Magneto-Electro-V iscoelastic Torsional Waves in Aeolotropic Tube under Initial Compression Stress

beigecakeUrban and Civil

Nov 16, 2013 (3 years and 8 months ago)

103 views

1

Abstract

Th
is study examines the

effect of
electric and
magnetic field on
to
r-
sional waves in
hetrogeneous

viscoelastic cylindrically aeolotropic
tube subjected to initial compression stresses.

A new equation of
motion and phase

velocity of torsional waves propagating in
cyli
n-
drically aeolotropic
tube
subjected to initial compres
sion stresses
,
nonhomogeneity, electric and magnetic field
have been derived.

The
study reveals that the initial stresses
,

nonhomogeneity, electric and
magnetic field

present

in the
aeolotropic
tube of viscoelastic solid
have a notable effect on the propagation of torsional waves.
The
results have been discussed graphically.
This investigation is very
significant for potential application
in various fields of science such
as detection of mechanic
al explosions in the interior of the earth.


Keywords

Aeolotropic Material
,

Viscoelastic Solids, Non
-
Homogeneous, Bessel

Functions
.


Magneto
-
Electro
-
V

iscoelastic Torsional Waves in
Aeolotropic

Tube

under Initial Compression Stress

















1
INTRODUCTION

The mutual interactions between an externally applied magnetic field and the elastic deformation in
the solid body, gi
ve rise to the coupled field of

magneto
-
elasticity. Since electric currents also give
rise to magnetic field and vice
-
versa, the combined effect is also s
o
metimes known as magneto
-
electro
-
elasticity.

It is evident that since many component fields are interacting, a large number of
unknowns are involved and the solution of even the most elementary problems becomes difficult and
cumbersome. We therefore almost always have to take certain assumptions to
solve the problems.
The interaction of elastic and electromagnetic fields has numerous applications in various field of
science such as detection of mechanical explosions in the interior of the earth.

In spite

of the fact that
Maxwell equations governing e
lectro
-
magnetic field have been known for long time, the
interest

in
the coupled field is
helpful

in the field such as geophysics, optics,
acoustics
, damping of
acoustic
waves in magnetic fields, geomagnetics and oil prospecting etc.

Much literature is av
ailable on torsional surface wave propagation in homogeneous elastic and visc
o-
elastic media. Pal (2000) presented a note on torsional body forces in a viscoelastic half space. Dey
et
al.

(1996, 2000, 2002, 2003) investigated the effect of torsional surface

waves in non
-
homogeneous
anisotropic medium, torsional surface waves in an elastic layer with void pores, torsional surface
waves in an elastic layer with void pores over an elastic half space with void pores and effect of grav
i-
ty and initial stress on to
rsional surface waves in dry sandy medium.

Kaliski (
1959) purposed
d
y-
namic equations of motion coupled with the field of temperatures and resolving functions for elastic
and inelastic bodies in a magnetic field, Narain

(
1978) discussed
magneto
-
elastic torsional waves in a
bar under initial stress, White
(
1981
)

studied c
ylindrical waves in transversely isotropic media
.
Das
et al.

(
1978
) investigated

a
xisymmetric vibrations of orthotr
opic shells in a magnetic field.
The co
n-
tribution of
var
ious researchers on
torsional wave propagation such as
Suhubi
(
1965
)
,
Abd
-
alla
Rajneesh Kakar


Principal. DIPS Polytechnic College, India
rkakar_163@rediffmail.com

2


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


(
1994
),

Dat
ta (
1985
) and
Se
lim (
2007
) cannot be ignored.

Kakar and Kakar (201
2
)
discussed

t
o
r-
sional waves in fiber reinforced medium subjected to magnetic field.

Kakar and Gupta

(2013) pr
e-
sented a note on torsional surface waves in a non
-
homogeneous isotropic layer over viscoelastic half
-
space.

Tang et al. (2010) discussed transient torsional vibration responses of finite, semi
-
infinite and
infinite

hollow cylinders.


Kakar

and K
umar

(2013)

investigated

surface waves in
electro
-
magneto
-
thermo

two layer heterogeneous
viscoelastic

medium involving time rate of change of strain and
r
ecently,
Kakar (2013) presented a note on
interfacial

waves in non
-
homogeneous

electro
-
magneto
-
thermoelastic orthotropic granular half space.

