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HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS RANA FOROUTAN A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of Doctor of Philosophy Department of Mechanical Engineering McGill University, Montreal December 2009 ©Rana Foroutan, 2009 To my children Ryan and Aiden “Great things are not done by impulse, but by a series of small things brought together.” Vincent Van Gogh ABSTRACT The thesis centres on the dynamic behaviour of woven composite materials. The increase in the use of these materials in the aerospace industry demands a reliable constitutive damage model to predict their response under high strain rate loading. The current available models do not include rate effects, are too complicated to use, are unable to provide a clear procedure for characterizing the model parameters, or are unable to accurately predict response when compared to experimental tests. The present work focuses on developing a rate-dependant continuum damage mechanics model which considers all in-plane damages, i.e., damage due to matrix cracks and fibre failure in normal and shear directions. A physical treatment of growth of damage based on the extensive experimental results is combined with the framework of continuum damage mechanics models to form the foundation of the model for materials whose response is governed by elastic deformation coupled with damage. The developed model simulates the non-linear and rate dependant behaviour of woven composite materials due to damage evolution. The model is implemented into a commercial finite element code (ABAQUS) as a two-dimensional user material subroutine. Detailed study of the behaviour of these materials under high velocity loading and several experiments were required to establish material model parameters. Uniaxial tension as well as bias extension shear tests were carried out at both static and dynamic rates on three different woven composite materials. A tensile version of the Hopkinson bar setup, in conjunction with the designed specimen and specimen fixture, are used for testing laminated woven composite materials at high rates of strain and a peak strain rate of up to 560/sec. Comparison of Hopkinson bar results with results of tests performed at a quasi-static rate using a servo-hydraulic testing machine i clearly show that strain rate has an effect on both the stress and the strain to failure for these carbon woven composite materials. A higher stress is observed for dynamic tests, whereas the strains at the maximum stress are higher for the static experiments. Also, an increase in the initial undamaged elastic modulus is observed for both tensile and shear response with increasing strain rate. Finally, the simulation results presented in this work demonstrate that the model can well predict the dynamic and the static response obtained from the experimental measurements. Simulation results for higher than tested strain rates are also presented to illustrate the response at these high rates. It is shown that the tensile strength increases with the increase in loading rate, and also a strain softening phenomenon is observed after the maximum tensile strength. The loading-unloading, as well as compression response of the materials are also simulated to ensure the capability of the model to capture these responses. ii RESUME Cette thèse se concentre sur le comportement dynamique des matériaux composites tissés. L’augmentation de l’utilisation de ces matériaux dans l’industrie aéronautique requiert le développement de modèles constitutifs fiables pour prédire leurs réponses sous des conditions de chargement à haute vitesse de déformation. Actuellement, les modèles disponibles n’incluent pas les effets de vitesse de déformation, sont compliqués à utiliser ou ne sont pas capables de prédire avec précision le comportement expérimental de ces matériaux. Ce travail focalise sur le développement d’un modèle de mécanique continue de la rupture tenant en compte la dépendance avec le taux de déformation. Ce modèle considère tous les dommages dans le plan : dommages provoqués par la fissuration de la matrice ainsi que ceux dus à la rupture de la fibre dans les directions normale et tangentielle. Un traitement physique de la propagation du dommage, basé sur un vaste ensemble de résultats expérimentaux, est combiné avec les modèles de mécanique continu de la rupture afin de développer un modèle pour ces matériaux tissés, dont la réponse comportementale est gouvernée par un couplage ente la déformation élastique et la rupture. Le modèle développé dans cette étude simule le comportement non-linéaire et dépendant du taux de chargement des matériaux composites tissés provoqué par l’évolution des dommages. Ce modèle est implémenté dans un logiciel commercial d’élément fini (ABAQUS) en tant qu’un sous-programme définissant les propriétés bidimensionnel des matériaux. Une étude détaillée du comportement de ces matériaux, soumis à des chargements à hautes vitesses, et plusieurs expériences ont été requises pour établir les paramètres du modèle. Des tests en tension uni-axiale et en cisaillement à bi-extension ont été réalisés iii à la fois pour des cas statiques et dynamiques sur trois matériaux composites tissés différents. Une version en tension du dispositif à barres de Hopkinson, utilisant les spécimens et fixations conçus, a été utilisée pour tester des laminés de matériaux composites tissés dans des conditions de haut taux de déformation. Un taux maximum de déformation a été obtenu à 560/sec. La comparaison des résultats obtenus avec les barres de Hokinson, avec ceux obtenus avec des tests quasi-statique utilisant une machine de test servohydraulique, démontre clairement l’effet du taux de déformation sur les contraintes et déformations à la rupture pour ces matériaux composite à fibres de carbone tissés. Des contraintes plus importantes sont observées pour les tests dynamiques, tandis que les déformations pour les contraintes maximales sont plus grandes pour les tests statiques. De plus, une augmentation du module élastique initial non endommagé est observée avec l’augmentation du taux de déformation, à la fois pour la réponse en tension et en cisaillement. Finalement, les résultats des simulations présentés dans ce travail démontrent que le modèle peut prédire correctement la réponse statique et dynamique obtenue par les mesures expérimentales. Les résultats des simulations, pour des taux de déformations plus grands que ceux testés expérimentalement, sont aussi présentés pour illustrer la réponse du matériau à ces haut taux de déformation. Cette étude montre que la résistance en tension augmente avec l’augmentation du taux de chargement. De plus, un phénomène d’adoucissement des déformations est aussi observé après la résistance maximum en tension. Le chargement-déchargement, ainsi que la réponse en compression de ces matériaux sont aussi simulés pour assurer la capacité du modèle à capturer ce type de réponses. iv ACKNOWLEDGEMENTS I would like to express my gratitude to all those who made it possible for me to complete this dissertation. It is difficult to write this page after so many busy years, so many memories, and so many people to remember. Many had a little word for me which kept me going. They are numerous to name all, but I like to mention my supervisor Prof. James. A. Nemes who has guided me through my life as a graduate student at McGill University with many fascinating discussions. He challenged me when needed, encouraged me when necessary and provided wisdom, insight and patience at all times. I was extremely fortunate to have him as my supervisor and a great friend. I certainly would like to extent my gratitude to Prof. Pascal Hubert, my co-supervisor, for his kindness and friendly support as well as his invaluable suggestions to this research. I would like to acknowledge the help and contribution provided by my research group at CRIAQ, in particular, the project leader Prof. Augustin Gakwaya from Laval University, Mr. Claude Boucher from Bombardier, Mr. Stephen Caufeild from Pratt and Whitney Canada, Mr. Alain Cole and Mr. Mathieu Ruel from Bell Helicopter, and Mr. Dennis Nandlall from DRDC Valcartier. I would like to acknowledge the financial support of the Consortium for Research and Innovation in Aerospace in Quebec (CRIAQ), the McGill University, and the generous support of Dr. James. A. Nemes and Dr. Pascal Hubert. I would like to thank my fellow students and the staff in the Department of Mechanical Engineering at McGill University, in Particular, Mr. George Tewfik, Mr. Francois Di Quinzio, Mr. Steve Kacani, and Dr. Lolei Khoun. v I am truly grateful to my considerate father, Mostafa, my wonderful mother, Simin, and my amazing brother, Shahram, without whose constant support, infinite love, encouragement and care, throughout my life this work would not have been possible. Finally, by no means the least, I specially would like to appreciate my loving and understanding soul mate, Masoud Roshan Fekr. I like to acknowledge his patience, support as well as the motivation he provided me for further studies. I also extend my appreciation to my children, Ryan, who came to this world during this work, and allowed so much time to be deprived from him while I was working and my second little one, who will be with us after the completion of this thesis. vi TABLE OF CONTENTS ABSTRACT ................................................................................................................. I RESUME .................................................................................................................. III ACKNOWLEDGEMENTS .............................................................................................. V TABLE OF CONTENTS .............................................................................................. VII LIST OF FIGURES ..................................................................................................... XI LIST OF TABLES.................................................................................................... XVII NOMENCLATURE ................................................................................................... XIX 1. INTRODUCTION ..................................................................................... 1 1.1 OVERVIEW ....................................................................................................... 1 1.2 OBJECTIVES AND STRATEGIES OF THIS RESEARCH ............................................. 4 1.3 STRUCTURE OF THIS THESIS ............................................................................. 5 2. LITERATURE REVIEW........................................................................... 7 2.1 CONTINUUM DAMAGE MECHANICS ................................................................... 9 2.1.1 INTERNAL STATE VARIABLES ........................................................................ 11 2.1.2 EFFECTIVE STRESS ....................................................................................... 15 2.1.3 STRAIN EQUIVALENCE HYPOTHESIS.............................................................. 18 2.2 DYNAMIC TENSILE EXPERIMENTS SETUPS ...................................................... 19 2.2.1 SPLIT HOPKINSON BAR SETUPS .................................................................... 21 2.3 STRAIN-RATE DEPENDANT EXPERIMENTS ...................................................... 25 2.4 REVIEW OF CDM MODELS FOR COMPOSITE MATERIALS ................................. 31 vii 3. EXPERIMENTAL SETUP ..................................................................... 41 3.1 INTRODUCTION ............................................................................................... 41 3.2 MATERIALS .................................................................................................... 42 3.3 DYNAMIC EXPERIMENTS ................................................................................. 45 3.3.1 HOPKINSON BAR SETUP AND THEORY ........................................................... 45 3.3.2 SPECIMENS AND FIXTURE DESIGNS ............................................................... 50 3.3.3 MEASUREMENT PROCEDURES ....................................................................... 56 3.4 STATIC EXPERIMENTS ..................................................................................... 62 3.4.1 UNIAXIAL TENSION TESTS .............................................................................. 63 3.4.2 BIAS EXTENSION SHEAR TESTS ....................................................................... 67 3.5 EXPERIMENTS SUMMARY ................................................................................ 72 4. EXPERIMENTAL RESULTS................................................................. 75 4.1 STATIC EXPERIMENTS..................................................................................... 75 4.1.1 TENSILE MODULUS AND TENSION TEST RESULTS ........................................... 76 4.1.2 SHEAR MODULUS AND BIAS EXTENSION TEST RESULTS................................... 85 4.2 DYNAMIC EXPERIMENTS ................................................................................. 91 4.2.1 EFFECTIVE GAGE LENGTH.............................................................................. 91 4.2.2 TENSION TEST RESULTS ................................................................................. 93 4.2.3 BIAS EXTENSION TEST RESULTS ..................................................................... 99 4.3 COMPARISON OF STATIC AND DYNAMIC RESULTS .......................................... 103 5. DEVELOPMENT OF A RATE DEPENDENT CONTINUUM DAMAGE MODEL ................................................................................................. 109 5.1 INTRODUCTION ............................................................................................. 109 5.2 RATE DEPENDENT FOROUTAN-NEMES DAMAGE MODEL ................................ 114 viii 5.2.1 MODEL ASSUMPTIONS ................................................................................. 115 5.2.2 INTERNAL STATE VARIABLES ....................................................................... 116 5.2.3 EVOLUTION OF DAMAGE.............................................................................. 120 5.2.4 RATE SENSITIVE CONSTITUTIVE CDM MODEL ............................................. 121 5.3 MATERIAL CHARACTERIZATION ..................................................................... 126 5.3.1 UNDAMAGED STIFFNESS AND STRENGTH CONSTANTS................................. 127 5.3.2 STIFFNESS REDUCTION FUNCTIONS ............................................................ 133 5.3.3 DAMAGE EVOLUTION FUNCTIONS ............................................................... 134 5.3.4 STRESS RATE FUNCTIONS ........................................................................... 136 6. IMPLEMENTATION OF THE DAMAGE MODEL INTO A FINITE ELEMENT CODE ................................................................................. 139 6.1 INTRODUCTION............................................................................................. 139 6.2 MODEL IMPLEMENTATION IN FINITE ELEMENT.............................................. 140 6.3 IMPLEMENTATION AND COMPARISON TO EXPERIMENTS ................................. 143 6.3.1 TENSILE TEST RESULTS ............................................................................... 143 6.3.2 BIAS EXTENSION SHEAR TEST RESULTS........................................................ 149 6.3.3 SIMULATION RESULTS FOR LOADING-UNLOADING-RELOADING .................... 154 6.4 SUMMARY .................................................................................................... 160 7. CONCLUSIONS ................................................................................... 161 7.1 SUMMARY AND CONCLUSIONS ....................................................................... 161 7.1.1 EXPERIMENTAL FINDINGS ........................................................................... 162 7.1.2 MATERIAL MODEL AND SIMULATION FINDINGS ............................................ 165 7.2 ORIGINALITY AND CONTRIBUTION TO KNOWLEDGE ........................................ 166 7.3 RECOMMENDED FUTURE WORK ..................................................................... 167 ix BIBLIOGRAPHY ............................................................................................................... 169 APPENDIX A: SUBROUTINE FOR ABAQUS/EXPLICIT ......................................... 183 x LIST OF FIGURES Figure 1.1 Bird strike impact on wing leading edge (Oxford 2004) ....................... 3 Figure 2.1 Meso-definition of damage ................................................................... 13 Figure 2.2 Schematic of Hopkinson bar configuration using direct tensile pulse22 Figure 2.3 Various Hopkinson bar setups using indirect tensile pulse adapted from (a) Ellwood et al. (1982), (b) Lindholm and Yeakley (1968), (c) Kawata et al. (1980), (d) Harding et al. (1960), (e) Eskandari and Nemes (2000), and (f) Li et al. (1993) ..................................................... 25 Figure 2.4 Stress-strain curves at dynamic and static strain rates in tension (a) Glass Satin Woven Cloth/ Polyester (b) Carbon Plain Woven Cloth/ Epoxy (adapted from Kawata et al. 1981) .............................................. 27 Figure 2.5 Stress-strain curves for woven carbon fibre epoxy laminates at different strain rates in tension (adapted from Harding et al. 1989) ... 28 Figure 2.6 Tensile stress-strain curves at different strain rates for (a) woven aramid/polypropylene (b) woven polyethylene/ polyethylene (Rodriguez et al. 1996 with permission of Elsevier) ................................................. 29 Figure 2.7 Stress-strain curves of glass woven fabric-reinforced composites subjected to different strain rates (adapted from Sham et al. 2000) .... 30 Figure 3.1 Schematic and picture of different weaves of woven composite: (a) plain weave, (b) 2×2 twill weave, (c) 8-harness satin weave ................. 43 Figure 3.2 Hopkinson bar setup ............................................................................ 45 Figure 3.3 Schematic representation of Hopkinson bar ....................................... 46 Figure 3.4 A specimen sandwiched between the input and output bars ............. 46 Figure 3.5 Schematic assembly of first specimen fixture design ......................... 51 Figure 3.6 Picture of the fixture with epoxy injection .......................................... 51 Figure 3.7 Schematic assembly of second fixture design (side and top views) .... 52 Figure 3.8 Schematic assembly of third fixture design ........................................ 53 xi Figure 3.9 Schematic diagram of the modified fixture.......................................... 53 Figure 3.10 Picture of the modified fixture ........................................................... 54 Figure 3.11 Initial dimensions of the composite specimen in millimetres ........... 54 Figure 3.12 Failure in the specimens (a) [0/45/90/-45]s (b) woven [0]6 ................. 55 Figure 3.13 Specimen’s dimensions in millimetres ............................................... 55 Figure 3.14 The aluminum jig used to cut the specimens .................................... 56 Figure 3.15 Half Wheatstone bridge ...................................................................... 58 Figure 3.16 Typical recorded strain gage signals on the input and output bars . 59 Figure 3.17 Typical dynamic behaviour of a woven composite material .............. 60 Figure 3.18 Schematic comparison of actual versus effective gage length .......... 61 Figure 3.19 Typical behaviour using different gage lengths ................................ 61 Figure 3.20 Woven composite specimen with mounted strain gage ..................... 62 Figure 3.21 Servo-hydraulic MTS testing machine, with the specimen fixed in place ......................................................................................................... 63 Figure 3.22 Correction for the extensometer arms length .................................... 64 Figure 3.23 Strain comparison using extensometer and strain gage ................... 65 Figure 3.24 Woven composite specimen with perpendicular strain gage ............ 66 Figure 3.25 Typical result to obtain the Poisson’s ratio........................................ 67 Figure 3.26 Loading direction and fibre direction coordinate systems ................ 69 Figure 3.27 Typical result for bias extension specimen to obtain Poisson’s ratio 71 Figure 4.1 Stress-strain curves for static tensile tests of material one (woven carbon/epoxy prepreg [0º]6) in the two fibre directions .......................... 77 Figure 4.2 Stress-strain curves for static tensile tests of material two (woven carbon/BMI prepreg [0º]6) in the two fibre directions ............................ 78 Figure 4.3 Stress-strain curves for static tensile tests of material three (woven carbon/epoxy prepreg [0º]8) in two fibre directions ................................ 79 Figure 4.4 Average static tensile strengths and their variability for the three materials in the two fibre directions....................................................... 81 Figure 4.5 Average static tensile modulus and their variability for the three materials in the two fibre directions....................................................... 82 xii Figure 4.6 Axial versus transverse strains for all the materials in the fibre direction ................................................................................................... 83 Figure 4.7 Poisson’s ratio of material one in 45º specimens ................................. 85 Figure 4.8 Poisson’s ratio of material two in 45º specimens................................. 86 Figure 4.9 Poisson’s ratio of material three in 45º specimens .............................. 86 Figure 4.10 Shear stress -strain curves for static bias extension tests of material one (woven carbon/epoxy prepreg [0º]6) in the 45º specimen ................. 88 Figure 4.11 Shear stress -strain curves for static bias extension tests of material two (woven carbon/BMI prepreg [0º]6) in the 45º specimen ................... 88 Figure 4.12 Shear stress-strain curves for static bias extension tests of material three (woven carbon/epoxy prepreg [0º]8) in the 45º specimen .............. 89 Figure 4.13 Comparison of shear stress-strain curves of the three materials .... 90 Figure 4.14 Typical comparison of different strain measurements ..................... 92 Figure 4.15 Stress-strain curves for dynamic tensile tests of material one in the two fibre directions .................................................................................. 94 Figure 4.16 Stress-strain curves for dynamic tensile tests of material two in the two fibre directions .................................................................................. 95 Figure 4.17 Stress-strain curves for dynamic tensile tests of material three in the two fibre directions ........................................................................... 96 Figure 4.18 Average dynamic tensile strengths and their variability for the three materials in the two fibre directions ...................................................... 98 Figure 4.19 Average dynamic strains at maximum stress and their variability for the three materials in the two fibre directions................................. 99 Figure 4.20 Stress-strain curves for dynamic bias extension tests of material one in the 45º specimen................................................................................ 100 Figure 4.21 Stress-strain curves for dynamic bias extension tests of material two in the 45º specimen................................................................................ 101 Figure 4.22 Stress-strain curves for dynamic bias extension tests of material three in the 45º specimen ...................................................................... 101 xiii Figure 4.23 Average dynamic shear strength and strains at maximum stress including their variability for the three materials in the 45º direction ................................................................................................................ 103 Figure 4.24 Comparison of static and dynamic behaviour in material one (a) 0º or 90º specimens, (b) 45º specimens .......................................................... 104 Figure 4.25 Comparison of static and dynamic behaviour in material two (a) 0º or 90º specimens, (b) 45º specimens .......................................................... 105 Figure 4.26 Comparison of static and dynamic behaviour in material three (a) 0º or 90º specimens, (b) 45º specimens ...................................................... 106 Figure 5.1 Schematic view of the six damage variables in a composite element (adapted from Dechaene et al. 2002) .................................................... 113 Figure 5.2 Typical stress-strain response showing zones of damage ................. 118 Figure 5.3 Dynamic stress-strain curve showing increment of stress due to damage ................................................................................................... 122 Figure 5.4 Typical stress-strain response of a woven composite material ......... 128 Figure 5.5 Typical shear stress-strain response of a woven composite material ................................................................................................................ 129 Figure 6.1 Flowchart for implementation of the CDM model into ABAQUS .... 142 Figure 6.2 Comparison between the displacement-time profiles obtained from dynamic test results and the equation 6.2 (a) material one, (b) material two, (c) material three ........................................................................... 144 Figure 6.3 Two dimensional element, tensile simulation in ABAQUS .............. 146 Figure 6.4 Comparison of the ABAQUS simulation with the average dynamic and static tensile test results (a) material one, (b) material two, (c) material three ........................................................................................ 147 Figure 6.5 Two dimensional element, shear simulation in ABAQUS ................ 149 Figure 6.6 Comparison between the shear displacement-time profiles obtained from dynamic test results and the equation 6.2 (a) material one, (b) material two, (c) material three ............................................................ 150 xiv Figure 6.7 Comparison of the ABAQUS simulation with the average dynamic and static shear test results (a) material one, (b) material two, (c) material three ........................................................................................ 152 Figure 6.8 Loading-unloading simulation results (a) material one, (b) material two, (c) material three ........................................................................... 155 Figure 6.9 Loading-unloading-reloading simulation results (a) material one, (b) material two, (c) material three............................................................ 157 Figure 6.10 Loading-unloading to compression-reloading simulation results (a) material one, (b) material two, (c) material three ............................... 159 xv xvi LIST OF TABLES Table 3.1 Summary of the materials used in this research .................................. 44 Table 3.2 Time to failure for the three woven composite materials ..................... 50 Table 3.3 Direction cosines .................................................................................... 69 Table 3.4 List of dynamic experiments performed................................................ 73 Table 3.5 List of static experiments performed .................................................... 74 Table 4.1 Summary of static tensile tests ............................................................. 80 Table 4.2 Poisson’s ratio measured in the fibre direction..................................... 83 Table 4.3 Summary of Poisson’s ratio and tensile modulus data from literature 84 Table 4.4 Poisson’s ratio measured in the 45º specimens ..................................... 87 Table 4.5 Summary of static bias extension tests ................................................. 90 Table 4.6 Summary of effective gage lengths ........................................................ 92 Table 4.7 Summary of dynamic tensile tests ........................................................ 97 Table 4.8 Summary of dynamic bias extension tests .......................................... 102 Table 4.9 Comparison of the average of static and dynamic results .................. 107 Table 5.1 Material constants present in the Foroutan-Nemes CDM model ...... 125 Table 5.2 Calculated threshold values for the three woven composite materials ................................................................................................................ 132 Table 5.3 Calculated constants for the three woven composite materials ......... 138 Table 6.1 Comparison of maximum dynamic tensile strength ........................... 148 Table 6.2 Comparison of maximum dynamic shear strength ............................. 154 xvii xviii NOMENCLATURE ROMAN LETTERS A0 cross sectional area of the specimen Ai1 to Ai6 material rate dependant constants, (i = 1, 4) - fibre and 45 degree direction Ab cross sectional area of the Hopkinson bars Bi1 to Bi4 material damage constants, (i = 1, 4) - fibre and 45 degree direction C0 undamaged stiffness tensor C1 to C5 material constants C(d ) damage dependant stiffness tensor C ij damaged elastic stiffness tensors E0 initial undamaged modulus E damaged modulus Eii Tensile and shear modulus components (i= 1, 2, 4) Eii0 initial undamaged tensile and shear modulus components (i= 1, 2, 4) Eii0H initial modulus components at high rates of strain (i= 1, 2, 4) Eb Young modulus of the Hopkinson bars F0 nucleation function Fi instantaneous tensile load GF gage factor of strain gage GL short specimen gage length xix ΔL displacement LS length of the striker tube LSF length of the specimen in between the two fixings M (d ) fourth order damage effect tensor P1 P2 , applied load on each side of the specimen Ri resistors in Wheatstone bridge (i= 1, 2 ,3 and 4) RG nominal resistance of the strain gage ΔR strain-induced change in resistance ΔR / R fractional change in electrical resistance S shearing yield strength in the 12-plane S cross sectional area S effective area Sd damaged area due to micro cracks and void growth VEX excitation voltage of Wheatstone bridge Vm volume element VO output voltage of Wheatstone bridge Wi1, Wi2,W13 material constants (i = 1, 4) - fibre and 45 degree direction X yield strengths under uniaxial loading in 11-direction X strength of the material associated with the damage mode Y yield strengths under uniaxial loading in 22-direction Z11, Z12, Z4 strain rate dependant parameters for evolution of damage cb longitudinal wave velocity in the Hopkinson bars xx cS longitudinal wave velocity in the specimen d scalar damage variable d second order damage tensor dij rate of damage, i = 1,2,4 - direction, and j = 1, 2 -damage type Δ d ij rate of damage in the incremental form di damage variables at their respective principle axes, i = 1,2,3 d ij damage variables, i = 1,2,4 - direction, and j = 1, 2 -damage type e base of the natural logarithm li , mi direction cosines ( i = 1,2 ) in (1,2) coordinated system l0 initial gage length of the short specimen li instantaneous gage lengths m, m1, m2 shape of the damage growth curve in Weibull distribution n unit vector t time Δt time increments ΔtI duration time of the incident pulse tf time to failure tS transit time needed for the incident wave to travel through the specimen once u1 , u2 displacements at the ends of the short specimen in HB u, v displacement in x- and y-direction xxi GREEK LETTERS Δ displacement Ωi principle values of the damage tensor α ,αi1 constants related to amount of damage ( i = 1,2,4 ) ε strain ε strain tensor ε effective strain tensor ε strain rate ε average strain rate Δε kl strain increments in the fibre coordinate system ( k = 1,2,3 l = 1,2,3 ) ε 0,i strains at initiation with damage equal to zero ε act actual strain ε ext extensometer strain εf nominal failure strain ε I , ε R , εT Incident, reflected, and transmitted strains measured ε ij strain components in the loading coordinate system ( i = x, y, z j = x, y, z ) ε kl strain components in the fibre coordinate system ( k = 1,2,3 l = 1,2,3 ) εm strain relative to maximum stress ε max,i strains at zero stress with damage equal unity εs average strain in the specimen γ 44 engineering shear strain xxii ν ij Poisson's ratio i ≠ j ρb Density of the Hopkinson bars ρS density of the specimen σ Cauchy stress tensor σ effective stress tensor σ stress rate σ01, σ02 scale parameters of the double Weibull distribution function σ d damaged stress rate σ0 threshold stress σ ii threshold stress (i = 1, 4) - fibre and 45 degree direction σ eng engineering stress σ ij stress components in the loading coordinate system 0 ( i = x, y, z j = x, y, z ) σ kl stress components in the fibre coordinate system ( k = 1,2,3 l = 1,2,3 ) Δσ kl stress increments in the fibre coordinate system ( k = 1,2,3 l = 1,2,3 ) σmax maximum stress xxiii xxiv HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 1 INTRODUCTION 1.1 Overview Composite materials combine the desirable properties or behaviour of two or more identifiable constituent materials in order to achieve improved strength, stiffness, and toughness at reduced weight. Composites are gradually replacing traditional metallic materials due to their high specific modulus, high specific strength, high stiffness to weight ratio, and their capability of being tailored for a specific application. These properties can be altered or modified by varying the constituents of the composite, i.e., the matrix and the fibres. 