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ABSTRACT

ELASTIC WAVE PROPfiGATIONS
IN TRANSVERSEIX ANISOTROPIC MEDIA

BY

Bahram Salagegheh

The subject of elastic wave propagations in general and
transversely anisotrOpic media is considered. Rayleigh surface
waves are also discussed for transversely anisotropic media.
Attention is mainly given to axi-symmetrical problems involving
transversely anisotrOpic materials. Free longitudinal and tor-
sional waves in an infinite cylindrical bar, made of such materials
are discussed and the general solutions are obtained.

An exact analysis of a semi-infinite, transversely aniso-
tropic, cylindrical bar with traction free lateral surface and
subjected to a suddenly applied pressure at the end is presented.
Asymptotic solutions are sought, through Fourier transforms. It
is shown that such explicit asymptotic solutions are easily obtained
whenever the elastic properties of the bar are known. An approx-
imate theory for the propagation of longitudinal waves in a
transversely anisotropic cylindrical bar is also developed and
applied to the same boundary value problem mentioned above and
explicit solutions are obtained.

Finally, general layered media are considered and their

correlations with transversely anisotropic media are discussed.

Bahram Salagegheh

The exact solution of the semi-infinite cylindrical bar problem is
further continued by assuming that the bar is made of a particular
class of layered media. Certain alternative approximate solutions

are also Obtained using a different approach than the asymptotic

method mentioned above.

ELASTIC WAVE PROPAGATIONS
IN TRANSVERSELX ANISOTROPIC MEDIA

BY

Bahram Salagegheh

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR 0F PHIIDSOPHY

Department of Metallurgy, Mechanics, and
Materials Science

1971

ACKNOWLEDGMENTS

Sincere appreciation is extended to all the individuals
who have assisted and encouraged me throughout the preparation
of this dissertation. Special thanks I owe to my thesis advisor,
Dr. David H. Y. Yen, who has given me extended help and guidance
as well as his invaluable time. The guidance of other committee
members, Dr. W. A. Bradley, Dr. G. E. Mase and Dr. Charles J.
Martin is also greatly appreciated.

Grateful thanks are extended to Dr. D. J. Montgomery,
Chairman of the Department of Metallurgy, Mechanics, and Materials
Science, who has given me substantial encouragement. I also wish to
give my appreciation to the Department of Metallurgy, Mechanics,
and Materials Science of Michigan State University, for providing

funds to finance my graduate work.

ii

Chapter 1

Chapter 2

Chapter 3

Chapter 4

TABLE OF CONTENTS

ACMOWIedgments OOOOOOOOOOOOOOOOO0.000......0..

List Of Figures OOOOOOOOOOOCO;;OOOOOOOO0.00....

INTRODUCTION AND LITERATURE SURVEY . . . . . . . . . . . .

1.1 Introduction OOOOOOOOOOCOOOOOO00.0.0.0...O
1.2.1 Isotropic Media OOOOOOOOOOOOCOOOOOOOOOO.

1.2.2 IlOtrOpiC CYlindriCal ROds ooooooooocooo.

1.2.3 General Anisotropic Media ..............
1.2.4 Transversely Anisotropic Media .........
1.3 Organization of Dissertation .............

DEFINITIONS. IONGITUDINAL AND TORSIONAL WAVES
IN A TRANSVLRSELY ANISOTROPIC CYLINDRICAL BAR .

2.1 Definitions and Basic Equations from
the Theory of Elasticity for a Trans-
versely Anisotropic Medium ...............

2.2 Free Longitudina1.Wayes in an Infinite
Cylindrical Bar with Traction Free
Lateral Surface ..........................

2.3 Free Torsiona1.Waves in an Infinite
Circular Cylindrical Bar with Traction
Free Lateral Surface .....................

AN APPROXIMATE THEORY FOR.IONGITUDINA1.WAVES
IN A TRANSVERSELY ANISOTROPIC BAR .............

3.1 The Classical Theory in which the Effects
of Radial Inertia are Included ...........

3.2 A First Order Approximate Theory for
Longitudinal Wave PrOpagation in a
Transversely Anisotropic Bar .............

3.3 Derivation of the Frequency Equation in
the Approximate Theory and Determination

of the Two Constants k,2 and k"2 ......

PROPAGATION OF ELASTIC WAVES IN A SEMI-INFINITE
CYLINDRICAL BAR SUBJECTED TO A SUDDENLY APPLIED
mssURE AT THEEND 00......00000000000.0.00...

4.1 The Exact Theory .........................

4.2 Asymptotic Approximations of the Integrals
in (4.66) ................................

4.3 Application of the Two-mode Approximate
Theory Developed in Chapter Three ........

iii

Page

ii

W)P‘G>G\h>h| P‘

P‘h‘

22

27

34

38

39

42

47

S6
57
8O

86

Chapter 5

Chapter 6

Page
APPLIMTION To THE [AYEHD mlm O O O O O O O O O 0 O O O 94

5.1 Definition of a Layered Medium, and Its
Correlation with a Transversely Aniso-

tropic Medium O00.00.000.000...0.0.0.0...O 95
5.2 Application to the Boundary Value Problem
Discussed in Sections 4.1 and 4.2 ........ 1005

SWYMDCONCIUSIW OOOOOOOOOOOOOOOOOOOOOOC. 128

6.1 summary 0 O O O O O O O O I O C O O O O C O O O O O O O O I O C O O O O O O 128
6. 2 cone 1‘18 ions 0 O O O O O O O O O O O O O O 0 O O 0 O O O O O O O 0 O O O 130
List Of References 0 O O O O O O O O O O O O 0 O O O O O O O O O O O O O O 133
Appendix A 00000 O 0 O O O O O O O O O O 0 O O O O O O O O O O O O O O I O O O 136

iv

Figure

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

6.1

6.2

Page

Contour of integration for evaluating integrals
(5.23) and (5.24), the two simple poles at A
and 31’ which are located slightly above the

1

real g-axis, are included inside the contour of
integration, whereaa,the other two located at
A. and B. are II‘OCQR-II.m ......OOOOOOOOOIOOOOOO 107

1 1
Contour of integration for evaluating integral

(5.36) for t > (22-)tz. Note that there are three

11
simple poles at A2, 0 and 32, R.» m ........... 110

Contour of integration for evaluating integral

(5.36) for t < (—9—)%z ................... . ...... 112
c11
Contour of integration for evaluating integral

(5.37) for t > (A'w2 - B')%z, with two simple

poles at D and D' and two branch points at

A and B. Note that line BE is the branch cut.

R a m ............................................ 114
Equivalent contour to that given by Figure 5.6 ,

note that line BA is the new branch cut. R e a ... 115
1 and L2, when

6' a 0, appearing in the integrals (5.42) and

(5.50), line AB is the branch cut ................ 118

The path of integration along L

Contour of integration for evaluating integral

(5.68) for t > (A'w2 - B')%z, with three simple

poles at D', 0 and D, and two branch points at

A and B. Note that line BE is the branch cut

and R aim ....................................... 124
Equivalent contour to that given by Figure 5.9

AB is the new branch cut, R «to .................. 125
An elementary volume of layered medium, given in

Figure (5.1), consisting of N alternating pairs

of isotropic layers .............................. 137
Tangential stress, and the corresponding strains

in (y’z)-plane ......U......OOOOOOOOOOOOOOOO0.000. 141

vi

Figure

2.1

4.2

4.3

4.4

5.1

5.2

LIST OF FIGURES

Page
The elastic constants of this transversely aniso-
tropic medium are independent of a1.The z-axis is
called the axis of transverse anisotropy ......... 23

Boundary conditions of the boundary value
problem given by equations (4.1-5) ............... 59
Contour of integration for evaluating integrals
(4.38). The two circles ci and c5 with radii
e' a 0 around the two branch points located
slightly above the real axis of g-plane, are
included inside the contour of integration, whereas
c5 and C; are not. Note that the two lines
o'A and o"B are the branch cuts, R14 m ........ 72
Contour of integration for evaluating the last
integral in equation (4.68), with a pole of order

four at the origin’R-‘m ......OOOOOOOOOOOOOOOOOO 79

llllustrating how to find the positions of the

saddle point 5' along the real axis of complex
w-plane. c1 represents g B §q(w), which is an
arbitrary root of m1(§,w) = 0, and c represents

3%.

dw ......OOOOOOOOOOO.....OOOOOOOO. ..... 0.00.00... 82

2

A layered medium consisting of n alternating
plane, isotropic layers with properties 11, “1
and p1 and x2, “2 and p2, and thicknesses

...........O0............OOOOOOOOOOOOO 96
d1 and d2

A semi-infinite cylindrical bar, with radius a,
consisting of alternating plane, isotrOpic layers,

Of thiChesses d and d2 00.000000000000000... 101

l

CHAPTER ONE

INTRODUCTION AND LITERATURE SURVEY

1.1 Introduction

The subject of elastic wave propagation in isotropic materials
has received a great deal of attention during the last few decades.

Many problems concerning wave motion have been considered, and the
solutions of a large number of such problems have been obtained. 0n
the other hand, very few correSponding problems involving anisotropic
media have been solved. The importance of the wave motion analysis

in anisotropic media such as those arising in the fields of soil me-
chanics, geophysical and seismological studies and nuclear blast prob-
lems is easily understood.

It is the purpose of this thesis to undertake the task of study-
ing wave prOpagation problems involving transversely anisotropic elastic
materials. More specifically, attention is given to axi-symmetrical
wave prOpagations in infinite cylindrical rods, and the solution of
a boundary value problem involving a semi-infinite cylindrical rod sub-
jected to a time-dependent pressure at the end.

A general review of wave motion in unbounded elastic isotropic
and general anisotropic media is presented. In sections 1.2.1 and 1.2.2,
isotropic unbounded media and cylindrical rods, and in sections 1.2.3
and 1.2.4, general anisotropic and transversely anisotropic media are

reviewed. In section 1.3 organization of the dissertation is pre-

sented.

1.2 A Survey of Free Wave PrOpagation in Infinite and Semi-Infinite
Elastic Media

1.2.1 Isotropic Media

The behavior of waves in isotropic materials is well understood.

*
It is well known [6] that the displacement equations of motion for
isotrOpic.materials can be separated in terms of a scalar potential
function m and a vector potential function E. These functions

satisfy the following three dimensional wave equation

2
2 -2
V m = c1 ‘34?
at
(1.1)
2»
24 -2
v (v = C2 5..le-
at

where

2 2 2 2
BX By az

and (x,y,z) are the Cartesian coordinates. The two constants c

 

l
and c2 are related to the material properties in the following
form

czzltjia
1 p
(1.2)
C2=u'-
2 p
*

Numbers in the brackets designate items listed in the reference
section.

where x,n are the elastic constants of the material, and p is

the mass density per unit volume. For detailed derivations of (1.1),
see [6], page 12. In view of (1.1), as Kolsky [6] has pointed out,
in any unbounded elastic isotropic medium two, and only two, types

of plane waves can prOpagate. The first type of wave,travelling

with phase velocity is called dilatational, and the second type

c1,
of wave,travelling with phase velocity c2, is called rotational.

Musgrave [15] has employed the following plane wave solution
to the general diSplacement equations of motion for an infinite, an-
isotropic medium

i zu(£x+my+nz-v't)
(u,v,w) = (A ,A ,A )e A (1.3)
x y z

where u,v and w are the components of the diSplacement vector
along x, y and z coordinates reSpectively, Ak, Ay and A.z are
the components of a nonzero vector, and L, m and n are the com-
ponents of a unit vector normal to the plane wave, i.e. Lx +-my +-
nz - v't= O. In equation (1.3) A is the wave length and v' is
the phase velocity which depends on L, m and n for the general
anisotropic material. However, if (153) is substituted into the
diSplacement equations of motion for the isotropic materials, (see
[6], page 12), it can be verified easily that the phase velocity v'

satisfies the following cubic equation

(v'2 - Ci) (v'2- C:)2 = O . (1.4)

From (1.4) it is obvious that the roots for v'2 are:

C
l 1 p
(1.5)
2 v2 2 E.
' = = =
V 2 V3 c2 p

which are independent of L, m and n. Hence, for isotropic materials

all directions are equivalent. The locus of the extremities of the

u u u
1, v2 and v3) forms a

surface which is called the "velocity surface". From (1.5) it is

phase velocity vector (with components v

seen that. this surface consists of three concentric Spherical sheets
of which two are coincident.

Following the argument given by [15], if a disturbing, un-
polarized point source is located inside an isotrOpic medium, the
disturbance spreads outward on two concentric Spherical fronts.

These two fronts move with two different velocities, and at any point
inside the medium, such surfaces separate the disturbed part of the
medium, which is in motion, from the undisturbed part of the medium,
which is motionless. Such surfaces, which are mathematically the

envelopes of all possible plane waves
Lx +‘my +-nz - v't= 0 at some fixed t (1.6)

which simultaneously passed through the origin at t = 0, are called

the "wave surfaces". Equation (1.6) for t = 1 can be re-written as
in +myi +-nzi I v; = constant, 1 = 1,2,3 (1.7)

where v; are the three roots of (1.4). Equations (1.7) are the
parametric equations of three spheres, with radius vi. and unit
normal (L,m,n) and two of them are coincident (see(1.5)) There-

fore, for isotropic materials the wave surfaces and the velocity

surfaces coincide. Furthermore, since the normal to the wave surface
is the same as the normal to the plane wave, we conclude that the
path along which energy is carried by a plane wave, with an
arbitrary unit normal, coincides with that normal.

If we make the following change of variables, i.e.

1
s = 3% (1.8)
and substitute it into equation (1.4), we have

1 2 2 _

2 1
(“'3 - GIN—2 - Cl) ‘ 0 (1-9)
s s
from which we obtain for the three roots s1, 32 and s3
2 _ L
31 — C2
1 (1.10)
2 _ 2 _
$2 — s3 —

_1_
C2 o

2
Equation (1.9) defines another type of surfacescalled the "slowness
surfaces". They consist of three Spherical sheets, of which two
coincide and their radii are (X§§;)% and (3)% which are the in-
verse of the radii of corresponding velocity surfaces. The three
displacement vectors with components '(Axi, Ayi, Azi), associated
with each phase velocity vi, form an orthogonal triad [15], in
which one is parallel to the plane wave unit normal (L,m,n) and
the remaining two components aretxnjlperpendicular t0 (L,man)o
Musgrave [15] points out that among the three surfaces, mentioned
above, the wave Surface possesses the clearestphysical significance,

since it represents the fronts on which a disturbance is observed

to Spread. However, as Synge [16] has pointed out, the most

importantcfi the three types of surfaces are the slowness surfaces.
The principal reasons for this are that their equationS,a sixth
degree one, is easily obtainable, and their polar reciprocal relation
to the wave surfaces allowsone to obtain the geometrical shape of the

wave surfaces.
1.2.2 Isotropic Cylindrical Rods

Davis [12]enu1Love [1] have given a full detail of
free wave propagation in an infinite elastic isotropic cylindrical
bar With a traction free lateral surface. Kolsky [6] has obtained
the general expressions for the component; u and v of the dis-
placement vector by employing a plane wave solution to the exact
displacement equations of motion in cylindrical coordinates. The
characteristic equation or the so-called Pochammer's frequency equa-
tion, (see equation(2.28)of Chapter two), is obtained upon using the
boundary conditions on the free lateral surface of the bar. This
frequency equation,which relates the diSpersion relationship be-
tween the phase velocity and the wave-length, has an infinite number
of roots. Bancroft [24] has given a full detailed investigation of
the above frequency equation, and the numerical results, for the
first three roots are given by Davis [12].

Semi-infinite elastic isotropic cylindrica1.barsare treated
by Curtis [11], Jones [14], Kennedy [25] and Redwood [26]. Rosenfeld
and Miklowitz [27] have considered the solution of a semi-infinite
cylindrical bar of arbitrary cross section, Subjected to a suddenly
applied end loading. They have employed a double Fourier and Laplace
transform to the equations of motion. Far field numerical information

is obtained with the aid of asymptotic analysis. Curtis [11] has

obtained the asymptotic solutions for a semi—finite isotropic
cylindrical bar of circular cross section, subjected to a time-
dependent pressure at the end. A double Fourier transform is applied
to the axi-symmetrical equations of motion. Stationary phase method
is used to obtain the solutions which are valid at stations far from

the end of the bar, and explicit results (in terns of Airy's integral)

is given for the Sum of (see +-ezz) corresponding to the funda-
mental mode of wave motion. Kennedy [25] has considered the in-
vestigation of wave motion in a semi-infinite circular cylindrical
bar, subjected to a time-dependent as well as radially varying
pressure. He also has used a double Fourier transform technique,
and obtained asymptotic far field solutions for the first mode.
The results obtained by direct numerical integration of the equa-
tions of motion. are in agreement with those obtained from the first
mode analysis at a distance equal to twenty times the diameter
away from the end of the bar. Skalak [28] has Studied the impact
of a semi-infinite isotropic cylindrical bar. Again use of a double
Fourier transform is made to obtain the transformed solutions of the
components u and v of the displacement vector. Asymptotic solu-
tions for large values of t are obtained. The numerical results
show' a continuous dispersion of the wave front and an oscillation
which is not found with usual classical theory solution.
Investigation of many other boundary value problems in in-
finite and semi-infinite elastic isotropic media, with different
load conditions, have been considered by several authors. For de-

tailed information see [29], [30], [31] and [34], among many others,

1.2.3 GENERAL ANISOTROPIC MEDIA

For the case of general anisotropic materials,however, the
basic equations are much more complicated and few results are known.
A full, detailed investigation of free wave motion in an unbounded
anisotropic medium is given by [15]. Referring to the same Cartesian
coordinates (x,y,z) as before Musgrave [15] has substituted the
plane wave solutions (1.3) into the general diSplacement equations
of motion for anisotropic material (see equation (20) of his paper).
The result of this substitution leads to a cubic equation for v'?,
which depends explicitly on the directional cosines (L,m,n), (see
(30) of his paper). Therefore, for the general anisotropic materials,
the phase velocities are not independent of direction, and none of
the three sheets of the velocity surfaces is Spherical, and no two
are coincident.

Synge [16] has substituted the following plane wave to the.
general equations of motion, (see (1.7) of [16]), for the aniso-
tropic materials

A iw(slx+szy+s3z-t)

(u,v,w) = (A ,A ,A )e (1.11)
x y z

where w is the wave frequency and S1, 82 and 33 are the com-
ponents of the position vector on the slowness surface. lflote that
in equations (1.11), u, v and w are, as before,the components of

the diSplacement vector in Cartesian coordinates (x,y,z) and

(Ax, Ay’ A2), are assumed not to vanish simultaneously.

H
F‘h

Note that '

(1.12)

4l5 <l8 <k5

is substituted into (1.11), we will arrive at that same equation as
(1.3), with w = v'%5 . The result of substitution of (1.11) into

the equations of motion leads to

0(Sl,sz,s3) = Det‘pépr - cpqrssqssl = O (1.13)

where 6pr is the Kroneckerdelta and c is the elastic coef-

pqrs
ficient tensor satisfying the following symmetry conditions

c = = = c

pqrs cqprs cpqsr rqu . (1.14)

Equation (1.13) defines a sextic surface in a Space in which 81,

1, $2 and

83 are complex, then (1.13) is a complex surface. On the other hand

32 and 83 are taken as rectangular coordinates. If s
if they are real, as in (1.12), then (1.13) gives us the slowness
surfaces. They consist of three sheets for the general anisotropic

case, and if a line is drawn from the origin, it intersects the sur-

1 1 1
face at three points, at distances 31','-7 and 37' from the origin,
1 2 3

where vi, v5 and vg are the correSponding points on the velocity

surfaces and satisfy the following equation

Det‘pveép - = o . (1.15)

c n n ‘
r pqrs q 8

Equation (1.15) is the same as that given by Musgrave, and can be

10

obtained by direct substitution of (1.12) into (1.13).

Following the argument given by Buchwald [17], if a dis-
turbance prOpagateS from the origin in an infinite elastic medium,
then the waves observed at any point on the wave Surface represented
by the position vector R in the (x,y,z) Space correSpond to

those points of the slowness surface 0(3 = 0 where the

1382983)

normal of this surface is parallel to R. If R is parallel to

the normal of slowness Surface at certain points, then obviously

R = A1 v 0(81’82’53) (1.16)
where A1 is a constant, and
(60) = 59— r = 1,2,3 . (1.17)
r aSr

Equation (1.16) can be re-written in the following equivalent form

-—o--.
—O = 80R
am

where 3 is a vector with components 31,52 and 33. Now, if the

wave surface is the envelope of the plane waves (see (1.11))

 

6h (1.18)

81X + 82y + 332 - t = O at t = 1, say

then

a-R = SIX +-szy + 332 = l . (1.19)

Substitution of (1.19) into (1.18) leads to

Ii =19 . (1.20)

dd

a-vfl
Now, if yr are the current Cartesian coordinates on the wave sur-

face, then from (1.20) we have

11
yr = 395/ 23 "si 3%.— , r = 1,2,3 (1.21)
r 1—1 1

which is the parametric equation of the wave surface. MuSgrave has
also shown that for anisotropic materials, the path along which
energy is carried by a plane wave does not coincide with the normal
to that plane, and that the three displacement vectors associated
with each phase-velocity v; form an orthogonal triad, none of
which is either parallel or perpendicular to the plane wave normal.
Therefore, the three types of waves, prOpagating in an anisotropic
medium, contain both dilatational and rotational components. Thus,
the dilatational and the rotational waves are generally coupled
within the medium. ‘Whereas in the isotropic case waves become coupled
only at the boundaries, in the anisotropic case coupling occurs
within the medium in addition to at the boundaries. Thus, there is
little to be gained in retaining the concept of dilatational and
rotational motions, though sometimes such terms as "quasi-dilatational"
and "quasi-rotational"are used to describe the behavior of such media.