In this study an attempt has been made to investigate the torsional wave propagation in non
-
homogeneous viscoelastic cylindrically aeolotropic material permeated by
an

electro
-
magneto field
.
The graphs have been plotted showing the effect of variation of elastic constants and the presence of
electro
-
magneto field
. It is observed that the torsional elastic waves in a viscoelastic solid body pro
p-
agating under the influence of a superimposed
el
ectro
-
magneto field

can be different significantly
from that of those propagating in the absence of
an

electro
-
magneto field
.



2

BASIC EQUATIONS

The problem is dealing with
electro
-
magnetoelasticity. Therefore the basic equations will be ele
c-
tromagnetism and elasticity.
The Maxwell equations of the electromagnetic field in a region with no
charges (ρ

=

0) and no currents (
J

=

0), such as in a vacuum, are
(
Thidé, 1997
)

0
 
,

(1a)


0
 

,

(1b)


t

  


,

(1c)

0 0
.
t


 


(1d)

w
here,

,

,
0

and
0

are electric field, magnetic field induction, permeability and
permittivity of the
vacuum.

For vacuum,
0

=
7
4 10



and
0

=
12
8.85 10


in SI units.
These equations lead directly to


and


satisfying the
wave equation

for which the solutions are linear combinations of
plane
waves

traveling at
the
speed of light
,
0 0
1
.
c



In addition,


and


are mutually perpendicular
to each other to the direction of wave propagation.

Also, the term
Ohm's law

is used to refer to various generalizations
.
The simplest example of this is:

,
J E



(2a)


3


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves


3

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


w
here
,

J

is the
current density

at a given location in a resistive material


is the electric field at that
location, and
σ

is a material dependent parameter

called the
conductivity
.
If an external
magnetic
field induction


is present and the conductor is not at rest but moving at velocity
V
, then an extra
term must be added to account for the current induced by the
Lorentz
force

on the charge carriers
(
Thidé,
1997
).


( ) ( ).
v
J E V B E B
t
 

     


(2b)


The
electromagnetic wave equation

is a second
-
order partial differential equation that describes the
propagation of
electromagnetic waves

through a vacuum. The homogeneous form of the equation,
written in terms of either the
electric field



or the magnetic field induction

, takes the form:
(
Thidé,
1997
)


2
2
0 0
2
0
t

 

   
 

 
ò
,


(3a)




2
2
0 0
2
0
t

 

   
 

 
ò
.


(3b)


w
here,



2 2
2
2 2 2
1 1
r r r r

  
   
  


The dynamical equations of motion in cylindrical coordinate


,,
r z


are (Love, 1944
)



w
here,
,,,.,,
rr r rz rr z zz
s s s s s s s
  
are the respective stress compone
nts,
,,
R Z
T T T


are the respective
body forces and
,,
u v w

are the respective displacement components.


The stress
-
strain relations are


2
2
1 1
( ),
r
rr rz
rr R
s
s s
u
s s T
r r z r t





 

     
   


(4a)



2
2
2
1
,
r z r
s s s s
v
T
r r z r t
   



  

    
   



(4b)



2
2
1
.
z
rz zz rz
Z
s
s s s
w
T
r r z r t




 

    
   


(4c)


4


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


0 0 0
11 12 13
,
rr rr zz
s e e e

  
  

(5a)



0 0 0
21 22 23
,
rr zz
s e e e
 
  
  


(5b)


0 0 0
31 32 33
,
zz rr zz
s e e e

  
  


(5c)


0
44
,
rz rz
s e




(5d)


0
55
,
z z
s e
 




(5e)


0
66
.
r r
s e
 




(5f)


w
here,

ij


elastic constants (
ij
= 1, 2……6).


The elastic constants of viscoelastic medium are
(Christensen, 1971)


2
0///
2
ij ij ij ij
t t
   
 
  
 
(
ij
= 1, 2……6).




