STATIC AND DYNAMIC ANALYSIS OF Bikash Kumar Majhi Roll No.:109ME0159

STATIC AND DYNAMIC ANALYSIS OF Bikash Kumar Majhi Roll No.:109ME0159
STATIC AND DYNAMIC ANALYSIS OF
FUNCTIONALLY GRADED FLAT PANELS
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
By
Bikash Kumar Majhi
Roll No.:109ME0159
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-769008
MAY 2013
STATIC AND DYNAMIC ANALYSIS OF
FUNCTIONALLY GRADED FLAT PANELS
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
By
Bikash Kumar Majhi
Roll No.:109ME0159
Under the guidance of
Prof. S. K. Panda
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-769008
MAY 2013
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
CERTIFICATE
This is to certify that the work in this thesis entitled “Static and
Dynamic analysis of Functionally Graded Flat Panels” by Bikash
Kumar Majhi, roll no 109ME0159 has been carried out under my
supervision in partial fulfilment of the requirements for the degree of
Bachelor of Technology in Mechanical Engineering during session 20122013 in the Department of Mechanical Engineering, National Institute of
Technology, Rourkela.
To the best of my knowledge, this work has not been submitted to
any other University/Institute for the award of any degree or diploma.
Dr. Subrata Kumar Panda
Date: 08/05/2013
(Supervisor)
Assistant Professor
Department of Mechanical Engineering
National Institute of Technology, Rourkela
ACKNOWLEDGEMENT
I wish to express my sincere gratitude to Dr. Subrata Kumar Panda for his inspiring
encouragement, guidance and efforts taken throughout the entire course of this work. His
constructive criticism, timely help, and efforts made it possible to present the work contained
in this Thesis.
I am grateful to Prof. S.K. Sarangi, Director, and Prof K.P Maity, Head of the Department,
Mechanical Engineering, for their active interest and support.
I would like to thank Mr. Girish Kumar Sahu, M.Tech and Mr. Pankaj Katariya, M.Tech
(Res) , Department of Mechanical Engineering, National Institute of Technology, Rourkela
for their constant help in understanding of the technical aspects of the project. I will also be
grateful to Ph.D scholar Mr. Vishesh Ranjan Kar, for his constant help in the successfully
carrying out the new results.
I express my deep sense of gratitude and reverence to my beloved parents for their blessings,
patience and endeavour to keep my moral high at all times. Last but not the least, I wish to
express my sincere thanks to all those who directly or indirectly helped me at various stages
of this work.
Bikash Kumar Majhi
109ME0159
CONTENTS
Pages
1. INTRODUCTION
1-3
1.1
Background
2
1.2
Application of FGM
3
1.3
Objective of the work
3
2. LITERATURE REVIEW
4-7
2.1
Static analysis
5
2.2
Dynamic analysis
6
3. FINITE ELEMENT FORMULATION
8-12
4. EFFECTIVE MATERIAL PROPERTY
13-16
4.1
Exponential law
14
4.2
Power law
14
4.3
FE modelling of FGM plates
16
5. STATIC ANALYSIS
17-26
5.1
FG model (ANSYS)
18
5.2
Convergence and validation
18
5.3
Numerical results
19
5.3.1
Aluminium/Zirconia plate
20
5.3.2
Different boundary conditions
21
5.3.3
Aluminium/Alumina plate
23
5.3.4
Silicon Nitride/Stainless steel plate
25
6. DYNAMIC ANALYSIS
27
6.1
FG model (ANSYS)
28
6.2
Numerical results
28
7. CONCLUSION
31
7.1
Conclusion
32
7.2
Future scope of work
32
8. REFERENCE
33
9. APPENDIX-A
36
ABSTRACT
Functionally graded materials have received a lot of interest in recent days by their
diversified and potential applications in aerospace and other industries. They have high
specific mechanical properties and high temperature capabilities which makes them special
over all the exiting advanced materials. The present work investigated static and dynamic
analysis of functionally graded plate. The material properties vary continuously from metal
(bottom surface) to ceramic (top surface). The effective material properties of functionally
graded materials for the plate structures are assumed to be temperature independent and
graded in the plate thickness direction according to a power law distribution of the volume
fractions of the constituents. The present model is developed using ANSYS parametric
design language code in the ANSYS platform. An eight noded isoparametric quadrilateral
shell element is used to discretise the present model for both static as well as dynamic
analysis. A convergence test has been done with different mesh refinement and compared
with published results. The parametric study indicates that the power-law indices, thickness
ratios, aspect ratios, support conditions and different material properties have significant
effect on non-dimensional mid-point deflection.
i
LIST OF FIGURES
Figure No
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Pages
Geometry and dimensions of the FG plate
Variations of volume fraction of ceramic through non-dimensional
thickness coordinate
Variations of volume fraction of metal through non-dimensional
thickness coordinate
FGM plate modelled in ANSYS 13.0 using SHELL281 element
Convergence study
Variation of non-dimensional mid-point deflection with different
mesh size in ANSYS SHELL 181 model for simply supported FGM
plate
Static analysis of Aluminum/Zirconia plate
Variation of non-dimensional mid-point deflection with different
mesh size in ANSYS SHELL 181 model for simply supported FGM
plate
Variation of non-dimensional mid-point deflection with different
thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply
supported FGM plate
9
15
Variation of non-dimensional mid-point deflection with different
aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply
supported FGM plate
Different boundary conditions
Non-dimensional deflection vs. Length to thickness ratio square
FGM plate with simply supported (SSSS) boundary condition
Non-dimensional deflection vs. Length to thickness ratio square
FGM plate with clamped (CCCC) boundary condition
Non-dimensional deflection vs. Length to thickness ratio square
FGM plate with clamped-simply supported (CSCS) boundary
condition
Non-dimensional deflection vs. Length to thickness ratio square
FGM plate with clamped-simply supported (SSCC) boundary
condition
Aluminum/Alumina plate
Variation of non-dimensional mid-point deflection with different
mesh size in ANSYS SHELL 281 model for simply supported FGM
plate
Variation of non-dimensional mid-point deflection with different
thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply
supported FGM plate
21
ii
15
16
19
20
20
21
22
22
23
23
24
LIST OF FIGURES
Figure No
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22
Figure 23
Pages
Variation of non-dimensional mid-point deflection with different
aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply
supported FGM plate
Silicon Nitride/Stainless steel plate
Variation of non-dimensional mid-point deflection with different
mesh size in ANSYS SHELL 281 model for simply supported FGM
plate
Variation of non-dimensional mid-point deflection with different
thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply
supported FGM plate
24
Variation of non-dimensional mid-point deflection with different
aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply
supported FGM plate
Dynamic analysis of Aluminum/Zirconia plate
Deflection of mid-point of simply supported FG flat panel with n=0
Deflection of mid-point of simply supported FG flat panel with n=1
Deflection of mid-point of simply supported FG flat panel with n=2
Deflection of mid-point of simply supported FG flat panel with n= ∞
Deflection of mid-point of simply supported FG flat panel with n=0
in time interval of 0.0001 to 0.012s
26
iii
25
25
28
29
29
30
30
LIST OF TABLES
Table No
Table 1
Table 2
Pages
Properties of the FGM plate constituents
Convergence study of non-dimensional mid-point deflection of simply
supported FG (Al2O3/ZrO2) flat panel with a/b = 1 and a/h = 5)
iv
16
19
NOMENCLATURE
a
Length of plate
b
Width of plate
h
Height of plate
a/h
Thickness ratio
a/b
Aspect ratio
u, v, w
Displacement field in x, y, z direction
u0, v0, w0
Mid-plane displacement in x, y, z direction
 x ,  y , z
Rotational displacement in x, y, z direction
 xx ,  xy ,  xz
Lateral strain
 xz ,  xy ,  yz
Shear strain
Rx, Ry, Rz
Radius of curvature in x, y, z direction
E(z)
Young’s modulus of material

