characterization of interlaminar fracture …

The Pennsylvania State University

The Graduate School

Department of Engineering Science and Mechanics

CHARACTERIZATION OF INTERLAMINAR FRACTURE

TOUGHNESS OF A CARBON/EPOXY COMPOSITE MATERIAL

A Thesis in

Engineering Mechanics

by

Ye Zhu

© 2009 Ye Zhu

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

May 2009

The thesis of Ye Zhu was reviewed and approved* by the following:

Charles E. Bakis Distinguished Professor of Engineering Science and Mechanics Thesis Advisor

George A. Lesieutre Professor of Aerospace Engineering Head of the Department of Aerospace Engineering

Kelvin L. Koudela Senior Research Associate Applied Research Laboratory

Judith A. Todd Professor of Engineering Science and Mechanics P. B. Breneman Department Head Chair Head of the Department of Engineering Science and Mechanics

*Signatures are on file in the Graduate School

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ABSTRACT

The primary objective of this investigation is to characterize the Mode I, Mode II,

and mixed Mode I/II interlaminar fracture of a proprietary carbon/epoxy composite

material system. A state-of-the-art review of the literature on quasi-static and cyclic test

methods for interlaminar fracture testing is given. The Mode I, Mode II, and mixed

Mode I/II interlaminar fracture behavior of the carbon/epoxy laminated material in quasi-

static and fatigue loadings was investigated using the double-cantilever-beam (DCB)

specimen, the end-notched flexure (ENF) specimen, and the single leg bending (SLB)

specimen, respectively. It was found that the Mode I interlaminar fracture toughness at

crack onset (GIc) was low for the investigated material system in comparison to results

reported in the literature for carbon/brittle epoxy material system. In addition, the Mode I

fracture toughness increased by about 40% after 50 mm crack extension. The Mode II

quasi-static tests were conducted with precracked and un-precracked specimens.

Compared to results reported in the literature, the Mode II fracture toughnesses (GIIc) of

the investigated material were in the common range for carbon fiber composites made

with brittle epoxies. The GIIc value of an un-precracked specimen was 44% -60% higher

than that of a precracked specimen. The mixed-mode fracture toughness (GTc) was found

to be low in comparison to the results in the literature and it increased by 11 to 53% after

20 mm crack extension. For all fatigue tests, the modified Paris’ law was used to fit the

experimentally determined crack growth rate per cycle (da/dN) versus the applied

maximum strain energy release rate (SERR, Gmax). The delamination growth rate

decreased rapidly with decreasing applied SERR, which gave rise to high exponents of

the Modified Paris’ law for Mode I, Mode II, and mixed Mode I/II fatigue tests, with the

highest in mixed Mode I/II. To assess the capability of commercial finite element

software in solving delamination growth problems, a crack propagation analysis of the

DCB specimen was carried out using the virtual crack closure technique (VCCT) for

Abaqus and Abaqus/Standard V6.7. Preliminary results showed good agreement of load

versus displacement behavior between the finite element analysis (FEA) and

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experimental results. However, the crack front shape predicted by FEA did not agree well

with experimental results.

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TABLE OF CONTENTS

LIST OF FIGURES ..................................................................................................... ix

LIST OF TABLES.......................................................................................................xvii

ACKNOWLEDGEMENTS.........................................................................................xix

Chapter 1 Introduction ................................................................................................1

1.1 Background..............................................................................................1 1.2 Introduction to Fracture Mechanics of Composite Materials..................2 1.3 Problem Statement and Research Objectives ..........................................6

Chapter 2 Literature Review.......................................................................................9

2.1 Mode I Interlaminar Fracture Toughness (IFT) Testing .........................9 2.1.1 Geometry and analysis of the Double Cantilever Beam (DCB)

specimen..........................................................................................9 2.1.1.1 Modified Beam Theory (MBT) method...............................11 2.1.1.2 Compliance Calibration (CC) method (Berry’s Method)...14 2.1.1.3 Modified Compliance Calibration (MCC) method.............15 2.1.1.4 Elastic Foundation Model (EFM) method..........................16

2.1.2 Experimental aspects of the DCB test...........................................18 2.1.2.1 Initial defect type.................................................................18 2.1.2.2 Definition of critical point for crack onset..........................21 2.1.2.3 Method of loading/unloading..............................................23 2.1.2.4 Crack resistance curve and fiber bridging...........................24

2.2 Mode II Interlaminar Fracture Toughness (IFT) Testing ........................26 2.2.1 Geometry and analysis of the End Notched Flexure (ENF)

specimen........................................................................................26 2.2.1.1 Classical Plate Theory (CPT) (Davidson et al. 1996) ........28 2.2.1.2 Beam Theory (BT) with shear correction (Carlsson et al.

1986) ........................................................................................31 2.2.1.3 Compliance Calibration (CC) method (Davidson et al.

1996) ........................................................................................33 2.2.2 Experimental aspects of the ENF test............................................34 2.2.3 Other test configurations of Mode II IFT testing ..........................36

2.3 The Mixed Mode I/II Interlaminar Fracture Toughness Testing.............38 2.3.1 Geometry and analysis of Single Leg Bending (SLB) specimen..39

2.3.1.1 Classical Plate Theory (CPT) Method (Davidson and Sundararaman 1996) ................................................................40

2.3.1.2 Beam Theory based analyses ..............................................42

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2.3.1.3 Compliance Calibration (CC) method (Polaha et al. 1996) ........................................................................................43

2.3.2 Experimental aspects of the SLB test............................................44 2.3.3 Other configurations of mixed Mode I/II IFT test ........................44 2.3.4 Mixed mode delamination failure criterion...................................46

2.4 Interlaminar Fracture Toughness Test Under Cyclic loading .................47 2.4.1 Fatigue delamination growth models ............................................51

2.4.1.1 Pure Mode I or II.................................................................51 2.4.1.2 Mixed-mode ........................................................................52

2.4.2 IFT fatigue test methods................................................................53 2.4.2.1 Fatigue threshold strain energy release rate

determination ...........................................................................53 2.4.2.2 Fatigue delamination growth test ........................................54

2.5 Finite Element Modeling of Crack Propagation......................................55 2.5.1 The Virtual Crack Closure Technique (VCCT) ............................56

2.5.1.1 The virtual crack closure technique formulation ................56 2.5.1.2 Crack growth criterion for VCCT.......................................60

2.5.2 Cohesive element (Abaqus 2007) .................................................62 2.5.2.1 Elastic behavior of the cohesive element ............................63 2.5.2.2 Damage initiation criteria of the cohesive element.............65 2.5.2.3 Damage evolution criteria of the cohesive element ............65

2.6 Preview of the Following Chapters .........................................................66

Chapter 3 The Mode I Interlaminar Fracture Toughness Testing...............................67

3.1 Material, Specimen and Test Configuration............................................67 3.2 The Mode I Quasi-static IFT Testing ......................................................69

3.2.1 Mode I quasi-static test method ....................................................70 3.2.2 Mode I quasi-static test results ......................................................72

3.2.2.1 Load-displacement curves...................................................73 3.2.2.2 Compliance calibration .......................................................78 3.2.2.3 GIc onset values ...................................................................81 3.2.2.4 Mode I resistance curve ......................................................84 3.2.2.5 Discussion of special issues ................................................85

3.3 The Mode I Fatigue IFT Testing .............................................................92 3.3.1 Mode I fatigue test method............................................................92 3.3.2 Mode I fatigue test results .............................................................95

3.3.2.1 Compliance calibration .......................................................95 3.3.2.2 Crack growth.......................................................................98 3.3.2.3 Crack growth rate (da/dN) vs. maximum SERR (GImax)

plots..........................................................................................99

Chapter 4 The Mode II Interlaminar Fracture Toughness Testing .............................102

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4.1 Material, Specimen and Test Configuration............................................102 4.2 The Mode II Quasi-static IFT Testing.....................................................107

4.2.1 Mode II quasi-static test method ...................................................107 4.2.2 Mode II quasi-static test results.....................................................111

4.2.2.1 Load-displacement curves...................................................111 4.2.2.2 Compliance calibration .......................................................114 4.2.2.3 GIIc onset values ..................................................................120 4.2.2.4 A short Mode II fracture resistance curve...........................122 4.2.2.5 Special issues on the ENF test ............................................126

4.3 The Mode II Fatigue IFT Testing............................................................129 4.3.1 Mode II fatigue test method ..........................................................129 4.3.2 Mode II fatigue test results............................................................132

4.3.2.1 Crack growth.......................................................................132 4.3.2.2 da/dN - GIImax plots..............................................................138

Chapter 5 Mixed-mode I/II Interlaminar Fracture Toughness Testing.......................141

5.1 Material, Specimen and Test Configuration............................................141 5.2 The Mixed-mode I/II Quasi-static IFT Testing .......................................144

5.2.1 Mixed-mode I/II quasi-static test method .....................................145 5.2.2 Mixed-mode I/II test results ..........................................................147

5.2.2.1 Load-displacement curves...................................................147 5.2.2.2 Compliance calibration .......................................................149 5.2.2.3 Mixed-mode I/II critical strain energy release rate (GTc)

onset value ...............................................................................152 5.2.2.4 Mixed-mode fracture resistance curve ................................156

5.3 Mixed-mode I/II Fatigue IFT Testing .....................................................158 5.3.1 Mixed-mode fatigue test method...................................................158 5.3.2 Mixed-mode fatigue test results ....................................................160

5.3.2.1 Crack growth.......................................................................160 5.3.2.2 Crack growth rate (da/dN) vs. maximum SERR (GTmax)

plots..........................................................................................162

Chapter 6 Finite Element Modeling of Crack Propagation in DCB Specimens.........165

6.1 Two-dimensional Modeling of the DCB Specimen ................................166 6.1.1 Geometry, loading and boundary conditions of 2D models..........166 6.1.2 Modeling techniques for 2D models .............................................166 6.1.3 Meshing of 2D models ..................................................................170 6.1.4 Results of 2D models ....................................................................171

6.1.4.1 Crack propagation ...............................................................171 6.1.4.2 Load-displacement curve ....................................................173 6.1.4.3 Stress distribution................................................................174

6.2 Three-dimensional Modeling of the DCB Specimen ..............................176

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6.2.1 Geometry, loading and boundary conditions of 3D models..........177 6.2.2 Modeling techniques for 3D models .............................................177 6.2.3 Meshing of 3D models ..................................................................181 6.2.4 Results of 3D Models ....................................................................183

6.2.4.1 Crack front shape observation.............................................183 6.2.4.2 Load vs. displacement curve behavior ................................188 6.2.4.3 Stress distribution................................................................190

6.3 Conclusions .............................................................................................191

Chapter 7 Conclusions and Recommendations...........................................................193

7.1 Mode I Interlaminar Fracture Toughness Characterization.....................193 7.2 Mode II Interlaminar Fracture Toughness Characterization ...................194 7.3 Mixed Mode I/II Interlaminar Fracture Toughness Characterization .....195 7.4 Preliminary Finite Element Modeling Results ........................................196

Bibliography ................................................................................................................198

Appendix A IFT Test Results From Literature...........................................................207

Appendix B Additional Specimen Information and Test Results...............................213

1. Dimensions of Specimens .........................................................................213 2. Mode I Quasi-static Fracture Toughness at Crack Onset ..........................216 3. Mode I Fatigue Crack Growth Rate vs. Maximum SERR Plots ...............217 4. Mode II Quasi-static Fracture Toughness at Crack Onset.........................221 5. Mode II Fatigue Crack Growth Rate vs. Maximum SERR Plots..............222 6. Mixed Mode I/II Quasi-static Fracture Toughness at Crack Onset...........225 7. Mode II Fatigue Crack Growth Rate vs. Maximum SERR Plots..............226

Appendix C Non-Technical Abstract..........................................................................229

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LIST OF FIGURES

Fig. 1-1: Load and displacement on a crack body .......................................................4

Fig. 1-2: The basic fracture modes. .............................................................................5

Fig. 2-1: Geometry of the ASTM D5528 Double Cantilever Beam (DCB) specimen ...............................................................................................................10

Fig. 2-2: A schematic of the DCB specimen (side view) ............................................12

Fig. 2-3: Determination of Δ in the modified beam method (MBT) ...........................13

Fig. 2-4: Determination of n in compliance calibration method..................................15

Fig. 2-5: Determination of α1 in the Modified Compliance Calibration (MCC) method ..................................................................................................................16

Fig. 2-6: An elastic foundation model of the DCB specimen, based on (Ozdil and Carlsson 1999) ......................................................................................................17

Fig. 2-7: A schematic of microscopic view of longitudinal section of specimen near the end of starter film....................................................................................20

Fig. 2-8: Crack initiation definitions............................................................................21

Fig. 2-9: Typical load-displacement curves for a DCB specimen with multiple loading/unloading cycles ......................................................................................23

Fig. 2-10: Typical delamination resistance curve (R curve) from a DCB test (ASTM D5528-01 2002) ......................................................................................24

Fig. 2-11: Picture showing fiber bridging (Mode I loading) .......................................25

Fig. 2-12: End-notched flexure test schematic ............................................................27

Fig. 2-13: A schematic of the ENF specimen (side view) ...........................................29

Fig. 2-14: Schematic of an ENF specimen subject to three-point bending .................32

Fig. 2-15: Typical load-displacement curves of an ENF test.......................................34

Fig. 2-16: ELS test configuration, based on (O'Brien 1998a) ....................................37

Fig. 2-17: The SLB specimen geometry ......................................................................39

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Fig. 2-18: A schematic of the SLB specimen geometry and detailed notation............40

Fig. 2-19: A schematic of MMB test configuration (Kim and Mayer 2003)...............45

Fig. 2-20: A schematic of the mixed mode end load split (MMELS) specimen, based on (Szekrényes and József 2006)................................................................45

Fig. 2-21: A schematic of the crack lap shear (CLS) specimen, based on (Tracy et al. 2003) ................................................................................................................45

Fig. 2-22: A total fatigue life model of composite materials, based on (Shivakumar et al. 2006).......................................................................................49

Fig. 2-23: The Mode I delamination onset SERR versus number of cycles, based on (ASTM standard D6115-97 1997(R2004)) .....................................................54

Fig. 2-24: Crack closure for VCCT, based on (Krueger 2002) ...................................57

Fig. 2-25: VCCT for four-node 2D element (plane strain or plane stress), based on (Krueger 2002)......................................................................................................58

Fig. 2-26: VCCT for eight-node solid element (3D view) (Krueger 2002).................59

Fig. 2-27: VCCT for eight-node solid element (top view) (Krueger 2002).................60

Fig. 2-28: Traction-separation response of cohesive element, based on (Abaqus 2007) .....................................................................................................................63

Fig. 3-1: Diagram of the DCB panel............................................................................68

Fig. 3-2: DCB specimen geometry and notation .........................................................69

Fig. 3-3: Photograph of DCB test set up......................................................................70

Fig. 3-4: Constant fracture toughness after certain length of crack extension.............74

Fig. 3-5: Load vs. displacement plot (Specimen 1-1)..................................................75

Fig. 3-6: Load vs. displacement plot (Specimen 1-4)..................................................76

Fig. 3-7: A C1/3 versus a plot for Mode I tests.............................................................79

Fig. 3-8: An a/2h versus (bC)1/3 plot for Mode I DCB tests........................................80

Fig. 3-9: Load vs. displacement curve -- small load drop occurred at crack onset (Specimen 1-1)......................................................................................................82

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Fig. 3-10: Load vs. displacement curve -- large load drop occurred at crack onset (Specimen 1-3)......................................................................................................82

Fig. 3-11: GIc onset values ...........................................................................................83

Fig. 3-12: Mode I IFT resistance curves for Specimen 1-2 .........................................84

Fig. 3-13: Overall Mode I IFT resistance curve for five DCB specimens, by MCC method ..................................................................................................................85

Fig. 3-14: Crack surfaces of a quasi-static DCB test specimen...................................87

Fig. 3-15: A schematic of crack propagation...............................................................88

Fig. 3-16: Fracture resistance curve for Specimen 1-4, with crack length calculated by compliance calibration by MCC method........................................90

Fig. 3-17: Fiber bridging observed through long distance microscope .......................91

Fig. 3-18: A schematic showing the loading and unloading procedures for the Mode I precrack test .............................................................................................93

Fig. 3-19: Mode I maximum SERR reduction as crack grows for a displacement controlled DCB fatigue test ..................................................................................95

Fig. 3-20: Crack growth by different methods (Specimen 4-2)...................................97

Fig. 3-21: Crack growth by various methods...............................................................99

Fig. 3-22: da/dN vs. GImax plots for four DCB fatigue specimens (with crack growth by visual measurement)............................................................................100

Fig. 3-23: da/dN vs. GImax plots for four DCB specimens (with crack growth calculated by compliance calibration) ..................................................................101

Fig. 4-1: The ENF and SLB panel diagram (Panel B).................................................103

Fig. 4-2: The ENF panel diagram (Panel AA).............................................................104

Fig. 4-3: A schematic of ENF specimen geometry and test configuration..................105

Fig. 4-4: Schematic of ENF un-precracked test configuration A ................................106

Fig. 4-5: Schematic of ENF precracked test configuration B......................................106

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Fig. 4-6: A photograph of the ENF test setup with un-precracked test configuration.........................................................................................................108

Fig. 4-7: A photograph of the ENF test setup with precracked test configuration ......109

Fig. 4-8: Markings on ENF specimen edge for compliance calibration ......................110

Fig. 4-9: A representative load vs. displacement plot for ENF crack onset test (Specimen 2-2, un-precracked, a0/L ≈ 0.5)...........................................................111

Fig. 4-10: A load vs. displacement plot for ENF crack onset test (Specimen 2-5, precracked, test configuration B)..........................................................................112

Fig. 4-11: A quasi-stable load vs. displacement plot for ENF crack onset test (Specimen 2-6, precracked) ..................................................................................113

Fig. 4-12: Construction method for the 5% compliance offset line.............................114

Fig. 4-14: C(8bh3) vs. a3 plots for all ENF specimens.................................................116

Fig. 4-15: A C vs. a plot for compliance calibration, by Eq. (2.39) ............................118

Fig. 4-16: GIIc onset values ..........................................................................................122

Fig. 4-17: Mode II resistance curve (Specimen 2-6, precracked)................................125

Fig. 4-18: A load vs. displacement plot for ENF test with constant G curves shown....................................................................................................................126

Fig. 4-19: A schematic of load vs. displacement curves for different states of crack growth .........................................................................................................127

Fig. 4-20: Plot of crack growth rate in a quasi-static ENF test versus normalized crack length...........................................................................................................129

Fig. 4-21: Maximum Mode II SERR (GIImax) vs. normalized crack length plot for an ENF test with fixed displacement amplitude (F1 is a factor related to initial crack length and maximum opening displacement) .............................................131

Fig. 4-22: Crack growth for ENF fatigue specimens...................................................133

Fig. 4-23: A microscopic view of a specimen edge while the specimen was in different loadings (Specimen 5-3) ........................................................................134

Fig. 4-24: Photographs of fracture surfaces.................................................................137

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Fig. 4-25: Crack growth rate against Mode II maximum SERR plot, with crack growth calculated by compliance calibration .......................................................139

Fig. 4-26: Crack growth rate against Mode II maximum SERR plot, with crack growth measured visually .....................................................................................140

Fig. 5-1: The SLB panel diagram ................................................................................142

Fig. 5-2: A schematic of SLB specimen geometry and test configuration ..................143

Fig. 5-3: SLB test configuration ..................................................................................144

Fig. 5-4: A photograph of SLB test set up ...................................................................145

Fig. 5-5: Markings on SLB specimen edge ................................................................146

Fig. 5-6: Load vs. displacement curve for SLB quasi-static tests................................147

Fig. 5-7: Load-displacement plot near the critical onset point (Specimen 3-2)...........148

Fig. 5-8: C vs. a plot for all SLB specimens...............................................................150

Fig. 5-9: C(8bh3) vs. a3 plot for all SLB specimens ...................................................150

Fig. 5-10: GTc values determined from SLB tests........................................................153

Fig. 5-11: The Reeder’s linear mixed mode failure locus and test data ......................155

Fig. 5-12: The B-K Law failure locus and test data.....................................................155

Fig. 5-13: Load vs. displacement plot for Specimen 3-4 with constant GTR curves shown....................................................................................................................157

Fig. 5-14: Fracture resistance curves for SLB specimens............................................158

Fig. 5-15: Crack growth of SLB fatigue specimens ....................................................160

Fig. 5-16: A sketch of opening crack...........................................................................161

Fig. 5-17: Fracture surfaces of a SLB specimen..........................................................162

Fig. 5-18: A da/dN vs. Gmax plot for SLB specimens (crack length by compliance calibration method)...............................................................................................163

Fig. 5-19: A da/dN vs. Gmax plot for SLB specimens (crack length by visual measurement)........................................................................................................164

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Fig. 6-1: Finite element analysis and the IFT test........................................................165

Fig. 6-2: Geometry and boundary conditions of 2-dimensional DCB models ............166

Fig. 6-3: Idealized Mode I fracture toughness with crack extension for Specimen 1-2 .........................................................................................................................168

Fig. 6-4: Mesh configuration for 2D Model #1 and #3................................................171

Fig. 6-5: Mesh configuration for 2D Model #2 ...........................................................171

Fig. 6-6: Crack growth versus opening displacement from test data and 2D finite element modeling..................................................................................................172

Fig. 6-7: Load vs. displacement curves from test data and 2D finite element modeling ...............................................................................................................173

Fig. 6-8: Contour plot of the stress in the longitudinal direction (σxx), in MPa, around the crack tip (2D Model #2) .....................................................................175

Fig. 6-9: Contour plot of the stress in the thickness direction (σ22), in MPa, around the crack tip (2D Model #2)..................................................................................176

Fig. 6-10: Geometry of 3D models of the DCB specimen ..........................................177

Fig. 6-11: Mesh configuration for 3D Model #1 .........................................................181



Fig. 6-14: Crack surfaces of a quasi-static DCB test specimen...................................183

Fig. 6-14: Crack fronts predicted by 3D Model #1......................................................184



Fig. 6-17: Load vs. displacement curves from 2D and 3D finite element analyses ....189

Fig. 6-18: Comparison of damping energy to total strain energy for 3D models........190

Fig. 6-19: Distribution of longitudinal stress (σ11) in a 3D DCB specimen ................191

Fig. A-1: Lay-up of specimens, based on(Polaha et al. 1996).....................................209

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Fig. B-1: da/dN - GImax plot for Specimen 4-2 (Crack growth measured visually.) ....217

Fig. B-2: da/dN - GImax plot for Specimen 4-2 (Crack growth calculated by compliance calibration.) .......................................................................................217







Fig. B-9: da/dN - GIImax plot for Specimen 5-3 (Crack growth rate was calculated with crack growth by compliance calibration. The points excluded from linear fit are near the crack growth arrest domain.)..............................................222

Fig. B-10: da/dN - GIImax plot for Specimen 5-3 (Crack growth rate was calculated with crack growth by visual measurement. The points excluded from linear fit are taken near the beginning of the test.) .........................................................222

Fig. B-11: da/dN - GIImax plot for Specimen 5-4 (Crack growth rate was calculated with crack growth by compliance calibration. The points excluded from linear fit are taken near the beginning of the test.) ...............................................223

Fig. B-12: da/dN - GIImax plot for Specimen 5-5 (Crack growth rate was calculated with crack growth by compliance calibration. The points excluded from linear fit are taken near the beginning of the test.) ...............................................224

Fig. B-13: da/dN - GIImax plot for Specimen 5-5 (Crack growth rate was calculated with crack growth by visual measurement. The points excluded from linear fit are taken near the beginning of the test.) .........................................................224

Fig. B-14: da/dN - Gmax plot for Specimen 6-1 (Crack growth was calculated by compliance calibration.) .......................................................................................226

Fig. B-15: da/dN-Gmax plot for Specimen 6-1 (Crack growth was measured visually.) ...............................................................................................................226

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Fig. B-16: da/dN-Gmax plot for Specimen 6-2 (Crack growth was calculated by compliance calibration.) .......................................................................................227

Fig. B-17: da/dN - Gmax plot for Specimen 6-2 (Crack growth was measured visually.) ...............................................................................................................227

Fig. B-18: da/dN-Gmax plot for Specimen 6-3 (Crack growth was calculated by compliance calibration.) .......................................................................................228

Fig. B-19: da/dN-Gmax plot for Specimen 6-3 (Crack growth was measured visually.) ...............................................................................................................228

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LIST OF TABLES

Table 2-1: DCB specimen dimensions required in ASTM D5528 ..............................10

Table 2-2: Some specific ENF specimen geometries from the literature ....................27

Table 3-1: Parameters of compliance calibration by the MBT method.......................80

Table 3-2: Parameters of compliance calibration by MCC method ............................80

Table 3-3: Mode I Fatigue test parameters ..................................................................94

Table 3-4: Crack growth prediction approaches for DCB fatigue tests.......................96

Table 4-1: Dimensions for ENF configuration A (un-precracked specimen)..............106

Table 4-2: Dimensions for ENF test configuration B (precracked specimen)............107

Table 4-3: Parameters A and B, determined by CC 1) for ENF specimens .................117

Table 4-4: Parameters C0, C1, C2, and C3, determined by CC 2) for ENF specimens..............................................................................................................119

Table 4-5: Testing parameters for ENF fatigue tests ...................................................130

Table 5-1: SLB test configuration dimensions ............................................................144

Table 5-2: Coefficients determined for compliance calibration by Eq. (5.1) .............151

Table 5-3: Coefficients determined for compliance calibration by Eq. (5.2) ..............152

Table 5-4: A summary of Mode I, Mode II, and Mixed-mode I/II fracture toughness at crack onset .......................................................................................154

Table 5-5: Testing parameters for SLB fatigue tests ...................................................159

Table A-1: DCB tests results from literature ...............................................................207

Table A-2: ENF test results from literature ................................................................210

Table A-3: Fatigue test results from literature............................................................211

Table B-1: Dimensions of DCB specimens used for quasi-static tests........................213

Table B-2: Dimensions of DCB specimens used for fatigue tests...............................213

xviii

Table B-3: Dimensions of ENF specimen (un-precracked, tested in quasi-static tests)......................................................................................................................214

Table B-4: Dimensions of ENF specimen (precracked, tested in quasi-static tests) ...214

Table B-5: Dimensions of ENF specimen (precracked, tested in fatigue tests) ..........214

Table B-6: Dimension of SLB quasi-static specimens ................................................215

Table B-7: Dimension of SLB fatigue specimens .......................................................215

Table B-8: A summary of Mode I quasi-static test results ..........................................216

Table B-9: A summary of Mode II quasi-static test results .........................................221

Table B-10: A summary of Mode I quasi-static test results ........................................225

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ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my advisor, Professor Charles E.

Bakis, for his invaluable guidance and encouragement along the course of my study at

Pennsylvania State University. Also, I would like to thank my lab colleagues for

assistance.

I am also grateful to Dr. Lesieutre and Dr. Koudela for serving on my committee

and taking the time to read this thesis and provide helpful advice.

Sincere thanks goes to my parents for their support and encouragement during my

study in Pennsylvania State University.

The financial support of this work is provided by Rhombus Consultants Group. I

would like to thank them for providing me the opportunity to work on this project.

1

Chapter 1

Introduction

1.1 Background

Composite materials are finding increased applications in many engineering fields.

In the aerospace industry, the use of composite materials in commercial and military

aircrafts has increased greatly over the last 20 years. For example, the usage of

composites has evolved from less than 5 percent of the structural weight in the Boeing

737 and 747 to about 50 percent in the Boeing 787 Dreamliner. By contrast, aluminum

will comprise only 12 percent of the Boeing 787 aircraft. According to Chambers (2003),

while the use of composites is less than 10% of the structural weight in the F14 fighter it

has increased to about 40% of the structural weight in the F22 fighter. In the ship-

building industry, thick-section glass and carbon fiber composites and sandwich

composites are more widely incorporated into ship structures than before to fulfill special

demands, such as light-weight, good insulation, low maintenance cost, and resistance to

corrosion (Daniel and Ishai 2006). In civil structures, such as bridges, the use of carbon

fiber-reinforced plastics (CFRP) has extended from only internal reinforcement in

structures to both internal and external reinforcement. In addition to structures, wide

applications of composite materials can be found in automobile parts and frames, trucks,

sports equipments, etc. Among these composite materials, the laminated fiber-reinforced

composite material is becoming commonplace in primary load bearing members of

structures and machines as a high performance material. Compared to metallic materials,

laminated fiber-reinforced materials can provide not only the primary advantage of high

strength to weight ratio, but also offer extra benefits of low coefficient of thermal

expansion (CTE), good resistance to corrosion, low maintenance cost, and low pollution.

These advantages together make the laminated fiber-reinforced material an attractive

candidate for modern structure and machine design.

2

Despite all their advantages, laminated fiber-reinforced composite materials have

certain disadvantages as well. The basic building block of a laminated fiber-reinforced

material is a lamina (ply). Within a lamina, high-strength fibers are combined with a

light-weight matrix. By selecting a bonded sequence of laminae with various orientations

of principal material directions and/or different materials, a wide range of mechanical

properties of laminated composites can be designed according to needs. With this special

material design methodology, the material anisotropy and heterogeneity are greatly

increased. As a result, many issues that do not exist for isotropic and homogeneous

materials arise when using laminated composites. One of the more prominent issues is the

large number of potentially interacting damage modes, such as fiber breakage,

intralaminar matrix cracking, fiber/matrix debonding, fiber pull-out, and delamination

(Daniel and Ishai 2006). Among these damage modes, delamination is one of the most

important and least understood. Delamination is especially important because it can cause

a catastrophic loss of compressive strength. Small delaminations cannot always be

detected by nondestructive inspections, and can potentially grow to unstable

configurations due to in-service loads. In some cases, the structure or machine can fail

catastrophically without any external warning signs. This failure scenario makes

delamination a major obstacle to wider utilization of advanced composite materials in

structures and machinery. Therefore, it is crucial to develop a better understanding of

delamination.

1.2 Introduction to Fracture Mechanics of Composite Materials

The linear elastic fracture mechanics (LEFM) approach has become a generally

accepted practice for the characterization of laminated composites behavior. The LEFM

approach was first developed for brittle homogeneous materials, such as certain types of

steel and ceramics, which exhibit no or small scale plastic deformation before fracture.

However, some of the theories within LEFM have been found to be applicable to predict

delamination in laminated composite materials.

3

From the LEFM approach, delamination growth kinetics are predicted by

comparing the crack driving force or energy release rate (ERR), G, to a critical value of

ERR, Gc. The crack driving force (G) is related to the loadings, geometry of the crack

body, and constraints, while the critical value is a property of the material, which is called

fracture toughness.

The energy approach for fracture was proposed by Irwin in 1956 (Irwin 1956).

The energy release rate (ERR, G) is a measure of the energy available for an increment of

crack extension. Generally, in mathematical form it is defined by Eq. (1.1), (Anderson

2005).

The potential of an elastic body, Π, is defined by Eq. (1.2).

where U is the strain energy stored in the body, F is the work done by external forces,

and A is area of crack surface.

Another expression for ERR derived from the definition and commonly used for

fracture toughness test specimens is given by Eq. (1.3) (Anderson 2005).

where C=u/P is the compliance of the crack body; b is width of the body, and a is the

crack length, as shown in Figure 1-1.

dGdAΠ

= − (1.1)

U FΠ = − (1.2)

2

2P dCG

b da= (1.3)

4

For a delamination growing under a constant displacement, which is usually the

case for a displacement controlled interlaminar fracture toughness (IFT) test, no work is

done by external forces and the energy released is only from the elastic strain energy in

the cracked body (U). Hence, the Strain Energy Release Rate (SERR), defined by

Eq. (1.4) , is commonly used as a measure of energy available for crack extension in an

IFT test specimen (ASTM D5528-01 2002).

Fracture Toughness is determined as the value of critical stress intensity factor (Kc)

or critical strain energy release rate (Gc) of a material. For an isotropic and homogeneous

material, the stress field in the vicinity of the crack tip can be characterized by a single

parameter, stress intensity factor (K). However, for composite materials, the stress field

near the crack tip is more complicated and sometimes shows oscillatory behavior. Thus

another parameter, ERR (G), based on energy released during the creation of new

surfaces, is more commonly used for composite materials. By comparing the energy

release rate, G, to the critical value, Gc, of a material, one can predict this material’s

capability to resist crack growth. In the JIS standard (JIS K 7086 1993), interlaminar

fracture toughness (IFT) is defined as the critical value of the energy required to create a

unit area of an interlaminar crack.

ba da

Pu

ba da

Pu

Fig. 1-1: Load and displacement on a crack body

1 dUGb da

= − (1.4)

5

According to the relative displacement crack surfaces, three modes of fracture

existing for laminated composite materials (Figure 1-2) are defined as (ASTM standard

D5528-01 2002):

crack opening mode (Mode I) — fracture mode in which the delamination faces

open away from each other and no relative crack face sliding occurs; the critical value of

G for delamination growth in this mode is named Mode I interlaminar fracture toughness

(IFT), denoted by, GIc;

crack sliding mode (Mode II)—fracture mode in which the delamination faces

slide over each other in the direction of delamination growth and no relative crack face

opening occurs (in the direction normal to the leading edge); Mode II interlaminar

fracture toughness is denoted by, GIIc;

crack tearing mode (Mode III)—fracture mode in which the delamination faces

slide over each other in the direction parallel to the leading edge.

The mixed mode fracture toughness, Gc, is defined as the critical value of strain

energy release rate, G, for delamination growth in mixed-mode.

