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Accepted Manuscript
Scattering of SH-waves by an elliptic cavity/crack beneath the interface between
functionally graded and homogeneous half-spaces via multipole expansion method
A. Ghafarollahi, H.M. Shodja
PII:
S0022-460X(18)30528-5
DOI:
10.1016/j.jsv.2018.08.022
Reference:
YJSVI 14313
To appear in:
Journal of Sound and Vibration
Received Date: 21 November 2017
Revised Date:
21 July 2018
Accepted Date: 13 August 2018
Please cite this article as: A. Ghafarollahi, H.M. Shodja, Scattering of SH-waves by an elliptic cavity/
crack beneath the interface between functionally graded and homogeneous half-spaces via multipole
expansion method, Journal of Sound and Vibration (2018), doi: 10.1016/j.jsv.2018.08.022.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to
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ACCEPTED MANUSCRIPT
Scattering of SH-waves by an elliptic cavity/crack
beneath the interface between functionally graded and
homogeneous half-spaces via multipole expansion
method
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A. Ghafarollahia , H. M. Shodjab,c,?
a
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Laboratory for Multiscale Mechanics Modeling, EPFL, CH-1015 Lausanne, Switzerland
b
Department of Civil Engineering, Sharif University of Technology, Tehran, Iran
c
Institute for Nanoscience and Nanotechnology, Sharif University of Technology, Tehran, Iran
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Abstract
In this study, based on multipole expansion method an analytical treatment is presented
for the anti-plane scattering of SH-waves by an arbitrarily oriented elliptic cavity/crack
which is embedded near the interface between exponentially graded and homogeneous half-
D
spaces. The cavity is embedded within the inhomogeneous half-space. The boundary value
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problem of interest is solved by constructing an appropriate set of multipole functions which
satisfy (i) the governing equation in each half-space, (ii) the continuity conditions across
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the interface between the two half-spaces, and (iii) the far-field radiation and regularity
conditions. The analytical expressions for the scattered elastodynamic fields are derived
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and the dynamic stress concentration factor associated with the elliptic cavity as well as
the dynamic stress intensity factor relevant to the case of a crack are calculated. In the
given numerical examples, the effects of such parameters as the incident wave number, angle
of the incident waves, the distance of the cavity to the bimaterial interface, and the aspect
ratio and the orientation of the elliptic cavity/crack on the scattered field are addressed in
?
Corresponding author.
Email address: shodja@sharif.edu (H. M. Shodja)
Preprint submitted to Journal of Sound and Vibration
August 16, 2018
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detail. It is seen that such parameters have significant effect on the dynamic response of
the medium.
Keywords: Bimaterial; Anti-plane shear wave; Elliptic cavity; Crack; Functionally graded
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material
1. Introduction
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The vital role of high performance composites and advanced materials in modern engineering, particularly in aerospace, high temperature, high strength and good wear resistance
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applications, has attracted the attentions of many investigators from various disciplines.
Functionally graded materials (FGMs) are a generation of high performance composites,
fabricated to have outstanding advantages over homogeneous materials and conventional
composite materials. FGMs were initially developed for use in ultrahigh temperature re-
D
sistant aircrafts, space shuttles, planes, and nuclear fusion reactors. Nowadays, these func-
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tionally graded materials are being widely used for various purposes such as sensors, wear
resistant coatings, joining dissimilar materials, dental and orthopaedic implants, among
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others. FGMs are designated as graded since their microscopic properties vary continu-
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ously and gradually with the spatial coordinates. This is achieved by gradually varying the
volume fraction of the constituent materials.
In recent years, the dynamic response of media made of functionally graded materials to elastic waves, especially those containing such flaws as cracks, holes, and fibers has increasingly
received much attention. For example, Ma et. al. [1] studied the dynamic stress intensity
factor at the crack-tips of two parallel interface cracks along the boundaries of a functionally
graded layer which is sandwiched between two different homogeneous half-planes. Ma et. al.
[2] investigated the dynamic behavior of a finite crack in an infinitely extended functionally
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graded medium subjected to incident elastic waves. The scattering problem of anti-plane
shear waves by a permeable crack in functionally graded piezoelectric/piezomagnetic materials was studied by Liang [3]. Yang et. al. [4] considered the anti-plane scattering of
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SH-waves by a circular cavity buried in a functionally graded surface layer which is bonded
to a homogeneous substrate. Their work was reconsidered by Fang et. al. [5], except the
circular cavity was replaced by a circular fiber. Based on the works of Gregory [6, 7] who
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proposed to solve the scattering problems involving circular cavity inside a half-space by
multipole expansion, Martin [8] obtained an analytical solution for the scattering of SH-
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waves by a circular cavity located beneath the interface of bonded functionally graded and
homogeneous half-spaces, inside the inhomogeneous half-space. Martin [8] solved the corresponding scattering problem by introducing an appropriate set of multipole functions.
These functions are singular on the axis of the cavity and satisfy the governing equation in
D
each half-space and, moreover, satisfy the continuity conditions across the interface between
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the two half-spaces. The coefficients of these functions are expressed as contour integrals.
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Alternatively, Liu et. al. [9] have employed the image and conformal mapping techniques
combined with the complex variable method to solve the same circular cavity problem. Most
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recently, by using multipole expansion method Ghafarollahi and Shodja [10] presented an
analytical solution for the scattering of SH-waves by a nano-fiber at the interface of two
perfectly bonded half-spaces, one made of a homogeneous material and the other made
of a FGM, in the framework of surface elasticity theory. Linton and Thompson [11] have
extended the work of Gregory [6, 7] to the problem of oblique Rayleigh wave scattering by
a cylindrical cavity. Their treatment required the field to be expressed in terms of three
coupled potentials.
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The paper is organized as follows. In Section 2, the problem statement is described and the
governing equations are presented. Section 3 is devoted to the formulations of anti-plane
scattering of SH-waves by an arbitrarily oriented elliptic cavity/crack within the inhomo-
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geneous half-space of a bimaterial. The pertinent boundary conditions are presented in
this section as well. By using the multipole expansion theorem of Gregory [6] the scattered
wave field is expanded in a series form which satisfies the governing equations, the conti-
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nuity conditions along the homogeneous-inhomogeneous media interface, and the far-field
The conclusion is given in Section 6.
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radiation and regularity conditions. Several descriptive examples are provided in Section 5.
2. Problem statement and the governing equations
Consider a cavity with elliptical cross section beneath the interface between two perfectly
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bonded semi-infinite media, ? = 1 and 2. The origin of the Cartesian coordinates (x, y)
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is set on the interface between the two half-spaces such that the x?axis points into the
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inhomogeneous medium and the y?axis is along the interface. Therefore, the homogeneous
and inhomogeneous half-spaces indicated, respectively, by ? = 1, 2 occupy the regions x < 0
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and x > 0, respectively. The elliptic cavity falls entirely inside region 2 with its center at a
depth of x = b beneath the bimaterial interface as depicted in Fig. 1. Let a1 and a2 denote
the semi-axes of the cavity and moreover, assume that the cavity?s orientation is indicated
by the angle, ? counterclockwise from the positive x?axis. A set of elliptic coordinate
system (?, ?) with origin at the center of the cavity is also considered. It is assumed that
the line ? = 0 makes an angle ? with the positive x?direction. The boundary of the elliptic
cavity is described as ? = ?0 , therefore the semi-major and minor axes of the circumscribing
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ellipse are a1 = d cosh ?0 and a2 = d sinh ?0 , respectively, where 2d is the foci distance. A
set of Cartesian coordinate system (x?, y?) is also attached to the elliptic cavity in such a
way that its origin coincides with the center of the cavity and a1 and a2 are along the x??
