VIBRATION ANALYSIS AND DAMPING CHARACTERISTICS OF Hybrid COMPOSITE PLATE USING FINITE ELEMENT ANALYSIS

VIBRATION ANALYSIS AND DAMPING CHARACTERISTICS OF Hybrid COMPOSITE PLATE USING FINITE ELEMENT ANALYSIS
VIBRATION ANALYSIS AND DAMPING
CHARACTERISTICS OF Hybrid
COMPOSITE PLATE USING FINITE
ELEMENT ANALYSIS
A Thesis submitted in partial fulfillment of the Requirements for the degree of
Master of Technology
In
Mechanical Engineering
Specialization: Machine Design and Analysis
By
SHIVAPRASAD BAAD
Roll No. : 212ME1272
Department of Mechanical Engineering
National Institute of Technology Rourkela
Rourkela, Odisha, 769008, India
June 2014
VIBRATION ANALYSIS AND DAMPING
CHARACTERISTICS OF Hybrid
COMPOSITE PLATE USING FINITE
ELEMENT ANALYSIS
A Thesis submitted in partial fulfillment of the Requirements for the degree of
Master of Technology
In
Mechanical Engineering
Specialization: Machine Design and Analysis
By
Shivaprasad Baad
Roll No. : 212ME1272
Under the Guidance of
Prof. Tarapada Roy
Department of Mechanical Engineering
National Institute of Technology Rourkela
Rourkela, Odisha, 769008, India
June 2014
Dedicated to…,
My parents , my sister,
My maternal aunt and uncles
DEPT. OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
ROURKELA – 769008, ODISHA, INDIA
Certificate
This is to certify that the work in the thesis entitled VIBRATION ANALYSIS AND
DAMPING CHARACTERISTICS OF HYBRID COMPOSITE PLATE USING
FINITE ELEMENT ANALYSIS by Shivaprasad Baad is a record of an original
research work carried out by her during 2013 - 2014 under my supervision and guidance in
partial fulfillment of the requirements for the award of the degree of Master of Technology
in Mechanical Engineering (Machine Design and Analysis), National Institute of
Technology, Rourkela. Neither this thesis nor any part of it, to the best of my knowledge,
has been submitted for any degree or diploma elsewhere.
Place: NIT Rourkela
Professor
Date: 02 June 2014
Dr. Tarapada Roy
DEPT. OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
ROURKELA – 769008, ODISHA, INDIA
Declaration
I certify that
a) The work contained in the thesis is original and has been done by myself under the
general supervision of my supervisor.
b) The work has not been submitted to any other Institute for any degree or diploma.
c) I have followed the guidelines provided by the Institute in writing the thesis.
d) Whenever I have used materials (data, theoretical analysis, and text) from other
sources, I have given due credit to them by citing them in the text of the thesis and
giving their details in the references.
e) Whenever I have quoted written materials from other sources, I have put them
under quotation marks and given due credit to the sources by citing them and giving
required details in the references.
Shivaprasad Baad
2nd June 2014
ACKNOWLEDGEMENTS
It is my immense pleasure to avail this opportunity to express my gratitude, regards and
heartfelt respect to Prof. Tarapada Roy, Department of Mechanical Engineering, NIT
Rourkela for his endless and valuable guidance prior to, during and beyond the tenure of
the project work. His priceless advices have always lighted up my path whenever I have
struck a dead end in my work. It has been a rewarding experience working under his
supervision as he has always delivered the correct proportion of appreciation and criticism
to help me excel in my field of research.
I also express my sincere gratitude to Prof. (Dr.) K. P. Maity, Head of the Department of
Mechanical Engineering for valuable departmental facilities.
I would like to make a special mention of the selfless support and guidance I received
from my senior Ashirbad Swain, Ph.D. scholar, Department of Mechanical Engineering,
NIT Rourkela during my project work.
Last but not the least; I would like to express my love, respect and gratitude to my
parents, younger sister and my maternal aunt and uncles, who have always supported me in
every decision I have made, guided me in every turn of my life, believed in me and my
potential and without whom I would have never been able to achieve whatsoever I could
have till date.
SHIVAPRASAD BAAD
[email protected]
i
ABSTRACT
Hybrid composite is a composite which consists of nanoparticles to enhance the strength
as compared to conventional composites. A model has been proposed to determine the
elastic properties of hybrid composite. The hybrid composite consists of conventional fiber
and nanocomposite as matrix. The first step here is to determine the properties of
nanocomposite which is done by using Mori – Tanaka method. The CNTs are considered
as cylindrical inclusions in polymer matrix in Mori – Tanaka method. Assuming perfect
bonding between carbon fibers and nanocomposite matrix, the effective properties of the
hybrid composite has been evaluated using mechanics of materials approach.
An 8 noded shell element has been used for the finite element analysis having 5 degrees
of freedom each node  u, v, w, x , y  . A 10 10 finite element mesh has been used to model
the shell element. The shell coordinates which are in Cartesian form are converted into
parametric form using two parameters 1 ,  2  . These parameters are again mapped into
isoparametric form  ,   . A 16 layered laminate with stacking sequence [0 -45 45 90]2S
has been used for vibration analysis of simply supported shell element. The dynamic
equations of shell are derived using Hamilton’s principle. As the damping characters of the
dynamic system are not available, for further investigation damping ratio of first mode and
last active mode are assumed. Using Rayleigh damping the damping ratios of intermediate
modes can be calculated. The time decay of the system from maximum amplitude to 5% of
the maximum amplitude has been used as a parameter to study various shell structures by
varying the volume fraction of CNTs in nanocomposite and by varying carbon fiber
volume fraction.
ii
CONTENTS
ACKNOWLEDGEMENTS.............................................................................................. I
ABSTRACT .................................................................................................................. II
CONTENTS ................................................................................................................. III
NOMENCLATURE....................................................................................................... V
LIST OF FIGURES .................................................................................................... VII
LIST OF TABLES ..................................................................................................... VIII
1 INTRODUCTION ...................................................................................................... 1
2 LITERATURE REVIEW AND MOTIVATION ............................................................ 5
2.1
Literature Review ...................................................................................................... 5
2.1.1 Material property – Hybrid composite ..................................................................... 5
2.1.2 Material Property – Hybrid composite ..................................................................... 6
2.1.3 Vibration analysis of plate........................................................................................ 6
2.1.4 Damping in composites ............................................................................................ 6
2.2
Motivation .................................................................................................................. 6
2.3
Objective ..................................................................................................................... 7
3 MATERIAL MODELING ........................................................................................... 8
3.1
CNT bases nanocomposite modeling ....................................................................... 8
3.2
Hybrid Composite modeling ................................................................................... 11
4 FINITE ELEMENT FORMULATION ....................................................................... 14
iii
4.1
Geometry of mid-surface of shell ........................................................................... 14
4.1.1 Isoparametric mapping ........................................................................................... 16
4.1.2 Transformation matrix used for isoparametric mapping ........................................ 17
4.2
Strain displacement relations ................................................................................. 18
4.2.1 In-plane/bending strain-displacement matrix ......................................................... 18
4.2.2 Transverse strain displacement matrix ................................................................... 20
4.2.3 ABBD matrix ......................................................................................................... 20
4.3
Equation of motion .................................................................................................. 22
4.3.1 Static finite equations ............................................................................................. 22
4.3.2 Dynamic finite equations ....................................................................................... 24
4.4
Calculation of non-dimensional frequencies ......................................................... 26
4.5
State space method for impulse response .............................................................. 26
5 RESULTS AND DISCUSSION ............................................................................... 28
6 CONCLUSION ........................................................................................................ 35
REFERENCES: .......................................................................................................... 37
iv
NOMENCLATURE
CNT
Carbon nanotubes
SWCNT
Single walled carbon nanotubes
DWCNT
Double walled carbon nanotubes
MWCNT
Multi walled carbon nanotubes
nc, m, hc
Nanocomposite, matrix, hybrid composite
Cp11, Cp12, Cp22, Cp23, Cp55
Elastic properties of constituent phase
nr , kr , pr , lr , mr
Hills elastic constant
, 
Euler angles for CNT orientation
 ,
Stress and strain
A
Strain concentration tensor
I
Identity matrix
S
Eshelby tensor
K
Bulk modulus
G
Shear modulus
E
Young’s modulus
vCNT
Volume fraction of CNT
vm
Volume fraction of matrix
vf
Volume fraction of fiber
1 ,  2
Curvilinear coordinates
s 1 , s 2
Isoparametric curves
r1 , r2
Tangent to isoparametric curves
A1 , A2
Lame’s parameters
R1 , R2
Normal curvatures of shell mid-surface
R12
Twist curvatures of shell mid-surface
a
Length of shell along x-axis
N
Shape function
u0i , v0i , w0i
Displacements of ith node in 1 ,  2 and z
directions
 ,
v
Isoparametric coordinates
J
Jacobian matrix

