VIBRATION, BUCKLING AND PARAMETRIC RESONANCE CHARACTERISTICS OF DELAMINATED COMPOSITE PLATES

VIBRATION, BUCKLING AND PARAMETRIC RESONANCE CHARACTERISTICS OF DELAMINATED COMPOSITE PLATES
Ph. D.
Thesis
VIBRATION, BUCKLING AND PARAMETRIC RESONANCE
CHARACTERISTICS OF DELAMINATED COMPOSITE PLATES
SUBJECTED TO IN-PLANE PERIODIC LOADING
VIBRATION, BUCKLING AND PARAMETRIC RESONANCE
CHARACTERISTICS OF DELAMINATED COMPOSITE PLATES
SUBJECTED TO IN-PLANE PERIODIC LOADING
Jayaram Mohanty
Jayaram Mohanty
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-769008, ODISHA, INDIA
DECEMBER 2012
NIT ROURKELA
2012
Vibration, Buckling and Parametric Resonance
Characteristics of Delaminated Composite Plates
Subjected to In-plane Periodic Loading
A thesis submitted to
National Institute of Technology, Rourkela
for the award of degree of
Doctor of Philosophy
in
Engineering
by
Jayaram Mohanty
Under the supervision of
Prof Shishir Kumar Sahu
National Institute of Technology
Rourkela-769008
&
Prof Pravat Kumar Parhi
College of Engineering & Technology
BPUT, Bhubaneswar-751003
Department of Civil Engineering
National Institute of Technology
Rourkela-769008, Odisha, India
December 2012
Dedicated
to my
Beloved Parents &
Spouse
Certificate
This is to certify that the thesis entitled “Vibration, buckling and
Parametric Resonance Characteristics of Delaminated Composite Plates
Subjected to In-plane Periodic Loading” being submitted to the National Institute
of Technology, Rourkela (India) by Jayaram Mohanty for the award of the degree of
Doctor of Philosophy in Engineering is a record of bonafide research work carried
out by him under my supervision and guidance. Jayaram Mohanty has worked for
more than three years and the thesis fulfills the requirements of the regulations of the
degree. The results embodied in this thesis have not been submitted to any other
university or institute for the award of any degree or diploma.
Date: December, 21, 2012
Prof. Shishir Kumar Sahu
Supervisor
Department of Civil Engineering
National Institute of Technology
Rourkela-769008
Odisha, India
iii
Certificate
This is to certify that the thesis entitled “Vibration, buckling and Parametric
Resonance Characteristics of Delaminated Composite Plates Subjected to Inplane Periodic Loading” being submitted to the National Institute of Technology,
Rourkela (India) by Jayaram Mohanty for the award of the degree of Doctor of
Philosophy in Engineering is a record of bonafide research work carried out by him
under my supervision and guidance. Jayaram Mohanty has worked for more than
three years and the thesis fulfills the requirements of the regulations of the degree.
The results embodied in this thesis have not been submitted to any other university or
institute for the award of any degree or diploma.
Date: December, 21, 2012
Prof. Pravat Kumar Parhi
Co-Supervisor
Department of Civil Engineering
College of Engineering & Technology
BPUT, Bhubaneswar-751003
Odisha, India
iv
Acknowledgement
I avail this unique opportunity to express my deep sense of gratitude and heartfelt
reverence to Dr. Shishir Kumar Sahu, Professor, Department of Civil Engineering,
National Institute of Technology, Rourkela for his valuable guidance, keen interest,
constructive criticism, painstaking effort and meticulous supervision during the entire
course of the investigation and preparation of the manuscript.
I acknowledge gratefully my indebtedness to Dr. Pravat Kumar Parhi, Professor
and Head, Department of Civil Engineering, College of Engineering and Technology,
Bhubaneswar for his constant support, generous advice, constructive suggestions, and
unstinted help in preparation of my manuscript.
I am highly obliged to Prof. S.K. Sarangi, Director, National Institute of
Technology, Rourkela for providing me in time support and facilities during my Ph.D.
programme.
I express my sincere gratitude to Prof. B.C. Ray, Head of the Metallurgy and
Materials Engineering, for extending the laboratory facility of the Department and his
suggestions at various stages of the work.
I express my gratefulness to Professor K.C. Patra, Prof. M. Panda and Prof. J.K.
Pani, Ex-Head of the Civil Engineering Department and Prof. N. Roy, Head of the
Department for their cooperation, valuable suggestions and inspiration during my Ph.D.
programme.
It is my fathomless pleasure to express my infinite gratitude to Prof. S. K. Das,
Department of Civil Engineering and Prof. K. K. Mohapatra, Department of
Electronics & Communication Engineering, National Institute of Technology,
Rourkela for their valuable advice, constant inspiration and unstinted help during the
course of the investigation.
I express my sincere gratitude to Professor Dr. O.N. Mohanty, Ex ViceChancellor, BPUT, Prof. Dr. J.K. Satpathy, Vice-Chancellor, BPUT; Prof. S.C. Mishra,
Ex Dean, Prof. S.K. Mishra, Ex Principal and Prof. P.K. Hota, Principal of CET and Prof.
P.K. Sahoo, Prof. A.C. Ray, Ex- Head of Civil Engineering Department for their valuable
advice and inspiration during the Ph.D. programme.
v
I would also like to thank Prof. M.R.Barik, Prof. K.C. Biswal , Prof. C.R.
Patra, Prof. S.P. Singh, Prof. A.V. Asha and Prof. U.K. Mishra of Civil Engineering
Department for their whole hearted suggestions at various stages of the work.
I acknowledge with thanks the help rendered to me by the staff members of
the Structural Engineering Laboratory and Metallurgy & Material Engineering
department Laboratory and other staffs of Civil Engineering department for their
continuous encouragement during the progress of my work.
The author is also thankful to Department of Science & Technology, Govt. of
India
for
their
financial
support
through
the
R
&
D
Project
No.
SR/S3/MERC/0009/2008 for the material and facility during the testing of
composites.
I record my deep sense of appreciation of adjustment of my son Sipun and
daughter Sikha and other family members for their love and affection despite the
difficulties they faced during my study due to insufficient attention.
I fall short of words to express my feeling to my wife Dr. Swarnalata Das for her
selfless sacrifice, inspiration, unstinting moral support and wholehearted cooperation in
achieving my goal.
At the nib but not neap, I bow my head before my late father, late mother, late
father-in-law, mother-in-law and Lord Jagannath who had shown a beam of spiritual light
in the darkness.
Date: December 21, 2012
(Jayaram Mohanty)
vi
ABSTRACT
The composite materials have significant applications over metallic materials in
different fields of structural engineering. Structural elements subjected to in-plane
periodic forces may lead to parametric or dynamic instability under certain
combination of load parameters which caused resonant transverse vibrations. The
spectrum of values of parameters causing unstable motion is referred to as the regions
of dynamic instability or parametric resonance. Delamination is a very serious
concern to composite applications and it arises as a consequence of impact loading,
stress concentration near geometrical or material discontinuity or manufacturing
defects. The study of dynamic stability itself requires a special investigation of basic
problems of vibration and static stability. So the present investigation deals with the
study of vibration, static and dynamic stability of delaminated composite plates.
However, some studies on static analysis of delaminated composites involving the
effects of different parameters on interlaminar shear strength (ILSS) are studied for
completeness. The influence of various parameters on the free vibration and static
stability (buckling) behavior of delaminated composite plates are investigated both
experimentally and numerically. The parametric instability behaviour of delaminated
composite plates is examined using finite element method.
For numerical analysis, a finite element model is developed with an eight noded
two dimensional quadratic isoparametric element having five degrees of freedom per
node based on first order shear deformation theory (FSDT). Element elastic stiffness,
geometric stiffness and mass matrices are derived using the principle of Stationery
potential energy. A simple two dimensional single delamination model proposed
earlier for vibration is extended in the present analysis for the vibration, static and
dynamic stability analysis of delaminated composite panels under in-plane uniaxial
periodic forces by multiple delamination modelling. A general formulation for
parametric resonance characteristics of delaminated composite plates under in-plane
periodic loading is presented.
Experimental investigations are conducted for ILSS, vibration and buckling
analysis of delaminated composite plates. Materials used for fabrication of laminates
are woven roving glass fiber as reinforcement, epoxy as resin, hardener, polyvinyl
vii
alcohol as a releasing agent and Teflon foil for introduction of artificial delamination.
Fiber and matrix are used in 50:50 proportion by weight. Material constants are
determined from the tensile test as per relevant ASTM standards. The FFT analyzer
B&K–3560 is used for modal testing of composite plates. To obtain the buckling result,
INSTRON 1195 machine of 100 KN capacities is used.
There is a very good agreement between numerical result and experimental
result in case of natural frequency and critical buckling loads of woven fiber
composite plates with delaminations. Both the results revealed that the fundamental
natural frequency and critical buckling load of delaminated composite plates decrease
with the increase in area of delaminations and fiber orientations. The instability
studies showed a good agreement with the results available in the open literature. The
onset of instability occurs at lower exciting frequency with the increase in
delamination size and static load factor. It is also observed that with the increase in
number of layers, aspect ratio and degree of orthotropy of delaminated plates, the
dynamic instability occurs at higher excitation frequency.
Thus the instability behavior of delaminated plates is influenced by the
geometry, material, ply lay-up, ply orientation and size of delamination. This can be
used to the advantage of tailoring during the design of delaminated composite
structure. This study can be used as a tool for structural health monitoring for
identification of delamination, its location and extent of damage in composites and
helps in assessment of structural integrity of composite structures.
Key words:
Delamination, dynamic stability, woven fiber, composite plate, inplane periodic loading, critical buckling load, natural frequency,
excitation frequency.
viii
CONTENTS
CHAPTER
1
2
PAGE
Abstract
vii
Contents
ix
List of Tables
xii
List of Figures
xiii
Nomenclature
xvi
List of publications
xviii
INTRODUCTION
1-4
1.1
Introduction
1
1.2
Importance of the stability of delaminated composite plates
2
1.3
Objectives of present study
3
REVIEW OF LITERATURE
5-17
2.1
Introduction
5
2.2
Static analysis of delaminated composites
5
2.3
Vibration of delaminated composite plates
8
2.4
Static stability of delaminated composite plates
10
2.5
Dynamic stability of delaminated composite plates
13
2.6
Critical discussion
15
2.6.1
Vibration of delaminated composite plates
15
2.6.2
Static stability of delaminated composite plates
16
2.6.3
Dynamic stability of delaminated composite plates
16
2.7
3
TITLE
Scope of the present study
17
THEORY AND FORMULATION
18-43
3.1
The basic problem
18
3.2
Proposed analysis
18
3.2.1
19
3.3
3.4
Assumptions of the analysis
Governing equations
19
3.3.1
Governing differential equations
19
3.3.2
Energy expressions
21
3.3.3
Formulation of static and dynamic problems
23
Finite element formulation
25
3.4.1
Displacement field and shape functions
26
3.4.2
Stress-strain relations
29
ix
CHAPTER
Delamination modeling
33
3.6
Strain displacement relations
38
3.7
Derivation of element matrices
39
3.7.1
Elastic stiffness matrix
39
3.7.2
Geometric stiffness matrix
40
3.7.3
Consistent mass matrix
42
Computer program
43
EXPERIMENTAL PROGRAMME
44-58
4.1
Introduction
44
4.2
Experimental programme for static analysis
44
4.2.1
Materials
44
4.2.2
Fabrication of specimens
45
4.2.3
Bending test
45
4.2.4
Scanning electron microscope (SEM) test
47
4.3
Determination of material constants
47
4.4
Experimental programme for vibration study
50
4.4.1
Fabrication of specimens
50
4.4.2
Equipments for vibration test
53
4.4.3
Procedure for free vibration test
55
4.5
5
PAGE
3.5
3.8
4
TITLE
Experimental programme for static stability (buckling) study
55
4.5.1
Fabrication of specimens
55
4.5.2
Experimental set-up and procedure for buckling test
56
RESULTS AND DISCUSSION
59-95
5.1
Introduction
59
5.2
Static analysis
60
5.3
Vibration analysis
65
5.3.1
Comparison with previous study
65
5.3.2
Numerical and experimental results
67
5.3.2.1 Effects of delamination area
67
5.3.2.2 Effects of boundary condition.
70
5.3.2.3 Effects of fiber orientations
71
5.3.2.4 Effects of number of layers of laminate
71
5.3.2.5 Effects of aspect ratio
72
5.3.2.6 Effects of multiple delaminations
73
x
CHAPTER
TITLE
5.3.3
5.4
6
75
Buckling/static stability analysis
77
5.4.1
Comparison with previous study
77
5.4.2
Experimental and numerical results
78
5.4.2.1 Effects of delamination area
79
5.4.2.2 Effects of fiber orientations
80
5.4.2.3 Effects of number of layers of laminate
81
5.4.2.4 Effects of aspect ratio
82
5.4.2.5 Effects of multiple delaminations
83
5.4.2.6 Effects of boundary conditions
84
Experimental determination of critical buckling load
from load v/s end shortening displacement graph
85
5.4.3
5.5
Pulse report
PAGE
Dynamic stability analysis
87
5.5.1
Comparison with previous study
88
5.5.2
Numerical results for dynamic stability
89
5.5.2.1 Effects of delamination size
89
5.5.2.2 Effects of number of layers
90
5.5.2.3 Effects of degree of orthotropy
91
5.5.2.4 Effects of aspect ratio
93
5.5.2.5 Effects of static loads
94
CONCLUSION
96-101
6.1
Introduction
96
6.2
Static analysis
96
6.3
Vibration analysis
97
6.4
Buckling analysis
98
6.5
Dynamic stability analysis
99
6.6
Further scope of research
100
REFERENCES
102-116
APPENDICES
117-123
xi
LIST OF TABLES
Table
Particular
Page
4.1
Size of the specimen for tensile test
47
4.2
Dimensions of composite plates with and without delamination
52
4.3
Dimensions of composite plates with and without delamination
56
5.1
Mean, SD & CV in ILSS (MPa) value of glass/epoxy composite
laminates at different delamination lengths and loading speeds
61
5.2
5.3
5.4
Percentage reduction in ILSS (MPa) value of 1 cm, 2.5 cm &
3.5 cm delaminated specimen
Comparison of fundamental frequency (Hz) for graphite/epoxy
composite plates with different boundary conditions
Comparison of fundamental frequency of cantilever composite
beams (127mm  12.7mm  1.016mm) with different mid–plane
delaminations
62
65
66
5.5
Material properties of composite plates used for vibration
66
5.6
Natural frequencies (Hz) of experimental and FEM results for
25% delaminated plate at different boundary conditions
70
Variation of frequencies of delaminated clamped and cantilever
composite plates with different % of delamination area
74
5.7
5.8
Comparison of buckling load (Newton) for laminated C-F-C-F
composite plates
77
5.9
Comparison of buckling load in (Newton/mm) for delaminated
C-F-C-F composite plates
78
5.10
Material properties of the plate for buckling analysis
79
5.11
Variation of buckling load (KN) of delaminated CFCF
composite plates
79
5.12
Comparison of numerical and experimental results of critical
buckling load (KN) of delaminated composite plates
83
5.13
Non-dimensional parameters of composite plates
88
5.14
Comparison of buckling loads for different mid plane
delamination length of the cantilever rectangular plates.
88
xii
LIST OF FIGURE
Figure
Particular
Page
3.1
Delaminated composite plate under in-plane periodic load
18
3.2a
Force resultants
20
3.2b
Moment resultants
20
3.3
Layer details of plate
25
3.4
Isoparametric co-ordinate
27
3.5
On-axis and off-axis configurations of lamina
30
3.6
Laminate geometry with multiple delaminations
33
3.7
Three arbitrary delaminations leading to four sub-laminates
34
4.1a
Three point bend test set-up and fixture
46
4.1b
Schematic diagram of three point bend test
46
4.2a
Diamond cutter for cutting specimens
48
4.2b
Specimens in “Y” direction
48
0
4.2c
Specimens in “45 ” direction
48
4.2d
Specimens in “X” direction
48
4.3
Tensile test of woven fiber glass/epoxy composite specimens
49
4.4
Failure pattern of woven fiber glass/epoxy composite specimens
49
4.5a
Application of gel coat on mould releasing sheet
51
4.5b
Placing of woven roving glass fiber on gel coat
51
4.5c
Removal of air entrapment using steel roller
51
4.5d
Teflon foil for artificial introduction of delamination
51
4.5e
Setup for fabrication of delaminated composite plate
51
4.6a
Iron frame for making different boundary condition setup
53
4.6b
Plate with cantilever boundary condition
53
4.6c
Plate with four sides clamped boundary condition
53
4.6d
Plate with four sides simply supported boundary condition
53
4.7
Modal Impact Hammer (type 2302-5)
54
4.8
Accelerometer (4507)
54
4.9
Bruel & Kajer FFT (spectrum) analyzer
54
4.10
Display unit
54
4.11a
Composite plate before buckling
57
4.11b
Composite plate after buckling
58
5.1
Variation of change in ILSS vs. delamination length
xiii
of
62
Figure
Particular
glass/epoxy at 2 mm/minute loading speed
Page
5.2
Variation of change in ILSS vs. delamination length
glass/epoxy at 50 mm/minute loading speed
of
62
5.3
Variation of change in ILSS vs. delamination length
glass/epoxy at 100 mm/minute loading speed
of
63
5.4
Variation of change in ILSS vs. delamination length
glass/epoxy at 200 mm/minute loading speed
of
63
5.5
Variation of change in ILSS vs. delamination length
glass/epoxy at 500 mm/minute loading speed
of
63
5.6
Scanning micrograph showing matrix cracking in laminated
composites
64
5.7
Scanning micrograph showing fiber
cracking in delaminated composites
64
5.8
Laminated composite plate with mid-plane delamination
67
5.9
Variation of fundamental natural frequency with delamination
area of woven fiber cantilever composite plates
68
5.10
Variation of natural frequency with delamination area of four
side clamped woven fiber composite plates
69
5.11
Variation of natural frequency with delamination area of four
sides simply supported woven fiber composite plate
69
5.12
Variation of natural frequency with fiber orientation for 25%
delaminated woven fiber cantilever composite plate
71
5.13
Variation of natural frequency with number of layers for 25%
delaminated woven fiber cantilever composite plate
72
5.14
Variation of natural frequency with aspect ratio for 25%
delaminated woven fiber cantilever composite plates
73
5.15
Variation of natural frequency of multiple delaminated cantilever and
clamped plate with different percentage of delamination area
75
5.16
Frequency response function spectrum
75
5.17
Coherence ( Response, Force)
76
5.18
Applied force Vs time curve
76
5.19
Variation of non-dimensional buckling load of single delaminated
CFCF composite plates with increasing delamination area
80
5.20
Variation of non-dimensional buckling load of 25% single
delaminated woven roving composite plate with fiber orientation
81
5.21
Variation of non-dimensional buckling load of 25% single
delaminated CFCF cross ply plate with number of layers
82
5.22
Variation of nondimensional buckling load of 6.25% delaminated
CFCF woven fiber composite plate with different aspect ratios
83
xiv
pullout and interfacial
Figure
Particular
Page
5.23
Variation of non-dimensional buckling load of multiple
delaminated CFCF plates with increasing delamination area
84
5.24
Variation of non-dimensional buckling load with different % of
delamination for different boundary conditions
85
5.25
Determination of critical buckling load of a plate with 25%
delamination from load v/s end shortening displacement graph
86
5.26
Determination of critical buckling load of a plate with 56.25%
delamination from load v/s end shortening displacement graph
86
5.27
Variation of instability region of [(0/90)2]s cross- ply plate with
different percentage of delamination area for L/t =125
90
5.28
Variation of instability region of 2-layer (0/90) composite plate
with different percentage of delamination
90
5.29
Variation of instability region of 4-layer (0/90)s composite plate
with different percentage of delamination
91
5.30
Variation of instability region for the degree of orthotropy
(E11/E22 =40) of composite plate with different percentage of
delamination
92
5.31
Variation of instability region for the degree of orthotropy
(E11/E22 =20) of composite plate with different percentage of
delamination
93
5.32
Variation of instability region of 0% delaminated cross ply plate
with different aspect ratio
93
5.33
Variation of instability region of 25% delaminated cross ply
plate with different aspect ratio
94
5.34
Variation of instability region of 6.25% delaminated rectangular
plate with different static load factor
95
xv
Nomenclature
The principal symbols used in this thesis are presented for easy reference. A single
symbol is used for different meanings depending on the context and defined in the
text as they occur.
English
a, b
Dimensions of plate along X and Y axis respectively
h
Thickness of the plate
[B]
Strain-displacement matrix of the element
[Bp]
Strain-displacement matrix of the element for plane stress
[D]
Flexural rigidity / elasticity matrix of plate
[Dp]
Flexural rigidity / elasticity matrix of plate for plane stress
dx, dy
Element length in x and y direction
dv
Volume of the element
E11, E22
Modulii of elasticity of lamina in both 1 & 2 direction respectively
G12, G13, G23
Shear modulii of rigidity
|J|
Jacobian
[K]
Global elastic stiffness matrix
[Kp]
Plane stiffness matrix
[Kg]
Global geometric stiffness matrix
kx, ky, kxy
Curvatures of the plate
[M]
Global consistent mass matrix
Nx, Ny, Nxy
In plane internal stress resultants of the plate
Mx, My, Mxy
Moment resultants of the plate
N(t)
In-plane harmonic load
Ns
Static portion of the load N(t)
Nt
Amplitude of dynamic portion of load N(t)
n
Number of layers of laminated panel
Qx , Qy
Transverse shearing forces
xvi
q
Vector of degrees of freedom
Ncr
Critical buckling load
{P}
Global load vector
T
Transformation matrix
u, v, w
Displacements in X, Y & Z direction
u0, v0, w0
Mid-plane displacements in X, Y & Z direction
U0
Strain energy due to initial in-plane stresses
U1
Strain energy associated with bending with transverse shear
U2
work done by the initial in-plane stresses and the nonlinear strain
X, Y, Z
Global coordinate axis system
Greek
óxx, óyy, óxy
,
,
Stresses at a point
Bending strains
í
Poisson’s ratio
èx , èy
Slopes with respect to Y and X axes
Natural frequency
Ù
Excitation frequency
á
Static load factor
â
Dynamic load factor
Mathematical Operators
[ ]-1
Inverse of a matrix
[ ]T
Transpose of a matrix
xvii
LIST OF PUBLICATIONS OUT OF THIS WORK
Papers in International Journals
1.
J. Mohanty, S. K. Sahu and P.K. Parhi (2012): Numerical and experimental
study on free vibration of delaminated woven fiber glass/epoxy composite
plates, International Journal of Structural Stability and Dynamics, Vol.12(2),
pp.377-394.
2.
J. Mohanty, S. K. Sahu and P.K. Parhi (2011): Numerical and experimental
study on buckling behaviour of multiple delaminated composite plates,
International Journal of Structural Integrity, (Accepted for publication).
3.
J. Mohanty, S. K. Sahu and P.K. Parhi (2012): Parametric instability of
delaminated composite plates subjected to periodic in-plane loading,
International Journal of vibration and control. (Recommended for publication)
Papers Presented in International Conferences
1.
S. K. Sahu and J. Mohanty (2012): Dynamic stability of delaminated
composite plates subjected to in-plane harmonic loading, International
Conference on Civil Engineering and Architecture (ICCEA 2012),August 0304 at Hong-Kong.
2.
J. Mohanty, S. K. Sahu and P.K. Parhi (2011): Effect of delamination on
interlaminar shear strength of laminated woven fiber glass/epoxy composite
plates, Proceedings of 5th International Conference on Advances in
Mechanical Engineering (ICAME-2011), June 06-08 at Sardar Vallabhbhai
National Institute of Technology, Surat, Gujarat, India, pp:114-118.
3.
J. Mohanty, S. K. Sahu , P.K. Parhi and B.C. Ray (2010): Vibration
analysis of delaminated woven fiber glass/epoxy composite plates,
Proceedings of International Conference on Challenges and Application of
Mathematics in Science & Technology (CAMIST) at NIT, Rourkela, India, pp:
168-176.
4.
J. Mohanty, S. K. Sahu, L Sinha and P.K. Parhi (2010): Buckling
behaviour of delaminated woven fiber glass/epoxy composite plates,
Proceedings of 5th International Conference on Theoretical, Applied,
Computational & Experimental Mechanics, Dec. 27-29 at IIT, Kharagpur,
India, pp: 320-322.
5.
P. Nayak, J. Mohanty and S. K. Sahu (2009): Vibration analysis of woven
fiber glass/epoxy composite plates, Proceedings of 9th International Conference
on Vibration Problems, January-2009 at IIT, Kharagpur, India, pp: 31
xviii
BIO-DATA
1
Name in Full
:
JAYARAM MOHANTY
2
Date of Birth
:
29.04.1967
3
Permanent Address
:
Jayaram mohanty,
Plot No. 333,
Barabari (Jagamara),
PO: Khandagiri
Bhubaneswar-751030, Odisha
4
Educational
Qualifications
:
Bachelor Degree in Civil Engineering from
Institution of Engineers (India)
M.Tech in Civil Engineering with
specialisation in Structural Engg. from
Biju Pattanaik University of Technology
Rourkela, Odisha, India
5
Research Experience
:
Five Years
6
Professional Experience
:
Twenty four years in
Civil Engineering Department,
College of Engineering & Technology,
BPUT, Bhubaneswar, Odisha
7
Publications
:
International Journal
National Journal
Reports
8
Book Published
:
Design of Rigid Pavement using High Volume
Fly Ash Concrete
3 Nos
3 Nos
3 Nos
Publisher : VDM, Verlag Dr. Muller GmbH &
Co. KG, Germany
xix
CHAPTER 1
INTRODUCTION
1.1
Introduction
Laminated composite structures are widely used in aircraft, automobile,
marine, nuclear, civil engineering structures and other industrial fields because of
the higher value of specific strength and stiffness. These can be tailored through
the variation of fiber orientation, number of layers, aspect ratio and stacking
sequence to obtain an efficient design. The optimum design of laminated
structures demands an effective analytical procedure. But the presence of varying
coupling stiffness and anisotropy complicates the problems of dynamic stability of
laminated plates in addition to inherent problems due to the diverse loading and
boundary conditions encountered for obtaining a suitable theoretical solution.
As the demand for composite materials increases, the defect problems in
composite structures become an important concern. Among them, delamination is
one of the common defects, which is induced due to improper handling in the
process of manufacturing, low velocity impact or excessive interlaminar stresses at
the free edges under loading. Delaminations may not be visible or barely visible
on the surface, since they are embedded inside the body of the composite
structures. However, the presence of delamination in composite laminates may
adversely affect the design parameters like interlaminar shear strength, vibration
characteristics, buckling strength of the structure, dynamic stability characteristics,
safety and durability of structures etc. Therefore, a comprehensive functional
understanding in respect of vibration, static and dynamic stability characteristics
of delaminated plates is very much essential.
1
1.2
Importance of the structural stability studies of delaminated
composite plates
The structural elements like beams, plates and shells are sometimes
subjected to periodic in-plane loading and become dynamically unstable with
increase in transverse vibration without bounds for certain combination of load
amplitude and disturbing frequency. The above phenomenon is called dynamic
instability or parametric resonance. Again the presence of delamination may
increase the complications associated with the parametric resonance of the
laminated plates. The dynamic instability of delaminated plates may occur below
the critical buckling load of the structure under compressive loading over a range
or ranges of excitation frequencies. Several methods used for combating
parametric resonance such as damping and vibration isolation may be insufficient
and sometimes produce dangerous effects. The parametric instability may arise not
merely at a single expectation frequency but even for small excitation amplitudes
and combinations of frequencies.
Presence of one or more delaminations in a composite laminate structure
may lead to a premature collapse of the structure due to buckling at a lower level
of compressive loading. So the effect of delamination on stability of composite
structures needs attention and thus constitutes a problem of current interest. The
location of dynamic instability region (DIR) is quite essential for the study of
dynamic stability. The calculation of dynamic instability region (DIR) spectra is
often provided in terms of the spectrum of natural frequencies and the static
buckling loads. Furthermore, composite plates with delamination may result in
significant changes of these characteristics. Therefore it is an important task to
calculate both of them with sufficiently high precision. Thus, the study of dynamic
stability itself requires a special investigation on basic problems of vibration and
static stability. Thus, the parametric instability characteristics are of great
technical importance for understanding the dynamic system of delaminated plates
under periodic loading.
2
1.3
Objectives of present study
The present investigation mainly focuses on the study of vibration, static
and dynamic stability of industry driven woven fiber glass/epoxy delaminated
composite plates. A study of variation of interlaminar shear strength (ILSS) of
composites with delamination is also studied for the sake of completeness. A first
order shear deformation theory based on finite element model is developed for
studying the free vibration, critical buckling load and parametric instability
characteristics of mid-plane delaminated composite plates. The effect of
delamination size on ILSS is studied experimentally. The influence of
delamination size, boundary conditions, number of layers, fiber orientations and
aspect ratio on the free vibration and static stability (buckling) behavior of
delaminated cross ply and angle ply laminates are investigated experimentally and
numerically. The effect of delamination size, number of layers, aspect ratio,
degree of orthotropy and static load factors on the parametric instability behaviour
of delaminated cross ply plates are also examined numerically.
This thesis contains six chapters. In Chapter 1, a brief introduction of the
importance of the study has been outlined.
In Chapter 2, a detailed review of the literature pertinent to the previous
research works made in this field has been listed. A critical discussion of the
earlier investigations is done. The aim and scope of the present study is also
outlined in this chapter.
In Chapter 3, a description of the theory and formulation of the problem
and the finite element procedure used to analyse the vibration, buckling and
parametric instability characteristics of homogeneous and delaminated composite
plates is explained in detail. The computer programme used to implement the