1 INTRODUCTION Among the different types of composite materials, the class of woven fabric composites has received attention in many structural applications because of the balanced in-plane properties, manufacturability, and high impact resistance. Reinforcement in all directions within a single layer, better impact resistance, better toughness and lower cost of fabrication are some of the reasons that the woven fabric composites have been recognized as more competitive than the unidirectional composites. Those properties make the woven fabric composites attractive for structural applications. The ability to simulate the material behaviour and to predict the survivability of structures becomes more and more important at the stage of the structure design. Woven composites are fabricated from strong and light fibres such as graphite, carbon, aramid, glass, nylon or other advanced polymeric fibrous materials. The fibres form yarns, which are woven in different textile architectures such as plain weave, twill, satin or crowfoot. Similar architectures could be achieved if the yarns are braided. The textile reinforcement is then impregnated by a resin and cured forming a lamina. It is also possible to stack several lamina in a laminate or stitch several textile layers together before impregnating and curing them. Generally, the fibres have transversely isotropic elastic properties and the matrix material is non-linear isotropic. Composite materials are being used in various fields including aerospace industries where aircraft structures may undergo high strain-rate dynamic loading. Examples of these loadings are bird strike or foreign object impact against wing leading edges, propellers, and engine rotor containments, as shown in Figure 1.1. Several examples are illustrated below where the aircraft structure experiences high strain rate impact loading. Consider a turbofan engine, in which air passes through several stages of rotating fan blades. If there is a flaw in the system, such as an unexpected obstruction, the fan blade can break, spin off, and if not contained, damage other critical components. Therefore, if the engine casing is made of the composite materials, it needs to be strong enough to contain broken blades and also should be damage-tolerant to withstand the impact of a loose blade-turned-projectile. 2 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Figure 1.1 Bird strike impact on wing leading edge (Oxford 2004) A recent example of a damage tolerant engine casing is the crash of the US Airways Flight 1549 making an emergency landing in the Hudson River. The plane took off from New York's LaGuardia Airport, colliding with a flock of geese approximately 2,900 feet above the ground. The most vulnerable part of the engine is the fan, which can be damaged by an ingested bird. Pieces of broken blade can then rip through the rest of the engine like shrapnel. Engines have been reinforced so that they can stay intact in their casing in the event of such a strike, but they usually cannot be restarted once they are damaged. Archie Dickey, an associate professor of aviation environmental science at Embry-Riddle Aeronautical University's campus in Prescott, Arizona has said that hits hard enough to cause a total failure are rare, only happening two or three times a year worldwide, and the chance of it hitting both engines, is less than one percent. 3 INTRODUCTION It is true that aircraft hit thousands of birds every year which usually bounce off without damaging the structure, nevertheless, it is important to consider such rare cases as well. In the mentioned case, the engine casing could withstand the dynamic impact, and as a result keep the rest of the plane intact from further damage. In the mentioned case, all the passengers were saved after the pilot miraculously landed the plane in the Hudson River. It is believed that composites can create lighter, more fuel-efficient engine casings that might be stronger and safer than those made with aluminum or other traditional materials. The use of composite materials for manufacturing many aerospace parts is an attractive, viable option for manufacturers. This is because of the low-cost of manufacturing as well as the added cost saving benefits of the weight reduction. However in order to safely use these materials, a more thorough understanding of the behaviour of composite materials under dynamic loadings coupled with damage is required. While numerous studies on composite materials have shown that they exhibit ratedependant behaviour (Gilat et al. 2002; Harding and Welsh 1983; Kawata et al. 1981; Rodriguez et al. 1996), most of the models available in the literature are not rate dependant. Among the few models which include rate dependency, some consider the rate effect only in the shear response (Marguet et al. 2007), and yet some models do not consider shear damage in the model at all (Iannucci et al. 2001). Characterizing the behaviour of woven composite materials under dynamic as well as static loading is necessary for developing a reliable constitutive model that can predict the response of these materials. 1.2 Objectives and strategies of this research The general objective of this research is to develop a rate-dependant constitutive damage model which is appropriate for various woven composite materials. A prerequisite for such a model is to characterize the static and dynamic behaviour of these materials by performing a series of experiments. It is essential to establish a methodology for the 4 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS determination of the damage parameters present in the model. The ultimate goal is to implement the developed model into a material user subroutine of a commercial finite element program that can be used in finite element simulations. Specifically, this research focuses on the characterization and material modelling of three woven composite materials, namely a 2×2 twill weave carbon/epoxy prepreg [0]6, a carbon/BMI prepreg [0]6 with 8 harness satin weave, and a plain weave carbon/epoxy prepreg [0]8. All the materials considered in this thesis are balanced woven fabrics with a zero direction stacking sequence. First, tensile and bias extension shear experiments are performed on the coupon specimens to obtain the static and dynamic behaviour of the materials. Then the Foroutan-Nemes rate dependant constitutive model is developed which governs the mechanical response of the woven composite material. The model includes damage variables that describe the mechanical effect of distributed micro defects, as well as formulations regarding the evolution of damage in the system. The third step is to develop a methodology for determining the material constants present in the constitutive model. And finally, the developed constitutive model is implemented into a material user subroutine of a commercial finite element program to solve boundary value problems in finite element simulations. 1.3 Structure of this Thesis A general introduction discussing the use of composite materials in the industry as well as the need to develop a rate dependant constitutive damage model has been presented in this chapter, Chapter 1. An extensive literature review is presented in Chapter 2, including a review of continuum damage mechanics. Different dynamic experimental set-ups as well as available ratedependant response of some composite materials are discussed. Also, a review of the available continuum damage mechanics models for composite materials is presented. 5 INTRODUCTION In Chapter 3, a detailed description of the experimental procedure is explained. A discussion about the theory behind the experiments and the related equations is presented in this chapter. A brief discussion about the materials under study has also been described here. The experimental results are presented and discussed in Chapter 4. The development of the rate-dependant continuum damage mechanics model is presented in Chapter 5. The characterization of material constants needed to make use of the model is explained in this chapter. Also, the methods to determine these constants from the experimental data are described. Numerical implementation and simulation procedures for justification of the material model are discussed in Chapter 6. The simulation results and findings are also presented. Finally, conclusions, comments, and recommendations for future work are presented in Chapter 7. 6 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 2 Equation Section (Next) LITERATURE REVIEW Impact loading on composite materials often causes a substantial amount of damage. The impact velocity, material properties, mass, and the geometry of the projectile are among several parameters that govern the type of damage. The most influential parameter is the impact velocity. Although low or intermediate impact velocities may not induce penetration, they could initiate surface micro cracks, delamination, and perhaps fibre fracture. However, high impact velocities, often called ballistic impact, may trigger complete penetration and local delamination. The penetration of a blade or blade fragment released from a failed rotor engine into the composite containment structure is an example of a ballistic impact. These fragments are dispersed circumferentially in all directions at very high velocities and could damage fluid lines, control systems hardware, and airframes if not contained. Such incidents can affect flying performance in a number 7 LITERATURE REVIEW of direct and indirect ways and can even lead to loss of aircraft and passenger fatalities. Therefore, a better understanding of the behaviour of composites under high velocity loading coupled with damage is required, particularly for the aerospace industries. When a material is loaded beyond its maximum strength, it experiences failure. In general, modelling the failure of materials is based on the following three approaches: • Failure criteria based on equivalent stress or strain • Fracture mechanics based on energy release rate • Continuum damage mechanics (CDM) The failure criteria based on the strength of materials approach only specifies the conditions under which failure occurs, but does not predict degradation in stiffness. A failure criterion is used to determine whether failure has occurred. Examples of failure criteria used by researchers are the Chang-Chang criterion (Chang and Chang 1987), the Hashin criterion (Hashin 1980), and that of Lee (Lee 1980). It is assumed that when damage occurs, the load carrying capacity of the material is lost. Depending on the type of failure, the modulus of elasticity in that direction is set to zero and in the case of delamination, the elastic modulus in the out of plane direction is set to zero. These models ignore the damage evolution leading to failure as well as energy dissipation mechanisms which are active during damage growth, and they have been mainly developed for modelling damage in unidirectional layers of laminated composites. These models can be used in low velocity loadings where the damage size is small. The objective of fracture mechanics is to study the fundamental mechanisms of fracture and to predict and control the process of fracture. The breakthrough in this field started at the beginning of the 20th century. Griffith (1920) is believed to be among the pioneers in this field in published papers related to the theory of fracture strength of glass. The theory assumed that the fracture strength is limited by the existence of initial cracks, which were subjected to an applied tensile stress. The basic concept behind his theory was the fact that solids possess surface energy, and in order to propagate a crack, the corresponding surface energy must be compensated through the externally added or internally released 8 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS energy. Griffith used Inglis (1913) solution, who had studied the stress analysis of a plate containing an elliptical cavity loaded by applied stress based on elasticity theory. His work is valuable in understanding the fundamental mechanism of fracture due to stress concentration at some points near the cavity or crack. A fracture mechanics approach is mainly used in composites when mechanics of macro cracks is under investigation. Many researchers (Chamis et al. 1996; Minnetyan et al. 1992) use this approach to model the progressive fracture in composite materials from initiation of crack to the final failure. However, it is worth mentioning, the application of this approach requires a prior knowledge of the crack specifications, e.g. the location of the crack. In continuum damage mechanics, deformation and micro failure in the material occurs simultaneously. A cracked composite body is treated as a homogeneous continuum containing representative internal damage variables. On loading, the composites develop multiple matrix cracks, which degrade the stiffness of the laminates and also give rise to more severe forms of damage such as delamination, interfacial debonding, sliding, fibre fracture etc. Continuum damage mechanics models can be used to predict stiffness degradation in composite materials. Continuum damage mechanics is also used to describe the strain softening behaviour of brittle or quasi-brittle materials such as geologic materials, concrete and wood as well as polymer composites. The aforementioned continuum damage mechanics is the adopted approach for the development of the damage constitutive model presented in this thesis, hence an extensive historical literature review of the knowledge and methodologies in this field that are relevant to this work is provided in this section. 2.1 Continuum Damage Mechanics The problems of idealized macro cracks are studied intensively in the framework of conventional fracture mechanics. Fracture mechanics is now an established science and has been a powerful tool in understanding the fracture of homogeneous solids and for extending the traditional design philosophy to include more accurate and realistic 9 LITERATURE REVIEW methods. However, in many more common cases, where defects are almost always at the micro-scale, especially prior to loading, and have complex geometries and topologies, the methodology employed in conventional fracture mechanics may lose its validity for modelling the evolution of a micro-crack system. For example, in materials such as ceramics, most composites, concrete, and rocks, the existing cracks are small and certainly far from being in simple shapes, such as a penny shape or planar. Cavities, and cracks with complex geometries can be assumed as damage. Accounting for every defect and assessing their interactions as well as their influence on the integrity and failure of the structure is not a task that can be approached using conventional fracture mechanics theories, such as linear elastic fracture mechanics and elastic-plastic fracture mechanics. For these cases, new theories and models needed to be developed to obtain a better understanding of fractures. Continuum damage mechanics (CDM) is a relatively new discipline that focuses on predicting the effects of progressive degradation of mechanical properties (damage evolution) on the macro-response and failure of solids. Damage mechanics deals with fracture in a more general way and seems to be a good approach for dealing with damage problems. Evolution of damage is related to nucleation and growth of micro-cracks and other micro-defects. The CDM approach can predict the full range of the deterioration of material, from the virgin material with no damage, to fully disintegrated material with full damage for different failure modes. CDM bridges the gap between traditional elasticity theory and fracture mechanics. The CDM-description of a defect-free material is equivalent to the application of elasticity theory. The CDM and elasticity theories will deviate as soon as micro defects start to nucleate and grow with increased loading. At the other end of the spectrum, when the growth of one or at most a few micro-cracks becomes the dominant aspect of the deformation of the material, a CDM-description will no longer suffice and the fracture mechanics theory is more suitable. 10 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS The CDM approach has been a subject of extensive research in the past decades. CDM smooths out the irregularities in shape and inhomogeneity of spatial distribution of microstructures of materials and thus make it possible to get a simple model that considers effects from the micro level while making predictions on the macro level. Kachanov (1958) and Rabotnov (1963) were the first to develop the concept of a scalar continuous damage variable to represent damage progression during creep of metals. They considered the ratio of void areas to total area in a cross section of a body as a measure of damage. Although their work was not directly in the field of CDM, the successful introduction of a separate internal state field variable, which provides a qualitative measure of the effects that randomly distributed micro-defects exert on the macro parameters of a structure, sparked the development of the concepts and theories which later became continuum damage mechanics (Chaboche 1988a; 1988b; Lemaitre and Chaboche 1990). The earliest research on applications of CDM can be found in works of Krajcinovic (Krajcinovic 1983; Krajcinovic 1984; Krajcinovic and Fonseka 1981; Krajcinovic and Silva 1982), Chalant (Chalant and Remy 1983), and Resende (Resende and Martin 1984). The CDM approach was later implemented for composite materials by Talreja (1985a). 2.1.1 Internal State Variables The type of internal state variables (damage variables) used in a CDM model greatly affects its success. The damage variables, besides being capable of representing the complex micro defect features, should be simple enough to make the model applicable to engineering applications. The area density definition of damage proposed by Kachanov and developed by Rabotnov is widely accepted in continuum damage mechanics (see Lemaitre and Chaboche 1990). Although Kachanov's approach is reasonable, it may be difficult to calculate the effective area of a cross-section directly from the crack geometry and distribution while taking stress concentration and interaction into account. It is known that the presence of a crack or defect causes a perturbation in the stress/strain and thus strain energy density in a certain volume around the crack or defect. That is to say, the effect of a crack on the strain energy density perturbation is volumetric in nature. 11 LITERATURE REVIEW Murakami (1983) defined damage as a micro structural change that results in a deterioration of material properties. It is therefore possible to define the macroscopic damage variables through a volumetric consideration to describe the mechanical state of the material. There are numerous ways to define the damage variable as discussed in Lemaitre (1996). Depending on the material, damage can be represented as a scalar, where a single variable can represent the complete damage state of a volume element of the material, as a vector, where the overall damage can be characterized by components in each co-ordinate direction, or as a higher order tensor. In composite materials, damage is usually related to crack density, however its definition and representation varies from model to model. For example, the pioneers in the field of CDM (Kachanov 1958; Rabotnov 1963), used the void area density in a cross-sectional area of a bar as the scalar damage variable for a one dimensional problem. Some other earliest proposed damage theories involved scalar damage variables based on the net reduction of load-bearing area due to micro-defect cavity growth. A scalar damage parameter was employed to describe the damage in composite materials by Pickett et al. (1990). They mentioned that the scalar representation was to simplify the model in the interest of computational efficiency. To explain a common damage definition, consider a volume element at macro scale, large enough to contain many defects, and small enough to be considered as a material point of mechanics of continua (Lemaitre 1996; Matzenmiller et al. 1995). Let S be the cross sectional area of this volume element, V m , defined by unit vector n , before any loading has occurred. This area becomes the effective area, S , after loading due to formation of micro cracks and void growth with an area of S d as illustrated in Figure 2.1. In the case of isotropy, that is when micro-cracks and cavities are distributed and oriented uniformly, 12 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS n S Sd Vm Figure 2.1 Meso-definition of damage there is no directionality involved, so the area no longer depends on n , hence the damage becomes an isotropic damage parameter represented by the following equation d= Sd S − S = S S (2.1) It’s worth mentioning that, the scalar damage variable, d , is defined in such a way that d equal to zero implies the initial undamaged state of the material, and when d is equal to one, it refers to the final failed state of the material, hence; 0 ≤ d ≤1 (2.2) Many models can be derived for damage, the difficulty being the choice of the analytic expression and its identification from experiments in each particular case of damage. Among several different methods, damage may come from the variation of Young’s modulus through elasticity law coupled with damage (Li et al. 1998; Zou et al. 2003). Relating damage to the influence it has on the stiffness is one of the most commonly used formulations. d = 1− E E0 (2.3) 13 LITERATURE REVIEW where E0 and E are the initial undamaged and current damaged stiffness, respectively. Li et al. (1998) also used a scalar damage parameter, however, its influence was related to the elastic constants of all directions. Budiansky and O'Connell (1976) assumed all cracks are elliptical and have the same aspect ratio, and derived an analytical expression for the effective elastic moduli as a function of the crack density parameter. When damage is isotropic or in other words it is randomly distributed, it can be represented through a scalar damage parameter. However, anisotropic damage requires a higher order tensor representation of damage. Due to the inevitable isotropic nature of a scalar damage variable, sometimes a more elaborate means to characterize the anisotropic damage observed experimentally in some materials is needed. In that case, more general states of damage can be captured using a second rank tensor. Second-rank tensor representation of damage was first proposed by Vakulenko and Kachanov (1971) and later developed by many other researchers (Kachanov 1980; Murakami and Ohno 1981). As such, a variety of symmetric second rank tensors were proposed for this purpose. A general formulation of this second order damage tensor, d , shown in principle directions n i , has the form 3 d = ∑ Ωi n i ⊗ n i (i = 1, 2,3; no summation) (2.4) i =1 where Ω i are the principle values of the damage tensor in a spectral representation. Importantly, if d is formulated such that Ω i = 1 /(1 − d ) for i = 1, 2, 3 , the resulting damage tensor provides a three dimensional generalization of the Kachanov-Rabotnov damage theory. The tensor d can be experimentally determined, and can be used to formulate a damage relation between the Cauchy and the effective stress tensors (Chow and Wang 1987). In general, some existing damage theories (Lemaitre and Chaboche 1990) use tensor variables to describe the state of the material (Allen et al. 1987). Nevertheless, as noted by Krajcinovic (1989), the mathematical complexity of the higher ranked tensors make them inexpedient for practical purposes. 14 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS In all the models damage is explained as a variable degrading a property of the original undamaged material. The most used type of variable for isotropic damage modelling is the scalar damage variable. Scalar variables mainly impose some constraints on the evolution of elastic properties. It is worth mentioning that isotropic damage does not mean that the original material is isotropic, it only indicates that the original isotropy or anisotropy of the material is preserved. In other words, in theory, it is possible for an orthotropic material to stay orthotropic after going through an isotropic damage process. In this work the state of damage at a point is described by a limited number of scalar damage variables, each representing a particular type of damage. They will be defined by the effect they have on the mechanical properties, in particular by the reduction in the elastic moduli. 2.1.2 Effective Stress The appearance of discontinuities leads directly to the concept of effective stress, that is, the stress calculated over the section which effectively resists the forces. If the stress reaches a certain value, damage starts to develop. This stress level is the damage threshold stress. After this, and due to presence of damage in the cross section, the resisting area is diminished and the effective stress will be acting on the resisting section. The mapping from the Cauchy stress tensor σ , to the effective stress tensor σ based on the area and effective area becomes straightforward as explained by Chow and Wang (1987), that is; σ=σ S σ = S 1− d (2.5) This is known as the Kachanov-Rabotnov damage theory, which was originally developed in one dimension, and formed the basis for the development of CDM. The effective stress concept states that any deformation behaviour, whether uniaxial, multi-axial, elastic, plastic or viscoplastic, of a damaged material is represented by the 15 LITERATURE REVIEW constitutive laws of the virgin material in which the applied stress σ is replaced by the effective stress σ . The deformation of damaged material depends not only on the damage parameters (net area reduction, for example), but also on their spatial arrangement. This implies that in the case of a second order damage tensor, the Cauchy stress tensor, σ , should be mapped to the effective stress tensor, σ , by a fourth order tensor. The fourth order damage effect tensor M( d ) , derived from the damage tensor , was developed for this purpose. The damage effect tensor provides a linear mapping of the stress tensor to the effective stress tensor, resulting in a redistribution of stresses over the reduced remaining area, and thus higher local stresses. σ = M (d ) σ (2.6) In the case of isotropic material behaviour, M( d ) should reduce to the scalar representation of the damage parameter as illustrated in the right hand side of equation (2.5). There are many proposed formulations for the damage effect tensor, M( d ) . Chow and Wang (1987) proposed an alternate form of the tensor M( d ) that was later discussed by Ju (1989) in the principle coordinate system. The 6x6 matrix representation of the damage effect tensor is as follows: 16 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS ⎡ 1 ⎢1 − d 1 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 M (d ) = ⎢⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢⎣ 0 1 1 − d2 0 0 0 0 0 0 0 0 0 1 1 − d3 1 0 0 0 0 0 0 0 0 0 (1 − d 2 )(1 − d 3 ) 1 (1 − d1 )(1 − d 3 ) 0 ⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 1 ⎥ (1 − d1 )(1 − d 2 ) ⎥⎦ 0 (2.7) In this formulation, d1 , d2 , and d3 are the damage variables at their respective principle axes and the effective stress tensor is used in its reduced format. It is important to note that the formulation in this equation does not depend on a priori knowledge of the principle stress directions. A special vector form of this formulation uses M( d ) as a 3x3 matrix of the damage variables represented in the principal form as follows: ⎡ 1 ⎢ ⎢1 − d1 ⎢ M (d ) = ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣ 0 1 1 − d2 0 ⎤ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 1 ⎥ ⎥ 1 − d3 ⎦ (2.8) where di ( i = 1,2,3 ) are the damage variables in their respective principle axes, and the effective stress vector is given as σ = [σ 1 σ 2 σ 3 ] . Note that the formulation does T reduce to isotropic material damage in the case of d1 = d2 = d3 = d . The damage effect matrix of equation (2.8) must be modified when the principle axes of the stresses are not known, and this restricts its use in general applications. 17 LITERATURE REVIEW Several other forms of the fourth order damage tensor M( d ) were discussed by Betten (1983) and Chow and Lu (1989). It is worth nothing that in each case the tensor M(d ) was defined by some seemingly arbitrary assumption aimed at making the effective stress tensor and researchers have used different forms which depend on the choice and form of the damage parameters. 2.1.3 Strain Equivalence Hypothesis A hypothesis of strain equivalence for isotropic damage was proposed by Lemaitre in which the Cauchy stress was replaced by the effective stress in the constitutive equation. In the hypothesis he states that (Lemaitre 1996): “Any strain constitutive equation for a damaged material may be derived in the same way as for a virgin material except that the usual stress is replaced by the effective stress.” For solving the damage problems, one must establish that the strain tensor is the same in the effective damaged and undamaged state, that is ε =ε (2.9) Combining the damage effect tensor, M( d ) , with the undamaged stiffness tensor, C 0 , results in a damage dependant stiffness tensor, C (d ) . The stress-strain relationship in a damaged material uses the effective stress and the damaged elastic stiffness tensor. A general constitutive relationship for a CDM model is represented as σ = C (d ) ε (2.10) It is worth mentioning that the damaged elastic stiffness tensor, C (d ) , is not guaranteed to be symmetric based on this principle. This elastic CDM model can be expanded to 18 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS consider the strain rate effects in the material behaviour (Armero and Oller 2000; Chaboche 1988a; 1988b; Dechaene et al. 2002; Nemes and Speciel 1996). σ = C (d , d ) ε + C (d ) ε (2.11) where the over dot represents the material time derivative. 2.2 Dynamic Tensile Experiments Setups The increasing use of composite materials in structures subjected to static and dynamic loading demands the full characterization of their mechanical properties under a wide range of strain rates. Unlike metals, which do not exhibit significant rate dependency at room temperature for rates below 1000/sec, composite materials may have this behaviour at much lower rates. The low rate tests are usually performed using hydraulic testing machines. On the other hand, various methods have been developed to characterize material behaviour at intermediate and high rate loading. Reliable data on the dynamic properties of composite materials is sparse, which could be due to the difficulties associated with the experimental procedures required in obtaining the dynamic response. Moreover, most of the results available in the literature are incomplete and as a result, cannot contribute to developing a reliable constitutive model for predicting the response of woven composite materials under dynamic loading. A review on the various methods used for characterizing the dynamic response of composite materials is provided in this section. The experimental method used to determine the dynamic properties depends on the range of strain rates concerned and on the load case. Some of the earliest systems are the Charpy pendulum (Harris et al. 1971; Hayes and Adams 1982), with fixtures adapted to perform tensile tests or to other test configurations, and Izod impact testing (Hancox 1971). These testing methods, in which a notched beam specimen is subjected to impact loading, were originally developed for use with isotropic materials, to obtain fracture energy. Even though high strain deformation can be achieved near the notches, 19 LITERATURE REVIEW fundamental formulation of strain rate effects on the material properties is not feasible due to the complex stress and strain fields as well as combined modes of failure involved in these methods. Drop weight, pendulum and explosively driven hammers can be employed to use unnotched specimens to obtain a uniform state of stress (Adams and Adams 1989; Armenakas and Sciammarella 1973; Lifshitz 1976; Rotem and Lifshitz 1971). These testing methods work more or less with the same principle, in which, a hammer driven by gravity, a spring or an explosive force hits an anvil connected, directly or indirectly, to one end of the specimen, as a result loading the specimen. Load in the specimen is usually measured by means of a load cell connected to the non-impacted side of the specimen. For strain measurements, however, different techniques have been used. Adam and Adam used pendulum impact tests, with strain rates of approximately 50/sec to test unidirectional graphite/epoxy and glass/epoxy under tensile loading. Lifshitz and his coworkers used a drop weight impact machine for angle ply specimens of glass/epoxy. Both tests need instrumentation of the specimen for strain measurements. Armenakas and Sciammarella used an explosively driven hammer to apply the load, and high speed photography and a Moire technique to measure the strain field. However their strain measurement was not uniform along the longitudinal and transverse axis, with as much as 100% difference. Therefore, they had to average their strain results to obtain the stressstrain curves. The non-uniformity of strain results is likely to hold true for the drop weight and the pendulum techniques due to the similarity of the methods. Moreover, these techniques are not free of stress wave reflection in the hammer as well as in the specimen. These waves are superimposed on the stress-time response of the specimen and create difficulties in interpreting the experimental results. The various experimental methods show that tests at high strain rates are quite difficult to perform and that they are not easy to interpret. The most popular method is the use of split Hopkinson bars, because they permit a uniaxial homogenous state of stress at very high strain rate. But this method is complex and needs to be suited to the range of materials to be studied. 