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and y??axis, respectively, as shown in Fig. 1. Both regions, ? = 1, 2 are isotropic linearly
elastic solids. Assume that the inhomogeneous half-space, ? = 2 is made of functionally
graded materials. Let �, ?1 and �(x), ?2 (x) denote the shear modulus and mass density
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of the homogeneous and inhomogeneous half-spaces, respectively. Here, it is assumed that
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�(x) and ?2 (x) vary exponentially with the coordinate x as:
�(x) = � e2?x ,
?2 (x) = ?02 e2?x ,
(1a)
(1b)
where � , ?02 are the shear modulus and mass density at the interface as x ? 0+ , respectively,
in(1)
(x, y, t) propagating with frequency ? toward the bimaterial
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harmonic shear wave, uz
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and ? is the inhomogeneity constant. Consider an incident SH-wave, anti-plane time-
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interface at an angle ? with the positive x-axis, as shown in Fig. 1. uz is the out-of-plane
component of the displacement field, in the z?direction. Hereafter, the superscript ?in?
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stands for incident, and the superscript ?(1)? refers to region ? = 1. t denotes time.
In the absence of body forces, the equation of motion for an inhomogeneous material is
given by:
?.? = ?(x)u?,
(2)
and the linear stress-strain relation pertinent to an isotropic body is:
? = ?(x) : II + 2�(x),
(3)
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y
1
m2 (x) , r2(x)
2
A2
B
p/2
x0
h=
g
a
a
x=
y
0
h=
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m ,r
1 1
x
d
A1
a2
p
h=
b
x
a1
/2
3p
h=
SC
a
Incident SH-wave
o
j
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Figure 1: A functionally graded half-space containing an arbitrarily oriented elliptic cavity/crack is bonded
to a homogeneous elastic half-space. Crack is modeled as the limiting case of the cavity as one of its axes
approaches zero. An incident SH-wave propagates through the bimaterial.
1
(?u + u?) .
2
(4)
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=
D
where
In the above equations, and ? are the strain and stress tensors, respectively. u and
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u? are the displacement and acceleration vectors, respectively. ?(x) and �(x) are Lame?
?
e
?x x
+
?
e
?y y
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parameters, and ?(x) is the mass density. I denotes the unit tensor and ? ?
is the spatial gradient operator, where ex and ey are the base vectors.
In view of the fact that the geometry of the scatterer is elliptic, it is beneficial to present
the corresponding formulation with respect to the elliptic coordinates (?, ?). The details of
this coordinate system together with the Cartesian coordinates (x?, y?) are displayed in Fig.
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2. The Cartesian components are related to the elliptic coordinates as:
(5a)
y? = d sinh ? sin ?,
(5b)
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x? = d cosh ? cos ?,
where 0 ? ? < ?, 0 ? ? < 2?. The delta-operator, ? and the Laplacian, ?2 = ? � ? with
respect to the elliptic coordinates are given by:
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SC
? 1 ?
,
,
?? h ??
2
1
?
?2
2
? = 2
,
+
h ?? 2 ?? 2
1
?=
h
with
h=d
q
cosh2 ? ? cos2 ?.
(6a)
(6b)
(7)
Further discussion in this section is restricted to anti-plane problems, so that the components
D
of the displacement field ux = uy = 0 and uz = uz (?, ?, t). Utilizing Eqs. (3), (4), and (6a)
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the non-zero components of the strain and stress fields are obtained as:
?z =
1 ?uz
,
2h ??
?z =
1 ?uz
,
2h ??
(8b)
??z =
�(x) ?uz
,
h ??
(8c)
??z =
�(x) ?uz
,
h ??
(8d)
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EP
(8a)
3. Formulations and the treatment using multipole expansion
As mentioned in the previous section, suppose a time harmonic anti-plane SH-wave
propagating through the homogeneous medium (? = 1) is incident upon an arbitrarily
7
h=
4
p/
p/
4
y
h=3
3
h=
p/8
h=5
h=p/2
p/8
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h=7
/8
h=p
x=0
(-d,0)
h=p
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p/8
x h=0
(d,0)
h=2p
h=1
5
p/8
p/8
1p/
8
h=
5
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h=1
p/8
3
h=1
h=3p/2
/4
7p
p/
4
h=
SC
h=9
Figure 2: Elliptic coordinate system.
oriented elliptic cavity which is embedded in the adjacent functionally graded half-space,
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? = 2 as shown in Fig. 1. The treatment of the defined problem is given in two parts as
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in the following Sections 3.1 and 3.2. In Section 3.1 attention is given to the formulations
corresponding to the reflected and transmitted waves, whereas Section 3.2 focuses on the
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treatment of the scattered wave field by an arbitrary oriented elliptic cavity. Later, in Sec-
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tion 5 the result corresponding to the crack problem is obtained through the consideration
of the limiting case of the elliptic cavity problem as a2 ? 0 (?0 ? 0).
3.1. Reflected and transmitted wave fields
When the incident waves impinge the bimaterial interface, the reflected and transmitted
waves are produced in domains ? = 1 and 2, respectively. Moreover, due to the presence
of the cavity, the scattered waves are generated outside of the cavity. Therefore, the total
(?)
displacement field within the domains ? = 1, 2 represented by u(?) = uz ez , where ez is
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the base vector along the z?axis, may be decomposed as:
(9a)
u(2) = utr(2) + usc(2) .
(9b)
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u(1) = uin(1) + ure(1) + usc(1) ,
In the above equation and the remainder of this paper, the superscripts ?in?, ?re?, ?tr?, and
?sc? over a field quantity indicate that the quantity corresponds to the incident, reflected,
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transmitted, and scattered wave fields, respectively. Utilizing Eqs. (1)-(4) and assuming
(?)
time harmonic solution uz (x, y, t) = W? (x, y) exp[?i?t], ? = 1, 2, where ? denotes the
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circular frequency, the governing equations of motion for the displacement field W (?) within
the homogeneous half-space (? = 1) and the inhomogeneous half-space (? = 2) become:
?2 + k12 W1 = 0,
?W2
?2 + k22 W2 + 2?
= 0.
?x
(10a)
(10b)
p
�/?1 and c2 =
p
� /?02 are the propagation velocity of shear waves
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to region ?; c1 =
D
In the above equations, k? = ?/c? , ? = 1, 2 denote the wave number of SH-waves pertinent
the substitution
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associated with the homogeneous and inhomogeneous half-spaces, respectively. By making
W2 (x, y) = e??x V2 (x, y).
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(11)
into Eq. (10b), we find that V2 (x, y) satisfies the following equation:
?2 + k22 ? ? 2 V2 = 0,
(12)
which may be represented as:
(?2 + k?22 )V2 = 0,
if
k22 > ? 2 ,
(13a)
(?2 ? k?� )V2 = 0,
if
k22 < ? 2 ,
(13b)
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where
k?2 =
k22 ? ? 2 > 0,
(14a)
q
? 2 ? k22 > 0.
(14b)
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k?�=
q
Suppose that the amplitude of the incident SH-wave traveling along an arbitrary direction,
making an angle ? with the positive x-direction (|?| < ?/2), is A1 . Moreover, let A2 and B
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be the amplitudes of the reflected and transmitted waves, respectively. These amplitudes
along with the angles of incident, reflected, and transmitted waves are schematically shown
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in Fig. 1. The incident and reflected displacement fields satisfying the equation of motion
(10a) inside the homogeneous medium (? = 1), may be expressed as:
(15a)
W1re (x, y) = A2 eik1 (?x cos ?+y sin ?) .