(6)



w
h敲攬

/
ij

慮搠
//
ij

慲攠瑨攠firs琠慮搠獥do湤nor摥爠d敲iv慴iv敳eof
.
ij




周T s瑲慩渠捯浰m湥湴猠慲攠


1
,
2
rr
u
e
r





(㝡)


1 1
,
2
v u
e
r r



 
 
 

 


(㝢)




1
,
2
zz
w
e
z





(7挩


1 1
,
2
z
w v
e
r r z

 
 
 
 
 
 


(7搩


1
,
2
rz
w u
e
r z
 
 
 
 
 
 


(7攩


1
,
2
zz
w
e
z





(7f)

周T ro瑡瑩o湡氠no浰o湥湴n 慲攠

5


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves


5

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


1 1
,
2
r
w v
r z

 
 
  
 
 
 


(8a)


1 1
,
2
u w
r z r

 
 
  
 
 
 



(8b)


1 ( )
.
z
rv u
r r

 
 
  
 
 
 



(8c)


Equations
governing the propagation of small elastic disturbances in a perfectly conducting visco
e-
lastic solid having electromagnetic force


J


(the Lorentz force,

J

is the
current density

and

being magnetic induction vector) as the only body force are (using Eq. (4))





2
2
1 1
( ),
r
rr rz
rr
R
s
s s
u
s s J
r r z r t





 

      
   



(9a)





2
2
2
1
,
r z r
s s s s
v
J
r r z r t
   



  

     
   



(9b)





2
2
1
.
z
rz zz rz
Z
s
s s s
w
J
r r z r t




 

     
   



(9c)




Let us assume the components of
magnetic field intensity

are
0
r

   
and
z
  
constant.
Therefore, the value of magnetic field intensity is
(
Thidé,
1997
).




0
0,0,
i
   

(10)






w
here,
0


is the initial magnetic field intensity along z
-
axis and
i


is the perturbation in the ma
g-
netic field intensity.


The relation between magnetic field intensity

and magnetic field induction

is


0

 

(For vacuum,
0

=
7
4 10




SI units.) (11)


From Eq. (1), Eq. (2), Eq. (3) and Eq. (10), we get


2
0
v
t t

 
 
 
 
    
 
 
 
 
 
 


(12)





6


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


The components of Eq. (12) can be written as
(
Thidé,
1997
).


2
,
0
1
r
r
t


  



(13a)




2
,
0
1
t




  



(13b)



2
0
1
.
z
t


  



(13c)



3 FORMULATION OF THE

PROBLEM


Let us consider a semi
-
infinite hollow cylindrical tube of radii
α

and
β
. Let the elastic properties of
the shell are symmetrical about z
-
axis, and the tube is placed in an axial magnetic field surrounded by
vacuum. Since, we are investigating the torsional waves in an aeolotropic
cylin
d
rical

tube therefore
the displacement ve
ctor has only
v

component. Hence,


0,
u


(14a)


0
w


(14b)


(,).
v v r z


(14c)


Therefore, from Eq. (14) and Eq. (7), we get,


0,
rr zz zr
e e e e

   

(15a)



1
,
2
z
v
e
z


 

 

 

(15b)




1
.
2
r
v v
e
r r


 
 
 

 


(15c)



From Eq. (14) and Eq. (8), we get,


1
,
2
r
v
z

 
 
 

 


(16a)



0,

 


(16b)




(16c)

7


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves


7

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


.
z
v v
r r

  





Using Eq. (14), Eq. (15) and Eq. (6), the Eq. (5) becomes


0,
rr zz rz
s s s s

   

(17a)


2
///
66 66 66
2
1
( ) ( ),
2
r
v v
s
t t r r

  
  
   
  



(17b)



2
///
55 55 55
2
1
( )( ).
2
z
v
s
t t r

  
  
   
  


(17c)



w
here,
/
ij

and
//
ij

are the first and second order derivatives of
.
ij



For perfectly conducting medium, (
i.e.




