PREDICTING THE FATIGUE BEHAVIOUR OF MATRICES AND FIBRE-COMPOSITES BASED UPON MODIFIED EPOXY POLYMERS

Transcription

1 PREDICTING THE FATIGUE BEHAVIOUR OF MATRICES AND FIBRE-COMPOSITES BASED UPON MODIFIED EPOXY POLYMERS A thesis submitted for the degree of Doctor of Philosophy of Imperial College London and the Diploma of Imperial College February 2012 By Jibumon B Babu Department of Mechanical Engineering Imperial College London

2 Abstract The present research work has studied the fatigue behaviour of matrices and fibre-composites based upon modified epoxy polymers. The basic epoxy polymer has been modified with (a) nano-silica particles, (b) micrometre-rubber particles, and (c) both of these additives, to give a hybrid modified epoxy. These modifications have been undertaken in order to try to increase the cyclic fatigue resistance of the fibre-composite material. The experimental work has used a linear elastic fracture mechanics (LEFM) approach to firstly ascertain the fatigue properties of the epoxy polymer matrices. Secondly, the unmodified (i.e. control) and the modified epoxy resins were used to fabricate glass fibre reinforced plastic (GFRP) composite laminates by a resin infusion under flexible tooling (RIFT) manufacturing method. Tensile cyclic fatigue tests were performed on these composites, during which the degree of matrix cracking and stiffness degradation were also monitored. The fatigue life of the GFRP composite was significantly increased due to the presence of the nano-silica particles and/or micro-rubber particles. Suppressed matrix cracking and a reduced crack propagation rate in the modified matrix of the fibre-composite were observed to contribute towards the enhanced fatigue life of the composites containing the nano-silica particles and/or micro-rubber particles. The theoretical studies employed an extended finite element method, coupled with a cohesive zone model, to predict the fatigue behaviour of the fibre composites based upon the unmodified (i.e. control) and modified epoxy polymer matrices. A user element subroutine has been developed in Abaqus to incorporate the extended finite element method and a mathematical model has been proposed to evaluate the constitutive laws for the cohesive zone model to simulate the growth of fatigue cracks. A fatigue degradation strategy based on the Paris law (determined from the fatigue tests on the matrix materials) has been adopted to change the constitutive law for the cohesive zone model as a function of the number of fatigue cycles that have been accumulated. The theoretical predictions for the fatigue behaviour have been compared to the experimental results, and very good agreement between the theoretical and experimental results was found to exist. i

3 Acknowledgment I am deeply indebted to Prof A.J. Kinloch, Prof Felicity Guild and Dr A. C. Taylor for their generous effort, useful advice and dedication with respect to the supervision of my Ph.D research. I would like to thank Prof A.J. Kinloch and Prof Felicity Guild for their relentless and continuous assistance regarding all aspects of the project. I would also like to thank Imperial College London for providing the opportunity that has enabled me to pursue my higher studies in one of the most prestigious institutions in the world. It is a pleasure to thank many people who have helped me during my Ph.D studies. I would like to give my special thanks to Dr. Manjunatha, Ms. Shamsiah, Dr. Hsieh and Dr. Masania for kindly sharing some of their data and for their guidance in the experimental aspects of my research. I am also grateful to Mr. Alvarez and Mr. Brett for many useful discussions. I am very grateful to the laboratory staff of the Mechanical Engineering and Aeronautics Departments for providing laboratory facilities. I am also grateful to my friends Giannis, Fendi, Hari, Sorates, Nanke, Idris, Catrin, Tim, Paul and Ruth in the research office for providing a good working environment and an amazing time. I also wish to thank Sandeep, Sahu, Linash, Davendu, Aditya, Anant and Ahmad, and many more friends whose names are not mentioned due to space constraints, for making my time at Imperial so enjoyable. Last, but not least, I would like to thank my family for their support during my studies. I would especially like to express my sincere gratitude to my parents and to my sister for their encouragement and support. Finally, I would like to thank my wife for her love and patience. ii

4 Contents Abstract... i Acknowledgment... ii Contents... iii List of Figures... ix List of Tables... xix Nomenclature... xxi Abbreviations... xxi English Alphabet...xxii Greek Alphabet... xxv 1. Introduction and Objectives 1.1 Introduction Damage in Composites Modelling Techniques Objectives of the Present Research Structure of the Thesis Literature Review 2.1 Introduction Fracture Mechanics Linear Elastic Fracture Mechanics (LEFM) Modes of fracture Griffith's criterion Stress intensity factor Damage Static damage Fatigue damage iii

5 2.4 Methods of Finite Element Analysis Introduction The Virtual Crack Closure Technique (VCCT) method The Cohesive Zone Model (CZM) method Introduction Pure mode loading Mixed-mode loading The Energy method Modelling Damage and the Life Time under Cyclic Fatigue Loading Quasi-static loading Fatigue loading Concluding Remarks Experimental Techniques 3.1 Introduction Materials Preparation of Epoxy Matrix Polymer Specimens Introduction Preparation of plates Single edge notched bending (SENB) specimens Compact tension (CT) specimens Preparation of GFRP Composite Specimens Introduction Resin infusion under flexible tooling (RIFT) Double cantilever beam (DCB) specimens Composite strip specimens Test Methods for the Epoxy Matrix Polymer Specimens iv

6 3.5.1 Introduction Single edge notched bending (SENB) tests Compact tension (CT) tests Test Methods for the GFRP Composite Specimens Introduction DCB tests: Quasi-static tests DCB tests: Cyclic-fatigue tests Composite strip laminate tests: Quasi-static and fatigue tests Concluding Remarks Theoretical Techniques 4.1 Introduction Quasi-Static Analysis The Virtual Crack Closure Technique (VCCT) The cohesive zone law Kinematics Constitutive laws Mathematical formulation A bi-linear cohesive zone law Norm of displacement jump tensor Damage Mixed-mode loading: onset of crack growth Mixed-mode loading: crack propagation Mode-mixity Fatigue Analysis Introduction Degradation strategies v

7 4.3.3 Static damage evolution under fatigue loading Fatigue damage evolution Damage Analysis The cycle jump strategy Displacement ratio and load ratio Concluding Remarks Experimental Results and Theoretical Modelling Studies 5.1 Introduction Elastic Properties of Materials Elastic properties of the lamina Elastic properties of the composite Elastic properties of DCB Elastic properties of composite strip Elastic properties of aluminium and steel Criteria for Cohesive Zone Modelling Mesh sensitivity analysis Initial value of the cohesive zone law parameters The cohesive zone length Quasi-Static Models The SENB test: Experimental and theoretical results SENB results: VCCT analysis SENB results: Cohesive contact analysis SENB results: Cohesive zone element analysis The DCB test: Experimental and theoretical analysis DCB results: VCCT analysis DCB results: Cohesive contact analysis vi

8 DCB results: Cohesive zone element analysis The composite material strip test: Experimental and theoretical results Normalised stiffness with crack density Normalised stiffness with number of fatigue cycles Quasi-static strength of the composite strip Toughening Mechanisms Fatigue Models User element subroutine Validation Fatigue analysis using the user element subroutine The CT test and user element analysis DCB test and user element analysis Strip test and user element analysis Concluding Remarks Conclusions & Recommendations for Future Work 6.1 Quasi-Static Fracture Properties Quasi-Static Modelling Fatigue Testing Fatigue Modelling The User Element Subroutine Analysis Fatigue Life Recommendations for Further Work Unidirectional composite analysis Delamination analysis Modelling process of failure The role of matrix properties vii

9 References Appendix.174 viii

10 List of Figures Figure 1.1 Stress-strain characteristics of unidirectional composites: (a) low stiffness fibres; (b) high stiffness fibres (Talreja [1])... 1 Figure 2.1 Different modes of failure in a material (mode I is the opening mode, mode II is the shear mode and mode III is the tearing mode) Figure 2.2 (a) Failed specimen (scaling factor, m s =1) and (b) detail of failed specimen edge (Hallett et al. [9]) Figure 2.3 Photograph and schematic of delaminations in ply level scaled specimen (Hallett et al. [9]) Figure 2.4 Photographs of the development of transverse cracking in (0/90/0) glass-fibre/epoxy specimens at different strain levels (Berthelot [10]) Figure 2.5 Fatigue load cycle parameters Figure 2.6 Typical growth rate curve Figure 2.7 Fatigue damaged edge of (0/90/45) carbon fibre epoxy laminates at: (a) 2% stiffness reduction (b) 4% reduction (c) 8% reduction (d) 12% reduction (e) 15% reduction (Reifsnider and Jamison [11]) Figure 2.8 Microscopic damage mechanisms resulting from a constant stress amplitude fatigue test observed via optical microscopy for (a) 10, (b) 100 and (c) 1500 cycles of loading (Hosoi et al. [12]) Figure 2.9 Different cohesive zone model laws. (a)triangular form (b) Constant stress form (c) Triangular form (bi-linear law) (d) Elastic constant linear damage form (e) Linear polynomial form and (f) Elastic constant form (Zou et al. [20]).. 21 Figure 2.10 The bi-linear cohesive zone law Figure 2.11 Fatigue degradation in a bi-linear cohesive zone law ix

11 Figure 2.12 The minimum load required to propagate a delamination of a certain size in its most detrimental location (Wimmer and Pettermann [28]) Figure 2.13 Comparison of theoretical equation for delamination growth rate with the test data (Shivakumar et al. [29]) Figure 2.14 Failure modes in laminate (Shivakumar et al. [29]) Figure 2.15 Transverse crack interference model (Boniface et al. [30]) Figure 2.16 Failure of a composite strip specimen (Huchette [32]) Figure 2.17 Stiffness reduction as a function of the average crack density in the case of two different laminates. E o is the initial elastic modulus of the laminate and E x is the elastic modulus with a given crack density (Leblond et al. [33]) Figure 2.18 Crack multiplication in transverse plies (Leblond et al. [33]) Figure 2.19 Fatigue life diagram for a (0/90 2 ) s carbon fibre/epoxy matrix laminate (Akshantala and Talreja [34]) Figure 2.20 Fatigue life diagram for a unidirectional composites under loading parallel to the fibres (Talreja [1]) Figure 2.21 The tapping mode atomic force microscopy (AFM) phase images of the hybrid-epoxy matrix polymer (Manjunatha et al. [39]) (CTBN: carboxytermianted butadiene acrylonitrile rubber) Figure 2.22 Transmitted light photographic images of matrix cracking in the GFRP composites after testing at stress of 150MPa. NR-Neat resin, NRR-Neat resin with rubber, NRS-Neat resin with silica and NRRS-Neat resin with rubber and silica (Manjunatha et al. [40]) Figure 2.23 Crack length versus cycles (Robinson et al. [41]) Figure 2.24 Experimental relation between the maximum SERR,, and the number of fatigue cycles,, for the onset of crack growth (Attia et al. [42]) x

12 Figure 2.25 Evolution of the interface/cohesive traction and the maximum interface/cohesive strength as a function of the number of cycles for a displacement jump controlled high-cycle fatigue test (Turon et al. [43]). (Here interfacial traction is the critical stress of the cohesive zone law at a given number of fatigue cycles and the interfacial traction is the traction at the cohesive zone at a given number of fatigue cycles.) Figure 3.1 Fabrication of a GFRP sheet using RIFT Figure 3.2 DCB specimen Figure 3.3 Experimental setup for a DCB test Figure 3.4 Fibre bridging in DCB test Figure 3.5 Experimental setup for a fatigue DCB test Figure 4.1 The VCCT model Figure 4.2 A four noded cohesive zone element in (a) undeformed state and (b) deformed state in a global coordinate system Figure 4.3 A four noded cohesive zone element Figure 4.4 The bi-linear cohesive zone law Figure 4.5 Cohesive law in mode II Figure 4.6 Variation of damage variable with displacement in a bi-linear cohesive zone law Figure 4.7 Flow chart of the fatigue analysis embedded in the user element subroutine Figure 4.8 Cohesive zone law and energy representation Figure 4.9 Representation of energy release under fatigue in a cohesive zone law xi

13 Figure 4.10 Fatigue degradation of cohesive zone law with time Figure 4.11 Resultant fatigue degradation of a cohesive zone law. Path 1-2 shows the fatigue damage evolution and path 1-3 shows the static and fatigue damage evolution (Robinson et al. [41]) Figure 4.12 The cycle jump strategy applied to a cohesive zone law approach to modelling fatigue (Van Paepegem and Degrieck [65]) Figure 4.13 Experimental and numerically applied displacement/stress in a displacement/stress controlled fatigue test. In the present study, displacement controlled fatigue tests were conducted on the CT bulk epoxy material and the DCB composite materials, whilst stress controlled fatigue tests were conducted on the composite material strip specimens Figure 5.1 Overview schematic of the work plan Figure 5.2 A section of a lamina with local coordinates Figure 5.3 Equivalent Abaqus and local coordinate system for a DCB composite specimen Figure 5.4 Equivalent Abaqus and local coordinate system for different lamina of the composite strip Figure 5.5 Cube with different faces (FF-Face front, FBk-face back, FR-face right, FL- face left, FT-face top, FB- face bottom). The control point (CP) boundary condition is at the origin of the coordinate system Figure 5.6 The modelled load versus displacement curves of the DCB composite specimen for different mesh sizes. The elastic properties of the unmodified (i.e. control) bulk epoxy matrix and the composite are used for the analysis (Table 5.13 and Table 5.4) Figure 5.7 Cohesive zone behaviour in (a) undeformed and (b) deformed state xii

14 Figure 5.8 Cohesive zone elements with cohesive zone length, Figure 5.9 SENB specimen Figure 5.10 Dimensions of the SENB bulk epoxy matrix specimen Figure 5.11 Loading and boundary condition applied on the SENB specimen the model Figure 5.12 The stress field around the crack tip in a SENB FEA model of the bulk epoxy matrix with cohesive contact in Abaqus Figure 5.13 The stress field around the crack tip in a SENB model with cohesive zone elements as modelled in FEA Abaqus Figure 5.14 Comparison of load-displacement curve for the SENB specimen based upon the unmodified (i.e. control) epoxy matrix Figure 5.15 Comparison of load-displacement curve for the SENB specimen based upon the micro-rubber modified epoxy matrix Figure 5.16 Comparison of load-displacement curve for the SENB specimen based upon the nano-silica modified epoxy matrix Figure 5.17 Comparison of load-displacement curve for the SENB epoxy specimen based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix Figure 5.18 Flow chart of quasi-static and fatigue analyses of the composite material DCB specimens Figure 5.19 DCB specimen Figure 5.20 Dimensions of the DCB composite material specimen Figure 5.21 Loading and boundary condition applied on the model Figure 5.22 The stress field around the crack tip in a DCB model with cohesive contact in Abaqus xiii

15 Figure 5.23 The stress field around the crack tip in a DCB model with cohesive zone elements in Abaqus Figure 5.24 Comparison of load-displacement curve for the DCB composite material based upon the unmodified (i.e. control) epoxy matrix Figure 5.25 Comparison of load-displacement curve for the DCB composite based upon the micro-rubber modified epoxy matrix Figure 5.26 Comparison of load-displacement curve for the DCB composite based upon the nano-silica modified epoxy matrix Figure 5.27 Comparison of load-displacement curve for the DCB composite based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix Figure 5.28 Flow chart of the life prediction modelling of the composite strip under cyclic fatigue loading Figure 5.29 Composite material strip with dimensions Figure 5.30 Section of the strip with transverse cracks (a) strip under loading (b) cross section of the strip with transverse cracks (c) symmetric cross-section of the strip with transverse cracks Figure 5.31 Dimension of a section of the modelled strip. The length, l, of the strip model depends on the crack density Figure 5.32 Boundary conditions applied on the symmetric cross-section of the composite material strip with transverse cracks Figure 5.33 Variation of ±45 o crack density with number of cycles in composite material strips based upon unmodified (i.e. control) and modified epoxy matrices Figure 5.34 Composite material strip with transverse cracks in the Abaqus FEA method xiv

16 Figure 5.35 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the unmodified (i.e. control) epoxy matrix Figure 5.36 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the micro-rubber modified epoxy matrix Figure 5.37 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the nano-silica modified epoxy matrix Figure 5.38 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix Figure 5.39 Comparison of the normalised stiffness versus the crack density for the composite strip, based upon the unmodified (i.e. control) and nano-silica modified epoxy matrices Figure 5.40 Comparison of the normalised stiffness versus the crack density for the composite strip, based upon the micro-rubber and with both nano-silica and micro-rubber modified epoxy matrices Figure 5.41 Normalised stiffness versus the number of cycles for the composite strip based upon the unmodified (i.e. control) epoxy matrix Figure 5.42 Normalised stiffness versus the number of cycles for the composite strip based upon the micro-rubber modified epoxy matrix Figure 5.43 Normalised stiffness versus the number of cycles for the composite strip based upon the nano-silica modified epoxy matrix Figure 5.44 Normalised stiffness versus the number of cycles for the composite strip based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix xv

17 Figure 5.45 Comparison of the global stress with percentage strain in the composite strip Figure 5.46 Comparison of global stiffness reduction with the percentage strain of composite strip Figure 5.47 A single cohesive zone element for testing Figure 5.48 Cohesive zone element testing in (a) mode I, (b) mode II and (c) mixed-mode Figure 5.49 Cohesive zone element in mode I. The cohesive zone law parameters used for the element is for the unmodified (i.e. control) bulk epoxy matrix ( =3900N/mm 2, =10.9 N/mm 2 and =75.8J/m 2 ) Figure 5.50 Cohesive zone element in mode II. The cohesive zone law parameters used for the element is for the unmodified (i.e. control) bulk epoxy matrix ( =3900N/mm 2, =10.9 N/mm 2 and =75.8J/m 2 ) Figure 5.51 Cohesive zone element in mixed-mode ( =0.5). The cohesive zone law parameters used for the element is for the unmodified (i.e. control) bulk epoxy matrix ( =3900N/mm 2, =10.9 N/mm 2 and =75.8J/m 2 for both modes) Figure 5.52 Compact tension specimen Figure 5.53 Dimension of the CT specimen Figure 5.54 Boundary condition applied on the CT specimen Figure 5.55 Stress field around the crack tip in a CT model with cohesive zone elements in Abaqus Figure 5.56 Growth rate curve for the CT specimen for the bulk unmodified (i.e. control) epoxy matrix Figure 5.57 Growth rate curve for the CT specimen for the bulk micro-rubber modified epoxy matrix xvi

18 Figure 5.58 Growth rate curve for the CT specimen for the bulk nano-silica modified epoxy matrix Figure 5.59 Growth rate curve for the CT specimen for the bulk nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix (experimental data from Lee [49]) Figure 5.60 Growth rate curve for the composite DCB specimen based upon the unmodified (i.e. control) epoxy matrix Figure 5.61 Growth rate curve for the composite DCB specimen based upon the micro-rubber modified epoxy matrix Figure 5.62 Growth rate curve for the composite DCB specimen based upon the nano-silica modified epoxy matrix Figure 5.63 Growth rate curve for the composite DCB specimen based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix Figure 5.64 Transmitted light photographs of GFRP composite with unmodified (i.e. control) epoxy matrix showing the sequence of matrix crack development with the number of cycles, N, under fatigue loading Figure 5.65 Applied maximum fatigue stress versus the number of cycles upon fatigue loading for the composite strip based upon the unmodified (i.e. control) epoxy matrix Figure 5.66 Applied maximum fatigue stress versus the number of cycles upon fatigue loading for a composite strip based upon the micro-rubber modified epoxy matrix Figure 5.67 Applied maximum fatigue stress versus the number of cycles upon fatigue loading for a composite strip based upon the nano-silica modified epoxy matrix xvii

19 Figure 5.68 Applied maximum fatigue stress versus the number of cycles upon fatigue loading for a composite strip based upon the nano-silica and microrubber (i.e. hybrid) modified epoxy matrix Figure 6.1 Fatigue crack growth rate curve for the CT epoxy specimen based upon the unmodified (i.e. control) epoxy matrix Figure 6.2 Stress versus number of cycles from the fatigue loading for a composite material strip based upon the micro-rubber modified epoxy matrix 165 xviii

20 List of Tables Table 4.1 Description of the material properties of the bulk epoxy matrix relevant to Figure Table 5.1 Elastic properties of bulk epoxies and glass fibre Table 5.2 Unidirectional elastic properties of the different composite lamina based on the various epoxy matrices Table 5.3 Derivation of homogenised elastic property of DCB from cube analysis Table 5.4 Elastic properties of the arms the DCB composite specimens for the various epoxy matrices Table 5.5 Equivalent local elastic properties of the lamina of the strip in the Abaqus coordinate system Table 5.6 Derivation of the elastic properties of 0 o lamina Table 5.7 Derivation of the elastic properties of 90 o lamina Table 5.8 Derivation of the elastic properties of ±45 o lamina Table 5.9 Elastic properties of 0 o lamina Table 5.10 Elastic properties of ±45 o lamina Table 5.11 Elastic properties of 90 o lamina Table 5.12 Elastic properties of aluminium-alloy and steel Table 5.14 Quasi-static cohesive contact/element parameters for the bulk epoxy matrices Table 5.15 Quasi-static cohesive contact/element parameters of the DCB lamina interface Table 5.16 Elastic properties of the composite material strip xix

21 Table 5.17 Fatigue parameters of the bulk epoxy matrix CT specimens from Paris law fit Table 5.18 Fatigue parameters obtained from the DCB composite material specimen Table 5.19 Threshold stress for the composite strip Table 6.1 Fracture energies of the different bulk epoxy matrices Table 6.2 Fracture energies of the composite materials based upon the different epoxy matrices xx

