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An investigation into the mixed mode delamination behaviour of brittle composite laminates Kanji, Karim Mohamed Jamal 2003

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AN INVESTIGATION INTO THE MIXED MODE DELAMINATION BEHAVIOUR OF BRITTLE COMPOSITE LAMINATES by Kar im Mohamed Jamal Kanji-B .A .Sc . (Mechanical Engineering), University of Waterloo, 1999 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E S T U D I E S (Department of Metals and Materials Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A May 2003 © Kar im Mohamed Jamal Kanji, 2003 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Abstract Delamination is a prevalent composite laminate failure mode. It is of particular concern to the aerospace industry where laminated composites have found widespread usage in critical applications. Delamination growth has been widely studied, with Linear Elastic Fracture Mechanics ( L E F M ) being the most common approach taken to predict delamination behaviour, typically through global parameters such as specimen geometry and applied load measured away from the actual crack tip. However, internal mechanisms, such as fibre bridging, can occur within the structure thus affecting the transfer of the globally applied conditions to the actual crack tip. Thus, measurements made at the actual crack tip, referred to as local measurements, can be compared to the delamination behaviour predicted from the global analysis. The objective of this thesis is to better understand the relationship between the globally predicted and the actual local crack tip behaviour under mixed mode loading. O f particular interest is the mixed mode region at low shear (mode, II) loads where the global tensile opening (mode I) loads exceed the pure mode I fracture toughness of the material. A n experimental loading j i g developed by Paris et al. (2001) is used in this work. The j ig can be used inside a scanning electron microscope (SEM) to allow for simultaneous local and global analysis of a specimen. Previous pure mode experiments (Paris, 1998) showed that in the absence of fibre bridging mode I global predictions match the actual local crack tip conditions. Global mode II loads, however, induce a local mixed mode condition where the mode I component is created due to the surface roughness as the crack surfaces slide over each other. In this work, mixed mode experiments on unidirectional AS4/3501-6 carbon fibre/epoxy laminates show that global applied shear loads are transferred directly to the crack tip. The crack surface roughness effects seen initially under pure mode II loading also manifest themselves under mixed mode loading resulting in a significantly higher local mode I component than predicted globally. A s such, current test practices and global data reduction schemes are inadequate and do not provide a complete picture of delamination behaviour. The total local critical strain energy release rate is also significantly higher than that determined globally and observed in the literature. The higher local failure loads observed are consistent with fractographic evidence in the literature that indicates failure at the mixed mode conditions examined here is similar to much tougher mode II dominated failure. 11 Table of Contents Table of Contents Abstract ii Table of Contents iii List of Tables vii List of Figures f viii Nomenclature xviii Acknowledgements xxi Chapter 1 Introduction 1 1.1 Building Block Approach 4 1.2 Global Approach . 5 1.3 Local Approach . 8 1.4 Summary 9 Chapter 2 Background and Literature Review 15 2.1 Pure Mode Delamination Behaviour 17 2.7.7 Mode I Delamination 7 7 2.1.2 Mode II Delamination 19 2.2 Mixed Mode Delamination Behaviour 21 2.2.1 Global Test Methods 22 2.2.2 Global Data Reduction Methods 23 2.2.2.1 Compliance Methods 23 2.2.2.2 Energy Based Area Method ^ 25 2.2.3 Mechanisms of Mixed Mode Delamination Growth 26 2.2.3.1 Mixed Mode Failure Envelopes 26 .2.2.3.2 Mixed Mode Fracture Processes 27 2.2.3.3 Crack Kinking and Preferred Crack Growth Directions 28 2.3 Summary 29 i i i Table of Contents Chapter 3 Experimental Method 43 3.1 Specimen Preparation 44 3.2 Data Acquisition and Test Control 45 3.3 Mechanical Testing 46 3.3.1 Pure Mode II Loading 46 3.3.2 Superposition Principle 46 3.3.3 Calculation of Global Strain Energy Release Rate 47 3.4 Image Acquisition 49 3.5 Image Analysis 49 3.5.1 Calculation of Local Strain Energy Release Rates 50 3.6 Modification to Specimen Configuration 51 3.7 Modification to Testing Method 52 3.7.1 Global Strain Energy Release Rate Calculation Method 52 3.7.2 Pure Mode II Loading of Specimen 52 3.8 Summary 52 Chapter 4 Experimental Results 61 4.1 Experimental Objectives 62 4.1.1 Parameters of Interest 63 4.1.1.1 Load History. : .. 63 4.1.1.2 Load Path 63 4.1.1.3 Crack Condition . _ _ 63 4.2 Specimen Descriptions 64 4.3 Test Descriptions 64 4.3.1 Testmx4 1212, mx4 1221, mx4 2121, mx4 2112, and mx4 12 Descriptions 65 4.3.2 Test mx3 1221 andmx3 12 Descriptions 65 4.3.3 Test mx4 w Description 65 4.3.4 Test mx4 rev 1221 Description 66 4.3.5 Test mx4 21 up Description 66 iv Table of Contents 4.3.6 Test mx4 cp and mx5 cp Descriptions 66 4.4 Test Results and Observations 67 4.4.1 Test mx4 1212, mx4 1221, mx4 2121, mx4 2112, and mx4 12 Results 67 4.4.2 Test mx3 1221 and mx3 12 Results 69 4.4.3 Test mx4 w Results 69 4.4.4 Test mx4 rev 1221 Results 69 4.4.5 Test mx4 21 up Results 69 4.4.6 Test mx4 cp and mx5 cp Results 69 4.5 Summary 70 Chapter 5 Analysis and Discussion 101 5.1 Specimen Load History 102 5.1.1 Damage Zone 102 5.1.2 Loading of Specimen With Crack Tip Damage 103 5.2 Test Load Path 104 5.3 Crack Conditions 105 5.3.1 Effect of Starter Film on Subcritical Specimen Response _106 5.3.2 Effect of Crack Surface Roughness on AG; 7 0 7 5.3.3 Effect of Crack Surface Roughness on the AGj- GIG Relationship 7 09 5.3.4 Transfer of Global and Internal Loads to the Crack Tip 110 5.3.4.1 Pure Mode I Loading . 110 5.3.4.2 Pure Mode II Loading . . 111 5.3.4.3 Mixed Mode Loading 111 5.3.5 Experimental Considerations 772 5.3.5.1 Current Global Data Reduction Practices 112 5.3.5.2 Using Global Parameters to Predict Actual Local Conditions 113 5.3.6 Variability 775 5.4 Increased Mode I Capacity Under Mixed Mode Loading 115 5.5 Failure Envelopes 117 5.6 Summary 118 v Table of Contents Chapter 6 Numerical Model and Results 135 6.1 Finite Element Model Definition 136 6.1.1 Element Type 136 6.1.2 Mesh _ _ 136 6.1.3 Boundary Conditions 136 6.1.4 Analysis Type_ 137 6.1.5 Model Limitations 137 6.2 Model Validation 137 6.2.1 Global Loads _137 6.2.2 Local Displacements 137 6.2.3 Local Stresses 139 6.3 Analysis of Experimental Measurement Technique 140 6.4 Mixed Mode Stress Field Analysis 141 6.5 Summary 141 Chapter 7 Conclusions and Future Work 154 7.1 Summary 155 7.2 Conclusions 155 7.3 Future Work 157 References 159 Appendix A Work Done During Fracture 165 A. 1 Analytical Approach 166 A.2 Theoretical Model 167 A.3 Test on Experimental Data 167 vi List of Tables List of Tables Table 2.1 Review of mixed mode global test methods (from Crews and Reeder, 1988). _ 30 Table 2.2 Assumptions of the general beam theory analysis that require modification for application to composite materials (from Davies and Benzeggagh, 1989). 31 Table 4.1 Characteristics of specimens mx3, mx4, and mx5. 71 Table 4.2 Review of Test Descriptions and Experimental Results 72 Table 4.3 Material Properties of AS4/3501-6 unidirectional laminate (Paris, 1998). 73 Table 5.1 Results of mx4 and mx5 mixed mode GIL partitioning into G/c, the globally applied component, and an equivalent AG/, the internally generated component. 119 Table 6.1 Comparison of mode I global response of F E A with analytical prediction. 143 Table 6.2 Comparison of mode II. global response of F E A with analytical prediction. 143 v i i List of Figures List of Figures Figure 1.1 Schematic of damage mechanisms in laminated composites (Paris, 1998). 11 Figure 1.2 Description of the failure that lead to the crash of American Airlines Fight 587 on November 12 t h , 2001 in New York (NTSB) . 11 Figure 1.3 N T S B photograph of failed composite connection on the tail section of A A Fit. 587. A t least five major delaminations can be observed in the highlighted portion of the photograph. 12 Figure 1.4 Range of scales of interest in the study of the failure of composite structures (Paris, 1998). 12 Figure 1.5 Schematic of an arbitrary body of width B containing a crack of length a subject to an external point load, P (Paris, 1998). 13 Figure 1.6 Schematic describing the three modes of loading. 13 Figure 1.7 Schematic of the displacement field of the crack faces near the crack tip under when subject to mixed mode loading (Paris, 1998). 14 Figure 2.1 Schematic of damage zone size in a brittle unidirectional fibre composite. The damage is limited to the resin rich region between lamina (Bradley, 1989). 32 Figure 2.2 Schematic showing fibre bridges across the crack opening. These fibre bridges can shield the crack tip from seeing the full applied load and can consume energy during the fracture process. 32 Figure 2.3 Example of R-curve behaviour for a brittle unidirectional AS4/3501-6 D C B specimen (Paris, 1998). 32 Figure 2.4 Finite element normal stress contour plots at the crack tip for an orthotropic split laminate under pure mode I elastic loading (Corleto et al, 1987). 33 Figure 2.5 Plot of COD profile as a function of r for an applied G / G of 34.9 J /m 2 (Ferguson, 1992). The combined global and local analysis shows the square root singularity at the crack tip. 33 Figure 2.6 Plot of Gn vs. GIG for unidirectional AS4/3501-6 D C B specimens, showing that GIL=GIG for all cases up to and including at crack growth but one. The exception occurs for the first loading from the insert and might be explained by the insert initially sticking to the crack faces (Paris et al, 2001). 34 Figure 2.7 Plot of Gn vs. G / G for unidirectional AS4/3501-6 specimen with significant fibre bridging (Paris, 1998). The dotted line represents a one-to-one correspondence as seen in the absence of fibre bridges (Figure 2.6). The fibre bridges act to shield the crack tip, G//,from experiencing the entire applied G / G . They can also absorb energy i f the fibre bridges are broken. 35 Figure 2.8 Plot of COD vs. r (distance from the crack tip) for G / G values of 99, 277 and 426 J/m 2 on a unidirectional AS4/3501-6 specimen with significant fibre bridging (Paris, 1998). Dashed lines show COD profile calculated from Gn vi i i List of Figures that gives the best fit. The point of inflection on the profile curve can be correlated directly to a fibre bridge acting to keep the crack tip closed. 36 Figure 2.9 Mixed mode delamination criterion for AS4/3501-6 (O'Brien, 1997 from Reeder, 1993). Gc is the critical total strain energy release rate and G is the total applied. Therefore G// /G = 0 is pure mode I loading and G// /G = 1 is pure mode II loading. Note that the scatter and variability associated with G//c is much greater than that associated with G/c. 36 Figure 2.10 Figure showing the mechanism of mode II fracture in brittle unidirectional AS4/3501-6 C F R P laminates (Paris, 1998). Hackles are formed ahead of the crack tip at an initial angle of 45 degrees due the maximum principal normal stresses in the area. These hackles then rotate and coalesce resulting in a mode II crack extension. ._. 37 Figure 2.11 Finite element shear stress contour plots at the crack tip for an orthotropic split laminate under pure mode II elastic loading (Corleto et al, 1987). 38 Figure 2.12 Plot of CSD vs. x (longitudinal position) for unidirectional AS4/3501 -6 C F R P specimen under applied pure shear loading (Paris, 1998). The origin of the CSD profile moves forward while the length of the square root behaviour diminishes with increasing shear load. 38 Figure 2.13 Plot of COD vs. x (longitudinal position) for the same unidirectional AS4/3501-6 C F R P specimen as Figure 2.12 under applied pure shear loading (Paris, 1998). The origin of the COD profile remains constant despite an increasing shear load and damage zone. 39 Figure 2.14 Local opening displacement created under global shear loading (Paris, 1998)._ 40 Figure 2.15 Effect of neglecting the mode I opening due to surface waviness on the mixed-mode failure envelope (Paris, 1998). 41 Figure 2.16 Schematic showing calculation of average global critical strain energy release rate using the area method. 41 Figure 2.17 Mixed mode failure envelope for a brittle AS4/3501 -6 unidirectional C F R P laminate (from Reeder, 1993). 42 Figure 3.1 Schematic of the complete experimental set-up showing integration of test control, data and image acquisition, and test analysis (Paris, 1998). 53 Figure 3.2 Photograph of the complete experimental system schematically described in Figure 3.1 (Paris, 1998). 53 Figure 3.3 Photograph of the loading j i g showing major components (Paris, 1998). 54 Figure 3.4 Specimen configuration used for all testing with aluminum loading tabs, steel clamping tabs, and a 2000 opening per square inch gold grid. 54 Figure 3.5 Photomicrograph montage of specimen surface showing the crack tip, and a gold grid created from a 500 opening per square inch mesh (Paris, 1998). 55 Figure 3.6 Photograph of the load cells and displacement sensors (Paris, 1998). 55 ix List of Figures Figure 3.7 Schematic of mode II loading conducted (a) by Paris (1998) and (b) in this work. Specimen is in pure mode II with 0 <f< 0.5 and in mixed mode with 0.5 < / < l . _ 56 Figure 3.8 Superposition of mode I and mode II on a D C B / E N C B specimen (Bradley and Cohen, 1985). 56 Figure 3.9 Photomicrographs showing method for determining COD and CSD (Paris, 1998). 57 Figure 3.10 Plot showing increased data density and reduced variability with finer grid spacing. The consistency of experimental results regardless of experimentalist is also evident. 58 Figure 3.11 Schematic of test boundary conditions for (a) original loading tab (3.175 mm pin hole diameter) and (b) modified loading tab (4.5 mm pin hole diamter). 58 Figure 3.12 Plot of CSD profile for mode II loading of original loading tab pin hole diameter of 3.175 mm w i t h / = 0.5, showing Gm « GUc- 59 Figure 3.13 Plot of mode II compliance for original (3.175 mm) and modified (4.5 mm) loading tab pin hole diameters w i t h / = 0.5, showing increased range of linear response with modified loading tab. 59 Figure 3.14 Plot of CSD profile for mode II loading of modified loading tab pin hole diameter of 4.5 mm, showing Gm = GUG- 60 Figure 3.15 Plot of CFD profile for pure mode II loading by one arm only (f= 0) and by two, equally loaded arms (f= 0.5). Profiles show no difference between the two conditions indicating negligible friction effects. 60 Figure 4.1 Schematic of test mx4 1212 loading path for plots presented in Figure 4.2 to Figure 4.4. The load was held constant at each point to allow for collection of slow scan images from the S E M . 74 Figure 4.2 Plot of CFD profile for pure mode I loading of mx4 1212 before application of mixed mode condition. A s expected GIL = GIG and no appreciable shear displacements are induced. 74 Figure 4.3 Plot of CFD profile for mixed mode loading of mx4 1212. Note that Gu > GIG, Gm = GUG, and an > ai. Gu is also greater than pure mode G/c. 75 Figure 4.4 Plot of CFD profile for pure mode II loading of mx4 1212 upon removal of mixed mode condition. Note that as expected some local opening is induced, G,,L = GUG, and au > a,. 75 Figure 4.5 Schematic of test mx4 2112 loading path for plots presented in Figure 4.6 to Figure 4.8. The load was held constant at each point to allow for slow scan imaging in the S E M . 76 Figure 4.6 Plot of CFD profile for pure mode II loading of mx4 2112 before application of mixed mode condition. Note that as expected some local opening is induced, Gm = GUG, and au > a;. 76 Figure 4.7 . Plot of CFD profile for mixed mode loading of mx4 2112. Note that GIL > GIG, GUL = GUG, and au > ah 77 x List of Figures Figure 4.8 Plot of CFD profile for pure mode I loading of mx4 2112 upon removal of mixed mode condition. Note that as expected local opening is induced, Gm = GIIG, and an ^ «/. 77 Figure 4.9 Schematic of test mx4 2121 loading path for plots presented in Figure 4.10 to Figure 4.12. The load was held constant at each point to allow for slow scan imaging in the S E M . 78 Figure 4.10 Plot of CFD profile for pure mode II loading of mx4 2121 before application of mixed mode condition. Note that as expected some local opening is induced, GUL = GIIG, and an > a/. 78 Figure 4.11 Plot of CFD profile for mixed mode loading of mx4 2121. Note that GIL = GIG, Cm = GIIG, and a„ > a,. : 79 Figure 4.12 Plot of CFD profile for pure mode I loading of mx4 2121 upon removal of mixed mode condition. A s expected GIL = GIG and no appreciable shear displacements are induced. 79 Figure 4.13 Schematic of test mx4 12 loading path for plot presented in Figure 4.14. The load was held constant at each point to allow for slow scan imaging in the S E M . 80 Figure 4.14 Plot of CFD profile for mixed mode loading of mx4 12. Note that Gn > GIG, GUL = GUG, and a„ > ah 80 Figure 4.15 Schematic of test mx4 1221 loading path for plots presented in Figure 4.16 to Figure 4.18. The load was held constant at each point to allow for slow scan imaging in the S E M . 81 Figure 4.16 Plot of CFD profile for pure mode I loading of mx4 1221 before application of mixed mode condition. A s expected Gn - GIG and no appreciable shear displacements are induced. 81 Figure 4.17 Plot of CFD profile for mixed mode loading of mx4 1221. Note that Gn > GIG, GUL = GUG, and an > a/. GIL is also greater than pure mode G/c. 82 Figure 4.18 Plot of CFD profile for pure mode I loading of mx4 1221 upon removal of mixed mode condition. A s expected GIL - GIG and no appreciable shear displacements are induced. 82 Figure 4.19 Schematic of test mx3 1221 loading path for plot presented in Figure 4.20. The load was held constant at each point to allow for slow scan imaging in : the S E M . 83 Figure 4.20 Plot of CFD profile for mixed mode loading of mx3 1221. Note that also for a shorter crack GIL > GIG, GUL = GUG, and an > a{. G/L is also greater than pure mode G/c- 83 Figure 4.21 Schematic of test mx3 12 loading path for plot presented in Figure 4.21. The load was held constant at each point to allow for slow scan imaging in the S E M . 84 Figure 4.22 Plot of CFD profile for mixed mode loading of mx3 12. Note that also for a shorter crack GIL > GIG, GUL = GUG, and an > ai. GIL is also greater than pure mode G/c. 84 x i List of Figures Figure 4.23 Schematic of test mx4 w loading path for plots presented in Figure 4.24 to Figure 4.31. The load was held constant at each point to allow for slow scan imaging in the S E M . 85 Figure 4.24 Plot of CFD profile for pure mode 1 wedge loading of mx4 w(0) before application of mixed mode condition. A s expected G/L = GIG and no appreciable shear displacement are induced. 85 Figure 4.25 Plot of CFD profile for mixed mode loading of mx4 w(42). Note that also for wedge opening G/L > GIG, Gm = GUG, and an > a/. 86 Figure 4.26 Plot of CFD profile for mixed mode loading of mx4 w(55). Note that also for wedge opening GlL > GIG, GIIL = GUG, and an > a/. G/L is also greater than pure mode G/c- 86 Figure 4.27 Plot of CFD profile for mixed mode loading of mx4 w(67). Note that also for wedge opening GIL > GIG, GUL = GUG, and an > a/. G!L is also greater than pure mode G/c- , 87 Figure 4.28 Plot of CFD profile for mixed mode loading of mx4 w(85). Note that also for wedge loading GIL >G/G, GUL = GUG, and an > a/. G!L is also greater than pure mode G/c- 87 Figure 4.29 Plot of CFD profile for mixed mode loading of mx4 w(97). Note that also for wedge loading G/L > GIG, GUL = GUG, and an > a/. G/L is also greater than pure mode G/c- 88 Figure 4.30 Plot of CFD profile for mixed mode loading of mx4 w(117). Note that also for wedge opening Gn > G/G, GUL = GUG, and an > a/. GIL is also greater than pure mode G/c. 88 Figure 4.31 Plot of CFD profile for mixed mode loading of mx4 w(u67). Note that also for wedge opening G/L > G/G, GUL = GUG, and an > a/. GiL is also greater than pure mode G/c. ; • 89 Figure 4.32 Schematic of test mx4 rev 1221 loading path for plots presented in Figure 4.33 to Figure 4.35. The load was held constant at each point to allow for slow scan imaging in the S E M . . 89 Figure 4.33 Plot of CFD profile for pure mode I loading of mx4 rev 1221 before application of mixed mode condition. A s expected GIL = G/G and no appreciable shear displacement are induced. 90 Figure 4.34 Plot of CFD profile for mixed mode loading of mx4 rev 1221. Note that even when shear loaded in the opposite direction G/L > G/c, GUL = GUG, and a// > a/. G/L is also greater than pure mode G/c. 90 Figure 4.35 Plot of CFD profile for pure mode I loading of mx4 rev 1221 upon removal of mixed mode condition. A s expected G / L = G/c and no appreciable shear displacements are induced. 91 Figure 4.36 Schematic of test mx4 21 up loading path for plots presented in Figure 4.37 and Figure 4.38. The load was held constant at each point to allow for slow scan imaging in the S E M . 91 Figure 4.37 Schematic of the load distribution in the upper arm and lower arm for test mx4 21 up. A s with most applied pure shear loads, both arms are equally Xll List of Figures loaded initially. A s the arms are opened during mode I loading the upper arm begins to carry a larger portion of the total load, as per normal. O f note in this test is that the mixed mode condition during primary data acquisition is set when the lower arm carries zero load (f= 1). 92 Figure 4.38 Plot of CFD profile for mixed mode loading of mx4 21up where /= 1. Note that for this case also G/L > G/G, G//L = GUG, and au > a/. G/L is also greater than pure mode G/c- _ _ 92 Figure 4.39 Schematic of test mx4 cp loading path for plots presented in Figure 4.40 to Figure 4.44. The load was held constant at each point prior to crack growth to allow for slow scan imaging in the S E M . Crack growth data point was analysed using video images. 93 Figure 4.40 Plot of CFD profile for pure mode II loading of mx4 cp(a) before application of mixed mode condition. Note that as expected some local opening is induced, GUL — GUG, and an > a/. 93 Figure 4.41 Plot of CFD profile for mixed mode loading of mx4 cp(b). Note that G/L > G/G, GUL = G I / G , and a,, > a,. 94 Figure 4.42 Plot of CFD profile for mixed mode loading of mx4 cp(c). Note that G/L > GIG, GUL = GUG, and an > a/. G/L is also greater than pure mode G/c. 94 Figure 4.43 Plot of CFD profile for mixed mode loading of mx4 cp(d). Note that G/L > G/G, GUL = GUG, and an > a/. G/L is also greater than pure mode G/c- In addition, significant subcritical mode I crack extension (ai) of 150 urn has occurred. 95 Figure 4.44 Plot of CFD profile taken from video for mixed mode loading of mx4 cp(e). Note that just prior to failure G/L > G/G and that both G/L and G/c are greater than pure mode G/c- 95 Figure 4.45 Schematic of test mx5 cp (from insert) loading path for plots presented in Figure 4.46 to Figure 4.49. The load was held constant at each point prior to crack growth to allow for slow scan imaging in the S E M . 96 Figure 4.46 Plot of CFD profile for pure mode II loading of test mx5 cp(a) before application of mixed mode condition. Note that as expected some local opening was induced, GUL = GUG, and a// > a/. 96 Figure 4.47 Plot of CFD profile for mixed mode loading of test mx5 cp(b). Note that G/L > G/G, GUL = GUG, and a// > a/. G/L is also greater than pure mode G/c for a crack loaded from the insert. 97 Figure 4.48 Plot of CFD profile for mixed mode loading of test mx5 cp(b). Note that G/L > G/G, GUL = GUG, and an > a/. G1L is also greater than pure mode G/c for a crack loaded from the insert. 97 Figure 4.49 Plot of CFD profile for mixed mode loading of test mx5 cp(b). Note that G/L > G/G, GUL = GUG, and an > a/. G/L is also greater than pure mode G/c for a crack loaded from the insert. 98 Figure 4.50 Plot of AG/ vs. G/G for all mixed mode conditions examined. There appears to exist general trends within specimens mx4 and mx5 of increased AG/ with increasing G/c- 98 xin List of Figures Figure 4 . 5 1 Plot of AGi vs. Gn for all mixed mode conditions examined. N o relationship between the variables is apparent. 9 9 Figure 4 . 5 2 Plot showing Gn vs. GIG for mixed mode data from specimens mx3 and mx4. Note that Gn regularly exceeds both GIG and pure mode G/c. Also G / G under mixed mode loading can also exceed G/c. 9 9 Figure 4 . 5 3 Plots showing progression of mode I and mode II crack tips during loading of mx4 w. Movement of both crack tips appears independent and irreversible. 1 0 0 Figure 5.1 Plots showing progression of mode I and mode II crack tips during loading of mx4 cp. Movement of both crack tips appears independent and irreversible. 1 2 0 Figure 5 .2 S E M image montage of mx4 w(117) under mixed mode loading ( G / G = 1 0 5 J/m 2 and GUG = 1 1 7 J/m 2). The mode I and mode II crack tip locations identified were determined from analysis of the CFD profile. Ahead of the mode I crack tip at (A), there is a distinct shear and opening displacement discontinuity across the crack plane. Behind the mode I crack tip there exists a zone similar to (B) where only a shear displacement discontinuity across the crack plane exists. The mode II crack tip location as determined from the CFD profiles appears to be behind the crack tip located from the image where no displacement discontinuities exist across the crack plane at (C). In the zone behind the original estimate of the mode II crack tip at (D) there is clearly no displacement discontinuity. 121 Figure 5.3 Plot showing equivalence of Gn and G / G for pure mode I conditions throughout specimen mx4 loading. Data labels indicate order of testing as described in Table 4 . 2 . " ' 1 2 2 Figure 5.4 Plot showing full transfer of GUG to the crack tip (GUL) for all mx4 pure mode II conditions applied. Data labels indicate order of testing as described in Table 4 . 2 . 1 2 2 Figure 5.5 Figure showing the variability and scatter in comparing AGj and load path. Data points are taken from mx3, mx4, and mx5 mixed mode loading conditions. No effect of loading path on AGi is apparent. 1 2 3 Figure 5.6 Plot showing AG; as a function of increasing G / G for tests mx4 cp and mx5 cp. Both tests were first loaded in pure mode II to similar levels and were then loaded in mode I while the shear load was held constant. Despite the loading similarities the resulting behaviour is markedly different. 1 2 3 Figure 5.7 Plot showing the development of AGi, as shear load is applied to a specimen with constant global mode I loading (mx4 w). 1 2 4 Figure 5.8 Plot showing partitioning of mx4 cp(d) local displacement field (a) into applied displacement (b) and internally generated displacement (c) components. Since G / varies as the square of displacement, a small internally generated displacement component results in large AGi under load. There is a similarity between the equivalent AG; for mx4 cp(d) of 4 J/m and the induced Gn under pure shear applied loading (mx4 cp(a)) of 5 J /m 2 that suggests a common cause of displacement. 1 2 5 xiv List of Figures Figure 5.9 Plot showing partitioning of mx4 cp(c) local displacement field into applied displacement (G/c) and internally generated displacement (equivalent AGj) components. The similarity between the normalized GIL for mx4 cp(c) and the induced GIL under pure shear applied loading (mx4 cp(a)) suggests a common cause of displacement. 126 Figure 5.10 Plot showing partitioning of mx5 cp(c) local displacement field into applied displacement (G/c) and internally generated displacement (equivalent AG/) components. The similarity between the normalized GIL for mx5 cp(c) and the induced GIL under pure shear applied loading (mx5 cp(aj) suggests a common cause of displacement. 126 Figure 5.11 Plot showing partitioning of mx5 cp(d) local displacement field into applied displacement (G/G) and internally generated displacement (equivalent AGj) components. The similarity between the normalized GIL for mx5 cp(d) and the induced GIL under pure shear applied loading (mx5 cp(a)) suggests a . common cause of displacement. 127 Figure 5.12 Plot showing AG/ as a function of increasing G/c for a constant internally generated displacement profile (equivalent AGj) of 5 J/m 2 . 127 Figure 5.13 Schematics describing (a) G/ and G// as a function of applied moments and (b) the key assumption in global analysis that the system is perfect and all global loads are transferred directly to the crack tip. 128 Figure 5.14 Schematic describing transfer of global loads to the local level for pure mode I loading with no local interactions such as fibre bridging. 128 Figure 5.15 Schematic describing transfer of global loads to the local level for pure mode I loading with fibre bridging. The internal mechanism of fibre bridging ( M F ) shields the crack tip from seeing the full applied load. 129 Figure 5.16 Schematic describing transfer of global loads to the local level for pure mode II applied loading. The interaction due to the crack face roughness as the two surfaces slide over each other under shear loading is modelled by M A which induces a local opening component. 129 Figure 5.17 Schematic describing transfer of global loads to the local level for mixed mode applied loading. The interaction due to the crack face roughness as the two surfaces slide over each other under shear loading is modelled as M A which increases the local opening component. 130 Figure 5.18 Plot of mode I load and displacement data for mx5 cp. Theoretically, the mode I component of work done in fracture can be determined by the difference in area between the loading and unloading curves. In this case the area is significantly greater than that expected based on local analysis. 130 Figure 5.19 Plot of mode II load and displacement data for mx5 cp. Theoretically, the mode II component of work done in fracture can be determined by the difference in area between the loading and unloading curves. In this case the area, as expected based on local analysis, is not large enough to be accurately determined. 131 xv List of Figures Figure 5.20 Plot for test mx4 cp showing evolution of GIL and its relation to G/c and G/c. Note that G/c and Gn both exceed pure mode G/c by approximately 10% and 105%> respectively at failure. 131 Figure 5.21 Plot for test mx5 cp from the insert showing evolution of Gn and its relation to G/c and G/c. Note that G / G and G/L both exceed pure mode G/c from the insert by approximately 15% and 100%) respectively at failure. 132 Figure 5.22 Micrographs of a crack grown from the insert under mixed mode loading (mx5 cp). The crack starts to grow straight ahead before immediately kinking due to the asymmetric stress field induced by the mixed mode loads. The crack continues to grow at the preferred angle until it reaches the fibre pack. The crack then grows adjacent to that fibre path as there is insufficient energy available to break the fibres. This fracture path is more tortuous than when the crack grows only in the resin rich region and is consistent with the higher than expected Gc values observed. 133 Figure 5.23 Plot showing the differences between the global and local mixed mode failure envelopes. The local mode I values should be higher than the global predictions throughout the mixed mode range. Currently, matched global and local data only exists in the highlighted regions and future examination throughout the mixed mode range is recommended. 134 Figure 6.1 Finite element mesh of specimen showing successive enlargements of the crack tip area. Elements defining the crack tip have been modified to simulate square root singularity. 