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Experimental investigation of a strain softening approach to predicting failure of notched composite… Kongshavn, Ingrid A. 1996

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Experimental Investigation of a Strain Softening Approach to Predicting Failure of Notched Composite Laminates by Ingrid A . Kongshavn B . A . S c , Mechanical Engineering, University of Waterloo, 1993 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E S T U D I E S Department of Metals and Materials Engineering We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A May 1996 © I. A. Kongshavn, 1996 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 rAETALS *t*b r4AT££ . l f l LS B ^ V ^ E G ^ I ^ The University of British Columbia | Vancouver, Canada Date DE-6 (2/88) Abstract This work describes an experimental investigation of a strain softening approach to the prediction of fracture in notched composite laminates. This approach has been found to more accurately predict fracture of large notched coupons or structures than traditional approaches. Strain softening accounts for the effect of damage in front of the notch tip, by the post-peak region of a strain softening curve. This region is traditionally defined by a stress that decreases with increasing strain. A physical understanding of the post-peak region of the curve is required to better examine the predictive capabilities of strain softening. For this to be achieved, a detailed physical understanding of the damage in the process zone is required. In this study, two carbon fibre reinforced laminate material systems were examined. A n overheight compact tension (OCT) specimen was developed to grow damage in a stable manner. The displacements in front of the notch tip indicated the progression of damage across the specimen width throughout the test. A detailed physical description of the damage, which consisted of a crack and process zone were obtained. The sequence of damage growth in the process zone was determined. Tensile specimens cut from the process zones indicated a preliminary shape of the strain softening curves for the two material systems. Finally, the experimental results were used to calibrate a preliminary F E M strain softening analysis of the O C T specimens. The preliminary F E M results suggest that a strain softening material response is necessary to capture the progressive damage growth across the specimen width. i i Sommaire Ce travail presente une etude experimental de la theorie de l'adoucissement afin de predire la rupture de composites stratifies contenant une entaille. E n comparaison avec les methodes conventionelles, cette theorie permet une meilleure prediction de la rupture d'echantillons ou de structures de grande taille contenant une entaille.. L a partie de la courbe d'adoucissement apres le maximum permet de tenir compte des effets de l'endommagement devant le bout de l'entaille. Cette partie de la courbe est caracterisee par une diminution de la contrainte lorsque la deformation augmente. A f i n de mieux evaluer les possibilites de prediction de la theorie de l'adoucissement, i l est necessaire de comprendre les phenomenes physiques associes a la partie de la courbe apres son maximum. Ains i , i l est necessaire de caracteriser la nature des dommages dans la zone d'endommagement confinee. Dans ce travail, deux types de stratifes renforces de fibres de carbone ont ete etudies. U n echantillon C T surdimensione en hauteur a ete developpe et rendommagement s'est propage de facon stable. Les deplacements mesures devant le bout de l'entaille indiquent revolution de rendommagement sur la largeur de l'echantillon. L'endommagement obtenu est caracterise par une fissure et une zone d'endommagement confinee. L'evolution sequentielle de l'endommagement dans la zone d'endommagement confinee a ete determinee. Des echantillons decoupes dans la zone d'endommagement confinee ont ete testes en tension et ont permis d'obtenir la forme preliminaire de la courbe d'adoucissement pour les deux materiaux etudies. Enfin, les resultats experimentaux ont ete utilises afin de calibrer des analyses par elements finis des echantillons C T modifies. Les i i i resultats preliminaries obtenus par les elements finis semble indiquer qu ' i l est necessaire de considerer l'adoucissement du materiau afin de simuler la progression de l'endomagement dans la largeur de l'echantillon. iv Table of Contents Abstract ii Sommaire iii Table of Contents v List of Tables viii List of Figures ix Acknowledgements xiv 1. CHAPTER ONE - INTRODUCTION 1 1.1 Motivation 1 1.1.1 Examples of Damage in a Composite Laminate 2 1.2 Linear Elastic Fracture Mechanics 2 1.2.1 The Process Zone in Quasi-Brittle Materials 4 1.3 Micromechanical Models Used to Predict Notched Behaviour 5 1.3.1 Predicting Fracture Based on a Fitted Parameter 5 1.3.2 Relating Fracture to a Physical Understanding of the Damage Evolution 6 1.4 Macromechanical Models Used to Predict Notched Behaviour 7 1.4.1 Fictitious Crack Model 8 1.5 A Strain Softening Approach to Predicting Notched Behaviour 8 1.5.1 Experimental Investigation of Strain Softening as a Material Property 10 1.5.2 Geometry of a Strain Softening Specimen 10 1.5.3 Experimental Investigation of Process Zone Softening 11 1.6 Statement of Objectives 13 2. CHAPTER TWO - OCT TEST DEVELOPED TO STUDY NOTCHED BEHAVIOUR 19 2.1 Introduction 19 2.2 The Overheight Compact Tension (OCT) Specimen 19 2.2.1 Development of a Specimen Which Exhibits Stable Damage Growth 19 2.2.2 The Standard C T Stress Intensity Factor 20 v 2.3 Description of the Experimental Setup 21 2.3.1 Loading Train and Specimen Preparation 21 2.3.2 Measured Specimen Displacements 21 2.3.3 Loading Train Deflection 22 2.4 Summary of OCT Tests Performed 23 2.4.1 Materials Studied 23 2.4.2 Tests Performed 23 2.4.3 Post-test Analyses 24 3. CHAPTER THREE - OCT TESTS AND DATA REDUCTION 27 3.1 Introduction 27 3.2 Cross Head and CMOD Displacements 28 3.2.1 General Shape of the Load vs Cross Head and Load vs C M O D Displacement Curves 28 3.2.2 Repeatability 29 3.3 Indications of Damage Growth 30 3.3.1 Surface Line Displacements 30 3.3.2 Cl ip Gauge Displacements 35 3.4 Discussion of Results 37 3.5 Summary of Results 38 4. CHAPTER FOUR - PHYSICAL DESCRIPTION OF THE DAMAGE 47 4.1 Introduction 47 4.2 Techniques 47 4.2.1 Deply 48 4.2.2 Sectioning and Micrograph Analysis 48 4.2.3 Pulse-Echo Ultrasonic Scan 49 4.3 Deply Results 49 4.4 Description of Damage in the Laminae 50 4.5 Sequence of Damage Progression Throughout the OCT Test 51 4.5.1 Development of Crack and Process Zone Height and Length 52 4.5.2 Temporal Sequence of Damage Growth in the Process Zone 53 4.6 Discussion 54 4.6.1 Notch-Sensitivity 54 4.6.2 Strain Softening Curve 56 4.7 Summary of Results 57 vi 5. CHAPTER FIVE - PROCESS ZONE TENSILE TESTS 70 5.1 Introduction 70 5.2 Background 70 5.3 Experimental Technique 71 5.4 Derivation of the Damaged Moduli 73 5.5 Experimental Results 73 5.5.1 Damage Observed Prior to Tensile Tests 73 5.5.2 Measured Modul i 74 5.5.3 Damaged Modul i 75 5.6 Discussion 78 5.7 Summary of Results 80 6. CHAPTER SIX - COMPARISON OF THE EXPERIMENTAL RESULTS WITH A PRELIMINARY FEM ANALYSIS 91 6.1 Introduction 91 6.2 Comparison of the Experimental and FEM Results 91 7. CHAPTER SEVEN - CONCLUSIONS AND FURTHER WORK 96 7.1 Introduction 96 7.2 Conclusions 96 7.3 Further Work 97 References 100 Appendix 1 103 Appendix 2 112 vii List of Tables Table 2.1 Overheight compact tension (OCT) specimen dimensions where the letters refer to the schematic in Figure 2.1. A l l dimensions are in mm. 25 Table 2.2 Summary of instrumentation and post-test analysis of each O C T specimen. 25 Table 3.1 Position in front of the notch tip at which the scribe line has zero displacement at a given load. The results are for the scribe line closest to the notch mid-plane for specimens A 3 , B l and B 2 , and refer to the specified figures. 39 Table 5.1 Tensile specimen dimensions. 82 Table 5.2 Measured and derived tensile specimen mechanical properties. 82 viii List of Figures Figure 1.1 Primary load-bearing structures in the Boeing 777 made of fibre reinforced composite laminates. (Courtesy of The Boeing Company) 14 Figure 1.2 Schematic of a fibre reinforced laminate. 14 Figure 1.3 Schematic showing some of the damage mechanisms that occur in fibre reinforced composite laminates. Taken from Anderson (1991). 15 Figure 1.4 Centrally cracked plate subject to a uniform far-field stress. 15 Figure 1.5 Stress distribution around two different sized holes in an infinite isotropic plate. This explains the hole size effect according to Whitney and Nuismer. A greater probability exists around the larger hole for a flaw to initiate failure, as the stress concentration is less localized. Reproduced from Whitney and Nuismer (1974). 16 Figure 1.6 Schematic of damage observed by Kortschot and Beaumont in cross-ply graphite reinforced epoxy laminates. The damage was observed in the 'terminal damage state' (TDS) just prior to failure of double edge-notched specimens. Reproduced from Kortschot and Beaumont (1990a). 16 Figure 1.7 Representation of the stress distribution in the process zone in Hillerborg's Fictitious Crack Model . The process zone is modeled as a fictitious crack, containing a linear cohesive stress distribution as shown. The preformed crack wi l l grow when the maximum stress at the tip of the fictitious crack reaches the unnotched tensile failure strength. Reproduced from Afaghi-Khatibi et al. (1995). 17 Figure 1.8 The uniaxial strain softening curve predicted by Matzenmiller et al. (1991), based on a Weibull distribution of fibre failure. Different degrees of strain softening are predicted by the parameter, m. 17 Figure 1.9 Different regions of a typical strain softening curve. 18 Figure 2.1 Schematic of the overheight compact tension (OCT) specimen and experimental setup used to monitor displacements during loading. 26 Figure 3.1 Load vs cross head displacement of specimens A l , A 2 , and A 3 . The cross head displacement is not a true measure of the displacement at the specimen loading pins due to loading train deflection as discussed in Chapter 2. 40 Figure 3.2 Load vs C M O D and clip gauge displacement of specimen A 4 . 40 Figure 3.3 Load vs C M O D displacement of specimens B I and B2 . 41 IX Figure 3.4 Load vs cross head and clip gauge displacement of specimen A 3 . The numbers on the curve indicate where the photos were taken throughout the test. The cross head displacement is not a true measure of the displacement at the specimen loading pins due to loading train deflection as discussed in Chapter 2. 42 Figure 3.5 Displacement of line #4 as a function of position in front of the notch tip for selected photos of specimen A 3 . 42 Figure 3.6 Load vs C M O D and clip gauge displacement of specimen B l . The numbers on the curve indicate where the photos were taken throughout the test. 43 Figure 3.7 Displacement of line #1 as a function of position in front of the notch tip for selected photos of specimen B 1 . 43 Figure 3.8 Load vs C M O D displacement of specimen B 2 . The numbers on the curve indicate where the photos were taken throughout the test. 44 Figure 3.9 Displacement of line #1 as a function of position in front of the notch tip for selected photos of specimen B 2 . 44 Figure 3.10 Displacement of line #3 as a function of position in front of the notch tip for selected photos of specimen B 1 . 45 Figure 3.11 R-Curve of specimens A 3 , B l and B2 , based on the line analysis estimation of the crack length. G I C is calculated for the peak loads corresponding to the crack lengths of photos: 1) specimen A 3 : 10, 12, 13, 14 in Figure 3.4, 2) specimen B l : a8, a l5 , a l9 , b3 in Figure 3.6, and 3) specimen B 2 : 9, 11, 12, 16 in Figure 3.8. 46 Figure 4.1 Deply schematic of specimen A l . The fibre direction in each layer is drawn in the top left or right corner of each ply. Positive angles are taken counter-clockwise to the 0° fibre direction to be consistent with the standard orientation used to specify lay-up fibre direction. 59 Figure 4.2 Example of a micrograph schematic, showing matrix cracking, fibre failure in the 0° plies and surface delamination. This shows the orientation to which all the micrographs have been redrawn for comparison purposes. The scales in the x and y direction are not the same. This example is taken at 18 mm in front of the notch tip in specimen A 3 . 60 Figure 4.3 Typical cross-section of specimen A 3 , taken at 20.25 mm in front of the notch tip. This shows delamination of the surface ply. The layup is[-45/90/45/0/-45/90/45/0] s 61 Figure 4.4 Typical cross-section of specimen B l , taken at 10.00 mm in front of the notch tip. The layup is [F0/-45/90/45/0/45/90/-45/F0]T , F0 = [0/90] weave. 61 Figure 4.5 Pulse-echo ultrasonic (PEU) scan of specimen A 3 . Darker regions correspond to a cleaner reflected signal, and most likely represent delamination grown between loads A to E (positions A ' - E ' on the P E U scan). 62 x Figure 4.6 Reconstructed profile of the damage in specimen A 3 , with the load vs cross head displacement curve from the O C T test presented in Chapter 3. 63 Figure 4.7 Cross-section at position A ' in Figure 4.6, taken at 1.85 mm in front of the notch tip in specimen A 3 . The layup is [-45/90/45/0/-45/90/45/0] s . 63 Figure 4.8 Cross-section at position B ' in Figure 4.6, taken at 4.45 mm in front of the notch tip in specimen A 3 . The layup is [-45/90/45/0/-45/90/45/0] s . 64 Figure 4.9 Cross-section at position C in Figure 4.6, taken at 11.45 mm in front of the notch tip in specimen A 3 . The layup is [-45/90/45/0/-45/90/45/0] s . 64 Figure 4.10 Typical cross-section between positions C and D ' in Figure 4.6. This micrograph is taken at 18.00 mm in front of the notch tip in specimen A 3 . The layup is [-45/90/45/0/-45/90/45/0] s. 65 Figure 4.11 Reconstructed profile of the damage in specimen B I , with the load vs C M O D displacement curve from the O C T test presented in Chapter 3. 65 Figure 4.12 Cross-section at position IT in Figure 4.11, taken at 1.00 mm in front of the notch tip in specimen B I . The layup is [F0/-45/90/45/0/45/90/-45/F0]T , F0 = [0/90] weave. 66 Figure 4.13 Cross-section at position V in Figure 4.11, taken at 2.00 mm in front of the notch tip in specimen B I . The layup is [F0/-45/90/45/0/45/90/-45/F0] x , F0 = [0/90] weave. 66 Figure 4.14 Cross-section at position W in Figure 4.11, taken at 5.65 mm in front of the notch tip in specimen B I . The layup is [F0/-45/90/45/0/45/90/-45/F0]T , F0 = [0/90] weave. 67 Figure 4.15 Typical cross-section between positions X ' and Y ' in Figure 4.11. This micrograph is taken at 12.00 mm in front of the notch tip in specimen B I . The layup is [F0/-45/90/45/0/45/90/-45/F0]T , F0 = [0/90] weave. 67 Figure 4.16 Cross-section in the process zone of specimen A 3 , taken at 28.00 mm in front of the notch tip. This micrograph shows damage in all but the centre 0° plies. The layup is [-45/90/45/0/-45/90/45/0] s . 68 Figure 4.17 Cross-section in the process zone of specimen B I , taken at 35.50 mm in front of the notch tip . This micrograph shows damage in all but the centre 0° ply and surface 0°/90° weave plies. The layup is [F0/-45/90/45/0/45/90/-45/F0]T , F0 = [0/90] weave. 68 Figure 4.18 Diagram of the failure stress vs the notch length for increasing degrees of notch-sensitivity. 69 Figure 5.1 Diagram of a strain softening curve back-calculated from process zone tensile specimens. 83 xi Figure 5.2 Schematic of tensile specimens cut from the process zones of O C T specimens A 4 and B 2 . 83 Figure 5.3 Jig to align end tabs parallel to tensile specimens while gluing, and the experimental setup to measure the specimen displacement during loading. 84 Figure 5.4 Simplification of the tensile specimen used to determine the damaged modulus, E D , using Eq . (5.3). 84 Figure 5.5 Schematic and micrographs of tensile specimen A 4 T 1 . These show the face closest to the notch tip. Layup is [-45/90/45/0/-45/90/45/0] s. 85 Figure 5.6 Schematic and micrographs of tensile specimen A4T1 before tensile test. This shows the face farthest from the notch tip, and is similar in damage to the face closest to the notch tip of A4T2 . Layup is [-45/90/45/0/-45/90/45/0] s. 85 Figure 5.7 Schematic and micrographs of tensile specimen A 4 T 3 . This shows the face farthest from the notch tip. Layup is [-45/90/45/0/-45/90/45/0] s . 86 Figure 5.8 Schematic and micrograph of tensile specimen B2T1 before tensile test. This shows the face closest to the notch tip. Layiip is [F0/-45/90/45/0/45/90/-45/F0]T where FO = [0/90] weave. 87 Figure 5.9 Schematic and micrograph of tensile specimen B2T2. These show the face closest to the notch tip. Layup is [F0/-45/90/45/0/45/90/-45/F0]T where FO = [0/90] weave. 88 Figure 5.10 Measured stress vs strain curves of A 4 tensile specimens. 88 Figure 5.11 Measured stress vs strain curves of B2 tensile specimens. 89 Figure 5.12 Plots of failure stress and strain of A 4 tensile specimens, derived using the damaged moduli. The most likely combination of effective load-bearing plies is indicated for each specimen. 89 Figure 5.13 Plots of failure stress and strain of B2 tensile specimens, derived using the damaged moduli. The most likely combination of effective load-bearing plies is indicated for each specimen. 90 Figure 5.14 Preliminary strain softening curve of systems A and B (shown by the solid lines), based on the damaged moduli of specimens A 4 and B2 . The arrows outline the possible range of the unloading portion of the curve. 90 Figure 6.1 Load vs C M O D displacement curve for O C T specimen A 4 predicted by Engels (1996) using several different F E M constitutive model responses. These curves can be compared to the experimental load vs C M O D displacement curve. The optimized strain softening curve is included in the figure. The F E M results are reproduced from Engels (1996). 94 x i i Figure 6.2 Load vs clip gauge displacement of O C T specimen A 3 , predicted by a F E M strain softening analysis and the experimental results. Points A ' to C and A to C are used to compare the predicted and experimentally estimated crack lengths at the given load, respectively. The optimized strain softening material response is included in the figure. The F E M results are reproduced from Engels (1996). 