), it can be seen that Eq. (2) becomes



0
,0,0
v
c t



 
  
 

 

(18)





Eq. (1) and Eq. (18), the Eq. (13) becomes,



0,,0
i
v
z

 
  
 

 

(19)





From the above discussion, the electric and magnetic components in the problem are related as



0
,0,0 0,,0
v v
c t z


 
 
 
  
 
 
 
 
 

(20)





Using Eq. (19) and Eq. (1) to get the components of body force in terms of
SI

system of units as:



2
2
2
0,,0
e
v
H
z

 


 

 


(21)







Eq.
(17)
and Eq.
(20) satisfy
the Eq.
(4a)

and Eq.
(4c)
, therefore, the remaining Eq.
(4b)
becomes


8


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


2 2
//////
66 66 66 55 55 55
2
2 2
2
2 2
///2 2
66 66 66
2 2
1 1
( ) ( ) ( )( )
2 2
2 1
( ) ( )
2 2
e e
v v v
v
r t t r r z t t r
t
v v p v
H E
r t t r r z
     

    
 
       
      
 

       
 

 

   
 
 
      
 
 
   
 
 


w
here,

p


is initial compression stress,
e

and
e

are the permeability and permittivity of the
material.





(22)




Let

//////
,,
l l l
ij ij ij ij ij ij
C r C r C r
  
  
and
0
m
r
 


(23)







w
here,
ij

,
/
ij

,
//
ij


and
0

are constants,
r

is the radius vector and
,
l m

are non
-
homogeneities
.


From Eq. (23), we get Eq. (17) as



2
///
66 66 66
2
1
( ) ( ),
2
l
r
v v
s r
t t r r

  
  
   
  


(24a)



2
///
66 66 66
2
1
( ) ( ),
2
l
r
v v
s r
t t r r

  
  
   
  



(24b)



Using Eq. (23), the Eq. (22) becomes


2 2
//////
66 66 66 55 55 55
2
2 2
0
2
2 2
///2 2
66 66 66
2 2
1 1
( ) ( ) ( ) ( )
2 2
2 1
( ) ( )
2 2
l l
m
l
e e
v v v
r r
v
r t t r r z t t r
r
t
v v p v
r H E
r t t r r z
     

    
 
       
      
 

       
 

 

   
 
 
      
 
 
   
 
 

(25)


w
here,

p


is initial compression stress,
e

and
e

are the permeability and permittivity of the material.


4 SOLUTION OF THE PR
OBLEM


Let
( )
( )
i z t
v r e
 




(Watson, 1944)

be the solution of Eq.

(25). Hence, Eq. (25) reduces to



2
2 2
1 2
2 2
( 1) ( 1)
0
l
l l
r r r r r
  
 
   
    
 


(26)








w
here,


9


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves


9

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


2///2 2
2
0 55 55 55
1
///2
66 66 66
2 ( )
,
i
i
     
   
  
 
 



(27a)



2 2
2
2
2
///2
66 66 66
2 2
.
( )
e e
H E
p
i
 

   
 
 
 
 
 
 




(27b)



Eq. (26) is in complex form, therefore we generalize its solution for
0
l


and
2
l




4.1 Solution for
0
l



For,
0
l


the Eq. (26) becomes,


2
2
2 2
1 1
( ) 0
r r r r
 

 
    
 


(28)



w
here,

2 2 2
1 2
   



(29)



The solution of Eq. (28) is


( )
1 1
{ ( ) ( )}
i z t
v PJ Gr QX Gr e
 

 


(30)


From Eq. (24) and Eq. (30)


///2 ( )
66 66 66 0 1 0 1
2 2
{ } { ( ) ( ) { ( ) ( )
2 2
i z t
r
P Q
s i GJ Gr J Gr GX Gr X Gr e
r r
 

   

 
     
 
 





(31)





5 BOUNDARY CONDITION
S AND FREQUENCY EQUA
TION


The boundary conditions that must be satisfied are


B1. For
r


α
, (
α

is
the internal radius of the tube)



0
( )
r r r
s
  
 
 


B2. For
r


β
, (
β

is the external radius of the tube)



0
( )
r r r
s
  
 
 



10


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


w
here
r


and
0
( )
r



are the Maxwell stresses in the body and in the vacuum, respectively. There will
be no impact of these Maxwell stresses. Hence,


0
( )
0
r r
 
 
 

(32)


On simplification, Eq. (18) and Eq. (30) gives


( )
0
1 1
{ ( ) ( )}
i z t
i PJ Gr QX Gr e
c
 




   


(33)

Let
,


( )
0
i z t
e
 

  


Hence, Eq. (3) becomes


2
2
2
1
0
r r r

  
  
 

(34)


w
here,

2
2 2
2
c

 
 


(35)


The solution of the Eq. (34) becomes


0 0
( ) ( )
RJ r SX r
 
 


(36)


w
here
0
J

and
0
X
are Bessel functions of order zero. R and S are constants.


