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Thermo-Mechanical Characterization of Hybrid Composites Using Finite Element Based Micromechanics

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Title:
Thermo-Mechanical Characterization of Hybrid Composites Using Finite Element Based Micromechanics
Creator:
Banerjee, Sayan
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (108 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
SANKAR,BHAVANI V
Committee Co-Chair:
IFJU,PETER G
Committee Members:
KUMAR,ASHOK V
ROQUE,REYNALDO
Graduation Date:
12/19/2014

Subjects

Subjects / Keywords:
Carbon ( jstor )
Composite materials ( jstor )
Compressive strength ( jstor )
Confidence interval ( jstor )
Glass fibers ( jstor )
Hybrid composites ( jstor )
Material concentration ( jstor )
Micromechanics ( jstor )
Tensile strength ( jstor )
Thermal stress ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
fea -- hybrid-composite -- micromechanics
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, Ph.D.

Notes

Abstract:
Multiphase composites also known as hybrids have major engineering applications where high strength to weight ratio and low cost of fabrication are required. Often times, the effective properties of the composites can be tailored by combining two different fibers. A high modulus fiber might be partially replaced owing to its brittle nature leading to sudden failure. For instance, by using Kevlar-graphite/epoxy systems considerable reduction in cost without significant loss of material properties can be achieved. This document outlines in detail the research performed to date on multiphase composites that includes theoretical prediction of elastic constants and strengths. Advances in computational micromechanics allow us to study hybrid multiphase systems using finite element simulations. A micromechanical analysis of the representative volume element (RVE) of a unidirectional hybrid composite is performed using the finite element method. The fibers are assumed to be circular, of equal size and packed in a hexagonal array. The overall fiber volume fraction is kept constant. However, the relative fiber volume fraction of each fiber is varied. The effects of the volume fraction of the two different fibers used as well as their random relative locations within the RVE are studied. It was observed that the effective elastic constants of the homogenized composite are a function of volume fraction of the fibers and matrix phases only, and shows no variability with random fiber locations for a given fiber volume fraction. Analytical results are obtained for all the elastic constants. It is observed that the Rule of Hybrid Mixtures (RoHM) can accurately predict the longitudinal modulus. However, for the transverse and shear moduli a modification has been proposed to the existing Halpin-Tsai relation for binary composites. The modified Halpin-Tsai relation can accurately predict the transverse and shear moduli. The effective coefficients of thermal expansion (CTE) have been calculated and the variation with the volume fraction of fibers is studied. It was observed that the longitudinal CTE from the micromechanics study can be accurately predicted from the analytical prediction. For the transverse CTE, modification has been proposed to the analytical relation for transverse CTE of binary composite proposed by Schapery. Thermal stresses arising due to mismatch of CTEs between the fibers and matrix has been quantified. It was observed that the peak thermal stress has significant variability due to random fiber locations. Thermal stresses are also higher for the hybrids when compared to binary composites. Uniaxial strengths for all composites were calculated using the Direct Micromechanics Method (DMM) which is similar to first ply failure for laminates. It was observed that longitudinal and compressive strengths (no instability) depend on the longitudinal modulus and the least failure strain member, and show no variability with random fiber locations for a given fiber volume fraction. However, the transverse tensile and compressive strengths are point functions, and depend on the relative location of the fibers inside the RVE. Also, the transverse strengths are lower for hybrids than binary composites due to the extra stress concentration created by the second fiber inclusion. Variation of shear strengths shows different trends for longitudinal and transverse directions. For the longitudinal shear strength, hybridization shows negligible effect on the shear strength indicating a strong dependence on matrix volume fraction. However, for transverse shear strength, the variation is very similar to transverse tensile and compressive strengths. Bi-axial failure envelopes were developed for all the composites and the results compared with envelopes for existing phenomenological theories. Finally, progressive damage behaviors for the composite were studied by considering a ductile damage model for the epoxy matrix. The transverse stress-strain curve was studied to calculate the 0.2% yield strength. The results show that, progressive damage strength significantly improves the transverse strength over the DMM strength. Also, all the composites show nearly the same transverse tensile strength, which shows that transverse strength could be a function on the overall fiber and matrix volume fraction only. There was no variability in transverse strength observed for random fiber locations from the progressive damage study. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: SANKAR,BHAVANI V.
Local:
Co-adviser: IFJU,PETER G.
Statement of Responsibility:
by Sayan Banerjee.