Shear strain
k
Curvature of the mid-surface
Bbe
In-plane strain displacement matrix
Bse
Transverse strain displacement matrix
N, M
Resultant force and moment per unit length
A
In-plane stiffness matrix
B
Coupling stiffness matrix
D
Bending stiffness matrix
z
Thickness of lamina
h
Thickness of laminate
Q
Reduced stiffness matrix
K
Shear correction factor

Total potential energy
U
Strain energy
T
Kinetic energy
W
Work done by external force
L
Lagrangian
M uue
Elemental mass matrix
K uue
Elemental stiffness matrix
de
Displacement vector
de
Acceleration vector
t
Time

Density

Natural frequency
 **
Non-dimensional natural frequency
i
Modal coordinates for ith degree of freedom
i
Modal damping ratio
1..... 2n
State vector
vi
LIST OF FIGURES
Figure 3-1 Ref: heshmati and yas (2012) ............................................................................. 8
Figure 3-2 Hexagonal RVE ................................................................................................ 11
Figure 4-1 Geometry of shell structure in Cartesian coordinates ..................................... 14
Figure 4-2 Isoparametric mapping .................................................................................... 16
Figure 4-3 Stacking sequence in laminate ......................................................................... 21
Figure 5-1 Variation of C11 w.r.t variation of carbon fiber and CNT volume fraction ..... 30
Figure 5-2 Variation of C12 w.r.t variation of carbon fiber and CNT volume fraction ..... 31
Figure 5-3 Variation of C23 w.r.t variation of carbon fiber and CNT volume fraction ..... 31
Figure 5-4 Variation of C22 w.r.t variation of carbon fiber and CNT volume fraction ..... 32
Figure 5-5 Variation of C55 w.r.t variation of carbon fiber and CNT volume fraction ..... 32
Figure 5-6 impulse response of cfrp composite for thick plate .......................................... 33
Figure 5-7 Impulse response of cfrp composite for thin plate ........................................... 33
Figure 5-8 Decay time for thick plate by varying the cnt volume fractions for different
volume fractions of carbon fiber ...................................................................... 34
Figure 5-9 Decay time for thin plate by varying the cnt volume fractions for different
volume fractions of carbon fiber ...................................................................... 34
vii
LIST OF TABLES
Table 5-1 Non-dimensional frequency [Ref:23] .................................................................. 29
Table 5-2 Non-dimensional frequency for the present formulation ..................................... 29
Table 5-3 Material properties of various constituents in hybrid composite ........................ 30
viii
1
INTRODUCTION
As long as there is development in the field of aerospace, automobile, healthcare,
electronics and consumer industry the demand for new materials will never seize. The
demand for new materials has led to continuous research and development of new
techniques to satisfy the needs.
1.1. Nanocomposites
Nanocomposites consist of reinforcements of nanoscale spread evenly or randomly in
polymer matrix. The commonly used polymeric matrix materials are:

Epoxy

Polystyrene

Nylon

Polyimide

PEEK – Polyether ether ketone
Carbon nanotubes – SWCNTs and

Nanotitanium oxide
MWCNTs

Nanosilica
The commonly used nano fillers are:


Nanoaluminium oxide
The reinforcements can be particles or fibres of size of few nanometers. The
nanocomposite has a wide range of materials from 3-D metal matrix composites, 2-D
laminated composites and nano-wires of small dimension representing variations of nano
reinforcements. Using nanoscale reinforcements was introduced by Usuki et al [1] who
1
built a nanocomposite using polyimide and organophilic clay. The nanocomposite formed
had twice the tensile modulus as compared to neat polyimide with just 2% volume fraction
of nano reinforcement. Nanocomposites have gained a wide popularity among researchers.
Researchers have discovered that the properties of the nanocomposite are better when
compared to the individual components of the composite. Properties such as increased
tensile strength, increased thermal conductivity are observed.
1.2.
Hybrid Composites
The important properties that are desired from any composite are strength, stiffness,
ductility, toughness, damping, energy absorption, thermal stability and low weight. With
conventional materials it is not possible to get all the desired properties, but with composite
materials we can tailor the properties of material as per our needs. By using reinforcements
of nanoscale in polymer composites there has been tremendous increase in mechanical
properties as compared to neat polymer matrix. Hybrid composites are new type of three
phase composites which have reinforcements of nanoscale in addition to conventional
reinforcing fiber in matrix or by growing reinforcements of nanoscale on the surface of
fiber. R.C.L. Dutra et al [2] defined hybrid composites as composites consisting of different
fibers. The main purpose of using hybrid composites is it increases the matrix dominated
properties.
1.3.
Classification of Plates [3]
Plates can be classified into two types:

Thin plates
A thin plate can further be classified as:

Plate with small deflection

Plates with large deflection
2

Thick plates
1.3.1.1.
Plates with small deflection
If the deflection of a plate when subjected to loading is less than or nearly equal to thickness, then
the plate is said to have small deflection. The necessary assumptions for developing theory for
plates having small deflection are:

The middle plane does not deform on loading.

Point on the plate initially normal to middle plane remain normal to middle plane even after
loading.

