaces which have as low as monoclinic symmetry. Kim [121] presented solutions for transient waves caused by line loads in infinite, semi-infinite and plate media also possessing monoclinic symmetries.

1.1.3

Fluid-loaded

solids

During the last two decades or so problems relating to the interaction of elastic waves with fluid-loaded solids have been widely studied. Besides the natural system of ocean bottoms, most loading conditions are introduced for the sake of ultrasonic inspection of the solid components. One of the methods used in the inspection of composite components is based upon the ultrasonic wave interactions with it. The available experimental methods on the ultrasonic wave interactions with general solid materials use either dry or water immersed techniques. Systems ranging from the relatively simple one of isotropic solid semi-spaces to the most general ones of multilayered anisotropic media have been considered. Undoubtedly, analytical solutions to the fluid-loaded systems constitute adaptations, modifications or extensions of solutions pertaining to the corresponding dry cases. Inspection applications include, but are not limited to, finding anomalies, properties, quality of bonds and even the morphology of the solid. The role of the fluid has been thought of in the past is to facilitate the experiments and very little, if any, to influence the behavior of the solid. It is premature at this stage to refute the absolute accuracy of this promise, but it is shown in chapter 11 that there are situations in which the liquid can substantially alter the behavior of the solid. The motivation behind the study of fluid-loading came about when it was shown that much relevant information about the solid can be extracted from the reflection of a finite width transducer beam from the fluid-solid interface. Other unexpected new phenomena were discovered when inspecting the total reflected field. A distortion of the otherwise specular reflection

10

C H A P T E R 1. I N T R O D U C T I O N

was experimentally observed and by now has been proven to be attributed to leakage of energy into the fluid. This is a manifestation of the "leaky wave" phenomenon. In particular, we refer to the recently emerging scanning acoustic microscopy which is entirely based upon the nonspecular reflection from fluid-solid interfaces (for references, see Briggs [29]). Among many other reasons, interest in the application of the leaky wave phenomena to the inspection of solids prompted extensive efforts on the study of the interaction of ultrasonic waves with fluid-loaded solids. Research in this critical area is current and by now extensive literature is already available. Assortment of the vast available literature on this subject is a difficult task but can be somewhat easily accomplished if aided with a discussion of the history and the current state-of-art of the leaky wave phenomena. Leaky wave history began with its discovery in the field of optics when in 1947 Goos and Hanchen [96] observed a lateral displacement of the actual reflected beam from the position predicted by geometric-optics considerations. A comprehensive review of the lateral shift phenomena was soon after given by Lotsch [137]. The null region and associated trailing properties seem to be first discussed in detail by Tamir and Oliner [235]. Parallel to the early development in the field of optics, Schoch [219,220] predicted and experimentally verified the beam-displacement effect for an acoustic beam incident on a fluid-solid interface. According to Schoch's predictions, the beam is nonspecularly reflected in that it is laterally displaced while retaining, more or less, its original profile. In contrast to these predictions, many more recent experiments by Neubauer and Dragonette [188], Neubauer [187], Ngoc and Mayer [190, 189, 191], Plona et al. [201, 202], Smith [228, 227] and Diachok and Mayer [70, 69] have revealed that the reflected beam may also suffer severe distortion if it is incident at, or near, coincidence angles. In the case of the Rayleigh coincidence angle, for example, resonant transfer of acoustic energy from a longitudinal wave in the fluid to a Rayleigh wave propagating along the fluid-solid interface occurs. As it propagates, the Rayleigh wave reradiates into the fluid at this angle because of the resonant coupling. The result is a redistribution of the reflected field intensity such that a sizable fraction of the acoustic energy seems linearly displaced along the interface. As in the case of optics, this energy redistribution includes, in addition to the lateral displacement, a null region and a trailing decaying field. Based upon their earlier analysis in optics, Bertoni and Tamir [19] have examined the reflection of acoustic waves for angles close to the Rayleigh angle and constructed a model which explains the distortion phenomena. Specifically, they pointed out that the suitably simplified reflection coeffi-

1.1. H I S T O R I C A L B A C K G R O U N D

11

cient from isotropic semi-spaces has a singularity which leads to solutions corresponding to radiating (leaky) Rayleigh waves. According to their analysis, the distortion is the result of interference between the geometrically reflected field and the field of a leaky Rayleigh wave created by the incident beam at the Rayleigh angle. Braezeale, Adler, and Scott [27] experimentally verified the Bertoni and Tamir model, while Pitts et al. [199] presented theoretical results for the case of a finite beam incident on a solid plate in a fluid. Their results show that leaky wave phenomena occur at the Lamb coincidence angles. Subsequently, many authors have demonstrated both analytically and experimentally the existence of the leaky wave phenomena for a large variety of multilayered anisotropic media. Reflection from fluid-anisotropic semi-spaces have been studied by Mott [150], Senneke and Jones [110], Rollins [210], Atalar [11], Arikan et al. [10] and iayfeh [160]. From this relatively simple situation, many important properties of the solid can be easily extracted from the reflection coefficient. These include identification of the surface and pseudo-surface wave speeds and also the reflected beam shifting. By far, most of the fluid-loading literature deals with plates totally immersed in fluid. Here, many analytical and experimental investigations have been carried out on single and multilayered anisotropic plates. Collectively, plates immersed in a fluid have been identified with the leaky Lamb wave problem. An extensive research program of theoretical and experimental studies on these anisotropic plates with various degrees of orientations was carried out by Nayfeh and Chimenti [172,169,167, 49, 48, 45, 44, 47, 46] and Chimenti and Rokhlin [51]. Theoretical and experimental studies on similar systems were conducted by Mal et al. [141]. Other studies on single plates were published by Dayal and Kinra [56]. Reflection from multilayered plates in contact with fluid on one side while the other side is either free or in contact with another solid substrate have also been studied extensively by iayfeh and Taylor [178, 177], Braga and Serrmann [26], Merkulov [145], Fiorito et al. [83], deBilly et al. [57], Guyott and Cawley [98], Dragonette [71], Sattunen and Luukkala [106], Fay and Fortier [81], Freedman [86] and Mampaert and Leroy [142].

1.1.4

Piezoelectric effects

Often layered materials in general and composites in particular contain components that exhibit some degree of piezoelectric coupling effects. The piezoelectric coupling is manifested by converting electrical energy to mechanical energy and visa-versa. Ever since their discovery in 1880 by the Currie

12

C H A P T E R 1. I N T R O D U C T I O N

brothers and their demonstrated success in the sonar systems by Langevin and his coworkers in 1916, piezoelectric materials have been applied in many important fields such as geophysics, electronics, communication, instrumentation and nondestructive evaluation and testing of materials. It is beyond the scope of this book to review the vast literature covering general piezoelectric effects and their various applications. Nevertheless, for general discussions of piezoelectric effects, we refer the reader to the standard books by Auld [12], Cady [34], Rosenbaum [213] and Tiersten [239]. Here, we limit ourselves, however to a review of literature pertaining to piezoelectric guided waves which are relevant to the scope of this book. Bleustein [20, 21] developed an "electroacoustic" surface wave, known as " Bleustein-Gulyaev wave" (B.-G. wave), which can be generated and detected on some classes of the piezoelectric materials. Herman et al. [92] studied generalized Rayleigh waves in cubic piezoelectric crystals. More recent literature relating to guided waves in plates and layered media can be found in Chien [42] and Nassar [157]. Nayfeh and Chien [165,166] presented unified analytical treatments of the interaction of ultrasonic waves with piezoelectric anisotropic half-space substrates and plates immersed in fluid. Simple analytical expressions for the reflection and transmission coefficients are derived from which all propagation characteristic are identified. Such expressions contain, as a by-product, the secular equation for the propagation of free harmonic waves on the piezoelectric substrate. It is found in [165, 166] that piezoelectric coupling, as well as water, influence both types of modes. Results for all higher symmetry materials, such as orthotropic, transverse isotropic and cubic, are contained implicitly in references [165, 166]. It is also demonstrated in these references that the motions of the sagittal and horizontal planes uncouple for propagation along axes of symmetry. As shall be shown in chapter 12, for such cases, however, piezoelectric coupling can influence one of these kinds of motions depending upon the type of piezoelectric model adopted.

1.1.5

Scattering from layered

cylinders

Compared with the voluminous literature on the scattering of elastic waves from coaxial isotropic media, we have not been able to identify many works that exist on similar, but anisotropic systems. The scattering from elastic isotropic cylinders, spheres and elliptic cylinders was first treated by Sezawa [224]. In his book "Vibration and Sound", Morse [148] discussed sound scattering from cylinders and spheres, and was the first to give polar diagrams that show the distribution-in-angle of the intensity of the scat-

1.1. H I S T O R I C A L B A C K G R O U N D

13

tered wave and also the total scattered intensity. Theoretical solutions to the problem of the scattering of sound by rigid, immovable cylinders, nonrigid cylinders in a fluid medium and small cylindrical obstacles in a solid medium were formulated by Rayleigh in 1945. The solutions he presented described geometries in which the diameters of the cylinders were small compared to the acoustic wavelength in the surrounding medium. He also outlined a more general method for finding the solution for large diameter cylinders in terms of cylindrical harmonics, since such harmonics are readily available. These pioneering works were followed by a large number of investigations, from which we mention the works of Faran [79], Zato [118] and Ying and Truell [258]. Scattering of elastic waves, incident obliquely on a circular cylindrical obstacle in a solid media, was formulated by White [251]. Various extensions and numerical investigations, based upon White's theory are given by Lewis and Kraft [132]. Other analytical and numerical methods have been used by Franssens et al [85] and Pao and Mow [194]. Comparatively speaking, very little work is available in the literature on the scattering of elastic waves from anisotropic cylindrical and spherical systems. Scattering from composite shell structures made up of thin shells, compliant coatings and annular fluid layers was treated by Akay [8]. The model he developed was for three concentric cylindrical shell. Chapter 14 is devoted to the study of S H wave scattering from multilayered coaxial anisotropic cylinders.

1.1.6

E l a s t i c p r o p e r t i e s of composites

In modeling the interaction of elastic waves with anisotropic layered media, the elastic properties of the individual layers are needed. For composite materials, these properties are not necessarily available and one needs to construct them. These properties will undoubtedly depend upon the virgin properties and volume fractions of the individual components comprising the composite. Over the last three decades or so, a large body of research has been expended upon modeling composite materials for the purpose of obtaining their effective elastic properties. The task is to obtain effective values of the stiffnesses Cijkl (or, for short, Cpq as per the adopted contraction notation of chapter 2). For layered and fibrous composites, certain anisotropy will result depending upon the geometric arrangement of the individual components. Perhaps the simplest composite consists of unidirectional fibers uniformly distributed in a host matrix. While the geometric simplicity of this model lends itself to obtaining exact results, in reality, only a limited num-

14

C H A P T E R 1. I N T R O D U C T I O N

ber of exact and directly obtained properties are possible to extract from even these primitive geometries. In cases exhibiting perfect periodicity, finite difference and finite element techniques have led to exact numerical results. Using such numerical techniques, a hexagonal geometry approximation of the unidirectional composite was first analyzed by Pickett [198] in 1968 and a square assemblage approximation was analyzed by Adams, Doner and Thomas [5] in 1967. In the meantime, several authors have attempted to derive, with limited success, exact analytical expressions for such geometric models. One of these is called the composite cylinder assemblage introduced by nashin [100, 99] and Hashin and Rosen [102] in 1964. Others are based on wave propagation models of composites where effective properties are extracted in the limit when the microstructural dimension is very small in comparison with incident wavelengths. Pioneering works based on wave propagation models include those of Rytov [215], Postma [203], Sun et al. [232], Bedford and Stern [16], Hegemier et al. [108, 107], Nayfeh et al. [174, 175, 163, 159] and Sabina and Willis [216]. Like the composite cylinder case, only a limited number of the effective properties can be extracted from the wave propagation models. The remaining properties are obtained by approximate methods. In short, there exists no analytical model which is capable of deriving exact expressions for all of the desired properties even for the simple geometries described above. As an alternative, researchers have sought to construct approximate models capable of predicting, with various degrees of accuracy, the effective properties of a wide range of composite architectures. The simplest ones are based on concepts borrowed from the strength of material. Others are based on simple or elaborate elasticity theories. The literature is almost saturated by the number of treatments which include books, archival and review articles. The pioneering works of Hashin et al. [101, 104, 103], Tsai et al. [244, 243], Adams and Tsai [6], Whitney et al. [254, 252, 253], Chamis and Sendeckyj [40], Sendeckyj [223], Hill [113, 114], Vincent and Choo [52], Christensen [53], Paul [195] and Jones [117] and, more recently Nemat-Nasser and Hori [181] are well documented and available for further references. It is, however, beyond the scope of this book to go in depth in reviewing this vast literature. Effective properties obtained for unidirectionaly reinforced fibrous composites constitute the foundation for finding properties of more complicated composite materials. Laminated plate theories are, for example, a means of obtaining effective properties for more complicated multilayered composite systems. Chapter 15 is devoted to describing some simple models for obtaining effective elastic properties of fibrous composites.

Chapter 2

FIELD E Q U A T I O N S A N D TENS OR ANALYSIS The dynamic behavior of a linear elastic, generally anisotropic solid can be described by the tensorial equations of motion written, in the reference orthogonal Cartesian system x i' = (xl, x~, x~) as

Oa~j = p '02u~ Ot 2

(2.1)

and the general tensorial stress-strain (constitutive) relations

(Tij'

-

-

(2.2)

' ' Cijklekl

or their dual inverses I

I

eij = Sijkla~kt

(2.3)

with the auxiliary strain-displacement relations

10u~ Ou~k ekl = 2(0-~x~ + -~x~)'

i , j , k , 1 = 1,2,3.

(2.4)

Here a~j and e~l are the stress and strain tensors, respectively; u i is the displacement vector and p' is the material density. The quantities C~jkl and ! 8ijkl are elastic constants known as the stiffness and compliance tensors, respectively. Since the rank (order) of a tensor depends upon the number of its free (unrepeated) indices, the quantities p' ,?.ti,' Oij' , and Cijkl constitute tensors of rank zero, one, two and four, respectively. In tensor analysis a repeated index in any term is designated as a contraction and reduces the rank of its 15

16

CHAPTER

2.

FIELD EQUATIONS

AND TENSOR

ANALYSIS

!

tensor by two. As Cijkl and e~l have ranks of order four and two, respectively, their product, as given by the right hand side of equation (2.2), is reduced to a tensor of order two in virtue of the contraction of the repeated indices k and I. The resulting second order tensor has the two free indices i and j consistent with the left hand side of the same equation. A similar contraction situation is also evident in the left hand side of equation (2.1). Here j is a repeated index leaving the free index i. By virtue of the summation convention, repeated indices imply summation over their index domain. As an example, for k - 1, 2, 3, one has !

=

+

(2.5)

+

a scalar (i.e., a tensor of order zero). Keeping the summation convention in mind, equation (2.1) is a first order tensor (vector) equation consisting of three equations for i - 1, 2 and 3, respectively.

2.1

T h e stiffness t e n s o r

Since each of the indices takes the values 1,2 and 3, any tensor of order ! n has 3n elements. Thus, a~j, e~l and Cijkl have nine, nine and eighty-one elements, respectively. However, since a~j and e~l are symmetric, namely a~j - a~i and e~l = e~k , then each has only six independent elements. These symmetry properties, if used in equation (2.2) reveal the following properties of the stiffness tensor C~jk I

I ! = cjikt = cijtk = c tj , k

(2.6)

!

thus reducing Cijkl to thirty-six independent elements. This, of course, is consistent with the fact that a linear combination of the six independent components of the stress tensor in terms of the six independent components of the strain tensor is specified by thirty-six coefficients. Additionally, by utilizing the following simple strain energy argument, we see that the number of independent elements in the stiffness tensor is twenty-one in number. To this end, the strain energy density is given by 1

V

=

I

I

-~o'ijeij

1

I

I

I

-- -~Cijklekleij

(2.7)

from which we obtain, by using straightforward differentiation ! Cijkl --

02U I

I

OeijOekl

(2.8)

2.2.

MATERIAL

SYMMETRY

17

Observe that interchange in the order of differentiation does not change t ! the relation (2.8); thus, we conclude that Cijkl = Cklij thereby leaving only twenty-one independent coefficients. This is also known by the reciprocity relationship. This is the maximum number of independent elastic constants that an anisotropic elastic material requires. Materials belonging to this class of anisotropy are called triclinic.

2.2

Material symmetry

Often, natural materials as well as man-made ones (such as structural composites) have material symmetries which can lead to further simplification in their constitutive relations (2.2). Simplification is achieved through a reduction in the number of independent coefficients. There are several methods available to describe material symmetries. The most widely known and used method is based upon strain energy considerations and is fully described in Love's classical book "A Treatise on the Mathematical Theory of Elasticity". In this present book, we adopt an alternative method which is based upon the properties of linear orthogonal transformations. In the following, we describe the general features of this transformation and show how it can be applied for identifying various symmetries in anisotropic media. 2.2.1

The transformation !

Noting that Cijkl is a fourth order tensor, its transformation from the reference coordinate system x ti to a global coordinate system xi is achieved by an orthogonal transformation in accordance with (see figure 2.1)

Cmnop = Z.~ Z,~j&k Zpt Cijkl. '

(2.9)

Here/3ij is a transformation tensor whose elements are the cosines of the ! angles between the xi and x j axes; for example,/33x - cos r of figure 2.1. The matrix of the transformation tensor/3ij is written as

[

Zll Z12 /~13]

~ij-

~21

/322 ~23

Z3~ ~32 /~33

9

(2.10)

Equation (2.9) is now applied in the definition of appropriate restrictions on the properties of anisotropic media possessing various material symmetries. As a reference to subsequent classifications, we conjecture that general anisotropic (triclinic) materials have no material symmetry.

CHAPTER 2. FIELD EQUATIONS AND TENSOR ANALYSIS

18

I

X 2

X2

X1 XPl

XP3

X3

Figure 2.1: Coordinate transformation. Starting with the triclinic case, we move to the slightly symmetric material, namely, the monoclinic one. Monoclinic materials possess a single plane of symmetry. Consequently, without loss in generality, and for the sake of convenience, we choose this plane to coincide with the x~ - x ~ plane. This implies that the x~ - x ~ plane constitutes a mirror plane and thus the material possesses a mirror symmetry with respect to it. Accordingly, the material occupying the semi-space x~ __ 0 has the same characteristics as that occupying the semi-space x~ ___0. This means that the material is ' , under the transformation invariant, namely Cijkl - - Cijkl

~ij=

1 0 0 0 1 0 J. 0 0 -1

(2.11)

Substituting equation (2.11) into equation (2.9) and requiring that all transformed properties be invariants dictate the vanishing of the entries C1123 ~ C2223 ~ C3323 ~ Ci 113~ C 213~ C3313~ C2312~ C1312 , , , , , ,

(2.12)

thereby leaving thirteen independent coefficients. As an illustration of how these results come about, we consider the transformed property C2312. From equation (2.9) we write c23~2 = Z 2 i Z 3 j Z ~ k Z 2 1 C 'i j k l .

(2.13)

2.2. MATERIAL S Y M M E T R Y

19

Summing over all repeated indices and using the special transformation (2.11) we conclude that C2312 -

' --C2312.

(2.14)

!

Now, since we require that these properties be invariants, i.e., r -" C2312, we see that C2312 must vanish. In performing the manipulations that lead to equation (2.14), advantage is taken of the fact that the contributing entries in Dij of equation (2.10) are the diagonal ones only. If the material has a second plane of symmetry, say the x~ - x~ plane, then, in like manner, the transformation

[100]

~ij-

0-1 0 0

0 1

(2.15)

leaves the material properties invariant. Starting now from the monoclinic case, applying equation (2.15) and following the above outlined procedure, t , c1323 ~ resulting lead to the further vanishing of the elements c~112, c2212, c3312 in an orthotropic material with only nine independent coefficients. We note that if two orthogonal planes constitute planes of symmetry, then any plane normal to them is also a plane of symmetry and will not contribute to further restrictions on the properties of the material. Materials with symmetries higher than orthotropic can be identified by using the same basic procedure that has been previously outlined. In order to facilitate further development, we shall specifically consider the case of a counterclockwise rotation through an angle r about the x~-axis, for which the appropriate transformation is described by

[

cos r -sine 0

~ij=

sin r cosr 0

0 0 1

]

.

(2.16)

When applied to orthotropic materials, t h e resulting collection of transformed coefficients becomes I G4 I S4 I $2G2 Cllll = Cllll -}- C2222 + 2(C~122 + 2C1212 ) , , ( S 4 + G 4) c1122 = (c~111 + c2222 - 4c~212)$2G2 + c1122 c1133 -~ r c~2

-

'

G2

, (c1~1~ ,

$4

c2222 -- cl 111

~- r

'

c ~, 2 2

,

$2 , ) S G 3 4-(c~122 2c~2~2

-

G4

+ c2222

I G2 ! $2 c2233 ~- c2233 ~ c1122

' c2222 + 2c'~2~)GS I , $2G2 + 2 (C1122 -]- 2C1212 )

3

20

C H A P T E R 2. FIELD E Q U A T I O N S A N D T E N S O R A N A L Y S I S C2212 --.

, , , (c]111 __ c1122 3 + ( CI122 __ 2c1212)GS

, - - C2222

, )SG 3 +2c1212

! C3333 - - C3333 ,

C3312

-c1133)SG

=

'

C2313 =

-

c1313)SG

! 2 ! 2 C2323 ---- c 2 3 2 3 G -+- c 1 3 1 3 S ! 2 ! 2 C1313 - c 1 3 1 3 G -+- c 2 3 2 3 S C1212 =

I

(c~111

+

! - 2c'1122 -4c'1212)$2G 2 + c1212 c2222

(2.17)

where, for convenience, we use G = cos r and S = sin r If there exists a rotational angle r under which the transformed stiffness coefficients are invariant, then further restrictions on the stiffnesses can be found. For example, if a rotation of r - 90 ~ leaves the stiffnesses unchanged, we can deduce from equation (2.17) that the directions x~ and x~ are completely interchangeable giving the restrictions I

I

C2222 - - C l l l l '

I

C2233 - - r

I

r

I

-- r

(2.18)

On the other hand, if the invariance exists for any angle r we can obtain, in addition to equation (2.18), further restrictions of the form !

!

CI111 - - C1122 ---- 2C~212

(2.19)

leading to complete isotropy in the x~ - x ~ plane. This last result can be easily seen by checking the entries c1112 and C2212 , for example. Under the invariance conditions these must be equated to the primed entries c1112' and / c2212', respectively. But since c~112 and c2212 are both zero for orthotropic materials to begin with, then c1112 and C2212 must also vanish, thereby recovering equation (2.18) and leading to equation (2.19). Under the conditions (2.18) and (2.19) the material becomes transversely isotropic, i.e., isotropic in the x~ - x ~ plane and has only five independent constants. Following the same line of thought, we further assume a transformation similar to that of equation (2.16) in the x~ - x~ plane through an angle 7. For the combined rotations ~/ - 90 ~ and r - 90 ~ if the stiffnesses are invariant, we have a cubic material which has three independent constants. In this case we see that the coordinates x~, x~ and x~ are completely interchangeable. Finally, if combinations of any arbitrary rotations through the angles ~/and r leave the stiffnesses unchanged we have complete spherical symmetry (isotropy) which has only two independent constants.

2.3. M A T R I X FORMS OF STIFFNESS

2.3

Matrix

21

forms of stiffness

Summarizing the above results and, in order to facilitate subsequent discussion, we now rewrite the stress-strain relations in their expanded matrix form. To illustrate this procedure, a representative stress component will be written in its expanded form. To this end, we choose a~l which, from equation (2.2) for a triclinic material, is given by !

I

!

I

I

(711 -- C~lllell + c1122e22 -t- c1133e~3 ! !

(2.20)

+2(cI~2e~2 + ci113e~a + cI123e~3).

To arrive at this expansion, we use the symmetry of the strain tensor and ! the various properties of CijkL as previously noted. By further adopting the contracted index notation 1-+11, 2--+22, 3--+33, 4-+23, 5-+13 and 6-+12 to the stiffness tensor and introducing the specific engineering shear strain components ~/[2 = 2e12, ~/~3= 2ela,

~/13= 2e~3

(2.21)

we rewrite the expression (2.20) as I

I

I

I

I

I

I

I

(2.22) O'il = C~lell -I- C12e22 + 613e~3 -l- C14")'23 + 615"/13 ~- C16~12 where we used upper case C's to differentiate the contracted form of the lower case ones. In adopting these notations, we can rewrite equation (2.2) in the expanded matrix form

Oll

C[1 C[2 C[3 C[4 6~5 616

a[a a~a a[a a~2

Ch C;~ C;6 sym C55 C56

a~2

ell (2.23)

% "/13

Equation (2.23) describes the most general stress-strain relations for a linear elastic anisotropic solid. Once again, it refers to a triclinic solid which has no material symmetry. Implementing the various property restrictions previously described we can now list the constitutive relations for the different symmetry classes as follows: M o n o c l i n i c h a v i n g x ~ - x~ as a p l a n e of s y m m e t r y :

c~, c~ ch ~= 3

c~

3

sym

2

o

o

cf~

o o c~ Ci4 Ci5 0 C~5

0

C~6

41

e~2 e~a ~23

(2.24) "

22

CHAPTER 2. FIELD EQUATIONS AND TENSOR ANALYSIS

Orthotropic :

1

C{I 6{2 C{3

0

0

0

e~l

~. 3 3 3

c;~

o 0 C~4

o 0 0 C~5

o 0 0 0

4~ e~3 ")'13 7~3

_=

ch Ci3

sym

(2.25)

"

Transversely isotropic in the x ~ - x~ plane : Cr~l ff22 a~ 3

C[1

C{2

C{3

C~1 6{3 =

C~3

~3

0 0 0

0 0

C~5

a~3

sym

0 0 0

0 o

e~2 e~3 723 7~3

0

6~5

~'12

41

0

ci~-cl..

(2.26)

"

2

Cubic: O'il

C~l

~2 a~3 a~3 a~3

C~2

C{2

0

0

0

c~

c~2

o

o

o

C{1

0

0

0

C~

0

0

__

sym

C~6

o~

0

c~

41 e~2 e~3 ~23 ~3

(2.27)

"

Isotropic :

o11

c{~ ch

~2 g33 g23 ai3 CrY2

__

ci~

o

o

C~l c{2

o

o

0 Cil-C122

0 0

C[1 sym

C~1-C[2

2

0 o o o o ci~-cl.. 2

41 e~2 e~3 ~3

(2.es)

2.4. ENGINEERING CONSTANTS 2.4

Engineering

23

constants

In this section we attempt to relate the various elastic stiffness coefficients to their corresponding conventional engineering constants. Engineering constants for orthotropic materials are generalized Young's moduli, Poisson's ratios and shear moduli. These constants are measured in simple tests such as uniaxial tension or pure shear tests, and thus have direct physical interpretations. Most simple tests are performed with a known load or stress. The resulting displacement or strain is then measured. Therefore, the components of the compliance tensor 8ijkl are determined more directly than those of the stiffness tensor Cijkl. For an orthotropic material, the strain-stress relations in terms of the engineering constants are given by

e~l

~1

e~2 e~ 3

=

/212

/2~1 E~ 1

/2~1 E'13 /232

E'11

~,' 12

E~

E~

E~

/213

E~

/223

1

0

0

0

0

0

0

0

0

0

, 9'23

0

0

0

1 G~ 4

"y~3

o

o

o

o

a~

o

o

o

o

o

] "~2

0 1

(2.29)

0

~3

o

~3

1 a~6

Here, E~,i = 1,2, 3 are the Young's moduli, v~j are the Poisson's ratio for transverse strain in the j-direction when the material is stressed in the idirection and G~4 , G~5, and G~6 are the shear moduli in the x ~ - x'3, X~ - - X~ and x'i-x'2, planes, respectively. By inverting equation (2.29), we obtain the engineering stiffness matrix which can be identified with the corresponding matrix (2.25) and hence establish appropriate relationships between the engineering constants and their stiffness coefficient counterparts. Since both the stiffness and compliance matrices are symmetric, one can identify in (2.29) three reciprocal relations that must be satisfied for an orthotropic material. Moreover, only v~2 , v[3 , and v~3 need be further considered since v;i, u~l, and v~2 can be expressed in terms of the first-mentioned Poisson's ratios and the Young's moduli.

24

C H A P T E R 2. FIELD E Q U A T I O N S A N D T E N S O R A N A L Y S I S

2.5

Transformed equations

I are tensors, in many cases Since ui,, aij, , e ~ , a n d Cijkl it convenient to conduct our analysis in the global rather than in the reference (crystollographical) one described, for any linear orthogonal transformation nonprimed system, we have

to follow, we will find coordinate system xi xi.' As was previously of the primed to the

Xi - /~ijX~ !

p-p

Ui ---- ZijU~

' (Tmn = Z.~Z.j i (Tij eop = ~okZplekl

Cm,wp = ~mi~nj~ok~ptc~jkZ.

(2.30)

Also, the equations of motion (2.1), the constitutive relations (2.2) and the strain-displacement relations (2.4) transform to (2.31)

COaij 02ui Oxj = p Ot 2

aij = Cijklekt

(2.32)

10ul

OUk ekt = 5(~--~Xk+ -~-x)

(2.33)

For the remainder of this book, we shall often conduct our analyses in an orthogonal transformed system obtained from a counterclockwise rotation through an azimuthal angle r about the x~-coordinate. Thus, x3 is identical with x~ and the appropriate transformation is given by equation (2.16). For the present situation the transformed coefficients for the monoclinic symmetry, using the contracted notation, take the form Vii = C~l G4 nt- 6 ; 2 S 4 -Jr-2(C12 -I- 2 C 6 6 ) $ 2 G 2

+4(C[6 a2 + C~6S2)SG C12 "- (C[1 -4- C;2 - 2C[2 +2(c~

- c;~)(s

~ -

4C;6)$2G 2 + C~2 a~)sa

2C~6Sa -{- 2c;6)sa 3 + (c;2- c~2

C13 -- C~3 a 2 -1-C;3 S2 + 616 --

(C12-C~l +3(c;~

-

-

c ~ ) s ~ a ~ + c[~a 4 - c;~s ~

C22 = C l l S 4 + C;2 G4 + 2(C[2 + 2c~6)s2a 2

-4(c~s ~+ c~a~)sa

2C~6)GS3

2.5. TRANSFORMED EQUATIONS

25

C23 = C~3G2 + C~3 $2 -- 2C~6SG

c ~ = (c~

+

2c~

C~)GS 3

-

+

(ci~

c~

-

-

2C~)SG ~

+3(C~6 -- C;6)$2G 2 - C~6$4 + C;6G4

caa = c~a C36 = (C; 3 - C[3)SG %.C;6(G 2 - S 2) C44 = Ct44G2 %.C;5 $2 - 2C45SG C45 -- ( C 4 4 - C 5 5 ) s a + c 4 5 ( a

2-s

2)

C55 = C55 a 2 %. C;4 S 2 %- 2C~5 SG C66 "-= C66 %- (C~1%- C;2 - 2C[2 - 4C~6)S2G2 + 2(c~

- c;~)(s

~ -

a~)sa.

(2.34)

These properties can be compared with those listed in equation (2.17) for orthotropic materials. They are collected in the matrix form all o22 0"33 023 013 (712

Cll __

C12 C22

C13 C23 C33

0 0 0 C44

sym

0 0 0 C45 C55

C16 C26 C36 0 0 C66

ell e22 e33 ")'23 ")'13 712

(2.35)

We note the similarity in appearance between this transformed matrix of orthotropic materials and that of the nontransformed matrix (2.24) belonging to monoclinic materials. 2.5.1

Advantages

of orthogonal

transformations

There are several advantages to conducting analyses in the transformed coordinate system as compared with the reference primed crystollographical coordinate system. Under the transformation (2.16), results are easily generated for any azimuthal angle including r = 0 ~ which describes the primed axes. Furthermore, as noted below equation (2.35), for any rotation in the plane of symmetry of monoclinic and higher symmetry situations (except for the isotropic case ), the transformed properties will resemble the form of the monoclinic material, i.e., no matter what rotational angle r is used, the zero elements in the matrix (2.24) will remain unchanged. For example, the vanishing of entries C~6 , C~6, C'36, and C~5 required to define an orthotropic material will not change the form of equation (2.35). However, if such entry vanishings are accompanied with the choice r = 0 ~ equation (2.35) reverts back to equation (2.25) which describes orthotropic symmetry. This

CHAPTER 2. FIELD EQUATIONS AND TENSOR ANALYSIS

26

remarkable property of the transformation will lead to important simplifications in all of the analyses that will be conducted in this book. Finally, for cases involving multilayers, a transformation of each layer's properties from its crystollographical axes to a common global system will also lead to great simplifications in the analyses.

Expanded field equations

2.6

Because the field equations (2.31) and (2.32) are the foundation of the remaining chapters in this book, it is of great heuristic value to recast them in their expanded form. In view of this, the equations of motion (2.31) are rewritten in the alternate expanded form

(~Gll _ 0G12 0G13 ox~ ~- ~

02Ul

Oa12 0a22 0a23

02u2

Ox---[ + ~

+ Ox---[ = p ot2

0613

0633 02U3 + OX3 = p Ot2

0a23

OXl + ~

(2.36)

+ ox---2 = p ot 2 (2.37) (2.38)

Substitution from the constitutive relations (2.23), after their transformation via equation (2.16), yields the following three coupled partial differential equations for the three displacement components u/

O~2Ul 02Ul 02Ul 02Ul 02Ul Vii 0x12 + C66 0x 2 + 655 0x 2 + 2C16 0Xl Ox--------~+ 2C15 (~Xl0x3 02Ul 02U2 C02U2 02U2 +2656 Ox20x3 + 616 0x 2 + 626 Ox2 + C45 Ox---~3

02u2 02u2 02u2 +(612 + 666) OXlOX2 + (C14 + C56) OXlOX3 + (646 + 625) Ox20x3 02~t3 02~t3 02~t3 02~t3 +615 0X21 + C46 Ox2 + 635 Ox~ + (C14 + 656)OXlOX2 02lt3

027"t3

+(C~3 + C5~) Ox~Ox3 + (636 +

02Ul

Oq2ltl

C4~)Ox20x3

02'/.tl

02ul = p Ot 2

0211,1

c~6 Ox~ + c26 Ox~ + c45 o ~ + (612 + c6~)ax~ax~ 02u2 02Ul 02Ul 02U2 +(614 + 656) OXlOX3 + (646 + 625) OX20X3 + 666 OX2 + 6 2 2 ~

(2.39)

2.6. EXPANDED FIELD EQUATIONS 02u2

02u2

02u2

27

02u~

+644 Ox2 + 2626 OXlOX--------~"1" 2646OXlOX3 "1" 2624 Ox20x3

02ua

02ua

+c~60x~ + 624 0 ~

02u3

+

02u3

ca40x~ + (c46+ c2~) OxbOw2

02u3 02u3 02u2 +(636 "1"645) OXlOX3 "1" (623 + 644) OX20X3 -- f) Ot 2

(2.40)

02Ul 02Ul 02Ul 02•1 615 oX 2 "1" 646 OX 2 "1" C35 Ox 2 "1" (C14 .1..C56)OXlOX2

02u2 02Ul 027/1 02t~2 +(C13 ,1,.655) OXlOX3 + (636 ,1,.645) oqx20x 3 + 656 Ox21 .1.. C 2 4o4 ~ 02u2 02u2 02u2 +634 OX2 + (646 ,1,.C25)C~XlOX2 .1.. (636 ,1,.C45)OXlOX3 02?/2 02?/3 02?/3 02U3 "1"(623 "1"644) OX20X3 + 655 OX 2 + 644 OX 2 "1" 633 OX----T 02ua 02u3 02u3 02ua +2645 OXlOX2 "1" 2635 OXlOX-----~"1" 2634Ox2Ox-----~= p Ot2 2.6.1

(2.41)

Monoclinic

Once again, equations (2.39)-(2.41) hold for triclinic materials with the choice of any azimuthal angle r These equations can be specialized to monoclinic materials by invoking the property restrictions (2.12). Taking advantage of the previously pointed out similarity in forms between the matrices of the original and transformed properties of monoclinic material symmetry, we conclude that the transformed properties will take the form (2.35). Thus, the equations describing the behavior of monoclinic materials for any azimuthal angle can be obtained directly from equations (2.39)-(2.41). Setting each of the properties C14,615, C24, C25,634, C35,646 and 656 to zero, results in

02Ul 02Ul 02Ul 02Ul Cll oqx2 "1" 666 Ox 2 "1" 655 oqx2 "1" 2616 OqXl OqX-~-'--

02u2 02u2 02u2 02u2 +616 Ox21 "1" 626 Ox 2 "1" 645 Ox 2 + (612 "1"666) OXl(~X2 02u3 02u3 02Ul +(C13 + 655) OXlOX3 ,1,. (636 -Jr-645) Ox20x3 = p Dt 2 02ul

02ul

02ul

02ul

c~6 ox~ + c~6 0x~ + c4~ Oxl + (c~ + c66) OXlOX2

(2.42)

CHAPTER 2. FIELD EQUATIONS AND TENSOR ANALYSIS

28

02u2 02u2 02u2 02u~ +666 Ox21 + C22 Ox2 + C44 Ox2 + 2626 OxlOx2 02u3

02u3

02u2

-t-(C36 -~- 645) OXlOX3 nu (C23 -+- C44) O x 2 0 x 3 --= fl Ot 2

(2.43)

and Oq2U l

D2 U l

(C13 "nt- 655 ) oqx 1oqx3 + (636 + 645 ) oqx 2 oqx 3

02u2

02u2

02u3

~-(636 -~- 645) OXlOX3 Jr- (623 nt- 644) O x 2 0 x 3 nt- C 5 5 ~

02u3 02u3 02ua 02u3 +644 Ox2 + C33 Ox2 + 2645 OXlOX-------~= p Ot2

(2.44)

Orthotropic

2.6.2

The vanishing of the entries C~6, C~6, C'36, and C~5 further reduces equations (2.42)-(2.44) to those applying to orthotropic media. However, such restriction will not change the form of these equations. Thus, we conclude that equations (2.42)-(2.44) are also applicable to orthotropic material subject to the above property restrictions. Implementing these restrictions leads to modifications of the transformed properties (2.34) in accordance with Cll = C~1G4 + C~2S4 + 2(612 + 2C~6)$2G 2 612 "- (611 + C;2 - 2C~2 - 4C~6)S 2G 2 + 6~2 c~a = c~aa 2 + c ; a s 2

C~6 = (C[2 + 2C~6 - C [ 1 ) S G a + (C~2 - C [ 2 - 2c;6)as a 622 =- C[1 s4 + C;2G 4 + 2(C12 + 2C~6)$2G 2 C2a = C~aa 2 + c~as 2 626 = (C[2 +

2C66-- C~I)GS 3 -~- (C;2

- C12 - 2C~6)SG 3

caa = c~a c3~ = ( c ; . - c ~ 3 ) s a c44 = c ; 4 a 2 + c ; ~ s 2

645 = ( C 4 4 - C;5 ) S G

C~5 = C~(;2 + C~4S2 =

+ (c 1

+

-

-

4c;

)s

a

(2.45)

29

2. 7. P L A N E S OF S Y M M E T R Y

2.7

P l a n e s of s y m m e t r y

We start with equations (2.42)-(2.44) and choose r = 0 ~ as required for propagation along a plane of symmetry for orthotropic or higher than orthotropic material symmetry. We can then conveniently revert to describing these equations in terms of the reference coordinate system x i. Here all transformed (nonprimed) properties, field variables, propagation directions and coordinates are identical with the reference (primed) counterparts. To identify such a situation we shall conduct its analysis by employing the reference system x~. Choosing x~ - x~ for the propagation plane implies that all field variables will be independent of the x~ coordinate, and hence the relevant equations respectively reduce to

02U'l

02u'3

02U~l

' ~ + c~ ,~ + ( c ~ + c ~ ) Cll Ox,2 Ox 3 OXll0xl3 = p at 2

02u'2

02u'2

02u'2

(2.47)

C66, OXll2 + 6144 0x!32 -- fl Ot2 ,

o2~'~

o2~'3

(2.46)

o2~'3

(C13 ~- 655)OXllOxl3 + C55 Ox? + C~3 bx '2

02u~3 =POt 2 9

(2.48)

Special notice can be given to equations (2.46)-(2.48), since they display partial uncoupling. Equation (2.47) has only one independent variable, namely u~ defining horizontally polarized ( S H ) motion. On the other hand, equations (2.46) and (2.48) constitute two coupled equations in the u~ and u~ displacement components describing a motion confined to the sagittal plane x ~ - x~. Situations like the present one will lead to significant simplifications in our subsequent studies.

This Page Intentionally Left Blank

Chapter 3

BULK WAVES 3.1

An overview

The fundamental mechanical behavior of anisotropic solids is best demonstrated through their response to propagating bulk waves. Bulk waves exist in infinite homogeneous bodies and propagate indefinitely without being interrupted by boundaries or interfaces. These waves can be decomposed into infinite plane waves propagating along an arbitrary direction g within the solid. The properties of these waves are determined by the relationships between the propagation direction and the constitutive properties of the medium. Generally speaking, three types of waves are possible. These are associated with the directions of the three particle displacement vectors u-'(k), k - 1, 2, 3 defined as acoustic polarizations. Typical polarizations are depicted together with the propagation direction in figure 3.1. The three polarization vectors are mutually orthogonal but, in general, none of them are necessarily parallel or normal to g. A bulk wave is termed "pure" if its polarization vector is directed either along or normal to the propagation direction. For isotropic materials, only pure modes are possible; one of these waves is known as longitudinal with polarization directed along the propagation direction g. The other two are known as shear waves with polarizations directed normal to the propagation direction. Thus ff x g - 0 and ft. g - 0 define purely longitudinal and shear waves, respectively. For an isotropic material with Lame elastic properties A and # and density p, it is known (and shall also be shown later on in this chapter) that longitudinal waves travel with speeds vl = x/(~ + 2 # ) / p , whereas shear waves travel with speeds vt = V/-~/p. In changing the propagation direction, the individual characteristics of these waves, and in particular their speeds, do not change. 31

C H A P T E R 3. B U L K W A V E S

32

X3

X2

XI

Figure 3.1: Schematic of typical polarizations in anisotropic media.

To further demonstrate the "clean" character of propagation in an infinite isotropic medium, we imagine the case of a spherical source in the medium. This specialized example also serves as a forum for introducing terminology of elastic waves in solids. Three spherical wave front surfaces emanate from the source. The fastest one travels with the longitudinal wave speed vl, while the two shear waves coincide and travel with the shear wave speed vt. These wave front surfaces are also normal to the propagation direction which, in this case, is the radial direction. More precisely, the propagation direction is normal to the tangents of these surfaces. Here the longitudinal wave is polarized along the radial direction, whereas the shear ones are polarized normal to it. Associated with wave front surfaces are two other kinds of surfaces: (i) slowness surfaces, defined by the inverse of the wave front speeds and (ii) energy flow surfaces, also known as group velocity surfaces, which define the direction of the flow of energy associated with the propagating waves. For isotropic media, energy flow surfaces coincide with the wave front and slowness surfaces. For anisotropic media however, as shall be demonstrated below, none of the three polarization vectors a priori satisfies either ~ • g - 0 or i f - g 0. If one of the polarization vectors satisfies the first condition, namely • g = 0, then both of the remaining two satisfy the second condition ~ . g = 0, thereby defining three pure modes. If on the other hand, one of the polarization vectors satisfies the second condition, there is no guarantee

3.2. THE C H R I S T O F F E L E Q U A T I O N

33

that the remaining two will satisfy either condition. Hence, for the later case we are assured of the existence of at least one single pure mode. Generally speaking, for anisotropic media, pure modes can occur for some propagation directions depending upon the degree of symmetry of the material under consideration. According to Auld [12], pure modes can also be defined as those associated with directions of propagation that are coincident with the flow of energy directions. These pure modes are not necessarily the same as the ones described above. Interesting situations can then arise when both sets of "pure" modes overlap. We conjecture that the intersection defines "true" pure modes. More fundamental differences exist between isotropic and anisotropic media. The propagation velocities of individual wave types in anisotropic media are intrinsically dependent upon the direction of propagation. Specific dependence is influenced by the level of the anisotropy of the medium. Furthermore, the energy and slowness surfaces are no longer coincident with each other or with the corresponding wave front surfaces. Also the two wave front surfaces associated with the shear motions are no longer coincident. Consequently, fundamental questions arise concerning the manner in which wave fronts and their associated energy surfaces travel. Answers to these questions are presented in the remainder of this chapter.

3.2

The Christoffel equation

Combining the momentum equations (2.31) with the general constitutive relations (2.32) and the strain-displacement relations (2.33) lead to

02ui 1 0 OUk Oul p Ot2 - -~Cijkl~xj(-~x l § ~Xk).

(3.1)

Taking into consideration the various symmetries of cijkl, by interchanging the indices k and l, we can reduce equation (3.1) to

02ui

02ul

P - - ~ - -- Cijkl OXjOXk"

(3.2)

For bulk waves, solutions of equation (3.2) are sought in the complex plane wave form Ui -- Ui ev/-~(~njxj-wt)

(3.3)

where r and nj represent the bulk wavenumber (scalar) and the propagation direction (unit vector) with the components nl, n2 and n3; ~ is the circular

34

CHAPTER3.

BULK WAVES

frequency and Ui is the displacement amplitude vector which also defines polarization. Substituting from equation (3.3) into equation (3.2) leads to the eigenvalue relation (3.4)

w2Ui _~ )~ijkl~2nknjUl

where we used (3.5)

)~ijkl : Cijkl/P.

Defining the phase velocity v = w/~ and using the Kronecker delta property Ui - UtSiz recast equation (3.4) into the characteristic equation

(3.6)

()~ijklnknj -- v25il)Ul -- O.

It is now convenient to introduce the second order tensor All, given by All = Aijklnjnk

(3.7)

and allowing equation (3.6) to be written in the more compact form (Air - v:(fit)Uz = 0.

(3.8)

Both equations (3.6) and (3.8) make up the well-known Christoffel equation which defines a set of three homogeneous linear equations for the displacement amplitudes Ut. Since the phase velocity v is still unknown, each of these equations constitutes an eigenvalue problem with its eigenvalues identified as v 2. Associated with each eigenvalue is an eigenvector Ut which also defines a polarization direction. Because Aij is symmetric with real elements, the eigenvalues are real and the associated eigenvectors are orthogonal. 3.2.1

General features of the

Christoffel equation

The transformation representation Now let us consider the solution of equation (3.8) in greater detail. To this end we rewrite the eigenvalue equation (3.8) in the expanded matrix form

(Aiiv A12 A13

A12 A13 A22 - v 2 A23 A23 A33 - v 2

U2 U3

= 0

(3.9)

3.2. THE CHRISTOFFELEQUATION

35

where the various elements Aij are given from combinations of equations (3.7) and (3.5) as

pAll --- 611n21+ 666 n2 + C55n2

+2C16nln2 + 2C15nln3 + 2C56n2n3 pAl2 --

C16n21-} C26rt 2 -I- C45 n2

nt- (C12 -~- C66)nln2

+(C14 + C56)nln3 + (C46 + C25)n2n3 pAl3 = C15n21+ C46n22+ C35n~ + (C14 + C56)nln2 +(C13 + C55)nln3 + (C36 + C45)n2n3 =

+

+

+2C26nln2 + 2C46nln3 + 2C24n2n3 ph23 = C56n~ + C24n~ + C34n~ + ((746 + C25)nln2 +(C36 + C45)nln3 + (C23 + C44)n2n3 pA33 - C55n21+ C44n2 + C33n2 +2C45nln2 + 2C35nln3 + 2C34n2n3.

(3.10)

In equation (3.10), the definitions of the elastic properties are used in accordance with the notation adopted in section 2.3. In principle, once the material properties and the direction of propaga2 k = 1 2,3. tion are chosen, equation (3.9) can be solved for its three roots Vk, For each of these roots, the same equations can be solved for the corresponding amplitude ratios:

U2(k) r(2k ) = U~k)

A23(A11 - v~) - A13A12

r~k)__ U(k)

A23(All - v~) - A12A13 A12(A33 - v~) - A13A23

k)

A13(A22 - v~) - A23A12

(3.11)

(3.12)

These amplitude ratios define the polarization directions with respect to the coordinate system xi. Specifically, the polarizations are directed along the three vectors (1, r~k), r~k)), k - 1, 2, 3. The normalized components with /-

to their respective lengths V/1 + r(2k)2+-r(3k)2 define their direction cosines with the coordinate axes.

respect

CHAPTER 3. BULK WAVES

36

Crystollographical r e p r e s e n t a t i o n An alternative representation of the Christoeffel equation can be given directly in terms of the crystollographical (reference) coordinate system. This representation would be identical to the transformation representation for r - 0 ~ As shall be demonstrated later on, each of the two representations has certain advantages over the other. The crystollographical representation will be advantageous when dealing with propagation along a priori known axis of symmetry. But this will require the existence of orthotropic or higher-symmetry situations. For propagation along an off axis of symmetry, the transformation representation will lead to significant algebraic simplifications. The crystollographical representation of the Christoffel equation can be written in a form that parallels the transformation representation (3.9)(3.12), leading to =o -

(3.13)

v

with t 2 t 2 P A ~ l -- C l l n l + C66 n2 -~- C 5 5 n 3

PA~2 -

, +2C16nln2 -~- 2C~5 n l n 3 + 2C~6n2n3 ' t 2 t 2 t 2 t C 1 6 n I + C 2 6 n 2 -k- C'45n 3 + (C12 + C~6)nln2 +(C~4 + C~6)n,n3 + (Ci6 + C;5)n2n3 t

2

t

2

t

2

t

t

PA~3 -- C15n1 + C'46n2 + C35n 3 + (C14 + C~6)nln2 +(C~3 + C~5)nln3 + (C~6 + C45)n2n3 pA~2 = C~6nl , 2 + C~2n2 , 2 + C44n32

+2C~6nln2 + 2C46nln3 + 2C&n2n3 pA~3

---- C 5,6 n l2 -+- C 2,4 n 22 +

C~4 n2 + (C46 -I- C ; 5 ) n l n 2

+(C~6 + C45)nln3 + (C~3 + C44)n2n3 PA~3 -- C55 , n 2i -~- C~14 , n22 + (733 , n32

+2C45nln2 + 2C~5nln3 + 2C~4n2n3

(3.14)

and t(k) r2

U2 (k) ----

U~(k)

A ~ 3 ( A ] 1 _ v~2) _ A13A12tt __

AIa(A~2 -- v~2) -- A~3A~2

r~3(k) = U~(k) = A23(A~1 - V~k2) - A~2A~3 U~(a) A~2(A~a- v~2) - AIaA~a

(3.16)

3.2. THE CHRISTOFFELEQUATION

37

Once again, these amplitude ratios define the polarization directions with respect to the coordinate axes x ii. Specifically, the polarizations are directed along the three vectors (1, r~2(k),r~3(k)),k = 1,2,3. The normalized components with respect to their respective lengths V/1 + r~(k)2+ r~3(k)2 define their direction cosines with respect to the primed axes. The above methods enable us to find both the velocity and the associated displacement vector (polarization) for any given propagation direction. Pure mode criteria As discussed earlier, a situation in which one of the polarization vectors coincides with the propagation direction defines three pure modes. The way to establish this is to take the dot product of each of the three polarization vectors with the propagation direction vector. For unit-length (normalized) polarization vectors, each of the three dot products will be less or equal to unity. The largest value is found to correspond to the largest eigenvalue (i.e., to the fastest wave), and is customarily associated with the quasilongitudinal wave (~(1) in figure 3.1). The two remaining waves are then termed quasi-shear ones. Accordingly, three pure modes are obtained only for the propagation directions where one of the dot products is unity. These modes can thus exist if combinations of material properties and propagation directions happen to satisfy the explicit condition described above. Consequently, the above description does not preclude the existence of one or more pure modes even in triclinic materials. If these dot products are not unity, any polarization corresponding to a zero value dot product will then define a single type pure mode. Generally, the above discussion reinforces the argument that solutions to equations (3.9) or (3.13) do not necessarily satisfy the relations ff • g = 0 or i f - g = 0, which define pure longitudinal or pure shear waves, respectively. 3.2.2

Limitations

of analytic solutions

For the general anisotropic medium, no simple analytical solutions exist for either the eigenvalues or their associated polarization ratios. Only numerical methods are useful for obtaining the required solutions. With the recent advances in computational and graphical illustration methods, solutions can be obtained and demonstrated with relative ease. In solving the cubic equation, one obtains three roots; generally, each root is associated with a sheet of the wave front surface. However, in order to determine the set of roots that belong to a specific surface branch, one has to either plot the com-

38

C H A P T E R 3. B U L K W A V E S

plete results or augment the computations with an a priori selected sorting subroutine. By virtue of considering all possible propagation directions, the loci of the computed velocities define the three complete wave front surfaces. Because the phase velocities depend upon the propagation direction, these surfaces are not necessarily uniformly shaped such as spheres. The degrees of deviation from spherical shapes depend upon the specific anisotropy. Of the three surfaces, the outermost surface (belonging to the largest velocity) defines the quasi-longitudinal surface, while the other two define quasi-shear ones. It follows that the innermost surface belongs to the slower quasi-shear wave. The dual slowness surfaces can be constructed from the wave surfaces by merely plotting the inverses of the phase velocities against the propagation directions.

3.3 3.3.1

Material s y m m e t r y Analytical

solutions

In some special cases requiring the existence of material symmetry, equations (3.9) and (3.13) can be factored out resulting in significant computational simplifications and leading to simple analytical solutions. As shall be shown in the sequel below, factorization is associated with the presence of pure modes. Potentially, the cubic equations in v 2 (3.9) and (3.13) can factor into two or three terms. We shall refer to two term factorization as partial, and to three term factorization as total. By inspection of these equations we see that factorization is associated with the vanishing of at least two of the three off-diagonal entries A12, A13, and A23; partial or total factorization is achieved depending upon whether two or all of these elements vanish. Let's now consider the various polarizations associated with individual factorizations. Inspection of the various elements in equation (3.10) or (3.14) shows no possibility of partial or total factorization for triclinic materials. Moving to materials with monoclinic symmetry (see equations (2.24) and (2.35)), concentrating on the transformation representation and implementing their material restrictions on the elements of equation (3.10), reduce them to pAll -- Clln21 nt- C66n22 nt- C55 n2 -1- 2C16nln2 pAl2 - C16n21 -k- C26 n2 -t- C45n32 -1- (612 nt- C66)nln2 pal3 = (613 -~- 655)nln3 -+- (636 -+- 645)n2n3 pA22 = C66n21 + C22n 2 + C44n 2 + 2C26nln2

3.3. M A T E R I A L S Y M M E T R Y

39

pA23 = (C36 + C45)nln3 + (C23 + C44)n2n3 pA33 = C55n21 + C44n~ + C33n 2 + 2645nln2

(3.17)

where the various elements Cij are as given in equation (2.35). Once again, inspecting the various elements (3.17) for possible factorization, we find two possibilities. Both involve partial factorization and are associated with the two specific propagation directions fi = (0,0,1) and fi = (nl,n2,0). For these two special cases, the required vanishing of the determinant in equation (3.9) reduces, respectively, to case (i)

C5~ - pv 2 C45 C4~ C44 - pv 2 O

0

0 0 C33 - pv 2

=0

(3.18)

case (ii) All - v 2 A12 0 A12 A22 - v 2 0 0 0 A33 - v 2

(3.19)

=0

with the reduced elements pAll = C l l n 2 + C66n 2 + 2C16nln2 pAl2 = C16n~ + C26n~ + (612 + C66)nln2 pA22 = C66n21 + C22n 2 + 2C26nln2

(3.20)

pA33 = C55n 2 + C44n 2 + 2C45nln2.

Due to the similarity in the algebraic structure of equations (3.18) and (3.19), we start with the analysis of case (ii). Results pertaining to case (i) can then be obtained by merely exchanging appropriate parameters. The characteristic equation (3.19) now factors out as [(All-

V2 ) ( A 2 2 - V2 ) - A 2 2 ] [ A 3 3 - v

2]=0

(3.21)

and thus admits the three solutions

Vl =

(3.22)

2 - ~[(All 1 v2,3 + A22)+ ~/(A22 _ All )2 + 4A22].

(3.23)

40

CHAPTER3.

