author lt col (retd) tanveer ahmed regn number 2011-nust
TRANSCRIPT
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Modelling of damage growth in FRP Composites with stress raisers such as holes and notches
Author
Lt Col (Retd) Tanveer Ahmed
Regn Number
2011-NUST-Dir PhD-Mech-36
Supervisor
Dr. Hasan Aftab Saeed
DEPARTMENT MECHANICAL ENGINEERING
COLLEGE OF ELECTRICAL & MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SCIENCES AND TECHNOLOGY
ISLAMABAD
JANUARY, 2019
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Copyright Statement
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vested in College of Electrical and Mechanical Engineering (CEME), NUST, subject to
any prior agreement to the contrary, and may not be made available for use by third parties
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(CEME), NUST, Islamabad.
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Acknowledgments
I am thankful to my Creator Allah Subhana-Watala to have guided me throughout this work
at every step and for every new thought which You setup in my mind to improve it. Indeed I could
have done nothing without Your priceless help and guidance. Whosoever helped me throughout
the course of my thesis, whether my parents or any other individual was Your will, so indeed none
be worthy of praise but You.
I am profusely thankful to my beloved parents who raised me when I was not capable of
walking and continued to support me throughout every department of my life.
I would also like to express special thanks to my supervisor Dr. Hasan Aftab Saeed for his
motivation throughout my research program.
I would like to express special thanks to Dr. Rizwan Saeed Choudhry for his shear guidance
and support through my research work. I can safely say that I haven't learned any other engineering
subject in such depth than the ones which he has taught.
I would also like to pay special thanks to my Co-Supervisor Dr. Atta Ur Rehman Shah for
his tremendous support and cooperation. Without his help, I wouldn’t have been able to complete
my thesis.
I would also like to thank Dr. Laraib Alam Khan, Dr. Imran Akhtar, Dr. Sajid Ullah Butt
and Dr. Raja Amer Azim for being on my thesis guidance and evaluation committee.
Finally, I would like to express my gratitude to all the individuals who have rendered
valuable assistance to my study.
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Dedicated to my beloved wife and adored children whose tremendous
support and cooperation led me to this wonderful accomplishment.
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Abstract
Fibre reinforced polymer composite panels are preferred in high performance structural panels
because they are strong, stiff and light. Stress raisers such as holes or notches (for accessibility,
mechanical joining, and routing of cables etc.) may be present in any engineering structure and
composite structures are no exception. Theoretically, a stress raiser is simply a localization of high
stress-strain concentrations quantified by the Stress Concentration Factor (SCF). It is well
established in literature and engineering practice that stress-strain concentrations due to holes or
notches, unless accompanied by local plastic strain hardening, reduce the apparent strength of the
panels. Since SCF is a function of elastic properties of the material, so in isotropic materials, the
SCF is defined with elastic SCF (entails elastic properties within the elastic range of material) and
plastic SCF (entails elastic properties in the plastic range of material). However, literature is scarce
of such definition for the case of anisotropic/orthotropic materials, where the SCF is also a function
of its elastic properties. Contrary to isotropic homogenous materials, composite panels offer a very
complex structure, where fibres are generally regarded as brittle which deform elastically to final
failure exhibiting either slight or no linear deformation. Whereas matrices generally experience
plastic deformation hence the failure strain in matrics is far higher than the fibres. Additionally,
once a composite panel containing a hole is subjected to tensile loading, tangential stress at the
periphery of the hole in a perpendicular direction to the load axis attains a magnitude three times
the far field stress under plane stress conditions. However, in a composite panel, the location and
magnitude of the maximum stress are at the periphery of the hole changes with the fibre orientation
and stacking sequence, therefore designers opt for large safety margins.
This study has been performed to investigate the pre-damaged SCF and progressive-damaged SCF
for anisotropic/orthotropic material analogous to elastic and plastic deformations in isotropic
material respectively. The study presents a novel technique of calculating progressive-damaged
SCF which evaluates the changing SCF in response to the progressive damage development within
the composite panel. Finite Element (FE) representations simulate delamination damage using
cohesive elements and in-plane damage using continuum damage mechanics. In the first part of
the study, test coupons have been formulated under static conditions to consider important
influencing factors on the SCF for the case of the composite panel containing a central circular
hole subjected to tensile loading and compared with the already published literature. Later, several
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FE coupons have been formulated to precisely investigate the pre-damaged SCF and progressive-
damaged SCF for the composite panel. During the study, the investigations of pre-damaged SCF
and progressive-damaged SCF have also been performed using analytical and experimental
approaches where applicable. The FE results are found in good agreement with the analytical and
experimental results. The study provides a novel systematic FE approach for the estimation of
progressive-damaged SCF for a composite panel, which has not been reported in the literature
before. Certainly, the study proposes a paradigm shift in design philosophy which at present is
limited to no-damage philosophy especially in aerospace, where the weight savings due to less
generous safety factors are significant.
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Table of Contents
Copyright Statement ............................................................................................................................................... viii
Acknowledgments .......................................................................................................................................................ix
Abstract .......................................................................................................................................................................xi
List of Figures ........................................................................................................................................................... xvi
List of Tables ............................................................................................................................................................. xix
CHAPTER 1: INTRODUCTION............................................................................................................................... 1
1.1 Overview ...................................................................................................................................................... 1
1.2 Composite Panel .......................................................................................................................................... 1
1.3 Elastic Properties of Composite Panel ......................................................................................................... 2
1.4 Damage in Composite Panel ........................................................................................................................ 3
1.4.1 Matrix Cracking ....................................................................................................................................... 3
1.4.2 Delamination............................................................................................................................................ 3
1.4.3 Fibre Damage ........................................................................................................................................... 4
1.5 Design Problem – An Open Hole ................................................................................................................. 4
1.6 Aim and objectives of the Study .................................................................................................................. 5
1.7 Structure of the Dissertation ......................................................................................................................... 5
CHAPTER 2: LITERATURE REVIEW .................................................................................................................. 8
2.1 Overview ...................................................................................................................................................... 8
2.2 Theory of Stress Concentration .................................................................................................................... 8
2.3 SCF in Isotropic Panel ................................................................................................................................. 9
2.4 SCF in Composite Panel ............................................................................................................................ 11
The net SCF is calculated using expression given in equation 2.10. ....................................................................... 13
2.5 Stiffness Reduction in Composite Panel .................................................................................................... 13
CHAPTER 3: MODELLING OF STRESS CONCENTRATION FACTOR ...................................................... 15
3.1 Overview .................................................................................................................................................... 15
3.2 Abaqus Software ........................................................................................................................................... 15
3.3 Element Type Selection ................................................................................................................................ 15
3.4 Modeling Scale .......................................................................................................................................... 17
3.5 Classical Laminate Theory ......................................................................................................................... 17
3.6 Failure of Composite Panel ........................................................................................................................ 21
3.7 Hashin failure criteria ................................................................................................................................. 22
3.7.1 Damage Initiation .................................................................................................................................. 23
3.7.2 Damage Evolution ................................................................................................................................. 25
3.8 Cohesive Law ............................................................................................................................................. 28
CHAPTER 4: RUDIMENTARY CALCULATIONS FOR STRESS CONCENTRATION FACTOR ............. 31
4.1 Overview .................................................................................................................................................... 31
4.2 Laminate Configurations ............................................................................................................................ 31
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4.2.1 Unidirectional (UD) Configuration ........................................................................................................ 31
4.2.2 Cross-ply (CP) Configuration ................................................................................................................ 32
4.2.3 Angle-ply (AP) Configuration ............................................................................................................... 32
4.2.4 Quasi-isotropic (QI) Configuration ........................................................................................................ 32
4.3 Model Description ...................................................................................................................................... 32
4.4 Mesh Convergence ..................................................................................................................................... 34
4.5 Material Properties and Approach .............................................................................................................. 35
4.6 Results ........................................................................................................................................................ 37
4.6.1 Isotropic Materials ................................................................................................................................. 37
4.6.2 Orthotropic Composite Panel ................................................................................................................. 38
4.6.2.1 Unidirectional (UD) Configuration ................................................................................................... 38
4.6.2.2 Cross-ply (CP) Configuration ............................................................................................................ 40
4.6.2.3 Angle-ply (AP) Configuration ........................................................................................................... 41
4.6.2.4 Quasi-isotropic (QI) Configuration ................................................................................................... 42
4.7 Summary .................................................................................................................................................... 43
CHAPTER 5: PRE-DAMAGED STRESS CONCENTRATION FACTOR ........................................................ 47
5.1 Overview .................................................................................................................................................... 47
5.2 Influencing Factors of Composite Panel on SCF ....................................................................................... 47
5.2.1 Laminae Orientation Effect.................................................................................................................... 47
5.2.2 Stress Distribution Effect ....................................................................................................................... 47
5.2.3 Decaying Effect ..................................................................................................................................... 48
5.2.4 Ligament Effect ..................................................................................................................................... 48
5.3 Model Description ...................................................................................................................................... 48
5.4 Mesh Sensitivity Analysis .......................................................................................................................... 50
5.5 Analytical models ...................................................................................................................................... 51
5.6 Results and Discussion ............................................................................................................................... 52
5.6.1 Lamina Orientation Effect ..................................................................................................................... 52
5.6.2 Stress Distribution Effect ....................................................................................................................... 53
5.6.3 Decaying Effect ..................................................................................................................................... 54
5.6.4 Ligament Effect ..................................................................................................................................... 55
5.7 Summary .................................................................................................................................................... 57
CHAPTER 6: PROGRESSIVE-DAMAGED STRESS CONCENTRATION FACTOR ................................... 58
6.1 Overview .................................................................................................................................................... 58
6.2 Model Description ...................................................................................................................................... 58
6.3 Influencing factors on Progressive-damaged SCF ..................................................................................... 60
6.4 Material Properties ..................................................................................................................................... 60
6.5 Results ........................................................................................................................................................ 62
6.5.1 Lamina Orientation Effect: .................................................................................................................... 62
6.5.2 Stress Distribution Effect: ...................................................................................................................... 66
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6.5.3 Decaying Effect: .................................................................................................................................... 66
6.5.4 Ligament Effect: .................................................................................................................................... 67
6.6 Analysis of Progressive-Damaged SCF ..................................................................................................... 68
6.7 Summary .................................................................................................................................................... 78
CHAPTER 7: EXPERIMENTAL EVALUATION OF STRESS CONCENTRATION FACTOR ................... 80
7.1 Overview .................................................................................................................................................... 80
7.2 Manufacturing of Composite Panel ............................................................................................................ 80
7.2.1 Prepreg ................................................................................................................................................... 80
7.2.2 Fabrication ............................................................................................................................................. 81
7.3 Material Properties of Composite Panel ..................................................................................................... 82
7.4 Preparation of Test Specimens ................................................................................................................... 84
7.5 Tensile Testing ........................................................................................................................................... 85
7.6 Experimental Results ................................................................................................................................. 87
7.7 Summary .................................................................................................................................................... 90
CHAPTER 8: CONCLUSION ................................................................................................................................. 91
8.1 Conclusion ................................................................................................................................................. 91
8.2 Future Work ............................................................................................................................................... 93
APPENDIX A ............................................................................................................................................................. 94
APPENDIX B: MATLAB PROGRAM FOR STRESS CCONCENTRATION FACTOR ................................. 95
B1: Material Transformation Program .............................................................................................................. 95
B2: Stress Concentration Factor Calculator ...................................................................................................... 97
APPENDIX C: Elastic Constants for Anisotropic Materials ................................................................................. 99
C.1 Anisotropic Material [5] ............................................................................................................................. 99
C.2 Orthotropic Lamina [5] ............................................................................................................................ 100
C.3 Transversely Isotropic Lamina [5] ........................................................................................................... 101
C.4 Estimation of Elastic Constants ................................................................................................................ 102
C.4.1 Tensile Modulus in loading direction (𝑬𝟏): ......................................................................................... 104
C.4.2 Transverse Modulus in 2 direction (𝑬𝟐): ............................................................................................. 104
REFERENCES ........................................................................................................................................................ 109
Instructions for Students ......................................................................................................................................... 117
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List of Figures
Figure 2. 1 Stress distribution in a central circular hole under tensile loading ............................... 9 Figure 2. 2 Stress Concentration Factor .......................................................................................... 9
Figure 2. 3 SCFs 𝐾𝑡𝑔 and 𝐾𝑡𝑛for central circular hole under tension [16] ................................. 10
Figure 3. 1 GUI editor composite panel layup of Abaqus .......................................................... 16 Figure 3. 2 Representation of actual and modeled laminae ......................................................... 18
Figure 3. 3 Laminae orientations .................................................................................................. 18 Figure 3. 4 Equivalent stress verses equivalent displacement ...................................................... 26
Figure 4. 1 Laminate configurations ............................................................................................ 32 Figure 4. 2 Dimensions of the STC .............................................................................................. 33 Figure 4. 3 Quarter STC showing boundary conditions and Tie MPC ......................................... 34
Figure 4. 4 Mesh Control .............................................................................................................. 34 Figure 4. 5 Close up view of fine mesh ........................................................................................ 34
Figure 4. 6 Mesh convergence using the smallest element size ................................................... 35
Figure 4. 7 d/w ratio effect vs tensile stress for an isotropic material .......................................... 37
Figure 4. 8 Hole size effect vs SCF for isotropic materials [75] .................................................. 38 Figure 4. 9 UD configuration ........................................................................................................ 39
Figure 4. 10 d/w ratio effect vs tensile stress for UD Laminate ................................................... 39 Figure 4. 11 d/w ratio effect vs SCF for UD laminate [76] .......................................................... 40 Figure 4. 12 CP configuration ....................................................................................................... 40
Figure 4. 13 d/w ratio effect vs SCF for CP configuration ........................................................... 41 Figure 4. 14 Contribution in SCF by Individual Lamina .............................................................. 41
Figure 4. 15 AP configuration ...................................................................................................... 42 Figure 4. 16 d/w ratio effect vs SCF for AP configuration ........................................................... 42 Figure 4. 17 QI configuration ....................................................................................................... 43
Figure 4. 18 d/w ratio effect vs SCF for QI configuration ............................................................ 43 Figure 4. 19 d/w ratio effect vs tensile stress both for isotropic and orthotropic materials .......... 44 Figure 4. 20 d/w ratio effect vs tensile stress for QI configuration .............................................. 44 Figure 4. 21 d/w ratio effect vs SCF both for isotropic and composite materials ........................ 45
Figure 5. 1 Dimensions of the STC .............................................................................................. 49 Figure 5. 2 Quarter STC showing boundary conditions and Beam MPC ..................................... 50 Figure 5. 3 Mesh sensitivity analysis ............................................................................................ 51 Figure 5. 4 QI configuration ......................................................................................................... 52
Figure 5. 5 Laminae orientation effect. ......................................................................................... 53
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Figure 5. 6 Mesh showing locations of the stress points. ............................................................. 53 Figure 5. 7 Ligament effect ........................................................................................................... 54 Figure 5. 8 Decaying effect. .......................................................................................................... 54 Figure 5. 9 Ligament effect ........................................................................................................... 55
Figure 5. 10 Ligament effect ......................................................................................................... 55 Figure 5. 11 Ligament effect (Analytical & FE)........................................................................... 56 Figure 5. 12 Ligament deflection .................................................................................................. 56
Figure 6. 1 Dimensions of the test coupon ................................................................................... 59
Figure 6. 2 Mesh details of the test coupon .................................................................................. 60 Figure 6. 3 Orientation effect verses progressive damaged SCF .................................................. 63 Figure 6. 4 Initial rise in progressive-damaged SCF .................................................................... 64
Figure 6. 5 Orientation effect for QI configuration of progressive-damaged SCF ....................... 65
Figure 6. 6 Initial rise in QI progressive-damaged SCF ............................................................... 65 Figure 6. 7 Stress Distribution effect of progressive-damaged SCF ............................................ 66
Figure 6. 8 Decaying effect of progressive-damaged SCF ........................................................... 66 Figure 6. 9 Ligament effect of progressive-damaged SCF ........................................................... 67 Figure 6. 10 Ligament effect of progressive-damaged SCF ......................................................... 68
Figure 6. 11 Location of 𝟎𝟎 lamina selected for analysis ............................................................ 69
Figure 6. 12 SCF vs Displacement of 𝟎𝟎for QI configuration ..................................................... 69
Figure 6. 13 Damage initiation and evolution at δ (1.33E-5mm) ................................................. 70 Figure 6. 14 Damage initiation and evolution at δ (5.33E-5mm) ................................................. 71
Figure 6. 15 Damage initiation and evolution at δ (5.67E-5mm) ................................................. 72 Figure 6. 16 Damage initiation and evolution at δ (1.01E-4mm) ................................................. 73 Figure 6. 17 Matrix damage at δ (1.01E-4mm) ............................................................................ 73 Figure 6. 18 Damage initiation and evolution at δ (1.20E-4mm) ................................................. 74 Figure 6. 19 Damage initiation and evolution at δ (1.63E-4mm) ................................................. 75 Figure 6. 20 Matrix damage at δ (1.63E-4mm) ............................................................................ 76
Figure 6. 21 Damage initiation and evolution at δ (3.01E-4mm) ................................................. 77 Figure 6. 22 Damage evolution at δ (3.01E-4mm) ....................................................................... 77
Figure 7. 1 Schematic view of the solvent impregnation process ................................................. 81 Figure 7. 2 Schematic view of the Autoclave Process .................................................................. 82 Figure 7. 3 Thermogravimetric analysis ....................................................................................... 83 Figure 7. 4 Dimensions of the test coupon ................................................................................... 84
Figure 7. 5 Drilling of Hole .......................................................................................................... 85 Figure 7. 6 Universal Tensile Machine (UTM) ............................................................................ 85 Figure 7. 7 Strain gauges for estimation of strain ......................................................................... 86 Figure 7. 8 Wheatstone bridge circuit with one active gauge ....................................................... 86 Figure 7. 9 Data Acquisition loop for localized strain measurement ........................................... 87
Figure 7. 10 Far field strain measured by UTM ........................................................................... 87
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Figure 7. 11 Localized strain measured by the strain gauge ......................................................... 88 Figure 7. 12 Localized strain (blue), far field (red) ...................................................................... 88 Figure 7. 13 Experimental SCF .................................................................................................... 89 Figure 7. 14 Experimental SCF (red), FE SCF (blue) .................................................................. 90
Figure C. 1 Orthotropic Lamina .................................................................................................. 100 Figure C. 2 Fibre distribution in cross-section of the lamina ..................................................... 101 Figure C. 3 Material Axis ........................................................................................................... 102 Figure C. 4 Thermogravimetric analysis .................................................................................... 104
Figure C. 5 Two strain gauges for Poisson’s Ratio .................................................................... 106 Figure C. 6 Tensile and compression response of two strain gauges for Poisson’s Ratio .......... 107
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List of Tables
Table 4. 1 Properties of composite material (IM7/8552) [70] ...................................................... 35 Table 4. 2 SCF vs d/w ratio .......................................................................................................... 38
Table 5. 1 Cases Description ........................................................................................................ 49 Table 5. 2 Mesh convergence analysis. ........................................................................................ 51
Table 6. 1 Case Description .......................................................................................................... 60
Table 6. 2 Properties of composite material (IM7/8552) [67] ...................................................... 61
Table 6. 3 Lamina properties for Hashin damage criteria ............................................................ 61 Table 6. 4 Cohesive element properties ........................................................................................ 62
Table 7. 1 Properties of Constituent Materials ............................................................................. 81
Table 7. 2 Elastic Properties of Composite Panel ......................................................................... 84
Table C. 1 Number of Constants Needed at Various Levels ........................................................ 99 Table C. 2 Elastic Constants of UD Transversely Isotropic Lamina .......................................... 102 Table C. 3 Properties of Constituent Materials ........................................................................... 103
Table C. 4 Mechanical Properties of FRPC Laminate ................................................................ 108
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CHAPTER 1: INTRODUCTION
1.1 Overview
Composite materials are preferred over conventional isotropic materials in many structural
applications. Comparing with the conventional materials like steel and aluminum, the composite
materials possess high strength to weight ratio and high stiffness to weight ratio along with other
advantageous properties. Consequently, these materials have become the preferred choice of
designers for structural applications. Earlier, due to the high cost associated with the composite
materials, the application of composite materials was limited with the high performance
applications such as aerospace industries. Currently, with a reduction in the cost of composite
materials, its application has been widened to many industries like automobile, defense, energy,
and sports. Advancement in composite manufacturing and processing techniques has resulted in a
wide variety of composite materials available for the designers.