The stresses in the thickness direction can be neglected.
1.3.1.2.
Thin plates with large deflection
If the deflections of the plate are large when subjected to lateral loading as compared to thickness,
then the plate is said to have large deflection. When plate is loaded mid plane strains are developed.
In plates with small deflection it is normally neglected, as a result the stresses are also neglected.
But if deflections are large as compared to thickness, the strains developed are large. So stresses
cannot be neglected. In this case we obtain nonlinear equations and analysis becomes complicated.
1.3.2. Thick plate
The above approximations for thin plates are not applicable. Thick plate theory must be used. The
thick plate theory involves 3-D theory of elasticity and calculation of stresses is quite complicated.
1.3.3. Difference between plate and shell

The major difference between plates and shells can be observed under the action of loading.
When a plate member is subjected to lateral load, equilibrium is possible by the action of
bending and twisting moments. In shells, when it is subjected to lateral loading, equilibrium
is possible by membrane stresses which act parallel to tangential plane at a point on middle
surface and are distributed uniformly over the thickness of shell.

1.4
Plates are plane member and shells are curved structural members.
Damping in composites
Damping is a very important parameter for vibration control, noise reduction, stability of system,
fatigue and impact resistance [4]. The damping in fibre reinforced composites is different from that
of metals.
The various forms of energy dissipation are [5]:
 Damping behaviour of matrix material.
The major contribution to damping is from matrix, the damping of fiber must also be
included for calculation of damping.

Damping behaviour of interphase
Interphase is the region between matrix and fiber. The type of interphase plays an
important role in damping. The interphase can be weak or strong.
3

Damping due to damage
a) Frictional damping due to delamination.
b) Damping due to energy dissipation of broken fibers or cracks in matrix.

Viscoplastic damping
At higher amplitudes of stresses, there is non-linear damping due to presence of high
stress and strain.
1.5.
Impulse response of linear time invariant system (LTIS)
Impulse force is a force which acts on the system for very short amount of time. Knowing the
impulse response of LTIS we can obtain by superposition the response of the same system to any
input provided the input conditions are zero in all cases. Unit impulse input has very short intervals
of time but very large amplitude and hence the effect of the behaviour of the system under study is
not negligible. Ex: Ball hitting the cricket bat, the ball is acted upon by very large force for a short
duration of time.
4
2
LITERATURE REVIEW AND MOTIVATION
2.1
Literature Review
Jian Ping Lu [6] Elastic properties of SWCNTs, MWCNTs and nanoropes are investigated using
force constant model.
Material property – Hybrid composite
2.1.1
Raifee et al [7] estimated mechanical properties of epoxy based nanocomposite with SWCNT,
MWCNT and graphene platelets were compared for weight fractions of 0.1%. The material
properties measured were Young’s modulus, fracture toughness, ultimate tensile strength. The
tensile strength of graphene based nanocomposites showed better properties as compared to CNT
based nanocomposites. F.H. Gojny et al [8] observed mechanical properties resulted in an increase
in Young’s modulus, strength at weight fractions of 0.1%. There was good agreement between
experimental observed data and results from modified Halpin-Tsai relation. Florian H [9] proposed
choosing appropriate type of CNTs (SWCNTs or DWCNTs or MWCNTs) has been a problem ever
since they are being used in composites. They evaluated the properties of nanocomposite for
different nano fillers. The nanocomposites exhibited greater strength, stiffness and fracture
toughness. They found that DWCNT based nanocomposite exhibited greater fracture toughness.
Seidel et al [10] estimated effective elastic properties of composites consisting of aligned SWCNTs
or MWCNTs using Mori-Tanaka method. The effects of an interphase layer between CNTs and the
polymer is also investigated using a multi-layer composite cylinders approach. Liu and Chen [11]
estimated effective elastic properties of the nanocomposite are evaluated using continuum
modelling and finite element method. The extended rule of mixture is used to determine the
properties of the continuum model.
5
Material Property – Hybrid composite
2.1.2
Dutra et al [2], made a hybrid composite consisting of carbon fiber and Polypropylene fiber and
mercapto-modified polypropylene blend fiber (PPEVASH). They found that hybrid composites
showed better impact resistance than CFRP composite. Mathur et al [12] CNTs were grown on
unidirectional carbon fiber. These fibers were used as reinforcements in matrix material. They
found that the mechanical properties improved with increase in amount of CNT deposition as
compared to neat CFRP composite. Garcia et al [13] CNTs were grown on alumina fiber cloth.
These fibers were used as reinforcements in matrix material. The growth of CNTs led to an increase
in inter-laminar shear properties of the order of 69% as compared to alumina cloth composite.
Kundalwal and Ray [14], they evaluated the elastic properties of FFRC (Fuzzy fiber reinforced
composite) using mechanics of materials approach and Mori-Tanaka method considering with and
without the interphase between CNT and polymer.
2.1.3
Vibration analysis of plate
Roy and Chakraborty [15] formulated layered shell finite element model for coupled
electromechanical analysis of curved smart composite structure.
2.1.4
Damping in composites
Gibson et al [4] used vibration used modal vibration response measurements to characterize the
mechanical properties of laminated structures. They showed that vibration in either first mode or
multiple modes can be used to determine the elastic properties and damping ratios. Modal testing
was done by impulse excitation methods. Kyriazoglou and Guild [17] found damping ratio using
experimental methods and by FEM. The FEM uses Rayleigh damping method and particularly mass
proportional damping. R. Verdejoan et al. [18] found that even a small volume fraction of CNT can
increase the sound absorption capabilities.
2.2
Motivation
Hybrid composites are new type of three phase composites which increase the matrix dominated
properties. The hybrid composite that is to be modeled here consists of nanocomposite matrix and
continuous long carbon fibers. The nanocomposite is made up of randomly distributed CNTs and
polymer matrix. The nanocomposite is modeled using Mori-Tanaka method. The hybrid composite
can be modeled using mechanics of materials approach. As the mobility of system goes on
increasing, modeling damping for such systems becomes complicated. Rayleigh damping model
has been used to model such multi degree of freedom systems. Further investigation has been
carried out by assuming suitable damping ratios for first mode and last significant mode where
6
mass is proportional to damping. Impulse response of the system has been carried out and a
comparative study has been made to know effect of damping in systems by varying the volume
fractions of carbon fiber and CNTs.
2.3

Objective
Material modelling and material characterization
1. Nanocomposite has been modelled using Mori – Tanaka method.
2. Hybrid composite consisting of carbon fiber and nanocomposite matrix has been
modelled using mechanics of materials approach.

8 noded shell element formulation
1. Mindlin theory of plates and shells has been used to model shell.

Modelling damping and Impulse response
1. Rayleigh damping has been used to model the damping of MDOF system.
2. Impulse response of the system has been carried out using the state space model.
7
3
MATERIAL MODELING
The material modeling is divided in two phases:

CNT based nanocomposite modeling.