144 Figure 6.2 Schematic of specimen and boundary conditions (a) during testing and (b) i n the finite element model. ; 145 Figure 6.3 Comparison between COD profile obtained experimentally, analytically using L E F M , and by finite element analysis in the region close to the crack tip for crack lengths of 25 mm. The experimental data eventually converges to the F E solution. • 145 Figure 6.4 Plot of COD vs. r showing the square root dependence of displacement fields behind the crack tip (singular zone) for four unique cracks; (a) a=19 mm (Paris, 1998), (b) a=21.5 mm (Paris, 1998), (c) a=24.5 mm, and (d) a=27 mm. The experimental singular zone size is variable for different crack tip conditions while the F E A singular zone size remains constant. The solid marker in each experimental data set indicates the end of the singular zone. _ 147 Figure 6.5 Plot of COD2 vs. r showing the square root dependence of displacement fields increased behind the crack tip from 125 um to 225 um due to the addition of a resin rich zone at the crack plane. The solid marker indicates the end of the singular zone. 148 Figure 6.6 Comparison between CSD profile obtained experimentally, analytically using L E F M , and by finite element analysis in the region close to the crack tip. The finite element results agree very well with experimental data. 148 Figure 6.7 Plot of CSD vs. r showing the square root dependence of displacement fields behind the crack tip. The higher order terms begin to have an effect much later experimentally than in the F E A . 149 xvi List of Figures Figure 6.8 Plot of (a) shear stress and (b) von Mises stress for pure mode I loading (G/ = G/c = 126 J/m 2). Note how the stress field ahead of the crack tip at crack propagation is dominated by the normal stress component (b). 150 Figure 6.9 Plot of (a) shear stress and (b) von Mises stress for pure mode II loading (G 2 = 95 J/m ). The crack flanks are to the left of the image. Note how the stress field ahead of the crack tip is dominated by the shear component (a). 151 Figure 6.10 Plot showing virtual measurement of COD profile at the crack face and at distances of 20 and 95 microns from the crack faces. 152 Figure 6.11 Plot showing virtual measurement of CSD profile at the crack face and at distances of 20 and 95 microns from the crack faces. 152 Figure 6.12 Plot of (a) shear stress and (b) von Mises stress for mixed mode loading (Gn = 220 J /m 2 and Gu = 95 J/m 2). The stress fields are for the local loads measured close to mixed mode crack propagation for mx4 cp. The asymmetric nature of the stress field is clear. The crack fracture path seen in Figure 5.21 is consistent with the stress distribution shown above. 153 n Figure A . l Plot showing theoretical model load-displacement data for (a) the left arm and (b) the right arm. 169 Figure A .2 Plot showing theoretical model load-displacement data for (a) mode I and (b) mode II. 170 Figure A.3 Plot showing mx5 cp experimental load-displacement data for (a) the left arm and (b) the right arm. The shapes of the loading and unloading curves are consistent with that expected based on the experimental model. 171 Figure A.4 Plot showing mx5 cp experimental load-displacement data for (a) mode I and (b) mode II. The shape of the mode II loading and unloading curves are consistent with the experimental model. A primary source of error, however, exists in using the global displacement data due to the slack in the loading pin setup. 172 xvi i Nomenclature Nomenclature a crack length a/ crack length in mode I an crack length in mode II atj unidirectional composite plane stress elastic constants A i mode I elastic function Au mode II elastic function A S T M American Society for Testing and Materials B specimen width C compliance C0 machine compliance CFD crack face displacement (COD or CSD for mode I and II components respectively) C F R P carbon fibre reinforced polymer C L S crack lap shear COD crack opening displacement CSD crack shear displacement D C B double cantilever beam E laminate flexural modulus Ei laminate longitudinal elastic modulus E2 laminate transverse elastic modulus E N C B end-notched cantilever beam F work performed by external load / fraction of total load in upper arm (pure mode II i f less than 0.5) F E finite element xv i i i Nomenclature F E A finite element analysis G strain energy release rate Gc critical strain energy release rate GG global strain energy release rate GGC critical global strain energy release rate G, mode I strain energy release rate G/c mode I critical strain energy release rate G/G mode I global strain energy release rate Gn mode II strain energy release rate Gnc mode II critical strain energy release rate G//G mode II global strain energy release rate Gm mode II local strain energy release rate G!L mode I local strain energy release rate GL local strain energy release rate GTGC total global critical strain energy release rate ( G / c c + GV/GC) GTLC total local critical strain energy release rate (G/LC + GULC) Gn laminate in-plane elastic shear modulus h half beam thickness K stress intensity factor K, mode I stress intensity factor Ku mode II stress intensity factor L specimen length L E F M linear elastic fracture mechanics L V D T linear voltage displacement transducer M M B mixed mode bending xix Nomenclature M M F mixed mode flexure N T S B National Transportation Safety Board P applied load Pi mode I applied load Pn mode II applied load Pi left arm load PR right arm load r distance from crack tip (behind crack tip is positive) S E M scanning electron microscope u longitudinal displacement U elastic energy v transverse displacement Vf fibre volume fraction W energy required for crack growth z perpendicular distance from crack face 8 displacement 8j mode I displacement 8u mode II displacement 8L left arm displacement 8R right arm displacement Aa distance between mode I and mode II crack tips AG; difference between local and global G/ V12 laminate Poisson's ratio xx Acknowledgements Acknowledgements First and foremost, I would like to express my deepest gratitude and thanks to my supervisor, Dr. Anoush Poursartip, for his guidance, support, wisdom, and, most importantly, patience during the course of this research. His willingness to continue to search for understanding and clarity has made my time here more educational and enjoyable. Thank you also to Dr. Isabelle Paris for the quality of her previous work and for her guidance and input in this work. I would also like to thank Dr. Reza Vazi r i and Dr. Goran Fernlund for their discussions and for their continued interest in me and in my education. I should also like to acknowledge the efforts and assistance of Roger Bennett for his help with the j i g and of Mary Mager for her help with the S E M . The past and present graduate and co-op students of the U B C Composites Group must also be thanked. Every one of you (yes, even those in A M P E L and C E M E ) has made a lasting impression on me and each of you has contributed to the completion of this thesis. A special mention must be made to my first co-op officemates in A M P E L 142 (Robert, Vincent, and Matt) and to those who graced the hollowed walls of F F 105 for some portion of my tenure here (Jason, Mike , Arafath, Wes, Scott, and Tomer). A thank you also to Anthony and Robert who were kind enough to assist me the many times I had 'technical' difficulties. I would also like to deeply thank all those who helped me find balance, from the volunteers with whom I have.spent countless hours working over the last few years to the R . U . S . H for giving me the opportunity to get back into 'game shape'. I am most grateful to all my friends and family for their support, to Slim and the Boys for their kindness and generosity, and to Zan for her encouragement and friendship. Finally, I would most importantly like to thank my parents for their continuous and unwavering love, support, encouragement, caring, and confidence that continue to bring me peace and give me strength. xx i "If it looks like a duck, and quacks like a duck, we have at least to consider the possibility that we have a small aquatic bird of the family anatidae on our hands. " - Dirk Gently -Introduction Chapter 1 Introduction The use of advanced composite materials today has not lived up to initial expectations. Specifically, the use of carbon fibre reinforced polymers (CFRP) in widespread applications has been limited. In some markets however, for example in military applications, aerospace applications, and sporting equipment, CFRPs have enjoyed some success. Entry into markets such as the automotive industry has taken longer and faced more challenges than the initial enthusiasm (Dharan, 1978) surrounding this class of materials had suggested. "The basic reason surely is the uncertainty that exists in determining the strength and safe-operating lifetime in service conditions - particularly when defects could be present" (Kanninen and Popelar, 1985). The technical promise of C F R P s still remains. Their substantial use in modern military aircraft and their increased use in primary civilian aircraft structures (rudder, empennage, etc.) along with increasing interest from the automotive industry indicate that the future of C F R P materials remains bright. Indeed, the challenges facing the composites industry have perhaps as much to do with non technical considerations, whether they be motivated by economics, politics, or a general resistance to change, than they do with technical deficiencies related to the material itself However, there do exist significant technical obstacles regarding understanding of the behaviour of composite materials that need to be overcome. These technical difficulties range from manufacturing and processing issues of CFRPs to issues surrounding the ultimate failure of these materials under service conditions. While many of these difficulties can be overcome, and have been overcome particularly by the aerospace industry, the solutions have come at a cost, a financial cost or overdesign cost, which limits the use of C F R P s in other applications. The scientific and engineering uncertainties surrounding the behaviour of CFRPs in the post elastic regime where damage is present is one of the greater technical difficulties faced by the composites community today. The Worldwide Failure Exercise (Hinton et al, 2002; Soden et al, 1998) showed that even the leading fibre composite failure theories in use and under development have difficulty predicting real composite behaviour. One of the more popular approaches to studying materials containing cracks is Linear Elastic Fracture Mechanics ( L E F M ) and it is this approach that is considered here. L E F M has been most successfully used to describe the behaviour of brittle, isotropic, homogeneous materials. 2 Chapter 1 Introduction The extension of this approach to anisotropic, heterogeneous materials is complex. "This is because fracture in composite materials is strongly dependent on lamination order, fiber orientation, and the constitutive relations that describe the mechanical properties of the fiber, matrix, and the interface" (Dharan, 1978). ' " - • Laminated fibre composites, such as those currently being employed in primary aircraft structures, have unique damage and failure mechanisms in different directions compared to their metal counterparts that display similar damage types regardless of orientation. Figure 1.1 shows the principal damage types in laminated composites; fibre breakage, matrix cracking, and interlaminar delamination. The difference between matrix cracking and delamination is that matrix cracking occurs within a lamina or layer while delaminations occur at the interface between layers and result in the separation of laminae. Although in real applications the damage mechanisms co-exist, each mechanism can be studied independently before it is modeled in conjunction with the others. A prevalent failure mode in aerospace composites is delamination and the current understanding of the complex behaviour of laminates containing delaminations is a key factor that limits the use of CFRPs . Due to their basic construction, laminated composites are inherently susceptible to delaminations throughout their entire life cycle. The region between laminae is rich in resin and provides a convenient low resistance fracture path. Material processing methods, manufacturing methods, and many service conditions all have the ability to trigger delaminations, that in turn can significantly reduce bending and compressive properties of laminates (Garg, 1988). The nature of delamination behaviour needs to be better understood and it is an increase in this understanding that is the driver behind the current project. A good example of the current state of the understanding of composite delamination failure behaviour is the crash of American Airlines Flight 587 on November 12 t h , 2001 in Belle Harbor, New York moments after takeoff. It was determined by the United States National Transportation Safety Board (NTSB) (www.ntsb.gov) that the C F R P composite vertical stabilizer had broken away from the airplane in flight (Figure 1.2). This was the first case of a catastrophic failure of a civilian aircraft composite primary structure. Figure 1.3 shows a picture of part of the tongue structure that tore away from the aircraft. Delaminations, many of which were caused during component failure, are clearly evident on this failed component. It is possible that some 3 Chapter 1 Introduction delaminations were present prior to failure but whether they were a cause of failure was questioned. In light of the accident, the N T S B removed from service a similar tail section, with a similar load history as the failed tail, for further investigation. The investigation revealed that a delamination did exist near one of the six attachment points of the vertical stabilizer to the main fuselage. According to design criteria, the vertical stabilizer remained acceptable despite this flaw and could have been returned to service without repair. Due to the circumstances of the accident, and the limited understanding of how such a delamination actually affected the structure in service, the vertical stabilizer was not returned to service but was the subject of extensive testing and investigation. It was determined that the structure did indeed still meet all design and certification criteria. It was also later determined by the N T S B that the cause of failure was not due to the composite material used in the tail section itself, but rather was likely due to loads in excess of those for which the structure had been designed. 1.1 Building Block Approach One of the primary advantages of composite materials is that large, complex geometries can be manufactured as monolithic structures. These types of structures typically carry complex loads. Figure 1.4 shows the range of scales of interest associated with composite materials. The range is quite large and spans the submicron level at the fibre/matrix interface up to the structural level that can be much greater that ten metres in some applications. The variety and number of possible failure modes in composite materials are such that neither testing nor analysis alone can efficiently and accurately predict the behaviour of composite structures. Testing specimens and structures for all the possible combinations of loads and failure mechanisms is prohibitively expensive and physically unrealistic (MIL-17, 2002). Analysis methods, on the other hand, are not yet capable of handling all possible conditions with confidence as was shown in the Worldwide Failure Exercise (Hinton et al., 2002). The building block approach is one method that is gaining acceptance in the composites industry as an approach that can be used to fully develop composite structures and new material systems at a minimum cost. In fact, the building block approach is currently in the process of being formally accepted into MIL-Handbook-17, The Composite Materials Handbook, an engineering handbook 4 Chapter 1 Introduction for structural applications of composite materials that includes standards for test and characterization methods. In the building block approach, testing and analysis methods are used in close combination to examine and study all levels of complexity and scales o f interest depicted in Figure 1.4. B y testing and understanding more completely the behaviour and response of lower scale and cheaper components, for example at the specimen level, the overall cost of design can be reduced by the need to test fewer full scale, expensive components. Despite the overall cost reduction, the building block approach maintains the rigour and outcomes required for certification ( M I L -17,2002). The building block approach can, in essence, be split into two streams. The first stream is the understanding of each block in isolation, for example the behaviour of the fibre/matrix interface or the behaviour of specimens under combined loading. The second stream is the integration of that understanding and knowledge over all the building blocks and over the entire range of length scales. The objectives of this project combine aspects of both of the above streams. The aim of the current project is to develop a greater understanding of the relationship between the predicted behaviour o f fracture test specimens using 'global ' L E F M analyses, and the actual experimental behaviour of the material at the ' local ' delamination tip (crack tip). 1.2 Global Approach In the global approach used to determine delamination specimen response to applied loads, global parameters such as applied load or deflection, specimen geometry, and crack length are used in conjunction with material properties to evaluate and predict the local conditions that prevail in the region around the crack tip. The global analysis technique used to predict the local conditions has been developed through the use of L E F M . In the analysis, global parameters can be combined into a similitude parameter that directly defines the local conditions (e.g. stress, strain, displacement). There are two primary similitude parameters that are widely used. The first, K, the stress intensity factor, directly relates the global, far field applied conditions to the local, crack tip area 5 Chapter 1 Introduction stress state. The other similitude parameter comes from the Griffith criterion that states a crack wi l l grow in a material i f sufficient energy exists (Broek, 1986). In this approach the similitude parameter is G, the strain energy release rate, and can be directly related to K. In a general body as shown in Figure 1.5, with a crack length of a and an applied load of P, the Griffith criterion is: da da where U is the elastic energy stored in the plate, F is the work performed by the external load, and W is the energy required for crack growth. The right hand side of the equation represents the material's resistance to crack growth. The left hand side of the equation is defined as the strain energy release rate, G , and is equivalent to the amount of energy the system would release per unit width per unit of crack growth at any given moment and has units of joules per square meter (J/m 2) (Broek, 1986). If at the moment of crack growth the ends of the specimen are fixed, as under displacement control, the external load can do no work and the energy for crack growth is provided by a release of elastic energy within the system. If the system is under an applied load and the ends of the specimen are free to move during crack growth, work is done by the applied load (Broek, 1986). B y calculating the work done by the external force and the elastic energy stored in the plate, the general expression for G becomes, G = - — (1.2) IB da where C is the specimen compliance and B is the width of the specimen. Due to the relationship between the load, P, the displacement, 8, and the compliance the above expression can also be written as either of the two equations below: 6 Chapter 1 Introduction f xY G = 2B i c y dC da (1.3) 2B C da The only parameter that needs to be evaluated to determine the strain energy release rate is dC/da. This can be evaluated experimentally, numerically, or analytically. Crack growth begins when the crack driving force, the left side of Equation 1.1, equals the energy necessary to create new surfaces, the right side of Equation 1.1 or in other words when G equals Gc, the critical strain energy release rate often referred to as the fracture toughness. A n y loading applied to a cracked specimen can be partitioned into the three constituent load components as shown in Figure 1.6. A crack is considered to be under mode I loading when it is subjected to tensile opening, mode II loading when it undergoes in-plane shearing, and mode III loading when it experiences anti-plane shearing. Traditionally, the most studied modes of loading have been mode I and mode II and the combination of those two loads wi l l be referred to as "mixed mode" loading for the remainder of this thesis. A l l modes of loading can be analysed using any of the Equations 1.2 to 1.4. The differences between the modes of loading manifest themselves in the compliance and in the dC/da term in the equations. The global approach is necessarily predictive in nature since the global applied loads and geometry are used to determine a similitude parameter that describes the local behaviour at the crack tip. The approach assumes, therefore, that the system transfers 100% of the global applied condition to the local crack tip. However, local perturbations, especially in fibrous composites, such as resin-rich regions, fibre bridging, microcracking, and crack path wandering have been observed (e.g. Davies and Benzeggagh, 1989). It is possible therefore, and perhaps even expected, that the global conditions are not transferred fully and directly into local crack tip conditions. A n added difficulty encountered in mixed mode loading, regardless of specimen type, is the partitioning of general mixed mode loading configurations into mode I and mode II constituents. Depending on the geometry and lay-up of the material, the partitioning can be performed using 7 Chapter 1 Introduction beam theory or various numerical methods. The approximations and simplifications made in order to divide the applied loads and calculate G/ and Gu may also lead to inaccuracies and variance between different techniques. 1.3 Local Approach In the local approach to evaluating delamination behaviour of specimens, quantities in the crack tip area are directly measured and related to the same similitude parameters used in the global approach. For example, the specimen crack face displacements (CFD) behind the crack tip can be measured and directly.related to the strain energy release rate. Figure 1.7 shows a schematic of the crack face displacements as expected under a mixed mode load. The mode I or opening component of the load causes the crack faces to move away from each other. These components of local displacement are called crack opening displacements (COD). Crack shear displacements (CSD), on the other hand, are caused by the in-plane shear or mode II component of loading as the crack surfaces slide over each other. A s in the global approach, the local analysis uses L E F M to relate the crack tip displacement fields to a similitude parameter. Sih et al. (1965), derived equations for crack tip displacement fields and their relationship to K for rectilinearly anisotropic (orthotropic) materials. Poursartip et al. (1998) showed that the crack face displacements behind the crack tip can be related to the mode I and mode II G components through the following: where A; and An are derived functions of the elastic properties of the laminate and r is the distance from the crack tip. Since these equations are only valid for rectilinearly (specially) orthotropic bodies, they pertain only to a small subset of C F R P systems, specifically unidirectional laminates, where lamina are stacked with all the fibres oriented parallel to the delamination (Figure 1.1). The equations also only use the first term in the more complicated general relationship that takes into account the boundary conditions of the specimen (Poursartip et al, 1998). A s a result, the CFD profiles are only valid in an area very close to the crack tip, before the higher order terms begin to influence the solution. (1.5) (1.6) 8 Chapter 1 Introduction j The values of G/ and GJI calculated through the local approach, are determined using measurements of the COD and CSD directly behind the crack tip. A s such, these strain energy release rate values are the actual, local strain energy release rates. G/ and Gn calculated by the global approach and the local approach are physically the same parameter. Therefore, the two approaches should ideally yield equivalent G solutions. If the globally predicted crack tip behaviour does not match that locally observed then some internal mechanism must be affecting the transfer of the global conditions to the local level (Paris, 1998). For example, the phenomenon of fibre bridging is well established in unidirectional laminates. Fibres can often cross the interlaminar region between laminae and resist opening of the crack tip (e.g. Davies and Benzeggagh, 1989). In this case, the local displacement profile would show a smaller local value of G than predicted by the global conditions. The energy release rate difference between the global and local conditions is due to the fibre bridges across the crack. Unfortunately, the global analysis method alone cannot predict the existence or the quantitative effect of not only fibre bridging but also of any other local mechanisms that may be present. 1.4 Summary The main objective of this thesis is to perform both global and local analyses simultaneously on a test specimen in order to compare directly the local and global behaviour. This method of determination and comparison wi l l provide a better general understanding of delamination behaviour in unidirectional C F R P composites in particular. The thesis is organised in the following manner: Chapter 2 : A review of the research presented in the literature is presented initially. Based on this literature review the objectives of this work are presented. Chapter 3: A n in-situ scanning electron microscope (SEM) experimental method that facilitates the comparison of the global and local approaches is described. The experimental apparatus and technique have been previously developed and used in pure mode I and mode II investigations (Paris, 1998). Chapter 4: Mixed mode loading experiments conducted on delaminations are described and the results from the global and local analyses are presented. 9 Chapter 1 Introduction Chapter 5: The results from the previous chapter are analysed and a discussion of their relevance to a better understanding of delamination behaviour in unidirectional composites is presented. Chapter 6: A simple numerical model is developed and is used to validate certain aspects of the experimental technique as well as to shed further insight into the experimental results and analysis. Chapter 7: In this final chapter the main conclusions from this work along with recommendations for further investigation are presented. 10 Chapter I Introduction interface between 2 plies Figure 1.1 Schematic of damage mechanisms in laminated composites (Paris, 1998). The tail is connected to the planes body with six sets of fittings. um SECTIONS T H A T mm T» I W/AV F R } « THE P i « E - MMmtfy Right engine Left engine Sowwr National Timspormm BomtL Tongues made of composite fiber materia! are bud! into the base of the tail " \ ' " " T h e s e tongues tit in metal connectors on the plane's spine. -• A bolt secures the connection. WHAT HAPPENED On Flight 587, the composite liber tongues on the tail tore apart, allowing the tail section to dislodge The connection assembly remained attached to the plane, Figure 1.2 Description of the failure that lead to the crash of Amer i can Air l ines Fight 587 on November 12 t h , 2001 in New Y o r k ( N T S B ) . 11 Figure 1.3 N T S B photograph of failed composite connection on the tail section of A A Fit . 587. A t least five major delaminations can be observed in the highlighted portion of the photograph. 10 100 1 mm 10 mm 100 mm 1 m 10 m 0.001 in. 0.01 in. 0.1 in. 1 in. 1 ft. 10 ft. Figure 1.4 Range of scales of interest in the study of the failure of composite structures (Paris, 1998). 12 777777 C = compliance Figure 1.5 Schematic of an arbitrary body of width B containing a crack of length a subject to an external point load, P (Paris, 1998). Mode I Mode II Mode III opening in-plane shear anti-plane shear Figure 1.6 Schematic describing the three modes of loading. 13 Chapter 1 Introduction 14 2 Background and Literature Review Chapter 2 Background and Literature Review The fracture of composite materials has been studied continuously over the past few decades. Investigators have attacked the problem of mode I, mode II, and mixed mode delamination behaviour of C F R P and other composite materials from a variety of perspectives. The majority have chosen to study, from a global perspective, the critical values of the strain energy release rates (Gc) for the various loading conditions. This approach results directly in the development of failure envelopes and design guidelines as well as predictions of the fracture of the material in structural applications. However, the global data alone is insufficient to allow for a thorough understanding of the delamination behaviour of composite materials. Indeed, as many researchers have noted, the global data for similar materials under similar conditions is quite variable compared with more traditional and more thoroughly studied materials. In addition, the global data often provides results that cannot be explained by global analysis alone. Consequently, other investigators have chosen different paths in the study of this subject. Some have attempted different methods of material testing and different data reduction schemes for existing testing methodologies in an attempt to gain further insight into the problem. Others have undertaken investigations into the micromechanisms involved in the fracture process either by observing the crack tip in-situ during the fracture event, with some taking local crack tip behaviour measurements, or through post failure fractographic inspection of the newly created crack surfaces. A limited number of researchers have combined the two approaches and used the local measurements to verify the predictions of the global approach. These combined investigations have confirmed the nature and variability of previous results while at the same time increasing the fundamental understanding of the problem. The most recent work in the field has tended to focus on the numerical modeling of delamination behaviour. This area of study is critical in applying the physical understanding of delamination behaviour to larger structures. Recently, though, less emphasis has been placed on efforts to increase that physical understanding itself. The following literature review wi l l first briefly review pure mode I and pure mode II delamination behaviour, with a focus on the global-local relationship, before undertaking a more detailed examination of mixed mode delamination behaviour. 16 Chapter 2 Background and Literature Review 2.1 Pure Mode Delamination Behaviour The following.section wi l l provide a brief overview of pure mode delamination behaviour in brittle unidirectional composites. The primary issues relating to mode I and mode II delamination behaviour wi l l be discussed from both a global and local perspective. 