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j.compstruct.2018.08.032

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Accepted Manuscript
Dynamic Response of A Functionally Graded Tube Embedded in an Elastic
Medium Due to Sh-Waves
Hasan Faik Kara, Metin Aydogdu
PII:
DOI:
Reference:
S0263-8223(18)31559-9
https://doi.org/10.1016/j.compstruct.2018.08.032
COST 10082
To appear in:
Composite Structures
Received Date:
Revised Date:
Accepted Date:
27 April 2018
20 July 2018
14 August 2018
Please cite this article as: Kara, H.F., Aydogdu, M., Dynamic Response of A Functionally Graded Tube Embedded
in an Elastic Medium Due to Sh-Waves, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.
2018.08.032
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DYNAMIC RESPONSE OF A FUNCTIONALLY GRADED TUBE
EMBEDDED IN AN ELASTIC MEDIUM DUE TO SH-WAVES
Hasan Faik Karaa and Metin Aydogdub*
a
Trakya University, Engineering Faculty, Civil Engineering Department,22030, Edirne ,TURKEY
b
Trakya University, Engineering Faculty, Mechanical Engineering Department,22030,
Edirne ,TURKEY
Abstract: Dynamic response of a cylindrical tube surrounded by an unbounded elastic
medium due to plane harmonic SH-Waves is studied. A two-dimensional mathematical model
is considered. Cylindrical coordinates are used for convenience. The surrounding medium is
assumed to be homogeneous, isotropic and linear elastic. The tube is assumed to be made of
linear elastic Functionally Graded Materials (FGMs) such that shear modulus and shear
wave velocity are assumed to change linearly from inner surface to outer surface. Material
properties are constant along circumferential direction. It is assumed that the inner surface of
the tube is traction-free and there is a welded contact between the tube and the surrounding
medium. Governing equations are slightly different in the tube region and the unbounded
region. Both of the governing equations are solved by applying Finite Fourier Transform in
circumferential direction. The exact solution series are presented in terms of Fourier-Bessel
series in the unbounded region and power series in the tube region. The presented numerical
results show that when the incoming wave lengths decrease, shear stresses at the tube
increase significantly. It was shown that for the shorter incoming wave lengths, tubes made of
FGMs are subjected to smaller shear stresses compared to the tubes homogeneously made of
outer surface material of the FG cases.
Keywords: Cylindrical Tubes, Functionally Graded Materials, SH-Waves
Corresponding author, metina@trakya.edu.tr, 902842251295
1
I.
Introduction
Functionally graded materials (FGMs) were invented by a group of Japanese scientists in
1984. FGMs are two-component composites in which that material properties change
gradually from one component to another in the desired direction. This structure of FGMs
enables to benefit from different advantages of two components. Therefore, FGMs are highly
convenient for many engineering applications. For example, it is possible to produce a
cylindrical tube that can withstand extremely high temperatures without compromising
structural integrity. Because of many other additional advantages, FGMs are widely used in
many industries such as civil engineering, automotive and aerospace applications.
Wave motion problems in FGMs are considered by many researchers to reveal the
dynamic behavior of FGMs. Han et al. [1] investigated elastic waves in FGMs by using
quadratic layer element method. A hybrid numerical method is presented by Han et al. [2] for
analyzing transient waves in a cylinder made of FGM.
SH-wave, also called secondary horizontal wave, is a type of elastic body wave together
with P and SV-waves. Polarization of SH-waves (direction of particle motion) is
perpendicular to the two-dimensional model. Scattered waves have also the same polarization
and therefore, unlike P and SV-waves, SH-waves are uncoupled. As a result, only SH-waves
needs to be considered in the complete analysis [29]. A computational method is presented by
Han and Liu [3] to investigate SH-waves in functionally graded material (FGM) plates. Han et
al. [4] studied characteristics of waves in a functionally graded cylinder. Shakeri et al. [5]
studied vibration and radial wave propagation velocity in functionally graded thick hollow
cylinder. Daros [6] presented a fundamental solution for SH-waves in a class of
inhomogeneous anisotropic media. Hu et al. [7] studied the scattering of shear waves from a
circular cavity buried in a semi-infinite slab of functionally graded materials. Scattering of SH
waves by a cavity in an exponentially graded half-space is analysed by Martin [8]. Baron [9]
studied propagation of elastic waves in an anisotropic functionally graded hollow cylinder in
vacuum. Wave propagation of functionally graded material plates in thermal environments is
studied by Sun and Luo [10]. Sun and Luo [11] also studied wave propagation and transient