BULK

WAVES

The eigenvectors corresponding to these three velocities are given by (0, 0, 1) and (1, r~k), 0), k = 2, 3, respectively, with rk -

-

Vk2 _ All _

A12

A12 - v k2 -- A22

(3.24)

By substituting for v k2 from equation (3.23) we can rewrite r k in the alternative form rk = d-(-1)kv/d 2 + 1

(3.25)

where d=

A22 -- All 2A12

(3.26)

Replacing p a l l , pA22, pA33, pal2 in the relations (3.21)-(3.26) with C55, C44, C33 and C45, respectively, we obtain a complete description pertaining to case (i). In both cases, we find that one mode is directed along the x3-(or equivalently x~-) direction while the other two are polarized in the xl - x 2 plane, at angles 7k measured from the xl-axis whose tangents are r~ k). Note the interesting relation 73 = 7r/2 + 72 which implies that the two inplane polarizations are orthogonal. This can be realized from the two algebraic identities t a n 7 tan(~/2 - 7) = - 1 (d-

v / d 2 + 1)(d + ~/d 2 + 1 ) = - 1

(3.27)

(3.28)

for any 7 and any d. Although results for both cases are algebraically similar, their physical significance and interpretation are different. To facilitate discussions, we depict the possible polarizations of cases (i) and (ii) in figures 3.23 and 3.2b, respectively. Upon closer inspection of the results pertaining to case (i), we see the existence of a pure mode situation. A pure longitudinal mode is found to propagate along the x3-direction with the wave speed Vl = x / C 3 3 / P . The other two define a pair of coupled quasi-shear modes propagating with the wave speeds v2 and v3 respectively. For case (ii), on the other hand, a pure shear mode is polarized along the x3-direction while the remaining two define a coupled quasi-longitudinal and quasi-shear waves, polarized in the Xl - x 2 plane.

3.3. M A T E R I A L S Y M M E T R Y

41

x 3

n

X3

~u~R U)

(a)

x2

x 1

x2

u 2) case (i)

case (ii)

Figure 3.2: Polarizations associated with cases (i) and (ii). 3.3.2

Higher symmetry

Total factorization will be possible only for orthotropic and higher-symmetry materials, such as transversely isotropic, cubic and isotropic. This is in virtue of the fact that these classes of materials possess three orthogonal axes of symmetry which are often identified with the reference (crystollographical) coordinate axes. The axes of symmetry are also known as the principal axes, and the three planes normal to them are termed principal planes. Except for the case of isotropy, complete factorization can only be achieved for propagation along directions that coincide with the axes of symmetry. For vanishing C[6 , C~6 , C~6 and C~5 , the form of equation (3.17) appropriate for orthotropic materials will not change as long as r r 0 ~ or r ~ 90 ~ and will thus be applicable to orthotropic media. For propagation along an axis of symmetry, we use the crystollographical representation of the Christoffel equation. Examination of the entries of equation (3.14) reveals that complete factorization is possible only for propagation along any of the principal axes, namely for n' = (1, 0, 0), (0, 1, 0) or (0, 0, 1) directions. In all three cases, pure, uncoupled modes exist with polarizations along the three principal axes. The modes can be identified once the propagation direction has been specified. To help facilitate our discussion, we specialize equation (3.14) to orthotropic materials and get pA~l = C l!l n l12 + ,-,! ~66n212 + C ~I n ~ 2

pA~2 = (C~2 + C~6)nlln~2

CHAPTER3.

42

BULK WAVES

pA~3 = (C[3 + C~5)nln3', pA~2 ~, ,2 ~, ,2 ,~, ,2 ~66nl -i- ~22n2 -[- ~44n3

pA~

(C~ + C~4)~

pA~3 --" ~, ~55 nl,2 -]- ,~, ~44 n2,2 -~- ,~, ~33 n3,2 9

(3.29)

For the propagation direction n' - (0, 1,0), for example, a longitudinal mode propagates with wave speed (C~2/p) 89along the x~-axis, and the remaining two modes are of shear type, polarized along the x~-and x~1 1 directions and propagate with speeds (C~6/p)~and (C~4/p)~, respectively. For propagation along an arbitrary direction in any of the principal planes, only partial factorization is possible and solutions will follow the formal procedure used above for the monoc]inic materials presented in cases (i) and (ii). Except for the case of isotropic symmetry where pure modes exist for all propagation directions, the procedure and results obtained for orthotropic symmetry can be directly adapted and applied to the highersymmetry cases. This is done by merely imposing the appropriate restrictions on the properties as fully exploited in chapter 2. For this reason we shall now move to the interesting and well-studied case of cubic materials. 3.3.3

Cubic symmetry

In higher than orthotropic symmetry materials, other symmetry axes besides the principal ones can potentially exist. In particular, we note the case of cubic materials. To facilitate the discussion pertaining to this case, we further impose the cubic material restrictions on the elements of equation (3.29) and get pA~l -- C[1

+ C~6(n~ 2 -+-

:

I

I

__

!

!

)

ph'~ (C~: + C~)n,n'~ pA'~ = C~(nl ~ + n'~~) + C~n'~ ~ pA'~

pA~

= _..

(C'~: + C ~ )n2n ' '3

C~(nl ~ + n~~) + . i l n ~ . ~,~I

! 2

(3.30)

For the transformation representation of the Christoffel equation in the cubic material, we first impose the appropriate material restrictions on equation (2.34) and get Cll = C[1 - 2(C11 - C[ 2 - 2C~6)$2G 2 C22 ~ C11

3.3. M A T E R I A L S Y M M E T R Y

43

C12 = C~2 + 2(C~1 - C ~ 2 - 2C~6)$2G 2 C2a = C l a = C~2 c )sa C16 C26 --" - C 1 6 C33 = c h = C36 = C 4 5 = 0 = C44 = C66 = C~6 -F 2(C[1 - C [ 2 - 2C~6)$2G 2. =

-

-

2c

)(s

-

(3.31)

These specialized properties result in simplifications of the various elements

Aij of equation (3.10) in accordance with pAll - C l l n 2 + C66n2 + C55n 2 + 2C16nln2 pAl2 = C16(n 2 - n 2) + (C12 + C66)nIn2 pAl3 - (C12 + C55)nln3 pA22 = C66n 2 + C l l n 2 + C44n~ + 2C26nln2 pA23 - (C23 + C44)n2n3 pA33 = C55n2 + C44n 2 + C33n2.

(3.32)

Although these representations appear to be algebraically different, they are equivalent in that they represent alternative Christoffel equations for cubic materials. The utility of these two representations can be demonstrated through a comparison of the manner in which they lead to the desired solutions. There are several propagation directions in cubic materials which can be identified that lead to factorization and hence to pure modes. These include the principal axes xi,' the face face diagonals and the body face diagonals. We start with the simplest situation of propagation along one of the principal axes, say x~. For this choice n' = (1, 0, 0) and, in light of r = 0, also fi = (1, 0, 0). Under these conditions, both representations immediately lead to total factorization of their respective Christoffel equations and hence to three pure modes. The longitudinal mode travels with the speed J C ~ l / p along the x~-direction and the remaining two shear modes are polarized along the x~ and x~-directions with the equal speeds of ~/C~6/p. Face d i a g o n a l Next, we consider propagation along face face diagonals, namely along bisectors to the in-plane angles of the principal planes. As a representative

44

CHAPTER

3.

BULK

WAVES

example, we choose the x] - x ~ face diagonal for the propagation direction. For this case, however, we have to identify the components of the propagation direction for both representations. For the crystollographical representation, it is obvious that n-~ - (1 / v/2, 1 / V~, 0). For the transformation representation on the other hand, this direction is uniquely specified by a combination of r = 45 ~ and g - (1, 0, 0). Under these conditions, the vanishing of the corresponding Christoffel determinants are given, respectively as

C~I -~- 6~6 - 2pv 2 6~2 + 6~6 0 6~2 + 6~6 C~I + 6~6 - 2pv 2 0 0 0 C~6 - pv 2

= 0

(3.33)

and Cll

0

0 C66 - p v 2

0 0

0

0

C66 - p v 2

-

pv 2

= 0

(3.34)

which display partial and total factorization, respectively. Expanding the determinant in equation (3.33) yields [(C~I + C~6 - 2pv2) 2 - (C~2 + C~6)2][C~6 - p v 2] = 0.

(3.35)

Recognizing that the first factor in equation (3.35) is a difference between two perfect squares, we can rewrite it as the product of the two terms (C~1 + C~2 + 2C~6 - 2pv2)(C~1 - C~2 - 2pv 2) = 0.

(3.36)

This allows equation (3.35) to yield the three solutions

(ci, + Vl

--

v2 =

vii

v3 -- ~ C ~ 6

-

2p

2p

+

(3.37) (3.38) (3.39)

The polarizations associated with equation (3.33) can be obtained from consideration of equation (3.25). Since here A~2 = All , application of equation (3.26) to the case of crystollographical representation, i.e., for the primed quantities, shows the vanishing of d implying two polarizations along 4-45 ~ from the x]-direction. These two modes propagate with the respective

3.3.

MATERIAL

45

SYMMETRY

velocities (3.37) and (3.38). Since the direction at 45 ~ coincides with the propagation direction, we have a situation of pure modes. The remaininfi mode is of shear type and propagates with the velocity (3.39) normal to n ~. By permutation of the axes, we can see that all bisectors of the in-plane angles constitute axes of symmetry for cubic materials. We now consider the solutions of equation (3.34). Notice the relative simplicity of the expression (3.34) revealing total trivial factorization and yielding three pure modes propagating with the speeds vl = x/C'll/p, v2 = v/C66/p and v3 = x/C55/p. The first of these belongs to a longitudinal mode polarized along the xl-direction, i.e., along the x~ - x ~ face diagonal, while the third is of shear type and is polarized in the x3- (also x~-) direction. This leaves the second one as a shear mode which is polarized along the x2-direction, namely at 90 ~ from the face diagonal. Specializing these results for r = 45 ~ in equation (3.31) identically recovers the results of the crystollographical representation, equations (3.37)-(3.39). Body diagonal Next, we show that propagation along all directions described by the rela12 12 12 tion n 3 = n 2 -- n 1 = 1/3 in the crystollographical axes, or by the relations n 2 - 2/3, n 2 - 0, n 2 = 1/3 with r = 45 ~ in transformed axes, define axes of symmetry in cubic materials. These coincide with the cubic body face diagonals. There are four such directions. We start with the crystollographical representation (3.30). The choice of any body diagonal for the propagation direction yields the restrictions A~3 = A~2 = A~I (all designated with A~I ) and A~3 - A~3 = A~2 (all designated with A'). These restrictions factor the characteristic equation (3.13) as -

A' - v 2 ) 2

'

+ 2h'-

= 0

(3.40)

which admits, with the help of equation (3.30), the three possible velocities

Vl = ~/h~l -~- 2At= v/(C~I --b 26~2 + 4C~6)/3p =

A/-- v/(C~I

6~2 .-[-C~6)/3p.

(3.41)

(3.42)

Thus far, we can only expect that the first solution belongs to a pure longitudinal wave, whereas the second belongs to the two remaining pure shear ones. The two shear waves are degenerate since their velocities are equal. To demonstrate this, we must examine the corresponding polarizations. In this instance, solutions depend upon the specific diagonal chosen for the propagation direction. We can however, treat all cases simultaneously by

46

CHAPTER3.

BULK

WAVES

substituting the cubic material restrictions into the amplitude ratios (3.15) and (3.16) leading to i(k)

=

_

(Ai

-

-

A'

-- ( n ~ l _ vk2) _ A/ -- 1

(3.43)

except when Vk 2 = A~I - A' where the expression is ill-defined. Examination of the expression (3.43) shows that vl of equation (3.41) corresponds to a longitudinal mode polarized along the propagation direction and hence defines a pure mode. Alternately, the choice of the other velocities, namely Vk -- v2 or v3, leads to ill-defined ratios and thus implies that any two orthogonal directions normal to n' define the two shear polarization directions. We finally show that the same results can be obtained by using the transformation representation. To this end, specializing equation (3.32) to n 2 = 2/3, n2 - 0 and n] - 1/3 leads to the characteristic equation 2Cll + C55 - 3pv 2 0

0

x/r2(c13 -t- C55)

2C66 + C44 - 3 p v 2

0

~/r~(C13 -{- C55)

0

2C55 -J- C33 - 3 p v 2

= 0. (3.44)

If this is followed by specializing the various elements in equation (3.31) to r = 45 ~ after some algebraic manipulation, one can show that the resulting solutions are identical with the results (3.41) and (3.42) obtained using the crystollographical representation.

3.3.4

The isotropic case

Thus far, we have dealt with situations where we identified specific directions along which pure modes exist. The question now arises whether there exist planes which admit pure modes for all propagation directions. This will undoubtedly require higher-symmetry materials. Indeed, as shall be demonstrated below, all propagation directions in isotropic media support pure modes. Thus, it follows that planes of transverse isotropy have similar properties. In this section we illustrate application of the procedures described so far to the case of isotropic materials. Returning to relation (3.6), we arbitrarily choose the wave amplitude vector as unity, namely UiUi - 1. Multiplying both sides of equation (3.6) by Ui, we obtain the interesting relation V 2 -- )~ijklnknjU1Ui.

(3.45)

Although it appears from equation (3.45) that the velocity v is dependent upon the displacement polarization, whereas the relations in equation (3.8)

3.3.

MATERIAL

47

SYMMETRY

do not show such a dependence, not withstanding, both relations are consistent. Since in equation (3.45) we normalized the polarization vector to unity, this equation also includes implicit information about the eigenvectors. The velocity v in equation (3.45) will be independent of the polarizations only if certain restrictions are applied to such polarizations. These restrictions will essentially result in the characterization of the required polarizations as would be obtained by the direct solution of equation (3.6). The simple example of propagation in isotropic media illuminates in best fashion the implications of the above argument. To this end, we specialize cijkl to that of an isotropic material as Cijkl = s

(3.46)

+ #(hikhjZ + 5iZhjk).

Then equation (3.45) can be written as (3.47)

pv 2 = ()~ + # ) n k n j U k V j + #

since nini -- 1 and ULUz = 1. For v to be independent of U, we must have one of the following two situations: (i) If [7 is directed along g, then n j U j - 1 which reduces equation (3.47) to (3.48)

p v 2 -- )~ + 2#

defining a pure longitudinal wave. (ii) If U is directed normal to g, then n j U j - 0 leading to (3.49)

pv 2 - #

which defines a pure shear wave. The same conclusions may be drawn by direct application of the eigenvalue equation (3.9). Substituting from equation (3.46) directly into equation (3.9) yields the characteristic equation (n2+v nln2 nln3

nln2 nln3)(U1) n 2 + "7 n2n3 U2 n2n3 n2+ ~ U3

= 0

(3.50)

where, for convenience, we introduced # -- pv 2

= ~ . A+#

(3.51)

Equation (3.50) can also be obtained from equation (3.9) by specializing the elements (3.10) to isotropic materials in accordance with Cll = 6'22 = C33 - - / ~ -~- 2 p , C12 -~ C13 - - C23 -- ~, C44 -- C55 - C66 - ]_t and such that all

C H A P T E R 3. BULK WAVES

48

remaining Cij properties vanish. For nontrivial solutions, the determinant of equation (3.50) must vanish. By expanding the determinant, using the condition n 2 + n 2 + n32 = 1

(3.52)

and following simple algebraic manipulations, we get ~/2(i + ~/) = 0

(3.53)

which, once again, yields the results (3.48) and (3.49). Notice that the relation (3.53) is independent of the propagation direction ~ signifying that only pure waves can propagate in isotropic media. This means that any choice of the propagation direction in equation (3.50) suffices to arrive at equation (3.53). An obvious example is the choice = (I,0,0). For this situation, equation (3.50) trivially recovers equation (3.53). For this case, the solution "7 = 0, if utilized in equation (3.50) leads to UI - 0 leaving U2 and U3 arbitrary, defining a shear wave. For "7 = - i , on the other hand, U2 and U3 vanish while UI is arbitrary and hence defining a longitudinal wave. Returning to the general propagation case described in equation (3.50), we see that for "7 - 0, consistent solutions of this equation require the relation

nl U1 + n2U2 + n3U3 - 0

(3.54)

which can only hold if U is normal to g describing a shear wave polarization. For "7 - - 1 , on the contrary, application of equation (3.50) leads to the relation U1 nl

.

.

.

U2 n2

.

U3 n3

(3.55)

which can only hold if U is parallel to ~ and thus describing a longitudinal polarization.

3.4

Computer aided analysis

For the symmetry classes considered so far, it was easy to analytically solve the Christoffel characteristic equation only for the simplest, usually easily identified symmetry cases. For most of these cases, solutions are obtained by inspection of the equation assisted by elementary algebraic manipulations. For the general case, however, numerical computation is required to calculate wave surfaces. In this section, we describe computer aided computational techniques, capable of treating the general anisotropic case, and also

3.4.

COMPUTER

49

AIDED ANALYSIS

p

X3

X3

ig,

X'2

3

p

X1

1(1

2

(a)

(b)

Figure 3.3: Strategy of the choices of propagation directions" (a) g~ in the reference coordinate system and (b) g in the transformed system.

of providing insight into the acoustic properties of symmetry classes such as those described above. Generally, good computational strategies are found to be of utility in the solution process. As a consequence of dealing with the representation of three-dimensional wave surfaces, it is of fundamental importance to adopt a systematic strategy for taking into account all propagation directions and the subsequent solutions of phase velocities and associated polarizations. Two methods are found to be particularly attractive. The first uses the reference coordinate system and is based upon the construction of wave curves for propagations in planes that are perpendicular to one pre-chosen axis, say x~ as shown in figure 3.3a. By stepping the component n~ of the propagation unit vector direction from- 1 to + 1, the whole three-dimensional surface can be constructed. The other technique is based upon the versatile properties of the linear orthogonal transformation of chapter 2. In this instance, solutions are first obtained for propagation in a plane that makes an azimuthal angle r with the x~-direction. This situation is illustrated schematically in figure 3.3b. Without any loss in generality, we can identify this plane with the Xl - x 3 plane obtained from a linear transformation of the original axes x i to the global one xi through a counterclockwise rotation of an angle r about the x~ axis; thus x2 is normal to this plane. With this choice of the coordinate system, being vertical, the propagation plane is often called the sagittal plane. Polarization normal to this plane defines

50

CHAPTER3. BULKWAVES

the horizontally polarized (SH) wave. Thus, the polarization in the sagittal plane constitutes a coupling between the quasi-longitudinal and the vertically polarized quasi-shear waves. By incrementally varying r from 0 ~ to 360 ~ we span the whole space. We now show how the calculations can be conducted in the transformed system xi with relative ease. To this end, let us examine the form of the formal solution (3.3) for the special case where n2 is zero. This means that we are solving the Christoffel equation in the Xl - x3 plane. But such a choice implies that all displacements, velocities and stresses are independent of the x2-direction. Consequently we can conduct the computations in two, rather than three dimensions. Since all field variables as well as the stiffness coefficients are tensors, by consistently applying the tensor transformation, we can conduct the analysis in the transformed coordinates as well. It turns out that the computations can thus be carried out independent of the x2-axis. Solutions can be obtained by replacing the properties by their respective transformed ones as described in section 3.2.1. As was indicated in chapter 2, we shall limit our computational illustrations to monoclinic and highersymmetry materials. We recall that the plane of symmetry of the monoclinic material is always chosen in this book to coincide with the x: - x 2 plane. Numerical identification of the pure modes can also be achieved in a rather simple and straightforward manner. First, we recall from section 3.2.1 that all wave modes are pure for situations where the longitudinal wave is polarized along the propagation direction g. Second, the dot products of the three normalized polarization vectors define the cosines of the angles between these individual vectors and the propagation direction. By taking the inverse cosines of the dot products, we obtain the skewing (deviation) angles of the polarization directions from the propagation direction. Guided by the fact that the pure shear modes propagate normal to the propagation directions, it is more appropriate to take the inverse sines of the dot products associated with the shear components and thus find the corresponding skewing angles from their pure directions. Numerical illustrations are now presented for selected materials chosen from the material menu listed in Appendix A. The choices of propagation planes are intended to exhibit the general features of slowness curves and to graphically isolate and illustrate the properties of the simplified cases discussed above. Figures 3.4a, b are samples of polar slowness and polarization skewing angle curves that belong to the cubic material InAs computed for r - 0 ~ In these, and in the remaining figures of this section, solid lines are reserved for the quasi-longitudinal wave component and the broken ones are reserved for

3.4. COMPUTER AIDED ANALYSIS .

..- .....

n/v ] .......

51 n~

.

- .." ;____,. : ,;....::.:. . . .

.-/

~,....,:, ,,,.,

".~'. / ,'"

,

. ~

(a)

,, ', ~.-

:,, .". .'~-.~- - -.,. . . ~. . .-. - ' ~, - ~ , ~ - ~ - -' -,. - ... .. .. . ~- .". ."

.~o

,..,,

,

,,

.,

o

,, ,,..:-

/

:

: ",%

"

oo.,, ,, ,, ::

; :

nJv

o

(b)

Figure 3.4: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation in the r - 0 ~ plane in InAs cubic material. Scale is (a) 0.15 sec/km-per-division and (b) 5~

the quasi-shear components. We start with the result depicted in figure 3.4a. The situation in this figure is similar to the case where propagation takes place in the plane of s y m m e t r y x~ - x~ where ~' - (n~, 0, n~). The presence of the perfect circle signifies a case of partial factorization that decouples the horizontally polarized shear wave. This wave component is polarized in the x~-direction and travels with the speed (v/C~6/p). The other two modes are coupled and are polarized in the sagittal plane. The specific polarizations of these two waves are governed by the relation (3.15) or (3.16). This figure shows two unique propagation directions along which the two shear waves are degenerate, namely, coalesce into a single one. Further inspection of this figure reveals three pure propagation directions. Two are along the principal axes x~ and x~ and the third is along the face diagonal. Specific results for these propagation directions confirm the analytical results discussed above. Using visualization of figure 3.4b, we first notice two rather than three curves exist. The third, which is associated with the perfect circle slowness curve in figure 3.4a, corresponds to a zero value dot product implying the existence of a pure mode for all propagation directions in the x l - x3 plane. As far as the other remaining two modes are concerned, we can identify four directions for possible propagation of pure modes in this system. We also notice the relatively smaller skewing angles associated with the quasi-longitudinal wave component. In figures 3.5a,b, the results depicted

CHAPTER3.

52

n]v

BULK WAVES

n~,

.g--'"~,i~~il---~ .... :~.

,' , ".,-

,

,

!

i

,,. 9 ~ .

~

,

-;; ',,

% ~,"... ',

.i

.~*~

,

g,

~'..~...~"ilk~ .....;."',

,,

', .

nJv

',

/

, ,i

"'.. % "'---]_ __',. . . . - : .."

(8.)

(b)

Figure 3.5: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation along r = 45 ~ in InAs cubic material. Scale is (a) 0.10 seclkm-per-division and (b) 5~ Horizontal axis corresponds to the face diagonal of the (x~ - x~) plane. The added solid line corresponds to the body diagonal n~ = n~ - n ~ = 0.577. in figure 3.4a,b are repeated for the azimuthal angles r = 45 ~ and 30 ~ respectively. Notice in both the appearance of the third curve in the skewing angle figures. The situation in figure 3.5ia,b is somewhat more involved in comparison with that of figure 3.4a,b. Here, the S H wave also uncouples but no longer defines a perfect circle. In fact its slowness curve defines an ellipse with major and minor a x e s (x/p/C66) and (X/p/C44), respectively. This figure also displays the results obtained for the propagation along all of the axes of symmetry discussed earlier for cubic materials. The vertical axis is still a principal axis of the unrotated (original) cube, the horizontal axis corresponds to a face diagonal direction and the direction shown in the added solid line arrow defines a body diagonal of the cube. Here there are three directions of degeneracy of the shear waves. Like in the case of propagation in the principal plane, four directions are identified as pure mode directions in the present system. Finally, figures 3.6a,b are examples of what might happen when propagation takes place in an arbitrary sagittal plane in a cubic material. Here no possible factorization exists and the three waves are completely coupled resulting in more complicated slowness curve textures. Since all propagation planes contain the original unrotated x3-axis, the figures, like all possible

3.4. COMPUTER AIDED ANALYSIS

53

n~v

n~v :

.-•

--,,...

"," ,"~-"9176 " ; -; 9

'

,,. . . . . . . ,

nJv

,

~ %-" %*.**

_%

,~~

(a)

"4

",

*'~

-

-..

"

9 -.. 9 . o . O - - . . , , ~o 9176 ', ~

o.4~

,.,.... ',,. 9. . . . ,

*,

-

~

',

', ,

',

,

,

u ;-.~...v ,,

,

,

,'

(b)

Figure 3.6: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation along r = 30 ~ in InAs cubic material. Scale is (a) 0.15 sec/km-per-division and (b) 5~

n~

n~

/:i:.::":::-'"'",,":::-ii:".. ~k

k

',,

,'

(a)

.

' iJ

n,/v

i

i

I

:**

9

',

',

nJv

(b)

Figure 3.7: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation along r -- 0 ~ in graphite-epoxy. Scale is (a) 0.15 sec/km-per-division and (b) 10~

CHAPTER3.

54

BULK WAVES

others, have a principal propagation direction, namely the x3-direction. Notice that only two directions of shear degeneracy exist for this case. For the 30 ~ sagittal plane orientation, only one direction along the x3-direction is possible for three pure modes to exist. Skewing angles ranging from 0 ~ to 25 ~ are possible for this situation with the smallest deviations belonging to the quasi-longitudinal mode. Some of the numerical results presented in figures 3.4a,b-3.6a,b are repeated for the unidirectional graphite-epoxy material and shown in figures 3.7a,b and 3.8a,b. Here uncoupling of the S H wave occurs for propagation in the principal plane r = 0 with its slowness curve being a perfect circle. The fact that only two curves are present in the skewing angle figure 3.7b once again suggests that a pure shear mode exists for all directions in this plane. This, of course, corresponds to the perfect circle associated with the horizontally polarized S H shear wave component. Notice also that only propagations along the principal directions Xl and x3 define pure modes. For any other sagittal plane, like r = 30 ~ of figure 3.8a,b, the three waves are completely coupled and only the principal direction x3 defines three pure modes. Skewing angles of up to 40 ~ for quasi-longitudinal modes and up to 65 ~ for quasi-shear modes are possible for these two cases. The significance and utility of the slowness surfaces can best be demonstrated through their dual relationship with the energy propagation surfaces (also known by the ray surfaces). Derivation of such relationships is the subject of the following section.

3.5

Group velocity

Having stated the phase velocity v - w/~, we now state the group velocity vector gj as

gJ-

Ow Ov = ~cgnj Onj

(3.56)

since ~ is a constant wavenumber. From equation (3.4) with UiUi = 1, we then deduce that

2~vdw = ~2~ijklUiUi(nkdn j + njdnk).

(3.57)

It can be easily shown, by interchanging j with k, i with 1 and using the symmetry relations

"klkji -'- ~lkij -- ,~ijkl

(3.58)

3.5.

55

GROUP VELOCITY

n~/v

n3/v ,-v-

u ........ -.,', "% ",,

/v

n1

",,,,'.... ..... :k .... Z ...... "

(a)

(b)

Figure 3.8: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation along r = 30 ~ in graphite-epoxy. Scale is (a) 0.15 sec/km-per-division and (b) 10~ that the second term is identical with the first and hence we obtain gj = ()~ijklnkUzUi) /v.

(3.59)

We can also recognize by multiplying both sides of equation (3.59) with nj and comparing with equation (3.45) that gjnj -- v.

(3.60)

From equation (3.60) we have gj dnj + nj dgj = dv

(3.61)

and through comparison with equation (3.56) we conclude that njdgj = O.

(3.62)

This remarkably simple result shows that the propagation direction fi is always normal to the propagating energy surface. Now we prove that the group velocity ~ is normal to the slowness surface. Since the slowness vector Sj is defined by Sj = n j ?2

(3.63)

56

CHAPTER

3.

BULK WAVES

we have v d S j + S j d v = dnj.

(3.64)

Multiplying both sides by gj, noting from equation (3.60) that dv - g j d n j , and from combinations of the equations (3.60) and (3.63) that g j S j - 1 leaves gjdSj = 0

(3.65)

which dictates that t7 be normal to the surface S. We finally show, in the form of an example, that the phase and group velocities coincide for isotropic materials. To this end, we start with equation (3.45) and rewrite it in terms of Cijkl by using the relation (3.5). Subsequently, we specialize it to the constitutive relation (3.46), multiply both sides by gj and use equation (3.60) to arrive at

pvgj -- [~(~ij(~kl + #((~ik(~jl + (~ilbjk)]nkU1Ui.

(3.66)

This equation can be contracted to pvgj = ()~ + # ) U j U k n k + # n j .

(3.67)

Once again, dependence upon the displacement amplitudes disappears in the following two circumstances. In the first, if the polarization is assumed along the normal, namely if Uk = nk and hence Uknk -- 1, then pvgj = (~ + 2~)nj.

Since for this case we showed earlier that pv 2 = (s reduces to gj = v n j

(3.68) then equation (3.68) (3.69)

which implies that ~ is directed along the propagation direction g. Furthermore, by multiplying both sides of equation (3.69) by gj and using the relation (3.60), namely g j n j -- v, we conclude that the energy speed [iT[ equals v. Thus, ~ is identical with vg. E n e r g y (Ray) surface The above described dual relationships between the slowness and the energy surfaces can be used to calculate the energy surfaces. To this end we rewrite equation (3.60) in the alternate manner ItTIcos r = v.

(3.70)

3.5.

57

GROUP V E L O C I T Y

4O

90

.

.

80

i

:

A

ID

3 0

"

-

=

70

-

~" eo

.o 2 0 o}

10

.

.

.

.

.

9 9: . . . . . .

/

~ ............

.:

,~ so : ............;,.......

,;,:: . . . .

:

.0i.!ii!.i ii.i.ii-,.-!!;!

_o

....... ..~

t.

/

c:: -10

~4o >,

.."

3o g . . . . . . . . . .

.............

~

i

:\

...... : . y ' : "

..... .......

20

.,

-20

0

10

20

30

incident angle (deg)

40

(a)

~

5

10

15

20

25

30

incident angle (deg)

35

40

(b)

Figure 3.9: Energy flow angle vs. incident angles for: (a) InAs cubic (r = 0~ (b)graphite-epoxy (r = 0~ Here ~ is the angle between t7 and g defined as the power flow angle. It is obvious that this angle depends upon the shape of the slowness curve and hence on the propagation direction g. From the duality relation (3.62) and (3.65), we conclude that cosr is equal to the dot product between and the normal to the slowness curve at that location. We can then find the power flow angle from the slowness curve by first finding the normal to it and then dot product this normal with ~ to find cos r By considering all propagation directions we can calculate the corresponding energy flow angles. Depending upon the shapes of the slowness curves, this angle can vary from one propagation to the other and sometimes can be quite large. The two sample calculations shown in figures 3.9a,b corresponding to the slowness curves of figures 3.4a and 3.7a clearly show a wide range of energy deviations from the propagation directions. Having done so, we can then proceed to calculate the energy curves by the following two steps: first we find ]gl from equation (3.60). In the second step we assign this value to the normal to the slowness curve. By doing so for all possible propagation directions we construct the energy flow surfaces. The corresponding energy curves to those of the slowness curves of figures 3.4a and 3.7a are plotted in figures 3.10a,b, respectively. The cusps appearing in these figures correspond to regions of null energy and will be elaborated upon in chapter 13 when dealing with the response of solids to transient sources.

CHAPTER 3. BULK WAVES

58

E3

E3

/,'//

====================== .... .

'~'~I ,l

~', , ,

,"/ I::I (a)

: I ,,

,, 1:1

(b)

Figure 3.10: Energy curves for: (a) InAs cubic (r - 0~ (b) graphite-epoxy (r = 0~ Background circles are in increments of 105 km/sec in (a) and 105 km/sec in (b).

3.6

Energy

flux

Any elastic deformation, and in particular that caused by the propagation of waves, is associated with the transfer of energy. At any instant in time, the total energy contained in the wave front as it propagates in a lossless media is given by the sum of kinetic and strain energies E -- -~

(p~igzi -+ a i j e i j ) d V .

(3.71)

Here, superposed dots denote differentiation with time t and V is the volume of the disturbed region. Let us concentrate a little more on the product aijeij. From the strain-displacement relations (2.33) we write 1 aij eij = -~ aij (ui,j -}- uj,i).

(3.72)

Since aij is symmetric, this reduces to aijeij -- aijui,j.

(3.73)

Furthermore, using the stress-strain relations (2.32) we get O'ijei j

-

-

1 -~Cijkl(~k, l nt- ~tl,k )Ui, j.

(3.74)

3.6. E N E R G Y FLUX

59

Since again, Cijki is symmetric with respect to k and l, it further reduces to

aijeij -- cijklUk,lUi,j

(3.75)

and hence we get

E -- -~

(puiui + CijklUk,lUi,j)dV.

(3.76)

Differentiating equation (3.75) with respect to time t gives

Oaijeij Ot -- Cijkl(Ui'ji~k'l + Uk,lUi,j).

(3.77)

Since Cijkl-~Cklij, this can be written as

Oaijeij Ot -- 2CijklUk,li~i,j = 2aij~ti,j.

(3.78)

By differentiating equation (3.76) with respect to t, we get ot =

(pit~ii~ + ~jit~,j)dV

(3.79)

Ot --

[pi~iui § (aiji~i),j -i~iaij,j]dV

(3.80)

or=

[(piii - aij,j)i~i + (aiji~i),jldV.

(3.81)

The term in the first bracket is the equation of equilibrium and hence vanishes reducing this equation to

OE Ot -- Iv (aiji~i),jdV.

(3.82)

Using Gaussian theorem, we replace the volume integral by the surface integral and get

OE Ot = IS aiji~injdS

(3.83)

which represents the energy crossing a wave surface in unit time. On introducing the energy flux (Poynting) vector Pj, we set

OE = - Is PjnjdS Ot

(3.84)

where

Pj = -aiji~i.

(3.85)

Thus the energy variation within volume V per unit time is represented as the flux of the Poynting vector over the boundary surface.

This Page Intentionally Left Blank

Chapter 4

GENERALIZED SNELL'S LAW AND INTERFACES If a propagating plane wave in a medium is interrupted by a boundary, certain conditions must be satisfied at that boundary. Since the incident wave cannot satisfy these conditions alone, a certain number of "scattered" waves will be generated. These include a number of reflected and transmitted waves propagating in the originating and the continuing media, respectively. The transmitted waves are also known as refracted waves. The number and nature of the scattered waves depend upon several important factors including the type of the incident wave, the properties of the two media sharing the boundary and the physical characteristics of the boundary (interface). In general, for two solid media in contact along a flat (planar) boundary, a maximum of six wave components are needed depending upon the specific situation encountered. The excitation (or lack of it) of the individual wave component also depends upon the incident wave characteristics and the various combinations of the properties of the sharing media. This idea is closely related to the concept of the critical angle phenomena. Beyond its critical angle, a wave component seizes to propagate and becomes evanescent (nonpropagating). Thus, the wave components can be either propagating or evanescent. All wave components have to work in concert in order to satisfy the interface conditions. The manner in which these waves interact constitutes the foundation of Snell's law. Due to the inherent complications in the behavior of the waves in anisotropic media, we choose to refer to it as the generalized Snell's law. In section 4.1, we discuss appropriate interface conditions for two media in contact along a flat interface. In section 4.2, we proceed to characterize 61

62

C H A P T E R 4.

GENERALIZED

S N E L L 'S L A W A N D I N T E R F A C E S

incident waves and introduce the generalized Snell's law criterion. Subsequently, in section 4.3, we discuss critical angles associated with the various wave components. Finally, in sections 4.4 and 4.5, we specialize the above concepts and results to two fluids and to two isotropic media, respectively.

4.1

B o u n d a r y conditions

The physical properties of the interface play important roles in the determination of the scattered field. Rigid contacts and smooth interfaces are widely encountered interface conditions. Intermittently connected and rough interfaces are also of special importance in laminated and fibrous composite media, as well as in geophysical systems. Rough interfaces are not unique and there have been many recent attempts to model and analyze them. Due to the diversity of rough interface conditions, only approximate models are now available. For these reasons and others, such interfaces will not be pursued in this book. However, various interfaces require different conditions of continuity or discontinuity on the field variables (displacements and stresses) occurring within both media. As an example, consider the situation of two different anisotropic solid media in contact along a flat interfaces as illustrated in figure 4.1. Also included is the unit normal vector z7 with respect to the coordinate system X i -- (Xl,X2,X3). Here we limit our discussion to the case where each medium consists of a monoclinic material with its plane of symmetry parallel to the interface. Without loss in generality, we choose this plane to coincide with the Xl - x 2 plane and thus z7 is directed along the x3-direction. As the figure shows, the originating and continuing media occupy the spaces x3 _> 0 and x3 _< 0, respectively. Subsequently, we identify the continuing medium with an overbar on all of its properties and field variables. 4.1.1

T y p e s o f i n t e r f a c e conditions

As mentioned above, the continuity conditions at the interface depend upon the physical nature of the interface. For rigid bonding, continuity is clearly required across the interface for the displacement components ui as well as for the stress vector Ti -- aijvj. For the present flat interface geometric arrangement, g - (0, 0, 1) and the conditions reduce to the continuity of the individual displacements and the stresses, namely ui -- ui,

o'i3 = ~i3,

i = 1, 2, 3.

(4.1)

4.1. BOUNDARY CONDITIONS

63

continuing medium

originating medium

V

Xa

Figure 4.1" Schematic of coordinate system showing originating and continuing media and the unit vector ~ normal to the interface. For smooth interfaces, the normal displacement and stress components are continuous U3 - - ~ 3 ,

0"33 --" 9 3 3

(4.2)

and the shear stress components vanish for both media, namely (713 :

913 - - ( 7 2 3 :

923 --0.

(4.3)

We note for smooth interfaces that no continuity is required on the tangential components Ul and u2 of the displacement. For the case in which the continuing medium is a vacuum, only reflected waves are generated. Thus, the appropriate boundary conditions require the vanishing of all three stress components

ai3=O,

i=1,2,3.

(4.4)

In the case where the continuing medium is a nonviscous fluid (such as water), the relevant interface conditions are similar to those applicable for smooth interfaces. Since the fluid does not support shear wave motions, the vanishing of its shear stresses as required by equation (4.3) is automatically satisfied.

64

CHAPTER 4. GENERALIZED SNELL'S L A W AND INTERFACES

n]v

:i ////

Figure 4.2: Typical slowness surface (for unidirectional graphite-epoxy with r = 0 ~ showing the three values of the slownesses for a given propagation direction. Scale is 0.15 sec/km-per-division.

4.2

Characterization

of incident

waves

As was described in chapter 3, three bulk waves can propagate in anisotropic media. Such waves have orthogonal polarizations, none of which are necessarily directed along or normal to the propagation direction 4. Each is associated with a slowness surface whose radius is dependent upon material properties and thus varies with the propagation direction. The displacement solutions listed in equation (3.3) are chosen for the case where the three wave components propagate in the same direction g as illustrated graphically in figure 4.2. These, of course, are the result of solving the Christoffel equation for the particular propagation direction g. We remind the reader that the slowness radii Sk as well as the polarization ratios r~k) of each are functions of g. Generally, each of the three wave components can be individually excited and propagates in a manner dictated by its slowness curve. Consistent with the solutions to the Christoffel equation, each wave component can propagate independent of the other two and hence the three constitute a roster (list of candidates) for the incident wave. As was previously discussed, the incident wave is supposed to be interrupted by another medium such as a vacuum, a fluid or a solid. A typi-

4.2. C H A R A C T E R I Z A T I O N OF INCIDENT WAVES

0.6

F //

-0'.6

;

:6

incident wave

65

transmitted waves

016 nl/V

'

-O.6 _

es

Figure 4.3: Half-slownesses of graphite-epoxy and InAs cubic material sharing the surface x3 - 0 and showing all scattered and an incident wave components. cal representative slowness figure for such combinations is shown in figure 4.3. This figure belongs to a unidirectional graphite-epoxy composite and a semi-space InAs cubic material combination. For the sake of demonstration, we arbitrarily choose the graphite-epoxy as the originating medium. Next, we must specify whether the incident wave is of quasi-longitudinal or of quasi-shear type. For the sake of clarity in the following discussion, we arbitrarily choose the slow quasi-shear wave u~3) as the incident one. We also arbitrarily reserve the third quadrant for the incident region. Referring to figure 4.3, if such a wave is assumed incident at an angle 8 (1) from the normal to the interface, then complete specification requires determination of its wavenumber and associated polarization. These two quantities are obtainable from solving the Christoffel equation subject to nl - sin 0(I),

n3 = cos 0(I).

We further note that S~I) is wave at ~(I). The incident as well as figure. A direct consequence quirement that, irrespective

(4.5)

the radius of the slowness curve of the incident all possible scattered waves are shown in the of satisfying the boundary conditions is the reof the material combinations across the bound-

C H A P T E R 4. G E N E R A L I Z E D S N E L L 'S L A W A N D I N T E R F A C E S

66

ary and regardless of the nature of the incident wave, all scattered waves have a unique intimate relationship with the incident wave in that they all have the same projected wavenumber along the boundary. This constraint is of great fundamental importance in finding the propagation directions of all scattered waves and is a restatement of the generalized Snell's law. The utility of Snell's law concept will be described below. To emphasize the fact that all of the scattered waves as well as the incident one have the same projected wavenumber along the x 1-direction as required by boundary conditions, we have also included in the figure dual mirror image vertical guide lines. These lines are equi-distant from the origin as required by Snell's law. The condition that must hold at the interface is then ~t) sin 0 (r) = ~(s)sin 0 (s) = ~

(4.6)

where ~ is the common projection wavenumber component along the interface and (s) labels the scattered components. Recalling that the incident and scattered wavenumbers are related to their slowness radii by

~(I) = wS(I),

~(s) = wS(S),

(4.7)

equality of the projected wavenumbers along the interface imply

S~ I) sin 0 (I) = S (s) sin 0 (s) = S,

s - 1, 2, ..., 6

(4.8)

which defines the generalized Snell's law condition for all incident and scattered waves. As it will be shown rigorously in the next chapters, (s) defines the six scattered waves (three reflected and three transmitted) and S = ~/w is the projected value of S~ I) on the horizontal axis. Alternatively, in terms of the velocities, equation (4.8) can be written as sin 0 (I) sin 0 (s) 1 = = v(I) v(s) c

(4.9)

where c - 1 / S is the common phase velocity along the interface of the incident and associated scattered waves.

4.3

Critical angles

Since the angle of incidence is measured from the vertical direction, for a zero angle of incidence, the two guide lines coincide with the vertical axis. By increasing the angle of incidence, the guide lines move apart but remain equi-distant. Depending upon the value of the angle of incidence, a scattered wave can either propagate or become evanescent. A scattered

4.3.

CRITICAL ANGLES

67

wave will propagate, for a given angle of incidence, if the guide line intersects its own slowness curve. For higher incidence angles, the guide lines move further and ultimately pass positions of tangency with respect to the various slowness curves. Angles of incidence corresponding to these locations are called critical angles. We choose the setup of figure 4.3 to illustrate this procedure. We recall that the incident wave is of the slow quasi-shear type. The first critical angle encountered is that of the quasi-longitudinal wave of the originating medium. Beyond this angle, this wave component becomes evanescent and decays exponentially from the boundary. As the angle of incidence further increases, a second critical angle is encountered. In this case, it belongs to the quasi-longitudinal wave of the continuing medium. Once again beyond this angle this wave component becomes evanescent. The next encountered critical angle belongs to the fast quasi-shear wave of the continuing medium. This sets up a pattern of identification of the various critical angles. The next two critical angles belong to the slow and fast quasi-shear waves in the continuing and originating media, respectively. Beyond this angle only one wave component, namely the reflected slow quasi-shear wave survives to propagate. Beyond all critical angles of the continuing medium, no energy will be transmitted and the incident wave suffers total internal reflection. Referring once again to figure 4.3, we see that the situation will be entirely different if the incident wave is of quasi-longitudinal type. For such case, no critical angles exist. For this situation, three transmitted and three reflected waves propagate besides the incident one. On the other hand, if the two media are interchanged and a quasi-longitudinal wave is incident, then one critical angle exists which is associated with the quasi-longitudinal component in the "new" continuing medium. Furthermore, such a situation will not admit total internal reflection no matter which component is incident, namely, there will always be some transmitted components. In short, the above discussion suggests that reflection, transmission and the associated critical angles are highly depend on the material combination, and incident wave characteristics. Although the above discussion was aided by the specific example of figure 4.3, the procedure can be applied to any situation involving material combinations and types of incident waves. Referring to figure 4.4, we notice the interesting situation in which the scattering angles will be the same if any one of the three possible incident waves that lie on the same guide line is chosen. This situation suggests the existence of a duality in the incidentscattered fields if we interchange the two media. The following examples of two fluid media and two solid isotropic media situations are chosen to

68

C H A P T E R 4.

GENERALIZED SNELL'S LAW AND INTERFACES

n#v 0.6rtransmitted waves

016 n~/v

I

-0.6

i

incident wave

- 0 . 6 L_

Figure 4.4: Same as figure 4.3 but demonstrating three possible incident waves each leading to the same scattering angles. illustrate some of the above features.

4.4

T w o fluid m e d i a

As a simple illustration, we consider the case of two different fluids whose half slowness curves are shown in figure 4.5. This is the simplest combination in which both media are isotropic and can support only pure longitudinal waves with constant phase velocities. With reference to this figure, Snell's law requires S (I) sin 0 (I) = S (R) sin 0 (R) = ~(T) sin 0 (T)

(4.10)

where we recall that the overbar denotes the continuing medium, i.e., the transmitting medium. Since both slowness portions consist of semi-circles, say of radii S1 and $1, respectively, then S (R) = S (I) = $1 and we conclude that 0(R) = 0(I) and that sin0 (T) = $1 sin0(I)/S1. It follows that 0(T) could be larger or small than 0(r) depending upon whether $1/$1 is larger or smaller than unity. In further reference to this figure we see that 0(T) increases with increasing 0(1). Of particular interest is the limit in which 0(T) reaches 7~/2, i.e., the transmitted wave becomes parallel to the interface.

4.5. TWO ISOTROPIC MEDIA

69

n~v i

i

i

i

i i

04

i

i

~

transmitted

o14 n J v

nt

Si,) ----- -o.4

Figure 4.5: Scattering field for two fluid semi-spaces sharing an interface. This can happen at the critical incidence angle 0c = sin -1 re.S1 S

(4.11)

Beyond this angle, the transmitted wave becomes evanescent, and the incident wave experiences total internal reflection.

4.5

T w o isotropic m e d i a

The concept of Snell's law and the associated critical angles introduced for the fluids are also applicable for cases of isotropic solids. The glass-steel slowness curve of figure 4.6 is chosen for illustration. Here however, one should take into consideration whether the incident wave is of longitudinal or of shear type. For longitudinal and shear wave type incidence, Snell's law requires, respectively,

sin 0~I) sin O~I)

= =

sin 0~R) -- ~$2 sin

S1 sin sin 0~R)_ - ~-1

0~T)_- ~$2 sin 0~T)

(4.12)

O~R) --- ~22 S1 sin O~R)_ $1 o~r)_- $22 $2 sin o~T). -- $22 sin

(4.13)

Since $1,$2, S1, and $2 are constants and since $2 > $1 and $2 > S1, we can easily examine these conditions for the existence of critical angles for

70

CHAPTER 4. GENERALIZED SNELL 'S L A W AND INTERFACES

n]v O.4

r-

transmitted w a v e s

' -0.4

0.4

n~/v

i -0.4

-

Figure 4.6: Glass-steel isotropic semi-spaces sharing the interface Shown are scattered waves for an incident longitudinal wave.

X3

:

0.

various material combinations. Critical angles associated with any of the scattered wave components occur if their wave slowness ratios are larger than unity. Inspection of the two equations (4.12) and (4.13) reveals the following possibilities: Longitudinal incidence S~ > S~

no critical angles

S,>82

2 critical angles (both transmitted)

1 critical angle (transmitted longitudinal)

Shear incidence

$2>S,>$2

1 critical angle (reflected longitudinal) 2 critical angles (reflected and transmitted

$2>$2

longitudinal) 3 critical angles (reflected longitudinal and both transmitted)

Chapter 5

FORMAL

SOLUTIONS

In this chapter we set the stage for the remainder of the book by presenting formal solutions of the field equations developed in chapter 2 in terms of unknown (constants of integration). Flat, uniform and finite thickness layers possessing various degrees of anisotropy are typical components of a layered system. As in previous chapters, we shall conduct analyses in terms of orthogonal Cartesian coordinate systems with the exception of chapter 14, where cylindrical systems will be used. We assign for each layer the local coordinate system x i such that its middle plane is oriented in the x~ - x~ plane with x~ normal to it. Constitutive relations for the layer will be given in terms of this system. A global transformed system xi will be used and is obtained from a counterclockwise rotation of the local system about the x~-direction through an angle r thus x3 and x~ coincide. Except for the stringent situation in which all layers possess orthotropic or higher symmetry and are oriented such that their principal axes coincide and that the waves travel along an axis of symmetry, we conduct analyses in the transformed system xi. The origin of the coordinate system will be located at an arbitrary location and, once solutions are obtained, appropriate solutions in terms of a different origin location can be presented by simple translation of the original system. Guided by the requirements of Snell's law, in section 5.1 we present the appropriate common form of solutions we seek. In section 5.2, we present formal solutions for a general triclinic layer. We proceed in section 5.3 to specialize these solutions to the case of monoclinic symmetry. Section 5.4 is devoted to cases involving orthotropic and higher symmetry materials where situations involving propagation along principal axes are treated in terms of the reference coordinate system x i. Due to its importance, section 5.5 is reserved for the derivation of formal solutions in fluid media. 71

72

CHAPTER&

5.1

FORMALSOLUTIONS

C o m m o n form of s o l u t i o n s

For problems involving interfaces, it will be desirable to cast the formal solution (3.3) in an equivalent form that automatically satisfies the Snell's law requirements (4.6), namely that the incident as well as all of the scattered wave components have the same phase at the interface x3 - 0. Since, as was previously pointed out in section (3.4), for plane wave propagation in the x2 = 0 plane, the motion is independent of x2, then guided by the relations (4.6)-(4.9), formal solutions for the three displacement equations (2.39)-(2.41) can now be sought in the alternative forms uj = Uje i~(xl+az3-ct),

j = 1, 2, 3

(5.1)

where i = x/~L--T, ~ is the x 1-component of the wavenumber, c as previously described in equation (4.9) is the phase velocity (= w/~) along Xl, w is the circular frequency, c~ is an unknown ratio of the wavenumber components along the x3- and xl-directions and Uj is the displacement amplitude. The choice (5.1) reflects a mere change of variables from the earlier form (3.3) used for the derivation of the Christoffel equation. Specifically, S = 1/c now stands for the horizontal axis of the slowness curves and c~/c defines the vertical one. This important relation will be examined in detail below in section 5.6.

5.2

Triclinic layer

Substituting from equation (5.1) into the displacement equations (2.39) (2.41) leads to the three coupled equations gij(ol)Uj -

0,

i , j = 1,2,3,

(5.2)

where summation on the index j holds and g i j is symmetric with Kll

=

Cll

-

pC2 "4- 2C15~ + 655ol 2

K12 = C,6 + (C14 + C~6)~ + C45~ 2 Kla = C15 + (Cla + C ~ ) a + Cas~ 2 /'(22

---- 6 6 6 --

pc 2 "4- 2646ot -4- 6 4 4 0 l

2

/(23 - C56 + (C36 + C45)cr + C34c~2 K33 : 655 - tic 2 '1 2635o~ @ C33o~2.

(5.3)

For the existence of nontrivial solutions in U1, U2 and U3, the determinant in equation (5.2) must vanish giving an algebraic equation relating c~ to c. This

5.2. TRICLINIC L A Y E R

73

is obviously an alternative representation of the Christoffel equation. The difference is that we are now solving for a in terms of c as compared with solving for v for a given propagation direction ~. Setting the determinant equal to zero, we obtain a sixth-degree polynomial equation in a written symbolically as

O~6 -4- B I ~ 5 + AI(~ 4 -4- B2o~3 + A2c~2 -4- B3c~ + A3 : 0

(5.4)

where the various coefficients are dependent upon c and the material properties; these coefficients need not be reported here for the general triclinic case. Equation (5.4) admits six distinct solutions for a which we arbitrarily label as aq, q - 1, 2, ..., 6. For each C~q, we use the relations (5.2) and express the displacement component ratios Vq = U2q/Ulq and Wq = U3q/Ulq as

Kli (O~q)K23 ((~q) - K13 (OLq)K12 (Olq) Vq - K13(O~q)K22(O~q) _ K12(O~q)K23(O~q)

(5.5)

K l l (~q)K23 (t~q) - K12(OLq)K13 (Ceq) Wq -- K12(O~q)K33 (O~q) - K23 (Ozq)K13 (C~q) "

(5.6)

Combining equations (5.5) and (5.6) with the stress-strain relations (2.23) in their transformed form via equation (2.30), and using the method of superposition, the formal solutions for the displacements and stresses become 6

(ul, u2, u3) - ~-~ (1, Vq, Wq)Vlqei~(xlW(~qx3-ct) q=l

(5.7)

6

(0"33, 0"13, 0"23) : E i~(Dlq, D2q, D3q)Ulqe i~(xl +aqX3-Ct) q=l

(5.8)

where

Dlq - C13 + 0~qC35 -4- (C36 --I-O~qe34)Vq q- (C35 q- C330~q)Wq D2q - C15 A- 0~qC55 -t-- (C56 -4- o~qe45 ) Vq -4- (C55 -3I- C350~q)Wq D3q - C14 -~- (~qC45 ~- (C46 -~- (~qC44)Vq ~- (C45 -~- C34(~q)Wq.

(5.9)

At this point, we have presented a formal solution of the field equations in a generally anisotropic medium, that is, one belonging to the triclinic symmetry group. The six displacement amplitudes Ulq are the unknowns.

74

C H A P T E R 5. F O R M A L S O L U T I O N S

5.3

The monoclinic

case

For monoclinic materials having x~ - x ~ ( or equivalently the Xl --X2 ) as a plane of mirror symmetry, the applicable equations of motion are given by equations (2.42)-(2.44). Application of the formal solution (5.1) to these equations recovers equation (5.2) but specializes the Kij elements to K11 :

C11 - PC2 + C550z 2

K12 ---- 616 -+- C45 O~2

K13 = (C13 + C55)a K22 = C66 - pc 2 + C44 0~2 t':23 = (C36 + C4s ),~

K33 :

(5.10)

C55 -- PC2 -5 C330~ 2.

The reduced complexity of these relations leads to the vanishing of the coefficients of odd powers in a of equation (5.4) resulting in Ol6 -~- AlO~ 4 + A20~2 -{- A3 - 0

(5.11)

where the algebraic coefficients A i, A2 and A3 are now listed as A1 -

[CllC33C44 - C123C44 -+- 2C13C36C45 - 2C13C44C55

+2C13C425 - 2C16C33C45 -5 C33C55C66 - C26C55 -- (C33C44 "1- C33C55 "t- C 4 4 C 5 5 -

C425)pc 2] / A

A2 -- [CllC33C66 - CllC26 - 2CllC36C45 Jr- CllC44C55

-CllC25 -+-C23C66 "+-2C13C16C36 + 2C13C16C45 -2C13C55C66 - C126C33+ 2C16C36C55 -- (CllC33 -+- CllC44 - C123 - 2C13C55 - 2C16C45 -+- C33 C66 - C26 - 2C36 C45 -~- C44 C55 - C25 + C55 C66) pc 2 + (C33 nt- C44 -+- C55) pc 4] / A A3 ~- [CIIC55C66-

C26C55- (CiiC55-f-CiiC66~-C126-~-C55C66)pc2

-+- (ell + C55 + C66) pc 4 - pc 6] / A with /k = C33 C44 C55 -- C33 C25 9

(5.12)

5.4. HIGHER S Y M M E T R Y M A T E R I A L S

75

Once again, equation (5.11) admits six solutions for a, or more precisely, three solutions for a 2. Thus, solutions for a occur in three pairs, each pair having two a's that are negative of each other. Here, and for the remainder of this book, we shall label these roots as c~q,q = 1, 2, ..., 6 and further adopt the convention Ot2 - - --O~1,

Ol4 -- --O~3,

O~6 -- --OL5.

(5.13)

These natural restrictions on c~'s do not, however, influence the forms of the amplitude ratios (5.5) and (5.6). The influence on these ratios is implicitly reflected through the new definitions of C~q and Kij in equation (5.10). However, the monoclinic stiffness restrictions reflected in equation (2.35) have direct and explicit influence on the stress amplitude ratios (5.9) simplifying them to

Dlq D2q D3q --

C36Vq awC330~qWq Wq) + C450~qVq C45(OLq -~- Wq) -~- C44 0tqVq.

C13 +

C55(0~q +

(5.14)

Inspection of the amplitude ratios (5.5) and (5.6), in light of the unique properties equation (5.13), reveals the following relations

w2--w~,

w4 = - w 3 ,

w6 = - w 5

v2

v4

v6

=

v~,

=

va,

=

(5.15)

vs.

Subsequent investigation of the stress amplitude ratios (5.14), in light of both (5.13) and (5.15), leads to the following properties D12 -

Dll,

D14 -

D13,

D16 -

D22 -- - D 2 1 ,

D24 -

-D23,

D26 -- -D25

D3z = - D 3 ~ ,

D34 = -D33,

D15

(5.16)

D36 - -D35.

