International Journal on Mechanical Engineering and Robotics (IJMER)
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NonlinearFree Vibration Response of Functionally Graded Materials
Cylindrical Shell in Thermal Environment.
1
P. D. Khaire, 2N. H. Ambhore, 3K. R. Jagtap
1,2
Department of Mechanical Engineering,Vishwakarma Institute of Information Technology, Kondhwa, Pune-48
3
Department of Mechanical Engineering,Sinhgad Institute of Technology and Science, Narhe, Pune -41
E mail:1Pradeep.khaire08@gmail.com, 2nitin.ambhore@gmail.com, 3krjagtap_sits@sinhgad.edu
Abstract- In this paper the nonlinear free vibration
response of functionally graded materials (FGMs)
cylindrical shell is investigated. The cylindrical shell is
subjected to uniform temperature distribution with
temperature independent (TID) material properties. The
basic formulation is based on higher order shear
deformation theory (HSDT) with Von-Karman nonlinear
strain kinematics using modified C0 continuity. A direct
iterative based nonlinear finite element method
(DIFEM)developed by last two authors for the
Functionally Graded Materials plate is extended for
cylindrical shell. The present outlined approach has been
validated with those available in literature.
Keywords:
FGM, cylindrical shell, nonlinear free
vibration, HSDT.
I. INTRODUCTION
cutouts in thermal environment by using higher order
shear deformation theory with C 0 continuity. K. R.
Jagtap et al. [2]presented stochastic nonlinear free
vibration analysis of functionally graded material plate
resting on elastic foundation in thermal environment by
using higher order shear deformation theory with von
Karman nonlinear strain kinematics with modified C 0
continuity. Achchhe Lal et al. [3] investigated nonlinear
bending response of laminated composite spherical shell
panel with system randomness subjected to hygrothermo-mechanical loading. A direct iterative based C 0
nonlinear finite element method combined with mean
centered first-order perturbation technique (FOPT) for
the plate is extended for the spherical shell panel
subjected to hygro-thermo-mechanical loading. Singh
B.N. et al. [4] investigated composite laminate plate
using finite element method. The formulation is based
on higher order shear deformation theory. Randomness
in the system properties are computed by using first
order perturbation technique. Hiroyuki Matsunaga
[5]studiedfree vibration and stability of functionally
graded shallow shells according to a 2D higher-order
deformation theory. Hui-Shen Shen and Hai
Wang[6]investigated the large amplitude vibration
behavior of a shear deformable FGM cylindrical panel
resting on elastic foundations in thermal environments.
The formulation is based on a higher order shear
deformation shell theory which includes shell panelfoundation interaction and the thermal effects. The
material properties of FGMs are assumed to be
temperature-dependent. The equations of motion are
solved by a two-step perturbation technique to determine
the nonlinear frequencies of the FGM cylindrical panel.
Functionally Graded Material (FGM) is a composite,
consisting of two or more phases, which is fabricated
such that its composition varies in some spatial direction
by changing the volume fraction index of constituent
materials. This design is intended to take advantage of
certain desirable features of each of the constituent
materials. For example, if the FGM is to be used to
separate regions of high and low temperature, thenat the
hotter end it may consist of pure ceramic as the ceramic
is having better resistance to higher temperatures. In
contrast, the cooler end may be pure metal because of its
better mechanical and heat transfer properties.Rapid
advances in the manufacturing techniques of bulk FGMs
have created exiting new possibilities of their
applications in large scale structural system such as
rocket heat shields, wear resistant lining in mineral
processing industry, thermoelectric generators, plasma
facing for nuclear reactors and electrically insulating
II. FORMULATION
metal/ceramic joints,thermoelectric generators, dental
implantation, andbone replacement and electrically
Consider schematic diagram a FGMs cylindrical shell
insulating metal/ceramic joints. A large number of
which consist of ceramic and metal at top and bottom
literatures have been reportedon linear and nonlinear
layer of length a, width b, and total thickness h.
free vibration of plates. K. R. Jagtapet al. [1]investigated
effect of random material properties on free vibration
response of functionally graded materials plate with
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International Journal on Mechanical Engineering and Robotics (IJMER)
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the mid surface with respect to x and y axes,  x and  y
are the slopes along x and y directions, θx =
θy =
dw
dy
dw
dx
and
.
The function f1 (z) and f2 (z) can be written as,
f1  z  = C1  z  - C2  z  ;
and f 2  z  = -C4  z  With
3
C1 = 1, C2 = C4 =
The properties of the FGMs shell are assumed to be
varying through the thickness. The effective mechanical
and thermal properties of the FGMs shell at an arbitrary
point within the shell domain are expressed as[2],
(1)
Where, t and b represents the ceramic and metal
constituents, E,α,ρ and k are the effective young
modulus, thermal expansion coefficient, density and
thermal conductivity, VC is the volume fraction index,
function of coordinate in the thickness direction (z),
z

