ls1be">



 

 
 


PP
(48)
 


1
1
1
*
34
1131 1411 1111
10
1111
0
31
11
0
51
0
4
(1) 1
2
1,for therma lly insulated
2
,for isothermal
1
12
sk
k
Bk
nn
sk
k
n
k
sk
k
n
k
n
cc
EnncQs cskAsiDs
skD sP
E
Ds P
E
 





 










 

 
k
P
 
1
1
111
1
0
7111 1
01
,
5,6,7, 8,and7, 8
1
12
sk
Bk
n
k
il il
sk
k
Bk
nn
skQs APP
EEfori l
EskQsAPP








 


 





(49)
Here the elements

0
11 2,3,4,5,6
0
k
ll
EE l
for k

,1
of determinant Equations (47) and (49) can be obtained
by just replacing l
s
l

1,3,5,7
il
Ei

2,4,6, 8
il
Ei
1
in with
l, while are ob-
tained by replacing
, 2,3,4,5sl,6
in with 2

1, 3, 5, 7
il
Ei
.
The element il and in (48) can be
obtained by replacing Bessel’s function of first kind
,Ei5,6,7,88l
J
with that of second kind Y
and the elements
and l can be obtained by replacing 6, 8
il
Ei7, 81
in 5,7
il
Ei
and 7,8l
with 2
respectively.
The Equation (44) holds iff each term vanishes sepa-
rately. This implies that
0
00for =0, 0
il Xkn
E

 (50)
0 for0,0
k
k
il Xkn
E


 (51)
The Equations (50) and (51) have a non-trivial solu-
tion iff
00, ,1,2,3,4,5,6,7,8
il
Eil for k = 0, (52)
0,,1,2,3,4,5,6,7,8
k
il
Eil for (53) 0,0kn
After lengthy but straightforward simplifications and
reductions, the determinant Equations (52) and (53) lead
to the following secular equations.
77 8878870EE EEforn 1 (54)

 
2d
10
d
n
n
PmPfor n





1
where
(55)

0
det0,,1,2 ,3,4, 5, 60,0
ij
Eil kn (56)

det0,,1,2, 3, 4, 5, 60,0
k
ij
Eilkn (57)
 


 
1
1
1
1
034
113410 11
1011
0
31
011
0
51110 11
4
(1) 2
1,for thermally insulated
2
,for isothermal
1
12
s
s
s
s
B
cc
Ennc csAs
sDs
E
Ds
EskQsAs



 











 




(58)
 
  


1
1
1
771111
*
34
1131 14 11 1111
1111
31
11
51
3
2
4
(1) 1
2
1,for thermally insulated
2
,for isothermal
s
k
k
kk
sk
k
k
sk
k
k
B
EJJ
cc
EnncBs cskAsiDs
skD s
E
Ds
EQ

 
 











 