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Source Institution:
UFRGP
Rights Management:
Copyright Banerjee, Sayan. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
974372487 ( OCLC )
Classification:
LD1780 2014 ( lcc )

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THERMO MECHANICAL CHARACTERIZATION OF HYBRID COMPOSI TES USING FINITE ELEMENT BASED MICROMECHANICS By SAYAN BANERJEE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014

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© 2014 Sayan Banerjee

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To Bishnupriya

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4 ACKNOWLEDGMENTS I would like to express my sincere gratitude to Dr. Bhavani V. Sankar, without whose tutelage it would have been impossible for me to complete this g igantic journey. His immense motivation and support kept me focused during the toughest of times for which I will be ever indebted to him . I would like to thank my committee members Dr. Peter Ifju, Dr. Ashok V. Kumar and Dr. Reynaldo Roque for their valuable comments, time and effort towards tailoring my work in the right direction . I would also like to thank my colleagues at the Center for Advanced Composites for their guidance and support whenever I reached out to them. And finally, t his work would not have been possible without th e moral support from my parents who believ ed in me t hroughout . I am grateful to have them in my life. I would also like to take this opportunity to thank all the teachers, I had the privilege to be mentored by, throughout my lifetime who urged me to ask the question l as the motivation for pursuing graduate studies. Last but not the lea st; I would like to thank my wife Bishnupriya , without whose patient presence , I cannot imagine surviving the ups and downs of graduate life. I cannot thank her enough for helping me t hroughout my journey both as a student working towards a doctoral degree but more importantly as a human trying to make sense of it all.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBREVIATIONS ................................ ................................ ........................... 12 ABSTRACT ................................ ................................ ................................ ................... 14 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 17 Hybrid Composites and its Applications ................................ ................................ .. 17 Literature Survey and previous work ................................ ................................ ...... 18 Motivation ................................ ................................ ................................ ............... 21 Objectives of this Dissertation Research ................................ ................................ 22 2 CALCULATION OF EFFECTIVE ELASTIC CONSTANTS ................................ ..... 26 Homogenized Stiffness Properties of a Composite ................................ ................. 26 Model for Hybrid Composite ................................ ................................ ................... 27 Micromechanics for Calculating the Elastic Constants ................................ ........... 28 Finite Element Analysis Methodology ................................ ................................ ..... 29 Results and Discussion ................................ ................................ ........................... 32 Conclusions from the Study of Elastic Constants ................................ ................... 37 3 CALCULATION OF EFFECTIVE THERMAL PROPERTIES ................................ .. 48 Homogenized Thermal Properties of a Composite ................................ ................. 48 Micromechanics for Calculating the Thermal Properties ................................ ......... 49 Thermal Stresses ................................ ................................ ................................ .... 50 Finite Element Model for Thermal Stresses ................................ ............................ 51 Results and Discussions ................................ ................................ ......................... 51 Conclusions from the Study of Thermal Properties ................................ ................. 55 4 FAILURE STRENGTHS OF HYBRID COMPOSITES ................................ ............ 62 Background on Calculation of Effective Strength Properties ................................ ... 62 Direct Micromechanics Method (DMM) ................................ ................................ ... 63 Failure Theories ................................ ................................ ................................ ...... 64 Results for Strength Properties ................................ ................................ ............... 67 Study of Longitudinal Strengths ................................ ................................ .............. 67

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6 Study of Transverse Normal and Shear Strengths ................................ ................. 72 Failure Envelopes and Off axis Strength Curves ................................ .................... 75 Progressive damage modeling ................................ ................................ ............... 79 5 CONCLUSIONS ................................ ................................ ................................ ... 100 Contributions of Dissertation ................................ ................................ ................. 100 Future Work ................................ ................................ ................................ .......... 102 LIST OF REFERENCES ................................ ................................ ............................. 103 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 108

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7 LIST OF TABLES Table page 2 1 Specimen nomenclature for the composites ................................ ....................... 44 2 2 Periodic Boundary Conditions for square unit c ell ................................ .............. 44 2 3 Elastic material properties of the fiber and matrix ................................ ............... 44 2 4 Longitudinal moduli, (GPa) for th e composites ................................ .............. 45 2 5 Statistical variability in and 95% Confidence Intervals of the mean ............... 45 2 6 Transverse moduli, (GPa) for the composites ................................ ........ 45 2 7 Statistical variability in and 95% Confidence Intervals of the mean .............. 45 2 8 for the composites ................................ ... 46 2 9 Longitudinal shear moduli, (GPa) for the composites ............................. 46 2 10 Statistical variability in and 95% Confidence Intervals of the mean .............. 46 2 11 Transverse shear moduli, (GPa) for the composites ................................ .... 46 2 12 Statistical variability in and 95% Confidence Intervals of the mean ............. 47 2 13 Statistical variability in and 95% Confidence Intervals of the mean .............. 47 2 14 Check for transverse isotropy in the 2 3 plane ................................ ................... 47 3 1 Longitudinal CTE, for the composites ................................ ................. 60 3 2 Statistical variability in ................................ ................................ ........ 60 3 3 Effective for the composites ................................ ................................ ....... 60 3 4 Transverse CTE, for the composites ................................ .................. 60 3 5 Statistical variation in and 95% Confidence Intervals of the mean ..... 61 3 6 Statistical variation in and 95% Confidence Intervals of the mean ..... 61