1.2 Composite Panel
The composite materials are formed by combining two or more different materials at the
macroscopic level. The main purpose of the composite material is to make a new material with
better mechanical properties in comparison with the constituent materials [1]. Generally, high
performance composite materials made of continuous fibre reinforced polymer composites. The
main component of these materials is fibre which provides high mechanical properties to the
composite materials like high strength and stiffness [2]. While the matrix is used to bind and hold
the fibres at their position and provide environmental protection to the fibres such as oxidation and
corrosion. The fibrous composite materials can be divided into the following two types:
1. Continuous fibre reinforced composites
a. Unidirectional (UD) fibre reinforced composites – in UD composites, all the fibres
are laid in one direction
b. Multidirectional fibre reinforced composites – the UD fibres are laid in different
directions
c. Woven fabric composites
d. Random fibre reinforced composites – the fibres are randomly distributed in the
composite
2
2. Discontinuous fibre reinforced composites
In this research work, the focus is on UD and multidirectional fibre reinforced composites
and would be termed as composite panels in this study. Among the variety of the composite
materials, the composite panels are extensively been used for high performance applications like
aerospace industry. Generally, composite panel is produced with continuous fibres in the form of
laminates. Laminates are fabricated by stacking of laminae on top of each other with a defined
laminate stacking sequence (LSS) to gain the desired mechanical properties. The mechanical
properties of the laminate largely associated with many factors like properties of the constituent
materials, laminae orientation, and LSS. Composite panels are often induced by external damages
in the form of holes and notches to address peculiar structural applications. These external damages
cause high stress-strain concentrations in the vicinity of the external damage which leads to
strength reduction of the composite panel. Consequently, the designers are constrained to have
high margins of safety (MoS) for composite panels to withstand service loads.
1.3 Elastic Properties of Composite Panel
The internal structure of the physical material either created by nature or artificially (e.g.
wood or composite) is often anisotropic. A generalized Hook’s Law is applied to anisotropic
materials, which demands additional independent elastic constants in comparison with the
isotropic materials. It is important to realize that actually, the independent elastic constants are not
strictly the constants since these may be the function of stress-strain and may change with time
[3]. The magnitude of the elastic constants would be constant till the material deformation is within
the elastic range of the material under loading. However, the magnitude of these elastic constants
would change once the material enters beyond the elastic range deformation under loading.
Therefore, the basic concepts of the mechanics and notational principles of anisotropic materials
[4][5] are considered important for the understanding of the stress-strain concentrations in a
composite panel.
Generally, anisotropic materials are described by three notational principles, which are the
stiffness tensor 𝐶𝑖𝑗𝑘𝑙 (a 9x9 matrix), stiffness matrix 𝐶𝑖𝑗 (a 6x6 matrix) or engineering constants
with respect to Young’s moduli 𝐸𝑖, Poisson’s ratios 𝜈𝑖𝑗and Shear moduli 𝐺𝑖𝑗. The most important
aspect related to the composite stiffness is the total number of independent elastic constants. An
3
isotropic material is defined by only two independent elastic constants. Whereas an orthotropic
material is defined by nine independent elastic constants such as three Young’s moduli (𝐸1, 𝐸2and
𝐸3), three Poisson’s ratios (𝜈12, 𝜈13and 𝜈23) and three shear moduli (𝐺12, 𝐺13and 𝐺23). Further, for
the case of transversely isotropic composite material, the number of independent elastic constants
are reduced to five such as two Young’s module (𝐸1and 𝐸2), Poisson’s ratio 𝜈12 two shear moduli
(𝐺12and 𝐺23). Therefore, for the modelling a transversely isotropic structure named as a composite
panel, all the five independent elastic constants would be required.
1.4 Damage in Composite Panel
Three common damage mechanisms associated with the composite panel are briefly
explained in this section. In isotropic materials, damage may be in terms of a single isolated crack
which propagates during its service period. Whereas the damage phenomena associated with the
composite panel is very complex. In the composite panel, a damage mechanism takes place with
damage initiation which generally starts from matrix cracking till final catastrophic failure
commonly by the fibre damage.
1.4.1 Matrix Cracking
Initial damage occurs in the composite panel in the form of matrix cracking. It appears
from the interfaces between fibre and matrix, locations of stress-strain concentrations, resin rich
pockets, and inadequacies within the composite panel. Matrix damage developed in the form of a
regular network of cracks. Generally, matrix cracks appear along the fibre direction. If longitudinal
shear stresses or/and transverse tension dominate, the matrix cracks appear perpendicular to the
plane of the lamina. If transverse shear or/and transverse compression stresses dominate, the matrix
cracks appear in an oblique direction. Though matrix damage has a limited effect on the composite
panel stiffness, it actually serves as a source of several damages. Actually, the tips of the cracks
become a source of delamination initiation.
1.4.2 Delamination
The delamination occurs between the resin rich regions with different laminae orientations.
Generally, delamination appears from the intralaminar matrix cracks at positions where matrix
cracks meet from two different laminae. The delamination initiation and growth is associated with
4
the out-of-plane normal and shear stresses. The most significant impact of delamination is the
reduction in compressive strength of the composite panel. Its effect on tensile stiffness is negligible
[3].
1.4.3 Fibre Damage
The last damage mechanism in the composite panel is of fibre failure. Fibre damage occurs
at positions where the local stress-strain concentrations exceed from the lamina strength. It often
happens where the fibres are themselves weaker or at tips of the matrix cracks [3]. Preliminary
isolated fibre breakage appears before the fibre damage shaped in a form of clusters. Fibre damage
significantly changes the initial (virgin) elastic properties of the composite panel.
1.5 Design Problem – An Open Hole
Considering a general design approach for a structural panel (rectangular plate), first of all,
structural requirements are established. The structural requirements constitute such as what would
be the strength/stiffness requirements of the panel, what would be the loading conditions,
environmental conditions in which the structural panel has to be used and more. Based on these
functional requirements the design analysis is performed by using stress-strain analysis. The
structural design analysis provides the ultimate design limit (UDL). The UDL encompasses the
mechanical behavior of the structural panel under all loading events in consideration with the
functional requirements. Further, a margin of safety (MoS) is added with the UDL to realize the
final structural panel design. The MoS is used to decrease the chance of failure by taking into
account all the uncertainties which are out of control of the designer. Generally, the uncertainties
related to such as the statistical distribution of the load, manufacturing processes, mechanical
properties of the material and more.
Once a structural panel contains holes or/and notches (externally induced damages or
irregularities), high stress-strain concentrations are produced in the close vicinity of the
holes/notches upon loading. Consequently, these holes or/and notches act as the “stress raisers”.
Therefore, a stress raiser can simply be defined as the localization of high stress-strain
concentrations which is quantified by the SCF. Generally, due to the presence of a central circular
hole in a structural panel, stresses at the periphery of the hole in a perpendicular direction to the
load axis reaches a value three times the far field reference stresses under plane stress conditions
5
[6]. Accordingly, the designers go for high MoS to strengthen the structural panel to withstand the
service loads. Therefore, from the design point of view, an adequate knowledge for precise
estimation of the stress-strain concentrations caused by the hole is essential [7][8]. Though the
concept of the elastic and plastic region does not exist for the case of composite panels, however,
the composite panels may demonstrate the plastic behavior [9]. Besides this the elastic properties
of composite properties highly influenced by the laminae orientation, laminate stacking sequence,
and laminate orthotropy ratios. Therefore, the composite panels present a very complex behavior
for the estimation of SCF. Consequently, the designers go for high MoS commensuration with the
service loads.
1.6 Aim and objectives of the Study
In this study, the response of SCF for the case of the composite panel has been considered
which is analogous to the elastic and plastic deformations in the isotropic material. The aim of the
study is to investigate the progressive-damaged SCF for the case of the composite panel containing
a central circular hole subjected to tensile loading. To achieve the aim, the research work has been
formulated into the following distinct objectives:
Provide initial insight into the effect on the SCF of a composite panel with varying hole
sizes “hole size effect” subjected to tensile loading.
Investigation of the SCF using various composite laminate configurations.
Determination of the pre-damaged SCF using various influencing parameters.
Determination of the progressive-damaged SCF using the same influencing parameters
used for the pre-damaged SCF.
Evaluation of the overall response of the SCF during pre-damaged and progressive-
damaged phase to the designers to make an economical design for load carrying structures
made of composite panels.
1.7 Structure of the Dissertation
The structure of the dissertation has been broken down into the following chapters with
regards to the aim of the study:
6
Chapter 2 extensively covers the literature review of SCF with special emphasis to the SCF
in composite panels. The concept of stress-strain concentrations with regards to a central circular
hole under tensile loading is discussed in length. How elastic SCF differs from the plastic SCF in
isotropic materials has been explained. The basis for the application of the same phenomena in
composite panels is build up and new terms have been introduced as pre-damaged SCF and
progressive-damaged SCF.
Chapter 3 covers in length the methodology being following for the investigation of pre-
damaged SCF and progressive-damaged SCF for the case of the composite panel using Abaqus.
In the first section, the influencing parameters with regards to the selection of elements and
modeling scale have been discussed. The application of classical laminate theory (CLT) in Abaqus
has been explained. Followed by Hashin’s damage criteria which are discussed in detail along with
the progressive damage law to account for the damage initiation and evolution in a composite
panel. In the last section, the influence of damaged variables on the loss of elastic properties of the
composite panel has been discussed.
Chapter 4 deals with the fundamental influencing factors associated with the SCF in a
composite panel containing a central circular hole under tensile load. In the first section, commonly
used configurations of the composite panels have been discussed followed by a detailed description
of the test coupon which will be used for the subsequent FE representations throughout the study.
In the central section, the methodology for the investigation of SCF is explained. Then the obtained
FE results have been evaluated. The FE analysis of the fundamental influencing factors affecting
the SCF will help the readers to understand the pre-damaged SCF and progressive-damaged SCF
presented in the subsequent chapters.
Chapter 5 deals with the analysis of pre-damaged SCF for composite panels under tensile
loading. Pre-damaged SCF is analogous to the elastic SCF in an isotropic material, where the
stress-strain concentrations remain within the range of elastic limits. Four significant influencing
factors i.e. laminae orientation effect, stress distribution effect, decaying effect and ligament effect
on SCF have been considered for the investigation in this section. Firstly, a brief description of
these influencing effects on the SCF of a composite panel has been given. Next, FE results are
discussed in detail for the influencing factors on SCF of the composite panel. Further, analytical
formulations have also been formulated to validate the FE results. FE results are found in good
agreement with the analytical results.
7
Chapter 6 covers the investigation of progressive-damaged SCF caused by the same
influencing factors such as the laminae orientation effect, stress distribution effect, decaying effect,
and ligament effect investigated for pre-damaged SCF in the previous chapter. For this case, the
FE representations are performed using Dynamic/Explicit approach. Hashin’s damage criteria
along with cohesive damage model are explained. Then FE results are discussed with regards to
the damage initiation and evolution within the FRPC laminate.
Chapter 7 covers the influencing factors on SCF of the composite panel using experimental
testing. The experimental investigation is carried out by the strain gauges to measure the strain
fields in the close vicinity of the hole boundary. In the first section of the chapter, a detailed
manufacturing root has been discussed for the development of composite panels. Then preparation
of the test coupons for experimental testing is explained followed by the material characterization
of the newly developed composite panel. The elastic properties obtained through material
characterization are used in the FE representations. In the last section, the experimental results are
discussed. The experimental results are found in good agreement with the FE results.
8
CHAPTER 2: LITERATURE REVIEW
2.1 Overview
Practically, the stress-strain concentrations exist in all the structural panels. Precise
knowledge of these stress-strain concentrations is extremely important for both isotropic and
anisotropic/orthotropic structural panels because the point at or near the maximum stress-strain
concentrations normally turn out to be the initiation location of the damage. The chapter starts with
the theoretical formulations related to the SCF. The concept of SCF in an isotropic structural panel
has been discussed and then proceeded with the accumulation of SCF in composite panels.
Extensive literature [10][11][12][13][14][15] can be found on SCF for the case of isotropic
structural panels. A comprehensive review of the literature related to the stress analysis for
composite panels is discussed in detail.
2.2 Theory of Stress Concentration
Theoretical formulations for calculating stress state in a structural panel are generally
derived with the assumption of uniform stress state within the cross-section of the structural panel.
As a general rule, the stresses should be transmitted point to point as uniformly as feasible. This is
not always possible once holes are incorporated in any structural panel. These holes act as stress
raisers in the structural panel and produce high stress-strain concentrations in the close vicinity of
the hole. Theoretically, any structural panel subjected to uniaxial loading experiences stresses
generally known as normal stress (or gross stress) [16]. Whereas the stress is defined as the
intensity of force per unit area.
σ = 𝐹
A =
F
w ∗ t (2.1)
In the above expression, the stress and applied load are denoted by 𝜎 and F respectively.
While width, area and thickness of the structural panel are denoted by w, A and t respectively.
Nevertheless, due to the presence of a hole in the structural panel, high localized stress-strain
concentrations are produced in the vicinity of the hole as shown in figure 2.1 [17][18]. Figure
indicates that the high stresses are passing through the central circular hole of the structural panel
9
under tensile loading. It is evident that due to the presence of a central circular hole, the net cross-
sectional area of the panel has been reduced (as the diameter of the hole is deducted from the
width). This reduction in cross-sectional area will increase the amount of the stresses near the
boundary of the hole.
Figure 2. 1 Stress distribution in a central circular hole under tensile loading
𝜎𝑛𝑒𝑡 = 𝐹
(𝑤 − 𝑑) 𝑡 (2.2)
In the expression, d denoted diameter of the hole.
2.3 SCF in Isotropic Panel
The SCF is expressed as the ratio of the maximum stress in the vicinity of the stress raiser
divided by the far field reference stress subjected to uniform uniaxial tensile loading as shown in
figure 2.2.
(i = 1,2,12 or θ)
Figure 2. 2 Stress Concentration Factor
𝐾𝑡 =σi
σx (2.3)
10
In expression, 𝐾𝑡 denotes the SCF, σi denote the maximum stress value at any point and
σx denotes the applied stress value in the x-axis. In figures points A, B and C indicates the angle
orientations, where at point A, θ = 0, point B, θ = 90 and point C, θ = 45. In the case of homogenous
elastic structures, the SCF 𝐾𝑡𝑔and 𝐾𝑡𝑛 given in equation 2.4 and equation 2.5 represents the gross
SCF and net SCF respectively based on the gross and net cross-sectional area for calculation of
the reference stresses. Generally, both the terms 𝐾𝑡𝑔and 𝐾𝑡𝑛are dissimilar. Like in the expressions
𝜎𝑔 and 𝜎𝑛 denotes the gross stress and net stress respectively. As shown in figure 2.3, once the d/w
ratio increases, the 𝐾𝑡𝑔 rises from 3 to infinity, while the 𝐾𝑡𝑛 declines from 3 to 2. Further, the
estimation of 𝐾𝑡𝑔 is difficult to read from the plot for the case of d/w ˃ 0.5. Contrary to this it is
easier to calculate the value of 𝐾𝑡𝑛. For that, only the calculation of cross-sectional area would be
essential.
𝐾𝑡𝑔 = 𝜎𝑖
𝜎𝑔 (2.4)
𝐾𝑡𝑛 = 𝜎𝑖
𝜎𝑛 (2.5)
Figure 2. 3 SCFs 𝐾𝑡𝑔 and 𝐾𝑡𝑛for central circular hole under tension [19]
11
The SCF is generally determined by using the theory of elasticity. The theory of elasticity
is considered appropriate within the region of elastic deformation since the stress-strain
relationship is linear within elastic deformation. However, once the stress state of the structural
panel enters into the plastic deformation region where the theory of elasticity is no more applicable.
The calculation of plastic SCF for the region of plastic deformation has been experimentally
investigated by Herbert. F et al [20] expressed in the expression given below:
𝐾𝑡𝑝 = 1 + (𝐾𝑡𝑒 − 1) 𝐸𝑠
𝐸∞ (2.6)
In the above expression, 𝐾𝑡𝑝 and 𝐾𝑡𝑒 denotes SCF for the plastic region and SCF for the
elastic region respectively, whereas 𝐸𝑠 and 𝐸∞ denotes elastic modulus at the point of maximum
stress (damaged material) and elastic modulus for the far field reference stress away from the stress
raiser. It has been found that the SCF with in the region of plastic deformation is less in comparison
with the SCF in the elastic region. The same results have been further validated in recent past by
S. Wakil et al [21] using a finite element approach.
2.4 SCF in Composite Panel
The stress-strain state in the composite panel presents a very complex behavior especially
with the presence of a stress raiser. The SCF in a composite panel containing a stress raiser is
influenced by the laminae orientation, laminae percentage in particular direction, laminate stacking
sequences, stiffness properties of the laminae and laminate orthotropy ratios besides the geometry
and loading conditions. Earlier in 1984 S. Lekhnitskii [22] presented a theoretical solution for
composite panels having elliptical holes. V. Ukadgaonker et al [23][24][25][26] carried out a
theoretical study on stress-strain distributions in a composite panel with triangular holes. Several
studies reveal that the SCF in composite panels depends not only on the geometry but also on the
elastic material properties [22][27][28]. This limit the possibility for the designers to use
approximate relationships [29] allowing rapid evaluation of SCF.
The increasing trend in structural applications is because of the high strength and stiffness to
weight ratio of the composites [30]. Composite panels are preferred in aerospace industry
containing notches of various sizes and shapes [31][32]. Whereas these notches in the composite
panels produce high stress-strain concentrations around the notch which adversely affects its
12
performance [33]. Past many years, the investigation of the SCF has been the major concern of the
researchers [34]. The evaluation of stress concentrations around the notch can be done using
experimental techniques, the theory of elasticity as well as by numerical techniques [35].
The effect of laminae orientations was experimentally investigated by A. Talib et al [36] using
Kevlar-29/epoxy composite laminates with and without holes and with varying lamina stacking
orientations. J. Awerbuch et al [37] highlighted that the notch sensitivity relates to the damage
occurred at the notch edge. F. Darwish et al [38] conducted a wide range of analysis on
carbon/epoxy (AS4/3501-6) to investigate the effects of varying geometric parameters at different
laminate configurations on the SCF. M. Caminero et al [39] performed the experimental study
using Digital Image Correlation (DIC) to investigate the damage around open hole composite
panels when loaded in tension. Reduction in strength with increasing the specimen size with and
without notches was investigated by many researchers in the past [40][41][42]. G. Belingardi et al
[43] emphasized the reduction of the load carrying capacity of the notched composite panel
because of the manufacturing parameters and itself the hole making process. FE analysis was
presented by Xiangqian. Li et al [44] to investigate the progressives damage mechanism on the
notched composite panels. J. Chen et al [45] developed FE model to investigate the combined
effect of delamination and in-lamina damage effects during progressive damage analysis of the
composite panel. A most recent investigation by J. Tan et al [46][47] highlights that the matrix
splitting and delamination which reduce the localized stress concentration at the notch tips in the
load bearing 00 laminae by redistributing the stress away from the notch; thereby delaying the final
failure. This is because of the crack blunting effect which strengthened the tensile specimen and
accounts for that the tensile strength of the notched specimen is not inverse in proportion to the
SCF.