95 x i i i Acknowledgements First and foremost, I would like to acknowledge Dr. Anoush Poursartip for the boundless energy he puts into his teaching and direction. His inspiration as an engineer and researcher, and kindness as a supervisor are to be greatly admired. I would also like to thank Dr. Reza Vazir i , Mr . K . Will iams and M r . H . Engels whose knowledge and perpetual optimism of the modeler have helped wrestle meaning from experimental chaos. Thanks to M r . Roger Bennett and M r . Serge Milaire whose technical assistance I could not have done without. Much inspiration and direction was provided by Dr. Larry Ilcewicz and Dr. Bernhard Dopker of the Boeing Company. I also wish to thank all the members of the Composites Group for their daily support and friendship, and for giving the lab a wonderful environment. I would like to acknowledge the Boeing Company, who provided the material for the project. A n d lastly, my heartfelt gratitude goes out to all my friends and family who turned my mountains into molehills and provided rest stops along the way. xiv Chapter One Introduction 1.1 Motivation Composite materials such as carbon fibre reinforced plastics (CFRJP) offer great advantages for aerospace applications due to their high modulus and strength combined with their light weight. For a given weight, the strength and modulus of a C F R P can be 10 times that of a metal. The strength of the final structure is important in design and typically these materials have a reasonably good strength. However, design potential is frequently sacrificed by an inability to effectively model and predict their structural strength. Structural failure frequently initiates at regions of high stress, such as at holes, notches or cracks. For simplicity, these discontinuities w i l l be referred to interchangeably in this chapter. Holes may be created by unexpected impact loadings such as collisions with flying debris and tools dropped during repair, which are a frequent hazard. Alternatively, holes may be introduced by design, such as at joints or fasteners. Notched failure is preceded by the onset of damage, which alters the stress concentration at the notch tip. Traditional means of modeling fracture in notched specimens do not properly account for the effect of this damage on the notch tip. The fracture strength of large coupons or structures is underestimated i f this damage is ignored. A critical design driver of the aerospace industry is to build damage tolerant aircrafts, with the focus of current research being to develop predictive tools best suited for composite structural failure. 1 1.1.1 Examples of Damage in a Composite Laminate One of the most promising applications of C F R P ' s in commercial transport aircraft is as the skin of primary load-bearing structures. For example, Figure 1.1 shows the primary structures in the Boeing 777 made of C F R P . The skin is made of a carbon fibre reinforced laminate. The laminate is made of individual laminae stacked at different orientations to give specific load bearing abilities as shown in Figure 1.2. The laminae consist of layers of long continuous fibres embedded in a matrix. A lamina is a specially orthotropic material when loaded in the fibre direction. If a balanced, symmetric laminate is loaded in a principle material direction, it is also a specially orthotropic material. Damage, shown in Figure 1.3, may grow within a lamina in the form of fibre pull-out, fibre bridging, fibre/matrix debonding, fibre failure, and matrix cracking (Figure 1.3a). Damage may also propagate as separation of the individual laminae, called delamination (Figure 1.3b) (i.e. Anderson, 1991). Due to the interaction of the many layers, damage frequently progresses as a complex combination of the individual damage mechanisms. While these mechanisms may increase the fracture resistance of a laminate by dissipating the concentration of stresses at the notch tip, they complicate the failure analysis. Several approaches have been developed to model damage evolution in notched laminate coupons, with the focus of current research in the aircraft industry being to extend these to predicting damage in large composite structures. 1.2 Linear Elastic Fracture Mechanics Linear Elastic Fracture Mechanics ( L E F M ) was one of the first approaches used to predict fracture in notched composite laminates. A classical stress concentration factor does not take into account the hole size when predicting fracture strength. In contrast, L E F M attempts to 2 account for the 'hole size effect', in which the tensile fracture strength of notched or centre-holed coupons decreases with increasing hole size (Whitney and Nuismer, 1974). Figure 1.4 shows a plate containing a preformed crack subjected to a load. L E F M predicts that the crack wi l l grow i f the energy release rate, G, is greater than the resistance to crack growth, R, such that: da where F is the work performed by the external forces, U is the elastic energy stored in the plate, and da is the increment of crack extension, and R-f (1.2) da where dW is the energy required for crack formation. It can be shown that prior to crack extension, G is the same whether the crack is grown under load or displacement control (Broek, 1986). If the plate is under load control, a fixed load is applied and the resultant displacement is dictated by the compliance of the system. In displacement control, a fixed displacement is applied, giving a resultant load. A rising R-curve occurs when the resistance to crack propagation increases with increasing crack length. The R-curve is considered not to be a material property but depends, for example, upon the specimen geometry and loading (Broek, 1986). A stress based L E F M approach predicts failure based on the stress intensity at the tip of the preformed crack. The elastic solution for the stress field at the tip of a crack in a body, subject to a load perpendicular to the flanks of the crack (Mode I loading) is: C7 ., = u V2 (i-3) 3 where K} is the Mode I stress intensity factor, r is the radial distance from the crack tip, and fy(Q) expresses the influence of the 0 direction on the stress magnitude. Solutions exist for the stress intensity factor for many geometries, and for finite-sized specimens, the effects of the edges are accounted for by a finite width correction factor, Y. For the plate shown in Figure 1.4, containing a centre crack of length 2a, subject to a nominal stress, a, the Mode I stress intensity factor is: K,=Yrsyfiza (1.4) For a given crack length, fracture occurs when a reaches a critical value, rjy and Kj is referred to as the fracture toughness of the material, K I C . L E F M therefore predicts that the fracture strength of the plate in Figure 1.4 is inversely proportional to the crack length, such that: CT/«-7= . (1.5) For a specially orthotropic material, Sih et al. (1965) found that K, is related to G 7 by: where an=l/E,, a22=l/E2, a12=-v12/Eh and a66=l/G12 for plane stress. Eh E2, v]2, G12 are the elastic constants of the material. 1.2.1 The Process Zone in Quasi-Brittle Materials When a notched specimen is loaded, Eq. (1.3) predicts a stress singularity at the notch tip. In a metal, however, once the stress reaches the yield stress, the material yields and forms a plastic zone. Analogous to the plastic zone in metals, the material at the notch tip in a quasi-brittle material forms a process zone, once the stress is high enough to initiate damage. The material in the plastic zone may strain harden, or conversely, the material in the process zone may strain soften. t22 + 2<?i2 + a 6 6 2a, (1.6) 4 The process zone consists of a region of discontinuous damage. In a fibre reinforced composite laminate, for example, the damage may be a combination of matrix cracking, fibre pull-out, fibre failure, fibre splitting or delamination. The large safety factors required in design of composite laminate structures are overconservative, as traditional predictive techniques do not properly account for the effect of the process zone on the stresses at the notch tip (Ilcewicz, 1995). Fracture mechanics predicts the notched strength to be inversely proportional to the notch size based only on elastic properties and geometry. Several alternate approaches have been developed which attempt to account for the effect of the process zone via a characteristic dimension or a physical description of the local failure mechanisms. 1.3 Micromechanical Models Used to Predict Notched Behaviour 1.3.1 Predicting Fracture Based on a Fitted Parameter A n initial attempt to account for the influence of the process zone on the hole size effect was made by Whitney and Nuismer (1974), in their point stress and average stress criteria. They proposed a failure criteria based on the stress or average stress at a characteristic distance in front of a hole. When the stress reaches the unnotched tensile stress, failure w i l l occur. They proposed that at the edge of a larger hole there is a higher probability that a flaw w i l l initiate failure, as the stress singularity is less concentrated than at a smaller hole, as shown in Figure 1.5. This would explain the hole size effect. Such a theory would be highly useful i f the characteristic distances were material properties. However, experimental evidence has since shown otherwise (Pipes et al., 1980). 5 Another technique which uses a fitted parameter to account for the hole size effect was proposed by Mar and L i n (Mar and L i n , 1977). They generalized the inverse square root dependence of notch strength on the crack size predicted by L E F M , to: a , = Hc(2r)-m (1.7) where oy is the fracture strength, r is the crack size, Hc is a fitted parameter called the composite fracture toughness, and the exponent, m, can be calculated from the elastic constants of the fibre and matrix. In a study of graphite epoxy laminates, Lagace (1986a) concluded that the Mar -L in criterion generally provided a better prediction of notched composite fracture strength than the point stress criterion. The point stress, average stress, and M a r - L i n criteria are still in use today. Rather than a physical description of the damage, they rely on a fitted parameter to account for the effects of the local failure mechanisms. 1.3.2 Relating Fracture to a Physical Understanding of the Damage Evolution Kortschot and Beaumont (1990a, 1990b, 1991c, 199Id) attempted to relate the notch size effect to a physical understanding of damage evolution at the notch tip. They developed a damage criterion based on the state of subcritical damage just prior to failure, called the 'terminal damage state' (TDS). Using radiography, they examined the T D S in double-edge notched (DEN) cross-ply graphite reinforced epoxy specimens. The TDS was incorporated into a finite element method ( F E M ) model to determine the notch tip stress distribution. They found that the remaining load-bearing plies in the T D S were the 0° plies, and a simple failure criterion based on the critical stress of the 0° ply was used to predict notched strength. In their model, the influence of layup and notch size on fracture strength was accounted for by their effect on the T D S . Three types of damage were identified, as shown in Figure 1.6, which 6 consisted of splits in the 0° plies (matrix splitting), transverse ply cracks in the 90° plies and triangular delamination zones at the 0 7 9 0 ° interface. They found that the greater amount of fibre splitting in the T D S of a [90° 2 , /0 o 2 ] s laminate than in a [ 9 0 7 0 ° ] 2 S laminate, doubled its fracture strength. Increasing the delamination height in the T D S increased the fracture strength regardless of layup, notch length or specimen width, due to its reduction of the notch tip stress concentration. Webb and Kortschot (1991) tried to identify a T D S in D E N quasi-isotropic carbon fibre reinforced epoxy laminates. However, they were unable to due to the complexity of the diffuse zone of cracking and local delaminations surrounding the notch immediately preceding failure. The micromechanical models represent the hole size effect in terms of mechanisms which operate at the scale of individual plies. They are useful in generating a good physical understanding of the damage. However, they cannot be directly extended to the more complex damage mechanisms found in the complex layups of a composite structure. 1.4 Macromechanical Models Used to Predict Notched Behaviour Rather than examining the hole size effect based on the scale of the individual plies, the size effect can also be accounted for by its influence on the scale of the structure. In L E F M , elastic finite width correction factors have been analytically derived for a given geometry. These factors are used to predict fracture of a structure from tests on small notched coupons. Bazant's empirical size effect law offers a means of predicting fracture of large structures in materials to which L E F M cannot be directly applied. This law, frequently used for quasi-brittle geomaterials, was also shown to apply to graphite fibre reinforced composite laminates (Bazant et al., 1995). It can be used to scale the nominal strength of geometrically similar structures of different sizes, based on a characteristic dimension, D , on the scale of the laminate. For example, 7 D may be taken to be the specimen width. This approach, although simple to use, requires empirical calibration and does not contribute to a physical understanding of the origins of the size effect. 1.4.1 Fictit ious C r a c k M o d e l A n F E M based approach to model the stress transfer capability of the process zone in quasi-brittle materials was initiated about 10 years ago, by Hillerborg et al. (1976). The model is called the Fictitious Crack Model ( F C M ) . Aronsson and Backlund (1986) later extended the model to fibre reinforced plastics, and called it the Damage Zone Model ( D Z M ) . In the F C M or D Z M , the fracture process zone ahead of a preformed crack is represented as a fictitious macroscopic crack, which transfers a stress distribution that varies with the crack opening displacement, as shown in Figure 1.7. The energy under the stress-displacement curve is equal to the energy absorbed per unit of newly-formed crack area. The preformed crack w i l l grow when the maximum stress at the tip of the fictitious crack reaches the unnotched tensile strength of the material. One draw-back to using this approach is that it is difficult to calibrate the stress-crack opening displacement curve. Either a linear shape is typically assumed, or a data fitting technique is used to calibrate the curve from the far-field load and crack mouth opening displacement. (Hillerborg, 1991; Mi l le r et al., 1991). 1.5 A Strain Softening Approach to Predicting Notched Behaviour In the last decade, a promising F E M based approach has surfaced to more accurately predict failure in notched composite laminates. This approach accounts for the effect of the process zone on the notch tip by incorporating a strain softening material response. 8 Continuum Damage Mechanics ( C D M ) can be used to describe a strain softening material response in fibre reinforced composite laminates. A state variable is defined which represents the state of damage in an element of material. The deterioration of mechanical properties due to damage is accounted for by an effective stress, based on an equivalent undamaged area. Matzenmiller et al. (1991) recently developed an anisotropic damage model for fibre reinforced composites. The model, here referred to as the M L T model, uses a damage variable based on a Weibull distribution to define the shape of a strain softening curve. The shape of the C D M strain softening curve is based on a Weibull distribution of fibre failure, and is shown in Figure 1.8. A s can be seen in the figure, the post-peak region is defined by a decreasing stress with increasing strain. For this reason, the curve is referred to as a strain softening curve, as opposed to strain hardening. Considerable controversy exists as to the fundamental validity of strain softening as a material response. Such a response assumes that the behaviour of the physically damaged material can be represented as a homogeneous continuum. A s well , a strain softening material response violates Drucker's stability postulate (Drucker, 1951). This postulate states that when an external agency applies additional surface and body forces to a volume of material (Chen and Han, 1988): • "The work done by the external agency on the changes in displacements it produces must be positive, and • the new work performed by the external agency on the changes in displacements it produces must be non-negative." According to this postulate, the post-peak region of the strain softening curve is unstable because the work done for an increment of strain, 5s, is negative (da is less than zero). To allay concerns of violating basic principles, a load-displacement curve w i l l be substituted for the strain softening stress-strain curve in this discussion. Stress and strain are parameters which 9 apply to a body that is a homogeneous continuum. However, once there is appreciable damage in the process zone, by definition there are voids and empty spaces. In other words, the material is no longer a homogeneous continuum. Therefore, defining the post-peak section of the curve by the load-displacement response avoids any implication of violating basic principles. However, to be consistent with current terminology, we w i l l refer to the load-displacement curve as the strain softening curve. Strain softening is a laminate property, and is particular to a given material system and loading. In a composite laminate, the material system represents the specific fibre and matrix used, placement (tow vs tape) and layup of the laminate. In a typical strain softening curve shown in Figure 1.9, the response is initially linear elastic between points A and B , whereupon it behaves nonlinearly due to damage initiation between points B and C. In a notched specimen, for example, damaged material in the process zone is subject to high local displacements, and at failure unloads to surrounding, less damaged material. This unloading is represented on the curve between points C and D . 1.5.1 Experimental Investigation df Strain Softening as a Material Property Materials which are considered to strain soften include rocks, concrete, soil, polymers and fibre reinforced plastics, (Read and Hegemier, 1984; Hasan et al., 1993). Read and Hegemier (1984) experimentally investigated the physical basis of strain softening in rocks, soil and concrete. They concluded that it is not a material property. They showed, for example, that the strain softening curves of concrete columns in compression were a function of specimen geometry. The load bearing area was reduced in columns which had a large length to diameter ratio, due to slabbing of the concrete walls. Scaling the force-displacement curve by the unfractured cross-sectional area, yielded a strain hardening behaviour. They also noted that the assumption of a homogeneous continuum inherent in the definition of a material property was invalidated by 10 inhomogeneous deformation. They concluded that a continuum mechanics representation of these materials is only justifiable i f the scale of the inhomogeneities is significantly smaller than the material element being considered, which is not the case in these materials. A n extensive discussion of using strain softening to predict the behaviour of structures can be found in a book by Bazant and Cedolin (1991). 1.5.2 Geometry of a Strain Softening Specimen Materials which strain soften wi l l appear brittle in specimens such as a dog-bone, due to limited opportunity for local load redistribution during failure. A special geometry is required to allow the gradual development of the damage zone. The most common specimens used are a centre-notched or edge-notched tensile specimen, subject to a uniform far-field load (Ilcewicz, 1995). A s the far-field stress increases, locally high displacements occur in the damaged region. The most damaged sections of the specimen w i l l have progressed to the largest displacements in the post-peak section of the strain softening curve. In contrast, the undamaged surrounding elastic medium w i l l be on the elastic loading section of the curve. 