From Eq. (37) and Eq. (40)


( )
0 0
{ ( ) ( )}
i z t
RJ r SX r e
 
 

 

(37)








The
boundary conditions B1 and B2

with the help of the Eq.
(31)
and
(32)
turn into:


0 1 0 1
{ ( ) 2 ( )} { ( ) 2 ( )} 0
P G J G J G Q G X G X G
     
   

(38)


0 1 0 1
{ ( ) 2 ( )} { ( ) 2 ( )} 0
P G J G J G Q G X Ga X G
    
   

(39)


Eliminating P and Q from Eq. (38) and Eq. (39)


0 1 0 1
0 1 0 1
( ) 2 ( ) ( ) 2 ( )
0
( ) 2 ( ) ( ) 2 ( )
G J G J G G X G X G
G J G J G G X Ga X G
     
    
 

 

(40)



On solving Eq. (40), we get the obtained frequency equation


11


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves


11

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


0 1 0 1
0 1 0 1
( ) 2 ( ) ( ) 2 ( )
0
( ) 2 ( ) ( ) 2 ( )
G J G J G G X G X G
G J G J G G X G X G
     
     
 
 
 

(41)



On the theory of Bessel functions, if tube under consideration is very thin i.e.
  
 

and n
e-
glecting
2 3
,........
 
 
, the
frequency equation can be written as (Watson [18])


3 2
1 0

  

(42)





w
here,


2 2
2///2 2 2
0 55 55 55
2
///2
66 66 66
2 ( )
2 2
e e
H E
i p
i
 
      
   
 
     
 
 
 
 


(43)




Putting the value of

in Eq. (42), the frequency

of the wave can be found. Clearly, frequency

is
dependent on magnetic field
, electric field
and initial pressure
.


Put ,



 

(44)


The phase velocity
1
/
c
 


can be written as


2 2
2
2
2
1
2///2
0 66 66 66
2 2
2
e e
H E
p
c
c i
 

    
 
 
 
 
 
  
 
 
 



(45)



w
here,

2
,
k



///2
55 55 55
///2
66 66 66
,
i
i
   
   
 
 
 


///2
2
66 66 66
0
0
2
i
c
   

 





(46)


The term
s

,
E


and
p

are

negative in Eq. (45) which
means that the combine effect of magnetic field,
electric field and initial pressure reduces

the phase velocity of torsional wave.


Case 1


Since the pipe under
consideration is made of an aeolotropic material, then

///
0
ij ij
 
 

(47)






12


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


Hence, from Eq. (42), Eq.

(44) and Eq. (47) the frequency equation becomes


3
0 0
0

   

(48)


Using Eq. (45) and Eq. (46), the phase velocity is


2 2
2
2 2
66 55
2 0
0 66 66
2 2
2 2
e e
H E
p
c
 
 

   
 
 
 
 
 
 
 
 
   
 
 
 
 
 
 



(49)




1
2 2
2
2
0
55
2
2
0 66 66
[ ]
2 2
2
[ ]
e e
H E
p
c
or
c
 



 

 
 

 
 
 
 
 
  
 
 
 
 




(
50)


w
here,

2
0 66 0
2
c
 






The term
s

,
E


and
p

are negative
in Eq. (49) which reduces the phase velocity of torsional wave.
This is in complete agreement with the corresponding classical results
given by
Chandrasekharaiahi
(
1972)
.


Case 2


If the pipe under consideration is made of an isotropic material, then


///
55 66
0,
ij ij
    
   

(51)



Using Eq. (49) and Eq. (50), the phase velocity is


2 2
2
2 2
2 0
0
2 2
1
2 2
e e
H E
p
c
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
 
 
 



(
52)


This is in complete agreement with the corresponding classical results
given by
Narain (
1978)
.



