Mechanics of hybrid polymer composites

Mechanics of hybrid polymer composites Rodrigo Paiva Tavares Supervisor: Prof. Dr. Pedro P. Camanho A Thesis submitted for the degree of Master of Science in Mechanical Engineering to the Faculty of Engineering, University of Porto Porto, June 2015 Abstract Composite materials, more specifically fibre-reinforced composites, play a huge role in structural applications, however their use is sill hampered partly due to the low toughness they exhibit. Fibre hybridisation is a strategy that can lead to improved composite properties and performance, as it not only changes the material properties but also changes the damage propagation mechanisms leading to failure. The failure of hybrid composites is, usually, less catastrophic than that of to nonhybrid composites. Predicting the tensile failure of unidirectional composites is a demanding task as there are multiple interacting failure mechanisms. The tensile behaviour of hybrid composites represents even more challenges as there are two types of fibres whose properties, including failure strain, differ. This thesis aims to increase the understanding of the tensile failure of unidirectional hybrid and non-hybrid composite when subjected to tensile loadings. To achieve this goal several models with increasing complexity have been developed and implemented and the effects of hybridization have been studied. A model for the tensile failure of dry tows of fibres, no matrix, based on the statistics of fibre strength has been developed and implement for hybrid and non-hybrid tows. This simple model considers the fibres to act independently of each other and cannot be used to predict the behaviour of composite materials. Nonetheless, it is useful to understand the effects of the fibre strength distributions, as well as the effects of hybridizing tows with different fibres using real distributions for the strength of the fibres parameters, in the tensile behaviour at the tow level. With careful selection of the hybridizing fibres it was possible to achieve a tow with increased ductility, whose failure was not catastrophic but was a progressive one. To account for the presence of the matrix in a composite material a model, based on the fibre fragmentation process, was extended for composite materials. This simplified model helps understanding the effects of hybridization in the tensile failure of composites. The concept of pseudo-ductile strain is introduced and a parametric study is done with the objective of maximizing this parameter and understanding i ii Abstract the main factors controlling the increase of ductility in composite materials. In order to accurately predict the tensile behaviour of unidirectional composite materials with a correct representation of the damage mechanisms direct numerical simulation is necessary. A material damage model that is able to account for the fibre strength variability was developed and implemented in a commercial finite element software. Alongside a material model for the matrix material a micromechanical model, with random fibre distributions, was developed to study the tensile failure of composite materials. The interfaces between these two constituents of the composite are modelled by considering a cohesive behaviour. Representative volume elements with random fibre distributions were created to study the sequence mechanisms leading to failure in composite materials and to validate the implemented models. Similar volume elements, with random fibre distributions, were developed for hybrid composites and the effects of the hybrid volume fraction on the tensile response of the materials was studied. The changes in the failure mechanisms due to the introduction of two types of fibres in a single unidirectional composite were also studied. These models allow to accurately represent the damage mechanisms in composites, including dynamic effects. Resumo Os materiais compósitos, mais especificamente os polímeros reforçados com fibras, têm um papel fundamental em aplicações estruturais, no entanto o seu uso ainda tem algumas limitações principalmente devido à baixa tenacidade que apresentam. A hibridização, principalmente através do uso de vários tipos de fibras, é uma estratégia que pode melhorar as propriedades e desempenho dos materiais, pois altera não só as propriedades do material mas também os mecanismos de propagação de dano que levam à rutura do compósito. A fratura dos compósitos híbridos é, usualmente, menos catastrófica do que a fratura dos compósitos não-híbridos. A previsão da fratura longitudinal de compósitos unidirecionais é uma tarefa difícil devido à interação entre múltiplos mecanismos de dano. Para compósitos híbridos esta previsão é ainda dificultada pela existência de dois tipos de fibras com propriedades diferentes, incluindo a deformação de rutura. O objetivo deste trabalho é aumentar o conhecimento sobre o comportamento de compósitos unidirecionais, híbridos e não híbridos, quando sujeitos a cargas longitudinais. Para alcançar este objetivo forem desenvolvidos e implementados modelos com crescentes níveis de complexidade, que permitiram não só estudar o comportamento dos compósitos não-híbridos mas também analisar os efeitos da hibridização. Um modelo de previsão da falha de tows de fibras, sem presença de matriz, baseado em distribuições estatísticas para a tensão de rutura das fibras foi desenvolvido e implementado para tows híbridos e não-híbridos. Este modelo simplificado considera que não existe interção entre as fibras e, como tal, não pode ser usado para a previsão do comportamento de materiais compósitos, onde está presente a matriz. Não obstante, o modelo é útil para perceber os efeitos dos parâmetros estatísticos da tensão de rutura das fibras, bem como o efeito da hibridização, usando distribuições reais, no comportamento dos tows. Através da cuidada seleção das fibras a hibridizar foi possível obter um tow com maior ductilidade, cuja falha é mais progressiva. De maneira a ter em conta a presença da matriz nos materiais compósitos foi desenvolvido um modelo baseado no processo de fragmentação das fibras. Este iii iv Resumo modelo, apesar de simplificado, permite perceber os efeitos da hibridização no comportamento à tração dos materiais compósitos. É também introduzido o conceito de pseudo-ductilidade e é feito um estudo paramétrico com a finalidade de maximizar este parâmetro e perceber os parâmetros que controlam a ductilidade nos materiais compósitos. Apesar destes modelos serem úteis, para ser possível prever o comportamento de compósitos unidirecionais à tração longitudinal, com uma representação correta dos mecanismos de dano, é necessário recorrer a simulação numérica. Um modelo material capaz de ter em conta a variação estocástica da tensão de rutura das fibras foi desenvolvido e implementado num software comercial de elementos finitos. Em conjunto com um modelo material para a matriz, um modelo micromecânico, com uma distribuição aleatória de fibras, foi desenvolvido para estudar a fratura longitudinal de materiais compósitos. Foi também considerado o comportamento da interface entre as fibras e matriz através do uso de modelos coesivos. Foram criados diversos elementos de volume representativos, com o objetivo de estudar a sequência de mecanismos de dano que levam à rutura dos materiais compósitos e validar os modelos materiais implementados. Foram também criados elementos de volume representativos para compósitos híbridos e o efeito da fração volúmica de cada tipo de fibras no comportamento à tração deste material foi estudado. Foram ainda estudas as alterações nos mecanismos de falha devido à introdução de dois tipos de fibras no compósito. Os modelos implementados são capazes de representar corretamente os mecanismos de dano nos materiais compósitos, incluindo os efeitos dinâmicos. Acknowledgements Firstly, I wish to express my gratitude to Prof. Dr. Pedro P. Camanho, supervisor of this thesis, for all the support, patience and availability during the development of this thesis, even at the most occupied times. I also would like to thank for the opportunity to work in such an interesting field and for all the encouragement. To Miguel Bessa, PhD candidate, for his help and cooperation during this work. Even though separated by an ocean he always managed to have time to help me in the problems I’ve encountered, even when time was hard to find. To Dr. António Melro for providing the tools that were the base of work and for helping me in the very early stages of this work and for giving me guidance when I didn’t even know what I was doing. To my workgroup, Giuseppe Catalanotti, Albertino Arteiro, Ricardo Pinto, Hélder Mata and Claudia Cardoso, I thank for the support and for all the valuables discussions, usually over a cup of coffee. I would also like to thank everyone that I met during this stage at FEUP. A special thanks for Rodrigo Carvalho and Luis Máximo for the friendship and support. To Carolina Furtado, for many deserved breaks and for all the shared knowledge and motivation. To my girlfriend, Rita Frade that, although being far during the development of this thesis, was always there to support me and push me forward. I would also like to thank Guilherme Pereira and all my friends whose friendship I can always count on. Lastly, a very special thanks to all my family. To my brother, Rafael, for always helping me move forward. To my parents, Magda and José, for everything, especially for always expecting the best of me. v vi Contents Contents Abstract i Resumo iii Acknowledgements v Contents vi List of symbols xi List of figures xv List of tables xx 1 Introduction 1 2 Mechanisms of longitudinal fracture 7 2.1 Distributions for fibre strength . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Weibull distribution . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Modified Weibull distributions . . . . . . . . . . . . . . . . . 8 2.1.3 The normal distribution . . . . . . . . . . . . . . . . . . . . . 9 2.2 Size effects in composites . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Stress redistribution after fibre failure . . . . . . . . . . . . . . . . . 11 2.4 Critical cluster size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Effects of the matrix and fibre-matrix interface . . . . . . . . . . . . 14 2.6 Modelling the tensile failure of unidirectional composites . . . . . . . 15 2.6.1 Deterministic rule of mixtures . . . . . . . . . . . . . . . . . . 15 2.6.2 Analytical fibre bundle models . . . . . . . . . . . . . . . . . 16 2.6.3 Micromechanical models based on Monte-Carlo simulations . 21 2.6.4 Continuum damage mechanic based models . . . . . . . . . . 25 2.6.5 Hierarchical scaling law for the strength of composite fibre bundles .............................. 26 2.7 Conclusion ................................ 28 vii xiv List of symbols αTTransverse thermal expansion coefficient; αm C,αm T, αm S Rate dependence parameter for the matrix model; δτMaximum displacement for the fibre-matrix interface; ε ε εStrain tensor; µτFriction at the fibre-matrix interface; νiPoisson coefficient in the i-direction; σ σ σStress tensor ˜σ ˜σ ˜σEffective stress tensor; σ0Weibull scale parameter; τiMatrix-fibre interfacial shear strength in the i-direction; φd fLoading function for the fibre model; ΨfEnergy dissipated per volume unit for the fibre model; List of Figures 2.1 Size effects in glass/epoxy composites [1]. . . . . . . . . . . . . . . . 10 2.2 Schematic illustration of fibre packings: (a) 1D regular packing, (b) 2D regular packing (c) 1D random packing and 2D random packing [2]. 12 2.3 Cluster of 14 fibres observed by Scott et al. [3] using synchrotron computedtomography........................... 13 2.4 Effect of matrix cracks in the ineffective length (left) and the stress concentration factors (right) [2]. . . . . . . . . . . . . . . . . . . . . . 15 2.5 Diagram of a fibre bundle model [4]. . . . . . . . . . . . . . . . . . . 18 2.6 Representation of the node lattice of a single-step spring-based model. 22 2.7 Schematics of the enhanced super-position stress redistribution: (a) stress concentration around a single break, (b) linear superposition results and (c) enhanced super-position [2]. . . . . . . . . . . . . . . 24 2.8 Flow chart of the model developed by Swolfs [2]. . . . . . . . . . . . 24 2.9 Stress profile in a fibre with multiple fractures, according to shear-lag model[5]................................... 26 2.10 Hierarchical fibre bundles [6]. . . . . . . . . . . . . . . . . . . . . . . 27 2.11 Definition of control region [6]. . . . . . . . . . . . . . . . . . . . . . 27 3.1 Hybrid configurations: (a) interlayer, (b) intralayer and (c) intrayarn configurations[7].............................. 32 3.2 Diagrams for the definition of the hybrid effect: (a) first definition proposed by Hayashi and (b) general definition based on the rule-ofmixtures[7]................................. 33 3.3 Dispersion in hybrid composites: the degree of dispersion increases from(a)to(d)[7]. ............................ 34 3.4 Hybrid effect for tensile failure strain. Information in Black was gathered by Kretsis and information in colour by Swolfs. The information inside the red line should be interpreted with care, due to errors [7]. 38 3.5 Bilinear rule-of-mixtures for tensile strength. Experimental data for carbon/glass hybrids [7]. . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6 Effects in the flexural modulus of hybridizing carbon fibre composites with glass fibres in the compressive layers [8]. . . . . . . . . . . . . 39 xv xvi List of figures 3.7 Effects in the flexural strength of hybridizing carbon fibre composites with glass fibres in the compressive layers [8]. . . . . . . . . . . . . 40 3.8 Fatigue response of several fibre reinforced composites [9]. . . . . . . 42 3.9 Schematic stress-strain diagrams for: (a) non hybrid composites, (b) typical hybrid composites and (c) pseudo-ductile hybrid composites [2]. 43 3.10 Failure modes as a function of absolute and relative layer thickness in carbon/glass hybrid composites [10]. . . . . . . . . . . . . . . . . . 44 3.11 Stress redistribution in hybrid composites with 50% carbon and glass fibres: (a) SCFs in both fibre types, considering the same and different radii; (b) ineffective length of the broken carbon fibre considering fibres with the same and different radii [11]. . . . . . . . . . . . . . 45 3.12 Stress concentration factors in glass fibres as a function of the distance from the broken fibre [11]. . . . . . . . . . . . . . . . . . . . . . . . 46 3.13 Stress concentration factors in carbon fibres as a function of the distance from the broken fibre [11]. . . . . . . . . . . . . . . . . . . . . 46 3.14 The ineffective length of carbon–glass hybrids for different hybrid volume fractions. The error bars indicates the 95% confidence interval based on five realisations[11]. . . . . . . . . . . . . . . . . . . . . . . 47 3.15 The ineffective length in carbon and HE fibres as a function of HE stiffness[2]. ................................ 47 3.16 Representation of the fibre packings used in Zweben’s model : (a) non-hybrid LE composite and (b) hybrid composite with alternating LEandHEfibres[2]. .......................... 48 3.17 Hybrid unit-cells with different hybridization ratios (NB) and hybridization degrees [12]. . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.18 Stress concentration factors, according to very local load sharing, around (a) a broken carbon fibre and (b) a broken glass fibre [2]. . . 52 3.19 Influence of failure strain ratio in the hybrid effect, for a hybrid composite with 50% of each fibre type [2]. . . . . . . . . . . . . . . . . . 53 3.20 Illustration of the (a) bundle-by-bundle and (b) layer-by-layer dispersion considered by Swolfs et al. (Adapted from[13]). . . . . . . . . . 56 4.1 Flowchart of the model for dry bundle failure. . . . . . . . . . . . . . 61 4.2 Effect of the scale parameter (σ) in the strength of the fibres. . . . . 62 4.3 Force-Strain diagrams for tows of fibres with different scale parameters (σ)................................... 63 4.4 Effect of the shape parameter (m) in the strength of the fibres. . . . 64 4.5 Force-Strain diagrams for tows of fibres with different shape parameters (m). ................................. 64 4.6 Effect of dispersion in the strength of the fibres, maintaining the same averagestrength.............................. 66 List of figures xvii 4.7 Force-Strain diagrams for fibres with different dispersions, but same averagestrength.............................. 66 4.8 Effect the gauge length (L) in the strength of the fibres. . . . . . . . 67 4.9 Force-Strain diagrams for tows of fibres with lengths (L). . . . . . . 67 4.10 Diagram of pseudo-ductile strain. . . . . . . . . . . . . . . . . . . . . 69 4.11 Failure strain distributions for different carbon fibres. . . . . . . . . 71 4.12 Stress-strain diagrams at various hybrid volume fractions for AS4/T300 hybridization. .............................. 71 4.13 Stress-strain diagrams at various hybrid volume fractions for AS4/M40S hybridization. .............................. 72 4.14 Stress-strain diagrams at various hybrid volume fractions for AS4/M50S hybridization. .............................. 74 4.15 Failure strain distributions for the carbon fibres used for hybridization. ................................... 75 4.16 Failure strain distributions for several glass fibres. . . . . . . . . . . 76 4.17 Stress-strain diagrams at various hybrid volume fractions for HD/HP AR glass hybridization. . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.18 Failure strain distributions for several kevlar fibres. . . . . . . . . . 78 4.19 Stress-strain diagrams at various hybrid volume fractions for kevlar 49/119 hybridization. . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.20 Stress-strain diagrams at various hybrid volume fractions for kevlar 119/129 hybridization. . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.21 Stress-strain diagrams at various hybrid volume fractions for T300 carbon and AR-HP glass hybridization. . . . . . . . . . . . . . . . . 81 4.22 Stress-strain diagrams at various hybrid volume fractions for 1000°C carbon and AR-HP glass hybridization. . . . . . . . . . . . . . . . . 82 4.23 Stress-strain diagrams at various hybrid volume fractions for AS4 carbon and kevlar 49 hybridization. . . . . . . . . . . . . . . . . . . 84 4.24 Stress-strain diagrams at various hybrid volume fractions for M40S carbon and kevlar 49 hybridization. . . . . . . . . . . . . . . . . . . 85 4.25 Stress-strain diagrams at various hybrid volume fractions for M43S carbon and kevlar 119 hybridization. . . . . . . . . . . . . . . . . . 86 5.1 Stress profile in a fibre with multiple fractures, according to shear-lag model[5]. ................................. 91 5.2 Stress-Strain diagrams for composites with different interfacial shear strength (τ). ............................... 95 5.3 Stress-Strain diagrams for composites with fibres with different scale parameters (σ)............................... 96 5.4 Stress-Strain diagrams for composites with fibres with different shape parameters (m), with τ= 40 MPa.................... 97 xviii List of figures 5.5 Stress-Strain diagrams for composites with fibres with different shape parameters (m), with τ= 10 MPa.................... 98 5.6 Stress-Strain diagrams for composites with fibres with different with different dispersion and the same average failure strength. . . . . . . 99 5.7 Stress-Strain diagrams for composites with fibres with different Weibull shape parameter (m) and same critical strength (σc).......... 100 5.8 Failure strain distributions for different carbon fibres. . . . . . . . . 103 5.9 Stress-strain diagrams at various hybrid volume fractions for AS4/T300 hybridization................................ 103 5.10 Stress-strain diagrams at various hybrid volume fractions for AS4/M40S hybridization................................ 105 5.11 Stress-strain diagrams at various hybrid volume fractions for AS4/M50S hybridization................................ 106 5.12 Failure strain distributions for several glass fibres. . . . . . . . . . . 108 5.13 Stress-strain diagrams at various hybrid volume fractions for HD/HP glasshybridization............................. 108 5.14 Stress-strain diagrams at various hybrid volume fractions for HD/Eglasshybridization............................. 109 5.15 Failure strain distributions for several kevlar fibres. . . . . . . . . . 111 5.16 Stress-strain diagrams at various hybrid volume fractions for kevlar 49/kevlar 119 hybridization. . . . . . . . . . . . . . . . . . . . . . . . 111 5.17 Stress-strain diagrams at various hybrid volume fractions for kevlar 119/kevlar 129 hybridization. . . . . . . . . . . . . . . . . . . . . . . 112 5.18 Stress-strain diagrams at various hybrid volume fractions for T300 carbon and AR-HP glass hybridization. . . . . . . . . . . . . . . . . 114 5.19 Stress-strain diagrams at various hybrid volume fractions for M50S carbon and AR-HP glass hybridization. . . . . . . . . . . . . . . . . 114 5.20 Stress-strain diagrams at various hybrid volume fractions for AS4 carbon and kevlar 49 hybridization. . . . . . . . . . . . . . . . . . . . 116 5.21 Stress-strain diagrams at various hybrid volume fractions for AS4 carbon and kevlar 129 hybridization. . . . . . . . . . . . . . . . . . . 117 5.22 Stress-strain diagrams at various hybrid volume fractions for AS4 carbon and kevlar 119 hybridization. . . . . . . . . . . . . . . . . . . 118 6.1 Flowchart of RAND_uSTRU_GEN algorithm [14]. . . . . . . . . . 122 6.2 Flowchart of the hard-core model for the generation of the microstructure. ................................... 124 6.3 Stress-strain diagrams for the fibre damage model for various element lengths. .................................. 129 6.4 Flowchart of the constitutive model for the fibres. . . . . . . . . . . 131 6.5 Distributions of tensile strength of all the elements present in the model, for two different random generators. . . . . . . . . . . . . . . 133 List of figures xix 6.6 Stress-strain curves for the AS4 carbon and epoxy matrix for a RVE of dimensions 15 ×15 ×15 fibre radius. . . . . . . . . . . . . . . . . 134 6.7 Stress-strain curves for the AS4 carbon and epoxy matrix: solid line - RVE 20 ×20 ×20; dashed line - RVE 15 ×15 ×15 . . . . . . . . . 135 6.8 Damage in the interface of a broken fibre. . . . . . . . . . . . . . . . 135 6.9 Damage progression in the RVE: (a) damage prior to fibre failure; (b) damage after first fibre failure; and (c) damage after failure of the compositematerial............................. 136 6.10 Broken fibre: (a) Location of the fibre break; (b) Stress field at fibre failure and (c) Stress field after dynamic effect. . . . . . . . . . . . . 137 6.11 Distributions of tensile strength of all the elements present in the model for the M50S fibres using both random generators. . . . . . . 138 6.13 Matrix crack developed after a fibre breaks (a) and stress concentrations in the neighbouring fibres of a broken one (b). . . . . . . . . . 138 6.12 Stress-strain curves for the M50S carbon and epoxy matrix. . . . . . 139 6.14 Distributions of tensile strength of all the elements present in the model for the M30S fibres using both random generators. . . . . . . 139 6.15 Stress-strain curves for the M30S carbon and epoxy matrix. . . . . . 140 6.16 Crack development in the matrix: (a) Fibre failure and (b) development of a crack in the matrix surrounding the broken fibre. . . . . . 140 6.17 Stress-strain curves for the hybridization between the AS4 and M50S carbonfibres................................ 142 6.18 Stress-strain curves for the hybrid composite with a volume fraction equal to 0.75 of AS4 carbon fibres and 0.25 of M50S (LE) carbon fibres.142 6.19 Multiple fractures of the M50S (LE) carbon fibres for a composite with a VLE = 0.5.............................. 143 6.20 Stress-strain curves for the hybrid composite with a volume fraction equal to 0.5of AS4 carbon fibres and 0.5of M50S carbon fibres. . . 143 6.21 Stress-strain curves for the hybrid composite with a volume fraction equal to 0.25 of AS4 carbon fibres and 0.75 of M50S (LE) carbon fibres.144 6.22 Fracture process of an high elongation fibre (AS4 carbon) for a hybrid composite.................................. 145 xx List of tables List of Tables 4.1 Effect of the shape parameter (m) in some reference properties. . . . 63 4.2 Effect of the shape parameter (m) in some reference properties. . . . 65 4.3 Effect of dispersion in some reference properties (distributions with the same average strength). . . . . . . . . . . . . . . . . . . . . . . . 65 4.4 Effect of the gauge length (L) in some reference properties. . . . . . 68 4.5 Mechanical properties for carbon fibres. . . . . . . . . . . . . . . . . 70 4.6 Stress-strain reference properties for AS4/T300 hybridization. . . . . 72 4.7 Stress-strain reference properties for AS4/M40S hybridization. . . . 73 4.8 Stress-strain reference properties for AS4/M50S hybridization. . . . 74 4.9 Mechanical properties for glass fibres. . . . . . . . . . . . . . . . . . 76 4.10 Stress-strain reference properties for HD/HP AR glass hybridization. 77 4.11 Mechanical properties for kevlar fibres. . . . . . . . . . . . . . . . . . 78 4.12 Stress-strain reference properties for kevlar 49/119 hybridization. . . 79 4.13 Stress-strain reference properties for kevlar 119/129 hybridization. . 80 4.14 Stress-strain reference properties for T300 carbon and AR-HP glass hybridization................................ 82 4.15 Stress-strain reference properties for 1000°C carbon and AR-HP glass hybridization................................ 83 4.16 Stress-strain reference properties for AS4 carbon and kevlar 49 hybridization. ................................ 84 4.17 Stress-strain reference properties for M40S carbon and kevlar 49 hybridization. ................................ 85 4.18 Stress-strain reference properties for M30S carbon and kevlar 119 hybridization................................ 86 5.1 Effect of interface shear strength (τin some reference properties. . . 95 5.2 Effect of the scale parameter (σ) in some reference properties. . . . . 96 5.3 Effect of the shape parameter (m) in some reference properties, with τ= 40 MPa................................. 97 5.4 Effect of the shape parameter (m) in some reference properties, with τ= 10 MPa................................. 98 5.5 Effect of the fibre strength dispersion in some reference properties. . 99 xxi xxii List of tables 5.6 Effect of the Weibull shape parameter (m) for composites with same critical strength (σc)............................ 100 5.7 Mechanical properties for carbon fibres. . . . . . . . . . . . . . . . . 102 5.8 Stress-strain reference properties for AS4/T300 hybridization. . . . . 104 5.9 Stress-strain reference properties for AS4/M40S hybridization. . . . 105 5.10 Stress-strain reference properties for AS4/M50S hybridization. . . . 106 5.11 Mechanical properties for Glass fibres. . . . . . . . . . . . . . . . . . 107 5.12 Stress-strain reference properties for HD/HP glass hybridization. . . 109 5.13 Stress-strain reference properties for HD/E-glass hybridization. . . . 110 5.14 Mechanical properties for kevlar fibres. . . . . . . . . . . . . . . . . . 110 5.15 Stress-strain reference properties for kevlar 49/kevlar 119 hybridization.112 5.16 Stress-strain reference properties for kevlar 119/kevlar 129 hybridization..................................... 113 5.17 Stress-strain reference properties for M50S carbon and AR-HP glass hybridization................................ 115 5.18 Stress-strain reference properties for AS4 carbon and kevlar 49 hybridization. ................................ 116 5.19 Stress-strain reference properties for AS4 carbon and kevlar 129 hybridization. ................................ 117 5.20 Stress-strain reference properties for AS4 carbon and kevlar 119 hybridization. ................................ 119 6.1 Fibre properties required for the material models. . . . . . . . . . . . 132 6.2 Matrix properties required for the material models. . . . . . . . . . . 132 6.3 Properties for the cohesive surfaces in the fibre-matrix interfaces. . . 