Hybrid composite modeling.
The nanocomposite consists of randomly distributed straight MWCNT as reinforcements and epoxy
as matrix. As the CNTs are randomly distributed the nanocomposite can be modeled as isotropic
material. The property of nanocomposite is evaluated using the Mori – Tanaka method. Assuming
perfect bonding between fiber and nanocomposite the hybrid composite can be modeled similar to
conventional composite using mechanics of materials approach.
3.1
CNT bases nanocomposite modeling
FIGURE 3-1 REF: HESHMATI AND YAS [18]
Fig. 3.1 shows a RVE of randomly distributed CNT in epoxy matrix.
The Mori – Tanaka method was used to estimate the elastic properties of the randomly
distributed MWCNT in matrix. The procedure to determine the isotropic properties of randomly
oriented MWCNTs dispersed in epoxy matrix is as follows:
8

The Hill’s elastic constants for the MWCNT can be obtained by equating the stress – strain matrix
of MWCNT with the Hill’s elastic matrix.
C11CNT
 CNT
C21
C31CNT
CCNT   
 0
 0

 0
 nr
l
 r
l
CCNT    0r

0

 0
C12CNT
CNT
C22
C32CNT
0
0
0
lr
kr  mr
kr  mr
0
0
0
C13CNT
CNT
C23
C33CNT
0
0
0
CNT
C44
0
0
lr
kr  mr
kr  mr
0
0
0
0
0
0
mr
0
0
0
0
0
0
0
0
0
C55CNT
0
0
0
0
0
pr
0








CNT
C66

0
0
0
0
0
0
0 
0

0
0

pr 
(1)
(2)
Eqn (1) and eqn. (2) are the stress – strain matrix of MWCNT and Hill’s elastic matrix.
The orientation of the CNT can be specified by two Euler angles  and  . The base vectors ei and
ei of the global  0  x1 x2 x3  and local co-ordinates can be related by the relation,
ei  gij e j
(3)
cos 
Where gij   sin 
 0

 cos  sin 
cos  cos 
sin 
sin  sin  
 sin  cos  

cos 
As the CNTs are randomly distributed in matrix it can be characterized by two Euler angles  and
 . The orientation distribution of CNTs in the composite is characterized by the probability density
function p  ,   satisfying the normalization condition and is given by the equation.
2  /2
  p  ,   sin  (d )(d  )  1
0

For completely randomly oriented CNTs,
p  ,   

(4)
0
1
2
(5)
From Mori – Tanaka method one can relate stress  CNT  ,   and strain  CNT  ,   of the CNT to
the stress in the matrix  m by,
 CNT  ,    CCNT A  ,    m  CCNT A  ,   Cm1   m
And
9
(6)
 CNT  ,    A  ,    m   A  ,   Cm1   m
(7)
Where strain concentration tensor is given by,
A   I  S (Cm )1 (CCNT  Cm ) 
1
(8)
Where S corresponds to Eshelby Tensor and is given by Li and Dunn [19] for cylindrical inclusion.
The
 CNT
average
stress
and
strain
for
the
randomly
oriented
CNTs
 2  /2

    p  ,   CCNT A  ,   Cm1  sin  d d    m
0 0

 2  /2
 CNT   
0
given
by,
(9)

 p  ,   A  ,   sin  d d   
is
(10)
m
0
Using rule of mixture one can get the stresses and strains in the nanocomposite.
As the CNTs are randomly distributed in the matrix, the nanocomposite behaves like an isotropic
material. The bulk modulus, shear modulus, Young’s modulus of the nanocomposite [20] are given
by,
K nc  K m 
vCNT ( CNT  3K m CNT )
3  vm  vCNT  CNT 
Gnc  Gm 
vCNT (CNT  2Gm CNT )
2  vm  vCNT CNT 
Enc 
(11)
9 K nc Gnc
3K nc  Gnc
Where,
 CNT 
3( K m  Gm )  kCNT  lCNT
3(Gm  kCNT )
2 Gm  3K m  Gm   Gm  3K m  7Gm   
4Gm
1  4Gm  2kCNT  lCNT

 

5  3(Gm  kCNT )
Gm  pCNT
Gm  3K m  Gm   mCNT  3K m  7Gm  
CNT  
1
3
 CNT   nCNT  2lCNT 
 2kCNT  lCNT  3K m  2Gm  lCNT  
Gm  kCNT


8Gm mCNT  3K m  4Gm 
2  kCNT  lCNT  2Gm  lCNT  
8G p
1 2

 nCNT  lCNT   m CNT 

5 3
Gm  pCNT 3K m  mCNT  Gm   Gm  7mCNT  Gm 
3  Gm  kCNT 

CNT  
(12)
vCNT  vm  1
(13)
vCNT and vm are volume fractions of CNT and matrix material, nc represents nanocomposite.
10
3.2
Hybrid Composite modeling
F IGURE 3-2
H EXAGONAL RVE
Fig. 3.2 shows a hexagonal RVE of hybrid composite consisting of carbon fibers distributed in
nanocomposite matrix.

Using the above calculated nanocomposite properties, the properties of the transversely isotropic
hybrid composite can be evaluated by the formulation of Kundalwal and Ray for fuzzy fiber [14].

Assuming perfect bonding between carbon fiber and nanocompostie, the normal strains in hybrid
composite, carbon fiber and nanocomposite are equal along the fiber direction and the transverse
stresses in the same phase are equal along the direction transverse to the fiber from isofield
conditions.

Using rules of mixture one can express the longitudinal and transverse stresses and strains in terms
of volume fractions of nanocomposite and carbon fiber.

Using isofield conditions and rule of mixture we can write,
v f  vNC  1
(14)
Where v f and vNC are volume fractions of carbon fiber and nanocomposite.
 1f   1NC   1HC 
 f   NC   HC 
 2   2   2 
 3f   3NC   3HC 
 f    NC    HC 
 23   23   23 
 13f   13NC   13HC 
 f   NC   HC 
 12   12   12 
(15)
HC represents hybrid composite.
 1f 
 1NC   1HC 
 f
 NC   HC 
 2 
 2   2 
  f 
  NC    HC 
v f  3f   vNC  3NC    3HC 
 23 
  23    23 
 13f 
 13NC   13HC 
 f
 NC   HC 
 12 
 12   12 
11
(16)
There the stress and strain in the hybrid composite lamina is given by,
   C    C  
(17)
   V    V  
(18)
HC
f
NC
1
HC
2
f
NC
1
2
But from iso-field conditions,
C3  f   C4  NC   0
(19)
By solving above equations we get,
   C   
HC
HC
HC
(20)
C HC  is the effective elastic matrix of the proposed Carbon fiber is reinforced polymer and is
given by,
C HC   C1 V3   C2 V4 
1
1
(21)
The various matrices appearing in the above equations are given below.
C11f C12f

0
 0
 0
0
C1   v f 
0
 0
 0
0

0
 0
vNC C11NC
 NC
 C12
 C12NC
 C2   
 0
 0

 0
 1
C f
 12
C13f
C3    0

 0

 0
12
0
C22f
C23f
0
0
0
C12f
0
0
0
0
0
vNC C12NC
C11NC
C12NC
0
0
0
0
C23f
C22f
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0
0

0 
vNC C12NC
C12NC
C11NC
0
0
0
0
0
0
C44p
0
0
0
0
0
0
C55p
0
(22)
0
0
0
C44NC
0
0
0 
0 
0 

0 
0 

C55p 
0
0
0
0
C44NC
0
0 

0 
0 

0 
0 

C44NC 
(23)
(24)
 1
C NC
 12
C12NC
C

 4  0

 0

 0
0
C11NC
C12NC
0
0
0
0 0
0 v
f

0 0
V1   0 0

0 0

0 0
0
0
vf
0
0
0
1 0
0 v
NC

0 0
V2   0 0

0 0

0 0
0
C12NC
C11NC
0
0
0
0
0
0
vf
0
0
0
0
vNC
0
0
0
0
0
0
C44NC
0
0
0
0
0
0
vf
0
0
0
0
vNC
0
0
0
0
0
0
C44NC
0
0
0 
0