Rayleigh Surface waves in a general anisotropic medium have
also been investigated by Synge [16].. The author has shown that
Rayleigh surface waves on the free surface of an anisotropic medium
either do not prOpagate at all, or they prepagate only along certain

directions.

1 .2.4 TRANSVERSELY ANISOTRO PIC MEDIA

For the case of transverse anisotrOpy, which is a particular
case of general anisotropy, the investigation of wave motions is
much easier than the general anisotropic case° Referring to the

Chrtesian coordinates (x,y,z), and taking the z-axis to be along

12

the axis of transverse anisotropy, then on any arbitrary plane
(x',y'), perpendicular to the z-axis, the medium behaves isotropically.
Hence, for this case there is a complete circular symmetry around
the z-axis, and all planes containing the z-axis are equivalent.

For the case of general anisotropy the displacement equations
are not separable by simple means. However, for the transverse
anisotropy one of the equations of motion can be separated from the

other two [17]. If we define the following change of variables, i.e.

U=M+ai

ax ay
v=§%-§i' (1.22)
w-BQ

52

and substitute them into the displacement equations of motion for
a transversely anisotropic medium, presented in Cartesian coordinates

(x,y,z), ‘see equations (9) in [18] , we obtain

c 2 c -c 2
[AAA_2+(_%_lZ)Vi-B—§]Vi¢+al-3§=0
C +C C c 2 2 2
13 44V2¢+E44V2+_3_35_-a_]a_§2+2£=o (1.23)
1 1 2 2 2 2
P P P a
52 at dz
c +c 2 c 2 c 2
13 44 2 44 11 2 Y
( p )la-_%+(p 2+ V1 37)“ +£+3§=0
az A2 at
where

2 a2. 33.
V1" 2+ 2
38 by

p, cll’ ch’ c13, c33 and c44 are the mass density per unit volume,

and the elastic constants of the material, and x, Y and z are

13

the components of the body force per unit volume.

From (1.23) it is

seen that the potential y appears only in the first equation and

is independent of ¢ and @, whereas, ¢ and g appear only in the

second and third equations in (1.23). Buchwald [19], in a different

paper, has also shown that the component of the rotation around the

z-axis, i.e. (§§'- 3%), can be separated from the displacement equa-

tions of motion for a transversely anisotropic material, and this

satisfies a three-dimensional wave equation.

Let us introduce the following change of variables,

A = a2._ a2.
ax BY

a 5!.

T 62
A1 ax by

50 50 50 2
XX+ 352+ X2+px=pau

ax BY 32

be 3° 0 2
._;§z.+.__xz.+.2_12.+.py g p 5.2.
ax ay 32

 

 

at

a 2
aoxz + a yz + 50'22 + pz = p u
ax a? 52 3:2

(1.24)
(a)
(b) (1.25)
(C)

along with the stress-d isplacement relationships fora transverse

anisotropic medium,

14

U

a a— a-E-j- a!
°gx c11 ax + C12 5y C13 52

o- a 52+c 53+ all

yy c12 ax 11 By C13 52

g 52. a_. HE
022 C13 ax +Ic c13 5y +c c33 az

(1.26)

o = @— + 21—)
yz BY 32

g 3!. as
Okz (a + )

= a_+ .a.

From (1.24) it is seen that u,v and w can be determined easily
when A, F and A1 are known. Now, if we differentiate both sides
of (1.258) and (1.25b) with reapect to y and x respectively,

we find, upon subtracting and using (1.26), that

c -c C 2 2
11 12V2A+__‘LB_A+B.Y._B.§=LA. (1.27)
29 1 p 522 ax ay 3,2

From (1.27) it is seen that A is independent of F and A1.
Similarly, if we differentiate both sides of (1.25c) with reSpect

to z, and use (1.26),we obtain

2
c +c a A c
13 c3; 52 5%

( 44) g +'(— 4 v1 + 2)r +-a;= (1.28)

Finally, differentiating both sieds of (1.258) and (1.25b) with

respect to x and y respectively, we find, upon adding and using

(1.26), that

2
c 2 c a A
c+c13 44 V2 44 a 11 2 ax aY 1
, P ( D 522 9 V1)A1 ax by at? ( )

15

where in (1.27), (1.28) and (1.29), as before,

From (1.28) and (1.29) it is seen that F and A1 are not independent
of each other, hence are not separated.

Mnsgrave [20] has given a full detailed investigation of
elastic free waves in transversely anisotropic materials. In his
study, upon substituting the plane wave solution (1.3) in the gen-
eral displacement equation of motion, the following cubic equation-

for phase velocity v' is obtained

n3 - [nzh + (1-n2)(a +-§)]n2 + (1-n2){(1-n2) %£'+-n2[h(a +-§)-d2]}n
- n2(l-n2)2 §(ah-d2) = o (1.30)

where

= ' -
H pv c44

'C

44

c a c - c12 - 2c44 (1.31)

°33 ' °44

2
From equation (1.30) it is seen that H = pv' - c depends on n

44
alone, showing that the sections of the phase velocity surfaces on‘
all planes which contain the z-axis are identical. Therefore the

three sheets of the phase velocity surfaces have circular symmetry

with respect to the z-axis. Synge [16] has shown that the cubic equa-

tion (1.30) can be factorized in the following form

16
H3 - [nzh + (l-n2)(a + @1112 + (1-n2){(1-n2) '32 + n2[h(a + §)-d2]}H
- n2(1-n2)2§(ah-d2) = [H - 5(1-n2)]{H2 - H[n2h + (l-n2)a] +
n2(l-n2)(ah-d2)} = o (1.32)

showing that the vanishing of the first factor on the right hand

side of (1.32), i.e.

2
H - % (l-n ) = 0

corresponds to a Spheroidal sheet for the slowness Surface.
From equation (1.30) it is concluded that the phase velocity
surfaces have circular synmetry around the z-axis. Therefore, we

need only find the intersections of Such surfaces with the (x,y)-

plane and any other plane that contains the z-axis. If this is
done, then we shall have complete information on the phase-velocity
surfaces which define the prOpagation of waves in such a mdeium.
MHSgrave [20] has shown that the intersections of velocity surfaces,
wave surfaces and the slowness surfaces with the (x-y)-plane are
all concentric circles. The sections of the velocity and the wave
surfaces coincide, and the waves prepagating on this plane, of
normal (L,m,0) are truly dilatational or truly rotational, de-

pending as to whether

(Ax, Ay) # 0
(1.33)
A = 0 ~
2
or
A. . A = o
x y
(1.34)

A2 # o

17

On any planes containing the z-axis, say the (y,z)-plane, where

IL = 0, the phase velocity satisfies a quadratic equation which de-
pends on both m and n. Therefore, the sections of velocity Sur-
faces as well as the sections of the wave surfaces and the slowness
surfaces on this plane will not be circles. The sections of the
wave surfaces and the velocity Surfaces do not coincide, except at
some isolated points, and the waves propagating on this plane are
either quasi-dilatational and quasi-rotational or truly rotational,

depending on whether

(Ay, AZ) ¥ 0
(1.35)
A = O
x
or
A350 .
x (1.36)
A = A = O .
y z

Musgrave has also plotted the sections of the velocity surfaces,
wave surfaces and the slowness Surfaces with the (y,z) plane - for
zinc and beryl. For details see Figures (1) and (2) of his paper [20].
Rayleigh surface waves have also been investigated in a
transversely anisotropic medium by several authors. Synge [16] has
shown that the progagation of Rayleigh waves on a free plane surface
of a transversely anisotropic medium is only possible when the free
plane surface is either normal or perpendicular to the axis of trans-
verse anisotropy. Buchwald [17] has investigated the free surface
waves in such a medium for two Special cases. In the first case he
takes the free surface to be perpendicular to the z-axis, and in the

second case he takes the free plane to be parallel with the z-axis.

18

He applies a three-dimensional Fourier transform to the three equa-
tions of motion (1.23) and obtains the transformed solutions for
W: ¢ and @. In order to satisfy the boundary conditions on the
plane 2 = 0, additional free waves with the same form Such as those
for W: ¢ and @ are added to W: ¢ and @. For the first case,
it is shown that the potential W makes no contributions to the
waves under consideration. The characteristic equation obtained
in this manner, which relates the phase velocity with the wave fre-
quency,is of order Six in v' and is independent of the frequency
and similar to the isotropic case. Therefore, there is no dis-
persion in the first case. He has also shown that the sections of
the slowness surfaces and the wave surface with the (x,y)-plane are
concentric circles in this first case. In the second case, when the
free surface is parallel to the z-axis, say plane y = O, the char-
acteristic equation is again independent of wave frequency, indicat-
ing that there is no dispersion. However, the equations for the slowness
surfaces, and the wave surfaces are too complicated for an analytical
discussion. Finally, he has also remarked that in the first case
the plane wave solution is sufficient to describe the properties of
the free surface waves in Such media. However, in the second case the
plane wave solution appears insufficient.

Investigation of wave motion in semi-infinite and infinite
transversely anisotropic media with different load conditions,such
as the point load and the line load, have also been considered.
Cameron [21] and Buchwald [19], have obtained the reSponse of an
infinite transversely anisotropic medium, subjected to a point load

at the origin, using two different approaches. The applied load is

l9

acting along the z-axis, around which there is complete circular
symmetry. Abubakar [22] has considered the reSponse of a semi-in-
finite transversely anisotropic mediumn which is excited by a buried
line source. He has obtained the explicit solution for this problem.
Ragge [23] has obtained the reSponse of a semi-infinite medium and

of an infinite slab Subjected to a moving uniform pressure over the
surface perpendicular to the z-axis. For detailed information and the
different approaches for attacking such problems, see [19], [21], [22]

and [23] respectively.

1.3 ORGANIZATION OF DISSERTATION~

In Chapter two, section 2.1,general definitions for a trans-
versely anisotrOpic medium”from the theory of elasticity and the dis-
placement equations of motion in cylindrical coordinates are pre-
sented. In section 2.2 , the so-called "longitudinal" waves in an
infinitely long cylindrical bar with a traction free lateral surface
are discussed. The characteristic equation which relates the dis-
persion relationship between the wave number and the wave frequency
is obtained, and the results are compared with the isotropic case.
In section 2.3 torsional waves in the same bar are discussed, and
the characteristic equation is obtained. The results are compared
with isotropic case, and it is shown that the characteristic of
torsional waves in transversely anisotropic materials is the same as
that for isotropic materials.

In Chapter three an approximate theory for axi-sylletrical
cylindrical bars is developed. This so-called "two-mode" theory,

follows closely that given by [7] for the isotropic case. In

20

section 3.1, an approximate theory which includes the effect of the
radial inertia, parallel to that given by [1] for isotropic materials
is presented. A two-mode theory, which includes the effect of the
radial inertia, as well as the effect of shear, is presented in
section 3.2. Assuming that the radial diSplacement component of the
displacement vector varies linearly with r and that, the axial

component is independent of r, a set of two partial differential

equations are obtained. In order to make this approximate theory
more consistent with the exact theory, two unknown constants k'
and k" are introduced into the equations of motion. In section
3,3, the approximate characteristic equation for an infinitely long
cylindrical bar with free lateral surface is obtained. The two
constants k' and k", are obtained by matching the correSponding
dispersion curves for the first modes of both the exact theory and
the approximate theory for their two limiting values of the wave-
length.

In Chapter four the solution of a semi-infinite cylindrical
bar with free lateral surface is discussed. The end of the bar is
subjected to a uniform but time-dependent pressure, and the radial
component of the displacement is assumed to be zero at the end of the
bar. Asymptotic solutions of the problem.are discussed in section
4.2, the results arediscussed and compared with- the isotropic case,
and it is shown that these aymptotic solutions are readily obtained
if the material properties of the bar are known. In section 4.3
the steady state solutiom of the same problem are obtained through

the use of the approximate two-mode theory developed in Chapter

three.

21

In Chapter five investigation of the wave motion in the same
cylindrical bar, as in Chapter four, made of alternating elastic,
isotropic, layers is considered. Definitions and the correlation
between a transversely anisotropic medium and a layered medium con-
Sisting of alternating, elastic, isotropic layers, are given in
section 5.1. The five elastic constants] for the above layered
medium are obtained in terms of the elastic constants and the
thickness of the layers. Two independent relationships are obtained
among the five elastic constants of the medium, by assuming certain
relationships among the elastic properties and the thicknesses of the
layers. In section 5.2, solution of the boundary value problem .
discussed in Chapter four is continued for a semi-infinite cylindrical
bar made of the particular layered medium discussed above. Approximate
solutions are obtained explicitly, upon expanding the various Bessél's
functions appearing in the solutions in their infinite series and
then considering the leading terms only.

In Chapter six, section 6.1-2,a summary and the conclusion
of what has been done in Chapters two, three, four and five is given.

Finally.in Appendix A, a complete detailed derivation of the
five elastic constants for a layered medium consisting of alternating
elastic, isotrOpic, layers in terms of the elastic constants and the

thickness of the layers, is presented.

CHAPTER TWO

DEFINITIONS,LONGITUDINAL AND TORSIONAI.WAVES
IN AN INFINITE TRANSVERSELY ANISOTROPIC CYLINDRICAL BAR

In this chapter the definition and basic equations from
the theory of elasticity for a tranversely anisotrOpic medium are
presented. In Section 2.2, free longitudinal elastic waves in an
infinite, circular, cylindrical bar with a traction free Surface
will be discussed. Also, the so-called frequency equation that
governs the diSpersion relation between the wave length and the
frequency for monochromatic waves will be presented. In Section
2.3, torsional waves in the same bar will be discussed.

2.1 Definitionsand Basic Equations from the Theory of Elasticity
for a Transversely AnisotrOpic Medium

Consider the medium referred to the Cartesian coordinates
(x,y,z) presented in Figure 2.1 . This medium is called trans-

versely anisotropic if for any arbitrary angle of rotation a

l
the following relationships hold [1] .
S = S S
o' = C e' (2.1)

where o and o are the Stress (column) matrices and e and e

are the strain (column) matrices, referred to the ‘(x,y,z) and the

(x',y',z') coordinates respectively, and c is a 6 x 6 symmetric

22

23

 

 

 

Figure '2.1 . The elastic constants of this transversely anisotropic
medium are independent of a1, z-axis is called the

axis of transverse anisotropy.

24

matrix, whose entries are known as the elastic moduli of the
medium [2]. In this case the z-axis is called the axis of trans-
verse anisotropy.

For a transversely anisotropic medium the matrix S has

five independent components, which may be shown in expanded form

as follows:

all C12 c13 0 0 0
C12 c11 C13 0 0 0
C = €13 C33 C33 0 0 O (2.2)
~ C44 0 O
0 C44 0
__ O O O C6E]

 

 

where c66 is related to c11 and c12 according to the follow-

1
ing equation C66 - 2(c11 - c12)'

In our Subsequent analysis it is more convenient to use a
set of cylindrical coordinates (r,e,z), in which z is taken

along the axis of transverse anisotropy. Now Suppose and

S1 51
are the stress and strain (column) matrices in the cylindrical
coordinates, we again have

= 2.
91 321 ( 3)

where c is the same as that given in (2.2) and will be independent

N

of e.

In cylindrical coordinates the stress equations of motion

are [2]

25

1 1 n
+-— + +~— - =
l°rr,r r Ore,e orz,z r(orr 088) +Fr pu
o +~l o + o + 2-0 +-F = pw (2.4)
re,r r 89,6 92,2 r r8 6
+-1 +1 +'1 +-F = "
o‘rz,r r 092,6 022,2 r orz z pv

where the 0's are the stress components and the F's are the
components of the body forces per unit volume. Also, u,w and
v are the components of the displacement vector along the r,
e and z coordinates reSpectively and p is the mass density
per unit volume. In addition, in equations (2.4) and henceforth,
subscripts preceded by a comma denote partial differentiations
with reSpect to the Spatial coordinates, and dots indicate partial
differentiations with reSpect to time.

The stress-Strain relationships, in the expanded form of

(2.3), are as follows:

- +
Orr C11 err C12 699 + C13 622

099 = c12 err + C11 699 + c13 ezz

zz c13 err +c13 699 + c33 ezz (2 5)

Q
ll

rz c44 erz
c’ez = C44 692
are = c66 6re

and the strain-displacement relationships are: [2]

26

err = u,r
e =l(w +u)
88 r ,8
ezz = v,z (2.6)
l w
= —’ + - —
er9 r u,e w,r r
erz = v,r +'u,z
l
e =W +—V
92 :Z r ,9

Substitutions of equations (2.6) and (2.5) into equations

(2.4) lead to the following three displacement equations of motion:

c
1 u 66
_ - _ + + -— +
C11(u’rr + r u’r r2) + (C13 + C44)v,rz C44 u,ZZ r2 [1,99
‘l‘ ( +' )w - ‘l"(3 ' ) + F = °' (2 73)
Zr c12 c11 ,re 2,2 C11 C12 w,e r pu .
(w +--‘£’- "‘1") + c w +-3:ll::lg u +':ll;:—:lg‘u
C66 ’rr r 2 [+4 ’22 2 :9 2r ’r9
r 2r
c C + C
+_1_1_w +_L3___4‘Lv +F =pw (2.7b)
r2 ’99 r 592 e
and
u 2 V r :LQQ
+ —L. +
“13 + c44)(u,rz + ) C44“, r + r r2 ) + C33 v.22 +
c + c
13 44 u
+ =
r w,ez Fz pv . (2.7c)

Equations (2.7a), (2.7b) and (2.7c) are the general diSplacement
equations of motion in cylindrical coordinates (r,e,z) for a
transversely anisotrOpic medium where,as mentioned previously,
2 is taken along the axis of the transverse anisotropy of the

medium.

27

In the next two sections we shall discuss the solutions
for particular cases of the above three equations.

2.2 Free Longitudinal Waves in an Infinite Circular Cylindrical
Bar with Traction Free Surface

Free longitudinal waves are characterized by periodic
extensions and contractions of the elements of the central axis
of the bar. For this type of wave motion, the particles move
symmetrically with reSpect to the axis of the bar. It then follows
that the components u and v of the displacement vector along
the r and z coordinates are independent of 8, and the component
w of the displacement vector along the e coordinate vanishes

identically. The above statements can be formulated mathematically

as follows:

“(rseaz:t) = “(r:z:t)

V(r,9,23t) =V(r,z,t) (2-8)

ll
0

w(r,9,z,t)

Consider a circular, cylindrical bar of infinite length
which is assumed to be free from stress on the lateral surface.
Let us choose the cylindrical coordinates (r,e,z) such that the
z coordinate along the axis of the cylindrical bar coincides with
the axis of the transverse anisotropy,in which case all of our
preceding equations would be applicable. Let us assume that the
bar occupies the following region:

-a$2$co
O s r s 8

0595211

28

where a is the radius of the bar.

The equations of motion for the present problem may be
obtained by substituting of equations (2.8) into equations (2.7a),
(2.7b) and (2.7c). If we substitute equations (2.8) into equations
(2.7a), (2.7b) and (2.7c), we see that equation (2.7b) is automatically
satisfied, and the results of equations (2.73) and (2.7c) in the
absence of the body forces are;

U

r u
u +—=—-
C11( ,rr r

—-+
r2) C44u,zz +

(C13 + C44)V = 9“

,rz

u V
,Z ,r = u

3

(C13 +'°44)(“,r

2

Now let us choose the two potential functions U and V, such that:

u(r,z,t) = U r
2.10
v(r,z,t) = V 2 ( )
In terms of U and V equations (2.9) become
+ + - - " =0
[C11A +’°44U,zz (C13 C44 C11)V.zz 9U],r
V (2.103)
_2£_ _ _ n :0
[(C13 +'°44)A + c44(v,rr+ r ) (613 + C44 C33)V,zz pv],z
where
U
u ,r
A=u +-+v =U +—+v . (2.1%)
,r r ,2 ,rr r ,zz

Equations (2.10s) are obviously satisfied by any U and

V which are solutions of

c11A + c44U,zz +"(C13 +’°44 ' Cll)v,zz ' P” = 0
V (2.11)

_L£ _ - _ "
(c13 +-c44)A + c44(V’rr +- r ) (c13 +c44 C33)V,zz pV

u
C)

We now seek the

equations (2.11) of the

U(r,z,t)

V(r,z,t)

where i = -1, and 5
frequency reSpectively.

Substitution of

29

so-called "plane wave" solutions for

following form:

U(r)el(§z+wt)

(2.12)
“magma
and w are the wave number and the wave

equations (2.12) into equations (2.11) yields:

K+plfi-p2\—I=O

 

- (2.13)
_ _ ,r _.
A+p3(V,r+—'—r )+p4V-0
where
_ _ 5.. 2
A = U,rr + F4-- 5 V (2.14)
and p1,p2,p3 and p4 are functions of g and w, defined as
follows:
p = pw - C44§
1 C11
:1
p2 = (C44 + c13 ' C11) cll
C44 (2.15)
p =--—--- = constant
3 c44 + c13
= [(c + c - c )§2 + 2]A@ + c )
94 44 13 33 pm 44 13 °

It is convenient to eliminate

0’ from equations (2.14) and (2.13)

and regard Z. and 3' as the two dependent variables. Thus:

30
._ E' 5'

_ — 2—
A,rr r +p1A p2(v,rr r ) p1g V

_ V r _ _ (2.16)
—L— = .-
p3(v,rr +- r ) + p4V A
where E. and V. now are functions of r only. Equations (2.16)

have solutions in the form of Bessel's functions. More specifically

we sat

<H
n

A J (kr)
° (2.17)

Dd
n

B Jo(kr)

where k, A and B are constants to be determined and A and
B do not vanish Simultaneously. Upon substituting equations

(2.17) into equations (2.16) we obtain the following equations

for k:
p (k2 - Bf’--)(1<2 - p ) - p (k2 +'El'§2) = 0 - (2-18)

Then it follows that
2

B = .-
A p3k 2 94
or B pl-k (2.19)
K = pzkzml:2 2
Equation (2.18) yields two roots for k , which will be denoted
2 2
d o
by k1 an R2 with
2

7"
II

1 f(§:w) ' [f2(§,w) ' 48(gsw)]%

2 a (2.20)
k2 = f(§aw) + [f (gaw) ' 48(gaw)]

and
2 2
f(§,w) = 3:11—97; [(c13 + C44) + c11(pw - c33§2) + 9,4011)2 - c4452”
(2.21)
8(§.w) = zz-lg- ’[(pw2 - c44§2)(pw2 - c33§2)] .

