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27N19 IJAET0319307 v7 iss1 226 241 .pdf



Original filename: 27N19-IJAET0319307_v7_iss1_226-241.pdf
Title: Wave propagation in a homogenous isotropic magento-thermo–elastic cylindrical panel
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International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963

WAVE PROPAGATION IN A HOMOGENEOUS ISOTROPIC
MAGNETO-THERMO-ELASTIC CYLINDRICAL PANEL
K. Kadambavanam1 and L. Anitha2
1

2

Department of Mathematics, Sri Vasavi College, Erode, India
Department of Mathematics, Nandha Arts & Science College, Erode, India

ABSTRACT
This work investigates the three dimensional wave propagation of a homogeneous isotropic magneto thermo
elastic cylindrical panel in the context of the linear theory of thermo elasticity. Three displacement potential
functions are introduced to uncouple the equations of motion. A Bessel function solution with complex
arguments is directly used to analyze the frequency equations with traction-free boundary conditions. The
special cases have also been deduced for magneto elastic, thermo elastic and elasto-kinetic at various levels
from the present analysis. The numerical example which demonstrates the present method is studied for the
material magneto-strictive cobalt iron oxide (CoFe2O4). The computed non-dimensional phase velocity and
attenuation coefficient are plotted in the form of dispersion curves. The coupling effect among thermal,
magnetic and elastic in magneto thermo elastic material provides a mechanism for sensing thermo mechanical
disturbances in the design of sensors and surface acoustic damping filters.

KEYWORDS: Wave propagation, isotropic cylindrical panel, Bessel function, attenuation coefficient.

I.

INTRODUCTION

The interaction between the magnetic and thermal fields plays a vital role in geophysics for
understanding the effect of Earth’s magnetic field on seismic waves. With the development of active
material systems, there is a significant interest in the coupling effects between the elastic, magnetic
and temperature for their application in sensing and actuation. The analysis of thermally induced
vibration of magneto elastic cylindrical panel is usually encountered in the design of structures,
atomic reactors, steam turbines, supersonic aircraft, space shuttle and other devices operating at
elevated temperature. In the field of non-destructive evaluation, laser-generated waves have attracted
great attention owing to their potential application to non-contact and non-destructive evaluation of
sheet materials. Thermoelectric currents in the presence of magnetic fields can cause pumping and
stirring of liquid metal coolants in nuclear reactors and molten metal in industrial metallurgy. In the
nuclear field, the extremely high temperatures and temperature gradients originating inside nuclear
reactors influence their design and operations. Moreover, it is well recognized that the investigation of
the magneto thermal effects on elastic wave propagation has bearing on many seismological
applications. This study may be used in applications involving non-destructive testing (NDT),
qualitative nondestructive evaluation(QNDE) of large diameter pipes and health monitoring of other
ailing infrastructure in addition to checking and verifying the validity of FEM and BEM for such
problems.
The static analysis cannot predict the behavior of the material due to the rapid thermal stress changes.
Green and Lindsay [1] and Lord and Shulman [2] modified the Fourier law and constitutive relations.
It is used to get hyperbolic equation for heat conduction by taking into account of the time needed for
acceleration of heat flow and relaxation of stresses. The theory of thermo elasticity is initially studied
by Nowacki[3]. A special feature of the Green-Lindsay model is that it does not violate the classical

226

Vol. 7, Issue 1, pp. 226-241

International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963
Fourier’s heat conduction law. The obtained results were compared by a special application of the
frozen stress technique of photo elasticity. Paul and Muthiayalu[4] studied magneto thermo elastic
free vibrations in an infinite plate by verifying the numerical result for aluminum alloy. Ponnusamy[5]
has obtained the frequency equation of free vibration of a generalized thermo elastic solid cylinder of
arbitrary cross section by using Fourier expansion collocation method. Sharma and Sidhu [6] studied
the propagation of plane harmonic thermo elastic wave in homogeneous isotropic, cubic crystals and
anisotropic materials in the context of generalized thermo elasticity. The three dimensional vibration
analysis of a transversely isotropic thermo elastic cylindrical panel has been investigated by Sharma
[7].The application of powerful numerical tools like finite element or boundary element methods to
these problems is also becoming important. The theory of magneto thermo-elasticity has aroused
much applications in many industrial appliances particularly in nuclear devices, where a primary
magnetic field exists.
Sherief and Ezzat[8] used the Laplace transform technique to find the distribution of thermal stresses
and temperature in a generally thermo elastic electrically conducting half-space under sudden thermal
shock and permeated by a primary uniform magnetic field. Wang and Dai [9] presented magnetothermo-dynamic stresses and perturbation of magnetic field vector in an orthotropic thermo elastic
cylinder.

II.

