PREDICTING THE FATIGUE BEHAVIOUR OF …...containing the nano-silica particles and/or micro-rubber particles. The theoretical studies employed an extended finite element method, coupled

PREDICTING THE FATIGUE BEHAVIOUR OF

MATRICES AND FIBRE-COMPOSITES

BASED UPON MODIFIED EPOXY POLYMERS

A thesis submitted for the degree of Doctor of Philosophy of Imperial College

London

and the Diploma of Imperial College

February 2012

By Jibumon B Babu

Department of Mechanical Engineering

Imperial College London

i

Abstract

The present research work has studied the fatigue behaviour of matrices and

fibre-composites based upon modified epoxy polymers. The basic epoxy

polymer has been modified with (a) nano-silica particles, (b) micrometre-rubber

particles, and (c) both of these additives, to give a ‘hybrid’ modified epoxy.

These modifications have been undertaken in order to try to increase the cyclic

fatigue resistance of the fibre-composite material. The experimental work has

used a linear elastic fracture mechanics (LEFM) approach to firstly ascertain the

fatigue properties of the epoxy polymer matrices. Secondly, the unmodified (i.e.

control) and the modified epoxy resins were used to fabricate glass fibre

reinforced plastic (GFRP) composite laminates by a resin infusion under flexible

tooling (RIFT) manufacturing method. Tensile cyclic fatigue tests were

performed on these composites, during which the degree of matrix cracking and

stiffness degradation were also monitored. The fatigue life of the GFRP

composite was significantly increased due to the presence of the nano-silica

particles and/or micro-rubber particles. Suppressed matrix cracking and a

reduced crack propagation rate in the modified matrix of the fibre-composite

were observed to contribute towards the enhanced fatigue life of the composites

containing the nano-silica particles and/or micro-rubber particles. The

theoretical studies employed an extended finite element method, coupled with a

cohesive zone model, to predict the fatigue behaviour of the fibre composites

based upon the unmodified (i.e. control) and modified epoxy polymer matrices.

A ‘user element subroutine’ has been developed in Abaqus to incorporate the

extended finite element method and a mathematical model has been proposed

to evaluate the constitutive laws for the cohesive zone model to simulate the

growth of fatigue cracks. A fatigue degradation strategy based on the ‘Paris law’

(determined from the fatigue tests on the matrix materials) has been adopted to

change the constitutive law for the cohesive zone model as a function of the

number of fatigue cycles that have been accumulated. The theoretical

predictions for the fatigue behaviour have been compared to the experimental

results, and very good agreement between the theoretical and experimental

results was found to exist.

ii

Acknowledgment

I am deeply indebted to Prof A.J. Kinloch, Prof Felicity Guild and Dr A. C. Taylor

for their generous effort, useful advice and dedication with respect to the

supervision of my Ph.D research. I would like to thank Prof A.J. Kinloch and

Prof Felicity Guild for their relentless and continuous assistance regarding all

aspects of the project. I would also like to thank Imperial College London for

providing the opportunity that has enabled me to pursue my higher studies in

one of the most prestigious institutions in the world.

It is a pleasure to thank many people who have helped me during my Ph.D

studies. I would like to give my special thanks to Dr. Manjunatha, Ms.

Shamsiah, Dr. Hsieh and Dr. Masania for kindly sharing some of their data and

for their guidance in the experimental aspects of my research. I am also grateful

to Mr. Alvarez and Mr. Brett for many useful discussions.

I am very grateful to the laboratory staff of the Mechanical Engineering and

Aeronautics Departments for providing laboratory facilities. I am also grateful to

my friends Giannis, Fendi, Hari, Sorates, Nanke, Idris, Catrin, Tim, Paul and

Ruth in the research office for providing a good working environment and an

amazing time. I also wish to thank Sandeep, Sahu, Linash, Davendu, Aditya,

Anant and Ahmad, and many more friends whose names are not mentioned

due to space constraints, for making my time at Imperial so enjoyable.

Last, but not least, I would like to thank my family for their support during my

studies. I would especially like to express my sincere gratitude to my parents

and to my sister for their encouragement and support. Finally, I would like to

thank my wife for her love and patience.

iii

Contents

Abstract ............................................................................................................ i

Acknowledgment ............................................................................................. ii

Contents ......................................................................................................... iii

List of Figures ................................................................................................. ix

List of Tables ................................................................................................ xix

Nomenclature ............................................................................................... xxi

Abbreviations ................................................................................................ xxi

English Alphabet ..........................................................................................xxii

Greek Alphabet ........................................................................................... xxv

1. Introduction and Objectives

1.1 Introduction ................................................................................................ 1

1.2 Damage in Composites ............................................................................. 3

1.3 Modelling Techniques................................................................................ 4

1.4 Objectives of the Present Research .......................................................... 5

1.5 Structure of the Thesis .............................................................................. 5

2. Literature Review

2.1 Introduction ................................................................................................ 7

2.2 Fracture Mechanics ................................................................................... 7

2.2.1 Linear Elastic Fracture Mechanics (LEFM) ......................................... 7

2.2.2 Modes of fracture ................................................................................ 8

2.2.3 Griffith's criterion ................................................................................. 9

2.2.4 Stress intensity factor ........................................................................ 10

2.3 Damage ................................................................................................... 11

2.3.1 Static damage ................................................................................... 11

2.3.2 Fatigue damage ................................................................................ 12

iv

2.4 Methods of Finite Element Analysis......................................................... 17

2.4.1 Introduction ....................................................................................... 17

2.4.2 The Virtual Crack Closure Technique (VCCT) method ..................... 17

2.4.3 The Cohesive Zone Model (CZM) method ........................................ 19

2.4.3.1 Introduction................................................................................. 19

2.4.3.2 Pure mode loading ..................................................................... 23

2.4.3.3 Mixed-mode loading ................................................................... 23

2.4.4 The Energy method .......................................................................... 25

2.5 Modelling Damage and the Life Time under Cyclic Fatigue Loading ....... 28

2.5.1 Quasi-static loading .......................................................................... 28

2.5.2 Fatigue loading ................................................................................. 31

2.6 Concluding Remarks ............................................................................... 39

3. Experimental Techniques

3.1 Introduction .............................................................................................. 41

3.2 Materials .................................................................................................. 42

3.3 Preparation of Epoxy Matrix Polymer Specimens.................................... 42

3.3.1 Introduction ....................................................................................... 42

3.3.2 Preparation of plates ......................................................................... 43

3.3.3 Single edge notched bending (SENB) specimens ............................ 43

3.3.4 Compact tension (CT) specimens ..................................................... 44

3.4 Preparation of GFRP Composite Specimens .......................................... 44

3.4.1 Introduction ....................................................................................... 44

3.4.2 Resin infusion under flexible tooling (RIFT) ...................................... 44

3.4.3 Double cantilever beam (DCB) specimens ....................................... 45

3.4.4 Composite strip specimens ............................................................... 46

3.5 Test Methods for the Epoxy Matrix Polymer Specimens ......................... 46

v

3.5.1 Introduction ....................................................................................... 46

3.5.2 Single edge notched bending (SENB) tests ...................................... 47

3.5.3 Compact tension (CT) tests .............................................................. 48

3.6 Test Methods for the GFRP Composite Specimens ................................ 49

3.6.1 Introduction ....................................................................................... 49

3.6.2 DCB tests: Quasi-static tests ............................................................ 50

3.6.3 DCB tests: Cyclic-fatigue tests .......................................................... 52

3.6.4 Composite strip laminate tests: Quasi-static and fatigue tests .......... 53


4. Theoretical Techniques

4.1 Introduction .............................................................................................. 55

4.2 Quasi-Static Analysis............................................................................... 55

4.2.1 The Virtual Crack Closure Technique (VCCT) .................................. 55

4.2.2 The cohesive zone law ..................................................................... 57

4.2.2.1 Kinematics .................................................................................. 57

4.2.2.2 Constitutive laws ........................................................................ 59

4.2.2.3 Mathematical formulation ........................................................... 60

4.2.3 A bi-linear cohesive zone law ........................................................... 66

4.2.3.1 Norm of displacement jump tensor ............................................. 68

4.2.3.2 Damage ...................................................................................... 69

4.2.3.3 Mixed-mode loading: onset of crack growth ............................... 70

4.2.3.4 Mixed-mode loading: crack propagation ..................................... 71

4.2.3.5 Mode-mixity ................................................................................ 73

4.3 Fatigue Analysis ...................................................................................... 74

4.3.1 Introduction ....................................................................................... 74

4.3.2 Degradation strategies ...................................................................... 76

vi

4.3.3 Static damage evolution under fatigue loading ................................. 77

4.3.4 Fatigue damage evolution ................................................................. 78

4.3.5 Damage Analysis .............................................................................. 84

4.3.5.1 The cycle jump strategy ............................................................. 84

4.3.5.2 Displacement ratio and load ratio ............................................... 84


5. Experimental Results and Theoretical Modelling Studies

5.1 Introduction .............................................................................................. 87

5.2 Elastic Properties of Materials ................................................................. 89

5.2.1 Elastic properties of the lamina ......................................................... 89

5.2.2 Elastic properties of the composite ................................................... 92

5.2.2.1 Elastic properties of DCB ........................................................... 94

5.2.2.2 Elastic properties of composite strip ........................................... 96

5.2.3 Elastic properties of aluminium and steel ........................................ 100

5.3 Criteria for Cohesive Zone Modelling .................................................... 100

5.3.1 Mesh sensitivity analysis ................................................................. 100

5.3.2 Initial value of the cohesive zone law parameters ........................... 101

5.3.3 The cohesive zone length ............................................................... 103

5.4 Quasi-Static Models .............................................................................. 104

5.4.1 The SENB test: Experimental and theoretical results ..................... 105

5.4.1.1 SENB results: VCCT analysis .................................................. 106

5.4.1.2 SENB results: Cohesive contact analysis ................................. 107

5.4.1.3 SENB results: Cohesive zone element analysis ....................... 108

5.4.2 The DCB test: Experimental and theoretical analysis ..................... 113

5.4.2.1 DCB results: VCCT analysis..................................................... 115

5.4.2.2 DCB results: Cohesive contact analysis ................................... 116

vii

5.4.2.3 DCB results: Cohesive zone element analysis ......................... 117

5.4.3 The composite material strip test: Experimental and theoretical results ................................................................................................................. 122

5.4.3.1 Normalised stiffness with crack density .................................... 128

5.4.3.2 Normalised stiffness with number of fatigue cycles .................. 133

5.4.3.3 Quasi-static strength of the composite strip.............................. 136

5.5 Toughening Mechanisms ...................................................................... 138

5.6 Fatigue Models ...................................................................................... 139

5.6.1 User element subroutine ................................................................. 139

5.6.1.1 Validation.................................................................................. 140

5.6.2 Fatigue analysis using the user element subroutine ....................... 143

5.6.2.1 The CT test and user element analysis .................................... 144

5.6.2.2 DCB test and user element analysis ........................................ 151

5.6.2.3 Strip test and user element analysis ......................................... 154

5.7 Concluding Remarks ............................................................................. 160

6. Conclusions & Recommendations for Future Work

6.1 Quasi-Static Fracture Properties ........................................................... 161

6.2 Quasi-Static Modelling........................................................................... 162

6.3 Fatigue Testing ...................................................................................... 162

6.4 Fatigue Modelling .................................................................................. 162

6.5 The User Element Subroutine Analysis ................................................. 164

6.6 Fatigue Life ............................................................................................ 164

6.7 Recommendations for Further Work...................................................... 165

6.7.1 Unidirectional composite analysis ................................................... 166

6.7.2 Delamination analysis ..................................................................... 166

6.7.3 Modelling process of failure ............................................................ 166

6.7.4 The role of matrix properties ........................................................... 167

viii

References .............................................................................................. 168

Appendix……………………………………………………………………….174

ix

List of Figures

Figure 1.1 Stress-strain characteristics of unidirectional composites: (a) low

stiffness fibres; (b) high stiffness fibres (Talreja [1]). ........................................... 1

Figure 2.1 Different modes of failure in a material (mode I is the opening mode,

mode II is the shear mode and mode III is the tearing mode). ............................ 9

Figure 2.2 (a) Failed specimen (scaling factor, ms=1) and (b) detail of failed

specimen edge (Hallett et al. [9]). ..................................................................... 11

Figure 2.3 Photograph and schematic of delaminations in ply level scaled

specimen (Hallett et al. [9]). .............................................................................. 12

Figure 2.4 Photographs of the development of transverse cracking in (0/90/0)

glass-fibre/epoxy specimens at different strain levels (Berthelot [10]). ............. 12

Figure 2.5 Fatigue load cycle parameters. ....................................................... 13

Figure 2.6 Typical growth rate curve. ............................................................... 14

Figure 2.7 Fatigue damaged edge of (0/90/45) carbon fibre epoxy laminates at:

(a) 2% stiffness reduction (b) 4% reduction (c) 8% reduction (d) 12% reduction

(e) 15% reduction (Reifsnider and Jamison [11]). ............................................. 15

Figure 2.8 Microscopic damage mechanisms resulting from a constant stress

amplitude fatigue test observed via optical microscopy for (a) 10, (b) 100 and (c)

1500 cycles of loading (Hosoi et al. [12]). ......................................................... 16

Figure 2.9 Different cohesive zone model laws. (a)Triangular form (b) Constant

stress form (c) Triangular form (bi-linear law) (d) Elastic constant linear damage

form (e) Linear polynomial form and (f) Elastic constant form (Zou et al. [20]).. 21

Figure 2.10 The bi-linear cohesive zone law. ................................................... 22

Figure 2.11 Fatigue degradation in a bi-linear cohesive zone law. ................... 23

x

Figure 2.12 The minimum load required to propagate a delamination of a

certain size in its most detrimental location (Wimmer and Pettermann [28]). .... 26

Figure 2.13 Comparison of theoretical equation for delamination growth rate

with the test data (Shivakumar et al. [29]). ........................................................ 27

Figure 2.14 Failure modes in laminate (Shivakumar et al. [29]). ...................... 28

Figure 2.15 Transverse crack interference model (Boniface et al. [30]). ........ 29

Figure 2.16 Failure of a composite strip specimen (Huchette [32]). ................. 30

Figure 2.17 Stiffness reduction as a function of the average crack density in the

case of two different laminates. Eo is the initial elastic modulus of the laminate

and Ex is the elastic modulus with a given crack density (Leblond et al. [33]). .. 31

Figure 2.18 Crack multiplication in transverse plies (Leblond et al. [33]).......... 31

Figure 2.19 Fatigue life diagram for a (0/902)s carbon fibre/epoxy matrix

laminate (Akshantala and Talreja [34]). ............................................................ 32

Figure 2.20 Fatigue life diagram for a unidirectional composites under loading

parallel to the fibres (Talreja [1]). ...................................................................... 33

Figure 2.21 The tapping mode atomic force microscopy (AFM) phase images of

the hybrid-epoxy matrix polymer (Manjunatha et al. [39]) (CTBN: carboxy-

termianted butadiene acrylonitrile rubber). ........................................................ 35

Figure 2.22 Transmitted light photographic images of matrix cracking in the

GFRP composites after testing at stress of 150MPa. NR-Neat resin, NRR-Neat

resin with rubber, NRS-Neat resin with silica and NRRS-Neat resin with rubber

and silica (Manjunatha et al. [40]) ..................................................................... 35

Figure 2.23 Crack length versus cycles (Robinson et al. [41]). ........................ 36

Figure 2.24 Experimental relation between the maximum SERR, , and the

number of fatigue cycles, , for the onset of crack growth (Attia et al. [42]). ... 37

xi

Figure 2.25 Evolution of the interface/cohesive traction and the maximum

interface/cohesive strength as a function of the number of cycles for a

displacement jump controlled high-cycle fatigue test (Turon et al. [43]). (Here

interfacial traction is the critical stress of the cohesive zone law at a given

number of fatigue cycles and the interfacial traction is the traction at the

cohesive zone at a given number of fatigue cycles.) ......................................... 38

Figure 3.1 Fabrication of a GFRP sheet using RIFT. ....................................... 45

Figure 3.2 DCB specimen. ............................................................................... 50

Figure 3.3 Experimental setup for a DCB test. ................................................. 51

Figure 3.4 Fibre bridging in DCB test. .............................................................. 51

Figure 3.5 Experimental setup for a fatigue DCB test. ..................................... 53

Figure 4.1 The VCCT model. ........................................................................... 56

Figure 4.2 A four noded cohesive zone element in (a) undeformed state and (b)

deformed state in a global coordinate system. .................................................. 58

Figure 4.3 A four noded cohesive zone element. ............................................. 61

Figure 4.4 The bi-linear cohesive zone law. ..................................................... 66

Figure 4.5 Cohesive law in mode II. ................................................................. 68

Figure 4.6 Variation of damage variable with displacement in a bi-linear

cohesive zone law. ............................................................................................ 70

Figure 4.7 Flow chart of the fatigue analysis embedded in the user element

subroutine. ........................................................................................................ 75

Figure 4.8 Cohesive zone law and energy representation. .............................. 79

Figure 4.9 Representation of energy release under fatigue in a cohesive zone

law. ................................................................................................................... 82

xii

Figure 4.10 Fatigue degradation of cohesive zone law with time. .................... 83

Figure 4.11 Resultant fatigue degradation of a cohesive zone law. Path 1-2

shows the fatigue damage evolution and path 1-3 shows the static and fatigue

damage evolution (Robinson et al. [41]). .......................................................... 83

Figure 4.12 The cycle jump strategy applied to a cohesive zone law approach

to modelling fatigue (Van Paepegem and Degrieck [65]). ................................. 84

Figure 4.13 Experimental and numerically applied displacement/stress in a

displacement/stress controlled fatigue test. In the present study, displacement

controlled fatigue tests were conducted on the CT bulk epoxy material and the

DCB composite materials, whilst stress controlled fatigue tests were conducted

on the composite material strip specimens. ...................................................... 86

Figure 5.1 Overview schematic of the work plan. ............................................. 88

Figure 5.2 A section of a lamina with local coordinates. ................................... 90

Figure 5.3 Equivalent Abaqus and local coordinate system for a DCB

composite specimen. ........................................................................................ 93

Figure 5.4 Equivalent Abaqus and local coordinate system for different lamina

of the composite strip. ....................................................................................... 93

Figure 5.5 Cube with different faces (FF-Face front, FBk-face back, FR-face

right, FL- face left, FT-face top, FB- face bottom). The control point (CP)

boundary condition is at the origin of the coordinate system............................. 94

Figure 5.6 The modelled load versus displacement curves of the DCB

composite specimen for different mesh sizes. The elastic properties of the

unmodified (i.e. control) bulk epoxy matrix and the composite are used for the

analysis (Table 5.13 and Table 5.4). ............................................................... 101

Figure 5.7 Cohesive zone behaviour in (a) undeformed and (b) deformed state.

........................................................................................................................ 102

xiii

Figure 5.8 Cohesive zone elements with cohesive zone length, . .............. 103

Figure 5.9 SENB specimen. .......................................................................... 105

Figure 5.10 Dimensions of the SENB bulk epoxy matrix specimen. ............... 106

Figure 5.11 Loading and boundary condition applied on the SENB specimen

the model. ....................................................................................................... 106

Figure 5.12 The stress field around the crack tip in a SENB FEA model of the

bulk epoxy matrix with cohesive contact in Abaqus. ....................................... 108

Figure 5.13 The stress field around the crack tip in a SENB model with

cohesive zone elements as modelled in FEA Abaqus. ................................... 110

Figure 5.14 Comparison of load-displacement curve for the SENB specimen

based upon the unmodified (i.e. control) epoxy matrix. ................................... 111


based upon the micro-rubber modified epoxy matrix. ..................................... 111


based upon the nano-silica modified epoxy matrix. ........................................ 112

Figure 5.17 Comparison of load-displacement curve for the SENB epoxy

specimen based upon the nano-silica and micro-rubber (i.e. hybrid) modified

epoxy matrix. ................................................................................................... 112

Figure 5.18 Flow chart of quasi-static and fatigue analyses of the composite

material DCB specimens................................................................................. 114

Figure 5.19 DCB specimen ............................................................................ 115

Figure 5.20 Dimensions of the DCB composite material specimen ................ 115

Figure 5.21 Loading and boundary condition applied on the model ............... 116

Figure 5.22 The stress field around the crack tip in a DCB model with cohesive

contact in Abaqus ........................................................................................... 117

xiv


zone elements in Abaqus ................................................................................ 119

Figure 5.24 Comparison of load-displacement curve for the DCB composite

material based upon the unmodified (i.e. control) epoxy matrix ...................... 120


based upon the micro-rubber modified epoxy matrix ...................................... 120


based upon the nano-silica modified epoxy matrix ......................................... 121


based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix

........................................................................................................................ 121

Figure 5.28 Flow chart of the life prediction modelling of the composite strip

under cyclic fatigue loading ............................................................................. 123

Figure 5.29 Composite material strip with dimensions ................................... 124

Figure 5.30 Section of the strip with transverse cracks (a) strip under loading

(b) cross section of the strip with transverse cracks (c) symmetric cross-section

of the strip with transverse cracks ................................................................... 125

Figure 5.31 Dimension of a section of the modelled strip. The length, l, of the

strip model depends on the crack density ....................................................... 126

Figure 5.32 Boundary conditions applied on the symmetric cross-section of the

composite material strip with transverse cracks .............................................. 127

Figure 5.33 Variation of ±45o crack density with number of cycles in composite

material strips based upon unmodified (i.e. control) and modified epoxy

matrices .......................................................................................................... 128

Figure 5.34 Composite material strip with transverse cracks in the Abaqus FEA

method ............................................................................................................ 129

xv

Figure 5.35 Comparison of the normalised stiffness versus the crack density for

the composite strip test. For the composite based upon the unmodified (i.e.

control) epoxy matrix. ...................................................................................... 130


the composite strip test. For the composite based upon the micro-rubber

modified epoxy matrix. .................................................................................... 130


the composite strip test. For the composite based upon the nano-silica modified

epoxy matrix. ................................................................................................... 131


the composite strip test. For the composite based upon the nano-silica and

micro-rubber (i.e. hybrid) modified epoxy matrix. ............................................ 131


the composite strip, based upon the unmodified (i.e. control) and nano-silica

modified epoxy matrices. ................................................................................ 132


the composite strip, based upon the micro-rubber and with both nano-silica and

micro-rubber modified epoxy matrices ............................................................ 132

Figure 5.41 Normalised stiffness versus the number of cycles for the

composite strip based upon the unmodified (i.e. control) epoxy matrix ........... 134


composite strip based upon the micro-rubber modified epoxy matrix ............. 134


composite strip based upon the nano-silica modified epoxy matrix ................ 135


composite strip based upon the nano-silica and micro-rubber (i.e. hybrid)

modified epoxy matrix ..................................................................................... 135

xvi

Figure 5.45 Comparison of the global stress with percentage strain in the

composite strip ................................................................................................ 137

Figure 5.46 Comparison of global stiffness reduction with the percentage strain

of composite strip ............................................................................................ 137

Figure 5.47 A single cohesive zone element for testing ................................. 140

Figure 5.48 Cohesive zone element testing in (a) mode I, (b) mode II and (c)

mixed-mode .................................................................................................... 141

Figure 5.49 Cohesive zone element in mode I. The cohesive zone law

parameters used for the element is for the unmodified (i.e. control) bulk epoxy

matrix ( =3900N/mm2, =10.9 N/mm2 and =75.8J/m2) .............................. 142

Figure 5.50 Cohesive zone element in mode II. The cohesive zone law

parameters used for the element is for the unmodified (i.e. control) bulk epoxy

matrix ( =3900N/mm2, =10.9 N/mm2 and =75.8J/m2) .............................. 142

Figure 5.51 Cohesive zone element in mixed-mode ( =0.5). The cohesive

zone law parameters used for the element is for the unmodified (i.e. control)

bulk epoxy matrix ( =3900N/mm2, =10.9 N/mm2 and =75.8J/m2 for both

modes) ............................................................................................................ 143

Figure 5.52 Compact tension specimen ......................................................... 145

Figure 5.53 Dimension of the CT specimen ................................................... 146

Figure 5.54 Boundary condition applied on the CT specimen ........................ 147

Figure 5.55 Stress field around the crack tip in a CT model with cohesive zone

elements in Abaqus ........................................................................................ 148

Figure 5.56 Growth rate curve for the CT specimen for the bulk unmodified (i.e.

control) epoxy matrix ....................................................................................... 149

Figure 5.57 Growth rate curve for the CT specimen for the bulk micro-rubber


xvii

Figure 5.58 Growth rate curve for the CT specimen for the bulk nano-silica


Figure 5.59 Growth rate curve for the CT specimen for the bulk nano-silica and

micro-rubber (i.e. hybrid) modified epoxy matrix (experimental data from Lee

[49]) ................................................................................................................. 150

Figure 5.60 Growth rate curve for the composite DCB specimen based upon

the unmodified (i.e. control) epoxy matrix ....................................................... 152


the micro-rubber modified epoxy matrix .......................................................... 153


the nano-silica modified epoxy matrix ............................................................. 153


the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix .............. 154

Figure 5.64 Transmitted light photographs of GFRP composite with unmodified

(i.e. control) epoxy matrix showing the sequence of matrix crack development

with the number of cycles, N, under fatigue loading ........................................ 155

Figure 5.65 Applied maximum fatigue stress versus the number of cycles upon

fatigue loading for the composite strip based upon the unmodified (i.e. control)

epoxy matrix .................................................................................................... 158


fatigue loading for a composite strip based upon the micro-rubber modified

epoxy matrix .................................................................................................... 158


fatigue loading for a composite strip based upon the nano-silica modified epoxy

matrix .............................................................................................................. 159

xviii


fatigue loading for a composite strip based upon the nano-silica and micro-

rubber (i.e. hybrid) modified epoxy matrix ....................................................... 159

Figure 6.1 Fatigue crack growth rate curve for the CT epoxy specimen based

upon the unmodified (i.e. control) epoxy matrix .............................................. 163

Figure 6.2 Stress versus number of cycles from the fatigue loading for a

composite material strip based upon the micro-rubber modified epoxy matrix 165

xix

List of Tables

Table 4.1 Description of the material properties of the bulk epoxy matrix

relevant to Figure 4.7. ....................................................................................... 76

Table 5.1 Elastic properties of bulk epoxies and glass fibre ............................ 91

Table 5.2 Unidirectional elastic properties of the different composite lamina

based on the various epoxy matrices ................................................................ 92

Table 5.3 Derivation of homogenised elastic property of DCB from cube

analysis ............................................................................................................. 95

Table 5.4 Elastic properties of the arms the DCB composite specimens for the

various epoxy matrices ..................................................................................... 95

Table 5.5 Equivalent local elastic properties of the lamina of the strip in the

Abaqus coordinate system ................................................................................ 97

Table 5.6 Derivation of the elastic properties of 0o lamina .............................. 97

Table 5.7 Derivation of the elastic properties of 90o lamina .......................... 98

Table 5.8 Derivation of the elastic properties of ±45o lamina ......................... 98

Table 5.9 Elastic properties of 0o lamina .......................................................... 99

Table 5.10 Elastic properties of ±45o lamina .................................................... 99

