MESOMECHANICAL MODEL FOR FAILURE STUDY OF TWO DIMENSIONAL TRIAXIAL BRAIDED COMPOSITE MATERIALS. A Dissertation. Presented to

Transcription

1 MESOMECHANICAL MODEL FOR FAILURE STUDY OF TWO DIMENSIONAL TRIAXIAL BRAIDED COMPOSITE MATERIALS A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Xuetao Li December 2010

2 MESOMECHANICAL MODEL FOR FAILURE STUDY OF TWO DIMENSIONAL TRIAXIAL BRAIDED COMPOSITE MATERIALS Xuetao Li Dissertation Approved: Advisor Dr. Wieslaw K. Binienda Committee Member Dr. Robert K. Goldberg Committee Member Dr. Ernian Pan Committee Member Dr. Gunjin Yun Accepted: Department Chair Dr. Wieslaw K. Binienda Dean of College Dr. George K. Haritos Dean of the Graduate School Dr. George R. Newkome Date Committee Member Dr. Xiaosheng Gao Committee Member Dr. Kevin Kreider ii

3 ABSTRACT Carbon fiber composite materials are being used in aerospace applications due to their excellent mechanical properties, such as high strength and stiffness as well as low density. Two dimensional triaxial braided polymer matrix composites have been shown to have improved performance under impact loads. Recently, many of the aircraft engine manufacturers have used such braided carbon fiber/epoxy composite for engine fan cases. A potential problem in application of triaxial braided composite is to understand the cracking, debonding and delamination. Simulation would reduce time and cost in the development of composite fan cases. Development of accurate computer model for simulation is crucial in predicting deformation and failure and to help understand experimental results. Multi-scale modeling is a well established approach to simulating textile composite behavior. This research focused on meso level modeling of triaxial braided composites. The unit cell scheme is used to take into account internal braiding architectures as well as mechanical properties of three phases: fibers tows, matrix and tow interfaces. Model requires local properties of the material so micromechanics approach is used to produce those material parameters for the model. Failure initiation and progressive damage concept has been implemented in the fiber tows by using the Hashin failure criterion and a damage evolution law. The weak/imperfect fiber tow iii

4 interface is modeled by using a cohesive zone approach, where a zero thickness cohesive element technique is used and a mixed mode cohesive law is adopted based on fracture mechanics principles to evaluate crack initiation and predict crack propagation. This meso scale modeling technique has been used to examine and predict the failure observed in coupon tests. The tensile deformation and damage response of braided specimens is simulated and the results compared to experimentally obtained data. The effects of the fiber tow interface were investigated based on the effective deformation response. Furthermore, the local damage computed by the simulations is compared to the damage patterns observed experimentally to attempt to quantify the causes of the local damage and to examine their effects on the effective deformation response. By comparing the analytical results to those obtained experimentally, the applicability of the developed model is assessed and the failure process is investigated. In order to enhance the model, through parameter studies were carried out. At the same time, by incorporation of classical laminate theory, a connection between models of meso scale and macro scale was set up. The latter could be used to predict response of braided composite structures under impact loading. In such a manner, a frame work of multi-scale modeling of braided composites has been set up. iv

5 ACKNOWLEDGEMENTS I would like to thank my academic supervisor Dr. Wieslaw Binienda for his continuously support and encouragement during the course of my PhD studies. I am very grateful to him for giving me the opportunity to work on this project. Acknowledgements to the committee member, Dr. Ernian Pan and Dr. Gunjin Yun from Department of Civil Engineering, Dr. Xiaosheng Gao from Department of Mechanical Engineering, Dr. Kevin Kreider from Department of Applied Mathematics. Thanks to taking time to be on my committee. Special Thanks to Dr. Robert Goldberg, also on my committee, for the valuable comments and suggests he supplied throughout the course of the research. I am deeply obliged to him for his availability and numerous advices. I would also like to thank NASA technician Dr. Gary Roberts for leading the projects and excellent experiment testing. Thanks are given to Drs. Justin Littell, Daihua Zheng and Jingyun Cheng, for their help and kindness. Thanks are also given to my colleagues, Lee Kohlman, Zhuopei Hu, and Brina Blinzer for discussing on the topic of research. Greatest thanks to my wife and parents for their love and support than I could ever expect. v

6 TABLE OF CONTENTS LIST OF TABLES... x Page LIST OF FIGURES xi CHAPTER I. BACKGROUND AND INTRODUCTION Background Introduction...3 II. LITERATURE REVIEW Composite Material Failure Modeling Multi-Scale Modeling of Textile Composites Micro-scale Modeling Approaches Meso-scale Modeling Approaches Macro-scale Modeling Approaches Composite Material Testing Current Research...30 III. MESOMECHANICAL MODELING METHODOLOGY OF TRIAXIAL BRAIDED COMPOSITE MATERIALS Studied Composite Material System Proposed Mesomechanical Model...35 vi

7 3.3 Model Material Properties Characterization Fiber Tows Pure Resin Interface Material Numerical Experiments on Mesomechanical Unit Cell CLT Based Macro-Mechanical Solutions Uni-strain Loading in Axial Direction Uni-strain Loading in Transverse Direction Conclusions...61 IV. FAILURE STUDY OF THE TRIAXIAL BRAIDED COMPOSITE MATERIAL RESPONSE IN SIX-PLY STRAIGHT SIDED SPECIMEN Experiment Test and Measurement Technique Finite Element Model Development Model Correlation Model Correlation of T700s/E862 Composites Model Correlation of T700s/PR520 Composites Numerical Investigation of Material Failure Axial Tension Test Axial Compression Test Transverse Tension Test Transverse Compression Test Conclusions...84 vii

8 V. NUMERICAL PARAMETRIC STUDY OF MESOMECHANCIAL TRIAXIAL BRAIDED COMPOSITE MATERIAL MODEL Summary of Model Parameters Qualitative Parameter Sensitivity Study Tow Interface Parameters Fiber Tow Longitudinal Tensile Strength Fiber Tow Longitudinal Compressive Strength Fiber Tow Transverse Tensile Strength Fiber Tow Transverse Compressive Strength Fiber Tow Shear Strength Fiber Tow Fracture Energy Viscosity Parameter Quantitative Parameter Sensitivity Study Strain Rate Dependence Investigation Strain Rate Variation in Single Test Strain Rate Effect Conclusions VI. NUMERICAL SIMULATION OF SINGLE LAYER SPECIMEN BEHAVIOR OF TRIAXIAL BRAIDED COMPOSITE Testing of Single Layer Specimen Simulation of Single Layer Specimen Finite Element Model Material Parameters Results and Discussion viii

9 6.3.1 Axial Tension Specimen Results Transverse Tension Specimen Results Local Deformation Investigation Advanced Numerical Analysis Conclusions VII. SUMMARY AND FUTURE WORK Summary Future Work REFERENCES ix

10 LIST OF TABLES Table Page 3.1 Mechanical property of Toray s T700s Mechanical property of matrix Fiber tow modulus Fiber tow strengths Failure criteria for fiber tows Fiber tow fracture energy Interface fracture toughness Fiber volume ratio and thickness of each integration point of unit cell Nominal axial stiffness of meso-model and CLT results Type of failure at different levels of axial load Nominal transverse stiffness of meso-model and CLT results Type of failure at different levels of transverse load Categorization of model parameters used in material models Influence factor of each parameter on four tests Homogenized laminar modulus of T700s/E862 under two strain rates Homogenized laminar strengthes of T700s/E862 under two strain rates Fiber tow stiffness and strength properties in sing layer specimens x

11 LIST OF FIGURES Figure Page /±60 0 tri-axially braided composite architecture Microscope image of cross section of triaxial braided composite Comparison of tow interface of six layer specimen before and after test Type and shape of solid element utilized to mesh the unit cell geometry Finite element model of unit cell of Triaxial Braided Composite Representative volume element for generalized method of cells Bilinear equivalent stress-displacement relationship Numerical study on fracture toughness parameter Behavior of pure resin phase used in the model Fracture mode and bilinear constitutive relationship Mixed mode behavior of cohesive law Coordination setup for one single unit cell testing Comparison of unit cells between meso and macro models Stress strain curves for unit cell and sub cells under axial loading Stress strain curves for unit cell and sub cells under transverse loading Illustration of straight sided tensile specimen dimensions Straight sided specimen compression test set up...64 xi

12 4.3 Optical measurement setup Boundary and load conditions of axial test modeling Boundary and load conditions of transverse test modeling Global stress strain curves of T700s/E862 in axial tension test Global stress strain curves of T700s/E862 in transverse tension test Global stress strain curves of T700s/E862 in axial compression test Global stress strain curves of T700s/E862 in transverse compression test Global stress strain curves of tension test on T700s/PR520 system Global stress strain curves of compression test on T700s/PR520 system Contour plots of failure modes in axial tension test Correlation between numerical contour plot and optical measurement Fiber splitting on fiber tows Contour plots of failure modes in axial compression test Contour plots of failure modes in transverse tension test Correlation between numerical contour plot and optical measurement Digital microscopic picture of a transverse tensile specimen Contour plots of failure modes in transverse compression test Parameter study on interface strengths Parametric studies on fiber tow longitudinal tension strength Parametric studies on fiber tow longitudinal compressive strength Parametric studies on fiber tow transverse tensile strength Parametric studies on fiber tow transverse compressive strength Parametric studies on fiber tow shear strength xii

13 5.7 Parametric studies on fiber tow fracture energy Parametric studies on viscosity parameters Strain rate histories over specimen span Strain rate histories and average strain rate in low scattering region Stress strain curves of resin E862 with strain rate Stress strain curves of resin E862 with strain rate Comparison between axial tension stress strain curves of model with different strain rate properties on resin only Comparison between transverse tension stress strain curves of model with different strain rate properties on resin only Comparison between axial tension stress strain curves of model with different strain rate properties on both resin and fiber tow Comparison between transverse tension stress strain curves of model with different strain rate properties on both resin and fiber tow Out of displacement of six layers specimen under transverse tension Finite element model and boundary conditions for axial tensile test Finite element model and boundary conditions for transverse tensile test Global stress strain curves of in axial tension test Contour of matrix tension damage on axial specimen at strain 0.8% Contour of fiber tensile failure and actual failed specimen on axial specimen at strain 2.2% Contour of interface debonding on axial specimen at strain 2.2% Global stress strain curves of in transverse tension test Contour of damages on transverse specimen at strain 0.4% Contour of damages on transverse specimen at strain 1.2% xiii

14 6.11 Contour of damages (low level) on transverse specimen at strain 1.2% Comparison of surface shear strain between measurement and simulation in an axial tension test at 2% global strain Comparison of surface shear stain between measurement and simulation in a transverse tension test at 1.2% global strain Comparison of out-of plane displacement between measurement and simulation in a transverse tension test at 1.2% global strain Locations of scissoring between two bias tows at free edges Region of finite element model cut out without edge damage for advanced investigation Stress strain curves for materials in the middle part and whole model xiv

15 CHAPTER I BACKGROUND AND INTRODUCTION 1.1 Background Carbon fiber/polymer matrix composites have been heavily utilized in recent years due to their excellent mechanical properties, such as high strength and stiffness, with a corresponding low weight. These properties make them particularly popular in aerospace applications. For example, the new Boeing 787, "Dreamliner", is made up of composite materials 50% by weight and 80% by volume (Boeing, 2007). Classically, laminated composites are used when in-plane properties are of primary importance. However, laminated composites generally have poor through thickness properties which will cause delamination under low levels of load. To avoid these difficulties, textile composites, based on three dimensional reinforcements produced using special equipment, such as woven and braided composites, have been developed in order to improve the through-thickness behavior. In particular, in two dimensionally triaxial braided composites, three sets of fiber tows with different orientations are intertwined to form a single layer of 0 0 /±θ 0 material. Bias tows alternatively undulate over and under each other while 0 0 tows (also called axial tows) are straight and define the axial direction of the composite. In particular, braided composites with a bias fiber orientation angle of ±60 0 fiber architecture are often 1

16 utilized due to the quasi isotropic nature of the material. Furthermore, it has been reported that braided composites with a 60 0 braiding angle can resist crack initiation and propagation as well as formation of delamination during impact (Roberts et al. 2003). Recently, new engine fan cases, which are required to contain a blade and fragments during an engine blade out event, have been made by using a two-dimensional triaxial braid composite architecture. Due to the complicated braided architecture, failure modes of the braided composites are quite complicated as well. More than one damage modes could be coupled at the same time even under uni-axial loading. A potential problem in this application of triaxial braided composite is to understand cracking and delamination when the composite material is subject to different levels of deformation. Full scale and subscale testing of an engine fan blade out event is expensive and time consuming. Simulation would reduce time and cost for development of such composite fan cases. Development of accurate computationally efficient models for simulation is crucial to predict deformation and failure and to help understand experimental results. Small scale specimen testing of braided composites provides a prior evaluation of damage initiation and fracture propagation mechanisms under expected loading environments. In addition, specimen testing determines basic effective properties and verifies analytical predictions of mechanical behavior. NASA Glenn Research Center is leading this type of testing on several braided composite systems with different combinations of carbon fiber and polymer matrixes. However, analyses of material and structural behavior just based on independent experimental study are also very challenging, computer models can be used as analysis tools to complement testing where necessary. 2

17 1.2 Introduction The material systems being examined are two-dimensional triaxial braided composites, with carbon fiber reinforcements and a polymer matrix. Braided composites are one kind of textile composites. The process of production takes place as follows: continuous fibers, called filaments, are firstly assembled into yarns, then winding into braids using machines. The resulting preform is then impregnated with a liquid polymer (thermosetting or thermoplastic) via resin transfer molding (RTM). From the processing, we can see that the textile composites are structured and hierarchical materials, usually three structural scales are defined as macro, meso and micro (Lomov et al. 2007). Process of homogenization links material properties between two neighbored scales. 1. The micro scale defines the packing pattern inside the yarns. It links the properties of fiber yarns or a single ply laminar to the properties of the constituents. So, local on micro scale means the constitutive properties of fiber and matrix and bonding condition on their interface. Homogenized global level properties of micro scale are used as the local parameters on meso scale. 2. The meso scale defines the internal architectures of the fiber yarns (variation of yarn orientation, yarns undulation as well as yarns contacts) and fiber volume ratio inside the fiber yarns. On meso scale, the local means the homogenization or averaging of properties on the scale of impregnated yarns in terms of fiber volume fraction inside fiber yarns. The global on meso scale means the local on macro. 3

18 3. The macro scale defines geometry of composites structures and the distribution of local material properties. On macro scale, the local means the homogenization or averaging of the properties on the scale of unit cells of textile composites, or meso-scale; and global means structural level properties (global fiber volume ratio, composite structure stiffness and strength, etc). The above three scales of homogenization form the idea of multi-scale modeling of textile composites. A key step is the meso scale calculations. When cross ply laminates are analyzed, classical laminate and plate theories are used. For braided composite, with much more complicated internal architectures, numerical solution is needed to examine details of material behavior. Finite element modeling of braided composite can satisfy this requirement and give maximum information on the geometry and local stress strain distribution. The proposed numerical model in this work has been used to examine tests on specimens of triaxial braided composites. The significance of meso scale modeling is that it serves as a tool to simulate composite material response, and then to investigate and explain many questions of great interests. One question that comes up during the experiments is that in some cases polymer matrix materials may have similar bulk properties, but when utilized within braided composites, the as fabricated braided composite materials having the different fiber/polymer combinations behave differently under identical loading conditions. One possible cause of this discrepancy is hypothesized to be related to the interaction or interface between the fibers and matrix in the various composites. Modeling of the interface behavior is of significance in the meso scale model. 4

19 Another point of great significance and interest in meso-scale modeling is failure process during various loading conditions, which could be extremely useful in understanding the behavior of the material. For example, identifying the initiation and propagation of mechanics such as local matrix micro cracking under static loading conditions could be significant. Third, the meso scale simulation results could be further used in developing more approximate numerical models of the material, such as macro scale models, which are less time consuming in simulating large scale structure responses, such as an engine case blade-out event. To accomplish these goals, a meso scale three dimensional finite element model was developed based on unit cell scheme. It takes into account the fiber yarn architecture and the fiber tow-matrix interface. Classical laminate theory (CLT) has also been applied to the braided composites in order to compare the predictions with the finite element model within the elastic region. CLT builds up the basic assumptions for the macromechanical model of textile composites, which could be further used to predict the response of braided composite structures under impact loading. In such a manner, a connection between models at the meso scale and macro scale was set up and a frame work for multi-scale modeling of braided composite has been set up. To examine the develop model, simulation of six ply thick straight sided coupon tests were conducted to examine the modes of failures such as intra laminar failure (local matrix micro cracking and breakage) and inter laminar delamination occurring in the composite materials under static loading conditions. By comparing the simulation results with measured data from the tests, the model has been characterized and enhanced. At the 5