k

1
11 111
1
2
sk
k
sskAs
3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method
World Journal of Mechanics, 2012, 2, 98-112
doi:10.4236/wjm.2012.22012 Published Online April 2012 (http://www.SciRP.org/journal/wjm)
3-D Exact Vibration Analysis of a Generalized
Thermoelastic Hollow Sphere with Matrix
Frobenius Method
Jagan Nath Sharma, Nivedita Sharma
Department of Mathematics, National Institute of Technology, Hamirpur, India
Email: jns@nitham.ac.in, niveditanithmr@gmail.com
Received December 28, 2011; revised January 29, 2012; accepted February 15, 2012
ABSTRACT
This paper presents exact free vibration analysis of stress free (or rigidly fixed), thermally insulated (or isothermal),
transradially isotropic thermoelastic hollow sphere in context of generalized (non-classical) theory of thermoelasticity.
The basic governing equations of linear generalized thermoelastic transradially isotropic hollow sphere have been un-
coupled and simplified with the help of potential functions by using the Helmholtz decomposition theorem. Upon using
it the coupled system of equations reduced to ordinary differential equations in radial coordinate. Matrix Frobenius
method of extended series has been used to investigate the motion along the radial coordinate. The secular equations for
the existence of possible modes of vibrations in the considered sphere are derived. The special cases of spheroidal
(S-mode) and toroidal (T-mode) vibrations of a hollow sphere have also been deduced and discussed. The toroidal mo-
tion gets decoupled from the spheroidal one and remains independent of the both, thermal variations and thermal re-
laxation time. In order to illustrate the analytic results, the numerical solution of the secular equation which governs
spheroidal motion (S-modes) is carried out to compute lowest frequencies of vibrational modes in case of classical (CT)
and non-classical (LS, GL) theories of thermoelasticity with the help of MATLAB programming for the generalized
hollow sphere of helium and magnesium materials. The computer simulated results have been presented graphically
showing lowest frequency and dissipation factor. The analysis may find applications in engineering industries where
spherical structures are in frequent use.
Keywords: Thermal Relaxation; Matrix Frobenius Method; Toroidal; Poloidal; Hollow Sphere
1. Introduction
The theory of thermoelasticity is well established, Nowa-
cki [1]. The governing field equations in classical dy-
namic coupled thermoelasticity (CT) are wave-type (hy-
perbolic) equations of motion and a diffusion-type (para-
bolic) equation of heat conduction. Therefore, it is seen
that part of the solution of energy equation extends to
infinity, implying that if a homogeneous isotropic elastic
medium is subjected to thermal or mechanical distur-
bances, the effect of temperature and displacement fields
is felt at an infinite distance from the source of distur-
bance. This shows that part of disturbance has an infinite
velocity of propagation, which is physically impossible.
With this drawback in mind, Lord and Shulman [2],
Green and Lindsay [3], modified the Fourier law of heat
conduction and constitutive relations so as to get a hy-
perbolic equation for heat conduction. These works in-
clude the time needed for the acceleration of heat flow
and take into account the coupling between temperature
and strain fields for isotropic materials. Dhaliwal and
Sherief [4] extended the generalized thermoelasticity [2]
to anisotropic elastic bodies. A wave-like thermal distur-
bance is referred as “second sound” by Chandrasekha-
raiah [5]. These theories are also supported by experi-
ments of Ackerman et al. [6] that exhibiting the actual
occurrence of second sound at low temperatures and
small intervals of time. The investigators Singh and Shar-
ma [7] studied the propagation of plane harmonic waves
in homogeneous anisotropic heat-conducting elastic ma-
terials. Sharma [8], Sharma and Sharma [9] presented an
exact analysis of the free vibrations of simply supported,
homogeneous, transversely isotropic cylindrical panel
based on the three-dimensional generalized thermoelas-
ticity.
The free vibrations of solid and hollow spheres have
been the subject of study for a long period, frequently
associated with interest in the oscillations of the earth. In
the late nineteenth century, Lamb [10] showed that two
basic types of free vibrations namely, 1) the vibrations
with zero volume change and zero radial displacement;
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL. 99
and 2) the vibrations with zero radial components of the
curl of the displacement, exist in an isotropic sphere.
These vibrations are referred as “vibrations of the first
and second classes” respectively. Lapwood and Usami
[11] named the first class of vibrations as “torsional or
toroidal” and the second class as “spheroidal or poloidal”.
Lapwood and Usami [11] presented an excellent treat-
ment of the vibration of a hollow sphere surrounding a
liquid core having finite normal and shear rigidity which
serves as an approximate model of the Earth. Lamb [10]
derived the equations governing the free vibration of a
solid sphere and subsequently Chree [12] obtained the
secular equations of free vibrations of a sphere in the
more convenient form. Much later, Sâto and Usami [13,
14] computed and tabulated the natural frequency pa-
rameters for an extensive set of modes of vibration for
the solid sphere. They provided equations and a com-
prehensive set of results for the distribution of displace-
ment within the vibrating sphere. Shah et al. [15,16]
studied the vibrations of hollow spheres by using two-
dimensional theory of elasticity to obtain natural fre-
quency parameters. Gupta and Singh [17] investigated
the problems of wave propagation in transradially iso-
tropic elastic sphere. They showed that, for a transradi-
ally isotropic sphere, the toroidal and spheroidal modes
of vibrations are independent of each other.
Bargi and Eslami [18] used Green-Lindsay theory of
thermoelasticity to study the thermo-elastic response of
functionally graded hollow sphere and investigated the
material distribution effects on temperature, displace-
ment and stresses. Sharma and Sharma [19], studied the
generalized transradially thermoelastic solid sphere.
We have not come across any systematic and exact
study on the effect of temperature variations on three di-
mensional vibration of heat conducting elastic genera-
lized hollow spherical structures. Therefore, the purpose
of this paper is to present the exact three dimensional vi-
bration analysis of transradially isotropic, thermoelastic
generalized hollow sphere subjected to stress free (or
rigidly fixed), thermally insulated (or isothermal) boun-
dary conditions. The secular equations governing three
dimensional vibrations in a generalized hollow sphere
have been derived by using Frobenius series method. The
derived secular equations for spheroidal (S) modes of
vibrations which are dependent on thermal variation, have
been solved numerically for zinc and cobalt materials in
order to compute lowest frequency and dissipation factor.
The obtained results in case of toroidal vibrations are
found to be in agreement with those of Cohen et al. [20].
2. Formulation and Solution
We consider the thermoelastic problem for homogeneous,
transradially thermally conducting, elastic generalized
hollow sphere of inner and outer radii R
1 and R2, re-
spectively, initially maintained at uniform temperature
0 in the undisturbed state. For generalized spherically
isotropic thermoelastic medium, in the spherical polar
coordinates
T
,,,rt
, the basic governing equations of
motion, heat conduction and constitutive relations can be
expressed as follows Sharma and Sharma [19].
,r,r,
r
11
rsin r
12cot
r
rr r
rr r
 
 

u
 


 


(1)
,,,
11
rsin r
132cot
r
rr
r

u
 




 


(2)

,,,
11
rsin r
13cot
r
rr
r

u
 

 


 


(3)


3,,1 ,,,
2222
0
2
001 13
2
21cot1
sin
rr r
e
lrr
KTT KTTT
rrrr
CTtT
Tt eee
tt

 
 

 






 




(4)
where


11121311 2
44
12111311 2
44
2
2
rr l
rr
rr l
rr
cececeTtT
ce
cececeTtT
ce
 

 





(5)

131333312
66
1
2, ,
1cot
sin
11
2sin
11
2
cot
11 1
2sin
rrrr l
rr
rr
r
r
r
r
r
cececeTtT
u
uu
ce ee
rrr
uu
eu
rrr
uu
u
errr
uu
u
errr
uu u
errr
 
 
 


























(6)