11 44

31

R? and k: as defined above obviously depend upon the

elastic constants of the material, as well as on the values of
w and g. It can be Shown that they are both real, positive
and satisfying ki < kg for most tranversely anisotropic crystals
encountered in practice.

Upon substituting equations (2.19) into equations (2.17)
and using equations (2.20) and (2.13) we will obtain the following

equations for U and VD;

 

 

p + p - 9 k2
- _ 2 4 3 1 §_
U(r) - C p1 Jo(k1r) - D k2 J0(k2r)
_' §(k: _ p1) (2.22)
V(r) = c.Jo(k1r) + D 2 Jo(k2r)
k2(p2 + E )

where C and D are two constants which have to satisfy the
boundary conditions on the free surface of the bar. It is noted
here that another set of independent solutions for O. and ‘V
will be in the form of Bessel's functions of the second kind. As
they are not finite at r = O, we shall not consider them. Now

by equations (2.10), (2.12) and (2.22), the following expressions

are obtained for u(r,z,t) and v(r,z,t),

 

 

2
p2 + 94 ' p31‘1 i(§z+wt)
u(r,z,t) = [D i J1(k2r) - Ck1 p1 J1(k1r)]e
2 2 (2.23)
D is (k - p ) .
v(r,z,t) = [c i g Jo(k]r) + 2 1 Joazrnel‘gmt)

2
k2<p2 + é)
where use has been made of the folloWing recurrence relations for

Bessel's functions [5],

32

n J;(n) (H)

n Jn(n) - n Jn+1

(2.24)

n J5(n) n Jn_1(n) - n Jn(n) .

Note that in equations (2.24) primes designate differentiations
with reSpect to the argument.
Now, on the lateral surface of the bar, which is assumed

to be traction free, the following relationships hold

ll
0

[Orr]r=a
(2.25)

[Orz]r=a 0 °

Using equations (2.5), (2.6) and (2.23) together with (2.25) we

obtain the following equations for C and D,

k
2 2 a 1
[(3 C11k1 ' C13g )Jo(k1a) "7§_(C11 + C12)J1(k1a)]C +

[E (C12 ' C11)J1(k2a) + gk2(C11 ' BC13-‘J’oa‘2‘3‘nD = 0 (2°26)

[§k1(o + 1)J1(k1a)]c + [(Bk: - g2)J1(kza)]D = 0

where

2
- k
p2 + p4 p3 1

 

a:
p1

2 2
_ a (k, - p1)

kficpz + 52)

(2.27)

 

Now if we set the determinant of the coefficients of C and D

in equations (2.26) equal to zero, the following equation is

obtained,

33

C - c
l 12 2 2
—-—-a—.—' (O’Bkz + g )k1J1(k28)J1(k-1a) "

cp<§.w> = 1
(k2 2)( k2 2 as (k >- (228)
B 2 ' g a C11 1 + C13g )Jl 2 Jo 1a '
gzklkze + 1><c11 - BC13)J1(k1a)Jo(k2a) = 0

and

§k1(o+l)J1(k1a)

 

D
E" . (2.29)

2 2
(g -Bk2)J1(kza)
Equation (2.28) is the so-called frequency equation for a trans-
versely anisotropic cylindrical bar, which relates the diSpersion
relationship between w and g.
If we now Specialize to the case of isotrOpy, i.e., we

set

c11 = C33 = 1 *‘2“
C12 - c13 = 1
C44 ' F

with k and n being the Lame's constants, then the results of

equations(2.28) become that given by [1] (page 289) with

2
k2 = fl. _ g2
1 1+2};

(2.30)

k2 fli 2
2 u ' g

and equation (2.28) reduces to the following well-known, Pbchhammer

frequency equation (See [1]: page 289)

34

2k1 2 2 2 2 2
u(ém) = T<k1 + E )J1(kla)J1(k28) - (k2 - § ) Jo(kla)J1(k2a)

2
- 4§.k1k211(k1a)10(k23) - 0 . (2.31)
Note that for the case of isotropy equations (2.27) yield

a = B = 1 (2.32)
which has been used in equation (2.31). The parameter g in
the present expressions is the same as y in [1].

2.3 Free Torsional.Waves in an Infinite Circular Cylindrical Bar
with Traction Free Lateral Surface

We consider now the torsional waves in the same bar as in
section 2.2. For this type of wave motion, the longitudinal and
the radial components u and v of the diSplacement vector are
zero identically, the only non-vanishing component of the dis-
placement vector being w, which is independent of e [1]. The

above statements can be expressed as follows:

u(r,z,t) = v(r,z,t) = O
(2.33)
w(r,e,z,t) = w(r,z,t) .

Substitutions of equations (2.33) into equations (2.7a),
(2.7b) and (2.7c) yield
w

_4£.- E_
C66(W,rr + r r2) + c

44w,zz = pw . (2.34)

We attempt again a plane wave solution for w(r,z,t) in the follow-
ing form

1(Ez'hnt)

w(r,z,t) = w(r)e

(2.35)

35

where g and w again are the wave number and the wave frequency

reSpectively, and t is time.

Substitution of equation (2.35) into equation (2.34) yields

 

 

C (a; +Zz—r--V7—)+(pu)2-C g2)§=0
66 ,rr r r2 44
or _
—- w r 2 l
w +—’—+(’fl -"’§')W=O (2.36)
,rr
r
with
2
Pw C E
n2 = 44 . (2.37)
C66
First we will consider the case where
2 2
pw - c E
02 = C 44 = 0 (2.38)
66

If we substitute this condition into equation (2.36), we arrive at

5 _
V7 +—-3-—r _ L: O . .(2.39)
,rr r 2
r
Equation (2.39) will be satisfied if we set
— -l
w(r) = A r +-B r (2.40)
3 3
where A3 is some non-zero constant and B3 must vanish. Substitution

of equation (2.40) into equation (2.35) yields the following equa-

~tion for w(r,z,t);
w(r,z,t) = A3r e1(§z+wt)

. (2.41)

Now on the free lateral surface of the bar, we have

W
Eor01r=a = C66[w.r- fjr=a = 0 9 (2.42)

36

Substitution of equation (2.41) into equation (2.42) shows that
the boundary condition on the free lateral surface of the bar is
satisfied for all non zero values of A3. Equation (2.41) Shows
that the amplitude of w(r,z,t) is proportional to r. There-
fore,the motion given there is a rotation of each transverse
section of the bar about its center. Moreover, from equation

(2.38) it can be seen that for n = O, we have,

c

w 44 %

—= _ (2.43)

§ [9]

which Shows that there is no diSpersion for this type of waves.
If n # 0, then equation (2.36) is in the form of Bessel's

equation of order one with the following general solutions,

{5(1) = A2J1(T]r) + B2Y1('flr) (2.44)

But, since, the second term of (2.44) is not finite at r = 0 we

set B2 = 0. Thus
Cur) = A2310)»

with A2 being some non zero constant. Setting

- = _ 2.. —
[are]r=a _ O [c66(w,r r )]r=a — 0 (2'45)

and using equations (2.35) and (2.44) we obtain

.1 (Hr)
a_ _._1
[ar J1(nr) r

 

A = .
]r=a 2 0

Finally setting the coefficient of A equal to zero and using

2

equations (2.24), we will arrive at the following frequency equation

which relates the diSpersion relationship between the wave number

37

g and the wave frequency w;

¢(§aw) = flaJO(na) - 2J1(na) = O (2.46)
or, equivalently
J (Na)
8032») = 35-1-73 = o (2.47)

which has an infinite number of rootsi11which the first one is

n = 0, but this is the same condition as was discussed before
(see equation 2.38). The other roots of equation (2.47) can be
determined numerically from the tables of Bessel functions. Each
value of .na, obtained in this way, correSponds to a different
mode of wave motion (see equation (2.38)).

We infer that the characteristics of the free torsional
waves in an infinite transversely anisotrOpic bar are exactly the
same as for the case of an isotropic bar, as it can be seen that
equation (2.47) is the same as that given by [1, page 288] for the
case of isotropy, the only distinctirnrbeing in the difference of

the values of c44 and c66 for the case of transverse anisotropy.

CHAPTER THREE

AN APPROXIMATE THEORY FOR IDNGITUDINA1.WAVES
IN A TRANSVERSELY ANISOTROPIC BAR

In this chapter we shall develop an approximate theory for
the propagation of longitudinal waves in a tranversely anisotropic
bar. For such waves in an isotropic bar there is already a well
known approximate theory, i.e., the classical, one-dimensional
theory. In its simplest form, the governing equation in the classical
theory is the one-dimensional wave equation, which also holds true
for a transversely anisotrOpic bar, as will be Shown in the next
section. For isotropic bars there is also an improved form of the
classical one-dimensional theory in which the effects of the radial
inertia of the bars are taken into account (see, for example, [1]
page 428). The approximate theory to be developed here, however,
will be an intermediate one between the exact theory of linear
elasticity, as described in Chapter two, and such classical one-
dimensional theories.

The complexity of the frequency equation derived in
Chapter two reveals the impracticability of attempting to employ
the general equations of elasticity in attacking problems of wave
prOpagation in transversely anisotrOpic bars. On the other hand,
the classical theory, even in its improved form in which the effects
of radial inertia are considered, is inadequate for providing all

the needed information, eSpecially when waves of high frequency,

38

39

or short wave-length are involved. An intermediate theory which
contains the essential physics of the short-wave phenomena and yet
remains simple enough for attacking wave propagation problems, is
thus highly desirable. Such an intermediate theory for isotropic
bar was deveIOped in [7], and the development here to be given in
sections 3.2 and 3.3 will follow essentially that of [7]. The
result will be the so-called first-order or two-mode theory. How-
ever, in section 3.1 we shall first develop an improved one-dimensional
theory for transversely anisotropic bars by including the effects

of the radial inertia. In sections 3.2 and 3.3 the so-called first-
order theory, or a two-modes approximation which is the main
objective of the present chapter, will be deve10ped.

3.1 The Classical Theory In Which the Effects of Radial Inertia
Are Included.

Consider the motions of a transversely anisotropic, circular,
cylindrical bar in which the axis of transverse anisotropy coincides
with the axis of the bar. We assume that the two components u and
v of the displacement vector along the r and z coordinates are
independent of the e coordinate, and w, the component of the dis-
placement vector along the 0 coordinate is identically zero. Also
as in the case of isotropic bars. [1], we assume that v is independent
of r, i.e., plane sections remain plane during the motions. In
consistence with the above assumptions we assume further that all
the stress components, with the exception of 022’ vanish identically,

i.e.,

o = o = o = o = o = 0 , o f 0 (3-1)

40

and that 022 is constant throughout the cross section of the bar.

The kinetic energy per unit volume of the bar in this case is
l .2 .2
T =§P(v +u) (3.2)

where p is the mass density. Substituting equations (3.1) into

equations (2.5) and using equations (2.6) yields

u
cllu,r + c12 ;-+ c13v’z -

(3.3)

c + c -2 + c = O
r

12“,r 11 13V,z

where v is a function of z and t only. Solution of equations

(3.3) for u leads to

u = - -——-l£L-- r v (3.4)

from which we obtain

C
u=--—-.1:3-—rv . (3.4a)
C11 c12 ’2

Substitution of equations (3.4a) into equation (3.2) and integrating

over the cross-sectional area of the bar yield

C

1 12 13 2 2 '2
T = -'pA [v + ( ) R v ] <3-5)
1 2 c11 + c12 ,z

where T1, A and R are, respectively, the kinetic energy per
unit length, the cross-sectional area, and the radius of gyration
of the bar.

The strain energy per unit volume of the bar is

W = %-o e . Using equations (2.5), (2.6) and (3.4), we obtain

22 22
2c2
2.1 - ____l§___. 2 (3.6)
w1 2 A(€33 )v,z

c11 +'°12

41

where W1 is the strain energy per unit length of the bar.

The total potential energy of the bar per unit length is

 

therefore
c 2c2
1 '2 13 - 2 13 2
E=T +w =—A{p[v +(———-Rv, )]+(c -——)v }.
1 l 2 c11 + c12 ,z 33 c11 + €12 ,
(3.7)
Now using the energy principle [2], we have
t L
8 dt j Edz = 0 . (3.8)
o
o
Substitution of equation (3.7) into (3.8) yields the following
equation of motion
c2 c2
13 2 u l --
p R v,zz + (2 c +3c + C33)v,zz = p v
(C11+C12) 11 12
or equivalently
p(§}-8282§; )=(2ac +c )v (3.9)
3 ,zz 3 13 33 ,zz
where
c
(13 = -—_]"_—3—- = constant . (3.10)
C11 C12

Equation (3.9) is the equation of motion in the z- direction
for a transversely anisotropic bar in which the effects due to the
radial inertia have been incorporated. It reduces to the following

one-dimensional wave equation when the effects of the radial inertia

are neglected.

G = ( )v . (3.11)

We note also that if we set

42

c A + 2H

11 = c33
C13 ‘ C12 = A
equations (3.9) and (3.11) reduce to the following two equations

respectively,

00 2 I.
P(v - v R v,zz) = E v;zz (3.12)
V=§vzz. on»
3

Equations (3.12) and (3.13) are given in [1] for the case of
isotropic bars (see page 428 [1]).

We now assume a plane wave solution of the form
v(z,t) = A1e1§(z'°t) (3.14)

where A1 is some nonzero constant and c is the phase-velocity.

We substitute (3.14) into (3.9) and obtain

C2 _ 20313 + c33
2 2 2
3 R 5

Equation (3.15) gives the dispersion relationship between the phase-

. (3.15)

 

1+a

velocity and the wave number. A similar substitution of (3.14)
into (3.11) will, of course, Show that the phasedvelocity c is
independent of g, and there will be no dispersion. These results
for transversely anisotropic bars are in consistence with those
known for isotropic bars.

3.2 A First Order Approximate Theory for Longitudinal Wave
PrOpagation in a Transversely AnisotrOpic Bar

In this section we shall develop the so-called first order

or two-mode theory for longitudinal wave prepagations in a transversely

43

anisotropic bar. The basic aSSumption in this development is that
the radial component u of the displacement vector varies linearly
with respect to r, whereas the axial component v along the z
coordinate is independent of r. As mentioned earlier, the develop-
ment here f0110W8 that of [7]. It should be remarked, however,
that a more systematic approach for the present theory as well as
any higher order theory has been presented in [8] for isotrOpic
bars, and could therefore be followed here.

The stress equations of motion in cylindrical coordinates,

as given in Chapter two, are

l
o + o +‘- (o
rr,r rz,z r r

r - 099) 3 p 11 (3-16)

0 +8 +—=p{;. (3.17)

2
Multiplying both sides of equations (3.16) and (3.17) by r dr
and rdr reSpectively, and integrating over the radius of the

cross-section of the bar result in

a 2 a 2 a a 2
d - d = "
[r Crr,rdr +-£r orz,z r +-£(grr gee)r r pgr u dr

3 a a a
r dr + dr + dr = 6'
g Orz,r gr 022,2 gorz Dir dr

Upon integrations by parts, one obtains

2 a a a 2 a a 2
- 2 d + d +» - = "
r Orr-Jo £1. Orr r {gr Orz r}.z {(Orr 099) rdr pit. u dr
a a a a a u
r orz]o - g arzdr +'{£ r ozzdr},z +'£ orzdr = pi r v dr . (3.18)

Now we assume that

44

u(r,z,t) = '2- E(zst)

(3.19)
v(r,z,t) = Ikz,t)
and define the following four quantities
a
Pr = g arrrdr
a
P = -d
8 {fear r
a (3.20)
P = I o rdr
z o 22
a o
P = I -££ r2 dr .
rz o a

Substitution of equations (3.19) and (3.20) into (3.18)

yieldS
a l‘(P +'P + P r 923 U
R - a r e) rz,z _ 4
a2 ;_ (3.21)
P +az=L-v
2,2 2
where
R = orr]r=a
(3.22)
z =°rz]r=a

are the normal and the tangential components of the surface traction
at the lateral surface of the bar.

To find the displacement equations of motion, we use the
stress-strain relationships in cylindrical coordinates (r,e,z)

as described in Chapter two, equations (2.5).

rr 11“,: c12

99 12u,r

°zz g C13(u, +' ) +'°33V, (3°23)
°rz = C44(“, + V,r)

= = =0.
0re 082 w

 

orr=%fi+:?fi+C13V,z (a)

088 = iii-2- U + C—il- 17 + (:13 17,2 0’) (3.2“)
ozz = 2:13 U'+c33 ng (c)

arz = C44 2 6,2 (d)

w = are = 092 = 0

We multiply equations (3.24s), (3.24b) and (3.24c) by rdr, and equa-
2
tion (3.24d) by r dr, and integrate over thr radius of the cross-

section of the bar to obtain

a c11_a c1 _a __ a
d =.___ d .___
jarrr r a U It r +- a U [rdr +~c13 V,z [rdr
o o o o
a c a a
o

[deerdr = c1——2-- U :rdr + —l— U
o

 

a 13 ,z
a d é 2:13 __ d + 6' a d (3°25)
gazzr r a U Wira3 r c33 ,z {r r
a

2
orz

We now use equations (3.20) and carry out the integrations of

equations (3.25) with reSpect to r. It follows that

P = C13 8U + T a V,z (3.26)

rz 4 ,z

Substitution of (3.26) into (3.21) results in

c 2 °
44 2— — — _ ga -
4 a U,zz ' (C11 + c:12)U " C13‘*V,z + a“ ' 4 U
— C33 2 — a2 1 (3°27)
+ .___ a + = P— .
C13aU,z 2 V,zz 82 2 V

Equations (3.27) are the approximate equations of motion in terms

of the "displacements" U and V1 We remark also that in equations
2

c448 ‘_

---— U represents the effect of the Shear

(3.27) the term 4 2 ,zz

a 2.
force and the term EZ_ U represents the effect of the radial

inertia.

In order to bring this approximate theory closer to the

2 2
exact theory. we introduce two adjustment constants k' and k"

into equation (3.27) in the following manner

c 2
44 2 "2 _ ' _ = 98—. _
4 a k U,zz k (c11 + c12)U c13aV,z + 8R 4 U
c 2 2 (3.28)
" 33 ‘- 98 ‘“
+ .___ =
c13aU,z 2 a ,zz +482 2 V °

Equations (3.28) are our final approximate equations of motion for
a transversely anisotropic bar with first order approximations on
the components. u and v of the diSplacement vector. The two
constants k'2 and k"2, introduced in equations (3.28), will be

determined in the next section by matching the phase»velocities

47

correSponding to equations (3.28) with those of the fundamental
mode of the exact theory in both the short and long wave-length

limits.

3.3 Derivation of the Frequency Equation in the Approximate 2
Theory and Determination of the Two Constants k'2 and k" .

Let us assume a plane wave solution of (3.28) in the form

A ei§(z-ct)

{112.0

(3.29)
B ei§(z-ct)

V(z,t)

where A and B are constants which we do not assume to vanish
simultaneously, g as before is the wave-number and c is the
phase velocity. Note that the assumed solutions in (3.29)
represent plane waves, traveling from left to right along the
z-axis with constant amplitudes.

Assuming that the surface of the bar is free from traction,

we have

fl
0

R = [Orr]r=a
(3.30)

Z = [crz]r=a = 0 -

Substitutions of equations (3.30) and (3.29) into equations (3.28)
and setting the determinant of the coefficients A and B equal

to zero yield
2 2 2 2 2 2 2 2 2
(pc - c33)[a c44k" g + 4k' (c11 + €12) - pa c g ] + 8c13 = 0. (3.31)

Dividing equation (3.31) by pzazg2 we obtain

48

2
2 C33 44

2
H 3
c k 4k (c11 + C12)

 

(C"p)[p + pa22

2
2 8C13

- c 3 + -————— ~ 0 . (3.32)

2 2 2 "
p a 5

Equation (3.31) or (3.32) is the frequency equation in the approx-

imate theory, which describes the diSpersion relationship between

the phase velocity c and the wave number g.

2
Equation (3.31) is a quadratic equation in c , and its

solutions for c2 in terms of g yield

C? = f . (€3k'23kl'2
l l

,5)

' = 1,2 (3.33)

where 6 is a function representing the material prOperties.

Equation (3.33),of course, corresponds to two different modes of

wave prOpagation in this approximate theory.

It remains to find the limiting values of equations (3.31)

and (3.32) for g a O and g a m, respectively. By matching these

values with the correSponding limits of the fundamental mode of

2

the exact theory the two constants k'2 and k" will be de-

termined. From equations (3.31) and (3.32) we have

. 2 _ 2 "2 2 ,2 _ 2 2 2 2 =
11m {(pc c33)[a C44k g +'4k (c11 + C12) pa c g 1+8c13 O}

 

 

 

2n
F—p 0
A
2 .2 2
- (pc - c33)k (c11 + C12) + 2c13 — O (3.34)
and
2 2 2
c c k" 4k' (c + c ) 8c
. 2 2
.23. p p g p a 5
H
2 c c k
= (C - :3)( 4: o (3.35)

where A in equations (3.34) and (3.35) is the wave length.

Equations (3.34) and (3.35) are the limiting forms of the frequency

49

equation for long and short wave-length reSpectively. Note that
k'2 appears only in equation (3.34) and k”2 appears only in
equation (3.35)° In order to find the two constants k'2 and
k"2 in equations (3.34) and (3.35), we shall match equations
(3.34) and (3.35) with the two limiting forms of the exact fre-
quency equation derived in Chapter two as g tends to zero and
infinity, reSpectively.