(15b)
D
W1in (x, y) = A1 eik1 (x cos ?+y sin ?) ,
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Furthermore, for the transmitted waves within the inhomogeneous region, depending on the
values of k2 and ?, the equations of motion may be governed by either Eq. (13a) or Eq.
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(13b). For the case of k22 > ? 2 the transmitted wave field is given by:
W2tr (x, y) = B e??x eik?2 (x cos ?+y sin ?) .
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(16)
In the above equation, |?| < ?/2 is the angle of the transmitted waves, which satisfies
Snell?s law, k?2 sin ? = k1 sin ?. By considering the continuity conditions across the interface
between the two half-spaces, the constants A2 and B will be obtained as:
A2 = A1
k1 �cos ? ? k?2 � cos ? ? i?�
,
k1 �cos ? + k?2 � cos ? + i?�
B = A1 + A 2 .
(17a)
(17b)
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Define the critical angle as ?cr = arcsin k?2 /k1 . If |?| > ?cr then the transmitted waves
will take on the following form:
W2tr (x, y) = B e?(?+p?)x eik?2 y sin ? ,
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(18)
where p? = k?2 (sin2 ? ? 1)1/2 > 0. It may be observed that the transmitted waves given by
Eq. (18) are the interfacial waves propagating along the plane x = 0. Likewise, in the case
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of ? = ?cr , Snell?s law yields ? = ?/2 and thus the transmitted waves propagate along the
bimaterial interface.
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On the other hand, in the case of k22 < ? 2 , the solution of the governing Eq. (13b) leads to
the following expression for the transmitted waves:
�
�
W2tr (x, y) = B e?(?+p?)x eik?2 y sin ? ,
(19)
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where p?� = k?�(1 + sin2 ?)1/2 . The transmitted waves given by Eq. (19) are similar to the
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interfacial waves described by Eq. (18). For this case, the amplitudes of the reflected and
transmitted waves are obtained as:
k1 �cos ? ? i � (p?� + ?)
,
k1 �cos ? + i � (p?� + ?)
(20a)
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A2 = A1
(20b)
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B = A1 + A 2 .
The scattered wave field is treated separately in the next section.
3.2. Scattered wave field
Up to this point, the incident, reflected, and transmitted wave fields associated with the
problem of interest have been presented. This section deals with the scattered wave field
generated due to the presence of the elliptic cavity which is embedded in the inhomogeneous
half-space. We begin by considering the equation of motion associated with the scattered
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field. In view of Eqs. (9), (10a), (11), and (12) we may write the equation of motion
pertinent to the scattered waves within both the homogenous and inhomogeneous halfspaces as:
if k22 > ? 2 ,
(?2 ? k?� )V2sc = 0,
if k22 < ? 2 ,
(21a)
(21b)
(21c)
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(?2 + k?22 )V2sc = 0,
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(?2 + k12 )W1sc = 0,
The continuity of the pertinent shear stress component, associated with the scattered field,
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across the bimaterial interface and the cavity?s wall, respectively, gives:
sc
sc
?xz2
? ?xz1
= 0,
on
x = 0,
tr
sc
??z2
+ ??z2
= 0,
on
? = ?0 .
(22a)
(22b)
D
sc
, ? = 1, 2 are the stresses induced by the scattered waves within
In the above equations, ?xz?
TE
tr
sc
the domain ?, ??z2
and ??z2
are the stresses induced within the inhomogeneous medium
(? = 2) by the transmitted and scattered waves, respectively. With the aids of Eqs. (1a),
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(8c), and (8d) the stress components appearing in Eqs. (22) can be readily expressed in
to:
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terms of the appropriate wave fields. Therefore, Eqs. (22a) and (22b), respectively, reduce
?W2sc
?W1sc
? �= 0, on x = 0,
?x
?x
?
W2tr + W2sc = 0, on ? = ?0 ,
??
�
(23a)
(23b)
moreover, the continuity condition associated with the scattered displacement fields across
the bimaterial interface yields:
W2sc ? W1sc = 0,
on
x = 0.
(24)
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Previously Martin [8] considered the current bimaterial problem, but for a circular cavity.
Martin [8] proposed an analytical solution using multipole expansion and constructing an
appropriate set of multipole functions. Since in the current work the cavity is considered to
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be elliptic, at the end the results pertinent to the case of the crack as the limiting case of
the elliptic cavity are also retrieved. It is also beneficial to treat the present boundary value
problem via multipole expansion. To this end, let the functions W1sc and W2sc be expressed
=
X
a?n
?,n
W2sc =
X
a?n
?,n
if
?
?
?
???�n
if
k22 < ? 2 ,
?
?
?
?
????n
if
k22 > ? 2 ,
?
?
?
???�n
if
n
k22 > ? 2 ,
x < 0,
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W1sc
?
?
?
?
????
x > 0,
D
?
?
?
(3)
� ,
?
?e??x M?n
(?; ?q?�)e?n (?; ?q?�) + ??
n
x > 0,
(26a)
x > 0,
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?
?
?
� ?,
???
n
(25b)
x < 0.
EP
??�n =
TE
?
?
?
(3)
??x
?
?
?e
M?n (?; q?2 )e?n (?; q?2 ) + ??n ,
?
?
?
????n ,
(25a)
k22 < ? 2 ,
in which
???n =
SC
as:
(26b)
x < 0,
where q?2 = k?22 d2 /4 > 0 and q?�= k?� d2 /4 > 0. Moreover, the following shorthand notions
have been employed and will be used in the subsequent developments:
X
?,n
=
?
XX
,
?=c,s n=0
(j)
(?; q) = M c(j)
Mcn
n (?; q),
ecn (?; q) = cen (?; q),
(j)
Msn
(?; q) = M s(j)
n (?; q), j = 1, 3,
esn (?; q) = sen (?; q).
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In the above equations cen and sen are known as the Angular Mathieu functions, while
(1)
(3)
(1)
(3)
Mcn , Mcn , Msn , and Msn are the Radial Mathieu functions. a?n are unknown constants.
sen (?; q?2 ) =
r=0
?
X
0
Anr (q?2 ) cos r?,
Brn (q�) sin r?,
r=1
?
X
cen (?; ?q?�) =
0
(?1)
n+r
2
Arn (q?�) cos r?,
(?1)
n+r
2
Brn (q?�) sin r?,
r=0
sen (?; ?q?�) =
?
X
0
(27a)
(27b)
(27c)
(27d)
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r=1
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cen (?; q?2 ) =
?
X
0
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Angular Mathieu functions are given by:
where the prime over the summation sign indicates that the summation is only over the even
values of r when n is even, and over the odd values of r when n is odd. Arn (q?�) ? Anr (q?�)
and Brn (q?�) ? Brn (q?�) when n is even, and Arn (q?�) ? ?Brn (q?�) and Brn (q?�) ? ?Anr (q?�) when
D
n is odd. Moreover, the coefficients Anr (q?2 ), Brn (q?2 ), Anr (q?�), and Brn (q?�) are determined
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by satisfying the recursion relationships which can be found in Ivanov [12]. As can be
seen from Eqs. (25), the scattered wave field is expanded in terms of the summation over
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multipole functions ???n and ??�n for the cases of k22 > ? 2 and k22 < ? 2 , respectively. We
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require that ???n and ???n satisfy the equations of motion (21a) and (21b), respectively. ???n
and ???n are to be chosen so that ???n satisfies the interface conditions along the bimaterial
� ? and ??
� ? must satisfy Eqs. (21a)
interface (x = 0), Eqs. (23a) and (24). Similarly, ??
n
n
and (21c), respectively, and are to be chosen so that ??�n satisfies the interface conditions
� ? , and ??