2.1.1 Mode I Delamination Delamination of brittle, unidirectional C F R P laminates occurs in the resin rich interface between different lamina in composite laminates (Figure 1.1 and Figure 2.1). It would be reasonable to expect, therefore, that the fracture mechanisms and fracture toughness of a composite would be closely linked to the properties of the resin. Hibbs and Bradley (1987) and Bradley and Cohen (1985) showed that for brittle resin systems, the composite delamination resistance is typically better than that of the neat resin. Ductile systems, on the other hand, do not behave in a similar fashion. In mode I situations two key factors lead to the increased toughness of the composite. The first factor is that unlike in the homogenous resin alone, the crack path is more complicated due to the influence and interference of the fibres. The thicker the resin rich layer is, the easier is the crack path and the closer the composite toughness is to the neat resin toughness (Hibbs and Bradley, 1987; Bradley and Cohen, 1985). Post failure fractographic investigations of brittle composites, such as AS4/3501-6, have shown that mode I fracture surfaces just after initiation tend to be "very flat indicating a brittle cleavage fracture" (Reeder, 1993). The second factor is that the bridging of fibres across the delamination (Figure 2.2) elastically works to keep the crack tip closed and also consumes energy as it breaks (Hibbs and Bradley, 1987). It was also observed that the process or damage zone is "very localized to the crack tip" (Bradley and Cohen, 1985) as is shown schematically in Figure 2.1. The concept of fibre bridging and its effect on the delamination toughness of unidirectional fibre composites, where it is primarily observed, has been well studied. The more fibres that bridge the crack, the greater is the resistance of the specimen to delamination, at least from a global analysis. A s a result, GQ can, under the correct conditions, increase as the crack grows and as more fibre bridges are created in its wake. Davies and Benzeggagh (1989) suggested that Gc reached a stable value when the creation of new fibre bridges occurs at the same rate as existing fibre bridges are broken. This type of behaviour, where the resistance to further crack growth 17 Chapter 2 Background and Literature Review increases with crack length, is known as R-curve behaviour (Figure 2.3). For unidirectional fibre composite laminates, this R-curve behaviour is primarily due to fibre bridging that is both variable and unpredictable. Another contributing factor to the increased composite toughness over the neat-resin toughness is provided by Corleto et al. (1987) where a finite element model is used to examine the stress distribution around the crack tip for an orthotropic material under pure mode loading. They observed that the normal stress distribution (Figure 2.4) around the crack tip is more distributed in an orthotropic material than in an isotropic material. The more diffuse stress distribution in a composite laminate, therefore, requires more applied energy to reach the neat resin fracture conditions. This observation was confirmed by Bradley (1989) where a local analysis method was used to measure and compare the strain field around neat-resin and composite crack tips. While many researchers have examined the local crack tip behaviour in a qualitative fashion, very few have been able to quantitatively study the crack tip area with sufficient resolution, in conjunction with the acquisition of global applied data to conduct a detailed investigation comparing the global and local approaches (Paris, 1998). One of the first to do so successfully were Ferguson et al. (1991) who were able to measure crack opening displacements very close to the crack tip, within the first few hundred microns, and compare these measurements with L E F M predictions determined from global applied conditions. From the mode I analysis conducted, it was found that the L E F M predictions regarding the shape of the COD profile behind the crack tip were accurate for a significant length behind the crack tip (on average approximately 400 - 500 microns) before the opening displacements began to deviate from the r dependence (Figure 2.5). This distance is commonly referred to as the singular zone. The size of the singular zone is primarily dependent on the specimen geometry and in particular on the distance from the crack tip to the nearest specimen boundary (Chona et al., 1983). One of the advantages of measuring the displacement field behind the crack tip as opposed to the strain field ahead of the crack tip is that the displacement field singular zone is three times greater than the stress or strain field singular zone in front of the crack tip (Poursartip et al., 1998). While the shape of the COD profile was consistent with L E F M predictions, the magnitude of the opening displacements were sometimes less than predicted from the global 18 Chapter 2 Background and Literature Review analysis. The authors attributed this discrepancy to the effects of fibre bridging (Ferguson et al, 1991). Paris et al. (2001) and Paris (1998), building upon the technique used by Ferguson et al. (1991), were able to obtain more detailed and higher resolution global and local data. Their examination of pure mode I delamination behaviour revealed that for brittle, unidirectional C F R P systems, where no fibre bridging existed, all the global loads were transferred fully and directly to the crack tip all the way up to failure (Figure 2.6). This investigation also was able to show the effects of fibre bridging in a more quantitative manner than previous studies. Despite the large global loads, the fibre bridges shielded the crack tip from seeing loads approaching critical conditions (Figure 2.7). Paris was also able to determine where fibre bridges occurred along the profile and to correlate that data with S E M images of the crack showing a fibre bridge (Figure 2.8) (Paris, 1998). 2.1.2 Mode II Delamination Global mode I testing for a limited class of continuous fibre reinforced polymers was standardized by the American Society for Testing and Materials ( A S T M ) in 1994 ( A S T M D 5528) and the double cantilever beam (DCB) specimen was selected as the standard to test for mode I fracture toughness. In a review of fracture toughness testing of fibre reinforced polymer laminates Brunner (2000) observed that mode I D C B test data generally agrees well , "within a few percent in-laboratory variation to around 10-15% inter-laboratory variation for identical material(s)". This level of variability, although often considered unavoidable, has been one of the inherent drawbacks of composite materials for critical structural applications. Mode II fracture toughnesses of brittle C F R P composites are significantly more variable than mode I fracture toughnesses. Brunner (2000) comments that global "fracture mechanics descriptions [are] sufficient for obtaining reproducible and consistent Gc values in the case of mode I loading but not for mode II". The variability and other issues associated with mode II testing, including difficulties achieving pure shear loads at the crack tip due to specimen and testing configurations, have resulted in no general standard being adopted by the international composites community (Brunner, 2000; O'Brien, 1997). 19 Chapter 2 Background and Literature Review This increased variability in Guc over G/c is shown for an AS4/3501-6 material system in Figure 2.9. In this case the variability results in a Guc of anywhere between 500 and 750 J /m 2 for the same material and specimen tested under the same conditions (O'Brien, 1997). In a tougher material system, A S 4 / P E E K , a brief compilation of the literature reflects that the differences between experiments conducted by different researchers results in a range of Guc from 1201 to 4250 J/m 2 (O'Brien, 1997). According to a review of composite material mode II literature by O'Br ien (1997), the mode II fracture toughness is always greater than that in mode I. The sources of this increased toughness over pure mode I delaminations is due to the different mechanisms involved in the mode II delamination fracture process. For brittle C F R P composites the fracture surfaces created in mode II growth are considerably rougher and show greater damage than the surfaces created in mode I. (O'Brien, 1997; Reeder, 1993; Corleto and Bradley, 1989; Hibbs and Bradley, 1987). The mechanism of fracture in these situations is through the creation of hackles; sigmoidal shaped microcracks that are created in front of the crack tip at an angle perpendicular to the principle normal stress. The hackles subsequently rotate and coalesce resulting in mode II crack growth (Figure 2.10) (e.g. Paris, 1998; Reeder, 1993; Bradley, 1989; Hibbs and Bradley, 1987). According to Morris (1979 from Reeder, 1993) hackles are "flake-like in appearance and ... overlap on top of one another similar to the shingles on the roof of a house". The greater energy required for shear crack growth can also be attributed to the creation of a much larger damage zone in mode II than in mode I. Local strain field measurements (Bradley, 1989) show that a longer, narrower strain field exists ahead of the crack tip in mode II loading. The resulting damage zone includes both shear deformation and microcracking. This results in a more diffuse load distribution and provides an energy dissipation mechanism. Finite element analysis of the stress field around the crack tip (Figure 2.11) (Corleto et al, 1987) is consistent with this type of stress distribution. This fracture process, therefore, requires much more energy than the cleavage type process under pure mode I loading and consequently Guc is significantly higher in brittle C F R P materials than G/c. In the Ferguson et al. study (1991), some of the CSD profiles were found to be equivalent to the global prediction while others were found to be greater. The authors attributed the difference between predicted and measured CSD profiles to local fibre volume fraction variations that lower 20 Chapter 2 Background and Literature Review the local modulus near the crack tip. The investigation also revealed that the size of the singular region under shear loading was comparable to the mode I singular zone. Paris' investigation (1998) into mode II loading of brittle unidirectional composites shed some insight into the complex and variable nature of the pure mode II fracture process. For low shear loads the size of the singular zone under mode II loading was found to be approximately 2.5 times greater than the average size of the mode I singular zone. A s the damage and the size of the damage zone increased, the origin of the CSD profile shifted ahead of the initial crack tip into the damage zone and the square root singularity behaviour diminished (Figure 2.12). It was also found that the CSD profile was in good agreement with the global L E F M predictions indicating no friction effects as the crack surfaces slid over each other (Paris, 1998). Perhaps the most significant of Paris' mode II results is that despite the application of a pure shear load at the global level, the local measurements indicate clearly that a local mixed mode condition prevails (Paris, 1998). The induced COD local profile (Figure 2.13) was attributed to the waviness of the crack and the opening induced as these wavy surfaces slid over each other (Figure 2.14). Paris (1998) suggested that the variable nature of the induced opening could explain the significant scatter seen in the Guc data in the literature. Figure 2.15 shows the effect of the previously unknown local opening component on a simple linear mixed mode failure envelope. The investigation also revealed that the nature of the damage zone created under applied shear loading was such that crack shear displacements were allowed while crack opening displacements were inhibited (Figure 2.12 and Figure 2.13). A s a result, for a local mixed mode condition there were two distinct crack profile origins; one for the COD that indicted the end of the zone where the hackles had coalesced and the beginning of the damage zone which ends at the origin of the CSD profile (Paris, 1998). 2.2 Mixed Mode Delamination Behaviour The following section examines the literature on mixed mode delamination behaviour for brittle, continuous fibre reinforced polymers. Different test configurations, data reduction methods, failure envelopes and the current understanding of the physical nature of mixed mode failure w i l l be discussed. The objectives for this work wi l l be formulated based on this discussion. 21 Chapter 2 Background and Literature Review 2.2.1 Global Test Methods A variety of mixed mode delamination fracture tests had been developed by the late 1980s. Table 2.1 shows some of the more common test configurations being used at the time (Crews and Reeder, 1988). Some of the tests, such as the crack lap shear (CLS) and the mixed mode flexure tests ( M M F ) , rely upon eccentricities in the load path to induce mixed mode loading at the crack tip. They require specimens with different thicknesses of the two arms to cause different mixed mode ratios at the crack tip. Also, these test configurations require finite element analysis in order to determine the actual mixed mode ratio at the crack tip (Crews and Reeder, 1988). The Bradley and Cohen (1985) method of asymmetrically loading a D C B specimen while clamping its tail combines the convenience of using standard specimens with constant arm thicknesses with the convenience of a simple superposition method to determine the global mixed mode ratio. The same specimen and loading j i g can load the specimen in an infinite number of mixed mode ratios including pure mode I and pure mode II. In the majority of the other tests, the mixed mode ratio is predetermined and cannot be adjusted during the loading of the specimen. Using the method proposed by Bradley and Cohen, the entire mixed mode loading process can be directly controlled. A s a result, mixed mode load application need not necessarily be conducted at a constant mixed mode ratio. Instead, a specimen can be loaded first in either pure mode before introducing the opposite mode to induce a mixed mode condition. The flexibility using this test method provides perhaps the best opportunity for detailed investigation into the various local mechanisms involved in the fracture process. The convenience and flexibility of the asymmetric D C B test method do come at a cost however. The other test methods described in Table 2.1 use standard uniaxial testing machines to apply the mixed mode loading either through applying a pure tension load or by applying a three point bend. The asymmetrically loaded D C B specimen, however, requires a more complex loading set-up including a clamp and independently controlled loading arms. This type of loading cannot be readily conducted on simple testing machines and is one of the primary reasons it is not widely used.. 22 Chapter 2 Background and Literature Review In an attempt to address many of the issues affecting the mixed mode test methods of the time, Crews and Reeder (1988) proposed a new test method known as the mixed-mode bending ( M M B ) test. The test used a standard D C B specimen but subjects it to a complicated load using the M M B test j ig . The test j ig was specifically designed for use in standard uniaxial testing machines. In the M M B method a lever is used to apply mode I and mode II loads simultaneously while the position of the lever arm determines the mixed mode ratio. Through this process, pure mode I and mode II loads can be applied as can almost any mixed mode ratio. The test configuration is also such that the mixed mode ratio is essentially constant during crack growth. Unlike the Bradley and Cohen mixed mode test method, the mixed mode conditions cannot be adjusted during testing and only constant mixed mode ratios can be applied. The M M B test is also convenient from a data reduction perspective as closed form, analytical solutions based on simple beam theory are capable of determining the mode I to mode II ratios without the added complexity of finite element analysis. A redesign of the M M B delamination test was later proposed (Reeder and Crews, 1991) to address the non-linearity caused by testing of tougher material systems. While the M M B test j ig requires a separate loading j ig , it is one of the more convenient methods of mixed mode testing due to its ease of use in standard uniaxial testing machines and its closed form, analytical data reduction scheme. The redesigned M M B j ig quickly became the de facto industry standard and very recently has been established as the A S T M standard for mixed mode fracture toughness testing of unidirectional fibre reinforced polymers ( A S T M D 6671). 2.2.2 Global Data Reduction Methods In general, there are two data reduction methodologies, compliance methods and energy based methods, that are used to interpret global test data. A s wi l l be discussed below, compliance methods are typically classic predictive, global methods and suffer from their inability to accurately predict the actual local crack tip conditions. Energy methods, the area method in particular, also use global test data but rely on data acquired before, during, and after the actual fracture process as opposed to data acquired up to the point of fracture initiation. 2.2.2.1 Compliance Methods A great variety of data reduction methods exist to determine G / G and G/cc as a function of global loads and displacements. For the M M B test configuration alone, Bhashyam and Davidson 23 Chapter 2 Background and Literature Review (1997) examined four common existing reduction methods while introducing a fifth variation in an effort to increase accuracy and reduce variability of the predictions. The most common techniques involve variations of the compliance method (Equation 1.2). The compliance term from the equation can be determined analytically using simple beam theory, experimentally using compliance calibration techniques, numerically, or using a combination of the techniques. Experimentally, compliance calibration or modified compliance calibration methods are often used (e.g. A S T M D 5528). In these situations, curve fits determine the compliance-crack length relationship and these equations are subsequently used in determining the strain energy release rates. This method, therefore, requires various crack lengths to be tested for each specimen configuration before G / can be determined for any one particular configuration. The analytic techniques are typically based on simple beam theory analysis of a specimen. These equations generally require modifications and correction factors to account for such effects as rotations at the delamination front due to the simplification in the analysis that the cantilevered arms are perfectly built-in (e.g. A S T M D 5528; Blackman and Kinloch, 2001; Hutchinson and Suo, 1992). Often, a combination of analytical and numerical methods (e.g. Hutchinson and Suo, 1992) or experimental considerations are used to determine the appropriate correction factors. Davies and Benzeggagh (1989) highlight some of the assumptions of general beam theory analysis that need to be modified for applicability to composite materials. Their discussion is summarized in Table 2.2. The beam theory reduction schemes rely heavily on the material properties of the specimen. The local heterogeneous nature of composites can result in local conditions that are quite different than global values of material properties. Consequently, even in ideal situations where no internal mechanisms such as fibre bridging exist, the global predictions of the local behaviour may still not be accurate. Despite some of the concerns surrounding many aspects of the compliance method, it is this method that is the most widely used for fracture testing. The A S T M has endorsed the compliance method in both the mode I ( A S T M D 5528) and the mixed mode test standards ( A S T M D 6671). The compliance method is simple to implement and provides many different analytical options, each with benefits and limitations, for the determination of G / G and G/GC-24 Chapter 2 Background and Literature Review 2.2.2.2 Energy Based A r e a Method In contrast to the compliance methods discussed above, the area method requires limited simplifying assumptions. Figure 2.16 shows the concept behind the area method approach (e.g. Hertzberg, 1996; Broek, 1986). The area between the loading and unloading curves of the specimen is used to measure net external work done on the specimen during loading, the fracture process, and unloading. Assuming that crack growth is the only energy absorbing mechanism the net work is then converted into a GGC once the area of the created fracture surfaces is determined. Consequently, i f the energy absorbing mechanisms can be properly quantified, the amount of energy absorbed at the crack tip can be determined. It is the quantification of these non-fracture related losses that is one of the drawbacks of this method. For example permanent deformation (Keary et al, 1984), hysteresis loading effects (Hashemi et al, 1990), and system losses are all examples of complications in GGC determination. Theoretically, for slow, stable crack growth the GGC calculated from the area method should be equivalent to that measured using a beam theory approach. Unlike the compliance method, the area method can be used for non-linear elastic materials and for non-linear specimen geometries. One of the key advantages of this method is that it does not require any knowledge of the elastic properties of the material to determine GGC (Corleto and Bradley, 1989). In the area method, the global load and displacement data are used to measure the average amount of energy used during the entireTracture process whereas in the compliance methods the global load and displacement data is used to determine the instantaneous strain energy release rate at the moment the fracture process is initiated (Corleto and Bradley, 1989). The area method, that provides an average strain energy release rate during fracture, therefore, is imprecise (Hashemi et al, 1990) i f initiation is a primary concern. This is one of the main reasons that the area method is not recommended by the A S T M mode I testing standard ( A S T M D 5528). The other reason is that the area method necessarily only examines dissipative mechanisms. Any elastic mechanisms such as fibre bridging, which affects compliance, would not be included in the determination of GGC but dissipative local mechanisms such as fibre bridge breakage would be accounted for even i f crack extension is not observed. Consequently, a classic delamination resistance curve cannot be determined by this method ( A S T M D 5528). 25 Chapter 2 Background and Literature Review One of the primary difficulties with the area method is that only critical values of the strain energy release rate can be determined. Unlike with the compliance method approach, Go cannot be monitored up to failure or evaluated under elastic loading conditions. The area method approach also requires that cracks be unloaded directly after crack growth in order to measure the amount of energy consumed during fracture. A s a result, continuous crack extension cannot be conveniently measured using this method. Due to the many complexities and difficulties involved in using the area method approach to determining GGC, it has not been widely used in the literature. However, in the global-local context, the approach has potential due to the fact that this method uses data derived from the actual fracture process. In addition, the lack of fundamental assumptions makes the technique desirable for situations where the effect of the assumptions made in other techniques is unknown. 2.2.3 Mechanisms of Mixed Mode Delamination Growth Much of the global data in the fibre reinforced polymer, mixed mode delamination literature is limited to the presentation of mixed mode results primarily in the form of failure envelopes. A very few researchers have attempted to understand the shape of the failure envelopes in terms of physical mechanisms that occur during the fracture process. The sections below wi l l examine the general shape of the mixed mode failure envelope for brittle unidirectional CFRPs . A n examination of the current hypotheses relating the shape of the failure envelope to fracture processes and mechanisms wi l l also be examined. 2.2.3.1 M i x e d Mode Fai lure Envelopes Figure 2.17 shows a typical mixed mode failure envelope for a brittle unidirectional C F R P , AS4/3501-6 (Reeder, 1993). The fact that mixed mode G/c values increase from the pure mode I value as shear load is applied is the most interesting aspect of the failure envelope. The applied shear load appears to interact in a positive manner with the opening load and increase, from a global perspective, the material's resistance to opening loads. A local analysis of mixed mode loading conducted by Ferguson et al. (1991) revealed that as mode II load was applied to an existing mode I situation, the magnitude of the COD profiles were observed to increase, decrease, or remain the same. This suggests that the mixed mode interaction could be shielding the crack tip from seeing the full global load as when the CODs 26 Chapter 2 Background and Literature Review decrease, or that the crack tip indeed is capable of withstanding larger openings. In the case where the CODs increase with the addition of mode II loading, even though global loads predict elevated CODs at failure, the actual COD profile at failure may be even larger than the global predictions. The hump in the curve is characteristic of the failure envelope for unidirectional, fibre reinforced, brittle polymer material systems (e.g. Guoyang and Guiquing, 2001; Greenhalgh and Matthews, 1996; Partridge and Singh, 1995; Sriram et al, 1993; Hashemi et al; 1991) including glass fibre composites (Ducept et al, 1997). This particular shape of the failure envelope appears limited to brittle material systems as composites that show evidence of ductile behaviour tend to have a more linear failure envelope (e.g. Greenhalgh and Matthews, 1996; Reeder, 1993; Sriram et al, 1993; Hashemi et al, 1991). 2.2.3.2 M i x e d Mode Fracture Processes At first glance, and in the absence of any local quantitative and even qualitative data, the notion of increased G/c with the addition of small amounts of shear load is questionable. A s Guoyang and Guiqiong (2001) state: "It is noteworthy that sometimes [mixed mode] G/ is even larger than [pure mode] G/c . . . which is unreasonable from the point of energy". If the fracture mechanisms and the fracture processes were the same for mixed mode conditions as they are for pure mode I crack growth, then indeed the above statement would have merit. However, the studies that have examined the fracture process in a qualitative manner, primarily through post mortem examination of the fracture surfaces, have shown that that differences between pure mode I fracture and mixed mode fracture, even with limited amounts of shear load, clearly exist. Greenhalgh and Matthews (1996), Partridge and Singh (1995), and Reeder (1993) all observed consistent changes in fracture morphology from pure mode I loading, to mode I dominated loading, to mode II dominated loading. The fractographic evidence from the studies mentioned above is very consistent. Pure mode I fracture surfaces were seen to be "very flat indicating brittle cleavage fracture" (Reeder, 1993). As the amount of mode II load was increased to a GIG I GUG ratio of 4/1, the fracture surface was observed to become rougher, resulting in a more tortuous crack path and in an increase in crack surface area. Reeder (1993) in particular suggests that this may "explain why the mode I component of fracture toughness rises as mode II is introduced." 27 Chapter 2 Background and Literature Review Very early in the mixed mode envelope, hackles (Figure 2.10), characteristic of a shear dominated failure mode, are evident. These hackles can be seen at a mixed mode ratio of 1/1. The amount of hackles seen with further addition of shear load continues to increase in the Greenhalgh and Matthews (1996) and the Partridge and Singh (1995) studies whereas Reeder's (1993) evidence indicates that the mode II fracture surface is very similar to the 1/1 ratio fracture surface. Regardless of the study however, there is a very strong indication that "the differences between [1/1 and pure mode II] fracture surfaces and those at the pure mode I and 4/1 case may indicate a change in the failure mechanism near the 1/1 ratio" (Reeder, 1993). This changing failure mechanism appears to be indicated globally by the hump seen on the failure envelopes. 2.2.3.3 C r a c k K i n k i n g and Preferred C r a c k Growth Directions Another aspect of mixed mode fracture that could help to explain the origin of the hump is the preferred direction of crack growth. In an isotropic material mixed mode stress fields at the crack tip wi l l result in the crack kinking away from the initial crack plane in a preferred direction. This preferred direction is, depending on the theory employed, the direction in which the total strain energy release rate is maximized or the direction in which the r-6 shear stress, representative of /Cw vanishes (e.g. Hutchinson and Suo, 1991; Nuismer, 1975). In other words, in an isotropic material the crack wi l l typically propagate in a direction such that the crack tip experiences a pure mode I stress field. Under both theories, maximum total G or nil Ku, the predicted crack growth angle is similar across the entire mixed mode regime (e.g. Hutchinson and Suo, 1991; Nuismer, 1975). Layered composite materials, especially continuous fibre reinforced polymer laminates, present a unique situation in that the crack is confined to the interlaminar region. It is much easier for the delamination to grow straight ahead in the region between laminae than to grow at an angle into the lamina on either side in which case the crack would have to break through fibres. As a result of this delamination path, the crack tip sees mixed mode stress fields that are not observed in isotropic materials since the crack is inhibited from growing in a direction such that only mode I stresses are experienced. The interlaminar region in which the crack tip originates is a resin rich zone. A crack tip under mixed mode stresses in that zone would still have the ability to grow at a certain angle to the 28 Chapter 2 Background and Literature Review crack plane within the resin rich zone until it reached the fibres in the adjacent lamina. A t this point, the crack would be inhibited from growing into the fibres as the resistance to that mode of failure is much higher than the resistance to failure along the fibre/matrix interface. Despite the fact that the crack would grow along the interface, which itself is not smooth, the crack would continue to attempt to grow into the fibre pack. This type of fracture process would be consistent with the fractographic evidence presented by the studies described in the previous section (e.g. Reeder, 1993) and would require more energy for crack growth than the pure mode I loaded case in which the crack growth occurs within the resin rich layer resulting in the flat fracture surface observed. 2.3 Summary A brief literature review of brittle unidirectional fibre reinforced polymer composite pure mode delamination behaviour has been provided with an emphasis on the global-local relationship. A more detailed literature review has been presented on composite mixed mode delamination behaviour. Based on many of the issues addressed, it is the aim of this thesis to examine, using a combined global-local approach, the mixed mode delamination behaviour of brittle unidirectional CFRPs in the region where the characteristic hump in the failure envelope occurs. It is hoped that this examination wi l l be able to shed further insight into the physical basis of the hump and the variability in the data observed (e.g. Figure 2.17). 