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Matrix cracking behaviour of off-axis plies in glass/epoxy composit laminates Lee, Pek Wah Pearl 1990

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MATRIX CRACKING BEHAVIOUR OF OFF-AXIS PLIES IN GLASS/EPOXY COMPOSITE LAMINATES  By Pek Wah Pearl Lee B.Eng.(Dist), Technical University of Nova Scotia, Halifax, 1985 M.A.Sc, Technical University of Nova Scotia, Halifax, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  THE FACULTY OF GRADUATE STUDIES (Department of Metals and Materials Engineering)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA February 1990 © Pek Wah Pearl Lee, 1990  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 of M e t a l s and M a t e r i a l s E n g i n e e r i n g The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  April  ?firh  lQQn  ii  ABSTRACT  The present work is a study of the matrix cracking behaviour of composite laminates which contain plies oriented at an angle to the loading axis. Incremental tensile tests were conducted on a set of glass-epoxy laminates having the [0/6] geometry where 8 takes the values s  of 45°, 60°, 75° and 90°. At each load increment, the stiffness reduction was measured and the cracking sequence was photographed. A novel technique using image analysis was used to measure the crack length and digitize the crack pattern in each photograph. The results were analysed in two ways - deterministically (using fracture mechanics) and statistically. In the first instance, the relationship between stiffness loss and crack length was used to calculate the strain energy release rate, G from a compliance expression. It was found that the overall stiffness loss for a given crack length increased with increasing 0. As G can also be viewed as the resistance to cracking, the calculated values were used to plot matrix cracking resistance curves (R-curves) for each lay-up. The R-curves showed that the overall resistance to cracking increased with increasing orientation angle, 0. For the [0/45], laminate, where cracking is driven by the highest proportion of G component, the least increase in n  resistance was observed. The differences in crack resistance in these lay-ups could be explained with results from the statistical analysis. A statistical analysis of the changes in distribution of crack length and number indicated that most of cracks in the [0/90] were short even at high loads. In addition, a calculation of the s  incremental growth with each incremental load showed that the amount of growth in that lay-up was limited. This implied that the process of crack initiation continually dominated crack  iii  propagation even late in the loading sequence. The opposite behaviour is seen as 9 decreases. In the [0/45] , [0/60],, and [0/75] lay-ups, the additional Mode II shear loading appeared to have s  s  assisted significantly the coalescence and growth of cracks. Hence, the overall crack resistance decreased as the proportion of the G component increased. H  Cracking in the off-axis plies is not uniform. In the [0/45] , [0/60] and [0/75] laminates, 8  s  s  cracking begins in distinct bands and are referred to as shear bands since they occur due to the presence of the Mode II shear loading. This phenomenon, however, has little effect on the stiffness. Although cracking is not uniform, the cracks tend to space themselves to within two ply thickness apart as crack density increases. In the shear band areas, the crack spacing can approach one ply thickness. It was also observed that crack tips stop growing either when they are two ply thickness apart or when they approach a stronger area in the laminate. Generally, the resistance to cracking is not affected when the crack density is low. However, as cracks begin to interact when they are spaced to within two ply thickness, the resistance increases dramatically.  iv  Table of Contents Abstract  ii  List of Tables  •  vii  List of Figures  •  viii  Acknowledgements  xii  Chapter 1 - Matrix Cracking In Composite Materials  1  1.1 1.2 1.3 1.4  Introduction Failure Mechanisms In Composite Laminates Purpose Of Research Review Of The Literature 1.4.1 Shear-Lag Methods 1.4.2 Mechanics Methods 1.4.3 L E F M Methods 1.4.4 Statistical Approach 1.4.5 Comments On Literature Review  Chapter 2 - Scope Of Research 2.1 Experimental Program 2.1.1 Testing of Angle-Ply Laminates 2.2 Theoretical Background 2.2.1 Brief Review of Linear Elastic Fracture Mechanics 2.2.2 Application of LEFM to Composite Laminates 2.2.3 Determination of G for Transverse Ply Cracks 2.3 Analysis 2.3.1 Crack growth resistance, R-Curves 2.3.2 Mode II Component in Off-Axis Laminates Chapter 3 - Fabrication 3.1 Materials 3.2 Fabrication Procedure 3.2.1 Production of Pre-preg  1 4 10 13 16 21 25 29 30 32 32 35 38 38 42 44 46 48 49 52 52 53 53  V  3.2.2 Autoclave Cure Characterization of Resin Properties  55 57 Voids and the Viscosity Cycle 3.2.3 Preparation for Cure 3.2.4 Description of Small and Large Autoclaves 3.3 Experimental Trials 3.3.1 Troubleshooting 3.3.2 Specimen Preparation  62 62 65 68 73 75  Chapter 4 - Experimental Procedure 4.1 Tensile Testing 4.1.1 Equipment Calibration 4.1.2 Data Acquisition 4.1.3 Determination of Material Properties 4.1.4 Verification of Toe Region in Load Displacement curve 4.1.5 Testing of Off-Axis Specimens 4.2 Measurement Of Crack Growth 4.2.1 Image Analysis  77 77 83 85 87 88 90 91 92  Chapter 5 - Results and Discussion 5.1 Material Properties 5.2 Deterministic Results 5.2.1 Stiffness and Crack Length Measurements 5.2.2 Normalised Stiffness as a Function of Stress and Crack Length 5.3 Progressive Crack Patterns  94 94 96 96 97 107  5.4 Statistical Results 5.4.1 Crack Length Distribution 5.4.2 Crack Spacing Distribution 5.4.3 Crack Tip Interaction and Growth 5.5 R-Curve Behaviour Of Off-Axis Laminates 5.5.1 Analysis of d(E/Eo)/da Results 5.5.2 Comparison of Present Experimental Results with Other models 5.5.3 Matrix Cracking Resistance Curves  118 118 125 134 143 143 149 150  vi Chapter 6 - Summary and Conclusions  155  References  159  Appendices.  165  Appendix A - Determination of Normal and Shear Stress Components  165  Appendix B - ASYST Program for Data Acquisition  166  Appendix C - Stress-strain Curves to Determine Material Properties  170  Appendix D - Secong Set of Stiffness Reduction and Stress Results  172  Appendix E - Program to Detect, Count and Identify Cracks  174  Appendix F - Program for Joining Up Cracks  175  Appendix G - Results of Cumulative Crack Length and Number  180  Appendix H - Program SPACE to Determine the Minimum Crack Spacing  182  Appendix I - Program TIP to Determine the Minimum Crack Tip Spacing  184  Appendix J - Program GROWTH to Determine the Incremental Crack Growth  186  Appendix K - Program ELIM to Determine Effective Crack Length  191  Appendix L - Magnitudes of Crack Length Eliminated  193  vii  Table of Tables Table 2.1: Relative proportions of Mode II in off-axis plies  50  Table 3.1: Summary of Autoclave Cure Trials  70  Table 3.2: Fibre volume and void content of specimens  75  Table 5.1: Lamina Properties  95  Table 5.2: Predicted and measured stiffnesses of laminates  95  Table 5.3: E/Eo, stress and crack length results for [0/90]  s  98  Table 5.4: E/Eo, stress and crack length results for [0/75]  99  Table 5.5: E/Eo, stress and crack length results for [0/60]  99  Table 5.6: E/Eo, stress and crack length results for [0/45]  99  Table 5.7: LPT predictions of stiffness losses  105  s  s  Table 5.8: Definition Of Crack Sizes  ,  123  viii  Table of Figures Figure 1.1: Delarnination between plies of a laminate.  5  Figure 1.2: Transverse cracking in a cross-ply laminate  7  Figure 1.3: Damage accumulation process in composites  11  Figure 1.4: The different routes to using fracture mechanics in ply cracking analysis  15  Figure 1.5: Geometry used for shear lag analysis  17  Figure 1.6: The relationship between crack spacing and applied stress  18  Figure 1.7: Matrix cracking resistance curves  28  Figure 2.1: Flow diagram showing course of present research work  33  Figure 2.2: Shear coupling ratio as a function of off-axis angle  36  Figure 2.3: The three modes of crack surface displacements  38  Figure 2.4: Crack in an infinite plate  39  Figure 2.5: Stresses acting on an off-axis element  49  Figure 2.6: The variation of the relative Mode II component with angle  51  Figure 3.1: Illustration of prepregging with a drum winder  54  Figure 3.2: Composite Cure Process Variables  56  Figure 3.3a: DSC thermal scan at 40 °C isotherm  58  Figure 3.3b: DSC thermal scan at 50 °C isotherm  59  Figure 3.3c: DSC thermal scan at 60 °C isotherm  59  Figure 3.4: Viscosity profile of resin at 50 °C  61  Figure 3.5a: Bagging technique used for large autoclave cure  64  Figure 3.5b: Bagged laminate ready for cure.  64  Figure 3.6: Schematic of small autoclave and vacuum system  66  Figure 3.7: Reusable vacuum bag frame for small autoclave  66  ix Figure 3.8: Side view of large autoclave  ~.  67  Figure 3.9: Initial cure cycle used in small autoclave  68  Figure 3.10: The optimum cure cycle used for processing  72  Figure 4.1: A typical loading and unloading curve.  78  Figure 4.2: Circuit diagram of LVDT  80  Figure 4.3: Schematic of LVDT with dimensions  80  Figure 4.4: Schematic of top and bottom blocks for mounting  82  Figure 4.5: Calibration curve for LVDT  84  Figure 4.6: Calibration of load cell for three load ranges  85  Figure 4.7: Equipment used for data acquisition  86  Figure 4.8: Stress-strain behaviour of thin specimens  89  Figure 4.9: Behaviour of foil gauges on thin specimens  89  Figure 4.10: Geometry of tensile specimens  90  Figure 4.11: Set-up used for photographing crack pattern  91  Figure 4.12: Coordinate system and typical division  93  Figure 5.1a: E/Eo as a function of crack length for [0/90]  100  s  Figure 5.1b: E/Eo and crack length as a function of stress for [0/90]  100  Figure 5.2a: E/Eo as a function of crack length for [0/75]  101  s  s  Figure 5.2a: E/Eo and crack length as a function of stress for [0/75]  101  Figure 5.3a: E/Eo as a function of crack length for [0/60]  102  s  s  Figure 5.3b: E/Eo and crack length as a function of stress for [0/60]  102  Figure 5.4a: E/Eo as a function of crack length for [0/45]  103  s  s  Figure 5.4b: E/Eo and crack length as a function of stress for [0/45]  103  Figure 5.5: Digitized crack patterns for [0/90] lay-up  109  Figure 5.6: Digitized crack patterns for [0/75] lay-up  110  Figure 5.7: Digitized crack patterns for [0/60] lay-up  Ill  s  s  s  s  X  Figure 5.8: Digitized crack patterns for [0/45] lay-up  Ill  Figure 5.9: Comparison between digitized and actual pattern.  112  Figure 5.10: Edge view of gauge length of [0/45] laminate  115  Figure 5.11: Continued edge view of [0/45] laminate  116  Figure 5.12: Thickness profile of the inner ply of Fig 5.11  117  Figure 5.13: Crack pattern of [0/45] lay-up at 321 MPa  117  Figure 5.14: Cumulative crack length for [0/90] lay-up  119  Figure 5.15: Cumulative crack length for [0/75] lay-up  119  Figure 5.16: Cumulative crack length for [0/60] lay-up  120  Figure 5.17: Cumulative crack length for [0/45] lay-up  120  Figure 5.18: Cumulative crack number for [0/90] laminate  121  Figure 5.19: Cumulative crack length for [0/75] laminate  121  Figure 5.20: Cumulative crack number for [0/60] laminate  122  Figure 5.21: Cumulative crack number for [0/45] laminate  122  Figure 5.22: Illustration of the zone of influence, r  126  s  s  B  s  s  5  s  s  s  s  s  s  Figure 5.23: Transformation from plane X to Xt  :  127  Figure 5.24: Flow diagram for determining the crack spacing  127  Figure 5.25: The variation of crack spacing for [0/90]  129  Figure 5.26: The variation of crack spacing for [0/75]  s  129  Figure 5.27: The variation of crack spacing for [0/60]  130  Figure 5.28: The variation of crack spacing for [0/45]  130  Figure 5.29: Histograms of frequency against crack spacing  132  Figure 5.30: Flow diagram of program to calculate incremental crack growth  135  Figure 5.31(a): The variation of crack tip distances for [0/90]  136  s  s  s  Figure 5.31(b): Incremental crack growth distribution for [0/90]  136  s  Figure 5.32(a): The variation of crack tip distances for [0/75]  s  ,  137  xi Figure 5.32(b): Incremental crack growth distribution for [0/75],  137  Figure 5.33(a): The variation of crack tip distance for [0/60],  138  Figure 5.33(b): Incremental crack growth distribution for [0/60],  138  Figure 5.34(a): The variation of crack tip distance for [0/45],  139  Figure 5.34(b): Incremental crack growth distribution for [0/45],  139  Figure 5.35: The elimination of very close cracks  144  Figure 5.36: Flow diagram to eliminate short cracks  144  Figure 5.37: Normalised stiffness as a function of a for all lay-ups  147  Figure 5.38: Normalised stiffness as a function of a for all lay-ups  147  Figure 5.39: Material constant c as a function of angle  148  Figure 5.40: Comparison of models with present results  149  Figure 5.41: Crack resistance curves for all lay-ups  151  Figure 5.42: Resistance curves using effective crack length  152  Figure 5.43: Crack resistance as a function of crack spacing  154  e  xii  Acknowledgements  The author expresses her deepest appreciation to her supervisor, Dr. A. Poursartip, for his invaluable help and guidance, without which this work would not have been possible. She also wishes to thank Mr. R. Bennett, Mr. N. Chinatambi, Ms. D. Rernmer and Ms. G. Riahi for their assistance in fabrication, and experimental set-up. She is grateful to Mr. A. Shook for his help with the ASYST program. Last but not least, she wishes to thank her husband Tom, for helping with the drawings in this thesis, his support and encouragement throughout the course of this work.  To my newly arrived Michael, my husband Tom, my parents and to the memory ofDrJS. Nadeau.  Chapter 1  1  CHAPTER 1  MATRIX CRACKING IN COMPOSITE MATERIALS  1.1 INTRODUCTION A composite material is defined as a macroscopic combination of two or more distinct materials, having a recognizable interface between them [1]. Since only high-performance composites are used for structural applications, the definition can be restricted to include those materials that consist of a binder or matrix material reinforced with strong and stiff fibres. These composites are fabricated by stacking layers or plies of parallel fibres, where each layer is selectively oriented to bear the expected loads. Hence, the term 'composite laminate' is used. Structural composites usually consist of polymer-based matrices reinforced with fibres such as carbon, Kevlar and E-glass. The fibres are the principal load bearers while the matrix serves to bind them together, transfer the load to the fibres and protect them against the environment. The use of high-performance composites in load-bearing applications occurred first in the aerospace field. The motivation was both technological and economic in origin. Prompted by demands from the aerospace industry for materials with better specific properties and improved corrosion resistance, they were first introduced in the late 1940's as filament-wound rocket motors [2], Nevertheless, their real development only began properly in the last two decades, mainly as a result of the influence of space technology. Their wide application in large cornmercial aircraft began rapidly in the 1970's because of weight considerations that were highlighted by the energy crisis. Composite components were substituted for metallic components by a part-by-part or piecemeal fashion, but without any attempt at overall re-design of the structure. It was later realized that a total redesign of structures using composites offered considerable potential for reducing costs, independent of the energy savings. The economic  Chapter 1  2  advantage is l a r g e l y d u e to the s i m p l i f i c a t i o n o f c o m p l e x structures, s u c h as i n aircraft w i n g s or helicopter p r o p u l s i o n systems, b y integrating several parts a n d e v e n several functions w i t h i n the b o d y o f a single c o m p o n e n t . In the 1980's the technology has e v o l v e d into o n e that p r o v i d e s a variety o f fibres a n d matrices that c a n be c o m b i n e d to f o r m vast range o f c o m p o n e n t s h a v i n g v e r y e x c e p t i o n a l properties. T h i s h a s l e d to their w i d e s p r e a d use i n c o m m e r c i a l a n d m i l i t a r y aircraft, space a n d m i s s i l e systems, a u t o m o t i v e a n d m a r i n e applications and f o r sports a n d recreational equipment. T r e n d s c o n t i n u a l l y indicate that c o m p o s i t e s are b e c o m i n g c h o i c e materials f o r p r i m a r y structural c o m p o n e n t s i n aircraft a n d aerospace applications [3]. In order to use c o m p o s i t e s to a n y s i g n i f i c a n t scale o r to f u l l y r e a l i z e their potential, the d e s i g n o f safe efficient structures must b e ensured. A t this stage, factors s u c h as l a c k o f u n i f o r m industry-wide specifications, test techniques, standards a n d d e s i g n allowables all c o m b i n e to create a degree o f conservatism. T h e r e is little agreement i n the d e s i g n a n d analysis c o m m u n i t y o n w h a t constitutes a structural failure a n d h o w to predict it [4]. T h e m a i n reason f o r this is the variety a n d c o m p l e x i t y o f failure m e c h a n i s m s that o c c u r i n c o m p o s i t e laminates u n d e r loading. T h e f a i l u r e m o d e s vary, d e p e n d i n g o n the loading, material properties, lay-up a n d e v e n the relative fibre-matrix c o m p o s i t i o n . In s o m e cases, o n l y o n e failure m e c h a n i s m is operative, while i n others, a m i x t u r e occurs simultaneously. In the event w h e r e a m i x t u r e o f failure m e c h a n i s m s are i n v o l v e d , it b e c o m e s difficult to estimate critical l o a d i n g c o n d i t i o n s since they are generally not r e p r o d u c i b l e [5]. T h i s uncertainty has l e d to a d i v e r s i t y o f failure d e f i n i t i o n s a n d f a i l u r e criteria, thus creating a multitude o f d e s i g n allowables f o r a g i v e n application. U n t i l a c o m m o n basis f o r d e f i n i n g f a i l u r e is f o u n d , d e v e l o p m e n t a n d certification o f c o m p o n e n t s w i l l h a v e to continue to r e l y u p o n extensive repetitive testing o f representative structures to c o n f i r m the a d e q u a c y o f a n y structural analysis [5],  Chapter 1  3  Currently, a damage tolerant design philosophy is applied to metallic aircraft components as part of certification of the structure under fatigue loading. This requires the structure to retain adequate strength and stiffness, despite the presence of damage (induced during manufacture or while in service), either until such damage is detected during scheduled inspection or until the aircraft is withdrawn from service [6]. Using Linear Elastic Fracture Mechanics analysis (LEFM) and experimental data on crack growth under cyclic loading, this requirement defines undetectable flaws and damage and determines the corresponding design allowables. The state of the art of fracture mechanics is highly developed for isotropic materials. In terms of safety, efficiency and economics it would be highly desirable that such procedures be available for composite structures. For this to be accomplished, considerably greater knowledge than exists at present is needed to define damage, the level of damage, the growth rate of damage and the strength of the component in the presence of damage. Compared to isotropic metallic materials where damage consists of a single crack growing in a self-similar manner, the development of a damage tolerant methodology for composites is a more difficult task. This is because the structural analysis is greatly complicated as composites are inherently anisotropic and heterogeneous. A strength or fatigue analysis must take into account the fibre-matrix properties, volume fractions, orientation angles, lamina or ply stacking sequence and applied load combinations. Analytical difficulties may arise in areas where geometric discontinuities and stress concentrations exist such as holes, joints, cutouts, edge effects and complicated support conditions. Although the problems are difficult, the designer can tackle them using the same overall approach used in metals. These analytical methods are based on principles of solid mechanics which are associated with states of stress in a body. For example, classical closed-form solutions or empirical relationships established by experiment [5] can be applied in regions of concentrated loadings and joints. In instances such as where through-the-thickness stress gradients are  Chapter 1  4  involved or in large structures having a number of elements, closed-form analyses are somewhat limited in defining the state of stress and strain. In such cases, numerical analysis techniques such as the finite element method can be used to provide approximate solutions [7]. The greatest problem is dealing with the different damage mechanisms which occur simultaneously, accumulate and interact. A slight change of laminate configuration or of state of loading can trigger a change in the sequence of these mechanisms and possibly a change in the failure mode. Moreover, the number of possible combinations of damage mechanisms are as numerous as there are types of lay-up systems. Damage accumulation thus becomes an extremely important design criterion, particularly in the definition of acceptable stress levels. In order to use composites more effectively and utilize a greater portion of their component life, the long-term objective is to increase the scale of acceptable damage. To achieve this, we must be able to quantify damage, model the process and develop sound predictive methods.  1.2 FAILURE MECHANISMS IN COMPOSITE LAMINATES As mentioned previously, laminated composites do not fail by the initiation and propagation of a single dominant flaw as in the case of metals. The failure process involves a host of damage mechanisms which accumulate and interact throughout the material. With increasing loads and/or time, stresses in the laminate continually redistribute until they reach some critical level, causing final failure. As these mechanisms are varied and complex, it is difficult to characterize damage in composite systems. However, for practical purposes, the damage can be classified into three distinct mechanisms: delamination, matrix cracking and fibre fracture.  Chapter 1  5  DELAMINA TION Delamination is the debonding or the separation of the plies of the laminate (Fig 1.1). The most common source of delamination is the laminate edge such as holes and joints, where high interlaminar or out-of-plane stresses (a , t „ in Fig 1.1) are developed due to the mismatch in z  Poisson contraction of the individual plies. A delamination that occurs under tensile loading may not pose a serious failure as there are alternative load paths through the laminate. In compression, however, a delaminated group of plies effectively reduces the laminate thickness, causing buckling and overall global instability of the laminate. For this reason, the study of its behaviour has received much attention particularly in the areas of durability and damage tolerance.  Figure 1.1 Delamination between plies of a laminate due to interlaminar stresses T81  6  Chapter 1 MATRIX CRACKING  Matrix cracking is the cracking of the resin within a ply. Because the matrix and the matrix-fibre interface are weaker than the fibre phase, cracking almost always occurs in the matrix phase. At a microscopic scale, the cracking may be along in the resin or along the fibre-resin interface. Macroscopically, cracking occurs parallel to the fibre direction. When cracking occurs in plies oriented transversely to the loading direction such as in [0/90] crossply systems, the mechanism is called transverse cracking. Cracking can also occur parallel to the loading direction. When this happens, it is usually termed matrix splitting. The process of transverse cracking can be described using a [0/90], (Figure 1.2) laminate loaded in uniaxial tension. As shown, cracking of the inner 90° layer is caused primarily by the in-plane stresses .particularly the tensile stress component (a ) normal to the fibres in that layer. x  Initially, cracking begins as one single crack which may or may not span the entire width of the layer. Then, another crack is formed at some distance away from the initial crack. This is subsequently followed by more cracks which occur between existing cracks. With further loading, the number of cracks multiplies in the same manner and as this happens, the distance between the cracks (crack spacing) becomes closer and closer. It has been observed that the matrix of polymer composites starts to crack at low strains under both static and cyclic loading. In particular, the transverse cracking strain is usually much less than the bulk matrix failure strain. This has been attributed to the existence of a localized strain magnification between the fibres which increases as the interfibre spacing decreases [9]. For fibres strained in a direction normal to the fibre axis, the strain magnification factor (SMF) in the matrix scales by Ef/E (where Ef and E,,, are Young's modulus of the fibre and matrix m  respectively). This implies that the onset of transverse cracking can be delayed by a closer  Chapter 1  7  Figure 1.2 Transverse cracking in a cross-ply laminate f8]. matching of E and E . Realistically, typical values of SMF for systems such as glass-epoxy are f  m  between 8 to 10. This is relatively high, considering that the S M F theoretically approaches 20 as the interfibre distance tends to zero [9]. While the S M F is a contributing factor, it is also believed that the most likely source of matrix cracking is from the debonding of the fibre-resin interface of the transverse fibres. Debonds have been observed at strains less than 0.1% [10]. Once debonding starts, less cross-sectional area is available for a uniform stress distribution, causing stress concentrations and further debonding. Another common hypothesis similar to that of debonding is the existence of basic material system flaws of microscopic dimensions such as voids. These microflaws act as local stress  Chapter 1  8  raisers when external loading is applied. At a certain location where the stress condition reaches a critical value, some dominantflawsbegin to coalesce and form a large crack. As the laminate is layered, the crack may be either arrested or blunted at some interface boundary. This permits an increase of the applied load without catastrophic failure. Consequently, some other flaws may become critical at a higher load; and a similar matrix crack is formed at another location. Thus, after a certain amount of loading, be it monotonic or cyclic, a distribution of matrix cracks is obtained.  FIBRE BREAKAGE The available high strength and stiffness in fibre composites is only developed in very fine fibres, which have diameters in the range of 7-15 \im and are usually very brittle. A fibre is inherently stronger than the bulk form because the flaw size is limited by the small diameter of the fibre. And since the breaking of fibres is a completely random process, it is unlikely that an entire assemblage of fibres will fail at the sametime.When fibre fracture occurs, it is generally considered to be a critical mode of failure since fibres are the main load-bearing components of in the composite structure. The load in the broken fibres is either transferred to the adjacent neighbouring fibres or distributed equally among the remaining unbroken fibres in a cross-section. As fibre breaks increase significantly, some cross-section of the laminate will become too weak to support an increased load causing complete failure of the laminate. Studies of the distribution of fibre breaks in unidirectional laminate under static loading have shown that a significant number of fibre breaks do not occur until about 70% of the ultimate strength [11]. Even then, about 80% of fibre breaks were singlets, and another 15-20% were doublets. Singlets are defined as fibre breaks bounded by two unbroken fibres while doublets are two adjacent breaks separated by a distance not greater then five fibre diameters. Higher  Chapter!  9  order multiplets are defined in the similar way. As the highest order multiplet observed at any load in unidirectional as well as cross-ply laminates was 4, it appears then that the critical multiplet size is of the order of 5 or less.  