FORMULATION OF THE PROBLEM

In cylindrical co-ordinates, the three dimensional stress equations of motions and strain displacement
relations and heat conduction equations, Maxwell equation for magnetic field, in the absence of body
force for linearly elastic medium are as follows:

rr,r  r 1r,  rz,z  r 1 (rr   )  

2u
t 2

2v
t 2
2 w

r,r  r 1,  z,z  2r 1 (r )  

rz,r  r 1z,  zz,z  r 1rz  

(1)

t 2

K T,rr  r 1T, r  r 2T,  T,zz 
2 

 C T,t  To  (err  e )  e zz  P3 ,z  o 2   C oT, tt
t 
 t



r 1 (rBr )  r 1 (B )  (Bz )  0
r

z
The comma in the subscripts denotes the partial differentiation with respect to the variables.
The stress-strain relations are

rr  (  2)e rr  e  e zz  T  d 31


z


z

 (  2)ezz  T  d31
z

  err  (  2)e  e zz  T  d 31
zz  err  e

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International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963
z   z  r 1d15

rz   rz  d15





r

r   r
Magnetic induction displacements are

Br  d15  rz  11


r

B  d15  z  r 111




Bz  d31 (err  e )  d 31ezz  33


 P3 T
z

The strain displacements are

err 

u
r

e  r 1u  r 1
ezz

v


w

z

(2)

Shear strain displacements are

 r 

v 1
u
 r v  r 1
r


v 1 w
r
z

w u
 rz 

r z
 z 

(3)

Substituting the equation (3) in equation(2), yields

u
v 
w


   r 1u  r 1   
 T  d31
r
 
z
z

u
v 
w


  
 (  2)  r 1u  r 1   
 T  d31
r
 
z
z

u
v 
w


zz  
   r 1u  r 1   (  2)
 T  d31
r
 
z
z

w  1 
 v
z     r 1
  r d15
 

 z

 w u 
rz   
   d15
r
 r z 
u 
 v
r     r 1 v  r 1 
 
 r
rr  (  2)

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International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963

 w u 
Br  d15 
   11
r
 r z 
w  1 
 r
B  d15   r 1
  r 11
 

 z
v 
w

 u
Bz  d31   r 1u  r 1   d31
 33
 P3T
 
z
z
 r

(4)

Substituting the equation (4) in equation (1), yields,

  2 u 1 u 2  2  2 u
 2 u 1
2v
(  2)  2  r
 r u   r  2   2  r (   )
r
r 

z
 r

 2 w 2

T
 2
 2u
 (   )
 r (  3)

 (d 31  d15 )
 2
rz

r
rz
t
2
2
 


 u
u
2v
  2  r 1  r 2 v   r 1 (  )
 r 2 (  )  r 2 (  2) 2
r
r


 r

2v
2w
T 1
 2
2v
 2  r 1 (  )
 r 1
 r (d 31  d15 )
 2
z

z
z
t

(5a)

(5b)

  2u
  2 w 1 w 2  2 w 
  2 v u  
2w
  (   ) 

r
r
 r 1 
    (  2)
 r 2
 z z  
r
 rz
2 
z 2



  2 1  2  2 
T
 2
2w


 d31
 d15 
r
r
 r 2
z
r
z 2
2 
t 2


(5c)

 T
  2T

T 2  2T  2T 
 2T
2  
k  2  r 1
r



C



C

T




o

o
o
2
2 
  t

r

t
2 z 2 

t

t
 r



  u 1
w
 
1 v 
  r  r u  r     z  P3 z   0

 


(5d)

 2w
 1  2 v
2w
w 
 2u 
1 u
d15  2  r 2 2  r 1

(d

d
)
r

r



13
15 
r 
z z 

 r
 z
  2
2w
 2
 2
 
d31 2  33 2  11  2  r 2 2  r 1   P3T  0
r 
z
z

 r

(5e)

To uncouple the equations (5a) to (5d), the mechanical displacement u, v and w along the radial,
circumferential and axial directions are assumed following:

u  r 1  ,  G, r
v  r 1 G ,   ,r
w  w ,z
  ,z
Hence

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International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963


2
2 
2w
 2
2
(


2

)





G

(



)

(d

d
)
 T  0


1
31
15
z 2
t 2 
z 2
z 2


2
2 
2
and  1   2   2    0
z
t 


(6a)

2
2
1 
2 

r

r
r
r 2
2

2
2 
2w
2
 (  2) 1   2   2  G  (  ) 2   T  0
z
t 
z


where 12 

(6b)

and
 2
2
2 
 1   2   2    0
z
t 

 2

2
2 
2 
2
2


(


2

)


w

(



)

G

d


d
 1

 15 1
   T  0
1
31
z 2
t 2 
z 2 



 2
2


2w
 2 
2
 k1  k 2  C  T  To  1 G   2  P3 2   0
t 
t 
z
z
z 



2 
2 
(d13  d15 )12 G   d1512  d31 2  w   1112  33 2    P3T  0
z 
z 



III.