The exact frequency equation from Chapter two, equation
(2.28),is

‘C

C 2 2 2
¢(§,w> = <—11;—l;)k1<aek2+§ )J1(k13)J1(k23) - J1(k23)JO(kla)(Bk2‘§2)

2 2 2
x (ac11k1+c13§ > - J1(kla)JO(k23)§ k1k29y+1>(c11-ec13) = o. (3.36)

To find the limit of m(§,w) as g tends to zero, we first re-

call that for the case of isotropic bars, the correSponding relation-
ship between the phase velocity c and the wave number g in the
classical approximate theory coincides with that for the fundamental
mode of the exact theory as A a w or g.» 0 [6]. Now, if we
compare equations (3.9) and (3.11) for the case of transverse
anisotrOpy, with equations (3,12) and (3.13) for the case of isotropy
respectively, and noticing the similarities in form between the

two corresponding frequency equations for the exact theoreis see
equations (2.28) and (2.31), it is reasonable to expect that a
similar relationship between c and g will also hold for the
fundamental mode of the exact theory for transversely anisotropic
bars as A34 m or g a 0. A justification of this observation is

given in [9] for aluminum bars. Thus, by comparing equation (3.34)

50

with the limiting form of equation (3.15) we obtain

 

Zac +C
2 2 2 2 3 13 33
2c + (pc - c )k' (c + c ) = lim {pc - = 0}. (3.37)
13 33 11 12 §~0 1 +’G§R2§2

Substitution of equation (3.10) into equation (3.37) yields

k'2 = 1 . (3.38)

To find constant k"2, we should compare equation (3.35)
with lim[¢(§,m) = 0] as g.» m, where ¢(§,uD is given by equa-
tion (3.36). To find the limit of equation (3.36) for 5 a m,
we first determine the limits of the parameters a,B,k1 and k2
appearing in (3.36), as g a m or A a 0.

First we introduce the following change of variables

u) = cg (3.39)

where c is the phase velocity. Substitution of equation (3.39)

into equations (2.15) of Chapter two leads to

11 (3.40)
C
= + '3 constant
c13 c44
2 - C
c 33)
+c44

p3

2
p), a <1 + .13

Now, from equation (2.18), in Chapter two, RE and k: are the

roots of the following quadratic equation in X:

2
pc - c

2 33 2
93x ’ (9193 +94 + "2)x +’91(c13 +-c

)g = o (3.41)
44

51

where use has been made of equation (3.39). If we substitute
equations (3.40) into equation (3.41) and solve equation (3.41)

for X, we will have

x1 = 521m) - (£202) - 4mm”) = ki
(3.42)
x2 = §21£<c> + «7(a) - 434M] = 1.:-

where f(c) and g(c) are functions of c and the material con-

stants defined by

f(c) = Ezlizzz'[(cl3 + C44)2 + c11(pc2 - c33) +c44(pc2 - c44)]
g(c) = ZEZiEII (pc2 - c44)(pc2 - c33) . (3.43)
From equation (3.41) the following equation is obtained for klk2
2 °°2 ' C44 2 %
klkz = g [(fixpc - .33” . (3.44)

Now from equations (3.42) and (3.44) we obtain the following results

2
lim (k1) = 1m §2[f(C) - (3(a) - 4g(c)>’5] = ..

g...» §-'°°
. 2 . 2 %
11m (k2) = 11m g [f(c) + (f(c) - 4g(c)) ] = m (3.45)
54” §-'°°
pc2 c
_ 2 ' 44 2 a _
éi:~(k1k2) ‘ 1:: § [(_;ll_EZZ—)(pc - C33)] - m
and k2

lim‘7%'= [f(c) - (f(c) - 4g(c))k] = constant
g-m 32

k2 %
lim -§-= [f(C) + (f(c) - 4g(c)) ] = constant (3.46)
§~°° E

2

k k pc - c

lim-l§g B [( 44)(pc2 - c33)]% I constant .

gam g c11 c44

52

Substitution of equations (3.40) into equations (2.27) yields the

following results for a and B as g a m

 

2
c +-c pc - c - c
13 44 33 44 2 3
11m (a) = c -———+ - [f(c) - (f (c) - 4g(c)) /
Ed” 11{ cll c13 + c44 C13 -l'-c44 ]}
(9.2 - c44)= constant (3.47)
and
pc2 - c
11m (3) = cumc) + «2(a) - mm}: - ——c——i‘-’-]/
§- 11
(c13 +-c44)[f(c) + (f2(c) - 4g(c))%] = constant. (3.48)

A
Now dividing both sides of equation (3.36) by g , we have

2
c - c k k
——11 a 12 31(k2a).11(k1a) 7}- ue -% + 1)
§ §
2 2
31‘2 k1
"' J1(k28)Jo(k18)(-—2' ' 1)(CYC11 T + C13) ' (3°49)
E
k k

J1(k13)Jo(k28) “22—2" (01 + 1>(c11 - ecu) = o .

Using equations (3.45), (3.46), (3.47) and (3.48) we see that the
first term of equation (3.49) vanishes as g a a. Therefore, we
need only consider the middle and last terms of equation (3.49)

for g a a, i.e.,
J1(k23)Jo(kla){B[f(C) + «2(a) - awn”) - macumc) -
<f2<c> - men") + .13}

2
+ (ks) (kaEL—‘fi 2 3+1 - )=o (350)
J1 1 JO 2 ) ( C11 (:44 )(pc " (333)] (0’ )(cll BC13 °

53

where a, B, f(c) and g(c) are given by equations (3.47), (3.48)
and (3.43) reSpectively, and use of (3.46) has been made.

If we define the two functions R(c) and S(c) as follows

R(c) = {s[f<c>+<£2<c)-4g<c>>%] - 1}{a.11[f(c)-(f2(c)-4g(c))*]+c13}

pc - c .
50:) = [(—E4—:ll)(pcz - c33)]%(a + 1) (e11 - Bc13) (3.51)

°11

and substitute them into equation (3.50), we obtain
Jo(kla)J1(k23)R(C) + Jo<k2a)J1<k1a)S(c> = o . (3.52)

Now, upon expanding the Bessel functions appearing in (3.52) in
their arguments and examining their asymptotic behavior as g a a,‘

we arrive at
k2R(c) + k18(c) = 0 . (3.53)

Upon using equations (3.42) and (3.35) we will arrive at the

following equation for k"2

[f - (£2 - 4g)”]"[a'[£ + (£2 - 49”] - l}{a'cu[f - (f2 - 43%] + .213}

H2
+ [f - (f2 - 4s)”]”[-95-;fi>- (cMIw2 - c33)]”(a' + 1) (c11 -B'c13) - o
(3.54)

where

54

 

 

 

-___1____ 2 "2 2 "2
f - 2C11C44 [(c13 + C44) + c11(c44k - C33) + C44 (k _ 1)]

-— 1 I32 "2-
g ' 4c11 (k ' 1)(‘2441‘ C33)

2 (3.55)
C c + c c k" - c c
2
c44(k" -1) ll 13 44 13 44

c11[f + (f2 - 4g)%] - c44(k"2 - 1)

 

2 3
(C13 +-c44)[f +-(f - 4g) ]
If we Specialize to the case of isotropy, it can be shown

from equations (3.55) that

a' = B' = l (3.56)
and
[f +> £2 - 4 )5 — k"2 - 1 953: - 1 (3 57)
_( g 1" 9 3+2“- - .

Substitution of equations (3.56) and (3.57) into equation (3.54)
will reduce to that given by [7] for the case of isotrOpic bars
(see equation (25) of [7]). Now if the material properties are
given equation (3.54) can be solved for k"2. This completes our
approximate theory.

It should be remarked finally that the obvious solution

k"2 = l of equation (3.54) must be neglected. For, by considering

equations (3.35), (3.51), and (3.52), for k"2 = 1, we have
R(c) = 0. Now from equations (3.51) together with equations (3.43)
and (3.48), R(c) = 0 if and only if the following relationship

among the material constants holds

C11(C33 ' C44) = C13(C13 + C44) (3°58)

55

which obviously need not be satisfied for general transversely
anisotrOpic crystals. Therefore, unless the relationship (3.58)
holds true among the elastic constants of a transversely anisotropic

bar, the solution k"2 = l of equation (3.54),which states that

no adjustment is needed, must be excluded.

CHAPTER FOUR
PROPAGATION OF ELASTIC WAVES IN A

SEMI-INFINITE CYLINDRICAL BAR SUBJECTED TO
A SUDDENIX APPLIED PRESSURE AT THE END

In this chapter we shall consider a semi-infinite cylindrical
bar, made of a transversely anisotropic elastic material, which is
subjected to 3 suddenly applied normal pressure at the end 2 = 0.

We take the z-axis to be the central axis of the bar, which also
coincides with the axis of transverse anisotropy. The applied normal
pressure, which is distributed uniformly over the cross-sectional
area of the bar, acts for a duration T and is then released. The
surface of the bar will be assumed to be free from traction, and the
radial displacement at the end of the bar, where the load is applied,
will be assumed to be zero for all time.

Exact displacement equations of motion in cylindrical co-
ordinates (r,e,z) will be used for the axi-symmetrical case here.

A double Fourier transform will be used for attacking the problem.
The solution will be represented as a sum of Fourier integrals, with
the integrands depending crucially on the frequency equation derived
in equation (2.28).

The complexity of the frequency equation makes the exact
evaluation of the integrals highly difficult, if not impossible.
Consequently, we shall aim at obtaining asymptotic solutions which
are valid for large values of z for several modes of the wave

motions. These asymptotic solutions will be obtained using the

56

57

saddle point method. As we shall see, the final solution will depend
on a total of seven parameters, namely, the five elastic constants
and the radius and the mass density of the bar. Naturally, when such
material parameters are Specified, numerical values of the asymptotic
solutions are readily obtained.

In section 4.3, the approximate theory developed in Chapter
three will be applied to find the steady-state or the non-transient
part of the solution of the same problem. A closed form solution of
the problem, which represents a harmonic motion with the same fre-
quency as that of the applied load, is possible without resorting to

the use of asymptotic analyses.

4.1 The Exact Solution

Consider the semi-infinite cylindrical bar which occupies the

following region

052500
0 s r s a (4.1)

0 s e S 2n

where (r,9,z) are the cylindrical coordinates and a is the radius
of the bar. For the case of axi-symmetry the component w of the
displacement vector along the a coordinate is identically zero,
whereas the radial diSplacement u along the r coordinates and

the axial diSplacement v along the z coordinate are independent
of e.

We assume that the initial conditions of the bar are

u=v=d=6=o at c=o (4.2)

58

where t is time. The boundary conditions of the problem are taken

to be
[Ozz]z=0 = - PO[H(t) - R(t - 1)], T > o (a)
(4.3)
U(r.O,t) = 0 (b)
and
[Orr a Orz]r=a = 0 (4'4)

where o , o and o are the stress components, P is the
22 rr rz o
intensity of the applied uniform normal pressure, R(t) is a unit
step function and T is the duration of the applied pressure. The
boundary conditions of the problem are illustrated in Figure '4-1 .
The equations of the axisymmetric motions for a transversely
anisotropic bar in cylindrical coordinates are (See Chapter two

equat iom (2 . 9)) .

cllA,r + c44”,“ + “‘13 + c44 ' °11)",rz = p“ (a)
(4.5)
v r
_a_ _
(“44 + C13)A,z + c44(v,rr + r )
(C13 + C44 ’ c33)v,zz = 9" (b)
where, as before:
A = u + 3 + v (4 6)
,r r ,2 °

and cll’ c12’ c13, c33, c44 and p are the elastic constants and

the mass density of the bar respectively.

59

 

0—0—0 “._O—O _."

Figure 4.1 . The boundary conditions of the boundary value problem
given by equations (4.1-5).

60

The boundary value problem defined by equations (4.1-5) will
be solved by a double transform method as follows. In this method,
we apply the appropriate Fourier sine, Fourier cosine and complex
Fourier transforms to the partial differential equations (4.5) in
such a way that, together with the available initial conditions (4.2)
and the boundary conditions (4.3), the transformed solutions can be
determined. We define the Fourier sine, the Fourier cosine and the

complex Fourier transform by the following formulas [10]

f(§,r,t) = I f(z,r,t)sin(§z)dz
O

§<§,r,t> =j g(z,r,t>cos(§z>dz (4.7)
O

fi<§,r.w) = j“ fl(§,r,t)eiwtdt
0

together with the following inversion formulas

f(z,r,t> = if f<§,r,t>sm<sz>d§
O
g(z.r,t) =§j §<§,r.t>cos<§z>d§ ' (4.3)
O
oo+ie ~ -i t
n(§,r.t) = 3” fl(§,r,u>)e w dw
-co+ie

where w is assumed to have a small, positive imaginary part.
Applying the Fourier sine and cosine transforms defined above,

to equations (4.5a) and (4.5b) reSpectively, we obtain

:21:

_. 2- a
C11A,r ' €44§ u ' (C13 + C44 - C11)§ V,r = p (4'9)

._9
V
,r

(“13 + C44)[§A ' Mr’o’tn + C44(v,rr + —) + “13 + C44 " C33)

1'

[V z(r:0,t) + 52;] = p: (4.10)

61

where, in obtaining equatitn1(4-10), use of the second end boundary

conditions (4.3) has been made. Also from equation (4.6) we have

0
Mnmo=urua¢)+%&4Q

r +-v,z(r,0,t).

However, from equations (4.3) we have

u(r,0,t) = u r(r,0,t) = O .
Therefore,
A(r,0,t) = v z(r,0,t) . (4.11)
3
Substitution of equation (4.11) into equation (4.10) yields
X - 2' + - - ) 3 = -
C11 ,r C44g ” (C11 c13 C44 §V,r 9“
a (4.12)

v
-' ” _L£ - - _ 2“ _
(“13+c44)§A + C449,11» + r ) (C33 c13 C44): V

C35V z(r,0,t) = pV ,

3

Applying the complex Fourier transform to equations (4.12)
and assuming that as t tends to infinity both u and v as well

as their partial derivatives with respect to time vanish, we obtain

~

2' 2 2 —
CllA,r + (C11 ‘ C13 ' C44>§v - (c445 - ow >u — o (a)

,r
( +-c )gi + c (3 +-:4£) + (c + c - c ) 2: 2: 4 13)

c13 44 44 V,rr 13 44 33 E v +'pw v ( -

m iwt
= C33 2E V’z(r303t)e (112 (b)
where

i=2 +E-f MIQ

U,r r v o o

62

Note that in arriving at equations (4.13), we have used the initial
conditions (4.2). To find F v’z(r,0,t)eiwtdt given in the right
hand side of equations (4.13:), let us consider the first end
boundary conditions given by equations (4.3a), i.e.

[022]“) = [C13(u,r + E) + 33322123); -PO[H(t) - H(t-T)].

Next, using equations (4.3b), we obtain

[C33V,z]z=0= -PO[H(t) - H(t-T)] . (4.15)

Now multiplying both sides of equation (4.15) by elwtdt, and integrate

from zero to infinity, we thus obtain

iwt ...Cl; _ in
-c33jv,z(r,o,t)e dt- w (1 e ). (4.16)

Substitution of equation (4.16) into equations (4.13) and
elimination of 5' between equations (4.14) and (4.13a) leads to

~
—

 

... A ... V
E,rr + '13-:— + 919 = F’2(V1,rr + ELL) ' p152V1
.: XlLE ipo(1-ein ) (4.17)
A + p3(Vl,rr + r ) + p4V1 = - (c13+c44)§w
where
v1 =2 , (4.18)

and p1, p2, p3 and p4 are given by (2.15) in Chapter two. The
homogeneous part of equations (4.17) is exactly the same as equations

(2.16). Therefore, for the homogeneous equation, we have the follow-

~
_

ing solutions for u and 3 (see equations (2.23) in Chapter two)

63

2
- k
92 + 94 p3

 

 

:.= i(r,§,w) = -( p1 1)k1J1(k1r) + EJ1(k2r)
(4.19)
2
g g (k; 91)
v = §Jo(k1r) + Jo(k2r)
k2(pz + 5 )

where equation (4.18) has been used in obtaining equations (4.19).

To find the particular solutions for equations (4.17) we set

(4. 20)

where D1 and D2

(4.20) into equations (4.17) then leads to

are merely constants. Substitution of equations

in§(1 - ele)

 

Z- = D =
l 2 2
(”(00) "C335 )
(4.21)

. 1_ imT
1P6( e )

2

v = D = 2
wg<¢33§ ‘OW )

 

Using equations (4.18) and (4.13a), we see that the particular solu-

~ ~
- _.
V

tions for u and are as follows

 

~ iP (l - ele)
a _ o
- 2 2
w(c33§ - pw )
(4.22)
'3 = o .

Combining equations (4.19) and (4.22) we arrive at the following

‘1 2'.
general solutions for u and 'v

 

2
=- k p2 +194 ' p3R1
u -A1 1 p1 J1(k1r) + B1§J1(k2r)
, (4.23)
2 2 . T
3 g (k, - p1) 1p <1-e‘w >

 

 

o
v A1§Jo(k1r) +-B J (kzr) +

1
k2(92 + 52) 0 w(c33§ - pwz)

64

where A and B

1 1 are constants as yet to be determined from the

boundary conditions at the lateral surface of the bar.

To find the two constants A1 and B1 in equations (4.23),

we proceed as follows: We begin by writing down the relations (see

equations (2.5) in Chapter two)
= + 2'+ c
Grr Cllu,r C12 r 13V,z
(4.24)

C = C44(U,z + v ) .

r2 ,r

We multiply both sides of these equations by elwt, and integrate

with reSpect to t from zero to infinity. Using the notation de-

fined in equations (4.6), we obtain

0!

+ c

_ ” 2
rr - Cllu,r 12 r

+ C13 V,z (a)

(4.25)

02

= C44(:,z +’; r) . (b)

rz ,

Multiplying both sides of equation (4.25a) by sin(§z)dz and equa-
tion (4.25b) by cos (gz)dz and integrating both equations from

zero to infinity, we obtain

N

OI!

“MCIz

+ c

rr C11“,r 12 ’ C13gv

(4.26)

012

=.+’;
rz c44(gu v,r)

where use has been made of equations(4.6). Now using equations (4.4)

together with equations (4.6) we obtain,

65

z a: :23 . t o -
(Orr r=a = I 1 (Orr)r=aelw S1n(§2)dzdt = O
O O
(4.27)
:1. co co . t
(Orz)r=a = 41‘ ‘1‘ (Orz)r=a 1w COS (gz)dZdt = O .
O O

Substitution of equations (4.27) into equations (4.26) then leads to

H

E
[C11“,r + C12 r

- C13§V1r=a = 0
(4.28)

[c44(§ui+-;,r)]r=a = 0 .

Finally, substitution of equations (4.23) into equations (4.28) leads

to the following equations for A and B

 

 

1 l
. 2 2 in
A = 1P0013§(§ ' Bk2)J1(kza)(1 - e )
2
1 w(pwz - c336, was)
. (4.29)
iP6c13(a + l)§2k1J1(kla)(l - ele)
B =
2 2
w(pw - C335 )¢(§.w)
where
C 'C
_ 11 12 2 2 2
¢(§,w) -'--;-- (5 + aBk2)k1J1(k2a)J1(kla) - E k1k2(a+1)(cll-Bcl3)

(4.30)
2 2 2 2
X J1(k13)Jo(k2a) ' (Bk2‘§ )(C13§ +9511k1)Jo(k1a)J1(k23)

Note that w (§,w) in equation (4.30) is the same as that appearing

in the frequency equation defined by equation (2.28).

Now if we substitute equations (4.29) into equations (4.23),

~ ~

we will obtain the following solutions for u and V

66

 

 

 

 

 

 

 

 

 

 

 

. 2 2 ' T 2
z 1Pc g(g -Bk )5 (k a><1-e1‘”> p +p -p k
_ o 13 2 1 2 2 4 3 1
w(pw - (133% )cp(§,w) 1
(4.31)
iPeC13§2k1J1(k1a)(l'ein) p2+p4'p3k1
+ 2 2 ( + 1)§J1(k2r)
w(pw - c335 )cp(§.w) p1
and
2 .
.. 11» c g (g2 - 3ka (k am - em)
~ _ o 13 2 l 2
V T 2 Jo(k1r)
w(pw " C335: )QP (5.911))
(4.32)
. 2 in . in
IPC Ԥ1<1<J(1<a)(1--e )(a+1) 1P(1-e )
o 13 l 2 1 l o
+ 8.] (k r) +
2 2 o 2 Z 2
w(pw ' C335 )w (§,w) w<C33§ “Pm )
where in equations (4.30), (4.31) and (4.32)
2
+ -
02 p4 p3R1
a =
p1
4.33)
2 2 (
B.fi( +2)
2 p2 5
and
2 2
p = pm ' C445
1 c11
p _ (C44 + C13 ' C11>§2
2 c11
c (4.34)
p 44
3 c13 + c44
2 2
p a (C13 +'°44 ' C33): + m”
4 C13 + C44
and k1 and k2 are the roots of the following quadratic equation
9(X-Eé')(X- )-p(X+p—1'§2)=0 (435)
3 p3 p1 2 92

67

Note that the expressions 0, 3, p1, p2, p3, p4, RI and k2 above
are the same as those introduced in equations (2.15), (2.20) and (2.27)
of Chapter two.

To find u(r,z,t) and v(r,z,t), we must invert :(r,§,w)

and ;(r,§,w) given by equations (4.31) and (4.32) reSpectively.