29 S © C3 90 00 ON 1-•a « <u Pi -o s « •si ii u u E © -a o 4> = JO "3D -a o S -a S3 H s o U 'cn -g c cd CD " O cn O > , CD £ " C _§ 7 3 § ^ * O cd CD " -a cd CD II c to a- £P —1 Cu, (U 5 ? S <D >-, 00 cd J 3 — O ft2 c 7 3 ^ t3 £> c = o s C T3 i— 00 •3 cd _o c o cd eu .c t/2 a. ^ cd c/5 CD w CJ cd u cn CD CD ~Q cn o cn c CD C ~ £ Cd CD T3 ,CD «= cd cn 7 3 ft CD 3 > a. ^ CD t 3 ,iS cd 3 <D <3" CD CD *s .3 cn 3 O a" \3 CD cd cn 3 >, c "> ^ SO c e -3. u 12 7 3 | g £ * g" * CD T 3 00 ft.£ -S 00 C T3 C CD .s 'o ft CD -a 2 « E T3 ? 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Classical beam theory analysis assumptions Modifications required for extension to general composite materials homogeneity and isotropy a generalized form of Hooke's law applies certain geometric conditions are respected Saint Venant's principle applies plane section remain plane after loading globally homogeneous but orthotropic satisfy a generalized Hooke's law have a stiffness matrix which, during loading such as bending or torsion, can promote large displacements and non-linear behaviour their mechanical behaviour depends strongly on load introduction conditions, /. e. end effects can be important their behaviour is also sensitive to shear phenomena which, e.g., in simple bending can provoke distortions of plane sections. The influence of transverse shear stress is closely related to the ratio of length to thickness. 31 Chapter 2 Background and Literature Review Figure 2.2 Schematic showing fibre bridges across the crack opening. These fibre bridges can shield the crack tip from seeing the full applied load and can consume energy during the fracture process. 80 -I 1 1 1 1 1 1 1 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Crack length (m) Figure 2.3 Example of R-curve behaviour for a brittle unidirectional AS4/3501-6 DCB specimen (Paris, 1998). 32 Chapter 2 Background and Literature Review Figure 2 . 4 Finite element normal stress contour plots at the crack tip for an orthotropic split laminate under pure mode I elastic loading (Corleto et al., 1 9 8 7 ) . r (microns) Figure 2 . 5 Plot of COD profile as a function of r for an applied G / G of 3 4 . 9 J/m2 (Ferguson, 1 9 9 2 ) . The combined global and local analysis shows the square root singularity at the crack tip. 33 Chapter 2 Background and Literature Review 140 x crack growth 120 + 100 + V from video E _ 80 + 60 + G I L - G I G / / 40 + 20 + Insert sticking B1 20 40 60 80 100 120 G| G (J/m2) 140 Figure 2.6 Plot of Gn vs. G/G for unidirectional AS4/3501-6 DCB specimens, showing that GJI=G/G for all cases up to and including at crack growth but one. The exception occurs for the first loading from the insert and might be explained by the insert initially sticking to the crack faces (Paris et al., 2001). 34 Chapter 2 Background and Literature Review 700 T 600 T E _ 500 + 400 + 300 4-200 + -•—R1 c1 - • - R 1 c2 - A - R 1 c3 - • - R 1 c4 -+-R1 c5 + B1 o R1 c1 failure • R1 c2 failure A R1 c3 failure o R1 c4 failure o R1 c5 failure G| L - G, G difficult to determine if G| L increasing or crack already growing 100 -F 100 200 300 400 500 600 700 G, G (J/m2) Figure 2.7 Plot of Gn vs. G/G for unidirectional AS4/3501-6 specimen with significant fibre bridging (Paris, 1998). The dotted line represents a one-to-one correspondence as seen in the absence of fibre bridges (Figure 2.6). The fibre bridges act to shield the crack tip, Gn from experiencing the entire applied GIG. They can also absorb energy if the fibre bridges are broken. 35 Chapter 2 Background and Literature Review 6 --5 --4 --3 --2 --1 ---100 -19--1 * WW G, G = 426 J/m' G , G = 277 J/m 2 + • G , G G|L 1— 99 J/m 2 G, c G =637 J/m 2 100 200 300 r (microns) 400 500 600 700 Figure 2.8 Plot of COD vs. r (distance from the crack tip) for G/c values of 99, 277 and 426 J / m 2 on a unidirectional AS4/3501-6 specimen wi th significant fibre br idging (Paris, 1998). Dashed lines show COD profile calculated from Gn. that gives the best fit. The point of inflection on the profile curve can be correlated directly to a fibre bridge acting to keep the crack tip closed. 1 0.8 kJ/m2 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 G/G II Figure 2.9 M i x e d mode delamination criterion for AS4/3501-6 ( O ' B r i e n , 1997 from Reeder, 1993). Gc is the cri t ical total strain energy release rate and G is the total applied. Therefore Gu/G = 0 is pure mode I loading and Gu/G = 1 is pure mode II loading. Note that the scatter and var iabi l i ty associated wi th Guc is much greater than that associated wi th <7/o 3 6 Chapter 2 Background and Literature Review 45° microcracks onset 20 microns images zone G„ (J/m2) \ \ W M 269 432 \^\\\W\i K - r 543 Figure 2.10 Figure showing the mechanism of mode II fracture in brittle unidirectional AS4/3501-6 C F R P laminates (Paris, 1998). Hackles are formed ahead of the crack tip at an ini t ial angle of 45 degrees due the max imum pr incipal normal stresses in the area. These hackles then rotate and coalesce resulting in a mode II crack extension. 3 7 Chapter 2 Background and Literature Review Figure 2.11 Finite element shear stress contour plots at the crack tip for an orthotropic split laminate under pure mode II elastic loading (Corleto et al., 1987). x (microns) Figure 2.12 Plot of CSD vs. x (longitudinal position) for unidirectional AS4/3501-6 C F R P specimen under applied pure shear loading (Paris, 1998). The origin of the CSD profile moves forward while the length of the square root behaviour diminishes wi th increasing shear load. 38 Chapter 2 Background and Literature Review 5.0 T -1.0 -1-x (microns) Figure 2.13 Plot of COD vs. x (longitudinal position) for the same unidirectional AS4/3501-6 C F R P specimen as Figure 2.12 under applied pure shear loading (Paris, 1998). The origin of the COD profile remains constant despite an increasing shear load and damage zone. 3 9 Chapter 2 Background and Literature Review B8 a 2 50 microns no load < global shear load Figure 2.14 Loca l opening displacement created under global shear loading (Paris, 1998). 4 0 Chapter 2 Background and Literature Review 140 T 120 100 80 60 40 + 20 I 9--100 B8 a 3 , B11 a-,, B13 a-. • including opening o neglecting opening real G M H O O K O- —I —I 1100 1300 100 300 500 700 -20 -L G„ (J/mz) 9 0 0 ^ apparent G M c Figure 2.15 Effect of neglecting the mode I opening due to surface waviness on the mixed-mode failure envelope (Paris, 1998). G Q C = net external work d o n e / c rack growth area ~ A A / B ( a 2 - a.) Figure 2.16 Schematic showing calculation of average global critical strain energy release rate using the area method. 41 Chapter 2 Background and Literature Review GIC'GIIC = 4 / 1 / 1 / 1 600 G I I C (J/m*) Figure 2.17 Mixed mode failure envelope for a brittle AS4/3501-6 unidirectional CFRP laminate (from Reeder, 1993). 42 3 Experimental Method Chapter 3 Experimental Method A n experimental set-up was developed by Paris et al. (2001) in order to investigate relationships between global applied conditions and local crack tip behaviours, a brief description of which w i l l be provided below. For more detail the reader is referred to Paris et al. (2001) and Paris (1998). Required modifications to the set-up for this work wi l l be covered in detail. In order to investigate relationships between global applied conditions and local crack tip behaviours, Paris et al. developed the experimental set-up shown schematically in Figure 3.1 (Paris et al, 2001; Paris, 1998). The complete experimental system, including data analysis, is composed of an instrumented, mechanical loading j ig , an electronic test control system, a data acquisition and analysis system, and an image acquisition and analysis system. Figure 3.2 shows a photograph of the actual experimental system. The instrumented loading j i g (Figure 3.3) provides the raw data for the 'global' analysis. Furthermore, the loading j i g is designed to replace the standard stage of a Hitachi S-2300 scanning electron microscope (SEM), which allows for in-situ observation of the crack tip. Digital still and video images obtained from the S E M provide the raw data for the ' local ' analysis. The real time combination of both local and global approaches in this experimental set-up allows for the investigation of a relationship between them. 3.1 Specimen Preparation The loading j i g is designed to test a D C B , split laminate specimen. When configured for mode II and mixed mode loading this specimen is commonly referred to as an end-notched cantilever beam ( E N C B ) . The loading j ig stage has been designed to accommodate standard D C B specimens within A S T M guidelines ( A S T M D 5528) and to apply mixed mode loading using the asymmetric D C B method (Table 2.1). A schematic of the D C B specimen used in this work is provided in Figure 3.4. The specimen length, L, is maintained at 70 mm, the specimen width, B, at approximately 20 mm, and 2h, the total thickness, is nominally 3.6 mm. The minimum crack length, a, is approximately 20 mm. The initial crack exists due to the placement of a starter film at the specimen midplane during fabrication. 44 Chapter 3 Experimental Method The aluminum loading tabs and steel clamping tabs (Figure 3.4) are bonded to the unidirectional C F R P using a j ig that ensures proper alignment. A 350°F, one hour cure, F M 300 epoxy film adhesive is used. Once the tabs are bonded, the specimen edge is polished using standard metallographic techniques. A thin carbon layer is then evaporated on to the specimen edge to minimize charging of the polymer phase under the electron beam of the S E M . In order to measure the crack tip displacements, a gold grid is applied to the specimen edge (Figure 3.4) through a 2000 openings per square inch nickel mesh produced by SPI Supplies. This mesh gives a grid spacing of 12.7 um. A j ig is used to assure alignment and contact of the nickel mesh while gold is evaporated onto the assembly. Once the j ig and mesh are removed only the gold grid remains (Figure 3.5). 3.2 Data Acquisition and Test Control The loading j ig is instrumented to provide all the necessary data for global analysis. The applied displacements, SR and SL, are continuously recorded by the data acquisition system on the computer, along with the loads induced in each loading arm, PR and PL. The displacements are measured with calibrated beryllium-copper cantilever beam displacement sensors (Figure 3.6) that are in contact with the loading pins. The stepper motors directly control the displacement of the loading pins and thus the specimen arms. These loading arms also act as load cells (Figure 3.6). Each displacement sensor and load cell has four strain gauges directly bonded to it and is wired in a Wheatstone full-bridge circuit. A linear voltage displacement transducer ( L V D T ) is used to measure the position of the electron beam in the S E M along the longitudinal axis of the specimen (Figure 3.3). In addition to measuring the crack length of the specimen at any given time, the L V D T system also allows for observation of the specimen at known distances from the crack tip. Loading of the specimen is achieved through two computer controlled stepper motors that independently displace each loading arm. Data acquisition and stepper motor control are accomplished simultaneously in custom designed software. The software allows for user control over testing. 45 Chapter 3 Experimental Method 3.3 Mechanical Testing The instrumented, loading j ig is shown in Figure 3.3. The platform that hosts the specimen can be manually manipulated parallel and perpendicular to the crack length to track the crack tip throughout testing. Testing is performed under displacement control by two stepper motors that control each loading arm independently. 3.3.1 Pure Mode II Loading For mixed mode loading both arms must be independently loaded. Paris showed that friction effects between the crack surfaces are negligible during applied pure shear loading (Paris, 1998). Figure 3.7 shows schematically the difference between pure mode II loading by one arm only (Paris' method) and pure mode II loading by both arms. Based on Paris' work regarding friction effects the two loading situations in Figure 3.7 should be equivalent for values of 0 < / < 0.5 at which point the loads in each arm are equal. Should f> 0.5, a pure mode II loading situation no longer exists as mixed mode loading develops. 3.3.2 Superposition Principle While Paris (1998) did not conduct any mixed mode tests herself, the loading j ig was designed to load D C B specimens in mode I/mode II mixed mode. The superposition principle described in Figure 3.8 (Bradley and Cohen, 1985) is implemented through the independent control of each loading arm. A s a result, mixed mode conditions can be achieved in any ratio and through any loading path desired. Paris demonstrated that the clamping system used in mode II and mixed mode loading induces minimal axial strains that can be reasonably neglected as they are less than 1% of typical bending strains (Paris, 1998). Once the mixed mode loading situation has been partitioned into mode I and mode II components using the principle described in Figure 3.8, the analysis of each component can proceed in isolation, independent of the other loading mode. Consequently, any analysis technique developed for pure mode loading can be used to analyse each component and the mixed mode condition subsequently reconstituted. Using the superposition principle, a mixed mode loading situation can be partitioned into constituent modes of loading based on the symmetric (mode I) and asymmetric (mode II) components. Using the notation of Figure 3.8 we have the following partitioning scheme: 46 Chapter 3 Experimental Method P, = (3.1) PIl=(PR+Pl},Sl (3.2) 2 where PR and PL are the right and left arm loads and SR and 8i are the right and left arm displacements as measured experimentally. The calculated parameters P\, Sj, Pu, 8u are the mode I and mode II components of load and displacement respectively. These calculated parameters along with geometric considerations constitute the global parameters required to predict the local conditions at the crack tip through linear elastic fracture mechanics ( L E F M ) . Specifically, the global parameters can be used to calculate the strain energy release rate, G, a similitude parameter that describes the local crack tip behaviour. 3.3.3 Calculation of Global Strain Energy Release Rate For an orthotropic D C B specimen under mode I and mode II loading as shown in Figure 3.8, the strain energy release rate for each mode has been obtained using finite element analysis in conjunction with analytical considerations by Hutchinson and Suo (1991). The equations determined by Hutchinson and Suo have been modified to include the effect of the machine compliance, Co. For this j ig , Paris (1998) found no mode I machine compliance and a constant machine compliance in mode II testing . The subscripts indicate mode of loading (I or II) and the approach used to obtain the strain energy release rate (G for global). A l l three equations for each mode of loading are theoretically equivalent. The differences between them are due to the different methods of calculating G , using either P only, 8 only, or both. In mode I: B2h3Et (3.3) * C 0 for mode II testing was determined to be 4.8.10"3 mm/N 47 Chapter 3 Experimental Method In mode II: where GIG = 3P,S, 2Ba,(l + ri) 35 2tiiE, 16^/(1 + Y l ) A GIIG = 9Pl2aI12(\ + y J 4E,B2h' GIIG = 9Sn2all2{l + rllfEy 2E,BtfC0 + U + 3an3(\ + y„ )3 9PIISnaII2(\ + yII)2 , , I G AE^tfCo + 25(z3 +3a / 7 3 ( l + / , , ) ' ) (3.4) (3.5) (3.6) (3.7) (3.8) a. (3.9) Yu = Y„l> a. Y, = 0.677 + 0 . 1 4 9 ( p - l ) - 0 . 0 U(p-\)2 Y„ = 0.206 + 0.078(yO -1 ) - 0.008(/7 - 1 ) 2 a„ (3.10) (3.11) (3.12) (3.13) au + *66 p where a\j are the plane stress elastic compliance constants for the laminate: au -v.. 1 , tC?22 — (3.14) a, 2 — 48 Chapter 3 Experimental Method . 3.4 Image Acquisition During testing the video image provided by the S E M is grabbed by the computer using a Coreco Occulus frame grabbing board. The image is instantly overlaid with data provided concurrently by the data acquisition system. The image overlaid with the data is then recorded on videotape. Due to vibrations generated by the stepper motors during loading, small pauses are incorporated into the loading path to ensure stable video images. A s the video image provided by the S E M can only focus on a small area around the crack tip at any one time, loading is routinely paused for a longer period of time to allow for digital recording of still images. Due to the depth of field of the S E M system, many images can be stitched together to provide high quality montages of areas of interest. These images are used to measure local crack face displacement profiles as wi l l be described in the following section. The still images are typically recorded at a magnification of 500x. This magnification provides a good compromise between resolution and the size of the viewed area. The images, which are immediately saved to a computer, have significantly better resolution and image quality than the video output. Although more resource intensive, the digital still images are used to obtain the best measurements of the crack face displacement profiles at specific load levels. The video images are primarily used to determine qualitatively and quantitatively what happens close to failure when events take place very quickly. 3.5 Image Analysis Image analysis software was developed to measure crack face displacements with respect to the crack tip location. The digital still images are stitched together to form a montage of the crack tip area (Figure 3.5). The software is then used to align the crack horizontally, to render the montage as a binary image thus highlighting the gold squares, and finally to calculate the centroid of each square. Before measurements are taken the image is calibrated using the grid pattern itself. The crack face displacements cannot be accurately measured directly from the crack face due to charging of the specimen. Therefore, COD profiles are determined by measuring the vertical change in the distance between two squares on either side of the crack before and after loading (Figure 3.9). 49 Chapter 3 Experimental Method Similarly CSD profiles are determined by measuring the horizontal change in distance (Figure 3.9). Measurements must be taken as close to the crack face as possible to minimize the effect of the high strains in the crack tip region. A detailed analysis of the impact of these strains on the CFD is provided in Section 6.3. Figure 3.10 shows the decreased variability from the finer gold grid used in this work compared with Paris' (1998) work. In addition to increased data density, the smaller grid spacing allows for measurements to be taken closer to the crack surface. Despite the difference in grid spacing however, the profiles generated by two different experimentalists, for two different specimens, but for similar crack lengths and applied G\ agree well with each other. 3.5.1 Calculation of Local Strain Energy Release Rates Applying the principle of superposition as described by Bradley and Cohen (1985) to a local approach, it is evident that any COD observed at the crack tip must be due to a mode I component of loading. Similarly any CSD observed must be due to a mode II component of loading. The local strain energy release rates can then be determined from the following equations (Poursartip et al, 1998): 4 1 (a , ,#22)4 4r-^G~ (3.15) 1 A4~rJG~^ (3.16) COD = 2an + a66 a 22 la. a. AJaZ i CSD = - ^ - 2 A 2an + 2a,, *22 a, The local values of G/ and Gn are determined by the COD and CSD profiles calculated from the equations above that give the best fit to the experimental profiles. The subscript notation is similar to the global values with the L indicating a locally determined strain energy release rate. The equations above were derived using only the first term of the elastic stress singularity, and away from the crack tip the higher order terms w i l l become significant. The region where this first term effectively represents the local conditions is the singular zone and is a function of r1/2 1/2 in the displacement field as opposed to f in the stress field. Paris (1998) observed that the size 50 Chapter 3 Experimental Method of the displacement singular zone is no greater than 500 microns for the specimen and crack dimensions used here. 3.6 Modification to Specimen Configuration In pure mode testing conducted by Paris et al. (2001) using the experimental system outlined above, the loading tab holes were just large enough to accommodate the loading pins (3.175 mm diameter). Under pure mode I loading there is no requirement for slack at the loading pins due to the symmetric nature of the loading. A s a result, the specimen can be loaded along a straight line perpendicular to its longitudinal axis as shown in Figure 3.11. Pure mode II loading, however, is more complex. In order to allow the crack surfaces to slide over each other as the shear load is applied, the loaded end of the specimen must be sufficiently free to allow for relative motion of the upper and lower beams. Paris kept the same loading tab configuration as for pure mode I testing (3.175 mm hole diameter) but only loaded the lower arm (f= 0) in order to induce the global shear load (Figure 3.7a). Thus, the top arm is completely free to move. When a specimen was loaded with two equally loaded arms (f= 0.5) as opposed to by the lower arm only, GUL calculated from the measured CSD profile was significantly less than that predicted from global conditions (Figure 3.12). In fact, regardless of the applied global loads, GUL did not exceed 45 J/m . Further examination revealed that the compliance of the specimen under this loading was non-linear (Figure 3.13). It was determined that the system was overconstrained as the crack faces were not being allowed to slide over each other freely. Therefore, the loading tab holes were increased to a 4.5 mm diameter and the specimen reloaded w i t h / = 0.5. A s shown in Figure 3.13 the modified specimen compliance is now linear and in good agreement with theoretical values. The CSD profile for the modified loading tab is also consistent with Paris' (1998) data in that GUL = GUG for pure mode II loading (Figure 3.14). A benefit of the combined global and local analyses of a test is that it is easy to identify the effects of boundary conditions. The analysis performed also highlights the fact that boundary conditions play a critical role in the local crack tip behaviour and that extra care should be taken 51 Chapter 3 Experimental Method to ensure that experimental boundary conditions are appropriate for the specific test being conducted. 3.7 Modification to Testing Method 3.7.1 Global Strain Energy Release Rate Calculation Method A consequence of the solution to the overconstrained system as described above is a loss of accuracy in the displacement data due to the slack in the loading tab. The effect is more pronounced in Si data as Gj values are much more sensitive to minor errors in Si than Gn is to minor errors in Su. Fortunately, Paris (1998) showed that experimental compliances were in good agreement with theoretical values. Consequently the strain energy release rate equations that are functions of load only, Equation 3.3 and Equation 3.6, can be used with confidence. 3.7.2 Pure Mode II Loading of Specimen Figure 3.15 shows the CSD profiles resulting from a specimen loaded w i t h / = 0, lower arm only, and w i t h / = 0.5, both arms loaded equally. There is no discernable difference between either of the two CSD profiles, or the two induced COD profiles. This is further confirmation that there is no effect of friction on the shear loading of these specimens. The shear load applied is fully transferred directly to the crack tip. 3.8 Summary Paris' experimental set-up has briefly described. The loading tab has been modified to allow for free motion of the crack faces, thus eliminating the overconstrained system induced by the original mixed mode loading set-up. The negligible effect of friction observed by Paris (1998) is confirmed by loading a specimen with one arm and two equally loaded arms. To facilitate mixed mode loading, all pure mode II loading conducted in this work wi l l have /= 0.5. 52 Chapter 3 Experimental Method m image analysis • V C R image marked with data image < data acquisition test control loading stage AW SEM Figure 3.1 Schematic of the complete experimental set-up showing integration of test control, data and image acquisition, and test analysis (Paris, 1998). Figure 3.2 Photograph of the complete experimental system schematically described in Figure 3.1 (Paris, 1998). 53 Figure 3.3 Photograph of the loading j i g showing major components (Paris, 1998). 1 pa gold grid loading tab 7* clamping tab Figure 3.4 Specimen configuration used for al l testing wi th a luminum loading tabs, steel clamping tabs, and a 2000 opening per square inch gold gr id . 54 Chapter 3 Experimental Method crack tip 100 [ i m gold grid Figure 3.5 Photomicrograph montage of specimen surface showing the crack tip, and a gold gr id created from a 500 opening per square inch mesh (Paris, 1998). 55 Chapter 3 Experimental Method O Figure 3.7 Schematic of mode II loading conducted (a) by Paris (1998) and (b) in this work. Specimen is in pure mode II with 0 <f< 0.5 and in mixed mode with 0.5 </< I-Figure 3.8 Superposition of mode I and mode II on a DCB/ENCB specimen (Bradley and Cohen, 1985). 56 Chapter 3 Experimental Method UNLOADED LOADED C O D = 5yL-5yu 20 microns C S D = 5XL-5Xu Figure 3.9 Photomicrographs showing method for determining COD and CSD (Paris, 1998). 57 Chapter 3 Experimental Method E Q o O 9 8 7 6 5 4 3 2 o G| G = 76 J / m - this work (a = 24.4 m m ) A G| G = 75 J / m 2 - Paris' work (a = 24.1 m m ) — G| L = G| G O A A oo<>£ A A 00° / 0°A><A A <AA O O •100 100 300 r ( f i m ) 500 700 Figure 3.10 Plot showing increased data density and reduced variability with finer grid spacing. The consistency of experimental results regardless of experimentalist is also evident. ////// z+ (a) ////// ////// ////// Zz. (b) S7s 77 /77 77 Figure 3.11 Schematic of test boundary conditions for (a) original loading tab (3.175 mm pin hole diameter) and (b) modified loading tab (4.5 mm pin hole diamter). 58 Chapter 3 Experimental Method x ( | i m ) Figure 3.12 Plot of CSD profile for mode II loading of original loading tab pin hole diameter of 3.175 mm w i t h / = 0.5, showing Gm « GUG-5 i 4 H ? 3 H E, 2 -I 1 H o Original Loading Tab Modified Loading Tab Theoretical Compliance a = 27 mm 50 100 P H ( N ) 150 200 Figure 3.13 Plot of mode II compliance for original (3.175 mm) and modified (4.5 mm) loading tab pin hole diameters w i t h / = 0.5, showing increased range of linear response wi th modified loading tab. 59 Chapter 3 Experimental Method E o <o o -250 8 7 6 5 -4 -3 -2 -1 --1 * x GIIG ~ 85 J / m G| | L = G | | G 250 500 750 1000 1250 1500 1750 2000 x (|j.m) Figure 3.14 Plot of CSD profile for mode II loading of modified loading tab pin hole diameter of 4.5 mm, showing Gm = GUG. 0 500 1000 1500 x ( f i m ) Figure 3.15 Plot of CFD profile for pure mode II loading by one arm only (f = 0) and by two, equally loaded arms (f= 0.5). Profiles show no difference between the two conditions indicating negligible friction effects. 60 4 Experimental Results Chapter 4 Experimental Results The global and local approaches to evaluating delamination behaviour in unidirectional C F R P s described in Chapter 1 were used to investigate mixed mode behaviour experimentally. A variety of mixed mode tests were conducted on AS4/3501-6, unidirectional, 24 ply C F R P D C B specimens. Experimental objectives and individual test results are presented initially, followed by analysis and discussion of the results and observations. 4.1 Experimental Objectives The primary objective for the current investigation draws motivation from Paris' (1998) examination of the pure mode delamination behaviour of unidirectional CFRPs . A n investigation into delamination behaviour under mixed mode loading is a natural progression from those results. Pure mode loading is of considerable scientific and academic interest particularly with respect to understanding delamination behaviour. Equally important for designers and users of composite materials, though, is the translation of this pure mode understanding into better understanding of mixed mode delamination behaviour. While much mixed mode data exists in the literature, there is a lack of physical understanding of the general material or specimen response. This is especially evident with respect to variability in mixed mode fracture toughness, in the shape of mixed mode failure envelopes, and in the establishment of failure criteria. The literature in this area has been examined in detail in Chapter 2. The work of Reeder (1993) is of particular interest as he explores failure envelopes and failure criteria and includes AS4/3501-6 as one of the examined material systems. A s many researchers have noted, in the presence of small amounts of mode II load, up to 150 J /m 2 for AS4/3501-6 (Reeder, 1993), there is an increase in the capacity of unidirectional C F R P s to sustain mode I load (Section 2.2.3.1). These mode I loads are stable and often in excess of those expected to cause failure under a pure mode condition. While the phenomenon of increased load carrying capability is well documented in the literature, a good understanding of why this occurs is lacking. The literature suggests that this phenomenon is associated in some capacity with a changing failure mode from tensile to shear crack propagation (Section 2.2.3.2). Considering the above, the primary objective of these experiments is to better understand, using the global/local analysis, the delamination behaviour of unidirectional C F R P laminates subject to 62 Chapter 4 Experimental Results mixed mode loading. Specifically, the investigation wi l l attempt to use the global/local approach to increase understanding of variability in mixed mode fracture toughness values and of the shape of mixed mode failure envelopes. The context of this investigation, though, is confined to examining the region early in the mixed mode regime where a mode I global load greater than pure mode G/c is sustainable (Figure 2.17). 4.1.1 Parameters of Interest To facilitate the experimental investigation into the primary objective, several broad parameters of interest were examined through the various tests performed. It is important to note that these parameters were examined specifically in the context of the mixed mode situation described above. 4.1.1.1 Load History It is of interest to determine whether or not subcritical loading of specimens affects the future response of the system. In essence, low cycle, moderate load, random mode testing was conducted to investigate the effect of load history of a cracked specimen. Specifically, how the creation of damage zones due to the stress concentrations at the crack tip may influence, in particular, the local response of the system was examined. 4.1.1.2 Load Path There is a need to examine the effect of various loading paths that result in mixed mode conditions. These loading paths included starting with a pure mode applied load, either mode I or mode II, before introduction of a mixed mode condition. A s discussed previously (Section 2.2.1), the mixed mode bending ( M M B ) test j ig developed initially by Crews and Reeder (1988) allows only for simultaneous mode I and mode II mixed mode load application. The specimen response to unloading from a mixed mode condition to pure mode conditions was also examined. The outcomes of this investigation wi l l allow for some comment on the generality of various test methodologies and the applicability of their results to real structures. 