Poisson’s ratio

Displacement vector

Strain vector
[B]
Strain-displacement matrix
[D]
Rigidity matrix
[K]
Stiffness matrix
[M]
Mass matrix
J
Jacobian matrix
F(t)
Time-dependent force
Ec
Young’s modulus of ceramic
Em
Young’s modulus of metal
c
Density of ceramic
m
Density of metal
c
Poisson’s ratio of ceramic
m
Poisson’s ratio of ceramic
v
Chapter I
Introduction
CHAPTER-1
INTRODUCTION
1.1 Background
Laminated composites have received a lot of interest in recent days by diversified and
potential applications in automotive and aerospace industry due to their strength to weight,
stiffness to weight ratio, low fatigue life and toughness and other higher material properties.
These are made from two or more constituent materials which have different chemical or
physical properties and produced a material having different behaviour from the individual.
These are used in buildings, storage tanks, bridges etc. Each layer is laminated in order to get
superior material properties. The individual layer has high strength fibres like graphite, glass
or silicon carbide and matrix materials like epoxies, polyimides. By varying the thickness of
laminas desired properties (strength, wear resistance, stiffness) can be achieved.
Although these materials have superior properties, their major drawback is the
weakness of laminated materials. This is known as delamination phenomenon which leads to
the failure of the composite structure. Residual stresses are present due to difference in
thermal expansion of the matrix and fibre. It is well known that at high temperature the
adhesive being chemically unstable and fails to hold the lamination. Sometimes due to fibre
breakdown it also prematurely fails.
Functionally Graded Material (FGM) is combination of a ceramic and a metal. A
material in which its structure and composition both varies gradually over volume in order to
get certain specific properties of the material hence can perform certain functions. The
properties of material depend on the spatial position in the structure of material. The effect of
inter-laminar stress developed at the laminated composite interfaces due to sudden change of
material properties reduced by continuous grading of material properties. Generally
microstructural heterogeneity or non-uniformity is introduced in functionally graded material.
The main purpose is to increase fracture toughness, increase in strength because ceramics
only are brittle in nature. Brittleness is a great disadvantage for any structural application.
These are manufactured by combining both metals and ceramics for use in high temperature
applications. Material properties are varies smoothly and continuously in one or many
directions so FGMs are inhomogeneous. FGM serves as a thermal barrier capable of
withstanding 2000K surface temperature. Fabrication of FGM can be done by different
processing such as layer processing, melt processing, particulate processing etc. FGM has the
ability to control shear deformation, corrosion, wear, buckling etc. and also to remove stress
2
concentrations. This can be used safely at high temperature also as furnace liners and thermal
shielding element in microelectronics and thermal protection systems for spacecraft,
hypersonic and supersonic planes and in combustion chamber also.
1.2 Application of FGM
i.
Engineering Application
a. Shafts
b. Engine parts
c. Blades of turbine
ii.
Aerospace Engineering
a. Rocket engine components
b. Aerospace parts and skins
iii.
Electronics
a. Sensor
b. Actuator
c. Integrated chips
d. Semiconductor
iv.
Biomaterials
a. Artificial bones
b. Drug delivery system
1.3 Objective of the work
The objective of the present study is to analyse the static and dynamic behaviour of
FG flat panel under different volume fraction indices, aspect ratios, thickness ratios and
different support conditions. ANSYS parametric design language (APDL) code is use to
develop the model in ANSYS13.0 platform and solve the problem using appropriate
algorithm. The present model is discretised by using an eight noded isoparametric
quadrilateral shell element (SHELL281), as defined in the ANSYS element library. Three
types of FG flat panels are used in this study namely, Aluminium/Alumina,
Aluminium/Zirconia and Silicon Nitride/Stainless steel. The effects of different parameters
on non-dimensional mid-point deflection are studied.
3
Chapter II
Literature review
CHAPTER-2
LITERATURE REVIEWS
FG material plates have created revolution in aerospace industry for its thermal properties,
multifunctionalities. It also provides opportunities to take the benefits of different material
system. Its static and dynamic analysis is necessary to estimate the properties of flat panels.
Many researchers reported static and dynamic behaviour of functionally graded plates based
on different theories and developed new methods of solutions.
2.1 Static and vibration analysis
Talha and Singh [1] investigated the free vibration and static analysis of rectangular
FGM plates using higher order shear deformation theory with a special modification in the
transverse displacement in conjunction with finite element models. Neves et al. [2] studied
the static deformations analysis of functionally graded plates by collocation with radial basis
functions, according to a sinusoidal shear deformation formulation for plates. Aragh and
Hedayati [3] studied the characteristics of free vibration and static response of a 2-D FGM
open cylindrical shell. Formulations are done by 2-D generalized differential quadrature
method (GDQM). Ferreira et al. [4] studied static deformations of functionally graded square
plates of different aspect ratios using meshless collocation method, the multiquadric radial
basis functions and a third-order shear deformation theory. Reddy[5] studied static and
dynamic analysis of FGM plates using third-order shear deformation theory. Navier solutions
are obtained for a simply supported square plate. Abrate[6] investigated static, buckling and