111121133
3131333661112
2,
cc c
ccccc


 
 2
(7)
where rr
is stress along radial direction and r
, r
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL.
100
are along tangential direction. Here
,,
r
uuu
u
is
the displacement vector, is the temperature
change, 11 1213 and 44 are five independent
isothermal elasticity; 13
,,,Tr t

c
,
33
,,,cccc
and 13
,
K
K are respec-
tively, the coefficients of linear thermal expansion and
thermal conductivities along and perpendicular to the
axis of symmetry,
and e
C are the mass density and
specific heat at constant strain and and 1 are ther-
mal relaxation times respectively 1l
0
t t
is Kronecker’s
delta in which for Lord-Shulman (LS) theory and
for Green-Lindsay (GL) theory of thermoelasticity.
The comma notation is used for spatial derivatives and
the superposed dot denotes time differentiation. It can be
proved thermodynamically Sharma and Sharma [9] that
13
and of course
1l
0K
2l
0,K0
0
e
C

12 11
33 1112
,cc
cc


0,
and 0. We
assume in addition that and the isothermal elas-
ticity are components of a positive definite fourth-order
tensor. The necessary and sufficient conditions for the
satisfaction of latter requirement are
0T
22
12
2
13
,c
c
2
rR
11 11
44
0,
0,
cc
cc
rR
0, 0,
r r

(8)
3. Boundary Conditions
We consider the free vibrations of a generalized hollow
sphere subjected to stress free (or rigidly fixed), ther-
mally insulated (or isothermal) boundary conditions at
the surfaces 1 (inner radius) and (outer ra-
dius). Mathematically, this leads to
1) For stress free, thermally insulated (or isothermal)
boundary of the sphere.
rr


R
0, 0,
r
uu


1
R
,0
r
T
R
0,u
,0
r
T
2
rR
(or ) (9a)
0T
0T
at (inner radius) and r (outer radius);
1 2
2) For rigidly fixed, thermally insulated (or isothermal)
boundary of the sphere.
r
r
(or ) (9b)
at (inner radius) and (outer radius).
4. Solution of the Problem
We define the dimensionless quantities
2
30
11
44
13
12
22 123
44 44
33 11 12
466
44
11
33 0
10
1
44
, ,
,
,,
,,,, ,
,,,
ij
ii ij
e
s
ss
ss
T
rrRuu RRR
cC
c
cc
RRRc c
ccc
c
cc
c
v
KTc R
c
44
11
44
,,
,,
2
T
R
cc
Kt
00
,
T
c
tT
1
,
c
K
RT
tt


v
t
v
Tvv
t
cRR


 
 
 


 
 
(10)
where 2
44s
vc
and 44 3e
Cc K
are shear wave
velocity and characteristic frequency of the generalized
hollow sphere respectively. The primes have been sup-
pressed for convenience.
It is advantageous to express the displacements ,uu
and r in terms of functions u,,Gw
and de-
fined by Sharma and Sharma [19] as.
T
1
sin
G
u

 
, 1
sin
G
u




, r
uw
(11)
Using Equation (11) in Equations (1)-(4), we find that
22
2
12 12
22222
22 0
2
cc cc
rr
rrrrt


 




(12)
22
2*
0
22 2
2
** 2
01 2
2
20
l
KtT
rr t
rr t
tw
trrr
t




 











 G








(13)

22
2
112
2222
*
12
312
2
2
2
120
l
ccc
G
rr
rrrt
Tt T
cccw
rr r
r
 


 
 








(14)


22
132 2
4222
2
3
312
2
*
12
2
21
111
2
22 0
l
ccc
cw
rr
rrr
cccc G
rrr
r
Tt T
rr r
2
t


 
 



 







 


(15)
where
22
2
22
1
cot sin
2

 
Assume spherical wave solution of the form
  

  

  

  

1
0
1
0
1
,,, cos
1
,,, cos
1
,,, cos
1
,,, cos
itm
m
nn
n
itm
m
nn
n
itm
m
nn
n
itm
m
nn
n
rtUrP e
r
wrtWrPe
r
GrtV rPe
r
TrtT rPe
r


 
 
 
 
(16)
where
cos
m
n
P
is the Legendre polynomial; n and m
are integers and
s
R
v
is the dimensionless fre-
quency.
Upon substituting solution (16) in Equations (12)-(15),
we obtain
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL. 101
2
2
22
1
n
U

 


0
(17)

2
2*
4
20 2
2
**
0
41 0
2
n
nn
aT
nn
WV
  

 




 




(18.1)

2
22
23
2
*
1
1
11
0
nn
n
a
Vc
T

 

 



2
1
2
a
W
(18.2)


22
231
42 3
22
*
1
1
111
34 0
2
nn
n
aa
cWnnc
T


 

 





 


V
1
(18.3)
where
1
00
1
102 001
,,
,
ll
rti
ti ti
 
 

 
22
(19)
The quantities and
1,2, 3,4
i
ai
used in Equa
tions (17) and (18) are defined as








12 3
2
1
2
121
2
2
2
41
2
3
2
2
4
22
12
32
2
94
4
48
4
14
4
192 2
4
ccc
a
cnn c c
a
cnnccc
a
Knn
a
nn cc





 