response of functionally graded material circular plates under a point impact load. A dynamic
solution for the propagating viscoelastic waves in functionally graded material plates
subjected to stress-free conditions is presented by Yu et al. [12]. By using Ritz method,
2
Uymaz et al. [13] analysed vibration of FGM plates with in-plane material inhomogeneity.
Daros [14] developed a Green’s function for SH-waves in inhomogeneous anisotropic elastic
solid with power-function velocity variation. Antiplane scattering of SH-Waves by a circular
cavity in an exponentially graded half space is studied by Liu et al. [15]. Elastic wave
propagation in functionally graded circular cylinders is studied by Dorduncu et. al. [16]. Filiz
and Aydogdu [17] analyzed wave propagation in functionally graded nanotubes conveying
fluid. By using a nonlocal strain gradient theory, Li et al. [18] analysed flexural wave
propagation in small-scaled functionally graded beams. Qiao et al. [19] presented an
analytical solution of characteristics of elastic waves in FGM spherical shells. Yang et al. [20]
studied scattering of out-plane wave by a circular cavity near the right-angle interface in the
exponentially inhomogeneous media. Aminipour at al. [21] developed a new model for wave
propagation in functionally graded anisotropic doubly-curved shells. Propagation of
axisymmetric waves in pressurized functionally graded elastomeric hollow cylinders is
studied by Wu et al. [22].
There are also many analytical studies on scattering and diffraction of plane harmonic SHWaves by homogeneous cylindrical tubes, pipes or tunnels surrounded by homogeneous
elastic mediums. By using wave function expansion method and image technique, an
analytical closed-form solution of dynamic response of cylindrical tunnels to incident SHWaves is presented by Lee and Trifunac [23]. By using a similar approach, Balendra and
Thambiratnam [24] presented the steady-state solution for two parallel underground tunnels of
circular cross-section subjected to incident plane harmonic SH-waves in closed form. A
closed-form solution of a cylindrical tunnel embedded in a quarter-space subjected to plane
harmonic SH-Waves is presented by Kara [25]. An analytical solution of scattering of plane
SH Waves induced by a semi-cylindrical canyon with a subsurface circular lined tunnel is
obtained by Gao et al. [26]. Although analytical studies of response of cylindrical tubes
surrounded by elastic mediums due to SH-waves are highly extensive, similar analytical
studies that consider tubes made of FGM do not exist. A possible reason may be that the
fundamental analytical solutions of cylindrical tubes subjected to SH-Waves had already been
developed before the invention of FGMs. Because the application areas of FGMs have been
expanding day by day since their invention, it is believed that developing an analytical
mathematical model to analyze the dynamic response of cylindrical tubes made of FGM to
SH-Waves would be a substantial contribution to the literature.
3
The aim of this study is to present an analytical closed-form solution of dynamic response
of a cylindrical tube made of FGM surrounded by an unbounded medium due to plane
harmonic SH-waves. Numerical results are also presented to demonstrate the dynamic
behavior.
4
II.
Model and Formulation of the Problem
The cross-section of the two-dimensional model considered in this paper is shown in
Figure 1. There is a cylindrical tube with inner radius ai and outer radius ao. The tube is
surrounded by an unbounded, homogeneous, isotropic and linear elastic medium which is
excited by out-of-plane harmonic incident SH-Waves with unit amplitude and  angular
frequency in +x direction. The surrounding medium is characterized by shear modulus s and
shear wave velocity s. The tube is also linear elastic but shear modulus and shear wave
velocity of the tube, represented by t andt, are linearly changing from the inner surface to
outer surface of the tube as a function of r. t andt take the values i andi at inner surface
and o ando at outer surface. The general form of the equation of motion of the problem is:
[27]
 ji
 2U i
  fi   2
x j
t
(1)
Here, Tji represents the stress tensor,  represents mass density of the material, fi represents
components of the body force vector and Ui represents components of the displacement vector.
In the considered problem, the system is excited by SH-Waves, so there are only out-ot-plane
displacements and the only displacement component is Uz, which is the displacement
component in the z direction. If the out-of-plane displacement function in the tube region is
denoted by Ut, mass density of the tube material is denoted by t and the body force per unit
mass is neglected, then Eq. (1) can be rewritten in terms of cylindrical coordinates as follows:
 (t )
1
1  (t )
2
 rz (r ,  , t )   (rzt ) (r ,  , t ) 
 z (r ,  , t )  t (r ) 2 U t (r ,  , t )
r
r
r 
t
(2)
where Trz(t) and Tz(t) are shear stress components inside the tube, defined by [29]:

U t (r ,  , t )
r
(3)
1 
U t (r ,  , t )
r 
(4)
 (rzt ) (r ,  , t )  t (r )
(tz) (r ,  , t )  t (r )
5
When Eqs. (3) and (4) are substituted into Eq. (2), there follows:
U t ( r ,  , t )
 2U t (r ,  , t ) t (r ) 
t ( r )
 t ( r )

U t (r ,  , t )
r
r 2
r r
1 
1 
2

t ( r )
U t (r ,  , t )  t (r ) 2 U t (r ,  , t )
r 
r 
t
'
(5)
Since steady-state wave propagation is considered, time dependent part of the displacement
can be separated as follows:
Ut (r ,  , t )  ut (r ,  )eit
(6)
where, i is the imaginary unit. Substituting Eq. (6) into Eq. (5) gives:
 (r ) 

2
ut (r ,  )  t (r ) 2 ut (r ,  )  t
ut (r ,  )
r
r
r r
1 
1 

t ( r )
ut (r ,  )   2 t (r )ut (r ,  )  0
r 
r 
t ' ( r )
(7)
Angular frequency () and shear wave velocity (t) in tube region are [29]:
  kt (r )t (r )
(8)
t (r )  t (r ) / t (r )
(9)
In identity (8), kt denotes wave number in the tube region. For convenience, a new
auxiliary function, tr is introduced such that:
tr (r ) 
1 d
 (r )
t (r ) dr t
(10)
 2ut (r ,  ) 1 ut (r ,  ) 1  2ut (r ,  )
2

 2
  kt (r ) ut (r ,  )  
Substituting Eqs. (8), (9) and (10) into (7) leads to:
2
2
r
r
r
r

To simplify Eq. (11), shear modulus t and shear wave velocity t are assumed to change
linearly with r, described by the following functions:
t (r )  o (r / ao )
(12)
t (r )  o (r / ao )
(13)
In this case, expression (10) and the wave number (kt) take the following forms:
6
tr (r ) 
1 d
1
d
t ( r ) 
 (r / ao )  1/ r
t (r ) dr
o (r / ao ) dr o
(14)
k
 1
1
 o o
k a /r
o (r / ao ) o (r / ao ) o o
(15)
 2ut (r ,  ) 2 ut (r ,  ) 1  2ut (r ,  )
2

 2
  ko ao / r  ut (r ,  )  0
2
2
r
r
r
r

(16)
kt (r ) 

t ( r )


o (r / ao )

Eq. (11) becomes:
The Finite Fourier Transform and its inverse are defined as:
1
ut ,n (r ) 
2
2
 in
 ut (r,  )e d
ut (r ,  ) 

u
t ,n
(r )ein
(17)
n 
0
Finite Fourier Transform of Eq. (16) is:
r
2
d 2ut ,n (r )
dr
2
 2r
dut ,n (r )
dr


  ko ao   n2 ut ,n (r )  0
2
(18)
For:

 n   ko ao   n2
2

(19)
Eq. (18) takes the form:
r
2
d 2ut ,n (r )
dr
2
 2r
dut ,n (r )
dr
  nut ,n (r )  0
(20)
Here, the solution of Eq. (20) is assumed to be in power series form as:
ut ,n (r )  r 
(21)
Substituting solution function (21) into Eq. (20) yields:

2

    n r  0
(22)
In Eq. (22), the solution function r cannot be zero so the coefficient of the solution
function must be zero so that Eq. (22) holds:
 2    n  0
7
(23)
The quadratic equation (23) has two solutions:


(24)


(25)
1  1  1  4 n / 2
 2  1  1  4 n / 2
Substituting the roots given in Eqs. (24) and (25) into Eq. (21) provides two solution
functions that both satisfy Eq. (20). A general solution would be a linear combination of these
two functions as follows:
ut ,n (r )  At1,n r 1  At 2,n r  2  At1,n r
 1
  A r  1
t 2, n
1 4 n /2

1 4 n /2
(26)
Inverse transform of Eq. (26) gives the static part of displacement function in tube region:
ut (r ,  ) 

 ut ,n (r )ein 
n 


At1,n r
 1

 e 
1 4 n /2 in
n 
A
t 2, n
r
 1
 e
1 4 n /2 in
(27)
n 
In the homogeneous unbounded region, shear modulus (s) and wave number (ks) are
constant and derivative of shear modulus is zero. As a result, governing equation for this
region is slightly simpler than Eq. (11):
 2us (r ,  ) 1 us (r ,  ) 1  2us (r ,  )