4.1.1.3 Crack Condition The behaviour of delaminations with different crack conditions was explored. Three crack lengths of nominally 21.2 mm (mx5), 23.8 mm (mx3), and 27 mm (mx4), were examined. 63 Chapter 4 Experimental Results __ Additionally, the 21.2 mm crack was tested directly from the starter film insert used to initiate the specimen delamination. The other two specimens were tested after mode I precracking. 4.2 Specimen Descriptions Three unidirectional 24 ply, AS4/3501-6, C F R P laminate, D C B specimens were used in the course of the investigation. Table 4.1 provides the key specimen characteristics. A l l three specimens are nominally equivalent with the primary differences being the crack length and crack condition. Specimens mx3 and mx4 were precracked in mode I from the insert before testing, while specimen mx5 was tested directly from the insert. Specimen preparation is detailed in Section 3.1. 4.3 Test Descriptions Table 4.2 provides all the required information in the first five columns to fully describe a test. The first column, 'Test', indicates the test name and most importantly identifies the specimen. The detailed loading is fully defined in the second column, 'Loading' . The loading is performed in stages and each comma demarks a distinct loading stage. Each line of the table coincides with the acquisition of a full complement of primary data. This primary data consists of global loads, global displacements, and slow scan images required for local analysis. After the crack length is presented in the third column, the next two columns provide the global values of G / G and GUG that exist at the conclusion of that loading stage. For example, test mx4 2112 was performed on specimen mx4 at a crack length of 26.9 mm. The loading took place in three distinct stages; first, a pure mode II load of GUG = 90 J/m 2 was applied and a full complement of primary data acquired; subsequently, a mode I load was applied 2 2 until a mixed mode condition of G / G = 66 J/m and GUG = 90 J/m existed and another set of primary data was acquired; finally, the mode I loading was removed and a pure mode II load of GUG = 90 J/m 2 remained when the final set of primary data was acquired. After each test the slow scan images were assembled into montages and from these images the CFD profiles were obtained as described in Section 3.5. The majority of the data regarding loading path and loading history came from the mx4 data set and was complemented by mx3 and mx5 data for validation. A l l tests performed on specimen mx4 were designed to specifically examine the effects of various loading paths as wi l l be 64 Chapter 4 Experimental Results described below. The collective mx4 data set for the 27 mm crack length provides the results necessary for an examination of load history on mixed mode delamination behaviour. 4.3.1 Test mx4 1212, mx41221, mx4 2121, mx4 2112, and mx412 Descriptions Tests mx4 1212, mx4 1221, mx4 2121, mx4 2112, and mx4 12 (a = 27 mm) were performed in the random order shown in Table 4.2. It is in this random order that the tests w i l l be presented below. These tests were conducted to examine the effect of various loading and unloading paths. For example, mx4 2121 and mx4 2112 are loaded identically at first. The difference in the tests lies with the unloading of the specimen from the mixed mode condition. In mx4 2121 the specimen is unloaded such that a pure mode I load remains whereas in mx4 2112 the specimen is unloaded such that a pure mode II load remains. One of the difficulties encountered during these tests was that due to the slack in the loading set-up it was very difficult to maintain a constant mode I load during mode II addition or subtraction due to the large displacements involved. A s a result these load paths involved simultaneous loading of mode I and mode II components from the pure mode I case to achieve the mixed mode conditions. For test mx4 1212 primary data was acquired at the following GIG'-GUG levels, in chronological order: 65:0, 125:72, and 0:63 J/m (Figure 4.1). For test mx4 2112 primary data was acquired at the following GIG'-GUG levels, in chronological order: 0:90, 66:90, and 0:90 J/m 2 (Figure 4.5). For test mx4 2121 primary data was acquired at the following GIG'-GUG levels, in chronological order: 0:85, 70:85, and 45:0 J/m (Figure 4.9). For test mx4 12 primary data was acquired at the following GIG'-GUG level: 60:90 J/m (Figure 4.13). For test mx4 1221 primary data was acquired at the following GIG'-GUG levels, in chronological order: 73:0, 94:63, and 53:0 J /m 2 (Figure 4.15). 4.3.2 Test mx3 1221 and mx3 12 Descriptions Tests mx3 1221 and mx3 12 (a = 23.8 mm) were used to examine mixed mode behaviour under a shorter crack length than mx4. For these two tests, primary data was acquired at GIG'-GUG levels of 110:62 (Figure 4.19) and 80:50 (Figure 4.21) J /m 2 respectively. 4.3.3 Test mx4 w Description The mx4 w (a = 27 mm) test was conducted using a fine wire wedge to provide the mode I opening and then loaded in mode II using the lower arm only. Figure 3.15 in Section 3.7.2 showed that there was no difference in pure mode II loading between one arm and two arm load 65 Chapter 4 Experimental Results : application. To verify these results in the mixed mode case, this test was done to ensure that the testing j ig was not affecting results due to the independent manipulation of each arm. In addition, it allowed for a constant mode I load during mode II loading which was difficult to maintain during independent arm manipulation due to the slack in the loading set-up. Primary data was acquired at the following eight GIG'-GUG levels, in chronological order: 105:0, 105:42, 105:55, 105:67, 105:85, 105:97, 105:117, and 105:67 J /m 2 (Figure 4.23). 4.3.4 Test mx4 rev 1221 Description Test mx4 rev 1221 (a = 27 mm) was conducted to determine i f mode II loading in an opposite direction impacted results. A l l other tests were conducted with the shear component of loading being directed to the left side of the stage. In this test, the mode II component was applied towards the right side of the stage. Primary data for this test was acquired at the following GIG'-GUG levels, in chronological order: 75:0, 110:75, and 85:0 J /m 2 (Figure 4.32). 4.3.5 Test mx4 21 up Description In most of the mixed mode conditions examined, the upper arm carried a high proportion of the load. Test mx4 21 up (a = 27 mm) was conducted to examine i f there was any difference in the results when only the upper arm was loaded, w h e n / = 1. Primary data was acquired at a GIG-GUG level of 85:75 J/m (Figure 4.36). Figure 4.37 shows the load in the upper and lower arms for this test. 4.3.6 Test mx4 cp and mx5 cp Descriptions Tests mx4 cp (27 mm) and mx5 cp (21.2 mm) were both conducted to failure. The loading for the tests was intentionally similar with an initial applied pure mode II load and subsequent application of mode I all the way to crack propagation. This allowed for a direct comparison between a test conducted from an insert versus a test conducted from mode I precracking. Primary data for mx4 cp was acquired at the following GIG'-GUG levels,, in chronological order: 0:95, 84:95, 100:95, 120:95, and 140:95 J /m 2 (Figure 4.39). The image for the 140:95 J /m 2 load level was acquired after testing from the video images obtained just prior to the moment of crack propagation. Primary data for mx5 cp was acquired at the following GIG'-GUG levels, in chronological order: 0:88, 36:95, 65:95, and 90:95 J/m 2 (Figure 4.45). Primary image data just prior to crack propagation that occurred at GIG'-GUG level of 110:95 J /m 2 was unavailable. 66 Chapter 4 Experimental Results 4.4 Test Results and Observations In addition to providing detailed test descriptions Table 4.2 also provides the summary results for all the tests presented below. These results include global and local values for both mode I and mode II strain energy release rates, Gic,Guc, G /L , and GUL- The difference between the global and local values, AG, for each mode is also presented. The column labeled ' A z ' shows the difference between the crack tips from the mode I and mode II crack face profiles as determined from the local analysis. In addition to Table 4.2, results for all mixed mode testing of the 24-ply unidirectional AS4/3501-6 C F R P D C B specimens are presented graphically. Each set of test results is preceded by a figure showing a schematic of the loading path along with the points where a full set of primary data was acquired. Graphical test results are presented using CFD profiles. Individual data points are acquired from analysis of the images taken as part of the primary data as described in Section 3.5.1. Local strain energy release rates are then calculated to match the profiles using Equation 3.15 for mode I components and Equation 3.16 for mode II components. The material properties used for the calculations are provided in Table 4.3. These calculated local values are presented as solid lines through the measured data points. 4.4.1 Test mx4 1212, mx41221, mx4 2121, mx4 2112, and mx412 Results Figure 4.2 shows the results of the local analysis for the first loading stage of test mx4 1212. The specimen is loaded initially in pure mode I to a G / G of 65 J/m 2 . The CFD profiles are all a function of an arbitrary distance, x, along the length of the specimen. Both the COD and CSD profiles are as expected and in agreement with Paris' pure mode I results (Paris, 1998). The applied mode I load is indeed 'pure' as no CSD local component is induced. Also, all the globally applied load is transferred directly to the crack tip as the COD profile shows that Gn -GIG-From Figure 4.3, the CSD profile for the mixed mode condition for this test, where GIG'-GUG is 125:72 J/m , is also as expected. As in pure mode II loading, (reference) all globally applied 67 Chapter 4 Experimental Results mode II load is transferred directly to the crack tip since Gm = GUG. However, the resulting COD profile in this mixed mode case is of interest. Under mixed mode loading Gn is greater than the applied G / G of 125 J /m 2 by 30 J/m 2 . Not only is G / G very close to the pure mode G/c determined by Paris of 126 J /m 2 (2001), but more significantly Gn is substantially in excess of the pure mode fracture toughness. Even with a very large Gn, the magnitude of the COD profile in the area of highest importance, the singular zone, is on the same scale as the fibre diameter - in this case approximately 8 um. Another key observation to be made is that according to the crack face displacement profiles, the mode I and mode II crack lengths, aj and an respectively, do not coincide. The mode II crack length, an, is 140 um greater than a/. This Aa between the two cracks is summarized for all the tests conducted in Table 4.2. Paris (1998) also showed a difference in crack tip locations between COD and CSD profiles in global shear loading. After the mixed mode loading is removed to retain a pure mode II load, the CFD profile as shown in Figure 4.4, is once again as expected. A s in Paris' observations, the applied shear load is fully transferred directly to the crack tip. In addition, what was applied as pure shear globally is transferred to a local mixed mode condition due to surface roughness interactions between the crack faces (Paris, 1998). Tests mx4 1221, mx4 2121, mx4 2112, and mx4 12 all showed similar tendencies as mx4 1212 as can be seen in Figure 4.5 to Figure 4.18. In all cases the pure mode results were in excellent agreement with Paris' (1998) results. With the exception of mx4 2121 where Gn was equivalent to G / G under mixed mode loading, all mixed mode conditions resulted in Gn exceeding G / G and in many cases also exceeding G/c. From the various CFD profiles, a/ and an grow a small amount during testing. This growth, on the order of tens of microns, indicates that subcritical crack growth occurs both in mode I and in mode II. The fact that the locations do not reverse position in any future analyses indicates that the subcritical crack growth is real and not an artefact of measurement. 68 Chapter 4 Experimental Results . 4.4.2 Test mx3 1221 and mx3 12 Results A s shown in Figure 4.20 and Figure 4.22, both tests conducted on shorter crack lengths, mx3 1221 and mx3 12, display the same mixed mode responses as the longer crack mx4 tests conducted and described above. From these tests no crack length effect could be determined. 4.4.3 Test mx4 w Results Test mx4 w which was loaded in pure mode I using a fine wire wedge before being loaded in shear using the lower arm only, also displays mixed mode behaviour as described above (Figure 4.23 to Figure 4.31). This indicates that the method of both tensile and shear load application does not affect the observed results. Consequently, any mixed mode loading, regardless of loading j i g or method of load application, should exhibit similar behaviour as noticed here. 4.4.4 Test mx4 rev 1221 Results The mx4 rev 1221 test results were indistinguishable from the tests that were shear loaded in the opposite direction (Figure 4.32 to Figure 4.35). There appears to be no directional effect in applying the shear load. 4.4.5 Test mx4 21 up Results Compared to the other mx4 tests conducted test mx4 21 up showed a larger difference between GIL and G / G (Figure 4.38). Whether or not this is due to the particular loading with the upper arm only as opposed to having some load in both arms as in the other tests w i l l be examined in Section 5.3.2. 4.4.6 Test mx4 cp and mx5 cp Results The two tests conducted to crack propagation, mx4 cp (Figure 4.39 to Figure 4.44) and mx5 cp (Figure 4.45 to Figure 4.49), also behave similarly to the other mixed mode tests. However, the AG] for mx5 cp is significantly greater than for the mx4 or the mx3 tests. The effect of the different crack conditions is discussed in detail in Section 5.3. Another result from these two tests is that in both cases, crack propagation occurred at GIL and G / G values in excess of those expected to cause pure mode I failure. For mx4 cp the mixed mode failure occurred at a GIL of 240 J /m 2 and G / G of 140 J /m 2 compared to a pure mode G/c of 126 J /m 2 (Paris et al., 2001). 69 Chapter 4 Experimental Results _ _ Similarly, mx5 cp failed at a mixed mode GIL of 200 J/m 2 and a G/c of 110 J /m 2 as opposed to the pure mode G/c of a test from the insert of 96 J/m (Paris, 1998). 4.5 Summary Mixed mode tests were conducted on AS4/3501-6 unidirectional laminates. A l l pure mode conditions, whether tensile or shear, whether prior to or after mixed mode conditions, behaved as expected and in agreement with previous results (e.g. Paris et ai, 2001; Paris, 1998; Ferguson, 1992; Ferguson et al., 1991). The consistency of the results indicates a robust experimental technique. The mixed mode test results show three key observations under mixed mode loading. First, GIL is typically much greater than G/c. Figure 4.50 and Figure 4.51 highlight the difference in GIL and G/c under mixed mode conditions through the use of the parameter AGi. In both plots, the data points identify the specimen used. From these two plots it is evident that above all there is significant variability in the amount of AGi for different tests under similar loading conditions. Despite the variability, some relationship between increasing G/c and increased AGi appears to exist. Further examination reveals that there is no apparent dependence of AG; on the applied shear component. It is also evident that specimen mx5 appears to have higher AG/S than the other two specimens examined. Secondly, G / L and G/c regularly exceed pure mode G/c- Figure 4.52 shows the relationship between G/c, G / L , and G/c for specimens mx3 and mx4. It is clear that G / L regularly exceeds G/c by considerable amounts. The third primary mixed mode result is that the crack lengths in each mode, a/ and an, do not coincide and subcritical growth of these cracks is evident. Figure 4.53 shows the difference in mode I and mode II crack lengths for test mx4 w. The individual and relative movement of these crack tips as determined from the crack face displacement profiles is noticeable. 70 Chapter 4 Experimental Results Table 4.1 Characteristics of specimens mx3, mx4, and mx5. mx3 mx4 mxS vf (%) 59 59 59 grid spacing (pm) 12.7 12.7 12.7 h (mm) 1.80 1.79 1.83 B (mm) 19.6 19.3 19.7 L (mm) 70 70 70 71 a Si I -£ cs s s °E a w C S3 en C O u CU H CU pi CU 15 S3 H u CU -a S-o ^ S be s • PM 93 O _ CU H I K <=>J C N o O N i l l o '—i «?1 oo lliisiliiii i lsi i i l i i n 00 O N CN CN O r i o CN © CN CN i n CN o O N CN O m l o A O o o oo r-- CN o i n i n c-~- m o m C N O i n i n m o o oo i n m oo i n oo oo CN 00 CN CN CN CN CN CN CN CN CN S CN R I 3 C s-•a O "53 i=5 re c -3 es o -so H C N C N C N C N C N C N C N C N C N e © C N H e o DO 03 -2 eg fi a .a a _ 03 <+H a 0 ° ^ . ^ 03 1—1 O H H . 1 o S <D -a -5. § _ a .s 001—1 3 . 0 <D <u |^ 1 S m 0) o OH S o "o3 00 —, a « o 1  ^ .2 g ^ '5b 1—1 o • S o s J 0 0 ON ON . 8 a 73 C a J2 "« s .© w w _&« •3 s s O «A T t C o CM — 3 93 H 03 CM o C N O 03 PH O ON , OS O K5 03 OH o 03 PH O in l<3 Chapter 4 Experimental Results 140 -| 120 -100 -60 H 0 # \ 40 A 20 H / / GIG 7 \ / / i d / / / / / / \ \ »IIG \ \ \ \ \ \ \ \ \ \ \ \ \ mx4 1212 Figure 4.1 Schematic of test mx41212 loading path for plots presented in Figure 4.2 to Figure 4.4. The load was held constant at each point to allow for collection of slow scan images from the SEM. 10 8 6 ? o 2 0 -2 J • G| G = 65 J/m' — G| L = G| G G, i r i = 0 J/m Dc5ta COD mx4 1212 a = 26.9 mm loading x Xv ,CSD 500 1000 1500 x (fj.m) Figure 4.2 Plot of CFD profile for pure mode I loading of mx41212 before application of mixed mode condition. As expected GIL = G/c and no appreciable shear displacements are induced. 74 Chapter 4 Experimental Results d) 500 1000 1500 x ((j,m) Figure 4.3 Plot of CFD profile for mixed mode loading of mx4 1212. Note that Gn > GIG, GUL = GUG, and au > a/. Gn is also greater than pure mode G/c. 1500 x ( | i m ) Figure 4.4 Plot of CFD profile for pure mode II loading of mx41212 upon removal of mixed mode condition. Note that as expected some local opening is induced, Gm = G/,G, and au > ah 75 Chapter 4 Experimental Results 100 90 -80 -70 -«<r 60 E 3 50 o CD 40 30 20 10 0 GIIG / / / s f R / \ U I G / \ / \ / \ / \ / \ / \ / \ / \ / \ / mx4 2112 \ Figure 4.5 Schematic of test mx4 2112 loading path for plots presented in Figure 4.6 to Figure 4.8. The load was held constant at each point to allow for slow scan imaging in the S E M . 6 -I 5 -4 -• G | G - 0 J/m — G | L = 10 J/m2 x G | | G - 90 J/m2 GIIL = GIIG 1500 x ([im) Figure 4.6 Plot of CFD profile for pure mode II loading of mx4 2112 before application of mixed mode condition. Note that as expected some local opening is induced, GUL = GUG, and an > ah 76 Chapter 4 Experimental Results x (|j.m) Figure 4.7 Plot of CFD profile for mixed mode loading of mx4 2112. Note that GIL > GIG, Gm = GUG, and au > «/. 7 6 5 ^ 4 E | 3 o 2 1 0 •1 • G | G = 0 J / m — G| U = 10 J / m 2 x G| | G = 90 J / m 2 G|IL = GIIG X x x > ^ ^ x< x>x CSD COD • mx4 2112 a = 26.9 mm loading = ll,l,-ll • • 500 1000 1500 x (|j.m) Figure 4.8 Plot of CFD profile for pure mode I loading of mx4 2112 upon removal of mixed mode condition. Note that as expected local opening is induced, GUL GUG, and au > «/. 77 Chapter 4 Experimental Results 90 80 70 6.0 CN E 50 Is 40 30 20 10 0 'IIG N \ N \ \ -d-mx4 2121 ^ \ \ \ \ \ \ \ \ \ \ t Figure 4.9 Schematic of test mx4 2121 loading path for plots presented in Figure 4.10 to Figure 4.12. The load was held constant at each point to allow for slow scan imaging in the S E M . 7 -i 6 -5 -LL o o -te -1 = 0 J/nV • G I G — G| L = 10 J/m x GlIG G|IL 85 J/m' G|IG x x ^ x CSD COD 0 500 1000 1500 x (um) Figure 4.10 Plot of CFD profde for pure mode II loading of mx4 2121 before application of mixed mode condition. Note that as expected some local opening is induced, GUL = GUG, and an > ah 78 Chapter 4 Experimental Results 12 -I 10 -8 -• G | G = 70 J/m' — G | L = G , G x GUG = 85 J/m GUL = GIIG COD CSD 1500 x (jam) Figure 4.11 Plot of CFD profile for mixed mode loading of mx4 2121. Note that GIL = GIG, Gm = GUG, and au > a,. x (\xm) Figure 4.12 Plot of CFD profile for pure mode I loading of mx4 2121 upon removal of mixed mode condition. As expected GIL = GIG and no appreciable shear displacements are induced. 7 9 Chapter 4 Experimental Results 100 -| 90 -80 -70 «C- 60 -E 3 50 -j o O 40 -30 20 -10 0 i 'IIG / \ ' X / x / / / ' G / \ \ \ \ / / / \ \ \ \ \ \ \ mx4 12 \ \ t Figure 4.13 Schematic of test mx412 loading path for plot presented in Figure 4.14. The load was held constant at each point to allow for slow scan imaging in the S E M . x (|im) Figure 4.14 Plot of CFD profile for mixed mode loading of mx412. Note that G,L > GT GUL = GUG, and au > ai. 80 Chapter 4 Experimental Results 100 -j 90 -80 -70 c<T 60 -E 3 50 -CD O 40-| 30 20 / / / s \ / \ \ • \ / / 10 -I / , / 0 / f G IIG \ \ mx4 1221 Figure 4.15 Schematic of test mx41221 loading path for plots presented in Figure 4.16 to Figure 4.18. The load was held constant at each point to allow for slow scan imaging in the S E M . 12 -j 10 -8 -GIG = 73 J / m 2 G , L = 73 J / m 2 GIIG = 0 J / m 2 COD 1500 x (|tim) Figure 4.16 Plot of CFD profde for pure mode I loading of mx41221 before application of mixed mode condition. As expected GIL = GIG and no appreciable shear displacements are induced. 81 Chapter 4 Experimental Results 16 -i x (\im) Figure 4.17 Plot of CFD profile for mixed mode loading of mx41221. Note that GIL > GrG, GnL = GUG, and au > a,. G,L is also greater than pure mode GIC. 12 -| 10 -8 -GIG = 53 Jim" G , L = 53 J/m2 GIIG = 0 J/m2 1500 x (jim) Figure 4.18 Plot of CFD profile for pure mode I loading of mx4 1221 upon removal of mixed mode condition. As expected G1L = GIG and no appreciable shear displacements are induced. 82 Chapter 4 Experimental Results 140 - i 120 -100 -ME 80 H 0 ° 60 H 40 20 0 c  / ^ / / G \ 'IIG / \ \ / / / \ \ \ mx3 1221 \ -m-Figure 4.19 Schematic of test mx3 1221 loading path for plot presented in Figure 4.20. The load was held constant at each point to allow for slow scan imaging in the S E M . E 3. Q LL o -300 1 1 OO CD T— T— • G , G — G I L 14 - X G|| G 12 - — G|| L 10 -8 -6 -4 -2 -110 J/m" 165 J/m 2 • 62 J/m 2 ; G| | G COD mx3 1221 a = 23.8 mm loading = I,II 1500 Figure 4.20 Plot of CFD profile for mixed mode loading of mx3 1221. Note that also for a shorter crack GIL > GlG, Gm = GI{G, and an > at. GIL is also greater than pure mode G/c-83 Chapter 4 Experimental Results 90 80 H 70 60 C M E 50 -5 "o 40 O 30 20 10 0 A G,G / / \ / / / / / / / / \ \ \ x G,,~ ^ 'IIG / \ \ \ \ \ / mx3 12 \ t Figure 4.21 Schematic of test mx3 12 loading path for plot presented in Figure 4.21. The load was held constant at each point to allow for slow scan imaging in the S E M . E 3. Q u_ O -300 16 -i 14 -12 -10 -8 -6 -4 -2 GIG = 80 J/m G , L = 140 J/m G | | G = 50 J/m GIIG = GIIL CSD x x mx3 12 a = 23.8 mm loading = 1500 Figure 4.22 Plot of CFD profile for mixed mode loading of mx3 12. Note that also for a shorter crack Gn > G/G, Gm = G//c, and a// > a/. G/L is also greater than pure mode G/c. 84 Chapter 4 Experimental Results o 120 100 80 60 H 40 20 H 0 ^ ~ G , G / *w(117) B — H - H - B - H - f S - \ H •w(97) \ \ \ GUG ' • •w(67) lw(55) w(42) w(85) • w(u67) \ / / / / / w(0) \ mx4 w t Figure 4.23 Schematic of test mx4 w loading path for plots presented in Figure 4.24 to Figure 4.31. The load was held constant at each point to allow for slow scan imaging in the S E M . x ( f i m ) Figure 4.24 Plot of CFD profile for pure mode I wedge loading of mx4 w(0) before application of mixed mode condition. As expected GTL = GIG and no appreciable shear displacement are induced. 85 Chapter 4 Experimental Results 14 -, . G| G = 105 J/m 2 — G| L = 125 J/m 2 G M G = 42 J/m 2 GIIL = GIIG 10 A COD CSD mx4 w(42) a = 27 mm loading = I,II 1500 x (|im) Figure 4.25 Plot of CFD profile for mixed mode loading of mx4 w(42). Note that also for wedge opening GIL > GIG, GUL = GI[G, and an > a,. mx4 w(55) a = 27 mm loading = I,II 1500 x (fim) Figure 4.26 Plot of CFD profile for mixed mode loading of mx4 w(55). Note that also for wedge opening G/L > G/G, GI/L = GUG, and an > a,. GIL is also greater than pure mode G / o 86 Chapter 4 Experimental Results 12 i 10 -8 -| 6 J a G , G = 105 J/m2 — G | L = 135 J/m2 G M G = 67 J/m2 GIIL ~ G M G COD x CSD mx4 w(67) a = 27 mm loading = I,II 1500 x (|um) Figure 4.27 Plot of CFD profile for mixed mode loading of mx4 w(67). Note that also for wedge opening GJL > G,G, G„L = G,/G, and an > af. GIL is also greater than pure mode Gic. • G,G= 105 J/m2 j) 500 1000 1500 x (\im) Figure 4.28 Plot of CFD profile for mixed mode loading of mx4 w(85). Note that also for wedge loading G,L > G,G, G„L = G,IG, and an > ah GLL is also greater than pure mode GIC. 87 Chapter 4 Experimental Results 14 12 10 ~ 8 E o 4 2 0 -2 -• G|G = 105J/m2 — G|L = 135 J/m2 x G M G = 97J/m2 — G||L = G M G COD CSD mx4 w(97) a = 27 mm loading = I,II 500 1000 1500 x (|j.m) Figure 4.29 Plot of CFD profile for mixed mode loading of mx4 w(97). Note that also for wedge loading GIL > GIG, GIIL = GIIG, and an > ah GIL is also greater than pure mode G/c. 16 14 12 10 - 2 * D G|G = 105J/m2 — G|L = 155 J/m2 x G„G = 117J/m2 = G M G QMSD ° COD CSD x< mx4 w(117) a = 27 mm loading = I,II 500 1000 1500 x (|j.m) Figure 4.30 Plot of CFD profile for mixed mode loading of mx4 w(ll 7). Note that also for wedge opening GIL > G/G, G/lL = GJ/G, and au > ah G,L is also greater than pure mode G/c. 88 Chapter 4 Experimental Results 14 -• G | G 12 - — G , L 10 -X GIIG GIIL ~ 8 -E LL ° 4 -2 -0 -( -2 J COD CSD x a = 27 mm loading = I,II 500 1000 1500 x ( f i m ) Figure 4.31 Plot of CFD profile for mixed mode loading of mx4 w(u67). Note that also for wedge opening G!L > GlG, Gm = G„G, and an > a,. GIL is also greater than pure mode G, 120 -j 100 -80 -I N E 3 60 -j o O 40 -\ 20 0^ ^ I G / \ / / / \ G M G \ / / / / / / mx4 rev 1221 t Figure 4.32 Schematic of test mx4 rev 1221 loading path for plots presented in Figure 4.33 to Figure 4.35. The load was held constant at each point to allow for slow scan imaging in the S E M . 89 Chapter 4 Experimental Results x (fim) Figure 4.33 Plot of CFD profile for pure mode I loading of mx4 rev 1221 before application of mixed mode condition. As expected GIL = GLG and no appreciable shear displacement are induced. E Q LL O -300 18 i 16 -14 -12 -• G l G = 110J/m 2 — G| L = 160 J/m 2 x G||G = 75 J/m 2 — G||L = G||G COD mx4 rev 1221 a = 27.2 mm loading = I,II 1500 Figure 4.34 Plot of CFD profile for mixed mode loading ofmx4 rev 1221. Note that even when shear loaded in the opposite direction G,L > GIG, G/IL = GIIG, and au > a{. GIL is also greater than pure mode G/C. 90 Chapter 4 Experimental Results 14 -12 10 A £ Q LL O -300 • G|G = 85 J/m — G|L = G|G G„G = 0 J/m2 C O D mx4 rev 1221 a = 27.2 mm loading = I,II,-I 1500 Figure 4.35 Plot of CFD profile for pure mode I loading of mx4 rev 1221 upon removal of mixed mode condition. A s expected GlL = GIG and no appreciable shear displacements are induced. 90 i 80 -70 60 E 50 fj 40 O 30 20 10 A o G IIG H / \ / / / ' G \ \ IG / / / / mx4 21 up t Figure 4.36 Schematic of test mx4 21up loading path for plots presented in Figure 4.37 and Figure 4.38. The load was held constant at each point to allow for slow scan imaging in the S E M . 91 Chapter 4 Experimental Results T 3 CO O CO O o CD D) ro c Oi o CU i-ro Q. 100 -j 90 -I 80 70 -60 -50 40 30 20 10 H 0 mode II both arms mixed mode / \ upper arm only \ \ lower arm mx4 21 up \ Figure 4.37 Schematic of the load distribution in the upper a rm and lower a rm for test mx4 21 up. A s with most applied pure shear loads, both arms are equally loaded ini t ial ly. As the arms are opened dur ing mode I loading the upper arm begins to carry a larger portion of the total load, as per normal . O f note in this test is that the mixed mode condition dur ing p r imary data acquisition is set when the lower a rm carries zero load (f= 1). x (p,m) Figure 4.38 Plot of CFD profile for mixed mode loading of mx4 21up w h e r e / = 1. Note that for this case also GIL > G/G, G„L = GnG, and an > ah G/L is also greater than pure mode G[G. 92 Chapter 4 Experimental Results 160 -j 140 -120 -_ 100 -CM E 3 80 C3 ° 6 o ^ 40 20 0 / / / ( e ) , /^^^ crack {dy ^ / growth 'NG /(a) / /Kb) \ / \ 'IG / / \ / / / \ \ \ \ mx4 cp t Figure 4.39 Schematic of test mx4 cp loading path for plots presented in Figure 4.40 to Figure 4.44. The load was held constant at each point pr ior to crack growth to allow for slow scan imaging in the S E M . C r a c k growth data point was analysed using video images. E Q u_ O 7 -j 6 -5 4 3 H -300 • G|Q = 0 J/m — G|L = 5 J/m2 x G||G = 95 J/m2 ¥S8 | | L ^m 11 G x / x x CSD COD Cs i^tftaffiWnfcnfe] m x 4 c p • a = 27.2 mm loading = II 300 600 x (jim) 900 1200 1500 Figure 4.40 Plot of CFD profde for pure mode II loading of mx4 cp(a) before application of mixed mode condition. Note that as expected some local opening is induced, G/IL = GUG, and an > af. 93 Chapter 4 Experimental Results 14 -, x (|um) Figure 4.41 Plot of CFD profile for mixed mode loading of mx4 cp(b). Note that G,L > Gt GUL = Guc, and au > ah ~ • i i 1 -300 0 300 600 900 x ( u m ) Figure 4.42 Plot of CFD profile for mixed mode loading of mx4 cp(c). Note that G1L > GK GUL = GUG, and an > af. GFL is also greater than pure mode G!C. 94 Chapter 4 Experimental Results E 3. G LL O -300 20 -18 - • G | G 16 - — G I L 14 -12 -x GlIG G| | L 10 -8 -6 -4 -120 J/m2 180 J/m2 = 95 J/m2 : GIIG 1500 Figure 4.43 Plot of CFD profile for mixed mode loading of mx4 cp(d). Note that GIL > GLG, GUL = GUG, and «// > a,. G/L is also greater than pure mode G1C. In addition, significant subcrit ical mode I crack extension (ai) of 150 pm has occurred. -100 0 100 200 300 400 500 x (|um) Figure 4.44 Plot of CFD profile taken from video for mixed mode loading of mx4 cp(e). Note that just pr ior to failure GIL > G,G and that both GLL and GJG are greater than pure mode G/c. 95 Chapter 4 Experimental Results 120 -| 100 -80 60 -40 -20 0 / / growth (a) A'*^ >^ crack \ \ / (d) GIG / / / ( b ) / / ( O / / mx5 cp \ v Figure 4.45 Schematic of test mx5 cp (from insert) loading path for plots presented in Figure 4.46 to Figure 4.49. The load was held constant at each point p r io r to crack growth to allow for slow scan imaging in the S E M . 8 -, 7 6 -5 -4 3 2 E Q LL O -300 1 -1 -> GIG = 0 J/m2 G , L = 15 J/m2 GIIG = 88 J/m2 G||L = GIIG X x * # . C S D x< • • 300 COD mx5 cp (from insert) a = 21.2 mm loading =ll 1200 1500 Figure 4.46 Plot of CFD profile for pure mode II loading of test mx5 cp(a) before application of mixed mode condition. Note that as expected some local opening was induced, GIIL = GUG, and an > ah 9 6 Chapter 4 Experimental Results x (p,m) Figure 4.47 Plot of CFD profile for mixed mode loading of test mx5 cp(b). Note that GIL > GIG, GUL = GuC, and an > ah Gu is also greater than pure mode G,c for a crack loaded from the insert. 12 -10 8 I 6 Q u . 4 -2 -0 -2 -• G|G = 65J/m — G,L = 150 J/m2 x G||G = 95J/m2 G||L = G||G -x'r i x 500 C O D — — C S D mx5 cp (from insert) a = 21.2 mm loading =ll,l 1000 1500 x (|um) Figure 4.48 Plot of CFD profile for mixed mode loading of test mx5 cp(b). Note that G/L > GIG, GUL - Gi/G, and an > au GtL is also greater than pure mode G / c for a crack loaded from the insert. 9 7 Chapter 4 Experimental Results 12 10 8 I 6 • G|G = 90 J/m — G|L = 180 J/m2 x G||G = 95 J/m — G|,L = G||G C O D x x * xx x C S D mx5 cp (from insert) a = 21.2 mm loading =ll,l 1500 x (um) Figure 4.49 Plot of CFD profile for mixed mode loading of test mx5 cp(b). Note that G,L > GIG, GUL = GNG, and au > ah G,L is also greater than pure mode G{C for a crack loaded from the insert. 