Vc  z    0.5   ,
h


0n
v w  y  x  y  x ]T (4)
2.2 Strain Displacement Relations
    l    nl    t  (5)
k  z   k b  [k t  k b ]Vc  z 
n
  [u
The strain vector consisting of strains in terms of midplane deformation, rotations of normal and higher order
terms associated with displacement for isotropic layer is,
E  z   E b   E t  E b  Vc  z 
ρ  z   ρ b  ρ t  ρ b  Vc  z 
4h2
.
3
The displacement vector for the modified model can be
written as,
Fig.1 Geometry of cylindrical shell
α  z   α b   α t  α b  Vc  z 
3
h
h
z ,
2
2
 
Where  l  ,  nl  and  t are the linear and nonlinear
strain vectors (Von-Karman sense), thermal strain vector
respectively. The nonlinear strain vector can be written
as,
1
2
 nl  [ Anl ]  (6)
Where
 (2)
Where, n is volume fraction index and is always
positive. For n=0, the shell is fully metal and when n=1,
the composition of metal and ceramic is linear. The
Poisson’s ratio  depends weakly on temperature
change and is assumed tobe a constant.
 w, x 0 
0 w 
,y 
 w, x 
1
Anl   w, x w, y  and    
2

 w, y 
0 0 
 0 0 
2.1 Displacement field model
The thermal strain vector  t is represented as,
 
0
Higher order shear deformation theory with C
continuity has been used to find displacement field
model. Displacement field is given as [4]
u  u  f1 ( z)x  f 2 ( z) x ,
v  v(1 
z
)  f1 ( z ) y  f 2 ( z ) y ,(3)
R
ww,
Where (u, v, w) denote the displacement of a point
along the (x, y, z) coordinates axe, (u, v, w) are
corresponding displacements of a point on the mid
plane,  x and  y are the rotations at z=0 of normal to
 
t
 x 
1 
 
 
 y 
2
 
 
  xy   T 12  (7)
 
0 
 yz 
 
 
0 
 zx 
Where 1 ,  2 and 12 are the coefficient of thermal
expansion in the x, y and z directions respectively which
can be obtained from the thermal coefficient in the
longitudinal 1 and transverse  2 directions of the
ceramic and metal using the transformation matrix and
T is the uniform and nonuniform temperature change.
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ISSN (Print) : 2321-5747, Volume-2, Issue-2,2014
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International Journal on Mechanical Engineering and Robotics (IJMER)
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The temperature field for nonuniform temperature
change is expressed as,
U 
T = T  z   T0 (8)

Where T  z  is expressed as,
1
2
1
   [ D]  dA  2    [ D ] A  dA
1
T
2
1
2
A
l
l
  [ D ] A  
T
A
l
T
4
T
5
T
Where [ D] , [ D3 ] , [ D4 ] and
Where T  z  is the temperature distribution along z
matrices and
surface, and parameter   z  is defined as,
1 
z

k
z

ktb2
z

 0.5  
h
(2n  1) kb2 
2 n 1
z

 0.5  
h

4 n 1
4
tb
k
(2n  1) kb4


shell
stiffness
  is the linear mid-plane vector. The
l
strain energy function calculated for each element above
can be summed to get the total strain energy.
e 1
n 1
 {q}T [ Kl  K nl (q)]q (13)
3 n 1
z

 0.5  
h

5 n 1
k
(2n  1) kb5
[ D5 ] are
NE
ktb3
z

 0.5  
h
(3n  1) kb3 
5
tb
(12)
U  U e
tb
 ( z )   0.5   
 0.5  
c 
h  (n  1)kb 
h