20 same time, by measuring the factors involved with the initiation and propagation of failure, discussions are included as to how local fiber tow properties affect the overall composite material response. As a point of great interest, the effect of the fiber-matrix interface on the overall response was also investigated. This main feature of the model has been studied through an extensive parameter study. The influence of each model parameter on the material response was analyzed qualitatively and quantitatively. As part of the parameter study, the strain rate effect was studied numerically as well. Finally, the proposed meso scale model was used to simulate testing on an alternative test specimen, a single layer straight sided specimen. It has just one layer in the thickness direction, allowing the single layer of material to deform in the thickness direction without constraints imposed in the multi-layer specimen. Localized and heterogeneous deformation and damage were observed by optical strain measurements in this type of test. The observed phenomena could be successfully simulated and understood by the finite element model and numerical results. At the end of this dissertation, the mesomechanical study of triaxial braided composite will be summarized and future work will also be discussed. 6

21 CHAPTER II LITERATURE REVIEW To give a background of the work those has been conducted in this area, and relate the previous research to the current work, an overview of relevant research is presented for the topics proposed in this study. Specifically, a review of methods for failure modeling of composite materials will be presented first. Next, an overview into multi scale modeling techniques for textile composites will be explored in three scales, macro, meso and micro. Finally, test techniques for composite materials will be reviewed. 2.1 Composite Material Failure Modeling Including failure modeling in composite constitutive models is one of the advanced issues and topic of interest in research nowadays. Differing from traditional metallic materials, the failure mechanisms of composite materials are considered to be composed of a series of cracks or fractures at different scales. There have been many advances toward a better understanding of the failure mechanisms and the mechanics of composites. Current approaches for predicting damage in advanced composite can be divided into four areas: failure criteria approach, plasticity approach, damage mechanics approach and fracture mechanics based approach. 7

22 1) Failure criteria approach Failure criteria theories for composites have been proposed by extending and adapting isotropic failure theories to account for in the anisotropic strength of composite materials. Damage of composite is distinguished into general types, including longitudinal fiber tensile breakage, compressive buckling or kinking, transverse matrix cracking and delamination, etc. Failure could be based on equivalent stress or strain. The failure surface is defined by a set of polynomial equations in terms of two or three dimensional stress and strain quantities with adjustable parameters to be determined. There are probably well over one hundred different theoretical failure criteria. Evaluation and characterization of the differences between all of these theories is out of the scope of the literature review in this work. The most recent surveys are by Christensen (2001), Paris (2001) and Hinton et al (2004). The well known and widely adopted theories applied on the lamina level are listed here: Tsai-Hill theory (1965), Tsai- Wu theory (1971), Hashin and Rotem failure criteria (1973), Hashin 3D failure criteria (1980), Chang-Chang criteria (1987), Christensen failure criteria (1997), Puck s criteria (1998). The well-known failure criteria on the lamina level fall short in predicting failure of textile composites, due to the high anisotropy of a single ply. Only textile composites with minimal fiber undulations and low 3D-reinforcement density can be described by failure criteria developed on the lamina level, such as the Puck and Tsai-Wu theories. For other textile composites, it is necessary to use appropriate failure criteria that mostly are still to be developed. Juhasz et al. (2001) developed a failure criterion for orthogonal 3D fiber reinforced plastics e.g. non-crimp fabrics. Xiao, et al. (2007) generalized the Hashin 8

23 failure criteria of a lamina to characterize the damage for a plain weave layer. The fill and warp fiber interaction are considered. Six failure modes were considered, including the delamination mode, and nearly 15 strength parameters were used. In addition, the effects of strain rate are considered in this constitutive model. Failure theory of anisotropic composite materials has been just used as a criterion for failure initiation for each observed failure mode in the progressive failure methodology. It signals the onset of damage in the composite. As discussed in some of the above literature, post failure behavior is assumed to act in an ideally brittle manner, e.g. the stiffness and stress are reduced to zero, or treated numerically as a progressive stiffness degradation in a certain time step, or a strength reduction using a scaling factor. It is suggested that degradation over a range of increasing strain, representing softening and fracture energy dissipation is more reasonable. Similar to how strain hardening has been successfully modeled by plasticity theory, strain softening behavior can be more rigorously dealt with through models based on continuum damage mechanics, which will be discussed later. 2) Plasticity approach In the plasticity approach, a phenomenological constitutive equation is developed based on the concepts of plasticity, such as a yield function, hardening parameters and flow rules. This approach has been applied specifically to thermoplastic composites, like G/PEEK, which have notable nonlinearities and rate dependent response. Due to these limits, only limited studies have been conducted in this area. Odegard et al. (2001) developed a continuum elastic plastic model for wovenfabric and polymer-matrix composite materials. The model was validated under bi-axial 9

24 stress loading conditions. Using elasto-viscoplastic theory, Ryon et al. (2007) analyzed woven composites (glass fabrics and epoxy resin), focusing on their nonlinear and rate dependent deformation behavior. Hardening parameters were determined by uni-axial tension and compression tests. The model was further used to verify three-point bending tests. Other notable efforts were done by Sun (1989 and 2001, etc) to characterize strain rate and nonlinear behavior of laminate and woven composites. 3) Damage mechanics approach Continuum damage mechanics has been applied more and more for predicting composite progressive failure. In this approach, a micro crack is replaced by a damage zone over a finite volume (e.g. a single element in a finite element model). Usually damage variables in scalar or tensor form are introduced, which determines the full range of deterioration in a composite material, from no damage to complete damage. Then, mathematical kinematic constitutive equations for the damaged composite are proposed, which depend on damage variables and lead to a strain softening behavior. Kachanov (1958) postulated that the loss of stiffness attributed to micro cracks can be measured by a macroscopic damage parameter. In the work done by Krajcinovic (1984), inhomogeneous micro cracking behavior was successfully averaged over a volume of material and described by a macroscopic plastic strain tensor. A review was made by Krajcinovic (2000) to highlight the accomplishments, weakness and trends on damage mechanics, and the research needs for further development. Research work on improving damage theory itself is out of scope of the current review. 10

25 One of the first applications of damage mechanics to composites was made by Talreja (1985). Two damage variables were proposed for a lamina model, each representing a major material orientation. The model was used to predict the stiffness reduction in angle ply laminates, and shows good agreement with experimental measured results. Another earlier application was by Frantziskonis (1988), who proposed a scalar damage model. The damage threshold and growth is a polynomial of strain. Tensile and compressive tests are needed to characterize the model, and predictions based on the proposed model show good agreement with laminate tests. Engblom and Yang (1995) developed their model straight forwardly based on damage mechanics to predict the effect of intralaminar damage (matrix failure only). As delamination is a critical failure modes, Randles and Nemes (1992) developed a composite model based on damage mechanics incorporating a delamination damage variable. Matzenmiller et al. (1995) developed a rigorous anisotropic damage constitutive model to describe the elastic-brittle behavior of fiber reinforced composites, under plane stress conditions. Four damage variables corresponding to four failure modes relating to the 2D Hashin failure criterion were introduced. A simple relationship between the effective stress and the nominal stress was built by a rank-four damage operator. The model is a strain controlled material model, so it is well suited for application and implementation in the standard finite element codes, such as MAT58, MAT59 and MAT161 in LSDYNA (LSDYNA 2007), and anisotropic damage model for fiber reinforcement composite in ABAQUS (ABAQUS 2005). Williams and Vaziri (2001) developed a progressive damage model to improve the prediction of impact damage on composite structures. In this work, the Chang-Chang 11

26 failure model and Matzenmiller s damage model were combined and programmed into LSDYNA as a user material model. Efforts have also been made to create damage model on composite laminate by Maimi, Camanho, Mayugo and Davila (2007). Hufner and Accorsi (2009) adopted a progressive failure approach and developed a phenomenological constitutive model applied to woven polymer-based composites. It is formulated to characterize the mechanical response for fully non-linear, rate dependent, anisotropic behavior under dynamic loading. A mechanistic fiber failure theory is implemented, while a degradation model describes the stiffness reductions after failure. The model is validated with experiments. Besides, The World Wide Failure Exercise (Hinton et al and 2004) provided an excellent comparison of various approaches. In combination with failure criteria theories, damage mechanics is used to predict the post failure behavior due to different composite failure modes, such as matrix cracking, fiber fracture, delamination and even compression behavior. The main difference between all of these models relates to damage modes and numerical degradation models. 4) Fracture mechanics based approach Due to the fact that the damage in brittle composites is composed of a series of cracks, the scale effect relating the length of cracks subject to the same stress field needs to be modeled. Failure criteria theories cannot model this correctly. Hence the use of fracture mechanics is attractive. One of the most recent literature reviews on the application of fracture mechanics to composites is made by Cotterell (2002). Basic fracture mechanics methodologies are 12

27 linear elastic fracture mechanics (LEFM) initiated by Griffith (1921), further developed to elasto-plastic fracture mechanics by Irwin (1956), and path-independent integration of J-integrals proposed by Rice (1968). Afterwards, notable research work falls in the class of the application of fracture mechanics in modeling the fracture process zone. Direct applications of elastic and plastic fracture mechanics have been successful, and incorporated into commercial finite element codes. However, they can be only used where an initial crack exists, and the crack size and location need to be known. Moreover, only small crack growth can be modeled. Modeling crack propagation prediction is computationally expensive, due to the need for mesh refinement and re-analysis. In order to predict crack propagation, new modeling techniques have been developed based on fracture mechanics, such as the virtual crack closure technique (VCCT) and the cohesive processing zone model (CPZ) or cohesive zone model (CZM). Because of the high level of numerical calculation required, crack propagation predictions are all conducted by the use of finite element analysis, using codes such as ABAQUS. The traditional VCCT is used for computing energy release rates or stress intensity factors based on results of continuum 2D or solid 3D finite element analysis to supply the mode separation required where a mixed mode fracture criterion is considered (Rybichi and Kanninen 1977), (Raju 1987), (Buchholz et al. 1988) and (Krueger 2002). The Cornell Fracture Group firstly implemented VCCT in their own specialized codes in conjunction with general finite element codes (Singh et al. 1998). Currently, the VCCTfor-ABAQUS (VFA) technology was developed by Boeing Commercial Aircraft Group as part of the Composite Affordability Initiative (CAI) in 2004, aiming at predicting crack 13

28 propagation in large integrated bonded structures, and evaluating interlaminar damage tolerance requirements. This new fracture analysis capability is available in ABAQUS v6.5 (ABAQUS 2005), but VFA is a separately licensed add-on. Unlike traditional VCCT, VFA assumes a crack interface and intermediate crack-tip location, re-meshing is not necessary. Additionally, it is relatively mesh-size independent. VCCT may be viewed more fundamentally based on fracture mechanics, especially on damage initiation criteria, when comparing with CPZ approach that will be talked about, while, the only drawback is the requirement of assuming an existing crack in the material, which means it cannot model crack initiation. With the capability to handle the above difficulties during simulation of fracture propagation, CPZ is becoming more and more popular. The basic idea of such a model can be traced back to the work of Dugdale (1960) and Barenblatt (1962). The cohesive element technique is the direct product of incorporating this approach into finite element codes. The cohesive zone can be of finite thickness or zero thickness depending on the specific application. It combines a stress based formulation for crack initiation with a fracture mechanics based formulation for crack propagation and is used to define the nonlinear constitutive law of the interface material or bonding media. Cohesive elements can still transfer load after crack initiation, until a critical value of the energy release rate is attained. The method was extended to model concrete by Hillerborg (1976) and Qiao (2008). Lawn (1993) provided a more through description of underlying theory of the cohesive elements by use of characteristic length theory, which was explained in detail recently by Taylor (2006). 14

29 Needleman (1987) recognized that cohesive elements are particularly attractive when interface strengths are relatively weak compared to the adjoining materials. This is the case for composites laminates or textile materials. Due to the fact that the ply to ply interface is relatively weak compared with the neighboring fiber tows, representing a potential crack propagation path (without pre-knowledge of crack initiation location), cohesive zone models are more than suitable and have beenwidely used to simulate debonding or delamination in laminated, woven or braided composites. Cohesive zone approaches are more accurate than just considering delamination as one failure mode in the failure criteria approaches. Meanwhile, in order to characterize the parameters in the constitutive models, double cantilever beam (DCB) tests, end notched flexure (ENF) tests and split DCB tests are used to measure single fracture mode behavior correspondingly. With the help of mixed mode tests such as mixed mode bending (MMB), the characterization of a mixed mode damage model was proposed (Reeder 1992). Because of this, the literature found is mostly about delamination simulation of those kinds of tests. The mode interaction relationship could be defined by the use of non-dimensional displacement (Tvergaard and Hutchinson 1993), traction components (Xu and Needleman 1994), and damage surfaces for relative displacement (Benzeggagh and Kenane 1996). Different types of element technologies have been selected to apply the cohesive zone model. Plane cohesive elements with zero thickness models were used to simulate the delamination of laminated composite by Moura et al. (1997 and 2000). A plane stress cohesive element with finite thickness was used to model crack initiation and growth by Reedy et al. (1997), where shell elements were connected by cohesive elements. A line 15

30 cohesion element, similar to spring element connecting nodes, was applied in work of Chen et al. (1999), Mi (1998) and Petrossian (1998). 3D eight node zero thickness cohesive element with a mixed mode damage law were used by Camanho and Davila (2001) to simulate stiffener-flange debonding. Later, Davila, Camanho and Turon (2007) presented their work on applying cohesive elements between stacked, non-coincident layers of shell elements, the result of which indicated that a computationally efficient shell element model can retain many of the necessary predictive attributes of 3D solid models. Impact damage prediction using shell elements and cohesive elements as an interface has been done by Johnson and Holzapfel (2006). Good agreement was shown for delamination prediction and penetration. Various cohesive constitutive relationships, also called cohesive laws, have been proposed by many researchers. The list includes a simple bilinear law by Reedy (1997) and Mi (1998); a mathematically continuous exponential law by Needleman (1987), and Xu and Needleman (1994); a trapezoidal law by Tvergaard and Hutchinson (1993); a perfectly plastic law by Cui and Wisnom (1993). Alfano (2006) gave an evaluation on how the cohesive law affects the application of the cohesive zone model. Li and coworkers (2006) proposed a different cohesive law for Mode I fracture by introducing cohesive strength and characteristic strength simultaneously, so that a law with multiple lines was shaped. Advanced issues of application of cohesive zone models have also been widely discussed. One of the most discussed issues is the mesh dependence. It is argued (Bazant and Oh 1983, Bazant and Jirasek 2002) that due to strain softening in the cohesive law, the solution is non-objective with respect to the mesh refinement and the computed 16

31 energy dissipated decreases with a reduction of the element size. A characteristic length of the finite element was introduced to eliminate this effect. One of the applications of this approach could be found in the work of Camanho et al. (2007), where simulation of failure of laminate plates with holes was conducted using a cohesive zone model. Alfano and Crisfield (2001) utilized damage mechanics to control strain softening in cohesive law, and compared the prediction results of CPZ and VCCT based on DCB tests. At the same time, a parameter sensitivity analysis was conducted on cohesive strength and stiffness, and mesh density. The conclusion was drawn that the solution is strongly dependant on cohesive strength. In the meantime, model robustness was discussed by using the tangent predictor procedure. Turon et al. (2007) discussed mesh size effects, and developed a methodology to determine the constitutive parameters for CZM, using a closed form expression for penalty stiffness or cohesive stiffness. Setting up the constitutive law using the proposed methodology allows for a coarser finite element mesh and facilitates large scale analysis. Borst (2003) and Chandra et al. (2002) addressed some of numerical issues in implementation of cohesive element models. Mesh dependence was also discussed by Zhou and Molinari (2004) for dynamic crack propagation. Delamination of laminated composite has been widely modeled. Zou et al. (2003) discussed the determination of parameter of a cohesive model during simulation of delamination growth of laminated beams. Allix and Blanchard (2006) included a cohesive zone in a meso scale laminate model to predict delamination damage of laminated plates containing circular holes, where delamination is supposed to be reproducible and stable around the hole. Guedes et al. (2008) proposed a 3D finite 17

32 element of sixteen layer laminates containing cohesive elements as interface between plies, and compared with test results of the high strain rate compression response of laminates. Mesh density was discussed and it is shown that there is no remarkable mesh sensitivity effect. Meanwhile, toughness was found to be the determinate factor to the compressive strength. Okabe and coworkers (2008) used the so called embedded progress zone (EPZ) for the transverse crack and delamination in cross ply laminates, and studied interaction between these two types of failure modes. The cohesive zone models represent a novel method to explicitly model interface behavior, and will be adopted in the current work to model the fiber tow interface response. 2.2 Multi-Scale Modeling of Textile Composite Textile composite represents a class of advanced materials which are reinforced with textile preforms. Although this work concentrates on modeling tri-axially braided composites, similarities can be found in the analysis of laminated, woven composites and other types of textile composites, hence, this review will also outline major studies in these fields to create a basis for the overall discussion. The modeling of textile materials is an active field currently. Since textile composites are structured and hierarchical, having different structure levels, the modeling of textile composites are going to different scales. According to different applications, models in macro, meso and micro scales have been developed. Meanwhile, there are two ways commonly to obtain material parameters in each scale, the bottom up approach and top down approach (Littell 2008). Bottom up approach refers to using homogenization theories and implementing constituents properties directly in the model 18