32
1
(20)
The uncoupling of equations for the displacement po-
tential n from n
and n indicates the exis-
tence of two distinct modes of vibrations. The solution of
Equation (11) for corresponds to the Toroidal mode.
There is no effect of temperature and generalization on
toroidal mode. Toroidal frequencies will be same as ob-
tained by Cohen et al. [20].
U,
n
VW T
The solution of the spherical Bessel Equation (17) is
given by
  

12
1
,
nn
n
n
BJ BY
Un



(21)
where


22
12
19220
4nncc

 

and
J
and Y
is Bessel function of first kind and
second kind. 1n and 2n are arbitrary constants de-
termined from the boundary conditions.
B B
Generalized Series Method
The system of Equations (18) has been solved with the
help of matrix Frobenius method. Clearly the point
0r
(i.e. 0
) is a regular singular point of Equa-
tions (18) and all the coefficients of the differential Equa-
tions (18) are finite, single valued and continuous in the
interval 12

where 11
R
and 22
R
.
The quantities satisfy all the necessary conditions to have
series expansions and hence the Frobenius power series
method is applicable to solve the coupled system of dif-
ferential Equations (18). Thus, we have took the solution
vector of the type
0
s
k
nk
k
YZ
(22)
where

nnnn
YWVT
,
kkkk
Z
ABD
, where s is a
constant (real or complex) to be determined and ,
kk
A
B
rR
,
k are unknown coefficients to be determined. We need
solution in the domain 12
, 1. The solu-
tion (22) is valid in some deleted interval
D
RrR R0
0
,
2
RR
(about the origin) where is the radius of
convergence.
R
Upon substituting solution (22) along with its deriva-
tives in Equations (18) and simplifying, we get
 
21
12
0
0
sk
k
k
HskH skHZ



 

(23)
where
*
0
1, 1,Hdiag
,




1
2
,, 1,2,3
, ,1,2,3
ij
il
HHskij
sk
HHskil
sk
 
 
(24)
The elements, ij
H
and ij
H
, of matrices 1
H
and
2
H
are defined as
 















22
11 43
2
123 1
2
213 1
22
22 2
22
33 4
*1
23
*
13 1
**
31 0
** 2
32 0
,
11
1,
,
34 ,
2
41
2
Hskcsk a
H
sknnc ska
Hsk cska
Hsksk a
Hsksk a
Hsk
Hsk sk
Hsk sk
Hnn



 

 


 

 
 


 





 



 
(25)
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL.
102
Equating to zero the coefficients of lowest powers of
in Equation (23), we obtain:
2
(.. 0)
s
ie


10
0HsZ (26)
where

0111
Z
ABD
,

1,1,2,
il
HsH sil3
(27)
For the existence of non-trivial solution of Equations
(26) we must have

10Hs, which results in the fol-
lowing indicial equations
42
22
4
0
0
sAsC
sa

 (28)
where the coefficients
A
and C are given by


2
22
342 3
4
11acann c
Ac

,
22 4
23 1
4
1aanna
Cc

The roots of indicial Equations (28) are given as
22
22
12
22
34
4
,
22
4
,
A
ACAAC
ss
sa
 

(29)
Clearly the roots j are related
through the relation 415263

1, 2,3, 4,5, 6
sj
,,
s
ss ss s, in
which 3
s
is real but the roots 1
s
and 2
s
may be, in
general, complex. In case the parameter, s, is complex,
then leading terms in the complex series solution (22) are
of the type:


00 00
0coslogsin log
RI
R
ss
s
s
II
ABD Z
Zsis

 


(30)
In order to obtain two independent real solutions, ac-
cording to Neuringer [21], it is sufficient to use any one
of the complex root in a part and taking its real and
imaginary parts. Also, the treatment of complex case is
unlike that of the real root case with the advantage that
the differential equation is required to be solved only
once in the former case rather than twice as in latter one.
For the choice of roots of the indicial equations, the sys-
tem of Equations (27) leads to following eigen vectors:
 



 

01 0410
02 0510
03 060
10
10
001
B
B
Zs ZsQsM
Zs ZsQsM
Zs ZsM
 





,
,
(31)
where




222
31 43
22 2
231
1
11
1, 2,3, 4,5,6
jj
Bj
jj
cs acsa
Qs sa nnc sa
j
 


 

and 0
A
is a constant. Thus we have


00
00
00
110 ,
120
001
BB
AM
BQ QM
DM
,
(32)
as the corresponding eigen vectors. Again equating to
zero the coefficients of next lowest degree term 1
s
in Equation (23) and noting that the matrix
1
j1
Hs
is nonsingular for each j, we obtain:

1*
11 201
1
jj
0
Z
HsHsZDZ
 (33)
where


1
*
11 2
1,
,1,2,3
jj
DHsHsA
il
 
il
(34)
The matrices
11
j
Hs
and

2
j
H
s
k
can be writ-
ten from the Equation (23) by setting and
1
il
A
are defined in the Appendix A1.
Now equating the coefficients of powers of
s
k
equal to zero, we obtain following recurrence relation:

1221
21
0,1,2
kkk
H
skZHsk ZHZ
k

 
(35)
where the matrices 12
,
H
H and
H
are defined in
Equation (23). This implies that


1
21
21
2
1
kj
j
kk
ZHsk
H
sk ZHZ
 

(36)
Now putting 0,1,2,3k
in Equation (36) suc-
cessively and simplifying, we get