90 80 70 60 E 50 CD 40 < 30 20 10 0 x mx3 • mx4 o mx5 15 O 35 O o X 55 75 G,G (J/m2) X —i— 95 115 135 Figure 4.50 Plot of AGi vs. (7/G for a l l mixed mode conditions examined. There appears to exist general trends wi th in specimens mx4 and mx5 of increased AGi wi th increasing C / o 98 Chapter 4 Experimental Results 90 80 70 60 50 40 30 20 10 0 x mx3 • mx4 O mx5 40 o X )P — i — 60 80 G„ (J/m2) 100 120 Figure 4.51 Plot of AG/ vs. G// for all mixed mode conditions examined. No relationship between the variables is apparent. 260 -I ! ' • 240 - ' ! 220 - j 200 - • | 180 - • 160 -140 - G|C • • • ••• i • 120 -100 - • . . . - " ' G I G - GIL ] 80 -60 -I Gic 1 1 1 - -T mx3 mx4 i 50 70 90 110 130 150 G,G (J/m2) Figure 4.52 Plot showing Gn vs. G/G for mixed mode data from specimens mx3 and mx4. Note that Gn regularly exceeds both G/c and pure mode G/c- Also G/G under mixed mode loading can also exceed G/c. 99 Chapter 4 Experimental Results G|L:G||L (J/m2) Test Name mode I crack tip • 105:0 mx4 w(0) mode II crack tip 125:42 mx4 w(42) • A 135:55 mx4 w(55) * -A 135:67 mx4 w(67) • A 135:85 mx4 w(85) • A -=ZZ 135:97 mx4 w(97) • A 155:117 mx4 w(117) • A 135:67 mx4 w(u67) • - A 0 100 200 300 x (|am) 400 500 Figure 4.53 Plots showing progression of mode I and mode II crack tips during loading of mx4 w. Movement of both crack tips appears independent and irreversible. 100 5 Analysis and Discussion Chapter 5 Analysis and Discussion The experimental results and observations from Chapter 4 wi l l be examined in the context of the objectives set out previously. In addition to an analysis of the effects of specimen load history, test load path, and the crack condition, the insight the results provide regarding the general mixed mode delamination behaviour of the material w i l l be discussed in this chapter. 5.1 Specimen Load History A s discussed in section 4.1, one of the objectives of this investigation was to determine i f the loading history of the specimen affected its response to further loading. The loading history of specimen mx4 was tracked and the order of loading is presented in Table 4.2. From examination of the images and from the local analysis, it was clear that loading caused damage and subcritical crack growth in some cases. The nature of that damage zone wi l l be discussed below as w i l l its effect on the specimen response to loading. 5.1.1 Damage Zone One of the key observations was the difference in the mode I and mode II crack lengths. The difference in the crack lengths, Aa, is present in all cases where local mixed mode conditions exist. Paris' (1998) results also show that the locations of the mode I and mode II crack tips differ under local mixed mode conditions. In her results, the distance between the two crack tips, Aa, tended to increase as GUG increased. At higher shear loads, the damage zone created in front of the crack tip resulted in Aa being several hundred microns long. The shear loads applied in the tests conducted here are lower than the Guc typically applied by Paris (1998) during her pure mode II investigation. Even at these low load levels though, the results from the CFD profiles indicate that a small damage zone does exist. More importantly, the nature of this damage zone is such that shear displacements are allowed while opening displacements are inhibited. This is consistent with Paris' result for much higher shear loads. Figure 4.53 and Figure 5.1 show the relative locations of a\ and au obtained from the CFD profiles presented previously for tests mx4 w (wedge opening) and mx4 cp (crack propagation) respectively. The schematics in these two figures are typical of similar data from other tests. In the case of Figure 4.53, test mx4 w is loaded in constant G/c by a wedge. Subcritical mode I crack growth however, still occurs under increasing shear loading. This is possibly due to the increased local opening loads in addition to the creation of a damage zone in front of the mode I 102 Chapter 5 Analysis and Discussion crack tip as the shear load is applied. A s would be expected, the mode II crack tip grows under increased shear loading and indicates an increasing damage zone. Test mx4 cp, on the other hand, is first loaded in constant mode II. A s the mode I component is applied we see a distinct jump in the location of the mode I crack tip. However, the mixed mode conditions also result in a small increase in the mode II crack length despite the constant shear load applied. Figure 5.2 is a photomicrograph of mx4 w(117) and shows that a distinct zone does indeed exist between ai and an where no COD is present while the shear load creates two distinct crack surfaces. The mode I and mode II crack tip locations identified in the figure were determined from analysis of the CFD profile. Behind the mode I crack tip at (A), there is a distinct shear and opening displacement discontinuity across the crack plane. Ahead of the mode I crack tip there exists a zone described above where at (B) only a shear displacement discontinuity across the crack plane exists. The mode II crack tip location as determined from the CFD profiles appears to be ahead of the crack tip located from the image at (C) where no displacement discontinuities exist across the crack plane. Despite the slight discrepancy in the location of the mode II crack tip, a zone clearly exists where CSD is present and COD is inhibited. 5.1.2 Loading of Specimen With Crack Tip Damage A s was noted in the results section, all pure mode results obtained here were as expected and consistent with Paris' (1998) results. This was true regardless of whether the pure mode condition existed before mixed mode loading was applied or upon removal of the mixed mode condition. Figure 5.3 shows all the pure mode I loads applied to specimen mx4 throughout testing. The data labels indicate the order of loading of specimen mx4 as presented in the last column of Table 4.2. It is evident that the first load applied to the specimen behaves similar to later pure mode I load applications. A s would be expected from Paris' data for brittle unidirectional C F R P systems without fibre bridging, in pure mode I loading, the global applied tensile load is transferred fully and directly to the crack tip (Paris et al, 2001). Specimen response to global shear loading is also consistent with expected behaviour throughout testing. In all cases, the global applied shear load is fully transferred directly to the crack tip (Figure 5.4). The induced mode I local component due to the surface interactions of the crack 103 Chapter 5 Analysis and Discussion faces under pure shear load application is also consistent with Paris' (1998) results that indicate a maximum induced GIL of 10 J/m under the mode II loads applied here. Throughout the course of loading specimen mx4, changes in local crack tip conditions as described in the previous section, though significant, did not affect the pure mode specimen response. Specifically, damage caused by tensile, shear, or mixed mode loading, with maximum 2 2 loads of GIL = 160 J/m* and G„L =117 J/iri , was insufficient to affect the specimen response to subsequent applied pure mode loads as Figure 5.3 and Figure 5.4 show. Since the pure mode response is unaffected, then any observed irregularities under mixed mode conditions such as the higher local mode I component than the applied global mode I component should be due to specific mixed mode interactions. 5.2 Test Load Path A s has been discussed previously in Section 2.2.1, the M M B test method allows for simultaneous, proportional mode I and mode. II loading of a D C B specimen. While the j i g can be used to perform pure mode testing, it can only apply mixed mode loads with a constant mode I to mode II ratio. Under the M M B method it is not possible to sequentially load a specimen first in either pure mode I or pure mode II, followed by applying further load to impose a mixed mode condition. With the asymmetric D C B methodology employed here a full complement of loading and unloading paths is possible. Several tests were conducted as outlined in Section 4.3 that allowed for investigation into the effect various loading paths might have on specimen response. Proportional loads were applied similar to load application in the M M B test but were also applied to specimens under pre-existing pure mode conditions. Specimens were also loaded sequentially where the two modes were applied separately to cause mode mixity. Unloading of specimens along various paths was also explored. Examination of the results indicates that there is no specimen response path dependence at the load levels considered in this work. Regardless of how a mixed mode condition was reached, the specimen response to that condition was consistent. Global shear loads were always transferred directly to the crack tip as can be seen in Table 4.2 where AGu is always zero while local crack opening displacement were, in all cases but one, greater than the applied load would predict. In 104 Chapter 5 Analysis and Discussion the one case where Gn was not greater than G/G, AG/ was zero suggesting no crack tip shielding takes place. While the mixed mode specimen response is consistent, the differences between local and global mode I components, AG/, is quite variable. A s shown in Figure 5.5, each load path examined displays a range of values of AG/. The only loading path that stands out as potentially being different is the II/I path. This path has a larger range of AG/ values associated with it including the minimum and maximum values. However, this loading path was also the most heavily tested and is the only path that contains results from the mx5 data set. This is significant to note as the Figure 4.50 shows that the mx5 specimen yields consistently higher AG/ values than specimens mx3 and mx4. The lack of a clear loading path effect, at least at the load levels examined here, suggests that the results produced by any test method and by any loading path can be used to consider what transpires in realistic situations where the loading path is not only variable but also unknown. 5.3 Crack Conditions The main objective of this section is to investigate the impact that crack conditions such as crack length or the nature of the crack surface have on specimen response to mixed mode loading. From Section 4.5, Figure 4.50 showed that mx3 and mx4 appeared to behave in a similar fashion with respect to AG/ while specimen mx5 appears to have consistently higher values of AG/. Most noticeably, these higher values of AG/ were present at lower values of G/c. If crack length were to be the primary reason for this difference in behaviour then some similar indication should be evident in the mx3 data set. However, mx3 data appears indistinguishable from the mx4 specimen behaviour. Consequently, crack length does not appear to be the main contributing factor to the differing mx5 response. Other than crack length, the main difference between the specimens is the location of the crack tip relative to the crack starter film. In the case of mx5 the crack tip coincides with the end of the starter film whereas with the other two specimens have crack tips approximately three and six millimetres away from the starter film. These cracks were created through mode I precracking. 105 Chapter 5 Analysis and Discussion 5.5.7 Effect of Starter Film on Subcritical Specimen Response To examine the effect of the starter film on the specimen response, two tests with similar loading paths, and both tested to crack propagation wi l l be directly compared. Figure 5.6 shows the response of mx4 cp and mx5 cp in terms of AG] as a function of G / G . Both tests were loaded first in shear before the mode I component was introduced. Comparison of these two tests, therefore, also allows an examination of the relationship between AG] and G / G under constant shear load. A s summarized by Brunner (2001), the starter film or insert can play a significant and undecided role in the actual fracture of specimens loaded directly from it. However, the role the insert plays in the global/local interaction of shear and tensile mode loading is not well understood. Under similar shear loads, the crack surface roughness results in an induced local COD profile 2 2 equivalent to a G / of only 5 J/m in the mode I precracked mx4 cp(a) test and 15 J/m in the starter film mx5 cp(a) test. In her mode II testing, Paris (1998) noted the opposite effect; that mode II tests from the insert resulted in lower induced local opening components than tests conducted from precracks. These contradicting results suggest that the insert or the precrack do not result in unique specimen global/local behaviours under shear load. The induced local opening is, strictly speaking, due to the surface characteristics of the crack faces and is not associated directly with the origin of those surface characteristics. From this work and Paris' (1998) results it does not appear that any general statement can be made relating the macroscopic crack condition to the roughness that induces local opening. Indeed, the physical amount of local opening that is induced is on the order of 1 - 2 microns approximately 1 mm from the crack tip, and its origin need not be directly related to the macroscopic crack condition. Essentially, the primary difference between the two specimens is the crack surface roughness that under similar shear loads in one case, mx4, induces a local opening of 5 J/m 2 and in the other case, mx5, induces a local opening of 15 J/m . The role the insert plays is, however, of considerable importance when analyzing testing methodologies and the mixed results from Paris' (1998) work and this work suggest that further study in this area is required. 106 Chapter 5 Analysis and Discussion 5.3.2 Effect of Crack Surface Roughness on AGj The effect of the roughness described above on pure shear applied loading is shown in Figure 4.40 (mx4 cp(a)) and Figure 4.46 (mx5 cp(a)). Whether a similar phenomenon exists in mixed mode loading wi l l be examined in this section. In order to determine the effect of the crack surface roughness on mixed mode behaviour, it is necessary to examine the nature of Gn in more detail. It is known from pure mode I analysis that in cases where no energy absorption mechanisms such as fibre bridging exist, all the applied opening load, G / G , is transferred fully to the crack tip (Paris et al, 2001). The mixed mode crack opening response of a specimen can then be partitioned into two constituent opening sources. The first COD source is the globally applied pure mode I load, G/c, which is fully transferred to the crack tip. The second COD source constitutes the remainder of the mixed mode crack opening displacement profile and manifests itself as AG/. Test mx4 w (Figure 5.7) showed that even with a specimen that was wedged open, and therefore had a constant global displacement profile during shear loading, a AG/ developed as the mixed mode condition grew. Consequently, this second displacement source must be generated within the specimen as no further external opening loads were applied. Figure 5.8 shows a graphical representation of the mixed mode COD profile partitioning described above. The decomposition is performed on mx4 cp(d) which has a AG; of 60 J /m 2 for 2 • 2 an applied load of GIG'-GUG of 120:95 J/m . The externally applied component, G / G - 120 J/m , is shown in Figure 5.8(b). The internally generated AG/ component of 60 J /m 2 has a COD profile that can conveniently be described by an equivalent AG/, in this case equivalent to 6 J /m 2 (Figure 5.8(c)). The equivalent AG/ is strictly a convenient parameter that can be used to describe the displacement profile that causes AGi and is not an energy release rate component in and of itself. There are two likely sources of this internal loading that causes the mixed mode AG;. One possibility is that loose particles or even fibres act as internal microwedges. These internal wedges could easily be created as the crack surfaces move relative to each other under shear loading. Examining Figure 5.8(c), which represents the internal displacement source, the size of 107 Chapter 5 Analysis and Discussion these wedges would have to be very small. For example a particle 1 micron in size located approximately 200 microns behind the crack tip would be sufficient to account for the 60 J /m 2 AGi observed. Similarly, a displaced fibre 8 microns in diameter could also act as wedge several millimetres behind the crack tip and have the same effect. It is also possible that the same crack surface interactions that induce a Gn under an applied pure shear load also have an effect in mixed mode situations. A s can be seen from the COD profiles, the actual applied COD is small, approximately 8 microns at a distance of 500 microns from the crack tip. The actual internal displacement component required to account for AGj is significantly smaller as was just described above. It seems quite reasonable, given the small CODs generated by the applied global load, that the amount of applied opening displacement is insufficient to negate the interactions due to crack surface roughness as the crack faces slide over each other. To summarize Figure 5.8, G/L can be partitioned into two constituent sources of displacement. The first is the globally applied displacement, equivalent to the displacement associated with the applied G / G . The second is the internally generated displacement that causes AG/. The displacement profile required to cause a AGi can be defined by the equivalent AG{. Strain energy release rates are not additive but displacements, that correspond to V G can be added. However, the equivalent AGi provides a very convenient description of the internal displacements that are required to account for the observed AGi. Figure 5.9, Figure 5.10, and Figure 5.11 show the same mixed mode G/L partitioning as Figure 5.8 for tests mx4 cp(c), mx5 cp(c), and mx5 cp(d) respectively. Similar G / partitioning was performed on all mx4 and mx5 mixed mode conditions. Table 5.1 provides a summary of these results and shows G / G , G/L, the actual AG/ observed, and the equivalent AG/ that would cause the observed AG/. It is evident from this table that the two specimens do indeed have unique behaviours. With the exception of mx4 21 up the equivalent AG/ values for the remainder of the mx4 mixed mode loadings are similar in magnitude (0 - 6.5 J/m ). The equivalent AG/ values determined for mx5 are an order of magnitude greater than the results for mx4 and are also self consistent (15.5 - 20 J/m 2). The equivalent AG/ values for each test are also consistent with the induced G/L under an applied pure shear load for the same specimen (mx4 cp(a) (5 J/m )) and 108 Chapter 5 Analysis and Discussion mx5 cp (a) (15 J/m )). This result would strongly suggest that the same surface roughness that induces G/L under pure shear loading is also responsible for the increased opening displacement that causes the mixed mode AG/. The unusually large equivalent AG/ for mx4 21 up could be due to the specific loading situation for this test. The main difference between this test and the others is the load in the lower arm. In all the other tests conducted there was always some load pushing the lower arm up, essentially resisting the opening mode. In the absence of such an opposing load in the mx4 21 up loading configuration it is possible that the roughness causes a larger local opening. 5.3.3 Effect of Crack Surface Roughness on the AGT - GIG Relationship Test mx5 cp, which has a crack surface containing a more interactive profile, shows a specimen response that in general includes higher AG/S than those observed in test mx4 cp (Figure 5.6). Also, these higher AG/S take place at relatively low values of G/c- Both data sets indicate that a relationship exists between higher G/c values and higher AG/ values. However, the nature and progression of these relationships differs as is highlighted in Figure 5.6. The relationship between the roughness and the resultant behaviour w i l l be examined below. A s was mentioned previously, strain energy release rates are not additive. Rather it is the physical displacements from the two mode I sources that are superposed to generate the mixed mode COD profile. Consequently, the relationship between G/ and COD needs to be examined. Equation 3.15 shows that G/ is a function of COD . Therefore, a COD profile, that in isolation is 2 2 equivalent to a G/ of 6 J/m as in mx4 cp(d), results in a AG/ of 60 J/m when that displacement is added to the displacement from an externally applied G/G of 120 J/m . It is evident that for the similar internally generated displacement fields that appear to exist within a specimen, a larger global displacement contribution wi l l result in a larger AG/. It is important to note here that under this theory, the internally generated displacements are expected to be similar only for a unique crack, and are not specimen properties. If the crack grows the growth wi l l result in new crack surfaces and those crack surfaces wi l l have a unique profile that may change the magnitudes of the internally generated displacements under shear load. For example, specimen mx4 was also fabricated with an insert but the crack was grown from that 109 Chapter 5 Analysis and Discussion insert before testing and the new crack behaved differently. In the tests conducted here, the specimens were all tested at a unique crack length and the generalization to a specimen is in fact a reference to a unique crack condition. It should be noted that the convexity and concavity of the curves shown in Figure 5.6 is a result of the varying equivalent AG/ that develops during loading. In the case of mx5 cp these values decrease with further loading while they increase in the case of mx4 cp. If the internal displacement profile were constant at an equivalent AG/ of 5 J/m for example, then the relationship between Gn and G/c would be similar to that shown in Figure 5.12. 5.3.4 Transfer of Global and Internal Loads to the Crack Tip In light of the above discussion regarding the origin of AG/, this section examines the effect of local perturbations such as crack surface roughness on the transmission of the global applied load to the local crack tip. Pure mode global loading is examined first followed by an examination of mixed mode loading. Previously, specimen loading has been defined as normal forces applied to the specimen ends. To facilitate the following discussion, global loading w i l l be is considered through applied moments. The advantage of considering moments is that they can be transferred simply to any position along the specimen length at a constant value. Figure 5.13(a) describes, in a general sense, G as a function of applied moments. The partitioning of the applied moments into mode I and mode II components is similar to the normal force partitioning described in Section 3.3.2 (Reeder, 1991; Will iams, 1988). Figure 5.13(b) shows schematically how the global analysis determines the local crack tip conditions. In the global analysis the crack is essentially assumed to be perfect in that no internal mechanisms that may affect the strain energy of the system are considered. The global analyses typically performed and recommended inherently assume that all the global loads applied are transferred fully and directly to the crack tip. 5.3.4.1 Pure Mode I Load ing Figure 5.14 shows schematically the transfer of the global load to the crack tip under pure mode I conditions. With no local effects such as fibre bridging the global moments in each arm are 110 Chapter 5 Analysis and Discussion transferred directly to the crack tip. A s a result the crack tip sees the same load that is applied externally to the specimen. In this case, the experimental evidence shows that the local and global evaluations are equivalent. A slightly more complex loading situation is described in Figure 5.15. In the case of mode I loading where some fibre bridging exists across the crack, the crack tip sees less of an opening load than is applied externally to the specimen. B y modeling the fibre bridging with an internal moment, M f , the analysis shows that the expected specimen behaviour should result in Gn less than G / G . It is not possible, however, to quantify a priori the magnitude of the fibre bridging effect and therefore quantitatively predict the ultimate specimen behaviour. 5.3.4.2 Pure Mode II Load ing In the case of pure mode II applied loading, there exists an interaction between the crack faces as they slide over each other that causes local opening at the crack tip in addition to the local shearing. The proposed mechanism of crack surfaces interacting to provide an internal opening load can be modeled similar to fibre bridging but with an opposite load direction. Instead of the fibre bridges acting to close the crack tip the rough crack surfaces interact to push the crack tip open. Figure 5.16 shows schematically, the effect of this type of internal mechanism on the transfer of the externally applied load to the crack tip. A s is seen experimentally the analysis shows that the load due to the crack surface roughness, MA, allows the entire shear load to be transferred fully to the crack tip (GUL = Guc) while inducing a local mode I component that was not present globally. It is not possible, however, to quantify a priori the magnitude of the surface roughness and therefore quantitatively predict the ultimate specimen behaviour. 5.3.4.3 M i x e d Mode Load ing A s shown in Figure 5.17 the mixed mode loading case is analytically quite similar to the pure mode II case described above. The only difference between them is that a non-zero global opening component exists in the mixed mode case. Again, the crack surface roughness in the mixed mode situation allows for full transfer of the externally applied shear load to the crack tip. At the same time, this roughness also adds to the opening provided by the external loads that leads to the local condition of G/L exceeding G/G-111 Chapter 5 Analysis and Discussion It is important to note here that i f there is no prior knowledge of the type of internal mechanism taking place, such as fibre bridging or the crack surface roughness profile, it is impossible to predict how the global loads wi l l be transferred to the crack tip. If, however, some knowledge or hypothesis exists as to the nature of the internal mechanism, as in the simple analyses performed here, a reasonable qualitative expectation of the specimen global/local behaviour can be determined. Due to the random and variable nature of the interactions, a quantitative prediction is generally not possible. 5.3.5 Experimental Considerations The local analysis provides the real crack tip response of the specimen. Combined with the information from the more standard global analysis, this local examination provides significant insight into mixed mode specimen behaviour. The local technique, however, is costly. The equipment required for such testing and analysis is not readily available, and the testing and analysis itself is difficult and time consuming. The discussion below examines whether or not, in light of the existence of internal mechanisms, some form of global analysis alone can be used to determine the local crack tip response. 5 . 3 . 5 . 1 Curren t Globa l Data Reduction Practices The most common data reduction methods, for pure and mixed mode loading, including those recommended by A S T M D 5528, A S T M D 6671, and those employed in this work, all involve using Equation 1.2 in some capacity. In this equation G is determined by taking the rate of change of the elastic strain energy, U, with respect to the crack length, a. This methodology assumes that the external applied load describes completely the elastic strain energy of the system. If internal interactions exist that are not known or that cannot be accounted for, this technique no longer provides a similitude parameter capable of defining the local conditions at the crack tip. Consequently, the global determination of G is only truly indicative of crack tip conditions when there are no internal mechanisms at play during loading. To date, the only situation encountered where this is the case is in pure mode I loading of a specimen with no fibre bridging. In all other cases, the GG provides an inadequate and incomplete prediction of the conditions at the crack tip. 112 Chapter 5 Analysis and Discussion 5.3.5.2 Using Global Parameters to Predict Actual Local Conditions A n alternative measurement of G is based on an energy conservation argument. The strain energy released during crack growth can be calculated by measuring the difference between the work done on the specimen during loading up to crack growth and the work recovered after crack growth when the specimen is unloaded. This can only be accomplished under stable crack growth conditions and with data acquisition schemes that include both load and displacement data. Figure 2.16 shows schematically this technique for the general case. The enclosed area of the curve represents the net work done by external loads on the specimen during loading, crack growth, and unloading. Assuming that the only energy dissipating mechanism is crack growth then this enclosed area is equivalent to the total energy consumed during crack growth and is directly related to the critical strain energy release rate: energy used for crack growth G c = crack growth area (5 1) Since this technique actually measures the energy consumed during the fracture event, it does not suffer from many of the weaknesses of the classical methods that are more predictive in nature. Internal mechanisms such as elastic fibre bridging or crack surface roughness interactions are automatically accounted for in the energy calculation. Ideally, the energy consumed during fracture would be calculated directly from the primary data. In the experimental set-up used here, that primary data is the independently measure load and displacement of each arm. These parameters provide directly the work done on each arm before and after crack extension. In order to determine the amount of energy required for fracture in each mode individually, it is still necessary to partition the measured loads and displacements as described in Section 3.3.2. Appendix A shows that the sum of the total work done by each arm is equal to the sum of the total work done by each mode of loading given the partitioning of Equation 3.1 and Equation 3.2. Consequently the enclosed area of the P\-5i curve is directly related to the mode I component of the work done during cracking, and similarly for the mode II component. 