1.5.3 Experimental Investigation of Process Zone Softening A s the load redistribution in the process zone cannot be measured directly, the strain softening curve must be indirectly determined. Despite the fundamental questionability of examining strains instead of displacements, many researchers have measured notch tip strains to investigate the influence of the process zone. Evidence of large strains in the process zone have been noted by Daniel (1978) in the 1970's. However, relating these strains to a strain softening model has only begun in the last decade. 11 Geers et al. (1995) monitored the 'Active ' strain localization ahead of the notch in a compact tension short glass fibre-reinforced polypropylene specimen, using a retro-reflective grid illuminated by halogen light. They defined the 'fictive strains' to be the measured displacements over the crack region. The maximum value of the notch tip fictive strain aligned with the load line was found to be 10%, while the maximum tensile test strain for the material was 3%. They found that during loading, matrix deformation created an elastically deformed zone prior to the peak load. After the peak load, the elastically deformed zone unloaded and fibre-matrix debonding and some fibre pull-out occurred. Other experimental techniques used to monitor displacements in the process zone were developed by Shah and Ouyang (1993) for cement-based materials. Laser holographic interferometry was used to map the surface displacements in a centre-notched plate subject to tension. The effect of the process zone was found by comparing the measured strain field to that predicted by L E F M . Behind the notch tip, the size of the process zone was found to increase with increasing load, while the size in front of the notch tip remained constant. It was suggested that the wake zone may thus more significantly affect concrete toughness. Instead of measuring the strain, Basham et al. (1993) proposed a means of back-calculating the stress in the process zone. B y measuring the applied load, crack mouth opening displacement ( C M O D ) , and the displacement directly ahead of the notch in a semicircular specimen, the local stress as a function of the displacement in the process zone can be back-calculated. However, several assumptions are inherent in their use of the J integral to back out the stress, such as self-similar crack growth. A more detailed investigation of the displacements in the process zone, coupled with the damage progression, is needed to get a clear physical understanding of the post-peak region of the strain 12 softening curve. Only once this physical understanding exists, can a powerful numerical technique such as the F E M based strain softening model provide a reliable tool for predicting notched structural failure. 1.6 Statement of Objectives The objectives of this work are to: 1. Develop a specimen which can be used to investigate a strain softening approach to predicting notched behaviour. 2. Generate a variety of experimental data with which to calibrate an F E M analysis of the specimen that uses a strain softening curve. 3. Develop a physical understanding of the damage evolution in the specimen, that can be used to examine the physical basis of strain softening. 13 Figure 1.1 Primary load-bearing structures in the Boeing 777 made of fibre reinforced composite laminates. (Courtesy of The Boeing Company) 14 Figure 1.3 Schematic showing some of the damage mechanisms that occur in fibre reinforced composite laminates. Taken from Anderson (1991). 2a a Figure 1.4 Centrally cracked plate subject to a uniform far-field stress. 15 Figure 1.5 Stress distribution around two different sized holes in an infinite isotropic plate. This explains the hole size effect according to Whitney and Nuismer. A greater probability exists around the larger hole for a flaw to initiate failure, as the stress concentration is less localized. Reproduced from Whitney and Nuismer (1974). transverse ply cracks delamination |J split — 90 ply 0 ply ply — 90 ply r - 0 Figure 1.6 Schematic of damage observed by Kortschot and Beaumont in cross-ply graphite reinforced epoxy laminates. The damage was observed in the 'terminal damage state' (TDS) just prior to failure, of double edge-notched specimens. Reproduced from Kortschot and Beaumont (1990a). 16 °"coh A fictitious crack Figure 1.7 Representation of the stress distribution in the process zone in Hillerborg's Fictitious Crack Model. The process zone is modeled as a fictitious crack, containing a linear cohesive stress distribution as shown. The preformed crack will grow when the maximum stress at the tip of the fictitious crack reaches the unnotched tensile failure strength. Reproduced from Afaghi-Khatibi et al. (1995). 6/x t A 1 + a = stress X t =strength s = strain X t IE = nominal failure strain m = fitted parameter e / ( x t / E ) Figure 1.8 The uniaxial strain softening curve predicted by Matzenmiller et al. (1991), based on a Weibull distribution of fibre failure. Different degrees of strain softening are predicted by the parameter, m. 17 AB: elastic behaviour BC: damage initiation C CD: damaged material sheds load to surrounding less damaged material ro B / w _ i D A* Displacement Figure 1.9 Different regions of a typical strain softening curve. 1 8 Chapter Two OCT Test Developed to Study Notched Behaviour 2.1 Introduction This chapter presents the development of an overheight compact tension (OCT) test to study notched behaviour. The specimen geometry and loading method is described, with the experimental setup devised to monitor the surface displacements ahead of the notch tip during loading. A description is given of the two material systems tested, with a summary of the tests and post-test analysis done on each specimen. 2.2 The Overheight Compact Tension (OCT) Specimen 2.2.1 Development of a Specimen Which Exhibits Stable Damage Growth To examine notch tip behaviour, a specimen geometry was required which would grow damage in a stable manner, in a small specimen size that would conserve material. A compact tension (CT) type specimen geometry was chosen, in which a load is applied perpendicular to the flanks of a preformed edge notch. Under displacement control, the load drops during crack extension, and the strain energy release rate, G, may drop below the crack growth resistance, R, of the material. According to fracture mechanics, a propagating crack is arrested i f G is less than R, and the specimen is said to exhibit stable crack growth. Initial increments of unstable crack extension are followed by crack arrest, leaving the remainder of the specimen intact for post-test examination. 19 To make efficient use of material, an investigation was done of the smallest specimen size which would allow a reasonable amount of stable crack growth. In the standard C T geometry typically used for testing metals, the specimen height is roughly equal to the width. A sizeable process zone size was expected in the material systems studied (Ilcewicz, L . , 1995). A s the exact dimensions were not known prior to testing, it was necessary to leave sufficient material available for the process zone to develop without the influence of the boundaries. With this in mind, the first C T geometry tested had a width roughly twice the height. Upon loading, a through-the-thickness crack grew several millimeters before deviating at an angle of 45°, tearing off one of the loading arms which were essentially acting as cantilever beams. The specimen height was then increased to twice that of the specimen width. A successful test was completed with a reasonable amount of damage grown in front of the notch tip, apparently without the influence of the boundaries. 2.2.2 The Standard C T Stress Intensity Factor For analysis purposes, the stress intensity factor of a standard C T specimen can be used to model the behaviour of the overheight C T specimen. The following equation is accurate to within 0.5% for a standard C T geometry, for 0.2 < a/W < 1 is: (Broek, 1986) a 2 + — W J Y = -P BW K, = * Y 0.886 + 4.641 — 1-13.32 W — +14.72 — -5.6 — \w) \W ) \W 3 / \ 4 (2.1) <0 1 - — W J where P is the load, B is the specimen thickness, W is the specimen width, and a is the preformed notch length. B, Wand a correspond to those dimensions shown in Figure 2.1c. 20 2.3 Description of the Experimental Setup The final specimen geometry and experimental setup is shown in Figure 2.1. 2.3.1 Loading Train and Specimen Preparation The specimen was loaded statically in an Instron testing machine under displacement control at a rate of 0.02 inch/minute, using a 20 000 lb load cell. A l l tests were performed at ambient temperature and humidity. A s thin laminates are prone to twisting, two narrow steel stiffener bars were suspended flush on the sides of the specimen to minimize out-of-plane bending (referred to as 'guide' in Figure 2.1a). Teflon tape was placed under the stiffener bars as well as the loading bars to minimize friction. The specimen was first prepared by cutting out the shape on a band saw. This gave a blunt notch tip. A starter notch and fatigue crack were not made as required by a standard C T test, so that the notch would be a more realistic representation of a through-the-thickness discontinuity found in a real structure. A s well , the stress concentration would be less localized, increasing the damaged area. 2.3.2 Measured Specimen Displacements Small modifications were then made to the specimen to measure the following displacements: 1. The displacement near the loading pins was recorded in later tests, by a 50% Instron gauge placed across the notch opening, referred to as a crack mouth opening displacement ( C M O D ) gauge. 2. The displacement at 1 cm in front of the notch tip was measured by a clip gauge. 21 3. The displacement in front of the notch, at fixed distances above and below it, were measured by lines inscribed across the width of the specimen surface. The displacements were measured using a photographic, image analysis technique. Lines were inscribed with a razor blade at successive heights on the front specimen surface, and colored with white correction fluid. This technique broke a few surface fibres, but when examined under a microscope after testing, did not appear to interfere with the damage grown during the test. Photographs were taken throughout the test with a camera using a 105 mm macrolens. Enlarged 8x10 inch photographs were then digitally scanned at an optical resolution of 100 pixels/inch, and imported into an image analysis program to measure the displacement of the lines in front of the notch tip. 2.3.3 Loading Train Deflection The cross head displacement of an Instron Universal testing machine can be calculated from the set cross head speed i f the time interval is known. In these experiments, the time interval was given by the clock of the microcomputer used for data acquisition of the load and gauge displacements. A n investigation was done of the stiffness of the Instron loading train to determine i f the pin hole displacement could be accurately calculated from the cross head deflection. The composite specimen was replaced by a very stiff steel plate, and the displacement of the plate measured by a clip gauge placed on the specimen surface. The cross head displacement was measured with a mechanical dial gauge. A t a load of 5000 N , the plate elongation was negligible (0.005 mm), while the cross head deflected 0.35 mm, as measured by the dial gauge and consistent with the set cross head speed. Therefore, it is clear that the applied deflection of the cross head is not equivalent to the specimen displacement at the loading pins. The calculated cross head 22 displacement therefore should only be used as a qualitative indication of the loading pin displacement. The C M O D gauge presents the only accurate measure of the displacement near the loading pins. In the test with the steel plate, the load vs cross head displacement curve was nonlinear, so no attempt has been made to back-calculate the true loading pin displacement for those tests where C M O D gauge results are not available. 2.4 Summary of OCT Tests Performed 2.4.1 Materials Studied Two material systems have been studied, both fabricated at The Boeing Company. System A is a thin laminate, which can be easily modelled in an F E M analysis. System B is a sandwich panel with an outer weave layer, which, although more difficult to model, is of interest as it represents a material system similar to that presently proposed for the fuselage skin of commercial aircrafts. The layup and components of each system are: 1. System A : • quasi-isotropic layup: [-45/90/45/0/-45/90/45/0] s • T300H carbon fibre/F593 epoxy, plate thickness = 3.35 mm 2. System B : • quasi-isotropic layup: [F0/-45/90/45/0/45/90/-45/F0]T where FO = [0/90] weave fabric A S 4 tow placed carbon fibre/8552 epoxy sandwhich panel, face thickness = 1.6 mm, core thickness =12.7 mm 2.4.2 Tests Performed A total of four O C T specimens of system A (specimens A 1 - A 4 ) were tested, and two of system B (specimens B l and B2). Table 2.1 lists the dimensions of each specimen according to Figure 2.1. 23 2.4.3 Post-test Analyses Several post-test analyses were done to qualitatively and quantitatively examine the damage grown during the test. Chapter 3 presents the damage progression throughout the test as indicated by the load vs cross head or C M O D displacements, and the load vs clip and surface line displacements. A qualitative examination of the damage is discussed in Chapter 4, as observed by post-test sectioning and deply. A quantitative investigation of the damage is given in Chapter 5, by a series of tensile tests of specimens cut from the damaged area. Table 2.2 summarizes the instrumentation used and post-test analyses done on each specimen. 24 Table 2.1 Overheight compact tension (OCT) specimen dimensions where the letters refer to the schematic in Figure 2.1. All dimensions are in mm. Specimen ID a W a/W H B b c d e f System A A l 28.3 87 .32 200 3.35 12.12 19.05 7.8 N/A N/A A2 29.1 87.85 .33 200 3.35 12.15 19.05 7.8 N/A 20 A3 25.26 85.86 .29 200 3.35 12.7 19.05 7.8 N/A 5f A4 17.63 79.8 .22 207 3.35 22.24 19.05 9.5 28.72 2.5* System B BI 17.51 78.77 .22 202.3 1.66* 22.74 27.98 15.9 33.7 2.5s B2 24.45 84.69 .29 204.8 1.66* 15.01 27.98 15.9 36.33 2.5s * thickness of each face sheet. Core thickness is 12.7 mm. ' 9 lines are inscribed above and 9 lines below notch mid-plane 20 lines are inscribed above and 20 lines below notch mid-plane Table 2.2 Summary of instrumentation and post-test analysis of each OCT specimen. Specimen ID CMOD clip line detailed deply* tensile gauge gauge analysis* sectioningT test" System A A l N Y N N Y N A2 N Y N N Y N A3 N Y Y Y N N A4 Y Y N N N Y System B BI Y Y Y Y N N B2 Y N N N N Y * presented in Chapter 3 f presented in Chapter 4 * presented in Chapter 4 ** presented in Chapter 5 25 (b) close-up of lines inscribed on specimen surface (a) OCT specimen in loading assembly prior to testing units: mm applied displacement, 8 CMOD gauge, slits ? applied displacement, S . , d r-7*T ; . . i>< K . . If. , f • • • Cl —^i. —+ * I £4.2) * l w L - / J notch mid-plane (n.m-p.) B .clip gauge holes on back surface lines inscribed on front surface a distance, f, apart (c) schematic showing locations at which displacements are measured and specimen dimensions Figure 2.1 Schematic of the overheight compact tension (OCT) specimen and experimental setup used to monitor displacements during loading. 26 Chapter Three OCT Tests and Data Reduction 3.1 Introduction This chapter discusses the evidence of damage progression throughout the O C T tests as indicated by the load vs cross head or load vs C M O D displacements, and the surface displacements in front of the notch tip. The displacement of lines inscribed on the specimen surface suggest the progression of damage across the specimen width throughout the test. A clip gauge provides a continuous measurement of a local displacement at a fixed location in front of the notch tip, close to the notch mid-plane. The clip gauge displacements support the results of the line analysis. Finally, the difference in the surface displacements of systems A and B is discussed. The following terms are defined: a notch represents a preformed through-the-thickness discontinuity in the specimen prior to testing, a crack represents through-the-thickness fibre failure, matrix cracking and some delamination which grows in front of the notch due to loading, and the process zone represents a zone in front of the crack, in which damage is visible in only some of the plies. This damage is observed in the post-test sectioning presented in detail in Chapter 4, but must be mentioned in conjunction with some the results of this chapter. The length of the crack refers to its dimension in the same direction as the specimen width, W, in Figure 2.1c. It includes the preformed notch length (dimension a in Figure 2.1c) and the distance the crack has grown in front of the notch tip during loading. 27 3.2 Cross Head and CMOD Displacements The far-field load vs cross head displacements of specimens A l to A 3 are given in Figure 3.1, together with the load vs C M O D displacement of specimen A 4 in Figure 3.2. Figure 3.3 shows the load vs C M O D displacements of specimens B l and B 2 . Specimens A l , A 2 and A 3 were tested prior to noting the inaccuracy of using the Instron cross head deflection, so a C M O D gauge was not used in the tests. The loads are presented as a function of the calculated cross head deflection. A complete test was done on specimen A 2 , such that the specimen fractured in two pieces. The remaining specimens were loaded until roughly the same point in the test, just after the first large load drop, so that a reasonable amount of damage was grown. The developing process zone could then be examined by post-test analysis. 3.2.1 General Shape of the Load vs Cross Head and Load vs C M O D Displacement Curves For both systems, the general shape of the load vs cross head displacement or load vs C M O D displacement curve is similar. Initially, the loading increases linearly, followed by a region in which small load drops occur. The load-carrying ability of the specimen increases until the maximum load is reached, followed by a large load drop. Several more load drops may then occur, with decreasing peak loads prior to each load drop, until complete fracture of the specimen i f the test is carried to completion (i.e. specimen A2) . The significance of the load drops as regards to damage progression throughout the test w i l l be discussed in section 3.3. 28 3.2.2 Repeatability The primary purpose of the O C T specimen was to grow a reasonable amount of damage as opposed to performing a valid fracture toughness test. However, an indication of the repeatability of the O C T test can be found by examining the expected results to the predicted behaviour of two specimens, using Eq. (2.1). Good repeatability is obtained between the measured and predicted peak loads in specimens B I and B 2 . Eq . (2.1) predicts a 17% higher peak load in specimen B I than specimen B2 for a given K I C , due to the different a/W. This is similar to the measured peak load in specimen B I , which is 12% higher than in specimen B 2 . Specimens A l and A 2 have a 3% difference in their a/W which corresponds to a 2% difference in their predicted peak load for a given K I C , using Eq . (2.1). However, a 19% difference exists in the experimentally measured peak loads. This variability is acceptable given the purpose of the O C T specimen and the bluntness of the notch tip. The approximate fracture toughnesses of systems A and B can be compared to a value in the literature by substituting the peak loads into Eq . (2.1). The fracture toughness of specimens A l , A 2 , A3 and A 4 are found to vary by 38%, and range from 33 MPa(m) 1 / 2 . to 53 MPa(m) 1 / 2 . A similar calculation for system B shows that the estimated fracture toughnesses is similar, to 1/2 1/2 system A . The fracture toughness are 56.2 MPa(m) and 56.8 MPa(m) for specimens B I and B2 , respectively. These fracture toughness values seem reasonable given that the fracture 1/2 toughness of a carbon fibre laminate is listed in the literature as ranging from 40 to 70 MPa(m) (Ashby, 1992). 