20 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 2.2.1 Split Hopkinson Bar Setups Kolsky (1949) was the pioneer of the split Hopkinson bar apparatus; he introduced the compression version in which he used detonators to launch compressive waves. Over the last several decades, shear, torsion and tensile versions of the apparatus have been developed. Nowadays, an axial impact from a striker bar fired by an air gun is used to achieve high strain rate loading. The Hopkinson bar apparatus is capable of achieving strain rates between 100/sec and 10000/sec, as compared to the quasi-static loading rate usually taken to be 0.001/sec. Despite the well established methods of the Hopkinson bar, at least for conventional materials, not much data on the high rate mechanical response of composite materials is available. Even though compression testing is the simplest to perform, for several reasons, tensile testing is preferred for obtaining the high rate response of composite materials. The compression failure is not as simple and well defined as tensile failure, and the compression of composite in the fibre direction is usually of little interest. This is due to the fact that failure occurs as a result of matrix cracking and interfacial debonding rather than by any type of fibre breakage. It is shown by many researchers that, even in the static strain rates, composite materials have different behaviour in compression and tension. Researchers have developed different tensile versions of the Hopkinson bar for testing composite materials. Generation of a tensile pulse as well as specimen gripping are the most noted difficulties in the tensile Hopkinson bar testing technique. Unlike metals, which can be fastened easily by the use of threaded ends, composite specimens, which are usually in the form of flat strips, cannot be threaded. In addition, to produce high strain rates, Hopkinson bar testing requires a short gage length in the specimens; hence the specimens cannot be designed according to standards for uniaxial static testing of composite materials (ASTM 1989). 21 LITERATURE REVIEW Several arrangements have been described in the literature for tensile testing. In general, tensile versions of the Hopkinson bar use two approaches for applying tensile pulse to the specimen. In one approach, the input bar is subjected to a direct tensile pulse which is in turn transmitted to the specimen. In the other approach, the input bar is subjected to an initial compressive pulse, which is subsequently transformed into a tensile pulse. APPROACH ONE, DIRECT TENSILE PULSE In this approach a clamp/load reaction assembly was used to pre-stress a portion of the input bar (Albertini and Montagnani 1974; Staab and Gilat 1991). Initially the clamp was tightened and a direct tension is applied at the end of the bar through a system of cables pulleys and hydraulic pump, which generates an input load stored in that section of the bar. On releasing the clamp, a tensile wave of half the magnitude of the stored force propagates towards the specimen as the tensile load, and a released wave propagates in the opposite direction (Gilat et al. 2006) (Figure 2.2). specimen clamp applied load Figure 2.2 Schematic of Hopkinson bar configuration using direct tensile pulse APPROACH TWO, INDIRECT TENSILE PULSE In the second approach, the researchers used a normal compressive version of Hopkinson bar in order to produce tension. This was first suggested by Nicholas (1981), and was later used by other researchers (Al-Mousawi et al. 1997; Ellwood et al. 1982; Peroni and Peroni 2008). In this design, a collar made of the same material as the set-up bars is placed over the specimen and firmly fitted between the incident and transmitted bars (Figure 2.3a). When the input bar is struck by a striker, the input compressive wave is transmitted almost entirely through the collar to the output bar, with little effect on the specimen. On reaching the free end of the transmitted bar, it is reflected back as a tensile 22 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS pulse. This tensile pulse loads the specimen, since the collar is unable to sustain any tensile load. Lindholm and Yeakley (1968) also used a compressive Hopkinson bar setup to obtain tensile behaviour. They fitted a special ‘hat’ shaped tensile specimen between a solid cylindrical incident bar and a tubular transmitter bar (Figure 2.3b). The applied loading and deformation of the specimen are derived from the strain-time measurements on the radial surface of the two bars using diametrically opposed strain gages. The internal wave propagation is neglected in this analysis. They investigated the effect of the unusual geometry of the ‘hat-shaped’ specimen by comparing its static stress strain behaviour with that of round tensile test specimens. Even though they found good agreement in their results, later Lindholm et al. (1971) have reported that the effect of the ‘hat shaped’ geometry of the specimen was the only source of error for the drop in the strength of an aluminum alloy. Kawata et al. (1980) developed a bar-to-block setup (Figure 2.3c), which consists of a hammer (rotating disc or pendulum type), impact block, specimen and output bar. The impact block is impacted by the hammer and the specimen is deformed. They believe that this is a simpler method as compared to the bar to bar method. In the other designs, the compression version of Hopkinson bar is significantly modified to produce tension in the specimen. The first design was developed by Harding et al. (1960) where the input loading bar is a hollow tube within which the output bar slides freely. The specimen connects the two bars at the yoke. Harding and Welsh (1983) modified this model some years later to include an instrumented input bar preceding the specimen (Figure 2.3d). Leblanc and Lassila (1993) used the same concept, however changed the hollow compression loading tube to a split compression tube in order to allow easy access to the incident and the transmitter bars during the installation of the sample grip assembly. Similar concept was used by Hauser (1966) and later by Eskandari and Nemes (2000) where instead of using the tube, two transmitter side bars was used to transform the compressive pulse into a tensile pulse (Figure 2.3e). 23 LITERATURE REVIEW specimen striker bar collar (a) specimen striker bar (b) applied load impact block specimen output bar (c) specimen transmitter bars transfer connection input bar specimen striker (d) (e) specimen impact block (f) 24 striker tube thin part thick part of input bar damping HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Figure 2.3 Various Hopkinson bar setups using indirect tensile pulse adapted from (a) Ellwood et al. (1982), (b) Lindholm and Yeakley (1968), (c) Kawata et al. (1980), (d) Harding et al. (1960), (e) Eskandari and Nemes (2000), and (f) Li et al. (1993) Some researchers have used a short striker tube around the input bar (Huh et al. 2002; Lataillade et al. 1995; Li et al. 1993). The striker loaded by a hammer or a gas gun impacts the end of the thicker part of the input bar. This produces a tensile pulse in the input bar which travels to the specimen (Figure 2.3f). The apparatus used in this research project is similar to those which use a striker tube around the input bar. This is the apparatus available at McGill University and its specifications will be explained in chapter three of this thesis. In that, a striker tube which is loaded with a gas gun impacts an anvil, which in turn produces a tensile pulse that loads the specimen in tension. 2.3 Strain-Rate Dependant Experiments Traditionally, composite materials have been assumed to have a linear elastic behaviour. However, results from the literature introduce doubts concerning this assumption. The stress-strain curves obtained by many researchers have shown significant non-linearity. In particular, at high strain rates the non-linear relationship between strains and stresses is more evident. A number of researchers have reported high strain-rate behaviour of composites, mainly noting that the failure stress and mechanism are strain-rate dependent. Abrate (1991; 1994; 2001) has presented a comprehensive review of the literature on impact on composite materials. As the rate of loading is increased, there is less time for damage to develop. Hence, the amount of accumulated damage at a particular strain level decreases as strain rate increases, and at high strain rates the material can withstand higher load and failure strain. 25 LITERATURE REVIEW Harding and Welsh (1983) performed impact tensile tests on both GRFP (Glass Reinforced Fibre Plastic) and CFRP (Carbon Reinforced Fibre Plastic), which are unidirectional composites. They found strain rate sensitivity in the experiments performed at the strain rate range of about 5E-5/sec, 7/sec, and 450/sec for CFRP and 1E-4/sec, 23/sec, and 870/sec for GFRP. Newill and Vinson (1993) tested various polymer matrix composites. The strain rates involved varied from static values up to 2000/sec and they found that strain rate effects are significant in general for all composite materials, or at least one should assume so until proven otherwise by numerous tests and subsequent analysis. Staab and Gilat (1995) explained the rate sensitivity of angle ply glass-epoxy laminates and in a later paper Gilat et al., (2002) showed the significant effect of strain rate on the response of carbon/epoxy composites. They conducted tensile tests on the same type of specimen over a wide strain rate range with tests performed at 5E-5/sec, 1/sec, and 400/sec. They showed that both unidirectional and angle ply composites are strain rate dependent. The shear results from the tests with [ ±45°]S specimen orientation also show considerable strain rate dependence (Goldberg et al. 2003). In general, with increasing strain rates, a stiffer material response is observed, that is, the stress-strain curve is steeper for higher strain rates. It is observed that, at all strain rates, the stress-strain curves have an initial linear response, a rounded transition to inelastic response followed by an inelastic range with a nearly constant hardening rate. Weeks and Sun (1998) and Jadhav et al.(2003) studied the rate dependent behaviour of angle ply composites of AS4-PEEK and graphite-epoxy, respectively at various strain rates of about 1E-5/sec for static and a moderate to dynamic strain rate range of 100/sec to 1000/sec. Rate dependency is confirmed in angle ply composites from their results. Their experiments were conducted with a servo-hydraulic testing machine and a tensile split Hopkinson pressure bar setup. 26 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Barre et al. (1996) have conducted experiments on five materials all reinforced with glass. They showed strain rate has a significant effect on tensile modulus and tensile strength, and also, noted significant discrepancies in the papers they reviewed. For a given material, some studies report an increase in tensile strength, some find negligible changes, while others find a decrease. Most of the available test data do not provide a complete characterization of the material under investigation. Kawata et al. (1981), Welsh and Harding (1985), Harding et al. (1989), Rodriguez et al. (1996), and Sham et al. (2000) studied the tensile behavior of woven composites. Excluding Kawata et al. who performed their dynamic tensile experiments using the one bar method of block-to-bar with a rotating disc or pendulum as the loading equipment, the other researchers used a tensile version of the Hopkinson bar apparatus for the dynamic experiments. The results from Kawata et al. show (Figure 2.4) that glass woven cloth/polyester and carbon woven cloth/epoxy composites are strain-rate sensitive. It can be seen that at the static strain rate the stress-strain curve is linear but at dynamic rates the behavior becomes non-linear and also the peak tensile stress increases. 80 Dynamic 70 50 Stress [Kg/mm 2 ] Stress [Kg/mm 2 ] 60 40 Dynamic 30 20 Static 10 60 50 40 30 20 Static 10 0 0 0 5 Strain % 10 15 0 5 Strain % 10 Figure 2.4 Stress-strain curves at dynamic and static strain rates in tension (a) Glass Satin Woven Cloth/ Polyester (b) Carbon Plain Woven Cloth/ Epoxy (adapted from Kawata et al. 1981) 27 LITERATURE REVIEW Harding and Welsh (1983; 1985) used a tensile Hopkinson pressure bar apparatus to characterize the response of composite materials. They obtained tensile stress strain curves at strain-rates of about 1E-4/sec, 100/sec, and 1000/sec for polyester resin specimens reinforced with satin-weave fabric using carbon, Kevlar and glass fibres. For all three composites, they observed rate-dependence of the tensile modulus. Harding et al. (1989) performed experiments on woven carbon fibre epoxy laminates at quasi-static and dynamic rates (9E-4/sec, 290/sec), and found that the laminates were strain-rate sensitive as illustrated in Figure 2.5. The stress-strain curve results show significant increase in the tensile failure strength with strain rate. 600 Impact test ~ 290/sec Stress [MPa] 500 400 300 Quasi-Static test ~ 9*10 -4/sec 200 100 0 0.0 0.5 1.0 1.5 Strain % Figure 2.5 Stress-strain curves for woven carbon fibre epoxy laminates at different strain rates in tension (adapted from Harding et al. 1989) Rodriguez et al. (1996) performed tensile tests at three strain rates of approximately 0.001/sec, 1/sec, and 1000/sec on aramid and polyethylene woven fabric composites using a tensile Hopkinson bar and conventional testing machine. Some of their experimental results are shown in Figure 2.6. Strain-rate sensitivity is apparent from these results as well. 28 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS (a) (b) Figure 2.6 Tensile stress-strain curves at different strain rates for (a) woven aramid/polypropylene (b) woven polyethylene/ polyethylene (Rodriguez et al. 1996 with permission of Elsevier) Sham et al. (2000) studied the tensile behaviour of glass woven fabric reinforced composites. Figure 2.7 shows the stress-strain curves at various strain rates ranging from about 1E-4/sec to 900/sec. According to their observation, the tensile strength and failure strain of GFRP increased with increasing strain rate due to the brittle-ductile transition of glass fibre, and also the strain rate sensitivity was higher in the warp direction than in the weft direction. Unlike the tensile response, not much data is available on the comparison of in-plane static and dynamic shear response of woven composite materials. However, it is worth mentioning that it is well understood in the literature that the behaviour in shear is strongly nonlinear and irreversible as was mentioned in many papers (Johnson et al. 2001; Marguet et al. 2007). This could be from the strain rate dependant results available for unidirectional [45°] specimens tested in static and dynamic rates. Many researchers (including Hosur et al. 2003b; Naik et al. 2007; Riendeau and Nemes 1996; Werner and Dharan 1986) have investigated the interlaminar shear response of composite materials. 29 LITERATURE REVIEW 800 Stress [MPa] 900/sec 600 600/sec 400 0.18/sec 1.67E-4/sec 200 0 0% 1% 2% 3% 4% 5% Strain % Figure 2.7 Stress-strain curves of glass woven fabric-reinforced composites subjected to different strain rates (adapted from Sham et al. 2000) Werner and Dharan (1986) obtained the interlaminar shear as well as transverse shear response of woven graphite epoxy laminates using split Hopkinson bar. They concluded that in transverse shear, the strain rate effect is significant but not so in interlaminar shear response. Chiem and Liu (1987) have obtained the shear response of woven carbon-epoxy composite materials using a torsional split Hopkinson bar. The strain rates varies from 6E-4 to 2E-3/sec for quasi static loading and from 2623/sec to 5410/sec for dynamic loading. In another paper (Chiem and Liu 1988) they compared the tensile and shear strength of these materials subjected to very high strain rates from 612-1368/sec for tensile and 1021-5410/sec for the shear loading. They concluded that these materials are strain rate dependent in both tension and shear, which according to them can be described by a power law function. Hsiao et al. (1999) conducted dynamic and static experiments from quasi static to 1200/sec on [45°] unidirectional off-axis carbon epoxy laminates specimen. They 30 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS investigated the strain rate effect on the in-plane shear behaviour of the unidirectional composite and from their results concluded that the shear stress-strain curves show high nonlinearity with a plateau region at a stress level that increases with increasing strain rate. Hou and Ruiz (2000) performed tensile and in-plane shear tests at different strain rates of 4/sec and 600/sec on woven CFRP T300/914. Their in-plane shear experiments were conducted with a split Hopkinson pressure bar on 5-ply [±45°] specimen. From their results, they concluded that the shear response is strain rate dependent; however, they observed linear elastic behaviour for the specimens under tension. From the above discussions it is evident that the advanced composite materials are strain rate dependent. The effect of strain rate is clearly distinct and significant as observed from the various stress-strain responses at different strain rates. It is worth noting that not much data is available in literature for characterizing the behaviour of carbon or graphite woven composite materials. However, in all the configurations tested, higher stiffness is observed with increasing strain rate. Since a rate-dependent constitutive relation is required for adequately modelling the failure in composites, in particular woven fabric composites which is the material used for this study, a review of the available CDM models is presented in the next section. 2.4 Review of CDM Models for Composite Materials The general theory of continuum damage mechanics is applied to the specific problem of fibre reinforced composites. Continuum damage mechanics is developed on the concept of material science and continuum mechanics. It is based on the thermodynamics of irreversible process, internal state variable theory and consideration of physical material response. Decisions must be made regarding the mathematical approach to formulate the damage parameters, as well as experimental approaches to characterize these parameters. Lemaitre (1996) postulated the strain equivalence principle. Alternative theories were postulated under this scenario; such as the elastic energy equivalence concept, the free 31 LITERATURE REVIEW energy function, and damage energy release rate. In any case, CDM introduces the effective stress tensor, to replace the traditional Cauchy stress tensor, in stress-strain relationships for solving boundary value problems. In general, the development of CDM models involves three primary issues: damage variables, damage evolution laws and the constitutive equations which are present in most CDM models. Since damage presents itself in the form of cracks in the material, it is a function of crack density. As explained before, there are numerous ways to model the damage variables. Due to different modes of damage in composite material, i.e. matrix cracking parallel to the fibres, fibre breakage, and fibre matrix debonding, each damage variable can represent a specific type of damage in the models. In the CDM models available in literature, different assumptions exist for defining the damage variables (Woo and Li 1993). The term damage evolution is commonly used to designate a non-linear deformation process during which two different phenomena take place, namely, nucleation of micro cracks and growth of the existing micro-cracks. The evolution equations of damage in the CDM models are perhaps the most arbitrary part of the model development. The evolution equations very much depend on how the defined damaged variables behave in a particular form of material. Various forms and logic are used by researchers which will be explained in more detail. Talreja (1985a; b) was one of the first researchers who used CDM for modelling composite materials. He assigned different damage parameters to the fibre and matrix directions in unidirectional lamina (Talreja 1989). The model was used to predict the stiffness reduction in angle ply laminates, however no detail was provided for the damage growth. The model was modified later to include the rate effects for predicting the viscoelastic behaviour of cross-ply laminates with transverse cracks (Kumar and Talreja 2003). 32 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS A simple CDM model was developed by Pickett et al. (1990). A number of assumptions were incorporated to have a computationally efficient model; however, this limited the capability of the model. Their model had linear damage development, i.e. a linear modulus reduction with a scalar representation of damage. Among other works, CDM models of Aboudi (1987) and Harris et al. (1988) also used a linear stiffness reduction with the damage parameters. Matzenmiller et al. (1995) constructed a CDM model based on the work of Talreja (1985a). The model consists of three damage parameters, two of which are related to the in-plane lamina directions, and one is associated with the effect of damage on shear. Their damage growth model was based on a Weibull distribution of strength. Williams and Vaziri (1995) implemented the model into a finite element code. Each damage growth parameter has the form ⎡ 1 ⎛ ε d = 1 − exp ⎢- ⎜ ⎢ me ⎜⎝ ε f ⎣ ⎞ ⎟⎟ ⎠ m⎤ ⎡ 1 ⎛ Eε ⎞m ⎤ ⎥ = 1 − exp ⎢- ⎜ ⎟ ⎥ ⎥ me X ⎝ ⎠ ⎥⎦ ⎢ ⎣ ⎦ (2.12) where E, ε f, and X are the elastic modulus, nominal failure strain, and strength of the material associated with the damage mode respectively, e is the base of the natural logarithm, and m defines the shape of the damage growth curve. The higher the m value, the more abrupt the descending branch of the stress-strain curve beyond the nominal peak stress and hence more instantaneous is the failure. The low m value represents a more gradual strain-softening response. The assumptions behind the formulation of their model are (Nandlall et al. 1998): (i) a quasi-isotropic laminated composite can be represented as an axisymmetric homogeneous continuum, irrespective of the damage state, (ii) applying linear elasticity provided the damage state does not change, (iii) damage is the only cause of non-linear behaviour of the composite lamina, that is no nonlinear elastic or plastic deformation of the constituents exist, (iv) damage takes the form of distributed disk-like cracks oriented 33 LITERATURE REVIEW parallel or normal to the principal material directions and as a result, symmetry of the lamina is preserved, and (v) The state of damage present in the material is characterized by a set of directionally-dependent damage parameters. Their work was further improved by Floyd (2004), who among other things included a technique that explicitly accounts for the energy dissipation by single elements. However, even though both works found that rate dependency is an important factor, it was not considered in their model. Wang and Xia (1997) used a bimodal Weibull distribution to describe the strength distribution of fibres and their damage growth parameter has the form of ⎡ ⎛ Eε ⎞m1 ⎛ Eε ⎞m2 ⎤ d = 1 − exp ⎢ − ⎜ ⎟ −⎜ ⎟ ⎥ σ ⎢ ⎝ σ 01 ⎠ ⎝ 02 ⎠ ⎥⎦ ⎣ (2.13) where m1, m2 and σ01, σ02 are the shape and scale parameters of the double Weibull distribution function. Wang and Xia (2000) assumed the unidirectional composite as a coated fibre bundle which, consists of a number of parallel coated fibres. Every coated fibre has the same cross-sectional area and length. They developed a one-dimensional damage constitutive equation for unidirectional composites. The model included strainrate effects in a double Weibull distribution and has the form, ⎡ ⎛ E ε ε ⎞m1(ε ) ⎛ E ε ε ⎞m 2 (ε ) ⎤ ( ) ( ) ⎥ σ = E ( ε ) ε exp ⎢− ⎜⎜ −⎜ ⎟⎟ ⎜ σ ( ε ) ⎟⎟ ⎢ ⎝ σ 01 ( ε ) ⎠ ⎥ ⎝ 02 ⎠ ⎣ ⎦ (2.14) It was found that with increasing strain rate, there is a tendency to increase Young’s modulus (E), maximum stress (σmax), and strain relative to maximum stress (εm). Therefore E, m1, m2, σ01, and σ02 are all function of strain rate. Several researchers have been concerned with the development of CDM laws for composites failure using finite element codes. Ladeveze and his coworkers (Ladeveze 1994; Ladeveze and Le Dantec 1992) developed a CDM model for unidirectional ply 34 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS composite materials. In their model, two scalar damage variables are considered; one normal to the fibre, that is 11- or 22-direction and the other in the shear 12-directions. The total strain is the summation of the elastic and plastic strains. They assume that no plastic deformation occurs in the fibre direction. Johnson et al. (2001) then extended Ladeveze’s model to fabric-reinforced composites under in-plane loads. Their model contains elastic damage in fibre directions and elasticplastic damage for inelastic shear effect. Neither of the models is rate-dependent; however, the delamination failure envelope is incorporated in their model. They defined three damage parameters; associated to damage in the two principal directions and inplane shear. Their damage parameters are in terms of the damage evolution function, which is a linear or log linear function of damage energy release rates. The plastic strains are associated only with the matrix dominated in-plane shear response, and the total strain is the summation of the elastic and plastic strains. Randles and Nemes (1992) and Nemes and Speciel (1996) developed a rate-dependent CDM model for thick composites subjected to high strain-rate deformation. It is worth noting that their model was shown to lead to unique and stable solutions and was mesh insensitive for a realistic range of damage propagation time constants. The model was used to simulate cantilever bending tests on graphite/epoxy at high strain rates. They did not show any comparison with real test data. Later, Eskandari (1998) developed a three-dimensional rate-dependent continuum damage model. The model considers a quasi-isotropic and laminate-based model in which the damage is assumed isotropic; i.e. the original isotropy or anisotropy is preserved. In these models, even though they seem to include strain rate effects in their formulation, the effect of various loading rates on the initial undamaged elastic coefficients is not considered. Dechaene et al. (2002) developed a constitutive damage model for woven glass fibre/ epoxy composite materials in which strain-rate effects are included into the model using a 35 LITERATURE REVIEW damage lag methodology. In this CDM model the effect of damage in the warp and weft are considered individually. One interesting innovation in their model is including rate effects in the evolution of damage equations. Their model is a stress rate formulation consisting of two parts. The first term is the elastic change in stress with reduced elastic modulus due to damage, and the second term is the drop in stress due to growth of damage resulting from matrix cracks as well as fibre failure (Iannucci et al. 2001). The latter term is a function of damage and rate of damage. σ = E ε − σ d (σ , d , d ) (2.15) Dechaene et al. proposed the rate of damage as an increasing function of stress and damage, which consists of the two stages of damage development: nucleation and growth. Since the rate of damage is expressed as a function of stress and damage, the damage lags behind the stress. In their model each fabric warp or weft layer is treated as an equivalent unidirectional layer, i.e., a 0/90 cross-ply composite, even though, the behaviour of 0/90 cross-ply composites can be different from that of the woven fabric composite. They also ignored the rate-dependent shear damage, which is reported in the literature by many researchers (Gilat et al. 2002). Quantitative comparisons to experiment results were not included. Iannucci (2006) continued the work of Dechaene et al. and proposed an in-plane failure model for a thin woven carbon composite. His approach is based on an unconventional thermodynamic maximum energy dissipation approach, which controls the damage evolution and hence energy dissipation per second, rather than damage. In the rate dependant model he assumes that behaviour of the woven carbon composite under tensile loading is linear until failure occurs in fibre bundles, that is, no damage is assumed for matrix cracks. Hence, in his model, the initial undamaged modulus is assumed unaffected by an increase in strain rates. However, many experimental results in the literature including Gilalt et al. (2002) and Harding et al. (1989) have shown the initial undamaged modulus is affected by increasing the strain rate from static to dynamic. For 36 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS the non-linear shear evolution response, the model predicts the failure at much higher strains than expected, and the use of a limited number of constants to change the shape of the curves, was blamed for the poor shear response. The energy based CDM model of Iannucci and Willows (2006) extends the previously published models (Dechaene et al. 2002; Iannucci 2006) to include evolution of damage as a function of strain, rather than damage energy rate. In the incremental form, the rate of damage related to the warp and weft fibre fracture case (i = 1, 2) is represented by Δdi = ⎡ ε 0,i ⎤ ε max,i ⎢ ⎥ Δε ii (ε max,i − ε 0,i ) ⎢ ε ii2 ⎥ ⎣ ⎦ (2.16) where ε 0,i and ε max,i are the strains at initiation with damage equal to zero, and at zero stress with damage equal to unity, respectively. Their model includes compression failure, and the approach uses a bilinear stress-strain law for the material response. However, again a linear behaviour until failure is assumed. An interface delamination model is also defined; however a prior knowledge of the propagation path is required to place the interface elements where delamination is expected. Marguet et al. (2007) have also developed a rate dependent constitutive model. Their damage model is coupled with viscoplasticity, that is, they split the total strain into an elastic part and a viscoplastic part. Their model assumes a brittle rupture of fibres under tensile loading, and hence does not consider the nonlinear strain rate effects in the normal direction. They do not have a real material response comparison with their model; hence a “pseudo experimental data” from their proposed model is generated. A material parameters identification method is developed which is based on a pattern search optimization algorithm, where the solution obtained is expected to give a satisfying behaviour to the model. To the author’s knowledge, there is no reliable accepted constitutive model to predict the detailed mechanical response of woven composite materials. Most of the models are not 37 LITERATURE REVIEW rate dependant, and some models include damage only in the shear direction (Johnson et al. 2001; Marguet et al. 2007). Some of the models, that include rate dependency, assume damage only in the strain softening part of the response (Eskandari 1998; Iannucci and Ankersen 2006; Nemes and Speciel 1996). Some consider damage mainly in the shear direction (Marguet et al. 2007), while others do not include shear damage in their model (Dechaene et al. 2002). Overall, to summarize, the available models do not include rate effects, are too complicated to understand, or are unable to provide a clear procedure for characterizing the model parameters, are unable to accurately match experimental tests, or require too much computational power, and so on. Consequently, as was mentioned in Chapter 1 in the objective section, a constitutive model is needed to predict the response of woven composite materials under high strain rate loading. To do so, the damage strain rate dependant model proposed by Dechaene et al. needs to be modified to include the shear damage besides providing a quantitative comparison to experimental results, which are performed in this research. Moreover, a methodology must be also established for determination of the damage parameters. In addition, numerical models are required to simulate the high velocity experiments using the material model. This is a complex problem that requires a thorough understanding of composite behaviour under dynamic loadings. It is believed that this research will prove directly useful for aerospace industries. Given the above theoretical framework, a system to solve problems can be established by means of material science, continuum mechanics and finite element computational methods using the following steps: 1. Perform experiments to obtain the behaviour of the material under different loading conditions, 2. Develop mechanical variables (internal damage state variables) describing the mechanical effects of distributed micro-defects, 3. Develop equations to govern the mechanical response of the damaged material (constitutive equation) generally by empirical means, 38 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 4. Formulate equations to describe the evolution of damage variables (evolution equations), and 5. Use the above equations to solve boundary value problems, commonly with FEA tools. 