formulation is briefly described.
In Chapter 4, the experimental investigation for ILSS, free vibration and
static stability are described in detail. This chapter includes fabrication procedures
3
for samples, test set-up, apparatus required for different tests and determination of
material constants.
In Chapter 5, the results and discussions obtained in the study have been
presented in detail. The effect of delamination size on ILSS, is studied
experimentally; the influence of delamination size, boundary conditions, number
of layers, fiber orientations and aspect ratio on the free vibration and static
stability (buckling) behavior of delaminated cross ply and angle ply plates are
investigated experimentally and numerically; the effect of delamination size,
number of layers, aspect ratio, degree of orthotropy and static load factors on the
parametric instability behaviour of delaminated cross ply plates are also examined
numerically.
Finally, in Chapter 6, the conclusions drawn from the above studies are
described. There is also a brief note on the scope for further investigation in this
field.
4
CHAPTER 2
REVIEW OF LITERATURE
2.1
Introduction
The presence of delamination in composite laminates may adversely affect
the safety and durability of structures. Therefore a comprehensive understanding
of the delamination behavior is of fundamental importance in the assessment of
structural performance of laminated composites. In structural mechanics, dynamic
stability has received considerable attention over the years which encompasses many
class of problems. Thus the dynamic stability characteristics are of great technical
importance for understanding the dynamic systems under periodic loads. Though the
present investigation is mainly focused on dynamic stability of delaminated composite
plates, some relevant researches on free vibration and static stability or buckling of
plates are also studied for the sake of its relevance and completeness. The studies in
this chapter are grouped as follows:
 Vibration
 Static stability/buckling
 Dynamic stability
However, some studies on static analysis of delaminated composites involving
effects of different parameters on inter laminar shear strength (ILSS) are also studied
for completeness.
2.2
Static analysis of delaminated composites
Some experimental studies involving effects of different parameters on the
static analysis i.e. interlaminar shear strength of laminated composites are studied.
Ruiz and Xia (1991) reported that relatively high values of interlaminar shear strength
(ILSS) appear under impact loading causing debonding. The interlaminar shear
strength is one of the most important parameter in determining the ability of a
composite material to resist delamination damage in laminates (Hallet et al.,1999).
Costa et al. (2001) studied the effect of porosity on interlaminar shear strength of
5
carbon/epoxy composite laminates. Kotek et al. (2001) examined the interlaminar
shear strength of textile reinforced carbon-carbon composite. Okutan (2002) discussed
the effects of geometric parameters on the failure strength for pin loaded fiberglass
reinforced epoxy laminates. Zhang et al. (2002) investigated the effect of fiber
orientation on the ILSS of graphite/epoxy laminated composites and showed that
ILSS depends on the fiber orientation of the neighboring plies around the interfaces.
Abadi and Poro (2003) measured interlaminar shear strength (ILSS) of fiber
reinforced carbon-carbon composites (CCC). Christensen and Teresa (2003) studied
the effect of transverse pressure on ILSS of a quasi isotropic laminate T300/F584.
They noted significant increase in ILSS of T300 carbon fiber and F584 epoxy matrix
with the increase in transverse pressure. Ray (2004) investigated the effects of thermal
and moist environments (hygrothermal) on ILSS of glass fibre/epoxy composites.
Teresa et al. (2004) examined the effect of through-thickness compression on the
interlaminar shear strength of laminated fiber composites. For both glass and carbon
fiber
composites,
through-thickness
compression
resulted
in
a
significant
enhancement in the interlaminar shear strength. Agrawal and Prasad (2005) discussed
the influence of environment (cathodic exposure) on inter laminar shear strength of
carbon/epoxy composite laminates and observed a significant reduction in the
interlaminar shear strength after cathodic exposure. Chakrabarti and Sheikh (2005)
made an analysis of laminated sandwich plates based on interlaminar shear stress
continuous plate theory. Nancy et al. (2005) used double notch shear (DNS) test to
characterize the ILSS of ceramic matrix woven composites. Pinho et al. (2005)
proposed that LaRC04 failure criteria can predict with accuracy interlaminar damage
in composite laminates and can be a good choice. Ray (2005) evaluated the ILSS of
glass fiber reinforced unsaturated polyster and epoxy resin composites by exposing
the composites to 75°C temperature gradient thermal shock, using 3-point bend test.
Yokoyama and Nakai (2006) studied the effects of deformation rate and specimen
thickness on the ILSS and failure mode. They found that the ILSS is independent on
the deformation rate up to nearly 1.6m/s, but depends on the specimen thickness.
Khashaba and Seif (2006) investigated on the effect of different loading conditions
on the mechanical behaviour of (0/+45/90)s woven composites and reported that the
woven composites performed better under bending loading than under tension
6
loading. Das et al. (2007) examined the effects of cross head velocities on ILSS of
woven FRP composites and found that the variation of ILSS of laminates of FRP
composites is significant for low loading speed and not so prominent for high speed
loading. Lopes et al. (2008) discussed the influence of porosity on interlaminar shear
strength of fiber-metal laminates both numerically and experimentally. They reported that
the interlaminar shear strength of fibre-metal laminates decreases considerably due to
porosity. Cox and Wilson (2008) evaluated ILSS of a graphite/epoxy composite with
carbon nanofiber reinforcement (CNF) and found that non-reinforced specimens exhibited
20% higher lap-shear strengths than CNF-reinforced specimens. Hong-yan et al. (2009)
investigated on the effects of voids (void content, void shape and size) on the ILSS of
[(+/-45)(4)/(0,90)/(+/-45)(2)]S and [(+/-45)/0(4)/(0,90)/0(2)]S composite laminates. Kong
and Wang (2009) examined the role of nano clays in the enhancement of interlaminar
shear strength (ILSS) of glass fiber reinforced diallyl phthalate (GFR-DAP) composites
and observed that only 2.5 wt% clay loading in DAP matrix increased the ILSS of
resulting GFR-DAP laminates by 7.64%. Aslan and Alnak (2010) performed
experimental and numerical analysis to study the interlaminar shear strength of laminated
woven E-glass/epoxy composites by four point bend shear test. Walter et al. (2010)
conducted monotonic, multi-step and cyclic short beam shear tests on 2D and 3D woven
composites. The test results were used to determine the effect of z- yarns on interlaminar
shear strength. The results showed that with the increase in z-yarns percentage there is
decrease in interlaminar shear strength. Jaeschke et al. (2011) reported that the magnitude
of heat affected zone at the cutting edge of the CF-PPS composite plate is correlated with
the interlaminar shear strength of the plate.
The above studies deals with static analysis of composites without considering
delamination. However studies involving delaminated composites are scarce in
literature. Kim and Donaldson (2006) made an experimental and analytical study on the
damage initiation in the form of interlaminar delamination of carbon fiber reinforced
polymer composite under thermal and mechanical loadings. Piotrowski et al. (2006)
studied the delamination behavior of a laminated E-glass reinforced polymer matrix
composite versus changes in the processing of the polymer matrix; changes in the resin
matrix rheology and in the e-glass finish chemistry and concentration. The ILSS of a
laminated polymer matrix composite was examined as functions of matrix phase
7
additions, the matrix mixing procedure, the glass fiber finish chemistry and concentration,
and the matrix rheology. Matrix phase additions, mixing, and glass finish were found
to significantly affect the ILSS, while the matrix rheology had less of an impact.
2.3
Vibration of delaminated composite plates
A considerable amount of investigations are available in the open literature on
vibration of laminated composite beams, plates and shells without delamination.
However, studies addressing delamination in composite laminates are very limited.
The delamination problem is generally more complex involving geometrical and
material discontinuity. Della and Shu (2007) reviewed the available mathematical
models for the vibration of delaminated composite laminates. As per the study, the
earliest reported model for the vibration analysis of composite beams was reported by
Ramkumar et al. (1979). Ostachowicz and Krawczuk (1994) analyzed the natural
frequency of composite beams with delamination using finite element method (FEM).
Lee et al. (2003) performed an analytical solution for multiple delaminated beams.
Brandinelli and Massabo (2003) developed an analytical model to investigate the
effect of bridging mechanisms between the delaminated interfaces on the vibration of
a delaminated composite beam. Park et al. (2004) developed a recurrent single
delaminated beam model for vibration analysis of multi-delaminated beams. Kim and
Hwang (2002) presented an analytical solution using the “constrained mode”
assumption for delaminated honey-comb sandwich beams. Shu and Della (2004)
investigated the vibration of sandwich beams with two delaminations at identical span
wise locations. Othman and Barton (2008) studied failure initiation and propagation
characteristics of honeycomb sandwich composites. Zhu et al. (2005) formulated the
reference surface element and its applications in dynamic analysis of delaminated
composite beams. Yuan et al. (2008) calculated the reflection and transmission
coefficients for time harmonic flexural waves in a semi-infinite delaminated beam
following analytical approach.
The above studies are based on one dimensional models. Two dimensional
models are also developed to predict the behaviour of delaminated composites in a
more realistic way. Ju et al. (1995) presented finite element formulation for the
analysis of free vibration of composite plates with multiple delaminations.
8
Parhi et al. (2000) investigated the dynamic behavior in the presence of single and
multiple delaminations of laminated composite plates. Zak et al. (2001) studied the
influence of the delamination length and position on changes in natural frequencies
and modes of vibration of the unidirectional composite plates by using finite element
method. Yam et al. (2004) proposed a finite element model to predict dynamic
behavior of multi-layer composite plates with internal delamination at arbitrary
locations. Oh et al. (2005) developed a four-noded finite element formulation based
on the efficient high-order zig-zag plate theory of laminated composite plates with
multiple delaminations to predict the natural frequencies, mode shape and time
response. Olsson et al. (2006) developed a model to predict delamination threshold
loads for dynamic impact on plates.
However, very few studies involving experimental investigations on vibration
of delaminated composite plates are available in literature. Champanelli and Engblom
(1995) presented limited vibration data of three numbers of delaminated
graphite/PEEK composite plates with delamination location at mid edge and corner of
the plate and compared with modeling results. Luo and Hangud (1996) conducted
modal analysis experiments on glass fiber/epoxy cantilever composite plates with
fixed size of strip delamination. Kessler et al. (2002) studied damage detection in
composite panels using frequency response method. Azouaoui et al. (2007) made an
experimental investigation to study the delamination behaviour of glass/polyester
composite plates subjected to low-energy compact fatigue.
Krawezuk et al. (1997) developed a finite element model to study the
dynamics of cracked composite material structures. Chang et al. (1998) studied
vibration of delaminated composite plates under axial load. Hou and Jeronimidis
(1999) made an experimental investigation on circular composite plates with an
impact induced delamination. Chattopadhyay et al. (2000) formulated a high order
theory for dynamic stability analysis of delaminated composite plates. Thornburgh
and Chattopadhyay (2003) used finite element method to study the vibration of
composite laminates with delamination and transverse matrix cracks. Suzuki et al.
(2004) used multilayered finite element numerical analysis for non-linear vibration
and damping characterization of delaminated CFRP composite laminates.
9
Sancho and Miravete (2006) developed a design for delaminated composite
structure considering the three dimensional stress field in the vicinity of free edges,
holes and changes in number of layers. Karmkar et al. (2006) investigated the effect
of delamination on free vibration characteristics of graphite-epoxy composite
pretwisted cylindrical shallow shells of various stacking sequences considering length
of delamination as a parameter. Hein (2006) performed free vibration analysis for
multiple delaminated beams and investigated the influences of delamination size and
position on the natural frequencies of the stepped beam numerically. Acharya et al.
(2007) studied free vibration of delaminated composite cylindrical shell roofs. Roy
and Chakraborty (2008) proposed three dimensional finite element analysis to
evaluate the response of graphite/epoxy laminates subjected to impact loading.
Shiau and Zeng (2010) investigated on the effect of delamination on free
vibration of a simply supported rectangular homogeneous plate with through-width
delamination by the finite strip method. Results showed that the delamination has
considerable effect on the natural frequencies of the plate. The aspect ratio of the plate
is also having significant effect on the natural frequency of the plate, especially on the
mode 2 frequency of the plate. Zhang et al. (2010) developed a Structural Health
Monitoring (SHM) system based on vibration monitoring to detect, locate and assess
delamination damage in laminated composite structures. Towards this end, finite
element modelling is employed to simulate the dynamic response of composite
laminates (beams and plates) with delamination and extract their vibration parameters.
Hadi and Ameen (2011) characterized the embedded delamination on the dynamic
response of composite laminated structures. A finite element model for geometrically
nonlinear large amplitude vibration of shallow cylindrical and delamination shell
analysis was presented using higher order shear deformation theory where the
nonlinearity was introduced in the Green-Lagrange sense.
2.4
Static stability of delaminated composite plates
Several theoretical studies involving analytical models and numerical analysis
were developed to study buckling behavior of delaminated structures. Use of
theoretical analysis in predicting buckling load of delaminated plate was difficult.
Therefore, numerical and experimental methods have become important in solving the
10
buckling problem of a laminated composite plate having delamination. The problem
of delamination buckling and growth was addressed by Chai et al. (1981) for a
laminated plate. They presented a one-dimensional model to describe the failure
mechanism. Sallam and Simitses (1985) investigated on the delamination buckling
and growth of flat cross-ply laminates using a one-dimensional model. Bruno and
Grimaldi (1989) analyzed delamination failure of layered composite plates loaded in
compression. Compression failure of carbon fiber-reinforced coupons containing central
delaminations was studied by Pavier and Chester (1990). An analytical and experimental
investigation on unidirectional graphite/epoxy composites was performed by Kutlu and
Chang (1992) to study the compression response of composites containing multiple
delaminations. Compressive buckling stability of composite panels with through-width,
equally spaced multiple delaminations were investigated experimentally and analytically
using Rayleigh-Ritz approximation technique by Suemasu (1993).
Yeh and Tan (1994) studied the buckling behaviour of laminated plates with
elliptical delamination under uniaxial compressive loading experimentally and
analytically. Kim and Hong (1997) presented a finite element model for buckling and
post buckling behavior of composite laminates with an embedded circular
delamination using degenerated shell elements. Gu and Chattopadhyay (1999)
investigated experimentally delamination buckling and postbuckling of composite
laminates. A buckling analysis of laminates with an embedded delamination was
conducted by Hu et al. (1999) by employing a finite-element method based on the
Mindlin plate theory. Hwang and Mao (2001) predicted the buckling loads of
delaminated unidirectional carbon/epoxy composites with strip delamination. The
buckling and postbuckling behaviors of carbon/epoxy composite laminates with
multiple delaminations were experimentally and numerically studied by Hwang and
Liu (2002). The effect of the strip delamination width on the buckling loads of the
simply supported carbon/epoxy woven laminated composite plates was investigated
by Zor (2003) following 3D finite element models. Kucuk (2004) established a threedimensional finite element model of the square laminated plates to study the effects of the
lateral strip delamination width on the buckling loads. Zor et al. (2005) developed a threedimensional finite element model to study the effects of the square delamination on the
buckling loads. Experimental measurements and numerical solutions using ANSYS on
11
the buckling of single centered strip delaminated woven glass-fiber composite laminates
were carried out on rectangular plates by Pekbey and Sayman (2006). Capello and
Tumino (2006) examined the buckling and post buckling behaviour of unidirectional
and cross-ply composite laminated plates with multiple delaminations.
Lee and Park (2007) investigated on the buckling behaviors of laminated
composite structures with a delamination using the enhanced assumed strain (EAS)
solid element three-dimensional finite element (FE) formulation. In particular, new
results reported in this paper are focused on the significant effects of the local
buckling for various parameters, such as size of delamination, aspect ratio, width-tothickness ratio, stacking sequences, and location of delamination and multiple
delaminations. Tumino et al. (2007) studied the role of delamination length, angle of
ply and stacking sequence on the buckling load of multidelaminated composite
specimens following finite element method with linear and nonlinear analysis. The
compressive behavior of composite laminates with through-the-width delaminations
was investigated analytically by Kharazi and Ovesy (2008).
Aran et al. (2009) examined the effect of delamination length, position
through thickness and stacking sequence of the plies on the buckling and postbuckling
of laminates with a single delamination using a three dimensional finite element
model. It was found that significant decrease occurs in the critical buckling load after
a certain value of the delamination length. The position of delamination and the fiber
orientation also affects the loads. Aslan and Sahin (2009) investigated the effects of
the delamination size on the critical buckling load and compressive failure load of
unidirectional E-glass/epoxy composite laminates with multiple large triangular
delaminations. Obdrzalek and Vrbka (2010) studied the buckling behavior of a small
delaminated plate subjected to compression loading by means of the finite element
analysis. Esfahani et al. (2010) made experimental and numerical analysis of
delaminated hybrid composite beam structures. Mohsen and Amin (2010) examined
numerically the buckling and post buckling analysis of composite laminated structures
with delaminations by using the generalized differential quadrature method. Kang et
al. (2011) followed numerical analysis to study the compressive buckling of
composite laminates with a delamination. Damghani et al. (2011) predicted the
critical buckling load of composite plates with through-the-length delaminations using
12
exact stiffness analysis. Chattopadhyay and Murthy (2011) investigated the elastic
buckling and postbuckling analysis of an axially loaded beam-plate with an acrossthe-width delamination, located at a given depth below the upper surface of the plate.
Tsouvalis and Garganidis (2011) made the buckling strength parametric study of
composite laminated plates with delaminations .
A critical review of above works show that most of the previous works deal with
numerical analysis of one dimensional delaminated unidirectional laminates using a
numerical technique including FEM based package. However, the numerical and
experimental investigations on buckling analysis of delaminated bidirectional industry
driven woven roving glass/epoxy composite plates are scarce in literature. Zhang and Fu
(2000) made a micro mechanical model of woven fabric for the analysis of buckling
under uniaxial tension. Xu et al. (2006) investigated on the buckling analysis of triaxial
woven fabric composite structures and parametric study-uniaxial loading.
2.5
Dynamic stability of delaminated composite plates
In modeling delamination, both, analytical as well as numerical methods have
been used in studying the dynamic and buckling behaviour of composite laminates. A
lot of studies are available on the use of numerical methods to predict the natural
frequencies and critical buckling load of delaminated composite plates using different
approaches. However studies involving dynamic stability of delaminated plates are
much less in literature. The earlier works on dynamic stability of structures are
reviewed by Simites (1987) and Sahu and Datta (2007). The dynamic stability of
rectangular isotropic plates under various in-plane forces has been studied by Bolotin
(1964) , Jagdish (1974), Hutt and Salam (1971). Deolasi and Datta (1995) used finite
element method based on first order shear deformation theory (FSDT) to study the
parametric instability characteristics of thin isotropic plates.
A number of researchers have investigated the dynamic stability characteristics
of rectangular composite plates.The dynamic stability of rectangular laminated
composite plate due to periodic in-plane load is studied by Srinivasan and Chellapandi
(1986) using finite strip method (FSM). Bert and Birman (1987) investigated the
dynamic instability of shear deformable antisymmetric angle ply plates. Yamaki and
Nagai (1975) examined the dynamic stability of rectangular plates under periodic
13
compressive forces. Dynamic stability of laminated composite plates due to periodic inplane loads is investigated by Chen and Yang (1990) using FSDT. Moorthy et al. (1990)
and Chattopadhyay and Radu (2000) predicted the dynamic instability boundaries of
rectangular composite plates. Kwon (1991) investigated the dynamic instability of
composite laminates following finite element method. Wang and Dawe (2002) examined
the dynamic instability of composite laminated rectangular plate and prismatic plate
structures. Liao and Cheng (1994) studied the dynamic instability characteristics of
stiffened isotropic and composite square plate. Srivastava et al. (2003) investigated the
dynamic instability of stiffened plates subjected to non-uniform harmonic edge loading.
Patel et al. (2009) performed the parametric study on dynamic instability behaviour of
laminated composite stiffened plate by using the FSDT. Lee (2010) studied the finite
element dynamic stability of laminated composite skew plates containing cutouts based
on higher order shear deformation theory (HSDT). Dey and Singha (2006) investigated
the dynamic stability characteristics of simply supported composite skew plates subjected
to a periodic in-plane load. The principal and second instability regions are identified for
different parameters such as skew angle, thickness- to- span ratio, fiber orientation and
static in-plane load. Dynamic instability behaviour of composite and sandwich laminates
with interfacial slips has examined by Chakrabarti and Sheikh (2010) by using RHSDT
(refined higher order shear deformation theory). Dynamic instability analysis of
composite laminated thin walled structures was carried out by Fazilati and Ovesy (2010)
by using two versions of FSM (finite strip method). Biswas et al. (2011) studied the static
and dynamic instability characteristics of curved laminates with internal damage
subjected to follower loading.
Park and Lee (2009) examined parametric instability of delaminated
composite spherical shells subjected to in-plane pulsating forces. Radu and
Chattopadhyay (2002) analyzed the dynamic stability of composite plates including
delamination using higher order theory and transformation matrix. They analyzed
composite plates with various thickness, delamination length and placement and
observed that delamination affects the instability regions by shifting them to lower
parametric resonance frequencies. Yeh and Tung (2006) investigated the dynamic
instability behavior of delaminated composite plates under transverse excitation
experimentally and analytically.
14
2.6
Critical discussion
The present review indicates that more studies are conducted on laminated
composite plate, beams and shells without delamination. However studies involving
delamination in composite laminates are very limited. As regards to methodology, the
researchers are more interested to use numerical methods instead of analytical
methods. With the advent of high speed computers, more studies are made using finite
element method. From the present review of literature, the lacunae of the earlier
investigations which need further attention of future researchers are presented below.
2.6.1 Vibration of delaminated composite plates
The review of the present work as cited in the literature indicated that a
considerable amount of analytical models and numerical analysis was reported for
the vibration analysis of unidirectional composite laminates with delaminations.
Many researchers followed one dimensional model for vibration behavior of
delaminated composites (Zhu et al., 2005; Othman & Barton, 2008). Two
dimensional models were also developed to predict the behaviour of delaminated
composites in a more realistic way. Ju et al.(1995), Zak et al.(2001) and Yam et
al.(2004) studied the free vibration of delaminated composites by using the finite
element formulation. The natural frequencies of composite plates with multiple
delaminations were predicted by Oh et al. (2005) following a four-noded finite
element formulation based on the efficient high-order zig-zag plate theory.
Experimental investigations on vibration of delaminated composite plates are very
scarce in literature. Campanelli and Engblom (1995), Luo and Hangud (1996) and
Azouaoui et al. (2007) made experimental investigation to study natural frequency of
delaminated graphite/PEEK, glass fiber/epoxy and glass/polyester composite plates. Hou
and Jeronimidis (1999) made an experimental investigation on circular composite plates
with an impact induced delamination.
The vibration of composite laminates with delamination and transverse matrix
cracks
using
finite
element
method
was
studied
by
Thornburgh
and
Chattopadhyay(2003). The effect of delamination on free vibration characteristics of
15
graphite-epoxy composite pretwisted cylindrical shallow shells of various stacking
sequences involving length of delamination as a parameter was investigated by
Karmkar et al.(2006).
2.6.2 Static stability of delaminated composite plates
The studies involving behavior of delaminated composite plates subjected to
in plane load are much less in literature. Several theoretical studies involving
analytical models and numerical analysis were developed to study buckling behavior
of delaminated structures. Numerical and experimental methods were attempted by
various researchers for predicting the buckling load of delaminated composite plates
(Yeh and Tan, 1994; Gu and Chattopadhyay 1999). Chai et al. (1981) and Sallam and
Simitses (1985) followed one one-dimensional model to describe the buckling
problem of delaminated composites.
Kutlu and Chang (1992) studied the compression response of unidirectional
graphite/epoxy composites containing multiple delaminations by using analytical and
experimental investigation. A three-dimensional FEM model was developed by
Kucuk (2004) and Zor et al. (2005) to study the effects of delamination on the
buckling loads. Tumino et al. (2007) studied the role of delamination length, angle of
ply and stacking sequence on the buckling load of multi-delaminated composite
specimens following finite element method with linear and nonlinear analysis.
2.6.3 Dynamic stability of delaminated composite plates
Most of the researchers followed analytical and numerical methods to predict
the natural frequencies and critical buckling load of delaminated composite plates.
Investigations on dynamic stability of delaminated plates are very scarce in literatures.
The parametric instability characteristics of thin isotropic plates was investigated by
Hutt and Salam (1971) and Deolasi and Datta (1995).
The dynamic stability characteristics of rectangular composite plates was
investigated by a number of researchers (Jagdish, 1974; Yamaki and Nagai,1975;
Srinivasan and Chellapandi, 1986; Wang and Dawe, 2002). Bert and Birman (1987)
investigated the dynamic instability of shear deformable antisymmetric angle ply
plates. Dynamic stability of laminated composite plates due to periodic in-plane loads is
16
investigated by Chen and Yang (1990). Liao and Cheng (1994) studied the dynamic
instability characteristics of stiffened isotropic and composite square plate. Srivastava et al.
(2003) examined the dynamic instability of stiffened plates subjected to non-uniform
harmonic edge loading. The parametric study on dynamic instability behaviour of
laminated composite plate was performed by using FEM (Patel et al., 2009; Lee, 2010).
However the study of instability behavior of composite plates subjected to delamination is
scarce in literature. Radu and Chattopadhyay (2002), Park and Lee (2009) and Yeh and
Tung (2006) investigated the dynamic instability behavior of delaminated composite
plates under transverse excitation experimentally and analytically.
2.7
Scope of the present study
An extensive review of the literature shows that a lot of work was done on the
vibration and static stability of delaminated composite plates. The woven composite is
a new class of textile composite and has many industrial applications. Very little work
has been done on dynamic stability of delaminated composite plates. The present
study is mainly aimed at filling some of the lacunae that exist in the proper
understanding of the dynamic stability of industry driven woven fiber delaminated
plates. Based on the review of literature, the different problems identified for the
present investigation are presented as follows.