Table 5.11 Elastic properties of 90o lamina ..................................................... 99

Table 5.12 Elastic properties of aluminium-alloy and steel ............................. 100

Table 5.14 Quasi-static cohesive contact/element parameters for the bulk

epoxy matrices ................................................................................................ 110

Table 5.15 Quasi-static cohesive contact/element parameters of the DCB

lamina interface ............................................................................................... 119

Table 5.16 Elastic properties of the composite material strip.......................... 124

xx

Table 5.17 Fatigue parameters of the bulk epoxy matrix CT specimens from

Paris law fit...................................................................................................... 148

Table 5.18 Fatigue parameters obtained from the DCB composite material

specimen......................................................................................................... 152

Table 5.19 Threshold stress for the composite strip ....................................... 160

Table 6.1 Fracture energies of the different bulk epoxy matrices ................... 161

Table 6.2 Fracture energies of the composite materials based upon the different

epoxy matrices ................................................................................................ 162

xxi

Nomenclature

Abbreviations

AEW Amine equivalent weight MMB Mixed-mode bending

AFM Atomic force microscopy NCF Non-crimp fibre

ASTM American Society for Testing and Materials

NL Non-linear

B-K Benzeggagh-Kenane NR Neat resin

CBT Corrected beam theory NRR Neat resin with rubber

CCT Crack closure technique NRRS Neat resin with silica

CFRP Carbon fibre reinforced plastic

NRS Neat resin with rubber and silica

CLT Classical laminate theory PTFE Polytetrafluoroethylene

CP Control point QI Quasi-isotropic

CT Compact tension RIFT Resin infusion under flexible tooling

CTBN Carboxyl-terminated butadiene-acrylonitrile

SENB Single edge notched bending

CZM Cohesive zone model SERR Strain energy release rate

DCB Double cantilever beam UD Unidirectional

DGEBA Diglycidyl ether of bis-phenol A

VCCT Virtual crack closure technique

DOF Degree of freedom

EEW Epoxide equivalent weight

ENF End notch flexure

EPFM Elastic plastic fracture mechanics

FB Face bottom

FBk Face back

FEA Finite element analysis

FF Face front

FL Face left

FR Face right

FRP Fibre reinforced plastic

FT Face top

GFRP Glass fibre reinforced plastic

ISO International Organization for Standardization

LEFM Linear elastic fracture mechanics

LVDT Linear variable displacement transducer

xxii

English Alphabet

Crack length

Area

Acz Area of cohesive zone

Ad Damaged cohesive zone area

Area of the element

Damaged area of the element

Ap Amplitude of sine wave

Breadth

B matrix

Paris law exponent

Damage variable

Global displacement

Components of stiffness tensor

Linear and non-linear element displacement in global coordinate system

Components of undamaged stiffness tensor

Components of undamaged stiffness tensor

Local stiffness tensor

Maximum damage

Static and fatigue damage

, Rate of change of static and fatigue damage

Damage at t and t+1 time

Tangent stiffness tensor

Components of tangent stiffness tensor

Elasticity modulus

Initial modulus of laminate

Modulus of elasticity in 1, 2 and 3 direction

Elastic modulus of fibre and matrix

Resultant elastic modulus

Transverse elastic modulus (through thickness direction)

Modulus of laminate with a given crack density

Modulus of elasticity in x, y and z direction

Differential of shape function matrix

Scalar factor

Correction factor

Forces in the x and y direction

Force vector

Function of x variable

Fracture energy

xxiii

Shear modulus in 1-2, 1-3 and 2-3 plane

Critical fracture energy

Shear modulus of fibre and matrix

, Energy release rate in mode I and mode II

Critical fracture energy in mode I and mode II

Maximum energy release rate in mode I

Energy release rate when growth is infinite/unstable

Maximum energy release rate

Total energy release rate

Threshold fracture energy

Shear modulus in P-T, P-P, T-P, T-T, T-L and L-T plane

Shear modulus in x-y, x-z and y-z plane

Shape/interpolation matrix

Factor

Variable

Identity matrix of size nxn

Variable

Jacobian matrix

Penalty stiffness of cohesive zone law

Approximate penalty stiffness of cohesive zone law

Compressibility modulus of fibre, matrix and composite

Stiffness of the element

Critical stress intensity factor in mode I

Maximum stress intensity factor

Length

Distance from the centre of the loading block to the mid-plane of the specimen

Cohesive zone length

Length of the element

Size of mesh

Exponent of Paris law

Parameter

Scaling factor

Number of nodes

Number of cycles

Number of elements

Number of cycles of fatigue life

Interpolation/shape function

Load

xxiv

Maximum load or 5% offset load

Maximum load in a fatigue cycle

Constant

Stress ratio

Displacement ratio

Spacing

Spacing between point A and B

Crack spacing before and after loading

t Time

Thickness of 0o and 90o lamina

Thickness of continuum element

Unit local coordinate vector

Traction vector

Local traction vector

Time period

Displacement

Energy stored

Displacements at c and d points in the u direction

Local displacements at the bottom and top of the crack/element

Local displacement

Minimum and maximum displacement in a fatigue cycle

Global displacements in the u and v direction (linear)

Global displacements in the u and v direction (non-linear)

Displacement

Displacements at c and d points in the v direction

Volume fraction

Width

Energy required

Variable

Differential in and direction of mid-plane coordinate in global coordinate system

Global coordinate in undeformed state

, Global displacement at the bottom and top of the crack/element

Mid-plane coordinate in global coordinate system

Global coordinate

xxv

Greek Alphabet

Power law factor

Factor

Power law factor

Displacement

Load-line displacement in DCB specimen

Displacement in 1 and 2 direction

, Kronecker delta

Displacement in mode I and II

Failure displacement in mode I and II

Opening displacement in mode I and II

Component of displacement jump

, Opening and failure displacement in cohesive zone law

Displacement at time t and t+1

Displacement jump threshold at time t

Δa Correction for crack tip rotation in DCB specimen

Δcoh Displacement in cohesive zone element

Strain in composite

Strain in matrix

Maximum strain

Resultant strain

Shape factor

Local coordinate in a cohesive zone

Mode-mixity factor

Factor

Transformation matrix

Poisson’s ratio

Poisson’s ratio in 1-2, 2-1, 2-3 and 1-3 direction

Poisson’s ratio of fibre and matrix in 1-2 direction

Poisson’s ratio in L-T, T-L, T-P, T-T and P-P direction

Poisson’s ratio in x-y, x-z and y-z direction

Difference in the energy

Local coordinate in a cohesive zone

Stress

Stress in the 1 and 2 direction

, Critical stress in 1 and 2 direction

Critical applied stress

Minimum, maximum and mean stress applied

Cohesive zone traction/stress

xxvi

Critical cohesive zone traction/stress

Cohesive zone traction/stress in the 1 and 2 direction

Component of cohesive zone traction/stress

Unit vectors in and the normal direction

Diameter

Energy at a given time and the initial energy

CHAPTER 1 INTRODUCTION AND OBJECTIVES

1

CHAPTER 1

1. INTRODUCTION AND OBJECTIVES

1.1 Introduction

Composite materials are engineered or naturally occurring materials formed by

mixing two or more constituent materials in different phases with different

physical or chemical properties. Composites are found in nature in the form of

bone, mollusc shell, wood etc. They have different constituents in different

proportions to form a new material with properties different from the individual

constituents.

The main constituents of the composite are the matrix and reinforcement. The

matrix of the composite helps to bind the reinforcement together in the

composite. The reinforcement is embedded in the matrix and the reinforcement

typically has a relatively high modulus and tensile strength, compared to the

matrix. The matrix has good binding properties and is less stiff than the

reinforcement. The proportion and the structure of the reinforcement greatly

influence the properties of the composite (see Figure 1.1).

Figure 1.1 Stress-strain characteristics of unidirectional composites: (a) low

stiffness fibres; (b) high stiffness fibres (Talreja [1]).


2

Typical reinforcements are fibrous in nature with high aspect ratios. Such

reinforcements are classified as long fibres and short fibres, depending on the

value of the aspect ratio. Short fibres are usually randomly orientated whereas

long fibres are orientated in different directions in the composite to design a

composite with the desired material properties. Typically, long, continuous fibres

are laid in sheets in unidirectional or multidirectional orientation within the

composite. Thus, the fibres are essentially oriented in different directions and

bonded together using a polymeric matrix to form a laminate. The unidirectional

fibres have all the fibres in the same direction in the laminate.

Glass fibre reinforced plastic (GFRP) and carbon fibre reinforced plastic (CFRP)

are commonly used as fibre reinforced laminates for the aerospace industry.

The main advantages of composite materials are their relatively low density and

high stiffness and strength compared to metallic materials. Composites also

provide good design flexibility, as they can be moulded into complex shapes

and geometries.

The composite for the present work is a multilayer and multidirectional quasi-

isotropic laminate of a polymeric matrix with glass fibres and the laminate is

known as GFRP. It is a quasi-isotropic laminate which has equivalent stiffness

properties in all directions in a given plane.

Epoxy polymers are widely used as the matrices for fibre reinforced composite

materials. They have good engineering properties, such as high modulus,

failure strength, low creep and good performance at elevated temperatures after

curing. The properties of the epoxy polymers can be improved by the addition of

a particulate phase which may increase the toughness of the epoxy polymer

matrix. The effectiveness of the addition of the particulate phase depends on

the dispersion of the particulate phase in the matrix as well as the adhesion of

the particle to the matrix. The most commonly used such additives are the

nano-silica and micrometre-rubber particles, which are used in the present

research.


3

1.2 Damage in Composites

Damage in composite materials occurs in different stages of their manufacture

and service life (Matthews and Rawling [2]). The main sources of defects in the

composite are

1. Defects in the fibres;

2. Defects introduced during manufacture;

3. Damage that develops during their service life.

Defects in the materials used for the manufacture of the composite material

causes the introduction of intrinsic damage in the composite. The defects in the

manufacture occur due to, for example, misalignment of fibres, inclusion of

impurities in the manufacture process, etc. The damage that develops during

the service life of the composite arises due to different types of loading, impact,

shocks etc. There are various types of damage that may be so introduced and

these invariably lead to a reduction in the strength, stiffness and fatigue life of

the composite material.

The failure of composite materials during their service life occurs mainly due to

delamination, debonding, transverse cracking and fibre failure (Sridharan [3]).

Delamination occurs in the lamina of composites and is often associated with

the prior formation of transverse cracks in the composite. Indeed transverse

cracking is one of the main failure mechanisms in composite materials and

essentially consists of the formation of matrix cracks in the lamina of composite

due to quasi-static and cyclic fatigue loading. The matrix cracks lead to the

stiffness of the laminate decreasing due to a lower degree of interaction

between the fibres and the matrix. They therefore lead to a lower stiffness due

to less transfer of the applied load by the matrix. Cycling loading causes the

evolution of transverse cracks due to the reversal of the stress with time.

Quasi-static damage is the damage in the composite material which arises due

to the application of a steady, or steadily increasing load. On the other hand,

cyclic fatigue damage is the progressive structural damage that occurs when a


4

composite material is subjected to cyclic loading. In fatigue study of composites,

the material is subjected to repeated loading and unloading cycles over a period

of time. The fatigue loading cycle is broadly divided into low cycle fatigue and

high cycle fatigue based on the number of cycles required for the failure of the

material and the level of stress applied on it. The low cycles fatigue regime is

characterised by a relatively high stress level, a very localised stress

concentration and a low number of cycles to failure. The high cycle fatigue is

characterised by a relatively low stress level, with damage developing at the

micro-scale, and large number of cycles to failure.

1.3 Modelling Techniques

Linear elastic fracture mechanics (LEFM) is a technique widely used to

characterise composite materials (Hull and Clyne [4]). The approach most

commonly adopted for the analysis of fracture is based on an energy balance,

which assumes energy released during fracture is at least equal to that

necessary to generate a new fracture surface when an existing crack

propagates.

An LEFM approach may be readily coupled with a finite element analysis (FEA)

where the damage in the form of transverse cracks and delamination may be

modelled (Cook et al. [5]). In a FEA approach, the growth of cracks and

delamination may be analysed based on the energy released during the

initiation and propagation of the crack. The energy release rate during fracture

in may be modelled using a virtual crack closure technique (VCCT) (Krueger

[6]) and this may be coupled with a cohesive zone model (CZM) (Camanho et

al. [7]). The VCCT method calculates the strain energy release rate (SERR)

based on the forces and displacements needed to advance the crack. A CZM

approach models the progressive damage and failure in the composite material

based a known cohesive zone law. The CZM approach also predicts the onset

of propagation and subsequent growth of a crack under different types of

loading, and the fatigue degradation of the composite material may be based on

a degradation and evolution law.


5

1.4 Objectives of the Present Research

The main objective of the present work is the analysis of transverse cracks in

composites under quasi-static and fatigue loading by developing a novel FEA

approach coupled with a CZM technique. The present research implements an

extended FEA method via the Abaqus software. The work investigates epoxies

and composites toughened with nano-silica and micrometre-rubber particles to

improve the quasi-static and fatigue performance. Different specimens of bulk

epoxy matrix and the composite are tested and modelled. The performance of

hybrid (with both nano-silica and micro-rubber particles) epoxy matrix and

composite is also studied and modelled. The fracture energy of the bulk epoxy

and interlaminar fracture energies of composite were determined experimentally

and modelled. The static behaviour of epoxy and the composite are modelled

using the fracture energy to obtain the cohesive zone parameters. The cohesive

properties of bulk epoxy and composite are modelled for fatigue condition and

growth rate curve is obtained for the fracture mechanics specimens. The

cohesive properties of the epoxy are used to obtain the S-N curve of the

composite and predict the fatigue life. The fatigue parameters of different

materials are compared to study the influence of addition of different additives

on the performance of composite.

In the analysis, a user element subroutine (Hibbitt [8]) is developed in Abaqus to

incorporate the extended finite element capabilities and a mathematical model

proposed in Chapter 4 to evaluate constitutive laws to simulate fatigue driven

transverse cracking in composite. A fatigue degradation strategy based on Paris

law is adopted for the analysis and is used to represent the behaviour of lamina

interface and transverse cracks in composites under fatigue loading.

1.5 Structure of the Thesis

The thesis is divided into six chapters with each Chapter describing different

aspects of work. The next Chapter deals with the literature study and the past

work reported on the quasi-static and fatigue studies on composites. The

Chapter also deals with the conventional and computational methods for the

modelling of composites.


6

Chapter 3 describes about the experimental techniques adopted in the present

work to obtain the experimental results. The Chapter explains about the

materials, manufacture of specimens, testing of specimens and interpretation of

the experimental data for the modelling studies.

Chapter 4 deals with the theoretical techniques adopted in the present study.

The Chapter describes about the theoretical formulation of the present problem

and the analysis using FEA method.

Chapter 5 presents the experimental and modelling results of the present work

on bulk epoxy matrix and composite. The results of the modelling are compared

with the experimental results to understand the quasi-static and fatigue

behaviour of bulk epoxy matrix and the composite. Chapter 6 concludes the

thesis with recommendations to study the behaviour of the composite materials.

CHAPTER 2 LITERATURE REVIEW

7

CHAPTER 2

2. LITERATURE REVIEW

2.1 Introduction

Static damage in a composite material arises from a constant applied load or

from a steadily increasing applied load and cyclic fatigue damage arises from

an applied oscillatory load. Such damage causes a reduction in the strength of

the composite material with increasing time. This fracture process can be

studied experimentally and modelled by analytical or numerical methods.

Fracture mechanics approaches have been widely used to study and model

such damage and the basis for this approach is firstly reviewed. The damage

that occurs in composites, and the methods of analysis used to study the

fracture processes, are then reviewed. Finally, the life prediction analysis

methods which have been proposed to describe the accumulation of such

damage and the eventual fracture of the composite material are discussed.

2.2 Fracture Mechanics

Fracture mechanics is the field of mechanics concerned with the study of the

propagation of cracks in materials. It uses methods of analytical solid

mechanics to calculate the driving force on a crack and those of experimental

solid mechanics to characterise the material's resistance to fracture. It applies

the physics of stress and strain using theories of elasticity and plasticity to

describe the growth of cracks.

2.2.1 Linear Elastic Fracture Mechanics (LEFM)

Linear elastic fracture mechanics (LEFM) is a theory of fracture mechanics

which helps to study the fracture in a material or a structure. The LEFM

technique assumes the material is linear elastic in the bulk of the material and

that the plastic or damage zone near the crack tip is relatively small. In LEFM,

most equations are derived for either plane stress or plane strain associated


8

with the three basic modes of loadings on a cracked body: opening, sliding and

tearing (Figure 2.1). The LEFM method is valid only when the inelastic

deformation zone is small compared to the size of the crack. In the case of large

zones of plastic deformation in the material associated with crack propagation,

then the elastic plastic fracture mechanics (EPFM) technique is used. However,

the EPFM techniques are not relevant to the present research, and therefore

will not be considered further.

LEFM is a method widely used for predicting the static failure in composites.

The LEFM technique is applied to materials or structures with an initial crack or

damage, as this method predicts the initiation of crack growth and the

subsequent propagation of a crack. It should be noted that the LEFM method

fails to predict the first initiation of a void or defect or crack.

2.2.2 Modes of fracture

Fracture in a composite laminate may occur due to the initiation of the growth of

various types of cracks. The growth of the crack may be classified as occurring

in the opening (mode I), shear (mode II) and tearing (mode III) modes. The

different modes (Figure 2.1) have different values of critical strain energy

release rate (SERR), , needed for crack propagation. Mode I failures are

more predominant in composite materials due to less energy being needed to

propagate a crack compared to the other modes. Transverse matrix crack

growth occurs mainly in mode I and the delamination in the composite occurs in

both in mode I and II. Transverse cracks are often precursors to the initiation of

delamination, but delamination often suppresses more transverse cracking

occurring due to more consumption of energy being required in mode II fracture.

This situation typically arises once the maximum transverse crack density has

been achieved. In composites, mode III failure is usually ignored due to

negligible fracture occurring in this direction. Hence, in typical analyses only

mode I and mode II failures are considered together with mixed-mode (I/II), and

mode III fracture is neglected.


9

Figure 2.1 Different modes of failure in a material (mode I is the opening mode,

mode II is the shear mode and mode III is the tearing mode).

2.2.3 Griffith's criterion

The Griffith criterion states that crack propagation will occur if the energy

released upon crack growth is sufficient to provide all the energy that is required

for crack growth. The condition for crack growth is

(2.1)

where is the energy required and is the energy stored which is available to

form a crack.

Griffith calculated the energy stored per unit plate thickness in an edge crack in

an isotropic infinite plate as

(2.2)

where is the elasticity modulus, is the crack length and is the applied

stress on the material. The term can be replaced to give

(2.3)

where is the critical fracture energy, also called the crack driving force and

the critical strain-energy release rate, and is the critical applied stress.


10

The energy required to grow the crack is dependent on the state of stress at the

crack tip. The energy release rate for crack growth can be calculated as the

change in elastic strain energy per unit area of crack growth, i.e.,

(2.4)

where is the elastic energy of the system and is the crack length. The value

of is evaluated from by either keeping the load, , or the displacement, ,

constant.

2.2.4 Stress intensity factor

The stress intensity factor is a different approach to measure the material’s

toughness. It describes the stress field around the crack tip. Crack extension

occurs when the stresses at the crack tip are such that a critical value of the

stress intensity factor is reached. The critical stress intensity factor in mode

I, , is given by the expression

(2.5)

where is a constant, is the initial crack length and is the critical applied

stress.

For a mode I crack, the critical fracture energy and the critical stress intensity

factor are related by the expression for plane strain as

(2.6)

where is the Young's modulus, is Poisson's ratio, and is the stress

intensity factor in mode I. The strain energy release rate (SERR), , of a crack

in a body can also be expressed in terms of the mode I, mode II and mode III

stress intensity factors. In composites, there is a significant difficulty in defining

the stress in the interlaminar layer and hence the fracture energy, i.e. the critical

SERR, , of the material is easier to define and use. The present work is

http://en.wikipedia.org/wiki/Fracture#Crack_separation_modes

http://en.wikipedia.org/wiki/Young%27s_modulus

http://en.wikipedia.org/wiki/Poisson%27s_ratio

http://en.wikipedia.org/wiki/Stress_intensity_factor






11

therefore based on the energy method to model fracture in the bulk epoxy

matrix polymers and the composite materials.

2.3 Damage

Damage in the composite material occurs due to different types of loading. The

two major types of damage occurring in the composites are static and cyclic

fatigue damage. The static damage occurs with a steady increase in the load

with time. The damage is characterised by the formation of transverse cracks

leading to delamination at relatively high stresses. Fatigue damage occurs in

composites due to cyclic loading over time. The damage is characterised by

progressive mechanisms of failure in which transverse cracking is accompanied

by delamination in the composites. The two types of damage are discussed and

reviewed below to provide a basis for the present research.

2.3.1 Static damage

Static damage occurs when a constant load is applied on the material or when a

load is applied which increases steadily with time. Static damage in the

composites occurs typically due to transverse cracking followed by interlaminar

delamination. Figure 2.2 and Figure 2.3 shows the failure of a composite strip

under a quasi-static load due to delamination and transverse cracking. Damage

arising from transverse cracking is one of the important mechanisms of failure in

composites. Figure 2.4 shows the development of transverse cracks in a strip

under quasi-static loading at different values of the applied strain.

Figure 2.2 (a) Failed specimen (scaling factor, ms=1) and (b) detail of failed

specimen edge (Hallett et al. [9]).


12

Figure 2.3 Photograph and schematic of delaminations in ply level scaled specimen (Hallett et al. [9]).

Figure 2.4 Photographs of the development of transverse cracking in (0/90/0) glass-fibre/epoxy specimens at different strain levels (Berthelot [10]).

2.3.2 Fatigue damage

Fatigue damage in composites occurs due to cyclic load reversals over time.

The fatigue damage occurs as transverse cracks and delamination in the plies

of the composite laminate. Fatigue driven cracking is governed by energy

release and the accumulation of damage in the composite material over time.

Usually, the fatigue load is applied as a sinusoidal stress wave-form of constant

amplitude. The stress cycles in the fatigue loading are expressed as the -ratio,

also called the stress ratio, which gives the ratio of the minimum to the


13

maximum stress in a fatigue cycle. A typical stress fatigue cycle can be

represented using the amplitude, maximum stress, , minimum stress,

, and the time period, as shown in Figure 2.5.

Figure 2.5 Fatigue load cycle parameters.

The Paris law is the most widely used law to describe crack growth curve under

fatigue loading. The growth rate in a composite material is related to the energy

release rate and has three regions. The crack growth in the first region (Region

I) of the curve growth rate curve (Figure 2.6) is zero as the energy released is

less than the threshold value, , required for crack growth. Region II of the

crack growth curve has a linear part in which crack growth depends on the

energy released. This linear part of the curve is described by the Paris law. The

last part of the curve (Region III) has an asymptote of the crack growth as the

energy release rate is now equal to the fracture energy, , of the material. The

Paris law describes the linear region of the growth rate curve using a power law

and can be expressed as

(2.7)


14

where the values of and are obtained experimentally. The value of

depends on the material, loading conditions, temperature etc and is the slope

of the growth rate curve when plotted logarithmically as shown in Figure 2.6.

Figure 2.6 Typical growth rate curve.

In fatigue loading, transverse cracking and delamination develop with an

increase in the number of fatigue cycles. Fibre breakage may also occur, and

this usually occurs during the final stages of fatigue cycles leading to further

transverse cracking and delamination. Coupling of the cracks also leads to

delamination, as well as fibre breakage in the laminate. Typical fatigue damage

observed in composite materials is shown in Figure 2.7. Transverse cracks and

delaminations may be seen at the edge of the strip and they give rise to

different degrees of stiffness reduction. Thus to summarise, in the damage

process, the initial mechanisms of damage lead to the formation of transverse

cracks in the composite and this progressive damage leads to delamination

occurring, and the damage that results is observed as a reduction in stiffness of

the composite material.

log

(d

a/d

N)

GcGth log (Gmax)

m

Region IIRegion I Region III


15

Figure 2.7 Fatigue damaged edge of (0/90/45) carbon fibre epoxy laminates at: (a) 2% stiffness reduction (b) 4% reduction (c) 8% reduction (d) 12% reduction (e) 15% reduction (Reifsnider and Jamison [11]).

The points are further illustrated in Figure 2.8 which shows the development of

the damage under fatigue loading and the development of transverse cracks,

matrix cracks and delaminations in a composite. In a multidirectional composite,

the transverse cracks occur in a 90o lamina and the matrix cracks occur in

angled ply lamina due to the loading directions.


16

Figure 2.8 Microscopic damage mechanisms resulting from a constant stress

amplitude fatigue test observed via optical microscopy for (a) 10, (b)

100 and (c) 1500 cycles of loading (Hosoi et al. [12]).


17

2.4 Methods of Finite Element Analysis

2.4.1 Introduction

Different methods are used for the analysis of composites which use either an

energy or a stress based criteria for the analysis of fracture. The most widely

used techniques are based upon a finite element analysis approach. They are

the virtual crack closure technique (VCCT) method and the cohesive zone

model (CZM) methods. The conventional energy method and other methods of

analysis are also used to study fatigue damage in composites. The different

analysis methods which have been used are described below.

2.4.2 The Virtual Crack Closure Technique (VCCT) method

Crack propagation studies using the VCCT method (Rybicki and Kanninen [13])

have been performed by many authors. In the VCCT method, the total energy

release rate is computed locally at the crack front. It involves determining the

energy release rate as a function of the direction in which the crack is extended.

By Irwin’s theory, the energy required for the crack propagation is directly

proportional to the crack length and the energy released during crack

propagation can be calculated from the nodal displacements.

In the VCCT method, the stress field around the crack is calculated based on

elasticity theory. In this analysis, the crack propagates when the strain energy

released is equal to the fracture energy of the material. The VCCT method is

computationally efficient due to a relatively low amount of computational time

being needed and due to its simplicity. The main disadvantage of this method is

its failure to be able to initiate crack propagation in an uncracked material, as it

depends on the nodal displacement ahead of crack tip in order to calculate the

strain energy release rate (SERR), .

Zou et al. [14] developed a model for the evaluation of energy release rate

using the VCCT method. Their study included the influence of the number of

laminae in determining the total energy release rate in composites. They

derived the energy and modes of failure in sub-laminates from nodal forces and


18

moments in the lamina and showed the oscillatory behaviour of the SERR at the

crack tip due to edge delamination occurring.

Krueger [6] has reviewed the application of the VCCT method in the analysis of

crack propagation in two dimensional and three dimensional problems

concerned with composite materials. He described the principles governing the

technique and the calculation of the energy release rate in three dimensional

space. In this study, equations for the energy release rate in the 2D shell and

3D solid elements were derived from the geometry of the structure. The model

gave an accurate prediction of crack propagation.

Krueger and O'Brien [15] used multi-point constraints in the VCCT method to

compute the SERR across the width of the composite specimen. They

computed the SERR for beams of different widths from classical laminate plate

theory and derived the equations for plate/shell and solid elements for crack

propagation based on the geometry of the composite specimens.