39 LITERATURE REVIEW 40 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 3 Equation Section (Next) EXPERIMENTAL SETUP 3.1 Introduction As discussed in chapter two, composite material structures are more and more often subjected to static and dynamic loading. At room temperature, composite materials, in general, exhibit rate dependency at much lower rates compared to metals (Eskandari 1998). Therefore, tests need to be conducted over a wide range of strain rates to properly characterize the mechanical properties of these materials. This chapter discusses the experimental procedure and the tests used to determine the mechanical properties of the materials used in this study, which are woven composite materials used in aerospace applications. 41 EXPERIMENTAL SETUP 3.2 Materials The aerospace need for lightweight, high performance vehicles subjects materials to extreme service. Textile composites offer unique combinations of properties that cannot be obtained using conventional tape laminates. The primary advantages of textile composites are their high speed preform manufacturing as well as their increased damage tolerance due to the tow interlacing. Also, there is the potential for large reductions in part count due to the ability to create complex preforms. They can be mass produced and are cost-effective as compared to conventional tape laminates. Textile composites are being used in applications ranging from prostheses for amputees to shrouds to capture debris from a failed engine. A number of textile manufacturing techniques are available to make fibre preforms. Braiding, weaving and knitting are the dominant forms of textile manufacturing techniques. They all share the characteristic that fibre tows are interlaced to create a preform that is impregnated with resin to make a composite. Woven fabric composites are a two dimensional class of textile composites where the warp and fill fibre tows are woven into each other to form a layer. Textile composites have complex microstructures characterized by tow undulation and interlacing. The tow consists of thousands of fibres. The woven structure is characterized by the orthogonal interlacing of two sets of tows called the warp and the fill tows. The fill tows, also called weft, run perpendicular to the direction of the warp tows which are along the fabric direction. Plain weave, twill weave, and satin weave are the dominant forms of woven architectures and are shown in Figure 3.1. In all the cases, the tows have both the undulated and straight regions except for the case of the plain weave in which the entire length of the tow of both fill and warp tows is undulated. But in the case of other weaves, the tows have some straight region before starting to undulate. 42 HIGH STRAIN IN RATE BEHA AVIOUR OF WOVEN O COMPO OSITE MATERIIALS (a) ( (b) (c) F Figure 3.1 Schematic S a picture of differentt weaves of woven and w comp posite: (a) plain p weave, (b) ( 2×2 twilll weave, (c)) 8-harness satin s weavee T plain weeave is the simplest forrm where warp The w and weeft yarns arre interlacedd in a reegular sequeence of ‘one over and onne under’ inn each directtion as show wn in Figure 3.1a. A Altering this sequence will w result in a differeent structuree. The plainn weave haas the m maximum fab bric stabilityy and firmneess with minnimum yarnn slippage. The T pattern gives unniform stren ngth in two directions d w when yarn sizze and countt are similarr in warp and fill. T This weave type is beliieved to bee most resisstant to in-pplane shear and is therrefore coonsidered to o be a stiff weave. T twill weeave is moree pliable thaan the plain weave andd drapes withh less resisttance. The H Hence, these structures are a better foor compoundd curves thaan plain weaave structurees. At thhe same timee they mainttain more fabbric stabilityy than a four or eight harrness satin weave. w T weave pattern The p is chharacterized by a diagonnal rib creatted by a ‘tw wo over andd two unnder’ sequen nce. As show wn in Figure 3.1b, the first f weft paasses over warps w 1 and 2 and unnder warps 3 and 4; thee next weft passes p over warps w 2 andd 3 and undeer warps 4 and a 5; annd so on. Tw will weaves feel f generallly tighter, orr more closelly woven, thhan plain weaaves. 43 EXPERIMENTAL SETUP A satin weave structure is one of the easiest fabrics to use and it is ideal for laying up contoured surfaces with minimal distortions. These weave patterns are most pliable and can comply with complex contours and spherical shapes. The ‘harness’ number (eg. 4harness, 5-harness, 8-harness, etc.) indicates the number of tows passed over or under before the tow repeats the pattern. For example, in an 8-harness satin weave, one warp yarn is carried over seven then under one weft yarn as shown in Figure 3.1c. This weave is less stable than plain or twill weaves. Also, anti-symmetry and coupling effects are present in these laminates and they generate undesired effects such as warpage and temperature sensitivity (Ishikawa and Tsu-Wei 1982). It is important to note that the mechanical properties of woven fabrics are governed by weave parameters such as weave architecture, yarn size, yarn spacing length, fibre volume fraction as well as by laminate parameters such as stacking orientation and overall fibre volume fraction. Woven fabrics in general have good dimensional stability in both the warp and fill directions but low in-plane shear stiffness. Three types of materials, with different characteristics, are used in this research. They are summarized in Table 3.1. The first composite material consists of 6 plies of carbon/epoxy prepreg with a 2×2 twill weave as shown in Figure 3.1b. The second material also consists of 6 plies and it is a carbon/Bismaleimide (BMI) composite with 8 harness satin weave as illustrated in Figure 3.1c. The third material as shown in Figure 3.1a, is a plain weave carbon/epoxy composite with 8 prepreg plies. All these materials can be considered as balanced woven fabrics with a zero direction staking sequence. Table 3.1 Summary of the materials used in this research Description Type of weave Staking sequence Thickness Material-1 carbon/epoxy prepreg 2×2 twill [0°]6 1.8 mm Material-2 carbon/BMI prepreg 8 harness satin [0°]6 2.3 mm Material-3 carbon/epoxy prepreg Plain [0°]8 1.6 mm 44 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 3.3 Dynamic experiments To perform dynamic experiments, a tensile version of Hopkinson bar was used. This apparatus was designed and fabricated at McGill University. It can produce an average strain rate of up to 500/sec on a woven composite material with a gage length of 10mm. 3.3.1 Hopkinson Bar Setup and Theory In this configuration of tensile Hopkinson bar, the high strain rate loading is achieved by an axial impact from a striker bar fired by an air gun. Figure 3.2 shows a picture of the Hopkinson bar and Figure 3.3 schematically represents the test set-up in which the specimen is held between two pressure bars and is loaded by a single traveling pulse in tension. The experimental set-up works as following: the striker, which is accelerated by the gas gun, hits the anvil, hence, generating a tensile pulse through the input bar towards the specimen located between input and output bars. On reaching the specimen, a part of this tensile incident pulse is reflected as a compressive pulse, and the rest is transmitted to the output bar. Figure 3.2 Hopkinson bar setup 45 EXPERIMENTAL SETUP Using the well established elementary linear elastic wave propagation theory, the stress, strain, and strain-rate versus time are calculated from the records of strain gages on the input and output bars. Based on this theory, the strain in the specimen is directly proportional to the time integral of the reflected pulse and stress is directly proportional to the amplitude of the transmitted pulse (Al-Mousawi et al. 1997). Fixings Output Bar Input Bar Pressure reading P Anvil Specimen Gas Gun Strain Gauges Transmitted Pulse Striker Incident Pulse Reflected Pulse Figure 3.3 Schematic representation of Hopkinson bar Taking a short specimen of initial length, l0 , (Figure 3.4) sandwiched between the two bars, and taking the displacements at the ends of the specimens as u1 and u2 , the average strain in the specimen can be expressed as l0 output bar input bar εI εR εT u2 P2 u1 P1 Figure 3.4 A specimen sandwiched between the input and output bars 46 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS εS = u1 − u2 l0 (3.1) where the displacements, in terms of incident, reflected and transmitted strains, are given by t t t u1 = cb ∫ ε I dt + (−cb ) ∫ ε R dt = cb ∫ (ε I − ε R )dt 0 0 (3.2) 0 t u2 = cb ∫ εT dt (3.3) 0 Here, cb is the elastic wave velocity in the bars. The applied load on each side of the specimen can be expressed as P1 = Eb Ab (ε I + ε R ) (3.4) P2 = Eb AbεT (3.5) where Eb and Ab are the Young’s modulus and cross sectional area of the bars, respectively. It is assumed that the short specimen is in equilibrium during the course of deformation, that is the stress across the specimen is constant, therefore P1 ≅ P2 ε I + ε R ≅ εT or (3.6) Hence, the average stress, strain, and strain rate in the specimen are obtained based on hypothesis of equilibrium using the following expressions: σS = P1 + P2 1 Eb Ab (ε I + ε R + ε T ) Eb ( Ab ) = ≅ εT 2 A0 2 A0 A0 (3.7) εS = u1 − u2 cb = l0 l0 (3.8) εS = cb (ε I − ε R − ε T ) cb ≅ εR l0 l0 t ∫0 (ε I − ε R − ε T ) dt ≅ cb l0 t ∫0 ε R dt (3.9) 47 EXPERIMENTAL SETUP Where, ε I , ε R , εT Incident, reflected, and transmitted strains measured Eb , Ab Young’s modulus and cross sectional area of the bars l0 , A0 Gage length and cross sectional area of the specimen cb Longitudinal wave velocity in the bars Eb ρ b To be able to use the one-dimensional theory of elastic wave propagation, the input and the output bars must remain elastic while the wave propagates throughout the test. The bars should be long enough so as to satisfy the one-dimensional theory, and also to avoid the overlap of the incident and reflected signals at the input bar strain gage. Besides, the bars must have enough mechanical strength not to deform plastically. The length to diameter ratio of the bars has to be greater than 20 (Follansbee 1978) to satisfy the one dimensional wave propagation which requires at least 10 bar diameter to dampen the end effects due to non-uniformities at the striker/incident bar interface. In this Hopkinson bar setup, the input and output bars are both 19.05 mm in diameter fabricated from high-yield steel material. The input bar is 4.88 m long and the output bar is 2.44 m long and they can move horizontally without any restriction. The striker tube is made of the same material as the bars and its length determines the duration time of the incident pulse, as expressed by Δt I = 2 LS cb (3.10) where LS is the length of the striker tube. The strain gages attached on the bars, at equal distances from each end of the bars, monitor the incident, reflected and transmitted pulses. It is worth mentioning that the strain rate varies during the test, and the stress, strain and strain rate versus time is calculated using equation (3.7) to (3.9). Although, it is theoretically possible to increase the average strain rate of the test by increasing the pressure in the gas gun, however the increase in the pressure has little effect on the strain rates calculated. 48 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Furthermore, for the split Hopkinson bar results to be taken as valid, there should at least be three to four, exactly π, reflections of the stress wave within the specimen prior to failure (Davies and Hunter 1963; Gama et al. 2004). This directly relates to the assumption of uniform stress field mentioned earlier. Let the wave velocity in the specimen be cS = ES / ρ S (3.11) where ρ S is an approximate value for the density of the carbon woven composite material and is taken to be 1600 kg/m3 and E is the Young modulus of the material. If LSF represents the length of the specimen between the two fixings, then the transit time, tS , that is needed for the incident wave to travel through the specimen once is given by tS = LSF cS (3.12) The time to failure, t f , should then be greater than the time required to obtain a uniform state of stress within the specimen, i.e. t f ≥ π t S = π ( LSF / cS ) (3.13) where π is the number of times the pulse is reflected back and forth. Using this principle, the minimum time to failure for the three materials is given in Table 3.2 and it shows that the dynamic tests are valid because all the calculated values of π tS are less than the time to failure of the experimental values taken from the shortest duration tests. For more comprehensive references on the theory and governing equations of Split Hopkinson bar set-up one can refer to Kawata et al. (1981), Harding and Welsh (1983), Staab and Gilat (1991), Al-Mousawi et al. (1997), Huh et al. (2002), Gama et al. (2004), and Yang and Shim (2005). 49 EXPERIMENTAL SETUP Table 3.2 Time to failure for the three woven composite materials Max. specimen Length between fixings (m) (LSF) Specimen wave velocity (m/s) cS = E S / ρ S Transit time (μs) tS = LSF cS (μs) Time to failure (μs) t f ≥ π tS [0°]6 0.03 6282 5 15 55 [90°]6 0.03 6094 5 15 56 [45°]6 0.03 1785 17 53 62 [0°]6 0.03 6456 5 15 62 [90°]6 0.03 6425 5 15 68 [45°]6 0.03 1962 15 48 57 [0°]8 0.03 6543 5 14 59 [90°]8 0.03 6455 5 15 57 [45°]8 0.03 1762 17 54 78 Description Material 1 Material 2 Material 3 π tS 3.3.2 Specimens and Fixture Designs As was explained in section 3.3.1, in a Hopkinson bar set up the specimen needs to be placed between the input and the output bars. To achieve that, a special fixture had to be designed which will be screwed to the bars while holding the specimen in between. There are two basic conditions that the specimen fixtures have to satisfy. Firstly, the gripping force should be large enough to prevent the specimen from slipping out of the fixture and secondly, change in the mechanical impedance of the fixtures must be as small as possible to prevent disturbance in the wave propagation. To satisfy the latter condition the material and the diameter of the fixture must be the same as the bars which is 19.05 mm. Rodriguez et al. (1996) and Gilat et al. (2002) have also used fixtures that maintain the cylindrical geometry of the Hopkinson bars. It is worth noting that for consistency, the same fixture is used for both static and dynamic tensile tests. Below is a description of all the designs that were considered. 50 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS The first design was adapted from Eskandari (1998) in which the specimen is fixed in a hollow cylindrical fixture by injecting epoxy resin through the holes as shown schematically in Figure 3.5. A collar was used in between the fixture to hold and align the specimen in position. The whole setup was left alone for 24 hours so that the resin would set. epoxy injection hole connected to the bar epoxy specimen fixture connected to the bar Figure 3.5 Schematic assembly of first specimen fixture design Even though this was a good fixture design for Eskandari’s work, it was not adequate for this research. This could be due to the fact that the materials used in this research have a higher strength compared to those in Eskandari’s work. The epoxy resin could not keep the specimen in the fixture, and during the experiment, at a loading below the tensile strength of the woven composite material, the composite sample would slip out of the fixture as can be seen in Figure 3.6. slipped out of fixture Figure 3.6 Picture of the fixture with epoxy injection 51 EXPERIMENTAL SETUP The next design was to sandwich the specimen directly between a half cylinder and a plate, as is presented schematically in Figure 3.7. This design was unsuccessful because the screws could not withstand the dynamic pulse applied to them during the experiment. When the tensile pulse reaches the fixture, the plate on top of the composite sample is stationary, placing a shear loading on the screws. As a result, all eight screws were cut off as soon as the dynamic pulse went through them. screws plate specimen half cylinder Figure 3.7 Schematic assembly of second fixture design (side and top views) The third design was to sandwich the composite specimen between two half cylindrical pieces that fits into a hollow cylinder. The two pieces have rough-teeth shaped grooves machined on their flat surfaces. On tightening the six screws on the hollow cylinder wall, the two gripping pieces are pushed towards each other and hence the composite specimen sandwiched tightly in the fixture as shown in the Figure 3.8. The specimen must be aligned with the fixtures to avoid eccentric loading. 52 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS screws connected to the bar teeth-shaped gripping pieces Figure 3.8 Schematic assembly of third fixture design Although this design was initially successful, after several dynamic tests, the teeth shaped surfaces were damaged. Also the loads that were transferred from the screws were not applied at the edge of the gripping pieces which resulted in their deformation. The design was then modified to use eight screws so that a better distributed load is applied on the griping pieces which have the composite sample sandwiched in between them. Also the teeth shaped gripping surfaces are replaced by two gripping pieces, one with a surface that has a smooth finish and the other with its surface machined in such a manner that the specimen can be fitted inside as illustrated in Figure 3.9. The presence of a barrier at the tip of this piece prevents the specimen from sliding under tensile loading. The picture of the modified fixture is shown in Figure 3.10. To avoid eccentric loading, it is essential that the specimen be aligned with the fixtures. screws connected to the bar gripping pieces gripping piece for fitting the specimen Figure 3.9 Schematic diagram of the modified fixture 53 EXPERIMENTAL SETUP specimen fitted in the gripping piece Figure 3.10 Picture of the modified fixture The key points for designing the shape and size of the specimen are the gage length and the cross section of the specimen. The specimen gage length must be short enough to allow stress equilibrium to be reached in the dynamic tests and also to achieve sufficiently high strain rates. Also, the specimens used in all the tests should have the same geometry so as to avoid any uncertainties related to size effect. The initial dog bone specimen’s geometry had the dimensions shown in Figure 3.11. This geometry was acceptable for dynamic tensile tests on [0/45/90/-45]s unidirectional graphite/ epoxy laminated composite, as all the specimen failed in the middle of the gage length as shown in Figure 3.12a. However, when the same geometry was used for the woven composite materials in this study, they mostly failed at either side of the transition area as shown in Figure 3.12b. This is believed to be mainly due to stress concentration at that point. Figure 3.11 Initial dimensions of the composite specimen in millimetres 54 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS (a) (b) Figure 3.12 Failure in the specimens (a) [0/45/90/-45]s (b) woven [0]6 Consequently, after trying different modifications a dog-bone specimen was designed with the shape and dimensions shown in Figure 3.13. The specimen width is designed to minimize stress concentration at its ends by selecting a smooth radius of curvature in the transition region. Other researchers such as Gilat et al.(2002) and Rodriguez et al. (1996) have also used a high radius of curvature in the transition region of the dog-bone specimen. Also the width at the gage section must be large enough to be a good representative of the woven composite material. The chosen size would ensure the presence of two rows of undamaged weaves in the gage section. Figure 3.13 Specimen’s dimensions in millimetres The specimens were cut from the plate using a CNC milling machine. As the composite plate thickness is very small, a special jig was designed to minimize damage to the specimen while being cut. The woven composite sheet was sandwiched between two aluminum plates which are firmly fixed to the base aluminum plate by means of screws. 55 EXPERIMENTAL SETUP The screws are positioned at the two ends of each specimen as shown in the Figure 3.14. Diamond-like coated end mill suitable for cutting carbon composite laminates was used to cut the dog-bone specimen out of the composite plate. Specimens in (0°), (90°), and (45°) directions were prepared from plates of the three materials discussed in section 3.2. The (0°) and the (90°) specimens were used to obtain static and dynamic tensile behaviour in the normal directions, whereas the (45°) specimens are used in biased extension shear tests to obtain the in-plane shear properties. screws aluminum plates dog-bone specimen Figure 3.14 The aluminum jig used to cut the specimens 3.3.3 Measurement Procedures As explained in section 3.3.1, strain gages are attached on the input and output bars of the Hopkinson bar setup to monitor the incident, reflected and transmitted pulses during the experiment. These 350-Ω strain gages are of type EA-06-062AQ-LE-350 (Measurement Group Inc) and have a gage length of 1.58mm. Their gage factor, GF, is equal to 2.105; 56 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS this is their sensitivity to strain expressed quantitatively, and is defined as the ratio of fractional change in electrical resistance to the fractional change in length (strain) that is GF = ΔR / R Δ R / R = ε ΔL / L (3.14) It is commonly known that the strain gage transforms the strain applied to it into a proportional change of resistance, and this small change in resistance can be measured using a Wheatstone bridge with a voltage or current excitation source (Dally et al. 1993; Nilsson and Riedel 1996). A general Wheatstone bridge consists of four resistive arms with an excitation voltage, VEX, that is applied across the bridge at terminals A and B as shown in Figure 3.15. The output voltage of the bridge, VO, is measured at the terminals C and D. The bridge is considered balanced when no current flows across C-D and VO is zero. Using Kirchhoff’s law for electrical circuits, it can be shown that the balanced circuit requires R1 R2 = R3 R4 . In general the output voltage VO is given by ⎡ R3 R2 ⎤ VO = ⎢ − ⎥ VEX R + R R + R 4 1 2⎦ ⎣ 3 (3.15) In order to use the bridge for strain gage measurements 1, 2 or 4 of the resistors in the bridge are replaced by strain gages. Figure 3.15 illustrates a half bridge configuration used in the lab experiment with two 350-Ω dummy resistors in place of R1 and R2 and two active strain gages in place of R3 and R4. These active strain gages are actually attached on opposite sides of the surface of the bar, and the deformation of the strain gage increases the resistance to (RG +ΔR) which also accounts for the bending. Any changes in the strain gages resistance will unbalance the bridge and produce a nonzero output voltage. If the nominal resistance of the strain gage is designated as RG, and the straininduced change in resistance as ΔR, then by simplifying equation (3.15) and substituting in equation (3.14), the strain experienced by the stain gage during the test can be written as ε =− 2VO ΔR =− GF ⋅ RG GF ⋅VEX (3.16) 57 EXPERIMENTAL SETUP A R1 + − VEX C R4 = RG + ΔR − VO + D R3 = RG + ΔR R2 B Figure 3.15 Half Wheatstone bridge The bridge is connected to a power supply and an oscilloscope. The power supply provides the required excitation voltage and the output voltage signals are recorded using a Nicolet Pro 40 digital oscilloscope. The oscilloscope simultaneously records the incident, reflected, and transmitted pulses versus time. The shapes of these pulses depend on the mechanical response of the specimen used. Figure 3.16 shows typical oscilloscope signals recorded in the experiment. It shows that most of the input pulse is reflected back on reaching the specimen, which results in a small transmitted signal for the samples being tested. This is due to the fact that the Hopkinson bar setup used is a bit strong for the material being used, nevertheless, as will be explained in chapter 4, very good results were obtained using this Hopkinson bar setup. 58 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 20 Incident+Reflected Transmitted Recorded signals [mV] 15 10 5 0 -5 -10 -15 -20 0 200 400 600 800 Time [μsec] 1000 1200 Figure 3.16 Typical recorded strain gage signals on the input and output bars The voltage recordings of the input bar consists of the incident and reflected pulses separated by the time it takes for the pulse to travel, and the voltage recording of the output bar is the transmitted pulse. These can be converted to strain experienced by each strain gage using equation (3.16). The stress, strain and strain rate in the specimen can be calculated using equations (3.7) to (3.9) and a typical stress versus strain curve is shown Figure 3.17. It is worth mentioning that in composite material characterization, strain measurements are always challenging. The dynamic tests, as mentioned above, are performed using a tensile version of the Hopkinson bar, and the dynamic strains are calculated using displacement of the bars (equation(3.7)-(3.9)). The primary difficulty arises because a smooth transition is required between the specimen gage length and the grips to avoid failure due to stress concentrations at the specimen ends, and so the specimen gage length is no longer well-defined. Due to the fact that the Hopkinson bar data analysis only provides data on the relative displacement between the input and the output bars, an effective gage length must generally be used (Davis 2004). 59 EXPERIMENTAL SETUP 800 Stress [MPa] 700 600 500 400 300 200 100 0 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% Strain Figure 3.17 Typical dynamic behaviour of a woven composite material Researchers have used different methods to obtain the effective gage length (Ellwood et al. 1982; Harding et al. 1960; Li and Ramesh 2007; Verleysen and Degrieck 2004). They have shown that the effective gage length was different from both the specimen gage length and the distance between the two bar ends. In this thesis, a series of specimens were tested using strain gages glued directly on the specimen to determine the effective gage length and as a result ensure the accuracy of dynamic strains. The effective gage length of the specimen is obtained using these instrumented tensile tests. It is worth noting that the effective gage length is greater than the actual gage length (10 mm). This increase in the effective gage length is due to the fact that the radius of curvature on the geometry of the specimen is very smooth, causing the occurrence of some deformation outside the actual gage length. A comparison between the dynamic strains of the Hopkinson bar and the instrumented strain gage results help obtain this effective gage length. Figure 3.18 schematically shows a typical effective gage length and the actual gage length on a dog-bone specimen. The procedure and calculations for obtaining the effective gage length will be explained in chapter 4. 60 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 10 mm Figure 3.18 Schematic comparison of actual versus effective gage length For the purpose of comparison, the stress strain relationship of the three results, one from the strain gage signal and the other two from the Hopkinson bar results using effective gage length and actual gage length, are plotted in Figure 3.19. It is interesting to note that accurate dynamic behaviour of the material can be calculated using the Hopkinson bar results when using the appropriate effective gage length. 1200 strain gage measurements HB- actual gage length HB-effective gage length Stress [MPa] 1000 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% Strain Figure 3.19 Typical behaviour using different gage lengths 61 EXPERIMENTAL SETUP The size of the strain gage used for this purpose is of great importance. If the strain gage is too small it might be bonded with only the fibre part or the matrix part of the composite and hence result in low strains. The strain gages have to be large enough to cover a good representative surface of the specimen as shown in Figure 3.20. The strain gages used were 350 Ω (EA-06-125BZ-LE-350 -Measurement Group Inc.) having an overall dimension of 3.3 mm by 7.4 mm. It is very important to prepare the surface of the specimen according to the instructions to maximize the bond between the strain gage and the specimen surface. Figure 3.20 Woven composite specimen with mounted strain gage 3.4 Static experiments Static tensile tests were conducted at strain rates of approximately 10-3/sec. These low strain rate loading tests were carried out using a servo-hydraulic MTS testing machine. Experiments under displacement control conditions yield constant static strain-rate tests. The force is measured by using a 5 kN load cell. It is worth noting that for consistency, the same fixture as the one used in dynamic tensile tests is used here. Also, the specimen’s geometry is kept the same so that no unknown inaccuracy might be introduced due to size effects. Figure 3.21 shows the hydraulic MTS testing machine used for the static tensile experiments. The specimen can be seen mounted on the machine with extensometer attached to it. 62 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Figure 3.21 Servo-hydraulic MTS testing machine, with the specimen fixed in place 3.4.1 Uniaxial tension tests The specimens cut in (0°) and (90°) directions are used to obtain the tensile behaviour in the normal directions. The strains in the specimen can be obtained in three ways: an extensometer, gluing strain gages directly to the specimen, and calculating the strains from the test machine actuator displacement. The strains obtained using machine actuator displacements are usually over estimated, as the measured displacements are not exclusive to the gage length of the specimen. The extensometer used has a 10mm gage length with maximum strain of 15%. As can be seen in Figure 3.21, the space available to attach the extensometer is limited; therefore the arms of the extensometer (MTS 632.13F-20) had to be replaced by longer ones. It is obvious that the increase in the arms length would alter the output strains, which is adjusted by applying the following correction. 