FINITE ELEMENT ANALYSIS OF SMART FUNCTIONALLY
GRADED COMPOSITE SHELL STRUCTURE
A THESIS SUBMITTED IN PARTIAL REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
By
ABHIJEET NAYAK
Roll No. – 10503004
Department of Mechanical Engineering
National Institute of Technology, Rourkela
May, 2009
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that this report entitled, “FINITE ELEMENT ANALYSIS OF SMART
FUNCTIONALLY GRADED COMPOSITE SHELL STRUCTURE” submitted by Abhijeet
Nayak in partial fulfillments for the requirements for the award of Bachelor of
Technology Degree in Mechanical Engineering at National Institute of Technology,
Rourkela (Deemed University) is an authentic work carried out by her under my
supervision and guidance.
To the best of my knowledge, the matter embodied in this report has
not been submitted to any other University / Institute for the award of any
Degree or Diploma
Date:
Rourkela
NIT
(Prof. T. Roy)
Dept. of Mechanical Engineering,
National Institute of Technology
Rourkela - 769008, Orissa
ACKNOWLEDGEMENT
I deem it a privilege to have been a student of Mechanical Engineering
stream in National Institute of Technology, Rourkela
I express my deep sense of gratitude and obligation to my project guide
Prof. T. Roy for his invaluable guidance and support. I am very grateful to him
for allowing me to do this project and for his constant help and support
throughout the making of this project. He always bestowed parental care upon us
and evinced keen interest in solving our problems. An erudite teacher, a
magnificent person and a strict disciplinarian, we consider ourselves fortunate to
have worked under his supervision.
Abhijeet Nayak
Roll No. – 10503004
8th Semester, B.TECH
Department of Mechanical Engineering
National Institute of Technology, Rourkela
ABSTRACT
Composite materials and structures are finding wide acceptance because of their stiffnessto-weight ratio that is particularly favorable. The main
drawback
of
laminated
composites, which is the weakness of interfaces between adjacent layers known as
delimitation phenomena, that may lead to structural failure has been partially overcome by
developing a new class of materials named Functionally Graded Materials.
Recently proposed (FGMs) have their various material properties vary through the thickness
in a continuous manner and thus free from interface weakness, typical of laminated
composites.
This project deals with the modeling of functionally graded doubly curved shells, with
the material properties graded in the thickness direction. Finite element modeling based on
first order shear deformation theory is used
The effect of grading on the deformation of the FGM shells in a given temperature boundary
conditions has been studied. Focus has also been put on Elemental model for static analysis
of smart functionally graded shells attached with distributed piezoelectric sensor and
actuator. Eight noded element with five degrees of freedom per node, three translational and
two rotations have been used. The electric field is applied in the thickness direction and
assumed to be constant through the thickness. The electrical potential is assumed to be
constant over the element. Also, effective coefficients of recently proposed Piezoelectric
Fiber Reinforced Composite
(PFRC) have been derived through micro-mechanical
analysis. The strength of materials approach has been employed to predict the coefficients
the static analysis of FGM shells has been done using this PFRC actuator.
i
CONTENTS
Chapter
Title
Page no.
Abstract
i
Contents
ii- vi
Nomenclature
vii-x
Chapter 1
Introduction
1-7
1.1 Composite materials
1
1.2 Drawbacks of laminated composites
1
1.3 Functionally graded materials
1
1.4 FGMs in nature
2
1.5 Effective properties of heterogeneous/multiphase materials
1.6 Smart Structures
3
3
1.7 Critical Elements in a smart structure
4
1.7.1 Actuators
4
1.7.2 Sensors
5
1.7.3 Control System
5
ii
1.7.4 Piezoelectric materials
5
1.8 Piezoelectric Fiber Reinforced Composites
5
1.9 Motivation of the present work
5
1.10 Applications of FGM/ smart FGM structures
Chapter 2
Background and literature review
2.1 General
6
8-12
9
2.2 Thermal stress analysis and applications
9
2.3 Fracture and creep analysis
10
2.4 Exact and analytical solutions
10
2.5 Vibration and control
10
2.6 Smart structures and Functionally Graded Materials
11
2.7 Piezoelectric Fiber Reinforced Composite (PFRC)
11
2.8 Objectives of present work
12
2.9 Layout of the thesis
Chapter 3
12
Determination of effective properties of PFRC layer
3.1 Assumptions involved in the analysis
13-19
13
iii
3.2 Strength of material approach
14
3.3Effective coefficients of PFRC layer
Chapter 4
17
Formulation
20-39
4.1Introduction
20
4.2 Geometry and kinematics of doubly curved shell element
21
4.2.1Assumptions in the model development
21
4.2.2 Shell geometry considerations
21
4.2.3 Discretization of shell global space to isoparametric space
4.2.4 Displacement field and strains for shell element
23
24
4.2.5 Isoparametric Finite Element approximation of displacement
field and electric field for plate element
4.2.6 Jacobian matrix (Transformation matrix)
24
26
4.3 Governing differential equations
27
4.4. Static finite element equations
28
4.4.1 Mechanical strain energy
28
4.4.2 Electrical potential energy
30
iv
4.4.3 Work done by the external forces and electrical charge
30
4.5 Dynamic finite element equations
32
4.6 Thermal load
37
4.7 Piezoelectric patches
37
4.8 Thermal analysis to determine temperature distribution in the
thickness direction
38
4.9 Element mass matrix
Chapter 5
39
Results and discussion
40-50
5 Validation of Finite Element Code of doubly curved shell element
5.1 Static mechanical shell element
40
41
5.2 Static thermo-mechanical analysis of the shell element
42
5.3 Validation of electromechanical coupling of the piezoelectric materials
using doubly curved shell element
43
v
5.3.1 PFRC actuator integrated on the surface of a doubly curved shell
43
5.4 Variations in response due to various types of loadings
45
5.4.1 Variations in response due to pure thermal loading
45
5.4.2 Variations in response due to pure mechanical loading
46
5.4.3 Variations in response due to thermo- mechanical loading
47
5.4.4 Variations in response due to electro-thermo- mechanical loading
5.5 Scope for future work
48
51
vi
NOMENCLATURE
σx ,σy ,σxy ,σxz ,σyz
Components of Stress vector in the global
or, σ1 ,σ2 ,σ12 ,σ13 ,σ23
Coordinate system
εx , εy , εxy , εxz , εyz
or, ε1 , ε2 , ε12 , ε13 , ε 23
Components of Strain vector in the global
Coordinate system
σ’x ,σ’y ,σ’xy ,σ’xz ,σ’yz
Components of Stress vector in the local
or, σ’1 ,σ’2 ,σ’12 ,σ’13 ,σ’23
Coordinate system
ε’x , ε’y , ε’xy , ε’xz , ε’yz
or, ε’1 , ε’2 , ε’12 , ε’13 , ε’23
αik
Components of Strain vector in the local
Coordinate system
Vectors defining the nodal coordinate
system at the node in degenerate shell element
ζ
A linear natural coordinate in the thickness
direction
x,y,z
Global space
α1 , α 2
Parametric space (in shell)
x’,y’,z’
Local co-ordinates system of layers
vii
ξ and η
Natural/isoparametric space of shell element
u, v, w
Displacement components of any point
in the shell space
Φx , Φy
{de},{d}
Rotation of yz and xz planes due to bending
Mechanical degrees of freedom vector at the
element and global level, respectively
cElastic matrix
v,µ
Poisson’s ratio
h
Thickness of the FG shell/plate
E1,E2
Modulus of elasticity at the top and bottom
of the FG layer, respectively
λ
Parameter of gradation on FG layer
{D}
Electric displacement vector
[Є]
Dielectric matrix at constant mechanical
strain
R 1, R 2 , R 3
Radii of curvature of doubly curved shells
A1 , A2
Lame’s parameters for doubly curved shell
eij
Piezoelectric coefficients
viii
{Ge} , {G}
Electric force vector resulting from the applied
charge density on the the actuator at the
element and the global level, respectively
F e ,FMechanical force vector resulting from the
vector resulting from the level, respectively
T
Kinetic energy