Shen et al. [16] developed a computational model of circular delamination for

the prediction of delamination growth in composites. They adopted the VCCT

method for modelling a circular delamination in different layers of a composite

laminate. Their study showed that the direction of delamination growth

coincided with the direction of maximum SERR. They obtained the distribution

of energy released in the delamination front to predict the resultant delamination

shape and direction. Their model also gave information about the shape of the

delamination under buckling loads.

Qian and Xie [17] developed a cohesive zone model using the VCCT method to

model the crack propagation under mixed-mode loading. They used a cohesive

zone model based on SERR principles in the finite element analysis package to

model mixed-mode crack propagation occurring at a constant crack velocity.

The cohesive zone model was incorporated using subroutines to model different

crack velocities. It should be noted that, by using the VCCT method together

with a cohesive zone model, it is possible to predict the initiation and

propagation of cracks. The approach worked well for quasi-static loading under


19

mixed-mode loading but failed to predict the fatigue behaviour due to the

absence of a degradation law in the models that they developed.

Rybicki and Kanninen [13] developed a technique to determine the stress

intensity factors based on the VCCT. The method they adopted measured the

stress intensity factors for a crack in a plate using a constant strain element in

mode I and II and gave good results which were comparable to the J-integral

method. The VCCT method, which uses calculations via FEA of the nodal force

and displacement, was shown to be a relatively simple method for the analysis

of stress intensity factors for different modes, mode I and II (see Section 2.2.2).

Mandell et al. [18] used the VCCT method to predict the life of skin stiffeners

bonded to a composite panel using the SERR method by predicting

delamination in wind turbines under static and fatigue loading. The SERR for

the different modes of loading was obtained from a FEA analysis. The tests

were undertaken for various crack growth rates and the energy release rate

determined. The method predicted delamination failure for different thickness of

the matrix resin, and also predicted the static delamination in wind turbine

blades for various mixed-mode loading conditions.

2.4.3 The Cohesive Zone Model (CZM) method

2.4.3.1 Introduction

Cohesive zone modelling is considered to be the most accurate method of

modelling the damage in composites. In this method, a cohesive zone model is

used to model the crack propagation. The cohesive zone model law relates the

traction and displacement at the crack tip to calculate the SERR under different

modes of loading. The cohesive zone model helps to overcome the complexity

of the singularity located at the crack tip. In the cohesive zone modelling

approach, the crack propagates in the material according to a defined cohesive

zone model law and damage evolution principles. The cohesive zone model

combines aspects of a strength based analysis and fracture mechanics to

predict the onset of damage and the propagation of the crack. The main


20

advantage of cohesive zone models is the ability to predict crack initiation and

propagation using the same cohesive zone model law.

Different shapes of traction, , versus displacement, , laws are proposed for

the cohesive zone law depending on the shape of the cohesive zone law: e.g.

bi-linear, linear-parabolic, exponential and trapezoidal; see Figure 2.9 and note

that the area under these shaped is equivalent to the fracture energy, . In a

cohesive zone model law, the shape of the cohesive zone law determines the

loading and the rate of degradation in the material. For example, Alfano [19]

studied the influence of the shape of the cohesive zone model law when

analysing delamination problems in composite materials. He studied bi-linear,

linear-parabolic, exponential and trapezoidal shapes for the cohesive zone

model law and compared them, assuming the same value of the initial penalty

stiffness (the initial penalty stiffness being the value of the initial gradient in the

cohesive zone law). He concluded that the trapezoidal law gave the poorest

results both in terms of the numerical stability and convergence and that the

exponential law gave the optimal results in terms of predicting accurately the

rate of degradation. His study also showed that the bi-linear law represented the

best compromise between computational cost and the prediction of the

degradation rate. He concluded that the influence of the shape of the cohesive

zone law curve depends on the penalty stiffness of the material.


21

Figure 2.9 Different cohesive zone model laws. (a)Triangular form (b) Constant

stress form (c) Triangular form (bi-linear law) (d) Elastic constant linear damage form (e) Linear polynomial form and (f) Elastic constant form (Zou et al. [20]).

The bi-linear cohesive zone law (see Figure 2.9 (c)) is widely used for the

analysis of fracture. It is the simplest of the cohesive zone laws as it represents

the elastic and degradation part linearly. The bi-linear cohesive zone law has a

discontinuity at the damage initiation point due to the sudden change in the

slope of the curve and hence different laws have been proposed to smooth out

δ

Gc

τ 0

(a)

τ

δ

Gc

τ 0

(b)

τ

δ

Gc

τ0

(c)

τ

δ

Gc

τ 0

τ

(d)

τ

Gc

τ 0

(e)

δ

Gc

τ 0

(f)

τ

δ


22

this discontinuity and hence possibly better predict the behaviour of the

material. A typical bi-linear cohesive zone model law is defined by a penalty

stiffness, , critical fracture energy, , and critical stress, , of the element, as

shown in Figure 2.10.

Figure 2.10 The bi-linear cohesive zone law.

Cohesive zone modelling is considered to be the most accurate method for the

analysis of cyclic-fatigue damage in composites. In this method, a cohesive

zone model is used to model crack propagation under fatigue loading. The

damage occurs in the cohesive zone by evolution of damage with time, and the

degradation of the cohesive zone law can be achieved by degrading the penalty

stiffness of the cohesive zone law with time. The degradation of the penalty

stiffness is computed based on the energy released at nodes of elements under

the fatigue loading. The relevant damage parameter, , in the cohesive zone

law is calculated based on the Paris law, which is obtained from experimental

tests on the composite material as described in Section 2.3.2 and as shown in

Figure 2.6. For example, the degradation of a bi-linear cohesive zone law due to

fatigue load with time is shown in Figure 2.11. The term and in Figure

2.11 are the fatigue damage at time and of the fatigue cycle. These

aspects are discussed in detail later in Chapter 4.

Tra

ctio

n,τ

Displacement, δ

Gc

K

δo

τ0

δf


23

Figure 2.11 Fatigue degradation in a bi-linear cohesive zone law.

2.4.3.2 Pure mode loading

Here the fracture in the material occurs when the energy released is equal to

the fracture energy in the particular mode. Therefore, the propagation criteria

can be expressed as

However, in real structure the loading is usually via a mixture of mode I and

mode II.

2.4.3.3 Mixed-mode loading

Mixed-mode loading occurs in most real structures. The mode-mixity in a

cohesive zone model analysis can be defined using the different criteria for the

mixed-modes. The simplest criterion considers that failure of the composite

occurs mainly in mode I due to the relatively high energy for fracture required in

mode II (Whitcomb [21]). The criteria are given by the relations

(2.8)

τ

δ

K

δt δf

τ0

K(1-dt)

δt+1

K(1-dt+1)


24

(2.9)

However, these very simple criteria are invariably found to be inadequate. So

another criterion for the prediction of the propagation of crack growth under

mixed-mode loading is to sum the individual fracture energies to give the critical

fracture energy in the mixed-mode condition (Wu [22]). This criterion can be

expressed as

(2.10)

And can be expressed in the normalised form as

(2.11)

Yet another, more generalised form of Equation 2.11 is a power law expression

(Camanho et al. [7]) given by where crack propagation occurs if

(2.12)

where the parameters and are the constants obtained from experimental

data, and and are the critical strain energy release rates in pure mode I

and II respectively.

Another criterion (Hahn [23]) which accounts for mode-mixity effects is the

expression

(2.13)

and yet another propagation criteria (Ramkumar [24]) is

(2.14)


25

The Benzeggagh-Kenane (B-K) criterion (Camanho et al. [7]) for accounting the

mode-mixity is given by

(2.15)

where and are given by the expressions

(2.16)

(2.17)

The mode-mixity factor, , is a factor which may be obtained from an

experimental fit. In the present work, B-K criterion is used for the mixed-mode

analysis as it depend only on one factor, which can readily be determined by

experimental fit, as discussed further in Chapter 4.

2.4.4 The Energy method

The energy method is the most conventional method used for the analysis of

damage in composites. Some of the important work undertaken using this

method is described below.

Rebière and Gamby [25] used the variational energy method to model

delamination and cracking behaviour of cross-ply laminates. The transverse and

longitudinal cracking of the laminates were modelled to determine the value of

SERR associated with the three modes of fracture. They modelled the

development of a triangular-shaped delamination and showed that the

delamination length is not uniform in the plane of the laminate.

Quantian Luo and Liyong Tong [26] developed a closed-form formula for

different mode energies for crack formation in a composite beam. The energy

release rate was expressed in terms of axial load, shear forces and bending

moments on the crack tip in the layered beam. An equation was derived for a


26

zero thick adhesive layer in the beam. The energy method predicted the static

crack growth in the material but was unable to account for the degradation of

material properties.

Kashtalyan and Soutis [27] developed a theoretical model based on stiffness

degradation and mechanical behaviour of symmetric laminates with off-axis ply

cracks and crack induced delaminations. They calculated the SERR based on

crack density, delamination area and ply orientation angle in symmetric

laminates. A shear-lag analysis was used to determine ply stresses in different

modes.

Wimmer and Pettermann [28] used a Griffith crack growth criterion to predict the

load required to propagate the crack and studied the stability of delamination

crack growth (Figure 2.12). Figure 2.12 shows the variation of force and

displacement with the delamination growth and the influence of delamination

size on the stability of crack growth. They also determined the critical size of

delamination where the growth changed from stable to unstable, and vice versa.

Their method could be applied to composites, as well as to other problems

where the crack path was known. They proved in their study that a small

delamination grew in an unstable manner while large delaminations grew in a

stable manner.

Figure 2.12 The minimum load required to propagate a delamination of a certain size in its most detrimental location (Wimmer and Pettermann [28]).


27

Shivakumar et al. [29] conducted experiments to determine the resistance to

crack growth as a function of delamination extension. They established an

equation for the delamination growth rate, , as a function of the maximum

cyclic energy release rate, (Figure 2.13). The increase of resistance due

to matrix cracking, fibre bridging, tow splitting, separation, bridging and breaking

(see Figure 2.14) were accounted for through normalisation of the equation by

the instantaneous resistance value, (i.e. the energy release rate when the

delamination growth was infinite, or unstable).

Figure 2.13 Comparison of theoretical equation for delamination growth rate

with the test data (Shivakumar et al. [29]).


28

Figure 2.14 Failure modes in laminate (Shivakumar et al. [29]).

2.5 Modelling Damage and the Life Time under Cyclic Fatigue Loading

2.5.1 Quasi-static loading

Quasi-static loading in composites causes damage due to an increase in the

applied load with time. The static damage in composites appears mainly as

transverse cracks which occur within the laminae of composites, as well as due

to interlaminar delaminations. The analysis of damage build-up, and the

associated loss of stiffness, in composites materials under quasi-static loading

has been undertaken by various authors using different methods. Some of the

important and relevant work undertaken is described below.

Boniface et al. [30] used a compliance, i.e. the inverse of stiffness, change

approach via a shear lag analysis to relate the crack growth rate with the SERR.

They studied the energy release and the interaction of transverse cracks

between each other, depending on the spacing of cracks. The study also

showed how the crack spacing affects (Figure 2.15) the interaction between the


29

cracks and confirmed the validity of the Paris curve to describe fatigue crack

growth.

Figure 2.15 Transverse crack interference model (Boniface et al. [30]).

Hallett et al. [9] analysed a quasi-isotropic composite plate to predict the failure.

In this work, the composite was analysed using both the VCCT technique and

the cohesive zone model approach to predict the delamination and transverse

cracking. It was shown that delamination, and its interaction with the transverse

cracks, affected the final failure of the laminate. The study also highlighted the

difficulty in accurately predicting the failure of composite materials.

Parvizi and Bailey [31] studied the growth of transverse cracks for different

stresses and ply thickness. They developed a shear lag analytical expression

using a exponential equation to model the reduction in strength with crack

spacing. They observed that a change in the stiffness of the composite due to

transverse cracking at relatively low strains was accompanied by a visual

whitening effect, due to the formation of the transverse micro-cracks. Their

method is difficult to adopt for non-isotropic materials, as their model cannot

take into account other directional material properties.


30

Huchette [32] illustrated the damage in the composite due to transverse

cracking and delamination. The different steps in the damage process of the

composite (02/902)s are shown in Figure 2.16. This study showed the interaction

of transverse cracks with the delamination in the failure of composites could be

successfully modelled using cohesive zone model elements.

Figure 2.16 Failure of a composite strip specimen (Huchette [32]).

Leblond et al. [33] used a finite element method to study cross-ply laminates.

They studied the longitudinal stiffness reduction due to cracks in the laminates

and modelled the laminates and obtained good agreement with the

experimental results. Their study showed the major reduction of stiffness that

occurred with an increasing crack density of transverse cracks in the cross-ply

laminates. Their theoretical predictions were in good agreement with the

experimental results (Figures 2.17 and 2.18).


31

Figure 2.17 Stiffness reduction as a function of the average crack density in the case of two different laminates. Eo is the initial elastic modulus of the laminate and Ex is the elastic modulus with a given crack density (Leblond et al. [33]).

Figure 2.18 Crack multiplication in transverse plies (Leblond et al. [33]).

2.5.2 Fatigue loading

Fatigue loading in a material occurs due to application of a periodic load and

causes the degradation of material properties with time. The load reversals


32

cause permanent damage in the material and the cyclic load applied in the

fatigue test is invariably considerably less than the ultimate load. There is also

usually a fatigue limit of the composite material (Figure 2.19), below which no

fatigue damage occurs.

Figure 2.19 Fatigue life diagram for a (0/902)s carbon fibre/epoxy matrix

laminate (Akshantala and Talreja [34]).

Turon et al. [35] developed a damage model for the simulation of progressive

delamination in composite materials under variable mixed-mode loading. The

model used constitutive laws for modelling the initiation and propagation of

delamination. The damage evolution for the cohesive zone model elements was

based on progressive degradation of the cohesive zone model parameters, as

will be described later in detail in Chapter 4.

Talreja [1] has developed a fatigue damage mechanism for the analysis of

composites. He proposed ‘fatigue life diagrams’ based on the strain in the fibres

and matrix (Figure 2.20). In his paper, he defined the fatigue ratio and defined

the fatigue limit for unidirectional, cross and angle plied laminates, as illustrated

in Figure 2.20. The study thus defined the fatigue resistance of the materials

from the strains in the composite material.


33

Figure 2.20 Fatigue life diagram for a unidirectional composites under loading

parallel to the fibres (Talreja [1]).

Manjunatha et al. [36] studied the fatigue behaviour of an epoxy matrix polymer

containing nano-silica particles and the corresponding GFRP composites. The

work quantified the development of transverse cracking with the number of

cycles of fatigue loading. The fatigue life of the GFRP composites was shown to

increase by about three to four times with the addition of the nano-silica

particles. The study also showed that reduced matrix cracking, due to the

nano-silica particles debonding and associated plastic void growth mechanisms,

contributed significantly to the increase in the fatigue life in the GFRP

composites with the modified matrix.

Tong et al. [37] studied the transverse cracking in the matrix in GFRP laminates

under fatigue loads. Their experimental observations of the fatigue crack growth

in the laminates were undertaken to study the fatigue degradation of the

strength of the material. Their work also studied the degradation of stiffness with

the increase of transverse cracks during the experiments. They observed

different crack densities in the various plies of composite and found that the

transverse crack density in the 90o fibres saturated after a certain cycles of


34

fatigue load. The experimental study also showed a relationship between the

crack density and material properties and gave a good insight into the

characteristic damage that occurs in composite materials.

Tong et al. [38] modelled the experimental results obtained in their previous

work on laminates to predict the reduction of stiffness of laminates with fatigue

cycles. They modelled open cracks in the laminates using a finite element

analysis inputting the value of crack density observed in the experiments. The

transverse cracks in different plies gave the distribution of the stresses around

the crack tip and their contribution to the total stiffness reduction. The model

was developed with plane strain elements and showed good agreement with

the experimental results for different values of crack density.

Manjunatha et al. [39] studied the fatigue life of GFRP composite materials

modified with both rubber and silica particles, see Figure 2.21. The fatigue life of

these hybrid (i.e. containing both micrometre-sized rubber and nano-silica

particles) epoxy matrix composites was about six to ten times higher than that

of the GFRP composites manufactured using the unmodified (i.e. control) epoxy

matrix polymer. They explained the increase in the fatigue life of the hybrid

epoxy composites as arising from the toughening micro-mechanisms caused by

the presence of both types of particles, such as cavitation of the rubber particles

and silica particle debonding. These effects both resulted in increased plastic

deformation of the epoxy matrix (Figure 2.21). Indeed, Manjunatha et al. [40]

observed less transverse cracking in the composites due to the addition of

rubber and silica particles (Figure 2.22).


35

Figure 2.21 The tapping mode atomic force microscopy (AFM) phase images of

the hybrid-epoxy matrix polymer (Manjunatha et al. [39]) (CTBN: carboxy-termianted butadiene acrylonitrile rubber).

Figure 2.22 Transmitted light photographic images of matrix cracking in the GFRP composites after testing at stress of 150MPa. NR-Neat resin, NRR-Neat resin with rubber, NRS-Neat resin with silica and NRRS-Neat resin with rubber and silica (Manjunatha et al. [40])


36

Camanho et al. [7] developed a cohesive zone model for crack propagation

under mixed-mode loading in composites. Their constitutive law proposed used

the relative displacement between the nodes for initiation and propagation of

the cracks. They used a mixed-mode criterion to determine the initiation and

propagation of delamination under the mixed-mode loading. They developed

the cohesive zone model formulation using a subroutine to simulate the double

cantilever beam (DCB), end notch flexure (ENF) and mixed-mode bending

(MMB) experimental tests. The model gave a good prediction of the fatigue life

of the composite material.

Robinson et al. [41] developed a cohesive zone model approach to predict

fatigue delamination growth in composite materials. They predicted the fatigue

growth curve for composites (Figure 2.23) using a novel degradation law which

had an exponential component to account for the degradation of the composite

material due to delamination occurring. The degradation law showed a similarity

to the Paris law for fatigue life prediction, see Figure 2.6.

Figure 2.23 Crack length versus cycles (Robinson et al. [41]).


37

Attia et al. [42] proposed a different method for predicting the growth of impact

damage in fibre composite skin structures when subjected to cyclic-fatigue

loading. The method that they developed used the experimental relationship

between the SERR and the number of fatigue cycles to initiate a fatigue crack,

which was included into a finite element analysis (Figure 2.24). The approach

uses the FEA model to deduce the SERR of the panel by deducing the energy

change for growth by approximately 5% of the original impact-damaged area.

Figure 2.24 Experimental relation between the maximum SERR, , and the

number of fatigue cycles, , for the onset of crack growth (Attia et

al. [42]).

Turon et al. [43] proposed a damage model for the simulation of delamination

propagation under high-cycle fatigue loading using a cohesive zone model

approach. They obtained the damage state as a function of the loading

conditions and determined the Paris law coefficients to use in the model in order

to degrade the cohesive zone law as a function of the number of fatigue cycles.

In their work the degradation of the material using the cohesive zone model

resulted in the degradation of the cohesive traction (i.e. stress) (Figure 2.25).

The model was validated by predicting the propagation rates in mode I, II and


38

mixed-mode tests and by observing that they obtained good agreement with the

experimental results.

Figure 2.25 Evolution of the interface/cohesive traction and the maximum interface/cohesive strength as a function of the number of cycles for a displacement jump controlled high-cycle fatigue test (Turon et al. [43]). (Here interfacial traction is the critical stress of the cohesive zone law at a given number of fatigue cycles and the interfacial traction is the traction at the cohesive zone at a given number of fatigue cycles.)

Iannucci [44] developed a cohesive zone modelling technique using a

formulation based on a damage mechanics approach and he used a stress

threshold and critical energy release rate for each particular delamination mode.

In his analysis, cohesive zone elements were placed where delaminations were

expected. (Thus, it should be noted that prior knowledge of their propagation

path was required.) The energy dissipated in different modes was used to

calculate the mode ratio. Again, this combination of a cohesive zone model with

a finite element analysis gave very good agreement with the experimental

results.


39

Khoramishad et al. [45] developed a bi-linear cohesive law to simulate damage

in adhesively bonded joints. The fatigue degradation in the model was based on

the degradation of the cohesive penalty stiffness, , with time. The cohesive

model predicted the fatigue life and strains in the adhesive joints. They also

derived a formulation for fatigue degradation using a mode-mixity criterion and

reducing the critical cohesive stress, instead of the penalty stiffness of the

element.

Mao and Mahadevan [46] have developed a mathematical model for the

degradation of composite materials under cyclic fatigue loading. A nonlinear

model was used for the damage evolution in the composite materials subject to

fatigue loading. The damage model accounted for the damage of the material

based on the degradation of elastic parameters. They used a curve fitting

method to predict the appropriate parameters for use in the fatigue law. The

damage accumulation law employed a power function for the number of cycles

and gave a reasonably good agreement with the experimental results.

2.6 Concluding Remarks

The present study has reviewed the literature on the prediction of fatigue life of

composite materials using various analysis methods. The finite element

analysis using a fracture mechanics concept is clearly a good tool for studying

the fatigue life of composite materials.

The present review also gives a good insight into the methods that may be

combined with such finite element analyses for the life prediction of composites

namely the VCCT, the cohesive zone model law, etc. The cohesive zone model

seems to be a very appropriate approach for the analysis of fatigue damage in

an uncracked specimen, because of its ability to predict the initiation and

propagation of a crack. The studies reviewed in the present Chapter also

highlight the difficultly in predicting the fatigue life of composites due to the

complex nature of the damage arising from both transverse cracking and

delamination occurring in the composite material.


40

The present literature review also reveals that little theoretical work has been

reported on the prediction of the fatigue life of nano-particle, rubber-particle and

hybrid modified epoxy matrix composites. It may also be seen from the present

review that there are very few reports available on the study of fatigue life in

transversely isotropic composite using a cohesive zone model law. Hence the

present work will mainly concentrate on modelling and predicting the fatigue life

of transversely isotropic composites with different formulations of epoxy

matrices using a cohesive zone model method, coupled with a finite element

analysis approach. The cohesive zone model formulations of Turon et al. [43],

Turon et al. [35] and Camanho et al. [7] will be used for the present analyses.

However, before the novel theoretical analyses developed in the present work

are described (Chapter 4), the experimental techniques will be given in the next

Chapter.

CHAPTER 3 EXPERIMENTAL TECHNIQUES

41

CHAPTER 3

3. EXPERIMENTAL TECHNIQUES

3.1 Introduction

In the present work, a glass fibre reinforced plastic (GFRP) composite is used to

study the fatigue behaviour of quasi-isotropic (QI) laminates. The properties of

the epoxy matrices used in the composite are also studied. The epoxy matrices

used for the present work are basically

a) A control formulation;

b) A modified formulation with a dispersion of nano-silica particles;

c) A modified formulation with a dispersion of micrometre-sized rubber particles;

d) A hybrid formulation with a dispersion of both micrometre-rubber and nano-

silica particles.

The bulk epoxies and the GFRP composite materials of the different

formulations have been tested. Such tests have been conducted in order to

determine the quasi-static and fatigue properties of the GFRP composite

laminates. The bulk epoxy matrices have been tested to ascertain the various

parameters needed for the theoretical modelling studies, which are later

developed to predict the fatigue life of the GFRP laminates (Chapter 4). The

tests undertaken on the epoxy are quasi-static single edge notched bending

(SENB) tests and the cyclic-fatigue test using compact tension (CT) specimens.

The tests undertaken on the GFRP composite are quasi-static and cyclic-fatigue

tests on double cantilever beam (DCB) and composite strip specimens. The

present experimental work is the continuation of previous studies on the

unmodified (i.e. control) and modified epoxy composites by Manjunatha et al.

[36], Manjunatha et al. [39], Manjunatha et al. [40]), Masania [47], Hsieh [48] and

Lee [49].


42

3.2 Materials

In this work, GFRP laminates based on the control and modified epoxy resin

matrices were prepared, i.e. the control resin, a resin with 9% wt. of micrometre-

rubber particles, a resin with 10% wt. nano-silica particles and a resin with 9%

wt. micrometre-rubber and 10% wt. nano-silica particles. The epoxy resin used

was ‘LY556’, a standard diglycidyl ether of bis-phenol A (DGEBA) with an

epoxide equivalent weight (EEW) of 185g/mol, supplied by Huntsman, Duxford,

UK. The curing agent was ‘Albidur HE 600’, an accelerated

methylhexahydrophthalic acid anhydride with an amine equivalent weight

(AEW) of 170g/mol and a stoichiometric amount of the curing agent was added

to the formulation to cure the epoxy resin. The nano-silica particles used in the

resin were based on ‘Nanopox F400’ where they were present in a

concentration of 40% wt. in a DGEBA epoxy resin with an EEW of 295g/mol

from Nanoresins, Geesthacht, Germany. The ‘Albipox 1000’, reactive liquid

carboxyl-terminated butadiene-acrylonitrile (CTBN) rubber was obtained as a

CTBN-epoxy adduct with a rubber concentration of 40% wt. in DGEBA epoxy

resin from Emerald, Cleveland, USA. The E-glass fibre sheet was a stitched two

layer of non-crimp fibre (NCF) arranged in a ±45° pattern with an areal weight of

450g/m2 from SP systems, Newport, UK.

3.3 Preparation of Epoxy Matrix Polymer Specimens

3.3.1 Introduction

Standard fracture mechanics specimens were prepared from the materials to

obtain the fracture mechanics data and the cohesive zone law parameters

needed for the theoretical modelling studies. The composite specimens used for

the fatigue analysis were also manufactured with the materials described in the

above section. The specimen preparation methods for the various types of test

specimens are described in the sections below.


43

3.3.2 Preparation of plates

Fracture mechanics testing was performed using epoxy matrix specimens

machined from bulk epoxy plates. The bulk epoxy specimens used for testing

were manufactured as plates. The epoxy was cured in a 6mm thick metal

mould, which was used for preparing the bulk epoxy plates. Initially silicone

gasket was placed around the mould to prevent the leakage of epoxy from the

mould. The moulds were first opened and the inside surface cleaned with

acetone. The surface of the mould was then coated with release agent to aid

the easy removal of the specimen. The moulds were then assembled and made

ready for pouring the resin. The epoxy resin matrix prepared as described

above, was poured into the mould. The resin was mixed with a stoichiometric

amount of the curing agent and degassed at 50oC and -1atm. The resin mixture

was then poured into the release-coated steel mould. The resin was poured

from the one side of the mould to prevent the development of bubbles in the

resin when pouring. The mould was taken to the oven for curing. The resin was

cured by increasing the temperature at 1oC/min and cured at 100oC for 2hr and

later post-cured at 150oC for 10hr. After curing, the plates were removed from

the mould and inspected for any defects or voids in the moulding. The cured

bulk epoxy plate was machined to obtain the required specimen dimensions.

The surface of the test specimens was made smooth by polishing with abrasive

papers, as surface irregularities might affect the fracture behaviour of these

relatively brittle materials.

3.3.3 Single edge notched bending (SENB) specimens

Standard tests were performed on the bulk epoxy materials to obtain the

fracture properties of the various epoxy formulations. The SENB test was

conducted to obtain the fracture energy, . The test was conducted according

to the standard ISO:13586:2000 [50]. The specimens were machined from the

epoxy plate, with the dimensions as in the standard. A sharp notch was

inserted in the specimen using a razor blade to act as a pre-crack, and the

specimen was tested. The load versus displacement curve of the specimen

under quasi-static load was obtained.