The various properties (5.13), (5.15) and (5.16) are critical in the simplification of solutions for monoclinic and higher material symmetry situations.

5.4

Higher symmetry materials

The above formal solutions obtained for the monoclinic case can, under appropriate restricted conditions, also hold for higher symmetry classes such as orthotropic, transversely isotropic, and cubic. Appreciating that these classes of materials are different from the monoclinic material in that each can possess two orthogonal symmetry axes in the plane of the layer, the formal solutions derived so far for the monoclinic case do apply if the wave

76

CHAPTER5.

FORMALSOLUTIONS

x 2 , x~

~

xr

.~._&_L_~_L ~_2~,~ ~=a__!._ ~ _ L . J _

~llr\l

!

I ~

_~__L~__L '_.,_Lq_S_~__Lj_I_L! I

~ ' % J _ ~ _ L _ L I I.L,.

~

~--~'--i1---77--~.

I

I

,

I

I

I

I~

X. I

- I--I--T--17 ~ .-~-~-~-~-/

(b)

Figure 5.1: Schematic of the propagation in orthotropic and higher symmetry materials (a) along an off-axis-of-symmetry where coupling between S H and sagittal plane motions occurs and (b) along an axis of symmetry where uncoupling occurs.

propagates along directions other than these principal axes. This is true because coupling of the S H wave motion with that of the sagittal plane will persist. On the other hand, decoupling of these equations also occurs for propagations along principal axes, resulting in further explicit simplified versions of equations (5.10)-(5.16). Both situations are illustrated graphically in figures 5.1a,b, respectively. On these figures we have added the grid background to designate the presence of in-plane symmetry axes characteristic of orthotropic and higher symmetry materials. In both cases, we also recognize that results for all symmetry classes higher than orthotropic are contained as special cases of the corresponding solutions for the orthotropic case. Once solutions for propagation directions that are either off or along principal axes are derived for orthotropic symmetry, the corresponding results for the higher symmetry materials can be obtained merely by applying the appropriate material restrictions on their properties. For this reason, we now concentrate on deriving results for orthotropic symmetry materials.

5.4.

5.4.1

HIGHER SYMMETRY

Propagation

MATERIALS

77

along off-principal-axes

For the off-principal-axis propagation directions of figure 5.1a, we need only to insure that further reduction in the number of nonzero elastic constants are properly exploited in equations (2.24) resulting in equation (2.25). Now if the x~- and x~- axes of the reference coordinate system are chosen to coincide with the in-plane principal axes for orthotropic symmetry, then their constitutive relations will be given by equation (2.25). For propagation directions that do not coincide with any of the principal axes directions, the transformed matrix of (2.25) will take the form (2.34) similar to the monoclinic symmetry case. For this situation, solutions to orthotropic media can be obtained as special cases of the solutions that belong to monoclinic materials. 5.4.2

Propagation

along an axis of s y m m e t r y

Returning to the case of orthotropic symmetry, we see that the axis x~ and x~ coincide with the azimuthal angles r = 0 ~ and r = 90 ~ respectively. For propagation along either of these directions, it is convenient to conduct analyses in the reference system x 'i. The applicable field equations are given by equations (2.46)-(2.48). These equations define two decoupled motions. The first defines motion in the sagittal plane and is described by the two equations (2.46) and (2.48), while the second defines a pure horizontally polarized ( S H ) shear wave and is described by equation (2.47). Formal solutions for each of the two motions is obtained independently. F o r m a l solutions for S H wave t y p e We start with the simple case of the S H wave. The behavior of this wave is governed by the equation of motion (2.47), namely ' 02u'2 02u'2 02u'2 666 OXtl 2 ~- 6144 0xl32 -- P Ot 2

(5.17)

and the corresponding stress-strain relation

Ou'2

a~3 = C44 Ox~3 .

(5.18)

A formal solution of equation (5.17) is sought in the form of equation (5.1) in accordance with u'2 -- U2e i~(xi +ax'3-ct).

(5.19)

78

CHAPTER5.

FORMAL SOLUTIONS

Substituting from equation (5.19) into equation (5.17) gives

Oil = --0/.2 -- ]~/ pc2 -- C66. V

c~

(5.20)

The principle of superposition leads to the formal solution 2

~tl2 -- E U2q ei~aqx~aei~(x~-ct)"

(5.21)

q=l

Combination of equations (5.21) and (5.18) results in

~

2

-- E i~D2qU2q ei~aqx'3ei~(x'l-ct)

(5.22)

q=l

where

D2q - CI44O~q.

(5.23)

Note from equations (5.20) and (5.23) that D22 - -D21. F o r m a l solutions for sagittal plane m o t i o n Formal solutions for the sagittal plane motion can be obtained by utilizing the coupled equations of motion (2.46) and (2.48) together with the stressstrain relations , , OU'l au'3 ~33 = 613-~x, ' + 6~3 Ox'3 , 0-13 --

Ou'3 OU'l C~5(~x ~ + ~x~)-

(5.24)

(5.25)

The construction procedure of the formal solutions parallels the previous general one pertaining to the monoclinic case and hence we only summarize its results as 4

(u~, u~) - ~--~(1,Wq)Ulqe i~(x~+aqx~3-ct) q=l 4 (0"33' 0"~3) -- E i~(Dlq, D3q)Ulqe i~(x'l+aqx'3-ct). q=l

(5.26)

(5.27)

5.4. H I G H E R S Y M M E T R Y

MATERIALS

79

The associated displacement and stress amplitude ratios are now given by t %2 pc 2 - C~1 - C55

Wq =

(C~3 + C ~ 5 ) ~ I

n l q - (C~3 -t- C~50lqWq) (5.28)

n3q = C~5 ( aq + Wq ) with a2 satisfying the quadratic equation A a 4 + B a 2 + C = 0.

(5.29)

The coefficients A, B and C are given by

A = c~c~

B = (C~ c -

pd)C~3

-

(c~

-

(c(, - pd)(c~

-

pd)C~

- ( c h

+ c~) ~ (5.30)

- pd).

Equation (5.29) admits four solutions for a having the properties a2 = - a l and O~4 - - --O~3. 5.4.3

Isotropic

media

Results for the isotropic case can be obtained by exploiting the degeneracies of the elastic properties C~j. Starting from the orthotropic case, and invoking the material relations C~3 = C~2 = C~I , C~3 --~ C ~ 3 - C~2 , C~6 - C~5 - C~4 with C ~ 1 - C~2 - 2C~6 , define an isotopic material. In terms of the Lame' constants A and #, C~1 = A + 2# and C~6 - #. These simplifications are adequate to reduce all of the previous results to the ones pertaining to isotropic media; they also lead to much simpler expressions for the various propagation parameters discussed in section 5.4.2. Specifically, implementing the isotropic restrictions reduces these parameters to

V/~l~

O~ 1 "-- - - O l 2 - -

- 1,

n2q

--

~Olq,

q = 1,2

(5.31)

for the S H wave and ~ f = c ~/~7 w1 = - w 2 Oll

-

l,

= al,

-~ D12 - - i ~ # ( o ~ 2 -

1),

D31 = D32 = 2i~pal, c~

-

(:~ + 2 v ) l p ,

for the sagittal plane motion.

o~2 = c2/c22 - 1 W3 = - W 4 = ~3

D13 = D14 - - 2 i ~ # 033 - D34 -i~#(o~] - 1)/c~3

(5.32)

CHAPTER5.

80

5.5

FORMALSOLUTIONS

F o r m a l s o l u t i o n s in fluid m e d i a

Utilizing the fact that the fluid does not resist shear deformation, its field equations are given by

Oa~f ) 02ul f) Oxj = pf Ot2

(5.33)

OU~f)

a}f ) - Af Oxk ~ij,

i, j = 1, 2, 3

(5.34)

where pf and Af are the fluid density and Lame's constant. For a motion confined to the sagittal plane, u2 vanishes and the remaining field variables are independent of the direction x2. For this situation, the field equations (5.33) and (5.34) are combined and written in the expanded form

02Ul

02U3

1 02Ul

(5.35)

OX21 + OXlOX3 -- c} Ot 2

02ul

OXlOZ3

+

02u3

Ox~

=

1 02u3

where c I = k/Ai/pl is the wave speed in the fluid. tions (5.35) and (5.36) in the form (5.1) yield =

=

v/c l

(5.36)

c~ Or2

-

Solutions of equa-

1

(5.37)

and the displacement amplitude ratios

Wfl -- - W f 2 =o/s.

(5.3s)

Thus, complete formal solutions in the fluid are summarized as 2

(u:,u3, a33): = y~ (1, Wfq, i:pfc 2) Vqei~[xl+(-1)q+aaIx3-ct].

(5.39)

q-1

5.6

T h e c~-c r e l a t i o n a n d t h e C h r i s t o f f e l e q u a t i o n

In section 5.1, we suggested that the a - c relation constitutes an alternative means of describing the Christoffel equation. With reference to the critical angle discussion of section 4.3, figures 4.4-4.6 graphically show that the common intersection of the guide line with the horizontal axis defines 1/c and the vertical projections of the interaction of the guide line with the various slowness curves define the corresponding values of aq/C, q = 1, 2 , . . . , 6. In

5.6. THE a - C R E L A T I O N AND THE CHRISTOFFEL EQUATION81 our subsequent description, we utilize the criteria used in the numbering of the various a's. Mathematically, the above discussion implies the following alternative description of the slowness curves 2 _ a2

Sq

1

--~ + --~

(5.40)

or equivalently

aq2

__C 2

Sq2 - 1.

(5.41)

Better appreciation of these relations can be gained by applying them to the case or isotropic media. Since, for isotropic media, Sq, q = 1, 3 define the slownesses of longitudinal and shear waves, respectively and since S1 = 1/Cl and 5'3 = 1/c2, we recover the expressions of al, a3 in equation (5.32). Furthermore, since c and Sq are both real, equation (5.41) implies that aq2 stays positive so long that c2S 2 > 1. In this region aq is positive and the corresponding wave component is propagating. For c2S2q < 1, aq2 becomes negative and thus the wave changes to a non-propagating behavior (i.e., it becomes evanescent). From this, we conclude that aq = 0 is associated with the condition for the propagating wave to become evanescent and hence defines an alternative critical angle identification.

This Page Intentionally Left Blank

Chapter 6

SCATTERED WAVE AMPLITUDES In chapter 4, we introduced the notion of reflection and refraction and discussed their relationships with the generalized Snell's law. We also outlined a procedure for identification of critical angles. In chapter 5 we suggested an alternative formal solution to that of (3.3) which a priori satisfies the interface requirement that all scattered wave components have the same projected wavenumber along the interface as that of the incident wave. The obvious question that presents itself now is, given the above information, what will be the relative amplitudes of the scattered wave components as compared with that of the incident one. This is what we intend to discuss in the present chapter. The scattering amplitudes depend upon the properties of both media sharing the common interface, the incident wave characteristics and the interface conditions. In the remainder of this chapter, the formal solutions (5.7) and (5.8) will be utilized to study the scattered field amplitudes for a variety of incident wave configurations. Solutions involving propagation along axes of symmetry can be treated in a similar manner if needed. In section 6.1, we lay down the rules for geometric arrangements, identification of the media sharing an interface and characterization of incident and reflected wave components. Section 6.2 is reserved for reflection from the free surface of a semi-infinite solid-space. In section 6.3 we treat reflection and transmission from fluid-solid interfaces. Finally, in section 6.4 we outline the procedure for obtaining scattered wave amplitudes from two solids in rigid contact at their common interface. 83

C H A P T E R 6. S C A T T E R E D WAVE AMPLITUDES

84

6.1

Notation

To assist in differentiating between the two media, and as was introduced in section 4.1, an overbar is assigned to designate the continuing medium. Inspection of the formal solutions (5.7) and (5.8) reveals that each displacement and stress component is a sum of six separate wave components. The relationships in equation (5.13) reveal that the six wave components consist of three pairs, each of which propagate in directions that are mirror images of each other about the Xl-axis. We arbitrarily assign al, c~3, c~5 to the waves that propagate in the positive x3-direction, while we reserve c~2,c~4, O~6 for the components that travel along the negative x3-direction. It is implicitly implied here that c~1, c~2 and c~5 are the ones with positive imaginary parts so that boundedness of the field variables is insured in situations involving semi-space substrates as x3 increases indefinitely away from the interface. It then follows that the remaining c~~s will have negative imaginary parts so that boundedness is also insured away from the interface in the negative x3-space. Furthermore, adopting the earlier introduced strategy of reserving the third quadrant (see figures 4.3 and 4.4) for the incidence region, we see that incident, as well as transmitted, waves propagate in the negative x3-direction while the reflected ones move in the positive x3-direction. According to this convention, components having even values of the subscript q in the originating medium define a population of possible incident waves. It then follows that components having even values of q in the continuing medium define the transmitted ones. To assist in facilitating our description, in most cases, we can identify q = (1,2), (3,4) and (5,6) with the quasi-longitudinal, fast and slow quasishear waves, respectively. As an example of using the above identification procedure, the incident slow quasi-shear wave in figure 4.3 is described by (Ul, u2, u3) (x) = (1, V6, W6)U~6x) e i~a6x3 9

(6.1)

For convenience, here and thereafter we recognize and suppress the common factor e i~(xl-ct). As illustrated in figure 4.4, an incident field of the form

(Ztl' Zt2' Zt3)(') --

Z (1, Vq, Wq)U}~ )e i~"qx3 q--2,4,6

(6.2)

will then constitute a superposition of three incident waves (having the same xl-component of the wavenumber vector) with amplitudes U~2/),U~4/), U~6/). Thus the choices

.1(4')=

: o.

.1(2')-

: o.

.1(2')=

o

(6.3)

6.2. REFLECTION FROM A FREE SURFACE

-0.6

0,0

85

0.6

nl/v

-0.6

Figure 6.1: Half-slowness of a graphite-epoxy in the plane r = 30 ~ showing an example of incident and reflected wave components.

define incident quasi-longitudinal, fast and slow quasi-shear waves, respectively. If we are able to solve for the total scattered field subject to the superposition of incident waves (6.2) in one strike, then scattering due to single incident wave component can be obtained by merely choosing from the menu (6.3).

6.2

Reflection

from a free surface

Let us start with the case of incidence at and reflection from the free boundary x3 = 0 of a monoclinic solid (i.e., the continuing medium is a vacuum). The situation is illustrated in figure 6.1 for a graphite epoxy semi-space generated for the azimuthal angle r = 30 ~ We first consider the case in which the slow quasi-shear wave is incident. This wave is completely characterized by equation (6.1). Such choice gives rise to the three reflected wave components shown in the figure. Accordingly, the reflected displacements are obtained from a superposition of the reflected components in accordance with equation (5.7) as

(it1, ~t2, it3) (R) --

E (1, Vq, Wq)U~ n) ei~aqx3. q=1,3,5

(6.4)

CHAPTER 6. SCATTERED WAVE AMPLITUDES

86

Hence, the total displacement field can be obtained from a direct sum of the incident and reflected contributions leading to (721, ~2, tt3) ---- (1, V6, W6)U~6I) e i(a~

(1, Vq, Wq)U~Rq)e i~aqxa.

+ ~

(6.5)

q=1,3,5

The total stress field is obtained from the stress-displacement relations (5.8) subject to the definitions (5.14) as (0"33, o"13, 0"23) -- i~(D16, D26, D36)U~6/)ei~aqx3

+ E

i~(Dlq'D2q'D3q)U~ R)ei~aqx3"

(6.6)

q--1,3,5

Since, as required by the stress free boundary conditions (4.4), the stresses must vanish at the free surface 13 = 0, we get the three algebraic equations that relate the reflected field amplitudes to that of the incident wave

(Oll o13 D21 O31

D23 033

.(R) '~13 U~R)

D25 D35

-_

(o16) D26

U~6I)

(6.7)

036

In virtue of the fact that the amplitude U~/) is known, the unknown reflection amplitudes can be obtained using Cramer's rule. It is tempting, however, to take advantage of the symmetry relationships of equation (5.16) and carry the algebra a little further in order to simplify the resulting expressions. To this end, using equation (5.16), the right hand side column matrix can be written as --

(o10)(o1) D26

--

D36

D25

9

(6.8)

D35

Replacing the right hand side column of equation (6.7) by the right hand side column of equation (6.8) and using the Cramer's rule lead, after minor algebraic manipulations of the determinant, to the results

U~R) -- - 2D15G-------A1U~I) 1

-

U~R) = 2D15G3u~I6) Av

U~R) - ( 1 -

(6.9)

Av

2D15G5 A------~-)U~6I)

(6 10) (6.11)

6.2. REFLECTION FROM A FREE SURFACE

87

where

Av = D11G1 - D13G3 + D15G5

(6.12)

with G1 = D23D35 -- D33D25

(6.13)

G3 = D21D35 - D31D25

(6.14)

G5 = D21D33 - D31D23.

(6.15)

If we start with the superposed incident field (6.2), we obtain a general solution from which the above result can be obtained as a special case. Using the incident field (6.2), the total displacements and stresses become (ul u2,u3) =

~

(1,1/q

W

~tr(I),~i~aqxa

q=2,4,6 ~7~ (1, l/q, Wq)U~R) ei~qz3 q--1,3,5 (a33, a13, a23) = ~ i~(Dlq, D2q, D3q)U~)e i~aqx3 q=2,4,6 + E i~(Dlq, D2q, Daq)U~Rq)e i~aqx3. q=1,3,5 +

(6.16)

Invoking the stress free conditions at x3 = 0, followed by using the stress amplitude relations (5.16) yields

(Oll o13 D21

D23

U13 (R) ~15rr(R)

D25

D31 D33 D35

Oll o13 =

D21

D23

D25

U~/)

D31

D33

D35

U~6/) (6.17)

Note that the results obtained above for the case of incident slow quasishear component can be easily obtained from equation (6.17) by setting U~ ) = 0 and U~4I) - 0 as required by equation (6.3). Using Cramer's rule, we finally obtain the reflected amplitudes for an incident quasi-longitudinal wave (U~4') - U~') = 0) ~s

U ~ ) - (1

-

2D11G1

Av

)U~ )

u~R) __ 2D11G3 U~/)

u(R)--15 -

(6.19)

A~

2D11G5

(I)

A-----~ U12

(6.18)

"

(6.20)

CHAPTER 6. SCATTERED WAVE AMPLITUDES

88

0

niv

transmitted wave

,

,

-0.7

.7

incident wave I -0.7

P

nl/v

reflected waves

L

Figure 6.2: Half-slowness of graphite-epoxy and water semi-space sharing the interface x3 - 0 ; shown are the incident and scattered waves. Similarly, for an incident fast quasi-shear wave (U}/) - U}/) - 0), we get

U~iR) = - 2Di3Gi

(6.21)

2D13G3 U}3R) = (1 + A------~)U}/)

(6.22)

U}5R) = _ 2Di3 G______.U}4 ~5 t) /k v

6.3

(6.23)

Scattering from fluid-solid interfaces

Explicit determination of the reflection and transmission coefficients for plane waves incident from the solid onto the solid-fluid interface requires the use of the formal solution (5.39) in the fluid. We proceed to derive the amplitudes of the scattered waves; these include one transmitted in the liquid and three reflected in the solid as illustrated in figure 6.2. The total field in the solid is once again given by equation (6.16). The required formal solution for the fluid is adapted from equation (5.39) as (~1, fi3, ~33) : (1,--O~f, i~flfC2)C~ T) e -i~alxa

(6.24)

6.3. SCATTERING FROM FL UID-SOLID INTERFACES

89

where the overbar is added for the consistency of identifying the fluid as a continuing medium. Invoking the continuity conditions (4.2) and (4.3), we obtain a system of four linear simultaneous equations in the amplitudes [~r~T),u~IR), U~3R) and

Dll D13 D15 _pfr

rr(R) '+13

O31 D33 D35

TT(T) '-"12

D21 D23 D25

=

-Dll

D21

0

0

U~R)

-D13 -D15 D23 D25

O31

033

D35

/ )rr(I) ~ 14 9

(6.25)

U~/)

Upon solving these equations, expressions for the reflection and transmission amplitudes are obtained. For an incident quasi-longitudinal wave these are U~IR) = (1 - 2D11G1 )U}/) Af

(6.26)

U~R3) - 2Dll a3 s~I) A/

(6.27)

U~5R) : _ 2Dll G______U~/) _~5 AI

(6.28)

u~T) = 2D11Y* U~I2) t~fAf with Af = Av + Y

Y = -PfC---~ZY*, Y* - WIG1 - W3G3 -t- W5G5. aI For an incident fast quasi-shear wave we get

(6.29)

(6.30) (6.31)

u~R1 ) -- -2013G~l u~I) Af 2D13G3 U}3R) -- (1+ Af )U~4I)

(6.32)

U}5R) = _ 2DlaG5 U~4I) AI

(6.34)

T~T) = 2DI3Y* U~/). afAf

(6.35)

(6.33)

CHAPTER 6. SCATTERED WAVE AMPLITUDES

90

Finally, for an incident slow quasi-shear wave, these amplitudes become

u~R) _

2nl5elU~I)

(6.36)

AI

(6.37)

A/ V~5R) = ( 1 - 2D1505)U~I6) AI

(6.38)

(7(1T) -- 2D15Y*u~I6)"

(6.39)

afAf

6.4

Scattering from solid-solid interface

Finally, for the case where the continuing medium is a solid, three transmitted waves are generated. A sample illustration was given in figure 4.3. For this case, the total displacement and stress fields in the originating medium are, once again, given by equations (6.5) and (6.6). For the continuing medium, on the other hand, the total fields will only involve the transmitted component contributions as

(~1, ~2, ~3) -- E (1, V'q,VVq)O~T) ei{aqza q=2,4,6

Z i~(blq, b2q,baq)O}T)ei~"qx3.

(#33, #13, #23) =

q--2,4,6

(6.40)

Upon invoking the continuity conditions (4.1), we get the algebraic relations 1

1

1

-1

V1

V3

V5

-- ~TZl --~P'3

-1

-1

U~3R)

-D13

-D15

~T)

/)23

/)25

U~4T) k~_~T) )

w1

Dll D13 n15 --f)ll O21 D23 025 /)21 D31 D33 D35 /7)31

b33

-i

-1

-1

_.c,<2)

-V1 W1

-g3 ~vV3

-Y5 W5

U~/)

-Dll

-D13

-DI5 D25 D35

U~/)

D21

D23

D31

033

( U~In)

-V'5

b35

(6.41)

6.4. S C A T T E R I N G FROM SOLID-SOLID INTERFACE

91

Once the incident wave is identified from equation (6.3), it is straight forward to use Cramer's rule and obtain all desired reflection and transmission amplitudes. For the sake of brevity we shall not pursue this any further.

This Page Intentionally Left Blank

Chapter 7

INTERFACE

WAVES

Interface waves are waves that propagate along the interface between two media and have the property that they decay away in both media, i.e., in directions normal to the interface. For two solid media in rigid (welded) contact, such waves are named after Stoneley [230] who discovered them in 1924. If one of the solids is replaced by a nonviscous fluid, then a "Stoneley" like wave exists which is named after Scholte [221] who discovered it in 1947. In contrast with the above two cases, " interface waves" that can propagate along a solid-vacuum interface, i.e., along the free surface of a solid semispace are termed surface waves; these are also named after Rayleigh [207] who discovered them in 1885. For interface waves to exist, certain conditions must be satisfied. With reference to the formal solution (5.1), a necessary but not sufficient condition is that c~ must be chosen such as to maintain boundedness far away from the interface. As adopted in chapter 6, this means that in most cases a must have a positive imaginary part for the medium occupying the space x3 > 0 and a negative imaginary part for the other medium (i.e., that occupies the space x3 < 0 ). In some cases involving propagation close to grazing angles, the boundedness should be based upon decaying of the energy field as related to the Pointing vector (see section 3.6). With reference to the discussion concerning critical angle phenomena in section (4.4), we see that true interface waves can only exist beyond the critical angles of all possible wave components. In a simplistic manner, we conclude that interface waves propagate with speeds that are lower than the lowest speeds of bulk waves in both media. Since beyond its critical angle, a wave component becomes evanescent, interface waves can only be of evanescent type and hence are often given the name inhomogeneous. Having given a qualitative description of interface waves, we proceed 93

CHAPTER 7. INTERFACE WAVES

94

to study them in a quantitative manner. We shall discuss them in the sequence of their algebraic complexities. In section 7.1 we study surface waves. Section 7.2 deals with the propagation of pseudo-surface waves. Finally, section 7.3 is devoted to the study of Scholte waves. Stoneley waves can be studied within the context of the scattering from solid-solid interfaces of section 6.4 and will not be elaborated upon any further.

7.1

Surface

waves

We start with the relatively simpler case of surface waves. Once again, we call upon the formal solution~ (5.7) and (5.8) as a start. Consistent with the convention of assigning al, OL3 and a5 to the wave components that propagate in the positive x3-direction, as adopted in chapter 6, we also require these a's to have positive imaginary parts in order to insure the existence of surface waves. Using these identification criteria, the formal solutions (5.7) and (5.8) are specialized to the semi-space as

(?~1, ~t2, ~t3) -(~733, a13, a 2 3 ) -

Z (1, Vq, Wq)Ulqei~aqx3 q--1,3,5

(7.1)

~ i~(Dlq,D2q,D3q)Ulqei~aqx3. q--1,3,5

(7.2)

These are also equivalent to the components identified with the superscript (R) in equations (6.5) and (6.6). For surface waves to exist, these solutions must satisfy the stress-free boundary conditions (4.4). By invoking such conditions, we obtain the three homogeneous equations

(o11 o1)(11) D21 D31

D23 D33

D25

V13

D35

U15

= 0.

(7.3)

For nontrivial solutions to exist, the determinant of equation (7.3) must vanish resulting in the characteristic equation Av : DllG1 -- D13G3 + D15G5 = 0

(7.4)

where G1, G3 and G5 are as given previously in equations (6.13)-(6.15). We immediately recognize that the expression Av is the denominator encountered in section 6.2 of the reflected field in the free solid semi-space. Hence the zeros of Av correspond to the poles of the reflection coefficients in these equations.

7.2. PSEUDO-SURFACE WAVES

95

In order to appreciate the algebraic complications (or lack of it) of the secular equation (7.4), it is instructive to refresh our memory and summarize its dependence upon the various parameters involved in its definition. With reference to the general relations (5.9) or to their specialization to monoclinic materials (5.14), the various Dij elements directly involved in the definition of Av are explicit functions of the material properties, the wave amplitude ratios Wq, Vq and aq. But the amplitude ratios are themselves functions of the properties and aq. Here we recall that the "material properties" refer to the transformed properties through the rotation of the azimuthal angle r about the x3-direction. Since the a's are direct functions of the properties and the phase velocity c, an implicit dependence of c on the azimuthal angle r is established. Hence, we conclude that Av depends solely on the properties and the varying velocity c. The specific value of c that satisfies the requirement Av ---- 0 thus defines the surface wave speed for each azimuthal angle r In general, finding these critical values of c needs numerical calculations. There exist many ways for numerically finding them, however. The most direct way is by inspection where values of Av for varyiflg c cross zero. If Av is complex, then the absolute value of Av must be monitored. By checking the numerical values of the corresponding a's for this critical c, we find that they are positive pure imaginary. These a's insure exponential decay of all field variables in the x3-direction. For graphical illustration of the above discussion, and to set the stage for other discussions to follow, we illustrate the above procedure for the representative example involving the InAs cubic material. The half slowness curves, corresponding to a semi-space situation, are shown for the azimuthal plane defined by r = 33 ~ in figure 7.1. Since the horizontal projection defines the variable 1/c, we find the critical value of c to be 1.95 km/s. Notice that the inverse of this speed, namely 0.51 s/km, is located outside the slowness curves. This signifies the fact that, for this value of c, all a's are positive pure imaginary and the corresponding three wave components are evanescent. Also shown in this figure the values of 1/c corresponding to the excitation of " pseudo-surface" waves which will be discussed in the next section.

7.2

Pseudo-surface

waves

Historically when the above primitive technique was first used to calculate the surface wave speeds for anisotropic materials, deep minima in the absolute value of Av were also found for some materials at selective propagation

CHAPTER 7. INTERFACE WAVES

96

-0.6 .

0,0 .

.

.

0.6 ,surfa 'cenlw/eVav

pseudosurfacewave

n3/v

-0.6

Figure 7.1: Half-slowness curves for the sagittal plane r - 33~ the cubic material InAs. Shown are the slownesses of the " pseudo-surface" and the surface waves. directions along the free surface. However, a search in the entire complex c plane does drive the boundary condition determinant Av to zero. Upon further investigation of the associated displacement and stress fields, it was found that such fields display many of the features encountered in the surface wave situation. For this reason these "secondary" waves have been named " pseudo-surface" waves. The common, as well as the different, features between the "surface" and " pseudo-surface" waves can be illustrated with reference, once again, to figure 7.1. Here, the pseudo-surface wave speed is found to be 2.21 km/s. Hence, its inverse of 0.45 s/km is located to the left of the curve belonging to the slowest bulk wave, namely the quasi-shear wave. This automatically implies that the pseudo-surface wave propagates with speeds that are larger than that of the slow quasi-shear one. One of the a's is now complex with a negative imaginary part and the remaining two have positive imaginary parts. Such modes contain a bulk wave component, that is propagating and conveying energy into the substrate. For each azimuthal angle r Av is searched for either its zero or absolute minima which correspond to surface or pseudo-surface wave speeds, respectively. Based upon this criterion, the variation of the wave speed with the azimuthal angle r is depicted in figure 7.2 for the InAs cubic material. The lower curve corresponds to the normal surface wave mode whereas the higher one is associated with the pseudo-surface mode. Along the xl-direction, the primary surface wave is polarized in the sagittal plane like an ordinary Rayleigh wave on the free surface of an isotropic solid.

7.2. PSEUDO-SURFACE WAVES

2.4

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

97

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

2.3 2.2 2.1F

r

2.0 1.9

1.80 ....

; ....

.... 1; .... 2'0 ....

(I) (deg)

.... 3'0 .... 3's .... 4'0

4;

Figure 7.2: Variation of wave speed with azimuthal angle r for InAs cubic material. Lower and upper curves correspond to the surface and pseudosurface modes, respectively.

As the azimuthal angle is increased, the displacement polarization changes from dominantly normal to dominantly parallel with respect to the surface. Along the face diagonal direction, the primary surface wave asymptotically approaches the horizontally polarized (slow) shear wave with polarization perpendicular to the sagittal plane. In this direction, there is also a pseudosurface wave polarized in the sagittal plane like the primary surface wave along the x~-direction. We shall see later that in the case of a fluid-solid interface the reflection coefficient is affected mainly by that mode which produces a significant normal displacement at the surface (figure 11.2). Figures 7.3a and 7.3b display sampl e comparisons of normalized spatial. distributions of the real parts of the displacement components u3, for the surface and pseudo- surface waves. These two figures correspond to the azimuthal angles 4)= 24~ and ~b = 32 ~ in the InAs material, respectively. We note that the fields, as measured by their intensity, are mostly confined to the vicinity of the surface. In comparison with the fields of the surface modes, we see the apparent persisting oscillations in the pseudo-surface modes. So far, we have presented our numerical illustrations for the InAs cubic material. Next, we demonstrate that pseudo-surface modes may or may not exist on other anisotropic media. In figures 7.4a and 7.4b, we plot the variation of the wave speed velocity c with azimuthal angle for graphite-epoxy

CHAPTER 7. INTERFACE WAVES

98

U3

U3 |

-s

j

lO

i

Secondary

Primary

Sec~ lX3

x3

(b) -2.~.o

-o.s

o.o

o.s

i~o

.o

-o'.s

o.o

o:s

i.o

Figure 7.3: Comparison of the profiles of Re(u3) displacement components corresponding to the excitation of the surface and pseudo-surface modes for (a) r = 24 ~ and (b) r = 32 ~ in the InAs cubic material.

2.4 r . . . . . . .

! .......... r

....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ssliiiiii ...............................................................

2.3I ...... i .........i ......i ............................... 2.1

"

...i

...... ~....

i .........................

~.......... i

..... !

..... ~......... ! ......... i ........!...... i

3.6

............................................................

3.4

..........................

~,32

~

3.0

0 2.8 ~.8

...... !............ ! ....... ! ....

1.5

....... i ......... i ........ i ...... i ...... i........... i ............. i ......... i

1"40

.I i 10 20

i......... !.......... ! .......... i ........ }

30 40 50 60 (deg)

..... !

' . . . . .... 70 80 90 ..... i

(a)

!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6 2.4

i! i il i i i fill Hil

2"20

........ i ............ ~

.........i .........i .

.

.

.

10 20 30 40 50 60 70 80 90 (deg) (b)

Figure 7.4: Variation of phase velocity with azimuthal angle r for (a) graphite-epoxy and for (b) centrifugally cast stainless steel.

7.3.

SCHOLTE WAVES

99

and for centrifugally cast stainless steel (CCSS) materials, respectively. Figure 7.4b shows the presence of double roots only for the CCSS material signifying the existence of pseudo-surface mode. Notice also, in contrast with the results shown in figure 7.2, the unusual situation where for r > 62 ~ both branches become practically indistinguishable. On the other hand, the calculations shown in figure 7.4a for the graphite-epoxy composite show no trace of such pseudo-surface modes.

7.3

Scholte

waves

In the above two sections, we found that the secular equation for the existence of surface waves on solid semi-spaces corresponds to the poles of the reflection coefficient of the medium. It then follows that interface waves between any two media, if they exist, are defined by the poles of their scattered fields. Since interface waves are a property of both media, the denominators of the scattered fields will be independent of the type or direction of propagation of the incident wave. In fact, the denominator is the same if the role of both media is interchanged. The situation in which one of the media is a nonviscous fluid (water) is of particular importance and has applications in several fields, one of which is the nondestructive evaluation of materials which will be discussed in later chapters. In the meantime, we proceed to discuss the characteristics of fluid-solid interface waves, namely, the Scholte waves. Guided by the above discussion, the Scholte wave characteristic equation is defined by the zeros of Af in equation (6.29), i.e., Af = A v + Y -- 0. As in the case of Av, Af is dependent upon the material properties, the azimuthal angle r and the speed c. For given material combinations and azimuthal angle, the zero of A f will then define the Scholte wave speed c. Since the Scholte wave decays exponentially away from the interface, a f is pure negative imaginary and, as before, al,C~3, and a5 of the solid must be pure positive imaginary. With reference to the definition of a f in equation (5.37), we conclude that Scholte wave requires c < c I. We note that the Scholte wave speed is always smaller than any of the involved wave speeds in total system. Figure 7.5 shows typical plots of the real parts of the displacement components u3 which clearly demonstrate the exponential decay behavior. The curves are normalized with respect to the common interface value. They are intended to show the qualitative behavior of the field variables of the Scholte wave within the system. A perhaps better display of interface wave behavior is shown in the form of two-dimensional spatial flooded displacement

CHAPTER 7. INTERFACE WAVES

100

U3

U3 i

X3

x3 -1

_21.0

-0.5

0.0

015

(a

1.0

0.0

(b0.1

Figure 7.5" Scholte wave displacements Re(u3) in (a) InAs cubic and (b) graphite-epoxy; both generated at r - 0 ~

0.20 0.15 0.10 0.05

0.00

-0.05 o0.10

-0.20

Figure 7.6: Snap shots demonstrating surface and Scholte wave amplitudes in water-graphite-epoxy semi-spaces sharing the interface x3 - 0.

7.3. SCHOLTE WAVES

101

1.50

1.45

"6" 0

~1.40 o 1.35

1.30 0

9 . . . .

I

10

.

.

.

.

.

.

.

20

.

I

30

. . . .

I

40

. . . .

(deg)

I

50

,

,

,

,

I

60

i

,

,

i

I

70

,

,

,

,

I

80

. . . .

I

90

Figure 7.7: Variation of the Scholte wave speed with azimuthal angle r The upper curve belongs to the InAs cubic material and the lower one belongs to graphite-epoxy. contours in figure 7.6. It is constructed for an InAs cubic material-water combination. The left hand side corresponds to the excitation of a surface mode in the solid while the right hand side corresponds to the excitation of a Scholte wave. In both figures, the water is at the bottom. The horizontal axis is xl and the vertical is x3. Both are constructed at the azimuthal angle r = 30 ~ Note the propagation in the water-decaying in the solid behavior of the surface wave situation in contrast to the decaying behavior for both media in the case of the Scholte wave excitation. Typical curves illustrating the variation of the Scholte wave speed c, with azimuthal angle r are depicted in figure 7.7. The upper curve belongs to a fluid-InAs cubic material whereas the lower one belongs to fluid-graphite-epoxy combination. Note the highly insensitive behavior of this material, its speed is practically constant at about 1.477 km/s which is strictly lower than 1.48 kin~s, the wave speed in the water.

This Page Intentionally Left Blank

Chapter 8

FREE WAVE IN PLATES Up until now, we have presented analyses for waves propagating in infinite homogeneous media and in infinite media possessing a single interface. We have shown that the presence of an interface gives rise to scattering (excitation) of other wave components necessary to satisfy appropriate interface conditions. This leads to the adoption of the formal solution type (5.1) which was chosen a priori to satisfy Snell's law. A detailed description of the formal solution for each homogeneous medium is found in chapter 5, where it is presented in terms of six unknown wave amplitudes. The task of solving for these unknowns is the central theme of wave propagation in solid systems. As was discussed in chapter 4, this depends upon the loading conditions, the properties of the media and on the physical conditions at the interfaces. Details of solutions for systems consisting of two semi-spaces sharing a flat connecting interface were discussed in chapters 4-7. In the remainder of this book, we investigate situations involving more than one interface, namely, the case of multilayered media. In order to do this, we use a building block approach in which we start with the relatively simple cases and then proceed to treat the more complicated ones. This particular chapter is then reserved for the study of free waves in homogeneous plates having finite but uniform thicknesses. Three distinct situations will be covered; these include: a plate in vacuum, i.e., having both faces stress free, a plate completely immersed in fluid and finally a plate in contact with a fluid semi-space on one side while the other side is free. In the general situation in which the plate consists of a triclinic material, complete coupling of the quasi-longitudinal and the quasi-shear wave components are expected. Complete coupling will also exist for all material symmetries (except in the case of isotropy) when propagation takes place in directions other than those of material symmetry. We recall that material symmetry 103

104

C H A P T E R 8. F R E E WAVE IN P L A T E S

-ci/2 fL~..._~_ /

i

s 9

/ I

i

I 7

d/2 '~-

I

I I I I

X2

Figure 8.1" A model of anisotropic plate showing the coordinate axes, thickhess and a typical orientation. axes are possible only in orthotropic and higher than orthotropic material symmetries. In situations where uncoupling occurs, two classes of motion are identified; the first (and simpler one) is that of a horizontally polarized S H type while the second is that of a sagittal plane motion which constitutes coupling between the quasi-longitudinal and the vertically polarized quasishear S V wave components. Drawing from isotropic nomenclature, the last coupled motion is designed as Lamb motion named in honor of Lamb who analyzed it in 1916. We note that Lamb considered propagation in isotropic media and thus did not encounter complications arising from anisotropy. We first develop in section 8.1 the representation for the propagation of free waves in a general anisotropic plate. Our starting point is a formal analysis for waves in a plate belonging to the triclinic material group. In section 8.2, the analysis is then carried out for a monoclinic plate, where the surface of the plate is parallel to the single plane of mirror symmetry in this material system. We derive the secular equation in closed form and isolate the mathematical conditions for symmetric and antisymmetric wave mode propagations separately. Generally speaking, results for material systems of higher symmetry, such as orthotropic, transversely isotropic, cubic, and isotropic are contained implicitly in those belonging to the monoclinic case. However, for orthotropic and higher symmetry materials where the remaining two principal axes lie in the plane of the plate, the particle motions for

8.1. FREE WAVES IN TRICLINIC PLATES

105

Lamb and SH modes uncouple under the condition that propagation occurs along either of these in-plane axes. Results for propagation along an axis of symmetry will be discussed separately in section 8.3. Section 8.4 is reserved for numerical computation strategies.

F r e e w a v e s in t r i c l i n i c p l a t e s

8.1

Consider an infinite, generally anisotropic plate, having thickness d, whose normal is aligned with the x~-axis of a reference Cartesian coordinate system x Ii as illustrated in figure 8.1. The midplane of the plate is chosen to coincide with the x~ - x ~ plane. With respect to this primed coordinate system, the equations of motion of the plate are given in equations (2.1)(2.4). As was pointed out in chapter 3, the response of the plate to such a wave is independent of the in-plane coordinate transverse to the propagation direction. This enables us to conduct analyses with relative ease by using a transformed coordinate system xi formed by a rotation of the orthogonal reference axes x i about the x~-direction through the azimuthal angle r This simple coordinate transformation leads to the transformed field equations (2.31)-(2.33). In the rotated system, the response of the plate for a plane wave propagating along xl is thus independent of x2. Nevertheless, the particle motion can generally have the three nonzero spatial components It 1 ~U2~ and U3. Formal solutions for the general triclinic case are given by equations (5.7) and (5.8) of section 5.1. Starting with these formal solutions, we first specialize the stresses a33, (~13 and a23 of equation (5.8) to the upper and lower faces x3 = -d/2 and x3 = d/2 of the plate. By setting these stresses equal to zero to invoke the stress-free boundary conditions, we obtain six equations relating the propagation partial amplitudes Ull, U12," ",/-/16. For nontrivial solution, we obtain the characteristic equation DllE1 D12E2 D13E3 D21E1 D22E2 D23E3 D31E1 D32E2 D33E3 DllE1 D12/~2 D13/~3 D21E1 D22/~2 D23/~3 D31/~1 D32/~2 D33/~3

D14E4 D24E4 D34E4 D14E4 D24/~4 D34/~4

D15E5 D25E5 D35E5 D15/~5 D25/~5 D35/~5

D16E6 D26E6 Da6E6 =o D16/~6 D26/~6 D36/~6

(s.~)

where Eq = E ~ 1 - e-i~,~d/2,

q = 1, 2, ..., 6.

(8.2)

106

CHAPTER 8. FREE WAVE IN PLATES

8.2

Free

8.2.1

waves

The dry

in monoclinic

plates

case

Thus far we have derived the characteristic equation for the propagation of free waves for a generally anisotropic plate; that is, one belonging to the triclinic material group. We see that it is applicable to propagation along any azimuthal direction in the X l - x2 or equivalently the x ~ - x~ plane. Although the characteristic equation (8.1) determines all possible propagation wavenumbers (or equivalently phase velocities), by virtue of its complexity, its utility is rather limited. For the slightly symmetric case of monoclinic material, where Xl - x2 is its single plane of symmetry, this characteristic equation collapses into a comparatively simpler set of expressions leading to simple interpretation. Results will still be valid for propagation along any arbitrary azimuthal direction in the X l - x 2 plane, subject to the restrictions presented in section 5.3. As a consequence, for its present importance and immediate applicability to other situations which are to follow, we now describe the manipulation procedure in some detail for a monoclinic plate. Implementing the restrictions resulting from equations (5.13)-(5.16) into the general result (8.1) results in DllE1 O21E1 D31E1 Dll.E1 D21E1 D31/~l

Dla/~l -D21/~l -D31E1 DllE1 -D21E1 -D31E1

DI3E3 D23E3 D33E3 D13/~3 D23E3 D33/~3

D13/~3 -D23E3 -D33/~3 D13E3 -D23E3 -D33E3

D15E5 D25E5 D35E5 D15E5 D25E5 D35/~5

D15/~5 -D25/~5 -D35/~5 = 0. D15E5 -D25E5 -D35E5

(8.3)

This leads the way for algebraic manipulations and reductions of this determinant characteristic equation. Listed below are the various steps taken in their logical order. For the first step, we add the second column to the first, the fourth to the third and the sixth to the fifth. This is followed by adding the original first column to the negative of the second, the third to the negative of the fourth and the fifth to the negative of the sixth, giving

8.2. FREE WAVES IN MONOCLINIC PLATES

107

DllC1 iDll S1 D 1 3 C 3 iD13S3 D 1 5 C 5 iD15S5 iD21S1 D 2 1 C 1 iD23S3 D 2 3 C 3 iD25S5 D25C5 iD31 S1 O 3 1 C 1 iD33$3 D 3 3 C 3 iD35$5 D35C5 =0 DIIC1 - i D l I S1 D 1 3 C 3 -iD13S3 D 1 5 C 5 -iD15S5 -iD21S1 D21C1 -iD23S3 D 2 3 C 3 -iD25S5 D25C5 -iD3i S1 D31C1 -iD33S3 D 3 3 C 3 -iD35S5 D35C5 (8.4) where we used the trigonometric identities

Eq + JEq = 2Ca = 2 cos(~c%d/2) Eq - Eq = 2iSq = 2isin(~c%d/2). ^

(8.5)

In the second step, we change the sign in the fourth row followed by adding the first, second and third rows to the fourth, fifth and sixth rows, respectively to obtain

DllC1

iDllS1 D 1 3 C 3 D21C1 iD23S3 iD31S1 D31C1 iD33S3 0 2iDl1S1 0 0 2D21C1 0 0 2D31C1 0 iD21S1

iD13S3 D 1 5 C 5 iD15S5 D23C3 iD25S5 D25C5 D33C3 iD35S5 D35C5 =0. 2iD13S3 0 2iD15S5 2D23C3 0 2D25C5 2D33C3 0 2D35C5

(8.6)

In a similar fashion, in the third step we divide the fourth, fifth and sixth rows by two and subsequently add their negatives, to rows one, two and three, respectively, to arrive at D11C1 0 D13C3 0 D15C5 0 0 D21S1 0 D23S3 0 D25S5 D31S1 0 D33S3 0 D35S5 0 = o. 0 D15S5 0 DllS1 0 D13S3 0 D25C5 0 D21 C1 0 D23 C3 0 D35C5 0 D31C1 0 D33C3

(8.7)

where we also isolated the nonzero factor i. In the fourth step, we divide the first, third and fifth columns by the common factors $1, $3, and $5 respectively, and the second, fourth and sixth columns by factors C1, C3 and C5, respectively. If this is followed by interchanging the various locations of

108

C H A P T E R 8. F R E E WAVE IN P L A T E S

the columns, then finally we rearrange the determinant matrix as DIIT1 D13T3 D15T5 0 0 0 D21 D23 D25 0 0 0 D31 D33 D35 0 0 0 0 0 0 D11T1 D13T3 D15T5 0 0 0 D21 D23 D25 0 0 0 D31 D33 D35

= 0

(s.s)

where Tq - tan(~aqd/2),

(8.9)

Tq = cot(~aqd/2).

It can be parted, leading to uncoupled characteristic equations S :- DllG1 cot(Tal) - D13G3 cot('7ol3) + DI5G5 cot(Ta5) = 0

(8.10)

A = DllG1 tan(val) - D13G3 tan(Ta3) + D15G5 tan(Ta5) = 0

(8.11)

corresponding to symmetric and anti symmetric modes, respectively. Here G1, G3 and G5 are, once again, as encountered in the semi-space case and are given in equations (6.13)-(6.15) and (8.12)

y = ~d/2 = wd/2c.

In terms of the frequency f = w/27r, "y = ~ f d / c . 8.2.2

M o n o c l i n i c p l a t e s i m m e r s e d in fluids

We next consider the case where the monoclinic plate is completely immersed in a nonviscous fluid, such as water. As before, we call upon the formal solutions (5.7) and (5.8) applicable for the solid and upon (5.39) for the fluid. Subject to the conditions (4.2) and (4.3), we get -W3/~3 W5E5 D13E3 D15E5 -D23/~3 D25E5 -D33E3 D35E5

W1E1

- Wl JE1

W3E3

DllE1 D21E1 D31E1 WIE1 Dll/~1 D21E1 D31/~1

D11E1 -D21E1 -D31E1

D13E3 D23E3 D33E3

-WI E1

W3E3 -W3E3

WsE5

D11E1 -D21E1

D13/~3 DI3E3 D23/~3 -D23E3 D33/~3 -D33E3

D15/~5 D25/~5 D35/~5

-D31E1

~f 0 D15E5-p$c 2 0 -D25-/~5 0 0 -D35/~5 0 0 =0 -WsE5 0 -aS D15E5 O-pfc 2 -D25E5 0 0 -D35E5 0 0

-ws/~

(8.13)

109

8.2. FREE WAVES IN MONOCLINIC P L A T E S

Following step by step the above outlined procedure for the free case, we obtain the modified version of the characteristic equations (8.10) and (8.11)

as (S+iY) = 0 (A - i r )

= 0

(8.14)

in which Y -~ pfc2 (WIG1 - W3G3 --[--W5G5).

(8.15)

af

Note that the presence of the fluid is totally reflected in the multiplicative factor pfc2/o~f of equation (8.15). By setting pf = O, we recover the free case results (8.10) and (8.11). 8.2.3

Fluid-monoclinic plate-vacuum system

Next we consider the situation where the plate is supported on one side by a fluid semi-space while its other side is kept free. Upon satisfying the appropriate stress-free boundary and the fluid-solid interface conditions, we obtain the following determinant characteristic equation W1E1

-WIEI

D11E1 DllE1 D21E1 -D21E1 D31E1 -D31/~1 Dll/~1 DllE1 D21/~1 -D21E1 D31/~1 -D31E1

W3E3 D13E3

-W3/~3 W5E5 D13Ea D15E5

-W5/~5

c~f

D15E5 - p f c 2 D23E3 -D23E3 D25E5 -D25/~5 0

D33E3 -D33/~3 D35E5 -D35/~5 D13/~3 D13E3 D15/~5 D15E5 D23/~3 -D23E3 D25/~5 -D25E5 D33/~3 -D33E3 D35/~5 -D35E5

0 0 0 0

=0

(8.16) This determinant is manipulated in accordance with the previous procedure resulting in A S + i Y (A - S)/2 = O.

(8.17)

which can not obviously be factored out to isolate symmetric or anti symmetric modes.

110

8.3

C H A P T E R 8. F R E E WAVE IN PLATES

Higher s y m m e t r y material plates

Results obtained for the monoclinic case can, under restricted conditions, also hold for higher symmetry classes such as orthotropic, transversely isotropic, and cubic. Since, as we established on several occasions, these classes of materials differ from monoclinic materials in that each can possess two orthogonal principal axes in the plane of the plate, the monoclinic results apply if the wave propagates along directions other than these axes. This happens in virtue of the fact that the coupling of the S H wave field equations with those of the classical Lamb wave persists. Uncoupling of these equations occurs, on the other hand, for propagation along the principal axes, resulting in simplified results. In this case, we also recognize that results for all symmetry classes higher than orthotropic are contained as special cases of the corresponding solutions for the orthotropic case. For these reasons, we concentrate on developing the results for orthotropic symmetry. Propagation a!ong-off-principal-axes For an off-principal-axis propagation, we need only to assure that further appropriate reductions in the number of nonzero elastic constants are exploited in the solution sets (8.10), (8.11), (8.14) and (8.17). Since x~ and x~ are chosen to coincide with the in-plane principal axes for orthotropic symmetry then, the further restrictions (2.25) specialize them to orthotropic symmetry situation. Results for higher than orthotropic material symmetries are obtained by employing their individual constitutive relations from equations (2.26)-(2.28). P r o p a g a t i o n a l o n g axes of rotational s y m m e t r y Returning to the case of orthotropic symmetry, we note that the axis x~ and x~ coincide with the azimuthal angles r = 0 ~ and r - 90 ~ respectively. For propagation along either of these symmetry axes, we need to employ the formal solutions of section 5.4.2, where we identified the two uncoupled motions. We start with the pure S H motion whose formal displacement and stress solutions are given by equations (5.21) and (5.22). The vanishing of the stress component a~3 at the upper and lower surfaces of the plate leads to the two equations D21 ei~1 d/2u21 ~- D22e i~a2d/2U22 -- 0

D21e-i~ald/2u21 -+-D22e-i~a2d/2u22 - O.

(8.18)

8.3. HIGHER S Y M M E T R Y M A T E R I A L P L A T E S

111

Using the relations (5.20) and (5.23), namely, a2 = - a l and D22 = -D12, these two equations can be combined to yield the single characteristic equation sin(2yal) = 0

(8.19)

with -7 as defined in equation (8.12). We next consider the motion confined to the sagittal plane with one of its axes coincident with one of the axes of symmetry. For such a motion, the formal solutions are those given by equations (5.26) and (5.27). Specializing the stresses (5.27) to the upper and lower surfaces of the plate and subsequently setting them equal to zero lead to a 4 x 4 determinant, that is a counterpart to the 6 x 6 one of equation (8.3) pertaining to the monoclinic case. Manipulating this determinant in a manner similar to that of section 8.2 leads to the two uncoupled characteristic equations S -- DllD23 cot('yal) - D13D21 cot(')'a3) - 0 A = D11D23 tan('yal) - D13D21 tan('),a3) = 0.

(8.20)

Here, the various Dij elements are as defined in equation (5.28), and a l , a3 are those described by equations (5.29) and (5.30). For the totally immersed plate in fluid, we have

(S + i Y ) ( A - iY) = 0

(8.21)

where Y now takes the reduced form

Y = pfc2 (W1D23 - W3D21). af

(8.22)

as compared with that encountered in equation (8.15) for the monoclinic case. Finally, for a plate supported by a fluid on one side and the other side is kept free, we recover the form of equation (8.17), consistent with the results (8.20)-(8.22), namely

A S + iY (A - S)/2 = 0.

(8.23)

The expressions (8.20) constitute the characteristic equations for symmetric and anti symmetric modes for Lamb waves propagating along an in-plane axis of symmetry of an orthotropic plate. Equation (8.19) is the characteristic equation of a horizontally polarized S H wave on the same plate. Furthermore, these relations contain implicitly results which correspond directly to materials possessing higher than orthotropic symmetry. In this development, we need only to appropriately exploit the material restrictions on the elastic properties as described in chapter 2.

CHAPTER 8. FREE WAVE IN PLATES

112

8.4

Numerical computation strategy

The various characteristic equations derived so far in this chapter will be of limited utility unless we understand their fundamental behavior, and also know how to extract relevant information from them. In the subsequent discussion, we shall treat them in the ascending order were we start with equations (8.10) and (8.11), namely the characteristic equations for a free monoclinic plate. Better understanding will result by adopting a "flow chart" like discussion.

Dry plates Once the plate material is specified, say unidirectional graphite-epoxy, we identify its properties (from the Appendix). Next, by specifying the azimuthal angle r say 45 ~ these properties are transformed, as is also given in the Appendix. We immediately recognize a monoclinic form and hence equations (8.10) and (8.11) are the appropriate characteristic equations. We then proceed to calculate the various a~s and the displacement and stress amplitude ratios Vq, Wq and Diq. These will depend upon the phase velocity c. By subsequent use of these parameters in the characteristic equations (8.10) and (8.11), we encounter the second independent variable ~. But ~/, by its definition ufd/c, directly depends upon the frequency f and inversely upon the phase velocity c. Hence both equations have the two independent variables c and f. Combinations of these two variables which satisfy the characteristic equations define Lamb wave dispersion relations. Two representative plots of these dispersion curves are shown in figures 8.2a and 8.2b for graphite-epoxy with the azimuthal angles r = 0 ~ and r = 45 ~ respectively. In figure 8.2a, the broken curves belong to the symmetric modes satisfying S - 0 The solid curves belong to the antisymmetric modes which satisfy A = 0. The lowest symmetric mode of figure 8.2a would eventually converge to the value c - 9.8 k m / s if the calculations were carried out for higher values of c (see figure 8.4b). In figure 8.2b, symmetric curves are also shown as broken lines and the antisymmetric as solid ones. Results are depicted in the form of variation of c with f d, rather than with f. It is often customary to describe the dispersion relations in terms of variations of the wavenumber ~ = w/c with ~v. After solving for ~, we calculate c from c = w/~. In the above example, either way is acceptable since here both w and ~ (also c) are real.

8.4. NUMERICAL COMPUTATION STRATEGY

113

8

6

~ 4 v

o

o

2 2

4

fd (MHz ram)

6

8

,;

8

fd (MHz mm)

Figure 8.2: Free modes for propagating along the azimuthal angle (a) r = 0 ~ and (b) r = 45 ~ of a unidirectional graphite-epoxy plate.

Wet plates Moving to the case of a plate totally immersed in fluid, we recognize the modification to the dry case results through the presence of Y. We note that, except for the multiplicative factor pfc2/af, all parameters involved in the definition of Y belong to the solid. Furthermore, like other parameters involved in the definitions of S and A, these depend upon the variable c. Here, however, for given real values of fd, equation (8.14) can only be satisfied if c becomes complex. Since c has been designated loosely as a phase velocity, this choice of designation is not appropriate except when it is real. The actual situation is that ~, rather than the phase velocity, is complex. For this reason we now think of c as a complex parameter from which we calculate the complex wave number ~ = w/c. Once ~ is calculated, we identify the phase velocity as w/Re(~) leaving the imaginary part of ~, namely, Im(~) to define attenuation. In fact Irn(~) is the foundation of the leaky wave phenomena which shall be extensively studied in chapter 11. Note that, in the absence of the fluid, namely for pf - 0 we recover the results of the free case. R o l e of fluid Of particular importance is the quantitative influence of the fluid on the phase velocity and attenuation characteristics of the Lamb wave. It has

C H A P T E R 8. FREE WAVE IN P L A T E S

114

6.0

...............