63 EXPERIMENTAL SETUP Consider the schematic diagram shown in Figure 3.22, where “c” is the length difference of the original and elongated extensometer arms. In a tensile test, the strains are obtained from the displacement of the lower arm of the extensometer. Let “ΔL1” be the displacement when using the original arms and “ΔL2” that of the elongated arms. The maximum values of these displacements, “ΔL1max and ΔL2max, can be measured by subtracting the gage length of 10 mm from the maximum opening at the tip of the extensometer arms. Using the similar triangle rule, one can write ΔL1max a = ΔL2 max b (3.17) Knowing the values of c = 15 mm , ΔL1max = 2 mm and ΔL2 max = 3.35 mm , the values of a = 22.22 mm and b = 37.22 mm can be calculated. b a c ΔL1 ΔL2 Figure 3.22 Correction for the extensometer arms length The extensometer strain and the actual strain can, respectively, be written as ε ext = ΔL1 ΔL1 = GL 10 ⇒ ΔL1 = 10ε ext (3.18) ε act = ΔL2 ΔL2 = 10 GL ⇒ ΔL2 = 10ε act (3.19) Using the similar triangle rule, one can write 64 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS ΔL1 a = ΔL2 b ⇒ ΔL2 = ΔL1 b b = 10ε ext a a (3.20) On substituting equation (3.19) in equation (3.20), the actual strain is obtained by b a ε act = ε ext = 1.675ε ext (3.21) In the third method which is using instrumented specimens, similar to the dynamic tests, the strain gages used have to be large enough to cover a good representative surface of the specimen. The strain gages used were 350 Ω (EA-06-125BZ-LE-350-Measurement Group Inc.) having an overall dimension of 3.3 mm by 7.4 mm. A half Wheatstone bridge and the Nicolet Oscilloscope were used to obtain the results of the instrumented static tensile tests. In these tests the extensometer was also attached simultaneously to compare the results. A typical comparison of these two methods which are presented in Figure 3.23 show little variation in the results. Therefore, for the static tests, strains are obtained using the extensometer. strain gage extensometer 1.2% 700 Stress [MPa] 1.0% 0.8% Strain strain gage extensometer 800 0.6% 600 500 400 300 0.4% 200 0.2% 100 0.0% 0 20 40 Time [Sec] 60 80 0 0.0% 0.4% 0.8% 1.2% Strain Figure 3.23 Strain comparison using extensometer and strain gage 65 EXPERIMENTAL SETUP It is also important to obtain the behaviour of the material in the direction normal to the loading direction. In a tensile test, while the strain in the loading direction increases, there exists a compressive strain in the normal direction to loading. To measure this transverse strain, the specimens need to be instrumented with strain gages mounted in the normal direction to loading as shown in Figure 3.24. The same type of strain gages could be used, as their gage length was small enough to cover the width of the dog-bone specimen. Figure 3.24 Woven composite specimen with perpendicular strain gage The Poisson’s ratio of a material is defined as the ratio of the transverse strain ( ε yy = ε 22 ) to the axial strain ( ε xx = ε11 ). In uniaxial tensile tests in which the loading (x,y,z) and the fibres (1,2,3) coordinate systems coincide, the ratio of the extensometer strain versus strain-gage strain represents the Poisson’s ratio of the material, as illustrated in Figure 3.25. The slope of this line is the Poisson’s ratio. In an orthotropic material, where the Poisson's ratio is different in each direction (1, 2 and 3 axis) the relation between Young's modulus and Poisson's ratio is described as follows: ν12 E11 = ν 21 E22 , ν13 E11 = ν 31 E33 , ν 32 E33 = ν 23 E22 (3.22) In the woven composite materials, if the weaves of the fabric are balanced, the Young's modulus and the Poisson’s ratio in the two fibre directions would be similar. 66 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Gage-Strain-ε 22 0.00% -0.05% -0.10% -0.15% 0.0% 0.5% 1.0% 1.5% Extensometer Strain-ε11 Figure 3.25 Typical result to obtain the Poisson’s ratio 3.4.2 Bias extension shear tests In order to characterize the shear response of woven composite materials, two test methods in general have received wide application, namely the picture frame test and the bias extension test. In a picture frame test, a tensile force is applied across diagonally opposite corners of the picture frame rig to move from an initially square configuration into a rhomboid. Consequently the sample clamped within the frame undergoes pure shear. Whereas, a bias extension test involves clamping a rectangular piece of woven material in such a way that the warp and weft directions of the tows are oriented at 45 degree to the direction of tensile loading (Lebrun et al. 2003) Some researchers like Leburn et al.(2003) and Harrison et al. (2004) have evaluated both the methods and have found comparable results in the two tests. In this research, it is needed to obtain the shear response in both static and dynamic loading. However due to the nature of the Hopkinson bar setup, only the bias extension shear test can be used for dynamic loading. Therefore, for consistency, the bias extension shear test is used for both static and dynamic tests to determine the nonlinear shear stiffness of woven composites 67 EXPERIMENTAL SETUP in pure shear (Lee et al. 2008). The specimens are cut in a way that the yarns ran at ±45° with respect to the loading direction. Samples of each of the materials are tested in tension using the same setup and fixture as the axial tensile tests. It is worth mentioning that the state of stress in the 45 degree cut specimen is not pure shear, and tensile normal stresses, “σ1” and “σ2”, in addition to the desired shear stress “σ12” is present in each lamina of the specimen. However, the bias extension test method is considered as a reliable method of obtaining the shear strength and modulus of the material for the following reasons. Firstly, the shear response of many types of woven composite materials is non-linear, and exhibit strain softening characteristics. As a result, even though the biaxial state of stress present in the specimen likely causes the value of shear strength to be lower than the true value, the reduction may be small due to the nonlinear softening response (Carlsson et al. 2002). Secondly, as the magnitudes of the normal stresses present are significantly lower than the ultimate stress in the fibre direction, the loading can be approximated to be one of pure shear. The dog-bone specimens are prepared and tested in tension to ultimate failure. Determination of the shear properties from tension test results uses a stress analysis of the 45 degree cut specimen. Using a transformation matrix, the results obtained in the loading direction can be transformed into the stresses and strains in the fibre direction. It is important to note that the x-direction is considered to be the loading direction throughout the thesis. Let (x,y) and (1,2) denote the two in-plane coordinate systems of loading and fibres directions, respectively, with a common origin as shown in Figure 3.26. The cosines of the angles between the coordinate axes (x,y) and the coordinate axes (1,2) are listed in Table 3.3. These direction cosines are all equal to either 1/√2 or -1/√2 simply because the angle between the two coordinates is 45 degrees. 68 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS y 2 1 σ xx x σ xx Figure 3.26 Loading direction and fibre direction coordinate systems Table 3.3 Direction cosines x y 1 l1 m1 2 l2 m2 The stress components of the (1,2) coordinated system is given by (Boresi et al. 1993) σ11 = l12σ xx + m12σ yy + 2l1m1σ xy σ 22 = l22σ xx + m22σ yy + 2l2 m2σ xy (3.23) σ12 = l1l2σ xx + m1m2σ yy + (l1m2 + l2 m1 )σ xy In the bias extension test, the axial stress in the loading direction is the only nonzero stress component; hence the stresses in the fibre direction can be obtained as σ11 = l12σ xx σ 22 = l22σ xx σ12 = l1l2σ xx = σ xx / 2 (3.24) Also, the transformation of strain tensor under a rotation from axes (x,y) to axes (1,2) is given by (Boresi et al. 1993) 69 EXPERIMENTAL SETUP ε11 = l12ε xx + m12ε yy + 2l1m1ε xy ε 22 = l22ε xx + m22ε yy + 2l2 m2ε xy (3.25) ε12 = 12 γ 12 = l1l2ε xx + m1m2ε yy + (l1m2 + l2 m1 )ε xy As it was shown by Adams et al. (2003) the shear strain is given by ε12 = (ε xx − ε yy ) / 2 (3.26) where the axial strain, ε xx , is measured using the extensometer, however, due to the fact that not all the tests are instrumented with strain gage in the transverse direction, the Poisson’s ratio obtained from the instrumented tests is used to calculate the transverse strains as ε yy = −ν xyε xx in the non-instrumented tests and hence the shear strain is calculated as ε12 = ε xx (1 +ν xy ) / 2 (3.27) It is worth mentioning that due to the small size of the specimens, small strain gages were used in these experiments. It was noted that for some of the materials tested in the 45degree direction, the transverse strain is higher than 6% at the maximum stress, which is greater than the gage capacity. It is interesting to note that the transverse strain measured before the gage failure, showed linear behaviour with respect to time. This linearity was used as an approximation to obtain the transverse strain at the onset of failure in the specimen, which is the time of drop in stress. Consequently, the Poisson’s ratio is obtained from the ratio of the transverse strain to axial strain at the time of stress drop. A typical graph showing the relationship of axial versus transverse strain up to the point of stress drop is illustrated in Figure 3.27 in which the slope of the curve gives the Poisson’s ratio in the 45 degree cut specimen. 70 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 0% Gage- Strain-εyy -1% -2% -3% -4% -5% -6% -7% 0% 5% 10% 15% Extensometer Strain-εxx Figure 3.27 Typical result for bias extension specimen to obtain Poisson’s ratio The in-plane undamaged shear modulus “G12” of the composite materials can be determined by plotting σ12 versus γ 12 and establishing the slope of the initial portion of the curve. It is believed that deviation from the linearity behaviour is the onset of damage. An alternate method of determining the undamaged shearing modulus is to measure the initial axial stiffness (Ex) and the Poisson’s ratio (νxy) of the 45-degree specimens, and then calculate the undamaged shear modulus according to the following relationship. G12 = Ex 2(1 +ν xy ) (3.28) It is worth mentioning that some researchers have assumed a nonlinear elastic behaviour for these materials (Hahn and Tsai 1973; Ogihara and Reifsnider 2002; Shokrieh and Lessard 2000). However, the nonlinear behaviour explained in these papers contribute to the very beginning of the curve, a little after deviation from linearity, and does not consider the results until failure. On applying a tensile load to the 45 degree cut specimen, the matrix crack due to presence of a scissoring effect. When unloading the sample, the stress-strain curve does not unload over the initial loading curve; hence the deviation from the linearity behaviour is attributed to the onset of damage. And researchers like 71 EXPERIMENTAL SETUP Johnson et al. 2001 (2001), Matzenmiller et al. (1995), and Marguet et al. (2007) have reported the nonlinear shear behaviour and included the shear damage parameter in their models. 3.5 Experiments summary Several types of experiments were discussed in this chapter. Static uniaxial tensile tests were conducted in order to obtain the static behaviour of the woven composite materials tested. Some of the tests were instrumented with strain gages in the axial direction to obtain the most accurate tensile strains. While some other tests were conducted with strain gages attached perpendicular to the loading direction in order to obtain the transverse strains, which is required to obtain the Poisson’s ratio of the materials. In addition, bias extension shear tests were conducted on 45-degree cut specimens to obtain the shear stress-strain curves. Some of these tests were also instrumented with strain gages in the perpendicular to loading direction in order to extract both the axial and transverse strains from the tests. The ratio of the two strains gives the Poisson’s ratio in the 45-degree cut specimens, which is required in order to convert the axial behaviour in the loading direction, to the shear behaviour in the fibre direction. Furthermore, dynamic tensile tests as well as dynamic bias extension tests were needed to capture the behaviour of these woven composite materials at high strain rates. These tests were performed using the tensile Hopkinson bar setup. Some of these tests were performed with strain gages attached to the specimen in order to obtain their effective gage length. It was confirmed that these materials are strain rate dependent. Moreover, these experiments are used as a tool to develop a constitutive damage model and its associated damage parameters which can predict the behaviour of woven composite materials. The constitutive damage model is implemented into a two dimensional VUMAT subroutine to be used for explicit dynamic finite element simulations. 72 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Finally, as a general summary, Tables 3.4 and 3.5 list all the experiments conducted, along with the number of repeats each experiment was performed. Due to the more repeatable nature of static tests, fewer numbers of repeat was needed compared to the dynamic tests. Based on the results, it was decided to have a minimum of three repeats for the static tests, and a minimum of four repeats for the dynamic tests, and at least one test was performed in the cases where instrumented results were required. The discussions of the test results are presented in chapter 4 of this thesis. Table 3.4 List of dynamic experiments performed Experiment type Dynamic uniaxial tension Material 6 90-degree 6 0-degree 5 90-degree 5 0-degree 5 90-degree 5 material 1 45-degree 6 material 2 45-degree 5 material 3 45-degree 5 0-degree 2 90-degree 1 0-degree 1 90-degree 1 0-degree 1 90-degree 1 material 3 Dynamic uniaxial tension with axial strain gage Number of test repeats 0-degree material 1 material 2 Dynamic bias extension Orientation of cut specimen material 1 material 2 material 3 Dynamic bias extension material 1 45-degree 2 with axial strain gage material 2 45-degree 1 material 3 45-degree 1 73 EXPERIMENTAL SETUP Table 3.5 List of static experiments performed Experiment type Static uniaxial tension Material 4 90-degree 3 0-degree 4 90-degree 3 0-degree 4 90-degree 4 material 1 45-degree 4 material 2 45-degree 3 material 3 45-degree 4 0-degree 1 90-degree 1 0-degree 1 90-degree 1 0-degree 1 90-degree 1 material 3 Static uniaxial tension transverse strain gage Number of test repeats 0-degree material 1 material 2 Static bias extension Orientation of cut specimen material 1 material 2 material 3 Static bias extension material 1 45-degree 1 transverse strain gage material 2 45-degree 1 material 3 45-degree 1 0-degree 1 90-degree 1 Static uniaxial tension axial strain gage in 74 material 1 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 4 Equation Section (Next) EXPERIMENTAL RESULTS 4.1 Static Experiments In a static tension test, the tensile force is recorded as a function of the displacement of the machine head. The deformation measured by the machine belongs to the whole system attached between the two heads of the machine, that is, the specimen as well as its gripping system. Therefore, force versus displacement would be of little value to accurately describe the material behaviour. Accordingly, normalization with respect to specimen dimensions would lead to more accurate representation of the material response. With that in mind, the engineering stress, σ eng , is given as σ eng = Fi A0 (4.1) 75 EXPERIMENTAL RESULTS where Fi is the instantaneous tensile load measured by the load cell during testing, and A0 is the initial cross sectional area in the gage section. The engineering strain, ε eng , can be obtained from the change in the gage section length, such that ε eng = li − l0 l0 where l0 (4.2) and li are the initial and instantaneous gage lengths, respectively. The engineering strain is measured directly during the test by attaching an extensometer to the gage section as explained in section 3.4 of this thesis. As a result, a record of instantaneous force and strain is obtained while loading. It is worth mentioning that the stress is calculated as a function of initial area. The true stress in a material is defined as a function of the instantaneous area, which is changing during loading, and the formulation used for the true stress and strain can be written as σ = σ eng (1 + ε eng ) ⎛ li ⎞ ⎟ = ln 1 + ε eng ⎝ l0 ⎠ ε = ln ⎜ ( ) (4.3) However, for the woven composite materials being tested in this research, the strain-tofailure is small contributing to very little difference between the engineering and true values. Therefore, in this study, the engineering strain on the test specimen is measured using an extensometer with 10mm gage length as the load is applied, and the engineering stress is calculated according to equation (4.1). 4.1.1 Tensile modulus and tension test results Static tensile tests are performed on the dog-bone specimens cut from both the fibre directions of all the three materials namely carbon/epoxy prepreg with 2×2 twill weave, carbon/BMI prepreg with 8 harness satin weave and carbon/epoxy prepreg with plain 76 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS weave. These results are presented in Figure 4.1 to Figure 4.3. And a summary of the static tensile results are presented in Table 4.1. 1000 Material - 1 M11S1 Static 11- direction M11S2 M11S3 Stress [MPa] 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain 1000 Material - 1 M12S1 Static 22- direction M12S2 M12S3 Stress [MPa] 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain Figure 4.1 Stress-strain curves for static tensile tests of material one (woven carbon/epoxy prepreg [0º]6) in the two fibre directions 77 EXPERIMENTAL RESULTS 1000 M21S1 M21S2 M21S3 Material - 2 Static 11- direction Stress [MPa] 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain 1000 M22S1 M22S2 M22S3 Material - 2 Static 22- direction Stress [MPa] 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain Figure 4.2 Stress-strain curves for static tensile tests of material two (woven carbon/BMI prepreg [0º]6) in the two fibre directions 78 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS M31S1 M31S2 M31S3 Material - 3 1000 Static 11- direction Stress [MPa] 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain M32S1 M32S2 M32S3 Material - 3 1000 Static 22- direction Stress [MPa] 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain Figure 4.3 Stress-strain curves for static tensile tests of material three (woven carbon/epoxy prepreg [0º]8) in two fibre directions 79 EXPERIMENTAL RESULTS Table 4.1 Summary of static tensile tests Material Specimen Type Static Test Name M11S1 Max. Stress MPa Strain at max. stress Modulus (GPa) 623 1.03% 60.36 M11S2 674 1.08% 60.91 M11S3 712 1.17% 60.83 Woven Average 670 1.09% 60.70 Carbon/ M12S1 653 1.12% 58.18 M12S2 681 1.18% 57.77 M12S3 689 1.21% 56.74 Average 674 1.17% 57.56 M21S1 611 1.01% 64.51 M21S2 726 1.18% 69.06 M21S3 812 1.30% 64.20 8-HS Woven Average 716 1.17% 65.92 Carbon/BMI M22S1 729 1.19% 62.14 M22S2 657 0.94% 68.58 M22S3 648 1.02% 61.27 Average 678 1.05% 63.99 M31S1 925 1.35% 68.75 M31S2 899 1.40% 64.03 M31S3 810 1.22% 66.29 Average 878 1.32% 66.36 M32S1 883 1.36% 64.77 M32S2 786 1.21% 65.10 M32S3 901 1.41% 63.87 Average 857 1.33% 64.58 Material-1: 2×2 Twill Epoxy Prepreg Material-2: Prepreg Material-3: [0°]6 [90°]6 [0°]6 [90°]6 [0°]8 Plain woven Carbon/ Epoxy Prepreg 80 [90°]8 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS It is observed that material three is the strongest material which has an average tensile strength of 867 MPa. This is approximately 180 MPa higher than the other two materials. In addition, material three has the maximum average strain to failure of about 1.33% whereas material two has the minimum average strain to failure of 1.05%. Looking at the variation of tensile strength between different repeats of the tests, one can observe that material one has the least variability and material two has the maximum. However, consistency is observed in the average tensile strength of the two fibre directions in each material. The variability in the tensile strength of the three materials is graphically illustrated in Figure 4.4 for both fibre directions. It is worth noting that variability has been observed by other researchers as well. For example Kim et al. (2004) have shown approximately 75 MPa variability in their tensile strength results for the plain weaves and about 35 MPa variability in both the 2×2 twill weave and 8 harness satin weaves. Average tensile strength Tensile strength [MPa] 1000 800 600 11-direction 400 22-direction 200 0 0 1 2 3 Material Number Figure 4.4 Average static tensile strengths and their variability for the three materials in the two fibre directions 81 EXPERIMENTAL RESULTS As presented in the graphs, all three materials show a brittle behaviour when loaded in the fibre directions. Hence, the static tensile modulus is calculated from the slope of the stress-strain curve as the tensile modulus is defined by E =σ /ε (4.4) It is worth mentioning that average tensile modulus of all the three materials has little variability in the two fibre directions (Figure 4.5). Furthermore, the variability is small even between the materials tested with different weaves, such that an average value of 63 ± 3 GPa is a good overall approximation. Kim et al. (2004) have shown approximately 18 GPa variation in the tensile modulus results of carbon-epoxy composites with all three weave patterns, namely the 2×2 twill weave and 8 harness satin weaves, and the plain weaves. Average tensile modulus Tensile modulus [GPa] 80 70 60 50 40 11-direction 30 22-direction 20 10 0 0 1 2 3 Material Number Figure 4.5 Average static tensile modulus and their variability for the three materials in the two fibre directions To calculate the Poisson’s ratio of the three materials, their behaviour in the loading direction as well as in the direction normal to the loading direction is required, as was explained in section 3.4.1. The two in-plane coordinate systems in this thesis, are (x,y) for 82 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS the loading direction and (1,2) for the fibre directions. The strains in the loading direction (ε xx = ε11 ) is measured by the extensometer and for the measurement of the transverse strain, (ε yy = ε 22 ) , the specimens are instrumented with strain gages mounted in the normal direction to loading, as was mentioned previously. The Poisson’s ratio of a material is the ratio of these two strains (ε 22 / ε11 ) . Figure 4.6 illustrates the axial versus the transverse strains measured in a tensile test for the three materials. Accordingly, the Poisson’s ratio of each material is calculated from the slope of a line fit to the data, and is tabulated in Table 4.2. Poisson's ratio Material-1 (Fibre direction) 0.00% Material-2 Material-3 Strain-yy -0.02% -0.04% ε yy = −0.06 ε xx -0.06% -0.08% ε yy = −0.08 εxx -0.10% εyy = −0.087 εxx -0.12% -0.14% 0.0% 0.5% Strain-xx 1.0% 1.5% Figure 4.6 Axial versus transverse strains for all the materials in the fibre direction Table 4.2 Poisson’s ratio measured in the fibre direction Poisson’s Ratio Material-1 0.087 Material-2 0.060 Material-3 0.080 83 EXPERIMENTAL RESULTS It is interesting to note that these materials possess a relatively low Poisson’s ratio. Due to the presence of little variability in the tensile modulus, as well as the low Poisson’s ratio in the fibre direction of these materials, it was decided that one test result would be sufficient in calculating this value. Small Poisson’s ratio was observed by many researchers as well. For example Kumangai et al. (2003), who tested a 5 harness satin weave carbon/epoxy material, measured a comparable value of 0.074 for their Poisson’s ratio. Table 4.3 summarizes the values of Poisson’s ratio obtained by other researches. Table 4.3 Summary of Poisson’s ratio and tensile modulus data from literature Poisson’s Ratio Tensile Modulus (GPa) 2 layer 0.041 66.9 4 layer 0.039 67.7 6 layer 0.035 68.2 ߥଵଶ =0.055 ܧଵଵ = 73.5 ߥଶଵ =0.036 ܧଶଶ = 63 ߥଵଶ =0.066 ܧଵଵ = 70.04 ߥଶଵ =0.05 ܧଶଶ = 66 ߥଵଶ =0.038 ܧଵଵ = - ߥଶଵ =0.077 ܧଶଶ = 69.6 x- direction 0.0346 ± 0.01 40.97 ± 2 y- direction 0.0666 ± 0.0203 47.30 ± 4.02 5HS weave carbon/epoxy 0.074 46.9 5HS weave carbon/epoxy 0.07 76 Paper Material Gao et al. (1999) 8HS weave CFRP ߝሶ = 1.2E-4 sec-1 Hou and Ruiz (2000) woven CFRP T300/914 ߝሶ = 4 sec-1 ߝሶ = 600 sec-1 Tan et al. (2000) Kumagai et al. (2003) Abot et al. (2004) 84 3D woven CFRP HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 4.1.2 Shear modulus and bias extension test results To determine the shear characteristics of the three woven composite materials, dog-bone specimens, cut from the plates with an off-axis angle of ±45°, are tested in tension using the same setup and fixture as the axial tensile tests. As was explained in section 3.4.2 the experimental values of the tensile stresses and tensile strains should be converted into the shear stress and strains using the transformation of stress and strain tensors. In order to perform this conversion, the strain experienced by the material in the transverse direction to loading is also required. Therefore some bias extension shear tests are performed using strain gages attached normal to the direction of loading. Consequently, the Poisson’s ratio, ν xy , of the materials in the 45 degree cut specimens are calculated from the slope of the transverse strain, ε yy , versus axial strain ε xx , as presented in Figure 4.7 to Figure 4.9, and the values are given in Table 4.4. Material - 1 (45° direction) 0.00% Strain-yy -0.40% εyy = -0.38 εxx -0.80% -1.20% -1.60% -2.00% 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% Strain-xx Figure 4.7 Poisson’s ratio of material one in 45º specimens 85 EXPERIMENTAL RESULTS Material - 2 (45° direction) 0.00% Strain-yy -0.20% εyy = -0.47 εxx -0.40% -0.60% -0.80% -1.00% 0.0% 0.5% 1.0% 1.5% 2.0% Strain-xx Figure 4.8 Poisson’s ratio of material two in 45º specimens Material - 3 (45° direction) 0.00% Strain-yy -1.00% -2.00% εyy = -0.52 εxx -3.00% -4.00% -5.00% -6.00% -7.00% 0.0% 5.0% 10.0% 15.0% Strain-xx Figure 4.9 Poisson’s ratio of material three in 45º specimens 86 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Table 4.4 Poisson’s ratio measured in the 45º specimens Poisson’s Ratio Material-1 0.38 Material-2 0.47 Material-3 0.52 It is interesting to note that other researchers have found similar values of Poisson’s ratio for 45 degree cut specimens of different composite materials. Gilesche et al. (2005) tested different structures of woven carbon/epoxy composites and reported a Poisson’s ratio of 0.70-0.75, Ogihara and Reifsnider (2002) found that the Poisson’s ratio in the 45 degree cut specimens of four-harness satin woven glass composites was approximately 0.543, and Kashaba (2004) tested cross-ply of GFRE with [45/-45]2S stacking sequence and reported a Poisson’s ratio of 0.67. Using the equations given in section 3.4.2 the shear stress and strains are calculated. Figure 4.10 to Figure 4.12 shows the shear results for the three materials each with three repeats of the test. From the curves it can be seen that all the materials show an initial linear elastic region followed by nonlinear deformation behaviour. This kind of nonlinear behaviour was observed by other researchers such as Gliesche et al. (2005), Abot et al. (2004), and Daniel et al. (2008). 87 EXPERIMENTAL RESULTS 120 Material - 1 M13S1 Static 12- direction M13S2 M13S3 Shear stress [MPa] 100 80 60 40 20 0 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% Shear strain Figure 4.10 Shear stress -strain curves for static bias extension tests of material one (woven carbon/epoxy prepreg [0º]6) in the 45º specimen 120 M23S1 Material - 2 M23S2 Static 12- direction M23S3 Shear stress [MPa] 100 80 60 40 20 0 0.0% 2.0% 4.0% 6.0% Shear strain Figure 4.11 Shear stress -strain curves for static bias extension tests of material two (woven carbon/BMI prepreg [0º]6) in the 45º specimen 88 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS M33S1 Material - 3 120 M33S2 Static 12- direction M33S3 Shear stress [MPa] 100 80 60 40 20 0 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% Shear strain Figure 4.12 Shear stress-strain curves for static bias extension tests of material three (woven carbon/epoxy prepreg [0º]8) in the 45º specimen As expected, material one with the 2×2 twill weaves is stiffer than material two with the 8HS weave (Kim et al. 2004). Material three shows the maximum strain to failure of about 12%, which is believed to be due to the fact that the weaves in this material are not as compact as the other two. For the purpose of comparison, one shear result from each material is presented in Figure 4.13. The average shear strength is between 81 to 101 MPa with material three having the maximum value. The initial undamaged shear modulus can be determined by plotting shear stress (σ12 = σ xx / 2) versus the engineering shear strain (γ 12 = 2ε12 ) and establishing the initial slope of the curve. An alternate method which would yield a similar undamaged shear modulus is using equation 3.28 together with the axial stiffness, Exx , and Poisson’s ratio, ν xy , of the 45 degree cut specimen. From the data, the undamaged shear modulus is between 5-6 GPa for the three materials. The summary of the shear results is presented in Table 4.5. 89 EXPERIMENTAL RESULTS Material-1 Static 12- direction 120 Material-2 Material-3 Shear stress [MPa] 100 80 60 40 20 0 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% Shear strain Figure 4.13 Comparison of shear stress-strain curves of the three materials Table 4.5 Summary of static bias extension tests Material Material-1: Material-2: Material-3: 90 Specimen Type [45°]6 [45°]6 [45°]8 Static Test Name Max. Shear Stress (MPa) Shear Strain at max. stress Modulus Initial (GPa) M13S1 83 2.73% 5.29 M13S2 88 2.40% 5.31 M13S3 91 2.25% 4.22 Average 87 2.46% 4.94 M23S1 79 1.50% 5.08 M23S2 81 1.33% 5.91 M23S3 84 1.42% 6.90 Average 81 1.42% 5.97 M33S1 103 7.48% 4.86 M33S2 102 7.36% 4.70 M33S3 98 6.68% 4.87 Average 101 7.17% 4.81 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 4.2 Dynamic Experiments The dynamic tests are performed on a tensile version of the Hopkinson bar. As discussed in section 3.31, a tensile wave is generated by the impact of the striker, and the propagation of this wave is analysed assuming the one dimensional theory, and measuring the elastic deformations produced in different positions on the bar system with strain gages. The stress, strain and strain rate in the specimen are determined with the traditional procedure based on the hypothesis of equilibrium and uniform strain along the specimen. It is worth mentioning that the same types of specimens are used in both static and dynamic experiments in order to eliminate any potential size effects. 4.2.1 Effective gage length Additional strain measurements are obtained for each material in the three directions following the procedure explained in section 3.3.3, that is, strain gages are glued directly on the dog-bone specimen in order to measure the exact axial strains. A comparison between the dynamic strains of the Hopkinson bar and the instrumented strain gage results can be used to calculate the effective gage length. As mentioned before, upon loading, the specimen undergoes some deformation outside its actual gage length. This could be due to the presence of a large and smooth radius of curvature on the dog-bone specimens as also observed by many other researchers such as Verleysen and Degrieck (2004) or due to the fact that the length between the grips is larger than the gage length and so the deformation between the grips is the sum of the deformation of the entire open length of the material under load. Therefore, a correction factor is used in order to calculate an effective gage length which would yield correct strains in the gage area. Figure 4.14 shows typical strains measurements from the strain gage results (the dotted line) as well as from the Hopkinson bar equations using the actual gage length (the dashed line). It can be seen that by increasing the gage length to an effective gage length (the solid line) the strain gage results coincide with the Hopkinson bar results. The summary of the calculated effective gage lengths for all the materials are presented in Table 4.6. 91 EXPERIMENTAL RESULTS 2.00% Strain strain gage measurements HB- actual gage length HB-effective gage length 1.00% 0.00% 0 10 20 30 Time [μ sec] 40 50 60 Figure 4.14 Typical comparison of different strain measurements Table 4.6 Summary of effective gage lengths Material Material-1: Material-2: Material-3: 92 Specimen Type Effective gage length (mm) [0°]6 25 [90°]6 25 [45°]6 20 [0°]6 18 [90°]6 18 [45°]6 14 [0°]8 17 [90°]8 17 [45°]8 12 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 4.2.2 Tension test results High strain rate tensile tests are performed on the dog-bone specimens cut from both the fibre directions of all the three materials using the Hopkinson bar setup. The strain rates in these experiments vary from approximately 200 /sec up to 550 /sec. These results are presented in Figure 4.15 to Figure 4.17. From the curves it can be seen that all the materials show an initial linear elastic region followed by nonlinear deformation until the ultimate stress which precedes a nonlinear strain softening behaviour. This kind of nonlinear behaviour was observed by other researchers such as Kawata et al. (1981), Welsh and Harding (1985), Harding et al. (1989), Rodriguez et al. (1996), and Sham et al. (2000) who studied the tensile behavior of different woven composites materials. A summary of the dynamic tensile results are presented in Table 4.7. 