5.1Solution for l=2


For,
2
l


the Eq. (26) becomes,

13


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves


13

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx



2
2
2
2
1
2 2
(3 )
3
( 0
r r r r
 


 
    
 


(53)


Putting
1
( )
r
r

 
in Eq. (53), one get


2 2
2
1
2 2
1
0
r r r r
 
   
    
 
 
 


(54)



w
here,


2 2
2
3
  



(55)


Soluti
on of Eq. (54) will

be (Watson, 1944
)


1 2
( ) ( )
RJ r SX r
 
    

(56)


Putting the value of


and


in Eq. (55),
we get


( )
1 1
1
{ ( ) ( )}
i z t
RJ r SX r e
r
 

 
    


(56)


From the Eq.

(24) and Eq. (56)


1 1 1 1
///2 ( )
66 66 66
1 1 1 1
{ ( ) ( 2) ( )}
2
( ) 0
{ ( ) ( 2) ( )}
2
i z t
r
R
rJ r J r
s i e
S
rX r X r
 

   
 

 
 
    
 
   
 
 
     
 
 



(57)



With the help of Eq. (32), Eq. (56) and boundary conditions B1 and B2, we get


1 1 1 1 1 1 1 1
{ ( ) ( 2) ( )} { ( ) ( 2) ( )} 0
2 2
R S
J J X X
     
   
           



(
58)




1 1 1 1 1 1 1 1
{ ( ) ( 2) ( )} { ( ) ( 2) ( )} 0
2 2
R S
J J X X
     
   
           


(
59)



Eliminating R and S from Eq. (58) and Eq. (59)


1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
0
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
J J X X
J J X X
     
     
   
   
         

         


(60)


On solving Eq. (60), we get


14


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
J J J J
X X X X
     
     
   
   
         

         


(61)


If η
1

is the root of the above equation, then



1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
J J FJ F J F
X X F X F X F
     
     
   
   
   

   




(62)



w
here,
1
F




On the theory of Bessel functions, if tube under consideration is very thin i.e.
  
 

and n
e-
glecting
2 3
,........
 
 
, the frequency equation can be written as (Watson
,

1944
)


2 2 2
1
1
1
( 2) 2 1 ( 2) 0

 
      
 

 


(63)



where
,

2 2
2
2 2 2
2
///2
66 66 66
2 2
3 3,
( )
e e
H E
p
i
 

   
 
 
 
 
     
 






(64a)



2///2 2
2
0 55 55 55
1
///2
66 66 66
2 ( )
.
i
i
     
   
  
 
 


(64b)




From the Eq. (62), Eq. (63) and Eq. (64), the phase velocity can be written as (same as above Eq. (45)
and Eq. (46))


2
///2
2
2
55 55 55
2///2
0 66 66 66
2
i
c
c i
   


    
 
 
 
 
 
 


(65)



Case 1


Since the pipe under consideration is made of an aeolotropic material, then


///
0
ij ij
 
 

(66)



The frequency equation is given by


15


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves


15

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


1 1 1 1
1 1 1 1
3 1 3 3 3 1 3 3
3 1 3 3 3 1 3 3
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
J J J J
X X X X
     
     
     
     
         

         


(67)




3
2 2
6 3 0
  
  






(68)


2 2
2
2
1
66
2 2
3,
e e
H E
p
 


 
 
 
 
  

2 2
2
0 55
3
66
2
,
 


 

2 3
 
 
1
1
at
 



(69)



Using Eq. (65), Eq. (66), Eq. (67) and Eq. (69), we get (calculations are done in the similar manner as
for the Eq. (48) to Eq. (50) for
0
l

case)


1
2
2
2
3 55
2
01 66
2
c
c






 
 
 
 
 
 
 
 
 
 
 
 
 


(70)



w
here,
2
01 66 0
/2
c
 



Case 2


If
the pipe under consideration is made of an isotropic material, then


///
55 66
0,
ij ij
    
   

(71)



The frequency equation (calculations are done as for the
l
=0 case) is




2 2 2 2
2 2 2 2
4 1 4 4 4 1 4 4
4 1 4 4 4 1 4 4
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
{ ( ) ( 2) ( )} { ( ) ( 2) ( )}
J J J J
X X X X
     
     
     
     
         

         






(72)



w
here


2 2
2
2
2
2 2
3,
e e
H E
p
 


 
 
 
 
  


2 2
2
0
4
2
.
 