~ 5 . 0

-

~. . . . . . . . . . . . .

~

~ .................

.

:-

!

So

:. . . . . . . . . . . . . . . . . .

::. . . . . .

i

,

10.0

.................

::

:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

so

imitl o.o

,.o

,:o

fd

;o

(a)

o.oo.o

1.o

,

,.oi fd

,

3:0

(b)

4:0

Figure 8.3: Modes in solid and fluid-solid for: (a) Steel and (b) graphite epoxy.

long been recognized that the influence of fluid on the phase velocity c is negligible. This is indeed the case involving metal plates. For this reason, in figure 8.3a dispersion curves belonging to the two fundamental symmetric and anti symmetric modes So and A0 for a steel plate are compared with and without the presence of fluid. Also shown is the true (nonleaky and nonattenuative) mode obtained from satisfying the anti symmetric mode A - 0 and converges for large f d to the Scholte limit discussed in the previous chapter. As can be seen, very little influence of the fluid is apparent. The generality of this common belief has been recently disputed when we [172] and others [51] have shown analytically and experimentally dramatic changes in the dispersion curves of composites such as graphiteepoxy due to the presence of fluid, especially for the lowest order symmetric modes. In the meantime, we shall defer discussion of this phenomenon to chapter 11 where the role of the loading fluid will be extensively studied. Nevertheless, a typical example is shown in figure 8.3b for a graphite-epoxy plate. The solid lines belong to the dry case and the broken ones to the wet plate. Notice the dramatic departure of the So mode in the presence of fluid loading. As shall be shown in chapter 11, the degree of departure in the behavior is attributed to the relative ratio of the plate density to that of the fluid; the smaller the density of the plate, the more pronounced the influence of the fluid. Again the true anti symmetric mode which converges

8.4. NUMERICAL COMPUTATION STRATEGY

115

to the Scholte limit is also included. Plates

wet

on

one

side

In the case of a plate in contact with fluid on one side and free on the other, the presence of fluid, has very little influence on its dispersion behavior. This again will be discussed further in chapter 11. However, this situation, strictly speaking, destroys the symmetry of the loading and as a consequence, we cannot isolate the symmetric from the anti symmetric modes as in the case of the totally immersed plate; i.e., equation (8.17) cannot be factored out. Finally, situations involving propagation along axes of symmetry, can, strictly speaking, be solved by using the appropriately reduced expressions of S,A and Y in equations (8.20) and (8.21). Alternatively, equivalent numerical solutions to such situations can also be obtained from the corresponding general cases outlined above. These are obtained as limiting, but very accurate values, however.

This Page Intentionally Left Blank

Chapter 9

GENERAL MEDIA

LAYERED

In this chapter, we utilize combinations of the linear transformation and the transfer matrix methods and extend the results of chapter 8 to the study of the interaction of harmonic waves with multilayered anisotropic media. Among the many factors that influence this interaction are the number of layers, their individual degree of anisotropy, relative orientations, interfacial conditions and the system's outer boundary conditions. We recall from chapter 1 that a layered medium consists of two or more material components attached at their interfaces in some fashion. A body made up of an arbitrary number of different material components and whose outer boundaries are either free or supported by semi-infinite media constitutes a general layered system. Here, we treat the class of layered anisotropic media that consists of an arbitrary number of layers, each of which possesses as low as monoclinic symmetry, that are arbitrarily oriented with respect to each other and rigidly bonded at their interfaces. A finite thickness layered plate then defines a unit cell and a repetition of the cell defines a periodic medium. Furthermore, the wave is allowed to propagate along an arbitrary angle from the normal to the system as well as along any azimuthal angle. By allowing each layer to possess monoclinic symmetry, results for higher-symmetry materials such as orthotropic, transversely isotropic, cubic and isotropic can also be obtained as special cases. These will be treated separately in chapter 10. Solutions are obtained by using the transfer matrix method. According to this method, the formal solutions (5.7) and (5.8) for each layer are derived and expressed in terms of wave amplitudes. Eliminating these amplitudes relates the displacements and stresses on one side of the layer to those on 117

118

C H A P T E R 9. G E N E R A L L A Y E R E D M E D I A

the other. By subsequently satisfying appropriate continuity conditions at interlayer interfaces, we construct a global transfer matrix which relates the displacements and stresses on one side of the layered plate to those on the other. By invoking appropriate boundary conditions on the plate's outer boundaries, we can solve a large variety of problems. In section 9.1, we describe the geometry of the system. Section 9.2 is devoted to the derivation of the "local" transfer matrix for the individual layers and also of the "global" transfer matrix of the system. In section 9.3, we describe the various properties of the transfer matrices. We then proceed in sections 9.49.6 to discuss results pertaining to free waves in the unit cell, in the periodic media and in the layered plate supported by a solid substrate, respectively.

9.1

Geometric description of unit cell

Consider a plate consisting of an arbitrary number n of monoclinic layers rigidly bonded at their interfaces and stacked normal to the x3-axis of a global orthogonal Cartesian system xi = ( X l , X 2 , X 3 ) . Hence the plane of each layer is parallel to the Xl-X2 plane which is also chosen to coincide with the upper surface of the layered plate. To maintain generality, we assume each layer to be arbitrarily oriented in the x l-x2 plane. In order to be able to describe the relative orientation of the layers, we assign for each layer k , k = 1 , 2 , . . . ,n, a local Cartesian coordinate (X~)k such that its origin is located in the upper plane of the layer with (x~3)k normal to it. Thus layer k occupies the region 0 < (x~3)k < d (k) where d (k) is its thickness. According to this notation, the total thickness of the layered plate d equals to the sum of the thicknesses of its individual layers and hence, the plate occupies the region 0 <_ x3 _< d. Equivalently, the orientation of the layer k in the xi-space can be described by a rotation of an angle Ck between (X~l)k and x l. Hence, once all orientation angles Ck are specified, the geometry of the plate will be uniquely defined. Repetition of this layered plate results in the periodic medium illustrated in figure 9.1.

9.2

Analysis

With respect to the primed coordinate system (X~)k, the elastic field equations of layer k are given by the momentum equation (2.1) and the constitutive relations for monoclinic media (2.24). In terms of the global coordinate system xi, the primed equations of motion for each layer are transformed through its orientation angle Ck leading to equations (2.31)-(2.35) or to their expanded forms (2.42)-(2.44). In this section~ we follow and extend

9.2.

ANALYSIS

119

',~.~..: ~xx

..../.....

Figure 9.1: Layered plate geometric model. the analytical procedure of chapter 5 in order to derive formal solutions for each layer. These will then be used to construct the local transfer matrix for each layer k. We propose an alternative formal solution to that of (5.1) in the form (ul, u2, ua) - (U1, U2, Ua) e i;(~ sinO+axa-vt)

(9.1)

where, as defined in chapter 3, ~ is the wavenumber, v is the phase velocity (= w/~), w is the circular frequency, c~ is still an unknown parameter, and Ui are the displacement amplitudes. The form of the solution (9.1) is different from its counterpart (5.1) due to the convenient inclusion of sinO in its exponential. This specific form is general and will be shown to include results belonging to a variety of layered systems. An important feature of the analysis concerns the manner in which the oblique propagation direction is introduced and the way it modifies the criterion necessary to insure periodicity. If we designate the angle 0 (measured from the normal to the interfaces) to define the propagation direction, then this will lead to an explicit dependence of the characteristic equations upon 0. We note that, although solutions (9.1) are explicitly independent of x2, an implicit dependence is contained in the transformation properties. Furthermore, we also note the nonvanishing of the transverse displacement component u2 in equation (9.1).

C H A P T E R 9. G E N E R A L L A Y E R E D MEDIA

120

Substitution from equation (9.1) into the displacement equations (2.42)(2.44) leads to the three coupled equations

gpq(Ol) Vq -- O,

p, q = 1, 2, 3,

(9.2)

where the summation convention on p and q is implied and Kpq is symmetric with K l l = Cll sin 2 0 - pv 2 + C55o~2

K12 = C16 sin 2 0 + C45c~2 K13 = (C13 + C55) c~sin0 K22 = C66 sin 2 0 - pv 2 + C44c~2 K23 = (C36 + C45) c~sin 0 K33 = C55 sin 2 0 - pv 2 + C33c~2.

(9.3)

Again, these expressions for the various Kij elements constitute a modification of the corresponding elements (5.10) through the presence of sin 0. The following algebraic steps also parallel, to a great extent, those of chapter 5. Nontrivial solutions for U1, U2 and U3 demand the vanishing of the determinant in equation (9.2) and yield the sixth-degree polynomial equation ct6 -+- Alc~4 + A2ct2 + A3 = 0

(9.4)

relating ct to v, where the coefficients A1, A2 and A3 are given as A1 -- [(C11C33C44 - C123C44 -t- 2C13C36C45 - 2C13C44C55 2 + 2C13C425 - 2C16C33C45 + 633655666 - C36C55 ) sin 2 0 -

+

+ c44c

-

A2 -- [ ( C 1 1 C 3 3 C 6 6 - C11C26 - 2Cl1C36C45 nt- Cl1C44C55

-CIIC25-k C123C66-k 2C13C16C36"Jr"2C13C16C45 - 2C 3c sc66 - c 6c33 + 2c 6c36c s) sin 4 0 (Cll C33 + Cll C44 - C123 - 2C13C55 - 2C16C45 + 633666 -- 626 -- 2636 645 + 644 655 -- 625 nt- 655666) pV 2 sin 2 0 + (633 + C44 ~- 655) pV 4]

/zx

A3 = [ ( C l l C 5 5 C 6 6 - C126C55)sin60 - (CllC55 -k Cll 666 - 6126 nt- 655 666) PV2 sin4 0

+ (Cll + C55 + C66) pv 4 sin 2 0 - pv 6 ] / A

(9.5)

121

9.2. A N A L Y S I S

with (9.6)

A = C33C44C55 - C33C~25 9

Equation (9.4) admits six solutions for a (three for a 2) which we label as a2 = - a l ,

a4 = - a 3 ,

a6 = - a 5 .

(9.7)

For each aq, q - 1 , 2 , . . . , 6 , the relations (9.2) express the displacement component ratios Vq = U2q/Ulq and Wq - U3q/Ulq in a manner similar to those of equations (5.5) and (5.6), namely

Kll (aq)K23(Olq)

-

-

(9.8)

K13(O~q)K12(O~q)

Vq = K13(aq)K22(aq) - K12(aq)K23(aq)

K11(Olq)K23(Ctq) - K12(Olq)K13 (Ctq) Wq - K12(Ctq)K33 (Ctq) _ K23 (Ctq)K13 (Ctq) "

(9.9)

Combining equations (9.8) and (9.9) with the stress-strain relations (2.24), and using superposition, we write the formal solutions for the displacement and stress components as 6 (Ul, u2, u3) = ~ ( 1 , Vq, Wq)Ulqe ir (9.10) q=l 6 ((7~3' crY3'crY3) = E (Dlq, D2q, D3q)Ulqe i~aqx3 (9.11) q=l where we note and suppress the common factor e i((xl sin O-vt). Here, for the convenience of Subsequent analyses, we introduced the change of variable % =

Utilizing the various relations (5.15) and (5.16), equations (9.10) and (9.11) can now be combined and written in their expanded matrix form Ul

u2 ?23 cry3 a~3 cry3 where Eq

=

__

1 V1 W1 Dll D21 D31

1 V1 -W1 Dll -D21 -D31

1 1 V3 V3 W3 -W3 D13 D13 D23 -D23 D33 -D33

1 1 V5 V5 W5 -W5 D15 D15 D25 -D25 D35 -D35

UI2EI

U12E2 U13E3 (9.12) U14E4 " U15E5 U16E6

e i~O~qx3

Dlq -- C13 sin 0 + C36Vq sin 0 + C330zqWq D2q - C55 (Ctq + Wq sin 0) + C45 CtqVq n3q = C45 (Ctq nt- Wq sin 0) -t- 644 OZqVq,

q - 1,2,... ,6.

(9.13)

CHAPTER 9. GENERAL LAYERED MEDIA

122 9.2.1

The local transfer matrix

Equations (9.12) and (9.13) hold for every layer k. They can be used to relate the displacements and stresses at (x~3)k -- 0 to those at (x~3)k -- d (k). To facilitate the discussion, we first adopt the formal solution (9.12) to each layer k and rewrite it in the compact form

Pk = XkDkUk

(9.14)

where Pk is the 6 x 1 displacement and stress matrix

Pk =

Ul u2 u3

a~ 3

(9.15)

a{a 0"~3

k

Xk is the 6x6 square matrix in equation (9.12), Dk is the 6x6 diagonal matrix whose entries are Eq, namely E1 Dk --

o

0 E2

0 0

0 0

0 0

0 0

0 0

0 0

E3 0

0 E4

0 0

0 0

0 0

0 0

0 0

0 0

E5 0

0 E6

(9.16) k

and Uk is the 6 • 1 displacement amplitude matrix U11

U12 Uk ":

(9.17)

U13 U14

U15 U16

k

We now specialize equation (9.14) to the upper and lower faces of layer k; this leads, respectively to

Pk = XkDkUk, P~ - XkD+Uk,

k-

1,2,...,n

(9.18)

9.2.

123

ANALYSIS

where D k and D + are obtained from the square matrix (9.16) when x~ is specialized to 0 and d (k), respectively. Noting that D k is the identity matrix I, it then follows, by eliminating the common amplitudes Uk, that k = 1,2,...,n,

P+ = A k P [ ,

(9.19)

where Ul u2 P~=

+

~t3 0.~3

(9.20)

0"~3 &~3

k

defines the field variables column specialized to the upper and lower surfaces of layer k, respectively, and Ak - X k D - ~ X ; 1.

(9.21)

The matrix Ak constitutes the local transfer matrix for layer k. 9.2.2

The global transfer matrix

By applying the above procedure for each layer, followed by invoking the continuity of the displacement and stress components at the layer interfaces namely, P/+I = Pk-, we finally relate the displacements and stresses at the upper surface of the layered plate, x3 = 0, to those at its lower surface, x3 = d, via the individual transfer matrix multiplication A = A n A n - 1 . . . A1.

(9.22)

This resells in P+ - A P -

(9.23)

where P - and P+ are now the displacement and stress column vectors specialized to the upper, x3 - 0, and lower, x3 = d, faces of the total plate, respectively and A is the global transfer matrix of the total plate.

CHAPTER 9. GENERAL LAYERED MEDIA

124

9.3

P r o p e r t i e s of the transfer m a t r i x

The global transfer matrix A has several properties which, if exploited, can ease the execution of the analysis and lead to simple analytical representation of the results. Before we proceed to list and discuss these properties, we indicate that such properties are also characteristic of the local transfer matrices Ak of the individual layers. In fact, since equation (9.21) holds for any number of layers n, then it holds for a single layer as well, and thus A can be represented by Ak for k - 1, 2 , . . . , n. Accordingly, we hypothesize that any general property of Ak is also a property of A. With this, we now concentrate on listing and discussing properties of the individual transfer matrix Ak. (a)

det Ak = 1.

(9.24)

This property can be easily proven by employing the well-known result that the determinant of the product is equal to the product of the determinants. From equation (9.21), we get det Ak = det Xk det X k I det D~= det XkX k I det D~= det D ~ - = 1

(9.25)

since D + is diagonal and its determinant is equal to the product of its elements which can be seen equal to unity. This conclusion can also be arrived at by noting that Ak and D + are similar and hence their determinants are equal (see reference [84]). In the subsequent analysis, and without confusion, we shall identify D + with Dk. (b) As a consequence of their similarity, Ak and Dk also have the same eigenvalues. This means that the six possible eigenvalues (say Aq, q 1 , . . . , 6 ) of Ak are given by the diagonal elements of Dk. By inspection, we see that these eigenvalues consist of three pairs with the elements of each pair being the inverse of each other. Thus, if A1, )k3 1 1 1 and A5 are eigenvalues of Ak, so are A2 = ~11' )~4 -- ~33 and )~6 -- ~---~(c) The results of (b), and the fact that the eigenvalues of Ak are the inverse of the corresponding eigenvalues of Ak 1, lead to the conclusion that Ak and Ak I have the same set of eigenvalues. As a consequence of items (a) - (c) and the definition (9.22), we conclude that:

9.3. P R O P E R T I E S OF THE T R A N S F E R M A T R I X (i)

125

det A = det An det A n - 1 . . . det A = 1.

(9.26)

(ii) The eigenvalues of A -1 are equal to the eigenvalues of A. To show this, let us assume that the eigenvalue of A is a. It follows then that the eigenvalue of A -1 is 7" 1 By substituting from equation (9.21) into equation (9.22) and carrying the inverse of A we get det ( X n D n X n l . . . X2D2X21X1D1X11 - a I ) = 0

det

(X1DllXllX2D21X21... XnDnlXn1-

1I)

(9.27)

- - O.

(9.28)

Now, using the fact that the products of the two equal rank square matrices M1M2 and M2M1 (although M1M2 ~- M2M1) have the same eigenvalues [84], by cyclic permutation, we can rewrite the relation (9.27) as det ( X 1 D I X l l X 2 D 2 X 2 1 .

. . X n D n X ; 1 - aI) -- 0

(9.29)

By inspecting equations (9.28) and (9.29), we conclude that each can be obtained from the other by merely inverting the diagonal matrices Dk and the eigenvalue a. Since the entries of Dk are made up of pairs which are inverse of each other, then it is obvious that Dk and D k 1 have the same entries (eigenvalues). Thus we conclude that the 1

eigenvalues aq, q = 1, 2 , . . . , 6 and m constitute the same set.

(iii) The property described under item (ii) can only imply that aq consists of three pairs, the entries of each pair are inverse of each other. In our subsequent analyses, we choose to arrange these six eigenvalues as al, 1 1 1 , ~73, m , r 5 and - - . (~1

gr3

gr5

(iv) In the degenerate case when all layers are made up of the same material (but not necessarily have equal thicknesses), Xk and the six values (C%)k are the same for all layers k. Substituting from equation (9.21) into equation (9.22) and recognizing that X~+Ixi -- I (identity) for i = 1, 2 , . . . n, the global matrix A collapses to

A-

X D X -1

(9.30)

126

C H A P T E R 9.

GENERAL LAYERED MEDIA

where we choose to represent each of the Xi, i - 1, 2 , . . . n with X and D = D n D n - I . . . D1

(9.31)

is a diagonal matrix whose entries are given by exp(i~o~qd), q = 1, 2 , . . . 6 spanning the space of all layers. Thus, we have shown that the global transfer matrix correctly reduces to the corresponding matrix of the single material plate when all layer properties are the same. (v) A very important consequence of the above listed properties is the resulting relations that exist between the invariants of A. To this end, if we expand the characteristic equation det(A - a I ) - 0, write it in terms of the invariants Iq of A, and compare the resulting expressions, we find the symmetric relations /5=I,,

/4=/2, / 6 = 1 .

(9.32)

The result/6 - 1 is consistent with the fact that det A = 1.

9.4

Free w a v e s on t h e layered cell

Equation (9.23) will now be used to present solutions for a variety of situations. In the first, we consider a single cell medium, namely a free nlayered plate. The characteristic equation for such a situation is obtained by setting 0 = 90 ~ and invoking the stress free upper and lower surfaces in equation (9.23), resulting in the characteristic equation A41 A51 A61

A42 A43 A52 A53 A62 A63

=0.

(9.33)

It is of interest to note that this result constitutes a generalization of the results (8.10) and (8.11) obtained for a single material plate. When the properties of all layers are the same, equation (9.33) is expected to reduce to equations (8.10) and (8.11). As an example numerical illustration of equation (9.33), figure 9.2 displays dispersion calculation for a ( - 6 0 ~ 0 ~ 60 ~ of graphite-epoxy composite layup of equal thickness layers totaling unity. The curves displayed on this figure are typical of free waves encountered in anisotropic plates. Results shown on this figure are plotted in the form of variation of phase velocity with the nondimensional wavenumber Q=r

9.5.

127

WAVES IN A PERIODIC MEDIUM

v >

3

S l

I

I

I

O Figure 9.2" Variation of phase velocity v with the nondimensional wavenumber Q for a (60 ~ 0 ~ - 6 0 ~ unidirectional graphite-epoxy layup free plate.

9.5

W a v e s in a p e r i o d i c m e d i u m

A second important situation is that of a periodic medium consisting of a repetition of the n-layered unit cell (plate). Here we generalize the classical Floquet periodicity condition to require (9.34)

P+ = P - e i;dc~

Combination of equations (9.34) and (9.23) yields the characteristic equation det ( A -

Ie i~dc~

= 0.

(9.35)

To pursue simplification of this equation, we first recognize that e Kdc~ is an eigenvalue of the matrix A, and, for the time being, we conveniently designate it with a. Since A is a 6 x 6 matrix, its characteristic equation can be formally written as a 6 -- I l a 5 + 12 a4 - 13 a3 + 14o 2 - I5a + 16 = 0

(9.36)

where I q , q - 1, 2 , . . . , 6 are the invariants of A. Implementing the various relationships (9.32) between these invariants further reduces equation (9.36) to 66 - I1 (a 1 + 65) + I2(a 2 + a 4) - I3a 3 + 1 = 0.

(9.37)

128

C H A P T E R 9. G E N E R A L L A Y E R E D M E D I A

Next we divide by

a 3

and rewrite equation (9.37) as

(a 3 + a -3) - 11(a 2 + a -2) + I2(a + a -1) - / 3 = 0. Finally, replacing a with its value e Kdc~ metric relations, an example of which e iCdc~

+ e -i~dc~

(9.38)

followed by using various trigono-

= 2 cos(~dcos 0)

(9.39)

casts the characteristic equation in the simple form

h

cos(3r cos 0) - I1 cos(2~d cos 0) + 12 cos(~d cos 0) - ~ = 0.

(9.40)

The characteristic equation (9.40) depends explicitly on the wavenumber ~, the total plate thickness d, the angle 0 and the invariants of the global matrix A. But the invariants depend upon the elements of A and hence, on the phase velocity, the various properties, orientations and thicknesses of the individual layers. Thus, once the properties and geometry of the system are specified, the characteristic equation relates the three variables consisting of the wavenumber ~, the phase velocity v and the propagation direction along 0. Due to the complicated dependence of the characteristic equation on the various parameters mentioned above, further analytical investigations will not be possible. For demonstration of the behavior of the characteristic equation, from now on, we revert to numerical illustrations. Two types of numerical illustrations are of particular interest. 9.5.1

Dispersion curves

In the first type, phase velocity v dispersion curves are plotted as functions of ~ for specified angles of incidence 0. Without any loss in generality, the thickness d of the representative unit cell is kept constant, and its constituents (layers) are assigned volume fractions adding to unity. Typical sample examples, which demonstrate the dependence of such curves on the number of layers and their orientations, are given for the representative angle 0 = 45 ~ in figures 9.3a,b. They correspond to (60 ~ 0 ~ -60 ~ and (0 ~ 15~ 30 ~ 45 ~ 60 ~ 75 ~ 90 ~ layups of periodic graphite-epoxy media configurations, respectively. Equal thicknesses are assigned to these layers. In these figures the phase velocity v is given in k m / s e c and the nondimensional wavenumber r = ~d. Close examination of these figures reveals several interesting features. At the zero wavenumber limit, each figure displays three values of wave speeds corresponding to one quasi-longitudinal and two quasi-shear waves. It is obvious that the largest value corresponds

9.5.

WAVES IN A PERIODIC M E D I U M

129

7

7

6

6

5

v >

3

>

2

10

2

2

4

Q

6

8

10

2

4

Q

6

8

Figure 9.3: Variation of phase velocity v with the nondimensional wavenumbet Q for a graphite-epoxy with the angle of incident 0 = 45 ~ for: (a) (60 ~, 0 ~, - 6 0 ~ and (b) (0 ~, 15 ~, 30 ~, 45 ~, 60 ~, 75 ~, 90 ~) layups. to the quasi-longitudinal mode. At relatively low values of the wavenumber, little change is seen to take place in these values. As Q increases, other higher order modes appear.

9.5.2

Dispersive slownesses

We demonstrate the variation of the phase velocity v with the angle of incidence 0 for specified frequencies f . This is done in order to demonstrate the dependence of wave front curves on frequency. For a varying frequency, "frequency-dependent wave front surfaces" can be obtained. Here different frequencies lead to different slowness surfaces; this is in contrast with the wave front surfaces for single homogeneous materials where the slowness surfaces are independent of the frequency. In figure 9.4 we depict, for the selected values f d = 0 and 4 M H z m m , wave front curves in the hi-n3 (i.e., c o s 0 - sin0) plane, using a (60 ~ 0 ~ -60 ~ graphite-epoxy layup periodic medium. These curves demonstrate wave front dispersion behavior. The complicated features shown in figure 9.4b are due to multivalued behavior shown in figure 9.3a especially at f d = 4 M H z . m m brought about by the presence of the higher order modes. Notice, in contrast, that the "clean" behavior displayed in figure 9.4a reflects the variations of effective wave speed v with the angle of incidence. The f d - 0 curves will thus constitute the wave fronts for an effective homogenized medium whose properties are volume fraction weighted properties of the individual layers. While the cases

130

CHAPTER

9.

GENERAL

LAYERED

-

'5

9o 9 ,.. 9 9 9 9 9

..-'" '"

2~

" 9

9 9

't ..-.".."--"~_

o OeOco~176/eoNooo.eeo

"'-..

"""

'"

I

L

5

-I0

,'o

: .

.

_~ eeee.eee

e

aboe

..: oee o~

e%oo

o

.....::. _- .::.::-.

9...

%

2''

t

,.9

e%

o

-

9

MEDIA

_

e.e o~

.

-'.......

.........:.~.~ eeee

-7

~ ~

"

%%.,q,.~,.~

....r

... ," .,

e.e~eeoeeo" ,e.oee~176

-8

Figure 9.4: Wave front curves for (60 ~ 0 ~ - 6 0 ~ graphite-epoxy layups for (a) f d - 0 M H z m m and (b) f d = 4 M H z m m . presented here are certainly typical, they are by no means exhaustive of the variety of the phenomenology contained in the analysis. 9.5.3

Specialization

to a single material

We now discuss the case in which all layers are the same. This is expected to result in a description of the behavior of single homogeneous anisotropic materials. As was discussed earlier, the global transfer matrix for such a situation collapses to the form given in equation (9.30). Using this matrix and noting that D and A are similar and hence have the same eigenvalues Eq, dictate that e i ~ d c ~ .~ e i~daq leading to aq2 = cos 2 ~9,

q = 1, 2 , . . . , 6.

(9.41)

This recasts the formal solution (9.1) into the form (3.1) appropriate for the single homogeneous medium; here, of course nl -- sin 8 and n3 = cos ~. With reference to equation (9.4) and for a fixed ~, the result (9.41) admits three roots for the phase velocity v corresponding to one quasilongitudinal and two quasi-shear motions. Thus, for a variable ~, equation (9.41) describes the variation of the three phase velocities with the incident angle and hence constitute wave front curves. For this specialized

9.6.

BOTTOM

BOUNDING

131

SOLID SUBSTRATE

single medium case, these curves will be independent of frequency, however. For an isotropic material, where the formal solutions are presented in section 5.4.3, equation (9.4) uncouples and gives ~2 = v2/v~ - sin 20;

a3,52 = v 2 / v 2 _ sin 20

(9.42)

where vl and vt are the longitudinal and shear wave speeds in the medium respectively. Thus, combination of equations (9.42) and (9.41) gives the roots v - vl and v - vt yielding two concentric spherical wave front curves as is expected. For the anisotropic case, however, the three solutions will be coupled resulting in nonspherical wave fronts.

9.6

B o t t o m b o u n d i n g solid s u b s t r a t e

If the layered plate is rigidly attached at its lower surface to a solid homogeneous substrate, the total system's free wave characteristic equation can be derived after constructing a formal solution for the substrate followed by satisfying the appropriate plate-substrate interface conditions. The substrate will then extend from x3 - d to c~. Here, we require continuity on all stress and displacement components. But the appropriate formal solution for the substrate was identified in chapter 5 in conjunction with the study of surface waves. It can also be specialized from equation (9.12) subject to the requirement that such solutions must satisfy the radiation condition, namely, that the field variables be bounded for large distances within the substrate. These requirements lead to the substrate formal solution adapted from the general solution (9.12) as ul u2

U3 a~3 a~3 a~3

1 V1

1

D31

-D31

V1 __ W1 -W1 Dll Dll D21 -D21 sub

1

1

1

1

V3 V3 W3 -W3 D13 D13 D23 -D23

V5 V5 W5 -W5 D15 D15 D25 -D25

D33

D35

-D33

-D35

UllE1

0 U13E3 0 U15E5 sub

0

sub

(9.43) where Eq - e i~aq(xa-d),

q = 1, 3, 5.

(9.44)

CHAPTER 9. GENERAL LAYERED MEDIA

132

We now recall the formal solution (9.23) for the layered plate and write it in the expanded matrix form Ul u2 u3 O'~3 ff~3 a~3

+

All A21 A31 A41 A51 A61

_--

A12 A22 A32 A42 A52 A62

A13 A23 A33 A43 A53 A63

A14 A24 A34 A44 A54 A64

A15 A25 A35 A45 A55 A65

A16 ul A26 u2 A36 u3 A46 a~3 A56 a~3 A66 plate a~3

(9.45)

Specializing the solution for the substrate at X3 -- d, followed by satisfying the plate-substrate interface condition, namely Ul

Ul

U2

U2

U3

a~3 a{3 a~3

__

~t3

at x3 : d

a~3 a~3 ~late

~3

(9.46)

sub

results in Ul u2 zt3 (7~3 ff~3 0"~3

__

Pll P21 g31 P41 P51 P61

/~ P22 P32 P42 P52 P62

P13 P23 P33 P43 P53 P63

P14 P24 P34 P44 P54 P64

P15 P25 P35 P45 P55 P65

P16 P26 P36 P46 P56 P66

Ull

0 (9.47)

U15 0

sub

where the 6• matrix Pij is the product of the inverse of the 6 • Aij of equation (9.45) and the 6x6 matrix in equation (9.43). By implementing the stress free boundary at x3 = 0, namely,

[] 0";3 a~3 CrY3

=0

matrix

(9.48)

and, for nontrivial solutions of the wave amplitudes

U13 g15

(9.49) sub

9.6. B O T T O M BOUNDING SOLID SUBSTRATE

133

we get the characteristic equation

P41 P43 P45 P51 P53 P55 = O. P61 P6~ P6~

(9.50)

This then constitutes a generalization of the surface wave characteristic equation (7.4). The important difference is that the present generalized equation representing a layered plate-substrate system yields frequency dependent surface wave speeds; i.e., it exhibits dispersive behavior. The surface wave speeds which satisfy this equation are highly dependent upon the architecture of the layered plate and the substrate. Numerical examples of this situation will be presented in the next chapter when we deal with the same problem but with propagation being restricted to take place along an axis of symmetry.

This Page Intentionally Left Blank

Chapter 10

PROPAGATION ALONG AXES OF SYMMETRY In this chapter we treat the simplest cases of propagation in multilayered anisotropic media. These include the two uncoupled wave types for propagation along axes of symmetry as described previously in chapter 5. The first, and by far the simplest of the two, is the case of propagating horizontally polarized S H waves, where a single displacement component is involved. The second corresponds to the motion that is confined to the sagittal plane; here two displacement components are involved. In order to generate either of these two uncoupled motions in the layered system, certain geometric arrangements and propagation direction restrictions must be met. Guided by the discussion in section 5.3.2, as a minimum, all layers must be composed of orthotropic or higher than orthotropic symmetry materials. Furthermore, the layers have to be stacked such that their material symmetry axes coincide. Lastly, either of the two wave types has to be restricted to propagate along a principal direction. Only when these stringent geometric and propagation directions are met, one can excite pure horizontally polarized or pure sagittal motion in the total system. In the subsequent sections of this chapter, we treat both types of motion separately. Solutions for both are obtained by following the general procedure of the previous chapter, however. Once again, this leads to results that are general and include those pertaining to free waves on finite thickness multilayered plates as well as on periodic media. Since no linear transformations are necessary to carry analysis in the present two cases, for consistency, we should revert to conducting analysis in terms of the crystollographical (primed) system. However, due to the limited number of possible orientations involved, namely those pertaining to the azimuthal 135

136

CHAPTER 10. PROPAGATION ALONG A X E S OF S Y M M E T R Y

~_/~,_'~

•

_

,

~

., . . . . . .

x2

,,,,, ~ -~'- - . . . . . . . . . .

x3

Figure 10.1: Geometry of an anisotropic plate showing propagation along an axis of symmetry. angles 0 ~ and 90 ~ instead, we shall be able to use the transformed system by affecting the necessary rotations through these two angles, wherever applicable.

10.1

Geometry

Consider a plate consisting of an arbitrary number n of orthotropic layers rigidly bonded at their interfaces and lined up such that their axes of symmetry coincide with each other and also with a global coordinate system x~ as illustrated for n - 2 in figure 10.1. Hence, the plane of each layer is parallel to the x l - x 2 plane which is also chosen to coincide with the upper surface of the layered plate. This specific stacking is necessary to allow for the existence of pure horizontally polarized shear or pure sagittal plane motions. Thus, for a plate constructed from several unidirectional fibrous composite laminae of the same material, combinations of only 0 ~ and 90 ~ layups are allowed. Similar to the layering description of the previous chapter, in subsequent sections, we refer to this plate as the representative unit cell of the medium. If this cell is repeated, we have an n-layered periodic medium. Alternatively a single cell medium constitutes a finite thickness plate. In order to carry out the analysis, we introduce for each layer k, k = 1, 2, ..., n, a local coordinate system x~(k) with the origin at the interface between layers k - 1 and k. Hence x3 coincides with x~ and Xl

10.2.

SH WAVES

137

can coincide with either x~ or x~ for 0 ~ or 90 ~ orientation, respectively. A similar role holds for x2. W i t h this description, layer k occupies the space 0 ~ x~ k) ~_ d (k) where d (k) is its thickness. Thus the unit cell has total thickness d = Y~'~=I d(k)" In the following section, we derive exact analytical expressions for the characteristic equations of a horizontally polarized shear waves S H propagating in the Xl - x 3 plane at an arbitrary angle 9 from the normal. In section 10.3, we conduct similar studies of the case involving sagittal plane motions.

10.2

SH

waves

With the above choice of the coordinate system and the restricted propagation directions for pure S H waves, the motion in the "transformed" system is reduced to a two-dimensional one. Here only the in-plane transverse displacement u2 exists for all layers which is independent of x2. The relevant field equations for each layer are those previously given by equations (5.17) and (5.18). These equations are supplemented with the interfacial continuity conditions

u~k) = u~k+l)

(k+l) 0"~k) -- 0"23

(10.1)

at x~ k) = d (k) or x3(k+l) -- 0. For each layer k, a similar formal solution to that of (9.1) is assumed, namely

u2 = U2e ~r

(10.2)

where we again note and suppress the common factor e i((xl sin O-vt) and where we recall that U2 is the displacement amplitude, ~ is the wavenumber, v is the phase velocity and a is the x~ component of the wavenumber. For nontrivial solutions of U2, when equation (10.2) is inserted in equation (5.17), we obtain a characteristic equation for a which admits the two solutions 1,2

+ / p v 2 -- 666 sin 2 0

(10.3)

V

By using superposition, we write formal solutions for the displacement and stress as ~3

6440 L _C44ol

a22e_i~alX3

,

(10.4)

10.2.

SH

139

WAVES

The general result (10.10) contains, as special cases, those pertaining to single and bilayered plates which are widely available, especially for isotropic media. By specializing this equation to n = 1 and n - 2 we get, D1 sin ")'1 --

0,

D2 cos "/1 sin 3'2 + D1 cos "/2 sin ")'1 -- 0,

for n = 1 for n = 2.

(10.11)

In order to isolate and quantify the relative importance of anisotropy we examine the behavior of equation (10.11) for the single layer plate. Substituting the definitions from equations (10.7) and (10.3) into the first relation in equation (10.11), we obtain the dispersion relation for the single layer plate as (pv 2 - C66 sin 20)~2d 2 = C44N27r 2.

(10.12)

where N is an integer. Since it is customary for free waves to propagate on a plate along directions parallel to its free faces, namely when 0 = 90 ~ then, for such a situation, equation (10.12) reduces to F 2 _ Q2

644 N 27r2

(10.13)

where we introduced the non-dimensional wavenumber Q = ~d, and the nondimensional circular frequency F = w d / v o with Vo = x / C 6 6 / p . Equation (10.13), through the ratio of C44/C66, demonstrates that anisotropy affects a shift in the optical modes, namely N >_ 1. Notice that the fundamental mode N - 0, the acoustic one, is not affected by the anisotropy since the solution of (10.13) yields F - Q resulting in a phase velocity v = Vo. 10.2.2

Periodic media

A second important situation is that of a periodic medium constructed from a repetition of the unit cell (layered plate). Similar to equation (9.34), periodicity is satisfied by invoking the condition

0"~3 x3--d

=[u2]

edcos0

0"~3 X3--O

,1014,

which, together with equation (10.8), yields d e t ( A - I e i~dc~176 = 0

(10.15)

140

C H A P T E R 10. P R O P A G A T I O N ALONG A X E S OF S Y M M E T R Y

namely, A11 A21

-- el(tic~

A12

--0.

A22 - e i(dc~

(10.16)

Expanding equation (10.16) and using the fact that det A = 1 as was proven in section 9.3, lead to e 2Kc~

-- e K C ~ 1 7 6

+ A22) +

1 = 0.

(10.17)

Dividing equation (10.17) by ei(c~176and using the trigonometric relation (9.39), we finally arrive at the characteristic equation cos[(d cos 0] = ~I ( A 11 + A22).

(10.18)

Once again, the general results (10.18) contain, as special cases, those pertaining to single and bilayered plates which are available, for isotropic media. By specializing this equation to n = 1 and n = 2, respectively, we get C~l 2 - cos 2 0, cosied cos 0] = cos 0/1 COS

for n = 1 sin',/1 sin 3'2 ( D2 D1 ')'2 - 2 Dll + D22)'

for n = 2. (10.19)

We recognize from examination of the first component of equation (10.19) namely, for n - l , the fact that infinite homogeneous materials are nondispersive, nevertheless the inclusion of anisotropy leads to a dependence of the phase velocity on the angle of propagation. This becomes immediately obvious, when we write this equation after substituting for c~ from equation (10.3), as PV2 = C44 cos 2 0 + C66 sin 2 0.

(10.20)

Explicit results for n > 2 can be just as easily obtained and need not be listed here. The results (10.19) pertaining to a periodic array of bilayered media (n = 2) are even themselves general and contain two sub-special cases which are available in the literature. These correspond to normal, 0 = 0 ~ and horizontal (wave guide), 0 = 90 ~ propagation directions. For 0 = 0, we get cos ~d = cos 71 COS 0'2 --

s i n 0/1

sin ~/2 (D2

2

D1

)'

for n = 2

(10.21)

10.2. SH WAVES

141

Table I

Material

644 • 101~ (dynes/cm 2)

(dynes/crn 2)

(gm/ m 2)

1

10 300

12.5 250

1 1.5

666

x

101~

0.4 0.6

with al,2 in equation (10.3) reduced to

~ / PV 2

(10.22)

al,2 = +V~44 For 0 - 90 ~ the characteristic equation (10.19) reduces to l = cos 71cos 72 -

sin 71 sin 72 ( D2 D1 2 ~1 +~ )'

forn=2

(10.23)

where al,2 now take the appropriate forms

~ / p v 2 -- 666

=

c- 2

(10.24)

Remark

Note that the results for isotropic media can be obtained by merely setting C44 = C66 = #, the shear modulus for each layer. Finally, by replacing C44 = C66 with A + 2# and setting 0 = 0 ~ we simulate the case pertaining to the propagation of longitudinal waves normal to the periodic array of multilayered media which we derived earlier in [164] in a much more complicated manner. For a periodically bilayered system, in figures 10.2a-d, the evolution of the frequency-wavenumber dispersion behavior for the four representative propagation directions 0 = 0 ~ 15 ~ 45 ~ and 90 ~ respectively, are demonstrated. The properties of the two materials used in the calculations are given, together with their arbitrarily chosen volume fractions, in Table I. As is seen, the thickness d of the unit cell is kept as constant, and its constituents (layers) are assigned volume fractions adding to unity. The most striking feature is the repetitive structure of figure 10.2a as compared with that of figure 10.2d. Figures 10.2b,c clearly display some of the general features of both figures 10.2a,d and thus demonstrate the manner of evolution as 0 increases. The periodic behavior displayed in figure 10.2a is a consequence of the fact that for 0 -- 0 ~ equation (10.22)

C H A P T E R 10. P R O P A G A T I O N A L O N G A X E S OF S Y M M E T R Y

142

12 10

8 6 4 2 5

10

15 Q

20

25

O0

30

10

10

8

8

6

5

10

15 Q

20

25

30

5

10

15 Q

20

25

30

6

1.1_

1.1_ 4

4

2

2 0

5

10

15 Q

20

25

30

0

Figure 10.2" Variation of nondimensional frequency F with nondimensional wavenumber Q for (a) 0 - 0 ~ (b) 0 - 15 ~ (c) 0 - 45 ~ and (d) 0 - 90 ~

implies periodicity of the solutions in multiples of 27r in Q. Common to all of these figures is the fact that the lowest order mode N - 0, which stems from the origin, is the acoustic mode. The remaining curves correspond to optical modes each possessing a cutoff frequency defined by its intercept with the frequency axis. At all cutoff frequencies, Q is zero and hence corresponding phase velocities that are effectively infinite. A glance at the general features of figures 10.2a-d may leave one with the impression that there exists a certain degree of similarities in their respective branches, especially of the slopes close to the origin. This is, of course, due to the normalization of the solutions. The actual behavior may be different depending upon the value of the normalizing phase velocity Vo in equation (10.20) as required by the normalization of F - cvd/vo. For comparisons, the values of Vo are calculated for 0 = 0 ~ 15 ~ 45 ~ and 90 ~ as 4.28, 5, 8.3, 10.92 km/sec, respectively. In figures 10.3a,b we depict, for the selected normalized frequencies F = 0 and 47 wave front curves in the n l - rt3 plane where n l - cos0 and

10.2. SH WAVES

143

12r-n3v

n3v

12

8

F

1,2nlv

-1'2

A

,

,

i

.

.ii

.

.

,

nIv

~ ~ _-a 7 -K,.) (a)

-12-

(b)

Figure 10.3: Variation of phase velocity with direction of propagation for (a) F = 0 a n d ( b ) F=4.

n2 = sin 0. These curves demonstrate the inverse of the slowness curves as functions of frequency and hence display and demonstrate dispersion behavior.

10.2.3

Effective elastic properties

Taking advantage of the relatively simple algebraic expressions pertaining to the present problem, below we outline a way of obtaining effective elastic properties for the medium. Such results lead to mixture expressions for the material density, properties and hence, for the effective wave speeds. As shall become obvious, similar results for the more complicated propagation situations, such as those pertaining to the sagittal planes or to arbitrary directions, are more difficult to obtain. Nevertheless, results of the following section will serve as an illustration of what may be expected. Effective mixture properties for the total medium can be obtained in the limit as the frequency tends to zero (or equivalently as the incident wavelength becomes very large compared with the micro-dimension d). For such a limit, an expansion in power series in ~ for each of the transfer matrices (10.6) yields, to the first order of approximation,

(

cos 7 i sin 7D

i sin 7/D cos 7

) (10)( 2,2 i ,o) k

,.m

0

1

k

+

i.yD

-

72

/2

k"

(10.25)

CHAPTER 10. PROPAGATION ALONG AXES OF SYMMETRY

144

To the same order of approximation then AkAk+l becomes

(1

-- "~k2/ 2

i "~k D k

I

(1

2 /2 --')'k+l

i "/k + l D k + l

1

Dk+l

1 -- ~(7~ + ~2+1 ) -- ~'k~/k+l Dk

i('~kDk + "/k+lDk+l)

i("/kDk+l + "/k+lDk+l) 1 Dk 1 - ~(3'~ + ~/~+1) - ")'k"Yk+lDk+l

\

)

(10.26) By carrying out the matrix multiplications for all layers leading to equation (10.18), while retaining terms of order ~ , it can be shown that equation (10.26) reduces to n

n

n

E ,.)/2 nt- E E "yp"yqnpq -- ")/2 Cos2O k=l p=l q=l

(10.27)

with the restriction that q > p and with

= ~d, Dpq = (

Dp

+ ~qq).

(10.28)

Dividing both sides of this equation by ,),2 and using the definitions (10.7) for the individual layers lead to n

n

n

E a~Vk + E E apaqViVjDpq = cos 2 O,

k=l

p=l q=l

(10.29)

where Vk -- d(k)/d is the volume fraction of layer k. If we substitute for ak and Dk from equations (10.3) and (10.7) into equation (10.29), collect common terms and define v~ ---- ~ : 1 ck6Vk ~-~= 1 pkVk 2 VV

Y'~= 1 (Vk/Ck44) -1 ~-'~~= 1 pkVk '

(lO.30a)

(10.30b)

we finally solve for the effective phase velocity v = Ve as 2 ve2 =v vcos 2 0 + v ~ s i n 20.

(10.31)

10.3. M O T I O N I N T H E S A G I T T A L P L A N E

145

Both components in equations (10.30) and (10.31) imply that the effective density of the medium is given as the volume fraction sum of the individual layer density. Furthermore, for propagation along the interface, namely for 0 - 90 ~ equation (10.31) yields v2 - v 2 which is the sum of the partial properties c6k6 divided by the effective density. This is a consequence of the fact that the layers are combined in parallel-like fashion. Finally, for propagation normal to the layers, namely for 0 = 0 ~ Ve2 = v2 corresponding to the case of combination by series as is expected.

10.3

Motion in the sagittal plane

The second situation of propagation along an axis of symmetry belongs to the motion that is confined to the sagittal plane. Here, the motion is described in terms of the two displacement components Ul and u3. Deriving characteristic solutions for this case is an easy task since it follows, to a great extent, the steps used in deriving the corresponding results for the general case of chapter 9. For consistency, we shall use, wherever possible, the same parameter definitions used in chapter 9. With respect to the primed coordinate system (X~)k, the elastic field equations for the sagittal motion of layer k are given by equations (2.46) and (2.48). However, we shall conduct our analysis, like in the last section, in terms of the transformed system. A modification of the general solution (9.1), which is applicable here takes the form

(ltl, •3) ---- (U1, U3)e i~ax'a,

(10.32)

where, for convenience, we suppress the common factor e (xlsinO-vt) and where the various parameters appearing here are as defined in conjunction with equation (9.1). This choice of solution leads to the two coupled equations Kpq(a)Uq = 0,

p, q = 1, 3

(10.33)

where now Kll = C11 sin 2 0 - pv 2 + C55a 2 K13 = (C13 + C55) a sin 0 K33 = C55 sin 2 0 - pv 2 + C33a 2.

(10.34)

The following steps parallel, to a great extent, those of chapter 9. The existence of nontrivial solutions for U1 and U3 demands the vanishing of the

C H A P T E R 10. P R O P A G A T I O N A L O N G A X E S OF S Y M M E T R Y

146

determinant in equation (10.33), and yields the fourth-degree polynomial equation A1 a4 -b A 2 a 2 + A3 - 0

(10.35)

relating a to v, with A1 -- 633655

A2 = C55(655 sin 2 0 - pv 2) + 6 3 3 ( C l l sin 2 0 - pv 2) - ( 6 1 3 + C55)2 sin 2 0 A3 = (655 sin 2 0 - pv2)(C11 sin 2 0 - pv2).

(10.36)

Equation (10.36) admits four solutions aq, q = 1, 2 . . . 4 in accordance with /..

- A 2 • ~/A~ - 4AIA3

O~I,3-

- -2A1 -

(10.37)

with the properties 0~2 :

--OZl,

Oz4 : --a3.

(10.38)

For each aq, we can use the relations (10.33) to express the displacement ratio Wq U3q/Ulq as =

K l l (OLq) _ Wq

=

K13(OZq) -

pv 2 -

C l l sin 2 0 - C550~2

C55)aqsinO

(C13 +

"

(10.39)

Using superposition, we finally write the formal solutions for the displacements and stresses in matrix form as 1 Wl

1 -Wl

1 W3

1 -W3

a~3

DII

DII

DI3

DI3

UIIEI U12E2 U13E3

a~3

D21

-D21

D23

-D23

U14E4

Ul

u3

=

(10.40)

where

Eq -- eiCaqx'3 Dlq = C13 sin 0 + C33aqWq D2q = C55 ( aq + Wq sin 0),

q = 1 , 2 , . . . ,4.

(10.41)

In arriving at the matrix equation (10.40), the relations (5.15) and (5.16), which also hold for the present case, have been used. Notice that the specific relations in the entries of the square matrix of equation (10.41), such as

10.4. F R E E WAVES ON THE L A Y E R E D CELL

147

W2 = -W1, for example, can be seen by inspection of the ratios (10.39) in conjunction with the restrictions (10.40). Equation (10.41) can be used to relate the displacements and stresses a t (X3) ' k : 0 to those at (x3) k -- d (k) . To facilitate the discussion, we first specialize the formal solutions (10.41) to each layer k and rewrite it in the compact form

Pk = XkDkUk

(10.42)

where Pk is the 4

x

1 displacement and stress matrix

~tl

~3 (7~3 ~h

(10.43)

k

Xk is the 4 x 4 square matrix in equation (10.40), Dk is the 4 x 4 diagonal matrix whose entries are Eq and Uk is the 4 x 1 displacement amplitude matrix Ull

ak=

U12

(10.44)

U13 U14

k

Repeating the steps commencing after equation (9.17) to the present situation finally yields to the local and global transfer matrices, respectively as

Ak = X k D k X k I

(10.45)

and

P+ = A P - ,

(10.46)

where now P+ and P - are the displacement and stress column vectors at the upper surface, x3 --- d, and the lower surface, x3 = 0, of the total plate, respectively.

10.4

Free waves on the layered cell

The characteristic equation for the propagation of free waves on the single layered plate is obtained by choosing 0 - 90 ~ and invoking the stress free

148

CHAPTER

10.

PROPAGATION

I

9

I

~

ALONG

AXES

OF SYMMETRY

I

I

I

I

4 0

5

I

6

I

7

5

A

E

4

0

10

1

2

~1

I

8

Figure 10.4: Dispersion curves (phase velocity versus nondimensional wavenumber Q) for a (0 ~ 90 ~ graphite-epoxy layup. upper and lower surfaces in equation (10.46) which lead to the characteristic equation A31 A32 A41 A42

= 0.

(10.47)

It is of interest to note that this result constitutes a specialization of the results (9.33) obtained for the multioriented monoclinic plate. When the properties of all layers are the same, it is expected that equation (10.47) then reduces to (8.10) and (8.11). For a numerical illustration of the free waves in a layered cell, we include in figure 10.4 a sample calculation based on equation (10.47) for (0 ~ 90 ~ graphite-epoxy equal thickness layup.

10.5

W a v e s in a p e r i o d i c m e d i u m

Invoking the Floquet periodicity condition (9.34) on equation (10.46) leads to the characteristic equation a 4 -- 1163 + 1262 -- 1 3 6 + 14 - - 0

(10.48)

where we recognize that a = e Kdc~ is an eigenvalue of the matrix A, and Iq , q - 1, 2 . . . 4 are the invariants of A. In accordance with the discussion leading to equation (9.32), the invariants of the present 4x 4 matrix A have

10.6.

149

BOTTOM BOUNDING SOLID SUBSTRATE

the properties /4 = 1 and /3 = I1. (10.48), after dividing by 0"2, to

These restrictions reduce equation

(~2 + o-2) _ h (~ + ~-~) +/2 = 0.

(10.49)

Finally, replacing a with its value e Kdc~ it can be easily shown that this characteristic equation takes the simple form

/2

(10.50)

cos[2~dcos O ] - I1 cos[~d cos O] + -~ = O.

We note that all discussions that followed equation (9.40) are also valid for this present case.

Bottom

10.6

bounding

solid

substrate

If the lower bounding medium of the layered plate happens to be a homogeneous substrate, pure sagittal plane motion can only exist if the substrate also possesses orthotropic or higher symmetry. Furthermore, the substrate has to be oriented such that its axes of symmetry coincide with those of the layers in the layered unit cell. Under these conditions, the total system's characteristic equation can be derived after constructing a formal solution for the substrate followed by satisfying the appropriate plate-substrate interface conditions. But the appropriate formal solution for the substrate was identified in chapter 5 in conjunction with the study of surface waves. In a manner similar to equation (9.43), the formal solution to the present substrate is Ul u3

=

0"~3 0"~3

sub

1 W~

1 -w~

1 w3

1 -w3

UllE1 o

DI1

Dll -D21

D13

O13

U13E3

D23

-D23

D21

sub

0

(10.51) sub

For the layered plate, we rewrite its global matrix solution (10.46) in the expanded form Ul u3 a~3

0"{3

__

All A12 A13 A14 A21 A22 A23 A24 A31 A32 A33 A34 A41 A42 A43 A44

Ul

U3 plate

a~3 0"~3

(10.52)

150

CHAPTER 10. PROPAGATION ALONG AXES OF S Y M M E T R Y

3.2

3.1

E 3.0 _~

v

r

o

Y ,

,

15

,

Figure 10.5" Variation of surface wave speed CR with ~t - wd/cs for a chromium plate of thickness d over a steel substrate. Cs is the shear wave speed in the steel, circles - experimental, solid lines - theoretical. By satisfying the continuity of displacement and stress components at the interface, we get the relation

Ul u3

_

0"]3 G~3

Pll

P12 P13 P14

Ull

P21

P22

0

P23

P24

P31 P32 P33 P34 V41 P42 P43 P44

(10.53)

U13 sys

o

sub

where the various Pij elements are elements of the 4 x 4 matrix constructed from the product of the inverse of Aij in equation (10.52) and the square matrix in equation (10.51). Invoking the stress free boundary condition at the upper surface of the plate, we finally get the desired characteristic equation P31

P33

P41 P43

- 0.

(10.54)

Equation (10.54) is the characteristic equation for the propagation of free waves on a layered plate attached to a solid substrate. This then constitutes a generalization of the surface wave characteristic equation (7.4).

10.6. B O T T O M BOUNDING SOLID S U B S T R A T E

151

3.~t 2.8

E 2.6 0

rr

2.4

2.2 [

2

I

4

l

6

I

8

I

10

I

12

14

Figure 10.6: Variation of surface wave speed CR with f~ '= wd/cs for a copper plate of thickness d over a steel substrate. Cs is the shear wave speed in the steel. The important difference is that the generalized equation yields frequencydependent surface wave speed; i.e., it exhibits dispersive behavior. The surface wave speeds which satisfy this equation are highly dependent upon the architecture of the layered plate as compared with the substrate. As pointed out by Nayfeh and Chimenti [170], the plate can either stiffen or load the substrate, depending upon whether the surface wave speed increases or decreases with increasing frequency. To best illustrate this behavior, we choose three isotropic materials for the system. These consist of steel, copper and chromium with properties collected in the Appendix. In figures 10.5 and 10.6, displayed are dispersion relations curves for a copper plate attached to a steel substrate and for a chromium plate attached to a steel substrate, respectively. Displayed on these figures are the variations of surface wave speeds with the nondimensional frquency f~ = wd/cs with cs being the shear wave speed in the steel substrate. Experimental data are also included in figure 10.6. Comparison with experimental data for the situation encountered in figure 10.5 will be appropriately deferred to section 11.4 when we discuss the leaky wave phenomena. For the copper-steel system, several modes besides the fundamental one exist whereas for the chromium-steel system, only the fundamental mode exists. In both cases the phase veloc-

152

CHAPTER 10. PROPAGATION ALONG AXES OF SYMMETRY

ity of the fundamental mode converges to the Rayleigh wave speed of the steel substrate at ~ - 0. As ~t increases, the phase velocity of the copper plate decreases to its limiting copper surface wave speed while other modes appear. The phase velocities of these higher order modes are bounded from below and above by the shear wave speeds in the steel substrate and the copper layer, respectively. This behavior is typical of softening (loading) materials. As ~ increases from zero, the phase velocity of the chromium plate increases towards the chromium's Rayleigh wave speed of 3.6km/sec. However, at certain values of ~, (corresponding to a cutoff frequency) where the phase velocity reaches the steel shear wave speed of 3.13km/s, the mode ceases to propagate. This behavior can be easily explained from the fact that, for surface waves to also exist in the steel substrate, the phase velocity cannot exceed its shear wave speed of 3.13km/s. This behavior is typical of stiffening materials.