133 Chapter 1 Introduction Composite materials are considered the materials of the future, manly in industries that have very high standards in terms of weight such as the aeronautical and aerospace industry. These materials can provide high level of performance with reduced weighs due to their excellent specific properties, namely strength and stiffness. A composite is a material that results of the combination of two or more macroscopic components, resulting in a new material with superior properties than the constituents by themselves. At the time composite materials are considered materials with a high stiffness and resistance reinforcing component - long fibres, short fibres or particlesinvolved in a matrix with weaker properties that acts as connector and protects the reinforcing material. Although composite materials are considered to be the materials of the future, they are widely available in the nature and have been used by humans since the beginning of civilization. The muscles in the human body are an example of a fibrous material. The arrangement of muscular fibres with different orientations allows the creation of a very adaptable material with outstanding properties in a preferential direction. Another example of a composite material is wood, whose arrangement of cellulose fibres provide the necessary strength while the matrix (lignin) provides the necessary connection between the fibres and gives the material its compressive resistance. Man-made composites have existed for a long time. The first evidences of a manmade composite appeared in the Egyptian era, where straw and mud were mixed and burnt together by Israelites in order to obtain tougher bricks for construction. Another example of composite materials from the Egyptian era is papyrus, where layers of stems of the papyrus plant were stacked in perpendicular directions in order to manufacture paper with enough resistance to be written on and handled without 1 8 Chapter 2. Mechanisms of longitudinal fracture 2.1 Distributions for fibre strength The tensile strength of a technical fibre cannot be represented by a single average value. Due to their brittle behaviour the fibre tensile strength is governed by surface or volume flaws [15] and exhibits weak-link characteristics. There are several statistical distributions that can be used to characterize the strength of fibres, being the most used the Weibull distribution, proposed by Weibull in 1951 [17]. 2.1.1 Weibull distribution The standard Weibull probability distribution can be written as: P(σ)=1−exp 1−L L0σ σ0m,(2.1) where Pis the failure probability at the applied stress σ,Lis the characteristic gauge length, L0is the reference gauge length (these can also be characterised as volumes [18]), σ0the scale parameter and mthe shape parameter or Weibull modulus [17]. This distribution leads to the underestimation of the fibre strength at short gauge lengths [2] and is very sensitive to the statistical parameters [19]. The discrepancy between the Weibull distribution and the experimental results for short gauge lengths can be attributed to variations in fibre diameter, variations of the Weibull distribution from fibre to fibre and presence of different flaw populations [2]. According Curtin [20] this distribution is not the most accurate to describe the strength of fibres, however is still the most used to characterize tensile strength of technical fibres. 2.1.2 Modified Weibull distributions Several authors found that the fibre strength is governed by more than one flaw population [21, 22] and therefore a bimodal Weibull distribution should be used: P(σ)=1−exp −L L0 σ σ01 m1 −L L0 σ σ02 m2,(2.2) where σ01 and σ02 are the scale parameters and m1and m2the Weibull moduli for both populations of flaws. The use a traditional Weibull distribution indicates that there is no threshold stress below which the failure probability is zero, which is 2.1 Distributions for fibre strength 9 common in brittle materials like fibres, however if such threshold exists the bimodal distribution is able to capture that limit [2]. As mentioned before, the traditional Weibull distribution fails to characterise the fibre strength at short gauge lengths, therefore was proposed a modified Weibull distribution that adds an exponent αto capture this dependency [23]: P(σf)=1−exp 1−L L0ασ σ0m.(2.3) This equation leads to the traditional Weibull distribution when αequals 1. Curtin [20] proposed a model, entitled "Weibull of Weilbulls" that considers that the strength along a fibre follows a traditional Weibull distribution (eq. 2.1), therefore it is possible to calculated the characteristic strength for the gauge length L. Curtin also states that the characteristic strength of each fibre are different and follow another Weibull distribution, leading to the "Weibull of Weibulls" distribution of fibre strength. There is still no consensus whether traditional Weibull or the modified Weibull distributions better represent the fibre strength. 2.1.3 The normal distribution M. R’Mili et al. [19] studied the normal distribution to characterize the distribution of flaw strengths. They compared both normal and Weibull distributions to the results of tensile testing with large sample sizes of 500-100 data per test. The tensile tests were done in tows and the fractures were determined by acoustic emission monitoring. In order to eliminate the variability associated with the fibre radius the probability functions were determined as a function of strain (ε), which leads to the following equation for the Gaussian probability density function: f(ε) = 1 S√2πexp"−(ε−µ)2 2S2#,(2.4) where µis the mean and Sis the standard deviation, which can be fitted to the experimental failure data. The Weibull distribution can also be written in terms of strain (ε) resulting in: PW= 1 −exp−V V0ε ε0m,(2.5) whith ε0=σ0/E. M. R’Mili et al. [19] concluded that the normal distribution is appropriate to describe the flaw strengths in brittle fibres, but also concluded that the Weibull distribution is a good approximation to this distribution and leads to simpler equations of failure probability. 10 Chapter 2. Mechanisms of longitudinal fracture As discussed there are several distribution to characterize the failure probability of fibres being the most used the traditional Weibull distribution (Equation 2.1). The determination of the parameters necessary for these distributions is not a simple one, meaning that large samples need to be tested in order to obtain representative parameters. The problems in obtaining the statistical parameters for fibre strength makes it hard to get a single distribution to characterise this property and, therefore, many statistical parameters can be found in the literature for the same type of fibre. 2.2 Size effects in composites Size effects affect, not only the strength of individual fibres, but also influencing the failure process and longitudinal strength of composite structures [1]. There are several factors that lead to this size effects, being that most authors agree that the statistics of fibre strength are essential for this. Several authors have experimentally proven this size scaling behaviour (see Figure 2.1). !"#$%& '%!'! 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'4% '6A%!; X%!6$'! 5%7% "(2@#7%& 53'4 A67!' '%!'! () =%!!%$! (? &3#2%'%7 A%'5%%) P/PR #)& K/,Y 22 53'4 3&%)'3"#$ $#B6@! #)& 2#'%73#$!; N4% 2#)6?#"'673)8 @7("%!!%! 5%7% #$!( !323$#7 ?(7 '4% ?6$$ !39%& =%!!%$! #)& 73)8!; N4% @7%!!67% '%!'! 3)&6"%& A3#F3#$ !'7%!!%!> #$'4(684 '43! 5#! \38; S; E39% %:%"' 3) DA7% &37%"'3() '%)!3() ?(7 8$#!!.%@(FB JKYL; "#$# %&'!() * +(),('&-.' /0&.!0. 1!2 3.04!(5(67 89 :;999< ;9=>?;98> P0,- Figure 2.1: Size effects in glass/epoxy composites [1]. The size effect is not only affected by statistical aspects, but is also by deterministic factors, that include the effects of the damage process zone and the change of the failure modes [24]. There are other influencing factors, namely the influence of manufacturing and testing. This means that in order to achieve a good design of large composite structures based on coupon testing, one has to take into account that the coupons should be representative of the manufacturing process of the large scale component and that larger structures have lower strength due to being more 2.3 Stress redistribution after fibre failure 11 probable the existence of a critical defect [1]. One of the factors that leads to size effect due to manufacturing is the fact have larger fibre waviness and overall defects present in larger components than in small coupons. 2.3 Stress redistribution after fibre failure The statistical distributions presented allow the determination of the strain at which a fibre will fracture. If a global load sharing rule is considered, then the stress that the fractured fibre carried is transferred equally to all the remaining intact fibres. This type of load sharing rule is able to predict the failure of lubricated tows, where the fibre interaction is low [18], but it’s not accurate for composite materials where the fibres are bonded by a matrix, where their is high interaction between fibres. The interaction between the fibres and between the fibres and the matrix results in a non uniform stress redistribution to the intact fibres, which is highly dependent on the composite geometry [25]. The models that consider a non uniform stress redistribution are said to consider a local load sharing rule. For composite systems the redistribution of stresses is a complex process that depends on several parameters, including the strength and sliding resistance of the fibre/matrix interface, the fibre to matrix moduli ratio, the matrix cracking or yield stress and the regularity of the fibre spacing [26]. This complex stress redistribution is often characterised by the stress concentration factor (SCF) and the ineffective length [2]. The SCF is the an adimensional parameter that is defined as the ratio between the longitudinal stress in an intact fibre after the failure of a neighbour fibre and the longitudinal stress in the absence of breaks. The stress in the absence of breaks is usually considered the stress in the intact fibre far from the plane of break, which simplifies the determination of this parameter. After a fibre breaks it locally looses the ability to carry stress, even so, away from the failure plane it is still able to carry loads, which means that a fibre doesn’t fully loose the ability to carry stress after it breaks. The ineffective length is a measure of the stress recovery length of the fibre and can be defined as twice the length at which the broken fibre can carry 90% of the applied stress [27]. These parameters are crucial in the modelling of composite materials as they will affect the stress redistribution and, therefore, the damage accumulation and the formation of clusters of broken fibres. The redistribution of stress is closely related with the fibre packing. There are several types of fibre packings that one can consider when modelling the microstructure of a composite material, some are represented in Figure 2.2. A 1D packing consists in a single row of fibres that can be equally spaced or 12 Chapter 2. Mechanisms of longitudinal fracture Figure 2.2: Schematic illustration of fibre packings: (a) 1D regular packing, (b) 2D regular packing (c) 1D random packing and 2D random packing [2]. randomly spaced. This type of packing is more common than 2D packings as it is easier to obtain, however it fails to give a accurate representation of the composite behaviour and leads to an overestimation of the stress concentration factors, making 2D packings more accurate to describe the micro-structure of composite materials. Being so, a 2D packing with random distribution of fibres is the model that most accurately represents the microstructure of a composite material, as it better relates with the real distribution of fibres in these materials [28]. The random distribution of fibres instead of a regular packing, leads to a varying fibre spacing and, therefore, changes the stress distribution. Obtaining this random distributions is more difficult and computationally expensive than regular ones, however the random fibre generator developed by Melro et al. [14] is able to do so in a reduced time. Another problem of using random fibre packings in FE analysis is that the variations of fibre spacing lead to some problems with the meshing. Nonetheless, if it is possible to use a random distribution, one should do so, as it translates better into the real microstructures and behaviour of fibrous composites. 2.4 Critical cluster size As already mentioned, the failure of a UD composite under tensile loads is due to the unstable propagation of a cluster of broken fibres. The clusters are formed due to stress concentrations in the intact fibres that neighbour a broken one. This increase in SCF causes the stress in the intact fibres to increase, thus increasing their probability of failure, making it more probable that fibres will fail in clusters. When a cluster of broken fibres is large enough, it propagates in an unstable manner, leading to the composite failure. Critical cluster size is, therefore, an important topic in understanding the failure of UD composites. Ibnabdeljalil and Curtin [29] studied this problem and derived an equation for 2.4 Critical cluster size 13 the critical cluster size (ncrit): ncrit = 403m−1.28 ,(2.6) where mis the Weibull modulus. This equation was derived from numerical simulations using Green’s function for stress redistribution, whose parameter Ωcharacterises the level of localization of the stress redistribution. The result presented in Equation 2.6 is for Ω=0.001, that represents a very local load sharing model (Ω→ ∞ corresponds to global load sharing). As the Weibull modulus presents some degree of dispersion, even for the same type of fibre, one can expect that, in a composite material, there are stronger and weaker regions [29], which will translate in variations of the critical cluster size and composite strength. Although with some difficulties, the critical cluster size was also tackled in an experimental way. Using synchrotron computed tomography, Scott et al. [3] found a cluster size of 14 fibres (14-plet) prior to failure (Figure 2.3). This cluster was found at 94% of the failure strain, which means that the critical cluster size should be superior to 14. CS32. The model makes the assumption that the composite has a periodic structure with the fibres arranged in a hexagonal manner in the ð~ x;~ yÞplane. It has been shown that one fibre break at the centre of the RVE has negligible effect on fibres outside of the 32 fibres considered. The RVE is a parallepiped having a square cross section normal to ~ zand with sides of length c= 0.05 mm. By using the work of Baxevanakis [27] and Hitchon and Phillips [38] the length of the RVE in the ~ zdirection is defined between the planes z= 0 and z=L= 4 mm. The length of the RVE represents the length in which a given fibre is assumed to only have one break along its length. The origin O m of R m loc is the geometric centre of the section contained in the z= 0 plane. On this scale, at M, the stress tensor is noted as r m and the strain tensor e m . The model allows the consequences of the accumulation of fibre failures to be determined as a function of the following (more details can be found in Blassiau et al. [7–9,12]): #the stochastic nature of fibre strength (described using a Weibull type distribution) and the position of failure along the fibre length; #the variation of the fibre Weibull modulus due to the small number of fibres (32) in the RVE, as the reliability of the Weibull modulus relies on the number of fibres in the population considered. To approach the deterministic value of the Weibull modulus a population of more than 300 fibres would have to be considered [39]; #the number of broken fibres in the RVE, which contains initially 32 intact fibres, considered for six different states of damage, allowing a complete description of the damage as it progresses from the undamaged to the failed state, represented by elementary cells designated C32, C16, C8, C4, C2 and C1. They contain, respectively, N= 1, 2, 4, 8, 16 and 32 broken fibres; Fig. 5. Fibre break accumulation in the same sample at different load increments. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 1 Fibre break cluster formation with increasing load [14]. Clusters/% of final failure 0 28 64 70 80 85 88 94 1-plet 0 1 6 10 15 58 98 151 2-plet 3 3 6 6 9 13 3-plet 1 3 7 4-plet 1 3 5-plet 6-plet 2 7-plet 8-plet 1 9-plet 10-plet 11-plet 14-plet 1 Fig. 6. Largest cluster of 14 fibre breaks. (a) (b) 2D slices on orthogonal planes. (c) 3D image (part of composite made transparent to reveal cluster) [14]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) A.E. Scott et al. / Composites: Part A 43 (2012) 1514–1522 1517 Figure 2.3: Cluster of 14 fibres observed by Scott et al. [3] using synchrotron computed tomography. Other authors have done similar studies. For example Aroush et al. [30] found a critical cluster size in the range 9-33 for quartz fibre reinforced epoxy resin. 14 Chapter 2. Mechanisms of longitudinal fracture 2.5 Effects of the matrix and fibre-matrix interface The tensile failure of composite materials is a fibre dominated phenomena, however the matrix also plays an important role. The matrix allows the stress recovery of a broken fibre due to shear stress transfer [2] and its properties affect the stress concentration factors and failure mechanisms. Several models consider the fibres and matrix to be perfectly bonded, which leads to a infinite stress concentration factor in the matrix around a fibre break. The matrix and the interface is unable to support such a high stress three scenarios can occur: (1) the matrix yields, (2) the interface debonds and (3) the matrix cracks in the break plane. [2] There can also be a combination of these. Zeng et al. [31] studied the influence of interfacial damage in the stress redistribution in UD composites and concluded that the stress concentrations increased with increasing the strength of the interface. He also concluded that matrix shear yielding resulted in lower stress concentration factors in intact fibres. The matrix yield strength also affects the ineffective length significantly. The lower the shear yield stress, the larger the ineffective length [32]. Interfacial debonding tends to occur in composites with weak interfacial bonds, and has a similar effect as matrix yielding. Both matrix yielding and interfacial debonding have been extensively studied in the literature, however the matrix cracking hasn’t been objective of focus. Recently, Swolfs et al. [33] studied the influence of matrix cracks in both the SCF and the ineffective length in a composite with a random distribution of fibres. The authors concluded that the matrix cracking increases the ineffective length, drastically changing the stress recovery profile. As one can see in Figure 2.4, the stress in the broken fibre rapidly increases to 35% when there isn’t a crack in the matrix, however, in the model with a crack, the stress slowly increases from zero, due to the presence of a crack in the matrix. Swolfs et al.[33] also showed that matrix cracks not only increases the ineffective length but, also the stress concentration factor, leading to an overall higher failure probability of the intact fibres. According to Hobbiebrunken et al. [34] the matrix strength is size dependent. In fibrous composites the matrix usually has a small thickness in the order of a few microns, thus having a strength higher than that of large test specimens. As the matrix strength affects the tensile behaviour of composites, not accounting for this size effect may lead less accurate results. Morais [35] studied the effect of the matrix shear modulus in the range 1.2− 1.6GPa and he concluded that the tensile strength of the composite material is practically insensitive to this parameter. However, he concluded the matrix shear strength is an important factor in the composite tensile strength. His results showed that increasing the shear strength from 40 to 100 MPa increased the composite 2.6 Modelling the tensile failure of unidirectional composites 15 Figure 2.4: Effect of matrix cracks in the ineffective length (left) and the stress concentration factors (right) [2]. strength from 1300 to 1600 MPa for T300 carbon fibre and from 2400 to 3100 MPa for T800 carbon fibre. This changes are due to modifications in the stress recovery profiles of the fibres. Pimenta and Pinho [6] reached similar conclusions using a different model. 2.6 Modelling the tensile failure of unidirectional composites There are several models to predict the tensile failure strength of UD composites in the literature. Mishnaevsky and Brøndsted [36] consider four categories of models: analytical models, fibre bundle models, fracture mechanics models and continuum damage mechanics models. The latter models usually lead to complex simulations, being therefore limited in the size of the models. In the next sections several models to predict the tensile failure of composites will be presented. The models are presented in five subsections. The first is about the deterministic rule of mixtures, the second presents the analytical fibre bundle models, the third presents the micromechanical models based on Monte-Carlo simulations, continuum damage mechanics based models and the last one presents a recent model based on a hierarchical scaling law. 2.6.1 Deterministic rule of mixtures The rule of mixtures is the simplest model to predict the tensile strength of UD composites. For the majority of composite materials the failure strain of the reinforcing fibres (εf) is lower than the failure strain of the matrix (εm), which means that the fibres will fail first. Usually composite materials have a fibre volume 16 Chapter 2. Mechanisms of longitudinal fracture fraction (Vf) around 50 to 70% and, after the fibres fail, the matrix is not able to carry the stress. This means that the failure strain of the composite (εC) is equal to the failure strain of the fibres (εf). This assumptions translate into the following equation for the composite tensile strength (Xt C): Xt C=VfXt f+ (1 −Vf)Em Ef Xt f;,(2.7) where Xt fis the fibres’ tensile strength and Efand Emare, respectively, the fibre and matrix stiffness. According to the already reviewed influencing factors of the tensile strength, this model fails in two aspects. Firstly, it considers the fibre strength to be a deterministic one, which was already proven to not be realistic, as the tensile strength of fibres follow a statistical distribution. The second flaw is that the model doesn’t consider interaction between components and therefore it fails to accurately predict the tensile strength of UD composites. More advanced and accurate models are presented in the next sections. 2.6.2 Analytical fibre bundle models Fibre bundle models (FBMs) consider a bundle of parallel fibres with stochastic tensile strength, but with the same elastic properties and loaded under uniaxial tension [36]. When the remote stress is high enough to make the weakest fibre fracture it breaks and the stress is redistributed towards the remaining intact fibres. If the stress concentration, due to this stress redistribution, is enough to make another fibre fail, it will fail and the stress is redistributed again, if not the remote stress is increased. This process is repeated until all fibres fail or until the material cannot withstand further load increments. FBMs have been developed for dry bundles (with no matrix) and for composite materials, considering the influence of the matrix. The matrix acts as a connector between fibres and alters the stress redistribution, affecting the ineffective length and the stress concentrations in the neighbouring fibres of a broken one. According to the stress distribution rule this models can be divided into global load sharing (GLS) models and local load sharing (LLS) models. The GLS models are able to predict the failure of a dry bundle where the interaction between the fibres is low [18]. The LLS models consider that there isn’t a uniform stress redistribution due to the presence of the matrix. Global load sharing models The first fibre bundle model was developed by Daniels [37] and was later developed by several authors. This model considers a bundle composed of Nfibres 2.6 Modelling the tensile failure of unidirectional composites 17 with a defined length (lr), and the fibre strength is considered to follow a Weibull distribution (Equation 2.1). He determined the following law relating the stress in the bundle (σ∞) with the applied strain (ε∞): σ∞=Efε∞·Sfσf,(2.8) where Efis the fibre longitudinal modulus, σfis the stress actuating in the intact fibres and Sf rσfrepresents survival probability under the stress σfof a fibre with length lr. In this equation the contribution of the matrix to the load carrying capacity has been neglected. This model allowed the conclusion that the tensile strength of a bundle with a large number of fibres can be represented by a normal distribution and the expected value for the strength of the bundle is given by: Xb r=σf r m1/me1/mwhere σf r=σf 0lr L0 −1/m .(2.9) σf 0,mand l0are the characteristic parameters of the Weibull distribution and eis the base of the natural logarithm. As this model was developed for dry bundles it considers that a broken fibre is no longer able to carry load, which is not accurate in the presence of a matrix. Rosen [27] considered the influence of the matrix trough a shear-lag model. This model considers that the stress is transferred, by the matrix, that is loaded in shear, back to the broken fibre. This means that there is a stress recovery in axial direction of the broken fibre. This leads to the definition of the ineffective length as the distance δfrom the break where the fibre recovered the ability to carry a percentage ξ(e.g. 90%) of the remote stress. Using his model he derived the following expression for the ineffective length δ: δ=φf 2s1−pVf pVf Ef Gm ln 1 1−ξ(2.10) where φfis the fibre diameter and Gmis the matrix shear modulus. Rosen considered that a bundle with length lrcould be divided into a chin of bundles width lengths lr/δand that the longer bundle will fail as soon as one of the sub-bundles fails, according to the weakest link theory. As the strength distributions of the sub-bundles are given by Daniels’ approach and, therefore, follow a normal distribution Fb δ(σ∞). This assumptions mean that the bundle follows weakest link theory and the strength distribution for a bundle with length lrcan be calculated by: Fb r(σ∞)=1−h1−Fb δ(σ∞)ilr/δ.