Many international organizations and groups are actively involved in carrying out

research on interlaminar fracture toughness (IFT) testing. Some of them are: i) ASTM

Subcommittee D30.06; ii) the Polymers & Composites Task Group of the European

Structural Integrity Group (ESIG, formerly the European Group on Fracture); iii) the

Japan High Polymer Center (JHPC). In 1990, an international round robin exercise was

v

uw

x y

z

Figure 1-2: The basic fracture modes

6

carried out under collaboration between the ASTM group, the ESIG group and the

Japanese Industrial Standards (JIS) group. International cooperation at a government

level, the Versailles Project on Advanced Materials & Standards (VAMAS), has also

dealt with IFT since 1986.

In fatigue, the delamination process of a composite material is often characterized

in terms of the relationship between the crack growth rate per cycle and the applied range

of ERR on a log-log plot. Generally, there are three regimes of crack growth when

plotting crack growth rate against ERR on a log-log scale, with the first regime showing a

fast decelerating growth rate with increased ERR, the second regime a linear relationship

between crack growth rate and ERR, and the third regime markedly increasing crack

growth rate with increasing ERR. The behavior of laminated materials in the second

regime is of great interest for developing a damage tolerance design approach. Applying

such an approach, a designer needs to take into consideration how fast an existing crack

can grow while the structure is in service. The Paris law ( ( )nGBdNda

Δ= ) and modified

Paris’ law ( ( )nGBdNda

max= ) between crack growth rate per cycle (da/dN) and the applied

ERR (G) are commonly used to characterize crack growth in the second regime .

1.3 Problem Statement and Research Objectives

Laminated fiber-reinforced composites are well-known to be susceptible to

delamination. Advanced stress analysis tools and validated failure criteria are needed to

determine conditions for crack onset and growth under design loads, and also the

delamination growth rate in fatigue loading. To apply the fracture mechanics based

criterion with confidence, standard test methods are needed to characterize the fracture

resistance as a generic property of a material and a database of such properties for

commonly used composite materials needs to be established for design and material

application purposes.

7

Based on previous research by the many IFT research organizations and

individual researchers, several test configurations have been proposed for studying

delamination fracture behavior under various kinds of loading. For Mode I tests, the

double-cantilever-beam (DCB) specimen is the most common specimen. Several national

and an international standard (ASD-STAN prEN 6033 1995; ASTM D5528-01 2002;

ISO 15024 2002; JIS K 7086 1993) already exist for the quasi-static Mode I IFT testing.

For Mode II, the end-notched-flexure (ENF) specimen is one of the most popular ones

but the unstable crack growth issue exists for the common configurations, and hence

other test configurations were proposed. A Japanese and an European standard (ASD-

STAN prEN 6034 1995; JIS K 7086 1993) are available for the quasi-static Mode II

testing using the ENF specimen. For mixed Mode I/II, more test configurations exist, e.g.

mixed mode bending (MMB), mixed mode end load split (MMELS), cracked lap shear

(CLS), and single-leg-bending (SLB). An ASTM standard using the MMB specimen

exists for the quasi-static mixed mode I/II IFT testing. Even though experts have more or

less agreed on a few test specimens for quasi-static Modes I, II, and I/II IFT testing, there

are still many practical issues being debated, and some should be treated differently

according to the material system of interest. For fatigue testing, no standard exists for

characterizing the crack growth law in the stable crack growth region, where the

relationship between the delamination growth rate and strain energy release rate (SERR)

follows a power law.

In addition to experimental methods, numerical methods, such as finite element

modeling, are valuable tools for validating the LEFM approach in predicting

delamination. However, performing crack propagation analysis using finite element

method is numerically intensive. Further, the ERR based failure criteria are not yet

available in most commercial finite element analysis software. Although a numerical

technique, the virtual crack closure technique (VCCT), has been proposed, research on

applying this technique to advanced structures is preliminary. Extensive efforts are

needed to develop efficient modeling techniques for crack propagation analysis.

With the need for better understanding of delamination, and establishing the IFT

property database for structure design, this thesis research aims to:

8

1) assess the interlaminar fracture toughness testing methods found in the

literature for static and fatigue testing of laminated composite materials;

2) characterize the Mode I, Mode II, and mixed-mode I/II interlaminar fracture

and interlaminar fatigue properties of a proprietary carbon/epoxy composite material

system using test methods from the literature or, if necessary, test methods tailored to suit

the current material;

3) critically evaluate the test results and make recommendations for future

investigations;

4) preliminarily assess the capability of the current commercial finite element

software to perform crack propagation analysis.

9

Chapter 2

Literature Review

In this chapter, the widely-used interlaminar fracture toughness (IFT) testing

specimens are introduced based on a wide survey of the literature. Experimental aspects,

data reduction and analysis methods regarding IFT testing are reviewed.

2.1 Mode I Interlaminar Fracture Toughness (IFT) Testing

To-date, the Double Cantilever Beam (DCB) specimen is the dominant Mode I

testing specimen. The specimen is easy to manufacture and a pure Mode I stress state at

the tip of the crack is easy to create with commonly available mechanical testing

equipment. Several analytical approaches for interpreting the results of DCB testing are

summarized in this section. Different views on typical practical issues involving Mode I

testing, e.g. the initial defect type, critical point definition, fiber bridging, etc., are

presented.

2.1.1 Geometry and analysis of the Double Cantilever Beam (DCB) specimen

Some national and international standards are available for reference regarding

Mode I IFT testing. ASTM standard (ASTM D5528-01 2002) using a DCB specimen for

Mode I IFT testing was first published in 1994. A DCB specimen is also used in Japanese

standard (JIS K 7086 1993) and in European standard (ASD-STAN preEN 6033 1995).

An ISO (ISO 15024 2002) standard is also available.

The geometry of the DCB specimen described in (ASTM D5528-01 2002) is

shown in Figure 2-1.

10

Specimen dimensions required in ASTM standard D5528 (ASTM D5528-01 2002) are listed in Table 2-1.

The ASTM DCB specimen shown in Figure 2-1 consists of a rectangular cross

section and uniform thickness and width. Opening forces are applied to the specimen by

piano hinges or loading blocks bonded to one end of the specimen. The ends of the

specimen are opened by controlling either the opening displacement or the crosshead

movement. The load, crosshead (or crack opening) displacement and delamination length

are recorded continuously during the test. Load versus displacement plots are generated

during or after the test. The delamination length is determined as the distance from the

loading line to the front of delamination. The initiation and propagation value of GIc can

be calculated based on these recorded data using beam theory and so-called compliance

calibration methods.

Several analytical models can be used to reduce the data for a DCB test. The three

data reduction methods recommended in (ASTM D5528-01 2002) are: (1) Modified

a0

l

2h

ba0

l

2h

b

Piano hinge Aluminum

block

Insert Insert

a0

l

2h

ba0

l

2h

b

Piano hinge Aluminum

block

Insert Insert

Fig. 2-1: Geometry of the ASTM D5528 Double Cantilever Beam (DCB) specimen

Table 2-1: DCB specimen dimensions required in ASTM D5528

Length Width Thickness Initial crack length

L, mm b, mm 2h, mm a0, mm

≥125 (5 in.) 20-25 (0.8-1.0in.) 3-5(0.12-0.2 in.) ~50 (2 in.)

11

Beam Theory (MBT) method, (2) Compliance Calibration (CC) method, and (3)

Modified Compliance Calibration (MCC) method. The main difference in these methods

lies in the different models used for the compliance vs. crack length relation. Both the

MBT and MCC methods assume that compliance (C) is related to crack length (a) by a

third order polynomial, but in different forms. According to the simplest of beam theories,

the delaminated portion of the DCB specimen can be treated as two Euler-Bernoulli

cantilever beams each built in at the crack tip and point-loaded near the opposite end.

Some inaccuracy can result from the perfect-built-in-end assumption, and the simple

beam model can be improved by considering two effects: i) end displacement and

rotation of the cantilever beams, and ii) transverse shear deformation of the delaminated

beams. The MBT method assumes that the end rotation effect can be incorporated simply

by adding a correction factor to the crack length and this correction factor is constant for

a certain material. Another beam analysis model, developed by Ozdil and Carlsson (Ozdil

and Carlsson 1999) and used for angle ply ([±θ]n) laminates, assumes that the un-

delaminated portions of the beams are bonded by an elastic medium of infinitesimal

thickness which allows beam end displacement and rotation. The effect of shear

deformation in a DCB specimen is typically small (<1%) and can be neglected, according

to one analytical study (Ozdil and Carlsson 1999). The most commonly used analytical

models for improving elementary beam theory in the analysis of DCB test results are

presented in more detail next.

2.1.1.1 Modified Beam Theory (MBT) method

This method assumes that the cracked part of the specimen consisting of an upper

arm and lower arm can be represented as two cantilever beams built-in at a distance Δ in

front of the crack tip (i.e., embedded in the un-delaminated part of the specimen), as

shown in Figure 2-2. (ASTM D5528-01 2002)

12

From Euler-Bernoulli beam theory, the predicted compliance can be written as a

third order polynomial, by Eq. (2.1).

Applying the SERR-compliance relation given by Eq. (2.2),

the Mode I SERR is then given by Eq. (2.3).

where P and δ are the load and total crack opening displacement as shown in Figure 2-2,

respectively. b is the specimen width, and a is the length of the delamination from

loading points to crack tip. C is the compliance and defined by δ/P. The parameter Δ is

determined experimentally by the abscissa intercept of a straight line fit by least squares

to C1/3 versus a data as shown in Figure 2-3.

P

P

δ 2h

a Δ

P, δ/2

P, δ/2a + Δ

Fig. 2-2: A schematic of the DCB specimen (side view)

( )33 Δ+= amC (2.1)

dadC

bPG2

2

= (2.2)

( )Δ+=abPGI 2

3 δ (2.3)

13

Eq. (2.3) can be modified to include the correction for the loading block

reinforcement effect and the large displacement effect. In this case, the expression for

SERR is shown in Eq. (2.4) (de Morais and Pereira 2007).

where, ΔI is a correction factor for crack tip displacement and rotation, F a correction

factor for large displacements, and N a correction factor for loading block effects.

Expressions for F and N are given in (ASTM D5528-01 2002).

Assume the experimentally obtained compliance C vs. a data points are

approximated by the linear polynomial of Eq. (2.5),

where, A0 and A1 are parameters obtained from a linear least-square fit to C/N vs. a data.

Then the end correction factor, ΔI, is given by Eq. (2.6),

and an estimate of the flexural modulus is given by Eq. (2.7),

a

C1/3

Δ0

Fig. 2-3: Determination of Δ in the modified beam method (MBT)

( )Ic3

2 I

P FGb a N

δ=

+ Δ (2.4)

( ) aAANC 103/1/ += (2.5)

1

0

AA

I =Δ (2.6)

14

On the other hand, if the material properties are known, the correction factor ΔI

can be predicted from the theoretical solution for a beam on an elastic foundation by

Williams (1989), Eq. (2.8),

where,

and E3 and G13 are the through-thickness Young’s modulus and transverse shear

modulus, respectively.

The end correction factor ΔI calculated by Eq. (2.6) and the estimated flexural

modulus E1 calculated by Eq. (2.7) are commonly found to be greater than the beam

theory prediction by Eq. (2.8) and the experimentally measured flexural modulus,

respectively (de Morais and Pereira 2007).

2.1.1.2 Compliance Calibration (CC) method (Berry’s Method)

From the simple beam theory model, the deflection, v, of the tip of a cantilever

beam with length a may be written as Eq. (2.10),

For a double cantilever beam, the end deflection is assumed to be related to the load by

Eq. (2.11). (ASTM D5528-01 2002; Davies et al. 1990)

( )311

8hAb

E = (2.7)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

Γ+Γ

−=Δ2

13

1

123

11GEhI (2.8)

13

3118.1G

EE=Γ (2.9)

EIPav3

3

= (2.10)

nRPa=2/δ (2.11)

15

Hence, the predicted compliance is in the form of Eq. (2.12).

The strain energy release rate GI may thus be written as Eq. (2.13).

where n is experimentally determined by the slope of straight line that has been fit to

lnC– lna data as shown in Figure 2-4. The logarithm of R is the intersection of the best-fit

straight line with the ordinate.

2.1.1.3 Modified Compliance Calibration (MCC) method

In the MCC method, the normalized compliance (bC) is related to the normalized

delamination length, a/2h, by Eq. (2.14) (JIS K 7086 1993).

where α1 can be determined experimentally by the slope of a straight line that has been fit

to a plot of a/2h versus (bC)1/3 data as shown in Figure 2-5.

nRaC = (2.12)

2 1

I 2 2

nnP nP RaGba bδ −

= = (2.13)

Fig. 2-4: Determination of n in compliance calibration method

03/1

1 )(2/ αα += bCha (2.14)

16

The expression for SERR by MCC method is given by Eq. (2.15).

An estimate of the coefficient α1 and α0 is given by Eq. (2.16) and Eq. (2.17),

respectively,

where, Γ is given by Eq. (2.9). E1 and G13 are the longitudinal Young’s modulus and

transverse shear modulus, respectively.

2.1.1.4 Elastic Foundation Model (EFM) method

Unlike the three methods illustrated above, where compliance versus crack length

relations are determined experimentally, the compliance of a DCB specimen as a function

of crack length is theoretically predicted based on a Euler-Bernoulli beam and Winkler

Fitted line

α1

(bC)1/3

a/2h

Fig. 2-5: Determination of α1 in the Modified Compliance Calibration (MCC) method

)2(23

21

3/22

hbCPGI α

= (2.15)

4

31

1

E=α (2.16)

2/12

13

10 1

23112

1⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

Γ+Γ

−−=GE

α (2.17)

17

elastic foundation model analysis. This model was derived by Ozdil and Carlsson (Ozdil

and Carlsson 1999) for angle ply laminates. Compliance and SERR of a DCB specimen

can be calculated based on material properties. Hence, by conducting this analysis, the

compliance calibration procedures can be avoided if the material properties are known.

The mathematical model of half of the DCB specimen is shown in Figure 2-6.

The governing differential equation for the DCB is given by Eq. (2.18),

where H(x) and λ is given by Eq. (2.19) and Eq. (2.20), respectively.

and

where ke is the foundation modulus and Ex is the effective bending modulus of the

laminate. Applying boundary conditions and solving the governing equation, the

expression for the deflection of the beam was obtained. Thus, the compliance of the beam

is given by Eq. (2.21).

a

δ/2h

x

zP

ke Figure 2-6: An elastic foundation model of the DCB specimen, based on (Ozdil and Carlsson 1999)

( ) ( ) ( ) 04 44

4

=+ xwxHdx

xwd λ (2.18)

( )⎩⎨⎧

<>

=0,00,1

xx

xH (2.19)

34 3

bhEk

x

e=λ (2.20)

18

and the expression for the Mode I SERR is given by Eq. (2.22),

where Ez is the Young’s modulus in the through-thickness direction. The term outside the

parentheses of Eq. (2.22) is identical to the SERR expression for a DCB specimen given

by the beam theory, which is strictly valid only for infinitely long crack lengths. The term

inside the parentheses is the elastic foundation correction factor for an angle-ply laminate

that accounts for the finite crack length by incorporating the through-thickness elasticity

of the uncracked region of the specimen.

2.1.2 Experimental aspects of the DCB test

Although the standard method for the Mode I IFT testing has been established

since the early 1990s, some experimental aspects are still not completely resolved. For

some issues, different recommendations could be made for different materials. Among

them, the issues of most concern for researchers are the effects of initial defect types,

crack initiation definition, effect of fiber bridging, etc.

2.1.2.1 Initial defect type

Many studies have shown the defect type may have a great influence on

experimental results (Brunner 2000). Generally, three approaches have been used for the

initial defect:

a. A thin film insert is put between two lamina during the layup process of a

laminated material and no precracking is performed prior to testing;

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

4/32/124/13

39.022.192.118

z

x

z

x

z

x

x EE

ah

EE

ah

EE

ah

ha

bEC (2.21)

1/ 4 1/ 222 2

I 2 3

12 1 1.28 0.41x x

x z z

E EP a h hGE b h a E a E

⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥= + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

(2.22)

19

b. A thin film insert is put between two lamina during layup, and a wedge is

driven into the initial crack to grow the delamination beyond the starter film tip,

or a brief Mode I test is performed before the test;

c. A thin film insert is put between two lamina during layup, and the test

specimen is precracked by performing a brief Mode II test to grow the

delamination beyond the starter film tip.

Method (a) is obviously the easiest method since no precracking is involved. The insert

film allows the formation of a macroscopically well defined shape of the starter crack tip

as well (usually straight across the specimen width, at least on a macroscopic scale).

However, one problem associated with this type of starter crack is that a resin pocket can

form beyond the tip of the insert film, as shown schematically in Figure 2-7. This

disturbance of local material distribution may become negligible as the thickness of the

insert film decreases. However, the limiting value of the thickness of the insert films

depends on the material type ((Hojo et al. 1995), (Murri and Martin 1993)). Early round-

robin tests have indicated that insert thickness has to be less than or equal to 13 μm to

yield results not affected by insert thickness in Mode I DCB testing, at least for

unidirectional carbon fiber reinforced polymer laminates. For glass fiber laminates, due

to the larger diameter of some types of glass fibers compared with carbon fibers, thicker

film inserts might still be suitable without significantly affecting the fracture toughness

measurement. Another disadvantage of using insert film as starter crack noticed by Hojo

et al. (1995) is that crack propagation from the starter film was unstable in brittle epoxy

unidirectional laminates without mechanically-induced precracks.

20

Precracking by mechanical loading (Methods b and c) has several advantages and

disadvantages. Precracking in Mode I has been shown to yield conservative values of

fracture toughness compared with insert films for a number of carbon and glass fiber

composites, in particular for Mode II testing (Hojo et al. 1995; Murri and Martin 1993).

Mode I or wedge precracking has been shown to yield starter tip shapes similar to those

observed during Mode I testing. However, precracking may form a damage zone ahead of

the crack tip and irregular crack front shape as well. An additional issue is that fibers tend

to bridge between interfaces with crack extension, which results in a higher initiation

value than an insert film defect type. Even so, it is suggested by the Japanese Standards

Association (JIS K 7086 1993) that a precrack should be done for a DCB specimen

before testing.

The ASTM DCB test (ASTM D5528-01 2002) suggests to use an insert film with

a thickness less than 13 μm for the initial defect of a DCB specimen so that an initiation

GIc value based on the insert film could be obtained. Then after the crack had initiated

and grown for 3-5 mm, the specimen shall be unloaded and reloaded so that a GIc value

based on Mode I precrack can be obtained as well.

End of insert film

Resin pocket Figure 2-7: A schematic of microscopic view of longitudinal section of specimen near the end of starter film

21

2.1.2.2 Definition of critical point for crack onset

Four approaches have been developed to define the critical crack onset point as


(a)

(b)

Figure 2-8: Crack initiation definitions

22

a. Non-linearity of the load-displacement plot (NL)

The nonlinear critical point is the first point that deviates from linearity on the

load vs. displacement curve. Work done at EMPA (Swiss Federal Laboratory for

Material Testing and Research) in Switzerland using in-situ micro focus radiography

has proved that the NL point is close to the real onset of delamination (Kalbermatten

et al. 1992). However, this definition is somewhat ambiguous, for the closer one looks

at the plots, the earlier nonlinear behavior can be detected. Additionally, nonlinear

behavior may be due to other reasons, for example, yielding of material at the crack

tip or local instead of global crack growth.

b. Maximum load point

The maximum load point is the critical point where load reaches its maximum

value during for the whole loading process.

c. 5% compliance increase (5% offset)

The 5% compliance offset point, is obtained by constructing a straight line whose

slope has decreased by 5% of the initial slope of load-displacement curve. Then the

critical point is intersection of this straight line and load-displacement curve. The

advantage of this definition is that it is less scattered than the NL method, but it

produces higher values.

d. Visual observation (VIS)

The visual observation point is the point where crack onset is observed visually.

The difficulty with this method is that it is operator dependent.

e. Acoustic emission (AE), and strain gages

These methods are relatively complicated compared to other methods and

require additional specialized equipment. They will not be discussed further in this

investigation.

Two typically observed types of load vs. displacement curves are shown in

Figure 2-8. For the first type, Figure 2-8 (a), it has a slight nonlinear behavior before the

load reaches the maximum, and the maximum load was reached before the compliance of

specimen increased by 5%. In this case, as suggested by the Japanese Standards

23

Association (JIS K 7086 1993), the maximum load point is usually taken as the critical

point. For the second type of curve, Figure 2-8 (b), a relatively greater degree of

nonlinearity is observed before the maximum load and the 5% compliance offset critical

point occurs before the maximum load. In the second case, the percentage compliance

increase criterion is commonly used for critical point definition for crack onset.

2.1.2.3 Method of loading/unloading

For a typical DCB test, the total amount of crack growth from the starter crack tip

is about 50-60 mm. Two kinds of loading/unloading methods have been widely used,

according to the literature. The first method includes only one or two loading/unloading

cycles, with the short loading/unloading cycle creating a crack extension of about 3-5 mm

and a second long loading/unloading creating a crack extension of about 50 mm. In the

second method, several loading/unloading cycles were conducted. A typical load against

opening displacement curve of loading and unloading procedures is shown in Figure 2-9.

The purpose of the loading/reloading process is to obtain a more accurate compliance

value for the crack lengths at which the specimen is unloaded.

Fig. 2-9: Typical load-displacement curves for a DCB specimen with multiple loading/unloading cycles

24

2.1.2.4 Crack resistance curve and fiber bridging

A typical critical strain energy release rate, GIc, against crack length, a, curve is


As shown in Figure 2-10, a resistance type of fracture behavior is commonly

found for laminated composite material (Brunner et al. 2006; Russell 1988). This

behavior commonly features a monotonically increasing GIc in the first few millimeters

(i.e., 3-5 mm) of Mode I crack extension, and then stabilizes with further crack growth.

According to many studies (Brunner et al. 2006; de Morais and Pereira 2007; Russell

1988), fiber bridging is the primary reason for this history dependent behavior. As the

crack starts to extend, fibers begin to pull out of the delaminated surfaces immediately

ahead of the crack tip, and gradually a zone of fibers bridging the gap is developed

between the delamination faces directly behind the crack tip, as shown in Figure 2-11. As

the delamination extends and the crack opening displacement increases, these bridged

fibers continue to pull out and in some cases break due to applied tensile stress.

Consequently, the bridging fibers divert some of the available strain energy away from

the crack tip.

Deviation from linearityVisual onset5% offsetPropagation

Crack length, a, mm

GIc, J/m2

Fig. 2-10: Typical delamination resistance curve (R curve) from a DCB test (ASTM D5528-01 2002)

25

As a consequence of the fiber bridging mechanism, the measured compliance of

the DCB specimen is less than that predicted by modified beam theory for the visually

observed crack length (Brunner et al. 2006). Additionally, if dense fiber bridging occurs

on the edge of the specimen, it can be difficult to visually locate the tip of the crack,

which results in unreliable crack length measurements.

Other effects that may be involved in the shape of a resistance curve include:

matrix cracking, tow cracking, multiple delamination, tow bridging and tow breaking in

the case of woven fiber composites, etc. The implication of R-curve behavior for

structural designers is that the initiation fracture toughness must be measured as well as

the increase of fracture toughness as a delamination grows if a damage tolerance criterion

is part of the design process.

Fig. 2-11: Picture showing fiber bridging (Mode I loading)

26

2.2 Mode II Interlaminar Fracture Toughness (IFT) Testing

For Mode II interlaminar fracture toughness testing, at least six types of test

specimens are available. They are the end notched flexure (ENF) specimen (Carlsson et

al. 1986; Ozdil et al. 1998), the stabilized end notched flexure (SENF) specimen (Tanaka

et al. 1995), the four point bend end notched flexure (4ENF) specimen (Davies et al. 2005;

Schuecker and Davidson 2000), the end load split (ELS) specimen (Hashemi et al. 1990a;

Hashemi et al. 1990b; Wang and Vu-Khanh 1996), the over notched flexure (ONF)

specimen (Szekrényes and Uj 2005; Tanaka et al. 1998; Wang and Qiao 2003), and the

tapered end-notched flexure (TENF) specimen (Qiao et al. 2004; Wang and Qiao 2003).

Among them, the ENF specimen has been most widely used for Mode II IFT testing of

fiber composites because of its simple test configuration. Many analytical models have

been developed for ENF specimens, and many discussions on experimental testing can be

found in the literature as well. No international standard exists for a Mode II Interlaminar

Fracture Toughness (IFT) test.

2.2.1 Geometry and analysis of the End Notched Flexure (ENF) specimen

In an ENF test, the specimen is placed in a three point bending fixture which

consists of two supporting points and one loading point. The generic ENF specimen

geometry is shown in Figure 2-12. Specific ENF specimen geometries found in the

literature are shown in Table 2-2. Two national standards were established in Japan (JIS

K 7086 1993) and Europe (ASD-STAN prEN 6034 1995) for Mode II testing using ENF

specimens.

27

In the ENF test, load is applied to the top of specimen at the mid-span in a

displacement controlled mode. The load, center point displacement and crack length are

measured and recorded during the test. Load versus displacement plots are generated

Fig. 2-12: End-notched flexure test schematic

Table 2-2: Some specific ENF specimen geometries from the literature

Material Half span length

Specimen thickness

Specimen width

Initial crack length

Reference

L, mm 2h, mm b, mm a0, mm

E-glass/epoxy 65 5 20 35 (Ducept et al. 1997)

Carbon/epoxy 40 4.2 20 N/A (Davies et al. 1990)

E-glass/polyester 50 4.4 20 25 (Ozdil et al. 1998)

Carbon/epoxy 50 3 20 20 (Tanaka et al. 1995)

28

during or after the test. The initiation and propagation GIIc can be evaluated based on

these recorded data using classical plate theory, beam theory and compliance calibration

methods.

Several analysis methods can be applied to an ENF test, including classical plate

theory, beam theories with shear correction, and the compliance calibration method.

These analysis methods are described next.

2.2.1.1 Classical Plate Theory (CPT) (Davidson et al. 1996)

The assumptions made in applying classical plate theory to the ENF test are:

i) the crack is at the midplane of the laminate;

ii) the laminate and two sublaminates created by the crack are specially

orthotropic and symmetric (A16 = A26 = Bij = D16 = D26 = 0), but not necessarily

homogeneous or unidirectional;

iii) as a result of the material symmetry, residual thermal stresses do not affect

the strain energy release rate;

iv) a multidirectionally reinforced ENF specimen may exhibit non-classical

bending behavior; however, with bending rigidities of the cracked and uncracked regions

chosen appropriately, 2D plate theory is assumed to be able to accurately predict

deflections and SERRs.

The ENF specimen is treated as a bending plate consisting of cracked and

uncracked regions as shown in Figure 2-13. Two extreme cases of constraint conditions

are considered by Davidson et al., with the first one referred to “generalized plane stress

condition” and the second one “generalized plane strain condition” (Davidson et al. 1996).

For generalized plane stress condition, it is valid for specimens where the cracked and

uncracked regions are long and narrow. In this case, the moments are zero on the edges of

the laminate, and the effective bending rigidity per unit width, D, of either region can be

expressed as Eq. (2.23),

222

1211 / DDDD −= (2.23)

29

where D11, D12, and D22 come from the D matrix of classical lamination theory (Daniel

and Ishai 2006).

The generalized plane strain condition is valid for specimens where the cracked and

uncracked regions are short and wide. In this case, the bending rigidity per unit width of

either region, D, can be expressed as Eq. (2.24).

where D11 comes from the D matrix of classical laminated plate theory (Daniel and Ishai

2006) for the region of interest in the specimen. The difference between the plane stress

and plane strain flexural rigidities may be characterized by the ratio, Dc, defined by

Eq. (2.25).

Small values of Dc indicate that the effect of finite width is relatively unimportant, while

the larger value of Dc indicate a greater influence of the three-dimensional effect as

explained in the following Section 2.2.2.

Using the moment-curvature relationship, the equation for compliance may be

derived using classical plate theory as Eq. (2.26),

L

P, δ

a

Uncracked region

Cracked region

L

Fig. 2-13: A schematic of the ENF specimen (side view)

11DD = (2.24)

2211

212

DDD

Dc = (2.25)

( )u

CPT

bDRaLC

2424 33 −+

= (2.26)

30

where, b is the specimen width, C is the compliance, defined as center point deflection

divided by load, R is the ratio of flexural rigidities of the uncracked and cracked regions,

such that cu DDR /= , and Dc is the bending rigidity of either of the identical uncracked

arms.

Substituting the expression for compliance into Eq. (2.2), expression for the

SERR of the ENF specimen may be obtained as Eq. (2.27)

or

For a homogeneous specimen, which could be a unidirectional or an angle-ply specimen

with a uniform through-the-thickness fiber distribution, for example, R = 8.

For a homogeneous orthotropic specimen configured as a narrow beam, the bending

rigidity may be written as,

where, E11 is the Young’s modulus in the x direction, I is entire specimen’s area moment

of inertia about its neutral axis, and h is the specimen’s half thickness. As a result, the

compliance of such a specimen is,

Substituting R = 8 and Eq. (2.30) into Eq. (2.27), yields the expression of SERR by

Eq. (2.31).

Normally, the accuracy of Eq. (2.28) is higher than Eq. (2.31) since material

properties are not required by Eq. (2.28). The accuracy of Eq. (2.28) in reducing data from

( )u

CPTII Db

RaPG 2

22

162−

= (2.27)

( )( )[ ]242

23

33

2

−+−

=RaL

RbaPGCPT

IIδ (2.28)

3/2/ 31111 hEbIEDu == (2.29)

311

33

832

bhEaLC HP

ENF+

=σ (2.30)

3211

22

169

hbEaPGCPT

II = (2.31)

31

an ENF test depends upon whether the experimental compliance, C, and its derivative,

∂C/∂a, obey the functional form of Eq. (2.26) (and its derivative).

For ENF tests with test fixtures that use loading pins, R will generally be less than

theory predicts due to the finite diameter of the pins ((Davidson et al. 1996) (O'Brien et

al. 1989)). For ENF fixtures that use “knife edge supports,” R may be either greater or

less than theory, and depends upon the fixture itself as well as the manufacturing process

for the specimen.

2.2.1.2 Beam Theory (BT) with shear correction (Carlsson et al. 1986)

Russell and Street presented a beam theory solution for the SERR of the ENF

specimen in 1985 (Russell and Street 1985). Based on their work, Carlsson et al.

(Carlsson et al. 1986) extended the beam theory solution to include the effect of shear

deformation on SERR.

This beam theory solution with shear correction can be applied to the ENF

specimen with the common geometry, for which delamination occurs at mid-thickness, as

shown schematically in Figure 2-14. The center point deflection can be treated

decomposed into three parts as by Eq. (2.32),

where, ΔAB, ΔBC, and ΔCD are defined in Figure 2-14. The beams BC and CD are treated

as cantilever beams and it is assumed that the cross-section at C does not warp because of

the line of load introduction is an approximate line of symmetry. Expressions for

deflection ΔBC, by Eq. (2.33), and ΔCD, by Eq. (2.34), may be obtained from Timoshenko

beam theory.

2CDBCAB Δ+Δ+Δ

=δ (2.32)

[ ] ( )⎥⎦

⎤⎢⎣

⎡ −+

++=Δ

bhGaLP

bhEaaLLP

BC13

311

323

3.08

32 (2.33)

32

where, E11 and G13 are the longitudinal Young’s modulus and the transverse shear

modulus of the composite laminate, respectively.

For the delaminated region, AB, the displacement, ΔAB, has two components, one

due to the bending and shearing deformations of the two delaminated beams and the other

due to the rotation of the cross-section at point B. The ends of the parallel beams are here

assumed to be allowed to deform freely under the action of shearing stress. Assuming

that each beam of the delaminated region carries the same load, P/4, the displacement

component due to bending and shearing, ΔAB,1, is obtained as Eq. (2.35) .

The rotation of the cross section at B is expressed by the slope of the cross section

with respect to the horizontal axis. The displacement component due to the slope, ΔAB,2, is

⎥⎦

⎤⎢⎣

⎡+=Δ

bhGPL

bhEPL

CD13

311

3

3.04

(2.34)

L L

a

2h

PA

P/2 P/2

B CD

LL

δ ΔCD

ΔAB

ΔBCA B C D

a

(a) Loading of the ENF specimen

(b) Deflection of the ENF specimen Fig. 2-14: Schematic of an ENF specimen subject to three-point bending

⎥⎦

⎤⎢⎣

⎡+=Δ

132

12

31

3

1, 831

GaEh

bhEPa

AB (2.35)

33

simply the slope at B multiplied by the length of the delaminated region, which yields the

following expression of Eq. (2.36).

The final expression for compliance and SERR of an ENF specimen incorporating

shear deformation is Eq. (2.37) and Eq. (2.38), respectively.

For geometries and material properties commonly in use (unidirectional lay-ups), the

error in the toughness value calculation induced by neglecting shear deformation is less

than 10% according to the analysis conducted by Carlsson et al. (Carlsson et al. 1986).

Ozdil et al. (Ozdil et al. 1998) pointed out that compliance and SERR determined for

relatively thin unidirectional and angle-ply laminate ENF specimens were in good

agreement with a previous classical plate theory formulation. For thicker laminates,

however, effects of shear deformation on the compliance of ENF specimen become

significant.

2.2.1.3 Compliance Calibration (CC) method (Davidson et al. 1996)

At any given crack length, the compliance can be obtained from the slope of a

linear least-squares curve fit of the deflection versus load data. A cubic polynomial,

shown in Eq. (2.39), can be used to fit the compliance vs. crack length data.

Other expressions typically used to fit compliance data may exclude the first and second

order terms, such as, Eq. (2.40).