Total potential energy of the entire structure
U
Strain energy of the entire structure
{E}Electric field vector
TΘ,T1 ,T2
Stress free temperature, temperature at
bottom, temperature at top, respectively
NMT Thermal load vector
N T, M T
Thermal force, thermal moment
ix
Abbreviations used
FOST
First Order Shear Deformation Theory
PVDF
Polyvinylidene fluoride
PZT
Lead Zirconate Titanate
PFRC
Piezoelectric Fiber Reinforced Composite
ITER
International Thermonuclear Energy Reactor

x
CHAPTER-1
INTRODUCTION
1.1 Composite materials
Composite materials are engineered materials made from two or more constituent materials
with significantly different physical or chemical properties which remain separate and distinct
on a macroscopic level within the finished structure.They have varied applications including
army and aerospace
vehicles,
nuclear
reactor
vessels, turbines, buildings ,
smart
highways as well as in sports equipment and medical prosthetics. Our area of interest
are Laminated composite structures consisting of several layers of different
reinforced
laminate
bonded
together
to
obtain
desired
fibre-
structural properties (e.g.
stiffness, strength, wear resistance, damping,etc). Varying the lamina thickness, lamina
material properties, and stacking sequence the desired structural properties can be achieved.
Composite materials exhibit high strength-to weight and stiffness-to-weight ratios, which
make them ideally suited for use in weight sensitive structures. This weight reduction
of structures leads to improvement of their structural performance especially in aerospace
applications.
1.2 Drawbacks of laminated composites
Though laminated composites have an edge over conventional materials, their major drawback
is the weakness of interfaces between adjacent layers, known as delamination phenomena
that may lead to failure of the structure. Additional problems include the presence of
residual stresses due to the difference in coefficient of thermal expansion of the fiber and
matrix. The effects of interlaminar stresses become more profound when laminated
composites are subjected to extreme temperatures which leads to failure of composite
structure due to delamination. In order to overcome these problems the sudden change of
material properties has to be taken care of.
1.3 Functionally graded materials
In “Functionally Grad e d Materials” (FGMs) the material properties are graded in a
1
predetermined manner. Thus, Functionally Graded Materials (FGMs) are an advancement of
composite materials where the composition or the microstructure is locally varied so
that a certain variation of the local material properties is achieved. FGM is also defined as,
composites in which the
volume fraction of
two or more
materials are achieved
continuously as a function of position along certain directions of the structure to achieve
properties as a required function. e.g. mixture of ceramic and metal. By grading the
material properties in a continuous manner, the effect of interlaminar stresses developed at
the interfaces of the laminated composite due to abrupt change of material properties
between neighbouring laminas is mitigated.
Thin walled members, i.e., plates and shells, used in reactor vessels, turbines and other
machine parts are susceptible to failure from buckling, large amplitude deflections, or
excessive stresses induced by thermal or combined thermo mechanical loading. Here
FGM’s can fill the void. Thus, FGMs are primarily used in structures subjected to extreme
temperature environment or where high temperature gradients are encountered. They are
typically manufactured from isotropic components such as metals and ceramics since they
are mainly used as thermal barrier structures in environments with severe thermal
gradients (e.g. reactor vessels, semiconductor industry).In such conditions ceramic provides
heat and corrosion resistance, while the metal provides the strength and toughness.
Presently intensive work is being done on for the manufacturing of a thermal plasma shield
made for the ITER (International Thermonuclear Energy Reactor). As FGMs are the only
possible materials to withstand the extreme temperatures developed within a reactor,
without failure.
1.4 FGMs in nature
Nature has designed all-biological load carriers such as stems of plants, trunks of trees,
bones and other hard tissues in such a way that they incorporate FGM structures by a natural
process of optimization and adaptation to their loading & boundary conditions. Their
constituents and geometry change continuously adjust to their physical environment. For
e.g. specific modulus and specific strength of pure bamboo fiber and shaft of a feather is
comparable to that of engineering alloy and ceramics.
2
1.5 Smart Structures
Essentially, a smart structural system is a multifunctional unit. It consists of a load
(electrical, thermal, magnetic, or mechanical) bearing part, which is usually passive, and
an active material part that performs the operations of sensing and actuating.
Newnham’s definition, says “The structures with surface mounted or embedded sensors
and actuators with the capability to sense and take corrective action” smart structures. E.g,
Vibration amplitudes in a flexible plate structure may be suppressed using the sensing
and actuation capabilities of piezoelectric or piezoceramic films by bonding them to the
surfaces of the plate. As the plate deforms due to external applied loads, the bonded
piezoelectric film (sensor) also deforms, and due to its constitutive behavior, it develops
a surface charge proportional to the applied force. The charge may be processed by a
control system, which supplies an appropriate voltage to the piezoelectric film (actuator) that
induces a counteractive deformation to the plate structure and suppresses the amplitudes of the
vibrations. The
commonly
used
smart
materials
are
piezoelectric
materials,
magnetostrictive materials, electrostrictive materials, shape memory alloys, fiber optics,
and electro rheological fluids. Each smart material has a unique advantage of its own.
1.6 Effective properties of heterogeneous/multiphase materials
As FGMs are heterogeneous materials, there is need for the determination of effective material
properties.
To
achieve
best
performance,
accurate
material
property estimation is
essential because associated analysis and design for selecting an optimal vol.-fraction depends
on its suitability.
Various modeling approaches used for FGMs are
1. Rules of mixtures

Linear rule of mixtures

Harmonic rule of mixtures
2. Variational approach
3. Micromechanical approaches
Rules of mixtures employ bulk constituent properties assuming no interaction
between phases. This approach derived from continuum mechanics and is free from
3
empirical
considerations.
In
variational
approach,
variational
principles
of
thermomechanics used to derive the bounds for effective thermophysical properties.
Micromechanical approaches include information about spatial distribution of the
constituent materials. Standard micromechanical approach is based on concept of unit cell or
Representative Volume Element (RVE) to represent the microstructure of composite.
1.7 Critical Elements in a smart structure
There are three important components of smart structures, namely

Actuators

Sensors

Control system.
1.7.1 Actuators
Actuator is generally the reverse of sensor. It converts electrical inputs to physical (thermal,
mechanical, etc) outputs. The ideal mechanical actuator would directly convert electrical
input into strain or displacement in the host structure. The principal actuating mechanism of
actuators is referred to as actuation strain.
1.7.2 Sensors
Sensors are mechatronics devices that can convert analogue physical values into electrical
impulses thus informing of their magnitude. The ideal sensor for smart structures converts
strain or displacement directly into electrical
output.
The
primary
functional
requirement of such sensors is their sensitivity to strain and displacement.
1.7.3 Control System
Implementing
various
control
methodologies
performs
the
control
of
intelligent
structures. Different type of control strategies are available.
1.7.4 Piezoelectric materials

Piezoelectric materials have the ability to generate electric potential in response to
applied mechanical stress.
4

This property is exhibited by certain materials like ceramics & some crystals.