Chapter 11

FLUID-LOADED

SOLIDS

This chapter is devoted to the study of the influence of fluid loading on the interaction of elastic waves with solids. Situations where such modelings are encountered are many and include, for example, under-water explosions such as earthquakes, sonar systems and nondestructive evaluation of materials. The industrial problem towards which the present treatment is directed is the nondestructive inspection of layered solids. Coupling ultrasound to the solid is most conveniently achieved either by immersion or by water column transmission. The premise is that details of the behavior of the solid are conveniently contained in and can be easily extracted from the reflected field in the fluid. The bulk of material covered in this chapter will therefore be concerned with the derivation of the reflection coefficient from fluid-solid interfaces. Within the context of our treatment, a general anisotropic medium is defined as a multilayered plate made up of an arbitrary but finite number of layers and constitutes a unit cell (the layered plate). This plate is general and may itself model a whole variety of finite thickness, solid component systems. One side of the plate will always be in contact with a semi-infinite fluid, thereafter designated as the upper fluid. To maintain generality, the other side of the plate can be supported with a similar or different fluid, another solid or can be left stress free (in vacuum). At this early stage, we alert the reader to the significance the fluid plays in the dispersion behavior of the propagation process. It had been believed for a long time that the fluid coupling can be considered as a minor perturbation to the dispersion behavior of the loaded solid. The unavailability of studies on the role of fluid-loading before the early eighties can perhaps suggest that this belief has been taken for granted. Recent discoveries indicate that, whereas this observation is still valid for most fluid153

154

C H A P T E R 11. FLUID-LOADED SOLIDS

loading situations, in cases involving finite thickness plates totally immersed in fluids, the fluid can, under certain conditions, play a decisive role on the dispersion process. This phenomenon will be discussed in detail later on in subsequent sections. In most practical situations, circular piston radiators are used to generate an acoustic wave which is subsequently allowed to propagate through the upper coupling fluid to the solid surface. The finite aperture transducer field can lead to a substantial distortion of the reflected field as will be discussed in detail in later sections, within the context of the leaky wave phenomenon. The current chapter is organized such that it follows the logical exposition adopted in the earlier chapters of this book. It begins with the derivation of the reflection coefficient for a fluid-loaded homogeneous substrate. Next, the analysis is carried further to the case of a single material plate and subsequently generalized to the multilayered one. These cover the plane wave solutions. Subsequent to these sections, we treat cases which involve finite aperture incident waves and introduce the leaky wave phenomenon. In all cases, we call upon formal solutions and results obtained in earlier chapters and modify them to reflect the influence of the loading fluid. To maintain generality, we treat the cases in which the solid is composed from components that possess monoclinic or higher symmetry. Throughout this chapter, we quote selective comparison with experimental results from previously published works conducted by us and others. The experiments on the completely immersed single and multilayered plates are conducted on graphite-epoxy samples. Descriptions of these samples are included below in section 11.5. For the numerical part of the comparison, the elastic properties needed for the graphite-epoxy composite are collected and written with respect to their crystollographical axes (namely, the primed coordinate system) in the Appendix. They are constructed from the individual properties of graphite and epoxy based upon the procedures of chapter 15. As listed, these properties also correspond to a defacto 0 ~ azimuthal angle r It is here implied that properties applicable to other azimuthal angles can be obtained via the transformation (2.16). For the free lower surface of the plate, experiments are available on an aluminum sample plate. Being isotropic should not be of concern, since the analytical models are general and include, as special cases, results pertaining to isotropic components. This point will be further elaborated upon in section 11.3 when dealing with higher symmetry materials. The geometric stacking of the system follows the convention adopted throughout the previous chapters. In accordance with this convention, the

11.1. R E F L E C T I O N FROM A S U B S T R A T E

155

upper fluid-solid interface is located at x3 - 0 such that the fluid extends infinitely in the negative x3-direction and that the solid extends in the positive x3-direction to a level depending upon the specific system under consideration. Typically, if the solid constitutes a finite thickness plate made up of either single or multilayers, we assign to it the arbitrary thickness d and hence the plate extends from x3 - 0 to x3 = d. Accordingly, the lower supporting medium will occupy the region x3 _> d.

11.1

Reflection

from a substrate

Let us first treat the case of incidence and reflection in the fluid from a fluid-monoclinic solid substrate as illustrated in figure 1t.1. This situation gives rise to four "scattered" waves, one reflected in the fluid and three transmitted in the solid as shown schematically in the figure. Explicit determination of the reflection and transmission coefficients requires formal solutions for the solid and for the fluid. Formal solutions for the displacements and stresses appropriate for the present solid substrate were given on several occasions in previous chapters. Following the notation and identification criteria of sections 6.1 and 6.2, by designating the the displacement and stress components with the superscript (T) to indicate transmission, we have (Ul , U2, U3) (T) ---- E (1 , Vq, Wq) Tr(T)e Vlq i~aqx3 9 q=1,3,5

(11 1)

Here, and in subsequent situations, the common factor ei~(zl-ct) is implied and conveniently suppressed. The corresponding stress components of the transmitted field are ((733'a13'f23)(T)--

E

q-1,3,5

TT(T)'~i~aqX3 i~(Dlq, D2q, D3q)~lq v .

(11.2)

Recalling the formal solutions (5..39) for the fluid, we adapt it to the present situation as (ul, u3, a33) = (1, a f , i~plc2)O~ I)e i~asz3 + ( 1 , - a l , i~pfc2)~f~ R) e -i~"fx3.

(11.3)

By arbitrarily choosing the incident wave amplitude C~I), here and in the remainder of this chapter, as unity, from now on, we can arbitrarily identify ~ R ) with the reflection coefficient R and the solid amplitudes U~T), u~3T),

CHAPTER 11. FLUID-LOADED SOLIDS

156

fluid

-> X 1

i

e

transmit

x3

Figure 11.1: Scattering from a fluid-solid substrate interface. U~5 T), with the transmission coefficients T1, T3 and T5, respectively. Also for convenience, we introduce the overbar to designate the fluid. Invoking, at x3 - 0, the continuity of the normal displacement and stress (i.e., ~3 -- U~T) and 933 = a~T)) and further setting a~3T) and a~3T) of the solid equal to zero, we obtain the following system of four linear simultaneous equations for the unknown transmission and reflection coefficients

w1

w3

w5

as

Dll D21 D31

D13 D23 D33

D15 D25 D35

--pfc 2 0 0

T1 T3

=

af p f c2

T5

0

R

0

"

(11.4)

Upon solving these equations, the following expressions for the reflection and transmission coefficients are obtained

R =

Av - y

Av+Y

T1 = 2pfc2G1

Av+r

T3 = 2PIc2G3

Av+y

(11.5) (11.6) (11.7)

11.1.

REFLECTION

FROM A SUBSTRATE

T5 -- 2p'fc2G5

Av + y

157

(11.8/

Here, A v = D l l G 1 - D13G3 + D15G5 y

~_

P / C 2 ( w I G 1 - W 3 G 3 + W5G5)

(11.9) (11.101

and

G1 -- D23D35 - D33D25 G3 - D21D35 - D31D25 G5 = D21D33 - D31D23

(11.11)

as were encountered previously in sections (6.2) and (7.1) of chapters 6 and 7, respectively. 11.1.1

Qualitative

discussion

The expressions (11.5)-(11.8) for the reflection and the transmission coefficients contain, as a by-product, the characteristic equation for the propagation of fluid-modified surface waves on the substrate. The vanishing of their common denominator, namely Av+Y=0

(11.121

defines the characteristic equation for such waves. In the absence of the fluid, i.e., when p/ (or Y ) = 0, equation (11.12) reduces to Av = 0 which is the secular characteristic equation (7.4) for propagation of surface waves on the dry substrate. For a given frequency, the real wavenumber solution ~ = ~R of Av = 0 03 defines the propagating Rayleigh surface mode whose wave speed is CR -- -~ that depends upon the azimuthal angle r Thus, for anisotropic substrates, the surface wave speed depends upon the propagation direction as was discussed in sections (7.11 and (7.2). In the presence of the fluid, these real wavenumber solutions are perturbed rather mildly and become complex. This, of course, is confirmed by equation (11.121 which, in general, admits the complex solution -- ~R + i5.

(11.131

158

C H A P T E R II.

FLUID-LOADED SOLIDS

From equation (11.13), the phase velocity of the Rayleigh wave is still given as cR -- -C~~and 5 is the attenuation coefficient. Note that (f vanishes in the absence of the fluid, and hence no attenuation (leaking of energy into the fluid) occurs. In the presence of a fluid these surface waves are usually called leaky waves. It is known that CR is hardly affected by the presence of the fluid, but (~ is significant. However, as shall be shown within the context of leaky waves, ~ is an important parameter because it also defines a measure of the lateral displacement of the reflected beam; in fact, the beam's actual 2 displacement along the interface is found to be equal to ~. Since we have concluded that the vanishing of Av defines propagating surface modes, it is clear from equation (11.5) that, as Av approaches 0, the real part of the reflection coefficient approaches - I and its imaginary part undergoes a simultaneous sharp change through zero. It is then obvious that such a condition defines an alternative method for identifying the wave modes in the dry substrate, and thus recovers the earlier results obtained in section 7.2. Furthermore, as shall be apparent from the figures listed below, a null (or at least a minimum) in the amplitude of the reflection coefficient corresponds to the excitation of a pseudo-surface wave on the interface. This also suggests an alternative and perhaps easier way of identifying the pseudo-surface mode. Sample calculations of the reflection coefficient for the InAs cubic material are given in figures l l . 2 a - c. In these figures the real and imaginary parts of R are plotted as functions of sin ~) (the incident angle) for the three representative azimuthal angles r 8~ 24~ and 32~ Also plotted to the right in figures 12.2d,e,f are the corresponding absolute Values of R. These figures clearly display all of the general features described above. In particular, the sharp dips farthest to the right in R corresponds to the excitation of the normal surface mode. Here the real part of R equals -1 and its phase suffers a sharp change through zero. We refer to these as the primary dips. Furthermore, the figures also contain other (secondary) dips to the left of the primary ones which do not exist for isotropic media and hence are a consequence of anisotropy. We have shown in [160] that these dips correspond to the excitation of pseudo-surface modes. For example, the second sharp dip which appears in figure ii.2c for r - 32~ very closely resembles that associated with the normal surface mode. As may be seen from figures 11.2b and I 1.2a, respectively, this secondary (pseudo-surface) mode degenerates and disappears as the azimuthal angles decrease below 24~. However, as these angles increase from 24~ the secondary mode slowly evolves until its features become practically indistinguishable from those of the normal surface mode. We also indicate that as increases, the secondary mode becomes much easier to identify whereas

159

11.1. R E F L E C T I O N F R O M A S U B S T R A T E

1.o 1.0 0.5

;-..

,

/,'!i 0.8

n" 0.0 . . . . . .

* . . . . . . . . . ;' ....

A mr v oo0.5 m <

:

0.3

-0.5

(d)

o18

"I"~.L3 ' 0'.4 ' 0'.5 ' 0'.6 ' 0.7

o.s-

0:9 sing

,i i

O.l

).3

0.4

0.5

0.6

0.7

0.8

o.Gsine

mr'-0"8 "

sing

-0.5

in '~.3

:11

0.4

o.s

0.6

/

.. . . . . . . . . .

0.7 "0'.8 ' 0:9 sine

0.3

~

0.4

o.s

If) 0.6

0.7

0:8

0:9 sine

Figure 11.2: Variation of complex (a,b,c) and absolute (d,e,f) reflection coefficient with sin 0 for the InAs cubic material: (a,d) r = 8 ~ (b,e) r = 24 ~ and (c,f) r = 32 ~ Solid curves are real parts and broken ones are imaginary parts. After Nayfeh [160].

160

C H A P T E R 11. FL UID-LOADED S O L I D S

fluid

-d12

"~ i

, _1.

plate > X1

i i

d/2

fluid v

X3 Figure 11.3: Geometry of scattering from fluid-solid plate interfaces; the case of total immersion. the normal one becomes very difficult, even numerically, to locate; nevertheless it is present for all angles but has very narrow angular aperture. Identification of the secondary modes can also be obtained by the help of the IRI figures. A sharp minimum, no matter how small, is associated with the excitation of a pseudo-surface mode.

11.2

P l a t e s c o m p l e t e l y i m m e r s e d in fluids

We next consider the case where a monoclinic plate is completely immersed in a nonviscous fluid as illustrated in figure 11.3. We recall that solutions for the propagation of free waves on this plate were discussed in detail in chapter 8. We now proceed to derive the reflection and transmission coefficients for such a system. Once again, we call upon the formal solutions applicable for the solid and for the fluid components. The formal solutions for the solid are those developed earlier in sections 5.1 and 5.2, repeated here for reference as 6

(~1, ~ , ~3) = ~ ( 1 , v~, w ~ ) u , ~ ~ ~ q--1

(11.14)

11.2. PLATES C O M P L E T E L Y IMMERSED IN FLUIDS

(cr33' crl3' cr23) :

161

6 E i { ( D l q , D2q, Daq)Ulqe i{aqxa q=l

(11.15)

where the various parameters are as defined in section 5.3. The formal solutions for the surrounding fluids are adapted from section 5.5 as follows. In the upper fluid, the formal solution is specialized and written with respect to the present coordinate of figure 11.3 system as

'~3 933

--

[1 OZf

i~ p f c 2

1 ][

--o~f

i{pfc 2

x3

Re-i{a/(xa+d/2)

1

(11.16)

where a} = (c2/~}) - 1 , and the superscript u denotes quantities belonging to the upper portion of the fluid. The continuity conditions at the plateupper fluid interface are given by ~

-- lZ3,

~3

--" ff33,

O"13 = O"33 = 0,

at

X3 =

(11.17)

-d/2.

For the lower portion of the fluid, the formal solution consists of the transmitted component and is given by

[ ] [ ] ~1

Zt3

0"33

1

--

O~f

(11.18)

T e i~a f (x3-d/2)

i~p f c 2

The continuity conditions at the plate- lower fluid interface are given by ?~/ -- lZ3'

~/3 -- ~r33,

O"13 --" O"33 --- 0,

at

X3 =

(11.19)

d/2.

Imposing the conditions at the solid-fluid interfaces, yields O WIE1 -WIE1 WaE3 -W3/~3 W5E5 -W5/~5 aI 0 DllE1 Dll/~71D 1 3 E 3 D l a E a D 1 5 E 5 D15/~5-pfc 2 0 0 D21E1 -D21 E1 D23E3 -D23E3 D25E5 -D25E5 0 0 D31E1 -D31 E1 D33E3 -D33E3 D35E5 - D35/~5 Wl& -WIE1 WaEa-WaEa W~E5-WsE5 0 -af 0 --pfc 2 D11E1 D11EID13/~3 D13E3 D15/~5 -D15E5 0 0 D21/~1 -D21 E1 D23E3 -D23E3 D25E5 -D25E5 0 0 D31E1 -D31 E1 D33E3 -D33E3 D35E5 -D35E5

.

u~21 u131

U14[ =

U~l U~6I R T

af

pfc 2

0 0

af

pfc 2

0 0

(11.20) This equation should be checked against equation (8:13) for the free waves on totally immersed plates. Using the Cramer rule, and following identically

162

C H A P T E R 11. FL UID-LOADED SOLIDS

the steps used in chapter 8 to derive the free wave characteristic equation, lead to the following expressions for the reflection and transmission coefficients AS-

R =

y2

(11.21)

(S+iY)(A-iY) iY(S+Y) T = (S + i Y ) ( A - i Y ) '

(11.22)

where S - DIIG1 cot(vai) - D11G3 cot(vc~3) + D15G5 cot(va5) A = DllG1 tan(v~l) - DllG3 tan(~a3) + D15G5 tan(~)'a5)

(11.23)

with

~d

wd

The functions S, A and Y, together with the parameters V, G1, G3 and G5 are the same as previously encountered in chapter 8 and partially in the previous section 11.1. 11.2.1

Cremer's

correspondence

principle

The expressions (11.21) and (11.22) for the reflection and the transmission coefficients contain, as a by-product, the characteristic equations for the propagation of fluid modified waves on the plate. The vanishing of their common denominators, namely

( s + iY) = 0

(A - iY) = 0

(11.24)

recovers the characteristic equations (8.14)for such waves. In the absence of the fluid, i.e., for Pl (or Y) = 0, equation (11.24) reduces to either A - 0 or S - 0 which are the secular characteristic equations describing the antisymmetric and symmetric free modes, respectively of the dry monoclinic plate. Thus, the poles of the reflection and the transmission coefficients correspond to the propagation of free waves on the fluid-loaded plate. Equivalently, dispersion relations of either the dry or the loaded plate can be constructed from satisfying A S = 0 and (S + i Y ) ( A - iY) = 0, respectively.

11.2. PLATES COMPLETELY IMMERSED IN FLUIDS

163

Based upon limited experiences, it has been widely accepted that the presence of the fluid contributes minor modifications to the dispersion curves of the dry case. For such a situation, it has also been observed that the dispersion relations happen to coincide with the onset of total transmission, where the absolute value of the reflection coefficient, namely I A S - Y21 vanishes. This is known as the Cremer's correspondence principle, discovered originally by Cremer [55] and subsequently put on firm theoretical ground by Schoch [219,220]. This principle consists of associating the occurrence of normal modes in the fluid-coupled plate with the onset of total transmission of the sound energy incident on it and has been used to deduce the velocity dispersion of Lamb waves within the plate. Due to the present interest in composite materials, recent attempts to apply Cremer's principle to fluid-loaded composite plates revealed unusual phenomena which have cast doubts on its universal applicability. In certain regions of the dispersion curves, it is found that results based upon solving the characteristic equations for the dry and loaded cases and those based upon the total transmission criterion were very different. It was conjectured that such disparity can either be related to the anisotropy inherent in composites or to their comparatively low densities. It was soon discovered that the low densities of the composites are responsible for such a disparity. Whereas the Cremer's principle is still applicable so long that the ratio of the fluid density p / t o that of the plate p is small, under conditions of heavy fluid loading (i.e., when the ratio of fluid to solid density approaches or exceeds unity), the fluid begins to play a decisive role in controlling both the propagation and reflection characteristics of waves in the immersed plate. The above findings have caused a contemporary reexamination of the applicability of Cremer's observation as a mode identification criterion. Numerous results are now available which confirm the above shortcoming of this observation. As shall be subsequently demonstrated, this peculiar behavior is typically found mainly in the low-frequency portion of the dispersion curves. A vivid illustration of this is seen in the behavior of dispersion curves for a unidirectional graphite-epoxy composite plate immersed in water as shown in figure 11.4. It displays sample numerical calculations of the secular equation for the fundamental So normal mode of propagation in the plate, both in the presence and absence of the fluid, together with the total-transmission prediction. Figure 11.5 depicts a portion of the results in figure 11.4 and includes experimental data. The density p of the graphiteepoxy sample is 1.6 g/cm 3. Experimental results like the current ones have been, in the past, compared to the secular equation solution in the absence of the fluid, presuming validity of Cremer's coincidence condition. As is

C H A P T E R 11. FL UID-LOADED SOLIDS

164

10

s'o ~"

6

v

4

E o

i II

t! i So , %%%

........

~

~

-

l.lo~oU

......

T

.4

fd (MHz mm)

4

Figure 11.4: Comparison of model calculations for the total transmission T, the symmetric Lamb mode So, and the leaky plate mode including the fluid S~. Unexpected effects are observed in the case of fluid coupling. After Nayfeh and Chimenti [169].

clearly seen, the total transmission loci in this case correspond neither to the free Lamb waves in vacuum or in the fluid-coupled plate. To further illustrate more clearly the behavior we have described, we now quote from Nayfeh and Chimenti [169] model calculations of equation (11.24) for a fluid-coupled aluminum plate (p = 2.Tg/cm 3) in which the fluid density has been arbitrarily adjusted to induce these unusual effects. Figure 11.6 demonstrates this behavior. The So mode dispersion has been calculated for fluid densities of 1.0, and 2.6 g / c m 3. The lower value is appropriate for water. In both cases, the speed of sound in water c I was kept constant at 1.485 k m / s . For p/ - 1, nothing unusual happens; the fluid influence is typical of what is expected and this specific dispersion is similar to those based on the total transmission criterion. However, with p / - 2.6, the phase velocity curve splits into two branches that approach each other closely. Simultaneously, the ratio of Im(~p ) to Re(~p) grows rapidly at this point instead of falling towards zero, as in the case for p / - 1.0. To calculate the numerical results of figures 11.4-11.6, careful extraction of the complex poles ~p of equation (11.24) has been accomplished by Muller's method. Either the pole may be inverted to yield the phase velocity of these modes, Cp - w/Re(~p), or the real and imaginary parts of the pole can be plotted to aid in the interpretation of the results.

165

11.2. P L A T E S C O M P L E T E L Y I M M E R S E D IN FLUIDS 4.5

i

'

i

i

t!

4.0 3.5

E

3.0

o

2.5 2.0 1.5 0

. js~

. ~

~

"<~'- ........==:

"- " ' " T

'

1

S~

:2

'

'

3

'

4

fd (MHz ram)

Figure 11.5: Expanded version of the dispersion curves shown in figure 11.4 including the data. Clearly, only the total transmission curve predicts the behavior of the data in the anomalous region. After Nayfeh and Chimenti [169].

,

6.0

,

~,

5.0

.

'

',

~

9

Pt=2-6

~ 1.0

-~06

~.1~ : "

E

v

o

4.0

3.0

0.0

/ ...........

=

1.0

: ='-'-

2.0

3.0

4.0

fd (MHzmm)

Figure 11.6: Fundamental S~ modes in an aluminum plate for arbitrarily adjusted fluid density p}. Discontinuous mode behavior is seen for p} = 2.6 analogous to that observed for the composite plate in water. Dashed curves are referred to the left-hand dashed axis. Phase velocity is plotted in solid curves, while damping term is represented as dashed curves. After Nayfeh and Chimenti [169].

C H A P T E R 11. FL UID-LOADED SOLIDS

166

i

!

!

~ f ~'MHz}

~

!

12 10 ~

...-

4

2

~

,

'

~

'

~

,'o

12

Figure 11.7: Reflection plate wave spectra for a graphite-epoxy plate at an incident angle ~9-12 ~ Normalized experimental measurement is shown as a solid curve; theoretical prediction, calculated from equation (11.21) is the dashed curve. Amplitude scale is in arbitrary units. After Nayfeh and Chimenti [169].

We now turn our attention on the general behavior of the reflection coefficient. Since the basis of the experimental data (see section 11.5) is an amplitude signal in the reflected field of the incident beam, meaningful comparisons can be achieved if we investigate the reflection coefficient for the same type of behavior, as is observed in the measurements. Figure 11.7 displays the reflection spectrum of a uniaxial graphite-epoxy plate in water for an incident angle of 12 ~ and wave vector along the fibers. The measurement is the solid curve, and the calculation is the dashed curve. The occurrences of the minima are well explained, and the line shape function follows the measurement closely. In contrast, figure 11.8 shows the measured and predicted spectra for the same incident angle of 12~ but for the azimuthal angle r 30 ~ The two curves have been vertically scaled, but in no other way adjusted. In this case, the solid curve is the theoretical and the dashed curve is the experimental data deconvolved to remove transducer response. Positions of the deep minima in the two curves are nearly coincident, as we have observed in figure 11.7 for propagation in the fiber direction. Apparently, the additional shallower sharp dips, some of which do not appear in the data, arise from the coupling between vertical and horizontal shear displacements which occur, for propagation in a general

11.2. PLATES COMPLETELY IMMERSED IN FLUIDS

12

lO

f)-'-- 12 d

'

'

'

"

'

167

'

~1~-30 ~ tI

'~ : , '~

o

,

;

o

o

9

4

2

f (MHz) Figure 11.8: Reflection plate wave spectra for ~ -- 12 ~ and r = 30 ~ Theory is solid curve and experiment is dashed curve. Data has been normalized to remove transducer response. After Nayfeh and Chimenti [172].

azimuthal direction. In addition to direct comparisons of amplitude spectra, the results can be expressed as dispersion-like curves, where the results correspond to functions conditioned by the reflection coefficient. The theoretical dispersion curves are based on the occurrence of total transmission for given f d and incident angle. Phase velocity is calculated from the incident angle via generalized Snell's law. Numerically, the curves are generated by searching equation (11.21) for the minima in the magnitude of the reflection coefficient. Typically, dispersion curves based upon the above discussed procedure can be experimentally obtained by taking the results of many dozens of experimental spectra and recording the minima as a function of the incident angle (expressed through generalized Snell's law as a phase velocity). We have demonstrated that the coincidence condition of Cremer concerning the occurrence of reflection minima and the excitation of Lamb wave modes is not a useful approximation in all regions of plate wave dispersion in graphiteepoxy composites. Therefore, although we present these data in the manner of a velocity dispersion curve, it must be stressed that it is the reflection properties which are being reported. Figure 11.9 shows comparison between data acquired at an azimuthal angle r = 0 ~ and the corresponding theoretical prediction derived by ex-

168

C H A P T E R 11. FL UID-LOADED SOLIDS

7

~s 0

E o

4

1

2

3

4

5

6

fd (MHz mm)

7

8

Figure 11.9: Reflection results expressed as phase velocity dispersion curves for r = 0 ~ Data are plotted as discrete open circles; theory presented by solid curves. After Nayfeh and Chimenti [172].

o

6

o

4

(D

v

1

0

1

2

3

4

5

fd (MHz mm)

6

7

8

Figure 11.10: Dispersion plot for r = 90 ~ Data are open circles; solid curves are theoretical. After Nayfeh and Chimenti [172].

11.2. PLATES COMPLETELY IMMERSED IN FLUIDS

169

5 o ..~m 4

E to

3 2

1

2

3

4

5

fd (MHz mm}

6

7

8

Figure 11.11" Dispersion-like plots derived from reflection spectra for r 60 ~ together with experimental data. After Nayfeh and Chimenti [172].

amining the behavior of the reflection coefficient, equation (11.21). The theoretical curves are constructed by identifying minima in the reflection coefficient, indicative of total transmission, recorded as a function of f d for many values of phase velocity. In this figure, the data, plotted as open circles, are in excellent agreement with the prediction of the model. Propagation in the other material symmetry direction r = 90 ~ is shown in figure 11.10. Here the results reflect the substantial effective softening of the composite as the fiber axes is rotated out of the plane of incidence. Therefore, we observe a marked reduction in the phase velocities at which certain features occur. The vertical intercept of the curve, which is similar to the So mode, is seen to be near c - 2 . 5 km/sec, in contrast to the corresponding situation of figure 11.9 that occurs near 9.8 km/sec which is not even visible on the figure. If we now depart from the principal axis directions, the reflection behavior becomes substantially more complicated, as can be seen in figure 11.11 for r - 60 ~ The simple structure of figure 11.9 is replaced by curves which split apart, rejoin, and cross over each other. Throughout the range of the measurement, relatively good agreement with the theory is apparent. Although interpretation of these results is delicate, we may note a few arguably consistent trends. As the fiber direction is rotated out of the plane of incidence, the effective material constants begin to decrease, causing the phase velocity intercept of the So-like curve to drop. In figure 11.11, the

CHAPTER 11. FL UID-LOADED SOLIDS

170

-d12

fluid plate >

d/2

X1

vaccum X3

Figure 11.12: The geometry of fluid-plate-vacuum. velocity is about 3.4 km/sec, whereas the value is 9.8 km/sec for propagation along the fibers. Moreover, an additional set of curves seems to have nucleated. These general features are typical of the results for values of r away from the axes of symmetry. The above described curves contain a rich variety of reflection phenomena, considering the relatively simple form of the r - 0 ~ and 90 ~ curves in figures 11.9 and 11.10. As we have stated earlier, these additional features arise from the coupling of the vertically and horizontally polarized waves, which are independent for propagation along principal axes. We note the unusual behavior of mode curving in the low-frequency domain of dispersion curves in figures 11.9-11.11. It is in these domains where the Cremer's principle does not seem to apply. This unusual behavior is less apparent in neighboring modes and diminishes continuously for larger values of f d. 11.2.2

Fluid-plate-vacuum

system

Next we consider the situation where the plate is supported on side by a fluid semi-space while its other side is kept free. This is illustrated in figure 11.12. To obtain the expressions for the reflection and transmission coefficients for the present case, we replace the solid- lower fluid interface conditions with the stress free boundary conditions, namely, we set a33, a13

11.2.

PLATES COMPLETELY

IMMERSED

and 6 2 3 of the solid equal to zero at algebraic equations

" W~E~ -W~E~ WaEa

X3 :

-Wa/~3 WsE~

171

IN FLUIDS

d/2. This leads to the system of

~:

-W~F.~ D l l E 1 D l l E 1 D13E3 D13/~3D15E5 D15/~5-pfc 2 D21 E1 -D21/~1 D23E3 -D23/~3 D25E5 -D25/~5 0 D31 E1 -D31/~1 D33E3 -D33/~3 D35E5 -D35/~5 0 Dll/~1 DllEID13E3 D13E3D15E5-DI5E5 0 D21/~1 -D21 E1D23/~3 -D23E3 D25/~5 -D25E5 0 D31/~1 -D31E1 D33/~3 -D33E3 D35/~5 -D35E5 0

U121

PfC 2

v~31 i o v~41 = o

V1~l U161 R

0

0

0

(11.25) These equations can be manipulated in accordance with the previous section resulting in the following expression A S - i Y (A - S ) / 2 R = A S + i Y (A - S)/2

(11.26)

We find that the reflection coefficient in equation (11.26) is expressed in parameters which are identical to the case of a fully immersed plate. Furthermore, equation (11.26) bears a strong formal resemblance to the reflection coefficient of the fluid-coupled semi-space defined in equation (11.5), in particular as far as the role of the fluid is concerned. Although somewhat less apparent, the real part of R approaches -1 and its phase suffers a sharp change through zero when either S or A vanishes. Also the vanishing of the denominator in this equations recovers the characteristic equation (8.17), namely A S + i Y (A - S ) / 2 = 0

(11.27)

for free waves on the plate that is supported on one side by the fluid. Unlike the previous case of total immersion, the fluid has much less influence on the dispersion behavior in the present situation. In fact, the idea of using Cremer's principle is meaningless since no transmission takes place especially in the absence of material viscosity which has not been, so far, included in the analytical model. Accordingly, for all incident angles and frequencies, all incident energy will be reflected in the fluid. Thus, theoretical results based on the reflection coefficient (11.26) can not be used, in the continued absence of material viscosity, to compare with experimental data obtained for reflected wave amplitudes. Since there is an inherent amount

172

[.1.

12

,

I

,

]

,

Xl=O

10

r', i;', } , <

CHAPTER 11. FL UID-LOADED SOLIDS

:; ,

,

,,

,, .......

,i

6

4

2

2.23

5

10

15

20

24.5

fd (MHz mm) Figure 11.13: Wave amplitude spectrum for an aluminum plate loaded by a fluid on only one side. Solid curve is experiment, and dashed curve is theory incorporating finite-beam and absorption. Incident angle 0 is 18~ After Chimenti and Nayfeh [47].

of viscosity in all material systems, the experimental data measuring the reflected wave amplitude (or energy) will exhibit certain features (minima) suggesting mode identification. To make realistic comparison between calculation of equation (11.26) and results of experimental measurements, we also include the influence of both the finite ultrasonic beam, as will be discussed later on in section 11.4, and material viscosity in the analysis. Each of these factors modifies the plane wave reflection coefficient in an important way. The latter contribution will be more significant here than for the totally immersed plate since, as just mentioned, the traction-free boundary condition at the lower plate surface implies that all incident energy will be either reflected or absorbed. Therefore, even small absorptive losses in the plate could affect the reflected field of the incident beam. Likewise, the finite beam mixes spectral components from a range of angles, further altering the behavior of the reflected beam. The experimental observations in section 11.5 confirm these suppositions. Introducing material viscosity to the analysis can be achieved in several ways. The easiest one is to perturb the properties Ckl by including a small dissipative part. Hence, in the analysis, we replace Ckl by the complex C~l

11.2.

PLATES COMPLETELY

IMMERSED IN FLUIDS

173

such that

C~l = Ckt + i~kl,

i = XflL~.

(11.28)

The only available experimental data on the lower free plate is that of our own [47] which was conducted on an aluminum plate. For the purpose of comparison, the reflection coefficient (11.26) can be easily specialized to the isotropic case by merely implementing the various relations that exist between the material constants Ckz. This will be discussed further in section 11.3 when we deal with higher symmetry materials. For the time being, we shall just present the comparison. With an incident angle of 18 ~ and the transducers 70 m m above the plate surface, the frequency spectrum of figure 11.13 has been obtained. The experimental measurement is the solid curve, whereas the theoretical calculation is the dashed curve. These data have been normalized to remove the frequency-dependent transducer response. In this figure, Xl = 0 and the transducers have a width of 9.5 m m and a center frequency of 2.5 M H z . Values of the damping terms have been chosen on a phenomenological basis to model the data, although they are consistent with the material we have studied. In deriving the theoretical curve of figure 11.13, we have combined the plane wave reflection coefficient of equation (11.26) with the beam model of equation (11.81) to be discussed later on. As expected, the minima in the two curves agree rather well, as do their shapes and relative amplitudes. The only scaling applied to the theory is in the vertical axis, since absolute reflection measurements have not been made. The slight reduction in the amplitude of the oscillatory character of the theoretical curve is a consequence of the presence of viscous damping and approximates the behavior of the data. A small slope in the data is attributable to the transducer normalization process and is not a significant aspect of the results.

11.2.3

The general layered media

The layered plate we consider next separates the upper fluid from the lower bounding media. To maintain generality, we consider a different fluid as in figure 11.14a, vacuum, or a solid substrate as in figure 11.17 as choices for the lower bounding medium. The formal solution for the layered plate is

174

CHAPTER

11.

FL UID-LOADED SOLIDS

upper fluid ~

. - - )

1

X1

2

layered plate n-1 n

lower fluid u

transmitted

xa

Figure 11.14: Geometry of fluid-loaded multilayered plate. given in terms of the global transfer matrix of equation (9.23) as Ul u2 ~t3

+ __

a~3

All A12 A13 A14 A15 A16 A21 A22 A23 A24 A25 A26 A31 A32 A33 A34 A35 A36

A41 A42 A43 A44 A45 A46 A51 A52 A53 A54 A55 A56 A61 A62 A63 A64 A65 A66

(7~3

a~3

Ul u2 u3

a~a a{a a~a

(11.29)

The displacements and stresses within the upper fluid are obtained by properly specializing equation (5.39) with respect to the coordinate system of figure 11.14 as

[Ol] [ 1 1 ]i eox3 ] ~t3

0"~3

=

OZu

-o~u

Pu c2 Pu c2

Re_i~a~,x 3

,

(11.30)

where a2u _ ( 2c / C2u ) - 1. Once again, the overbar describes fluids and the superscript and subscript u denote quantities belonging to the upper fluid. The continuity conditions at the plate- upper fluid interface are u~=u3,

~3=a33,

a13=a33-0,

at

x3-0.

(11.31)

175

11.2. P L A T E S C O M P L E T E L Y I M M E R S E D IN FLUIDS

If the lower bounding medium of the plate is a different fluid, then its formal solution consists of the transmitted component and is given by

[ ]z [ ] fil

1

~t3 a~3

Now,

=

at ptc 2

Te i~at(x3-d)

(11.32)

the appropriate interface conditions at the lower plate surface are

~/ -- U3+,

#/3 "- 0"+33'

0"+ = 0"+ = 0,

at

x3 = d.

(11.33)

Equations (11.30) and (11.32) are now specialized to the locations x3 - 0 and x3 = d, respectively, as a~31ux3=0 -" puc2(1 + R)

u31;3:o -- au(1 - R) -, l 0"33 ]x3=d -- plc2T

(11.34)

U3l x3=d I = azT.

Since the shear stresses on both sides of the solid vanish, equation (11.29) reduces to Ul u2 u3 a~3

+

All __

0 0

A21 A31 A41

A12 A22 A32 A42

A13 A23 A33 A43

A14 A24 A34 A44

_ Ul u2 u3 a~3

A51 A52 A53 A54 A61 A62 A63 A64

"

(11.35)

Implementing the fluid-solid continuity relations (11.31) and (11.33) and using equation (11.34) result in u+

All

A12 A13 A14

u2+

A21 A31

A22 A32

alT plc2T 0 0

=

A23 A33

A24 A34

u1 u2

a u ( 1 - R) puc2(1 + R)

A41 A42 A43 A44 A51 A52 A53 A54 A61 A62 A63 A64

(11.36)

By rearranging, we obtain B~3 B~3

B~3 B~3

__

A31 A41 A51 A61

A32 A42 A52 A62

B33 - a t B43 - D l B53 0 B63 0

u~ sys

R T

(11.37)

176

C H A P T E R 11. FLUID-LOADED SOLIDS

where Du = puc 2 Dl = plc

2

B33 = A34Du

-- Aa30~u

B43 = A44Du

- A430Lu

B53 = A54Du

- A53 OLu

B63 = A64Du

- A63 OLu

(11.38)

and Bi~ is obtained from the corresponding Bij by replacing Du with - D u . Using Cramer's rule, and after minor algebraic manipulations, we solve equation (11.37) for the reflection and transmission coefficients as R =

(M21 --I- Q u M 2 2 ) - Q I ( M l l

+ QuM12)

(M21 - Q u M 2 2 ) - Q I ( M l l

-

T =

QuM12)

(11.40)

-2puc21AI

al[(M21 -

Q,~M22)-

(11.39)

QI(Mll

- QuM12)]

where IA[ is the determinant of the 4 x 4 matrix in equation (11.35) and

Mll

-

A31 A32 A33 A51 A52 A53 A61

M21 -

A62

,

M12

=

A63

A41 A42 A43 A51 A52 A53 , A61 A62 A63

M22 =

A31 A32 A34 A51 A52 A54 A61 A62 A64

(11.41)

A41 A42 A44 A51 A52 A54 A61 A62 A64

(11.42)

with

= p cJ, OLu

= P c2 Oq

(11.43)

The expressions (11.39) and (11.40) for the reflection and transmission coefficients contain, as a by-product, the characteristic equation for the propagation of modified waves on the corresponding structures. The vanishing of the common denominator, namely, (/l//21 - QuM22) - Qt(M11 - QuM12) = 0

(11.44)

defines the characteristic equation for such waves on a plate separating two different fluids. By setting Pl = Pu, we get results for the free waves on

11.2.

P L A T E S C O M P L E T E L Y I M M E R S E D I N FLUIDS

177

1.0 0.8 "~ 0.6

~0.4 0.2

0.01

2

3

4

5

6

7

8

f (iHz) Figure 11.15: Reflection spectrum for [02,902]s laminate with 0 - 16~ and r - 29 ~ Experiment is solid curve, while theory is dashed curve. After Nayfeh and Chimenti [167]. a layered plate immersed in a single fluid. Furthermore, in the absence of both fluids, equation (11.44) reduces to

J~/'21 -- 0

(11.45)

which defines the characteristic equation for the propagation of free waves on a dry multilayered plate. By setting Pt - 0 , equation (11.44) reduces to M21 - Q u M l l : 0

(11.46)

which defines the characteristic equation for free waves on the plate with one side being traction free. Sample comparison with experimental data for a biaxial graphite-epoxy composite plate of [02, 902 Is layup is shown in figure 11.15. The solid curve is the measurement, while the calculation is represented as a dashed curve. Only the relative amplitudes of the two curves have been scaled, since absolute reflectance has not been measured. An incident angle of 16~ is selected, and the fiber direction in the upper layer makes an angle of 29 ~ with respect to the incident plane. Note that nearly all details of the data are reproduced in the calculation. Figure 11.16 shows the result of rotating the incident plane so that it makes an angle of 50 ~ with the fiber direction in the uppermost layer, while maintaining a 16 ~ incident angle. The experimental reflection spectrum has

178

C H A P T E R 11. FL UID-LOADED SOLIDS

1.0 0.8

"~

2

0.6

~ 0.4 0.2 0.0

2

3

4

f (MHz)

5

6

7

8

Figure 11.16: As figure 11.15, but with r - 50 ~ After Nayfeh and Chimenti [167].

changed markedly compared to that of figure 11.15. In fact, even small azimuthal variations can significantly change the appearance of the reflection spectrum. While agreement between experiment and prediction in these two figures is not perfect, nearly every feature in the measurement has a corresponding expression in the theoretical curve. Among other factors, the complicated morphology of the sample, and t h e coupling of the S H motion wave with that of the sagittal plane contribute to the lack of detailed agreement.

11.2.4

Bottom substrate

If the lower bounding medium is a solid substrate, as illustrated in figure 11.17, then the response of ~he total system can be obtained by extending the transfer matrix of the layered plate (11.29) to include the substrate. Thus, we first develop the formal solution for the substrate which insures the boundedness of its displacements and stresses for large values of x3. This can be done by specializing equation (9.12) to the substrate followed by insuring boundedness of the field variables by selecting only the non-zero

179

11.2. P L A T E S C O M P L E T E L Y I M M E R S E D I N FLUIDS

"•~

fluid ~

--> X1

1 2

layered plate

n-1

n

rate

i\, \i

9

i transmitted I

v t

Xa Figure 11.17: Geometry of fluid-multilayered plate-substrate system. appropriate amplitudes leading to Ul

1

1

1

1

1

1

u2

Vl

Vl

V3

V3

V~

V~

W1

W3 D13 D23

-W3 D13 -D23

W5 D15 D25

-W5

O21

-W1 Dll -D21

D31

-D31

D33 -D33

U3 (7~3 a~3

a~3

__

Dll

sub

TI E 1

0 T3E3

o

D15 -025

D35 -D35

(11.47)

TsE5

sub

o

sub

where Eq -- e i~aq(x3+d),

q = 1, 3, 5.

(11.48)

We now recall the formal solution for the layered plate equation (11.29). Specializing the solution for the substrate to x3 - d, and satisfying the plate-substrate interface condition, namely Ul ?.t2 U3

§

__

a~a a~a 0"~3

Ul u2 ~t3

,

a~3 a~3 x --d

0"~3

sub

at x3 = d

(11.49)

180

C H A P T E R 11. FLUID-LOADED SOLIDS

result in

Ul

Pll

u2

P21 P22 P23 P24 P25 P26

0

P31 P32 P33 P34 P35 P36 P41 P42 P43 P44 P45 P46

T3 0

a~3

P51 P52 P53 P54 P55 P56

T5

~7~3

g61 P62 P63 P64 P65 P66

0

?-t3 0:~3

__

P12 P13 P14 P15 P16

T1 (11.50)

sub

where the 6x6 matrix Pij is the product of the inverse of the 6x6 matrix Aij of (11.29) and the 6x6 matrix in equation (11.47). By implementing the upper fluid-solid interface condition we get

u~-

P,1 P13 P15

U2

P21 P23 P25

O~u(1 - R) puc 2 (1 + R) 0

_

P31 /)33 P35 P41 P43 P45 P51 P53 P55

0

P61 P63 P65

T3 T5

(11.51) sub

sys

which can be rearranged as P31

~ Pu c2

O 0

_.

P33

P35

P41 P43 P45 P51 P53 P55 P61 P63 P65

~

T1

- P u c2

T3

0 0

T5 sys

R

(11.52) sub

Using Cramer's rule, we solve equation (11.52) and obtain the reflection and transmission coefficients as Z21 - QuZ22 n = Z21 + QuZ22

(11.53)

T1 =

2Qu (P65P53 - P55P63) Z21 + QuZ22

(11.54)

T3 =

2 Q u (Ps ~P61 - P~s P51) Z21 + QuZ22

(11.55)

T5 =

2Q~ (P63P~1 - P53P61) z2, + Q~z22

(11.56)

where

Z21 --

P41 P43 P45 P51 P53 P55 , P61 P65 P65

Z22=

P31 P33 P35 P51 P53 P55 P61 P65 P65

(11.57)

11.3. HIGHER S Y M M E T R Y CASES

181

The vanishing of the denominator in equation (11.53), namely,

Z21 + QuZ22 = 0

(11.58)

defines the characteristic equation for the propagation of fluid-modified surface waves on the multilayered plate bonded to a solid substrate. In the absence of the fluid, equation (11.59) reduces to Z21 -- 0

(11.59)

recovering the characteristic equation (9.50) for the layered half-space without the fluid.

11.3

Higher symmetry cases

As mentioned on many occasions before, all results obtained in the previous sections for the monoclinic case can, under restricted conditions, also hold for higher symmetry classes such as orthotropic, transversely isotropic, and cubic. These equations apply if the wave propagates along directions other than these axes. For example, this, strictly speaking, implies that equations (11.5)-(11.8) do not hold in their present forms if propagation takes place along an axis of symmetry. Alternatively, results can be obtained from these general expressions asymptotically in the limit. By choosing azimuthal angles arbitrarily close (but not strictly equal) to those of the principal axis we can avoid all numerical difficulties and obtain highly accurate results for such cases.

Biaxial C o m p o s i t e Plate Ultrasonic reflection behavior in biaxial composites is dependent on details of the layering. We present results primarily as frequency spectra from which dispersion curves could also be developed. An example of experimental and theoretical comparison for a four-layer biaxial composite [02,902]s is shown in figure 11.18. The dashed curve is the measurement, while the calculation is demonstrated as a solid curve. Once again, the relative amplitudes of the two curves have been scaled, since absolute reflectance has not been measured. An incident angle of 16 ~ is selected, and the fiber direction in the upper layer is in the incident plane. Nearly all details of the data are reproduced by the model calculation. Figure 11.19 shows the result of rotating the incident plane by 90 ~ while maintaining a 16 ~ incident angle, so that the upper fiber layer is perpendicular to the wave propagation direction. The experimental reflection spectrum is markedly different, both in appearance

CHAPTER 11. FLUID-LOADED SOLIDS

182

lO

, [%902]

,

, "',

,~,-,

/,,- "~*,

/0=16 0

,

~_

+I,00 f~l~"~,.

'}1I.L.i } f" :

,9

<

~

:-,

.

4

O0

1

I , :

',

I

9l

i

2

%,

i

3

f (iHz)

:

4

5

Figure 11.18: Experimental (dashed) and theoretical (solid) reflection spect r u m for the [02,902]s laminate for ~ = 16 ~ and r = 0 ~ After Chimenti and Nayfeh [48].

1.2 1.0

.,

o.e


0.4

i

[0=190=] i O= 160

+=900

,/I

.-"-'_1

/

/

./

~

,.

~

f (MHz)

i

!

" \'

, ,

o+

~176

.

!

o",

:/

I

V I ~

~

".

;

Figure 11.19: As figure 11.18 but with ~ = 16 ~ and r = 90 ~ After Chimenti

and Nayfeh [48].

183

11.4. L E A K Y W A V E S

TRANSMITTER

ECEIVER

t

Null

SOLID P

Leaky Wave Figure 11.20: Schematics of the leaky wave phenomena.

and precise location of the minima. Yet the model calculation follows the data very well. This case, and many others we have examined, exemplify the complexity and non-intuitive nature of the results in layered orthotropic materials. Clearly, a simpler model in which the mechanical properties are averaged through the plate thickness would be unable to account for the difference between the data in these two figures.

11.4

Leaky waves

Leaky wave phenomenon has been playing major roles in the ultrasonic inspection of a large variety of solid structural components. Distortion of the otherwise commonly expected specular reflection of incident finite beam from fluid-solid interfaces defines the foundation of the leaky wave phenomenon. It is now well understood that this distortion consists of a lateral shift (along the interface) of the reflected beam, as well as a distortion of its profile which consists of the appearance of a null zone and a trailing field which becomes weaker as it extends along the interface away from the incident beam location as illustrated schematically in figure 11.20.

C H A P T E R 11. FLUID-LOADED SOLIDS

184

J

•

FLUID SOLID

I

J

•

X3

Figure 11.21: Geometry of incident Gaussian beam on fluid-solid interfaces.

11.4.1

Field of t h e i n c i d e n t finite b e a m

In this section, we present a summary of the theoretical description of the leaky wave as applied to the field of acoustics. For illustration, we consider the situation in figure 11.1 consisting of a fluid-loaded solid substrate having a flat interface occupying the Xl - x2 plane of the Cartesian coordinate system x/. As for the bounded beam that originates in the fluid and strikes the interface, we choose a Gaussian type as shown in figure 11.21. Without loss in generality, further simplicity can be achieved by assuming a two-dimensional variation such that the incident and reflected fields are independent of the x2-coordinate. To facilitate the analysis, we also include a local coordinate system x/' (the incident beam coordinate). To this end, we consider a two-dimensional, finite-width Gaussian beam generated in a fluid at an arbitrary location x3 = - L , and propagating toward the interface at an angle ~9 from the normal to the surface. The profile of the beam is characterized by an effective width 2w that is large compared to the wavelength in the fluid. Therefore, the acoustic field has significant amplitude only for a distance w on either side of the beam axis. We further assume that the beam consists of rays that are parallel to the axis of the beam. The propagation of the beam in the fluid is then described

11.4.

LEAKY

185

WAVES

in terms of its local coordinates by

(11.6o)

Cs(x ,x ) =

where we have arbitrarily chosen unit amplitude and suppressed the time harmonic factor e i•t. Here k f - w / c f is the wavenumber in the fluid. By using the coordinate transformations (X3 -Jr-L)sin 0

X~ ---- Xl COS 0 -

x~ = xl sin0 + (x3 + L)cos0,

(11.61)

the field at the beam location x3 - - L is also des.cribed by Cf(xl,-L)

-

el-- (X1/w0) 2+ikiXl ] x / ~ w0cos0 "

(11.62)

Here ki - k f sin 0 and Wo - w sec 0 is the half-width acoustic beam along the xl-axis. Since the incident beam field is known at the position x3 - - L , it can be determined at any location by solving the field equation in the fluid subject to the boundary conditions (11.62). In order to do this, we refer to the fluid's field equations (5.35) and (5.36). Since the fluid is assumed to be homogeneous, both equations can be combined into a single one by introducing the displacement potential Cf leading to o92r 0x21

02r 1 02r +- 0x23 - c 2f Ot 2 .

(11.63)

For harmonic excitation, this equation can be formally solved by the Fourier transform method, subject to zero initial conditions. The function el(x1, x3) and its Fourier transform Cf(~,x3) are related by dual relations Cf (~, x3) =

J_ (X)

Cf(Xl,X3) = ~

Cf(xl,x3)e

dxl

c~

(11.64) (11 65)

Application of the Fourier transform (11.64) to equation (11,63) leads to

d2 ~)f dx 2

~- (}el = 0

(11.66.)

where ~] = (k} - : 2 ) . Equation (11.66) admits the solution Cf - O f ( ~ ) e iCs(x3+L)

(11.67)

C H A P T E R 11. FL UID-LOADED SOLIDS

186

which, when inverted in accordance with equation (11.65), results in

1/2

Cf(xl,x3) = ~

Of(~)ei~Zl+iCf(z3+L)d~,

(11.68)

oo

where =

/?

(11.69)

Substitution from equation (11.62) into equation (11.69) yields

e-[(~-ki)wo/2]e Of(:)

-

cosO

(11.70)

which, if used in equation (11.68), gives Cf (xl, x3) - ~1

/?

~ e_[(~_k~)wo/2]2ei~Zl+i~f(z3+L ) cos0"d~

(11.71)

Examination of equation (11.71) reveals that it cannot be inverted to give exact analytic results. However, the principal contribution to the integral comes from values of ~ lying in the.vicinity of the incident wavenumber ki. Accordingly, approximate values of Cj'(Xl,X3) can be obtained by expanding ~f in a power series about ~ = ki. Retaining three terms in the expansion leads to -

-

2

r = (k~ - k~)~ - (k~ - k2) 89- 2(k~ - k~)3/2"

(11.72)

Since ki = kf sin 0, ~f becomes ~f = kf cos O - (~ - ki)tan 0 - ( ~ - ki)2/(2kf COS3 0).

(11.73)

Introducing the change of variable

= ki + r/cos O,

(11.74)

equation (11.73) transforms to 772 ~f = kf cosO - rl sinO - ~ . 2kf cos 0

(11.75)

Let us now examine the term gl = i[~xl + ~f(x3 + L)] appearing in the exponential of equation (11.71). Substituting from equations (11.74) and (11.75) and collecting terms yield

gl = ikf[xl sin 0 + (X3 -4- L)cos O] T}2(X3 + L) +ir/[Xl c o s O - (x3 + L ) s i n O ] - i 2kf cosO "

(11.76)

11.4. L E A K Y WAVES

187

With reference to the transformation relations (11.61), we recognize that

gl - i[kfx'3 + ~lX~l-

?]2(X3 -i- L) 2k I cosO ]"

(11.77)

Substituting into equation (11.71) and collecting terms, we finally write

eikfx'3 f_x~ 2~ e-(~we/2)2e~Z'~&7

r

(11.78)

oo

where we used the fact that d~ - c o s O&7 and

[

2_w 2 1+

We

2i(x3 + L) ] . k f w 2 cos 0

(11.79)

By inspection, we recognize equation (11.78) as the Fourier transform of

r

, X'3 ) =

)' e x/,_~w

.

(11.80)

which is also Gaussian and propagates in the x~-direction with an effective complex width of 2We. Note that if two, rather than three, terms are retained in the expansion of equation (11.72), the width of the beam We would reduce to w ; hence, the beam would only be shifted without a change in shape (spreading and reduction in amplitude). The inclusion of the third term in the expansion is responsible for some spreading of the Gaussian beam. 11.4.2

Field of the reflected beam

Upon reflection of the incident beam into the fluid, the beam profile of equation (11.68) is modulated by the reflection coefficient R(~). The reflected beam is then described by

Cf(Xl,X3) - ~I

S

~ y(~)R(~)ei~Zl-i;s(z3-L)d~

(11.81)

where X 3 + L has been replaced by - - X 3 "~- L to indicate reflection. The evaluation of the integral in equation (11.81) is more complicated than the corresponding integral of equation (11.68). This complication is due to the additional dependence of R(~) on ~. In the case of a rigid solid, the reflection coefficient will be unity. For this case, equation (11.81) resembles that of the incident beam except for

188

CHAPTER

11. F L U I D - L O A D E D

the minor change in the sign of X3. Affecting the change from the expression for gl yields the new definition

X3

SOLIDS to - x 3 in

gl = i k / [ x l sin 0 + (L - x3)COS 0] ~72 (k/cos----~" L - x3) +iy[xl cos 0 - (L - x3) sin 0] - i--~-

(11.82)

We now recognize the new coordinates Xl -- Xl cos/~- (L - x3) sint~ X3

- - Xl

sin0 + ( L -

x3) cos ~

(11.83)

and rewrite gl as r]2 ( L - x3) gl = ik/23 + i~721 - i - ~ k/ cos-----7"

(11 84)

The coordinates in equation (11.83) constitute a mirror image system of the incident beam with respect to the interface. These will then lead to a reflected beam that is a mirror image of the one described by equation (11.80) with the modified beam spreading function 2i(L - x3) -2 w2[1 + ] we = kfw 2cos0 "

(11.85)

With these approximations, the reflected field (11.81) takes the form

Cf(Xl'X3)= 11.4.3

eikf x'3 / / 27r

oo

R(~)e-(V~e/2)2eiVX'~d~?

(11.86)

An overview of the reflection coefficient

The reflection coefficient is not, however, dependent on the characteristics of the incident beam. It only depends upon combinations of the properties of the solid and the surrounding fluids. In fact, it is based on the reflection of an infinite train of harmonic waves incident from the fluid on the fluidsolid interface. In most situations, the fluid is chosen as water and hence is modeled as a homogeneous medium. On the other hand, the solid medium sharing the interface with the fluid can be quite complicated especially in its geometrical, local and global properties. Naturally, such complications will be reflected in the properties of the reflected field. In short, the "sirepler" the solid, the simpler the corresponding behavior of its reflected field. By far, the simplest solid, as far as the present application is concerned, is

11.4.

LEAKY

WAVES

189

the homogeneous, isotropic linear elastic semi-space. Significant complications can be brought about in its reflected field by merely assuming it to be anisotropic. The next more complicated case will then belong to finite thickness single material plates. Following this scheme of identification, the next will belong to layered solids which includes a wide variety of geometric and material property combinations. The previous chapters were devoted to the study of these complications. For our subsequent demonstration of the general features of the leaky wave, we choose an anisotropic substrate. The reflection coefficient of which is given by equation (11.5). We have shown previously on several occasions that the vanishing of Av defines propagating surface modes, and that, as Av --+ 0, the real part of the reflection coefficient approaches - 1 and its phase suffers a simultaneous sharp change through zero. Since this characteristic behavior of the reflection coefficient is also typical of that of multilayered plates attached to a solid substrate, the following discussion is also applicable to this more complicated situation. Using this property, at the critical location of surface waves, R(~) can be approximated as R(~) ,~ e is(~)

(11.87)

insuring that [R(~)[ = 1 where S(~) is an arbitrary real function of ~. Following the expansion procedure of equation (11.72), we now expand R(~) in power series about ~ = ki and, by retaining the first three terms, we obtain R(~) ~ r i[S(ki)+(~-ki)S'(ki)+2(~-ki)2s''(ki)].