⎥⎦

⎤⎢⎣

⎡+−≅Δ

13

12

323

12, 8

3G

EahaaLbhEP

AB (2.36)

( )( ) ⎥

⎦

⎤⎢⎣

⎡++

++

=13

331

2

31

33

, 329.02.121

832

GaLEhaL

bhEaLC SHENF (2.37)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=

2

13

132

1

22

, 2.0116

9ah

GE

hbEPaG SHENF (2.38)

2 30 1 2 3C C C a C a C a= + + + (2.39)

34

Substituting expression for C given by Eq. (2.2) yields,

2.2.2 Experimental aspects of the ENF test

Similar to Mode I testing, the issues of initial defect type and critical point

definition exist as well for Mode II testing with ENF specimens. However, in many cases

the definition of crack onset critical point is quite clear for an ENF test because the crack

growth immediately after crack onset is often unstable. This instability is manifested as a

large load drop as shown in a typical stroke-controlled test result as shown in Figure 2-15.

Other issues concerning ENF testing include the effect of initial defect type, the

effect of friction between the crack surfaces, and three dimensional effects. They are

described below.

(a) Effect of initial defect type for Mode II testing

330 aCCC += (2.40)

22

1 2 3( 2 3 )2IIcPG C C a C ab

= + + (2.41)

Fig. 2-15: Typical load-displacement curves of an ENF test

35

The apparent inconsistency between GIIc values measured by growing the

crack front from a thin midplane insert and GIIc values measured by growing the

crack from an initial shear precrack was noticed by many researchers (Murri and

Martin 1993; O'Brien 1998a). For Mode I testing, a characteristic toughness value

may be obtained as the insert thickness is decreased because the GIc value reaches an

asymptotic limit as the insert film thickness decreases. However, according to some

researchers (Murri and Martin 1993; O'Brien 1998a), GIIc values decrease

continuously with decreased insert thickness, never reaching an asymptotic value that

may be considered a characteristic of the composite material. Furthermore, GIIc

values measured from the insert are sometimes greater than and sometimes less than

GIIc measured from a shear precrack. In most cases, the precrack value is lower than

the insert value. However, for two materials (S2/SP250 glass/epoxy and IM7/F3900

graphite/epoxy) the reverse has been shown to be true (O'Brien 1998a).

(b) Effect of friction between fracture surfaces

Friction between the crack surfaces is inevitable for Mode II testing.

According to finite element analyses by Davies (Davies et al. 1996), this effect is not

significant for typical specimen geometries. However, it is recommended that a

PTFE film be placed between crack faces to reduce the influence of friction (JIS K

7086 1993). According to Carlsson et al. (1986), an analysis shows that, for

reasonable values of coefficients of friction (0.25-0.5), the error in GII induced by

neglecting friction is only 2-4%.

(c) Three-dimensional effects

Davidson et al. (1995a) noticed that significant 3-D effects, in the form of

concentrations in the Mode II SERR and occurrence of a Mode III SERR component

at specimen’s free edges, may occur during an ENF test with multidirectional

specimen. To a lesser extent, this 3-D effect has also been shown to occur in

unidirectional specimens.

36

According to Davidson et al. (Davidson et al. 1995a), 3-D effects may occur

in ENF geometries due to two causes: 1) the bending-stretching or bending-twisting

behavior of sublaminates; 2) the finite width of an ENF specimen. As a result of it,

the SERR and/or the mode ratio varies across the delamination front. The bending-

stretching or bending-twisting behavior may cause local gradients in the SERR and

mode ratio across the delamination front, and may be avoided by choosing specimen

geometries in which the cracked and uncracked sections are individually specially

orthotropic. A finite-width specimen will exhibit a local increase in the Mode II

SERR at each of its free edges. Also, due primarily to the anticlastic curvature of the

individual sublaminates, a local Mode III SERR component will occur at the

specimen’s edges. This effect is small in unidirectional laminates, but significant for

some other layups with a large value of Dc.

2.2.3 Other test configurations of Mode II IFT testing

One disadvantage of using the ENF specimen is that the crack growth is unstable

until the crack grows to a certain point where the crack length to half span ratio, a/L, is

about 0.7 (Davies et al. 2001). Hence a fracture resistance versus crack length curve (R

curve) may be not generated from an ENF test. Many other test configurations are able to

develop stable crack growth, such as: SENF, 4ENF, ELS, but these have their own

advantages and disadvantages which are reviewed next.

Stabilized End Notched Flexure (SENF) test uses the same test configuration as the

ENF test. But rather than using a fixed cross-head displacement rate as in the ENF test,

the SENF test is controlled by a feedback signal from the test so that the crack growth

can be stabilized. The feedback can be the crack shear displacement (CSD) measured

between the top and bottom halves of the specimen at the insert end, or a coordinate

conversion control (Davies et al. 1998; Tanaka et al. 1995) signal which is a function of

the load and displacement output. This configuration allows generation of a R curve, but

requires a relatively more sophisticated test setup and procedure.

37

The 4ENF test uses a four-point-bending fixture rather than the more typical 3 point

fixture. It was shown by Paris et al. (2003) and Zile and Tamusz (2005) that crack growth

is stable in this test configuration and a resistance curve similar to Mode I can be

obtained. Schuecker and Davidson (2000) show that if the crack length and the

compliance are measured accurately, the ENF and 4ENF give the same fracture

toughness values. The primary difficulty with the 4ENF test is the nonlinear effect, such

as, effect of friction. As a result, the values of GIIc measured by 4ENF tests are often

significantly larger than those obtained from 3ENF tests. Values of GIIc obtained by the

4ENF test has been found to be 9-60% higher than those obtained by the 3ENF test

(Martin and Davidson 1999; Martin et al. 1998; Schuecker and Davidson 2000).

The End Load Split (ELS) specimen is shown in Figure 2-16. The condition for

stable crack growth for the ELS test specimen is a/L > 0.55, whereas for the ENF

specimen it is a/L > 0.7 (Davies et al. 2001). Hence, the ELS test specimen offers an

advantage over the ENF specimen in promoting stable propagation. This enables several

values of compliance and critical SERR to be obtained from one specimen at different

crack lengths and allows an experimental compliance calibration to be performed, as in a

DCB test. The disadvantage of the ELS test is that it requires a complicated fixture to

ensure a clamped end condition.

P

2h

aL

Fig. 2-16: ELS test configuration, based on (O'Brien 1998a)

38

2.3 The Mixed Mode I/II Interlaminar Fracture Toughness Testing

For mixed mode I/II testing, many specimens have been developed, such as the

mixed mode bending (MMB) specimen, the mixed mode ELS specimen, the single leg

bending (SLB) specimen, the mixed-mode flexure (MMF) specimen, the cracked-lap

shear (CLS) specimen, the asymmetric double cantilever beam (ADCB) specimen. These

test methods are compared in (Szekrényes and Uj 2007). ASTM has a standard for mixed

mode testing by MMB unidirectional specimen (ASTM standard D6671 2006). Among

these specimens, the MMB specimen is likely to be the most widely used specimen for

mixed mode testing, for the wide range of mode mixity it can create. The single leg

bending specimen is also an attractive candidate since it requires a relatively simple

loading apparatus and test configuration. Davidson et al. performed extensive theoretical

and experimental investigations using SLB coupons (Davidson et al. 1997; Davidson et

al. 2000; Davidson and Koudela 1999; Davidson and Sundararaman 1996; Davidson et

al. 1995c). Later, based on this SLB specimen, Tracy et al. proposed a single-leg four

point bend (SLFPB) geometry (Tracy et al. 2003), and Szekrényes et al. introduced the

over leg bending (OLB) specimen (Szekrényes and Uj 2007). One advantage of the SLB

specimen over the MMB specimen is that the compliance calibration method can be used

for the SLB specimen to obtain more accurate critical SERR values. In a MMB test,

SERR calculation involves some material properties. Large discrepancies commonly

exist between theoretically predicted and as-manufactured values of material properties,

such as bending rigidities, shear modulus and through thickness modulus (especially for

multidirectional laminates). As a result, the MMB test is not suitable for multidirectional

lay-ups. Considering all these factors, the single leg bending (SLB) specimen was

selected as the mixed mode IFT testing specimen in the current investigation. Hence,

further details on the SLB specimen and associated experimental aspects are described

next.

39

2.3.1 Geometry and analysis of Single Leg Bending (SLB) specimen

The SLB specimen has a beam type of geometry, as shown in Figure 2-17. A

portion of the lower arm is removed from the cracked end in an SLB specimen, so that

the reaction force act directly to the upper arm. The removed part can be used as a

“spacer” under the cut end, so that the upper surface of the specimen is horizontal before

loading. A standard three point bending fixture is used in this test. The load, center point

displacement and crack length are measured and recorded during the test. Compliance

and critical strain energy release rate, GTc, can be calculated by compliance calibration

and classical plate theory methods, and mode ratios can be obtained by finite element

analysis incorporating a crack tip element.

Fig. 2-17: The SLB specimen geometry

40

2.3.1.1 Classical Plate Theory (CPT) Method (Davidson and Sundararaman 1996)

A schematic of side view of the SLB specimen geometry and detailed notation is

shown in Figure 2-18. By classical plate theory, the expressions of compliance, C,

derived by Davidson (Davidson and Sundararaman 1996) for the SLB specimen is given

by Eq. (2.42),

and total critical strain energy release rate GTC is given by Eq. (2.43).

Alternatively, the expression for total SERR in the case where bending rigidity per unit

width of the uncracked region (Du) is unknown, is given by Eq. (2.44).

where,

C = the compliance, defined by center-point deflection divided by the load;

a = crack length;

b = specimen width;

L = half span length as indicated in Figure 2-18;

( )3 32 112

CPTSLB

u

L a RC

bD+ −

= (2.42)

( )2 2

, 2

18

CPTTc SLB

u

P a RG

b D−

= (2.43)

t1

La

L

t2

P, δUncracked region

Cracked region

Top plate

Fig. 2-18: A schematic of the SLB specimen geometry and detailed notation

( )⎥⎦⎤

⎢⎣

⎡−+

−=

121

23

33

2

, RaLR

baPGCPT

SLBTcδ (2.44)

41

R = the ratio of the flexural rigidity of the uncracked region, Du, to the bending

rigidity of the top plate of the cracked region, Dc, as given by Eq. (2.45).

The bending rigidity of the uncracked region can be estimated by considering the

appropriate constraint condition as described in Section 2.2.1.1.

For a unidirectional specimen configured so that delamination occurs at the

midplane, the ratio R is 8. In this case, the expression for the total SERR can be

simplified to Eq. (2.46).

Mode decomposition can be achieved by a finite element approach using a crack

tip element (Davidson et al. 1995a; Davidson et al. 1995b). A crack tip element analysis

predicts the SERR components as Eq. (2.47) and Eq. (2.48).

where Nc and Mc are the concentrated crack-tip force and moment per unit width,

respectively. These parameters are given by Eq. (2.49) and Eq. (2.50), respectively.

where Mp is the moment resultant at the crack tip, i.e.,

The constants c1, c2, a12, and a22 are related to the plate rigidities as indicated by Eq. (2.52)

and Eq. (2.53), Eq. (2.55), Eq. (2.54):

/u cR D D= (2.45)

33, 727

23

aLbPGCPT

SLBTc +=

δ (2.46)

2

22 c

IMc

G = (2.47)

2

21 c

IINc

G = (2.48)

pc MaN 12= (2.49)

Pc Mata

M ⎟⎠⎞

⎜⎝⎛ −= 22

112

2 (2.50)

bPaM P 2

= (2.51)

42

where, t1 and t2 are the thickness of top plate and bottom plate respectively. A1 and D1 are

terms from the stiffness matrix of classical lamination theory (Daniel and Ishai 2006). A′1,

D′ and D′1 are terms from the compliance matrix of classical lamination theory.

Specifically, A1 is the in-plane rigidity of the top plate; A′1 is its in-plane compliance; D1

is the flexural rigidity of the top plate; D′1 is the flexural compliance of the top plate, and

D′ is the flexural compliance of the uncracked region.

2.3.1.2 Beam Theory based analyses

Szekrényes and Uj have applied Euler-Bernoulli and Timoshenko beam theories

in conjunction with a Winkler-Pasternak Foundation analysis, a Saint-Venant effect

analysis at the crack tip, and a crack tip shear deformation analysis to determine the

compliance and SERR of an SLB specimen with an initial crack at the midplane

(Szekrényes and Uj 2007). The result for compliance is shown in Eq. (2.56),

while the Mode I and II components of SERR are given by Eq. (2.57) and Eq. (2.58),

respectively,

2'

'2211

11tD

Ac += (2.52)

12 '2Dc = (2.53)

2' 21

12tDA

a−

= (2.54)

'122 DDa = (2.55)

1/ 2 23 3 311 11

3 311 13 11 33 33

1/ 2 1/ 4 1/ 222 311 11 11

2 311 33 11 33 33

7 2 2 0.98 0.438 8 8

1 3 5.07 8.584 8

SLB E Ea L a L a h hCbh E bhkG bh E a E a E

E E Ea a h hbh E E bh E a E a Eπ

⎡ ⎤⎛ ⎞ ⎛ ⎞+ + ⎛ ⎞ ⎛ ⎞⎢ ⎥= + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎢+ + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣

3/ 4311

33

2.08 Eha E

⎤⎛ ⎞⎛ ⎞ ⎥+ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎥⎝ ⎠ ⎦

(2.56)

43

where,

k = 5/6, shear correction factor;

E11 = flexural modulus;

E33 = through thickness modulus;

G13 = longitudinal shear modulus.

2.3.1.3 Compliance Calibration (CC) method (Polaha et al. 1996)

A compliance versus crack length curve can be obtained by fitting a cubic

polynomial in the form of Eq. (2.59), to the compliance versus crack length data obtained

experimentally.

Other expressions typically used to fit compliance data may exclude the first and second

order term, as indicated by Eq. (2.60).

Therefore, the critical SERR can be obtained using either Eq. (2.61) or Eq. (2.62).

1/ 4 1/ 222 211 11

2 311 33 33

1/ 2 211 11

33 33

12 1 0.85 0.7116

0.32 0.1

SLBI

E EP a h hGb h E a E a E

E Eh ha E a E

⎡ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎢= + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎝ ⎠ ⎝ ⎠⎣

⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎥+ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎥⎝ ⎠ ⎝ ⎠⎦

(2.57)

1/ 2 22 211 11

2 311 33 33

9 1 0.22 0.04816

SLBII

E EP a h hGb h E a E a E

⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥= + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

(2.58)

33

2210 aCaCaCCCSLB +++= (2.59)

330 aCCCSLB += (2.60)

( )2321

2

, 322

aCaCCb

PG c

SLBTc ++= (2.61)

( )23

2

, 32

aCb

PG c

SLBTc = (2.62)

44

2.3.2 Experimental aspects of the SLB test

Since in a mixed-mode I/II test the specimen is subject to a combination of Mode

I and II loading, the experimental issues identified previously as potentially important in

pure mode tests, such as initial defect type, fiber bridging, and three-dimensionality, may

also be present during an SLB test to some extent depending on the specific mode ratio

and material type. Regarding the definition of critical point on the load-deflection curve,

the method used in Mode I and Mode II tests can as well applied as well for the SLB test.

2.3.3 Other configurations of mixed Mode I/II IFT test

For mixed Mode I/II IFT testing, another widely used test configuration is mixed

mode bending (MMB). The mixed-mode bending (MMB) test apparatus was designed by

Reeder and Crews (Reeder and Crews 1990), and redesigned later to reduce nonlinear

effects (Reeder and Crews 1992). Although the MMB test was created for thin,

unidirectional, symmetric laminates, it has been used for thick, asymmetric, off-axis ply

laminates as well (Kim and Mayer 2003). A schematic of the MMB test configuration is

shown in Figure 2-19. A great advantage of using the MMB test is that it provides the

ability to characterize delamination onset and growth for a wide range of mode ratios.

The loading in this test is a simple combination of the double cantilever beam Mode I and

the end notch flexure Mode II tests. This test method allows the generation of a wide

range of mode ratio by changing the lever length c as shown in Figure 2-19.

45

Many other mixed mode test configurations can be found in (Szekrényes and

József 2006; Tracy et al. 2003), including, mixed mode end load split (MMELS) shown

in Figure 2-20, and crack lap shear (CLS) in Figure 2-21.

Pδ

2h

a

L L

c

Base

Level arm

Specimen

Fig. 2-19: A schematic of MMB test configuration (Kim and Mayer 2003)

P

ha

L

h

Fig. 2-20: A schematic of the mixed mode end load split (MMELS) specimen, based on (Szekrényes and József 2006)

P P

Fig. 2-21: A schematic of the crack lap shear (CLS) specimen, based on (Tracy et al. 2003)

46

2.3.4 Mixed mode delamination failure criterion

In most realistic situations, both tensile and shear stresses can be present at the

delamination front, which result in a mixture of Mode I and Mode II loading. It is

commonly observed that Mode I fracture dominates the fracture failure process of

isotropic materials, however, both tensile and shear fracture can be significant in

laminated composites. Many delamination failure criteria proposed by researchers are

based on the decomposition of the total SERR, GT, into separate modes, GI, GII, and GIII,

and an empirical relation between the applied SERRs and the critical value of each mode

(GIc, GIIc, and GIIIc). However, there is some debate over whether Mode II fracture

toughness is a true material property (O'Brien 1998b; Tay 2003). Based on microscopic

examination, it has been suggested that the so called pure Mode II tests may be in fact

mixed-mode tests with variable local Mode I contributions. This complicates the

establishment of a general delamination failure criterion which can be suitable for all

materials at various mode ratios. Hence, there are many different forms of delamination

onset criterion found for different materials. Among them, the most commonly used ones

are listed below:

• The linear criterion (Reeder and Crews 1991), by Eq. (2.63):

• The power law criterion (Reeder 1993), by Eq. (2.64):

• The B-K law (Benzeggagh and Kenane 1996), by Eq. (2.65) and Eq. (2.66):

• The bilinear criterion (Reeder 1993), by Eq. (2.67) and Eq. (2.68):

1=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

IIc

II

Ic

I

GG

GG

(2.63)

1=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛βα

IIc

II

Ic

I

GG

GG

(2.64)

IIIT GGG += (2.65)

( )m

T

IIIcIIcIcTc G

GGGGG ⎟⎟

⎠

⎞⎜⎜⎝

⎛−+= (2.66)

47

where, ξ and ζ are the slopes of the two line segments used in the bilinear criterion.

• Kim and Mayer (Kim and Mayer 2003) used a relation by Eq. (2.69) :

Some other criteria addressing Mode I and Mode II interaction were developed by

Hashemi et al. (Hashemi et al. 1990a; Hashemi et al. 1991). Reeder and Crews (Reeder

and Crews 1991) found that a linear form is adequate for AS4/PEEK composites. Also,

Rhee (Rhee 1994) obtained experimental results indicated a linear form GI-GII behavior.

Ducept et al. (Ducept et al. 1997) developed a nonlinear failure locus for glass/epoxy.

Nonlinear failure loci was also developed by Kinloch et al. (Kinloch et al. 1993; Reeder

1993), and Singh et al.(Singh and Partridge 1995) for carbon/epoxy. Mathews and

Swanson (Mathews and Swanson 2007) found that both the linear and power law criteria

considered give a reasonable representation of the experimental delamination fracture

data for AS4/3501-6 carbon/epoxy. The fact that so many different mixed-mode criteria

have been suggested and used indicates that there is still much to be understood about the

mixed mode failure mechanism.

2.4 Interlaminar Fracture Toughness Test Under Cyclic loading

For metallic materials, it is common practice to use Paris’ law (Paris 1964) to

predict crack growth rate under a certain stress intensity factor range, ΔK. It is also

assumed that there is a threshold level in the stress intensity amplitude below which no

crack growth will take place. Above this threshold value, fatigue crack propagation is

governed by Eq. (2.70),

IcmIIc

mIc GGG += ξ (2.67)

IIcmIIc

mIc GGG ζζ −= (2.68)

( )k

IIIcIIcIcc G

GGGGG ⎟

⎠⎞

⎜⎝⎛⋅−+= 2 (2.69)

( )pKAdNda

Δ= (2.70)

48

where da/dN is the fatigue crack growth rate, A and p are material property parameters.

For fiber reinforced composites, the stress field near the crack tip is oscillatory,

and hence the SERR, or crack driving force, G, is more commonly used for composite

materials. A relationship similar to Eq. (2.70), based on SERR, was proposed by Wilkins

et al. (Wilkins et al. 1982) to predict the crack growth rate for composite materials, as in

Eq. (2.71).

For tests at small R, Gmin is small compared to Gmax, and hence the expression for ΔG can

be written as, Eq. (2.72),

By this approach, the modified Paris’ law becomes Eq. (2.73),

Based on the modified Paris’ law, a total fatigue life model of composite materials was

suggested by Shivakumar et al. (Shivakumar et al. 2006). A brief description of this

model is given next.

Similar to metallic materials, the fatigue delamination process of a composite

material is often characterized in terms of the relationship between the crack growth rate

per cycle and the applied SERR as visualized on a log-log plot (Figure 2-22). The first

region of the crack growth process occurs at low SERR values and consists of either no

crack growth below a certain threshold of SERR or a decelerating growth rate with

increased SERR on the log-log scale. The second region, occurring at increased SERR

values, shows a linear relationship between crack growth rate and SERR on the log-log

scale. The third region, occurring at still higher SERR values, corresponds to markedly

increasing crack growth rate with increasing SERR, leading eventually to unstable crack

growth in a single loading cycle as SERR approaches its critical value. According to

Shivakumar et al., the delamination growth rate depends on microscopic details of

interactions between the fibers and resin in Region 1, crack driving force (strain energy

( )nGBdNda

Δ= (2.71)

maxminmax GGGG ≈−=Δ (2.72)

( )nGBdNda

max= (2.73)

49

release rate G or ΔG) in Region 2, and on interlaminar fracture characteristics of the

laminate in Region 3. In Figure 2-22, the delamination growth rate curve is bounded on

the left by Gmax = Gth and on the right by Gmax = GR, where Gmax is the maximum cyclic

strain energy release rate, Gth is the threshold strain energy release rate, and GR is the

interlaminar fracture toughness resistance as the delamination grows.

The total life model for Mode I fatigue growth proposed by Shivakumar et al.

(Shivakumar et al. 2006) is given by Eq. (2.74).

Regime

1Regime

2Regime

3

m

Log A

Log(Gmax/GR)

Log(da/dN)

Gmax= Gth

Gmax= GR

Fig. 2-22: A total fatigue life model of composite materials, based on (Shivakumar et al. 2006)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

Im

ImIm

1

1

D

IR

ax

D

ax

Ithm

IR

ax

GG

GG

GGA

dNda (2.74)

50

According to the model, the parameters needed to characterize interlaminar crack growth

under cyclic loading include: 1) fatigue interlaminar fracture property parameters: A, m,

D1, D2 and threshold SERR, Gth, at or below which the delamination growth rate, da/dN,

is nearly zero; 2) static interlaminar fracture parameters, GR (as a function of Mode I

interlaminar fracture toughness and delamination length). In Region 1, the delamination

growth rate, da/dN, is dominated by D1 if the maximum strain energy release rate Gmax is

greater than Gth. In Region 2 (stable crack growth domain), the crack growth rate, da/dN,

follows the power law dependency on either the maximum strain energy release rate,

Gmax, or range of strain energy release rate, ΔG, under cyclic loading. The fatigue

interlaminar properties of material can be characterized by parameters in this power law

relation (A and m in Eq. (2.74)). In Region 3, the dominant parameter is D2 as long as the

maximum strain energy release rate Gmax is less than the interlaminar fracture toughness

GR.

Substantial research work (Asp et al. 2001; Blanco et al. 2004; Gregory and

Spearing 2005; Gustafson and Hojo 1987; Hojo et al. 1994; Hojo et al. 2006; Matsubara

et al. 2006; Nakai and Hiwa 2002; Ramkumar and Whitcomb 1985; Tanaka 1997;

Vinciquerra et al. 2002) has contributed to the characterization of delamination laws in

Region 2. For carbon fiber composites, the threshold toughness is often found to be

important since the slope of the stable crack growth region is very high. Hence, research

work has also focused on determining the threshold SERR (Asp et al. 2001; Gregory and

Spearing 2005; Hojo et al. 1987; Tanaka 1997). The fatigue fracture resistance of

laminated fiber composite materials has been found to be closely related to matrix

properties, matrix toughness and the strength of the fiber/matrix interface. The effect of

matrix material on delamination growth was studied by Hojo et al. (Hojo et al. 1994).

The effect of fiber surface treatment was studied by Hojo et al. (Hojo et al. 2001). For the

same laminate material, the delamination growth law has been found to depend on the

stress ratio (Matsubara et al. 2006), the loading frequency and environment (Gustafson

and Hojo 1987; Nakai and Hiwa 2002), and mode ratio (Gustafson and Hojo 1987).

51

2.4.1 Fatigue delamination growth models

Since the delamination mechanism is considered to be different for different

modes of crack tip stress state, the analysis of the crack growth rate under cyclic loading

should be considered in terms of the individual operative modes.

2.4.1.1 Pure Mode I or II

In the stable crack growth region, fatigue crack growth rate in pure Mode I or

Mode II loading is typically related to range of strain energy release rate by the modified

Paris’ law, Eq. (2.75):

where, i = I or II for Mode I and Mode II, respectively, Bi and ni are material constants, a

is the crack length, N is loading cycle, and max minG G GΔ = − is the range of strain energy

release rate.

In the case of homogeneous materials, such as metals and plastics, Eq. (2.75) is

often found to be in the form of Eq. (2.70), where, the range of stress intensity factor, ΔK,

instead of strain energy release rate is often used (Gustafson and Hojo 1987).

Assuming that the material is homogeneous and isotropic, the stress intensity factor,

Ki, can be related to the strain energy release rate, Gi, (i = I, II) by Eq. (2.76).

where Hi is a function of the elastic constants of the material (Eij, νij) (Hojo et al. 1987).

The stress ratio, R, is defined by Eq. (2.77) (Hojo et al. 1994):

When R is positive, substituting Eq. (2.76) into Eq. (2.77) yields, Eq. (2.78) :

( ) ini GB

dNda

Δ= (2.75)

2iii KHG = (2.76)

max

min

KKR = (2.77)

52

When R is small, Gmin is small compared to Gmax, and thus the power law relation can be

rewritten as Eq. (2.73) (Asp et al. 2001).

Many studies have found that the exponent in the power law relationship for

composite materials is much higher than that for metals. This indicates that small

uncertainties in applied loads will lead to large uncertainties in predicted delamination

growth rate.

2.4.1.2 Mixed-mode

Several models exist for relating loading and delamination growth rate. Some

typical models used are given from Eq. (2.79) to Eq. (2.82).

Ramkumar and Whitcomb, by Eq. (2.79) (Ramkumar and Whitcomb 1985):

Gustafson and Hojo, by Eq. (2.80) (Gustafson and Hojo 1987):

Russell and Street, by Eq. (2.81) and Eq. (2.82) (Russell and Street 1989):

where,

In these equations, B and n are material parameters.

max

min

GGR = (2.78)

III n

IIc

IIII

n

Ic

II G

GB

GG

BdNda

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= (2.79)

( ) ( ) III nIIII

nII GBGB

dNda

Δ+Δ= (2.80)

mn

cm G

GBdNda

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ= (2.81)

IIc

II

Ic

I

c GG

GG

GG Δ

+Δ

=Δ (2.82)

53

2.4.2 IFT fatigue test methods

2.4.2.1 Fatigue threshold strain energy release rate determination

The fatigue crack growth onset test can be used to determine threshold strain

energy release rate values. The threshold strain energy release rate value is defined as the

maximum SERR value below which no delamination onset occurs after a certain number

of cycles (e.g. 106 cycles in (Shivakumar et al. 2006), 1,800,000 cycles in (Asp et al.

2001)). This SERR value can be expected to depend on the fidelity of the crack length

measurement technique. By conducting fatigue onset tests, a plot of the maximum SERR

values, Gmax, versus number of cycles at delamination onset, N, can be obtained. A curve

is then fitted to the data. Based on this curve, the cyclic strain energy release rate to cause

delamination onset after a specified number of cycles can be predicted. A typical G-N

plot from Mode I delamination onset tests is given in Figure 2-23. A threshold SERR,

GIth, can thus be determined for a selected number of loading cycles for no crack growth.

ASTM D6115 (ASTM D6115-97 1997(R2004)) was established for Mode I fatigue

delamination onset testing. For many kinds of composite materials, the exponent of the

Paris’ law relation by Eq. (2.71) or Eq. (2.73) is very high. This implies that a small error

in SERR calculation will result in a large error in crack growth and service life prediction,

if such a material is used in a structure. Hence, for such cases, the no-growth design

criterion is suggested and the threshold SERR is proposed to be used as the fatigue

interlaminar fracture toughness.

54

2.4.2.2 Fatigue delamination growth test

Fatigue delamination growth tests are used to establish a relation for crack growth

rate prediction in the form of Eq. (2.71) or Eq. (2.73). Before conducting a fatigue test, a

quasi-static test of the corresponding deformation mode is usually needed to obtain an

estimate of static fracture properties, e.g. critical load and displacement at crack onset,

and critical strain energy release rate for the type of material investigated. The same

specimen type and test configuration are then used for fatigue tests.

Fatigue tests can be either displacement controlled (Blanco et al. 2004; Gregory

and Spearing 2005) or load controlled (Bureau et al. 2002; Nakai and Hiwa 2002;

Vinciquerra et al. 2002). However, the displacement control method is more commonly

used because it is considered to be more likely to create stable crack growth. To obtain a

single power law relation for crack growth prediction as by Eq. (2.71) or Eq. (2.73),

stress ratio, min max/R G G= , (or maxmin /δδ for displacement controlled tests; for linear

elasticity and small deflections (δ/a < 0.4), maxmin /δδ is identical to the R-ratio), are kept

GImax

J/m2

GI

NaN, cycles to delamination growth onset

GImax

J/m2

GI

NaN, cycles to delamination growth onset

Fig. 2-23: The Mode I delamination onset SERR versus number of cycles, based on (ASTM standard D6115-97 1997(R2004))

55

constant during a test. Typical load ratios used are within the range of 0.1-1, and typical

cyclic loading frequencies are within 1-10 Hz, for Mode I, Mode II, and mixed-mode I/II

tests. Typical test configurations for each type of fatigue test are the same as those

presented for quasi-static testing in Section 2.1-2.3.

For data acquisition, the crack growth rate and SERR values need to be obtained

on a regular basis. The crack growth rate is obtained by recording cycles and crack

growth length during fatigue test. Two methods are generally used to measure the

delamination length and data are usually taken either at certain intervals of delamination

length or certain intervals of number of cycles. One way to measure delamination length

is to obtain all the data optically using a microscope. The other way is to use the

compliance calibration method. In the second method, a compliance versus crack length

relationship is obtained for this material either based on quasi-static tests or fatigue tests.

Then the crack length at certain number of cycles is calculated based on the measured

compliance. Hence load-displacement curves are recorded to obtain the compliance at

selected cycle intervals. Measurements can be taken periodically until the crack

propagation rate reaches a certain level, e.g. below 10-8 m/cycle, or the number of load

cycles reaches 105 (this criterion was used in (Gregory and Spearing 2005)). The SERR is

then calculated based on the maximum load, load amplitude, crack length, displacement,

etc., at certain numbers of cycles.

2.5 Finite Element Modeling of Crack Propagation

The widely used techniques for modeling crack propagation interface in

composite materials include: modeling the crack interface with the virtual crack closure

technique (VCCT), with a tiebreak contact, and with a cohesive layer (Meo and Thieulot

2005). Among them, only the virtual crack closure technique is a fracture mechanics

based method. This technique is discussed in more detail in the following paragraph. In

the second technique, coincident nodes are tied together with a constraint relation and

remain joined until the maximum interlaminar stresses are reached. Once this maximum

value is exceeded, the nodes associated with the constraint are released to simulate the

56

initiation of delamination. Meo and Thieulot found that poor results were obtained when

using the tiebreak contact technique to model the delamination behavior of the double

cantilever beam specimen (Meo and Thieulot 2005). The cohesive element technique,

which is capable of simulating the initial elastic behavior, damage initiation, and damage

evolution of the material, is used to model the cohesive layer. However, in order to

simulate the delamination behavior correctly, criteria and parameters defining each

loading stage of the cohesive element need to be selected appropriately. In the following

sections, fundamentals and basic principles for using VCCT and cohesive elements are

briefly reviewed.

2.5.1 The Virtual Crack Closure Technique (VCCT)

The virtual crack closure technique (VCCT) implemented in finite element

analyses is gaining popularity. This is especially true in aerospace engineering, because

of the practical computation time and suitability of the technique for modeling the

behaviors of anisotropic materials. VCCT predicts crack propagation by comparing local

SERR to the critical SERR (fracture toughness), and gives results of SERR from node to

node in a finite element model. VCCT is available as an add-on capability of Abaqus

V6.7-1 to efficiently simulate delamination behaviors for composite materials. Using

VCCT for Abaqus, an engineer can simulate the crack propagation process and estimate

the residual strength of a composite structure after damage. VCCT for Abaqus is based

on patent-pending technology by Abaqus, Inc. (VCCT for Abaqus Manual 2007).

2.5.1.1 The virtual crack closure technique formulation

Using VCCT, the crack propagation is modeled with a contact interaction defined

between a pair of crack surfaces. Therefore, the crack path is predefined using VCCT.

Crack onset or growth prediction is achieved by comparing the calculated SERR (G) with

the interlaminar fracture toughness property (Gc) of the material. VCCT is based on the

57

assumption that the energy released when the crack is extended from a to a+Δa is

identical to the energy required to close the crack (Figure 2-24, (Krueger 2002)).