(2.11) 24 Chapter 2. Mechanisms of longitudinal fracture Chapter 4: Strength model for UD non-hybrid composites 205 Enhanced superposition principle In view of this, an enhanced superposition principle is developed that does maintain force equilibrium. This principle first applies linear superposition to single break solutions, but then additionally distributes the SCFs that the fibre breaks exert on each other. This redistribution is performed proportionally to the SCFs obtained from linear superposition. This procedure is further clarified in Figure 4-22c. For the linear superposition in Figure 4-22a, an SCF of 33% is missing, which would mean that force equilibrium is not maintained. The enhanced superposition principle solves this problem by redistributing the missing SCF proportional to the SCFs from linear superposition (see Figure 4-22c). This proportional redistribution causes a larger portion of this stress redistribution to end up on the two fibres that previously carried 33%. These fibres receive an additional 6.7%, while this additional SCF is only 3.3% for the other 6 intact fibres (see Figure 4-22c). Figure 4-22: Illustration of the superposition principles in the fibre break plane: (a) a single fibre break solution (b) linear superposition of two coplanar fibre breaks, and (c) enhanced superposition of the same two fibre breaks. The white crosses indicate fibre breaks. The numbers inside the fibres indicate the value of the SCF as the percentage by which it exceeds unity. A hexagonal packing is assumed to simplify the situation, and the SCFs are assumed to be concentrated on the nearest neighbours only. This principle was illustrated for two coplanar fibre breaks, but can easily be extended to multiple non-coplanar fibre breaks. In case of non-coplanar fibre breaks, the stress in the broken fibre is not zero, which causes the SCFs on the intact fibres to be lower. The sum all of SCFs on intact fibres hence has to be equal to the percentage of load that is lost in the broken fibres. The extension towards more than two fibre breaks follows the same procedure. Validation A set of four FE models was created to analyse the stress redistribution around three fibre breaks and validate the enhanced superposition principle. Figure 4-23 illustrates these four models, which all have exactly the same random packing realisation and mesh. Three models with single fibre breaks and one with all three fibres broken were created by changing the boundary conditions. Linear and enhanced superposition results are computed based on the individual fibre break solutions. Their relative error in the maximum SCF is computed by comparison of the FE model with three fibre breaks. (a) (b) 16.7 33.3 16.7 16.7 16.7 16.7 33.3 16.7 20 40 20 20 20 20 40 20 16.7 16.7 0 16.7 16.7 0 16.7 16.7 0 (c) Figure 2.7: Schematics of the enhanced super-position stress redistribution: (a) stress concentration around a single break, (b) linear superposition results and (c) enhanced super-position [2]. simulation. This model was developed with the objective of incorporating different types of fibres in a RVE in order to study hybrid composites and his able to do so due to its versatility. According to the author [2] the discrepancies to the experimental data of this model are due to: (1) errors in the Weibull distribution, (2) neglecting dynamic stress concentrations and (3) averaging of the SCFs over the entire cross section of the fibre. Nonetheless, this model was used to predict the tensile behaviour and cluster formation in different composite materials with good results. Chapter 4: Strength model for UD non-hybrid composites 210 No new element failed New element failed Find break clusters Update SCFs for interacting breaks Update SCFs assuming non-interacting breaks Compare element stress and strength Apply Weibull strength Increase strain Calculate element stresses Create RVE Shape & dimensions Element length Packing type V f Fibre properties Matrix properties Boundary fibres Weibull parameters Cluster information Final failure? No Stop model Yes Stress-strain diagram SCF trend lines from FE model Figure 4-26: Flow chart of the strength model. The dashed rectangles indicate inputs and outputs. If the composite failure criterion is not satisfied and a new fibre element has failed, then the model updates the break-clusters. Two fibre breaks are considered to be part of the same break-cluster if: (1) the lateral distance between the fibre centres is smaller than 4 fibre radii, and (2) the axial distance between them is less than 10 fibre radii. This definition is illustrated in Figure 4-27. Figure 4-27a displays a cluster of only two fibre breaks, as the other two fibre breaks are too far apart, either in the axial or lateral direction. Figure 4-27b illustrates a cluster of five fibre breaks, where the outer fibre breaks on their own would not satisfy the definition. These breaks are still considered to be part of the same cluster. Figure 4-27: Illustration of the definition of a cluster: (a) a cluster of two fibre breaks as the other two fibre breaks are too far away, and (b) a cluster of five fibre breaks even though the fibre breaks on the left and right side on their own are too far apart. This illustration is not made to scale. (a) Maximum axial distance Maximum lateral distance Cluster Cluster (b) Figure 2.8: Flow chart of the model developed by Swolfs [2]. Two-scales FE models Finite elements are extensively used in modelling composite materials, however, the prediction of micromechanical behaviour requires extremely refined meshes, mak- 2.6 Modelling the tensile failure of unidirectional composites 25 ing the models computationally costly. This fine mesh is required in order to accurately capture the stress redistribution, especially when a random packing is considered. In order to avoid the use of such refined meshes in full scale models, coupled two-scale FE models have been developed [48, 49]. This separation of micro-macro scales allows the simulation of composites specimens with millions of fibres. In the model developed by Thionnet et al. [49] the micro-scale model considers a Unit Cell (UC) with a length of 4 mm and 32 linear-elastic fibres arranged in a square packing. This UC are used to study the stress distribution due to different number of broken fibres (2,4,8 or 16 broken fibres). This micro-scales model are used to generate a library with the stiffness of the UC and the stress concentrations in the fibres for different damage states [4]. The macro-scale FE model is composed of several UCs with one integration point to which is attributed a fibre strength given by a Weibull distribution. The damage state of the UC is considered to evolve from no damage to 2, 4, 8, 16 and 32 broken fibres, and the library of micro-scale models is used to predict the stress concentrations and stiffness reduction as function of the damage. It is considered that the specimen fails when there is a numerical instability, due to a rapid increase of the strain in constant stress. This model was used to predict the behaviour of UD composites under different loading conditions [49]. Also, it was considered the effect of time in the simulations, which lead the authors to conclude that time is an important factor due to visco-elastic behaviour of the matrix. Monte-Carlo based models are extensively used in the literature, however they require a very fine mesh and large number of simulations in order to accurately capture the material response. Nonetheless, they are very versatile and allow to take into account several factors that most models aren’t capable of. 2.6.4 Continuum damage mechanic based models The failure mechanisms of UD composites can be described in the framework of continuum damage mechanics. This type of modelling uses simple definitions of internal damage variables, formulated in the framework of the thermodynamics of irreversible process [36]. These damage variables are then used to alter the mechanical properties of the constituents, namely the reduction of stiffness. This type of modelling has been tackled by several authors and can be divided into three stages: (1) definition of a suitable norm for the damage variable, (2) definition of a damage criterion and (3) definition of the evolution law for the damage variable [5]. Matzenmiller et al. [50] developed a model that relates the effective elastic properties and the damage state of the composite material, and studied the 26 Chapter 2. Mechanisms of longitudinal fracture influence of the material parameters in the stress-strain diagrams. More recently, Turon et al. [5] developed a progressive damage model based on fibre fragmentation for UD composites. This model proposes a degradation of composite effective stiffness based on fibre fragmentation. The fibre fragmentation model considers that the fibre strength follows a Weibull distribution, and when the applied stress reaches the fibres tensile strength it will break. A broken fibre is still capable of carrying stress, according to a shear lag theory, and therefore it can fracture again at a certain distance from the original fracture. A fibre will fracture into shorter fragments until the shear stress transfer across the interface is no longer able to cause another fracture (Figure 2.9). where Kis the number of breaks per unit length, and it is computed from Eq. (5) K¼hNi L¼1 L0 r r0 !" q .ð7Þ This assumption is valid for an infinitely long fibre and at the initial fragmentation stages. As mentioned above, at advanced fragmentation stages, some flaws will be obscured in the load recovery region, and then, the break density will be lower than the predicted by Eq. (7). Many authors [9–11] introduce this phenomenon into their formulation obtaining other distribution functions which take into account higher break densities. Other authors [15,16] modify Eq. (7) and obtain an expression of the break density without the possible obscured potential breaks. In fact, the exact mathematical solution for the problem was provided by Hui et al. [11]. These distributions are more complex than Eq. (7) and, in some cases, numeric techniques are required for evaluating the expression. As the purpose of the present work is to develop a stiffness degradation model for the initial stages of damage, the influence of not considering the flaws obscured in the load recovery region is rather negligible. In Section 4, the numerical simulations from the present model based on Eq. (7) are compared to the more refined models cited above, and a very small difference for the first stages of fibre breakage is obtained. Moreover, at advanced fragmentation stages, the localized stress transfer in the composite has an important influence on the evolution of fibre breaks and the degradation and final failure is controlled by the formation and growth of clusters which require 2D models to be accounted for. In order to reach a mathematical expression of the apparent stiffness of the composite, it is necessary to compute the average fibre stress when some fractures have occurred. 2.3. Average fibre stress It has been shown in previous subsections that some flaws in a fibre will grow to a fully formed crack under the applied load. These cracks, or fibre breaks, cause a new stress redistribution along the fibre which may cause further breaks. When a fibre breaks, the load carried by the fibre drops down to zero at the position of the break and the load is carried by the shear stress between the fibre and the matrix (see Fig. 2). The stress in a broken fibre, r F , as a function of the distance from the break can be written as drF dz¼2s R;ð8Þ where Ris the radius of the fibre, sis the maximum shear stress and zis the distance from a break. This causes a stress redistribution near fibre breaks (see Fig. 2) which has been widely studied. Cox [17] was the pioneer to predict the real stress near the breaks by using a shear-lag model. The formulation of Cox!s model is quite complex and other simplified shear-lag approaches have been derived. One of the most widely used is the shear-lag model which was first introduced by Kelly and Tyson [18], and which assumes a linear increase of the axial stress from a fibre break, until a certain distance from it. At this distance, called the load recovery region, the stress reaches the far-field stress, see Fig. 2. According to the Kelly–Tyson shear-lag model, the length of this load recovery region, l ex , is obtained from the far-field stress, E F e(where E F is the fibre Young!s modulus and ethe composite strain), the radius of the fibre, R, and the maximum shear stress, s, between the fibre and the matrix before fibre debonding or matrix yielding occurs lex ¼R s EFe 2.ð9Þ From this stress redistribution, the average fibre stress along the fibre, r m , can be computed by integrating the axial stress over all of the fibre fragments along the fibre length rm¼NRðLÞ hi ¼N1 LZxRðxÞfðxÞdx;ð10Þ where f(x) is the fragment length distribution given in Eq. (6), and R(x) is the average stress in a fibre of length x. This integral is worked out in two steps. First, the average stress corresponding to the axial stress profile along a fibre fragment of length xis computed. Then, this axial average stress is integrated over all fibre fragments. In order to compute the axial average stresses for a fibre fragment of length x, it is necessary to distinguish whether the fibre fragment is greater to two times the length of the load recovery region (2l ex ), or not. (a) Average stress in a fibre of length 2l ex 6x. In a fibre of length x, greater than two times the stress recovery region, the stress profile assuming a linear shear-lag Fig. 2. Kelly–Tyson!s shear lag model. Stress profile at a fragment of broken fibre. At a break the axial stress is zero and it increases until it reaches the far-field stress (E F Æe). This region is called load recovery region, and has a length of l ex . 2042 A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048 Figure 2.9: Stress profile in a fibre with multiple fractures, according to shear-lag model[5]. Based on the multiple fracture of the fibres, the authors formulate a damage model that considers a global load sharing rule and the influence of each component of the material is accounted for by using the mixity law. The authors used this model, implemented in a FE model to study the effect of several parameters on the stress-strain curve and were able to accurately capture the stiffness loss in UD composites. 2.6.5 Hierarchical scaling law for the strength of composite fibre bundles Pimenta and Pinho [6] developed a model that considers a scaling law for composite strength. They consider the composite to be composed of bundles with different levels, and that a bundle of the level i+ 1 is composed of two level-ibundles (see Figure 2.10). The level-0 bundle is composed of a singular fibre (embedded in the matrix), which means that a level-ibundle has number of fibres niequal to 2i. The model firstly considers a level-1 bundle of length lrwhich is remotely loaded with a tensile stress σ∞. When both fibres are intact their stress is considered 2.6 Modelling the tensile failure of unidirectional composites 27 Laffan et al. (2010) and Pimenta et al. (2010) reported self-similar or quasi-fractal fracture surfaces in thin (under 0.5 mm) UD laminas and fibre bundles; this provides experimental evidence that the length-scale of the failure process increases with the number of fibres involved. Moreover, such observations suggest a hierarchical failure process, hence supporting the use of hierarchical models —e.g. Newman and Gabrielov’s (1991) model for dry bundles. Here, considering that a bundle of level [iþ1] is composed of two sub-bundles of level [i], strength distributions were calculated recursively as F ½iþ1# ð s Þ¼F ½i# ð s Þ'½2'F ½i# ð2' s Þ(F ½i# ð s Þ# ð3Þ where F ½i# ð s Þis the failure probability of a level-[i]bundleunderanappliedstress s .Therecursivenatureofthisscalinglawalso allowed its efficient implementation, so that large-scale bundles could be computed. However —being a model for dry bundles —it does not consider the effect of an embedding matrix, and does not include any characteristiclength(whichisparamount for quasi-brittle materials, Baz ˇant, 1999); the model is also inconsistent with the WLT for length scaling. Altogether, a comprehensive explanation of the micromechanics and statistics of tensile failure in composites is yet to be provided, as are validated quantitative predictions over a complete range of scales. Still, FBMs surface as one of the most promising approaches to overcome this knowledge gap. This paper presents the development, implementation and validation of a FBM for predicting size effects on the longitudinal tensile strength of composite bundles. Following Newman and Gabrielov’s (1991) work, bundles are hierarchically organised; however, the role of the matrix (or fibre–matrix interface) is now considered through a simplified shear-lag model, in which the characteristic length scales hierarchically as well. This paper is organised as follows: Section 2 presents the analytical model for predicting strength distributions of FRP bundles of different dimensions. Section 3 explores modelling results (including experimental validation), subsequently discussed in Section 4. Finally, Section 5 draws the main conclusions. 2. Model development 2.1. Fibre bundle geometry and shear-lag boundary This model is based on hierarchical fibre–matrix bundles (Fig. 1a). These are generated by pairing individual fibres (level-[0]) into level-[1] bundles, and then sequentially grouping two level-[i] bundles into one level-[iþ1] bundle (Newman and Gabrielov, 1991). The number of fibres (n ½i# ) in a level-[i] bundle is therefore: n ½i# ¼2 i 3i¼log 2 n ½i# ð4Þ The fibres (superscript f,diameter f f ,circumferenceC f and area A f )areembeddedinthematrix(withvolumefractionV f )in asquarearchitecture(Fig. 1b). During hierarchical failure of a large composite bundle (Fig. 2a), shear-lag stresses will be transferred between the (unbroken) surrounding material and a broken level-[i] bundle through the shear-lag boundary, with perimeter C ½i# . Considering preferential interfacial debonding (Fig. 2b), C ½i# ¼3'C f þ4'ð ffiffiffiffiffiffi n ½i# p(1Þ's Q þð ffiffiffiffiffiffi n ½i# p(2Þ'C f 2 "# with s Q ¼ffiffiffiffi p p 2'ffiffiffiffiffiffi V f p(1 $% ' f f ð5Þ This expression is strictly valid only for even values of i, but used for any bundle size so that C ½i# is a smooth function of n ½i# . Other geometries for bundles and their boundaries (e.g. hexagonal fibre arrangement with fractal boundary, preferential matrix failure, free-edge effects) are considered in Appendix A. These variations are shown to have a minor influence on calculated bundle strength distributions (as already suggested by Curtin and Takeda, 1998). 2.2. Stress field around a fibre break and definition of the control region Consider a level-[1] bundle of reference length l r , composed of two level-[0] fibres (Aand B) in a soft matrix (i¼1 in Fig. 1a). The bundle is loaded in tension by a progressively increasing remote stress s 1 , so that each fibre undergoes Fig. 1. Hierarchical bundles in square fibre arrangement. (a) Bundle hierarchy. (b) Fibre arrangement. S. Pimenta, S.T. Pinho / J. Mech. Phys. Solids 61 (2013) 1337–1356 1339 Figure 2.10: Hierarchical fibre bundles [6]. equal and equal to σ∞. The strength of the fibres is consider to follow a Weibull distribution. When the weakest fibre in the level-1 bundle fails, the broken fibre is assumed to follow a perfectly-plastic shear-lag formulation, with shear strength τSL and, therefore, the broken fibre linearly recovers its capacity to carry load (in the length le). Has the broken fibre locally looses its ability to carry stress, there is a stress concentration in the remaining intact fibre, increasing its probability of failure. A level-1 bundle is considered to fail if both fibres fail in nearby locations, close enough to promote complete yielding of the matrix. Is so possible to define a control region, where if the second fibre breaks leads to the failure of the bundle (see Figure 2.11). This region has a length lc= 2le. a uniform stress state s A ¼ s B ¼ s 1 . Note that longitudinal stresses are expressed as fibre stresses, i.e. normalised by the area of fibres in the cross section. Assume that fibre Afails at the location x¼0 under a given s 1 (Fig. 3a). Shear-lag models have been shown to accurately reproduce the resulting stress fields, as validated by more complex Finite Element analyses (Landis and McMeeking, 1999;de Morais, 2001). The in situ response of the matrix/interface to this event is complex, as for instance epoxy is usually brittle in bulk, but actually ductile and much stronger in situ (Gulino et al., 1991;Hobbiebrunken et al., Fig. 2. Shear-lag boundary (assuming preferential interfacial failure). (a) Longitudinal view. (b) Section view. Fig. 3. Stress fields and length scales in a level-[1] fibre bundle. (a) Stress fields after first fibre failure. (b) Definition of critical distance between fibre breaks: the bundle fails only if fibre Bbreaks at a distance smaller than l c =2 from the break in fibre A. (c) Definition of the control region and fibre segments. S. Pimenta, S.T. Pinho / J. Mech. Phys. Solids 61 (2013) 1337–13561340 Figure 2.11: Definition of control region [6]. A level-1 bundle with length lcsurvives an applied remote stress σ∞in the following situations: •If the stress σ∞is lower than the strength of both fibres. Which means that the four elements with length Leremain intact. If the survival probability of a level-0 bundle is S0 U,e, then the survival probability of the four regions is hS0 U,ei4; 28 Chapter 2. Mechanisms of longitudinal fracture •If the weakest fibre fails in the stress σ∞and the surviving fibre is able to support the stress concentration. The stress is the intact fibre (Figure 2.11) consists of half a length of constant stress and the other half with linear stress concentrations. If the surviving probability of a element with length leunder linear stress concentrations is S0 K,e, the probability of the bundle to survive in this conditions is given by: 21−S0 U,e2S0 U,eS0 K,e. Considering both events, the failure probability of a level-1 bundle with length lc, under the stress σ∞can be expressed as S1 U,c =hS0 U,ei4+ 2 1−S0 U,e2S0 U,eS0 K,e .(2.21) The authors consider a self-similar hierarchical failure process and generalise the failure events to higher order bundles, obtaining the following hierarchical scaling law: Si+1 U,c =hSi U,ei4+ 2 1−Si U,e2Si U,eSi K,e ,(2.22) where the probabilities with the superscript irefer to the failure modes of a level-i bundle, the same for the superscript i+. The subscript Urefers to uniform stress in the bundle region and the subscript Kto linear stress concentrations in that region. The survival probability distributions Si U,e and Si K,e can be calculated analytically, taking into account that: li e= 2 niAf CiτSLσ∞and li c=li−1 e,(2.23) where Cithe shear-lag perimeter for a level-i bundle, τSL is the yielding strength of the matrix (or sliding resistance) and Afis the area of a single fibre. It should be noted that the length of the damage zone increases, not only with the level of the bundle, but also with the applied stress. This means that the cluster of broken fibres has influence over a larger zone than a single fibre break. This model is able to make predictions of tensile behaviour of composites with large number of fibres in short amount of time and is able to capture the size effects of composite materials. 2.7 Conclusion In this chapter several models for tensile behaviour of UD composites have been presented, as well as the main influencing parameters in these materials’ behaviour. It is well agreed that the failure of UD composites is a progressive one and that 2.7 Conclusion 29 fibre will fracture progressively forming clusters that will grow until a critical size is reached. This, in turn, results in an unstable propagation leading to failure. This process in governed mainly by the fibre strength statistics and the micromechanical stress redistribution, which has been proven to be affected by several parameters. Several models, with different backgrounds and formulations, are able to accurately predict the behaviour of UD composites, however there are still improvements to be made. Most of the models don’t include the increase in influence area of a cluster in relation that of a single broken fibre, which has been proven by experimental data [4]. Another aspect that most models fail to capture is the dynamic effects of the loading, which has been proven to affect the matrix properties. The dynamic effects are also present when a fibre fails, leading to dynamic stress concentrations that vary with time [39]. In the next chapter models will be presented for hybrid UD composites, and the main influencing factor in hybrid behaviour will be accessed. 30 Chapter 2. Mechanisms of longitudinal fracture Chapter 3 HybridizationState-of-the-art The previous chapter focused in understanding the mechanisms of failure in unidirectional composites. While it is important to understand the models and physical mechanisms of UD failure, the main goal of this thesis is the understanding of the failure mechanisms of hybrid unidirectional composites. To do so, it is necessary to understand the main aspects regarding hybrid composites and the main parameters that influence their behaviour. In this chapter, the state-of-the-art of hybrid composites will be reviewed, firstly focusing on the general aspects of hybrid composites, including the main effects of hybridization under different loadings. Later on, the models for tensile failure of UD hybrid composites will be reviewed and conclusions will be drawn regarding the most influential parameters of their behaviour. 3.1 Hybrid composites Fibre-reinforced composites play a fundamental role in aircraft structural applications, however their optimal use is still hampered partly due to the relatively low toughness they exhibit. Hybridisation is a strategy that can lead to improved composite properties. Hybrid composites can be defined as a composite material that contains more than one of type of fibre and/or matrix system [51]. The main focus of this work will be in the hybrid composites with more than a single type of fibres, the so called fibre-hybrid composites. 31 32 Chapter 3. HybridizationState-of-the-art 4.3. Impactresistance ............................................................................................... 192 4.3.1. Positioning ............................................................................................. 192 4.3.2. Dispersion.............................................................................................. 193 4.3.3. Conclusion.............................................................................................. 193 4.4. Fatigueresistance............................................................................................... 193 4.5. Conclusion..................................................................................................... 194 5. Currenttrends ....................................................................................................... 194 5.1. Pseudo-ductility ................................................................................................ 194 5.2. Ductilefibres................................................................................................... 195 5.3. Naturalfibrehybrids............................................................................................. 196 6. Conclusionandoutlook................................................................................................ 197 Acknowledgments.................................................................................................... 197 References .......................................................................................................... 197 1. Introduction Lightweight design is becoming increasingly important in various industries, particularly in aerospace, wind energy and automotive applications. Fibre-reinforced composites are attracting more interest for these weight-sensitive applications as their excellent stiffness and strength are combined with a low density. Unfortunately, the high stiffness and strength of these composites come at the expense of their limited toughness. Like most materials, fibre-reinforced composites also face the strength versus toughness dilemma. Over the years, toughening of fibre-reinforced polymer composites has been a highly active research area. Many different strategies have been proposed to make these materials more damage resistant and less brittle. One of the most researched strategy is toughening of the polymer matrix by tuning the polymer chemistry or by rubbers, thermoplastics or nano-scale reinforcements. In this strategy, the increased matrix toughness has a beneficial effect on the matrix-dominated composite properties [1–3]. In search of new toughening mechanisms, there has been an increasing interest in structure–property relations of biological composites that are exceptionally resilient to failure [4–6]. The failure strain and toughness can be dramatically increased if brittle fibres are replaced by ductile fibres. In this respect, metal fibres have the potential of high stiffness and large failure strain, but they are hampered by their high densities. Polymer fibres, on the other hand, do have low densities and can be ductile, but are limited by their low stiffness and limited temperature resistance. Because of the drawbacks of these toughening strategies and the strong need for new lightweight materials with improved toughness, the research interest in ‘‘hybridization’’, is reviving. The term ‘hybrid composite’ is generally used to describe a matrix containing at least two types of reinforcements, but this review is restricted to hybrid composites containing two types of reinforcing fibres. Such composites are also called ‘fibre hybrids’ or ‘fibre hybrid composites’. This review focuses on polymer matrix composites, though some references to hybrid composites with ceramic or metal matrices will be made. Research on fibre hybrid composites started several decades ago. After the invention of carbon fibres in the sixties [7,8], the high price was their main drawback. In an attempt to reduce the price, while still exploiting the exceptional properties of carbon fibre, hybridization became a highly active research area in the seventies and eighties. Afterwards, the price dropped [9] and the focus shifted towards production technologies and understanding the mechanical behaviour of non-hybrid composites. The last review paper on hybrid composites was written in 1987 by Kretsis [10]. Since then, a much wider range of materials is available and several processing technologies have been invented and improved. This resulted in a renewed interest in hybrid composites as a possible strategy for toughening fibre-reinforced composites. In general, the purpose of bringing two fibre types in a single composite is to maintain the advantages of both fibres and alleviate some disadvantages. For instance, replacing carbon fibres in the middle of a laminate by cheaper glass fibres can significantly reduce the cost, while the flexural properties remain almost unaffected. If a hybrid composite is loaded in the fibre direction in tension, then the more brittle fibres will fail before the more ductile fibres. This fracture behaviour can be used for health monitoring purposes [12] or as a warning sign before final failure [13]. The two fibre types are typically referred to as low elongation (LE) and high elongation (HE) fibres. The first fibre to fail is normally the LE fibre. The HE fibre does not necessarily have a large failure strain, but it is always larger than the one of the LE fibre. This is also the reason why the terminology brittle/ductile fibres instead of LE/HE fibres can lead to confusion. The LE and HE fibres can be combined in many different configurations. The three most important configurations are visualised in Fig. 1. In the interlayer configuration, see Fig. 1a, the layers of two fibre types are stacked onto each other. This is the simplest and cheapest method for producing a hybrid composite. In the Fig. 1. The three main hybrid configurations: (a) interlayer or layer-by-layer, (b) intralayer or yarn-by-yarn, and (c) intrayarn or fibre-by-fibre. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 182 Y. Swolfs et al. / Composites: Part A 67 (2014) 181–200 Figure 3.1: Hybrid configurations: (a) interlayer, (b) intralayer and (c) intrayarn configurations [7]. There are three main types of fibre-hybrid composites, defined according to the configurations of both fibre types: •Interlayer or layer-by-layer hybrids have different types of fibres in different layers, being that each layer only has a single fibre type (Figure 3.1a); •Intralayer or yarn-by-yarn hybrids have both types of fibres in a single layer (Figure 3.1b), layers that can be stacked in different configurations; •Intrayarn or fibre-by-fibre hybrids have both fibre types in a single tow (Figure 3.1c). This type of configuration is the one that leads to a better dispersion of both fibre types. The different types of hybridization will lead to different properties and different mechanisms of failure. The study of hybrid composites started in the 70s. Due to the high price of the recently invented carbon fibres, cheaper fibres, like glass fibres, would be added to the carbon fibre composites in order to reduce the material’s price and still take advantage of the better properties of carbon fibres. Afterwards, and with the reduction of carbon fibre prices, the hybridization of composite materials became a secondary topic and the focus became in modelling non-hybrid composites. Nowadays the topic of hybridisation is growing in interest due to the fact that it allows the compensation of some of the disadvantages while maintaining part of the advantages of each fibre. This is valid in several loading cases, for instance, in flexural loadings it is advantageous changing the inner carbon layer for carbon fibres allows the reduction of the price of the material while maintaining its flexural properties [7]. In hybrid composite materials it is usual to refer the two types of fibres as: high elongation (HE) and low elongation (LE) fibres. The HE fibres are the ones that have the highest failure strain, while LE have the lowest. It should be noted that high and low are relative terms and that in a hybrid system glass fibres can be the 3.1 Hybrid composites 33 HE fibres while in another they can be the LE fibres. In the topic of hybrid composites it is usual to refer to the hybrid effect, whose definition and characteristics are presented next. 3.1.1 Hybrid effect The hybrid effect was firstly defined by Hayashi [52], in 1972, when he noticed that the apparent failure strain of the carbon fibres in a carbon/glass hybrid was enhanced in relation to that of the non-hybrid carbon composite. This experimental fact lead to the creation of the first definition of hybrid effect, defined as the apparent failure strain enhancement of the LE fibres in a hybrid composite compared to the failure strain of the LE fibress in the non-hybrid reference composite (Figure 3.2a). The application of this definition requires an accurate determination of the failure strain of the reference non-hybrid composite as this value is the baseline in the determination of the hybrid effect. The determination of the failure strain can be affected by the experimental set-up, namely the stress concentrations at the grips. These stress concentrations are usually higher for non-hybrids than for hybrid composites [7], leading to a reduction of the baseline failure strain, which can result in an overestimation of the hybrid effect. intralayer hybrid, the two fibre types are mixed within the layers. This is illustrated in Fig. 1b, where different yarns are co-woven into a fabric. Other intralayer configurations such as parallel bundles are also possible. The two fibre types can also be mixed or comingled on the fibre level, resulting in an intrayarn hybrid (see Fig. 1c). More complex configurations can be obtained by combining two of these three configurations. For example, an intrayarn hybrid can be woven together with a homogeneous yarn. A crucial aspect in hybrid composites is the dispersion of the two fibre types. This is a measure for how well the two fibre types are mixed and is defined as the reciprocal of the smallest repeat length [10,14].Fig. 2 schematically illustrates the degree of dispersion. Fig. 2a shows a hybrid with a low degree of dispersion, as the two fibre types are in two distinct layers. This can be improved by increasing the number of layers or decreasing the layer thickness, as illustrated in Fig. 2b. Another way to increase the dispersion is by hybridising on the fibre bundle level, see Fig. 2c. The best dispersion is achieved if the two fibre types are completely randomly distributed, as in Fig. 2d. The present paper is split up into six sections, of which the first one is this introduction. In the second section, the synergy between the two fibres, the so-called hybrid effect, will be discussed. The third section reviews the existing models for the hybrid effect and failure development of UD hybrid composites and provides suggestions for future model developments. The fourth section describes the mechanical properties of composites and how they can be improved by fibre hybridisation. The fifth section gives an overview of the most recent trends in fibre hybridisation. The final section gives conclusions as well as recommendations for future work. 2. The hybrid effect 2.1. Introduction In 1972, Hayashi [15] reported that the failure strain of the carbon fibre layers in a carbon/glass hybrid composite was 40% higher than in the reference carbon fibre composite. As will be shown in ‘‘4.1.2 Failure strain’’, typical values for this remarkable synergistic effect are typically in the range 10% to 50%. Various definitions have been coined for this hybrid effect. The most basic definition of the hybrid effect is the apparent failure strain enhancement of the LE fibre in a hybrid composite compared to the failure strain of a LE fibre-reinforced non-hybrid composite. This definition is schematically illustrated in Fig. 3a and corresponds to Hayashi’s observations [15]. This definition requires an accurate determination of the failure strain of the reference carbon fibre composite. Fig. 2. Illustration of the various degrees of dispersion (a) two layers, (b) alternating layers, (c) bundle-by-bundle dispersion, and (d) completely random dispersion. Displacement Load Hybrid composite Posi!ve hybrid effect LE failure 0% 100% Vol% LE fibre composite Property Posi!ve hybrid effect Nega!ve hybrid effect Property (a) (b) Fig. 3. Illustration of the definitions of the hybrid effect: (a) the apparent failure strain enhancement of the LE fibres, under the assumption that relative volume fraction is 50/ 50 and that the hybrid composite is twice as thick as the reference composites and (b) a deviation from the rule of mixtures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Y. Swolfs et al. / Composites: Part A 67 (2014) 181–200 183 Figure 3.2: Diagrams for the definition of the hybrid effect: (a) first definition proposed by Hayashi and (b) general definition based on the rule-of-mixtures [7]. The definition proposed by Hayashi [52] for the hybrid effect refers only to the enhancement of apparent failure strain, however, hybridization introduces changes in other mechanical properties [53]. This lead to a new, more general definition for the hybrid effect. Hybrid effect was then defined has a deviation from the linear rule of mixtures [54]. This definition is more general and allows it to be applied to several mechanical properties and allows the existence of positive or negative hybrid effect (Figure 3.2b) if there is, respectively, an improvement or a deterioration of the property in question. 40 Chapter 3. HybridizationState-of-the-art is found between the experiments and FEA, while the CLT gives much lower flexural moduli. Both the experimental and FEA results show that flexural modulus decreases with increasing percentage of S-2 glass fiber. The FEA results show that the flexural modulus of the G 1 C 4 configuration is 21.1% lower than that of the full carbon configuration but 34.3% higher than the full glass configuration. Likewise, the flexural modulus of the G 2 C 3 configuration is 23.3% lower than that of the full carbon configuration but 30.7% higher than the full glass configuration. Both the experimental and FEA results suggest that the flexural moduli of these two hybrid configurations are not significantly different, and no significant hybrid effects are found. Flexural Strength The flexural strengths from the experiments, FEA and CLT are shown in Fig. 14. It is seen that for the C 6 , G 2 C 3, and G 5 configurations, the flexural strength from the experiments is in reasonable agreement with the FEA prediction. However, the flexural strength of the G 1 C 4 configuration from the experiments is significantly higher than the FEA prediction. This huge difference is due to the occurrence of delamination, since the flexural strength is predicted based on the assumption that composites fail by microbuckling. The CLT overestimates the flexural strengths of the C 6 ,G 1 C 4, and G 5 configurations while underestimating that of the G 1 C 4 configuration. Both the experimental and FEA results show that positive hybrid effects exist for both the G 1 C 4 and G 2 C 3 configurations, and the G 1 C 4 configuration yields the highest flexural strength. The experiments suggest that the average flexural strength of the G 1 C 4 configuration is 90.6 and 48.2% higher than that of full carbon and full glass configurations, respectively. The FEA prediction suggests that the flexural strength of the G 1 C 4 configuration is 25.8 and 20.7% higher than that of full carbon and full glass configurations, respectively. The average flexural strength of the G 2 C 3 configuration from the experiments is 40.2 and 9.2% higher than that of full carbon and full glass configurations, respectively. The FEA prediction shows that the flexural strength of the G 1 C 4 configuration is 21.6 and 16.6% higher than that of full carbon and full glass configurations, respectively. It can be concluded from both the experimental and FEA results that positive hybrid effects exist by substituting carbon fibers with glass fibers. The G 1 C 4 configuration yields the highest flexural strength, which is in agreement with a recent study by Sudarisman et al. [12], who noted a positive hybrid effect, with smaller amounts of glass fiber substitution (approximately up to 25%) in a glass/carbon composite resulting in greater increases of flexural strength. The experimental flexural strength of the G 1 C 4 configuration is significantly higher than the FEA prediction. Although the reason is unclear, it is noticed that delamination occurs with this unusual high flexural strength, and further study is needed. In summary, it is seen that positive hybrid effects exist for both hybrid configurations. The G 1 C 4 configuration yields the highest flexural strength. Although the thicknesses were not constant due to the hand-layup process, the experimental results were sufficient to validate our modeling approach. The future work will use the modeling approach to investigate the effects of hybrid ratio, V f , etc. on flexural strength. CONCLUSIONS A study on the flexural properties of hybrid composites reinforced by S-2 glass and TR30S carbon fibers is presented in this article. Specimens were made by the hand lay-up process in an intra-ply configuration with varying degrees of glass fibers added to the surface of a carbon laminate. Specimens were then tested in the three point bend configuration in accordance with ASTM D790-07 at a span to depth ratio of 32. The failure modes were examined under an optical microscope. The results show that the dominant failure mode is compressive failure. The flexural behavior was also simulated by FEA, and the flexural modulus and flexural strength were calculated. FIG. 14. Flexural strengths and hybrid effects from experiments and FEA 180 370 mm (300 3300 DPI). 780 POLYMER COMPOSITES—-2012 DOI 10.1002/pc Figure 3.7: Effects in the flexural strength of hybridizing carbon fibre composites with glass fibres in the compressive layers [8]. simple FE analysis. Other authors [66] reached similar conclusions in hybridizing carbon composites with glass fibres. These results mean that an symmetrical layup may not be optimal when there are flexural loads and that there is an optimal hybrid ratio to improve flexural properties wich, according to Dong et al.[65], is 12.5% of glass fibres. 3.2.3 Impact resistance One of the main goals of hybridizing fibrous composites is improving the toughness of these materials, making impact resistance properties important, as they are related with the toughness of the material. Impact resistance can be characterized by three parameters: (1) energy absorbed during penetration impact, (2) damaged area after a non-penetrating impact and (3) post impact properties. These parameters are governed by different mechanisms and hybridization may affect differently each one of them [7]. In impact tests, the material behaviour is highly dependent on the ply configuration and, therefore, hybridizing different plies will have different influence in the impact resistance. Similarly to the other mechanical properties, the dispersion of both fibres types is important in impact resistance, in this case due to changes in the damage mechanisms [2]. For interlayer hybrids,the positioning of the layers is important because it not only changes the flexural properties (as seen in Section 3.2.2), but also the damage mechanisms to dissipate the impact energy. Sayer et al. [67] tested asymmetric interlayer carbon/glass hybrids. With this asymmetric laminate it was possible to study the effect of having the LE in the impact side or in the other side of the laminate, which is subjected to tensile loadings. The authors found that if the carbon fibres (LE) were on the impacted side, the impact resistance was increased by 30%. 3.2 Mechanical properties of hybrid composites 41 Jang et al. [68] studied several hybrid composites with different fibre types. For a two-layer carbon/aramid hybrid, the dependence of which layer is in the compressive side was reduced, fact that was attributed to similar impact behaviour of the aramid and carbon reference composites. However, replacing the carbon fibres with polyethylene (PE) resulted in diferent behaviours if the aramid fibres were on the compressive or tensile side of the impact. If the PE fibres (HE fibres) were on the compressive side, the impact resistance increase by 50% in comparison with the aramid fibres in the compressive side, which suggests that HE preform better when in the tensile side of the impact. These results contradict the ones of Sayer et al. [67], however thheys can be attributed to differences in the damage mechanisms which are related to fibre and fibre/matrix interface properties [2]. Naik et al. [69] tested the impact behaviour and post-impact properties of carbon/glass symmetrical hybrids and reported that the compression-after-impact strength of the hybrid material was higher than that of both reference composites (non-hybrid). As seen, the positioning of the different plies affects the impact properties of the material however dispersion also influences these properties. Sarasini et al. [70] tested glass/basalt hybrid composites and concluded that the well dispersed specimens showed smaller damaged area and higher post-impact flexural strength, which was attributed to the presence of high amount of small delaminations in the well dispersed composites compared to extensive fibre breaks and delaminations in the less dispersed ones. De Rosa et al. [71] got similar results for the same hybrid material. Park and Jang [72] studied aramid/polyethylene hybrids and observed that the interlayer hybrids had a higher penetration impact resistance than the intralayer hybrids, which means that less dispersed composites had better impact resistance. In terms of damaged area it was found that intralayer hybrids presented a smaller damaged zone and therefore should have better post-impact properties (which were not determined). 3.2.4 Fatigue resistance Although fatigue resistance is an important property for many applications, the effects of hybridization in this property have not been extensively studied [7]. In principle, hybridization should lead to improved fatigue properties, as HE can act as bridging points in a crack and stop crack propagation. Wu et al. [9] studied the fatigue properties of several materials, including hybrid composites. 42 Chapter 3. HybridizationState-of-the-art basalt hybrid (C1B1) was, thus, stronger than that of the glass hybrid (C1G1). Due to the effect of the low moduli of the GFRP and BFRP composites, the adhesive layer of the hybrid composites attracted more stresses (i.e., transverse cracks), as shown in Fig. 7c and d. The propagated transverse cracks induced local damage concentrations in the hybrid FRP composites, as typically shown in Fig. 7d. 4.3. Damage accumulation Fig. 8 shows a change of the tensile moduli of FRP composites, depending upon the normalized fatigue cycles. The damage represented by the reduced modulus was permanent. The fatigue failure of tested coupons, thus, occurred when the total accumulated damage reached a critical limit. Although there were notable scatters in the modulus reduction (Fig. 8), the critical limit was approximately 60–80% of the initial modulus for all types of fibres. Table 2 Details of the experimental program. FRP Specimens Max load ratio (%) Failure cycles Remarks FRP Specimens Max load ratio (%) Failure cycles Remarks CFRP C-1 93 4574 BFRP B-1 93 852 C-2 93 3318 B-2 93 4191 C-3 93 145,305 B-3 93 5191 C-4 88 20641 B-4 81 3107 C-5 88 724,917 B-5 81 30,295 C-6 88 1623 B-6 81 45,397 C-7 82 924,575 B-7 70 171,255 C-8 82 306,419 B-8 70 1,029,247 C-9 82 1,485,196 B-9 70 1,324,600 C-10 77 2,000,000 Not failed B-10 55 2,000,000 Not failed C-11 77 2,000,000 Not failed B-11 55 2,000,000 Not failed C-12 77 2000000 Not failed B-12 55 2,000,000 Not failed PBO P-1 85 12,779 C1G1 C1G1-1 89 957 P-2 85 4618 C1G1-2 89 596 P-3 85 80,235 C1G1-3 89 1096 P-4 82 453,391 C1G1-4 79 1454 P-5 82 12,779 C1G1-5 79 2622 P-6 80 2,000,000 Not failed C1G1-6 65 48,731 P-7 80 1,585,520 C1G1-7 65 43,086 P-8 80 1,331,56 C1G1-8 58 2,000,000 Not failed P-9 80 21,464 C1G1-9 58 1,662,511 P-10 80 40,100 C1G1-10 58 6,638,04 P-11 70 2,000,000 Not failed C1B1 C1B1-1 90 1546 P-12 65 2,000,000 Not failed C1B1-2 90 2278 GFRP G-1 73 7095 C1B1-3 90 619 G-2 73 20,925 C1B1-4 80 98,816 G-3 73 816,000 C1B1-5 80 801,890 G-4 64 198,419 C1B1-6 80 794,722 G-5 64 464,602 C1B1-7 70 2,130,572 G-6 64 2,000,000 Not failed C1B1-8 70 1,364,690 G-7 64 2,000,000 Not failed C1B1-9 70 2,000,000 Not failed G-8 55 2,000,000 Not failed G-9 55 2,000,000 Not failed G-10 55 2,000,000 Not failed Table 3 Mechanical properties of composite coupons tested in monotonic load. FRP Measured properties a Tensile strength (MPa) Rupture strain (%) Tensile modulus (GPa) fu r CV (%) f u !3 r CFRP 4214 258 6.12 3440 1.74 242 PBO 4250 250 5.86 3503 1.60 266 GFRP 2121 178 8.39 1587 2.45 87 BFRP 2332 58 2.49 2158 2.56 91 C1G1 b 3305 288 8.71 2441 2.04 162 C1B1 b 2771 223 8.05 2102 1.67 166 a Average of four test coupons. b Average strength counting for the entire hybrid FRP sheet. 40% 50% 60% 70% 80% 90% 100% 110% -0.5 0.5 1.5 2.5 3.5 4.5 5.5 6.5 Lo g N S=P max /P avg CFRP PBO GFRP BFRP C1G1 C1B1 S=1.001-0.026logN S=1.011-0.042logN S=1.004-0.062logN S=0.994-0.036logN S=1.015-0.069logN S=0.997-0.071logN Fig. 5. Fatigue response of FRP sheets. Z. Wu et al. / Composites: Part B 41 (2010) 396–402 399 Figure 3.8: Fatigue response of several fibre reinforced composites [9]. As one can see in Figure 3.8 for the in the same conditions (same fraction of maximum and medium loads), the addition of carbon fibres, whose base behaviour is represented by CFRP, in a basalt composite (BFRP) increases the number of cycles to rupture of the hybrid material (C1B1). The authors justified this increase with the reduction of the stress in the basalt fibres due to the addition of the carbon fibres, that have a higher modulus, improving the fatigue life of the basalt fibres. The addition of carbon to a glass composite (GFRP) didn’t have the same effect as the previous material (see Figure 3.8-C1G1), which was attributed to the superficial properties of glass fibres. Peijis and de Kok [73] studied the fatigue resistance of PE/carbon hybrids and found that hybridization resulted in flatter S-N curves, meaning that the fatigue life of the material was improved. They also reached the conclusion that hybrid composites have a less scattered fatigue life and that increasing the dispersion improves it. 3.2.5 Pseudo-ductile behaviour Fibrous composites are widely used, however, the longitudinal failure of this material is catastrophic and without any previous warning. Another problem is that the material can be damaged in its interior without it being noticeable in the outer layers. This can lead to mechanical properties lower than expected, leading to a premature failure of the component. Hybridization can tackle some of these challenges by developing a more gradual 3.2 Mechanical properties of hybrid composites 43 failure of the material. The usual behaviour of a composite material is showed in Figure 3.9a, which represents a catastrophic failure. Hybridization leads to a diagram like the on in Figure 3.9b but, if the material is designed correctly, a pseudoductile behaviour can be achieved (Figure 3.9c), which is considered to be the ideal response of a hybrid composite [74]. possible to achieve a more gradual failure and hence pseudo-ductility, as illustrated in Fig. 9c[13,116]. There is a growing interest in pseudo-ductile material systems. This is driven by a strong need to reduce the safety factor in the design of composites and the corresponding need for increased toughness. Pseudo-ductility can also be achieved by controlling the damage mechanisms in non-hybrid composites [127,128], but the focus here is on pseudo-ductility in hybrid composites. Czél et al. [13] sandwiched a 29 l m thin layer of unidirectional carbon fibre-epoxy in between thicker layers of glass fibre-epoxy on each side. By making the carbon fibre layer thin enough, a change in the material behaviour was observed. The carbon fibre layer is able to break several times along the length of the sample, before the glass fibre layers break. For their specific material combination, an upper limit of 84 l m for the carbon fibre layer thickness was determined both experimentally and theoretically. Further understanding of this phenomenon was performed by Jalalvand et al. [129], who developed a finite element model for these thin ply hybrid composites. This led to the development of damage mode maps with relative thickness and absolute thickness on x and y-axis (see Fig. 10), showing four quadrants, each of which represent a different failure behaviour of the hybrid composite. Jones and Dibenedetto [62] achieved pseudo-ductile behaviour by finely dispersing carbon fibres with glass or aramid fibres. They calculated an upper limit of 92% improvement in the apparent strength of the carbon fibres if all carbon fibres acted independently from each other. This high value could only be achieved at carbon fibre volume fraction below 6%. The importance of fine dispersion for pseudo-ductility is also shown by Bakis et al. [66] on pultruded rods. Pseudo-ductility was only achieved for their most finely dispersed carbon/glass hybrid, while lower dispersion resulted in two distinct peaks as shown in Fig. 9b. Somboonsong et al. [130] achieved pseudo-ductility in hybrid bars, by braiding and pultruding carbon and aramid yarns. The various stress drops were attributed to yarns breaking and transferring their stress to the other yarns. Based on their models, Somboonsong et al. could show that the braiding architecture was important in achieving this pseudo-ductility. Liang et al. [67] demonstrated that carbon/glass rods break at the failure strain of the carbon fibres when the fibres are well dispersed. Some degree of pseudo-ductility is claimed when all the glass fibres were put on the inside. Their tensile diagrams resemble the one in Fig. 9b and therefore should not be called pseudoductile. Interestingly, however, the lower dispersion did allow the glass fibres to continue carrying load after the carbon fibre failure. Liang et al. suggest that damage to the glass fibres by the failure of the carbon fibres was limited by the lower dispersion. Pseudo-ductility has so far only been achieved in composites with a low LE fibre volume fraction. Bunsell and Harris [31] and Manders and Bader [14] did succeed in achieving pseudo-ductility at relatively high carbon fibre fractions, but this was mainly due to the weak carbon fibres at that time. The carbon fibre peak in their hybrids was lower than their glass fibre peak, making it easier to achieve pseudo-ductility. With the strength of the state-of-theart carbon fibres, the easiest way to reduce the height of the carbon fibre peak in a hybrid is to reduce the carbon fibre volume fraction. The major challenge for the pseudo-ductility concept in the future is hence to develop strategies for achieving it at higher volume fraction of the LE fibre. Bonding in general is seen as a crucial parameter for achieving pseudo-ductility. Bunsell and Harris [31] showed that a minimal bonding strength between carbon and glass layers is required to achieve pseudo-ductility. The importance of the interlaminar fracture toughness was shown in the analytical equation developed by Czél and Wisnom [13]. Similar work in fibre-reinforced concrete also showed that the fibre–matrix adhesion was a crucial parameter to obtain pseudo-ductile concrete [131,132]. It has not yet been proven that improved tensile behaviour also leads to improvements in other mechanical properties, such as fatigue or impact resistance. So far, the research has focused on tensile behaviour. 5.2. Ductile fibres An alternative way of achieving higher failure strains in hybrid composites is to combine brittle fibres with ductile fibres. As explained in ‘‘3.3 Influencing parameters’’, a large difference in failure strain of the fibres may lead to larger hybrid effects. It may also lead to increases in energy absorption. In the early literature on hybrid composites, however, carbon fibres were hybridised with either glass or aramid fibres. While these fibres indeed have a larger failure strain than carbon fibres, it is still relatively low. In the past decades, however, ductile fibres for polymer composites have become increasingly popular. Examples include steel [133,134], PP [135–137], PE [73,138], polyamide [139], polyvinyl alcohol (PVA) [140], coir [141–143] and silk [96] fibres. (a) (b) (c) σ εεε σσ Fig. 9. Schematic stress–strain diagrams for (a) non-hybrid composites, (b) typical hybrid composites, and (c) pseudo-ductile hybrid composites. Fig. 10. Damage mode map for carbon/glass hybrid composites. The experimental data points are marked with an additional square marker (reprinted from [129], with permission from Elsevier). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Y. Swolfs et al. / Composites: Part A 67 (2014) 181–200 195 Figure 3.9: Schematic stress-strain diagrams for: (a) non hybrid composites, (b) typical hybrid composites and (c) pseudo-ductile hybrid composites [2]. The increase in interest in a pseudo-ductile behaviour can be attributed to the high safety factors that are used in composite materials, which can be reduced if the materials showed a pseudo-ductile behaviour (like in metals) [2]. Czél et al. [75] and Jalalvand et al. [10] achieved a pseudo-ductile behaviour for carbon/glass hybrid with thin plies. The authors considered carbon/glass hybridization due to: the compatibility between materials, the existence of ultra-thin carbon prepregs, the difference in failure strain of the considered materials allows the alteration of the reference properties by hybridization and the transparency of the glass allows the detection of failure mechanisms in the ultra-thin carbon layer. The authors [10, 75] presented both experimental and numerical evidence of the pseudoductile behaviour. It was noted that the pseudo-ductile behaviour was closely connected with the failure mechanisms and that the total and relative thickness of the layers lead to different behaviours (see Figure 3.10). It was noted that when there was multiple fragmentation of the carbon layer the material showed a pseudo-ductile behaviour. This multiple fragmentation could be achieved when the carbon layer thickness was small (see Figure 3.10). Other authors have tackled the pseudo-ductile topic. Jones and Dibenedetto [74] demonstrated pseudo-ductile behaviour of carbon/glass and carbon /aramid composites when each carbon fibre was surrounded by carbon or aramid fibres, reducing the interaction between the carbon fibres. Liang et al. [76] demonstrated some degree of pseudo-ductility in carbon/glass rods when the glass fibres were all put in the core of the rod. However, the stress-strain diagram is more close to the one from Figure 3.9b than the pseudo-ductile one in Figure 3.9c. Czél et al. demonstrated a pseudo-ductile behaviour in unidirectional discontinuous carbon/glass fibre composites [77] and for composites with discontinuous carbon fibres and continuous glass 44 Chapter 3. HybridizationState-of-the-art The laminates [G/C/G], [G 2 /C 2 /G 2 ] and [G 2 /C/G 2 ] have some more carbon layer fragmentation randomly spread over their length before the glass failure. The final failure of the laminates [G 2 /C 3 /G 2 ] and [G 2 /C 4 /G 2 ] does not happen before the delamination is complete. All of the predicted damage modes in each laminate are in agreement with the observed experimental behaviour [7]. Fig. 10(a–f) indicates the obtained stress–extension curves of the different laminates (black line with a bold dot at the end) against the experimental results (grey lines). The early glass failure of the laminates with one single glass layer on each side is well predicted in the FE results. In the laminates [G 2 /C/G 2 ] and [G 2 /C 2 /G 2 ], a stress deviation from the linear elastic response is distinguishable in both experimental and numerical results before glass failure. In the laminates with 3 and 4 central carbon layers, there is a load drop after the first carbon layer failure due to rapid initial delamination propagation. The delamination propagation then becomes stable and since the value of G IIc of the interface is assumed constant, the load stays constant until the delamination extends over the whole glass/carbon interface. Glass fibre failure then happens when the delamination is complete and the load is only carried by the glass layers in these two laminates. As mentioned in Section 2, only the point of first glass fibre failure is predicted (the progressive damage was not modelled) and therefore, the load drops during glass fibre failure were not captured in the analysis. Table 2 gives the numerical results of all of the modelled laminates in this paper including both tested and a number of additional non-tested specimens. The tested specimens are specified by their layup configuration which is mentioned in the first column of the table. The damage modes are mentioned in the order they were observed in the numerical modelling and the predicted glass failure strain and also the difference from the experimental results are given in the last column for the tested specimen. The predicted glass failure of the tested specimens is less than 5% different from the average measured glass failure in the experiments, except the one for the laminate [G 2 /C/G 2 ]. The glass failure in this laminate has been predicted 11.5% earlier. It is believed that this difference is mainly because of non-uniformity of the carbon fragmentation across the width, which particularly affected this laminate. The proposed two-dimensional FE approach assumes that all of the tips of the fragmented carbon layer are aligned across the width, so the stress concentration is higher and glass failure is predicted earlier. In this respect the proposed approach is conservative. Fig. 9 indicates the contours of stress in the fibre direction in the [G 2 /C/G 2 ] and [G 2 /C 2 /G 2 ] laminates between first carbon layer fragmentation and final glass failure. Around the fragmented carbon layer, the stress drops in the carbon layer at the middle and increases in the glass layer. Due to the shorter process zone around the fragmented fibres in the thinner laminate, the crack density is also higher in this laminate. The average crack spacing of these two laminates is 1.0 and 0.3 mm !1 over the 50 mm length of the model which is in agreement with the experimental observations. The unstable delamination after carbon layer fragmentation of the laminate [G 2 /C 3 /G 2 ] is shown in Fig. 11. In fact, the sudden load drop in Fig. 10(e) is due to this unstable partial delamination of the specimen. 4. Damage mode domain maps After validating the modelling approach with the experimental results, other new hybrid combinations can be analysed with the same numerical tool. To investigate the variation of damage modes with respect to the glass and carbon layer thicknesses, new hybrid combinations as indicated in the Table 2 were modelled. The material properties and the strength distribution of the embedded cohesive elements were the same as in the previously modelled specimens. The only difference between all of these new models and the previous ones is that the variation of glass and carbon layer thickness was not constrained by the ply thickness. Therefore, the number of possible hybrid configurations is increased which is helpful in distinguishing the dependency of the damage process on the geometry of the hybrid. The damage modes after first carbon fragmentation along with the glass failure strain obtained from the proposed approach are also included in Table 2. Fig. 12 shows all of the analysed hybrid specimens on a chart showing the absolute and relative thickness of the carbon layers. Each point on the graph relates to a specific hybrid configuration and from the damage modes obtained from the model, different areas have been associated with different damage processes and divided schematically. The experimentally tested configurations are also distinguished with an additional bigger square marker. With such a plot, it is possible to predict the damage modes of a particular hybrid or to design a hybrid for a certain desired characteristic. To increase the pseudo-ductile part of the stress–strain response, it is necessary to avoid single delamination and premature glass layer failure. Additionally, it is important to increase the carbon proportion to increase the potential of larger stiffness variation during the damage process. But to have both carbon fragmentation and diffuse delamination in the damage process, an upper limit exists for the carbon ratio. Furthermore, there are lower and upper bands on the carbon thickness in laminates with the same carbon ratio to achieve the desired diffuse delamination. This map can also be produced for other material combinations and used to help to design hybrid laminates with the desired damage process and characteristics. 5. Conclusion In this paper, two modelling approaches for the damage process of UD hybrid laminates have been discussed. In the first approach, Fig. 11. The stress distribution just before carbon layer fracture and after unstable delamination in the laminate [G 2 /C 3 /G 2 ]. Fig. 12. Categorisation of different damage modes as a function of absolute and relative thickness of carbon layers. 46 M. Jalalvand et al. / Composites Science and Technology 94 (2014) 39–47 Figure 3.10: Failure modes as a function of absolute and relative layer thickness in carbon/glass hybrid composites [10]. fibres [78]. Swolfs et al. [79] also demonstrated that pseudo-ductility can be achieved by controlling the failure mechanisms in composites. This was done for carbon fibre and self-reinforced polypropylene composites and the pseudo-ductility was achieved when the carbon fibre layers were able to fracture in multiple locations before the failure of the composite. Yu et al. [80] also achieved a pseudo-ductile behaviour for hybrid carbon/glass composites with highly aligned discontinuous fibres. This hybridization was done at the ply level and each ply was constituted by both types of fibres (intralayer hybridization). Overall, the pseudo-ductile behaviour has only been achieved for low fractions of LE fibres (e.g. carbon fibres) and therefore their mechanical properties are reduced. New strategies need to be developed to achieve this behaviour in higher LE fibre fractions [7]. 3.3 Failure development and stress redistribution in UD hybrid composites The failure development of hybrid composites follows the same guide lines of nonhybrid composites (presented in chapter 2). The same base principles can be applied: (1) the strength of the fibres is not deterministic and (2) the stress previously carried by a broken fibre is redistributed among the intact ones in a complex manner. As in a hybrid composite, there is presence of two fibre types (HE and LE), the 3.3 Failure development and stress redistribution in UD hybrid composites 45 failure development will be more complex than in non-hybrid composites. As the LE fibres have a lower mean failure strain, they will break first, causing stress concentrations in the remaining fibres. The failure of the LE fibres causes the initiation of cracks in the matrix, which will extended with increasing applied stress. As the HE fibres have a higher failure strain, they will act as crack arresters, bridging the cracks formed by broken LE fibres [2, 74, 81]. This will lead to a delay in damage development and failure of hybrid composites. Nonetheless, the increase in applied strain/stress will cause the creation of clusters of broken fibres, constituted by LE and HE fibres. These clusters will grow and, when they reach a critical size, the failure of the composite will occur. As previously stated for non-hybrid composites, the stress redistribution after a fibre breaks is crucial to understanding the behaviour of UD composites under tensile loadings. Swolfs et al. [11] did an extensive study of stress redistribution in hybrid composites. Using a 3D FE model with a random fibre packing with a broken LE fibre (in this case carbon) in the middle, the authors were able to study the effects of several parameters in the stress redistribution in hybrid composites. Firstly, Swolfs et al. [11] studied the effect of the having fibres with different radii in the SCFs and ineffective lengths (see Figure3.11). They considered the carbon to have a radius of 3.5µmand the glass fibres to have a radius of 3.5or 6µm. recovered by shear loads in the surrounding material, it is related to the homogenised shear stiffness. Less fibrous material nearby the broken fibre, results in a slower stress recovery and longer ineffective length for the models with different radii. From this discussion, it is clear that the assumption of the same radii for both fibres introduces large errors. This should be avoided in future models for hybrid composites. Packings with different fibre radii will be used in the rest of this paper. 3.2. Influence of the hybrid volume fraction The hybrid volume fraction, which is defined as the volume of carbon fibres over the total volume of fibres, is an essential parameter for hybrid composites. In literature, it is commonly stated that lower hybrid fibre volume fractions, which is equivalent to low carbon fibre content, result in a higher hybrid effect. To further understand this effect, carbon fibres will be hybridised with glass fibres in five different hybrid volume fractions: 0%CF, 20%CF, 50%CF, 80%CF and 100%CF. Five realisations of the microstructure are generated for each hybrid volume fraction and one of those realisations for each fraction is illustrated in Fig. 4. It should be noted that 0%CF contains one carbon fibre in the middle, which is broken and surrounded by glass fibres only. For the sake of clarity, the results are split up into glass fibre (see Fig. 5) and carbon fibre data points (see Fig. 6). Results are plotted for all five realisations of each of the five hybrid volume fractions. For glass fibres, the SCFs decrease slightly with increasing carbon fibre content. This is due to the increased longitudinal composite stiffness. The stiffer composite takes up more stress and hence reduces the stress carried by the glass fibres. Upon close Fig. 3. Stress redistribution for 50%CF packings, with the same and different radii: (a) the stress concentration factor as a function of the normalised distance from the broken fibre, and (b) the ineffective length. Five realisations were calculated for both cases. Fig. 4. Example of one of the five realisations for each hybrid volume fraction. Y. Swolfs et al. / Composites Science and Technology 85 (2013) 10–16 13 Figure 3.11: Stress redistribution in hybrid composites with 50% carbon and glass fibres: (a) SCFs in both fibre types, considering the same and different radii; (b) ineffective length of the broken carbon fibre considering fibres with the same and different radii [11]. The authors concluded that having fibres with different radius affects both the SCFs and the ineffective length. For the model with different fibre radii the SCFs, in the carbon and glass fibres follow the same trend-lines. This was attributed to the fact that considering glass fibres with higher diameter causes the SCFs to decrease due to an increase in cross section, compensating, therefore, the difference in the stiffness of the fibres. In terms of ineffective length, the authors attribute the 46 Chapter 3. HybridizationState-of-the-art increase of ineffective length in the model with fibres with different radii to the fact that, in packings with different radii, there is less fibrous material surrounding the broken fibre, reducing the homogenized shear stiffness of the material in this region of the composite. As the stress is transmitted to the broken fibre in shear, reducing the homogenized shear stiffness causes an increase of the ineffective length. investigation, a similar, but smaller decrease can be observed for CF in Fig. 6. This confirms the results in [22], in which a small influence of the fibre stiffness is observed for non-hybrid composites. The influence of the hybrid volume fraction on the ineffective length is illustrated in Fig. 7. The ineffective length is expressed relative to the radius of the broken carbon fibre. Similar to the SCF results, the ineffective length slightly decreases with increasing hybrid volume fraction. Two effects are counteracting each other in this case. The first effect is the lower shear modulus of the carbon fibre, resulting in a lower composite shear stiffness at higher fibre volume fractions. This results in slower stress recovery at higher hybrid volume fractions and hence a higher ineffective length. This trend is not observed, as the second effect appears to be stronger. In the 0%CF model, the small, broken carbon fibre is surrounded by larger glass fibres, resulting in a less efficient packing than in the 100%CF model. Hence, the latter model has more fibrous material in the vicinity of the broken fibre. Since the stress recovery is dominated by the material nearby the broken fibre, the 100%CF model locally has higher shear stiffness, which results in faster stress recovery and lower ineffective length. This trend is actually observed in Fig. 7, but is small due to the two counteracting effects. 3.3. Influence of the hybridisation fibre Most literature on hybrid composites investigates carbon–glass hybrids. Nevertheless, the hybridisation fibre is not always glass, as aramid fibres are also a popular choice. These fibres have a wide range of possible mechanical properties, out of which two common grades were chosen: Kevlar 29 (K29) and Kevlar 49 (K49). The longitudinal stiffness E L of K29 is two times lower than the E L of K49 (see Table 1). The five 50%CF packings, which were used in Section 3.2 for CF/GF hybrids, were copied and the engineering constants of aramid fibre were applied. This way, the mesh is exactly the same. The results are again split up into data points for glass or aramid fibres (see Fig. 8) and carbon fibres (see Fig. 9). Fig. 8 illustrates the importance of an adequate choice of the hybridisation fibre, or more specifically its elastic properties. The higher longitudinal stiffness of K49 results in lower SCFs than in K29 and GF hybrids. Similar to the SCF decrease with increasing hybrid volume fraction, this is also caused by the increased longitudinal composite stiffness. The influence of the choice of hybridisation fibre on the carbon fibre SCFs is smaller than the influence on the hybridisation fibre SCFs (see Fig. 9). The use of aramid fibres in a hybrid results in slightly higher SCF on the carbon fibres. This is related to the low shear stiffness of the aramid fibres, which results in more shear deformation of the fibres. This increased shear deformation transfers more stress onto the intact fibres, resulting in a higher SCF. Fig. 5. Stress concentration factors as a function of the distance from the broken fibre for packings with different fibre radii. The influence of hybrid volume fraction is shown for glass fibres. Fig. 6. Stress concentration factors as a function of the distance from the broken fibre for packings with different fibre radii. The influence of hybrid volume fraction is shown for carbon fibres. Fig. 7. The ineffective length of carbon–glass hybrids for different hybrid volume fractions. The error bars indicates the 95% confidence interval based on five realisations. Fig. 8. Stress concentration factors on the hybridisation fibres as a function of the distance from the broken fibre for 50%CF packings. 14 Y. Swolfs et al. / Composites Science and Technology 85 (2013) 10–16 Figure 3.12: Stress concentration factors in glass fibres as a function of the distance from the broken fibre [11]. investigation, a similar, but smaller decrease can be observed for CF in Fig. 6. This confirms the results in [22], in which a small influence of the fibre stiffness is observed for non-hybrid composites. The influence of the hybrid volume fraction on the ineffective length is illustrated in Fig. 7. The ineffective length is expressed relative to the radius of the broken carbon fibre. Similar to the SCF results, the ineffective length slightly decreases with increasing hybrid volume fraction. Two effects are counteracting each other in this case. The first effect is the lower shear modulus of the carbon fibre, resulting in a lower composite shear stiffness at higher fibre volume fractions. This results in slower stress recovery at higher hybrid volume fractions and hence a higher ineffective length. This trend is not observed, as the second effect appears to be stronger. In the 0%CF model, the small, broken carbon fibre is surrounded by larger glass fibres, resulting in a less efficient packing than in the 100%CF model. Hence, the latter model has more fibrous material in the vicinity of the broken fibre. Since the stress recovery is dominated by the material nearby the broken fibre, the 100%CF model locally has higher shear stiffness, which results in faster stress recovery and lower ineffective length. This trend is actually observed in Fig. 7, but is small due to the two counteracting effects. 3.3. Influence of the hybridisation fibre Most literature on hybrid composites investigates carbon–glass hybrids. Nevertheless, the hybridisation fibre is not always glass, as aramid fibres are also a popular choice. These fibres have a wide range of possible mechanical properties, out of which two common grades were chosen: Kevlar 29 (K29) and Kevlar 49 (K49). The longitudinal stiffness E L of K29 is two times lower than the E L of K49 (see Table 1). The five 50%CF packings, which were used in Section 3.2 for CF/GF hybrids, were copied and the engineering constants of aramid fibre were applied. This way, the mesh is exactly the same. The results are again split up into data points for glass or aramid fibres (see Fig. 8) and carbon fibres (see Fig. 9). Fig. 8 illustrates the importance of an adequate choice of the hybridisation fibre, or more specifically its elastic properties. The higher longitudinal stiffness of K49 results in lower SCFs than in K29 and GF hybrids. Similar to the SCF decrease with increasing hybrid volume fraction, this is also caused by the increased longitudinal composite stiffness. The influence of the choice of hybridisation fibre on the carbon fibre SCFs is smaller than the influence on the hybridisation fibre SCFs (see Fig. 9). The use of aramid fibres in a hybrid results in slightly higher SCF on the carbon fibres. This is related to the low shear stiffness of the aramid fibres, which results in more shear deformation of the fibres. This increased shear deformation transfers more stress onto the intact fibres, resulting in a higher SCF. Fig. 5. Stress concentration factors as a function of the distance from the broken fibre for packings with different fibre radii. The influence of hybrid volume fraction is shown for glass fibres. Fig. 6. Stress concentration factors as a function of the distance from the broken fibre for packings with different fibre radii. The influence of hybrid volume fraction is shown for carbon fibres. Fig. 7. The ineffective length of carbon–glass hybrids for different hybrid volume fractions. The error bars indicates the 95% confidence interval based on five realisations. Fig. 8. Stress concentration factors on the hybridisation fibres as a function of the distance from the broken fibre for 50%CF packings. 14 Y. Swolfs et al. / Composites Science and Technology 85 (2013) 10–16 Figure 3.13: Stress concentration factors in carbon fibres as a function of the distance from the broken fibre [11]. Swolfs et al. [11] also studied the effect of hybrid volume fraction in the stress redistribution, for carbon/glass hybrids. By analysing Figure 3.12 and 3.13 one can see that the hybrid volume fraction has a low influence in the SCFs in both fibre types, however, increasing the volume of carbon fibres, slightly decreases the SCFs due to an increase in the composite longitudinal stiffness. The influence of the hybrid 3.3 Failure development and stress redistribution in UD hybrid composites 47 volume fraction in the ineffective length is similar to that in the SCFs (see Figure 3.14), the increase of the carbon volume content slightly decreases the ineffective length. investigation, a similar, but smaller decrease can be observed for CF in Fig. 6. This confirms the results in [22], in which a small influence of the fibre stiffness is observed for non-hybrid composites. The influence of the hybrid volume fraction on the ineffective length is illustrated in Fig. 7. The ineffective length is expressed relative to the radius of the broken carbon fibre. Similar to the SCF results, the ineffective length slightly decreases with increasing hybrid volume fraction. Two effects are counteracting each other in this case. The first effect is the lower shear modulus of the carbon fibre, resulting in a lower composite shear stiffness at higher fibre volume fractions. This results in slower stress recovery at higher hybrid volume fractions and hence a higher ineffective length. This trend is not observed, as the second effect appears to be stronger. In the 0%CF model, the small, broken carbon fibre is surrounded by larger glass fibres, resulting in a less efficient packing than in the 100%CF model. Hence, the latter model has more fibrous material in the vicinity of the broken fibre. Since the stress recovery is dominated by the material nearby the broken fibre, the 100%CF model locally has higher shear stiffness, which results in faster stress recovery and lower ineffective length. This trend is actually observed in Fig. 7, but is small due to the two counteracting effects. 3.3. Influence of the hybridisation fibre Most literature on hybrid composites investigates carbon–glass hybrids. Nevertheless, the hybridisation fibre is not always glass, as aramid fibres are also a popular choice. These fibres have a wide range of possible mechanical properties, out of which two common grades were chosen: Kevlar 29 (K29) and Kevlar 49 (K49). The longitudinal stiffness E L of K29 is two times lower than the E L of K49 (see Table 1). The five 50%CF packings, which were used in Section 3.2 for CF/GF hybrids, were copied and the engineering constants of aramid fibre were applied. This way, the mesh is exactly the same. The results are again split up into data points for glass or aramid fibres (see Fig. 8) and carbon fibres (see Fig. 9). Fig. 8 illustrates the importance of an adequate choice of the hybridisation fibre, or more specifically its elastic properties. The higher longitudinal stiffness of K49 results in lower SCFs than in K29 and GF hybrids. Similar to the SCF decrease with increasing hybrid volume fraction, this is also caused by the increased longitudinal composite stiffness. The influence of the choice of hybridisation fibre on the carbon fibre SCFs is smaller than the influence on the hybridisation fibre SCFs (see Fig. 9). The use of aramid fibres in a hybrid results in slightly higher SCF on the carbon fibres. This is related to the low shear stiffness of the aramid fibres, which results in more shear deformation of the fibres. This increased shear deformation transfers more stress onto the intact fibres, resulting in a higher SCF. Fig. 5. Stress concentration factors as a function of the distance from the broken fibre for packings with different fibre radii. The influence of hybrid volume fraction is shown for glass fibres. Fig. 6. Stress concentration factors as a function of the distance from the broken fibre for packings with different fibre radii. The influence of hybrid volume fraction is shown for carbon fibres. Fig. 7. The ineffective length of carbon–glass hybrids for different hybrid volume fractions. The error bars indicates the 95% confidence interval based on five realisations. Fig. 8. Stress concentration factors on the hybridisation fibres as a function of the distance from the broken fibre for 50%CF packings. 14 Y. Swolfs et al. / Composites Science and Technology 85 (2013) 10–16 Figure 3.14: The ineffective length of carbon–glass hybrids for different hybrid volume fractions. The error bars indicates the 95% confidence interval based on five realisations[11]. Swolfs [2] also presented the results for SCF and ineffective length for carbon hybrids hybridized with HE fibres with different stiffnesses. HE concluded that increasing the stiffness of the HE fibres increased the SCF in these fibres. However, the effect is opposite in the SCFs in the carbon fibres, but this effect is reduced. The effect of the stiffness of HE fibres in the ineffective length can be seen in Figure 3.15. As one can see, increasing the stiffness of the HE fibres slightly reduces the ineffective length of a broken carbon fibre, but increases that of a broken HE fibre. Chapter 5: Strength model for UD hybrid composites 261 10 40 70 0 10 20 30 40 Ineffective length z/R HEfibrestiffness(GPa) Brokencarbonfibre BrokenHEfibre Figure 5-10: The ineffective length for broken carbon and HE fibres as a function of the HE fibre stiffness. The overall Vf was 50%. 