3
dA
T  z   Tb  Tt  Tb   z 
direction, Tt and Tb , are temperature of top and bottom
l
dA 
  A [ D ] A  
A
T
A



Where [ Kl ], [ Knl ] and {q} are defined as global linear,
nonlinear stiffness matrix and displacement vector
respectively.
2.5 Work done
With ktb  kt  kb and k is defined as thermal
conductivity. The uniform temperature change Eq. can
be written as,
Where, T0 is initial temperature.
Because of uniform and nonuniform temperature
change, pre-buckling stresses in FGM shell are
generated the in-plane pre-buckling stress resultant per
unit length are reason for buckling. The work done (W)
by in-plane stress resultants in producing out of plane
displacements ‘w’ can be expressed as,
2.3 Stress strain relation
W
T ( z)  T0  (Tt  Tb ) (9)
The stress strain relation accounting thermal effect can
be written as,
1
[ N x ( w, x ) 2  N y ( w, y ) 2
2 A
2 N xy ( w, x ) 2 ( w, y ) 2 ]dA
T

 x 
 
 y
   Q   or  xy  (10)
 
 yz 
 xz 
1  w, x   N x
  
2 A  w, y   N xy

 Q11 Q12

Q 21 Q 22

 0
0
 0
0

 0
0

0
0
0
0
Q 66
0
0
Q 44
0
0
N xy   w, x 
  dA
N y   w, y 
Where N x , N y and N xy are thermal in plane,
thermal compressive stress resultant per unit length.
Using finite element method and summing over the
entire element above equation can be written as,
0 

0 

0   l    nl    t
0 
Q 55 
 
 
Where Q ij ,   and   are the transformed stiffness
matrix, stress and strain vectors for isotropic shell
respectively.
2.4 Strain energy of the shell
The strain energy of the FGM shell is given by,
U
(14)
T
1
    dv (11)

v
2
NE
NE
e 1
e 1
W  W ( e )  {(e ) }T T [ K g(e ) ]{(e ) }  T {q}T [ K g ]{q} (15)
Where T and [ K g ] are defined as critical thermal
buckling temperature and global geometric stiffness
matrix.
2.6 Kinetic energy of FGM shell
The kinetic energy (T) of the vibrating FGM shell can
be expressed as
(k )
T    {uˆ}T {uˆ}dV (16)
V
Where  and {uˆ}  { u v w } are the density and
velocity vector of the shell respectively, above equation
can be expressed as,
Above equation can be expanded as,
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ISSN (Print) : 2321-5747, Volume-2, Issue-2,2014
39
International Journal on Mechanical Engineering and Robotics (IJMER)
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NE
NE
v  w   y   y  0, at x  0, a;
e 1
e 1
u  w   x   x  0 at y  0, b
T   T ( e )   {}T [m]{} (17)
 {q}[ M ]{q}
4.1 Validation of fundamental frequency and parametric
study
Where, [M] is the global mass matrix.
III. EQUATION OF MOTION AND ITS
SOLUTION
The governing equation for thermally induced nonlinear
free vibration of the shell analysis can be derived using
Lagrange’s equation of motion.
t2
  (U  W  T )dT  0 (18)
t1
Substituting the values and obtaining in the form of
nonlinear generalized eigenvalue problem as,
[ K ]{q}  [M ]{q}  0 (19)
Where, [ K ]  {[ Kl ]  [ K nl (q)]  T [ K g ]}
The above equation is nonlinear free vibration equation
which can be solved as a linear eigenvalue problem
assuming that the shell is vibrating in its principal made
in each iteration, the above equation can be expressed as
generalized eigenvalue problem as,
[[ K ]  [M ]]{q}  0 (20)
Where    2 with  is natural frequency of the shell.
The nonlinear eigenvalue problem is solved by
employing a direct iterative based C 0 nonlinear finite
element method in conjunction with perturbation
technique.
IV. RESULT AND DISCUSSION
A nine noded Lagrange isoparametric element with 63
DOFs per element for the present HSDT model has been
used for discretizing, (4 × 4) mesh has been used for the
study.The results are compared with those in literatures.
The dimensionless nonlinear fundamental frequencyof
the FGMs cylindrical shell is,