The VCCT method assumes that a crack extension of Δa from a+Δa (node i) to

a+2Δa (node k) does not significantly alter the state at the crack tip. Therefore, the

displacements behind the crack tip at node i are approximately equal to the displacements

behind the original crack tip at node l. The basic formulae for SERR calculation by using

VCCT are presented next.

X1l

Z1l

l i

a Da Da

Crack closed

x, u, X

z, w, Z

li

a Da Da

Crack extended

x, u, X

z, w, Z

Du2l

Dw2l

Fig. 2-24: Crack closure for VCCT, based on (Krueger 2002)

58

(1) 2D 4-node element

In a two-dimensional plane stress or plane strain model, the Mode I and II

components of SERR, GI and GII, for a four node element are calculated by Eq. (2.6) and

Eq. (2.7), (Figure 2-25, (Krueger 2002)):

where, Δa is the length of the elements immediately in front of the crack along the crack

growth direction. Xi and Zi are the forces at the crack tip (node point i). ul and wl are the

nodal displacements behind the crack tip of the upper crack face, and ul* and wl* are the

nodal displacements behind the crack tip of the lower crack face. (Figure 2-25). GIII = 0

for two-dimensional cases. The crack is represented as a one-dimensional discontinuity

by a line of nodes on the top and bottom fracture surfaces. Nodes on bottom and top

fracture surfaces initially have identical coordinates before loading. As loading proceeds,

the bonded nodes are released along the crack sequentially.

( )*21

lliI wwZa

G −⋅⋅−= (2.83)

( )*21

lliII uuXa

G −⋅⋅−= (2.84)

Fig. 2-25: VCCT for four-node 2D element (plane strain or plane stress), based on (Krueger 2002)

59

(2) 3D solid 8-noded element

The Mode I, Mode II and Mode III components of SERR, GI, GII and GIII, for an

eight node element are calculated by Eq. (2.85), Eq. (2.86) and Eq. (2.87), respectively.

where, baA ⋅Δ=Δ as shown in Figure 2-26. Other notations are defined in Figure 2-26

and Figure 2-27.

( )*21

LlLlLiI wwZA

G −⋅⋅Δ

−= (2.85)

( )*21

LlLlLiII uuXA

G −⋅⋅Δ

−= (2.86)

( )*21

LlLlLiIII vvYA

G −⋅⋅Δ

−= (2.87)

wLlb vLluLl

wLl* vLl*uLl*

ZLi

XLi

YLi

L

l

l* i

aΔa

Δa

x’, u’, X’

z’, w’, Z’

z, w, Z

x, u, X

y, v, Y

Global system

Local crack tip system

Fig. 2-26: VCCT for eight-node solid element (3D view) (Krueger 2002)

60

2.5.1.2 Crack growth criterion for VCCT

Crack onset or growth is predicted by comparing the calculated SERR, G, with

interlaminar fracture toughness properties of material, Gc. The SERR along the

delamination front is calculated at the end of a converged increment. Once the SERR

exceeds the critical SERR, the node at the crack tip is released at the following

increment, which allows the crack to propagate. To avoid sudden loss of stability, the

force at the crack tip before advance is released gradually during succeeding increments

in such a way that the force is brought to zero no later than the time at which the next

node along the crack path begins to open.

For pure Mode I case, debonding of nodes occurs when: (VCCT for Abaqus

Manual 2007)

b/2

b/2

bb

l i

l

a Δa Δa

i

vLl

uLl XLi

YLi

y’, v’, Y’

x’, u’, X’L

Delamination front

Delaminated area

x, u, X

y, v, Y

Global system

Fig. 2-27: VCCT for eight-node solid element (top view) (Krueger 2002)

61

For mixed mode, debonding of nodes occurs when:

VCCT for Abaqus releases a bonded node at the crack tip when Eq. (2.89) is satisfied,

however, when the ratio of the difference between the computed and critical SERR to the

critical SERR exceeds the specified release tolerance, as described by Eq. (2.90), a

cutback operation (rerun the current iteration with smaller time increment) is requested

by VCCT for Abaqus. (VCCT for Abaqus Manual 2007)

In the above two equations, Gequiv is the equivalent SERR calculated at a node in the

general mixed mode case and GequivC is the critical SERR calculated based on the user-

specified mode-mix criterion and the critical SERR of the interface.

For mixed mode fracture, VCCT for Abaqus provides three mixed-mode

criterions to model failure. The users can choose from three of the most commonly used

mixed mode fracture criterion, the B-K law by Eq. (2.91), the Power law by Eq. (2.92),

and the Reeder law by Eq. (2.93).

GI > GIc (2.88)

Gequ > GequC (2.89)

tolerancerelease >−

equivC

equivCequiv

GGG

(2.90)

η

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+−+=

IIIIII

IIIIIICIICICequivC GGG

GGGGGG )( (2.91)

ao

IIIC

III

an

IIC

II

am

IC

I

equivC

equiv

GG

GG

GG

GG

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= (2.92)

( )

( )η

η

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+−+=

IIIIII

IIIII

IIIII

IIIIICIIIC

IIIIII

IIIIIICIICIcequivC

GGGGG

GGG

GG

GGGGG

GGGG

(2.93)

62

For the B-K law and the Reeder law, GIC, GIIC, GIIIc and η are material properties to be

specified by users to define the model. For the Power law, GIC, GIIC, GIIIc η, am, an, and

ao are material properties need to be specified.

2.5.2 Cohesive element (Abaqus 2007)

Besides VCCT, cohesive elements are sometimes used to model fracture behavior

of laminated composites. The advantage of cohesive elements over VCCT is that the

former does not require the existence of an initial crack in the material. However, extra

material property parameters for defining elasticity, damage initiation and evolution

criterion, are needed for cohesive elements. This makes the usage of cohesive element

more versatile, but more complicated as well.

Cohesive elements are capable of modeling the initial elastic loading, the

initiation and the propagation of damage leading to eventual failure at a bonded interface.

Unlike the usual elements, the thickness of a cohesive element can be as small as zero

before loading is applied. The model need not have any crack to begin with. However,

the cracks are restricted to propagate along the layer of cohesive elements.

Unlike usual elements, the constitutive response of cohesive elements can be

defined using a traction-separation law. A typical traction-separation response of a

cohesive element, as shown in Figure 2-28, involves three stages: initial elastic loading,

damage initiation and damage evolution. The constitutive thickness used for traction-

separation response is generally different from the geometric thickness, which is typically

close or equal to zero. By default, if the traction-separation law is specified, Abaqus sets

the constitutive thickness of the element to be 1.0, so that, the nominal strain is equal to

the relative normal displacements (separation) between the top and bottom faces. The

nominal stresses are the force components divided by the original area at each integration

point, while the nominal strains are the separations divided by the original thickness at

each integration point.

63

2.5.2.1 Elastic behavior of the cohesive element

The nominal traction stress vector, t, consists of three components in three-

dimensional problems: tn, ts, and tt, which represent the normal and the two shear

tractions, respectively. The corresponding separations are denoted by δn, δs, and δt.

Denoting by To the original thickness of the cohesive element, the nominal strains are

defined as Eq. (2.94),

The elastic behavior can then be written as,

( )000 , tsn ttt

( )000 , tsn δδδ ( )ft

fs

fn δδδ ,

Traction

Separation

Initial loading

Damage initiation

Damage evolution

Fig. 2-28: Traction-separation response of cohesive element, based on (Abaqus 2007)

o

tt

o

ss

o

nn TTT

δε

δε

δε === , , (2.94)

Kε=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

t

s

n

ttstnt

stssns

ntnsnn

t

s

n

KKKKKKKKK

ttt

tεεε

(2.95)

64

The terms in the stiffness matrix, K, are material properties. The off-diagonal terms in the

stiffness matrix are zero if uncoupled behavior between the normal and shear components

is desired.

The traction-separation response can be better understood by comparing it to the

load-displacement response of a rod, as described by Eq. (2.96).

or Eq. (2.97)

where the end displacement of the truss is δ. The length is denoted by L, the elastic

stiffness by E, the original area by A, and the axial load by P. APS /= is the nominal

stress and LEK /= is the stiffness that relates the nominal stress to the displacement. If

the bulk adhesive material of the cohesive elements has stiffness Ec, the stiffness of the

interface is given by ccc TEK /= . When the constitutive thickness of the cohesive layer

is “artificially” set to 1.0, ideally Kc should be specified as the material stiffness as

calculated with the true thickness of the cohesive layer. As the thickness of the interface

layer is often very small, the equation ccc TEK /= implies that the stiffness, Kc, tends to

infinity. This stiffness is sometimes chosen as a penalty parameter. Very large penalty

stiffnesses may result in ill-conditioning of the element operator in Abaqus/Standard. To

ensure the cohesive elements will have no adverse effect on the stable time increments,

the Abaqus manual suggests to choose, ec KK 1.0= , where Kc is the cohesive element

stiffness, and Ke is the stiffness of surrounding material.

AEPL

=δ (2.96)

KS

=δ (2.97)

65

2.5.2.2 Damage initiation criteria of the cohesive element

Four criteria can be used for defining damage initiation: maximum nominal stress,

maximum nominal strain, quadratic nominal stress, and quadratic nominal strain criterion.

The mathematical forms of these criteria are given by Eq. (2.98) to Eq. (2.101).

In the above equations, tn0, ts

0, and tt0 are ultimate stresses in the normal, first and second

shear directions, respectively, as shown in Figure 2-28. εn0, εs

0, and εt0 are ultimate strains

in the normal, first and second shear directions, respectively, as shown in Figure 2-28.

2.5.2.3 Damage evolution criteria of the cohesive element

For damage evolution many criteria are available for Abaqus. Generally, they can

be divided into linear and nonlinear categories. For nonlinear damage evolution criteria,

the Power Law form, given by Eq. (2.102), and the B-K form, given by Eq. (2.103), are

commonly used.

Maximum nominal stress: 0 0 0max , , 1n s t

n s t

t t tt t t

⎧ ⎫=⎨ ⎬

⎩ ⎭ (2.98)

Maximum nominal strain: 0 0 0max , , 1n s t

n s t

ε ε εε ε ε

⎧ ⎫=⎨ ⎬

⎩ ⎭ (2.99)

Quadratic nominal stress: 2 2 2

0 0 0 1n s t

n s t

t t tt t t

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

(2.100)

Quadratic nominal strain: 2 2 2

0 0 0 1n s t

n s t

ε ε εε ε ε

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

(2.101)

1n s tC C Cn s t

G G GG G G

α β γ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

+ + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(2.102)

( )C C C CSn s n

T

GG G G GG

η⎛ ⎞

+ − =⎜ ⎟⎝ ⎠

(2.103)

66

In the Power law form, Gn, Gs, and Gt are the work done by the traction and its conjugate

relative displacement in the normal, the first and the second shear directions, respectively.

The quantities GnC, Gs

C, and GtC are the critical fracture energies required to cause failure

in the normal, the first, and the second shear directions, respectively. In the B-K form,

S s tG G G= + , T n sG G G= + , and η is a material parameter. The mixed mode fracture

energy Cn s tG G G G= + + . The user needs to specify Gn

C, GsC, and η.

2.6 Preview of the Following Chapters

Similar to using the traditional failure criteria of materials, crack onset and growth

is predicted based on comparing the local crack driving force (G) with the material

fracture resistance (Gc). The local drive force, or named strain energy release rate

(SERR), G, represents the local loading conditions, while the fracture resistance, Gc,

represents the material strength. To predict delamination behavior accurately, firstly, a

repeatable and conservative value of the fracture toughness needs to be characterized as a

material property. Secondly, a reliable and effective method for evaluating local SERR is

required. The interlaminar fracture toughness (IFT) test is the type of test to characterize

the capability of a laminated composite material to resist fracture. The finite element

method (FEM) is potential to accurately predict local SERR distribution and

delamination behavior based on a fracture mechanics approach. For most of the currently

popular commercial FE software, the SERR based failure and crack propagation analysis

methods have not yet been incorporated, except for Abaqus. During this investigation, the

fracture behavior in Mode I, Mode II, and mixed Mode I/II of a proprietary carbon/epoxy

composite material system were characterized under static and fatigue loadings. The

methods and results for Mode I, Mode II, and Mixed Mode I/II testing are presented in

Chapter 3, 4 and 5, respectively. Additionally, in an attempt to preliminary assess the

capability of Abaqus in solving crack propagation problems, several finite element

models of the DCB specimen were analyzed using VCCT for Abaqus. The modeling

technique and results obtained for finite element modeling are presented in Chapter 6.

67

Chapter 3

The Mode I Interlaminar Fracture Toughness Testing

Mode I interlaminar fracture toughness (IFT) tests were conducted under quasi-

static and cyclic loading conditions with the Double Cantilever Beam (DCB) specimen.

In this chapter, DCB test configurations and methods used in this investigation are

introduced. The Mode I interlaminar fracture toughness at crack onset and fracture

resistance with crack extension were characterized with DCB quasi-static specimens.

Mode I fatigue tests were conducted in displacement control with displacement ratio R =

0.2. Crack growth rate (da/dN) was related to maximum Mode I strain energy release rate

(GImax) by the power law relation ( ) InaxI GBdNda Im/ = .

3.1 Material, Specimen and Test Configuration

The specimens used in this investigation are machined from a flat carbon/epoxy

panel of [0]12 lay-up as shown in Figure 3-1. A Teflon film of 12.7 μm thickness was

inserted at the mid-plane during the panel lay-up process to define the initial starter crack.

The DCB specimens were cut from the panel at the Penn State Composites Lab using a

water-cooled diamond abrasive cut-off wheel. The distribution of tested DCB specimens

in the panel is shown in the panel diagram, Figure 3-1.

68

The DCB specimen geometry and notations are shown in Figure 3-2. Five

specimens (Specimen 1-1, 1-2, 1-3, 1-4 and 1-5 in Figure 3-1) were tested in quasi-static

loading and four specimens (Specimen 4-2, 4-3, 4-4, and 4-5 in Figure 3-1) in fatigue.

The specimen dimension of length × width × thickness is approximately 150 × 25.4 × 3.9

mm, with an initial artificial crack created by the embedded thin film of about 76.2 mm

length. Specific dimensions of each specimen are listed in Appendix A Table A-1 and

Table A-2, for static and fatigue case, respectively.

00.0005’’ Teflon film

0-degree fiber direction

Specimen 1-1

Specimen 1-2

Specimen 1-3

Specimen 1-4

Specimen 1-5

Specimen 4-2Specimen 4-3


Fig. 3-1: Diagram of the DCB panel

69

3.2 The Mode I Quasi-static IFT Testing

Although an ASTM standard (ASTM D5528-01 2002) has been established for

the Mode I quasi-static IFT testing, certain issues are still debated in the literature, such

as whether to use an initiation GIc value from a Mode I precrack or from an insert film,

and whether it is advantageous to use several reloading/unloading cycles or to use just

one reloading/unloading cycle. The test methods used in this investigation follow the

general suggestions by the ASTM standard; however, certain modifications were made

for the special properties of the tested material and also to compare the results of different

methods.

a0 l

2h

b

Piano hinge

Insert film

Fig. 3-2: DCB specimen geometry and notation

70

3.2.1 Mode I quasi-static test method

The Mode I interlaminar fracture toughness tests using DCB specimens were

performed according to ASTM standard D5528 (2002). The specimens were loaded by a

13.5 kN (3 kip) servo-hydraulic MTS machine. The load was measured using a 110 N (25

lb) load cell and the displacement was measured by the MTS LVDT built into the

actuator. A photograph of the test setup is shown in Figure 3-3.

Loading was applied to the specimen through piano hinges. Before the test, the

piano hinges were bonded to specimen surfaces using M-bond adhesive. A thin layer of

white correction fluid (slightly transparent) was painted on one specimen edge for

convenience of crack growth observation. The position of the end of the insert film was

marked on the painted edge before testing, and pencil markings were made by hand at

Top Grip (stationary)

110 N Load Cell

Universal Joint

Specimen

Bottom Grip (actuated)

Optical Fiber Light Source

Not shown:- Servo-hydraulic actuator- Long distance instrumented stage microscope

Top Grip (stationary)

110 N Load Cell

Universal Joint

Specimen

Bottom Grip (actuated)

Optical Fiber Light Source

Not shown:- Servo-hydraulic actuator- Long distance instrumented stage microscope

Fig. 3-3: Photograph of DCB test set up

71

every one millimeter in the first 5 mm beyond the insert film, and every 2 or 3 mm in the

next 45 mm. The exact distances from the first marking (insert film tip marking) to the

following markings were measured through a Questar® QM-1 long distance telescope

before the test.

One modification to the loading/unloading method was made to the ASTM

standard D 5528 test method. In the current investigation, a DCB specimen was subjected

to five successive loading and unloading cycles, instead of one initial loading/unloading

cycle and one reloading/unloading cycle as specified in the ASTM standard. For the first

loading cycle, loading was stopped and unloading started after the crack growth had

reached 5 mm or unstable crack growth had arrested. For each of the next four loading

cycles, the loading was stopped and unloading started after 10-15 mm crack growth had

been created in a loading cycle.

During the test, the specimen was loaded at a constant displacement rate of 1

mm/min and unloaded at 2 mm/min. Crack tip position was pinpointed with a Questar®

QM-1 long distance telescope mounted on an instrumented stage with 0.001 mm

resolution. Load and displacement data were recorded continuously for each loading

cycle, together with the elapsed time from the beginning of the test, by a digital data

acquisition system. The distance from each subsequent marking to the first marking,

where the initial crack is, was measured by the Questar® telescope before the test. The

crack length at each subsequent marking was calculated by the sum of the initial crack

length and the distance to the first marking (initial crack front marking). The elapsed time

from the beginning of test to the moment at which the crack propagates to each marking

was recorded during the test, so that after the test the crack length could be matched with

the concurrent load and displacement based on the data recorded by the data acquisition

system.

72

3.2.2 Mode I quasi-static test results

The characterization of the fracture resistance is based on the load-displacement

plot generated after the tests. From this plot, the compliance values of the specimen at

several crack lengths were obtained and related to crack lengths by several models. This

compliance vs. crack length relation was then used for critical strain energy release rate

(SERR) calculation, together with the load and/or displacement recorded during the test.

Some general observations of crack growth during the tests are: the crack growth

was typically unstable immediately after initiation in the first loading cycle, while the

crack growth in other reloading/unloading cycles was typically quasi-stable and very fast,

especially at shorter crack lengths. As the crack extended, the growth rate tended to slow

down. This trend coincides with the prediction by the crack length vs. displacement

relationship from the modified beam theory, Eq. (3.1) and Eq. (3.2) (assuming constant

fracture resistance with crack extension).

where, aeff is the effective crack length taking into consideration rotation of the cantilever

beam at the built-in end. Also, GIR is the fracture resistance as the crack grows; m is the

coefficient from the compliance-crack length relation in Eq. (2.1); b is the specimen

width, and δ is the opening displacement. For a displacement controlled test, d(aeff)/dδ is

proportional to the speed of crack growth (d(aeff)/dt, where t is time). As implied by

Eq. (3.2), d(aeff)/dδ and thus the speed of crack growth decreases as the loading

displacement increases.

2/14 32

3 δIR

eff Gbmaa =Δ+= (3.1)

( ) 2/14

3 21

23 −== δ

δδ IR

eff

Gbmdda

dad

(3.2)

73

3.2.2.1 Load-displacement curves

The load-displacement curve consists of two parts. The first part is the linear part,

where load is linearly related to displacement. During this portion of test, there is neither

crack extension nor nonlinear material behavior in the specimen as loading proceeds. For

this part, simple or advanced beam theory can be used to predict the loading curve. The

second part is the nonlinear part, where there is either nonlinear material behavior or

local/global crack growth in the specimen. If nonlinear material behavior or local crack

growth occurs in the specimen, the loading curve deviates slightly from the straight line

that can be fitted into the initial linear part of loading curve. However, if global crack

growth occurs, the loading curve exhibits a marked deviation from the straight line fitted

into initial loading curve. For the global crack growth regime, fracture mechanics

approach can be used to predict loading curve.

For the linear regime, the slope of the load-displacement curve is mainly related

to the modulus in the longitudinal direction of specimen, E1, and the crack length, as by

Eq. (3.3), which is derived from MCC expression of compliance (Section 2.1.1.2).

where α0 is given by Eq. (2.17), and involves the material properties E1, E3, and G13.

For the nonlinear regime, if the nonlinearity is caused by global crack growth, the

trend of the loading curve is dependent on fracture resistance of the specimen with crack

extension. Derived from the modified beam theory (Section 2.1.1.1), the load of the DCB

specimen is related to the loading displacement by Eq. (3.4) if the material’s fracture

toughness with crack extension is GIR.

A special case when the material demonstrates constant fracture toughness after a

certain length of crack extension (as shown in Figure 3-4) is considered next. Assume a

specimen demonstrates a constant fracture toughness IR IcG kG= , after the crack extends

( )

31

30

18 2

E bhP

a hδ α=

⎡ ⎤−⎣ ⎦ (3.3)

3/ 41/ 2IR2

3bGP

mδ −⎡ ⎤= ⎢ ⎥⎣ ⎦

(3.4)

74

the distance of Δa1, where GIc denotes the fracture toughness at crack initiation. At crack

initiation, Eq. (3.5) is predicted by MBT.

where Pcr and δcr are load and displacement at crack onset, respectively. After the crack

length reaches 1 0 1a a a= + Δ , we have,

Comparing Eq. (3.5) to Eq. (3.6), one obtain, Eq. (3.7),

and thus the loading curve is predicted by, Eq. (3.8).

when,

3/ 41/ 2 Ic2

3cr crbGPm

δ ⎡ ⎤= ⎢ ⎥⎣ ⎦ (3.5)

3/ 41/ 2 IR2

3bGP

mδ ⎡ ⎤= ⎢ ⎥⎣ ⎦

(3.6)

3/ 41/ 2IR

1/ 2cr cr Ic

GPP Gδδ

⎡ ⎤= ⎢ ⎥⎣ ⎦

(3.7)

( ) ( )3/ 4

1/ 2 1/ 2 3/ 4 1/ 2 1/ 2IRcr cr cr cr

Ic

GP P k PG

δ δ δ δ− −⎛ ⎞= =⎜ ⎟⎝ ⎠

i i i i (3.8)

21/ 2 1

10

craka

δ δ δ⎛ ⎞+ Δ

≥ = ⋅ ⋅⎜ ⎟+ Δ⎝ ⎠ (3.9)

Crack length, a, mm

G, J/m2

GIc

a0 a1

IR IcG kG=

Fig. 3-4: Constant fracture toughness after certain length of crack extension

75

In Figure 3-5 and Figure 3-6, load vs. displacement data from two tests were

plotted along with constant GIR curves, as predicted by Eq. (3.8). The initial linear portion

of the load-displacement curve from experimental data coincides with the theoretical

prediction. For the nonlinear portion (fracture portion) of the P-δ curve, the data

intersects with curves of increasing G as the crack extends. The higher order constant GIR

curves plotted coincide with the experimental data fairly well for the second, third and

fourth loading cycles of Specimen 1-1, and the third, fourth and fifth cycles of Specimen

1-4. This indicates that for the tested material, the fracture resistance tends to stay

constant after certain distance of crack extension, and a somewhat flat fracture resistance

vs. crack length curve may be obtained for this material.

0

10

20

30

40

0 2 4 6 8 10 12 14 16Displacement, mm

Load

, N

G IR=G Ic

G IR=1.6G Ic

Fig. 3-5: Load vs. displacement plot (Specimen 1-1)

76

Two types of behavior not predicted by the theories were also observed in the

load-displacement plots. The first one is the sudden load drop immediately after crack

initiation in the first loading cycle for most tested specimens. The second one is the

“stick-slip” behavior of the fracture portion of load-displacement curve. These two types

of behavior are described in more details as follows.

a) Sudden load drop after crack initiation

The sudden load drop at the crack initiation point of the load vs. displacement

curve often corresponds to an unstable crack growth at crack initiation. However, this

unstable crack growth is not predicted by basic theories in the fracture mechanics

approach. Based on the amount of load drop and unstable crack growth, two types of

load-displacement behavior exist at crack onset among the tests conducted were observed.

In the first type, as loading proceeds, a relatively small amount of load drop occurs

immediately after the crack onset of the first loading cycle, as shown in Figure 3-5, and

“stick-slip” behavior occur after crack starts to propagate. In this case, the unstable crack

growth at crack onset is less than 5 mm. In the second type, a relatively large amount of

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14Displacement, mm

Load

, N

G IR=G Ic

G IR=1.1G Ic

Fig. 3-6: Load vs. displacement plot (Specimen 1-4)

77

load drop occurs immediately after crack onset, with a noticeably unstable crack growth

observed through the telescope and sometimes an audible noise. For this case, the

unstable crack growth arrests after 5-10 mm crack extension. As loading proceeds, the

load-displacement curve becomes linear and then “stick-slip” behavior occurs with

further crack extension. The second type of load-displacement curve is shown in

Figure 3-6.

For the first loading cycle, four out of five specimens showed a load-displacement

behavior of the second type at initial crack onset, with dramatically unstable crack growth

from the insert film. However, for the reloading cycles, the crack growth after onset was

stable but a few cases. This unstable crack growth from the insert film might imply the

formation of a resin pocket near the front of insert film, as was shown schematically in

Figure 2-7 (Chapter 2). Further, the “less than 13 μm thickness of insert film” criterion

for initial defect type, proposed by ASTM to eliminate the resin rich zone beyond the

insert film, might not be appropriate for the material system of the current investigation.

This thickness limit might still not be small enough to eliminate the effect of resin rich

zone just in front of the insert film for some materials. If a resin rich zone exists in front

of the insert film, the perceived onset GIc value may be not indicative of the fracture

toughness of the laminated material. If there exists a big difference of the fracture

resistance of the local material and the delamination resistance of the laminate, a rather

large amount of load drop might occur at the instance of initial crack onset.

Additionally, unlike load-displacement curves observed for some other materials,

e.g. some GFRP material, in which significant nonlinear behavior occurs before the load

starts to decrease, for the investigated material the loading curve is particularly linear

before the initial and subsequent crack onset.

b) “Stick-slip” behavior

The “stick-slip” behavior of the load vs. displacement curve can be explained as

follows. The load vs. displacement curve is continuous and smooth as predicted

theoretically if the crack growth proceeds slowly and continuously across the width in

infinite small increments. However, in a real test, the crack growth proceeds step by step

in uneven increments. If the crack growth jumps ahead by a small amount, a small load

78

drop will be observed in the load vs. displacement curve, which is called “slip” behavior

of the loading curve. Since the crack length has increased, the crack driving force (G) is

less than the material fracture toughness at the new crack length in a displacement

controlled test. As loading proceeds, the load vs. displacement curve is linear again, until

the crack driving force (G) exceeds the fracture toughness of the material and the crack

starts to propagate again. If the crack advances in finite small steps progressively, the

corresponding loading-displacement curve exhibits small “zigzag” shape or even is

possibly close to a smooth shape. On the other hand, if the crack grows in a large sudden

increment, an unstable crack growth behavior may be observed.

3.2.2.2 Compliance calibration

Compliance values of each specimen at certain measured crack lengths were

obtained by two methods. During the test, each DCB specimen was unloaded and

reloaded at four crack lengths in addition to an initial loading at the initial crack length.

At these crack lengths, compliance values were obtained by inverting the slope of the

linear portion of the loading curves. Meanwhile, compliance values at other crack lengths

measured without unloading/reloading were obtained by dividing the instantaneous

displacement by the instantaneous load at the selected crack lengths.

Compliance calibration was conducted individually for every test specimen. Two

expressions were used to relate compliance (C) to crack length (a). The first one by the

Modified Beam Theory (MBT) (ASTM standard D5528-01 2002), Eq. (2.1),

and the other is based on the Modified Compliance Calibration method, (JIS standard

(JIS K 7086 1993), Eq. (2.14)).

( )33 Δ+= amC (2.1)

03/1

1 )(2/ αα += bCha (2.14)

79

To determine the coefficients for compliance calibration, two plots were made as shown

in Figure 3-7 and Figure 3-8, respectively. In Figure 3-7, the cubic root of compliance

(C1/3) was plotted against crack length (a) along with a straight line fitted by least squares

for each specimen, so that the coefficient m of Eq. (2.1) can be determined by the slope of

the fitted line, and Δ can be determined by the abscissa intersection. Coefficients of

Eq. (2.1) determined for each specimen are shown in Table 3-1. In the second plot,

Figure 3-8, normalized crack length, a/2h, was plotted against compliance normalized by

width, (bC)1/3, along with a straight line fitted by least squares for each specimen. For

each specimen, coefficient α1 is determined by the slope of the corresponding fitted line,

and α0 determined by the ordinate intersection. In the second type of plot, the coordinates

are normalized by specimen dimensions, and thus the fitted lines are not subject to

variations of specimen dimensions. Coefficients determined by line-fitting in the second

plot are listed in Table 3-2.

0.000.10

0.200.30

0.400.50

0.600.70

0.800.90

1.00

-20 0 20 40 60 80 100 120Crack length, a , mm

C1/

3 , (m

m/N

)1/3

Specimen 1-1Specimen 1-2Specimen 1-3Specimen 1-4Specimen 1-5

( )33 Δ+= amC

Fig. 3-7: A C1/3 versus a plot for Mode I tests

80

Table 3-1: Parameters of compliance calibration by the MBT method

Specimen Δ, mm

m, 1/(mm2/3N1/3)

1-1 9.594 7.580E-03 1-2 6.626 7.315E-03 1-3 6.238 7.196E-03 1-4 8.378 7.368E-03 1-5 7.386 7.479E-03

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

0.5 1.0 1.5 2.0 2.5 3.0(bC )1/3, (mm2/N)1/3

a/2

h

Specimen 1-1Specimen 1-2Specimen 1-3Specimen 1-4Specimen 1-5

03/1

1 )(2/ αα += bCha

Fig. 3-8: An a/2h versus (bC)1/3 plot for Mode I DCB tests

Table 3-2: Parameters of compliance calibration by MCC method

Specimen α0 α1 1-1 -2.5720 12.240 1-2 -1.6155 11.872 1-3 -1.5513 12.094 1-4 -2.0847 11.681 1-5 -1.8749 11.730

81

Based on the two compliance calibration plots, an interesting observation was

made. If the initial compliance vs. initial crack length data were also plotted, in the first

plot by the MBT method, this data point will fall below the fitted straight line for the

corresponding specimen, while it will fall above the fitted straight in the second plot by

the MCC method. Noticing that the compliance is involved in the y-coordinate in the

first plot and x-coordinate in the second plot, both plots seem to consistently indicate that

the specimen at the initial crack length is stiffer than that predicted by the fitted

compliance calibration relation. However, in fact, this disagreement is caused by the

dissimilarity of crack front shapes at the initial crack length and at the subsequent

propagated crack lengths, which will be explained in detail in Section 3.2.2.5.

3.2.2.3 GIc onset values

In the first loading cycle, the specimen was loaded until the crack had propagated

for about 5 mm or unstable crack growth had arrested. Then, the DCB specimen was

unloaded and subject to another four loading/unloading cycles. The onset value of SERR,

GIc, was determined for the crack onset from the insert film, at the critical (maximum

load) point of the first loading cycle. The crack onset from a Mode I precrack was

determined from the critical point of the first reloading cycle (second loading cycle). The

critical points were found from load vs. displacement curves. The loading curve is

typically linear up to the maximum load point, which is typically the visual crack onset

point as well. As discussed before, two kinds of load-displacement behavior were

observed at crack onset of first loading cycle, with the first one having a small amount of

load drop at the critical onset point, and the second one a large amount of load drop at the

instance of unstable crack growth of about 3-7 mm in one step. These two kinds of

behaviors correspond to the load vs. displacement curves for first loading/unloading cycle

in Figure 3-9 and Figure 3-10, respectively.

82

y = 13.427x - 0.1079R2 = 0.9994

y = 9.1699x - 1.9322R2 = 0.9998

0

10

20

30

40

50

0 1 2 3 4 5Displacement, mm

Load

, NLoadingUnloading1st PmaxLinear (Loading)Linear (Unloading)

P cr

Fig. 3-9: Load vs. displacement curve -- small load drop occurred at crack onset. (Specimen 1-1)

y = 11.03x + 0.4456R2 = 0.9996

y = 16.549x + 1.6032R2 = 0.9997

0

10

20

30

40

50

0 0.5 1 1.5 2 2.5 3 3.5Displacement, mm

Load

, N

LoadingUnloading1st PmaxLinear f it (Unloading)Linear f it (Loading)

Fig. 3-10: Load vs. displacement curve -- large load drop occurred at crack onset. (Specimen 1-3)

83

The onset GIc values from starter film and precrack (first reload) were calculated

with MBT (Eq. (2.3)) and MCC (Eq. (2.15)) methods, respectively. The results from

different specimens are compared in Figure 3-11 and numerical values are available in

Appendix Table B-8. It was obtained that the onset value from the insert film calculated

by MCC was always higher than that by MBT. For Specimens 1-1 and 1-2, for which a

small load drop occurred at the crack onset, the GIc value based on the precrack is higher

than that based on the insert film. The opposite holds for the other cases, for which

significant unstable crack growth occurred in the initial loading cycle. From the DCB test

results found from the literature (Appendix A, Table A-1), the GIc values for crack onset

from an insert film range from 100 to 290 J/m2. For the investigated material, the GIc

onset values based on an insert film determined by different calculation methods range

from 110 to 150 J/m2. Compared to the results from the literature, the Mode I quasi-static

fracture toughness of the investigated material is near the low end of the common range

for carbon/brittle epoxy materials.

0

20

40

60

80

100

120

140

160180

200

1-1 1-2 1-3 1-4 1-5Specimen No.