5.2.4 Conclusion Some minor modifications to the FE methodology were necessary to analyse the stress redistributions in hybrid composites. The stress redistribution around a broken carbon fibre was hardly affected by the hybrid volume fraction. These are the first results ever to prove this important aspect of the failure behaviour of hybrid composites. The HE fibre stiffness did have a significant influence on stress redistribution around broken fibres. Around a broken carbon fibre, the SCFs on the intact fibres increased with decreasing HE fibre stiffness. Around a broken HE fibre however, the SCFs on the intact fibres decreased with decreasing HE fibre stiffness. These trends were explained based on the lower load carrying capacity of the HE fibres, in combination with a lower load released by a broken HE fibre. The ineffective length of a broken carbon fibre did not depend on the HE fibre stiffness. The decreased HE fibre stiffness did result in a smaller ineffective length for a broken HE fibre, as less stress needs to be build up in such a fibre. These results will be used as input data for the strength model for hybrid composites. This model can then predict whether the hybrid volume fraction and HE fibre stiffness influence the failure development and hybrid effect in hybrid composites. Figure 3.15: The ineffective length in carbon and HE fibres as a function of HE stiffness[2]. The cluster development in composite materials is affected by hybridization. Swolfs et al. [13] studied the how the hybrid ratio affected the cluster develop- 48 Chapter 3. HybridizationState-of-the-art ment, concluding that, increasing the volume fraction of HE fibres would lead to an delay in the cluster formation. This means that to achieve the same level of clusters of broken fibres the applied strain needs to be higher. The authors also found a relation between the critical cluster size and the hybrid ratio, concluding that increasing the volume fraction of HE fibres leads to a reduction of the critical cluster size. Due to theses results in the stress redistribution, Swolfs et al. [11] concluded that bridging of the broken carbon fibres by the intact hybridisation fibres is the major contribution to the hybrid effect. 3.4 Modelling the tensile failure of UD hybrid composites The first author to model the tensile failure of hybrid composites was Zweben [58], in 1997, with the intention of predicting the hybrid effect for failure strain. Zweben extended a shear-lag model for UD hybrid composites and considered localload-sharing for stress redistribution after a fibre break. This model considered a 1D fibre packing with alternating HE and LE fibres (Figure 3.16b) which represents a simplification of the complex geometry of hybrid composites. The hybrid composite behaviour was compared with the non-hybrid composite composed only with LE fibres (Figure 3.16a) to determine the hybrid effect. Chapter 5: Strength model for UD hybrid composites 246 5.1.1 Zweben’s model In 1977, Zweben [133] was the first author to extend shear-lag models for unidirectional composites to hybrid composites and model the hybrid effect for failure strain. His model is based on local load sharing instead of very local or global load sharing (see “4.1.3 Strength models for unidirectional composites”). Zweben modelled 1D fibre packings, consisting of a single row of LE fibres (see Figure 5-2a). This was modelled and compared to a similar packing with alternating LE and HE fibres, as illustrated in Figure 5-2b. This type of packing is common in models for hybrid composites [135,394-396], as it is the most straightforward way to simplify the geometrical complexity of hybrid composites. Figure 5-2: Schematical representation of 1D fibre packings used in Zweben’s model: (a) a non-hybrid composite with only LE fibres, and (b) a hybrid composite with alternating LE and HE fibres. Zweben derived analytical expressions for the strain concentrations and ineffective length in both packings. The strain concentration factor kwas defined as the ratio of the strain in a fibre next to a single broken fibre over the applied strain. Since all fractures were assumed to occur in a single plane, this parameter was only defined in the plane of fibre break. The strain concentration factor for hybrid composites h k only depends on E A R, which is the ratio of normalised stiffnesses of both fibre types: L ELE EA H EHE EA R EA   , (5-1) in which L E E and H E E are the Young’s moduli of the LE and HE fibres, respectively, and L E A and H E A are the cross-sectional areas of the LE and HE fibres, respectively. For the exact relationship between h k and E A R, the reader is referred to Zweben [133]. The factor h k monotonically increases with E A R and is larger than k for E A R-values above 1. (a) (b) LEfibre HEfibre Figure 3.16: Representation of the fibre packings used in Zweben’s model : (a) non-hybrid LE composite and (b) hybrid composite with alternating LE and HE fibres [2]. As the model considered fibres with different cross sections and elastic modulus, Zweben considered that when a LE fibre breaks the neighbouring HE fibres would be subjected to a strain concentration, rather than a stress concentration. The strain concentration factor can be defined as the ratio between the strain in a fibre next 3.4 Modelling the tensile failure of UD hybrid composites 49 to a single broken fibre over the applied strain. Zweben also considered that after a fibre breaks there is a length at which the fibre is not capable of fully carry stress (ineffective length) and derived analytical expressions for both strain concentration factors and ineffective length for hybrid and non hybrid composites. For non-hybrid composite, the strain concentration factor (kLE) is equal to 1.293 and for hybrid composites the strain concentration factor (kh) only depends on ρ, which is the ration of normalised stiffness of both fibres: ρ=ELEALE EHEAHE ,(3.2) where ELE and EHE are the Young’s modulus of the low elongation (LE) and high elongation (HE) fibres, and ALE and AHE are the cross-sectional areas of both fibre types. The ineffective lengths for the non-hybrid (δLE) and hybrid composite (δh) can be determined as: δ=NELESLEdm Gmtm1/2 ,(3.3) where Gmis the matrix shear modulus, tmand dmare, respectively the matrix thickness and the fibre spacing. the factor Nis equal to 1.531 for non-hybrid composites and is a function of ρfor the hybrid ones. Zweben considered that the hybrid composite fails when the first HE fibre breaks, resulting in a lower bound for composite strength [58]. Zweben assumed that the failure of a HE fibre would trigger the unstable failure of all the other LE fibres, therefore, the failure strain used to determine the hybrid effect is according to the definition of hybrid effect presented by Hayashi [52] and explained in Section 3.1.1. Combining the equations for ineffective length, strain concentration factors and the Weibull distributions for fibre strain, Zweben derived the following expression for the hybrid effect (Rhyb): Rhyb =rεHE εLE δh(km h−1) 2δLE (kLE −1) −1/2m ,(3.4) where εLE and εHE are the mean failure strains of the LE and HE fibres at the considered gauge length and mis the Weibull modulus of the fibres, which was considered to be equal in both fibre types. According to Swolfs [2] the main conclusions to draw from this model are: •The strain concentration factor depends only on the normalised stiffness ratio of the two fibres (ρ), which means that in the case of a hybrid composite with both fibres with the same factor [E×A], the stress concentration in the hybrid and non-hybrid composite will be the same; 56 Chapter 3. HybridizationState-of-the-art Composites Part A: Applied Science and Manufacturing 69 (2015) p. 279-287 doi:10.1016/j.compositesa.2014.12.001 14 the number of fibre bundles in each model would have perhaps been more intuitive. Unfortunately, the circular cross-section of the model leads to incomplete fibre bundles. Figure 10: Illustration of bundle-by-bundle dispersion, where black circles are carbon fibres and red denotes glass fibres. The influence of the bundle size on the hybrid effect and triplet evolution is shown in Fig. 11. For the 2 bundles model, the hybrid effect is only 1.5%, while it increases to 7% for 16 bundles, as can be seen in Fig. 11a. The latter effect approaches the 9% hybrid effect found for random dispersion at 50% hybrid volume fraction. Fig. 11b proves that increased dispersion leads to a delay in break-cluster development. A similar delay was also found for other cluster sizes, but is not shown here. 2bundles 4bundles 8bundles 16bundles Composites Part A: Applied Science and Manufacturing 69 (2015) p. 279-287 doi:10.1016/j.compositesa.2014.12.001 15 Figure 11: (a) The hybrid effect for bundle-by-bundle fibre dispersion, and (b) the evolution of triplets (break-clusters of 3 fibres) as a function of strain. The result for random dispersion was added to facilitate comparisons. The second dispersion type is layer-by-layer, as shown in Fig. 12. The fibre dispersion is labelled according to the number of fibres across the thickness of each layer. The corresponding hybrid effects and sequences of triplet evolution are shown in Fig. 13. Even though these layer-by-layer hybrids seem less dispersed than randomly dispersed hybrids, they are able to reach a higher hybrid effect. For the single fibre layer case, the hybrid effect is 16%, which is significantly higher than the 9% found for random dispersion. (a)(b) 0% 5% 10% Hybrid effect 0 100 200 1 1.2 1.4 1.6 Average number of3‐plets Strain(%) 2bundles 4bundles 8bundles 16bundles Random Eightfibrelayer TwofibrelayerSinglefibrelayer Fourfibrelayer (a) (b) Figure 3.20: Illustration of the (a) bundle-by-bundle and (b) layer-by-layer dispersion considered by Swolfs et al. (Adapted from[13]). authors didn’t study the limit case of bundles with just one fibre. The authors also reported that dispersion had a small influence in the critical cluster size. 3.5.6 Matrix properties Has in non-hybrid composites, the matrix properties are only expected to have a secondary effect in the composite properties, by influencing the SCFs and the ineffective lengths. The matrix shear modulus has an influence on the ineffective length, however its effect on the SCFs is usually not represented in the models due to shear-lag assumptions. 3.6 Conclusion Hybrid composites are attracting an ever growing attention from both academia and industry, due to their potential. The interactions between the components in hybrid composites are hard to predict, however they may lead to better resulting properties that those of the non-hybrid composites of reference, leading to the existence of positive hybrid effects. This hybrid effects have been reported under several loading conditions and in several hybrid materials. Modelling the tensile behaviour of hybrid composites has been shown to be a difficult task. The earlier model to do so was Zweben’s model [58], which considered a 1D packing of fibres. This model was able to predict the existence of hybrid effect, but due to its simplicity wasn’t able to fully predict the full composite’s behaviour, nor fully justify the hybrid effect. Other models have since then been presented, however there is still no model that is able to fully predict the hybrid composite’s behaviour up to failure. 3.6 Conclusion 57 The models and the experimental work done in this area allow us to determine the main influencing parameters in the behaviour of hybrid composites, being the most important the dispersion of the fibres, the hybrid volume fraction and the fibre strength distributions. 58 Chapter 3. HybridizationState-of-the-art Chapter 4 Model for the tensile failure of dry tows Failure of UD composites under tensile loading is a phenomena mainly dominated by the fibres, which means that fibres and fibre tows are fundamental entities in composite materials. The tensile testing of dry tows (bundles of fibres without the presence of the matrix) is important to determine the strength distributions of the fibres [18]. This is a more adequate testing method than single fibre testing, because it is more representative as there is no selection of the fibres and the results are obtained from the average behaviour of a large number of fibres [87]. As fibre strength is of high importance in UD composites and hybridization implies an interaction between fibres that present different distributions of strength, the study of hybrid dry bundles is important. This study will allow the understanding of the effects of tow hybridization and the main parameters controlling the tow’s behaviour. 4.1 Model development To study the tensile failure of hybrid tows, a model based on the works of Calard and Lamon [18] was developed. The model was firstly implemented for non-hybrid tows (tows with a single type of fibres), and it considers a bundle composed of Nt parallel fibres with radius Rfand length L. The fibre strengths are considered to follow a Weibull distribution: 59 60 Chapter 4. Model for the tensile failure of dry tows P(σ)=1−exp −L L0σ σ0m,(4.1) where P(σ)is the failure probability of a fibre with length Lwhen subjected to a tension σ.L0is the reference length at which the Weibull parameters σ0and m were calculated. The Weibull distribution for fibre strength leads to an average fibre strength (hσi) given by hσi=σLΓ1 + 1 m,(4.2) where Γ() is the gamma function, that can be defined as: Γ(x)=(x−1)! ,(4.3) if xis a positive integer, or Γ(x) = Z∞ 0 tx−1e−tdt , (4.4) if xis an complex number with positive real part. σLis the reference tension at gauge length Land is related to σ0and L0by: σL=σ0L0 L1/m .(4.5) The model assumes global-load-sharing of the stress after a fibre breaks, meaning that there is no interaction between fibres, as there is no presence of a matrix and it is considered that there is no friction between the fibres. The load previously carried by a broken fibre is equally redistributed among the remaining fibres. The model assumes strain-controlled conditions and the strain is incremented from zero with a pre-defined value of ∆ε. The stresses in the fibres are calculated and compared to the tensile strength assigned to each fibre. This assignment is done by generating a random number (X) between 0 and 1 for each fibre that represents the failure probability P() in Equation 4.1. The strength of the fibre is then calculated by the following expression: σf=σ0−L0 Lln (1 −X)1/m .(4.6) If the stress in a fibre (i) reaches the fibre’s strength (σi f) that fibre breaks and the number of broken fibres (Nf) is incremented. The force and strain can be related by the following expression: F=ε(Nt−Nf)SfEf(4.7) 4.2 Non-hybrid tow behaviour 61 BEGINBEGIN Generate random strengths σi f Generate random strengths σi f Strain increment ∆ε Strain increment ∆ε Calculate the total force F Calculate the total force F Determine the stress in the fibres σ Determine the stress in the fibres σ σ > σi f σ > σi f Update Nf Update Nf Nf=Nt Nf=Nt ENDEND N Y N Y Figure 4.1: Flowchart of the model for dry bundle failure. where εis the applied strain, Sfis the fibre sectional area and Efis the Young modulus of the fibres. The simulation is concluded when all fibres have failed. The algorithm of the model is shown in Figure 4.1. 4.2 Non-hybrid tow behaviour In this section a parametric study will be done in order to determine the influence of the Weibull distribution parameters on the tow behaviour. As the fibre strengths are calculated using a random distribution at least 10 simulations are performed for each case, in order to obtain an average response. As base information for fibre properties, the AS4 carbon fibres will be considered. These fibres have a mean radius (R) of 3.5µm, Young’s modulus (E) equal to 234 GPa and the parameters for the Weibull distribution are: σ0= 4275 MPa and m= 10.7at L0= 12.7 mm [88]. This values will serve as a baseline for the parametric study of the effect that the strength distributions of the fibres have in 62 Chapter 4. Model for the tensile failure of dry tows the tow behaviour. In this study the tow is considered to have 500 fibres with a length (L) equal to 75mm, unless other parameters are specified. One of the statistical parameters of the Weibull distribution is the Weibull strength scale parameter (σ0), which relates linearly with the average fibre strength (Equation 4.2). Figure 4.2 shows the effect that varying σ0has in the failure strength of the fibres. It can be seen that varying the scale parameter translates the curves horizontally, which means that the average fibre strength is modified but the strength dispersion is constant. 1500 2000 2500 3000 3500 4000 4500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Stre ss (MPa) Fai lure Probability (%) σ0= 3800 σ0= 4000 σ0= 4275 σ0= 4500 Figure 4.2: Effect of the scale parameter (σ) in the strength of the fibres. Figure 4.3 shows the stress-strain diagrams for the Weibull distributions with different scale parameters, with the same shape parameter. It is observed that reducing the shape parameter leads to a reduction in the maximum load that the tow can withstand. The curves, however, seem to be very similar, as it is expected, due to the same dispersion of fibre strength. The key parameters for the characterization of the tow behaviour are presented in Table 4.1. It is possible to see a direct relation between the scale parameter (σ0) and the maximum load in the load-strain diagrams. As the elastic modulus of the fibres was not altered, increasing the scale parameter results in an increase of the tow’s strength and failure strain (strain at maximum force). 4.2 Non-hybrid tow behaviour 63 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 Strain (%) Load (N) σ0= 3800 σ0= 4000 σ0= 4275 σ0= 4500 Figure 4.3: Force-Strain diagrams for tows of fibres with different scale parameters (σ). Table 4.1: Effect of the shape parameter (m) in some reference properties. σ0(MPa) mhσi(MPa) Maximum load (N) Strain at max. load (%) 3800 10.7 3070.9 45.8 1.11 4000 3232.5 48.0 1.16 4275 3454.7 51.2 1.24 4500 3636.6 53.9 1.3 The second Weibull distribution parameter is the shape parameter (m), that is a measure of the dispersion of fibre strength, influencing the width of the distribution. As one can see in figure 4.4, increasing the shape parameter will lead to a wider distribution of fibre strengths, however, the shape parameter will also change the average failure probability, which is expected from Equation 4.2. This modification of the fibre strength will translate into different stress-strain diagrams (Figure 4.5). As one can see, as the average failure strength of the tows is reduced, by reducing the shape parameter, so is the maximum load in the stressstrain diagram. Another effect of the reduction of the shape parameter is that the strain at which the first fibres fail is reduced, which can be seen in Figures 4.4. This translates into the tow behaviour in Figure 4.5, where the non-linear behaviour starts at lower strains if the shape parameter is reduced. This is the result of a earlier failure of the weakest fibres, which reduces the tow’s effective stiffness. At higher values of mthis non-linear behaviour is reduced as the majority of the fibres 64 Chapter 4. Model for the tensile failure of dry tows 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Stre ss (MPa) Fai lure Probability (%) m = 12.0 m = 10.7 m = 7.5 m = 5.0 Figure 4.4: Effect of the shape parameter (m) in the strength of the fibres. breaks in a small range of strains, due to a reduced dispersion. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 Strain (%) Load (N) m = 12.0 m = 10.7 m = 7.5 m = 5.0 Figure 4.5: Force-Strain diagrams for tows of fibres with different shape parameters (m). Table 4.2, presents the relation between the Weibull shape parameter and the tow’s average failure strength (hσi), the maximum load and the strain at maximum load. As explained in the previous paragraphs, changing σ0changes the average failure strength of the fibres without changing the dispersion of this parameter, however, changing the shape parameter (m), not only changes the dispersion, but also the average strength of the fibres. To study the effect that dispersion has in the behaviour of the tows it is necessary to change both the shape and scale parameter, in 4.2 Non-hybrid tow behaviour 65 Table 4.2: Effect of the shape parameter (m) in some reference properties. σ0(MPa) mhσi(MPa) Maximum load (N) Strain at max. load (%) 4275 12.0 3533.1 53.4 1.29 10.7 3454.7 51.5 1.24 7.5 3167.0 44.0 1.10 5.0 2751.7 34.7 0.93 order to have the same average failure strength, as given by Equation 4.2. Taking as the baseline the combination σ0= 4275 MPa and m= 10.7, which gives an average failure strength (hσi) equal to 3454.7 MPa, and varying the both the shape and scale parameter we obtain several distributions. The main results are presented in Table 4.3. Table 4.3: Effect of dispersion in some reference properties (distributions with the same average strength). σ0(MPa) mhσi(MPa) Maximum load (N) Strain at max. load (%) 4180.1 12.0 3454.7 52.2 1.3 4275.0 10.7 3454.7 51.2 1.2 4663.4 7.5 3454.7 47.8 1.2 5367.1 5.0 3454.7 43.5 1.2 The strength distribution for the fibres are presented in Figure 4.6 and it can be seen that the average failure strength (strength at a failure probability of 50%) is the same in all distributions, but the dispersion is different, resulting in broader distributions if the shape parameter is reduced. The effect of dispersion in the tow’s behaviour is shown in Figure 4.7. Increasing the dispersion, by reducing m, reduces the strain at which the first fibres will fail but increases the failure at which the last fibres will fail. This leads to a reduced maximum force but creates a broader stress-strain diagram. Another aspect that can be concluded from analysing the results in Figure 4.7 and Table 4.3 is that, as the average failure strength is the same in all distributions, the strain at maximum strength will be identical in all distributions, the differences are due to the random generation of the vector of fibre failure probabilities. As seen in Chapter 2, the tensile strength of brittle fibres is dominated by surface 72 Chapter 4. Model for the tensile failure of dry tows the failure strain is reduced and, as the fibres have similar elastic moduli, so is the maximum load. These parameters can be found in Table 4.6. The maximum pseudoductile strain is achieved for a volume fraction of T300 fibres equal to 0.25, however, the tow’s behaviour does not represent a pseudo-ductile behaviour, as there is no gradual failure. The pseudo-ductile strain is, mainly, due to the non-linearity cause by the failure of the weakest fibres. Table 4.6: Stress-strain reference properties for AS4/T300 hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2657.05 1.24 0.11 0.125 2387.96 1.22 0.20 0.25 2152.55 1.17 0.24 0.5 1813.23 0.98 0.20 0.75 1650.80 0.85 0.14 1 1551.89 0.82 0.15 AS4/M40S carbon hybridization Figure 4.11 shows the failure strain distributions for the AS4 [88] and the M40S [91] are quite different. The M40S fibres are, in this case, the LE fibres. The stressstrain curves for several hybrid volume fraction are presented in Figure 4.13. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 500 1000 1500 2000 2500 3000 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 4.13: Stress-strain diagrams at various hybrid volume fractions for AS4/M40S hybridization. In Figure 4.13 it can be seen that for a LE fibre volume fraction of 0.25, there 4.3 Hybrid tow behaviour 73 is a pseudo-ductile behaviour and a gradual failure of the tow. The values of the pseudo-ductile strain and other properties are shown in Table 4.7. Table 4.7: Stress-strain reference properties for AS4/M40S hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2650.92 1.25 0.12 0.125 2260.18 1.24 0.35 0.25 1889.69 1.23 0.53 0.5 1688.61 0.73 0.18 0.75 1807.68 0.67 0.14 1 2023.12 0.66 0.12 In this table it is possible to note that the pseudo-ductile strain εdis higher for a LE fibre volume fraction of 0.25 and is equal to 0.53%, which is expected from the interpretation of the Figure 4.7. For a LE fibre volume fraction of 0.125 there is a reduction of stiffness when the LE fibres start to fail, which leads to a second almost linear (0.75 <ε>1.22) behaviour until the rupture of the tow. This type of hybridization may have potential in composite materials as some degree of pseudo-ductility is already present in the dry tow behaviour. The pseudoductile behaviour comes at a cost of strength, as the maximum force the tow can withstand is reduced, from 2651 MPa in the AS4 carbon tow to 1889 MPa for a M40S fibre volume fraction equal to 0.25. AS4/M50S carbon hybridization As the AS4 and M50S failure strain distributions are quite different (see Figure 4.11), the results of their hybridization may be interesting. Figure 4.14 shows the stress-strain curves for AS4/M50S carbon hybrids. Table 4.5 shows that the difference of average failure strain hεibetween the AS4 and M50S fibres is higher than that between AS4 and M50S fibres. This, in addition to the differences in the elastic properties, leads to the different behaviours found in Figures 4.13 and 4.14. Table 4.8 shows that for LE fibres volume fraction of 0.25 the pseudo-ductile strain is maximum and equal to 0.58%. However, analysing Figure 4.14, one can see that there is a drop in load bearing capacity of the tow when the LE fibres fail, resulting in a load drop. For a LE volume fraction of 0.125 there is an increase in 74 Chapter 4. Model for the tensile failure of dry tows 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 500 1000 1500 2000 2500 3000 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 4.14: Stress-strain diagrams at various hybrid volume fractions for AS4/M50S hybridization. Table 4.8: Stress-strain reference properties for AS4/M50S hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2647.83 1.26 0.13 0.125 2266.18 1.24 0.38 0.25 1904.89 1.23 0.58 0.5 1887.11 0.63 0.10 0.75 2182.22 0.62 0.10 1 2603.82 0.61 0.07 ductility in relation to the non-hybrid composites, with a pseudo-ductile strain equal to 0.38% without having the load drop that is present in the composite with a LE fibre volume fraction of 0.25. Analysis of the results From the previous sections it is concluded that hybridizing dry tows can lead to a drastic changes in the tow’s behaviour under tensile loadings, even for hybridizations using only carbon fibres. To understand the behaviours previously presented for the tows, it is necessary to understand the difference in failure strain distributions (Figure 4.15). The AS4 fibres are present in the three hybridizations studied and should be 4.3 Hybrid tow behaviour 75 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Strai n % Fai l ure Probability (%) AS4 (Curtin 1998) T300 (Curtin 1998) M40S (Tanaka 2014) M50S (Tanaka 2014) Figure 4.15: Failure strain distributions for the carbon fibres used for hybridization. used as reference. The T300 carbon fibres have the most similar failure strain distribution to that of the AS4 carbon fibres. This leads to a behaviour like the one shown in Figure 4.12, where it can be seen a more gradual failure with the hybridization, however, it is far from the desired pseudo-ductile behaviour. As the distributions between the hybridized fibres are further apart, it is shown that there is a less catastrophic failure (Figures 4.13 and 4.14). For the AS4/M50S hybridization, the distributions are the furthest apart and there is no continuity in fibre failure, leading to a load drop, as seen in Figure 4.14 for VLE = 0.25 or VLE = 0.5. For the AS4/M40S hybridization there is a continuity in the fibre failure and there is a more gradual failure, leading to a pseudo-ductile behaviour. 4.3.2 Glass-Glass hybridization The previous section analysed the hybridization of tows only with carbon fibres. This section will a similar study for glass fibre hybrid tows, whose fibre properties are in Table 4.9. These properties result in the failure strain distributions present in Figure 4.16. The main objective of this study is to verify whether the phenomena observed in the carbon-carbon hybridization can also be found in glass-glass hybrids. It is also important to understand that the conclusions, from the study of the carboncarbon hybrid tows, still hold for the glass hybrids. As it was mentioned in the previous section, the separation in the strain failure distributions may be necessary for the existence of a pseudo-ductile behaviour, therefore, the first hybridization to be studied will be the High Performance (HP) and High Dispersion (HD) AR glass fibres [92] hybridization. 