free vibration deflections of FGM plates by using classical plate theory, FSDT model and
HSDT model. Zenkour[7] studied the static behaviour of a rectangular FG plate under simply
supported condition and subjected to uniform transverse load. Ferreira et al. [8] studied static
deformations of simply supported functionally graded plate by using HSDT and multiquadric
radial basis functions. Vel and Batra[9] investigated the exact 3-D elasticity solutions of
simply supported rectangular FG plates under thermo-mechanical load. The author has
assumed power law for material volume fractions. The exact solutions of displacements and
stresses are used to find out the accuracy of the solutions. Qian et al.[10] investigated plain
strain static thermostatic deformations of simply supported thick rectangular FG elastic plate.
Displacement and stress are computed and validated from the 3D exact solutions of the
problem. Ramirez et al.[11] studied static analysis of 3D, elastic, anisotropic FG plates. The
5
author has taken simply supported graphite/epoxy material for analysis. Zenkour [12] further
studied the static response of FG plates using shear deformation plate theory using power law
for grading. Bhangale and Ganesan[13] investigated static analysis of simply supported FG
plates which are exponentially graded in the thickness direction. Aghdam et al.[14] studied
static analysis foe bending of FG clamped thick plates. The solutions are compared with the
solutions of finite element code ANSYS, power law is used for grading the properties in
thickness direction. Neves et al.[15] investigated the static deformations of FG square plates
using radial basis function. Talha and Singh [16] investigated the static and free vibration
analysis using C0 finite element with 13 degrees of freedom per node and formulated by
HSDT. Nguyen-Xuan et al.[17] studied the static, free vibration and mechanical/thermal
buckling problems of FG plates by Reisnner/Mindlin plate theory.
2.2 Dynamic Analysis
Yang and Shen [18] studied dynamic response of FGM thin plates under initial stress
and partially distributed impulsive lateral loads. The author used silicon nitride/ stainless steel
rectangular plates, assumed temperature dependent material properties clamped on two
opposite edges, used power law for grading and used Modal superposition method for
transient response. In 2001 Yang and Shen [19] studied free and forced vibration analysis for
the same plate and found functionally graded plate with material properties intermediate to
isotropic material do not necessarily have intermediate natural frequency if thermal effects
are considered. Liew et al. [20] investigated dynamic stability of symmetrically laminated
FGM rectangular plates under uniaxial plane load. Formulation is done by Reddy’s thirdorder shear deformation theory and material is silicon nitride and stainless steel. [21]Kim
studied vibration characteristics of rectangular FGM plate under initial stress. Third-order
shear deformation plate theory is adopted and Rayleigh-Ritz procedure is applied for getting
frequency equation. Lanhe et al. [22] investigated dynamic stability of thick FGM plate under
aero-thermo-mechanical loads and used novel numerical solution technique. The equations
for dynamic analysis are derived by Hamilton’s principle. For different parameters dynamic
instability regions are studied. Ansari and Darvizeh [23] investigated vibrational behaviour of
functionally graded shells, based on first-order shear deformation shell theory. The grading
functions are power law, sigmoid and exponential distribution. Behjat et al. [24] studied
dynamic response, static bending of functionally graded piezoelectric material plate (PZT4/PZT-5H), formulated by using potential energy and Hamilton’s principle. Effects of
material composition and boundary conditions on dynamic response are also studied. Sladek
6
et al. [25] investigated dynamic analysis of FG plates by MPLG method. For displacement
field author used Reissner-Mindlin plate bending theory. Simply supported and clamped
boundary conditions are taken in to consideration. Wen et al. [26] studied 3-D analysis of
isotropic and orthotropic FG plates with simply supported edge under dynamic loads. The
equations formulated is based on state-space approach in Laplace transform domain and
solved by RBF method. Grading has done by exponential method as well as volume fraction
law. Shariyat [27] studied the vibration and dynamic buckling response of rectangular FG
plates under thermo-mechanical loading. A nine noded second-order formulation has done
and graphs are studied under temperature dependent material properties.
From the above study it has been seen that very few researcher studied the dynamic
analysis of FG plates. Since most of the practical cases deals with transient or dynamic load,
its responses has to be analysed with different parameters like volume fraction index. This
work analysed dynamic responses of aluminium/zirconia flat panel under step load with
different volume fraction index (n=0, 1, 2, ∞);
7
Chapter III
Finite element
formulation
CHAPTER-3
FINITE ELEMENT FORMULATION
In the present analysis, a FG plate of uniform thickness h with rectangular base of sides a and
b is established through APDL code and shown in the Figure 1. The FG plate model is
developed in ANSYS based on the inbuilt FSDT kinematics. The displacements field u, v and
w at any point along x, y and z axes can be written as follows:
u ( x, y, z )  u 0 ( x, y )  z x ( x, y )
v( x, y, z )  v 0 ( x, y )  z y ( x, y )
(1)
w( x, y, z )  w0 ( x, y )  z z ( x, y )
where, u, v and w denote displacements; u 0 , v 0 and w0 are the mid-plane displacements in
x, y, z axes respectively and  , 
and  are the shear rotations.
z
a
y
b
Ceramic rich
x
+h/2
-h/2
Metal rich
Figure 1: Geometry and dimensions of the FG plate
The linear stains corresponding to the displacement field is
 xx 
u w