 2
 ks2us (r ,  )  0
2
2
r
r
r
r

(28)
Here, us denotes static part of displacement function in the homogeneous region. Applying
Finite Fourier Transform to Eq. (28) gives:
r
2
d 2us , n ( r )
dr
2
r
dus ,n (r )
dr


  rks   n 2 us ,n (r )  0
2
(29)
For convenience, the following variable transformation is made:
  rks
(30)
d
d d
d

 ks
dr d  dr
d
(31)
d2
d d  d 
2 d

   ks
2
dr
d  dr dr
d
(32)
With this variable transformation, Eq. (29) takes the form:
8
2
d 2us ,n ( / ks )
d
2

dus ,n ( / ks )
d


  2  n2 us ,n ( / ks )  0
(33)
The ordinary differential equation (33) is the well-known Bessel differential equation and
the solution is described as [28]:
us ,n ( / ks )  Asj ,n J n ()  Asy ,nYn ()
(34)
Inverse variable transformation yields:
us ,n (r )  Asj ,n J n (ks r )  Asy ,nYn (ks r )
(35)
Here, Jn and Yn are the first and second kind of Bessel functions of order n. In our case,
describing solution function (35) in terms of the first and second kind of Hankel functions is
more convenient:
us ,n (r )  Ash1,n H n(1) (ks r )  Ash 2,n H n(2) (ks r )
(36)
where the first and second kind of Hankel functions are defined as [28]:
H n(1) (ks r )  J n (ks r )  iYn (ks r )
(37)
H n(2) (ks r )  J n (ks r )  iYn (ks r )
(38)
Inverse transform of Eq. (36) is:
us ( r ,  ) 

 us,n (r )ein 
n 


Ash1,n H n(1) (ks r )ein 
n 

A
sh 2, n
H n(2) (ks r )ein
(39)
n 
Solution function (39) corresponds to the scattered waves from the tube. In this function,
terms multiplied by Hn(1) and Hn(2) represents outgoing and incoming waves respectively. Due
to Sommerfeld Radiation Condition, there will not be incoming waves from infinity, which
will reduce Eq. (39) to:
us ( r ,  ) 

A
sh1, n
H n(1) (ks r )ein
(40)
n 
In the unbounded homogeneous region, there are also incident SH-Waves with unit
amplitude,  angular frequency and +x direction. Incident waves are denoted by Ui and they
can be easily described by the following function:
Ui  eiks ( xst )  eiks x eit  ui eit
9
(41)
By using identity (42) [28], spatial part of incident waves (ui) can be expressed in FourierBessel series form:
e
1
z ( t 1/ t )
2

t

k
(42)
J k ( z)
k 
ui (r ,  ) 

e 
in /2
J n (ks r )ein
(43)
n 
Resultant displacement function in the unbounded region is the sum of the scattered waves
(us) and the excitation (ui):
usT (r ,  )  us (r,  )  ui (r,  )
(44)
Displacement function that corresponds to scattered waves from the cavity (us) has one set
of unknowns (Ash1,n). Displacement function that corresponds to the waves inside the tube (ut)
has two sets of unknowns (At1,n and At2,n). Totally three sets of unknowns will be determined
from the following three boundary conditions:
Zero stress at inner side of the tube:
 rz(t )  r  ai ,    i

ut  r  ai ,    0
r
(45)
Continuity of stress and displacement at outer side of the tube:
 rz(t )  r  ao ,     rz( s )  r  ao ,    o


ut  r  ao ,    s usT  r  ao ,    0
r
r
ut  r  ao ,    usT  r  ao ,    0
(46)
(47)
Substituting displacement functions (27) and (44) into boundary conditions (45), (46) and
(47) yields:
 A

t1, n
n 

 1 

1  4 n ai

At 2,n 1  1  4 n ai


 / 2
1 1 1 4 n /2
10

 / 2 ein  0
1 1 1 4 n /2
(48)
A
 t

  
n 
 1 
t1, n
s


1  4 n ao

At 2,n 1  1  4 n ao


H


 /2
1 1 1 4 n /2

 /2 
1 1 1 4 n /2
(49)
ks in /2
e
 J n1 (ks ao )  J n1 (ks ao ) 
2
Ash1,n
(1)
n 1
(k s ao )  H n(1)1 (k s ao )
   A

a
 1
  e
in
  A a  1
t 2, n o
1 4 n /2
t1, n o
n 
e
in /2
0


1 4 n /2
J n (ks ao )  Ash1,n H (k s ao ) e
(1)
n
in

(50)
0
Because of periodicity conditions, Eqs. (48), (49) and (50) reduce to:
At 2,n

 1 



1  4  a
At1,n 1  1  4 n ai
n

 / 2

 /20
1 1 1 4 n /2
1 1 1 4 n /2
i
t
1  1 1 4  /2
At1,n 1  1  4 n ao
/ 2
s
1  1 1 4  /2
A 1  1  4 a
/2 
t 2, n



n

H

(51)
n
n
o

(52)

(53)
ks in /2
e
 J n1 (ks ao )  J n1 (ks ao ) 
2
Ash1,n
A
(1)
n 1
a
 1
t1, n o


(k s ao )  H n(1)1 (k s ao )  0
  A a  1
t 2, n o
1 4 n /2

1 4 n /2

ein /2 J n (ks ao )  Ash1,n H n(1) (ks ao )  0
Unknown constants are determined by solving Eqs. (51), (52) and (53) simultaneously, as
follows:
 Ash1,n   0