76 Chapter 4. Model for the tensile failure of dry tows Table 4.9: Mechanical properties for glass fibres. Material Reference σ0 (MPa) L0(mm) mE (GPa) R (µm) hεi @75mm (%) E-Glass T.Okabe 2001 [43] 1550 24 6.34 76 6.5 1.59 E-Glass Feih 2005 [93] 1649 20 3.09 66.9 7.8 1.44 E-Gkass Pauchard 2002 [94] 2300 10 3.6 70 5 1.69 AR-HP Foray 2012 [92] 1363 60 9.6 70 7 1.81 AR-HD Foray 2012 [92] 876 60 4.8 70 7 1.09 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Strai n % Fai l ure Probability (%) E−Glass (Okabe 2001) E−Glass (Feih 2005) E−Glass (Pauchard 2002) AR−HP (Foray 2012) AR−HD (Foray 2012) Figure 4.16: Failure strain distributions for several glass fibres. HD/HP AR glass hybridization The HP fibres have a higher average failure strain (hεi) and will referred as HE fibres, while the HD fibres as the LE fibres. The stress-strain diagrams for this type of hybridization can be seen in Figure 4.17. The fibres in study in this section have the same elastic moduli and the same radii, therefore, the difference in behaviour due to hybridization may only be attributed to the differences in strength distributions. In this hybridization it is possible to see a progressive failure, which results in a maximum pseudo-ductile strain equal to 0.4%. However, the behaviour seen in Figure 4.17 is quite different of that for the AS4/M40S carbon (Figure 4.13). This may be attributed to the glass fibres having the same elastic properties and radii, while in the carbon hybridization the elastic modulus of the AS4 carbon (HE fibres) is inferior to that of the M40S carbon. 4.3 Hybrid tow behaviour 77 0 0.5 1 1.5 2 2.5 0 100 200 300 400 500 600 700 800 900 1000 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.25 VL E = 0.35 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 4.17: Stress-strain diagrams at various hybrid volume fractions for HD/HP AR glass hybridization. Table 4.10: Stress-strain reference properties for HD/HP AR glass hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 953.24 1.50 0.14 0.25 738.89 1.45 0.40 0.35 660.87 1.34 0.40 0.5 588.33 1.07 0.23 0.75 528.39 0.94 0.18 1 498.60 0.89 0.18 Trying to hybridize a tow with any other two types of glass fibres (from those in Table 4.9) results in a tow behaviour similar to that of Figure 4.12, meaning there is no pseudo-ductile behaviour. Analysing the fibre failure strain distributions (Figure 4.16) it is possible to reinforce what was concluded in Section 4.3.1: to have a pseudo-ductile behaviour there has to be a continuity in fibre failure and the start of failure of the HE fibres must be for strains close to the failure strains of the strongest LE fibres. The Feih [93] and Pauchard [94] E-glass strength distributions have a high dispersion and their hybridization shows high values for pseudo-ductile strain, however, there is no pseudo-ductile behaviour. The high dispersion of fibre strengths leads to a high non-linearity in the tow’s tensile behaviour, justifying the high results of the pseudo-ductile strain. 78 Chapter 4. Model for the tensile failure of dry tows 4.3.3 Kevlar-Kevlar hybridization Kevlar fibres are polymeric fibres used in technical composites. The fibre properties and strength distributions are shown in Table 4.11 and Figure 4.18. This section presents the study of the influence of hybrid tows based on kevlar fibres. Table 4.11: Mechanical properties for kevlar fibres. Material Reference σ0 (MPa) L0(mm) mE (GPa) R (µm) hεi @75mm (%) Kevlar 29 Naito 2013 [95] 3445.8 25 11.8 85.3 6.895 3.52 Kevlar 49 Naito 2013 [95] 4083.3 25 8.2 149.1 5.135 2.26 Kevlar 119 Naito 2013 [95] 3101.2 25 11.8 61.4 5.46 4.41 Kevlar 129 Naito 2013 [95] 3433 25 10.3 99 5.79 2.97 This fibres have, in general, an average failure strain superior to that of carbon and glass fibres. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Strai n % Fai l ure Probability (%) Kevlar 29 (Naito 2013) Kevlar 49 (Naito 2013) Kevlar 129 (Naito 2013) Kevlar 149 (Naito 2013) Figure 4.18: Failure strain distributions for several kevlar fibres. Kevlar 49/119 hybridization The hybridization of Kevlar 49 and Kevlar 119 is the hybridization between the Kevlar fibres with the highest and lowest average failure strain, among the data found. The stress-strain diagrams for this hybridization can be found in Figure 4.19 and the most significant properties in Table 4.12. 4.3 Hybrid tow behaviour 79 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 500 1000 1500 2000 2500 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 4.19: Stress-strain diagrams at various hybrid volume fractions for kevlar 49/119 hybridization. Table 4.12: Stress-strain reference properties for kevlar 49/119 hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2120.98 3.27 0.24 0.125 1853.77 3.27 0.95 0.25 1704.65 3.23 1.24 0.5 1905.42 1.96 0.22 0.75 2183.09 1.91 0.22 1 2457.20 1.87 0.22 As it can be seen from the data presented this hybridization shows a high value for the pseudo-ductile strain (εd= 1.24%), and the failure strain distributions show a similar behaviour as the previous ones whose behaviour as pseudo-ductile like (the failure of the stronger LE fibres coincide with the beginning of failure of the HE fibres). The higher pseudo-ductile strain, in comparison with the carbon-carbon and glass-glass hybridization may be related to the higher failure strain of the Kevlar fibres. Kevlar 119/129 hybridization The Kevlar 119 and 129 fibres have failure distributions whose average failure strains are less apart than that in the Kevlar 49/119 hybridization. This hybridization results in the stress-strain curves and major properties in Figure 4.20 and Table 80 Chapter 4. Model for the tensile failure of dry tows 4.13. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 500 1000 1500 2000 2500 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 4.20: Stress-strain diagrams at various hybrid volume fractions for kevlar 119/129 hybridization. It can be seen that the failure of this hybrid tow is more catastrophic than that of the previous hybridization. This leads us to similar conclusions to those from carbon-carbon and glass-glass hybridization: to achieve a pseudo-ductile behaviour in tow failure the HE fibres need to start failing near the end of the failure of the LE fibres. This causes a progressive failure of the fibres leading to an pseudo-ductile behaviour. Table 4.13: Stress-strain reference properties for kevlar 119/129 hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2118.78 3.27 0.24 0.125 1984.39 3.01 0.32 0.25 1957.57 2.87 0.28 0.5 2023.90 2.65 0.25 0.75 2115.10 2.56 0.25 1 2251.12 2.48 0.21 4.3.4 Carbon-Glass hybridization In the previous sections the effects of hybridizing tows with fibres of the same type were studied. As previously stated in Chapter 3, the hybridization of composite materials started between carbon and glass fibres,with the objective of achieving a 4.3 Hybrid tow behaviour 81 reduced price of the composite while maintaining some of the desired properties of the carbon fibres. The hybridization between fibres of different materials enables the creation of a wider range of material properties, that can be tuned to a specific application. In the next sections, with the aim of maximizing the pseudo-ductile behaviour, the hybridization of tows with fibres of different materials will be studied. Some different behaviours are expected due to a more diverse range of fibre properties. This section is dedicated to carbon-glass hybrids, whose properties are presented in Tables 4.5 and 4.9. T300 carbon/AR-HP glass hybridization The T300 carbon fibres [88], whose elastic modulus is E= 232 GPa and Weibull parameters are σ0= 3170 MPa and m= 5.1at L0= 25 mm, was hybridized with the AR-HP glass fibres [92], whose elastic modulus is E= 70 GPa and Weibull parameters are σ0= 1363 MPa and m= 9.6at L0= 60 mm. In this hybridization, the T300 carbon fibres are the LE fibres, while the AR-HP glass are the HE fibres. The stress-strain diagrams for several hybrid volume fractions are presented in Figure 4.21. 0 0.5 1 1.5 2 2.5 0 200 400 600 800 1000 1200 1400 1600 1800 Strain (%) Stress (MPa) VLE = 0.0 VLE = 0.125 VLE = 0.25 VLE = 0.5 VLE = 0.75 VLE = 1.0 Figure 4.21: Stress-strain diagrams at various hybrid volume fractions for T300 carbon and AR-HP glass hybridization. As it is possible to see in Table 4.14, the maximum value for the pseudo-ductile strain is equal to 0.74% and is achieved at a volume fraction of the LE fibres equal to 0.125. This pseudo-ductility comes at the cost of the maximum stress that the tow can withstand, as it is reduced from 1552 MPa, for full carbon, to 685 MPa 88 Chapter 4. Model for the tensile failure of dry tows Chapter 5 Progressive damage model for hybrid composites The effects of fibre hybridization on the tensile response of dry tows were studied in the previous chapter. In the dry tows there is no matrix and the fibres are not connected, which does happen in composite materials. The presence of the matrix changes the material’s behaviour and this needs to be taken into account in a model to predict the tensile failure of composite materials. The present chapter aims to extend the progressive damage model for unidirectional fibre-reinforced composites based on fibre fragmentation, developed by Turon et al. [5] to hybrid composites. The model will be used to predict the tensile behaviour of hybrid fibre reinforced composites to understand the factors controlling the pseudo-ductile behaviour of hybrid composites. 5.1 Model development The model developed by Turon et al. [5] is based on the multiple fragmentation of the fibres in single fibre fragmentation tests, whose mechanisms are present in the failure of multiple fibre composite materials. 5.1.1 Fibre break density The failure probability of the fibres is considered to follow a Weibull distribution [17], given by P(σ)=1−exp −L L0σ σ0m,(5.1) 89 90 Chapter 5. Progressive damage model for hybrid composites where P(σ)is the failure probability of a fibre with length Lwhen subjected to a tension σ.L0is the reference length at which the Weibull parameters σ0and m were determined. From Equation 5.1 it is possible to show that the number of breaks in a fibre follows a Poisson distribution, therefore, it is possible to determine the average number of breaks (hNi) in a fibre as a function of the applied stress: hNi=L L0σ σ0m .(5.2) This equation is not fully accurate as some defects will be located in the stress recovery region of the fibres, where the stress is lower than the applied stress, which is not taken into account by Equation 5.2. If the number of breaks in a fibre follows a Poisson distribution then the distance between fibre breaks can be described by an exponential law: f(x)=Λe−Λx,(5.3) where Λis the number of breaks per unit length, given by Λ = hNi L=1 L0σ σ0m .(5.4) With the presented equations, the fibre break distribution, as a function of the applied strain, is fully characterized. 5.1.2 Fibre stress As previously mentioned, a broken fibre does not loose the totality of the capacity to carry stress. After a fibre breaks, the matrix is loaded in shear and is able to transfer the stress back to the broken fibre, which will cause the fibre to have a stress profile similar to the one in Figure 5.1, described by the shear-lag theory. The length recovery region (lex), distance from the fibre break to the location where the fibre is fully able to carry stress, can be defined as: lex =Rf τ Efε 2,(5.5) where τis the interfacial shear strength, Rfthe fibre radius, Efthe elastic modulus of the fibre and εis the applied strain. The average stress in a fibre of length Lcan be computed by integrating the axial stress in all the fibre fragments along the fibre length: 5.1 Model development 91 where Kis the number of breaks per unit length, and it is computed from Eq. (5) K¼hNi L¼1 L0 r r0 !" q .ð7Þ This assumption is valid for an infinitely long fibre and at the initial fragmentation stages. As mentioned above, at advanced fragmentation stages, some flaws will be obscured in the load recovery region, and then, the break density will be lower than the predicted by Eq. (7). Many authors [9–11] introduce this phenomenon into their formulation obtaining other distribution functions which take into account higher break densities. Other authors [15,16] modify Eq. (7) and obtain an expression of the break density without the possible obscured potential breaks. In fact, the exact mathematical solution for the problem was provided by Hui et al. [11]. These distributions are more complex than Eq. (7) and, in some cases, numeric techniques are required for evaluating the expression. As the purpose of the present work is to develop a stiffness degradation model for the initial stages of damage, the influence of not considering the flaws obscured in the load recovery region is rather negligible. In Section 4, the numerical simulations from the present model based on Eq. (7) are compared to the more refined models cited above, and a very small difference for the first stages of fibre breakage is obtained. Moreover, at advanced fragmentation stages, the localized stress transfer in the composite has an important influence on the evolution of fibre breaks and the degradation and final failure is controlled by the formation and growth of clusters which require 2D models to be accounted for. In order to reach a mathematical expression of the apparent stiffness of the composite, it is necessary to compute the average fibre stress when some fractures have occurred. 2.3. Average fibre stress It has been shown in previous subsections that some flaws in a fibre will grow to a fully formed crack under the applied load. These cracks, or fibre breaks, cause a new stress redistribution along the fibre which may cause further breaks. When a fibre breaks, the load carried by the fibre drops down to zero at the position of the break and the load is carried by the shear stress between the fibre and the matrix (see Fig. 2). The stress in a broken fibre, r F , as a function of the distance from the break can be written as drF dz¼2s R;ð8Þ where Ris the radius of the fibre, sis the maximum shear stress and zis the distance from a break. This causes a stress redistribution near fibre breaks (see Fig. 2) which has been widely studied. Cox [17] was the pioneer to predict the real stress near the breaks by using a shear-lag model. The formulation of Cox!s model is quite complex and other simplified shear-lag approaches have been derived. One of the most widely used is the shear-lag model which was first introduced by Kelly and Tyson [18], and which assumes a linear increase of the axial stress from a fibre break, until a certain distance from it. At this distance, called the load recovery region, the stress reaches the far-field stress, see Fig. 2. According to the Kelly–Tyson shear-lag model, the length of this load recovery region, l ex , is obtained from the far-field stress, E F e(where E F is the fibre Young!s modulus and ethe composite strain), the radius of the fibre, R, and the maximum shear stress, s, between the fibre and the matrix before fibre debonding or matrix yielding occurs lex ¼R s EFe 2.ð9Þ From this stress redistribution, the average fibre stress along the fibre, r m , can be computed by integrating the axial stress over all of the fibre fragments along the fibre length rm¼NRðLÞ hi ¼N1 LZxRðxÞfðxÞdx;ð10Þ where f(x) is the fragment length distribution given in Eq. (6), and R(x) is the average stress in a fibre of length x. This integral is worked out in two steps. First, the average stress corresponding to the axial stress profile along a fibre fragment of length xis computed. Then, this axial average stress is integrated over all fibre fragments. In order to compute the axial average stresses for a fibre fragment of length x, it is necessary to distinguish whether the fibre fragment is greater to two times the length of the load recovery region (2l ex ), or not. (a) Average stress in a fibre of length 2l ex 6x. In a fibre of length x, greater than two times the stress recovery region, the stress profile assuming a linear shear-lag Fig. 2. Kelly–Tyson!s shear lag model. Stress profile at a fragment of broken fibre. At a break the axial stress is zero and it increases until it reaches the far-field stress (E F Æe). This region is called load recovery region, and has a length of l ex . 2042 A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048 Figure 5.1: Stress profile in a fibre with multiple fractures, according to shear-lag model [5]. σm=hNi1 LZxΣ(x)f(x) dx , (5.6) where f(x)is the fragment length distribution function (Equation 5.3) and Σ(x)is the average stress in a fragment of length x. The average stress in a fragment will be dependent on the relation between the fragment length (x), the fibre length (L) and the recovery region length (lex), and three cases are possible: 1. The fragment length is lower than two times the recovery region (Equation 5.7a); 2. The fragment length is higher than two times the recovery region but lower than the fibre length (Equation 5.7b); 3. The fragment length is higher than the fibre length (Equation 5.7c). As there are different cases for the average stress in a fragment, the function Σ(x) can be described as a piecewise function given by: Σ(x) =              Efεx 4lex , x ≤2lex (5.7a) Efε1−lex x,2lex ≤x≤L(5.7b) Efε , x ≥L(5.7c) Using Equations 5.4 and 5.7 in Equation 5.6, the average stress in a fibre can be written as: σm(ε) = EfεΛ Z2lex 0 x2 4lex f(x) dx+ZL 2lex (x−lex)f(x) dx+Z∞ L xf (x) dx!. (5.8) 92 Chapter 5. Progressive damage model for hybrid composites The analytical solution of the previous equation gives the average stress in a fibre as a function of the applied strain (ε) and is given by: σm(ε) = Efε 1−e−2lexΛ 2lexΛ+ Λlexe−LΛ!.(5.9) In this type of models, based on fibre fragmentation, it is usual to define the critical strength parameter (σc) [96] . This critical strength relates the statistical parameters for fibre strength and the matrix properties. Imagine that the composite material is loaded with an applied stress (σi), which causes the fibres to fracture. If the spacing between the fibres is greater than twice the recovery region length (lex), then there are locations where the stress in the fibres is equal to the far field stress (σi), therefore the fibres are still able to fracture into smaller fragments, by increasing the applied stress. The critical stress (σc) is defined as the stress that causes the average fragment spacing to be equal to twice the recovery region and is equal to: σc= σm 0τl0 Rf!1/1 + m .(5.10) In order to better relate the different types of fibres, with different elastic moduli, it is better to define the critical strain: εc=σc Ef =1 Ef σm 0τl0 Rf!1/1 + m .(5.11) 5.2 Composite damage model To develop the damage model for the composite material it is necessary to define how a fibre failure affects the loads in the remaining intact fibres and how to assemble the mechanical behaviour of the constituents in the composite. The model considers global load sharing (GLS) and, therefore, the stress is equally distributed among the intact fibres after one fails. This causes some limitations to the model as the longitudinal failure of UD composites is caused by the propagation of a cluster of broken fibres, which the model is not able to predict. The second aspect is resolved considering the rule-of-mixtures [5]. In order to avoid some physical incompatibilities the damage model is developed in the framework of the thermodynamics of irreversible processes. The free ernergy of the model is obtained by adding the free energy of the constituents as: ψ= (1 −df1)ψ0 f1(εf1)Vf1+(1 −df2)ψ0 f2(εf2)Vf2+(1 −dm)ψ0 m(εm)Vm,(5.12) 5.2 Composite damage model 93 where dNis the damage variable, VNthe volume fraction and εNthe strain of the constituent N, with Nequal to f1,f2or mdepending on the constituent (type one fibre, type two fibre or matrix). The variable ψ0 Nrepresents the free energy of the undamaged material given by: ψ0 N=1 2εN ij CN ijklεN kl ,(5.13) where Cijkl is the constitutive tensor of the constituent N. Equation 5.13 is a function of the strains in the constituent in cause, however it is necessary to define the damage model as a function of the composite deformation. To do so is necessary to resort to the rule of mixtures considering a serial-parallel behaviour [97]. This allows to define the influence tensors Tijkl as: εN ij =Tijklεkl ,(5.14) where εN ij is the strain in the constituent Nand εkl is the strain of the composite. Using Equation 5.14 it is possible to determine the free energy as a function of the composite strains as ψ0 N=1 2εmnTN mnijCN ijklTN klopεop .(5.15) The rate of dissipation Ξcan be written as: Ξ = σij ˙εij −˙ ψ= σij −∂ψ ∂εij !˙εij −X∂ψ ∂dN ˙ dN≥0.(5.16) The constitutive equation for the damage model can be written as: σmn =∂ψ ∂εmn =hX(1 −dN)TN mnijCN ijklTN klopiεop .(5.17) With the derivatives of the free energy respect to the damage variables and respect to the strains it is possible to write Equation 5.16 as: Ξ = Vf1ψ0 f1˙ df1+Vf2ψ0 f2˙ df2+Vmψ0 m˙ dm≥0.(5.18) From this equation is can be shown that, in order to guarantee the thermodynamic consistency, the derivatives of the damage variables must be positive: ˙ df2≥0, ˙ df2≥0and ˙ dm≥0 As we are interested in studying the longitudinal failure of UD composites the model can be simplified by considering only the stresses in the longitudinal direction 94 Chapter 5. Progressive damage model for hybrid composites due to an applied longitudinal strain. The constitutive equation, for the simplified case, can be written as: σ(ε) = X(1 −dN)ENVNε , (5.19) where EN,VNand dNare, respectively, the elastic modulus, volume fraction and damage variable of the constituent N.Ncan be the matrix or the fibres and there can be more than one type of fibres. The damage variables for the constituents can be obtained from different damage models. For the fibres it was considered the damage variable that results from Turon’s et al. model [5], given by: df= 1 − 1−e−2lexΛ 2lexΛ+ Λlexe−LΛ!.(5.20) As the longitudinal failure of the composite is a fibre dominated process the damage in the matrix was not considered, therefore, the damage variable for the matrix dM is considered 0for all the applied strains. 5.3 Non-hybrid composite behaviour To better understand the effects of hybridization on the longitudinal failure of UD composites, it is important to understand the effects that the fibre and matrix properties have in the material response. In this section, the composite materials are considered to have a fibre volume fraction (Vf) of 60 % and a gauge length (L) of 75 mm. As base information for fibre properties, the AS4 carbon fibres will be considered. These fibres have a mean radius (R) of 3.5µm, Young’s modulus (E) equal to 234 GPa and the parameters for the Weibull distribution are: σ0= 4275 MPa and m= 10.7at L0= 12.7 mm [88]. This values will serve as a baseline for the parametric study of the effect that the strength distributions of the fibres have in the tow behaviour. The matrix is considered to have a elastic modulus (Em) equal to 7 GPa and the maximum shear stress in the fibre-matrix interface is considered to be 40 MPa. The presented parameters should be considered unless other parameters are specified in the specific sections. 5.3.1 Matrix properties The matrix is considered to have a secondary influence in the longitudinal failure of UD composites, however, the interfacial shear strength (τ) affects the recovery length of the fibres, therefore, affecting the average stress in the fibres. The influence 5.3 Non-hybrid composite behaviour 95 influence of the matrix elastic modulus (Em) is very reduced, as the fibres are the main load carrying constituent. The effects of the interfacial shear strength in the longitudinal response of the composite is shown in Figure 5.2 and Table 5.1. The parameter hσiis the average failure strength of the fibres. Table 5.1: Effect of interface shear strength (τin some reference properties. σ0(MPa) mhσi(MPa) τ(MPa) Maximum load (N) Strain at max. load (%) 4275 10.7 3454.73 10 2487.08 1.89 3454.73 20 2638.83 2.01 3454.73 40 2799.88 2.13 3454.73 60 2898.62 2.21 0 0.5 1 1.5 2 2.5 3 3.5 0 500 1000 1500 2000 2500 3000 Strain (%) Load (MPa) τ= 10 M Pa τ= 20 M Pa τ= 40 M Pa τ= 60 M Pa Figure 5.2: Stress-Strain diagrams for composites with different interfacial shear strength (τ). From the analysis of Figure 5.2 and Table 5.1 it is possible to understand that increasing the interfacial shear strength has a positive effect in the composite strength. This can be attributed to a reduced recovery length of the fibres, which will decrease the length of the region where the fibre is not fully able to carry load and, therefore, increase the average stress in the fibres. As the elastic properties of the fibres remain the same, as the composite strength increases, so will the failure strain. 96 Chapter 5. Progressive damage model for hybrid composites 5.3.2 Weibull scale parameter The effects of the scale parameter (σ0) of the Weibull distribution, which is a measure of the average fibre strength, is shown in Figure 5.3 and the principal results from the composite response are presented in Table 5.2. 0 0.5 1 1.5 2 2.5 3 3.5 0 500 1000 1500 2000 2500 3000 Strain (%) Load (MPa) σ0= 3800 σ0= 4000 σ0= 4275 σ0= 4500 Figure 5.3: Stress-Strain diagrams for composites with fibres with different scale parameters (σ). Table 5.2: Effect of the scale parameter (σ) in some reference properties. σ0(MPa) mhσi(MPa) Maximum load (N) Strain at max. load (%) 3800 10.7 3070.9 2514.0 1.91 4000 3232.5 2634.7 2.01 4275 3454.7 2799.9 2.13 4500 3636.6 2934.4 2.24 As expected, the scale parameter affects the average failure strength of the composite material. As the average failure strength of the fibres is increased by increasing the scale parameter, so is the failure strength of the composite material. This results are similar to the ones from Chapter 4.2. 5.3.3 Weibull shape parameter The Weibull shape parameter (m) affects the dispersion of the strength of the fibres, however, as previously stated, it also affect the average failure strength of the 5.3 Non-hybrid composite behaviour 97 fibres (see Equation 4.2). Having as base the AS4 carbon fibre properties and varying the shape parameter results in the stress-strain curves shown in Figure 5.4. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 500 1000 1500 2000 2500 3000 3500 Strain (%) Load (MPa) m = 12.0 m = 10.7 m = 7.5 m = 5.0 Figure 5.4: Stress-Strain diagrams for composites with fibres with different shape parameters (m), with τ= 40 MPa. Table 5.3: Effect of the shape parameter (m) in some reference properties, with τ= 40 MPa. σ0(MPa) mhσi(MPa) Maximum load (N) Strain at max. load (%) 4275 12.0 3533.13 2758.15 2.08 10.7 3454.73 2799.88 2.13 7.5 3167.01 2984.57 2.35 5.0 2751.74 3346.93 2.78 The results present in Figure 5.4 and Table 5.3 are not in accordance with those from the model for the dry tows (Chapter 4.2). It was expected that, as we increased the Weibull shape parameter, the failure of the composite would start at lower strains, the stress-strain curve would be wider and the materials resistance would be reduced. With this model the results show that increasing the shape parameter leads to a wider stress-strain curve, but also increases the composite strength. This increase in composite strength can partially be explained by an increase of the fibre strength, however this effect is higher for composite materials than for the dry tows, meaning that the increase in fibre strength does not fully explain the increase in the strength of the composite material. As the interfacial shear strength has an influence in the composite strength, it may also affect the results from the influence of the Weibull shape parameter. 