33 development; while, the top down approach refers to developing different methods for determining material properties by using material parameters obtained from material testing. Some researchers attempted to set up a systematic work as multi steps or multi scales analysis of textile composites. Theory and computational frame work on multiscale (three scales) damage modeling for textile composite has been done by Fish and Yu (2001), to extend a two scale (micro-macro) non-local damage theory (Fish and Yu 2001). A closed form asymptotic expression of nonlocal piecewise damage model was derived. Meanwhile, there are some other work aiming at developing the model on specific level and working out the corresponding methods to incorporate the model parameters Micro-scale modeling approaches In the frame of multi scale modeling of textile composite, micro scale focus on constitution levels, defining the arrangement of fibers in representative volume element (RVE) of impregnated yarn or ply. Most of the micro scale approaches have been applied for laminar property predictions. Great success has been achieved in predicting the stiffness of laminar. Strength prediction has been less successful due to the complexity of composite failure at the local level. The mechanics of materials approach is the simplest and most intuitive one (Daniel and Ishai 2006). Chamis (1984) developed an independent program to improve the prediction in transverse properties. Sun and Chen (1991) used square cell RVE with assumptions of full plane stress status to deduct a closed form and explicit relationship between laminar strain and stress, which is suitable for strain incremental based finite 19

34 element implementation, and plastic and strain rate behavior of matrix can be included as well. This is method was further extended by Roberson and Mall (1994). The method of cell is another approach aimed at setting up directly compliance matrix of composite based on constituents properties (Aboudi 1991). Pindera and Bednarcyk (1999) derived their reformulation of Generalized Method of Cell. Goldberg and Stouffer (2002) utilized a micromechanical model similar to above and incorporated with strain rate matrix material model and failure criteria to develop a material model for analysis of polymer matrix composite material under high strain rate impact loading. Pecknold and Rahman (1994) proposed a similar model with different presentation of geometry for RVE. In the models mentioned above, appropriate uniform stress and uniform strain assumptions were applied to the model. Constitutive model was applied to simulate the response of the fiber and matrix phases. The multi-continuum method (Garnich and Hansen 1997) can be used to determine strains and stresses in the individual constituents given a total strain state in the composites, those values can be used further for plasticity analysis. Beside the above analytical methods, numerical finite element methods (e.g. Barbero 2007) were used to explicitly model the composite RVE. By applying appropriate periodical boundary conditions, effective stiffness properties can be determined. Also, by using point failure criterion, strength and failure phenomena can be studied. In multi-scale modeling of textile composites, the methods cited above for single ply laminar property prediction are usually used as first level homogenization to obtain material parameters for fiber tows. However, great efforts have attempted to develop straight forward micro-scale approaches for textile fabric composites. To lead a micro 20

35 scale analysis of textile composite, the consideration has to be taken into account for defining fabric architecture. Compared with meso scale models, the definition of fabric architecture in micro scale models is more simplified and a lot of assumptions are made. As will be discussed later, different levels of approximation are assumed, some use laminate analogy (Halpin, Jerina and Whitney 1971) by taking advantage of existing laminate theory; some use more sophisticated methods. In each way, repeating volume cell (RVC) is always selected as a smallest unit to start with. Ishikawa and Chou (Ishikawa 1981, Ishikawa and Chou 1982, Ishikawa and Chou 1983) adopted and modified classical laminated plate theory to model thermo and stiffness properties of woven fabric composites. Three different models were built: the mosaic model, the fiber undulation model and the bridging model. Then by Yang, Ma and Chou, this method was applied further for two dimensional braided (1984) and three dimensional braided composites (1985), where the proposed model is called fiber inclination model or diagonal brick model. It treats the RVC as an assemblage of inclined unidirectional laminar. Karkkainen and Sankar (2006) used a laminated model (considering moment terms) to model plain weave textile composites. At the same time, a finite element model was employed as substitution of physical experiment to produce input for laminated model. Naik and Shembekar (1992) and Raju and Wang (1994) developed models for woven composites which accounted for yarn undulations. The overall stiffness of woven composites was computed based on iso-strain or iso-stress assumption, or a combination of them, and then analytically integrating through the volume of RVC. Naik (1994) developed analytical models to predict thermal and mechanical properties of textile composites, including predefined geometrical models for 21

36 woven (plain, 5-harness satin, 8-harness satin weave) and braided (2D, 2X2, 2-D triaxial, 3D, multi-interlock, 5-layer) composites. Classic laminate theory was applied to average orientated yarn properties. Moreover, a computer based program, TEXCAD (1994), was implemented to provide a user friendly design interface. Its capability later expanded into failure analysis of textile composites (Naik 1994). Tabiei (1999 and 2001) and his coworkers developed a micromechanical model which is suitable to implement into a finite element code as a nonlinear constitutive model for woven composites. The incremental constitutive model considered nonlinearity of both matrix and yarns. In the model, average strain and perfect interface bonding conditions were assumed. Construction of subcells with different combination of warp/fill yarns and matrix could represent woven fabric architecture. Bednarcyk et al. (2003) discretized woven composites into a number of subcells; a two step generalized method of cells (GMC) was utilized then to determine the in-plane effective properties of the composites. Elastic prediction models have also been developed for 3D textile composites by Wang and Wang (1995), Pandy and Hahn (1996), and Cox and Dadkhah (1995). Ayranci and Carey (2008) made a literature review on applications of braided composites as well as relevant analytical stiffness predictive models. Marrey and Sankar (1997) developed a finite element based micromechanical method for computing the plate stiffness matrix (A, B, and D matrix) and thermal coefficients of a textile composite model, which was considered as a homogenous material. Byun (2000) developed an analytical model based on a unit cell for prediction of geometric influence and engineering constants of 2D triaxially braided composites. The 22

37 unit cell was decomposed into four parts, two braider yarns, one axial yarn and matrix. Coordination transformation was used to average the compliance matrix of crimp yarn into axial orientation. Constant strain was assumed and volume averaging technique was applied to obtain effective stiffness of braided composites. Yan and Hoa (2002) obtained a closed form expression for effective stiffness of 2D braided composites by the analysis of elastic deformation energy based on a RVC. The path of braider yarns and their cross section shape were all taken into account when effective stiffness matrix was integrated. The impregnated yarn was assumed to be a transversely isotropic material, and an iso-strain assumption was again applied. Zebdi et al. (2009) proposed an inverse approach based on plate laminate theory to back calculate the virtual ply properties, which would be used further to predict elastic properties of any braiding orientation. Experiments were necessary to measure 4 composite properties (Ex, Ey, Gxy, vxy). The basic assumption in this approach that a woven or braided composite is equivalent to a stack of angle plied laminates. The problem for this approach, and all other classis laminate theories applied on textile composites, is that undulation and strand shear effect are neglected. From above review of micro scale modeling approaches, we can see the limitation of their application, either incapability to account for detailed fabric geometry, or failure to detailed stress strain field with sticking at iso-stress or strain assumptions, or failure of strength prediction. However, they can deliver a fast and approximate way to simulate complicated textile composite and can be used as the first level homogenization in multiscale modeling. 23

38 2.2.2 Meso-scale modeling approaches Meso scale models of textile composite define internal structures and variations of orientation of fiber yarns. Different from micro scale models of textile composite, meso scale models represent textile fabric geometry explicitly, moreover, volumes of fabric reinforcement and matrix are distinguished and specific material properties will be assigned accordingly. The dimensions of the meso scale model are related to actual textile structures. As far as unit cell is concerned, periodical boundary conditions need to be carefully defined. At this sense, more elaborate numerical solutions are preferable in meso scale modeling of textile composites. The analysis of textile composites in meso scale will lead to a result of ununiform stress distribution over the RVC, which is different from the results obtained by most of micro scale approaches. Furthermore, by implementing specific failure behavior into distinguished phases, damage initiation and development are able to be investigated. Whitcomb (1994) developed techniques for performing three dimensional finite element analyses of plain weave composites. It was used to study effects of variation of microstructure and properties on composite effective moduli. Lomov and coworkers in Composite Material Group developed a software tools, WiseTex (Lomov et al. 2001), for geometry modeling of internal structure of textile reinforcement, such as 2D/3D woven, two- and three- axial braided and knitted etc, and transferring data into general FE codes. Fiber yarns cross section and undulations could be explicitly modeled and also controlled as required. The material properties used in the meso scale finite element model are calculated using micro scale homogenization discussed in last section. Damage modeling was included by adopting damage mechanics 24

39 approached discussed in Section 2.1. The model was verified to successfully predict local stress strain field by comparison with full field measurement (Lomov 2005). Meanwhile, a deep discussion on issues of meso FE modeling was made (Lomov 2007). Similar work has been done by Textile Composites Research Group at the University of Nottingham. Python based open source software, TexGen (Sherburn 2007), combined geometry building and volume meshing algorithms together, delivering a user friendly tool. Both of these two software packages could handle orthogonal textile structures, like woven, but they have difficulties to handle issues like intersections of flat yarns in non-orthogonal structures, such as braided, especially when high global fiber volume fraction is desired. Miravete (2001) carried out meso-mechanical FE analysis using solid elements to mesh fiber yarns and matrix, failure was taken into account by using Hashin failure criteria. Three axial braided composites with two bias angles 20 0 and 30 0 were studied in tension and shear tests and compared with proposed analytical model which is based on principle of superposition. Zeng et al. (2004) used a brick element model to study the local stress distribution of 3D braided composites and damaged mechanical properties. Song and Wass et al. (2008) predicted compression strength of 2D braided textile composites based on mesomechanical finite element unit cell (single and multi) model. Micromechanics homogenization model was used for fiber tows, boundary conditions and imperfection imposing was discussed in eigenmode and response analysis. Fang (2009) analyzed damage development of 3D four directional braided composites based on meso scale finite element model by implementing anisotropic 25

40 damage model. Other attempts of developing geometry modeling technique have been made such as Sun (2001). Most recent work on meso scale analysis of triaxial braided composites has been conducted by Ivanov et al (2009). WiseTex model was utilized to obtain the geometrical model of studied material, damage of yarns was implemented by a stiffness degradation model; quasi static tests were simulated to compare with experiments. At the same time, crack density has been examined by acoustic emission and X-ray investigation. In order to capture damage in meso-mechanics approach, finite element mesh with correct geometry is the first step, while implementing damage mechanics is as critical. The developed models could predict overall response at certain degree of success. One of the most common damage mechanisms, debonding or delamination between various orientated fiber tows, however, is usually not studied by these models. Some work implemented a thin layer of matrix between fiber tows, which finally will lead to a model with large numbers of elements and cause numerical difficulties since they are easy to be distorted under deformation. These are where current work is going to improve Macro-scale modeling approaches Macro-scale modeling approaches are aiming at building up a constitutive model for the proposed composite material. Anisotropic constitutive equations are usually obtained to describe composite behavior. Detailed formulation of constitutive model requires knowledge of a set of effective material parameters, such as material moduli, Poisson s ratios, and strengths in different material directions. In some cases, in order to define progressive failure or softening behavior, failure strain or artificial scaling factors 26

41 are needed. Determination of these parameters needs support from corresponding experiment tests, which are tedious and expensive. Due to this reason, majority of the composite macro scale models are applied on unidirectional composite lamina with relatively less material parameters. However, very few macro scale models could be found on textile composites, due to the difficulties to include complex undulation and crossing over of fiber yarns. Xiao et al. (2007) developed a material model (MAT161/162) for use in LSDYNA (2007) that can be sued to simulate onset and progression of damage in woven composite under 3D stress fields. In order to determine four damage parameters, the center for composite modeling (CCM) in University of Delaware has developed a methodology of coupling Quasi-static Punch Shear Test (QS-PST) and model based simulations. This macro model is capable of modeling failure modes including tensile, compression, shear as well as delamination without model the physical interface. Iannucci (2006) attempted to apply damage modeling methodology on woven carbon composite, to predict structure failure under impact loading. Rate dependence was included in the constitutive model. Shell element with through the thickness integration points allows representing each ply in the laminate. Damage information was calculated on a ply-ply basis. Three damage variables are considered, fiber fracture in the local warp bundles, fiber fracture in the local weft bundles and fiber-matrix deterioration due to inplane shear. Specimen and impact modeling showed good agreement between prediction of the model and experiment measurement. Similarly, Littell (2008) adopted shell elements with through thickness integration point scheme to modeling triaxially braided composite, and then used top-down approach to characterize material parameters for 27

42 input to damage material model. By comparison with full field measurement data, damage properties were also determined. Xie and coworkers (2006) treated triaxial braided composite (0 0 /±45 0 ) as elastic plastic orthotropic homogenized material. The inplane effective mechanical properties were measured by ASTM tests, while the plastic behavior was characterized by static off-axis compression tests from which a plastic parameter could be determined. In the same work, cohesive zone model was used to simulate mode I and mix mode fracture (2006). Zeng (2005) used an available composite material damage model in LSDYNA, MAT59, as macroscopic model to simulate response of 3D braided composite tube under compression. What need to be mentioned is that the material parameters used as input for MAT59 are based on meso scale model proposed cited in last section (Zeng 2004). As to macro scale models, they are phenomenological based. Empirical material models are adopted. Very simple models could be used and they are usually applied in the impact simulations due to computational efficiency. However, these models failed to predict of behavior of fiber tows, resin effects or interface explicitly. Developing a phenomenological macro scale model of textile composite required averaging local deformation into material global response. Meso scale modeling, as discussed in last section, can supply insight into material behavior that could be used at macro scale modeling. 2.3 Composite Material Testing Since this work concentrates on modeling of braided composites at meso level, small scale specimen static testing technique is mostly discussed. As far as effective material properties, such as overall tensile, compressive and shear strength and modulus, 28

43 are concerned, ASTM standardized tests are generally suggested. While, there are also some not-standard test methods developed specially for braided composite to quantify its behavior under different load conditions. Meanwhile, in order to collect test data and identify the damage mechanism, advanced measurement techniques are utilized. Master et al. (1993 and 1996) did extensive investigation on two dimensionally triaxially braided composite materials with notched and un-notched specimen uni-axial tension tests. Modulus and strength were measured; meanwhile, full field deformation on the surface was measured by Moire interferometry to demonstrate the heterogeneity on the strain field. Damage initiation and development was studied by acoustic emission during the test, and X-ray and microscopy after the test. In the work of Lomov et al (2008), experimental methodology of damage investigation in textile composites was summarized. Beard and Chang (2002) did crush tests on braided composite tubes to study the energy absorption character. Warrior and Fernie (2004) tested braided carbon/vinylester composites with 30 0, 45 0 and 60 0 bias angles at different stain rate, and found that transverse properties were more sensitive to strain rate. Compression responses of braided composite were investigated by experiments by Quek and Waas et al. (2004). Xie (2006) carried out fracture tests with single edge notch bend specimen to characterize fracture toughness as input to constitutive level material model. Littell (2009) at NASA Glenn Research Center carried out an extensive experimental characterization of triaxially braided composite with straight sided and H shape specimens. Four braided materials, consisted of the carbon fiber system but impregnated with different types of polymer epoxy, were tested under uni-axial tension and compression as well as shear loading conditions. Optical strain measurement 29

44 techniques were applied in the testing. Some of the experimental data obtained in his work will be referred in this work. Different epoxies induce different fiber/matrix bonding conditions, and further difference on global behavior can be observed. By using the meso scale model proposed later in this work, the interface effect can be investigated. Tests for interface delamination fracture toughness measurement, double cantilever beam (DCB) for pure mode I (Whitney 1989), end notched flexure (ENF) or center notch test (CNT) for pure mode II (Maikuma 1989), and MMB for mixed mode (Reeder 1990), are commonly for unidirectional laminar or laminated composites, where a weak plane exists between plies for self-similar cracks to propagate. Due to the complexity, few references could be found on charactering interface properties of textile composite which control delamination mechanism. Currently, NASA Glenn Research Center is leading some other static tests on triaxial braided composites to measure both in-plane properties (single ply specimen in different shape and composite tube of multi-axial stress tests) and interlaminar properties (three point bending of short beam). Simulations are made to characterize models by comparison with available data. 2.4 Current Research Based on the above literature review on available work relevant to triaxially braided composites, meso scale numerical modeling technique in this study is developed to improve the capability of capturing braid fabric geometry, individual constituent s behavior and fiber tow interface effects, and the feasibility to apply in small scale specimen tests and even subscale tests. 30

45 Micro scale homogenization approaches will be firstly used to predict material parameters of reinforced fabric which is assumed to be unidirectional laminar with transversely isotropic behavior. Extensive parameter studies will then be applied to correlate the model with experimental test results. At the same time, it will be shown how the developed model is used to investigate interface effects and explain test results. Interlaminar properties of the proposed model will be examined by modeling short beam bending tests, which induce delamination between undulated and crossover fiber tows. Finally, preliminary results will be presented on panel damage investigation under impact based on the proposed numerical model. The model developed in this research has built a bridge to link micro and macro models of braided composites, falling in the multi-scale modeling of braided composites. And adoptability of this model allows it to be adjusted for other types of textile fabric composites. 31