(6c)

(6d)

(6e)

SOLUTION TO THE PROBLEM

The equations (6a) to (6e) are coupled partial differential equations with three displacements, heat
conduction components and magnetic potential. To uncouple these equations, assume the transverse
wave along the axial direction z to be zero. gives a purely transverse wave. Hence the solutions of
the equations (6a) to (6e) can be presented in the following form:

 n 
G(r, , z, t)  G(r) sin(m)z cos    eiwt
  
 n 
w(r, , z, t)  w(r) sin(m)z sin    eiwt
  
 n 
(r, , z, t)  (r) sin(m)z cos    eiwt
  
 n 
T(r, , z, t)  T(r) sin(m)z cos    eiwt
  
(7)

 n 
(r, , z, t)  (r) sin(m)z cos    eiwt
  
Introducing the non-dimensional quantities

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International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963
r1 

r 1 z
T
n
mR
w 2 R 2
;z  ;T ;
; tL 
; 2 
;
R
L
To

L


P (2  )
ab


; 
  1; K  1;  '  ; P  3
2

o
 d31 
d  d31
d
d1  15
d 2  15
(8)
d31 
d31
R

1 

11 (2  )
 d312

 d31 o
 To R 2
2 To 
*
; p 
;  
; 1 
2
2
 C (2  )

Rewriting the equations (6a) to (6e) and (7) in a convenient form of equations as follows:

( 22  g1 ) G  (1   )g 2 w   p g 5   *T  0
where  22 

2
2
1 

r

r ' r '2
r '2

where
g1  C2 ( 2  t L2 ); g 2  c3 t L2
g3  2 

C1 2
tL
C2

g 5  d1t 2L

C2 (22  g3 )w  (1  )C322 G  d 2p (22  g6 ) *T  0
1 c2 2 2
2 G   t 2L w  Pp t12    *(22  g 4 )T  0
i *









d1 22G  d 2 ( 22  g 6 )w  1 p  22  g 7    * PT  0
 (2  )
1  11 2
;
 d31

(9)

g 7    t L2

The determinant form of the system of equations in equation (9) as follows:

 22  g1

(1  )g 2

 p g 5

*

C3 (1  ) 22

C2 ( 22  g3 )

d 2  p ( 22  g 6 )

 * 

d 2 ( 22  g 6 ) 1 p ( 22  g 7 )

 * P

d1 22
1 C2 2 2
2
i*

t 2L

P p t L2

(G, w, , T)  0 (10)

 * ( 22  g 4 )

Equation (10) on simplification reduces to the following differential equation

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International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963

[28  A26  B24  C22  D] G  0
where

1
[C21g1  C21g 4  C21g3  d 2 2g 4  2d 2 2g 6  C21g1g 7


A

 C21g1g3  d 2 2g 4g1  d 2 2g1g 6  C2 2 1  21 / i* ]
1
B  [C21g3g 7  C21g3g 4  C2 1p 2 t L 2  2d 2 2g 4 g 6  d 2 pt L 2  d 2 2g 6g 4


 d 2 2g 6 2  d 2 p  1t L 2  C2 p 2 t L 2g1  C 21g1g 3g 7  C21g1g 3g 4
2

 d 2 2g1g 4g 6  d 2  t L 2 pg1  d 2 2g1g 6  d 2C3 (1  ) g 4g 5  d 2C3 (1  )g 4g 6
 d 2C3g 4g 5g 6  C3 (1  )g 5 t L 2  d 2C3 g 4g 5  d 2 C3 g 4 g 6  d 2C3 g 4g 5g 6
 C2 d1g 4g 52  C2 2 p 1  2g 5 / iX * C 2d1g 3g 5  C 2d 2 1  2g 5 
 d 2 C3 (1  )pt L 21  C3 (1  )t L 21p  d1C 2 pt L 2  C 2 2 1  21g1 / i *
 C2 2 1 g 3 21 / i * d1d 2 t L 2  d 2 2C2  2 t1g 6 / i*]
2
1
C  [C2 1g 4 g 7  C11g3g 4 g7  C2 p 2 t L 2  d 2 2g 4 g 6 2  2d 2 pt L 2 g6   1t L 2g 7

 d 2 g1g 4 g 7  C2 1g3g 4 g1g 7  C2 p 2 t L 2 g1  d 2 2g1g 4 g 6  d 2 2g 6 2 g1  d 2  pg1 g11t L 2
 d 2g 4g5C3 (1  )  C2g 5 p 1 2g 3 / i * d1g 5 2 t L 2  d 2g 5  1  2C2g 6 / i *
 C3 (1  )d 2 t L 2g 6  C3 (1  ) 1t L 2g 7  d1C2 pt L 2  C2d 2 1  2 g 5
 C2 2 1 2g3 g 7 1 / i * d 2 2 t1 1  2g 6 / i * d1d 2  t L 2g 6 ]