,
x Rx
 yz   y 
w v

,
y Ry
 yy 
v w

,
y Ry
 xz   x 
w u
 ,
y Rx
 zz   z ,
 xy 
For the flat panel i.e. plate, Rx = Ry = Rxy = 
9
u v 2w
 
y x Rxy
(2)
It can be rewritten as
   
,  yy ,  zz ,  yz ,  xz ,  xy  ,
T
xx
 k x' 
 xx    x0 
 ' 
   0 
 ky 
 yy    y 
0
 k z' 
  zz    z 
l      0   z  ' 
 k yz 
 yz   yz 
0
k ' 
 xz   xz 
 xz' 
   0


 xy   xy 
 k xy 
0
where  x 
(3)
(4)
w0 u 0
w0 v 0 0
u 0 w0 0 v 0 w0

,  z  0,  yz0   y 

,  xz   x 
 ,

, y 
y Ry
y Ry
y Rx
x Rx
y

u 0 v 0 2w0 '  x  z '  y  z '


, kx 

, k y   z , k yz'  z 
,

, ky 
y
x
Rxy
y Ry
y Ry
x Rx
 y  z



'

k xz'  z  x , k xy  x 
y
x Rxy
y Rx
 xy0 
The linear constitutive relations are
 xx   Q11 Q12
  
 yy  Q21 Q22
 zz  Q31 Q32
 
0
 yz   0
 xz   0
0
  
0
 xy   0
Q13
Q23
Q33
0
0
0
0
0
0
Q44
0
0
0
0
0
0
Q55
0
0   xx 
 
0   yy 
0   zz 
 
0   yz 
0   xz 
 
Q66   xy 
(5)
where,
E ( z )(1  v 2 )
(1  3v 2  2v3 )
E ( z )v(1  v)
Q12  Q13  Q23 
(1  3v 2  2v3 )
E( z)
Q44  Q55  Q66 
2(1  v)
Q11  Q12  Q13 
The modulus of elasticity E(z) and elastic coefficients Qij vary through the plate thickness.
For the implementation of finite element method, the developed model is discretised
by using an eight noded isoperimetric quadrilateral shell element (SHELL281), as defined in
the ANSYS element library. This element is suitable for analysing thin to moderately-thick
shell structures with six degrees of freedom.(three translations and three rotations) at each
10
node in the x, y and z directions. The displacements are expressed in terms of interpolation
functions (Ni)
 