Interlaminar shear strength of delaminated composites

Free vibration of delaminated composite plates

Buckling/ static stability of delaminated composite plates

Dynamic stability of delaminated composite plates
The present study mainly focuses on the parametric resonance characteristics
of homogeneous and delaminated composite plates. The influence of various
parameters such as delamination size, aspect ratio, number of layers, degree of
orthotropy and static load factor on the instability behaviour of delaminated plates are
examined numerically using Bolotin’s approach and finite element method. A special
investigation of vibration and buckling of delaminated industry driven composite
plates are also conducted both numerically and experimentally.
17
CHAPTER 3
THEORY AND FORMULATION
3.1
The basic problem
This chapter represents the theory and finite element formulation (FEM) for
free vibration, static stability and dynamic stability analysis of the composite plate of
various geometry with and without delamination. The basic configuration of the
problem considered here is a composite laminated plate with mid-plane single
delamination subjected to in plane periodic load as shown in Figure 3.1. The
boundary conditions are incorporated in the most general manner. The details of
delamination is shown in Figure 8.1 through 8.7 of Appendix-I.
N(t)
Figure 3.1: Delaminated composite plate under in-plane periodic load
3.2
Proposed analysis
The governing equations for the dynamic stability of delaminated composite
plates subjected to in-plane periodic loading are developed. The presence of external
in-plane loads induces a stress field in the structure.
This necessitates the
determination of the stress field as a prerequisite for the solution of problems like
vibration, buckling and dynamic stability behaviour of plates. As the thickness of the
18
structure is relatively smaller, the determination of stress field reduces to the solution
of a plane stress problem. The governing differential equations have been developed
using the first order shear deformation theory (FSDT). The assumptions made in the
analysis are given below.
3.2.1 Assumptions of the analysis
1.
The analysis is linear, in line with previous studies on the dynamic stability
of panels (Bert and Birman, 1988; Sahu and Datta, 2003) with a few
exceptions. This implies both linear constitutive relations (generalized
Hooke’s law for the material and linear kinematics) and small
displacements to accommodate small deformation theory.
2.
The delaminated panels are of various shapes with no initial imperfections.
The consideration for imperfections is less important for dynamic loading and
is consistent with the work of Bert and Birman (1988).
3.
The straight line that is perpendicular to the neutral surface before
deformation remains straight but not normal after deformation (FSDT).
Normal stress in the Z-direction is neglected.
4.
The loading on the delaminated panel is considered as axial with a simple
harmonic fluctuation with respect to time.
5.
3.3
All damping effects are neglected.
Governing equations
The Governing differential equations, the strain energy due to loads, kinetic
energy and formulation of vibration, buckling and dynamic stability problems are
derived on the basis of principle of Potential Energy and Lagrange’s equation and are
presented as follows.
3.3.1 Governing differential equations
The equation of motion is obtained by taking a differential element of plate as
shown in Figure 3.2(a) & (b). The figure 3.2(a) shows an element with internal forces
like membrane forces (
.The Figure 3.2(b) shows shearing forces (Qx
and Qy) and the moment resultants (Mx, My and Mxy).
19
Figure 3.2 (a): Force resultants
Figure 3.2 (b): Moment resultants
The governing differential equations of equilibrium for free vibration of a
shear deformable laminated plate
subjected to external in-plane loading can be
expressed as (Chandrashekhara,1989; Sahu and Datta, 2003):
N x N xy
 2 x
 2u

 P1 2  P2
x
y
t
t 2
N xy
x

N y
y
 P1
 2 y
 2v
P

2
t 2
t 2
Q x Q y
2w
2w
2w

 N x0 2  N y0 2  P1 2
x
y
x
y
t
20
(3.3.1)
 2u
M x M xy
 2 x

 Qx  P3 2  P2 2
x
y
t
t
M xy
x
where N x0

M y
y
 Q y  P3
 2 y
 P2
t 2
 2v
t 2
and N y0 are the external loading in the X and Y directions
n
respectively. ( P1 , P2 , P3 ) 
zk
  ( )
k
(1, z, z 2 )dz
k 1 z k 1
where n = number of layers of the laminated composite panel, (ñ)k = mass density of
kth layer from the mid-plane.
3.3.2 Energy expressions
The delaminated composite plate is subjected to initial in-plane edge
loads N x0 , N y0 and N xy0 . These in-plane loads cause in-plane stresses of óx0, óy0 and
óxy0inducing a plane stress problem. The delaminated composite plates with the initial
stresses undergo small lateral deformations. The total stress at any layer is the sum of
the initial stresses plus the stresses due to bending and shear deformation. The strain
energy Uo due to initial in-plane stresses is written as
1
0
2 
T
0
   dV
U0 = U0 =
(3.3.2)
where
0 T
   