 



Using Eq. (71) and Eq. (72), the phase velocity for this case is (same as above Eq. (45) and Eq. (46))


16


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


02
2
2
2
4
2
2
2
1
c
c




 
 
 
 
 
 
 
 
 
 
 
 
 




(73)



w
here,

02
2
0
2
c
 




7

NUMERICAL RESULTS


The effect of
non
-
homogeneity
, electric field and magnetic field

on torsional waves in an aeolotropic
material made of viscoelastic solids has been studied. The numerical computation of phase velocity
has been made for homogeneous and non
-
homogeneous pipe. The graphs ar
e plotted for the two
cases (l=0 and l=2). Different values of
α/λ (diameter/wavelength) for homogeneous in the pre
s-
ence of
electro
-
magneto

field and
non
-
homogeneous

case in the absence of
electro
-
magneto field

are
calculated from Eq. (49) and Eq. (65) wi
th the help of MATLAB. The variations elastic constants and
presence of
electro
-
magneto field

in two curves have been obtained by choosing the following p
a-
rameters for homogeneous and non
-
homogeneous aeolotropic pipe

(table 1)
.

The curves obtained in
fig.
1 clearly show that the phase velocity for homogeneous as well as non
-
homogeneous case d
e-
creases inside the aeolotropic tube. The presence of
electro
-
magneto field

also reduces the speed of
torsional

waves in viscoelastic solids. These curves justify the r
esults obtained in Eq. (50) and Eq.
(52) mathematically
given by
Narain (
1978)
and
Chandrasekharaiahi (
1972)
.





Table 1
:

Material

parameters



l


0




E (Volt/m)

H (Tesla)

P
(Pascal)


55 66
/
 


Homogeneous Pipe

0

2.33

15

0

0

0

0.9

Inhomogeneous
Pipe

2

2.33

15

50

0.32x10
4

0.1

0.9


Table 2:

Shows values of

2
0
c
c
(l =0) and
0
c
c
(l = 2) for
different

values of
α/λ (
diameter

/

wavelength)


α/λ

2
0
c
c

0
c
c

0.2

1.9849

2.5680

0.4

1.1662

1.5243

0.6

0.9393

1.2380

0.8

0.8455

1.1206

1.0

0.7985

1.0619

1.2

0.7717

1.0286

1.4

0.7557

1.0080

1.6

0.7441

0.9944

1.8

0.7365

0.9850

2.0

0.7310

0.9782

17


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves


17

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx




Fig
ure 1:

Torsional wave dispersion curves



We see that for homogeneous case when
electro
-
magneto field

is present and for non
-
homogeneous
case when
electro
-
magneto field

is not present the variation i.e. shape of the curves is same. For non
-
homogeneous case, the elastic constants and the density of the tube are varying as the square of the
radius vector.



6

CONCLUSIONS


The above problem deals with the interaction of
elastic and electromagnetic fields in a viscoelastic
media. This study is useful for detections of mechanical explosions inside the earth. In this study an
attempt has been made to investigate the torsional wave propagation in non
-
homogeneous viscoela
s-
tic
cylindrically aeolotropic material permeated by a
electric and
magnetic field. It has been observed
that the phase velocity decreases as the magnetic field
and
electric

field
increases.



ACKNOWLEDGEMENTS


The authors are thankful to the referees for their

valuable comments.


References


Abd
-
alla
,

A.N.,
(
1994
).

Torsional wave propagation in an orthotropic magnetoelastic hollow circular cylinder, Applied
Mathematics and Computation, 63: 281
-
293
.

Chandrasekharaiahi
,

D.S.,
(
1972
).

On the propagation

of
torsional waves

in magneto
-
viscoelastic solids,

Tensor, N.S.,
23: 17
-
20.

Christensen
,

R.M.,
(
1971
). Theory of Viscoelasticity,
Academic Press.