� ? must satisfy the far-field radiation and
(23a) and (24). Moreover, ???n , ???n , ??
n
n
regularity conditions. Although, the use of elliptic coordinates is convenient for handling
the boundary condition on the elliptic cavity, it is inconvenient when trying to impose the
pertinent boundary conditions along the interface between the two half-spaces, x = 0. To
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overcome this difficulty, we convert from elliptic coordinates to Cartesian coordinates by
using an integral representation. This leads to the following integral representation for
(3)
(3)
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M?n (?; q?2 )e?n (?; q?2 ) and M?n (?; q?�)e?n (?; q?�) (Ivanov [12]):
Z ?
in + 2 ?i? ?ik?2 (x?b) cos t ?ik?2 y sin t
(3)
M?n (?(x, y); q?2 )e?n (?(x, y); q?2 ) =
e
e
? ? ?2 +i?
� e?n (t ? ?; ?q?�) dt,
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� e?n (t ? ?; q?2 ) dt,
x < b, |y| < ?,
Z
(?1)n +i? k?�(x?b) cos t k?�y sin t
(3)
M?n (?(x, y); q?�)e?n (?(x, y); q?�) =
e
e
2i
?i?
x < b, |y| < ?,
(28a)
(28b)
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which can conveniently be used along the interface x = 0. Subsequently, the following
EP
TE
D
� ? , and ??
� ? are considered:
integral representations for ???n , ???n , ??
n
n
?
Z
in + 2 ?i?
???n (x, y) =
A?(t)eik?2 x cos t eik?2 b cos t?ik?2 y sin t e?n (t ? ?; q?2 ) dt, x > 0,
(29a)
? ? ?2 +i?
Z ?
in + 2 ?i?
�
?
??n (x, y) =
B?(t)ex?(t) eik?2 b cos t?ik?2 y sin t e?n (t ? ?; q?2 ) dt, x < 0,
(29b)
? ? ?2 +i?
Z
�
�
�
(?1)n +i? �
?
�
??n (x, y) =
A?(t)e?k?2 x cos t e?k?2 b cos t+k?2 y sin t e?n (t ? ?; ?q?�) dt, x > 0, (29c)
2i
?i?
n Z +i?
�
�
�
� ? (x, y) = (?1)
� (t)ex?(t)
??
B?
e?k?2 b cos t+k?2 y sin t e?n (t ? ?; ?q?�) dt, x < 0. (29d)
n
2i
?i?
� (t) are the unknown functions to be deterIn the above equations, A?(t), B?(t), A?�(t), and B?
AC
C
mined through the enforcement of the pertinent boundary conditions along the bimaterial
interface, x = 0. By the consideration of Eqs. (21a), (25a), (25b) (26a), (26b), (29b), and
(29d), one can readily show that:
� 2 (t) = k? 2 sin2 t ? k 2 ,
?
2
1
(30a)
� 2 (t) = ?k?�sin2 t ? k 2 .
?
2
1
(30b)
�
�
Depending on the values of k?2 , k?�, k1 , and t, ?(t)
and ?(t)
may be pure imaginary or
� ? must be bounded as x ? ?? and thus
real. It should be emphasized that ???n and ??
n
15
ACCEPTED MANUSCRIPT
the integrands in Eqs. (29b) and (29d) must be bounded for all values of t. Therefore,
� )] ? 0 and, moreover, to ensure that the scattered waves propagate
� )], Re[?(?
Re[?(?
�
�
away from the bimaterial interface, it is required that Im[?(t)],
Im[?(t)]
< 0.
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PT
In view of Eqs. (23a), (24)-(26), (28), and (29), the following expressions for A?(t), B?(t),
� (t) are found:
A?�(t), and B?
�
�?(t)
+ � ? + i k?2 � cos t
,
�
�?(t)
+ � ? ? i k?2 � cos t
2 i k?2 � cos t
,
�
�?(t)
+ � ? ? i k?2 � cos t
�?(t) + � ? ? k?�� cos t
�
A?(t) = ?
,
�
�?(t)
+ � ? + k?�� cos t
2 k?�� cos t
� (t) =
.
B?
�
�?(t)
+ � ? + k?�� cos t
M
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B?(t) = ?
SC
A?(t) = ?
(31a)
(31b)
(31c)
(31d)
It should be noted that the case of k22 > ? 2 has yielded the expressions (31a) and (31b) for
D
A?(t) and B?(t), while the consideration of the case k22 < ? 2 has led to the expressions for
TE
� (t) given by Eqs. (31c) and (31d), respectively.
A?�(t) and B?
EP
Now, let
(32a)
�
� (i k?� sin t),
�?(t)
+ � ? + k?�� cos t ? H?
2
(32b)
AC
C
�
�?(t)
+ � ? ? i k?2 � cos t ? H?(k?2 sin t),
where
H?(??) = �
� (??�) = �
H?
1
q
�
q
(33a)
q
q
0
0
�
2
2
?? ? k1 + �? + �??�+ k?� .
(33b)
?
k12
�
?? 2 ? k?22 ,
?? 2
+
?+
� (??�) have only two zeros each, �? and �?� , respectively,
It can be shown that H?(??) and H?
0
0
where ??0 and ??�are real and positive. Therefore, H?(??) has two simple poles, ??/2 + i t?0
16
ACCEPTED MANUSCRIPT
and ?/2 ? i t?0 on the contour of integration, where t?0 = cosh?1 (??0 /k?2 ), Fig. 3(a). Similarly,
� (??�) has simple poles ?i t?� and i t?� on the contour of integration, where t?� = sinh?1 (??� /k?� ),
H?
0
0
0
0
2
Fig. 3(b).
(b)
- p2 + i t0
Im(t)
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PT
Im(t)
(a)
i t0
2
- p2
Re(t)
0
p -i t
0
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2
Re(t)
SC
p
0
-i t0
??
Figure 3: Contour of integration and the simple poles pertinent to the integral expressions of (a) f痬n
and
D
??
(b) f痬n
.
TE
It remains to determine the coefficients a?n appeared in Eqs. (25) by imposing the boundary
condition along the cavity?s wall, Eq. (23b). To this end, at first, the functions W2tr , ???n ,
EP
� ? are represented in elliptic coordinates. Thus, W tr which is given by Eqs. (16) and
and ??
2
n
AC
C
(19) when k22 > ? 2 and k22 < ? 2 , respectively, may be written as:
W2tr = Be??x eik?2 a(cosh ? cos ? cos[???]+sinh ? sin ? sin[???])
= 2 B e??x
X
(1)
im M?m
(?; q?2 )e?m (?; q?2 )e?m (? ? ?; q?2 ),
k22 > ? 2 ,
(34a)
?,m
�
�
W2tr = Be?(?+p?)x eik?2 a sin ?(sinh ? sin ? cos ?+cosh ? cos ? sin ?)
�
= 2 B e?(?+p?)x
X
(1)
im M?m
(?; q? )e?m (?; q? )e?m (?/2 ? ?; q? ),
k22 < ? 2 ,
(34b)
?,m
where the substitution B = B exp[ik?2 b cos ?] has been used and q? = (k?�sin ?)2 d2 /4. Fur-
17
ACCEPTED MANUSCRIPT
thermore, Eqs. (34) may be easily reduced to:
W2tr (?, ?) = 4 B? e??b
XX
(1)
(?; q?2 )M?(1)
(R? )p im ?? M?m
p (?; ?q? )
?,m ?,p
� e??b
W2tr (?, ?) = 4 B?
XX
k22 > ? 2
RI
PT
� e?m (?; q?2 )e? p (?; ?q? )e?m (? ? ?; q?2 )e? p (?; ?q? ),
(35a)
(1)
(R? )p im ?? M?m
(?; q? )M?(1)
p (?; ?q? )
?,m ?,p
k22 < ? 2 .