.0
x
.0
y
 
.0 T
xy

 u 0 v 0 u 0 v 0 



x 
 x y y
T
(3.3.3)
and the stresses are
{ 0 }  [ D p ]{ 0 }
(3.3.4)
The strains can be expressed in terms of initial in-plane deformations u0, v0 as
21
{ 0 }  [ BP ]{q 0 }
(3.3.5)
Substituting the values of stress and strain in the equation (3.3.2), we get
U0 
1
{q 0 }T [ BP ]T [ DP ][ BP ]{q 0 }dA

2
(3.3.6)
The strain energy is expressed as
1
U 0  {q 0 }T [ K P ]{q 0 }
2
(3.3.7)
where
K P    [ BP ]T [ D P ][ BP ]dA
(3.3.8)
Considering the prestressed state as the initial state, the strain energy stored due to
bending and shear deformation in the presence of initial stresses (neglecting higher
order terms) is given by
U = U1+ U2
(3.3.9)
where U1 = Strain energy associated with bending with transverse shear,
U2 = Work done by the initial in-plane stresses and the nonlinear strain
U1 
1
[{ l }T [ D ]{ l }]dV

2
(3.3.10)
where the strains can be expressed in terms of deformations as
{ l }  [ B ]{ q 0 }
and
U2 
(3.3.11)
1
[{ 0 }T { nl }]dV

2
(3.3.12)
The method of explicit integration is performed through the thickness of the
panel and thus the generalized force and moment resultants can directly be related to
22
the strain components through the laminate stiffness. The kinetic energy V of the plate
can be derived as
2
2
  2
 y 
h  u
v 2 w 2  h 3   x
V    






dxdy
 2  t
t
t  12  t
t 

 

(3.3.13)
Now, the various energies can be expressed in matrix form as
1
U0 = {q}T [ K P ]{q}
2
1
U 1  {q}T [ K ]{q}
2
(3.3.14)
1
U 2  {q}T [ K g ]{q}
2
1
V  {q}T [M ]{q}
2
where [Kp] = Plane stiffness matrix
[K] = Bending stiffness matrix with shear deformation
[Kg] = Geometric stiffness or stress stiffness matrix
[M] = Consistent mass matrix
3.3.3 Formulation of static and dynamic problems
The equation of motion for vibration of a delaminated composite panel, subjected to
in-plane loads can be expressed as:
(3.3.15)
[ M ]{ q}  [[ K ]  N (t )[ K g ]]{ q}  0
Here,‘q’ is the vector of degrees of freedom u, v, w, x and y. The in-plane load
‘N (t)’ may be harmonic and can be expressed in the form:
N (t )  N s  N t Cos  t
(3.3.16)
23
where N s is the static portion of the load N(t), N t is the amplitude of the dynamic
portion of N(t) and  is the frequency of the excitation. Considering the static and
dynamic components of load as a function of the critical load,
N s  N cr , N t  N cr
(3.3.17)
where á and â are the static and dynamic load factors respectively. Using equation
(3.3.16), the equation of motion under periodic loads in matrix form may be obtained
as:
[ M ]{q}  [[ K ]   N cr [ K g ]   N cr [ K g ]Cos  t ]{q}  0
(3.3.18)
The above equation (3.3.18) represents a system of differential equations with
periodic coefficients of the Mathieu-Hill type. The development of regions of
instability arises from Floquet’s theory which establishes the existence of periodic
solutions of periods T and 2T. The boundaries of the primary instability regions with
period 2T, where T = 2 /Ù are of practical importance [Bolotin, 1964] and the
solution can be achieved in the form of the trigonometric series:

q (t ) 
 [{a }Sin ( kt / 2)  {b }Cos ( kt / 2)]
k
k
(3.3.19)
k 1, 3 , 5 ,..
Putting this in equation (3.3.18) and if only the first term of the series i.e., k =
1 is considered, and equating coefficients of Sin Ùt/2 and Cos Ùt/2, the equation
(3.3.19) reduces to
1
2
[[ K ]  N cr [ K g ]  N cr [ K g ] 
[ M ]]{q}  0
2
4
(3.3.20)
Equation (3.3.20) represents an eigenvalue problem for known values of ,
 and Ncr. The two conditions under the plus and minus sign correspond to the two
boundaries of the dynamic instability region. The eigen values are , which give
the boundary frequencies of the instability regions for given values of  and . In
this analysis, the computed static buckling load of the panel is considered as the
reference load in line with many previous investigations (Ganapati et al., 1994;
24
Moorthy et al. 1990). This equation (3.3.20) represents a solution to a number of
related problems as follows,
(1)
Free vibration: á = 0, â = 0 and ù = Ù/2
[[K] −ù² [M]]{q}=0
(2)
(3.3.21)
Vibration with static axial load: â = 0 and ù = Ù/2
[[K] − áNcr [Kg] −ù² [M]]{q}=0
(3)
(3.3.22)
Static stability: á = 1, â = 0, Ù = 0
[[K] − Ncr [Kg]] {q}=0
3.4
(3.3.23)
Finite element formulation
A delaminated composite plate of length a, width b and thickness h consisting
of n arbitrary number of anisotropic layers is considered as shown in Figure 3.1. The
layer details of the plate are shown in Figure 3.3. The global coordinate system is
considered with respect to the mid-plane of the plate with the Z-axis perpendicular to
the X-Y plane and  is the angle of fiber orientation, measured anticlockwise with
respect to X-axis. In the present investigation, the delaminated composite plate is
discretised in to a mesh of 8×8 with total 64 elements. An eight noded two
dimensional quadratic isoparametric element having five degrees of freedom (u0,
v0,w, èx, ,èy) per node is chosen.
Z
Figure3.3: Layer details of the plate
25
3.4.1 Displacement field and shape functions
The displacement field of any point at a distance z
from the mid surface is
assumed to be in the form of
u(x,y,z) = u0(x,y) + zèx(x,y)
(3.4.1)
v(x,y,z) = v0(x,y) + zèy(x,y)
(3.4.2)
w(x,y,z) = w0(x,y)
(3.4.3)
where u, v, w are displacements in the X, Y, Zdirections respectively for any
point, u0,v0 , w0 are those at the middle plane of the plate. èx, èy are the rotations of
the cross section normal to the Y and X axis respectively. The middle plane of the
plate is considered as the reference plane of the plate. The mid plane strains of the
laminate are given by
åxx0 = u0,x; åyy0 = v0,y; ãxy0 = u0,y + v0,x; ãxz0 = èx + w,x;, ãyz0 = èy + w,y
(3.4.4)
Assuming small deformations, the generalized linear in-plane strains of the
laminate at a distance z from the mid-surface are expressed as
{åxxåyyãxyãxzãyz }T={å0xx å0yyã0xy ã0xz, ã0yz}T+z{ kxx kyy kxy kxz kyz}T
 u 0



 x

0

0xx   v

 0   y



 yy 
 0   u 0 v 0 

where  xy   

x 
 0   y
 xz  
 
 0    x 

x 
 yz  
 

  y  y 


26
(3.4.5)
  x

 x




k xx    y

   y

k yy  





 
y 
and k xy    x 

y 
   x
k xz   0

k  

 yz   0







where å0xx, å0yy, ã0xy are the mid-plane strains and kxx kyy kxy are the
curvatures of the laminated plate .
The element has 4 corner nodes and 4 mid side nodes. In the displacement
model, simple functions are assumed to approximate the displacements for each
element. For the present isoparametric element, the shape functions which are used to
represent the geometry as well as the displacements within the element are expressed
by the shape functions Ni.
8
8
x   Nixi
i 1
8
,
y   Ni yi
i 1
,
8
8
u 0   N i u i0
v 0   N i v 0i
i 1
8
,
i 1
8
w   N i w i è x   N i è xi è y   N i è yi
i 1
i 1
i 1
,
,
(3.4.6)
where xi , yi ,are the co-ordinates of the ith node and ui0, vi0,wi , èxi, èyi are the
displacement functions for different nodes .
Figure 3.4: The element in isoparametric co-ordinates
27
Ni for different nodes as shown in Figure 3.4 is defined as,
At corner nodes (i.e. for node 1, 3, 5, 7)
1
N i  (1  îî i )(1  ççi )(î i  ççi  1 )
4
At middle nodes (i.e. for nodes 2, 6)
Ni 
1
(1  î 2 )(1  çç i )
2
At middle nodes (i.e. for nodes 4, 8)
Ni 
1
(1  îî i )(1  ç 2 )
2
(3.4.7)
where î and ç are the local isoparametric co-ordinates of the element and îi and çi are
the respective values at node i. The correctness of the shape function Ni is checked
from the relations
N
i
1
 N , î  0  N ,  0
i
i
(3.4.8)
The derivatives of the shape functions Ni with respect to x and y are expressed
in terms of their partial derivatives with respect to î and ç by the relationships:
N i N i  N i 


x
 x  x
N i N i  N i 


y
 y  y
 N i   
 x  
x
 N    
 i 
 y   y
  N i 


x   
  N 
 i 
y   


28
 N i, x 
1  N i , 
 N   J   N 
 i, y 
 i , 
(3.4.9)
 x,  y ,    N i , x i
where [J] = 

 x, y ,   N i , x i
 N  y 
N y 
i,
i
i,
i
is the Jacobian matrix.
3.4.2 Stress strain relations
A macromechanical analysis is carried out to establish the relationship
between the forces and strains of a laminate. The elastic behavior of each lamina is
essentially two dimensional and orthotropic in nature. So the elastic constants for the
composite lamina are given below.
E11 = Modulus of elasticity of lamina along 1-direction
E22 = Modulus of elasticity of lamina along 2-direction
G12 = Shear modulus
í12 = Major Poisson’s ratio
í21 = Minor Poisson’s ratio
The on-axis elastic constant matrix [Qij]k corresponding to material axes 1-2
for kth layer is given by
Q 
ij k
Q11 Q12 0 
 Q12 Q22 0  for i, j = 1, 2, 6


 0
0 Q66 
Q44
 0
and [Qij]k  
0 
for i,j =4,5
Q55 
(3.4.10)
(3.4.11)
For obtaining the off-axis elastic constant matrix, [ Q ij ]k corresponding to any
arbitrarily oriented reference X-Y axes for the kth layer ,appropriate transformation is
required.
29
Figure 3.5: On-axis and off-axis configurations of lamina
Hence as shown in Figure 3.5, the off-axis elastic constant matrix is obtained from the
on axis elastic constant matrix by the relation
Q 
ij k
Q11 Q12

 Q12 Q22
Q16 Q26
Q16 

Q26 
Q66 
for i ,j =1,2,6
[Qij]k = [T]-1[Qij]k[T]T
 m2
 m2
n2
n2
mn 
 2mn 
 2

 2

2
2
m
2mn  [Qij]k  n
m
 mn 
=n
mn  mn m 2  n 2 
 2mn 2mn m 2  n 2 




k
k
(3.4.12)
 m2 n 2
2mn 
 2

2
m
- 2mn 
where [T] = Transformation matrix =  n
- mn mn m 2  n 2 


Q11 
E11
;
1  í12 í 21
Q12 
k
E11í 21
E 22 í12
E 22
; Q21 
; Q22 
1  í12 í21
1  í12 í 21
1  í12 í 21
Q66 = G12; Q44 = G13; Q55 = G23
30
For i ,j =4,5
Q 
ij k
Q Q 
  44 45 
 Q45 Q55  k
m  n  Q44
= 
 
n m k  0
0   m n
Q55  k  n m  k
(3.4.13)
where m = cosè, n = sinè , è is the angle from the X-axis to the 1-axis measured
anticlockwise.
The stress strain relationship for a laminate at a distance z is given by
 xx  Q11 Q12 Q16   xx 
  
  
 yy   Q12 Q22 Q26   yy 
  Q Q
Q66  z  xy 
26
 xy  z  16
z
 0 

Q11 Q12 Q16   xx  k xx 

  0    
 Q12 Q22 Q26   yy   z k yy 
Q16 Q26 Q66   0  k xy 
 
z 
 xy 

 xz  Q44
 
 yz  z Q45
and 
(3.4.14)
Q45   xz 
  
Q55  z  yz 
(3.4.15)
Here óxx and óyy are the normal stresses along X and Y directions respectively
and ôxz and ôyz are the shear stresses in xz, yz planes respectively.
The force and moment resultants are obtained by integrating the stresses and
their moments through the laminate thickness as given by

 N xx 
 zk
n
 
 N yy    k 1 Qij z
N 
 k 1
xy
 z

 xx0 
zk
 0
 yy dz  z
k 1
 0 
xy
 

 M xx 
 zk
n


M yy   k 1 Qij z
M 
 k 1
 xy  z

 xx0 
zk
 0
 yy  zdz  z
k 1
 0 
 xy 
 
 
31
k xx  
  
k yy  zdz 
k  
 xy  

k xx 
  2 
k yy  z dz 
k 

 xy 

(3.4.16)
(3.4.17)
 z
Qxz 
n
   k 1 Qij z

Qyz  z
 xz0  
 0 dz i , j  4 ,5
 yz  
k
k 1
(3.4.18)
The above 3 equations are combined together to obtain the force, moment and
transverse shear resultants.
The relationship among stress resultants and the deformations are given by
 N xx   A

  11
N
 yy   A12
 N xy   A

  16
M
 xx   B11


M yy   B12
M   B16
 xy  
Q xz  0

 0
Q yz  
A12
A16
B11
B12
B16
A22
A26
B12
B22
B26
0
0
A26
A66
B16
B26
B66
0
B12
B16
D11
D12
D16
B22
B26
D12
D22
D26
0
0
B26
B66
D16
D26
D66
0
0
0
0
0
0
S 44
0
0
0
0
0
S 45
 0 
0   xx 
 0
0   yy 
 0 
0   xy 

0   K xx 


0   K yy 

0   K xy 


S 45   0 
 xz
S 55   0 
 yz 
(3.4.19)
where Aij =∑nk=1 ( Q ij ) k (zk – zk-1)
Bij =
1 n
∑ k=1 ( Q ij )k (z2k – z2k-1)
2
Dij =
1 n
∑ k=1 ( Q ij ) k (z3k – z3k-1) for i ,j=1,2,6
3
Sij = ê ∑nk=1 ( Q ij )k (zk – zk-1)
for i,j=4,5
(3.4.20)
Aij = inplane stiffness terms relating the in-plane forces with inplane strains.
Bij= Coupling stiffness terms relating the in-plane forces with curvature and
moments with in-plane strains.
Dij = bending stiffness terms relating moments with curvature.
Here, [ Q ij ]k is the off axis elastic constant matrix for the kth lamina and the
shear correction factor ê is assumed as 5/6. It accounts for the non-uniform
distribution of transverse shear strain across the thickness of the laminate.
32
Aij , Bij , Dijare the extension, bending stretching coupling
and bending
stiffness respectively. Sij is the transverse shear stiffness of the laminate. The elastic
properties of each lamina are generally assumed to be constant through its thickness
as these laminae are considered to be thin.
3.5
Delamination modeling
A simple two dimensional single delamination model proposed by Gim (1994)is
extended by Parhi et al. (2000) for the vibration of delaminated composite panels. In the
present analysis, it is further extended for static and dynamic stability analysis under inplane uniaxial periodic forces by multiple delamination modelling. In order to satisfy the
compatibility and equilibrium requirements at the common delamination boundary, it is
assumed that the in-plane displacement, transverse displacement and rotation at a
common node for all the three sublaminates including the original one are identical
applying multiple constraint condition at any arbitrary delamination boundary. It can be
applicable to any general case of a laminated composite plate having multiple
delaminations at any arbitrary location. Here, the delaminated area is assumed as the
interface of two separate sub laminates bonded together along the delamination surface.
1
2
1st
h/2
2
Z1
Z0
nd
k
k+1
Zk
h/2
Z n-1 Zn
Pth
n-1
n
ZZ
Figure 3.6: Laminate geometry with multiple delaminations
33
r
hs/2 h/2C
s
Z hs/2
hs
zr0
z0s
z0t
t
h/2
z0u
u
ZZ
Figure 3.7: Three arbitrary delaminations leading to four sub-laminates
Typical composite plate of uniform thickness ‘h’ with ‘n’ number of layers
and ‘p’ number of arbitrarily located delaminations is considered for the analysis as
shown in Figure 3.6. The principal material axes of each layer are arbitrarily oriented
with respect to the mid-plane of the plate. Let z0sbe the distance between the midplane of the original laminate and the mid-plane of the arbitrary sth sub-laminate as
shown in Figure 3.7.
Considering the sub-laminates as a separate plate, the displacement field
within it is expressed as:
u s  u s0  z  z s0  xs ,


vs  vs0  z  zs0  ys


(3.5.1)
where u0s and v0s are the mid-plane displacements of the sth sub-laminate along
X and Y direction and z s0 is distance between mid-plane of sth sub-laminate and the
mid-plane of the laminate in Z direction
The mid-plane strains of the sub-laminate are
å
0
xx
å
0
yy
0
ã xy
T

s
 u 0
= s
 x
v s0
y
u s0 v s0 


y
x 
where å0xx å0yy ã0xy are mid-plane strains.
34
T
(3.5.2)
From equation (3.5.2) the strain components within the sub-laminate s can be
expressed as
 u 0
= s
 x

= å xx0
å
å
0
yy
0
ã xy
T

s
 u
= s
 x
v s
y
u s v s 


y x 
u s0 v s0 
0

 + ( z  zs
y
x 
0
T
ã xy + (z  z s0 ) k xx

s

k yy
T
T
T
v s0
y
å 0yy
0
xx
 è
) x
 x
è y
y
è x èy 


y
x 
T
k xy s
(3.5.3)
where kxx kyy kxy are curvatures of the laminated plate.
In order to satisfy the compatibility and equilibrium requirements at the
common delamination boundary, it is assumed that the in-plane displacements,
transverse displacement and rotations at a common node for all the three
sub-laminates including the original one are identical. Applying multiple constraint
condition at any arbitrary delamination boundary c, the in-plane displacements at ‘c’
at a distance ‘z’ from the mid-plane of the laminate can be written as
uc = u0 +zèx ,vc = v0+zèy
From equation (3.5.1), the displacement at any point, c is given by
usc = us0 + (z-zs0)èx , vsc=vs0 + (z-zs0)èy
Equating uc with usc and vc with vsc , the mid-plane displacements of the
sub-laminate can be expressed in the form of the mid-plane displacements(u0, v0) of
the original un-delaminated laminate as,
us0 = u0 + zs0èx, vs0 = v0 + zs0èy
(3.5.4)
From equation (3.5.4), the mid-plane strain components of the sth sub-laminate
can be derived as:
35

0
xx
T
T
T
 yy0  xy0 s   xx0  yy0  xy0   z s0 k xx k yy k xy 
(3.5.5)
The strain components within the sub-laminate can be written as

T
 
  
xx yy xy s
0
  xx
 yy0  xy0

T

0 0 0 T
xx yy xy s
 
0
s
  ( z  z )k
 z s0 k xx k yy k xy

T

k k
xx yy xy
T

 ( z  z s0 ) k xx k yy k xy

T

(3.5.6)
For any lamina of sth sub-laminate, the in-plane and shear stresses are found
from the relation
T
xxyyxy
xzyzT
Q11 Q12

 Q12 Q22
Q16 Q26

Q
  44
Q45
Q45 
  xz  yz`
Q5 

Q16 

Q26   xx yy xy
Q66 

T

s
(3.5.7)
T

s
(3.5.8)
where óxx and óyy are the normal stresses along X and Y directions respectively and
ôxz and ôyz are the shear stresses in XZ, YZ planes respectively.
Integrating these stresses over the thickness of the sub -laminate, the stress and
moment resultants of the sub- laminate are derived which lead to the elasticity matrix
of the sth sub-laminate [D]s in the form
 Aij
Ds   Bij
0

z s0 Aij  Bij
0
s
z Bij  Dij
0
0

0
S ij 
(3.5.9)
[D]s is the elasticity matrix of the sth sub-laminate
hs
 z s0
2
where,
A 
ij s
 Q  dz

,
ij s

hs
 z s0
2
36
hs 0
 zs
2
B 
ij s
hs 0
 zs
2
ij s

ij s
hs 0
 zs
2
0 2
s
ij s

0 2
s
2
ij s
hs 0
 zs
2
hs
 2 z s2 2h
 z s0
s
2
 z s0
ij s
 hs 0
 zs
2
 Q  z  z  dz   Q  z  z 

hs 0
 zs
2
Q  zdz   z  A 
hs
 z 0s
2
hs
  z 0s
2
0 2
s
ij s
ij s

 2 zzs2 dz 
 hs 0
 zs
2
S   
ij s
0
s
ij s
hs 0
 zs
2
hs 0
 zs
2
D 
0
s
 Q  z  z dz   Q  zdz  z A 