46 CHAPTER III MESOMECHANICAL MODELING METHODOLOGY OF TRIAXIAL BRAIDED COMPOSITE MATERIAL The objective of the current work is to set up an analytical tool, which takes into account the effects of the failure mechanics of the constituents and geometry, to investigate the failure process on a local level of textile composite materials. For this study, a meso scale model, also called mesomechanical model was used. Specifically, a detailed finite element model was applied. This chapter will introduce the proposed mesomechanical model in detail. Furthermore, in order to illustrate the behavior of the proposed mesomechanical model, numerical experiments are performed on a representative unit cell and then compared with classical laminate theory based solution. 3.1 Studied Composite Material System For this work, the focus was on a particular textile composite system, a 0 /±60 two dimensional triaxial braided composite, see Figure 3.1. The modeling process starts with a representative volume cell, also called unit cell. A unit cell is the smallest portion of a composite whose behavior is representative of the overall behavior of the composite. For braided composites, the unit cell size depends on the dimensions of the fiber tows and the magnitude of the bias angle (Chou and Ko, 1989). 32

47 a) Braided composite dry fabric and dimension of unit cell b) Model representation of triaxial braided composite Figure /±60 0 tri-axially braided composite architecture Figure 3.2 Microscope image of cross section of triaxial braided composite For the typical materials studied here, the axial fiber tows consist of 24,000 fibers, the bias fiber tows have 12,000 fibers, and the global fiber volume ratio is approximately 56%. Based on scanning electron microscope measurements (Littell, 2008), the average unit cell size was determined to be inch (w) 0.2 inch (l) inch (t) or mm (w) 5.08 mm (l) 0.53 mm (t). Meanwhile, the cross section dimension of fiber tows was also averaged based couple of measurement, and their shape is assumed to be lenticular. In reality from Figure 33

48 3.2 (Littell, 2008), the cross-sectional shape is not perfectly uniform throughout fiber tow longitudinal direction, actually unpredictable. To simplify the problem, they are assumed to be identical in the model. Furthermore, by comparison between cross section views between before tests specimen and after tests specimen, shown in Figure 3.3 (Littell, 2008), it is observed that the closed tow interfaces (Figure 3.3a) turn into open gap (Figure 3.3b), which indicates debonding of interface. Such geometrical information has been captured by proposed unit cell model and shown in Figure 3.5. Axial tows Site of debonding (a) bonded interface before test (b) debonding after transverse tension test Figure 3.3 Comparison of tow interface of six layer specimen before and after test For this study, the composites which were examined both have the same fiber system, Toray s T700s (Toray Carbon Fibers America, Inc), but are infused with different polymer resin systems Cytek Industries PR520 and Epon s 862. The Toray fiber is a high strength intermediate modulus fiber with elastic brittle failure behavior. Cytec s PR520 is a toughened thermo-set resin. Studies by Littell (Littell, 2009) have indicated that the T700s/PR520 composite most likely has a very strong fiber matrix interface. Epon s 862 resin is a low viscosity and high flow thermo-set resin. The T700s/E-862 composite has been found to have intermediate interface cohesion strength (Littell, 2008). Table 3.1 and Table 3.2 list the mechanical properties of fiber and each resin, as reported by the manufacturer. 34

49 Density (g/cm 3 ) Table 3.1 Mechanical property of Toray s T700s Tensile Strength (MPa) Longitudinal Modulus (GPa) Transverse Modulus (GPa) ρ S FT E F11 E F22 G F Table 3.2 Mechanical Property of Matrix Density (g/cm 3 ) Tensile Strength (MPa) Young s Modulus (GPa) ρ S MT E M Cytec PR Epon In-plane Shear Modulus (GPa) 3.2 Proposed Mesomechanical Model The proposed mesomechanical model of unit cell is a finite element model. The main feature of finite element mesh is that it utilizes six nodes pentahedron elements as shown in Figure 3.4. Figure 3.4 Type and shape of solid element utilized to mesh the unit cell geometry Figure 3.5 illustrates the finite element model for the braided composite unit cell in meso scale. Three phases are introduced, fiber tows, pure resin and tow-to-tow interface. In Figure 3.5a, the elements representing the matrix material are not shown in order to clearly show the braid architecture. Fiber tows cross section area and bias tow 35

50 undulation are modeled explicitly. In the isometric view of the unit cell shown in Figure 3.5b, the tow cross sectional shape and bias tow undulation can be observed (a) Top view of unit cell (b) Iso-view of unit cell Top view lay up region cross over region lay up region cross over region Front view (c) Cohesive Zone for fiber tow-to-tow interface Real material Model Pure resin Bias tows Axial tows (d) Comparison on cross section view of real material and model Figure 3.5 Finite element model of unit cell of Triaxial Braided Composite The interfaces between fiber tows observed in Figure 3.3 were modeled by using interface elements around axial fiber tows and between bias fiber tows, shown in Figure 3.5c. The location of interface has been categorized into two types, lay-up region which is around 0 0 fibers tows, and cross over region which is between bias tows. Comparison 36

51 between model and real material on a cross section view of whole unit cell is shown in Figure 3.5d, where similarities could be found on pure resin pocket, fiber tow cross section shape and bias tow undulation. The mesh of proposed finite element model could be easily adjusted. There are some advantages. Firstly, by modeling the braided unit cell in this manner, the tow undulation and cross-sectional shape were controllable, and a high global fiber volume fraction could be achieved, which is limitation of many of other mesh codes. The second, the tow to tow interface was able to be modeled using cohesive zone elements in a fairly straightforward manner, which will allow for more flexibility in simulating failure modes; and this is the main difference between current mesomechanical model and any other models. Furthermore, the finite element models of braided composites with other braiding angles can be easily generated by adopting top and bottom surfaces of pentahedron elements with different triangles. Finally, the number of elements used in the model was relatively small, which results in the model being reasonable computationally efficient. 3.3 Model Material Properties Characterization As discussed previously, three phases (fiber tows, pure resin and interface) have been explicitly shown from the above finite element model. The material behavior of each phase is defined separately. The failure behavior is considered to simulation local failure progress. In following, material modeling approaches will be discussed according to each of these three phases. 37

52 3.3.1 Fiber Tows During manufacturing process, the resin transfer molding (RTM) technique is widely used during resin impregnation of fiber tows, in such a way that consistency in fiber volume fractions and hence mechanical properties may be achieved. Therefore, in this paper, the material properties are assumed to be uniform within all fiber tows. The fiber tows are further assumed to be unidirectional transversely isotropic material under plane stress conditions since the thickness of fiber tows are still very small compared with in-plane dimensions. The available fiber reinforced composite material failure model in ABAQUS (ABAQUS 2005) is used to simulate fiber tow behavior. In this model, the fiber tows are assumed to behave linear elastically before damage initiation. The model combines the Hashin failure criteria (Hashin, 1973) and an anisotropic damage evolution law (Matzenmiller, 1995) to simulate the onset of damage and its subsequent evolution. Plane stress continuum shell elements with a three-dimensional geometry are used to model the fiber tows. Due to the presence of matrix only regions in the composite, the fiber volume fraction of the composite V C f towsv tow f. The tow fiber volume fraction unit cell model by using Equation 3.1, is lower than the fiber volume fractions within the fiber tow V f is calculated based on the geometry of the V Volume = (3.1) Volume Unitcell tow C f Vf tows where Volume Unitcell and Volume tows are the volume of the unit cell and the volume of the fiber tows in the unit cell, respectively. tow V f is calculated to be about 80%, which will be 38

53 further used to calculate the effective mechanical properties of the fiber tows based on a micromechanical model. Figure 3.6 Representative volume element for generalized method of cells To compute the effective stiffness properties of the fiber tows, the generalized method of cells (GMC) developed by Aboudi (Aboudi, 1991) was utilized. In this micromechanical model, assumed representative volume element (RVE) adopts four subcells, shown in Figure 3.6, stress averaging technique was applied under uniform strain assumption over subcells. The strength parameters were much more difficult to determine since they may be related to local defects, local variations and fiber matrix interface bond quality. The simplified micromechanics formulations, from Equation (3.2) to Equation (3.6), included in Chamis s Model (Chamis, 1984) were used to obtain a first approximation of the required strength values. The approximation may have large discrepancy from one material to another, while they will be further correlation with experimental test results. F 1T EM = S FT [ VF + (1 VF ) ] (3.2) E F11 39

54 F F (1) 1C (2) 1C EM = S FC [ VF + (1 VF ) ] (3.3.1) E F11 GM = (3.3.2) GM 1 VF (1 ) G F12 G G 13 S (3.3.3) (3) M 12 F 1 C = β ( α 1 + ) S MS + GF12 α GM MC F E E F T = β (1 VF ) S MT (3.4) EM E F 22 VF E22 F E E F C = β (1 VF ) S MC (3.5) EM E F 22 VF E22 F S G G F12 12 = β (1 VF ) S MS (3.6) GM GF12 VF G12 where, the computed F 1T, F 1C, F 2T, F 2C and Fs are composite longitudinal tension and compression, transverse tension and compression, and shear strengths correspondingly; and V f is the fiber volume ratio; E M and G M define matrix Young s modulus and shear modulus; E F11, E F22 and G F12 define longitudinal, transverse and shear modulus; E 22 and G 12 define composite transverse and shear modulus, which are computed based on GMC; meanwhile, S FT and S FC are fiber tensile and compressive strengths, which are assumed to be same; S MT, S MC and S MS are matrix tensile, compressive and shear strengths, as reported in testing results of pure resin by Littell (2008); additionally, two coefficients are π 1 EM calculated as α = and β = ( α ). 4 V F α 1 EM E F 22 1 VF (1 ) E F 22 40

55 To help in determining the accuracy and applicability of these values, later in this work in Chapter 5, a numerical parameter study which examined the sensitivity of the composite global behavior to the values of the fiber tow strength parameters will be described. The effective stiffness values determined for the fiber tows (for both of the composite systems examined in this study) are shown in Table 3.3, and the effective strength values for the fiber tows are given in Table 3.4. What needs to be mentioned here is that the determination of F 1C. Failure mechanisms of unidirectional composites under longitudinal compression are complicated. There are three different comprehensive failure modes: 1) fiber rupture under compression, by Equation (3.3.1); 2) buckling or kinking, by Equation (3.3.2); 3) fiber matrix debonding or matrix cracking, by Equation (3.3.3). Strong interfacial bonding (Madhukar, 1992) between the fiber and matrix will lead to fiber rupture failure, while weak bonding causes failure due to debonding or matrix cracking. The failure stresses due to each of these failure modes can be predicted by using the Chamis equations (Chamis, 1984). The fiber tow longitudinal compression strengths were firstly selected as an approximation by averaging the values predicted for each of the three failure modes, and further correlated with test results. As discussed in next chapter, F 1C was correlated with straight sided specimen axial compression test. Those values of F 1C listed in Table 3.4, for the two different composite systems, are correlated ones. Note that subscript 1 is for fiber longitudinal direction, 2 for transverse direction, and T for tension, C for compression, S for shear. 41

56 Table 3.3 Fiber tow modulus E 1 (GPa) E 2 (GPa) G 12 (GPa) v 12 G 23 (GPa) T700s/E T700s/PR Table 3.4 Fiber tow strengths F 1T (MPa) F 1C (MPa) F 2T (MPa) F 2C (MPa) F S (MPa) T700s/E T700s/PR For the materials which compose the fiber tows, developing methods to predict the initiation and propagation of damage are required. An existing ABAQUS fiber reinforced composite material model was used in this paper to predict the damage and failure of the fiber tows. To predict the initiation of damage, the two dimensional plane stress Hashin failure criterion (Hashin, 1973) was utilized. As listed in Table 5, in the Hashin criterion four different failure modes are considered, which correspond to tensile or compressive failure of the fiber and matrix. The material strength parameters used here are based on Chamis s Model. Degradation is assumed to begin once one of the Hashin failure criteria functions identified in Table 3.5 reaches a value greater than or equal to one. 42

57 Table 3.5 Failure criteria for fiber tows Tensile fiber mode σ 11 > 0 e t f 2 σ σ α = + F1 T FS 2 Compressive fiber mode σ 11 < 0 e c f σ 11 = F1 C 2 Tensile matrix mode σ 22 > 0 e t m σ 2 σ = + F2 T FS 2 Compressive matrix mode σ 22 < 0 e c m σ 22 F 2C σ 22 σ 12 = FS 2FS F2 C F S After the initiation of damage, the progression of damage is characterized by material stiffness degradation, leading to material failure (Matzenmiller, 1995), which is also available in adopted ABAQUS composite failure model. Imposing stiffness degradation over a range of increasing strain, representing softening and fracture energy dissipation, as opposed to an abrupt failure of the elements, has been found to be a more reasonable approach (Hufner, 2009). The implementation of a degradation model is designed to more accurately simulate the process of failure and to improve the convergence of the numerical analysis in the implicit finite element solver by avoiding non-positive definite stiffness matrix (negative or zero stiffness matrixes determinate), (ABAQUS, 2005). This model could model damage evolution for elastic brittle materials with anisotropic behavior, such as fiber-reinforced materials. An anisotropic damage evolution law is used to take into account the four different failure modes (fiber or matrix driven tensile or compressive failure). Four 43

58 damage variables, t d f, t c d, d and d, describe damage for each failure mode based c f m m on energy dissipation. The stiffness matrix for the damaged material is computed as follows (1 d f ) E1 (1 d f )(1 dm) v21e1 0 1 Cd = (1 d f )(1 dm ) v12e2 (1 dm ) E2 0 D 0 0 (1 ds ) GD (3.7) where D= 1 (1 d f )(1 dm) v21v, and 12 d f, d m and ds are the internal variables characterizing fiber, matrix and shear damage, and could be computed by derived from four damage variables, corresponding to four failure modes as previously. Once the damage occurred for at least one mode, it will affect damage initiation of other modes. d f d = d t f c f σ11 > 0 σ < 0 11 (3.8) t d m σ 22 > 0 d m = (3.9) c d m σ 22 < 0 t c t c d = 1 (1 d )(1 d )(1 d )(1 d ) (3.10) s f f m m The damage variable for a particular mode is computed by using the following expression: f 0 eq ( eq eq ) f 0 eq ( eq eq ) δ δ δ d = δ δ δ (3.11) δ eq, defined as the equivalent displacement, is calculated by the characteristic length of the element and related strain component (ABAQUS, 2005). Next, the material stiffness matrix, Equation 3, is expressed as a stress-displacement relation, show as Figure 3.7 (ABAQUS, 2005), in order to alleviate mesh dependency during the energy dissipation 44

59 0 process. δ eq is the initial equivalent displacement at which the damage initiation criterion for that mode was met. The values of 0 δ eq depend on the elastic stiffness and strength. is the equivalent displacement at which the material is completely damaged. The value of f δ eq for each mode depends on the respective fracture toughness Gc. f δ eq Figure 3.7 Bilinear equivalent stress-displacement relationship A simple numerical study on fracture toughness was performed. In this numerical study, a single element is tested under unidirectional loading along longitudinal direction. Different values of fracture toughness were used in separated simulations, and the element longitudinal stress strain curves shown in Figure 3.8. It could be seen that this parameter numerically controls the slope of post failure curves. Large value of parameter gives slower softening behavior, indicating more energy would be dissipated. Smaller value gives a sharp drop of stress; it is more realistic for failure behavior of carbon fiber material. 45

60 Figure 3.8 Numerical study on fracture toughness parameter Due to an absence of experimental data, the values for the fracture energies for corresponding four failure modes were obtained from literature (Alfano, 2001) for similar materials, T300s fiber/epoxy resin, and were used as a first approximation for the current numerical analysis, see Table 3.6. Table 3.6 Fiber tow fracture energy Mode I Mode II Mode III Mode IV G Lt (J/mm 2 ) G Lc (J/mm 2 ) G Tt (J/mm 2 ) G Tc (J/mm 2 ) Pure Resin The in-situ material properties of pure resin phase in mesomechanical model is assumed to be the same as those of the bulk material. Strain rate effect is not considered in a single simulation. Effects of strain rate on specimen response will be discussed in Chapter 5. Based on results for the stress-strain response of the epoxy resins examined in this study which were obtained by Littell (Littell, 2008), the one at low strain rate was utilized to obtain the material properties used in the analysis. For example, among E862 46

61 resin testing curves in Figure 3.9 (Littell, 2008), the one with strain rate approximately 1x10-5 /s at room temperature is utilized. For this study, the matrix material was simply assumed to be an elastic-perfectly plastic material. Red dot line in Figure 5 shows how the yield stress was approximated as the ultimate flow stress of the material. The material properties utilized for the two matrix materials used in this study are tabulated in Table 3.2. Perfectly plastic Figure 3.9 Behavior of pure resin phase used in the model Interface Material Interfacial debonding between fiber tows and between fiber tows and regions of pure resin has been identified as a potential key damage mode in braided composites (Littell, 2008). The high stress gradients occurring near geometric discontinuities promote initiation of the debonding, which may further cause significant loss of structure integrity. Three potential methods for simulating this delamination have been found in the literature. The first method is to include delamination as one type of failure mode in 47