Chapter 1  10  1.3 PURPOSE OF RESEARCH Design requirements, which are the criteria against which designs are checked, are application specific, but the key concerns are generally similar. In designing composite structures, these concerns usually include avoiding (i) initiation or growth of cracks from existing material flaws, (ii) local failure caused by overloads from enlarging with normal flight loads and (iii) propagation of local damage caused by impact and handling [12]. From the perspective of (i), the predominant mechanism in the initiation stage is the formation of intralaminar (within ply) matrix cracks in the off-axis plies. Generally in the damage process (Figure 1.3), matrix cracking occurs at low loads and its effects are usually not accounted for in the design stage. On the other hand, fibre breakage and delamination are significant at relatively high loads which usually exceed the design allowable. The criterion which is generally used for ultimate failure is the maximum strain in the 0° plies. When the local strain reaches the ultimate strain for the ply, with the appropriate statistical distribution, failure occurs. In many cases, most of the fibre failure and damage localization occur in the last 10% of the life of the laminate. Fatigue tests on both polymer and metal matrix laminated plates indicate that stiffness and residual static strength reductions by cracking in the plies alone may be as high as 10-50% [13]. This means that it is necessary for the designer to consider the effects of matrix cracking on composite properties. The existence of visual cracks in the off-axis plies of laminates is the first obvious failure event of the damage process. In the initial stages, matrix cracking itself is not critical although it can lead to a loss in strength, stiffness and stability, leakage of fluids in vessels, and changes in vibrational characteristics. The long term effects, however, are detrimental to the reliability and durability of the composite structure as a whole. This is because these cracks:  Chapter 1 (i)  11  are either arrested or blunted along the interfaces such as in the 0790° cross-ply systems (Fig 1.2). There, they cause interfacial failures which in turn provide initiation sites for delamination which is critical,  (ii) initiate fibre breakage as the crack tips propagate across the interface to the adjacent fibres, (iii) change the internal stress distribution of the laminate. This will result in local stress concentrations which may lead to premature cracking in neighbouring plies, (iv) reduce the load transfer capability of the matrix phase. Since the damage process is progressive, the entire process could be delayed if matrix cracking is prevented at low stresses.  D A M A G E M O D E S DURING FATIGUE LIFE 1 - Matrix Cracking  3 - Delamination  5 - Fracture  Fibre Breaking  P E R C E N T O F LIFE  Figure 1.3 The damage accumulation process in composite laminates fl41.  Chapter 1  12  Given the complexity of the damage process, any attempt to study all the failure mechanisms collectively would be an intractable task. A more pragmatic approach would be to isolate each damage mechanism and understand the physical mechanisms that cause its occurrence. It is believed in this work that a primary consideration to understanding the laminate failure problem is to first understand how matrix cracking behaves. The purpose here is to study the cracking behaviour of the off-axis plies for a set of laminates with a [0/8] type geometry. s  The orientation of the cracking plies 6, are 45°, 60°, 75° and 90°. The experimental program consists of static tension tests and a continuous monitoring of the cracking pattern and distribution. A systematic investigation of such a nature has yet to be documented. Most of the work in ply cracking, as will be seen, has used the [0/90] family of laminates for simplicity. s  However, most realistic laminates usually have plies oriented at different angles (such as 45° and 60°) to the loading axis. Using the experimental results, general relationships on matrix cracking behaviour using an energy method available from classical fracture mechanics are derived. The intent here is also to observe how the cracking pattern differs in each of the different ply orientations, in particular, the distribution of crack sizes, growth behaviour and crack interaction. The ultimate goal in the study of matrix cracking behaviour is the ability to describe analytically the crack formation and distribution characteristics by a general methodology. Only then will it be possible to assess its various effects on laminate mechanical properties.  Chapter 1  13  1.4 REVIEW OF T H E LITERATURE Existing design practices generally apply a two-dimensional laminate analysis (Laminated Plate Theory - LPT) whereby the stresses and strains in the individual laminae or ply are calculated. Knowing these stresses and strains and by applying some strength criteria at the lamina or ply level, the onset of laminate failure can be predicted. Commonly used failure criteria include the maximum stress, maximum strain and maximum work or Tsai-Hill theories [15] which are modified from similar theories for isotropic materials. Using this procedure, the ply which fails first or the load at which first-ply-failure (FPF) occurs may be determined. After the first ply fails, the laminate may still be able to carry a higher load, which means additional analysis of subsequent ply failures is required. One approach is the frequently used 'ply discount scheme' [16] which assumes that the failed ply can no longer carry any load. The load which it carried before failure is distributed to the remaining plies. The ply or plies that fail are deleted or discounted by reducing their transverse and in-plane shear stiffnesses to zero in those plies. A new round of analysis is then conducted, and so on. Using this method the progressive ply-to-ply failure; from first to last-ply failure, design limits and ultimate strengths can be derived. This scheme has been universally applied for over 15 years [17] and has been found to agree with experimental values to within 10-15%. Using LPT analysis together with some suitable failure criteria has been generally useful for predicting allowable stresses and strains for a given design since it can be optimized for all possible combinations of service stress conditions. When dealing with laminates having multidirectional plies, the Tsai-Hill maximum work criterion and other higher order failure theories [18,19] are particularly helpful since they take account of the normal and shear stress interactions between the plies. However, the overall method suffers from a major limitation in that no consideration is given to the different failure modes which can occur or to the effects  Chapter 1  14  of their interaction. For instance, the ply discount scheme breaks down when edge delamination dominates the failure process. This is because LPT assumes that the interlaminar stresses which are responsible for delamination failure are equal to zero. Also, in L P T analysis, failure in a ply is assumed to be an independent event and is solely governed by the inherent ply strength properties. Failure criteria are determined empirically and hence, they cannot be readily related to failure modes [20]. The physical phenomena of failure of composites are much too complicated to be described by any one failure criterion. This is true even with the best efforts made to determine which criterion correlates the closest with experimental results. Ideally, irrespective of the failure criterion used, FPF should be able to predict the onset of transverse ply cracking in cross-plied systems with fairly good accuracy given the simplicity of the [0/90] type configuration. Since transverse cracks are caused by in-plane tensile stresses normal to the principal loading direction, FPF would predict that the onset is determined by the tensile strength of the 90° ply. Contrary to this, experiments have shown that the initiation stress, and thus strain in the 90° ply vary greatly with the thickness of the ply itself. Also, even after the first transverse cracks appear, the cracked ply still continues to carry a considerable load whilst the FPF method discounts it completely. In an attempt to study the transverse cracking process, Hahn and Tsai [21] presented a simplified post-failure theory to model a cracked 90° ply within a laminate. It could predict crack spacing and the overall stress-strain behaviour of a [0/90] glass-epoxy laminate. This s  method considered that any failed lamina will support its load, which it was carrying when it failed, until total laminate failure occurs. However, like the FPF method, it only considered the interaction between plies and could not fully describe the failure process. Nevertheless, this was one of the earliest work that associated stiffness degradation with transverse cracking.  Chapter 1  15  Over the last decade or so, a number of different approaches have been used to explain and predict matrix crack initiation and growth as well as characterize their effects on stiffness reduction. These methods can be grouped into two distinct categories, namely; the energy methods and the statistical methods. In the energy approach, the studies have used shear-lag theory, continuum mechanics and empirical methods to explain this failure mechanism. As shown in Fig 1.4, some of the work in both the deterministic and statistical approaches involves the use of classical fracture mechanics. In this view it is reasoned that the actual amount of strain energy stored in the plies of the laminate and the manner of its release during a crack process play an important role in the crack initiation and its growth behaviour.  ANALYSIS OF P L Y C R A C K I N G  DETERMINISTIC  STATISTICAL  CONTINUUM  SHEAR-LAG  MECHANICS  E N E R G Y (G)  STRESS INTENSITY FACTOR (K)  LINEAR ELASTIC FRACTURE MECHANICS  Figure 1.4 The different routes to using fracture mechanics when analyzing ply cracking. It will be also become apparent in this review that most workers have used stiffness degradation as a parameter for measuring damage due to matrix cracking in their formulations.  Chapter 1  16  There are a number of reasons for this. Unlike strength or life measurements, stiffness can be measured frequently without further damaging the material. The method is non-destructive and directionally sensitive since it is a fourth rank tensor. Perhaps more importantly, while other techniques such as ultrasonic attenuation and acoustic emission are available, moduli measurements are quantitative.  1.4.1 Shear-Lag Methods Shear lag theory models the load transfer of shear stress to normal stress or vice versa. In studies of transverse ply cracking, shear-lag analysis is used to approximate the stress distributions near the matrix crack. Figure 1.5 shows the typical cross-ply geometry used in formulating shear-lag equations. These equations are based on equilibrium of a material element in the cracked ply. In the plane of the crack, the stress in the 90° ply is zero since it can no longer carry a load, but away from the crack, the stress is non-zero; load is transferred back into the inner ply from the adjacent outer 0° plies by shear. One of the most significant contributions in modelling the transverse cracking process was a series of studies by Garrett, Bailey, Parvizi and co-workers [22-27]. A one-dimensional shear-lag analysis was used to determine the magnitude of the additional stress Aa in the longitudinal plies after the first crack had occurred [22,23]. This additional stress Aa, was found' to decay from a maximum of Aa , with distance y as the load is reintroduced back into the 90° 0  ply (see Fig 1.5). Cracks were assumed to occurmidway between existing cracks as the shear stress would build up from these cracks. The value of Aa was evaluated in terms of the applied stress as well as the crack spacing and transverse ply thickness. This theory could predict the crack spacings as a function of applied stress for different transverse ply thicknesses and correlated well with experimental results.  Chapter 1  Figure 1.5 Typical geometry used for shear lag; analysis of transverse plv cracking,  17  Chapter 1  18  One set of results is shown in Figure 1.6 where it can be seen that the crack spacing decreases as the applied stress is increased. At high stresses, however, crack spacing approaches a limiting value which was found to be dependent on the inner ply thickness. The thinner the transverse ply, the smaller the final crack spacing.  14  12  E ^  o> c o  10  <B Q.  o  m  O)  2  3  100  200  300  400  500  600  -2 ,  Applied stress (MN m  F i g u r e 1.6  C o m p a r i s o n o f e x p e r i m e n t a l r e s u l t s w i t h t h e o r e t i c a l c u r v e s o f c r a c k s p a c i n g s as a f u n c t i o n o f  a p p l i e d stress f o r a 1 9 0 m m l o n g g l a s s f i b r e - e p o x y r e s i n c r o s s - p l y l a m i n a t e w i t h a t r a n s v e r s e t h i c k n e s s o f 1.2 m m . T h e s t e p p e d c u r v e s s h o w t h e c r a c k s p a c i n g w h e n the f i r s t c r a c k o c c u r s i n t h e m i d d l e o f t h e s p e c i m e n o f this p a r t i c u l a r l e n g t h , a n d the u p p e r a n d l o w e r c o n t i n u o u s c u r v e s i n d i c a t e the r a n g e o f c r a c k s p a c i n g s f o r c r a c k s p e c i m e n s o f a n y l e n g t h w i t h a n a r b i t r a r y p o s i t i o n o f t h e f i r s t c r a c k [ 2 2 ].  The thickness effect was further studied by Parvizi et al. [24] who observed that the strain required to initiate transverse cracking increased when the ply thickness decreased. They accounted for this constraint effect using an earlier theory by Aveston and Kelly [28] which showed that when the inner ply is very thin, the onset of cracking is partially or completely suppressed because insufficient elastic strain energy is released by crack growth.  Chapter 1  19  The concept of energy constraints was also studied by Flaggs and Kural [29] who conducted an experimental program using a class of [±0/9O] laminates to verify the above s  findings. They found that the initiation and development of transverse cracking were indeed subjected to two sources of constraints: one, the thickness of the cracking plies which sets the limits to the size of the transverse crack which it can attain before being arrested by the constraining plies, and two, the stiffness of the constraining plies relative to the stiffness of the cracking plies. It was shown that as the angle 0 increases, the constraint on 90° ply cracking tends to decrease. It was suggested by Bailey et al. [27] that initial cracking is also controlled by inhomogeneities within the transverse ply. Sheard and Jones [30] extended their findings by studying the effect of local fibre variations in fibre volume fraction and ply dimensions on the cracking pattern. Using a modified version of their shear-lag equation, the ply failure strain was determined. A computer simulation based on a Monte Carlo type search routine [31] was designed to determine the ply failure strain and hence the onset and location of transverse cracks in 0/90/0 glass-epoxy laminates. It was found that initial cracks tend to form in fibre rich areas but post-curing could change their location to resin rich cracks due to the increase in thermal strain produced in these areas. The shear-lag analysis was also employed by Reifsnider [32] who reported the observation of periodic saturation crack patterns in static and fatigue loading. It was noticed that after sufficient loading, the off-axis plies in a laminate reach a saturation state, wherein the distance between consecutive cracks in a particular ply is nearly uniform throughout the specimen. Reifsnider and co-workers call this the Characteristic Damage State (CDS). The saturation crack spacing in these distinctive patterns was found to be about one ply thickness. The CDS condition was said to be characteristic of a particular laminate configuration and is independent of load history and environmental factors. Reifsnider and Talug [33], as well  Chapter 1  20  as Reifsnider and Jamison [ 17], elaborated on this finding and devised a simple shear-lag method to evaluate stiffness loss due to cracking. Using the same [0/90] configuration as in Figure 1.5, s  the analysis was based on the following assumptions: (a) The normal stress in the applied load direction is constant over the ply thickness. (b) A n interlaminar shear lag zone of unknown thickness exists between the 0/90 plies, Shear stresses develop only within this boundary layer. (c) Cracks, remain sufficiently far apart so that the mutual interaction can be neglected. This simple analysis yielded reasonably good predictions of stiffness reduction (Highsmith and Reifsnider [34]) and like the analysis offered by Bailey et al., is of considerable conceptual value. The method, however, is not sufficiently accurate. The thickness of the boundary layer was estimated using specimen edge replication technique which is somewhat arbitrary and inconsistent. Furthermore, the assumption of no interaction of cracks is inconsistent with observed experimental data.  Chapter 1  21  1.4.2 Mechanics Methods The analyses of failure in composite laminates have been examined by both micro-mechanical or macro-mechanical models. The former approach analyzes the individual contribution of fibre, matrix and interface to composite strength. Generally, this has not been as successful as the latter approach and is not commonly used. The macro-mechanical approach considers the composite as a homogeneous system which has overall material properties of the macrostructure and applies the principles of continuum mechanics of homogeneous materials. It usually ignores local failure modes and develops a tensorial stress criterion for ultimate strength of the laminate based on the inplane stress state of each ply. The first application of continuum mechanics to treat microstructural damage is attributed to Kachanov [35]. In this method it is recognized that an attempt to analyze each individual crack in a multi-layered material with numerous cracks is hopelessly complex. Since the scale of the cracks is usually very small compared to the body, a more pragmatic approach is to treat the effects of the microcracks on the physical response using one or more internal state variables (ISV's). It is assumed the body remains continuous after cracking and the ISV's can be introduced as damage parameters to reduce its elastic constitutive properties. Using this method, Allen, Harris, Groves and Novell [36-42], developed a constitutive model for predicting stiffness loss as a function of damage in composite laminate. The phenomenological model utilizes a set of second order tensor valued ISV's to describe the internal damage state. Using thermodynamics, the model represents a set of damage-dependant laminated plate equations. The stress-strain relations in each ply are given by : ^ =C where  1 > u  e +/^ w  [ L 1 ]  Chapter 1  22  a is the stress tensor y  Eu is the strain tensor C  /  i j k l  ijkl  is the undamaged elastic modulus tensor represents the damage modulus tensor  au represents the internal state variables All subscripts range from 1 to 3. The superscript T| ranges from one to the number of damage modes. It is the internal state variable a* which quantifies damage state of each ply. 1  As they are derived from energy principles, the local energy loss is directly related to the strain energy release rate for crack propagation during loading. Using expressions for the strain energy release rate, the ISV for matrix cracking in cross-ply laminates was developed. Compared with experimental results, the model could predict axial stiffness loss as a function of matrix cracking damage in a family of [0/90] graphite/epoxy laminates fairly well. s  This group of workers has since attempted to extend the model to describe damage in the presence of both matrix cracking and interply delamination. Their efforts to develop a cumulative damage model incorporating classical Kirchoff plate theory is still in the early stages and needs verification using other sets of data. They have also attempted to predict the response of laminates having angle or off-axis plies as cracking plies. Although the predictions were promising, the model could not fully illustrate the effects of damage orientation on laminate behaviour. A similar approach was employed by Nuismer and Tan [43,44], who formulated constitutive equations for a cracked lamina using an approximate solution to the generalized plane strain elasticity problem. The compliances are given for general inplane laminate loading and are written in typical laminated plate form with damage state variables included. Using a three-dimensional elasticity model, they could derive a direct relationship between crack density or spacing and the damage state variable. Conditions for the formation of new cracks were  Chapter 1  23  specified. Experimental and predicted values for the reduced laminate stiffnesses agreed reasonably well for a family of [0/90 ] and [0/9OJ, and the model could explain the 'constraint' n s  effect reported by Ref [29]. However, it was algebraically too complicated to be extended to a more general problem. The work, however, explicitly showed that the ply constitutive behaviour is laminate dependent i.e., cracking behaviour is strongly dependent on the laminate in which the damaged ply is contained. The problem of stiffness reduction resulting from transverse cracking was also considered by Talreja [45] in terms of continuum damage theory utilizing a vector field description of damage for a two-dimensional solid containing oriented crack arrays. By assuming the energy density in a representing volume to be a function of the strain tensor and the damage vector set, he reconstructed the constitutive equations with the observable strains and the effective stiffness tensor. The stiffness reduction was mathematically described by a set of unknown ply damage constants which had to be determined experimentally. Another mechanics approach that has been used to analyze the effect of matrix cracking systems on the properties of the composite laminate is a technique commonly used to evaluate elastic constants of composite materials, that is the Self-Consistent Scheme (SCS). Essentially, the stiffness and compliance changes in the laminate are approximated by replacing the cracked layer with an effective homogeneous medium which contains many elliptical cracks. The "new" uniform overall stresses and strains are related by the usual constitutive equations. For an isotropic material with stacked cracks, the SCS retrieves the average crack opening displacement which may be directly obtained from the fracture mechanics solution to the problem of a single crack embedded in an infinite medium. Laws, Dvorak and Hejazi [46-49] employed the SCS approximation together with classical laminated plate theory to estimate the thickness effect on stiffness reduction in a family of [0/9CL/0],. laminates. It is believed that cracks are generally initiated at pre-existing flaws that  24  Chapter 1  are created by localized fibre debonding and microcracking. Although the cracks are randomly distributed in the ply, it is assumed that the crack distribution is statistically homogeneous. With this assumption, the three-dimensional model can predict the strength of transverse plies in a laminate as a function of ply thickness and applied stress state, provided experimental values of strain energy release rate for crack extension G are known. lc  Furthermore, their model gives a direct relationship between stiffness reduction and crack density p\ for a cracked laminate:  [1.2]  where d and b are the thicknesses of the 90° and 0° plies respectively, E , E, are the transverse t  and longitudinal Young's modulus and £ is a non-dimensional shear-lag parameter. The determination of £ has been argued in depth and is suggested to be determined by measuring the first ply failure stress. Another well known mechanics approach known as the variational method was used by - Hashin [50] to analyze the cracking in crossply [0° ,90°J laminates for tensile and shear m  s  membrane loading. In the analyses, admissible stress systems which satisfy equilibrium and all boundary and interface conditions were constructed. Then utilizing a relatively simple procedure based on the principle of complimentary energy, lower bounds for the elastic constants as well as ply stresses were obtained in an explicit way. The analysis allowed for crack interaction and could show the stress distribution between two neighbouring cracks. At the same time, the interlaminar out-of-plane stresses that were produced from cracking could be evaluated. For a set of [0 /90 l type of laminates studied, the results for effective stiffness agreed very well with m  n  s  experimental results. However, difficulties arise when the model is generalized for multi-layered  25  Chapter 1 laminates.  The variational technique was used earlier by Gottesman, Hashin and Brull [51] who considered a lamina in plane stress condition having cracks parallel to the fibres. This work, however, assumed that the cracks were non-interacting. Using the expressions for the elastic energy associated with a single crack in an infinite homogeneous orthothropic sheet, they derived expressions for the overall elastic constants of a cracked lamina. They also gave the estimates for the elastic constants by the SCS and gave the bounds to these constants by a variational method. A basis for estimating stiffness changes due to transverse cracking for a two-dimensional case was established. However, unlike the later work by Hashin [50], the accuracy of this method remains to be experimentally verified.  1.4.3 L E F M Methods It was suggested by Bailey etal. [26] that the failure modes associated with matrix cracking are in fact governed by fracture mechanics criteria and not by strength. As discussed earlier, many investigators have incorporated the concept of the strain energy release rate G, from LEFM into parts of their analyses. Meanwhile, others have strictly confined their analysis strictly to the principles of classical linear elastic fracture mechanics (LEFM). The use of LEFM to predict failure is based on the existence of a material property, fracture toughness, defined by either the critical stress intensity factor K , or the critical strain energy release rate, G . These properties e  c  are related and correspond to the onset of unstable crack growth. Due to the analytical difficulties in determining the crack tip conditions in a heterogeneous, layered, orthothropic material, the use of G rather than K as a criterion for crack growth is preferred. This point will be elaborated further in the next chapter. Based on the fundamental assumption that a microcrack exists in the matrix phase, Wang and Crossman [52-53] used LEFM criteria to analyze the conditions for its growth. They  26  Chapter 1  calculated the strain energy release rate associated with crack growth using a generalized plane strain finite element analysis and crack closure technique. Assuming that the crack growth would become unstable when the energy release rate reaches the critical value for the material, they calculated the threshold stress. The analysis was tested on transverse cracking in 90° plies constrained by ±25° and 0° plies. Although good agreement was found, the conceptual characteristics of the analysis is open to question. Crack growth in the transverse ply is actually three-dimensional, that is, it occurs through the thickness as well as across its width of a test specimen. The former is immediate and insignificant [54] whereas the latter is gradual under loading. It is therefore of interest to study crack growth as it gradually moves across the transverse ply width before being terminated at the free-edge. Wang and Crossman, on the other hand, modelled the through-the-thickness crack growth (two-dimensional) and implicitly assumed that the matrix crack extends across the laminate width. The strain energy release concept was also used by Poursartip [55] to model matrix cracking and correlate stiffness reduction due to transverse cracking with peak fatigue stress for a GFRP cross-ply laminate. The associated strain energy release rate G, considered to be the controlling factor, was calculated by using the compliance expression:  G=  P^d£ It da  [1.3]  where P is the load on the laminate, t the cracking ply thickness and dClda is the rate of change of compliance with crack length. He assumed that all the matrix cracks in the laminate could be treated as a single equivalent flaw. Caslini, Zanotti and O'Brien [56] applied the concept of crack resistance curves (R-curves) borrowed from the L E F M to characterize the cracking behaviour of cross-ply and quasi-isotropic  Chapter 1  27  [0/+45/-45/0/90], under quasi-static and fatigue loadings. This was based on earlier work by O'Brien [57-58] who successfully used the R-curve for the edge delamination problem. The critical strain energy release rate for formation and accumulation of matrix cracks was calculated using the relation [57]:  G=-V  £dE 2dA  [1-4]  where e is the applied strain, V is the volume of the laminate and dEldA is the rate of stiffness change as the flaw extends by an area dA. Using the same assumption as Poursartip for the crack length, G values were plotted for a range of crack densities in the transverse plies (Figure 1.7). It was found that the laminate resistance to formation of new cracks progressively increased as the number of these cracks increased. This increased crack growth resistance was not attributed to other energy absorbing mechanisms, but to the constraint and load transfer capability of the neighbouring plies adjacent to the cracked ply. The relationship for the change in stiffness as a function of change in crack density, dEIdD was also calculated using a numerical stress analysis program, NASTRAN. Results generated from the three-dimensional finite element model was found to agree well with experimental results for the multidirectional laminates. Han, Hahn and Croman [59] proposed a method for calculating the strain energy release rate based on a second order polynomial to represent the crack opening displacement (COD). It was earlier shown by Hahn and Johannesson [60] that the energy release rate associated with the widthwise growth of a long through-the-thickness flaw is equal to the work of crack closure per unit width of the fully developed flaw. The energy release rate is independent of the flaw length and hence the growth of the flaw is stable. The calculated G per ply crack is then used in conjunction with the energy balance principle to predict the multiplication of cracks. Based  28  Chapter 1 600  • •  •  •  500  •  CM  E  rr  A  A  •  400  •  4  300  200  Gc=  *  • • ""^  —k 100  •/>  0  [0/90]s  < N s  i  0  [+45/-45/0/90]s  i 0.2  i .  