(35b)
SC
� e?m (?; q? )e? p (?; ?q? )e?m (?; q? )e? p (?; ?q? ),
� = B exp[?p?痓], q = ? 2 d2 /4, q = (? + p?�)2 d2 /4, ? = 1 if ? ? c
In the above relations, B?
?
?
?
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and ?? = ?1 if ? ? s, R? = 1 if ? < 0 and R? = ?1 if ? > 0, and R? = 1 if ? + p?� < 0 and
R? = ?1 if ? + p?� > 0. In arriving at the expressions (35a) and (35b), the following series
representation of e?%x for constant % has been employed (Ivanov [12]):
e?%x = 2e?%b
X
(R% )p ?? M?(1)
p (?; ?q% )e? p (?; ?q% )e? p (?; ?q% ),
D
?,p
(36)
TE
in which R% = 1 if % < 0, R% = ?1 if % > 0, and q% = %2 d2 /4. In a similar way, utilizing the
EP
relations:
eik?2 x cos t?ik?2 y sin t = 2eik?2 b cos t
�
�
�
X
(37a)
?,m
e?k?2 x cos t+k?2 y sin t = 2e?k?2 b cos t
AC
C
(1)
im ?? M?m
(?; q?2 )e?m (?; q?2 )e?m (t + ?; q?2 ),
X
(1)
(?1)m ?? M?m
(?; ?q?�)e?m (?; ?q?�)e?m (t + ?; ?q?�),
?,m
(37b)
together with Eqs. (29a) and (29c), we obtain:
???n (?, ?) = 2
X
??
(1)
im ?? f痬n
M?m
(?; q?2 )e?m (?; q?2 ),
(38a)
??
(1)
(?; ?q?�)e?m (?; ?q?�),
(?1)m ?? f痬n
M?m
(38b)
?,m
� ? (?, ?) = 2
??
n
X
?,m
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ACCEPTED MANUSCRIPT
where
in
??
=
f痬n
?
+ ?2 ?i?
Z
A?(t)e2ik?2 b cos t e?m (t + ?, q?2 )e?n (t ? ?; q?2 )dt,
(39a)
? ?2 +i?
Z
+i?
�
A?�(t)e?2k?2 b cos t e?m (t + ?, ?q?�)e?n (t ? ?; ?q?�)dt.
(39b)
?i?
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PT
(?1)n
??
=
f痬n
2i
??
??
A close scrutinization of the integral expressions for f痬n
and f痬n
reveals that, by taking the
following considerations into account an accurate evaluation of the corresponding integrals
+ ?2 ?i?
Z
? ?2 +i?
??
f痬n
=
Z
i?
?i?
? ?2
+ ?2 ?i?
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??
I痬n
(t)
dt
? ?2 +i? H?(k?2 sin t)
Z ??
Z +?
Z
2
2
=
. . . dt +
. . . dt +
??
f痬n
=
SC
is possible:
. . . dt,
Z i?
Z 0
??
(t)
I痬n
� (i k?� sin t) dt = ?i? . . . dt + 0 . . . dt,
H?
2
(40a)
+ ?2
(40b)
� (ik?� sin t) are given by Eqs. (32a) and (32b), respectively. It
where H?(k?2 sin t) and H?
2
D
??
??
(t) are analytic functions. H?(k?2 sin t) has one simple
(t) and I痬n
should be noted that I痬n
TE
pole, ??/2 + i t?0 in the interval (??/2 + i?, ??/2] and one simple pole, ?/2 ? i t?0 in the
EP
� (i k?� sin t) has one simple pole, ?i t?� in the interval (?i?, 0]
interval [?/2, ?/2 ? i?). H?
2
0
and one simple pole, i t?�in the interval [0, i?). Thus, the integrals associated with the
AC
C
intervals (??/2+i?, ??/2], [?/2, ?/2?i?), (?i?, 0], and [0, i?) can be readily calculated
numerically as:
Z ??
??
??
2
I痬n
(t)
I痬n
(t)
I�? (??/2 + i t?0 )
dt = ?
dt ? i ? mn
,
Z?(??/2 + i t?0 )
? ?2 +i? H?(k?2 sin t)
? ?2 +i? H?(k?2 sin t)
Z ? ?i? �?
Z ? ?i? �?
2
2
I�? (?/2 ? i t?0 )
Imn (t)
Imn (t)
dt = ?
dt + i ? mn
,
?
?
Z?(?/2
?
i
t?
)
H?(
k?
sin
t)
H?(
k?
sin
t)
0
2
2
2
2
Z 0
Z 0
??
??
??
�
Imn (t)
I痬n
(t)
I痬n
(?i t?�)
dt
=
?
dt
?
i?
,
� �
� �
Z?� (?i t?�)
?i? H? (i k?2 sin t)
?i? H? (i k?2 sin t)
Z i?
Z i?
??
??
??
I痬n
(t)
I痬n
(t)
I痬n
(i t?�)
dt
=
?
dt
+
i?
,
� (i k?� sin t)
� (i k?� sin t)
Z?� (i t?�)
H?
H?
0
0
2
2
Z
? ?2
19
(41a)
(41b)
(41c)
(41d)
where
R
?
ACCEPTED MANUSCRIPT
. . . d? denotes the Cauchy principal value of the integral, and:
(42)
h
i
� ?1 (t) .
Z?� (t) = ?k?�sin t � + k?��cos t ?
(43)
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� ?1 (t) ,
Z?(t) = k?2 sin t i � + k?2 �cos t ?
Employing Eqs. (25b), (26), and (38) along with the expansion relation (36), we find the
following expressions for the scattered displacement pertinent to the functionally graded
a?n
(
X
?,n
?,p
(3)
(R? )p ?? M?n
(?; q?2 ) M?(1)
p (?; ?q? )
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W2sc (?, ?) = 2e??b
X
SC
half-space:
� e?n (?; q?2 ) e? p (?; ?q? ) e? p (?; ?q? )
+2
XX
??
(3)
(R? )p im ??? f痬n
M?m
(?; q?2 ) M?(1)
p (?; ?q? )
?,m ?,p
� e?m (?; q?2 ) e? p (?; ?q? ) e? p (?; ?q? )} ,
X
a?n
(3)
(R? )p ?? M?n
(?; ?q?�) M?(1)
p (?; ?q? )
?,p
TE
?,n
(44a)
D
W2sc (?, ?) = 2e??b
(
X
k22 > ? 2 ,
� e?n (?; ?q?�) e? p (?; ?q? ) e? p (?; ?q? )
XX
EP
+2
??
(3)
(R? )p (?1)m ??? f痬n
M?m
(?; ?q?�) M?(1)
p (?; ?q? )
AC
C
?,m ?,p
� e?m (?; ?q?�) e? p (?; ?q? ) e? p (?; ?q? )} ,
k22 < ? 2 , (44b)
where ??? = 1 if ? = ? and ??? = ?1 if ? 6= ? . Subsequently, utilizing Eqs. (23b), (35a),
(35b), (44a), and (44b) then the boundary condition along the free surface of the elliptic
20
ACCEPTED MANUSCRIPT
cavity becomes:
(
X
X
? (3)
a?n
(R? )p ??
M?n (?, q?2 )M?(1)
p (?; ?q? )
??
?,n
?,p
+2
(3)
?? ?
(R? )p im ??? f痬n
M?m (?; q?2 )M?(1)
p (?; ?q? )
??
?,p
XX
?,m
� e?m (?; q?2 )e? p (?; ?q? )e? p (?; ?q? )}
XX
(R? )p im ??
?,m ?,p
? (1)
M?m (?; q?2 )M?(1)
p (?; ?q? )
??
SC
+ 2 B?