The piezoelectric effects can be seen as transfer between electrical and mechanical
energy.

Such transfers can only occur if the material is composed of charged particles and can
be polarized.

For a material to exhibit an anisotropic property such as piezoelectricity, its crystal
structure must have no centre of symmetry.
1.8 Piezoelectric Fiber Reinforced Composites
Active control of smart structures depends on the magnitude of electric potential difference
for a given mechanical stress. This subsequently depends on the piezoelectric stress/strain
constants. The existing monolithic piezoelectric materials being used in smart structures posses
low
control
authority as
magnitude. Because,
their piezoelectric
tailoring
of
these
characteristics of the smart structures.
Composites
(PFRC)
applications.
These
is
stress/strain
properties
constants
may
improve
are
the
of
small
damping
Currently, Piezoelectric Fiber Reinforced
being
effectively
composites
show
used
in
improved
underwater
mechanical
and
medical
performance,
electromechanical coupling characteristics, and acoustic impedance matching with the
surrounding medium over the piezoelectric material alone.
1.9 Motivation of the present work
Functionally Graded Materials are a new breed of materials which are the answer to many
structural problems demanding self control and flexible characteristics involving mechanical
and thermal stresses. The technological implications of this class of materials (FGMs) are
immense, as they are especially useful in remote operations, expensive space operations
subjected
to extreme thermo-mechanical loadings, aerospace skins, protective shields,
components in reactor vessels, machine tools, and medical applications, to name only a few.
As the advent of steel changed the last century, similarly FGM’s are the materials which will
revolutionize the 21st century. These material systems have charecteristics such as thermoelectro-mechanical coupling, functionality, intelligence, and gradation at micro and nano
scales. The reliability and integrity of these systems are the main challenges before us. They
5
can be customized to operate under varying conditions covering the whole spectrum of
electro-thermo-mechanical conditions. The conditions can vary across a wide range of
temperature, magnetic & electric fields, pressure and mechanical load, and/or a combination
of two or many. Experimental investigations of both these systems & materials although
possible, are prohibitively expensive, and therefore must be complemented with simulations
and theoretical analyses.
1.10 Applications of FGM/ smart FGM structures
A wide variety of applications exist for smart FGM structures.
1. Aerospace

Aerospace skins

Rocket engine components

Vibration control

Adaptive structures
2. Engineering

Cutting tools

Wall linings of engines

Shafts

Engine components

Turbine blades
3. Nuclear energy

Nuclear reactor components

First wall of fusion reactor

Fuel pellet
4. Optics

Optical fiber

Lens
5. Electronics

Graded band semiconductor

Substrate
6

Sensor

Actuator

Integrated chips
6. Chemical plants

Heat exchanger

Heat pipe

Reaction vessel

Substrate
7. Energy conversion

Thermoelectric generator

Thermoionic converter

Fuel cells, solar cells
8. Biomaterials

Implants

Artificial skin

Drug delivery system

Prosthetics
9. Commodities

Building material

Sports goods

Car body

Casing of various materials

Air Conditioning temperature control
7
CHAPTER 2
BACKGROUND AND LITERATURE REVIEW
2.1 General
Advanced
composite
materials
offer
numerous
superior
properties
to
metallic
materials, like high specific strength and high specific stiffness. This has resulted in the
extensive use of laminated composite materials in aircraft, spacecraft and space structures.
In an effort to develop the super heat resistant materials, Koizumi [1] first proposed the
concept of FGM. These materials are microscopically heterogeneous and are typically
made from isotropic components, such as metals and ceramics. After the concept of FGMs
was set by the Japanese school of material science, see e.g. [1] for an earlier contribution,
and confirmation of their potentials found even in natural materials [2], several branches of
research originated and are still being broadened by research groups all over the world.
2.2 Thermal stress analysis and applications
Abrupt transition in material properties of laminated composites across the interface between
discrete materials can result in large interlaminar stresses and lead to plastic deformation or
cracking. Teymur [2] carried out the thermomechanical analysis of materials, which are
functionally
graded
in
two
directions,
and demonstrated that the onset of
delamination could be prevented by tailoring the microstructures of the composite piles.
Thus, the use of FGM may become an important issue for developing advanced
structures. Feldman and Aboudi [3] studied the elastic bifurcational buckling of functionally
graded plates under in-plane compressive
nonuniform
distribution
load.
They concluded
that
with
optimal
of reinforcing phases, the buckling load can be significantly
improved for FG plate over the plate with uniformly distributed reinforcing phase. Mian and
Spencer [4] derived the exact solutions for functionally graded plates with zero surface
traction. Gasik [5] developed an efficient micromechanical model for FGMs with an
arbitrary non-linear 3D-distribution of phases. This model has been reported to provide
accurate estimates of the properties of the FGMs. The model is also capable of computing
8
thermal stresses, evaluating dynamic stress/strain distribution and inelastic behavior of
FGMs. Praveen and Reddy [6] investigated the nonlinear thermoelastic behavior of
functionally graded ceramic metal plates.
2.3 Fracture and creep analysis
In 2000, Wang et al.[ 7] proposed a method to determine the transient and steady state
thermal stress intensity factors of graded composite plate containing noncollinear cracks
subjected to dynamic thermal loading. Yang [8] presented an analytical solution for
computing the time-dependent stresses in FGM undergoing creep. Yang and Shen [9]
studied the dynamic response of initially stressed functionally graded thin plates subjected to
partially distributed impulsive loads.
2.4 Exact and analytical solutions
An elasticity solution for functionally graded beams was provided by Sankar [10] in which
the beam properties are graded in the thickness direction according to an exponential law.
Batra and Vel [19] have presented the exact solutions for thermoelastic deformations of
thick FG plates subjected to both thermal and mechanical loads. Woo and Meguid
[11] presented an analytical solution for the large deflections of plates and shallow shells
made of FGMs under the combined action of thermal and mechanical loads. The exact
solutions for thermoelastic deformations of thick FG plates subjected to both thermal and
mechanical loads have been presented by Batra and Vel [12]. Zhong and Shang [22]
presented three dimensional exact analysis of a simply supported functionally gradient plate.
2.5 Vibration and control
Loy [13] studied the vibration of cylindrical shells made of a functionally graded material,
which was composed of stainless steel and nickel. Aboudi et al. [14] further developed a more
general higher-order theory for functionally graded materials and illustrated the utility of
functionally graded microstructures in tailoring the behavior of structural components in
various applications.
9
2.6 Smart structures and Functionally Graded Materials
The continuing research on materials being lightweight yet having high strength & flexibility,
which would constitute self-controlling & self-monitoring capabilities. For such requirements
piezoelectric materials are essential. They are also the main constituents of actuators &
sensors mounted or embedded in the system.
Such structures capable of demonstrating self-control and adaptability are called smart
structures.
The concept of developing smart structures has been extensively used for active control
of flexible structures during the past decade. Reddy and Cheng [20] presented three
dimensional solutions of smart functionally graded plates. He and Liew K M [21] presented
active control of FGM plates with integrated piezoelectric sensors and actuators.
Very recently, Huang and Shen [18] investigated the dynamics of a functionally
graded (FG) plate coupled with two monolithic piezoelectric layers at its top and bottom
surfaces undergoing nonlinear vibrations in thermal environments.
2.7 Piezoelectric Fiber Reinforced Composite (PFRC)
The major bottleneck in the development of smart structures is the small magnitude of control
exhibited due to the small strain in piezoelectric. Hence the structure cannot demonstrate
sufficient amount of control and hence damping (In vibration damping systems)
and the flexibility of the system is also not up to the mark. The solution is
improving the piezoelectric stress/strain coefficients
which will increase t hei r control
authority and hence the damping (By decreasing the vibration amplitude as well as oscillation
decay time. In an effort to tailor the piezoelectric properties, Mallik and Ray [23]
proposed the concept of longitudinally Piezoelectric Fiber Reinforced Composite (PFRC)
materials and investigated the effective mechanical and piezoelectric properties of these
composites. The main concern of their investigations was to determine the effective
piezoelectric coefficient (e31) of these new concept PFRC materials, which quantifies the
induced normal stress in the fiber direction due to the applied electric field in the
direction transverse to the fiber direction. They observed that this effective piezoelectric
coefficient was significantly larger than the corresponding co-efficient of piezoelectric
material of the fibers. Ray and Sachade [24] have recently derived the exact solutions for
10
the linear analysis of the simply supported functionally graded plates integrated with a
layer
of
this
new Piezoelectric Fiber Reinforced Composite (PFRC) material.
Subsequently, they also developed a finite element model for the linear analysis of
simply s upport ed functionally graded plates integrated with the layer of this PFRC material
[25].
Still breakthroughs are awaited in the field of functionally graded shell structures subjected to
combined electromechanical loading under temperature field
2.8 Objectives of present work
The objective of the present work has been to:

Develop a finite element code for smart functionally doubly
curved
shells
subjected to a coupled electro-thermo-mechanical loading integrated with layers
of the piezoelectric sensor -actuator patches on its surface.

Develop a finite element code to determine the free vibrations of a FG layer.

To study the effect of
grading on the static response and fundamental
frequency of a FG layer.
2.9 Layout of the thesis
The thesis is organized in various chapters as mentioned below.
Chapter 1 gives an introduction to FGM and smart FGM structures, their advantages over
conventional laminated composites.
Chapter 2, is an extensive literature review on the developments of theoretical analysis
of functionally graded materials and their finite element modes. The literature review
concerning the use of FGM structures as smart structures is also discussed. New class of
sensor-actuator (PFRC) for smart control is also studied.
Chapter 3 includes the determination of effective properties of a PFRC using micromechanical analysis is presented.
Chapter 4 includes the formulations for static and free vibration analysis for the
functionally graded doubly curved shells.
11
Chapter 5
 Covers the numerical examples to validate the models developed for static analysis.
 Few examples are given for electro-thermo-mechanical analysis of smart FGM
structures.
 Subjected to similar conditions, the performance of plate with the shell element
is compared.
 The performance of piezoelectric patches and PFRC for deflection control is
compared.
 Effect of gradation on the static and frequency response is studied. Conclusions from
the investigations of the finite element analysis of smart FGM structures, the proposed
work to be done in next phase is presented in Chapter 6.
12
CHAPTER-3
EFFECTIVE COEFFICIENTS OF PIEZOELECTRIC FIBERREINFORCED COMPOSITES
3.1 Assumptions involved in this analysis

Fibers in the to-be developed composite are parallel and continuous

Fiber and composite are linearly elastic.

Matrix material is piezoelectrically inactive.

Fiber and matrix are bonded firmly.

A constant electric field exists in a direction at right angles to the fiber direction.
(Made using Paint)
13
3.2 Strength of material approach
The micromechanical analysis is confined to a Representative element (RVE) that
includes both fiber and the surrounding matrix. The piezoelectric fibers are oriented
longitudinally, i.e. along the global x-axis.
Fig.3.2 A longitudinal cross-section of a representative volume of PFRC(made usingPaint)
The constitutive relations for the fibers (piezoelectrically active) are
14
And those for the Matrix are
As there is perfect bonding assumed, therefore the strains are same in x-direction
Also, lateral stresses are same in the y and z direction
By rule of mixtures the lateral strains in y and z direction can be written as
vf , vm
are the volume fractions of fiber and matrix, respectively.
From eq. (3.16)
σ ym = σ yp
15
Similarly we have
σzm = σzp
from eq (3.16). Equating eq. (3.3) and (3.10) and using eq
(3.15), and after rearranging the terms
where
16
Now, axial equilibrium of the composite requires that the composite stresses in the axial
direction can be written using the rule of mixture as
Putting (3.1) and (3.8) in the above equation, and rearranging
If we define the resultant coefficients of composite as
Then on comparing the coefficients of eq. (3.32) and (3.34)
3.3 Effective coefficients of PFRC layer
Resultant effective coefficients of a PFRC layer are written as
17
Also, putting eq. (3.20) and (3.21) in (3.2) and using
and equating the coefficients, the rest of the effective coefficients of a PFRC layer are
obtained as
Also, resultant permittivity in the PFRC composite can be written using the rule of
mixtures as
Putting the values of Dxp and D x m from eq (3.17) and (3.14) and using eqs (3.17), (3.20)
and (3.21) and after rearranging the terms, effective permittivity of the PFRC composite are
obtained as
18
Thus effective coefficients of a PRFC are determined in the above micromechanical
analysis, which would be used in the present work. They can be used now to compare their
effectiveness in controlling the deflections of a mechanical structure, as compared to
existing monolithic PZT actuators.
19
CHAPTER 4
FINITE ELEMENT MODELLING OF SMART FGM SHELL
4.1 Introduction
Study of physical systems frequently results in partial differential equations, which either
cannot be solved analytically, or lack an exact analytic solution due to the complexity of
the boundary conditions or domain. For a realistic and detailed study, a numerical method
must be used to solve the problem. The finite element method is often found the most
adequate. Over the years, with the development of modern computers, the finite element
method has become one of the most important analysis tool in engineering. It has penetrated
successfully many areas such as heat transfer, fluid mechanics, electromagnetism, acoustics
and fracture mechanics. Finite element packages are now widely available on personal
workstations.
Studying functionally graded materials require a numerical technique to solve, as the
variation of material properties would be very difficult to analyze analytically. As FGM
materials are used primarily at high temperature environments, therefore they may be of
arbitrary shape and sizes, geometry and loading thus making it almost impossible to
obtain analytical or exact solutions for the real life conditions. The finite element method is
very much suited for the analysis of plates and shells of general shape because of its
flexibility in accounting for arbitrary geometry, loadings and variation in material
properties. In finite element analysis, the structure is subdivided into a finite number of
elements of simple geometry, and the physical fields are interpolated inside these
elements using shape functions and nodal values of the field variables.
In this
work, the
equations of
motion are
described using a first-order shear
deformation theory (FSDT) based on the Reissner – Mindlin assumptions. Eight- noded
serendipity plate element have been used. Linear elastic behavior of materials
is assumed
throughout this analysis and temperature field is assumed to be known. The top and bottom
surfaces are bonded with piezoelectric films
20
4.2 Geometry and kinematics of doubly curved shell element
4.2.1 Assumptions in the model development
In developing the working model we have taken into account some assumptions
They are:

FGM shell is assumed to be graded in thickness direction,

FGM shell is isotropic in other two directions.

Linear elasticity is assumed in the formulation

The deformations follow Mindlin’s hypothesis, i.e. normal to the middle
surface
only
of the shell before deformation may not remain normal after
deformation but remains straight and inextensional.

The in-plane displacement components are assumed to vary linearly along the
thickness direction to yield constant transverse shear strain.

The piezoelectric patches are thin and are perfectly bonded to the FGM layer.
4.2.2 Shell geometry considerations
The present work deals with regular doubly curved shells where the shell midsurface Ω ϵ R3
has been mapped into parametric space (α 1, α 2 )ϵ A:R 2 →R 3 through a suitable exact
parameterization. Two independent coordinates (α 1, α 2 ) parametric space have been
considered as the midsurface curvilinear coordinates of the shell as shown in Fig.4.3 The
normal direction coordinate to the middle surface of the shell has been represented by ζ. The
reference surface or shell midsurface thus defined can be described as
21
For the analysis of the shell the Lame’s parameters neglecting the trapezoidal effect of the
shell cross-section can be computed as
Fig 4.1 A layered composite doubly curved shell element[39].
(4.1)
The comma denotes the partial differentiation. Unit tangent vectors of the midsurface can be
expressed as
(4.2)
The unit normal vector to the tangent plane of any point on the reference surface has been computed
using the following relation
(4.3)
22
The physical components of the normal and twist curvatures of the shell midsurface can be
expressed as:
(4.4)
4.2.3 Discretization of shell global space to isoparametric space
We have taken into account the parametric space A as an assembly of sub domains or
elements which are quadrilateral in nature. Here sample space A is summation of individual
elements in the parametric space. We have approximated any point within an element in the
parametric space by performing‘isoparametric mapping’.The concept used is(ξ,η) є [1,1]Х[-1,1]|→Ae as shown in Fig 4.4. so the curvilinear coordinates (α1,α2) of any point
within an element [36] may be expressed as follows:
Figure4.2(a). Global space  x, y, z  (b) parametric space α1,α2 ; and (c) isoparametric
spaceξ,η[39]
23
4.2.4 Displacement field and strains for shell element
The displacement components on shell midsurface S at any point within an element may be
expressed as
(4.5)
Here ui,vi,Φ1i,Φ2i are the unknown displacement components of the ith node. Ni (ξ,η)
is the interpolation function corresponding to the ith
node of an element and ne
is the
number of nodes per element. In this analysis, programming code is developed in such
a way that the interpolation functions are that of a Serendipity element[39]
4.2.5 Isoparametric Finite Element approximation of displacement field and electric
field for shell element
The strain vector of a doubly curved shell may be expressed as
(4.6)
(4.7)
(4.8)
24
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
25
(4.14)
The stress-strain relationship becomes
(4.15)
Where [D] is the rigidity matrix
4.2.6 Jacobian matrix (Transformation matrix)
(4.16)
(4.17)
(4.18)
26
4.3 Governing differential equations
Equation of equilibrium
(4.19)
Equation of electrostatics
(4.20)
One dimensional steady state heat transfer coefficient
(4.21)
Finite element analysis can be done either, by finding weak form of above governing
differential equations and then extremizing the functional obtained w.r.t elemental
displacement vector {d e }or from the virtual work principle
4.4 Static finite element equations
(4.22)
27
4.4.1 Mechanical strain energy
(4.23)
(4.24)
Putting in eq (4.23) [39]
(4.25)
After putting FE approximation for displacement vector eq.(4.55) and using FSDT
theory eq.(4.46-4.48)
(4.26)
(4.27)
28
(4.28)
(4.29)
4.4.2 Electrical potential energy
Using constitutive relations, strain displacement and electric field electric-potential
relations, the element electrical energy can be written as
(4.30)
(4.31)
(4.32)
29
4.4.3 Work done by the external forces and electrical charge
The virtual work done by the external forces due to the applied surface traction and the
applied electrical charge density is given by
(4.33)
where {f(x,y)} and {g(x,y)}the surface force intensity and surface electrical charge density
respectively. Ω1 and Ω2 are the surface areas where the surface forces and electric charge are
applied, respectively.
Discretisizing above equation
(4.34)
The work done by the body force and point forces are not considered for simplicity.
Substituting the elastic strain energy eq and electrical potential energy eq.
in internal potential energy eq
(4.35)
Substituting the internal potential energy and the external virtual work done in total
potential energy and setting its first variation to zero, the following system of equations for
the element are obtained[39].
.
30
(4.36)
where the superscript e refers to the parameter at the element level and [K] matrices with
subscripts uu , uφ , φu , φφ are defined below.
(4.37)
(4.38)
(4.39)
(4.40)
The element mechanical force {Fe} and {NMTe} in the eq and the element electrical force
vector {Qce} in eq are defined as
(4.41)
(4.42)
(4.43)
4.5 Dynamic finite element equations
The dynamic equations of a piezo-laminated composite shell can be derived from the
Hamilton’s principle
31
(4.44)
where Le represents the Lagrangian, and δW e is the virtual work of external forces
The Lagrangian is to be properly adapted in order to include the contribution from the electrical field
besides the contribution from the mechanical field
Le=Te–Ue
(4.45)
Above eq. can be further expressed by
(4.46)
in which T is the kinetic energy and can be written as
(4.47)
d is the velocity vector and  ρ is the mass density matrix. t1and t2 define the time interval.
1
All variations must vanish at t = t1 and t = t2 .The individual parts of the Hamilton equation
(Eq. (4.109)) can be written as
Kinetic energy
(4.48)
On integrating by parts
(4.49)
As all variations vanish at t  t1 and t  t2
32
(4.50)
(4.51)
The potential energy terms and virtual work done by external forces can be written from
that of static situation, as they would remain the same
(4.52)
Putting eq. (100) and eq. (101) in Hamilton’s eq. (94)
(4.53)
which must be verified for any arbitrary variation of the displacements and electrical
