Table Of Contentvi
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 .01 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 21 extends the results of the previous chapters
to include piezoelectric coupling. In chapter ,31 the techniques introduced
for harmonic wave motions are modified to study transient motions in un-
bounded and in semi-space media. In chapter ,41 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 ,51 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 ,01 1995
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 nonde-
structive 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 geomet-
ric 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 sim-
ple 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 un-
doubtedly 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 applica-
tions. Smooth, as well as rough, interfacial conditions which require mod-
ifications 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 indefi-
nitely 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 sub-
ject has been focused on phenomena that strictly deal with isotropic ma-
terials 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, Kol-
sky 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 SH 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 prop-
agation 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 wher-
ever the phrase " Lamb waves" is mentioned, it precisely refers to the above
problem. In 1924 Stoneley 230 generalized Rayleigh's surface wave prob-
lem by studying the propagation of free waves at the interface between two
solid semi-spaces. Such waves have since then been referred to as " Stone-
ley 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, 95
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 ma-
trix transfer method in order to facilitate his analysis. A small error in his
work was subsequently corrected by Haskell 105. According to this tech-
nique 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 re-
late 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 neigh-
bor. 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 re-
sults 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 con-
ditions 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 me-
dia; 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 contribu-
tors 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
fd (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, 73 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 poten-
tim 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 com-
ponents, the direct approach is appropriate ( Achenbach and Keshava ,4
Nayfeh et al. 173, Chimenti et al. 50, and Bogy and Gracewski 22). Al-
though 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 801
1.1. HISTORICAL BACKGROUND 5
studied similar model and derived exact and asymptotic results for the prop-
agation of waves having arbitrary frequencies in such system. Subsequently,
iayfeh 164 generalized the results of 108 to the case of multilayered me-
dia. iemat-Nasser 179 and iemat-Nasser et al. 180, 183, 184 studied har-
monic 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, 76 and others 23, 24, 95, 112,214. Ap-
proximate 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 media
In comparison to the extensive literature on the interaction of plane har-
monic waves with isotropic media, much less work is available on the re-
sponse 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 radia-
tion, 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 con-
sider 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 displace-
ments 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, alge-
braic 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 applica-
tions 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 homo-
geneous 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 (mi-
crostructure) 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 com-
plicated 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. HISTORICAL BACKGROUND 7
Dry solids
As compared with isotropic cases, solutions to the anisotropic problem are
much more difficult to obtain. This is because of the added algebraic com-
plications in handling of the pertinent equations even in simple cases in-
volving 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 re-
ceived attention and is well understood, comparatively fewer quantitative
results have been reported on solutions of problems containing either in-
ternal or external boundaries, which may classify them as layered me-
dia. Here, we mention that several authors including Merkulov 146, Mus-
grave 155, Gates 91, Joel 115, Henneke 111, Lim and Musgrave 135,
Johnson 116, Chadwick and Currie 38 and Rokhlin et al. 209 have dis-
cussed, 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 re-
ceived 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 var-
ious 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, Verevk-
ina et al. 247, Diachok et al. 68, Farnell 80, Every 76, Burridge 23
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 me-
dia, respectively. Fryer and Frazer 88, 78 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 Nemat-
Nasser 256, Helbig 109, Braga and Herrmann 25, Ting and Chad-
wick 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).
Transient waves and anisotropy
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 classi-
cal work of Lamb 128 who obtained the exact solution for the disturbances