62 the failure criterion, like the 3D failure criterion adopted in MAT162 in the LSDYNA material library (Xiao et al, 2007). This method is computationally efficient, but the interface behavior is not independent and not modeled explicitly. The second approach is based on the direct use of fracture mechanics, such as the virtual crack closure technique (Raju, 1987), and the J-integral (Rice, 1968). The drawback of this method is that the initial position of the crack needs to be known and the computational burden increases significantly for three-dimensional problems. The third method involves an indirect use of fracture mechanics, combined with a strength-based failure criterion and damage evolution procedure. This technique initiated from Hilleborg s cohesive zone model (Hilleborg 1976), and was further applied in conjunction with interface elements (cohesive element) by Camanho (Camanho, 2002). By using this method, no information on an initial crack needs to be known, and the onset of crack initiation can be predicted within a preset cohesive zone, which is considered to be a potential crack propagation path. This method is particularly suitable to problems such as braided composites, where the fiber tow interface is relatively weak when compared with the adjoining material. For this work, the cohesive elements formulation and cohesive constitutive models available in ABAQUS are utilized. Zero thickness cohesive elements are proposed in this paper to simulate the fiber tow interface. It is helpful to consider cohesive elements as top and bottom surfaces that are initially well bonded. When they deform together with surrounding elements, there is no membrane stiffness. As illustrated in Figure 3.10, the constitutive equation of the cohesive element, also called the cohesive law, is established in terms of three components of tractions and separations between two surfaces, one through thickness 48

63 component, and two transverse shear components. Each component represents a corresponding fracture mode, such as opening, sliding and shearing modes. In this sense, tractions are considered to be interfacial stresses. (a) Mode I (b) Mode II or III Figure 3.10 Fracture mode and bilinear constitutive relationship It has been shown that a cohesive law can be related to a theory of fracture if the area under the traction-separation curve is equal to the corresponding fracture toughness, for a cohesive element with unit mid-surface area (Rice, 1968). The effects of the shape of the traction separation curve have been investigated by Alfano (Alfano, 2006). For behavior representing a pure fracture mode, a bilinear traction-displacement law represented in Figure 3.10 (ABAQUS, 2005) is used. A very high initial stiffness, so called penalty stiffness, is used to hold the top and bottom surface together. After the traction reaches a peak value, which is considered to be the interface strength, the traction decreases with increasing separation. At the moment traction is equal to zero, debonding between the top surface and the bottom surface occurs. For completeness, the unloading 49

64 behavior is also defined such that a linear curve unloads towards the origin with a degraded stiffness. The area under the bilinear curves is the respective fracture toughness. Two critical separations in the cohesive constitutive relationship could be obtained once the interface parameters are known. c 0 ti δ i = (3.12) K i f i 2G c i c ti δ = (3.13) where c t i is the interfacial strength, fracture toughness. K i is the penalty stiffness and G i is the interface For the triaxially braided composite being investigated in this study, the three interfacial strengths are assumed to be equal to respective fiber tow transverse strengths in Table 3.4, since they are all considered as matrix dominated properties. A very large number of penalty stiffness K (10 6 ) was used to keep top and bottom face together in the elastic region. The fracture toughness can be measured by a single mode fracture test carried out on simple specimens, such as double cantilever beam test (DCB) for opening mode, end notched flexure test (ENF) for sliding mode and split DCB test (SDCB) for tearing mode. Due to lack of the relevant experimental data for this material, literature values in Table 3.7 obtained by Alfano (Alfano 2006), for composite of carbon fiber impregnated by epoxy resin, were used for this study. In a realistic loading situation, the fracture propagation is most likely to occur under mixed mode conditions. Therefore, an element formulation which accounts for mixed mode behavior is required; shown as Figure 3.11 (ABAQUS, 2005). 50

65 51 Table 3.7 Interface fracture toughness (J/mm 2 ) Mode I Mode II Mode III G n G s G t Figure 3.11 Mixed mode behavior of cohesive law The crack is initiated when value of quadratic interaction equations of traction components reaches 1, see Equation = + + c t t c s s c n n t t t t t t (3.14) where, t n, t s, and t t are stress components corresponding to opening, shearing and tearing modes; the critical stress on the denominator are indicated by superscript c. Damage evolutions are defined based on fracture energy. Linear softening behavior is utilized. The dependency of fracture energy on mix mode is expressed by the widely used BK formulation (Benzeggagh and Kenane, 1996). η } ){ ( C t C s C n C t C s C n C s C n C G G G G G G G G G = (3.15)

66 where, Gn, Gs and Gt are the work done by tractions and their conjugate relative displacements corresponding to opening, shearing and tearing modes respectively. The power, η, is a material parameter, here selected to 1.45 for carbon fiber composite (Benzeggagh and Kenane, 1996). 3.4 Numerical Experiments on Mesomechanical Unit Cell In order to further illustrate the behavior of triaxial braided composite, including failure within the unit cell, proposed finite element mesomechanical model will be firstly tested on a unit cell level under specific boundary conditions, see Figure Figure 3.12 Coordination setup for one single unit cell testing This result will provide insights for multi-scale modeling. Since failure behavior is included in material models, the local failure process could be obtained within a single unit cell. Moreover, it is extremely useful for development of macro-mechanical model, since the local material behavior will be explicitly exhibited and examined, which could not be directly pulled out from physical experimental tests. In the other words, the mesomechanical model will be used as virtual test to provide necessary information for higher level modeling methodology. 52

67 3.4.1 CLT Based Macro-Mechanical Solution A classic-laminate-theory (CLT) based macro-mechanical model was created as a comparison to mesomechanical model. Comparison between meso and macro model will be made on effective properties of local laminar level. For CLT based macro model, failure model is not included yet, and its unit cell strategy is discussed below. (a) Unit cell scheme of meso-mechanical model (b) Unit cell scheme of macro-mechanical model Figure 3.13 Comparison of unit cells between meso and macro models In CLT based macro model, it is assumed that each of four sub cell consists of three or two layers of unidirectional laminas, show in Figure 13. Each local layer is represented in different shade. It is further assumed that local layers are of uniform thickness and constant local fiber volume ratio, which can be significantly higher or lower than global fiber volume ratio. Each local laminate is named by capital letters from A to D that indicate different lamination sequence: /0 0 /+60 0, /+60 0, /0 0 /-60 0, /-60 0, respectively. Based on the measurements of braided composite coupon with 53

68 0 0 /±60 0 architecture, the average sub cell size is the same and equal to inch (l) 0.2 inch (w) inch (t). Since in-plane dimensions are 8~10 larger than thickness dimension, the CLT assumptions are satisfied. Then, the individual thickness and local fiber volume ratio of each ply could be obtained from geometry mesh of proposed mesomechanical model at corresponding volume. Table 3.8 shows the assumed local ply thicknesses and fiber volume ratios. Note that all plies with bias fiber tows have 50% fiber volume ratio and the axial layers must have 80% fiber volume ratio, meanwhile thickness of those layers in sub cells A and C are half of axial layers. Consequently, unidirectional lamina modules properties will be different between 0 0 and ±60 0 plies and could be further calculated through generalized method of cells (Aboudi, 1991). Table 3.8 Fiber volume ratio and thickness of each integration point of unit cell 0 fiber tow Fiber volume ratio ±60 fiber tow Fiber volume ratio 0 fiber tow thickness ±60 fiber tow thickness Sub Cell A 80% 50% 0.5 w 0.25 w Sub Cell B N/A 50% N/A 0.5 w Sub Cell C 80% 50% 0.5 w 0.25 w Sub Cell D N/A 50% N/A 0.5 w Two-dimensional constitutive relationship is assumed for each ply, see Equation (3.16). For each sub cell, classic laminate theory is utilized to relate middle plane strain and curvature to surface load and moment, see Equation (3.17). Unit cell behavior will be summation of four sub cells based on certain assumptions. 54

69 C C C σ x ε x σ y = C12 C22 C26 εy σ C C C γ xy xy (3.16) N A B M = B D κ 0 ε (3.17) To perform the numerical study based on unit cell in next two sections, uni-strain fields are applied on unit cell both axially and transversely since it is used as a common assumption in many modeling approaches and it is easy to realize. Comparison on unit cell and sub cell stiffness is made between meso and macro model; while failure progress is investigated based meso mechanical model Uni-strain Loading in Axial Direction Based on the given coordination in Figure 3.11, the boundary conditions below will apply on unit cell to achieve uni-strain axial loading conditions. When such boundary conditions are applied, each sub cell is under multi-axial stress status. u (0, y, z) = 0; u (l, y, z) = 0; w (x, y, 0) = 0; w (x, y, t) = 0; v (x, 0, z) = 0; v (x, w, z) = Δ. To solve based on CLT, some assumptions are required. 1. ε y (A) = ε y (B) = ε y (C) = ε y (D) = constant; Strains in Y direction of each sub cell are the same as unit cell global strain. 2. κ(a) = κ(b) = κ(c) = κ(d) = 0; Mid-plane curvatures of each sub cell are zero. 3. ε x (A, B, C, D) 0; ε x (A) + ε x (B) + ε x (C) + ε x (D) = 0 since ε x (unit cell) = 0; 55

70 X direction strains of each sub cell is not zero, but summation is equal to zero since unit cell X strain is zero noting that lengths of four sub cells are equal. 4. N x (A) = N x (B) = N x (C) = N x (D) Transverse forces on four sub cells are equal. 5. N y (A) + N y (B) + N y (C) + N y (D) = N y (Unit cell); Total Y force is equal to summation of Y forces of four sub cells. Equation 3.18 gives relations between consultants and strains for each sub cell. A, B and D matrixes are known for this case. i i N x A11 A12 A16 B11 B12 B16 εx( εy, A22, A12 ) N y A12 A22 A26 B12 B22 B 26 ε y N xy A16 A26 A66 B16 B26 B 66 0 = M x B11 B12 B16 D11 D12 D16 0 M y B12 B22 B26 D12 D22 D 26 0 M xy B16 B26 B66 D16 D26 D66 0 (3.18) Based assumption 3 and 4, equation (3.18) will be used firstly to solve out ε x for each sub cell as a function of known Y strain and two items of A matrix. Then, submit solved values of ε x and known ε y back to Equation (3.18) for each sub cell to obtain all forces and moments values. Finally total Y force acting on unit cell is computed based on assumption 5. Table 3.9 Nominal axial stiffness of meso-model and CLT results (GPa) Type of model Object Meso-Model CLT Unit cell Sub cell A & C Sub cell B & D

71 For the purpose of comparison, nominal stiffness σ y /ε y of unit cell and sub cell are computed and listed in Table 3.9. Since uni-strain boundary conditions result in multiaxial stress status for each layer in each sub cell, the stiffness obtained is called nominal stiffness here. It can be seen that two methodologies give close predictions. Figure 3.14 shows stress strain curves generated for unit cell and each of four sub cells based on mesomechanical modeling results. It is obvious that more loads are carried by unit cell A and C since axial fiber tows are contained and aligned with loading direction. From the curves, it could be also seen that the failure of whole unit cell occurred when sub cell A and C failed. It means that in this loading situation, the ultimate failure is caused by axial fiber tow failure. Figure 3.14 Stress strain curves for unit cell and sub cells under axial loading Table 3.10 Type of failure at different levels of axial load Sub cell A&C Sub cell B&D ε = 0.5% III III III III ε = 1.8% I 57

72 Table 3.10 shows the local failure modes on each fiber tows during the loading process. The roman character in the table indicates the failure modes: I for fiber tensile failure mode, II for fiber compressive failure mode, III for matrix tensile failure mode, and IV for matrix compressive failure mode. Detailed failure criteria have been shown in Table 3.5. The numerical results shows that at lower strain level, 0.5%, the matrix tensile failure transverse to fiber longitudinal direction were common on bias fiber tows; and at 1.8% strain level, axial fibers failed and lead whole composite material failure Uni-strain Loading in Transverse Direction Similar to axial direction, the boundary conditions below will apply on unit cell to achieve uni-strain transverse loading conditions. u (0, y, z) = 0; u (l, y, z) = Δ; w (x, y, 0) = 0; w (x, y, t) = 0; v (x, 0, z) = 0; v (x, w, z) = 0. Again, to solve based on CLT, some different assumptions are required here for transverse loading situation. 1. ε y (A) = ε y (B) = ε y (C) = ε y (D) = 0; Strains in Y direction of each sub cell are all zero. 2. κ(a) = κ(b) = κ(c) = κ(d) = 0; Mid-plane curvatures of each sub cell are zero. 3. ε x (A) ε x (B) ε x (C) ε x (D) 0; but, {ε x (A) + ε x (B) + ε x (C) + ε x (D)}/4= ε x (unit cell); 58

73 X direction strains of each sub cell are not zero and different from each other. But, summation of X deformation of each sub cell is equal to total X deformation of unit cell. 4. N x (A) = N x (B) = N x (C) = N x (D) = N x (Unit cell); X force is equal to Y force of each of four sub cells, as four sub cells are in parallel style in transverse direction. Equation 3.19 gives relations between consultants and strains for each sub cell, with A, B and D known. N x A11 A12 A16 B11 B12 B16 ε x ( ABCD,,, ) N y A12 A22 A26 B12 B22 B 26 0 N xy A16 A26 A66 B16 B26 B66 0 = M x B11 B12 B16 D11 D12 D16 0 M B B B D D D 0 y M xy B16 B26 B66 D16 D26 D66 0 (3.19) Equation 3.19 gives direct relationship between N x and ε x. Noting the fact that sub cell A and C have the same value of A 11, so do B and D, a ratio relationship of ε x (A,C) and ε x (B,D) could be found based on assumption 4. Then, equation in assumption 3 will compute the value of ε x for each sub cell. Then, submit solved values of ε x back to Equation 3.19 for each sub cell to obtain all forces and moments values. Comparison of nominal stiffness σ y /ε y of unit cell and sub cells are computed and listed in Table Close prediction are found again for two methodologies. Figure 3.15 shows stress strain curves generated for unit cell and each of four sub cells. Since the four sub cells were in parallel style, the loads on each were equal. Seen from the curves and Table 3.11, the transverse stiffness of sub cells B&D were larger 59

74 than that of sub cells A&C. Failure occurred within the region of sub cell C, since it exhibited a softening behavior after peak stress. Table 3.11 Nominal transverse stiffness of meso-model and CLT results (GPa) Type of model Object Meso-Model CLT Unit cell Sub cell A & C Sub cell B & D Figure 3.15 Stress strain curves for unit cell and sub cells under transverse loading Table 3.12 Type of failure at different levels of transverse load Sub cell A&C Sub cell B&D σ = 165 MPa III σ = 250 MPa III III σ = 375 MPa III III σ = 1015 MPa I I 60

75 The failure process is summarized in Table 3.12, according to different level of stresses. It shows that matrix tension failure mode on axial fiber tow caused the first damage within unit cell; then the damage propagated into bias tows neighboring to axial tows in sub cell A and C; and then matrix failure were on all fiber tows; finally fiber tows in bias directions broke under longitudinal tension, which lead to final failure of unit cell. What need to emphasize here is that the boundary conditions applied here on one single unit cell forced bias tows to be totally tensioned until failure. However, for coupon level simulation as discussed in next chapter, actual boundary conditions are different, and explain why the material strengths obtained are different. It again emphasizes the significant of boundary conditions during investigating textile composite material behaviors. 3.5 Conclusions A finite element based approach has been developed for examining the behavior of triaxial braided polymer matrix composites. The architecture of the braided composite unit cell has been explicitly modeled. A transversely isotropic constitutive, damage and failure model has been utilized for the fiber tows. A cohesive element approach has been applied to model the imperfect interfaces between fiber tows and between the fiber tows and regions of pure matrix. Numerical experiments on one single unit cell with certain boundary conditions exhibits the ability of proposed mesomechanical models on predicting material behavior and capturing local damage. Classic laminate theory solutions are used for comparison. The reason for this numerical work is based on the fact that a lot of macro scale material 61

76 model in current literature are based on CLT. This work could be extended to be a way to connect meso and macro models, falling into the frame of multi-scale modeling. 62

77 CHAPTER IV FAILURE STUDY OF THE TRIAXIAL BRAIDED COMPOSITE MATERIAL RESPONSE IN SIX-PLY STRAIGHT SIDED SPECIMEN Triaxial braided carbon fiber composites, as opposed to traditional laminate composites, have enabled structures with complex shapes to be designed and fabricated with components reinforced with a fiber architecture optimized locally for overall performance. One potential mechanism which may account for the difference in performance between braided composites and traditional composites is the interaction between fiber tows in triaxial braided composites. Hence, a more detailed numerical investigation of the deformation, local stress distribution and failure processes in triaxial braid composites is presented in this chapter. 4.1 Experiment Test and Measurement Technique Qusi-static tensile tests on 0 0 /±60 0 braided composite specimens were performed by Littell (Littell, 2008) using an MTS axial-torsional test machine under displacement control. Straight sided tensile specimens were cut with 12 inch long by inch wide with six braided layers, see Figure 4.1. These dimensons were chosen such that the width of specimen contained at least two unit cells in length and seven unit cells in width, and length of specimen conformed to ASTM length to width ratio. Compression test specimens were also prepared, with specimen dimensions of 6 inch in length, and 63