104 Chapter 5. Progressive damage model for hybrid composites Table 5.8: Stress-strain reference properties for AS4/T300 hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2799.88 2.13 0.18 0.125 2799.55 2.15 0.19 0.25 2800.51 2.16 0.20 0.5 2808.15 2.20 0.23 0.75 2829.91 2.27 0.28 1 2893.32 2.41 0.38 The T300 carbon fibres have a higher critical strain than the AS4 carbon, therefore the T300 carbon fibre composite has a higher failure strain. In terms of pseudoductility the results are in accordance to those of the dry tow model, where there was no significant increase in the pseudo-ductile strain (εd) due to the hybridization of these two fibre types. AS4-M40S carbon hybridization This section studies the hybridization of AS4 and the M40S [91] carbon fibres. The M40S fibres have a lower average failure strain than the AS4 fibres and than the T300 carbon fibres used in the precious hybridization. This leads to a greater difference in failure strains in the hybridized fibres. The difference between the shape parameters between these two fibres is similar to that between the AS4 and the T300 carbon fibres, since the shape parameter for the M40S carbon fibres is equal to 5.1. The tensile stress-strain curves resulting from this hybridization are shown in Figure 5.10. As it can be seen from analysing Figure 5.10 there is no pseudo-ductile effect in the hybridization with these two types of fibres, which can be confirmed from the results in Table 5.9. These results are not in accordance with those from the dry tow model (Figure 4.13), where some pseudo-ductile behaviour was achieved for a low elongation fibre volume fraction equal to 25%. The differences in theses results may lead to the conclusion that the pseudo-ductile behaviour in composite materials are affected by different parameters than those that affect the behaviour of dry tows. Analysing the critical strains for both fibre types it is possible to see that they are very similar, 2.47% for the AS4 and 2.23% for the M40S fibres. This causes the non-hybrid composites to have similar failure strains, which causes the tensile 5.4 Hybrid composite behaviour 105 0 0.5 1 1.5 2 2.5 3 3.5 0 500 1000 1500 2000 2500 3000 3500 4000 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 5.10: Stress-strain diagrams at various hybrid volume fractions for AS4/M40S hybridization. Table 5.9: Stress-strain reference properties for AS4/M40S hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2799.88 2.13 0.18 0.125 2888.33 2.10 0.23 0.25 2983.37 2.07 0.26 0.5 3194.03 2.00 0.29 0.75 3429.11 1.93 0.29 1 3682.06 1.88 0.29 response of the hybrid composites to not be pseudo-ductile. AS4-M50S carbon hybridization This section deals with the hybridization between the AS4 and the M50S [91] carbon fibres. The M50S fibres have a high modulus (E= 480 MPa), a low failure strain and a low fibre strength dispersion, with a Weibull shape parameter equal to 9. This shape parameter is the closer to the one for the AS4 carbon fibres. The resulting stress-strain curves for the hybridization of these two types of fibres are shown in Figure 5.11. For this hybridization there is a tendency for a pseudo-ductile behaviour. For a low elongation fibre volume fraction (VLE) equal to 0.25 the pseudo-ductile strain 106 Chapter 5. Progressive damage model for hybrid composites 0 0.5 1 1.5 2 2.5 3 3.5 0 500 1000 1500 2000 2500 3000 3500 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 5.11: Stress-strain diagrams at various hybrid volume fractions for AS4/M50S hybridization. reaches its maximum value and is equal to 0.94 (see Table 5.10). This values represents a large percentage of the composite total failure strain, that is equal to 2.12%. As in all the cases that shown a pseudo-ductile behaviour, the increase of the pseudo-ductile strain is accompanied by a reduction in the maximum load of the material. Analysing the properties of the hybridized fibres it is possible to conclude that there is a significant difference between their critical strain, which can explain the pseudo-ductile behaviour achieved. Table 5.10: Stress-strain reference properties for AS4/M50S hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2799.88 2.13 0.18 0.125 2465.91 2.13 0.60 0.25 2132.19 2.12 0.94 0.5 2379.40 1.22 0.12 0.75 2708.83 1.18 0.12 1 3052.04 1.16 0.11 Analysis of the results From the previous results it is concluded that the behaviour due to hybridization changes drastically from when we consider only a tow of fibres (dry tow) to when we consider the composite material. The introduction of the matrix changes the 5.4 Hybrid composite behaviour 107 behaviour of the material and, therefore, the factors controlling its behaviour. In the previous chapter (Model for the tensile failure of dry tows), and for a carbon-carbon hybridization, it was seen that there was a pseudo-ductile behaviour for the hybridization between the AS4 and M40S fibres and for the AS4 and M40S fibres, which was explained by the differences in the failure strain distributions. In this section, using the damage model for the composite, it was concluded that there was only pseudo-ductility for the AS4/M50S hybridization. Analysing the properties of the hybridized fibres, namely the critical strain (εc) and the average failure strain (hεi) of the fibres , it is possible to conclude that the main factor controlling the pseudo-ductility is not the difference between the average failure strains, but is the difference between the critical strains, which control the failure strain of the non-hybrid reference composites. This is the case for the AS4 and M50S hybridization, that have critical strains, respectively, 2.47% and 1.36%. Although not present here, it was verified that other combinations of carbon fibres with different critical strains lead to similar results, but this hypotheses will be studied for other types of hybridization in the following sections. 5.4.2 Glass-glass hybridization This section presents the study of hybridizing composite materials with different types of glass fibres, whose properties are shown in Table 5.11 and failure strain distributions in Figure 5.12. This study will try to support the conclusions withdrawn from the carbon-carbon hybridizations. Table 5.11: Mechanical properties for Glass fibres. Material Reference σ0 (MPa) L0 (mm) mE (GPa) R (µm) hεi @75mm (%) σc (MPa) εc E-Glass T.Okabe 2001 [43] 1550 24 6.34 76 6.5 1.59 2883.72 3.79 E-Glass FEIH 2005 [93] 1649 20 3.09 66.9 7.8 1.44 4526.85 6.77 E-Glass Pauchard 2002 [94] 2300 10 3.6 70 5 1.69 4975.15 7.11 AR-HP Foray 2012 [92] 1363 60 9.6 70 7 1.81 2295.99 3.28 AR-HD Foray 2012 [92] 876 60 4.8 70 7 1.09 2451.92 3.50 108 Chapter 5. Progressive damage model for hybrid composites 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Strai n % Fai l ure Probability (%) E−Glass (Okabe 2001) E−Glass (Feih 2005) E−Glass (Pauchard 2002) AR−HP (Foray 2012) AR−HD (Foray 2012) Figure 5.12: Failure strain distributions for several glass fibres. HD/HP glass hybridization The hybridization between the HD and HP glass fibres lead to good results in terms of pseudo-ductility in the previous chapter where dry tows were analysed. With the presence of the matrix and using the model implemented here, the results of hybridizing a composite material are shown in Figure 5.13 and Table 5.12. 0 1 2 3 4 5 6 0 200 400 600 800 1000 1200 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 5.13: Stress-strain diagrams at various hybrid volume fractions for HD/HP glass hybridization. From the analysis of this results it is possible to conclude that there is no pseudoductile response in the hybridization of these two types of fibres, which contradicts the results from the previous chapter. This supports the conclusion that the results of the dry tows cannot be transferred for the composite material. If we analyse the critical strain of both fibre types, we conclude that they are very similar, which may explain why the hybridization leads to this type of response. 5.4 Hybrid composite behaviour 109 Table 5.12: Stress-strain reference properties for HD/HP glass hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 1118.02 2.99 0.49 0.125 1120.43 2.95 0.44 0.25 1123.79 2.92 0.41 0.5 1132.10 2.88 0.35 0.75 1141.57 2.85 0.30 1 1151.71 2.83 0.26 HD/E-glass hybridization As the previous hybridization did not lead to the expected results other hybridization needed to be studied. Following the conclusions from the carbon-carbon hybridization, that fibres with similar critical strain do not have a pseudo-ductile behaviour, the hybridization between the HD glass fibres [92] and the E-glass fibres [93] is presented next. The results from this hybridization are shown in Figure 5.14 and Table 5.13. 0 2 4 6 8 10 12 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.25 VL E = 0.4 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 5.14: Stress-strain diagrams at various hybrid volume fractions for HD/Eglass hybridization. The present hybridization and for a low elongation volume fraction (VLE) equal to 0.4the pseudo-ductile strain is maximum and equal to 2.63%. From the analysis of the stress-strain curve in Figure 5.14 we can see that the failure is progressive and that from a strain equal to 3.5% to 6% the failure occurs at a constant stress. Analysing the properties of these two fibres it is possible to see that their critical 110 Chapter 5. Progressive damage model for hybrid composites strain is quite different, 7.11% for the E-glass and 3.5% for the HD glass fibres. This reinforces the conclusion that the fibres have to have a different critical strain in order to result in a pseudo-ductile behaviour when hybridized. Table 5.13: Stress-strain reference properties for HD/E-glass hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 1969.91 5.95 1.37 0.25 1548.82 5.82 2.26 0.4 1298.82 5.60 2.63 0.5 1241.77 3.52 0.69 0.75 1163.17 3.17 0.55 1 1118.02 2.99 0.49 5.4.3 Kevlar-kevlar hybridization This section focuses on the study of the hybridization of composite materials with different types of kevlar fibres, whose charecteristics are shown in Table 5.14 and Figure 5.15. Table 5.14: Mechanical properties for kevlar fibres. Material Reference σ0 (MPa) L0 (mm) mE (GPa) R (µm) hεi @75mm (%) σc (MPa) εc kevlar 29 Naito 2013 [95] 3445.8 25 11.8 85.3 6.895 3.52 4615.08 5.41 kevlar 49 Naito 2013 [95] 4083.3 25 8.2 149.1 5.135 2.26 6215.15 4.17 kevlar 119 Naito 2013 [95] 3101.2 25 11.8 61.4 5.46 4.41 4264.92 6.95 kevlar 129 Naito 2013 [95] 3433 25 10.3 99 5.79 2.97 4855.81 4.90 5.4 Hybrid composite behaviour 111 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Strai n % Failure P robability (%) Kevlar 29 (Naito 2013) Kevlar 49 (Naito 2013) Kevlar 129 (Naito 2013) Kevlar 149 (Naito 2013) Figure 5.15: Failure strain distributions for several kevlar fibres. Kevlar 49/119 hybridization The hybridization between Kevlar 49 and 119 showed great potential in the dry tow model, having a maximum pseudo-ductile strain equal to 1.24% (Figure 4.19). The kevlar 49 are, for this hybridization, considered the LE fibres, as they have an average failure strain equal to 2.26% while the kevlar 119 have an average failure strain equal to 4.41%. The results from the hybridization of these two types of fibres is shown in Figure 5.16 and Table 5.15. 0123456789 0 500 1000 1500 2000 2500 3000 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 5.16: Stress-strain diagrams at various hybrid volume fractions for kevlar 49/kevlar 119 hybridization. Analysing these results it is possible to see that for a LE fibre volume fraction equal to 0.125 the pseudo-ductile strain is maximum and equal to 1.72%. Combining this information to the stress-strain curve from Figure 5.16 gives to the conclusion that this type of hybridization has a pseudo-ductile behaviour. From Table 5.14 it is possible to see that the critical strains for the hybridized fibres are 4.17% and 6.95% for kevlar 49 and kevlar 119, respectively. 112 Chapter 5. Progressive damage model for hybrid composites Table 5.15: Stress-strain reference properties for kevlar 49/kevlar 119 hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2218.43 6.06 0.46 0.125 1984.58 6.02 1.72 0.25 1865.13 4.02 0.49 0.5 2192.81 3.73 0.41 0.75 2560.16 3.62 0.39 1 2939.01 3.56 0.37 Kevlar 119/129 hybridization The kevlar 119 and 129 fibres have a closer average failure strain that those for the previous hybridization, as well as a close Weibull shape parameter, respectively 10.3and 11.8. The stress-strain curves for this hybridization are shown in Figure 5.17. 0123456789 0 500 1000 1500 2000 2500 Strai n (%) Stress (MPa) VL E = 0.0 VL E = 0.125 VL E = 0.25 VL E = 0.5 VL E = 0.75 VL E = 1.0 Figure 5.17: Stress-strain diagrams at various hybrid volume fractions for kevlar 119/kevlar 129 hybridization. Analysing both Table 5.16 and Figure 5.17 it is possible to see that this hybridization leads to a pseudo-ductile behaviour at a LE fibre volume fraction equal to 0.125. Analysing the properties of both hybridized fibres from Table 5.15 it is possible to see that the critical strains of the fibres are equal to 6.95% and 4.9%, for the kevlar 119 and kevlar 149, respectively. This analysis helps supporting the conclusion that it is essential that the fibres have different critical strains in order to achieve a pseudo-ductile behaviour with their hybridization. 5.4 Hybrid composite behaviour 113 Table 5.16: Stress-strain reference properties for kevlar 119/kevlar 129 hybridization. VLE Maximum stress (MPa) Strain at max. stress (%) εd(%) 0 2218.43 6.06 0.46 0.125 2008.39 5.90 1.17 0.25 1950.20 4.79 0.48 0.5 2067.84 4.46 0.40 0.75 2231.20 4.32 0.38 1 2409.35 4.24 0.37 5.4.4 Carbon-glass hybridization Similarly to Chapter 4, the effects of hybridization with fibres of the same base (carbon, glass and kevlar) were studied. This allowed to achieve some important conclusions about the key parameters to obtain a pseudo-ductile behaviour in hybrid composite materials. This section and the following will address the study of hybridizing composite materials with fibres of different bases, in this case carbon and glass. The properties for the fibres are shown in Table 5.7 and 5.11. T300 carbon/AR-HP glass hybridization In the previous chapter (Section 4.3.4) the hybridization of tows with T300 carbon and AR-HP glass fibres was studied and the results demonstrated a pseudo-ductile behaviour. However, as stated in the previous sections, the results from tow hybridization cannot be extrapolated for the composite material as the presence of the matrix affects the material’s response. Figure 5.18 presents the stress-strain curves for this hybridization. Although the failure strain distributions for this two types of fibres are quite different, the fibres have similar critical strains and, therefore, a pseudo-ductile behaviour is not expected for the composite material, which is shown in Figure 5.18. In this figure it is possible to see that the failure strains of the reference composite materials (VLE = 0 and VLE = 1) are quite similar, in spite of the fibres having quite different average failure strains. This reinforces the conclusion that it is not possible to predict the composite response based, only, in the tow behaviour and that the critical strains of the hybridized fibres control the response of hybrid composite. 120 Chapter 5. Progressive damage model for hybrid composites Chapter 6 Micromechanical models The previous chapters focused on the study of the effects of hybridization using simple analytical models. These models are useful in understanding some of the main effects of hybridization, however, due to their simpleness, they are not able to take into account all the mechanisms that lead to the longitudinal failure of UD composites. To do so, it is necessary a more complex model that can take into account the more complex mechanisms such as the damage in the matrix, fibres and interface. To study this complex mechanism it is necessary to develop micromechanical models that are able to take into account the damage mechanisms in the longitudinal failure of composite materials. These models need to be able to correctly represent the behaviour of each of the constituents in the composite. As the fibres and matrix have different characteristics and behaviour two damage models need to be implemented. In order to connect these constituents it is necessary to define the interface between them, which has a large influence the behaviour of the material as the interfacial separation is one important failure mechanism. The next sections focus on the development and implementation of these models and are proceeded by the results of the micromechanical analysis of hybrid and non-hybrid composites. 6.1 RVE generation As previously stated in Chapter 2, the fibre arrangement in the material is an important factor in the composite’s behaviour, as square and hexagonal fibre arrangements usually lead to unrealistic results. In order to overcome this difficulty it is necessary to generate a Representative Volume Element (RVE) that is represen121 122 Chapter 6. Micromechanical models tative of the material, which means having a correct size and being able to represent the microstructure of the composite material. The algorithm used to generate the random distribution of fibres is based on the algorithm developed by Melro et al. [14], with some changes to allow the RVE to be hybridized, i.e., having multiple types of fibres. The algorithm is composed of three steps, of which only one had major changes, the first step. The flowchart of the algorithm is shown in Figure 6.1. The variable Nmax iis the maximum allowed number of iterations of the overall algorithm and Niis the current number of iterations. 2.2. MATLAB!SCRIPT TO GENERATE A SRVE 29 BEGIN ❄ Input Variables ❄ Hard-core model STEP 1 ❄ First Heuristic ❄ STEP 2 Second Heuristic ❄ STEP 3 Ni⩽Nmax i N Y ❄ ✛ vcur f<v req f Y N ❄ ✲ Output Results ❄ END Figure 2.7: Flowchart of algorithm RAND uSTRU GEN. delta – Value of δ,definedbyequation(2.3). Vol fibre req – Fibre volume fraction requested by user, vreq f. DISTMIN – Minimum distance between any two fibre centres, ∆min. N guesses max – Number of attempts of fibre placement in step one, Nmax g. N cycles max – Maximum number of iterations the algorithm is allowed to perform, Nmax i. N change – Number of iterations before changing criteria in step two, Nc. Square size – Distance of square size in step three, So. Figure 6.1: Flowchart of RAND_uSTRU_GEN algorithm [14]. The changes that were required to do in the hard-core model (STEP 1) are related with the necessity of guaranteeing that the algorithm is able to generate fibres with different diameters and that it is able to achieve the specified volume fraction of each fibre. The flowchart of the modified hard-core model is shown in Figure 6.2. The variables Vcurr fand Vreq fdefine the current and required fibre volume fraction, and with the subscript 1and 2represent the volume fractions of each type of fibre 6.1 RVE generation 123 in the hybrid composite. The variable Ngis the current number of iterations in this step and Nmax gis the maximum number of iterations before executing the step two of the algorithm. The changes in this algorithm allowed the generation of a RVE with different fibre volume fractions and hybrid volume fractions, with fibres with different radii. This algorithm also guarantees material symmetry in order to correctly implement the boundary conditions to the RVE. Another important aspect is guaranteeing a minimum distance between fibres in order to inhibit poorly meshed regions in the RVE. As previously stated, an important factor in the composite material’s behaviour is the interface between the fibres and the matrix, therefore, it necessary to correctly simulate this interface. This is done by introducing cohesive surfaces in all the interfaces between fibres and matrix. To correctly simulate the behaviour of UD composite materials under tensile loadings it is necessary to have a RVE with an adequate size to capture the mechanisms of failure of these materials, which leads to the necessity of having a large RVE. The dimensions of the RVE in the direction perpendicular to the fibres directly affect the number of fibres in the RVE. The in-plane dimensions of the RVE need to be such as it is possible to capture the damage mechanisms and cluster formation prior to the failure of the composite. Thefore, the number of fibres must be higher than the expected critical cluster size as this is the main mechanism of failure of UD composites. In terms of the size in the fibre direction, the RVE needs to be long enough to simulate the ineffective length of the fibres. As seen in Section 2.3 is about 15-20 fibre radius. Taking this into account two dimensions of RVEs were studied: 15 ×15 ×15 and 20 ×20 ×20 fibre radius. As the RVE needs to be representative of the material we are trying to model it is necessary to define correct boundary conditions. The characteristics of a composite material make it necessary to define periodic boundary conditions for the RVE. These boundary conditions define constraints in the displacements and rotations in the faces, edges and vertices that are opposed to each other [98]. As this type of boundary conditions is computationally very expensive, a simplified version of these boundary conditions was used by constrainning the displacements of opposite faces in the RVE, using ABAQUS tie constraints. 124 Chapter 6. Micromechanical models BEGINBEGIN Randomly generate first fibre position Randomly generate first fibre position Determine Vcur f Determine Vcur f Vcurr f< V req f Vcurr f< V req f Y N Vcurr f1< V req f1 Vcurr f1< V req f1 Randomly generate type one fibre position Randomly generate type one fibre position Randomly generate type two fibre position Randomly generate type two fibre position Ng=Ng+ 1Ng=Ng+ 1 Compatibility Check Compatibility Check New fibreNew fibre Update Vcurr f,Vcurr f1,Vcurr f2 Update Vcurr f,Vcurr f1,Vcurr f2 Vcurr f< V req f Vcurr f< V req f Ng< Nmax g Ng< Nmax g Output Results Output Results ENDEND Y N N Y NY Y N STEP TWOSTEP TWO Figure 6.2: Flowchart of the hard-core model for the generation of the microstructure. 6.2 Damage models 125 6.2 Damage models To simulate the micromechanical behaviour of composite materials it is necessary to develop material models that are able to represent the behaviour of the constituents in the composite. As the composite materials is constituted by two types of materials, the matrix and the fibres, it is necessary to develop two material models and one additional material model for the interface. The material model used for the matrix is the one developed by Bai et al. [99], which is a damage model with plasticity based on a modified paraboloid yield criteria, able to capture the thermal and strain-rate dependency in the matrix behaviour. As no modifications were made to this model in order to be implement in this work it will not be presented here. The material model for the fibres is based on the work of Melro [98] and is presented in the following section. 6.2.1 Damage model for the fibres The damage model implemented to characterize the behaviour of fibres was based on Melro’s work [98] with the modifications needed in order for the fibre strength to be characterized by a Weibull distribution. It is considered that the fibres possess a transversely isotropic behaviour, however, the damage model considers only one damage variable and is solely activated by the longitudinal stress component. In order to guarantee a thermodynamically consistent model it is firstly necessary to define the complementary free energy of the material: Gf=σ2 11 2E1(1 −df)+σ2 22 +σ2 33 2E2(1 −df)−ν12 E1 (σ11σ22 +σ11σ33) −ν23 E2 σ22σ33 +σ2 12 +σ2 13 2G12 (1 −df)+σ2 23 2G23 (1 −df), (6.1) where E1and E2are the longitudinal and transverse Young’s moduli, G12 and G23 the longitudinal and transverse shear moduli and dfis the damage variable for the fibres. To ensure that the damage process is irreversible it is necessary to guarantee that the rate of change of the complementary free energy density is greater than the externally applied stress: ˙ Gf−˙σ:ε ˙σ:ε ˙σ:ε≥0,(6.2) which can be written as 126 Chapter 6. Micromechanical models ∂Gf ∂σ σ σ−ε ε ε:˙ σ σ σ+∂Gf ∂df ˙ df⩾0.(6.3) To ensure a positive dissipation of mechanical energy it is necessary that the strain tensor to be equal to the derivative of the complementary free energy density with respect to the stress tensor, ε ε ε=∂Gf ∂σ σ σ.(6.4) The compliance tensor (Hf Hf Hf) can be defined as: Hf Hf Hf=∂2Gf ∂σ σ σ2.(6.5) Inverting the compliance tensor results in the stiffness tensor Cf Cf Cf= 1−df ∆             E11−β2E2ν12 (1 −df)(1 + β)E2ν12 (1 −df)(1 + β) 0 0 0 E2[1 −γ(1 −df)] E2(1 −df)(ν23 +γ) 0 0 0 E2[1 −γ(1 −df)] 0 0 0 G12∆ 0 0 sym. G12∆ 0 E2∆ 2(1+ν23)             , (6.6) where β=ν23 (1 −df),(6.7a) γ=ν12ν21 (1 −ff),(6.7b) ∆ = (1 −β) [1 −β−2γ(1 −df)] .(6.7c) The damage activation function can be defined as: Fd f=φd f−rf≤0,(6.8) where φd fis the loading function φd f=˜σ11 Xt f ,(6.9) and rfthe internal variable rf= max 1,max t−→∞ nφd f,to.(6.10) 6.2 Damage models 127 The loading function is a function of the fibre tensile strength (Xt f), which has a stochastic value and will vary from element to element. The loading function is also a function of the effective longitudinal stress ˜σ11. This is a component of the effective stress tensor given by: ˜σ ˜σ ˜σ=H0 f H0 f H0 f −1:ε ε ε , (6.11) where H0 f H0 f H0 fis the compliance tensor of the undamaged material, which can be obtained by forcing the damage variable to be null. To avoid mesh dependency problems and control the energy dissipated in the fracture process, Bažant’s crack band model [100] was implemented. The dissipated energy for the fibres is defined as: Ψf=Z∞ 0 Yf˙ dfdt=Z∞ 1 ∂Gf ∂df ∂df ∂rf drf=Gff le,(6.12) where Gff is the fracture toughness the fibres, lethe characteristic length and Yfis the thermodynamic force associated with the variable df. Using the complementary free energy for the fibres, given by Equation 6.1, it is possible to define Yfas: Yf=∂G ∂df =1 (1 −df)2"σ2 11 2E1+σ2 22 +σ2 33 2E2 +σ2 12 +σ2 13 2G12 +σ2 23 2G23 #,(6.13) that is always positive. The damage evolution law defined for the fibres is given by: df= 1 −eAf(1−rf) rf ,(6.14) where Afis a parameter that must be determined by solving Equation 6.12 as a function of the characteristic length, therefore, this parameter must be computed for each element of the mesh. The derivative of the damage law in order to rfis given by: ∂df ∂rf =eAf(1−rf) rf Af+1 rf!.(6.15) In order to solve Equation 6.12 it is necessary to define the relation between the real stress tensor and the effective stress tensor. This is done by imposing the principle of strain equivalence: σ σ σ=Cf:ε Cf:ε Cf:ε ˜σ ˜σ ˜σ=C0 f:ε C0 f:ε C0 f:ε)σ σ σ=Cf Cf Cf:C0 f C0 f C0 f −1: ˜σ ˜σ ˜σ=Cf Cf Cf:H0 f H0 f H0 f: ˜σ ˜σ ˜σ , (6.16) 128 Chapter 6. Micromechanical models where C0 f C0 f C0 fis the undamaged stiffness tensor and H0 f H0 f H0 fthe undamaged compliance tensor. If the particular case of uniaxial tensile loading is considered, the effective stress tensor (˜σ ˜σ ˜σ) is given by: ˜σ ˜σ ˜σ=            ˜σ11 0 0 0 0 0            .(6.17) For this tress state the three normal components of the real stress tensor are: σ11 =1−df ∆h1−β2−2γ(1 + β)i˜σ11 ,(6.18a) σ22 =σ33 =−1−df ∆ν12 (1 + β)df˜σ11 ,(6.18b) the remaining shear components of the tress tensor are equal to zero. Using Equations 6.18 in Equation 6.13 results in: ∂G ∂df UN =(1 + β)2 2E1∆2h(1 −β−2γ)2+ 2ν12ν21d2 fi˜σ2 11 ,(6.19) for the uniaxial tensile state. The damage activation function for the uniaxial tensile state is given by: Fd f UN =˜σ11 Xt f−rf≤0,(6.20) and for the damage to propagate, this equation needs to be equal to zero. Solving this equation in order to the applied effective stress results in ˜σ11 =Xt frf.(6.21) Using Equations 6.19, 6.18 and 6.21 in in Equation 6.13 results in: Z∞ 0 Xt f 2r2 f(1 + β)2 2E1∆h(1 −β−2γ)2+ 2ν12ν21d2 fi∂df ∂rf drf=Gff le,(6.22) which needs to be solved numerically along the damage evolution law (Equation 6.14) to determine the parameter Af. This variable is forced to be lower than zero in the initial conditions to force the initialization of the algorithm. The tensile strength assigned to each element with is generated using a process similar to that presented in Chapter 4, resorting to the generation of random numbers. A random number Xis generated for each element, using Fortran random 6.2 Damage models 129 number generator, in the interval 0 to 1. This number represents the failure probability in the Weibull distribution: X=P(σ)=1−exp 1−L L0σ σ0m.(6.23) Solving Equation (6.23) for σ, which will be the assigned tensile strength for the fibre (Xt f) results in: Xt f=σ0−L0 Lln (1 −X)1/m .(6.24) The fibres used in fibre reinforced composites have a very low energy release rate [91] which means that the maximum element length is very small. If the element length is higher than a critical value (le max) there is a snap-back effect, represented by the red line in Figure 6.3. le↑le↑ εε σσ Xf t Xf t Gcrit ff le Gcrit ff le Figure 6.3: Stress-strain diagrams for the fibre damage model for various element lengths. In order to avoid this issue, without reducing the element’s size to values that are too computationally expensive, a strategy was implemented that changes the effective energy release rate of the material to guarantee that there is no occurrence of the snap-back effect. Although this changes the material properties, its effect is reduced as the energy released in fibre failure is very small. To implement this strategy it is necessary to determine the minimum energy release rate (Gcrit ff ) that is required to avoid snap-back in an element of length le. This is done by considering the gray triangle in Figure 6.3, which represents minimum energy that needs to be released in the model: Gcrit ff le=1 2 Xt f EXt f.(6.25)