i 0.4  i  i 0.6  i  i  i  0.8  1  Crack density * crack width, Dw Figure 1.7 Matrix cracking resistance curves. Redrawn from Ref T561. on an experimental correlation of the analytical result, a resistance curve was proposed to be used as a measure of the resistance to crack multiplication. Similar to results by Caslini et al. [56] this resistance was found to increase with increasing crack density. Ogin, Smith and Beaumont [61-63] approached the crack growth problem by attempting, to calculate the stress intensity factor K, for a transverse crack in a (0/90/0) laminate using a simple, essentially one-dimensional model. Away from the crack tip, the load is assumed to be reintroduced into the 90° plies by shear at the 0/90 interface, whilst a stress intensity is said to be present due to the localized stress disturbance at the crack tip and is derived as : [1.5]  Chapter 1  29  where 2d is the transverse ply thickness and o, is the stress in the transverse layer acting on the crack tip. The value of K is derived essentially by arguing that it must take the above form, with the transverse ply thickness being the only relevant dimension. To define a , they introduced t  a dimensionless parameter £2(s), which allows for the crack spacing effect.  1.4.4 STATISTICAL A P P R O A C H The statistical viewpoint is based on the premise that there is a certain random distribution of microdefects in each ply and at the ply interface. Each of the flaws can act like a small crack. In the analysis, the fracture of the cracking ply is assumed to be a result of statistically distributed defects of different size. Thus, the explanation of the thickness effect; increasing the thickness of the transverse layer increases the probability of crack formation. This reasoning, however, can only explain the observed thickness effect qualitatively. Manders, Chou, lones and Rock [64] measured the distributions of crack spacings developed in cross-ply laminates and showed that the 90° ply has a variable strength. It was found that the probability of crack formations close to existing cracks was lower than in the other areas of the 90° ply. This finding could not be fully accounted by deterministic theories. Their approach was further refined by Fukunaga, Chou, Peters and Schulte [65] who considered an inhomogeneous stress distribution in the cracked laminates. Using a two-parameter Weibull distribution of transverse strength, together with a shear-lag concept, they determined the onset of cracking and subsequent multiple cracking. A comparison of the analytical results with experimental data from other workers showed reasonable agreement. It was found that the first cracking strain, not the ultimate, was very sensitive to the thickness of the 90° layer. Lim and Hong [66-67] provided a modified version of their one-dimensional model in which the boundary conditions were satisfied as the crack spacings became smaller. Their approach on the other hand, was a deterministic one using an energy criterion to predict onset and multiplication of  Chapter 1  30  transverse cracking. Peters [68-69] estimated the strength distribution of the 90° plies in [±0/9OJ laminates s  and found that the Weibull shape parameter and the fracture strain were thickness dependent. This was attributed to the suppression of growth of defects in the 90° ply which were lying close to the 0790° interface. Wang, Chou and Lei [70] attempted to model the cracking process for static and cyclic loads by combining the G concept from fracture mechanics with a probabilistic representation of the material flaw distribution. They introduced the concept of 'effective' flaws to replace the conventional constant ply strength criterion found in ply-elasticity. The flaw distribution was represented by a normal distribution and the strain energy release rate was calculated using a numerical method. Using a Monte-Carlo simulation routine [71], the model tried to determine the order of occurrence of the cracks and the entire load- sequence of the multiple crack formation. The usefulness of simulation results has yet to be verified due to the lack of experimental results. Recent work by Swanson [72] used a shear-lag to determine the stress distribution around the matrix cracks and the associated stiffness and shear modulus reductions. Shear loading of the cracked laminates was considered, along with the more usual case of cross-ply tension. By incorporating the calculated stress distributions into a Weibull failure criterion that included combined stresses (taken from Tsai-Wu quadratic expression), the interaction of crack spacing and applied stress was obtained. The analysis predicts directly stiffness and shear stiffness reductions as a function of crack spacing. Although the analytical results were reported to agree within 10% with those of Laws et al.[49], they have yet to be compared with any experimental results.  Chapter 1 1.4.5  COMMENTS O N LITERATURE  31 REVIEW  From this survey of the various approaches taken, it is difficult to lean towards either the energetic approach or statistical methods since both have their merits. Perhaps, both approaches should be considered in any satisfactory argument. Generally, most of the models were effective in predicting the first failure of the inner cracking ply in cross-ply laminates and its dependence on thickness, but few have considered crack accumulation. The mechanics models are often too complicated mathematically even for predicting failure in simple crossply systems. One of the problems in assessing the various approaches is that they are based on a limited amount of experimental data. Unless comprehensive sets of data are available to verify and compare all these models, there will be little possibility of improving or utilizing any one of them as a general methodology for predicting ply cracking.  32  Chapter 2  CHAPTER 2  SCOPE OF RESEARCH  2.1 E X P E R I M E N T A L P R O G R A M The mechanism of matrix cracking is influenced by a number of parameters. The most important parameters are listed below: 1. the fibre properties 2. the matrix properties 3. the fibre volume fraction 4. the defect distribution 5. the constraining influence of the neighbouring layer 6. the quality of the interface 7. the stacking sequence and lay-up 8. the orientation of the cracking ply In this work, the first seven parameters are constants since the same materials are used and the laminates are assumed to rank equally in terms of interfacial quality and defect distribution. This is a reasonable assumption as the fabricated specimens have nearly identical fibre volume fractions and negligible difference in their void contents. Also, a similar stacking sequence whereby the inner cracking ply is stacked between two adjacent 0° longitudinal plies is used in this set of experiments. The objectives of the research program were two-fold. One was to investigate the influence of the orientation of the inner cracking ply on matrix cracking behaviour. The changes in axial  Chapter 2  33  stiffness were monitored as matrix cracking progressed under tensile loading. At the same time, the actual cracking sequence and distribution was photographed and quantified. This allowed the mechanisms of initiation and growth in a set of laminates to be identified. The second objective was to study the matrix cracking phenomena both deterministically (using fracture mechanics) and statistically. In the first instance, our interest was to examine the strain energy release rate, G, as a parameter for predicting crack growth. In the latter instance, our intent was to observe how crack tips interact and crack spacings change as matrix cracks multiply in the off-axis plies. To accomplish these objectives, the course outlined in the flow diagram in Figure 2.1 was taken.  FABRICATION OF LAMINATES  CRACK GROWTH  STATIC TENSILE TESTS  MEASUREMENTS Image Analysis  - load vs deflection - stress vs strain - stiffiiess E  -  crack number and size distribution  -  crack tip interaction  Crack length a = f( a ,E/Eo, e )  t  dE/da for each 9  R-curve, Cracking Behaviour = f ( e )  Figure 2.1 Flow diagram showing course of present research work,  Chapter 2  34  The study used a limited set of glass/epoxy laminates having a simple configuration [0/9]  s  where 0 is the off-axis angle taking the values 90°, 75°, 60° and 45°. The largest 0 was chosen to determine the 90° ply matrix cracking behaviour which is driven by a pure tensile Mode I loading (see section 2.2.1). In the smallest 9 ([0/45] ), cracking in the inner plies occurs mainly s  by shear (Mode II) although both Modes I and II are present. The other two sequences enable any distinct trends between the first and last sequence to be readily recognized since they represent a combination of both fracture modes. The proportions of Modes I and II in each respective laminate which are calculated from the material properties (section 5.1) are given in section 2.3.2 of this chapter. The laminates were fabricated using an epoxy resin system which has a refractive index closely matched to the fibreglass after curing so that the specimens would be optically translucent. Hence, when a high intensity light is illuminated through the specimen, the cracking pattern is visible and can be photographed. As shown in Figure 2.1, the experimental work consisted of tensile tests accompanied by crack growth measurements. A correlation of the data obtained enabled two sequences of analytical steps to be taken: A l - stress, strain and stiffness E, were related to matrix crack length for the 9 values chosen. A2 - once dElda was obtained, G was calculated and crack growth resistance or R-curves were generated for each value of 9. A3 - having constructed R-curves for each 9, relationships between crack resistance and dElda could then be derived. B l - measurement of the coordinates and length of each crack by image analysis provided the number, size and location of all the cracks. A statistical frequency distribution in terms of size could be related to the applied strain for the different 0s.  Chapter 2  35  B2 - using the coordinates of the cracks, an analysis of crack growth behaviour from one load to another was performed. The crack tip interaction as cracks grew closer was also analysed. A correlation between applied strain and the crack spacing was also found.  2.1.1 Testing of Angle-Ply Laminates As noted previously in the literature review, the outer longitudinal 0° plies present a constraint to the cracking in the center 90° plies. Hence, by varying the orientation of the center plies, the influence of this constraining effect can be studied. The off-axis or angle plies have their unidirectional fibres neither parallel nor perpendicular to the direction to the applied tensile forces. In a uniaxial tension test, the applied tension direction is not coincident with the principal elastic axis of the inner angle plies of the laminate. Thus, a rotation is induced. This arises because the [0/9] laminate is balanced but not symmetric and a shear coupling exists in the s  laminate when subjected to tension. The shear coupling causes the flat specimen to undergo a rotation that is restricted by the stationary grips. Hence, a slightly non-uniform state of stress is applied inducing shear and bending forces at the specimen ends. The other alternative to using the unbalanced geometry was to choose the next simplest lay-up, the [O/±0] . This would eliminate the shear coupling problem but introduces a host of s  other problems, the most important of which is the inability to uncouple the cracking behaviour in the +9 and -9 plies. According to Pagano and Halpin [73], the non-uniform stress problem can be minimized if a large specimen aspect ratio (length-to-width greater than 6) is used. Before any experimental work began, it was necessary to determine how this shear coupling would affect the tensile test results. If only off-axis specimens ([9]J were tested in tension, coupling would be most pronounced when 0°<9<45° but would disappear as 9 tends to 9 0 ° . As illustrated graphically in the Figure 2.2 [74], the worst case for typical glass/epoxy systems occurs when 9 is about 15° where the shear coupling ratio, T[ reaches -1.0. xy  Chapter 2  36  In the present work, since the off-axis plies constitute only half the [O/0] specimen volume s  and are constrained by the outer 0° plies, the end constraint effects were not expected to be severe. In order to alleviate the effects of shear coupling, two measures were also taken. Firstly, the value of 0 was restricted to between 45° to 90°. Secondly, a specimen with an aspect ratio of about 8 was used. Furthermore, a simple analyis was carried out to estimate the magnitude of the possible shear coupling effects.  Figure 2.2 The variation of shear coupling ratio as a function of off-axis angle T741  37  Chapter 2 Analysis of Shear Coupling Effects The shear coupling ratio r i ^ , can be determined experimentally since it is defined as  Ixy  where y  xy  is the shear strain and e , the longitudinal strain. This approach was not taken. x  Alternatively, r\  xy  may be calculated from the expression T)^ =S /S 16  n  where S  n  and S  are  16  elements of the compliance matrix for the laminate. Calculating T| for the worst case, which xy  is the [0/45] laminate gives a value of 0.2763. To determine if this is significant, we can use s  an expression derived by Halpin and Pagano [73] who analytically solved the stresses and strains induced by shear coupling in a constrained specimen. The expression relates the actual axial Young's modulus E* to E , the measured Young's modulus, x  x  . E E =• 1-C  [2.2]  z  in which  3  5  " 3S {  6 6  ^  -  6  + 2S (^)  [  2  '  3  ]  2 }  1 1  and LG/W is the specimen length-to-width ratio. The value of £ in equation [2.2] is the error between the actual and measured moduli. For the [0/45] lay-up, £ is 0.00165 (0.165% error) s  when LG/W is 8. The negiglible error implies that the stresses in the grip area of the test specimen are fairly uniform.  Chapter 2  38  2.2 T H E O R E T I C A L BACKGROUND 2.2.1 Brief Review of Linear Elastic Fracture Mechanics K, THE STRESS INTENSITY  FACTOR  The reader is directed to standard texts for a review of L E F M principles (e.g., Ref [75]). The following assumptions are relevant to this work : a.  Crack growth is co-planar and self-similar: the crack always advances along the original crack direction.  b.  The crack tip displacement can be separated into three different modes: crack-opening mode I, in-plane shearing mode II and anti-plane shearing mode i n (Figure 2.3).  MODE I - OPENING  ^ M O D E II - SLIDING M O D E III - S H E A R I N G  Figure 2.3 The three modes of crack surface displacement. c.  The crack tip stress and displacement equations for the above modes can be derived by a number of methods, both analytical and numerical. For example, using Westergaard's equations [76], the problem of an infinite two-dimensional homogeneous plate with a central crack 2a (Figure 2.4), and subjected to Mode I loading can be solved.  39  Chapter 2  t  t  t  t  t  I  I  I  I  I I  Figure 2.4  t  Crack in an infinite plate T751.  K the M o d e I stress intensity factor which describes all the crack tip stresses takes the h  form o f :  K, = oyfc  £2.4]  where a is the applied tensile stress and a is the crack half-length. Similarly, stress intensity factors K and K, can also be determined. It can be seen from Westergaard's equations that the u  u  crack tip stress distributions for isotropic materials are independent of material orientations, and material properties. A l o n g any radial direction (see Figure 2.4) they are only functions of ll^r.  If the same assumptions (a), (b) and (c) are made for an orthotropic medium, K, has to  Chapter 2  40  take the same dimensional form as Equation 2.4. The critical values of Kj which corresponds to the start of unstable crack growth is a material constant called the fracture toughness, denoted by**,  G, THE STRAIN ENERGY RELEASE  RATE  The condition for unstable crack growth, can also be expressed in the form of its energy release rate. The energy balance of the plate shown in Figure 2.4, per unit thickness, can be expressed as : d(U-F  + W)  =  [2.5]  da  where F = work done by external forces W = energy for crack formation U = stored elastic energy of the plate  According to Griffith, at the instant of crack extension (by amount da), there is a release of energy equal to the difference between the rate of work done dFlda and the rate at which elastic strain energy is stored dUlda; this energy is assumed to be converted entirely into dW/da, the energy required for crack growth. A balance of energy during stable crack extension leads to the criterion d(F-U)_dW da da  [2.6]  Chapter 2  41  Equation (2.6) defines the general limiting condition under which the existing crack begins unstable self-similar propagation. The strain energy release rate, G, which can alternatively be viewed as the crack extension force, is defined as G = d(F-U)/da per unit crack extension. The energy consumed in creating the crack is denoted by R = dWIda which is called the crack resistance (force). To a first approximation, it can be assumed that for a brittle system the energy required to produce a crack (the decohesion of atomic bonds) is the same for each increment da. This means that R is a constant for typical brittle materials. The energy condition for crack growth in a plate (Eqn 2.6) states that G must be at least equal to R, the material's fracture resistance, before crack propagation can occur. If R is a constant, this means that G must exceed a certain initial value G . Hence, the crack grows when c  G>G . C  Although satisfying the energy criterion is a necessary requirement for crack extension, it is not sufficient. Even if sufficient energy for crack growth can be provided, the crack will not propagate unless the material at the crack tip cannot cope with local stresses and is ready to fail. By considering the crack tip stress and strain distributions, it can be shown that for Mode I loading, the two criteria are related by the expression: [2.7] In an isotropic material, E' in Equation 2.7 (under plane stress) is the Young's modulus commonly termed E. However, E ' for an orthotropic medium [77] is [2.8]  Chapter 2  42  where a , a , etc. are the elastic compliances associated with the principal material directions. n  12  Manipulation of equation [2.8] will show that E' may also be expressed in terms of the elastic moduli E , E , etc. giving n  12  £  '  =  >  / 2 ^  v « . _ i) - A _ _ ^ yE E n *G (  l  B  +  P- * 9  L  n  22  n  2.2.2 Application of L E F M to Composite Laminates  As already pointed out in the preceding chapter (section 1.4.3), it is mathematically very difficult to determine the value of the singularity at the crack tip in a composite laminate. For delamination problems in particular, the singularity at the crack tip is not necessarily of the simple  form. It is found to be a function of the orientation and moduli of the adjacent  layers, and may have an oscillatory component [78]. Generally, it is not possible to define the stress intensity factors K AT and K for cracks between dissimilar anisotropic composite layers h  w  UI  in the same manner as for cracks in homogeneous material. In the same manner, the orthotropic, heterogeneous and layered nature of the composite would have to be considered when tackling the ply cracking problem. A three-dimensional analysis which involves geometric and material discontinuities is often necessary. In the case of off-axis plies, the crack tip displacements are mixed mode consisting of Mode I and Mode II. It is extremely difficulty to determine and satisfy the boundary conditions, particularly when crack multiplication occurs. This is needed if mutual interaction between cracks are to be considered. Thus, for these and other reasons, the use of G as a criterion for crack growth has been preferred over K. The strain energy release rate concept begins with the assumption that the strain energy is released when a crack surface is created in a strained elastic body. Hence, the problem is  Chapter 2  43  simplified since crack growth is dealt with from a global energetic view rather than by addressing the complex stress fields near the crack tip. The rationale here is that by determining the actual amount of strain energy trapped in the plies of a laminate and by understanding the manner of its release, the crack initiation and growth behaviour can be quantified and explained. Furthermore, the strain energy release is controlled at least partially by the structural interaction between the plies during the loading of the laminate and this interaction can be altered by the distribution of the cracks. G has been determined for composite laminates using both compliance solutions and numerical virtual crack extension techniques [52,57,79-81]. Numerical analyses usually employ the finite-element method in conjunction with a procedure such as the Irwin crack closure integral [82]. This procedure involves the introduction of a virtual crack of known dimensions, and computing the work done to close it. The work done is assumed to be equal to G and is expressed in terms of nodal forces and displacements. Thus, the crack problem is considerably simplified because it is not required to know the singular stress field near the crack tip. Homogeneity and orthotropy is usually assumed. Reasonably simple models such as that in References [52,57] compare well with experimental results but are often specific to the problem analysed. Moreover, these simple analyses determine total strain energy release rates, G rather than the components G G and M  GJJJ which  H  N  are difficult to calculate. It is not known if such models will adequately predict the  interactive behaviour of damage since the use of G , may not be justified in all cases. Unless M  G  M  is the driving force, it will need to be decomposed into its mode I, II and III components  which will complicate the modelling considerably.  44  Chapter 2 2.2.3 Determination of G for Transverse Ply Cracks  The Griffith energy criterion for fracture [83-84] states that unstable crack growth can occur if the energy required to form an additional crack of size da can just be delivered by the system. There are two loading conditions to be considered. In the case of a plate with fixed ends, the external force cannot do work, and the energy for crack growth must just be delivered by the release of elastic energy. If the plate ends are free to move during crack extension, work is done by the external load, and the elastic energy of the plate increases. Consider a crack in a plate of thickness B. If it grows by da under constant load (as shown in Figure 2.4), a displacement of dv will be produced in the loading direction. Hence, the work done by the external force is P dv. From equation 2.6, it follows that d(F-U) l( dv dU da B\ da da  [2.10]  p  J  where U is the total elastic energy in the plate. As the deformation is elastic, the compliance C of the plate is C = vIP or v = CP. The elastic energy contained in the cracked plate is then  U=\PV  2  =\cP  [2.11]  2  2  By combining equations 2.10 and 2.11, we have the total strain energy release rate, G „ (where w  G , = G,+G„+G ) to be t0  m  Chapter 2  45  and [2.13]  '•251 da , As the terms with dPIda cancel, this means that G , is independent of whether or not the load l0  is constant, for small da. Before G, , can be calculated for off-axis ply cracking, the crack extension da must first 0  be defined. Although the geometry is different from the conventional case of a single crack growing in a plate, two new free surfaces, each of length da, are still being created in a brittle material (off-axis ply). As proposed by Poursartip [55], the crack length can be defined as the sum of all the transverse crack lengths. So WL  a = 2s  [2.14]  where W = the width of the specimen L = the gauge length 2s= the crack spacing For the case of off-axis plies, where the cracking ply is oriented at some angle 0, W is the maximum cracking width of the specimen W , i.e. W = Wl sin 0°. The unconventional geometry is reflected in the compliance-crack length relationship. By definition, the compliance C is inversely proportional to the stiffness E. With [2.15]  It follows then  46  Chapter 2 PL Av  [2.16]  „ L AC  [2.17]  B  E=  E=  where A is the cross-sectional area of the laminate. If we let LIA = K , then C = KJE and differentiating, we have [2.18]  d£ _K_d£ =  da  E da 2  Using the same geometry as in Figure 1.5, it can be seen that if A is the cross-sectional area of the laminate and if A = W2(b +d), then  K  L 2W(b+d)  [2.19]  2.3 ANALYSIS  Equations 2.13 and 2.18 show that dElda must be determined in order to calculate G  tat  This function may be predicted analytically or numerically using approximate methods (as seen in section 2.2.2), or alternatively it may be determined empirically. In this work, the latter method is chosen. The stress-strain and stiffness values calculated from load-displacement plots obtained is correlated with crack length for each value of 6. As dElda is the relationship of interest, it is obtained from a plot of normalised stiffness EIE against a, the crack length. It is noted that the 0  normalised stiffness - with respect to the original stiffness - is used rather than the absolute values. By doing so, any minor differences between the test specimens can be accounted for, and hence more accurate comparisons of stiffness changes can be made.  47  Chapter 2  The relationship between E and a for cross-ply lay-ups is known to be linear initially but drops off asymptotically as the crack spacing becomes very close. Using shear-lag analysis, Steif [85] developed a closed-form expression for stiffness loss in a [0/90] laminate as a function s  of crack spacing, 2s, in which [2.20]  where  [2.21]  3G E (b+d) £i E db l2  0  2  2  E , E are the moduli of the cracked and the uncracked laminate, respectively. E E and 0  x  L  2  G  l 12  are the lamina longitudinal, transverse, and shear moduli, and b and d are the thicknesses of individual longitudinal and transverse plies, respectively. Assuming that the crack spacings are large, (i.e. tanh (ks) ~ 1), Ogin et al. [61] used a series expansion for (1 + x)" to simplify equation [2.20] and showed that the axial modulus is 1  related to D, the average crack density (D= l/2.s) by the expression [2.22]  E=E (\-cD) 0  with [2.23]  1 The shear modulus should actually be G  23  but G  12  and G  23  are very close numerically.  Chapter 2  48  for a [0/90], lay-up. Accordingly, if the E/E versus a function is linear for any [0/0],, the 0  expression in [2.22] can be applied giving E -(0)=l-c(0)a  [2.24]  where c(6) would be the [0/6], laminate constant and a, the crack length. Once dElda is available, the strain energy release rate, G  m  can be calculated using  equation [2.13] and used to plot the R-curve for each 0. For cross-ply laminates, G  WI  is equal  to G, (opening mode). For angle-plied laminates, G will have contributions from G, and G„ w  (in-plane shear mode).  2.3.1  C r a c k growth resistance, R-Curves A common practice in determining the resistance of a material to fracture is by using crack  growth resistance curves or R-curves. An R-curve is a continuous plot of toughness development in a material under loading, in terms of G (orK)  against a, the crack extension. During slow-stable  crack growth, the crack extension force, G applied to the specimen must just be equal to the developed crack growth resistance G . Otherwise, the crack would grow even further. The crack R  can be driven by increments of load or displacement. As measurements are made at each increment, the G values calculated become the individual data points on the R-curve for the material.  