0
0

v f 
0
0
0
0
vNC
0
0 
0 
0 

0 
0 

C44NC 
(25)
(26)
0 
0 
0 

0 
0 

vNC 
(27)
V3   V1   V2 C4  C3 
(28)
V4   V2   V1 C3  C4 
(29)
1
1
13
4
FINITE ELEMENT FORMULATION
4.1
Geometry of mid-surface of shell
F IGURE 4-1
G EOMETRY
OF SHELL STRUCTURE IN
C ARTESIAN
COORDINATES
The shell geometry used in the present formulation has been developed using an orthogonal
curvilinear coordinate system with the mid-plane of the shell assumed to be the reference surface as
shown in Fig.4.1. The shell mid-surface in the Cartesian rectangular coordinate system has been
first mapped into a parametric domain through the suitable exact parameterization. Two
independent coordinates (1 ,  2 ) in the parametric space have been considered as the mid-surface
curvilinear coordinates of the shell. The normal direction coordinate to the middle surface of the
shell has been represented by z. The reference surface or the shell mid-surface can be described in
the global Cartesian coordinates in terms of the position vector as,



r (1 , 2 )  X (1 , 2 ) i  Y (1 , 2 ) j  Z (1 , 2 ) k


(30)

Where, i , j and k are unit vectors along the X, Y and Z axis, respectively.
The tangent to the isoparametric curves s 1 and s 2 respectively are
r1 
r
;
1
r2 
r
 2
(31)
The vector joining two points on the middle surface (1 ,  2 ) and (1  d1 ,  2  d 2 ) is given as
14
ds  r1d1  r2 d 2
(32)
Scalar product of ds,
ds.ds  (r1.r1 )d12  (r2 .r2 )d 22
(33)
Lame’s parameters can be defined as,
A1  r1.r1
(34)
A2  r2 .r2
Eqn. (4) can be written as
ds 2  A12 d12  A22 d 22
(35)
Since the 1 and  2 are independent coordinates
ds 2  ds21  ds2 2
Where,
(36)
ds 1  A1d1
ds 2  A2 d 2
The unit tangent vectors to the isoparametric curve s 1 and s 2 can be expressed respectively as,

e1 
r1
;
A1

e2 
r2
A2
(37)
The unit normal vector to the tangent plane of any point on the reference surface can be expressed
as

en 
r1  r2
r1  r2
(38)
The normal curvatures and twist curvatures of the mid-surface of shell can be expressed as:

e .r
1
  n 211
R1
A1

e .r
1
  n 222
R2
A2
(39)

e .r
1
  n 12
R12
A1 A2
Where R1 , R2 are the normal curvatures of the midsurface of the shell and R12 is the twist curvature
of the mid-surface of the shell.
15
4.1.1 Isoparametric mapping
F IGURE 4-2
I SOPARAMETRIC
MAPPING
Fig.4.2 shows Cartesian coordinates are converted into curvilinear coordinates and it is mapped into
isoparametric form.
The shell midsurface in the rectangular cartesian coordinate system has been mapped into the
parametric space (1 ,  2 ) and the midsurface in the parametric space has been divided into required
number of quadrilateral elements or sub-domains. The reference coordinates ( , ) map the
quadrilateral element in the curvilinear coordinates (1 ,  2 ) into the reference coordinates that is a
square as shown in Fig. 4.2. Any point within an element in the parametric space has been
approximated by the isoparametric mapping.
Hence the curvilinear coordinates (1 ,  2 ) of any point within an element may be expressed as
nd
1   N i1i
i 1
(40)
nd
 2   N i 2i
i 1
(1 ,  2 ) is the
coordinate of
midsurface at i th node in curvilinear
coordinate system.
u0i , v0i and w0i are the deflection of midsurface at i th node in 1 ,  2 and z directions respectively.
1i is the rotation of normal at i th node about  2 axis and  2i is the rotation of normal at i th node
about 1 axis.
The displacement components on the shell midsurface at any point within an element may be
expressed as
16
u0i 
u0 
v 
v 
0i
 0  nd
 
 w    N i  wi 
  i 1  
 1
 1i 
 2 
 2i 
(41)
Where, nd is the number of nodes in an element, N i is the shape function corresponding to the i th
node and shape functions of 8 noded serendipity element are given below
1
N1  (1   )(1   )(1     )
4
1
N 2  (1   2 )(1   )
2
1
N 3  (1   )(1   )(1     )
4
1
N 4  (1   )(1   2 )
2
1
N 5  (1   )(1   )(1     )
4
1
N 6  (1   2 )(1   )
2
1
N 7  (1   )(1   )(1     )
4
1
N8  (1   )(1   2 )
2
(42)
4.1.2 Transformation matrix used for isoparametric mapping
Since the integration is to be done in natural coordinates ( , ) , the element is mapped into the
isoparametric space ( , ) using the isoparametric shape functions. The transformation matrix used
is given below.
The relation between the shape function derivatives in parametric space (1 ,  2 )
isoparametric space ( , ) are given as
 N i 
 N i 
  


 1
 1   

   J  

 N i 
 N i 
  2 
  
17
(43)
and in
The jacobian matrix can be expressed as
 1
 
 J    
 1
 

4.2
 2 
 

 2 
 
(44)
J  a J*
(45)
d   J d d
(46)
Strain displacement relations
Neglecting normal strain component in the thickness direction, the five strain components of a
doubly curved shell may be express as
 xx   xx0 
 k xx 
   0 
k 
 yy   yy 
 yy 
 xy    xy0   z  k xy 
   0
 
 yz   yz 
0
0
 xz   xz 
 0 
 
(47)
Where  xx0 ,  yy0 and  xy0 is the in-plane strains of the midsurface in the cartesian coordinate system
and k xx , k yy and k xy are the bending strains (curvatures) of the midsurface in the cartesian
coordinates system. After incorporating the effect of transverse stain in Koiter’s shell theory,
inplane and transverse strain-displacement relations may be expressed as described in the following
subsections.
4.2.1 In-plane/bending strain-displacement matrix
The strain components on the midsurface of shell element are
    xx0  yy0  xy0
kxx
k yy
k xy 
T
(48)
By using isoparametric 8-noded shell element, the displacement component on the shell midsurface
at any point within an element can be expressed as
u0 
v 
 0 
e
 w    N  d 
 
 1
 2 
18
(49)
The mid-surface strains and curvatures from Koiter’s shell theory are:
 xx0 
1 u
v A1 w


A1 1 A1 A2  2 R1
(50)
 yy0 
1 v
u A2 w


A2  2 A1 A2 1 R2
(51)
 xy0 
1 v
1 u
u A1
v A2 2w




A1 1 A2  2 A1 A2  2 A1 A2 1 R12
(52)
k xx 
 A1
1 1
1  1 v
1 u
u A1
v A2 
 2






A1 1 A1 A2  2 2 R12  A1 1 A2  2 A1 A2  2 A1 A2 1 
(53)
k yy 
 A2
1  2
1  1 v
1 u
u A1
v A2 
 1