8
N
i
i 1
(6)
i
T
where,  i  u 0 , v0 , w0 , x , y , z  . The interpolation functions for eight noded isoperimetric
quadrilateral shell element in natural (ξ-η) coordinates are given as [15]
1
N1  (1   )(1   )(    1) ,
4
1
N 2  (1   )(1  )(   1) ,
4
1
N3  (1   )(1   )(    1) ,
4
1
N 4  (1   )(1   )(    1) ,
4
1
N5  (1   2 )(1   ) ,
2
N6 
1
1    1  2  ,
2
1
1   2  1    ,
2
N8 
1
1    1  2 
2
N7 
(7)
The strain vector in terms of nodal displacement vector can be written as
 

 B 
(8)
where,  B  is the strain-displacement matrix containing interpolation functions and derivative
operators and { } is the nodal displacement vector.
The generalized stress-strain relation with respect to its reference plane can be written
as
{ }   D{ }
(9)

where     x  y  z  xy  yz  xz  and     x
T
 y  z  xy  yz  xz  are
T
the linear stress and strain vector, respectively and  D  is the rigidity matrix.
[Q]  [T ]T [Q][T ]
[ D]  
h /2
 h /2
[Q]dz
(10)
(11)
The elemental stiffness matrix  K  e and the mass matrix [M]e are integrated by using
Gauss-quadrature integration over the domain to obtain the global stiffness and mass matrices
and this can be conceded as
K 
e
M 
e