 
 At1,n    M 21
 At 2,n   M 31


M 12
M 22
M 32
M 13 
M 23 
M 33 
1
0


k

 s ein /2  J n 1 (ks ao )  J n 1 (ks ao )  
2

in /2


e J n (ks ao )



where
11
(54)


1 1 1 4 n /2


1 1 1 4 n /2
M12  1  1  4 n ai
M13  1  1  4 n ai
M 21  

 /2
(55)

 /2
(56)

ks
H n(1)1 (ks ao )  H n(1)1 (ks ao )
2
M 22 
t
1  1
1  1  4 n ao
2 s
M 23 
t
1  1
1  1  4 n ao
2 s




M 31  H n(1) (ks ao )
M 32  ao
M 33  ao
III.
 1
1 4 n /2
 1
1 4 n /2

(57)

(58)

(59)
1 4 n /2
1 4 n /2
(60)

(61)

(62)
Numerical Results and Discussions
Displacement functions (27) and (44) given in the previous section are out-of-plane
displacements in complex form in the tube and the unbounded regions respectively.
Maximum displacements in a full period (displacement amplitudes) at any point can be
calculated by evaluating absolute value of these functions. Displacements as a function of
time can also be obtained by multiplying Eqs. (27) and (44) by exp(-it) and calculating
either real or imaginary part [29]. Stress components can also be derived from displacements
as given in Eqs. (3) and (4). In this section, some numerical results are demonstrated for
normalized problem parameters. Incident wave length (s) and inner radius of the tube (ai) are
normalized with respect to tube outer radius (ao). Material coefficients of the tubes outer
surface (o, o) are normalized with respect to material coefficients of the unbounded region
(s, s). Plot ranges are in between =0o and =180o since the displacements or stress
components are perfectly symmetric with respect to x axis.
Since there is no study on two-dimensional diffraction and scattering of SH-Waves by a
FGM tube embedded in an elastic unbounded space, the results obtained in this study cannot
12
be directly compared with the existing studies. However, there are many analytical studies on
two-dimensional diffraction and scattering of SH-Waves by homogeneous tubes, tunnels or
pipes (including but not limited to [23-26]). In these studies, none of the surrounding
mediums is unbounded and therefore, there are additional reflected waves. These additional
reflected waves significantly affect displacement variations. Consequently, a comparison with
these results is also uninformative. In order to verify the results of this study, an analytical
solution of a similar problem where the tube material is homogeneous, is developed.
Displacement functions that correspond to incident waves (Eq. (43)) and scattered waves from
the tube (Eq. (40)) remain in the same form. Displacement function in the homogeneous tube
region is given below:
ut( hom) (r ,  ) 



Ath1,n H n(1) (kt r )ein 
n 
A
th 2,n
H n(2) (kt r )ein
(63)
n 
where

0


 Ash1,n  
H n(1) (k s r )

 
A



 th1,n   s
r
r  ao
 Ath 2,n  


(1)
  H n (ks ao )


1
H n(1) (kt r )
r
r a
i
o
H (kt r )
r
r a
(1)
n
o
(1)
n
H (kt ao )

H n(2) (kt r )

r

r  ai  
0



H n(2) (kt r )

 (64)
in /2 J n ( k s r )

o
s e



r
r
r  ao
r  ao

 
in /2
H n(2) (kt ao )   e J n (ks ao ) 


Note that Eq. (63) is in exactly same form as Eq. (39) since the governing equations are the
same. In Eq. (39), the terms that correspond to incoming waves are omitted because of
Sommerfeld Radiation Condition. In the tube region, however, there are both incoming and
outgoing waves, so, none of the terms is omitted. The solution function given in Eq. (63) is
convenient to be compared with the solution function of a homogeneous cylinder embedded
in an unbounded medium subjected to SH-Waves. Pao and Mow [29] presents the analytical
displacement function in the homogeneous cylinder as:

ut( P&M ) (r ,  )   Cn J n (kt r )Cos  n 
n0
where
13
(65)
Cn 




i n  n k s s H n(1) (k s ao ) J n 1 (k s ao )  J n 1 (k s ao )  J n (k s ao ) H n(1)1 (k s ao )  H n(1)1 (k s ao )



kt t H n(1) (k s ao ) J n 1 (kt ao )  J n 1 (kt ao )  k s s J n (kt ao ) H n(1)1 (k s ao )  H n(1)1 (k s ao )