22 Nomenclature Abbreviations AEW Amine equivalent weight MMB Mixed-mode bending AFM Atomic force microscopy NCF Non-crimp fibre ASTM American Society for Testing NL Non-linear and Materials B-K Benzeggagh-Kenane NR Neat resin CBT Corrected beam theory NRR Neat resin with rubber CCT Crack closure technique NRRS Neat resin with silica CFRP Carbon fibre reinforced plastic NRS Neat resin with rubber and silica CLT Classical laminate theory PTFE Polytetrafluoroethylene CP Control point QI Quasi-isotropic CT Compact tension RIFT Resin infusion under flexible tooling CTBN Carboxyl-terminated SENB Single edge notched bending butadiene-acrylonitrile CZM Cohesive zone model SERR Strain energy release rate DCB Double cantilever beam UD Unidirectional DGEBA Diglycidyl ether of bis-phenol A VCCT Virtual crack closure technique DOF Degree of freedom EEW Epoxide equivalent weight ENF End notch flexure EPFM Elastic plastic fracture mechanics FB Face bottom FBk Face back FEA Finite element analysis FF Face front FL Face left FR Face right FRP Fibre reinforced plastic FT Face top GFRP Glass fibre reinforced plastic ISO International Organization for Standardization LEFM Linear elastic fracture mechanics LVDT Linear variable displacement transducer xxi

23 English Alphabet Crack length Area A cz Area of cohesive zone A d Damaged cohesive zone area Area of the element Damaged area of the element A p Amplitude of sine wave Breadth B matrix Paris law exponent Damage variable Global displacement Components of stiffness tensor Linear and non-linear element displacement in global coordinate system Components of undamaged stiffness tensor Components of undamaged stiffness tensor Local stiffness tensor Maximum damage Static and fatigue damage, Rate of change of static and fatigue damage Damage at t and t+1 time Tangent stiffness tensor Components of tangent stiffness tensor Elasticity modulus Initial modulus of laminate Modulus of elasticity in 1, 2 and 3 direction Elastic modulus of fibre and matrix Resultant elastic modulus Transverse elastic modulus (through thickness direction) Modulus of laminate with a given crack density Modulus of elasticity in x, y and z direction Differential of shape function matrix Scalar factor Correction factor Forces in the x and y direction Force vector Function of x variable Fracture energy xxii

24 Shear modulus in 1-2, 1-3 and 2-3 plane Critical fracture energy Shear modulus of fibre and matrix, Energy release rate in mode I and mode II Critical fracture energy in mode I and mode II Maximum energy release rate in mode I Energy release rate when growth is infinite/unstable Maximum energy release rate Total energy release rate Threshold fracture energy Shear modulus in P-T, P-P, T-P, T-T, T-L and L-T plane Shear modulus in x-y, x-z and y-z plane Shape/interpolation matrix Factor Variable Identity matrix of size nxn Variable Jacobian matrix Penalty stiffness of cohesive zone law Approximate penalty stiffness of cohesive zone law Compressibility modulus of fibre, matrix and composite Stiffness of the element Critical stress intensity factor in mode I Maximum stress intensity factor Length Distance from the centre of the loading block to the mid-plane of the specimen Cohesive zone length Length of the element Size of mesh Exponent of Paris law Parameter Scaling factor Number of nodes Number of cycles Number of elements Number of cycles of fatigue life Interpolation/shape function Load xxiii

25 Maximum load or 5% offset load Maximum load in a fatigue cycle Constant Stress ratio Displacement ratio Spacing Spacing between point A and B Crack spacing before and after loading t Time Thickness of 0 o and 90 o lamina Thickness of continuum element Unit local coordinate vector Traction vector Local traction vector Time period Displacement Energy stored Displacements at c and d points in the u direction Local displacements at the bottom and top of the crack/element Local displacement Minimum and maximum displacement in a fatigue cycle Global displacements in the u and v direction (linear) Global displacements in the u and v direction (non-linear) Displacement Displacements at c and d points in the v direction Volume fraction Width Energy required Variable Differential in and direction of mid-plane coordinate in global coordinate system Global coordinate in undeformed state, Global displacement at the bottom and top of the crack/element Mid-plane coordinate in global coordinate system Global coordinate xxiv

26 Greek Alphabet Power law factor Factor Power law factor Displacement Load-line displacement in DCB specimen Displacement in 1 and 2 direction, Kronecker delta Displacement in mode I and II Failure displacement in mode I and II Opening displacement in mode I and II Component of displacement jump, Opening and failure displacement in cohesive zone law Displacement at time t and t+1 Displacement jump threshold at time t Δ a Correction for crack tip rotation in DCB specimen Displacement in cohesive zone element Strain in composite Strain in matrix Maximum strain Resultant strain Shape factor Local coordinate in a cohesive zone Mode-mixity factor Factor Transformation matrix Poisson s ratio Poisson s ratio in 1-2, 2-1, 2-3 and 1-3 direction Poisson s ratio of fibre and matrix in 1-2 direction Poisson s ratio in L-T, T-L, T-P, T-T and P-P direction Poisson s ratio in x-y, x-z and y-z direction Difference in the energy Local coordinate in a cohesive zone Stress Stress in the 1 and 2 direction, Critical stress in 1 and 2 direction Critical applied stress Minimum, maximum and mean stress applied Cohesive zone traction/stress Δ coh xxv

27 Critical cohesive zone traction/stress Cohesive zone traction/stress in the 1 and 2 direction Component of cohesive zone traction/stress Unit vectors in and the normal direction Diameter Energy at a given time and the initial energy xxvi

28 CHAPTER 1 INTRODUCTION AND OBJECTIVES CHAPTER 1 1. INTRODUCTION AND OBJECTIVES 1.1 Introduction Composite materials are engineered or naturally occurring materials formed by mixing two or more constituent materials in different phases with different physical or chemical properties. Composites are found in nature in the form of bone, mollusc shell, wood etc. They have different constituents in different proportions to form a new material with properties different from the individual constituents. The main constituents of the composite are the matrix and reinforcement. The matrix of the composite helps to bind the reinforcement together in the composite. The reinforcement is embedded in the matrix and the reinforcement typically has a relatively high modulus and tensile strength, compared to the matrix. The matrix has good binding properties and is less stiff than the reinforcement. The proportion and the structure of the reinforcement greatly influence the properties of the composite (see Figure 1.1). Figure 1.1 Stress-strain characteristics of unidirectional composites: (a) low stiffness fibres; (b) high stiffness fibres (Talreja [1]). 1

29 CHAPTER 1 INTRODUCTION AND OBJECTIVES Typical reinforcements are fibrous in nature with high aspect ratios. Such reinforcements are classified as long fibres and short fibres, depending on the value of the aspect ratio. Short fibres are usually randomly orientated whereas long fibres are orientated in different directions in the composite to design a composite with the desired material properties. Typically, long, continuous fibres are laid in sheets in unidirectional or multidirectional orientation within the composite. Thus, the fibres are essentially oriented in different directions and bonded together using a polymeric matrix to form a laminate. The unidirectional fibres have all the fibres in the same direction in the laminate. Glass fibre reinforced plastic (GFRP) and carbon fibre reinforced plastic (CFRP) are commonly used as fibre reinforced laminates for the aerospace industry. The main advantages of composite materials are their relatively low density and high stiffness and strength compared to metallic materials. Composites also provide good design flexibility, as they can be moulded into complex shapes and geometries. The composite for the present work is a multilayer and multidirectional quasiisotropic laminate of a polymeric matrix with glass fibres and the laminate is known as GFRP. It is a quasi-isotropic laminate which has equivalent stiffness properties in all directions in a given plane. Epoxy polymers are widely used as the matrices for fibre reinforced composite materials. They have good engineering properties, such as high modulus, failure strength, low creep and good performance at elevated temperatures after curing. The properties of the epoxy polymers can be improved by the addition of a particulate phase which may increase the toughness of the epoxy polymer matrix. The effectiveness of the addition of the particulate phase depends on the dispersion of the particulate phase in the matrix as well as the adhesion of the particle to the matrix. The most commonly used such additives are the nano-silica and micrometre-rubber particles, which are used in the present research. 2

30 CHAPTER 1 INTRODUCTION AND OBJECTIVES 1.2 Damage in Composites Damage in composite materials occurs in different stages of their manufacture and service life (Matthews and Rawling [2]). The main sources of defects in the composite are 1. Defects in the fibres; 2. Defects introduced during manufacture; 3. Damage that develops during their service life. Defects in the materials used for the manufacture of the composite material causes the introduction of intrinsic damage in the composite. The defects in the manufacture occur due to, for example, misalignment of fibres, inclusion of impurities in the manufacture process, etc. The damage that develops during the service life of the composite arises due to different types of loading, impact, shocks etc. There are various types of damage that may be so introduced and these invariably lead to a reduction in the strength, stiffness and fatigue life of the composite material. The failure of composite materials during their service life occurs mainly due to delamination, debonding, transverse cracking and fibre failure (Sridharan [3]). Delamination occurs in the lamina of composites and is often associated with the prior formation of transverse cracks in the composite. Indeed transverse cracking is one of the main failure mechanisms in composite materials and essentially consists of the formation of matrix cracks in the lamina of composite due to quasi-static and cyclic fatigue loading. The matrix cracks lead to the stiffness of the laminate decreasing due to a lower degree of interaction between the fibres and the matrix. They therefore lead to a lower stiffness due to less transfer of the applied load by the matrix. Cycling loading causes the evolution of transverse cracks due to the reversal of the stress with time. Quasi-static damage is the damage in the composite material which arises due to the application of a steady, or steadily increasing load. On the other hand, cyclic fatigue damage is the progressive structural damage that occurs when a 3

31 CHAPTER 1 INTRODUCTION AND OBJECTIVES composite material is subjected to cyclic loading. In fatigue study of composites, the material is subjected to repeated loading and unloading cycles over a period of time. The fatigue loading cycle is broadly divided into low cycle fatigue and high cycle fatigue based on the number of cycles required for the failure of the material and the level of stress applied on it. The low cycles fatigue regime is characterised by a relatively high stress level, a very localised stress concentration and a low number of cycles to failure. The high cycle fatigue is characterised by a relatively low stress level, with damage developing at the micro-scale, and large number of cycles to failure. 1.3 Modelling Techniques Linear elastic fracture mechanics (LEFM) is a technique widely used to characterise composite materials (Hull and Clyne [4]). The approach most commonly adopted for the analysis of fracture is based on an energy balance, which assumes energy released during fracture is at least equal to that necessary to generate a new fracture surface when an existing crack propagates. An LEFM approach may be readily coupled with a finite element analysis (FEA) where the damage in the form of transverse cracks and delamination may be modelled (Cook et al. [5]). In a FEA approach, the growth of cracks and delamination may be analysed based on the energy released during the initiation and propagation of the crack. The energy release rate during fracture in may be modelled using a virtual crack closure technique (VCCT) (Krueger [6]) and this may be coupled with a cohesive zone model (CZM) (Camanho et al. [7]). The VCCT method calculates the strain energy release rate (SERR) based on the forces and displacements needed to advance the crack. A CZM approach models the progressive damage and failure in the composite material based a known cohesive zone law. The CZM approach also predicts the onset of propagation and subsequent growth of a crack under different types of loading, and the fatigue degradation of the composite material may be based on a degradation and evolution law. 4

32 CHAPTER 1 INTRODUCTION AND OBJECTIVES 1.4 Objectives of the Present Research The main objective of the present work is the analysis of transverse cracks in composites under quasi-static and fatigue loading by developing a novel FEA approach coupled with a CZM technique. The present research implements an extended FEA method via the Abaqus software. The work investigates epoxies and composites toughened with nano-silica and micrometre-rubber particles to improve the quasi-static and fatigue performance. Different specimens of bulk epoxy matrix and the composite are tested and modelled. The performance of hybrid (with both nano-silica and micro-rubber particles) epoxy matrix and composite is also studied and modelled. The fracture energy of the bulk epoxy and interlaminar fracture energies of composite were determined experimentally and modelled. The static behaviour of epoxy and the composite are modelled using the fracture energy to obtain the cohesive zone parameters. The cohesive properties of bulk epoxy and composite are modelled for fatigue condition and growth rate curve is obtained for the fracture mechanics specimens. The cohesive properties of the epoxy are used to obtain the S-N curve of the composite and predict the fatigue life. The fatigue parameters of different materials are compared to study the influence of addition of different additives on the performance of composite. In the analysis, a user element subroutine (Hibbitt [8]) is developed in Abaqus to incorporate the extended finite element capabilities and a mathematical model proposed in Chapter 4 to evaluate constitutive laws to simulate fatigue driven transverse cracking in composite. A fatigue degradation strategy based on Paris law is adopted for the analysis and is used to represent the behaviour of lamina interface and transverse cracks in composites under fatigue loading. 1.5 Structure of the Thesis The thesis is divided into six chapters with each Chapter describing different aspects of work. The next Chapter deals with the literature study and the past work reported on the quasi-static and fatigue studies on composites. The Chapter also deals with the conventional and computational methods for the modelling of composites. 5

33 CHAPTER 1 INTRODUCTION AND OBJECTIVES Chapter 3 describes about the experimental techniques adopted in the present work to obtain the experimental results. The Chapter explains about the materials, manufacture of specimens, testing of specimens and interpretation of the experimental data for the modelling studies. Chapter 4 deals with the theoretical techniques adopted in the present study. The Chapter describes about the theoretical formulation of the present problem and the analysis using FEA method. Chapter 5 presents the experimental and modelling results of the present work on bulk epoxy matrix and composite. The results of the modelling are compared with the experimental results to understand the quasi-static and fatigue behaviour of bulk epoxy matrix and the composite. Chapter 6 concludes the thesis with recommendations to study the behaviour of the composite materials. 6

34 CHAPTER 2 CHAPTER 2 LITERATURE REVIEW 2. LITERATURE REVIEW 2.1 Introduction Static damage in a composite material arises from a constant applied load or from a steadily increasing applied load and cyclic fatigue damage arises from an applied oscillatory load. Such damage causes a reduction in the strength of the composite material with increasing time. This fracture process can be studied experimentally and modelled by analytical or numerical methods. Fracture mechanics approaches have been widely used to study and model such damage and the basis for this approach is firstly reviewed. The damage that occurs in composites, and the methods of analysis used to study the fracture processes, are then reviewed. Finally, the life prediction analysis methods which have been proposed to describe the accumulation of such damage and the eventual fracture of the composite material are discussed. 2.2 Fracture Mechanics Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterise the material's resistance to fracture. It applies the physics of stress and strain using theories of elasticity and plasticity to describe the growth of cracks Linear Elastic Fracture Mechanics (LEFM) Linear elastic fracture mechanics (LEFM) is a theory of fracture mechanics which helps to study the fracture in a material or a structure. The LEFM technique assumes the material is linear elastic in the bulk of the material and that the plastic or damage zone near the crack tip is relatively small. In LEFM, most equations are derived for either plane stress or plane strain associated 7

35 CHAPTER 2 LITERATURE REVIEW with the three basic modes of loadings on a cracked body: opening, sliding and tearing (Figure 2.1). The LEFM method is valid only when the inelastic deformation zone is small compared to the size of the crack. In the case of large zones of plastic deformation in the material associated with crack propagation, then the elastic plastic fracture mechanics (EPFM) technique is used. However, the EPFM techniques are not relevant to the present research, and therefore will not be considered further. LEFM is a method widely used for predicting the static failure in composites. The LEFM technique is applied to materials or structures with an initial crack or damage, as this method predicts the initiation of crack growth and the subsequent propagation of a crack. It should be noted that the LEFM method fails to predict the first initiation of a void or defect or crack Modes of fracture Fracture in a composite laminate may occur due to the initiation of the growth of various types of cracks. The growth of the crack may be classified as occurring in the opening (mode I), shear (mode II) and tearing (mode III) modes. The different modes (Figure 2.1) have different values of critical strain energy release rate (SERR),, needed for crack propagation. Mode I failures are more predominant in composite materials due to less energy being needed to propagate a crack compared to the other modes. Transverse matrix crack growth occurs mainly in mode I and the delamination in the composite occurs in both in mode I and II. Transverse cracks are often precursors to the initiation of delamination, but delamination often suppresses more transverse cracking occurring due to more consumption of energy being required in mode II fracture. This situation typically arises once the maximum transverse crack density has been achieved. In composites, mode III failure is usually ignored due to negligible fracture occurring in this direction. Hence, in typical analyses only mode I and mode II failures are considered together with mixed-mode (I/II), and mode III fracture is neglected. 8

36 CHAPTER 2 LITERATURE REVIEW Figure 2.1 Different modes of failure in a material (mode I is the opening mode, mode II is the shear mode and mode III is the tearing mode) Griffith's criterion The Griffith criterion states that crack propagation will occur if the energy released upon crack growth is sufficient to provide all the energy that is required for crack growth. The condition for crack growth is (2.1) where is the energy required and is the energy stored which is available to form a crack. Griffith calculated the energy stored per unit plate thickness in an edge crack in an isotropic infinite plate as (2.2) where is the elasticity modulus, is the crack length and is the applied stress on the material. The term can be replaced to give (2.3) where is the critical fracture energy, also called the crack driving force and the critical strain-energy release rate, and is the critical applied stress. 9

37 CHAPTER 2 LITERATURE REVIEW The energy required to grow the crack is dependent on the state of stress at the crack tip. The energy release rate for crack growth can be calculated as the change in elastic strain energy per unit area of crack growth, i.e., (2.4) where is the elastic energy of the system and is the crack length. The value of is evaluated from by either keeping the load,, or the displacement,, constant Stress intensity factor The stress intensity factor is a different approach to measure the material s toughness. It describes the stress field around the crack tip. Crack extension occurs when the stresses at the crack tip are such that a critical value of the stress intensity factor is reached. The critical stress intensity factor in mode I,, is given by the expression (2.5) where is a constant, is the initial crack length and is the critical applied stress. For a mode I crack, the critical fracture energy and the critical stress intensity factor are related by the expression for plane strain as (2.6) where is the Young's modulus, is Poisson's ratio, and is the stress intensity factor in mode I. The strain energy release rate (SERR),, of a crack in a body can also be expressed in terms of the mode I, mode II and mode III stress intensity factors. In composites, there is a significant difficulty in defining the stress in the interlaminar layer and hence the fracture energy, i.e. the critical SERR,, of the material is easier to define and use. The present work is 10

38 CHAPTER 2 LITERATURE REVIEW therefore based on the energy method to model fracture in the bulk epoxy matrix polymers and the composite materials. 2.3 Damage Damage in the composite material occurs due to different types of loading. The two major types of damage occurring in the composites are static and cyclic fatigue damage. The static damage occurs with a steady increase in the load with time. The damage is characterised by the formation of transverse cracks leading to delamination at relatively high stresses. Fatigue damage occurs in composites due to cyclic loading over time. The damage is characterised by progressive mechanisms of failure in which transverse cracking is accompanied by delamination in the composites. The two types of damage are discussed and reviewed below to provide a basis for the present research Static damage Static damage occurs when a constant load is applied on the material or when a load is applied which increases steadily with time. Static damage in the composites occurs typically due to transverse cracking followed by interlaminar delamination. Figure 2.2 and Figure 2.3 shows the failure of a composite strip under a quasi-static load due to delamination and transverse cracking. Damage arising from transverse cracking is one of the important mechanisms of failure in composites. Figure 2.4 shows the development of transverse cracks in a strip under quasi-static loading at different values of the applied strain. Figure 2.2 (a) Failed specimen (scaling factor, m s =1) and (b) detail of failed specimen edge (Hallett et al. [9]). 11

39 CHAPTER 2 LITERATURE REVIEW Figure 2.3 Photograph and schematic of delaminations in ply level scaled specimen (Hallett et al. [9]). Figure 2.4 Photographs of the development of transverse cracking in (0/90/0) glass-fibre/epoxy specimens at different strain levels (Berthelot [10]) Fatigue damage Fatigue damage in composites occurs due to cyclic load reversals over time. The fatigue damage occurs as transverse cracks and delamination in the plies of the composite laminate. Fatigue driven cracking is governed by energy release and the accumulation of damage in the composite material over time. Usually, the fatigue load is applied as a sinusoidal stress wave-form of constant amplitude. The stress cycles in the fatigue loading are expressed as the -ratio, also called the stress ratio, which gives the ratio of the minimum to the 12

40 CHAPTER 2 LITERATURE REVIEW maximum stress in a fatigue cycle. A typical stress fatigue cycle can be represented using the amplitude, maximum stress,, minimum stress,, and the time period, as shown in Figure 2.5. Figure 2.5 Fatigue load cycle parameters. The Paris law is the most widely used law to describe crack growth curve under fatigue loading. The growth rate in a composite material is related to the energy release rate and has three regions. The crack growth in the first region (Region I) of the curve growth rate curve (Figure 2.6) is zero as the energy released is less than the threshold value,, required for crack growth. Region II of the crack growth curve has a linear part in which crack growth depends on the energy released. This linear part of the curve is described by the Paris law. The last part of the curve (Region III) has an asymptote of the crack growth as the energy release rate is now equal to the fracture energy,, of the material. The Paris law describes the linear region of the growth rate curve using a power law and can be expressed as (2.7) 13

41 log (da/dn) CHAPTER 2 LITERATURE REVIEW where the values of and are obtained experimentally. The value of depends on the material, loading conditions, temperature etc and is the slope of the growth rate curve when plotted logarithmically as shown in Figure 2.6. Region I Region II Region III m G th log (G max ) G c Figure 2.6 Typical growth rate curve. In fatigue loading, transverse cracking and delamination develop with an increase in the number of fatigue cycles. Fibre breakage may also occur, and this usually occurs during the final stages of fatigue cycles leading to further transverse cracking and delamination. Coupling of the cracks also leads to delamination, as well as fibre breakage in the laminate. Typical fatigue damage observed in composite materials is shown in Figure 2.7. Transverse cracks and delaminations may be seen at the edge of the strip and they give rise to different degrees of stiffness reduction. Thus to summarise, in the damage process, the initial mechanisms of damage lead to the formation of transverse cracks in the composite and this progressive damage leads to delamination occurring, and the damage that results is observed as a reduction in stiffness of the composite material. 14