Chapter 2  49  2.3.2 Mode II Component in Off-Axis Laminates The relative magnitudes of the Mode Et component with respect tp the combined Modes (I+H) in each laminate can be determined either in terms of the stress intensity factor, K or the strain energy release rate, G. From equation [2.4], K, = trjna and similarly, K„ = W T W where cr and T are the normal and shear stresses in principal material axes, Figure 2.5. It can also be seen that a 2 is the normal stress responsible for crack opening in the off-axis ply whereas T is the shear stress along the fibre direction.  X  2  Figure 2.5 Stresses acting on an off-axis ply element. It follows then that the relative magnitude of Mode II component for K can be expressed as  Chapter 2  50 K„ _ T K,+Ki,~ T + a  [2-25] 2  To express the proportion of Mode II in terms of G, using equations [2.7] and [2.4] it can be seen that Gi^{K IE') or G^irflE').  A similar expression also applies for x such that  2  Gn^tflE).  This then gives the expression ± * *  G„ Gi + G  [2.26]  E  n  E"  E'  where E " is the Young's modulus for an orthotropic medium for shear opening Mode II [77]. The values of E' and E " are calculated from the material properties and a and x in each lay-up 2  are determined using Laminated Plate Theory (Appendix A). The calculated values in expressions [2.25] and [2.26] are listed in Table 2.1.  Table 2.1 RELATIVE PROPORTIONS OF MODE II IN OFF-AXIS PLIES  Lay-up  IlL  [0/45],  .56  .476  [0/60],  .42  .224  [0/75]  .25  .052  [0/90],  0  0  s  Chapter 2  51  It can be shown by plotting the above Mode LT proportions against 6, Figure 2.6, that the proportion of Mode II decreases almost linearly with increasing 9 for the K definition. In terms of G, the proportion of Mode II drops dramatically from 45° to 60° and then decreases gradually. While it is permissible to add the energies from the different modes (G,, G„, G ) to obtain m  the total strain energy release rate, G in combined mode cracking, the different stress intensity Mt  factors (K K K ) are not additive. Hence, the use of G to represent the relative Mode II ratio h  lh  UI  (equation [2.26]) is more meaningful for combined loading conditions.  0.8  •  0.7  in terms of  + in terms of  K G  0.6 -  Mode II Mode l+l  0 5  0.4 0.3  0.2 -  0.1 -  30  70  90  Off-axis Angle  Figure 2.6 The variation of Mode II proportion as a function of off-axis angle.  52  Chapter 3  CHAPTER 3  FABRICATION 3.1 MATERIALS The material system used in this work is E-glass/epoxy. The E notation represents "electrical grade" fibreglass roving which is used as a general-purpose fibre with medium strength (as compared to other fibreglass grades). Typically, E-grade glass has a tensile modulus of about 70 GPa and fails at about 5% strain. The glass roving used was supplied by Fibreglass Canada and is identified by batch number 475 AA 1100. When received from the supplier, the fibre filaments are already sized, that is to say chemically coated to make them compatible with epoxies as well as polyesters. The resin to be reinforced consists of a combination of epoxy and hardener (curing agent). Epoxies are thermosetting resins where the liquid resins are converted to a hard brittle material by curing. Curing refers to the chemical reaction whereby the epoxide group is cross-linked with the hardener forming atightlybonded three-dimensional network of polymer chains. The characteristic of epoxies, as thermosets, is that they do not soften on reheating and cannot be reformed. The epoxy used is commercially known as EPON 828 which is an industrial grade suitable for structural purposes. Its chemical composition is that of diglycidyl ether of bisphenol A (DGEBA). The hardener is an amine-based liquid known as Ancamine K1784. In a cured state, this resin has a tensile modulus of about 2.2 GPa and failure strain of about 3%. As already stated, the reason for choosing this particular resin system was so to obtain an optically translucent end-product. Such a material is not available commercially in prepreg form at present. (Prepreg is a sheet of unidirectional fibres impregnated with resin).  Chapter 3  53  For fabrication, the desired resin is obtained by adding 40 parts by weight of hardener to 100 parts by weight of epoxy. Although this resin system can harden or cure at room temperature, the normal practice is to cure at a higher temperature to accelerate the process. Since this is usually done in an enclosed vessel (an autoclave), the quality of the final composite part is generally much better controlled.  3.2 FABRICATION PROCEDURE The fabrication of glass-epoxy laminates involves two main processes: (i) production of prepreg and hand lay-up (ii) autoclave cure.  3.2.1 Production of Pre-preg The first stage of fabrication is to make the prepreg. A drum prepregger is used to wind the fibres onto a 61 cm diameter mandrel lined with release paper (Figure 3.1). Fibre roving is first passed over a rotating wheel immersed in a resin bath, then immediately passed through glazed ceramic rollers to remove the excess resin, and finally wound onto the rotating mandrel. In order to obtain a precise overlap of fibres, the drum is set to rotate at 25 RPM and travel with a crosshead speed of about 5 rnm/s. When winding is completed, the sheet of prepreg is quickly removed from the mandrel and placed onto a long table for cutting. A typical prepreg sheet is 1.83 m long and can range from 30.5 to 40.6 cm in width, depending on the lay-up desired. First, 30 x 20 cm sheets of unidirectional 0° are cut parallel to the longitudinal fibre direction. Then, 90° and off-axis plies of the same size are cut out from the remaining prepreg. To obtain an accurate orientation of the 75°, 60° and 45° plies, pre-cut pieces of release paper are placed on top of the prepreg to serve as a guide for cutting.  Chapter 3  54  HAND LAY-UP Once the plies are cut, they are stacked according to the desired sequence. As each ply is laid down, it is hand rolled with a 15 cm wide roller to remove bubbles and even out the resin. The release paper is then carefully removed. The blunt side of a surgical knife is used to pull the fibres taut and close gaps. After each four-ply laminate is stacked, it is placed inside polyethylene bags and stored overnight in a freezer at about -40 °C.  Chapter 3  55  3.2.2 Autoclave Cure The next stage of fabrication is to cure the prepreg inside an autoclave to form a solid laminate. Curing involves the application of heat and pressure to consolidate the plies. The autoclave provides both (a) heat for curing (initiating and maintaining the chemical reaction) the resin and (b) pressure for compacting and squeezing out excess resin. A typical autoclave cycle consists of an initial heat-up followed by an isotherm and the application of pressure and vacuum at predetermined times. As there is no 'universal' cure cycle, the magnitudes and durations of temperature and pressure for our chosen resin system had to be established first. They are the most important variables since they affect the quality of the end product significantly. In order to find an appropriate cure cycle, a temperature cycle is first selected and then the times for applying pressure and vacuum are determined. They are based on the thermal, chemical and physical properties of the resin under that selected temperature cycle. As estimates are made, small specimens are cured and the "quality" of these specimens are evaluated after each cure. Since the method is based on trial and error, numerous experiments and measurements have to be performed before the optimum cure cycle is found. The final objective of these trials is therefore to obtain a cure cycle that meets the following requirements: Void Elimination: The curing process must prevent unacceptable porosity levels by compressing entrapped air and preventing boiling of water. The final void content must be less than 1%. Fibre Wet-out: The viscosity of the resin must be low enough during cure to coat every fibre and reduce porosity of the laminate. Only then can the desired translucent quality be achieved.  56  Chapter 3  Consolidation: The laminate must be uniformly compacted and each cure should give a reproducible final thickness. Resin Removal: The excess resin must be removed to produce a part that has a minimum fibre volume fraction of 6 0 % . There are three main variables which can be manipulated to achieve the above requirements. They are the autoclave temperature and pressure, and the vacuum applied on the part. The purpose of applying vacuum is to aid the removal of entrapped air and volatiles. At the same time, the use of vacuum also applies additional pressure to the laminate. As illustrated in Figure 3.2, both the temperature and pressure profiles in a cure cycle have major effects on the final laminate quality since they directly affect the controlled variables which are the resin temperature and pressure.  MANIPULATIVE VARIABLES  CONTROLLED VARIABLES  INTERACTION  HEAT GENERATION REACTION RATE AUTOCLAVE TEMPERATURE  RESIN TEMPERATURE DEGREE OF CURE VISCOSITY  RESIN FLOW AUTOCLAVE PRESSURE  RESIN PRESSURE LAMINATE THICKNESS VOIDS  C O M P O S I T E C U R E P R O C E S S VARIABLES  Figure 3.2 The effects of autoclave temperature and pressure on laminate quality T861  Chapter 3  57 Characterization of Resin properties  The four items mentioned above emphasizes the need to understand the rheological or flow properties of the resin before a proper cure cycle can be determined. Hence, the first step is to characterize the resin properties. Two techniques are used to determine the flow behaviour and the operating limits of the resin; (1) thermal scans using differential scanning calorimetry (DSC) and (2) viscosity measurements. (1) THERMAL SCANS USING DSC The differential scanning calorimetry (DSC) method utilizes the fact that the cross-linking reaction between the resin and curing agent is exothermic. Hence, the extent of the reaction can be monitored by measuring the heat of reaction evolved. A DSC analysis usually involves plotting the heat (energy) of reaction either as a function of temperature or time for isothermal conditions. These plots are known as thermal scans. In this work, this technique is used as a guide to qualitatively estimate the nature of the resin at various stages in a cure cycle. Our interest is to determine where the cure begins and compare how rapidly the resin and hardener react at different operating temperatures. The instrument used consists of a basic 910 DuPont DSC cell and thermal analyzer (calorimeter). In a thermal scan, the behaviour of a sample is compared with that of a reference material. To begin the test, a fresh mixture of epoxy and hardener weighing between 2-3 mg is encapsulated in a shallow aluminium pan about 6 mm in diameter. The encapsulated sample is then placed in a DSC sample holder and the heat of reactions from the sample are measured as a function of time. An empty aluminium pan and cover is used as a reference. In each test, the temperature is ramped from room temperature to the cure isotherm at about 5°F/min, similar to the ramp rate being used in the selected autoclave thermal cycle. After the DSC heater reaches its set temperature, the tests are performed under isothermal scanning where  Chapter 3  58  the sample was kept at constant temperature. They were performed at three different temperatures; 38°C, 50°C and 60°C. Figures 3.3a,b and c show the thermal scans that were generated. The most visible characteristic feature in these scans is the onset of cure which defines the time at which the resin begins to cure. The onset of cure manifests itself in a rapid increase in the heat measured in the DSC cell. This is illustrated by points C at the base of the curves which also indicates thetimeto apply the pressure. At this point, the resin viscosity is sufficiently low and when pressure is applied to the system, the laminate is compacted and excess resin squeezed out. Gelation occurs when curing is considered to be completed and the laminate can no longer be compacted. However, the cross-linking reaction may not necessarily have completed and hence post-curing is usually done to ensure this.  0.8--  '0.0  20.0  40.0  80.0  80.0  100.0  120.0  140.0  180.0  180.0  200.0'  Tlm« (min)  220.0  DuPont 1090  Figure 3.3a DSC heat of reaction as a function of time at 40 C isotherm. e  59  Chapter 3 0.8 +  0.0  20.0  *a.e  aa. a  aa. a  100.0  120.0  140.0  IBC.  a  IBB.  a  200.2  Time Cmlr>>  ??n.a  DuPont 1090  Figure 3.3b DSC heat of reaction as a function of time at 50 C isotherm. a  &0  20.0  40I0  ' Sal0  ' 80.0  100.0 Tlm«  120.0  140.0  180.0  180.0  (min)  200.0  220.0 DuPont  Figure 3.3c DSC heat of reaction as a function of time at 60 C isotherm. g  1090  Chapter 3  60  (2) VISCOSITY M E A S U R E M E N T S During cure, the chemical cross-linking reaction is accompanied by a physical change in the resin molecular structure. As the epoxy and hardener molecules react, larger molecules are formed resulting in a significant increase in the viscosity. The increase in the molecular weight and viscosity is a function of the extent of the curing reaction. Hence, viscosity characterization is a direct method of determining the flow process. A complete viscosity profile enables one to estimate the resin flow behaviour and the time before curing begins. It also allows one to estimate the pot life of the resin if a higher than ambient temperature is used during prepregging. The viscosity of the present resin system upon mixing at room temperature is approximately 6.63 Pa.s. As little change occurs before the onset of cure, it is usually not necessary to obtain a complete viscosity profile for the entire cure cycle. Using Figure 3.3b, it was estimated that curing did not occur during the first 18 minutes of the cycle. This meant that at the ramp rate of 5 °F/min, the temperature of the resin after 18 minutes was about 32 °C. A viscosity profile of the resin beginning at 32 °C and extending to the 50 °C isotherm and beyond was determined to estimate the time before gelation occurs. A Brookfield R V T spindel-type viscometer was used to monitor the viscosity changes. The instrument was first calibrated and tested using a standard consisting of glycerine having a viscosity of 1.013 Pa.s. These tests showed that the readings using spindels #4 and #6 were reasonably accurate - about 5% higher than the actual - for this range. Measurements were then made on a 500 ml batch of the present epoxy resin-hardener mixture using spindel #4 at 100 rpm. The pouring temperature of the epoxy was about 33 °C and upon mixing with the hardener the temperature rose to about 38 °C within the first five minutes. At this point (t=0 on Figure 3.4), the first reading was taken. Further readings were taken every five minutes. Simultaneously, the temperature of the resin was ramped at about 5 °F/min (similar  Chapter 3  61  to the cure cycle) using a hot water bath until it reached 50 °C. Beyond this point, the temperature in the hot water bath was not raised. However, this temperature rose sharply to about 120 °C within the next 12 minutes due to the exothermic reaction of gelation. As shown in Figure 3.4, the viscosity increased rapidly with increasing temperature until it approached the gelation stage. The epoxy resin then immediately changed from a liquid to a rubbery state within 8 minutes at 50 °C. Within seconds the gelation point was reached and the resin solidified. Thus in an autoclave cure, little resin movement or flow will occur even if more pressure is applied once gelation occurs. The cure is almost completed and the final shape and thickness of the composite is fixed.  VISCOSITY-TIME-TEMPERATURE CURVE 0.4 0.39  FOR EPON 828 + 40phr K1784  -  0.38 0.37 0.36  - 38°C  1\  43 "C  \  0.35 0.34  CL  applied 50°C isotherm  0.33  \48 °c  0.32 0.31  CO  O o  0.3 0.29  1  I  °°c  0.28  GO  0.27  >  0.26 0.25 0.24 0.23  -  gelation  0.22 0.21  02 0  1  i  4  i  i  8  i  i  12  i  i  i  16  i 20  i  i 24  i  i 28  TIME (min)  Figure 3.4 Viscosity profile of resin before and after reaching SO^C. ( N o t e t h a t b e y o n d 5 0 ° @ t= 13 m i n , t h e r e s i n t e m p e r a t u r e is i n f a c t r i s i n g d u e t o t h e  exotherm.)  Chapter 3  62 Voids and the Viscosity Cycle A good cure results in a total void volume substantially less than 1 % . Conventionally, voids are considered to be a consequence of gaseous emanation of solvents retained in the prepreg and/or entrapped air. Large amounts of voids are inevitably formed during the mechanical swirling of the resin during prepregging and trapped in even the most careful handling and lay-up procedures. Some void removal can be obtained by the use of vacuum stripping in the cure cycle. In this technique, vacuum is repeatedly applied and removed several times before the cure cycle is started. Nonetheless, there is a third source of voids which is more subtle and is due to a mismatch between interfibre spacing and final resin content.  3.2.3  Preparation for Cure The prepreg must be prepared before it is cured in the autoclave. The preparation procedure  controls the resin content of the cured part and ensures proper application of the pressure to the lay-up. Two methods of assembly were used. Initially, a small autoclave was used to cure the prepreg panels. Later, a much larger one was installed and used. In both cases, the procedures used to wrap the panel were similar. The major difference lay in the "bagging" technique. When the prepreg is taken out of the freezer, it is left inside the polyethylene bag to thaw. The top and bottom release papers are removed and the laminate is wrapped over with polytetrafluoroethylene (Teflon) coated release cloth. This is to prevent lateral movement of the fibres at the edges when pressure is applied. The release cloth allows volatiles to escape and excess resin to be bled out from the panel into the bleeder plies during cure. The perforated material gives a slightly textured surface. After wrapping, the panel is then flattened by gentle rolling.  Chapter 3  63  The next step is to "bag" the panel, that is, to seal the panel from the autoclave interior. This procedure is usually known as vacuum bagging since its purpose is to ensure a tight seal to contain the vacuum and to transmit the external pressure to the part for compaction. The bag allows the laminate to be subjected to a differential pressure in the autoclave without being directly exposed to the autoclave atmosphere. It also prevents any gaseous pressurizing medium used in the autoclave (air or inert gas) from permeating the part. The "bagging" procedure for processing in the small autoclave mainly involves wrapping the prepreg since the vacuum sealant is provided by the vacuum cart, which will be described in the following section. First, one or two layers of bleeder cloths are placed on top of the panel which already has been wrapped with release cloth. The purpose of the bleeder cloths is to absorb excess resin from the panel during cure, thereby producing the desired volume fraction. The amount of bleeder determines its absorbency, the desired fibre volume fraction and the resin content of the prepreg used. It has been found in the past that direct placement of the bleeder cloths onto the panel causes over-bleeding. The alternative method was to 'edge-bleed' instead. To do this, a caul plate is placed directly on top of the prepreg to give it a flat surface and then followed by bleeder cloths which folded over the four edges. Hence, the excess resin can only be bled out from the edges of the panel. The entire 'package' is then wrapped with thin temperature resistant paper and placed inside the vacuum cart. When curing in the large autoclave, the prepreg is wrapped with release cloth and placed onto a tool plate already lined with polyethylene film. Two layers of bleeders are then placed directly on top of the panel. Bagging simply consists of placing a nylon film over the entire assembly and securing it to the tool plate with a bag sealant tape that adheres to both the tool and the bag. The expendable bag is stretched so that large wrinkles are not left in the bag causing undesirable wrinkle marks in the cured part. Once, the bag is tightly sealed it is ready for processing in the autoclave, Figure 3.5a and b.  Chapter 3  64  Figure 3.5b Bagged laminate ready for cure.  Chapter 3  65  3.2.4 Description of Small and Large Autoclaves Two autoclaves were used to process the laminates; initially, a small one equipped with a simple heating, vacuum and control system and later a large production quality system with complete plumbing, instrumentation and vacuum systems installed. The small autoclave consists of a cylindrical vessel, 30 cm in diameter and 46 cm in length. As shown in Figure 3.6, the vessel shell is fabricated from 1 cm thick stainless steel while the inside is insulated with preformed fibreglass and lined with a 24 cm diameter cylindrical aluminium wall. The laminate to be processed is placed on a 6.2 mm thick aluminium base plate fitted with a heating element tape underneath. This allows direct heating by conduction with minimal lag. Power is supplied to the heating element through leads provided with plug-in connectors which mate to sockets mounted onto the aluminium liner. A vacuum system for evacuating the immediate area around the laminate is provided by a top vacuum bag frame consisting of flexible silicon rubber mounted onto an aluminium frame and a riser. The riser acts as a dam and holds the vacuum port which accesses to the external vacuum pump by quick connect fittings. The laminate is sealed by bolting the top frame, riser and base plate together into a 'vacuum cart', Figure 3.7. A thermocouple is inserted into the laminate via the base plate to monitor the laminate temperature. Heat output is controlled with a Leeds and Northrop microprocessor based unit which simply controls by a preset program that includes all ramps and plateaus required in a cure cycle. Vacuum and pressure are adjusted with regulator valves and any excess gas pressure is bled by hand.  Chapter 3  66  Figure 3.6 Schematic of small autoclave and auxiliary equipment  Figure 3.7 Reusable vacuum bagframeused for curing panels in small autoclave T871  Chapter 3  67  The large autoclave is a modified pressure vessel with external dimensions 2.13 m (7 ft) long by 0.91m (3 ft) in diameter (Figure 3.8). For easy loading of parts, tracks are installed in the front portion of the vessel. The dome-shaped door is an interlocking-lug type which is hinged on one side. The heating system consists of a 18 kW electric heater and a 1.5 kW electric blower fan mounted at the rear. Nitrogen and/or compressed air are used as a heating medium as well as for pressurizing the composite part. As the gas is introduced from the top through solenoid valves it is drawn into the blower where it is heated and directed to the work space. There are six available vacuum lines operated by a rotary-type vacuum pump. Cooling is achieved with a circulating heat exchanger located in front of the blower. During a cure cycle, the temperature is controlled by the microporocessor based Leeds and Northrop Controller with feedback from the thermocouples. Although the solenoid valves, vacuum chamber pump, vacuum pump and fan (like the heater output) can be controlled using relays interfaced with a PC-computer, they were controlled manually in this work. The system is also equipped with a network of safety features involving sensors and relieving devices to avoid overpressure and overtemperature.  r-  8V  H  Figure 3.8 Schematic showing side view of large autoclave T881.  Chapter 3  68  3.3 E X P E R I M E N T A L TRIALS A n initial cure cycle based on past experience as well as on the available viscosity profiles and DSC scans was first used to fabricate the panels. As shown in Figure 3.9 , the cure cycle consists of pressurizing the small autoclave to 45 psi(g) and applying the vacuum pump to 20 inHg from the very beginning of the temperature cycle (@ time = 0 mins) and keeping them constant throughout the cure. After 5 similar trials, a panel of acceptable quality was produced. (A panel of acceptable quality is one that meets "the requirements outlined in section 3.2.2.) After that, however, it was not possible to fabricate such a panel even after numerous runs. The subsequent panels were dry and full of tiny "white lines" and voids. It was then decided that a step-by-step experimental program had to be conducted to solve the problem.  Estimated Laminate Temperature Profile  t=0  10  20  30  40  50  Time (min) Figure 3.9 Initial cure cvcle used for processing in small autoclave.  180  Chapter3  69  As many as 45 runs were done before another acceptable panel was made, but only 15 of them are summarized in Table 3.1. This is because most of them were repeats since reproducibility was a major difficulty. Fortunately, by this time, the large autoclave was in full operation and the rest of the panels were made after only 4 trials. Hence, we were convinced that the inconsistencies in quality were largely due to the fact that the process control hardware and vacuum pump system for the small autoclave were both unreliable and defective. In addition to this, another major contributing factor was the relatively low viscosity resin which is very sensitive to slight changes in the operating variables. To begin the experimental program, 12-ply [0] unidirectional panels were made in the 12  small autoclave since thicker prepregs are richer with resin. The interest at this initial part of the program was to determine how the changes in isothermal temperature and vacuum affected the resin content of the panels. As shown in runs 1-4 in Table 3.1, the selected temperature range had only a slight effect. However, it is usually good practice to use the lowest operating temperature possible when curing room temperature resins to avoid boiling. As should be expected, there was a definite relationship between the level and time of vacuum applied and the final panel quality. Resin rich panels are made with lowest available vacuum i.e., about 10 inHg(g) but this leads to unacceptably low fibre volume fractions. Higher vacuum, on the other hand, tend to overstrip the resin leaving a dry panel with little " white lines". Another visible problem with the use of a vacuum was the formation of froth near the edges of the panel. The next step was to attempt to eliminate the dryness problem by either eliminating the vacuum suction or lowering the applied pressure. As seen in runs 5-8 for 4-ply [0] panels, there 4  was definite improvement with both options but the lack of vacuum and pressure meant that the tiny voids trapped in the center of the panel could not be drawn out.  70  Chapter 3 T A B L E 3.1  SUMMARY OF A U T O C L A V E CURE TRIALS  TCQ  P(psi)  V (inHg)  isotherm  @time (mins)  @time (mins)  (fibre)  [0]i2-l  60  45 @ 0  20 @ 0  56.6%  voids.dry  2  50  45 @ 10  10 @ 10  37.6%  froth  3  60  45 @ 10  13 @ 10  46.0%  voids <1  4  40  45@45  15@45  45.1  dry  50  45 @ 10  13 @ 10  37.7  voids  6  50  45 @ 0  0  50.0%  no froth  7  40  45 @ 45  15@ 45  60.9%  froth  8  40  25 @ 10  0  60.75  tiny voids  [0/90],-9  40  25 @ 10  15@0  60.0%  dry  10  40  25 @ 10  0  NA  voids  11  40  25 @ 5  0  NA  voids  12  40  20@ 10  15@ 18  57.0%  improved  13  40  10 @ 0  15 @ 10  NA  voids  EXP.NO.  . [0]«-5  v  f  30 @ 10  Results  dry  14  40  25 @ 14  15@20  NA  clear.voids  15  40  30@10  21 @ 10  60.5%  excellent!  45 @ 15  (leaking?) NA  dry  16  40  30 @ 10  18 @ 10  45 @ 15  (leak fixed)  LARGE AUTOCLAVE NI  40  45 @ 10  10 @ 12  NA  improved  N2  40  45(2)20  5 @ 35  NA  improved  N3  40  61%  excellent  N4  40  20@ 0  +15 psi(a)  35 @ 15  (in bag)  35 @ 0  +25 psi(a)  45 @ 15  (in bag)  thick 61%  perfect!  Chapter 3  71  With the improved panel clarity andfibrevolume fractions, the next stage was to fabricate the cross-ply panels. From run nos. 9-14, the applied pressure of 25 psi(g) with low or no vacuum gave fairly good results repeatedly. However, an attempt to apply the pressure at two stages (run no. 15); 30 psi(g) first at 10 minutes into the cycle and then five minutes later increase it to 45 psi(g), gave excellent results. The panel was very clear and had a fibre volume fraction of 60.5%. The variable settings were however open to question since a repeat run gave a very dry panel. It is suspected that the vacuum gauge reading is inaccurate in run no. 15 due to a small leak in the bagging when the screws were insufficiently tightened. The problems with the small autoclave were not pursued once the large one became operational. This was to avoid deviating from the objective of the work which was to determine the optimum cure variables for fabricating the best quality panels possible. As illustrated in runs N1-N4 under the heading of LARGE AUTOCLAVE this was quite easily achieved. The first two runs, N l and N2 were conducted to compare them with those already done in the small autoclave. The quality of the panels were not acceptable, but were a significant improvement over similar runs in the small one. The large autoclave has a number of important advantages, particularly in the fact that it can provide accurate, reliable and controlled conditions. Also, it enables us to use a technique which helped to markedly reduce the void content and produce the desired volume fractions. The technique consists of "pressurizing the bag" instead of applying vacuum during the cure. The cure cycle is illustrated in Figure 3.10. Before the cure is started, the bag is tested with the vacuum line to ensure that there are no leaks. The vacuum is applied to the maximum and then quickly removed. The "sucking" action 'pops' any air bubbles locked inside the panel. This is repeated a few times and the vacuum is then isolated. At the start of the temperature cycle, the autoclave is pressurized to 20 psi(g). When the isotherm is reached in 15 minutes, the pressure is gradually increased to 35 psi(g). At the same time, the bag is also pressurized to +15 psi(g)  72  Chapter 3  using nitrogen in the same 'vacuum' line. The process must be done gradually and the pressure difference between the inside and outside of the bag should not exceed 5 psi(a) at any time or it will rip open.  