2
 Q  z dz
ij s
 hs 0
 zs
2
for i, j = 1, 2, 6

Q dz
for i,j =4,5
ij
(3.5.10)
The in-plane stress and moment resultants for the sth sub-laminate can be
expressed in a generalized manner as:
 N xx   A11
N  
 yy   A12
 N xy   A
16


 N xx   A11
 N yy   A12

 
 N xy   A16
A12 A16 z s0 A11  B11 z s0 A12  B12
A22 A26 z s0 A12  B12 z s0 A22  B22
A26 A66 z s0 A16  B16 z s0 A26  B26
A12 A16 z s0 A11  B11 z s0 A12  B12
A22 A26 z s0 A12  B12 z s0 A22  B22
A26 A66 z s0 A16  B16 z s0 A26  B26
 0 
z s0 A16  B16   xx 
0

z s0 A26  B26   yy 
 0
z s0 A66  B66   xy 
   (3.5.11)
z s0 A16  B16  k xx 
z s0 A26  B26  k yy 
  
z s0 A66  B66  s k 
 xy  s
Similarly, the transverse shear resultants for the sth sub-laminate are presented as
Qxz   S 44
  
Q yz  s  S 45
S 45   xz 
 
S 55  s  yz 
(3.5.12)
After finding the elastic stiffness matrices separately for different sublaminates along the thickness, the sum of all the sub-laminate stiffnesses represents
the resultant stiffness matrix.
37
3.6
Strain displacement relations
Green-Lagrange’s strain displacement is used throughout the structural
analysis. The linear part of the strain is used to derive the elastic stiffness matrix and
non-linear part of the strain is used to derive the geometrical stiffness matrix.
{å}={ ål}+{ ånl}
(3.6.1)
The linear strains are defined as
 xl 
u
 Zk x
x
 yl 
u
 Zk y
y
 xyl 
u v
  Zk xy
y x
 xz 
u w

z x
 yz 
w v

y z
(3.6.2)
where the bending strains kj are expressed as
Kx 
 x
x
Ky 
 y
y
K xy 
 x  y

y
x
(3.6.3)
Assuming that w does not vary with Z, the non-linear strains of the plate are
expressed as
xnl= [ (∂u/∂x)2 +(∂v/∂x)2 + (∂w/∂x)2]/2,
38
ynl= [ (∂v/∂y)2 +(∂u/∂y)2 + (∂w/∂y)2]/2,
xynl = [ (∂u/∂x) (∂u/∂y) + ((∂v/∂x) (∂v/∂y) + (∂w/∂x)(∂w/∂y)],
xznl = [ (∂u/∂x) (∂u/∂z) + (∂v/∂x) (∂v/∂z) + (∂w/∂x) (∂w/∂z)],
xznl= [(∂u/∂y) (∂u/∂z) + (∂v/∂y) (∂v/∂z) + (∂w/∂y) (∂w/∂z)],
The linear strain can be described in term of displacements as
{} =[B] {de}
(3.6.4)
where {de} = [u1 v1 w1x1y1 u2 v2 ………….. u8 v8 w8x8y8]T
[B] = [[B1] …… [B7] [B8]]
 Ni
 x

 0
 Ni

y
Bi    0

 0
 0

 0
 0

3.7
0
0
0
0
Ni
y
Ni
x
0
0
0
0
0
0
0
0
Ni 0
x Ni
0
y
Ni Ni
y x
Ni
0
0
0
0
0
0
0
Ni
x
Ni
x
0
Ni














 i 1to 8
(3.6.5)
Derivation of element matrices
3.7.1 Elastic stiffness matrix
The potential energy of deformation for the element is given by
∫∫{ }T[ó]dA
Ue =
{å} =
(3.7.1)
{å0xx å0yy ã0xy kxx kyy kxy ãxz ãyz}T
(3.7.2)
where {}= [B]{de}=[ [B1] ……………… [B8] ]{de}
with {de}= u10 v10 w1 x1 1y ...........................u0 v0 w  x  y

39
T

(3.7.3)
(3.7.4)
Then Ue =
1
1
∫∫{de}T[B]T[D][B]{de}dxdy =
{de}T[ Ke] {de}
2
2
(3.7.5)
where the element stiffness matrix
[Ke] = 
1
1
1

1
[B]T[D][B]│J│dîdç
(3.7.6)
[B] is called the strain displacement matrix.
In equation (3.7.6)
[Bi] =
8

i 1
 N i, x
 0

 N i, y

 0
 0

 0
 0

 0
0
0
0
N i, y
N i, x
0
0
0
0
0
0
N i, x
0
0
0
0
0
N i, y
0
N i, x
Ni
0
N i, y
0
0 
0 
0 

0 
N i, y 

N i, x 
0 

N i 
│J│dîdç, is the determinant of the jacobian matrix. The element stiffness matrix can
be expressed in local natural co-ordinates of the element. The integration of equation
(3.7.6) is carried out using the Gauss quadrature method.
3.7.2 Geometric stiffness matrix
The element geometric stiffness matrix is derived using the non-linear in-plane
Green’s strains. The strain energy due to initial stresses is
U2    0
v
T
  
n1
dV
(3.7.7)
Using non-linear strains, the strain energy can be written in matrix form as
U2 
1
T

f  S  f dV

2 v
(3.7.8)
 u u v v w w  x  x  y  y 
{f} =  , , , ,
, ,
,
,
,

 x y x y x y x y x y 
40
T
(3.7.9)
s
0

[S] = 0

0
0
0 0 0 0
s 0 0 0
0 s 0 0

0 0 s 0
0 0 0 s 
 x  xy  1  N x
 
 xy  y  h  N xy
[s]= 
N xy 
N y 
The in-plane stress resultants Nx , Ny, Nxy at each Gauss point are obtained by
applying uniaxial stress in X-direction and the geometric stiffness matrix is formed
for these stress resultants.
[f] = [G] [e]
(3.7.10)
where, [äe ] = [u, v, w, èx, èy ]T
The strain energy, U2 becomes
U2 =
1
 e T GT S G  e dV  1  e T K g  e 

2
2
(3.7.11)
where element geometric stiffness matrix
[Kg] =
1

 N i,x
N
 i, y
 0

 0
 0
[G] = 
 0
 0

 0
 0

 0
1
1

1
[G]T [S] [G]│J│dd
0
0
0
0
0
0
N i,x
0
0
Ni, y
0
0
0
0
N i,x
Ni, y
0
0
0
0
N i,x
0
0
N i, y
0
0
0
0
0
0
0 
0 
0 

0 
0 

0 
0 

0 
N i,x 

N i , y 
(3.7.12)
(3.7.13)
41
3.7.3 Consistent mass matrix
The consistent element mass matrix [Me] is expressed as
[Me] =
1
1

[N]T [P] [N]│J│dd

1
1
(3.7.14)
where [N], the shape function matrix and [P], the inertia matrix
 Ni
0

[N] =  0

0
 0
 P1
0

[P] =  0

 P2
 0
0
0
0
Ni
0
0
0
0
Ni
0
0
Ni
0
0
0
0
0
P2
P1
0
0
0
P1
0
0
P2
0
0
P3
0
0
0 
0  i  1 to 8

0
Ni 
0
P2 
0

0
P3 
where,
(P1, P2, P3)=
n

k 1
zk
   (1, z, z
z k 1
k
2
) dz
The derivatives of the shape function Ni with respect to x,y are expressed in term of
their derivatives with respect to î and ߟ by the following relationship
 N i ,x 
 N i , 
1
N   [ J ]  N 
 i , 
 i ,y 
 x
 
where [J] = 
 x
 
(3.7.15)
y 
 

y 
 
42
3.8
Computer program
A computer program is developed by using MATLAB environment to perform
all the necessary computations. The element stiffness, geometric stiffness and mass
matrices are derived using the formulation. Numerical integration technique by
Gaussian quadrature is adopted for the element matrices. The overall matrices [K],
[Kg], and [M] are obtained by assembling the corresponding element matrices.
Reduced integration is used to avoid possible shear locking. The boundary conditions
are imposed restraining the generalized displacements in different nodes of the
discretized structure. The further details of program features and flow charts, used in
this study are presented in Appendix II.
43
CHAPTER 4
EXPERIMENTAL PROGRAMME
4.1 Introduction
This chapter deals with the details of the experimental works conducted on
the static analysis involving interlaminar shear strength (ILSS), free vibration and
buckling of industry driven woven roving delaminated composite plates. Therefore
composite plates are fabricated for the aforementioned experimental work and the
material properties are found out by tensile test as per ASTM D3039/ D3039M (2008)
guidelines to characterize the delaminated composite plates. The experimental results
are compared with the analytical or numerical predictions. The experimental work
performed is categorized in four sections as follows:
 Static analysis
 Determination of material constants
 Vibration study
 Buckling study
4.2
Experimental programme for static analysis
4.2.1 Materials
The following constituent materials were used for fabricating the laminate:

Woven roving glass fiber as reinforcement

Epoxy as resin

Hardener

Polyvinyl alcohol as a releasing agent
 Teflon foil for artificial introduction of delamination
44
4.2.2
Fabrication of specimens
In the present investigation, the glass:epoxy laminate was fabricated in a
proportion of 50:50 by weight fractions of fiber: matrix. Araldite LY-556, an unmodified
epoxy resin based on Bisphenol-A and hardener (Ciba-Geig, India) HY-951, aliphatic
primary amine were used with woven roving E-glass fibers treated with silane based
sizing system (Saint-Gobain Vetrotex) to fabricate the laminated composite. Woven
roving glass fibers were cut into required shape and size for fabrication. Epoxy resin
matrix was prepared by using 8% hardeners. Contact moulding in an open mould by
hand lay-up was used to combine plies of woven roving (WR) in the prescribed
sequence. A flat plywood rigid platform was selected. A plastic sheet i.e. a mould
releasing sheet was kept on the plywood platform and a thin film of polyvinyl alcohol
was applied as a releasing agent. Laminating starts with the application of a gel coat
(epoxy and hardener) deposited on the mould by brush, whose main purpose was to
provide a smooth external surface and to protect the fibers from direct exposure to the
environment. Subsequent plies were placed one upon another with the matrix in each
layer to obtain sixteen stacking plies. The laminate consisted of 16 layers of identically 090o oriented woven fibers as per ASTM D2344/ D2344M (2006) specifications.
Delaminations were introduced at 1, 2.5 and 3.5 cm lengths by providing Teflon film at
the mid plane of the laminates through full width and equidistant from both ends of the
specimen. The mould and lay up were covered with a release film to prevent the lay up
from bonding with the mould surface. Then the resin impregnated fibers were placed in
the mould for curing. The laminates were cured at normal room temperature under a
pressure of 0.2 MPa for three days. After proper curing of the delaminated plates, the
release films were detached. From the laminates the specimens were cut for three-point
bend test (Figure 4.1a & 4.1b) by brick cutting machine into 45 x 6mm (Length x
Breadth) size as per ASTM D2344/ D2344 specification and the thickness was taken as
per the actual measurement. The average thickness of specimens for bend test is 4.8mm.
4.2.3
Bending test
The most commonly used test for ILSS is the short beam strength (SBS) test
under three point bending. The SBS test was done as per ASTM D 2344/ D 2344 M
(2006) by using the INSTRON 1195 material testing machine. The specimens were
tested at 2, 50, 100, 200 and 500 mm/minute cross head velocities with a constant span of
34 mm to obtain interlaminar shear strength (ILSS) of intact and delaminated samples.
45
Before testing, the thickness and width of the specimens were measured accurately. The test
specimen was placed on the test fixtures and aligned so that its midpoint was centered and
it’s long axis was perpendicular to the loading nose. The load was applied to the specimen
at a specified cross head velocity. Breaking load of the sample was recorded. About five
samples were tested at each level of experiment and their average value along with standard
deviation (SD) and coefficient of variation (CV) were reported in result part.
The interlaminar shear strength was calculated using the formula,
S = (0.75Pb)/bd as per ASTM D 2344
Where Pb is the breaking load in kg; b is the width in mm and d is the thickness in mm.
Figure 4.1 (a): Three point bend test setup and fixture
.
Figure 4.1 (b): Schematic diagram of three point bend test
4.2.4
Scanning electron microscope (SEM) test
After failure in bending test, 14 samples were selected for S.E.M. test. The test
was conducted by Scanning electron microscope (JEOL-JSM-6480 LV) for each
46
selected sample at 3 different magnifications i.e., X700, X500 and X300 to study the
crack pattern at the interface.
4.3
Determination of material constants
Laminated composite plates behave like orthotropic lamina, the characteristics
of which can be defined completely by four material constants i.e. E1, E2, G12, and 12
where the suffixes 1 and 2 indicate principal material directions. For material
characterization of composites, laminate having eight layers was fabricated to evaluate
the material constants.
The constants are determined experimentally by performing unidirectional tensile tests on specimens cut in longitudinal and transverse directions, and at 45° to the
longitudinal direction, as described in ASTM standard: D 3039/D 3039 M (2008).
The tensile test specimens are having a constant rectangular cross section in all the
cases. The dimensions of the specimen are mentioned below in Table 4.1.
Table 4.1: Size of the specimen for tensile test
Length(mm)
200
Width(mm)
25
Thickness(mm)
3
The specimens were cut from the plates themselves by diamond cutter or by hex
saw as per requirement as shown in Figure 4.2 (a). Four replicate sample specimens
were tested and mean values were adopted. The test specimens are shown in Figure
4.2. (b) to Figure 4.2(d).
47
Figure 4.2(a)
Figure 4.2(b)
Figure 4.2(c)
Figure 4.2(d)
Figure 4.2(a): Diamond cutter for cutting specimens, (b) Specimens in “Y”
direction, (c) Specimens in “450” direction, (d) Specimens in
“X” direction.
Coupons were machined carefully to minimize any residual stresses after they
were cut from the plate and the minor variations in dimensions of different specimens
are carefully measured. For measuring the Young's modulus, the specimen was
loaded in INSTRON 1195 universal testing machine (as shown in Figure 4.3)
monotonically to failure with a recommended rate of extension (rate of loading) of 0.2
mm/minute. Specimens were fixed in the upper jaw first and then gripped in the
movable jaw (lower jaw). Gripping of the specimen should be as much as possible to
prevent the slippage. Here, it was taken as 50mm in each side for gripping. Initially
strain was kept at zero. The load, as well as the extension, was recorded digitally with
the help of a load cell and an extensometer respectively. Failure pattern of woven
fiber glass/epoxy composite specimen is shown in Figure 4.4. From these data,
engineering stress vs. strain curve was plotted; the initial slope of which gives the
48
Young's modulus. The ratio of transverse to longitudinal strain directly gives the
Poisson's ratio by using two strain gauges in longitudinal and transverse direction.
But here Poisson’s ratio is taken as 0.17.
The shear modulus was determined using the following formula from Jones [1975]as:
The values of material constants finally obtained experimentally for vibration
and buckling are presented in Chapter-5.
Figure 4.3: Tensile test of woven fiber glass/epoxy composite specimens
Figure 4.4: Failure pattern of woven fiber glass/epoxy composite specimen
49
4.4
Experimental programme for vibration study
4.4.1 Fabrication of specimens
The fabrication procedure for preparation of the plate in case of vibration
study was same as in ILSS. Specimens are fabricated by hand layup technique and
cured under room temperature. The laminate consisted of eight layers of identically 090o oriented woven fibers. The artificial delaminations have been introduced at
6.25%, 25% and 56.25% area of composite plate by providing Teflon film centrally at
mid-plane of the plate during fabrication. After completion of all the layers, again a
plastic sheet was covered on the top of last ply by applying polyvinyl alcohol inside the
sheet as releasing agent. Again one flat ply board and a heavy flat metal rigid platform
was kept at the top of the plate for compressing purpose. The plates were left for a
minimum of 48 hours before being transported and cut to exact shape for testing.
Figure 4.5 (a-e) shows the fabrication process of delaminated composite plates. All
the specimens are tested for free vibration analysis. The geometrical dimensions (i.e.
length, breadth, and thickness), ply orientations and percentage of delamination of the
fabricated plates are shown in Table-4.2.
All the specimens described in Table 4.2 were tested for its vibration
characteristics. To study the effect of boundary condition on the natural frequency of
delaminated plates, the plates were tested for three different boundary conditions
(B.C) i.e. for four sides simply supported, fully clamped and cantilever. For different
boundary conditions, one iron frame was used. Some of the test specimens with
different boundary conditions are shown in Figure 4.6 (a-d).
50
Figure 4.5(a)
Figure 4.5(b)
Figure 4.5(c)
Figure 4.5(d)
Figure 4.5 (e)
Figure 4.5 (a): Application of gel coat on mould releasing sheet, (b) Placing of
woven roving glass fiber on gel coat, (c) Removal of air
entrapment using steel roller, (d) Teflon foil for artificial
introduction of delamination, (e) Set-up for fabrication of
delaminated composite plate
51
Table 4.2: Dimensions of composite plates with and without delamination
Size of plate
in meter
No. of
% of
Ply
No. of
layers delamination orientation delamination
No.
of
plates
0.237X0.237X0.003
8
0
(0/90)4
0
5
0.237X0.237X0.003
8
6.25
(0/90)4
1
5
0.237X0.237X0.003
8
25
(0/90)4
1
5
0.237X0.237X0.003
8
56.25
(0/90)4
1
5
0.237X0.237X0.0015
4
0
(0/90)2
0
5
0.237X0.237X0.0015
4
25
(0/90)2
1
5
0.237X0.237X0.0021
6
0
(0/90)3
0
5
0.237X0.237X0.0021
6
25
(0/90)3
1
5
0.237X0.237X0.003
8
0
[(30/-30)2]s
0
5
0.237X0.237X0.003
8
25
[(30/-30)2]s
1
5
0.237X0.237X0.003
8
0
[(45/-45)2]s
0
5
0.237X0.237X0.003
8
25
[(45/-45)2]s
1
5
0.240X0.240X0.003
8
25
(0/90)4
1
5
0.240X0.120X0.003
8
25
(0/90)4
1
5
0.240X0.160X0.003
8
25
(0/90)4
1
5
0.237X0.237X0.003
8
6.25
(0/90)4
1
5
0.237X0.237X0.003
8
25
(0/90)4
1
5
0.237X0.237X0.003
8
56.25
(0/90)4
1
5
52
Figure 4.6 (a)
Figure 4.6 (b)
Figure 4.6 (c)
Figure 4.6 (d)
Figure 4.6 (a): Frame for different boundary condition,( b) Cantilever plate, (c)
Four sides simply supported plate (d) Four sides clamped plate
4.4.2 Equipments for vibration test
In order to achieve the right combination of material properties and service
performance, the dynamic behavior is the main point to be considered. To avoid the
typical problems caused by vibrations, it is important to determine natural frequency
of the structure and the modal shapes to reinforce the most flexible regions or to
locate the right positions where weight should be reduced or damping should be
increased. The fundamental frequency is a key parameter. The natural frequencies are
sensitive to the orthotropic properties of composite plates and design-tailoring tools
may help in controlling this fundamental frequency. Due to the advancement in
computer aided data acquisition systems and instrumentation, experimental modal
analysis or free vibration analysis has become an extremely important tool in the
hands of an experimentalist.
53
The apparatus which are used in free vibration test are

Modal hammer ( type 2302-5)

Accelerometer (type 4507)

FFT Analyzer (Bruel Kajer FFT analyzer type –3560)

Notebook with PULSE software.