However, the literature is scarce of the studies to investigate the behavior of SCF for
composite panels with regards to the elastic and plastic deformation upon loading like for the case
of isotropic panels. The probable justification for this could be that composite panels have not been
fully treated with elastic and plastic range deformations. Consequently, the composite panels
which are designed based on the value of the highest stress in the vicinity of the stress raiser would
be much heavier than composite structures without the stress raisers. Generally, with the presence
of a hole, the stresses at the periphery of the hole in a perpendicular direction to the loading axis
achieve a magnitude three times the far field reference stress under tensile loading in-plane stress
13
conditions. This would be critical especially for engineering applications like designing of the
composite panel for the aerospace industry and certainly, the designers would go for high safety
margins.
The analytical formulation of SCF for the case of the composite panel was presented by S.
Lekhnitskii [48] based on a complex variable method for an infinite thin UD homogenous laminate
panel with a central circular hole shown below:
𝐾𝑡𝑔∞ = 1 + √2 (√
𝐸𝑥
𝐸𝑦 − 𝜈𝑥𝑦) +
𝐸𝑥𝐺𝑥𝑦
⁄ (2.7)
Here 𝐾𝑡𝑔∞represents the gross SCF and 𝐸𝑥, 𝐸𝑦, 𝐺𝑥𝑦 and 𝜈𝑥𝑦 are the longitudinal modulus,
transverse modulus, shear modulus and Poisson’s ratio respectively. So far no direct formulation
is available to calculate the SCF for a finite composite panel. However, S. Tan [32] presented an
approximate solution for the finite composite panel by using a finite-width correction factor
expressed in equation 2.8. In this expression w and d denotes width of the panel and the diameter
of the hole respectively and 𝐾𝑡𝑔 denotes the gross SCF for a finite thin composite panel. While M
is the magnification factor expressed in equation 2.9.
𝐾𝑡𝑔
∞
𝐾𝑡𝑔=
3(1−𝑑
𝑤)
2+(1− 𝑑
𝑤)3
+1
2(
𝑑
𝑤𝑀)
6
(𝐾𝑡𝑔∞ − 3) [1 − (
𝑑
𝑤𝑀)
2
] (2.8)
M2 =
√1 − 8 [3(1 − d/w)
2 + (1 − d/w)3 − 1] − 1
2(d/w)2
(2.9)
The net SCF is calculated using expression given in equation 2.10.
𝐾𝑡𝑛 = 𝐾𝑡𝑔 (1 − 2 𝑎 𝑤⁄ ) (2.10)
2.5 Stiffness Reduction in Composite Panel
Stiffness reduction in the composite panel is highly influenced by the damage initiation
and propagation mechanisms developed within the composite panels. Basically, these damage
mechanisms become a cause to deplete the original (virgin) stiffness properties of the composite
14
panels. Failure in composite panels may follow different sequences of damage mechanisms
depending on the loading and composite panel configuration. Few damage sequences described in
DNV Standards [49] are as follows:
Matrix cracking – delamination – fibre failure
Matrix cracking and debonding – fibre buckling – fibre failure
Delamination – crack progression because of fatigue – global buckling
These failure sequences indicate the significance of the onset of matrix failure. Stiffness
degradation is linked to these damage mechanisms. Stiffness degradation subjected to tensile
loading can be seen in literature [50][51][52][53]. A detailed study was conducted by S. Adden
et al [54] on stiffness degradation under loading using classical laminate theory. P. Smith et al
[55] explained the reduction in stiffness upon bending loads for the case of cross-ply laminates.
Further, the design of composite panels has been discussed by many researchers in the past
based on the first ply failure approach and variation approach [56][57][58][59]. While the two
researchers can be found in the literature [60][61] who argued the design of composite panels with
an elliptical hole subjected to in-plane loading.
15
CHAPTER 3: MODELLING OF STRESS CONCENTRATION FACTOR
3.1 Overview
In this chapter, a detailed modeling scheme is discussed for the construction of FE
representations. The structure of the composite panel is very complex especially once the damage
is induced like in the form of a central circular hole. In the first section, Abaqus software packages
are introduced for the FE analysis. Then the importance of appropriate element type selection along
with the modeling scale is emphasized. In the central section, the application of CLT is explained
followed by the application of Hashin’s damage criteria which is discussed in detail to capture the
damage initiation and evolution mechanisms. Then cohesive law for the interface between the
laminae is explained for the evaluation of delamination between the adjacent laminae.
3.2 Abaqus Software
Abaqus software package is equipped with efficient modeling capabilities. It contains
material models for anisotropic/orthotropic materials model and armed with a graphical user
interface (GUI) lamina layup editor. The lamina layup editor provided in Abaqus GUI can be used
to define the element sets for the composite panel properties shown in figure 3.1. The lamina layup
editor facilitates the designer to assign appropriate material properties, thickness, and lamina
orientation of each lamina.
3.3 Element Type Selection
Abaqus offers three different categories of elements which can be used for the meshing of
the test coupon of the composite panel. Composite panel geometry can be modeled with 3
dimensional (3D) solid part, the designer can mesh it with 3D solid stress elements or using
continuum shell elements [62]. If a composite panel geometry defined with a planar surface, the
test coupon can be meshed with continuum shell elements. The continuum shell elements are
useful over conventional shell elements where the physical thickness of the composite panel
influences the other instances in the model assembly.
16
FE results are highly influenced by the type of element. The selected type of elements must
be commiserating with the scenario. The element type can be determined based on the required
accuracy of FE results, computational cost, various symmetries, and facets of the coupon
geometry, likely displacements and more. Consequently, the appropriate type of element would
improve the degree of accuracy of FE representations. In engineering applications reducing and
managing the computational cost for finite element analysis is very important for proper decision
making. Computational cost can be reduced using a simpler type of elements, reducing the number
of elements and taking into account the loading and geometrical symmetries. The total number of
elements used in the meshed coupon generally influences the accuracy of finite element simulation
and also directly proportional to the central processing unit (CPU) time required for the simulation.
Further 2D element types would require lesser computational time in comparison with the 3D
element types. Using symmetry conditions is also important due to the reduction of a total number
of elements by half to quarter elements depending on the plane symmetry conditions.
Figure 3. 1 GUI editor composite panel layup of Abaqus
17
3.4 Modeling Scale
It is extremely important to choose an appropriate modeling scale keeping in view the
intended results especially once dealing with the composite panels. Normally the smaller modeling
scale would yield highly accurate results. Three modeling scales are defined for FE representations
of composite panels such as micro-scale, macro-scale and a meso-scale, which is somewhere
within the two extreme scales [63]. Meso-scale is considered as a lamina by lamina scale and also
known as lamina scale. Macro-scale is also known as a laminate level modeling and it is
considered to be the least complex as compared to the other two scales. However, macro-scale
does not provide any useful information with regards to the fibre matrix interaction and
interlaminar responses.
In this study, a meso-scale is used to perform lamina level FE representation. It facilities
allocation of single lamina properties or set of laminae within a laminate. It also allows defining
interfacial properties between the two laminae. By defining interfaces the designers can use these
representations to ascertain delamination and interlaminar stresses. These interfacial regions in the
representation are defined using cohesive zone elements with the traction-separation law for
initiation of damage and its propagation.
3.5 Classical Laminate Theory
The CLT offered in Abaqus is used to ascertain the elastic response of a composite panel
with the assumption that each lamina is defect free, have a separate row of fibres and displays an
orthotropic material behavior. Representation of actual lamina in comparison with the modeled
lamina [64] of the composite panel is illustrated in the figure: 3.2.
18
In the composite panel, the laminae are assembled to articulate a laminate where each
lamina has its own material properties, thickness, and orientation but it is assumed that the strain
in each lamina is equivalent to the global laminate strain. In the composite panel, the principal
directions 1, 2 and 3 against each lamina denotes the fibre direction, in-plane direction
perpendicular to the fibres and out-of-plane direction perpendicular to the fibres respectively. The
laminae orientation with regards to the global reference system [64] is shown in figure 3.3.
Positive 휃 Negative 휃
Figure 3. 3 Laminae orientations
For the simplicity, the CLT is defined with plane stress conditions. Whereas, the four
independent elastic material parameters are defined as the initial starting point. These independent
elastic material parameters are denoted with 𝐸1, 𝐸2, 𝐺12 and 𝜈12.
A stiffness matrix [𝑄] expressed in equation 3.1 [64], will be formulated for each lamina
in the composite panel. These stiffness matrices are defined in the principal axis direction for the
respective lamina.
a. Modeled b. Actual
Figure 3. 2 Representation of actual and modeled laminae
19
[𝑄] = [𝑄11 𝑄12 0𝑄12 𝑄22 00 0 𝑄66
]
(3.1)
here,
𝑄11 = 𝐸1
1 − 𝜈12 𝜈21 (3.2)
𝑄22 = 𝐸2
1 − 𝜈12 𝜈21 (3.3)
𝑄12 = 𝜈12 𝐸2
1 − 𝜈12 𝜈21 =
𝜈21 𝐸1
1 − 𝜈12 𝜈21 (3.4)
𝑄66 = 𝐺12 (3.5)
Thereafter, the [Q] matrices must be transformed into the global directions keeping in view
of their orientations. Here the [�̅�] denotes the global stiffness matrices of the laminae given in
equation 3.6 [64].
[�̅�] = [
�̅�11 �̅�12 �̅�16
�̅�12 �̅�22 �̅�26
�̅�16 �̅�26 �̅�66
] (3.6)
here,
�̅�11 = 𝑄11 𝑚4 + 2(𝑄12 + 2𝑄66) 𝑚
2 𝑛2 + 𝑄22 𝑛4 (3.7)
�̅�12 = (𝑄11 + 𝑄22 − 4𝑄66) 𝑚2 𝑛2 + 𝑄12 (𝑚
4 + 𝑛4) (3.8)
20
�̅�16 = −𝑄22 𝑚 𝑛3 + 𝑄11 𝑚3 𝑛 − (𝑄12 + 2𝑄66) 𝑚 𝑛 (𝑚2 − 𝑛2) (3.9)
�̅�22 = 𝑄11 𝑛4 + 2(𝑄12 + 2𝑄66) 𝑚
2 𝑛2 + 𝑄22 𝑚4 (3.10)
�̅�26 = −𝑄22 𝑛 𝑚3 + 𝑄11 𝑛3 𝑚 − (𝑄12 + 2𝑄66) 𝑚 𝑛 (𝑚2 − 𝑛2) (3.11)
�̅�66 = (𝑄11 + 𝑄22 − 2𝑄12) 𝑚2 𝑛2 + 𝑄66 (𝑚
2 + 𝑛2) (3.12)
In the above expressions, m and n described as m = cos θ and n = sin θ respectively.
Whereas, the θ is calculated counterclockwise from the x-axis with the 1-axis. Subsequently, after
conversion [Q] of each lamina into [�̅�], the global stiffness matrix [�̅�]𝑡𝑜𝑡𝑎𝑙 is formed by summing
up all the [�̅�] matrices. Then this final global stiffness matrix can be used to ascertain the strain in
a composite panel. For example, by using the expression given in equation 3.13 [64], the strains
in the composite panel subjected to tensile loading can be calculated.
{
𝜖1
𝜖2
𝜖6
} = [�̅�]𝑡𝑜𝑡𝑎𝑙−1 [
𝑁1
𝑁2
𝑁3
] (3.13)
Considering the strain in each lamina is equivalent to the total global strain, therefore the
previous strain could be used to ascertain the stress in each lamina of the composite panel.
Consequently, the stress value can be computed by using the expression given in equation 3.14
[64].
{
𝜎𝑥
𝜎𝑦
𝜖𝑥𝑦
} = [
�̅�11 �̅�12 �̅�16
�̅�12 �̅�22 �̅�26
�̅�16 �̅�26 �̅�66
] {
𝜖1
𝜖2
𝜖3
} (3.14)
The constitutive relationship between the force resultants and the strain fields for the
CLT is shown in equation 3.15.
21
{𝑁𝑀
} = [𝐴 𝐵𝐵 𝐷
] {𝜖0
𝑘} (3.15)
Whereas the [𝐴], [𝐵] and [𝐷] are the extensional, coupling and bending rigidity matrices
respectively and defined as:
𝐴𝑖𝑗 = ∑ �̅�𝑖𝑗𝑘 (𝑍𝑘 − 𝑍𝑘−1)
𝑁
𝑘−1
(3.16)
𝐵𝑖𝑗 = 1
2 ∑ �̅�𝑖𝑗
𝑘 (𝑍𝑘2 − 𝑍𝑘−1
2 )
𝑁
𝑘−1
(3.17)
𝐷𝑖𝑗 = 1
3 ∑ �̅�𝑖𝑗
𝑘 (𝑍𝑘3 − 𝑍𝑘−1
3 )
𝑁
𝑘−1
(3.18)
3.6 Failure of Composite Panel
The structural panels tend to fail when loaded beyond their loading capacity. A stress raiser
in composite panel increases the stress-strain concentrations within the composite panel leading to
a significant strength reduction in comparison with the composite panel without a stress raiser. The
stress reduction can be attributed to a number of failure mechanisms. The multiple failure
mechanisms in the composite panel may be designated as fibre breakage/rupture, fibre pull-out,
fibre kinking/buckling, matrix cracking/crushing, lamina buckling, delamination and progressive
failure. The Abaqus software package has a built-in damage model based on two parts, damage
initiation and damage propagation.
The delamination is considered to be a common failure mode in a composite panel.
Delamination is actually expressed as a debonding or separation of two adjacent laminae. Most
commonly the meso-scale (lamina level) FE representation is used to simulate the delamination.
Many researchers in the past have been using cohesive zone elements to calculate the delamination
22
against various practical situations [65][66][67]. These cohesive zone elements are used to
describe the interplay between laminae based on the stiffness, strength, and failure evolution
properties [68]. In this study, the layers of cohesive zone elements have been used throughout the
coupon thickness. These cohesive zone elements in the coupon accurately captured the onset
progression of delamination. Hashin damage model is used to predict the four different failure
mechanisms in a composite panel. These failure mechanisms include fiber rupture in tension, fiber
kinking/buckling in compression, matrix cracking in transverse tension/shearing and matrix
crushing in transverse compression/shearing.
3.7 Hashin failure criteria
The Hashin’s failure criteria [69][70] uses a homogenized representation of a lamina and
identifies four failure modes. These failure modes are longitudinal tension, longitudinal
compression, combined transverse tension and shear, and combined transverse compression and
shear. Although there is no possibility in the model of directly relating these failure modes with
failure in constituent, these can be referred to in terms of constituent failure mode based on the
assumption of dominant failure mechanisms. Thus Abaqus describes these modes as:
a. Fibre rupture in tension
b. Fibre buckling and kinking in compression
c. Matrix cracking under transverse tension and shearing; and
d. Matrix crushing under transverse compression and shearing
The damage modeling technique which is discussed here applied to the test coupon as the
progressive damage of the composite panel. The Hashin failure criteria for the tensile and/or
compressive damage can be initiated in the fiber or matrix when the respective F (failure function)
equals to one as expressed in the following expressions [62]:
Fibre Tensile (�̂�11 ≥ 0)
𝐹𝑓𝑡 = (
�̂�11
𝑋𝑇 )2
+ 𝛼 (�̂�12
𝑆𝐿)2
(3.19)
23
Fibre Compression (𝜎11 < 0)
𝐹𝑓𝑐 = (
�̂�11
𝑋𝑐)2
(3.20)
Matrix Tensile (�̂�22 ≥ 0
𝐹𝑚𝑡 = (
�̂�22
𝑌𝑇)2
+ 𝛼 (�̂�12
𝑆𝐿)
2
(3.21)
Matrix Compressive (�̂�22 < 0)
𝐹𝑚𝑐 = (
�̂�22
2 𝑆𝑇)
2
+ [(𝑌𝐶
2 𝑆𝑇)
2
− 1] �̂�22
𝑌𝐶 + (
�̂�12
𝑆𝐿)
2
(3.22)
Here, 𝑋𝑇 & 𝑋𝐶 denotes the longitudinal tensile and longitudinal compressive strengths,
𝑌𝑇 𝑎𝑛𝑑 𝑌𝐶 denotes the transverse tensile and transverse compressive strengths, 𝑆𝐿 𝑎𝑛𝑑 𝑆𝑇 denotes
the longitudinal and transverse shear strengths, 𝐹𝑓𝑡 𝑎𝑛𝑑 𝐹𝑓𝑐 denotes the longitudinal tensile and
longitudinal compressive fracture energies and 𝐹𝑚𝑡 𝑎𝑛𝑑 𝐹𝑚𝑐 are transverse tensile and transverse
compressive energies of the lamina properties respectively.
3.7.1 Damage Initiation
Damage initiation is estimated using components of an effective stress tensor �̂�. The
effective stress tensor is calculated through a tensor operation from true stress, σ. The damage
tensor (M) would be revised after each step of the finite element simulation
with the associated damage variables of fiber, matrix, and shear. The expressions [62] of these
relationships given in equations 3.23 and 3.24. It is evident from the expressions that the
effective stress value would be equal to the true stress value once there is no damage initiation and
the value of damage variables would be equal to zero.
�̂� = 𝑀 𝜎 (3.23)
24
𝑀 =
[
1
(1 − 𝑑𝑓)
1
(1 − 𝑑𝑚)1
(1 − 𝑑𝑠)]
(3.24)
The damage variables associated with the damage tensor (M) can be changed against the
responses of element loading types. As an example, once the effective stress equal or greater than
zero in line with fibre direction, the tensile criteria will be applied on the fiber damage variable
(df) and when the stress is less than zero, the compressive criteria will be applied on the damage
variable (df). The same concept would be applicable to the case of matrix damage
variable (dm), except for the case of in-plane effective stress and perpendicular to the fibres is
measured. Further, the shear damage variable (ds) can be calculated using (df) and (dm). The
relationship of these damage variables [62] is defined using expressions given in equations 3.25 to
3.27.
𝑑𝑓 = {𝑑𝑓
𝑡 𝑖𝑓 �̂�11 ≥ 0
𝑑𝑓𝑐 𝑖𝑓 �̂�11 < 0
} (3.25)
𝑑𝑚 = {𝑑𝑚
𝑡 𝑖𝑓 �̂�22 ≥ 0𝑑𝑚
𝑐 𝑖𝑓 �̂�22 < 0} (3.26)
𝑑𝑠 = 1 − (1 − 𝑑𝑓𝑡) (1 − 𝑑𝑓
𝑐) (1 − 𝑑𝑚𝑡 ) (1 − 𝑑𝑚
𝑐 ) (3.27)
The output variables assigned for the damage initiation are as follows:
a. DMICRT: All damage initiation criteria components.
b. HSNFTCRT: Maximum value of the fibre tensile initiation criterion experienced during
the analysis.
c. HSNFCCRT: Maximum value of the fibre compressive initiation criterion experienced
during the analysis.
d. HSNMTCRT: Maximum value of the matrix tensile initiation criterion experienced
during the analysis.
e. HSNMCCRT: Maximum value of the matrix compressive initiation criterion
experienced during the analysis.
25
If any of the above variables is greater or equals to one, the failure initiates. For finite
element models with damage evolution, the maximum value remains fixed at one.
3.7.2 Damage Evolution
After the initiation of damage process, the Abaqus software will compute the material
response by using modified stiffness matrix (reduced stiffness) and loading of the damaged
material for the case of plane stress orthotropic material. It is based on the energy dissipation
during the damage process. Prior to the damage initiation, the material is linearly elastic, with the
stiffness matrix of a plane stress orthotropic material. The stress-strain relationship [62] for the
case of modified tensor expression is given in equation 3.28. Further, it can be seen in plane stress
tensor (𝐶𝑑) expressed in equation 3.29 where with the increase of the damage variables the
stiffness matrix would become less stiff (means a reduction in stiffness). Moreover, if the value of
any of the damage variable equals to one, the stress will not be supportive in that direction as the
value of stiffness will be zero.
𝜎 = 𝐶𝑑 휀 (3.28)
here 𝜎 and 휀 are the stress and strains respectively.