Laminate temperature = 40deg C ^ 35 psig /  / /  45 psig autoclave pressure  25 psig inside bag pressure  Viscosity Profile  s  - ^ \ ambient \ pressure \  'gelation  \ \ \ \ \  \ \  J  t=0  10  I  20  30  40  50  7^-  time (min)  Figure 3.10 The optimum cure cycle used for processing in large autoclave.  180  Chapter 3  73  3.3.1 Troubleshooting As the experimental trials have indicated, it was very difficult to fabricate thin panels that were clear and void free since there was little latitude in processing. A variety of problems were encountered ranging from dry-looking panels having good fibre volume fractions to panels which appear translucent but are full of voids. The most common problem was high void content. It has been observed from results of the trials that this may be attributed to several causes which may be remedied to some extent. Resin Starvation: To a small extent, misalignment of fibres create spaces which are resin short and only considerable pressure can reduce these spaces. Voids due to resin shortage are most likely to arise in resin-rich areas. The problem is caused by excessive bleeding (too many bleeders or too high a vacuum) or too much pressure. A good quality laminate requires that the spaces between the fibres be completely filled with resin. To achieve this, the viscosity profile needs to be evaluated to determine a suitable pressurization point past the minimum. Entrapment ofVolatiles: Voids may also result from volatiles remaining in the resin at the point of gelation. Sometimes a slower heat-up rate or an intermittent hold at a time of low viscosity permits these volatiles to diffuse out of the composite. Solvents are the main source of volatiles but as they were not used during prepregging, this did not pose a problem. The most severe problem was with the entrapment of air bubbles which were formed by the rotation of the wheel in the resin bath. It was discovered that an effective method for reducing the air bubbles was to heat-up the resin bath to about 33°C and maintaining it throughout the prepregging. This, however, also meant that the pot life of the resin was limited to about 15 minutes and the prepreg had to be cut and laid-up very quickly. Another alternative, although it would cause other problems, would be to hold a vacuum throughout the cure.  Chapter 3  74  Incorrect Pressurization Point: Voids can be caused by waiting too long to apply pressure during the cure cycle. As the resin advances, viscosity increases and flow decreases. It the resin has advanced too far, the application of pressure will not give the compaction necessary to produce a good part. The remedy is simply to apply pressure at an earlier point during the cure. Variability in Prepreg Quality: One of the methods that could alleviate the problem of variability in fibre volume fraction and resin bleed-out was to ensure a quality control of the prepreg. By following the same procedures during prepregging and measuring the prepreg weight per unit area during each run, a more consistent prepreg with the desired volume fractions can be attained.  75  Chapter 3 3.3.2 Specimen Preparation  Only one very high quality panel of each lay-up was needed for this research. The lay-up configurations and volume fractions of the panels that were selected for testing are listed in Table 3.2. Fibre volume and void fractions were determined from pieces cut from the panel. Using the guidelines given in ASTM standard D2734-70, weight fractions were converted to volume fractions using densities. The density for cured resin as determined experimentally is 1.15 g/cm while the fibre density as quoted by the supplier is 2.715 g/cm . Void content was 3  3  computed from the discrepancy between theoretical density and the measured specific gravity of the composite. The average thickness of one ply (t) is 0.175 mm. T A B L E 3.2 FIBRE V O L U M E AND VOID CONTENT OF SPECIMENS LAY-UP  PROCESSING AUTOCLAVE  VOLUME FRACTION FIBRE  VOID THICKNESS FRACTION (mm)  small .  58.0%  0.5%  0.72  [90]  small  58.5%  0.5%  0.72  [±45]*  small  58.0%  0.7%  1.38  [0/90]  small  60.9%  0.3%  0.71  [0/60]  s  small  60.7%  0.3%  0.66  [0/75],  large  61.4%  0.4%  0.68  [0/45],  large  61.0%  0.3%  0.68  [0]  4  4  s  The panels, which were 19 cm wide by 30 cm long, were postcured for 72 hours at 60°C in a postcure oven. This ensured that all chemical reactions in the resin are completed. Then, using a water-cooled diamond saw, the panels were cut oversize into tensile specimens 2.54 cm  Chapter 3  76  by 25.4 cm. The parallel edges of these specimens were polished to size on 600 grit silicon carbide paper and then finished with optical grit size cerium oxide. Once the specimens were cleaned and dried, they were ready for testing.  Chapter 4  77  CHAPTER 4  EXPERIMENTAL PROCEDURE  4.1 TENSILE TESTING The mechanical testing was conducted at room temperature using an Instron testing machine furnished with wedge action friction grips. All tests were performed under monotonic loading at a constant cross-head displacement rate of 0.02 mm/s. In the gripping area, 25.4 cm wide silicon carbide abrasive strips were inserted between the specimen and the grip pads to prevent slippage. The free specimen length between the grips was 203 mm giving an aspect ratio of about 8. In order to monitor the axial stiffness loss due to matrix cracking, the loading was increased incrementally in a step-wise manner as shown in Figure 4.1. Initially, the specimen was loaded to about 90 MPa to measure its initial stiffness. The specimen was then step loaded at about 500 N (30 MPa) increments. After each increment such as A in Figure 4.1, the loading was interrupted. The specimen was then reloaded to a lower value between 90-100 MPa, as indicated by E , during which load-displacement response was measured. The stiffness (E ) was A  A  calculated by taking the slope of the stress-strain curve in the linear region. Following this, the off-axis ply cracking pattern was photographed and the specimen was again reloaded to 90-100 MPa for another measurement. This portion is not shown in Figure 4.1. The second measurement was taken to ensure that the stiffness (E' ) remained the same as E . The whole procedure was A  A  repeated, eachtimeto a higher load such as to B and C until the inner ply was fully cracked but before ultimate failure. This was done so that the specimen would be still intact and the edges could be examined after testing.  Chapter 4  78  Typical Load-Displacement Curves 5  -  2  -  o o o  < O  i  =r 0.2  T—i  0.4  r—i  0.6  i  1  0.8  i  1  r  1  DISPLACEMENT (mm) Figure 4.1 A typical loading and unloading curve to determine stiffness. (The occurrence of toe regions in these curves is due to the takeup of slack in thin specimens, see section 4.1.4.)  MEASUREMENT OF DISPLACEMENT The axial displacement in the specimen during loading was measured using a linear variable differential transducer (LVDT). Essentially, the LVDT measures voltage or emf in terms of displacement of the ferromagnetic core of a transformer. The basic components of the LVDT are shown in Figure 4.2. The oscillator converts the DC input to AC, exciting the primary winding of the differential transformer. The transformer consist of a single primary winding and two secondary windings which are placed on either side of the primary. The secondary circuits are connected in series opposition so that the emfs induced in the coils oppose each  Chapter 4  79  other. Voltage is induced in the secondary windings by the axial core position. The resultant output is therefore a DC voltage proportional to the core displacement from the center, or reference position. When the core is in the reference position, the induced emf s in the secondaries are equal, and since they oppose each other, the output voltage will be 0 V. The polarity of the voltage is a function of the direction of the core displacement with respect to the reference. Figure 4.2 illustrates the type of transducer used in this work. This particular model has dimensions as indicated in Figure 4.3. The recommended working range is about ±12.7 mm. The size and working range are considerably larger than desired for this work but this model was used because it was available. This, however, caused some difficulties when mounting onto the specimen. Normally, the LVDT is mounted by clamping around the housing to a physical reference point. The dynamic member to be monitored is coupled to the threaded connecting rod of the core assembly. In this work, the displacement to be measured was between the two far end points of the specimen gauge length. This meant that the LVDT had to be mounted on the specimen itself. Since the LVDT was rather large and heavy, while the specimen was light, thin and flexible, a special fixture had to be designed to ensure that the transducer was mounted rigidly in a perfectly upright position. There were a number of factors that had to be considered in the design. Firstly, the LVDT had to be removed and remounted several times during the course of testing for each specimen. This was to allow the crack pattern to be photographed in between loadings. It was also necessary that the fixture be moved out of the way to allow the total area of the gauge length be photographed. Secondly, the LVDT had to be mounted onto the specimen by point contact. This would ensure that the measured displacement is strictly between two defined points. Also, the LVDT had to be mounted onto the exact same gauge length throughout the testing of each specimen. This was absolutely necessary as the stiffness changes are small and a slight change in the gauge length would affect the stiffness measurements significantly.  80  Chapter 4 DISPLACEMENT  DC OUTPUT  Figure 4.2 Circuit diagram of L V D T T891.  1.9 cm O.D.  9.42 cm  REFERENCE POSITION  0.3 cm  0.95 cm  4.75 cm  6 cm  LVDT ASSEMBLY WEIGHT 83.8 grams Figure 4.3 Schematic of linear variable differential transducer with dimensions.  Chapter 4  81  The fixture that was developed consisted of two parts, top and bottom blocks, which are shown in Figure 4.4. They were made from acrylic block (non-magnetic material) to keep them as light as possible. For easy removal and remounting, the LVDT is mounted onto the top block by a front face piece held by two screws. Once the face piece is unscrewed, the LVDT and connecting rod can be easily removed. The two blocks are mounted onto the specimen by a pair of cylindrical rods which can be tightened against each other by a set of back screws. To mount these blocks, the specimen is slipped in between the rods before it is gripped. After the gauge length is marked, the blocks are levelled and the back screws are tightened. This clamps the rods onto the marked lines. The clamping action in the top block can hold both the block and the LVDT firmly but not upright. In order to secure the whole set-up in an upright position, another set of rods are inserted in the upper half of the top block. This set of rods are spring loaded. As these rods are not tightened, in principle they only provide support for the heavy load and do no not make any contact with the specimen. When the gauge length is to be photographed between load intervals, the LVDT is removed in the manner already described and the blocks can easily slide upwards (and downwards) once the back screws are loosened. As expected, it is difficult to remount the fixtures onto their exact original positions visually. To help locate the exact positions, two 'stopper blocks' consisting of acrylic blocks with grooves were made. One having dimensions 50 mm x 47.8 mm x 22.4 mm was used as a back-stopper to position the bottom block first, using the bottom grip surface as a reference point since it does not change. Once the bottom block was mounted, the other block (49 mm x 24 mm x 12.3 mm) was then used to position the top block. This technique was found to be reliable and stiffness measurements were repeatable to within 1%.  82  Chapter 4  Spring Loaded R o d  Removable front face  LVDT  Specimen  Bottom block  Figure 4.4 Schematic of top and bottom blocks - fixture for mounting LVDT.  Chapter 4  83  4.1.1 Equipment Calibration The two variables that were monitored simultaneously during the tensile tests were the applied load and the LVDT (specimen gauge length) displacement. Before testing began, both the load cell and LVDT output voltages were calibrated and rechecked after testing each specimen. LVDT VOLTAGE CALIBRATION The allowable input voltage for the LVDT was between 6 to 30 V. A stable signal source of 10V supplied by a signal conditioning instrument with a digital indicator was used. This would give a maximum saturation LVDT output voltage of 10 V. As the working range was ±12.7mm, an absolute displacement of 0.254 mm in either direction would produce an output voltage of 1 V. Since very small displacements were expected in these tests, the output signal was amplified 10X using a Bascom-Turner isolation amplifier. The amplifier also significantly reduced the high electrical noise level which was present in the output signal. Using a digital extensometer calibrator, the LVDT output voltage was calibrated against displacement and the calibration factor was 1 V per 0.252 mm of downward displacement, Figure 4.5. The estimated maximum gauge length displacement before failure was 1.8 mm or about 3% strain. With the resolution of about 4 V/mm, this would allow measurements of all displacements below the expected maximum without exceeding more than 80% of the saturation voltage. However, the working range corresponded to 25% of the calibration curve when measuring stiffnesses where displacements were no more than 0.5 mm, or about 0.8% strain.  84  Chapter 4  Vertical Displacement (mm)  Figure 4.5 Calibration curve for LVDT.  LOAD CELL CALIBRATION The Instron GR-load cell was calibrated for the 0 - 8896 N (0 - 2000 lb) range as given in Figure 4.6. Again, to improve the resolution for data recording, the output signal was amplified, using an amplifier-filter, by a factor of 6. The first 100 lbs (444.8 N) were calibrated with dead weights and at 500 lbs load, the machine was electronically calibrated. It can be seen in Figure 4.6 that the load is linearly related to the output voltage but a residual voltage exists as indicated by the non-zero intercept.  85  Chapter 4  (x 1000 N )  B  > Q.  ••—»  o  Load (lbs) Figure 4.6 Calibration of load cell voltage.  4.1.2 Data Acquisition The analog voltage output from both the load cell and LVDT were converted into digital form using an analog-to-digital (A/D) converter. The converter hardware consisted of an IBM Data Acquisition and Control Adapter (DACA) Board which is installed in an IBM PC. Being a 12-bit converter, it can detect a minimum voltage difference of 2.4 mV when set for a range of 0 to 10 V. This is equivalent to about 4 N or 2.6 kPa for the load cell signal and 0.6 Lim or 10 microstrain (0.001% strain) for a specimen gauge length of 60.52 mm. Using a laboratory software package, AS YST (A Scientific System), a computer program was written to translate the digital signals and store the data. The sampling rate for both input  Chapter 4  86  voltages or channels was 5 readings per second. Readings were recorded for each loading and unloading interval. The input voltages were directly converted into their respective load (in N) and displacement (in mm) values. Concurrently, the program also plotted the load readings against displacement after each incremental load was reached and calculated the slope of the curve using the least squares fit method. This feature was included as a quick check of the stiffness change which is indicated by the slope. A complete listing of the program is given in Appendix B. The entire laboratory set-up for testing and data acquisition is shown in Figure 4.7.  LOAD CELL  10 V INPUT A/D CONVERTER I 9999 I  IBM-DACA SIGNAL CONDITIONER BASCOM-TURNER ISOLATION AMPLIFIER  O O  IBM-PC  Figure 4.7 Illustration of experimental set-up and data acquisition equipment.  87  Chapter 4 4.1.3 Determination of Material Properties  The first step in characterizing the glass-epoxy composite was to establish the basic lamina elastic and strength properties. These properties were determined from testing unidirectional [0] and [90] and angle-ply [+45] * specimens. The properties that were measured included: 4  4  Young's modulus in the axial fibre direction, E  t  Young's modulus transverse to the fibre direction, E2 Major Poisson's ratio, v  12  Minor Poisson's ratio, v  21  In-plane shear modulus, G  1 2  where subscript 1 refers to the direction parallel to the fibres and subscript 2 refers to the transverse direction. The stress-strain response of these specimens were monitored with electrical resistance strain gauges as recommended by the ASTM Standard D3039-76. To determine the first four properties, foil gauges Type 6/120LY61 with 120Q ±0.2% resistance and a gauge length of 3 mm were used. For each test, a foil gauge was bonded in the longitudinal direction on the center of the specimen. Determination of the shear modulus required the use of rosette gauges Type 6/120XY11 which consist of two foil gauges, one mounted in the longitudinal and the other in the transverse direction. This enabled the recording of strains in both directions which was needed to determine the shear modulus. During all the tests, the output voltage from the strain gauges (which gives strain directly) and the load cell was recorded with an ORION datalogger. Before the tests were conducted, the load was calibrated for three load ranges (500,1000,2000 lbs) to cover the wide range of ultimate strengths of these specimens.  Chapter 4  88  4.1.4 Verification of Toe Region in Load Displacement curve After a number of preliminary tests using unidirectional as well as cross-ply laminates, it was observed that the resulting stress-strain curves had a toe region such as that shown in Figure 4.8. This feature was defined in the ASTM Standard D 882-81 ' Tensile Properties of Thin Plastic Sheeting' as an artifact which does not represent a property of the material. The artifact is most predominant when testing specimens which are less then 1.0 mm in thickness. It is caused by a takeup of slack, and alignment or seating of the specimen. In order to confirm this, an experiment was carried out using foil strain gauges mounted onto the front and back faces of a cross-ply specimen . The specimen was loaded up to about 200 MPa ( 0.8% strain) and unloaded. As shown in Figure 4.9, the average of gauges 1-2 which were in the front face showed a distinct curved or toe region, implying that this side of the specimen was originally in a compressive state. On the other hand, the stress-strain curves for the gauges 3-4 were linear implying that the back face was in tension. Hence, the initial curved regions which were observed in our tests were indeed the toe-regions. According to the ASTM standard, in order to obtain correct values of modulus and strain, this artifact must be compensated for to give the corrected zero point on the strain or displacement axis. In the case of our material which shows a region of linear behaviour, the linear region CD in Figure 4.9 is extrapolated through the zero stress axis. This intersection point B is the corrected zero-point from which all displacements or strains must be measured.  Chapter 4  89  STRESS (MPa)  Figure 4.9 Behaviour of front and back foil gauges mounted on thin glass/epoxy specimen  90  Chapter4 4.1.5 Testing of Off-Axis Specimens  Since the off-axis 9 inner plies of the [0/90] , [0/75] , [0/60], and [0/45], laminates are s  s  weaker than the outer 0°, the cracking pattern seen is at an angle 9. The off-axis angle is measured from the 0° loading axis. Figure 4.10 shows the dimensions and tolerances of a typical test specimen used in the present experimental work.  Figure 4.10 Geometry of a typical tensile specimen.  Chapter 4  91  4.2 MEASUREMENT OF C R A C K GROWTH After much experimentation, a method for recording the cracking process without removing the specimen from the test rig was found. As explained previously, the LVDT and mounting fixture were removed after each incremental load and before photographing the cracking pattern. The set-up used for capturing the cracking over the entire gauge length is shown in Figure 4.11.  Figure 4.11 Set-up used for photographing the cracking pattern. A high intensity light was placed behind the gripped specimen. With the aid of hinged metal flaps surrounding the light source, a beam of light was focussed onto the entire gauge length of the specimen. When viewed from the front, one could see a very distinct pattern consisting of dark solid "crack" lines spanning over a bright translucent "column". A Polaroid  Chapter 4  92  camera equipped with a polarizer-filter was placed in front to capture this pattern. During processing, the 10.2 x 12.7 cm (4 x 5 in) Polaroid negatives were enlarged about four times into positives using a high contrast filter. The final photographs were 10.2 x 25.4 cm while the size of the gauge length image was 9.7 x 2.3 cm.  4.2.1 Image Analysis The Leitz TAS Plus Image analyzer is an opto-electronic instrument system for automatic processing and evaluation of optical images. It is capable of detecting, counting, measuring and providing size distributions for almost any type of image. The processing screen consists of an electronic hexagonal point raster which divides the image field into 256 x 256 image points or pixels. Hence, the instrument can give considerable resolution if used together with a microscope and enlarging lens. In this work, the imaging camera is directly focused onto the enlarged photographs. The picture quality in the processing screen is hence limited by the photograph. One of the reasons for enlarging the photographs was to improve the resolution, particularly for the patterns where the crack spacing was very small. The photograph was then divided into 10 frames about 48.5 cm by 55.5 cm each. Each frame was placed under the analyzer camera and every crack present in that frame was detected and the length measured using the analyzer computer. To do this, the computer was programmed to detect each crack and to identify it by a set of x-y coordinates. The reference coordinate system used to identify each crack consisted of choosing the gauge length as the x-axis and the width of the specimen as the y-axis, Figure 4.12. A difficulty that was faced in this technique was with the poor contrast between the crack and the background. The analyzer could not detect the difference between the darker shades of  Chapter 4  93  the background and the actual cracks which were sometimes faint. The only way to resolve this was to trace each crack by hand using a light pen detector and then program the computer to read the already traced cracks and then assign their respective coordinates.  Frames  A-J Photograph  97 mm Y  0,0  Figure 4.12  F  G  H  1  J  A  B  C  D  E 230 mm  Coordinate system and typical frame-division used for each photograph.  94  Chapter 5  CHAPTER 5  R E S U L T S A N D DISCUSSION The results of this work are presented and discussed in three parts. First, the material properties, stiffness and crack length measurements from tensile tests are given. Then, a statistical analysis of the crack sizes and spacings is shown. Aspects of crack tip interaction and growth are also included in this portion. Finally, the R-curve behaviour characteristics of the four different lay-ups are presented.  5.1 M A T E R I A L PROPERTIES The basic lamina properties of this glass/epoxy system were determined from the average of three specimen tests for the tensile properties, and from one test for the shear modulus. The results are given in Table 5.1. The Poisson's ratios v  1 2  and v  2 1  were found by taking the ratio  of (-e^Ei) from the stress-strain curve of a [0] and [90] tensile test respectively, where 6! and 4  4  £2 are the longitudinal and transverse strains. It is noted that the results agree well with the known relationship between these four constants given by v i s = v j£\ (e.g., Ref [75]). The I2  2  2  shear modulus G , was determined by testing a [+45]* laminate and establishing the initial 1 2  slope of the a /2 versus lej + le^l curve [90]. Typical curves for the three tests are shown in x  Appendix C. Using the material properties in Table 5.1, the stiffnesses of the four different laminates in the loading direction were calculated by employing Laminated Plate Theory (LPT). These values, as shown in Table 5.2, agree well with the actual measured stiffnesses, E . 0  95  Chapter 5  T A B L E 5.1 L A M I N A PROPERTIES  Lay-up  Property  Values (GPa)  Average (sd) (GPa)  UTS (MPa)  [0]  E„  34.96 35.40 34.81  35.00 (0.306)  755  2.17%  10.79 10.10 11.43  10.80 (0.66)  20  0.18%  0.288 0.289 0^291  0.29 (0.001)  0.086  0.086  6.13  6.13  NA  NA  4  [90] -  E22  4  [0]  V12  4  [90]  v  [±45],,  G  4  21  12  T A B L E 5.2 PREDICTED AND MEASURED STIFFNESSES OF LAMINATES (GPa)  LAMINATE  MEASURED E  [0/90]  L.P.T.  % diff  23.00  23.09  0.3%  [0/75]  22.46  23.23  3.1%  [0/60]  23.54  23.82  1.1%  [0/45]  24.41  25.39  3.7%  s  s  s  s  0  Chapter 5  96  5.2 DETERMINISTIC RESULTS 5.2.1 Stiffness and Crack Length Measurements The axial stiffness or Young's modulus was measured before (E) and after (E') photographing each crack pattern. This was to ensure that the stiffness readings did not deviate significantly after removing and remounting the LVDT and fixture. The procedure was repeated at incremental load intervals of about 30 MPa where each increment was denoted by numbers 1, 2,..I. Stiffness was determined by taking the initial slope of the 'corrected' (section 4.4) stress-strain curves calculated from the load-displacement curves obtained. To ensure consistency, the slope was taken over a range between 0.2 to 0.4% strain for all tests. The measured absolute stiffnesses were then normalised by dividing by the original measured stiffness E for each specimen. This compensates for any small differences and allows for easier 0  comparisons between the different lay-ups. Two specimens of each lay-up were tested where their stiffness changes were measured. The results for the first set of specimens, one for each of the four lay-ups, are given in the next section while those for the second set are listed in Appendix D. However, crack length measurements using image analysis were only performed for the first set of specimens. This is because the measurements were extremely time-consuming. As the crack patterns in both sets were quite similar, one set of crack length data is considered adequate to determine any general trends. Another limitation was encountered when making measurements of the limiting crack density. With increasing applied load, the crack density increased significantly causing the crack spacing to become extremely small at high load levels. It became very difficult to resolve the different cracks on the image analyzer screen. Because of this, only 3 to 4 photographs (corresponding to an equal number of load levels) of each lay-up were processed with the image  97  Chapter 5  analyzer. This is not a serious limitation, since the principal interest lay mainly in the initiation and growth stages, where the crack multiplication processes occur. At higher stresses, where the cracks are very closely spaced, the total crack length of each photograph was measured visually with a micrometer, where the total crack length a is the sum of all the crack lengths measured in the gauge length. 5.2.2 Normalised Stiffness as a Function of Stress and Crack Length Normalised stiffness reduction for the [0/90] , [0/75] , [0/60] , and [0/45] lay-ups is plotted s  s  s  s  against the total crack length a, in part (a) of Figures 5.1, 5.2, 5.3 and 5.4 respectively. The normalised stiffness reduction is cross-plotted with the total crack length as a function of increasing applied laminate tensile stress in part (b) of the same figures. Tables 5.3,5.4,5.5 and 5.6 list the values that are plotted in these graphs. As will be seen in section 5.3, tiny voids or cracks are present in the virgin specimens prior to testing. Hence, the original 'true' modulus E is not necessarily the measured initial ot  modulus E but perhaps some higher value as though the specimen were flawless. However, 0  as shown in Table 5.1 to 5.4, the total length of these minute initial cracks is less than 2% of the total crack length at the saturation stage i.e., when fully cracked. This is negligible and E  0  can be assumed to be representative of E . ot  Error bars are included when plotting the values of normalised stiffness. The lower limits of these bars are the values measured before the LVDT was removed (E) and the upper limits are those measured after the LVDT was remounted (E'). As there was little difference between them, only (E) values were used in curve fitting. In cases where the total crack length a was measured with a micrometer, the values are marked by an asterisk (*). As preliminary tests were performed to determine the failure strength of the specimens, the last crack photographs were taken at about 80% of the failure strength.  