potentials compatible with the essential boundary conditions. For an element above eq. can
be written as
(4.54)
33
Where
(4.55)
After assembly of the elemental matrices eq. (4.119) - (4.120), the global sets of
equations are obtained as follows
(4.56)
In this work, the plate or shell is assumed to be of constant thickness and isotropic in the
plane perpendicular to the thickness direction. Structural stiffness can be separated into
bending stiffness and transverse stiffness matrices as they are not coupled and thus are
independently calculated. [39]
(4.57)
(4.58)
on converting to natural coordinates,
(4.59)
(4.60)
34
 A,  B, D are given by eq. (4.27).For a FG shell
(4.61)
(4.62)
(4.63)
(4.64)
And E(z) represents the elastic Young’s modulus and is function of z .v is poison’s ratio and
is assumed constant throughout the thickness. On integrating we get the effective bending
stiffness matrix.
For power law, variation
(4.65)
(4.66)
35
(4.67)
(4.68)
(4.69)
(4.70)
(4.71)
4.6 Thermal load
(4.72)
36
4.7 Piezoelectric patches
Uniform electric field and displacement across the thickness and aligned on the normal
to the mid-plane (direction z or  ). It is assumed that no shear strain is induced by a
transverse electric field, which is the case for most commonly used piezoelectric
materials in laminar design (PZT and PVDF). If constant potential across each element
is assumed, one degree of freedom for electrical potential per layer is defined [35].
(4.73a) ,(4.73b)
Variation of electric potential function across the thickness may be considered linear, then at
any point, if potential is taken as a nodal variable [25].
(4.74a), (4.74b)
4.8 Thermal analysis to determine temperature distribution in the thickness direction
For the evaluation of temperature induced load N T and moment M T  it is required
to determine the temperature distribution T
(ζ,η,ξ) subjected to Dirichlet boundary
conditions.
Since present work deals with transversely isotropic material, therefore temperature will be
a function of z only. The temperature distribution can be obtained by solving one
dimensional steady state heat conduction equation
37
(4.75)
Closed form of equation is not possible as k is also a function of thickness coordinate. Hence,
it solved using Ritz method with Langrage polynomials. Temperature distribution is
approximated by trial functions as a sixth-order polynomial [31]
(4.76)
This
results
in
pretty
good
approximation
without
excessive
computational
overheads. The unknown parameters are calculated from the boundary conditions.
4.9 Element mass matrix
Element mass matrix in an element is defined as
(4.77)
(4.78)
(4.79)
ρk is the density of the kth lamina.
38
CHAPTER-5
RESULTS AND DISCUSSIONS
5 Validation of Finite Element Code of doubly curved shell element
Doubly curved shell is obtained from the surface of a sphere. The dimensions of the spherical
shell obtained after cutting from a sphere are as mentioned.
5.1 Static mechanical shell element
In the element the plate element if its radius of curvature is very high as compared to its
dimensions. Due To lack of existing literature on FG shells, the response of the shell element is
compared with that of FG plate. To achieve this, very high radii (of the order of 106 meters) has
been taken and the response obtained has been compared with the corresponding response of
the plate element under same loading, geometry, boundary conditions, gradation and material
properties. The lower shell surface is assumed to be aluminum while the top surface is assumed
to be zirconia. Material properties vary with the power law defined in chapter 5.
P(z)=(P2-P1){(2z+h)/2h}λ+P2
and the values of λ=0,1,2,106, assuming (∞=106), are considered. Physical material properties
are given in table 5.1.
The dimensions of the shell
Length a=200mm, width b=200mm and thickness h=10mm
The radius of curvature R1, R2=106
Material Properties
Aluminum
Zirconia
Young’s Modulus
70GPa
151GPa
Poisson’s Ratio
0.3
0.3
Thermal Conductivity
204W/mK
2.09W/mK
Thermal Expansion
23x10-6/oC
10x10-6/oC
Table 5.1: Material properties of FG shell
40
Boundary conditions applied
Simply supported boundary conditions:
At x=0 and x=a,
u=0, w=0, ϕy=0 for all nodes in that particular plane.
At x=0 and x=a,
u=0, w=0, ϕx=0 for all nodes in that particular plane.
Uniform load q given by P= (a4q/t4Ebottom) where P is the load applied along z direction.
Power
Load parameter
-6
Center
deflection
w(in mm) of
present shell
element
-1.23576
Center
deflection
w(in mm)
Corce, Venini
[28]
-1.2
coefficient
P=(a4q/t4Ebottom)
-10
-0.205960
-2.05
-13
-2.67744
-2.7
-6
-1.86573
-1.75
-10
3.10955
-3.15
-13
-4.04243
-4.09
-6
-2.6653
-2.2
-10
4.44274
-4.45
-13
-5.77551
-5.86
λ
0
2
106(∞)
Table 5.2: Comparison of center displacement of doubly curved shell (large radii),
With a plate element using (4x4) elements under mechanical loading only.
Since the results for shell and plate agree well, within limits, as shown in Table 5.2, therefore
the element predicts the mechanical behavior accurately.
41
5.2 Static thermo-mechanical analysis of the shell element
In addition to a uniformly distributed mechanical normal load on the top surface, the shell has
been subjected to a thermal field where the ceramic rich top surface is held at 3000C ant the
metal rich bottom surface is held at 200C. a stress free temperature of 00C is assumed.
Power
Load parameter
coefficient
P=(a4q/t4Ebottom) deflection
λ
Center
w(in
Center
mm)
present
0
2
5
106(∞)
deflection
of
shell
w(in mm)
Corce, Venini
element
[28]
0
1.45186
1.3
-5
0.422042
0.43
-10
-0.607756
-
0
1.02927
0.998
-5
-0.525495
-0.5
-10
-2.08026
-
0
1.32473
1.3
-5
-0.373779
-0.39
-10
-2.07229
-
0
2.30241
2.46
-5
0.208969
0.2
-10
-2.01239
-
Table 5.3: Comparison of center displacement of doubly curve (large radii), with a plate
element using (4x4) elements under thermo-mechanical loading
Above results in Table5.3 show that the complete thermo-mechanical modeling of the doubly
curved FGM shells are accurate. It can be seen that the response of graded shells is not
intermediate to the metal and ceramic shells. The center deflections of both the metallic and the
42
ceramic shells are higher then those of the graded shells. This is in agreement with the result
obtained for FG shells.
5.3 Validation of electromechanical coupling of the piezoelectric materials using doubly
curved shell element
5.3.1 PFRC actuator integrated on the surface of a doubly curved shell