A2  2 A1 A2 1 2 R12  A1 1 A2  2 A1 A2  2 A1 A2 1 
(54)
1 A1
 2 A2
 1  2 1 1

 A   A   A A   A A  

1
1
2
2
1 2
2
1 2
1


k xy 
 1  1 1  1 v

1 u
u A1
v A2 



   

 2  R1 R2  A1 1 A2  2 A1 A2  2 A1 A2 1  
(55)
By using 8-noded isoparametric shape functions from Eqn. (13), the strain components at any point
on
the
shell
midsurface
Ni A1

1 Ni

A1 1
A1 A2  2

Ni A2

1 Ni

A1 A2 1
A2  2

 1 Ni
Ni A1
Ni A2
1 Ni



A2  2 A1 A2  2
A1 1 A1 A2 1
8 
     1 1  1 N N A  1 1  1 N N A 
i 
i
i 
i
1
2
i 1  




 2 R12  A2  2 A1 A2  2  2 R12  A1 1 A1 A2 1 

 1 1  1 Ni  Ni A1   1 1  1 Ni  Ni A2 
 2 R  A  A A  
 2 R  A 
12  2
2 A1 A2  2 
12  1 1
1 2
1






A

A

N
N

N
N

1
1
i
i
i 
i
1
2
 C0 
 C0  A   A A  
 A  A A  
 2 2 1 2 2
 1 1 1 2 1

can
be
expresse
Ni
0
0
R1
Ni
0
0
R2
2 Ni
0
0
R12
0
1 Ni
A1 1
0
Ni A2
A1 A2 1
0
Ni A1
1 Ni

A2  2 A1 A2  2
Ni
A1 A2  2
1 Ni
A2  2
Ni
1 Ni

A1 1 A1 A2
(56)
    Bbe  d e 
1 1
1 
 Bbe  is the element in plane stain- displacement matrix and Co   

2  R1 R2 
19
A1
(57)






 u0i 
 v0i 
  
  wi 
  
  1i 
  2i 



A2 

1

4.2.2 Transverse strain displacement matrix
According to the FSDT, the transverse shear strain vector of a doubly curved shell element may be
expressed as
1

2 

A2
 yz  
 
 xz     1
 1 A1
w
u
v 

 
 2 R12 R2 

w u
v 
 
1 R1 R12 
(58)
And hence the transverse shear strain at any point on the shell mid surface can be expressed as
 Ni
 R
nd
 yz 
12
   

 xz  k 1   Ni
 R
 1
N
 i
R2
1 Ni
A2  2
N
 i
R12
1 Ni
A1 1
u0i 
 
0 Ni  v0i
 
  wi 

Ni
0  1i 
 
 2i 
 yz 
e
e
    Bs  d 
 xz 
(59)
(60)
 Bse  is the element transverse stain- displacement matrix.
4.2.3 ABBD matrix
The resultant force per unit width is
N   N xx
N xy 
N yy
T
(61)
The resultant moment per unit width is
M    M xx
M yy
M xy 
T
(62)
In plane strains of mid-surface is
   
0
0
xx
 yy0  xy0 
T
(63)
Curvatures of mid-surface is
k  kxx
k yy
kxy 
T
(64)
The strain at any point on an element
 
( xy )
  0   z k
(65)
The resultant force per unit width can be expressed as
h /2
 N     
( xy )
 h /2
dz
 N    A 0    B k
20
(66)
The resultant moment per unit width
M  
h /2
  
( xy )
zdz
(67)
 h /2
M    B   o   D  k
Eqs. (37) and (38) can be expressed in matrix form as
  N    A


M   B 
 B    0 
 D  k 
(68)
 A  B  
   
  
 B   D 
(69)
F IGURE 4-3
Qb k
Q11 
Q22 
Q12 
Q11 Q12
 Q12 Q22
 0
0
S TACKING
SEQUENCE IN LAMINATE
0 
0 
Q66 
(70)
E1
(71)
 12  21

E1 E2
(75)
E2
(72)
 12  
2
1
(76)
(73)
 21  
1
2
(77)
1  12 21
1  12 21
 12 E2
 E
 21 1
1  12 21 1  12 21
Q66  G12
(74)
The transformation matrix
 cos 2 
1
 RT k   sin 2 
sin  cos 

sin 2 
cos 2 
 sin  cos 
2sin  cos  

2sin  cos  
cos 2   sin 2  
(78)
The transformed elasticity matrix for bending
Qb    RT 1 Qb   RT T
k
k
k
 k
21
(79)
 KG23
 0
0 
KG13 
QS   
(80)
Where K denotes shear correction factor
The transformation matrix
cos 
 sin 
 sin  
cos  
 Rs   
(81)
The transformed elasticity matrix for transverse shear
Q s    Rs T QS  Rs 
 
(82)
n
 D s    Q s  ( Z k  Z k 1 )
  k 1  
(83)
The extensional stiffness matrix
n
 A   Qb   Z k  Z k 1 
k 1
(84)
k
The extensional-bending coupling stiffness matrix
 B   Qb   Zk2  Zk21 
2
1
n
k 1
(85)
k
The bending stiffness matrix
 D   Qb   Z k3  Z k31 
3
1
n
k 1
4.3
(86)
k
Equation of motion
The elastic finite element formulation has been derived for static and dynamic analysis.
4.3.1 Static finite equations
The total potential energy of the element is given by
  U W
(87)
U and W are the strain energy of the entire structure and work done by the external force.
Strain energy of the shell structure is given by,
U
1
T
    dV

2V
U
1
T
  C   dV

2V
T
T
1
U  d e    Bue  C   Bue  dV d e 
2
V
U
1 e T
d   Kuue  d e 
2
Work done by external force is given by,
22
(88)
W
 d   f  x, y  dA
e T
e
s
A1
W  d e 
T
  N   f  x, y  dA
T
e
s
(89)
A1
W  d
 F 
e T
e
Substituting the values of U and W in total potential energy Eqn.58, we have
 Kuue  d e   F e 
(90)
Eqn. (61) is the elastic equation for one element.
Where,
e
 Kuu
 is the element structural stiffness matrix is given by
 Kuue     Bue  C   Bue  dV
(91)
e
e
 Kuu
   Kbb
   K sse 
(92)
  Bbe   0 
 

 B   
  0  Bse  
 

(93)
 D   0 
C    0b D 
    s 
(94)
T
V
e
u
 Db  is the in-plane/bending constitutive matrix and  Ds  is the transverse shear constitutive matrix.
e
 Kbb
 , is the element in plane/bending stiffness matrix.
 Kbbe     Bbe 
T
 Db   Bbe  dV
V
 A
 K     B  

 B 
e T
b
e
bb
 B   Be  d 
 D   b 
(95)
 K sse  , is the element transverse shear stiffness matrix
 K sse     Bse 
T
 Ds   Bse  dV
V
 K sse     Bse   D s   Bse  d 
T
(96)