Chapter 5  TABLE 5.3 S T I F F N E S S , S T R E S S A N D C R A C K L E N G T H D A T A F O R [0/90]s Eo = 23.00 GPa S T R E S S (MPa)  E/Eo  E'/Eo  C R A C K L E N G T H (m)  E1  102  0.997  0.997  0.162  E2  125  0.997  0.997  0.162  E3  154  0.983  0.984  0.180  E4  180  0.964  0.970  0.911  E5  202  0.921  0.920  1.874  E6  229  0.890  0.891  2.286  E7  246  0.870  0.872  2.638  E8  298  0.863  0.864  3.000  E9  350  0.847  0.850  3.310  E10  395  0.830  0.836  3.562  E11  443  0.829  0.818  3.750  TABLE 5.4 S T I F F N E S S , S T R E S S A N D C R A C K L E N G T H D A T A F O R [0/75]s Eo = 22.46 GPa  S T R E S S (MPa)  E/Eo  E'/Eo  C R A C K L E N G T H (m)  E1  102.74  1.0040  1.0040  0.055  E2  127.60  0.9987  0.9987  0.055  E3  157.30  0.9982  1.0000  0.173  E4  179.00  0.9875  0.9850  0.411  E5  207.00  0.9780  0.9780  0.820  E6  234.00  0.9690  0.9680  1.547  E7  261.70  0.9250  0.9330  2.602  E8  285.60  0.9120  0.9210  3.333  E9  366.40  0.8820  0.8800  3.442  E10  413.80  0.8860  0.8610  3.683  E11  451.80  0.8730  0.8730  3.898  Chapter 5  TABLE 5.5 STIFFNESS, STRESS AND CRACK LENGTH DATA FOR [0/60]s Eo = 23.54 GPa  E1 E2 E3 E4 E5 E6 E7 E8 E9  E10 E11 E12  STRESS (MPa)  E/Eo  E'/Eo  CRACK LENGTH(m)  154  1.052  1.052  0.072  166  1.047  1.049  0.072  190  1.032  1.028  0.072  223  1.003  1.011  0.144  254  0.984  0.982  283  0.977  0.980  307  0.950  0.950  1.355  335  0.945  0.943  2.367  359  0.943  0.943  ,-  385  0.931  0.926  3.409  421  0.912  0.914  455  0.903  0.906  -  0.511  -  3.609  TABLE 5.6 STIFFNESS, S T R E S S AND CRACK LENGTH DATA FOR [0/45]s Eo = 24.42 GPa  E1  S T R E S S (MPa) 104.2  E/Eo 1.026  E'/Eo 1.026  C R A C K LENGTH(m) 0.106  E2  152.6  1.020  1.015  0.106  E3  186.8  1.026  1.007  0.106  E4  210.5  1.008  1.007  0.106  E5  240.8  1.009  1.000  0.106  E6  268.6  1.000  1.003  0.205  E7  295.8  1.000  1.000  -  E8  321.0  0.999  1.010  0.392  E9  353.8  0.980  0.975  0.732  E10  400.0  0.980  0.980  1.752  E11  437.2  0.985  0.986  2.974  E12  456.8  0.966  0.970  3.970  E13  487.8  0.970  0.970  -  Chapter 5  100  Figure 5. Kb) Stiffnessreduction and crack length as a function of applied stress for rO/901 laminate.  Chapter 5  101  0  200  400  600  A P P L I E D S T R E S S (MPa)  Figure 5.2(b) Stiffness reduction and crack length as a function of applied stress for [U/75L laminate.  102  Chapter 5  C/3  Q  <! 2  CRACK LENGTH (m)  Figure 5.3(a) Stiffness reduction as a function of crack length for fQ/601- laminate.  Figure 5.3(b) Stiffnessreduction and crack length as a function of applied stress for 10/601 laminate.  103  Chapter 5  1.1  [0/45]s  1.08  o  ty LU CO to  1.06 1.04  1  0.98  UJ  0.96  CO  0.94  o  UJ CO  rr O  -  1.02  uc  0.92 0.9  0.88  -  0.86 0.84 0.82 O.B  -i  1 1  1  1  1  2  1  r  3  C R A C K LENGTH (m)  Figure 5.4(a) Stiffness reduction as a function of crack length for rO/451, laminate.  APPLIED S T R E S S (MPa)  Figure 5.4(b) Stiffness reduction and crack length as a function of applied stress for rO/451, laminate.  104  Chapter 5  In comparing parts (a) in Figures 5.1 to 5.4, it is seen that the relationship between E and a changes from one which is directly linear in the [0/90] system towards a pronounced S-type s  curve at 8 = 60°, and then reverting back to an almost linear curve as 0 = 45°. In Figure 5.1(a), the stiffness decreases linearly with crack length and the total reduction is about 18% when the inner 90° ply is fully cracked. It should be noted again that total reduction is defined here as the difference between the last measurement (specimen has not totally failed) and the original one. In the case of the off-axis plies, Figure 5.2 to 5.4 (a), there is an initial increase in the stiffnesses ranging from 0.4% for the [0/75],, 5% for the [0/60] , and 2.6% for the [0/45] laminates s  s  respectively, before a general decrease is seen. This is probably due to the rotation of the off-axis plies in tension since it is not observed in the [0/90] lay-up, or an initial straightening of fibres s  in the axial layers. The total stiffness reductions in these cases are 12.7%, 9.7% and 3% respectively. However, if the total reduction is taken from the highest measured stiffnesses for each case, they increase to 13%, 15% and 5.6% respectively. Except for the slightly anomalous results in the [0/60] lay-up, the general trend seems to show that the total overall stiffness s  reduction decreases with decreasing 0. The differences in total stiffness reduction in these lay-ups may be explained using Laminated Plate Theory. The normal method for calculating the total stiffness reduction (generally called the ply discount method [16]) consists of setting the transverse modulus E , 2  the Poisson's ratio v , and the shear modulus G , of the cracked ply to zero. These values are 12  12  then substituted into the standard LPT analysis which assumes that N is non-zero, but all the x  other laminate forces, N and N are zero. The laminate strains are then calculated. The laminate y  x y  modulus is then E=N^/he^, where h is the laminate thickness. This can be shown to be equivalent x  to the inverse of the first term of the laminate compliance matrix (1/S ). If this approach is n  taken, then the stiffness reduction predicted for the [0/45] laminate is larger than the [0/90] s  s  Chapter 5  105  laminate. The results for the two other laminates lie in between. These values are presented in Table 5.7, and marked 'unconstrained' (uc) meaning that the strains are not constrained to any fixed value. In reality, in the case of a symmetric but unbalanced laminate, application of an axial load gives rise to a shear strain y . Given the nature of the loading by fixed grip, in practice y xy  xy  =  zero. Therefore, the laminate modulus is no longer (1/S ) but must be calculated using N^/h^ n  where y  xy  is constrained to zero. In this case, N  x y  is non-zero. These values are presented in  Table 5.7 as 'constrained' (c) values. In comparing the above calculations with the experimental results, a good agreement is observed.  TABLE 5.7 LPT PREDICTIONS OF STIFFNESS REDUCTION UNDER CONSTRAINED AND UNCONSTRAINED CONDITIONS -  Lay-up  Unconstrained (uc)  Constrained (c)  original  cracked  E  original  cracked  E  E (GPa)  E (GPa)  E  E„(GPa)  E (GPa)  E  [0/90].  23.09  17.86  0.774  23.07  17.86  0.774  [0/75],  23.23  18.33  0.790  23.24  18.64  0.802  [0/60],  23.82  19.88  0.837  23.95  21.00  0.876  [0/45],  25.39  22.82  0.900  25.95  24.77  0.955  0  * where E , v 2  12  x  x  0  x  z  0  = 0 and G 4 0. I 2  The good agreement is especially true if only E and v of the cracking ply are discounted, 2  but G  1 2  1 2  is left at its original value. For example, for the [0/45], laminate, if G  1 2  is set to zero  Chapter 5  106  then the unconstrained E = 18.22 GPa and the constrained E = 18.78 GPa, both of which x  x  exceed the observed values considerably. If G  1 2  is not set to zero, then there is good agreement  for all the laminates. It is noteworthy that the same amount of stiffness reduction is predicted if a balanced [0/±45] laminate having the same inner ply thickness as the [0/45] laminate is analysed. s  s  Therefore, the amount of stiffness reduction that will be seen in an off-axis ply is roughly dependent on the degree of constraint. In general, boundary conditions are such that the off-axis layers are constrained, and thus stiffness reductions are correspondingly smaller. In comparing the curves in graphs (b) of Figures 5.1 to 5.4, it can be seen that the onset of cracking occurs at increasingly higher loads as 6 decreases. In the [0/90], laminate, cracking is observed after 125 MPa and is followed by a gradual increase in crack length before reaching a plateau. The limiting crack length indicates that cracking does not occur indefinitely. The onset of cracking was accompanied by a rapid drop in stiffness before becoming gradual. Comparing this behaviour with that of the off-axis laminates, the onset of cracking occurred beyond 127.6 MPa, 190 MPa, and 240 MPa for the [0/75] , [0/60], and [0/45], respectively. But s  as 8 decreases, stiffness reductions become more and more gradual. However, the corresponding load versus crack length curve becomes sharper before reaching a saturation value. The differences in the curves suggest that in the [0/90], system, where the 90° cracking ply is relatively weaker, cracking is initiated at lower loads and the Mode I crack growth due to the transverse tensile stress, a (Figure 2.5), is gradual. On the other hand, as the inner ply 2  becomes stronger with decreasing 8, it becomes increasingly harder to initiate cracks. Crack growth in the 75° and 60° are predominantly in Mode I even though a small Mode II shear component is present. In the 45° lay-up where Modes I and II are almost in equal ratio, cracks grow much faster after initiation than in the case of pure Mode I ([0/90],).  Chapter 5  107  5.3 PROGRESSIVE C R A C K PATTERNS As described in section 4.2.1, the photographs were divided into 10 frames before the cracks were digitized on the image analyzer. Each crack was identified by a pair of x-y coordinates that correspond to the crack tips, using a computer program (Appendix E). As cracking progresses, cracks grow across the width of the specimen and may occupy more than one frame in a photograph. Hence after the x-y coordinates were acquired, the next task was to rejoin all ten frames. A computer program was written to join visually the cracks which straddled the frames, Appendix F [91]. The new coordinates were then plotted for each load level. It was discovered at this stage that the cracks were misaligned in some of the plots, i.e. slight misalignments occurred while moving the photographs from frame to frame. In order to ensure the accuracy of each plot, the only recourse was to check visually each crack in each plot. To do this, transparencies were made of all the plots for each lay-up. The transparencies of each consecutive load level were then superimposed on the original photographs and each crack was checked visually. In cases where the cracks in one plot did not align with the rest, their coordinates were corrected. This was repeated until all the plots aligned exacdy. Although the procedure was laborious, it was necessary to ensure accuracy. These 'corrected' coordinates were used to plot the final crack patterns as shown in Figures 5.5 through to 5.8. An example of the fidelity of the digitized data can be seen in Figure 5.9, where the digitized and original crack patterns for the [0/45] s pattern are compared. The corresponding applied stresses and strains for each crack pattern are also shown in these figures. It is observed that the onset of cracking in the inner transverse ply of the [0/90]  5  laminate occurred at a much higher strain (0.81%) than that which was measured for the [90]  4  lay-up (0.18% in Table 5.1). The very much higher cracking strain indicates that the onset of cracking was significantly delayed due to the constraining effects of the outer 0° plies.  Figure 5.5 Digitized crack pattern for rO/901,. laminate with increasing stress and strain  o 00  Figure 5.6 Digitized crack pattern for \0/15\ laminate with increasing stress and strain  o  Figure 5.7 Digitized crack pattern for r0/601 laminate with increasing stress and strain. £  t—»  o  Figure 5.8 Digitized crack pattern for rO/451. laminate with increasing stress and Strain  Chapter 5  Scale -  2.8 : 1  Figure 5.9 Comparison between actual crack pattern and the digitized equivalent  Chapter 5  113  Figures 5.5 to 5.8 show that there are distinct similarities as well as differences in the cracking patterns of the four different lay-ups. The most obvious similarity is that cracking occurs preferentially, resulting in some areas being more heavily cracked than others. Even at the high loads, some areas in the inner ply do not crack at all and these areas occur randomly. This observation agrees with the findings of Refs [64-70] who showed that there is a variable strength distribution along the cracking ply. It can also be seen that in all the lay-ups, cracking does not necessarily originate from existing flaws or voids (seen at zero load) and that the areas of extensive cracking do not have any more flaws than others. This suggests that crack initiation is probably influenced more by other factors which are on a smaller scale, such as local fibre volume fraction and fibre-matrix bond strength. If so, then the nuclei for initiation sites are likely provided by mechanisms such as fibredebonding [10] and matrix strain magnification effects [9]. If the manner of crack propagation is observed, it can be seen that in this material system, cracks do not necessarily propagate across the entire width of the specimen and that the crack spacing is not uniform as assumed by most idealized models [23]. Generally, however, a larger number of short cracks are observed in the cross-ply lay-up than in others. The distribution of crack sizes is examined statistically in the next section. Like the crack size, the crack spacing is not a unique value but varies along the gauge length. However, the manner in which it varies is distinctly different between the transverse ply and the off-axis plies. In the transverse ply, cracks tend to form between existing cracks but they do not necessarily form midway. In the off-axis plies, the cracks confine themselves to form bands before spreading out to other areas at higher loads. These will be referred to as shear bands.  114  Chapter 5  The appearance of shear bands in the off-axis plies, particularly in the [0/75] and [0/60] s  s  may explain the S-shape of the curves (EIE against a) which is evident in Figures 5.2(a) and 0  5.3(a). In the [0/75] laminate, points E5 to E6, which correspond to stiffnesses in loading from s  179 MPa to 234 MPa show little reduction although there was considerable crack growth. In the same way, in loading the [0/60] laminate from 307 MPa to 335 MPa which corresponds to s  points E7 to E8 in Figure 5.3(b), the stiffness only dropped by 0.5% for a crack growth of 1 m. This is compared to points E6 to E7 where a stiffness drop of 2.7% is seen for a 0.84 m increase in crack length, when the shear bands have yet to form. The large differences in stiffness reduction indicate that the formation of shear bands, where the crack spacing become extremely close, does not affect the stiffness significantly. It is unknown as to why the cracks grow so closely at first in these off-axis plies, but the phenomenon must be due to the existence of the shear stress component, t , which is zero in the [0/90] system. 12  s  As the thickness dependence of ply cracking is well known [10], the variation in crack distribution and shear band formation in these laminates may be attributed to local differences in the inner ply thickness. To investigate this, the edge of a [0/45] laminate was polished and s  examined under an optical microscope. The edge of the gauge length was photographed and alphabetically marked at about 3 mm intervals, Figures 5.10 and 5.11. The thicknesses were measured off these markings and are plotted in Figure 5.12. Then using the digitized pattern at 400 MPa where the shear bands are distinct, the local crack density was measured. These measurements are also plotted in Figure 5.12. It can be seen that the thickness profile shows no significant variation in the thickness along the inner ply. Also, comparing this profile with the local crack density profile and the crack pattern (Figure 5.13), the heavily cracked areas are not confined to any particularly thicker (or even thinner) area, as might be expected.  Figure  5.10 The edge view of gauge length of a rO/451. laminate gauge length after loading to 488 MPa (continued).  Chapter 5  Figure 5.11 The edge view of gauge length continued from Figure 5.10  116  117  Chapter 5  THICKNESS AND CRACK DENSITY PROFILE OF 45-DEG LAYER m  0.38  mm  0.37-  m  0.36-  m  E E,  0.35-  CO CO UJ  0.34-  L  0.33-  K f _ /\  z  o  I r-  > _l  o LU o  in  0.320.31 0.3-  G  D  H  m m  E E o  A/ \  B  55  ,M  IV / \A E I  C  )  •  0.29-  m  s  CE  o  I \N  F  3 _*V  0.28-  u •  I  1  40  1  1  1—  i  •  \  ir  / o  0.270.26  o <  R  \r  B  m m m  i  i  160  i  i  o  i  200  240  DISTANCE ALONG GAUGE LENGTH (mm)  Figure 5.12 Thickness and crack density profile of the inner ply shown in Figs 5.10-11.  U  W  Figure 5.13 Crack pattern of the rO/451. laminate at 400 MPa.  Y  118  Chapter 5  5.4 STATISTICAL RESULTS The aim of this portion of the study was to investigate how cracking occurs in terms of crack size distribution and crack spacing. At the same time, the manner in which crack tips interact as they grow and approach each other was also studied. 5.4.1 C R A C K L E N G T H DISTRIBUTION As the cracking pattern and distribution (shown in Figures 5.5 to 5.8) differ in all the four lay-ups, it is necessary to examine statistically how crack initiation and growth, in terms of crack sizes, vary between them as the laminates are loaded. In order to make comparisons, the results are presented in two ways using stacked bar graphs. First, the lengths of individual cracks for each strain level are classified into six intervals; each interval pertains to a percentage of the total cracking width W , of the inner ply. For example, W is 25.4 mm for the [0/90] lay-up s  and 35.92 mm for the [0/45] lay-up. The intervals which were selected are <0.0625W', s  0.0625-0.125W, 0.125-0.25W, 0.25-0.5W, 0.5-0.75W and 0.75-1W. Comparisons are made in terms of strains rather than applied stresses since the stresses are different in each ply of the laminate. The stacked bar graphs for each lay-up represents the cumulative crack length distribution. These distribution are plotted as a function of strain, as shown in Figures 5.14 to 5.17. The second scheme for presenting that data was to classify the number or frequency of cracks that fall into the same six intervals. The frequency distribution is plotted against the corresponding strain as shown in Figures 5.18 to 5.21. It should be noted that the y-axes of all stacked bar graphs have the same scale for easier comparison. A l l the results for plotting the stacked bar graphs are listed in Appendix G.  Chapter 5  3  E,  x  2.8  i-  2.4  Z  2.2  o <  1.8  (3 LU I  -  2 -  1.6  DC  1.4 -  o  1.2  LU >  [0/90]S  2.6  1  3  O  0.85  1.12  0.94  STRAIN (%)  Figure 5.14 Cumulative crack length distribution for rO/901 laminate.  x r—  Z  LU _l  O <  o  LU >  5 O 0.81  0.94  1.07  1.26  STRAIN (%) LEGEND  •i<oo625(w) ESgSa  0.062S-0.12S(W)  [ 0.125-0.25(W)  0.25-O.S(W) PT>71 0.5-0.75(W) K/l 0.75-1 (W)  Figure 5.15 Cumulative crack length distribution for r0/751, laminate.  1.48%  1.3%  LEGEND  1.67%  STRAIN (%) <O.0625(W)  | 0.25-0.5(W)  0.0625-0.125(W)  | O.W>.75(W)  0.125-0.25<W)  P"7T  0.75-1<W)  Figure 5.17 Cumulative crack length distribution for f0/451s laminate,  Chapter 5  (fi o < or o u_ O O z UJ  > 5 ID  O 0.81  1.12  0.94  STRAIN (%)  Figure 5.18 Cumulative number of cracks for r0/901s laminate.  (fi O < rr o  LL.  O  LU > 3  3 O  LEGEND  0.81  0  mM  0.94  1.07  1.26 STRAIN (%)  | 0.25-0. 5(W)  <0.D625(W)  | 0.0625-0.125(W)  EO?1 0.5^>.7S<W)  | 0.125-0.25(W)  PJZ]  0.75-1 (W)  Figure 5.19 Cumulative number of cracks for r0/751s laminate,  Chapter 5  CO  o < rr o LL  O  rr  LU CD  LU  ^  200  5 3 3  ioo  o STRAIN (%)  Figure 5.20 Cumulative number of cracks for rO/60^ laminate.  CO  [0/45]s  o < rr o LL  O LU > I-  3 O 1.3%  LEGEND  1.48%  1.67% STRAIN (%)  | 0.25-0.5<W)  <0.0625(W) 0.0625-0.125(W)  KXl  0.125-0.25(W)  P"/H 0.75-1 (W)  0.5-0.75(W)  Figure 5.21 Cumulative number of cracks for rO/451, laminate.  123  Chapter 5  As the crack sizes in the intervals vary depending on W for each lay-up, they will be defined as initiated, short, medium and long according to the size ranges (for all the laminates) listed in Table 5.8 below. T A B L E 5.8 DEFINITION OF C R A C K SIZES NAME CODE  INTERVAL  SIZE RANGE (mm)  Initiated  < 0.0625 W  >0 - 2.3  Short  0.0625-0.125 W  2.3-4.5  Medium  0.125- 0.5 W  4.5-18.0  Long  0.5-iW  18.0-35.92  The two sets of stacked bar graphs show that although the total crack length, a, increases with increasing strain, the number of cracks actually decreases or remain the same. Such a trend is expected since cracks tend to join up as they grow hence decreasing their numbers. It is also seen that the maximum number of cracks measured in the cross-ply is about 650 while the other laminates have between 300-500 cracks for approximately the same total crack length. In Figure 5.18, although the combined initiated and short cracks make up between 60 to 85% of the total number, their combined contribution to the total length a, is never higher than 40% at any strain level, Figure 5.14. The medium cracks contribute mostly to a, at about 55%, even though they only represent 35% of the total number. In the [0/75] lay-up, it is seen that the long cracks contribute more than 60% to the crack s  length (Figure 5.15) although all the different sizes are equal in number (Figure 5.19). The combined initiated and small cracks increase with strain but the increase is not as dramatic as  124  Chapter 5  the [0/90] lay-up. Also, the number of initiated and small cracks drop by half from 1.07% to s  1.26% strain, which is not seen in the [0/90] lay-up. This implies that the shorter cracks have 5  joined up to form more medium and long cracks. The same behaviour is exhibited in the [0/60]  s  lay-up (Figure 5.20), although the medium and long cracks contribute equally to the total length, a. In the [0/45] lay-up, the total number of cracks (Figure 5.21) did not show significant increases s  with increasing strain. However, the total crack length more than doubled from 1.48% to 1.67% strain. This indicates that crack initiation only occurs in one stage and beyond that the cracks only grow and join up to form longer cracks. This is in contrast to the behaviour in the [0/90]  s  laminate where crack initiation occurs continually with increasing strain. Comparing specifically the crack numbers between the [0/90] and [0/75] lay-ups at 1.12% s  s  and 1.26% strain, we see that initiated and short cracks constitute about 60% of the total number in the [0/90] lay-up whereas they only make up 35% of the total in the [0/75] lay-up. This and s  s  earlier observations may be explained in terms of how a crack behaves depending on the type of cracking mode involved. It is likely that the crack path ahead of a tip is obstructed by fibre bridging or fibre bunching since the fibres in these plies are usually not perfectly aligned and parallel with each other. If this is the case, a crack growing in pure Mode I (transverse ply) will be hindered and blunted when it encounters a fibre bridge and a new crack forms ahead. In the [0/75] and other off-axis lay-ups, where there is a shear stress component, it is possible for the s  cracks to slide along the fibre-matrix interface and continue growing.  125  Chapter 5  5.4.2 C R A C K SPACING DISTRIBUTION As shown earlier in the digitized crack patterns of Figures 5.5 to 5.8, crack multiplication is not uniform in the off-axis laminates and a distribution of crack spacings exists. Even though the cracking is not entirely uniform in the [0/90] , the crack spacing S for that configuration s  may still be defined by equation [2.15] in which S = WLIa for any crack length a. In the off-axis plies, cracking occurs in discrete shear bands at lower load levels and then tends to become more uniform as the load is increased. It is also of interest to investigate how closely spaced these cracks are within the shear bands. The value of crack spacing S determined from equation [2.15] is essentially the perpendicular distance between one crack plane to another. Based on this definition, the actual crack distances for the off-axis plies were computed using a computer program and applying the following analysis: (a)  The crack spacing is the average value of the minimum perpendicular distances between cracks. This infers that the actual minimum perpendicular distance between each crack must be determined.  (b)  However, for each crack, i, this minimum distance must be associated with a neighboring crack which has some influence on its growth behaviour. Figure 5.22 shows neighbouring cracks a,b,c,d,g and h. These cracks either span or have crack tips lying between the two tips Xjy! and x y of crack i. Although they are located in close vicinity, they obviously 2  2  have less influence than either crack e or f. If r is the radius of the zone of influence of crack i, then e and f fall into that criterion. This zone of influence is determined from the crack tip interaction analysis, which will be described in the next section, and is found to be about 1 ply thickness, t (0.175 mm).  126  Chapter 5  It has been considered that at low load levels, there may not be any cracks lying in region r. In a case such as that shown in Figure 5.22, the closest vertical crack spacing will be between i and d. However, as the load increases, S will be defined as the distance between crack i and any neighbouring crack which falls into region r.  a b  i  e  h  9 Figure 5.22 Illustration of crack i and its zone of influence, r It is mathematically complicated to compute the crack spacings using directly the assigned x-y coordinates of the cracks in the off-axis plies since they are oriented at 9° from the loading axis X. It would be easier to calculate if they were oriented perpendicularly to another axis X as shown in Figure 5.23. The new transformed coordinates would consist t  of finding the intersection point of any crack with the X axis and then taking the cosine value of (90-0) to find the new X . Hence, the entire crack will lie in the plane of X . Using t  t  the transformed coordinates and a computer program SPACING, the distribution of 5 for all the laminates was computed. The flow diagram for the analysis is given in Figure 5.24  127  Chapter 5  while the program itself is listed in Appendix H.  off-axis cracks  Figure 5.23 Transformation form planes X to X  r  Select crack I 1-1.1  Select crack i+j 1-1.1-1  NO Compare crack I with crack l+j Is crack i+j an Interacting crack?  NO  If not, does crack i+j fall into crack i's zone of influence?  (a) YES (b) YES Calculate the perpendicular distance [XI-(Xi+j)j from parts(a) and (b) and place values into array [S] Find the minimum value of [S]  continue with next crack  Figure 5.24 Flow diagram for determining the crack spacing for each crack i.  Chapter 5  128  The actual decrease in crack spacing with increasing load is shown in Figures 5.25 to 5.28 where the results are presented as the frequency or cumulative number of cracks as a function of crack spacing for the different strain levels. It is noted that the frequency plotted here is determined by subtracting the original number of cracks (flaws) prior to loading from the number of cracks computed. Since most cracks do not necessarily grow from existing flaws, this gives a better representation of the crack spacing as they multiply and grow.  Chapter 5  129  Figure 5.26 The variation of crack spacing with increasing strain for T0/75L laminate.  Chapter 5  0  0.2  0.4  0.6  0.8  1  CRACK SPACING (mm) Figure 5.27 The variation of crack spacing with increasing strain for rO/601. laminate.  Figure 5.28 The variation of crack spacing with increasing strain for fO/451. laminate  Chapter 5  131  A comparison between Figures 5.25 to 5.28 shows that over a 0.35% increase in strain level, the shape of the curves in the [0/90] and [0/45] laminates shifts abruptly from being s  s  gradual to become much steeper. The other two laminates, however, have curves which remain fairly gradual over the same increase but there is a tendency to become steeper. All the curves tend to converge towards some limiting value which lies between t and 2t, the values for one and two ply thicknesses respectively. If the frequency of the cracks for each ply are plotted for the highest available strain, as in Figure 5.29, it can seen that the limiting value is about 2t for the [0/90] lay-up and t for the [0/45] lay-up. Assuming that the cracking plies are near or have s  s  reached saturation, then these results can be interpreted as being the minimum crack spacing for these laminates. The observation for the [0/90] laminate is in agreement with the findings s  of Refs [23,32]. Histograms for the other two lay-ups show that the saturation crack spacing is not as well-defined and can range between t and 2t. A closer examination of the curves in Figures 5.26 to 5.28 reveals that cracks which are equal or closer than r, one ply thickness, constitute between 10 to 30% of the total cumulative number. This corresponds to strains where shear bands are formed in the [0/75] and [0/60] s  s  lay-ups (Figure 5.7 and 5.8) and thus imply that the crack spacing in these bands is of the order of t. On the other hand, in the [0/90] system where no noticeable bands occur, the proportion s  of crack spacing less than t is never higher than 10% for any strain level. If the shear bands are indeed caused by the presence of the shear component x, then one would expect to see a large number of cracks having spacings in this t range for the [0/45] laminate. Appropriately, this is s  seen in Figure 5.29.  