Dey, S., Gupta, A., Gu
pta, S. and Prasad, A. (2000).
Torsional surface waves in nonhomogeneous anisotropic mediu
m
under initial stress,
J. Eng. Mech
.,
126
(11)
:
1120
-
1123.

Dey, S., Gupta, A., Gupta, S., Karand, K. and De, P.K. (2003)
.
Propagation of torsional surface waves in an

elastic layer
with void pores over an elastic half space with void pores,
Tamkang J. Sci.
Eng
.,
6
(4)
:

241
-
249.

Dey, S., Gupta, A. and Gupta, S. (2002), “Effect of gravity and initial stress on torsional surface waves in

dry sandy
medium”,
J. Eng. Mech
.,
128
(10), 1115
-
1118.

Dey, S., Gupta, A. and Gupta, S. (1996)
.
Torsional surface waves in
nonhomogeneous and anisotropic

medium,
J.
Acoust. Soc. Am
.,
99
(5)
:

2737
-
2741.

0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.5
1
1.5
2
2.5
3
3.5
4
Diameter/Wavelength
Phase Velocity
l=0
l=2
18


R. Kakar
/
Electro
-
Magneto
-
Viscoelastic Torsional Waves

Latin American Journal of Solids and Structures xx (20
13
) xxx
-
xxx


Kakar, R., Gupta, K.C., (2013). Torsional surface waves in a non
-
homogeneous isotropic layer over viscoelastic half
-
space
,
Interact. Multiscale Mech.,

6
(1): 1
-
14.

Kakar, R.
,

Kakar,
S
.,
(201
2
).
Torsional waves in prestressed fiber reinforced medium

subjected to magnetic field
,

Jou
r-
nal of Solid Mechanics
,

4

(
4)
:

402
-
415
.

Kakar, R., Kumar, A., (2013).

A mathematical study of electro
-
magneto
-
thermo
-
Voigt viscoelastic surface wave

prop
a-
gation under gravity involving time rate of change of strain,

Theoretical Mathematics & Applications, 3(3): 87
-
106.

Kakar, R., (2013). Theoretical and numerical study of interfacial waves in non
-
homogeneous electro
-
magneto
-
thermoelastic orthotropic g
ranular half space,
Int. J. of Appl. Math and Mech.
9
(14): 90
-
115.

Kaliski
,

S., Petykiewicz
,

J.,
(
1959
).

Dynamic equations of motion coupled with the field of temperatures and resolving
functions for elastic and inelastic bodies in a magnetic field, Proceedings
Vibration
Problems
,

1
(2):17
-
35.

Love, A.E.H.,
(
1944
)
, Mat
hematical Theory of Elasticity,
Dover Publ
ications, Forth Edition.

Narain, S.
,
(
1978
).

Magneto
-
elastic torsional waves in a bar under initial stress, Proceedings Indian Academic.
Science
,
87 (5): 137
-
45.

Pal, P.C. (2000).
A note on torsional body forces in a viscoelastic half space”,
Indian J. Pur
e Ap. Mat
.,
31
(2)
:
207
-
213.

Selim, M.,
(
2007
).

Torsional waves propagation in an initially stressed dissipative cylinder, Applied Mathematical
Sciences, 1(29): 1419


1427.

Suhubi
,

E.S.,
(
1965
).

Small torsional oscillations of a circular cylinder with
finite electrical conductivity in a

constant
axial magnetic field, International Journal of Engineering Science, 2: 441.

Tang
,

L., Xu
,

X. M.,
(
2010
).

Transient torsional vibration responses of finite, semi
-
infinite and infinite

hollow cylinders,
Journal of

Sound and Vibration, 329(8): 1089
-
1100.

Thidé
,

B.,
(
1997
)

Electromagnetic Field Theory
,

Dover Publications.

Watson
,

G.N.,
(
1944
).

A Treatise on the Theory of Bessel Functions: Cambridge University Press, Second Edition.

White
,

J.E., Tongtaow
,

C.,
(
1981
).

Cylindrical waves in transversely isotropic media, Journ
al of Acoustic Society. Amer
i-
ca,
70(4):1147
-
1155.