2
1**
121
*
20
21
k
jjk
k
Z
0k
H
skHsk DHDZ
DZ
  
(37)
It can be easily shown that the matrix has simi-
lar form to that of
*
2k
D
12
j
Hsk
for even values of
and it is alike
k
22
j
Hs k
for odd values of k.
Thus we have:
**
22220 21210
,,
0, 1,2,3
kk kk
ZDZZDZ
k
 

(38)
where
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL.
Copyright © 2012 SciRes. WJM
103
*
*
,
j
3
**






1
**
221221 233
1
**
21122 2133
22 21
21 2
kj jkki
kj jkkij
DHskHskDHDK
DHskHskDHDK










(39)
Here the elements are given
by Equations (A2)-(A3) as defined in the Appendix.
Moreover, it can be shown that
,,1,2,
ij ij
KK ij
in Equation (22) is analytic and hence can be differenti-
ated term by term. Moreover, the derived series are also
analytic functions.
Thus the general solution of the system of Equations
(18) has the form

*2
22k
Dok
*
D

*1
21k
DokD
, (40)
where



6
1
,,
,,
nnn
s
jk
jk kjkjkj
jko
WVT
CAsBsDs


 (42)

**
*1
4
**
101and
is33 null matrix
Ddiag
c
D

(41)
where
1,2,3,4,5,6
j
sj are the eigen-values and
,,
kjkj kj
A
sBsDs are eigenvectors correspond-
ing to the eigen-values
j
s
and integer . The quanti-
ties
k
j
k are arbitrary constants to be evaluated. Conse-
quently, the potential functions and T are writ-
ten from Equations (12) by using (42) as under:
C
,wG
Noting that both the matrices and
*
22 0
k
D
*
21 0
k
D
lim (
k
PP
, as and using the fact that
k
k , if each component sequence
converges (Cullen [22]), we can conclude that the series
(22) are absolutely and uniformly convergent with in-
finite radius of convergence. Therefore, the considered series
k

P P 2
k

 

 

  

16
2
1
1
2
12
1
,, ,,,,,cos
,,,() cos
sj k
im t
m
njkk jkjk jn
njko
im t
m
nnn
n
wGTr trCAsBsDsPe
rtrBJBYP e


 
  




(43)
5.1. Stress-Free Generalized Hollow Sphere
The unknowns , 2n and
1n
B B
1,2,3, 6
njk
Cj4,5, can be evaluated by using
boundary conditions (9) at the inner and outer boundaries
of the generalized hollow sphere.
Upon employing stress free and thermal boundary condi-
tions (9a) at the surface 1

and 2

of the
sphere and simplifying we have
 

00
0
1
0
,1,2,3,4,5,6,7,8
k
il il
nk
XX
EE
il


 
 


 (44)
5. Secular Dispersion Relation
For a generalized spherically isotropic, thermally con-
ducting hollow sphere the stress free (or rigidly fixed),
thermally insulated (or isothermal) conditions (9) hold. where
0 10203040506012nnnn nnnn
X
CCCCCCBB (45)
1234561knknknknknknknn2
X
CC CC CCBB (46)
 


 

1
1
1
1
034
11314 1011
1011
0
31
011
0
5111011
0
711 1
4
(1) 2
1,for thermally insulated
2
, for isothermal
3
1,
2
3
12
s
B
s
s
s
B
B
cc
EnncQscsAs
sk DsP
E
Ds P
EsQsAsPP
EsQs



 












 





 



P
1
011
s
As PP



(47)
J. N. SHARMA ET AL.
104
  
00
571111581111
11
33
,
22
nn
EJJPPEY Y
 









(59)
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL. 105
The elements of determinant
Equations (58) andby just replacing

02,3,4,5,6
il
Ej
(59) can be obtained
,1
l
s
l
while
in

1, 3, 5 with ,
l

,6 are obtained by
0
il
Ei
2,4
2,3,4,5,6sl
replacing 1
0
il
Ei
in

5
01,
il
Ei3, with 2
. The element
Bessel’s function
78
E can be ob-
of first kind tained by replacing
J
with that of second kind Y
and th
replacing 1
e elements 87
E
can be obtained by
in 87
E and 88
E with
2
respectively.
ularThe sec dispersion Equation (54) provides us first
class vibrations called Toroidal vibrations (T-modes) as
discussed by Cohen et al. [20]. Clearly these modes do
ot depend on thermal variations as expected. These are
h absence of radial c
pheroidal Mo
secular Eq
es
ristics of the Sp
roi
dal Mode
uation (54) provides us
n
c aracterized by theomponent of dis-
placement.
5.1.1. Sde
The uations (56) and (57) govern the second
class vibrations called Spheroidal vibrations (S-mod)
for 0n, 0k and 0,0kn and 0n respec-
tively. These relations contain complete informtion re-
garding frequency and other charactehe-
a
dal modes of vibrations in a transradially generalized
isotropic hollow sphere. The detail of Spheroidal vibra-
tions on neglecting the thermal effects and considering
only the radial vibrations have been discussed by Ding et
al. [23] in case of elastokinetics.
5.1.2. Toroi
The secular dispersion Eq
 
2
3*
tan *
3*
t
t
t
, for 1n (60)

1/2 3/2
1** *0Jt tJt



 , for 1n
(61)
where t*(=h/R) is thickness to mean radius ratio, where
thickness of the sphere is defined as 21
hR R and
mean radius as 21
2
RR
R
.
Clearly these modes do not depend on thermal varia-
quations (60) agree with Ding et
f radial
otropic
solid we have
tions as expected. The E
al. [23] and are characterized by the absence o
component of displacement. For homogeneous is
11 33121344
131 3
2,, ,
,.
ccccc
KKK