113 Chapter 5 Analysis and Discussion The area method is widely used when introducing the concept of strain energy release rates (e.g. Hertzberg, 1996; Broek, 1986) but is rarely used experimentally. While theoretically there is no fundamental drawback to this G determination method since there are no simplifying assumptions made, there exist key experimental concerns that may limit the potential of this technique. The first drawback, is that this method only provides critical values of G since a crack must grow and the specimen fully unloaded for there to be any usable data. A s a result there is no indication of the true crack tip conditions during loading or unloading, only during the fracture event. Secondly, an accurate length of crack extension becomes extremely important. The calculation of Gc is directly related to the crack growth area (Equation 5.1). Therefore a 10% error in the crack growth area w i l l lead to a 10% error in Gc- While the crack width can be accurately calculated the typical methods for crack extension determination are relatively crude. The precise location of the crack tip is difficult to determine under low magnification and varies across the width of the specimen. The third difficulty with this technique deals with data acquisition. Figure 5.18 and Figure 5.19 show the mx5 cp global load-displacement data for mode I and mode II respectively. The data resolution, though acceptable for classical data reduction techniques, is insufficient to accurately measure the work done in fracture. The crack extension observed for mx5 cp was 700 microns. Given the local data obtained for Gnc and Gmc the total amount of energy used in the mode I and mode II components of fracture is approximately 2.75 and 1.31 N .mm respectively. Given the amount of energy in the system, these values correspond to an area equivalent to 0.3% of the area under the mode II load-displacement curves and of 5.5% of the area under the mode I load-displacement curve. In both cases, although especially in mode II (Figure 5.19), the degree of accuracy required of the raw data is not currently achievable. Finally, as is shown in Appendix A , the values of G/c and G//c determined from the global experimental data are significantly greater than those determined from the local data. In addition to the above factors, there are likely some system losses that occur during the test that are not accounted for in the analysis. The reduced accuracy of the mode I component of displacement due to the enlarged loading pin holes required to allow for free movement of the specimen also affects the energy calculation significantly. While this technique does initially show promise 114 Chapter 5 Analysis and Discussion when compared to other global methods (as the assumptions about the nature of the crack are minimal), there are major challenges that would need to be overcome before it can be used with confidence. 5.3.6 Variability Paris noted that the induced crack opening under global shear loading could account for some of the generally recognized and accepted variability in pure mode II fracture toughness values (Paris, 1998). The proposed sources of the internally generated displacements, crack surface roughness and other internal wedges, may provide cause for much of the variability that is seen in mixed mode data sets in literature (e.g. Figure 2.17). The variable nature of the interactions results in varying local conditions for similar global loads. It is possible that local values of G/c may show less variability than their global counterparts and this is an area that requires future examination. Theoretically, i f a specimen could be produced with initially perfectly flat crack faces, then a reduction in variability should be observed. However, cracks in real applications would always have some surface waviness, as would any cracks grown from the above theoretical specimen. Indeed a greater understanding is required of the exact nature of the surface roughness especially in terms of typical size and distribution of the profile as well as an understanding of the differences between the roughness found in laboratory specimen delaminations and those in real structures. 5.4 Increased Mode I Capacity Under Mixed Mode Loading A plausible hypothesis as to the origin of the increased mode I load carrying capability of the specimen under mixed mode conditions would have been that the introduction of Guo somehow depresses local opening. Therefore, additional global mode I load application would be required for local conditions to meet pure mode I fracture conditions similar to the fibre bridging effect. A s the experimental results have demonstrated though, this is not the case. Rather, the material actually has a capability to withstand much greater local levels of Gn than are suggested by the global analysis alone. Figure 5.20 and Figure 5.21 show the relationship between G/L, G/G, and G/c for the two tests conducted to crack growth, mx4 cp and mx5 cp. O f interest from these two plots is the absolute 115 Chapter 5 Analysis and Discussion magnitude of Gn relative to G/c at failure. In both cases the Gn is on the order of twice the pure mode G/c-A consensus appears to exist in the literature as to the cause of elevated G/cc under mixed mode loading of these materials. Among others, Reeder suggests that post failure fractographic evidence indicates that with the addition of shear loading, what is a tensile failure under pure mode I conditions quickly transitions to a shear dominated failure. In his analysis, by the time a GIG'.GUG ratio of 1:1 exists in loading, the resulting crack surfaces created are similar to those created under more dominant globally applied shear loads (Reeder, 1993). Section 2.2.3.2 discusses this issue further. A s a result of this transition to shear dominated crack propagation, greater energy is required for crack growth and the material can withstand larger opening loads near the crack tip. It was noted previously that the cracks grew in a self similar manner in the tests conducted here. This type of mixed mode crack extension is possible in certain layered, heterogeneous, orthotropic materials. In homogenous, isotropic materials, such as common metals, tested under similar conditions as the tests conducted here, the mixed mode crack growth would not be self similar. Instead of extending along the specimen midplane, the crack would kink to a preferred direction. This preferred direction coincides with the direction in which K/j at the crack tip is zero or the direction in which the total G is maximized. These two directions are essentially equivalent (e.g. Nuismer, 1975). Fundamentally, the crack grows at an angle such that the crack tip experiences pure mode I conditions, or the lowest resistance to fracture. With layered materials, specifically unidirectional C F R P specimens, the crack does not grow at an angle to the specimen midplane but extends essentially straight ahead. The material heterogeneity and orthotropy are such that it is exceedingly difficult for the crack to grow out of plane. Physically, it is easier for the crack to extend straight ahead than to kink and break through fibres in the adjacent material. If the crack tip were initially located at the midplane of the resin rich region between the two central plies, then theoretically the crack should angle within that resin layer as the stress intensities would dictate. However, once the crack tip reaches the interface between the resin rich layer and the fibres, further growth in that direction would be inhibited by the increased 116 Chapter 5 Analysis and Discussion energy required to break the fibres. In the laminates used here, while the stress intensities continue to direct the crack tip into the fibre pack, the energy required to break the fibres is too great for any crack growth to occur in that direction. Consequently, the increased energy required to grow the crack at the interface as opposed to within the resin rich region alone would account for some of the increase in the ability of the material to withstand larger Gj. Figure 5.22 shows the crack path of specimen mx5 cp. The crack path described above is shown in this micrograph. The crack immediately deviates from the midplane of the specimen, stopping only when it reaches the adjacent fibre bed. The crack then grows directly along the interface between the resin rich region and the fibre region. This crack growth along the interface is a . much more complicated path then directly through the resin rich region and can account for some of the additional energy required for crack propagation compared to pure mode I crack growth. The fracture surface from this type of failure would be consistent with Reeder's post mortem analysis of low mode II mixed mode fracture surfaces (Reeder, 1993). The significantly higher than expected Guc values indicate that at a global mode I to mode II ratio of 1:1 the local ratio at failure is actually closer to 2:1. 5.5 Failure Envelopes A s was discussed in Section 2.2.3.1, there is general agreement in the literature that for brittle unidirectional C F R P laminates, addition of small amounts of shear load increases the materials ability to withstand opening loads. Consequently the condition of the mixed mode G/c exceeding G/c is generally accepted and the results obtained here for G/c values before failure are consistent with the literature. More importantly however, is that the mixed mode G/L values consistently and significantly exceed pure mode G/c. Under these circumstances G/c is merely one contributing opening source to the local crack tip condition. Therefore the mixed mode envelopes currently available in the literature which examine G/cc vs. G//GC are different in magnitude and maybe in shape than perhaps the more fundamental G/LC vs. G//LC envelopes. From the results of this work the G/LC V S . G/ILC plot should show a much greater rate of increase from pure mode I to low G// mixed mode conditions. When combined with the work done previously by Paris the results suggest a local failure envelope as shown in Figure 5.23. The internal mechanism identified to date should act over the entire mixed mode range and consequently, in the absence of fibre bridging or other crack closing mechanisms, w i l l result in the local envelope being significantly larger than the global envelope. 117 Chapter 5 Analysis and Discussion 5.6 Summary The experimental results presented in Chapter 4 were examined in detail. The loads applied to specimen mx4 did not result in any specimen load history effect on the specimen behaviour. The damage zone that was created did not affect the pure mode loading situations indicating no local plasticity effects. The specimen load path also does not appear to have an effect on the specimen behaviour either during loading or unloading. The nature of the crack surfaces, or the crack condition, plays the most significant role in determining the specimen response. Surface roughness interactions affect mixed mode conditions in a similar fashion to their impact on pure mode II global loading. As the crack surfaces slide over each other, an internal opening displacement is induced that affects the local levels of G/. The applied global opening is not sufficient to negate the interactions under shear loads. A s a result, although classical analysis suggests that in mixed mode loading a small increase in the allowable G/c before mixed mode failure exists, the local conditions are such that Gn can be approximately double the local loads expected. Some of the variability seen in literature data can be attributed to the crack to crack variation in local interactions such as fibre bridging or surface roughness effects. Standard global data reduction practices are insufficient to capture these local effects. The key to global data reduction schemes is the underlying theory that the local conditions at the crack tip can be predicted using global data alone. A s a result they have difficulty accounting for the possibility of internal mechanisms that may exist to varying degrees. In addition, these test methods only use data up to the point of fracture but do not take advantage of any data acquired during or after the fracture process. Determining G/c and G//c from the actual measurements of work done during fracture does appear to be possible. 118 Chapter 5 Analysis and Discussion Table 5.1 Results of mx4 and mx5 mixed mode G,L partitioning into GIG, the globally applied component, and an equivalent AG/, the internally generated component. Specimen GIG (J/m2) G,L (J/m2) AG, (J/m2) equivalent AG, (J/m2) mx4 1212 125 155 30 1.6 mx4 2112 66 75 9 0.5 mx4 2121 70 70 0 0 mx4 12 60 70 10 0.5 mx4 1221 94 150 56 6.5 mx4 w(42) 105 125 •20 0.9 mx4 w(55) 105 135 30 1.9 mx4 w(67) 105 135 30 1.9 mx4 w(85) 105 135 30 1.9 mx4 w(97) 105 135 30 1.9 mx4 w(117) 105 155 50 5 mx4 w(u67) 105 135 30 1.9 mx4 rev 1221 110 160 50 4.5 mx4 21 up 85 165 80 13 mx4 cp (b) 84 105 21 1.2 mx4 cp (c) 100 145 45 4 mx4 cp (d) 120 180 60 6 mx5 cp (b) 36 110 74 20 mv> cp (c) 65 150 85 17.5 mx5 cp (d) 90 180 l>0 15.5 119 Chapter 5 Analysis and Discussion G | L : G , | L (J/m 2) Test Name 5:95 mx4 cp(a) 105:95 mx4 cp(b) 145:95 mx4 cp(c) 180:95 mx4 cp(d) mode II crack tip mode I crack tip — i 1 --200 0 x (um) -600 -400 200 400 Figure 5.1 Plots showing progression of mode I and mode II crack tips during loading of mx4 cp. Movement of both crack tips appears independent and irreversible. 120 Chapter 5 Analysis and Discussion mode I mode II (A) (B) (C) (D) Figure 5.2 S E M image montage of mx4 w(117) under mixed mode loading (GIG = 105 J / m 2 and GUG =117 J /m ). The mode I and mode II crack tip locations identified were determined from analysis of the CFD profile. Ahead of the mode I crack tip at (A) , there is a distinct shear and opening displacement discontinuity across the crack plane. Behind the mode I crack tip there exists a zone similar to (B) where only a shear displacement discontinuity across the crack plane exists. The mode II crack tip location as determined from the CFD profiles appears to be behind the crack tip located from the image where no displacement discontinuities exist across the crack plane at (C). In the zone behind the original estimate of the mode II crack tip at (D) there is clearly no displacement discontinuity. 121 Chapter 5 Analysis and Discussion 100 i 90 80 -| CM E 3 70 -I 60 -50 -40 -•' G | L = G | G 24 11 22 13 a = 27 mm 40 — i — 50 — i — 60 — i — 70 80 90 100 G I G (J /nO Figure 5.3 Plot showing equivalence of G]L and GIG for pure mode I conditions throughout specimen mx4 loading. Data labels indicate order of testing as described in Table 4.2. 100 90 80 70 60 50 40 30 -20 -10 -0 0 4,6 B 2 6 ' 7 G||L ~ GIIG a = 27 mm 20 40 60 80 100 G„G (J/m2) Figure 5.4 Plot showing full transfer of GUG to the crack tip (Gm) for all mx4 pure mode II conditions applied. Data labels indicate order of testing as described in Table 4.2. 122 Chapter 5 Analysis and Discussion 90 -I Ol.ll 80 - • II,I Al.l/ll 70 - • l/ll CM E 50 -5 4 0-< 30 -A 20 - O 10 - 1 0 -A o o o o o o 10 F r e q u e n c y Figure 5.5 Figure showing the variability and scatter in comparing AG/ and load path. Data points are taken from mx3, mx4, and mx5 mixed mode loading conditions. No effect of loading path on AG/ is apparent. Figure 5.6 Plot showing AG/ as a function of increasing G/G for tests mx4 cp and mx5 cp. Both tests were first loaded in pure mode II to similar levels and were then loaded in mode I while the shear load was held constant. Despite the loading similarities the resulting behaviour is markedly different. 123 Chapter 5 Analysis and Discussion E 3 CD 80 0 ^ -e- -s- - 6- e> 20 40 60 80 100 G„ (J/m') mx4 w a = 27 mm loading = I,I 120 140 Figure 5.7 Plot showing the development of AG], as shear load is applied to a specimen with constant global mode I loading (mx4 w). 124 Chapter 5 Analysis and Discussion -100 100 300 500 x (um) + 2 i — G I L = 6 J/m2 -100 100 300 500 x (um) Figure 5.8 Plot showing partitioning of mx4 cp(d) local displacement field (a) into applied displacement (b) and internally generated displacement (c) components. Since G7 varies as the square of displacement, a small internally generated displacement component results in large AG\ under load. There is a similarity between the equivalent AGi for mx4 cp(d) of 4 J/m2 and the induced GIL under pure shear applied loading (mx4 cp(a)) of 5 J/m2 that suggests a common cause of displacement. 125 Chapter 5 Analysis and Discussion 0 4 — — 1 1 1 1 1 100 200 300 400 500 600 x (Lim) Figure 5.9 Plot showing parti t ioning of mx4 cp(c) local displacement field into applied displacement (G/G) and internally generated displacement (equivalent AGj) components. The similarity between the normalized G/L for mx4 cp(c) and the induced G/L under pure shear applied loading (mx4 cp(a)) suggests a common cause of displacement. 500 600 700 800 900 1000 x(urn) Figure 5 . 1 0 Plot showing partit ioning of mx5 cp(c) local displacement field into applied displacement (GIG) and internally generated displacement (equivalent AGj) components. The similarity between the normalized GIL for mx5 cp(c) and the induced G/L under pure shear applied loading (mx5 cp(a)) suggests a common cause of displacement. 126 Chapter 5 Analysis and Discussion 500 600 700 800 900 1000 x (Lim) Figure 5.11 Plot showing parti t ioning of mx5 cp(d) local displacement field into applied displacement (GIG) and internally generated displacement (equivalent AGi) components. The similari ty between the normalized G,L for mx5 cp(d) and the induced G,L under pure shear applied loading (mx5 cp(a)) suggests a common cause of displacement. 200 n 0 50 100 150 G I G (J/m2) Figure 5.12 Plot showing AG, as a function of increasing GIG for a constant internally generated displacement profile (equivalent AG,) of 5 J / m 2 . 127 Chapter 5 Analysis and Discussion General loading M /2 = My - M L Gj a M,2 M,/2 = My + M L G„ a M||2 Global Analysis System M L = M G G L = G G Figure 5.13 Schematics describing (a) G/ and G// as a function of applied moments and (b) the key assumption in global analysis that the system is perfect and all global loads are transferred directly to the crack tip. Mode I M G L = -M G U M M LU LL I ) ) M } M ML U = M G U ML L = M G L = -M G U MIL/2 = ML U - ML L = 2MG U MIG/2 = M G U - M G L = 2MG U GIL = GIG MIIL/2 = ML U + ML L = 0 MIIG/2 = M G U + M G L = 0 G I I L ~ G I I G = 0 Figure 5.14 Schematic describing transfer of global loads to the local level for pure mode I loading with no local interactions such as fibre bridging. 128 Chapter 5 Analysis and Discussion Mode I With fibre bridging M G L = - M G U ; M F U = -M F L M LU J 1 M FU M L L ) ) M F L ) M, } M, M L U = M G U + M F U =MGU - M F L MLL = MGL+.MFL = -M G U + M F L MIL/2 = M L U - M L L = 2MGU - 2MFL MIG/2 = 2MGU G I L < G / G MIIL/2 = M L U + M L L = 0 MIIG/2 = M G U + M G L = 0 G I I L ~ G I I G - ° Figure 5.15 Schematic describing transfer of global loads to the local level for pure mode I loading with fibre bridging. The internal mechanism of fibre bridging (Mp) shields the crack tip from seeing the full applied load. Mode II M G L = M G U ; IVV = -M A U M L U ) ) M A U _ ) M, ) M, M L U = M G U + M A U M L L = M G L + IVV = M G U - M A U MIIL/2 = M L U + M L L = 2MG U MnG/2 = M G U + M G L = 2MG U GIIL =  GIIG MIL/2 = M L U - M L L = 2MAU MIG/2 = M G U - M G L = 0 G I L >  GIG Figure 5.16 Schematic describing transfer of global loads to the local level for pure mode II applied loading. The interaction due to the crack face roughness as the two surfaces slide over each other under shear loading is modelled by M A which induces a local opening component. 129 Chapter 5 Analysis and Discussion Mixed Mode Mr,,, > M r , ; IVL = -M •AU M M LU ) ) M A U M GU LL 3 ) M A L 1) M GL M L U = M G U + M A U M L L = M G L + IVV = M G L - M A U MIL/2 = M L U - M L L = M G U MIG/2 = M G U - M GIL > GIG M G L + 2MA U GL MIIL/2 = M L U + M L L = M G U + M G L MIIG/2 = M G U + M G L GIIL = GIIG Figure 5.17 Schematic describing transfer of global loads to the local level for mixed mode applied loading. The interaction due to the crack face roughness as the two surfaces slide over each other under shear loading is modelled as M A which increases the local opening component. z CL -0.2 70 60 50 •10 J — loading unloading 0.2 0.4 0.6 5| (mm) 0.8 Figure 5.18 Plot of mode I load and displacement data for mx5 cp. Theoretically, the mode I component of work done in fracture can be determined by the difference in area between the loading and unloading curves. In this case the area is significantly greater than that expected based on local analysis. 130 Chapter 5 Analysis and Discussion 150 130 -110 -90 70 50 30 10 -10 •loading unloading • 8,| (mm) mx5 cp a = 21.2 mm Figure 5.19 Plot of mode II load and displacement data for mx5 cp. Theoretically, the mode II component of work done in fracture can be determined by the difference in area between the loading and unloading curves. In this case the area, as expected based on local analysis, is not large enough to be accurately determined. 0 50 100 150 G|G (J/m2) Figure 5.20 Plot for test mx4 cp showing evolution of G,L and its relation to GIG and GIC. Note that GJG and GjL both exceed pure mode Gtc by approximately 10% and 105% respectively at failure. 131 Chapter 5 Analysis and Discussion GIIG = GUL = 95 J/m extrapolated Figure 5.21 Plot for test mx5 cp from the insert showing evolution of Gn and its relation to GIG and Gic. Note that GIG and Gn both exceed pure mode G/c from the insert by approximately 15% and 100% respectively at failure. 132 c .g >) >1 a • -~ Q s I a 3 § 6 ss l i s b E Qi JS 5 6 i H o g o • -t- vt « — © ss 8 cu "O H S • eu i "9 cu M l 3 S f •3 .5 O 2 eu a> ig T 3 ^ o o S3 o 8 V 1 1 £ CU S £ CU & § v? *— CQ U CU pfl cu 3 S 7 3 O M £ .s 3 -9 | 2 UD j£» CJ S3 U "3 U cu £ W S3 * vi 3 O £ 2 ° o 5 =u 3 M © •= 3 ** 5 3 » f € o ^ S T 3 £ « « 2 S3 g" £ a 8 * 5 ^ 3 .e CU t v tS 3 es L cu _ J3 C .2 M 5 -a 2 « H t J 3 V. S ** cu _ cu J2 .S • s « ^ S3 =U * -eu u3 8 I . -M CU — £ 'K • — ^ r-6 k S 8 - S ° £ cu s x: = S3 S3 S3 2 " -a CU CU cu bB _ 2 § u 8 S3 «•* 8 O . 3 % cu u3 s-g cu O -o S S3 f S ir! cu u 3 DID S ^ 8 ce p j - cu p I S 3 8 •— 0 .5 w .2 cu 1 £ .5 W 3 e o 8 *-O vi . CU C3 M ) Chapter 5 Analysis and Discussion Figure 5.23 Plot showing the differences between the global and local mixed mode failure envelopes. The local mode I values should be higher than the global predictions throughout the mixed mode range. Currently, matched global and local data only exists in the highlighted regions and future examination throughout the mixed mode range is recommended. 134 6 Numerical Model and Results Chapter 6 Numerical Model and Results A finite element model of a mixed mode D C B specimen was created to validate key aspects of the experimental technique as well as to examine further some of the experimental observations. The actual local measurement technique is examined to determine i f the technique itself can give rise to AG/. Also explored is whether or not the F E model is capable of predicting the size of the singular zone. The mixed mode stress distribution around the crack tip is examined to determine i f it provides any additional insight into the specimen response to mixed mode loading. 6.1 Finite Element Model Definition 6.1.1 Element Type The D C B specimen is modeled using 2 n d order, 6-noded triangular elements. These elements are specifically required at the crack tip in order to satisfy L E F M conditions. The midside node of each element directly connected to the crack tip is moved to the quarter point closest to the crack tip. This ensures a square root stress singularity at the crack tip of the form r'U2, a requirement of L E F M (Barsoum, 1976). 6.1.2 Mesh To account for the complexity of the analysis due to the induced stress-singularity at the crack tip, a significantly refined mesh is required in this region. Figure 6.1 shows successive enlargements of the crack tip area. The elements at the crack tip have a length of 2 urn. This element length increases as elements get further away from the singular zone. The model o f the entire D C B specimen with a 25 mm crack length consists of 37164 nodes in 18018 elements. 6.1.3 Boundary Conditions Figure 6.2 shows schematics of a D C B specimen along with imposed boundary conditions as they exist (a) during testing and (b) in the model. The slack in the loading pin holes along with the allowance for axial motion by the clamping system is modeled by a fixed clamping system and free arms. The boundary conditions in the model are equivalent to those used in the L E F M analysis of the D C B specimen for mixed mode conditions. To simplify the analysis, a contact surface between the crack faces is not defined. Instead pure mode II loading is achieved by loading each arm equally (f= 0.5). This is justified by Paris' 136 Chapter 6 Numerical Model and Results (1998) observation and by Figure 3.15 that confirm the negligible effect of friction between the two surfaces in pure shear loading. 6.1.4 Analysis Type A static, linear, plane stress analysis is employed using A B A Q U S / S T A N D A R D Version 5.8. Paris (1998) and Corleto et al. (1987) both noted that for orthotropic D C B specimens, there is negligible difference between plane stress and plane strain analyses and that both are representative of conditions across the specimen width. The material properties used for the analysis are provided in Table 4.3. A s in the experiments, loading is achieved through an applied displacement to the tips of the crack flanks. 6.1.5 Model Limitations The finite element (FE) analysis is used primarily to investigate stress and displacement fields around the crack tip and only takes into account the elastic response of the system. Consequently, no crack extension formulations have been used. In addition, the development of a damage zone ahead of the crack tip as observed in mode II and mixed mode loading is not included in the model. 6.2 Model Validation 6.2.1 Global Loads A finite element analysis (FEA) was conducted for pure mode I and pure mode II loading of the specimen. In Table 6.1 the global mode I response of the system is shown to be in good agreement with the analytical L E F M prediction. Paris' numerical analysis also showed good agreement between F E A and experiments for the mode I displacement field (Paris, 1998). The global mode II response of the specimen shows an error in the specimen response on the order of 7.5% (Table 6.2). Since the model was run using an applied displacement, the slight error in compliance should not affect the local displacement profiles. The local stress field should not be significantly affected. 6.2.2 Local Displacements The resulting crack face displacements, u and v, from the F E analysis are used to determine the COD and CSD profiles as a function of r, the distance from the crack tip. Figure 6.3 shows a 137 Chapter 6 Numerical Model and Results comparison between the COD profile obtained analytically using L E F M , experimentally, and from the F E analysis. Initially all three methods coincide. However, the experimental data matches the L E F M predictions for a longer period of time before it eventually converges to the F E results. The zone where the experimental data or the F E results coincide with the L E F M predictions is the singular zone described in Section 2.1.1. Although the experimental displacements eventually converge to the F E model away from the crack tip, an appreciable difference that exists between the F E analysis and the experiments is the size of that singular zone. Since the COD profile follows an rU2, curve a plot of COD2 vs. r w i l l show the singular zone as having a linear dependence. The plots in Figure 6.4 show the relationship between F E A , L E F M , and experimental data for four unique crack conditions. Each of the four plots is from a different specimen, two of which are taken from Paris' (1998) data. It is evident that the singular zone according to all the F E A analyses is consistently 125 um. On the contrary, the experimental data suggests varying singular zone sizes anywhere from 150 urn to 500 urn. The size of the singular zone is a strong function of the specimen geometry (Chona et al, 1983), specifically the distance from the crack tip to the nearest free edge. In the D C B specimens, this parameter is the arm thickness, h. Since h is constant for all F E analyses conducted, the consistency of the F E A singular zone size is justifiable. However, the singular zone size can also be influenced by the varying local conditions at the crack tip. In the F E analysis the heterogeneous, orthotropic C F R P material is modeled as a homogenous continuum with effective material properties. While this is a standard practice for determining global responses of composites structures, when examining mechanical responses on a scale comparable to material heterogeneities, it is does not appear to be appropriate. In composites fabricated through autoclave processing using preimpregnated fibre sheets (prepreg), a resin rich layer exists between the plies. Figure 5.21 shows a clear image of that region at the specimen midplane. O f importance in this analysis is the resin rich crack plane. When a thin resin rich layer is added to the model at the crack plane, the size of the singular zone 138 Chapter 6 Numerical Model and Results is increased to 225 mm long (Figure 6.5). This is a good indication that the cause for the variability in the singular zone sizes observed experimentally is linked to the high variability in local crack tip conditions. If a direct quantitative comparison with experimental data on a local scale is required, modeling the composite material as a continuum does not appear to be sufficient. The smearing of material properties is not ideal for analysis on this scale. Despite the difference in the actual size of the singular zone, the stress field resulting from the F E analysis should qualitatively still be as predicted by L E F M . In mode II loading a similar situation arises. A s shown in Figure 6.6 the F E analysis agrees well with the L E F M prediction and the experimental data. The size of the F E analysis singular zone is also smaller than that observed experimentally (Figure 6.7). However, the F E CSD profile deviates quite gradually from the L E F M prediction in comparison to the COD profile. Similar to the experiments, the F E A predicted mode II singular zone size is larger than that in mode I. 6.2.3 Local Stresses Figure 6.8 shows the von Mises and shear stress fields resulting from the pure mode I F E analysis of the D C B specimen. Similarly the stress fields from the pure mode II analysis are shown in Figure 6.9. The shape of the stress field is similar to that observed by Corleto et al. (1987) for pure mode analysis of D C B specimens and agrees with L E F M predictions. A s expected the pure mode II shear stress distribution and the pure mode I normal stress distribution share a similar shape. The mode I stress fields shown in Figure 6.8 are for a G/c = Gn = Gic = 126 J/m 2 . These stress fields at failure clearly indicate the tensile nature of the fracture that occurs under pure mode I loading. Figure 6.8(a) shows that there are negligible shear stresses induced directly ahead of the crack tip under pure mode I loading. The normal stresses seen in Figure 6.8(b) dominate. Figure 6.9 shows the mode II stress fields for a relatively low value of Gn = 95 J/m . However, even from this low level of loading the nature of mode II crack propagation is evident. In this case, Figure 6.9(a) shows the dominant stress field in front of the crack tip and is, as expected, 139 Chapter 6 Numerical Model and Results very similar in shape to the stress field shown in Figure 6.8(b). Given that the shear failure under mode II loading takes place at a significantly higher load level, the shear stresses wi l l dominate the normal stresses shown in Figure 6.9(b) as Gu increases towards Guc. The F E analysis performed here is not sufficient to examine high Gn loads due to the lack of modeling of the damage zones created ahead of the crack tip. Even at the low Gu loads examined here there is a damage zone created as previously shown. 