𝐶𝑑
= 1
𝐷 [
(1 − 𝑑𝑓) 𝐸1 (1 − 𝑑𝑓) (1 − 𝑑𝑚) 𝜈21 𝐸1 0
(1 − 𝑑𝑓) (1 − 𝑑𝑚) 𝜈12 𝐸2 (1 − 𝑑𝑚) 𝐸2 0
0 0 (1 − 𝑑𝑠) 𝐺 𝐷
] (3.29)
𝐷 = 1 − (1 − 𝑑𝑓) (1 − 𝑑𝑚) 𝜈12 𝜈21 (3.30)
Here D is the damage variable shows the existing state of 𝑑𝑓 , 𝑑𝑚 and 𝑑𝑠 which are referred
to the current fibre damage, matrix damage and shear damage variables respectively. 𝐸1 and 𝐸2
denotes the Young’s moduli against directions 1 and 2 respectively. G and 𝜈12denotes shear
modulus and Poisson’s ratios in 1-2 axes respectively.
Once the load increases beyond the damage initiation, a damage evolution model will be
required to calculate the material response. The linear damage evolution model [62] is described
with the help of the plot illustrated in figure 3.4.
26
Figure 3. 4 Equivalent stress verses equivalent displacement
In the damage evolution model, the damage initiated point is represented by a point A.
Whereas the material response is shown by the leg OA prior to any of the damage, which indicates
a linear rise in stress till point A. The damage evolution model at the damage initiation point will
start to estimate the decrease in stress as the displacement of the element increases. The stress
value will decrease until it approaches to zero, the total displacement needed to reach this point is
specified by the fracture energy GC. The area under the curve OAC represents the fracture energy.
Consequently, for the case of consistent strength properties with increased fracture energy, the
element would experience a large displacement until it reaches to the point of ultimate failure. The
expressions given in equations 3.31 to 3.38 are used to determine the equivalent stress and the
equivalent displacement.
Fibre Tension(�̂�11 ≥ 0):
𝛿𝑒𝑞𝑓𝑡
= 𝐿𝐶 √⟨𝜖11⟩2 + 𝛼 𝜖122 (3.31)
𝜎𝑒𝑞𝑓𝑡
= ⟨𝜎11⟩ ⟨𝜖11⟩ + 𝛼 𝜏12 𝜖12
𝛿𝑒𝑞𝑓𝑡
𝐿𝐶⁄ (3.32)
Fibre Compression(�̂�11 < 0):
𝛿𝑒𝑞𝑓𝑐
= 𝐿𝐶 ⟨− 𝜖11⟩ (3.33)
𝜎𝑒𝑞𝑓𝑐
= ⟨− 𝜎11⟩ ⟨− 𝜖11⟩
𝛿𝑒𝑞𝑓𝑐
𝐿𝐶⁄ (3.34)
27
Matrix Tension(�̂�22 ≥ 0):
𝛿𝑒𝑞𝑚𝑡 = 𝐿𝐶 √⟨𝜖22⟩2 + 𝜖12
2 (3.35)
𝜎𝑒𝑞𝑚𝑡 =
⟨𝜎22⟩ ⟨𝜖22⟩ + 𝜏12 𝜖12
𝛿𝑒𝑞𝑚𝑡 𝐿𝐶⁄
(3.36)
Matrix Compression(�̂�22 < 0):
𝛿𝑒𝑞𝑚𝑐 = 𝐿𝐶 √⟨− 𝜖22⟩2 + 𝜖12
2 (3.37)
𝜎𝑒𝑞𝑚𝑐 =
⟨− 𝜎11⟩ ⟨− 𝜖11⟩ + 𝜏12 𝜖12
𝛿𝑒𝑞𝑚𝑐 𝐿𝐶⁄
(3.38)
Here the ⟨ ⟩ bracket set represents the Macaulay bracket operator, which is used to define
for every x ∈ R as ⟨𝑥⟩ = (𝑥 + |𝑥|) 2⁄
Additionally, the damage evolution is a scale dependent scheme. Consequently, the
variable mesh size will effect on the damage propagation in the FE representation. In order to
reduce this mesh dependency, a characteristic length Lc is introduced by the Abaqus Software
which determines the characteristic length for every element based on its formation and geometry
(Lc = square root of the in-plane area of the element for shell elements). The expression [62] used
to determine varying damage variables against each failure mechanism is expressed in equation
3.39.
𝑑 = 𝛿𝑒𝑞
𝑓 (𝛿𝑒𝑞 − 𝛿𝑒𝑞
0 )
𝛿𝑒𝑞𝑓
(𝛿𝑒𝑞𝑓
− 𝛿𝑒𝑞0 )
(3.39)
There will be a constant increase in the element damage until it is completely damaged,
this will happen once the damage variable equals to one. Consequently, when the element is
completely damaged it will not support any load. The leg AC in figure 3.4 represents the material
response against damaging element. In case the FE representation unloaded prior to the 100
percentage damage, the unloading will be linear to zero stress and displacement. However, the
unloaded element will lose strength and stiffness. Certainly, upon reloading, it will represent a
different behavior as illustrated in the loading cycle OBC in figure 3.4.
28
3.8 Cohesive Law
A technique used to add interface elements into a 3D mesh that occupy the interfacial
regions between the adjacent laminae. The cohesive elements are commonly used to determine the
delamination between the laminae. It enhances the accuracy of the finite element results once
delamination is considered to be the main damage mode. The cohesive interface relationship
applied with a traction separation law which assumes at prior linear elastic behavior followed by
the damage initiation and progression. Practically the interface regions between the laminae
present a finite thickness and accordingly the interface elements are modeled. Consequently, the
cohesive elements used to represent the interfaces may have zero thickness. The progressive
damage model can be applied to determine the damage of cohesive elements. The damage model
is represented as the traction-separation law, which has two pre-damage and post-damage stages.
The linear elastic response is presented by a penalty stiffness matrix [62] expressed in equation
3.40, where 𝐾𝑛𝑛, 𝐾𝑠𝑠and 𝐾𝑡𝑡 represents the stiffness in normal, first shear and second shear
direction. In the representation, traction is shown by the nominal stress {𝑡} and separation is shown
by the nominal strains {𝜖} across the cohesive elements.
{
𝑡𝑛𝑡𝑠𝑡𝑡
} = [𝐾𝑛𝑛 𝐾𝑛𝑠 𝐾𝑛𝑡
𝐾𝑛𝑠 𝐾𝑠𝑠 𝐾𝑠𝑡
𝐾𝑛𝑡 𝐾𝑠𝑡 𝐾𝑡𝑡
] {
𝜖𝑛
𝜖𝑠
𝜖𝑡
} (3.40)
Here 𝑡𝑛 denotes the normal stress and 𝑡𝑠 and 𝑡𝑡 denotes the shear stress components
perpendicular with the normal stress. Further, the three nominal strains would be needed to
calculate the stress vector which are determined by the expressions given in equations 3.41 to 3.43.
𝜖𝑛 = 𝛿𝑛
𝑇0 (3.41)
𝜖𝑠 = 𝛿𝑠
𝑇0 (3.42)
𝜖𝑡 = 𝛿𝑡
𝑇0 (3.43)
Here 𝛿 denotes the separation and 𝑇0 denotes the initial elemental thickness. The Abaqus
Software allows the designers to choose the post failure response that is appropriate for the
29
simulation. As the damage initiation points will be needed to capture the post failure responses.
The several damage initiation criterion integrated with traction separation law [62] are expressed
in equations 3.44 to 3.47.
Maximum nominal stress criterion:
max(⟨𝑡𝑛⟩
𝑡𝑛0 ,
𝑡𝑠
𝑡𝑠0 ,
𝑡𝑡
𝑡𝑡0) = 1 (3.44)
Maximum nominal strain criterion:
max(⟨𝜖𝑛⟩
𝜖𝑛0 ,
𝜖𝑠
𝜖𝑠0 ,
𝜖𝑡
𝜖𝑡0) = 1 (3.45)
Quadratic nominal stress criterion:
(⟨𝑡𝑛⟩
𝑡𝑛0 )
2
+ (𝑡𝑠
𝑡𝑠0)
2
+ (𝑡𝑡
𝑡𝑡0)
2
= 1 (3.46)
Quadratic nominal strain criterion:
(⟨𝜖𝑛⟩
𝜖𝑛0 )
2
+ (𝜖𝑠
𝜖𝑠0)
2
+ (𝜖𝑡
𝜖𝑡0)
2
= 1 (3.47)
Thereafter, the damage evolution will start where the strength of the material drops which
indicates the degradation rate in stiffness of the material once the respective damage initiation
criteria have been met. The initiation of the post damage response is represented by a scalar
damage expression as shown in equation 3.48.
𝜎 = (1 − 𝐷) 𝜎 (3.48)
Here D and 𝜎 represents the damage variable and effective stress tensor. The value of the
damage variable lies between zero and one. Once the D is zero means no damage and when D is
one which indicates a fully damaged condition. Further, the damage evolution is evaluated using
displacement and energy based schemes. For the case of displacement based evolution, the
designers have to prescribe the variation between the effective displacements at hundred percent
30
damage and damage initiation. The damage variable is determined by the expressions [62] given
in equations 3.49 and 3.50.
Displacement based linear damage evolution:
𝐷 = 𝛿𝑚
𝑓 (𝛿𝑚
𝑚𝑎𝑥 − 𝛿𝑚0 )
𝛿𝑚𝑚𝑎𝑥 (𝛿𝑚
𝑓− 𝛿𝑚
0 ) (3.49)
Displacement based exponential damage evolution:
𝐷 = 1 − (𝛿𝑚
0
𝛿𝑚𝑚𝑎𝑥)
(
1 −
1 − exp(− 𝛼 (𝛿𝑚
𝑚𝑎𝑥 − 𝛿𝑚0
𝛿𝑚𝑓
− 𝛿𝑚0
))
1 − exp(− 𝛼)
)
(3.50)
For the case of energy based damage evolution which is usually calculated by using criteria
based on energy release rate and the fracture toughness under mixed-mode loading. Traditionally,
two energy based criteria are being used as shown in equations 3.51 and 3.52.
Power law criteria:
(𝐺𝐼
𝐺𝐼𝐶)
𝛼
+ (𝐺𝐼𝐼
𝐺𝐼𝐼𝐶)𝛼
+ (𝐺𝐼𝐼𝐼
𝐺𝐼𝐼𝐼𝐶)𝛼
= 1 (3.51)
BK (Benzeggagh Kenane) criteria:
𝐺𝐶 = 𝐺𝐼𝐶 + (𝐺𝐼𝐼𝐶 − 𝐺𝐼𝐶) [𝐺𝑆
𝐺𝑇]𝜂
(3.52)
Where 𝐺𝑇 is the work performed by the traction interface, 𝐺𝑆
𝐺𝑇⁄ is the fraction of the
cohesive energy dissipated by the shear tractions, 𝐺𝑆 is the work performed by the shear
components of the of the interface tractions, whereas the 𝐺𝐼𝐶 and 𝐺𝐼𝐼𝐶are the critical energy rates
in mode I and mode II and η is the material insensitivity.
31
CHAPTER 4: RUDIMENTARY CALCULATIONS FOR STRESS
CONCENTRATION FACTOR
4.1 Overview
The motivation of this chapter is to deliberate on fundamental aspects associated with the
SCF which facilitates to gain a deeper understanding of rudimentary calculations for the SCF. In
the first section, general configurations of the composite panel are explained in detail followed by
the description of a standard test coupon required for the investigation SCF. Then the methodology
to investigate the responses on the SCF using several laminate configurations are explained for a
basic understanding of the readers. FE representations used for the analysis based on the
Static/General approach offered in Abaqus/CAE 6.13-1. The FE results are compared with the
previously published results in the literature. The FE analysis is also performed on the isotropic
material using the same approach. The analysis of fundamental influencing factors on the SCF will
help the readers to understand methodology and techniques followed for the estimation of pre-
damaged SCF and progressive-damaged SCF in subsequent chapters.
4.2 Laminate Configurations
Part of this chapter describes the general structure of a composite panel laminate. A
composite panel structure is fabricated by using prepregs commonly regarded as a semi-cured
lamina. Stacking of these laminae in various directions are defined by the designers keeping in
view the imminent service loads. Several common use lamina stacking configurations are
discussed below:
4.2.1 Unidirectional (UD) Configuration
UD configuration of composite panel laminate consists of multiple laminae layers stacked
on top of each other in the same direction (i.e. 08 or 908) as shown in figure 4.1 (a). These
laminates exhibit highest strength and stiffness in the parallel direction of the applied load but
exhibit low strength and stiffness in the perpendicular direction of the applied load. The UD (08)
and UD (908) laminates are configured parallel and perpendicular to the loading axis.
32
4.2.2 Cross-ply (CP) Configuration
CP configuration of composite panel laminate consists of laminae stacking both parallel
(00) and perpendicular (900) direction to the applied load shown in figure 4.1 (b). CP configuration
could be symmetric or antisymmetric. However, in the case of symmetric CP configuration, the
composite panel exhibit balance strength, and stiffness in both parallel and perpendicular direction
of the applied load because fibre orientations are at 00and 900 hence the �̅�16 = �̅�26 = 0 see
equation 3.6 for both laminae.
a. UD configuration b. CP configuration c. AP configuration d. QI configuration
Figure 4. 1 Laminate configurations
4.2.3 Angle-ply (AP) Configuration
AP configuration of composite panel laminate shown in figure 4.1 (c) has laminae
orientation at 휃 and – 휃 with laminate stacking orientation other than 00 and 900. Further AP
configuration could be symmetric or antisymmetric. The symmetric AP configuration will have an
odd number of laminae. While the antisymmetric AP configuration will have an even number of
laminae.
4.2.4 Quasi-isotropic (QI) Configuration
QI configuration of composite panel laminate shown in figure 4.1 (d) represents the
stacking of laminae in such a manner that generally a QI configuration will yield an effect of the
isotropic material under applied load. The stacking sequence followed for QI configuration in this
study is [450/900/−450/00]4𝑠 for the evaluation of SCF.
4.3 Model Description
A number of FE representations are modeled to replicate a rectangular structural composite
panel containing a central circular hole by using shell planer elements offered in Abaqus/Standard
33
of commercial software Abaqus/CAE 6.13-1. The FE representation is formulated on a standard
test coupon (STC) illustrated in figure 4.2. The dimensions of the STC are 128 mm, 32 mm and 2
mm in length, width and thickness respectively [71][72]. The hole sizes was scaled up simply by
using even numbers in the STC starting from STC starting with the smallest hole size of diameter
1 mm followed by 2, 4, 6, 8, 10, 12 and 14 mm for the investigation of their effect on the SCF,
which is commonly known as “Hole Size Effect” like other researchers have followed the similar
trend on this account found in the literature [71][72]. The STC is constructed with 16 layers of
laminae. The thickness of each lamina is of 0.125 mm to achieve an overall thickness of 2 mm by
the STC.
Figure 4. 2 Dimensions of the STC
To gain the advantage of the symmetry conditions of the STC, a quarter model is
constructed to save three fourth of the computational resources. Consequently, the boundary
conditions are imposed by containing x-displacement at y = 0 and y-displacement x = 0, as shown
in figure 4.3 (a). In the figure U2 = UR1 = UR3 = 0 implies that the translation on y-axes is zero
and rotation about x-axes and z-axes are set on zero. The test coupon is loaded with a constant
displacement rate of 1 m/min using a Tie multi-point constraint (MPC) through a dummy node as
shown in figure 4.3 (b). The dummy node is placed sufficiently far away from the STC so that a
parallel loading effect could be achieved on the STC.
34
a. Boundary conditions b. Tie multi-point constraint
Figure 4. 3 Quarter STC showing boundary conditions and Tie MPC
4.4 Mesh Convergence
The accuracy of the FE results highly depends on the mesh refinement level of the FE
representation. While the extremely fine mesh can be a burden on computational resources.
Therefore refined mesh is only preferred in the vicinity of the hole boundary and the coarse mesh
is used away from the hole. The partition scheme is applied to define fine mesh and the coarse
mesh regions. Fine mesh is created by applying Quad, Free and Medial element shape, technique
and algorithm respectively using mesh control options as shown in figure 4.4. In this FE
representation 4-node (S4R) quadrilateral, stress displacement elements have been used. A close-
up view of the fine mesh around the periphery of the hole in the STC is illustrated in figure 4.5.
Figure 4. 4 Mesh Control
Figure 4. 5 Close up view of fine mesh
35
Mesh sensitivity analysis is carried out by using the smallest element size versus the SCF
as illustrated in figure 4.6 (a) and also convergence results using percentage difference have been
graphically shown in figure 4.6 (b).
a. Mesh convergence b. Percentage difference
Figure 4. 6 Mesh convergence using the smallest element size
4.5 Material Properties and Approach
Both isotropic (metal) and orthotropic (composite panel) materials have been used in these
FE representations. Steel is chosen for the case of isotropic materials. The properties of the steel
used in the investigation of SCF are as E = 209 GPa (modulus of elasticity) and 𝜇 = 0.3 (poisson’s
ratio). Whereas the material chosen for the composite panel is Carbon/Epoxy (IM7/8552). The
material properties of IM7/8552 [44][73] are given in table 4.1.
Table 4. 1 Properties of composite material (IM7/8552)
Name Symbol Value Unit
Density of the composite 𝜌 1610 Kg/m3
Tensile Modulus 𝐸11 161 MPa
Transverse Modulus 𝐸22 11.4 MPa
Poisson’s ratio in the 1-2 direction 𝜇12 0.32
Shear modulus in the 1-2 direction 𝐺12 5.17 MPa
Shear modulus in the 1-3 direction 𝐺13 5.17 MPa
Shear modulus in the 2-3 direction 𝐺23 3.98 MPa
36
The values of maximum tensile stress and maximum transverse stress at the intended points
are obtained from the element mesh from the FE representation. Moreover, the magnitude of the
reaction force is obtained using unique nodal point for the estimation of SCF. For composite panel
laminate, the average maximum stress value is calculated by averaging the individual laminae
maximum stress values of the laminate through FE representation. The maximum tensile and
transverse stresses are calculated both for the material axis (laminae orientation) and for the
loading axis orientations (obtained by using the transformation method available in Abaqus
Software). Additionally, the values of stress concentrations are also calculated by using effective
laminate properties. The effective laminate properties are also obtained with the help of online
software of computer-aided design environment for composites (CADEC). Since properties of
composite materials highly influenced by the fibre orientations. Consequently, the transformation
of composite properties has been evaluated with respect to the fibre directions. Using basic
properties such as E11 and E22 etc, the plan strain and stress transformation, stiffness
transformations have been computed. Elastic laminate properties are defined by the expressions in
A, B, D, H matrices [74]as appended below:
𝐴𝑖,𝑗 = ∑(�̅�𝑖𝑗)𝑘𝑡𝑘; 𝑖, 𝑗 = 1, 2, 6
𝑁
𝑘=1
(4.1)
𝐵𝑖,𝑗 = ∑(�̅�𝑖𝑗)𝑘𝑡𝑘�̅�𝑘; 𝑖, 𝑗 = 1, 2, 6
𝑁
𝑘=1
(4.2)
𝐷𝑖,𝑗 = ∑(�̅�𝑖𝑗)𝑘 (𝑡𝑘�̅�𝑘
2 + 𝑡𝑘3
12) ; 𝑖, 𝑗 = 1, 2, 6
𝑁
𝑘=1
(4.3)
𝐻𝑖,𝑗 =5
4∑(�̅�𝑖𝑗)𝑘
[𝑡𝑘 − 4
𝑡2 (𝑡𝑘 �̅�𝑘
2 + 𝑡𝑘3
12)] ; 𝑖, 𝑗 = 1, 2, 6
𝑁
𝑘=1
(4.4)
Here 𝑄𝑖𝑗 represents the effective stiffness of the laminae. Subsequently, to compute the
deformation response, the values of these matrices are entered into the Abaqus software of CADEC
as shown by Auto-Generated formulation of CADEC for one of the case in Appendix A.