29 3.3 Indications of Damage Growth 3.3.1 Surface Line Displacements The surface line displacements indicate the progression of damage across the specimen width throughout the test. The results are presented as the total displacement between a pair of lines inscribed at the same height above and below the notch mid-plane. The lines presented in this section are the lines closest to the notch mid-plane not affected by surface delamination. The line displacement is plotted as a function of the position in front of the notch tip, at increasing loads, or photographs, throughout the test. The line displacements are presented in conjunction with the load vs cross head or load vs C M O D gauge displacements, and the load vs clip gauge displacements. The photograph number corresponding to each load is marked. The line displacements for selected photographs for systems A and B are presented in Figure 3.5, Figure 3.7, and Figure 3.9. The figures in Appendix 1 show all the displacements that have been measured from photos taken throughout the tests. 3.3.1.1 Repeatability and Accuracy A n investigation was done of the variability of repeat measurements of the line displacements measured by the image analysis technique. The average image resolution of the line analysis of specimen A 3 in Figure 3.5 is 0.045 mm/pixel, giving repeat measurements a variability of +0.045 mm. Improving the image resolution to 0.02 mm/pixel was not found to be beneficial, however, as the scribe line thickness was roughly 0.1 mm, and human error was introduced in repeating a measurement at the same location on the scribe line. 30 The clip gauge displacements confirm the displacements measured by the line analysis. For example, the displacement of line #4 of specimen A 3 at 10 mm in front of the notch tip (Figure 3.5) can be compared to the clip gauge displacement (Figure 3.4) at the same load. The damage grown after the largest load drop (between photos 15 to 19), corresponds to an opening of both the line displacement height and the clip gauge of roughly 0.3 mm. A s the crack extends during loading, the material on either side of the open crack rotates slightly about the crack tip. This rotation affects the given positions in front of the notch tip at which measurements are taken of each scribe line displacement. A marked specimen indicated that in the worst case, the shift of a given position in front of the notch tip due to specimen rotation was 0.5 mm. This rotation was therefore ignored when measuring the scribe line displacements. 3.3.1.2 Indication of Damage Length In order to interpret the line analysis, the sectioning results presented in detail in Chapter 4 w i l l be briefly mentioned. The results indicate that the point of zero displacement of each line in front of the notch tip roughly corresponds to the length of the internal damage. For example, stopping the test of specimen A 3 at photo 19 in Figure 3.4 followed by deply and sectioning, reveals fibre failure running in each layer across the laminate as a crack perpendicular to the load, roughly 25 mm in length. In front of this is a zone 7 mm in length, in which there is fibre failure and matrix cracking in the outer layers, while the centre layers are undamaged. Inspection of the displacement of line #4 at the same point in the test (photo 19 in Figure 3.5), shows that the line has zero displacement at 30 mm in front of the notch tip. Thus, the point of zero displacement indicates the length of the internal damage. 31 The line analyses of specimens B I and B2 confirm this hypothesis, and indicate a final crack length of 38 to 40 mm. Post-test sectioning indicates a crack 35 to 40 mm long, with a process zone between 3.5 to 4 mm in front of it, respectively. For example, at photo b9 of specimen B I in Figure 3.7, line #1 has zero displacement at roughly 38 mm in front of the notch tip. Stopping the test at this point in the test (photo b9 in Figure 3.6), followed by sectioning, reveals a 35 mm long crack with a 3.5 mm long process zone in front of it. Similar results are obtained in specimen B2 . Stopping the test and sectioning at photo 21 in Figure 3.8, indicates a crack 40 mm long, with a process zone 4 mm long in front of it. Line #1 on specimen B 2 has zero displacement at roughly 40 mm in front of the notch tip (Figure 3.9). Table 3.1 lists the approximate position in front of the notch tip at which the line closest to the notch mid-plane has zero displacement, for specimens A 3 , B I and B 2 . 3.3.1.3 Damage Progression as a Function of the Far-field Load The line analysis describes the general sequence of damage progression in both systems throughout the test. During the initial linear elastic load up, the line analysis indicates that negligible crack growth occurs, as the position of zero displacement of the line corresponds to the tip of the notch. A small amount of crack growth then follows, during the small load drops or nonlinear section of the curve of 5 to 10 mm. This crack growth does not seem to significantly affect the load carrying ability of the specimen, as after each drop the load rises to a larger peak value. Once the maximum peak load sustained by the specimen is reached, a large load drop occurs during which the crack extends significantly, between 15 to 30 mm. The test is stopped shortly after, with a few specimens haying several millimeters of crack extension during the small load drops that occur prior to unloading. 32 3.3.1.3.1 System A The damage progression in specimen A 3 follows the above damage sequence, and the displacement of line #4 in specimen A 3 (Figure 3.5) indicates that negligible crack growth occurs prior to the first load drop, as line #4 in photos 10 and 11 has zero displacement at the notch tip. U p to 10 mm of crack extension occurs prior to the large load drop (photo 14), as the line has zero displacement at roughly 10 mm in front of the notch tip. Once the load has dropped (photo 15), the slope of the line increases, and has zero displacement at 25 mm in front of the notch tip. Therefore, the largest load drop corresponds to the largest growth increment of 15 mm, after which the crack grows only another 5 mm during the small load drop between photos 18 and 19. 3.3.1.3.2 System B 3.3.1.3.2.1 Specimen B1 The line analysis indicates a slightly different sequence of damage growth in system B . According to the analysis of line #1 in Figure 3.7, once the load rises nonlinearly between photos a8 and b3, the crack extends from 1 to 10 mm in front of the notch tip. During the largest load drop, which occurs in two sections (photos b3 to b8), there is a notable difference in the behaviour of the surface lines of specimen B I compared to specimen A 3 . In specimen A 3 , during the largest load drop, the intersection of scribe line #4 with the x axis jumps from 10 to 25 mm in front of the notch tip, representing approximately 15 mm of crack extension. In contrast, the largest load drop in specimen B I occurs in two sections, photos b3 to b4, and b7 to b8 in Figure 3.6. In the first section of the load drop, photos b3 to b4, the position at which line #1 has 33 zero displacement remains constant at 10 mm in front of the notch tip, yet the slope and shape of the line changes significantly. In the second stage of the load drop, photos bl to b8, the position of zero displacement jumps from 12 mm to 38 mm in front of the notch tip. However, unlike the first section of the load drop, the shape of the line in the second section remains approximately linear. Thus it appears that local deformation without crack extension is occuring in the first section of the load drop, while in the second section primarily crack extension occurs. 3.3.1.3.2.2 Specimen B2 Although the sequence of crack growth in specimen B2 is slightly different than in specimen B1 the final crack length is the same. In both specimens, the largest load drop occurs in two sections. However, during the first part of the load drop significant deformation does not occur in specimen B 2 prior to crack extension as was observed in specimen B l . Instead, the crack grows 18 mm during the first part of the load drop, Figure 3.9, and 7 mm during the second section of the load drop, giving a final crack length of 40 mm. 3.3.1.4 Displacement of Lines Farther Away from the Notch Mid-plane Lines farther from the notch mid-plane show a similar damage progression to the line closest to the notch mid-plane. In specimen B l , for example, line # 3 (Figure 3.10) is 7.5 mm above and below the notch mid-plane, while line #1 (Figure 3.7) is 2.5 mm above and below it, and both indicate a similar sequence of crack extension. Prior to the peak load, the crack has extended to roughly 10 mm in front of the notch. In the first section of the large load drop, negligible crack extension is observed, and after the second part of the load drop, the final crack length measured by both lines is roughly 37 to 38 mm. 34 Thus lines not directly adjacent to the notch mid-plane give a reasonable description of the damage progression, and the displacements of line #4 in specimen A 3 may be assumed to give a reasonable representation of the damage growth. 3.3.1.5 R-Curve Behaviour The line displacements can be used to estimate the amount of crack growth at increasing loads throughout the test, so that a crack growth resistance or R-curve, can be drawn. The critical mode I strain energy release rate, G1C, is plotted as a function of the total amount of crack extension in front of the notch tip, Aa, for specimens A 3 , B I and B2 in Figure 3.11 (note that KIC is calculated using Eq. (2.1), and GIC is calculated from KJC using Eq . (1.6)). The results are plotted up to the largest load drop. Beyond this drop the crack is sufficiently large that edge effects may influence the R-curve behaviour. The line analysis is used to estimate the crack length at the photo corresponding to the load, indicated in Figure 3.11. In specimen A 3 , GIC is calculated for the four peak loads prior to photo 15 in Figure 3.4. However, a photo was not taken at each peak load, so the crack length was estimated from the photo immediately preceding the peak load. Both systems exhibit a rising R-curve, and thus an increasing resistance to crack growth prior to the peak load. It appears that system B has a higher resistance to crack growth, and may be a tougher system, as GIC for specimen B2 rises to a value 2 to 5 times greater than specimen A 3 . 3.3.2 C l i p Gauge Displacements The clip gauge displacements confirm the results of the line analysis. For example, during the large load drop in specimen A 4 , (points A and B in Figure 3.2) the clip gauge displacement 35 increases significantly, opening 0.54 mm, while the C M O D gauge opens less (0.2 mm). The line analysis of specimen A 3 indicated that the largest load drop corresponded to 15 mm of crack growth in front of the notch tip. One would expect the displacement of the C M O D gauge to be negligible during the load drop as it is a displacement controlled test. However, the low stiffness of the loading train, discussed in Chapter 2, would allow some deflection at the pin holes as the compliance of the specimen increases during crack growth. During the smaller load drop, the clip gauge displacement approximates that of the C M O D , as the damage front has passed ahead of the gauge. In light of the results of system A , one would expect a relatively larger opening of the clip gauge (0.45 mm) relative to the C M O D gauge (0.39 mm) during the first part of the largest load drop in specimen B l , Figure 3.6. However, as their damage sequence is different, in specimen A 3 (Figure 3.4) significant crack extension occurs during the largest load drop, while in the first section of the load drop in specimen B l , damage other than crack extension occurs. Thus, in specimen A 3 fibre failure would allow free opening of the crack flanks within the gauge length, whereas in specimen B l intact fibres restrain opening. When these fibres break and the crack extends, during the second part of the load drop in B l , (photos b7 to b8) one would expect a large increase in the clip gauge displacement. However, the line analysis indicates that the clip gauge is now behind the tip of the damage, due to the 26 mm of crack extension which occurs. A clip gauge was placed on specimen B 2 , however it popped out during the test, invalidating the results. 36 3.4 Discussion of Results The line analysis and clip gauge results provide a consistent explanation of the damage progression in both systems. The line analysis indicates the progression of damage across the specimen width, with the position of zero displacement of the line corresponding to the front of the crack, as w i l l be confirmed by the post-test sectioning presented in Chapter 4. For both systems, the onset of crack growth occurs once the relationship between the specimen load and C M O D displacement or the load and cross head displacement either becomes nonlinear or small load drops are observed. The largest crack growth increment occurs during the largest load drop, corresponding to approximately 15 mm of crack growth in system A , and 25 mm in system B . During this drop, a significant opening of the crack flanks is measured by the line analysis and clip gauge, of 0.5 to 0.8 mm, for systems A and B respectively. The most significant difference that exists in the line analysis of systems A and B is that crack extension occurs during the largest load drop in system A , while process zone development, most likely due to matrix cracking and a small amount of fibre failure, may occur prior to crack extension during the largest load drop in system B . This was observed in the line analysis of specimen B l . This suggests that system B may have a greater capacity to redistribute load during failure than system A . To continue, an investigation to physically map the nature of the damage in the laminate is needed, and is presented in Chapter 4. Once a spatial description of the damage is known, it can be used in conjunction with the line analysis to create a picture of the sequence of damage growth throughout the test. 37 3.5 Summary of Results The following summarizes the results of this chapter: 1. The inscribed line and clip gauge displacements offer a consistent explanation of the damage progression throughout the test. 2. Discrete load drops indicate damage growth ahead of the notch, primarily as crack extension. 3. Large displacements in front of the notch tip are captured by both the line analysis and clip gauge during specimen unload. The largest load drop corresponds to roughly 15 to 25 mm of crack extension in systems A and B respectively, with an opening of 0.5 to 0.8 mm measured by the surface displacements. 4. System B may have a greater ability to redistribute load during failure than system A . Evidence of this is given by the line analysis of specimen B I which shows significant local displacement ahead of the notch, suggestive of matrix cracking and only a small amount of fibre failure, during specimen unloading prior to crack extension. In system A , predominantly crack extension occurs during unloading. 5. The R-curves suggest that system B may be a tougher system, with greater ability to absorb energy, as G1C rises more steeply and is significantly higher than system A up to the largest load drop. 38 Table 3.1 Position in front of the notch tip at which the scribe line has zero displacement at a given load. The results are for the scribe line closest to the notch mid-plane for specimens A3, B l and B2, and refer to the specified figures. System A: specimen A3 System B: specimen Bl System B: specimen B2 line #4, Figure 3.5 line #1, Figure 3.7 line #1, Figure 3.9 Photo # Position in Photo # Position in Photo # Position in Front of Front of Front of Notch Tip Notch Tip Notch Tip (mm) (mm) (mm) 10 0 4a 0 5 0 11 0 8a 2 8 1 12 1 17a 5 9 2 13 4 18a 9 11 6 14 10 3b 10 12 10 15 25 4b 10 15 15 16 25 7b 12 16 15 18 25 8b 38 17 33 19 30 9b 38 18 33 19 33 20 40 21 40 39 9000 specimen A1 a/W = 0.32 specimen A2 a/W = 0.33 specimen A3 a/W = 0.29 0 1 2 3 4 5 Cross Head Displacement (mm) Figure 3.1 Load vs cross head displacement of specimens A l , A2, and A3. The cross head displacement is not a true measure of the displacement at the specimen loading pins due to loading train deflection as discussed in Chapter 2. 40 41 T3 03 O 7000 6000 5000 4000 3000 0 Clip Gauge Cross Head -12 12 •I / 1 3 14 11 \ / 11 710 \ \ 1718 15 19 15 1 19 1— i Numbers refer to photo # 0 0.5 1 1.5 2.5 Displacement (mm) Figure 3.4 Load vs cross head and clip gauge displacement of specimen A3. The numbers on the curve indicate where the photos were taken throughout the test. The cross head displacement is not a true measure of the displacement at the specimen loading pins due to loading train deflection as discussed in Chapter 2. E E 3 (U c: d) E 0) o _ro q . •photo 10 -photo 11 -photo 12 -photo 13 photo 14 -photo 15 -photo 16 -photo 18 -photo 19 notch clip gauge Position in Front of Notch Tip (mm) Photo numbers refer to Figure 3.4. Displacement is measured relative to photo 5. Image resolution is .045 mm/pixel. Line #4 is 20 mm above and 20 mm below the notch mid-plane. ^ Figure 3.5 Displacement of line #4 as a function of position in front of the notch tip for selected photos of specimen A3. 42 12000 TJ 03 O Clip Gauge $3— CMOD Gauge Numbers refer to photo # a = roll #1, b = roll #2 1 0 0.5 1 1.5 2 Displacement (mm) 2.5 Figure 3.6 Load vs C M O D and clip gauge displacement of specimen BI. The numbers on the curve indicate where the photos were taken throughout the test. -s-photo a4 photo a8 - i — photo a17 - s - photo a18 photo b3 photo b4 - * - photo b7 - ° - photo b8 —•—photo b9 clip gauge Position in Front of Notch Tip (mm) Photo numbers refer to Figure 3.6. Displacement is measured relative to photo al. Image resolution is .048 mm/pixel. Line #1 is 2.5 mm above and 2.5 mm below the notch| mid-plane. Figure 3.7 Displacement of line #1 as a function of position in front of the notch tip for selected photos of specimen BI. 43 12000 TJ CO O 2000 t 0 9 11i>Jf B 9 T ^ 15 \ / \ 19 5 / 1 7 ^ \ J f 21 1 / 20 1 Numbers refer to photo # 1 1 1 n 0 0.5 1 1.5 2.5 CMOD Displacement (mm) Figure 3.8 L o a d vs C M O D displacement of specimen B2. The numbers on the curve indicate where the photos were taken throughout the test. E E, CD C C CD E CD O _ r o C L </) Q .clip gauge ' line #1 -•-photo 5 -s- photo 8 photo 9 - x - photo 11 -* - photo 12 -©—photo 15 -•-photo 16 — 1 — photo 17 - « - photo 18 photo 19 -* - photo 20 photo 21 notch ip gauge Position in Front of Notch Tip (mm) Photo numbers refer to Figure 3.8. Displacement is measured relative to photo 1. Image resolution is .042 mm/pixel. Line #1 is 2.5 mm above and 2.5 mm below the notch| mid-plane. Figure 3.9 Displacement of line #1 as a function of position in front of the notch tip for selected photos of specimen B2. 