93 EXPERIMENTAL RESULTS 1200 Material - 1 M11D1 Dynamic 11- direction M11D2 M11D3 1000 Stress [MPa] M11D4 800 600 400 200 0 0.0% 1.0% 2.0% 3.0% Strain 1200 Material - 1 M12D1 Dynamic 22- direction M12D2 M12D3 1000 Stress [MPa] M12D4 800 600 400 200 0 0.0% 1.0% 2.0% 3.0% Strain Figure 4.15 Stress-strain curves for dynamic tensile tests of material one in the two fibre directions 94 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 1200 Material - 2 M21D1 Dynamic 11- direction M21D2 M21D3 1000 Stress [MPa] M21D4 800 600 400 200 0 0.0% 1.0% 2.0% 3.0% Strain 1200 Material - 2 M22D1 Dynamic 22- direction M22D2 M22D3 1000 Stress [MPa] M22D4 800 600 400 200 0 0.0% 1.0% 2.0% 3.0% Strain Figure 4.16 Stress-strain curves for dynamic tensile tests of material two in the two fibre directions 95 EXPERIMENTAL RESULTS 1200 Material - 3 M31D1 Dynamic 11- direction M31D2 M31D3 1000 Stress [MPa] M31D4 800 600 400 200 0 0.0% 1.0% 2.0% 3.0% Strain 1200 Material - 3 M32D1 Dynamic 22- direction M32D2 M32D3 1000 Stress [MPa] M32D4 800 600 400 200 0 0.0% 1.0% 2.0% 3.0% Strain Figure 4.17 Stress-strain curves for dynamic tensile tests of material three in the two fibre directions 96 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Table 4.7 Summary of dynamic tensile tests Material Specimen Type [0°]6 Material-1 [90°]6 [0°]6 Material-2 [90°]6 [0°]8 Material-3 [90°]8 (MPa) Strain at max. stress Strain Rate at max. stress M11D1 764 0.63% 274 M11D2 M11D3 760 742 0.57% 0.59% 200 251 M11D4 726 0.68% 314 Average 748 0.62% 260 M12D1 M12D2 868 814 0.52% 0.50% 215 200 M12D3 690 0.51% 189 M12D4 657 0.46% 209 Average 757 0.50% 203 M21D1 1043 1.22% 457 M21D2 930 1.15% 424 M21D3 924 0.87% 347 M21D4 913 0.92% 311 Average 953 1.04% 385 M22D1 1074 1.32% 565 M22D2 1039 1.15% 460 M22D3 923 0.87% 336 M22D4 921 1.16% 465 Average 989 1.13% 456 M31D1 1074 0.83% 308 M31D2 1022 0.90% 331 M31D3 963 0.76% 284 M31D4 Average 819 0.88% 346 969 0.84% 317 M32D1 951 0.80% 300 M32D2 877 0.81% 319 M32D3 M32D4 872 854 0.88% 0.79% 340 294 Average 889 0.82% 313 Test Name Max. Stress 97 EXPERIMENTAL RESULTS It is observed that material two has the highest average tensile strength of 979 MPa followed by materials one and three with 929 MPa and 752 MPa average tensile strength, respectively. Looking at the variation of dynamic tensile strength between different repeats of the tests, it is observed that material two has the least variability and material three has the maximum. However, consistency is observed in the average tensile strength of the two fibre directions in each material. The variability in the tensile strength of the three materials is graphically illustrated in Figure 4.18 for both fibre directions. Average dynamic tensile strength Tensile strength [MPa] 1200 1000 800 600 11-direction 22-direction 400 200 0 0 1 2 3 Material Number Figure 4.18 Average dynamic tensile strengths and their variability for the three materials in the two fibre directions Furthermore, the highest average strain to failure of about 2.45% is observed for material two and the lowest (1.1%) is detected for material one. Looking at the variation of average strain at ultimate stress between different repeats of the tests, it is observed that material two has the maximum variability, whereas the other two materials have minimal variability. Moreover, variability in the average strain at ultimate stress is also observed between the two fibre directions in each material. It is important to point out that the strain rate, at which each test is performed, cannot be precisely controlled. As a result 98 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS these test repeats are each performed at a somewhat different strain rate, which does contribute to these inconsistencies. Also, the values presented in Table 4.7 shows that the tests with the higher maximum stress, are mostly performed at higher strain rate. Figure 4.19 illustrates graphically the average dynamic strains variability in the three materials for both fibre directions. It is interesting to see that dynamic results of the two normal directions are comparable for each material. This fact was also observed in the static results, confirming that these materials are balanced, which justifies the use of the biased extension shear test to obtain the in-plane shear properties. Average strain Strain at max. stress 1.5% 1.0% 11-direction 22-direction 0.5% 0.0% 0 1 2 3 Material Number Figure 4.19 Average dynamic strains at maximum stress and their variability for the three materials in the two fibre directions 4.2.3 Bias extension test results To determine the high strain rate shear characteristics of the three woven composite materials, dog-bone specimens, cut from the plates with an off-axis angle of ±45°, are tested using the same Hopkinson bar setup and fixture as the axial tensile tests. The strain rates in these experiments vary from about 170 /sec up to 580 /sec. As was explained in section 3.4.2 the experimental values of the tensile stresses and tensile strains are 99 EXPERIMENTAL RESULTS converted into the shear stress and strains using the transformation of stress and strain tensors. The Poisson’s ratio, ν xy , of the materials in the 45 degree cut specimens, obtained in section 4.1.2, along with the equations given in section 3.4.3 are used to calculate the dynamic shear stress and strains. Figure 4.20 to Figure 4.22 represents the dynamic shear behaviour of the three materials with four test repeats each. The behaviour of all the three materials is similar, that is, an initial linear elastic region followed by nonlinear deformation until the ultimate stress which precedes a nonlinear strain softening behaviour. Table 4.8 summarizes the dynamic bias extension test results. In the bias extension test results, as expected, the material can withstand less strength as compared to the normal directions; however they fail at much higher strains. Material three has the highest average strength of 171 MPa, followed by material one with 164 MPa, and material two with 132 MPa. 200 Material - 1 M13D1 Dynamic 12- direction M13D2 Shear stress [MPa] M13D3 150 M13D4 100 50 0 0.0% 1.0% 2.0% 3.0% 4.0% Shear strain Figure 4.20 Stress-strain curves for dynamic bias extension tests of material one in the 45º specimen 100 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 200 Material - 2 M23D1 Dynamic 12- direction M23D2 Shear stress [MPa] M23D3 150 M23D4 100 50 0 0.0% 1.0% 2.0% 3.0% 4.0% Shear strain Figure 4.21 Stress-strain curves for dynamic bias extension tests of material two in the 45º specimen 200 Material - 3 M33D1 Dynamic 12- direction M33D2 Shear stress [MPa] M33D3 150 M33D4 100 50 0 0.0% 1.0% 2.0% 3.0% 4.0% Shear strain Figure 4.22 Stress-strain curves for dynamic bias extension tests of material three in the 45º specimen 101 EXPERIMENTAL RESULTS Table 4.8 Summary of dynamic bias extension tests Material Material-1 Material-2 Material-3 Specimen Type [45°]6 [45°]6 [45°]8 (MPa) Strain at max. stress Strain Rate at max. stress M13D1 178 0.51% 198 M13D2 175 0.51% 190 M13D3 160 0.47% 186 M13D4 144 0.46% 171 Average 164 0.49% 186 M23D1 136 0.61% 279 M23D2 134 0.44% 263 M23D3 134 0.53% 269 M23D4 125 0.36% 243 Average 132 0.49% 263 M33D1 181 1.22% 516 M33D2 171 1.39% 589 M33D3 166 1.07% 423 M33D4 165 1.15% 446 Average 171 1.21% 493 Test Name Max. Stress Furthermore, material three has the highest average strain to failure of about 3.60%, followed by material two and one with 1.43% and 1.08%, respectively. From the test results, it can be deducted that material one has the maximum variability in the shear strength, whereas material two has the maximum variability in the shear strain (Figure 4.23). It is worth mentioning, even though not much data is available on the comparison of inplane static and dynamic shear response of woven composite materials, it is well understood in literature that the behaviour in shear is strongly nonlinear and irreversible as was mentioned in many papers for example by Marguet et al. (2007) and Johnson (2001). This could be due to the fact that in shear experiments, the nonlinear properties of the resin constituent of the composite materials is no longer insignificant, and while the 102 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS fibre in a [45°] specimen are trying to align themselves in the loading direction, nonlinearity is evident in both the static and dynamic results. 2.0% 150 1.5% 100 1.0% 50 0.5% Stress Strain 0 Shear strain at max. stress Shear strength [MPa] Average dynamic shear result 200 0.0% 0 1 2 3 Material Number Figure 4.23 Average dynamic shear strength and strains at maximum stress including their variability for the three materials in the 45º direction 4.3 Comparison of Static and Dynamic Results The experimental results presented in the previous sections clearly show that strain rate has an effect on both the stress and the strain. Figure 4.24 to Figure 4.26 graphically illustrate the comparison between typical static and dynamic test results for the three materials in the specimens cut in both the fibre as well as the 45 degree directions. This strain rate sensitivity confirms the need for a reliable constitutive rate dependent damage model capable of modelling the response of these woven composite materials under high velocity loadings. 103 EXPERIMENTAL RESULTS 1200 Material - 1 M1FD Fibre direction M1FS Stress [MPa] 1000 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain (a) Shear stress [MPa] 200 Material - 1 M13S 12- direction M13D 150 100 50 0 0.0% 1.0% 2.0% 3.0% 4.0% Shear strain (b) Figure 4.24 Comparison of static and dynamic behaviour in material one (a) 0º or 90º specimens, (b) 45º specimens 104 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 1200 Material - 2 M2FD Fibre direction M2FS Stress [MPa] 1000 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Strain (a) Shear stress [MPa] 200 Material - 2 M23D 12- direction M23S 150 100 50 0 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% Shear strain (b) Figure 4.25 Comparison of static and dynamic behaviour in material two (a) 0º or 90º specimens, (b) 45º specimens 105 EXPERIMENTAL RESULTS 1200 Material - 3 M3FS Fibre direction M3FD Stress [MPa] 1000 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% 2.0% Strain (a) Shear stress [MPa] 200 Material - 3 M33D 12- direction M33S 150 100 50 0 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% Shear strain (b) Figure 4.26 Comparison of static and dynamic behaviour in material three (a) 0º or 90º specimens, (b) 45º specimens 106 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS A look at the initial portion of the stress-strain curves reveal that the dynamic curves are typically steeper than the static ones. This confirms the increase in the initial undamaged elastic modulus with increasing strain rate. In general, on comparing the static and dynamic results, a higher stress is observed for dynamic tests, whereas the strains at the maximum stress are higher for the static tests. Looking at the dynamic results in the normal direction, it is evident that material two experiences the maximum increase (about 38%) in the tensile strength and the minimum decrease in the strain at maximum stress relative to quasi-static results. On the other hand, the dynamic versus static shear behaviour of the three materials reveal a large increase (between 63% to 89%) in the shear strength and a large decrease (between 66% to 83%) in the strains obtained at the maximum stresses. These are presented in Table 4.9. Table 4.9 Comparison of the average of static and dynamic results Average stress [MPa] Description Material 1 Material 2 Material 3 Static Dynamic Increase Average strain at max. stress Static Dynamic Decrease Normal 672 752.5 12% 1.13% 0.56% 51% Shear 87 164 89% 2.46% 0.49% 80% Normal 704 971 38% 1.12% 1.08% 3% Shear 81 132 63% 1.42% 0.49% 66% Normal 868 929 7% 1.33% 0.83% 37% Shear 101 171 69% 7.17% 1.21% 83% 107 EXPERIMENTAL RESULTS 108 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 5 Equation Section (Next) DEVELOPMENT OF A RATE DEPENDENT CONTINUUM DAMAGE MODEL 5.1 Introduction This chapter focuses on the development of a stress rate continuum damage mechanics (CDM) based model for composite materials. A physical treatment of growth of damage based on the extensive experimental results is combined with the frame work of continuum damage mechanics models to form the foundation of the model for materials whose response is governed by elastic deformation coupled with damage. 109 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL The basis for the stress-strain relationship in many CDM approaches is the concept of stress (or strain) equivalence (Lemaitre (1996) among others). The effective stress concept states that any deformation behaviour, whether uniaxial, multi-axial, elastic, plastic or viscoplastic, of a damaged material is presented by the constitutive laws of the virgin material in which the applied stress σ is replaced by the effective stress σ by the following equation; σ = M (d ) σ (5.1) where M(d ) is a fourth order damage effect tensor developed for this purpose which provides a linear mapping of the stress tensor to the effective stress tensor, resulting in a redistribution of stresses over the reduced remaining area, and thus higher local stresses. A hypothesis of strain equivalence for isotropic damage was proposed by Lemaitre in which the Cauchy stress was replaced by the effective stress in the constitutive equation. For solving the damage problems, one must establish that the strain tensor is the same in the effective and damaged state, that is ε =ε (5.2) The notation used for a second order tensor is bold letter, and the one used for a fourth order tensor is italic bold letter. The relationship between the effective stress and strain in linear elasticity has the simple form of σ = C 0ε (5.3) 0 where C is the original undamaged stiffness tensor. The stress-strain relationship in a damaged material can be obtained by rearranging the terms of equations (5.1) and (5.3) as follows: σ = M ( d ) −1 C 0 ε 110 (5.4) HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS The damaged elastic stiffness tensor C (d ) can be defined as the combination of the damage effect tensor with the undamaged stiffness tensor C (d ) = M (d ) −1 C 0 (5.5) Therefore, a general constitutive relationship for a CDM model is obtained as σ = C (d ) ε (5.6) On performing the time derivative on both sides of equation (5.6), this elastic CDM model can be expanded to consider the rate effects, ε , for the materials that are strain rate dependant (Armero and Oller 2000; Chaboche 1988a; 1988b; Dechaene et al. 2002; Nemes and Speciel 1996), such that; σ = C (d ) ε + C (d , d ) ε (5.7) From the formulation it can be clearly seen that the change in stress results not only from the increments of strain, but also from the rate of damage development, which is responsible for the stress softening that occurs. The main role of a CDM model is to provide a mathematical description of the dependence of elastic coefficients on the damage state as well as the damage evolution. The key to the success of all CDM models is to maintain a reasonable link with the physical and experimental observations of damage growth and material response. As a preliminary step in the development of a new CDM model for woven composite materials, an existing composite damage model by Dechaene et al. (2002) was considered. The selection of this model over other models presented in the literature is due to existence of rate dependency in the material behaviour. The Dechaene et al. model is developed for woven glass fibre/ epoxy composite materials in which the fabric warp or weft layer is treated as an equivalent unidirectional layer, that is, the fabric layer is treated 111 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL as a 0/90 cross-ply composite, even though, the behaviour of 0/90 cross-ply composites can be different from that of the woven fabric composite. Their model is a stress rate formulation, similar to equation (5.7), consisting of two parts. The first term is defined as the elastic change in stress with reduced elastic stiffness due to damage, and the second term is the drop in stress due to growth of damage resulting from matrix cracks as well as fibre failure (Iannucci et al. (2001). The latter term is a function of damage and rate of damage. σ = C (d ) ε + σ d (σ ,d , d ) (5.8) Dechaene et al. (2002) introduce six damage parameters to track the damage growth. The damage state of the lamina is divided into three categories, matrix cracks between fibre in the warp (tracked by d1 in the x-direction and d4 in the y-direction), fibre breakage and pull out (tracked by d2 in the x-direction and d5 in the y-direction), and shear matrix cracks representing the horizontal component of matrix crack, or small scale delamination (tracked by d3 in the xz-plane and d6 in the yz-plane). Figure 5.1 illustrates all these damage variables. 112 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Z d3 Y X d2 d6 d5 d1 d4 Figure 5.1 Schematic view of the six damage variables in a composite element (adapted from Dechaene et al. 2002) In their model, the evolution of damage is expressed as an increasing function of stress and damage which includes two stages, nucleation and growth. Nucleation refers to initiation of cracks and it is assumed that the rate of nucleation depends only on the level of stress. The rate of evolution of the damage, which is the growth of the existing cracks, depends on the amount of cracks already present, or in other words, on the existing level of damage. In addition, they have introduced a threshold stress value, σ 0 , below which there is no nucleation nor growth of damage. For growth of damage, the equation suggested by Dechaene et al. (2002) has the form, d1 = ( A0 + A1d1 ) σ xx σ 0 (1 − c1d1 )(1 − c2 d 2 ) 2 −1 (5.9) 113 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL They mention that similar expressions apply for the other damage rates. On the other hand, in an earlier paper by Iannucci et al. (2001), a different form for damage evolution is suggested. It is worth mentioning that the purpose here is not to judge their model but rather point out the existence of various empirical forms to represent the growth of damage and hence obtain the best form that can characterize the behaviour of the woven composite materials based on the accomplished experimental results. While in those papers the authors mention that their proposed damage model can predict the damage modes observed in the laboratory experiments, a number of issues were raised during this investigation which should be addressed by a new damage model. These include: 1. Effect of strain rate on the undamaged elastic stiffness 2. Presence of rate-dependent shear damage in the xy-plane 3. Effect of strain rate on the undamaged shear modulus 4. Material characterization and quantitative comparisons to experimental results 5. A methodology to obtain the material constants in the new CDM model The first three items were not only observed by many researchers (Gilat et al. 2002; Harding et al. 1989; Johnson et al. 2001; Marguet et al. 2007), but also were confirmed by the experimental results presented in chapter 4. Later in this chapter all these items will be dealt with in detail. Nonetheless, it is important to point out that the new CDM model concentrates on the in-plane response of the woven carbon composite materials. It is believed that the concept of delamination can be used to consider the effects of damage on the out-of plane or through-thickness direction. Nevertheless, this concept is deemed to be beyond the scope of the present work. This is not to say, however, that its importance is being downplayed. Rather, a reasonable structure for the model must first be developed before through-thickness damage can be adequately addressed. 5.2 Rate dependent Foroutan-Nemes damage Model The main approach in developing a complete 3D rate dependant composite constitutive damage model is to divide the model into two sections; one concentrating on the in-plane response , which is the subject of this thesis, and the other on the out-of plane or through114 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS thickness response, which can be considered as the inter-laminar response. In the following sections the formulation for the Foroutan-Nemes stress rate CDM model for woven composite materials is presented. 5.2.1 Model assumptions The proposed composite CDM model is capable, but not limited to, predicting the development of damage due to high strain rate loading, i.e., the model is able to predict the behaviour of the woven composite materials undergoing dynamic as well as static loading. Details and material constants of the developed model may be different for every type of woven composite material; however, the model is independent of the geometry and boundary conditions of the structure for the particular composite material. The constitutive damage model is developed for woven carbon fibre fabric with an epoxy or BMI matrix. However, it is believed that the model can be used to predict the behaviour of other strain rate dependant woven composite materials which have similar response. The Foroutan-Nemes rate dependant CDM model is based on the following assumptions: 1. Plastic strain in these materials is negligible; hence the constitutive equation is divided into an elastic part and a damage part, therefore, eliminating the need for a plastic part. 2. The stress and strain are considered as continuous functions within an element. This means that the discontinuous displacements due to crack opening are smeared out, or in other words damage is assumed to be evenly distributed throughout the element. 3. A limited number of scalar damage variables, dij, characterize the state of damage at a point, where i variables represent the particular direction and j variables represent the type of damage in that particular direction. These variables are defined by their effect on the composite mechanical properties and are always ascending, as the damage process is irreversible and no healing 115 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL occurs during a test. Clearly this principle can only apply if the damage is made up of a large number of minute cracks evenly distributed throughout the element. 4. The nucleation and growth of damage should be considered as two distinct stages of damage development. Nucleation means that damage starts from nowhere and it is assumed that the rate of nucleation depends on the level of stress as well as rate of loading. In addition, the growth of damage depends on the existing amount of damage already present. This dependence on loading rate contributes to the difference between the behaviour under the static and under the dynamic loading. Besides, the rate of damage growth creates a time lag between damage and stress resulting in strain rate sensitivity. 5. Start of damage is apparent when the stress level exceeds certain threshold values. The threshold for nucleation and growth of damage are assumed to be equivalent. 6. The onset of damage at the threshold stress level results in the reduction of the elastic stiffness. There is no absolute need for damage variables to have a geometric meaning, such as size or number of cracks of a particular type. Only clear relations between the damage variables and the components of stress and strains are required. The value of the damage variables can never decrease, as there is no healing of the material. For the ease of understanding and calculations, it is possible to restrict these variables within the interval [0,1], zero meaning no damage and unity meaning that the damage is complete as far as that particular type is concerned. 5.2.2 Internal state variables The internal state variables (damage variables), besides being capable of representing the complex micro defect features, should be simple enough to make the model applicable to engineering applications. Traditionally, damage parameters are associated with cracking (e.g. reduction in the cross sectional area of the load bearing material) in the principal 116 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS lamina directions, parallel and perpendicular to the fibre direction, corresponding to matrix and fibre dominated damage modes. In the Foroutan-Nemes damage model three zones are assumed in the stress-strain response. Figure 5.2 illustrates a typical stress-strain response with the three zones indicated. The first zone is the elastic zone where no damage is present; hence the stress in this zone is below the threshold stress. Zone II starts when the stress exceeds the particular threshold stress; by definition, this is the nucleation stress level below which no damage is present in the material. The damage di1, which is the damage due to matrix cracks and some fibre failure, is present in this zone. In the third zone, the damage di2 is in effect. This damage can be related to the fibre pullout and fibre breakage which causes the softening of the material beyond the ultimate tensile stress. Therefore, six damage variables can be introduced in the developed 3D stress-rate formulation with in-plane damage, which are defined as following: d11 – damage due to matrix cracks, due to x- direction loading d12 – damage due to fibre breakage and pullout, due to x- direction loading d 21 – damage due to matrix cracks, due to y- direction loading d 22 – damage due to fibre breakage and pullout, due to y- direction loading d 41 – damage due to shear matrix cracks, due to shear loading d 42 – damage due to fibre breakage and pullout, due to shear loading 117 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL Ultimate stress Stress Zone II, di1 = 0 to di1max di 2 = 0 Zone III, di1 = di1max di 2 = 0 to di 2 max Threshold stress Zone I, di1 = di 2 = 0 0 0.0% Strain Figure 5.2 Typical stress-strain response showing zones of damage In a general three dimensional case for a linear elastic material, each of the stress components is a linear function of the strain tensor; resulting in 36 elastic coefficients, Cij . As explained in Boresi et al. (1993) these coefficients are constants that are characteristics of the material and it is mathematically shown, with double differentiation from the strain energy density function, that the elastic coefficients are symmetrical, i.e. Cij = C ji reducing the elastic coefficients to 21. However, woven composite materials are orthotropic in nature, that is, they possess three orthogonal planes of material symmetry and three corresponding orthogonal axes. It is shown in Boresi et al. that the most general orthotropic material contains nine elastic coefficients relative to the orthotropic axes. In an in-plane two dimensional case, the number is hence reduced to 4 elastic coefficients. In the Foroutan-Nemes damage model the notation Eij is used instead of Cij to represent the elastic coefficients. In the first zone where no damage is present, the elastic coefficients considered are the initial undamaged coefficients, which are characteristics of the material. 118 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Since damage reduces the elastic stiffness of the specimen, a set of equations has to be developed which describes this reduction in the elastic moduli due to the types of damage considered. The developed set of equations would serve as the definition of the damage variables. A survey of approaches taken to this problem in other CDM models shows a somewhat varied treatment (Chow and Wang 1987; Dechaene et al. 2002; Matzenmiller et al. 1995; Yazdchi et al. 1996). However, the modulus of elasticity formulation adopted here considering the six damage parameters associated with modulus reduction can be written as 0 E11 = E11 0 E22 = E22 0 E44 = E44 [(1 − α11 d11 )(1 − α d12 )][(1 − α 21 d21 )][(1 − α 41d 41 )(1 − α d42 ) ] [(1 − α 21 d21 )(1 − α d 22 )][(1 − α11 d11 )][(1 − α 41 d 41 )(1 − α d42 )] [(1 − α 41 d41 )(1 − α d 42 )][(1 − α11 d11 )(1 − α d12 )][(1 − α 21 d21 )(1 − α d 22 )] (5.10) 0 0 where E11 and E22 are the initial undamaged Young’s modulus in the x- and y-directions, 0 E44 is the initial undamaged shear modulus in the xy-plane, αi1 is a constant that controls the maximum value of damage di1 at the end of zone II, and α is a constant usually set to 0.9999 prevent the value of (1 − α 2 di 2 ) going to zero when di2 are equal to unity. It is important to point out that the value of initial undamaged modulus is dependent on the rate of loading; therefore the following expression is used to obtain the undamaged rate dependant modulus: 0 E11 = C 2 (1 + C1 ε11 ) C3 0 E22 = C2 (1 + C1 ε22 ) C3 0 E44 = C4 (1 + C1 ε44 ) C5 (5.11) where C 1 to C5 are the material constants which are determined using the modulus at quasi static and high rates of strain. 119 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL 5.2.3 Evolution of Damage In a rate dependant damage model, the presumption of the evolution of damage equations is perhaps the most arbitrary part of the model development, and various forms can be used for these equations. Nevertheless, an attempt is made to use a logical and convincible method in developing a set of successful evolution equations. The rate of damage is expressed as a function of stress, damage and strain rate. The empirical form for the rate of growth of damages in the x-direction for the damages d11 and d12 - damage due to matrix cracks and damage due to fibre fracture respectively - are given as ⎛ ⎞ σ 11 0 ⎜ σ (1 − α d )(1 − α d ) ⎟⎟ 11 11 12 ⎠ ⎝ 11 0.5 d11 = Z11 [ A11 + A12 d11 ] ⎜ ⎛ ⎞ σ11 d12 = Z12 [ A13d11 + A14 d12 ] ⎜ 0 ⎜ σ (1 − α d )(1 − α d ) ⎟⎟ 11 11 12 ⎠ ⎝ 11 (5.12) 0.5 (5.13) where A11 to A14 are the material constants, σ 110 is the loading rate independent, threshold stress below which there is no damage, and Z11 and Z12 are functions of the strain rates with the following equations Z11 = ⎡ (W11 ε11 ) W12 ⎤ ⎣ Z12 = ⎡ Z 11 ⎣⎢ ⎦ W13 ⎤ ⎦⎥ (5.14) where W11 to W13 are the material constants. It is important to mention that the constant A11 in equation (5.12) contributes to the nucleation of damage, as it allows the rate of damage to start from a state of no damage. Also, equation (5.13) is chosen such that evolution of damage depends on the existing damage present in zone II of the stress-strain response, with A13 being the responsible constant. 120 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Similar expressions apply for d21 and d22 -damage rates related to damages in the ydirection. However, the rate of damage in the shear xy-direction is expressed as ⎛ ⎞ σ 44 d41 = Z 4 [ A 41 + A 42 d 41 ] ⎜ 0 ⎜ σ (1 − α d )(1 − α d ) ⎟⎟ 41 41 42 ⎠ ⎝ 44 0.5 ⎛ ⎞ σ 44 d42 = Z 4 [ A 43d 41 + A 44 d 42 ] ⎜ 0 ⎜ σ (1 − α d )(1 − α d ) ⎟⎟ 41 41 42 ⎠ ⎝ 44 (5.15) 0.5 (5.16) where A41 to A44 are the material constants, Z 4 is a function of strain rate with the following expression: Z 4 = ⎡ (W 41 ε44 ) W42 ⎤ ⎣ ⎦ (5.17) 0 where W41 and W42 are the material constants, and σ 44 is the rate dependent threshold shear stress. 5.2.4 Rate sensitive constitutive CDM model Having defined the damage parameters as well as the evolution of damage equations, it is possible to derive the Foroutan-Nemes rate dependant CDM model. For ease of understanding the features of the model, the resulting stress-strain response for a one dimensional case is considered initially. Figure 5.3 shows the basic idea for the evolution of stress and strain. If there was no growth of damage, the stress increment corresponding to the stain increment Δε which takes place in the time interval Δt , would be Δσ = E1Δε , shown as AD in the figure. However, during the same time interval, the presence of damage causes a drop in stress by an increment value of Δσ d as shown by BD, and this depends on damage, rate of damage, as well as stress. Hence, the general form of the stress rate formulation in one-dimensional form is σ = E ε − σ d (σ , d , d ) (5.18) 121 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL Δε d D C Stress σ2 A σ1 Δσ d B Δσ Δε E2 E1 0 0.0% ε1 ε2 Strain Figure 5.3 Dynamic stress-strain curve showing increment of stress due to damage In equation (5.18) the first term is the elastic change in stress with reduced elastic modulus due to damage, and the second term is the drop in stress due to growth of damage resulting from matrix cracks as well as fibre failure. Due to presence of two types of damage, it is wise to divide the drop in stress increment due to damage into two components, one mainly dealing with the damage due to matrix cracks, di1, and the other dealing with the damage due to fibre breakage and fibre pullout, di2. As a result, the following equation is used for the stress rate formulation in the 11- direction. σ11 = E11ε11 − d11 ( B11 + B12σ 11 ) d12 ( B13 + B14σ11 ) (1 − α11d11 ) − (1 − α d12 ) (5.19) where B11 to B14 are the material damage constants, related to the 11-direction, to be obtained from experiments. In the equation, the damaged rate of stress is dependent on the stress, the rate of damage, as well as the damage itself. 