42 CHAPTER 2 LITERATURE REVIEW Figure 2.7 Fatigue damaged edge of (0/90/45) carbon fibre epoxy laminates at: (a) 2% stiffness reduction (b) 4% reduction (c) 8% reduction (d) 12% reduction (e) 15% reduction (Reifsnider and Jamison [11]). The points are further illustrated in Figure 2.8 which shows the development of the damage under fatigue loading and the development of transverse cracks, matrix cracks and delaminations in a composite. In a multidirectional composite, the transverse cracks occur in a 90 o lamina and the matrix cracks occur in angled ply lamina due to the loading directions. 15

43 CHAPTER 2 LITERATURE REVIEW Figure 2.8 Microscopic damage mechanisms resulting from a constant stress amplitude fatigue test observed via optical microscopy for (a) 10, (b) 100 and (c) 1500 cycles of loading (Hosoi et al. [12]). 16

44 CHAPTER 2 LITERATURE REVIEW 2.4 Methods of Finite Element Analysis Introduction Different methods are used for the analysis of composites which use either an energy or a stress based criteria for the analysis of fracture. The most widely used techniques are based upon a finite element analysis approach. They are the virtual crack closure technique (VCCT) method and the cohesive zone model (CZM) methods. The conventional energy method and other methods of analysis are also used to study fatigue damage in composites. The different analysis methods which have been used are described below The Virtual Crack Closure Technique (VCCT) method Crack propagation studies using the VCCT method (Rybicki and Kanninen [13]) have been performed by many authors. In the VCCT method, the total energy release rate is computed locally at the crack front. It involves determining the energy release rate as a function of the direction in which the crack is extended. By Irwin s theory, the energy required for the crack propagation is directly proportional to the crack length and the energy released during crack propagation can be calculated from the nodal displacements. In the VCCT method, the stress field around the crack is calculated based on elasticity theory. In this analysis, the crack propagates when the strain energy released is equal to the fracture energy of the material. The VCCT method is computationally efficient due to a relatively low amount of computational time being needed and due to its simplicity. The main disadvantage of this method is its failure to be able to initiate crack propagation in an uncracked material, as it depends on the nodal displacement ahead of crack tip in order to calculate the strain energy release rate (SERR),. Zou et al. [14] developed a model for the evaluation of energy release rate using the VCCT method. Their study included the influence of the number of laminae in determining the total energy release rate in composites. They derived the energy and modes of failure in sub-laminates from nodal forces and 17

45 CHAPTER 2 LITERATURE REVIEW moments in the lamina and showed the oscillatory behaviour of the SERR at the crack tip due to edge delamination occurring. Krueger [6] has reviewed the application of the VCCT method in the analysis of crack propagation in two dimensional and three dimensional problems concerned with composite materials. He described the principles governing the technique and the calculation of the energy release rate in three dimensional space. In this study, equations for the energy release rate in the 2D shell and 3D solid elements were derived from the geometry of the structure. The model gave an accurate prediction of crack propagation. Krueger and O'Brien [15] used multi-point constraints in the VCCT method to compute the SERR across the width of the composite specimen. They computed the SERR for beams of different widths from classical laminate plate theory and derived the equations for plate/shell and solid elements for crack propagation based on the geometry of the composite specimens. Shen et al. [16] developed a computational model of circular delamination for the prediction of delamination growth in composites. They adopted the VCCT method for modelling a circular delamination in different layers of a composite laminate. Their study showed that the direction of delamination growth coincided with the direction of maximum SERR. They obtained the distribution of energy released in the delamination front to predict the resultant delamination shape and direction. Their model also gave information about the shape of the delamination under buckling loads. Qian and Xie [17] developed a cohesive zone model using the VCCT method to model the crack propagation under mixed-mode loading. They used a cohesive zone model based on SERR principles in the finite element analysis package to model mixed-mode crack propagation occurring at a constant crack velocity. The cohesive zone model was incorporated using subroutines to model different crack velocities. It should be noted that, by using the VCCT method together with a cohesive zone model, it is possible to predict the initiation and propagation of cracks. The approach worked well for quasi-static loading under 18

46 CHAPTER 2 LITERATURE REVIEW mixed-mode loading but failed to predict the fatigue behaviour due to the absence of a degradation law in the models that they developed. Rybicki and Kanninen [13] developed a technique to determine the stress intensity factors based on the VCCT. The method they adopted measured the stress intensity factors for a crack in a plate using a constant strain element in mode I and II and gave good results which were comparable to the J-integral method. The VCCT method, which uses calculations via FEA of the nodal force and displacement, was shown to be a relatively simple method for the analysis of stress intensity factors for different modes, mode I and II (see Section 2.2.2). Mandell et al. [18] used the VCCT method to predict the life of skin stiffeners bonded to a composite panel using the SERR method by predicting delamination in wind turbines under static and fatigue loading. The SERR for the different modes of loading was obtained from a FEA analysis. The tests were undertaken for various crack growth rates and the energy release rate determined. The method predicted delamination failure for different thickness of the matrix resin, and also predicted the static delamination in wind turbine blades for various mixed-mode loading conditions The Cohesive Zone Model (CZM) method Introduction Cohesive zone modelling is considered to be the most accurate method of modelling the damage in composites. In this method, a cohesive zone model is used to model the crack propagation. The cohesive zone model law relates the traction and displacement at the crack tip to calculate the SERR under different modes of loading. The cohesive zone model helps to overcome the complexity of the singularity located at the crack tip. In the cohesive zone modelling approach, the crack propagates in the material according to a defined cohesive zone model law and damage evolution principles. The cohesive zone model combines aspects of a strength based analysis and fracture mechanics to predict the onset of damage and the propagation of the crack. The main 19

47 CHAPTER 2 LITERATURE REVIEW advantage of cohesive zone models is the ability to predict crack initiation and propagation using the same cohesive zone model law. Different shapes of traction,, versus displacement,, laws are proposed for the cohesive zone law depending on the shape of the cohesive zone law: e.g. bi-linear, linear-parabolic, exponential and trapezoidal; see Figure 2.9 and note that the area under these shaped is equivalent to the fracture energy,. In a cohesive zone model law, the shape of the cohesive zone law determines the loading and the rate of degradation in the material. For example, Alfano [19] studied the influence of the shape of the cohesive zone model law when analysing delamination problems in composite materials. He studied bi-linear, linear-parabolic, exponential and trapezoidal shapes for the cohesive zone model law and compared them, assuming the same value of the initial penalty stiffness (the initial penalty stiffness being the value of the initial gradient in the cohesive zone law). He concluded that the trapezoidal law gave the poorest results both in terms of the numerical stability and convergence and that the exponential law gave the optimal results in terms of predicting accurately the rate of degradation. His study also showed that the bi-linear law represented the best compromise between computational cost and the prediction of the degradation rate. He concluded that the influence of the shape of the cohesive zone law curve depends on the penalty stiffness of the material. 20

48 CHAPTER 2 LITERATURE REVIEW τ τ τ 0 τ 0 (a) G c (b) G c δ δ τ τ τ 0 τ 0 (c) (d) G c G c δ δ τ τ τ 0 τ 0 (e) (f) G c G c δ δ Figure 2.9 Different cohesive zone model laws. (a)triangular form (b) Constant stress form (c) Triangular form (bi-linear law) (d) Elastic constant linear damage form (e) Linear polynomial form and (f) Elastic constant form (Zou et al. [20]). The bi-linear cohesive zone law (see Figure 2.9 (c)) is widely used for the analysis of fracture. It is the simplest of the cohesive zone laws as it represents the elastic and degradation part linearly. The bi-linear cohesive zone law has a discontinuity at the damage initiation point due to the sudden change in the slope of the curve and hence different laws have been proposed to smooth out 21

49 Traction, τ CHAPTER 2 LITERATURE REVIEW this discontinuity and hence possibly better predict the behaviour of the material. A typical bi-linear cohesive zone model law is defined by a penalty stiffness,, critical fracture energy,, and critical stress,, of the element, as shown in Figure τ 0 K G c δ o Displacement, δ δ f Figure 2.10 The bi-linear cohesive zone law. Cohesive zone modelling is considered to be the most accurate method for the analysis of cyclic-fatigue damage in composites. In this method, a cohesive zone model is used to model crack propagation under fatigue loading. The damage occurs in the cohesive zone by evolution of damage with time, and the degradation of the cohesive zone law can be achieved by degrading the penalty stiffness of the cohesive zone law with time. The degradation of the penalty stiffness is computed based on the energy released at nodes of elements under the fatigue loading. The relevant damage parameter,, in the cohesive zone law is calculated based on the Paris law, which is obtained from experimental tests on the composite material as described in Section and as shown in Figure 2.6. For example, the degradation of a bi-linear cohesive zone law due to fatigue load with time is shown in Figure The term and in Figure 2.11 are the fatigue damage at time and of the fatigue cycle. These aspects are discussed in detail later in Chapter 4. 22

50 CHAPTER 2 LITERATURE REVIEW τ τ 0 K K(1-d t ) K(1-d t+1 ) δ t δ t+1 δ f δ Figure 2.11 Fatigue degradation in a bi-linear cohesive zone law Pure mode loading Here the fracture in the material occurs when the energy released is equal to the fracture energy in the particular mode. Therefore, the propagation criteria can be expressed as However, in real structure the loading is usually via a mixture of mode I and mode II Mixed-mode loading Mixed-mode loading occurs in most real structures. The mode-mixity in a cohesive zone model analysis can be defined using the different criteria for the mixed-modes. The simplest criterion considers that failure of the composite occurs mainly in mode I due to the relatively high energy for fracture required in mode II (Whitcomb [21]). The criteria are given by the relations (2.8) 23

51 CHAPTER 2 LITERATURE REVIEW (2.9) However, these very simple criteria are invariably found to be inadequate. So another criterion for the prediction of the propagation of crack growth under mixed-mode loading is to sum the individual fracture energies to give the critical fracture energy in the mixed-mode condition (Wu [22]). This criterion can be expressed as (2.10) And can be expressed in the normalised form as (2.11) Yet another, more generalised form of Equation 2.11 is a power law expression (Camanho et al. [7]) given by where crack propagation occurs if (2.12) where the parameters and are the constants obtained from experimental data, and and are the critical strain energy release rates in pure mode I and II respectively. Another criterion (Hahn [23]) which accounts for mode-mixity effects is the expression (2.13) and yet another propagation criteria (Ramkumar [24]) is (2.14) 24

52 CHAPTER 2 LITERATURE REVIEW The Benzeggagh-Kenane (B-K) criterion (Camanho et al. [7]) for accounting the mode-mixity is given by (2.15) where and are given by the expressions (2.16) (2.17) The mode-mixity factor,, is a factor which may be obtained from an experimental fit. In the present work, B-K criterion is used for the mixed-mode analysis as it depend only on one factor, which can readily be determined by experimental fit, as discussed further in Chapter The Energy method The energy method is the most conventional method used for the analysis of damage in composites. Some of the important work undertaken using this method is described below. Rebière and Gamby [25] used the variational energy method to model delamination and cracking behaviour of cross-ply laminates. The transverse and longitudinal cracking of the laminates were modelled to determine the value of SERR associated with the three modes of fracture. They modelled the development of a triangular-shaped delamination and showed that the delamination length is not uniform in the plane of the laminate. Quantian Luo and Liyong Tong [26] developed a closed-form formula for different mode energies for crack formation in a composite beam. The energy release rate was expressed in terms of axial load, shear forces and bending moments on the crack tip in the layered beam. An equation was derived for a 25

53 CHAPTER 2 LITERATURE REVIEW zero thick adhesive layer in the beam. The energy method predicted the static crack growth in the material but was unable to account for the degradation of material properties. Kashtalyan and Soutis [27] developed a theoretical model based on stiffness degradation and mechanical behaviour of symmetric laminates with off-axis ply cracks and crack induced delaminations. They calculated the SERR based on crack density, delamination area and ply orientation angle in symmetric laminates. A shear-lag analysis was used to determine ply stresses in different modes. Wimmer and Pettermann [28] used a Griffith crack growth criterion to predict the load required to propagate the crack and studied the stability of delamination crack growth (Figure 2.12). Figure 2.12 shows the variation of force and displacement with the delamination growth and the influence of delamination size on the stability of crack growth. They also determined the critical size of delamination where the growth changed from stable to unstable, and vice versa. Their method could be applied to composites, as well as to other problems where the crack path was known. They proved in their study that a small delamination grew in an unstable manner while large delaminations grew in a stable manner. Figure 2.12 The minimum load required to propagate a delamination of a certain size in its most detrimental location (Wimmer and Pettermann [28]). 26

54 CHAPTER 2 LITERATURE REVIEW Shivakumar et al. [29] conducted experiments to determine the resistance to crack growth as a function of delamination extension. They established an equation for the delamination growth rate,, as a function of the maximum cyclic energy release rate, (Figure 2.13). The increase of resistance due to matrix cracking, fibre bridging, tow splitting, separation, bridging and breaking (see Figure 2.14) were accounted for through normalisation of the equation by the instantaneous resistance value, (i.e. the energy release rate when the delamination growth was infinite, or unstable). Figure 2.13 Comparison of theoretical equation for delamination growth rate with the test data (Shivakumar et al. [29]). 27

55 CHAPTER 2 LITERATURE REVIEW Figure 2.14 Failure modes in laminate (Shivakumar et al. [29]). 2.5 Modelling Damage and the Life Time under Cyclic Fatigue Loading Quasi-static loading Quasi-static loading in composites causes damage due to an increase in the applied load with time. The static damage in composites appears mainly as transverse cracks which occur within the laminae of composites, as well as due to interlaminar delaminations. The analysis of damage build-up, and the associated loss of stiffness, in composites materials under quasi-static loading has been undertaken by various authors using different methods. Some of the important and relevant work undertaken is described below. Boniface et al. [30] used a compliance, i.e. the inverse of stiffness, change approach via a shear lag analysis to relate the crack growth rate with the SERR. They studied the energy release and the interaction of transverse cracks between each other, depending on the spacing of cracks. The study also showed how the crack spacing affects (Figure 2.15) the interaction between the 28

56 CHAPTER 2 LITERATURE REVIEW cracks and confirmed the validity of the Paris curve to describe fatigue crack growth. Figure 2.15 Transverse crack interference model (Boniface et al. [30]). Hallett et al. [9] analysed a quasi-isotropic composite plate to predict the failure. In this work, the composite was analysed using both the VCCT technique and the cohesive zone model approach to predict the delamination and transverse cracking. It was shown that delamination, and its interaction with the transverse cracks, affected the final failure of the laminate. The study also highlighted the difficulty in accurately predicting the failure of composite materials. Parvizi and Bailey [31] studied the growth of transverse cracks for different stresses and ply thickness. They developed a shear lag analytical expression using a exponential equation to model the reduction in strength with crack spacing. They observed that a change in the stiffness of the composite due to transverse cracking at relatively low strains was accompanied by a visual whitening effect, due to the formation of the transverse micro-cracks. Their method is difficult to adopt for non-isotropic materials, as their model cannot take into account other directional material properties. 29

57 CHAPTER 2 LITERATURE REVIEW Huchette [32] illustrated the damage in the composite due to transverse cracking and delamination. The different steps in the damage process of the composite (0 2 /90 2 ) s are shown in Figure This study showed the interaction of transverse cracks with the delamination in the failure of composites could be successfully modelled using cohesive zone model elements. Figure 2.16 Failure of a composite strip specimen (Huchette [32]). Leblond et al. [33] used a finite element method to study cross-ply laminates. They studied the longitudinal stiffness reduction due to cracks in the laminates and modelled the laminates and obtained good agreement with the experimental results. Their study showed the major reduction of stiffness that occurred with an increasing crack density of transverse cracks in the cross-ply laminates. Their theoretical predictions were in good agreement with the experimental results (Figures 2.17 and 2.18). 30

58 CHAPTER 2 LITERATURE REVIEW Figure 2.17 Stiffness reduction as a function of the average crack density in the case of two different laminates. E o is the initial elastic modulus of the laminate and E x is the elastic modulus with a given crack density (Leblond et al. [33]). Figure 2.18 Crack multiplication in transverse plies (Leblond et al. [33]) Fatigue loading Fatigue loading in a material occurs due to application of a periodic load and causes the degradation of material properties with time. The load reversals 31

59 CHAPTER 2 LITERATURE REVIEW cause permanent damage in the material and the cyclic load applied in the fatigue test is invariably considerably less than the ultimate load. There is also usually a fatigue limit of the composite material (Figure 2.19), below which no fatigue damage occurs. Figure 2.19 Fatigue life diagram for a (0/90 2 ) s carbon fibre/epoxy matrix laminate (Akshantala and Talreja [34]). Turon et al. [35] developed a damage model for the simulation of progressive delamination in composite materials under variable mixed-mode loading. The model used constitutive laws for modelling the initiation and propagation of delamination. The damage evolution for the cohesive zone model elements was based on progressive degradation of the cohesive zone model parameters, as will be described later in detail in Chapter 4. Talreja [1] has developed a fatigue damage mechanism for the analysis of composites. He proposed fatigue life diagrams based on the strain in the fibres and matrix (Figure 2.20). In his paper, he defined the fatigue ratio and defined the fatigue limit for unidirectional, cross and angle plied laminates, as illustrated in Figure The study thus defined the fatigue resistance of the materials from the strains in the composite material. 32

60 CHAPTER 2 LITERATURE REVIEW Figure 2.20 Fatigue life diagram for a unidirectional composites under loading parallel to the fibres (Talreja [1]). Manjunatha et al. [36] studied the fatigue behaviour of an epoxy matrix polymer containing nano-silica particles and the corresponding GFRP composites. The work quantified the development of transverse cracking with the number of cycles of fatigue loading. The fatigue life of the GFRP composites was shown to increase by about three to four times with the addition of the nano-silica particles. The study also showed that reduced matrix cracking, due to the nano-silica particles debonding and associated plastic void growth mechanisms, contributed significantly to the increase in the fatigue life in the GFRP composites with the modified matrix. Tong et al. [37] studied the transverse cracking in the matrix in GFRP laminates under fatigue loads. Their experimental observations of the fatigue crack growth in the laminates were undertaken to study the fatigue degradation of the strength of the material. Their work also studied the degradation of stiffness with the increase of transverse cracks during the experiments. They observed different crack densities in the various plies of composite and found that the transverse crack density in the 90 o fibres saturated after a certain cycles of 33

61 CHAPTER 2 LITERATURE REVIEW fatigue load. The experimental study also showed a relationship between the crack density and material properties and gave a good insight into the characteristic damage that occurs in composite materials. Tong et al. [38] modelled the experimental results obtained in their previous work on laminates to predict the reduction of stiffness of laminates with fatigue cycles. They modelled open cracks in the laminates using a finite element analysis inputting the value of crack density observed in the experiments. The transverse cracks in different plies gave the distribution of the stresses around the crack tip and their contribution to the total stiffness reduction. The model was developed with plane strain elements and showed good agreement with the experimental results for different values of crack density. Manjunatha et al. [39] studied the fatigue life of GFRP composite materials modified with both rubber and silica particles, see Figure The fatigue life of these hybrid (i.e. containing both micrometre-sized rubber and nano-silica particles) epoxy matrix composites was about six to ten times higher than that of the GFRP composites manufactured using the unmodified (i.e. control) epoxy matrix polymer. They explained the increase in the fatigue life of the hybrid epoxy composites as arising from the toughening micro-mechanisms caused by the presence of both types of particles, such as cavitation of the rubber particles and silica particle debonding. These effects both resulted in increased plastic deformation of the epoxy matrix (Figure 2.21). Indeed, Manjunatha et al. [40] observed less transverse cracking in the composites due to the addition of rubber and silica particles (Figure 2.22). 34

62 CHAPTER 2 LITERATURE REVIEW Figure 2.21 The tapping mode atomic force microscopy (AFM) phase images of the hybrid-epoxy matrix polymer (Manjunatha et al. [39]) (CTBN: carboxy-termianted butadiene acrylonitrile rubber). Figure 2.22 Transmitted light photographic images of matrix cracking in the GFRP composites after testing at stress of 150MPa. NR-Neat resin, NRR-Neat resin with rubber, NRS-Neat resin with silica and NRRS-Neat resin with rubber and silica (Manjunatha et al. [40]) 35

63 CHAPTER 2 LITERATURE REVIEW Camanho et al. [7] developed a cohesive zone model for crack propagation under mixed-mode loading in composites. Their constitutive law proposed used the relative displacement between the nodes for initiation and propagation of the cracks. They used a mixed-mode criterion to determine the initiation and propagation of delamination under the mixed-mode loading. They developed the cohesive zone model formulation using a subroutine to simulate the double cantilever beam (DCB), end notch flexure (ENF) and mixed-mode bending (MMB) experimental tests. The model gave a good prediction of the fatigue life of the composite material. Robinson et al. [41] developed a cohesive zone model approach to predict fatigue delamination growth in composite materials. They predicted the fatigue growth curve for composites (Figure 2.23) using a novel degradation law which had an exponential component to account for the degradation of the composite material due to delamination occurring. The degradation law showed a similarity to the Paris law for fatigue life prediction, see Figure 2.6. Figure 2.23 Crack length versus cycles (Robinson et al. [41]). 36