By equations (4.8) we have

1 “3+“ 2 ° =
u(r,z,t) = 57L” {;£ u(r.§,w>sin<§z>d§} dw
e (4.36)
1 co'l'ie 2 co :3
V(r,z,t) = '2: Lie{;£ V(r,§,w) COS(§Z)d§ do.) '

Considering equations (4.31) and (4.32), together with equations
(4.30), (4.33), (4.34) and (4.35), we see that C(r,g,w) is an odd
function of g and ;(r,§,w) is an even function of g. Therefore,

equations (4.36) can be re-written as

G'I'ie . m a . W
1
u(r,z,t) = EL16{- fflL u(r.§.w>e1§zd§jdw
(4.37)
—Lw+le 1m 3 i§2d d
v(r,z,t) - 211 Lie T-TL v(r,§,(n)e g u) .

It is more convenient to integrate equations (4.37) first with reSpect

to g. Following the expressions given by equations (4.8) we have

30.2.1») = - if :<r.§.w)ei§zd§
(4.38)

v(razam) 3 3(1’ 3E ’w)ei§zd§ '

dih-
éc—Ds

We shall apply Cauchy's residue theorem in the evaluation of the
integrals in (4.38). However, we shall first study the behavior of

the integrands appearing in (4.38) before attempting to apply such

68

theorems. We substitute equations (4.33) and (4.34) into equations

N
b

(4.31) and (4.32) to obtain the following expressions for u and

N
-o

 

 

 

V
" =w£1cv1(§.w) [k1(m2_§2+°11'+°°11°44k1)(‘43 11k2)J1(k2°)J1(k1°) +
(c11L1+mL3 -c11c44k1)m§2k.Jl(kla)J1(k2r)] (4.39)
and,
3 h(w)k22§ 2
V = lecp1(§,w) [mi/203° 11 k2)1‘2J1(k2 a” 00‘ 1") + (°11°1*“‘£3 °11°44k1)
2 h
x @1154.sz (k loan (k 20] - L (4.40)
(M.
°13 1
where
. 1 T
h<w> = 1p0c13<1 - e w >
_ 2 2
L1 _ 9‘” ‘ C33§
2 2
‘2 = P‘” ‘ C445 (4.41)
2 2
‘3 = p‘” "' °13§
m = C13 + C44 = constant
and
(c -c )k
_ 11 12 2 2 2 2 2

k1k231(k1°‘)J 10°23) '

2 2
§2Ck11‘2[(°11 44k2+°13°2) (°11*’1"”°3'°11°44k1)111(kia)Jo(kza)
(4.42)

2 2
“2“ 2(°11k 2 2'Wmg (°13"2*‘“°11k1)+°11k1(41°44k1)] x

Jo(k1ai:1<k2a)

69

note that

2 2
CP1(§ ,UJ) = [w(g 303)]m sz2 (4.428)

2 2
where m(§,w) is given by (4.30), and k1 and k2 are the roots

of the following quadratic equation

2 1 2 2 2 1
X - -—————'[(c +2c c -c c )§ +(c +c )pw ]X +--————'L L = 0 .
c11044 13 13 44 11 33 11 44 c11C44 Z
(4.43)

From equations (4.39) through (4.43) it is seen that for
both i. and 3, the degree of the denominator in g is at least
two higher than that of the numerator. Therefore,we may close the
path of integration along the real axis of the complex g-plane with
the addition of a semicircle of infinite radius in the upper half
plane. As the contributions from the semicircle vanish , the in-
tegrals can thus be evaluated by invoking Cauchy's residue theorem.

To find the branch points and the poles of the integrands,
we proceed as follows:

‘Multiplying the numerator and the denominator of the equations

(4.39) and (4.40) by k and replacing the various Bessel's functions

2’

by their respective infinite series, we see that only even powers

N

of RI and k2 appear in the expressions for 5' and ‘3. However,

since ki and k: are the roots of a quadratic equation (4.43),

2
branch points occur. Solving equation (4.43) for ki and k2, we

obtain

1 2 2 2
k2 = eagwlé +c3w > + [(61% +c3w2>2 - 4.2.1.213)
(4.44)

1 2 2
k1 = 52;):(C1g +C3W2) - [(Clg +c3w2)2 " 4C2L1L215’}

70

where c1, and c

°2

tions

2
c1 — c13 + 2c13c44 - C11C33

2 = °11°44

c3 = p(c11 + c44) .

3 are constants defined by the following equa-

(4.45)

From equations (4.44) we see that branch points in the complex @-

plane satisfy the equation

2 2 2 % _
: [(c1§ + c3w ) - 4c2LIL2] - O

or equivalently

2 2 2
(e15 + c3w ) = 4c2LIL2 .

(4.46)

(4.47)

Substituting equations (4.41) for L1 and L2 and then solving

equation (4.47) lead to the following values for g

x l l
-= 9.35.; 42 4.335%
é i<2>wiu1i[(p1) + 1}

 

 

 

“1
where x1, K2 and ”1 are constants defined as follows
4(c +c )
= 29 2 _ 44 33
)1 2(°13+2°13°44 °11°33)(°11+°44) + c c
(c11c44) 33 44
2
)2 g [°11°44 (°11 ‘ °44)1
2

u = 4°33 _ (°13+2°13°44'°11°33)2

1 °11 °11C44

(4.48)

(4.49)

If we compute the values for X1: X2 and “1 we see that for a wide

class of transversely anisotropic cyrstals, the two values inside

the curly brackets appearing in equation (4.48) are both small, but

71

4

positive, as )1 and p1 are both negative and the quantity -

H1

is negative and small compared to (El)2. Now we define the follow-
1

ing constants

K x l
2.3 .1 .1 2 2 2 2
. (2) {41 +-[(u ) 4-4 1] } — 1 > o
% l1 )\1 2 >\ 12
(g) {Eb-1- [(-—) + 4 21%} — d2 > 0

g = :;d w
§=idw .

The position of any branch points given by equations (4.50) in the
complex fi-plane depends on the values of w. However, the values
of w to be considered here have a small, positive imaginary part
is. Therefore, by giving a small variation is to m, such as

w = mo +‘ie where we is real, and substituting into equations
(4.50), we see that two of the branch points are located slightly
above the positive real §.-axis in the complex g- plane, whereas
the other two are located slightly below the real axis in g-plane.
Since our path of integration is closed in the upper half plane, we

shall consider the first two of the branch points. The path of

(4.50)

integration, branch points and the appropriate branch cuts are shown

in figure ‘4.2 .
Considering equations (4.38) and figure 4.2 , and invoking
Cauchy's residue theorem, we obtain the following expressions for

N

E and v

72

Im § ¢

S‘Plane

 

+Re§

 

 

 

 

 

 

Figure 4.2 . Contour of integration for evaluating integrals (4.38).

C. I
1 2
around the two branch points located slightly above

The two circles and c with radii e' 4 0

the real axis of g-plane, are included inside the con-
tour of integration, whereas c5 and c4 are not.
Note that the two lines o'A and o"B are the branch

73

U(r,z,w) = - fiiZni 2 residues - I E eigzdg - f 5 eigzdg]
I I

1 °2 (4.51)
v(r,z,w) ={%[2ni 2 residues - I I: Wigz -I %_ei§zd§]

I

1 c2

where 2 residues in equations (4.51) indicate the sum of the residues

C
C

of the integrands 4; and '3 given in (4.39) and (4.40), at all poles,
reSpectively,in the upper half of complex -plane.We introduce the

following notations

_ i 2- '
81 " + ;I u elgzdg
Ci +c'
.. .1. 3 i§z (4.51s)
32 - "I v e dg
I
c1 +c2
so that equation (4.51) may be rewritten as
u(r,z,w) B 2 2 residue +lel
(4.52)

V(r,z,w) = 2i 2 residues + 32 .

If we expand the different Bessel's functions,appearing in the
expressions (4.39) and (4.40) in terms of their infinite series,and sub-
stitute them into (4.5]a) it can be verified that equations (4.51s) can

be rewritten as

f1(§,w)+sl(§.w)[(§idl m) (Eidzwfls‘;
+c;_ F(§,w)fl(§:w)[(§id1w)(5231240)]?!
f2(§.a<1’)‘*"82(§5m)".(‘23.4'41111)@4621»)?i
F(§.w)+G(§:w)[(§id1w)(§+ (1219)]?

 

eigzdg

€1-5LJ3
n ci

(4.515)

 

1
(‘2: T? {,h, eigzdg

where f's g's, F's and G's are power series in § andu.of which do not
vanish at :1;de andi-dzu) , where dlm and dzm are the positions of the
branch points. Thus, by examining the degree of the branch points atqdfg
and<huu appearing in the integrands of (4.51b),it follows the integrals

defined in (4.51b) vanish. Therefore el and e2 appearing in (4.52) are

74

both zero,and will be eliminated from further consideration.

To find the locations of the poles in the complex:§-plane we shall
set the denominator of the integrands of the equations (4.38) equal to zero,
i,e.,

= 2 2 _

£91054») = 0 . (4.53)

where m1(§,w) is given by (4.42). From equations (4.53) and (4.42)
it can be verified that there are an infinite number of roots for

g satisfying equations (4.53). We denote such roots by

sq = §q(w)- (4.54)

For a real value of w, say wo, let us consider the subset S for
which gq are real. As the integrations in equations (4.38) are
to be taken along the real axis of g, as shown in Figure 4.2 ,
there is the question of whether one should take all the poles in the
subset S into consideration. In other words in taking the sum of

the residues, whether we should include all or just a portion of the
poles located on real §-axis inside our path of integration. The
above question may be answered again by considering the inverse Fourier
transforms in the w-plane in which w has a small, positive imaginary
part ie, as was discussed above on the branch points. Since the

branch points have been excluded from the path of integration (see
Figure 4.2 ), then each §n(m) belonging to the subset s is analytic.

Therefore,it can be expanded as a Taylor's series about mo as

‘15
gun») a §n(u)o) + 5;“ Au.) + (4.55)

Substitution of is for Am leads to

75

‘1';

§n(w> = gnoeo) +~353 is + ... (4.56)

Equation (4.56) shows that the 5n in the Subset S will have a

352.

dw

is positive. Hence, among all poles belonging to the subset 8,
d5

we shall include gnly those poles for which 552 > O, and neglect
5

those for which 552- is negative [11].

From the first equation of (4.53), we have the following

small but nonzero imaginary part whose sign is positive if

two roots for g corresponding to the two simple poles

a = :;<;°—>3w .
33
. . gs .
From these two poles we con81der only the one for which dw > 0, i.e.,
E = 69-)?» . (4.57)
°33

'The contributions of this pole, as can be verified easily from equa-
tions (4,39), (4.40) and (4.43), to fi is zero, and to s is
Hgggfiwz
" “:9 ° . (4.58)
°13‘” (c _p_3')f(°33 °44)
Considering the poles occurring in the second equation of (4.53),
and evaluating the last part in the right hand side of (4.40) and
using equations (4.52), we arrive at the following expressions for
U and s

h(w)k1k2
ficr.z.w) = 2 zt----[(m2 g H+°1r¢1 11c44k1><c11kzur3>31<k a)J1(k r)

eZiE

v §}§=§q (w) (“'59)

+ (c MIL1+mL3 11c44k1)m§2J1(k12a)J1(k r)]e

76

and

2
s<r.z.w>= 2 mflg—(mkz(L3-c11:2)11(k2an0(k1r) +

 

 

_1__e§z
(4. 60)
+ g(waz)
where
1(EL)%mz 103%")!5 (1)2
g(z ,w) = h(wL%--2 e 33 _ 119)) e (4.61)

and g = §q(w) appearing in the equations (4.59) and (4.60) are the
roots of ¢1(§,w) = 0. Note that the comma followed by g in the
above equations indicates partial differentiation of ¢1(§,w) with
respect to g. Note also that the first term in the right hand side
of (4.61) is the result of the integration of the last term in the
right hand side of (4.40) over the same contour of integration as
shown in Figure 4.2 without the branch cuts.

In order to find u(r,z,t) and v(r,z,t), we should apply

the inverse Fourier transform to equations (4.59) and (4.60), i.e.,

«+16 -iwt
u(r,z,t) 3 %I1 G e dw
(4.62)
1 ”is -iu)t
v(r,z,t) = 35.4 v e dw .
‘Q'l'ie

Since the integration of g(z, m) given by equation (4.61 ) in the
complex valane is quite standard,we shall now concentrate on (4.59)

and the first part v1(r,z,t) say,of (4.60).

77

v1(r,z,t) is defined by

art-is _
v1(r,z,t) = v(r,z,t) ”I g(zsw) e
-Q+le

iwt (4.63)
dw

For convenience, we also introduce the following notations

 

Zih(w)k1k2 2 2 2 2
F1(r’§q’w) = wL1¢1,§ [(m g +C11‘1‘C11C44k1)(°11k2'43)J1(k2a)J1(k1r) +
(clltiimL3-c11c44ki)m§2J1(k1a)Jl(k2r)i} (4.64)
§=§q(w)

2
= {-2h (,0 2
F2<r.§q,w> 1W—[mkz“3"‘11k2)~’1(kza”o(k1‘) +

2 2
(°1I41+m*3’°11°44k1)(°11k2'42)k1J1(k18)Jo(kzr)l} (4’65)
§=§q (0))

Substituting equation (4.63) into equations (4.62) and using equa-

tions (4.64) and (4.65), together with equations (4.59) and (4.60)

yield the following equations for u (r,z,t) and v1(r,z,t)

1 f+ie i(§qz-wt)
U (r,z,t) = 2 —-' F (rag #09 d0)
q 2T” -o+ie 1 q
(4.66)
1 n+1; i(§qz-U.)t)
v (r,z,t) = z: —— F (r,§ ,w)e dw -
1 q 2ni £m+ie 2 q

Now the complete solutions of v(r,z,t) is

78
wI-ie
1 .-
v(r,z,t) = v1(r,2,t) + Z—TI-J‘ . g(zsw)e
'°'*1€

iwtdw . (4.67)
The last term in the right hand side of equation (4.67) can be evaluated
easily. If we close our path of integration in the complex m-plane
as that shown in Figure 4.3 for t > (ER—9&2, and use Cauchy's
residue theorem, we see that 33
co+ie .
Re I g(Z,w)e-lwtdm = o , (4.68)
«so-Pie

Therefore, this term gives no contributions to v(r,z,t) and hence-
forth will be neglected.Hence, v1(r,z,t)=v(r,z,t).

In order to evaluate integrals in equations (4.66), the path
of integration in the complex w-plane will be chosen as shown in
Figure 4.3 for the case wt > §qz. However, the exact evaluation of
such integrals is far from straightforward because of the complexity
of expressions F1(r,§q,w) and F2(r,§q,w) appearing in the inte-
grands. We shall therefore obtain only asymptotic solutions for
u and v. for large values of z, i.e., at sections far from
the end of the bar. From equations (4.66) it is seen that, for a
given value of w, there will be an infinite number of g = §q(w) sat-
isfying the second equation of (4.53). These values for g may be
real or complex. However, by restricting our analysis to large dis-
tances from the end of the bar, only real gq need be considered [14].

Following Kelvin's principles [12] the results for u and
v , as given by equations (4.66), can be regarded as the superposition
of an infinite number of wave trains of equal amplitudes, which are
all in phase at time t e 0. At any subsequent time t > 0, these

various wave components become out of phase and interfere, i.e., they

79

Im 03 A (1)-plane

 

b—m—u

 

 

’Rew

Figure 4.3 . Contour of integration for evaluating the last integral

in equation (4.68), with a pole of order four at the

origin. R -o co.

80

either cancel or reinforce one another. The solution at any point
inside the bar may be obtained by summing up all the wave components
each traveling at its own phase velocity inside the medium. Because
phase differences exist between the various wave components, and they
interfere, the main effect at any point z and any time t, is pro-
duced only by a small group of waves whose phase velocities and wave
lengths are nearly the same, and are in phase at z and t.

The waves in such a group that reinforce- one another for a
given 2 and t may be determined by the condition that this phase,

(qu - wt), as given in equation (4.66),must be stationary, or
£1—(‘§.z-wt>=0 (469)
do) q ° '

More details will be given in the next section.

4.2 Asymptotic Approximations of the Integrals in (4.66).

In the previous section we concluded that among the infinite
number of trains of waves appearing in equations (4.66), only those
waves for which ga-(ng - wt) = 0 will give the main contributions
to the integrals for u (r,z,t) and v (r,z,t). The standard saddle
point method [13]. will now be applied to evaluate the integrals in
(4.66). In this method, the original path of integration shown in
Figure ’4.3‘ is deformed into a new path going through the saddle
point in the direction of steepest descent ihn(i(§qz - mt))= constant

To find the saddle point in the complex w-plane, we set
d
30)" (Eq2 - wt) = 0

from which we obtain

81

(4.70)

D- Q4
elm“
ll
NIH

where z in equation (4.70) is assumed to be large and fixed.

Those values of gq and w satisfying equations (4.70) and con-
sistent with the second equation of (4.53) are labeled EH and

E, and a point 5' in the complex w-plane identifies a saddle point.
Therefore, to find the saddle points in the complex w-plane, we solve
the equation m1(§,w) = 0, and plot the correSponding diSpersion
relationship 5 = §q(w) in (g-m)-plane for real values of g and

w for each mode of the wave motion. Then from each of these plots

g versus m we determine the correSponding plot of 9S, versus

dw
. . d_§ . . t .
w. By intersecting the plots Wlth the quantity 2, we find

dw
the position of the saddle point 5, and EH is then found from the
corresponding g versus w plot. This process is illustrated in
Figure 4.4 for some arbitrary root of ¢1(§,w) = O.

The main contributions to the integrals in (4.66) come from
the portions of the paths near the saddle points and from poles that
are crossed in deforming the contour of integration. The contribu-
tions from the poles are equal to Zni times the residues of the
integrands at the poles.

From equation (4.42) it is seen that m1(§,w) is an even
function of g and w. Consequently if 5, EH are associated with
one saddle point, then so are -$' and -§A. Thus the saddle points
appearing on the real axis of the complex w-plane and are located
symmetrically with reSpect to the origin.

The approximate contributions from the saddle points are obtained

by expanding F1(r,§q,m), F2(r,§q,m) and (ng - wt) about 5’ and

82

(g aw) 'Plane

WI

 

 

 

2%
(dm

- w)-plane

 

 

————
Nln

’0)

Figure 4.4 . Illustrating how to find the positions of the saddle
point 5' along the real axis of complex w-plane. c

represents g = §q(w), which is andgrbitrary root of
$1053.09) = 0. and c

1

represents '—Jl .

2 dw

'83

considering only the leading terms [13]. Assuming that there are

no poles or zeros near m, we write [11]

F1(r9§q 3w) 5 F1(r :Eq ,CU‘)

F2(r:§q3w) E F2(r:§q:;) _ (4.71)
_ ._ d i ._
ng - wt E ng - wt +'% z —$§3-(w - w)2.

When approximations given in (4.71) are substituted into equations

(4.66), we then have for u (r,z,t) and v (r,z,t) (see [13], page 11)

G) . —- —- n
r,z,t .=.-:A Sln z- t+—
U ( ) q (gq 0-) _4)
(4.72)
. (2) _. — " E.
V (r,z,t) E Aq SLU<§qZ ' wt :24
where
2....
d g
(1) _ " " __g_ ~15
Aq — 2F1(r,§q,w)[2nz|d;2 ‘]
2__ (4.73)
(2) - - dig -%
A =2F (r,§ 46mm) __ H .
q 2 q dwz
2—
d E
In equations (4.72) the plus sign is used if -——Jl is positive, and
—- -Q
d2§ dw
the minus sign is used if -:33- is negative. We assume here
dZE- dw
‘f:§&'# 0, otherwise standard modifications must be made, see [13].
dw

To the u (r,z,t) and v (r,z,t) appearing in (4.72), we must add
the contributions due to the poles of. F1(r,E§,5) and F2(r,E€,ab
respectively. Upon summing up the contributions due to all the
saddle points, we have the following results for u (r,z,t) and

V (r,z,t)

‘84

.. (1)
u (r,z,t) a---G1 +2 Aq

sin(§ z - at +13)
q q _'

(4.74)

v (r,z,t) s (:2 + z A512) sinéqz - 5t 1 1%)
q

where G1 and G2 are the contributions due to the poles of
F1(r,ga,$) and F2(r,E§,;), reSpectively.

From the above analysis, it may be concluded that explicit
asymptotic solutions for the boundary value problem under considera-
tion can be obtained once the physical parameters characterizing
the transversely anisotrOpic medium are given. More Specifically,
if the medium is defined the frequency equation (4.42) can be solved,
as is the isotropic case. For any particular mode of wave motions,
one can proceed to find the asymptotic solutions (4.72) without too
much additional efforts over those similar problems in isotrOpic case.

Now, if we specialize to the case of isotropic bars by

setting
C11 = c33 = A + 2”
c12 = (:13 = x (4.75)
%4=“

and substitute them.into equations (4.39) and (4.40x*we obtain

‘3 a h(w)§k1

2 2
uwllpw -().+2u-)§ ]<p(§ .w)

 

2
[(1‘9— - 2:2)J1<k2a)J1<k1r)

+ 2§2J1(k1a)J1(k2r)] (4.76)

85

 

 

and
~ h 2 2 2 -
:7 = 4m, 2 [(2% - W-nlmzanomlr)
quDw -(i+28)§ ]¢(§.w) u
iP6(l-ein)
+ 2k R J (k a)J (k r)] + (4.77)
2 2
l l l o 2 w[pw2-(A+2u)§ ]
where
1166) = 1p0)(1 - eiwm) (4.78)

and

Zkl 2 2

(4.79)

2 2 2 2
- (k2-§ ) Jo(k1a)J1(k2a) - 4g k1k2J1(k1a)Jo(k2a) .