 
 
(62)
so that 1
2
n
 and the secular Equation (61) reduces
to
 
1/2
1
nn
nJ tJ

3/2
** *t t
 
0
for
(63)
It can be shown that Equation (63) is identical to the
one obtained by Love [24], [page 284, Equation (38)].
These modes have also been discussed in detail by Cohen
et al. [20] and Ding et al. [23] and the corresponding
frequencies of such Toroidal modes are same as in case
of elastokinetics. The analysis in case of coupled ther-
moelasticity (CT) can be obtained by setting
1n
01
0tt
) by taking and for uncoupled thermoelasticity (UCT
0
, 01
0tt
nd all othe
theories of dynam
d from the a
in the present study. Th
r relevant results in case of L
ic generalized thermoelas
bove analysis by taking
e secular equ-
S and GL
ticity can be
1
l
ations a
obtaine
and
2
l
, respectively, in Equations (4) and (
th
ly fixed and thermal boundary conditions
(9b) at the surface
5) then using
e resulting values of these parameters in different rela-
tions at various stages.
5.2. Rigidly Fixed Sphere
Taking the rigid
and 2

rmin
1
of the sphere on
displacements we get following deteantal equations
0, 1,2,3,4,5,6
ij
dij (64)

0YJ Y
 
12 21
J

(65)


2d
10
d
n
n
PmP

 (66)
here cos
,


w


1
1
1
1
1111
3111
111
51
11
1,for thermally insulated
2
,for isothermal
sk
k
sk
k
sk
k
sk
k
dAs
dBs
sDs
d
Ds



(67)
The elements
2,3,4,5,6
il
dl of determinantal
Equation (64) can be obtained by just replacing,1
l
s
l
in
1,3,5
il l with , 2,3,4,5,6
il
sl, while
d
'2, 4,6
il s
di are obtained by replacing 1
in
1,3 ,5
il
di with 2
.
5.2.1. Toroidal Vibrations
Equation (65) corresponds to first class vibrations (To-
roidal mode). Clearly these modes do not depend on ther-
mal and relaxation time variations as expected.
5.2.2. Spher oidal Vibrations
The secular Equations (64) govern the Spheroidal vibra-
tions (S-modes) in case of and
respectively in a transradGene
moelastic hollow sphere subjected to stress fr
conditions. These relations contain complete information
0, 0nk
ial isotropic
0, 1k, n
ralized ther-
ee boundary
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL.
106
regardin cy ag frequennd other characteristics of the Sphe-
roidal modes of vibrations in a transradially isotropic
Generalized thermoelastic hollow sphere.
6. Numerical Results and Discussion
We consider the case of free vibrations of a transradially,
isotropic generalized thermoelastic hollow spheres made
up of solid helium and magnesium materials whose
physical data is given in Table 1. As given by Sh
and Dhaliwal and Sin
general
plex transcens co
plex values of the fre
quency (ω) giv frequency
arma
d Sharma [19] angh [26].
Due to the presence of dissipation term in heat con-
duction Equation (4), the secular equations are in
comdental equations which provide um-
quency (ω). The real part of fre-
es us lowest(

1/2
44RRc

 )
and imaginary part provide us dissipation factor
(

1/2
44I
DRc

), where

Re
R
and

Im
I
, for fixed values of n and k. The computer
simulated profiles of lowest frequency and dissi-
presented res 1 to 1