44

3.3.4 Compact tension (CT) specimens

Compact tension specimens were used to determine the cyclic-fatigue

properties of the bulk epoxy materials. The CT specimens were manufactured

from the plates of the bulk epoxy as specified in the ASTM:E647 [51] standard.

A sharp notch was inserted in the CT specimen by machining and then

sharpened using a razor blade. The surface of the specimen was again

polished to remove any defects. The compact specimen was loaded in tension-

tension fatigue.

3.4 Preparation of GFRP Composite Specimens

3.4.1 Introduction

Composite specimens were manufactured from the materials described in

Section 3.2. The preparation of the composite specimens was based on the

manufacturing method described below.

3.4.2 Resin infusion under flexible tooling (RIFT)

The laminated plates for the specimens were prepared from a multidirectional,

high strength, glass fibre epoxy pre-preg, and a resin infusion under flexible

tooling (RIFT) method was used to prepare the fibre reinforced epoxy

composites (Figure 3.1). In this method, the woven fibres were laid up and

placed in a vacuum bag and the resin was made to infuse through the fibre

layup using the vacuum pressure that was applied. The epoxy resin was

therefore forced to spread throughout the fibre layup, and the layup was then

cured to form the laminate. The glass fibre sheet was cut into 330x330mm2

squares.

To give further details, then to produce the laminates a temperature controlled

plate surface was set to the required infusion temperature. The surface of the

plate was cleaned and made smooth and an infusion stack was built. A

polyamide film was laid over the plate and fixed using adhesive tape. The

polyamide film formed the first and the outer layer of the vacuum bag. A sealant

tape was bonded on to the polyamide film along the plate border to form the


45

vacuum bag. A flow media, i.e. a sheet of plastic net to help uniform flow of the

epoxy resin, was placed next to the polyamide film to help the infusion of the

matrix resin. The inlet pipes and outlet pipes were fixed on the ends of the plate.

The fibre layup was placed over the peel ply, and the same procedure was

repeated on the other side of the fibre layup. The vacuum bag was sealed with

the sealant tape and a vacuum pump was connected to the so-formed vacuum

bag. The resin was infused into the vacuum bag through the inlet pipe and the

temperature of the plate was controlled. The resin flowed through the dry fibre

layup and reached the other end of the plate. The inlet pipe was then closed to

prevent the infusion of the resin into the vacuum chamber. The resin was cured

by ramping the temperature to 100oC at 1oC/min, cured for 2hr, again ramped to

150oC at 1oC/min and post-cured for 10hr. After the curing cycle was complete,

the laminate was taken out of the vacuum bag and machined around the edges.

The composite laminate was visually checked for voids. Composite plates of

330X330X5.4mm3 were therefore manufactured by the above method.

Figure 3.1 Fabrication of a GFRP sheet using RIFT.

3.4.3 Double cantilever beam (DCB) specimens

Composite plates were prepared using the glass fibre sheet manufactured as

described above laid up in the sequence of [(-45/45)s(90/0)s]2 [(0/90)s(45/-45)s]2

to give a 0o/90o lamina interface (mid plane) across the fracture plane. A


46

12.5μm thick polytetrafluoroethylene (PTFE) film was inserted in the mid-plane

of the plate to act as a pre-crack. The specimens were designed to be

sufficiently stiff to avoid large displacements, plastic deformation, intra-ply

damage and to reduce elastic couplings. Each plate had sixteen layers of

woven glass fibre. Using this method, standard composite plates of

300x300x5.4mm3 were made.

The DCB specimens were cut from the plates using a wet-saw cutting machine

with nominal dimensions as recommended by the standard (ISO:15024:2001

[52]). Each laminate plate was cut into ten specimens. The pre-crack lengths

were approximately 60mm for the specimens. Machined aluminium alloy blocks

of the same width as the specimens were bonded onto the end of the DCB test

specimen using an epoxy adhesive, which was cured at room temperature. One

of the edges of the DCB specimen was coated with white ink and was

graduated at 1mm interval to monitor the crack growth.

3.4.4 Composite strip specimens

Composite plates of 300x300x2.7mm3 in dimension were prepared using the

RIFT method, as described above, with a layup sequence of [(45/-45/0/90)s]2.

Each plate had eight layers of woven glass fibres. The strip specimens were cut

from the plates using the wet-saw cutting-machine with nominal dimensions of

150x25x2.7mm3. The strips were grit blasted where the end tabs were to be

fixed and the tabs were fixed to the specimen using an epoxy adhesive, which

was cured at room temperature. The strip specimens were finished by lightly

abrading the edges of the test specimens to remove any major defects.

3.5 Test Methods for the Epoxy Matrix Polymer Specimens

3.5.1 Introduction

Tests were undertaken on the bulk epoxy matrix specimens at 20o±2oC to

determine the fracture properties of the material. These are described below.


47

3.5.2 Single edge notched bending (SENB) tests

SENB specimens were tested to determine the mode I fracture energy, , of

the bulk epoxy matrix polymer. This test measures the resistance to the

initiation and propagation of a crack in the bulk epoxy polymer under mode I

loading. A standard quasi-static test was conducted to determine the mode I

fracture toughness, , according to the standard (ISO:13586:2000 [50]). The

specimen was loaded under displacement control at a rate of 0.05mm/min. The

start of the crack was observed visually and the growth of the crack was noted

using the event marker, the length of the propagating crack being recorded on

the load versus displacement trace. A linear variable displacement transducer

(LVDT) was used to measure the displacement during the loading of the

specimen. The fracture toughness, , of the material may be calculated from

the expression

(3.1)

where is the maximum load or 5% offset load, the breadth, the width and

the shape factor of the specimen where

(3.2)

The fracture energy, , can be calculated from the equation

(3.3)

where is the elasticity modulus (Table 5.1 ) and is the Poisson’s ratio (Table

5.1) of the bulk epoxy.


48

3.5.3 Compact tension (CT) tests

The CT specimen test was conducted to determine the cyclic-fatigue properties

of the epoxy matrix polymers under cyclic tension-tension loading. This test also

ascertains the threshold fracture energy of the epoxy matrix material under

fatigue loading. The fatigue parameters obtained from these tests will be used

for the modelling analyses of the composite strip specimens (Chapter 5).

A cyclic-fatigue test (based on the ASTM:E647 [51] and ISO:15850 [53]

standards) was conducted to determine the rate of crack growth, , per cycle

as a function of the maximum value of the applied strain energy release rate,

. The load was applied as a sinusoidal function with a maximum

displacement less than the displacement required for the initiation of crack

growth under quasi-static loading. The test was conducted using a 1kN

computer-controlled servo-hydraulic test machine under displacement control

loading. A fatigue ‘Krak gauge’ was bonded on the side of the specimen using a

standard M-bond adhesive resin, which was cured at 25oC for 10hrs using a

curing agent. The ‘Krak gauge’ is used to monitor the crack growth in the

specimen.

Cyclic-fatigue tests were carried out for the different bulk epoxy matrix

polymers. The specimens were subjected to displacement-controlled fatigue

loading with the frequency of loading kept at 5Hz. As well as using the ‘Krak

gauge’ method the crack growth under the fatigue loading was also monitored

using an optical microscope focussed on the crack front. This test therefore

measured the maximum load and crack growth under fatigue loading. It also

determined the load below which there was no propagation of the crack, and

therefore the threshold fracture energies of the different bulk epoxy matrices

were determined from the data. The maximum fracture toughness, , for a

given cycle can be calculated from the expression

(3.4)


49

where is the maximum load in the fatigue cycle, the breadth, the width

and is a shape factor of the specimen where . The shape factor is

given by the expression

(3.5)

The maximum fracture energy may be calculated from the equation

(3.6)

where the elastic modulus, , and the Poisson’s ratio, , of material are

obtained from Table 5.1. The crack growth curve can be obtained from the

secant method and the incremental polynomial method (ASTM:E647 [54]). In

the incremental polynomial method, the growth rate curve, , is obtained by

fitting a polynomial between a set of points. In the present work, the secant

method was used for obtaining the growth rate curve. In the secant method, the

slopes of the adjacent points are used to calculate the growth rate curve.

3.6 Test Methods for the GFRP Composite Specimens

3.6.1 Introduction

Tests were undertaken the composite material at 20o±2oC under both quasi-

static and cyclic fatigue loading to determine the fracture mechanics properties

of the material. Tension-tension cyclic fatigue tests were also conducted on

strips of the GFRP composite material to determine the lifetime of the material

under fatigue conditions. Details of the tests conducted on the specimens are

given below.


50

3.6.2 DCB tests: Quasi-static tests

A standard quasi-static test was conducted to determine the mode I interlaminar

fracture energy, , under quasi-static load for the GFRP composites using a

DCB specimen (ISO:15024:2001 [52]). During the test the load, displacement

and crack length required for the initiation and growth of the crack were

measured in the DCB specimen. The values of for initiation and propagation

of the crack were obtained from the load versus displacement trace. The DCB

specimen was loaded under displacement control at a rate of 1mm/min. The

start of the crack was observed visually and the growth of the crack was noted

using the event marker to indicate the crack length on the load versus

displacement trace. The laminate sequence of the composite was [(-

45/45)s(90/0)s]2 [(0/90)s(45/-45)s]2, with the initial crack located at the mid-plane

of the laminate. The test is shown in Figures 3.2 and 3.3.

Figure 3.2 DCB specimen.

The load versus displacement trace of the composite was typical of brittle matrix

fibre laminates. The fracture surfaces showed evidence of tow splitting, fibre-

breakage and fibre-matrix interfacial fracture (see Figure 3.4). The non-linearity

(i.e. the 5% offset) or the maximum load criteria was used to define the point of

crack initiation. The 5% offset is the intersection of the load versus

displacement curve with a line corresponding to a value of the compliance

which is 5% higher than the initial slope (ISO:15024:2001 [52]).

Aluminium loading block

Composite

Initial crack


51

Figure 3.3 Experimental setup for a DCB test.

Figure 3.4 Fibre bridging in DCB test.

The value of the interlaminar fracture energy, , was calculated using the

‘Corrected beam theory’ (CBT) method as defined in the ISO standard

(ISO:15024:2001 [52]) and which may be expressed by


52

(3.7)

where , , , and are the load, load-line displacement, crack width, crack

length and the crack-tip rotation correction factor of the specimen, respectively.

The factor is given by

(3.8)

where is the distance from the centre of the loading pin to the mid-plane of

the specimen beam.

3.6.3 DCB tests: Cyclic-fatigue tests

The tests were conducted according to the prescribed standard (ASTM:E647

[51]). The load was applied as a sinusoidal function with a maximum

displacement of 50% of the displacement required for the initiation of the crack

in the corresponding quasi-static tests of the DCB specimens. The frequency of

the loading was 1~3Hz. The laminate sequence of the composite strip was [(-

45/45)s(90/0)s]2[(0/90)s(45/-45)s]2, which is identical to that of the DCB

specimens described above.

The cyclic-fatigue DCB test was conducted using a 1kN computer-controlled

servo-hydraulic test machine (Figure 3.5). The tests were conducted under

displacement control loading. A travelling microscope was mounted on a

traversing stand to monitor the growth of delamination. The number of cycles

required for the growth of the crack was noted for different amplitudes of the

applied displacement. The maximum load was recorded when the visual onset

of delamination growth was observed on the edge of the specimen. The

threshold energy of the strain energy release rate was also obtained from the

fatigue test. The maximum energy release rate, , was calculated using the

above equations for the DCB quasi-static test. The crack growth rate, , and


53

the for the experiments were plotted using logarithmic scales to obtain the

typical growth rate curve.

Figure 3.5 Experimental setup for a fatigue DCB test.

3.6.4 Composite strip laminate tests: Quasi-static and fatigue tests

A quasi-static test was conducted on the composite strip to determine the quasi-

static strength and elastic properties of the laminate. The tests were conducted

based on the ASTM:D3039 [55] standard. The tensile test was performed using

a 100kN computer-controlled screw-driven test machine and the specimen was

loaded at a rate of 1mm/min. The load versus displacement curve was obtained

to determine the elastic property of the composite.

Cyclic-fatigue tests were conducted on the composite strip to study the fatigue

life of the composite laminate (ASTM:D3479M [56]). Composite strips were

loaded using a stress controlled cyclic-fatigue test to different stress levels, and

the number of cycles to failure was noted. During this test the growth, the

initiation and propagation of transverse cracks in the composite laminate strip

under fatigue loading were also observed. The size of the strips used for the

testing was 150x25x2.7mm3 and the load was applied as constant amplitude

sinusoidal stress with a stress ratio of 0.1, using a 25kN computer-controlled


54

servo-hydraulic test machine. The frequency used for the low-cycle fatigue (i.e.

a high applied stress) tests was 1Hz and for the high-cycle fatigue (i.e. a low

applied stress) was 4Hz. The tests were conducted using different specimens at

different stress levels and the number of cycles to failure of the specimen was

noted. The crack density was also noted for a given number of cycles and the

variation of crack density on the surface of the specimen was plotted against

the number of fatigue cycles. The crack density was observed on the surface of

the specimens using an optical microscope, and the stiffness reduction of the

strip was also measured as a function of the number of fatigue cycles.


Tests on the bulk epoxy and the GFRP composite were undertaken to measure

their behaviour under quasi-static and fatigue loading. From the quasi-static

tests the values of the fracture energy, , for both the bulk epoxy matrices and

the GFRP laminates were measured. The cyclic-fatigue tests enabled the rate

of the crack growth per cycle, , and the corresponding maximum applied

stain-energy release rate, , to be determined, again for both the bulk epoxy

matrices and for the corresponding GFRP laminates. All these data are needed

for the modelling studies described in Chapter 5. To validate the model, the

applied stress versus the number of fatigue cycles to failure was measured,

using composite strip specimens of the GFRP laminates of the same lay-up

sequence. The next Chapter describes the theoretical modelling work

developed during the current research.

CHAPTER 4 THEORETICAL TECHNIQUES

55

CHAPTER 4

4. THEORETICAL TECHNIQUES

4.1 Introduction

In the present chapter, theoretical models and formulations based upon finite

element analysis (FEA) methods are developed to simulate the experimental

results. Different methods of analysis, such as virtual crack closure technique

(VCCT), cohesive contact and cohesive zone elements are used, with the FEA

approach, for modelling the fracture of the composite materials. Analysis of the

FEA models is undertaken using the Abaqus software program employing

continuum elements and cohesive zone elements. All the analyses of the

continuum elements is done using 2D plane-strain elements. In a typical

cohesive zone element model of the specimens, plane-strain elements are used

to model the continuum elements and the cohesive zone elements are used

solely to model the fracture path. It should be noted that in Abaqus the cohesive

law is represented by stress/traction versus strain in the cohesive zone.

4.2 Quasi-Static Analysis

A quasi-static analysis formulation is derived in the following sections employing

the different methods as listed below. The underlying principles of the FEA

approach are also explained and the details of the methodology are shown.

4.2.1 The Virtual Crack Closure Technique (VCCT)

The VCCT method is derived from the crack closure technique (CCT), see

Section 2.4.2. This technique determines the energy released during crack

propagation from the geometry of the crack. The energy released can be

calculated from the displacement and forces in each direction at the nodes of

crack tip. The energy released by the crack extension, , is the work required


56

to close the crack by the same amount to its original length, keeping the

external load constant.

Figure 4.1 The VCCT model.

The energy released can be calculated using the displacement and nodal forces

in the different directions (Krueger [6]). The crack is represented by a one

dimensional discontinuity of a line of nodes which have the same coordinates at

the top and bottom surfaces (Figure 4.1). The energy released due to mode I

and II is due to the opening and shear displacements between the contact

surfaces. The energy released in different directions is the energy release

associated with that given mode, e.g. mode I or II.

The different mode components of the strain energy release rate (SERR) can

be obtained by combining two analysis methods. In the first analysis, nodal

forces are calculated at the nodes prior to crack growth and in the second

analysis the crack nodes are released to obtain the displacements at the crack

tip. The SERR is calculated by multiplying half of the nodal forces from the first

analysis with the displacements obtained in the second analysis. The energy

released in different modes due to crack opening can be expressed as

(4.1)

(4.2)

y, v

x, u

a

Δa

b

c

d

ef


57

(4.3)

where are the SERR at the nodes, and are the nodal

forces acting at the nodes with displacements . The above equations

are presented for the mixed-mode SERR for a two dimensional model in plane

stress or plane strain.

4.2.2 The cohesive zone law

Cohesive zone laws are powerful tools used for modelling the failure of

composites, such as delamination, shear cracks, matrix cracking, fibre failure or

micro-buckling (e.g. kink-band formation), friction between the plies, bridging by

through-thickness reinforcement and oblique crack-bridging fibres. In this

method, a cohesive zone is used to model the crack propagation, in mode I

mainly. The cohesive zone law helps to overcome the complexity of considering

a singularity at the crack tip. The cohesive zone law relates the traction and

displacement at the crack tip to the energy release rate of the material when

loaded in the different modes. The concept behind such a law is that crack

propagates in the material according to a defined cohesive zone law, and the

law itself may change according to well defined damage evolution principles.

4.2.2.1 Kinematics

The kinematics of the cohesive zone can be developed from a crack present in

a material (Ortiz and Pandolfi [57]). The crack in the material causes the

formation of new surfaces, which can be assumed to possess a top and a

bottom. The crack in the material can be represented using a cohesive zone

element (Figure 4.2) of zero thickness. The relative displacement across this

cohesive zone element can be written as

(4.4)

where and are the displacements at the top and bottom surface of the

cohesive zone element.


58

Figure 4.2 A four noded cohesive zone element in (a) undeformed state and (b)

deformed state in a global coordinate system.

The coordinates of the cohesive zone element can be represented using a

global coordinate system, as

(4.5)

(4.6)

The global coordinate system for the mid-plane of the cohesive zone element

can be represented as

(4.7)

The vector which defines the normal and the tangential surface of a deformed

cohesive zone element is given by

(4.8)

(4.9)

where and are the normal and tangential direction in the numerical

coordinate system and and are the coordinates in the local system. The unit

vector normal to the local coordinate system can obtained as

(4.10)

X1

X2

ξ

η

1 2

3 4

(a)

X1

X2

(b)


59

The unit vectors tangential to the local coordinate system can be represented

as

and

(4.11)

(4.12)

where , and are the direction cosines of the local coordinate system in

the global coordinate system

4.2.2.2 Constitutive laws

The relation between the relative displacement and the traction for an cohesive

zone is given by the relation of the traction versus the separation (Turon et al.

[35]). The traction in the cohesive zone law for a 2D cohesive zone model is a

function of a displacement jump norm and can be written as

(4.13)

(4.14)

where is tangent stiffness tensor, is the cohesive stress and is the

norm of the displacement jump. The energy in the cohesive zone law is related

to the traction and displacement jump. The free energy per unit surface of the

layer can be expressed as

(4.15)

where is the damage variable. The energy can be expressed as

(4.16)

where is the initial stiffness tensor. It shall be noted that the negative values

of are eliminated to avoid the interpenetration of the different surfaces. Thus,

the expression for the free energy is given by


60

(4.17)

where is the MacAuley bracket defined as and is the

Kronecker delta. The equation for the cohesive surface is obtained by

differentiating the free energy with respect to the displacement jump as

(4.18)

where is the cohesive traction. The undamaged stiffness tensor is defined

as

(4.19)

where is the penalty stiffness of the cohesive zone element. The penalty

stiffness of the element is selected in order to have a high penalty stiffness

condition being used to simulate the cohesive surface. The constitutive equation

can be written in Voigt notation as

(4.20)

The energy at a given period can be expressed as

(4.21)

4.2.2.3 Mathematical formulation

A 2D cohesive zone element is made up of two linear-line elements connected

to the fracture surface (Feih [58]). The two surfaces of the cohesive zone

element initially lie together in the unstressed deformed state and separate as

the adjacent elements deform. The relative displacements of nodes of the

cohesive zone element in the normal and shear direction create element

stresses. A four noded cohesive zone element (Figure 4.3) is considered for the

present study, but it can be implemented here to develop the formulation for


61

higher degrees of cohesive zone element. The thickness of the cohesive zone

element is assumed to be zero.

Figure 4.3 A four noded cohesive zone element.

The present element formulation is derived for a linear-line element for 2D

simulations. The 2D element has two degrees of freedom (DOF) at each node

and hence the number of degrees of freedom is twice the number of nodes in

the element. Hence the linear cohesive element has four nodes ( and eight

(2x ) degrees of freedom; four on the top surface and four on the bottom

surface. The displacement at the nodes of cohesive element is expressed as a

vector, . The nodal displacement vector in a global coordinate system is given

by

(4.22)

where and are the displacement at the node in the direction and

direction, respectively. The relative displacement between the paired nodes is

used to derive the cohesive formulation. The relative displacement between the

linked pair of nodes can be obtained by operating the displacement vector with

a matrix, where is the identity matrix. Hence, the relative

displacement, , vector is given by

(4.23)

The displacements at the nodes are used to obtain the integration point

functions. The different integration point functions for an element can be

ξ

η

ξ=-1 ξ=11 2

3 4

ξ =0thickness=0

u

v


62

obtained from the relative displacement at the nodes using a shape or an

interpolation function. The interpolation functions, , (Cook et al. [5]) for

each node are obtained in the local coordinate system ( ) of the element. The

degree of the interpolation function depends on the number of node pairs in a

given element. The relative displacement between the paired nodes in an

element is given by

(4.24)

where

(4.25)

(4.26)

where is

(4.27)

In case of large deformations, the mid-plane of the element is taken as the

coordinate of the element. The mid-plane coordinate is calculated to determine

the deformation of the element. The mid-plane coordinate for the element under

deformation can be obtained as given below

(4.28)

where is the coordinate of the element in the undeformed state in a given

coordinate system. Hence, the relative displacement between paired nodes in a

coordinate system is given by

(4.29)


63

This local coordinate vector with unit length is obtained by differentiating the

global position vector with respect to the local coordinates.

(4.30)

(4.31)

where can be written as

(4.32)

where is the derivative of the shape function matrix given by

(4.33)

The length of the element is given by the modulus of the and is obtained

as

(4.34)

The transformation matrix, , which relates the local and global displacement

is given by the relation

(4.35)

The global displacement, , and local displacement, , can be related

using the transformation matrix, , as


64

(4.36)

The force vector for an element is then given by the expression

(4.37)

(4.38)

where is the width of the cohesive zone element, or the through thickness of

the model and is the traction vector. The above integration can be achieved

by a Newton-Raphson numerical integration technique (Cook et al. [5]). The

above expression can be integrated using the numerical integration technique

as given below.

(4.39)

where is the local traction vector The determinant of a Jacobian matrix,

, is given by the expression

(4.40)

The value of should be positive and it transforms the local coordinate

system to a global coordinate system. The stiffness matrix of an element is

given by the relation

(4.41)

By integrating numerically as before, is given by the expression


65

(4.42)

(4.43)

(4.44)

(4.45)

The local stiffness tensor is given by the relation

(4.46)

(4.47)

where and are the penalty stiffnesses of the cohesive zone law in the

directions 1 and 2, and is the coupling term which is assumed to be zero.

The numerical implementation of the above formulation is undertaken by using

the tangent stiffness tensor, , of Equation 4.14 which can be derived as

(see Turon et al. [59] for the derivation)

(4.48)

where is the displacement jump threshold in the loading history and the

factor is given by

(4.49)


66

4.2.3 A bi-linear cohesive zone law

A bi-linear cohesive zone law is defined by the traction and displacement

between adjacent cohesive nodes in a cohesive zone. A cohesive constitutive

law relates the traction to displacement jumps at the cohesive surface. The area

under the traction-displacement jump curves is the respective fracture energy,

, for the given mode. A typical cohesive zone model is characterized by a bi-

linear, rate-independent, damage-dependent failure law across the cohesive

surfaces. The cohesive zone law is represented by the penalty stiffness, ,

critical displacement for failure, , and the final failure displacement, , as

shown in Figure 4.4. The value of the critical stress is ideally a characteristic

property of the material.

The propagation of the crack is dependent on the strain energy release rate

(SERR) corresponding to the different modes. The energies released in the

different modes are combined to determine the critical SERR, as was discussed

in Section 2.4.3.3. Different laws may be used to combine the different modes

of the energy to find the critical SERR. The propagation of the crack occurs, of

course, once the strain energy released is more than the critical strain energy

for propagation. The parameters for the cohesive zone law are determined by

calibrating the theoretical model using the experimental results, and the critical

SERR, , is the criterion used in linear elastic fracture mechanics (LEFM) for

the propagation of the crack.

Figure 4.4 The bi-linear cohesive zone law.

Tra

ctio

n, τ

Displacement, δ

Gc

K

δo

τ0

δf


67

In the present work, a bi-linear cohesive zone law is assumed for each of the

fracture modes. The parameters required to define a bi-linear cohesive zone

law are the critical displacement, , critical stress, , and critical SERR, as

shown in Figure 4.4. The three parameters in the bi-linear law are independent

of each other and depend on the material properties. The bi-linear cohesive

zone law is divided into the linear elastic region, the linear stiffness degradation

region and the failure zone. The first part of the cohesive zone law defines the

behaviour between the elastic limit and the critical displacement. The elastic

limit coincides with the maximum stress value and, once the elastic limit is

exceeded in the zone, the cohesive zone starts to degrade. The last part of the

cohesive zone law defines a relative displacement value that is equal, or larger,

than the critical displacement value.

The main characteristic of the cohesive zone models is that the cohesive

surface can still transfer load after the onset of damage. When the critical value

of displacement jump norm (Section 4.2.3.1), i.e. , (Turon et al. [43]) is

reached or exceeded, the element fails. When formulating this cohesive

constitutive law for mode I, any negative relative displacement is avoided, to

prevent interpenetration of surfaces, by adopting a cohesive zone law as in

Figure 4.4. In mode II, negative relative displacements may readily exist and

therefore a symmetrical bi-linear constitutive law is adopted (see Figure 4.5).


68

Figure 4.5 Cohesive zone law in mode II.

A cohesive zone law can be implemented in a FEA analysis using either (a) a

cohesive contact analysis, or (b) a cohesive element analysis. In a cohesive

contact analysis, the fracture surfaces are connected together by nodes of the

fracture surface and the displacement between the nodes are used to

determine the cohesive zone law. The displacements at the element nodes of

the surface are employed for the calculation of the cohesive zone law, and to

determine the failure behaviour. The cohesive zone law can also be

implemented using cohesive zone elements which represent the fracture

surface. Here, the displacements at the adjacent nodes of the cohesive

elements are used to implement the cohesive zone law.

4.2.3.1 Norm of displacement jump tensor

The displacement jump (Turon et al. [43]) is a function representing the

resultant displacement at the nodes in a mixed-mode analysis. The norm of the

displacement jump tensor is used to compare different stages of the

displacement jump state. The displacement jump norm, , is a continuous

function accounting for the other modes and can be expressed using mode I

and II displacements as

Gc

δo

τ0

δf δ

Gc

-δo-δf

τ


69

(4.50)

(4.51)

where is the shear displacement jump in mode II and is the displacement

jump in mode I. The value of the displacement jump norm is always greater

than, or equal to zero, and is used to avoid the interpenetration of the

cohesive zone elements.