GIc, J

/m2

MCC, insert film MBT, insert filmMCC, precrack Average, by MCCAverage, by MBT Average, precrack, by MCC

Fig. 3-11: GIc onset values

84

3.2.2.4 Mode I resistance curve

During the test, the crack lengths were measured visually at certain points using

an instrumented-stage telescope with 0.001 mm resolution. After the test, strain energy

release rates were calculated at these points with the measured crack length and

compliance calibration relations. The modified beam theory (Eq. (2.3)) and modified

compliance calibration method (Eq. (2.6)) were used for SERR calculations. However,

the SERR values obtained by the two methods are very similar except for the first point

(crack onset), as shown in a typical case in Figure 3-12. An overall Mode I IFT resistance

curve fit to the data from all five specimens is shown in Figure 3-13.

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60Δa, mm

GIc

, J/m

2

MCC methodMBT methodOnset value by MBT methodOnset value by MCC methodPow er (MCC method)Pow er (MBT method)

Fig. 3-12: Mode I IFT resistance curves for Specimen 1-2

85

A relatively flat resistance curve was obtained for the investigated material,

compared to other laminated material system. For example, for a glass/vinyl ester

material, Shivakumar et al. (2006) obtained that GIR value increased by about 140%

comparing to onset value GIc after 50 mm crack extension. For all the DCB specimens in

this investigation, the toughness increased by less than 40% after about 50 mm crack

growth. The flat resistance curve is typical of brittle matrix laminates as was found in

some round robin tests (Hojo et al. 1995).

3.2.2.5 Discussion of special issues

Some special issues concerning Mode I testing are discussed in this section based

on experimental observations and results obtained for the currently investigated material.

These issues, either unresolved in the literature or not reported for carbon/brittle epoxy

material, include: i) whether to measure critical SERR at crack onset (GIc) based on a

Fig. 3-13: Overall Mode I IFT resistance curve for five DCB specimens, by MCC method

86

starter film or a precrack, ii) accuracy of SERR calculation at crack onset using the MBT

or MCC method, iii) effect of reloading/unloading, and iv) fiber bridging. They are

discussed as follows.

i) Whether to measure GIc based on a starter film or precrack?

The advantages of measuring GIc based on the starter film is that it forms an

identical straight shape of crack front, and additionally, it is not dependent on the loading

history. The latter advantage is important for materials with an increasing fracture

resistance curve with crack extension because GIc measured from a precrack is higher

than that measurement from the starter film, and the difference between the two depends

on the precrack length and precracking methods. However, for the material system in this

investigation, the fracture resistance curve is very flat after ~ 5 mm crack extension. Even

though a very thin (13 μm-thickness) insert film was used as the starter film, it is still

likely to create a resin rich pocket and unstable crack growth occurs at crack onset. For

this reason, as shown in the GIc results by different methods in Figure 3-11, the average

GIc based on the precrack and MCC is even more conservative than GIc based on the

insert film and MCC. Furthermore, the variation of GIc values based on the precrack is

less than that based on insert film. Therefore, the GIc value based on a Mode I precrack is

recommended to be a more conservative and repeatable material property for the material

system in this investigation.

ii) Accuracy of SERR calculation of the MBT and MCC methods at crack onset

and propagation

Comparing MBT and MCC methods (Figure 3-12), the results of critical SERR

calculations by these two methods are very similar except for the onset value (GIc). The

onset value (GIc) obtained using the MBT method is always less than that obtained using

the MCC method. This disagreement is a result of the dissimilarity of crack front shapes

at the initial crack length and at the subsequent propagated crack lengths, as mentioned in

the earlier section. The theoretical methods assume plane strain or plane stress condition,

and the crack front shape across the specimen width with crack extension is always

straight. However, in reality, the crack front shape in a DCB specimen develops from an

87

initially straight shape into a curved shape after certain length of propagation, as shown

in Figure 3-14. This implies that, shortly after crack initiation, the crack growth at the

center of specimen is more than that on the edge. The differentiation of compliance to

crack length, dC/da, predicted by theories is somewhat inaccurate. Hence, the SERR at

crack onset predicted by either MBT or MCC is somewhat inaccurate because the crack

front changes from a straight to a convex shape.

A comparison of how the crack propagates theoretically and experimentally is

shown in Figure 3-15. Theoretically, the crack starts from the straight front formed by

insert film (shown in solid blue line), and advances with a same straight front shape as

shown in dashed blue lines. However, during a test, the crack propagates in a curved

shape rather than straight, as shown in solid black lines.

Approximated crack front

Fig. 3-14: Crack surfaces of a quasi-static DCB test specimen

88

Assume that the beam theory can be still used to predict the compliance versus

crack length relationship if the crack length measured on the edge, a, is replaced by an

average crack length, aavg, which is given by, Eq. (3.10)

Parameter k1 is positive since the crack length is longer in the center than on the edge of

specimen. If the crack front shape during propagation is self-similar, then k2 = 0. If the

crack front shape becomes flatter as crack propagates, then k2 > 0; if it is the opposite,

then k2 < 0.

Based on this assumption, the compliance versus crack length relation can be

written as,

From Eq. (3.12), C1/3 is still in a linear relationship with respect to crack length measured

on the specimen edge, as assumed by the MBT and MCC methods. Comparing Eq. (3.12)

to the compliance vs. crack length relationship by the MCC method, by Eq. (2.5)

(neglecting the effect of loading block, N=1),

a0

ai

aj

Insert film

Loading line(aavg)i

(aavg)jEnd of insert film

Straight crack front assumed by theories

Propagated curved crack front

Fig. 3-15: A schematic of crack propagation

avg 1 2a a k k a= + − (3.10)

1/30 1 avg 0 1 1 2' '( ) ' '( )C A A a A A a k k a= + = + + − (3.11)

( )1/30 1 1 1 2' ' '(1 )C A A k A k a= + + − (3.12)

89

if a curved front shape exists in a DCB specimen, parameter A0 will be larger than that

predicted by theoretical solution with straight front shape assumption . In the MBT

compliance calibration method, it will result in a higher value of end correction factor Δ

than the theoretical prediction (Eq. (2.8)). However, since C1/3 is still linearly related with

respect to crack length, as assumed by the MBT and MCC method, the SERR value for

crack propagation calculated by both methods are still accurate.

Based on the proceeding discussion, since the calculated GIc values onset from an

insert film is not accurate by both MBT and MCC methods, it is again recommended to

use the initiation value based on a short Mode I precrack.

iii) Effect of reloading/unloading

From Figure 3-5 and Figure 3-6, one can see that after a loading/unloading cycle,

when the load returns to zero, there is a small positive offset in displacement relative to

the original displacement before loading. This might be caused by the plastic deformation

occurring near the crack tip, and/or fibers pulling out of the material after the formation

of new fracture surfaces. In addition, when another loading cycle is applied, the critical

load and displacement for crack onset is more than the load and displacement for crack

growth at the end of the last loading cycle, respectively. As a result, the unloading and

reloading procedure may have an effect of increasing the perceived toughness value (GIR).

In Figure 3-16, the critical SERR during crack extension calculated by MCC method, is

plotted against the crack length calculated by the MCC compliance calibration

relationship for Specimen 1-4. As shown in the circled part of the resistance curve, at the

same crack length the critical SERR for crack growth is increased after an unloading-

reloading procedure.

1/30 1C A A a= + (3.13)

90

iv) Fiber bridging

Small scale fiber bridging was observed during tests, as shown in Figure 3-17.

Comparing to laminates with tougher matrix, the effect of fiber bridging is small for the

material system in this investigation, as was indicated by the flat resistance curve as well.

0

20

40

60

80

100

120

140

160

40 50 60 70 80 90 100 110

Crack length, a , mm

GIR

, J/m

2

Crack onset

where reloading-unloading occurs

Fig. 3-16: Fracture resistance curve for Specimen 1-4, with crack length calculated by compliance calibration by MCC method

91

(a)

(b)

Fig. 3-17: Fiber bridging observed through long distance microscope

Fiber bridging

Fiber bridging

92

3.3 The Mode I Fatigue IFT Testing

The DCB fatigue tests use the same test configuration as the quasi-static tests. The

test method involves a brief precrack process, fast cyclic loading and some slow

compliance calibration cycles. Test results are presented as crack growth with increasing

cycles and the relation between SERR and crack growth rate. No test standard exists for

characterizing crack growth behavior using the modified Paris law.

3.3.1 Mode I fatigue test method

The Mode I fatigue test uses the same setup as the Mode I quasi static test as

shown in Figure 3-3. Before the fatigue test, the specimen was precracked by a short

Mode I quasi-static test according to the following procedures:

1) The specimen was loaded by increasing crack opening displacement (COD)

until crack onset was visually observed. As seen in some of the previous quasi-static

tests, the crack growth was unstable immediately after crack onset during the test.

The actuator movement was stopped after the visual onset of crack growth was

observed.

2) After the unstable crack growth was arrested, the specimen was unloaded

slightly and then loaded again until stable crack propagation was observed. The total

precrack growth (Δapr) was measured by a long distance microscope and recorded at

the maximum crack opening displacement. The maximum crosshead opening

displacement was recorded and used as the critical displacement (δIc) at the crack

length of a0 + Δapr for the fatigue test.

3) The specimen was unloaded to zero opening displacement.

The loading procedures for the precrack test are shown schematically in Figure 3-18.

93

The fatigue test started right after the Mode I precrack test. The test was carried

out in displacement control. The opening displacement was cycled between the maximum

displacement (δImax) and minimum displacement (δImin) using a sine wave. The maximum

opening displacement was set to Ic 00.8( ( ))pra aδ + Δ , corresponding to 80% of the Mode

I critical SERR at this crack length. A loading frequency of 10 Hz was used for cyclic

loading and a slower loading frequency of 0.0028 Hz was used for compliance

measurement at certain cycle numbers (N =1, 100, 200, 500, 1000, 2000… 50,000).

Maximum and minimum displacements were recorded for every cycle. During slow

cycles, crack growth was measured by a QM-1 long distance telescope. Additionally,

load-displacement data were recorded throughout every slow cycle to obtain the

specimen’s compliance. Testing parameters for the four fatigue tests are summarized in

Table 3-3. The displacement ratio R was selected such that the minimum SERR is much

smaller than maximum SERR, but the minimum displacement is still large enough to

avoid crack closure.

δIc(a0 + Δapr)

P

δ

Fig. 3-18: A schematic showing the loading and unloading procedures for the Mode I precrack test

94

The maximum SERR reduces very quickly as the Mode I crack grows in a

displacement controlled DCB test, and it is predicted by beam theory to be inversely

proportional to the fourth order of the effective crack length, (a + Δ)4, as expressed in

Eq. (3.14).

A plot of normalized maximum SERR vs. crack length predicted by simple beam theory

for the current nominal specimen geometry (a0 = 50 mm) is shown in Figure 3-19.

Table 3-3: Mode I Fatigue test parameters

Specimen No. 4-2 4-3 4-4 4-5 Displacement ratio: R 0.2 0.2 0.2 0.2 Loading frequency (Hz): f 10 10 10 10 Maximum displacement (mm): δImax 3.6 3.85 4.12 3.687 Minimum displacement (mm): δImin 0.72 0.77 0.82 0.737 Critical displacement before fatigue test (mm): δIc(a0+Δapr) 4.02 4.30 4.60 4.12

Precrack length (mm): Δapr 9.716 12.89 12.866 12.852 * Except for periodic crack and compliance measurements, where f = 0.0028 Hz.

( )( )

2 40Imax Imax

4Ic Ic

aGG a

δδ

+ Δ⎛ ⎞= ⎜ ⎟

+ Δ⎝ ⎠i (3.14)

95

3.3.2 Mode I fatigue test results


The purpose of compliance calibration for fatigue tests is to 1) theoretically

predict crack growth based on the compliance obtained from the load-displacement curve,

and 2) obtain the correction factor Δ for SERR calculation by the MBT method, or the

multiplier A1 for SERR calculations by the MCC method. Similar to quasi-static tests,

compliance calibration could be done based on two analytical methods, the MBT method

(Figure 2-3) and the MCC method (Figure 2-5). However, based on these two

compliance calibration methods, crack growth could be predicted by at least four

alternative approaches:

0%

20%

40%

60%

80%

100%

50 55 60 65 70 75 80Crack length, a , mm

GIm

ax/G

Ic, %

δ Imax = 1.1δ Ic(a 0)




Fig. 3-19: Mode I maximum SERR reduction as crack grows for a displacement controlled DCB fatigue test

96

i) obtain C-a relationship by MBT from quasi-static test results, 1/3 ( )s sC m a= + Δ ; predict crack growth by 1/3

1 ( ) / sa C mΔ = Δ ;

ii) obtain C-a relationship by MBT method from current fatigue test result, 1/ 3 ( )f fC m a= + Δ ; predict crack growth by 1/ 3

2 ( ) / fa C mΔ = Δ ;

iii) obtain C-a relationship by MCC method from quasi-static test results,

ss BCAha += 3/112/ ; predict crack growth by 1/ 3

3 12 ( )sa hA CΔ = Δ ;

iv) obtain C-a relationship by MCC method from current fatigue test result:

sf BCAha += 3/112/ ; predict crack growth by )(2 3/1

14 ChAa f Δ=Δ ;

where the subscript “s” indicates a parameter from static tests, and “f” indicates one from

a fatigue test. Details of calculation methods are listed in Table 3-4.

Table 3-4: Crack growth prediction approaches for DCB fatigue tests

Approach i) Approach ii) Approach iii) Approach iv)

C-a data for calibration from

five quasi-static tests

one fatigue test with the specimen of interest

five quasi-static tests

one fatigue test with the specimen of interest

Theory, plotting method

MBT, Figure 2-3

MBT, Figure 2-3

MCC, Figure 2-5

MCC, Figure 2-5

Calibration parameter obtained

ms, Δs mf, Δf 1 0, s sα α 1 0, f fα α

C-a relation obtained

1/3 ( )s sC m a= +Δ 1/3 ( )f fC m a= +Δ 1/31 0( )

2 s sa bCh

α α= + 1/31 0( )

2 f fa bCh

α α= +

Crack growth predicted by

1/31 ( ) / sa C mΔ =Δ 1/3

2 ( ) / fa C mΔ =Δ 1/33 12 ( )sa h bCαΔ = Δ 1/3

4 12 ( )fa h bCαΔ = Δ

SERR calculated by ( )

32I

s

PGb a

δ=

+ Δ

( )3

2If

PGb a

δ=

+ Δ 2 2 / 3

21

32 (2 )I

s

P CGb hα

= 2 2 / 3

21

32 (2 )I

f

P CGb hα

=

97

Crack growth calculations based on the compliance measured at certain cycles

were done using these four approaches for one specimen (Specimen 4-2), as an example.

The results of crack growth predicted by these four approaches are compared to crack

growth measured visually (Δa) in Figure 3-20. The crack growth predicted by the MBT

method is very similar to that predicted by the MCC method. However, the crack growth

calculated using the compliance calibration relation obtained from quasi-static tests

(approach i) and iii)) under-predict the crack growth by 0-1 mm relative to the crack

growth calculated from the compliance calibration relation from the fatigue test

(approach ii) and iv)). One possible explanation for this could be: at a same crack length,

the compliance of a fatigue specimen is less than the compliance of a quasi-static

specimen because more severe fiber bridging occurs in a fatigue specimen and thus make

the fatigue specimen stiffer. The difference between the MBT and MCC method is small

as shown in Figure 3-20.

0

1

2

3

4

5

6

7

8

0 20,000 40,000 60,000 80,000

N, cycle

Δa

, cra

ck le

ngth

, mm

a measured visually

a calculated by Eqn. (1)




1/31 ( ) / sa C mΔ = Δ

1/32 ( ) / fa C mΔ = Δ

1/33 12 ( )sa hA CΔ = Δ

)(2 3/114 ChAa f Δ=Δ

, measured visuallyaΔi)

ii)iii)iv)

MBT

MCC

Fig. 3-20: Crack growth by different methods (Specimen 4-2)

98

Since the compliance calibration by approaches i) and iii) usually under-predict

the crack growth and the results by approaches ii) and iv) are similar, crack growth

calculation for other specimens was done using approach iv) only.

3.3.2.2 Crack growth

Crack growth measured visually and that calculated by the MCC method

(approach iv)) based on the measured compliance were plotted against number of cycles

in Figure 3-21 for all Mode I fatigue specimens. From Figure 3-21, the crack growth by

visual measurement is relatively “noisy” compared to the crack growth calculated by the

MCC compliance calibration method. This is caused by the fact that the visual

measurement focuses on the crack growth on one specimen edge, while the compliance

calibration method is reflective of crack growth across the entire specimen width. Hence,

the error in visual measurements could be easily introduced by a small scale fiber

bridging in the edge area, or a thicker white painting in some local areas.

99

3.3.2.3 Crack growth rate (da/dN) vs. maximum SERR (GImax) plots

The crack growth rate, da/dN, with crack growth obtained by visual measurement,

was plotted against Mode I maximum strain energy release rate, GImax, for all Mode I

fatigue specimens, as shown in Figure 3-22. Also, the crack growth rate, da/dN, with

crack growth calculated by compliance calibration of MCC method, was plotted against

Mode I maximum strain energy release rate, GImax, for all Mode I fatigue specimens, as

shown in Figure 3-23. The power law relationship by Eq. (2.73) was used to fit the data

from all test specimens. The exponent of the power law relation found by using visually

measured crack growth and by using calculated crack growth are 11.12 and 12.29,

respectively. Additional crack growth rate (da/dN) vs. maximum SERR (GImax) plots for

individual specimens are given in Appendix B (Figure B-1 to Figure B-8).

60

62

64

66

68

70

72

74

0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000Cycle No., N

Cra

ck le

ngth

, a, m

m

4-2, by CC 4-3, by CC4-4, by CC 4-5, by CC4-2, by visual 4-3, by visual4-4, by visual 4-5, by visual

Fig. 3-21: Crack growth by various methods

100

Fig. 3-22: da/dN vs. GImax plots for four DCB fatigue specimens (with crack growth by visual measurement)

101

Comparing Figure 3-22 to Figure 3-23, there is more scatter in the plot for which

crack growth is obtained by visual means, which is expected because of the inherent

uncertainty in crack length measurement. The exponents of the Modified Paris law

( ( ) InaxI GBdNda Im/ = ) are similar in the two plots. Comparing to the exponents found in

literature (Appendix Table A-3), which are in the range of 3.6 to 15, the exponents found

in this investigation (11.12 and 12.29) are near the high end of the common range. This

indicates that the crack growth rate decreases very quickly as the crack driving force

decreases. Also, the high exponent indicates that a small error in the crack driving force

prediction will result in a large error in the crack growth prediction, and thus large error

in life prediction of a structure made of this material.

Fig. 3-23: da/dN vs. GImax plots for four DCB specimens (with crack growth calculated by compliance calibration).

102

Chapter 4

The Mode II Interlaminar Fracture Toughness Testing

Mode II interlaminar fracture toughness tests under quasi-static and cyclic

loadings were conducted with the End Notched Flexure (ENF) specimen. ENF test

configurations, methods, and results are presented in this chapter.


The specimens used in this investigation are machined from two flat

carbon/epoxy panels of [0]12 lay-up as shown in Figure 4-1 and Figure 4-2. A thin Teflon

film of 12.7 μm thickness was inserted at the mid-plane during the lay-up process of each

panel to define the initial starter crack. The ENF specimens were cut from the panels at

the Penn State Composites Lab using a water-cooled diamond abrasive cut-off wheel.

Distributions of tested ENF specimens in the panels are shown in Figure 4-1 and

Figure 4-2.

103



Specimen 2-1


Specimen 2-4 Specimen 3-1

Specimen 3-3

Fig. 4-1: The ENF and SLB panel diagram (Panel B)

104

The ENF specimen geometry and notations are shown in Figure 4-3. For the

three-point bending fixture used in this investigation, the three loading and supporting

pins are attached to the bending fixture by three springs, so that the loading pin can rotate

about its centerline and the two supporting pins can roll along the longitudinal direction

of the bending fixture. The quasi-static tests were conducted with seven ENF specimens.

In four of these (Specimen 2-1, 2-2, 2-3, and 2-4 in Figure 4-1) the crack initiated from

an insert film (un-precracked specimens) and in three (Specimen 2-5, 2-6, and 2-7 in

Figure 4-2) the crack initiated from a short Mode I precrack that extended beyond the

insert film (precracked specimens). Fatigue tests were conducted with three ENF

specimens, namely Specimen 5-3, 5-4, and 5-5 in Figure 4-2. The dimensions of ENF

specimens in terms of length × width × thickness are approximately 150 × 25.4 × 3.9 mm,

with an initial artificial crack created by the embedded thin film of about 50.8 mm length



Specimen 2-5


Specimen 5-4


Fig. 4-2: The ENF panel diagram (Panel AA)

105

for the un-precracked specimens. For the precracked specimens, the length of the

embedded thin film is about 76.2 mm length and this artificial crack was extended further

by about 3-5 mm in Mode I loading before testing. Dimensions of un-precracked

specimens tested quasi-statically are listed in Appendix B Table B-3 , while dimensions

of precracked specimens tested quasi-statically and in fatigue are listed in Appendix B

Table B-4 and Table B-5, respectively.

The ENF tests were conducted with two kinds of configurations, since specimens

with two different initial crack lengths were used. Because the precracked specimens had

longer initial delamination lengths, in order to create enough room for crack growth, the

three loading pins were shifted a certain distance away from the initially cracked end of

specimen. The un-precracked test configuration, denoted Configuration A, is described in

Figure 4-4 and Table 4-1. The precracked test configuration, denoted Configuration B, is

described in Figure 4-5 and Table 4-2.

l

b

2h

a0

Support

Loading direction

Support Fig. 4-3: A schematic of ENF specimen geometry and test configuration

106

L L crcl

l

Supporting line Supporting line

r1

r2

a0Test specimenLoading line

Loading direction

Fig. 4-4: Schematic of ENF un-precracked test configuration A

Table 4-1: Dimensions for ENF configuration A (un-precracked specimen).

Notation Parameter measured Dimension, mm (in.) l Specimen length 152.4 (6.0) L Distance from support to loading point 50 (2.0) r1 Radius of loading nose 3.2 (0.125) r2 Radius of support 6.4 (0.25) cl Left overhang 25.4 (1.0) cr Right overhang 25.4 (1.0)

L L crcl

l

r1

r2

ap

Loading direction

Loading line


Test specimen

L L crcl

l

r1

r2

ap

Loading direction

Loading line


Test specimen

Fig. 4-5: Schematic of ENF precracked test configuration B

107

4.2 The Mode II Quasi-static IFT Testing

The Mode II quasi-static IFT tests were conducted with ENF precracked and un-

precracked specimens. Since it was found that a large amount of unstable crack growth

occurred at initiation in both quasi-static and fatigue tests with ENF un-precracked test

configuration, an ENF precracked test configuration was used in an attempt to solve the

instability issue. The result did show stable crack growth with ENF test configuration B

(precracked specimen) in the fatigue tests. From quasi-static tests, the initiation value of

GIIc obtained for precracked specimens is much lower than that obtained for un-

precracked specimens.

4.2.1 Mode II quasi-static test method

Photographs of the ENF test setup are shown in Figure 4-6 for the un-precracked

test configuration and in Figure 4-7 for the precracked test configuration. A three-point

bending fixture with a maximum span length of 203.2 mm was used as the loading fixture.

The specimen was loaded by a servo-hydraulic MTS 810 machine. Load was measured

by a 13.5 kN MTS load cell using the 2.2 kN (500 lb) load range, and a 1.1 KN (250 lb)

Table 4-2: Dimensions for ENF test configuration B (precracked specimen)

Notation Section of test specimen measured Dimension of specimen, mm (in.)

l Specimen length 152.4 (6.0) L Half span length of bending fixture 50 (2.0)

r1 Radius of loading roller 6.4 (0.25) (same as configuration A)

r2 Radius of supporting rollers 3.2 (0.125) (same as configuration A)

cl Left overhang 43.4 (1.709) cr Right overhang 9.2 (0.362)

108

load cell was also connected into the load frame in order to check the accuracy of 500 lb

load cell measurement. The loading point displacement was measured by the MTS LVDT

built into the actuator.

MTS built-in load

cell (stationary)

1.1 kN (250 lb) load cell

Bending fixture

Specimen

MTS built-in load cell (stationary)


Bending fixture

Specimen

Fig. 4-6: A photograph of the ENF test setup with un-precracked test configuration

109

Generally, the Mode II interlaminar fracture toughness test was conducted in three

steps: 1) precracking and marking (or just marking for tests with un-precracked

specimens), 2) compliance calibration, and 3) crack initiation test. For compliance

calibration and crack initiation, the test was conducted under displacement control with a

constant displacement rate of 0.5 mm/min for loading and unloading. Details of the

testing procedures are as follows:

(1) Before any testing, the specimens slated for precracking were precracked in

Mode I by driving a thin blade into the manufactured crack created by the embedded

thin film. The specimen was clamped completely across the width at the position of

the intended crack front. The blade was taken out from the specimen before the

specimen was unclamped.

Fig. 4-7: A photograph of the ENF test setup with precracked test configuration

110

(2) To obtain the compliance versus crack length relationship, compliance

calibration procedures were conducted before the crack initiation test at crack lengths

of a0, a0 ± 10, and a0 ± 5 mm (a0 is the initial crack length based on the insert film)

for test configuration A, and at crack lengths of ap + 9, ap + 6, ap + 3, ap and ap - 3

mm for test configuration B. Marks with increment of Δa were made on the edge of

the specimen for locating the corresponding positions of the rollers for compliance

calibration at each crack length, as shown schematically in Figure 4-8. The crack

length increment, Δa, was 5 mm for ENF test configuration A, and 3 mm for ENF test

configuration B. For each compliance calibration test, the specimen was loaded to a

load point displacement of about 50% - 60% of the estimated critical displacement

and then unloaded. Five initial crack lengths were achieved by sliding the specimen in

the bending fixture in the longitudinal direction of specimen. Care was taken to

prevent crack onset during the compliance calibration procedures.

(3) For the crack initiation test, the specimen was placed in the bending fixture as

shown schematically in Figure 4-4 for ENF test configuration A and Figure 4-5 for

ENF test configuration B. The specimen was loaded until crack initiation was

observed, and then unloaded.

Δa

ap

Fig. 4-8: Markings on ENF specimen edge for compliance calibration

111

4.2.2 Mode II quasi-static test results


Typical load vs. displacement plots for ENF crack initiation tests with un-

precracked and precracked specimens are shown in Figure 4-9 and Figure 4-10,

respectively. These two plots show some differences between the un-precracked

specimen with shorter initial crack length (a0/L ≈ 0.5), and precracked specimen with

longer initial crack length (ap/L ≈ 0.7).

y = 441.81x - 40.558R2 = 0.9999

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3Displacement, mm

Load

, N

LoadingUnloading

Pcr

Fig. 4-9: A representative load vs. displacement plot for ENF crack onset test (Specimen 2-2, un-precracked, a0/L ≈ 0.5)

112

In the un-precracked cases tested, the amount of load drop immediately after

crack initiation ranged from 497 to 731 N. The unstable crack in some tests grew to a

point that was about 10 mm past the center loading roller. In the test for which the load-

displacement plot is shown in Figure 4-9, immediately after the unstable crack onset the

load dropped about 630 N. Additionally, before the crack onset critical point, the loading

curve is particularly linear (except for the initial nonlinear load take-up portion). Hence

for the un-precracked configuration, the critical point for crack onset critical SERR

calculation is unambiguously the maximum load point.

In the precracked cases tested, the amount of load drop immediately after crack

initiation ranged from 52 to 232 N. The unstable crack growth typically stopped

somewhere near the center loading roller. In one test, the crack growth was quasi-stable,

showing both unstable and stable crack growth behaviors before the crack observed

through the telescope reached the center loading roller, and the load vs. displacement

curve for this test is shown in Figure 4-11. Additionally, in contrast to the un-precracked

specimens, nonlinearity in the load-displacement curve was observed for the precracked

Linear fit: y = 399.47x - 15.871R2 = 0.9987

0

100

200

300

400

500

600

700

800

0.0 0.5 1.0 1.5 2.0Displacement, mm

Load

, N

P_max loadP_nonlinearP_5%offset

5% Complianceoffset

Fig. 4-10: A load vs. displacement plot for ENF crack onset test (Specimen 2-5, precracked, test configuration B)

113

specimens. In most pre-cracked specimens, the maximum load occurred before the

compliance increased 5% relative to the initial compliance, and after nonlinearity became

obvious. Hence, the critical point for crack onset critical SERR was defined as the

maximum load point in precracked specimens, according to JIS standard (JIS K 7086

1993).

In Figure 4-10, Figure 4-11 and the later plots in which the 5% compliance offset

construction line is shown, the 5% compliance offset construction line is defined by

assigning the x-intercept and slope as follows. Firstly, the initial compliance straight line

is constructed by fitting a straight line through the load vs. displacement test data where

the load-displacement relation is observed to be linear (usually from the point where

displacement is greater than 0.24 mm in an ENF test or 0.22 mm in an SLB test to the

point fracture initiates). Then, the 5% compliance offset straight line is constructed in

such a way that, it intersects with the x-axis at the same point as the initial compliance

straight line and has a slope which is 1/1.05 times that of the initial compliance line. The

construction method is shown schematically in Figure 4-12.

Linear fit: y = 391.6 x - 29.158R2 = 0.999

0

100

200

300

400

500

600

700

0.0 0.5 1.0 1.5 2.0 2.5Displacement, mm

Load

, N

P_maxP_5% offsetP_nonlinear

5% complianceoffset

Fig. 4-11: A quasi-stable load vs. displacement plot for ENF crack onset test (Specimen 2-6, precracked)

114


Some typical load-displacement plots for a compliance calibration test are shown

in Figure 4-13. The compliance value at each initial crack length was obtained by taking

the inverse of the slopes of straight lines best-fitted to the linear portion of loading curves

(where displacement is greater than 0.24 mm or so). Similar compliance calibration

procedures were conducted for both ENF quasi-static specimens and fatigue specimens.

Hence, the results are presented together.

Load

, P

2 1

1 11.05S S

=

Fig. 4-12: Construction method for the 5% compliance offset line

115

Two forms of polynomial were used to correlate compliance and crack length.

Firstly, it is predicted by classical plate theory that compliance of an ENF specimen is

given by Eq. (2.30). Hence, a polynomial in the form of Eq. (4.1) was used to fit the

compliance vs. crack length data.

Parameters A and B were determined from a linear least squares fit of the C vs. a3 plot.

Parameter B is only related to the flexural modulus of the specimen. The ratio A/B is only

related to half span of the bending fixture, L. The first compliance calibration method is

denoted by CC 1) in the following discussion. A plot of C(8bh3) vs. a3 for all specimens

is given in Figure 4-14. The parameters determined are listed in Table 4-3.

L inear fit (a=43.7 mm):

y = 295.61x ‐ 21.618, R 2 = 0.9988


y = 321.36x ‐ 20.274, R 2 = 0.9985


y = 351.22x ‐ 21.559, R 2 = 0.9985


y = 383.56x ‐ 22.888, R 2 = 0.9988


y = 412.71x ‐ 26.888, R 2 = 0.9986

0

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1 1.2 1.4

D is placement, mm

Load

, N

a=43.7a=40.7a=37.9a=35.0a=32.0

Fig. 4-13: Load vs. displacement curves for compliance calibration (precracked Specimen 2-6, at initial crack lengths of a = 32.0, 35.0, 37.9, 40.7 and 43.7 mm)

33)8( BaAbhC += (4.1)

116

0.0E+00

1.0E-15

2.0E-15

3.0E-15

4.0E-15

5.0E-15

6.0E-15

7.0E-15

8.0E-15

-2.0E-05 0.0E+00 2.0E-05 4.0E-05 6.0E-05 8.0E-05 1.0E-04 1.2E-04

Crack length, a3 , m

C(8bh

3 ), (

m5 /N

)

2-1 2-2 2-32-4 2-5 2-62-7 5-3 5-45-5

33 )8( BaAbhC +=

Fig. 4-14: C(8bh3) vs. a3 plots for all ENF specimens

117

Secondly, another polynomial in the form of Eq. ((4.2) was used to produce

compliance vs. crack length relationship as well. Parameters, C0, C1, C2, and C3, were

determined by a cubic least squares fit. The second compliance calibration method is

denoted by CC 2) in the following discussion.

The second plot made for compliance calibration is shown in Figure 4-15. Parameters

determined by the second compliance calibration methods are listed in Table 4-4.