pation factor

D in spheres of solid helium and mag-
nesium materials have beenin Figu 2
for different values of thickness to mean radius ratio (t*)
in the context of LS, GL, and CT theories of coupled
thermoelasticity. The values of the thermal relaxation
time has been estimated from Equation (2.5) of Chan-
drasekharaiah [27] 02
44
3
.
iz
e
K
vt Cc
and 1
t has been taken
proportional to that of . The sameerical technique
19].
es linearly and re
of sphe
nd CT,
neralized theorieoelasticity. It has been ob-
served that
(LS) > (C
ee
low
Heliu
0
has been used as given by Sharma and Sharma [
t num
Figures 1 and 3 show the variations of lowest fre-
quency () with thickness to mean radius ratio (t*) for
different values of degree of spherical harmonics (n) for
solid helium and magnesium respectively in case of
stress free generalized hollow sphere. From Figure 3, it
is observed that the profile of lowest frequency increases
parabolically with the increase of t* and the order with
respect to generalized theories of thermoelasticity is
(CT) > (GL) >(LS) for n = 1 and n = 2. Figure 3
reveals that the lowest frequency vari-
mains dispersionless at all values of the degree-
rical harmonics (n) in the context of LS, GL a
ges of therm
the order for lowest frequency interlaces is
T) > (GL) for n = 1 and n = 2. It can be
concluded that with the increase of degrof spherical
harmonics (n),est frequency increases. From both the
Figures we conclude that with the increase of t* lowest
frequency increases.
Figures 2 and 4 show the variations of damping factor
(D) with thickness to mean radius ratio (t*) for different
values of degree of spherical harmonics (n) for solid he-
Table 1. Physical data for helium and magnesium crystals.
QuantityUnitsm Magnesium
11
c 2
Nm
10
0.404010 10
5.974 10
12
c 2
Nm
10
0.212010 10
2.624 10
13
c 2
Nm
10
0.0105010 10
2.17 10
33
c 2
Nm
10
0.553010 10
6.17 10
44
c 2
Nm
10
0.1245 10 10
3.278 10
1
3
21
Nm deg
21
Nm deg
6
2.3620 10
6
2.641 10
6
2.68 10
6
2.68 10
1
K
3
K
11
Wm deg
11
Wm deg
2
0.3 10
1
0.2 10
2
1.7 10
2
1.7 10
0.04162 0.0202
-1
s 13
1.9890 10 11
3.58 10
0
t s 13
0.0000091 10
11
0.27910
0
0.02
0.04
01 0.0250.055 0.07
85 0.1
t*
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0. 0.040.0
LS
GL
CT
LS
GL
CT
Hollow ball (n = 1)
Without ball (n =2)
cy ()
Lowest equen
Figure 1. Variation of lowest frequencwith thickness to
mean radius ratio for different values of degree of spherical
harmoics (n) for lium material ine of stress free
boundary condition.
y
nhe cas
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.01 0.0250.04 0.055 0.070.0850.1
t*
Damping factoer (D
)
LS
GL
CT
LS
GL
CT
Hollow ball (n = 1)
Without ball (n =2)
Figure 2. Variation of damping factor with thickness to
mean radius ratio for different values of degree of spherical
harmonics (n) for helium material in case of stress free
boundary condition.
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL. 107
0
0.05
0.1
0.15
0.2
0.25
0.01 0.025 0.04 0.055 0.070.0850.1
t*
Lowest equency ( )
LS
GL
CT
LS
GL
CT
Hollow ball (n = 1)
Without ball (n =2)
Lowest equency ()
Figure 3. Variation of lowest frequency with thickness to
mean radius ratio for different values of degree of spherical
harmonics (n) for magnesium material in case of stress free
boundary condition.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.010.0250.040.055 0.07 0.0850.1
t*
Damping factor (D)
LS
GL
CT
LS
GL
CT
Hollow ball (n = 1)
Without ball (n =2)
Figure 4. Variation of damping factor with thickness to
mean radius ratio for different values of degree of spherica
m crystals respectively in case of
stress free boundary condition. From Figures 2 and 4 it
is observed that the trend of profile of damping increases
linearly with t* and the order is
l
harmonics (n) for magnesium material in case of stress free
boundary condition.
lium and magnesiu
 
DLSDGLDCT
 
L DCT for n = 2 re- for n = 1 and
spectively.
An examination is made on the variations of dimen-
sionless lowest frequency with respect to thickness to
mean radius ratio (t*) ranging from thin spherical shell
(t* = 0.05) to the thick spherical shell (t* = 0.1) of iso-
tropic materials. The findings confirm that the variation
of the Spheroidal frequencies increases with t* and for
degree of spherical harmonics (n), the same trend of pro
ickness to mean radius ratio (t*) for
ifferent values of degree of spherical harmonics (n) for
two materials solid helium and magnesium respectively
in case of rigidly fixed boundary condition. For both the
materials similar behaviour has been observed i.e. the
lowest frequency increases with the increase of t*. In
 
DLS DG
-
file for damping has been observed i.e. damping in-
creases with the increase of t*.
Figures 5 and 7 show the variations of lowest fre-
uency () with thq
d
0
0.005
0.01
0.015
0.02
0.025
0.01 0.025 0.04 0.0550.07 0.0850.1
t*
Lowest frequency ( )
LS
GL
CT
LS
GL
CT
Hollow ball (n = 1)
Without ball (n =2)
Lowest frequency ()
Figure 5. Variation of lowest frequency with thickness to
mean radius ratio for different values of degree of spherical
harmonics (n) for helium material in case of rigidly fixed
oundary condition.
b
0
0.01
0.02
0.01 0.025 0.04 0.0550.07 0.0850.1
t*
0.03
0.
Damping fa
04
0.05
0.06
0.07
ctor (D)
LS
GL
CT
LS
GL
CT
Hollow ball (n = 1)
Without ball (n =2)
Figure 6. Variation of damping factor with thickness to
mean radius ratio for different values of degree of spherical
harmonics (n) for helium material in case of rigidly fixed
boundary condition.
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
west frequency (
0.01 0.0250.04 0.055 0.070.0850.1
t*
Lo )
LS
GL
CT
LS
GL
CT
Hollow ball (n = 1)
Without ball (n =2)
west frequency () Lo
Figure 7. Variation of lowest frequency with thickness to
mean radius ratio for different values of degree of spherical
harmonics (n) for magnesium material in case of rigidly
fixed boundary condition.
Figure 5 the trend of profiles of lowest frequency inter
laces is
-