78 allowed for a long gripped region with short gauge section equal to 1 inch long, as shown in Figure 4.2. Both tensile and compression specimens were cut in two ways for testing. Axial tensile tests were conducted using specimens with the 0 0 fibers aligned along the applied load direction. Transverse tensile tests were conducted using specimens where the 0 0 fibers were perpendicular to the direction of the applied load. Figure 4.1 Illustration of straight sided tensile specimen dimensions Grip area Gauge Area 1 Figure 4.2 Straight sided specimen compression test set up All test were conducted under low loading rate at inch/min. All strain measurement were taken by use of optical measurement system, which is also capable of 64

79 capturing load from the MTS machine. Figure 4.3 shows a schematic of the optical measurement setup (Littell, 2008). Figure 4.3 Optical measurement setup The field of view of specimen surface deformation can be captured on the prepainted specimen surface areas which cover entire specimen width and approximately 2 inch length of specimen length. Stress strain curves are calculated from the recorded load and strain. Meanwhile, in the experimental program conducted by Littell (Littell, 2008), based on full field strain maps on the surface of the triaxial braided composite specimens generated by use of an optical strain measurement system, an overlay technique was then applied to correlate the full field strain results to the geometry of the braided composite. In this manner, the relative location (axial fiber, bias fiber) of the damage in the composite could be quantified. The composite test methods described above not only examine overall material response by presenting a series of stress vs. strain curves, but also provide insight into the nature of progressive composite failure by developing techniques to directly measure the local failure mechanisms. The ability of proposed mesomechanical model will be 65

80 examined in the following sections by model correlation to match the experimental effective stress-strain curves and the sequence of damage. 4.2 Finite Element Model Development In the numerical simulation of the straight sided specimen tests, the actual loading and boundary conditions of the mid-span of the test specimen was approximated. A finite element model was created with three unit cells aligned along the axial direction, shown as Figure 4.4b and Figure4.5b. Symmetric boundary conditions were applied to generate a plate size of 1.2 by with two layers in the thickness direction, shown as Figure 4.4a and Figure 4.5a. The model dimensions are comparable to the middle part of the tensile specimens shown as the shadow region in Figure 4.1, and similar to the size of the actual compressive specimens. The free edge boundary conditions were also explicitly simulated according to the actual test conditions, it is critical to capture material response under current specimen design, since they turns out to influence material behavior by causing edge damages and affecting local failure mechanisms on fiber tows. This free edge effect will also be discussed later. To obtain the global stress strain curves shown later in this paper, the forces and displacements along the loading surface were utilized. The stresses are calculated by dividing force over surface area, and strain by dividing surface displacement by original length. 66

81 (a) Sketch of model and boundary conditions (b) Finite element model Figure 4.4 Boundary and load conditions of axial test modeling (a) Sketch of model and boundary conditions Figure 4.5 Boundary and load conditions of transverse test modeling 67

82 (b) Finite element model Figure 4.5 Boundary and load conditions of transverse test modeling (Cont d) The fact that the actual test specimen boundary conditions were simulated is significant since, as discussed by Littell (Littell 2009), there are deficiencies in the transverse tensile tests. Since a number of the bias fibers in the gage section are ungripped due to the free edges in the specimen design, and this free edge boundary situations may not exist for real composite structures, the stress-strain curves and ultimate strengths obtained in the transverse tensile test may not be truly representative of the actual composite material response. In order to capture these effects, a model should be built in such a way that it could discretely model the complete architecture but not a homogenized presentation. Specifically, the stress-strain curves obtained in the transverse tensile test may be artificially soft and the predicted strength may be low. This point will be important later in this section as several of the material properties were correlated based on the results of the experimental transverse tensile data. While valid for the particular test geometry presented, the results may not be truly indicative of the actual composite material properties and results. 68

83 NASA Glenn Research Center tested several braided composite material systems, which have same fiber constituents but infused with different resin (Littell, 2008). Significant discrepancy of material strengths were observed for different material systems. This implies different failure mechanisms in different material, and this different is hypothesized to be related to interface bonding effect. The fiber tow-to-tow bondings are simulated explicitly in proposed mesomechanical model, in order to investigate the effects of the interface on the effective response of the model, three different interfacial conditions were simulated; perfect, moderate and weak interfaces. The moderate and weak interfaces were designed to investigate the effects of a non-perfect bond, and are differentiated by the interfacial strengths of the cohesive elements. The strength of the weak interface was assumed to be equal to the transverse strength of the fiber tows. Since both interface strengths and tow transverse strengths are matrix dominated, they would be in the same order of magnitude, here they are initially assumed to same for weak interface. The interfacial strength of the model with a moderate interface was set to be double the value used for the weak interface model, in order to investigate the effects of varying interfacial strength. In the perfect interface model, meant to simulate the case of a perfect bond, there were no cohesive elements or interface zone applied; instead node sharing was used on the contacting surfaces of the fiber tows. 4.3 Model Correlation The goal of the model correlation was to match the experimental effective stressstrain curves and the sequence of damage, and understand how material property parameters affect effective properties of models. The study was separated into two parts. In the first part, model correlation was conducted on the T700s/E862 composite, which 69

84 experimental evidence indicated (Littell, 2008) has a moderate interface. By simulating the axial and transverse tensile response of this material, the interface effect was also investigated by comparing the results obtained by applying all three of the previously discussed interfacial conditions. In the second part of the study, the axial and transverse tensile response of the T700s/PR520 system, which experimental evidence indicated had a strong interfacial bond (Littell 2008), was simulated by using the perfect interface model Model Correlation of T700s/E862 Composites The model correlations for the T700s/E862 composite are shown in Figures Figure 4.6 shows the experimental results and computed values from an axial tension test, Figure 4.7 shows the results from a transverse tension test, Figure 4.8 shows the results from an axial compression test and Figure 4.9 shows the results from a transverse compression test. Figure 4.6 Global stress strain curves of T700s/E862 in axial tension test 70

85 Figure 4.7 Global stress strain curves of T700s/E862 in transverse tension test Figure 4.8 Global stress strain curves of T700s/E862 in axial compression test 71

86 Figure 4.9 Global stress strain curves of T700s/E862 in transverse compression test In order to match computed effective stress strain curves close to testing results, the fiber tow longitudinal compressive strength was adjusted as discussed in Section For all the correlations shown in Figures , the linear portions of the simulated curves correlate very well with the experimental results. It indicates that elastic constants, which were initially calculated by using of micromechanics, are accurate. For the axial tension tests shown in Figure 4.6 and the axial compression tests shown in Figure 4.8, the nonlinear portion of the computed stress-strain curves and the computed ultimate strength correlated well with the experimental results, indicating that the damage and failure model applied for the fiber tows was reasonably accurate. Further discussions on the simulated nonlinearity in the stress-strain curves will be given in the next section. The extended unloading observed in the computed stress-strain curves is most likely a numerical artifact, as the experimental specimens most likely had a more brittle fiber failure, while the computed curves had a more gradual unloading to promote numerical stability. Varying the interfacial conditions also had a minor effect on the 72

87 computed results, which is reasonable since under axial loading conditions the 0 0 fibers should take the majority of the load, and interfacial effects should not be significant when 0 0 fibers are loaded axially. While, when interface is very strong (as perfect interface bonding condition), the interface will keep bonding axial and bias tows together, which make bias tows contribute more to loading carrying compared with weak interface conditions. For the transverse tension tests shown in Figure 4.7 and the transverse compression tests shown in Figure 4.9, the nonlinearity in the stress-strain curves observed in the experimental results was once again captured reasonably well in the simulations. This indicates again that the failure models which have been implemented into the mesomechanical model are reasonably accurate. As discussed in later this chapter, the nonlinearity is due to the matrix failure mode, which is related to fiber tow transverse strength, F 2T or F 2C, and shear strength, F S. Since these strength parameters are related to matrix properties and relatively small comparing with fiber tow axial strengths, the matrix failure modes came in early and kept accumulated during the loading progress, resulting in nonlinear effective material response. In these tests, varying the strength of the interface had a significant effect on the predicted ultimate strength of the composite, with lower interfacial strengths resulting in a weaker ultimate strength of the material. These results are reasonable since under a transverse loading condition both the axial and bias fibers carry significant portions of the load transverse to the fiber axis, which would result in the interface playing a significant role in the material response, with a stronger interface being able to carry more load. The results computed using the moderate interface appeared to correlate the best with the experimental values, which is reasonable 73

88 since this material has been identified as having a moderately strong interface through experimental investigations (Littell, 2008). Overall, the developed model appears to have the capability to capture the effects of varying interfacial strength on the global stressstrain response of a braided composite Model Correlation of T700s/PR520 Composites In order to verify the capability of the finite element model to simulate the effective response of triaxial braided composites without regard to interface effects, the effective response of the T700s/PR520 system, which is known to have a strong fiber tow interface (Littell, 2008), was computed. Since this system has a strong interfacial bond, only the perfect interface condition described above was applied for these simulations. Experimental and correlated effective stress-strain curves from axial tension and transverse tension tests are shown in Figure 4.10, and results from axial compression and transverse compression tests are shown in Figure Figure 4.10 Global stress strain curves of tension test on T700s/PR520 system 74

89 Figure 4.11 Global stress strain curves of compression test on T700s/PR520 system In order to match computed effective stress strain curve close to testing results, the fiber tow longitudinal compressive strength was adjusted again. All correlated material properties have been tabulated in Table 3.3 and Table 3.4. Overall, the correlation between the experimental and analytical results is reasonably good, indicating that the modeling approach is reasonably accurate, and that the perfect interface bond captures the expected response of the material. The ultimate strengths under transverse loading are somewhat over-predicted. The axial tensile strength is somewhat underpredicted, but this result could be due to the fact that the method used to compute the fiber tow axial longitudinal strength based on a mechanics of materials approach (Chamis, 1984) is not accurate for all kinds of combination of composite systems (Kaw, 2005). In axial tension test, nonlinearity exhibited in numerical results, but not seen in testing results. It is due to the nature of damage model used in the numerical simulation. As indicated in the stiffness matrix of damaged material of equation 3.7 (ABAQUS, 75

90 2005), fiber failure mode and matrix failure mode are coupled. Matrix tensile failure, occurring at the point of nonlinearity, will also cause longitudinal stiffness reduction on fiber tows. This damage mechanism may be just suitable for some material system, e.g. T700s/E862. To summarize, for both of the materials discussed in this section, several model parameters, such as fiber tow stiffness and strength, interface strength and toughness, were correlated based on the results of the four specimen tests. Firstly, the elastic properties were correlated based on the initial stiffness of the four testing curves; the tow strengths were correlated based both on tests conducted with a material with a perfect interface and indirectly by correlating based on axial tension and compression tests of a material with an imperfect interface. From the transverse tension tests, the interface plays a major role in these tests so the tow strengths could not be correlated just from these tests. However, the interface strengths were correlated based on the transverse tests. From the numerical parameter studies, the fiber tow fracture energy was found to mostly affect the post peak part of the stress strain curves. More detailed parameter sensitivity analysis will be addressed in Chapter 5. An important point to note is that, as discussed above, there are deficiencies in the transverse tension test that result in the corresponding curves most likely not being representative of the actual material response. Therefore, the correlated model parameters are correct for the test geometry as given, but are most likely not representative of the actual material response. In the actual case, the boundary conditions might be different and the bias fiber tow would be working thoroughly, which is different from the case in straight sided specimen. However, the procedure as shown in this paper demonstrates 76

91 how test data could be used to correlate the various model parameters. Furthermore, the general trends of how the interfacial strength affects the simulated results are valid. 4.4 Numerical Investigation of Material Failure A primary motivation for utilizing a high fidelity finite element approach for analyzing the response of composites is to gain insight into the local mechanisms which cause damage and failure in the material. The numerical model developed in this study, due to its explicit modeling of the composite architecture, has the capacity to capture damage mechanisms such as fiber tow failure, matrix cracking and interfacial failure. Since the individual failure variable corresponding to a certain failure mode can be reported for each element, with the assistance of the post processing function of ABAQUS code, contour plots can be generated as field information for all elements at each computational step for different levels of global stress or strain. In order to correlate the identified failure modes to particular damage mechanisms, the contour plots relating to failure patterns are overlaid over the undamaged mesh (Figure 4.4b and Figure 4.5b). For all the contour plots afterwards, each plot represents one particular failure mode, which is indicated under the plot; the darkness of the color defines the status of damage, which corresponds to the value of the particular damage variable ranging from 0 to 1, shown in the left picture in Figure A value of 0 means damage has not occurred yet, while a value of 1 means the element is totally damaged. Meanwhile, the global strain and stress values at which the contour plots were generated are shown. 77

92 Another point that needs to be emphasized is that all contour plots are shown with axial direction going vertically, and the load direction for axial and transverse specimen are same as indicated in Figure 4.4b and Figure 4.5b respectively. For this study, the T700s/E862 material was selected. The results of moderate interface were utilized in the numerical investigation since this interfacial condition resulted in the best correlation with the experimental results. Furthermore, by investigating a system with an imperfect interface, the capability of the model to capture interfacial failure events could be quantified. Axial tension, axial compression, transverse tension and transverse compression tests were simulated and discussed individually below Axial Tension Test a. Matrix Tensile b. Fiber Tensile c. Interface failure 1 - totally failed ε = 0.6% ε = 1.9% ε = 1.9% 0 no failure σ = 300 MPa σ = 750 MPa σ = 750 MPa Figure 4.12 Contour plots of failure modes in axial tension test In the simulations of the axial tension tests, at a global strain level of 0.6%, matrix tensile failure in the bias fiber tows, which exhibits like fiber splitting (a single fiber tow split into several tows in smaller size) due to transverse tensile stress and shear stress, was observed as Figure 4.12a. These results correlate to observations made in the 78

93 experiment (Littell, 2008), Figure 4.13b, in which high strains in the bias fibers were identified at the corresponding global strain level. It has been confirmed that the high strains in the bias fiber identified experimentally are due to fiber splitting within the fiber tow (Littell, 2008), shown in Figure Simulation Optical measurement a) tensile matrix failure on bias tows b) high strain region show subsurface cracking Figure 4.13 Correlation between numerical contour plot and optical measurement Figure 4.14 Fiber splitting on fiber tows Furthermore, as seen in the experimental axial tensile stress-strain curve shown in Figure 4.6, at this global strain level, 0.6%, the stress-strain curve became nonlinear, which appears to indicate and confirm that the nonlinearity observed in the experimental global stress strain curve (and predicted numerically) is due to fiber splitting within the bias fiber tows. At a global strain level of 1.9%, fiber tow failure due to tensile fiber failure, as well as failure of the interface elements was identified, Figure 4.12b. As observed in stress strain curve of Figure 4.6, in the experimental axial tensile test the composite 79

94 failure was identified as occurring at a global strain level of 1.9%. The results obtained here indicate that the ultimate failure of the composite is a result of failure of the axial fibers followed by a corresponding failure in the interface at geometry discontinuities, which was also reflected in the computed effective composite response reaching its ultimate strength at this global strain level. These results are reasonable since for an axial tension test one would expect the ultimate failure of the composite to be governed by the failure of the axial fibers Axial Compression Test a) Tensile Matrix b) Compressive Matrix ε = 0.5% ε = 0.5% c) Compressive Fiber ε = 1.0% d) Interface failure ε = 1.0% σ =200 MPa σ = 200 MPa σ = 340 MPa σ = 340 MPa Figure 4.15 Contour plots of failure modes in axial compression test In the detailed damage contour plots obtained in the simulations of the axial compression tests shown in Figure 4.15, at a global strain level of 0.5% significant levels of damage of tensile matrix failure in the axial fiber tows (due to Poisson s ratio effect) and significant levels of damage in compressive matrix failure mode in the bias fiber tows were identified. By comparison to the global stress-strain curve shown in Figure 4.8, at this global stress and strain level the effective curve became nonlinear. These results 80

95 indicate that nonlinearity in the effective stress-strain curve is a result of the identified damage in the fiber tows. At a global strain level of 1.0%, compressive fiber failure in the axial fiber tows, along with slight amounts of damage in the interfacial elements, was identified. Since the strain level of 1.0% corresponded to the strain level at which the ultimate compressive strength occurred in the effective stress-strain curve, these results indicate that the ultimate failure in an axial compression test is due to compressive failure in the axial fiber tows Transverse Tension Test a) Tensile Matrix b) Compressive Fiber ε = 0.4% ε = 0.6% c) Tensile Matrix d) Interface failure ε = 1.2% ε = 1.2% σ = 150 MPa σ = 220 MPa σ = 400 MPa σ = 400 MPa Figure 4.16 Contour plots of failure modes in transverse tension test In the simulation of the transverse tension test, multiple damage modes were identified as seen in Figure Direct correlations between the damage modes identified in these simulations and the global stress strain curves could not be made due to identified deficiencies in the transverse tensile test methodology, which result in complexities in the detail of the obtained results. First, at a relatively low strain level 0.4%, damage due to tensile matrix cracking within the axial fiber tows was observed 81