FG shell, as in plates is an exponential function of thickness (z), measured from the mid plane
of the FG layer, and is given by
E (z ) = E1eλ(z+h/2)
Where, E1=200GPa and μ=0.3 in which μ is the Poisson’s Ratio of the FGM.
λ is the exponential gradient parameter depending on the ratio of Young’s modulus at the top
to that at the bottom.
Dimensions of shell structure,
h=3 mm for FG plate and h=250μm of the PFRC layer.
Length a=60mm
The radius of curvatures of the shell element R1, R2=106
Fiber/
C11
C12
C13
C33
C44
e31=e32
e33
ϵ33
Matrix
GPa
GPa
GPa
GPa
GPa
GPa
GPa
GPa
PZT-5H
151
98
96
124
14
-5.1
27
13.27x10-9
Epoxy
3.86
2.57
2.57
3.86
2.57
0
0
0.079x10-9
Table 5.4: Material properties of fibers and matrix of PFRC layer
The piezoelectric fiber and the matrix of the PFRC layer are made of PZT5H and epoxy,
respectively. The elastic and piezoelectric co-efficient for piezoelectric fiber and matrix of the
43
PFRC layer are given in Table-5.4. Effective material and piezoelectric constants are obtained
by using the micro-mechanics model derived earlier in chapter-2, and are used for computing
the numerical results. Fiber volume fraction vf =0.4 is taken for numerical analysis.
Power law
Potential
s=a/h=10
s=a/h=10
applied
coefficient
λ
on PFRC
Surface ϕ
wХ10
wХ10
-8
(shell)
-8
(Ray &
Sachade [25])
Φ=nodal
wХ10
-8
(shell)
Φ=nodal
wХ10
-8
(Ray &
Sachade [25])
dof
dof
λ=767.528
λ=767.528
-100
-1.55353
-1.623291
-6.47042
-6.5976096
0
-0.00404
-0.0089736
-0.13117
-0.1401312
100
1.54542
1.6053324
6.20804
6.3173376
-100
-26.275
-28.455
-114.508
-114.9024
0
-0.0544
-0.0820716
-1.20644
-1.2778944
100
26.2037
28.2917
112.1
112.3488
Table 5.5: Center deflection of the FGM shell bonded to a curved PFRC actuator on its
top
It could be observed that the response of the smart FG shell match with that of the plate
element and that in [25], in the limit of large radius. Thus the formulation of the shell element
for the functionally graded material is correct and can be used to determine static response of
FG materials.
44
5.4 Variations in response due to various types of loadings
r/a = 1,2,5,10,100,200
a/h = 10,100
Special Case that we have considered
h=3.5x10-3, a=3.5x10-1,r/a=10,a/h=100,Distributed loading in z-direction only.
5.4.1 Variations in response due to pure thermal loading
Serial no
λ
Tfree_surface
0
C
Tbottom
0
Ttop
0
C
C
Mid-surface
Sensor Voltage
Deflection
Volts
(in meter)
1
767.582
0
300
25
0.0031652m
168.78
2
767.582
25
300
30
0.0031701m
173.45
3
767.582
30
250
30
0.0031801m
161.5
4
767.582
10
280
35
0.004529m
201.025
5
767.582
20
300
35
0.004422m
205.57
6
767.582
25
300
30
0.004371m
189.2
Table 5.6: Variations during Thermal loading
On subjecting the structure to different temperature conditions, we find a regular variation in
the mid-surface deflection and induced voltages in the nodes. Studying these responses we can
find the range of temperatures to which the layers are subjected to. This will give us an idea
about the temperature range in which the structure can have maximum performance and
optimized results.
45
5.4.2 Variations in response due to pure mechanical loading
Serial
no
Mechani
-cal
Loading
N/mm
Tfree_surface
0
Tbottom
0
C
Ttop
0
C
C
Angle of
Piezoelctreic
Fiber
Mid-surface
θ
(in meter)
2
Deflection
Sensor
Voltage
Volts
(in radian)
1
-100
0
0
0
0
-2.982x10-6m
0.0193
2
-80
0
0
0
0
-1.91176x10-6m
0.1243
3
-50
0
0
0
0
-1.491x10-6m
0.0096
4
-100
0
0
0
30
-2.979x10-6m
0.0192
5
-100
0
0
0
45
-2.9839x10-6m
0.0191
6
-100
0
0
0
60
-2.2984x10-6m
0.01911
Table5.7: Variations during pure Mechanical Loading
When the structure is subjected to mechanical loading, due to deflections at the mid-surface of
different nodes different voltages are induced.
We note down these deflections and corresponding voltages for further in-depth study. We also
tabulate the midpoint deflection of the central node and the highest induced voltage and try to
find out if a pattern is present.
46
5.4.3 Variations in response due to Thermo- mechanical loading
Thermo-Mechanical Loading
Serial
Mechani-
no
cal
Tfree_surface Tbottom
0
Loading
C
0
Ttop
0
C
C
λ
Mid-surface
(ratio)
Deflection
(in meter)
Sensor
Voltage
Volts
N/mm2
1
-100
0
300
35
767.582
0.00521m
224.53
2
-100
10
300
35
767.582
0.00481m
215.8373
3
-100
30
300
35
767.582
0.004025m
198.362
4
-50
0
300
35
767.582
0.005211m
224.54
5
-50
10
300
35
767.582
0.004817m
215.827
Table5.8: Variations during Thermo-Mechanical Loading
Above we are trying to build a relationship between the different conditions that affect the final
response to thermal and mechanical loading taken simultaneously. The values obtained from
simulation tell us about the direct response of these parameters to the response.
47
5.4.4 Variations in response due to Electro-Thermo- mechanical loading
Sl
no
Mechanical
Loading
N/mm2
Tfree_
Ttop
Tbottom
surface
0
C
0
C
0
Applied
actuator
Voltage
Volts
Angle
of
Piezoelectric
Fiber
C
λ
(ratio)
Midsurface
Sensor
Voltage
deflection
Volts
(in meter)
θ
(in
radian)
1
-100
25
30
300
0
0
767.58
0.0409
202.13562
2
-100
25
30
300
20
0
767.58
0.004
202.132
3
-100
25
30
300
50
0
767.58
0.004
202.08
4
-100
25
30
300
100
0
767.58
0.0041
202
5
-100
20
35
300
20
0
767.58
0.0042
157
6
-100
20
35
300
50
0
767.58
0.0042
147
7
-100
25
30
300
50
30
767.58
0.0042
139
8
-100
25
30
300
20
30
767.58
0.00409
200.14
9
-100
25
30
300
50
60
767.58
0.0041
190
10
-100
25
30
300
20
60
767.58
0.0041
137
Table5.9Variations during Electro-Thermo-Mechanical Loading
Applying all three conditions simultaneously we try to reach a conclusion on how to optimize
the deflections and voltages induced. The results obtained by changing the three loading
parameters give us a better understanding of the weightage of each of the parameters in
determining the overall response.
48
5.5 Scope for Future Work
Smart materials are all set to occupy the center stage. In the consistent efforts to develop them
we can recommend some measures which are being implemented throughout by scientists but
is out of scope for us due to constraints in infrastructure and laboratory equipments.
Still we propose the following
1. Joining the sensors and actuators through a control circuit will result in reducing the
deflection and hence stresses produced in smart structures. This control setup tries to
minimize the deflection in the structure. The underlying principle behind this is that