CTGL CT
for n = 1 and n = 2.
In Figure 7 the profile of lowest frequency interlaces is
CTLS GL in context of linear theories
of generalized thermoelasticity.
Figures 6 and 8 show the variations of damping factor
D) with degree of spherical harmonics (n) for two mate- (
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL.
108
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.01 0.025 0.040.055 0.07 0.0850.1
t*
Damping factor (D
)
LS
GL
CT
LS
GL
CT
Hollow ball (n = 1)
Without ball (n =2)
of damping factor with thickness to
ean radius ratio for different values of degree of spherical
terial in case of rigidly
fixed boundary condition.
rials solid helium and magnesium respectively, for n = 1
and 2. From both the Figures it is observed that the trend
of profile of damping factor increases linearly with t*. In
Figure 6 the order of damping is D(LS) > D(GL) >
D(CT) for n = 2 and the profile of damping interlaces as
D(GL) > D(LS) > D(CT) for n = 1. In Figure 8 the order
of damping is D(LS) = D(CT) > D(GL) for n = 2 and
D(LS) > D(CT) > D(GL) for n = 1.
Figures 9 and 11 show the variations of lowest fre-
be significantly affected with
solid and magnesium respectively. Figure 10
shows tmagnitude of dissipation is constant in the
region
Figure 8. Variation
m
harmonics (n) for magnesium ma
quency

with time ratio for different values of
thickness to mean radius ratio (t*) for solid helium and
magnesium respectively. The profiles of lowest fre-
quency are observed to
thermal relaxation time in both the materials with in-
creasing values of thickness to mean radius ratio (t*).
From both the Figures it is observed that with the in-
crease of time ratio lowest frequency is nearly constant
value, also lowest frequency has more value for thick
shell as compared to thin shells. Figures 10 and 12 show
the variations of damping factor (D) with time ratio for
different values thickness to mean radius ratio (t*) for
helium
hat the
10
00tt .2and there is sharp decrease in the
region 10
0.2 0.4tt and then for 100.4tt there
is slowing. In Figure 12 there is sharp
decrease to
decrea
se in damp
in damping up100.6tt and for
10 0.6
ed
ttreases. From both the figures it
is obwith the increase of time ratio the mag-
nitude of damping factor is nearly constant value, also
lowest frequency has more value for thick shells as com-
pared to thin shells.
7. Conclusions
zed thermoelastic hollow sphe-
has been investigated in the context of LS and GL
damping inc
serv that
The effect of thermal variations and thermal relaxation
time on lowest frequency and dissipation factor of sphe-
rical vibrations in generali
re
theories of thermoelasticity with the help of Matrix Frö-
benius method. The numerical computations have been
done with the help of MATLAB files.
It is noticed that the first class vibrations are not af-
fected by temperature change, thermal relaxation time
and remain independent of rest of the motion. The ob-
tained results are similar with the corresponding results
0
2
4
6
8
10
12
14
18
00.2 0.40.6 0.81
16
qy(
)
t*= 1/20
t*= 1/10
10
/tt
Lowest frequency ()
Figure 9. Variation of lowest frequency with time ratio for
various value of thickness to mean radius ratio (t*) for he-
lium material.
0
0
1
2
3
4
5
Damping FactD)
6
0.2 0.4 0.6 0.81
or (
t*= 1/20
t*= 1/10
10
/tt
Figure 10. Variation of damping factor with time ratio for
various value of t* for helium material.
0
5
10
15
20
25
00.2 0.4
0.6
0.81
t*= 1/20
t*= 1/10
st frequency ()
t
1
/t
0
Lowe
Figure 11. Variation of lowest frequency with time ratio for
various value of t* for magnesium material.
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL. 109
g
0
1
2
3
4
5
00.2 0.4 0.6 0.81
Dampinf Factor (D)
t*= 1/20
t*= 1/10
10
/tt
Damping Factor (D)
Figure 12. Variation of damping factor with time ratio for
[24], Lamb [10] and Cohen et al. [20]
elastokinetics. The lowest frequency of spheroidal
vibrations is noticed to be significantly affected due to
both temperature variations and relaxation time, in hol-
low spheres of both helium and magnesium materials.
The lowest frequency and dissipation factor have shown
strong dependency on the degree of spherical harmonics
(n) and hence the importance of the degree of spherical
harmonics must be taken into consideration while de-
signing a spherical structure.
The effect of relaxation time ratio on lowest frequency
and dissipation factor of vibration under consideration is
also observed in hollow spheres of helium and m-
ies where spherical structures are in frequent use.
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J. N. SHARMA ET AL. 111
Appendix
The quantities used in Equations (34) are defined as

,, 1,2,3
il j
As il
 
 
 
 




11
12
2213 1223
13
11 221221
21 12
2113 1123
23
11 221221
31
31
33
31
32
33
33
0
0
11
11 1
0, 0
11
11 1
1
,
1
0
jjjj
jj jj
jjj
jj jj
j
j
j
j
A
A
HsHs HsHs
AHsHsHs Hs
AA
Hs HsHs Hs
AHsHsHs Hs
Hs
AHs
Hs
AHs
A


1
1
j
 


 
 
(A1)
The quantities

,,1,
ij jijj
Ks Ksij
2,3
used in Equations (39) are by
















22 31
11 13
*
33
1221 11
2313 23
***
22 32
12 13
*
22
22 2
21 1
22 21
2221 21
22 2
2221 21
22 2
21
22
jj
j
jj
jjj
jj
jjj
jj
j
j
Hs kHs k
KHsk
HskH sk
Hs kHs kHs k
Hs kHs kH
Hs kHs kHsk
Hs kHs k
KHsk
Hs kH

 



 


 


 

 

 


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1223 32
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Hs kHs kHs k
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23
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j
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
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 
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23
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32 0
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jj
j
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kHs k
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 
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
 

(A2)
Copyright © 2012 SciRes. WJM
J. N. SHARMA ET AL.
Copyright © 2012 SciRes. WJM
112
and

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2
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,
21 21
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H
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
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
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 
(A3)