(11.88)

Here S'(ki) = OS(~)/O~ evaluated at ~ - ki; the same holds for S" (ki). In terms of the ~ - r] relation (11.74), we rewrite equation (11.88) as R (~ ) ,~ ei[ S (k~)+ S' (k~)~ c~ O+} s'' (k~)'72 c~ el.

(11.89)

Multiplying this equation with the expression e gz of equation (11.84) and collecting similar terms lead to the new modified definition of gl as gl = i k f 2 3 + i~[cos 0(x] + S') - (L - x3) sin 0] ~]2 (L - x3) _ cos20S"]. - i - ~ [ kf cos 0

(11.90)

We recognize the expression in the first square bracket as a modified 21 through the appearance of the shift factor S'(ki). We also recognize from the last term, a modification of the beam spreading function as 2i(L - x3) 2i cos 2 0S" w~2 = w211 + k f w 2 cos0 w2 ]"

(11.91)

C H A P T E R 11. FL UID-LOADED SOLIDS

190

The inclusion of the second term in the expansion was also retained by Brekhovskikh [28]. From the above modifications we deduce that, apart from a constant multiplier, the influence of R(~) is twofold; first, it accounts for the lateral displacement of the reflected beam along the x 1-axis through the relation

AS -- - S ' ( k i )

(11.92)

and, second, the beam width is modified through an axial shift [18] to the new complex width w~. Both of these effects exist regardless of the value of x3. The subscript S on the displacement parameter As refers to , Schoch the developer of this formalism for isotropic materials; we retain this terminology because of its widespread use in the literature. Close to the surface wave excitation, and guided by the relation (11.13), a specific approximation of R can be conveniently adopted as R ~ ~ - ~R + i(f -- ~R -- i(f"

(11.93)

We note that the denominator is the complex conjugate of the numerator. This expression clearly shows that the pole of R is given by ~ = ~R § i(f, consistent with the notation and conclusions arrived at in section 11.1. It now remains to derive specific expressions for S'(ki) and S"(ki). This can be easily done by differentiating equation (11.87) with respect to ~. Noting that here IR(~)I is unity, we have

S'(~) = iR(~)

(11.94)

which, together with equation (11.93), lead to -2(f S'(~) -- (~ _ ~R)2 -t- (f2"

(11.95)

In particular, for incidence at the Rayleigh wave angle, this equation reduces to 2 S'(~)-- 5. (11.96) 11.4.4

Rayleigh pole

Because of the presence of the pole ~n + i(f, a simplified canonical form of R(~) must account for the pole whenever ~ is near ~n 9 We can then conveniently rewrite equation (11.93) as (see Bertoni and Tamir [19]) R(~) = ~ - ~p

(11.97)

11.4. L E A K Y WAVES

191

where ~p is the pole of R and ~p is its conjugate. Comparing with expression (11.93), we see that ~p - ~R + i5. It is also convenient to rewrite equation (11.97) as the sum of two components in accordance with R(~) - R0 + n l (~)

(11.98)

with k i - ~p R0 = k i - ~ p

(11.99)

and R~(~) = ~p ; • ~~ -_ ~p. ki ~p -- ~ ki

(11.100)

Since R0 is independent of ~, it contributes to the reflection part that is associated with specular reflection. Here R1 (~) is regarded as a correction term deviating from the specular reflection and thus describing the leaky wave contribution. Substituting from equations (11.98)-(11.100) into equation (11.86), we find that the reflected field is the sum of two field components which follow the decomposition of R. These are given by

C / = Csp + @w

(11.101)

defining the specular and leaky wave components

Csp -- ~Ro e i k f e 3 / /

e-

(rpDe/2)

2

e *vxl d~, "

-

(11.102)

(X)

@w = _~_~Aeikse3 / / ~ : _- ki e-(~/2)2eine: drl

(11.103)

where A = :p - ~

.

(11.104)

Equation (11 9103) can be split into two parts @w = ,~(1) v'lw + a(2) wlw" With

,a(1)

A eikfe3

Wlw ~ - ~

/? e-(n~/2)2einel drl o~

(11.105)

and (2) A eiks~3 f / ~p -- ki _(n~/2)2 9lw = ~ o~ 7/cos 0 -- (~p -- ki) e eZ~Z~dr~

(11.106)

C H A P T E R 11. F L U I D - L O A D E D SOLIDS

192

where we have used equation (11.74). By inspection, we recognize Csp(Xl

X3) --

R~ x / ~

e

eikyx3

(11.107)

and (1)

lw ( X l , X 3 )

--

Ae -(~1/~e)2 eikf~a ~/r~?~ge

.

(11.108)

The integral in equation (11.106) is not as easily determined. This is due to the presence of an r/-dependent term in the denominator of its integrand. Inversion of this component can bd'accomplished by using the residue theorem. A simplified version of this expression, namely, when x3 is specialized to L, has been inverted by Tamir and Bertoni [234]. Using similar steps we obtain the present general expression as (2) iAe-(~/~')2({P - ki)eiksXae~2erf c(7) lw(Xl'X3) = 2cosO

(11.109)

where

ave i~e(ki - {R) Y - 2cosO ~ 2cosO

5:1 @e

(11.110)

and erfc(7) is the complementary error function. The specialized expression obtained by Bertoni and Tamir [19] which holds at the interface can be obtained from the general results (11.106) and (11.107) as follows: Setting x3 - L reduces We to w. Also 5:1 and 5:3 reduce to Xl cos 0 and x3 sin 0, respectively. Finally, using the relation w0 = w~ cos 0 recovers the results derived in reference [19]. Further simplifications result for incidence beam At the Rayleigh angle; here {R -- ki, 7 reduces to w o / A s - x l / w o and R0 to-1. Implementing these specialized conditions, the total reflected field consists of the specular and leaky wave components e-(Xl/WO) 2

r

r

= g'~wo cosO e < ~ , X3) = - 2r

~ wA---Te'Rerfc(TR)], 0 ~2 - Vrr

(11.111) (11.112)

where

w0 VR = As

Xl WO"

(11.113)

11.4. LEAKY WAVES

193 w 9

r

0'00

,.' 0.5 ss 9

-..~,.,

i 1 \

~

-o.5

2

3

4

5 X/Wo

/"r //

Figure 11.22: Variation of reflected field components along the interface, for the case wo/As = 0.25. To show this variation, the factor Roe(ik~z)/(Trl/2WoCOSO) in the functions has been replaced by unity. After Bertoni and Tamir [19]. 11.4.5

Reflected beam profile

To obtain a better understanding of the interference between Csp and r we note from equation (11.113) that VR is a function of only two quantities: xl/wo and wo/As. The latter is a constant for any given beam and interface, so that it may be taken as a fixed parameter. For any value of WO/As, 'TR changes sign only once as xl/wo varies between - c ~ and +c~. 1 As an illustration, Csp and r normalized to Ro/(Tr~wo cos 0) are plotted in figure 11.22 for wo/As = 0.25, the phase term e i(kixl) being suppressed for simplicity. Because of the change in sign discussed above, Czw consists of two portions separated by a null; the left-hand portion is in anti-phase and the right-hand one is in phase with Csp, as shown in figure 11.22. As a result, the superposition of Csp and r yields a net reflected field whose peak is displaced to the right. Furthermore, because the amplitude of CLw is larger than Csp for most negative values of x l, the total reflected field exhibits a null and a small secondary peak on the left. On the right, the total field decreases more slowly than Csp. Hence the reflected beam appears to be accompanied by a weaker trailing field, which was seen to be due to the leaky Rayleigh wave excited by the incident beam. As was mentioned in subsection 11.4.3, once R(~) for a given fluid-solid system is specified, the task

194

C H A P T E R 11. F L U I D - L O A D E D S O L I D S 3.0

2.8

2.6

rO~ 2.4

2.2

2Oo

~

~

~

Q

~

~

Figure 11.23: Dispersion of the fundamental mode of a copper plate over a steel substrate. Discrete points are experimental data. After Chimenti, Nayfeh and Butler [50]. of calculating the reflected beam profile becomes straightforward. We recall that the above analyses of leaky waves were conducted for semi-infinite solid systems consisting of either a single substrate or a multilayered plate attached to a substrate. The reflection coefficient R(~) for these systems have all of the characteristics covered in section 11.4. At the coincidence angle of Rayleigh surface waves, we calculate (either analytically or numerically) the pole ~p from which we obtain the Rayleigh wave speed CR and the beam displacement parameter A s as, cR = co/Re({p)

(11.114)

As = 2 / I m ( { p ) .

(11.115)

The value of the calculated Rayleigh wave speed as a function of Q = 2wd/cs is shown in figure 11.23 together with the experimental results on a sample of copper-coated stainless steel. Here Cs is the shear wave speed in the stainless steel and d is the thickness of the copper coating. Data are collected by searching for the characteristic minimum, in the reflected field distribution which indicates the Rayleigh angle. Agreement with the theoretical model is quite good from Q = 0 to Q - 3, where the Rayleigh wave speed has nearly reached the asymptotic value appropriate for the elastic properties of the layer.

11.4. LEAKY WAVES

195

50

Copper/Steel 0

40

30 O O

200

'

,

2

,

,

,

;

,

6

Q

Figure 11.24: Normalized beam displacement parameter vs Q. Data are plotted as discrete points, while solid curve is theory. After Chimenti [43].

Figure 11.24 shows the beam displacement parameter As normalized by the acoustic wavelength in the fluid as a function of Q. The quantity As/)~f is not explicitly frequency dependent and would be a constant at all frequencies if the copper coating was absent. The strong dependence of As/)~f on Q, particularly between Q = 0 and Q = 2, indicates the substantial effect of the layer on the leaky wave. For Q - 0 or Q >> 1, As assumes limiting values corresponding respectively to a halfspace of the substrate material alone or the layer material alone. The reflected field amplitude distribution for a 2.5 M H z bounded beam incident at the Rayleigh coincidence angle of 37.5 ~ on a copper layer-steel substrate is shown in figure 11.25. Experimental points are indicated by the open circles, and the solid curve is the model calculation. From this figure, we see that agreement between the model calculation and the experiment is rather good. The shapes of the displaced leaky wave beam and the remnant specular reflection to the left of Xl = 0 are both accurately reproduced by the theory. Some disparity can be noted in the tails of the distribution, where diffraction effects, perhaps caused by finite x2 extent of the transducer together with a small misalignment, appear in the data. Also, the degree of cancellation of the specular reflection and leaky wave in antiphase near Xl = 0 shown by the numerical results is less than that observed experimentally. This can perhaps be attributed to poor resolution of the numerical procedure.

C H A P T E R 11. FLUID-LOADED SOLIDS

196

1.0

.

,

9

!

,

0.8

-~ m

0.6

e~

E <:

0.4

fi :

0.2

o.%

-2

0

2

4

Position (cm) Figure 11.25: Reflected field amplitude distribution as a function of receiver position at a frequency of 2.5 MHz and an incident angle of 37.5 ~ layer thickness is 0.37 m m and Q - 1.86. Solid curve is theoretical predication and open circles are data. After Chimenti, Nayfeh and Butler [50].

=

1.0

r

'

i

i

i

2

4

6

(

0.8 0

o 9 0.6 . . .

~

o

0.0 0" //. -

"

"2

0

Position (cm)

Figure 11.26: Reflected field amplitude distribution as a function of receiver position at 2.5 MHz and 0.2 ~ below the Rayleigh critical angle 0R. Layer thickness is 0.14 m m and Q = 0.68. Solid curve is theory, and open circles are data. After Chimenti, Nayfeh and Butler [50].

11.5. EXPERIMENTAL TECHNIQUE

197

Figure 11.26 shows the results for a different sample of copper-loaded stainless steel, where the beam is not incident at the Rayleigh angle, and Q = 0.68. Away from the Rayleigh angle, coupling to the leaky wave weakens, and the two components of the reflected field are nearly equal. Still farther from OR, the peaks would merge into one, which would then be Gaussian in shape and centered on xl - 0. Reasonably good agreement is evidenced in the data of this figure, particularly in the modeling of the relative peak amplitudes, their separation, and the depth of the null.

11.5

Experimental technique

As has been demonstrated so far, the analysis of wave propagation in terms of reflection properties of the fluid-coupled structure under study has been so far discussed. The remainder of this chapter will describe experimental procedure used to compare with the analytical models. Experimentally, the following description is taken from Chimenti and Nayfeh [49]. To obtain the reflection characteristics of fluid-coupled plates we direct a sound beam from an ultrasonic piston transducer onto the surface of the plate at a selected angle. The plate normal vector and the incident sound wavevector define the incident plane of the ultrasound. A second transducer of nominally identical frequency bandwidth and sensitivity characteristics, whose axis is also in the incident plane, receives the reflected signal from the plate. The receiver is oriented at the negative incident angle, and the point of intersection of its axis with the surface of the plate may either coincide with that of the incident transducer or may be displaced from it in the propagation direction. The situation is illustrated schematically in figure 11.20, where the typical distortion and displacement of the reflected field from its specular value is shown. These effects occur when conditions are favorable for the generation of propagating waves in the plate. Both transducers are positioned a distance above the plate which places it in their far field. To remove the effects of the frequency-dependent transducer response and the acoustic attenuation in the fluid, a series of reference curves are recorded by measuring the ultrasonic reflection at several incident angles from a thick homogeneous isotropic plate, considered to be a halfspace. These results are later combined with the experimental data to produce spectra which may be compared to the theory. We have chosen to excite the transmitting transducer with radio- frequency (r f) tone bursts of 20-40 # sec duration, repeated at intervals of about 0.4-1.0 msec. Between 0.75 to 12 MHz this burst length is sufficient

198

C H A P T E R 11. F L U I D - L O A D E D SOLIDS

to define the r f accurately, while largely avoiding potential confusion and artifacts from overlapping signals. In addition, this mode of excitation has the advantage that it permits effective use of the skirts of the transducer response curve. After interaction with the composite plate the experimental signal is amplified from 60 to 80 dB in several low-noise, wide bandwidth amplifiers and is video detected to reveal the r f envelope of the tone burst. The amplitude of the video signal is synchronously measured using a gated, integrating amplifier, further improving the signal-to-noise ratio. At the end of the amplifier chain, the principal contribution to amplitude uncertainty is the overlap echoes from the water surface, and these can be effectively suppressed by a judicious choice of tone-burst repetition rate. The resultant dc amplitude is then sampled by a 13-bit analog-to-digital converter and sent over IEEE-488 bus lines to computer storage for further off-line processing. To obtain frequency spectra with our system, the tone burst r f frequency is scanned under computer control in discrete steps across the frequency range of interest, halting at each value long enough to allow the boxcar integrator to settle before sampling its output. Referring once again to figure 11.20, the geometry of the experiment consists of wideband transducers transmitting and receiving ultrasound through the fluid coupling medium, which in all cases is distilled water, to the plate completely immersed in the fluid. At the incident angles and frequencies where reflection minima occur (also corresponding generally to mode conversion between the acoustic and plate wave modes), we expect, under certain conditions, a characteristic displacement and distortion of the reflected field of the incident beam. The shaded regions in figure 11.20 illustrate this behavior schematically. For composites, the reflected field profile may deviate somewhat from this idealized behavior. Also, the transmitted field below the plate has been omitted for clarity. To position the receiving transducer or scan the reflected field, the receiver coordinate is mechanically varied by a precision ball nut and screw combination, giving 0.025 m m resolution with a linear accuracy of one part in 10,000 over 30 cm. We establish the transducer angles through the use of optical rotation stages having 0.001 ~ resolution. The plate normal is identified ultrasonically to within + 0.05 ~ A tungsten foil pressed tightly against the sample surface is used to make an approximate determination of receiver position corresponding to specular reflection. A detailed description of the specimens used in the experiments can be found in Chimenti and Nayfeh [49].

11.5. EXPERIMENTAL TECHNIQUE

199

Signal analysis and data reduction The data obtained using the above technique require very little manipulation for direct comparison to the model calculations. As mentioned above, normalization of the frequency spectra removes the transducer response function from the data. This step is performed by dividing the interpolated video-detected output of a quasi-halfspace reflection into the experimental video signal. Conventional Wiener filtering is not performed in this case since the signals have been stripped of their phase content in the envelope detection. The experimental use of finite-aperture piston transducers implies that an additional analysis procedure should be performed to compare the measured spectra with the calculated plane-wave reflection coefficients. A finite ultrasonic beam is composed of an angular spectrum of plane wave components which define its real-space profile. After their interaction with the solid, each component of the incident beam will contribute to the reflected field weighted by the appropriate value of the reflection coefficient for that 0. Then, the total reflected field will be the sum of these contributions, in accordance with equation (11.86). In most of the comparisons presented in the previous sections, we have found little significant difference between the finite-beam and plane-wave results, so long as the transducer sensitivity axes intersect near the plate surface. Therefore, this additional computational step, for most of the data on the composite plates, can often be deleted. However, the finite beam effect should be taken into account in the calculations whenever the plane-wave energy reflection coefficient is unity. This includes solid substrates above the second critical angle or bottom free plates.

This Page Intentionally Left Blank

Chapter 12

PIEZOELECTRIC EFFECTS In this chapter we extend many of the results presented in previous chapters to include the effects of piezoelectricity. The importance of these effects can be easily appreciated when one realizes that piezoelectric coupling provides the physical basis for almost all practical applications of acoustic fields. They provide an effective means for electrically generating and detecting acoustic deformations. For these reasons and others, it is necessary to establish a mathematical formalism for coupled electromagnetic and acoustic fields. The coupling is simply based on the fact that certain materials become electrically polarized when they are mechanically excited and that they are strained when placed in an electric field. This leads to a coupling between the electromechanical field equations governing the behavior of such materials. This interesting property of piezoelectric materials has recently given them the name of " smart materials". Besides their traditional use as transducer materials, they are now widely investigated for possible use in other applications such as controllers and actuators. In the subsequent sections we follow the logical expositions dealt with in the previous chapters and start with the derivation of the characteristic equation for surface waves on a semi-space. This will be followed by obtaining the characteristic equation for a free plate made up of a single material and subsequently carried further to the multilayered plate case. We exploit situations which give rise to phenomena absent in the purely elastic case. The influence of loading fluids will finally be included and the reflection and transmission coefficients for several solid systems will be derived. We close this chapter with some remarks on the general layered piezoelectric media. The material covered in this chapter can be found in expanded forms in 201

202

C H A P T E R 12. P I E Z O E L E C T R I C E F F E C T S

Nayfeh and Chien [165, 166] and Chien [42].

12.1

Basic relations of piezoelectric materials

The linear piezoelectric description of solids involves combinations of Newton's law of motion (see, for example, Tiersten [239]) Oaij C02Ui OXj = p Ot 2

(12.1)

and the Maxwell's equations, in the absence of conduction currents and free charges, OEj _ eij k OXk

OBi

-

Ot

cOHj _ ODi eij k COXk

-

(12.2) (12.3)

Ot

OB~ = 0 Oxi OD~ = 0. Oxi

(12.4) (12.5)

The associated constitutive relations for piezoelectric materials are given by the coupled relations aij -- CijklSkl -- ekijEk

(12.6)

Dk = ekij~ij q- ekiEi,

(12.7)

where Sij, Cijkl, ekij and eki representing the mechanical strains, the elastic stiffness constants, the piezoelectric stress constants, and the dielectric permittivities, respectively According to reciprocity and symmetry conditions, the elastic material constants satisfy the conditions described in chapter 2 and the piezoelastic and electric constants satisfy ekij -- ekji,

s

=Eik.

(12.8)

As is easily seen from the above relations, the fundamental physical laws describing these fields are not altered by piezoelectricity. Piezoelectric effects enter only through coupling of their constitutive relations. Furthermore, it has been commonly known that the presence of acoustic waves does not cause electromagnetic radiation, and, as a consequence, this does

12.2. SIMPLIFIED FIELD E Q U A T I O N S

203

not allow a coupling between the electric and the magnetic fields in equations (12.2) and (12.3). This quasi-static approximation leads to significant simplification in the mathematical formalism of the piezoelectric system. The above quoted equations constitute the foundation for the study of the influence of piezoelectric effects on the acoustic field. We shall develop formal solutions of the simplified model. In keeping with the spirit of the previous chapters, we limit our study to cases involving monoclinic and higher symmetry classes of materials. We demonstrate in subsequent sections that, as a consequence of the piezoelectric coupling effects, two types of monoclinic materials are possible to exist. This will be followed by utilization of the formal solutions to the study of surface and plate waves, as well as a variety of guided waves in piezoelectric systems.

12.2

Simplified field equations

Consider a piezoelectric anisotropic medium possessing monoclinic symmetry and oriented such that its crystollographical axes coincide with the global reference Cartesian coordinate system xi - (Xl,X2, X3). For consistency in our differentiation between the primed (crystolograpical) and the transformed (global) coordinate systems of chapter 2, this means that both are now identical. This can also be interpreted in the sense that we are now conducting analyses in the "transformed" system via the trivial rotation r - 0~ The plane Xl - x 2 is chosen to coincide with the implied single plane of symmetry. It thus follows that the x3-coordinate is the medium's normal to the plane of symmetry as illustrated on several occasions in previous chapters (see, for example, figures 11.1 and 11.3). With respect to this coordinate system, the coupled piezoelectric field equations are given by the equations of motion and electrostatic charge

02uk 02r 02Ui Cijkl OXlOXj + ekij OXkOXj = p Ot 2 02ui

02r

ekij OXkOXj -- eki OXkOXi

and the associated relations

0r

O'ij = CijklSkl + ekij OXk

or

Dk -- ekijSij -- eki o x i =

10ui

+

Ouj )

-- 0

(12.9)

204 Ek =

CHAPTER 12. PIEZOELECTRIC EFFECTS

0r OXk

(12.10)

reflecting the fact that we introduced the scalar potential r in accordance with the last component in equation (12.10). In arriving at these specific expressions, the various properties and symmetries of the involved terms are exploited as previously introduced in chapter 2.

12.3

Analysis

For the monoclinic symmetry, as was demonstrated in previous chapters, we expect dramatic algebraic simplifications in the analysis and in the final results. For this reason, we shall limit the treatment to cases involving monoclinic and higher symmetry materials. However, due to the presence of piezoelectric coupling, there are two classes of such materials systems: these belong to a "2" or "m" groups, described separately as (711 (722 (733 o'23 (713 (712 DI

D2 D3

Cll C12

C13 623

0 0

0 0

C16 C26

0 0

0 0

-e31 -e32

C13 C23 C33

0

0

C36

0

0

-e33

C44 C45 0 el4 e24 0

C45 C55 0 el5 e25 0

0 0 C66 0 0 e36

-el4 -e15 0 (~11 el2 0

-e24 -e25 0 ([12 ~22 0

0 0 -e36 0 0 ~33

0 0 C16 0 0 e31

C12 622 0 0 C26 0 0 e32

0 0 C36 0 0 e33

"$11 $22 1 $33 ")'23 ~13 '~12 E1

E2 E3 (12.11)

(711 (722 (733 (723 (713 = (712

D1 D2 D3

"Cli

612

C13

0

C12 C22 C23 0 C13 C23 C33 0 0 0 0 C44 0

C16 ell e21 0

0 626 e12 e22 0

0 636 e13 e23 0

C45 0 0 0 e34

0

C16

-611

-e21

0

"Sll"

0

C26 -e12 C36 -e13

-e22 -e23

0

$22 $33

0 0 C66 e16 e26 0

0 0 - e26 s e22 0

0 C45 C55 0 0 0 e35

0 0 - e 16 ell el2 0

0 -e34 -e35 0 0 0 e33

"/23 ~/13 ")'12 E1

E2 E3 (12.12)

Here "~ij = 2Sij, i ~ j are the shear strains as previously encountered in chapter 2. Also, the contracted subscript notation for the elastic properties

12.4. F O R M A L S O L U T I O N S

205

are those described in chapter 2. We adopt the same contracting procedure for the piezoelectric coupling coefficients ekij, thus e14 stands for e123, for example. By inspecting equations (12.11) and (12.12), we note that the purely elastic or electric portions of these two classes are identical whereas the coupled portions are different. In fact, by further examination, we conclude that the vanishing elements in one correspond to the nonvanishing elements in the other, namely, there are no common nonvanishing coupling terms. Such unique property has important consequences in the manner in which the various waves interact. For this reason, these two cases can be treated separately. It is demonstrated that, upon presenting solutions for one case, results for the second case can be identified by inspection. Accordingly, in the present treatment, we limit "the study to the case of monoclinic-2 symmetries.

12.4

Formal solutions

For the monoclinic-2 class, the particle motion has the three non-zero spatial displacement components u l, u2 and u3 and the electric potential r Formal solutions of the coupled piezoelectric equations can be obtained by following the same procedure used in the previous chapters in the absence of piezoelectric coupling effects. Also, the treatment follows, to a large extent, those of references [165, 166]. For plane waves propagating along the xl-axis and independent of the x2-coordinate, a formal solution for ui and r can be written as (see equation (5.1))

(ltl, U2, it3, r ---- (U1, U2, U3, U4) exp i~(xl+ax3-ct)

(12.13)

where a is still an unknown parameter, and (U1, U2, U3, U4) are displacement and electric potential amplitudes. From now on, we recognize and suppress the common factor e i~(xl-ct). Substituting from equation (12.13) into equation (12.9) yields four linear homogeneous coupled equations in the amplitudes, namely

Kmn (a)Un = 0,

m, n = 1, 2, 3, 4,

(12.14)

where Un = (U1, U2, U3, u4)T, the summation convention is implied and

K l l (c~) - Cll - pc 2 + C55a 2 K12(a) - C16 + C45 0~2

K13(a) -- (C13 -}- C55)oL

CHAPTER 12. PIEZOELECTRIC EFFECTS

206 K 1 4 (c~) -- ( e l 5 -+- e31)ol K 2 2 (ol) - - C66 -

tic2 -~- C44 0~2

K23( ) = (C36 + C45) K24(0l)

- - ( e l 4 -~- e36)oz

K33(a) = C5~ K34(a)

-

C33a 2

pc 2 +

- - e l 5 -+- e33 a 2

K 4 4 (c~) - - - ( e l l

"~- e33ot2).

(12.15)

subject to the symmetry property Kmn = Knm. Nontrivial solutions for [71, U2, U3, and U4 demand the vanishing of the determinant in (12.14) and yield an algebraic equation relating a to c. This equation results in an eighth-degree polynomial equation in a which can be written formally as oL8 -+- AlO~ 6 -+- A20L 4 -+- A30~ 2 -+- A 4 :

(12.16)

0,

where the coefficients A1, A2, A3 and A4 can be easily extracted from equation (12.14) and, for the sake of saving space, do not need to be repeated here. Equation (12.16) admits four solutions for c~2 and leads to eight solutions for c~ having the properties Ol2 - - --O~1,

Ol4 = --C~3,

C~6 = --C~5,

C~8 = --aT.

(12.17)

For each C~q,we use equation (12.14) to relate the field ratios, Vq = U2q/Ulq, Wq = U3q/Ulq and Oq = U4q/Ulq as Kll

(K33K24-K23K34)+K12(K13K34-K14K33)+K13(K14K23-K13K24)

K 12 ( K23 K34 - K24 K33 ) + K 13 (K23 K24 - K22 K34 ) + K 14 ( K22 K33 - K23 K23 )

Wq

K11(K22K34- K24K23)K12(KI 3K24- KI2 K34)~-K14(KI2K23- KI3 K22)

K 12 ( K23 K34 - K24 K33 ) -{-K 13 ( K23 K24 - K22 K34 ) + K 14 (K22 K33 - K23 K23 )

~q __ Kll(K23K23-K22K33)+K12(K12K33-KI3K23)+K13(KI3K22-K12K23) (12.18) K 12 (K23 K34 - K24 K33 ) + K 13 (K23 K24 - K22 K34 ) + K 14 ( K22 K33 - K23 K23 )

Combining equations (12.18) with the constitutive relations (12.11), and using superposition, we write the formal solutions for the displacements, stresses, electric potential and electric displacement as 8 ( U l , U2, U3, (~) - -

~ ( 1 , Vq, Wq, ~Pq)Ulq exp i~q=3

(12.19)

q:l 8 (ff33, a 1 3 , a23, D 3 )

:

E q=l

i~(Dlq, D2q, D3q, D4q)Ulq expi~aqx3

(12.20)

12.4. F O R M A L SOLUTIONS

207

where

Dlq = C13 + C36Vq + C33o~qWq + e33O~qOq D2q = C55o~q + C45oLqVq + C55Wq + e15Oq

D3q -- C450Lq+ 6440zqVq -4--645Wq -4- el4(I)q D4q = e31 + e36Vq + e33aqWq - e33aqOq.

(12.21)

With reference to the relations (12.17), by inspection of equations (12.18) and (12.21), we deduce the important properties

Yj---~ Yj+l,

Wj---Wj+l,

D13 = Dlj+I, D2j = -D2j+I, D3j = - D 3 j + I , D4j = D4j+I,

Oj---Oj+l, j = 1,3,5,7.

(12.22)

The formal solutions (12.19) and (12.20) with their unknown wave amplitudes Ulq, together the various parameter definitions and restrictions expressed in equations (12.17)-(12.22), are the foundation for the study of a wide variety of piezoelectric systems. Using these formal solutions, with appropriate boundary and interface conditions, lead to final desired results. Fortunately, we find the algebraic procedures for finding the required solutions to be identical to those used in the preceding chapters. For this reason, we can proceed to identify the desired cases and then refer to their previous counterparts (i.e., in the absence of piezoelectric effects) for a guidance to the algebraic reductions and manipulations. We shall attempt to maintain or modify the definitions of involved parameters to reflect the influence of piezoelectricity. 12.4.1

Surface waves

Consider now a semi-space whose free surface coincides with the x l-X2 co.ordinates such that x3 increases awav from this surface into the solid. For surface waves to exist, the total wave field (displacements, stresses and electric potential) have to stay bounded as x3 increases indefinitely. This is known as the radiation condition. Due to the properties (12.17), only four of the eight wave components in equations (12.19) and (12.20) can satisfy this requirement. These are obtained by choosing the four a's which have positive imaginary parts, arbitrarily identified, for convenience, as al, a3, a5, and aT. Thus, the applicable formal solutions for surface waves are

(ltl, it2, it3, r

--

~ (1, Vq, Wq, Cq)Ulq exp i~aqx3 q=1,3,5,7

(12.23)

CHAPTER 12. PIEZOELECTRIC EFFECTS

208

i~(Dlq, D2q, D3q, D4q)Ulq exp i~aqx3 9 (12.24)

(0"33, O"13, 0"23, D3) -q-1,3,5,7

At this stage, we identify two types of free waves that can possibly propagate along the free surface. The first requires, besides the vanishing of the stress components, the vanishing of the electric potential r at the free surface, namely at x3 - 0 . This situation is referred to as the "shorted" condition. The second situation requires the vanishing of the electric displacement rather than the electric potential. This situation is known as the "free" case. With reference to the formal solutions (12.19) and (12.20), it is obvious that, once solutions are obtained for one case, say for the shorted case, results for the free case can be obtained by replacing (I)q with D4q. Choosing the shorted case, the vanishing of the three stress components a33, Crl3, 623 and r on the free surface x3 = 0 results in the characteristic equation DII D21 D31 (I)1

D13 D15 D17 D23 D25 D27 D33 D35 D37 (I)3 (1)5 (I)7

- 0.

(12.25)

This can be expanded to give the alternative form (12.26)

Av = D11GI - D13G3 -4- DI5G5 - D17G7 -- 0 where Gq are given as

el -

G5 --

(I)3

(I)5

(I)7

D23 D33

D25 D35

D27

(I)l

(I)3

D21 D23

~7 D27

D31

D37

D33

,

G3 -

,

G7 -

D37

(I)1 D21 D31

(I)1

D21 D31

(I)5 D25 D35

(I)7 D27 D37

(I)3

(I)5

D23 D25 D33

D35

The solution that satisfies the characteristic equation (12.26) is not dependent upon the frequency but is still a function of the azimuthal angle. This implies that the surface wave is not dispersive but its wave speed depends upon the azimuthal angle.

12.5.

HIGHER SYMMETRIC

12.4.2

MATERIALS

209

Free plate modes

For the case of a finite thickness plate, the radiation conditions are no longer required and all wave components of the formal solutions (12.19) and (12.20) are needed. Once again, choosing for illustration the shorted conditions, by requiring the vanishing of the stress components a13,623, and 633 and the electric potential r on both of the plate's free faces x3 = =kd, we obtain a system of eight linear simultaneous equations for the amplitudes Ulq. We immediately recognize that this equation is similar to, and constitutes a modification to the characteristic equation derived in section 8.2 for the purely acoustic case. By implementing the various wave amplitudes ratios (12.18) into this equation and subsequently following, step by step, the algebraic manipulations outlined in section 8.2 we finally arrive at the simple characteristic equation (12.27)

AS = 0

where S -- D11Gl~'l - D13G3~'3 -!- D15GsT~ - D17GT~'7 A = DllG1T1 - D13G3T3 -t- D15G5T5 - D17G7T7

(12.28)

with Tj = t a n ( T a j ) ,

Tj - 1 / T j ,

7=~d/2

(12.29)

consistent with the format adopted in chapter 8. Like in the purely acoustic case, S and A, represent the symmetric and anti symmetric modes for the free waves in the piezoelectric plate.

12.5

Higher s y m m e t r i c materials

The next higher symmetry class belongs to orthotropic materials. Results for orthotropic and higher symmetry classes are undoubtedly contained in the previous results derived for the monoclinic symmetry class and should, at least in principle, be obtained as special cases. Care is however required when trying to extract specialized solutions from the general ones. Since, as was discussed on several previous occasions for the purely acoustic case, orthotropic and higher symmetry materials differ from monoclinic materials in that they posses extra two axes of symmetry in the plane, singularity might be encountered for situations involving propagation along one of these axes of symmetry. This, of course, suggests that, for propagation along any direction other than the symmetry axes, the use of linear transformation

210

CHAPTER 12. PIEZOELECTRIC EFFECTS

can be used in conjunction with the general solutions to obtain specialized results. 12.5.1

Orthotropic-222

In the case of purely acoustic materials, we found that the horizontally polarized S H wave uncouples from the one associated with the sagittal plane (namely the coupled quasi-longitudinal and vertically polarized quasi-shear S V waves) when propagation takes place along an in-plane axis of symmetry. The obvious question which now needs to be answered is whether such uncoupling occurs in piezoelectric materials and, if it does, what will be the role of the piezoelectric coupling. In order to answer this question, we first specialize the constitutive relation (12.11) to the corresponding orthotropic case; referred to as the orthotropic-222 class. The constitutive relations for orthotropic-222 material can be obtained by requiring the vanishing of the elements C16,626,636,645,631,632,633,624, e15, s

in the monoclinic-2 constitutive relations. For clarity in the subsequent analysis and discussions, it is worth while to implement these material restrictions in the expanded matrix (12.11) and reduce it to Cll

C12

C13

0

0

0

0

0

0

a22 a33

C12 C13

C22 C23

(723 C33

0 0

0 0

0 0

0 0

0 0

0 0

$22 S33

0 0 0

0 0 0

0 0 0

644 0 0

0 C55 0

0 0 666

--614 0 0

0 -625 0

0 0 --eS6

"/23 ~13 "Y12

D1

0

0

0

e14

0

0

611

0

0

E1

D2

0

0

0

0

625

0

0

0

L.D3.

0

0

0

0

0

e36

0

E2 E3

a23 !'cr13 -a12

s

0

s

-

"$11

!'all"

(12.30) Representative material properties of this kind are collected in Tables Ia-Ic. of the Appendix. Table Ia gives the material properties of GaAs, a cubic43m crystal. It is a higher symmetry material belonging to the orthotropic222 group which belongs to the monoclinic-2 group. Table Ib lists the material properties of PZT-65/35, an artificially polarized ceramic of uniaxial crystal. It is a transversely isotropic material (i.e., isotropic in the xl - x 2 plane) also belonging to the monoclinic-2 group. Table Ic depicts properties belonging to a monoclinic-m group. Interestingly, it is once again a

12.5.

HIGHER SYMMETRIC

211

MATERIALS

PZT-65/35 material obtained from those shown in Table Ib by interchanging direction 1 and 3. This seemingly simple transformation will be shown to dramatically change the behavior of the response of the plate. For the orthotropic-222 case with its propagation taking place along an axis of symmetry, say along x l, the wave components uncouple into two distinct sets as follows: (i) A pure elastic wave that propagates in the sagittal plane, namely, the Xl-X3 plane. This situation was discussed earlier in section 7.1 for surface waves and in section 8.3 for plate waves. (ii) A horizontally polarized S H wave coupled with the electric potential. This later case presents an interesting situation with important consequences. For example, the coupling of electric to S H field will allow the horizontally polarized shear wave to have a surface wave-like character which does not exist in the purely acoustic case. This wave was discovered by Bleustein [21] and is known as the B.-G. wave. 12.5.2

B.-G. waves

The characteristic equation for such a wave will be derived subsequent to obtaining formal solutions for the piezoelectrically coupled S H wave. We now seek solutions to the coupled displacement u2 and electric potential r as

(it2, (~) -- (Vl, V2)exp i~nx3

(12.31)

where we remind the reader that we suppressed the common factor e i~(xl-ct) . The above choice of solution leads to the characteristic equation C44012+C66-pc

2

(e14 + e36)~

(e14~-e36)o~ - ~ 3 3 ~ 2 - ~11

~VI'[ ~, f V 2

_0.

(12.32)

The nonvanishing determinant of equation (12.32) yields the fourth-degree polynomial c~4 + Alc~2 + A2 = 0

(12.33)

which admits four solutions having the properties c~2 - -C~l, c~4 = -c~3 with the coefficients A1 and A2 being easily identifiable. For each C~q, the electric potential, the shear stress and electric displacement amplitudes are normalized with respect to Vq as

644 O~2 -- 666 (el4 -~- e36)Olq

p c 2 --

(I)q ----

(12.34)

212

CHAPTER 12. P I E Z O E L E C T R I C EFFECTS

D3q D4q

-

( C 4 4 q- e l 4 ) o ~ q

-

e36-

(12.35)

e330~qe~q 9

Investigation of these parameters reveals the further properties (I)j =

D3j

-(I)j+l,

-- -D3j+I,

D4j = D4j+I,

j = 1, 3.

(12.36)

Using superposition, the formal solutions to the present case are thus given by 4

= y~'(1, d2q)Vlqexp ifaqxa

(u2, r

(12.37)

q=l 4

(0"23, D3) = E i~(D3q, D4q)Vlq exp i~aqx3

(12.38)

q=l

These formal solutions are now utilized to derive the characteristic equations for propagation on the free surface of a half-space and also on the finite thickness plate. For the semi-space, the radiation condition has to be satisfied. This implies that only two of the four wave components are allowed to satisfy such condition. These can be arbitrarily identified with al and a3 having positive real parts. The characteristic equations of the B.-G. wave can now be derived. For the shorted case, we set 0"23 = r = 0 at x3 = 0 and get D 3 1 (1)3 -

D33(1)1 -

(12.39)

0.

For the free situation, we set D31D43

-

D33D41

-

0"23 - -

D3 = 0 at

X 3 --

0.

0 and arrive at (12.40)

For the plate, invoking the appropriate boundary conditions on the outer surfaces of the plate we get the characteristic equations for the shorted case as

S A - ((I)1D33 - (b3D31) 2 sin(2~/al) sin(2~ya3) - 0

(12.41)

and for the free case as S = D31D43 cot(~/c~3) - D33D41 cot(~/cq) = 0

(12.42)

A - D31D43 tan(vc~3) - D33D41 tan(~/al) - 0.

(12.43)

213

12.6. R E M A R K S ON THE M O N O C L I N I C - M CASE

12.6

Remarks

on the monoclinic-m

case

Results based upon the monoclinic-m modeling of piezoelectric media can be easily obtained by following, step by step, the procedure used in deriving the corresponding results obtained above for monoclinic-2 materials. Undoubtedly, similar solutions with appropriate parametric redefinitions will be expected. No apparent differences of particular importance will be easily identified, if they exist. As was also the case encountered in the modeling of the monoclinic-2 model, for orthotropic or higher symmetry monoclinicm materials, the wave components uncouple into sagittal and horizontally polarized types of motion for propagation along an axis of symmetry. Here, however, the electric field is found to couple with the sagittal motion rather than with the horizontally polarized motion of the monoclinic-2 model. As a consequence of this, monoclinic-m piezoelectric materials cannot support B.-G. waves. To further argue this point, we present in figure 12.1 a slowness plot for the propagation in the xl - x 3 plane of PZT-65/35. By carefully examining this figure, we can identify the potential role that the piezoelectric coupling will play in the behavior of the medium. This figure shows that piezoelectric effect couples with the sagittal plane for propagation along the x3-axis and couples with the S H wave for propagation along the x 1-axis as are implied by the monoclinic-2 and monoclinic-m models of the PZT-65/35. This automatically illustrates that monoclinic-m piezoelectric materials do not support the propagation of the B.-G. waves.

12.7

Reflection

and transmission

coefficients

To determine the reflection and transmission coefficients for a piezoelectric substrate or a plate subjected to plane waves incident from the fluid onto the solid surface at an arbitrary angle 0 we need to utilize the general formal solutions as introduced in chapter 11 for the involved fluids. In these solutions, we have not accounted for the piezoelectric coupling in the fluid. In the subsequent analysis, we maintain this restriction and ignore piezoelectric coupling in the fluid. This implies that the electric potential r of the solid vanishes at the interface. The geometric arrangements of the systems, together with their chosen coordinate systems, are identical with their nonpiezoelectric counterparts shown respectively in figures 11.1 and 11.3 of chapter 11.

CHAPTER 12. PIEZOELECTRIC EFFECTS

214

Figure 12.1: Slowness curves of PZT-65/35 on the Xl-X3 plane (x2-cut). Solid: without piezoelectricity; Dashed: with piezoelectricity. 1 sec/km per division. 12.7.1

Reflection and transmission from a substrate

For this case, the relevant field equations for the fluid are given in equation (11.3). By invoking the continuity of the normal displacement and stress and setting the solid shear stresses, 4 3 3 and a 1 3 and the electric potential r equal to zero at the interface x3 - 0, we obtain, by following identical steps to those of section 11.1, A v -

y

R = Av + Y

(12.44)

T1 = 2pfc2G1 /~ + y

(12.4~)

T3 = 2plc2G3 Av + y

(12.46)

T5 = 2plc2G5 Av + Y '

(12.47)

where, except for the function Y, all of the various parameters are those obtained in the pervious section, namely in equation (12.26). and Y is

12.8.

SAMPLE ILLUSTRATIONS

215

written in terms of these parameters as Y = pyc2 (W1G1 - W3G3 + W5G5 - W7G7). af

12.7.2

(12.48)

Reflection and transmission f r o m a plate

For the plate system, the geometric setup is shown in figure 11.3. The plate's surfaces are located at x3 - + d / 2 and the two fluid portions extend to infinity away from these faces. The field equations for the fluids are given in equations (11.16) and (11.18). By invoking the appropriate interface conditions, we carry similar algebraic manipulations to those of the pure acoustic case described in section 11.2 and finally arrive at the familiar expression n =

AS-

y2

(S + i Y ) ( A - i Y ) iY(S+Y) T = (S + i Y ) ( A - iY)"

(12.49) (12.50)

where S , A and Y are as given in equations (12.28), (12.29) and (12.49), respectively. So far we have casted the expressions of the reflection and transmission coefficients in the standard forms which we derived earlier for the purely elastic cases. The only differences are in the new definitions of the parameters S, A and Y. In general, the number of terms appearing in the expression of S (or A) reflects the number of contributing wave components. In the present case, four terms are present which represent the contributions of one quasi-longitudinal, two quasi-shear and one electric potential wave components. It should be understood that each of these terms depends upon the characteristics of the four wave components through the coupled solutions of various a's in equation (12.17).

12.8

Sample illustrations

In this section, we illustrate the analytical results derived above with some numerical examples drawn from the several material symmetry classes belonging to the monoclinic-2 and monoclinic-m as listed in equations (12.11) and (12.12).

Plate Considering a GaAs plate with a wave propagating along the xl-direction on Xl - x 2 plane, we see that the wave decouples into a pure elastic wave

216

CHAPTER 12. PIEZOELECTRIC EFFECTS

6.0 5.0

0 (D

4.0

E

3.0

eo

2.0

I l l

I

1.0 0.0 0.0

1.0

2.0

3.0

4.0

5.0

fd (MHz mm) Figure 12.2" Dispersion curves for a GaAs plate with wave propagation on Xl - x2 plane along r - 0 ~ azimuthal angle. Solid: anti symmetric modes; Dashed: symmetric modes. After Nayfeh and Chien [165]. propagating in sagittal plane and a piezoelectrically stiffened S H wave propagating in the plane parallel to the plate. The dispersion curves of these two components are shown in the figures 12.2 and 12.3, respectively. For propagation in the xl - x3 plane of a PZT-65/35 plate, the wave decouples into a piezoelectrically stiffened Lamb wave propagating in the sagittal plane and a pure elastic S H wave in the plane perpendicular to the sagittal plane. The dispersion curves of the piezoelectrically stiffened Lamb wave are depicted in figure 12.4. Substrate For a PZT-65/35 substrate, we calculate from equation (12.45) the reflection coefficient profiles in the presence and absence of piezoelectric coupling and compare them in figure 12.5. This figure displays variation of the real parts of the reflection coefficients with the phase velocity for propagation in the sagittal plane defined by 0 ~ azimuthal angle. The solid line corresponds to the piezoelectric case and the broken one to the purely elastic case. This figure clearly shows the rather large influence of piezoelectric coupling. Other situations involving different materials such as quartz showed much smaller

12.8.

217

SAMPLE ILLUSTRATIONS

6.0

',

\

-

5.5t

\

\

",

5.0

4.5

4

4.0

~'~

3.5

0.0

3.0

0.5

~ ~ ~ i ml i t .

1.0

,

1.5

~

~

2.0

Q

~

2.5

,

3.0

Normalized Thickness, 6/;~

Figure 12.3: Dispersion curves for piezoelectrically stiffened SH-modes of a GaAs plate with wave propagation on x z - x2 plane along r = 0 ~ azimuthal angle. Solid: symmetric modes; Dashed: anti symmetric modes. After Nayfeh and Chien [165].

0

9

6

E v 0

mmlllmII~Q~ 4

0

0

1

2

3

4

5

fd (MHz ram) Figure 12.4: Dispersion curves for piezoelectrically stiffened plate-modes of a PZT-65/35 plate with wave propagation on X l - x 2 plane and along r = 0 ~ azimuthal angle. Solid: shorted case; Dashed: free case. After Nayfeh and Chien [165].

CHAPTER 12. PIEZOELECTRIC EFFECTS

218

1.0 o o

0.5

|

0.0

"-

,

i

-0.5

|

9

i i | |

rr

-1.0 2.0

:9

.

| 9 I

9 9

I

:

9

m

m e

tl,

2.1

2.2

2.3

2.4

2.5

2.6

c (Km/sec)

Figure 12.5: Variation of reflection coefficient with phase velocity for x2-cut PZT-65/35. Solid and broken lines correspond to the purely elastic and piezoelectrically coupled cases, respectively. After Nayfeh and Chien [166]. difference in the phase velocity. The corresponding comparisons of the displacement and electric potential profiles obtained for the same parameters used in the generation of figure 12.5 are collected in figure 12.6. The broken lines belong to the purely elastic case where the electric potential is clearly zero. The solid lines on this figure reflect the presence of the coupling. Only the real parts of the displacements ul and U3 and r are shown.

12.9

R e m a r k s on l a y e r e d p i e z o e l e c t r i c m e d i a

In the previous sectionsof this chapter, we have demonstrated that the procedures used to obtain solutions for purely acoustic waves were identically carried over to the case of piezoelectric materials once their formal solutions were derived. For the case involving multilayered piezoelectric media there will be no exception to the rule. In fact, with the availability of the formal solutions (12.19) and (12.20) results for layered media can be obtained following identically, step by step, the transfer matrix procedures of their purely acoustic counterparts. The minor differences are of course include the change in the order of these matrices from 6x6 to 8x8 due to the presence of the extra two variables r and Dk from the electric field. Also, the interface conditions imposed on the pure acoustic field increase to eight to

12.9. R E M A R K S ON LAYERED PIEZOELECTRIC MEDIA

219

0.2

(D -~ :3 ~_

0.0

m

~.

|

E

|

"r (D

-0.2

E

(D

._~

a

-o.4

0.0

,

I

0.5

,

I

1.0

I

1.5

,,,

2.0

Normalized Depth X 3

Figure 12.6: Profiles of the real parts of the displacement components Ul and u3 and the electric potential r in the x3-direction. Solid and broken lines correspond to coupled and pure elastic cases, respectively. After Nayfeh and Chien [166]. include continuity on both r and Dk. Other than these changes, the general properties of both the local and global transfer matrices will carry over to the present case and hence a wide variety of situations including piezoelectric effects can be easily constructed following the procedure of their purely acoustic counterparts. For these reasons we shall not pursue solutions for the layered piezoelectric media.

This Page Intentionally Left Blank

Chapter 13

TRANSIENT

WAVES

In this chapter, we extend the formal developments in previous chapters and study the response of two anisotropic systems to transient buried line loads. This includes infinite and semi-infinite systems. We carry out the analysis on anisotropic media possessing monoclinic or higher symmetry. The load is taken in the form of a normal stress acting at an arbitrary direction in the plane of symmetry of the material. We use a building block approach in which we start by deriving results for an infinite medium. Subsequently, we obtain the results for the semi-space by employing superposition of the infinite medium solution together with a scattering solution from the boundary. The sum of both solutions is required to satisfy stress free boundary conditions thereby yielding complete solutions. We obtain explicit solutions for the particle displacements in both systems using the Cagniard-de Hoop method. The analysis is facilitated by using the linear transformation approach in which we identify the line load with the x2-direction. This implies that all involved field variables are independent of the x2-direction. Nevertheless, and in general, three nonvanishing particle displacements are present. Material systems of higher symmetry, such as orthotropic, transversely isotropic, cubic, and isotropic are contained implicitly in the analysis.

13.1

Theoretical development

We consider an anisotropic elastic medium possessing monoclinic symmetry. The medium is oriented with respect to the reference Cartesian coordinate system x~ such that the x~ is assumed normal to its plane of symmetry as shown in figure 13.1. The plane of symmetry defining the monoclinic material is thus coincident with the x ~ - x~ plane. With respect to this 221

222

CHAPTER

13.

TRANSIENT

WAVES

Xl

!

X2

X3, X 3 J

Figure 13.1: An applied line load in an anisotropic infinite media. primed coordinate system, the equations of motion which consist of the momentum equations and associated constitutive relations were discussed in detail in chapter 2. The medium is subjected to a uniform time dependent line load applied along a direction that makes an arbitrary azimuthal angle r with the x~-axis. That is, the direction r - 0 ~ coincides with the reference coordinate x~. Since, as was pointed out in earlier chapters, the response of the medium to such a wave is independent of the coordinate parallel to the applied line direction, the analysis is conducted in a transformed coordinate system Xk formed by a rotation of the plane x~ - x ~ through the angle r about the x~-direction. For convenience, the choice of r is made so that the direction x2 coincides with the line load direction. The transformed equations describing the present problem are also given in chapter 2. In terms of the rotated coordinate system Xk, the momentum equations, including a time dependent load taken in the form of a body force, are written compactly as

OakJ OXj

02Uk

F f k -- P Ot 2 9

(13.1)

with its expanded form, in the absence of fk, given in equations (2.42)(2.44). Even though, for propagation in the x l - x3 plane, the motion is independent of x2, the particle can generally have the three nonzero displacement components ul, u2, and u3. In fact, the u2 displacement component

13.2. SOURCE C H A R A C T E R I Z A T I O N

223

is identified as belonging to the horizontally polarized S H wave. In its expanded form, equation (13.1), when specialized to monoclinic materials and accompanied by dropping differentiation with respect to x2, is rewritten as 02

02

02

02

(611~x12 ~- C55~x~)%tl+ (C16~x12-~ C45~x~)tt2 0 0 02Ul ~-~x ((613 ~-C55)~xl)tt3 = p Ot2 -fl 02

02

02

(13.2)

02

[C16~1x21-+-C45~2x32]ttl-+-[C66~1x21-~-C44~x~]%t2 + b -0~ [(c36 + c45) 0~1 ]~3 = p 02u20t2 - f2 o [(c13 + c5~) Ox3 02 oz7

(13.3)

]~1 + ~

02 oz5

02ua Ot2

(13.4)

Here, the forcing function fk is chosen in the form

fk = Qk6(Xl)6(xa)F(t)

(13.5)

which means that a line load is acting along the x2-direction and is situated at the arbitrary location defined by x l - 0 and x3 - 0.

13.2

Source characterization

Solutions to the present problem are accomplished by recognizing that the infinite space can be thought of as consisting of two semi-spaces whose hypothetical interface contains the applied load. Thus the lower semi-space occupies the region x3 _ 0 whereas the upper semi-space occupies the region x3 _< 0. As a result of this, appropriate conditions must be specified at the hypothetical interface. To specify these conditions, we consider a very thin "interface" layer extending from x3 = - 0 to x3 = +0. We then require the following continuity on the displacement components and their time derivatives

~ ( ~ , x3, t)l_+~ = 0

a~i(x~,xz, t) +o Otn ]_ -- O,

(13.6~)

n = 1,2,...

(13.65)

CHAPTER 13. TRANSIENT WAVES

224

Integrating equations (13.2)-(13.4) across the interface layer and using the above continuity conditions (13.6a,b) lead to the following discontinuities in the displacement spatial derivatives w 5 5 ~ x3 + C450~x3) I_ = -Q15(xl)F(t)

O%tl

(0~t2 ]+ 0

( C 4 5 ~ x 3 -~- C44~3x3 ) _ : C 3 3 ~ x 3 _ ~-

-Q25(xl)F(t)

-Q35(xl)F(t).

(13.7a) (13.7b) (13.7c)

From now on, we consider the special case where Q1 = Q2 = 0 and Q3 = Q. For this situation equations (13.7a)-(13.7c) reduce to

0?-tl 0~t2 +0 (655~3x3 +645~x3)1_ =0

(13.8a)

OUl OU2 +0 (C45~x3 -~-C44~3x3)1_ =0

(13.8b)

Ou3 +0

+0

C330-~x3l_ -- a33]_ = -QS(xl

)F(t) .

(13.8c)

Examination of the relations (13.8a,b) reveals that, if (C44C55-C25) does not vanish (which is the case), then Ul and u2 must be at least constant and, in accordance with equation (13.6), equal. This constant is then chosen to be zero and thus conclude that Ul and u2 vanish at the interface. Finally, the normal stress discontinuity condition (13.8c) is satisfied by assigning

(~U3

(13.9a)

C33 ~x3 I+0 = -QS(xl)F(t)/2

(~U3 ox3

C33~-:--1-o = QS(xl)F(t)/2.

(13.9b)

Collecting the above relations, we summarize the conditions at the hypothetical interface as

Ul ----0 u2----O

OU3 __ 633 G.)X3 --

1QS(xl)F(t)

2

'

for x3 > 0 at x3 = 0 -

(13.10a)

13.3. I N T E G R A L T R A N S F O R M S OF FORMAL SOLUTIONS

225

and Ul--0

u2=O Ou3 1 C33 Ox3 = ~QS(xl)F(t),

for x3 <_ 0 at x3 = 0.

(13.105)

Employing the stress-strain relations (2.35), we conclude that the third relations in equations (13.10a,b) are equivalent to the normal stress conditions 1 a33 = --~Q(~(xl)F(t), 1

~

= ~Q~(x~)F(t),

13.3

Integral

f o r x 3 _>

f o r x 3 <__ 0

0

at

at

transforms

(13.11a)

X 3 ----0.

(13.11b)

X3 --0.

of formal

solutions

Next, we outline the steps leading to formal solutions of equations (13.2) -(13.4) for each of the two semi-spaces. Since the body force has been replaced by the " hypothetical interface" condition, we drop fk from these equations. Applying Laplace and Fourier transforms to equations (13.2)(13.4) in accordance with

f~k -~k =

f?

Uke-ptdt

(13.12a)

~ke -~p'7~ dxl

(13.12b)

9

followed by assuming solutions of the resulting ordinary differential equations in the form ~k = Uke - p ~ ,

k = 1, 2, a

(la.13)

leads to the characteristic equation

(KIIK12K13(gl) ) K12 K13

K22 K23 / / 2 3 K33

U2 -0. U3

Here, the various elements Kij are given by KII = C55~2 - C11~2 _ p

K12

- - C450~ 2 - C161] 2

(13.14)

226

CHAPTER

13.