Table 4-3: Parameters A and B, determined by CC 1) for ENF specimens

Specimen A, m4/N B, m/N A/B, m3 R2 2-1 2.56E-15 3.32E-11 7.69E-05 0.992 2-2 3.01E-15 3.16E-11 9.55E-05 0.998 2-3 2.67E-15 2.93E-11 9.11E-05 0.996

Un-precracked

2-4 2.37E-15 3.15E-11 7.54E-05 0.999 STD * 2.69E-16 1.61E-12 1.01E-05 0.003 Mean 2.65E-15 3.14E-11 8.47E-05 0.996 Statistical

results COV ** 10.15% 5.12% 11.87% 0.33%

Specimen A, m4/N B, m/N A/B, m3 R2 2-5 2.58E-15 2.89E-11 8.94E-05 1.000 2-6 2.78E-15 2.96E-11 9.40E-05 0.999 2-7 2.57E-15 3.01E-11 8.53E-05 0.999 5-3 2.62E-15 2.98E-11 8.79E-05 0.999 5-4 2.74E-15 3.36E-11 8.15E-05 1.000

Precracked

5-5 2.42E-15 3.00E-11 8.07E-05 1.000 STD 1.31E-16 1.69E-12 5.06E-06 0.0003 Mean 2.62E-15 3.03E-11 8.65E-05 0.999 Statistical

results COV 4.99% 5.56% 5.85% 0.03%

Note: * standard deviation ** coefficient of variation, STD/COV

2 30 1 2 3C C C a C a C a= + + + (4.2)

118

1.0E-06

1.5E-06

2.0E-06

2.5E-06

3.0E-06

3.5E-06

4.0E-06

0 0.01 0.02 0.03 0.04 0.05Crack length, a , m

Com

plia

nce,

C, m

/N

2-2 2-3 2-42-1 2-5 2-62-7 5-3 5-45-5

2 30 1 2 3C C C a C a C a= + + +

Fig. 4-15: A C vs. a plot for compliance calibration, by Eq. (2.39)

119

Comparing the two compliance calibration methods, CC 1) the compliance

calibration by Eq. (4.1) and CC 2) by Eq. (2.39), the second method fits the experimental

data better based on R2 values. Since there are two more parameters in Eq. (2.39) than

Eq. (4.1), the relation by CC 2) should be better fitted to test data. However, the two

additional parameters (C1 and C2) are not involved in any physical property of the ENF

specimen. The better fit provided by CC 2) might be a result of including the errors of the

crack length measurement, the effect of the diameter of the loading and supporting pins,

effect of friction and others, into the final compliance calibration relation. With these

additional effects incorporated, it is uncertain whether the resulting CC relation is

reflective of the compliance vs. crack length relation of the ENF specimen or the

compliance vs. crack length relation of the whole loading system. Additionally, large

specimen-to-specimen variation exists in the parameters, C0 to C3, determined by CC 2).

This creates difficulties in comparing the CC relation from different specimens and

evaluating the accuracy of this compliance calibration method. On the other hand,

Table 4-4: Parameters C0, C1, C2, and C3, determined by CC 2) for ENF specimens

Specimen C0, m/N C1, 1/N C2, 1/(Nm) C3, 1/(Nm2) R2 2-1 2.07E-06 -1.15E-05 -2.38E-04 3.57E-02 0.998 2-2 2.43E-06 -6.20E-05 2.83E-03 -1.95E-02 1.000 2-3 1.55E-06 2.19E-05 -5.17E-04 2.15E-02 1.000

Un-precracked

2-4 1.31E-06 5.67E-05 -2.53E-03 5.83E-02 1.000 STD 5.07E-07 5.05E-05 2.21E-03 3.27E-02 Statistical

results Mean 1.84E-06 1.28E-06 -1.13E-04 2.40E-02 0.999

Specimen C0, m/N C1, 1/N C2, 1/(Nm) C3, 1/(Nm2) R2 2-5 -1.15E-06 2.40E-04 -6.68E-03 8.03E-02 1.000 2-6 1.30E-05 -9.09E-04 2.45E-02 -1.99E-01 1.000 2-7 6.10E-06 -3.43E-04 8.83E-03 -5.49E-02 1.000 5-3 1.39E-05 -9.20E-04 2.29E-02 -1.69E-01 1.000 5-4 1.13E-05 -7.66E-04 2.06E-02 -1.60E-01 1.000

Precracked

5-5 5.35E-06 -2.90E-04 7.43E-03 -4.30E-02 1.000 STD 5.75E-06 4.54E-04 1.20E-02 1.05E-01 Statistical

results Mean 8.08E-06 -4.98E-04 1.29E-02 -9.10E-02 1.000

120

calculating SERR using the compliance calibration relation directly involves the

differentiation of the compliance to crack length, dC/da, instead of compliance, C, at a

certain crack length. Hence, the accuracy of the dC/da vs. a relation derived from the

compliance calibration relation is also important. It is hard to judge the accuracy of this

relation derived from CC 2) since large specimen-to-specimen variation exists in the

parameters and no physical meaning is involved in these parameters. Therefore, so far the

compliance calibration method CC 2) does not show obvious advantage to CC 1). For

convenience and a more repeatable compliance calibration relation, it is suggested to use

the relation determined by CC 1). For fatigue tests, it is more convenient to use the first

approach CC 1) to predict crack length based on the compliance obtained during certain

cycles and hence it was used for fatigue tests in this investigation.

4.2.2.3 GIIc onset values

The GIIc onset values were calculated by three methods: the classical plate theory

(CPT) using Eq. (4.3) and two compliance calibration (CC) methods using Eq. (4.5) and

Eq. (4.6), respectively.

In the previous two equations, a1 is crack length calculated for the critical point, by

Eq. (4.4). The initial crack length, a0, is replaced by the precracked crack length, ap, if a

precracked specimen is used. C0 is the compliance of the initial elastic portion, which is

taken as the inverse of the slope of the initial linear portion of the load-displacement

curve; C1 is the compliance at the critical point; Pc is the load at the critical point.

The expressions for SERR using two compliance calibration relations are:

2 21 0

IIc 3 30

92 (2 3 )

ca P CGb L a

=+

(4.3)

1/3

3 31 11 0

0 0

2 13

C Ca a LC C⎡ ⎤⎛ ⎞

= + −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(4.4)

121

where in the previous two equations, B is a parameter determined by CC 1), Eq. (4.1),

and C1, C2 and C3 are parameters determined by CC2) , using Eq. ((4.2).

The results of the GIIc values at crack onset using three methods are shown in

Figure 4-16. Comparing the average GIIc onset values for precracked and un-precracked

specimens as shown in Figure 4-16, the values calculated by the two compliance

calibration methods are very similar, however, for precracked specimens, the average

value calculated by CPT is about 10% higher than that by CC methods. The average GIIc

value of un-precracked specimens is about 44% higher than that of precracked specimens

by CPT method, and about 60% higher by CC methods. The numerical values of GIIc for

crack onset are given in Appendix B, Table B-9. From the ENF test results found from

the literature (Appendix A, Table A-2), the GIIc values based on an insert film for

carbon/brittle epoxy materials range from 300 to1500 J/m2, while the GIIc values based on

a Mode I precrack range from 250 to 1000 J/m2. During this investigation, the average

GIIc values based on an insert film determined by different calculation methods ranged

from 782 to 801 J/m2, while the average GIIc values based on a Mode I precrack ranged

from 498 to 545 J/m2. Compared to results in the literature, the Mode II quasi-static

fracture toughnesses of the investigated material are in the middle of the common range

for carbon/brittle epoxy materials.

22

IIc 03

32 8

cP BG ab bh

= (4.5)

22

IIc 1 2 0 3 0( 2 3 )2

cPG C C a C ab

= + + (4.6)

122

4.2.2.4 A short Mode II fracture resistance curve

For the quasi-stable ENF test for which the load-displacement curve is shown in

Figure 4-11, a short Mode II fracture resistance curve can be generated based on the load-

displacement curve, the compliance calibration results and the critical SERR value at

crack onset. The methods for fracture toughness and crack length calculations are

discussed next.

From Classical plate theory, the compliance of an ENF specimen can be written in

the form of Eq. (4.7).

0100200300400500600700800900

1000

2-1 2-2 2-3 2-4 2-5 2-6 2-7

Specimen No.

GII

c, J/m

2

by CPTby CC 1)by CC 2)average by CPTaverage by CC 1)average by CC 2)

un-precracked precracked

Fig. 4-16: GIIc onset values

3 33

31

2 3 ' '8L aC A B aE bh+

= = + (4.7)

123

Thus, A’ and B’ are only related to half span length L, specimen dimensions, b and h, and

Young’s modulus in the longitudinal direction, E1. They are given by,

The crack length can thus be predicted by, Eq. (4.9)

The SERR can be written as Eq. (4.10),

Substituting Eq. (4.9) into Eq. (4.10), results in,

Substituting C = δ/P into Eq. (4.11) and rearranging terms, yields,

Substituting load and displacement at the critical crack onset point, Pcr and δcr, yields,

Denoting the propagated critical SERR by GIIR, and dividing Eq. (4.12) by Eq. (4.13),

results in,

Parameter A’ is related to the compliance calibration parameter A (Eq. (4.1)) by,

3

31

2'8

LAE bh

= , 31

3'8

BE bh

= (4.8)

1/3''

C AaB−⎛ ⎞= ⎜ ⎟

⎝ ⎠ (4.9)

223 '

2IIPG B a

b= (4.10)

( )2 /32 2

2/31/33 ' 3' ' '2 ' 2IIP C A PG B B C Ab B b

−⎛ ⎞= = −⎜ ⎟⎝ ⎠

(4.11)

( ) ( )2 1/3 2/32 /31/3 2 33 3 '' / ' '

2 2IIP BG B P A P A Pb b

δ δ= − = ⋅ − (4.12)

( )1/3 2 /32 33 ' '

2IIc cr cr crBG P A P

bδ= ⋅ − (4.13)

( )( )

2 /32 3IIR

2/32 3IIc

'

'cr cr cr

P A PGG P A P

δ

δ

⋅ −=

⋅ − (4.14)

3'8

AAbh

= (4.15)

124

As a result, an expression for propagated critical SERR is given by,

The crack length can be calculated based on instantaneous load and displacement by,

Eq. (4.17)

where parameters B’ is related to the compliance calibration parameter B (Eq. (4.1)) by,

Eq. (4.18)

Using Eq. (4.16) and Eq. (4.17), a short fracture resistance curve based on the

load vs. displacement data for the quasi-stable ENF test was generated as shown in

Figure 4-17. The crack length was calculated based on the compliance calibration relation

1). From Figure 4-17, the fracture resistance in Mode II changes slightly with crack

extension.

2/32 3 3

IIR IIc2/32 3 3

/(8 )

/(8 )cr cr cr

P A bh PG G

P A bh P

δ

δ

⎡ ⎤⋅ − ⋅⎣ ⎦=⎡ ⎤⋅ − ⋅⎣ ⎦

(4.16)

1/3 1/3' / '' '

C A P AaB B

δ− −⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(4.17)

3'8

BBbh

= (4.18)

125

On the other hand, if a constant Mode II fracture resistance with crack extension

is assumed, a theoretical load vs. displacement curve after crack initiation can be

predicted based on Eq. (4.14). Load P can be solved as a function of displacement δ and

material Mode II fracture toughness, GIIR. Assuming GIIR equals crack initiation value

GIIc, constant G curves generated for an ENF specimen from Eq. (4.14) is shown in

Figure 4-18.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

30 35 40 45 50Crack length, a, mm

GIIR

/GIIc

Fig. 4-17: Mode II resistance curve (Specimen 2-6, precracked)

126

4.2.2.5 Special issues on the ENF test

(1) Load drop immediately after crack initiation

The steep slope of the constant G curve (G = GIIc) after crack initiation as shown

in Figure 4-18 somewhat explains the large amount of load drop at the crack initiation

point of an ENF test. The steep slope indicates that a small amount of crack growth

greatly reduces the stiffness of specimen and thus initiates a significant load drop. The

theoretical prediction assumes the crack continuously advances in infinite small

increments as shown in the double dot centerline in Figure 4-19. However, in reality, the

crack advances in bigger increments than theoretical prediction, which results in a load-

displacement curve as shown in the blue dashed line in Figure 4-19. If the crack advances

in even bigger increments, then it could result in a load vs. displacement curve close to a

real test as shown in the sold line in Figure 4-19.

0

100

200

300

400

500

600

0.0 0.5 1.0 1.5 2.0 2.5Displacement, mm

Load

, N

P_maxP_5% offsetP_nonlinear

G = G IIc

G = 1.1G IIc

Fig. 4-18: A load vs. displacement plot for ENF test with constant G curves shown

127

(2) Crack instability after crack initiation

It was predicted by theory solution that the crack growth in an ENF specimen is

unstable when a/L< 0.7. However, during this investigation, even when using precracked

specimens for which a0/L> 0.7, the crack growth immediately after onset was unstable.

This instability can be explained by the crack growth rate for a quasi-static ENF test

derived from CPT theories. Derived from Eq. (4.10), alternative expressions for Mode II

SERR could be,

Solving for displacement from Eq. (4.19), one obtain,

Fig. 4-19: A schematic of load vs. displacement curves for different states of crack growth

( )2 2 2

2 2 222 3

3 ' 3 ' 3 '2 2 2 ' '

IIRPG B a B a B ab bC b A B a

δ δ= = =

+ (4.19)

( )3IIR

' '23 '

A B abGB a

δ+

= ⋅ (4.20)

128

Assuming a constant GIIR with crack extension, results in,

and, Eq. (4.22)

In a displacement controlled test with constant loading speed (dδ/dt=0.5 mm/min),

da/dδ is proportional to the crack propagation speed assuming the crack advance in

infinite small increments as assumed theoretically. Substituting 3'/ ' 2 / 3A B L= , obtained

from Eq. (4.8), dδ/dt=0.5 mm/min, and using 3'/(8 )B bh =3.03E-11 and GIIR=500 J/m2,

the calculated da/dδ is plotted against crack length as shown in Figure 4-20. It is

predicted that the crack propagation speed near a/L= 0.7 is extremely high. The high

speed crack propagation, or high sensitivity of crack growth in response to the loading

displacement near a/L= 0.7, could be the cause of an observed unstable crack growth

immediately after crack initiation during a precracked ENF test.

IIR IIR2 3

2 2 '' '2 ' 23 ' 3 'bG bG Bd A AB a a

da B a B aδ ⎛ ⎞ ⎛ ⎞= ⋅ − + = ⋅ − +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (4.21)

IIR3

12 ' ' 2

3 '

dad bG B A a

B aδ=

⎛ ⎞⋅ − +⎜ ⎟⎝ ⎠

(4.22)

129

4.3 The Mode II Fatigue IFT Testing

The Mode II fatigue IFT tests were conducted with precracked ENF specimens as

shown in Figure 4-5. The aim of the tests is to determine the crack growth length versus

loading cycles and the crack growth rate with respect to cycles versus the SERR relation.

4.3.1 Mode II fatigue test method

Mode II fatigue tests were conducted with ENF precracked specimens, following

the same procedures as was applied in the ENF quasi-static tests. The same compliance

calibration procedures used for the ENF quasi-static tests were applied as well for the

fatigue tests.

0

5

10

15

20

25

30

35

0.6 0.7 0.8 0.9 1a/L

da/d

t, m

m/se

c

a 0 /L

Fig. 4-20: Plot of crack growth rate in a quasi-static ENF test versus normalized crack length

130

The fatigue test was carried out in displacement control. The critical displacement

for crack onset in a quasi-static test at the initial length range tested, δIIcr, is

approximately 1.6 mm. The loading point displacement was cycled between maximum

displacement (δIImax) and minimum displacement (δIImin) using a sinusoidal wave. Testing

parameters for the fatigue test are summarized in Table 4-5.

The maximum cycling displacement used for this fatigue test was approximately

75% of the estimated critical opening displacement for ENF quasi-static test at the same

initial crack length. As a result, the maximum SERR to critical SERR ratio (GIImax/ GIIc)

at the beginning of the fatigue test is approximately 0.56, where the critical SERR is the

estimated critical SERR for crack onset in a quasi-static ENF test. The test was set up in a

way such that the initial crack length to half span ratio, ap/L, for the precracked specimen

is greater than 0.693. For a fatigue test with a constant maximum loading point

displacement, the maximum SERR is predicted by Eq. (4.23) (Classical Plate Theory),

and its trend with crack extension is shown in Figure 4-21.

Table 4-5: Testing parameters for ENF fatigue tests

Displacement ratio: R 0.2 Loading frequency: f 10 Hz Maximum displacement: δIImax 1.2 mm Minimum displacement: δIImin 0.24 mm Estimated critical displacement of static test: δIIcr 1.6 mm

( )( )

22 3 3 2,max ,max

22 3 3

2 3

2 3pII II

IIc IIcr p

L aG aG a L a

δδ

+⎛ ⎞= ⋅ ⋅⎜ ⎟

+⎝ ⎠ (4.23)

131

In Figure 4-21, the normalized maximum SERR (F1*GIImax/ GIIc) is plotted against

normalized crack length (a/L) for a constant displacement amplitude fatigue test as

predicted by classical plate theory. The maximum to critical SERR ratio (GIImax/ GIIc)

occurs at the point where a/L ≈ 0.693. When a/L > 0.693, we have dGIImax/da < 0 and

hence stable crack growth is predicted till the crack reaches the loading line where a/L =1,

assuming the fracture resistance of material is not decreasing as crack length increases.

To obtain the crack length as a function of the number of loading cycles, two

methods were used: visual measurement and compliance calibration. A slower loading

frequency of 0.086 Hz was used for loading cycle # 1, 100, 500, 1000, 1500, 2000, 2500,

3000, 4000, 5000, 6000, 7000, and etc. Load-displacement data were recorded for each

slow cycle. A load-displacement curve was obtained for every slow cycle. Compliance

was obtained by taking the inverse of the slope of the load-displacement curve. The crack

length was then calculated using the compliance-crack length relationship obtained from

a compliance calibration done earlier on the specimen. Additionally, crack growth was

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1a/L

F1*G

IIm

ax/G

IIc

Predicted stable crackgrowth range

Fig. 4-21: Maximum Mode II SERR (GIImax) vs. normalized crack length plot for an ENF test with fixed displacement amplitude (F1 is a factor related to initial crack length and maximum opening displacement)

132

measured visually by a Questar QM-1 long distance telescope during each slow cycle.

The ENF test was stopped when the crack grew close to but not beyond (about 3-5 mm

away from) the loading line.

4.3.2 Mode II fatigue test results

The results of compliance calibration of ENF specimens were presented in

Section 4.2.2. In this section, the compliance calibration relation obtained was used for

crack length calculation, and compared to the crack length measured visually. Crack

growth rate was related to the maximum SERR by the modified Paris’ law, and

exponents were found to be high in comparison to literature.


Crack growth measured visually and calculated by compliance calibration is

shown in Figure 4-22. The crack length calculated by compliance calibration is about

between 0 and 3 mm larger than that by visual measurement. Possible reasons for this

could be: a) the crack length was hard to measure visually when the specimen was loaded

in Mode II and hence large error could exist for visual measurement; b) the crack length

is measured visually on one specimen edge, and it could be more or less than the crack

length calculated based on the compliance, which is theoretically an average crack length

across the width.

133

A microscopic view of an ENF specimen edge when the same specimen was

subject to different loading conditions is shown in Figure 4-23 (a) to (c). Comparing (a)

to (b) and (c), the crack was hard to discriminate from the surface when the specimen was

loaded in Mode II and crack surfaces were forced in contact with each other. The reason

one can observe the crack in Mode I opening is that the crack appears to be a black line

on a white background. However, in Mode II, one can observe the crack only because of

the relative shear displacement of the crack surfaces, which is very small even far away

from the crack tip when the specimen is loaded. As a result, for Mode II loading, it is

difficult to visually detect the crack.

30

32

34

36

38

40

42

44

46

48

50

0 2,000 4,000 6,000 8,000 10,000

Cycle No., N

Cra

ck le

ngth

, a, m

m

5-3, by CC5-4, by CC5-5, by CC5-3, by visual5-5, by visual

Fig. 4-22: Crack growth for ENF fatigue specimens

134

(a) Crack opened in Mode I

Fig. 4-23: A microscopic view of a specimen edge while the specimen was in different loadings (Specimen 5-3)

135

~1 mm

(b) specimen loaded to the mean loading point displacement of a Mode II fatigue cycle

(c) specimen loaded to the maximum loading point displacement of a Mode II fatigue

cycle

Fig. 4-23 (continued): A microscopic view of a specimen edge while the specimen was in different loadings (Specimen 5-3)

136

After the ENF fatigue tests, the specimens were split apart by hand. Photographs

of crack surfaces of three specimens are shown in Figure 4-24. While the crack front

shapes created by Mode I precrack appear to be irregular, the crack front shapes created

by Mode II fatigue test are consistently slightly convex relative to the front created by the

insert film. This indicates that there is a trend that the crack length on the edge of

specimen is smaller than the average crack length across width. Additionally, it could be

easily examined that the crack front shape is not always perfectly symmetric about the

center-width of the specimen as the crack proceeds progressively. Due to the local

property of the material, the crack can proceed faster or slower on one specimen edge at

certain stages of a test compared to the progress elsewhere across the width. Only

focusing on the progress of crack growth on one specimen edge, the visual measurement

of crack growth can be somewhat “noisy”, particularly if crack growth from one

measurement point to the next is very small.

137

Crack surface created by Mode I precrack

Crack surface created by an ENF fatigue test Crack surface created during precrack

Crack surface created during the fatigue test


Crack surface created by an ENF fatigue test


Crack surface created by an ENF fatigue test

Fig. 4-24: Photographs of fracture surfaces

138

4.3.2.2 da/dN - GIImax plots

The crack growth rate, da/dN, with crack growth calculated by compliance

calibration, was plotted against Mode II maximum strain energy release rate, GIImax, as

shown in Figure 4-25. Additionally, da/dN, with crack growth measured visually, was

plotted against GIImax for two tests, as shown in Figure 4-26. The power law relation

between da/dN and GIImax, ( ) II

II IImax/ nda dN B G= , was used to fit the data from each ENF

specimen. With the crack growth lengths calculated from compliance calibration, the

exponent for the power law relation was found to be in the range of 7.7 to 12.8. With the

crack growth lengths measured visually, the exponent was in the range of 7.1 to 6.8. In

the literature, the exponents of this power law relation for ENF fatigue tests with

carbon/epoxy or glass/epoxy material mostly range from 4.3 to 6.5 (Appendix Table A-3).

However, in one case, this value was found to be between 13 and 15 for an unidirectional

carbon/epoxy material (Hojo et al. 2006). Compared to results in the literature, the

exponent for the material investigated is somewhat high. From the literature, for the same

material the exponent of the power law relation for Mode I is usually higher than that for

Mode II. This is also the case for the investigated material. Additional crack growth rate

(da/dN) vs. maximum SERR (GIImax) plots for individual specimens are given in

Appendix B from Figure B-9 to Figure B-13.

139

Fig. 4-25: Crack growth rate against Mode II maximum SERR plot, with crack growthcalculated by compliance calibration

140

Fig. 4-26: Crack growth rate against Mode II maximum SERR plot, with crack growthmeasured visually

141

Chapter 5

Mixed-mode I/II Interlaminar Fracture Toughness Testing

Mixed-mode I/II interlaminar fracture toughness (IFT) tests under quasi-static and

cyclic loading were conducted with single-leg-bending (SLB) specimens. The Mode II to

total strain energy release rate (SERR) ratio (GII/GT) achieved with mid-thickness

delaminated SLB specimens was about 0.43. The mixed-mode interlaminar fracture

toughness for crack onset was characterized under this mode ratio, and the fracture

resistance curve was obtained based on the calculated crack length. SLB fatigue tests

were conducted in displacement control with displacement ratio R = 0.2. The results

show that the exponent of the power law relation ( ) ini GBdNda max/ = is higher for

mixed mode loading than that for Mode I or Mode II loading.


The specimens used in this investigation are from two flat carbon/epoxy panels of

[0]12 lay-up as shown in Figure 4-1 and Figure 5-1. The mixed-mode I/II IFT tests were

conducted with SLB specimens. The SLB specimen geometry and notations are shown in

Figure 5-2. Quasi-static tests were conducted with four SLB specimens, of which

Specimens 3-1 and 3-3 are from the panel shown in Figure 4-1 and Specimens 3-2 and 3-

3 from the panel shown in Figure 5-1. All the specimens were tested without precracking.

Fatigue tests were conducted with three SLB specimens (Specimen 6-1, 6-2, and 6-3 in

Figure 5-1). The dimensions of the SLB specimens in terms of length × width × thickness

are approximately 150 × 25.4 × 3.9 mm, with an initial artificial crack created by the

embedded thin film of 13 μm thickness and ~50.8 mm long. Specific dimensions of each

SLB specimen tested quasi-statically and cyclically are listed in Appendix Table B-6 and

Table B-7, respectively.

142



Specimen 3-2

Specimen 3-4


Specimen 6-1

Fig. 5-1: The SLB panel diagram

143

The SLB test is essentially an un-symmetric three-point-bending test using a

three-point bending fixture as the loading apparatus. Part of one delaminated leg of the

specimen was removed before testing so that a crack opening displacement can be

applied to the other leg. The SLB test configuration is shown in Figure 5-3 and test

dimensions are listed in Table 5-1. For the three-point bending fixture used in this

investigation, the three loading and supporting pins are attached to the bending fixture by

three springs, so that the loading pin can rotate about its centerline and the two supporting

pins can roll along the longitudinal direction of the bending fixture.

l

b

2h

a 0 L

L

Loading direction

Support

Support

Fig. 5-2: A schematic of SLB specimen geometry and test configuration

144

5.2 The Mixed-mode I/II Quasi-static IFT Testing

The goal of the SLB quasi-static test is to characterize the critical SERR value for

crack onset and growth, and also obtain the load and displacement at the critical onset

point. The test setup used for a SLB test is similar to the setup for an ENF test; however,

while for a quasi-static ENF test the crack growth was usually unstable at crack onset, the

crack growth was stable but very fast for a quasi-static SLB test. Since the mode ratio

GII/GTc is about 0.43 for the SLB test configuration used and the Mode II fracture

toughness is about 5-8 times higher than the Mode I fracture toughness, the SLB test is

considered to be more governed by Mode I behavior than Mode II behavior.

Loading roller r1 Test specimen

Loading direction

c c

Left supporting roller

Right supporting roller

r2r2

L LSpacer

Loading roller r1 Test specimen

Loading direction

c c

Left supporting roller

Right supporting roller

r2r2

L LSpacer

Fig. 5-3: SLB test configuration

Table 5-1: SLB test configuration dimensions

Notation Section of test specimen measured Dimension, mm (in.) L Half span length of bending fixture 63.5 (2.5) r1 Radius of loading roller 6.4 (0.25) r2 Radius of supporting rollers 3.2 (0.125) c Overhang 12.8 (0.5)

145

5.2.1 Mixed-mode I/II quasi-static test method

A photograph of the SLB test set up is shown in Figure 5-4. The SLB specimen

was placed in a three point bending fixture and loaded by a 13.5 kN (3 kip) MTS

machine. The load was measured by a 0.448 kN (100 lb) capacity load cell. Loading

point displacement was measured by the MTS LVDT built into the actuator.

Due to the special geometry of the SLB specimen, certain attention should be paid

to the fabrication of the specimen. Before testing, the majority of the lower cracked

region needs to be removed using a water-cooled diamond abrasive cut-off wheel. The

specimen was marked at a certain distance (~ 35 mm) from the delaminated end on the

top surface before cutting. To prevent accidental cutting of the upper “leg”, a thin blade

was inserted between the upper and lower crack region, and pushed forward until it went

close to the desired cutting point. Care was taken to ensure the crack growth did not

occur during the cutting process.

Compliance calibration procedures were conducted before the crack initiation test

at crack lengths of a0, a0 ± 3 mm, and a0 ± 6 mm (a0 is the initial crack length of the

MTS built-in load

cell (stationary)


SLB Specimen

Bending fixture

Not shown: -Servo-hydraulic actuator -long-distance instrumented stage microscope

MTS built-in load cell (stationary)


SLB Specimen

Bending fixture

Not shown: -Servo-hydraulic actuator -long-distance instrumented stage microscope

Fig. 5-4: A photograph of SLB test set up

146

crack initiation test). For the compliance calibration procedure and the crack initiation

test, a constant displacement rate of 0.5 mm/min was used for loading and unloading.

Marks were made on the edge of the specimen to locate the positions of the rollers

for compliance calibration tests at each crack length, as shown in Figure 5-5. The

distance between two neighboring marks (Δa) was approximately 3 mm and was

measured through the instrumented telescope before testing. Additionally, the initial

crack length was measured before testing. For each compliance calibration test, the

specimen was loaded to a load point displacement of about 60% of the estimated critical

displacement and then unloaded. Five initial crack lengths were achieved by sliding the

specimen in the bending fixture in the longitudinal direction of specimen.

For the crack initiation test, the specimen was placed in the bending fixture as

shown schematically in Figure 5-3. The specimen was loaded until the crack propagated

for about 20 mm, and then unloaded.

a0

Δa

Fig. 5-5: Markings on SLB specimen edge

147

5.2.2 Mixed-mode I/II test results


Load-displacement curves from four SLB quasi-static tests are shown in Figure 5-

6. In a manner similar to Mode I and Mode II IFT tests, the typical loading curve shows

increasing load up to its maximum value followed by decreasing load with crack

extension. However, unlike Mode I and Mode II testing, a slight nonlinear behavior

before the maximum load point was observed for the mixed mode tests, as shown in the

closer view of the loading curve near the critical crack onset point for one test in

Figure 5-7.

0

50

100

150

200

250

0.0 0.5 1.0 1.5 2.0Displacement, mm

Load

, N

3-2 3-3

3-4 3-1

Fig. 5-6: Load vs. displacement curve for SLB quasi-static tests

148

The slight non-linear behavior seen in the loading curve before load decreases

raises the question on the definition of the critical crack onset point. Four alternate

definitions are typical for defining the critical point in an interlaminar fracture toughness

test. They are: 1) the nonlinear point, where nonlinear behavior of load-displacement

curve is first observed; 2) the visual onset point, where crack onset is visually observed;

3) the maximum load point, where the maximum load occurs; and 4) the 5% compliance

offset point, where the compliance increases by 5%. For the material tested in this

investigation, the visual crack onset point was very close to the maximum load point;

however, it is difficult to capture the exact point of visual crack onset and discriminate it

from the maximum load point. From the nonlinear point to the maximum load point, the

compliance had increased by a small amount e.g., in the case shown in Figure 5-7, the

compliance increased by 0.7% from the nonlinear point to the maximum load point. This

may be due to a local crack growth within the center of specimen. It is estimated that the

maximum load point is the closest to the point where global crack growth onset among

all the critical points could be defined. For the material tested, it was always the case that

the maximum load was reached before the compliance increased by 5% relative to the

y = 166.4x - 3.8998R2 = 0.9999

150

170

190

210

230

250

1.1 1.2 1.3 1.4Displacement, mm

Load

, N

Max. load point

Fitted straight line

5% complianceincrease constructionline

Nonlinear point

Fig. 5-7: Load-displacement plot near the critical onset point (Specimen 3-2)

149

initial compliance. Therefore, the maximum load point was determined to be suitable for

the critical crack onset point definition for the material investigated.


Compliance calibration procedures were conducted for each specimen at five

initial crack lengths of a0, a0 ± 3 mm, and a0 ± 6 mm. Two relations were used to

correlate compliance vs. crack length data. One is a third order polynomial with four

terms, as expressed by Eq. (5.1), denoted by CC method (1) in the later discussion,

and the other one is also a third order polynomial but omits the first and second order

terms, as expressed by Eq. (5.2), denoted by CC method (2) in the later discussion, which

is in a similar form to compliance predicted by classical plate theory.

In these equations, CSLB is the compliance of a SLB specimen and a is the crack length.

Parameters q0, q1, q2, and q3 are parameters determined fitting a third order polynomial

by least squares in the C vs. a plot. Parameters β1 and β2 are parameters determined by a

least square fit of a straight line in the C(8bh3) vs. a3 plot. The plot for Eq. (5.1) is show

in Figure 5-8, while that for Eq. (5.2) is shown in Figure 5-9 for all the available SLB

data.

33

2210 aqaqaqqC SLB +++= (5.1)

( ) 321

38 abhC SLB ββ += (5.2)

150

4.0E-06

4.5E-06

5.0E-06

5.5E-06

6.0E-06

6.5E-06

7.0E-06

7.5E-06

8.0E-06

8.5E-06

3.0E-02 3.5E-02 4.0E-02 4.5E-02 5.0E-02a , m

C,

m/N

3-2 3-3 3-43-1 6-1 6-26-3

33

2210 aqaqaqqC +++=

Fig. 5-8: C vs. a plot for all SLB specimens

0.0E+00

4.0E-15

8.0E-15

1.2E-14

0.0E+00 2.0E-05 4.0E-05 6.0E-05 8.0E-05 1.0E-04 1.2E-04a 3 , m3

C(8bh

3 ), m

5 /N

3-2 3-3 3-43-1 6-1 6-26-3

( ) 321

38 abhC SLB ββ +=

Fig. 5-9: C(8bh3) vs. a3 plot for all SLB specimens

151

From the two compliance calibration plots, one can see that both expressions fit

the data very well based on the R2 values. Theoretically, the shape of each fitted curve in

the C-a plot is related to each specimen’s flexural modulus in the longitudinal direction

of specimen, E1f, the span length (2L) and specimen dimensions (b, 2h). For the second

C(8bh3) vs. a3 plot, according to classical plate theory(Eq. (5.3)), the slope of each fitted

straight line is only related to longitudinal flexural modulus, E1f, and x-intercept of the

straight line is only related to support distance, 2L.

Hence, it is reasonable that the data from different test specimens look more collapsed in

the second plot than in the first one since in the first plot the curve shape depends on

additional specimen dimensions. Furthermore, the variation in the slope of different

curves in the second plot indicates the variation in flexural modulus from specimen to

specimen. The coefficients determined for the first and second compliance calibration

relation are listed in Table 5-2 and Table 5-3, respectively. Since it is more convenient to

use Eq. (5.2) for crack length calculations based on compliance, Eq. (5.2) was used for

crack growth calculation for fatigue tests.

31

33

872

bhEaLC

f

+= (5.3)

Table 5-2: Coefficients determined for compliance calibration by Eq. (5.1) (Load was measured in Newton and displacement was measured in meter.)