RI
PT
� e?n (?; q?2 )e? p (?; ?q? )e? p (?; ?q? )
on
? = ?0 ,
(45)
on
? = ?0 ,
(46)
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� e?m (?; q?2 )e? p (?; ?q? )e?m (? ? ?; q?2 )e? p (?; ?q? ) = 0,
for the case of k22 > ? 2 , and
(
X
X
? (3)
a?n
(R? )p ??
M?n (?, ?q?�)M?(1)
p (?; ?q? )
??
?,n
?,p
(3)
?? ?
(R? )p im ??? f痬n
M?m (?; ?q?�)M?(1)
p (?; ?q? )
??
?,p
XX
?,m
TE
+2
D
� e?n (?; ?q?�)e? p (?; ?q? )e? p (?; ?q? )
� e?m (?; ?q?�)e? p (?; ?q? )e? p (?; ?q? )}
? (1)
M?m (?; q? )M?(1)
p (?; ?q? )
??
EP
�
+ 2 B?
XX
(R? )p im ??
?,m ?,p
AC
C
� e?m (?; q? )e? p (?; ?q? )e?m (? ? ?; q? )e? p (?; ?q? ) = 0,
for the case of k22 < ? 2 . It should be emphasized that the Angular Mathieu functions
e?n (?; q?2 ) and e? p (?; ?q? ) appearing in Eq. (45) and the functions e?n (?; ?q?�) and e? p (?; ?q? )
appearing in Eq. (46) are not orthogonal to each other since q?2 6= ?q? and q?�6= q? . Therefore, in order to calculate the unknown coefficients an approximate method is used to replace
these functions by their series representation provided by Eqs. (27). Then the orthogonality
conditions associated to sin r? and cos r? are employed.
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ACCEPTED MANUSCRIPT
The dynamic stress concentration factor, Sd along the boundary of the cavity, ? = ?0 is
defined as:
sc
tr
|
+ ??z2
|??z2
,
?0
on
? = ?0 ,
(47)
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Sd =
tr
in which ?0 is the maximum value of |??z2
|. It is worth mentioning that the solution for
the special case where the elliptic cavity is replaced by a crack in the functionally graded
SC
half-space will be obtained by taking the limit as a2 ? 0. In this case Sd is redefined as the
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dynamic stress intensity factor, Kd .
4. Verification of the current formulation
In order to verify the current formulation the solution for the special case where the
elliptic cavity is replaced by a circular one in the exponentially graded half-space is obtained
D
by letting a2 ? a1 . The normalized absolute values of the stress component |??z |/(�A1 k1 )
TE
along the boundary of the circular cavity for different values of the normalized distance
between the center of the cavity and the bimaterial interface b/a1 = 1.5, 3, 5, and 8 are
EP
recovered in the present work and compared with the result of Liu et. al. [9] in Fig. 4. In
AC
C
arriving at this plot it is assumed that � = �, ?02 = ?1 , ? = 0, ?a1 = 0.2, and k1 a1 = 2.
It is observed that the result obtained as a special case of the current work is in complete
agreement with the result reported by Liu et. al. [9].
5. Numerical results
Based on multipole expansion method, a formulation for the anti-plane scattering of SHwaves by an elliptic cavity/crack beneath the interface of two perfectly bonded half-spaces,
one made of a homogeneous material and the other made of a FGM, has been developed in
22
SC
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ACCEPTED MANUSCRIPT
Figure 4: Regeneration of the result of Liu et. al. [9] for the normalized magnitude of the stress component
M
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??z /(�A1 k1 ) along the cavity?s wall for different values of the normalized distance b/a1 = 1.5, 3, 5, 8 as a
special case of the present formulation. In these plots � = �, ?02 = ?1 , ? = 0, ?a1 = 0.2, and k1 a1 = 2.
Section 3. The scattered fields pertinent to an embedded elliptic cavity and the dynamic
stress intensity factor pertinent to an embedded crack are examined in Sections 5.1 and 5.2,
5.1. Elliptic cavity
TE
D
respectively.
EP
In this section, several descriptive examples are provided to examine the elastodynamic
AC
C
response of the defined bimaterial system in which one of the half-spaces is made of a
functionally graded material and contains an arbitrarily oriented elliptic cavity. In particular, the displacement field along the bimaterial interface, the stress field along the cavity
periphery just inside the inhomogeneous medium, and the dynamic stress concentration
factor will be examined in Sections 5.1.1, 5.1.2, and 5.1.3, respectively. The displacement
is normalized by the amplitude of the incident SH-wave, A1 and the stresses within the
inhomogeneous medium are normalized by � A1 k2 . Moreover, in all the examples of this
section, we assume that � = �and ?02 = ?1 .
23
ACCEPTED MANUSCRIPT
5.1.1. Displacement field along the bimaterial interface
The examination of the effects of the incident angle, incident wave number, and the
distance between the center of the cavity and the bimaterial interface on the variation
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of the z?component of the displacement field, W2 (0, y) along the bimaterial interface is
of particular interest. Throughout the examples given in this section, it is assumed that
?a1 = 0.2 and ? = ?/2. In order to examine the influence of the incident angle, ? on the
SC
displacement along the interface between the two half-spaces, we assume that the normalized
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incident wave number is fixed at k1 a1 = 1, the normalized distance is b/a1 = 1.5, and the
aspect ratio is a2 /a1 = 0.4. Under these conditions, for different values of the incident angle
? = 0, ?/24, ?/6, and ?/3 the normalized absolute value of the displacement component
|W2 |/A1 ? |W1 |/A1 along the bimaterial interface for ?20 6 y/a1 6 20 is plotted in Fig.
D
5. It is seen that, when the incident waves propagate along the directions making an angle
TE
of ? = ?/24, ?/6, ?/3 with the positive x?axis, the distribution of |W2 |/A1 is asymmetric
with respect to y = 0. In contrast, for the incident waves propagating along the positive
EP
x?axis (i.e., ? = 0), |W2 |/A1 is symmetric with respect to the plane y = 0, as expected.
It can also be observed that, for the values of y/a1 within the range of ?20 to 0, the
AC
C
displacement distributions show notable oscillations along the interface for all four cases of
? = 0, ?/24, ?/6, ?/3. It is evident that, by increasing the incident angle, the displacement
becomes more oscillatory. On the other hand, when 0 6 y 6 20 the displacement curves
pertaining to two cases of ? = ?/6 and ?/3 do not show much oscillatory behavior, as
compared to cases of ? = 0 and ?/24. In all the considered cases, the maximum values of
|W2 |/A1 occur in the vicinity of the origin.
Next, by holding ? = 0, b/a1 = 1.5, and a2 /a1 = 0.4 fixed, the influence of the incident wave
24
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y
x
a1
y
o
b
a2
SC
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a
x
Figure 5: Variation of the normalized displacement moduli, |W2 |/A1 along the bimaterial interface for
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different values of the incident angle, ?.
number on the distribution of |W2 |/A1 along the bimatrial interface can be examined from
Fig. 6. It is observed that, the normalized displacement |W2 |/A1 pertaining to the case
of k1 a1 = 0.5 smoothly varies along the interface. Whereas, the normalized displacements
D
stemming from the incident wave numbers k1 a1 = 1, 2, 5 show notable oscillations. The
TE
frequency of these oscillations strongly depends on the wave number. In fact, the increase
EP
in the incident wave number, causes the displacement |W2 |/A1 to oscillate with higher
frequency. Again, the peak of |W2 |/A1 occurs in the neighborhood of the origin. For the
AC
C
case of k1 a1 = 0.5 it is exactly at the origin.