Specimens to be tested
The apparatus which are used in the vibration test are shown in Figure 4.7 to
Figure 4.10.
.
Figure 4.7: Modal Impact Hammer
(type 2302-5)
Figure 4.8: Accelerometer (4507)
Figure 4.9: Bruel & Kajer FFT
analyzer
Figure 4.10: Display unit
54
4.4.3 Procedure for free vibration test
The setup and the procedure for the free vibration test are described
sequentially as given below. The test specimens were fitted properly to the iron frame.
The connections of FFT analyzer, laptop, transducers, modal hammer, and cables to
the system were done. The pulse lab shop version-10.0 software key was inserted to
the port of laptop. The plate was excited in a selected point by means of small impact
with Impact hammer (Model 2302-5) for cantilever plates. The input signals were
captured by a force transducer, fixed on the hammer. The resulting vibrations of the
specimens on the selected point were sensed by an accelerometer. The accelerometer
(B&K, Type 4507) was mounted on the specimen by means of bees wax. The signal
was then processed by the FFT Analyzer and the frequency spectrum was also
obtained. Both input and output signals are investigated by means of spectrumanalyzer (Bruel & kajer) and resulting frequency response functions are transmitted to
a computer for modal parameter extraction. The output from the analyzer was
displayed on the analyzer screen by using pulse software. Various forms of frequency
response functions (FRF) were directly measured. However, the present work
represents only the natural frequencies of the plates. For FRF, at each singular
point the modal hammer was struck five times and the average value of the
response was displayed on the screen of the display unit. At the time of striking
with modal hammer to the points on the specimen precaution were taken for
making the stroke to be perpendicular to the surface of the plates.
Then by
moving the cursor to the peaks of the FRF graph the frequencies are measured.
4.5
Experimental program for buckling study
4.5.1 Fabrication of specimens
Materials required and fabrication procedure followed for preparation of plates
in case of buckling study was same as that used for vibration study. The geometrical
dimensions (i.e. length, breadth, and thickness), ply orientations and percentage of
delamination, number of delamination and number of the fabricated plates are shown
in Table 4.3.The single delamination in composite plates (No of delamination:1) is
provided in mid plane only for all number of layers as shown in figure 8.6 of
Appendix I. The multiple delamination (No of delamination:3) is provided in 2nd, 4th
55
and 6th layer of 8 layer composite plates as shown in figure 8.7 of Appendix I. The
size of delamination for different percentage of delamination is shown in figure 8.2,
8.3 and 8.4 of Appendix I.
Table 4.3: Dimensions of composite plates with delamination
Size of plate
In “mm”
No. of
layers
% of
delamination
240*190*3.5
8
0
Ply
stacking
sequence
[0]8
240*190*4.5
12
0
240*190*6.5
16
200*150*3.0
No. of
delamination
No. of
plates
0
5
[0]12
0
5
0
[0]16
0
5
8
0
[0]8
0
5
200*150*3.0
8
0
[30/-30]2s
0
5
200*150*3.0
8
0
[45/-45]2s
0
5
190*160*3.5
8
0
[0]8
0
5
190*120*3.5
8
0
[0]8
0
5
190*80*3.5
8
0
[0]8
0
5
240*190*3.5
8
25
[0]8
1
5
240*190*4.5
12
25
[0]12
1
5
240*190*6.5
16
25
[0]16
1
5
200*150*3.0
8
25
[0]8
1
5
200*150*3.0
8
25
[30/-30]2s
1
5
200*150*3.0
8
25
[45/-45]2s
1
5
190*160*3.5
8
6.25
[0]8
1
5
190*120*3.5
8
6.25
[0]8
1
5
190*80*3.5
8
6.25
[0]8
1
5
240*190*3.5
8
6.25
[0]8
1
5
240*190*3.5
8
25
[0]8
1
5
240*190*3.5
8
56.25
[0]8
1
5
240*190*3.5
8
6.25
[0]8
3
5
240*190*3.5
8
25
[0]8
3
5
240*190*3.5
8
56.25
[0]8
3
5
56
For a plate of size 240mm×190mm with 6.25% delamination, the size of delamination
adopted is 60 mm×47.5 mm.
4.5.2 Experimental set-up and procedure for buckling test
To obtain the experimental buckling result, the specimens were loaded in axial
compression using INSTRON 1195 machine of 100 KN capacity. The specimen was
clamped at two ends and kept free at the other two ends. A dial gauge was mounted at
the centre of the specimen to observe the lateral buckling deflection. All specimens
were loaded slowly until buckling took place. Clamped boundary conditions were
simulated along the top and bottom edges, restraining 2.5cm length. For axial loading,
the test specimens were placed between the two extremely stiff machine heads, of
which the lower one was fixed during the test, whereas the upper head was moved
downwards by servo hydraulic cylinder. All plates were loaded at constant cross-head
speed of 0.5mm/minute. The test set up was shown in Figure 4.11 (a & b). As the load
was increased the dial gauge needle started moving, and at the onset of buckling there
was a sudden large movement of the needle. The load v/s end shortening
(displacement) curve was plotted. The displacement is plotted on the X -axis and load
is plotted on the Y- axis. The load, at which the initial part of the curve deviated
linearity, was taken as the critical buckling load in line with previous studies.
Figure 4.11 (a): Composite plate before buckling
57
Figure 4.11 (a): Composite plate before buckling
Figure 4.11(b): Composite plate after buckling
58
CHAPTER 5
RESULTS AND DISCUSSION
5.1
Introduction
The present chapter deals with the determination of interlaminar shear
strength, natural frequency, buckling load and excitation frequency of composite
plates with delamination. The vibrations, buckling and parametric resonance
characteristics of delaminated composite plates are numerically studied by using the
formulation given in the Chapter 3. The influence of various parameters like
delamination size, number of layers, fiber orientation and aspect ratio on vibration,
buckling and parametric resonance characteristics of delaminated composite plates are
presented using numerical model. The experimental results on vibration and buckling
of industry driven woven fiber glass/epoxy delaminated composite panels are also
used to support the numerical predictions. The experimental static results involving
the effect of different parameters on interlaminar shear strength (ILSS) of delaminated
composite plates are also presented for completeness. The various studies made are
presented below.

Static analysis

Vibration analysis
i. Comparison with previous studies
ii. Numerical and experimental result


Static stability analysis
i.
Comparison with previous studies
ii.
Numerical and experimental result
Dynamic stability analysis
i.
Comparison with previous studies
ii.
Numerical result
59
5.2
Static analysis
Delamination is one of the most critical failure modes in composite laminates.
Interlaminar shear strength (ILSS) is an important parameter in determining the
ability of a composite material to resist delamination damage in laminates. Therefore,
there is a need for accurate prediction of the interlaminar shear strength of the
delaminated composites in order that they may be properly designed to overcome
failure. In the present investigation, ILSS of delaminated woven glass/epoxy
laminates are studied at different loading speeds and the results are presented.
The inter laminar shear strength of undelaminated ( 0 cm), 1 cm, 2.5 cm and
3.5 cm delaminated woven glass/epoxy composite laminates at different loading
speeds is presented in Table 5.1. The ILSS values of undelaminated specimen at 2,
50,100, 200 and 500 mm/minute. loading speeds were 27.93, 28.34, 28.67, 26.78 and
26.49 MPa; standard deviations (SD) were 0.55, 0.93, 1.05, 0.65 and 0.98 MPa and
coefficient of variation (CV) were 1.97%, 3.28%, 3.69%, 2.43% and
3.70%
respectively. For the 1 cm delaminated composite laminates the ILSS values at 2,
50,100, 200 and 500 mm/min. loading speed were 26.38, 27.64, 27.93, 25.41 and
24.98 MPa with the coefficient of variation of 2.35%, 1.88%, 1.40%, 2.84% and
2.88% respectively. The standard deviation of the 2.5 cm and 3.5 cm delaminated
composite laminates varied from 0.61 to 1.24 MPa and 0.37 to 1.28 MPa respectively.
This represents acceptable data correlation within the tests.
The figures 5.1 to 5.5 show the variations of changes-in-inter laminar shear strength
of delaminated woven glass/epoxy composite laminates as a function of delamination
length at different loading speeds. At 2 mm/minute loading speed, the ILSS value of
1cm delaminated glass/epoxy composite specimen (Figure 5.1) is found to be less
than the laminated specimen. But the ILSS value of 2.5cm delaminated composite
specimen is more than 1 cm delaminated laminate and 3.5cm delaminated specimen
has the least ILSS value. At 50, 100, 200 and 500 mm/minute loading speeds (Figure
5.2, 5.3, 5.4 & 5.5) the change in ILSS values of delaminated specimen gradually
decrease with the increase of delamination length. The discrepancy is only observed
at 2mm/minute loading speed. The probable reason for this happening may be that at
2mm/minute loading speed the interfacial bonding of 1cm delaminated specimen is
60
affected by the presence of Teflon ends, which may prematurely nucleate weakening
effect. Thus, the change in ILSS value is reduced. However, for 2.5cm delaminated
specimen, the increase in the ILSS value may be due to the increase in interfacial
bond strength between Teflon, fiber and polymer and subsequently there is an
increase in ILSS value. The reduction in ILSS values (averaged over all the loading
speeds, Table 5.2) for 1cm and 2.5cm delaminated specimens are 4.23% and 7.24%
respectively with respect to control and non significant whereas for 3.5cm
delaminated specimen the reduction is significant (12.37%). The present investigation
clearly indicates that the change in ILSS value of delaminated specimens gradually
decreases with the increase in delamination length.
Table 5.1: Mean, SD & CV in ILSS (MPa) value of glass/epoxy composite
laminates at different delamination lengths and loading speeds
Delamination Length(cm)
Loading speed (mm/min)
Control
(0 cm)
1 cm
2.5 cm
3.5 cm
27.93
0.55
1.97%
26.38
0.62
2.35%
27.67
1.24
4.48%
24.61
1.18
4.80%
ILSS
SD
CV
28.34
0.93
3.28%
27.64
0.52
1.88%
27.44
1.12
4.08%
26.51
1.28
4.83%
ILSS
SD
CV
28.67
1.05
3.69%
27.93
0.39
1.40%
24.10
0.61
2.53%
22.96
0.37
1.61%
ILSS
SD
CV
26.78
0.65
2.43%
25.41
0.72
2.84%
24.02
0.67
2.79%
23.01
0.40
1.74%
ILSS
SD
CV
26.49
0.98
3.70%
24.98
0.72
2.88%
24.96
0.83
3.32%
24.03
0.93
3.87%
2 mm
ILSS
SD
CV
50 mm
100 mm
200 mm
500 mm
Note: SD = Standard deviation, CV = Coefficient of variation
61
Table 5.2: Percentage reduction in ILSS (MPa) value of 1 cm, 2 cm & 3.5 cm
delaminated specimen
Loading speed (mm/minute)
Delaminated
specimen
%
2
50
100
200
500
Mean
0 cm
27.93
28.34
28.67
26.78
26.49
27.64
1 cm
26.38
27.64
27.93
25.41
24.98
26.47
4.23
2.5 cm
27.67
27.44
24.10
24.02
24.96
25.64
7.24
3.5 cm
24.61
26.51
22.96
23.01
24.03
24.22
12.37*
reduction
*indicates significant at 5% probability level
Figure 5.1: Variation of change in ILSS vs. delamination length of
glass/epoxy at 2 mm/minute loading speed
Figure 5.2: Variation of change in ILSS vs. delamination
length of glass/epoxy at 50 mm/minute loading
speed
62
Figure 5.3: Variation of change in ILSS vs. delamination length
of glass/epoxy at 100 mm/minute loading speed
Figure 5.4: Variation of change in ILSS vs. delamination
length of glass/epoxy at 200 mm/minute
loading speed
Figure 5.5: Variation of change in ILSS vs. delamination length of
glass/epoxy at 500 mm/minute loading speed
63
Scanning electron microscope test result
The SEM micrographs for glass/epoxy show (Figure 5.6 & 5.7) that matrix
cracking, fiber pull out, cohesive failure and interfacial cracking are dominating
failure modes that nucleate damage in fractured surface of delaminated composite
plates. The cleaner fibers and fiber breakage are prevalent in fractured surface of
laminated glass/epoxy composites.
Matrix cracking
Figure 5.6: Scanning micrograph showing matrix
cracking in laminated composites
Fiber pull out
Interfacial cracking
Figure 5.7: Scanning micrograph showing fiber
pullout and interfacial cracking in
delaminated composites.
64
5.3
Vibration analysis
Delamination in the composite plates, greatly affect the dynamic behavior of
structures. So in the present investigation, natural frequency of delaminated industry
driven woven fiber glass/epoxy composite plates were determined both numerically
and experimentally. The effects of various parameters like delamination area,
boundary conditions, fiber orientation, aspect ratio, number of layers and multiple
delaminations were studied critically. Numerical and experimental results are
presented for free vibration of delaminated composite plates after comparison with
previous investigations.
5.3.1
Comparison with previous study
Based on the finite element formulation and delamination modeling
mentioned in Chapter 3, programs are developed as per flow chart (given in appendix)
for numerical computations. To validate the programs, the results for free vibration of
laminated composite plate obtained by the present finite element formulation are
compared with the results of Ju et al. (1995). As shown in Table 5.3, it is observed
that there is an excellent agreement between two results.
Table 5.3: Comparison of frequency (Hz) for graphite/epoxy composite plates
with different boundary conditions
E11=132 GPa, E22=5.35 GPa, G12=2.79 GPa, í12= í13=0.291, í23=0.3, ñ=1446.20
kg/m3, a=b=0.25m, h=0.00212m No. of layers=8, Lay up = (0/90/45/90)2
Boundary Condition
Mode
Four sides simply
supported
1st
2nd
3rd
Results of Ju et
al. (1995)
164.370
404.380
492.290
658.400
346.590
651.510
781.060
1000.200
41.350
60.660
221.52
258.72
4th
Four sides clamped
1st
2nd
3rd
4th
Cantilever
1st
2nd
3rd
4th
65
Present FEM
result
163.651
400.918
494.141
650.089
342.543
635.641
766.589
963.542
41.162
60.520
220.461
257.709
Similarly the fundamental frequencies for a single delaminated composite
cantilever beam, based on the present delamination modelling are compared with the
analytical results of Shen and Grady (1992), FSDT results of Hu(1999) and HSDT of
Hu (2002).
The width of delamination is 12.7mm which is width of beam. As
observed in Table 5.4, there exists excellent agreement between the present FEM
results with literature.
Table 5.4:
Comparison of frequency of cantilever composite beams (127mm
 12.7mm  1.016mm) with different mid–plane delaminations
E11=132 GPa, E22 =10.3 GPa, G12=5.0 GPa,
ply orientation = ((0/90)2)s
í12= 0.33, ñ =1480 kg/m3,
Delamination
length
Analytical
(Shen and
Grady 1992)
FSDT
(Hu, 1999 )
HSDT
(Hu, 2002)
Present
FEM
Intact
82.042
-
-
82.13
25.4 mm
80.133
-
-
81.97
50.8mm
75.285
76.643
75.369
78.41
76.2mm
66.936
-
-
64.55
Determination of material constants
The composite laminates of eight layers are fabricated to evaluate the material
constants. Tensile tests on samples are performed following the procedure described in
ASTM D2309/ D2309M (2008) standard and the characteristics of woven fiber
glass/epoxy composite plate used for numerical study are presented in Table 5.5.
Table 5.5:
Material properties of plates used for vibration
Lay-up
N
E1(GPa)
E2(GPa)
E45(GPa)
G12(GPa)
í12
ñ(kg/m3)
WR
8
7.7
7.7
7.04
2.81
0.17
1661.25
N : - Number of layers, E45 :- Tensile modulus obtained in 45° tensile test
E1, E2 :- Elastic modulus in longitudinal (1) and transverse direction(2) respectively.
G12 :- In-plane shear modulus ,
:- Poisson’s ratio
í12
ñ :- Density
66
5.3.2 Numerical and experimental results
In the present investigation, both the numerical computation and experimental
study are carried out for an eight-layered (0/90)4 woven roving glass/epoxy composite
plate. The geometrical dimensions considered for the woven roving composite plates
are: length, a = width, b = 0.24m, thickness, h = 0.003m. The material properties of
the woven roving glass/epoxy composite plates are considered as given in Table 5.5.
Square size delamination was introduced at the mid-plane as shown in Figure 5.8. In
this study, the effects of delamination area, boundary conditions, fiber orientations,
number of layers and aspect ratio on the natural frequencies are investigated.
Y, v0
Square size delamination
h
Z, w0
b