   b2 / h

m / Em , (21)
Table 2 shows the dimensionlessnonlinear fundamental
frequency of FGM(Si3N4/SUS304)simply supported
cylindrical shell in thermal environments with
temperature independent material properties and
a/b=b/R=1, a/h=20. Clearly, it is observed that the
present results using C0 DIFEM are in good agreement
with the available semi analytical method published
results[6].
Table 2 validation of mean dimensionless fundamental
frequency of cylindrical shell subjected uniform
temperature distributionwith Wmax/h=1, a/h=20.
T (K)
n
Present
Shen H S [6]
Tc=400K
Tm=400K
1
14.8713
15.9915
5
11.9025
12.8445
Table 3 showsthe effect of temperature change (∆T),
thickness ratio, volume fraction index (n), amplitude
ratio(Wmax/h)and uniform temperature distribution on
the dimensionless nonlinear fundamental frequency of
SSSS supported FGM(ZrO2/ Ti-6Al-4V) cylindrical
shell with R/a=10, ∆T=30K.
Table 3The effect of temperature change (∆T), thickness
ratio, volume fraction index (n), amplitude ratio(W max/h)
and uniform temperature distribution on the
dimensionless nonlinear fundamental frequency of SSSS
supported FGM (ZrO2/ Ti-6Al-4V) cylindrical shell
with R/a=10, ∆T=30K.
a/h
0
10
Table 1The following temperature independent material
properties are used
Types
of
material
E
(N/m2)
ZrO2
151e+9
Ti-6Al4V
α
(1/C)
n
K
(W/mK)
( Kg / m3 )
204
2707
18.591e-6
1

0
70e+9
6.941e-6
Boundary Condition:
All edges simply supported (SSSS):
2.09
3000
20
1
Wmax/h
0.3
0.6
0.9
1
l
0.3
0.6
0.9
1
l
0.3
0.6
0.9
1
l
0.3
0.6
0.9
nl
6.8151
8.1318
9.8770
10.4496
6.2423
5.5676
6.6761
8.1393
8.6132
5.0831
6.8300
8.2435
10.0708
10.7317
6.1694
5.5736
6.7692
8.3069
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ISSN (Print) : 2321-5747, Volume-2, Issue-2,2014
40
International Journal on Mechanical Engineering and Robotics (IJMER)
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1
l
8.8617
5.0113
[2]
Jagtap K. R., Achchhe Lal, Singh B. N.,
stochastic nonlinear free vibration analysis of
elastically supported functionally graded
materials plate with system randomness in
thermal environment , composite structures
2011;93:3185-3199.
[3]
Achchhe Lal, Singh B. N., Soham Anand,
Nonlinear bending response of laminated
composite spherical shell panel with system
randomness
subjected
to
hygro-thermomechanical loading, International Journal of
Mechanical sciences 2011;53:855-866.
[4]
Singh B. N., Yadav D., Iyengar N.G.R., A C 0
element for free vibrationof composite plates
withuncertain material properties, advanced
composite material 2003;11:331-350.
[5]
Hiroyuki Matsunaga, free vibration and stability
of functionally graded shallow shells according
to a 2D higher-order deformation theory,
composite structures 2008; 84:132-146.
[6]
Hui-Shen Shen, Hai Wang, Nonlinear vibration
of shear deformable FGM cylindrical panels
restingon elastic foundations in thermal
environments, Composites 2014; 60:167–177.
V. CONCLUSION
A C0 nonlinear finite element method based on direct
iterative procedure [DIFEM] is used to compute the
nonlinear fundamental frequency simply supported
FGM cylindrical shell in thermal environment. Higher
order shear deformation theory with von-Karman
nonlinearity with modified C0 continuity is used for
basic formulation. The nonlinear fundamental frequency
of vibration increases with increase in the amplitude
ratio. It is also observed that for same thickness ratio and
the amplitude ratio, the fundamental frequency of
vibration increases. It is also observed that for same
thickness ratio and amplitude ratio, the nonlinear natural
frequency decreases with increase in volume fraction
index because of decrease in the stiffness.
REFERENCES
[1]
Jagtap K. R., Achchhe Lal,Effect of random
material properties on free vibration response of
functionally graded materials plate with cutouts
in thermal environment. International Conference
on Modern Trends in Industrial Engineering,
2011.

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