64 CHAPTER 2 LITERATURE REVIEW Attia et al. [42] proposed a different method for predicting the growth of impact damage in fibre composite skin structures when subjected to cyclic-fatigue loading. The method that they developed used the experimental relationship between the SERR and the number of fatigue cycles to initiate a fatigue crack, which was included into a finite element analysis (Figure 2.24). The approach uses the FEA model to deduce the SERR of the panel by deducing the energy change for growth by approximately 5% of the original impact-damaged area. Figure 2.24 Experimental relation between the maximum SERR,, and the number of fatigue cycles,, for the onset of crack growth (Attia et al. [42]). Turon et al. [43] proposed a damage model for the simulation of delamination propagation under high-cycle fatigue loading using a cohesive zone model approach. They obtained the damage state as a function of the loading conditions and determined the Paris law coefficients to use in the model in order to degrade the cohesive zone law as a function of the number of fatigue cycles. In their work the degradation of the material using the cohesive zone model resulted in the degradation of the cohesive traction (i.e. stress) (Figure 2.25). The model was validated by predicting the propagation rates in mode I, II and 37

65 CHAPTER 2 LITERATURE REVIEW mixed-mode tests and by observing that they obtained good agreement with the experimental results. Figure 2.25 Evolution of the interface/cohesive traction and the maximum interface/cohesive strength as a function of the number of cycles for a displacement jump controlled high-cycle fatigue test (Turon et al. [43]). (Here interfacial traction is the critical stress of the cohesive zone law at a given number of fatigue cycles and the interfacial traction is the traction at the cohesive zone at a given number of fatigue cycles.) Iannucci [44] developed a cohesive zone modelling technique using a formulation based on a damage mechanics approach and he used a stress threshold and critical energy release rate for each particular delamination mode. In his analysis, cohesive zone elements were placed where delaminations were expected. (Thus, it should be noted that prior knowledge of their propagation path was required.) The energy dissipated in different modes was used to calculate the mode ratio. Again, this combination of a cohesive zone model with a finite element analysis gave very good agreement with the experimental results. 38

66 CHAPTER 2 LITERATURE REVIEW Khoramishad et al. [45] developed a bi-linear cohesive law to simulate damage in adhesively bonded joints. The fatigue degradation in the model was based on the degradation of the cohesive penalty stiffness,, with time. The cohesive model predicted the fatigue life and strains in the adhesive joints. They also derived a formulation for fatigue degradation using a mode-mixity criterion and reducing the critical cohesive stress, instead of the penalty stiffness of the element. Mao and Mahadevan [46] have developed a mathematical model for the degradation of composite materials under cyclic fatigue loading. A nonlinear model was used for the damage evolution in the composite materials subject to fatigue loading. The damage model accounted for the damage of the material based on the degradation of elastic parameters. They used a curve fitting method to predict the appropriate parameters for use in the fatigue law. The damage accumulation law employed a power function for the number of cycles and gave a reasonably good agreement with the experimental results. 2.6 Concluding Remarks The present study has reviewed the literature on the prediction of fatigue life of composite materials using various analysis methods. The finite element analysis using a fracture mechanics concept is clearly a good tool for studying the fatigue life of composite materials. The present review also gives a good insight into the methods that may be combined with such finite element analyses for the life prediction of composites namely the VCCT, the cohesive zone model law, etc. The cohesive zone model seems to be a very appropriate approach for the analysis of fatigue damage in an uncracked specimen, because of its ability to predict the initiation and propagation of a crack. The studies reviewed in the present Chapter also highlight the difficultly in predicting the fatigue life of composites due to the complex nature of the damage arising from both transverse cracking and delamination occurring in the composite material. 39

67 CHAPTER 2 LITERATURE REVIEW The present literature review also reveals that little theoretical work has been reported on the prediction of the fatigue life of nano-particle, rubber-particle and hybrid modified epoxy matrix composites. It may also be seen from the present review that there are very few reports available on the study of fatigue life in transversely isotropic composite using a cohesive zone model law. Hence the present work will mainly concentrate on modelling and predicting the fatigue life of transversely isotropic composites with different formulations of epoxy matrices using a cohesive zone model method, coupled with a finite element analysis approach. The cohesive zone model formulations of Turon et al. [43], Turon et al. [35] and Camanho et al. [7] will be used for the present analyses. However, before the novel theoretical analyses developed in the present work are described (Chapter 4), the experimental techniques will be given in the next Chapter. 40

68 CHAPTER 3 CHAPTER 3 EXPERIMENTAL TECHNIQUES 3. EXPERIMENTAL TECHNIQUES 3.1 Introduction In the present work, a glass fibre reinforced plastic (GFRP) composite is used to study the fatigue behaviour of quasi-isotropic (QI) laminates. The properties of the epoxy matrices used in the composite are also studied. The epoxy matrices used for the present work are basically a) A control formulation; b) A modified formulation with a dispersion of nano-silica particles; c) A modified formulation with a dispersion of micrometre-sized rubber particles; d) A hybrid formulation with a dispersion of both micrometre-rubber and nanosilica particles. The bulk epoxies and the GFRP composite materials of the different formulations have been tested. Such tests have been conducted in order to determine the quasi-static and fatigue properties of the GFRP composite laminates. The bulk epoxy matrices have been tested to ascertain the various parameters needed for the theoretical modelling studies, which are later developed to predict the fatigue life of the GFRP laminates (Chapter 4). The tests undertaken on the epoxy are quasi-static single edge notched bending (SENB) tests and the cyclic-fatigue test using compact tension (CT) specimens. The tests undertaken on the GFRP composite are quasi-static and cyclic-fatigue tests on double cantilever beam (DCB) and composite strip specimens. The present experimental work is the continuation of previous studies on the unmodified (i.e. control) and modified epoxy composites by Manjunatha et al. [36], Manjunatha et al. [39], Manjunatha et al. [40]), Masania [47], Hsieh [48] and Lee [49]. 41

69 CHAPTER 3 EXPERIMENTAL TECHNIQUES 3.2 Materials In this work, GFRP laminates based on the control and modified epoxy resin matrices were prepared, i.e. the control resin, a resin with 9% wt. of micrometrerubber particles, a resin with 10% wt. nano-silica particles and a resin with 9% wt. micrometre-rubber and 10% wt. nano-silica particles. The epoxy resin used was LY556, a standard diglycidyl ether of bis-phenol A (DGEBA) with an epoxide equivalent weight (EEW) of 185g/mol, supplied by Huntsman, Duxford, UK. The curing agent was Albidur HE 600, an accelerated methylhexahydrophthalic acid anhydride with an amine equivalent weight (AEW) of 170g/mol and a stoichiometric amount of the curing agent was added to the formulation to cure the epoxy resin. The nano-silica particles used in the resin were based on Nanopox F400 where they were present in a concentration of 40% wt. in a DGEBA epoxy resin with an EEW of 295g/mol from Nanoresins, Geesthacht, Germany. The Albipox 1000, reactive liquid carboxyl-terminated butadiene-acrylonitrile (CTBN) rubber was obtained as a CTBN-epoxy adduct with a rubber concentration of 40% wt. in DGEBA epoxy resin from Emerald, Cleveland, USA. The E-glass fibre sheet was a stitched two layer of non-crimp fibre (NCF) arranged in a ±45 pattern with an areal weight of 450g/m 2 from SP systems, Newport, UK. 3.3 Preparation of Epoxy Matrix Polymer Specimens Introduction Standard fracture mechanics specimens were prepared from the materials to obtain the fracture mechanics data and the cohesive zone law parameters needed for the theoretical modelling studies. The composite specimens used for the fatigue analysis were also manufactured with the materials described in the above section. The specimen preparation methods for the various types of test specimens are described in the sections below. 42

70 CHAPTER 3 EXPERIMENTAL TECHNIQUES Preparation of plates Fracture mechanics testing was performed using epoxy matrix specimens machined from bulk epoxy plates. The bulk epoxy specimens used for testing were manufactured as plates. The epoxy was cured in a 6mm thick metal mould, which was used for preparing the bulk epoxy plates. Initially silicone gasket was placed around the mould to prevent the leakage of epoxy from the mould. The moulds were first opened and the inside surface cleaned with acetone. The surface of the mould was then coated with release agent to aid the easy removal of the specimen. The moulds were then assembled and made ready for pouring the resin. The epoxy resin matrix prepared as described above, was poured into the mould. The resin was mixed with a stoichiometric amount of the curing agent and degassed at 50 o C and -1atm. The resin mixture was then poured into the release-coated steel mould. The resin was poured from the one side of the mould to prevent the development of bubbles in the resin when pouring. The mould was taken to the oven for curing. The resin was cured by increasing the temperature at 1 o C/min and cured at 100 o C for 2hr and later post-cured at 150 o C for 10hr. After curing, the plates were removed from the mould and inspected for any defects or voids in the moulding. The cured bulk epoxy plate was machined to obtain the required specimen dimensions. The surface of the test specimens was made smooth by polishing with abrasive papers, as surface irregularities might affect the fracture behaviour of these relatively brittle materials Single edge notched bending (SENB) specimens Standard tests were performed on the bulk epoxy materials to obtain the fracture properties of the various epoxy formulations. The SENB test was conducted to obtain the fracture energy,. The test was conducted according to the standard ISO:13586:2000 [50]. The specimens were machined from the epoxy plate, with the dimensions as in the standard. A sharp notch was inserted in the specimen using a razor blade to act as a pre-crack, and the specimen was tested. The load versus displacement curve of the specimen under quasi-static load was obtained. 43

71 CHAPTER 3 EXPERIMENTAL TECHNIQUES Compact tension (CT) specimens Compact tension specimens were used to determine the cyclic-fatigue properties of the bulk epoxy materials. The CT specimens were manufactured from the plates of the bulk epoxy as specified in the ASTM:E647 [51] standard. A sharp notch was inserted in the CT specimen by machining and then sharpened using a razor blade. The surface of the specimen was again polished to remove any defects. The compact specimen was loaded in tensiontension fatigue. 3.4 Preparation of GFRP Composite Specimens Introduction Composite specimens were manufactured from the materials described in Section 3.2. The preparation of the composite specimens was based on the manufacturing method described below Resin infusion under flexible tooling (RIFT) The laminated plates for the specimens were prepared from a multidirectional, high strength, glass fibre epoxy pre-preg, and a resin infusion under flexible tooling (RIFT) method was used to prepare the fibre reinforced epoxy composites (Figure 3.1). In this method, the woven fibres were laid up and placed in a vacuum bag and the resin was made to infuse through the fibre layup using the vacuum pressure that was applied. The epoxy resin was therefore forced to spread throughout the fibre layup, and the layup was then cured to form the laminate. The glass fibre sheet was cut into 330x330mm 2 squares. To give further details, then to produce the laminates a temperature controlled plate surface was set to the required infusion temperature. The surface of the plate was cleaned and made smooth and an infusion stack was built. A polyamide film was laid over the plate and fixed using adhesive tape. The polyamide film formed the first and the outer layer of the vacuum bag. A sealant tape was bonded on to the polyamide film along the plate border to form the 44

72 CHAPTER 3 EXPERIMENTAL TECHNIQUES vacuum bag. A flow media, i.e. a sheet of plastic net to help uniform flow of the epoxy resin, was placed next to the polyamide film to help the infusion of the matrix resin. The inlet pipes and outlet pipes were fixed on the ends of the plate. The fibre layup was placed over the peel ply, and the same procedure was repeated on the other side of the fibre layup. The vacuum bag was sealed with the sealant tape and a vacuum pump was connected to the so-formed vacuum bag. The resin was infused into the vacuum bag through the inlet pipe and the temperature of the plate was controlled. The resin flowed through the dry fibre layup and reached the other end of the plate. The inlet pipe was then closed to prevent the infusion of the resin into the vacuum chamber. The resin was cured by ramping the temperature to 100 o C at 1 o C/min, cured for 2hr, again ramped to 150 o C at 1 o C/min and post-cured for 10hr. After the curing cycle was complete, the laminate was taken out of the vacuum bag and machined around the edges. The composite laminate was visually checked for voids. Composite plates of 330X330X5.4mm 3 were therefore manufactured by the above method. Figure 3.1 Fabrication of a GFRP sheet using RIFT Double cantilever beam (DCB) specimens Composite plates were prepared using the glass fibre sheet manufactured as described above laid up in the sequence of [(-45/45) s (90/0) s ] 2 [(0/90) s (45/-45) s ] 2 to give a 0 o /90 o lamina interface (mid plane) across the fracture plane. A 45

73 CHAPTER 3 EXPERIMENTAL TECHNIQUES 12.5μm thick polytetrafluoroethylene (PTFE) film was inserted in the mid-plane of the plate to act as a pre-crack. The specimens were designed to be sufficiently stiff to avoid large displacements, plastic deformation, intra-ply damage and to reduce elastic couplings. Each plate had sixteen layers of woven glass fibre. Using this method, standard composite plates of 300x300x5.4mm 3 were made. The DCB specimens were cut from the plates using a wet-saw cutting machine with nominal dimensions as recommended by the standard (ISO:15024:2001 [52]). Each laminate plate was cut into ten specimens. The pre-crack lengths were approximately 60mm for the specimens. Machined aluminium alloy blocks of the same width as the specimens were bonded onto the end of the DCB test specimen using an epoxy adhesive, which was cured at room temperature. One of the edges of the DCB specimen was coated with white ink and was graduated at 1mm interval to monitor the crack growth Composite strip specimens Composite plates of 300x300x2.7mm 3 in dimension were prepared using the RIFT method, as described above, with a layup sequence of [(45/-45/0/90) s ] 2. Each plate had eight layers of woven glass fibres. The strip specimens were cut from the plates using the wet-saw cutting-machine with nominal dimensions of 150x25x2.7mm 3. The strips were grit blasted where the end tabs were to be fixed and the tabs were fixed to the specimen using an epoxy adhesive, which was cured at room temperature. The strip specimens were finished by lightly abrading the edges of the test specimens to remove any major defects. 3.5 Test Methods for the Epoxy Matrix Polymer Specimens Introduction Tests were undertaken on the bulk epoxy matrix specimens at 20 o ±2 o C to determine the fracture properties of the material. These are described below. 46

74 CHAPTER 3 EXPERIMENTAL TECHNIQUES Single edge notched bending (SENB) tests SENB specimens were tested to determine the mode I fracture energy,, of the bulk epoxy matrix polymer. This test measures the resistance to the initiation and propagation of a crack in the bulk epoxy polymer under mode I loading. A standard quasi-static test was conducted to determine the mode I fracture toughness,, according to the standard (ISO:13586:2000 [50]). The specimen was loaded under displacement control at a rate of 0.05mm/min. The start of the crack was observed visually and the growth of the crack was noted using the event marker, the length of the propagating crack being recorded on the load versus displacement trace. A linear variable displacement transducer (LVDT) was used to measure the displacement during the loading of the specimen. The fracture toughness,, of the material may be calculated from the expression (3.1) where is the maximum load or 5% offset load, the breadth, the width and the shape factor of the specimen where (3.2) The fracture energy,, can be calculated from the equation (3.3) where is the elasticity modulus (Table 5.1 ) and is the Poisson s ratio (Table 5.1) of the bulk epoxy. 47

75 CHAPTER 3 EXPERIMENTAL TECHNIQUES Compact tension (CT) tests The CT specimen test was conducted to determine the cyclic-fatigue properties of the epoxy matrix polymers under cyclic tension-tension loading. This test also ascertains the threshold fracture energy of the epoxy matrix material under fatigue loading. The fatigue parameters obtained from these tests will be used for the modelling analyses of the composite strip specimens (Chapter 5). A cyclic-fatigue test (based on the ASTM:E647 [51] and ISO:15850 [53] standards) was conducted to determine the rate of crack growth,, per cycle as a function of the maximum value of the applied strain energy release rate,. The load was applied as a sinusoidal function with a maximum displacement less than the displacement required for the initiation of crack growth under quasi-static loading. The test was conducted using a 1kN computer-controlled servo-hydraulic test machine under displacement control loading. A fatigue Krak gauge was bonded on the side of the specimen using a standard M-bond adhesive resin, which was cured at 25 o C for 10hrs using a curing agent. The Krak gauge is used to monitor the crack growth in the specimen. Cyclic-fatigue tests were carried out for the different bulk epoxy matrix polymers. The specimens were subjected to displacement-controlled fatigue loading with the frequency of loading kept at 5Hz. As well as using the Krak gauge method the crack growth under the fatigue loading was also monitored using an optical microscope focussed on the crack front. This test therefore measured the maximum load and crack growth under fatigue loading. It also determined the load below which there was no propagation of the crack, and therefore the threshold fracture energies of the different bulk epoxy matrices were determined from the data. The maximum fracture toughness,, for a given cycle can be calculated from the expression (3.4) 48

76 CHAPTER 3 EXPERIMENTAL TECHNIQUES where is the maximum load in the fatigue cycle, the breadth, the width and is a shape factor of the specimen where. The shape factor is given by the expression (3.5) The maximum fracture energy may be calculated from the equation (3.6) where the elastic modulus,, and the Poisson s ratio,, of material are obtained from Table 5.1. The crack growth curve can be obtained from the secant method and the incremental polynomial method (ASTM:E647 [54]). In the incremental polynomial method, the growth rate curve,, is obtained by fitting a polynomial between a set of points. In the present work, the secant method was used for obtaining the growth rate curve. In the secant method, the slopes of the adjacent points are used to calculate the growth rate curve. 3.6 Test Methods for the GFRP Composite Specimens Introduction Tests were undertaken the composite material at 20 o ±2 o C under both quasistatic and cyclic fatigue loading to determine the fracture mechanics properties of the material. Tension-tension cyclic fatigue tests were also conducted on strips of the GFRP composite material to determine the lifetime of the material under fatigue conditions. Details of the tests conducted on the specimens are given below. 49

77 CHAPTER 3 EXPERIMENTAL TECHNIQUES DCB tests: Quasi-static tests A standard quasi-static test was conducted to determine the mode I interlaminar fracture energy,, under quasi-static load for the GFRP composites using a DCB specimen (ISO:15024:2001 [52]). During the test the load, displacement and crack length required for the initiation and growth of the crack were measured in the DCB specimen. The values of for initiation and propagation of the crack were obtained from the load versus displacement trace. The DCB specimen was loaded under displacement control at a rate of 1mm/min. The start of the crack was observed visually and the growth of the crack was noted using the event marker to indicate the crack length on the load versus displacement trace. The laminate sequence of the composite was [(- 45/45) s (90/0) s ] 2 [(0/90) s (45/-45) s ] 2, with the initial crack located at the mid-plane of the laminate. The test is shown in Figures 3.2 and 3.3. Aluminium loading block Composite Initial crack Figure 3.2 DCB specimen. The load versus displacement trace of the composite was typical of brittle matrix fibre laminates. The fracture surfaces showed evidence of tow splitting, fibrebreakage and fibre-matrix interfacial fracture (see Figure 3.4). The non-linearity (i.e. the 5% offset) or the maximum load criteria was used to define the point of crack initiation. The 5% offset is the intersection of the load versus displacement curve with a line corresponding to a value of the compliance which is 5% higher than the initial slope (ISO:15024:2001 [52]). 50

78 CHAPTER 3 EXPERIMENTAL TECHNIQUES Figure 3.3 Experimental setup for a DCB test. Figure 3.4 Fibre bridging in DCB test. The value of the interlaminar fracture energy,, was calculated using the Corrected beam theory (CBT) method as defined in the ISO standard (ISO:15024:2001 [52]) and which may be expressed by 51

79 CHAPTER 3 EXPERIMENTAL TECHNIQUES (3.7) where,,, and are the load, load-line displacement, crack width, crack length and the crack-tip rotation correction factor of the specimen, respectively. The factor is given by (3.8) where is the distance from the centre of the loading pin to the mid-plane of the specimen beam DCB tests: Cyclic-fatigue tests The tests were conducted according to the prescribed standard (ASTM:E647 [51]). The load was applied as a sinusoidal function with a maximum displacement of 50% of the displacement required for the initiation of the crack in the corresponding quasi-static tests of the DCB specimens. The frequency of the loading was 1~3Hz. The laminate sequence of the composite strip was [(- 45/45) s (90/0) s ] 2 [(0/90) s (45/-45) s ] 2, which is identical to that of the DCB specimens described above. The cyclic-fatigue DCB test was conducted using a 1kN computer-controlled servo-hydraulic test machine (Figure 3.5). The tests were conducted under displacement control loading. A travelling microscope was mounted on a traversing stand to monitor the growth of delamination. The number of cycles required for the growth of the crack was noted for different amplitudes of the applied displacement. The maximum load was recorded when the visual onset of delamination growth was observed on the edge of the specimen. The threshold energy of the strain energy release rate was also obtained from the fatigue test. The maximum energy release rate,, was calculated using the above equations for the DCB quasi-static test. The crack growth rate,, and 52

80 CHAPTER 3 EXPERIMENTAL TECHNIQUES the for the experiments were plotted using logarithmic scales to obtain the typical growth rate curve. Figure 3.5 Experimental setup for a fatigue DCB test Composite strip laminate tests: Quasi-static and fatigue tests A quasi-static test was conducted on the composite strip to determine the quasistatic strength and elastic properties of the laminate. The tests were conducted based on the ASTM:D3039 [55] standard. The tensile test was performed using a 100kN computer-controlled screw-driven test machine and the specimen was loaded at a rate of 1mm/min. The load versus displacement curve was obtained to determine the elastic property of the composite. Cyclic-fatigue tests were conducted on the composite strip to study the fatigue life of the composite laminate (ASTM:D3479M [56]). Composite strips were loaded using a stress controlled cyclic-fatigue test to different stress levels, and the number of cycles to failure was noted. During this test the growth, the initiation and propagation of transverse cracks in the composite laminate strip under fatigue loading were also observed. The size of the strips used for the testing was 150x25x2.7mm 3 and the load was applied as constant amplitude sinusoidal stress with a stress ratio of 0.1, using a 25kN computer-controlled 53