37
4.6 Results
4.6.1 Isotropic Materials
The maximum tensile stress occurs at a point located at an angle of 𝜋 2⁄ on the periphry of
the hole [19] for the case of rectangular plate loaded in tension. The stress value of these locations
are obtained from the FE mesh used in the test coupons. Several coupons have been constructed
with varying hole size’s to capture the hole size effect. In this study, the hole size effect is presented
in the form of d/w ratio effect. The length, width and thickness of the sample is 128 mm, 32 mm
and 2 mm respectively. Initially, the d/w ratio effect is investegated with repect to the maximum
tensile stress of referd points obtained from the FE representations and plotted in figure 4.7. Results
illustrate a very small tensile stress with a minimum d/w ratio (1 mm diameter hole size). Then a
consistent upward trend in tensile stress can be seen from 2 mm hole size till 14 mm hole size.
Therefore, with the increase of hole size, the tensile stress also increases. However, the presence
of very small tensile stress at 1 mm hole size indicates the stress fileds are confined to a very
limited area. Therefore, stress-strain concentration caused by a much smaller stress raiser may be
called as a localized stress-strain concentrations.
Figure 4. 7 d/w ratio effect vs tensile stress for an isotropic material
The SCF is calculated by dividing the localized stress over the far filed reference stress.
While the reference stress 𝜎𝑛𝑒𝑡 is calculated by dividing the reaction force (obtained through a
unique nodal point from the dummy in the FE representation) over the net cross-sectional area of
the coupon (w x t of the coupon). The FE results of SCF relative to the d/w ratio are shown in
38
figure 4.8 along with the results of previous literature [75]. The results illustrate a sudden rise in
the SCF value from 1 mm to 2 mm hole size after that till 14 mm hole size a consistent minor
decline in the SCF value is observed. The reason for apparent unexpected rise indicates that the
intensity of SCF is more for the case of small size holes in comparison with the relatively large
size holes due to the fact that the SCF is highly sensitive to the d/w ratio [7]. The results are also
shown in tabular form in table 4.2.
Table 4. 2 SCF vs d/w ratio
Hole Diameter d/w Ratio SCF 1 0.031 1.808 2 0.063 2.408 4 0.125 2.447 6 0.188 2.417 8 0.250 2.356
10 0.313 2.277 12 0.375 2.222 14 0.438 2.172
Figure 4. 8 Hole size effect vs SCF for isotropic materials [75]
4.6.2 Orthotropic Composite Panel
4.6.2.1 Unidirectional (UD) Configuration
The LSS of the UD configuration is [00]16𝑠 where all the laminae are alligned in direction
of the applied load. Mesh constituting a tatal of 4153 elements and lamina stack plot of the UD
configuration of 4 mm diameter hole is shown in figure 4.9. The maximum tensile stress value
points are shown in figure 4.10 obtained from the FE representations. These maximum tensile
39
stress points are plotted against their d/w ratios. An exponential upward trend can be seen till 8
mm hole size then an almost smooth trend is observed for the case of tensile stress.
Figure 4.11 illustrates the SCF against the d/w ratio alongwith the results of previous
literatur [76]. A consistent upward trend can be seen till 0.25 d/w ratio after that a minor decline
in the value of SCF is observed.
a. Mesh b. Lamina stack plot
Figure 4. 9 UD configuration
Figure 4. 10 d/w ratio effect vs tensile stress for UD Laminate
40
Figure 4. 11 d/w ratio effect vs SCF for UD laminate [76]
4.6.2.2 Cross-ply (CP) Configuration
The LSS of the CP configuration is [00 / 900 ]8𝑠. Mesh constituting a total of 4153
elements and lamina stack plot of the CP configuration of 4 mm diameter hole is shown in figure
4.12. The SCF against varying d/w ratios is plotted in figure 4.13. The figure shows the SCF for
the case of lamina orientation (labeled as “theta”) and the global orientation (transformed
magnitude in the direction of applied force, labeled as “tx”). A consistent upward trend can be
seen with the increase of d/w ration till 0.25 d/w ratio and then almost a smooth trend in SCF can
be seen for the case of CP configuration. The lamina level i.e. 00 and 900 contribution on the SCF
can be seen in figure 4.14. The figure illustrated that the main influence in the SCF is of 00 lamina.
The highest 6.6 and lowest 0.5 SCF found in 00 and 900 lamina, respectively.
a. Mesh b. Lamina stack plot
Figure 4. 12 CP configuration
41
Figure 4. 13 d/w ratio effect vs SCF for CP configuration
Figure 4. 14 Contribution in SCF by Individual Lamina
4.6.2.3 Angle-ply (AP) Configuration
The LSS of the AP configuration is [450/ −450]8𝑠. Mesh constituting a total of 4153
elements and lamina stack plot of the AP configuration of 4 mm diameter hole is shown in figure
4.15. The SCF against various d/w ratios are plotted in figure 4.16. It can be seen that against the
initial rise only upto the d/w ratio of 0.063, the SCF increase for both the laminae (450 and -450
laminae) orientations after that with further increase in d/w ratio the SCF drops. The SCF valuse
of both the laminae orientations come closer due to the dominace of shearing forces. The highest
2.4 and lowest 1.4 SCF found in -450 and 450 lamina, respectively.
42
Figure 4. 16 d/w ratio effect vs SCF for AP configuration
4.6.2.4 Quasi-isotropic (QI) Configuration
The LSS of the QI configuration is [450/900/−450/00]4𝑠. Mesh constituting a total of
5377 elements and lamina stack plot of the QI configuration of 4 mm diameter hole is shown in
figure 4.17. The plots of SCF against varying d/w ratios are shown in figure 4.18 (a) for the case
of material orientation (labeled as “theta”) and figure 4.18 (b) for the case of global orientation
(transformed magnitude in the direction of applied force, labeled as “tx”), respectively. It is evident
that the main contribution in the SCF is due to the 00 laminae orientation.
a. Mesh b. Lamina stack plot
Figure 4. 15 AP configuration
43
a. Laminae orientation b. Global orientation
Figure 4. 18 d/w ratio effect vs SCF for QI configuration
4.7 Summary
The plot showing d/w ratio effect versus maximum tensile stress (𝜎𝑚𝑎𝑥) is shown in figure
4.19 for the case of isotropic panel and composite panel (various laminate configurations). For the
case of the composite panel the highest tensile stress (𝜎𝑚𝑎𝑥) is found in UD configuration and
lowest in AP configuration. Further, the plot showing d/w ratio effect against maximum tensile
stress value for the case of QI configuration of the composite panel is illustrated in figures 4.20.
a. Mesh b. Lamina stack plot
Figure 4. 17 QI configuration
44
Figure 4. 19 d/w ratio effect vs tensile stress both for isotropic and orthotropic materials
Figure 4. 20 d/w ratio effect vs tensile stress for QI configuration
FE results are compatible with the previous research work such as
[77][78][33][79][75][10]. Comparing all the laminate configurations, the maximum values of SCF
observed 6.3, 3.5, 2.4 and 1.7 in UD, CP, QI and AP configurations, respectively. Considering the
tensile stress for the case of composite panels, based on the results following conclusion can be
drawn:
1. The tensile stress mainly intensify due to the 00 laminae orientation in direction of the
applied load.
2. The more number of laminae stacking in 00 orientation will increase the tensile strength
of the composite panel.
45
3. Once the d/w ratio is too low (i.e. 1 mm hole size), the global stress-strain
concentrations of the panel would be the same excluding the localized portion of the
stress raizer. Therefore, referred to as the localized stress concentrations [19].
4. With increased d/w ratio (i.e. 2 mm and above hole size), the stress-strain field
increases and produce nonlocal stress-strain concentrations. Therefore, referred as the
nonlocalized stress concentrations.
The d/w ratio effect on the SCF is graphically shown in figure 4.21 for all the composite
panel configurations i.e. UD, CP, AP and QI along with the isotropic panel.
Figure 4. 21 d/w ratio effect vs SCF both for isotropic and composite materials
Considering the composite panels following judgments could be drawn:
1. The SCF will increase with respect to the increase in d/w ratio of the composite panel,
which is commonly known as the hole size effect.
2. The highest SCF (𝐾𝑡𝑛(𝑚𝑎𝑥) = 4.203) is found in UD configuration and lowest
(𝐾𝑡𝑛(𝑚𝑎𝑥) = 1.785) is found in the AP configuration of the composite panel.
3. The SCF in UD configuration rises from 1.509 to 4.203 (which is the maximum SCF)
against the increase d/w ratio of 0.031 to 0.25.
4. Once the hole size is too small, the SCF will also be small, this is because the stress-
strain concentrations produce only in close vicinity of the stress raiser and the
phenomena is termed as the localized stress concentration.
5. Once the hole size (diameter) approaches to the width of the panel, the stress fields of
the hole boundary interact with the stress fields of the boundary as in the case of 5 mm
46
hole size and above in STC, SCF starts decreasing. So with 0.3 d/w ratio and above,
the SCF is not only influenced by the hole rather also influenced by the stress field of
the width boundary. This phenomenon may be termed as hole-boundary interaction
effect.
47
CHAPTER 5: PRE-DAMAGED STRESS CONCENTRATION FACTOR
5.1 Overview
This chapter deals with the analysis of pre-damaged SCF of the composite panel under
tensile loading. Pre-damaged SCF is analogous to the elastic SCF in isotropic materials, where the
stress-strain concentrations remain within the range of elastic limits. Four significant influencing
factors i.e. laminae orientation effect, stress distribution effect, decaying effect and ligament effect
which effects on the SCF have been considered for the investigation in this section. Initially, a
brief description of these influencing factors on SCF of the composite panel has been given for
better understanding of the readers. Afterward, the brief details on the construction of STC for the
FE representations have been provided, which is used for the investigation of all the influencing
factors on the SCF. Then FE results are discussed in detail considering all the influencing factors
on SCF of the composite panel. Further, analytical models have also been formulated to validate
the FE results. FE results are found in good agreement with the analytical results.
5.2 Influencing Factors of Composite Panel on SCF
Salient influencing factors on SCF of the composite panel has been briefly discussed in
this section. The influencing parameters on SCF mentioned below are formulated based on the
past research work [80][72][81] and books [19] on the topic.
5.2.1 Laminae Orientation Effect
A principal feature peculiar to the composite panel is the laminae orientations relative to the
loading direction. The influence of varying laminae orientations on SCF has been investigated in this
section. Ten STC of UD configuration containing each a hole of 2.667 mm diameter have been
modeled with laminae orientations at angles of 00, 100, 200, 300, 400, 500, 600, 700, 800and 900
for the investigation of “laminae orientation effect”. The laminae orientations were chosen to cover
the complete range of probable options available for practical applications.
5.2.2 Stress Distribution Effect
Another fundamental concept in mechanics of isotropic materials is the varying stresses
48
around the periphery of the hole generally known as “stress distribution effect”. The maximum SCF
occurs around the periphery of the hole along π/2 and 3π/2 angles subjected to uniaxial loading in an
infinite rectangular plate [19]. An STC of UD configuration having a hole size of 4 mm diameter has
been constructed for the investigation of these varying stresses around the periphery of the central
circular hole for the case of the composite panel.
5.2.3 Decaying Effect
Another well-known phenomenon in the mechanics of isotropic materials is the reduction
in stress concentrations away from the peak value point to the lowest value which is known as
“decaying effect”. Although, the “decaying effect” is basically a stress gradient, where SCF
decreases away from the edge of the hole along the same angle from its peak value. A UD laminate
model of STC with a 4 mm diameter of the hole was modeled for the analysis of this phenomena.
5.2.4 Ligament Effect
Another important phenomenon in the structural panel containing a hole is the effect of
d/w ratio commonly known as the “ligament effect”. Owing to the presence of the central circular
hole, the composite panel is separated by two equal halves where each half is recognized as a
ligament of the composite panel. Large ligament (i.e. w/d ratio) indicates the presence of a small
size hole and small ligament indicates a large size hole in the composite panel. Several STC of UD
configuration has been modeled with varying diameters i.e. 1.778, 2, 2.286, 2.667, 3.175, 4, 5.33
and 8 mm to realize the w/d ratios of 9, 8, 7, 6, 5, 4, 3 and 2 respectively for the analysis of this
phenomena. Summary of all the cases has been tabulated in table 5-1. The results have been
presented graphically and discussed in succeeding paragraphs.
5.3 Model Description
FE analysis has been performed by using Abaqus/Standard in commercially available
software Abaqus/CAE 6.13-1. A meso-scale FE representation using STC has been constructed,
where each lamina is modeled discretely using layers of continuum shell elements (SC8R) stacked
on top of each other as per the LSS defined for each case. The continuum shell elements (SC8R)
are capable of capturing through-the-thickness deformations and considered highly accurate to
49
account for contact and thickness changes in the test coupon. All FE representations constitute 16
laminae having a 0.125 mm thickness of each individual lamina to achieve an overall 2 mm
thickness of the composite panel. The dimensions of the test coupon are shown in figure 5.1 with
a varying central circular hole for a rectangular plate of the composite panel.
Figure 5. 1 Dimensions of the STC
To gain the advantage of the symmetry conditions of the STC, a quarter model is
constructed to save three fourth of the computational resources. Consequently, the boundary
Table 5. 1 Cases Description
Case Description Models
1 Laminae orientation effect:
Laminae angle effect is evaluated
Ten STC of UD configurations are modeled wit
h lamina orientations at an angle of 00, 100,
200, 300, 400, 500, 600, 700, 800and 900.
2
Stress Distribution effect:
SCF intensity is evaluated around
3600 the angle at the periphery of the
hole
One STC of UD configuration is modeled with
00 laminae orientation.
3 Decaying effect:
Evaluation of SCF gradient
One STC of UD configuration is modeled with
00 laminae orientation.
4
Ligament effect:
SCF against varying width to diameter
(w/d) ratios
Thirty two STCs of UD configuration are modeled
with 00, 100, 200 and 900 laminae orientations ha
ving w/d ratios of 2, 3, 4, 5, 6, 7, 8 and 9.
50
conditions are imposed by containing x-displacement at y = 0 and y-displacement x = 0, as shown
in figure 5.2 (a). The test coupon is loaded with a constant displacement rate of 1 mm/min (0.16667
E-6 m/s) using a rigid body Beam multi-point constraint (MPC) through a dummy node shown in
figure 5.2 (b). The dummy node is placed sufficiently far away from the STC so that a parallel
loading effect could be achieved on the STC.
a. Boundary conditions b. Beam MPC
Figure 5. 2 Quarter STC showing boundary conditions and Beam MPC
The material selected for the test coupon is Carbon/Epoxy (IM7/8552) which is a widely used
composite material for high performance applications. The material properties of the individual
lamina are same used for the case of rudimentary SCF in preceding chapter shown in table 4.1.
5.4 Mesh Sensitivity Analysis
A mesh sensitivity analysis has been performed for the accuracy of the FE results. The
percentage difference method has been used to evaluate the optimal mesh as expressed in equation
5.1. The SCF is determined and compared with the subsequent fine mesh value to improve the mesh
convergence at each mesh level. The results of FE representations used for mesh sensitivity analysis
are tabulated in table 5-2. Similarly, the mesh convergence results have been graphically illustrated
in figure 5.3, where precisely the result of model number 10 having 8696 elements, which exhibits
the converged value.
1
1
% i i
i
SCF SCFDifference
SCF
x 100 (5.1)
Where i indicates model number.
51
Table 5. 2 Mesh convergence analysis.
Model # No of Elements SCF % Difference
1 1736 1.68 4.00%
2 2264 1.75 2.78%
3 2760 1.8 2.70%
4 3320 1.85 1.60%
5 4136 1.88 2.08%
6 4824 1.92 1.03%
7 5384 1.94 2.51%
8 6680 1.99 1.00%
9 7560 2.01 1.47%
10 8696 2.04 0.00%
11 9224 2.04
Figure 5. 3 Mesh sensitivity analysis
5.5 Analytical models
Analytical models have also been used for the calculation of SCF against varying test
coupons mentioned in the preceding section for the validation of FE results. The analytical
calculations are carried out using Lekhnitskii expression and Tan.’s finite width correction factor
given in equations (2.7) and (2.8) respectively. The analytical SCF predominantly depends on the
geometric shape, loading, laminate stacking sequence and stiffness properties of the homogeneous
composite laminate. Matlab codes generated to calculate the SCF against the same STC used for
the FE representations are given in Appendix B.
52
5.6 Results and Discussion
5.6.1 Lamina Orientation Effect
Mesh constituting a total of 9160 elements and lamina stack plot of the UD configuration
along 00 lamina orientation containing 4 mm diameter hole is shown in figure 5.4. The influence
of laminae orientation on SCF is presented in figure 5.5. Maximum SCF has been observed with
laminae orientation of 00 degree followed by a steady drop till 400 degree orientation of laminae.
A consistent trend has been seen until 900 degree orientation of laminae. The presence of high
SCF along 00 degree laminae orientation indicates a high loading capacity of the composite panel.
Conversely, the decrease in SCF on a deflection of 00 degree orientation along the loading
direction means a reduction in loading capacity. This happens owing to the fact that 00 degree
laminae orientation presents more strength under tensile loading against the other laminae
orientations. Since fibres are the main load bearing constituents in the composite panel which
results in more damage (cutting) to the fibres which results in a significant SCF. Therefore, from
the design point of view the effective laminae orientation range would always lie between 00
degree to 400 degree to gain desirable strength in a composite panel. The FE results are found in
good agreement with the analytical results.
a. Mesh b. Lamina stack plot
Figure 5. 4 QI configuration
53
Figure 5. 5 Laminae orientation effect.
5.6.2 Stress Distribution Effect
Figure 5.6 indicates the probable locations on the mesh having 9160 elements from where
the stress values are measured. The FE results in figure 5.7 (a), represent the pattern of SCF around
the periphery of the central circular hole, the pattern matches with the results of the previous
literature [76] as shown in figure 5.7 (b). The FE representation looks like the stress distribution
pattern found for an isotropic materials [82]. The periphery of the half circle is marked with
protractor angles and the SCF values relative to those angles are given on the vertical scale as
shown in the figure. The maximum SCF occurring at an angle of π/2 in figure 5.7 shows that the
intensity of SCF starts falling with the angle deflection towards zero and π angles. However, to
measure the stress distribution effect for the composite panel no analytical formulation is available.
Figure 5. 6 Mesh showing locations of the stress
points.
54
a. Numerical b. Literature [76]
Figure 5. 7 Ligament effect
5.6.3 Decaying Effect
Figure 5.8 indicates the probable locations on the mesh having 9160 elements from where
the stress values are measured. The FE representation for the case of decaying effect in the
composite panel is shown in figure 5.9 (a), the trend matches with the results of previous literature
[76] as shown in figure 5.9 (b). FE analysis has been performed along the angle π/2 of the circular
hole in this case. This is the line perpendicular to the loading direction. The peak value SCF
occurred at the periphery of the central circular hole at a π/2 angle. Afterward, the intensity of SCF
declined away from the periphery of the hole. It has also been observed that the decaying effect is
higher near the peak stress value point and is lesser away from the peak stress value point as
reflected in the figure. No analytical solution is equipped for the representation of this decaying
effect for the case of composite panel other than FE formulations. Consequently, from the design
point of view, it is possible to strengthen the composite panel at the desired location.
Figure 5. 8 Decaying effect.
55
c. Numerical d. Literature [76]
Figure 5. 9 Ligament effect
5.6.4 Ligament Effect
The fourth case investigates the influence of ligament effect on SCF for a composite panel.
The ligament effect has been investigated with laminae orientations along 00, 100, 200 and
900angles. Both analytical and FE results are presented in figure 5.10 (a) and 5.10 (b) respectively.
Results reveal high SCF for the case of small ligaments (large w/d ratio) and low SCF for the case of
large ligaments (small w/d ratio).
e. Analytical f. Numerical
Figure 5. 10 Ligament effect
56
Figure 5. 11 Ligament effect (Analytical & FE)
The comparison of analytical results with FE results is illustrated in figure 5.11, for the
case of 00 laminae orientation. Additionally, it can also be seen in FE results, where the reduction
in SCF is large for the case of small ligaments as compared to the large ligaments. This extra drop
in SCF is produced because of the compressive flexural component, which reduces the tensile
stress component. Accordingly, the composite panel with small ligaments would depict a profile
of an elliptical hole as graphically illustrated in figure 5.12. The inner bend in small ligament yields
flexural stress and suppresses the tensile stress, which causes further reduction in SCF. This extra
drop can only be seen in FE results, however, no analytical formulation is available to show the
same behavior.