F  , is the element external mechanical force vector
e
F     N   f  x, y dA
T
e
e
s
A
Hence all the element stiffness matrices can be expressed in its final form as
23
(97)
1 1
 K e      K he  J d d
(98)
1 1
After assembling the elemental stiffness matrices, the global set of elastic equations is given by
 Kuu d  F
(99)
4.3.2 Dynamic finite equations
The dynamic finite element formulation has been derived by using Hamilton’s principle as
t2
  L   W dt  0
(100)
t1
where, the Lagrangian, L  T  U
T is the kinetic energy of the system, U is the elastic strain energy, W is the external work done by
the force on the structure.
t2
  T  U    W dt  0
(101)
t1
where t1 and t2 defines the time interval.
Kinetic energy T is given by,
1
T   d
2
V
  d dV
T
(102)
 is the mass density.
Individual parts of the Hamilton equation can be written as follows:
Part 1: The kinetic energy of the system
t2
t2
t1
t1 V
1
d
2
  Tdt      
t2
t2
t1
t1 V
t2
t2
t1
t1
t2
t2
t1
t1
  d dVdt
T
  Tdt      d d  dVdt
T


V

e
T
e
  Tdt =    d      N   N dV  d  dt
T
  Tdt    d 
e T
 
 M uue  d e dt
(103)
Part 2: The total internal energy of the system
t2
t2
t2
t2
t1
t1
t1
t1


m
e
e
e
 Udt   V dt  Udt    d   Kuu  d  dt
T
Part 3: The work done by the external forces
24
(104)
t2
t2

e
 We dt      d 
t1
t1
 A1
t2
t2

T
e
 We dt    d 
t1
t1
e
s


  N   f  x, y  dAdt
T
T

t2

 f  x, y  dAdt
e
s

A1
 W dt   d  F 
e T
e
(105)
e
t1
Substituting eqns. (74), (75), (76) in eq. (71) gives
t2
  d 
e T
t1
  M  d    K  d   F dt  0
e
uu
e
e
uu
e
e
(106)
Since  d e  can be any arbitrary values,  d e   0 and therefore the eq. (43) is zero only if
 
 M uue  d e   Kuue  d e   F e 
(107)
Eqn. (78) is dynamic finite element equations of one element.
Where the structural mass matrix is given by,
 M uue      N T   N dV
(108)
V
The displacement at any point in the laminate cane be expressed as
 u  u0  z x 
  

 v   v0  z y 
 w  w 
  

 u  nd  N i
 

 v    0
 w i 1 0
 

(109)
0
0
 zN i
Ni
0
0
Ni
0
0
u0i 
 
0  v0i 
 
 zN i   wi 
0   xi 
 
 yi 
(110)
u 
 
e
 v    N d 
 w
 
(111)
 M uue      N T   N dV
(112)
V
 M uue   
h
2
   N   N  dzd 
 h
2
25
T
(113)
Mass matrix in its final form can be expressed as
1 1 h /2
 M uue    
   N   N  dz J d d
T
(114)
1 1  h /2
4.4
Calculation of non-dimensional frequencies
In order to verify the above formulation for spherical shell having mesh 10 10 , the various nondimensional frequencies of the present formulation are compared with existing ones.
 **  
a2
h

E2
(115)
Where  ** is the non-dimensional frequency,  is the natural frequency, a is the length of shell
structure along x-axis, h is the thickness of the laminate,  is the density of composite, E2 is the
transverse modulus of the lamina having 00 orientation.
4.5
State space method for impulse response
The equation of motion of system considering damping in modal form is given as:
 i    2 ii  i   i2   i    fi 
(116)
Where  i ,  i , i and fi are the modal coordinates, modal damping ratio, natural frequency and force
vector in modal form for i=1, 2, 3…., dof.
For damped systems which are modeled using Rayleigh proportional damping, it is difficult to
determine the Rayleigh constants. Calculating Rayleigh damping coefficients for large degree of
freedom system has been provided with detail in Chowdhury and Dasgupta [21]. Assuming 3% of
the total modes to be active modes or which participate in mass proportional damping. As there is
no data regarding the damping ratios for the above formulated hybrid composite we proceed the
investigation by assuming the damping ratio of the first mode to be 0.01 and damping ratio of the
last active mode to be 0.05. The constants  and  can be calculated using the damping ratios and
natural frequencies of the first mode and the last active mode. The damping ratio for the i th
intermediate mode  i can be calculated once the value of  and  are known. Thus
i 

 i
2i
26
(117)
The state space and output equations in modal form for n number of modes can be written as,
 1    0 
n n
  
 
2
     n n n
 2n 
  1  
 T
 0 dof 1


  dof  dof  Fdof 1 (118)
   T
1
  2 nn n n      dof  dof  M dof
 dof 

  2n 
 I n  n
 1 
 1 
 
 
T
    I 2 n  2n     02 n  2 n  dof  dof  Fdof 1
 
 
 2n 
 2n 
(119)
 