81 CHAPTER 3 EXPERIMENTAL TECHNIQUES servo-hydraulic test machine. The frequency used for the low-cycle fatigue (i.e. a high applied stress) tests was 1Hz and for the high-cycle fatigue (i.e. a low applied stress) was 4Hz. The tests were conducted using different specimens at different stress levels and the number of cycles to failure of the specimen was noted. The crack density was also noted for a given number of cycles and the variation of crack density on the surface of the specimen was plotted against the number of fatigue cycles. The crack density was observed on the surface of the specimens using an optical microscope, and the stiffness reduction of the strip was also measured as a function of the number of fatigue cycles. 3.7 Concluding Remarks Tests on the bulk epoxy and the GFRP composite were undertaken to measure their behaviour under quasi-static and fatigue loading. From the quasi-static tests the values of the fracture energy,, for both the bulk epoxy matrices and the GFRP laminates were measured. The cyclic-fatigue tests enabled the rate of the crack growth per cycle,, and the corresponding maximum applied stain-energy release rate,, to be determined, again for both the bulk epoxy matrices and for the corresponding GFRP laminates. All these data are needed for the modelling studies described in Chapter 5. To validate the model, the applied stress versus the number of fatigue cycles to failure was measured, using composite strip specimens of the GFRP laminates of the same lay-up sequence. The next Chapter describes the theoretical modelling work developed during the current research. 54

82 CHAPTER 4 CHAPTER 4 THEORETICAL TECHNIQUES 4. THEORETICAL TECHNIQUES 4.1 Introduction In the present chapter, theoretical models and formulations based upon finite element analysis (FEA) methods are developed to simulate the experimental results. Different methods of analysis, such as virtual crack closure technique (VCCT), cohesive contact and cohesive zone elements are used, with the FEA approach, for modelling the fracture of the composite materials. Analysis of the FEA models is undertaken using the Abaqus software program employing continuum elements and cohesive zone elements. All the analyses of the continuum elements is done using 2D plane-strain elements. In a typical cohesive zone element model of the specimens, plane-strain elements are used to model the continuum elements and the cohesive zone elements are used solely to model the fracture path. It should be noted that in Abaqus the cohesive law is represented by stress/traction versus strain in the cohesive zone. 4.2 Quasi-Static Analysis A quasi-static analysis formulation is derived in the following sections employing the different methods as listed below. The underlying principles of the FEA approach are also explained and the details of the methodology are shown The Virtual Crack Closure Technique (VCCT) The VCCT method is derived from the crack closure technique (CCT), see Section This technique determines the energy released during crack propagation from the geometry of the crack. The energy released can be calculated from the displacement and forces in each direction at the nodes of crack tip. The energy released by the crack extension,, is the work required 55

83 CHAPTER 4 THEORETICAL TECHNIQUES to close the crack by the same amount to its original length, keeping the external load constant. Δa y, v c a x, u f e d b Figure 4.1 The VCCT model. The energy released can be calculated using the displacement and nodal forces in the different directions (Krueger [6]). The crack is represented by a one dimensional discontinuity of a line of nodes which have the same coordinates at the top and bottom surfaces (Figure 4.1). The energy released due to mode I and II is due to the opening and shear displacements between the contact surfaces. The energy released in different directions is the energy release associated with that given mode, e.g. mode I or II. The different mode components of the strain energy release rate (SERR) can be obtained by combining two analysis methods. In the first analysis, nodal forces are calculated at the nodes prior to crack growth and in the second analysis the crack nodes are released to obtain the displacements at the crack tip. The SERR is calculated by multiplying half of the nodal forces from the first analysis with the displacements obtained in the second analysis. The energy released in different modes due to crack opening can be expressed as (4.1) (4.2) 56

84 CHAPTER 4 THEORETICAL TECHNIQUES (4.3) where are the SERR at the nodes, and are the nodal forces acting at the nodes with displacements. The above equations are presented for the mixed-mode SERR for a two dimensional model in plane stress or plane strain The cohesive zone law Cohesive zone laws are powerful tools used for modelling the failure of composites, such as delamination, shear cracks, matrix cracking, fibre failure or micro-buckling (e.g. kink-band formation), friction between the plies, bridging by through-thickness reinforcement and oblique crack-bridging fibres. In this method, a cohesive zone is used to model the crack propagation, in mode I mainly. The cohesive zone law helps to overcome the complexity of considering a singularity at the crack tip. The cohesive zone law relates the traction and displacement at the crack tip to the energy release rate of the material when loaded in the different modes. The concept behind such a law is that crack propagates in the material according to a defined cohesive zone law, and the law itself may change according to well defined damage evolution principles Kinematics The kinematics of the cohesive zone can be developed from a crack present in a material (Ortiz and Pandolfi [57]). The crack in the material causes the formation of new surfaces, which can be assumed to possess a top and a bottom. The crack in the material can be represented using a cohesive zone element (Figure 4.2) of zero thickness. The relative displacement across this cohesive zone element can be written as (4.4) where and are the displacements at the top and bottom surface of the cohesive zone element. 57

85 CHAPTER 4 THEORETICAL TECHNIQUES 3 η 4 X ξ X 2 X 1 X 1 (a) (b) Figure 4.2 A four noded cohesive zone element in (a) undeformed state and (b) deformed state in a global coordinate system. The coordinates of the cohesive zone element can be represented using a global coordinate system, as (4.5) (4.6) The global coordinate system for the mid-plane of the cohesive zone element can be represented as (4.7) The vector which defines the normal and the tangential surface of a deformed cohesive zone element is given by (4.8) (4.9) where and are the normal and tangential direction in the numerical coordinate system and and are the coordinates in the local system. The unit vector normal to the local coordinate system can obtained as (4.10) 58

86 CHAPTER 4 THEORETICAL TECHNIQUES The unit vectors tangential to the local coordinate system can be represented as (4.11) and (4.12) where, and are the direction cosines of the local coordinate system in the global coordinate system Constitutive laws The relation between the relative displacement and the traction for an cohesive zone is given by the relation of the traction versus the separation (Turon et al. [35]). The traction in the cohesive zone law for a 2D cohesive zone model is a function of a displacement jump norm and can be written as (4.13) (4.14) where is tangent stiffness tensor, is the cohesive stress and is the norm of the displacement jump. The energy in the cohesive zone law is related to the traction and displacement jump. The free energy per unit surface of the layer can be expressed as (4.15) where is the damage variable. The energy can be expressed as (4.16) where is the initial stiffness tensor. It shall be noted that the negative values of are eliminated to avoid the interpenetration of the different surfaces. Thus, the expression for the free energy is given by 59

87 CHAPTER 4 THEORETICAL TECHNIQUES (4.17) where is the MacAuley bracket defined as and is the Kronecker delta. The equation for the cohesive surface is obtained by differentiating the free energy with respect to the displacement jump as (4.18) where is the cohesive traction. The undamaged stiffness tensor is defined as (4.19) where is the penalty stiffness of the cohesive zone element. The penalty stiffness of the element is selected in order to have a high penalty stiffness condition being used to simulate the cohesive surface. The constitutive equation can be written in Voigt notation as (4.20) The energy at a given period can be expressed as (4.21) Mathematical formulation A 2D cohesive zone element is made up of two linear-line elements connected to the fracture surface (Feih [58]). The two surfaces of the cohesive zone element initially lie together in the unstressed deformed state and separate as the adjacent elements deform. The relative displacements of nodes of the cohesive zone element in the normal and shear direction create element stresses. A four noded cohesive zone element (Figure 4.3) is considered for the present study, but it can be implemented here to develop the formulation for 60

88 CHAPTER 4 THEORETICAL TECHNIQUES higher degrees of cohesive zone element. The thickness of the cohesive zone element is assumed to be zero. η 3 4 thickness=0 ξ =0 ξ v ξ=-1 ξ=1 1 2 u Figure 4.3 A four noded cohesive zone element. The present element formulation is derived for a linear-line element for 2D simulations. The 2D element has two degrees of freedom (DOF) at each node and hence the number of degrees of freedom is twice the number of nodes in the element. Hence the linear cohesive element has four nodes ( and eight (2x ) degrees of freedom; four on the top surface and four on the bottom surface. The displacement at the nodes of cohesive element is expressed as a vector,. The nodal displacement vector in a global coordinate system is given by (4.22) where and are the displacement at the node in the direction and direction, respectively. The relative displacement between the paired nodes is used to derive the cohesive formulation. The relative displacement between the linked pair of nodes can be obtained by operating the displacement vector with a matrix, where is the identity matrix. Hence, the relative displacement,, vector is given by (4.23) The displacements at the nodes are used to obtain the integration point functions. The different integration point functions for an element can be 61

89 CHAPTER 4 THEORETICAL TECHNIQUES obtained from the relative displacement at the nodes using a shape or an interpolation function. The interpolation functions,, (Cook et al. [5]) for each node are obtained in the local coordinate system ( ) of the element. The degree of the interpolation function depends on the number of node pairs in a given element. The relative displacement between the paired nodes in an element is given by (4.24) where (4.25) (4.26) where is (4.27) In case of large deformations, the mid-plane of the element is taken as the coordinate of the element. The mid-plane coordinate is calculated to determine the deformation of the element. The mid-plane coordinate for the element under deformation can be obtained as given below (4.28) where is the coordinate of the element in the undeformed state in a given coordinate system. Hence, the relative displacement between paired nodes in a coordinate system is given by (4.29) 62

90 CHAPTER 4 THEORETICAL TECHNIQUES This local coordinate vector with unit length is obtained by differentiating the global position vector with respect to the local coordinates. (4.30) (4.31) where can be written as (4.32) where is the derivative of the shape function matrix given by (4.33) The length of the element is given by the modulus of the and is obtained as (4.34) The transformation matrix, is given by the relation, which relates the local and global displacement (4.35) The global displacement,, and local displacement,, can be related using the transformation matrix,, as 63

91 CHAPTER 4 THEORETICAL TECHNIQUES (4.36) The force vector for an element is then given by the expression (4.37) (4.38) where is the width of the cohesive zone element, or the through thickness of the model and is the traction vector. The above integration can be achieved by a Newton-Raphson numerical integration technique (Cook et al. [5]). The above expression can be integrated using the numerical integration technique as given below. (4.39) where is the local traction vector The determinant of a Jacobian matrix,, is given by the expression (4.40) The value of should be positive and it transforms the local coordinate system to a global coordinate system. The stiffness matrix of an element is given by the relation (4.41) By integrating numerically as before, is given by the expression 64

92 CHAPTER 4 THEORETICAL TECHNIQUES (4.42) (4.43) (4.44) (4.45) The local stiffness tensor is given by the relation (4.46) (4.47) where and are the penalty stiffnesses of the cohesive zone law in the directions 1 and 2, and is the coupling term which is assumed to be zero. The numerical implementation of the above formulation is undertaken by using the tangent stiffness tensor,, of Equation 4.14 which can be derived as (see Turon et al. [59] for the derivation) (4.48) where factor is the displacement jump threshold in the loading history and the is given by (4.49) 65

93 Traction, τ CHAPTER 4 THEORETICAL TECHNIQUES A bi-linear cohesive zone law A bi-linear cohesive zone law is defined by the traction and displacement between adjacent cohesive nodes in a cohesive zone. A cohesive constitutive law relates the traction to displacement jumps at the cohesive surface. The area under the traction-displacement jump curves is the respective fracture energy,, for the given mode. A typical cohesive zone model is characterized by a bilinear, rate-independent, damage-dependent failure law across the cohesive surfaces. The cohesive zone law is represented by the penalty stiffness,, critical displacement for failure,, and the final failure displacement,, as shown in Figure 4.4. The value of the critical stress is ideally a characteristic property of the material. The propagation of the crack is dependent on the strain energy release rate (SERR) corresponding to the different modes. The energies released in the different modes are combined to determine the critical SERR, as was discussed in Section Different laws may be used to combine the different modes of the energy to find the critical SERR. The propagation of the crack occurs, of course, once the strain energy released is more than the critical strain energy for propagation. The parameters for the cohesive zone law are determined by calibrating the theoretical model using the experimental results, and the critical SERR,, is the criterion used in linear elastic fracture mechanics (LEFM) for the propagation of the crack. τ 0 K G c δ o Displacement, δ δ f Figure 4.4 The bi-linear cohesive zone law. 66

94 CHAPTER 4 THEORETICAL TECHNIQUES In the present work, a bi-linear cohesive zone law is assumed for each of the fracture modes. The parameters required to define a bi-linear cohesive zone law are the critical displacement,, critical stress,, and critical SERR, as shown in Figure 4.4. The three parameters in the bi-linear law are independent of each other and depend on the material properties. The bi-linear cohesive zone law is divided into the linear elastic region, the linear stiffness degradation region and the failure zone. The first part of the cohesive zone law defines the behaviour between the elastic limit and the critical displacement. The elastic limit coincides with the maximum stress value and, once the elastic limit is exceeded in the zone, the cohesive zone starts to degrade. The last part of the cohesive zone law defines a relative displacement value that is equal, or larger, than the critical displacement value. The main characteristic of the cohesive zone models is that the cohesive surface can still transfer load after the onset of damage. When the critical value of displacement jump norm (Section ), i.e., (Turon et al. [43]) is reached or exceeded, the element fails. When formulating this cohesive constitutive law for mode I, any negative relative displacement is avoided, to prevent interpenetration of surfaces, by adopting a cohesive zone law as in Figure 4.4. In mode II, negative relative displacements may readily exist and therefore a symmetrical bi-linear constitutive law is adopted (see Figure 4.5). 67

95 CHAPTER 4 THEORETICAL TECHNIQUES τ τ 0 G c -δ f -δ o δ o δ f δ G c Figure 4.5 Cohesive zone law in mode II. A cohesive zone law can be implemented in a FEA analysis using either (a) a cohesive contact analysis, or (b) a cohesive element analysis. In a cohesive contact analysis, the fracture surfaces are connected together by nodes of the fracture surface and the displacement between the nodes are used to determine the cohesive zone law. The displacements at the element nodes of the surface are employed for the calculation of the cohesive zone law, and to determine the failure behaviour. The cohesive zone law can also be implemented using cohesive zone elements which represent the fracture surface. Here, the displacements at the adjacent nodes of the cohesive elements are used to implement the cohesive zone law Norm of displacement jump tensor The displacement jump (Turon et al. [43]) is a function representing the resultant displacement at the nodes in a mixed-mode analysis. The norm of the displacement jump tensor is used to compare different stages of the displacement jump state. The displacement jump norm,, is a continuous function accounting for the other modes and can be expressed using mode I and II displacements as 68

96 CHAPTER 4 THEORETICAL TECHNIQUES (4.50) (4.51) where is the shear displacement jump in mode II and is the displacement jump in mode I. The value of the displacement jump norm is always greater than, or equal to zero, and is used to avoid the interpenetration of the cohesive zone elements Damage A damage variable,, is employed with respect to the cohesive zone law to define the three states of the cohesive zone law: the elastic state, the damaged state and the failure state. for the elastic state: (4.52) for the damaged state: (4.53) for the failure state: (4.54) The damage variable,, increases rapidly once the critical damage in the cohesive zone element is reached. This rapid increase in the damage variable is due to the definition of the damage variable in the cohesive zone law. The variation of the damage variable with an increase in the displacement is shown in Figure

97 Damage CHAPTER 4 THEORETICAL TECHNIQUES (δ/δ f ) Figure 4.6 Variation of damage variable with displacement in a bi-linear cohesive zone law Mixed-mode loading: onset of crack growth An initiation criterion derived from the Benzeggagh-Kenane (B-K) fracture criterion, see Section , gives a sound basis for a cohesive zone law for mixed-mode loading based on the mode-mixity factor, (Benzeggagh and Kenane [60]). The opening displacement under mixed-mode loading is given by the relation (4.55) where is the energy release rate in mode I and is the energy release rate in mode II. The mode-mixity factor,, depends on the mode-mixity and is determined experimentally. The equivalent failure displacement for mixed-mode loading can be calculated from the expression 70

98 CHAPTER 4 THEORETICAL TECHNIQUES (4.56) The traction in the cohesive zone after the elastic limit has been exceeded may be described in terms of the critical traction,, the opening displacement,, and the failure displacement,, as (4.57) where. The stress in the cohesive zone element is zero when the displacement jump is equal to or more than the failure displacement is given by ; (4.58) The damage variable for mixed-mode loading is given by (4.59) Mixed-mode loading: crack propagation The propagation criteria for a mixed-mode crack growth are derived based on the components of the energy release rate in the different modes. The critical SERR,, is derived for mixed-mode loading and is used to predict crack propagation. The crack growth occurs when the strain energy released is more than the critical strain energy release rate for a given mixed-mode load. Hence, the criterion can be written as (4.60) The expression for the can be derived from the mode-mixity of the problem and the energy release rate,, is the resultant of the energy released in mode I and II. The critical strain energy release rate,, for mixed-mode crack growth may be obtained as 71

99 CHAPTER 4 THEORETICAL TECHNIQUES (4.61) The energy release rate in different modes for a mode-mixity, expressed as (Travesa [61]), can be (4.62) (4.63) The mode-mixity factor can be obtained from the relation (Camanho et al. [7]) (4.64) The mode-mixity factor can be written based on the displacement as (4.65) is the Mac Auley bracket defined as =. The displacement jump in the different modes can be expressed as (4.66) Using the Equation 4.66, the opening and shear displacements under mixedmode loading are related by the expression (4.67) (4.68) 72

100 CHAPTER 4 THEORETICAL TECHNIQUES Using the above equation, the ratio of energy released in mode II to the total energy released rate can be obtained as (4.69) It should be noted that under cyclic fatigue loading, crack propagation occurs when is greater than the threshold value,. In fatigue loading, crack growth occurs and is stable if the energy released is more than and less than of the material Mode-mixity A mixed-mode criterion is used to establish the interaction between the different components of strain energy release rate for mixed-mode loading. The criterion simulates the onset of crack propagation and failure under mixed-mode loading. The B-K criterion (Camanho et al. [7]) which accounts for mode-mixity is given by (4.70) where and are given by the expressions (4.71) (4.72) The mode-mixity factor,, is the factor obtained from an experimental fit. Under mixed-mode loading, damage onset may occur before any of the critical stresses involved reach their respective critical limits. In typical industrial applications of composites, crack growth occurs mainly under mixed-mode conditions. 73

101 CHAPTER 4 THEORETICAL TECHNIQUES 4.3 Fatigue Analysis Introduction The analysis of fatigue driven crack growth in composite materials using FEA is tedious and is dependent on the interactions between the variables involved. In the present work, a mathematical model is developed for mode I and mode II fracture, with the mode II parameters assumed to be equivalent in value to mode I, due to mode I failure being dominant failure mode observed in composite materials. This assumption was shown to be valid concept by Harper and Hallett [62]. Further, the variation of the test frequency and displacement, or stress, ratio are not considered to be significant factors, as observed by Yang et al. [63] and Manjunatha et al. [40], and hence the rate dependence of material need not to be taken into account. The subroutine for the fatigue degradation is written in FORTRAN. The degradation of the cohesive zone law for the composite materials, using the Paris law constants determined from the experiments conducted using the corresponding epoxy polymer matrices, can be described using the following flowchart. 74

102 CHAPTER 4 THEORETICAL TECHNIQUES Input parameters for an user element subroutine for fatigue analysis (, and (see Table 4.1) Calculate Calculate Calculate static damage, Calculate fatigue damage rate, where Damage= static damage + x time increment Stiffness= (1-Damage) Run the model with the degraded cohesive zone element to get the complete degradation of the model with time Figure 4.7 Flow chart of the fatigue analysis embedded in the user element subroutine. 75

103 CHAPTER 4 THEORETICAL TECHNIQUES Table 4.1 Description of the material properties of the bulk epoxy matrix relevant to Figure 4.7. Parameter Meaning How obtained Explained Fracture energy of the bulk epoxy matrix Penalty stiffness of the cohesive zone model Critical stress (i.e. traction) of the cohesive zone model Threshold strain energy release rate of bulk epoxy matrix Paris law parameter of bulk epoxy matrix Paris law parameter of bulk epoxy matrix Quasi-static SENB experiments on the bulk epoxy matrix By fitting the CZM model to the bulk epoxy matrix load versus displacement curve from quasistatic SENB tests By fitting the CZM model to the bulk epoxy matrix load versus displacement curve from quasistatic SENB tests Fatigue CT experiments on the bulk epoxy matrix Fatigue CT experiments on the bulk epoxy matrix Fatigue CT experiments on the bulk epoxy matrix Section Sections & Sections & Section Section Section Degradation strategies The degradation under cyclic fatigue loading of the cohesive zone law parameters can be achieved using different strategies based on the degradation of the penalty stiffness,. Indeed, the present work uses degradation of the penalty stiffness as a strategy for the degradation of the constitutive law embedded in the cohesive zone law upon fatigue loading. The penalty stiffness degradation of the bi-linear law can be achieved using the evolution of a damage variable in the cohesive zone law. To initiate this modelling approach, it should be noted that the stress in the cohesive zone law can be expressed as 76

104 CHAPTER 4 THEORETICAL TECHNIQUES (4.73) where (4.74) and where is the damage variable Static damage evolution under fatigue loading The evolution of static damage occurs due to the reduction of the penalty stiffness of the cohesive zone, which leads to the development of further static damage. The increase in such defined static damage due to fatigue loading can be derived from the rate of damage evolution with time. The evolution of the static damage variable (Robinson et al. [41]) under fatigue loading can be derived from Equation 4.59 as (4.75) The static damage evolution for a given number of cycles,, is the given by (4.76) where is the time to cycles and is the time corresponding to cycle. Integrating the above equation, the evolution of the static damage with the number of cycles is given by (4.77) 77