Specimen q0 q1 q2 q3 R2 3-1 1.61E-05 -1.01E-03 2.65E-02 -1.87E-01 0.998 3-2 2.16E-05 -1.43E-03 3.69E-02 -2.70E-01 1.000 3-3 -1.69E-05 1.55E-03 -3.86E-02 3.67E-01 1.000 3-4 2.27E-05 -1.50E-03 3.92E-02 -2.94E-01 1.000 6-1 -1.97E-05 1.76E-03 -4.44E-02 4.14E-01 1.000 6-2 5.53E-06 -1.65E-04 4.61E-03 2.19E-03 1.000 6-3 6.52E-06 -2.19E-04 5.36E-03 3.81E-03 1.000

STD 1.90E-05 1.46E-03 3.74E-02 3.16E-01 Mean 4.89E-06 -1.33E-04 4.01E-03 5.60E-03 1.000

152

5.2.2.3 Mixed-mode I/II critical strain energy release rate (GTc) onset value

The mixed-mode I/II critical SERR for SLB specimen was calculated by classical

plate theory and two compliance calibration methods with compliance vs. crack length

relation by Eq. (5.1) and Eq. (5.2), respectively. The expression used for total critical

SERR calculations are written as Eq. (5.4) by classical plate theory, Eq. (5.5) by CC

method (1) and Eq. (5.6) by CC method (2).

where, C0 is the compliance of specimen at the initial crack length.

The GTc results by these three methods are shown in Figure 5-10. Numerical

results are shown in Appendix B Table B-10. The average GTc value is 185 J/m2 by CPT,

185 J/m2 by CC method (1), and 181 J/m2 by CC method (2). Within the three methods

for SERR calculation investigated, the GTc value by CPT method yields the least

Table 5-3: Coefficients determined for compliance calibration by Eq. (5.2) (Load was measured in Newton and displacement was measured in meter.)

Specimen β1 β2 β1/β2 R2 3-1 4.90E-15 6.36E-11 7.70E-05 0.998 3-2 4.89E-15 6.68E-11 7.33E-05 1.000 3-3 5.06E-15 6.88E-11 7.35E-05 0.999 3-4 5.39E-15 6.51E-11 8.29E-05 1.000 6-1 4.84E-15 6.14E-11 7.89E-05 0.999 6-2 5.30E-15 6.60E-11 8.03E-05 1.000 6-3 5.24E-15 6.92E-11 7.57E-05 1.000

Mean 5.09E-15 6.58E-11 7.74E-05 0.999 STD 2.22E-16 2.80E-12 3.55E-06 COV 4.4% 4.2% 4.6%

2 20 0

Tc 3 30

212 (2 7 )

ca P CGb L a

=+

(5.4)

22

Tc 1 2 3( 2 3 )2

cPG q q a q ab

= + + (5.5)

222

Tc 3

32 8

cPG ab bh

β= (5.6)

153

variation (4.8%) from specimen to specimen, and the CC (2) method results in the

maximum variation (8.7%).

A summary of the Mode I, Mode II, and Mixed-mode fracture toughness at crack

onset by one or two representative calculation method(s) is given in Table 5-4. In this

table, the fracture toughness shown at each mode ratio is the average toughness for all the

specimens tested at this mode ratio during this investigation. Mode I and Mode II SERR

in the mixed mode test was calculated using the mode ratio GII/GT = 0.43 and average

fracture toughness GT = 185J/m2. It was obtained by Asp et al. (Asp et al. 2001) that: for

a carbon/epoxy material, the Mode I and Mode II toughness (GIc, GIIc) were 260 J/m2 and

1002 J/m2, respectively, and the mixed mode fracture toughness (GTC) was 447 J/m2 at

the mode ratio GII/G=0.5. Comparing to their results, the fracture toughnesses of the

currently investigated material are low.

Two possible mixed-mode delamination failure criterion for the investigated

carbon/epoxy material are proposed based on the test results at the mode ratio

GII/GT=0.43. At this mode ratio, Reeder’s linear mixed mode failure criteria by Eq. (2.63)

gives fairly good prediction of the failure locus, as shown in Figure 5-11. Choosing m =

2.5 in Eq. (2.66), the B-K law also gives a good prediction of the failure locus at the

0

50

100

150

200

250

3-1 3-2 3-3 3-4Specimen

GTc

, J/m

2

by CPTby CC (1)by CC (2)

Fig. 5-10: GTc values determined from SLB tests

154

investigated mode ratio, as shown in Figure 5-12. However, to decide which criteria gives

a better prediction of the failure locus at all mode ratios, more mixed-mode test results at

other mode ratios are needed. Summarizing the current IFT test results for the AS4/3501-

6 carbon/epoxy material, Mathews and Swanson (Mathews and Swanson 2007) found

that the Reeder’s linear law and power law by Eq. (2.64) gave good representation of the

test data. Also, using the B-K law to represent the failure locus, they found the parameter

m to be 4.78 by performing a least squares fit.

1=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

IIc

II

Ic

I

GG

GG

(2.63)

( )m

T

IIIcIIcIcTc G

GGGGG ⎟⎟⎠

⎞⎜⎜⎝

⎛−+= (2.66)

Table 5-4: A summary of Mode I, Mode II, and Mixed-mode I/II fracture toughness at crack onset

GTc, by CPT: 185 J/m2 GI: 105.45 J/m2 GII: 79.55 J/m2

GIc, for the non-precracked case by MBT: 125 J/m2

GI/GIc for the SLB test using GIc by MBT, non-precrack: 0.84

GIc, for the precracked case by MCC: 131 J/m2

GI/GIc for the SLB test using GIc by MCC, precrack: 0.80

GIIc, for the non-precracked case by CPT: 782 J/m2

GII/GIIc for the SLB test using GIIc by CPT, non-precrack: 0.10

GIIc, for the precracked case by CPT: 545 J/m2

GII/GIIc for the SLB test using GIIc by CPT, precrack: 0.15

by CC 1): 498 J/m2 GII/GIIc for the SLB test using

GIIc by CC 1), precrack: 0.16

155

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0G I/G Ic

GII/

GIIc

nonprecrack nonprecrack precrack precrack by CPT) precrack precrack by CC

G Ic G IIc

Reeder'sfailure locus

Fig. 5-11: The Reeder’s linear mixed mode failure locus and test data

0100

200300

400500

600700

800900

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4G II/G T

GTc

, J/m

2

G IIc, nonprecrackG IIc, precrack byCPT

G IIc, precrack byCC

m=1m=2.5

test data

Fig. 5-12: The B-K Law failure locus and test data

156

5.2.2.4 Mixed-mode fracture resistance curve

The mixed-mode fracture resistance curve can be generated based on the load-

displacement curve, the compliance calibration results and the critical SERR value at

crack onset. The methods for fracture toughness and crack length calculations are

discussed next.

The crack length is solved from the compliance calibration relation. Starting from

the expression for compliance (Eq. (5.3)) from classical plate theory, and equating the

compliance expression to a third order polynomial that can be obtained from compliance

calibration, yields,

Solving for crack length, a, we have,

where )8/( 312 bhA β= and )8/( 3

22 bhB β= . β1 and β2 are coefficients obtained from

compliance calibration (Eq. (5.2)).

The load-displacement behavior of the SLB specimen after crack initiation can be

derived from classical plate theory. Substituting Eq. (5.8) and C = δ/P into the expression

for SERR by CPT (Eq. (5.4)), and rearranging terms, the δ-P relation is obtained as in

Eq. (5.9).

If a material demonstrates constant mixed-mode fracture toughness (GTR) as crack

extends, post-initiation load-displacement curves with various GTR values can be

generated based on Eq. (5.9). By observing the intersection of constant GTR curves with

the loading curve from a test, the trend of GTR as crack extends can be estimated. In

Figure 5-13, the constant GTR curves were plotted along with the loading curve for

3223

1

33

872 aBA

bhEaLC

f

+=+

= (5.7)

3/1

2

2⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

BACa (5.8)

( )3/ 2

2 3TR21/3

2

23bG P A PB

δ⎛ ⎞

= ⋅ −⎜ ⎟⎝ ⎠

(5.9)

157

Specimen 3-4. The loading curve of Specimen 3-4 intersects with constant GTR curves of

increasing order as crack extends, beginning with the GTR = GTc curve and ending with

the GTR = 1.4 GTc curve. This indicates that the fracture toughness of the Specimen 3-4

was increasing with crack extension and had increased by about 40% at the end of test.

The critical SERR expression without involving crack length can be derived

based on Eq. (5.9). Normalizing Eq. (5.9) by the values at the critical point, Eq. (5.10) is

determined. With the crack length calculated by the compliance calibration relation,

Eq. (5.11), a fracture resistance curve can be generated for an SLB test.

where, GTc is critical SERR value for crack onset, and GTR is SERR as crack extends. δcr

and Pcr are the displacement and load at crack onset, respectively.

0

50

100

150

200

250

0.0 0.5 1.0 1.5 2.0 2.5Displacement, mm

Load

, N

G TR=G Tc

G TR=1.1G Tc

G TR=1.4G Tc

Fig. 5-13: Load vs. displacement plot for Specimen 3-4 with constant GTR curves shown

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⋅−⋅

=⎟⎟⎠

⎞⎜⎜⎝

⎛3

22

32

22/3

crcrcrTc

TR

PAPPAP

GG

δδ (5.10)

( )( )

3/1

32

31

3/1

2

2

8/8//

⎥⎦

⎤⎢⎣

⎡ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

bhbhP

BACa

ββδ (5.11)

158

Fracture resistance curves were obtained for four SLB tests as shown in Figure 5-

14. In Figure 5-14, one can see that for all the specimens the fracture resistance curve

rises slightly as the crack length increases. The fracture toughness increases slightly in

the first 10 mm crack extension. After about 20 mm crack extension, the fracture

toughness had increased approximately 11-53% for three of the specimens. For

Specimen 3-1, the crack growth at the end of the test was less than 20 mm.

5.3 Mixed-mode I/II Fatigue IFT Testing

5.3.1 Mixed-mode fatigue test method

The mixed mode fatigue crack propagation test was conducted using SLB

specimens. The test setup for the fatigue test is the same as for quasi-static test shown in

Figure 5-4. The SLB specimen was placed in a three point bending fixture and loaded by

a 13.5 kN (3 kip) MTS machine. The load was measured by a 0.448 kN (100 lb) capacity

0

50

100

150

200

250

300

350

30 40 50 60 70

Crack length, a , mm

GTR

, J/m

2

3-13-23-33-4

Fig. 5-14: Fracture resistance curves for SLB specimens

159

load cell. Loading point displacement was measured by the MTS LVDT built into the

actuator.

Before the fatigue test, the same compliance calibration procedures as used for the

SLB quasi-static tests were applied for SLB fatigue specimens as well. The fatigue test

was carried out in displacement control. The loading point displacement was cycled

between maximum displacement (δmax) and minimum displacement (δmin) using a

sinusoidal wave. Testing parameters for the fatigue test are summarized in Table 5-5.

The maximum cycling displacement used for the fatigue tests was approximately

88% of the estimated critical opening displacement for an SLB quasi-static test at the

same initial crack length. As a result, the maximum SERR to critical SERR ratio (Gmax/

GTc) at the beginning of the fatigue test is approximately 0.78, where the critical SERR is

the estimated critical SERR of crack onset in a quasi-static SLB test.

To obtain the crack length as a function of the number of loading cycles, two

methods were used: visual measurement, and compliance calibration. A slower loading

frequency of 0.0093 Hz was used for loading cycle # 1, 100, 200, 500, 1000, 1500, 2000,

2500, 3000 … and so on. Load-displacement data were recorded for each slow cycle. A

displacement-load plot was obtained for every slow cycle and compliance was obtained

from the slope of a straight line fit by least-squares. Crack length was then calculated by

a compliance-crack length relationship obtained from a compliance calibration performed

earlier on the specimen. Additionally, crack growth was measured visually by a Questar

QM-1 long distance telescope when the loading point displacement reached the

maximum value of each slow cycle.

Table 5-5: Testing parameters for SLB fatigue tests

Displacement ratio: R 0.2 Loading frequency*: f 10 Hz Maximum displacement: δmax 1.12 mm Minimum displacement: δmin 0.224 mm Estimated critical displacement for crack onset of quasi-static test: δcr 1.27 mm

* Except for periodic crack growth and compliance measurements, where f = 0.0093 Hz.

160

5.3.2 Mixed-mode fatigue test results


Crack growth was observed on one specimen edge and back calculated by a

compliance calibration relation derived from Eq. (5.2), and written as Eq. (5.12).

Crack growth vs. number of cycles by visual measurement and compliance calibration

method for all fatigue SLB specimens is shown in Figure 5-15.

As shown in Figure 5-15, the crack growth predicted by compliance calibration is

always more than that measured visually. For Specimen 6-1, after cycle No. 2000 (after

which cycle crack growth was visually observed), the crack length predicted by

compliance calibration was about 2.8-3.5 mm greater than that measured visually. For

3/1

2

1

2

3 )8(⎥⎦

⎤⎢⎣

⎡−=ββ

βCbha (5.12)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 20,000 40,000 60,000 80,000 100,000

Cycle No., N

Cra

ck g

row

th, Δa

, mm

6-1, by CC6-1, by visual6-2, by CC6-2, by visual6-3, by CC6-3, by visual

Fig. 5-15: Crack growth of SLB fatigue specimens

161

Specimen 6-2, after cycle No. 1500, the crack growth predicted by compliance

calibration was about 1.7-2.4 mm greater than crack growth measured visually. For

Specimen 6-3, it was observed that after cycle No. 1500, the crack length predicted by

compliance calibration was approximately 2.3-2.7 mm greater than crack length

measured visually.

Possible explanations for this difference between crack growth by visual

measurement and compliance calibration are proposed as follows. Firstly, for the visual

measurements, because the crack tip created during the SLB test is very sharp, how well

the tip can be seen depends strongly on crack opening displacement and on the thickness

of the edge coating. Assuming that the crack can be visually detected through a

microscope when the distance between the crack surfaces is more than a critical value, dc,

the real crack length will be larger than the crack length visually measured, by an amount

εa. If the visual measurement of crack length is always taken at the maximum crack

opening displacement (δmax) of a fatigue loading cycle, which is constant through a

fatigue test, the error of visual measurement, εa, is increasing as the crack length increases,

as shown schematically in Figure 5-16. Additionally, from this sketch as the opening

displacement (δmax) increases the error of visual measurement (εa) decreases.

a1

a2

εa1

εa2dcdc

a1 < a2

εa1 < εa2

Crack tip position 1


Crack can not be visually detected

a1

a2

εa1

εa2dcdc

a1 < a2

εa1 < εa2



Crack can not be visually detected

Fig. 5-16: A sketch of opening crack

162

In the SLB fatigue test, the crack opening displacement is very small, and thus

there should be a certain length of crack (εa) behind the crack tip that can not be visually

detected. Additionally, if crack closure occurs at the crack tip, the error of visual

measurement is even greater. However, for the compliance calibration procedure, the

crack tip position can be accurately measured because of the different color of the insert

film defining the initial crack.

Secondly, because of the curvature of the crack front shape created by a SLB test,

the crack length is greatest at the mid-width of the specimen as shown in Figure 5-17.

While the visual method gives a measure of crack growth on the edge, the compliance

calibration method predicts the average crack length across the specimen width.

5.3.2.2 Crack growth rate (da/dN) vs. maximum SERR (GTmax) plots

The crack growth rate was calculated based on the crack growth measured

visually and predicted by the compliance calibration method. The maximum SERR was

calculated by the compliance calibration method according to Eq. (5.6). In Figure 5-18,

the crack growth rate calculated with crack length obtained using the compliance

calibration method is plotted against maximum total SERR. Additionally, in Figure 5-19,

the crack growth rate calculated with crack length by visual measurement is plotted

Crack surface created by an SLB fatigue testCrack surface created by an SLB fatigue test

Crack front at the end of a test

Fig. 5-17: Fracture surfaces of a SLB specimen

163

against maximum total SERR. More da/dN-Gmax plots for individual specimens are given

in Appendix B, Figure B-14 to Figure B-19. In these graphs, the power law relation,

given in Eq. (2.75), was used to correlate crack growth rate and maximum total SERR.

( ) ini GB

dNda

max= (2.75)

Fig. 5-18: A da/dN vs. Gmax plot for SLB specimens (crack length by compliance calibration method)

164

As shown in Figure 5-18 and Figure 5-19, a power law relation can fit the da/dN-

Gmax test data from individual specimens very well. This indicates that among all the

fatigue crack growth models presented in literature review from Eq. (2.79) to Eq. (2.81),

the modified version of the Russell and Street’s model, ( )max/ nda dN B G= , can provide

a good fit to crack growth rate versus SERR data for the investigated material system.

The exponent of this modified Paris’ law for mixed-mode I/II, which is in the range from

16 to 22, is higher than that for Mode I (11-13) and Mode II (7-13). It was found by Asp

et al. (Asp et al. 2001) that, for a carbon/epoxy material, the exponent of modified Paris’

law was 5.5 under Mode I loading, 4.4 under Mode II loading, and 6.3 under mixed-mode

loading at the mode ratio GII/GT = 0.5. Compared to their results, the exponent of the

modified Paris’ law for the currently investigate material is very high under loadings at

various mode ratios.

Fig. 5-19: A da/dN vs. Gmax plot for SLB specimens (crack length by visual measurement)

165

Chapter 6

Finite Element Modeling of Crack Propagation in DCB Specimens

Two-dimensional (2D) and three-dimensional (3D) finite element models of the

DCB specimen were built with Abaqus using the virtual crack closure technique (VCCT)

and/or cohesive elements, in order to further analyze delamination behavior and evaluate

the experimental results. Geometries, loadings, modeling techniques, and results of three

2D models and three 3D models are presented in this chapter. Performing the crack

propagation analysis by the finite element method (FEM) is the opposite of conducting an

IFT test as described in Figure 6-1. That is, in an IFT test, the critical SERR (fracture

toughness) is characterized based on certain information obtained from the test, such as,

the load vs. displacement response and crack growth vs. displacement. However, in a

crack propagation analysis with VCCT, the fracture resistance property of the material

(critical SERR) is part of the input information of the finite element model. Based on this

input information, the load vs. displacement response and crack growth are predicted by

performing finite element analysis.

Fig. 6-1: Finite element analysis and the IFT test

166

6.1 Two-dimensional Modeling of the DCB Specimen

6.1.1 Geometry, loading and boundary conditions of 2D models

The geometry of the 2D models is shown in Figure 6-2. The specimen length, l, is

100 mm and thickness is 3.9 mm. The initial crack is located at the mid-thickness of

specimen. Equal but opposite displacements are applied to the ends of the two

delaminated beams. During analysis, the displacement is increased linearly with the

loading time. The end opposite to the loading end is restricted from moving in the

thickness and longitudinal directions of the specimen.

6.1.2 Modeling techniques for 2D models

The objectives of 2D modeling are to: 1) compare the experimentally measured

crack lengths with those predicted by FE models; 2) compare the load-displacement

behavior predicted by FE models with the experimental results; 3) determine the effect(s)

of element size and release tolerance for VCCT; and 4) obtain detailed stress distribution

near the crack tip, through the thickness of the specimen. To fulfill these purposes, three

2D models were built under the plane strain condition with differences in element size,

release tolerance, and/or fracture interface type. In 2D Models #1 and #2, the fracture

δ/2, P

δ/2, P

Fig. 6-2: Geometry and boundary conditions of 2-dimensional DCB models

167

interface incorporates only the VCCT interaction, i.e., the VCCT subroutine for Abaqus

controls the debonding of the fracture surfaces by comparing the current and critical

SERR. However, 2D Model #2 uses a more refined mesh and a smaller release tolerance

compared to Model #1. Thus, the first two models were designed to reveal the effect(s) of

release tolerance and element size. In 2D Model #3, the same meshing scheme as in

Model #1 is used; however, in addition to the VCCT interaction, the fracture interface in

2D Model #3 incorporates a cohesive layer which continues to apply tensile forces

between the pre-bonded nodes after a debonding reaction is requested by the VCCT

subroutine. From the experimental results of one DCB test specimen (Specimen 1-2), the

fracture toughness for the investigated material slightly increased within the first 10 mm

crack extension, and stabilized after that. This “toughening” behavior of the material

could be a result of fibers bridging between crack surfaces. The bridging fibers can

transmit stresses across the crack faces, and thus absorb a portion of applied energy

before breaking and/or pulling out of the crack faces. It is assumed that the mechanical

behavior of the bridging fibers can be simulated by the cohesive elements, and thus in the

third model, a cohesive layer is added to bond the fracture interaction surfaces. The load

vs. displacement response predicted by the third model can then be compared with those

given by the first two models and experimental results. For convenience, the fracture

resistance curve of the DCB test was idealized as a bi-linear curve, as shown in Figure 6-

3. To implement the increasing resistance curve to the FE model, the user needs to

specify spatially varying critical SERRs, i.e., critical SERR value is specified from node

to node along the slave fracture surface.

168

The main features of the three 2D models are summarized as follows:

a) 2D Model #1:

Initial crack length is 51 mm;

The element size in the length direction is about 1 mm; 4 elements comprise

the thickness direction for each delamination leg;

Maximum crack opening displacement is 8 mm;

Release tolerance is 0.01 (the release tolerance is a parameter of the VCCT

subroutine for controlling the debonding of nodes, more details are available

in Section 2.5.2);

Assumed elastic properties of composite material are: (E11 was back-

calculated from the DCB test results, and other parameters, which were

considered to be less important for the current model, are found from the

literature for similar type of material (Krueger and O’Brien 2001).)

E11 = 110.0 GPa; E22 = 10.16 GPa; E33 = 10.16 GPa; ν12 = ν13 = 0.3; ν23 = 0.436; G12 = G13 = 4.6 GPa; G23 = 3.54 GPa.

Mode I fracture toughness linearly increases from 110 N/m to 130 N/m in the

first 10 mm crack extension, as shown in Figure 6-3.

0.060.070.080.090.100.110.120.130.14

50 60 70 80 90 100a , mm

GIc

, N/m

m

Fig. 6-3: Idealized Mode I fracture toughness with crack extension for Specimen 1-2

169

b) 2D Model #2:

Initial crack length is 50.5 mm;

Refined mesh with 0.5 mm element size in the length direction; 4 elements

comprise the thickness direction for each delamination leg;

Maximum crack opening displacement is 8 mm;

Release tolerance is 0.002;

The same elastic and fracture toughness material properties as 2D Mode #1

are used.

c) 2D Model #3

The fracture surfaces are bonded by a cohesive layer in addition to the VCCT

contact interaction;

Initial crack length is 51 mm;

The same meshing discretization as 2D Model #1 is used;

Release tolerance is 0.01, as in 2D Model #1;

Assumed fracture toughnesses of the laminates material are: GIc = 110 N/m;

GIIc = 600 N/m; GIIIc = 1200 N/m.

The elastic properties of the cohesive layer material are specified in terms

of the traction separation response with stiffness values. To ensure the cohesive

elements will have no adverse effect on the stable time increments, it is suggested

to choose, ec KK 1.0= , where Kc is the cohesive element stiffness, and Ke is the

stiffness of surrounding material. Based on this principle, elastic properties of the

cohesive layer chosen are:

E = 1.016 GPa; G1 = G2 = 1.016 GPa

where E is the Young’s modulus and G1 and G2 are shear moduli of the bulk

material of the cohesive elements.

170

The above elastic properties are used to construct diagonal terms in the

stiffness matrix as described in Section 2.5.3.1. The stiffness values are chosen

using the same scheme as used in the cohesive element application example in

Abaqus Benchmark Manual. In this example the stiffness values are chosen to be

1/10 that of the stiffness E2 of the composite material.

The quadratic traction-interaction failure criterion is selected for damage

initiation in the cohesive elements; and a mixed-mode, energy based damage

evolution law based on a B-K law criterion is selected for damage propagation.

Ultimate normal stress (tn0) and shear stress (ts

0) are:

tn0 = 0.55 MPa, ts

0 = 0.55 MPa

(A range of ultimate normal and shear stress values (0 - 7.5 MPa) were tried for

3D Model #3. The values listed above were found to result in closest load vs.

displacement response to the experimental data.)

The B-K form of damage evolution criterion was used, and the parameters

are:

GnC = 110 N/m, Gs

C = 600 N/m, η = 1.

Here, GnC and Gs

C are the critical fracture energies required to cause failure in the

normal, and the first shear directions, respectively. They are specified to

approximately equal to the Mode I and Mode II fracture toughness of the

composite material respectively. η is the parameter for B-K form of damage

evolution criterion (details on damage evolution criterion of cohesive elements are

available in Section 2.5.3.3).

6.1.3 Meshing of 2D models

All 2D models were meshed with 4-node bilinear plane strain elements (CPE4).

For 2D Model #1 and #3, a coarser mesh was used with a total element number of 800 as

shown in Figure 6-4. For 2D Model #2, a refined mesh is used with a total element

171

number of 1600, as shown in Figure 6-5. The cohesive layer for 2D Model #3 was

meshed with COH2D4 elements.

6.1.4 Results of 2D models

6.1.4.1 Crack propagation

The bond state of nodes within the slave fracture surface can be written as an

output parameter by VCCT for Abaqus. As a result, the crack front can be identified after

each loading increment, and crack growth versus opening displacement curve can be

obtained after a crack propagation analysis. The crack growth results obtained for each

model is shown in Figure 6-6, along with experimental results obtained using visual

measurements on one edge of the specimen.

Fig. 6-4: Mesh configuration for 2D Model #1 and #3

Fig. 6-5: Mesh configuration for 2D Model #2

172

Comparing the results between finite element models (Figure 6-6), the

experimental and FE crack growth curves almost coincide with each other. However, the

FE result of crack growth is about 2-4 mm more that obtained from the experiment. This

could be explained by the curved crack front usually created during a DCB test. As

summarized in Chapter 3, it is commonly observed during DCB tests that the crack

length in the center-width of specimen is 2 to 5 mm more than that on the edge. It is the

crack growth on the specimen edge that can be measured during experiments. However,

for the 2D FE models presented here, the plane strain condition is assumed and the crack

growth is therefore the same across specimen width. The crack growth predicted by 2D

FE models should be close to the average crack growth across specimen width.

Therefore, the FE models are expected to over-predict the crack growth relative to the

visual experimental results.

40

50

60

70

80

90

0 2 4 6 8Opening displacement, δ , mm

Cra

ck le

ngth

, a, m

m

IFT Test2D Model #12D Model #22D Model #3

Fig. 6-6: Crack growth versus opening displacement from test data and 2D finite element modeling

173

6.1.4.2 Load-displacement curve

The load vs. displacement curve is indicative of the residual strength of a DCB

specimen as crack size grows. The load vs. displacement curves obtained from 2D FE

models are plotted in Figure 6-7 and along with the curve from the DCB test, which the

FE models are simulating.

From Figure 6-7, the following conclusions can be made:

1) The load vs. displacement curve predicted by any of the 2D models is not

smooth after crack initiation, as was previously pointed out in (Krueger 2008).

However, the curve is smoother in 2D Model #2 which has finer mesh and smaller

release tolerance than the other FE models. A “stick-slip” behavior is also observed in

the load-displacement curves of FE models, similar to that of a test specimen. This

behavior of FE models can be explained as follows: the “slip” (steep load drop)

portion is observed immediately after a pair of bonded nodes has just been released

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8Applied opening displacement, δ , mm

Load

, P, N

IFT test2D Model #12D Model #22D Model #3

Fig. 6-7: Load vs. displacement curves from test data and 2D finite element modeling

174

after the release criterion has been met. As the opening displacement further increases,

the load increases as well, until the applied SERR of the node at crack front reaches

the release criterion again. During this linear loading period, a “stick” (linearly

increasing) behavior of the load-displacement curve is observed. Comparing the

post-initiation load vs. displacement curve from 2D Model #1 and that from Model

#2, it can be concluded that the amplitude of load drop and increase encountered in

one “stick-slip” step is related to the element size and node release tolerance used by

VCCT.

2) The FEM prediction by 2D Models #1 and #2 agrees well with the

experimental results. This indicates that the toughening behavior of this material

could be better modeled by specifying spatially varying critical SERR than using a

cohesive layer.

3) Using a cohesive layer combining with the VCCT interaction, 2D Model #3 at

first over-predicts the load shortly after crack initiation and then as loading proceeds

it coincides with the experimental data.

6.1.4.3 Stress distribution

To observe the distribution of stresses over the thickness of a DCB specimen, values

of the stresses in the longitudinal and thickness direction are plotted for the area around

the crack tip as shown in Figure 6-8 and Figure 6-9. Stress concentration was observed in

the area surrounding the crack tip.

175

Crack tip

Fig. 6-8: Contour plot of the stress in the longitudinal direction (σxx), in MPa, around the crack tip (2D Model #2)

176

6.2 Three-dimensional Modeling of the DCB Specimen

While the 2-dimensional (2D) models described above give good predictions of

load vs. displacement behavior of the DCB specimen with relatively small computation

cost, some details are ignored because of the plane strain assumption. Based on this

assumption, it is expected that the 2D finite element analysis results agree well with beam

theory, which is also based on the plane strain assumption. To simulate SERR

distribution along specimen width and thus analyze finite width effects, three 3-

dimensional (3D) models with straight and curved initial crack front shapes and/or

different fracture interaction schemes were built.

Crack tip

Fig. 6-9: Contour plot of the stress in the thickness direction (σ22), in MPa, around the crack tip (2D Model #2)

177

6.2.1 Geometry, loading and boundary conditions of 3D models

The 3D model geometry is shown in Figure 6-10. Specimen length (l) is 100 mm

and initial crack length (a0) is about 50 mm. Specimen width (b) is 25 mm and total

thickness is 3.9 mm. The initial crack is located at the mid-thickness position of the

specimen. Loading is applied by specifying the linearly increasing opening displacements,

δ/2, at the ends of initially delaminated arms. At the other end, the specimen is restricted

from moving in the width (y) and thickness (z) directions.

6.2.2 Modeling techniques for 3D models

Crack propagation analysis using VCCT is a numerically intensive and expensive

process. The VCCT subroutine in Abaqus actively checks the bond state of nodes within

the fracture interfaces, and hence tracks the crack front along the fracture region. Only

nodes along the delamination front are allowed to debond during any given increment.

Because of the nonlinear property of the crack propagation analysis, very small time

Initial crack length, a0

Specimen length, l

δ/2

δ/2

Fracture interface

x

y

z

Width, b

Fig. 6-10: Geometry of 3D models of the DCB specimen

178

increments are required for convergence of the solution. The crack propagation is

realized by the debonding of nodes along fracture interface. To avoid losing stability

during the analysis, the crack needs to propagate sequentially from node to node through

small time increments. In addition, the element size along the crack front needs to be

small enough so that a dramatic change of energy will not occur if the one or several

pairs of nodes debond(s) in the next increment. Since the 3D analysis requires a small

time increment and fine mesh, the crack propagation analysis simulation is very time

consuming. Thus, efficiency and convergence are two major considerations for 3D

models. For convergence of the solution after the crack initiates, stabilization techniques

are required. For crack propagation analysis using VCCT, the following stabilization

technique can be used (Abaqus 2007):

(i) Contact stabilization—applied across only selected contact pairs and used to

control the motion of two contact pairs while they approach each other in multi-

body contact. Damping is applied when bonded contact pairs debond and move

away from each other.

(ii) Automatic or static stabilization—applied to the motion of entire model,

commonly in models that exhibit statically unstable behavior such as buckling.

(iii) Viscous regulation—only applied to the nodes on contact pairs that have just de-

bonded. The viscous regulation damping causes the tangent stiffness matrix of

the softening material to be positive for sufficiently small time increments.

After the application of stabilization, it is expected that the crack propagation behavior

may not be physically correct, having been modified by the damping forces. It is

suggested in the Abaqus Manual to plot the damping energy and compare the results to

the strain energy in the model. If the damping parameter has been set properly, the value

of damping energy should appear to be a small portion of the total strain energy.

To simulate the crack propagation of specimen with an artificial defect, 3D Model

#1 was designed to have an initially straight crack front, as in a DCB test specimen with a

crack defined by an insert film. In a DCB test specimen, as the crack propagates, a

convex crack front forms under Mode I loading. However, to experimentally track the

crack front shape of the DCB test specimen in real time is very difficult. Hence, it is

179

interesting to find out whether the finite element method with VCCT is capable of

simulating the change in crack front shape as the crack propagates, and 3D Model #1 was

built for this purpose. To reveal the effect of the initial crack front shape on the

development of the subsequent crack front and compare with 3D Model #1, 3D Model #2

was built with an initially curved crack front. While with 3D Model #1, one design

initiative is to examine whether the initially straight crack front will become curved after

the crack propagates, with 3D Model #2, the initiative is to examine whether the opposite

will occur, i.e., the initially curved crack front will become flatter as the crack

propagates. In 3D Model #1 and 3D Model #2, only the fracture interaction defined by

VCCT was applied to the fracture surfaces and a constant fracture toughness of each

mode was specified. To determine the effect(s) of material toughening, in 3D Model #3,

an adhesive layer was added to bond the fracture surfaces in order to simulate the

toughening behavior. The release tolerance for VCCT was set to be a relatively larger

value (0.2) for all 3D models compared to 2D models to facilitate convergence of the

solutions. Details of the 3D models are listed as follows:

a) 3D Model #1

Initially straight crack front; the initial crack length is 50.5 mm;

Elastic properties of composite materials are:

E11 = 110.0 GPa; E22 = 10.16 GPa; E33 = 10.16 GPa;

ν12 = ν13 = 0.3; ν23 = 0.436;

G12 = G13 = 4.6 GPa; G23 = 3.54 GPa;

Fracture toughness: GIc = 110 N/m; GIIc = 600 N/m; GIIIc = 1200 N/m. No

toughening is assumed as the crack grows;

For global automatic stabilization, the damping factor used is 2E-8.

b) 3D Model #2

Initially curved crack front. Initially, crack length at width-center of specimen

is 54 mm, and on the edge is 50.5 mm;

Element size along crack propagation direction in the region where fracture can

occur is about 0.5 mm;

180

Material constitutive and fracture properties are as described for 3D Model #1.