44 E E co c c CD E CD O _ r o o. w b notch ti Position in Front of Notch Tip (mm) photo a4 photo a8 photo a17 photo a18 photo b3 photo b4 photo b7 photo b8 photo b9 Photo numbers refer to Figure 3.6. Displacement is measured relative to photo al . Image resolution is .048 mm/pixel. Line #3 is 7.5 mm above and 7.5 mm below the notch mid-plane. Figure 3.10 Displacement of line #3 as a function of position in front of the notch tip for selected photos of specimen B l . 45 specimen A3 specimen B1 specimen B2 0 2 4 6 8 10 12 14 16 A a (mm) Figure 3.11 R-Curve of specimens A3, B l and B2, based on the line analysis estimation of the crack length. G I C is calculated for the peak loads corresponding to the crack lengths of photos: 1) specimen A3: 10,12,13,14 in Figure 3.4, 2) specimen B l : a8, al5, al9, b3 in Figure 3.6, and 3) specimen B2: 9,11,12,16 in Figure 3.8. 46 Chapter Four Physical Description of the Damage 4.1 Introduction This chapter presents a physical description of the damage grown during the O C T test. Fibre failure in each lamina is revealed by post-test deplies, in which the resin is removed in an oven. Lamina matrix cracking, fibre failure and delamination between laminae are observed from post-test sectioning. A crack, containing damage in all plies, is identified growing in front of the notch tip, in front of which is a process zone, in which only some of the plies contain damage. The final dimensions of the crack and process zone are reconstructed for systems A and B . The line analysis of Chapter 3 is used to approximate the load at which the damage was created. The spatial sequence of damage growth in the process zone indicates the temporal sequence of damage growth. This information helps to physically explain the notch-sensitivity of both systems, and provides a tool with which to calibrate their respective strain softening curves. 4.2 Techniques Three techniques were used to identify the post-test damage in the O C T specimens. Deply showed the path of fibre failure in each ply. Cross-sections cut perpendicular to the notch revealed through-the-thickness matrix cracking, fibre failure and delamination, at a given position in front of the notch tip. Finally, a pulse-echo ultrasonic (PEU) scan mapped delaminated areas, as viewed from one side of the specimen. 47 4.2.1 Deply One specimen of system A , specimen A l , was deplied. It was first placed in an oven at 400°C to burn off the resin. After roughly 4 hours, sufficient resin had been removed to allow separation of the individual layers for inspection. The layers were carefully separated, and placed side by side in the same order as the laminate lay-up sequence, to recreate the fibre failure through-the-thickness. A transparency was placed on top of each ply, and an outline traced of the visible fibre failure. These outlines are drawn in Figure 4.1. 4.2.2 Sectioning and Mic rog raph Analysis One O C T specimen each of system A and B , specimens A 3 and B l respectively, were sectioned parallel to the height of the specimen, at increasing distances in front of the notch tip as shown in Figure 4.2. A slow-speed saw was used with a diamond-finished blade to minimize additional damage. The cross-section surfaces were then polished. The cross-section surfaces were examined under an optical microscope. Photographs were taken, called micrographs. A schematic of the damage patterns observed in the microscope has been drawn next to each micrograph for comparison purposes, to the same scale and orientation as that shown in Figure 4.2. The scales in the x and y direction are not the same. A s some of the micrographs are not the best quality, some detail may be drawn on the schematic which is not clearly seen in the micrograph. Selected micrographs are included at the end of the chapter, and the complete set of micrographs are presented in Appendix 2. A s system B was a sandwich panel, the honeycomb core of specimen B l was first removed on a band saw before sectioning of each laminate, labeled the front (F) and back (B) laminate in Figure 4.2. Each laminate has an outer (O) and inner (I) side, where the inner (I) side represents the side that was attached to the honeycomb core before sectioning. 48 4.2.3 Pulse-Echo Ultrasonic Scan A map of the delaminated areas in specimen A 3 is given by a pulse-echo ultrasonic (PEU) scan, Figure 4.5 (performed by Integrated Technologies, Inc. of Bothell, Washington). Darker regions indicate a cleaner signal reflection, thus large delamination surfaces are more easily picked up than, for example, small isolated pockets of fibre failure or matrix cracking which scatter the reflected signal. The P E U transducer was placed on the back side of the specimen, as the surface delamination emanating from the notch tip was smaller on that side. 4.3 Deply Results Post-test deplies confirm that fibre failure has been grown in front of the notch during the O C T test. For example, the deply of specimen A l , outlined in Figure 4.1, reveals the following: 1. Localized fibre failure exists in each ply, and the combination of these failures through all the plies creates a through-the-thickness crack in the specimen. 2. Large visible separations of the fibres indicate that matrix splitting, in which the matrix cracks parallel to the load, occurred in the -45°, 90°, and 0° plies during the O C T test. During loading, the splitting reduced the stress concentration at the notch tip, effectively blunting the notch tip. 3. The direction of fibre failure in the centre 0° plies is at -90°, perpendicular to the far-field load. (Positive angles are taken as counter-clockwise to the loading direction to be consistent with the standard orientation used to specify lay-up fibre direction.) 4. In the surface -45° plies, matrix splitting is observed starting at the notch tip. This splitting is followed by fibre failure at roughly 10 mm in front of the notch tip. Fibre failure occurs once it becomes energetically more favourable for the fibres to break than more splitting and delamination. Delamination was observed on the surface plies accompanying the matrix splitting. 49 5. The angle of fibre failure and matrix splitting in layers sandwiched between the surface -45° 's and centre 0° layers shifts from -45° to -90° with increasing proximity to the centre 0° plies. For example, as mentioned, the surface -45° plies show matrix splitting at an angle of -45°. However, in the -45° plies closer to the centre 0° plies, smaller amounts of matrix splitting occur, followed by fibre failure, giving a general direction of propagation at -75°. A second example is given by the 0° plies. Fibre failure propagates at roughly -75° in the outer 0° plies and at -90° in the centre 0° plies. The direction of fibre splitting and fibre failure in each layer, is both a function of the fibre orientation and constraint imposed by the direction of propagation in the adjoining plies. 6. Fibre failure extends to roughly 15 mm in front of the notch tip in the two centre 0° plies, and to roughly 22 mm in front of the notch tip in the remaining plies. Notch tip shielding from damage mechanisms such as the matrix splitting observed in the centre 0° plies, w i l l retard damage propagation in the centre layers. A s well , damage growth w i l l be influenced by the state of stress which varies through-the-thickness. The centre layers approach a state of plane strain, making the process zone smaller in the in-plane lamina direction than in the outer layers. The outer layers approach a state of plane stress. 7. The angle of fibre failure and fibre splitting in each layer is symmetrical about the centre 0° plies, as expected from the symmetry of the laminate lay-up. 4.4 Description of Damage in the Laminae Figure 4.3 and Figure 4.4 represent typical damage observed in specimens A 3 and B I respectively. The following three types of damage are visible: 1. Matrix cracking is observed in the off-axis plies. Slightly more matrix cracking is observed in system B than in system A . This is shown in the close-up of the micrograph of Figure 4.4. 50 2. Fibre failure is observed in the 0° plies, and in the [0790°] weave layers of system B . 3. Delamination is observed in both systems. In the cross-sections, delamination is shown as a vertical split between two adjoining plies. A map of the delamination in specimen A 3 is given by the P E U scan shown in Figure 4.5. A dark triangular region is shown between positions A ' and C , extending up to 10 mm in front of the notch tip. Most likely, this dark region corresponds to the surface delamination observed on the back surface of specimen A 3 at the same location in front of the notch tip. This delamination is grown during the small load drops prior to the peak load in the O C T test, between loads A and C in the load vs cross head displacement curve given in the figure. Between positions C and D ' in Figure 4.5, a dark band is observed with a width of 15 mm in the x direction and a height of 4 mm in the y direction. This band most likely represents the crack grown during the largest load drop (loads C and D), which contained some delamination of roughly the same height as the dark band, and extends roughly the same distance in front of the notch tip. A large triangular area appears between positions D ' and E ' in the P E U scan, which was grown between loads D and E , and corresponds to the development of the process zone. This area most likely also represents delamination, as the sectioning showed some delamination about the centre 0° plies in the process zone. 4.5 Sequence of Damage Progression Throughout the OCT Test The following results relate the damage created in each specimen to the corresponding approximate load in the O C T test. The sequence of damage progression throughout the test is recreated. 51 A s defined in the introduction, damage in the laminate consists of a crack, emanating from the notch tip, which contains through-the-thickness fibre failure, matrix cracking, and delamination. In front of the crack is a process zone, with visible damage in only some of the plies. 4.5.1 Development of Crack and Process Zone Height and Length Profiles of the final length and height (x and y directions in Figure 4.2 respectively) of the crack and process zone are given for specimen A 3 in Figure 4.6 and for specimen B I in Figure 4.11. Cross-sections at several positions in front of the notch tip are shown for specimen A 3 in Figure 4.7 to Figure 4.10, and for specimen B I in Figure 4.12 to Figure 4.15. The corresponding load vs cross head or C M O D displacement is included in the figure, to show the approximate load at which the damage was created. The points of zero displacement of the inscribed lines taken from the line analysis presented in Chapter 3 are used to approximate the length of the damage at a given load. This gives the approximate load at which the damage in the cross-section was created. 4.5.1.1 Specimen A3 A reconstructed profile of the crack and process zone in specimen A 3 is shown in Figure 4.6. Figure 4.7 to Figure 4.9 show cross-sections at positions A ' to C in Figure 4.6, and Figure 4.10 shows a typical cross-section between positions C and D ' in Figure 4.6. The damage in each cross-section was created at loads A to D in the load vs cross head displacement curve in Figure 4.6. The following observations can be made: 1. The final length of the crack is 25 mm. During the damage grown between loads A to C and prior to the big load drop, the height of the crack increases to 5 mm (cross-sections at positions A ' to C ) , at an angle, 6, of roughly 30° to the notch mid-plane. During the large 52 load drop, loads C to D , the crack height remains roughly constant at an average height of 5 mm. A typical cross-section between positions C and D ' is shown in Figure 4.10. 2. The final length of the process zone is 7 mm, and the average height is 5 mm. The process zone is grown between loads D and E , and is shown by the shaded area between positions D ' and E ' in Figure 4.6. 3. Delamination about the centre 0° plies is observed between positions C and D ' , and in the process zone. 4.5.1.2 Specimen Bl The dimensions of the crack and process zone in specimen B1 are slightly different to those of specimen A 3 . Cross-sections are shown at positions U ' to W (Figure 4.12 to Figure 4.14), as well as a typical cross-section between positions X ' and Y ' (Figure 4.15), corresponding to loads U to Y in the load vs C M O D displacement curve, Figure 4.11. The reconstructed profile of the damage in specimen B l is shown in Figure 4.11, and the following observations can be made: 1. The final length of the crack is 35 mm. During the initiation of damage between loads U to X , the crack height grows to 12 mm (cross-sections at positions U ' to X ' ) , at an angle, 0, of roughly 50°. During the large load drop from X to Y , the average crack height is 12 mm. A typical cross-section between positions X ' and Y ' is shown in Figure 4.15. 2. The process zone is 3.5 mm long, and the average height is 12 mm. 3. Delamination is observed at each stage of the test, joining matrix cracking and fibre failure typically between the -45790° and + 4 5 7 9 0 ° plies, to give a staircase-like crack profile. 4.5.2 Temporal Sequence of Damage Growth in the Process Zone The spatial development of fibre failure, matrix cracking and delamination in the process zone indicates the temporal sequence of damage growth. In both systems, it appears that the centre 0° 53 plies, and in system B the [0°/90°] weave layers as well , are the last to fail. For example, Figure 4.16 shows a cross-section of specimen A 3 in the process zone. Damage is visible in all plies but the centre 0° plies. A cross-section in the process zone of specimen B I , Figure 4.17, shows damage in all but the centre 0° ply and outer [0790°] weave layers. (Due to poor resolution in the micrographs, the schematic is more clear.) 4.6 Discussion The deply, sectioning, and P E U scan offer a consistent picture of the damage grown during the O C T test. This damage consists of a narrow band of fibre failure, matrix cracking and delamination, propagating as a crack and process zone in front of the notch. 4.6.1 No tch -Sens i t i v i t y Damage mechanisms in the process zone affect the notch-sensitivity of the laminate by redistributing the stress concentration at the notch tip. The degree of notch-sensitivity may range from being notch-sensitive, in which L E F M predicts a square root dependence of the notched fracture strength on the crack length, to notch-insensitive, in which the only effect of the notch is to reduce the net section area. A diagram showing the failure stress as a function of notch length for increasing degrees of notch-sensitivity is given in Figure 4.18. A n example of how the damage in the process zone may alter the stresses at the notch tip is given by a [±45° ] s laminate, in which damage initiates as matrix splitting, followed by decoupling of the +45° and -45° layers. This decoupling dissipates the stress concentration at the notch tip, and makes the laminate effectively notch-insensitive. Systems A and B are quasi-isotropic laminates, which according to a study by Wells and Beaumont (1987), typically show intermediate notch-sensitivity. Thus the sequence and distribution of damage in the process zone w i l l affect the fracture strength. 54 A greater emphasis is placed in this work on the role of fibre failure in the O C T specimens than by others studying damage in notched carbon fibre laminates (Kortschot and Beaumont, 1990a; Webb and Kortschot, 1991). In this work, the damage is considered to develop as a localized band of fibre failure in front of the notch, with relatively little matrix cracking and delamination. In contrast, Webb and Kortschot (1991) have described the development of a diffuse zone of matrix cracking and delamination prior to failure in quasi-isotropic double edge-notched (DEN) carbon fibre laminates. This different emphasis on fibre failure is due in part to the different investigative techniques used to examine the damage mechanisms. Webb and Kortschot used the absorption of a radiographic dye to reveal the damage prior to failure. However, detailed cross-sectioning was used on the O C T specimens. The sectioning and deply results suggest that the dominant damage mechanism in the development of the O C T process zones is fibre failure. The geometry of the specimen allows a gradual development of the process zone, from which the sections can be cut to reveal the gradual onset of fibre failure., It is not clear whether the D E N specimen used elsewhere is large enough to allow this gradual fibre failure without catastrophic failure of the specimen. The weave in system B may increase notched strength by resisting surface delamination. Lagace (1986b) noted that surface delamination greatly decreased notched strength in a quasi-isotropic carbon fibre reinforced laminate. The laminates failed by fibre splitting followed by delamination of the outer +45° and -45° plies. However, when the +45° and -45° plies were replaced by a [±45°] fabric, and the centre 0° and 90° plies by a [0790°] weave, no delamination occurred and the strength was comparatively higher. 55 A greater ability to carry and redistribute load during failure may exist in system B due to the greater height and staircase pattern observed in the sectioning. Damage in system B initiates as fibre failure and matrix cracking in the off-axis plies at a greater height difference between adjoining plies than in system A . A t a given height, therefore, a greater amount of undamaged material exists through the thickness to sustain load. A greater number of load path combinations are therefore available in which to redistribute the load, and dissipate the notch tip stress concentration. 4.6.2 Strain Softening Curve The post-peak region of the strain softening curve accounts for how damage evolution in the process zone affects the notch-sensitivity of the material. The physical description of the damage given in this chapter provides useful information with which to determine the strain softening curve of the O C T specimens. In the F E M model, the element size must be sufficiently large to include the effects of strain softening within each element. The steady-state height and final length of the process zone in systems A and B might define the minimum element size needed to model the O C T specimen behaviour. A qualitative description of the damage in the process zone has been given in this chapter, which can be used to calibrate the F E M mesh and explain differences in the post-peak shape of the strain softening curve for each system. These results, in conjunction with the load vs surface displacement curves of Chapter 3, may be used to calibrate a F E M model of the O C T specimens. However, the actual shape of the strain softening curve is unknown. Therefore, a preliminary shape with which to begin modeling the O C T specimens would be useful, as well as any additional knowledge of the evolution of damage in the process zones. This is presented in Chapter 5. 