TRANSIENT

WAVES

K13 = -i~?a(C13 -[- 655) K22 : C44oL2 - 666?-]2 - p K23 = - i n a ( C 3 6 + 645) 1<33 =

C33a 2 -

C~5~72 - p

(13.15)

with p and ~ being the Laplace and Fourier transform parameters, respectively. The algebraic steps needed to obtain formal solutions for Uk are identical with those used throughout this book to obtain other formal solutions. However, for further clarity, we shall include some of them here. For the existence of nontrivial solutions in Uk, the determinant in equation (13.14), must vanish giving an algebraic equation of a. Notice that this is an alternative presentation, of the Christoffel equation (3.9). The difference is that we now solve for a as compared with solving for the phase velocity for a given propagation direction. Upon setting the determinant of equation (13.14) equal to zero, we obtain a sixth-order equation in a (cubic in a 2) which is written symbolically as A l a 6 -+- A 2 a 4 + A3 a2 q- A4 -- 0

(13.16)

with its coefficients given by A1 -- D1

A2 = D2r12 + D3p A3 = D4rl 4 + D5P~72 + D6p 2 A4 -- D7r] 6 q- D8p~ 4 -'k D9p2r] 2 -b p3,

(13.17)

where D1 = 633 C25 - 633 644 655 D2 - Cl1 633644 q- 655F1 - C16633645 - C45F2

+(613 + C55)F3 D3 = [655(633 + 644) ~- 633644 -- 6425] p D4 - [(C13 q- C55)F4 -t- C45C16655 -+- C16F2 - C525666 - CllF1] D5 = [(C134- C55) 2 § 2645616 - F1 - C55(655 + C66) - C l l (633 -[- C44)]p

D6 = - (633 + C44 + C55)p2 0 7 - (611655666 - C26655) D8 = [611 (655 + C66) - C216]P D9 = (Cll + C55 + C66)P 2 and El = ([]'33666 -[- 644 C55 - (C36 -I- C45)2

(13.18)

13.3. I N T E G R A L

TRANSFORMS

OF F O R M A L S O L U T I O N S

F2 = C33 C, 6 + C45 C55 - (C36 + C45) (C~ 3 + C ~ ) F3 : C45(C36 -~- 645) - C44(613 -+- 655) F4 -- 666(613 -~- 655) - C16(636 -~- 645).

227

(13.19)

Equation (13.16) admits six solutions for a. These a's have the further properties that c~2 = -c~1, c~4 = -c~3,

c~6 = -c~5.

(13.20)

Furthermore, for each c~, equation (13.14) yields the partial displacement amplitude ratios Vq = U2q/Ulq, Wq - U3q/Ulq as

K l l (O~q)K23 (O~q) - gl2(O~q)gl3(O~q) Vq - - K12(C~q)K23 (6~q) - K22((~q)Kla (6~q)

(13.213)

K12(~176 - K13(~q)K22(~q) Wq -~ - K23 ( (~q ) K23 ( ~q ) _ K22 ( (~q ) K33 ( (~q ) .

(13.21b)

Finally, invoking superposition, we write the formal solutions for the displacements of equations (13.2)-(13.4) and their associated stress components using equation (2.35) as 6 (~1, it2, ~3) -- ~ ( 1 , Vq, Wq)Ulqe -paqx3 q=l

(13.22)

6 (c}33, (}13, c}23) -- E p(Dlq, D2q, D3q)Ulqe -paqx3 q----1

(13.23)

where

Dlq - ir/(613 + C36Vq)

-- C 3 3 0 L q W q

D2q = C55 (i~lWq -- (~q) - C45~qVq Daq -- C45 (i~lWq - c~q) - C44(~qVq,

q-

1,2,..,6.

(13.24)

With reference to the relations (13.20) and to the Kij elements in equation (13.15) and, by inspection of equations (13.21) and (13.24), we recognize the relations

V2 = V1, W2 = - W ~ ,

D12 -- D l l ,

Vn= V3,

V~= V5

Wn = - W 3 ,

D14 - D13,

W6 = - W ~

D16 = D15

D22 - -D21,

D24 - -D23,

D26 - -D25

D32 = -D31,

D34 = -D33,

D36 = -D35.

(13.25)

CHAPTER 13. TRANSIENT WAVES

228

These properties have been encountered in several locations in the previous chapters. The above solutions with their various properties can now be specialized to both hypothetical semi-spaces by the following steps. Inspection of the solutions (13.22) indicates that each displacement consists of three pairs of wave components, each pair propagating in mirror image fashion with respect to the interface, namely along positive and negative x3-directions. Since propagation is expected to emanate from the interface into both media, we arbitrary reserve q = 1, 3, 5 for the lower semi-space; the remaining ones, namely those described with q = 2, 4, 6 for the upper one. We thus list the formal solution in the lower and upper semi-spaces according to (?~1, ?~2, ?~3) -(~r33, ~13, (~23) ----

(1, Vq, Wq)Ulqe -paqx3 q=1,3,5

E p(Dlq,D2q,D3q)Ulqe-paqxa

q-1,3,5

(13.26)

for x3 > 0 and (~tl, ?~2, ?~3) -((~33, ~13, ~23) --

(1, Vq, Wq )Ulqe -pOtqx3 q=2,4,6 E p(Dlq,D2q,D3q)Ulqe-paqxa q=2,4,6

(13.27)

forx3 <0.

Specialization of formal solutions to infinite media At this point, formal solutions of the field equations in a monoclinic medium have been presented. The partial amplitudes Ulq are the unknowns "constants of integration". These unknowns will now be determined by implementing the hypothetical interface conditions (13.10a)-(13.10b). To this end, if equation (13.26) is subjected to the conditions (13.10) and equation (13.27) to the conditions (13.11), we finally solve for the unknowns as Ull = -U12 = (V5 - V3)F(p)Q/(2C33pAum)

(13.283)

_

U13 = -U14 - (V1 - V5)F(p)Q/(2C33PAum)

(13.28b)

U15 = -U16 = (V3 - V1)F(p)Q/(2C33PAum)

(13.28c)

where /x~

= v1(a3w3 - a~ws) + Y3(asw5 -

+vs(alW~ - 33w3).

alW~) (13.29)

13.3. I N T E G R A L T R A N S F O R M S OF F O R M A L S O L U T I O N S

229

We note that Aura ---- 0 defines an equivalent Christoffel characteristic equation for the propagation of bulk waves in the medium. With these solutions for the wave partial amplitudes, using the various relations in equation (13.25), solutions in the upper region can be written in terms of q - 1, 3, 5 as

(~1,/~2, 1~3) -- Z (--1,-Uq, q-1,3,5 (~33, ~13, 023) --- Z p(-Dlq, q-1,3,5

Wq)Ulqepaqx3

D2q,D3q)Ulqepaqxa.

(13.30)

In summary, solutions (13.26) and (13.30), with the amplitude solutions (13.28), uniquely define the propagation fields in the lower and upper "hypothetical" semi-spaces, respectively. In other words, their combination constitutes the total solutions for the infinite medium. 13.3.1

Methods

of inverting the transforms

The inverse Fourier transformations of solutions (13.26) are

=

pS

(~33, 0"13, 0"23) ---- ~P

/2

~

(1, Vq, Wq)Ulqe-P/q('l)d 0

Z (Dlq,D2q,D3q)Ulqe-pfq('7)dO cc q=1,3,5

(13.31) (13.32)

where fq(O) = O~qX3 - iOXl.

(13.33)

The solutions of the problem can be obtained in two steps: in the first, the partial amplitudes Ulq are substituted from equation (13.28) into equation (13.31) and the resulting integrals are evaluated for Uk; in the second, the displacements Uk are subsequently obtained by taking the inverse of the Laplace transforms. Such a procedure is very complicated and, for all practical purposes, impossible to do, especially since most of the transformed functions have complicated dependence on the transform parameters p and 0. Attempts to evaluate the resulting integrals by the above procedure may lead to little success. As an alternative, in the following we shall utilize the elegant technique introduced by de Hoop [61] who modified and used a method originally

CHAPTER 13. T R A N S I E N T WAVES

230

presented by Cagniard [35]. This method, known as the Cagniard-de Hoop method, inverts the double transforms, namely the Laplace and' the Fourier transforms, in one strike. Mathematically, it is based on the rather elementary observation that if

Uk =

Uk(Xl,X3,t)e -ptdt

(13.34)

then its inverse is given by Uk(Xl,x3, t ) H ( t - tq), where H ( t - tq) is the Heaviside step function. The Cagniard-de Hoop method has been found to apply well for the idealized case of a homogeneous isotropic elastic space subjected to internal line and point loads. This is again a consequence of the fact that only pure mode waves exist for isotropic materials. Within the context of the present analysis, this leads to a factorization of the characteristic equation for the involved c~s in equation (13.16). The lack of complete factorization of the corresponding characteristic equation for anisotropic materials is the foundation of the difficulty of direct application of the Cagniard-de Hoop technique for such materials. Simply speaking, no closed form analytical results can be obtained for anisotropic material using this technique. With advanced numerical computation capabilities, one can however obtain, with relative ease, numerical solutions using such a method. In order to appreciate the method and its utility, it is of particular importance to go through the exercise of applying it to isotropic media. This will lead to a clear understanding of the method and will shed a light on the roles of the various parameters involved, thus making the method easier to apply to anisotropic media. Before we proceed to apply the Cagniard-de Hoop method to anisotropic media, however, we shall now digress a little and discuss its application to isotropic media.

13.4

Isotropic media

Consider an infinite isotropic medium subjected to a normal stress line load in the direction x3 and distributed uniformly along the x2-direction. As a consequence of isotropy, the horizontally polarized S H wave along the x2-direction will uncouple from the sagittal Xl - x 3 plane motion. Since no load is applied in the x2-direction, the displacement u2 and the shear stresses a12 and a23 vanish. Thus, the motion of the medium will be confined to the sagittal plane with the two nonvanishing displacement components u l and u3. Hence, the problem becomes two-dimensional with its basic field

13.4. ISOTROPIC MEDIA

231

equations given by

02

02

0

[(A + 2#)~-~x2 + #~2x2]ul + ~-~x.[(A + #)

~,~1~

02 U2

02

02u30t 2

]u3 = p

(13.35)

v

0

Ox3 [(/~ -}- , )

~ -

.

02

]ltX + [#~-~X~ + (A + 2#) ~-~x~]U3 = p

(13.36)

subjected to the hypothetical interface conditions (13.10), specialized to the isotropic case as Ul=O Olt 3

_

(A + 2#)0X3

--

1QS(xl)F(t)

--2

'

for

X3 > 0 --

at

X3 :

0

(13.37)

Ul=O

Ou3 _ 1QS(xl)F(t)

(A + 2#)Ox3 - -2

'

for x3 < 0 at x3 = 0.

-

(13.38)

Applying the integral transforms (13.12) to equations (13.35) and (13.36) leads to -i?)a(A + p)

(A + 2#)c~ 2 - p~2 _ p

U3

For nontrivial solutions of U1 and U2, the determinant of the 2 • 2 matrix in equation (13.39) must vanish leading to [pa 2 - (A + 2#)?) 2 - p][(A + 2p)a 2 - #?)2 _ p ] _ a2?)2(A + p)2 = 0.(13.40) With minor algebraic manipulations, equation (13.40) factors out in accordance with [(A + 2#)(a 2 - ?)2) _ p][#(o~2 _ ?)2) _ p] = 0

(13.41)

and yields four roots for a that we label and identify as OL1 .._ __0~2 =

(?)2 _~_ ~12)1/2

(13.42)

and

~3 = -~4 = (?)2 + $2) 1/2.

(13.43)

These are written compactly as

aq2 = ?)2 + Sq2

(13.44)

C H A P T E R 13. T R A N S I E N T W A V E S

232 where P )1/2 S] = (~ + 2# '

$3 = (__P)1/2 #

(13.45)

are the slownesses of the longitudinal and shear waves in the isotropic material, respectively. Furthermore for each a, equation (13.39) yields the displacement amplitude ratios Wq = U3q/Ulq as w~ =

iv/a1 ()~ + #)

(~ + 2 , ) . ~

(13.46)

- ,~2 _ p

W3 = #a] + (~ + 2p)~2 - p

(13.47)

which, with the help of equation (13.44) reduce, respectively to ial W~ = - - , rl

iv] W3 = - c~3

(13.48)

Finally, invoking superposition, we write the formal solutions for the displacements of equations (13.35) and (13.36) and their associated stress components as 4 (~1, %t3) -- ~ ( 1 , Wq)Ulqe -paqx3 (13.49) q=l 4 (533,513) -- E p(Dlq, D2q)Vlqe -paqx3 (13.50) q=l where

Dlq = i~7)~- ()~ + 2#)aqWq D2q - #(i~lWq - ~q).

(13.51)

The above solutions, with their various properties can now be specialized to both hypothetical semi-spaces by the following steps. Inspection of the above solutions indicate that each consists of two pairs of wave components, each pair propagating in mirror image fashion with respect to the interface, namely along positive and negative x3-directions. Since propagation is expected to emanate from the interface into both media, we arbitrarily reserve q = 1, 3 for the lower semi-space; the remaining ones, namely described with q - 2, 4, for the upper one. We list the formal solution in the lower and upper semi-spaces according to (ul,u3)

=

~ (1, Wq)Ulqe -paqx3 q=1,3

(533,513) -- E p(Dlq, D2q)Ulq e-paqx3 q=l,3

x3 >_ 0

(13.52)

13.4. ISOTROPIC MEDIA

233

and (~1, ~3)

--

E (1, Wq)Uiqe -paqx3 q-2,4

(&33, ~13)

=

E p(Dlq, q=2,4

D2q)Ulqe-paqxa

x3 <_ 0.

(13.53)

Remark

With special care, the formal solutions obtained above for the isotropic case can be obtained from the derived solutions for the monoclinic medium. First, the fact that the SH wave uncouples is evidenced from the vanishing of the properties C16, C36 and C45 in equation (13.15). This results in the vanishing of the elements Ki2 and/(23. Furthermore, since there is isotropy, the property restrictions C33 - Cil - A + 2p and Cli - Ci3 - 2C55 with C13 and C55 replaced respectively by A and #, reduce equation (13.14) to equation (13.39), and hence to the steps leading to the final formal solutions (13.51) and (13.52). S p e c i a l i z a t i o n of f o r m a l s o l u t i o n s t o i n f i n i t e m e d i a At this point, a formal solution of the field equation in an isotropic medium has been presented. The amplitudes Ulq are the unknowns. These amplitudes Uiq will now be determined by implementing the hypothetical interface conditions (13.10) and (13.38). These, together with the formal solutions (13.52) and (13.53), lead to the displacement amplitudes as U l l - - U 1 2 -- F(p)Q/[2p()~ + 2#)(c~iWi -c~3W3)] V13 ---- - U 1 4 --~ -F(p)Q/[2p()~ + 2p)(c~iWi -c~3W3)]. _

(13.53a) (13.53b)

With the help of equations (13.44) and (13.48), the expression oLIW1- 0t3W3 reduces to iS2/~7 and gives

Ull -

-U12 -

-iF(p)~Q

(13.54a)

iF(p)~Q 2pp

(13.54b)

_

Ui3 - -Ui3 -

2pp

C H A P T E R 13. T R A N S I E N T

234

WAVES

With these expressions for the wave amplitudes, solutions in the upper region can be written in terms of q = 1, 3 as (?~1, ?~3) ---- E (--1, Wq)Ulqepaqxa q=1,3 (~33, (Y13) = E p (-Dlq, D2q)glq epaqx3 q=1,3

(13.55)

In summary, solutions (13.52) and (13.55) with the amplitude solutions (13.54a,b) uniquely define the propagation fields in the lower and upper hypothetical semi-spaces. In other words, their combination constitutes the total solutions for the infinite isotropic medium. The inverse Fourier transformation of the displacements in (13.52) are given by (~1, u3) =

~ (1, Wq)Ulqe-P.fq('7)d~ c~ q=1,3

f q(rl) = OlqX3 - irlXl. 13.4.1

(13.56a) (13.56b)

The Cagniard-de Hoop transformation

As a prelude to introducing the Cagniard-de Hoop method, we observe that if the functions fq07) in (13.56b) are identified with time t, then the integrals can potentially be inverted by inspection as per equation (13.34). Since t is real and positive, ~ must then become complex. The Cagniardde Hoop scheme then consists in considering ~? as a complex number and so deforming the path on integration from the real ~-axis to a convenient one in the complex fl-plane such that the integrals can be recognized as the Laplace transforms of certain explicit functions of time. This will allow us to write down the inverse transforms by inspection. Since the entire complex rl-plane is now involved, we seek the path of integration that makes the quantities fq07), q = 1, 3, which appear in the exponential of equation (13.56a), real and positive. Furthermore, the integrands of these equations have the branch points ~q - iSq defined by the zeros of C~q. A necessary condition for the convergence of the integrals is that OL1 and Ct3, namely (rl2 + $12)1/2 and (rl2 + $2) 1/2 be single-valued functions, with positive real parts on the path of integration. To allow for subsequent deformation of the contour, we insure that this condition holds everywhere in the rl-plane by introducing branch cuts along the imaginary axis from the branch points +iS1 to +ioo and from +iS3 to +ioo. No other singularities exist in these integrals. The integrands are analytic in the entire plane of cut in this manner.

13.4. I S O T R O P I C M E D I A

235

I I

E I

!

I

I

I

I

-

~D Of.

...... I

I

I

$1 r .~,,,, t

0.0

J

I

I

I

I

I

I

"L

x,.l I_

,

I,

O.2

I

L I

.

,

_

I

I

I

0.4

I

I

O.6

Figure 13.2: R(r/) and Im(~) versus time for steel at r = 1 mm and 0 = 5~ Following de Hoop, we set

aqX3 - i~lxi = t

(13.57)

and deforming the original path of integration in such a way that the corresponding path of integration in the t-plane is the positive real axis from 0 to oc. Solving this equation for ~? leads to the new path of integration +

Tlq

t2

"-

i(-~

-- Sq

2 i/2

)

it

sin 0 + --r cos 0

(13.58)

corresponding to Sq, where r 2= x 2i + x 23 and ~ = t a n - i (x3/xi), with 0 _ 8 < 7r. These paths are demonstrated graphically in figures 13.2 and 13.3. Figure 13.2 shows variations of Re(rl) and Im(rl) as functions of t for given values of r and O. In generating the plots in these figures, we used stainless steel with r = 1 and 8 -- 5~ Figure 13.3, on the other hand, gives a plot of Re(~lq) versus Im(rlq). These representative figures display many important features of the Cagniard-de Hoop scheme. Since t is positive real, we see that T/q is pure imaginary for 0 < t < Sqr Beyond this value of t, ~q becomes complex with its imaginary part varying linearly with t in accordance with Im(~lq) = (t/r)cos0. Simultaneously, from the expression for ~q, the real part defines two symmetric parabolas

CHAPTER 13. TRANSIENT WAVES

236

Im(Tl)

f /[,

~

l/I

~

%~

II

iI

{},

iI

iI I II iI z

I II

lI

~

%~ ~

~ ~ ~x x

Figure 13.3: The Cagniard contour for steel. with vertices located at t = Sqr. As t approaches c~, the two parabolas assume the common asymptotic values of +(t sin O/r). Thus Sqr define the arrival times for the waves. The shapes of the curves in these figures depend upon 0. For Re(r}) the curves narrow as 0 increases and the curves of the imaginary part rotate in a counterclockwise manner as 0 increases. An outstanding feature of these figures is that, independent of 8, locations of the projections of tips of the Irn(~q) curves on the t-axis do not move signifying the fact that, for isotropic materials, the wave speeds are uniform and independent of the propagation direction. Furthermore, these arrival times can also be deduced from the Re(r}q) plots at the location where it becomes non-zero. But these locations are identified with d%/dt = c~ and hence present alternative means of calculating arrival times. Examination of figure 13.3 shows that the Re(r}q) versus Irn(r}q) curves also define two parabolas emanating from the imaginary axis at the locations iS1 cos 0 and iS3 cos 0 and having the common asymptote (as t -+ c~) defined by argument 0 = + t a n 0. Also, arrival times are obtained at the moment when ~q seizes to be pure imaginary. Since the integrands in (13.56a) approach zero exponentially as ~ tends to infinity, the contribution from the arcs of ~k vanishes at infinity. Referring to equation (13.58), we deduce that for q = 1, the relevant contour in the ~?-plane has one possible configuration as shown in figure

13.4. I S O T R O P I C M E D I A

~.

237

Im01)/

is,coso" .-'~.~,,~",,.

Re(T1)

Figure 13.4: Cagniard contour showing extra deformation of the path of integration. 13.3. For q = 3, the particular contour has two possible configurations in the 7-plane depending upon the values of 0" if 5'3 cos 0 < $1, the relevant contour is simply given by 73 in (13.58) and a similar situation prevails to that of figure 13.3. However, if 5'3 cos 0 > $1, then the vertex of the parabola 73 lies on the branch cut between iS3 and iS1 and the appropriate contour is shown in figure 13.4. This contour is given by 73 of equation (13.58) plus an additional path that consists of a circle of radius E, (e - 0), centered at 7 - iS1, and two segments represented by t2 t 7 = i[-( $2 - _~)1/2 sin 0 + - c o s 0 ] ~:e r within the following range of t

SlrCOSO + r(S 2 - $2) 1/2 sin0 <_ t <_ S3r.

(13.59)

(13.60)

These equations are applicable in the range 0 _< 0 <_ cos -1 ($1/$3). 13.4.2

Displacement

distribution

For the special loading described by the function F(t) - H(t), the Heaviside step function,/~(p) then equals p. Using this loading, inspection of equation

CHAPTER 13. TRANSIENT WAVES

238

(13.56a) reveals that the first two terms correspond to the longitudinal and the shear motions, respectively 4~rp Q

~1 -

0?]1+

[?]+1- ~

- ?]1

0ri~-

--~-]g(t

- tl)

-[?]+-~- - ?]3--~-]g(t-

t3)

(13.61)

where tl and t3 are the arrival times of the longitudinal and transverse wavefronts. In other words, they are the times that correspond to the values of the imaginary ?]-axis intercepts of the two branches of 77. The notation ?]+ denotes the branch of ?]q to the right side of the imaginary ?]-axis and ?]q denotes the branch of ?]q to the left side of the imaginary ?]-axis. By using the same interpretation, the displacement u3 is obtained by

4~rp Q

=

07]+

0?]1

-[?]+W3(?]+)--~ - - ?]3W3(?]3)---~-~-]g(t - t3).

(13.62)

In cases involving the extra contour in figure 13.4, its contribution gives rise to the well known "head waves". For further discussion of these waves in isotropic materials we refer the interested reader to Fung [89].

13.5

Anisotropic media

We now return to the anisotropic case. The Laplace transforms of the displacements and stresses which need to be inverted are given in equations (13.31) and (13.32). A comparison of the integrals appearing in these transforms with the corresponding ones belonging to the isotropic case reveals similarity in appearance. Nevertheless, in the case of anisotropic media, due to the absence of explicit dependence of C~q on ?], the functions fq(?]) are themselves non-explicit functions of ?], in contrast with the situation encountered for isotropic media. Therefore, this does not, lend itself to direct application of these schemes. However, this difficulty does not preclude the applicability of the method to anisotropic media. Moreover, it is to be recalled that the plots of the ?] function as displayed in figures 13.2 and 13.3 for isotropic media were naturally uncoupled due to the factorization of their characteristic equation (13.41). The lack of similar factorization in the anisotropic case, leads to coupling of the ?] functions and can only be individually identified using a priori some sorting subroutines, or by visual identification of the graphical mapping of their total numerically obtained

13.6.

CAGNIARD-DE

HOOP TRANSFORMATION

239

solutions. Since, in both isotropic and anisotropic cases, solving for O~q c o n s t i t u t e d an intermediate step in achieving the solutions for 7, we can avoid this step and directly solve for ~ as a function of time t and the spatial coordinates Xl and x3.

13.6

Cagniard-de Hoop transformation

Guided by the form (13.57) appropriate for isotropic media, we introduce the general transformation (13.63)

t = ax3 - i~Txl

with t real and positive. Solving this equation for a as c~ -

t + i~xl

(13.64)

X3

and directly substituting into the characteristic equation (13.16), yield the sixth order polynomial with complex coefficients in ~ as B6~?6 + B5~ 5 + B4r/4 -~- B3~?3 + B2r/2 + B l r / + B0 = 0

(13.65)

where the various coefficients are given by

B6 -

D 2 X 23 X4l - D 4 x 4 x 2

Dlx6+DTx~

B5 = iCl t B4 = C2t 2 + D3x3x 41 - D x4x

+

B3 = iC3t 3 + iC4t B2 = C5t 4 + C6t 2 - D 6 x 4 x 2 + D 9 x 6 B1 = 6 i D l x l t 5 + 4 i D 3 x 2 x l t 3 + 2 D 6 x 4 x l t Bo = D l t 6 + D3x4t 4 + D6 x4 + p3x6

(13.66)

and C1 = 6 D l x 5 + 2 D 4 x 4 x l - 4 D 2 x 32x l3 C2 = 1 5 D l x 4 - 6 D 2 x 2 x 2 + D 4 x 4 C3 = 4 D 2 x 2 x l - 2 0 D l x 3 C4 = -403x2x31 + 2 0 5 x 4 x l C5 = D 2 x 2 - 1 5 D l x 2 C6 = -603x23x21 + D 5 x 4.

(13.67)

CHAPTER 13. TRANSIENT WAVES

240

n~

!

0.0 Figure 13.5: ReO?) and m m and 0 = 5 ~

!

~

O.

i

l

t(/J,s)

i

I

0.4

|.,.

i

,'

0,6

Irn(~) versus time for the cubic InAs material r = 1

The six complex roots of r/in (13.65) are functions of the spatial variables Xl and x3, time t and material properties. These roots are not expected to lead to a generic curve types similar to those of figures 13.2 and 13.3 which we can rely upon to describe all common features encountered in anisotropic materials. In fact, different curves can exist which are highly dependent not only on the specific materials involved, but also on the monitoring location of the wave. For this reason, meaningful discussions of phenomena can be understood only when referred to specific situations. To aid in our subsequent discussion, we choose an InAs cubic material whose properties are listed in Appendix A. We choose, for illustration, the location described by the azimuthal angle r = 30 ~ with r = 1ram making an angle 0 - 5 ~ with the normal x3 as the monitoring direction. As will become obvious below, this specific choice of parameters constitutes a good representation of most common features encountered in anisotropic materials. For this specific combination of parameters, the r/-roots, when numerically determined and graphically displayed show an array of six curves. These curves are displayed in the forms of variations of Re(~?) and Irn(~?) with t and also variations of Re(r~) with Irn(rl) as shown in figures 13.5 and 13.6, respectively. These figures should be compared with the ones corresponding to isotropic materials, namely, figures 13.2 and 13.3.

13.6. CAGNIARD-DE HOOP TRANSFORMATION

241

Im( )

I

,

Figure 13.6: Cagniard contour corresponding to figure 13.5. All curves exhibit complicated texture when compared with the corresponding ones belonging to isotropic materials. The most striking difference can best be inferred from comparisons of the variations of Im(~) with t. In the anisotropic case, the curves exhibit various degrees of interference among each other as is evidenced from their crossing and touching regions. In some regions, these curves change curvatures. Furthermore, the lines of degeneracy of the double valued curves are no longer coincident or even parallel. In fact, the supposedly straight line ab follows a "detour" in the region between b and c, travels along one of the double valued curves and reappears beyond point c. Using the arrival time identification criteria described graphically above for the isotropic media, we here identify five possible arrival times defined by the horizontal coordinates (along the t-axis) of the points a, b, c, e and f. These also correspond to locations where the tangents to the Re(~7) curves are vertical. During the time increment between the arrival times corresponding to points b and c, i.e., the detour segment, no energy arrives to the point of abservation (here all field variables vanish). This is known as the lacuna region. Lacunas are found to depend not only on the material properties but most importantly on the monitoring angle 0. This suggests that, for

CHAPTER 13. TRANSIENT WAVES

242

anisotropic materials, there is no guarantee that the arrival times will be independent of either the propagation or monitoring directions. Furthermore, the extent of the presence or absence of the lacuna regions will also depend upon the propagation and monitoring directions. To show both of these effects, we demonstrate the evolution of the curves in figure 13.5 and 13.6 with the monitoring angle 0 in figure 13.7. Here we arbitrarily choose the angles 9 = 0 ~ 10 ~ and 40 ~ The arrival times are identified for all 9 between 0 ~ and 360 ~ for the cubic material and collected in figure 13.8. It is recognized that this figure displays the inverse of the energy ( group velocity) curve. Hence, we conclude that the arrival times can be thought of as the " energy slowness". To show how the number of arrival times depends upon the presence of lacuna regions which in turn are dependent upon the monitoring angle 9, we isolate and expand in figure 13.9 the narrow region between the dotted lines of figure 13.8. The dotted straight lines in figure 13.9 correspond to the monitoring angles 0 = 5~ and 12.9 ~ respectively. The intersections of these lines with the solid portions of the curves correspond to the arrival times at that particular monitoring angle. Thus, for 9 = 5 ~ we identify five arrival times. These are actually the ones we have identified from figure 13.5. The dotted line at 12.9 ~ is intentionally chosen as the critical angle beyond which the lacuna region disappears. Also, beyond this critical angle only three distinct arrival times are found. These correspond to the three bulk waves, the innermost belongs to the quasi-longitudinal component and the other two belong to the quasi-shear components. Notice that along the 0~ only three distinct arrival times are recognized. Thus, a maximum of five arrival times are possible to exist for this cubic interval.

13.6.1

Displacement solutions

Once the roots of ~ are found, The transformed equations (13.26) together with (13.28) then lead, in typical regions away from lacunas, to the sample result 4~rC33

~ % t

Q

I ~- [

-

-

o t

+[(V3Aura - Vl)(T/+ ) 0/13+ Ot

ti) -

(V3 - V1)(?73 ) 0?73 Aura --~-]H(t

t3) (13.68) -

13.6.

243

CAGNIARD-DE HOOP T R A N S F O R M A T I O N

i~(n)

g_ 0

0.0

"

"

0.'2

tO,~)

0.4

' Re(r/)

f

0.6

I~(n)

.~,.~~-~ II 0.0

'

'

'0.'2

t(•s)

04

Re(11)

06

*~(n)

\/ _

|

o.o

.

0.'2

t(~s)

0.4

0.6

0

/~ \

Figure 13.7: Evolution of the ~/behavior for various monitoring angles for the cubic InAs material.

C H A P T E R 13. T R A N S I E N T WAVES

244

tcos8

i

0.6

tsin~ .6

0.6

0.6 Figure 13.8: Polar description of arrival times for the cubic InAs material, r - 1 mm.

tcos8

0~--~2.9~

'

O. 15

~t

Lt

,

- 0 15

- ~ -

0.00

.

~, tsinO

0.15

Figure 13.9" E x p a n d e d part of figure 13.8.

13. 7. SEMI-SPACE M E D I A

245

where tl, 't2, and t3 are the arrival times of the various wavefronts. By using the same procedure, the displacement components u2 and u3 can be obtained and need not to reported here. Meaningful results can be extracted from these expressions in two categories. In the first, variations of field variables, i.e., stresses and displacements (also velocities) with time can be readily obtained for fixed spatial positions. In the second, at a given instant of time, the complete spatial distributions of the field variables (in the forms of " snap shots") can be obtained. Typical velocity fields monitored at the location r - l m m and 0 - 5 ~ are shown in figure 13.10 for the InAs cubic material with the line load described by F(t) = H(t) and directed along the azimuthal angle r - 30 ~ Also included in the figure is the variation of Im(~) with t. The choice of 5 ~ for the monitoring angle is intended to show the details of the propagation process when monitoring is well inside a lacuna region. Although small, the zero intensity regions within the lacuna region can be identified, together with the various arrival times from these figures. In figures 13.11a,b, respectively, we collected snap shots at t - 0.2#s for two loading situations. These correspond to the cases where the load is directed along the azimuthal angles r = 0 ~ and 20 ~ of the InAs cubic material, respectively. Notice the appearance of the third wave front in figure 13.11b belonging to the S H quasi-shear wave. Its absence in figure 13.11a is due to the fact that the line load is directed along an axis of symmetry, namely, r = 0 ~ and hence S H will uncouple and will not be excited. In both equations, a spatial grid of 100xl00 points is generated for the first quadrant. The remaining quadrants are then generated by the mirror of the first quadrant. The vertical line load is located at the origin which is at the center of the picture.

13.7

Semi-space media

We next extend the solutions obtained so far for infinite media to the case of a semi-space. For convenience, we assume the source to be located in the plane x3 - 0 and that the free boundary being located at x3 = - d as depicted in figure 13.12. The implication here is that the free boundary is located in the upper region of the infinite space treated above and thus can only interfere with the propagation fields in the negative x3-direction. For this case, the solution (13.27) or equivalently (13.30) will constitute an incident wave on free surface. As a result, waves will reflect from the free boundary and propagate in the positive x3-direction. Thus, appropriate formal solutions for the

CHAPTER 13. TRANSIENT WAVES

246

g..

E I .......

0.2

0.3

o o~ &

0.3

i

,

I

t(,us)

,

i,

_

0.4

___1

0.5

E

o 0

I

.

.

.

.

.

I

,

1

t(~s)-V r

0.5

9~ _ 4 ~

o oh

|

hJ~

i

0.3

t(~,,)-v o4

0.3

t(/~s)

~

,

,,

|

o15

E

u O

~

9" -

V

x

0

" !

'--v

0.4

0.5

Figure 13.10: Distribution of displacements with t for the cubic InAs material at r -- 1 mm, 8 = 5 ~ showing amplitudes and arrival time. After Kim [121].

13.7.

SEMI-SPA CE MEDIA

247

:::: :.. * ' ~ < :::::::

.~,

....~:.:.........+..:•

::::'b',-,~:"

I

"

~x

N~

"<':~:~:::~,~:::::::::

(a)

r

~:~..<..-..,~,.-.-<<-<< x-x.- .... % 2 < . ~ .~:.-

~<,,.~ . . . . . . . . . . . . .

~.:~

:"~ii~iiiiiiiii!!ii!ii:

1

...~

(b)

Figure 13.11" Snap shots of displacement fields at t = 0.2 microsec of an InAs cubic material for (a) r = 0 ~ and (b) r = 20 ~ After Kim [121].

CHAPTER 13. TRANSIENT WAVES

248

free surface

X 3 - -d x3-O

Figure 13.12: Geometry of a semi-space with applied line source. reflected waves can be adapted from the solution (13.26) in accordance with (~, ~, ~) =

~

(1, Vq, Wq)U~qe-p~qxa

(13.69)

p(Diq'D2q'Daq)U~qe-p"qxa"

(13.70)

q=1,3,5

13' 5~3) =

E q=1,3,5

With this, the total solution for the semi-space (designated with superscript 's') which is required to satisfy the stress free boundary condition is obtained by superposing the incident and reflected waves in accordance with (fi~,fi~,fi~) =

(-1,-Vq, Wq)U~qepaqx3

~ q=1,3,5

(1, Vq, Wq)U~qe-paqxa

+ ~

(13.71)

q=1,3,5

(5~3,5~3,5~3) =

p(-nlq, n2q, D3q)U~qevaqz3

~ q=1,3,5

+ ~

p(nlq, D2q,D3q)Ufqe-paqz3.

(13.72)

q=1,3,5

The free surface boundary conditions are given by (}~3 -- 5~3 - - &:~3 ---- O,

at

X 3 --

-d.

(13.73)

249

13.7. SEMI-SPACE MEDIA

By imposing the boundary conditions (13.73) on equation (13.72), we get

(Oll o13 o15)( 1 1)(R1) D21 D31

D23 D33

D25 D35

U~3E3 U~5E5

=

R3 R5

,

(13.74)

r

(13.76a) (13.76b) (13.76c)

where 9

"

r

i

R1 - (U~1DIIE ~ + U~3D13E 3 + U15D15E5) R3 = - (U~ID21E[ + U~3023E ~ -+- U~5025E~) ' D ~ r . R5 = - (U~lD31E[ + U~3 33E~ + U~5D35E5) Using Cramer's rule, we solve for the reflected amplitudes as 1

~(R1Gll U~, = AvE1 -1 ~/ k(vRE I3G 1 3 Uf3= 1

R1G15 u 5= ~A(E5

- R3G21 + R5G31)

(13.77a)

- R3G23 + R5G33)

(13.77b)

- R3G25 + R5G35).

(13.77c)

In equations (13.76) and (13.77) we have Eq -- e p~

Eq - e -p~

q=1,3,5

Av = DII G11 - D21G21 -+-D31G31

(13.78)

with Gll

-

D23D35 - D33D25

G12 -- D 3 3 D 1 5 - D 1 3 D 3 5 G13 -- D 1 3 D 2 5 - D 1 5 D 2 3

G21 = D31D25 - D21D35 G22 "- DIID35 - D31DI5 G23 :

DI5D21 - D I I D 2 5

G31 -- D21D33 - D31D23 G32 = D31DI3 - D I I D 3 3 G33 = DIID23 - DI3D21.

(13.79)

Note that Av = 0 is as given previously by equation (7.4) and defines the characteristic equation for the propagation of the Rayleigh wave on the free

C H A P T E R 13.

250

~

~

,

.

~.... ~ ~ ~

9

. ~..~...'i~!!!!.-:::.-"::!'..:.'::i::~. :-:~!.....;-:$..~.~::i:::~:~:~:::i::.~::..'~i'.'.-:~::.'.i:::$$::. ~:~..'.:..:.'~iii:~:.:.'~'i:.'.'i:i:~".-':i.'.-~

~'~.'.,:~$.-'.~ ~!:~ ""

~

~

~.~'i::':::'-'!-':~:..'::.-:~.-..:~

T R A N S I E N T WAVES

.......... ~ ...................... ::::::::::::::::::::::::::::::::::::::::::: ~:!:~:i:i:i:!~:i':'..'~:~i::.:..':-:; ================================

~..'-'..'.-'~.-!-i$'-!-i-!'--~!$:~

9~

::::'.:.~:~::::::::;::~ 9 .,~..~..r162

9

~:~

~.~!$~:i~:]

$'~

i i i ~ i i i ~ ! ~ i' ~i i'i:i:i:~' "' : : i ~ .. -~~= ~= .=' .=-"~=i i~=i:"::::":::~: =====================

~:.:.~.~.:e.:.~.;~

"

. . ~4!~.~ -~

~. ~':-:-~-

(a)

:::::::::::::::::::::::::::::::::::::::::::...... ~:'.':i:':-":::.:-:i:i~'-::i.!~:i:i:~:i:i:~:i~% .........:tr :':':':-:':':':'.":~-i:':-:':-:".'.:'.'::-: !:.::'~" ~ . ~ $ ~

~

(b)

~:.'; ...::.,:.'.'-~. :.-.y.'...'.:.::'.. ;..:.::.-.-...f... ......

u-..~.:...

.....

-.-~:;,;~f...-

~ ~ i

..r

,~..':

...... -".'.:'." ~-- k~:--.:.:.-.:.:..:.:-.-.'-z.'-.......~.....:.:::-..

..::::.:::::::::::::::.:.'.

(c) F i g u r e 13.13: S n a p shots of t h e d i s p l a c e m e n t fields in a steel s u b s t r a t e caused by a surface line load of Dirac d e l t a t y p e at t = 0.2 ItS for: (a) f = 1, (b) f = 1.5 w i t h r = 0 ~ a n d (c) f = 1.5 w i t h r = 30 ~ . After K i m [121].

13.7. SEMI-SPACE MEDIA

251

surface. Having solved for the reflected field amplitudes (13.77), the remaining task of obtaining solutions for the displacement and stress components becomes a relatively straightforward matter. Analytical expressions of the solutions can be obtained by following steps similar to those used in the previous section. However, such expressions would be of limited value and need not be reported here. Instead, we limit our presentation to numerical illustrations. We present numerical illustrations of the snap shots for the displacement fields generated by Dirac delta line loads. Figure 14.13a presents a snap shot of radial displacement at the fixed time t - 0.2#s for an isotropic steel substrate. We next present a snap shot for an anisotropic ficticious substrate constructed from the isotropic steel material by introducing the parameter f - (Cll -C12)/2C66 with varying f. Note that f = 1 defines isotropic steel. For the snap shot of figure 14.13b, we chose f - 1.5 with r = 0 ~ Figure 14.13c repeats figure 14.14b with f = 1.5 and r - 30 ~ A spatial grid of 100xl00 points is generated for the each figure. In these pictures we can clearly recognize the three bulk waves, the surface wave and the head wave curves. The surface wave attribution is shown at the inside of the longitudinal wave front and near to the surface.

This Page Intentionally Left Blank

Chapter 14

SCATTERING FROM LAYERED CYLINDERS This chapter is devoted to the study of scattering of elastic waves from multilayered anisotropic cylinders. The present problem differs from the previous ones, treated earlier in this book, in that it requires description in cylindrical, rather than in Cartesian, coordinates. In most situations of interest, the combined anisotropy and polar coordinate description cause unusual difficulties in the analysis. This is due to two fundamental reasons: the first, and by far the most critical, is due to the fact that the field equations describing the behavior of anisotropic solids do not uncouple into pure longitudinal and shear modes as in the case of isotropic media. The second is a consequence of the fact that the equilibrium equations in cylindrical coordinates are partial differential equations with linear variable coefficients, which are usually more complicated to solve than those with constant coefficients as occur in Cartesian coordinates. In this chapter we treat the simplest case of scattering of horizontally polarized S H waves from multilayered anisotropic cylinders embedded in isotropic solids. Their relative simplicity stems from the fact that only one displacement component exists, with motion polarized along the axis of the cylinder. In order to generate only pure horizontally polarized S H waves in this system, certain geometric arrangements and propagation direction restrictions have to be met. Guided by the discussion in section 4.3.2, as a minimum, all layers must be composed of orthotropic or higher than orthotropic symmetry materials. Furthermore, the layers defining the concentric cylindrical system have to be stacked such that their material symmetry axes coincide. Lastly, a pure S H wave has to be incident from within the host medium. Only when these stringent geometric and propaga253

254

CHAPTER 14. S C A T T E R I N G F R O M L A Y E R E D C Y L I N D E R S

tion directions are met, one can excite pure S H waves in the total system. The geometric model is displayed in figure 14.1.

Y

\

r

?

Figure 14.1: Layered cylindrical model with an incident S H wave. In the subsequent sections, we seek to derive exact solutions for the scattered field due to the incident horizontally polarized shear waves on the multilayered cylinder. The wave strikes the cylinder in a direction normal to its generator, i.e., normal to the interfaces. The solution procedure uses the matrix transfer method. First, formal solutions are derived for the host medium as well as for each layer. These formal solutions are then used to relate the field variables at the outer surface to those at the inner surface in each layer. Secondly, the response of the total cylinder proceeds by satisfying appropriate interracial conditions across the layers. In section 14.1, we discuss the geometric arrangement of the system and write down the field equations and associated interface conditions in each component. Section 14.2 is devoted to deriving formal solutions in isotropic media. In section 14.3, characterization of incident waves is discussed in detail. Formal solutions for the individual anisotropic layers are given subsequently in section 14.4. These include the derivation of local and global transfer matrices and a discussion of their properties. In section 14.5, we discuss scattering amplitudes and then proceed to study scattering from solid cores and cavities. The following treatment is an adaption of the work

14.1. FIELD EQUATIONS

255

host solid o

Figure 14.2: Geometrical description of the layered system. introduced originally by Qian [205].

14.1

Field equations

Consider a multilayered cylinder consisting of an arbitrary number p of orthotropic layers rigidly bonded at their interfaces and lined up such that their axes of symmetry coincide with each other and also with the reference coordinate axis z, such that r is normal to their interfaces. This specific stacking is necessary to allow for the existence of pure shear modes. Thus, for a cylinder constructed from several unidirectional fibrous orthotropic composite laminae of the same material, only 0 ~ and 90 ~ lay-ups are allowed. The layers are numbered 1, 2...p consequently from the host solid as illustrated in figure 14.2. In our subsequent discussions, we reserve the superscript and subscript "0" to designate the host material. The outer surface of a typical layer l is located at the location r = rl measured from the center of the cylinder. It follows that the thickness of layer 1 will be r l - rz+l and hence the total thickness of the cylinder will be rl, i.e., the distance to the location of the host-cylinder interface. With the above restrictions on the geometry and on the propagation directions for pure S H waves, the motion is essentially reduced to a twodimensional problem. Here only the in-plane transverse displacement Uz (hereafter referred to as u) for each layer 1 will exist, and will be independent

256

CHAPTER

14.

SCATTERING

FROM LAYERED

CYLINDERS

of z. The relevant m o m e n t u m equation which holds for each layer and also for the solid host is Oarz Or

10aoz r O0

t

1 r

t--arz

-- p

02u Ot 2

(14.1) 9

Since the other two displacement components Ur and uo are zero here, the stress-strain relations are the simple ones OU

arz = C55~rr,

aoz =

644 OU r

(14.2)

--

Notice that, for isotropic media C44 = C55 - # which is the shear modulus. Equations (14.1) and (14.2) are supplemented with the interfacial continuity conditions U (l) - -

U(/§

,

a r(l) ----vrrr z(z+I) z

(14.3)

at r = rl+l. By combining equations (14.1) and (14.2) we obtain the wave equation

02u 1 0u 644 02u p 02u ---~ 0r + -r ~rr 4 C55 r2 002 = C55 Ot 2 .

14.2

(14.4)

Formal solutions in isotropic cylinders

We start by deriving formal solutions for the isotropic host material, namely when C44/C55 - 1. For this case, equation (14.4) reduces to 0 2u

10u

---y Or + -r ~

1 02u

-~. r.2002 . .

1

0 2U

c2t Ot 2'

(14.5)

where c 2 = C55/P which, for isotropic materials, is also given by # / p . Equation (14.5) can be solved for the harmonic case by letting

~(~, o, t) = ~(~, o)~ - ~

(14.6)

resulting in 02 (I)

1 00

1 02 (I)

w2

Or--~ + -r ~ r + -r-2- -002 + - J 9 = 0

where w is the circular frequency. variables as = R(~)O(O)

(14.7) This can be solved by separation of

(14.8)

14.2. FORMAL SOLUTIONS IN ISOTROPIC CYLINDERS

257

which leads to the separate equations

r2

d2R

d20

dO2

+

dR

(14.9)

+ (k2 2 - n 2 ) n = o

+ n 2 0 = 0,

(14.10)

where k = w/c is the wavenumber and n is a separation number. We recognize equation (14.9) as a Bessel equation; its solution is given by

R(r) = CnJn(kr) + DnYn(kr)

(14.11)

or equivalently by the form

R(r)

"--"

AnH 0) (kr) + B n H n(2) (kr)

(14.12)

where Jn (kr) and Yn(kr) are Bessel functions of the first and second kinds, H 0) (kr) and H (2) (kr) are Hankel functions of first and second kinds and An, Bn, Cn and Dn are constants. These two types of functions are related by H (1)'(2) = Jn ~ iYn. The solutions to equation (14.10) are the harmonic ones

0(0) = e •

(14.13)

The terms H(1)(kr)e -i~t and H(2)(kr)e -i~t describe progressive waves diverging and converging, respectively from the z-axis. Thus, the products of H O)'(2) (kr)e -i~t represent outward and inward propagating simpleharmonic cylindrical waves, respectively. Therefore, the general solution for cylindrical waves is obtained by superposition as

u = ~ [ A n H O ) ( k r ) + B n H(2)(kr)] Tt

cos nO sinn0

e

-i,~t

(14.14)

n--O

Or, in terms of ,In (kr) and Yn (kr) as

oo {c~ u = E [ C n J n ( k r ) + DnYn(kr)] sinn0 n--O

"

(14.15)

258

CHAPTER

14.3

14.

SCATTERING

Characterization

FROM LAYERED

of incident

CYLINDERS

waves

Since we need to deal with plane waves in cylindrical coordinates, the relation between plane and cylindrical waves is important. Consider the plane wave with displacement u traveling along the x-direction in an infinite host medium as shown in figure 14.1. Its motion is described by (14.16)

u - e i(k~

where we arbitrarily choose the wave amplitude to be unity and k0 is the wavenumber. In cylindrical coordinate description, we replace x with r cos and write -- e i(k~ cos O-wt).

u -- e i(k~

(14.17)

The exponential function e ik~176 is periodic in ~ with period 2~. It can be expanded into the Fourier series OG eik~176

--

Cn(r)einO

E

(14.18)

Tt---- CX:)

with

~(~)

1 [2~

=

lfo

e ik~ cos~e-ingd~

,,i U

eikor cos 0 COSnOdO.

__ _

From the integral definition of the Bessel function

i~g~(~) = 1 fo ~r eiko r cos 0 cos nOdO

(14.19)

we identify (14.20)

cn(r) = i n J n ( k o r )

and thus equation (14.18) becomes OG

e ik~176

~

i n J n ( k o r ) e ine.

(14.21)

n----O0

Since J - n ( k o r ) = ( - 1 ) n J n ( k o r ) , we finally write the displacement u as a superposition of infinite number of cylindrical waves as u -- ~ n:0

~ n i n J n ( k o r ) c o s n ~ e -iwt

(14.22)

14.3.

CHARACTERIZATION

OF INCIDENT

259

WAVES

with ~n={

1 n=O 2 n>0.

The incident shear wave u i from the host medium can now be written in the form ui - ~

~ n i n J n ( k o r ) c o s n O e -iWt.

(14.23)

n--O

In order to represent the scattered wave from the interface into the host, we only need the outgoing wave; this leads to (x)

us - ~

A n H ( 1 ) ( k o r ) c o s n O e -i~t.

(14.24)

n--O

The total field in the host will then consist of the sum of the incident and scattered components, namely, u = u i + u s resulting in (x)

u = ~[~,~i'~Jn(kor) + A n H n ( k o r ) ] cosnOe -i~t

(14.25)

n:O

where, for convenience, we have dropped the superscript (1) from the Hankel function; thus, from now on, Hn stands for H O). By introducing the parameter Un to write equation (14.25) compactly as (x)

u = E

Un cos nOe -i~t

(14.26)

n--0

we recognize that Un = ~ninJn(kor) + A n H n ( k o r ) .

(14.27)

Using the stress-strain relations (14.2), and referring to the shear stress component arz thereafter as a we get a = ~[E~/oJ~n(kor) + An~/oH~n(kor)] cosnOe -iWt

(14.28)

n=0

where the primes designate differentiation with respect to r and where 9/0 - C~~ )/ko,

E - ~ni n.

(14.29)

This completes the formal solution for the host solid in terms of its unknown amplitude An.

CHAPTER 14. SCATTERING FROM LAYERED CYLINDERS

260

By introducing the stress parameter an, we also write equation (14.28) compactly as oo

a = ~ an cosnOe -i~t

(14.30)

n--O

with

an = E3'oJ~n(kor) + An~/oH~n(kor).

(14.31)

The two equations (14.26) and (14.31) are collected in the matrix form

[ ] Un fin

0

= [X]0

An

'

,14

where

[ Jn(kor) Hn(kor) 1 7OJ~n(kor) ~/og~n(kor) 9

[X]0=

14.4

(14.33)

F o r m a l s o l u t i o n s for a l a y e r

When C44/C55 is not equal to 1, which is the case pertaining to orthotropic materials, we again attempt solutions of equation (14.4) in the form oo

u -- ~ R(r) cos nOe -iWt.

(14.34)

n--0

We substitute equation (14.34) into equation (14.4), recall that c2 - C55/p and arrive at the following equation for R(r)

202R OR r ~ + r-~-~r + (k2r 2 - v2)n = 0,

(14.35)

where we set v 2 = C44n2/C55 . We recognize equation (14.35) as the Bessel equation of order v, which, in general, is a non-integer; its general solution is given by

R(r) - BnJ~(kr) + C~Yv(kr)

(14.36)

where Bn and Cn are still undetermined coefficients and Jv and Yv are the first and second kind Bessel functions of order v, respectively. Accordingly, we take the general solution for the displacement in any layer 1 (except the most inner one, p) as (x)

ut -- ~'~[BnJv(kr) + CnYv(kr)]l cosnOe -i~t n--O

(14.37)

14.4. F O R M A L S O L U T I O N S FOR A L A Y E R

261

which, when used together with the constitutive relations (14.2), yields the corresponding expression for the stress as (x)

al = ~

~/l[BnJ~(kr) + C~nY~(kr)]l cos nOe -iWt.

(14.38)

n--O

The subscript l of the square brackets implies that the parameters ~ and k and the amplitudes Bn and Cn are specialized to layer 1 such that r-~(1) 1.

~l -- w55 ~l ,

r~(l) ~44 Y~ -- ~ - ~ n2 .

(14.39)

~55

By introducing the displacement and stress functions Uq) and a q), we can rewrite equations (14.37) and (14.38) as 0o

U(1) cos nOe -i~t

(14.40)

ol = ~ ~(1)cos ~0~ -i~,

(14.41)

ul - ~ n-O

and (x)

n--0

where Uq) = [BnJv(kr)+ CnYv(kr)]l

(14.42)

and

~(n~) = [ B ~ J ' ( k ~ ) +

Cn~r'(k~)]z.

(14.43)

Equations (14.42) and (14.43) can be combined, for each layer l, into the matrix form

[ ] Un ~n

_[X]l

l

[~

Cn

(14.44) l

where the 2x2 square matrix [X]i is given by [ Jv(kr)

Y~(kr) ]

[x]~= ~J'(k~) .yr'(k~)

(14.45)

CHAPTER 14. SCATTERING FROM LAYERED CYLINDERS

262 14.4.1

Local transfer

matrix

Equation (14.44) can now be used to specialize the displacement and stress at the inner and outer radii r = rz+l and r = rl of layer l, respectively as

+

O'n

l

~Tn

l

[ ]

(14.46)

= [X]i- Bn C~ ] '

(14.47)

=[X] +

B~ Cn

l

and

where

[ Jv(krl+l) Yv(krl+l) I ~J~(krl+l) ~/Y~(krl+l) l

[X]l+

(14.48)

and

Jv(krt) Yv(krl)

[X]l= 7J"(k~z) Vr'(k~)

]

(14.49) "

Here we use + superscripts to label the inner and outer boundaries of the layer l, respectively. We next eliminate the common amplitudes Bn and Cn, namely,

Cn

t

from equations (14.46) and (14.47) to arrive at

[ Un ] ~Tn

= [M]l

l

, r

(14.50)

l

where [M]z

-

IX]}-([X]+) -1.

(14.51)

The 2 x 2 matrix [M]l constitutes the transfer matrix of layer l (i.e., the local transfer matrix) which relates the displacement and stress at its outer surface to those at its inner one.

14.4. FORMAL SOLUTIONS FOR A LAYER 14.4.2

263

G l o b a l transfer m a t r i x

By invoking the continuity conditions (14.3) across each of the interfaces we relate the displacement and stress at the outer radius r = rl of the total system to those at the inner radius r - rp via the global transfer matrix relation 6rn

~n

1

p-1

'

(14.52)

where

[M] = [M]I[M]2-.- [M]p-1

(14.53)

is the global transfer matrix of the total system. 14.4.3

Properties

of the transfer

matrices

Properties of the global and local transfer matrices relevant to the subsequent analysis will now be revealed. The properties of transfer matrices for planar geometries have been extensively discussed in previous chapters. These properties also hold for the present case. For instance, by inspection of equation (14.51) we see that the determinant of its square matrix, namely [M]l is unity. Furthermore, since the determinant of the product of matrices is equal to the product of their individual determinants, we conclude that the determinant of the global transfer matrix [M] is also unity. Let us next investigate situations in which neighboring layers are made up of the same material. In particular, let us examine the term [M]t[M]l+l when the material properties of layers l and l + 1 are identical. To this end, we consider the bilayer transfer matrix [M]I[M]L+I

=

[X]-[- ([X]-~-)-l[xll+l ([X] 1+1)+ 19

(14.54)

When kl = kl+l, 71 = ~/1+1, and vl = VZ+l, in other words, when layer l and layer 1 + 1 are made up of the same material, by inspection, the matrices [X]~- and [X]/-+I are found identical. This means that the middle two terms of equation (14.54) compact to the unity matrix [I] leading to the reduced form

[M]I[M]I+I=[X][-([X]+/+1 )-1

(14.55)

which is the transfer matrix for a single material spanning the region of both layers 1 and 1+ 1. By permutation of the above, we can see that, when all layer materials are the same, the global transfer matrix [M] reduces to a single "local" transfer matrix for the single material occupying the whole cylindrical region.

CHAPTER 14. SCATTERING FROM LAYERED CYLINDERS

264

14.5

Scattering amplitudes

We now specialize equation (14.32) of the host material to the location rl and get An

an

= (Ix]+

0

'

(14.56)

where [X]+ = [

Jn(krl) Hn(krl)] ~J~n(krl) VH~n(krl) o

(14.57)

and where the superscript + designates quantities specialized to the location r = rl 9We notice that the continuity conditions at the host-layered cylinder interface are satisfied by requiring that

an

0

an

1

Using this condition between equations (14.56) and (14.52) results in

An

-

[Q]

an

,

(14.59)

p-1

where [Q] = ([X]+)-I[M].