every deflection causes a simultaneous change in voltage. Sensing this change if we
apply a somewhat opposite voltage through the control circuitry this deflection would
be minimized as a contradictory effect is produced.
2. Free vibration analysis of Smart structure
Gradation can be taken assuming power law type.
P(z)=(P2-P1){(2z+h)/2h}λ+P2
Formulation can be developed for developing the first natural frequency of the FGM
plate putting the conditions λ=0. For a FGM plate the grading parameter λ is varied
from 0 for determining the first two fundamental natural frequencies. The frequencies
can be plotted with λ(exponent of gradation) to reach the optimized thickness.
3. Again thermo-mechanical analysis can be performed along dynamic temperature
conditions and load variations.
51
REFERENCES
1. Koizumi M, 1993, “Concept of FGM”, Ceram. Trans., 34 3–10
2. Teymur M, Chitkara N R, Yohngjo K, Aboudi J, Pindera M J and Arnold S M
1996,”Thermoelastic Theory for the Response of Materials Functionally
Graded in Two Directions”, Smart Materials and structures, vol 33, pp 931–66
3. Feldman E and Aboudi J 1997, “Buckling analysis of functionally graded plates
subjected to uniaxial loading”, Compos. Struct. 38 29–36
4. Mian A M and Spencer A J M 1998, “Exact solutions for functionally graded and
laminated elastic materials”, J. Mech. Phys. Solids 46 2283–95
5. Gasik M.M., 1998, “Micromechanical modeling of functionally graded materials.”
Computational Materials Science 13 (1), 42–55.
6.
Praveen G N and Reddy J N 1998, “Nonlinear transient thermoelastic analysis of
functionally graded ceramic metal plates”, Int. J. Solids Struct. 35 4457–76
7. Wang, B.L., Han, J.C., Du, S.Y., 2000, “Crack problems for Functionally Graded
Materials under transient thermal loading”, Journal of Thermal Stresses 23 (2), 143–
168.
8. Yang, 2000, “Time-dependent stress analysis in functionally graded materials”,
Int. J. Solids Struct. 37 7593–608
9. Yang J and Shen H S 2001, “Dynamic response of initially stressed
functionally graded rectangular thin plates”, Compos. Struct. 54 497–508
10. Sankar B V 2001, “An elasticity solution for functionally graded beams:
Compos. Sci. Technol. 61 689–96
11. Woo J and Meguid S A 2001, “Nonlinear analysis of functionally graded plates and
shallow shells” Int. J. Solids Struct. 38 7409–21
12. Batra R C and Vel S S 2001, “Exact solution for thermoelastic deformations of
functionally graded thick rectangular plates” AIAA J. 40 1421–33
13. Loy C T, Lam K Y and Reddy J N 1999, “Vibration of functionally graded cylindrical
shells”, Int. J. Mech. Sci. 41 309–24
14. Aboudi, J., Pindera, M.J., Arnold, S.M., 1999, “Higher-order
functionally graded materials” Composites. Part B: Engineering 30 (8), 777–
theory
for
832.
15. Bailey T and Hubbard J E 1985, “Distributed piezoelectric polymer active vibration
control of a cantilever beam” J. Guidance, Control Dyn. 8 605–11
16. Miller S E and Hubbard J E 1987, “Observability of a Bernoulli–Euler beam using
PVF2 as a distributed sensor MIT Draper Laboratory Report” (July 1,
1987)
17. K.M. Liew, X.Q. He, T.Y. Ng, S. Sivashanker, “(126) FSDT, Int. J. Numer. Methods
Engrg. 52 (2001) 1253–1271.
18. Huang X-L and Shen H-S 2006, “Vibration and dynamic response of
functionally graded plates with piezoelectric actuators in thermal environments”
J. Sound Vib. 289 25–53
19. Vel S S and Batra R C 2003, “Three-dimensional analysis of transient thermal stresses in
functionally graded plates”, Int. J. Solids Struct. 40 7181–96
20. Reddy J N and Cheng Z Q 2001, “Three-dimensional solutions of smart
functionally graded plates” ASME J. Appl. Mech. 68 234–41
21. He X Q, Ng T Y, Sivashanker S and Liew K M 2001, “Active control of FGM
plates with integrated piezoelectric sensors and actuators” Int. J. Solids Struct.
38 1641–55”
22. Zhong Z and Shang E T 2003, “Three dimensional exact analysis of a simply supported
functionally gradient plate”, Int. J. Solids Struct. 40 5335–52
23. Mallik N and Ray M C 2003, “Effective coefficients of piezoelectric fiber reinforced
composites” AIAA J. 41 704–10
24. Ray M C and Sachade H M 2006, “Exact solutions for the functionally graded plates
integrated with a layer of piezoelectric fiber-reinforced composite” ASME J. Appl. Mech.
73 622–32
25. Ray M C and Sachade H M 2006, “Finite element analysis of smart
functionally graded plates”, Int. J. Solids Struct. 43 5468–84
26. Hwang, W.S. & Park, H.C., 1993, “Finite element modeling of piezoelectric sensors and
actuators”, AIAA Journal, 31(5): 930-937
27. Tzou, H.S. and Ye, R., “Analysis of piezoelectric structures with laminated
piezoelectric triangle shell elements,” AIAA Journal, 34, 110-115, 1996.
28. Croce, L.D.; Venini, P., “Finite elements for functionally graded Reissner
Mindlin plates”. Comput Methods Appl Mech Eng 193 (2004) 705-725
29. Reddy, I.N,”Analysis of FG plates.” Int J Numer Meth Eng47 (2000) 663-684
30. Tiersten HF. Linear Piezoelectric Plate Vibrations. Plenum Press: New York,
1969
31. Naghdabadi, R.; Hosseini Kordkheili, 8.A.”A finite dement formulation based on 3-D
degenerated shell dement for analysis of FG shells. In: Proceedings of the ICCE2003 (2003)
421-428
32. Reddy, J. N. and Liu, C. F. (1985),” A Higher-order Shear Deformation
Theory of Laminated Elastic Shells, International Journal of Engg. Science,
23(3): 319–330.
33. Zienkiewicz, O. C. (1979). The Finite Element Method, 3rd edn, Tata
McGraw-Hill Publishing Company Limited, New Delhi.
34. Jones, R.M., “Mechanics of composite materials”, Second edition,
1999,Taylor and Francis.
35. Vincent Piefort, “FE modelling of piezoelectric Active Structures”, Thesis,
2000-01, Doctor in Applied Sciences, Deptt. Of Mechanical Engg. and
Robotics, U.L.B.
36. Latifa,SK; Sinha,PK,2005, “Improved FEA of Multilayered Doubly Curved
Composite Shells”, J. of Reinforced Plastics & Composites, 2005:24,385.
37. MH Kargarnovin, MM Najafizadeh and NS Viliani, 2007, “Vibration control of a
functionally graded material plate patched with piezoelectric actuators and sensors under a
constant electric charge”, Smart Materials and Structures,
16(2007), 1252-1259.
38. Reddy JN, “Mechanics of Laminated Composite Plates and Shells: Theory and
Analysis”, CRC Press, Second edition,1997
39. Anil Kumar(Roll no-06410306) & Debabrata Chakraborty {IIT Guwahati}. Finite element
modeling of smart functionally graded composite structures. First phase MTP project submitted
as M.Tech thesis for course completion.