Fig. 7 compares the variation of the normalized displacement moduli, |W2 |A1 associated
with SH-waves having incident angle ? = 0 and normalized wave number k1 a1 = 2 for
various values of b/a1 = 1.5, 5, 10, 20. It should also be mentioned that, a2 /a1 = 0.4 is kept
constant in this figure. It is seen that, the displacement curves suffer strong oscillations
for all the values of the normalized distance, b/a1 between the center of the cavity and the
bimaterial interface. When the cavity is close to the interface (b/a1 = 1.5) the displacement
25
ACCEPTED MANUSCRIPT
y
x
a1
y
x
o
a2
SC
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b
Figure 6: Variation of the normalized displacement moduli, |W2 |/A1 along the bimaterial interface for
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different values of the normalized incident wave number, k1 a1 .
oscillates with higher amplitude and lower frequency as compared with the case in which
the cavity is placed farther away from the bimaterial interface, b/a1 = 10 or 20. This clearly
demonstrates the interaction between the cavity and the bimaterial interface.
D
y
x
a1
y
TE
o
a2
AC
C
EP
b
x
Figure 7: Variation of the normalized displacement moduli, |W2 |/A1 along the bimaterial interface for
different values of the normalized distance between the center of the elliptic cavity and the interface, b/a1 .
The plot of the displacement |W2 |/A1 along the bimaterial interface for various aspect ratios
of the elliptic cavity a2 /a1 = 1, 0.6, 0.3, and 0.1 are compared in Fig. 8. In arriving at these
curves, it is assumed that k1 a1 = 0.5, b/a1 = 1.5 and ? = 0. It is observed that |W2 |/A1
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attains its maximum value at y = 0 for different values of a2 /a1 . Decreasing the aspect ratio
causes the peak to decrease. Moreover, for the case of a2 /a1 = 1 where the elliptic cavity
collapses to a circular cavity, the plot of |W2 |/A1 is in agreement with the corresponding
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plot obtained by Liu et. al. [9].
y
x
a1
y
b
x
a2
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Figure 8: Variation of the normalized displacement moduli, |W2 |/A1 along the bimaterial interface for
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different values of the aspect ratio, a2 /a1 .
5.1.2. Stress distribution along the cavity?s wall
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In this section, the effects of the dimensionless parameters ?a1 , b/a1 , and a2 /a1 on the
distribution of the absolute value of the stress component normalized as |??z2 |/(� A1 k2 )
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along the cavity-FG matrix interface (? = ?0 ) is studied. Throughout this section, it is
assumed that the incident waves propagate along the positive x?axis, ? = 0 and moreover
? = 0. In order to examine the effect of the dimensionless inhomogeneity constant, ?a1 we
keep k1 a1 = 1, b/a2 = 2, and a2 /a1 = 0.8 fixed. Then, the distributions of |??z2 |/(� A1 k2 )
along the cavity?s wall for different values of ?a1 = ?0.3, ?0.2, 0, 0.1, 0.2, 0.3 are plotted in
Fig. 9. It is observed that, the inhomogeneity parameter ?a1 has significant effect on the
stress distribution. As it can be seen, the increase in the value of the inhomogeneity constant
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causes |??z2 | to increase. Note that the variations of the normalized stress component for
different values of ?a1 follow nearly a similar trend.
y
y
a2
x x
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a1
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Figure 9: Distribution of the normalized magnitude of the stress component, |??z2 |/(� A1 k2 ) along the
cavity?s wall for different values of the normalized inhomogeneity constant, ?a1 .
To assess the effect of the distance between the center of the cavity and the bimaterial
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interface on the stress along the cavity?s wall, we hold k1 a1 = 2, ?a1 = 0.2, and a2 /a1 =
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0.5 fixed. Subsequently, for various values of b/a1 = 1.5, 3, 4.5, and 6 the corresponding
distributions of |??z2 |/(� A1 k2 ) along the cavity-FG matrix interface are plotted in Fig. 10.
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It is observed that the normalized magnitude of the stress component increases with b/a1 .
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It is noteworthy to mention that the distributions of |??z2 |/(� A1 k2 ) along the cavity?s wall
for different values of b/a1 follow a similar trend.
Next, to examine the effect of the aspect ratio, a2 /a1 on the absolute value of the stress
component, |??z2 |/(� A1 k2 ) along the boundary of an elliptic cavity with its center at a
depth of b/a1 = 1.5 beneath the bimaterial interface, we keep k1 a1 = 2 and ?a1 = 0.2 fixed.
Then, for different values of a2 /a1 = 1, 0.8, 0.5, and 0.2 the distributions of the normalized
magnitude of the stress component, |??z2 |/(� A1 k2 ) along the boundary of the cavity are
calculated and plotted in Fig. 11. As it can be seen, the stress distributions pertaining
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y
y
a2
x x
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a1
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Figure 10: Distribution of the normalized magnitude of the stress component, |??z2 |/(� A1 k2 ) along the
cavity?s wall for different values of the normalized distance between the center of the elliptic cavity and the
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interface, b/a1 .
to various values of a2 /a1 have different trends. Moreover, the discrepancy between the
location of the maximum value of the stresses stemming from different aspect ratios is
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apparent.
y
y
a2
x x
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a1
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Figure 11: Distribution of the normalized magnitude of the stress component, |??z2 |/(� A1 k2 ) along the
cavity?s wall for different values of the aspect ratio, a2 /a1 .
5.1.3. Dynamic stress concentration factor
The expression for the dynamic stress concentration factor, Sd for an arbitrarily oriented
elliptic cavity beneath the interface between two perfectly bonded half-spaces one made of
a homogeneous material and the other made of a FGM was derived in Section 3.2. In this
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example the effect of the aspect ratio of the elliptic scatterer, a2 /a1 on the variation of Sd
with the normalized incident wave number, k1 a1 is studied by keeping b/a1 = 1.5, ? = ?/2,
and ? = 0, fixed. For various values of a2 /a1 = 1, 0.8, 0.6, 0.5, and 0.4 the variation of Sd
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at ? = 0 with the normalized wave number, k1 a1 is calculated and shown in Fig. 12(a) for
the inhomogeneity constant ?a1 = 0, and Fig. 12(b) for ?a1 = 0.2. It should be noted that
for the case of ?a1 = 0 both half-spaces become identical and hence the problem collapses
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to an unbounded homogeneous medium containing an elliptic cavity. Among the results
plotted in Figs. 12(a) and 12(b), the curves pertinent to the case of a2 /a1 = 1 corresponds
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to the special case of a circular cavity and are in complete agreement with the classical
results presented in Ghafarollahi and Shodja [10]. From Figs. 12(a) and 12(b), it is seen
that for a given wave number, the increase of the aspect ratio lowers the dynamic stress
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concentration factor at ? = 0.
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5.2. Crack
The examples of this section are devoted to the crack problem. The corresponding
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result is obtained through the consideration of the limiting case of the elliptic cavity as
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a2 ? 0 (?0 ? 0). Using this concept, Shodja and Ojaghnezhad [13] have presented a
general unified treatment of lamellar inhomogeneities. Let Kd and Ks , respectively, define
the dynamic and static stress intensity factors at the crack-tip. For different values of the
normalized inhomogeneity constant, ?a1 = 0, 0.1, 0.15, 0.2, and 0.25 the distributions of
Kd /Ks at ? = 0 as a function of the incident angle, ? are plotted in Fig. 13. It should be
mentioned that in Fig. 13, k1 a1 = 0.5, b/a1 = 1.5, and ? = ?/2 are kept fixed. Moreover, it
is assumed that � = �and ?02 = ?1 . Note that Ks denotes the intensity factor for a crack
located in an unbounded homogeneous medium (?a1 = 0) under static anti-plane loading
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(a)
x
h=0
a1
y
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(a)
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a2
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(b)
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a1
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a2
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(b)
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Figure 12: Variation of the dynamic stress concentration factor, Sd with the normalized wave number, k1 a1
at ? = 0 pertaining to cases of (a) ?a1 = 0 and (b) ?a1 = 0.2, and different values of the aspect ratio,
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a2 /a1 .