1
1

1
1
1
  B  D B J d d
T
1
1
  N  m N  J d d
1
11
(12)
T
(13)
where, J is the determinant of the Jacobian matrix and [N] is the interpolation function
matrix. The jacobian is used to map the domain from natural coordinate to the general
coordinate.
The governing equation of static analysis of FG plate under force F can be expressed as
follows:
 K { }  F  0
(14)
where, [K] is the global stiffness matrices. Eqn. (8) is a generalized eigenvalue problem and
non-dimensional central point deflection can be found from this equation.
The governing equation of dynamic analysis of FG plate under dynamic load F (t) can be
expressed as follows:
 K { }   M { }  F (t )
12
(15)
Chapter IV
Effective material
properties
CHAPTER-4
EFFECTIVE MATERIAL PROPERTIES
The effective material properties of the FGM plate are assumed to be varying continuously
along their thickness direction as discussed earlier and are obtained by using a simple powerlaw distribution or exponential law which counts the volume fraction of each constituent.
4.1 Exponential law
Exponential law of grading FGM states that for a FGM structure of uniform thickness
‘h’, the material properties ‘P(z)’ at any point located at ‘z’ distance from the mid-plane
surface is given by:
P( z )  Pe
t
  2z 
   1  
  h 
, where,
1 P 
  ln  t 
2  Pb 
(16)
P( z ) denotes material property like Young’s modulus of elasticity (E), shear modulus of
elasticity (G), Poisson’s ratio (  ), material density (  ) of the FGM structure. Pt and Pb are
the material properties at the top (z=+h/2) and bottom (z=-h/2) surfaces.  is the material
grading indexes which depend on the design requirements.
4.2 Power law
The power-law distribution of a panel considered from the mid-plane reference plane can be
written as
 z 1
Vf    
h 2
n
(17)
where, n is the power-law index, 0  n   . The variations of volume fraction of the
ceramic and metal phase through the non-dimensional thickness coordinate are plotted in
Figure 2 and 3 for five different values of power-law indices (n = 0.2, 0.5, 1, 2 and 10). The
functionally graded material with two constituents and their properties such as, Young’s
modulus E and the mass density ρ have been obtained using the following steps.
n
 z 1
E   Ec  Em      Em
h 2
(18)
n
 z 1
   c   m       m
h 2
(19)
n
 z 1
  c  m      m
h 2
14
(20)
In the present work, the power- law distribution is used for the continuous gradation of
material properties in thickness direction.
The effective material properties are calculated based on the Eqns. (18), (19) and (20),
when z = - h/2, E= Em , ρ = ρm and  =  m similarly, when z = + h/2; E= Ec , ρ = ρc and  =
c i.e., the material properties vary continuously from metal at the bottom surface to ceramic
at the top surface. The Poisson’s ratio ν is assumed to be constant throughout the thickness of
the shell panel. The properties of the FGM constituents at room temperature (27oC) are used
for the analysis and presented in Table 1. The different material properties are used to analyse
the responses for throughout the study.
Volume fraction of ceramic
1.0
0.8
n = 0.2
n = 0.5
n=1
n=2
n = 10
0.6
0.4
0.2
0.0
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Non-dimensional thickness coordinate
Figure 2: Variations of volume fraction of ceramic through non-dimensional thickness coordinate
1.0
Volume fraction of metal
0.8
0.6
0.4
0.2
0.0
-0.6
n = 0.2
n = 0.5
n=1
n=2
n = 10
-0.4
-0.2
0.0
0.2
0.4
0.6
Non-dimensional thickness coordinate
Figure 3 Variations of volume fraction of metal through non-dimensional thickness coordinate
15
Table 1: Properties of the FGM plate constituents
Properties
Young's Modulus
E (GPa)
Poisson's Ratio
Density
(Kg/m3)
Aluminium (Al)
70
0.3
2707
Alumina (Al2O3)
380
0.3
3800
Zirconia (ZrO2)
151
0.3
3000
Silicon Nitride (Si3N4)
348.43
0.28
2370
Steel (SUS304)
201.04
0.28
8166
Materials
4.3 FE Modelling of FGM plates:
FGM plates with different length to thickness ratio(
) , aspect ratio (a/b) are
analysed in this experiment. The analysis is performed in commercially available software
(ANSYS 13.0). The loading conditions are assumed to be static. The element chosen for this
analysis is SHELL281, which is a layered version of the 8-node structural shell model. This
is suitable for analysing thin to moderately-thick shell structures. This shell element has six
degrees of freedom at each node namely three translations and three rotation in the nodal x, y
and z directions respectively. The FGM plate is modelled in ANSYS 13.0 as shown in the fig
4.
Figure 4: FGM plate modelled in ANSYS 13.0 using SHELL281 element
16
Chapter V
Static analysis
CHAPTER-5
STATIC ANALYSIS
5.1 FG model (ANSYS)
The static responses of the FG plates are analysed using ANSYS 13.0 under static surface
load for simply supported boundary condition for Aluminium/Zirconia FG flat panel. The
computed results are validated and compared with those available in the literature. The
analysis is carried out for thickness ratio (a/h) = 5, aspect ratio (a/b=1) with different volume
fraction indices. APDL code has been developed in ANSYS 13.0 for analysing the above
panel. The following non-dimensional parameters are used:
Central deflection
̅
Load parameter
(21)
5.2 Convergence and validation:
The static analysis of FG plates is analysed by ANSYS 13.0 using APDL program. The
aluminium/zirconia material is analysed for different mesh size under simply supported
boundary condition. These boundary conditions are there to reduce the no of unknowns from
the final equation and in order to avoid rigid body motion.
Simply-supported (SSSS): v0  w0  y   z  0 at x=0 and a
u 0  w0   x  z  0 at y = 0 and b
For validation of obtained data and the efficiency of present finite element model, the results
obtained using FG flat panel model is compared with the published literature. The nondimensional mid-point deflection ̅
of square simply supported FG flat panel (a/h =5)
are computed for five different power-law indices (n=0, 0.5,1,2, ∞) and tabulated in Table. It
is observed from the results that convergence satisfies at a (16 x 16) mesh and the differences
between present and published results are negligible.
18
Table 2: Convergence study of non-dimensional mid-point deflection of simply supported FG (Al2O3/ZrO2) flat
panel with a/b = 1 and a/h = 5)
Mesh size
n
Ref.[4]
6x6
8x8
10x10
12x12
14x14
16x16
18x18
Ceramic
0.024875
0.024855
0.024845
0.02484
0.024835
0.024835
0.024835
0.0247
0.5
0.031245
0.031215
0.0312
0.031195
0.03119
0.031185
0.03118
0.0313
1
0.03459
0.034555
0.03454
0.03453
0.03452
0.034515
0.034515
0.0351
2
0.037955
0.03792
0.0379
0.03789
0.03788
0.037875
0.03787
0.0388
Metal
0.05366
0.053615
0.053595
0.053585
0.053575
0.05357
0.05357
0.0534
0.055
Ceramic
n=0.5
n=1
n=2
Metal
Non dimensional deflection
0.050
0.045
0.040
0.035
0.030
0.025
6x6
8x8
10 x 10 12 x 12 14 x 14 16 x 16 18 x 18 Ref [4]
Mesh size
Figure 5: Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 181
model for simply supported FGM plate