 (66)
and
1, n  0
2, n  1, 2,...
n  
(67)
As the inner radius of the homogeneous tube approaches to zero, the dynamic behavior of
the homogeneous tube must be similar to the homogeneous cylinder. Figure 2 demonstrates
this comparison. In this Figure, displacement amplitudes at cylinders and tubes outer surfaces
are plotted. When ai/ao=0.05, displacement differences are barely distinguishable. For smaller
values of ai/ao, the differences become unnoticeable. So, the analytical solution of the
homogeneous tube given in Eq. (63) converges to the analytical solution of the homogeneous
cylinder given in Eq. (66) presented by Pao and Mow when inner radius of the tube goes to
zero. Figure 2 also demonstrate another comparison. According to the presented mathematical
model for the FGM tube, when the tube thickness decreases (or inner radius ratio,
ai/ao ,increases), the difference of material properties of outer and inner side of the tube also
decrease. Therefore, for a thin FGM tube, the dynamic behavior must be similar to the
homogeneous tube. In Figure 2, displacement amplitudes at homogeneous and FGM tubes
outer surfaces are plotted for ai/ao=0.96. It can be seen that there is a perfect agreement with
these results. Briefly, the results presented in this study for the homogeneous tube converges
to the results of the analytical formulation of a homogeneous cylinder presented by Pao and
Mow as the inner radius goes to zero. When inner radius ratio (ai/ao) approaches to 1, the
results for the homogeneous and FGM tubes coinside.
In Figures 3-14, rz stress amplitudes are given for three different outer shear modulus
ratios (4, 8 and 12) and a fixed shear wave velocity ratio (1.6). Stress amplitudes in
homogeneous cases are also included for a fixed shear modulus ratio, which is the average of
inner and outer shear modulus ratios of FG cases (t=(i+o)/2). Shear modulus ratios of the
homogeneous cases are chosen to be same as the minimum inner shear modulus ratio and the
maximum outer shear modulus ratio among the FG cases, respectively.
Within the range of used parameters, the stress distributions of FG cases are generally in
between these homogeneous cases as expected. In general, peak stresses increase as the
rigidities increase. In Figures 4 and 7, peak stresses are higher in some FG cases, but the
14
differences are small. In these two cases, incoming wave lengths are relatively long. In the
rest of the Figures, peak stresses are always higher in the homogeneous cases with highest
rigidity. There are also considerable increases in stress amplitudes when incoming wave
lengths decrease because of increased spatial variations of displacements. As an example,
when the wavelength fall in half, rz, approximately doubles for all thicknesses. These results
show that the FG tubes are subjected to less rz shear stresses compared to the tubes
uniformly made of outer surface material of the FG case for shorter wave lengths which cause
significantly higher stresses. Fluctuations in stress distributions also increase for shorter wave
lengths. In the most of the cases, the peak stresses occur at =180o. But in some cases (Figure
3 and 7), peak stresses occur at the other side of the tube (at =0o). It is observed that the
difference between the compared cases increase by the thickness which also determine the
material coefficients of the inner side of the FGM according to the proposed mathematical
model.
In Figures 15-26, z stress amplitudes are compared. z stresses are always zero at =0o
and =180o because of the displacement symmetry. In between these points, the z stresses
are higher than the rz stresses. Similar to the rz stresses, z stresses increase remarkably as
incoming wave lengths decrease and stresses generally increase by rigidities. Stresses for FG
cases are usually in between homogeneous cases. For relatively shorter incoming wave
lengths, peak stresses usually occur in the most rigid homogeneous cases. The point where the
peak stresses occur changes with incoming wave lengths and thicknesses. Same as rz stresses,
fluctuations increase for shorter wave lengths. The difference between FG and homogeneous
cases increase for higher thicknesses also here.
It should be noted that actual stresses in FGM tube changes between positive and negative
values of these stress amplitudes. Periodic behavior of stresses is an important effect on
fatigue life FGM tubes.
Tables 1-4 demonstrate the displacement amplitudes at tubes outer surface for various
incident wave lengths. It can be seen that average displacement amplitudes usually increase as
the incoming wave lengths increase. Fluctuation in the displacement amplitudes also increase
as the incoming wave lengths decrease as expected.
IV.
Conclusions
15
Scattering and diffraction of plane SH-Waves by a functionally graded cylindrical tube
surrounded by an unbounded homogeneous medium is studied. Governing equations of the
two-dimensional wave propagation problem in each region are solved analytically and
displacement functions are obtained in closed form. The presented closed-form solutions are