Figure 5. 12 Ligament deflection
57
5.7 Summary
The results of the four influencing factors on SCF of composite panel infer the following
conclusions:
1. The maximum SCF has been observed with laminae orientation of 00 degree followed
by a steady drop till 400 degree orientation of laminae, then a consistent trend has been
seen till 900 degree orientation of laminae.
2. The decrease in SCF on the deflection from 00 degree orientation along the loading
direction means a reduction in loading capacity. Therefore, from the design point of
view the effective laminae orientation range would always lie between 00 degree to
400 degree under tensile load.
3. The maximum SCF occurs at an angle of π/2. Whereas the intensity of SCF starts
falling with the angle deflection towards zero and π angles.
4. The peak value SCF occurs at the periphery of the central circular hole at a π/2 angle.
Afterward, the intensity of SCF starts declining away from the periphery of the hole.
5. The high SCF for the case of small ligaments (large w/d ratio) and low SCF for the
case of large ligaments (small w/d ratio).
6. Considering numerical and analytical results shown in figure 5.11, the reduction in SCF
is large for the case of small ligaments as compared to the large ligaments.
58
CHAPTER 6: PROGRESSIVE-DAMAGED STRESS CONCENTRATION
FACTOR
6.1 Overview
In the previous chapter, the FE representations were formulated to investigate the pre-
damaged SCF caused by the influencing factors such as the laminae orientation effect, stress
distribution effect, decaying effect and ligament effect of the composite panel containing a central
circular hole. In this chapter, the investigation of the progressive-damaged SCF has been
performed under the same influencing factors. FE representations for the evaluation of
progressive-damaged SCF are performed using Dynamic/Explicit approach. In the first part, FE
models are discussed in detail wherein the Hashin’s damage criteria is applied to predict the
damage initiation and evolution in the test coupon. Further cohesive zone elements are also used
to predict the delamination between laminae is explained. Afterward, the FE results of progressive-
damaged SCF have been discussed which are computed by using the impaired elastic properties
of the material obtained from the progressive damage initiation and evolution models.
6.2 Model Description
3D FE representations are formulated using explicit time integration method offered in
Abaqus/CAE 6.13-1 to evaluate the progressive damage response in the test coupon. The
configuration and boundary conditions defined for the test coupon are same used for the analysis
of pre-damaged SCF as shown in figure 6.1. However, in these FE representations, the Hashin
damage criteria [69] is used which offers tools to investigate the progressive damage through an
iteration process. During the iteration process, the damaged elements are characterized by impaired
elastic properties of the composite panel until its final catastrophic failure. The Hashin’s damage
criteria predict following four different damage modes as elaborated in section 3.6 above:
1. Fibre tensile failure
2. Fibre compressive failure
3. Matrix tensile failure
4. Matrix compressive failure
59
For instance, once a lamina experiences matrix damage, the elastic properties of the
damaged lamina are multiplied with the associated degradation factor. Consequently, the FE
representations are capable to predict the information of the pre-damaged or initial (virgin) elastic
properties of the material and the progressive-damaged or degraded elastic properties of the
material as well. The same degraded (impaired) elastic properties of the material are used to
compute the progressive-damaged SCF.
Figure 6. 1 Dimensions of the test coupon
In addition, the cohesive law is implemented to capture delamination between the adjacent
laminae. The cohesive law generally contains two parts describing the response of delamination
initiation and its propagation. Prior to the onset of the delamination, the linear elastic traction-
separation displacement law defines to hold the adjacent surfaces of the laminae. After the
initiation of delamination, the response of the interface layers is governed by the softening law.
The cohesive law applied using the same approach by R.S Choudhry [83], where every lamina is
modeled with a single layer of 8-noded reduced integration continuum shell element (SC8R) and
the interfaces between the neighboring laminae are modelled with a single layer of 1𝜇𝑚 thickness
of cohesive element (COH3D8) as shown in figure 6.2. The cohesive interface elements are
generated by using offset mesh, means that they share the mating surface nodes. Every continuum
shell element (SC8R) have one element integration point and three section integration points. The
mesh sensitivity is addressed on two accounts. One because of the contact between every lamina
is controlled by the interface elements rather than by defining the exclusive contact surfaces.
Second the mesh sensitivity with regards to the energy dissipation during the softening phase of
progressive damage in continuum shell elements. This has been fixed by introducing a
characteristic length in the formulation of interface elements. Because of this characteristic length,
it have been possible to describe the damage propagation by using a stress-displacement relation.
60
Figure 6. 2 Mesh details of the test coupon
6.3 Influencing factors on Progressive-damaged SCF
The progressive-damaged SCF is evaluated considering the same influencing factors
defined for the pre-damaged SCF in the preceding chapter as shown in table 6-1.
6.4 Material Properties
The material selected for the FE representations is same Carbon/Epoxy (IM7/8552)
which is used for the investigation of pre-damaged SCF. The material properties of the individual
Table 6. 1 Case Description
Case Description Models
1 Laminae orientation effect:
Laminae angle effect is evaluated
Six STC of UD configurations are modeled with
lamina orientations at an angle of 00, 400, 450,
500, 700, and 900.
2
Stress Distribution effect:
SCF intensity is evaluated around
3600 the angle at the periphery of
the hole
One STC of UD configuration is modeled with
00 laminae orientation.
3 Decaying effect:
Evaluation of SCF gradient
One STC of UD configuration is modeled with
00 laminae orientation.
4
Ligament effect:
SCF against varying width to dia
meter (w/d) ratios
Thirty two STCs of UD configuration are modeled
with 00, 100, 200 and 900 laminae orientations ha
ving w/d ratios of 2, 3, 4, 5, 6, 7, 8 and 9.
61
lamina are obtained from previously available data in the literature [44][73] as shown in table 6-2.
Table 6. 2 Properties of composite material (IM7/8552)
Name Symbol Value Unit
Density of the composite 𝜌 1610 Kg/m3
Tensile Modulus 𝐸11 161 MPa
Transverse Modulus 𝐸22 11.4 MPa
Poisson’s ratio in the 1-2 direction 𝜇12 0.32
Shear modulus in the 1-2 direction 𝐺12 5.17 MPa
Shear modulus in the 1-3 direction 𝐺13 5.17 MPa
Shear modulus in the 2-3 direction 𝐺23 3.98 MPa
Additionally, due to the application of the Hashin damage criteria further, ten material
properties are required which are obtained from previous research work [44][73] as shown in table
6.3.
Table 6. 3 Lamina properties for Hashin damage criteria
Name Symbol Value Unit
Longitudinal tensile strength 𝑋𝑇 2.806 GPa
Longitudinal compressive strength 𝑋𝐶 1.4 GPa
Transverse tensile strength 𝑌𝑇 60 MPa
Transverse compressive strength 𝑌𝐶 185 MPa
Longitudinal shear strength 𝑆𝐿 90 MPa
Transverse shear strength 𝑆𝑇 69.7 MPa
Longitudinal tensile fracture energy 𝐺𝑓𝑡 1.127 E5 J/m2
Longitudinal compressive fracture energy 𝐺𝑓𝑐 3.72 E4 J/m2
Transverse tensile fracture energy 𝐺𝑚𝑡 2.73 E2 J/m2
Transverse compressive fracture energy 𝐺𝑚𝑐 6.31 E2 J/m2
The cohesive zone elements (COH3D8) has been introduced among the neighboring
laminae to capture the delamination effect. The cohesive zone element properties shown in table
6.4 are obtained from previous research work [73]. The thickness of the cohesive element used is
1𝜇𝑚 with a density of 1085 Kg/m2.
62
Table 6. 4 Cohesive element properties
Name Symbol Value Unit
Elastic modulus in 3-direction (stiffness
penalty in normal direction = 50E33/t)
𝐾𝑛𝑛 4.55 E15 N/m3
Shear modulus in 1-3 plan (stiffness penalty
in shear direction = 50G13/t)
𝐾𝑠𝑠 2.07 N/m3
Shear modulus in 2-3 plan (stiffness penalty
in shear direction = 50G23/t)
𝐾𝑡𝑡 1.59 N/m3
Nominal stress in the normal direction 𝜎𝑛 40 MPa
Nominal stress in the 1-3 direction 𝜎𝑠 50 MPa
Nominal stress in the 2-3 direction 𝜎𝑡 50 MPa
Mode I fracture energy 𝐺𝐼𝐶 2 E-4 N/m
Mode II fracture energy 𝐺𝐼𝐼𝐶 1 E-4 N/m
Model III fracture energy 𝐺𝐼𝐼𝐼𝐶 1 E-4 N/m
Power 𝛼 1
6.5 Results
6.5.1 Lamina Orientation Effect:
The pre-damaged SCF is valid only within the elastic range. However, once the strains
around the periphery of the hole enter into the damage zone, the elastic properties degrade which
changes the SCF. In the first case, progressive-damaged SCF is investigated on various lamina
orientations i.e. all the laminae oriented at 00, 400, 450, 500, 700 and 900 angles. The results of
progressive-damaged SCF against these laminae orientations are shown in figure 6.3. Results
indicate a consistent drop of SCF from its initial value. A detailed justification of the results is
given in following section 6.6.
63
Figure 6. 3 Orientation effect versus progressive damaged SCF
64
It has also been observed that prior to the significant decrease in the progressive-damaged
SCF there is a slight rise in the progressive-damaged SCF from its initial value which is shown in
figure 6.4 against all the laminae orientations. This slight rise in the SCF is because of the fact that
upon loading no immediate matrix cracking or fibre/matrix debonding takes place so there is a
slight rise in the SCF, these defects gradually accumulate within the composite panel which leads
to degrading elastic properties progressively.
Figure 6. 4 Initial rise in progressive-damaged SCF
Similar, behavior has been observed with regards to the laminae orientation effect on
progressive-damaged SCF for the case of a QI configuration of LSS [450/ 900/−450/00]4𝑠 as
illustrated in figure 6.5. Initially a, slight rise in progressive-damaged SCF and then a consisted
drop as illustrated in figure 6.6.
65
Figure 6. 5 Orientation effect for QI configuration of progressive-damaged SCF
Figure 6. 6 Initial rise in QI progressive-damaged
SCF
66
6.5.2 Stress Distribution Effect:
The FE results imitate the stress distribution pattern found for the pre-damaged SCF in the
preceding chapter. The maximum progressive-damaged SCF occurring at angles of π/2 and 3π/2
and after that, the intensity of SCF starts falling with the deflection of angle towards zero and π
angles. FE results of progressive-damaged SCF are illustrated in figure 6.7.
Figure 6. 7 Stress Distribution effect of progressive-damaged SCF
6.5.3 Decaying Effect:
A similar decaying trend like in pre-damaged SCF is observed in the progressive-damaged
SCF which can be seen as the intensity of progressive-damaged SCF decreases away from the hole
along the same axis in figure 6.8.
Figure 6. 8 Decaying effect of progressive-damaged SCF
67
6.5.4 Ligament Effect:
FE results presented for the investigation of ligament effect on the progressive-damaged
SCF are shown in figure 6.9 using w/d ratios of 3, 4, 5 and 6. A leaning trend is found in the
progressive-damaged SCF where the w/d ratio is 5 and above. Whereas an unexpected response is
observed for w/d ratios of 3 & 4. This is probably because of the surface effects where the stress
raiser is no more represents the localized stress concentrations. This happens once stress fields of
boundary edge interact with the stress fields in the vicinity of the hole. FE results showing
progressive-damaged SCF for all the w/d ratios are shown in figure 6.10.
Figure 6. 9 Ligament effect of progressive-damaged SCF
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Figure 6. 10 Ligament effect of progressive-damaged SCF
6.6 Analysis of Progressive-Damaged SCF
Damage often occurs in terms of intralaminar matrix cracks like matrix splitting. The
matrix splitting may not always reduce the strength but may support other types of damage. During
loading, these cracks may spread into the adjacent laminae. When these cracks penetrate through
the thickness of numerous off-axis laminae, it can produce a stress concentration in the 00 load
bearing laminae, resulting in loss of elastic properties and tensile strength [84]. These intralaminar
cracks may also link with the interlaminar matrix damage (delamination), causing completely
debonding of one or more laminae. Delamination is more likely to initiate from the free edges of
the stress raiser.
To gain insight into the effect on stress concentration, a QI configuration with LSS of
[450/ 900/−450/00]4𝑠 is selected where the load bearing axis is along the direction of 00
laminae. The damage initiation and damage evolution variables can be seen at the key points on
the plot. The lamina selected for analysis is oriented at 00angle along the direction of the load as
shown in figure 6.11. The adjacent laminae one on right is oriented at −450angle and on the
opposite side is at 00 angle. The progressive-damaged SCF at key points relative to their
displacements are indicated on the plot shown in figure 6.12.
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Figure 6. 11 Location of 𝟎𝟎 lamina selected for analysis
Figure 6. 12 SCF vs Displacement of 𝟎𝟎for QI configuration
The first significant drop in SCF occurring at a displacement of 1.33E-5 is a point where
signs of matrix damage appear shown in figure 6.13. This is the point where nonlinearity in the
curve starts which can be seen in subsequent figures. It is because of a large number of damaged
variables. Corresponding to the increase in the magnitude of the damage variables, the elastic
properties of the laminate degrades.
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(-45 / 0) (0 / 0)
Interface splitting
Matrix Fibre
Initiation of damage in 00 lamina
Figure 6. 13 Damage initiation and evolution at δ (1.33E-5mm)
Next point at displacement 5.33E-5 initiation of matrix damage can be seen in the may be
due to widespread matrix splitting between two adjacent 00with 00 laminae and between two
adjacent 00 with −450 laminae in figure 6.14.
71
(-45 / 0) (0 / 0)
Interface splitting
Matrix Fibre
Initiation of damage in 00 lamina
Figure 6. 14 Damage initiation and evolution at δ (5.33E-5mm)
Further, at displacement 5.67E-5 initiation of matrix damage and fibre damage can be seen
in the 00 lamina and growth matrix splitting between the interfaces of adjacent 00with 00 laminae
and between the adjacent 00 with −450 laminae in figure 6.15.
72
Further, matrix damage and the degradation can be observed at the displacement 1.01E-4
in figure 6.16.
(-45 / 0) (0 / 0)
Interface splitting
Matrix Fibre
Initiation of damage in 00 lamina
Figure 6. 15 Damage initiation and evolution at δ (5.67E-5mm)
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(-45 / 0) (0 / 0)
Interface splitting
Matrix Fibre
Initiation of damage in 00 lamina
Figure 6. 16 Damage initiation and evolution at δ (1.01E-4mm)
Additionally, the matrix damage evolution can be seen in figure 6.17.
Figure 6. 17 Matrix damage at δ (1.01E-4mm)
Further development of excessive matrix splitting can be seen in both the neighboring
laminae marked at displacement 1.20E-4 and more prominent signs of initiation of matrix damage
and fibre damage can be seen at an angle of 45 degrees in the lamina which shown in figure 6.18.
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(-45 / 0) (0 / 0)
Interface splitting
Matrix Fibre
Initiation of damage in 00 lamina
Figure 6. 18 Damage initiation and evolution at δ (1.20E-4mm)
Complete delamination can be observed in both the neighboring laminae marked at
displacement 1.63E-4 shown in figure 6.18. Also, extensive matrix damage can be seen in figure
19.
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(-45 / 0) (0 / 0)
Interface splitting
Matrix Fibre
Initiation of damage in 00 lamina
Figure 6. 19 Damage initiation and evolution at δ (1.63E-4mm)
Further development in the matrix damage evolution can be seen in figure 6.20 at
displacement 1.63E-4.
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Figure 6. 20 Matrix damage at δ (1.63E-4mm)
Complete failure can be observed in the next point marked at the displacement of 3.01E-4
in the form of complete debonding of lamina from the adjacent laminae complete matrix damage
and fibre damage of the lamina shown in figure 6.21 and figure 6.22.
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(-45 / 0) (0 / 0)
Interface splitting
Matrix Fibre
Initiation of damage in 00 lamina
Figure 6. 21 Damage initiation and evolution at δ (3.01E-4mm)
Matrix damage Fibre damage
Figure 6. 22 Damage evolution at δ (3.01E-4mm)
Therefore, the failed elements in the coupon will experience a reduction in the elastic
properties of the material according to the damage initiation and evolution laws followed in the
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FE representation. In response to the reduced magnitude of the local stiffness, which will
redistribute the stresses in the surrounding elements to recapture equilibrium in the coupon
structure [85]. This phenomenon is repeated again and again or the same calculations are
performed over and over at the same load until no more elements fail.
The analysis for the designing of a composite panel with the help of the proposed
computational approach would be very effective. The matrix debonding and matrix splitting will
cause a reduction in local stress concentration at the periphery of the hole along the load bearing
00 laminae by redistributing the stress away from the periphery, which causes a delay of the
catastrophic failure of the 00 fibres. The redistribution of the stress, in fact, strengthen the
composite panel and indicates that the tensile strength of the composite panel containing a central
circular is not in inverse proportion with the SCF.
Further, with regards to the designing of a composite panel, the study provides simpler
conclusions for the reduction of stress-strain concentrations in the critical regions.
6.7 Summary
FE representations are formulated using an STC, which comprises the Hashin’s
damage criteria to capture the responses of damage initiation and evolution under tensile loading.
Also, cohesive zone elements are incorporated between the laminae to capture the delamination.
During the damage initiation and evolution process degraded values of elastic constants are being
used for the estimation of progressive-damaged SCF. The results of the four influencing factors
on the progressive-damaged SCF of composite panel infer the following conclusions:
1. A consistent decrease in the value of SCF from its base (initial) value has been observed
for all the laminae orientations.
2. It is also observed that prior to the significant decrease in the value of SCF, there is a
slight rise in the SCF value from its initial SCF for all the laminae orientations.
3. The slight rise in the SCF occurs due to the fact that on the onset of matrix splitting
upon loading stress redistribution among the laminae takes place which causes a slight
rise in the SCF.
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4. The same effect occurs for the case of stress distribution as for the pre-damaged SCF
such as the maximum SCF occurs at angles of π/2 and 3π/2 and its intensity reduces
with the deflection of angle towards zero and π angles.
5. A similar decaying trend like in pre-damaged SCF is observed in the progressive-
damaged SCF, where the value of the SCF decreases away from the periphery of the
hole.
6. Four w/d ratios such as 6, 5, 4 and 3 used for the analysis of ligament effect on the
progressive-damaged SCF. The outcomes of the analysis are as below:
a. A leaning trend is found in the SCF for w/d ratios of 5 and 6.
b. An unexpected response in the SCF is observed for w/d ratios of 3 & 4. This occurs
due to the combined effect of the stress fields at the edge of the hole and the stress
fields occurred at the edge of width boundary. This effect may be termed as hole-
boundary interaction effect.
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CHAPTER 7: EXPERIMENTAL EVALUATION OF STRESS
CONCENTRATION FACTOR
7.1 Overview
In previous chapters, the influencing factors on SCF of the composite panel containing a
central circular hole upon tensile loading have been evaluated by FE representations. In this
chapter experimental investigation is performed by using the same test coupon dimensions for the
validation of FE results. During experimental testing, strain gauges were used to measure the
strains at the point of interest i.e. in close vicinity of the hole boundary. Through the experimental
approach, a localized SCF is analyzed keeping w/d ratio well within the range of local stress-strain
fields. The structure of this chapter is as follows. Initially, a detailed manufacturing root for the
development of composite panel has been discussed. Then preparation of the test specimen for
tensile testing is described followed by the estimation of properties of the elastic constants of the
newly developed composite panel. In the last section, the analysis of the experimental results has
been done. The experimental results are found in good agreement with the FE results.
