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: NIT

Rourkela (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 stiffness-

to-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 3

1.6 Smart Structures 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 6

Chapter 2 Background and literature review 8-12

2.1 General 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 12

Chapter 3 Determination of effective properties of PFRC layer 13-19

3.1 Assumptions involved in the analysis 13

iii

3.2 Strength of material approach 14

3.3Effective coefficients of PFRC layer 17

Chapter 4 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 23

4.2.4 Displacement field and strains for shell element 24

4.2.5 Isoparametric Finite Element approximation of displacement

field and electric field for plate element 24

4.2.6 Jacobian matrix (Transformation matrix) 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 39

Chapter 5 Results and discussion 40-50

5 Validation of Finite Element Code of doubly curved shell element 40

5.1 Static mechanical shell element 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 48

5.5 Scope for future work 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 Components of Strain vector in the global

or, ε1 , ε2 , ε12 , ε13 , ε 23 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 Components of Strain vector in the local

or, ε’1 , ε’2 , ε’12 , ε’13 , ε’23 Coordinate system

αik 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 Rotation of yz and xz planes due to bending

de,d Mechanical degrees of freedom vector at the

element and global level, respectively

cElastic

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

R1, R2 , R3 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 ,FMechanical 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

EElectric field

vector

T 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 fibre-

reinforced laminate bonded together to obtain desired 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 Graded 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 their piezoelectric stress/strain constants are of small

magnitude. Because, tailoring of these properties may improve the damping

characteristics of the smart structures. Currently, Piezoelectric Fiber Reinforced

Composites (PFRC) is being effectively used in underwater and medical

applications. These composites show improved mechanical 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

thermo-electro-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

Substrate6

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 load. They concluded that with

optimal nonuniform distribution 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 thei 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 for10

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 supported 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.

Chap t er 1 gives an introduction to FGM and smart FGM structures, their advantages over

conventional laminated composites.

Chap t er 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.

Chap t er 3 includes the determination of effective properties of a PFRC using micro-

mechanical analysis is presented.

Chap t er 4 includes the formulations for static and free vibration analysis for the

functionally graded doubly curved shells.

11

Chap t er 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 FIBER-

REINFORCED 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)

σym = σy

p

15

Similarly we have σzm = σz

p 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 Dx

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 films20

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, only

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 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 (α1,α2)ϵ A:R2→R3 through a suitable exact

parameterization. Two independent coordinates (α1,α2) 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

spaceξ,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. 1t1 and t2 define the time

interval.

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 equation37

(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)/2hλ+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

coefficient

λ

Load parameter

P=(a4q/t4Ebottom)

Center deflection

w(in mm) of present shell

element

Center deflectionw(in mm)

Corce, Venini[28]

0 -6 -1.23576 -1.2

-10 -0.205960 -2.05

-13 -2.67744 -2.7

2 -6 -1.86573 -1.75

-10 3.10955 -3.15

-13 -4.04243 -4.09

106(∞) -6 -2.6653 -2.2

-10 4.44274 -4.45

-13 -5.77551 -5.86

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

coefficient

λ

Load parameter

P=(a4q/t4Ebottom)

Center

deflection

w(in mm) of

present shell

element

Center

deflection

w(in mm)

Corce, Venini

[28]

0 0 1.45186 1.3

-5 0.422042 0.43

-10 -0.607756 -

2 0 1.02927 0.998

-5 -0.525495 -0.5

-10 -2.08026 -

5 0 1.32473 1.3

-5 -0.373779 -0.39

-10 -2.07229 -

106(∞) 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/

Matrix

C11

GPa

C12

GPa

C13

GPa

C33

GPa

C44

GPa

e31=e32

GPa

e33

GPa

ϵ33

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

coefficient

λ

Potential

applied

on PFRC

Surface ϕ

s=a/h=10 s=a/h=10

wХ10-8

(shell)

Φ=nodal

dof

wХ10-8

(Ray &

Sachade [25])

wХ10-8

(shell)

Φ=nodal

dof

wХ10-8

(Ray &

Sachade [25])

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

λ=767.528

-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

0C

Tbottom

0C

Ttop

0C

Mid-surface

Deflection

(in meter)

Sensor Voltage

Volts

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/mm2

Tfree_surface

0C

Tbottom

0C

Ttop

0C

Angle of Piezoelctreic

Fiber

θ(in radian)

Mid-surface

Deflection

(in meter)

Sensor Voltage

Volts

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

noMechani-

cal

Loading

N/mm2

Tfree_surface

0C

Tbottom

0C

Ttop

0C

λ

(ratio)

Mid-surface

Deflection

(in meter)

Sensor

Voltage

Volts

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

Mechani-cal

Loading

N/mm2

Tfree_

surface

0C

Ttop

0C

Tbottom

0C

Applied actuator Voltage

Volts

Angle of

Piezo-electric Fiber

θ(in

radian)

λ

(ratio)

Mid-surface

deflection

(in meter)

Sensor Voltage

Volts

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)/2hλ+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

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