Wave Propagation in Layered Anisotropic Media with Applications to Composites

347 Pages·1995·11.877 MB·English

by  Adnan H. Nayfeh

WAVE PROPACIAI'ION IN LAYERED ANISOTROPIC MEDIA
with Applications to Composites
NORTH-HOLLAND SERIES IN
APPLIED MATHEMATICS
AND MECHANICS
EDITORS:
J.D. ACHENBACH
Northwestern University
B. BUDIANSKY
Harvard University
H.A. LAUWERIER
University ofA msterdam
EG. SAFFMAN
California Institute of Technology
L. VAN WIJNGAARDEN
Twente University of Technology
J.R. WILLIS
University of Bath
VOLUME 39
ELSEVIER
AMSTERDAM (cid:12)9L AUSANNE (cid:12)9N EW YORK (cid:12)9O XFORD (cid:12)9 SHANNON (cid:12)9T OKYO
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 (cid:12)9L AUSANNE ~ NEW YORK (cid:12)9O XFORD ~ SHANNON ~ TOKYO
ELSEVIER SCIENCE B.V.
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ISBN: 0-444-89018-1
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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 dy-
namic behavior of layered materials in general, and laminated fibrous com-
posites 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 materi-
als 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 dispersiv-
ity 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 macro-
scales. 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 dif-
ferent 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 pro-
duced 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 me-
chanical 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 de-
pending 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 solu-
tions 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 applica-
tions. 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 half-
space). 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 con-
tact with fluid is covered in chapter 8. In chapter 9, we present solutions for
the interaction of elastic waves with multilayered anisotropic media. Spe-
cialization 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 flu-
ids 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 un-
bounded and in semi-space media. In chapter 14, we present an example of
wave interaction with layered coaxial systems. Specifically, we discuss scat-
tering of horizontally polarized shear waves from multilayered anisotropic
cylinders. The book concludes by presenting, in chapter 15, some model cal-
culations 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 compu-
tations. 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 analyti-
cal models brought about by the meticulous experiments of Dale Chimenti
is acknowledged. The invaluable help over the years of my graduate stu-
dents 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 col-
leagues 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
INTRODUCTION 1
1.1 Historical background . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Mostly isotropic media . . . . . . . . . . . . . . . . . . 2
1.1.2 Mostly anisotropic media . . . . . . . . . . . . . . . . 6
1.1.3 Fluid-loaded solids . . . . . . . . . . . . . . . . . . . . 9
1.1.4 Piezoelectric effects . . . . . . . . . . . . . . . . . . . . 11
1.1.5 Scattering from layered cylinders . . .. ......... 12
1.1.6 Elastic properties of composites ............. 13
2 FIELD EQUATIONS AND TENSOR ANALYSIS 15
2.1 The stiffness tensor . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Material symmetry . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 The transformation . . . . . . . . . . . . . . . . . . . . 17
2.3 Matrix forms of stiffness . . . . . . . . . . . . . . . . . . . . . 21
2.4 Engineering constants . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Transformed equations . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Advantages of orthogonal transformations ....... 25
2.6 Expanded field equations . . . . . . . . . . . . . . . . . . . . 26
2.6.1 Monoclinic . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6.2 Orthotropic . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Planes of symmetry . . . . . . . . . . . . . . . . . . . . . . . 29
BULK WAVES 31
3.1 An overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The Christoffel equation . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 General features of the Christoffel equation ...... 34
3.2.2 Limitations of analytic solutions ............ 37
3.3 Material symmetry . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Analytical solutions . . . . . . . . . . . . . . . . . . . 38
3.3.2 Higher symmetry . . . . . . . . . . . . . . . . . . . . . 41
ix