(14.60)

Furthermore, applying equation (14.44) to the core layer p followed by using the continuity relations between layers p - 1 and p results in

An

Cn p'

,1461,

where IT] = [Q][X]p

(14.62)

and IX]p=

[ Jv(krp) Y,(krp) ] . VJ~(krp) vY~(krp) p

(14.63)

14.5. SCATTERING AMPLITUDES 14.5.1

Scattering

265

from a solid core

If the innermost component p is solid with no cavity, its Bessel function solution Yv(kpr) becomes singular as kpr --+ O, and hence, its coefficient Ca must vanish. Therefore, the solution in the core component takes the reduced form oo

Up -- E [BnJv(kr)]p cos nOe -i"n

(14.64)

n=0

Since E is known, equation (14.61) has only the two unknowns An and [Bn]p determined as [gn]p-

14.5.2

-

E

--, Tll

ET21 An = ~Tll.

Scattering

(14.65)

from an inner cavity

If the inner component p is a complete cavity, then the reflected amplitude An can be derived directly from equation (14.59) under the condition that [an];_l : 0 leading to

EQ21 An = ~ .

(14.66)

Qll

14.5.3

S t r e s s e s in

the

host

medium

After the unknown coefficient An is determined, we use the stress-strain relation to solve for the stresses in the host material. Under excitation by the S H wave, these stresses are then given by (2O

arz = ~/o ~_~[EJ'n(kor) + AnH'n(kor)] cosn tge-i~t

(14.67)

n=0

We next define the dynamic stress concentration factor nij for a particular stress component a i j as

tZij --

ai____A_J

(14.68)

which are obviously functions of time. Here a~)~n is the stress in the host in the absence of the obstacle, namely, the layered cylinder.

266

CHAPTER 14. SCATTERING FROM LAYERED CYLINDERS

14.5.4

Scattering

cross section

If we consider an obstacle irradiated by an incoming plane harmonic wave, the time average power flux of the scattered field, over a given time period T, is obtained from equation (3.84) as

is =

1 T OE dt = 1 T T fo --~ T foo I s / P ~ n k d S d t '

(14.69)

where the superscript s denotes the quantity solely associated with a scattered field. Here P~ is the Pointing vector described in section 3.6. This quantity describes the energy scattered into all directions and, thus lost by the incident beam due to its interaction with the obstacle. The above surface S is associated with the surface of the scattering obstacle. The magnitude of I s is often normalized with respect to the average incident energy intensity P. The intensity per unit area in the direction of propagation ni is defined as i~ = ~1 ~0T Pinidt

(14.70)

We introduce the ratio Is V = ~7

(14.71)

to characterize the sole effect of the obstacle on the power withdrawn from the incident elastic wave. The quantity V(w) is referred to as the total elastic scattering cross section.

Chapter 15

ELASTIC PROPERTIES OF C O M P O S I T E S In the previous chapters, analyses were conducted in terms of material properties that are presumed to be known for all materials involved. In practice, however, these properties are not always available, especially for structural fibrous composites, and that such properties have to be determined a priori. Traditionally, material properties for conventional materials have been obtained experimentally and are subsequently compiled in handbooks. Such an approach is not practical for fibrous composites due to the large variety of the parametric tailoring that can exist in the construction of these man-made materials. Also, it is obvious that such variability will present difficulties in the construction of universal analytical models for the prediction of their physical behavior. For these reasons, many modeling attempts have emerged during the last three decades or so that were directed towards the prediction of the various physical properties of laminated and fibrous composite materials. This chapter is devoted to the task of developing models for constructing the effective composite elastic properties of straight as well as undulated fibers.

15.1

General description of fibrous composites

Typically, structural fibrous composites are composed of brittle, stiff fibers embedded in a matrix of a more ductile material which bonds the fibers together and acts as a load-transfer medium. In principle, we can think of an infinite medium constructed from such combinations. This is referred to as the bulk composite medium. Out of this large medium, we isolate a thin 267

268

C H A P T E R 15. E L A S T I C P R O P E R T I E S OF COMPOSITES

x

1

Figure 15.1: The circular and square fibrous models.

layer, bounded by two planes that are parallel to those of the fiber direction. This layer is called a coupon or a lamina. Across its thickness, a lamina can have one or more fibers. Even if the fiber and matrix constituents of the lamina are assumed to be isotropic, the lamina will be anisotropic and the fiber direction constitutes a principal material direction. From the basic lamina cell, we can define and construct a laminate. A laminate is a stack of laminae with various orientations of principal material directions that are attached at their interfaces in some quantifiable fashion. For the construction of desired effective material properties, the geometry commonly employed belongs to an infinite elastic medium reinforced with circular cross-sectional fibers of different material, aligned along the x 1-direction of the reference coordinate system xi. This geometry, together with its alternative one with square cross-sectional fibers are shown in figure 15.1. In most situations of interest in structural applications, the matrix component is isotropic. The fiber material, on the other hand, often exhibits some degree of anisotropy. If anisotropy is present in the fiber, it is mostly in the form of transverse isotropy where the cross section of the fiber is isotropic. In light of the above restrictions on the "virgin" properties of both the fiber and the matrix, the symmetry exhibited by geometries similar to those displayed in figure 15.1 leads to imported simplifications in the overall behavior of the composite system. In actual situations, it is inevitable for such man-made materials to have

15.2.

THEMODEL

269

a large variety of defects. These include, but are not limited to, variability in fiber cross section, misalignment, poor bonding, delamination and porosity, to name a few. These defects can influence the behavior of the composite to various degrees. In idealized situations, where perfect geometry and interface conditions are encountered, the dominant factors that control the overall properties include the properties of both the fiber and the matrix, their volume fractions and their specific arrangements. The result is a homogenized anisotropic material whose effective properties are volume fraction weighted. Traditionally, and as a first step in estimating their mechanical properties, idealized situations are assumed such that the influence of defects is ignored.

15.2

The model

One simple model which we find to give good results is based upon approximating the circular fiber with a square one having the same volume fraction as shown in figure 15.1. Confidence in this modeling procedure has been established by the fact that comparisons between the theory and the experiments presented in chapter 11 were all based upon properties calculated using this approach. Undoubtedly, the resulting expressions are approximate, but highly accurate. This chosen geometric approximation still insures, as a minimum, that properties belonging to the x2-direction are completely interchangeable with those corresponding to the x3-direction. Keeping this observation in mind, we can then proceed to employ a building-block approach in order to arrive at the desired results. First, a layered structure, composed of fiber and matrix layers, is analyzed thereby, deriving from this parallel-layer model the properties of a compound layer 1. In the second step, intermediate results for the composite are established by treating it as consisting of the compound layer 1 stacked in series with the matrix layer 2. Guided by the interchangeability of properties in the x2 and x3-directions, in the third step, we interchange the results pertaining to the directions x2 and x3. In the final step, we take the average of both properties to arrive at the desired results.

15.3

T h e layered m o d e l

Consider the bilayered composite of figure 15.2a, extracted from figure 15.1 and representing an extended (in the x3-direction) infinite medium. It consists of a periodic array of two laminae of fiber and matrix with the thicknesses hi and h2, respectively in welded contact and stacked in the

270

CHAPTER 15. E L A S T I C PROPERTIES OF COMPOSITES

x3

x3 x~ = h 1

X2

Matrix Layer, 2

(a)

Co)

Figure 15.2: The layered model. x2-direction. The transitional invariance (periodicity) of the problem permits us to isolate the repeating unit cell shown in figure 15.2b. For the convenience of the following analysis, we choose the local coordinate system xi with its origin located at the interface. With reference to this system, figure 15.2b shows that materials 1 and 2 occupy the regions 0 _ x2 < hi and - h 2 _< x2 <_0, respectively. We now seek to replace this layered system with the homogenized compound material shown in figure 15.3a. The constitutive relations for each of the two assumed isotropic materials are written as al = (A + 2#)el + A(e2 + e3)

(15.1)

62 -- ()k -~- 2 # ) e 2 ~- ~ ( e l + e3)

(15.2)

63 = (A + 2#)e3 + A(el + e2)

(15.3)

623 = #~/23

(15.4)

613 = #~/13

(15.5)

612 = #712.

(15.6)

If in this intermediate step, we arbitrarily identify the effective properties of the resulting compound material with the elements Dpq, then, with reference

15.3. THELAYEREDMODEL

271

to figure 15.2a, and to the fact that both materials are assumed isotropic, the resulting layered medium is expected (but not yet proven) to possess transverse isotropy with isotropy confined to the xl-13 plane. This will also be realized from the subsequently derived results. In the meantime, we assume the constitutive relations for the layered model to take the form of the less symmetric orthotropic material, namely ~1 ~2 a3 023 ~13 ~12

=

Dll D12 D13 0 0 0

D12 D22 023 0 0 0

D13 D23 D33 0 0 0

0 0 0 D44 0 0

0 0 0 0 D55 0

0 0 0 0 0 D66

el e2 e3 "~23 ~13 "712

(15.7)

Here we introduce the overbar to indicate effective field variables (stresses and strains). We now proceed to derive analytical expressions for these property elements. As just mentioned, in the final analysis, it is expected that not all of the nine entries of equation (15.7) are independent of each other. Essential to the subsequent determination of these properties, is the ability to accurately account for the appropriate deformation compatibilities that exist between the fiber and matrix layers.

15.3.1

Averaging

If equations (15.1)-(15.6) are averaged for each component across their respective thicknesses according to (~(a) = 1

ha

foh~()(a)dx~a)~

a= 12

(15.8)

we get ~.~a) = (A + 2 # ) a ~ a) + Aa(~ a) + ~,~a))

(15.9)

6"~a) -- (A + 2 # ) a ~ a) + Aa(~ a) + C~a))

(15.10)

-- (A + 2#)a~ ~) + A,~(~ a) + ~a))

(15.11)

-(o/ - #~723

(15.12)

-- #a713

(15.13)

= #a712 ,

(15.14)

~)

where a = 1 and 2 refer to material 1 (fiber) and material 2 (matrix), respectively.

272

C H A P T E R 15. E L A S T I C P R O P E R T I E S OF C O M P O S I T E S

15.3.2

Strain and stress compatibilities

An important consequence of the periodicity and the uniformity of loadings of the present layered medium is that well characterized compatibilities of deformation and stress distributions exist. For example, under uniform inplane loading, due to symmetry, all the deformation compatibility relations for the in-plane strain components are the equal effective ones ~I = ~,~1) = ~2) ~'3 -- ~ I ) = ~_,~2) ~(2) '~13 ~ ~(1) I13 ~-I13 9

(15 15)

For those belonging to the out-of-plane strains, the effective components have the properties ( + =(2) e2 -- Vle 89 -~" v2v2

= Vl+ 1> _ ~(1) + =(2) ")'23 -- V1123 -~- v2"Y23

(15.16)

where vl = ( h i / h ) and v2 - ( h 2 / h ) are the linear volume fractions of layer 1 and layer 2, respectively, (h = h l + h2. These deformation compatibilities are supplemented with appropriate effective stresses oij defined as follows: those belonging to the in-plane are the effective mixture ones (~1 -- ( ~ I ) v l - ~ - ~ 2 ) V 2 = _

+ _(1)

(2

a13 = ~13 Vl -~-~i3)V2

(15.17)

and, for the out-of-plane, as the equal effective ones

(15.18) This sets the stage for relating the effective average stresses to the effective average strains and thus for identifying the various stiffness coefficients Dvq. 15.3.3

Analysis

To this end, by substituting from equations (15.9), (15.11) and (15.13) into equations (15.17), and using the compatibility relations (15.15), we get +

=(2)

(~1 -- [()~ + 2 # ) l V l +(/~ + 2~)2V2]el-+-()~lVl + )~2v2)e3+~lVle~ 1) -+-)k2u2~2

15.3.

THE LAYERED

273

MODEL

0-3 -- [(z~ %- 2#)1V 1 -'[-(z~ %- 2~)2V2]e3-['- (z~lVl %- z~2V2)e 1 -['-z~lVle~1) -'[-z~2~2L'~2)

(15.19)

0"13 -- (~lVl %-#2V2)~13 9

Examination of the first two components of equation (15.19) reveals that, except for the terms involving ~1) and g~2), they relate the effective average stresses to the effective average strains. It now remains to eliminate ~1) and ~2) from these two equations. In order to do so, we first use equation (15.10) together with the first of equations (15.18) and arrive at (~ %- 2~)le~ 1) -- ()~ %- 2#)2e~ 2) --" ()~2 -- /~l)(el %- e3)"

(15.20)

This equation, together with the first of equations (15.16) constitute two linear algebraic equations for the two unknowns ~1) and ~2). Solving for these strains gives ~1) :

()~ %- 2~)2e2 %- v2()~2 -- )~l)(el %- e3)

(A %-2#)2Vl %- (A %-2#)1v2 ~2) _ (A %-2#)1e2 %-Vl(A1 -- A2)(el %-e3) (A %-2#)2Vl %- (~ %-2~)1v2 "

(15.21)

Finally, substituting from equation (15.21) into equation (15.19) and collecting similar terms lead to 01 -~ D11e1%- D12e2 %- D13e3 ~3 = D13el %- D23e2 %- D33e3

0"13 = D55"~13,

(15.22)

where D33 - D l l - [(A %- 2#)lVl %- (A %- 2 # ) 2 v 2 ] - (A2 (A %- 2#)2v1A1%- (A %- 2#)1v2A2

D12 -

A1)2/E

VlV2E

D13 -- (AlVl %- A2v2) - (A2 - A1)2/E

D55 = #1vl + #2v2

(15.23)

and E is given by E = (A %-2#)1%- (A + 2#)2 . Vl

v2

(15.24)

Next, we derive expressions for the remaining properties, namely, D22, D44 and D66. In order to do so, we call, once again, upon the components of equation (15.10) and explicitly solve them for g~l) and g~2) as ~1) __ 0"~1) __ )~1(~1) %- ~1))

(A %-2#)1

CHAPTER 15. ELASTIC PROPERTIES OF COMPOSITES

274

(A + 2#)2

"

(15.25)

Using the first component of equation (15.18) and the first two components of equation (15.15) further reduce equations (15.25) to e l) = a2 -

1(el + e3) (A -+- 2#)1

(A + 2#)2

"

(15.26)

Next, constructing from these expressions the effective strain ~2 as per the first component of equation (15.16), followed by collecting similar terms and rearranging, we finally write #2 -- D12el q- D22e2 -b D23e3

(15.27)

where we recognize that D23 = D12, which has already been derived above in equation (15.23) and + + 2,)2 9 (15.28) Vl (,~ -[- 2#)2 + v2(A + 2#)1 It is a simple matter to apply the above procedure, which we used to calculate D22, in order to derive expressions for D44 and D66. Using equations (15.12) and (15.14) in conjunction with the strain relations (15.16) and the average stresses (15.18) yields 022 ---

#12 = D66~12 #23 -- D44~23,

(15.29)

where D44 = D66 =

#1#2

#lV2 -F- ~2Vl

.

(15.30)

This completes the derivation of the layered model properties Dpq, namely those belonging to the compound layer 1. For convenience, we collect them in the expanded matrix form #1 #2 #3

Dll D12 D13 0 0 0 el D12 D22 D12 0 0 0 e2 = D13 D12 Dll 0 0 0 e3 (15.31) #23 0 0 0 D44 0 0 ~23 ' #13 0 0 0 0 D55 0 ~13 #12 0 0 0 0 0 D44 "712 which reflects the various relationships that exist between them. These clearly display the transverse isotropic property of the layered model as we conjectured earlier. Note from equations (15.25) that Dll - D 1 3 - 2D55.

15.4. THE SQUARE FIBROUS CASE

275

Figure 15.3: The fibrous transformation.

15.4

The square fibrous case

We now proceed to derive effective properties for the "layered" system composed of a periodic combination of the compound layer 1 and the matrix layer 2 stacked normal to the x3-direction as illustrated in figure 15.3b. The properties of the compound layer 1 are given by equation (15.31) and those of the matrix layer 2 by equations (15.1)-(15.6). It is then obvious that the construction procedure will follow the one used in the previous section for the layered model. The only difference is that layering now is normal to x3 rather than to x2 as in the previous case. Starting afresh, we consider the compound material 1 as a new "virgin" material which is to be stacked together with the matrix layer 2 in a periodic manner normal to the x3-direction. To this end, we drop, for the time being the overbar from equation (15.31) and proceed to average the new combinations of constituents across their thicknesses in accordance with

()- = V.1 ~0 ha ()(~)ex~)'

a = 1,2

This leads to

(1) (71

_ ~11~1) + ~1241) + ~13~1)

41/_ ~1~1/+.~41) + ~41)

(15.32)

CHAPTER 15. E L A S T I C PROPERTIES OF COMPOSITES

276

=(1) ~1) _ D13g,~l)+ D12~l)..k n*-'11~3

(15.33)

~3) -- D44723 -(1) ~1) n =.(1) = .u55- Y13 ~.{1) ____D 4 4 ~ ) .

(15.34)

Equations (15.9)-(15.14) with c~ = 2 hold for the matrix layer 2 with averaging is now understood to take place with respect to x3 rather than with respect to x2.

Compatibilities

15.4.1

The deformation compatibility relations for the in-plane strains now require

el--

~1) __~2)

e2 = ~1)_. ~2)

_(1) = ~ ) ~12 -- i ")'12

(15.a~)

and, for those belonging to the out of plane strains, the effective mixture ones m(1)

e3 --- Ul~3 _

_(1)

~2) -~-V2 .

=(2)

~13 ~---VI")113 -}- v2"YI3 . =(2) ")'23 -- Vl x(l) /23 ~- "u2"Y23 _

(15.36)

where Vl = (hi/h) and v2 - (h2/h) are, once again, the linear volume fractions of layer 1 and layer 2, respectively. These deformation compatibilities are supplemented with appropriate effective stresses ~ij. For the in-plane components, we construct the effective mixture ones (~I -- (7~I)vl -I- ~2)V2

~

= 41)v1 + 4 ~)~

and, for the out-of-plane, we require

O'3 = ~1) _= ~2) -

)

(15.38)

15.4. THE SQUARE FIBROUS CASE

277

Equations (15.36) and (15.38), together with the constitutive relations for the compound layer 1 and matrix laver 2 can be combined in order to solve for the individual averaged strains; these lead to ~1) _- [v2(A2 - D13)el 4- v2(A2 - D12)~2 + (A + 2#)2a3] v2Dll 4- Vl (A 4- 2#)2 ~2) __ [vl(D13 - ~2)el 4- vl(D12 - )~2)e2 4- Dlle3] (v2D11 4- Vl(A 4- 2#)2 ,~(1) _ #2 _ 13 #2Vl 4- D55v2 ~/13

(2) = 13

D55

_

#2Vl 4- D55v2")'13

(i)

#2

(2)

D44

_

23 -- #2Vl 4- 944 V2 ")/23 23 -- #2Vl 4- D44v2 "~23

(15.39) As an example, the expressions for ~1) and g~2) can be easily obtained by satisfying ~1) = ~2) and using the first of equations (15.36). 15.4.2

Analysis

By following the steps commencing immediately after equation (15.18) and ending with equation (15.31) we obtain (~1 ~2 O'3 (~23 ~13 o'12

__

Gll Gi2 G13 0 0 0

G12 G22 G23 0 0 0

G13 G23 G33 0 0 0

0 0 0 G44 0 0

0 0 0 0 G55 0

0 0 0 0 0 G66

where Gii = [Diivi + (A 4- 2#)2v2]- (A2 - Di3)2/Ev Gi2 - (Di2vi 4- A2v2) - (A2 - D13)2/Ev G22 -

[D22vl 4- (~ 4- 2 # ) 2 v 2 ] - (~2 - D12)2/Ev

G13 = [D13(A 4- 2#)2/v2 4- A2Dll/Vl]/Ev G23 - [Di2(A 4- 2#)2/v2 4- A2Dii/vi]/Ev D11(A 4- 2#)2 G33 = Dilv2 4- (A 4- 2#)2Vl

E1 e2 e3 "~23 ~13 "~12

(15.40)

CHAPTER 15. ELASTIC PROPERTIES OF COMPOSITES

278

G44 -

D44~2 D44v2 + #2Vl

D55~2 G55 -- D55v2 + #2Vl

(15.41)

G66 -- D44Vl + #2v2

with Ev given by

Ev = (~ + 2#)2 § D ll . V2

(15.42)

Vl

The results obtained so far are based upon adopting the procedure in which we arbitrarily choose the x2-direction to be normal to the fiber and matrix layers, followed by choosing the x3-direction to be normal to the compound layer 1 and matrix layer 2 assembly. Due to the symmetry of the fiber cross-section, we now interchange the roles of the x2 and x3-axes in the previous two steps and obtain corresponding properties. This, of course, results in interchanging Gij elements corresponding to x2 and x3, respectively. Better approximations of the overall properties will then be obtained by taking the average of these two results, finally leading to ~1 ~2 ~3 ~23

__

~3 0"12

Cll C12 C13 0 0 0

C12 622 623 0

C13 623 633 0

0 0 0 644

0 0 0 0

0 0 0 0

~1 e2 e3 "~23

0

0

0

0

0

0

C~5

0

9~3

0

C66

'~12

(15.43)

Cll - Gll 623 - G23 C44 = G44 C33 = C22 =

G22 + G33

2 G12 -I- G13 C13 -- C12 -2 G55 + G66 655 -- 6 6 6 . 2

(15.44)

To illustrate the degree of accuracy of the above modeling procedure, figure 15.4 displays a sample comparison between results obtained via the present model and by standard finite element codes. For illustration, we

15.5.

279

A N I S O T R O P I C FIBER A N D M A T R I X

15

............................... :

12

......

1~

! ............

.

:

:

::

............

FEM .circularfiber i ........... i [-7 FEM square fiber :: '-theoreticalrn o ~ ~ Q.

.

.

.

.

.

:.

o-- 7

2 :

0

0.0

.

.

.

.

I

0.2

. . . .

I

:

0.4

,

,

,

,

I

0.6

:

,

,

,

,

!

0.8

.

.

.

.

I

1.0

Vf

Figure 15.4: Comparison of analytical model with finite element results. choose the property relations ($ + 2#)1/($ + 2#)2 = #1/#2 = 10. The finite element results were obtained for both circular and square fiber cross sections, having the same volume fraction. As can be seen, the model very well predictsthe results of the finite element calculations.

Anisotropic

15.5

fiber

and

matrix

For the case in which the fiber and the matrix are made up of orthotropic materials, the above construction procedure of the effective properties holds in its entirety. Specifically, all deformation compatibilities and the definitions of the effective stresses and strains are identical. The only differences are encountered in the forms of the fiber and matrix "virgin" constitutive relations. We now designate the orthotropic properties of the fiber and matrix, by their expanded forms as

al a2 a3 0"23

--

fll f12 f13

f12 f22 f23

f13 f23 A3

0 0 0

0 0 0

0 0 0

0

0

0

f44

0

0

O.13

0

0

0

0

f55

0

0"12

0

0

0

0

0

f66

el e2 e3 723

713 712

(15.45)

280

C H A P T E R 15. E L A S T I C P R O P E R T I E S OF C O M P O S I T E S m::

al 02

m12

(73 0"23

0"13

0"12

_

m13 0 0 0

m12 m22 m23 0 0 0

m13

0

0

0

el

m23

0

0

0

e2

m33 0 0 0

0 m44 0 0

0 0 m55 0

0 0 0 m66

e3 "f23 ")/13 ")'12

(15.46)

Next, we summarize the steps needed to obtain intermediate and final effective properties. The steps are identical with those encountered in sections 14.3 and 14.4. 15.5.1

The layered model

Here, the deformation and stress compatibility relations (15.15)-(15.18) of the previous section are applicable in their entirety. It remains to express the remaining strains associated with the direction x2 in terms of the effective composite strains. As explained in more detail in subsection 15.2.1, but summarized here, we first use the stress-strain relations (15.45) and (15.41) together with the stress and strain compatibilities (15.18) and (15.14), respectively to obtain ~1) =

v2

v2 f 22 + vim22

Vl v2 f 22 + v: m22

[(m12 -- f12)~'1 +

[(1:2 -

m121:1 +

m22 e2 + (m23 -- f23):,3] v2

f22:2 + (f2a - m2a) 31 Vl

(15.47)

_

m66v: + f 66v2 y:2 --

~(23:) = ,~(2) 23 =

m66 Vl + f66 V2 m44 m44vl +

')'12

f44v2")'23

f44

m44v: + f44v2

(15.48)

~/23

Next we substitute from equations (15.45) and (15.46) into equation (15.17), use the relations (15.15) and (15.47) and arrive at the formal relations (15.22) with the new definition of Dij as Oil

-

( f l l V l + m l l V 2 ) - (f12 -

ml2)2/Ea

15.5. ANISOTROPIC FIBER AND MATRIX O12 =

281

f12m22vl + m12f22v2 m22vl + f22v2

D13 = (f13vl + m13v2) + (f12 - m12)(f23 -

m23)/Ea

(m23 - f23)2/Ea f23m22vl + m23f22v2

D33 -- (f33vl + m33v2) D23 --

m22vl + f22v2 D55 = f 55vl + m55v2 E~ -

f22 Vl

m22

+ ~.

(15.49)

v2

To obtain new definitions for the remaining elements D22, D44 and D66, we follow the steps commencing below equation (15.24) to obtain D22 :

f22m22

m22vl + f22v2 f44m44

D44 = fa4v2 + m44v1 D66

-

15.5.2

f66m66 . f66v2 "4- m66vl The

fibrous

(15.50) case

The steps followed in section 15.4 for the is0tropic cases can be applied to the anisotropic case leading to the new definitions of the Gij elements G l l -- ( D l l V l + r a l l y 2 ) - (D13 -

m13)2/Ee

G12 -- (D12vl + m12v2) + (D13 - m13)(m23 - D23)/Ee G13 =

D13m33v1 + m13D33v2 m33vl + D33v2

G22 = (D22vl + m22v2) - (m23 G23 -" G33 = G44 =

D23)2/Ee

D23m33vl + m23D33v2 m33vl + D33v2 D33m33 m33vl -4- D33v2

D44m44 D44v2 + m44v1

D55m55 G55 = D55v2 + m55vl G66 --= D66Vl + m66v2 D33 m33

E~=

Vl

~--.

v2

(15.51)

282

C H A P T E R 15. E L A S T I C P R O P E R T I E S OF C O M P O S I T E S

From these elements, the Cij properties can be constructed by averaging elements belonging to directions 2 and 3 in accordance with equation (15.44).

Strain energy approach

15.6

The effective elastic properties derived so far for the composite model can also be obtained by using the strain energy approach. In this approach, the strain energy of the total system consists of the sum of the contributions from the fiber and matrix components. The appropriate compatibility conditions of the strain and stress components found in the proceeding sections can be used to write the total strain energy in terms of effective strain components of the composite. Differentiating this expression with respect to the individual effective stress components yields the constitutive relations. For orthotropic materials, the strain energy density is given in terms of the average stresses and strains in accordance with 1

1

(15.52)

U "-- -~O'ijeij "-- -~Cijklgijekl

15.6.1

T h e layered m o d e l

The total strain energy density of the system is given by (15.53)

U - Uf +Um

which, when using the tress-strain relations (15.45) and (15.46), is written in the expanded form gf -

1 )2 + Vlfl2 g,{1)~1) + ~Vlfll(~l)

Vlf13~1)~1)

1 )2 . . _(I)_(1) 1 [-(I)~2 q-~Vlf22(g'~1) q--vla23e2 e3 +-~Vlfaave a ;

+~Vlf44,(1) )2 + ~Vlf551,-(1) 1 I,'~23 (,~f13 ) 2 + 1 Um=

1 ))2 + ~v2mll(~l

v2m12~l)~1) +

2

v2m13 ~1) ~,~1)

+~v2m221 (~2))2 + V2m23~2)~(33) + 1 1 1 1 ,_(2) 2 .+_~v2m44,(2) 2 -{- ~v2m551,"/13 ,_(1) )2 -{- ~v2m66("Y12 ('/:23) ) (15.54) where Vl and v2 are fiber and matrix volumes, respectively.

15.6. STRAIN ENERG Y APPROACH

283

In order to express the strain energy density of the system in terms of the effective strains of the system, certain compatibility conditions must be used which relate the average strains in the fiber and the matrix to the effective composite strains. For the layered system these are given in equations (15.15), (15.18) and (15.54). The appropriate relationships between the various stress components are also the same as given by equations (15.17) and (15.18). Implementing these various compatibilities into the expression for the strain energy density gives

1

Vf

-

-

~Vlfllg'2 -'k Vlfl2e.1 1

q-~Vlf22(

~1)

~1) 2

q- Vlfl3ele3

) -+-Vlf23

1

~1)

1

1

2

g'3 "Jr-~vlf33e-3 1

+~vlf44/-(1) t,'Y23 ) 2 q- ~Vl f55'~23-+- vlf66('~)) 2 1 -2 -- ~v2mlle I + v2ml2el e~l) q- v2ml3ele3

Um

1

~2) 2

i

)2

+~v2m44 '-(2) %a

8~2)e3 +

1

v2m33e2

i 1 + ~v2m55(~123 q- ~v2m66('~2)) 2

(~5.55) where g~l), ~2), ~2) ' etc. are given in terms of the average strains in equation (15.47). Either direct substitution from equations (15.47) and (15.48) into equations (15.55) to obtain explicit dependence of U on effective average strains or implicit combination of these equations finally insures writing the strain energy density in terms of the purely effective strain components. Constitutive relations are then obtained from the strain energy density by straightforward differentiating with respect to the effective stress leading to

OU al

:

Oel

= (v~f~ + v~m~)< + ~ f ~ 4 ~ + v~m~4 ~ 0 ~ 2) +(vlf12 Oe~) 0~1 + v2m~2 0~1 )e~ + (vlAa + v2m~a)~a

0g'l "k v2m22 4

0~ 1) + (VlS 0e.1 0gl +

Oq~2) 0gl

(~5.56)

Carrying differentiation with respect to the remaining effective stress components and using equation (15.47) finally recovers the forms of equations

284

CHAPTER 15. ELASTIC PROPERTIES OF COMPOSITES

(15.22), (15.27) and (15.29) with the elements Dij as given by equations (15.49) and (15.50). 15.6.2

The fibrous model

Starting afresh, we assign the Dij properties to the compound layer 1. If this layer type is combined with the matrix layer 2 in series normal to the x3-direction, we simulate the fibrous case. For this system, the strain energy density is now expressed as 1 1 ,-(1) 2 V = ~vlDll~2 + 2lv1D22~ 5~ + ~Vl D33(e 3 ) = =(1) n =(1) +vlD12el el2 -+- vlD23c2c 3 -+- VlJJl3V 3 1 ,-(1)~2 1 ,-(1))2 1 +~Vl D44 (~/23 J + ~VlD55 (,~13 + ~Vl D66~22 +"~lv2mlle21 -+- v 2 m l 2 e l e2 + v2m13 e 1e3

1 2 +-~lv2rn22~2 + v2rn23~:~2) + ~v2m33(~2))

1

,-(2/ ) 2 + 1

('Yla

)2

1

+ -~v2m66

)2 (15.57)

Utilizing the compatibility relation os section 15.4.1, followed by differentiation with respect to the effective stress components, finally recovers relations (15.51) for the elements Gij.

15.7

U n d u l a t e d fiber

In many applications weakness in the transverse direction, especially the transverse shear, renders the composite susceptible to damage and hence limits its application. There are many ways in which we may be able to achieve an increase in the transverse stiffnesses. One way is to alter the basic design (texture) of the composite. Rather than adding extra stiffening fibers in the transverse direction, we can use the concept of woven composites in which the fibers are assumed to be curved with a certain periodicity as illustrated in figure 15.6. As will be demonstrated later, however, an increase in the transverse stiffnesses will be achieved at the expense of a simultaneous reduction in the longitudinal ones. In fact, we can, by using appropriate tailoring, achieve an almost isotropic material.

15.7.

UNDULATED FIBER

285

Figure 15.5: Unidirectional undulated fibrous composite system.

We start with the case of a straight fiber. Its constitutive relations were derived in section 15.4 and 15.5. If the fiber happens to be oriented in the Xl - x 3 plane at angle ~ with respect to the xl-direction, then its contribution to the xi-direction is governed by the transformation (2.9) applied to x l - x 3 plane. In what follows we present a simple model based upon microlaminates plate theory to obtain effective elastic properties for an arbitrarily undulated fibrous system. We first recall that the linear transformation in the x l - x 3 plane leads to C~1 = C11G 4 + C33S 4 + 2(C13 + 2C55)$2G 2

C~3 = (Cll + C33 - 2C13 - 4C55)$2G 2 + C13 C~2 = C12G 2 + C23S 2

C~5 - (C13 + 2C55 - C 1 1 ) S G 3 + (C33 C13 - 2C55)GS 3 C~3 = C l l s 4 + C33G 4 + 2(C13 + 2C55)$2G 2 C~3 = C23G 2 + C~2S 2

C~5 - (C13 + 2C55 - C 1 1 ) G S 3 + (C33 - C 1 3 - 2C55)SG 3

6~2 = 622 C~5 = (C23 - C12)SG C'44 = C44G 2 + C66S 2

CHAPTER 15. ELASTIC PROPERTIES OF COMPOSITES

286

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f f

/-

* .

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.

.

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.

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.

.

.

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.

.

.

L

.

.

.

.

.

.

.

.

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.

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.

.

.

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.

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.

.

.

.

:/ / .~. . . . . . . . . . . . . . . . .

X!p+I) 2L

X!p)

iX1

h

x~+2L

.....................................

Figure 15.6: Geometry of the undulated fiber. C~5 -- (644 - C66)SG C~6 = C66G 2 + C44 S 2 C~5 = C55 + (Cll + C33 - 2C13 - 4 C 5 5 ) $ 2 G

2,

(~5.58)

where C=cos0 and S=sin0. 15.7.1

Discretization

Let us imagine a periodic fiber with period extending from xl to Xl + 2L is approximated by an arbitrary number n of equal length straight segments p, p - 1,2, ..... n as shown in figure 15.6. The typical straight segment p then occupies the region between x~p) and x~p+I) defined by

x~~) = x~ + ph,

~§

= ~1 + (p + 1)h

(15.59)

where h = 2L/n is the length of each segment. We assume that segment p is inclined at an angle Cp from the x 1-axis such that tan ~)p ~-

df (x~P))/h,

(15.60a)

df (x~p)) - f (x~p+I)) - S(x~P)),

(15.60b)

where

From this relation we can easily define Gp = cos r and Sp = sin r Furthermore, since all of the straight segments have the same "local" properties,

15.7.

UNDULATED FIBER

287

the overall properties of the composite can be obtained by the sum of the averaged contribution of the individual straight segments. The contribution to the global coordinates can be obtained from the transformation (15.58) subject to the new definitions n

n

p=l

p=l

G4 - ~-~G4p/n, S 4-- ~ S 4 / n ,

$2G2

n

=

E Spap/n, 2 2 SG

p=l

n

G2= ~ G 2 / n , p:l

n

S2- ~S2/n p=l

n

= ~

n

p=l

Spap/n

n

SG3 = E SPG3/n' aS3 = E GpS]/n" p=l

p=l

(15.61)

Better accuracy of the properties can be obtained by increasing the number of divisions n. In fact, for very large values of n, one can replace the summations in equation (15.61) by their integration limits. This can be easily achieved by identifying h with dxl , df(xl)/h by f'(xl) and finally getting tan r

= f'(xl).

(15.62)

Implementing these new variables, the summations (15.61) are exchanged by their integration limits G4 = 12L f02L G4(xl) dxl

$4 = ~2L ]i 2L S4(xl) dxl G2S2 -- 2L1 ~2L G 2 (xl)S2(xl) dxl

G3S -- ~1 fO2L G3 (xl)S(xl) dxl 1 fo 2L G(xl )S 3 (Xl) dxl GS 3 - -~

(15.63)

where now a(Xl) =

S(xl)

-

[1 -4- f'(xl)] 1/2 f'(Xl) [1 + f'(xl)]l/2

(15.64)

(15.65)

288

C H A P T E R 15. E L A S T I C P R O P E R T I E S OF COMPOSITES

The transformed properties (15.58) subject to the definitions (15.63)(15.65) constitute the basic building block for a wide variety of woven (including straight) fibrous composites. By applying appropriate linear transformations to these "basic" curved fibers, we can construct properties for a wide variety of woven composites. The above results are general and hold for any periodic function f(Xl). Of particular importance are the situations where f (Xl) is a sinusoidal function. A variety of typical situations involving sinusoidal fibers have been recently modeled by Nayfeh and Chimenti [171].

Appendix

289

Appendix Key Definitions and Units 9 Listed stiffnesses Cij are •176

dynes/cm 2.

9 Listed piezoelectric constants eij are x l05 columb/cm 2, the permitivities % given in nondimensional forms as eij/eo where e0 is the dielectric permitivity of free space - 8 . 8 5 4 • 10 -12 Farad/m. 9 Units of material density are gm/cm 3. 9 r is the azimuthal angle in degrees. 9 r is the fiber angle defined in chapter 15.

I s o t r o p i c steel, p-- 7.9: 278.74 a~2 a~3 a~3 a~3 a~2

114.91 278.74

114.91 114.91 278.74

0 0 0 81.91

0 0 0 0 81.91

sym

0 0 0 0 0 81.91

e~i e~2 e~3 713

(I)

")'~3

7~2

Isotropic Aluminum, p-- 2.7:

dl

a~2 a~3 a~3 a~3 a~2

107.5

54.59 107.5

54.59 54.59 107.5 sym

0 0 0 26.45

0 0 0

0 0 0

e' 1 e! 2 e 3

o

o

3

26.45

0 26.45

7 7

3 2

(2)

Appendix

290 I s o t r o p i c C o p p e r , p-- 8.9: 201.60 a~2 (733

107.44 201.60

__

a~3 a~3 a~2

107.44 107.44 201.60

0 0 O 47.08

sym

0 0 0 O 47.08

0 0 0 0 0

e~l e~:2 e~3 7~3

(3)

~'~3

47.08

~

I s o t r o p i c C h r o m i u n , p = 7.2: 313.63 a~2 (733

83.23 313.63

__

a~3 a~3 a~2

83.23 83.23 313.63

0 0 0 o

0 0 0 o

e~, e~2 e~3 %

115.20

0 115.20

7~3 7~2

0 0 0 115.20

sym

(4)

C u b i c InAs, p = 5.67, r = 0~ a~l (722 (733

83.29

45.26 83.29

~_

a~3 (7~3 a~2

45.26 45.26 83.29

0 0 0 39.59

sym

0 0 0 0 39.59

0 0 0 0 0 39.59

e~2 e~3

7~3 7~3 7~2

(5)

C u b i c InAs, p----- 5.67, r = 45~ 103.86

(711 (722 (733 (723 (713 (712

--__

24.68 103.86

45.26 45.26 83.29 sym

0 0 0 39.59

0 0 0 0 39.59

0 0 0 0 0 19.01

ell e22 e33 ")/23 ~13 ")'12

(6)

Appendix

291

C u b i c I n A s , p= 5.67, r = 30~ all a22 (733

98.72

29.83 98.72

45.26 45.26 83.29

_

0 0 0 39.59

0.23

sym

0.13

0 0 0 0 39.59

-8.91 8.91 0 0 0 24.16

0"12

eii e22 e33

723 713

(7)

~i2

G r a p h i t e , p-- 1.79, r = 0 ~ 0.~i ff22 0.33

235.0

3.69 26.0

__

0.23 a~3 a~2

3.69 3.32 26.0

0 0 0 5.52

sym

0 0 0 0 28.2

0 0 0 0 0 28.2

e~2 e~3

(8)

")/23

Epoxy, p-- 1.26: a~i a~2 a~3 a~3 a~3 a~2

7.67

3.75 7.67

=

0 0 0 1.96

sym

65~ Graphite-35~ a~l a~2 0.33 a~3 a~3 a~2

3.75 3.75 7.67

155.43 =

0 0 0 0 1.96

41

0 0 0 0 0 1.96

e~2 e~3

(9)

~!3 13

epoxy, p-- 1.6, r = 0~

3.72 16.34

3.72 4.96 16.34 sym

0 0 0 3.37

0 0 0 0 7.48

0 0 0 0 0 7.48

e~ 1

ei 2

3 3 7: 3 7 2

(10)

Appendix

292

65% G r a p h i t e - 3 5 % epoxy, p = 1.6, r = 30~ 95.46

all

28.93 25.91

a22 (733 0"23 a13

_--

4.03 4.65 16.34

0 0 0 4.4

0 0 0 -1.78 6.45

sym

(712

44.67 15.56 0.54 0 0 32.68

ell e22 e33 ~23 ")/13 ")/12

(11)

65% G r a p h i t e - 3 5 % epoxy, p = 1.6, r = 45~ all O'22 a33 a23

52.28

37.33 52.28

_--

a13

4.34 4.34 16.34

0 0 0 5.43

sym

0 0 0 -2.05 5.43

O'12

34.77 34.77 0.62

ell

0 0

e22 e33 ")/23 ~13

41.08

~/12

(12)

C C S S , p-- 8, r = 0~ a~l a~2 (:r~3 (:r~3

242

140 282

_

a~3 a~2

140 1006 282

0 0 0 91

sym

0 0 0 0 135

41

0 0 0 0 0 135

e~2 e~3 "~23

(13)

65% G r a p h i t e - 3 5 % epoxy, p = 1.6, r = 0 ~ a n d r = 10~ 145.79

all

a~2 a~3 G~3

a~3 a~2

_

3.77 16.34

8.33 4.91 16.77 sym

0 0

0 0

0 0

e~l e~2

0

0

0

e~3

3.52

0 12.08

0

7~3

0

7~3

7.33

"Y~2

(14)

Appendix

293

65% G r a p h i t e - 3 5 ~ 122.77 a~2

e p o x y , p = 1.6, r = 0 ~ a n d r = 20~

3.88 16.34

18.88 4.79 28.3

a~3 a~3 a~3 a~2

sym

65% G r a p h i t e - 3 5 ~ a~l a~2 crY3 a~3

96.97

4.84 16.34

___

a~.1 !2 ~3 !3 3 2

26.17 4.64 29.9

65.41

4.29 16.34

__

0 0 0 0 0 6.94

e~l e~2 e~3 "7~3

(15)

')'~3

0 0

0 0

0 0

e~l e~2

0

0 0

0 0

e~3 ~23

6.41

7~2

sym

Graphite-35~

0 0 0 0 21.83

e p o x y , p = 1.6, r = 0 ~ a n d r = 30~

4.44

a~3 a~2 65~

0 0 0 3.91

0

29.94

(16)

"Y~3

e p o x y , p = 1.6, r = 0 ~ a n d r = 45~ 26.17 4.39 54.990

0 0 0 5.27

sym

0 0 0 0 33.96

0 0

e~l e~2 e~3 ~3 7~3 ~12

0 0 5.58

(17)

T h e following is a s y m b o l i c a l l y w r i t t e n c o n s t i t u t i v e r e l a t i o n for a piezoelectric material. S u b s e q u e n t to it, d u e to t h e lack of space, o n l y n u m e r i c a l e n t r i e s in t h e 9 x 9 s q u a r e m a t r i x as listed in T a b l e s Ia-c. or11 cr22 cr33 ff23 cr13 ~r12

D1 D2 .D3.

C l l C12 C13 C12 C22 C23

0 0

C13 623 633 0 0 0 0 644

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 e24 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 -e24

0 0

655

0

0

--e25

0

")/13

0 0 e22 0

--e36 0 0 s

"712 E1

0 C66 0 0 0 {[11 e25 0 0 0 e36 0

"Sl1"

&2 S33 '723

E2 E3

(18)

Appendix

294 T a b l e Ia: M a t e r i a l P r o p e r t i e s of G a A s

"118.8 53.8 53.8 0 0 0 0 0 0

53.8 118.8 53.8 0 0 0 0 0 0

53.8 53.8 119.0 0 0 0 0 0 0

0 0 0 59.4 0 0 0.15 0 0

0 0 0 0 59.4 0 0 0.15 0

0 0 0 0 0 59.4 0 0 0.15

0 0 0 0.15 0 0 0.11 0 0

0 0 0 0 0.15 0 0 0.11 0

0 0 0 0 0 0.15 0 0 0.11

T a b l e Ib: M a t e r i a l P r o p e r t i e s of P Z T - 6 5 / 3 5 159.4 73.9 73.9 0 0 0 0 0 -6.13

73.9 159.4 73.9 0 0 0 0 0 -6.13

73.9 73.9 126.1 0 0 0 0 0 10.7

0 0 0 38.9 0 0 0 8.39 0

0 0 0 0 38.9 0 8.39 0 0

0 0 0 0 0 42.8 0 0 0

0 0 0 0 8.39 0 5.66 0 0

T a b l e Ic: M a t e r i a l P r o p e r t i e s of x - C u t 126.1 73.9 73.9 0 0 0 10.7 0 0

73.9 159.4 73.9 0 0 0 -6.13 0 0

73.9 73.9 159.4 0 0 0 -6.13 0 0

0 0 0 42.8 0 0 0 0 0

0 0 0 0 38.9 0 0 0 8.39

0 0 0 0 0 38.9 0 8.39 0

0 0 0 8.39 0 0 0 5.66 0

-6.13 -6.13 10.71 0 0 0 0 0 2.24

PZT-65/35

10.7 -6.13 -6.13 0 0 0 2.24 0 0

0 0 0 0 0 8.39 0 5.66 0

0 0 0 0 8.39 0 0 0 5.66

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Subject Index A

Christoffel equation, 33, 34, 4143, 64, 65, 72, 73, 80 coefficient reflection, 11, 94, 97, 99, 153156, 158, 159, 163, 166, 167, 169, 172, 173, 187189, 194, 199, 216, 218 transmission, 88, 155, 156, 162, 170, 176, 180, 213, 215 condition boundary, 3, 63, 65, 66, 86, 94, 96, 105, 117, 118, 150, 170, 172, 185, 221, 248, 249 interface, 3, 61-63, 83, 103, 109, 131, 132, 149, 170, 175, 179, 180, 218, 228, 231,233, 254, 269 constitutive relation, 17, 21, 24, 26, 77, 110, 118, 202, 206, 210, 222, 261, 270, 271, 277, 279, 285 coordinate system global, 24, 118 local, 71, 136, 184, 270 Cramer's rule, 86, 87, 91,176, 180, 249 Cremer's coincidence condition, 163 correspondence principle, 162 cylindrical coordinate, 253, 258

acoustic emission, 5 energy, 10 wave, 10, 13, 154, 195, 202, 218 angle azimuthal, 25, 27, 49, 52, 77, 85, 95-99, 101, 105, 110, 112, 113, 117, 135, 154, 157, 158, 166, 167, 181, 208, 216, 217, 222, 245 coincidence, 10, 11, 194, 195 critical, 61, 62, 67, 69, 70, 80, 83, 93, 196, 199, 242 incident, 57, 130, 158, 166, 167, 171, 173, 177, 181, 196198 attenuation, 113, 158, 197

B beam displacement, 194, 195 finite, 11, 172, 183, 184, 199 branch cuts, 234 points, 234

C Cagniard method, 6 Cagniard-de Hoop method, 9,221, 230, 234 325

326

Subject Index

D diagonals face, 43, 45 dispersion relation, 112, 139, 151, 163 dispersion curve, 112, 114, 128, 163, 165, 167, 170, 181,216 Dispersive slowness, 129

E effective properties, 14, 269, 270, 275, 279 electric displacement, 206, 208, 211 potential, 205-208, 211, 213215, 218, 219 energy curves, 57 flow angle, 57 flow surface, 32 flux, 59 slowness, 242 strain, 16, 17, 282-284 surfaces, 33, 55, 56 engineering constants, 23 equations of motion, 15, 24, 26, 74, 78, 105, 118, 222

F fiber undulated, 267, 284, 286 Floquet, 127, 148 fluid lower, 161, 170 upper, 153, 155, 161,173, 174, 180

G general anisotropic, 1, 17, 37, 48, 104, 153 geophysical, 3, 8, 62 geophysics, 1, 12 group velocity, 7, 54, 55, 242 guide line, 66, 67, 80

H host medium, 253, 254, 258, 259, 265 hypothetical interface, 223-225

L lacunas, 241-242 linear transformation, 135, 209, 285

M media anisotropic, 6-9, 11, 17, 32, 33, 61, 64, 97, 117, 135, 230, 238 periodic, 118, 135 mode antisymmetric, 108, 109, 111, 114, 115, 209, 216, 217 optical, 139, 142 symmetric, 108, 109, 111,112, 114, 115, 209, 216, 217 monoclinic-2, 205, 210, 213, 215 monoclinic-m, 210, 213, 215

N nondestructive evaluation, 1, 12, 99, 153 null, 10, 57, 158, 183, 193, 197

O optical modes, 139, 142

Subject Index

327

P piezoelectric coupling, 11,201,203205, 210, 213, 216 polarization vectors, 31, 32, 37, 50 Poynting vector, 59 propagator matrix, 4 pseudo-surface, 8, 11, 95-99, 158, 160 pure mode, 31-33, 37, 38, 40, 42, 43, 46, 50-52, 54, 230

R radiation condition, 131, 207, 209 refraction, 7, 83 representation Crystollographical, 36 transformation, 34, 36, 38, 42, 44, 46

T total internal reflection, 67, 69 transfer matrix global, 3, 118, 123, 124, 130, 138, 174, 263 individual, 123, 124 local, 119, 122, 123 transform Fourier, 185, 187, 225, 226, 229, 230, 234 Laplace, 229, 238 transversely isotropic, 20, 41, 75, 104, 110, 117, 181, 221 triclinic, 1, 17, 18, 21, 27, 37, 38, 71, 73, 103-106

V viscosity, 171, 172

S sagittal motion, 135, 145, 213 plane, 3, 29, 49-52, 54, 7680, 96, 104, 111, 135-137, 143, 145, 149, 178, 211, 213, 216 Scattering cross section, 266 scattering cross section, 266 seismology, 5, 6 smart material, 201 snap shot, 245, 251 Snell's law, 61, 62, 66, 68, 69, 71, 72, 83, 103, 167 source line, 2, 248 transient, 5, 57 strain energy density, 16, 282, 283 stress concentration factor, 265

W wave

SH, 2, 13, 54, 76, 77, 79, 110, 111, 135, 137, 210, 211, 213, 216, 223, 233, 253255, 265 SV, 104, 210 B.-G., 211-213 bulk, 3, 7, 31, 33, 64, 93, 96, 229, 242, 251 evanescent, 4 Lamb, 3, 8, 11, 110-113, 163, 164, 167, 216 ultrasonic, 9, 10, 12 waves harmonic, 2, 5, 8, 12, 117, 188 interface, 93, 99 leaky, 158 pseudo-surface, 97 Rayleigh, 7, 11, 12

328

Subject Index

Scholte, 3, 94, 99 Stoneley, 3, 7, 94 surface, 5, 8, 93-97, 99, 131, 149, 152, 157, 158, 181, 189, 194, 201, 207, 211 transient, 1, 2, 9

Author Index C

A

Cady, W. G., 12 Cagniard, L., 5, 6, 230, 236 Castaings, M., 4 Cawley, P., 11 Chadwick, P., 7, 8 Chamis, C .C., 14 Chamuel, J. R., 3 Chien, H. T., 12, 202, 216-219 Chimenti, D. E. , 4, 8, 11, 151, 164-169, 172, 177, 178, 182, 194-198, 288 Choo, V. K. S., 14, 29, 208 Christensen, R. M., 14 Crean, G. M., 8 Cremer, L., 163 Currie, P. K., 7

Abo-Zena, A., 4 Abubakar, I., 8 Achenbach, J. D., 2, 4 Adams, D. F., 13, 14 Adler E. L., 4 L., 11 Akay, A., 13 Aki, K., 2 Arikan, O., 11 Atalar, A., 11 Auld, B. A., 2, 8, 12, 33

B Backus, G. E., 4 Barnett, D. M., 8 Baylis, E. R., 8 Bedford, A., 5, 14 Beltzer, A. I., 2 Bertoni, H. L., 10, 11, 190, 192, 193 Bleustein, J. L., 12, 211 Bogy, D. B., 4 Braga, A. M. B., 8, 11 Brekhovskikh, L. M., 2, 190 Briggs, G. A., 10 Brooke, G. H., 3 Buchwald, V. T., 8 Burridge, R., 8, 9 Butler, D. L., 194, 196

D Datta, S. K., 8 Davis, A., 8 Dayal, V., 11 de Hoop, A. T., 6, 229 Delph, T. J., 5 Diachok, O. I., 8, 10 Doner, D. R., 14 Dragonette, L. R., 10, 11 Dunkin, J. W., 4

E Engan, H., 8 Evans, R. B., 4 329

330

Author Index

Every, A. G., 8 Ewing, W. M., 2, 5

F Fahmy, A. H., 4 Faran, J. J., 13 Farnell, G. W., 8 Fay, R. D., 11 Fedorov, F. I., 7 Fiorito, R., 11 Fortier, O. V., 11 Franssens, G. R., 13 Frazer, L. N., 8 Freedman, A., 11 Fryer, G. J., 8 Fung, Y. C., 238

G G arvin, W. W., 6 Gates, E., 7 G avazza, S. D., 8 Gazis, D. C., 8 Gilbert, F., 4 Goos, F., 10 Gracewski, S. M., 4 Graft, K. F., 2 Green, W. A., 8 Guyott, C. C. H., 11

H Hanchen, H., 10 Hashin, Z., 14 Haskell, N. A., 3 Hattunen, M., 11 Hegemier, G. A., 4, 14 Helbig, K., 8 Henneke, H. E. G., 7, 11 Herman, R., 12 Herrmann, G., 8, 11 Hill, R., 14

Hori, M., 14, 52 Hosten, B., 4

I Ingard, K. U., 2

J Jardetsky, W. S., 5 Jensen, F. B., 4 Joel, N., 7 Johnson, W. W., 7 Jones G.L., 11 R.M., 14

K Kato, K., 13 Kaul, R. K., 8 Kennett, B. L. N., 2 Keshava, S. P., 4 Kim, Y. Y., 9, 246, 247 Kind, R., 4 Kinra, V. K., 11 Knopoff, L., 4 Kolsky, H., 2 Kosevich, Yu. A., 8 Kraft, D. W., 13 Kraut, E. A., 8

L L~vesque, D., 4 Lamb, H., 2, 5, 8, 104, 105, 110 Lee, E. H., 8 Leroy, O., 11 Lewis, T. S., 13 Lim, T. C., 4, 7, 8, 37 Lothe, J., 8 Lotsch, H. K. V., 10 Love, A. E., 2, 17 Lowe, M. J. S., 4, 97

331

Author Index

Luukkala, M., 11

M Mal, A. K., 4, 11 Mampaert, K., 11 Markus, S. A., 8 Mayer, W. G., 10 Menke, W., 4 Merkulov, L. G., 7, 11 Miklowitz, J., 2 Mindlin, R. D., 8 Morse, P. M., 2, 12 Mott, G., 11 Mow, C. C., 13 Murakami, H., 5, 8 Musgrave, M. J. P., 7

N Nassar, A. A., 12 Nayfeh, A. H., 4, 5, 8, 11, 12, 14, 151,159, 164-169, 172, 177, 178, 182, 194, 196-198, 202, 216-219, 288 Nemat-Nasser, S., 5, 8, 14 Neubauer, W. G., 8, 10 Ngoc, T. D. K., 10

O Officer, C. B., 2 Oliner, A. A., 10

P Pao, Y. H., 13 Paul, B., 14 Phinney, R. A., 4 Pich~, L., 4 Pickett, G., 13 Pilant, W. L., 2 Pitts, L. E., 11 Plona, T. J., 10

Postma, G. W., 14 Press, F., 2, 4, 5

Q Quentin, G., 3

R Randall, M. J., 4 Rayleigh, Lord, 2, 12, 93 Richards, P. G., 2 Rokhlin,. S. I., 7, 11 Rollins, F. R., 8, 11 Rose, J. L., 8 Rosen, M., 14 Rosenbaum, J. F., 12 Rytov, S. M., 4, 14

$ Sabina, F. J., 14 Schmidt, H., 4 Schoch, A., 10, 163, 190 Scholte, J. G., 3, 93 Schwab, F. A., 4 Scott, G. W., 11 Sendeckyj, G. P., 14 Sezawa, K., 12 Shah, A. H., 8 Shaikh, N., 8 Smith G. D., 8 J.H., 10 R. F., 10 Solie, L. P., 8 Stern, M., 5, 14 Stoneley, R., 3, 7, 93, 94 Sun, C. T., 5, 14 Synge, J. L., 7 Syrkin, E. S., 8

T

Author Index

332

Tamir, T., 10, 11, 190, 192, 193 Taylor, T. W., 4, 9, 11 Thomas, R. L., 14 Thomson, W. T., 3, 8 Thrower, E. N., 4 Tiersten, H. F., 12, 202 Ting, T. C. T., 8 Toledano, A., 8 Truell, R., 13 Tsai, S. W., 14

V Velasco, V. R., 8 Verevkina, L. V., 8 Viktorov, I. A., 2

W Waintal, A., 8 Wasley, R. J., 2 Watson, T. H., 4 White, R. M., 13 Whitney, J. M., 14 Willis, J. R., 14 Woodhouse, J. H., 8

Y Yamada, M., 8 Yang, W. H., 8 Ying, C. F., 13