56 4.7 Summary of Results The following lists the results presented in this chapter: 1. The deply, sectioning and P E U scan indicate that damage exists as a narrow band of fibre failure, surrounded by a small amount of matrix cracking and delamination. 2. This damage grows as a crack, containing damage in all the plies, in front of which is a process zone, containing damage in only some of the plies. 3. The final length of the crack in specimen A 3 is 25 mm, and the average height is 5 mm, while the final crack length in specimen B I is 35 mm, and the height is 12 mm. 4. The final length of the process zone in specimen B I is half that of specimen A 3 (3.5 vs 7 mm respectively), while the height in specimen B I is twice that of specimen A 3 (12 vs 5 mm respectively). We may assume that as the height has reached a steady-state value, it represents the fully-developed process zone height for different a/W of the O C T specimens. 5. In the process zone, the spatial sequence of fibre failure and matrix cracking mirrors the temporal sequence of damage growth. The sectioning shows that the centre 0° plies in both systems and the [0790°] weave in system B , are the last to fail. 6. The physical description of the damage provides quantitative information with which to calibrate the strain softening curve for each system. For example, the final steady-state height of the process zone can be used in conjunction with a final length, to determine the minimum dimensions of the elements needed to include the effects of strain softening. 7. This physical understanding of the damage also provides qualitative information with which to generate a physical understanding of the strain softening curve for each system. For example, the larger height and staircase-like profile of the damage in system B may indicate a 57 greater ability to redistribute load during failure than in system A . This might influence the shape of the strain softening curve of system B . 58 Front Side Figure 4.1 Deply schematic of specimen A l . The fibre direction in each layer is drawn in the top left or right corner of each ply. Positive angles are taken counter-clockwise to the 0° fibre direction to be consistent with the standard orientation used to specify lay-up fibre direction. 59 x = 18.00 mm Layup: [-45/90/45/0/-45/90/45/0]s delamination Top Top Back CH. =4.20 mm units: mm CH. = crack height P.Z.H. = process zone height N.M-P.: notch mid-plane F = front side of laminate B = back side of laminate Top = top side of laminate 0 = 0° ply O = outer side £4 N.M-P.H B - 10 - 9 fibre - 7 "failure - 6 - 5 ,, matrix ^ 3 cracking - 2 - 1 - 0 - -1 - -2 - -3 - -4 - -5 - -6 - -7 - -8 - -9 L-10 3.35 Figure 4.2 Example of a micrograph schematic, showing matrix cracking, fibre failure in the 0° plies and surface delamination. This shows the orientation to which all the micrographs have been redrawn for comparison purposes. The scales in the x and y direction are not the same. This example is taken at 18 mm in front of the notch tip in specimen A3. 60 3.35 Figure 4.3 Typical cross-section of specimen A3, taken at 20.25 mm in front of the notch tip. This shows delamination of the surface ply. The layup is[-45/90/45/0/-45/90/45/0]s . x = 10 Front C H . = units: n Figure 4.4 Typical cross-section of specimen B l , taken at 10.00 mm in front of the notch tip. The layup is [F0/-45/90/45/0/45/90/-45/F0]T , F0 = [0/90] weave. 61 Back 100 mm notch tip 7000-6000-z 5000-"D 4000-CO O _ J 3000-2000-1000-0 -orientation of P E U scan T ° P Fron Back o 3 o f x 3 o -0 0.5 1 1.5 2 2.5 Cross Head Displacement (mm) Figure 4.5 Pulse-echo ultrasonic (PEU) scan of specimen A3. Darker regions correspond to a cleaner reflected signal, and most likely represent delamination grown between loads A to E (positions A ' - E ' on the PEU scan). 62 to scale: angle, 8 N. M-P. 7000 6000 j 5000 ^ 4 0 0 0 5 3000-1 k ~ 1 25 mm w = 0 2 8 7 mm 5 mm 0 0.5 1 1.5 2 2.5 Cross Head Displacement (mm) Figure 4.6 Reconstructed profile of the damage in specimen A3, with the load vs cross head displacement curve from the OCT test presented in Chapter 3. x = 1. C H . = units: i 3.35 Figure 4.7 Cross-section at position A' in Figure 4.6, taken at 1.85 mm in front of the notch tip in specimen A3. The layup is [-45/90/45/0/-45/90/45/0]s . 63 x = 11.45 mm C H . = 6.1 mr units: mm 3.35 Figure 4.9 Cross-section at position C in Figure 4.6, taken at 11.45 mm in front of the notch tip in specimen A3. The layup is [-45/90/45/0/-45/90/45/0]s . 64 x = 18.00 mm Top C H . =4.20 mm units: mm N.M-P. B -10 - 9 " 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 - 0 - -1 - -2 - -3 - -4 - -5 - -6 - -7 - -8 - -9 -10 F T o P B Figure 4.10 Typical cross-section between positions C and D' in Figure 4.6. This micrograph is taken at 18.00 mm in front of the notch tip in specimen A3. The layup is [-45/90/45/0/-45/90/45/0]s. to scale: W V 35 mm — = 0 22 W -angle, 6 3.5 mm 12000 10000 8000 £ 6000f o 0.5 1 1.5 2 2.5 CMOD Displacement (mm) Figure 4.11 Reconstructed profile of the damage in specimen Bl , with the load vs CMOD displacement curve from the OCT test presented in Chapter 3. 65 x = 1.0 mm Front Side Top O Top I O C H . = 1 mm units: mm N.M-P — H 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 - -3 - -4 - -5 - -6 - -7 - -8 - -9 -10 -11 -12 -13 14 1.66 Figure 4.12 Cross-section at position U' in Figure 4.11, taken at 1.00 mm in front of the notch tip in specimen BI. The layup is [FO/-45/90/45/0/45/90/-45/FO]T , F0 = [0/90] weave. x = 2.0 mm Front Side Top O C H . = 3 mm units: mm N.M-P—H 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 O Top I 1.66 Figure 4.13 Cross-section at position V in Figure 4.11, taken at 2.00 mm in front of the notch tip in specimen BI. The layup is [F0/-45/90/45/0/45/90/-45/F0]T, F0 = [0/90] weave. 66 1.66 Figure 4.14 Cross-section at position W in Figure 4.11, taken at 5.65 mm in front of the notch tip in specimen Bl . The layup is [F0/-45/90/45/0/45/90/-45/F0]T , FO = [0/90] weave. Figure 4.15 Typical cross-section between positions X' and Y' in Figure 4.11. This micrograph is taken at 12.00 mm in front of the notch tip in specimen Bl . The layup is [F0/-45/90/45/0/45/90/-45/F0]T , FO = [0/90] weave. 67 P.Z.H. = units: mr Figure 4.16 Cross-section in the process zone of specimen A3, taken at 28.00 mm in front of the notch tip. This micrograph shows damage in all but the centre 0° plies. The layup is [-45/90/45/0/-45/90/45/0]s. Figure 4.17 Cross-section in the process zone of specimen BI, taken at 35.50 mm in front of the notch tip . This micrograph shows damage in all but the centre 0° ply and surface 0790° weave plies. The layup is [F0/-45/90/45/0/45/90/-45/F0]T , F0 = [0/90] weave. 68 r j f = failure stress G y = yield stress in a metal, or the stress at which damage initiates in a fibre reinforced composite Figure 4.18 Diagram of the failure stress vs the notch length for increasing degrees of notch-sensitivity. 69 Chapter Five Process Zone Tensile Tests 5.1 Introduction This chapter examines the effect of damage in tensile specimens which have been cut from the process zone in systems A and B . A simple equation is derived to extract the modulus of the damaged section of each tensile specimen from the measured modulus. Laminated Plate Theory is used to reconstruct possible combinations of the load-bearing plies according to the damaged moduli. The change in load-bearing combinations indicates the sequence of damage growth in the process zone. Finally, the damaged moduli are used in conjunction with the measured failure stress to back-calculate several points in the post-peak region of the strain softening curve. A preliminary shape of the strain softening curve for systems A and B is presented. 5.2 Background A preliminary strain softening curve is presented in this chapter, which is back-calculated from the response of tensile specimens of systems A and B . For reasons discussed in Chapter 1, it is preferable to represent the strain softening curve as the load-displacement response of a given material system. However, for ease of implementation into a Finite Element Method ( F E M ) model, it is numerically convenient to represent the response in terms of stress and strain. These material responses are numerically equivalent for a constant element size. The preliminary results of an F E M analysis of the O C T specimen are discussed in Chapter 6. Tensile specimens cut from the process zone provide a quantitative measure of the strain softening curve. In the O C T specimen, the material in the fully developed process zone is 70 subject to a high stress concentration, and large local displacements result from the damage presented in Chapter 4. A s wi l l be shown in this chapter, the displacements in the process zone translate into strains which exceed the laminate failure strain. The growth of the zone is represented by the post-peak region of the strain softening curve, shown between points A and C in Figure 5.1. If the material in the process zone is loaded to point B in Figure 5.1, for example, it w i l l follow the path O A B . During unloading, i f strain softening is caused by damage, the material w i l l unload along the path B O . If this material is then removed, and made into a tensile specimen, it w i l l reload along the path O B and wi l l fail at point B . This gives the location of point B on the strain softening curve, which was previously unknown. This technique provides an experimental back-calculation of points on the post-peak region of the strain softening curve, which cannot be directly measured from the O C T specimen. Many tensile specimens cut at different locations in the process zone should thus give the overall shape of the curve. It should be noted that during unloading from point B , a strain softening material model predicts a different unloading path than does an elastic-plastic model. The elastic-plastic model predicts that material initially at point B unloads along the path B O ' , with the same modulus as the undamaged modulus, EUD. However, the tensile tests presented in this chapter indicate that the material would unload along the path B O , with a new damaged modulus ED. 5.3 Experimental Technique A process zone similar to that presented in Chapter 4, was reproduced in one O C T specimen of each system (specimens A 4 and B2). The zones were sectioned perpendicular to the notch mid-plane into thin strips roughly 2 mm wide. A diamond-finished blade slow-speed saw was used to minimize additional damage. Figure 5.2 shows the approximate location of the tensile 71 specimens, and Table 5.1 lists their dimensions. The specimens are numbered A4T1 to A 4 T 4 , and B2T1 to B2T3, where a higher number represents increasing distance in front of the notch tip. The sandwich core of specimen B2 was first removed before cutting strips out of the front and back laminates. The damage before and after the tensile test was recorded in a schematic, and as the specimens were very delicate, only a few faces were polished and micrographs taken (Figure 5.5 to Figure 5.9). The orientation of the micrographs is the same as that used for the sectioning results of Chapter 4 (Figure 4.2). However, the position relative to the notch mid-plane is not included as the micrographs were taken primarily to show the type of damage in the plies. A scale is included to indicate the approximate damage height. The ends of the specimens were first embedded in hollow cylindrical steel tabs and bonded with an epoxy, as shown in Figure 5.3, before mounting them in the Instron grips. A special j ig was built to align the end tabs parallel to the specimen, as shown in the figure, to minimize eccentric loads during testing. To glue the tabs onto the specimen, each tab was temporarily taped to the specimen, enclosed in the j ig placed upright, and the glue poured into the larger end of the hollow cylinder. The epoxy was then cured for 16 hours at room temperature. This procedure was repeated with the other tab, and the specimen was placed in an oven at 100° C for three hours to complete the cure cycle. A clip gauge was placed on one side of the specimen to measure the displacement during loading, as shown in Figure 5.3. In order to ensure that the gauge was within a region of uniform loading, each arm of the gauge was placed at least one specimen thickness away from the application of the load. The tensile specimens were then statically tension loaded in an Instron testing machine until failure. 72 5.4 Derivation of the Damaged Moduli The moduli of the damaged sections of the tensile specimens were back-calculated from the measured moduli. A s shown in Figure 5.4, the tensile specimen is simplified to contain a uniform region of damaged material with modulus ED, surrounded by a region of undamaged material with modulus EUD. If the tensile specimen is subject to- an applied stress, a, then by compatibility: where 5M is the total displacement measured by the clip gauge, 5 D is the displacement of the damaged region, and 5 ^ is the displacement of the undamaged region. Substituting Hooke's law, Eq . (5.1) becomes: ° " M L < * D H D + E D CTuD(L-hD) "UD (5.2) 1 " "V" " ( L - h D ) / " = / L + / L •^UD where hD is the height of the damaged section, and L is the clip gauge length, as shown in Figure 5.4. A s oM = oD = oUD, Eq . (5.2) is simplified to: (5.3) The damaged modulus, ED, can then be back-calculated using Eq . (5.3), where hD, is determined by visible inspection. The undamaged modulus, EUD, is the measured undamaged tensile specimen modulus. 5.5 Experimental Results 5.5.1 Damage Observed Prior to Tensile Tests Slightly more damage is visible with increasing proximity to the notch tip, in the tensile specimens taken from O C T specimen A 4 , given in Figure 5.5 to Figure 5.7. The face farthest 73 from the notch tip of A4T1 (Figure 5.6) is similar to the damage observed in A 4 T 2 , for which no micrograph was obtained. The schematics show that: 1. In all specimens, matrix cracking is observed in all the +45°, -45° and 90° plies. 2. None of the specimens show fibre failure in the centre 0° plies. Fibre failure in the outer 0° plies is visible in specimens A4T1 (specimen closest to the notch tip) and A 4 T 2 , but not in specimen A 4 T 3 . 3. Significant delamination about the centre 0° plies is apparent in A4T1 and A 4 T 2 , and only a small amount of delamination to one side of the centre 0° plies is evident in A 4 T 3 . The following damage is visible in the tensile specimens of specimen B 2 , given in Figure 5.8 and Figure 5.9: 1. Specimen B2T1 (taken from the front laminate) appears to be the most damaged, as it has damage in all the plies. This damage was observed in the face closest to the notch tip, and the specimen was still able to carry load. 2. Specimen B2T2 (taken from the back laminate) appears to have damage in all but the centre 0° ply. 5.5.2 Measured Moduli The measured stress vs strain curves of the tensile specimens for system A and B are presented in Figure 5.10 and Figure 5.11, respectively. The measured moduli are listed in Table 5.2. From the graphs, the following conclusions can be made: 1. The measured moduli of specimens A4T1 to A4T3 decrease with increasing proximity to the notch tip. This is consistent with the observed damage, where more damage is visible in specimens nearer the notch tip. A s only one damaged specimen was retrieved from each laminate's process zone in specimen B2 , a similar conclusion cannot be made. However, 74 B 2 T 1 , which had greater visible damage prior to testing, had a lower measured modulus than B2T2, as shown in Figure 5.11. 2. In most specimens, the stress increases linearly with the strain until failure. However, small drops in stress are observed which significantly lower the modulus in specimen A 4 T 2 . Two significant drops in the measured modulus occur, Figure 5.10, where EM drops from 16 GPa between points A and B to 6 GPa between points B and C. During loading, popping noises were heard and delamination observed during the load drops, indicating damage growth. For analysis, the two different moduli are represented as two separate specimens. A4T2a represents the behaviour of specimen A 4 T 2 between points A and B and A4T2b the behaviour between points B and C. 3. In specimens A4T1 and A 4 T 2 , the failure strain exceeds the undamaged failure strain. For example, the failure strain of specimen A 4 T 2 is 2.45% while the undamaged failure strain is 1.1%. Such large strains must be made possible by the damage in the specimen. A measure of this damage is given by the damaged moduli, which are presented in the following section. 5.5.3 Damaged Moduli Various combinations of the undamaged plies were reconstructed such that the modulus, determined by Laminated Plate Theory (LPT) , was similar to the calculated damaged modulus of Eq. (5.3). Coupled with the visible damage prior to and post-testing, this provided a measure of the effective load-bearing plies remaining in each specimen. The damage height, hD, was estimated from the visible damage prior to and post-testing. In the specimens where the damage height increased during the test, an average of the initial and final damage height was used for hD. This yields a higher damaged modulus in Eq . (5.3) than using the initial height only, and therefore may underpredict the damage in the specimen. However, 75 the average height more accurately represents the effective damage in the specimen, as the visible damage prior to testing may not show all the existing damage. A graph of the failure stress and strain of each specimen based on the damaged moduli is shown in Figure 5.12 and Figure 5.13 for specimens A 4 and B2 respectively. The failure stress, oy is the measured specimen failure stress, and the failure strain, Sy, is back-calculated from the damaged moduli, ED and oy according to: (5.4) 5.5.3.1 Specimen A4 A s indicated in Figure 5.12, the following are the most likely combinations of effective load-bearing plies in each tensile specimen: 1. The specimen farthest from the notch tip, specimen A 4 T 3 , has roughly 2 0° plies (out of 4) which effectively bear load. This combination would give a laminate with an approximate modulus of 14 GPa, which is in the range of the damaged A4T3 modulus of 13 GPa. This is the only load-bearing combination which is consistent with the damaged modulus and the damage observed prior to testing. (The micrograph showed damage in all +45°, -45° and 90° plies, with only some fibre splitting in the 0° plies, before testing, Figure 5.7.) 2. The damaged modulus of specimen A4T2a (10 GPa), suggests that only between 1 and 2 of the 0° plies are load-bearing. This combination would give a laminate that has a modulus between 7 and 14 GPa respectively. Either of these load-bearing combinations are consistent with the damage observed prior to testing, in which damage was visible in all but the centre 0° plies. 