very compact, so they are remarkably convenient for programming and can be used directly in
many engineering applications. Presented results can also be very useful for verification of
subsequent numerical studies that may consider nonlinearity. The solution technique proposed
in this study can be easily applied to many similar problems. For instance, by using Addition
Theorems, dynamic response of twin functionally graded tubes surrounded by an infinite or
semi-infinite medium can be solved analytically. The numerical results presented in this study
can be outlined as:
1) Stress distributions at a FG cylinder are usually in between homogeneous tubes made
of outer surface and inner surface material of the FG tube.
2) The behavior of FG tubes are close to the homogeneous tubes when material
coefficients of outer and inner side are close. Otherwise, the behaviors of FG cylinders
are significantly different. These findings indicate the necessity of further research on
dynamic response of FG tubes.
3) The decrease in incoming wave lengths result in remarkably higher shear stresses
inside the FG tube.
4) In most of the cases, FG tubes are subjected to smaller shear stresses compared to the
most rigid homogeneous cases, especially when the incoming wave lengths are short.
5) Locations of peak stresses change by the values of problem parameters and they are
highly unpredictable.
6) Fluctuations in stress distributions increase as the incoming wave lengths decrease.
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FIGURE CAPTIONS
Figure 1: Geometry of the problem
Figure 2: Displacement Amplitude at tubes and cylinders outer surface for s/ao=2, o/s=1.5
and o/s=10
Figure 3: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=4
Figure 4: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=2
Figure 5: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=1
Figure 6: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=0.5
Figure 7: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=4
Figure 8: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=2
19
Figure 9: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=1
Figure 10: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=0.5
Figure 11: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=4
Figure 12: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=2
Figure 13: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=1
Figure 14: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=0.5
Figure 15: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=4
Figure 16: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=2
Figure 17: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=1
Figure 18: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=0.5
Figure 19: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=4
Figure 20: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=2
Figure 21: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=1
Figure 22: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=0.5
Figure 23: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=4
Figure 24: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=2
Figure 25: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=1
Figure 26: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=0.5
20
Figure 1: Geometry of the problem
Figure 2: Displacement Amplitude at tubes and cylinders outer surface for s/ao=2, o/s=1.5 and o/s=10
21
Figure 3: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=4
Figure 4: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=2
Figure 5: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=1
22
Figure 6: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=0.5
Figure 7: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=4
Figure 8: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=2
23
Figure 9: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=1
Figure 10: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=0.5
Figure 11: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=4
24
Figure 12: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=2
Figure 13: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=1
Figure 14: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=0.5
25
Figure 15: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=4
Figure 16: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=2
Figure 17: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=1
26
Figure 18: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=0.5
Figure 19: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=4
Figure 20: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=2
27
Figure 21: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=1
Figure 22: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=0.5
Figure 23: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=4
28
Figure 24: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=2
Figure 25: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=1
Figure 26: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=0.5