5.3 Numerical results:
In this section some new problems have been solved and new data, graphs are given for
different parameters and responses are discusses. The variations of non-dimensional midpoint deflection of FG plate for simply supported boundary condition (SSSS) with different
mess size in ANSYS SHELL 181 element are plotted in figure 6. The results are obtained
using other geometric properties i.e., thickness ratio, aspect ratio for five fraction indices (n =
0, 0.5,1,2 and ∞). And also there are results for different material properties and different
mesh size in ANSYS. Different material properties of FGM plates are given in table 1.
19
Graphs varying mesh size, thickness ratio and aspect ratio are plotted for aluminium/
zirconia, aluminium/ alumina, silicon nitride/stainless steel respectively. And also graphs are
plotted for different boundary conditions for aluminium, zirconia FGM plates. Figure 7 and
figure 8 shows the variation of thickness ratio and aspect ratio and central deflection varied.
Figure 9-12 shows the central deflection varied under different boundary conditions for
aluminium/zirconia plate. Figure 13-15 shows the behaviour of mid-point deflection if mesh
size, thickness ratio and aspect ratio varies respectively for aluminium/alumina plate. For
aluminium/stainless steel flat panel, the same behaviours are studied and plotted in figure 1618.
5.3.1 Aluminium/Zirconia plate:
0.055
Ceramic
n=0.5
n=1
n=2
Metal
Non dimensional deflection
0.050
0.045
0.040
0.035
0.030
0.025
6x6
8x8
10 x 10
12 x 12
14 x 14
16 x 16
18 x 18
Mesh size
Figure 6: Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 181
model for simply supported FGM plate
0.30
0.28
Ceramic
n=0.5
n=1
n=2
Metal
0.26
Non dimensional deflection
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0
5
10
15
20
a/h ratio
Figure 7: Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in
ANSYS SHELL 281 model for simply supported FGM plate
20
Ceramic
n=0.5
n=1
n=2
Metal
500
450
Non dimensional deflection
400
350
300
250
200
150
100
50
0
-50
0
5
10
15
20
a/b ratio
Figure 8: Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS
SHELL 281 model for simply supported FGM plate
5.3.2 Different boundary conditions:
Non Dimesional Center Deflection
SSSS
0.045
n = 0.5
n=1
0.040
n=2
n=5
0.035
0.030
0.025
0
20
40
60
80
100
Length to thickness ratio
Figure 9: Non-dimensional deflection vs. Length to thickness ratio square FGM plate with simply supported
(SSSS) boundary condition
21
CCCC
Non Dimesional Center Deflection
0.020
n = 0.5
n=1
0.015
n=2
n=5
0.010
0.005
0
20
40
60
80
100
Length to thickness ratio
Figure 10: Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped (CCCC)
boundary condition
CSCS
Non Dimesional Center Deflection
0.030
n = 0.5
n=1
0.025
n=2
n=5
0.020
0.015
0.010
0
20
40
60
80
100
Length to thickness ratio
Figure 11: Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped-simply
supported (CSCS) boundary condition
22
SSCC
0.030
Non Dimesional Center Deflection
n = 0.5
n=1
0.025
n=2
n=5
0.020
0.015
0.010
0
20
40
60
80
100
Length to thickness ratio
Figure 12: Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped-simply
supported (SSCC) boundary condition
5.3.3 Aluminium/Alumina plate:
Ceramic
n=0.5
n=1
n=2
Metal
0.055
Non dimensional deflection
0.050
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
(6 x 6)
(8 x 8)
(10 x 10) (12 x 12) (14 x 14)
(16 x 16) (18 x 18)
Mesh size
Figure 13: Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 281
model for simply supported FGM plate
23
Ceramic
n=0.5
n=1
n=2
Metal
0.30
Non dimensional deflection
0.25
0.20
0.15
0.10
0.05
0.00
0
5
10
15
20
a/h ratio
Figure 14: Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in
ANSYS SHELL 281 model for simply supported FGM plate
500
Ceramic
n=0.5
n=1
n=2
Metal
Non dimensional deflection
400
300
200
100
0
0
5
10
15
20
a/b ratio
Figure 15: Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS
SHELL 281 model for simply supported FGM plate
24
5.3.4 Silicon Nitride/Stainless steel plate:
0.055
Non dimensional deflection
0.050
Ceramic
n=0.5
n=1
n=2
Metal
0.045
0.040
0.035
0.030
0.025
0.020
(6 x6)
(8 x 8)
(10 x 10) (12 x 12) (14 x 14) (16 x 16) (18 x 18)
Mesh size
Figure 16: Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 281
model for simply supported FGM plate
Ceramic
n=0.5
n=1
n=2
Metal
0.30
0.28
Non dimensional deflection
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
2
4
6
8
10
12
14
16
a/h ratio
Figure 17: Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in
ANSYS SHELL 281 model for simply supported FGM plate
25
Ceramic
n=0.5
n=1
n=2
Metal
Non dimensional deflection
500
400
300
200
100
0
0
2
4
6
8
10
12
14
16
a/b ratio
Figure 18: Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS
SHELL 281 model for simply supported FGM plate
26
Chapter VI
Dynamic analysis
CHAPTER-6
DYNAMIC ANALYSIS
6.1 FG model (ANSYS)
Rectangular simply supported Aluminium/Zirconia FG flat panel has been developed in
ANSYS13.0 platform. Time dependant step load has been taken for transient dynamic
analysis. Step type loading has been taken in to consideration. From time 0 to 0.001s force is
zero and from 0.001 to 0.002s force is 10kN. APDL code has been developed in ANSYS 13.0
for analysing the above panel.
6.2 Numerical results:
The analysis is carried out for different volume fraction indices (n=0, 1, 2, ∞). Dynamic
behaviour of FG flat panel can be seen in figure 19-22. By time step of 0.0001s analysis has
been performed and displacement has been plotted. . An enlarged view of dynamic response
has been shown in figure 23 for ceramic flat panel.
Figure 19: Deflection of mid-point of simply supported FG flat panel with n=0
28
Figure 20: Deflection of mid-point of simply supported FG flat panel with n=1
Figure 21: Deflection of mid-point of simply supported FG flat panel with n=2
29
Figure 22: Deflection of mid-point of simply supported FG flat panel with n=∞
Figure 23: Deflection of mid-point of simply supported FG flat panel with n=0 in time interval of
0.0001 to 0.012s
30
Chapter VII
Conclusion
CHAPTER-7
CONCLUSION
7.1 Conclusion:
In this study, static and dynamic responses of FGM plates are analysed. The effective
material properties of functionally graded materials for the plate structures are assumed to
vary continuously through the plate thickness and are graded in the plate thickness direction
according to a volume fraction power law distribution. Various boundary conditions have
been considered to check the efficacy of ANSYS model. Convergence tests and comparison
studies have been carried out with the commercially available software (ANSYS). An eight
noded layered shell element (SHELL281) is used throughout the problem. The obtained
results have illustrated a good agreement with those available in the literature for different
volume fraction indices, thickness ratios, aspect ratios and different support conditions. The
following points revealed the concluded remarks for thin to thick FGM plates are:

For all the boundary conditions, the non-dimensional central deflection increases as
the volume fraction index increases.

For all the boundary conditions, the non-dimensional central deflection increases as
the aspect ratio increases

For all the boundary conditions, the non-dimensional central deflection increases as
the thickness ratio increases

For simply supported boundary condition vibration amplitude increases as the volume
fraction index increases.
7.2 Future Scope of work

Different geometric structures can be modelled such as cylindrical, spherical,
conical, hyperboloid etc.

Temperature dependent material property can be considered.

Different type of analysis like buckling, post buckling, free vibration, forced
vibration etc. can also be performed using the presented model.
32
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35
Appendix-A
36
APPENDIX-A
u ( x, y, z )  u 0 ( x, y )  z x ( x, y )
v( x, y, z )  v 0 ( x, y )  z y ( x, y )
w( x, y, z )  w0 ( x, y )  z z ( x, y )
 xx 
u0

w
u
w



 z x  0  z z  ( 0  0 )  z( x  z )
x
x Rx
Rx
x Rx
x Rx
 yy 
 y w0
 y  z
u0
v
w

z

 z z  ( 0  0 )  z(
 )
x
y Ry
Ry
y Ry
y Ry
 zz   z
 yz   y 
y
y
w0
v
w
v


z z  0 z
 ( y  0  0 )  z ( z  )
y
y Ry
Ry
y Ry
y Ry
 xz   x 
w0
u

w u



 z z  0  z x  ( x  0  0 )  z ( z  x )
y
y Rx
Rx
y Rx
y Rx
 xy 
 y 2w0
 y  z

v


y
y v0 2w0
z x  0 z

z z (


)  z( x 

)
u0
y
x
x
Rxy
Rxy
u0 x Rxy
y
x Rxy
1
0

0
l  
0
0

0
0 0 0 0 0 z 0 0 0 0
1 0 0 0 0 0 z 0 0 0
0 1 0 0 0 0 0 z 0 0
0 0 1 0 0 0 0 0 z 0
0 0 0 1 0 0 0 0 0 z
0 0 0 0 1 0 0 0 0 0
37
  x0 
 0
 y 
  z0 
 0
0   yz 
 0
0   xz 
0
0   xy 
 
0   k x' 
0   k y' 
 
z   k z' 
 ' 
 k yz 
 k xz' 
 ' 
 k xy 
 
 x


 0

 0

0
x  
 0  0
 y  
  z0   1
 0  
 yz   Rx
 0   
 xz  
 xy0   y
 ' 
 kx   0
 k'  
 y'  
 kz   0
 '  
 k yz  
 k xz'   0
 '  
 k xy 
 0


 0


 0

 Q11
Q
 21
 Q31

 0
 0

 0
Q
zQ
 11
 zQ21
 zQ
 31
 0

 0
 0
0
1
Rx
0
0

y
1
Ry
0
0
0
0
0
0
1
Ry

y
0
1
0

y
1
0

x
2
Rxy
0
0
0
0

x
0
0
0
0

y
0
0
0
0
0
0
0
0
0
0
0


1
Ry
1
Rx
0

y

x


0 


0 

1 

0 



0 
  u0 
 v 
0   0
 .  w0 
 
1   x 

Rx   y 
 
1    z 

Ry 
0 

 
y 
 

y 
1 

Rxy 
Q12
Q13
0
0
0
zQ11
zQ12
zQ13
0
0
Q22
Q23
0
0
0
zQ21
zQ22
zQ23
0
0
Q32
Q33
0
0
0
zQ31
zQ32
zQ33
0
0
0
0
Q44
0
0
0
0
0
zQ44
0
0
0
0
Q55
0
0
0
0
0
zQ55
0
0
0
0
Q66
zQ12
zQ22
zQ32
0
zQ13
zQ23
zQ33
0
0
0
0
zQ44
0
0
0
0
0
0
0
0
z Q12
2
2
22
zQ
2
2
z Q11
z Q21
z Q31
0
0
2
0
2
z Q32
0
0
0
0
2
z Q13
0
0
2
0
0
0
0
z Q23
2
z Q33
0
2
z Q44
0
2
0
0
0
zQ55
0
0
0
0
0
z Q55
0
0
0
0
zQ66
0
0
0
0
0
38
0 
0 
0 

0 
0 

zQ66 
0 

0 
0 

0 

0 
z 2 Q66 
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