3. The damaged moduli of both specimens A4T2b and A4T1 effectively represent less than one 0° load-bearing ply. The damaged moduli are 6 and 4 GPa respectively, which represent a 76 laminate containing only 85% and 57% of one 0° ply, respectively. The damaged moduli indicate that slightly more damage exists than that observed prior to testing. A t that time, no fibre failure was observed in either of the two centre 0° plies in either specimen, Figure 5.5 and Figure 5.6. Thus the damaged moduli indicate that the development of the process zone in specimen A 4 represents the progressive failure of the 0° plies. The number of intact 0° plies predicted by L P T represent an effective load-bearing ability of the 0° plies in the specimen. For example, in specimen A4T2b, 85% of the fibres in one 0° ply cannot be completely intact, as the strain in the fibres would exceed the fibre failure strain by a factor of roughly 2.5. The 0° fibres must therefore contain damage, but the load is redistributed through the undamaged portion of the fibre by the adjoining layers. Furthermore, one effective load-bearing 0° ply may actually represent the load-bearing ability of several 0° fibres dispersed throughout the specimen. 5.5.3.2 Specimen B2 The following represents the most likely combination of effective load-bearing plies in the B 2 tensile specimens, Figure 5.13, prior to testing: 1. The damaged modulus of B2T2 (37 GPa), indicates that the centre 0° ply is undamaged, and the [0790°] weave layers and +45° and -45° layers are at least partially intact. This indicates a greater load-bearing ability than that expected based on the visible damage prior to testing. The micrograph shows damage in all but the centre 0° ply, Figure 5.9. However, i f only the centre 0° ply carried load, L P T underpredicts the damaged modulus by 55%. 2. The visible damage prior to testing and the low damage modulus of specimen B2T1 (5 GPa), indicate that roughly only 30% of the 0° fibres of one 0° ply bear load. Prior to testing, 77 damage is visible in all plies, Figure 5.8, and the damage length did not change during testing. The most likely failure sequence of system B is failure of the off-axis plies, followed by the [ 0 7 9 0 ° ] weave layers and finally, the centre 0 ° ply. The visible damage in the process zone sectioning of specimen B I , presented in Chapter 4 (Figure 4.17), shows that the off-axis layers fail first. The centre 0 ° ply is the last to fail, as it is the only visibly undamaged ply before testing in tensile specimen B2T2. However, the damaged modulus of B2T2 indicates that, although damage may exist in all but the centre 0 ° ply, the damaged plies are still able to carry and redistribute a significant amount of load. 5.5.3.3 Preliminary Strain Softening Curves for Systems A and B The damaged moduli give a preliminary shape for the strain softening curve for specimens A 4 and B 2 , which is shown in Figure 5.14. The failure stress and strain are plotted for those specimens which did not fail prematurely. Although these curves are derived from only one notched specimen, they provide a general preliminary shape for the F E M softening curve for systems A and B . They represent the strain softening behaviour of the steady-state process zone in an O C T specimen. 5.6 Discussion The tensile specimens offer a quantitative means of explaining the damage evolution in the process zone. Crack growth is dictated by the failure of the 0 ° plies, as these are the primary load-bearing plies. In specimen A , L P T predicts that the growth of the process zone primarily represents progressive failure of the 0 ° plies. This would explain the large drops in the measured moduli of the tensile specimens with increasing proximity to the notch tip. Although the 0 ° fibres must contain damage to exceed a nominal strain greater than the true fibre failure strain, 78 load redistribution must occur such that the fibres are still able to carry a significant amount of load. The spread of damage in the process zone is unique to the O C T specimen, in that failure of the 0° plies occurs gradually, giving a gradual loss in load-bearing ability. In contrast, a dog-bone style specimen fails instantaneously once the 0° fibres reach the fibre failure strain. A notched specimen is thus needed to grow the type of damage represented by the post-peak region of the strain softening curve. The tensile specimens provide a preliminary shape of the strain softening curve to input into an F E M model. However, several difficulties are apparent when trying to determine the strain softening curve with this technique. For example: 1 . The technique is labor intensive. Many specimens must be tested to obtain an average value for each point on the strain softening curve. Detailed knowledge of the damage grown prior to and during the test is required. 2. Many sources of error exist. For example, the small specimen size increases the risk of damage being introduced during specimen preparation. A s well , surface imperfections and other stress risers increase the risk of premature specimen failure. Finally, care must be taken to properly align the specimens with the load to minimize eccentric loading. 3. The failure stress is based on the cross-sectional area, yet as damage progresses during failure, the actual load-bearing area decreases. 4. The properties of the damaged material must be extracted from the tensile specimens. The specimen must be simplified analytically, and the damage height, h D , must be known. Care must be taken when determining a preliminary shape for the curve from the tensile specimens, as the shape wi l l be affected by the dimensions of the process zone. The process 79 zone of system A is twice the width of system B , and consequently more specimens, may be retrieved to plot points on the strain softening curve. Unless sufficient specimens of system B are tested, the shape of the curve may be underdeveloped, and the contrast in shape between the two systems not noticed. Regardless of the exact shape of the post-peak region of the curve, the tensile specimens suggest that a gradual post-peak loss in stiffness is needed to model failure of the O C T specimen. F E M analyses of notched composite laminate failure traditionally model an instantaneous failure response. This assumes that once failure initiates in a lamina the stiffness and stress of the ply drop to zero. However, the tensile specimens are proof that once damage initiates, a loss in stiffness only gradually occurs. 5.7 Summary of Results The following results were presented in this chapter: 1. Tensile specimens were cut from the process zones of an O C T specimen of system A and B . 2. A simple equation was derived to back-calculate the modulus of the damaged section of the tensile specimens from the measured modulus of each tensile specimen. 3. Large drops in the measured moduli occur with increasing proximity to the notch tip. 4. The large drops in measured moduli are explained by the sequence of damage growth in the process zone. This sequence was determined using Laminated Plate Theory in accordance with the damaged moduli. In both specimens, the off-axis plies fail first followed by failure of the 0° plies. In system A , the development of the process zone represents the progressive failure of the 0° plies. The number of intact 0° plies decreases from 2 plies (out of 4) in the tensile specimen farthest from the notch tip, to less than one in the specimen closest to the 80 notch tip. In system B , the off-axis plies fail first, followed by the [0790°] plies, and lastly the centre 0° ply. 5. Preliminary shapes for the strain softening curve of systems A and B were back-calculated from the tensile specimens, based on the damaged moduli. 6. The type of damage grown in the O C T specimen is different from that grown in a dog-bone style specimen. Systems A and B are 0° dominated systems, and failure of the 0° plies occurs gradually as the load is redistributed. Strains on the order of 2.5 times the fibre failure strain have been measured. In contrast, a dog-bone style specimen would fail instantaneously once the 0° plies reach the fibre failure strain. 7. The tensile specimens provide physical proof of why a F E M instantaneous failure material response underpredicts the strength of a notched laminate. This response assumes that the lamina stiffness and stress drop to zero once failure initiates. However, the growth of damage in the process zone leads to a gradual loss in load-bearing ability. A strain softening material response models this loss in the post-peak section of the curve. 81 Table 5.1 Tensile specimen dimensions. Specimen Width* Clip Gauge Length Damage Length Before Damage Length After (mm) (mm) Testing (mm) Testing (mm) Specimen A4 Undamaged A4T4 1.39 35 - -Damaged A4T3 1.43 29.2 7.5 22 A4T2 1.24 29.5 15 29.5 A4T1 2.70 40 18.5 18.5 Specimen B2 Undamaged B2T3 (back laminate) 1.65 29.7 - -Damaged B2T2 (back laminate) 1.66 30 10.8 28.4 B2T1 (front laminate) 1.95 30.2 3.7 3.7 * Indicates width at the mid-height of the tensile specimen. **Represents L, used in Eq. (5.3) and drawn in Figure 5.4. Note: Specimen thickness for system A is 3.35 mm, and for system B is 1.66 mm for each front and back laminate. Table 5.2 Measured and derived tensile specimen mechanical properties. Measured Failure Stress (MPa) Measured Failure Strain (%) Measured Modulus (GPa) Calculated Damaged Modulus T (GPa) Damage Height, hD\ Used in Eq. (5.3) (mm) Specimen A4 Undamaged A4T4 399 1.10 36 - -Damaged A4T3 132 0.60 19 13 14.8"" A4T2a 140 1.10 16 10 15 A4T2b 170 2.45 6 6 29.5 A4T1 98 1.18 8 4 18.5 Specimen B2 Undamaged B2T3 (back laminate) 467 1.16 40 - -Damaged B2T2 (back laminate) 408 1.08 38 37 19.5 B2T1 (front laminate) 125 .62 22 5 3.7 Y Calculated damaged modulus, ED, derived from Eq. (5.3), and drawn in Figure 5.4. * Damage height, hD, used in Eq. (5.3) and drawn in Figure 5.4. **Based on the average of the damage height observed prior to and post testing (listed in Table 5.1). 82 E U D = undamaged modulus E D = damaged modulus Strain (%) Figure 5.1 Diagram of a strain softening curve back-calculated from process zone tensile specimens. Specimen A4 Specimen B2 Front Laminate Back Laminate o o C M A4T1-3 W o o C M O O C M B2T2-3 \ O O < 100 100 100 units: mm Process zone positions, measured as distances in front of the notch tip: • specimen A4: 34.5 to 43.85 mm • specimen B2, front laminate: 42.56 to 45.36 mm • specimen B2, back laminate: 41.09 to 45.98 mm Figure 5.2 Schematic of tensile specimens cut from the process zones of OCT specimens A4 and B2. 83 cylindrical hollow tab V tensile specimen 2 mm epoxy E E oo CO ±T4 50-65 mm 82 mm applied displacement, 5 t .damaged material clip gauge J ^ U \ applied displacement, 5 Figure 5.3 Jig to align end tabs parallel to tensile specimens while gluing, and the experimental setup to measure the specimen displacement during loading. 1 CTf -UD -UD a f is the failure stress • E U D is the undamaged specimen modulus • E D is the damaged specimen modulus • L is the gauge length • h D is the damage height Figure 5.4 Simplification of the tensile specimen used to determine the damaged modulus, E D , using Eq. (5.3). 84 Face closest to notch tip: : Top A4T1 x = 34.5 mm P.Z.H. = 3.06 mm units: mm B 3.35 before tensile test after tensile test Figure 5.5 Schematic and micrographs of tensile specimen A4T1. These show the face closest to the notch tip. Layup is [-45/90/45/0/-45/90/45/0]s. Face farthest from notch tip: Top A4T1 x = 37.2 mm P.Z.H. = 3.60 mm units: mm 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 B Top F 3.35 Figure 5.6 Schematic and micrographs of tensile specimen A4T1 before tensile test. This shows the face farthest from the notch tip, and is similar in damage to the face closest to the notch tip of A4T2. Layup is [-45/90/45/0/-45/90/45/0]s. 85 Face farthest from the notch tip: A4T3 x = 41.57 mm P.Z.H. =4 mm units: mm Figure 5.7 Schematic and micrographs of tensile specimen A4T3. This shows the face farthest from the notch tip. Layup is [-45/90/45/0/-45/90/45/0]s . 86 Face closest to notch tip: Figure 5.8 Schematic and micrograph of tensile specimen B2T1 before tensile test. This shows the face closest to the notch tip. Layup is [F0/-45/90/45/0/45/90/-45/F0]x where FO = [0/90] weave. 87 Face closest to notch tip: Top B2T2 Back Laminate x = 41.09 O P.Z.H. = 10.76 units: mm 2 8 ' 2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 1 1 0 9 8 7 6 5 4 3 2 1 0 1.66 close up before tensile test Top O 1.66 after tensile test Figure 5.9 Schematic and micrograph of tensile specimen B2T2. These show the face closest to the notch tip. Layup is [F0/-45/90/45/0/45/90/-45/F0]T where FO = [0/90] weave. 400 ro CL C/5 W CU co ^ / M T T " " A4T2 f A4T3 — C ^ — A4T1 i _i i 0.5 1 1.5 Strain (%) 2.5 Figure 5.10 Measured stress vs strain curves of A4 tensile specimens. 88 Figure 5.12 Plots of failure stress and strain of A4 tensile specimens, derived using the damaged moduli. The most likely combination of effective load-bearing plies is indicated for each specimen. 89 Figure 5.13 Plots of failure stress and strain of B2 tensile specimens, derived using the damaged moduli. The most likely combination of effective load-bearing plies is indicated for each specimen. Figure 5.14 Preliminary strain softening curve of systems A and B (shown by the solid lines), based on the damaged moduli of specimens A4 and B2. The arrows outline the possible range of the unloading portion of the curve. 90 Chapter Six Comparison of the Experimental Results with a Preliminary FEM Analysis 6.1 Introduction The experimental results are compared with an F E M analysis of the O C T specimen. The experimental load vs C M O D displacement of specimen A 4 is compared to that predicted by an F E M analysis by Engels (1996). In addition, the experimentally determined crack growth in front of the notch tip for specimen A 3 is compared to that predicted by a strain softening material model. 6.2 Comparison of the Experimental and FEM Results The experimental results presented in the previous chapters have been used by Engels (1996) to investigate whether a strain softening material response can predict the damage progression observed in the O C T specimen. Based on the experimental work, an F E M analysis of the O C T specimen was performed by Engels (1996). He used the ABAQUS™/Standard Code, in which he modeled a strain softening material response using an anisotropic piece-wise linear elasto-plastic material model. Engels calibrated the strain softening curves for O C T specimens A 3 and A 4 , using the experimental load vs C M O D displacement and load vs clip gauge displacement curves presented in Chapter 3. He assumed a linear post-peak unload in the strain softening curves, as shown in Figures 6.1 and 6.2. 91 The F E M results showed that of the three different material responses considered in the study, a strain softening response was necessary to model the gradual progression of damage observed in the O C T specimens. He compared the load vs C M O D displacement behaviour of specimen A 4 predicted by a plasticity based strain softening response, an elastic-instantaneous failure response, and an elastic-perfectly plastic response. These curves are reproduced in Figure 6.1, as well as the experimental results for the specimen. The elastic-instantaneous failure response resulted in severe mesh distortions at the notch tip, raising doubts as to the validity of the solution. The elastic-perfectly plastic material response did not capture the load drops that occur once failure initiates. However, the strain softening material model predicts drops in load similar to the experimental results. During the largest experimentally measured load drop (between points A and B in Figure 6.1), which corresponded to roughly 15 mm of instantaneous crack growth, the F E M model did not converge to a solution. This numerical instability is analogous to the unstable crack growth observed experimentally. A difference exists between the plasticity based strain softening model used by Engels and a C D M based strain softening model, such as the M L T model (Matzenmiller et al., 1991). The F E M strain softening curve used by Engels implies that during unloading the modulus of increasingly damaged material in the process zone does not change from the undamaged modulus. In the M L T model, however, the modulus of material unloading in the post-peak section of the curve decreases. The M L T model corresponds more closely to the tensile test results of Chapter 5, in which the modulus of the material in the process zone decreased with increasing proximity to the notch tip. (i.e. increasing damage) Despite this difference, the strain softening material model used by Engels provides a reasonable description of the damage growth across the specimen width. The crack length at different loads 92 throughout the test was estimated by Engels by plotting the plastic strain in the elements. Figure 6.2 shows the predicted and experimental load vs clip gauge displacement of specimen A 3 . A predicted plastic strain greater than 3.15% translates experimentally into the presence of damage (i.e. the crack). The F E M results predicted that no crack growth occured prior to point A ' in Figure 6.2. This was followed by 12.5 mm of crack growth just prior to the largest load drop at point B ' . A total of 30 mm of crack growth had occured just after the largest load drop, roughly at point C . This agrees well with the experimental results, which indicated no crack growth prior to point A , 10 mm of crack growth at point B and 25 mm of crack growth (with a process zone 7 mm long) at point C. 93 FEM strain softening curve: 390 — A "co / \ / \ / \ s (%) 4 T3 ro o 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 F E M elastic - perfectly plastic F E M strain softening ' • A experimental \ F E M elastic -instantaneous failure 'B 0 0.5 1 1.5 2 2.5 CMOD Displacement (mm) 3.5 Figure 6.1 Load vs CMOD displacement curve for OCT specimen A4 predicted by Engels (1996) using several different F E M constitutive model responses. These curves can be compared to the experimental load vs CMOD displacement curve. The optimized strain softening curve is included in the figure. The F E M results are reproduced from Engels (1996). 94 7000 6000+ 5000+ 4000+ T3 § 3000+ 200OH 10004 0.2 0.4 0.6 0.8 1 1.2 Clip Gauge Displacement (mm) FEM strain softening experimental 1.4 Figure 6.2 Load vs clip gauge displacement of OCT specimen A3, predicted by a F E M strain softening analysis and the experimental results. Points A' to C and A to C are used to compare the predicted and experimentally estimated crack lengths at the given load, respectively. The optimized strain softening material response is included in the figure. The F E M results are reproduced from Engels (1996). 95 Chapter Seven Conclusions and Further Work 7.1 Introduction Conclusions are drawn from the results presented in Chapters One to Six. Further work is suggested. 7.2 Conclusions The following conclusions can be drawn from the results of the previous chapters: 1. A specimen and test methodology have successfully been developed which can be used to study notched composite laminate behaviour. The specimen is an overheight compact tension (OCT) specimen which has the following advantages: • it exhibits stable damage growth, allowing a post-test investigation of the damage evolution. Information can be gathered concerning how the damage develops both spatially and temporally. • it provides several different types of results with which to calibrate the strain softening curve of a given material system. • it does not require a large amount of material. 