29
Table 1: Displacement Amplitude at tubes outer surface for ai/ao=0.75, o/s=1.6 and o/s=8
θ=
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
100°
110°
120°
130°
140°
150°
160°
170°
180°
s/ao=4
1.218153
1.204439
1.163781
1.097877
1.01038
0.908191
0.803255
0.714128
0.66398
0.669311
0.72591
0.812751
0.907842
0.996258
1.070176
1.126705
1.165768
1.188457
1.195873
s/ao=2
0.973206
0.922322
0.776354
0.557975
0.331377
0.323084
0.569618
0.832862
1.024632
1.107547
1.073863
0.94523
0.768765
0.609413
0.531754
0.548774
0.608174
0.659233
0.678306
s/ao=1.5
0.688102
0.621512
0.437673
0.195951
0.220595
0.482936
0.691131
0.779322
0.723277
0.556808
0.409158
0.44663
0.566772
0.620808
0.582168
0.483103
0.37262
0.292502
0.26437
30
s/ao=1
0.715233
0.556847
0.168109
0.33927
0.661981
0.718211
0.505537
0.310209
0.615245
0.849159
0.758499
0.481381
0.482478
0.638759
0.610781
0.414292
0.187628
0.153835
0.202529
s/ao=0.75
0.584825
0.361854
0.156127
0.542195
0.527176
0.185532
0.422737
0.639843
0.463997
0.410676
0.695063
0.568586
0.294133
0.421468
0.407551
0.217987
0.072766
0.22639
0.300231
s/ao=0.5
0.481908
0.1026
0.442727
0.29406
0.34281
0.437653
0.21839
0.493464
0.220533
0.561158
0.438245
0.568668
0.435595
0.337594
0.26246
0.20713
0.345673
0.237038
0.299292
Table 2: Displacement Amplitude at tubes outer surface for ai/ao=0.75, o/s=1.6 and o/s=12
θ=
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
100°
110°
120°
130°
140°
150°
160°
170°
180°
s/ao=4
1.29263
1.276979
1.230366
1.153964
1.050267
0.923991
0.783622
0.644149
0.531341
0.481221
0.513622
0.605156
0.718403
0.828879
0.924435
0.999891
1.053622
1.08559
1.09618
s/ao=2
0.984711
0.931523
0.778255
0.544642
0.278494
0.232452
0.502463
0.772441
0.964396
1.044823
1.004558
0.860165
0.651746
0.442812
0.335611
0.393481
0.508667
0.595519
0.626539
s/ao=1.5
0.551038
0.496753
0.345948
0.137978
0.150438
0.367996
0.531736
0.594921
0.536675
0.377004
0.216866
0.265623
0.392712
0.445488
0.412536
0.329768
0.248646
0.207054
0.198787
31
s/ao=1
0.696482
0.541031
0.151876
0.323796
0.638703
0.682531
0.448156
0.200256
0.536251
0.764648
0.661704
0.340787
0.369138
0.585695
0.577656
0.365726
0.092684
0.189948
0.278451
s/ao=0.75
0.501829
0.30875
0.129515
0.462488
0.443503
0.120841
0.348825
0.527024
0.337934
0.275727
0.539487
0.41269
0.135824
0.313152
0.337272
0.203393
0.024068
0.222598
0.313721
s/ao=0.5
0.438615
0.090233
0.402868
0.257993
0.307319
0.384256
0.177845
0.4346
0.129455
0.467968
0.309136
0.461507
0.343321
0.296181
0.257448
0.192884
0.328207
0.153198
0.258657
Table 3: Displacement Amplitude at tubes outer surface for ai/ao=0.5, o/s=1.6 and o/s=8
θ=
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
100°
110°
120°
130°
140°
150°
160°
170°
180°
s/ao=4
1.291011
1.275305
1.228318
1.150584
1.043494
0.91012
0.756611
0.59483
0.448567
0.364768
0.390738
0.497831
0.628422
0.752191
0.856979
0.938512
0.995995
1.029982
1.041208
s/ao=2
0.94616
0.895442
0.748833
0.522996
0.25399
0.183764
0.461351
0.731688
0.921341
0.994337
0.938756
0.770335
0.529436
0.2818
0.201494
0.358934
0.5182
0.620998
0.65605
s/ao=1.5
0.43091
0.388933
0.272139
0.107303
0.105246
0.28336
0.420717
0.474865
0.420188
0.268898
0.121174
0.214206
0.31957
0.324998
0.236586
0.095513
0.04822
0.151928
0.189456
32
s/ao=1
0.632024
0.490642
0.134081
0.287053
0.572776
0.617531
0.411155
0.157073
0.481551
0.700766
0.576385
0.189585
0.286221
0.487351
0.41276
0.283819
0.506792
0.776897
0.882732
s/ao=0.75
0.598331
0.45319
0.32182
0.524355
0.554421
0.415966
0.526205
0.69269
0.552602
0.251342
0.163941
0.255653
0.68955
0.85439
0.678128
0.468606
0.50493
0.648837
0.717213
s/ao=0.5
0.202543
0.120168
0.138951
0.229286
0.303313
0.067994
0.471361
0.315389
0.306326
0.678814
0.331885
0.842008
0.601984
0.487994
0.464441
0.123711
0.391839
0.189825
0.140725
Table 4: Displacement Amplitude at tubes outer surface for ai/ao=0.5, o/s=1.6 and o/s=12
θ=
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
100°
110°
120°
130°
140°
150°
160°
170°
180°
s/ao=4
1.255664
1.239465
1.191057
1.111094
1.000963
0.863295
0.702828
0.528137
0.357137
0.240776
0.273677
0.410748
0.564393
0.705543
0.824593
0.917957
0.984548
1.02433
1.03755
s/ao=2
0.893182
0.843934
0.70168
0.482367
0.215338
0.14236
0.42109
0.675747
0.853019
0.922411
0.872318
0.714217
0.480511
0.223545
0.150248
0.352197
0.531115
0.645377
0.684425
s/ao=1.5
0.302584
0.272584
0.189179
0.07049
0.071182
0.197569
0.292999
0.3295
0.288266
0.175549
0.062314
0.15379
0.235249
0.238273
0.169442
0.060199
0.053103
0.133858
0.163236
33
s/ao=1
0.562083
0.435648
0.11525
0.257675
0.510429
0.54153
0.344406
0.096334
0.412866
0.608694
0.505496
0.161386
0.256432
0.45893
0.3986
0.206457
0.370472
0.636638
0.742277
s/ao=0.75
0.501687
0.411113
0.307337
0.380594
0.383998
0.340344
0.480103
0.596578
0.477827
0.249466
0.130093
0.172446
0.510228
0.627047
0.486704
0.312377
0.307965
0.401502
0.450574
s/ao=0.5
0.059595
0.078622
0.079163
0.138231
0.26864
0.011845
0.384868
0.285977
0.241082
0.546909
0.218147
0.633203
0.436422
0.348327
0.368591
0.137592
0.33956
0.097768
0.105755
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