No toughening is assumed as the crack grows;

For global automatic stabilization, the damping factor used is 2E-8.

c) 3D Model #3

Initially straight crack front shape; Initial crack length is 50 mm;

Fracture interfaces are bonded by a cohesive layer in addition to the definition

of VCCT fracture interaction;

Elastic properties of the cohesive layer are:

E = 1.016 GPa; G1 = G2 = 1.016 GPa

The above cohesive element properties are the same as 2D Model #3.

Ultimate normal stress and shear stresses in two directions are:

tn0 = 0.75 MPa, ts

0 = 0.75 MPa, tt0 = 0.75 MPa

These values were specified to be somewhat higher than those in 2D Model #3

to reduce the amount of load drop resulting from energy consumption by

stabilization.

The B-K form of damage evolution criterion was used, and the parameters are:

GnC = 110 N/m, Gs

C = 600 N/m, Gt

C = 1200 N/m, η = 1.

Here, GnC, Gs

C and GtC are the critical fracture energies required to cause

failure in the normal, the first and second shear directions, respectively. They

are specified to approximately equal to the Mode I, Mode II and Mode III

fracture toughness of the composite material respectively. η is the parameter

for B-K form of damage evolution criterion.

Material is toughened as the crack grows.

• Constitutive properties are as described for the other 3D Models;

• For global automatic stabilization, the damping factor used is 2E-11.

181

6.2.3 Meshing of 3D models

For all 3D models, the composite material was modeled with 8-node linear brick,

reduced integration elements (C3D8R). For 3D Model #1 (Figure 6-11), the total number

of elements is 15120. For 3D Model #2, shown in Figure 6-12, the total number of

elements is 16254. For 3D Model #3, as shown in Figure 6-13, a coarser mesh was used

with total element number of 13450, including 1000 cohesive elements. The cohesive

layer for 3D Model #3 was meshed with COH3D8 elements.

x, length

y, width

x, length

z, thickness

Initial crack front


182

x, length

y, width

x, length

z, thickness

3.5 mm

Crack frontInitial crack front


x, length

y, width

x, length

z, thickness

Cohesive layer


183

6.2.4 Results of 3D Models

6.2.4.1 Crack front shape observation

The crack front shapes obtained from 3D Models #1 and #2 are compared to that

of a DCB test specimen next. In a DCB test specimen, the crack fronts are outlined by the

dashed lines in Figure 6-14, repeated here for convenience. In a FE model, the crack front

of a DCB specimen is implied from the bond state of nodes on the bottom and top crack

surfaces. This bond state can be requested as an output parameter for each node in the

bottom crack surface during an Abaqus analysis, and varies from 0 (fully debonded) to 1

(fully bonded). The bond states of the crack surfaces at several loading increments of 3D

Model #1 and 3D Model #3 are shown in Figure 6-14 and Figure 6-15, respectively. In

both figures, the red color indicates fully debonded area, and blue color indicates fully

bonded area, and the crack front is the intersection between them.

Approximated crack front

Fig. 6-14: Crack surfaces of a quasi-static DCB test specimen

184

Increment #650:

Increment #200:

After crack initiation, increment #1:

Fig. 6-14: Crack fronts predicted by 3D Model #1

185

In the following, the crack front shape predicted by 3D Model #1 and Model #3

are compared. For 3D Models #1 and #3, they both have an initially straight crack front;

however, material toughening is simulated by adding a cohesive layer bonding the

fracture surfaces. In Figure 6-14, which is obtained from 3D Model #1, it is shown that

the crack initiates at the center-width region of specimen. As the crack advances, the

crack front shape nearly stays straight. In Figure 6-15, which is obtained from 3D Model

Increment #1240:

Increment #300:


Increment #1240:

Increment #300:



186

#3, it is shown that the crack initiates from center width region as well. In contrast to 3D

Model #1, as crack advances, the crack front becomes slightly convex relative to the left

end in 3D Model #3. This might indicate that a different crack front shape during crack

propagation might be caused by the toughening behavior of material as crack grows,

which is modeled by adding an adhesive layer between the crack surfaces in 3D Model

#3. Another difference between 3D Model #1 and #3 is that the element size in the width

direction is relatively smaller in 3D Model #1. Comparing the FE results to the

experimental results, the convexity predicted by FE analysis is less than that shown in the

DCB test specimen.

On the other hand, 3D Model #2 was built with an initially curved crack front

with no toughening behavior assumed. The contour plot of bond states of crack surfaces

at several increments are shown in Figure 6-16. It is shown that in this case the crack

initiates at the specimen edges. As the loading proceeds, the curved crack front tends to

grow flatter compared to the original crack front shape.

187

Increment #660:

Increment #300:



188

6.2.4.2 Load vs. displacement curve behavior

The load-displacement curves from the three 3D models are plotted in Figure 6-17

along with those from two 2D models. From Figure 6-17, the following conclusions can

be made:

i) The initial stiffness of the specimen (slope of the initial linear portion of the

curve) is slightly smaller in 3D models than in 2D models. This difference

might be a result of different constraint conditions, i.e.—the plane strain

condition is assumed for 2D models. The initial stiffness for 3D Model #2 is

the smallest among all models because the curved crack front makes the

average initial crack length longer for 3D Model #2 than for other models.

ii) For all 3D models, a sudden load drop occurs at the crack initiation point.

This could be because damping is suddenly applied to the system

immediately after crack initiation. Global stabilization was used for all 3D

models. As a result, the damping energy increased as the crack grew. The

total strain energy of whole model and damping energy versus loading time

is plotted in Figure 6-18 for all 3D models. It is shown that the damping

energy is always less than 25% that of total strain energy. Immediately after

crack initiation, the load for 3D Model #1 or Model #3 is about 5 N lower

than that for 2D models. As the damping energy increases with crack

propagation in 3D models, the difference in load between 3D models and 2D

models increases.

iii) Comparing the result from 3D Model #3 to 3D Model #1, the post-initiation

load-displacement curve for 3D Model #3 appears to be shifted upward from

that of 3D Model #1, which indicates a material toughening in 3D Model #3.

This is contributed by the adhesive layer added between the crack surfaces in

3D Model #3.

189

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3 3.5Opening displacement, δ , mm

Load

, P, N

3D Model #13D Model #23D Model #32D Model #32D Model #2

Fig. 6-17: Load vs. displacement curves from 2D and 3D finite element analyses

190

6.2.4.3 Stress distribution

The stress distributions of all 3D models are very similar. As an example, the

distribution of longitudinal stress (σ11) in the specimen is plotted in Figure 6-19.

Total strain energy, 3D Model #1Damping energy, 3D Model #1Total strain energy, 3D Model #2Damping energy, 3D Model #2Total strain energy, 3D Model #3Damping energy, 3D Model #3

Fig. 6-18: Comparison of damping energy to total strain energy for 3D models (The loading speed is 2 mm/sec. Unit for time is second and for energy is N·mm.)

Crack initiation

Crack initiation

191

6.3 Conclusions

The virtual crack closure technique (VCCT) was used to model the crack

propagation behavior of the DCB specimen using Abaqus/Standard V6.7. In two-

dimensional models, two methods were used to simulate the slightly toughening behavior

of DCB specimens. For the first method, critical SERR value was specified as a spatially

varying material property. For the second method, a cohesive layer was used to bond the

fracture interaction surfaces in addition to the constant fracture toughness specified by

VCCT. The results show that the load-displacement curve predicted by the first method

match the experimental data better. For three-dimensional modeling, a stabilization

technique has to be used in order to obtain a converged solution. Possibly because of the

energy consumed by stabilization, an accurate prediction of load-displacement behavior

was not obtained. In terms of crack front shape, one 3D model predicts a slight convexity

Fig. 6-19: Distribution of longitudinal stress (σ11) in a 3D DCB specimen

192

of the crack front shape after crack propagation. However, compared to experimental

results, the curvature is less than that of a DCB test specimen.

193

Chapter 7

Conclusions and Recommendations

7.1 Mode I Interlaminar Fracture Toughness Characterization

Mode I interlaminar fracture toughness tests under quasi-static and cyclic loadings

were conducted with the Double Cantilever Beam (DCB) specimens. For the investigated

carbon/epoxy material system, from quasi-static tests critical SERR values onset from a

thin insert film (GIc) determined by different calculation methods ranged from 110 to 150

J/m2. Compared to the results from the literature, GIc of the investigated material is near

the low end of the common range for carbon/brittle epoxy materials. Based on a

comparison of alternative methods of creating the initial crack for Mode I testing, to base

the GIc value on a short Mode I precrack is suggested for the investigated material system

for three reasons. First, there was larger variation in the critical SERR value based on

an insert film than that based on a Mode I precrack. In addition, in several cases the

critical SERR value based on an insert film was higher (unconservative) than that based

on a short Mode I precrack. Such behavior could be attributed to a resin rich pocket of

material existing just ahead of the inert film. Second, when using either the modified

beam theory or modified crack compliance method to calculate GIc, the value based on

initiation at an insert film could be inaccurate because of the dissimilarity in crack front

shapes formed by an insert film and by the Mode I loading. Third, compared to the

resistance curves obtained for other types of unidirectional laminated material in the

literature, a relatively flat fracture resistance curve was obtained with crack extension for

the material system investigated, (the increase in toughness after 50 mm crack extension

is less than 40% of the GIc value). The often-cited disadvantage of creating a much

higher critical SERR value based on a Mode I precrack does not exist for the material

system investigated. Based on these three arguments, the critical SERR value onset from

a short Mode I precrack is considered to represent a more repeatable and accurate crack

194

onset fracture toughness value for the currently investigated material system. Based on

quasi-static test results, the modified beam theory and modified compliance calibration

method are compared. These theories gave very similar results for the compliance

calibration and SERR value except for the results related to crack growth from the insert

film.

For Mode I fatigue tests, the modified Paris law ( ( ) InaxI GBdNda Im/ = ) was used

to fit the experimentally determined crack growth rate per cycle versus the maximum

applied Mode I SERR. Comparing to the exponents found in literature, which are in the

range of 3.6 to 15, the exponents found in this investigation (11.12 and 12.29) are near

the high end of the common range. For structural design, this material characteristic

implies that a small error in the crack driving force prediction results in a larger than

usual error in the crack growth prediction. Hence, in this case, it might be suitable to

consider a no-growth design criteria, which uses the threshold SERR value, GIth, as the

fatigue fracture toughness for design purposes.

7.2 Mode II Interlaminar Fracture Toughness Characterization

The Mode II interlaminar fracture toughness tests under quasi-static and cyclic

loadings were conducted with the End Notched Flexure (ENF) specimen. For the quasi-

static tests, un-precracked and precracked ENF specimens were used. For Mode II testing,

the crack growth is usually unstable after onset if the common ENF test configuration is

used. Furthermore, even if an alternative test configuration were to be able to produce

stable crack growth, it is very hard to visually measure crack extension accurately under

mode II loading by ENF because of the extended crack tip being held tightly closed.

Therefore, the critical SERR value at crack onset is of main interest. During this

investigation, the average GIIc values based on an insert film determined by different

calculation methods ranged from 782 to 801 J/m2, while the average GIIc values based on

a Mode I precrack ranged from 498 to 545 J/m2. Compared to results in the literature, the

195

Mode II quasi-static fracture toughnesses of the investigated material are in the middle of

common range for carbon fiber composites made with brittle epoxies. The average GIIc

value of un-precracked specimens is about 44% higher than that of precracked specimens

according to the classical plate theory method, and about 60% higher by the compliance

calibration methods. For compliance calibration, the third order polynomial relation in

the form of 33)8( BaAbhC += is recommended.

As was done in the Mode I fatigue tests, the Modified Paris law was fit to the

experimentally determined crack growth rate versus the maximum SERR for each Mode

II specimen. With the crack growth lengths calculated from compliance calibration, the

exponent for the power law relation was found to be in the range of 7.7 to 12.8. With the

crack growth lengths measured visually, the exponent was in the range of 7.1 to 6.8.

Compared to results in the literature (4.3-15), the exponent for the material investigated is

somewhat high. From the literature, for the same material the exponent of power law

relation for Mode I is usually higher than that for Mode II. This is also the case for the

investigated material.

7.3 Mixed Mode I/II Interlaminar Fracture Toughness Characterization

The mixed Mode I/II interlaminar fracture toughness (IFT) tests under quasi-static

and cyclic loading were conducted with single-leg-bending (SLB) specimens. The Mode

II to total strain energy release rate (SERR) ratio achieved with mid-thickness

delaminated SLB specimens was about 0.43. Given this mode ratio GII/GTc and the fact

that the Mode II fracture toughness is about 5-8 times higher than the Mode I fracture

toughness, the SLB test results can be expected to be dominated by Mode I behavior.

From quasi-static tests, critical SERR values at crack onset from an insert film (GTc)

calculated by three methods were in the range of 181 to 185 J/m2. The fracture resistance

curves were obtained for SLB tests with the crack length calculated from compliance

calibration methods. In all cases, the fracture toughness increased slightly with crack

196

extension. In particular, after about 20 mm crack extension, the fracture toughness

increased by about 11 to 53%.

Among all the fatigue crack growth models presented in literature review for

mixed mode crack growth in fatigue loading, the model given by ( )max/ nda dN B G=

provides a good fit to crack growth rate per cycle versus maximum SERR data for the

investigated material system. The exponent of this modified Paris’ law for mixed-mode

I/II, which is in the range from 16 to 22, was higher than that for either Mode I (11-13) or

Mode II (7-13).

7.4 Preliminary Finite Element Modeling Results

The virtual crack closure technique (VCCT) was used to model the crack

propagation behavior of the DCB specimen using ABAQUS/Standard V6.7. In two-

dimensional (2D) models, by specifying the critical SERR value as a spatially varying

material property, a good match of load-displacement curves between the finite element

modeling results and experimental data was obtained. Meanwhile, crack length versus

opening displacement curves by 2D modeling and experiment were similar. However,

the model over predicts the crack length by 2-4 mm, compared to the crack length

measured on the specimen edge during the DCB test. For three-dimensional modeling,

stabilization technique(s) should be used in order to obtain a converged solution. Because

of the energy consumed by stabilization, an accurate prediction of load-displacement

behavior was not obtained. In terms of the crack front shape, the 3D model with an

initially straight crack front predicts a slight convexity of the crack front shape after crack

propagation. However, compared to experimental results, the curvature is less in the

model. For future work, the stabilization technique of 3D modeling needs to be improved

to give better prediction of load versus displacement behavior. Also, it is of interest to

study how the material properties are related to the curvature created by Mode I loading,

so that the inaccuracy of SERR calculations and compliance calibrations resulting from

197

the curved crack front shape can be corrected. Further, the accuracy of SERR calculations

for Mode II and mixed Mode I/II specimens and mode decomposition for mixed-mode

specimens predicted by classical theories should be examined by finite element methods.

198

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207

Appendix A

IFT Test Results From Literature

Specimen dimension Material Fiber Matrix

Loading speed,

mm/min Length, L, mm

Thickness,2h, mm

Width, b, mm

Initial crack

length, a0, mm

Insert type Precrack GIc (J/m2) Source

UD FRP E-glass Epoxy 1 150 5 20 35 and 50 Polypropylene 8 μm NO

a0 = 35 mm, NL:243, 5%: 317, AE:246; (initiation) a0 = 50 mm, NL:268, 5%: 343, AE:178; (initiation)

(Ducept et al. 1997)

UD FRP E-glass M11 2 150 6 20 23-55 Teflon film N/A Initiation value: 118.02±2.72

(Benzeggagh and Kenane 1996)

UD FRP Carbon Epoxy 1-5 4.2 20

PTFE 60 and 12.6 μm; aluminum foil 20 and 40 μm

with and without

100-200 (initiation), 150-250 (propagation)

(Davies et al. 1990)

UD FRP HR

carbon (T300)

Epoxy (914) 0.5 190 4.16 20 30 N/A N/A 165 (initiation) (Hug et al. 2006)

Table A-1: DCB tests results from literature

208

Table A-1. Continued

Specimen dimension Material Fiber Matrix

Loadingspeed,

mm/minLength, L, mm

Thickness,h, mm

Width, b, mm

Initial crack

length, a0, mm

Insert type Precrack GIc (J/m2) Source

IM7 8552 220±10 (initiation) HTA7 6376 230±3 (initiation) UD FRP T300 914

220 3.12 20 50 Release film A4000, 10 μm

112±10 (initiation)

(Schőn et al. 2000)

UD FRP IM7 8551-7A 3.2 25 40 PTFE 25.4 and 12.7 μm

None and wedge open

220-284 (25.4 μm insert); 144-176 (12.7 μm insert)

(Walsh and Bakis 1995)

[0]6 4.38 NO 282±42 (initiation)

[±30]5 7.3 NO 214±14 (initiation); 500 (steady state)

[±45]5

E-Glass Isophthalic

Polyester 0.76

7.3

20 33 Polypropylene

8 μm NO 176±32 (initiation);

300 (steady state)

(Ozdil and

Carlsson 1999)

See Figure A-1

Ciba-Geigy

graphite

Epoxy, R6376 0.75 4.06 25.4 57.2 Teflon with and

without See Figure A-2 (Polaha et al. 1996)

209

mean

150

200

250

300

350

400GIc,

J/m2

GIc, of non-precracked DCB specimens

[ ]0++ 1 5⎡ ⎤⎣ ⎦

+- 15⎡ ⎤⎣ ⎦

++30 cD⎡ ⎤⎣ ⎦

+- 30 cD⎡ ⎤⎣ ⎦

++30 tB⎡ ⎤⎣ ⎦

+- 30 tB⎡ ⎤⎣ ⎦

mean

150

200

250

300

350

400GIc,

J/m2

GIc, of precracked DCB specimens

[ ]0++ 1 5⎡ ⎤⎣ ⎦

+- 15⎡ ⎤⎣ ⎦

++30 cD⎡ ⎤⎣ ⎦

+- 30 cD⎡ ⎤⎣ ⎦

++30 tB⎡ ⎤⎣ ⎦

+- 30 tB⎡ ⎤⎣ ⎦

Fig. A-2: GIc values of different layups from (Polaha et al. 1996).

Specimen notation Layup [ ]0 [ ]32

0 ++ 1 5⎡ ⎤⎣ ⎦

+ -- 4 +15/0/-15/0/15/0 /15 / 0 / 15 / 0 / 15

s⎡ ⎤−⎣ ⎦

+- 15⎡ ⎤⎣ ⎦

+ -- 4 +

- ++ 4 -

[ 15/0/-15/0/15/0 /15 / 0 / 15 / 0 / 15/ /

15/0/15/0/-15/0 / 15 / 0 /15 / 0 / 15]

d−

−

++30 cD⎡ ⎤⎣ ⎦

+ -- 4 +30/0/-30/0/30/0 /30/0/ 30/0/ 30

s⎡ ⎤−⎣ ⎦

+- 30 cD⎡ ⎤⎣ ⎦

+ -- 4 +

- ++ 4 -

[ 30/0/-30/0/30/0 / 30 / 0 / 30 / 0 / 30/ /

30/0/30/0/-30/0 / 30 / 0 / 30 / 0 / 30]

d−

−

++ 30 tB⎡ ⎤⎣ ⎦

+ -- 12 +30/0 / 30

s⎡ ⎤⎣ ⎦

+- 30 tB⎡ ⎤⎣ ⎦

+ - - +- 12 + + 12 -30/0 / 30/ / 30/ 0 / 30d⎡ ⎤⎣ ⎦

Fig. A-1: Lay-up of specimens, based on (Polaha et al. 1996).

210

Material Fiber Matrix Loading speed,

mm/min

Half Span Length, L,

mm

Thickness, 2h, mm

Width, b, mm

Initial crack

length, a0, mm

Insert type Precrack GIIc (J/m2) Source

UD FRP E-glass Epoxy 1 65 5 20 35 Polypropylene 8 μm no

1460 (NL); 2185 (5%); 1595 (AE)

(Ducept et al. 1997)

UD FRP Carbon Epoxy 1-5 40 4.2 20 PTFE, 60 and

12.6 μm; aluminum foil,

20 and 40 μm

w/wo

~500 (mode I precrack);~1000-1500 (starter film)

(Davies et al. 1990)

[0]6 4.4 20 25 no 496±135 [±30]5 7.3 20 25 no 976±71 [±45]5

E-Glass Isophthalic polyester 1 50

7.3 20 25

Polypropylene 8 μm

no 1485±158

(Ozdil et al. 1998)

PI 25 μm folded w/wo 372/1036PI 12.5 μm w/wo 417/480 PI 7.5 μm w/wo 257/304

UD CFRPCF

(Besfight HTA)

Brittle epoxy

≈3 mm (22 plies)

PTFE 12.5 μm w/wo 320/370 PI 25 μm folded w/wo 920/890 PI 12.5 μm w/wo 560/740 PI 7.5 μm w/wo 610/460

Woven CFRP

CF (Besfight T400HB)

Brittle epoxy

≈3 mm (16 plies)

PTFE 12.5 μm w/wo 570/640 Toughened UD CFRP

CF (IM-600)

Toughened epoxy PI 12.5 μm w/wo 930/991

UD CFRP AS4 CF PEEK

0.5 50

25 25

PI 7.5 μm w/wo 1530/1477

(Tanaka et al. 1995)

Table A-2: ENF test results from literature

211

Specimen Fatigue loading Paris' law parameters Material

Lay-up Specimen GII/G Dimension (L×b×2h)

Initial crack

length, a0 Frequency Load

ratio, R Gc** Coefficient, B

Exponent, n

Reference

mm mm Hz J/m2 mm/cycle UD carbon/epoxy, HTA/6376C

MMELS * 0.5 150×20×3.3 N/A 2 0.1 447 1.68×10-7 6.28 (Blanco et

al. 2004)

UD graphite/epoxy, IM7/977-3

[0]28 DCB 0 150×25×7.5 50 5 0.1 200 15 (Gregory and Spearing 2005)

AT400/epoxy [0]26 ENF 1 170×25×3 span length

100 25 10 0.5 610 6.5 and 4.6 (Hojo et al.

2001)

UD graphite/epoxy [0]26 ENF 1 200×25×NA 31.75 10 0.1 (Tanaka

1997) Woven E-glass/vinyl ester DCB 0 254×38×5 38 1-4 0.1 350 5.4 (Shivakumar

et al. 2006) UD graphite/epoxy, T300/914C

DCB 0 145×N/A×8 N/A 0.1, 0.3, 0.5 1.32×10-25 8.81

(Gustafson and Hojo 1987)

Glass/polypropylene ENF 1

130×20×4, total span

110 26-51 5 0.1 1.4×10-5-

1.1×10-3 4.3-4.8 (Bureau et al. 2002)

UD carbon/epoxy

[0]16 T800H/3900-2 DCB 0 140×20×3 20- 25 10 0.1- 0.5 180 3.6-7.3 (Hojo et al.

2006b)

Table A-3: Fatigue test results from literature

( )[ ]s85 0,5//0 ±

( )[ ]s820 0,5//0 ±

212

Table A-3. Continued

Specimen Fatigue loading Paris' law parameters Material

Lay-up Specimen GII/G Dimension (L×b×2h)

Initial crack

length, a0 Frequency Load

ratio, R Gc** Coefficient, B

Exponent, n

Reference

mm mm Hz J/m2 mm/cycle

[0]16 T800H/3900

-2 0.1- 0.5 180 3.6-7.3 UD

carbon/epoxy [0]24 UT500/111

DCB 0 140×20×3 20- 25 10

0.2- 0.5 150

6-8

(Hojo et al. 2006b)

DCB 0 140×20×3 25 160 4.6 and 5.0 UD

CF/epoxy, UT 500/111

ENF

160×20×3, total span

100 25

10 0.1 and 0.5 470

13 and 15

(Hojo et al. 2006a)

DCB 0 1.2×10-7 5.5 ENF 1 7.5×10-7 4.4

carbon/epoxy, HTA/6376C

[012//(± 5/04)S]

MMB 0.5 150×20×3.1 35 5 0.1

2.5×10-8 6.3

(Asp et al. 2001)

* Mixed mode end load split specimen ** Critical strain energy release rate from static tests

213

Appendix B

Additional Specimen Information and Test Results

1. Dimensions of Specimens

Table B-1: Dimensions of DCB specimens used for quasi-static tests

Specimen Dimension Notation1-1 1-2 1-3 1-4 1-5

Length: l 152.5 152.8 153.2 152.4 152.7 Nominal width: b 24.5 25.2 25.2 25.1 25.1 Nominal thickness: 2h 3.7 3.9 3.9 4.0 3.9 Initial crack length: a0 49.3 51.4 52.4 50.2 51.1

Table B-2: Dimensions of DCB specimens used for fatigue tests

Specimen Dimension Notation4-2 4-3 4-4 4-5

Length: l 153.0 153.1 153.2 153.3 Nominal width: b 25.1 25.0 25.0 25.1 Nominal thickness: 2h 4.0 4.0 3.9 3.8 Initial crack length: a0 51.8 53.0 52.9 50.0

214

Table B-3: Dimensions of ENF specimens (un-precracked, tested in quasi-static tests)

Specimen Dimension (mm) Notation

2-1 2-2 2-3 2-4 Length: l 152.5 153.1 152.9 152.6 Nominal width: b 25.0 25.0 25.0 25.1 Nominal thickness: 2h 3.8 3.9 3.9 3.8 Initial crack length: a0 24.6 25.1 23.8 24.2

Table B-4: Dimensions of ENF specimens (precracked, tested in quasi-static tests)

Specimen Dimension (mm) Notation

2-5 2-6 2-7 Length: l 152.5 152.6 152.7 Nominal width: b 25.3 25.5 25.3 Nominal thickness: 2h 3.9 3.9 3.9 Initial crack length created by insert film: a0 32.3 32.1 31.1

Initial crack length after precracking: ap 35.1 35.0 33.6

Table B-5: Dimensions of ENF specimens (precracked, tested in fatigue tests)

Specimen Dimension (mm) Notation 5-3 5-4 5-5

Length: l 152.6 152.7 152.7 Nominal width: b 25.3 25.3 25.5 Nominal thickness: 2h 3.9 3.9 3.9 Initial crack length created by insert film: a0 33 31.1 31.1

Initial crack length after precracking: ap 36.8 34.5 34.7

215

Table B-6: Dimension of SLB quasi-static specimens

SLB, Quasi-static Specimen

Dimension (mm) Notation3-1 3-2 3-3 3-4

Length: l 152.6 151.6 153.3 151.7 Nominal width: b 25.1 25.1 25.2 25.1 Nominal thickness: 2h 3.9 3.9 3.9 3.9 Initial crack length: a0 38.4 38.7 38.4 38.2

Table B-7: Dimension of SLB fatigue specimens

SLB, Fatigue Specimen Dimension (mm) Notation6-1 6-2 6-3

Length: l 151.8 151.6 152.2 Nominal width: b 25.2 25.2 25.0 Nominal thickness: 2h 3.9 3.9 3.9 Initial crack length: a0 40.0 38.4 38.9

216

2. Mode I Quasi-static Fracture Toughness at Crack Onset

Table B-8: A summary of Mode I quasi-static test results

Critical point Mode I Critical SERR Initial crack length Displacement Load MCC,

initial onsetMBT,

initial onset Precrack, by MCC

Specimen No.

a0, mm δcr, mm Pcr, N GIc, J/m2 1-1 49.285 2.824 38.1 120 111 142 1-2 51.441 2.675 40.4 124 111 129 1-3 52.447 2.852 48.7 161 145 118 1-4 50.246 2.750 42.8 136 120 130 1-5 51.080 3.010 44.7 146 137 134

Average 2.822 42.9 137 125 131 STD * 17.0 15.5 8.7

COV ** 12.4% 12.4% 6.7%

* Standard deviation; ** Coefficient of variation, STD/mean.

217

3. Mode I Fatigue Crack Growth Rate vs. Maximum SERR Plots

Fig. B-1: da/dN - GImax plot for Specimen 4-2 (Crack growth measured visually.)

Fig. B-2: da/dN - GImax plot for Specimen 4-2 (Crack growth calculated by compliance calibration.)

218



219



220

Fig. B-7: da/dN - GImax plot for Specimen 4-5. (Crack growth measured visually.)


221

4. Mode II Quasi-static Fracture Toughness at Crack Onset

Table B-9: A summary of Mode II quasi-static test results

Critical point Mode II Critical SERR Initial crack length Displacement Load by CPT by CC (1) by CC (2)Specimen

No. a0, mm δcr, mm Pcr, N GIIc, J/m2

2-1 24.6 2.013 980 733 837 801 2-2 25.1 2.294 1086 808 745 812 2-3 23.8 2.308 1108 851 821 833 2-4 24.2 2.156 1007 736 801 740

STD 57.6 40.1 39.9 Mean 24.4 2.193 1045.3 782 801 797 COV 7.37% 5.01% 5.01%

2-5 35.1 1.624 610.4 576.0 518.0 505 2-6 35.0 1.567 561.6 497.0 436.0 484 2-7 33.6 1.629 630.7 563.0 539.0 511

STD 42.4 54.4 14.2 Mean 34.6 1.607 600.9 545 498 500 COV 7.77% 10.94% 2.84%

222

5. Mode II Fatigue Crack Growth Rate vs. Maximum SERR Plots

Fig. B-9: da/dN - GIImax plot for Specimen 5-3 (Crack growth rate was calculated with crack growth by compliance calibration. The points excluded from linear fit are near the crack growth arrest domain.)

Fig. B-10: da/dN - GIImax plot for Specimen 5-3 (Crack growth rate was calculated with crack growth by visual measurement. The points excluded from linear fit are taken near the beginning of the test.)

223

Fig. B-11: da/dN - GIImax plot for Specimen 5-4 (Crack growth rate was calculated with crack growth by compliance calibration. The points excluded from linear fit are taken near the beginning of the test.)

224

Fig. B-12: da/dN - GIImax plot for Specimen 5-5 (Crack growth rate was calculated with crack growth by compliance calibration. The points excluded from linear fit are taken near the beginning of the test.)

Fig. B-13: da/dN - GIImax plot for Specimen 5-5 (Crack growth rate was calculated with crack growth by visual measurement. The points excluded from linear fit are taken near the beginning of the test.)

225

6. Mixed Mode I/II Quasi-static Fracture Toughness at Crack Onset

Table B-10: A summary of Mode I quasi-static test results

Critical point Mixed Mode I/II Critical SERR Initial crack length Displacement Load by CPT by CC (1) by CC (2) Specimen

No. a0, mm δcr, mm Pcr, N GTc, J/m2

3-1 38.379 1.249 208.6 177 173 167 3-2 38.718 1.358 218.5 196 207 198 3-3 38.447 1.290 207.9 178 173 181 3-4 38.172 1.325 215.0 187 187 178

Mean 185 185 181 STD 38.4 1.306 212.5 8.9 16.1 12.8 COV. 4.8% 8.7% 7.1%

226

7. Mode II Fatigue Crack Growth Rate vs. Maximum SERR Plots

Fig. B-14: da/dN - Gmax plot for Specimen 6-1 (Crack growth was calculated by compliance calibration.)

Fig. B-15: da/dN-Gmax plot for Specimen 6-1 (Crack growth was measured visually.)

227

Fig. B-16: da/dN-Gmax plot for Specimen 6-2 (Crack growth was calculated by compliance calibration.)

Fig. B-17: da/dN - Gmax plot for Specimen 6-2 (Crack growth was measured visually.)

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Fig. B-18: da/dN-Gmax plot for Specimen 6-3 (Crack growth was calculated by compliance calibration.)

Fig. B-19: da/dN-Gmax plot for Specimen 6-3 (Crack growth was measured visually.)

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Appendix C

Non-Technical Abstract

In laminated fiber-reinforced composite materials, high-strength fibers are

combined with a ductile, light-weight matrix to achieve good mechanical properties in

the fiber directions. According to engineering needs, thin layers with various

unidirectional orientations of fibers are selected and bonded together to achieve the

desired properties. With this manufacturing method, high mechanical performance and

great design flexibility are achieved. This type of material is ideal for load bearing

components in many engineering fields—particularly those where low weight is

important. One weakness of this type of material is delamination or interlaminar fracture,

where adjacent layers separate from each other. Delamination is especially important

because it could cause a catastrophic loss of compressive strength of a structural

component if undetected in nondestructive inspections. In developing new materials, it is

essential to characterize the material’s resistance to delamination, which is named

interlaminar fracture toughness (IFT). The purpose of this investigation is to characterize

the IFT properties of a carbon/epoxy laminated material system in quasi-static and fatigue

loadings. Additionally, it is the goal of this investigation to examine the state-of-the-art

IFT test methods, tailor them to suit the current material, and make recommendations for

standardizing the test methods for similar material systems. According to the relative

displacement between the fracture surfaces, the delamination behavior can be

decomposed into three modes: Mode I, Mode II, and Mode III. The Mode I, Mode II, and

Mixed-mode I/II interlaminar fracture toughnesses of the investigated material were

characterized with the DCB specimen, ENF specimen, and SLB specimen, respectively.

Under Mode I loading, the DCB quasi-static test results showed that for the investigated

material system the fracture toughness is low in comparison to results in the literature.

Under Mode II loading, the quasi-static ENF test results showed that the fracture

toughness for the investigated material system is in the middle of common range. Under

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fatigue loading, tests in all modes showed that the delamination growth rate decreased

greatly as the crack driving force decreased. The test results can be used to complete the

database of the properties of the investigated material system. Some controversial test

methods found in the literature were examined for the investigated material system, and

recommendations were made to give more repeatable and accurate characterization of

fracture toughness for similar material systems.

characterization of interlaminar fracture …

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