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(k1 a1 = 0). It is evident that for a fixed incident angle, ? fixed, the ratio Kd /Ks increases
with ?a1 . It is seen that, the discrepancy between the results pertaining to different values
of ?a1 becomes more notable as the absolute value of the incident angle increases.
Next, by holding k1 a1 = 1, b/a1 = 1, ?a1 = 0.3, and ? = 0 fixed, the variation of
Kd /Ks at ? = 0 with the angle of crack orientation, ? for various values of G = �/� =
?1 /?02 is calculated and plotted in Fig. 14. It is seen that, for a fixed value of ?, Kd /Ks
increases as the normalized material parameter G decreases. Moreover, corresponding to
G = 0.1, 0.5, 1, 2, and 10, M ax(Kd /Ks ) = 1.88, 1.58, 1.43, 1.33, and 1.26 occurs, respectively,
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y
x
h=0
a1
y
x
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Figure 13: Variation of the normalized dynamic stress intensity factor, Kd /Ks at the crack-tip, ? = 0 with
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the incident angle, ? for different values of the normalized inhomogeneity constant, ?a1 .
at ? = 81.3? , 73.5? , 60.0? , 37.2? , and 23.1? .
h=0
j
y
a1
x
y
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x
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Figure 14: Variation of the normalized dynamic stress intensity factor, Kd /Ks at the crack-tip, ? = 0 with
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the orientation parameter, ? for different values of the normalized material parameter, G = �/� = ?1 /?02 .
The variation of Kd /Ks at ? = ? with the normalized wave number, k1 a1 for different
crack orientations, ? = 0, ?/4, ?/3, 5?/12, and ?/2 for two cases of ?a1 = 0 and 0.3 are,
respectively, shown in Figs. 15(a) and (b). In arriving at these figures, it is assumed that
� = �, ?02 = ?1 , b/a1 = 1.5, and ? = 0. When ?a1 = 0 then the problem describes a
crack within an unbounded homogeneous medium. When the crack plane is parallel to the
y?axis, ? = ?/2 then it is subjected to a normally incident SH-wave. The result of this
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problem obtained by the current calculations is in complete agreement with the result of
Mal [14]. It is interesting to note that, the decrease in the orientation parameter, ? causes
the peak of the curves to decrease in both cases of ?a1 = 0 and 0.3.
x
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(a)
y
j
a1
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h=p
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j
j
j
j
j
(a)
x
y
a1
y
j
b
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x
h=p
j
j
j
j
j
(b)
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(b)
Figure 15: Variation of the normalized dynamic stress intensity factor, Kd /Ks at the crack-tip, ? = ? with
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the normalized wave number, k1 a1 pertaining to cases of (a) ?a1 = 0 and (b) ?a1 = 0.3, and different
values of the orientation parameter, ?.
6. Conclusion
Anti-plane scattering of SH-waves by an arbitrarily oriented elliptic cavity/crack located
beneath the interface of two perfectly bonded half-spaces, one made of a homogeneous material and the other made of a FGM is addressed using multipole expansion method. The
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scattered displacement field is expanded by summation over multipole functions which satisfy: (I) the governing equation of motion in each half-space, (II) the continuity conditions
along the interface between the two half-spaces, and (III) the far-field radiation and reg-
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ularity conditions. By satisfying some appropriate boundary conditions peculiar to the
considered problem, the analytical expressions for the pertinent elastodynamic fields are
derived. The dynamic stress concentration factor along the cavity periphery as well as the
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dynamic stress intensity factor at the crack-tips are also calculated. In the section on the
descriptive examples, the effects of several parameters have been examined. More specifi-
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cally, it is realized that the inhomogeneity constant, ?a1 pertinent to the inhomogeneous
medium, the dimensionless wave number and the angle of the incident wave, the distance
between the center of the cavity/crack and the bimaterial interface, and the orientation of
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the cavity/crack have significant effects on the dynamic response of the proposed model.
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References
[1] Ma, L., Wu, L. Z., Zhou, Z. G. (2004). Dynamic stress intensity factors around two
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parallel cracks in a functionally graded layer bonded to dissimilar half-planes subjected
to anti-plane incident harmonic stress waves. International Journal of Engineering Science, 42(2), 187-202.
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[2] Ma, L., Wu, L. Z., Guo, L. C., Zhou, Z. G. (2005). Dynamic behavior of a finite crack
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in the functionally graded materials. Mechanics of materials, 37(11), 1153-1165.
[3] Liang, J. (2007). Scattering of harmonic anti-plane shear stress waves by a crack in functionally graded piezoelectric/piezomagnetic materials. Acta Mechanica Solida Sinica, ,
20(1), 75-86.
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[4] Yang, Y. H., Wu, L. Z., Fang, X. Q. (2010). Non-destructive detection of a circular cav-
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ity in a finite functionally graded material layer using anti-plane shear waves. Journal
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of Nondestructive Evaluation, 29(4), 233-240.
[5] Fang, X. Q., Liu, J. X., Zhang, L. L., Kong, Y. P. (2011). Dynamic stress from a
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subsurface cylindrical inclusion in a functionally graded material layer under anti-plane
shear waves. Materials and structures, 44(1), 67-75.
[6] Gregory, R. D. (1967). An expansion theorem applicable to problems of wave propagation in an elastic half-space containing a cavity. In Mathematical Proceedings of the
Cambridge Philosophical Society, vol. 63, no. 4, pp. 1341-1367. Cambridge University
Press.
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[7] Gregory, R. D. (1970). The propagation of waves in an elastic half-space containing a
cylindrical cavity. Mathematical Proceedings of the Cambridge Philosophical Society,
vol. 67, no. 3, pp. 689-710. Cambridge University Press.
Journal of Applied Mechanics. 76, 031009. 4 pages.
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[8] Martin, P. A. (2009). Scatttering by a cavity in an exponentially graded half-space.
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[9] Liu, Q., Zhao, M., Zhang, C. (2014). Antiplane scattering of SH waves by a circular cavity in an exponentially graded half-space. International Journal of Engineering
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Science, 78, 61-72.
[10] Ghafarollahi, A., Shodja, H. M. (2017) Scattering of SH-waves by a nano-fiber beneath
the interface of two bonded half-spaces within surface/interface elasticity via multipole
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expansion. International Journal of Solids and Structures, 130-131, 258-279.
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[11] Linton, C. M., Thompson, I. (2015). Oblique Rayleigh wave scattering by a cylindrical
cavity. The Quarterly Journal of Mechanics and Applied Mathematics, 68(3), 235-261.
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Springfield.
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[12] Ivanov, E. A. (1970). Diffraction of electromagnetic waves on two bodies, Washington
[13] Shodja, H. M., Ojaghnezhad, F. (2007). A general unified treatment of lamellar inhomogeneities. Engineering Fracture Mechanics, 74(9), 1499-1510.
[14] Mal, A. K. (1970). Interaction of elastic waves with a Griffith crack. International
Journal of Engineering Science, 8(9), 763-776.
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SH-waves scattered by an arbitrarily oriented elliptic cavity/crack
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Multipole expansion of the scattered displacement field
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Exponentially graded half-space with an arbitrarily oriented elliptic cavity/crack
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Dynamic stress intensity factor at the crack-tip
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An arbitrarily oriented elliptic cavity/crack near the interface of a bimaterial
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