D

1 2
[d 2 g1g 4g62  2d 2  pg1g6 t L 2  1t L 2g1g 7  d 22 1 C22g 62 / i*]


(11)

and
  C22 1  d 22  C2 1g1  d 22 g1
The solution of the equation (11) are obtained as,

G(r)



4

[Ai J (i r)  Bi Y (i r)]
i 1

232

Vol. 7, Issue 1, pp. 226-241

International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963
4

 Li [Ai J (i r)  Bi Y (i r)]



W(r)
(r)

i 1
4

 Mi [Ai J (i r)  Bi Y (i r)]



i 1
4

 Ni [Ai J (i r)  Bi Y (i r)]



T(r)

i 1

(r)



A5 J (Kr)  B5 Y (Kr)

(12)

where K =  –
The arbitrary constants Li, Mi and Ni are obtained from
2

2

tL2

 g 2 (1  )
g5 p
*   L    2 2  g1 
i




 C2 ( 2 2  g3 ) d 2 p ( 2 2  g 6 ) *    Mi   C3 2 2 (1  ) 

  

2
2
2
*
 d 2 ( 2  g 6 ) 1 p ( 2  g 7 )  p   Ni   d1 2

1
Li

(i 2  g1 )[pd 2 (i  g 6 )  1 (i 2  g 7 )]




g5 [pC3 (1  )i 2  1 (i 2  g 7 )]



1[1C3i 2 (i 2  g 7 )  d1d 2i 2 (i 2  g 6 )]


Mi



1
g 2 (1  )[pC3 (1  )i 2  d1i 2 ]
 p
(i 2  g1 )[pC2 (i 2  g 3 )  d 2 ( i 2  g 6 )]



 [d1C2i 2 (i 2  g 3 )  C3 (1  )i 2d 2 (i 2  g 6 )]



Ni



1
g 2 (1  )[d 2 2 (i 2  g 6 ) 2  1C3 (1  ) (i 2  g 7 )i 2 ]
*

g5 [d 2 (i 2  g3 )(i 2  g 6 )C2  C3 (1  )d 2i 2 (i 2  g 6 )]



(i 2  g1 )[1C2 (i 2  g3 )(i 2  g 7 )  d 2 2 (i 2  g 6 ) 2 ]

IV.

BOUNDARY CONDITIONS

The secular equations for the three dimensional vibration of cylindrical panel subjected to the traction
free boundary conditions at the upper and lower surfaces at r = a, b listed below
(i)
The traction free non-dimensional mechanical boundary conditions for a stress free edge are
given by

rr  0;

233

r  0;

rz  0

Vol. 7, Issue 1, pp. 226-241

International Journal of Advances in Engineering & Technology, Mar. 2014.
©IJAET
ISSN: 22311963
(ii)

Thermal Condition

T,r hT  0
Where h is the surface heat transfer co-efficient, h  0 thermally insulated surface and h refers to
an isothermal one.
(iii)
Magnetic Condition
B,r = 0
If rr = 0,

  

i
((i r ') 2 R 2  2 )
(2


)A
J
(

r
')

J
(

r
')

J  (i r ') 

i  ' 2 
i
1
i
2
r'
r'
  r
i 1

4



 (  1)

  ' 2 J  (i r ')  i J 1 (i r ')    t L 2 Li W  1 TR 2 N i  d 31M i   
r'
 r


K
(  1)


 A5 (  2)
J 1 (Kr ') 
J  (Kr ') 
2
r'
r'


K
 (  1)

 
J  (Kr ')  J 1 (Kr ')   0
2
r'
 r'


If

r  0 ,

2
 2(i )

J 1 (i r ')  2 (  1)J  (i r ') 
r'
r'


4

 Ai 
i 1

 K R 2 r '2

2(  1)
 A5  1 2 J  (Kr ') 
J  (Kr ') 
2
r'
 r'

K

  J 1 (Kr ') 
r'

If rz  0 ,



4



0







t

 Ai  t n (1  Li )  r ' J (i r ')  i J1(i r ')   A5  rL' J  (Kr ') 

 0

i 1

If T,r = 0,



4



 Ni  r ' J (i r ')  i J1(i r ') 

 0

i 1

If B,r = 0,
4



i 1







 

t

 A5  L J  (Kr ')   0
 r'



 d15t L (1  11Mi  Li )  r ' J (i r ')  i J1(i r ')  
V.

FREQUENCY EQUATIONS

| aij| = 0 for i, j = 1, 2, 3, … 10

234

Vol. 7, Issue 1, pp. 226-241


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