4.2.3.2 Damage

A damage variable, , is employed with respect to the cohesive zone law to

define the three states of the cohesive zone law: the elastic state, the damaged

state and the failure state.

for the elastic state:

(4.52)

for the damaged state:

(4.53)

for the failure state:

(4.54)

The damage variable, , increases rapidly once the critical damage in the

cohesive zone element is reached. This rapid increase in the damage variable

is due to the definition of the damage variable in the cohesive zone law. The

variation of the damage variable with an increase in the displacement is shown

in Figure 4.6.


70

Figure 4.6 Variation of damage variable with displacement in a bi-linear

cohesive zone law.

4.2.3.3 Mixed-mode loading: onset of crack growth

An initiation criterion derived from the Benzeggagh-Kenane (B-K) fracture

criterion, see Section 2.4.3.3, gives a sound basis for a cohesive zone law for

mixed-mode loading based on the mode-mixity factor, (Benzeggagh and

Kenane [60]). The opening displacement under mixed-mode loading is given by

the relation

(4.55)

where is the energy release rate in mode I and is the energy release rate

in mode II. The mode-mixity factor, , depends on the mode-mixity and is

determined experimentally. The equivalent failure displacement for mixed-mode

loading can be calculated from the expression

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(δ/δf)

Da

ma

ge


71

(4.56)

The traction in the cohesive zone after the elastic limit has been exceeded may

be described in terms of the critical traction, , the opening displacement, ,

and the failure displacement, , as

(4.57)

where . The stress in the cohesive zone element is zero when the

displacement jump is equal to or more than the failure displacement is given by

; (4.58)

The damage variable for mixed-mode loading is given by

(4.59)

4.2.3.4 Mixed-mode loading: crack propagation

The propagation criteria for a mixed-mode crack growth are derived based on

the components of the energy release rate in the different modes. The critical

SERR, , is derived for mixed-mode loading and is used to predict crack

propagation. The crack growth occurs when the strain energy released is more

than the critical strain energy release rate for a given mixed-mode load. Hence,

the criterion can be written as

(4.60)

The expression for the can be derived from the mode-mixity of the problem

and the energy release rate, , is the resultant of the energy released in mode I

and II. The critical strain energy release rate, , for mixed-mode crack growth

may be obtained as


72

(4.61)

The energy release rate in different modes for a mode-mixity, , can be

expressed as (Travesa [61])

(4.62)

(4.63)

The mode-mixity factor can be obtained from the relation (Camanho et al. [7])

(4.64)

The mode-mixity factor can be written based on the displacement as

(4.65)

is the Mac Auley bracket defined as = . The displacement jump

in the different modes can be expressed as

(4.66)

Using the Equation 4.66, the opening and shear displacements under mixed-

mode loading are related by the expression

(4.67)

(4.68)


73

Using the above equation, the ratio of energy released in mode II to the total

energy released rate can be obtained as

(4.69)

It should be noted that under cyclic fatigue loading, crack propagation occurs

when is greater than the threshold value, . In fatigue loading, crack growth

occurs and is stable if the energy released is more than and less than of

the material.

4.2.3.5 Mode-mixity

A mixed-mode criterion is used to establish the interaction between the different

components of strain energy release rate for mixed-mode loading. The criterion

simulates the onset of crack propagation and failure under mixed-mode loading.

The B-K criterion (Camanho et al. [7]) which accounts for mode-mixity is given

by

(4.70)

where and are given by the expressions

(4.71)

(4.72)

The mode-mixity factor, , is the factor obtained from an experimental fit.

Under mixed-mode loading, damage onset may occur before any of the critical

stresses involved reach their respective critical limits. In typical industrial

applications of composites, crack growth occurs mainly under mixed-mode

conditions.


74

4.3 Fatigue Analysis

4.3.1 Introduction

The analysis of fatigue driven crack growth in composite materials using FEA is

tedious and is dependent on the interactions between the variables involved. In

the present work, a mathematical model is developed for mode I and mode II

fracture, with the mode II parameters assumed to be equivalent in value to

mode I, due to mode I failure being dominant failure mode observed in

composite materials. This assumption was shown to be valid concept by Harper

and Hallett [62]. Further, the variation of the test frequency and displacement, or

stress, ratio are not considered to be significant factors, as observed by Yang et

al. [63] and Manjunatha et al. [40], and hence the rate dependence of material

need not to be taken into account. The subroutine for the fatigue degradation is

written in FORTRAN.

The degradation of the cohesive zone law for the composite materials, using the

Paris law constants determined from the experiments conducted using the

corresponding epoxy polymer matrices, can be described using the following

flowchart.


75

Figure 4.7 Flow chart of the fatigue analysis embedded in the user element subroutine.

Input parameters for an user element subroutine for fatigue analysis

( , and (see Table 4.1)

Calculate

Calculate

Calculate static damage,

Calculate fatigue damage rate,

where

Damage= static damage + x time increment

Stiffness= (1-Damage)

Run the model with the degraded cohesive zone element to

get the complete degradation of the model with time


76

Table 4.1 Description of the material properties of the bulk epoxy matrix relevant to Figure 4.7.

Parameter

s

Meaning How obtained Explained

Fracture energy of

the bulk epoxy matrix

Quasi-static SENB experiments on the bulk epoxy matrix

Section 3.3.3

Penalty stiffness of the cohesive zone

model

By fitting the CZM model to the bulk epoxy matrix load

versus displacement curve from quasi-static SENB tests

Sections 5.4.1.2 & 5.4.1.3

Critical stress (i.e.

traction) of the cohesive zone model

By fitting the CZM model to the bulk epoxy matrix load

versus displacement curve from quasi-static SENB tests

Sections 5.4.1.2 & 5.4.1.3

Threshold strain energy release rate of bulk epoxy matrix

Fatigue CT experiments on the bulk epoxy matrix

Section 3.5.3

Paris law parameter of bulk epoxy matrix


Section 3.5.3

Paris law parameter of bulk epoxy matrix


Section 3.5.3

4.3.2 Degradation strategies

The degradation under cyclic fatigue loading of the cohesive zone law

parameters can be achieved using different strategies based on the degradation

of the penalty stiffness, . Indeed, the present work uses degradation of the

penalty stiffness as a strategy for the degradation of the constitutive law

embedded in the cohesive zone law upon fatigue loading. The penalty stiffness

degradation of the bi-linear law can be achieved using the evolution of a

damage variable in the cohesive zone law. To initiate this modelling approach, it

should be noted that the stress in the cohesive zone law can be expressed as


77

where

(4.73)

(4.74)

and where is the damage variable.

4.3.3 Static damage evolution under fatigue loading

The evolution of static damage occurs due to the reduction of the penalty

stiffness of the cohesive zone, which leads to the development of further static

damage. The increase in such defined static damage due to fatigue loading can

be derived from the rate of damage evolution with time. The evolution of the

static damage variable (Robinson et al. [41]) under fatigue loading can be

derived from Equation 4.59 as

(4.75)

The static damage evolution for a given number of cycles, , is the given by

(4.76)

where is the time to cycles and is the time corresponding to

cycle. Integrating the above equation, the evolution of the static damage

with the number of cycles is given by

(4.77)


78

4.3.4 Fatigue damage evolution

The fatigue crack growth may be defined as the extent of growth of the crack

per cycle, which is represented as . The crack growth rate curve is usually

represented as versus . The Paris law is then typically used to

describe the linear region of the growth curve and which relates the maximum

energy release rate, , in a fatigue cycle to by the relation

(4.78)

where and are constants that depend on the material and the mode ratio.

The constants and are given by the vertical intercept and slope of the linear

region of the growth rate curve, respectively, when using logarithmic scales.

The values of both constants are obtained by fitting the Paris law equation to

the experimental results.

The damage which develops under fatigue loading is the sum of the quasi-static

and the fatigue damage (Muñoz et al. [64]). The evolution of the damage

progresses with time, and the rate of change of damage with time can be

expressed as

(4.79)

The static damage evolution can be calculated from Equation 4.75 for the

cohesive zone element. The damage evolution with the number of cycles, ,

can be related to the crack growth rate curve in the fatigue loading as

(4.80)

where is the growth rate of the crack and is the damaged area. The

growth rate of the curve depends on the material properties. The expression for


79

the can be derived from the strain energy released during the propagation

of the crack. The ratio of the energy released, to the critical fracture energy,

, can be expressed as (Figure 4.8)

(4.81)

where is the area of the cohesive zone element.

Figure 4.8 Cohesive zone law and energy representation.

The expression for the can be obtained from the above equation as

(4.82)

The increase in the growth of the damaged area with the number of cycles is

the sum of the damaged area growth in the entire cohesive zone. Hence, the

growth of the damaged area with the number of cycles can be represented as

δ

τ

δo

τ0

δf

Gc

δ

Θ

(1-d)K


80

(4.83)

where is the total area of the cohesive zone and is the damaged area in

a given cohesive zone element. The average damaged area in the cohesive

zone element can be taken as . The number of cohesive zone elements in

the cohesive zone can be obtained as

(4.84)

The expression for the growth rate can then be simplified as

(4.85)

The terms can next be rearranged to get the expression for as

(4.86)

The damage evolution for the cohesive zone element can now be obtained by

substituting from Equation 4.82 and can be expressed as

(4.87)

The area of the cohesive zone (Turon et al. [43]) for mode I loading can be

written as

(4.88)

The crack growth rate arising from fatigue damage is dependent on the energy

released and can be expressed using the Paris law as


81

(4.89)

where and are the constants to be determined experimentally, as described

in Chapter 2. The damaged area in the cohesive zone can be expressed as

(4.90)

where is the width of the crack. The total change in the energy released in a

fatigue cycle is the difference in the maximum and minimum energy release rate

in a cycle and can be expressed as

(4.91)

where and are the maximum and minimum strain energy release rates

during a fatigue loading cycle (Figure 4.9).


82

Figure 4.9 Representation of energy release under fatigue in a cohesive zone law.

The maximum energy released is given by the expression

(4.92)

where and are the maximum displacement jump and damage in the

whole cyclic loading history. The constitutive relationship derived is independent

of the element formulation. It is important to note that the fatigue degradation of

the cohesive zone law with time occurs as shown in Figure 4.10 where and

are the values of the damage variables at time and . As noted above,

the damage in the cohesive zone law occurs due to the combined static and

fatigue loading. The actual degradation of the cohesive zone law with time due

δo

τ0

δf

Gmin

δ

δ

τ

τ

δ max

δ min

δo

τ0

δf

Gmax


83

to both the static and fatigue damage evolution can be illustrated as shown in

Figure 4.11. In these figures, and in the present modelling studies, the fatigue

damage variable, , can be calculated from Equation 4.87 based on the

number of cycles accumulated with time. The term can then be used to

account for the total damage in Equation 4.74.

Figure 4.10 Fatigue degradation of cohesive zone law with time.

Figure 4.11 Resultant fatigue degradation of a cohesive zone law. Path 1-2

shows the fatigue damage evolution and path 1-3 shows the static and fatigue damage evolution (Robinson et al. [41]).

τ

δ

K

δt δf

τ0

K(1-dt)

δt+1

K(1-dt+1)

Gc

δo

τ0

δfδ 1

1

32

τ

δδ 2


84

4.3.5 Damage Analysis

When using the above modelling approach the analysis is not undertaken cycle

by cycle due to the computational effort needed. The complete cycle of fatigue

loading is undertaken based on a ‘cycle jump strategy’ as described below.

4.3.5.1 The cycle jump strategy

The cycle jump strategy in the fatigue analysis is employed to limit the number

of individual analyses needed in modelling high cycle fatigue. The cycle jump

strategy controls the accuracy of the damage variable for a given cycle jump.

The accuracy of the degradation modelling is controlled by limiting the

maximum change in the damage variable for a given jump of cycles (Van

Paepegem and Degrieck [65] and Muñoz et al. [64]). The cycle jump principle is

illustrated in Figure 4.12. In the present work, the cycle jump strategy is adopted

in the model of the fatigue life to limit the maximum time increment employed in

the analysis.

Figure 4.12 The cycle jump strategy applied to a cohesive zone law approach

to modelling fatigue (Van Paepegem and Degrieck [65]).

4.3.5.2 Displacement ratio and load ratio

The displacement ratio, , is defined as the ratio of the minimum

displacement, , to the maximum displacement, , applied in a cycle of

fatigue loading (Lee [49]). Higher values of the displacement ratio cause a

decrease in the fatigue damage due to less change occurring in the strain

energy released. The value of may be defined by


85

(4.93)

The load ratio, , is defined as the ratio of the minimum stress, , to the

maximum stress, , applied in a cycle of fatigue loading (Turon et al. [43]).

Higher values of the load ratio also cause a decrease in the fatigue damage due

to a smaller change occurring in the strain energy released, . The load ratio

for a fatigue cycle can be expressed as

(4.94)

Now, the cyclic loading is applied as a sinusoidal load with a given frequency

and a displacement, or stress amplitude. Numerically in the modelling studies

the load is applied as a constant displacement, or stress, which is equivalent to

the maximum displacement, or stress, applied in the experiment. The minimum

to the maximum displacement for the fatigue cycle depends on the

displacement ratio, , and the minimum to the maximum stress for the fatigue

cycle depends on the stress ratio, , relevant to the fatigue cycle. Hence, in the

numerical model, the displacement, or stress, is applied in the first cycle of

loading and the displacement, or stress, is kept constant for the remainder of

the cycles being modelled (Robinson et al. [41]), as shown in Figure 4.13.


86

Figure 4.13 Experimental and numerically applied displacement/stress in a displacement/stress controlled fatigue test. In the present study, displacement controlled fatigue tests were conducted on the CT bulk epoxy material and the DCB composite materials, whilst stress controlled fatigue tests were conducted on the composite material strip specimens.


The present Chapter has described the theoretical methods which have been

developed to model and predict the quasi-static and fatigue behavior of the

composite materials. The values needed for these modelling studies are

ascertained from experiments conducted upon the bulk epoxy matrices, and

validated using the DCB composite material test results, as will be described in

Chapter 5. The model will then be used to predict the cyclic fatigue behavior

and lifetime of the composite material strips. The model represents a novel

method to predict such behaviour. It builds upon the research of Robinson et al.

[41] & Turon et al. [43] but contains several novel features, including an

important new user element subroutine.

The following Chapter will describe the experimental results obtained in the

present research and compare these results to the theoretical predictions which

have been obtained using the models developed in the present Chapter.

CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES

87

CHAPTER 5

5. EXPERIMENTAL RESULTS AND

THEORETICAL MODELLING STUDIES

5.1 Introduction

In the present work, the material and cohesive zone law parameters of the bulk

epoxy matrices and the composite materials are obtained from the various

experimental tests. The tests were undertaken on the bulk epoxy matrices and

the corresponding composites, as explained in Chapter 3. The quasi-static and

the fatigue test experimental results are described in the sections of the present

Chapter.

The predicted theoretical results are then discussed and compared to the

experimental results. The flowchart for the different analyses of the various

specimens is given in Figure 5.1, which gives an overview of the experimental

and modelling work undertaken in the present research.


88

Figure 5.1 Overview schematic of the work plan.

Model the DCB fatigue composite

specimen and obtain the growth

rate curve to match the test (Section

5.6.2.2)

Model the quasi-static strip specimen and obtain the normalised

stiffness as a function of crack density and number of cycles (Sections 5.4.3.1 & 5.4.3.2)

Model the composite strip specimen under

fatigue loading with the crack density observed from the experiments

(Section 5.6.2.3)

Test the composite strip under quasi-static load to obtain strength of the

laminate (Section 3.6.4)

Test the composite strip under fatigue

loading to obtain the crack density,

stiffness and the number of cycles to

failure for the laminate

(Section 3.6.4)

Test the DCB composite specimen under fatigue load to

obtain the crack growth rate curve and

fatigue parameters

(Section 3.6.3)

Prediction and initial validation of the

fatigue life of the DCB composite specimen under fatigue loading

(Section 5.6.2.2)

Prediction of the fatigue life of strip specimen under fatigue loading

(Section 5.6.2.3)

Obtain the elastic properties of the bulk epoxy and the

composite material (Sections

5.2.1 & 5.2.2)

Model the quasi-static DCB composite

specimen and obtain the

parameters of the cohesive zone law (Sections 5.4.2.2 &

5.4.2.3)

Model the quasi-static SENB specimen of bulk

epoxy and obtain the cohesive zone law

parameters of the bulk epoxy (Sections 5.4.1.2

& 5.4.1.3)

Model the CT fatigue specimen of bulk epoxy and obtain the fatigue parameters of the bulk epoxy (Section 5.6.2.1)

Test the SENB specimen of bulk

epoxy under quasi-static load to obtain the

load-displacement

curve (Section 3.5.2)

Test the CT specimen of bulk

epoxy under fatigue loading to obtain the

crack growth rate curve and fatigue

parameters (Section 3.5.3)

Test the DCB composite specimen

under quasi-static loading to obtain the load-displacement

curve (Section 3.6.2)


89

Notes to Figure 5.1:

a) Sections 5.4.1.2 & 5.4.1.3: The cohesive zone law parameters from the bulk

epoxy matrix are used to model the development of transverse cracks in the

composite, and hence are used in Section 5.6.2.3 of the flow chart.

b) The fatigue parameters determined in Section 5.6.2.1 are used for the

composite strip modelling in Section 5.6.2.3.

c) In Section 5.6.2.2, the DCB fatigue composite sample is modelled so as to

validate the novel user element subroutine analysis.

d) In Section 5.6.2.3, it should be noted, it is not necessary to undertake fatigue

experiments to find the crack density, but one can use the saturated crack

density as explained later in Figure 5.28.

5.2 Elastic Properties of Materials

The elastic properties of the different materials are derived as detailed below in

Sections 5.2.1, 5.2.2 and 5.2.3. The different directional elastic properties of the

composite are derived from the basic equations for composite materials.

5.2.1 Elastic properties of the lamina

The elastic properties of a unidirectional lamina are derived to determine the

properties of the composite material. The fibre properties of the 0o ply can then

be calculated from the general expression for the lamina. The volume fractions

of the composite material (Manjunatha et al. [40]) are also used to determine

the properties of the lamina. The elastic properties of the lamina are calculated

from the expressions for the unidirectional properties of the lamina. The lamina

is assumed to be oriented in the 1-2 planes with the fibre direction in the 1 axis,

as shown in Figure 5.2.


90

Figure 5.2 A section of a lamina with local coordinates.

The different material properties of the lamina in different directions can be

obtained from the expressions below. The equations are obtained from

Khashaba [66].

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

1

23


91

(5.10)

(5.11)

(5.12)

where , , , , , and are the elastic moduli of the fibre and

matrix, volume fraction of plies, shear moduli of fibre and matrix, bulk moduli of

composite, fibre and matrix, respectively. The factor is the shape factor, which

has a value of two for a circular fibre (Khashaba [66]). The matrix properties of

the different bulk epoxy matrices (Manjunatha et al. [67], Pegoretti et al. [68])

and the glass fibre (Pegoretti et al. [68]) used in the modelling studies are given

in Table 5.1. The unidirectional elastic properties of the lamina obtained from

the equations above are given in Table 5.2.

Table 5.1 Elastic properties of bulk epoxies and glass fibre


92

Table 5.2 Unidirectional elastic properties of the different composite lamina based on the various epoxy matrices

5.2.2 Elastic properties of the composite

The elastic properties for the composite are obtained from the unidirectional

properties for the different coordinate system of the composite material. The

elastic properties of the composite are homogenised for simplicity so as to

readily model the composite material in Abaqus. The homogenised elastic

properties of the composite material are used for the subsequent analyses. It

should be noted that the material coordinate system in Abaqus, the local

material coordinate system of the DCB (Figure 5.3) and the unidirectional

lamina are different (Figure 5.4). Hence, the elastic properties of the composite

are obtained by a cube (i.e. a 3D representation with solid elements) analysis of

the composite in Abaqus using the composite layup of the specimen to

determine the homogenised property of the composite material (Figure 5.5).

The layup of the composite cube has the same stacking sequence as that of the

composite specimen, so as to obtain the appropriate homogenised elastic

properties. The lamina properties of the composite material are analysed for the

different orientations using both classical laminate theory (CLT) and FEA (via

Abaqus) to obtain the homogenised elastic properties.

Control Rubber Nano Hybrid

E1 (GPa) 42.39 42.18 42.59 42.28

E2(GPa) 10.88 9.05 12.40 9.92

E3 (GPa) 10.88 9.05 12.40 9.92

ν12 0.24 0.24 0.24 0.24

ν13 0.24 0.24 0.24 0.24

ν23 0.28 0.28 0.28 0.28

G12 (GPa) 3.25 2.67 3.74 2.94

G13 (GPa) 3.25 2.67 3.74 2.94

G23 (GPa) 4.26 3.55 4.86 3.89

PropertiesComposite


93

Figure 5.3 Equivalent Abaqus and local coordinate system for a DCB

composite specimen.

Figure 5.4 Equivalent Abaqus and local coordinate system for different lamina of the composite strip.

x

y

z

P

T

P

Abaqus coordinate Local coordinate

DCB

x

y

z

x

y

z

x

y

z

T

L

T

T

T

L

T

P

P

Abaqus coordinate Local coordinate

0o fibre

90o fibre

±45o fibre


94

Figure 5.5 Cube with different faces (FF-Face front, FBk-face back, FR-face right, FL- face left, FT-face top, FB- face bottom). The control point (CP) boundary condition is at the origin of the coordinate system.

5.2.2.1 Elastic properties of DCB

The elastic properties of the DCB composite in the Abaqus coordinate system

are determined from the local coordinate system of the composite (Figure 5.3).

The homogenised elastic properties for the DCB analysis are derived from a 3D

solid cube (Figure 5.5) analysis in Abaqus, with different boundary conditions as

described in Table 5.3. The layup of the DCB composite in the cube analysis is

[(-45/45)s(90/0)s]2[(0/90)s(45/-45)s]2. The unidirectional properties of lamina of

Table 5.2 are used to model the composite cube. The homogenised elastic

properties obtained from the cube analysis are given in Table 5.4.

y

FT

FR

FBk

FL

FF

FB

z


95

Table 5.3 Derivation of homogenised elastic property of DCB from cube analysis

Table 5.4 Elastic properties of the arms the DCB composite specimens for the

various epoxy matrices

Properties Loading Boundary condition Calcualtion

Ex

Abaqus analysis of the cube with

load in the x direcrtionFL(x=0), FR(x=1% strain)

force in the x direction/area of

cube

Ey


load in the y directionFB(y=0), FT(y=1% strain)

force in the y direction/area of

cube

Ez

Same as the undirectional fibre

property E3

- -

νxy

Abaquq analysis of the cube with


strain in y direction/strain in x

direction

νxz



strain in z direction/strain in x

direction

νyz


load in the y direcrtionFB(y=0), FT(y=1% strain)

strain in z direction/strain in y

direction

Gxy


shear force in the x plane in the y

direction

FR(y=1% strain), FL(y=-1% strain),

FT(x=1% strain), FB(x=-1% strain)

shear force in the y direction

in the x plane/

(shear strain x area of cube)

Gxz


shear force in the x plane in the z

direction

FR(z=1% strain), FL(z=-1% strain),

FF(x=1% strain), FBk(x=-1% strain)

shear force in the z direction

in the x plane/


Gyz


shear force in the y plane in the z

direction

FF(y=1% strain), FBk(y=-1% strain),

FT(z=1% strain), FB(z=-1% strain)

shear force in the z direction

in the y plane/


Abaqus properties Local properties Control Rubber Nano Hybrid

Ex EP 18.41 17.37 19.30 17.86

Ey ET 10.88 9.05 12.40 9.92

Ez EP 18.41 17.37 19.30 17.86

νxy νPT 0.20 0.20 0.29 0.22

νxz νPP 0.30 0.31 0.39 0.33

νyz νTP 0.12 0.10 0.11 0.10

Gxy GPT 3.99 3.30 4.57 3.63

Gxz GPP 8.22 9.81 10.68 10.04

Gyz GTP 4.09 3.40 4.66 3.73


96

5.2.2.2 Elastic properties of composite strip

The elastic properties of each lamina of the strips are derived to model the

transverse cracks within the lamina. The elastic properties of the lamina in the

Abaqus coordinate system are determined from the local coordinate system of

the unidirectional lamina (Figure 5.4). The equivalent properties of the lamina

for different orientations (in Abaqus) are derived by considering a coordinate

system of the strip in Abaqus.

The elastic properties of the 0o and 90o degree layers in the composite material

strip are obtained by considering the fibre direction as the longitudinal direction,

L, and the other direction as the transverse direction, T, in the local coordinate

system (Figure 5.4). The elastic properties of the ±45o layer in the strip are

obtained by considering the plane of the lamina as the plane direction, P, and

the normal direction perpendicular to the plane as the transverse direction, T, in

the local coordinate system (Figure 5.4). Table 5.5 shows the equivalent local

elastic properties of the lamina of the strip in the Abaqus coordinate system for

different orientations. The different lamina properties of each lamina are

obtained from the cube analysis in Abaqus. The loads and boundary conditions

are applied on the cube (Figure 5.5), on the different faces, to derive the elastic

properties. The derivations of elastic properties of the lamina are given in Table

5.7 to Table 5.8, and the elastic properties obtained are given in Tables 5.9 to

5.11.

The calculated elastic modulus values were validated by calculating the

modulus of the composite strip layup based on the unmodified (i.e. control)

epoxy matrix. The agreement was very good between the calculated and

measured values, being with in ±3% (Manjunatha et al. [40]).