122 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Having defined the one dimensional formulation, it is now possible to return to the general constitutive equation and present the final form of the stress rate constitutive equation. As explained previously, the general CDM model has the form of σ = C (d ) ε + C (d , d ) ε (5.20) which can be written as σ = C (d ) ε + σ d (σ ,d , d ) (5.21) where the coefficients of the constitutive secant stiffness tensor, C(d), are functions of the 0 0 0 damage state, d ij , and the undamaged elastic constants, E11 , E22 , and E44 (as presented in equation (5.10)). Therefore, the full 3D stress rate constitutive equation is written as σ11 = C11ε11 − d11 ( B11 + B12 σ 11 ) − d12 ( B13 + B14 σ 11 ) + C ε + C ε 12 22 13 33 (1 − α d12 ) (1 − α11d11 ) d ( B σ + B22 ) d22 ( B23 + B24σ 22 ) − σ 22 = C22ε22 − 21 21 22 + C21ε22 + C23ε33 (1 − α 21d 21 ) (1 − α d 22 ) σ 33 = C13ε11 + C23ε22 + C33ε33 σ 44 = C44ε44 − d41 ( B41 + B42σ 44 ) (1 − α 41d 41 ) (5.22) − d42 ( B43 + B44σ 44 ) (1 − α d 42 ) σ 55 = C55ε55 σ 66 = C66ε66 where Bij are the material damage constants, d ij and dij are the damage and rate of damage evolution, respectively and Cij are damaged elastic stiffness tensors presented by the following for an orthotropic material: 123 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL ⎡ C11 C12 ⎢C ⎢ 12 C22 ⎢C13 C23 ⎢ 0 ⎢ 0 ⎢ 0 0 ⎢ 0 ⎢⎣ 0 C13 0 C23 0 C33 0 0 C44 0 0 0 0 0 0 0 0 C55 0 0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ C66 ⎥⎦ (5.23) In the current 3D formulation, the through thickness stresses and stiffness are not affected by the damage propagation. Models dealing with the inter-laminar failure known as delamination can be combined with this model to include damage that might be present through the thickness. Hence this model is reduced to an in-plane stress rate formulation. A summary of the material constants present in the model along with their units as well as the equations is presented in Table 5.1. It can be seen that overall 29 constants are present in the Foroutan-Nemes rate dependant CDM model. 124 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Table 5.1 Material constants present in the Foroutan-Nemes CDM model Damaged No. modulus constants Unit 1 C1 sec 2 C2 Pa C 0 E11 = C 2 (1 + C1 ε11 ) 3 3 C3 unit less 0 E22 = C2 (1 + C1 ε22 ) C3 4 C4 Pa 0 E44 = C4 (1 + C1 ε44 ) C5 5 C5 unit less 6 α unit less 0 E11 = E11 7 α11 = α 21 unit less 0 E22 = E22 8 α41 unit less 0 E44 = E44 Equations [(1 − α11 d11 )(1 − α d12 )] [(1 − α 21 d 21 )(1 − α d 22 )] [(1 − α 41 d 41 )(1 − α d 42 )] Damage rate No. dependent constants Unit Equations 9 W11 sec 10 W12 unit less Z11 = ⎡ (W11 ε11 ) W12 ⎤ 11 W13 unit less 12 W41 sec 13 W42 unit less 14 A11 sec-1 15 A12 sec-1 16 A13 sec-1 17 A14 sec-1 18 A41 sec-1 19 A42 sec-1 20 A43 sec-1 21 A44 sec-1 ⎣ Z12 = ⎡ Z 11 ⎦ ⎤ ⎣⎢ ⎦⎥ Z 4 = ⎡ (W 41 ε11 ) W42 ⎤ ⎣ ⎦ W13 0.5 ⎛ ⎞ σ11 d11 = Z11 [ A11 + A12 d11 ] ⎜ 0 ⎜ σ (1 − α d )(1 − α d ) ⎟⎟ 11 11 12 ⎠ ⎝ 11 0.5 ⎛ ⎞ σ 11 d12 = Z12 [ A13d11 + A14 d12 ] ⎜ 0 ⎜ σ (1 − α d )(1 − α d ) ⎟⎟ 11 11 12 ⎠ ⎝ 11 0.5 ⎛ ⎞ σ 44 d41 = Z 4 [ A 41 + A 42 d 41 ] ⎜ 0 ⎜ σ (1 − α d )(1 − α d ) ⎟⎟ 41 41 42 ⎠ ⎝ 44 0.5 ⎛ ⎞ σ 44 d42 = Z 4 [ A 43d 41 + A 44 d 42 ] ⎜ 0 ⎜ σ (1 − α d )(1 − α d ) ⎟⎟ 41 41 42 ⎠ ⎝ 44 125 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL Table 5.1 (continues) Material constants present in the Foroutan-Nemes CDM model No. Stress rate constants Unit 22 B11=B21 Pa 23 B12=B22 unit less 24 B13=B23 Pa 25 B14=B24 unit less 26 B41 Pa 27 B42 unit less 28 B43 Pa 29 B44 unit less 5.3 Equations σ11 = E11ε11 − d11 ( B11 + B12 σ 11 ) − d12 ( B13 + B14 σ 11 ) (1 − α d12 ) (1 − α11d11 ) d ( B σ + B22 ) d22 ( B23 + B24σ 22 ) σ 22 = E22ε22 − 21 21 22 − (1 − α d 22 ) (1 − α 21d 21 ) d ( B + B42σ 44 ) d42 ( B43 + B44σ 44 ) σ 44 = 2 E44ε44 − 41 41 − (1 − α d 42 ) (1 − α 41d 41 ) Material characterization The determination of the material parameters required in the available CDM models has been one of the most difficult tasks, and for most cases, has not been covered in detail in the literature. The damage laws are frequently abstract and the means of determining the parameters required are often not clear. In the current approach, the growth of damage as well as the stiffness reduction functions have been chosen to be representative of the performed experimental observations, even though they might be more complex than many other CDM models. It is believed that this complexity is of little importance as long as a proper procedure is explained for obtaining the required material constants. It is possible to divide the material parameters required by the Foroutan-Nemes model into four groups. The first group consists of the material properties such as the tensile and shear stiffness and the threshold stresses which are required by many CDM models. The other three groups are the specific requirements of the Foroutan-Nemes damage model. These groups contain the parameters that define the reduction in stiffness, (equation set 126 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS (5.10)), the rate of nucleation and growth of damage, (equations (5.12) to (5.16)), and the stress rate formulation (equation set (5.22)). 5.3.1 Undamaged Stiffness and Strength Constants The first and foremost step required for measuring material elastic and strength constants is to perform a series of static and dynamic tests as was explained in chapter 4 of this thesis. It is important to recall that these woven composite materials show a high variability in the tensile test results. Therefore a typical average result should be considered in determining the constants. Figure 5.4 illustrates a typical dynamic as well as static result in the fibre direction. Also for the ease of calculations a smooth curve is considered for each of the static and dynamic responses which show the respective approximate behaviour. The slope of the approximate static response is considered as the static tensile modulus of the material and the initial slope of the smooth dynamic curve, obtained from the average of Hopkinson bar response, is defined as the undamaged tensile stiffness at high rates of strain, E110 H . This is possible because in the initial part of the dynamic response the elastic undamaged tensile modulus is represented by: 0H E11 = σ 11 / ε 11 (5.24) As explained in section 5.2.2, this corresponds to zone I of the stress-strain curves, where no damage is present. By definition, the initiation of damage, causes a reduction in the tensile stiffness, which is the start of zone II, and the stress level at this initiation is 0 . defined as the threshold stress, σ 11 127 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL Dynamic Result approx. Dynamic Static Result approx. Static 800 700 Stress [MPa] 600 500 400 300 200 100 0 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% Strain Figure 5.4 Typical stress-strain response of a woven composite material It is worth mentioning that the woven composite materials tested were all balanced, meaning that the tensile responses in the two fibre directions are very similar, hence, one set of constants is sufficient for the 11- and 22- directions. In the same context, Figure 5.5 represents the shear response of the woven composite material with both the smooth approximation curves. The undamaged static and dynamic shear modulus, is half the initial slope of the corresponding shear stress–shear strain curves as it is characterized by 0 E44 = σ 44 / 2ε 44 = σ 44 / γ 44 (5.25) And the reduction in the shear modulus marks the start of zone II, with the threshold 0 . Therefore, it is worth mentioning that the undamaged tensile and shear stress, σ 44 0 0 and σ 44 ) are modulus as well as the tensile and shear threshold stress levels ( σ 11 calculated from the initial portion of the stress-strain curves. 128 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Dynamic Result approx. Dynamic Static Result approx. Static 200 Shear stress [MPa] 180 160 140 120 100 80 60 40 20 0 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% Shear strain (ε 44 =γ 44 /2) Figure 5.5 Typical shear stress-strain response of a woven composite material The rate dependant undamaged modulus defined in the developed CDM model is calculated using equation (5.11) which is 0 E11 = C 2 (1 + C1 ε11 ) C3 0 E22 = C2 (1 + C1 ε22 ) C3 0 E44 = C4 (1 + C1 ε44 ) C5 (5.26) where the value of C1 is equal to unity and the constants C2 to C5 are obtained using data fit algorithm between the strain rate data and the modulus at quasi static and high rates of strain. In the material model, it is necessary to define a threshold or nucleation criterion which can determine the nucleation of damage. Clearly, if the nucleation function is greater than unity, growth of damage has initiated in the material. Several failure criteria exist in the literature. A brief description of some of these failure criteria are presented here. One of the most known criteria is the von Mises criterion, 129 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL which is used to define yield for a combined state of stress in isotropic materials (Boresi et al. 1993). However, woven composite materials are orthotropic in nature and hence the von Mises criterion cannot be used. The yield criterion proposed by Hill (1983) predicts yielding of anisotropic materials and accounts for six different material properties in the various material direction. In order to determine the material constants, it is necessary to measure the yield stresses corresponding to the three normal and three pure shear stresses in the principal directions and orthogonal planes of anisotropy, respectively (Boresi et al. 1993). The Hill criterion is the generalization of the von Mises criterion and under plane stress, the theory predicts yielding would initiate when the magnitude of stresses reaches the following condition 2 2 2 1 ⎞ ⎛ σ 11 ⎞ ⎛ σ 22 ⎞ ⎛ 1 ⎛ σ 44 ⎞ ⎜ ⎟ +⎜ ⎟ − ⎜ 2 + 2 ⎟ σ 11σ 22 + ⎜ ⎟ =1 Y ⎠ ⎝ X ⎠ ⎝ Y ⎠ ⎝X ⎝ S ⎠ (5.27) where “X” and “Y” are the yield strengths under uniaxial loading in the two in plane directions, and “S” is the in plane shearing yield strength in the xy-plane (Lessard 2002). Azzi and Tsai (1965) extended the Hill theory to anisotropic unidirectional fibre reinforced composites. According to Tsai-Hill criterion, failure occurs when the following equation is equal to or greater than one. 2 2 2 ⎛ σ 11 ⎞ ⎛ σ 22 ⎞ ⎛ σ11σ 22 ⎞ ⎛ σ 44 ⎞ + ⎜ ⎟ +⎜ ⎟ −⎜ ⎟ =1 2 ⎟ ⎜ ⎝ X ⎠ ⎝ Y ⎠ ⎝ X ⎠ ⎝ S ⎠ (5.28) The Tsai-Wu failure criterion is a failure theory which uses different strengths in tension and compression (Tsai and Wu 1971). For the case of plane stress, the Tsai-Wu failure criterion reduces to 2 2 ⎛ σ 11 σ 11 ⎞ ⎛ σ 22 σ 22 ⎞ ⎛ σ 11 ⎞ ⎛ σ 22 ⎞ ⎛ σ 11σ 22 ⎜ t − c ⎟ + ⎜ t − c ⎟ + ⎜ t c ⎟ + ⎜ t c ⎟ − ⎜⎜ X ⎠ ⎝Y Y ⎠ ⎝ X X ⎠ ⎝ Y Y ⎠ ⎝ Xt X c Yt Y c ⎝X 2 ⎞ ⎞ ⎛ σ 22 ⎟⎟ + ⎜ 2 ⎟ = 1 (5.29) ⎠ ⎝ S ⎠ where the superscripts t and c refer to the tensile and compression strengths, respectively. 130 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Hashin (1980) developed a three dimensional failure criteria for unidirectional lamina that recognized two distinct and uncoupled failure modes. While the failure criterion was based on composite stresses, it constructs a piecewise continuous failure form based on failure modes. Hashin identified two failure modes, fibre versus matrix. He developed separate equations based on the failure mode to determine a failure state for unidirectional composite laminates. For the case of plane stress, the Hashin failure criterion has the following two conditions in tension: 2 2 ⎛ σ 11 ⎞ ⎛ σ 44 ⎞ ⎜ t ⎟ +⎜ ⎟ =1 ⎝X ⎠ ⎝ S ⎠ 2 Tensile fibre mode (5.30) 2 ⎛ σ 22 ⎞ ⎛ σ 44 ⎞ ⎜ t ⎟ +⎜ ⎟ =1 ⎝ Y ⎠ ⎝ S ⎠ Tensile matrix mode In this thesis, it is intended to use the failure criterion concept to determine the nucleation of damage. Therefore, instead of using the corresponding strengths, the threshold stresses are used. 2 2 ⎛ σ 11 ⎞ ⎛ σ 44 ⎞ ⎜ 0 ⎟ +⎜ 0 ⎟ =1 ⎝ σ 11 ⎠ ⎝ σ 44 ⎠ 2 Tensile fibre mode (5.31) 2 ⎛ σ 22 ⎞ ⎛ σ 44 ⎞ ⎜ 0 ⎟ +⎜ 0 ⎟ =1 ⎝ σ 22 ⎠ ⎝ σ 44 ⎠ Tensile matrix mode And a nucleation function, F 0 , can be defined as 2 2 ⎛ σ ⎞ ⎛ σ 22 ⎞ ⎛ σ 44 ⎞ +⎜ 0 ⎟ +⎜ 0 ⎟ F = ⎜ 11 0 ⎟ ⎝ σ 11 ⎠ ⎝ σ 22 ⎠ ⎝ σ 44 ⎠ 2 0 (5.32) 0 0 0 where σ 11 , σ 22 , and σ 44 are the threshold stress values that are obtained from experiments. Therefore, if the nucleation function is greater than unity, damage initiates in the material. In the special cases of uniaxial tension or pure shear, equation (5.32) reduces to one of the following equations: 131 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL ⎛σ ⎞ F = ⎜ 11 0 ⎟ ⎝ σ 11 ⎠ 2 0 ⎛σ ⎞ F = ⎜ 22 0 ⎟ ⎝ σ 22 ⎠ 2 ⎛σ ⎞ F = ⎜ 44 0 ⎟ ⎝ σ 44 ⎠ 2 (5.33) 0 0 Table 5.3 tabulates the nucleation threshold values for the three materials using experimental data presented in Figure 4.15 to 4.17 and Figure 4.20 to 4.22. Table 5.2 Calculated threshold values for the three woven composite materials Constants 0 0 σ 11 and σ 22 [MPa] 0 σ 44 [MPa] Material 1 Material 2 Material 3 109 124 165 25 23 15 It is worth mentioning that the combined stress nucleation function can predict the initiation of damage in a material under multi-dimensional loading. To understand this better, consider an example in which material one experiences a multi-dimensional loading such that the two normal and shear stress are as following σ11 = 70 MPa, σ 22 = 70 MPa, σ 44 = 15 MPa (5.34) Using equation (5.32) and Table 5.2, the nucleation function is calculated to be 1.09 which is greater than unity, implying that damage has initiated in the material. It can be seen that even though the individual stress components are each lower than the corresponding threshold stress, damage has initiated in the material. Also, it is important to point out that the failure criteria that were briefly described at the beginning of this section each have a physical justification, and not all can be used as a 132 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS threshold criterion for woven composite materials. To understand this better, consider another case of multi-axial loading on material one that result in the following stress components: σ11 = 110 MPa, σ 22 = 110 MPa, σ 44 = 0 (5.35) If the Hill criterion, equation (5.27), is used to calculate the nucleation function, the value calculated would be zero, that is damage is not present. This cannot be justified since both normal stress components have already exceeded the threshold limit, which means damage has initiated in the material. On the other hand, if the combined stress criterion defined in equation (5.32) is used, the nucleation function would equal 1.4, implying that damage is present in the material. 5.3.2 Stiffness Reduction Functions Considering the one dimensional tensile case, it becomes possible to simplify the stiffness reduction equation (5.10) to the following: 0 E11 = E11 [(1 − α11 d11 )(1 − α d12 ) ] (5.36) However, beyond the threshold stress, the reduction in the stiffness is due to the presence of damage d11. This corresponds to zone II (Figure 5.2) where, due to absence of fibre failure and fibre pullout, the damage variable d12 is zero. Hence the value of damage d11 which is mainly due to matrix cracks can be calculated from α11 d11 = 1 − ( E11 / E110 ) 2 at Zone II (5.37) It is known that the value of damage, d11, varies between zero at the threshold stress and unity at the maximum tensile stress. Therefore, it is possible to obtain the constant α11 such that a value of 0.9999 is obtained at the end of zone II. This constant determines the start of the strain softening in the stress-strain response. 133 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL In zone III, the damage variable d12, which is mainly due to fibre failure and fibre pullout, can be calculated by rearranging the equation (5.36) in the following form α d12 = 1 − 0 2 ( E11 / E11 ) (1 − α11 d11 ) at Zone III (5.38) Here the damage variable d11 has a known value close to unity, and α, is a constant usually set to 0.9999 which prevents the value of (1 − α di 2 ) going to zero when di2 are equal to unity. Following the same procedure for the shear response, one can obtain the constant α41 as well as the increments of damage related to the shear response, that is, d41 and d42, with the following equations: 0 2 α 41 d 41 = 1 − ( E44 / E44 ) α d 42 = 1 − 0 2 ( E44 / E44 ) (1 − α 41 d 41 ) at Zone II (5.39) at Zone III (5.40) 5.3.3 Damage Evolution Functions The rate of damage in the one dimensional case can be calculated from the increments of damage at the time increment, Δ t , for both the matrix failure as well as fibre breakage damages, such that Δd d11 = 11 Δt Δd d12 = 12 Δt (5.41) ′ and d12 ′ In order to simplify the evolution of damage equations (5.12) and (5.13), let d11 be defined and calculated as the following 134 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS ⎛ σ 0 (1 − α11d11 ) ⎞ ′ = d11 ⎜ 11 d11 ⎟⎟ ⎜ σ 11 ⎝ ⎠ 0.5 0 ⎛ σ 11 (1 − α11d11 )(1 − α d12 ) ⎞ ′ d12 = d12 ⎜ ⎟⎟ ⎜ σ 11 ⎝ ⎠ 0.5 (5.42) Hence, the evolution of damage equations (5.12) and (5.13) can be rewritten as ′ − Z11 ⎣⎡ A11 + A12 d11 ⎦⎤ = 0 d11 (5.43) ′ − Z12 ⎡⎣ A13d11 + A14 d12 ⎤⎦ = 0 d12 where the Z constants are related to loading rate of the specimen. At high loading rate at which Hopkinson bar experiments are performed, the Z values are equal to unity, and decreases according to equation (5.14) as the loading rate decreases to static loading. The two sets of rate of damage parameters ( A11 , A12 and A13 , A14 ) are obtained from the best-fit curve of d1 j verses d1′ j , where j denotes the type of damage. Similarly, from the shear response, one can calculate two sets of rate of damage parameters ( A 41 , A42 and A 43 , A44 ) from the best-fit curve of d 4 j verses d4j′ , using the following equations: ′ − Z 4 ⎡⎣ A 41 + A 42 d 41 ⎤⎦ = 0 d41 (5.44) ′ − Z 4 ⎡⎣ A 43d 41 + A 44 d 42 ⎤⎦ = 0 d42 where ⎛ σ 0 (1 − α 41d 41 ) ⎞ ′ = d41 ⎜ 44 d41 ⎟⎟ ⎜ σ 44 ⎝ ⎠ 0.5 0 ⎛ σ 44 (1 − α 41d 41 )(1 − α d 42 ) ′ = d42 ⎜ d42 ⎜ ⎝ σ 44 ⎞ ⎟⎟ ⎠ 0.5 (5.45) 135 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL Z4 is related to loading rate of the specimen, calculated according to equation (5.17). It has a value of unity at the experimented high strain rate which decreases with a decrease in the strain rate. 5.3.4 Stress Rate Functions The next step is to determine the constants in the stress rate CDM model presented in equation (5.22). First, consider the one dimensional case in which the stress rate formulation can be simplified as σ11 = E11ε11 − d11 ( B11 + B12 σ 11 ) (1 − α11d11 ) − d12 ( B13 + B14 σ 11 ) (1 − α d12 ) (5.46) As was explained in section 5.2.4, the first part is the elastic change in stress with a reduced elastic modulus E11, the second term is the drop in stress due to the growth of damage d11 which is mainly in the form of matrix cracks, and the third term is the drop in stress due to growth of damage d12 resulting from fibre breakage and fibre pullout. In the zone II region of the stress-strain response, due to absence of the growth of damage d12, the third term is not considered. Hence, for zone II, equation (5.46) can be simplified and rearranged as follows: B11 + B12 σ11 = ( E11ε11 − σ11 )(1 − α11d11 ) d11 at Zone II (5.47) The stress rate constants B11 and B12 can be calculated from a linear fit between the stress and incremental values obtained from the right hand side of equation (5.47) using a datafit algorithm. Here, the rate of stress is calculated from the stress increments Δσ11 at time increments Δ t as: σ11 = 136 Δσ 11 Δt (5.48) HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS The values of d11 and d11 at different time intervals can be obtained using the equations explained previously. The two remaining unknown stress rate constants of equation (5.46), i.e. B13 and B14 can be obtained from the zone III region of the stress-strain response. In this zone, the rate of damage d11 is assumed to be zero; hence equation (5.46) can be simplified as follows: B13 + B14 σ11 = ( E11ε11 − σ11 )(1 − α d12 ) d12 at Zone III (5.49) and the stress rate constants are calculated from a linear fit between the stress and incremental values obtained from the right hand side of equation (5.49) using a data-fit algorithm. Similarly, it is possible to calculate the shear stress rate constants B 41 , B 42 and B 43 , B44 using a data-fit algorithm and the following equation σ 44 = 2 E44ε44 − d41 ( B41 + B42 σ 44 ) (1 − α 41d41 ) − d12 ( B43 + B44 σ 44 ) (1 − α d42 ) (5.50) Consequently, using this methodology, it is feasible to easily obtain all the constants present in the Foroutan-Nemes constitutive damage model. Table 5.3 tabulates the calculated values for these material constants for the three materials presented in this thesis. 137 DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL Table 5.3 Calculated constants for the three woven composite materials No. Constants 0 1 σ 11 [Pa] Material 1 Material 2 Material 3 1.09E+08 1.24E+08 1.65E+08 2 0 σ 44 [Pa] 2.50E+07 2.30E+07 1.50E+07 3 ν 12 4 5 6 7 8 C1 [sec] C2 [Pa] C3 C4 [Pa] C5 9 10 11 α α11 = α 21 α41 12 13 14 15 16 W11 [sec] W12 W13 W41 [sec] W42 A11 [sec-1] A12 [sec-1] A13 [sec-1] A14 [sec-1] A41 [sec-1] A42 [sec-1] A43 [sec-1] A44 [sec-1] B11=B21 [Pa] B12=B22 B13=B23 [Pa] B14=B24 B41 [Pa] B42 B43 [Pa] B44 0.087 1 8.35E+10 0.2 8.04E+09 0.45 0.99999 0.73 0.775 0.0056 1.045 0.7 0.005 1.07 18000 -12000 21000 -21000 25000 -15500 13000 -13000 -1.00E+06 0.25 1.50E+05 0.35 -1.30E+06 0.15 2.00E+06 0.35 0.06 1 6.08+10 0.14 1.00E+10 0.3 0.99999 0.7 0.775 0.0029 0.97 0.7 0.004 1.05 15000 -10000 17000 -17000 32000 -19000 10000 -10000 4.00E+06 0.35 -5.00E+07 0.55 -1.50E+06 0.12 3.00E+06 0.7 0.08 1 8.58E+10 0.17 6.40E+09 0.3 0.99999 0.73 0.7 0.0038 1 0.85 0.0025 1.04 16000 -10000 18000 -14000 17000 -13500 7000 -7000 -7.00E+06 0.38 7.00E+07 0.5 -1.40E+06 0.50 1.40E+07 0.65 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 138 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 6 Equation Section (Next) IMPLEMENTATION OF THE DAMAGE MODEL INTO A FINITE ELEMENT CODE 6.1 Introduction The ultimate criterion for a successful model is its ability to predict results comparable with those of experiments, regardless of how complex the developed mathematical model is in describing the material behaviour. Currently, the finite element method is one of the most powerful tools for analysing complex structures subjected to different loading conditions. Therefore, in order to use this technique for any developed material model, it must be incorporated into a finite element code. Although in many engineering 139 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE applications, complicated structural geometry may be used, they mostly consist of some fundamental structural units. This makes it reasonable to test models at high loading rates dealing with damage analysis using simple structural units. 6.2 Model implementation in finite element The Foroutan-Names damage model for woven composite materials developed and parameterized in the previous sections is next implemented in a VUMAT Material User Subroutine of the commercial finite element program ABAQUS/Explicit. This subroutine is compiled and linked with the finite element solver and enables ABAQUS/Explicit to obtain the needed information regarding the state of the material and the material mechanical response during each time step, for each integration point of each element. The basic difference between an explicit and an implicit analysis is that the former requires no information regarding the state variables at the end of the increment. Rather it relies on sufficiently small time increments such that system’s future state can be extrapolated from the current state. The state variables are updated after the increment. An implicit approach accounts for the change in state variables that occur over the step. Hence, they require that the stress Jacobian be defined. Implicit solutions tend to be more time consuming, however, much larger time increments are possible. In the current situation, where short duration loading, such as impact, is of concern and total time of the event is a fraction of a second, the explicit solution method is the reasonable analysis technique to carry out. For a detailed review of the ABAQUS theories or their implementation the reader is referred to the ABAQUS theory manual (ABAQUS 2006). Hence, the model defined in Chapter 5 was programmed in FORTRAN language and supplied as a VUMAT subroutine to the ABAQUS/Explicit finite element software following the instruction provided in the ABAQUS documentation. The analysis procedure involves three elements: the main program, input file and the subroutine. 140 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Figure 6.1 illustrates the interaction between these main three elements and the subroutine procedure. In the flowchart, nblock is a term used in ABAQUS that refers to the number of integration points to be processed in one call to the VUMAT. The FORTRAN code for the VUMAT subroutine is provided in Appendix A. The essential features of the coupling between the ABAQUS/Explicit finite-element solver and the VUMAT Material User Subroutine at each time increment at each integration point of each element can be summarized as follows: (a) The corresponding previous time-increment, stresses, and material state variables as well as the current time-step strain increments are provided by the ABAQUS/Explicit finite element solver to the material subroutine. In the present work, the strain components, the damage variables, the secant modulus and a variable defining the deletion status of the element in question are used as state variables; and (b) The material stress state as well as values of the material state variables at the end of the time increment are determined within the VUMAT and returned to the ABAQUS/Explicit finite-element solver, using the information provided in (a), the nucleation function define in Section 5.3.1 for damage initiation, and the CDM material model presented in Section 5.2. 141 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE FE data base Mesh, BC & loading Main Program ABAQUS/Explicit User-supplied material properties & constants VUMAT subroutine From k = 1 to nblock Calculate total strain & nucleation function Is F >1? 0 NO Calculate stress σii NO Calculate di2, Eii, σii YES Calculate di1, Eii, σii Is di1 < 0.9999 ? YES Is di2 < 0.9999 ? NO Element is deleted YES NO Is k > nblock YES ? Next time step t = Δt + t YES Is t < total time ? NO STOP Figure 6.1 Flowchart for implementation of the CDM model into ABAQUS 142 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 6.3 Implementation and comparison to experiments The material model for woven composite materials developed and parameterized in the previous chapter is justified in this section by carrying out a series of dynamic tensile analyses on a plane stress single element and by comparing the computational results with the experimental results. In the remainder of this section, a brief description of loadingunloading as well as compression tests and the nature of the results obtained in these simulations are presented. This is followed by the description of the computational procedure used to simulate the test. 6.3.1 Tensile test results For the tension test, a two dimensional plane stress element is subjected to dynamic tensile displacement at one end. The element used is a four node bilinear plane stress quadrilateral with reduced integration point denoted by CPS4R in ABAQUS. In order to consider the rate effect, the amplitude toolset is used which allows the user to specify a time varying displacement for the loading, which is to say, the displacement history extracted from the results of the Hopkinson bar testing can be applied to the simulation. From the dynamic experimental results it was observed that the displacement, Δ , is a function of time, which can be approximated by Δ = A t2 (6.1) Figure 6.2 compares the actual time-displacement data obtained from a Hopkinson bar experiment with the profile obtained using the equation (6.1) for the three materials, and it can be concluded that the profile from the equation is a representation. Also A is found to be a function of average strain rate, that is, A = Bε 2 (6.2) where B is a constant obtained from a data fit algorithm using the displacement versus time data.. Therefore, a displacement versus time profile can be applied at different average strain rates to simulate the loading of the material. 143 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE Material One Displacement [mm] 1 Fibre Direction 0.8 0.6 0.4 Mat-1 Test Mat-1 eq 0.2 0 0 10 20 30 40 50 60 Time [μ sec] (a) Material Two Displacement [mm] 2.5 Fibre Direction 2 1.5 1 Mat-2 Test Mat-2 eq. 0.5 0 0 10 20 30 40 50 60 70 Time [μ sec] (b) Figure 6.2 Comparison between the displacement-time profiles obtained from dynamic test results and the equation 6.2 (a) material one, (b) material two, (c) material three 144 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Material Three Fibre Direction Displacement [mm] 1.5 1 Mat-3 eq 0.5 Mat-3 Test 0 0 10 20 30 40 50 60 Time [μ sec] (c) Figure 6.2 (continued) Comparison between the displacement-time profiles obtained from dynamic test results and the equation 6.2 (c) material three Due to symmetry only a quarter of specimen gage section is modeled. Figure 6.3 illustrates the tensile loading in the fibre direction where, the nodes on the left are restrained in x-direction, and the nodes on the bottom are restrained in the y-direction. The results for the two fibre direction will be similar as the properties in the two directions were the same for all the three materials in this thesis. 145 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE y u=0 u = Δx(t ) x v=0 Figure 6.3 Two dimensional element, tensile simulation in ABAQUS Using the VUMAT subroutine for the material model, the load is applied at different strain rates and the stress-strain behaviour is obtained. The simulated strain rates vary from quasi static value of 1E-3/sec to a value equivalent to the experimental average strain rates for each material. The dynamic average strain rates obtained using the Hopkinson bar setup are 230/sec, 420/sec and 315/sec for materials one to three, respectively. For the purpose of comparison, the tensile simulation results at different strain rates is plotted with the average experimental dynamic and static results for each material as illustrated in Figure 6.4. The figure shows that the model can well predict the dynamic experimental result. Table 6.1 summarizes the maximum dynamic tensile strength obtained from the simulation and compares them with the ones from the test data. It is found that the percentage difference is less than 5% for all three materials. It is worth mentioning that due to presence of symmetry in the material, that is similar behaviour in both fibre directions, the same results can be postulated for the perpendicular y-direction. 146 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 1000 Material One 900 Fibre Direction Dy-Test St- Test ABQ 1000 ABQ 230 Stress [MPa] 800 ABQ 100 700 ABQ 50 600 ABQ 5 500 ABQ 1 ABQ 1E-3 400 300 200 100 0 0.0% 0.5% 1.0% 1.5% Strain (a) 1400 Dy-Test Material Two St-Test Fibre Direction 1200 ABQ 1000 ABQ 420 Stress [MPa] ABQ 200 1000 ABQ 100 ABQ 50 800 ABQ 5 ABQ 1 ABQ 1E-3 600 400 200 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Strain (b) Figure 6.4 Comparison of the ABAQUS simulation with the average dynamic and static tensile test results (a) material one, (b) material two, (c) material three 147 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE 1400 St-Test Fibre Direction 1200 Stress [MPa] Dy-Test Material Three ABQ 1000 ABQ 315 ABQ 100 1000 ABQ 50 800 ABQ 5 600 ABQ 1E-3 ABQ 1 400 200 0 0.0% 0.5% 1.0% 1.5% 2.0% Strain (c) Figure 6.4 (continued) Comparison of the ABAQUS simulation with the average dynamic and static tensile test results (c) material three As can be seen from the graphs, a much higher strain rate of 1000/sec, is also chosen to illustrate the prediction of the simulation, and also compare the result with the available dynamic test data. The model cannot perfectly predict the static behaviour of the material; however an approximate response can be obtained. Since the main purpose of the developed model is to predict the behaviour under dynamic loading, an approximate prediction for the static response is accepted. Table 6.1 Comparison of maximum dynamic tensile strength Max. dynamic tensile stress [MPa] 148 Material 1 (232 sec-1) Material 2 (420 sec-1) Material 3 (315 sec-1) Simulation 755 990 1014 Test 752 971 969 % difference 0.3% 2.0% 4.5% HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 6.3.2 Bias extension shear test results For the shear test, a two dimensional plane stress element is subjected to dynamic shear displacement at the top left corner node, as illustrated in Figure 6.5. Due to the nature of the bias extension shear test, in which the fibres are at 45 degree to the loading direction, the whole square part of the specimen gage section is modeled. As shown in the Figure 6.5, the bottom left corner node is restrained in x-direction and y-direction. In a similar manner to the tensile simulations, the amplitude toolset is used to specify various displacement times for the loading, in order to consider different average strain rates as illustrated in Figure 6.6. y v = Δ y (t ) u = Δx(t ) u=v=0 x Figure 6.5 Two dimensional element, shear simulation in ABAQUS 149 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE Material One Displacement [mm] 0.6 12- Direction 0.4 Mat-1 Test 0.2 Mat-1 eq 0 0 10 20 30 40 50 60 70 Time [μ sec] (a) Material Two Displacement [mm] 0.8 12- Direction 0.6 0.4 Mat-2 Test Mat-2 eq. 0.2 0 0 10 20 30 40 50 60 70 Time [μ sec] (b) Figure 6.6 Comparison between the shear displacement-time profiles obtained from dynamic test results and the equation 6.2 (a) material one, (b) material two, (c) material three 150 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Material Three 2 12- Direction Displacement [mm] 1.8 1.6 1.4 1.2 1 0.8 Mat-3 eq 0.6 Mat-3 Test 0.4 0.2 0 0 20 40 60 80 Time [μ sec] (c) Figure 6.6 (continued) Comparison between the shear displacement-time profiles obtained from dynamic test results and the equation 6.2 (c) material three Using the VUMAT subroutine for the material model, the above mentioned load is applied at different strain rates, varying from quasi-static value of 1E-3/sec to a value equivalent to the experimented average strain rates for each material, and the stress-strain behaviour is obtained. Using the Hopkinson bar setup, the dynamic average strain rates for the bias extension shear experiments are calculated as 180/sec, 260/sec and 490/sec for the materials one to three respectively. For the purpose of comparison, the shear simulation results at different strain rates is plotted with the average experimental dynamic and static bias extension shear results for all the three material as illustrated in Figure 6.7. It can be seen that the model can well predict the dynamic behaviour, and the percentage difference between the simulation and test data for the maximum shear strength is approximately between 3-5% for the three materials as tabulated in Table 6.2. A simulation for a much higher strain rate of 1000/sec, is also performed to illustrate the prediction of the response, and also compare the result with the available dynamic test data. It is worth mentioning that the model cannot perfectly predict the static shear 151 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE behaviour of the material; however an approximate response can be obtained. Since the main purpose of the developed model is to predict the behaviour under dynamic loading, an approximate prediction for the static response is accepted. 