132  Chapter 5  [0/60]s  2t  iii.llllii.. 0.1 0.1ft 0,17 U  OM OJ  0,3» CM 0.1 0.11 0.17 U  O A OJ  a«  IU OM OA OJU OJ  0.M 0.7 0.7S  [0/75]S 110 • 100 -  21  .Imlllliii i  i  i  0.1 0-16 0.17 U  i  i  OJS U  i  i i II O A IU Q M U  i i i i D i l O J D.M 0.7 H i t  i 1  CRACK SPACING (mm) Figure 5.29 Histograms showing the crack frequency as a function of crack spacing near saturation.  It is also noteworthy to point out that in Figure 5.29, the histograms for the off-axis plies have two peaks, one at t and the other at 2t. This suggests that as the cracks reach saturation, there will be two distinct crack spacings where one, t, is associated with the earlier formation of shear bands while the other is the spacing, 2t, for cracks which multiply in the more uniform  Chapter 5  133  and conventional manner. The evidence so far indicates that the formationof shear bands have little effect on the stiffness of the laminate and since these cracks are so closely spaced, they probably do not interact and have little influence on each other.  134  Chapter 5  5.4.3 C R A C K T I P I N T E R A C T I O N A N D G R O W T H The manner in which crack tips interact as the cracks grow was studied using the available x-y coordinates for each crack. To analyze the interaction behaviour, the relevant parameter was considered to be the closest tip-to-tip distance of one crack from another. This limit implies two possible events. Firstly, a crack tip stops growing due to the presence of another nearby crack tip. Secondly, this is the limiting distance between two cracks in the same plane before they join to form a longer crack. A computer program TIP (Appendix I) was written to calculate the closest tip-to-tip distance from its closest neighbour. The program takes the ith crack and computes the closest tip-to-tip distance from the next (i+1) crack. The closest value is determined from four tip-to-tip distances for each calculation since each crack has two tips. To assist in the interpretation of the crack tip interaction results, it was necessary to determine how the cracks grew from one incremental load/strain to another. A program, GROWTH (Appendix J) was written to calculate the incremental growth, Aa of each crack as it was loaded at increasing strains. As shown in the flow diagram of Figure 5.30, the easiest way of calculating Aa is to find the difference in crack length while going from a high load to a lower one. The value of Aa is then calculated by taking each crack i at say, load L (a higher load) and subtracting from it the length at load L - l (a lower one). Since it is possible that crack i may not be perfectly aligned from one load level to another, a tolerance of ± 0.08 mm in crack 1  spacing was introduced. The accuracy of the alignment is believed to be within that tolerance. The program also takes into account that a crack may have grown from two or more shorter  1 based on random manual checks of alignment.  135  Chapter 5  ones and thus finds the total incremental growth of all of them. As the crack coordinates of the original material flaws (zero load) are known, the program also allows the number of cracks which grew from them to be determined.  *~  For LAST load level L Select crack i, i=1 ,l  Find all the cracks in Load (L-1),(L-2),..etc. which lie inside the coordinates of crack i i.e. within 0.08 mm tolerance crack spacing t  Calculate growth in crack i from loads [L-(L-1 j], [(L-1)-(L-2)],etc... Tabulate t& from one incremental load to another.  Figure 5.30 Flow diagram to calculate the incremental crack growth for each crack. For each lay-up, the cracks at each strain level were analyzed and the results from the TIP. analysis were referred to as the closest  crack tip distance .  The results from the GROWTH  analysis were also classified into five intervals of incremental growth. They are < 1/16 W, 1/16 - 1/4 W, 1/4 - 1/2 W, 1/2 - 3/4 W and 3/4 - 1 W. Both results for each lay-up are shown together in parts (a) and (b) of Figures 5.31 to 5.34 respectively. Parts (a) are in terms of the cumulative number (frequency) of cracks closer than the closest crack tip distance. Parts (b) 2  show the incremental crack growth from one strain level to another.  2 by closest we mean the minimum of four tip distances  Chapter 5  MINIMUM C R A C K TIP DISTANCE (mm)  Figure 5.31(a) Cumulative number of cracks having a crack closer than the closest crack tip distance for increasing strains in the rO/901. lay-up.  INCREASE IN STRAIN (%)  Figure 5.31(b) Incremental crack growth distribution for r0/90i lay-up.  Chapter 5  MINIMUM CRACK TIP DISTANCE (mm) Figure 5.32(a) Cumulative number of cracks having a crack closer than the closest crack tip distance for increasing strains in the fO/751. lay-up.  0-0.81  0.81-0.94  0.94-1.08  1.08-1.26  INCREASE IN STRAIN (%) Figure 5.32(b) Incremental crack growth distribution for \0^5^ lay-up. 1  Chapter 5  MINIMUM C R A C K TIP DISTANCE (mm)  Figure 5.33(a) Cumulative number of cracks having a crack closer than the closest crack tip distance for increasing strains in the r0/60h lay-up.  0-0.94  0.94-1.23  1.23-1.37  1.37-1.5  INCREASE IN STRAIN (%)  Figure 5.33(b) Incremental crack growth distribution for ["0/601.. lay-up.  Chapter 5  0  02  0.4  06  08  1  MINIMUM CRACK TIP DISTANCE (mm)  Figure 5.34(a) Cumulative number of cracks having a crack closer than the closest crack tip distance for increasing strains in the fO/451, lay-up.  0-1.3  1.3-1.48  1.48-1.67  INCREASE IN STRAIN (%)  Figure 5.34(b) Incremental crack growth distribution for rO/451 lay-up  140  Chapter 5  The discussion in this section incorporates these results with those from Figures 5.18 to 5.21, where the cumulative crack number distribution were given. The analysis from program GROWTH showed that about 36% of the cracks in the [0/90]  s  grew from existing flaws while in the [0/75] , [0/60] and [0/45] lay-ups, they were about 20%, s  8  s  12% and 11% respectively. It should be noted that the results in parts (b) are the incremental crack growth distribution of ail the cracks which therefore includes those which grew from the existing flaws as well as those newly created. This was done because the distributions in both types of cracks were found to be similar. Also, parts (a) only show the distribution of cracks closer than 1 mm apart since cracks which are far apart can be assumed not to interact. In comparing all the parts (b) of the above Figures, it can be seen that the majority (68%-94%) of cracks in the first 0.9% strain only grew by less than 1/16 W. This is approximately 1.6 mm in the [0/90] lay-up and 2.2 mm in the [0/45] lay-up. With a further increase of about s  s  0.25% strain, the number which grew by this small increment decreased by 14% in the [0/90]  s  and by 26% in the [0/45] lay-up. This supports our earlier observation from Figures 5.18 and s  5.21 (cumulative crack number) which indicated that new short cracks are continually created in the [0/90] s whereas most of the cracks in the [0/45] are initiated at the onset strain. Also in s  Figure 5.31(a), 50% of the cracks which grew over a 1.12% strain increase were within 2t, two ply thickness apart. It can be argued that this large proportion is due to crack growth at the specimen edges, hence giving the minimum crack spacing of 2t. According to Figure 5.14, the total number of cracks in this lay-up at 1.12% strain is 650. By analysing the x-y coordinates of all these cracks, it was found that only 38% of this total were edge cracks. Moreover, Figure 5.18 shows that most of the cracks at that strain level were either short or medium cracks which in turn implies that most cracks, including the ones at the edges, stopped growing as they came closer than 2t to another nearby crack.  Chapter 5  141  Even with a further strain increase from 0.95 to 1.12% in the [0/90] lay-up, the largest s  proportion of cracks are initiated cracks. For the same increase in the [0/45] .lay-up from 1.48 s  to 1.67%, the number of cracks which grew by less than 1/16 W was about the same as those which grew between 1/16 - 1/2 W. The overall incremental growth in cross-ply lay-up seem to be between < 1/6 W which corresponds to <1.59 mm. The larger proportion of cracks growing between 1/16 -1/2 W (2.25-8.98 mm) for the [0/45], is also reflected in Figure 5.21 where they make up 60% of the total number and 73% of the crack length, a, in the specimen at 1.67% strain. In Figure 5.34(a) for the [0/45]„ only 30% of the crack tips are within 2t distance apart while the remaining are further. The results in Figure 5.34(a) are inconclusive as to the value of the tip-to-tip interacting distance since the curves have no sharp cut-offs compared to the other lay-ups. Furthermore, the digitized crack patterns in Figure 5.8 show that the inner ply is far from being fully cracked. Since we do not have results for the [0/45] at a higher strain, we can examine the curves s  in Figure 5.33(a) for the [0/60] lay-up. The curve at last strain level (1.5%) shows that about s  40% of the cracks have tip distances closer than 2t. This implies that there is a tendency for the same behaviour as in the [0/90] where the cracks stop growing when they approach a neighbour s  within 2t distance away. However, the amount of incremental growth was considerably longer than in the [0/90] before the cracks stop. In part (b), about 46% of the cracks grew between s  3.67-7.33 mm (1/4 - 1/2 WO and 12% grew between 7.33-14.7 mm (1/2 - 3/4 W). The larger incremental growth hence accounts for the higher percentage (23%) of cracks having lengths longer than 1/4 W, (Figure 5.20). This is compared to the [0/90] lay-up where only 16% of the s  cracks are longer than 1/4 W, (Figure 5.18). As discussed previously, these results again suggest that it is easier to propagate a crack under mixed Modes I and II loading than it is under pure  142  Chapter 5  Mode I. The incremental growth behaviour of the [0/75] lay-up has elements of that shown by s  the [0/90] and the [0/60] lay-ups. First, we see a large proportion of cracks are initiated at the s  s  first 0.94% strain as in the [0/90], lay-up but beyond that the cracks grow by increments of 6.6-13.13 mm (1/4 - 1/2 W) to form a large number of medium and long cracks, Figure 5.19. Of the four lay-ups, this laminate has the largest number of cracks which grew between 3/4 - 1 W, which is almost across the total cracking width. It is interesting that we see some cracks in the digitized patterns (Figures 5.5 to 5.8) of all four lay-ups which just stop growing even when there are no other cracks nearby. Since these cracks are not within interacting distance from the others, the only possible explanation is that they are located in the stronger areas of the specimen.  143  Chapter 5  5.5 R-CURVE BEHAVIOUR OF OFF-AXIS LAMINATES 5.5.1 Analysis of d(E/Eo)/da Results The plots of normalised stiffness against crack length in Figures 5.1 to 5.4 show that the relationship between (E/EJ and a for the [0/90] lay-up is directly linear but the relationship is s  not as clear for the off-axis lay-ups. However, as long as the cracks in the cracking 6 plies are non-interacting, which is true initially, we should expect the same linear form for them. As discussed in section 5.3, the S shape of these (E/EJ as a function of a curves is associated with the formation of shear bands. The multiplication of small cracks very near the longer ones in the band area does not directly affect the stiffness but is reflected as an increase in crack length. It follows then that if the effect of these short cracks can be eliminated, it would be possible to determine the effective crack length, a which leads to the measured stiffness reduction. e  A simple analysis was performed to determine the value of a . We first consider a long t  crack i in the vicinity of the shear band area. Any crack which has a crack spacing closer than the interacting distance, r from crack i should have little or no influence on the stiffness. As shown in section 5.4.2, cracks in the shear band region tend to be spaced within the distance of t, one ply thickness (0.178 mm). Hence, the value of t will be used for r. At the same time, the length of crack (i+j) to be eliminated should only be limited to the portion which is directly influenced by crack i. Given these assumptions as shown in Figure 5.35, all the cracks for each load level will be eliminated accordingly and the effective crack length a calculated. The e  computer program ELIM (Appendix K) having the algorithm shown in Figure 5.36, was used to carry out this analysis.  Chapter 5  not eliminated at all since longer than i  '  totally eliminated  x  i  v  X  i I I \  £f  2  I I I  \^  i+G+2)  partly eliminated Figure 5.35 The guideline used to eliminate very close short cracks.  Get crack i=1.1  Find next crack i+1 which is within 1 ply thickness away  Is that crack longer than i ? YES  NO subtract from it the portion of length a which has a direct influence on i.  Figure 5.36 Flow diagram for program E L I M to eliminate short cracks.  145  Chapter 5  As only 3-4 photographs were analysed using the image analysis, crack patterns at higher strain levels (*) were not available for this ELIM analysis. In these instances, the magnitude of the total crack length eliminated was calculated based on the proportion of (aja) determined from analyzing the last strain level which was available. Such an approach would mean that the end results for the higher strain levels would be conservative. The magnitudes of the reduction of a to obtain a are listed in Appendix L. The results from this analysis for all four lay-ups are e  plotted in Figure 5.38 while the actual measured results of normalised stiffness as a function of crack length, a are shown in Figure 5.37. A linear regression method was used to fit the results for both graphs using equation [2.25] and E/E„ were found to be Lay-up  (Original crack length, a)  (Effective crack length, a )  E/E =  EIE =  e  [0/90].  1-0.0486a  1-0.0528a,  [0/75].  1-0.032a  l-0.0377a  [0/60],  1-0.0192a  l-0.022a  [0/45],  1-0.0081a  1 -0.0088a.  e  e  As shown above, the elimination of the short cracks closer than distance r caused the (EIEJ versus a slopes to become steeper. This is because a larger percentage of a was eliminated as the shear band cracks form and cracks grew closer with increasing strain. The largest reduction in the slope occurred in the [0/75], and [0/60] lay-ups at about 15% followed by 9% in the s  [0/45], lay-up and 8% in the [0/90] . This agrees well with the crack patterns shown in Figures s  146  Chapter 5  5.6 to 5.8 where shear bands are most obvious in the first two instances. Even with the analysis, however, the (E/EJ versus a curves for these two lay-ups in Figure 5.38 remain in the S-shape form. According to equation [2.25] [E = E (l -c(Q)a)], the coefficients of a in the above 0  expressions are the material constants, c(9) for the different lay-ups. It can be shown by plotting c(9) as a function of 8 that they are linearly related when 6 is between 45° and 90°, Figure 5.39. This trend is expected since the final crack length is similar in all the lay-ups, and the stiffness changes are linearly related to 0. For both sets of results, Figure 5.39 shows that as 0 is less than 37°, c(0) is equal to 0. This implies that the laminate stiffness will not decrease when matrix cracking occurs in the off-axis plies oriented less than 37°.  Chapter 5  Figure 5.38 Normalised stiffness as a function of effective crack length. a  147  v  148  Chapter 5  0.06  Off-axis Angle 0 Figure 5.39 Material constant c(Q) as a function of off-axis angle.  149  Chapter 5  5.5.2 Comparison of present experimental results with other models The present experimental results for the [0/90] laminate are compared with the models s  by Ogin et al. [61], Laws and Dvorak [48] and Hashin [50] in Figure 5.40. These three models were chosen since they give explicit expressions for the stiffness loss against matrix crack length. As shown in the figure, the shear-lag model given by Ogin et al. gives the best prediction followed by the self-consistent model by Laws and Dvorak. The variational model by Hashin overestimates the stiffness reduction by as much as 5% of the total but this is reasonable since the model predicts the lower bounds for the stiffnesses.  Ogin et. al.  o yj  Laws and OvoraK  LU  Hashin  CO CO LU  [0/90]s experimental results  z  LL.  CO Q LU CO  \  s  E/Eo = [1 -0.0492a]  DC  O  CRACK LENGTH a(m)  Figure 5.40 Comparison of stiffness reduction models with present experimental results.  150  Chapter 5  5.5.3 Matrix cracking Resistance Curves The values for d(E/EJ/da were determined by differentiating equation [2.25] for both a and a . Then using Equation [2.14], the values of G were calculated and used to plot the G e  R  curves for each lay-up as shown in Figure 5.41 and 5.42. The calculated values of G are G, „ 0  which in the off-axis laminates is a combination of both (G,) and (G ) components. n  The G curves in both figures show that the resistance to matrix cracking increases with R  crack length in the [0/90]s [0/75], and [0/60], lay-ups. In the [0/45], lay-up, however, there is a slight increase after the onset of cracking but the resistance approaches a limiting value. The shape of the G curves in the first three instances show two distinct regions. Initially, the curve R  starts as a plateau but later G increases dramatically and the amount of increase per unit crack R  length increases with 6. Generally, there is greater resistance to matrix crack growth as 0 tends from 45° to 90°. The shape of the resistance curves may be explained using the results of our previous statistical analysis in terms of crack size distribution and growth. At the onset of cracking in the [0/90],, [0/75], and [0/60], respectively, cracks initiate and grow in the weaker areas first. The resistance in the [0/60], and [0/75], laminates then remain constant at about 70-80 J/m as 2  cracks coalesce to form longer cracks, which is shown by the flat plateau region. However, in the [0/90], laminate, there is still a 'gradual climb' in the plateau region to about 100 J/m which 2  corresponds to the continual initiation of short cracks. The later portion where there is a sharp increase in the resistance to about 500 J/m is due to the initiation and growth of cracks in the 2  stronger areas of the specimen. In the [0/45], laminate, the strain energy release rate or the resistance to crack growth increases very gradually after the onset of cracking and approaches  151  Chapter 5  Figure  5.41  Matrix cracking resistance curves for all  rO/91.  using a  Figure 5.42 Matrix cracking resistance curves for all TO/Glj using a  v  153  Chapter 5  a saturation value of about 55 J/m . There does not seem to be a second region of increase in 2  resistance as seen in the other laminates. This is possibly because the crack length data at higher crack densities are not available. The relatively low resistance to the onset of cracking and the overall flatness of its G  R  curve suggest that the onset and propagation of cracks in the [0/45] lay-up does not absorb s  much energy. From the work so far, it is not possible to determine the relative amounts of the Gf and G„ components of these curves. However, it can be said that the G curves for any angle R  8, in decreasing order, will lie between that of the [0/90]„ and [0/45], lay-ups. As shown in Figure 5.41, the most visible effect of eliminating the shorter cracks in the shear band region is to move the G curves of the off-axis plies closer towards the crossply R  curve. Also, all the curves show an overall higher resistance to cracking. The plateau regions of all the lay-ups tend to overlap which suggests that the growth of the cracks greater than two ply thickness spacing apart are driven by the same rate of strain energy release irrespective of their lay-up. It has been suggested by some workers [63,59] that the stress intensity factor related to crack growth is independent of crack length but only depends of crack spacing. Furthermore,, according to Reference [92], K, = CP/7 while K„ = W3T for the isotropic case, where s is the crack spacing. If our resistance curves were to be plotted against the crack spacing (Figure 5.43), it can be seen that the strain energy release rate does not increase until the crack spacing becomes small enough to allow interaction. The crack spacing here is defined by the expression given in Equation [2.15]. Even though this definition does not accurately represent the true value of s at low crack densities, as there is a distribution, the curves show that there is almost no change in resistance when cracks are far apart. But as crack density increases and crack spacing reaches its limiting value of 2t, the resistance climbs dramatically.  154  Chapter 5  CM  E  CC  o  LU  o  < I-  200  -  co  CO UJ  cc  DECREASING CRACK SPACING (mm)  Figure 5.43 Crack resistance as a function of crack spacing. This behaviour is seen in all the lay-ups, even for the [0/90] lay-up where we see a large s  number of cracks being continually initiated. It can also be seen that the resistance curves tend to overlap, although the [0/90] lay-up still shows the highest overall resistance and the [0/45] s  s  lay-up the least. From the presently available results, the strain energy release rate for the [0/45]  s  is not affected by the multiplication of cracks and decreasing crack spacing. Again, this may be due to the fact that the cracking has not reached saturation.  Chapter 6  155  CHAPTER 6  SUMMARY AND CONCLUSIONS  The scope of the present work involved fabricating and studying the cracking behaviour of a class of glass/epoxy laminates having the [0/8], geometry, where 0 is equal to 45°, 60°, 75° and 90°. The purpose was to investigate the ply cracking behaviour as a function of the orientation angle, 0 both deterministically (using fracture mechanics) and statistically. In the first instance, the onset and stable crack growth of the off-axis plies (0) was characterized using the strain energy release rate parameter, G and matrix cracking resistance curves (R-curves). In the second instance, the manner in which cracks multiplied and grew was studied in terms of crack size and number distribution, crack spacing and crack tip interaction. Laminates were fabricated from a medium strength fibreglass and epoxy resin which produced optically translucent specimens when cured. They were made using a drumprepregger and cured in an autoclave using a cure cycle which was experimentally determined. These four-ply laminates had a fibre volume fraction of 0.6 and were about 0.7 mm thick. The experimental work consisted of performing incremental tensile tests and monitoring the axial stiffness loss due to the inner ply cracking. At the same time, the crack growth pattern at each load increment was photographed. Using image analysis, each crack in the photographs was traced and identified with a pair of x-y coordinates. The results were then analysed in two ways. The first analysis used the stiffness reduction and crack length results to determine the strain energy release rate, G from a compliance expression. The crack length was defined as the total crack length of all the cracks measured at each load level. The G values which can also  Chapter 6  156  be viewed as the resistance to cracking were used to plot the crack resistance G curves for each R  lay-up. In the second analysis, the changes in the distribution of crack length and number with increasing strain was studied. An analysis of the crack tip interaction as the cracks grew and the magnitude of growth with each incremental strain was also performed. By combining both the analyses, the following conclusions can be drawn (1) The stress at onset of cracking increases as the orientation angle 8 is changed from 90° to 45°. With increasing stress, the crack length increases more rapidly with decreasing 6. This is because a large proportion of the length increase in the off-axis [0/45],, [0/60], and [0/75], lay-ups is due to the growth and coalescence of initiated cracks. In the [0/90], transverse ply, crack initiation continually dominates coalescence at all stages and hence a larger number of cracks are short. The difference in behaviour is attributed to the additional Mode II shear loading which exists in the other lay-ups but not in the [0/90], lay-up. (2) The overall loss in stiffness due to matrix cracking increases with increasing 8. A stiffness loss of as much as 18% was found in the [0/90], lay-up while the loss in the [0/45], lay-up was about 5%. The much smaller stiffness reduction in the [0/45], lay-up is due to the ability of the cracked layer to carry the load in shear even when cracking is near saturation, and the boundary conditions imposed onto the cracked layer. (3)  The stiffness reduction is linearly related to crack length in the [0/90], laminate but the relationship is not as clear for the others. An S-shape curve was found when plotting the stiffness loss as a function of crack length for the off-axis plies. Some of this behaviour is due to the existence of shear bands in the initial stages of cracks initiation and growth. The formation of shear bands has no significant effect on the stiffness but manifests itself as an increase in crack length.  157  Chapter 6  (4)  Cracking in the inner ply does not necessarily originate from existing flaws. Initiation is more influenced by other local factors, perhaps such as the fibre debonds and variations in the fibre volume fraction. Local crack densities could not be correlated with any locally measured parameter in this work.  (5) The inner cracking ply has variable strength and hence the weaker areas crack first. Initially, crack multiplication is generally not uniform leading to a distribution of crack sizes and spacings. But as cracking reaches saturation, the crack spacing tends to a value equal to two ply thickness in the [0/90], lay-up. In the other lay-ups, the crack spacing can get even closer, to within 1 ply thickness. This is particularly true in the shear band areas. (6)  As the applied strain is increased, the incremental crack growth is considerably greater in the off-axis lay-ups than in the [0/90], lay-up. Due to the combined Mode I and II loading, crack initiation is quickly followed by crack coalescence and growth. In the [0/90], lay-up where crack growth is driven by Mode I loading, cracks are continually being initiated but the amount of growth is limited. When crack propagation does occur, the cracks tend to stop when the crack tips get within two ply distances of each other. The same behaviour is also found for the other lay-ups. However, cracks in the shear bands can get within one ply thickness apart before they interact.  (7)  The behaviour of matrix cracking in the [0/6], laminates can be characterized by crack resistance, G curves. It is found that the overall resistance to cracking increases with 9. R  Initially, the resistance after the onset of cracking in the [0/90],, [0/75], and [0/60], laminates remains constant. Later, however, the resistance gradually increases as cracking occurs in the stronger areas. The sharpest and largest increase is seen in the [0/90], plies where crack initiation always dominates coalescence and growth. Such an increase is not seen in the [0/45], lay-up where crack growth is the dominating mechanism and is driven  Chapter 6  158  equally by Modes I and II loading. In this lay-up, the crack resistance only increases at the onset of cracking and approaches a limiting value with increasing crack length. (This is probably because the inner ply has not cracked completely). 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[75] Broek,D., Chapter 5, Elementary Engineering Fracture Mechanics, Fourth Revised Edition, Martinus Nijhoff Publishers, 1986.  