97

Table 5.5 Equivalent local elastic properties of the lamina of the strip in the Abaqus coordinate system

Table 5.6 Derivation of the elastic properties of 0o lamina

0 90 ±45

Ex ET ET ET

Ey EL ET EP

Ez ET EL EP

νxy νTL νTT νTP

νxz νTT νTL νTP

νyz νLT νTL νPP

Gxy GTL GTT GTP

Gxz GTT GTL GTP

Gyz GLT GTL GPP

PropertiesPly orientation (

o)

Global material

properties

Local material

propertiesDescription Loading condition

Boundary

Condition

Ex ET Unidirectional property, E2 - -

Ey EL Unidirectional property, E1 - -

Ez ET Unidirectional property, E1 - -

νxy νTL

Derived from unidirectional

property (=ν12(E2/E1))- -

νxz νTT

Abaqus analysis of the cube

with load in the y direcrtion FT(y=1% strain) FB(y=0)

νyz νLT Unidirectional property, ν12 - -

Gxy GTL


with shear force in the y

plane in the x direction

FT(x=1% strain), FB(x=-1% strain),

FR(y=1% strain), FL(y=-1% strain) CP(z=0)

Gxz GTT



plane in the z direction

FT(z=1% strain), FB(z=-1% strain),

FF(y=1% strain), FBk(y=-1% strain) CP(x=0)

Gyz GLT Unidirectional property, G12 - -


98

Table 5.7 Derivation of the elastic properties of 90o lamina

Table 5.8 Derivation of the elastic properties of ±45o lamina

Global material

properties

Local material


Boundary

Condition


Ey ET Unidirectional property, E2 - -

Ez EL Unidirectional property, E1 - -

νxy νTT


with load in the y direcrtion FT(y=1% strain) FB(y=0)

νxz νTL


property (=ν12 (E2/E1))- -

νyz νTL


property (=ν12(E2/E1))- -

Gxy GTT



plane in the z direction

FT(z=1% strain), FB(z=-1% strain),

FF(y=1% strain), FBk(y=-1% strain) CP(x=0)

Gxz GTL






Gyz GTL






Global material

properties

Local material


Boundary

Condition


Ey EP

Laminator analysis of (45)8

layup sequence- -

Ez EP

Laminator analysis of (45)8

layup sequence- -

νxy νTP


with load in the z direcrtion FF(z=1% strain) FBk(z=0)

νxz νTP


with load in the z direcrtion FF(z=1% strain) FBk(z=0)

νyz νPP


with load in the x direcrtion FR(x=1% strain) FL(x=0)

Gxy GTP


with shear force in the z


FF(x=1% strain), FBk(x=-1% strain),

FR(z=1% strain), FL(z=-1% strain) CP(y=0)

Gxz GTP


with shear force in the z


FF(x=1% strain), FBk(x=-1% strain),

FR(z=1% strain), FL(z=-1% strain) CP(y=0)

Gyz GPP


with shear force in the x

plane in the y direction

FR(y=1% strain), FL(y=-1% strain),

FT(x=1% strain), FB(x=-1% strain) CP(z=0)


99

Table 5.9 Elastic properties of 0o lamina

Table 5.10 Elastic properties of ±45o lamina

Table 5.11 Elastic properties of 90o lamina


100

5.2.3 Elastic properties of aluminium and steel

The elastic properties used to model the aluminium-alloy blocks and steel pins

in the experiments are given in Table 5.12 (Callister [69]).

Table 5.12 Elastic properties of aluminium-alloy and steel

5.3 Criteria for Cohesive Zone Modelling

A cohesive zone analysis is dependent on many factors for the accurate

analysis of the failure and the important factors are discussed below.

5.3.1 Mesh sensitivity analysis

The results of the cohesive zone element analysis maybe mesh dependent, and

an accurate and reproducible analysis needs a refined mesh (Turon et al. [70]),

as such a mesh gives an accurate representation of the stress field around the

crack tip. Also the convergence of the cohesive zone element analysis depends

on the mesh size, and a more refined mesh is required to obtain convergence.

The convergence test is conducted by studying the convergence using different

mesh sizes for the specimens, and the mesh which shows no deviation of the

results with further refinement is then used for the subsequent modelling

studies. The mesh sensitivity analysis is therefore done to understand the size

of the mesh required for the analyses. In the present study, meshes with

different degrees of refinement are used for the analysis of the DCB specimen

and the predicted load-displacement curves are compared to the experimental

curves (Figure 5.6). From the results shown in Figure 5.6, in all the present

analyses a 0.1mm size mesh has been adopted for the cohesive zone

elements.

Properties

Elastic modulus (GPa)

Poisson's ratio

Aluminium Steel

69 207

0.33 0.30


101

Figure 5.6 The modelled load versus displacement curves of the DCB composite specimen for different mesh sizes. The elastic properties of the unmodified (i.e. control) bulk epoxy matrix and the composite are used for the analysis (Table 5.13 and Table 5.4).

5.3.2 Initial value of the cohesive zone law parameters

Different methods are used to find the approximate value of the penalty stiffness

of the cohesive zone. The initial value of the cohesive zone penalty stiffness is

obtained from an expression which equates the transverse stress in the

cohesive zone and in the adjacent material (Figure 5.7). The stress in the

cohesive zone and in the adjacent material is made equal to get the

approximate value of the cohesive zone penalty stiffness (Turon et al. [70]).

Thus

(5.13)

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25

2 mm

1.25 mm

1 mm

0.5 mm

0.25 mm

0.2 mm

0.1 mm

Displacement (mm)

Forc

e (

N)


102

where is the through-thickness elastic modulus of the material and )

is the strain in the adjacent material. The cohesive zone penalty stiffness

is the approximate penalty stiffness of the cohesive zone and is the

relative displacement between the surfaces.

Figure 5.7 Cohesive zone behaviour in (a) undeformed and (b) deformed state.

The total strain in the whole model is given by

(5.14)

Hence, from the above equation, the resultant stress is the same and is given

by the expression

(5.15)

The resultant modulus of the composite can be written as

tct

tct+εtrtct

Δcoh

tct

Continuum

elements

(a) (b)

Continuum

elements

Continuum

elements

Continuum

elements

tct+εtrtct


103

(5.16)

It should be noted that the resultant properties of will be not be affected if

is much greater than . Indeed, the value is chosen in such a

way that the overall stiffness in the through thickness direction is not

significantly affected. From Equation 5.16, the expression for the cohesive zone

penalty stiffness is given by

(5.17)

where is a proportionality factor. The size of the cohesive zone should

provide a reasonable penalty stiffness but be small enough to avoid numerical

problems due to oscillations of the stresses at the crack tip. The value of the

penalty stiffness should also be fixed in such a way that it is small as possible to

avoid ill-conditioning of the stiffness matrix. For relatively high values of , the

loss of stiffness is considerably less. The expression for takes into

consideration the elastic properties of the adjacent material and hence it is

relatively accurate.

5.3.3 The cohesive zone length

The length of the cohesive zone is defined as the distance from the crack tip to

the point where the critical cohesive zone stress is attained (Figure 5.8).

Figure 5.8 Cohesive zone elements with cohesive zone length, .

Continuum elements

Continuum elements

lcz

Cohesive

elements


104

A necessary condition for obtaining a good solution using a cohesive analysis is

the size of the element (Harper and Hallett [62]). The cohesive zone length of

the element should be less than the cohesive zone at the crack tip. The

expression to derive the cohesive zone length, , is

(5.18)

where is the elastic modulus of the material, is the critical energy release

rate, is the critical stress in the cohesive zone and the parameter depends

on the cohesive zone model. The number of elements required for a given

mesh length is

(5.19)

where is the size of the mesh. The accuracy of the results increases when

value is the least. The minimum number of elements required for predicting the

initiation and propagation is two as the crack tip stress variation is high due to

the initiation of the crack. Hence, more than two elements were also used in the

present research.

5.4 Quasi-Static Models

The epoxy matrix SENB and the composite material DCB specimens were

tested experimentally and then modelled in FEA Abaqus, using 2D plane strain

elements, to obtain the cohesive zone law parameters. Structured meshing is

adopted for the analysis and the size and type of the mesh employed were

based upon the meshes defined from the mesh convergence study, see Section

5.3.1 above.


105

5.4.1 The SENB test: Experimental and theoretical results

A three point bending test is conducted using the SENB specimen. The load

versus displacement at the middle point of the application of the load is

obtained from the test. The test is conducted for the different bulk epoxy

matrices and the value of the fracture energy, , of the bulk epoxy is directly

obtained. The mean load versus displacement curves obtained from the test are

shown in Figures 5.14 to 5.17. The mean fracture energies of the different

epoxies obtained from the test are given in the Table 5.14.

The SENB test is conducted to study the quasi-static behaviour of the bulk

epoxy matrices. The cohesive zone properties of the bulk epoxy which are then

derived are used to model the fatigue behaviour of the composite material strip.

The transverse cracks in the strip develop due to matrix cracking, and hence

the cohesive zone properties obtained from the SENB test are very appropriate

to employ to model the transverse cracks in the composite strip. The values of

the fracture energy, , obtained directly from the experiments are also used for

modelling studies.

Figure 5.9 SENB specimen.

Steel roller support

Steel loading roller


106

Figure 5.10 Dimensions of the SENB bulk epoxy matrix specimen.

Figure 5.11 Loading and boundary condition applied on the SENB specimen the model.

5.4.1.1 SENB results: VCCT analysis

A model of the SENB specimen is developed using a FEA approach in Abaqus.

The dimensions of the specimen are given in Figure 5.10. The specimen is

modelled as two parts and a VCCT criterion is applied on the surface where

crack growth occurs. The three point supports in the experiment are modelled

as semi circles and a ‘hard contact’ criterion is used in Abaqus between the

surfaces to avoid any interpenetration. The friction between the surfaces is

assumed to be zero. The elastic properties of the bulk epoxy (Table 5.1) and

steel rollers (Table 5.12) are used for the modelling studies. The load is applied

as a displacement on the top middle roller and the load versus displacement

curve for the specimen is obtained from the reaction forces and displacement at

the top support. The boundary conditions applied on the model are shown in

60 mm

30 mm

6 mm

12 mm

6 mm

1 mm

x

y

z

6 mm


107

Figure 5.11. The analysis is conducted for the different bulk epoxy matrices and

the load versus displacement curve from the analysis is compared with the

experiments, see Figures 5.14 to 5.17.

5.4.1.2 SENB results: Cohesive contact analysis

The SENB specimen of the bulk epoxy is modelled using FEA with cohesive

contact, via Abaqus. The dimensions of the specimen are given in Figure 5.10.

The SENB specimen is modelled in two parts and the ‘cohesive behaviour’

contact option in Abaqus is adopted between the surfaces. The elastic

properties of the bulk epoxy (Table 5.1) and steel rollers (Table 5.12) are used

for modelling the specimen. The three point bending supports are modelled as

semi circles in Abaqus and a ‘hard contact’ criterion in Abaqus is adopted

between the surfaces to avoid any interpenetration. The friction between the

surfaces is assumed to be zero. The load in the model is applied as a

displacement on the middle roller (Figure 5.11) and the load versus

displacement curve for the specimen is obtained from the reaction forces at the

support. The cohesive contact parameters of the bulk epoxy are obtained using

the same procedure as in the cohesive contact analysis of the DCB model, see

later in Section 5.4.2.2. The stress field in a cohesive contact analysis in

Abaqus is shown with the stress contours in Figure 5.12. The load versus

displacement curves of the modelling for different materials are shown in

Figures 5.14 to 5.17.


108

Figure 5.12 The stress field around the crack tip in a SENB FEA model of the

bulk epoxy matrix with cohesive contact in Abaqus.

5.4.1.3 SENB results: Cohesive zone element analysis

The SENB specimens of the bulk epoxy matrices are modelled in Abaqus with

the dimensions of the specimen as given in Figure 5.10. The SENB specimen is

modelled as one part, and the part is partitioned as consisting of continuum

elements and cohesive zone elements. The elastic properties of the bulk epoxy

matrix (Table 5.1) and steel rollers (Table 5.12) are used for modelling the

specimen. The thickness of the cohesive zone element was adopted as 0.001

mm, as brittle fracture is observed in the experiments.

The three point bending supports are modelled as semi circles in the Abaqus

programme and ‘hard contact’ criterion in Abaqus is adopted between the

surfaces to avoid any interpenetration. The friction between the surfaces is

assumed to be zero. The load is applied as a displacement on the middle


109

support and the load versus displacement curve for the specimen is obtained

from the reaction force and displacement at the support. The parameters of the

cohesive contact are obtained by the procedure described in the DCB cohesive

contact analysis, see later in Section 5.4.2.2. The analysis is run for the

different bulk epoxy matrices using Abaqus, to obtain the cohesive contact

parameters needed to accurately model the SENB test.

The elastic properties used for modelling the SENB specimen are given in

Table 5.1. The values of the elastic properties and fracture energies, , of the

different bulk epoxies are obtained from the experiments (Table 5.14). The

values of the penalty stiffness and the critical stress for the different epoxies are

obtained from matching the experimental results with the modelling results, and

are given in Table 5.14. The load versus displacement curves obtained from the

modelling are compared with the experiments, and are given in Figures 5.14 to

5.17 for the different materials. The fracture energies of the bulk epoxies are

obtained directly from the experiments (Section 3.5.2). The stress contours of a

SENB cohesive zone element analysis in Abaqus is shown in Figure 5.13.


110

Figure 5.13 The stress field around the crack tip in a SENB model with cohesive zone elements as modelled in FEA Abaqus.

Table 5.14 Quasi-static cohesive contact/element parameters for the bulk epoxy matrices


111

Figure 5.14 Comparison of load-displacement curve for the SENB specimen based upon the unmodified (i.e. control) epoxy matrix.


based upon the micro-rubber modified epoxy matrix.

0

5

10

15

20

25

30

35

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Experiment

VCCT

Cohesive contact

Cohesive element

Displacement (mm)

Load (

N)

0

10

20

30

40

50

60

70

80

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Experiment

VCCT

Cohesive contact

Cohesive element

Displacement (mm)

Load (

N)


112


based upon the nano-silica modified epoxy matrix.

Figure 5.17 Comparison of load-displacement curve for the SENB epoxy

specimen based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix.

0

5

10

15

20

25

30

35

40

45

50

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Experimental

VCCT

Cohesive contact

Cohesive element

Displacement (mm)

Load (

N)

0

10

20

30

40

50

60

70

80

90

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Experiment

VCCT

Cohesive contact

Cohesive element

Displacement (mm)

Forc

e (

N)


113

5.4.2 The DCB test: Experimental and theoretical analysis

A flowchart for the DCB testing and modelling of the composite material

specimens is given in Figure 5.18, which schematically illustrates an overview

of the experimental and modelling work.

The DCB specimen is tested under quasi-static conditions to obtain the load

versus displacement curve of the composite material. The fracture energy of the

composite material is also obtained from the experiments, as described in

Section 3.6.2. The mean load versus displacement traces of the DCB

specimens for the various composite materials, based on the different epoxy

matrices, are shown in Figures 5.24 to 5.27.

A FEA 2D model of the DCB (Figure 5.19) specimen is modelled in Abaqus to

validate the models and subroutine which will be used to predict the fatigue life

of the composite material strips, see Section 5.6.2.3 below. The laminate

sequence of the composite DCB test specimen is [(-45/45)s(90/0)s]2 [(0/90)s(45/-

45)s]2. The initial crack in the DCB specimen is located between the 0o and 90o

lamina interface at the mid-plane of the laminate sequence. The dimensions of

the DCB specimen are shown in Figure 5.20. The size of the aluminium alloy

end-blocks used is 20x12x20mm3 and the initial length of the crack is 60 mm.

The elastic properties in Table 5.4 and Table 5.12 are used in the modelling

studies. The load versus displacement curves obtained from the analyses are

compared with the experiments. The model use VCCT, cohesive contact and

cohesive element methods to obtain the parameters of the lamina interface for

the cohesive zone law.


114

Figure 5.18 Flow chart of quasi-static and fatigue analyses of the composite material DCB specimens

Prediction of the fatigue life of DCB specimen under a displacement controlled

fatigue test

Hence validation of proposed modelling method

Fatigue parameters , and of the lamina interface

from DCB fatigue tests Subroutine uses the Paris

law to degrade the cohesive law, see Figure 5.1

of the lamina interface of interest

from the DCB

quasi-static test

Fatigue modelling of the DCB specimen to match the growth rate curve using the

subroutine

Quasi-static modelling of the DCB specimen

using the VCCT method

Quasi-static modelling of the DCB specimen using

cohesive contact and

cohesive element analyses

Obtain and for the cohesive

zone law


115

Figure 5.19 DCB specimen

Figure 5.20 Dimensions of the DCB composite material specimen

5.4.2.1 DCB results: VCCT analysis

The fracture energy of composite material, which fractured at the 0o/90o lamina

interface, obtained from the experiment is used to model the DCB using the

VCCT technique. The two arms of the DCB are modelled as separate parts and

the surfaces are connected using the VCCT criterion. The elastic properties of

the composite and aluminium blocks are taken from Table 5.4 and Table 5.12,

respectively. The loads are applied as a displacement in the opposite direction

at the centre of the two aluminium blocks, as in Figure 5.21. The boundary

conditions are applied at the top and bottom arms (Figure 5.21) at the centre of

the block, as the hole in the aluminium block is assumed to be filled. The result

Aluminium block

Composite

Initial crack

5.4 mm

Fracture plane

20 mm

60 mm

12 mm6 mm

120 mm

x

y

z

Initial crack


116

obtained from the analysis is half the displacement in the y direction of the top

arm. The modelling results are compared with the experimental results to

validate the stiffness and failure displacement of the DCB experiment. The load

versus displacement curves of the modelling for the different composite

materials based upon the different epoxy matrices are shown in Figures 5.24 to

5.27.

Figure 5.21 Loading and boundary condition applied on the model

5.4.2.2 DCB results: Cohesive contact analysis

The analysis of the quasi-static test of the DCB model is undertaken to obtain

the cohesive zone law parameters of the lamina interface of interest in the

composite material. The model used for the analysis is the same as that of the

VCCT analysis but with the surface contact option, as embedded in the Abaqus

software. The elastic properties of the composite and aluminium blocks in the

model are taken from Table 5.4 and Table 5.12, respectively. The loads are

applied as a displacement at the centre of aluminium blocks and the arms are

given an equal displacement in the opposite directions, as shown in Figure

5.21. The boundary conditions are applied at the top and bottom arms at the

centre of the block (Figure 5.21).

The cohesive zone law parameters of the lamina interface are obtained by a

trial and error method by fitting to the experimental load versus displacement

curve to determine these parameters. Initially an approximate value of the

cohesive zone penalty stiffness is assumed in the analysis, with a critical stress

equal to the yield stress of the bulk epoxy matrix. The load versus displacement

x

y

z


117

curve predicted by the analysis is then compared with the experiments to match

the slope of the experimental curve. In particular, the penalty stiffness of the

cohesive contact theoretical analysis is varied to match the slope of the

experimental curve. Once good fit of the penalty stiffness of the cohesive

contact behaviour is obtained, the critical stress of the cohesive contact analysis

is next varied to match the failure displacement in the experiment. The cohesive

contact parameters are then obtained for the given DCB specimen. The stress

contours obtained at the crack tip of the model are shown in Figure 5.22. The

cohesive zone law parameters obtained from the analysis are given in Table

5.15. The load versus displacement curves of the modelling for different

materials are shown in Figures 5.24 to 5.27.


contact in Abaqus

5.4.2.3 DCB results: Cohesive zone element analysis

The quasi-static analysis of the DCB is undertaken in the FEA Abaqus software

to obtain the cohesive zone law parameters of the composite materials. The


118

cohesive zone parameters of the lamina interface of interest are obtained by a

trial and error method, as described above for the cohesive contact analysis

approach. The elastic properties of the composite and aluminium blocks are

taken from Table 5.4 and Table 5.12, respectively. The loads are applied as

displacement at the centre of the aluminium blocks, and the arms are given an

equal displacement in the opposite directions as in Figure 5.21. The boundary

conditions are applied at the top and bottom arms at the centre of the block

(Figure 5.21).

The model for the DCB is developed with the same arm thickness as that in the

experiments. The thickness of the matrix in-between the lamina is of order 10-2

mm (Masania [47]) and hence the thickness of the cohesive layer is adopted as

0.01 mm. The penalty stiffness and critical stress for the cohesive zone element

are obtained, described above for the cohesive contact analysis approach. The

DCB model is developed as a single part and the part is partitioned as cohesive

zone elements and continuum elements. The load is applied in the model as a

displacement and the boundary condition adopted is the same as that of the

cohesive contact analysis. The boundary condition for the DCB model is applied

at the arms and the load is applied as a displacement in the opposite directions.

The penalty stiffness and critical stress of the lamina interface obtained from the

analysis are given in Table 5.15. The load versus displacement curves from the

modelling for the different composite materials based upon the different epoxy

matrices are shown in Figures 5.24 to 5.27. In these figures the VCCT method

gives a relatively poor fit to the experimental results. This is suggested to arise

from the fibre bridging which is seen to occur behind the crack tip during the

testing of the DCB composite specimens. Unlike the cohesive contact and

cohesive element methods, the VCCT method does not take such fibre bridging

into account. The contours of the stress variation in a DCB cohesive zone

element analysis is shown in Figure 5.23.


119


zone elements in Abaqus

Table 5.15 Quasi-static cohesive contact/element parameters of the DCB

lamina interface


120


material based upon the unmodified (i.e. control) epoxy matrix


based upon the micro-rubber modified epoxy matrix

0

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10 12 14 16 18 20

Experimental

VCCT

Cohesive contact

Cohesive element

Displacement (mm)

Lo

ad

(N

)

0

20

40

60

80

100

120

0 5 10 15 20 25

Experiment

VCCT

Cohesive contact

Cohesive element

Displacement (mm)

Load (

N)


121


based upon the nano-silica modified epoxy matrix


based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25

Experiment

VCCT

Cohesive contact

Cohesive element

Displacement (mm)

Load (

N)

0

20

40

60

80

100

120

0 5 10 15 20 25

Experiment

VCCT

Cohesive contact

Cohesive element

Displacement (mm)

Load (

N)


122

5.4.3 The composite material strip test: Experimental and theoretical

results

The flowchart for the composite strip testing and analysis of the specimen are

given in Figure 5.28, which gives an overview of the experimental and modelling

work.


123

Figure 5.28 Flow chart of the life prediction modelling of the composite strip under cyclic fatigue loading

Fatigue modelling of the composite with the

saturation/experimental crack density using the subroutine (see Figure

4.7)

Determine the maximum crack density for maximum stiffness reduction in the fatigue

test of the strip

OR

The saturation crack density; obtained by predicting the crack

density required for the maximum stiffness reduction using trial and error method of

static modelling. (Can be validated from the

experiments.)

Prediction of the fatigue life of composite for different applied stresses. (Life of the composite is assumed to be equivalent to

the number of cycles required for the maximum stiffness reduction for crack

density adopted earlier.)

Subroutine (see Figure 4.7) validated with

growth curve matching with the CT fatigue test

Fatigue parameters , and of the

epoxy from the CT bulk epoxy matrix

fatigue test Subroutine (see Figure 4.7) uses the Paris law

to degrade the cohesive zone law

Fatigue modelling of the CT bulk epoxy matrix specimen to match the growth rate curve

using the subroutine (see Figure 4.7)

Obtain of the cohesive zone law

of the epoxy matrix

from the SENB bulk

epoxy matrix quasi-static

test

Quasi-static modelling of the SENB bulk epoxy matrix

specimen using the VCCT method

Quasi-static modelling of the SENB bulk epoxy matrix

specimen using the cohesive contact and cohesive element

analysis

Obtain of the cohesive zone law

of the epoxy matrix


124

The quasi-static test of the composite strip is undertaken to determine the

quasi-static strength of the composite material strip. The strip is loaded quasi-

statically and the load versus displacement graph is obtained. The ultimate

tensile strength of the composite strip may be deduced, and the change in

stiffness with loading is also obtained from the test. The results of the test are

shown in Table 5.16. The research on the composite strip has been undertaken

together with Dr. Manjunatha [40].

Table 5.16 Elastic properties of the composite material strip

The quasi-static analysis of the strip is performed to study the failure behaviour

of the GFRP composite. The composite strips are modelled to study the

influence of transverse cracks in the stiffness reduction of the composite. The

dimensions of the strip are as shown in Figure 5.29. The symmetric model of

the strip is analysed using the symmetric boundary condition for the length of

strip (Figure 5.30).

Figure 5.29 Composite material strip with dimensions

The strip is modelled with cracks having a crack density as experimentally

measured in the experiments (Figure 5.31). The transverse crack density of the

strip is obtained from the fatigue experiments for different cycles of loading

(Figure 5.33). The elastic properties in Tables 5.9, 5.10 and 5.11 are used to

model the different laminae of the strip. The normalised stiffness of the

composite material strip as a function of the crack density is also obtained from

the experiments (Figures 5.35 to 5.38).

50 mm

End tabs

150 mm

Composite

2.7 mm


125

Figure 5.30 Section of the strip with transverse cracks (a) strip under loading

(b) cross section of the strip with transverse cracks (c) symmetric

cross-section of the strip with transverse cracks

x

y

z

(b)

s

s

(a)

(c)

s s

Load

Load


126

Figure 5.31 Dimension of a section of the modelled strip. The length, l, of the strip model depends on the crack density

The transverse cracking within the laminae is modelled using a relation between

the cohesive traction and the relative displacements. The effect of transverse

crack density on the stiffness degradation is studied and the model simulates

the critical stress above which the crack starts propagating, and also gives the

strain energy release rate in the system. The damage model is able to simulate

crack onset and propagation.

The transverse cracks in the strip are modelled as continuum elements (with

very low elasticity modulus, i.e. =1x10-9N/mm2 and a Poisson’s ratio, ν=0.01)

of thickness 0.001mm (i.e. the thickness of the cohesive zone element in the

SENB and CT models). The load in the strip is applied as a displacement at the

top of the model and the boundary conditions adopted for the model are shown

in Figure 5.32.

1.35 mm

l mm


127

Figure 5.32 Boundary conditions applied on the symmetric cross-section of the

composite material strip with transverse cracks

The variation of the crack density of a laminae consisting ±45o fibres for

different fatigue cycles observed in the experiments is shown in Figure 5.33.

The 90o crack density is assumed to be constant at a value of 0.64/mm (Tong et

al. [37]) for all the cycles of fatigue. As may be seen from the results shown in

Figure 5.33, the composite strips based upon the control matrix exhibit the

highest crack densities whilst those based upon the hybrid matrix exhibit the

lowest values of crack density. From the previous results in Chapter 5, this is as

would be expected.


128

Figure 5.33 Variation of ±45o crack density with number of cycles in composite material strips based upon unmodified (i.e. control) and modified epoxy matrices

5.4.3.1 Normalised stiffness with crack density

In the quasi-static analysis of the strip, the normalised stiffness of the composite

is compared with the experimental values of crack density obtained for different

cycles of fatigue (Figure 5.33). The normalised stiffness reduction is also

obtained for the corresponding crack density for different cycles of fatigue

cycles in the experiment (Figures 5.35 to 5.38).

Different models with different crack densities are then run and the normalised

stiffness of the strip is obtained. The models are run with continuous increments

of crack density to obtain a smooth reduction of the stiffness with crack density

in the composite (Figures 5.35 to 5.38). The normalised stiffness as a function

of crack density obtained from the model is compared with the normalised

stiffness with an increase in crack density (for different number of cycles in the

0

0.5

1

1.5

2

2.5

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Number of cycles

Control

Nano

Rubber

Hybrid

Cra

ck d

ensity (

mm

-1)


129

fatigue experiment.) The comparison of normalised stiffness with crack density

from the modelling and experimental studies is shown in Figures 5.35 to 5.38.

Comparisons of the normalised stiffness of the composite with crack density

from the modelling and experimental studies are also shown in Figure 5.39 to

5.40, where further comparison are made between the different composite

materials based on the different epoxy matrices. The strip analysis in Abaqus

with transverse cracks is shown in Figure 5.34.

Figure 5.34 Composite material strip with transverse cracks in the Abaqus FEA method

The agreement between the modelling results and the experimental results is

very good for all the different composite materials, with the poorest agreement

being seen for the composite materials based upon the hybrid epoxy matrix at

the relatively very high crack densities.


130

Figure 5.35 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the unmodified (i.e. control) epoxy matrix.


the composite strip test. For the composite based upon the micro-rubber modified epoxy matrix.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00 2.50

Experiment

Modelling

Crack density (mm-1)

Norm

alis

ed

stiff

ness

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Experiment

Modelling


Norm

alis

ed s

tiff

ness


131

Figure 5.37 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the nano-silica modified epoxy matrix.