Where  0 n  n is null matrix,  I n  n is the identity matrix, n2
matrices, 1.....  2n are the state vectors.
27
n n
and  2 nn n n are diagonal
5
RESULTS AND DISCUSSION
Summary of the above chapters
Hybrid composite is a composite which consists of nanoparticles to enhance the strength as
compared to conventional composites. A model has been proposed to determine the elastic
properties of hybrid composite. The hybrid composite consists of conventional fiber and
nanocomposite as matrix. The first step here is to determine the properties of nanocomposite which
is done by using Mori – Tanaka method. The CNTs are considered as cylindrical inclusions in
polymer matrix in Mori – Tanaka method. Assuming perfect bonding between carbon fibers and
nanocomposite matrix, the effective properties of the hybrid composite has been evaluated using
mechanics of materials approach.
An 8 noded shell element has been used for the finite element analysis having 5 degrees of freedom
each node  u, v, w, x , y  . A 10 10 finite element mesh has been used to model the shell element.
The shell coordinates which are in Cartesian form are converted into parametric form using two
parameters 1 ,  2  . These parameters are again mapped into isoparametric form  ,   . A 16
layered laminate with stacking sequence [0 -45 45 90]2S has been used for vibration analysis of
simply supported shell element. The dynamic equations of shell are derived using Hamilton’s
principle. As the damping characters of the dynamic system are not available, for further
investigation damping ratio of first mode and last active mode are assumed. Using Rayleigh
damping the damping ratios of intermediate modes can be calculated. The time decay of the system
from maximum amplitude to 5% of the maximum amplitude has been used as a parameter to study
various shell structures by varying the volume fraction of CNTs in nanocomposite and by varying
carbon fiber volume fraction.
28
Validation of formulation
Free vibration analysis is done to validate the above formulation. A 2, 3 and 4 layered cross ply
laminate has been used to carry out free vibration analysis by varying the a/h ratio and R/a ratio of
the spherical shell.
Reference [23]
Panel
a/h
R/a
1
2
5
10
20
50
100
500
0/90
10
14.481
10.749
9.2302
8.9841
8.9212
8.9035
8.9009
8.9001
0/90
100
125.93
67.369
28.826
16.706
11.841
10.063
9.7825
9.6873
0/90/0
10
16.115
13.382
12.372
12.215
12.176
12.165
12.163
12.162
0/90/0
100
125.99
68.075
30.993
20.347
16.627
15.424
15.244
15.183
0/90/90/0
10
16.172
13.447
12.437
12.280
12.240
12.229
12.228
12.226
0/90/90/0
100
126.33
68.294
31.079
20.38
16.638
15.426
15.245
15.184
T ABLE 5-1
N ON - DIMENSIONAL
FREQUENCY
[R EF :23]
Present Formulation
Panel
a/h
R/a
1
2
10
20
50
100
500
0/90
10
0/90
100 136.0635 73.90925 31.89584 18.20593 12.12838 9.353029 8.72118
8.358714
0/90/0
10
10.75471
0/90/0
100 142.7383 76.47821 32.7317
0/90/90/0
10
15.51217 11.1521
5
15.5545
8.906713 8.373873 8.166252 8.061078 8.027878 7.990367
12.22585 11.01655 10.82998 10.78238 10.76802 10.7647
19.15716 13.78834 11.85253 11.54031 11.43834
14.8861
11.99612 10.95903 10.79977 10.75936 10.74652 10.74316 10.73102
0/90/90/0 100 134.838
72.67357 31.29011 18.50931 13.52047 11.74193 11.45988 11.35508
T ABLE 5-2
29
N ON - DIMENSIONAL
FREQUENCY FOR THE PRESENT FORMULATION
Material Properties
Properties of various constituents of Hybrid composite
Constituent
C11(GPa)
C12(GPa)
C22(GPa)
C23(GPa)
C55(GPa)
Density(kg/m3)
Carbon
236.4
10.6
24.8
10.7
25
1800[24]
1180
146
411
133
189
1740[25]
13.4615
5.7692
13.4615
5.7692
3.8462
1150
fiber[22]
(25,25), 5
walled
CNT[6]
Epoxy[18]
T ABLE 5-3
M ATERIAL
PROPERTIES OF VARIOUS CONSTITUENTS IN HYBRID COMPOSITE
C11
F IGURE 5-1
V ARIATION
OF
C 11
W . R . T VARIATION OF CARBO N FIBER AND
CNT
VOLUME FRACTION
From Fig. 5.1, it can be seen that as the carbon fiber volume fraction increases the longitudinal
elastic properties increase. At lower volume fractions of carbon fiber (10%) it can be observed that
for CFRP composite the elastic modulus is around 35GPa, but with the increase in volume fraction
of CNT from 1% to 5% the elastic modulus has increased from 38GPa to 47.3GPa. With the
increase in carbon fiber volume fraction, the volume fraction of nanocomposite goes on decreasing;
as a result the elastic properties almost converge at higher volume fractions of carbon fiber.
30
C12 and C23
F IGURE 5-2
V ARIATION
OF
C 12
W . R . T VARIATION OF CARBO N FIBER AND
CNT
VOLUME FRACTION
F IGURE 5-3
V ARIATION
OF
C 23
W . R . T VARIATION OF CARBO N FIBER AND
CNT
VOLUME FRACTION
From Fig. 5.2 and Fig. 5.3, similarly as C11, the elastic properties along 1-2 direction also increase
with the increase with in volume fraction of carbon fiber. At lower volume fractions (10%) it can be
observed that CFRP has elastic modulus of 6Gpa, but with increase in CNT volume fraction from
1% to 5%, the elastic properties have increased from 6.8Gpa to 10Gpa. With the increase in carbon
fiber volume fraction, it can be observed that composites having lower CNT volume fractions show
steep increase in elastic properties than compared to composites having higher CNT volume
fractions.
31
C22
F IGURE 5-4
V ARIATION
OF
C 22
W . R . T VARIATION OF CARBO N FIBER AND
CNT
VOLUME FRACTION
From Fig. 5.4, it can be seen that the transverse elastic properties have improved with the addition
of CNT in matrix material for lower volume fractions of carbon fiber. This increase in elastic
properties can be attributed to the randomly distributed CNTs. With the increase in carbon fiber
volume fraction the composites having lower volume fractions of CNT have shown an increase in
transverse elastic modulus, but for composites having higher volume fractions of CNT with the
increase in carbon fiber volume fraction there is decrease in transverse elastic modulus.
C55
F IGURE 5-5
V ARIATION
OF
C 55
W . R . T VARIATION OF CARBO N FIBER AND
CNT
VOLUME FRACTION
From Fig. 5.5, it can be observed that the in-plane shear properties have increased with increase in
volume fractions of CNT and carbon fiber. Composites normally fail due to shear. So in order avoid
failure it is advantageous to use hybrid composite in place of conventional CFRP composites.
32
Impulse response
F IGURE 5-6
F IGURE 5-7
IMPULSE RESPONSE OF CFRP COMPOSITE FOR THICK PLATE
I MPULSE
RESPONSE OF CFRP COMPOSITE FOR THIN PLATE
Fig. 5.6 and Fig. 5.7 indicate the response of thick plate and thin plate in modal coordinates for the
first mode of vibration.
33
F IGURE 5-8
D ECAY
TIME FOR THICK PLATE BY VARYING THE CNT VOLUME FRACTIONS FOR DIFFERENT VOLUME
FRACTIONS OF CARBON FIBER
F IGURE 5-9
D ECAY
TIME FOR THIN PLATE BY VARYIN G THE CNT VOLUME FRACTIONS FOR DIFFERENT VOLUME
FRACTIONS OF CARBON FIBER
Fig. 5.8 and Fig. 5.9 show the decay time for thick and thin plates. At lower volume fractions
carbon fiber, the decay time of the system goes on decreasing with increase in CNT volume
fraction. As the volume fraction of carbon fiber increases the decay time also decreases.
34
6
CONCLUSION
This chapter presents important observations on the material properties, Rayleigh damping in
composite materials, impulse response, and decay time.
Conclusions
The hybrid composite has been modeled using Mori-Tanaka method and mechanics of materials
method. It is found that
1. The longitudinal properties C11 of the hybrid composite increase with the increase in volume
fraction of CNT at lower volume fractions of carbon fiber. As the volume fraction of carbon
fiber goes on increasing the longitudinal modulus tends to converge because the volume fraction
of CNT goes on decreasing.
2. There is tremendous increase in elastic properties C12 and C23 of the hybrid composite with the
increase in volume fraction of CNT at lower volume fractions of carbon fiber. As the volume
fractions of carbon fiber goes on increasing there is slow increase in composites having higher
volume fractions of CNT as compared to composites having lower volume fractions of CNTs.
3. The transverse modulus C22 of the hybrid increases with the increase in CNT volume fraction
but as the volume fraction of carbon fiber increases the composites having lower CNT volume
fractions show increase in transverse modulus and composites having higher CNT volume
fraction show gradual decrease in transverse modulus.
4. The in-plane shear modulus C55 increases with increase in CNT and carbon fiber volume
fraction.
5. The amplitude goes on decreasing with increase in damping ratio.
6. As the volume fraction of CNT increases the decay time goes on decreasing.
7. The decay time of thick plate is less than the decay time of thin plate.
35
Future Scope

Estimate temperature dependent and hygrothermal properties.

Buckling analysis of hybrid composite laminated shell structure.

Active vibration control of the laminated shell structure.

Delamination analysis.

Nonlinear analysis of laminated shell structure.
36
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38
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