96 since axial tows are under tensile in this situation. This contributed to initiation of nonlinearity on effective stress strain curves. Meanwhile, in the optical strain measurements (Littell, 2008), at approximately the same global strain level, see comparison in Figure 4.17, high strain regions was identified experimentally in the subsurface fiber tows. The digital microscopic picture Figure 4.18 (Littell, 2008) confirmed that the data, seen by the optical measurement in transverse tensile tests, correlated to fiber splitting damage occurring in the axial fiber tows, which is a major damage causing mechanism. Simulation Optical measurement a) tensile matrix failure on axial tows b) sub-layer fiber splitting Figure 4.17 Correlation between numerical contour plot and optical measurement Figure 4.18 Digital microscopic picture of a transverse tensile specimen At a slightly higher strain level, 0.6%, some regions of the axial fiber tows in the simulations exhibited compressive fiber failure due to Poisson ratio effects. At a global strain level of 1.2%, failure occurred in the simulations, which according to the damage 82

97 contour plots, was due to a combination of tensile matrix cracking in all of the fiber tows, which would be expected in a transverse tensile test, and interfacial failure, which would be expected in a material with a weaker interfacial bond Transverse Compression Test a) Compressive matrix ε = 0.5% σ = 150 MPa b) Compressive fiber ε = 0.9% σ = 300 MPa c) Interface failure ε = 0.9% σ = 300 MPa Figure 4.19 Contour plots of failure modes in transverse compression test In the simulations of a transverse compression test shown in Figure 4.19, damage due to matrix compressive failure in the fiber tows was identified as occurring first at 0.5% strain level. By comparison to the global stress-strain curve shown in Figure 4.9, the effective curve became nonlinear at this global stress and strain level, indicating that nonlinearity in the effective stress-strain curve is a result of the identified damage in the fiber tows. At the strain level 0.9%, compression fiber failure was found on bias tows mostly at the locations where bias tows go around axial tows. Meanwhile, significant amount of interface debonding were observed around the boundaries and free edges in the simulation results. Since 0.9% strain corresponds to the strain level at which ultimate failure occurred, these results indicate that both damage modes contribute to specimen ultimate failure under transverse compression. 83

98 4.5 Conclusions From the detailed analysis of failure process for each test in Section 4.4, together with Section 4.3 of global stress strain correlation, it shows that the presented model gives good local failure prediction and can be used to explain and correlate with experimental measurement. The global stress-strain response of two representative triaxial braided composite systems has been simulated, with reasonably good correlations obtained with experimentally obtained results. The effects of varying the strength of the interface on the simulations of the global response have been identified. By examining detailed contour plots of the simulated damage and failure within the composite, the progression of damage and failure within the composite has been identified, and in many cases correlations between the local damage mechanisms and details of the effective stress-strain response could be made. By utilizing a modeling approach of this type, numerical experiments could be performed on a braided composite system for situations where experimental data is not available, such as tests at high strain rates. Furthermore, by studying the details of the local damage and failure within the finite element model, significant insight could be gained into the details of the material response, which could be extremely useful in understanding the behavior of the material and developing more approximate numerical models of the material behavior. 84

99 CHAPTER V NUMERICAL PARAMETERIC STUDY OF MESOMECHANICAL TRIAXIAL BRAIDED COMPOSITE MODEL The proposed model has been verified to be able to successfully predict material global behavior and failure process of triaxial braided composite in straight sided specimen test. This chapter further discusses the feathers of the model by applying thoroughly parameter. 5.1 Summary of Model Parameters Table 5.1 Categorization of model parameters used in material models Group I Group II Group III Group IV Group V Fiber tow modulus E 11, E 22, G 12 v 12, v 23 Fiber tow strength F 1T, F 1C F 2T, F 2C, F S Fiber tow toughness G IT, G IC G IIT, G IIC Interface parameters C C C t I, tii, tiii G I, G II, G III Resin parameters E m, v m, σ y The model input parameters, material properties for each of three phases, have been introduced thoroughly in Chapter 3. Some are directly from experimental test results; some are calculated from well established material mechanics based on vendor publish data; some are referred from current literatures. Some have specific physical meanings, while some are just for numerical purpose. Table 5.1 gives a summary on model parameters used in this mesomechanical model for three material phases. They are 85

100 categorized into four groups, and will be investigated in following numerical studies. During the model correlation process in Chapter 4, it turns out that the value of these parameters in Chapter 3 gave a good first guess. However, a thorough parameter study will help us understand the main feature of proposed FE model and how composite responses on effective stress-strain curve just by tuning one of these material parameters each of them will affect material behavior; it is also a procedure to understand the ability of the model and useful for further model development. 5.2 Qualitative Parameter Sensitivity Study Parameter studies are based on previously FE model in Chapter 4 developed for simulating six ply straight sided specimens testing. For each material property parameter, we will decrease and increase certain amount based on correlated value (or base value, shown as 1.0X in the picture) to produce two more numerical predictions for each specimen test. Then, stress strain curves for each of four tests are generated and compared in on chart. In such a way that the influence on entire global behavior could be investigated, so here name it qualitative parameter study. In the parametric study, fiber tow strengths and interface parameters will be investigated, and fiber tow modulus are not investigated here since they give good prediction on initial stiffness of all four specimens, as shown in the numerical results in Chapter 4. 86

101 5.2.1 Tow Interface Parameters 1) axial tension 2) axial compression Figure 5.1 Parametric studies on interface strengths 87

102 3) transverse tension 4) transverse compression Figure 5.1 Parametric studies on interface strengths (Cont d) Figure 5.1 shows comparison between effective stress strain curves generated from experiment test and three simulations with different interface strengths (t C I, t C II, t C III ), 0.2 times base value, base value and 5.0 times base value. These three interface strengths are scaled up and down simultaneously. 88

103 Varying the interfacial adhesive strength conditions has a negligible effect on the computed results of axial tests, which is reasonable since under axial loading conditions the 0 0 fibers should take the majority of the load, and interfacial effects should not be significant when 0 0 fibers are loaded axially. However, varying the strength of the interface has a significant effect on the predicted ultimate transverse strength of the composite, with lower interfacial strengths resulting in a weaker ultimate strength of the material. These results are reasonable since under a transverse loading condition both the axial and bias fibers carry significant portions of the load transverse to the fiber axis, which would result in the interface playing a significant role in the material response, with a stronger interface being able to carry more load. Tuning of tow interface toughness, Gc, will give a similar tendency but less amount of influence on four specimen tests as interface strengths, the results will not be shown here. A sense of such effect could be obtained from Table 5.2 as a quantitative parametric study Fiber Tow Longitudinal Tensile Strength Varying the fiber tow longitudinal tension strength, F 1T, affects just the effective stress strain curves of axial tension tests, shown in Figure 5.2. From the micromechanics calculation, the laminar longitudinal tension strengths are based on the properties of two constituents (more influence by fiber properties than resin) and the bonding conditions. Increasing or decreasing F 1T means the changing of fiber constituents properties, or changing of the bonding conditions, or even changing of the fiber volume ratios in the fiber tows. 89

104 1) axial tension 2) axial compression Figure 5.2 Parametric studies on fiber tow longitudinal tension strength 90

105 3) transverse tension 4) transverse compression Figure 5.2 Parameter study on fiber tow longitudinal tension strength (Cont d) In the Hashin failure criteria, F 1T is included in the fiber tensile failure mode. This failure mode will be triggered once the longitudinal stresses are beyond the strength, F 1T. It could be seen from Figure 5.2 that numerical strength of an axial tensile specimen 91

106 increases proportionally with increasing F 1T. It is reasonable since the fiber tow longitudinal tension failure mode controls ultimate failure of axial tensile specimen. Meanwhile, all other three specimen responses are not affected at all, which means the fiber axial tensile failure does not occur in any of these three tests, even in the transverse tensile tests, where the bias tows are under axial tensile stress status. This explains why the transverse tensile specimen strength is lower than axial tensile strength. Actual value of F 1T relies on properties of constitution as well as fiber/matrix bonding conditions. In order to predict more accurately, more advanced micromechanics methodology is required Fiber Tow Longitudinal Compressive Strength Tuning of the fiber tow longitudinal compressive strength parameter, F 1C, has more significant influence on compression test, both axial and transverse, shown in Figure 5.3. The fiber longitudinal compression failure is common in these two numerical simulations. As discuss in Chapter 3, the real failure situations of a fiber tow under compression are very complicated, which have been studied by Madhukar (Madhukar 1992). In the braided composite materials, as modeled with the proposed mesomechanical model where the detailed the architectures of braided composites are shown explicitly, the tow undulations and misalignment, shear effects and out of plane bending effects need to be taken into account, which makes it more complicated. Furthermore, the bonding conditions and interactions between different fiber tows played important role on determining failure mode of tows under compression, especially for the transverse straight sided specimen where the interface are transferring load between fiber 92

107 tows. All of these will affect the determination of the fiber tow longitudinal compressive strengths. As stated in Chapter 4, this parameter needs to be correlated with testing results. 1) axial tension 2) axial compression Figure 5.3 Parametric studies on fiber tow longitudinal compressive strength 93

108 3) transverse tension 4) transverse compression Figure 5.3 Parametric studies on fiber tow longitudinal compressive strength (Cont d) 94

109 5.2.4 Fiber Tow Transverse Tensile Strength 1) axial tension 2) axial compression Figure 5.4 Parametric studies on fiber tow transverse tensile strength 95

110 3) transverse tension 4) transverse compression Figure 5.4 Parametric studies on fiber tow transverse tensile strength (Cont d) The fiber tow transverse strength (either tensile or compressive) is matrix dominated property, which depends on the strength of matrix and fiber-matrix bonding conditions. Varying it means changing of matrix properties. From the numerical results shown in Figure 5.4, varying fiber tow transverse tensile strength, F 2T, affects more on 96

111 tensile specimens, like the nonlinearity rising region of stress-strain curve of the axial and transverse tension test simulation. Very little influence on compressive specimens is found before the peak stress. For effective stress strain responses of axial and transverse tensile specimens, the nonlinearity is observed to relate to local damage such as matrix cracking or fiber splitting, which is corresponding to matrix tensile failure mode based on strengths F S and F 2T. This type damage is primary in the composite before ultimate failure (Littell 2008). It breaks down the integrity of composite and induces the nonlinearity (stiffness degradation) of effective stress-strain curves. The higher numerical strength F 2T is, the less transverse damage is induced before ultimate failure. One interesting point is that, in an axial tension simulation, a larger value of F 2T produced lower degraded specimen stiffness and lower numerical specimen ultimate strength. The reason can be understand from the interactions between fiber tows in different orientations. In axial tensile test, matrix tensile failure controlled by F 2T occurs on bias tows as the first damage of specimen. A higher value of F 2T delays the matrix tensile failure on the bias tows and transfers more load onto axial tows, which carry majority of the external loads. Such transferred load creates shear stresses that contribute to an earlier axial tensile failure on axial tows, causing lower specimen strength. This interaction effect does not affect transverse tensile specimen, since here majority of the loads are carried by bias tows itself, and both axial and bias tows are under matrix tensile failure mode. The higher of F 2T, the less nonlinearity and the higher specimen strength is for the effective specimen response. 97

112 5.2.5 Fiber Tow Transverse Compressive Strength 1) axial tension 2) axial compression Figure 5.5 Parametric studies on fiber tow transverse compressive strength 98

113 3) transverse tension 4) transverse compression Figure 5.5 Parametric studies on fiber tow transverse compressive strength (Cont d) Shown in Figure 5.5, varying of fiber tow transverse compressive strength has influence on numerical compression test as well as numerical transverse tension test. The reason is that the compressive matrix failure mode is violated in those simulations, but not on axial tension simulation. The actual compressive strength is affected by 99

114 fiber/matrix interfacial bonding. It could be seen that the increase of compressive strength would delay the associated failure and increase the numerical ultimate strength Fiber Tow Shear Strength 1) axial tension 2) axial compression Figure 5.6 Parametric studies on fiber tow shear strength 100

115 3) transverse tension 4) transverse compression Figure 5.6 Parametric studies on fiber tow shear strength (Cont d) Due to the braiding architectures, the neighbored axial and bias tows tend to rotate from each other under external loadings, in both axial and transverse directions. Such scissoring effect causes shear stresses existing everywhere with the composite materials. 101

116 Since shear stresses contribute to both axial and transverse failure modes, varying of fiber tow shear strength has influence on all simulation curves as shown in Figure 5.6. The influences of fiber tow shear strengths on axial and transverse tensile specimens are similar to those of fiber tow transverse tensile strengths. They can be explained by the interactions of fiber tows and associated failure modes. The differences are varying of fiber tow shear strengths also affects specimen compressive strengths Fiber Tow Fracture Energy In the progressive damage constitutive model of fiber tows, shown in Figure 3.7, fiber tow fracture energies, as parameter group III, are used to govern the softening behavior of the material after failure initiation. It indicates the amounts of energy dissipated during the damage process of elements. The application has been discussed in Chapter 3. In reality, smaller fiber tow fracture energy corresponds to a more fragile composite failure behavior; larger value indicates a more ductile material behavior, a longer energy dissipation process with increasing strain. Varying fiber tow fracture energy parameter has influence on all numerical results as show in Figure 5.7. Larger fiber tow fracture energies lead to higher simulated specimen strengths, since material degradation is slower showing delaying phenomena in the numerical results. In the numerical material model, fiber tow fracture energies are used to calculate the corresponding failure strain for four failure modes. As shown in Table 3.6, fracture energies are assumed to be the same in fiber tow tensile and compressive failure modes. Due to the fact that fiber tow tensile strength is much larger compressive strength (see Table 3.4), the tensile failure strain is much smaller than compressive failure strain. This 102

117 explains why effective stress strain curves of axial compression specimens exhibit a wide plateau of peak stress. 1) axial tension 2) axial compression Figure 5.7 Parametric studies on fiber tow fracture energy 103

118 3) transverse tension 4) transverse compression Figure 5.7 Parametric studies on fiber tow fracture energy (Cont d) 104

119 5.2.8 Viscosity Parameter 1) axial tension 2) axial compression Figure 5.8 Parametric studies on viscosity parameters 105

120 3) transverse tension 4) transverse compression Figure 5.8 Parametric studies on viscosity parameters (Cont d) This parameter is not listed in Table 5.1, since it is a purely for numerical purpose. In the process of computation on softening material models by ABAQUS/Standard implicit solver, the convergence is a big issue that sometimes lead to numerical converging difficulties. Viscous parameters are then introduced to artificially 106

121 improve the rate of convergence. As to material behavior from numerical results, the numerical strength values were influenced by viscous parameters as shown in Figure 5.8. Larger value of viscosity leads to higher numerical strength. Another difference caused by viscous parameters is the post-peak part of macroscopic stress-strain curve. The smaller the value of viscosity is, the stiffer the post-peak descending curve is, and the more it approaches to brittle failure. In order to improve the rate of convergence as well as decrease the influence on the numerical strength, a constant value of the viscous parameter equal to was used in Chapter 4 for model correlation. 5.3 Quantitative Parameter Sensitivity Study This section gives quantitative parameter study. A concept of influence factor is defined here as percentage change of specimen strengths based on 100% change of model parameters. The value of influence factor indicates how sensitive it is on certain test. For example, if doubling one parameter doubles the strength of the material response, then that value would be 1 (or 100%). If doubling the value leads to minimal change, then that value would be approximately 0. This helps understand which parameters affect material response more than others, and would be a good addition to qualitative parameter sensitivity study. In the numerical studies, one pair of data is selected and tabulated in Table 5.2 for comparison. What need to mention here is that material behavior is nonlinear, one data point may be not sufficient. To explain the result, it could be done either by column to see how one parameter affects four tests, or by row to see which parameter affect most for each test. The most or most two affecting parameters are underlined in the table. For example, axial strengths 107

122 are mostly affected by fiber tow tensile strength F 1T and compressive strength F 1C for tension test and compression test respectively; and transverse strengths are affected more by interface strengths. This matches the investigation of failure in Chapter 4. Table 5.2 Influence factor of each parameter on four tests Parameters Tests F 1T F 1C F 2T F 2C F S G F t C G C axial tensile test axial compressive transverse tensile transverse compressive Strain Rate Dependence Investigation In aerospace application, the composite would be loaded at strain rate up to hundreds per second. High strain rate tests on composite structures are not easy to realize, often expensive and unreliable. There are several factors affect the rate-dependence of the composites such as their configuration and loading conditions, but first of all depends on the rate-dependence of each constituent. As for fiber phase in the composite, the rate sensitivity depends on the type of the fiber. Of two most commonly used fibers, carbon fiber is known to be rate insensitive, while the glass fiber is rate sensitive. As for matrix phase, polymers are known to have a strain rate dependent response. As to the polymers discussed in current work, E862, the rate dependence testing results were reported by Littell (Littell 2008). A reliable composite model is an alternative approach to determine rate dependence response of fiber reinforced polymer matrix composite. A lot of sophisticated 108