1 and k2 here are the same as those given in equa-

tion (2.30), i.e.,

Note that k

2
k2 = 2a.. _ g2
1 A+2M
. (4.80)
2 _ 293 2
k ‘ ' §
2 u

where A, M: as before, are the elastic constants for the case of

isotropy.
From equations (4.76) through (4.80), we see that, as in the
case of transverse anisotropy, E is an odd function of g, whereas

~

3 is an even function of g. Furthermore if we multiply the numerator
and the denominator of expressions (4.76) and (4.77) by k2 respectively,
as was done in the previous analysis, and replace the various Bessel

functions by their reSpective infinite series, we see that only even

86

powers of k1 and k2 appear in the expressions (4.76) and (4.77).
Now from equations (4.80), it is obvious that no even powers of k1
or k2 can be the square root of any function of 5. Therefore,

there is no branch points in the contour of integration,such as that

shown in Figure 4.2 for the case of isotrOpic barsoThe procedure of the
complete asymptotic analysis for the case of isotropic bars is the

same as that discussed previously, except equations (4.39), (4.40)

and (4.42) have to be replaced by (4.76), (4.77) and (4.79)

reSpectively, and the subsequent analysis follows through.

4.3 Application of the Two-mode Approximate Theory DevelOped in
Chapter Three

In this section we consider the steady state solution of the
preceding boundary value problem, where the restriction on u(r,0,t) = 0
is removed, and since any step function of the form PO[H(t)-H(t-T)]
can be represented in terms of an infinite Fourier sine series, we
consider the reSponse due to one term of this infinite series. In

this case the end boundary conditions are

gzz = -P0 Sin mnt (a)

at z = o (4.81)

OI'Z = 0 (b)

whereas the initial conditions and the boundary conditions on the
free lateral surface of the bar are the same as equations (4.2)

and (4.4) respectively.

87

The equations of motion in terms of 5' and V', as given

in Chapter three equations (3.28), are

22.

2— - - _
U z ' 4(C11 + C12)U ' 4C133V,z ’ 93 U

C4432k"

’

(4.82)

2C13U,z + C33av,zz = 98V

where 6' and W. are the r- independent factors of the radial and
axial components of the displacement vector. Note that use of the
boundary conditions at the free lateral surface of the bar and equa-
tion (3.38) have been made, and k"2 is given by equation (3.54).

For the steady state solutions of (4.82) we set

3(z,t)

II
C
A

N
v

m

(4.83)

I
<H
A
N
v
(‘D

7(z,c) —

where mm is a known number. Substitution of (4.83) into (4.82)

leads to

2 2: 22: :
n _ - - =
c44a k U,zz [4(c11 + 012) pwna ]U 4c133V,z 0

(4.84)

.... : 2:-
2C13U,z + c33a'V’zz + apan — O .

Now we define the following quantities
= 2 "2
oz c44a k > O
2 2
32 4(c11 + €12) - pa mn > 0

v2 = 4c13a > 0 (4.85)

c a
6 = 33 > 0
2 2c1

Pawn

 

> 0

 

2c13

88

We substitute (4.85) into equations (4.84) to obtain

a U 'Bzu ' Y1 V,Z = 0

2 ,zz

(4.86)

C”
+
c»
<3“
a:
6'
H
C)

,2 2 ,zz

The general solutions of the second order ordinary differential

equations (4.86) are

C"
f\
N
y!
H
>
m

(4.87)

6(2) = B3epz

where A3 and B

are nonzero constants, and p is a parameter

3

independent of 2. Upon substitution of (4.87) into (4.86), we obtain

2
(“2p ' e2M3 ' Y2p B3 ' 0

(4.88)
2
A l =
p 3 + (62p + 3 )B3 0 .
Setting the determinant of the coefficients A3 and B3 equal to
zero we obtain
2
X-MX-N=O (4.89)
where
2
X = p (4.90)

and M and N are positive quantities for most transversely anisotrOpic

crystals, defined as follows

6 B - a s' - Y
M a 2 2 2 2 > 0
0’252

 

(4.91)
623'

 

N‘ >0.

“252

89

Solutions of equations (4.89) and (4.90) lead to the following four

values for p

2
2
P1=[%+<%+N>%1%>0

2
P2 = -v1=-E%+ 329+“)? < o
2 (4.92)

2

-p3 = {1%- (’41— + Mfg];é .

Pa
From equations (4.92), it is obvious that p3 and p4 are

purely'imaginaryu Substitution of equations (4.92) into equations

(4.87) yields

:3. — 1 P12 P22 3 P32 P42
U(z) — A3 e +-A3 e +A3 e +A3 e
= 1 P12 2 P22 3 P32 P42 (4.93)
V(z)=B3e +33e +83e +B3e
*

where A's and B's are constants. Upon substitution of (4.93)
into equations (4.83) we obtain the following expressions for

fi(z,t) and sz,t)

_. 1 P12 2 p22 3 p32 4 p42 iw t
U(z,t) = (A3 e +A3 e + A3 e +-A3 e )e n
v(2,t7 = (133 e + B3 e +133 e +33 e )e .

Now as 2 a m, we have fi.= V’= 0, from which we conclude

A; = B;- = O (4.95)

 

*
Superscripts on A's and B's do not imply powers.

90

P32
1(T + (out)
On the other hand, since e is a plane wave traveling

to the left, and is not of our concern, we set
A =BB=O (4 96)
3 3 O O

Substitution of equations (4.95) and (4.96) into equations (4.94)

and using equations (4.92) lead to

_. -pz 'PZ iwt

U(z,t) = (A; e 1 + A3 e 3 )e

— 2 -plz 4 -p32 iwnt (4.97)

V(z,t) = (B3 e +'B3 e )e

2 4 2 4 . . .
where A3, A3, B3 and B3 are constants which will be determined

by the end boundary conditions and equations (4.88). From equations

(4.88) we have

A2
_3_ ___ Y2 p2 _ szl
2 2_ ’
B3 o’2"2 E‘2 E’2 012131
(4.98)
A4
_9_ ___ Y2"4 _ Y2P3
4 2
B3 “294'52 B2 “293
from which we obtain
2
32 = (52’0291M2
3 szl 3
2 (4.99)
4 B2"0’293 4
B3 THE °
Y2 3

Substitution of (4.99) into (4.97) leads to

‘_ -p z -p z t
U(z,t) = (A; e 1 + A? e 3 ) lwn
2 2 . 4.100
-— 2 B2'0’291 ‘Piz+'i‘”nt 4 B2‘0’2p3 “P3Z+l‘”n’t ( )
V(z,t) = A (-—————— e +.A (———————
3 szi 3 Y2P3

To find the two constants Ag and Ag, we use the boundary conditions

at z = 0. Now from equations (2.5) and (4.81b) we have
[Crz]z=0 = [044(u,z +‘V,r)]z=0 = 0 - (4.101)

If we substitute equation (4.100) into equation (4.101) and use equa-

tions (3.19) of Chapter three, we obtain

2 + A4

A3P1 3P3 = 0
from which we have
P
4 _ _l 2
A3 p3 A3 . (4.102)
Upon substitution of (4 102) into (4 100) we obtain
._ 2 -p z p -p z iw t
U(z,t) = A (e 1 --l e_ 3 )e n
3 p3
(4.103)

2

B- p -p z 82 -a 92 -9 1w t

V(z t) = A2 2 0’2 1 e 1 -C—')( 2 3 e 3218 n
3LY2p1 Vzpa

Finally, using equations (2.5) and (4.813), we have
. u _ _ ,
[Ozz]z=0 = [c11(u,r + r) + C33v,z]z=0 - Po Sin wnt (4.104)

Now we substitute equations (3.19) into (4.104) to obtain

2c
11
[

 

U+s9214=-P.nn%t <4ww

92

Upon using equations (4.103) and (4.105), we obtain the follow-

2
ing result for A3

2 P y2p3 -Ziwnt 2
= —— 1 - - -
3 21 ( e )/@C13Y2(p1 P3) + a°33p3(82 “2"?

(4.106)
2
+ ac331°1(CY2P3'E’2):}'

Now we substitute (4.106) into equations (4.103), and use equations
(3.19) of Chapter three, to obtain the following expressions for

u(r,z,t) and v(r,z,t)

 

 

- rP Y P ’9 Z P -p
r —' o 2 3 l 1 3 .

, ,t = -U z,t = Re .______,_. - -—-e s t 4.107
“(r z ) a ( ) [af(p1ap3) (e p3 7% 1n U)“ ( )
and

PYP (B 92) 92 p(8 92) 92
— ‘0! ' -a -
v(r,z,t) = V(z,t) = Re fo 2 3 ) 2 2 l e 1 _ % 2 2 3 e 3;] X
(P19P3 szl p Y
3 2
sin wnt (4.108)
where
2 2
f(p1,p3) = 2C13Y2(P1‘P3) + ac33£P3(82'0’2P1)“p1(62'dzp3)] (4.109)

' are functions of the material

and p1, p3, oz, 62, Y2’ 62 and 3
property and mu as given by equations (4.92) and (4.85).

Equations (4.107) and (4.108) are the steady state solutions
of the radial and the axial components of the displacement vector
due to one component of the infinite Fourier sine series represents
ing the step-function po[H(t) - H(t-T)]. If we expand
po[H(t) - H(t-T)] in terms of a Fourier sine series, the following

expressions result for u(r,z,t) and v(r,z,t), which are the complete

steady state solutions of our problem in the two-mode approximate theory.

93

 

 

 

 

u(r z t) - Re{.l:9.:!2. Z p3(n) le-p1(n)z
9 a "' \ -
a " =1,3,5,...“fn(91’p3)
p1(n) -p3(n)z
p3(n) e )81n wnt (4.110)
and
4p 2
v(r,z,t) = Re —;2- Z p3(n) r52(n)‘azpl(n) e-p1(n)z
_ L -
n—1,3,5,...nfn(p1,p3) p1(n)
2
P1(n)(82(n)'azp3(n)) -p3(n)z
2 e ]sin m t (4.111)
n
p3(n)
where
w = BLT. 3 n = 133,430.. (40112)

and T, as before,is the duration of the applied pressure. Note
that the subscript n appearing in equations (4.110) and (4.111)
indicates the dependence of the different parameters upon n, which

comes from equations (4.85), (4.91) and (4.92).

CHAPTER FIVE

APPLICATION TO THE LAYERED MEDIUM

In this chapter we shall consider the propagation of longi-
tudinal waves in a semi-infinite cylindrical bar consisting of
alternating plane, parallel, isotropic and homogeneous elastic layers.
The axis of the bar is taken to be perpendicular to the plane of
each layer. Furthermore, we assume that the connection between each
pair of layers is made in such a way that the whole combination of
layers can be treated as a single continuous body. Such a medium,
as will be shown in Section 5.1, can be treated as a transversely
anisotropic one. The problem to be considered here is exactly the
same as that defined in Chapter four, Section 4.1, with the same
initial and boundary conditions.

The first objective of this chapter is to consider the
correlatidflibetween transversely anisotrOpic media and layered media,
and to develop relationships between the five elastic constants
Cll’ c12, c13, C33 and c44 and the elastic properties of the layers.
The second objective of this chapter is to consider a special class
of layered media for which the exact solution of the semi-infinite
cylindrical bar problem presented in the preceding chapter can be

further simplified and made more explicit.

94

95

5.1 Definition of a Layered Medium, and its Correlation with a
Transversely Anisotropic Medium. '

Consider a medium such as that shown in Figure ‘5.1 . This
medium is assumed to consist of alternating plane, isotropic layers.
The material properties of each pair of layers are designated by
pl, )1 and p1, and pz, )2 and p2 reSpectively, where x and
u are the elastic constants for each layer and p is the mass
density. The layer thickness for the first material is d1, and that

for the second material is d We take the z-axis perpendicular to,

2.

and the x- and y-axes parallel to the layers. If we pass through

the medium with an arbitrary plane perpendicular to the z-axis, it

is obvious that over this plane of intersection the medium behaves
isotrOpically. That is, the elastic prOperties of the com-

posite medium do not change through any arbitrary angle of rotation

of the x- and y- coordinate axes about the z-axis. This suggests that

this composite medium may be treated as a transversely anisotropic

uedium. Clearly, the z-axis is the axis of transverse anisotropy,

and the connections between each pair of adjacent layers must be

such that the whole combination of layers may be treated as a con-

tinuous body. We shall now develop the expressions for the five

and

elastic constants 4, which characterize

°11’ C12’ C13’ C33 c4
this elastic medium, as a general transversely anisotropic one,in
terms of the material properties and the thicknesses of the layers.

Following the arguments given in [181 the following relation-

ships hold among the five elastic constants Cll’ ch’ c13, C33

and c449 and X1: “'19 d1: K2: “'2 and d2:

96

 

  
    
 

2! ”2: P a 2

  
   

  
 

XI’ “19 Pl: 1

Figure 5.1 . A layered medium consisting of n alternating plane,
isotropic layers with properties )1, p1 and p1 and

12: p2 and p2, and thicknesses <11 and d2.

97

°11 = %{(d1+d2)2(11+201)(12+202)+4d1d2(01-92)[(11m1)-(i2+02)]} (a)

 

(:12 = 611.(d1fi2)2)\1).2+20\1d1+7\2d2)(dluzwzu1)] (b)
c13 = Bit(d1H2)[ild1(12+202)+12d2(11+201m (c) (5.1)
C = 1[(d +d )20. +2 )(i +2 ) (d)
3313121912142]

(difizmlp'z
C44 '3 d «1 (6)

2”1 1“2
and

c - 511.332 = ”1‘5 + ”2d2 (5 2)
66 2 d1 + d2 °

 

where

D = (d1 + d2)[d1(x2 + 2H2) + (120‘1 + 2%)] (5.3)

and K1: “1 and d1 and l2; “2 and (12 are the elastic constants
and the thickness of the first and the second layer reSpectively.
The details of derivation leading to equations (5.1) will be pre-
sented in Appendix A.

From equations (5.1), it is seen that the five elastic con-
stants depend in general on the six parameters )1, “1, )2, ”2, <11
and d2. However, under appropriate choice of the parameters, certain
further relationships may exist among the five constants. As a

first illustration we assume the elastic constants of the materials

of the adjacent pairs of the layered medium to be related as follows

11+p1ex2+u2. (5.4)

98

We remark that this equation holds for a large number of isotropic
materials. Now, upon substuting (5.4) into equations (5.1a) and

(5.1d), we obtain

~
—

cu - C33 = %[(d1 + d2)2()\1 + 2111) (12 + 21.2)] . (5.5)

It is noted that the above relation holds for any arbitrary choice

of (11 and d2. We now set the ratio dl/d2 as

G.
H

= 0/” > 0 (5.6)

Q.

2

and substitute it into equations (5.1). It then follows that

2
(a +1) (11+201)(12+202)
33 " W+ntamz+zuz>+<il+201n (a)

ll!

 

11 °

(om+l)2x1l2+2(amll+l2)(omp2+u1)
C ='—1W ‘12 (b)
12 (oz +1)[oz (12+242>+(11+201>]

 

m 5.7
CY X1(7\2+2H2)+7\2(A1+2U1) ( )

c13 ' W12+202)+(11+201) (C)

 

(8’41) (0 102)
44 g ”7 ° (d)
H1+p20

 

We further choose a“ in such a way that

N

11‘“13"’C

14 . (5.8)

To determine such a value for a” we substitute (5.8) into (5.7)

to Obtain

(0! +1)<11+201)<12+.202> .. a 11(12+2u,)+12<11+201> + (a +0010,
7(k2+2u2)+(11+201) WX2+ZU2)+()\1+2LL1) [$1 + 1120'

 

 

or equivalently

99

A all/2 + B all] _ C 0 (5.9)

III

from which we obtain the following two roots for a

,,,=;_ _ 2 a
0,1 2A[B+(B +440]

(5.9a)
III 1 2
dz = -2K [B + (B + 4AC)%]
where A, B and C are constants defined as follows
A = (11 + 20901102 + 11(02 - 01)]
B = uluzifll + 20.1) + (712 + 2142)] (5-9b)

C a (7&1 + 2H1)[)\2(H2 " U1) ' “'11-‘21 '

From (5.9b), it is seen that B > 0 for all isotrOpic materials.
However, A and C can be either negative or positive depending

on K1: p1, l2 and p2. From (5.9a) it can easily be verified

that one of two following conditions must hold true in order to have

a positive root for am, either

A>0
"XQJ‘ ‘u)>p.p. (5.9c)
c>0 2 2 1 12
or
A < 0 .. 11011 - 02) > 11102 (5.9d)
C < 0 1.202 - 111) < p.192 °

Conditions (5.9c) and (5.9d) are satisfied reSpectively, if

Ill

”2 2H1 (a)

(5.10)
“1 E 292 ' (b)

100

Note that if (5.10s) holds true, then a? > 0, and if (5.10b) holds
true, then a; > 0. It is also noted that, on arriving at (5.10),

we have assumed that
(3)141: 1:192

which is true for most isotropic materials. Otherwise (5.9c) is a

sufficient condition for a?

to be greater than zero. Hence, for
the particular layered medium under consideration we assume that
(5.4), (5.9) and either (5.103) or (5.10b) hold true.

For the class of layered media characterized by equations
(5.4), (5.9) and either (5.108) or (5.10b) there are only three
independent elastic constants instead of the usual five. Equations
(5.5) and (5.8) are the two relationships among the five elastic
constants. Such layered media will be considered further in section
’5.2 .

5.2 Application to the Boundary Value Problem Discussed in
Sections 4.1 and 4.2

Consider now a semi-infinite cylindrical bar, made of alternating
plane, isotropic layers as shown in Figure 5.2 . We again choose
the cylindrical coordinates (r,e,z), where the z-axis, the axis of
the bar, is perpendicular to the layers. We shall treat the bar as
a transversely anisotropic one. Furthermore, we assume that for this
layered medium equations (5.4), (5.9) and either one of (5.10) hold
true.

We now continue the solution of the boundary value problem
presented in Chapter 4. We substitute equations (5.5) and (5.8)

into equations (4.34), to obtain

101

<D-'*

Q—dl—u d2 L—-

I 11,111 IXZ’
eigod—»_

 

 

 

 

 

 

 

 

 

 

.h—INy—Jv

 

Figure 5.2 . A semi-infinite cylindrical bar, with radius a, con-

sisting of alternating plane, isotropic layers, of

thicknesses d1 and d2.

102

 

2 2
pm - C44§
p =
1 C11
92 = 0
c (5.11)
p3 = _&&.= constant
c11
2
p =.E£_
4 c11

Next, we substitute (5.11) into equations (4.35) and obtain
p4
(X - —)(X - 91) = 0 (5.12)
p3

from which we have

02 _ g2
ka2‘EF’ c44 =9
C11 1

 

(5.13)

2 2
x2 3 k1 c.29—
c44

Substitution of equations (5.11) into equations (4.33) leads to the

following expressions for a, B

2
p4 + p2 ' p3R1
a a = 0
p1

 

5.14)
2 2 (

g (k2 - p1)

' 2 2 7 0 '

k2(pz +'§ )
Finally, we substitute equations (5.14), (5.13) and (5.11) into equa-
tions (4.31) and (4.32) to obtain the following expressions for 5'
and '3

1

h (w)k1§J1(kla)J1(k2r)

2 .
(pm -c11§2>¢ (5.4)

 

c|z

(5.15)
2
h'(w)§ J (k a)J (k r) h'
2 122 o 1 +, £9) 2
(Duo ‘C115 )cp'(§.w) (611’; -pw )

<11

 

 

103

 

where
11> c13(1 - em“)
h'(u>) - (5.16)
U)
and
, :11 c1___2_
cp (5.4») =‘11c1J ('lklafl (kza) - cnklkZJl ('oklan (122a)
(5.17)

' c44§2Jo(kla)Jl(k2a)

From equations (5.15), (5.16) and (5.17) it is seen that, as before,
:- is an odd function of g and '3 is an even function of 5.
Furthermore, if we unltiply both the numerators and the denominators
in (5.15) by k2, and substitute for the various Bessel's functions
their respective infinite series, we see that only even powers of
k1 and k2 will appear in equations (5.15).

Now, using equations (4.7) and (4.8) together with (5.15),

we obtain the following expressions for u(r,z,t) and v(r,z,t)

01-13

°° h ((1))ng (11¢ a)Jl (k2 1')
u(rnzt)=1—fl~[fi uféj‘ 1 1

 

e1§Zd§}e'iwtdw (5.18)
(pwf-c11§2)(p' (g 9‘”)

and
«+1. 1.. h'<w>§2J1(k2a>Jo<k1r)

1
v(r,z,t) - ‘—
2“ [adie n -a (pwz-c11§2)m'(§,w)

 

eigzd§}e-iw

(5 .19)
”+16

1 l i
.1. ‘Z—nj _ HEEL—2'8 §Zd§}e'iw

-m+ie 11' -co C 121§ 'pwz

where ¢'(§,m) and h'(m) are given, reSpectively, by (5.17) and

(5.16).

104

In order to evaluate integrals (5.18) and (5.19), we proceed
exactly in the same way as in section 4.1 . However, in integrating
first with respect to g, it should be noted that there are no branch

points along the path of integration as those shown in Figure 4.2 .

This is because, as was mentioned earlier, the integrands of
(5.18) and (5.19) involve only even powers of k1 and k2. Thus,
by the expressions for k2 and k2 in (5.13) there are no branch

1 2
points.

The exact evaluation of integrals (5.18) and (5.19) remains
difficult as before, due to the infinite number of poles in the complex
§-plane which are roots of m'(§,w) = 0. We remember that, as in
section 4.1 , among all poles appearing on the real g-axis, we in-
clude only those for which gE-> 0. Also, as was shown in Chapter
4, we note that the real part of the last integral on the right hand
side of equation (5.19) is zero after performing the integration
on both complex g- and w-planes and shall henceforth be eliminated
from consideration. Asymptotic approximations for the integrals
(5.18) and (5.19) can then be obtained more easily than those dis-
cussed in section 4.2 as the integrals here are simpler. However,
we shall obtain approximations for u(r,z,t) and v(r,z,t) here
by a different procedure as described below.

We expand the different Bessel's functions in terms of their

reSpective series

- - X—
X X
J1(x) " 2 ' '6'" +

for any argument X.