Figure 5.38 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix.

0

0.2

0.4

0.6

0.8

1

1.2

0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800

Experiment

Modelling


Norm

alis

ed s

tiff

ness

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Experiment

Modelling


Norm

alis

ed

stiff

ness


132


the composite strip, based upon the unmodified (i.e. control) and nano-silica modified epoxy matrices.


the composite strip, based upon the micro-rubber and with both nano-silica and micro-rubber modified epoxy matrices

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.5 1 1.5 2 2.5 3


Control Modelling

Control Experiment

Nano Modelling

Nano Experiment

Norm

alis

ed s

tiff

ness

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.5 1 1.5 2 2.5


Rubber Modelling

Rubber Experiment

Hybrid Modelling

Hybrid Experiment

Norm

alis

ed s

tiff

ness


133

5.4.3.2 Normalised stiffness with number of fatigue cycles

The normalised stiffness with the number of cycles may also be plotted for the

different crack densities observed in the experiments. The analysis is performed

with a crack density for a given number of cycles, and the normalised stiffness

is then obtained for the strip. The normalised stiffness with the number of cycles

of fatigue is obtained from the experiments, which are modelled using the same

crack density for different cycles of loading (Figure 5.33) to obtain the

normalised stiffness. The 90o crack density is again assumed to be constant at

a value of 0.64/mm (Tong et al. [37]). The model is analysed with the crack

density as measured in the experiments and the normalised stiffness is plotted

versus the number of cycles.

The normalised stiffness as a function of the number of cycles, (with crack

density also being measured) is obtained from the experiments. Different

models with the different crack densities (for different cycles of fatigue loading)

obtained are run and the normalised stiffness is derived from the analysis. The

values of normalised stiffness as a function of the number of cycles of loading

for the different composite materials are shown in Figures 5.41 to 5.44.

Again the agreement between the results from the modelling studies and the

experimental results is very good. Although, the composite strip specimens

based upon the hybrid epoxy matrix do show a somewhat larger discrepancy

between the experimental results and the modelling studies than for the

composite strips based upon the other matrices. This may arise due to the

presence of the fibre changing somewhat the morphology, and hence the

mechanical properties, of the hybrid epoxy matrix in the composite strip

specimens compared to the bulk hybrid epoxy polymer. However, even for

Figure 5.44, the agreement between the experimental and theoretical modelling

studies is still relatively good. This is very encouraging for the work on

modelling the fatigue life of the composite strips, which is discussed later.


134

Figure 5.41 Normalised stiffness versus the number of cycles for the composite strip based upon the unmodified (i.e. control) epoxy matrix

Figure 5.42 Normalised stiffness versus the number of cycles for the composite strip based upon the micro-rubber modified epoxy matrix

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2000 4000 6000 8000 10000 12000 14000 16000

Experiment

Modelling

Number of cycles

Norm

alis

ed

stiff

ness

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Experiment

Modelling


Norm

alis

ed s

tiff

ness


135

Figure 5.43 Normalised stiffness versus the number of cycles for the composite strip based upon the nano-silica modified epoxy matrix

Figure 5.44 Normalised stiffness versus the number of cycles for the composite strip based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10000 20000 30000 40000 50000

Experiment

Modelling

Number of cycles

Norm

alis

ed s

tiff

ness

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Experiment

Modelling

Number of cycles

Norm

alis

ed s

tiff

ness


136

5.4.3.3 Quasi-static strength of the composite strip

The quasi-static strength of the composite is obtained from modelling composite

with maximum crack density (Figure 5.33) observed in the experiments. The 90o

crack density is assumed to be constant at 0.64/mm (Tong et al. [37]). The

cracks are modelled as cohesive zone elements with the cohesive zone

parameters of the epoxy material (Table 5.14). A quasi-static model (Figure

5.31) is run and the global stiffness reduction with increase in displacement is

noted. The cracks in the strip are modelled using cohesive zone elements and

the reduction of stiffness with displacement is plotted with percentage strain

applied in the model. The damage in cohesive zone element due to quasi-static

loading of composite causes reduction of global stiffness of the model. The

global stiffness of model attains a plateau with increase in strain and the final

failure of the composite is uncertain as the model fail to predict the abrupt

failure of glass fibre in 0o lamina.

The Figures 5.45 and 5.46 shows the global stress with increase in strain and

the corresponding reduction of global stiffness in the composite model. If the

model was able to predict the failure strain in the global stiffness versus strain

curve (Figure 5.46), the corresponding failure global stress for the composite

can be obtained for the failure strain from the Figure 5.45.


137

Figure 5.45 Comparison of the global stress with percentage strain in the composite strip

Figure 5.46 Comparison of global stiffness reduction with the percentage strain of composite strip

0

100

200

300

400

500

600

700

800

0 1 2 3 4 5 6

Control

Rubber

Nano

Hybrid

% strain

Glo

bal s

tress (

N/m

m2)

3.0E+05

3.2E+05

3.4E+05

3.6E+05

3.8E+05

4.0E+05

4.2E+05

4.4E+05

0 1 2 3 4 5

Control

Rubber

Nano

Hybrid

% strain

Glo

bal s

tiff

ness

(N/m

m)


138

5.5 Toughening Mechanisms

The fracture energy, , of the unmodified (i.e. control) and modified bulk epoxy

matrices are found to be different, with the modified epoxies having significantly

higher fracture energies, see Table 5.14. Indeed, the hybrid bulk epoxy matrix

has the highest fracture energy of all the epoxy polymers. The increase in the

fracture energy in the epoxy, due to the addition of the nano-silica and micro-

rubber particles, occurs from to the energy-dissipating toughening mechanisms

developing in the structure of the epoxy. Now, epoxies are highly cross-linked

thermosetting polymers. Hence, they have a poor resistance to initiation and

growth of cracks. This is due to the fact that when the epoxy is polymerised it is

amorphous and highly-crosslinked in structure (Kinloch [71]) resulting in a

relatively high modulus and strength. This structure of the epoxy matrix also

leads to the development of the brittle nature of the epoxy. Hence, the addition

of micro-rubber particles dispersed in the bulk epoxy helps to increase the

fracture energy by the dissipation of energy. The microstructure, and hence the

mechanical properties of the micro-rubber modified epoxy, depends on the

dispersion of the rubber particles and on the adhesion of the particle to the

epoxy matrix. Kinloch et al. [72] and Kinloch et al. [73] observed that the plastic

deformation of the modified epoxy matrix causes energy dissipation, and hence

an increase in the fracture energy. The increase in the fracture energy is

attributed to the interaction of the stress field with the micro-rubber particles

around the crack tip. The micro-rubber particles have a lower shear modulus,

but a comparable bulk modulus, than the epoxy matrix. This leads to stress

concentrations and volume constraint in the matrix around the rubber particles.

Hence, the rubber particles in the epoxy matrix act as stress concentrators, as

well as transferring the load. This leads to cavitation of the micro-rubber

particles in the epoxy matrix due to the triaxial stress state ahead of the crack

tip. (The residual stress after the curing cycle of the epoxy also causes the

development of such voids in the rubber particle.) The void in the micro-rubber

particles enables the development of extensive local plastic void growth in the

epoxy at the crack tip, and reduces the triaxiality which leads to even more

plastic deformation occurring at the crack tip in the epoxy polymer. Hence, the


139

presence of rubber particles in the epoxy increases the local plastic deformation

and so significantly increases the value of .

The fracture energy of the modified epoxy with nano-silica particles is found to

be significantly higher than the umodified epoxy matrix. As for micro-rubber

particle modified epoxies, the toughening of the epoxy with nano-silica particles

is dependent on the values of glass transition temperature, molecular weight

between cross-links of the epoxy polymer and the adhesion at the interface of

nano-silica particles with the epoxy (Hsieh et al. [74] and Hsieh et al. [75]). The

toughening mechanisms are somewhat similar to that described above.

Namely, localised shear bands are initiated by the stress concentrations around

the nano-silica particles and the debonding of the nano-silica particles leads to

plastic void growth. These toughening mechanisms lead to the increase in the

fracture energy of the modified epoxy.

5.6 Fatigue Models

The fatigue analysis is undertaken in Abaqus using the subroutine, see Figure

4.7. The degradation of the cohesive zone element is based on the Paris law,

and the subroutine degrades the penalty stiffness of the cohesive zone element

with time, as described in Section 4.3.4.

5.6.1 User element subroutine

The fatigue analysis of the composite is performed using the user element

subroutine (Hibbitt [8]) and the subroutine is written in FORTRAN (see the

Appendix). The subroutine uses the theoretical formulations to calculate the

damage in the material due to fatigue cycling. The subroutine is firstly validated

against the quasi-static and fatigue behaviour using a unit cell element with

given material properties. The quasi-static analysis of the unit cell of the

cohesive zone element using the subroutine is matched with the standard

cohesive zone element in Abaqus. The testing of the user cohesive zone

element is done, and the results plotted to validate the element, as described

below.


140

5.6.1.1 Validation

The unit cell cohesive zone element analysis is done using a standard

continuum element and hence the user cohesive zone element is tested. The

cohesive zone element to be tested is placed between the continuum elements

as shown in Figure 5.47. The 2D cohesive zone element has 0.001mm

thickness (i.e. same as that of the SENB and CT quasi-static cohesive zone

element analyses used for the bulk epoxy matrix studies) and the breadth and

through-thickness of the cohesive zone element is taken as unity. The unit cell

has a displacement loading applied at the nodes and the load versus

displacement curve is compared with the standard Abaqus cohesive zone

element with the same geometry.

Figure 5.47 A single cohesive zone element for testing

The quasi-static test of the cohesive zone element is undertaken to validate the

user element (Figure 5.48). The element is tested in mode I and mode II using

the cohesive zone parameters of the SENB specimen based on the unmodified

(i.e. control) epoxy matrix. The values of the critical displacement, critical stress

and failure displacement are checked to validate the cohesive zone element.

The material properties adopted for the continuum element (Table 5.1) and

continuum

element

cohesive

elements

x

y

z

node


141

cohesive zone element (Table 5.14) are, of course, that of the unmodified (i.e.

control) bulk epoxy. As noted above, the subroutine cohesive zone element is

subjected to mode I, mode II, as well as mixed-mode loading, and is compared

with the standard Abaqus cohesive zone element analysis. The behaviour of the

subroutine cohesive zone element and standard cohesive zone element are

shown in Figures 5.49 to 5.51. The analyses show that the subroutine cohesive

zone element has the same behaviour as that of standard cohesive zone

element in Abaqus, and hence the new user cohesive zone element proposed

in the current research is validated.

Figure 5.48 Cohesive zone element testing in (a) mode I, (b) mode II and (c) mixed-mode

The critical displacement for cohesive zone element is given by

(5.20)

The failure displacement is given by the relation

(5.21)

The critical stress at each node of a unit length cohesive zone element is .


142

Figure 5.49 Cohesive zone element in mode I. The cohesive zone law parameters used for the element is for the unmodified (i.e.

control) bulk epoxy matrix ( =3900N/mm2, =10.9 N/mm2 and =75.8J/m2)

Figure 5.50 Cohesive zone element in mode II. The cohesive zone law parameters used for the element is for the unmodified (i.e.

control) bulk epoxy matrix ( =3900N/mm2, =10.9 N/mm2 and =75.8J/m2)

0

1

2

3

4

5

6

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Abaqus element

User element

Displacement (mm)

Load (

mm

)

0

1

2

3

4

5

6

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Abaqus element

User element

Displacement (mm)

Load (

N)


143

Figure 5.51 Cohesive zone element in mixed-mode ( =0.5). The cohesive

zone law parameters used for the element is for the unmodified (i.e. control) bulk epoxy matrix ( =3900N/mm2, =10.9 N/mm2 and =75.8J/m2 for both modes)

5.6.2 Fatigue analysis using the user element subroutine

The user element subroutine may now be employed with confidence for the

fatigue analysis of the composite material strips. The subroutine is used to

modify the penalty stiffness of the cohesive zone law element with number of

cycles of fatigue loading. The damage variable in the cohesive zone law is

calculated in the subroutine for each cycle of analysis and the penalty stiffness

of the cohesive zone element is varied according to the change in the damage

variable. The initial quasi-static penalty stiffness of the cohesive zone element is

varied according to the cohesive zone law and the new penalty stiffness is

calculated from the formulation. The equivalent penalty stiffness is calculated in

the subroutine to account for both static and fatigue damage. The static damage

in the cohesive zone element is given by the Equation 4.77 and the fatigue

damage rate is given by the Equation 4.87 and the resultant damage is applied

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Abaqus element

User element

Displacement (mm)

Load (

N)


144

on the cohesive zone by reducing the penalty stiffness of the cohesive zone

element in the stiffness matrix. The total damage and equivalent penalty

stiffness is calculated in each element, and hence the continuous degradation

of the cohesive zone law for the element takes place.

The strain in the cohesive zone element and the number of cycles are the two

parameters which determine the penalty stiffness of the cohesive zone element

after an analysis step. For each step of the analysis, the penalty stiffness of the

individual cohesive zone element is calculated based on the strain of the

element and also accounting for the fatigue damage from the fatigue cycles.

The subroutine is called up for each analysis step, and for each cohesive zone

element. The strain in mode I and II are requested in the subroutine to calculate

the damage in the element. Using the equation for the damage variable, the

penalty stiffness is updated for each element to get the stiffness matrix of the

element during each step of the analysis to therefore have a continuous

degradation of the cohesive zone element in the fatigue analysis.

5.6.2.1 The CT test and user element analysis

The fatigue test is conducted on the CT specimen of the bulk epoxy matrix. The

load is applied to the specimen as a sinusoidal constant-amplitude

displacement. The frequency of the periodic load and the displacement ratio

(ratio of minimum displacement to maximum displacement in fatigue cycle) is

kept constant. The threshold fracture energy, , of the bulk epoxy and the

growth rate curve of the epoxy are obtained for the various epoxy matrices from

the test data. The growth rate curves obtained from the test are shown in

Figures 5.56 to 5.59. The threshold fracture energy for the bulk epoxies are

obtained from the growth rate curve for the specimen.

The CT specimen (Figure 5.52) is analysed in the Abaqus programme to model

the fatigue behaviour. The dimensions of the specimen are as shown in Figure

5.53. The quasi-static cohesive zone element parameters (Table 5.14) of the

SENB test are used to model the CT specimen. The elastic properties used for

the specimen are same as that of the SENB specimen (Table 5.1 and Table

5.12). The thickness of the cohesive zone element is kept same as that of the


145

quasi-static SENB model. The fatigue growth in the CT specimen is modelled

using the subroutine and the growth rate curve of the specimen is compared

with the experimental results.

Figure 5.52 Compact tension specimen

Steel pin

Bulk epoxy


146

Figure 5.53 Dimension of the CT specimen

The load in the fatigue model is applied as a constant displacement, as

described in Section 4.3.5.2, and the load is kept constant throughout whole

cycle of fatigue analysis (Figure 5.54). The growth rate curve of the experiment

is matched with that from the modelling studies obtained from cyclic fatigue

studies.

23 mm

20 mm

50 mm

48 mm

10 mm

12 mm

Ф 8 mm

x

y

z


147

Figure 5.54 Boundary condition applied on the CT specimen

The fatigue parameters of the CT specimens of the bulk epoxy matrices are

obtained from the Paris law fit for the different materials and are given in Table

5.17. The fatigue parameters of the specimen are used in the subroutine

analysis to determine the growth rate curve of the specimen. The growth rate

curve obtained from the modelling and experimental studies for the different

epoxies are shown in Figures 5.56 to 5.59. It should be noted that there is some

scatter in the modelling data shown in Figures 5.56 to 5.59. This occurs

because the calculation of the crack growth rate is not exact between a period

of cycles. This is due to the inability of the numerical method to find the exact

growth rate between adjacent two analysis steps during modelling of the

fracture process. The stress contours at the crack tip in the CT model is shown

in Figure 5.55.


148

Figure 5.55 Stress field around the crack tip in a CT model with cohesive zone

elements in Abaqus

Table 5.17 Fatigue parameters of the bulk epoxy matrix CT specimens from Paris law fit

Matrix fatigue properties

c

m

Gth (J/m2)

13.29 16.48 7.77 5

20.5 49 49 100

Control Nano Rubber Hybrid

22.26 1.05 2.91 0.042


149

Figure 5.56 Growth rate curve for the CT specimen for the bulk unmodified (i.e.

control) epoxy matrix

Figure 5.57 Growth rate curve for the CT specimen for the bulk micro-rubber

modified epoxy matrix

-7

-6

-5

-4

-3

-2

-1

-0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2

Experiment

Modelling

Paris law fit

log (Gmax/Gc)

log (

da/d

N)

-8

-7

-6

-5

-4

-3

-2

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4

Experiment

Modelling

Paris law fit

log (Gmax/Gc)

log (

da/d

N)


150

Figure 5.58 Growth rate curve for the CT specimen for the bulk nano-silica

modified epoxy matrix

Figure 5.59 Growth rate curve for the CT specimen for the bulk nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix (experimental data from Lee [49])

-7.2

-6.2

-5.2

-4.2

-3.2

-2.2

-1.2

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05

Experiment

Modelling

Paris law fit

log (

da/d

N)

log (Gmax/Gc)

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4

Experiment (Paris law)

Modelling

Paris law fit

log (Gmax/Gc)

log (

da/d

N)


151

5.6.2.2 DCB test and user element analysis

The DCB fatigue test is undertaken to find the lamina interface fracture energy

of the composite material. The cohesive zone properties of the lamina are

obtained from the fatigue test of the composite material. The growth rate curves

obtained from the tests are shown in Figures 5.60 to 5.63. The threshold

fracture energy for the composite material is found from the growth rate curve.

The fatigue analysis of the DCB is performed using a FEA approach in the

Abaqus software. The analysis of the fatigue damage in the composite is

modelled using a cohesive zone element. The subroutine is developed in

Abaqus and accounts for the fatigue damage of cohesive zone element with

time. The thickness of the cohesive zone element is kept the same as in the

quasi-static model at 0.01mm. The model is analysed using the subroutine to

obtain the fatigue parameters of the cohesive zone element. The load in the

fatigue model is applied as a constant displacement, as described in Section

4.3.5.2, and the load is kept constant throughout the fatigue cycles.

The growth rate curve predicted by the model is compared with the growth rate

curve from the experimental tests. The parameters for the fatigue model are

obtained by comparing the slope and intercept of the growth rate cure and

matching them with the experimental results. The experimental growth rate

curve of the different materials is used to validate the fatigue parameters. (Since

the cohesive zone element parameters are derived from the quasi-static work

are used to model the DCB test in fatigue.) The growth rate curve of the DCB

specimen is matched with the experiments to obtain the fatigue parameters of

the lamina interface (Figures 5.60 to 5.63). The fatigue parameters and

obtained from the Paris law fit are given in Table 5.18. The fatigue parameters

are then used in the user element subroutine to degrade the cohesive zone

element according to the Paris law, as described in Section 4.3.4.


152

Table 5.18 Fatigue parameters obtained from the DCB composite material specimen

Figure 5.60 Growth rate curve for the composite DCB specimen based upon the unmodified (i.e. control) epoxy matrix

Matrix fatigue properties

c

m

Gth (J/m2)

3.88 3.27 3.54

39.5 57 54 118

2.71


0.0123 0.0084 0.01740.0243

-8

-7

-6

-5

-4

-3

-2

-1

0

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Experiment

Modelling

Paris law fit

log (Gmax/Gc)

log (

da/d

N)


153

Figure 5.61 Growth rate curve for the composite DCB specimen based upon the micro-rubber modified epoxy matrix

Figure 5.62 Growth rate curve for the composite DCB specimen based upon the nano-silica modified epoxy matrix

-7

-6

-5

-4

-3

-2

-1

0

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Experiment

Modelling

Paris law fit

log (Gmax/Gc)

log (

da/d

N)

-7

-6

-5

-4

-3

-2

-1

0

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Experiment

Modelling

Paris law fit

log (

da/d

N)

log (Gmax/Gc)


154


the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix

5.6.2.3 Strip test and user element analysis

The fatigue test on the composite strip is undertaken to determine the fatigue

life of composite strip specimens. The composite strip is tested under a

constant fatigue load to determine the fatigue life of the strip. The transverse

crack density of the composite strip is obtained for different cycles of fatigue

and the number of cycles to failure of the strip is also found out. A typical

photograph of the crack density observed in the composite for different cycles of

fatigue is shown in Figure 5.64. The graphs of stress versus number of cycles

for failure of the composite strip are shown in Figures 5.65 to 5.68; and values

of the crack density for different cycles of fatigue obtained for the different

composite materials are as shown previously in Figure 5.33.

-6

-5

-4

-3

-2

-1

0

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Experiment

Modelling

Paris law fit

log (

da/d

N)

log (Gmax/Gc)


155

Figure 5.64 Transmitted light photographs of GFRP composite with unmodified

(i.e. control) epoxy matrix showing the sequence of matrix crack development with the number of cycles, N, under fatigue loading.

The composite strips are now finally modelled to predict the fatigue life of the

composite material based on the different epoxy matrices. The composite strip

has a layup of [(-45/45/0/90)s]2 and dimensions of 50x2.7x25mm3 is used to

study the fatigue behaviour of the composites. The strip is modelled considering

a small section of the strip from the cross-section, as shown in Figure 5.30. The

strip is then modelled with transverse cracks with a density as observed in the

experiments (Figure 5.33). The 90o lamina crack density is assumed to be

constant at a value of 0.64/mm (Tong et al. [37]). The fatigue analysis is

undertaken with different crack densities, and with the different epoxy

composites.

The load is applied as constant stress on the strip as shown in Figure 4.13. The

nodes on the applied stress surface are tied with each other to have an equal

displacement in the y direction of the strip (in Figure 5.32 the top surface nodes

are made to move equally in the y direction). The width of the model is half the

thickness of strip and the length of the model depends on the crack density

(Figure 5.31). In the model, a symmetric section (Figure 5.30) of the strip is


156

modelled. The strip is modelled as shown in Figure 5.31. The boundary

condition for the strip is shown in Figure 5.32.

The cohesive degradation law is used to model the effect of the accumulation of

damage upon fatigue loading of the composite. The cohesive zone elements

are placed in the transverse cracks and the maximum transverse crack density

is selected to study the fatigue crack growth in the strip. The Paris law

parameters obtained from the CT specimen (Table 5.17) and the quasi-static

cohesive zone law parameters obtained from the SENB test (Table 5.14), both

of which are measured on the corresponding bulk epoxy matrix, are used to

model the fatigue life of composite.

The threshold value of the composite is taken as of the corresponding bulk

epoxy matrix (Table 5.14). The composite is modelled with the highest crack

density observed in the experiments. However, the value of the maximum crack

density, or the saturation crack density for maximum stiffness reduction, for the

fatigue modelling of the strip can also be attained by achieving a plateau in the

normalised stiffness with an increase in crack density via a quasi-static

modelling. Now, the saturation crack density which causes the maximum

stiffness reduction may be determined from a quasi-static model. Thus, when

the experimental value of the crack density is not available, the fatigue model

proposed in the present research can still be employed.

The failure criteria for the composite strip in fatigue can be described in two

ways for different levels of fatigue stress. In low and high stress analysis, the

threshold energy of the bulk epoxy is kept the same as that of the bulk epoxy

CT specimen fatigue test. However:

1. For higher stresses, the failure criterion is defined by the number of

cycles required for the maximum reduction of stiffness with the failure of

cohesive zone elements occurring.

2. The failure criteria for the cohesive zone element in the strip under a low

applied fatigue stress can be defined as the number of cycles required

for the stiffness to achieve a plateau value; and no energy is now


157

available for any further stiffness reduction caused by the growth of

cracks. The cohesive zone element in this case never fails completely,

and the growth of the crack stops as the energy available is less than the

threshold energy required for the growth of crack in fatigue.

The model is run until the maximum stiffness reduction is obtained for the crack

density. The stress versus number of cycles for failure of the composite strip is

shown in Figures 5.65 to 5.68 for the modelling and experimental results. The

modelling matches very well with the experimental results for the relatively low

applied fatigue stress values. At the higher fatigue stresses, the failure of the

composite is mainly due to delamination and fibre failure. Hence the life of the

composite is over predicted at these higher stresses.

It shall be noted that the very good agreement between the experimental results

and the theoretical predictions at the relatively low levels of applied fatigue

stress is of major industrial importance since the fatigue life at the threshold

stress is a typical design criterion.


158

Figure 5.65 Applied maximum fatigue stress versus the number of cycles upon fatigue loading for the composite strip based upon the unmodified (i.e. control) epoxy matrix

Figure 5.66 Applied maximum fatigue stress versus the number of cycles upon fatigue loading for a composite strip based upon the micro-rubber modified epoxy matrix

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Figure 5.67 Applied maximum fatigue stress versus the number of cycles upon fatigue loading for a composite strip based upon the nano-silica modified epoxy matrix


fatigue loading for a composite strip based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix

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The threshold stress for the composite strip is obtained by progressively

applying even lower fatigue stresses and checking the stiffness reduction with

time. The stress at which there is no further reduction of the stiffness from the

model is determined and taken as the threshold stress below which no fatigue

failures of the composite material strip will occur. The values of the threshold

stress so obtained for the different composite strip models are given in Table

5.19.

Table 5.19 Threshold stress for the composite strip

The improvement in the value of the threshold stress for cyclic fatigue failure of

the composite materials based upon the modified epoxy matrices should be

especially noted.


The modelling of the different specimens has been undertaken using a FEA

model constructed using Abaqus together with a novel user subroutine. The

fatigue life of the composite materials based upon the different epoxy matrices

has been found to be predicted accurately. It is very note worthy that the input

into the model developed in the present research is that needed from tests

undertaken on the bulk epoxy matrices. Thus, the model does not require prior

knowledge of the behaviour of the composite material under cyclic fatigue

loading.

Strip properties

Stress (MPa) 80 100 105 110


CHAPTER 6 CONCLUSIONS & RECOMMENDATIONS FOR FUTURE WORK

161

CHAPTER 6

6. CONCLUSIONS & RECOMMENDATIONS

FOR FUTURE WORK

6.1 Quasi-Static Fracture Properties

The quasi-static testing of the bulk epoxy matrices, using the single edge

notched bending (SENB) specimen, gives the fracture properties of the bulk

epoxy matrices. It is shown that the modified bulk epoxy matrices have a higher

fracture energy, and their failure strength is also increased, compared with the

unmodified (i.e. control) epoxy matrix. The fracture energies for the different

modified epoxy matrices are compared with that of the unmodified (i.e. control)

epoxy matrix in Table 6.1.

Table 6.1 Fracture energies of the different bulk epoxy matrices

The quasi-static testing of the double cantilever beam (DCB) specimens gives

values of the fracture energy of the GFRP composite materials based upon the

different epoxy matrices, for the 0o/90o lamina plane. The fracture energy of the

composite material based on the epoxy ‘hybrid’ matrix ( i.e. modified with the

nano-silica and micro-rubber particles) is found to be higher than those based

upon the epoxy matrices modified only with the nano-silica or only with the

micro-rubber particles (Table 6.2). Thus, as may be seen from the summary of

results given in Table 6.1 and 6.2, the addition of nano-silica or micro-rubber

particles, and especially the combination of both particles to give a hybrid

modified matrix, results in tougher bulk epoxy matrices, and tougher GFRP

composite materials (Table 6.2). The toughening mechanisms responsible for

these observations are discussed in Section 5.5.