6.3 Analysis of Experimental Measurement Technique When describing the experimental technique, Paris et al. (2001) indicated that local measurements should be taken as close to the crack surfaces as possible. The high strains in the crack tip area wi l l have a greater effect on the displacement profiles the further the measurements are taken from the free edge at the crack surface. The F E model allows for a direct comparison of displacement measurements taken at the crack surface with measurements taken at varying distances from the crack face (z). To generate COD and CSD profiles, virtual measurements were taken from the F E results to simulate the measurement technique used during image analysis. Measurements were taken at three locations: z = 0 urn (crack surface), z = 20 urn, and z = 95 urn. Figure 6.10 and Figure 6.11 show the results of these virtual measurements for the opening and shear displacement profiles respectively. The COD measurements were taken using all nodes within ±5 urn of the defined z. The variability within that 10 um is small and the COD measurements are relatively insensitive with respect to z (Figure 6.10). O f significance is that the further one measures away from the crack face (greater z) the more difficult it is to pick up the square root singularity; the profile flattens out as it approaches the crack tip. Since the majority of measurements surrounding the crack tip are taken within 20 urn of the crack faces, the resulting profile does provide a clear indication of the local G. Consequently, the difference between Gn and G/c discussed earlier cannot be attributed to the measurement technique itself. Examination of the COD profiles presented in Chapter 4 reveals that the difference between the measurement profile and the actual crack face profile at the crack tip is seen in experimental data. 140 Chapter 6 Numerical Model and Results The CSD measurements are much more sensitive than the opening profiles. When profiles were taken using a ±5 um tolerance the results were quite scattered. A tighter tolerance of ±1 um was eventually used and the results for that measurement are provided in Figure 6.11. There is still some discernable variability even with this tight tolerance. However, as long as experimental measurements are taken within approximately 20 um of the crack face, as is typically the case, the results wi l l provide a profile from which the local Gu can be determined. Examination of the CSD profiles presented in Chapter 4 reveals that the difference between the measurement profile and the actual crack face profile near the crack tip is seen in experimental data. The variability seen in the experimental data appears to be caused by the high strains in the region. 6.4 Mixed Mode Stress Field Analysis Figure 6.12 shows the F E prediction of the local shear and von Mises stresses near the crack tip close to fracture. The asymmetric stress field is as expected based on L E F M . The stress fields are also consistent with the crack growth path described in Section 5.4. The preferred direction of crack growth under an asymmetric stress state is the direction of minimum Ku. A s Figure 6.12(a) shows this would cause the crack to grow at an angle away from the midplane. A s shown in Figure 5.21 the crack does indeed being to grow by kinking away from the midplane until it reaches the adjacent fibres. 6.5 Summary The F E model created has been shown to predict well the pure and mixed mode global and local elastic behaviour of the specimen for truly 'pure' load applications where no internal mechanisms exist to shield or intensify the local loads. A s a result, without the benefit of the local analysis to identify the local loads, the model would suffer the same downfalls as any other global based analysis. Using the known local strain energy release rates the model was used to examine the mixed mode stress fields close to fracture. The results indicate that the mixed mode crack path behaviour observed experimentally is consistent with the likely initial crack growth predicted by the F E local stress fields. The model was unable to predict accurately the size of the singular zone observed experimentally. The singular zone size is variable between cracks and is heavily dependant on highly local effects such as the size of the resin rich region that the model does not take into 141 Chapter 6 Numerical Model and Results account. Since the many local interactions that take place cannot yet be modeled in a predictive manner the results from F E fracture models based simply on global data and perfect transfer of global load to the crack tip should be treated cautiously. The experimental technique used to measure local crack face displacement profiles was also examined in detail and shown to be acceptable in its current form. However, i f not performed carefully, the strain fields near the crack surfaces can significantly affect the local crack face displacement measurements. A s a result measurements, especially those taken in the singular zone, should be taken as close to the crack surfaces as possible. Measurements taken within 20 microns of the crack tip appear to be minimally affected by the adjacent strains. 142 Chapter 6 Numerical Model and Results Table 6.1 Comparison of mode I global response of FEA with analytical prediction. 5, (mm) P, - LEFM (N/mm) P, - FEA (N/mm) difference (%) 0.66 2.308 2.312 0.17 0.86 3.007 3.008 0.03 Table 6.2 Comparison of mode II global response of FEA with analytical prediction. 8n (mm) P„ - LEFM (N/mm) P„-FEA (N/mm) difference (%) 1.5 4.50 4.16 -7.5 2.1 6.33 5.83 -7.9 143 Chapter 6 Numerical Model and Results 1.8 mm 22 urn Figure 6.1 Finite element mesh of specimen showing successive enlargements of the crack tip area. Elements defining the crack tip have been modified to simulate square root singularity. 144 Chapter 6 Numerical Model and Results ( ) ' — 1 ( ) (a) (b) Figure 6.2 Schematic of specimen and boundary conditions (a) dur ing testing and (b) in the finite element model. 12 -I 10 -8H E Q O O 6A 2 0 D Experiment — LEFM FEA 200 -OrP G, = 105 J/rrf a = 25 mm 400 r ( L i m ) 600 800 Figure 6.3 Compar ison between COD profile obtained experimentally, analytically using L E F M , and by finite element analysis in the region close to the crack tip for crack lengths of 25 mm. The experimental data eventually converges to the F E solution. 145 Chapter 6 Numerical Model and Results 146 Chapter 6 Numerical Model and Results 250 200 A FEA — LEFM • Experiment G, = 105 J/m" a = 27 mm — i 1 0 200 400 600 800 1000 1200 1400 r ( j i m ) (d) Figure 6.4 Plot of COD2 vs. r showing the square root dependence of displacement fields behind the crack tip (singular zone) for four unique cracks; (a) a=19 m m (Paris, 1998), (b) a=21.5 mm (Paris, 1998), (c) a=24.5 m m , and (d) a=27 mm. The experimental singular zone size is variable for different crack tip conditions while the F E A singular zone size remains constant. The solid marker in each experimental data set indicates the end of the singular zone. 147 Chapter 6 Numerical Model and Results r(^m) Figure 6.5 Plot of COD2 vs. r showing the square root dependence of displacement fields increased behind the crack tip from 125 um to 225 p,m due to the addition of a resin r ich zone at the crack plane. The solid marker indicates the end of the singular zone. x Experiment — LEFM r(nm) Figure 6.6 Compar ison between CSD profile obtained experimentally, analytically using L E F M , and by finite element analysis in the region close to the crack tip. The finite element results agree very well wi th experimental data. 148 Chapter 6 Numerical Model and Results •"(Um) Figure 6.7 Plot of CSD vs. r showing the square root dependence of displacement fields behind the crack tip. The higher order terms begin to have an effect much later experimentally than in the FEA. 149 Chapter 6 Numerical Model and Results propagation is dominated by the normal stress component (b). 1 5 0 Chapter 6 Numerical Model and Results (b) Figure 6 . 9 Plot of (a) shear stress and (b) von Mises stress for pure mode II loading (G„ = 9 5 J /m 2 ) . The crack flanks are to the left of the image. Note how the stress field ahead of the crack tip is dominated by the shear component (a). Chapter 6 Numerical Model and Results x (|j,m) Figure 6.10 Plot showing virtual measurement of COD profile at the crack face and at distances of 20 and 95 microns from the crack faces. x (|nm) Figure 6.11 Plot showing virtual measurement of CSD profile at the crack face and at distances of 20 and 95 microns from the crack faces. 152 Chapter 6 Numerical Model and Results Figure 6 . 1 2 Plot of (a) shear stress and (b) von Mises stress for mixed mode loading ( G / L = 2 2 0 J / m 2 and Gu = 9 5 J /m 2 ) . The stress fields are for the local loads measured close to mixed mode crack propagation for mx4 cp. The asymmetric nature of the stress field is clear. The crack fracture path seen i n Figure 5 . 2 1 is consistent with the stress distribution shown above. 7 Conclusions and Future Work Chapter 7 Conclusions and Future Work __ The main objective of this work was to better understand mixed mode delamination behaviour in composite laminates. Mixed mode conditions where the mixed mode G / G exceeded pure mode G / G C were of particular interest. To achieve this objective, the local crack tip behaviour was quantified and compared to global behaviour. A brief summary of the work performed in this thesis is first provided. The main conclusions that can be drawn from this work are then presented followed by recommendations for future work. 7.1 Summary 1. A n introduction to L E F M based global and local approaches to examining delamination behaviour was presented. 2. A brief review of pure mode C F R P delamination behaviour was provided before a more detailed review of mixed mode delamination testing and behaviour was presented. A n emphasis was placed on brittle, unidirectional composite laminates. 3. A n existing testing (Paris, 1998) and analysis technique (Ferguson, 1992; Paris, 1998) that allowed for a combined, quantitative global and local evaluation of delamination behaviour was evaluated and modified to allow for mixed mode loading and analysis. 4. Several mixed mode experiments were conducted on brittle AS4/3501-6, unidirectional split laminate specimens. In particular, mixed mode ratios in the region where mixed mode G / G exceeded pure mode G / G C were examined. Different crack conditions (crack length and crack starter), different loading and unloading paths, and the specimen load history were considered. 5. A simple numerical finite element model of the experiments was constructed to examine the stress fields at the delamination tip as well as to validate the experimental local measurement technique. 7.2 Conclusions 1. Under the levels pf mixed mode loading studied in this thesis it was clear that internal mechanisms exist that result in a difference between local and global mixed mode conditions. These mechanisms do not affect shear loading as in all cases examined Gm 155 Chapter 7 Conclusions and Future Work was equal to GUG- However, except for one case where it was determined that Gn = GIG, all mixed mode loading conditions showed that the local component of mode I loading was greater than that applied globally (Gn > GIG)- The same surface roughness interactions that induce a local mixed mode condition under pure applied shear also appear to increase the local mode I component under mixed mode global conditions. When the results of this work are combined with previous work by Paris (1998) for pure mode loading, it is evident that traditional global analysis techniques, including those recommended by fracture toughness testing standards, do not accurately depict the actual conditions at the crack tip. Variable internal mechanisms, such as fibre bridging and surface roughness interactions, have the potential to affect delamination behaviour across the entire mode I/mode II spectrum and indeed may be responsible for much of the variability in fracture toughness values observed in the literature. As such, current global data reduction schemes are incomplete and inadequate due to some of the assumptions made and cannot be used in isolation to gain a thorough understanding of delamination behaviour. However, rigorous treatment of global data in an energy balance wi l l result in accurate global predictions. 2. Under the mixed mode conditions examined in this thesis it was determined that the local conditions at fracture were significantly more severe than those determined using a global analysis. The total local critical strain energy release rate (GTLC) was significantly greater than the total global critical strain energy release rate (GTGC) due directly to the fact that the mixed mode Gnc was significantly greater than expected globally. These total local critical strain energy release rates are perhaps more consistent with the post mortem fractography found in the literature (e.g. Reeder, 1993) than the global values alone would suggest. The similarity between the fracture surfaces created under similar mixed mode conditions as those examined in this work and those created under a pure shear load suggests that higher total critical strain energy release rates could be expected. 3. Under all local mixed mode conditions the mode I and mode II crack tips do not coincide. The resulting Aa is representative of a damage zone, even at the low load levels considered here, that allows shear displacements but inhibits opening displacements. Related effects, such as load path and specimen load history, do not appear to affect the 156 Chapter 7 Conclusions and Future Work delamination behaviour. The shear loads applied here that cause the damage are small, however, and the observed lack of effect on delamination behaviour may not continue over the entire mixed mode range. 4. The role of boundary conditions plays a significant role in the transfer of global loads to the crack tip. The local analysis of the CSD profile under applied shear load for the original experimental set-up revealed that Gm « GUG- The global analysis also revealed that the specimen compliance became non-linear. When the loading set-up was modified to allow for free movement of the specimen arms under shear load, it was found that, as expected, Gm = GUG and the specimen compliance remained linear throughout loading. In general, compliances should be examined during testing to ensure that the system is not overconstrained. 5. A specimen loaded in pure shear by one arm only or by two arms together behaves the same. A local analysis performed on one specimen that was loaded in pure shear by the lower arm only (f= 0) and then by two, equally loaded arms (f= 0.5) showed that the CFD profiles were indistinguishable. It does not appear, therefore, that the actual method of shear load application affects the local specimen behaviour. This reinforces Paris' (1998) results that showed no friction effect between the crack surfaces under applied shear load. 7.3 Future Work 1. This work has shown that mixed mode delamination behaviour of brittle composite laminates is considerably more complex than is generally believed in the composites community. It is also evident that traditional global evaluation methods are insufficient to provide accurate predictions of the actual crack tip behaviour. The full significance and impact of these results on current toughness testing and design methodologies involving composites should be investigated. 2. A global evaluation technique capable of providing similar insight into actual crack tip behaviour as the local evaluation technique used here should be developed. A good starting point for such a technique would be to examine rigorously the area method for 157 Chapter 7 Conclusions and Future Work fracture toughness determination which is shown in Appendix A to have theoretical promise. 3. A simplified local evaluation technique should be developed that can be used with various global test methods, especially the A S T M recommended M M B test j i g ( A S T M D 6671). This would allow for local analyses to be conducted on standard test jigs instead of on the specialized j i g used here. 4. Using a similar material as that studied here the tests and analyses conducted in this work should be mirrored at more mixed mode ratios. This would result in a complete global/local analysis and hopefully understanding of delamination behaviour in brittle, unidirectional composites throughout the entire failure envelope. 158 References References A S T M D 5528, 2001, "Standard Test Method for Mode I Interlaminar Fracture Toughness of Unidirectional Fiber Reinforced Polymer Matrix Composites", In: 2002 Annual Book of A S T M Standards, Volume 15.03 Space Simulation; Aerospace and Aircraft: High Modulus Fibres and Composites, American Society for Testing and Materials, West Conshohocken, P A , pp. 292-302. A S T M D 6671, 2001, "Standard Test Method for Mixed Mode I-Mode II Interlaminar Fracture Toughness of Unidirectional Fiber Reinforced Polymer Matrix Composites", In: 2002 Annual Book of A S T M Standards, Volume 15.03 Space Simulation; Aerospace and Aircraft; High Modulus Fibres and Composites, American Society for Testing and Materials, West Conshohocken, P A , pp. 409-420. Barsoum, R.S. , 1976, "On the use of isoparametric finite elements in linear fracture mechanics", International Journal for Numerical Methods in Engineering, V o l . 10, pp. 25-37. Bhashyam, S. and B . D . Davidson, 1997, "Evaluation of data reduction methods for the mixed mode bending test", American Institute of Aeronautics and Astronautics Journal, V o l . 35, pp. 546-552. Blackman, B ; R . , and A . J . Kinloch, 2001, "Fracture Tests for Structural Adhesive Joints", In: Fracture Mechanics Testing Methods for Polymers, Adhesives and Composite, eds. A . Pavan, D . A . Moore, and J.G. Williams, Elsevier Science, Amsterdam, pp. 225-263. Bradley, W . L . , 1989, "Relationship of matrix toughness to interlaminar fracture toughness", In: Application of Fracture Mechanics to Composite Materials, ed. K . Friedrich, Elsevier Science, Amsterdam, pp. 159-187. Bradley, W . L . and R . N . Cohen, 1985, "Matrix deformation and fracture in graphite-reinforced epoxies", In: Delamination and Debonding of Materials, A S T M STP 876, ed. W.S. Johnson, American Society for Testing and Materials, Philadelphia, P A , pp. 389-410. 159 References Broek, D . , 1986, Elementary Engineering Fracture Mechanics, Martinus Nijhoff Publishers, Dordrecht. Brunner, A . J . , 2000, "Experimental aspects of mode I and mode II fracture toughness testing of fibre-reinforced polymer-matrix composites", Computer Methods in Applied Mechanics and Engineering, V o l . 185, pp. 161-172. Chona, R., G.R. Irwin, and R.J . Sanford, 1983, "Influence of Specimen Size and Shape on the Singularity-Dominated Zone", In: Fracture Mechanics: Fourteenth Symposium - Volume I: Theory and Analysis, A S T M STP 791, eds. J.C. Lewis and G . Sines, American Society for Testing and Materials, Philadelphia, P A , pp. 1-3-1-23. Corleto, C , W . L . Bradley, and M . Henriksen, 1987, "Correspondence Between Stress Fields and Damage Zones Ahead of Crack Tip of Composites Under Mode I and Mode II Delamination", In: Volume 3 of the conference proceedings of the Sixth International Conference on Composite Materials and Second European Conference on Composite Materials, Elsevier Applied Science Publishers Ltd., pp. 3.378-3.387. Corleto, C R . and W . L . Bradley, 1989, "Mode II Delamination Fracture Toughness of Unidirectional Graphite / Epoxy Composites", In: Composite Materials: Fatigue and Fracture, Second Volume, A S T M STP 1012, ed. P .A . Legace, American Society for Testing and Materials, Philadelphia, P A , pp. 201-221. Crews, J .H. and J.R. Reeder, 1988, " A mixed-mode bending apparatus for delamination testing", Technical Memorandum, N A S A TM-100662, N A S A , Langley Research Center, Hampton, V A . Davies, P. and M . L . Benzeggagh, 1989, "Interlaminar mode-I fracture testing", In: Application of Fracture Mechanics to Composite Materials, ed. K . Friedrich, Elsevier Science, Amsterdam, pp. 81-112. Dharan, C . K . H . , 1978, "Fracture Mechanics of Composite Materials", Journal of Engineering Materials and Technology, V o l . 100, pp. 233-247. 160 References Ducept, F., P. Davies, and D . Gamby, 1997, " A n experimental study to validate tests used to determine mixed mode failure criteria of glass/epoxy composites", Composites Part A , V o l . 28A, pp. 719-729. Ferguson, J.S., 1992, "Measurement of delamination crack tip displacements in G / E laminates using scanning electron microscopy", M . A . S c . Thesis, The University of British Columbia, Vancouver, B C . Ferguson, J.S., L . R . Gambone, and A . Poursartip, 1991, "The influence of mode of loading on the local crack tip behaviour of delaminatations", In: Conference proceedings of the First Canadian International Composites Conference and Exhibition ( C A N C O M '91). Garg, A . C . , 1988, "Delamination - A damage mode in composite structure", Engineering Fracture Mechanics, V o l . 29, No . 5, pp. 557-584. Greenhalgh, E.S. and F . L . Matthews, 1996, "Characterization of mixed-mode fracture in unidirectional laminates", In: Realising their commercial potential: E C C M - 7 , Seventh European Conference on Composite Materials, Woodhead Pub. Ltd., pp. 135-140. Guoyang, G . and J. Guiqiong, 2001, "The mixed-mode interlaminar fracture toughness of composite on hygrothermal conditions", In: Conference proceedings of 13 t h International Conference on Composite Materials ( ICCM-13), Paper 1503. Hashemi, S., A . J . Kinloch, and G . Will iams, 1991, "Mixed-Mode Fracture in Fiber-Polymer Composite Laminates", In: Composite Materials: Fatigue and Fracture (Third Volume), A S T M STP 1110, ed. T . K . O'Brien, American Society for Testing and Materials, Philadelphia, P A , pp. 143-168. Hashemi, S., A . J . Kinloch, and G . Will iams, 1991, "The analysis of interlaminar fracture in uniaxial fibre-polymer composites", Proceedings of the Royal Society of London, Series A , V o l . 427, pp. 173-199. 161 References Hertzberg, R .W. , 1996, Deformation and Fracture Mechanics of Engineering Materials. John Wiley & Sons, Inc., Toronto, O N . Hibbs, M . F . and W . L . Bradley, 1987, "Correlations Between Micromechanical Failure Processes and the Delamination Toughness of Graphite/Epoxy Systems", In: Fractography of Modern Engineering Materials: Composites and Metals, A S T M STP 948. eds. J.E. Masters and J.J. A u , American Society for Testing and Materials, Philadelphia, P A , pp. 68-97. Hinton, M . J . , A . S . Kaddour, and P.D. Soden, 2002, " A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence", Composites Science and Technology, V o l . 62, pp. 1725-1797. Hutchinson, J .W. and Z . Suo, 1992, "Mixed mode cracking in layered materials", In: Advances in Applied Mechanics, eds. J.W. Hutchinson and T . Y . Y u , pp. 63-191. Kanninen, M . F . and C H . Popelar, 1985, "Fracture Mechanics Models for Fiber Reinforced Composites", In: Advanced Fracture Mechanics, Oxford University Press, New York, N Y , pp. 392-436. Keary, P.E. , L . B . Ilcewicz, C. Shaar, and J. Trostle, 1985, "Mode I Interlaminar Fracture Toughness of Composites Using Slender Double Cantilevered Beam Specimens", Journal of Composite Materials, V o l . 19, Iss. 2, pp. 154-177. MIL-17 , 2002, MIL-Handbook-17: The Composite Materials Handbook, chaired by L . B . Ilcewicz and D. Granville, www.mill7.org. Nuismer, R.J . , 1975, " A n energy release rate criterion for mixed mode fracture", International Journal of Fracture, V o l . 11, No . 2, pp. 245-250. O'Brien, T .K . , 1997, "Composite Interlaminar Shear Fracture Toughness, GIJc: Shear Measurement or Sheer Myth?", Technical Memorandum, N A S A TM-110280, N A S A , Langley Research Center, Hampton, V A . 162 References Paris, I., R. Bennett, M . Mager, and A . Poursartip, 2001, " A Technique for Measuring Delaminaton Crack Tip Behaviour in Composite Laminates Inside a Scanning Electron Microscope", Journal of Composites Technology and Research, V o l . 23, pp. 173-185. Paris, I., 1998, "In-Situ Measurements of Delamination Crack Tip Behaviour in Composite Laminates Inside a Scanning Electron Microscope", Ph.D. Thesis, The University of British Columbia, Vancouver, B C . Poursartip, A . , L . R . Gambone, J.S. Ferguson, and G . Fernlund, 1998, "In-situ S E M Measurements of Crack Tip Displacements in Composite Laminates to Determine G in Mode I and Mode II", Engineering Fracture Mechanics, V o l . 60, No . 2, pp. 173-185. Reeder, J.R., 1993, " A Bilinear Failure Criterion for Mixed-Mode Delamination", In: Composite Materials: Testing and Design, Eleventh Volume, A S T M STP 1206, American Society for Testing and Materials, Philadelphia, P A , pp. 303-322. Reeder, J.R. and J .H. Crews, 1991, "Nonlinear Analysis and Redesign of the Mixed-Mode Bending Delamination Test", Technical Memorandum, N A S A TM-102777, N A S A , Langley Research Center, Hampton, V A . Sih, G.C. , P .C. Paris, and G.R. Irwin, 1965, "On cracks in rectilinearly anisotropic bodies", International Journal of Fracture Mechanics, V o l . 1, pp. 189-202. Singh, S. and I.K. Partridge, 1995, "Delamination Failure in Unidirectional Carbon Fibre/Epoxy Under Mixed-Mode Loading", Polymers & Polymer Composites, V o l . 3, pp. 35-39. Soden, P.D. , Hinton, M . J . , and A . S . Kaddour, 1998, " A comparison of the predictive capabilities of current failure theories for composite laminates", Composites Science and Technology, V o l . 58, pp. 1225-1254. Sriram, P., Y . Khourchid, and S.J. Hooper, 1996, "The Effect of.Mixed-Mode Loading on Delamination Fracture Toughness", In: Composite Materials: Testing and Design, Eleventh 163 References . Volume, A S T M STP 1206, American Society for Testing and Materials, Philadelphia, P A , pp. 291-302. Wiil iams, J.G., 1998, "On the calculation of energy release rates for cracked laminates", International Journal of Fracture, V o l . 36, pp. 101-119. 164 A Work Done During Fracture Appendix A: Work Done Purine Fracture The global primary test data that is acquired from the instrumented loading j ig is the load and displacement of each loading arm. The work done for fracture can be determined by the area between the loading and unloading curves for each arm. However, this method only determines Gc total and does not distinguish between the energy associated with mode I and mode II fracture. The following examines whether or not the superposition method of mode partitioning can provide the modal components of the work done during fracture. A.l Analytical Approach The total work done in fracture can be examined from the primary arm data alone as described below where the path includes loading, crack growth, and complete unloading. WD, = JP, d8, WDR = JPR ddR WDT(L+R)=WDL+WDR WDr{l+R) = j(PLdSL+PRdSR) In order for the modal analysis to give correct energy release rates the total work done in fracture using this data, which invokes superposition, must be consistent with any other similar measure of the total work done in fracture. Since the mode components are simply mathematic manipulations of the arm data based on the principle of superposition, the total work done by the modes should be equivalent to the total work done by the arms. A simple check of this assumption is provided below. WD, = JP, dS, WD„ = JP„ d5„ using superposition (Equation 3.1 and Equation 3.2) WD, = j Up _p \ d8. P -P dSR J ) WD„=j((PL+PR)d^ + (PL+PR)d^-WDT{l+IJ)=WDl+WDu FFPL-PR , PL+PRVS ( PL~PR , PL+PR^ ' dS, + + • ddR J J WDni+ll) = j(PLdSL+PRdSR) 166 Appendix A: Work Done During Fracture WDT{MI) = WDT{L+R) A s expected the total work by the modes is indeed equivalent to the total work done by the arms. The benefit of using the modal data as opposed to the primary arm data is that the modal data provides the amount of work done in mode I fracture and in mode II fracture without any further manipulations or assumptions. A.2 Theoretical Model A simple model was constructed in a spreadsheet that generates virtual global primary data. A sequential loading path was selected that started with application of mode II load before mode I application until crack growth. After crack growth the mode I component of the load was removed first before, finally, the mode II component of load was removed. The virtual data was then reduced in the same manner as actual experimental data. Figure A . 1 and Figure A.2 show the load-displacement plots of the arms and of the modes. The area method was used to determine the theoretical work done in fracture using both data sets. From the arm data the left arm performed 2.53 N . m m of virtual work while the right performed 2.14 N . m m of work for a total work done in fracture of 4.67 N . m m . The modal data shows that the mode I component does 2.56 N . m m of work while the mode II component does 2.11 N . m m of work in the fracture process. A s expected from the results of the previous section, the total work done by the modes during fracture, 4.67 N . m m , is the same as the total work done by the arms. Theoretically then, this method has potential for use on real data to determine, from a global level, the actual G/cs and G//cs. A.3 Test on Experimental Data The experimental load-displacement data for test mx5 is presented in Figure A.3 and Figure A .4 . The arm behaviour is essentially as expected based on Figure A . 1 although some nonlinearity exists during the mode I component of loading and unloading. The areas determining the work done in fracture for each arm and for each mode were calculated using an image analysis program. From Figure A . 3 , the left arm performed 3.7 N . m m of work while the right arm performed 5.6 N . m m for a total work done in fracture of 9.3 N . m m . The mode data shows that the mode I component does 7.12 N . m m of work while the mode II component does 2.9 N . m m of work in the fracture process. Although the total work done in fracture as determined by the 167 Appendix A: Work Done During Fracture modal analysis is approximately 0.8 N . m m greater than that determined by the arm analysis, the inherent noise in the data likely accounts for much of the difference. Based on the local experimental data the total amount of work done during fracture was 4.06 N . m m , with 2.75 N . m m attributed to the mode I component of fracture and the remaining 1.31 N . m m due to the mode II component. These values were determined based on Equation 5.1. Clearly a significant disparity exists between the results obtained from this global method for determining G/c and Guc and the local method. One of the dominant sources of error is likely the slack in the loading pin set up. The slack required to allow for free motion of the specimen also leads to measurement errors of the actual arm displacement that in turn leads to errors in the energy calculation. This is also the reason that the displacement data is not used in the global determination of G resulting in the use of Equation 3.3 and Equation 3.6. 168 Appendix A: Work Done Purine Fracture 0 -I 1 r 0 1 2 3 4 8 L ( m m ) (a) a = 20 mm 0 1 2 3 4 S R ( m m ) (b) Figure A . l Plot showing theoretical model load-displacement data for (a) the left a rm and (b) the right arm. 169 Appendix A: Work Done During Fracture 180 -| 160 -140 -120 -z 100 -CL 80 -60 -40 -20 -0 -•loading unloading a = 20 mm ~i 1 1 1 1 1 1 1 0 0.5 1 1.5 2 2.5 3 3.5 on (mm) (b) Figure A . 2 Plot showing theoretical model load-displacement data for (a) mode I and (b) mode II. 170 Appendix A: Work Done During Fracture 5 R (mm) (b) Figure A.3 Plot showing mx5 cp experimental load-displacement data for (a) the left a rm and (b) the right arm. The shapes of the loading and unloading curves are consistent wi th that expected based on the experimental model. 171 Appendix A: Work Done During Fracture 70 -| 60 -50 40 -30 -20 -10 — loading unloading 150 - i 130 -110 -90 -70 -50 -30 -10 -— loading unloading s y mx5 cp a = 21.2 mm -1 -10 rj 1 2 Su ( m m ) (b) Figure A .4 Plot showing mx5 cp experimental load-displacement data for (a) mode I and (b) mode II. The shape of the mode II loading and unloading curves are consistent wi th the experimental model. A pr imary source of error, however, exists in using the global displacement data due to the slack in the loading pin set up. 172 

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