105

Let us take the first two terms from the first equation and only
the first term from the second of (5.20). On substituting them into

equations (5.18) and (5.19) we obtain

 

 

1 ”is i 6313(1‘eiwt)prw§ eigz
u(r,z,t) g__J‘ {-_J‘ dg}
2n . 2 2 2
«+1.: “ «2c:41<§2c11-p02>[§ -w (A'w -B'>]
.‘ t
X e 10) dw (5.21)
and
1 «4:13 1 on 1P c13(1- -ei‘”T)§2(4c "2940 r 2)e £2
a — _ O 4: d
v(r,z,t) - 2n I {n 2 E}
-84... .. 4.4462116 -pw 22)[§ -w 22m» -B 91»
e-lwtdw (5.22)
where
A' = 223—4: constant
44
C11 + €12 (5.22s)
B' = __§_————— = constant .
C44

Note that use of equations (5.13), (5.16) and (5.17) has been made
in obtaining equations (5.21) and (5.22) and that the last term in
the right hand side of equation (5.15» has been neglected.

It is more convenient to evaluate integrals (5.21) and (5.22)
5 first with reSpect to g. Using equations(4.8), (5.21) and (5.22),

we obtain

 

. m iPo c (l-ew T)prw§ e i§z
36:52:90) 2 'fi-I 13 2 dg (5’23)
‘” 2C44(C11§2 'Pw H)[§ 'w 2(A'w ‘B')]
and
th 2 2 2 igz
m iP c (l-e )§ (4c -pw r )e
V(r,z,m) ‘='= 1 ° 13 4“ dg . (5.24)

 

n -m 4c44w(c11§2-pw2)[§2'w2(Adwz-B')]

106

We choose our path of integration along the real g-axis and add to
it a semi-circle, of infinite radius, in the upper half plane of the
complex g-plane as shown in Figure f5.3 . From equations (5.23)

and (5.24), it is seen that the degree of g in the denominator

is at least two higher than that of the numerator for both integrands.
Therefore, by invoking the Cauchy residue theorem, and noting that
the contributions in (5.23) and (5.24) from integration along curve

C vanish as R a m, we have

hflf i z
m iPoc13(1-e )prwge §

 

2 2 2 2 2 dg I 2ni 2 residues (5.25)
-°° 2c44(cu§ -pw )E"; -w (A'w -B')]

. T 2 2 2 i z
1Poc13(l-eiw )g (hcéh-pw r )e §

 

dg = 2ni 8 residues (5.26)

5‘3—18

4c44w(c11§2-pw2)[§2-w2(A'wz-B')]
where 2 residues in equation (5.25) and (5.26), indicates the sum
of the residues at all poles of each integrand. From the denominator
of (5.25) and (5.26) it is seen that there are four simple poles
appearing on the real axis of §-plane, namely

§ = i;w(:2-)%

11 (5.27)

§ = iw(A'u)2 - B')25 .
However, in consistence with the general discussion given previously,
we shall include only those poles for which g&->»0, i.e. those
associated with the "+” sign in (5.27). Upon computing the sum of
the residues of eadh of the integrand in equations (5.25) and (5.26),
and substituting the results into equations (5.23) and (5.24), we

obtain the following results for G and v

107

E-plane

 

4” Re g

 

Figure 5.3 . Contour of integration for evaluating integrals (5.23)
and (5.24), the two simple poles at A1 and 31’

which are located slightly above the real.§-axis, are

included inside the contour of integration, whereas,

the other two located at Ai and Bi are not. Rea a.

i-(EL) (DZ . ' 2 ' $5
a s u(r,z,w) e f(u),r)[e 11 - e“A w “B > “’21 (5.28)
and
1(E£_° wz . . 2 . 5
V 2 v(r,z,w) E g(r,m)[(;2-i)t e 11 (A'u)2-B')L‘5el(A w ‘3 ) wz]
1
(5.29)

where

f(w,r) = iPoCIBr p(l - ein)/c44w(p +-cllb'-c11A'w2) (5.30)
and

2 2 i T 2 2
g(r,w) = poc13(pw r - 4c44)(l-e w )/4c44w (p+c11B'-c11A'w ). (5.31)

In order to find u(r,z,t) and v(r,z,t), we must perform
the inverse Fourier transform, as defined by (4.8), to equations

(5.28) and (5.29), i.e.

°°+le .
... - t
u(r,z,t) = %;:I u e 1w dw (5.32)
-oo+ie
and
0°+16 .
1 ~ - t
v(r,z,t) = EF'I v e 1w dm (5.33)
«+13

where G and v are given reSpectively by (5.28) and (5.29).
We first evaluate the integral in (5.32). Substitution of

(5.28) into (5.32) leads to the following expression for u(r,z,t)

. _g_ %
ca+i lu)[(c ) 24;] . . 2 3!
u(r,z,t z %_'I e f(r,w)[e ll _ e1w[(A w -B') z-t]]dw
TT -m+ie
= u1(r,z,t) + u2(r,z,t) (5.34)

where

109

iw[(-Q- %z-t]

1 ”+16 (:11
u1(r,z,t) = -—j f(r,u))e du)
211 -co+ie
(5.35)
.1 1 ”+16 in (A.w2_B ')%Z-t]
u2(r,z,t) = " Ell-I f(raw)e do.) °

-aflie

Now, we substitute equation (5.30) into equations (5.35) to obtain

. .2. 32 ..
m+ie iP c r p . lMic ) z t]
_ 1 o 13 in 11
-co+ie C44w(p+cllB -C11A'u) )

 

(5.36)

and

1 m+ie ipoc13r p %

u2(r,z,t) = ---

2
' T ‘ A. _ '
2n (1-elw)elw[( w B)

2 z-t]dw .
-w+ie 044w(p+c113'-c11A'w )

(5.37)
In order to evaluate integrals in (5.36) and (5.37), we choose our
path of integration in the complex m-plane as shown in Figure 5.4 .
First we observe that the degree of w in the denominator is three
higher than that in the numerator. Hence, the contribution of
integration along curve c vanishes as R aim. Furthermore, the
integrand for. u1(r,z,t) contains no branch points in the complex
w-plane. The integrand for u1(r,z,t) has three simple poles in

the integrand of equation (5.36), namely,

w = 0 (a)
, (5.38)
0) = + (w);é (b)
... C11 AI °

The path of integration and the positions of the above three simple

poles are shown in Figure 5.4 for t > (22-952. Computing the
11

110

Im w A

w-plane

 

t;~ 44;, :lr-Re w

 

O

 

Figure 5.4 . Contour of integration for evaluating integral (5.36)

for t > (En—982' Note that there are three simple
ll

poles at A2,

0 and 32‘ R ~>o.

lll

sum of the residues, and using the Cauchy residues theorem, we obtain

the following expression for u1(r,z,t)

 

 

2-_Poc13pr
u (r z,t)- cos[5'(z- -(— 11) 35t)
1 ’ [299.4% 11(‘211"“=12)]L ]

- cos[6' (z (— %)%(t-T))]} (5.39)
where

2 c + 2 + c c

. _ 2 , p 44 c11 11 12 5
6 -ac \ 2 ) (5.40)
11 P
.235
and the use of equations (5. 22a) has been made. For t < (c ) 2,
C11

the path of integration will be taken as that shown in Figure 5.5 .
Since the three poles of the integrand are located on the real w-
axis, and there are no other poles along, or inside, the contour of
integration,we conclude that,

0+1;

f si(r,w)dw +'f si(r,w)dw = 0

«SH-is C

where gi(r,w) is the integrand of (5.36). But

I gi(r,w)dw = O as R a m
c

therefore
”+13
I g'(r,m)dw = 0 -* u1(r,z,t) = 0 .
«+13
In order to evaluate u2(r,z,t) as given in equation (5.37),

we notice that there are two branch points located along the real

m-axis, namely

112

Im.w
w-plane

 

v. 1: 4R80.)

 

Figure 5.5 . Contour of integration for evaluating integral (5.36)

for t < (‘2'9%z.
cll

113

First we expand (l - ele) to eliminate the simple pole w = 0

from the origin. This results in

 

art-i3 iPc pr 2 . 2 3
1 l . T 1w T
u2(r,z,t)=-2-1;J‘ ° 3 +c B. (-1T+‘£2—+——6—...) x
C44 11 w cuA' ]

. . 2 . is
8““ “’ ‘B ) “3 dw . (5.41)

The contour of integration, the two simple poles as given by (5.38b),
and the appropriate branch cuts are shown in Figure 5.6 , for

2 2
t > (A'w - B')%z. For t < (A'w - B')%z, the path of integration

is taken as that shown in Figure 5.5 . Since the reSult of integra-

tion along this contour vanishes, we conclude that,

k

2
u2(r,z,t) = 0, for t < (A'w - B') z .

It can be easily verified that the integrand appearing in equation
(5.41) is single-valued along the real m-axis, to the left side of
point A [33]. Hence, the contributions of the integral along the
line segmenusto the left of point A cancel each other. We may there-
fore take our contour of integration as that shown in Figure '5.7 ,

which is equivalent to Figure 5.6 .

Following the contour in Figure 5.7 and invoking the Cauchy

residue theorem, we have

I8'(r.w)dw +-Ig'(r,m)dw +-I g'(r,w)dw +-I g'(r,w)dm +-I g'(r,w)dm +
C L c3 c2 c1

{ g'(r,w)dw +~£ g'(r,m)dw = 2ni(Residue of g'(r,m),
l 2

at point D). (5°42)

114

DmcuA

w-plane

 

1'"
O
...:
‘H—d

 

 

Figure 5.6 . Contour of integration for evaluating integral (5.37)

2
for t > (Adm - B')kz, with two simple poles at D
and D' and two branch points at A and B. Note

that line BB is the branch cut, R «in.

115

w-plane

 

 

Figure 5.7 .

ssJF—‘ 4+u—-Re w
0 L2 \. (J

 

Equivalent contour to that given by Figure (5.6), note
that line BA is the new branch cut. R dim.

116

p + c113.
Note that at point D, u) = (T) , and g'(r,w) is the
11

integrand of (5.41) defined by the following equation

 

 

ip c pr 2 2 3
g'(r.w) = ° 13 .5 (-iT +-QI-+ i w T +...) x
' 2 p+cllB 2 6
°44CuA [w '( “cl'l'T )3
. . 2_ . 5 _
elw[(A U) B ) Z t] . (5.43)
Now, if we set
1 .
w i (i_')% = p = eIele
and (5.44)

dw = dP = ie'elede

and substitute them into equation (5.42) and perform the integration,

we obtain the following contributions due to c1 and c2 respectively
n iP c r 2 3
I g'(r,w)dw = lim e' I -—2—£§— (-iT +-B" Ié'+'8"2 16 +,,,) x
c1 e'ao ~n c44
-i 5"t
e
d9 = 0 (5.45)
and
Zn iP c r 2 3
I s'(r.w)dw - lim e' I -—%—l§- (-rr - a" §—-+'5"2 %—-+...) x
c2 a'aO o 44
-' H
e16 td9 = o (5.46)
where
I
B" ‘ (‘3‘?)35 " -1-(2c + 20 )k = constant . (5.47)
A pa 11 12

With a similar substitution such as (5.44), we obtain

117
-ia"[(;-‘-’—>%z-t1

 

 

P c p ra .
I 8'(r,w)dw - 0 13 2 (l-ela'lT)e 11 x
c3 2c44(p a+c11(2c11+2c12) )
2n _.
I e 1989 = o (5.48)
o
where
p-l-c 8' 4c p+2c (c +c )
a" ‘ (‘2‘;iT—9k = l"[ 44 11 11 12 ]35 = constant . (5.49)

ap c

11 11

If we compute the residue of (5.42) at w = a" and use equations

(5.42), (5.45), (5.46) and (5.48) and let R a m, we obtain

 

 

- n _L%_
m+ie ia"T Id [(611) z t]
js'(r.w)dw - I 8'(r.w)dw = G(1L')(1--e )e
L *16
- {.8'(r.w)dw - f g'(r,w)dw (5.50)
2 L1
where
2nP c13c44r
G(r) = - ° 2 (5.51)
°11(2°44p+°11+°11°12) 5
a[ 2 1
and the path of integration along L1 and L2 is shown in Figure

5.8 . Finally, we set

w +'B" = p
(5.52)

dw = dp

and substitute them in (5.43). Performing the integration along 1,1

and.‘L2, as shown in Figure 5.8 , we obtain

P c pr 28" -ixt
1
g'<r.w>dw= ° 3 e

r —— —2——2—(eiKT-l)s in f1(n,z)dK
1.2 "C44 0 51.01 -or")

113 (r,w)dw -

(5.53)

118

w-plane

 

 

 

tr - :; ‘<§‘// e—*4h-'Re w

 

 

Figure 5.8 . The path of integration along L1 and L2, when
6"* 0, appearing in the integrals (5.42) and (5.50),

line AB is the branch cut.

119

where

f1(n,z) = u(nz - B"2)$5 -EEE-g (5.54)
2(”44)

and n is a real valued parameter. Upon using equations (5.41),
(5.50) and (5.53), we then obtain the following result for u2(r,z,t)

. ’5
mn[(_9_) 24:]
u2(r,z,t) a (Lg? (1 _ eiomT 011

)e +
P c pr 28" -int .
o 13 8—7—7 (GMT-l)sin f1(n,z)dn . (5.55)
2n c44 o u(n -a" )
The total solution for u(r,z,t), is the sum of equations (5.39) and

(5.55), i.e.

u(r,z,t) g “1(razat) + “2(rgzat) =-

2Poc13pr

r 1 c_1]_'_ % ' fill %
2pc44+c11(c11+012)tcos[6 (z‘( p ) t)]-Cos[6 (z-( p ) (t-T))]}
+ (i?)- {cosia"<<;;L1)J’z-t)1-cos[a"(<;i)52-0-1))1}

+ Pocl3pr f8" Lc08(x(t-T))-cos at]

2
2n c44 o u(n - 01"2)

sin f1(n,z)du (5.56)

where 5', C(r) and f1(n.z) are given by (5.40), (5.51) and (5.54)
respectively, and the imaginary part of (5.55) has been neglected.

In order to evaluate v(r,z,t), we substitute equations (5.29)

and (5.31) into (5.33) and use (5.49) to obtain

 

1

’5
2 2 1w[(-2—) z-t]
an-l-ie pc (4c -pu)r) c
v(r z t) g %"f o 13 442 2 2 (1_eLwT){(_2_)%e
" -O+1€ hc44c11A'w (u) ~a" ) c:11
. 2 ..2 k . 1:
[(w2_a,,2)A.]%eiw[(m 'B ) (A ) z-t]}dm 3 v +V2 (5.57)

120

where k
2 g in . Q 5
«.16 Cl3pr (€11) (1-e ) 1w[(c11) z-t]

v = v (r,z,t) = - -—‘I e dm
1 2 2 2
1 TI «Io-+13 4C44011A'QD '0!" )

 

 

 

 

. 2 2 "2 5 imT
+1.?“ P 0°13pr (w "B l (I: ) w((w2-en2)%(m%z-e3d
e
211’ 491-13 4C44c11(A.)E(U) -o" ) w
a V1 *‘V1 (5.58)
and
(Jake-em) iw[(-L>r"z-c
co-i-iePoc13cc11 ]
v2 = v2(r,z,t) =-§-‘f 2 e dw
‘1 «n+1; A'c11w2(w2-o" )
2 2
1 F16 POC13((D “B" )%(1_ein) iw[(u)2-e"2 )%(A')%Z-t]
2" '“fiie (A')%C1W2(w2-o"2) e dw
= v; + v: . (5.59)

From equation (5.58), it is seen that there are two simple poles
appearing in the integrand of vi, and there are no branch points.
We choose our path of integration as shown in Figure 5.4 , and
compute the sum of the residues of the integrand at m = +'o" to

obtain

952
90°13‘29) r

 

vi(r,z,t) =

% {sin[a"((;£-)%z-t)] -
2ac11[2pc44+c11(c11+c12)] 11

sin[a"((-E-)%z-(t-T))]} . (5.60)
c11
However, for vi, as can be seen from.the integrand of the last
integral of (5.58), there are two simple poles at w - i:a", and

two branch points at m '51 B". The path of integration will be

121

taken as shown in Figure 5.7 . The procedure of evaluation for
vi is exactly the same as for u2(r,z,t). Computing the sum of

the residues and the contribution due to L1 and L2, we have
2 g
P C13pr ( )35
2< c) o c” { °["<<—9—>
V r,z, = " 1 ll '51“ 0’
1 8c11°44A “ c11

5

 

z-(t-T))]

+-sin[a"((;£;?%z-t)]}
1

2
p c pr 23" 2 2 k _
+- o 13 E 2 2 cos f1(x,z)[e
4c c U(A') o n - a"
11 44

 

int_e-in(t-T)]dn
(5.61)

If we now add equations (5.60) and (5.61), we obtain the following

equation for v1(r,z,t)

l 2
v1(r,z,t) = v1(r,z,t) + v1(r,z,t) =

2
Poc13(2p)kr

 

g{sin£a"((;2-9%z-t>] -
48c11E2°°44+°11(°11+°12)] 11

sin[a"((;fi;)*z-(e-T))]}

2 u
+_Pocl3r 25 (n2_B"2)%
2

 

2 cos[f1(n,z)(cos nt - cos n(t¢T))]dn
2c11n(c44) o n '0"

(5.62)
where f1(n,z) is given by (5.54).
Note that in (5.62), n is a reaidvalued parameter as was
mentioned earlier and use of equations (5.22a), (5.47) and (5.49)
has been made. Note also that the imaginary part of the integral
in the right hand side of (5.61) has been neglected.

In order to evaluate v2(r,z,t) in (5.59), we have

v2(r,z,t) = v:(r,z,t) +-v:(r,z,t) (5.63)

122

 

where
_2__% in . _g_ 3
Pc ( ) (l-e ) 1w[( )z-t]
1 1 m+ie o 13 c11 c11
v2(r,z,t = 3;.f 2 2 2 e dw (5.64)
«afie A'cllm (w pa" )
and
. 2 2 ‘
"He Po°13(w -B" )sifl-em’T

 

v:(r,z,t) = - “'

2n 2 2

) iw[(w2-B"2)%(A')%z-t]
49+ie (A')%C11@2(w -a" ) e

dw

(5.65)

From equation (5.64) it is seen that there are no branch points in

the complex w-plane. However, there are two simple poles at w = j;a"
as well as a pole of order two at the origin. Therefore, we perform
the integration in equation (5.64) over the same path as shown in
Figure 5.4 . Computing the sum of the residues of the integrand

and using Cauchyb residue theoremAwe obtain the following expression

for v;(r,z,t)

 

%
1 2P6°13°44(2fi;? 1
v2(r.2.t) = - 29°44+°11(°11+C12) {T +-;;{81n f2(z,t) - s16 f3(z,t)]}
(5.66)
where
f2(z,t) = a"((‘2—9%Z't)

C11

(5.67)

£3(z.t> = a"(<-2—>“‘z-(t-T>> .
C11

2
In order to find v2(r,z,t), we notice that from equation (5.65)
there are two branch points on the real w-axis at w = j a", as well
as two simple poles at w - 110": and one pole of order two at the

hfir

origin. We expand l-e in an infinite series, and then divide

both the numerator and the denominator by w to obtain

123

 

2 3
2 "2 2 - Q1. _ 2.1.
1 °°+ie Poc13(w -B ) [1T- 2 id) 6 +...) x
v2(r, z ’0: 211 _LME (A?c cuwflnz - 01"2)
. 2 .
eiw[(w ~B"2)%(A )%z-t]dw . (5.68)

The path of integration, locations of the poles and branch points,
and the equivalent path of integration [33] are shown in Figures 5.9 and
5.10 .From equation (5.65) it is seen that the contribution from

integrating around C 'vanishes as R a w. Then, if we set

and substitute these relations into equation (5.68) and perform the
integration around c4 and c5, we see that their contributions
are zero as e' a 0. Similarly the contributions of the integral
and

around c3 are zero, as was shown in calculating

°1' c2
u2(r,z,t). Therefore, by calculating the residue of the integrand
at w B a", and performing the integration along L1 and 12, as shown

in Figure ,5.8 , and finally invoking the Cauchy residue theorem,

we obtain the following expression for 2v§(r,z,t)

ch: 23" T3 2 4
v2(r,z,t) a ___gj' $_L)_(-T2 .2— - m— +L— T +...)
2 nc (A ) 24
11
x cos f “hide-int“
. P c T 26"
+ _°1_3_£ L—ii—L— CDS f1(z,n) 8111 nt d”.
nc11(A') o u(n -a' 2)
12°13(EL)’5 2 3 4 1°"[(ZL)%z't]
+ 11 (_ L _ ia"—- L + (1"2 .1:— + )[e 11 J
__r_w____2A a °11 6 24

(5.69)

124

 

’Rew

 

 

m-plane
i L 4
C c c T
c3 2 4. 1 Q
D' A ‘0 J
o
W
q.
Figure '5.9 . Contour of integration for evaluating integral (5.68)

for t > (Adwz - B')%z, with three simple poles at
D', 0 and D, and two branch points at A and B.
Note that line BE is the branch cut and R «>w.

125

w-plane

 

 

 

 

 

 

Figure 5.10 . Equivalent contour to that given by Figure (5-9k
AB is the new branch cut. R ~>a.