7.2 Manufacturing of Composite Panel
7.2.1 Prepreg
Composite panels are developed initially by manufacturing prepregs (short form of
preimpregnated) which is a semi-cured form of laminae. The prepregs can be manufactured by
combining more than one distinct materials which are chemically insoluble with each other
through various manufacturing roots. For this study, the prepregs are manufactured by resin
impregnation in the fibres by using a solvent impregnation method [86]. During solvent
impregnation process, the continuous fibres are passed through a resin container where fibres are
thoroughly impregnated by the resin as illustrated in the schematic view in figure 7.1 (a) and allied
apparatus shown in figure 7.1 (b). These continuous fibres are properly guided on to the take-up
reel generally known as prepreg take-up drum. These prepregs are then stored in a freezer maintain
a temperature of −180𝐶 prior to the fabrication of composite panel.
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a. Schematic View b. Apparatus
Figure 7. 1 Schematic view of the solvent impregnation process
Constituent materials used for the manufacturing of prepregs are carbon fibre (TC 36S –
12K of Tairyfil) as reinforcement [87] and Epoxy (ESP – 135 A / B of EPORITE) [88] as Matrix.
The properties of these constituent materials have been tabulated in Table 7-1.
Table 7. 1 Properties of Constituent Materials
Carbon Fibre Epoxy Matrix
Tensile
Modulus
Density Poisson’s
Ratio
Tensile
Modulus
Density Poisson’s
Ratio
𝐸𝑓 (GPa) 𝜌𝑓 (Kg/m3) 𝜐𝑓 𝐸𝑚 (GPa) 𝜌𝑚 (Kg/m3) 𝜐𝑚
250 1.81 0.2 3.45 1.2 0.36
7.2.2 Fabrication
Composite panels have been fabricated by stacking of prepregs one top of each other in a
defined stacking LSS. The resin used between the two laminae is the same Epoxy matrix used in
the manufacturing of the prepregs obtained from the manufacturer. For this study, the UD
configuration of LSS [00]16 has been used to fabricate the composite panels having dimensions of
300 mm, 300 mm and 2 mm in length, width and thickness respectively. The composite panel is
fabricated by using 16 layers of laminae to achieve an overall thickness of 2 mm (thickness each
lamina is of 0.125 mm) through an autoclave process [89].
During the autoclave process, all 16 layers of the prepregs have been stacked on top of
each other in UD configuration. After the stacking of the prepregs, the entrapped air between the
layers is removed such as a vacuum bagged process. The process is performed while the uncured
composite panel is placed inside the autoclave chamber. The schematic view of the autoclave
process has been shown in figure 7.2. In the autoclave chamber, the required temperature and
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pressure are achieved for a defined period of time. During this process, uniformly matrix is
distributed and close contact is achieved by proper bonding between the reinforcement and the
matrix. Then the autoclave chamber is cooled down as per the defined cooling rate and the
composite panel is removed from the chamber.
Figure 7. 2 Schematic view of the Autoclave Process
7.3 Material Properties of Composite Panel
The properties of the composite material highly influenced by the properties of constituent
materials. Further, the composite panel properties are influenced by the fibre packing arrangement
as well as fibre volume fraction. For relatively quick and simple calculations, the “Rule of
Mixture” has been applied [5]. In this study, where the behavior of the composite panel is of a
transversely isotropic material, the total number of independent elastic constants will reduce to
four such as𝐸1, 𝐸2, 𝜈12 and 𝐺12by using composite laminate theory. Strictly, the composite panel
may not be manufactured as per the governing principals related to the mechanics of composite
materials due to the following uncertainties to manufacture the composite panel:
1. The internal structure of the laminate such as voids, uneven laminate thickness, non-
uniform lamina, resin rich regions and more.
2. Alignment of the fibres in the lamina.
The Rule of Mixture requires the values of density and fibre volume fraction 𝑉𝑓 as an input
for the estimation of the elastic properties of the composite panel. Density of the composite panel
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is determined 1473 Kg/m3 by using a hydrostatic method. The fibre volume fraction 𝑉𝑓of the
composite panel is found to be 65 % calculated from the burn-off method as per the ASTM
Standard D2584 [90]. The fibre volume fraction 𝑉𝑓is also calculated by using Thermogravimetric
analysis (TGA) procedure using Shimadzu DTG-60/60H apparatus [91] as shown in figure 7.3.
Through the TGA method, the Weight Fraction (𝑊𝑓) of Carbon Fibre for the composite panel is
determined as 55 % against the total weight of the composite panel. Then fibre volume fraction
𝑉𝑓 is determined by using the expression given in equation 7.1. The fibre volume fraction 𝑉𝑓
determined by the TGA method is found in close proximity with the burn-off method as
graphically illustrated in figure 7.3.
𝑉𝑓 =𝑊𝑓 𝜌𝑓
𝑊𝑓 𝜌𝑓 + 𝑊𝑚 𝜌𝑚 (7.1)
Shimadzu DTG-60 / DTG-60H TGA Graph
Figure 7. 3 Thermogravimetric analysis
The detailed procedure for the estimation of independent elastic constants using “Rule of
Mixture” is provided in Appendix C and obtained values are given in table 7-2.
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Table 7. 2 Elastic Properties of Composite Panel
Name Symbol Value Unit
Density of the composite 𝜌 1473 Kg/m3
Tensile Modulus 𝐸11 173.57 MPa
Transverse Modulus 𝐸22 17.54 MPa
Poisson’s ratio in the 1-2 direction 𝜇12 0.28
Shear modulus in the 1-2 direction 𝐺12 6.50 MPa
Shear modulus in the 1-3 direction 𝐺13 6.50 MPa
Shear modulus in the 2-3 direction 𝐺23 5.83 MPa
7.4 Preparation of Test Specimens
All the tensile test specimens are prepared from already fabricated composite panels. The
cutting of tensile specimens has been carried out by the slitting process on Universal Milling
Machine (XQ6232WA) using Slitting Saw of 2.5 mm. The tensile specimens have been prepared
by strictly following the (Standard Test Method for Open-hole Tensile Strength of Polymer Matrix
Composite Laminates) [92]. The dimensions of the tensile specimens are 260 mm in length
(including 128 mm of gauge length), 32 mm in width and 2 mm in thickness having a central
circular hole of 3.175 mm diameter as shown in figure 7.4.
Figure 7. 4 Dimensions of the test coupon
The drilling of the hole is carried out on a computerized numerically controlled (CNC)
milling machine (MV1060) with carbide drill bits as shown in figure 7.5. During drilling an
aluminum plate is used as a backup plate to avoid interlaminar delamination and bending of the
85
composite panel especially in the last lamina. Spindle speed for drilling of holes is maintained as
2000 rev/min with a feed rate of 0.1 mm/rev.
Figure 7. 5 Drilling of Hole
7.5 Tensile Testing
Tensile testing is performed in accordance with ASTM D3039 Standard (standard test
method for tensile properties of polymer matrix composite materials) [93] by using Universal
Testing Machine (WDW 100E) as illustrated in figure 7.6. The tensile test is conducted to evaluate
the elongation (or the change) in gauge length under the applied load. The tensile specimen is
loaded with a constant displacement rate of 0.18 mm/min. The far-field reference strain has been
calculated using the displacement rate and the gauge length until failure of the tensile specimen.
Figure 7. 6 Universal Tensile Machine (UTM)
The localized strain has been determined at locations precisely close with the periphery of
the hole at an angle 𝜋 2⁄ along the direction of load. In this study, the strain gauges are used to
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determine the strains. Since strain is a measure of change in length to the original length ratio,
therefore it is a dimensionless quantity. However, its magnitude is determined in microstrains (με).
The strain gauge (CEA-06-032UW-120) [94] with the gauge resistance of 120±1Ω connected in
full Wheatstone bridge circuit has been used for the measurement of localized strain and bonded
with M-200 bond at the periphery of the hole at an angle of 𝜋 2⁄ as illustrated in the figure 7.7. The
Wheatstone bridge circuit and the data acquisition loop has been illustrated in figures 7.8 and 7.9
respectively.
Figure 7. 7 Strain gauges for estimation of strain
Figure 7. 8 Wheatstone bridge circuit with one active gauge
87
Figure 7. 9 Data Acquisition loop for localized strain measurement
7.6 Experimental Results
The tensile test specimen is speckled in an ultimate tensile machine (UTM) and loaded
with a constant displacement rate during which strain measurements are performed by using SM
1010 Strain Gauge Meter [95]. Strain Gauge Meter measures the strain fields around the vicinity
of the hole subjected to tensile loading. The strain values obtained from the UTM are plotted with
respect to time scale which is graphically shown in figure 7.10. These strain values are generally
the overall strain values which developed in the tensile specimen and therefore these values are
considered as the normalized or far field reference strain values of the tensile specimen. These are
not those strain values which are caused in the vicinity of the stress raiser (central circular hole in
this case).
Figure 7. 10 Far field strain measured by UTM
The strains developed in the vicinity of the central circular hole under tensile loading is
measured by using strain gauges shown in figure 7.11 (a) and the obtained values are graphically
plotted with respect to time scale as shown in figure 7.11 (b). Certainly, the magnitude of these
localized strain values is higher than the far field reference strain values in the test specimen.
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a. Strain measurement b. Localized strain values upon loading
Figure 7. 11 Localized strain measured by the strain gauge
Both the strain values localized (measured by the strain gauges) and far field (measured by
the UTM) are graphically represented in one plot are shown in figure 7.12.
Figure 7. 12 Localized strain (blue), far field (red)
Since the SCF is defined as the ratio of the localized stress-strain concentrations caused
due to the presence of the stress raiser over the far field reference stress-strain concentrations in a
structural panel, so the SCF is calculated for this case and graphically shown in figure 7.13.
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Figure 7. 13 Experimental SCF
The experimentally calculated SCF plotted with the FE SCF obtained from the FE
representations shown in figure 7.14. Large displacement can be seen in the case of experimental
SCF, which is mainly caused by the slippage of the test specimen. Further, in FE representations,
the strain concentration is measured exactly at the periphery of the hole, whereas this was not
possible due to the width of the strain gauge itself. However, the pattern for both the experimentally
calculated SCF and obtained from the FE representations is similar. It is also evident that the
reduction in the magnitude of the progressive-damaged SCF is mainly because of the fact that
material crosses its elastic range and this happens due to degradation in the elastic properties of
the material.
90
Figure 7. 14 Experimental SCF (red), FE SCF (blue)
7.7 Summary
Experimental testing is performed by using tensile test specimens of newly manufactured
composite panels. The prepregs were manufactured by using a solvent impregnation method and
then composite panels were fabricated by the autoclave process. All the composite panels are
fabricated with 16 layers of laminae in UD configuration. Material properties of this newly
developed composite are also calculated using the Rule of Mixture method. The cutting of test
specimens is carried out with Slitting Saw of 2.5 mm on the Universal Milling Machine
(XQ6232WA) and then drilling of holes is performed with carbide drill bits on CNC milling
machine (MV1060). The test specimen is speckled in an ultimate tensile machine (UTM) and
loaded with a constant displacement rate of 0.18 mm/min. The strain values obtained from the
UTM are the overall or far field reference strain values. Whereas the strains developed in the
vicinity of the central circular hole is measured by using strain gauges. Certainly, the magnitude
of these localized strain values is higher than the far field reference strain values in the test
specimen. The SCF is calculated by using these two strain values (localized strain values over the
far field stain values). The experimental results are comparable with the FE results. However, the
slight variation observed due to the manufacturing / fabrication processing limitations of the
composites which further augmented due to the limitations associated with the specimen cutting
and experimental setup.
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CHAPTER 8: CONCLUSION
8.1 Conclusion
The current study presents a novel systematic FE approach for the estimation of
progressive-damaged SCF of the composite panel containing a central circular hole. In past, the
researchers have investigated the SCF based on the geometrical and loading parameters within the
elastic region. However, the progressive damage response along with its dependency on different
parameters has not yet been explored properly. The apparent reason for this study is because the
composite panels have not been fully treated with elastic and plastic range deformations.
Theoretically, the fibres in a composite panel show a brittle behavior which deforms elastically to
final failure exhibiting either very little or no linear deformation. Whereas matrices generally
experience plastic deformation hence the failure strain in matrice is far higher than the fibres. This
elastoplastic nature of the composite panels presents a very complex behavior for the estimation
of SCF. To capture the elastic and plastic behavior in composite panels, the SCF is categorized
with pre-damaged SCF (analogous to elastic range deformation) and progressive-damaged SCF
(analogous to plastic range deformation) like in an isotropic material. FE representations have been
formulated using Hashin’s damage criteria along with physically based traction separation
relationship within the FE framework to estimate the progressive-damaged SCF. During damage
evolution, the damaged elements will experience a reduction in stiffness properties with regards
to the virgin stiffness properties.
Therefore, the response of a notched composite panel under tensile loading categorized
into three distinct regions. Firstly, the linear elastic region which is characterized by the initial
(virgin) elastic moduli’s of the material. Second, the region where the damage initiation begins
and the material behavior loses its linearity. And the final region is of increased nonlinearity till
the final damage of the composite panel. The study reveals that strength reduction due to the
presence of a stress raiser is not truly proportional to the reverse of the SCF due to the stiffness
reduction in the composite panel. Further, in this study, an effort is made to correlate the
progressive damage response on SCF with other influencing factors such as the hole size, laminate
configurations, laminae orientations, stress distribution around the periphery of the hole, decaying
effect and ligament effect. The study draws the following conclusions:
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1. based on the pre-damaged SCF:
a. The SCF is proportional with the hole size both for anisotropic (composite)
and isotropic panels and the phenomenon is known as the “hole size effect”.
b. The maximum SCF has been observed with 00 laminae orientation followed
by a steady drop till 400 degree orientation of laminae, then a stable trend
has been seen till 900 degree orientation of laminae.
c. The decrease in SCF from deflection of 00 degree orientation along the
loading direction means a reduction in loading capacity. Therefore, from
the design point of view the effective laminae orientation range would
always lie between 00 degree to 400 degree under tensile load.
d. The peak value SCF occurs at an angle of π/2 at the periphery of the central
circular hole. By shifting of angle towards π/2 or 0 around the periphery of
the hole, the intensity of SCF will start declining.
e. The high SCF would be in small ligaments (large w/d ratio) and low SCF
would be in large ligaments (small w/d ratio).
f. The reduction in SCF is large for the case of small ligaments as compared
to the large ligaments.
2. Based on the progressive-damaged SCF:
a. The SCF decreases from its initial (baseline) magnitude on the onset of the
damage evolution process against all the laminae orientations.
b. Prior to the significant decrease in the SCF, there is a very small rise in the
SCF from its initial (baseline) SCF for all the laminae orientations.
c. The same effect occurs for the case of stress distribution as for the pre-
damaged SCF such as the maximum SCF occurs at angles of π/2 and 3π/2
and its intensity reduces with the deflection of angle towards zero and π
angles.
d. A similar decaying trend like in pre-damaged SCF is observed in the
progressive-damaged SCF, where the intensity of SCF decreases away from
the hole.
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e. Four w/d ratios such as 6, 5, 4 and 3 used for the analysis of ligament effect
on the progressive-damaged SCF. The outcomes of the analysis are as
below:
(1) A leaning trend is found in the SCF for w/d ratios of 5 and 6.
(2) An unexpected response in the SCF is observed for w/d ratios of 3
& 4. This occurs due to the joint of stress fields of the hole edge with
stress fields on the edge of boundary width. This effect may be
termed as hole-boundary interaction effect.
Analytical and experimental analysis has also been performed for the comparison of FE
results where applicable. The FE results are found in good agreement with the analytical and
experimental results. Certainly, the study proposes a paradigm shift in design philosophy which at
present is limited to no-damage philosophy especially in aerospace, where the weight savings due
to less generous safety factors can be significant. This paradigm shift based on the fact that in
composites even after the appearance of the internal failure, its propagation detained by the internal
arrangement of the composites. Consequently, the composites could withstand an ample higher
load even after the occurrence of damage.
8.2 Future Work
Future work on the progressive-damaged SCF of the composite panel should concern:
1. Investigation of progressive-damaged SCF under multi-axial loading scenario.
2. Analysis progressive-damaged SCF with different geometry of the stress raiser.
3. Investigation of progressive-damaged SCF with other types of composite materials.
4. Experimental investigation progressive-damaged SCF with different laminate
configurations.