2. A comprehensive picture of the damage evolution in the O C T specimen has been described in two material systems. A good understanding of the crack length at increasing loads 96 throughout the test has been given, as well as a picture of the sequence "of damage evolution in the process zone. 3. The material in the process zone softens. The damage differs from that developed in a dog-bone style tensile specimen, in that: • it has a gradual decrease in load-bearing ability. • the failure strain of the damaged material can exceed the fibre failure strain by as much as two times. • the dominant sequence of damage growth consists of progressive failure of the 0° plies. 4. A strain softening material response in an F E M analysis can be used to model failure of the O C T specimen. Preliminary results indicate that a strain softening curve, calibrated from the experimental load vs displacement results, captures the progressive growth of damage across the specimen width. 7.3 Further Work The experimental results indicate that despite substantial damage formed in the process zone of the O C T specimen, the damaged material is still able to carry a significant amount of load. With increasing amounts of damage, this material has a progressive decrease in load-bearing ability. Physically, therefore, the concept of a strain softening curve is sound. A clearer understanding of the predictive capabilities and fundamental validity of the strain softening curve must now be developed. 97 A useful feature of the O C T specimen is that it provides detailed information with which to calibrate the strain softening curve of a given material system. For example, after assuming a shape for the curve, the curve can then be calibrated using the far-field load and C M O D displacement of the specimen. The influence of modifications to the shape, such as altering the slope of the tip and tail region of the curve, can then be examined by comparing the predicted damage growth to that indicated by the line analysis. Once the strain softening curve is found which best predicts failure in the O C T specimen, the curve can then be used to model failure in an F E M analysis of a large aircraft structure. Detailed knowledge of the damage growth from sectioning results provide physical evidence with which to make the F E M strain softening analysis physically meaningful. For example, using the experimental results presented in this thesis, the following future work is suggested: 1 . A n investigation should be performed to see how a different element size alters the optimized strain softening curve of an O C T specimen. The appropriate element size is a function of the damage created in the process zone. For example, the minimum element size required to capture the effects of strain softening may be a function of the steady-state process zone dimensions. 2. The numerical strain softening curve of system B should be determined and compared to system A . Different shapes of the curve should be examined to compare the predicted and experimental surface displacements and damage growth. Differences in the curves for the two systems may then be related to the physical understanding of the differences in the damage pattern. For example, the stair-case like crack profile and greater steady-state damage height may translate into a greater ability to redistribute load during failure in system B . This increased load-redistribution would be reflected in the post-peak shape of the curve. 98 Following this, other tests should be performed to address the useful range of application of the strain softening curve. For example, different O C T coupon sizes could be tested to examine the expected capabilities of the strain softening curve to predict the fracture strength of small and large coupons and structures. A s well , the usefulness of the strain softening curve to other types of loading should be examined. Tests such as these w i l l help to determine how powerful a technique strain softening is for predicting failure in these materials. In so doing, the fundamental validity of strain softening may be better understood. 99 References Ashby, M . F . , "Materials Selection in Mechanical Design". 1992, Pergamon Press, New York Afaghi-Khatibi, A . , Y e , L . , and M a i , Y . W . , "Progressive Damage and Residual Strength of Notched Composite Laminates", eds. A . Poursartip and K . Street, vol . 1, 1st edition, 1995, pp. 383-390, Proceedings of the Tenth International Conference on Composite Materials, Woodhead Publishing Ltd. , Vancouver, Canada. Aronsson, C - G . and Backhand, J., "Tensile Fracture of Laminates with Cracks", Journal of Composite Materials, vol . 20, 1986, pp. 287-307. Basham, K . D . , Chong, K . P . , and Boresi, A . P . , " A New Method to Compute Size Independent Fracture Toughness Values for Brittle Materials", Engineering Fracture Mechanics, vol . 46, no. 3, 1993, pp. 357-363. Bazant, Z.P. , and Cedolin, L . , "Stability of Structures: Elastic. Inelastic. Fracture, and Damage Theories". 1991, Oxford University Press Inc., N e w York. Bazant, Z . B . , Daniel, I . M . , and L i , Z . , "Size Effect and Fracture Characteristics of Composite Laminates", Report 94-4/475s, 1995, Department of C i v i l Engineering, Northwestern University, Evanston, IL. Broek, D . , "Elementary Engineering Fracture Mechanics". 1986, Martinus Nijhoff Publishers, Boston. Chen, W.F. , and Han, D.J . , "Plasticity for Structural Engineers". 1988, Springer-Verlag New York Inc., New York. Daniel, I . M . , "Strain and Failure Analysis of Graphite/Epoxy Plates with Cracks", Experimental Mechanics, 1978, pp. 246-252. Drucker, D . C , " A More Fundamental Approach to Plastic Stress-Strain Relations", Proceedings of the First U.S. National Congress of Applied Mechanics, A S M E , 1951, pp. 487-491. Geers, M . G . D . , Peijs, T., de Borst, R., and Brekelmans, W . A . M . , "Experimental Dynamic Analysis of Damage Evolution in Short Fibre-Reinforced Composite Materials", eds. A . Poursartip and K . Street, vol . 1, 1st edition, 1995, pp. 755-762, Proceedings of the Tenth International Conference on Composite Materials, Woodhead Publishing Ltd. , Vancouver, Canada. 100 Hasan, O .A. , Boyce, M . C . , L i , X . S . , and Berko, S., " A n Investigation of the Y ie ld and Postyield Behavior and Corresponding Structure of Poly(methyl methacrylate)", Journal of Polymer Science, vol . 31, 1993, pp. 185-197. Hillerborg, A . , "Application of the Fictitious Crack Model to Different Types of Materials", International Journal of Fracture, vol . 51, 1991, pp. 95-102. Hillerborg, A . , Modeer, M . , and Petersson, P-E. , "Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements", Cement and Concrete Research, vol . 6, 1976, pp. 773-782. Ilcewicz, L . , The Boeing Company, personal communication, 1995. Kortschot, M . T . and Beaumont, P .W.R. , "Damage Mechanics of Composite Materials: I-Measurements of Damage and Strength", Composites Science and Technology, vol . 39, 1990a, pp. 289-301. Kortschot, M . T . and Beaumont, P .W.R. , "Damage Mechanics of Composite Materials: II-A Damaged-Based Notched Strength Model" , Composites Science and Technology, vol . 39, 1990b, pp. 303-326. Kortschot, M X , Beaumont, P .W.R. and Ashby, M . F . , "Damage Mechanics of Composite Materials: Ill-Prediction of Damage Growth and Notched Strength", Composites Science and Technology, vol . 40, 1991c, pp. 147-165. Kortschot, M . T . and Beaumont, P.W.R. , "Damage Mechanics of Composite Materials: IV-The effect of Lay-up on Damage Growth and Notched Strength", Composites Science and Technology, vol . 40, 1991d, pp. 167-179. Lagace, P. A . , "Notch Sensitivity and Stacking Sequence of Laminated Composites", Composite Materials: Testing and Design (Seventh Conference), ASTMSTP 893, J . M . Whitney, Ed. , American Society for Testing and Materials, Philadelphia, 1986a, pp. 161-176. Lagace, P .A . , "Notch Sensitivity of Graphite/Epoxy Fabric Laminates", Composites Science and Technology, vol . 26, 1986b, pp. 95-117. L i n , G . , Q i , C-S. and Zhou, H-T. , " A n Analytical Approach of the Fictitious Crack Model for Concrete", Engineering Fracture Mechanics, vol . 47, no. 2, 1994, pp. 269-280. Mar, J.W. and L i n , K . Y . , "Fracture Mechanics Correlation for Tensile Failure of Filamentary Composites with Holes", J. Aircraft, vol . 14, 1977, pp. 703-704. Matzenmiller, A . , Lubliner, J., and Taylor, R . L . , " A n Anisotropic Damage Model for Fiber-Composites", Report UCB-SEMM-91/09, Department of C i v i l Engineering, University of California at Berkeley, Berkeley, C A . 1991. Also see Mechanics of Materials, vol . 20, 1995, pp. 125-152. 101 Miller , R . A . , Castro-Montero, A . , and Shah, S.P., "Cohesive Crack Models for Cement Mortar Examined Using Finite-Element Analysis and Laser Holographic Measurements", J. Am. Ceram. Soc, vol . 74, no. 1, 1991, pp. 130-138. Pipes, R . B . , Wetherhold, R . C . and Gillespie, J . M . Jr., "Macroscopic Fracture of Fibrous Composites", Materials Science and Engineering, vol . 45, 1980, pp. 247-253. Read, H .E . and Hegemier, G . A . , "Strain Softening of Rock, Soil and Concrete - A Review Article", Mechanics of Materials, vol . 3, 1984, pp. 271-294. Shah, S.P. and Ouyang, C , "Toughening Mechanisms in Quasi-Brittle Materials", Transactions of the ASME, vol . 115, 1993, pp. 300-307. Sih, G.C. , Paris, P .C. , and Irwin, G.R. , "On Cracks in Rectilinearly Anisotropic Bodies", International Journal of Fracture Mechanics, vol . 1, 1965, pp. 189-202. Webb, G . M . and Kortschot, M . T . , "Damage Controlled Failure in Quasi-Isotropic Carbon Fibre Composites", eds. Hoa, S.V., and Gauvin, R., 1991, pp. 644-651, Proceedings of the First Canadian International Composites Conference ( C A N C O M 91), Elsevier Applied Science, Montreal, Quebec. Wells J .K. , and Beaumont, P .W.R. , "The Prediction of R-Curves and Notched Tensile Strength for Composite Laminates", Journal of Materials Science, vol . 22, 1987, pp. 1457-1468. Whitney, J . M . and Nuismer, R.J . , "Stress Fracture Criteria for Laminated Composites Containing Stress Concentrations", J. Composite Materials, vol . 8, 1974, pp. 253-265. 102 Appendix 1 Additional Surface Line Analyses The figures in Appendix 1 show additional surface line measurements of all analysed photos taken during the O C T test of specimen A 3 and specimens B l and B 2 . 103 Table Al . l Table of the position in front of the notch tip at which scribe line #4 has zero displacement at a given load, for system A, specimen A3. The values refer to Figure A1.2. line #4, specimen A3 Photo # Position in Front of Notch Tip (mm) 9 0 10 0 11 0 12 1 13 4 14 10 15 25 16 25 18 25 19 30 20 23 21 30 22 30 104 Table A1.2 Table of the position in front of the notch tip at which the scribe line has zero displacement at a given load, for system B, specimen Bl. The values refer to Figures A1.4 and A1.5. line #1, specimen Bl line #3, specimen Bl Photo # Position in Front of Notch Tip (mm) Photo # Position in Front of Notch Tip (mm) 2a 0 2a N/A 4a 0 4a N/A 6a 2 6a 2 8a 2 8a 2 10a 2 10a 2 12a 5 12a 5 14a 5 14a 5 16a 5 16a 5 18a 10 18a 10 3b 10 3b 10 4b 10 4b 12 5b 10 5b 12 6b 12 6b 20 7b 12 7b 20 8b 38 8b 37 9b 38 9b 37 105 Table A1.3 Table of the position in front of the notch tip at which the scribe line has zero displacement at a given load, for system B, specimen B2. The values refer to Figures A17 and A1.8. line #1, specimen B2 line #2, specimen B2 Photo # Position in Front of Notch Tip (mm) Photo # Position in Front of Notch Tip (mm) 5 0 5 0 8 1 8 3 9 1 9 4 11 6 11 8 12 10 12 8 15 15 15 20 16 15 16 20 17 33 17 35 18 33 18 35 19 33 19 35 20 40 20 40 21 40 21 40 106 Clip Gauge Cross Head Disp lacement (mm) Figure A l . l Load vs cross head and clip gauge displacement of specimen A3, showing all the analysed photos of the OCT test. -®— photo 9 - a - photo 10 photo 11 - * - photo 12 - • - p h o t o 13 - • - photo 14 —•— photo 15 - • - p h o t o 16 photo 18 - * - photo 19 photo 20 photo 21 photo 22 notch clip gauge Posit ion in Front of Notch Tip (mm) Photo numbers refer to Figure A l . l . Displacement is measured relative to photo 5. Image resolution is .045 mm/pixel. Line #4 is 20 mm above and 20 mm below the notch mid-plane. Figure A.2 All displacements of line #4 as a function of position in front of notch tip analysed for specimen A3. 107 12000 10000 8000 6000 X J 03 . _ l 4000 2000 clip gauge a20 CMOD gauge — a 1 5 a 2 0 b 3 Numbers refer to photo # 0.5 1 1.5 2 Displacement (mm) 2.5 Figure A1.3 Load vs CMOD and clip gauge displacement of specimen BI, showing all the valid photos taken during the OCT test. E E cu c c CD E CD O ro CL 40 £0 notch tidf T* clip gauge Position in Front of Notch Tip (mm) -•— photo a2 photo a4 - * - photo a6 - * - photo a8 -°—photo a10 photo a12 - * - photo a14 - • - photo a16 — ^ photo a18 -o-photo b3 photo b4 photo b5 - B - photo b6 -«—photo b7 photo b8 ^f—photo b9 Photo numbers refer to Figure A1 . 3 . Displacement is measured relative to photo al. Image resolution is .048 mm/pixel. Line # 1 is 2.5 mm above and 2.5 mm below the notch mid-plane. Figure A1.4 AH displacements of line #1 as a function of position in front of notch tip analysed for specimen BI. 108 E E co c c 0 E 0 o m a. <n b notch Position in Front of Notch Tip (mm) -•-photo a2 - * - photo a4 - * - photo a6 -•-photo a8 photo a10 photo a12 - * - photo a14 - ° - photo a16 —•—photo a18 - ® - photo b3 - a - photo b4 photo b5 -B-photo b6 photo b7 photo b8 photo b9 Photo numbers refer to Figure A 1.3. Displacement is measured relative to photo al . Image resolution is .048 mm/pixel. Line #3 is 7.5 mm above and 7.5 mm below the notch mid-plane. Figure A1.5 All displacements of line #3 as a function of position in front of notch tip analysed for specimen B l . 109 T3 CO O 1 2 0 0 0 1 0 0 0 0 8 0 0 0 + 6 0 0 0 4 0 0 0 2 0 0 0 0 g 1 1 1 2 ^ i j j 6 V ^ O 1 5 fl \ 19 / 5 / 5 / 4 2 2 2 0 2 1 5^ 1 1 r — Numbers refer to photo # 1 1 1 0 . 5 1 1 .5 2 CMOD Displacement (mm) 2 . 5 Figure A1.6 Load vs CMOD displacement of specimen B2, showing all the valid photos taken during the OCT test, (the clip gauge results are invalid as the gauge popped out during loading) 1 .8 -E CD c c cu E 0 o m a. w Q •photo 5 •photo 8 -photo 9 •photo 11 -photo 1 2 -photo 1 5 -photo 1 6 -photo 1 7 -photo 1 8 -photo 1 9 -photo 2 0 -photo 2 1 ip gauge Position in Front of Notch Tip (mm) Photo numbers refer to Figure A 1.6. Displacement is measured relative to photo 1. Image resolution is .042 mm/pixel. Line #1 is 2.5 mm above and 2.5 mm below the notch mid-plane. Figure A1.7 All displacements of line #1 as a function of position in front of notch tip analysed for specimen B2. 110 notch tipT - • - p h o t o 5 - o - photo 8 - * - photo 9 - * - photo 11 - * - photo 12 photo 15 - • - p h o t o 16 - ^ - p h o t o 17 - e - photo 18 - * - photo 19 - * photo 20 - ° - p h o t o 21 clip gauge Posit ion in Front of Notch Tip (mm) Photo numbers refer to Figure A 1 . 6 . Displacement is measured relative to photo 1 . Image resolution is . 0 4 2 mm/pixel. Line # 2 is 5 mm above and 5 mm below the notch mid-plane. Figure A1.8 All displacements of line #2 as a function of position in front of notch tip analysed for specimen B2. I l l Appendix 2 Additional Micrographs These figures show all the micrographs taken of O C T specimen A 3 and O C T specimen B 1 . 1 1 2 A2.1 Micrographs of Specimen A3 Figure A2.1 Reconstructed profile of the damage in specimen A3, with the load vs cross head displacement curve from the OCT test presented in Chapter 3. A2.11 Damage Grown Prior to Load C x = 1. C H . = units: r 3.35 Figure A2.2 Cross-section at 1.85 mm in front of the notch tip. 1 1 3 Figure A2.3 Cross-section at 4.45 mm in front of the notch tip. A2.12 Damage Grown Between Loads C-D Figure A2.4 Cross-section at 11.45 mm in front of the notch tip. 114 x = 12.10 mm CH. = 5.59 mm units: mm 3.35 Figure A2.5 Cross-section at 12.10 mm in front of the notch tip. x = 18.00 n CH. = 4.20 units: mm Figure A2.6 Cross-section at 18.00 mm in front of the notch tip. 115 Figure A2.7 Cross-section at 20.25 mm in front of the notch tip. 116 A2.13 Damage Grown Between Loads D-E Figure A2.10 Cross-section at 25.37 mm in front of the notch tip. 117 118 x = 32.00 mm P.Z.H. = 1.11 mm units: mm N.M-P — B - 1 0 " 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 - 0 - -1 - -2 - -3 - -4 - -5 - -6 - -7 - -8 - -9 --10 B Top 3.35 Figure A2.14 Cross-section at 32.00 mm in front of the notch tip. 119 A2.2 Micrographs of Specimen B1 - Front Side Layup: [F0/-45/90/45/0/45/90/-45/F0]s, FO = [0/90] weave to scale: 0 _ 35 mm — = 0 22 W 3.5 mm 0 0.5 1 1.5 2 2.5 CMOD Displacement (mm) Figure A2.15 Reconstructed profile of the damage in specimen B l , with the load vs CMOD displacement curve from the OCT test presented in Chapter 3. A2.21 Damage Grown Prior to Load X x = 1.00 mrr Top O C H . = 1 mm units: mm N.M-P — H 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 h -3 -4 -5 -6 -7 -8 -9 10 11 12 13 14 O Top 1.66 Figure A2.16 Cross-section at 1.00 mm in front of the notch tip. 120 Figure A2.17 Cross-section at 2.00 mm in front of the notch tip. 1.66 Figure A2.18 Cross-section at 5.65 mm in front of the notch tip. 121 A2.22 Damage Grown Between Loads X-Y X = 10 C H . = units: r Figure A2.19 Cross-section at 10.00 mm in front of the notch tip. 122 x = 11.00 mm Top 0 C H . = 17.80 mm units: mm N.M-P-14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 I Top O » f 1.66 Figure A2.20 Cross-section at 11.00 mm in front of the notch tip. x = 12.00 mm O Top C H . = 20.65 mm units: mm N.M-P -r 14 - 13 - 12 - 11 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 s - 0 - -1 j - -2 - -3 - -4 - -5 - -6 - -7 Q - -o - -9 - 1 0 - 1 1 - 1 2 - 1 3 L-14 I Top O 1.66 Figure A2.21 Cross-section at 12.00 mm in front of the notch tip. 123 Figure A2.22 Cross-section at 19.50 mm in front of the notch tip. Figure A2.23 Cross-section at 20.89 mm in front of the notch tip. 124 Figure A2.24 Cross-section at 34.62 mm in front of the notch tip. 125 A2.23 Damage Grown Between Loads Y-Z 126 A2.3 Micrographs of Specimen B1 - Back Side A2.31 Damage Grown Prior to Load X 127 x = 2.00 mm C H . = 3 mm units: mm N.M-P — U Top 0 1.66 Figure A2.27 Cross-section at 2.00 mm in front of the notch tip. x = 5.65 mm Top T.C.H. = 8 mm units: mm O N.M-P — r 1 4 - 13 - 12 - 11 - 10 - 9 - 8 o - 7 - 6 - 5 - 4 - 3 - 2 - 1 - 0 - -1 - -2 -o - -4 - -5 - -6 - -7 - -8 - - 9 - 1 0 - 1 1 - 1 2 - 1 3 - 1 4 I Top O 1.66 Figure A2.28 Cross-section at 5.65 mm in front of the notch tip. 128 A2.32 Damage Grown Between Loads X-Y 129 x = 11.00 mm Top 0 Top I O CH . = 11.62 mm units: mm N.M-P.H V- 14 f- 13 h 12 11 h 10 9 8 7 6 5 - 4 - 3 - 2 - 1 - 0 :i -3 -4 -5 -6 -7 -8 r 9 1—10 |~11 1-12 13 14 1.66 Figure A2.30 Cross-section at 11.00 mm in front of the notch tip. x = 12.00 mm| Top O CH . = 3.92 mm units: mm N.M-P — 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 r 9 H O I—11 H 2 -13 -14 O Top I 1.66 Figure A2.31 Cross-section at 12.00 mm in front of the notch tip. 130 Figure A2.33 Cross-section at 20.89 mm in front of the notch tip. 131 Figure A2.35 Cross-section at 35.50 mm in front of the notch tip. 132 133 A2.33 Damage Grown Between Loads Y-Z x = 38.22 mm Top 0 Tip | O P.Z.H. = 11.19 mm units: mm N.M-P.H 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 12 13 14 15 16 1.66 Figure A2.37 Cross-section at 38.22 mm in front of the notch tip. O Tip | x = 39.06 mm Top O P.Z.H. = 12.30 mm units: mm N.M-P — 10 9 8 7 6 n 2 li -5 -6 -7 - 1 0 - 1 1 - 1 2 - 1 3 - 1 4 - 1 5 16 I 1.66 Figure A2.38 Cross-section at 39.06 mm in front of the notch tip. 134 

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