126

Now, adding equations (5.66) and (5.69), we obtain

~ 1 2
v2(r,z,t) = v2(r,z,t) +-v2(r,z,t) =

Zl’c ‘2—)%
c

o 13c44( 11 1

_. r'1‘ + —-{,Sin f (z t)-'Sin f (z t)]}
I I l I ’ ’

2pc:44 11(c11 C12) 0" 2 3

 

is
21». Me > 7-6" 2 .26
a oC13 44 Sn £§__%_.cps f1(z,n)sin at d“
"p 11 0 u(n ~a" )

 

 

 

%
2pc (c ) 25" 2 "2a; 2 3 24
o 13 44 SK -§ 2 I___ . l;_ n T
+ Re{ a-npc11 o nz-a"2 (- 2 1K 6 +’ 24 +%") X
1130C13(.<-:-p__)%E 2 3
e-intcns f (z n)dn + 11 (- 1_._ id" i—,+
1 ’ 2A'a"c11 2 6
1a"[ (—L) *2 -t]
"2 T4 C11
0 '2; +...)e } . (5.70)

Finally, we add equations (5.62) and (5.70) to obtain the following

approximation for v(r,z,t)

v(r,z,t) 5 v1(r,z,t) +-v2(r,z,t) =

 

 

 

P c r 28" 2_ "2 k

0 13 1,1 Luz—L2)— 6°8{fl(n,2)[c08 nt-cos u(t-T)]}dn
2nc11(c44) o n -a"

k 2
Poc13(29) r [ _L. 95 E _L)15
' in "(( ) -t - ' " z-(t-T))
4acn}:2pc44-l-c11(c1]_-«l-c12)]_35is 0’ cll z )1 sm 0((c11 1}
35
2P6c13c44(2i1) 1
2pc44+c11(c11+c12){T +';F[sin f2(z,t)-sin f3(z,t)]}

55
21’ c T(C ) 28" 2- "2 k
+ o 13 44 S31, E 2

anpcll 2 2

cos f (z,n)sin xt dn
u 1
0 “(K ‘0! )

+ (continued on next page)

127

 

 

 

5 n
2 2
2P0c13(c44) 23 in _ "2:5 j . 1:: + T4 +
+'Re{[ an e l 2 2 (' 2 ' 1” 6 24 °°°)
p 11 o n -a"
”(flaky—9% 2 3
'int 11 T c n T
6 cos f1(z’“)d“] + ZATa"C11 - 2— - la '6— +
4 id”[ (L) tz _t]
2 T C11
a" ..27: +...)e 1 (5071)

where, A', a" and B" are constants given by equations (5.22a),
(5.49) and (5.47), and f1(z,n), f2(z,t) and f3(z,t) are given
by equations (5.54) and (5.67) reSpectively. It is understood that
the expressions (5.60), (5.61), (5.66) and (5.69) obtained for
vi(r,z,t), vi(r,z,t), v%(r,z,t) and v:(r,z,t) are valid for

t > (-2-%z

c11
and (5.72)

t > (1")350»2 - 8"2)%z -

However, for t smaller than the above quantities given in (5.72),
the contour of integration is the same as that shown in Figure 5.5 .
As was discussed in obtaining u(r,z,t), the contributions of
integrations along this path are zero for vi(r,z,t), vi(r,z,t),
v;(r,z,t) and v§(r,z,t). Therefore, if (5.72) does not hold true,
then v(r,z,t) 8 0.

Equations (5.56) and (5.71) represent the approximate solutions
to u(r,z,t) and v(r,z,t). From the above solutions it is seen that
the dependence of u(r,z,t) upon r is linear, whereas the dependence

of v(r,z,t) upon r is nonlinear. From the preceding analysis, it

is also noted that the the phase velocities of the majarparts of

c
the waves are (_g'lfi.

CHAPTER SIX

SUMMARY AND CONCLUSION S

6.1 Summary

The investigation of wave motions in transversely aniso-
trOpic elastic media has been the main objective of this disserta-
tion.

Free wave motions in unbounded, transversely anisotropic
as well as general anisotropic elastic media were reviewed in
Chapter 1, where the notions of wave surfaces, slowness surfaces
and the velocity surfaces were introduced and investigated.

Attention was also given to Rayleigh's surface waves in transversely
anisotropic media.

In Chapter 2, the basic equations from the theory of elasticity
for a transversely anisotropic media were presented. An alternative
form of the equations of motions, in terms of two potential functions,
was also given. These equations were applied to study the propaga-
tion of longitudinal and torsional waves in an infinite cylindrical
bar with a traction free lateral surface. The characteristic equation that
governs the dispersion relation between the phase velocity and the
wave-length was obtained.

In Chapter 3, an approximate theory was deve10ped for axi-
symmetrical bars, in which the effects of the radial inertia and

shear were included. In order to further improve the accuracy of

128

129

this approximate theory, two adjustment parameters were introduced
into the equations of motion. These two parameters were then de-
termined by matching the corresponding diSpersion relations between
the phase velocities and wave-lengths of the fundamental modes in
the approximate and in the exact theories for short and long wave-
lengths limits.

Chapter 4 was devoted to the boundary value problem of a semi-
infinite, transversely anisotropic cylindrical rod subjected to a
time-dependent pressure at the end. Solutions based on the exact
equations of motion were obtained with the aid of Fourier transforms'
and the detailed inversions were carried out as far as possible.
Asymptotic solutions were then obtained that are valid at stations
away from the end of the bar.

Explicit approximate solutions of the same boundary value
prdblem, using the approximate theory developed in Chapter 35were
also presented. However, the results were not explicit because of
the large number of unknown material properties involved.

Chapter 5 was devoted to the study of layered media consisting
of alternating plane, isotrOpic layers and their correlations with
transversely anisotropic media. It was shown how the five elastic
constants Characterizing a general, transversely anisotropic
medium might be determined in terms of the elastic properties and
the thicknesses of the layers. A Special class of layered media,
in which two specific relations on the elastic prOperties and the
thicknesses of a pair of adjacent layers were made, was then con-
sidered and the solution of the boundary value problem for semi-

infinite cylindrical bar, carried out in Chapter 4, was specialized

130

to this class of layered media. The inversions of the transformed
solutions were further carried out in this case and asymptotic solu-

tions were obtained explicitly.

6.2 Conclusions

As is to be expected, the behavior of free elastic waves in
a transversely anisotropic medium is quite different from the iso-
tropic case. The wave Surfaces, slowness surfaces and velocity Sur-
faces .in a transversely anisotropic medium are no longer Spherical
surfaces. They are more irregular and»in fact, singular points
exist on the wave surfaces. For the case of the so-called longitudinal
waves in an infinite cylindrical rod, with the axis of rod coinciding
with the axis of transverse anisotrOpy, the behavior of waves in a
transversely anisotropic medium also differs from that in an isotropic
medium. A generalized version of the "POchhammer" equation governing
the general diSpersion relations between the phase velocities and
the wave-lengths was developed in Chapter 2. It is found that the
prOpagation of longitudinal waves in a transversely anisotropic bar
is more complicated, due to the coupling of the quasiedilatational
waves and quasi-shear waves inside the bar. However, the behavior
of torsional waves propagating in an infinite transversely aniso-
tropic bar remains the same as those propagating in an infinite
isotrOpic bar. Also, in analogy with the isotropic case, approx-
imate theory for the propagation of longitudinal waves in a cylindrical
bar was developed.

As mentioned above Chapter 4 was devoted entirely to the
exact solution of the semi-finite, transversely anisotropic

cylindrical rod subjected to a suddenly applied pressure at the

131

end. Transformed solutions were obtained for this boundary value
problem, which may have practical applications in the fields of
soil mechanics, geophysics and seismology. It was found that the
transformed solutions were difficult to invert. It might be

worth mentioning that such exact evaluations of the same boundary
value problems have not been carried out, even for isotropic
cylindrical rods. Nevertheless, asymptotic solutions which are
valid at stations away from the end of the bar were given. Such
asymptotic analysis in the present problem were carried out as far
as possible, without explicitly specifying the material constants,
using the method of stationary phase. As compared to the isotropic
case, it was found that additional branch points now exist in the
complex plane of transformed parameters that do not occur in the
case of isotropic bars. However, such branch points may be dealt
with quite easily and the resulting contour of integration coincides
with that for isotropic case.In fact, it was shown that the contri-
butions of the branch points are zero.Thus, a class of asymptotic
solutions for the boundary value problems involving transversely
anisotropic media may be obtained without too much difficulties,

as those for isotropic material.

In order to give the present theory wider application, the
correlations between transversely anisotropic media and layered
media were studied in detail in Chapter 5. It was found that a
layered medium can be regarded as a transversely anisotropic medium
insofar as their dynamical behaviors are concerned and the five
elastic constants for such layered media were obtained in terms of

the properties of the layers.

132

Finally, a special class of layered media in which certain
approximate relations exist among the properties of the layers was
considered, and the solution of the boundary value problem involving
a semi-infinite cylindrical rod mentioned above was further con-

tinued for such layered media and more explicit results were obtained.

9.

10.

11.

12.

13.

LIST OF REFERENCES

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Elasticity, Dover Publications, Inc., New York, 1944.

1.8. Sokolnikof, Mathematical Theory of Elasticity,
MeGraw-Hill Book Co., Inc., New York, 1956.

W.M. Ewing, W.S. Jardetzky and F. Press, Elastic Wave in
Layered Media, McGraw-Hill Book Co., Inc., New York, 1957.

 

G.F.D. Duff, "Cauchy Problems for Elastic Waves in an
Anisotropic Medium", Phil. Trans., Roy. Soc. of London,
(A) 252, 1960, pp. 397-430.

G.N. Watson, A Treatise on the Theory of Bessel Functions,
Cambridge at the University Press, 1966.

H. Kolsky, Stress Waves in Solids, Dover Publications, Inc.,
New York, 1963.

R.D. Mindlin and G. Herrmann, "A One-dimensional Theory of
CompressionaJ.Waves in an Elastic Rod", J. Appl. Mech. Trans.
(ASME), 1952, pp. 187-191.

R.D. Mindlin and R.D. McNiven, "Axially Symmetric Waves in
Elastic Rods", J. Appl. Mech., 27, 1960, pp. 145-151.

Frank C. Eliot and G. Mott, "Elastic Wave PrOpagation in
Cylindrical Bar, having Hexagonal Symmetry", J. Acous. Soc.
Amero, 44, 1967’ pp. 423-4300

I.N. Sneddon, Fourier Transforms, McGraw-Hill Book Co., Inc.,
New York, 1951.

R. Folk, G. Fox, C.A. Shook and C.W. Curtis, "Elastic Strain
Produced by Sudden Application of Pressure to one end of a
Circular Bar", J. Acoust. Soc. Amer., 30, 1958, pp. 552-568.

R.M. Davis, "A Critical Study of the Hopkinson Pressure Bar",
Phil. Trans. Roy. Soc. of Iondon, (A) 240, 1948, pp. 375-457.

MlV. Carrillo, "An Elementary Introduction to the Theory of
the Saddle Point Method of Integration, (Research Laboratory
of Electronics, Massachusetts Institute of Technology, 1950),
Technical Report 55: 2a.

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14.

15.

16.

17.

18.

19.

20.

21.

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23.

24.

25.

26.

27.

28.

134

O.E. Jones and F.R. Norwood, "Axially Symmetric Cross-sectional
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M.J.P. Musgrave, "PrOpagation of Elastic Waves in Aeolotropic
Media", Proc. Roy. Soc. of Iondon, (A) 226, 1954, pp. 339-355.

J.L. Synge, "Elastic Waves in AnisotrOpic Media", J. Math.
Phys. (35), 1955, pp. 323-334.

V-T- Buchwald, "REVIEigh‘Waves in Transversely Isotropic Media",
Quar. J. Mech. and Appl. Math., Vol. (14), 1961, pp. 298-317.

G.W. Postma, "Wave Propagation in a Stratified Medium", J.
GeOpysics, Vol. 20, No.4, 1955, pp. 780-806.

V.T. Buchwald, "Elastic Waves in Anisotropic Media", Proc.
Roy. Soc. of London, (A) 253, 1959, pp. 563-580.

M.J.P. Musgrave, "On the Propagation of Elastic Waves in
Aeolotropic Media, Part 11. Media of Hexagonal Symmetry",
Proc. Roy. Soc. of London, (A) 226, 1954, pp. 355-363.

N. Cameron and G. Eason, "Wave Propagation in an Infinite
Transversely Isotropic Elastic Solid", Quart. J. Mech. and
Appl. Math., Vol. (20), 1967, pp. 23-40.

Iya Abobakar, "Motion of the surface of a Transversely
Isotropic Half-Space Excited by a Buried Line Source", J.
Geophysics, 1961, pp. 87-101.

T.R. Rogge, "Stable Elastic Waves in AnisotrOpic Materials",
Zamp., Vol. 19, 1968, pp. 761-770.

D. Bancroft, "The Velocity of Longitudinal Waves in Cylindrical
Bars", Physical Review,‘Vol. (59), 1941, pp. 588-593.

IHW. Kennedy and O.E. Jones, "Longitudinal.Wave PrOpagation
in a Circular Bar loaded Suddenly by a Radially Distributed .
End Stress", J. Appl. Mech. Trans. (ASME), 1969, pp. 470-477.

M. Redwood, "Velocity and Attenuation of a Narrow Band, High-
Frequency Compressional Pulse in a Solid Wave Guide", J. Acoust.
Soc. Amer., 31, 1959, pp. 442-448.

R.L. Rosenfeld and J. Miklowitz, "Elastic Wave PrOpagation in

Rods of Arbitrary Cross Section", J. Appl. Mech. Trans. (ASME),
1952), pp. 187-191.

R. Skalak, "Longitudinal Impact of a Semi-Infinite Circular
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135

D.W. Jordan, "The Stress Waves From a Finite Cylindrical
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S.C. das Gupta, "Waves and Stresses Produced in an Elastic
Medium due to Impulsive Radial Forces and Twist on the Sur-
face of a Spherical Cavity", Geofis. pura Appl. Milano, 27,
1954, pp. 3-8, AMR, 11, 1958, Rev. 395.

M. Mirta, "Solution of the Buried Source Problem for an
Extended Two-Dimens ional Source in an Elastic Medium",
Proc. Nat. Sci. India, Part A, 1959, pp. 236-242.

M.J. Lighthill, "Studies on Magneto-Hydrodynamic Waves and
Other Anisotropic Wave Motions", Phil. Trans. Roy. Soc. of
London, (A) 252, 1960, pp. 397-430.

N.W. McLachlan, Complex Variable Theory and Transform jalculus,
Cambridge at the University Press, 1963.

A. Ben-Menahem, "Radiat ion of Seismic Surface-Waves from a
Finite Moving Source", Bull. Seism. Soc. Amer., 51, 1961, pp.
401-435.

APPENDIX A
DERIVATION OF FORMUIAS 5.1

A.1 Let us consider a parallelopiped, with height N(d1 + d2)
and the basal dimensions la and b, from.the layered medium given
in Figure 5.1. The faces of this elementary volume are taken to be
parallel with the coordinate planes and the z-axis is taken to be
perpendicular to the layers, see Figure 6.1.

Suppose that on the faces perpendicular to the z, x and

y axes, respectively, the state of stress is as follows,

0 i 0
22
(A.1)
oxz = oyz a O
and
(Oxxl’ Oxx2) 5‘ 0
(A.2)
Oxyl a o‘xy2 B c,le = C4x22 g 0
and
(Owl, nyz) ¢ 0
(A.3)

° 1=° 2=0 1=0 2'=0

where in equations(AnZ) and (A.3), the subscripts l and 2, indicate
that the above stresses are applied to a pair of adjancent layers

with thickness <11 and d2. The normal stresses oxxl’ Oxe’ ogyl

136

137

N

 

 

 

 

 

 

 

 

 

 

1 1’ “’1! p1 /

 

T
d
_L

 

 

Figure 6.1 . An elementary volume of layered medium, consisting

of N alternating pairs of isotropic layers.

138

and are such that
°yy2
exxl =-. e:xx2 a exx
(A.4)
eyyl g eyyZ ‘1 eyy

where exxl’ exx2’ eyyl and eyy2 are the components of the strain
in the above pair of adjacent layers. This restriction is necessary
to insure the continuity of the displacement as a function of the
radius vector. The components of the strain along the z-axis, in
two different layers, need not be equal, i.e. in general, 6221 i 3222’
In the first isotropic layer with thickness d1, from Hook's

law we have:

0
ll

xxl “1+2“? 6and + 11(eyy1+€zzi) ... 0‘1‘“21’1)’sxxflleyy'H‘fiZZ1

Oyyl = 0‘l+2““1)eyy1 + xl(€xx1+€zzl) B (41+2pl)eyy+klexx+mlezzl (A.5)

O

221 ul+2u1>ezzl + 11(exx1+€yy1) = 0‘1‘”1‘1)"=z.v.1+"1(€ )

xx+€yy

and in the second isotropic layer with thickness d2, we have

Oxxz a (2.2-52142) 63x2 + *2 (€yy2+€zzz) (2x2+2IJ2) exx‘l'kz (ny+€zzz)
Oyyz a.- (2324,2152) eyyz + XZ (€xe+€zzz) a (AZ-+2142) eyy+)\2 (exx+€zzz) (A° 6)
0222 ‘ (12+2u2)szzz +'12(exx +'eyy)

where use of equations (A.4) has been made. The average stress on

faces perpendicular to the x and y axes become:

139

dIQXxl +d2°xx2

O 3
xx d1 + d2

 

(A.7)
G : d1°y11 +d2°yy2

yy 2

while the average stress on the face perpendicular to the z-axis

is 022. Combining equations (A.5), (A.6) and (A.7), we obtain
(d1 + d2)axx = exx[d1(),1 + 2,1,1) + d2()\2 + 2%)] +
eyyuldl + 42‘12) + 622141‘11 + 6.2227.de
(‘11 + d2)°yy ... enydel + 2All) + d2”? + 2”2)1 + exx “1‘11 + k2‘12)”
ezzlxldl + 6222.4de (A-B)
(d1 + d2)ozz == enoud1 + i262) + eyy(),1d1 + x262) +
6221(11 +.2”1)d1 + e2220‘2 +'2”2)d2 °
We define 622 by

(d1 +'d2)ezz = dlezzl + d26222 (A.9)

where 622 is the over-all elongation of a linear element parallel

to the z-axis, which contains an equal number of sections through

the layers (11 and d2.

From the last equations of (A.5) and (A.6), and (A.9), we
find:

6221 = [(dl + dz) (12 + 2H2)€zz ' (11 " X2) (sxx + eyy)d2]/[d1()\2+zpz) +

(1201 + 261)] (A.10)

140
6222 = [(d, + d,)(1, + 211,)622 + (1, - 1,)(exx + eyy)d,]/[d,(1,+2u,) +

d2(),1 + 2u1)] . (A.11)

Substituting equations (A.10) and (A.11) into (A.8) we arrive at:

=-—-{[(d 1«12) (11+2u1) (12+Zu,)+d 1,6,1“). +-2u,) (12+2u,)) 2411-1921}
2
+ “6111112“1“? +2(1,d,+1,d,) 6111124421119] +

e
—;—z{ <d,«I,>[1,d,<1,+2u,)+1,d,<1,+211,>11 (11.12)

2
'Oyy g ‘12)l{ (d1fi2)2()\1+2U-1) (212+2H2)4'd1q2[((211+2H1)'(212+2P2)) '(AI'A2)2]}

exx
+ _XX'D‘ "1’12‘d1 “2) 2d+201‘11“? 2) (*‘21""‘1‘12)J +

-,z,—z{<d,«I «I,)[1,d 1(12+2u2)+>.2d2(11 +2691} (1.13)

+
(cxx e

on = —31H<d,«I ,>[1,d,<1, +Zuz)+12<12(11 +2191}

4,01 «I,) 2(1, 211,) (1,126,) (11.14)

where

D . (d1 + c12)[c11(12 + 2,12) + d2(),1 + 2%)] . (11.15)

Now, we apply a tangential stress a to the upper face

and the lower face of an adjacent pair of layers as shown in Figure

 

6.2 . The average strain 6 z is
de +'d2e
1 eyzl 2ey22
eyz- (11 + (12 (A°16)

141

 

 

 

Figure 6.2 . Tangential stress, and the corresponding strains

in (y,z)-p1ane.

142

the stress Oyz on the upper and the lower faces respectively is

Oyz a “'1 e:yzl
(A.17)

o = ”2 e

yz yzZ °

Combining equations (A.16) and (A.17), we obtain

a = (d1 +Ad2)”1”2 e (A 18)
yz “1‘12 + pzdl yz

 

With a similar manner we obtain

. (d1 +'dz)“‘1““2 (A ,9)
0x2 ,1,le + pzdl exz ° °

 

Finally applying two tangential forces oxyladl and
OxyZadZ’ reSpectively, to the two layers with thickness (11 and

d2, and realizing that 6 must be equal to g in order to

xyl xy2

insure the continuity of the diSplacement, we find that:

I"'1 exy = nyl

 

 

(A.20)
p'2 6xy = °ky2
taking the average tangential stress as oxy’ we have
0 = d102,, +Id205¥2 . (A 21)
xy d1 +d2
Combining (A.20) and (A.21), we obtain
“1d1'*“2d2

If we compare equations (A.12), (A.13), (A.14), (A.18), (A.19)

and (A.22), with the corresponding stress-strain equations in Cartesian

143

coordinates (x,y,z), for a transversely anisotropic medium, see

equations (1) and (2) of [18], we find that:

c11 = %1 (‘1 1M2)2 (2.144112 1) 0.2+2H2)+4d1d2 (141142”; (41‘1“ 1) ' (424142)]1 (a)

 

 

2

612 = %[(d1+d2) 1112+2(1ld,+1262)(uldzmzdln (b)
C13 g 51% (d1fi2)[xld1(212+2|42)+212d2 (211+2H1)]} (C)

1 2
C33 g B[(d1+d2) (211+2p1) (2124-21-52)] ((1) (A023)
C - (dl‘mzh‘fl‘z (e)
4“ d1*"2 "’ d2M1

p, d + p. d
c . 1 1 2 2 (f)
66 (11 + (12

which are the same as those given by equations (5.1). Note that
if we subtract (A.23b) from (A.23a), and divide the result by 2,

we obtain the obvious relationship

c‘11 " c12 .. “1‘11 + “2‘12 _

c
2 d1 +-d2 66

 

   

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