105 CHAPTER 4 THEORETICAL TECHNIQUES Fatigue damage evolution The fatigue crack growth may be defined as the extent of growth of the crack per cycle, which is represented as. The crack growth rate curve is usually represented as versus. The Paris law is then typically used to describe the linear region of the growth curve and which relates the maximum energy release rate,, in a fatigue cycle to by the relation (4.78) where and are constants that depend on the material and the mode ratio. The constants and are given by the vertical intercept and slope of the linear region of the growth rate curve, respectively, when using logarithmic scales. The values of both constants are obtained by fitting the Paris law equation to the experimental results. The damage which develops under fatigue loading is the sum of the quasi-static and the fatigue damage (Muñoz et al. [64]). The evolution of the damage progresses with time, and the rate of change of damage with time can be expressed as (4.79) The static damage evolution can be calculated from Equation 4.75 for the cohesive zone element. The damage evolution with the number of cycles,, can be related to the crack growth rate curve in the fatigue loading as (4.80) where is the growth rate of the crack and is the damaged area. The growth rate of the curve depends on the material properties. The expression for 78

106 CHAPTER 4 THEORETICAL TECHNIQUES the can be derived from the strain energy released during the propagation of the crack. The ratio of the energy released,, can be expressed as (Figure 4.8) to the critical fracture energy, (4.81) where is the area of the cohesive zone element. τ τ 0 Θ (1-d)K G c δ o δ δ f δ Figure 4.8 Cohesive zone law and energy representation. The expression for the can be obtained from the above equation as (4.82) The increase in the growth of the damaged area with the number of cycles is the sum of the damaged area growth in the entire cohesive zone. Hence, the growth of the damaged area with the number of cycles can be represented as 79

107 CHAPTER 4 THEORETICAL TECHNIQUES (4.83) where is the total area of the cohesive zone and is the damaged area in a given cohesive zone element. The average damaged area in the cohesive zone element can be taken as. The number of cohesive zone elements in the cohesive zone can be obtained as (4.84) The expression for the growth rate can then be simplified as (4.85) The terms can next be rearranged to get the expression for as (4.86) The damage evolution for the cohesive zone element can now be obtained by substituting from Equation 4.82 and can be expressed as (4.87) The area of the cohesive zone (Turon et al. [43]) for mode I loading can be written as (4.88) The crack growth rate arising from fatigue damage is dependent on the energy released and can be expressed using the Paris law as 80

108 CHAPTER 4 THEORETICAL TECHNIQUES (4.89) where and are the constants to be determined experimentally, as described in Chapter 2. The damaged area in the cohesive zone can be expressed as (4.90) where is the width of the crack. The total change in the energy released in a fatigue cycle is the difference in the maximum and minimum energy release rate in a cycle and can be expressed as (4.91) where and are the maximum and minimum strain energy release rates during a fatigue loading cycle (Figure 4.9). 81

109 CHAPTER 4 THEORETICAL TECHNIQUES τ τ 0 G max δ o δ max δ f δ τ τ 0 G min δ o δ min δ f δ Figure 4.9 Representation of energy release under fatigue in a cohesive zone law. The maximum energy released is given by the expression (4.92) where and are the maximum displacement jump and damage in the whole cyclic loading history. The constitutive relationship derived is independent of the element formulation. It is important to note that the fatigue degradation of the cohesive zone law with time occurs as shown in Figure 4.10 where and are the values of the damage variables at time and. As noted above, the damage in the cohesive zone law occurs due to the combined static and fatigue loading. The actual degradation of the cohesive zone law with time due 82

110 CHAPTER 4 THEORETICAL TECHNIQUES to both the static and fatigue damage evolution can be illustrated as shown in Figure In these figures, and in the present modelling studies, the fatigue damage variable,, can be calculated from Equation 4.87 based on the number of cycles accumulated with time. The account for the total damage in Equation term can then be used to τ τ 0 K K(1-d t ) K(1-d t+1 ) δ t δ t+1 δ f δ Figure 4.10 Fatigue degradation of cohesive zone law with time. τ τ 0 1 G c 2 3 δ o δ 1 δ 2 δ f δ Figure 4.11 Resultant fatigue degradation of a cohesive zone law. Path 1-2 shows the fatigue damage evolution and path 1-3 shows the static and fatigue damage evolution (Robinson et al. [41]). 83

111 CHAPTER 4 THEORETICAL TECHNIQUES Damage Analysis When using the above modelling approach the analysis is not undertaken cycle by cycle due to the computational effort needed. The complete cycle of fatigue loading is undertaken based on a cycle jump strategy as described below The cycle jump strategy The cycle jump strategy in the fatigue analysis is employed to limit the number of individual analyses needed in modelling high cycle fatigue. The cycle jump strategy controls the accuracy of the damage variable for a given cycle jump. The accuracy of the degradation modelling is controlled by limiting the maximum change in the damage variable for a given jump of cycles (Van Paepegem and Degrieck [65] and Muñoz et al. [64]). The cycle jump principle is illustrated in Figure In the present work, the cycle jump strategy is adopted in the model of the fatigue life to limit the maximum time increment employed in the analysis. Figure 4.12 The cycle jump strategy applied to a cohesive zone law approach to modelling fatigue (Van Paepegem and Degrieck [65]) Displacement ratio and load ratio The displacement ratio,, is defined as the ratio of the minimum displacement,, to the maximum displacement,, applied in a cycle of fatigue loading (Lee [49]). Higher values of the displacement ratio cause a decrease in the fatigue damage due to less change occurring in the strain energy released. The value of may be defined by 84

112 CHAPTER 4 THEORETICAL TECHNIQUES (4.93) The load ratio,, is defined as the ratio of the minimum stress,, to the maximum stress,, applied in a cycle of fatigue loading (Turon et al. [43]). Higher values of the load ratio also cause a decrease in the fatigue damage due to a smaller change occurring in the strain energy released,. The load ratio for a fatigue cycle can be expressed as (4.94) Now, the cyclic loading is applied as a sinusoidal load with a given frequency and a displacement, or stress amplitude. Numerically in the modelling studies the load is applied as a constant displacement, or stress, which is equivalent to the maximum displacement, or stress, applied in the experiment. The minimum to the maximum displacement for the fatigue cycle depends on the displacement ratio,, and the minimum to the maximum stress for the fatigue cycle depends on the stress ratio,, relevant to the fatigue cycle. Hence, in the numerical model, the displacement, or stress, is applied in the first cycle of loading and the displacement, or stress, is kept constant for the remainder of the cycles being modelled (Robinson et al. [41]), as shown in Figure

113 CHAPTER 4 THEORETICAL TECHNIQUES Figure 4.13 Experimental and numerically applied displacement/stress in a displacement/stress controlled fatigue test. In the present study, displacement controlled fatigue tests were conducted on the CT bulk epoxy material and the DCB composite materials, whilst stress controlled fatigue tests were conducted on the composite material strip specimens. 4.4 Concluding Remarks The present Chapter has described the theoretical methods which have been developed to model and predict the quasi-static and fatigue behavior of the composite materials. The values needed for these modelling studies are ascertained from experiments conducted upon the bulk epoxy matrices, and validated using the DCB composite material test results, as will be described in Chapter 5. The model will then be used to predict the cyclic fatigue behavior and lifetime of the composite material strips. The model represents a novel method to predict such behaviour. It builds upon the research of Robinson et al. [41] & Turon et al. [43] but contains several novel features, including an important new user element subroutine. The following Chapter will describe the experimental results obtained in the present research and compare these results to the theoretical predictions which have been obtained using the models developed in the present Chapter. 86

114 CHAPTER 5 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES 5. EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES 5.1 Introduction In the present work, the material and cohesive zone law parameters of the bulk epoxy matrices and the composite materials are obtained from the various experimental tests. The tests were undertaken on the bulk epoxy matrices and the corresponding composites, as explained in Chapter 3. The quasi-static and the fatigue test experimental results are described in the sections of the present Chapter. The predicted theoretical results are then discussed and compared to the experimental results. The flowchart for the different analyses of the various specimens is given in Figure 5.1, which gives an overview of the experimental and modelling work undertaken in the present research. 87

115 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Test the SENB specimen of bulk epoxy under quasistatic load to obtain the load-displacement curve (Section 3.5.2) Model the quasi-static SENB specimen of bulk epoxy and obtain the cohesive zone law parameters of the bulk epoxy (Sections & ) Obtain the elastic properties of the bulk epoxy and the composite material (Sections & 5.2.2) Test the CT specimen of bulk epoxy under fatigue loading to obtain the crack growth rate curve and fatigue parameters (Section 3.5.3) Model the CT fatigue specimen of bulk epoxy and obtain the fatigue parameters of the bulk epoxy (Section ) Test the DCB composite specimen under quasi-static loading to obtain the load-displacement curve (Section 3.6.2) Model the quasistatic DCB composite specimen and obtain the parameters of the cohesive zone law (Sections & ) Test the composite strip under quasi-static load to obtain strength of the laminate (Section 3.6.4) Test the DCB composite specimen under fatigue load to obtain the crack growth rate curve and fatigue parameters (Section 3.6.3) Model the DCB fatigue composite specimen and obtain the growth rate curve to match the test (Section ) Test the composite strip under fatigue loading to obtain the crack density, stiffness and the number of cycles to failure for the laminate (Section 3.6.4) Model the quasi-static strip specimen and obtain the normalised stiffness as a function of crack density and number of cycles (Sections & ) Model the composite strip specimen under fatigue loading with the crack density observed from the experiments (Section ) Prediction of the fatigue life of strip specimen under fatigue loading (Section ) Prediction and initial validation of the fatigue life of the DCB composite specimen under fatigue loading (Section ) Figure 5.1 Overview schematic of the work plan. 88

116 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Notes to Figure 5.1: a) Sections & : The cohesive zone law parameters from the bulk epoxy matrix are used to model the development of transverse cracks in the composite, and hence are used in Section of the flow chart. b) The fatigue parameters determined in Section are used for the composite strip modelling in Section c) In Section , the DCB fatigue composite sample is modelled so as to validate the novel user element subroutine analysis. d) In Section , it should be noted, it is not necessary to undertake fatigue experiments to find the crack density, but one can use the saturated crack density as explained later in Figure Elastic Properties of Materials The elastic properties of the different materials are derived as detailed below in Sections 5.2.1, and The different directional elastic properties of the composite are derived from the basic equations for composite materials Elastic properties of the lamina The elastic properties of a unidirectional lamina are derived to determine the properties of the composite material. The fibre properties of the 0 o ply can then be calculated from the general expression for the lamina. The volume fractions of the composite material (Manjunatha et al. [40]) are also used to determine the properties of the lamina. The elastic properties of the lamina are calculated from the expressions for the unidirectional properties of the lamina. The lamina is assumed to be oriented in the 1-2 planes with the fibre direction in the 1 axis, as shown in Figure

117 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure 5.2 A section of a lamina with local coordinates. The different material properties of the lamina in different directions can be obtained from the expressions below. The equations are obtained from Khashaba [66]. (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) 90

118 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES (5.10) (5.11) (5.12) where,,,,, and are the elastic moduli of the fibre and matrix, volume fraction of plies, shear moduli of fibre and matrix, bulk moduli of composite, fibre and matrix, respectively. The factor is the shape factor, which has a value of two for a circular fibre (Khashaba [66]). The matrix properties of the different bulk epoxy matrices (Manjunatha et al. [67], Pegoretti et al. [68]) and the glass fibre (Pegoretti et al. [68]) used in the modelling studies are given in Table 5.1. The unidirectional elastic properties of the lamina obtained from the equations above are given in Table 5.2. Table 5.1 Elastic properties of bulk epoxies and glass fibre 91

119 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Table 5.2 Unidirectional elastic properties of the different composite lamina based on the various epoxy matrices Composite Properties Control Rubber Nano Hybrid E 1 (GPa) E 2 (GPa) E 3 (GPa) ν ν ν G 12 (GPa) G 13 (GPa) G 23 (GPa) Elastic properties of the composite The elastic properties for the composite are obtained from the unidirectional properties for the different coordinate system of the composite material. The elastic properties of the composite are homogenised for simplicity so as to readily model the composite material in Abaqus. The homogenised elastic properties of the composite material are used for the subsequent analyses. It should be noted that the material coordinate system in Abaqus, the local material coordinate system of the DCB (Figure 5.3) and the unidirectional lamina are different (Figure 5.4). Hence, the elastic properties of the composite are obtained by a cube (i.e. a 3D representation with solid elements) analysis of the composite in Abaqus using the composite layup of the specimen to determine the homogenised property of the composite material (Figure 5.5). The layup of the composite cube has the same stacking sequence as that of the composite specimen, so as to obtain the appropriate homogenised elastic properties. The lamina properties of the composite material are analysed for the different orientations using both classical laminate theory (CLT) and FEA (via Abaqus) to obtain the homogenised elastic properties. 92

120 CHAPTER 5 Abaqus coordinate y EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Local coordinate T z x DCB P P Figure 5.3 Equivalent Abaqus and local coordinate system for a DCB composite specimen. Abaqus coordinate Local coordinate y L 0 o fibre z x T T y T z x 90 o fibre L T y P ±45 o fibre z x P T Figure 5.4 Equivalent Abaqus and local coordinate system for different lamina of the composite strip. 93

121 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES y FT FL FF FBk FR FB z Figure 5.5 Cube with different faces (FF-Face front, FBk-face back, FR-face right, FL- face left, FT-face top, FB- face bottom). The control point (CP) boundary condition is at the origin of the coordinate system Elastic properties of DCB The elastic properties of the DCB composite in the Abaqus coordinate system are determined from the local coordinate system of the composite (Figure 5.3). The homogenised elastic properties for the DCB analysis are derived from a 3D solid cube (Figure 5.5) analysis in Abaqus, with different boundary conditions as described in Table 5.3. The layup of the DCB composite in the cube analysis is [(-45/45) s (90/0) s ] 2 [(0/90) s (45/-45) s ] 2. The unidirectional properties of lamina of Table 5.2 are used to model the composite cube. The homogenised elastic properties obtained from the cube analysis are given in Table

122 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Table 5.3 Derivation of homogenised elastic property of DCB from cube analysis Properties Loading Boundary condition Calcualtion E x Abaqus analysis of the cube with load in the x direcrtion FL(x=0), FR(x=1% strain) force in the x direction/area of cube E y Abaqus analysis of the cube with load in the y direction FB(y=0), FT(y=1% strain) force in the y direction/area of cube E z Same as the undirectional fibre property E ν xy Abaquq analysis of the cube with load in the x direcrtion FL(x=0), FR(x=1% strain) strain in y direction/strain in x direction ν xz Abaqus analysis of the cube with load in the x direcrtion FL(x=0), FR(x=1% strain) strain in z direction/strain in x direction ν yz Abaqus analysis of the cube with load in the y direcrtion FB(y=0), FT(y=1% strain) strain in z direction/strain in y direction G xy Abaqus analysis of the cube with shear force in the x plane in the y direction FR(y=1% strain), FL(y=-1% strain), FT(x=1% strain), FB(x=-1% strain) shear force in the y direction in the x plane/ (shear strain x area of cube) G xz Abaqus analysis of the cube with shear force in the x plane in the z direction FR(z=1% strain), FL(z=-1% strain), FF(x=1% strain), FBk(x=-1% strain) shear force in the z direction in the x plane/ (shear strain x area of cube) G yz Abaqus analysis of the cube with shear force in the y plane in the z direction FF(y=1% strain), FBk(y=-1% strain), FT(z=1% strain), FB(z=-1% strain) shear force in the z direction in the y plane/ (shear strain x area of cube) Table 5.4 Elastic properties of the arms the DCB composite specimens for the various epoxy matrices Abaqus properties Local properties Control Rubber Nano Hybrid E x E P E y E T E z E P ν xy ν PT ν xz ν PP ν yz ν TP G xy G PT G xz G PP G yz G TP

123 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Elastic properties of composite strip The elastic properties of each lamina of the strips are derived to model the transverse cracks within the lamina. The elastic properties of the lamina in the Abaqus coordinate system are determined from the local coordinate system of the unidirectional lamina (Figure 5.4). The equivalent properties of the lamina for different orientations (in Abaqus) are derived by considering a coordinate system of the strip in Abaqus. The elastic properties of the 0 o and 90 o degree layers in the composite material strip are obtained by considering the fibre direction as the longitudinal direction, L, and the other direction as the transverse direction, T, in the local coordinate system (Figure 5.4). The elastic properties of the ±45 o layer in the strip are obtained by considering the plane of the lamina as the plane direction, P, and the normal direction perpendicular to the plane as the transverse direction, T, in the local coordinate system (Figure 5.4). Table 5.5 shows the equivalent local elastic properties of the lamina of the strip in the Abaqus coordinate system for different orientations. The different lamina properties of each lamina are obtained from the cube analysis in Abaqus. The loads and boundary conditions are applied on the cube (Figure 5.5), on the different faces, to derive the elastic properties. The derivations of elastic properties of the lamina are given in Table 5.7 to Table 5.8, and the elastic properties obtained are given in Tables 5.9 to The calculated elastic modulus values were validated by calculating the modulus of the composite strip layup based on the unmodified (i.e. control) epoxy matrix. The agreement was very good between the calculated and measured values, being with in ±3% (Manjunatha et al. [40]). 96

124 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Table 5.5 Equivalent local elastic properties of the lamina of the strip in the Abaqus coordinate system Properties Ply orientation ( o ) 0 90 ±45 E x E T E T E T E y E L E T E P E z E T E L E P ν xy ν TL ν TT ν TP ν xz ν TT ν TL ν TP ν yz ν LT ν TL ν PP G xy G TL G TT G TP G xz G TT G TL G TP G yz G LT G TL G PP Global material properties Table 5.6 Derivation of the elastic properties of 0 o lamina Local material properties Description Loading condition Boundary Condition E x E T Unidirectional property, E E y E L Unidirectional property, E E z E T Unidirectional property, E ν xy ν TL Derived from unidirectional property (=ν 12 (E 2 /E 1 )) ν xz ν TT Abaqus analysis of the cube with load in the y direcrtion - - FT(y=1% strain) ν yz ν LT Unidirectional property, ν G xy G TL with shear force in the y Abaqus analysis of the cube plane in the x direction G xz G TT with shear force in the y Abaqus analysis of the cube plane in the z direction FT(x=1% strain), FB(x=-1% strain), FR(y=1% strain), FL(y=-1% strain) FT(z=1% strain), FB(z=-1% strain), FF(y=1% strain), FBk(y=-1% strain) G yz G LT Unidirectional property, G FB(y=0) CP(z=0) CP(x=0) 97

125 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Table 5.7 Derivation of the elastic properties of 90 o lamina Global material properties Local material properties Description Loading condition Boundary Condition E x E T Unidirectional property, E E y E T Unidirectional property, E E z E L Unidirectional property, E ν xy ν TT Abaqus analysis of the cube with load in the y direcrtion ν xz ν TL Derived from unidirectional property (=ν 12 (E 2 /E 1 )) ν yz ν TL Derived from unidirectional property (=ν 12 (E 2 /E 1 )) G xy G TT with shear force in the y Abaqus analysis of the cube plane in the z direction G xz G TL with shear force in the y Abaqus analysis of the cube plane in the x direction G yz G TL with shear force in the y Abaqus analysis of the cube plane in the x direction FT(y=1% strain) FB(y=0) FT(z=1% strain), FB(z=-1% strain), FF(y=1% strain), FBk(y=-1% strain) FT(x=1% strain), FB(x=-1% strain), FR(y=1% strain), FL(y=-1% strain) FT(x=1% strain), FB(x=-1% strain), FR(y=1% strain), FL(y=-1% strain) CP(x=0) CP(z=0) CP(z=0) Table 5.8 Derivation of the elastic properties of ±45 o lamina Global material properties Local material properties Description Loading condition Boundary Condition E x E T Unidirectional property, E E y E P Laminator analysis of (45) 8 layup sequence E z E P Laminator analysis of (45) 8 layup sequence ν xy ν TP Abaqus analysis of the cube with load in the z direcrtion FF(z=1% strain) FBk(z=0) ν xz ν TP Abaqus analysis of the cube with load in the z direcrtion ν yz ν PP Abaqus analysis of the cube with load in the x direcrtion G xy G TP with shear force in the z Abaqus analysis of the cube plane in the x direction G xz G TP with shear force in the z Abaqus analysis of the cube plane in the x direction G yz G PP with shear force in the x Abaqus analysis of the cube plane in the y direction FF(z=1% strain) FR(x=1% strain) FF(x=1% strain), FBk(x=-1% strain), FR(z=1% strain), FL(z=-1% strain) FF(x=1% strain), FBk(x=-1% strain), FR(z=1% strain), FL(z=-1% strain) FR(y=1% strain), FL(y=-1% strain), FT(x=1% strain), FB(x=-1% strain) FBk(z=0) FL(x=0) CP(y=0) CP(y=0) CP(z=0) 98

126 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Table 5.9 Elastic properties of 0 o lamina Table 5.10 Elastic properties of ±45 o lamina Table 5.11 Elastic properties of 90 o lamina 99

127 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Elastic properties of aluminium and steel The elastic properties used to model the aluminium-alloy blocks and steel pins in the experiments are given in Table 5.12 (Callister [69]). Table 5.12 Elastic properties of aluminium-alloy and steel Properties Elastic modulus (GPa) Poisson's ratio Aluminium Steel Criteria for Cohesive Zone Modelling A cohesive zone analysis is dependent on many factors for the accurate analysis of the failure and the important factors are discussed below Mesh sensitivity analysis The results of the cohesive zone element analysis maybe mesh dependent, and an accurate and reproducible analysis needs a refined mesh (Turon et al. [70]), as such a mesh gives an accurate representation of the stress field around the crack tip. Also the convergence of the cohesive zone element analysis depends on the mesh size, and a more refined mesh is required to obtain convergence. The convergence test is conducted by studying the convergence using different mesh sizes for the specimens, and the mesh which shows no deviation of the results with further refinement is then used for the subsequent modelling studies. The mesh sensitivity analysis is therefore done to understand the size of the mesh required for the analyses. In the present study, meshes with different degrees of refinement are used for the analysis of the DCB specimen and the predicted load-displacement curves are compared to the experimental curves (Figure 5.6). From the results shown in Figure 5.6, in all the present analyses a 0.1mm size mesh has been adopted for the cohesive zone elements. 100