Matrix Properties

Gc (J/m2) 75.8 415.9 123.4 859.0



162

Table 6.2 Fracture energies of the composite materials based upon the different epoxy matrices

6.2 Quasi-Static Modelling

The modelling results from using a FEA approach coupled with a cohesive zone

model (CZM) works very well for both the bulk epoxy matrices and the

composite materials. The quasi-static model predicts the static failure of the

SENB fracture mechanics tests on the bulk epoxy matrix relatively accurately.

The modelling defines the parameters of the cohesive zone law, which are then

used subsequently for the modelling of the composite strip materials, see

Section 6.6. The modelling of the static failure of the DCB fracture mechanics

tests on the composite materials gives results which are also in very good

agreement with the experimental results, and this helps to validate the

modelling methods developed in the present studies.

6.3 Fatigue Testing

Fatigue testing of the compact tension (CT) specimen is undertaken to

determine the cyclic fatigue properties of the bulk epoxy matrices. The growth

rate curves of the maximum strain energy release rate, , versus the rate of

crack growth per cycle, , for the bulk epoxy matrices are obtained from

these tests. The fatigue parameters of the bulk epoxy matrices are obtained

from the linear fit to the test data, i.e. the Paris law parameters, are readily

obtained from these experimental results.

6.4 Fatigue Modelling

The novel user element subroutine developed for the fatigue analysis is written

in FORTRAN, see Section 6.5 below. The analysis predicts the stiffness

degradation of the cohesive zone law as a function of time, i.e. as a function of

time taken to accumulate damage developed, via the number of fatigue cycles,

in the cohesive zone due to the cyclic fatigue loading. The degradation in the

cohesive zone law is implemented by using a bi-linear constitutive law for the

Matrix Properties

Gc (J/m2) 910 1080 1790 1894



163

CZM, with the degradation of this constitutive law governed by the Paris law

parameters, which are determined as noted above.

To illustrate the experimental results, and the predictions from the modelling

studies, a typical fatigue growth curve is shown again in Figure 6.1. In this case

for the unmodified (i.e. control) epoxy matrix, and the results were obtained

using the CT specimen. The linear region is fitted to the Paris law, as shown in

Figure 6.1. The FEA model, incorporating the CZM, has been run with the

cohesive zone law parameters degraded via employing the user element

subroutine, see Section 6.5. The modelling results are also shown in Figure 6.1.

The agreement between the results from the proposed model and the

experimental results is very good. This good agreement is taken to provide a

further validation of the modelling procedures proposed in the present research.

Figure 6.1 Fatigue crack growth rate curve for the CT epoxy specimen based upon the unmodified (i.e. control) epoxy matrix

-7

-6

-5

-4

-3

-2

-1

-0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2

Experiment

Modelling

Paris law fit

log (Gmax/Gc)

log (

da/d

N)


164

The fatigue crack growth rate curves of the DCB specimens using the

composite materials are also measured, and predicted from the modelling

studies. As for the bulk epoxy matrices, the agreement between the

experimental and modelling results for the values of versus for the

composite materials is found to be excellent. This provides a further validation

of the model proposed in the present research.

6.5 The User Element Subroutine Analysis

A special mention must be made of the novel user element subroutine

developed in the present research, since it is a crucially important feature of the

proposed model, and is given in the Appendix. This subroutine modifies the

penalty stiffness of the cohesive zone law with time equivalent to the number of

fatigue cycles, and hence a model of the fatigue degradation in the material is

achieved. The degradation of the penalty stiffness is based upon knowing the

values of Paris law parameters which are determined as described above. The

user element subroutine analysis was found to run very well for the fatigue

analyses of the bulk epoxy matrices and the composite material specimens.

Indeed, it provides a good prediction of the fatigue crack growth rate data which

are experimentally determined from the bulk epoxy matrix CT specimens and

composite material DCB specimens, which therefore provide a validation of the

proposed model and the user element subroutine as noted above.

6.6 Fatigue Life

Finally, this validated FEA/CZM model is used as a new approach to undertake

predicting the failure of the composite materials under cyclic fatigue loading.

The proposed fatigue model of the composites is modelled with multiple

transverse cracks, since this is the main form of damage observed in the fatigue

tests. These transverse cracks are modelled using CZM with a constitutive law

which is degraded as described above. The criterion for stopping the further

degradation of the constitutive law of the CZM is based upon attaining a

saturation matrix crack density. However, it is not necessary to directly measure

this parameter. The model determines the loss of stiffness of the composite


165

material strip, which arises from the matrix cracking, until no further decrease in

stiffness is predicted by the model. At this point the fatigue life of the composite

material is assumed to have been reached. A typical result is summarised in

Figure 6.2 for the composite material strip based on the epoxy matrix containing

micrometre-sized rubber particles. Here the maximum stress applied in the

fatigue cycle is plotted against the number of cycles to failure. As for all the

composite materials studied, the agreement between the experimental results

and the predictions from the model developed in the present research is very

good.

Figure 6.2 Stress versus number of cycles from the fatigue loading for a

composite material strip based upon the micro-rubber modified epoxy matrix

6.7 Recommendations for Further Work

Future work that could be undertaken to extend further the present modelling

studies for the fatigue behaviour of composite materials is discussed below.

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166

6.7.1 Unidirectional composite analysis

A modelling analysis of unidirectional fibre composite materials needs a

different approach to developing a sound predictive analysis, as the transverse

crack density in the matrix is not readily available. Further, the analysis of the

unidirectional composite may depend on the mode II matrix parameters, and the

prediction of the fibre failure needs to be based on knowledge of the failure

strain. Indeed, the failure of a unidirectional composite material mainly occurs

due to interlaminar damage, and hence a different model has to be developed

for the analysis. This has been outside the scope of the present research but

represents a worthwhile future research goal.

6.7.2 Delamination analysis

Delamination in composite materials of various layups may occur due to fatigue

loading, and it occurs at the lamina interfaces. Delamination can be analysed in

the composite using the mode II quasi-static and fatigue properties. At higher

cycles, delamination occurs when the degree of transverse cracking saturates

in the lamina. Modelling delamination at the lamina interface may give better

results for the prediction of the fatigue life of composite materials, as it may give

more accurate values of the associated stiffness reductions. Delaminations

could be incorporated into the proposed model along the lamina interface by

employing a thin layer of cohesive zone elements between the laminae.

6.7.3 Modelling process of failure

The present work is unable to model accurately the development of the damage

mechanisms which are actually observed during the fatigue failure of composite

materials; whereby in reality the cracks develop one after another, i.e. in a

sequential manner. (In the current modelling, the highest crack density is used

for the analysis and all cracks grow simultaneously.) Modelling of the composite

with such transverse cracks growing sequentially represents a limitation in the

proposed model. Models could therefore be developed in order to overcome

this limitation.


167

6.7.4 The role of matrix properties

The present research has confirmed that the addition of a particulate phase of

nano-silica or micrometre-size rubber particles improves the toughness, , of

the bulk epoxy matrix, and that this improvement is transferred to the

corresponding GFRP composite material. Further, a hybrid toughened matrix, or

corresponding composite material, exhibits the highest values of . However,

further properties of the composite materials based upon these modified epoxy

matrices need to be evaluated. For example, the abrasion resistance of such

nano-silica modified composite materials might be significantly improved

compared with composites based on the unmodified (i.e. control) epoxy matrix

material. On the other hand, the hygrothermal properties of such composite

materials might be adversely affected. Thus, the hygrothermal properties of

such matrices and corresponding composite materials are worthy of further

study. Indeed, the models proposed in the current research project could be

extended to attempt to predict the hygrothermal properties of the composite

materials.

REFERENCES

168

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173


APPENDIX

174

Appendix

User Element Subroutine

SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,

PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,DTIME,

KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,PREDEF,

NPREDF,LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,JPROPS,NJPROP,

PERIOD)

INCLUDE 'ABA_PARAM.INC'

PARAMETER (NGAUS = 2)

DIMENSION RHS(MLVARX,*),AMATRX(NDOFEL,NDOFEL),PROPS(*),

SVARS(NSVARS),ENERGY(8),COORDS(MCRD,NNODE),U(NDOFEL),

DU(MLVARX,*),

V(NDOFEL),A(NDOFEL),TIME(2),PARAMS(*),JDLTYP(MDLOAD,*),

ADLMAG(MDLOAD,*),DDLMAG(MDLOAD,*),PREDEF(2,NPREDF,NNODE),

LFLAGS(*),JPROPS(*)

DIMENSION TAU(NNODE/2,MCRD),POSGP(2)

DIMENSION ASDIS(NNODE/2,MCRD)

DIMENSION DERIV(MCRD-1,NNODE/2),SHAPEE(NNODE/2),

BMATX(MCRD,NDOFEL),WEIGP(NGAUS),DBMAT(MCRD,NDOFEL),

DMATX(NNODE/2,MCRD,MCRD),XJACM(MCRD-1,MCRD),

KF(NNODE/2),SCALAR(NNODE/2),BMAT(MCRD,NDOFEL/2),

BMATXT(NDOFEL,MCRD),AAMATRX(NDOFEL,NDOFEL),VP(MCRD,MCRD),

BMATXB(MCRD,NDOFEL),VN(NDOFEL/2,NDOFEL)

IF (JTYPE.EQ.2) THEN

SVARS(1)=1.D0

SVARS(2)=1.D0

END IF

CALL NUMPROP(GIC,GIIC,T1,T2,PEN,ETA,THICK,NLGEOM,PROPS,JPROPS,

COORDS,MCRD,NNODE)

DO IEVAB=1,NDOFEL

DO JEVAB=1,NDOFEL

AMATRX(IEVAB,JEVAB)=0.0D0

END DO

RHS(IEVAB,1)=0.0D0

END DO

IF(LFLAGS(4).NE.0) THEN

STOP 771

END IF

IF(LFLAGS(3).EQ.1)THEN

CALL GAUSSQ(NGAUS,POSGP,WEIGP)

!--------------------------------------

! Numerical integration

!--------------------------------------

KGASP=0

DO IGAUS=1,NGAUS

DO JGAUS=1,1

KGASP=KGASP+1

R=POSGP(KGASP)

CALL SFR3(DERIV,R,MCRD,SHAPEE,NNODE,KGASP)

CALL JACOBT(DERIV,DNORMA3,COORDS,KGASP,MCRD,NNODE,

SHAPEE,VN,XJACM,JELEM,NDOFEL,U,THICK,NLGEOM,VP)

DAREA=0.0D0

DAREA=DNORMA3*WEIGP(IGAUS)*WEIGP(JGAUS)*THICK

APPENDIX

175

CALL BMATT (SHAPEE,NDOFEL,MCRD,BMATX,VN,NNODE,BMAT,VP,KGASP,

BMATXB)

!--------------------------------------

! Relative displacements

!--------------------------------------

DO ISTRE=1,MCRD

ASDIS(KGASP,ISTRE)=0.0D0

DO IEVAB=1,NDOFEL

ASDIS

(KGASP,ISTRE)=ASDIS(KGASP,ISTRE)+(BMATXB(ISTRE,IEVAB)*U(IEVAB))

END DO

END DO

!--------------------------------------¬

! [D] matrix

!--------------------------------------¬

CALL MODT (MCRD,DMATX,KGASP,JELEM,

PEN,ASDIS,T1,T2,GIC,GIIC,ETA,SVARS,KF,SCALAR,NNODE)

!---------------------------------------

! Traction vector

!---------------------------------------

DO ISTRE=1,MCRD

TAU(KGASP,ISTRE)=DMATX(KGASP,ISTRE,ISTRE)*ASDIS(KGASP,ISTRE)

END DO

!---------------------------------------¬

! Force vector

!---------------------------------------¬

BMATXT=TRANSPOSE(BMATX)

DO IEVAB=1,NDOFEL

DO ISTRE=1,MCRD

RHS(IEVAB,1)=RHS(IEVAB,1)-

BMATXT(IEVAB,ISTRE)*TAU(KGASP,ISTRE)*DAREA

END DO

END DO

!--------------------------------------¬

! Stiffness matrix

!--------------------------------------¬

CALL STIFF (MCRD,NDOFEL,NNODE,DMATX,BMATX,KGASP,ASDIS,

KF,SCALAR,DAREA,AAMATRX,JELEM,KINC,KSTEP,PNEWDT,TIME,DTIME,VP)

DO IEVAB=1,NDOFEL

DO JEVAB=1,NDOFEL

AMATRX(IEVAB,JEVAB)=AMATRX(IEVAB,JEVAB)+AAMATRX(IEVAB,JEVAB)

END DO

END DO

END DO

END DO

ELSE

Write(7,*)'*****WARNING LFLAGS(3)=',LFLAGS(3)

END IF

RETURN

END

!*****************SUBROUTINE NUMPROP*************

! Materials properties

!************************************************

SUBROUTINE NUMPROP(GIC,GIIC,T1,T2,PEN,ETA,THICK,NLGEOM,PROPS,

JPROPS,COORDS,MCRD,NNODE)


DIMENSION COORDS(MCRD,NNODE),PROPS(*),JPROPS(*)

GIC= PROPS(1)

GIIC= PROPS(2)

T1= PROPS(3)

APPENDIX

176

T2= PROPS(4)

PEN= PROPS(5)

ETA = PROPS(6)

THICK = PROPS(7)

NLGEOM = JPROPS(1)

RETURN

END

!*****************SUBROUTINE GAUSSQ*******

! Integration points

!*****************************************

SUBROUTINE GAUSSQ(NGAUS,POSGP,WEIGP)


DIMENSION POSGP(2),WEIGP(NGAUS)

POSGP(1)=-1.0D0

POSGP(2)=1.0D0

WEIGP(1)=1.0D0

WEIGP(2)=1.0D0

RETURN

END

!*****************SUBROUTINE MODT**************

! [D] matrix calculation

!**********************************************

SUBROUTINE MODT (MCRD,DMATX,KGASP,JELEM,

PEN,ASDIS,T1,T2,GIC,GIIC,ETA,SVARS,KF,SCALAR,NNODE)


DIMENSION DMATX(NNODE/2,MCRD,MCRD),ASDIS(NNODE/2,MCRD),

SVARS(*),KF(NNODE/2),SCALAR(NNODE/2),PBEF(2)

KF(KGASP) = 0

thickness= !VALUE

ODI = T1/(PEN/thickness)

ODII = T2/(PEN/thickness)

DDI = ASDIS(KGASP,2)

DII = ABS(ASDIS(KGASP,1))

IF (ABS(DDI).LT.1.0D-20) THEN

BETA = 1.0D0

CYCDI=DII

ELSE

BETA = DII/(ABS(DDI)+DII)

CYCDI = SQRT(DDI*DDI+DII*DII)

END IF

A =(BETA**2/(1.0d0+2.0d0*BETA**2-2.0d0*BETA))**ETA

OD=SQRT(ODI**2+(ODII**2-ODI**2)*A)

FD = 2.0d0*(GIc+(GIIc-GIc)*A)/((PEN/thickness)*OD)

TR=OD*(PEN/thickness)

GCR=0.50d0*(PEN/thickness)*OD*FD

GTH= !INPUT VALUE

GR=GCR

WIDTH= !INPUT VALUE

PBEF(1)=!INPUT VALUE

PBEF(2)=!INPUT VALUE

E33=!VALUE

DISPOLD=SVARS(KGASP+8)

DAMFATOLD=SVARS(KGASP+4)

IF (DISPOLD.LE.CYCDI.AND.CYCDI.GE.0.0d0) THEN

DI=CYCDI

END IF

IF (DISPOLD.GE.CYCDI.AND.CYCDI.GE.0.0d0) THEN

DI=DISPOLD

END IF

IF (CYCDI.LT.0.0d0) THEN

APPENDIX

177

DI=DISPOLD

END IF

IF (DI.LE.OD.AND.DI.GE.0.0d0) THEN

DAMSTATT(1)=0.0d0

END IF

IF (DI.GT.OD.AND.DI.LE.FD.AND.DI.GT.0.0d0) THEN

DAMSTATT(1)=FD/DI*(DI-OD)/(FD-OD)

END IF

IF (DI.GT.FD) THEN

DAMSTATT(1)=1.0d0

END IF

IF (DI.LT.0.0D0) THEN

DAMSTATT(1)=SVARS(KGASP)

END IF

IF (DAMSTATT(1).LE.SVARS(KGASP)) THEN

DAMSTATT(1)=SVARS(KGASP)

END IF

IF (DAMSTATT(1).GT.1.0d0) THEN

DAMSTATT(1)=1.0d0

END IF

IF (DI.LE.OD.AND.DI.GE.0.0d0) THEN

GMAX=(PEN/thickness)*DI**2.0d0/2.0d0

END IF

IF (DI.GT.OD.AND.DI.LE.FD.AND.DI.GT.0.0d0) THEN

GMAX=(PEN/thickness)*OD*OD/2.0d0 +

(((1.0d0-DAMSTATT(1))*(PEN/thickness)*DI+TR)/2.0d0*(DI-OD) )

END IF

IF (DI.GT.FD.OR.DI.LE.0.0d0) THEN

GMAX=0.0d0

END IF

IF (GMAX.GE.GTH.AND.GMAX.LE.GCR) THEN

RATE=WIDTH*PBEF(1)*(GMAX/GR)**PBEF(2)

ELSE

RATE=0.0d0

END IF

ACZ=WIDTH*9.0d0*22.0d0*E33*GMAX/(32.0d0*7.0d0*TR**2.0d0)

IF (ACZ.GT.0.0d0) THEN

DRATE=(FD*(1.0d0-DAMSTATT(1))+

DAMSTATT(1)*OD)**2.0d0*(RATE)/

(FD*OD*ACZ)

ELSE

DRATE=(FD*(1.0d0-DAMSTATT(1))+

DAMSTATT(1)*OD)**2.0d0*(RATE)/

(FD*OD*0.1d0*WIDTH)

END IF

DAMSTATT(2)=DAMFATOLD + DTIME*DRATE

DAMTOT= DAMSTATT(1)+DAMSTATT(2)

IF (DAMTOT.GE.1.0d0) THEN

DAMTOT=0.999999999d0

END IF

IF (DAMTOT.GT.(SVARS(KGASP)+SVARS(KGASP+4))) THEN

KF(KGASP) = 1

ELSE

KF(KGASP) = 0

END IF

IF (DAMTOT.GT.1.0D0) THEN

DAMTOT = 1.0D0

KF(KGASP)=0

END IF

DO I=1,MCRD

APPENDIX

178

DO J=1,MCRD

DMATX(KGASP,I,J)=0.D0

END DO

DMATX(KGASP,I,I)=(1.0D0-DAMTOT)*(PEN/thickness)

END DO

IF (DI.LT.0.D0) THEN

DMATX(KGASP,MCRD,MCRD)=(PEN/thickness)

END IF

IF(KF(KGASP).EQ.1) THEN

SCALAR(KGASP) = FD*OD*(PEN/thickness)/(DI**3*(FD-OD))

END IF

SVARS(KGASP)= DAMSTATT(1)

SVARS(KGASP+4)= DAMSTATT(2)

SVARS(KGASP+8)= CYCDI

RETURN

END

!***************** SUBROUTINE STIFF *************

! Stiffness matrix

!************************************************

SUBROUTINE STIFF (MCRD,NDOFEL,NNODE,DMATX,BMATX,KGASP,ASDIS,

KF,SCALAR,DAREA,AAMATRX,JELEM,KINC,KSTEP,PNEWDT,TIME,DTIME,VP)


PARAMETER (ITMAX=15)

DIMENSION AAMATRX(NDOFEL,NDOFEL),DMATX(NNODE/2,MCRD,MCRD),

SCALAR(NNODE/2),ASDIS(NNODE/2,MCRD),BMATX(MCRD,NDOFEL),

DTANG(NNODE/2,MCRD,MCRD),DBMAT(MCRD,NDOFEL),KF(NNODE/2),

TIME(2),VP(MCRD,MCRD),BMATXTT(NDOFEL,MCRD)

COMMON /KZERO/ KSEC

COMMON /KINCREMENTZ/ KINC1

COMMON /KCOUNTINGZ/ KCOUNT

COMMON /KAUX/ KELEMENT

COMMON /KELLABEL/ KLABEL

COMMON /KCHSTEP/ KSTEP1

COMMON TI

IF(KELEMENT.NE.1) THEN

KLABEL = JELEM

KELEMENT = 1

END IF

IF(KINC.NE.KINC1.AND.KSTEP1.NE.KSTEP) THEN

IF(JELEM.EQ.KLABEL) THEN

KCOUNT = KCOUNT+1

KINC1 = KINC+1

KSTEP1 = KSTEP+1

END IF

ELSE

KCOUNT = 0

KINC1 = KINC+1

KSTEP1 = KSTEP +1

KSEC = 0

TI = DTIME

END IF

IF(KCOUNT.GE.ITMAX*NNODE/2) THEN

IF(DTIME.NE.TI) THEN

KCOUNT=0

TI=DTIME

KSEC=1

END IF

END IF

APPENDIX

179

IF

(JELEM.EQ.KLABEL.AND.KCOUNT.GE.ITMAX*NNODE/2.AND.KSEC.NE.1)THEN

PNEWDT = 0.9D0

END IF

DO I=1,MCRD

DO J=1,MCRD

DTANG(KGASP,I,J)=DMATX(KGASP,I,J)

END DO

END DO

IF (KCOUNT.LT.(ITMAX+1)*NNODE/2) THEN

IF (KF(KGASP).EQ.1) THEN

DO I=1,MCRD

DO J=1,MCRD

DTANG(KGASP,I,J)=DMATX(KGASP,I,J)-

SCALAR(KGASP)*ASDIS(KGASP,J)*ASDIS(KGASP,I)

END DO

END DO

IF (ASDIS(KGASP,MCRD).LT.0.D0) THEN

DO I=1,MCRD

DTANG(KGASP,MCRD,I)=DMATX(KGASP,MCRD,I)

DTANG (KGASP,I,MCRD)=DMATX(KGASP,I,MCRD)

END DO

END IF

END IF

ELSE

END IF

!--------------------------------------

!calculate [DTAN]x[B]

!--------------------------------------

DO I=1,MCRD

DO J=1,NDOFEL

DBMAT(I,J)=0.0D0

DO K=1,MCRD

DBMAT(I,J)=DBMAT(I,J)+(DTANG(KGASP,I,K)*BMATX(K,J))

END DO

END DO

END DO

DO IEVAB=1,NDOFEL

DO JEVAB=1,NDOFEL

AAMATRX(IEVAB,JEVAB)=0.0D0

END DO

END DO

!------------------------

!calculate [BT]x[DTAN]x[B]dA

!-------------------------

BMATXTT=TRANSPOSE(BMATX)

DO IEVAB=1,NDOFEL

DO JEVAB=1,NDOFEL

DO ISTRE=1,MCRD

AAMATRX(IEVAB,JEVAB)=AAMATRX(IEVAB,JEVAB)+BMATXTT(IEVAB,ISTRE)*

DBMAT(MCRD,JEVAB)*DAREA

END DO

END DO

END DO

RETURN

END

APPENDIX

180

!***************** SUBROUTINE BMATT *************

! [B] matrix

!************************************************

SUBROUTINE BMATT

(SHAPEE,NDOFEL,MCRD,BMATX,VN,NNODE,BMAT,VP,KGASP,

BMATXB)


DIMENSION

SHAPEE(NNODE/2),BMATX(MCRD,NDOFEL),BMAT(MCRD,NDOFEL/2),

VN(NDOFEL/2,NDOFEL),BMATXB(MCRD,NDOFEL),VP(MCRD,MCRD)

DO I=1,MCRD

DO J=1,NDOFEL/2

BMAT(I,J)=0.0D0

END DO

END DO

DO I=1,MCRD

K=0

DO J=1,(NDOFEL/2),MCRD

K=K+1

BMAT(I,J+I-1)=SHAPEE(K)

END DO

END DO

DO I=1,MCRD

DO J=1,NDOFEL

BMATXB(I,J)=0.0D0

DO M=1,NDOFEL/2

BMATXB(I,J)=BMATXB(I,J)+BMAT(I,M)*VN(M,J)

END DO

END DO

END DO

DO I=1,MCRD

DO J=1,NDOFEL

BMATX(I,J)=0.0D0

DO M=1,MCRD

BMATX(I,J)=BMATX(I,J)+VP(I,M)*BMATXB(M,J)

END DO

END DO

END DO

RETURN

END

!*****************SUBROUTINE JACOBT********

! Jacobian matrix

!******************************************

SUBROUTINE JACOBT(DERIV,DNORMA3,COORDS,KGASP,MCRD,NNODE ,

SHAPE,VN,XJACM,JELEM,NDOFEL,U,THICK,NLGEOM,VP)


DIMENSION COORDS(MCRD,NNODE),SHAPEE(NNODE/2),

VN(NDOFEL/2,NDOFEL),XJACM(MCRD-1,MCRD),U(NDOFEL),VP(MCRD,MCRD),

VPP(MCRD,MCRD),DERIV(MCRD-1,NNODE/2)

DO I=1,NDOFEL/2

DO J=1,NDOFEL

VN(I,J)=0.0D0

END DO

END DO

DO I=1,NDOFEL/2

VN(I,I)=-1.0D0

END DO

APPENDIX

181

PAD=0

DO I=NDOFEL/2+1,NDOFEL

PAD=PAD+1

VN(PAD,I)=1.0D0

END DO

DO IDIME=1,MCRD-1

DO JDIME=1,MCRD

XJACM(IDIME,JDIME)=0.0D0

DO INODE=1,NNODE/2

XJACM(IDIME,JDIME)=XJACM(IDIME,JDIME)+DERIV(IDIME,INODE)*

(0.50D0*(COORDS(JDIME,INODE+2)+COORDS(JDIME,INODE))

+0.50D0*(U((INODE-1)*MCRD+JDIME)+

U((INODE+NNODE/2-1)*MCRD+JDIME)))

END DO

END DO

END DO

DNORMA3=SQRT(XJACM(1,1)**2+XJACM(1,2)**2)

VPP(1,1)=XJACM(1,1)/DNORMA3

VPP(1,2)=-XJACM(1,2)/DNORMA3



VP=VPP

RETURN

END

!*****************SUBROUTINE SFR3***********

! Shape functions

!*******************************************

SUBROUTINE SFR3(DERIV,R,MCRD,SHAPEE,NNODE,KGASP)


DIMENSION SHAPEE(NNODE/2),DERIV(MCRD-1,NNODE/2)

RP=1.0D0+R

RN=1.0D0-R

DO I=1,NNODE/2

SHAPEE(I)=0.0D0

DO J=1,MCRD-1

DERIV(J,I)=0.0D0

END DO

END DO

SHAPEE(1)=RN/2.0D0

SHAPEE(2)=RP/2.0D0

DERIV(1,1)=-0.5D0

DERIV(1,2)=0.5D0

RETURN

END

Documents

PREDICTING THE FATIGUE BEHAVIOUR OF …...containing the nano-silica particles and/or micro-rubber particles. The theoretical studies employed an extended finite element method, coupled