123 constitutive laws have been used to describe the deformation response of polymer material. Meanwhile, for a laminar point of view, the polymer constitutive equations were incorporated into the composite micromechanical models to predict rate dependent response of homogenized composite materials (Zheng, 2006). The mesomechanical model proposed in this work provides a platform for strain rate effect investigation for textile composite with complicated internal geometries, as three phases of material have been explicitly distinguished and defined Strain Rate Variation in Single Test To investigate the strain rate variation over the specimen span, a preliminary numerical analysis was carried out to investigate the strain rate variation over the span of specimen. Quasi-static loading was simulated by LSDYNA explicit code (LSDYNA, 2007), and element local strains were output and then differentiated to obtain strain rate on selected characteristic elements. For the preliminary analysis, an axial tension test was simulated, loaded by displacement control to level of deformation dl = 0.3 mm in dt = 0.2 seconds. Since the length of specimen is L = mm, the strain rate equal to 0.1/sec can be calculated as below. Engineering strain rate (dl/l)/t = /sec True strain rate = ln (1+dL/L)/t = /sec Then, five elements were selected over the specimen span and examined. Firstly, as shown in Figure 5.9, three different levels of strain rates scattering were observed. 109

124 low medium high Figure 5.9 Strain rate histories over specimen span 0.1/sec Figure 5.10 Strain rate histories and average strain rate in low scattering region This phenomenon could be understood as energy releasing from local damage. It is similar to the acoustic transmission caused by cracking within the specimen. When strain rate scattering is low, see enlarge picture in Figure 5.10, the average strain rates on all selected elements are equal to global specimen strain rate, 0.1/sec (Waving curve is due to numerical differentiation based on finite data points). Medium strain rate scatting region is corresponding to fiber splitting failure model, and large strain rate scattering 110

125 region is corresponding to fiber breakage, which suppose to release more energy. Although these results cannot be directly related to any physical meaning, it is still indicating level of damage numerically. It could be concluded that the strain rate is spatially uniform upon the specimen; and that it is reasonable for previous numerical study to exclude strain rate effect in single case, since the results in the region with large strain rate scattering are for post failure and not adopted for model correlation Strain Rate Effect The numerical study is based on specific braided composite of carbon fiber reinforced polymer matrix composites, T700s/E862. Since carbon fiber is rate insensitive, polymer matrix contribute to strain rate effect of composites. In proposed mesomechanical, the material response of all three phases (pure resin, fiber tows and interface) is resin related. Developing a sophisticated rate dependent material model for each of three phases will be a large amount of work. The numerical study will stick on the material models available in ABAQUS/Standard which have been discussed in Chapter 3, but specific material properties from corresponding strain rate level testing results will be used for separated numerical cases. First of all, resin properties used for elastic perfect plastic material model are obtained from experimental testing results under different strain rate, 10-5, 10-3 and 10-1 (Littell, 2008). Based on Equation (5.1) and (5.2), engineering stress and strain obtained from experiment test could be transfer into true stress and true strain which will be used in ABAQUS/Standard. ε T = ln ( 1 + ε E ) (5.1) 111

126 σ T = σ E ( 1 + ε E ) (5.2) Figure 5.11 Stress strain curves of resin E862 with strain rate 10-5 Figure 5.12 Stress strain curves of resin E862 with strain rate 10-1 Figure 5.11 and Figure 5.12 show curves of engineering stress strain, true stress strain, and elastic perfect plastic model behavior in ABAQUS. The material behavior of elastic perfect plastic model used in ABAQUS is indicated in the dashed curves; the values of Young s modulus and plastic yield stress are also indicated on the figures. With this information, using micromechanics methods discussed in Chapter 3, laminar properties of fiber tows could be computed again based on different rate properties of 112

127 resin. Results are tabulated in Table 5.3 and Table 5.4. These values will be used as input for ABAQUS fiber reinforced composite model. Since the relationship of strain rate dependence between interface and resin cannot be found so far, here, just material properties of matrix and fiber tows phases will be varied in the following numerical study. However, the parameter study in earlier part of this chapter shows the influence of interface parameters on specimen behavior. Table 5.3 Homogenized laminar modulus of T700s/E862 under two strain rates E 1 (GPa) E 2 (GPa) G 12 (GPa) v 12 G 23 (GPa) Rate = Rate = Table 5.4 Homogenized laminar strengthes of T700s/E862 under two strain rates F 1T (MPa) F 1C (MPa) F 2T (MPa) F 2C (MPa) F S (MPa) Rate = Rate = Two comparisons will be made based on numerical simulation results of axial and transverse tension tests. One comparison is between models with different strain rate properties on resin only, and the other one is between models with different strain rate properties on both resin and fiber tow. Two sets of comparison could help us distinguish the effects coming from two phases, fiber and pure resin. 113

128 Figure 5.13 Comparison between axial tension stress strain curves of model with different strain rate properties on resin only Figure 5.14 Comparison between transverse tension stress strain curves of model with different strain rate properties on resin only 114

129 Figure 5.15 Comparison between axial tension stress strain curves of model with different strain rate properties on both resin and fiber tow Figure 5.16 Comparison between transverse tension stress strain curves of model with different strain rate properties on both resin and fiber tow From Figure 5.13 and Figure 5.14, it could be seen that varying properties of pure resin only will not change specimen modulus, which again indicates the fact that most of the load are carrying by fiber tows (axial specimen) and by tows and bonding interface 115

130 (transverse specimen). While, there are a little bit influence on the specimen strength (more on transverse test) are observed. If varying properties of both resin and fiber tow, as shown in Figure 5.15 and Figure 5.16, not only strengths but also stiffness are affected. Together with the first comparison, it could be seen that strain rate dependent resin properties will more affect global behavior of composites by contributing to the properties of fiber tows; and again transverse tests are affected more compared with axial tests on straight sided specimen. In axial tests, fiber tows are taking more than 90% of the load, carbon fiber contribute a lot; while, in transverse tests of straight sided specimen, all fiber tows will be loaded transversely with certain combination of shear, which are matrix dominated. 5.5 Conclusions In this paper, a meso scale finite element model has been proposed for examining the behavior of triaxial braided polymer matrix composites. The architecture of the braided composite unit cell has been explicitly modeled. A transversely isotropic constitutive, damage and failure model has been utilized for the fiber tows. A cohesive element approach has been applied to model the interfaces between fiber tows and between the fiber tows and regions of pure matrix. In such a way, various damage mechanisms that arise within the composite are modeled individually which is necessary to capture the effects of and the interactions between these mechanisms. Based on proposed model, the effective stress-strain response of selected triaxial braided composite systems has been simulated. Also, an extensive parameter study has been carried out. With reasonably good correlations obtained with experimental results, the correlated meso scale finite element model presented in this paper appears to be a 116

131 promising approach for investigating damage mechanisms of triaxial braided composite materials. The effects of each model parameter on the effective stress strain curves have been investigated and discussed, which enhances the strength of the developed model and explore the main feature of the numerical model. 117

132 CHAPTER VI NUMERICAL SIMULATIONS OF SINGLE LAYER SPECIMEN BEHAVIOR OF TRIAXIAL BRAIDED COMPOSITE The work presented in previous chapters is focusing on fabricated composite panels having a thickness of 1/8 inch (31.75 mm) which contains six layers of braided preform through the thickness. In order to fully understanding the complex nature of triaxial braided carbon fiber composites, alternative specimen designs have been or are being investigated. This chapter will apply proposed modeling methodology on simulation of one of these tests, such that the capability of the model will be further verified and material behavior of triaxial braided carbon fiber composites could be more investigated. 6.1 Testing of Single Layer Specimen One type of alternative specimen designs is single layer specimen. The objective of this specimen design is to remove the interactions between the braided preform layers since there is just one layer of braided preform through the thickness, and expose and investigate more localized deformations and damages since constrains in the thickness direction are released. For example, Figure 6.1 shows the out of plane displacement plot on transverse tensile specimen with six layers in the thickness direction (Littell, 2008). The large displacement region in circles indicates edge damages such as delamination, 118

133 and that material in these edge areas do not fully contribute to the overall composite material response. Meanwhile, from modeling point of view, testing one layer of braid will also validate assumptions made in the composite modeling section. Figure 6.1 Out of displacement of six layers specimen under transverse tension Quasi-static tensile tests on 0 0 /±60 0 braided composite single layer specimens of the material system T700s/E862 were performed in NASA Glenn Research Center (Roberts and Kohlman, 2010). Specimens were cut into straight sided with dimensions mm (12 inch) long by 35.8 mm (1.409 inch) wide, which is same as six layer specimen. But there is just one layer of braids in the thickness direction. Again, axial tensile tests were conducted using specimens with the 0 0 fibers aligned along the applied load direction, and transverse tensile tests were conducted using specimens where the 0 0 fibers were perpendicular to the direction of the applied load. 6.2 Simulation of Single Layer Specimen Finite Element Model Based on the proposed unit cell methodology, finite element models were created to simulate the single layer specimen tests, see Figure 6.2 for axial tension specimen and 119

134 Figure 6.3 for transverse tension specimen. The dimensions of one single unit cell are also indicated in the picture (left bottom corner of finite element models), it could be seen that 22 unit cells are utilized to generate axial specimen and about 24 unit cells for transverse specimen. The dimensions of the finite element models for both the axial and transverse test specimens were selected such that the width was the same as in the real specimen, and the length was long enough to capture more local deformations and compare with the full-field surface displacement measurement region by digital image correlation during the tests. The size of generated finite element model takes into account of computational load as well. (1) Dimensions (2) Boundary conditions Figure 6.2 Finite element model and boundary conditions for axial tensile test 120

135 Figure 6.3 Finite element model and boundary conditions for transverse tensile test In the numerical simulation, the load is applied on the top end of the specimen by displacement control, meanwhile the symmetric boundary conditions are applied on the other end to generate a larger span of specimen, and free edge is also simulated explicitly on right and left sides such that are it is similar to actual specimen boundary conditions Material Parameters The material models used for three phases are same as the ones applied in six layer specimen simulation. They have been discussed in section 3.3 in detail. Values of material properties of pure resin phase use the bulk properties again, and interface material phase are kept the same as the correlated values, which is obtained in Chapter 4. Due to the fact that the global fiber volume ratio in single layer specimen is lower than that of six layer specimen, the local fiber volume ratio of each fiber tow was computed to be 75% in this calculation, based on the finite element mesh. Then the 121

136 effective laminar level stiffness and strength values for the composite are calculated again based on micromechanics. Again, Method of Cells (Aboudi, 1991) was used to compute the effective modulus, and, Chamis simplified micromechanics equations (Chamis, 1984) were used to give a baseline of strength values. The computed values of effective modulus and strengths are tabulated in Table 6.1. Table 6.1 Fiber tow stiffness and strength properties in sing layer specimens Modulus: GPa Strength: MPa E 1 E 2 G 12 v 12 G 23 F 1T F 1C F 2T F 2C F S Results Discussion Axial Tension Specimen Results Figure 6.4 Global stress strain curves of in axial tension test The macroscopic stress strain curves of testing (Roberts and Kohlman, 2010) and finite element simulation are shown in Figure 6.4. By comparing the numerical results to 122

137 the experimental data, the linear part of the stress-strain curves before damage initiation matches the experiments very well. While, the nonlinear portions of the predicted stressstrain curve is off the testing results, actually softer than observed in the experimental data. The predicted stress stain curves exhibit nonlinearity along with the development of damage with increasing strain. Simulation results indicate the damage initiated at 0.8% global strain on bias fiber tows by matrix tension failure mode as shown in Figure 6.5 (the label on the left shows the colors on contour plot on the right hand side according to level of damage, the same label will be applied on the following contour plots as well). It is again coming from the nature of material failure models adopted for fiber tow phase, in which the damage of the matrix failure mode in transverse direction will also degrade the stiffness in fiber longitudinal direction. Figure 6.5 Contour of matrix tension damage on axial specimen at strain 0.8% The damage will continue to accumulate until the global stress reaches the peak value. At this moment, the longitudinal tensile stresses on axial fiber tows reach failure strength, cause fiber tension failure shown in Figure 6.6a. As discussed before, in an axial 123

138 tension test the majority of the load is carried by the 0 0 fiber tows, failure of axial tows leads to final failure of the axial specimen. Figure 6.6b shows the damage region on a failed axial specimen. Fiber breakage can be observed, and missing of material fragments indicates large amount of energies have been released. 0 0 fiber tow orientation (a) Fiber tensile failure (b) Final failure on actual specimen Figure 6.6 Contour of fiber tensile failure and actual failed specimen on axial specimen at strain 2.2% The interface debonding failure can be investigated from numerical results. Figure 6.7 shows the contour plot of interface damage at the final failure. It can be seen that there is little damage over the specimen. 124

139 Figure 6.7 Contour of interface debonding on axial specimen at strain 2.2% Transverse Tension Specimen Results Figure 6.8 Global stress strain curves of in transverse tension test From Figure 6.8, the predicted stress strain curve of transverse tensile specimen is close to experimental testing results (Roberts and Kohlman, 2010). The curves shows that 125

140 the initiation of nonlinearity starts early of loading process. Various damage mechanisms get involved in the whole process. Based on the numerical results, the failure process can be investigated. Firstly, damage occurred as early as a global strain is ε=0.4%, which is matrix tensile failure on axial tows which is under transverse tension stress status, shown as Figure 6.9a. Meanwhile, debonding of interface starts at the edges, shows as Figure 6.9b. Along with increasing strain, these types of damage accumulate and introduce the nonlinearity of the global stress strain curve accordingly. The ultimate strength of transverse specimen was reached at a global strain level about 1.3%. At the point of material failure, matrix tension failure is common on all fiber tows, and interface debonding failure is also significant on the free edges, shown in Figure (a) Matrix tension failure (b) Interface debonding Figure 6.9 Contour of damages on transverse specimen at strain 0.4% 126

141 (a) Matrix tension failure (b) Interface debonding Figure 6.10 Contour of damages on transverse specimen at strain 1.2% (a) Fiber tows tension failure (b) Fiber tows compression failure Figure 6.11 Contour of damages (low level) on transverse specimen at strain 1.2% Meanwhile, another two types of damage, shown in Figure 6.11 as the fiber tension failure on bias tows and fiber compression failure on axial tow (the latter is due to Poisson s ratio effect), get involved in and contribute to final failure, although the levels are not significant. 127

142 6.3.3 Local Deformation Investigation Because the global response of the composite is tightly correlated to local damage mechanisms, the understanding of strain variations on the specimen surfaces from optical measurement results can be used to identify the types of damage modes occurring in the composite. a) Optical measurement b) Finite element simulation Figure 6.12 Comparison of surface axial tensile strain between measurement and simulation in an axial tension test at 2% global strain Figure 6.12 shows the local distribution of axial tensile strains in an axial tension test at global strain 2%, obtained from simulation results (Figure 6.12a) and optical measurement (Figure 6.12b), (Roberts and Kohlman, 2010). It showed that the highest axial tensile strain was around 3.0%, which is above the average tensile strain 2.0%. The predicted high strain pattern is similar to the optical measurement. As can be seen in the above figure, the highest local strains track along the directions of the bias fiber tows. 128

143 (a) Optical measurement (b) Finite element simulation Figure 6.13 Comparison of surface shear stain between measurement and simulation in a transverse tension test at 1.2% global strain (a) Optical measurement (b) Finite element simulation Figure 6.14 Comparison of out-of plane displacement between measurement and simulation in a transverse tension test at 1.2% global strain Figure 6.13 shows in-plane shear strain plots in a transverse tension specimen at material failure strain 1.2%. At both right and left free edges, high shear strain regions are observed, they are spaced between positive and negative as indicated by the red and blue. At the same where, it is also observed the out of displacement going up and down, shown in Figure 6.14 (Roberts and Kohlman, 2010). The patterns are repeated periodically along 129

144 with the free edges. The plots compare reasonably well between simulation results and optical measurements. Figure 6.15 Locations of scissoring between two bias tows at free edges The locations of these large deformation regions are indicated on the original FE mesh shown in Figure On the figure, the circle with color in red or blue indicates the specific location of positive and negative out of displacement. The mesomechanical model clearly shows that the locations are where there are only the bias fiber tows at the free edges. When specimen loaded transversely, bias tows in two different orientations tends to rotate and create scissoring. Due to cutting of the specimen, the right and left free edges are coincide with each other, that is the reason when right side has positive shear, the left side in the corresponding height has the negative one, see Figure And this effect is even out within the specimen. Furthermore, when the in-plane shear cannot be balanced, out of plane displacement is induced, see Figure

145 6.3.4 Advanced Numerical Analysis As mostly discussed previously, in a transverse tension specimen, there are damage regions present on the free edges, which may make the measured effective transverse properties not good for design application. Real material behavior in the large structures may have different behaviors since such edge damage is not present. Such an issue can be investigated by proposed mesomechanical model. Figure 6.16 Region of finite element model cut out without edge damage for advanced investigation Based on the previous modeling result, the transverse stress strain curves of materials that are affected by edge damage can be generated. The strategy is shown in Figure The area is cut in the middle of model and the away from the delamination areas. Deformations and forces in the transverse directions on such area can be obtained from numerical results, which are further used to generate stress strain curves. The curves are shown in Figure Testing and simulation results of effective global stress strain curves are shown as well for comparison. 131