128 Force (N) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES mm 1.25 mm 1 mm 0.5 mm 0.25 mm 0.2 mm 0.1 mm Displacement (mm) Figure 5.6 The modelled load versus displacement curves of the DCB composite specimen for different mesh sizes. The elastic properties of the unmodified (i.e. control) bulk epoxy matrix and the composite are used for the analysis (Table 5.13 and Table 5.4) Initial value of the cohesive zone law parameters Different methods are used to find the approximate value of the penalty stiffness of the cohesive zone. The initial value of the cohesive zone penalty stiffness is obtained from an expression which equates the transverse stress in the cohesive zone and in the adjacent material (Figure 5.7). The stress in the cohesive zone and in the adjacent material is made equal to get the approximate value of the cohesive zone penalty stiffness (Turon et al. [70]). Thus (5.13) 101

129 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES where is the through-thickness elastic modulus of the material and ) is the strain in the adjacent material. The cohesive zone penalty stiffness is the approximate penalty stiffness of the cohesive zone and is the relative displacement between the surfaces. t ct +ε tr t ct Continuum elements Continuum elements t ct Δ coh Continuum elements t ct t ct +ε tr t ct Continuum elements (a) (b) Figure 5.7 Cohesive zone behaviour in (a) undeformed and (b) deformed state. The total strain in the whole model is given by (5.14) Hence, from the above equation, the resultant stress is the same and is given by the expression (5.15) The resultant modulus of the composite can be written as 102

130 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES (5.16) It should be noted that the resultant properties of will be not be affected if is much greater than. Indeed, the value is chosen in such a way that the overall stiffness in the through thickness direction is not significantly affected. From Equation 5.16, the expression for the cohesive zone penalty stiffness is given by (5.17) where is a proportionality factor. The size of the cohesive zone should provide a reasonable penalty stiffness but be small enough to avoid numerical problems due to oscillations of the stresses at the crack tip. The value of the penalty stiffness should also be fixed in such a way that it is small as possible to avoid ill-conditioning of the stiffness matrix. For relatively high values of, the loss of stiffness is considerably less. The expression for takes into consideration the elastic properties of the adjacent material and hence it is relatively accurate The cohesive zone length The length of the cohesive zone is defined as the distance from the crack tip to the point where the critical cohesive zone stress is attained (Figure 5.8). Continuum elements l cz Cohesive elements Continuum elements Figure 5.8 Cohesive zone elements with cohesive zone length,. 103

131 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES A necessary condition for obtaining a good solution using a cohesive analysis is the size of the element (Harper and Hallett [62]). The cohesive zone length of the element should be less than the cohesive zone at the crack tip. The expression to derive the cohesive zone length,, is (5.18) where is the elastic modulus of the material, is the critical energy release rate, is the critical stress in the cohesive zone and the parameter depends on the cohesive zone model. The number of elements required for a given mesh length is (5.19) where is the size of the mesh. The accuracy of the results increases when value is the least. The minimum number of elements required for predicting the initiation and propagation is two as the crack tip stress variation is high due to the initiation of the crack. Hence, more than two elements were also used in the present research. 5.4 Quasi-Static Models The epoxy matrix SENB and the composite material DCB specimens were tested experimentally and then modelled in FEA Abaqus, using 2D plane strain elements, to obtain the cohesive zone law parameters. Structured meshing is adopted for the analysis and the size and type of the mesh employed were based upon the meshes defined from the mesh convergence study, see Section above. 104

132 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES The SENB test: Experimental and theoretical results A three point bending test is conducted using the SENB specimen. The load versus displacement at the middle point of the application of the load is obtained from the test. The test is conducted for the different bulk epoxy matrices and the value of the fracture energy,, of the bulk epoxy is directly obtained. The mean load versus displacement curves obtained from the test are shown in Figures 5.14 to The mean fracture energies of the different epoxies obtained from the test are given in the Table The SENB test is conducted to study the quasi-static behaviour of the bulk epoxy matrices. The cohesive zone properties of the bulk epoxy which are then derived are used to model the fatigue behaviour of the composite material strip. The transverse cracks in the strip develop due to matrix cracking, and hence the cohesive zone properties obtained from the SENB test are very appropriate to employ to model the transverse cracks in the composite strip. The values of the fracture energy,, obtained directly from the experiments are also used for modelling studies. Steel loading roller Steel roller support Figure 5.9 SENB specimen. 105

133 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES 60 mm 6 mm 12 mm y 6 mm 30 mm 1 mm 6 mm z x Figure 5.10 Dimensions of the SENB bulk epoxy matrix specimen. Figure 5.11 Loading and boundary condition applied on the SENB specimen the model SENB results: VCCT analysis A model of the SENB specimen is developed using a FEA approach in Abaqus. The dimensions of the specimen are given in Figure The specimen is modelled as two parts and a VCCT criterion is applied on the surface where crack growth occurs. The three point supports in the experiment are modelled as semi circles and a hard contact criterion is used in Abaqus between the surfaces to avoid any interpenetration. The friction between the surfaces is assumed to be zero. The elastic properties of the bulk epoxy (Table 5.1) and steel rollers (Table 5.12) are used for the modelling studies. The load is applied as a displacement on the top middle roller and the load versus displacement curve for the specimen is obtained from the reaction forces and displacement at the top support. The boundary conditions applied on the model are shown in 106

134 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure The analysis is conducted for the different bulk epoxy matrices and the load versus displacement curve from the analysis is compared with the experiments, see Figures 5.14 to SENB results: Cohesive contact analysis The SENB specimen of the bulk epoxy is modelled using FEA with cohesive contact, via Abaqus. The dimensions of the specimen are given in Figure The SENB specimen is modelled in two parts and the cohesive behaviour contact option in Abaqus is adopted between the surfaces. The elastic properties of the bulk epoxy (Table 5.1) and steel rollers (Table 5.12) are used for modelling the specimen. The three point bending supports are modelled as semi circles in Abaqus and a hard contact criterion in Abaqus is adopted between the surfaces to avoid any interpenetration. The friction between the surfaces is assumed to be zero. The load in the model is applied as a displacement on the middle roller (Figure 5.11) and the load versus displacement curve for the specimen is obtained from the reaction forces at the support. The cohesive contact parameters of the bulk epoxy are obtained using the same procedure as in the cohesive contact analysis of the DCB model, see later in Section The stress field in a cohesive contact analysis in Abaqus is shown with the stress contours in Figure The load versus displacement curves of the modelling for different materials are shown in Figures 5.14 to

135 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure 5.12 The stress field around the crack tip in a SENB FEA model of the bulk epoxy matrix with cohesive contact in Abaqus SENB results: Cohesive zone element analysis The SENB specimens of the bulk epoxy matrices are modelled in Abaqus with the dimensions of the specimen as given in Figure The SENB specimen is modelled as one part, and the part is partitioned as consisting of continuum elements and cohesive zone elements. The elastic properties of the bulk epoxy matrix (Table 5.1) and steel rollers (Table 5.12) are used for modelling the specimen. The thickness of the cohesive zone element was adopted as mm, as brittle fracture is observed in the experiments. The three point bending supports are modelled as semi circles in the Abaqus programme and hard contact criterion in Abaqus is adopted between the surfaces to avoid any interpenetration. The friction between the surfaces is assumed to be zero. The load is applied as a displacement on the middle 108

136 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES support and the load versus displacement curve for the specimen is obtained from the reaction force and displacement at the support. The parameters of the cohesive contact are obtained by the procedure described in the DCB cohesive contact analysis, see later in Section The analysis is run for the different bulk epoxy matrices using Abaqus, to obtain the cohesive contact parameters needed to accurately model the SENB test. The elastic properties used for modelling the SENB specimen are given in Table 5.1. The values of the elastic properties and fracture energies,, of the different bulk epoxies are obtained from the experiments (Table 5.14). The values of the penalty stiffness and the critical stress for the different epoxies are obtained from matching the experimental results with the modelling results, and are given in Table The load versus displacement curves obtained from the modelling are compared with the experiments, and are given in Figures 5.14 to 5.17 for the different materials. The fracture energies of the bulk epoxies are obtained directly from the experiments (Section 3.5.2). The stress contours of a SENB cohesive zone element analysis in Abaqus is shown in Figure

137 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure 5.13 The stress field around the crack tip in a SENB model with cohesive zone elements as modelled in FEA Abaqus. Table 5.14 Quasi-static cohesive contact/element parameters for the bulk epoxy matrices 110

138 Load (N) Load (N) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment VCCT Cohesive contact Cohesive element Displacement (mm) Figure 5.14 Comparison of load-displacement curve for the SENB specimen based upon the unmodified (i.e. control) epoxy matrix Experiment VCCT Cohesive contact Cohesive element Displacement (mm) Figure 5.15 Comparison of load-displacement curve for the SENB specimen based upon the micro-rubber modified epoxy matrix. 111

139 Force (N) Load (N) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experimental VCCT Cohesive contact Cohesive element Displacement (mm) Figure 5.16 Comparison of load-displacement curve for the SENB specimen based upon the nano-silica modified epoxy matrix Experiment VCCT Cohesive contact Cohesive element Displacement (mm) Figure 5.17 Comparison of load-displacement curve for the SENB epoxy specimen based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix. 112

140 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES The DCB test: Experimental and theoretical analysis A flowchart for the DCB testing and modelling of the composite material specimens is given in Figure 5.18, which schematically illustrates an overview of the experimental and modelling work. The DCB specimen is tested under quasi-static conditions to obtain the load versus displacement curve of the composite material. The fracture energy of the composite material is also obtained from the experiments, as described in Section The mean load versus displacement traces of the DCB specimens for the various composite materials, based on the different epoxy matrices, are shown in Figures 5.24 to A FEA 2D model of the DCB (Figure 5.19) specimen is modelled in Abaqus to validate the models and subroutine which will be used to predict the fatigue life of the composite material strips, see Section below. The laminate sequence of the composite DCB test specimen is [(-45/45) s (90/0) s ] 2 [(0/90) s (45/- 45) s ] 2. The initial crack in the DCB specimen is located between the 0 o and 90 o lamina interface at the mid-plane of the laminate sequence. The dimensions of the DCB specimen are shown in Figure The size of the aluminium alloy end-blocks used is 20x12x20mm 3 and the initial length of the crack is 60 mm. The elastic properties in Table 5.4 and Table 5.12 are used in the modelling studies. The load versus displacement curves obtained from the analyses are compared with the experiments. The model use VCCT, cohesive contact and cohesive element methods to obtain the parameters of the lamina interface for the cohesive zone law. 113

141 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Quasi-static modelling of the DCB specimen using the VCCT method Obtain and for the cohesive zone law Quasi-static modelling of the DCB specimen using cohesive contact and cohesive element analyses of the lamina interface of interest from the DCB quasi-static test Subroutine uses the Paris law to degrade the cohesive law, see Figure 5.1 Fatigue modelling of the DCB specimen to match the growth rate curve using the subroutine Fatigue parameters, and of the lamina interface from DCB fatigue tests Prediction of the fatigue life of DCB specimen under a displacement controlled fatigue test Hence validation of proposed modelling method Figure 5.18 Flow chart of quasi-static and fatigue analyses of the composite material DCB specimens 114

142 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Aluminium block Composite Initial crack Figure 5.19 DCB specimen 20 mm 12 mm 6 mm Initial crack Fracture plane 5.4 mm 120 mm y 60 mm x z Figure 5.20 Dimensions of the DCB composite material specimen DCB results: VCCT analysis The fracture energy of composite material, which fractured at the 0 o /90 o lamina interface, obtained from the experiment is used to model the DCB using the VCCT technique. The two arms of the DCB are modelled as separate parts and the surfaces are connected using the VCCT criterion. The elastic properties of the composite and aluminium blocks are taken from Table 5.4 and Table 5.12, respectively. The loads are applied as a displacement in the opposite direction at the centre of the two aluminium blocks, as in Figure The boundary conditions are applied at the top and bottom arms (Figure 5.21) at the centre of the block, as the hole in the aluminium block is assumed to be filled. The result 115

143 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES obtained from the analysis is half the displacement in the y direction of the top arm. The modelling results are compared with the experimental results to validate the stiffness and failure displacement of the DCB experiment. The load versus displacement curves of the modelling for the different composite materials based upon the different epoxy matrices are shown in Figures 5.24 to y z x Figure 5.21 Loading and boundary condition applied on the model DCB results: Cohesive contact analysis The analysis of the quasi-static test of the DCB model is undertaken to obtain the cohesive zone law parameters of the lamina interface of interest in the composite material. The model used for the analysis is the same as that of the VCCT analysis but with the surface contact option, as embedded in the Abaqus software. The elastic properties of the composite and aluminium blocks in the model are taken from Table 5.4 and Table 5.12, respectively. The loads are applied as a displacement at the centre of aluminium blocks and the arms are given an equal displacement in the opposite directions, as shown in Figure The boundary conditions are applied at the top and bottom arms at the centre of the block (Figure 5.21). The cohesive zone law parameters of the lamina interface are obtained by a trial and error method by fitting to the experimental load versus displacement curve to determine these parameters. Initially an approximate value of the cohesive zone penalty stiffness is assumed in the analysis, with a critical stress equal to the yield stress of the bulk epoxy matrix. The load versus displacement 116

144 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES curve predicted by the analysis is then compared with the experiments to match the slope of the experimental curve. In particular, the penalty stiffness of the cohesive contact theoretical analysis is varied to match the slope of the experimental curve. Once good fit of the penalty stiffness of the cohesive contact behaviour is obtained, the critical stress of the cohesive contact analysis is next varied to match the failure displacement in the experiment. The cohesive contact parameters are then obtained for the given DCB specimen. The stress contours obtained at the crack tip of the model are shown in Figure The cohesive zone law parameters obtained from the analysis are given in Table The load versus displacement curves of the modelling for different materials are shown in Figures 5.24 to Figure 5.22 The stress field around the crack tip in a DCB model with cohesive contact in Abaqus DCB results: Cohesive zone element analysis The quasi-static analysis of the DCB is undertaken in the FEA Abaqus software to obtain the cohesive zone law parameters of the composite materials. The 117

145 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES cohesive zone parameters of the lamina interface of interest are obtained by a trial and error method, as described above for the cohesive contact analysis approach. The elastic properties of the composite and aluminium blocks are taken from Table 5.4 and Table 5.12, respectively. The loads are applied as displacement at the centre of the aluminium blocks, and the arms are given an equal displacement in the opposite directions as in Figure The boundary conditions are applied at the top and bottom arms at the centre of the block (Figure 5.21). The model for the DCB is developed with the same arm thickness as that in the experiments. The thickness of the matrix in-between the lamina is of order 10-2 mm (Masania [47]) and hence the thickness of the cohesive layer is adopted as 0.01 mm. The penalty stiffness and critical stress for the cohesive zone element are obtained, described above for the cohesive contact analysis approach. The DCB model is developed as a single part and the part is partitioned as cohesive zone elements and continuum elements. The load is applied in the model as a displacement and the boundary condition adopted is the same as that of the cohesive contact analysis. The boundary condition for the DCB model is applied at the arms and the load is applied as a displacement in the opposite directions. The penalty stiffness and critical stress of the lamina interface obtained from the analysis are given in Table The load versus displacement curves from the modelling for the different composite materials based upon the different epoxy matrices are shown in Figures 5.24 to In these figures the VCCT method gives a relatively poor fit to the experimental results. This is suggested to arise from the fibre bridging which is seen to occur behind the crack tip during the testing of the DCB composite specimens. Unlike the cohesive contact and cohesive element methods, the VCCT method does not take such fibre bridging into account. The contours of the stress variation in a DCB cohesive zone element analysis is shown in Figure

146 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure 5.23 The stress field around the crack tip in a DCB model with cohesive zone elements in Abaqus Table 5.15 Quasi-static cohesive contact/element parameters of the DCB lamina interface 119

147 Load (N) Load (N) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experimental VCCT Cohesive contact Cohesive element Displacement (mm) Figure 5.24 Comparison of load-displacement curve for the DCB composite material based upon the unmodified (i.e. control) epoxy matrix Experiment VCCT Cohesive contact Cohesive element Displacement (mm) Figure 5.25 Comparison of load-displacement curve for the DCB composite based upon the micro-rubber modified epoxy matrix 120

148 Load (N) Load (N) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment VCCT Cohesive contact Cohesive element Displacement (mm) Figure 5.26 Comparison of load-displacement curve for the DCB composite based upon the nano-silica modified epoxy matrix Experiment VCCT Cohesive contact Cohesive element Displacement (mm) Figure 5.27 Comparison of load-displacement curve for the DCB composite based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix 121

149 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES The composite material strip test: Experimental and theoretical results The flowchart for the composite strip testing and analysis of the specimen are given in Figure 5.28, which gives an overview of the experimental and modelling work. 122

150 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Obtain of the cohesive zone law of the epoxy matrix Obtain of the cohesive zone law of the epoxy matrix Subroutine (see Figure 4.7) uses the Paris law to degrade the cohesive zone law Quasi-static modelling of the SENB bulk epoxy matrix specimen using the VCCT method Quasi-static modelling of the SENB bulk epoxy matrix specimen using the cohesive contact and cohesive element analysis Fatigue modelling of the CT bulk epoxy matrix specimen to match the growth rate curve using the subroutine (see Figure 4.7) from the SENB bulk epoxy matrix quasi-static test Fatigue parameters, and of the epoxy from the CT bulk epoxy matrix fatigue test Subroutine (see Figure 4.7) validated with growth curve matching with the CT fatigue test Fatigue modelling of the composite with the saturation/experimental crack density using the subroutine (see Figure 4.7) Prediction of the fatigue life of composite for different applied stresses. (Life of the composite is assumed to be equivalent to the number of cycles required for the maximum stiffness reduction for crack density adopted earlier.) Determine the maximum crack density for maximum stiffness reduction in the fatigue test of the strip OR The saturation crack density; obtained by predicting the crack density required for the maximum stiffness reduction using trial and error method of static modelling. (Can be validated from the experiments.) Figure 5.28 Flow chart of the life prediction modelling of the composite strip under cyclic fatigue loading 123

151 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES The quasi-static test of the composite strip is undertaken to determine the quasi-static strength of the composite material strip. The strip is loaded quasistatically and the load versus displacement graph is obtained. The ultimate tensile strength of the composite strip may be deduced, and the change in stiffness with loading is also obtained from the test. The results of the test are shown in Table The research on the composite strip has been undertaken together with Dr. Manjunatha [40]. Table 5.16 Elastic properties of the composite material strip The quasi-static analysis of the strip is performed to study the failure behaviour of the GFRP composite. The composite strips are modelled to study the influence of transverse cracks in the stiffness reduction of the composite. The dimensions of the strip are as shown in Figure The symmetric model of the strip is analysed using the symmetric boundary condition for the length of strip (Figure 5.30). Composite End tabs 2.7 mm 50 mm 150 mm Figure 5.29 Composite material strip with dimensions The strip is modelled with cracks having a crack density as experimentally measured in the experiments (Figure 5.31). The transverse crack density of the strip is obtained from the fatigue experiments for different cycles of loading (Figure 5.33). The elastic properties in Tables 5.9, 5.10 and 5.11 are used to model the different laminae of the strip. The normalised stiffness of the composite material strip as a function of the crack density is also obtained from the experiments (Figures 5.35 to 5.38). 124

152 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Load s s s s (b) (c) y Load z x (a) Figure 5.30 Section of the strip with transverse cracks (a) strip under loading (b) cross section of the strip with transverse cracks (c) symmetric cross-section of the strip with transverse cracks 125

153 l mm CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES 1.35 mm Figure 5.31 Dimension of a section of the modelled strip. The length, l, of the strip model depends on the crack density The transverse cracking within the laminae is modelled using a relation between the cohesive traction and the relative displacements. The effect of transverse crack density on the stiffness degradation is studied and the model simulates the critical stress above which the crack starts propagating, and also gives the strain energy release rate in the system. The damage model is able to simulate crack onset and propagation. The transverse cracks in the strip are modelled as continuum elements (with very low elasticity modulus, i.e. =1x10-9 N/mm 2 and a Poisson s ratio, ν=0.01) of thickness 0.001mm (i.e. the thickness of the cohesive zone element in the SENB and CT models). The load in the strip is applied as a displacement at the top of the model and the boundary conditions adopted for the model are shown in Figure

154 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure 5.32 Boundary conditions applied on the symmetric cross-section of the composite material strip with transverse cracks The variation of the crack density of a laminae consisting ±45 o fibres for different fatigue cycles observed in the experiments is shown in Figure The 90 o crack density is assumed to be constant at a value of 0.64/mm (Tong et al. [37]) for all the cycles of fatigue. As may be seen from the results shown in Figure 5.33, the composite strips based upon the control matrix exhibit the highest crack densities whilst those based upon the hybrid matrix exhibit the lowest values of crack density. From the previous results in Chapter 5, this is as would be expected. 127

