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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
iii
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
iv
experimental results. However, the crack front shape predicted by FEA did not agree well
with experimental results.
NAVAIR Public Release 08-1184 Distribution: Statement A – “Approved for public release; distribution is unlimited”
v
TABLE OF CONTENTS
LIST OF FIGURES ..................................................................................................... ix
LIST OF TABLES.......................................................................................................xvii
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
vi
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)
Chapter 4 The Mode II Interlaminar Fracture Toughness Testing .............................102
vii
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
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
6.2 Three-dimensional Modeling of the DCB Specimen ..............................176
viii
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
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
ix
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. 2-9: 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
x
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
xi
Fig. 3-10: Load vs. displacement curve -- large load drop occurred at crack onset (Specimen 1-3)......................................................................................................82
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
xii
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. 4-17: 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
xiii
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
xiv
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. 6-12: Mesh configuration for 3D Model #2 .........................................................182
Fig. 6-13: Mesh configuration for 3D Model #3 .........................................................182
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. 6-15: Crack fronts predicted by 3D Model #3......................................................185
Fig. 6-16: Crack fronts predicted by 3D Model #2......................................................187
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
Fig. B-8: da/dN - GImax plot for Specimen 4-5 (Crack growth calculated by compliance calibration.) .......................................................................................220
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
xvi
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
xvii
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
xix
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
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.
4.1 Material, Specimen and Test Configuration
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
00.0005’’ Teflon film
0-degree fiber direction
Specimen 2-1
Specimen 2-2Specimen 2-3
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
00.0005’’ Teflon film
0-degree fiber direction
Specimen 2-5
Specimen 2-6Specimen 5-3
Specimen 5-4
Specimen 5-5Specimen 2-7
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
Supporting line Supporting line
Test specimen
L L crcl
l
r1
r2
ap
Loading direction
Loading line
Supporting line Supporting 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)
1.1 kN (250 lb) load cell
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
4.2.2.1 Load-displacement curves
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
4.2.2.2 Compliance calibration
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
L inear fit (a=40.7 mm):
y = 321.36x ‐ 20.274, R 2 = 0.9985
L inear fit (a=37.9 mm):
y = 351.22x ‐ 21.559, R 2 = 0.9985
L inear fit (a=35.0 mm):
y = 383.56x ‐ 22.888, R 2 = 0.9988
L inear fit (a=32.0 mm):
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)
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.
4.3.2.1 Crack growth
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 Mode I precrack
Crack surface created by an ENF fatigue test
Crack surface created by Mode I precrack
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
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.
5.1 Material, Specimen and Test Configuration
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
00.0005’’ Teflon film
0-degree fiber direction
Specimen 3-2
Specimen 3-4
Specimen 6-3Specimen 6-2
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)
0.44 kN (100 lb) load cell
SLB Specimen
Bending fixture
Not shown: -Servo-hydraulic actuator -long-distance instrumented stage microscope
MTS built-in load cell (stationary)
0.44 kN (100 lb) load cell
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
5.2.2.1 Load-displacement curves
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.
5.2.2.2 Compliance calibration
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
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
5.3.2.1 Crack growth
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 tip position 2
Crack can not be visually detected
a1
a2
εa1
εa2dcdc
a1 < a2
εa1 < εa2
Crack tip position 1
Crack tip position 2
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).)
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
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
Bibliography
Abaqus (2007). Abaqus User’s Manual, Version 6.7.
Anderson, T. L. (2005). Fracture Mechanics Fundamentals and Applications, third edition, Tayler & Francis Group, Boca Raton.
ASD-STAN preEN 6033. (1995). "Aerospace series carbon fibre reinforced plastics test method determination of interlaminar fracture toughness energy mode I - GIc." AeroSpace and Defense Industries Association of Europe - Standardization.
ASD-STAN prEN 6034. (1995). "Aerospace series carbon fibre reinforced plastics test method determination of interlaminar fracture toughness energy mode II-GIIc." AeroSpace and Defence Industries Association of Europe-standardization.
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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
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.)
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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