References 164 [76] Westergaard,H.M., "Bearing Pressure and Cracks," Journal of Applied Mechanics, Vol.61,1939, pp A49-53. [77] Sih,G.C, Paris,P.C, and Irwin,G.R., "On Cracks in Rectilinearly Anisotropic Bodies," International Journal of Fracture Mechanics, Vol.1 ,1965, p. 189. [78] Wang,S.S., "Fracture Mechanics for Delamination Problems in Composite Materials," Journal of Composite Materials, Vol.17,1983, pp 210. [79] Chai,H., "The Characterization of Mode I Delamination Failure in Non-woven, Multidirectional Laminates," Composites, Vol.15, No.4, 1984, pp 277-290. [80] Rybicki.E.F., Schmueser.D.W., and Fox,J., "An Energy Release Rate Approach for Stable Crack Growth in the Free-Edge Delamination Problem", Journal of Composite Materials, Vol. 11, 1977, pp 470-87. [81] Wilkins.D.J., Eisemann,J.R., Camin.R.A., Margolis,W.S., and Benson,R.A., "Characterizing Delamination Growth in Graphite-Epoxy," Damage in Composite Materials, ASTM STP 775, 1982, p. 168. [82] Irwin,G.R., "Fracture", Hanbuch der Physik, Vol. V, Springer-Verlag, 1958, p.551. [83] Griffith,A.A., "The Phenomena of Rupture and Flow in Solids ", Phil. Trans. Royal Society of London A221, 1921, pp 163-197. [84] Griffith, A.A., "The Theory of Rupture", Proceedings of the 1st International Congress on Applied Mechanics, 1924, pp 55-63. [85] Steif,P.S., "Stiffness Reduction Due to Fibre Breakage", Journal of Composite Materials, Vol. 18, No.2, 1984, pp 153-172. [86] Lee,C.W., "Composite Cure Process Control by Expert Systems," .American Society for Composites, pp 187'-196. [87] Radford,D.W., Fracture Toughness of a Carbon Fibre-Epoxy Composite Material, M.ASc. Thesis, University of British Columbia, Vancouver, August 1982. [88] Dorosh.M., and Poursartip,A., "Optimized Processing of Composite Materials," Report No. AGAR 77 (RC-17), April 1989, University of British Columbia. [89] Trans-tek Inc., "Displacement transducer DC-DC Series 240," Bulletin SO12-0030-KHD. [90] Whitney, J.M., Daniel, J.M., Pipes, R.B., "Experimental Mechanics of Fibre Reinforced Composites, rev. ed. Society for Experimental Mechanics, Englewood, Cliffs,New Jersey, Prentice Hall, 1984. [91] Poursartip, A., private communication, 1989. [92] Tada,H., Paris,P., and Irwin.G., The Stress Analysis of Cracks Handbook, DEL Research Corp. Pennsylvania, 1973.  165  Appendices  APPENDICES APPENDIX A  D E T E R M I N A T I O N OF N O R M A L A N D SHEAR STRESS COMPONENTS I N LAMINATES For any applied Force N , (i.e N and N = 0) x  [A] = 2f[Q] + 2r[Q] e  y  xy  0  Transforming [A] ==> [A]'  1  mid-plane strains  [0/45],  [0/75].  [0/60].  [0/90],  (. 631 B - •> f+ 5.911 s - i\ f+ IJ» « - i \m. - 1.0 s - tion. 1- i.7o B - tlun. {- *Jt B 9} [ 2.9J B 9) { - 1.64 « - « J  /  -  0  6.13 £ 8.13 £  -  »\ 9 [Kl.  V  stresses (°:  [(21 =  (+ {+  937.7 "| 3.55 m 100.5;  (+ 766.2" (+ 695.8" - 45.8 IN], - 89.95 U 51.1 J 1+ 17.59J  ( + 645.4" 103.6 m, 1,0.00  [T] =  (+ + [ -  567.6^ 366.6 470.6;  (+ 201.4> + 518.9 I" 377.1;  28.5 ' 634.4 211.7J  f 103.6 + 645.6 [N] 1,0.00  (V  x  T  |V , 12, T  lYzy  (+ {-  s  x  J  Modell Model +11 In terms  ofK  0.562  0.45  0.25  0  In terms  ofG  0.476  0.224  0.052  0  Appendices  APPENDIX B ASYST PROGRAM FOR DATA ACQUISITION  :IT; INTEGER DIM[ 1500 , 0 ] ARRAY LOAD.DISP IBM.DACA DAS.INIT 10. CONVERSION.DELAY DAS.INIT REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL REAL  DIM[ DIM[ DIM[ DIM[ DIM[ DIM[ DIM[ DIM[ DIM[ DIM[ DIM[  1500 ] ARRAY HC.O 1500 ] ARRAY HC. 1 1500 ] ARRAY LOAD.N 1500 ] ARRAY DISP.M 1500 ] ARRAY HC.0.DEC 1500 ] ARRAY LOAD.DEC 1500 ] ARRAY XLO AD 1500 ] ARRAY XLOAD.DEC 1500 ] ARRAY HC.0.NEW 3 ] ARRAY FTT.INC 3 ] ARRAY FIT.DEC  0 HC.0.NEW := 0 HC.O := 0HC.1 := XLAOD []RAMP XLOAD.DEC [] RAMP XLOAD.DEC REV[ 1 ] XLOAD.DEC := INTEGER INTEGER INTEGER INTEGER INTEGER INTEGER INTEGER INTEGER INTEGER  SCALAR SCALAR SCALAR SCALAR SCALAR SCALAR SCALAR SCALAR SCALAR  NUM NUMINC NUMLNC2 NUMINC 1 NUMDEC DECST NUMDEC 1 DEC.COUNT FITTER  \ASSIGN ORDER OF POLYNOMIAL FIT (LINEAR = 1) 2 FITTER := 0 N U M := 0 NUMINC := 15 STRING FILE.NAME 20 STRING OPEN.NAME 35 STRING OPEN.IT 5 STRING M M 5 STRING N  Appendices 0 1 A/D .TEMPLATE PEARL.TEMPLATE PEARL.TEMPLATE LOAD.DISPL TEMPLATE BUFFER PEARL.TEMPLATE 1 A/D GALN : SS.CLEAR ODO 7DROP LOOP \SETTING UP MULTITASKING : SETUP.TASKS 50 TASK PERIOD PEARL.TEMPLATE A/D.INTT 1 TASK A/D.LN>ARRAY 10 1 TASK.MODULO PRIME TASKS \START ACQUIRING DATA :GO.FOR.IT "ORDER OF POLYNOMIAL FIT =" FITTER. CR 250 MSEC DELAY SETUP.TASKS TRIGGER.TASKS SCREEN.CLEAR "ACQUIRING DATA WHILE TEST IS RUNNING" CR BEGIN ?KEY UNTIL 7BUFFER.LNDEX 3 - 2 / NUMINC :=\GET DATA FOR LOADING PORTION 1 1 TASK.STATE SCREEN.CLEAR "WAITING FOR PEARL TO INTERRUPT TEST" PAUSE 2 1 TASK STATE SCREEN.CLEAR " CONTINUE TEST FOR UNLOADING PORTION" BEGIN ?KEY UNTIL 7BUFFER.LNDEX 3 - 2 / NUM:= \GET DATA FOR UNLOADING PORTION SCREEN.CLEAR  Appendices "ACQUISITION DONE"  168  STOP.TASKS CLEAR.TASKS \TO CONVERT VOLTAGES INTO LOADS AND DISPLACEMENTS :CONVERT.IT LOAD.DISPL XSECT[! , 1 ] 0. 10. A/D.SCALE HC.O := LOAD.DISPL XSECT[! , 2 ] 0. 10. A/D/SCALE H C . l := \CONVERT H C . l TO DISPLACEMENT AND HC.O TO LOAD H C . l .252* DISP.M:= \SMOOTH THE LOADING PORTION XLOAD SUB[ 1 , NUMINC1 ] HC.O SUB[ 1 , NUMINC 1 ] FITTER LEASTSQ.POLY.FIT FIT.INC := \ STORE COEFFICIENTS INTO FIT.LNC ARRAY NUMINC 1 + NUMINC1 := XLOAD SUB[ 1 , NUMINC1 ] FIT.INC POLY[X] HC.O.NEW SUB[ 1 , NUMINC ] := HC.O.NEW 374 * 37 - 4.448 * LOAD.N := \PLOT LOAD AGAINST DISPLACEMENT DISP.M SUB[ 1 , NUMINC ] LOAD.N SUB[ 1 , NUMINC ] XY.AUTO.PLOT N U M NUMINC - NUMDEC := NUMDEC 1 + NUMDEC 1 := NUMINC 1 + DECST := \SMOOTH UNLOADING PORTION XLOAD.DEC SUB[ DECST , NUMDEC ] HC.O SUB[ DECST , NUMDEC ] FITTER LEAST.SQ.POLY.FIT FIT.DEC \STORE COEFFICIENT INTO FIT.DEC XLOAD.DEC SUB[ DECST , NUMDEC 1 ] FIT.DEC POLY[X] HC.O.DEC SUB[ DECST ,NUMDEC1 ] := HC.O.DEC SUB[ DECST , NUMDEC ] 374. * 36 - 4.448 LOAD.DEC SUB[ 1, NUMDEC ] := \PLOT UNLOADING PORTION DISP.M SUB[ DECST , NUMDEC ] LOAD.DEC SUB[ 1, NUMDEC ] XY.DATA.PLOT PAUSE  Appendices :STORE.DATA NORMAL DISPLAY "NEW FILENAME= " "INPUT FILE.NAME ":= " OUT>FILE " OPEN.NAME ":= OPEN.NAME FILE.NAME "CAT OPEN.IT ":= OPEN.IT "EXEC " N " N ":= " mm" M M ":= -1 3 FIX. FORMAT CR NUMINC 1 + NUMINC := NUMLNC 1 DO HC.O [ I ] . LOAD.N [ I ] . N "TYPE 5 SPACES DISP.M [ I ] . M M "TYPE CR LOOP ."NUMBER OF LOADING INPUT READINGS; " NUMINC. CR N U M 1 + NUM:= NUMDEC 1 + NUMDEC := CR DECST DEC.COUNT := NUMDEC 1 DO LOAD.DEC [ I ] . ." N " 5 SPACES DISP.M [ DEC.COUNT ] . . " M M " CR DEC.COUNT 1 + DEC.COUNT := LOOP . " NUMBER OF UNLOADING INPUT READINGS =" NUMDEC . CR CR FIT.INC . CR FIT.DEC . CR \FTND THE BEST FIT FOR THE SECOND 50% OF THE LOADING CURVE NUMINC 2. / NUMINC := DISP.M SUB[ NUMINC2 , NUMINC2 ] LOAD.N SUB [ NUMINC2 , NUMINC2 ] 1 LEAST.SQ.POLY.FIT "PLOT THE LOAD VERSUS DISPLACEMENT:". OUT>FILE.CLOSE :RUN.IT GO.FOR.IT CONVERT.IT STORE.IT  Appendices APPENDIX C STRESS-STRAIN CURVES TO DETERMINE MATERIAL PROPERTIES  STRAIN (%)  Appendices  TENSILE R E S P O N S E O F[ 9 0 L G L A S S - E P O X Y  E22= 1 0 . 8 G P a v  21  =0.09  -i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—r  STRAIN  (%)  Shear stress-strain curve for a [+45]2s glass/epoxy s p e c i m e n under tension.  to Q. to to  CO  k_ 55  20  Glass/Epoxy  «5  G , =6.0 GPa 2  CD _C GO  -1—i—i—i—i—i—r 0.2  0.4  0.6  0.8  T 1  1  \  1  1.2  Shear Strain (%)  1 1.4  1  1 1.6  1 1— 1.8  APPENDIX D STIFFNESS R E D U C T I O N A N D STRESS R E S U L T S NORMALISED STIFFNESS AND STRESS DATA FOR [0/90]s EO = 22.80 GPa  APPLIED STRESS (MPa) EO  E/Eo  .0.0  1.000  101.0  0.995  127.0  0.996  153.0  0.983  177.0  0.973  203.8  0.924  E6 E7  226.0  0.895  249.0  0.872  E8 E9  299.0  0.865  361.0  0.850  E10 E11  400.0  0.837  440.0  0.818  E1 E2 E3 E4  E5  NORMALISED STIFFNESS AND STRESS DATA FOR [0/75]s EO = 22.43 GPa  EO E1 E2  APPLIED STRESS (MPa)  E/Eo  0.00  1.0000  100.00  1.0420  127.00  1.0200  E3  151.30  1.0090  E4  178.70  0.9780  E5  203.30  0.9470  E6  230.90  0.9337  E7  263.05  0.8960  E8  309.30 '  0.9170  E9  361.50  0.9068  E10  417.10  0.8990  471.00  0.8740  E11  Appendices NORMALISED STIFFNESS AND STRESS DATA FOR [0/60]s EO = 23.60 GPa APPLIED STRESS (MPa)  E/Eo  EO  0  1.000  E1  152  1.042  E2  170  1.035  E3  191  1.030  E4  220  1.002  E5  251  0.980  E6  280  0.970  E7  300  0.955  E8  337  0.950  E9  350  0.945  E10  390  0.924  E11  422  0.900  E12  466  0.880  NORMALISED STIFFNESS AND STRESS DATA FOR [0/45]s EO = 24.41 GPa APPLIED STRESS (MPa) EO E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 e15  E/Eo  0.0  1.000  103.3  1.037  130.0  1.034  160 5  1.027  185.0  1.025  213.7  1.027  238.4  1.023  267.5  1.024  290.0  1.024  322.0  1.022  345.0  1.019  373.0  1.005  404.0  0.997  4290  0.972  453.0  0.960  482.0  0.943  174  Appendices APPENDIX E  PROGRAM TO DETECT. COUNT AND IDENTIFY CRACKS WRITTEN IN TASIC LBL (**CRACK.PRM** TO DETECT CRACKS FOR LAY-UPS LN A N Y ORIENTATION) TSC 0 DIM HO, R3, JI(5), Xl(1000) FNC J1 (1 )=105, J1 (2)=106, J1 (3)= 107, J1 (4)=108 ,J 1 (5)=5 8 FNC 116=0 PDF5,J1(1) PNT (' $LNPUT FILENAME=') ASN.ll LBL M PCT CLR X I FNCI11=1,X1(I11)=0 INP (' $DISTANCE FROM ORIGIN?')R2 INP (' $LF FRAMES ITOJ, ADD 48.5')R3 LBL N PTI 1,0,I3,I4,5,J1(1),X1(I11) IFC I4,E,E,+ FNC X I (II1) = X l ( I l l ) * . 2176 + R2 FNS 112 = 111+1 FNC X1(I12) = X1(I12)*.190 + R3 FNC 113 = 111+2 FNC X1(I13) = X1(I13)*.2176 +R2 FNC 114 = 111+3 ,115 = 111+4 FNC X1(I14) = X1(I14)*.190 + R3 FNC 117 = 113+116 WRT 11,DATA,X1(I11), X1(I12), X1(I13), X1(I14), X1G15), 117 FRM DATA (6F8.2) FNC 111 =111 +5 GTON LBL E FNC 116 = 117 CMA PNT(' $MORE FIELDS Y/N?') Y O N 115 IFC I15,B,+,M LBL B CLO 11 IEL HME 1  Appendices APPENDIX F PROGRAM FOR JOINING UP CRACKS 1080' I N I T I A L I Z E 1090 ' 1100 DEFINT A-Z 1110 DIM CURSORQ5.1) 1120 K E Y OFF 1130 GOSUB 3000 1140 SCREEN 9 1150 COLOR 15,0 1160 CLS 1165 X.OFFSET=0 : Y.OFFSET=0 1170' 1270 ' Determine mouse driver location, if not found, quit. 1280 ' 1290 DEF SEG=0 1300 MSEG=256*PEEK(51*4+3)+PEEK(51*4+2) ' Get mouse segment 1310 MOUSE=256*PEEK(51*4+l)+PEEK(51*4) ' Get mouse offset 1320 IF MSEG OR (MOUSE-2) THEN 1370 1330 PRLNT'Mouse driver not found" ' Not found, so print error. 1340 PRINT 1350 PRINT'Press any key to return to system" 1360 I$=INKEY$ : IF 1$="" THEN 1360 ELSE SYSTEM 1370 DEF SEG=MSEG : MOUSE=MOUSE+2 ' Set mouse segment 1373 IF PEEK(MOUSE-2) = 207 THEN 1330 '207 is iret 1376 'Mouse Driver is there, continue. 1380 M l = 0 : C A L L MOUSE(Ml,M2,M3,M4) ' Initialize the mouse 1390 ' 1400 ' Set mouse sensitivity 1410' 1420 M l = 15 : M3=4 : M4=8 1430 C A L L MOUSE(Ml,M2,M3,M4) 1440 ' 1450 ' Define the "logical and" cursor mask 1460' 1470 CURSOR( 0,0)=&H3FFF 1480 CURSOR( 1,0)=&H1FFF 1490 CURSOR( 2,0)=&HFFF 1500 CURSOR( 3,0)=&H7FF 1510 CURSOR( 4,0)=&H3FF 1520 CURSOR( 5,0)=&H1FF 1530 CURSOR( 6,0)=&HFF 1540 CURSOR( 7,0)=&H7F 1550 CURSOR( 8,0)=&H3F 1560 CURSOR( 9,0)=&H1F 1570 CURSOR(10,0)=&H1FF 1580 CURSOR(11,0)=&H10FF 1590 CURSOR(12,0)=&H30FF 1600 CURSOR(13,0)=&HF87F 1610 CURSOR(14,0)=&HF87F 1620 CURSOR(15,0)=&HFC3F 1630' 1640 ' Define the "exclusive or" cursor mask  Appendices 1650' 1660 CURSOR( 0,1)=&H0 1670CURSOR( 1,1)=&H4000 1680 CURSOR( 2,1)=&H6000 1690 CURSOR( 3,1)=&H7000 1700 CURSOR( 4,1)=&H7800 1710 CURSOR( 5,1)=&H7C00 1720 CURSOR( 6,1)=&H7E00 1730 CURSOR( 7,1 )=&H7F00 1740 CURSOR( 8,1)=&H7F80 1750 CURSOR( 9,1)=&H78C0 1760 CURSOR(10,l)=&H7C00 1770 CURSOR(11,1)=&H4600 1780 CURSOR(12,1)=&H600 1790 CURSOR(13,1)=&H300 1800 CURSOR(14,1)=&H300 1810 CURSOR(15,1)=&H180 1820' 1830 ' Set the mouse cursor shape 1840 ' 1850 M l =9 :M2=-1 : M3 =-1 1860 C A L L MOUSE(M1,M2,M3,CURSOR(0,0)) 1870' 2210 ' 2220 ' Set mouse cursor location, then turn on cursor 2230 ' 2240 M l = 4 : M3 = 320 : M4 = 160 : C A L L MOUSE(Ml,M2,M3,M4) 2250 M l = 1 : C A L L M0USE(M1,M2,M3,M4) 2260 ' 2270 GOSUB 4000 ' INTT LNDOWS 2280 ON KEY(l) GOSUB 5000 : KEY(l) ON 2290 ON KEY(2) GOSUB 6000 : KEY(2) ON 2300 ON KEY(3) GOSUB 7000 : KEY(3) ON 2310 ON KEY(4) GOSUB 8000 : KEY(4) ON 2320 ON KEY(5) GOSUB 9000 : KEY(5) ON 2330 ON KEY(12) GOSUB 10000 : KEY(12) ON 2340 ONKEY(13) GOSUB 11000 : KEY(13) ON 2350 O N K E Y ( l l ) GOSUB 13000 : K E Y ( l l ) ON 2360 ON KEY(14) GOSUB 12000 : KEY(14) ON 2370 ON KEY(10) GOSUB 14000 : KEY(10) ON 2380 ON KEY(30) GOSUB 15000 : KEY(30) ON 2385 ON KEY(31) GOSUB 4000 : KEY(31) ON 2390 'loop and wait for button call as well as FI key 2400 Ml=3 : C A L L MOUSE(Ml,BT,MX,MY) 2410 IF (BT AND 1) THEN GOSUB 5020 2420 IF (BT AND 2) THEN GOSUB 6000 2430 GOTO 2390 2999 STOP 3000 INPUT "filename for input ",FILEIN$ 3010 OPEN FILEIN$ FOR INPUT AS #1 3020 INPUT "filename for row-wise output ",FILEROW$ 3030 OPEN FILEROW$ FOR OUTPUT AS #2 3040 INPUT "filename for column-wise output ",FILECOL$ 3050 OPEN FILECOL$ FOR OUTPUT AS #3  Appendices 111 3060' 3070 'read in data 3080 ' 3090 DIM HEADER$(5) 3100 DIM CLN!(6,1000) 3110' 3120 FOR 1=1 TO 5 3130 INPUT #1,HEADER$(I) 3140 NEXT I 3150 ' 3160 FOR 1=1 TO 1000 3170 IF EOF(l) THEN 3220 3180 CLN!(1,I)=I: IN.COUNT=I 3190 INPUT #l,CIN!(2 r),CLN!(3 I),CIN!(4,I),CIN!(5,D,CIN!(6,I) 3200 NEXT I 3210 STOP 3220 CLOSE #1 3230 RETURN 4000 ' init start 4010 WINDOW (0,0)-(225,100) 4020 GOSUB 4500 4060 RETURN 4500 'sbr to plot data that is valid 4510 CLS 4520 FOR 1=1 TO LN.COUNT 4530 IF CLN!(2,I)=-1 GOTO 4550 4540 LINE (CLN!(3,I),CLN!(4,I))-(CLN!(5,I),CIN!(6,I)) CLN!(2,I) 4550 NEXT 4560 RETURN 5000 'choose 5010 Ml=3 : C A L L MOUSE(Ml,BT\MX,MY) 5020 LOCATE 24,1 : PRINT "WAIT "; 5025 X.MULT!=639/(X.HIGH-X.LOW): Y.MULT!=349/(Y.HIGH-Y.LOW) 5030 FOR 1=1 TO LN.COUNT 5040 IF CLN!(2,I)=-1 GOTO 5080 5045 X.CAND=X.MULT!*(CLN!(3,I)-X.LOW): Y.CAND=Y.MULT!*(Y.HIGH-CIN!(4,I)) 5050 IF ABS(MX-X.CAND)<5 AND ABS(MY-Y.CAND)<5 THEN SIDE=5 : GOSUB 20000: GOTO 5090 5055 X.CAND=X.MULT!*(CLN!(5,I)-X.LOW) : Y.CAND=Y.MULT!*(Y.fflGH-CIN!(6,I)) 5060 IF ABS(MX-X.CAND)<5 AND ABS(MY-Y.CAND)<5 THEN SIDE=3 : GOSUB 20000: GOTO 5090 5080 NEXT 5090 LOCATE 24,1 : PRINT "READY";: RETURN 6000 'connect 6010 IF MATCH.COUNT<>2 THEN BEEP: BEEP : RETURN 6070 LINE (CLN!(3,C1),CIN!(4,C1))-(CIN!(5,C1),CIN!(6,C1)),0 6080 LINE (CLN!(3,C2),CIN!(4,C2))-(CIN!(5,C2),CIN!(6,C2)),0 6090 CLN!(3,C1)=CIN!(C1.SLDE,C1) 6100 CLN!(4,C1)=CIN!((C1.SIDE+1),C1) 6110 CLN!(5,C1)=CIN!(C2.SLDE,C2) 6120 CIN!(6,C1)=CIN!((C2.SIDE+1),C2) 6130CLN!(2,C2)=-1 : CIN!(2,Cl)=COLOURl 6140 LINE (CLN!(3,C1),CIN!(4,C1))-(CIN!(5,C1),CIN!(6,C1)),CLN!(2,C1) 6150 GOSUB 14000 )  )  )  Appendices 178 6160 RETURN 7000 RETURN 8000 'horizontal plot 8005 X.LOW=X.OFFSET : X.fflGH=X.OFFSET+60 : Y.LOW=45 : Y.HIGH=55 8010 WINDOW (X.LOW,Y.LOW)-(X.HJGH,Y.HIGH) 8015 MODE=l ' HORIZONTAL 8020 GOSUB 4500 8030 RETURN 9000 'vertical plot 9005 X.LOW=X.OFFSET+45 : X.HIGH=X.OFFSET+65 : Y.LOW=Y.OFFSET : Y.HIGH=Y.OFFSET+60 9010 WINDOW (X.LOW,Y.LOW)-(X.HIGH,Y.HIGH) 9015 MODE=2 ' VERT 9020 GOSUB 4500 9030 RETURN 10000 'left move 10010 X.OFFSET=X.OFFSET-55 10020 IF X.OFFSET<0 THEN BEEP : X.OFFSET=0 10030 IF MODE=l THEN GOSUB 8000 ELSE IF MODE=2 THEN GOSUB 9000 10040 RETURN 11000 'right move 11010 X.OFFSET=X.OFFSET + 55 11020 IF X.OFFSET>225 THEN BEEP : X.OFFSET =220 11030 IF MODE=l THEN GOSUB 8000 ELSE IF MODE=2 THEN GOSUB 9000 11040 RETURN 12000 'up move 12010 Y.OFFSET=Y.OFFSET - 50 12020 IF Y.OFFSET<0 THEN BEEP : Y.OFFSET=0 12030 IF MODE=l THEN GOSUB 8000 ELSE IF MODE=2 THEN GOSUB 9000 12040 RETURN 13000 'down move 13010 Y.OFFSET=Y.OFFSET + 50 13020 IF Y.OFFSET>100 THEN BEEP : Y.OFFSET=60 13030 IF MODE=l THEN GOSUB 8000 ELSE IF MODE=2 THEN GOSUB 9000 13040 RETURN 14000 'cancel 14005 IF MATCH.COUNT>2 THEN STOP 14010 MATCH.COUNT=0 14040 IF CIN!(2,C1)<>-1 THEN CIN!(2,Cl)=COLOURl ELSE STOP 14050 IF CIN!(2,C2)<>-1 THEN CIN!(2,C2)=COLOUR2 ELSE GOTO 14070 14060 LINE (dN!(3,C2),CIN!(4,C2))-(CIN!(5,C2),CIN!(6,C2)),CIN!(2,C2) 14070 LINE(dN!(3,Cl),CIN!(4,Cl))-(CIN!(5,Cl),CIN!(6,Cl)),CIN!(2,Cl) 14080 RETURN 15000 LOCATE 23,1 : PNPUT;"output files ? (Y/N)",AN$ 15010 IF LEFT$(AN$,l)="y" OR LEFT$(AN$,1)="Y" THEN 15020 ELSE LOCATE 23,1 : PRINT" no action ";: BEEP : RETURN 15020 FOR 1=1 TO 5 15030 PRINT #2, HEADER$(I) 15040 PRINT #3, HEADER$(I) 15050 NEXT I 15060 ' 15070 FOR 1=1 TO IN.COUNT 15080 IF CIN!(2,I)=-1 THEN 15130 15090 PRINT #2, CIN!(2,I),CIN!(3,I),CIN!(4,I),CIN!(5,I),CIN!(6,I)  Appendices 179 15100 PRINT #3, dN!(3,I),CIN!(4,I) 15110 PRINT #3, CIN!(5,I),CIN!(6,I) 15120 PRINT #3,"" 15130 NEXT 15140 CLOSE 15150 LOCATE 23,1 : PRINT " done "; 15160 RETURN 20000 BEEP: 'mark 20005 IF MATCH.COUNT>2 THEN STOP 20010 IF MATCH.COUNT=2 THEN CIN!(2,C2)=COLOUR2 : C2.SIDE=SJX>E : OLD.C2=C2 : C2=CIN!(1,I): IF CIN!(2,I)<>14 THEN COLOUR2=CIN!(2,I): CIN!(2,I)=14 20020 IF MATCH.COUNT=l THEN C2=CIN!(1,I) : C2.SIDE=SIDE : MATCH.COUNT=2 : IF CEsf!(l,I)<>14 THEN COLOUR2=CIN!(2,I) : CIN!(2,I)=14 20030 IF MATCH.COUNT=0 THEN C1=CIN!(I,I) : C1.SIDE=SIDE : MATCH.COUNT=l : IF CIN!(2,I)<>14 THEN COLOURl=CIN!(2,I): CIN!(2,I)=14 20050 IF CIN!(2,C1) o - l THEN LINE (CIN!(3,C1),CIN!(4,C1))-(CIN!(5,C1),CIN!(6,C1)),CIN!(2,C1) 20060 IF CIN!(2,C2) <>-1 THEN LINE (CIN! (3,C2),CIN! (4,C2))-(CIN! (5,C2),CIN! (6,C2)),CIN! (2,C2) 20065 IF CIN!(2,OLD.C2) o - l THEN LINE (CIN!(3,OLD.C2),CIN!(4,OLD.C2))-(CIN!(5,OLD.C2),CIN!(6,OLD.C2)),CIN!(2 OLD.C2) 20070 RETURN )  Appendices  180  APPENDIX G RESULTS OF CUMULATIVE CRACK LENGTH AND NUMBER T A B U L A T I O N OF C U M U L A T I V E C R A C K L E N G T H DATA E  <0.625 .062-. 125 .125-.25  [0/90]s  Percentage of total cracking width W* TOTAL .25-.50 .50-.75 .75-1 (mm)  EO  92  42  11  0  0  0  145  E2  92  42  11  0  0  0  145  E4  220  219.4  217  67  0  0  723.4  E5  295.8  417  641.4  288  43.5  0  1685.7  E6  198.5  438.3  752.7  661  335.3  189.5  2575.4  [0/75], EO  120  38.9  8  0  0  0  166.9  E3  120  38.9  8  0  0  0  166.9  141.2  94.6  74  36.2  28.42  21.2  395.6  152  114  E4  139  160  156.3  48  769  E6 . 157.3  192  223  307  282.3  272.9  1434.3  E7  54  327  555  535.1  950.6  2567.6  7.3  0  0  0  0  75.9  0  75.9  E5  145.5  [0/60]s E0 E3  68.6 68.6  7.3  0  0  0  0  0  0  151.2  E4  119.7  26.1  5.42  E6  260.3  146.5  83.7  34.3  16  0  540.8  108  29.6  1463.3  E7  272.7  308.6  472  272.4  E8  126.8  260.3  701.5  688.2  431.7  356  2564.5  [0/45], E0  100.3  4.9  0  0  0  0  105.2  E6  150.5  14.4  30.3  0  0  0  195.2  E8  227.2  68.43  49.1  36.6  0  0  381.3  E9  257  148  204.9  65.45  64  0  740  E10  176.8  281  530.7  482.3  191  101  1762.8  Appendices  e%  TABULATION OF C R A C K FREOEIJNCY DATA  Percentage of total cracking width, W <0.06 ..06-. 125 .125-.25 .25-.50 .50-.75 0.75-1.0 TOTAL  [0/90] 0 113 0.4 113 0.81 235 0.94 283 1.12 181  20 20 100 187 196  3 3 51 143 167  0 0 8 35 76  0 0 0 3 22  0 0 0 0 8  136 136 394 651 650  [0/75] 0 155 0.3 155 0.81 174 0.94 181 1.07 160 1.26 61  18 18 44 50 83 49  2 2 17 30 48 69  0 0 4 17 32 56  0 0 2 10 17 33  0 0 1 2 11 38  175 175 242 290 351 306  [0/60] 0 106 0.2 106 0.94 170 1.23 276 1.37 260 1.5 115  3 3 10 57 118 106  0 0 1 16 94 133  0 0 0 4 30 70  0 0 0 1 6 24  0 0 0 0 1 13  109 109 181 354 509 461  [0/45] 0 96 1.1 180 1.3 245 1.48 208 1.67 124  0 4 20 47 87  0 6 8 34 79  0 0 3 4 38  0 0 0 3 9  0 0 0 0 3  100 190 276 296 340  s  s  s  s  181  Appendices  APPENDIX H PROGRAM SPACE TO DETERMINE MINIMUM CRACK SPACING C C C  PROGRAM TO FIND THE MINIMUM CRACK SPACING FOR EACH CRACK  PROGRAM SPACE IMPLICIT R E A L M (A-H.O-Z) REAL V I DIMENSION XMLN(700), XDIST(700,700),CRACK(700,4) CHARACTER*(64) FNAME2, OUT1 INTEGER I,J,L,M,N,IT,IJ,NT 112 FORMAT(A) WRITE(*,113) 113 FORMATC X-DISTANCES FILENAME ?') READ (*, 112) FNAME2 WRiTE(*,116) 116 FORMAT (' NAME OF OUTPUT FILE ?') READ(*,112) OUT1 WRITE(*,117) 117 FORMAT ('ZONE OF INFLUENCE') READ(*,118) V I 118 FORMAT(F6.2) C OPEN (UNiT=3,FLLE=FNAME2) OPEN (UNIT=4,F1LE=0UT1) C WRrTE(4,*) ' ZONE OF INFLUENCE =', V I C TO INPUT DATA: C 1=1 20 READ(3,100,ERR=23,END=25) (CRACK(I,J),J=1,4) 100 FORMAT(4F8.2) 1=1+1 GOTO 20 23 PRINT 103 103 FORMATC I/O ERROR IN FLLE2') 25 PRINT 105 105 FORMATC END-OF-FILE2') CONTINUE C INITIALIZE A L L VALUES; DO 99 n=1,700 DO 9811=1,700 XDIST(IT,U) =400.00 98 CONTINUE 99 CONTINUE 1=1-1 L =I M= 1 30 IF (M.GT.L) GO TO 40 DO60N=l,L IF (N.EQ.M) GO TO 60 C C CALCULATE THE XDISTANCES C1Y1=CRACK(M,2)  Appendices C1Y2=CRACK(M,3) C2Y1=CRACK(N,2) C2Y2=CRACK(N,3) C ARE THEY WITHIN INTERACTING DISTANCE?... DIF1=ABS(C1Y2-C2Y1) DIF2=ABS(C1Y2-C2Y2) DIF3=ABS(C1Y1-C2Y2) DIF4=ABS(C1Y1-C2Y1) C IF(((C2Yl.GT.ClY2).AND.(C2Y2.GT.ClY2)).OR. + ((C2Y1.LT.C1Y1).AND.(C2Y2.LT.C1Y1))) THEN C IF(((DIFl.LT.Vl).OR.(DIF2.LT.Vl)).OR. + ((DIF3.LT.Vl).OR.(DIF4.LT.Vl)))THEN XDIST(M,N)=ABS(CRACK(M, 1 )-CRACK(N, 1)) ELSE XDIST(M,N)=400.00 ENDIF ELSE XDIST(M,N)=AB S (CRACK(M, 1 )-CRACK(N, 1)) C ENDIF 60 CONTINUE M=M+1 GOTO 30 C FIND OUT THE MINIMUM CRACK DISTANCE C 40 CONTINUE M=l L=I 50 XMIN(M)=300.00 IF (M.GT.L) GO TO 90 DO 55NT=1,L IF (XDIST(M,NT).EQ.0.0) GO TO 55 XMIN(M) = MIN(XMIN(M),XDIST(M,NT)) 55 CONTINUE M=M+1 GO TO 50 90 CONTINUE M=l DO80M=l,L WRITE(4,300) CRACK(M,1), XMIN(M) 80 CONTINUE 300 FORMAT(2F10.2) STOP END  Appendices APPENDIX I PROGRAM TIP TO DETERMINE CLOSEST CRACK SPACING C C C  Program TIP to generate crack tip distances  Program TIP IMPLICIT REAL*4 (A-H.O-Z) REAL CRACK(700,4),TIPMIN(700),TIPDIS(7(X)),YFL(700,700) REAL YL1 (700),YL2(700),YL3(700),YL4(700),XFL(700) CHARACTER*64 FNAME, OUT1 INTEGER I,J,K,L,IT,U,M,Z write(*,lll) 111 format('Input filename ?') read(*,112) fname 112 format(A) write(*,113) 113 formatC tip-distances filename ?') read(*,112) outl C Open (2,file=fname,Status='old') Open (3,fde=outl,status='new') C C To input data: 1=1 C 10 Read (2,100,err=22,end=24) (crack(I,J),J=l,4) 1=1+1 100 Format (4F8.2) goto 10 C 22 print 101 101 formatC i/o error') C 24 print 102 102 formatC end-of-file') C continue C 1 = 1-1 C write (3,*) 'total no. of cracks = ', I C C INITIALIZE ARRAYS.... DO 99IT=1,I DO 98 LT= 1,1 YFL(IT,IJ)=300.00 98 CONTINUE 99 CONTINUE Z =I M= 1 30 IF (M.GT.Z) GO TO 40 C DO60K=l,Z  Appendices IF (M.EQ.K) GO TO 60 C C calculate x-distances... XFL(K)=ABS(CRACK(M, 1 )-CRACK(K, 1)) IF (XFL(K).EQ.O.O) GOTO 60 C C ...calculate y-distances.. YL1(K)=ABS(CRACK(M,2)-CRACK(K,2)) YL2(K)=ABS(CRACK(M,2)-CRACK(K,3)) C YL3(K)=ABS(CRACK(M,3)-CRACK(K,2)) YL4(K)=ABS(CRACK(M,3)-CRACK(K,3)) C ....find rrunimum crack distance... YFL(M,K)=MIN(YL1(K),YL2(K),YL3(K),YL4(K)) C C ...find the tip distances... TLPDIS(K)= SQRT((XFL(K)**2)+(YFL(M,K)**2)) 60 CONTINUE TLPMIN(M)=300.00 DO 61 L=1,Z IF (L.EQ.M) THEN TLPDIS(L)=300.0 TLPMIN(M)=MLN(TIPMIN(M),TIPDIS(L)) ELSE TLPMIN(M) = MLN(TLPMIN(M),TIPDIS(L)) ENDIF 61 CONTINUE M=M+1 GO TO 30 40 CONTINUE D 0 65M=1,Z WRITE(3,300) M,CRACK(M,1),TIPMIN(M) 65 CONTINUE 300 FORMAT (I4.2F8.2) STOP END  Appendices APPENDIX J C C C  PROGRAM GROWTH TO DETERMINE INCREMENTAL CRACK GROWTH PROGRAM TO CALCULATE THE INCREMENTAL CRACK GROWTH  PROGRAM GROWTH IMPLICIT REAL*4 (A-H.O-Z) DIMENSION STRES1(700,4),STRES4(700,4),STRES3(700,4),STRES2(700,4) REAL SLAST(700,4),A1(700,50),B1(700,50),C1(700,50),SUMLS(700) REALD1(7(X),50),SUM1(700) SUM2(7(X)),SUM3(700),SUM10(700) TGR4(700) REALTGR1(700),TGR2(700),TGR3(700),SUMLS(700),SUM9(700),SUM4(700) REAL SUMI 1(700) CHARACTER*(64) FNAME1, FNAME2, FNAME3, FNAME4, FNAME5,OUTl CHARACTER*(64) OUT3,OUT2,OUT3,OUT4 INTEGER I,J,K,L,M,N,MT,KT,IT,JG,KG,LG,LL,JJ INTEGER MS,NS,IN,JN,MZ,MX,KS,KH,KX,NZ,NG WRITE(*,111) 111 FORMATC FILENAME OF LAST LOAD LEVEL?') READ (*,112) FNAME1 112 FORMAT(A) WRITE(*,113) 113 FORMATC FILENAME OF 1ST LOAD LEVEL?') READ(*,112) FNAME2 WRITE(*,114) 114 FORMATC FILENAME OF 2ND LOAD LEVEL?') READ(*,112) FNAME3 WRITE(*,115) 115 FORMATC FILENAME OF 3RD LOAD LEVEL?') READ(*,112) FNAME4 WRITE(*,119) 119 FORMATC FILENAME OF 4TH LOAD LEVEL?') READ(*,112) FNAME5 WRITE(*,116) 116 FORMATC OUTPUT FILE 1?') READ(*,112)OUTl WRITE(* 117) 117 FORMATC OUTPUT FILE 2?') READ (*,112) OUT2 WRITE (*,118) 118 FORMATC OUTPUT FILE 3?') READ(*,112)OUT3 WRITE(*,120) 120 FORMATC OUTPUT FILE 4?') READ(*,112)OUT4 C OPEN(UNIT=2,FILE=FNAMEl STATUS='OLD') OPEN (UNIT=3,FILE=FNAME2,STATUS='OLD') OPEN (UNIT=4,FILE=FNAME3,STATUS='OLD') OPEN (UNIT=5,FILE=FNAME4,STATUS='OLD') OPEN (UNIT=12,FILE=FNAME5,STATUS='OLD') OPEN (UNIT=7,FILE=OUTl,STATUS='NEW') OPEN (UNIT=8,FILE=OUT2,STATUS='NEW') OPEN (UNIT=10,FILE=OUT4,STATUS='NEW ) OPEN(UNIT=9,FILE=OUT3,STATUS= NEW') )  )  )  ,  ,  Appendices C C INPUT DATA: 1=1 10 READ (2,202,ERR=22,END=24) (SLAST(I,J),J=1,4) 1=1+1 GOTO 10 22 WRITE (*,23) 23 FORMAT(' I/O ERROR IN FNAME1') 24 WRITE (*,25) 25 FORMATC END-OF-FILE IN FNAME 1') C K=l 11 READ (3,202,ERR=26,END=27) (STRES1(K,L),L=1,4) 202 FORMAT (4F8.2) K = K+1 GOTO 11 26 WRiTE(*,28) 28 FORMATC I/O ERROR IN FNAME2') 27 WRITE (*,29) 29 FORMAT (' END-OF-FILE LN FNAME2') C C M=l 12 READ(4,202,ERR=30,END=31) (STRES2(M,N),N=1,4) M=M+1 GOTO 12 30 WRITE (*,32) 32 FORMATC I/O ERROR IN FNAME3') 31 WRiTE(*,33) 33 FORMAT (' END-OF-FILE LN FNAME3') C MS=1 13 READ(5,202,ERR=34,END=35) (STRES3(MS,NS),NS=1,4) MS=MS+1 GOTO 13 34 WRITE (*,36) 36 FORMATC I/O ERROR IN FNAME4') 35 WRITE (*,37) 37 FORMATC END-OF-FILE IN FNAME4') C KS=1 14 READ(12,202,ERR=38,END=39) (STRES4(KS,KH),KH=1,4) KS=KS+1 GOTO 14 38 WRLTE(*,85) 85 FORMATC I/O ERROR IN FNAME5') 39 WRITE(*,86) 86 FORMATC END-OF-FILE IN FNAME5') C 1=1-1 M=M-1 K=K-1 MS=MS-1 KS=KS-1  Appendices WRrrE(7,220)' 1=', I 220 FORMAT(A,I3) WRITE (7,220)' K=',K WRITE (7,220)'M=',M WRITE (7,220)' MS=',MS WRITE (7,220)' KS=',KS DIST1A=0.0 DIST1B=0.0 DIST1C=0.0 DIST1D=0.0 C ... INITIALIZE A L L ARRAYS... DO 400 IN=1,50 DO 401 JN=1,50 A1(IN,JN) = 0.0 BiaN,JN) = 0.0 C1(IN,JN) = 0.0 D1(IN,JN) = 0.0 SUM1(JN)=0.0 SUM2(JN)=0.0 SUM3(JN)=0.0 SUM4(JN)=0.0 TGR1(JN)=0.0 TGR2(JN)=0.0 TGR3(JN)=0.0 TGR4(JN)=0.0 SUM9(JN)=0.0 SUM10(JN)=0.0 •SUMLS(JN)=0.0 SUM11(JN)=0.0 401 CONTINUE 400 CONTINUE IT=1 40 CONTINUE IF (IT.GT.I) GO TO 90 LG=1 JG=1 KG=1 NG=1 DO 50KT=1,K C C ..GROWTH FOR THE SECOND HIGHEST LOAD... DISTlA=ABS(SLASTaT,l)-STRESl(KT,l)) IF (DIST1A.GT.0.3) GO TO 50 C IF ((STRES1(KT,2).GE.SLAST(IT,2)).AND. + (STRES1(KT,3).LE.SLAST(IT,3))) THEN A1(IT,JG) = SLAST(IT,4)-STRES1(KT,4) SUMLS(iT)=SLAST(iT,4) SUM 1 (IT)=STRES 1 (KT,4)+SUM 1 (IT) JG=JG+1 C GROWTH FOR THE THIRD HIGHEST LOAD... DO60MT=l,M DIST1B= ABS(SLAST(TT,1)-STRES2(MT,1)) IF (DIST1B.GT.0.3) GO TO 60  Appendices  C  IF ((STRES2(MT,2).GE.STRES1(KT,2)).AND. + (STRES2(MT,3).LE.STRES1(KT,3))) THEN B1 (TXLG) = STRES1 (KT,4)-STRES2(MT,4) SUM9(1T)=STRES 1 (KT,4)+SUM9(IT) SUM2(IT)=STRES2(MT,4)+SUM2(IT) L G = LG+1 C GROWTH FOR THE FOURTH HIGHEST LOAD... D 0 65MX=1,MS DIST1C= ABS(SLAST(IT,1)-STRES3(MX,1)) IF (DIST1C.GT.0.3) GO TO 65 C IF((STRES3(MX,2).GE.STRES2(MT,2)).AND. + (STRES3(MX,3).LE.STRES2(MT,3))) THEN C1(IT,KG)= STRES2(MT,4)-STRES3(MX,4) SUM10(IT)=SUM10(IT)+STRES2(MT,4) SUM3(IT)=SUM3(IT)+STRES3(MX,4) K G = KG+1 C GROWTH FOR THE LOWEST LOAD DO 66 KX=1,KS DIST1D= ABS(SLAST(IT,1)-STRES4(KX,1)) IF (DIST1D.GT.0.3) GO TO 66 C IF((STRES4(KX,2).GE.STRES3(MX,2)).AND. + (STRES4(KX,3).LE.STRES3(MX,3))) THEN D1(IT,NG)= STRES3(MX,4)-STRES4(KX,4) SUMI 1(IT)=SUM11(IT)+STRES3(MX,4) SUM4(IT)=SUM4(IT)+STRES4(KX,4) NG = NG+1 C ENDIF 66 CONTINUE TGR4(IT)=SUM11 (IT)-SUM4(IT) ENDIF 65 CONTINUE TGR3(IT)=SUM10(rT)-SUM3(IT) ENDIF 60 CONTINUE TGR2(IT)=SUM9(IT)-SUM2(IT) ENDIF 50 CONTINUE TGR1 (IT)=SUMLS (IT)-SUMl (IT) JG=JG-1 LG=LG-1 KG=KG-1 NG=NG-1 WRITE (7,300) IT,SLAST(IT,4),SUM1(IT),TGR1(IT),(A1(IT,JJ),JJ=1,JG) WRITE (8,300) IT,SUM9(rT),SUM2(IT),TGR2(IT),(Bl(IT,LL),LL=l,LG) WRITE (9,300) IT,SUM10(iT),SUM3(IT),TGR3(IT),(Cl(IT,MZ),MZ=l,KG) WRITE (10,300) IT,SUM11(IT),SUM4(IT),TGR4(IT),(D1(IT,NZ),NZ=1,NG) 300 FORMAT (13,10F8.2) IT=IT+1 GOTO 40  Appendices  C 90  CONTINUE STOP END  Appendices APPENDIX K PROGRAM ELIM TO DETERMINE EFFECTIVE CRACK LENGTH C PROGRAM ELIM TO ELIMINATE SMALL CLOSE CRACKS THAT ARE PRESUMED TO HAVE C NO AFFECT ON STIFFNESS REDUCTION C C PROGRAM ELIM IMPLICIT REAL*4 (A-H,0-Z) DIMENSION CRACK(700,4) CHARACTER* (64) FNAME1, OUT1 INTEGER I,J,K,L,M,MT,Z WRITE (*,111) 111 FORMAT ('INPUT FILENAME?') READ (*,112) FNAME1 112 FORMAT (A) WRITE (*,113) 113 FORMAT (' OUTPUT FILENAME ?') READ(*,112)OUTl C WRTTE(*,114) 114 FORMAT (' PLY THICKNESS ?') READ (*,115) PLY 115 FORMAT (F4.2) C OPEN (UNIT=2,FILE=FNAME1 ,STATUS='OLD') OPEN(UNIT=5,FILE=OUTl STATUS='NEW') C C TO INPUT DATA: 1=1 10 READ (2,100,ERR=22,END=24) (CRACK(I,J),J=1,4) 100 FORMAT(4F8.2) 1=1+1 GO TO 10 22 WRITE (*,23) 23 FORMATC I/O ERROR IN FNAME 1') 24 WRITE (* 25) 25 FORMAT (' END-OF-FILE IN FNAME 1') C 1=1-1 Z=I WRITE(5,120) T= ',1 120 FORMAT (A,I3) WRITE (5,121) ' PLY THICKNESS =', PLY 121 FORMAT (A.F10.2) C INITIALIZE A L L ARRAYS SMALL=0.0 SUMCRK=0.0 SUMSML=0.0 EFFEC=0.0 XDIST=0.0 M=0 L=l )  Appendices DO 50MT=1,Z SUMCRK=SUMCRK+CRACK(MT,4) 50 CONTINUE C 30 M=M+1 IF (M.GT.Z) GO TO 90 C DO40K=l,Z IF (K.EQ.Z) GOTO 40 XDIST= ABS(CRACK(M,1)-CRACK(M+K,1)) IF (XDIST.GT.PLY) GOTO 40 C C1Y1=CRACK(M,2) C1Y2=CRACK(M,3) C2Y1=CRACK(K,2) C2Y2=CRACK(K,3) C IF(((C2Y1.GT.C1Y2).AND.(C2Y2.GT.C1Y2)).0R. + ((C2Y1.LT.C1Y1).AND.(C2Y2.LT.C1Y1))) GOTO 40 C IF(CRACK(M, 1).NE.CRACK(K, 1))THEN IF (CRACK(M,4).LT.CRACK(K,4))THEN IF((CRACK(M,2).GT.CRACK(K,2)).AND. + (CRACK(M,3).LT.CRACK(K,3)))THEN SMALL = CRACK(M,4) SUMSML = SUMSML +SMALL ELSEIF((CRACK(M,3).GT.CRACK(K,2)).AND. + (CRACK(M,2).LT.CRACK(M+K,2))) THEN SMALL=CRACK(M,3)-CRACK(K,2) SUMSML = SUMSML + SMALL ELSE SMALL=CRACK(K,3)-CRACK(M,2) SUMSML= SUMSML + SMALL ENDIF ENDIF ENDIF 40 CONTINUE GO TO 30 90 CONTINUE C FIND THE EFFECTIVE CRACK LENGTH... EFFEC = SUMCRK - SUMSML WRITE(5,121) ' SUM OF A L L CRACKS = ', SUMCRK WRITE(5,121) ' SUM OF SMALL ONES = ' , SUMSML WRITE(5,121) ' EFFECTIVE LENGTH =', EFFEC CONTINUE STOP END  193  Appendices  APPENDIX L MAGNITUDES OF C R A C K L E N G T H ELIMINATION  [0/90]  a  s  t  •  % eliminated  a  io  E4 E5 E6 E7,E8,E9,E10,E11  099 0.99 0.97 0.93 0.93  1 1 3 7 7  [0/75] EO E3 E4 E5 E6 E7 E8,E9,E10,E11  0.986 0.958 0.933 0.925 0.858 0.858 0.858  1.4 4.2 6.7 7.5 14.2 14.2 14.2  [0/60] EO E4 E6 E7 E8 E10,E12  0.993 0.986 0.99 0.95 0.935 0.935  0.7 1.4 1 5 6.5 6.5  [0/45] EO E6 E8 E9 E10 E11,E12  0.997 0.997 0.957 0.938 0.923 0.923  0.3 0.3 4.3 6.2 7.7 7.7  s  s  s  


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