a
X, u0
Figure 5.8: Laminated composite plate with mid-plane delamination
5.3.2.1 Effects of delamination area
To study the effects of delamination area on the natural frequencies of an eight
layered delaminated plate, square size mid-plane delaminations were introduced at
6.25%, 25% and 56.25% of total plate area. The fundamental natural frequency of the
delaminated (0/90)4 plate is depicted in Figure 5.9 as a function of delamination area
for a cantilever specimen. The result for natural frequency obtained from numerical
analysis is found to be in a good agreement with the experimental result. The
experimental fundamental frequencies of 6.25%, 25% and 56.25% delaminated plates
67
are found to decrease by 10%, 14% and 22 % respectively as compared to the
laminated plate. This may be due to reduction in stiffness of the laminates.
Figure 5.9: Variation of fundamental natural frequency with
delamination area of woven fiber cantilever composite
plates
The same study was extended to the composite plates with four sides
clamped and four sides simply supported boundary conditions, the results of which
are presented in Figure 5.10 & 5.11, respectively. The numerical results showed a
good agreement with the experimental results for both clamped and simply supported
boundary conditions. The experimental fundamental natural frequencies of 6.25%,
25% and 56.25% delaminated plates are found to decrease by 14%, 19.5% and 35 %
for four sides simply supported boundary condition and 5.23%, 22% and 32% for four
sides clamped condition as compared to the intact plate. This result reveals that at
low delamination area (6.25%) the natural frequency of four sides clamped boundary
condition is least affected as compared to cantilever boundary condition and four
sides simply supported condition.
68
Figure 5.10: Variation of natural frequency with delamination area of
four sides clamped woven fiber composite plates
Figure 5.11: Variation of natural frequency with delamination area of
four sides simply supported woven fiber composite plate
At high delamination area (56.25%) the natural frequency is least affected for
cantilever boundary condition as compared to other two boundary conditions. But
relatively large delamination area has considerable effect on the fundamental natural
frequency of all the three boundary conditions. From the present investigation, it is
noticed that the natural frequency decreases in general with the increase of
delamination area invariably for all the three boundary conditions.
69
5.3.2.2 Effects of boundary condition
To investigate the influence of boundary conditions on natural frequencies of
delaminated plates, three types of boundary conditions are considered, namely, S-SS-S(four edges simply supported),where u=w=èy=0, at x=0, a and v=w=èx=0, at x=0,
a; C-C-C-C (four edges clamped), where u= v=w= èx= èy=0, at x=0, a and y=0, b; CF-F-F (cantilever), where u= v=w= èx= èy=0, at x=0. The specimen taken for the
study was of eight layered composite plate having stacking sequence of (0/90)4 with
25% of delamination area.
Natural frequencies of 25% delaminated composite plates for experimental
and numerical results under different boundary conditions are given in Table 5.6.
From this Table it is observed that the numerical and experimental results are in good
agreement for all the boundary conditions. The 1st, 2nd and 3rd mode natural
frequencies are found to be the least (21.5, 47.0 & 135.7 Hz) for C-F-F-F (cantilever)
condition and the highest (145.0, 285.0 & 450.0 Hz) for C-C-C-C (four sides
clamped) condition. The experimental fundamental natural frequency of 25%
delaminated plate with four sides simply supported and cantilever boundary
conditions are decreased by 44.48% and 85.17% respectively with respect to four
sides clamped condition.
Table 5.6:
Natural frequencies (Hz) of experimental and FEM results for
25% delaminated plate at different boundary conditions
Experimental result
1
2nd
3rd
mode
mode
mode
Four sides
clamped
145.0
285.0
Four sides
simply
supported
80.5
21.5
Boundary
conditions
Cantilever
1
mode
FEM result
2nd
mode
3rd
mode
450.0
156.4
295.2
471.2
186.0
330.0
79.9
200.6
349.4
47.0
135.7
22.7
48.5
128.4
st
70
st
This experimental result implies that the natural frequencies of delaminated
plates are greatly dependent on the boundary conditions, i.e. the more strongly the
plate is restrained, the greater is the effect of the delamination on the natural
frequencies for all the cases.
5.3.2.3 Effects of fiber orientations
In order to know the effect of fiber orientations on natural frequencies of 25%
delaminated plate (8-layers), three types of fiber orientations i.e. [(0/90)2]s, [ (30/30)2]s, [ (45/-45)2]s. are considered. In this study the changes in the natural frequency
as a function of fiber orientation are presented in Figure 5.12 for cantilever boundary
condition. The results obtained from free vibration of the plates of both experimental
and present FEM analysis are in good agreement. From Figure 5.12, it is observed
that the experimental fundamental natural frequency of 25% delaminated plate with
30° and 45° orientation is decreased by 2.32% and 9.30% respectively with respect to
the 0° orientation. This reveals that the fundamental natural frequency of delaminated
plate decreases with the increase in fiber orientation but the decrease in the
fundamental natural frequency is not conspicuous.
Figure 5.12: Variation of natural frequency with fiber orientation for
25% delaminated woven fiber cantilever composite plate
5.3.2.4 Effects of number of layers of laminate
To examine the effects of number of layers on natural frequency of 25%
delaminated [(0/90)4] plate, three different types of laminate are fabricated, i.e. made
71
up of 4, 6 and 8 layers, respectively . All the geometrical and material properties of
the laminates are same as given in Table 5.5 except the density. The density of the
laminates taken for the study was 1402 kg/m3 and 1599 kg/m3 for 4 layers and 6
layers respectively. The natural frequencies for free vibration as obtained from
experimental study and numerical analysis for cantilever boundary condition show a
good agreement as shown in Figure 5.13. The change in natural frequencies as a
function of number of layers as depicted in Figure 5.13 reveals that the fundamental
natural frequency of delaminated composite plate increases with the increase in
number of layers. The increase in the experimental fundamental natural frequency of
25% delaminated plate is 29% and 79% for 6 layers and 8 layers laminate respectively
as compared to a 4 layered laminate. This result indicates that relatively more number
of layers have considerable positive effect on the fundamental natural frequency of
delaminated composite plate.
Figure 5.13: Variation of natural frequency with number of layers
for 25% delaminated woven fiber composite plate
5.3.2.5
Effects of aspect ratio
To investigate the influence of aspect ratio on natural frequencies of an eight
layered 25% delaminated (0/90)4 plate, three different types of aspect ratios i.e. a/b=
1.0, 1.5 and 2.0 are considered. For different aspect ratios, the plate dimension varied,
keeping the thickness of the plate (h=0.003m) unchanged. For the aspect ratio of 1.0,
a=0.24m and b=0.24m; for 1.5, a=0.24m and b=0.16m; for 2.0, a=0.24m and
b=0.12m. The variation in the fundamental natural frequencies as a function of aspect
72
ratio is given in Figure 5.14 for cantilever boundary condition. A good agreement is
observed between numerical and experimental results. The experimental fundamental
natural frequency of 25% delaminated plate with 1.5 & 2.0 aspect ratio is found to
increase by 2 times and 4 times respectively as compared to 1.0 aspect ratio. This
indicates that increase in the aspect ratio increases the natural frequency of a
delaminated composite plate.
Figure 5.14: Variation of natural frequency with aspect ratio for 25%
delaminated woven fiber cantilever composite plates
5.3.2.6 Effects of multiple delaminations
To investigate the effects of multiple delaminations, three types of composites
plates with delaminations are considered. Each plate consists of eight layers with a
stacking sequence of (0/90)4.The delamination is presumed to be located at 2nd , 4th
and 6th layer. The percentage of delamination size is of 6.25%, 25% and 56.25%. The
results are found out numerically and are given in Table 5.7 for cantilever boundary
condition and four sides clamped boundary conditions.
The variation of natural frequencies with increased percentage of delimitation
area for multiple delaminated clamped and cantilever composite plates is shown
graphically in Figure 5.15. It is noted that there is a decrease in fundamental natural
frequency of multiple delaminated plates with the increase in delamination area as
73
compared to single delaminated plate for both the boundary conditions. At 6.25 %
delamination area the decrease is quite more (28%) for cantilever boundary condition
as compared to clamped boundary condition. But at higher delamination area the
decrease is more or less same for both the boundary conditions. From the present
numerical results (Table 5.7) it is also observed that, in comparison to single
delamination, the fundamental frequencies of vibration of delaminated composite
plates reduce significantly with multiple delamination.
Table 5.7:
Variation of natural frequency for delaminated clamped and
cantilever composite plates with different % of delamination area
Clamped BC
(natural frequency in Hz)
% of
delamination
area
mode
Control (0)
6.25
25
56.25
Cantilever BC
(natural frequency in Hz)
Multiple
delamination
% reduction in
multiple
delamination
w.r.t
control
25.1
25.1
--
390.37
54.8
54.8
576.82
576.82
149.36
149.36
1st
174.38
166.74
24.63
18.26
2nd
373.18
358.70
53.15
43.719
3rd
556.59
550.35
144.62
107.52
1st
156.38
133.0
22.65
15.92
2nd
295.22
239.25
48.47
38.20
3rd
471.24
364.19
128.42
88.06
1st
131.64
81.27
18.55
11.38
2nd
258.96
166.45
40.95
29.06
3rd
388.35
265.65
107.09
60.79
Single
delamination
Multiple
delamination
1st
190.54
190.54
2nd
390.37
3rd
% reduction in
multiple
Single
delamination
delamination
w.r.t
control
--
12.0
30.0
57.0
74
28.0
36.0
56.0
Figure 5.15: Variation of natural frequency of multiple delaminated clamped
and cantilever plates with different percentage of delamination area
5.3.3 Pulse report
The Natural frequencies of the free vibration analysis are found out
experimentally by using pulse software. Typical pulse reports for the delaminated
composite plate (for a/b ratio 1.5, and 25% of delamination) are shown in Figure 5.16 to
Figure 5.18. The peaks of the FRF shown in Figure 5.16 give the different natural
frequencies of vibration. The coherence shown in Figure 5.17 gives indication of the
accuracy of measurement. The variation of the applied force with time (Figure 5.18) gives
indication of the magnitude of applied force with time and also number of hits.
Figure 5.16: Frequency response function spectrum (In X-axis: Frequency in
Hz, In Y-axis: Acceleration per force (m/s2) ∕ N)
75
Coherence (response, Force)-input
Working: Input: Input: FFT Analyzer
Figure 5.17: Coherence ( Response, Force)
Figure 5.18: Applied force Vs time curve (In X-axis: Time
in‘s’, In Y-axis: force in N)
76
5.4
Buckling/static stability analysis
The presence of delamination may significantly reduce the stiffness and
strength of the structures and may affect some design parameters such as the buckling
strength of the structure. So in the present investigation the critical buckling load of
delaminated glass/epoxy composite plates were determined both numerically and
experimentally. The effects of various parameters like delamination area, boundary
conditions, fiber orientations, aspect ratio, number of layers and multiple
delaminations were studied critically. Numerical and experimental results are
presented for buckling load of delaminated composite plates after comparison with
previous investigations.
5.4.1 Comparison with previous study
Based on the finite element formulation and delamination modeling, programs
are developed for numerical computations. To validate the programs, the results for
buckling of laminated composite plate, obtained by the present finite element
formulation are compared with the results of Baba (2007) for clamped-free-clampedfree boundary condition. The rectangular plate had eight layers of E-glass/epoxy
composites. As shown in Table 5.8, it is observed that there is an excellent agreement
between two results.
Table 5.8:
Comparison of buckling load (Newton) for laminated C-F-C-F
composite plates
E1 =39.0 GPa, E2=E3=8.2 GPa, G12=G13=G23=2.9 GPa , í12= í23= í31=0.29,
Length=150 mm, width=25 mm, thickness=1.5 mm
Fiber orientation
Baba (2007)
Present FEM
% error
[0]8
482.42
481.71
0.14%
[90]8
106.33
101.42
4%
[0/90]2s
366.52
364.78
0.47%
[(0/90)2]as
290.22
287.46
0.96%
77
The buckling loads of delaminated composite plate are computed using
present formulation and are also compared with those results published by Pekbey
and Sayman (2006) for clamped-free-clamped-free boundary condition. The results
obtained from both are presented in Table 5.9. The Comparison results show that
there exist an excellent agreement between the present FEM and the previously
published results of other investigators.
Table 5.9:
Comparison of buckling load (Newton/mm) for delaminated C-FC-F composite plates
E1 =27.0 GPa, E2 =21.5 GPa, G12 =7.5 GPa, í12=0.15
Dimension of
plate in mm
Fiber orientation
Pekbay and
Sayman(2006)
Present FEM
200 x 160 x 1.7
[0]8
6.42
6.919
200 x 160 x 1.
[30/-30]2s
5.64
6.251
200 x 160 x 1.7
[45/-45]2s
5.38
5.795
5.4.2
Experimental and numerical results
After validating the formulation with the existing literature, both the
experimental and numerical results for non-dimensional buckling load (as per Table
5.13) of delaminated composite plates are carried out for the eight-layered (0/90)4
woven roving glass/epoxy composite plate. The material properties (determined by
tensile testing of specimens) of the woven roving composite plates which are used for
numerical studies are presented in Table 5.10. Square size delamination was provided
at the mid-plane. In this study, the effects of delamination area, fiber orientations,
number of layers, aspect ratio and multiple delamination on the critical buckling load
are investigated under clamped free clamped free boundary condition.
78
Table 5.10:
Material properties of the plate for buckling analysis
Lay-up
N
E1 (GPa)
E2 (GPa)
E45 (GPa)
G12 (GPa)
í12
ñ (kg/m3)
WR
8
7.4
7.4
5.87
2.18
0.17
1661.25
5.4.2.1 Effects of delamination area
The variation of non-dimensional buckling load of single delaminated
composite plate (0/90)4 with increasing delamination area is shown in Figure 5.19. To
study the effects of delamination area on the critical buckling loads of delaminated
plates, mid-plane delaminations were introduced at 6.25%, 25% and 56.25% of total
plate area of an eight layered laminated plate. The size of plate is taken as 240mm
x190mm x3.5mm. Five identical specimens for each specimen design are tested to get
an average buckling load. The non-dimensional buckling loads of the delaminated
plates are depicted in Figure 5.19 as a function of delamination area for clamped-freeclamped-free boundary condition (B.C). The result for critical buckling loads obtained
from numerical analysis is found to be in a good agreement with the experimental
result. The experimental critical buckling load of 6.25%, 25.0% and 56.25%
delaminated plates decreased by 11%, 28% and 52% respectively from the plate
without delamination (Table 5.11). From this study it is observed that for increasing
percentage of delamination area there is a decrease in non-dimensional buckling load
because of the reduction of stiffness.
Table 5.11:
Variation of buckling load (KN) of delaminated CFCF composite
plates
% of
delamination
Present FEM
results
Experimental
result
% reduction
0
7.079
7
-
6.25
6.238
6.75
11.0
25
5.079
5.75
28.0
56.25
3.358
3.9
52.0
79
Figure 5.19: Variation of non-dimensional buckling load of single
delaminated CFCF composite plate with increasing
delamination area
5.4.2 .2 Effects of fiber orientations
Fiber orientation angle is the main parameter for controlling buckling load
capacity of composite plates. To investigate the effect of fiber orientations on nondimensional buckling load of 25% delaminated plate, three types of fiber orientations
are considered. i.e. [0]8, [(30/-30)2]s, [(45/-45)2]s . To orient the fiber in a specified
angle, the fabric is initially given a rotation equal to ply orientation and cut to the
desired size. The size of plate is taken as 200mm x150mm x3.0mm. The variation of
non-dimensional buckling load as a function of fiber orientation for both experiment and
numerical solution is presented in Figure 5.20. From this figure it is shown that there is
good agreement between experimental and FEM results for 25% delaminated plate. It is
observed that as the fiber orientation increases, the non-dimensional buckling load
decreases for 25% delaminated plate. It is greatest for fiber orientation angle 0°. The
experimental critical buckling load of 25% delaminated plate with 30° and 45° orientation
is decreased by 8% and 19% with respect to 0° orientation. So, 25% delaminated plate
has a more reliable buckling load when it has [0]8 fiber orientation. It may be observed
that, the plates yield highest stability resistance when fibers are aligned along the load
direction. From the above results, it is understood that the fiber orientation of the lamina
may be used as an index for quality control and a safety factor for the laminated
composites.
80
Figure 5.20: Variation of non-dimensional buckling load with fiber
orientation for 25% single delaminated woven roving
composite plate
5.4.2.3
Effects of number of layers of laminate
To examine the effects of number of layers on 25% delaminated plate with
single mid-plane delamination, three different types of laminate are fabricated, which
are made up of 8, 12 and 16 layers. The stacking sequence of each plate are [0]8, [0]12,
[0]16. Variation of non-dimensional buckling load with number of layers of composite
laminate with 25% delamination (both numerical and experimental results) is
graphically presented in Figure 5.21. It is shown that there is good agreement between
experimental and FEM results for 25% delaminated plate under clamped-freeclamped-free boundary condition. From Figure 5.21, it is observed that as the number
of layers increases the non-dimensional buckling load also increases both
experimentally and numerically. The increase in the experimental buckling load of
delaminated plate is 2.1 times (108.7%) for 12 layers and 5.91 times (491.3%) for 16
layered laminate as compared to an 8 layered laminate. From this study, it is evident
that relatively more number of layers may increase the stability of delaminated
composite structures. This result clearly indicates that number of layers had
tremendous positive effect on the non-dimensional buckling load of delaminated plate.
So this parameter must be considered with due emphasis for safety factor of laminated
composite plates with delaminations.
81
Figure 5.21: Variation of non-dimensional buckling load of 25%
single delaminated CFCF cross ply plate with number of
layers
5.4.2.4
Effects of aspect ratio
To investigate the influence of aspect ratio on non-dimensional buckling load
of an eight layered 6.25% single delaminated (0/90)4 plate, four different types of
aspect ratios i.e. a/b = 0.79 (a =190mm and b =240mm), 1.18 (a =190mm and
b=160mm), 1.58 (a =190mm and b =120mm) and 2.38 (a = 190mm and b = 80mm)
are considered. For different aspect ratios, the plate dimension varied, keeping the
thickness of the plate (h=0.0035m) unchanged. The variation of non-dimensional
buckling load as a function of aspect ratio is given in Figure 5.22. It is observed that
buckling load of delaminated composite plate decreases with the increase in aspect
ratio. A good agreement is observed between numerical and experimental results. By
increasing the aspect ratio from 0.79 to 1.18, 1.58 & 2.38, the critical buckling load of
delaminated plate is found to be decreased by 33.3%, 44.4% and 63.0% respectively.
This indicates that increase in the aspect ratio decreases the non-dimensional buckling
load of a delaminated plate.
82
Figure-5.22: Variation of non-dimensional buckling load with different aspect
ratios of 6.25% delaminated CFCF woven fiber composite plate
5.4.2.5 Effects of multiple delaminations
To investigate the effect of multiple delaminations, three types of composites
plates with delaminations are considered. Each plate consists of eight layers with a
stacking sequence of [0]8. The size of plate is taken as 240mm x190mm x3.5mm.The
delamination is located at midplane, 2nd and 6th layer. The percentage of delamination is
6.25%, 25% and 56.25%. The results are found out numerically and experimentally.
Numerical and experimental results of buckling analysis of composite plate with different
% of delamination and without delamination for multiple delaminated composite plates
with clamped-free-clamped-free boundary condition are presented in Table 5.12.
Table 5.12: Comparison of numerical and experimental results of buckling load
(KN) of delaminated composite plates
% of
delamination
area
Present FEM result
Multiple
Single
delaminated
delaminated
plate
plate
Experimental result
Multiple
Single
delaminated
delaminated
plate
plate
0
7.079
7.079
7
7.0
6.25
5.672
6.24
6.4
6.75
25
2.746
5.08
3.25
5.75
56.25
1.155
3.36
1.5
5.9
83
The variation of non-dimensional buckling load of multiple delaminated
plates as a function of delamination area is depicted in Figure 5.23 for clamped-freeclamped-free boundary condition.
Figure 5.23 shows good agreement between
experimental and FEM result for multiple delaminated composite plate. From Figure
5.23 and Table 5.12, it can be seen that the non-dimensional buckling load of multiple
delaminated plate decreases as percentage of delamination increases both numerically
and experimentally. Also with the increase in the number of delamination from one to
three, there is significant decrease in critical buckling load because of the reduction of
stiffness. The decrease in critical buckling load is 9%, 45% and 65% from single to
multiple delaminated plates with 6.25%, 25% and 56.25% delamination area
respectively. So the increase in number and size of delaminations has in general, a
deteriorating effect on the stiffness of the plate.
Figure 5.23: Variation of nondimensional buckling load of multiple
delaminated CFCF plate with increasing delamination area
5.4.2.6
Effects of boundary conditions
To investigate the influence of boundary conditions on non-dimensional buckling
load of delaminated composite plates, three types of boundary conditions i.e. all sides
simply supported (SSSS), all sides clamped (CCCC) and two sides clamped two sides
free (CFCF) are assumed for numerical analysis. The size of the composite plate assumed
is (240x190x3.5) mm3 with 0%, 6.25%, 25% and 56.25% of delamination and stacking
sequence is [0]8. From the Figure 5.24 it is evident that the non-dimensional buckling
load of 6.25%, 25% and 56.25% delaminated plates along with the laminated plate (0%
84
delamination) are highest under all sides clamped boundary condition followed by CFCF
boundary condition and the lowest load is observed under simply supported boundary
condition. This happens because of the rigidity of the clamped boundary condition as
compared to simply supported boundary condition. The critical buckling load of 6.25%,
25% and 56.25% delaminated plates under CCCC boundary condition decreased by 23%,
44% and 63% respectively as compared to laminated plate. In case of CFCF boundary
condition the reductions are 11%, 28% and 52% and for simply supported boundary
condition the reductions are 19%, 43% and 58% respectively as compared to laminated
plate. These results clearly indicate that boundary conditions have immense influence on
non-dimensional buckling load of delaminated plates.
Figure 5.24: Variation of Non dimensional buckling load with different %
of delamination for different boundary conditions (BC)
5.4.3 Typical experimental determination of critical buckling load from load
v/s end shortening displacement graph
The critical buckling load from buckling analysis of delaminated composite
plates is found experimentally by load v/s end shortening displacement graph. The
load is plotted in y-axis and end shortening displacement in x-axis. For the
determination of critical buckling load, the point where left from the straight line is
determined on the graphics and the value of this point on the y axis is called as the
critical buckling load. Determination of critical buckling load of a plate with 25% and
56.25% delamination from load v/s end shortening displacement graph is shown in
Figure 5.25 and 5.26 respectively.
85
Load in KN
Displacement in mm
Load in KN
Figure 5.25: Determination of critical buckling load of a plate with 25%
delamination from load v/s end shortening displacement graph
Displacement in mm
Figure 5.26: Determination of critical buckling load of a plate with 56.25%
delamination from load v/s end shortening displacement graph
86
5.5
Dynamic stability analysis
Dynamic stability analysis is an integral part of most engineering structures.
Delaminations reduce the stiffness of the plates. It is therefore important to
understand the performance of delaminated composites in a dynamic environment.
The subject of predicting the dynamic stability of delaminated structures has thus
attracted considerable attention. The dynamic stability of plate structures with
delamination is presented by using FEM formulation mentioned in chapter 3. The
instability regions are determined for composite plates with and without
delaminations. A detailed parametric investigation is carried out to study the influence
of delamination area, number of layers, degree of orthotropy, aspect ratio and static in
plane load on the parametric resonance characteristics of delaminated cross ply plates.
Boundary conditions
Numerical results are presented for delaminated composite plates with
different combination of boundary conditions. Further descriptions of boundary
conditions are as follows:

Simply supported boundary
v=w=

=0 at x=0, a and u=w=
Clamped boundary
u=v=w=

=0 at y=0, b
=
= 0 at x=0, a and y=0, b
Free edges
Non-dimensionalisation of parameters
Table 5.13 shows the non-dimensional parameters used for vibration,
buckling and excitation frequency considered for dynamic stability analysis with the
reference to Bert and Birman(1988).
87
Table 5.13: Non-dimensional parameters of composite plates
No
parameter
1
Frequency of vibration ( )
2
Buckling load ( )
3
Frequency of excitation ( )
Where
and
Composite plates
a 2  / h 2 E 2
Nx
b2
E2 h 3
 a 2  / h 2 E2
are in radian.
5.5.1 Comparison with previous study
The numerical validation of the governing equation is performed by solving
the corresponding free vibration and buckling eigenvalue problems. The natural
frequency and critical buckling load results are compared with the results available in
the existing literature. To validate the program, the natural frequency for mid-plane
delaminated plate is already compared with the result by Shen and Grady (1992) as
shown in Table 5.4. The results on buckling with different delamination length of
cross-ply composite plates due to dynamic load is compared with results by Radu and
Chattopadhyay (2002) using higher order shear deformation theory and are shown in
Table 5.14. It is observed that there is good agreement between the two results.
Table 5.14:
Comparison of buckling load for different mid plane delamination
length of the cantilever rectangular plates
E11 =134.4 GPa, E22 =10.34 GPa, G12 = G13 =4.999 GPa, í12=0.33, ñ=1600 kg/m3
,a=127 mm, b=12.7 mm, h=1.016 mm, stacking sequence = (0/90)2s, a/h=125
Delamination length
(mm)
Critical buckling load (N)
Radu & Chattopadhyay (2002)
Present FEM
0
16.336
16.3296
25.4
16.068
15.8292
50.8
15.054
14.9085
88
After validation of the present formulation, investigation is performed on the
dynamic instability characteristics of the composite plates. The present study is
carried out considering graphite/epoxy rectangular plates made out of eight identical
plies with material properties: E11 =134.4 GPa, E22 = 10.34 GPa, G12 = G13 =4.999
GPa, í12=0.25, ñ=1600 kg/m3 ; length a= 127 mm, width b=12.7 mm, thickness
t=1.016 mm, stacking sequence= (0/90/0/90/90/0/90/0). Where E11 and E22 are
Young’s modulus, G12 and G13 are Shear modulus and í12 is Poisson’s ratio. The
effects of various parameters like delamination area, number of layers, degree of
orthotropic, static load factor, length-thickness ratio and aspect ratio on the dynamic
instability characteristics are studied.
In this study, the boundary condition in one of the short edge is fixed and
opposite edge is loaded with dynamic buckling force. The non-dimensional excitation
frequency
Where
(=
a2
) is used throughout the dynamic instability studies.
is the excitation frequency in radian /second. Instability regions are plotted
in the plane having non-dimensional excitation frequency as abscissa and dynamic
load factor as ordinate.
5.5.2
Numerical results for dynamic stability
5.5.2.1
Effects of delamination size
To study the effect of delamination size on the dynamic instability region (DIR),
single mid-plane delaminated graphite/epoxy plates with different delamination sizes
like 0%, 6.25%, 12.5% and 25% of total plate area are considered. Taking the lower
and upper boundary of primary instability region the graph is depicted in Figure 5.27
for length to thickness ratio L/t =125. From the Figure it is observed that the onset of
instability occurs earlier with the increase in delamination size from 0% to 6.25%. But
the introduction of 6.25% delamination has no remarkable effect in lowering the nondimensional excitation frequency. Further increase in delamination from 6.25% to
12.5% and from 12.5% to 25%, the instability region is found to shift to a lower and
lower excitation frequency. This result reveals that with the increase in delamination
size, the non-dimensional excitation frequency decreases and therefore the dynamic
instability regions (DIR) are shifted to lower excitation frequencies.
89
Figure 5.27: Variation of instability region of [(0/90)2]s cross- ply plate
with different percentage of delamination for L/t =125
5.5.2.2 Effects of number of layers
The variation of instability region of 2-layer (0/90) delaminated (0%, 6.25%,
12.5% and 25%) composite plates with length to thickness ratio L/t =125 is shown in
Figure 5.28. From the Figure it is observed that, the dynamic instability occurs at
3.238, 3.214, 3.147 and 2.962 non-dimensional excitation frequency for 0%, 6.25%,
12.5% and 25% delaminated plates respectively. Instability occurs later for the
laminated plate than the delaminated plates. Up to 12.5% delamination the decrease in
excitation frequency is not so conspicuous like 25% delamination.
Figure 5.28: Variation of instability region of 2-layer (0/90) composite
plate with different percentage of delamination
90
The variation of instability region of 4-layers (0/90)s composite plate with
different percentage of delamination (0%, 6.25%, 12.5% and 25% ) is shown in
Figure 5.29. Dynamic instability regions (DIR) are plotted for 4- layers cross-ply
rectangular plate with length to thickness ratio L/t =125. It is noted that the dynamic
instability of 6.25%, 12.5% and 25% delaminated plates having 4-layers starts at
4.396, 4.211 and 3.974 instead of at 3.214, 3.147 and 2.962 non-dimensional
excitation frequency as in case of 2-layer composite plates with 6.25%, 12.5% and
25% delamination respectively. The results reveal that by increasing the number of
layers of the delaminated plates the instability regions occur at higher excitation
frequencies. The observed behavior is attributed due to the effect of bendingstretching coupling for the case of delaminated composite plates. This indicates that
the delaminated plates with more number of layers may impart better structural
stability and safety.
Figure 5.29: Variation of instability region of 4-layer (0/90)s composite
plate with different percentage of delamination
5.5.2.3 Effects of degree of orthotropy
The effects of degree of orthotropy on dynamic instability region (DIR) of
single mid-plane delaminated graphite/epoxy plates with delamination sizes 0%,
6.25%and 25% is studied for E11/E22=40 and 20 keeping other material parameters
constant. Figure 5.30 shows the variation of instability region for the degree of
orthotropy, E11/E22 = 40 of composite plate with different percentage of
delamination. For the intact plate (0% delamination) instability starts at a higher
91
excitation frequency. The introduction of 6.25% delamination in laminated plate
induces instability at a lower excitation frequency. Further increase in delamination
from 6.25% to 25% causes instability earlier. This indicates that the dynamic
instability occurs earlier for 25% delaminated plates than 6.25% delaminated plates.
Figure 5.30: Variation of instability region for the degree of
orthotropy (E11/E22 =40) of composite plate with
different percentage of delamination
Variation of instability region for the degree of orthotropy (E11/E22 =20) of
composite plate with different percentage of delamination is shown in Figure 5.31. As
expected, the onset of instability of composite plates with the degree of orthotropy
E11/E22 = 20 occurs earlier with the introduction of delamination from 0% to 6.25%.
With further increase of delamination from 6.25% to 25%, the excitation frequency
reduces significantly. Comparison of Figure 5.30 and Figure 5.31 reveals that the
effect of delamination is more pronounced for the composite plates with lower degree
of orthotropy (E11/E22 = 20) than the plates with higher degree of orthotropy (E11/E22 =
40). These results indicate that with the increase in degree of orthotropy of
delaminated plates, the excitation frequency increases and thus the instability regions
shifted to a higher frequency due to increase in stiffness.
92
Figure 5.31: Variation of instability region for the degree of
orthotropy (E11/E22 =20) of composite plate with
different percentage of delamination
5.5.2.4 Effects of aspect ratio
Figure 5.32 shows the variation of instability region of 0% delaminated
simply supported cross ply plate (L/t =10, E11/E22= 25) for different aspect ratio (a/b =
0.5, 1.0 and 1.5) on the dynamic instability characteristics. For the laminated plate
with a/b = 0.5, the instability occurs at a lower non-dimensional frequency (11.069).
By increasing the aspect ratio from 0.5 to 1.0 and from 1.0 to 1.5 the instability
regions are shifted to a higher and higher excitation frequency.
Figure 5.32: Variation of instability region of 0% delaminated
cross ply plate with different aspect ratio
93
The variation of instability region of 25% delaminated simply supported cross
ply plates (L/t =10, E11/E22= 25) with different aspect ratio is shown in Figure 5.33. In
case of 25% delaminated plate, the onset of dynamic instability occurs much later for
a/b = 1.5 and earlier for a/b = 0.5 and the width of the instability region increases
from a/b = 0.5 to a/b = 1.5. The comparative study of both the Figures 5.32 & 5.33
reveals that with the introduction of 25% delamination in the intact plates with a/b =
0.5, a/b = 1.0 and a/b = 1.5 the excitation frequency is found to shift from 11.069 to
9.992, from 15.062 to 14.225 and from 26.931 to 23.794 respectively. This indicates
that in the presence of delamination in a composite laminate for a particular aspect
ratio, instability occurs at a lower excitation frequency than the intact plate and by
increasing the aspect ratio of the delaminated plates the region of instability may be
shifted to a higher excitation frequency.
Figure 5.33: Variation of instability region of 25% delaminated
cross ply plate with different aspect ratio
5.5.2.5 Effects of static loads
The effect of static component of load (for á = 0.0, 0.2 and 0.4) on the
dynamic instability region of a 6.25% delaminated cross ply rectangular plate with
clamped-free-clamped-free boundary condition is demonstrated in Figure 5.34. The
instability occurs earlier for higher static load factor (á = 0.4) and later for lower static
94
load factor (á = 0.0). This study reveals that with the increase in static in-plane load,
the lowest natural frequency decreases and the instability regions occur at a relatively
lower excitation frequency. The width of the instability zone also increases with the
increase in static in-plane load.
Figure 5.34: Variation of instability region of 6.25% delaminated
rectangular plate with different static load factor
95
CHAPTER 6
CONCLUSION
6.1
Introduction
The present work deals with the study of the vibration, buckling and parametric
resonance characteristics of delaminated composite plates. The formulation is based on
the first order shear deformation theory. A finite element procedure using an eight-node
isoparametric quadratic plate element is employed in the present analysis with five
degrees of freedom per node. The development of regions of instability arises from
Floquet’s theory developed by Bolotin (1964) and the boundaries of the primary
instability regions have been determined to study the effect of various parameters on the
dynamic instability regions of the delaminated composite plates.
Results are presented for the interlaminar shear strength (ILSS), free vibration,
buckling and parametric resonance characteristics of delaminated composite plates.
The effects of various geometrical parameters like delamination size, boundary
conditions, number of layers, fiber orientations, aspect ratios, degree of orthotropy
and static load factor on the free vibration and stability characteristics of delaminated
composite plates have been analysed. The conclusions drawn in respect of different
studies are presented below.
6.2
Static analysis
Effects of different delamination lengths on interlaminar shear strength of
woven fiber glass/epoxy woven fiber composite plates at different cross head
velocities are investigated experimentally. Three point bending tests are also
conducted to assess the interlaminar shear strength of the laminates. Following
conclusions are drawn from the present research work.

The ILSS decreases with the increase in delamination length.

The ILSS of delaminated glass/ epoxy composite plates decreases with
the increase of crosshead velocities unlike the plates without
delamination.
96

SEM test result reveals that the variations of delamination length and
crosshead velocity might have strong influence in changing the
predominating mode of failure of glass/epoxy laminates.
6.3
Vibration analysis
The main thrust of the present study is the modal analysis of delaminated
composite plates, because of its inherent link with stability analysis. The effect of
different parameters on the frequencies of vibration are observed and compared with
numerical prediction using FEM. The conclusions are summarized as given below.

There is a good agreement between the experimental and numerical
results.

The frequencies of vibration decrease with introduction and further
increase of size of delamination in woven fiber composite plates.

Numerical and experimental results show that the effect of delaminations on
the modal parameters of delaminated composite plates is dependent not only
on the size but also on the boundary conditions i.e. the more strongly a plate
is restrained along its edges, the greater the effect of delmination on the
modal parameter.

The natural frequencies of delaminated plates also vary with different
ply orientation. For cantilever boundary condition, there is a decrease in
natural frequency with increase in the orientation angle.

It is observed that natural frequency increases with increase in number
of layers and aspect ratios for delaminated plate.

Variation of natural frequency is observed for number of multiple
delaminations. The fundamental natural frequency reduces with
increasing number of delaminations. Thus increase in the number and
sizes of delaminations have a deteriorating effect on the plate dynamic
stiffnesses. The effect of multiple delaminations on the natural
frequencies of delaminated plate is also greatly dependent on boundary
conditions as observed in single delamination.
97
6.4
Buckling analysis
The present study includes numerical and experimental study of buckling
analysis of delaminated composite plates with clamped-free boundary condition. The
finite element analysis allowed the mechanics of delamination to be investigated and
understood more easily than would be possible from experimental data alone. The
influences of various parameters like effects of delamination area, ply orientation,
number of layers, aspect ratio, multiple delaminations and different boundary
condition on the buckling load of woven roving delaminated plates are studied. The
experimental buckling load is in good agreement with the predictions using FEM.
Main sources of discrepancy differ between finite element analysis and experimental
results are imperfect in specimen geometry, specimen material and experimentally
achieved boundary conditions. The conclusions are summarized as given below.

For delaminated composite plate, with the increase of percentage of
delamination area, the non-dimensional buckling load decreases as the
stiffness decreases. The rate of decrease of buckling load is not uniform
for increase in percentage of delamination.

The different fiber orientation angles affected the non-dimensional
buckling load. When the fiber orientation angle increases, the nondimensional buckling load decreases. So the delaminated composite
plate with [0]8 layup has highest buckling load and with [45/-45]2s layup
has lowest buckling load.

For delaminated composite plate, with the increase of number of layers,
the non-dimensional buckling load also increases.

With the increase in aspect ratio of delaminated composite plates the
non-dimensional buckling load decreases.

For multiple delaminated composite plates, with the increase of
percentage of delamination area compared to single delaminated plate
the buckling load decreases more because of the reduction in stiffness of
composite plate.

The effect of boundary condition on buckling load of composite plates is
studied numerically. The clamped boundary conditions show the highest
98
buckling load in the context of considered edge conditions. This can be
explained by the rigidity of clamped boundary conditions. The buckling
loads for delaminated composite plates under clamped free and simply
supported boundary conditions are much lower than those under clamped
boundary condition.
6.5
Dynamic stability analysis
A first order shear deformation theory based on finite element model has been
developed using Bolotin’s approach for studying the instability region of mid-plane
delaminated composite plate. The results of dynamic stability studies are summarized
as follows:

The onset of instability occurs at lower excitation frequency with the
increase in delamination size.

With the increase in number of layers of delaminated plate, the dynamic
instability occurs at higher excitation frequency.

The dynamic instability region is shifted to a lower excitation frequency
with decrease of degree of orthotropy of delaminated composite plates.

The onset of instability occurs at higher excitation frequency with increase
of aspect ratio and width of the instability region increase with increase of
aspect ratio for delaminated cross ply plate.

With increase of static load factor the instability region tends to shift to
lower excitation frequencies and become wider showing destabilizing effect
on the dynamic stability behaviour of delaminated composite plate.
The significant contributions on the behavior of delaminated composite plates are:
A general formulation dealing with vibration, buckling and dynamic stability of
multiple delaminated composite plates is presented. Suitable finite element
formulations for single and multiple delaminations are presented using eight noded
isoparametric element. A program is developed in MATLAB environment to compute
99
the natural frequency, critical buckling load and instability regions of delaminated
composite plates. From the results, it is clear that delamination causes the reduction of
natural frequency and critical buckling load. It also shifts the dynamic instability
region of composite materials to a lower excitation frequency. The presence of one or
more delamination reduces the stiffness of the structure from the point of view of
buckling behavior and causes early resonance due to reduction in natural frequency.
So, delaminations play a critical role on the vibration behavior of the structures. The
instability behavior of delaminated plates is influenced by the geometry, material, ply
lay-up, ply orientation, boundary conditions, the type and position of loads and size of
delamination. So the designer has to be careful while dealing with structures subjected
to in-plane periodic loading. This can be used to the advantage of tailoring during
design of delaminated composite structure. Finally the figures dealing with the
variations in natural frequency, buckling load and instability regions can be used as
design aids for delaminated composites subjected to in plane loads.
The parametric resonance characteristics of the delaminated composite plates
can be used as a tool for structural health monitoring for identification of
delamination location and extent of damage in composites and also helps in
assessment of structural integrity of composite structures.
6.6
Further scope of research
The present investigation has been confined on free vibration, buckling and
dynamic stability of delaminated composite plates. This can be further extended for
force vibration and stability studies.

The present study deals with square delamination. This may be extended
for different shapes of delamination.

For single delaminated plate, the delamination is considered here at the mid
plane. So arbitrary location may be taken into account for further extension.

Dynamic stability of multiple delaminated shell can be studied.
100

The present study is based on linear range of analysis. It can also extend
for nonlinear analysis.

Dynamic stability of composite plates with circular, elliptical, triangular
shaped delamination can be studied.

The effects of damping on instability regions of delaminated composite
plates and shells can be studied.

The effects of piezo electric system in delaminated composites can be
studied.
101
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APPENDIX- I
Multiple delamination modeling
For Single mid-plane delamination and multiple delaminations including the
mid plane with different sizes like 0, 6.25%, 25%, 56.25% of the total plate
area is considered. The delamination sizes are assumed to increase from the
centre of the laminate and can be located anywhere along the thickness of the
laminate. In case of three delaminations (multiple delamination), the
delamination interfaces are located after 2nd, 4th and 6th ply in a 8 layered
laminate. The composite plates with different percentage of delamination,
single and multiple delaminations are shown in Figure 8.1 to 8.7.
Figure 8.1: 64 Elements, 225 nodes Plate with no delamination
117
Figure 8.2: 6.25% central delamination
Figure 8.3: 25% central delamination
Figure 8.4: 56.25% central delamination
118
H/2
H/2
X
H
Z
Figure 8.5: Eight layered laminate without delamination
H/2
H
X
Z
Figure 8.6: Midplane delamination
119
X
h
h/2
Z
Figure 8.7 Eight layered laminate with three delaminations
120
APPENDIX- II
Program Features and Flow Charts
For the present analysis, codes are developed in MATLAB environment for vibration,
buckling and dynamic stability analysis of delaminated composite panels using the
present formulation. The codes consist of a main program and several functions. The
finite element procedures involve three basic steps for computation in line with the
formulation which may be termed as:

Preprocessor

Processor

Post Processor
The different functions of these steps are elaborated as below.
Preprocessor
This module of the program reads the necessary information about the geometry i.e.
length, breadth and thickness of the panels including percentage of delamination,
mesh divisions, material properties, boundary conditions, static and dynamic load
factors in case of instability analysis. The nodal connectivity is generated out of the
dimensions of the panels and mesh divisions by discretization of the structures
through automatic mesh generation. This also identify the elements undergone
delamination based on the percentage of delamination and the scheme of
delamination. The nodal coordinates of each element is generated and each degrees of
freedom is identified for imposing boundary conditions.
Processor
This module of the program performs the following tasks:

Generation of constitutive matrices for both laminated and delaminated
elements using the online and offline stiffness of lamina

Generation of strain displacement matrices from the shape function derivatives
121

The element elastic stiffness matrices are generated for both laminated and
delaminated elements using both constitutive matrix and strain displacement
relations after identification of delaminated elements.

The element mass matrix is generated using the density parameters and shape
function of the element

Generation of geometric stiffness matrix from the in plane stress distribution
using

All the element elastic stiffness, mass and geometric stiffness are assembled to
form global or overall stiffnesses.

Determination of eigenvalues using inbuilt ‘eigen’ function in MATLAB i.e
natural frequencies, critical load for the vibration and buckling analysis of
delaminated composite panels.

The lowest critical load is taken as a reference load and static and dynamic
load as input parameters, the different points of instability regions are
computed as a eigenvalue problem as in equation
POST PROCESSOR
In this part of the program, all the input data are echoed to check for their accuracy.
The output data is subsequently processed to get the frequency in Hz and also nondimensionalised wherever desired. The results are stored in a series of output files for
each category of problems and these are used to prepare tables and graphs. The flow
charts used in this study are presented below.
122
Read plate geometry
Material properties, boundary conditions
Delamination parameters
Generate nodal connectivity
Identification of DOF
Delaminated constitutive matrix
Derivation of shape functions
Strain displacement matrix
Element stiffness matrix of delaminated composite plates
Element mass matrix
Element geometric stiffness matrix due to mechanical loads
of delaminated composite plates
Assembling
Overall stiffness matrix of delaminated composite plates
Assembling
Overall mass matrix
Overall geometric stiffness matrix due to mechanical loads
Boundary conditions
Eigenvalue solver
Natural frequency /Critical buckling load/
Excitation frequencies
Flow chart of program in MATLAB for instability of delaminated composite
plates subjected to in-plane periodic loading
123
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