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APPENDIX A
Computer-Aided Design Environment for Composites (CADEC)
To calculate the effective stiffness’s of the laminae a pictorial view is shown obtained as
an Auto-Generated formulation of Computer-Aided Design Environment for Composites
(CADEC) as appended below:
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APPENDIX B: MATLAB PROGRAM FOR STRESS CCONCENTRATION
FACTOR
B1: Material Transformation Program
The material properties of each lamina are entered as per the respective lamina orientation
and to calculate the overall response of the composite panel along the loading direction. The
Matlab Code is appended below:
%Transformed principle engineering constant into Laminate constant % But only for 8 ply clc; clear all; disp('Enter all values in GPa') E11= input('Enter the value of E11= '); E22= input('Enter the value of E22= '); G12 = input('Enter the value of G12= '); v12= input('Enter the value og v12= '); v21=v12*(E22/E11);
% for ply-1 theta1= input('Enter the value of Ply-1 Angle = '); m1=cosd(theta1); n1=sind(theta1); E_1x= (m1^2/E11)*(m1^2-n1^2*v12)+(n1^2/E22)*(n1^2-m1^2*v21)+(m1^2*n1^2/G12); Exx1=1/E_1x; E_1y= (n1^2/E11)*(n1^2-m1^2*v12)+(m1^2/E22)*(m1^2-n1^2*v21)+(m1^2*n1^2/G12); Eyy1=1/E_1y; G_1xy= (4*m1^2*n1^2/E11)*(1+v12)+(4*m1^2*n1^2/E22)*(1+v21)+(m1^2-n1^2)^2/G12; Gxy1=1/G_1xy; v_xy1=Exx1*[(m1^2/E11)*(m1^2*v12-n1^2)+(n1^2/E22)*(n1^2*v21-
m1^2)+(m1^2*n1^2/G12)];
%For ply-2 theta2= input('Enter the value of Ply-2 Angle = '); m2=cosd(theta2); n2=sind(theta2); E_2x= (m2^2/E11)*(m2^2-n2^2*v12)+(n2^2/E22)*(n2^2-m2^2*v21)+(m2^2*n2^2/G12); Exx2=1/E_2x; E_2y= (n2^2/E11)*(n2^2-m2^2*v12)+(m2^2/E22)*(m2^2-n1^2*v21)+(m2^2*n2^2/G12); Eyy2=1/E_2y; G_2xy= (4*m2^2*n2^2/E11)*(1+v12)+(4*m2^2*n2^2/E22)*(1+v21)+(m2^2-n2^2)^2/G12; Gxy2=1/G_2xy; v_xy2=Exx2*[(m2^2/E11)*(m2^2*v12-n2^2)+(n2^2/E22)*(n2^2*v21-
m2^2)+(m2^2*n2^2/G12)];
%For ply-3 theta3= input('Enter the value of Ply-3 Angle = '); m3=cosd(theta3); n3=sind(theta3); E_3x= (m3^2/E11)*(m3^2-n3^2*v12)+(n3^2/E22)*(n3^2-m3^2*v21)+(m3^2*n3^2/G12); Exx3=1/E_3x;
96
E_3y= (n3^2/E11)*(n3^2-m3^2*v12)+(m3^2/E22)*(m3^2-n3^2*v21)+(m3^2*n3^2/G12); Eyy3=1/E_3y; G_3xy= (4*m3^2*n3^2/E11)*(1+v12)+(4*m3^2*n3^2/E22)*(1+v21)+(m3^2-n3^2)^2/G12; Gxy3=1/G_3xy; v_xy3=Exx3*[(m3^2/E11)*(m3^2*v12-n3^2)+(n3^2/E22)*(n3^2*v21-
m3^2)+(m3^2*n3^2/G12)];
%For ply-4 theta4= input('Enter the value of Ply-4 Angle = '); m4=cosd(theta4); n4=sind(theta4); E_4x= (m4^2/E11)*(m4^2-n4^2*v12)+(n4^2/E22)*(n4^2-m4^2*v21)+(m4^2*n4^2/G12); Exx4=1/E_4x; E_4y= (n4^2/E11)*(n4^2-m4^2*v12)+(m4^2/E22)*(m4^2-n4^2*v21)+(m4^2*n4^2/G12); Eyy4=1/E_4y; G_4xy= (4*m4^2*n4^2/E11)*(1+v12)+(4*m4^2*n4^2/E22)*(1+v21)+(m4^2-n4^2)^2/G12; Gxy4=1/G_4xy; v_xy4=Exx4*[(m4^2/E11)*(m4^2*v12-n4^2)+(n4^2/E22)*(n4^2*v21-
m4^2)+(m4^2*n4^2/G12)];
%For ply-5 theta5= input('Enter the value of Ply-5 Angle = '); m5=cosd(theta5); n5=sind(theta5); E_5x= (m5^2/E11)*(m5^2-n5^2*v12)+(n5^2/E22)*(n5^2-m5^2*v21)+(m5^2*n5^2/G12); Exx5=1/E_5x; E_5y= (n5^2/E11)*(n5^2-m5^2*v12)+(m5^2/E22)*(m5^2-n5^2*v21)+(m5^2*n5^2/G12); Eyy5=1/E_5y; G_5xy= (4*m5^2*n5^2/E11)*(1+v12)+(4*m5^2*n5^2/E22)*(1+v21)+(m5^2-n5^2)^2/G12; Gxy5=1/G_5xy; v_xy5=Exx5*[(m5^2/E11)*(m5^2*v12-n5^2)+(n5^2/E22)*(n5^2*v21-
m5^2)+(m5^2*n5^2/G12)];
%For ply-6 theta6= input('Enter the value of Ply-6 Angle = '); m6=cosd(theta6); n6=sind(theta6); E_6x=(m6^2/E11)*(m6^2-n6^2*v12)+(n6^2/E22)*(n6^2-m6^2*v21)+(m6^2*n6^2/G12); Exx6=1/E_6x; E_6y= (n6^2/E11)*(n6^2-m6^2*v12)+(m6^2/E22)*(m6^2-n6^2*v21)+(m6^2*n6^2/G12); Eyy6=1/E_6y; G_6xy= (4*m6^2*n6^2/E11)*(1+v12)+(4*m6^2*n6^2/E22)*(1+v21)+(m6^2-n6^2)^2/G12; Gxy6=1/G_6xy; v_xy6=Exx6*[(m6^2/E11)*(m6^2*v12-n6^2)+(n6^2/E22)*(n6^2*v21-
m6^2)+(m6^2*n6^2/G12)];
%For ply-7 theta7= input('Enter the value of Ply-7 Angle = '); m7=cosd(theta7); n7=sind(theta7); E_7x=(m7^2/E11)*(m7^2-n7^2*v12)+(n7^2/E22)*(n7^2-m7^2*v21)+(m7^2*n7^2/G12); Exx7=1/E_7x; E_7y= (n7^2/E11)*(n7^2-m7^2*v12)+(m7^2/E22)*(m7^2-n7^2*v21)+(m7^2*n7^2/G12); Eyy7=1/E_7y; G_7xy= (4*m7^2*n7^2/E11)*(1+v12)+(4*m7^2*n7^2/E22)*(1+v21)+(m7^2-n7^2)^2/G12; Gxy7=1/G_7xy;
97
v_xy7=Exx7*[(m7^2/E11)*(m7^2*v12-n7^2)+(n7^2/E22)*(n7^2*v21-
m7^2)+(m7^2*n7^2/G12)];
%For ply-8 theta8= input('Enter the value of Ply-8 Angle = '); m8=cosd(theta8); n8=sind(theta8); E_8x=(m8^2/E11)*(m8^2-n8^2*v12)+(n8^2/E22)*(n8^2-m8^2*v21)+(m8^2*n8^2/G12); Exx8=1/E_8x; E_8y= (n8^2/E11)*(n8^2-m8^2*v12)+(m8^2/E22)*(m8^2-n8^2*v21)+(m8^2*n8^2/G12); Eyy8=1/E_8y; G_8xy= (4*m8^2*n8^2/E11)*(1+v12)+(4*m8^2*n8^2/E22)*(1+v21)+(m8^2-n8^2)^2/G12; Gxy8=1/G_8xy; v_xy8=Exx8*[(m8^2/E11)*(m8^2*v12-n8^2)+(n8^2/E22)*(n8^2*v21-
m8^2)+(m8^2*n8^2/G12)];
%output commands
Ex= (Exx1/8)+ (Exx2/8)+( Exx3/8)+( Exx4/8 )+
(Exx5/8)+(Exx6/8)+(Exx7/8)+(Exx8/8); Ey= (Eyy1/8)+ (Eyy2/8)+( Eyy3/8)+( Eyy4/8 )+
(Eyy5/8)+(Eyy6/8)+(Eyy7/8)+(Eyy8/8); Gxy =
(Gxy1/8)+(Gxy2/8)+(Gxy3/8)+(Gxy4/8)+(Gxy5/8)+(Gxy6/8)+(Gxy7/8)+(Gxy8/8); Vxy=
(v_xy1/8)+(v_xy2/8)+(v_xy3/8)+(v_xy4/8)+(v_xy5/8)+(v_xy6/8)+(v_xy7/8)+(v_xy8/
8);
fprintf('The value of Ex = %f Gpa\n',Ex) fprintf('The value of Ey = %f Gpa\n',Ey) fprintf('The value of Gxy = %f Gpa\n',Gxy) fprintf('The value of Vxy = %f Gpa\n',Vxy)
B2: Stress Concentration Factor Calculator
The Matlab Code is generated to calculate the SCF at the point of interest in the FRPC
laminate STC containing a central circular hole.
%To find stress concentration factor
clc;
clear all;
disp('Enter all values in GPa')
Exx= input('Enter the value of Exx= ');
Eyy= input('Enter the value of Eyy= ');
Gxy = input('Enter the value of Gxy= ');
vxy= input('Enter the value og vxy= ');
R= input('Enter the ratio 2a/w=');
M2= (sqrt(1-8*(((3*(1-R))/(2+(1-R)^3))-1))-1)/(2*(R)^2);
M=sqrt(M2);
fprintf( 'The magnification factor is = %f\n',M)
KT_inf= 1+ sqrt(2*(((sqrt(Exx/Eyy)))-vxy+(Exx/(2*Gxy))));
98
fprintf('The infinite stress concentration factor is = % f\n',KT_inf)
K_T= (3*( 1-R )/(2+(1-R)^3)) + (0.5*( R*M )^6)*(KT_inf-3)*( 1-(R*M)^2);
k_t= (1/K_T)* KT_inf;
fprintf('The stress concentration factor is = % f\n',k_t)
k_t_net=(k_t)*(1-R);
fprintf('The net stress concentration factor is= %f\n', k_t_net')
99
APPENDIX C: Elastic Constants for Anisotropic Materials
C.1 Anisotropic Material [5]
The purpose of this appendix is to describe the elastic behavior of the composite panel. The
number of independent elastic constants required to fully describe at various levels for the
materials including isotropic is tabulated in table C-1.
Table C. 1 Number of Constants Needed at Various Levels
General Anisotropic
81
36 With symmetry of stress and strain tensors
21 With strain energy considerations
Orthotropic 9
Transversely Isotropic 5
Plane Stress Lamina 4
Isotropic 2
The generalized Hooke’s law represents a stress state at a point in general continuum with
81 elastic constants, however, due to the stress-strain symmetries, these elastic constants are
reduced to 36 entities. For composite materials generally contracted notations 𝜏 for the shear stress
and engineering strains 𝛾𝑖𝑗 = 2 𝜖𝑖𝑗 are used. Then the (6 x 6) stiffness matrix [C] with 36 entities
is used to describe the stress strain relationship as given below:
[ 𝜎1
𝜎2𝜎3
𝜏23𝜏13
𝜏12]
=
[ 𝐶11 𝐶12 𝐶13 𝐶14 𝐶15 𝐶16
𝐶21 𝐶22 𝐶23 𝐶24 𝐶25 𝐶26
𝐶31
𝐶41
𝐶51
𝐶61
𝐶32
𝐶42
𝐶52
𝐶62
𝐶33
𝐶43
𝐶53
𝐶63
𝐶34
𝐶44
𝐶54
𝐶64
𝐶35
𝐶45
𝐶55
𝐶65
𝐶36
𝐶46
𝐶56
𝐶66]
[ 𝜖1
𝜖2𝜖3
𝛾23𝛾13
𝛾12]
(C.1)
Also, the compliance matrix [S] is the inverse of the stiffness matrix as expressed below:
𝜎𝑖 = ∑𝐶𝑖𝑗 𝜖𝑗 and 𝜖𝑖 = ∑ 𝑆𝑖𝑗 𝜎𝑗, here i, j = 1, 2, 3, 4, 5, 6 (C.2)
100
Since the stiffness and compliance matrices are symmetric, therefore only 21 independent
elastic constants would be required under generalized Hooke’s Law.
C.2 Orthotropic Lamina [5]
The basic building block in a composite panel is lamina. Orthotropic materials constitute
three mutually perpendicular planes of symmetry shown in figure C-1. Therefore, for the
representation of orthotropic lamina, a total of 9 independent elastic constants will be needed.
a. UD Lamina b.
c. Planes of Symmetry
Figure C. 1 Orthotropic Lamina
The components of the compliance and stiffness matrices for the case of orthotropic
material principal directions are related with the engineering constants as shown below:
[𝑆] =
[
1
𝐸1−
𝜈21
𝐸2
− 𝜈31
𝐸3
− 𝜈12
𝐸1
1
𝐸2
− 𝜈32
𝐸3
− 𝜈13
𝐸1−
𝜈23
𝐸2
1
𝐸31
𝐺231
𝐺131
𝐺12]
(C.3)
101
[𝐶] = [𝑆]−1 =
[
1 − 𝜈23 𝜈32
𝐸2 𝐸3 Δ
𝜈12 + 𝜈13 𝜈32
𝐸1 𝐸3 Δ
𝜈13 + 𝜈12 𝜈23
𝐸1 𝐸2 Δ𝜈21 + 𝜈31 𝜈23
𝐸2 𝐸3 Δ
1 − 𝜈13 𝜈31
𝐸1 𝐸3 Δ
𝜈23 + 𝜈21 𝜈13
𝐸1 𝐸2 Δ𝜈31 + 𝜈21 𝜈32
𝐸2 𝐸3 Δ
𝜈32 + 𝜈12 𝜈31
𝐸1 𝐸3 Δ
1 − 𝜈12 𝜈21
𝐸1 𝐸2 Δ 𝐺23𝐺13
𝐺12]
(C.4)
Whereas:
Δ = 1 − 𝜈12 𝜈21 − 𝜈23 𝜈32 − 𝜈31 𝜈13 − 2 𝜈21 𝜈32 𝜈13
𝐸1 𝐸2 𝐸3 (C.5)
Whereas 𝐸𝑖 represents tensile and compressive modulus of elasticity along i direction. The
𝐺𝑖𝑗represents shear modulus on the plane of axis i – j. the 𝜈𝑖𝑗represents or expansion or contraction
in j direction when the applied load in i direction. Here a vital relationship exists between the
Young’s modulus and Poisson’s ratios shown below:
𝐸𝑖
𝜈𝑖𝑗=
𝐸𝑗
𝜈𝑗𝑖 Here i, j = 1, 2, 3 and 𝑖 ≠ 𝑗 (C.6)
C.3 Transversely Isotropic Lamina [5]
The continuous fibres considered in this study are of circular cross-sections. The
distribution of fibres can be typically idealized as cross-sectional arrays with square, rectangular,
hexagonal or layer-wise distribution against the actual fibre distribution as shown in figure C.2.
a. Idealized Hexagonal b. Actual (Random)
Figure C. 2 Fibre distribution in cross-section of the lamina
The 𝑉𝑓fibre volume fraction for the case of hexagonal arrangement [5] is expressed as:
102
𝑉𝑓 = 𝜋
2 √3 (
𝑟
𝑅)2
(C.7)
Here r denotes the fibre radius and 2R indicated the distance between the fibres. The 0.907
is considered to be the maximum possible volume fraction against the hexagonal arrangement. For
the case of a hexagonal arrangement, it needs only five independent elastic constants. Considering
fibre direction along axis 1 and axes 2′- 3′ obtained by the rotation about axis 1 as shown in figure
C.3. Hence the plane 2 – 3 is considered to be the plane of isotropy by rotation about axis 1 along
the fibre direction. Then the independent elastic constants will be 𝐸1, 𝐸2, 𝜈12, 𝐺12 and 𝐺23 as shown
in table C-2.
Figure C. 3 Material Axis
C.4 Estimation of Elastic Constants
Generally, the properties of the composite panel are based on the constituent materials such
as matrix properties and fibre properties. The intended elastic properties are commonly reduced to
four in-plane independent elastic constants as𝐸1, 𝐸2, 𝜈12 and 𝐺12using the composite laminate
Table C. 2 Elastic Constants of UD Transversely Isotropic Lamina
Nomenclature Independent Constants Dependent Constants
Young’s Moduli 𝐸1
𝐸2 𝐸2 = 𝐸3
Poisson’s Ratio 𝜈12 𝜈12 = 𝜈13
Shear Moduli 𝐺12 𝐺12 = 𝐺13
𝐺23 𝐺23 =
𝐸2
2 (1 + 𝜈23)
103
theory. Further the composite properties influenced by the fibre packing arrangement and the fibre
volume fraction. For relatively simple formulations and quick calculations the “Rule of Mixture”
is considered appropriate based on the considerations of mechanics of material. The rule of mixture
is highly dependent on the fibre volume fraction 𝑉𝑓 and matrix volume fraction 𝑉𝑚.
In this research work, constituent materials used for the manufacturing of composite panel
are Carbon fibres (TC 36S – 12K of Tairyfil) as reinforcement and Epoxy (ESP – 135 A / B of
EPORITE) as Matrix. Prepregs are manufactured by using solvent impregnation method of an
average thickness of 0.125 mm. Then UD configuration is fabricated by 16 layers of laminae to
achieve an overall thickness of 2 mm through the autoclave process. Mechanical properties of the
constituent materials generally obtained from the manufacturers are given in table C-3.
Table C. 3 Properties of Constituent Materials
Carbon Fibre Epoxy Matrix
Tensile
Modulus
Density Poisson’s
Ratio
Tensile
Modulus
Density Poisson’s
Ratio
𝐸𝑓 (GPa) 𝜌𝑓 (Kg/m3) 𝜐𝑓 𝐸𝑚 (GPa) 𝜌𝑚 (Kg/m3) 𝜐𝑚
250 1.81 0.2 3.45 1.2 0.36
The fibre volume fraction 𝑉𝑓of the composite panel is found to be 65 % calculated from
the burn-off method as per the ASTM Standard D2584 [90]. Further for the validation, the fibre
volume fraction 𝑉𝑓is evaluate using Thermogravimetric analysis (TGA) procedure using
Shimadzu DTG – 60, DTG – 60H apparatus as shown in figure C.4. In that Weight Fraction (𝑊𝑓)
of Carbon Fibre has been determined, which is estimated to be 60 % of the total composite panel
weight. Then using expression given in equation (C.8), the fibre volume fraction 𝑉𝑓 found in close
proximity of 65 % as graphically shown in figure C.4.
104
Shimadzu DTG-60 / DTG-60H TGA Graph
Figure C. 4 Thermogravimetric analysis
𝑉𝑓 =𝑊𝑓 𝜌𝑓
𝑊𝑓 𝜌𝑓 + 𝑊𝑚 𝜌𝑚 (C.8)
Density of the composite laminate is determined 1473 Kg/m3 by using a technique for the
hydrostatic method. Further to fully model a composite panel in Abaqus six mechanical properties
𝐸1, 𝐸2, 𝜈12, 𝐺12, 𝐺13, and 𝐺23 are required.
C.4.1 Tensile Modulus in loading direction (𝑬𝟏):
The Tensile Modulus in the loading direction (𝐸1) has been calculated using well-known
Rule of Mixture as given in equation C.9.
𝐸1 = (1 − 𝑉𝑓)𝐸𝑚 + 𝑉𝑓 𝐸𝑓 (C.9)
C.4.2 Transverse Modulus in 2 direction (𝑬𝟐):
Transverse Modulus (𝐸2) is calculated using semi empirical Halpin – Tsai relationship
given in equation C.10.
𝐸2 = 𝐸𝑚 (1 + 휁 휂 𝑉𝑓)
1 − 휂 𝑉𝑓 (C.10)
105
For the Carbon Fibre Laminate (휁 = 1), whereas (휂) is calculated using expression given
in equation C.11.
휂 =
𝐸𝑓
𝐸𝑚− 1
𝐸𝑓
𝐸𝑚+ 휁
(C.11)
To calculate the Shear Modulus in the 1-2 direction(𝐺12), the semi empirical Halpin – Tsai
relationship given in equation C.12, is used.
𝐺12 = 𝐺𝑚 1 + 휁 휂 𝑉𝑓
1 − 휂 𝑉𝑓 (C.12)
Whereas, (휁 = 1) and (휂) has been calculated using expression given in equation C.13.
휂 =
𝐺𝑓
𝐺𝑚− 1
𝐺𝑓
𝐺𝑚+ 휁
(C.13)
Here Shear Modulus of Fibre (𝐺𝑓) and Shear Modulus of Matrix (𝐺𝑚) are calculated using
expressions given in equation C.14 and equation C.15, respectively.
𝐺𝑓 = 𝐸𝑓
2 (1 + 𝜈𝑓) (B.14)
𝐺𝑚 = 𝐸𝑚
2 (1 + 𝜈𝑚) (B.15)
Further, being an in-plane composite panel(𝐺13) is taken equivalent to (𝐺12) and (𝐺23) is
calculated using semi empirical Halpin – Tsai relationship as given in equation C.16.
𝐺23 = 𝐸2
2 (1 + 𝜈23) (C.16)
Here Poisson’s Ratio (𝜈23) is calculated expression given in equation C.17.
𝜈23 = 1 − 𝜈21 − 𝐸2
3𝐾 (C.17)
Here (𝜈21) is calculated expression given in equation C.18.
106
𝜈21 = 𝜈12 (𝐸2
𝐸1) (C.18)
Whereas, (𝜈12) is calculated expression given in equation C.19, as well as experimentally
by expression given in equation C.20 using Universal Testing Machine (UTM) WDW-100E (UTM
100kN) with strain gages placed in the loading direction and transverse direction as shown in figure
C.6 and graphically illustrated in figure C.7. Both analytical and experimental results are fund in
good agreement.
Specimen UTM
Figure C. 5 Two strain gauges for Poisson’s Ratio
107
Figure C. 6 Tensile and compression response of two strain gauges for Poisson’s Ratio
𝜈12 = 𝑉𝑓 𝜈𝑓 + (1 − 𝑉𝑓) 𝜈𝑚 (C.19)
𝜈12 = 휀2
휀1 (C.20)
The Bulk Modulus for the composite panel (K), the given in equation C.21 is used.
𝐾 = [𝑉𝑓
𝐾𝑓+
(1 − 𝑉𝑓)
𝐾𝑚]
−1
(C.21)
Whereas, the values for Bulk Modulus Fibre (𝐾𝑓) and Bulk Modulus Matrix (𝐾𝑚) are
calculated using expressions given in equation C.22 and equation C.23, respectively.
𝐾𝑓 = 𝐸𝑓
3 (1 − 2 𝜈𝑓) (C.22)
𝐾𝑚 = 𝐸𝑚
3 (1 − 2 𝜈𝑚) (C.23)
108
The mechanical properties of the composite panel are given in table C-4.
Table C. 4 Mechanical Properties of FRPC Laminate
𝐸1 𝐸2 𝜈12 𝐺12 𝐺13 𝐺23
GPa GPa GPa GPa GPa
173.57 17.54 0.28 6.50 6.50 5.83
For the Progressive Damage analysis using Hashin Failure Criterion further following
strength and fracture properties for the composite panel are required.
109
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