250 St-Test Material One Dy-Test Shear stress [MPa] 12- Direction ABQ 1000 200 ABQ 165 ABQ-50 ABQ 5 150 ABQ 1 ABQ 1E-3 100 50 0 0.0% 1.0% 2.0% 3.0% 4.0% Shear strain (a) Figure 6.7 Comparison of the ABAQUS simulation with the average dynamic and static shear test results (a) material one, (b) material two, (c) material three 152 Shear stress [MPa] HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 180 Material Two 160 12- Direction Dy-Test St-Test ABQ 1000 140 ABQ 260 ABQ-50 120 ABQ 5 100 ABQ 1 ABQ 1 E-3 80 60 40 20 0 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% Shear strain (b) 200 Dy-Test Material Three St-Test Shear stress [MPa] 12- Direction ABQ 1000 ABQ 490 150 ABQ 100 ABQ-50 ABQ 5 100 ABQ 1 ABQ 1 E-3 50 0 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% Shear strain (c) Figure 6.7 (continued) Comparison of the ABAQUS simulation with the average dynamic and static shear test results (b) material two, (c) material three 153 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE Table 6.2 Comparison of maximum dynamic shear strength Max. dynamic shear stress [MPa] Material 1 (186 sec-1) Material 2 (263 sec-1) Material 3 (493 sec-1) Simulation 172 136 165 Test 164 132 171 % difference 4.9% 3.0% 3.5% 6.3.3 Simulation results for loading-unloading-reloading During a typical dynamic impact, a part of the impacted material can also go under compressive loading after initially experiencing tensile loading. Hence, it is important to check the behaviour of the material model when the material goes under a loadingunloading-reloading cycle. For this purpose, a constant strain rate is considered which is equal to the average strain rate at which dynamic tests are performed, that is, 230/sec for material one, 420/sec for material two and 315/ sec for material three. First the loadingunloading where the unloading starts at different time intervals is considered. In these simulations the specimen is loaded to a fraction of the total time followed by an unloading interval as illustrated in Figure 6.8. It is interesting to note that the response returns to a point of zero stress and zero strain when the specimen is unloaded. 154 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Material One 900 Fiber Direction 800 Stress [MPa] 700 600 500 400 300 200 100 0 0.0% 0.5% 1.0% Strain (a) Material Two 1200 Fiber Direction Stress [MPa] 1000 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% Strain (b) Figure 6.8 Loading-unloading simulation results (a) material one, (b) material two, (c) material three 155 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE 1200 Material Three Fiber Direction Stress [MPa] 1000 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain (c) Figure 6.8 (continued) Loading-unloading simulation results (c) material three In the loading-unloading-reloading simulation, after the specimen is loaded to a fraction of the total time, two unloading scenarios are possible. In the first case the specimen is unloaded to remove the entire initial tensile load on the specimen as illustrated in Figure 6.9, which is then followed by applying the load again with the same strain rate. It can be seen that due to the presence of damage already present in the specimen, the reloading response has lower initial tensile modulus. Also it is worth nothing that the tensile strength is much lower than the initial one step loading tensile simulation due to damage during the first loading cycle. 156 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Material One 900 Fiber Direction 800 Stress [MPa] 700 600 500 400 300 200 100 0 0.0% 0.5% 1.0% Strain (a) Material Two 1200 Fiber Direction Stress [MPa] 1000 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% Strain (b) Figure 6.9 Loading-unloading-reloading simulation results (a) material one, (b) material two, (c) material three 157 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE 1200 Material Three Fiber Direction Stress [MPa] 1000 800 600 400 200 0 0.0% 0.5% 1.0% 1.5% Strain (c) Figure 6.9 (continued) Loading-unloading-reloading simulation results (c) material three In the second case the specimen is unloaded for a longer duration and hence the specimen would experience a compressive load as illustrated in Figure 6.10, which is followed by reapplying the load with the same constant strain rate until failure. It can be seen that the damage present in the specimen is larger than the previous case and the tensile strength is much lower than first case simulation. It is important to note that the material experiences damage even in the compression response, and as soon as there is load reversal from compression, the growth of damage becomes zero. Hosur et al .(2003a) have shown the high strain rate response of woven carbon epoxy materials under compression. From their experimental results it can be concluded that the peak stress and the slope of the stressstrain response increases with strain rate in compression, which is similar to the tensile response. It is worth mentioning that in the developed constitutive damage model, it is assumed that the material behaviour is similar in both tension and compression. 158 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Material One 900 Fiber Direction Stress [MPa] 600 300 0 -0.5% 0.0% 0.5% 1.0% -300 -600 Strain (a) Material Two 1200 Fiber Direction 900 Stress [MPa] 600 300 0 -1.2% -0.6% 0.0% -300 0.6% 1.2% 1.8% 2.4% -600 Strain (b) Figure 6.10 Loading-unloading to compression-reloading simulation results (a) material one, (b) material two, (c) material three 159 IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE Material Three Fiber Direction 1200 Stress [MPa] 800 400 0 -0.6% 0.0% 0.6% 1.2% -400 -800 Strain (c) Figure 6.10 (continued) Loading-unloading to compression-reloading simulation results (c) material three 6.4 Summary The developed Foroutan-Names damage model was implemented in a VUMAT Material User Subroutine of the finite element program ABAQUS/Explicit. Uniaxial tension and shear simulations were performed on the three materials at different loading rates. The dynamic response was well simulated, and the simulation for the static loading was shown to be approximately close to the test data. Simulation results for higher than tested strain rates were also presented to illustrate the response at these high rates. It was shown that the tensile strength increases with the increase in the loading rate, and also a strain softening phenomenon was observed after the maximum tensile strength. The loadingunloading, as well as compression response of the materials were also simulated to ensure the capability of the model to capture these responses. 160 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 7 Equation Section (Next) CONCLUSIONS 7.1 Summary and conclusions A summary of the major findings drawn from this research along with its important conclusions are provided in this section, which is divided into two parts. The first part discusses the experimental work needed to develop a rate dependant constitutive damage model capable of predicting the response of woven composite materials under high strain rate loading and the second part discusses the developed CDM model that is implemented into a finite element code along with the simulation results. 161 CONCLUSIONS 7.1.1 Experimental findings In this work, four types of experiments were performed, namely static uniaxial tension, static bias extension shear, dynamic uniaxial tension, and dynamic bias extension shear experiments. The tests were conducted on three composite materials, namely a 2×2 twill weave carbon/epoxy prepreg [0]6, a carbon/BMI prepreg [0]6 with 8 harness satin weave, and a plain weave carbon/epoxy prepreg [0]8, all of which are balanced woven fabrics with a zero direction staking sequence. The static experiments were performed using a servo-hydraulic MTS testing machine, where as the dynamic tests were conducted using a tensile version of Hopkinson bar setup. For the dynamic experiments, the shape and size of the specimen was designed such that, it is short enough to reach stress equilibrium but large enough to characterize the mechanical properties respective of the material. Also to insure failure in the gage section, the specimen’s cross sectional area must be reduced. This is done using a smooth radius of curvature in the transition region to prevent stress concentration. For consistency, the same specimen size and geometry are used for all static and dynamic experiments. The test specimens are cut from composite plate in the (0°), (90°) and (45°) directions. The (0°) and the (90°) specimens are used to obtain static and dynamic tensile behaviour in the normal directions, whereas the (45°) specimens are used in biased extension shear tests to obtain the nonlinear in-plane shear properties. The specimens cut in (45°) directions are tested in tension to ultimate failure. A stress analysis of the 45 degree cut specimen uses the transformation matrix to transform the results obtained in the loading direction into the stresses and strains in the fibre direction. A special fixture was designed to hold the dog bone specimen in between the Hopkinson bar setup while performing the experiments. The designed fixture has the following specifications. Firstly, a large gripping force is needed to prevent the specimen from slipping out of the fixture, secondly, it is made of the same material as the Hopkinson 162 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS bars to prevent disturbance in the wave propagation, and thirdly, the specimen must be aligned with the fixtures to avoid eccentric loading. In static experiments, the strains are measured using a reduced extensometer in the loading direction, and the transverse strains are measured from the results of the strain gages attached on the specimens perpendicular to the loading direction. In the Hopkinson bar experiments, a calibration factor is introduced to acquire the specimen strains using the equations of Hopkinson bar theory. This calibration factor is needed to calculate the effective gage length of the dog-bone specimen. Due to the smooth radius of curvature in the dog bone specimens, the specimen gage length is no longer well-defined. Hence, a series of specimens are tested using strain gages glued directly on the specimen to determine the calibration factor to calculate the effective gage length and as a result ensure the accuracy of dynamic strains. Static tensile tests are performed and a linear brittle behaviour is confirmed for all the three materials. It is observed that material three is the strongest with an average tensile strength of 867 MPa. This is approximately 180 MPa higher than the other two materials. In addition, material three has the maximum average strain to failure of about 1.33%. The average static tensile modulus of all the three materials has little variability in the two fibre directions. Even the variability between the materials tested with different weaves is very small, hence an average value of 63±3 GPa is a good approximation. Also a very low Poisson’s ratio is obtained for all three materials. In the static bias extension shear results for all the materials, an initial linear elastic region followed by nonlinear deformation behaviour is observed. Material three has the maximum strength, and the average shear strength for the three materials is 81-101 MPa. And the material with the 2×2 twill weaves is stiffer than material two with the 8HS weave. Material three shows the maximum strain to failure of about 12%, about twice the other two materials. The initial undamaged shear modulus obtained from the initial slope of the shear stress-strain curve is approximately 5-6 GPa for the three materials. 163 CONCLUSIONS High strain rate tensile tests are performed in both fibre directions of all three materials using the Hopkinson bar setup. The strain rates in these experiments vary from approximately 200 /sec up to 550 /sec. All the materials show an initial linear elastic region followed by nonlinear deformation until the ultimate stress which precedes a nonlinear strain softening behaviour. It is observed that material two has the highest average tensile strength of 979 MPa followed by materials one and three with 929 MPa and 752 MPa average tensile strength, respectively. Furthermore, material two has the highest average strain to failure of about 2.45%, and material one has the lowest of about 1.1%. Also, it was observed that the tensile results of the two fibre directions are comparable for each material, which confirms that these materials are symmetric and justifies the use of the biased extension shear test to obtain the in-plane shear properties. The dynamic bias extension shear experiments are performed using the same Hopkinson bar setup and fixture. The strain rates in these experiments vary from about 170 /sec up to 580 /sec. The behaviour of all the three materials is similar, that is, an initial linear elastic region followed by nonlinear deformation until the ultimate shear stress which precedes a nonlinear strain softening behaviour. It was observed in these experiments that the shear strength is much lower than the tensile strength, as expected. However, the materials fail at much higher strains. Material three has the highest average shear strength of 171 MPa, followed by material one with 164 MPa, and material two with 132 MPa. Comparisons of the static and dynamic experimental results clearly show that strain rate has an effect on both the stress and the strain. A higher stress is observed for dynamic tests, whereas the strains at the maximum stress are higher for the static experiments. Also an increase in the initial undamaged elastic modulus, for both tensile and shear response, with increasing strain rate is observed. This strain rate sensitivity confirms the need for a reliable constitutive rate dependent damage model capable of modelling the response of these woven composite materials under high velocity loadings. 164 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS 7.1.2 Material model and simulation findings A rate dependant constitutive damage model is developed in the framework of continuum damage mechanics which exhibits elastic deformation coupled with damage. The proposed Foroutan-Nemes stress rate CDM model concentrates on the in-plane response of the woven carbon composite materials. The model is mainly developed to predict the behaviour of woven composite materials undergoing dynamic loading. It cannot perfectly predict the static brittle behaviour of the material; however, an approximate static response can be obtained. Details and material constants of the developed model may be different for every type of woven composite material; however, the model is independent of the geometry and boundary conditions of the structure for the particular composite material. In the Foroutan-Nemes CDM model the reduction in the elastic moduli is attributed to damage. And the evolution of damage is expressed as a function of stress, damage and strain rate. The developed stress rate CDM formulation includes two parts, one is related to the elastic change in stress with reduced elastic modulus, and the second is the drop in stress due to growth of damage. The damaged rate of stress is dependent on the stress, the rate of damage, as well as the damage itself. A comprehensive methodology is presented in this thesis for obtaining all the damage parameters and material constants present in the developed CDM model. The developed Foroutan-Names damage model is implemented into a VUMAT Material User Subroutine of the finite element program ABAQUS/Explicit. Uniaxial tension and shear simulations are performed on the three materials at different loading rates. It is shown that the model simulates well the dynamic and the static response. Simulation results for higher than tested strain rates are also presented to illustrate the response at these high rates. It is shown that the tensile strength increases with the increase in the loading rate, and also a strain softening phenomenon is observed after the maximum tensile strength. The loading-unloading, as well as compression response of the materials are also simulated to ensure the capability of the model to capture these responses. This is 165 CONCLUSIONS essential as during a typical dynamic impact, a part of the impacted material can go under compressive loading after initially experiencing a tensile loading. 7.2 Originality and contribution to knowledge In this work the following can be considered as a contribution to the literature: 1. A modified continuum damage mechanics model that includes shear damage is developed for woven composite materials. The model simulates the non-linear, rate dependant behaviour due to damage evolution. The model is implemented into a material user subroutine of a finite element program, and the simulation results are presented in this work. 2. A comprehensive simple methodology is established for determination of damage parameters and the material constants present in the developed model. This methodology simplifies the procedure for obtaining all constants present in the model using tensile and bias extension shear experiments. 3. Characterization of the response of three differently woven composite materials, all of which use carbon fabric as the constituent. The comparison between the static and dynamic results obtained from both uniaxial tension and bias extension shear experiments are also presented in this work. The stress-strain results clearly show strain rate sensitivity in these materials. 4. The tensile version of Hopkinson bar setup, in conjunction with the designed specimen and specimen fixture, is shown to be suitable for testing laminated woven composite materials in uniaxial tension and bias extension shear at high rates of strain. 5. With the increase in the use of the woven composite materials in the aerospace industry, the developed model can be used with the ones dealing with the interlaminar failure known as delamination to simulate the response of the material under impact. 6. The following publications resulted from the current work: a- Foroutan, R., and Nemes, J.A., (2009) High Strain Rate Characterization of Woven Carbon/Epoxy Composites. In: American Society for Composites and 166 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS The Canadian Association for Composite Structures and Materials, 24th ASC Conference 2009, Newark, Delaware, United states. b- Foroutan, R., and Nemes, J.A., (2008) Damage Modeling and Characterization of Woven Composite Materials, May 2008, (Technical Report-8) c- Foroutan, R., and Nemes, J.A., (2007b) Woven Composite Damage Model, November 2007, (Technical Report-7) d- Foroutan, R., and Nemes, J.A., (2007a) Instrumented Tensile Tests, May, 2007, (Technical Report-6) e- Foroutan, R., and Nemes, J.A.,(2006a) Material Modeling and Instrumented Tensile Tests, November 2006, (Technical Report-5) f- Foroutan, R., and Nemes, J.A., (2006b) Tensile Tests II, April 2006, (Technical Report-4) g- Foroutan, R., and Nemes, J.A., (2005b) Tensile Tests I, November 2005, (Technical Report-3) h- Foroutan, R., and Nemes, J.A., (2005a) Development of Tensile Test Procedures, April 2005 (Technical Report-2) i- Foroutan, R., and Nemes, J.A., (2004) A Review on Impact Modeling of Composite Structures, October 2004, (Technical Report-1) 7.3 Recommended future work The following can be considered as potential future tasks that can be done to enhance the prediction of the response of high strain rate materials subjected to dynamic loading: 1. The experiments in this study were mainly conducted to determine the in-plane behaviour of the woven composite material. In order to explore the concept of delamination, it is suggested to perform Mode I and Mode II fracture tests, to determine the inter-laminar behaviour of these materials. 2. The developed model can be combined with theory behind cohesive elements available in finite element codes to simulate the three-dimensional response of the 167 CONCLUSIONS material under impact. With the help of cohesive elements the inter-laminar behaviour of the material can be included in the simulations. 3. In this study, the focus of the thesis is on the high strain rate tensile behaviour of woven composite materials and it is assumed that the behaviour is similar in compressive loading. It would be interesting to perform compressive high strain rate experiments in order to compare the two responses. 4. At the moment, the Foroutan-Nemes model does not include permanent deformation, so in case of load reversal, the damage stays constant, however the material returns to zero stress at zero applied strain. It would be interesting to investigate the presence of permanent strain in the response of these materials in case of load reversal. 5. The developed model is shown to be very effective for predicting rate-dependant response of the woven composite materials subjected to tension and bias extension shear. The next interesting step is to validate the developed model which includes the following two steps. First a high strain rate loading experiment like drop tower experiment, punch experiment using Hopkinson bar theory, or ballistic impact experiment must be performed on woven composite panels. Then, the performed experiment must be simulated using the developed Foroutan-Nemes damage model combined with the cohesive elements theory. 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Journal of the Mechanics and Physics of Solids, 51(2), 333-356. 182 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS APPENDIX A: SUBROUTINE FOR ABAQUS/EXPLICIT C USER SUBROUTINE VUMAT subroutine vumat( C Read only 1 nblock, ndir, nshr, nstatev, nfieldv, nprops, lanneal, 2 stepTime, totalTime, dt, cmname, coordMp, charLength, 3 props, density, strainInc, relSpinInc, 4 tempOld, stretchOld, defgradOld, fieldOld, 3 stressOld, stateOld, enerInternOld, enerInelasOld, 6 tempNew, stretchNew, defgradNew, fieldNew, C Write only 5 stressNew, stateNew, enerInternNew, enerInelasNew ) C include 'vaba_param.inc' C C All arrays dimensioned by (*) are not used in this algorithm dimension props(nprops), density(nblock), 1 coordMp(nblock,*), 2 charLength(*), strainInc(nblock,ndir+nshr), 3 relSpinInc(*), tempOld(*), 4 stretchOld(*), defgradOld(*), 5 fieldOld(*), stressOld(nblock,ndir+nshr), 6 stateOld(nblock,nstatev), enerInternOld(nblock), 7 enerInelasOld(nblock), tempNew(*), 8 stretchNew(*), defgradNew(*), fieldNew(*), 183 ABAQUS VUMAT SUBROUTINE 9 stressNew(nblock,ndir+nshr), stateNew(nblock,nstatev), 1 enerInternNew(nblock), enerInelasNew(nblock) C character*80 cmname CHARACTER*100 WRKDIR, FILE5 C Sig0H1 = props(1) Sig0H4 = props(2) E0H11 = props(3) E0H44 = props(4) xnu = props(5) C1 = props(6) C2 = props(7) C3 = props(8) C4 = props(9) C5 = Props(10) C6 = Props(11) alpha = props(12) alpha11= props(13) alpha41= props(14) W11 = props(15) W12 = props(16) W13 = props(17) W41 = props(18) W42 = props(19) A11 = props(20) A12 = props(21) A13 = props(22) A14 = props(23) A41 = props(24) A42 = props(25) 184 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS A43 = props(26) A44 = props(27) B11 = props(28) B12 = props(29) B13 = props(30) B14 = props(31) B41 = props(32) B42 = props(33) B43 = props(34) B44 = props(35) C WRKDIR='C:\abq test' FILE5='R1.TXT' OPEN(91,DEFAULTFILE=WRKDIR,FILE=FILE5,STATUS='UNKNOWN') C C Parameters C txnu = 1 C IF (StepTime.EQ.0) Then C do I = 1,nblock C E11 = E0H11 *(C2* (1 + C1 * abs(strainInc(i,1)/dt) )**C3) E22 = E0H11 *(C2* (1 + C1 * abs(strainInc(i,2)/dt) )**C3) E44 = E0H44 *(C4* (1 + C1 * abs(strainInc(i,4)/dt) )**C5) stressNew(i,1) = sig1 + E11*(strainInc(i,1)+ xnu*strainInc(i,2)) stressNew(i,2) = sig2 + E22*(strainInc(i,2)+ xnu*strainInc(i,1)) stressNew(i,4) = sig4 + E44* strainInc(i,4) C C 185 ABAQUS VUMAT SUBROUTINE End Do C Else C do 100 i = 1,nblock C C State variables C d11 = StateOld(i,1) d12 = StateOld(i,2) d21 = StateOld(i,3) d22 = StateOld(i,4) d41 = StateOld(i,5) d42 = StateOld(i,6) E11 = StateOld(i,7) E22 = StateOld(i,8) E44 = StateOld(i,9) C C Strains C strain11 = StateOld(i,10) + strainInc(i,1) strain22 = StateOld(i,11) + strainInc(i,2) strain33 = - xnu * (strain11 + strain22 ) strain44 = StateOld(i,12) + strainInc(i,4) C F1 = StateOld(i,13) F2 = StateOld(i,14) F4 = StateOld(i,15) C E011 = StateOld(i,16) E022 = StateOld(i,17) 186 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS E044 = StateOld(i,18) C dout = 1 C C Trial stress sig1 = stressOld(i,1) sig2 = stressOld(i,2) sig4 = stressOld(i,4) C C Delta strain Rate (DSR) C DSR1 = strainInc(i,1)/dt DSR2 = strainInc(i,2)/dt DSR4 = strainInc(i,4)/dt DSR = max(abs(DSR1),abs(DSR2), abs(DSR4)) C C Damage rate constant (DRC) is for damage non-heeling property C If (strain11.GE.0) then If (DSR1.GE.0) Then DRC1 = 1 Else DRC1 = 0 End If Else If (DSR1.LE.0) Then DRC1 = 1 Else DRC1 = 0 End If End If 187 ABAQUS VUMAT SUBROUTINE C If (strain22.GE.0) then If (DSR2.GE.0) Then DRC2 = 1 Else DRC2 = 0 End If Else If (DSR2.LE.0) Then DRC2 = 1 Else DRC2 = 0 End If End If C If (strain44.GE.0) then If (DSR4.GE.0) Then DRC4 = 1 Else DRC4 = 0 End If Else If (DSR4.LE.0) Then DRC4 = 1 Else DRC4 = 0 End If End If C C2 E011 = E0H11 *(C2* (1 + C1 * DSR )**C3) C2 E022 = E0H11 *(C2* (1 + C1 * DSR )**C3) 188 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS C2 E044 = E0H44 *(C4* (1 + C1 * DSR )**C5) C Sig01 = Sig0H1 Sig02 = Sig0H1 Sig04 = Sig0H4 C Z11 = ( (W11 * abs(DSR1))**W12 ) Z12 = ( abs(Z11)**W13 ) Z4 = ( (W41 * abs(DSR4))**W42 ) C If (abs(sig1).LT.sig01.and.F1.EQ.0) Then C E011 = E0H11 *(C2* (1 + C1 * abs(DSR1) )**C3) E11 = E011 sig1dE = E11/txnu *(strainInc(i,1)+ xnu*strainInc(i,2)) sig1d = sig1dE go to 20 End If C If (abs(sig1).GE.sig01.OR.F1.NE.0) Then C F1 = 1 C If (d11.LT.0.9999) Then C d11d = max ( 0.000 , dt * Z11 * (A11 + A12 * d11 )* 1 sqrt(abs(sig1) / sig01 / (1 - alpha11 * d11)) * DRC1) d11 = min(d11 + d11d, 0.9999) d12d = 0 d12 = d12 + d12d C 189 ABAQUS VUMAT SUBROUTINE E11d = - alpha11 * E011 / 2 * d11d * 1 sqrt (1/(1- alpha11 * d11 )) C sig1dE = E11/txnu*(strainInc(i,1)+ xnu* strainInc(i,2)) sig1dD = - d11d* (B11 + B12 * sig1)/(1 - alpha11 * d11) sig1d = sig1dE + sig1dD C If (strain11.GT.0) Then If (DSR1.GE.0) Then E11 = E11 + E11d Else E11 = (sig1 + sig1d)/strain11 End If Else If (DSR1.LE.0) Then E11 = E11 + E11d Else E11 = (sig1 + sig1d)/strain11 End If EndIf C go to 20 End If C If (d11.EQ.0.9999) Then C d11d = 0 d11 = 0.9999 d12d = max ( 0.0000, dt * Z12 * (A13 * d11 + A14 * d12)* 1 sqrt( abs(sig1) / sig01 / (1 - alpha11 * d11) / 1 (1-alpha*d12)) * DRC1) 190 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS d12 = min (d12 + d12d , 0.9999) C E11d = - alpha * E011 / 2 * d12d * 1 sqrt ((1-alpha11 * d11)/(1-alpha*d12)) C sig1dE = E11/txnu *(strainInc(i,1)+ xnu* strainInc(i,2)) sig1dD = - d12d *( B13 + B14 * sig1)/(1-alpha*d12) sig1d = sig1dE + sig1dD C If (strain11.GT.0) Then If (DSR1.GE.0) Then E11 = E11 + E11d Else E11 = (sig1 + sig1d)/strain11 End If Else If (DSR1.LE.0) Then E11 = E11 + E11d Else E11 = (sig1 + sig1d)/strain11 End If EndIf C go to 20 C End If C End If C 20 Continue C 191 ABAQUS VUMAT SUBROUTINE If (abs(sig2).LT.sig02.and.F2.EQ.0) Then C E022 = E0H11 *(C2* (1 + C1 * abs(DSR2) )**C3) E22 = E022 sig2dE = E22/txnu *(strainInc(i,2)+ xnu*strainInc(i,1)) sig2d = sig2dE go to 30 End If C If (abs(sig2).GE.sig02.OR.F2.NE.0) Then C F2 = 1 C If (d21.LT.0.9999) Then C d21d = max ( 0.000 , dt * Z11 * (A11 + A12 * d21 )* 1 sqrt(abs(sig2) / sig02 / (1 - alpha11 * d21)) * DRC2) d21 = min(d21 + d21d, 0.9999) d22d = 0 d22 = d22 + d22d C E22d = - alpha11 * E022 / 2 * d21d * 1 sqrt (1/(1- alpha11 * d21 )) C sig2dE = E22/txnu*(strainInc(i,2)+ xnu* strainInc(i,1)) sig2dD = - d21d* (B11 + B12 * sig2)/(1 - alpha11 * d21) sig2d = sig2dE + sig2dD C If (strain22.GT.0) Then If (DSR2.GE.0) Then E22 = E22 + E22d 192 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS Else E22 = (sig2 + sig2d)/strain22 End If Else If (DSR2.LE.0) Then E22 = E22 + E22d Else E22 = (sig2 + sig2d)/strain22 End If EndIf C go to 30 End If C If (d21.EQ.0.9999) Then C d21d = 0 d21 = 0.9999 d22d = max ( 0.0000, dt * Z12 * (A13 * d21 + A14 * d22)* 1 sqrt( abs(sig2) / sig02 / (1 - alpha11 * d21) / 1 (1-alpha*d22)) * DRC2) d22 = min (d22 + d22d , 0.9999) C E22d = - alpha * E022 / 2 * d22d * 1 sqrt ((1-alpha11 * d21)/(1-alpha*d22)) C sig2dE = E22/txnu *(strainInc(i,2)+ xnu* strainInc(i,1)) sig2dD = - d22d *( B13 + B14 * sig2)/(1-alpha*d22) sig2d = sig2dE + sig2dD C If (strain22.GT.0) Then 193 ABAQUS VUMAT SUBROUTINE If (DSR2.GE.0) Then E22 = E22 + E22d Else E22 = (sig2 + sig2d)/strain22 End If Else If (DSR2.LE.0) Then E22 = E22 + E22d Else E22 = (sig2 + sig2d)/strain22 End If EndIf C go to 30 C End If C End If C 30 Continue C If (abs(sig4).LT.sig04.and.F4.EQ.0) Then C E044 = E0H44 *(C4* (1 + C1 * abs(DSR4) )**C5) E44 = E044 sig4dE = E44 * strainInc(i,4) sig4d = sig4dE go to 40 End If C If (abs(sig4).GE.sig04.OR.F4.NE.0) Then 194 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS C F4 = 1 C If (d41.LT.0.9999) Then C d41d = max ( 0.000 , dt * Z4 * (A41 + A42 * d41 )* 1 sqrt(abs(sig4) / sig04 / (1 - alpha41 * d41)) * DRC4) d41 = min(d41 + d41d, 0.9999) d42d = 0 d42 = d42 + d42d C E44d = - alpha41 * E044 / 2 * d41d * 1 sqrt (1/(1- alpha41 * d41 )) C sig4dE = E44 * strainInc(i,4) sig4dD = - d41d* (B41 + B42 * sig4)/(1 - alpha41 * d41) sig4d = sig4dE + sig4dD C If (strain44.GT.0) Then If (DSR4.GE.0) Then E44 = E44 + E44d Else E44 = (sig4 + sig4d)/strain44 End If Else If (DSR4.LE.0) Then E44 = E44 + E44d Else E44 = (sig4 + sig4d)/strain44 End If EndIf 195 ABAQUS VUMAT SUBROUTINE C go to 40 End If C If (d41.EQ.0.9999) Then C d41d = 0 d41 = 0.9999 d42d = max ( 0.0000, dt * Z4 * (A43 * d41 + A44 * d42)* 1 sqrt( abs(sig4) / sig04 / (1 - alpha41 * d41) / 1 (1-alpha*d42)) * DRC4) d42 = min(d42 + d42d, 0.9999) C E44d = - alpha * E044 / 2 * d42d * 1 sqrt ((1-alpha41 * d41)/(1-alpha*d42)) C sig4dE = E44 * strainInc(i,4) sig4dD = - d42d *( B43 + B44 * sig4)/(1-alpha*d42) sig4d = sig4dE + sig4dD C If (strain44.GT.0) Then If (DSR4.GE.0) Then E44 = E44 + E44d Else E44 = (sig4 + sig4d)/strain44 End If Else If (DSR4.LE.0) Then E44 = E44 + E44d Else E44 = (sig4 + sig4d)/strain44 196 HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS End If EndIf C go to 40 C End If C End If C 40 Continue C If(d12.GT.0.999) go to 60 If(d22.GT.0.999) go to 60 If(d42.GT.0.999) go to 60 C StateNew(i,19) = StateOld(i,19) C Go to 70 C 60 dout = 0 Strain11 = strain11 - StrainInc(i,1) Strain22 = strain22 - StrainInc(i,2) Strain33 = - xnu * (strain11 + strain22 ) Strain44 = strain44 - StrainInc(i,4) C C Update the stress C 70 Continue C stressNew(i,1) = (sig1 + sig1d) * dout stressNew(i,2) = (sig2 + sig2d) * dout 197 ABAQUS VUMAT SUBROUTINE stressNew(i,4) = (sig4 + sig4d) * dout C C Update the state variables StateNew(i,1) = d11 StateNew(i,2) = d12 StateNew(i,3) = d21 StateNew(i,4) = d22 StateNew(i,5) = d41 StateNew(i,6) = d42 StateNew(i,7) = E11 StateNew(i,8) = E22 StateNew(i,9) = E44 StateNew(i,10) = strain11 StateNew(i,11) = strain22 StateNew(i,12) = strain44 StateNew(i,13) = F1 StateNew(i,14) = F2 StateNew(i,15) = F4 StateNew(i,16) = E011 StateNew(i,17) = E022 StateNew(i,18) = E044 StateOld(i,19) = dout C WRITE(91,*) nblock C 100 continue C End If C return end 198