155 Crack density (mm -1 ) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Control Nano Rubber Hybrid Number of cycles Figure 5.33 Variation of ±45 o crack density with number of cycles in composite material strips based upon unmodified (i.e. control) and modified epoxy matrices Normalised stiffness with crack density In the quasi-static analysis of the strip, the normalised stiffness of the composite is compared with the experimental values of crack density obtained for different cycles of fatigue (Figure 5.33). The normalised stiffness reduction is also obtained for the corresponding crack density for different cycles of fatigue cycles in the experiment (Figures 5.35 to 5.38). Different models with different crack densities are then run and the normalised stiffness of the strip is obtained. The models are run with continuous increments of crack density to obtain a smooth reduction of the stiffness with crack density in the composite (Figures 5.35 to 5.38). The normalised stiffness as a function of crack density obtained from the model is compared with the normalised stiffness with an increase in crack density (for different number of cycles in the 128

156 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES fatigue experiment.) The comparison of normalised stiffness with crack density from the modelling and experimental studies is shown in Figures 5.35 to Comparisons of the normalised stiffness of the composite with crack density from the modelling and experimental studies are also shown in Figure 5.39 to 5.40, where further comparison are made between the different composite materials based on the different epoxy matrices. The strip analysis in Abaqus with transverse cracks is shown in Figure Figure 5.34 Composite material strip with transverse cracks in the Abaqus FEA method The agreement between the modelling results and the experimental results is very good for all the different composite materials, with the poorest agreement being seen for the composite materials based upon the hybrid epoxy matrix at the relatively very high crack densities. 129

157 Normalised stiffness Normalised stiffness CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment Modelling Crack density (mm -1 ) Figure 5.35 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the unmodified (i.e. control) epoxy matrix Experiment Modelling Crack density (mm -1 ) Figure 5.36 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the microrubber modified epoxy matrix. 130

158 Normalised stiffness Normalised stiffness CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment Modelling Crack density (mm -1 ) Figure 5.37 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the nanosilica modified epoxy matrix Experiment Modelling Crack density (mm -1 ) Figure 5.38 Comparison of the normalised stiffness versus the crack density for the composite strip test. For the composite based upon the nanosilica and micro-rubber (i.e. hybrid) modified epoxy matrix. 131

159 Normalised stiffness Normalised stiffness CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Control Modelling Control Experiment Nano Modelling Nano Experiment Crack density (mm -1 ) Figure 5.39 Comparison of the normalised stiffness versus the crack density for the composite strip, based upon the unmodified (i.e. control) and nano-silica modified epoxy matrices Rubber Modelling Rubber Experiment Hybrid Modelling Hybrid Experiment Crack density (mm -1 ) Figure 5.40 Comparison of the normalised stiffness versus the crack density for the composite strip, based upon the micro-rubber and with both nano-silica and micro-rubber modified epoxy matrices 132

160 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Normalised stiffness with number of fatigue cycles The normalised stiffness with the number of cycles may also be plotted for the different crack densities observed in the experiments. The analysis is performed with a crack density for a given number of cycles, and the normalised stiffness is then obtained for the strip. The normalised stiffness with the number of cycles of fatigue is obtained from the experiments, which are modelled using the same crack density for different cycles of loading (Figure 5.33) to obtain the normalised stiffness. The 90 o crack density is again assumed to be constant at a value of 0.64/mm (Tong et al. [37]). The model is analysed with the crack density as measured in the experiments and the normalised stiffness is plotted versus the number of cycles. The normalised stiffness as a function of the number of cycles, (with crack density also being measured) is obtained from the experiments. Different models with the different crack densities (for different cycles of fatigue loading) obtained are run and the normalised stiffness is derived from the analysis. The values of normalised stiffness as a function of the number of cycles of loading for the different composite materials are shown in Figures 5.41 to Again the agreement between the results from the modelling studies and the experimental results is very good. Although, the composite strip specimens based upon the hybrid epoxy matrix do show a somewhat larger discrepancy between the experimental results and the modelling studies than for the composite strips based upon the other matrices. This may arise due to the presence of the fibre changing somewhat the morphology, and hence the mechanical properties, of the hybrid epoxy matrix in the composite strip specimens compared to the bulk hybrid epoxy polymer. However, even for Figure 5.44, the agreement between the experimental and theoretical modelling studies is still relatively good. This is very encouraging for the work on modelling the fatigue life of the composite strips, which is discussed later. 133

161 Normalised stiffness Normalised stiffness CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment Modelling Number of cycles Figure 5.41 Normalised stiffness versus the number of cycles for the composite strip based upon the unmodified (i.e. control) epoxy matrix Experiment Modelling Crack density (mm -1 ) Figure 5.42 Normalised stiffness versus the number of cycles for the composite strip based upon the micro-rubber modified epoxy matrix 134

162 Normalised stiffness Normalised stiffness CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment Modelling Number of cycles Figure 5.43 Normalised stiffness versus the number of cycles for the composite strip based upon the nano-silica modified epoxy matrix Experiment Modelling Number of cycles Figure 5.44 Normalised stiffness versus the number of cycles for the composite strip based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix 135

163 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Quasi-static strength of the composite strip The quasi-static strength of the composite is obtained from modelling composite with maximum crack density (Figure 5.33) observed in the experiments. The 90 o crack density is assumed to be constant at 0.64/mm (Tong et al. [37]). The cracks are modelled as cohesive zone elements with the cohesive zone parameters of the epoxy material (Table 5.14). A quasi-static model (Figure 5.31) is run and the global stiffness reduction with increase in displacement is noted. The cracks in the strip are modelled using cohesive zone elements and the reduction of stiffness with displacement is plotted with percentage strain applied in the model. The damage in cohesive zone element due to quasi-static loading of composite causes reduction of global stiffness of the model. The global stiffness of model attains a plateau with increase in strain and the final failure of the composite is uncertain as the model fail to predict the abrupt failure of glass fibre in 0 o lamina. The Figures 5.45 and 5.46 shows the global stress with increase in strain and the corresponding reduction of global stiffness in the composite model. If the model was able to predict the failure strain in the global stiffness versus strain curve (Figure 5.46), the corresponding failure global stress for the composite can be obtained for the failure strain from the Figure

164 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Global stiffness (N/mm) Global stress (N/mm 2 ) Control Rubber Nano Hybrid % strain Figure 5.45 Comparison of the global stress with percentage strain in the composite strip 4.4E E E E E+05 Control Rubber Nano Hybrid 3.4E E E % strain Figure 5.46 Comparison of global stiffness reduction with the percentage strain of composite strip 137

165 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES 5.5 Toughening Mechanisms The fracture energy,, of the unmodified (i.e. control) and modified bulk epoxy matrices are found to be different, with the modified epoxies having significantly higher fracture energies, see Table Indeed, the hybrid bulk epoxy matrix has the highest fracture energy of all the epoxy polymers. The increase in the fracture energy in the epoxy, due to the addition of the nano-silica and microrubber particles, occurs from to the energy-dissipating toughening mechanisms developing in the structure of the epoxy. Now, epoxies are highly cross-linked thermosetting polymers. Hence, they have a poor resistance to initiation and growth of cracks. This is due to the fact that when the epoxy is polymerised it is amorphous and highly-crosslinked in structure (Kinloch [71]) resulting in a relatively high modulus and strength. This structure of the epoxy matrix also leads to the development of the brittle nature of the epoxy. Hence, the addition of micro-rubber particles dispersed in the bulk epoxy helps to increase the fracture energy by the dissipation of energy. The microstructure, and hence the mechanical properties of the micro-rubber modified epoxy, depends on the dispersion of the rubber particles and on the adhesion of the particle to the epoxy matrix. Kinloch et al. [72] and Kinloch et al. [73] observed that the plastic deformation of the modified epoxy matrix causes energy dissipation, and hence an increase in the fracture energy. The increase in the fracture energy is attributed to the interaction of the stress field with the micro-rubber particles around the crack tip. The micro-rubber particles have a lower shear modulus, but a comparable bulk modulus, than the epoxy matrix. This leads to stress concentrations and volume constraint in the matrix around the rubber particles. Hence, the rubber particles in the epoxy matrix act as stress concentrators, as well as transferring the load. This leads to cavitation of the micro-rubber particles in the epoxy matrix due to the triaxial stress state ahead of the crack tip. (The residual stress after the curing cycle of the epoxy also causes the development of such voids in the rubber particle.) The void in the micro-rubber particles enables the development of extensive local plastic void growth in the epoxy at the crack tip, and reduces the triaxiality which leads to even more plastic deformation occurring at the crack tip in the epoxy polymer. Hence, the 138

166 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES presence of rubber particles in the epoxy increases the local plastic deformation and so significantly increases the value of. The fracture energy of the modified epoxy with nano-silica particles is found to be significantly higher than the umodified epoxy matrix. As for micro-rubber particle modified epoxies, the toughening of the epoxy with nano-silica particles is dependent on the values of glass transition temperature, molecular weight between cross-links of the epoxy polymer and the adhesion at the interface of nano-silica particles with the epoxy (Hsieh et al. [74] and Hsieh et al. [75]). The toughening mechanisms are somewhat similar to that described above. Namely, localised shear bands are initiated by the stress concentrations around the nano-silica particles and the debonding of the nano-silica particles leads to plastic void growth. These toughening mechanisms lead to the increase in the fracture energy of the modified epoxy. 5.6 Fatigue Models The fatigue analysis is undertaken in Abaqus using the subroutine, see Figure 4.7. The degradation of the cohesive zone element is based on the Paris law, and the subroutine degrades the penalty stiffness of the cohesive zone element with time, as described in Section User element subroutine The fatigue analysis of the composite is performed using the user element subroutine (Hibbitt [8]) and the subroutine is written in FORTRAN (see the Appendix). The subroutine uses the theoretical formulations to calculate the damage in the material due to fatigue cycling. The subroutine is firstly validated against the quasi-static and fatigue behaviour using a unit cell element with given material properties. The quasi-static analysis of the unit cell of the cohesive zone element using the subroutine is matched with the standard cohesive zone element in Abaqus. The testing of the user cohesive zone element is done, and the results plotted to validate the element, as described below. 139

167 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Validation The unit cell cohesive zone element analysis is done using a standard continuum element and hence the user cohesive zone element is tested. The cohesive zone element to be tested is placed between the continuum elements as shown in Figure The 2D cohesive zone element has 0.001mm thickness (i.e. same as that of the SENB and CT quasi-static cohesive zone element analyses used for the bulk epoxy matrix studies) and the breadth and through-thickness of the cohesive zone element is taken as unity. The unit cell has a displacement loading applied at the nodes and the load versus displacement curve is compared with the standard Abaqus cohesive zone element with the same geometry. continuum element cohesive elements y node z x Figure 5.47 A single cohesive zone element for testing The quasi-static test of the cohesive zone element is undertaken to validate the user element (Figure 5.48). The element is tested in mode I and mode II using the cohesive zone parameters of the SENB specimen based on the unmodified (i.e. control) epoxy matrix. The values of the critical displacement, critical stress and failure displacement are checked to validate the cohesive zone element. The material properties adopted for the continuum element (Table 5.1) and 140

168 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES cohesive zone element (Table 5.14) are, of course, that of the unmodified (i.e. control) bulk epoxy. As noted above, the subroutine cohesive zone element is subjected to mode I, mode II, as well as mixed-mode loading, and is compared with the standard Abaqus cohesive zone element analysis. The behaviour of the subroutine cohesive zone element and standard cohesive zone element are shown in Figures 5.49 to The analyses show that the subroutine cohesive zone element has the same behaviour as that of standard cohesive zone element in Abaqus, and hence the new user cohesive zone element proposed in the current research is validated. Figure 5.48 Cohesive zone element testing in (a) mode I, (b) mode II and (c) mixed-mode The critical displacement for cohesive zone element is given by (5.20) The failure displacement is given by the relation (5.21) The critical stress at each node of a unit length cohesive zone element is. 141

169 Load (N) Load (mm) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Abaqus element User element Displacement (mm) Figure 5.49 Cohesive zone element in mode I. The cohesive zone law parameters used for the element is for the unmodified (i.e. control) bulk epoxy matrix ( =3900N/mm 2, =10.9 N/mm 2 and =75.8J/m 2 ) Abaqus element User element Displacement (mm) Figure 5.50 Cohesive zone element in mode II. The cohesive zone law parameters used for the element is for the unmodified (i.e. control) bulk epoxy matrix ( =3900N/mm 2, =10.9 N/mm 2 and =75.8J/m 2 ) 142

170 Load (N) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Abaqus element User element Displacement (mm) Figure 5.51 Cohesive zone element in mixed-mode ( =0.5). The cohesive zone law parameters used for the element is for the unmodified (i.e. control) bulk epoxy matrix ( =3900N/mm 2, =10.9 N/mm 2 and =75.8J/m 2 for both modes) Fatigue analysis using the user element subroutine The user element subroutine may now be employed with confidence for the fatigue analysis of the composite material strips. The subroutine is used to modify the penalty stiffness of the cohesive zone law element with number of cycles of fatigue loading. The damage variable in the cohesive zone law is calculated in the subroutine for each cycle of analysis and the penalty stiffness of the cohesive zone element is varied according to the change in the damage variable. The initial quasi-static penalty stiffness of the cohesive zone element is varied according to the cohesive zone law and the new penalty stiffness is calculated from the formulation. The equivalent penalty stiffness is calculated in the subroutine to account for both static and fatigue damage. The static damage in the cohesive zone element is given by the Equation 4.77 and the fatigue damage rate is given by the Equation 4.87 and the resultant damage is applied 143

171 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES on the cohesive zone by reducing the penalty stiffness of the cohesive zone element in the stiffness matrix. The total damage and equivalent penalty stiffness is calculated in each element, and hence the continuous degradation of the cohesive zone law for the element takes place. The strain in the cohesive zone element and the number of cycles are the two parameters which determine the penalty stiffness of the cohesive zone element after an analysis step. For each step of the analysis, the penalty stiffness of the individual cohesive zone element is calculated based on the strain of the element and also accounting for the fatigue damage from the fatigue cycles. The subroutine is called up for each analysis step, and for each cohesive zone element. The strain in mode I and II are requested in the subroutine to calculate the damage in the element. Using the equation for the damage variable, the penalty stiffness is updated for each element to get the stiffness matrix of the element during each step of the analysis to therefore have a continuous degradation of the cohesive zone element in the fatigue analysis The CT test and user element analysis The fatigue test is conducted on the CT specimen of the bulk epoxy matrix. The load is applied to the specimen as a sinusoidal constant-amplitude displacement. The frequency of the periodic load and the displacement ratio (ratio of minimum displacement to maximum displacement in fatigue cycle) is kept constant. The threshold fracture energy,, of the bulk epoxy and the growth rate curve of the epoxy are obtained for the various epoxy matrices from the test data. The growth rate curves obtained from the test are shown in Figures 5.56 to The threshold fracture energy for the bulk epoxies are obtained from the growth rate curve for the specimen. The CT specimen (Figure 5.52) is analysed in the Abaqus programme to model the fatigue behaviour. The dimensions of the specimen are as shown in Figure The quasi-static cohesive zone element parameters (Table 5.14) of the SENB test are used to model the CT specimen. The elastic properties used for the specimen are same as that of the SENB specimen (Table 5.1 and Table 5.12). The thickness of the cohesive zone element is kept same as that of the 144

172 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES quasi-static SENB model. The fatigue growth in the CT specimen is modelled using the subroutine and the growth rate curve of the specimen is compared with the experimental results. Steel pin Bulk epoxy Figure 5.52 Compact tension specimen 145

173 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES 10 mm 20 mm 50 mm 23 mm 12 mm 48 mm Ф 8 mm y z x Figure 5.53 Dimension of the CT specimen The load in the fatigue model is applied as a constant displacement, as described in Section , and the load is kept constant throughout whole cycle of fatigue analysis (Figure 5.54). The growth rate curve of the experiment is matched with that from the modelling studies obtained from cyclic fatigue studies. 146

174 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure 5.54 Boundary condition applied on the CT specimen The fatigue parameters of the CT specimens of the bulk epoxy matrices are obtained from the Paris law fit for the different materials and are given in Table The fatigue parameters of the specimen are used in the subroutine analysis to determine the growth rate curve of the specimen. The growth rate curve obtained from the modelling and experimental studies for the different epoxies are shown in Figures 5.56 to It should be noted that there is some scatter in the modelling data shown in Figures 5.56 to This occurs because the calculation of the crack growth rate is not exact between a period of cycles. This is due to the inability of the numerical method to find the exact growth rate between adjacent two analysis steps during modelling of the fracture process. The stress contours at the crack tip in the CT model is shown in Figure

175 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure 5.55 Stress field around the crack tip in a CT model with cohesive zone elements in Abaqus Table 5.17 Fatigue parameters of the bulk epoxy matrix CT specimens from Paris law fit Matrix fatigue properties c m G th (J/m 2 ) Control Nano Rubber Hybrid

176 log (da/dn) log (da/dn) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment Modelling Paris law fit log (G max /G c ) Figure 5.56 Growth rate curve for the CT specimen for the bulk unmodified (i.e. control) epoxy matrix Experiment Modelling Paris law fit log (G max /G c ) Figure 5.57 Growth rate curve for the CT specimen for the bulk micro-rubber modified epoxy matrix 149

177 log (da/dn) log (da/dn) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment Modelling Paris law fit log (G /G ) Figure 5.58 Growth rate curve for the CT specimen for the bulk nano-silica modified epoxy matrix Experiment (Paris law) Modelling Paris law fit log (G max /G c ) Figure 5.59 Growth rate curve for the CT specimen for the bulk nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix (experimental data from Lee [49]) 150

178 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES DCB test and user element analysis The DCB fatigue test is undertaken to find the lamina interface fracture energy of the composite material. The cohesive zone properties of the lamina are obtained from the fatigue test of the composite material. The growth rate curves obtained from the tests are shown in Figures 5.60 to The threshold fracture energy for the composite material is found from the growth rate curve. The fatigue analysis of the DCB is performed using a FEA approach in the Abaqus software. The analysis of the fatigue damage in the composite is modelled using a cohesive zone element. The subroutine is developed in Abaqus and accounts for the fatigue damage of cohesive zone element with time. The thickness of the cohesive zone element is kept the same as in the quasi-static model at 0.01mm. The model is analysed using the subroutine to obtain the fatigue parameters of the cohesive zone element. The load in the fatigue model is applied as a constant displacement, as described in Section , and the load is kept constant throughout the fatigue cycles. The growth rate curve predicted by the model is compared with the growth rate curve from the experimental tests. The parameters for the fatigue model are obtained by comparing the slope and intercept of the growth rate cure and matching them with the experimental results. The experimental growth rate curve of the different materials is used to validate the fatigue parameters. (Since the cohesive zone element parameters are derived from the quasi-static work are used to model the DCB test in fatigue.) The growth rate curve of the DCB specimen is matched with the experiments to obtain the fatigue parameters of the lamina interface (Figures 5.60 to 5.63). The fatigue parameters and obtained from the Paris law fit are given in Table The fatigue parameters are then used in the user element subroutine to degrade the cohesive zone element according to the Paris law, as described in Section

179 log (da/dn) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Table 5.18 Fatigue parameters obtained from the DCB composite material specimen Matrix fatigue properties c m G th (J/m 2 ) Control Nano Rubber Hybrid Experiment Modelling Paris law fit log (G max /G c ) Figure 5.60 Growth rate curve for the composite DCB specimen based upon the unmodified (i.e. control) epoxy matrix 152

180 log (da/dn) log (da/dn) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment Modelling Paris law fit log (G max /G c ) Figure 5.61 Growth rate curve for the composite DCB specimen based upon the micro-rubber modified epoxy matrix Experiment Modelling Paris law fit log (G max /G c ) Figure 5.62 Growth rate curve for the composite DCB specimen based upon the nano-silica modified epoxy matrix 153

181 log (da/dn) CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Experiment Modelling Paris law fit log (G max /G c ) Figure 5.63 Growth rate curve for the composite DCB specimen based upon the nano-silica and micro-rubber (i.e. hybrid) modified epoxy matrix Strip test and user element analysis The fatigue test on the composite strip is undertaken to determine the fatigue life of composite strip specimens. The composite strip is tested under a constant fatigue load to determine the fatigue life of the strip. The transverse crack density of the composite strip is obtained for different cycles of fatigue and the number of cycles to failure of the strip is also found out. A typical photograph of the crack density observed in the composite for different cycles of fatigue is shown in Figure The graphs of stress versus number of cycles for failure of the composite strip are shown in Figures 5.65 to 5.68; and values of the crack density for different cycles of fatigue obtained for the different composite materials are as shown previously in Figure

182 CHAPTER 5 EXPERIMENTAL RESULTS AND THEORETICAL MODELLING STUDIES Figure 5.64 Transmitted light photographs of GFRP composite with unmodified (i.e. control) epoxy matrix showing the sequence of matrix crack development with the number of cycles, N, under fatigue loading. The composite strips are now finally modelled to predict the fatigue life of the composite material based on the different epoxy matrices. The composite strip has a layup of [(-45/45/0/90) s ] 2 and dimensions of 50x2.7x25mm 3 is used to study the fatigue behaviour of the composites. The strip is modelled considering a small section of the strip from the cross-section, as shown in Figure The strip is then modelled with transverse cracks with a density as observed in the experiments (Figure 5.33). The 90 o lamina crack density is assumed to be constant at a value of 0.64/mm (Tong et al. [37]). The fatigue analysis is undertaken with different crack densities, and with the different epoxy composites. The load is applied as constant stress on the strip as shown in Figure The nodes on the applied stress surface are tied with each other to have an equal displacement in the y direction of the strip (in Figure 5.32 the top surface nodes are made to move equally in the y direction). The width of the model is half the thickness of strip and the length of the model depends on the crack density (Figure 5.31). In the model, a symmetric section (Figure 5.30) of the strip is 155