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8 3 7 Mean maximum thermal stress (KPa) in the matrix for ........................... 61 3 8 Statistical variability in the maximum thermal stress (KPa) and 95% Confidence intervals of the mean for ................................ ....................... 61 4 1 Strength properties of fiber materials (MPa) ................................ ....................... 96 4 2 Strength properties of matrix materials (MPa) ................................ .................... 96 4 3 Normal and shear strength for the interfaces (MPa) ................................ ........... 96 4 4 Comparison of longitudinal failure strains for the composites ............................. 96 4 5 Longitudinal tensile strength, (MPa) for the composites with low strength epoxy ................................ ................................ ................................ .................. 96 4 6 Variability in (MPa) for low strength epoxy and 95% Confidence Intervals of the mean ................................ ................................ ................................ ......... 97 4 7 Longitudinal tensile strength, (MPa) for the composites with high strength epoxy ................................ ................................ ................................ .................. 97 4 8 Variability in (MPa) for high strength epoxy ................................ .................. 97 4 9 Longitudinal compressive strength, (MPa) for the composites ....................... 97 4 10 Variability in (MPa) and 95% Confidence Intervals of the mean ................... 98 4 11 Variability in (MPa) and 95% Confidence Intervals of the mean .................... 98 4 12 Variability in (MPa) and 95% Confidence Intervals of the mean .................... 98 4 13 Variability in (MPa) and 95% Confidence Intervals of the mean .................... 98 4 14 Variability in (MPa) and 95% Confidence Intervals of the mean .................... 99 4 15 0.2% offset transverse yield strengths for composites ................................ ........ 99

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9 LIST OF FIGURES Figure page 1 1 Unidirectional intraply hybrid composite ................................ ............................. 24 1 2 Effects of hybridization on longitudinal strength ................................ .................. 24 1 3 glass graphite epoxy system ................................ ................................ ........................ 25 1 4 Variation of axial tensile strength with relative fiber volume fraction of graphite glass epoxy hybrids ................................ ................................ .............. 25 2 1 Cross sectional area of a composite with volume fraction of carbon and glass 18.75% and 6.25% respectively ................................ ................................ ......... 39 2 2 RVE for Hybrid composi te ................................ ................................ .................. 39 2 3 Finite element model of the RVE and mesh of the repetitive block ..................... 40 2 4 Variation of with volume fraction of carbon ................................ ..................... 40 2 5 Variation of with volume fraction of carbon ................................ ................... 41 2 6 Variation of and with volume fraction of carbon and comparison with prediction using rule of hybrid mixtures ................................ .............................. 41 2 7 Vari ation of and with volume fraction of carbon ................................ ........ 42 2 8 Variation of and with volume fraction of carbon ................................ ..... 42 2 9 Variation of with volume fraction of carbon ................................ .................. 43 2 10 Variation of with volume fraction of carbon ................................ ................... 43 3 1 Variation of with volume fraction of carbon ................................ .................... 57 3 2 Variation of with volume fraction of carbon ................................ ................. 57 3 3 Variation of with volume fraction of carbon ................................ .......... 58 3 4 Variation of with volume fraction of carbon (magnified) ....................... 58 3 5 Variation of with volume fraction of carbon ................................ ....... 59

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10 3 6 Variation of mean maximum thermal stress in the matrix with volume fraction of carbon for ................................ ................................ ............................ 59 4 1 Flowchart for the Direct Micromechanics method (DMM) ................................ ... 82 4 2 Schematic of interface normal and shear stress ................................ ................. 82 4 3 Variation of (MPa) with volume fraction of carbon ................................ ........ 83 4 4 Variation of (MPa) with volume fraction of carbon ................................ ......... 83 4 5 Variation of (MPa) with volume fraction of carbon (without interface failure) ................................ ................................ ................................ ................ 84 4 6 Variation of & (MPa) with volume fraction of carbon (without interface failure) ................................ ................................ ................................ ................ 84 4 7 Variation of & (MPa) with volume fraction of carbon (without interface failure) ................................ ................................ ................................ ................ 85 4 8 Variation of (MPa) with volume fraction of carbon ................................ ......... 85 4 9 Variation of (MPa) with volume fraction of carbon ................................ ......... 86 4 10 Variation of (MPa) with volume fraction of carbon ................................ ......... 86 4 11 Failure envelopes for the composites in the plane ................................ .. 87 4 12 Failure envelopes for the composites in the plane ................................ . 87 4 13 Failure envelopes for the composites in the plane ................................ . 88 4 14 Failure envelopes for the composites in the plane ................................ . 88 4 15 Failure envelopes for the composites in the plane ................................ . 89 4 16 Interface effects on the failure envelopes for GFRP in the plane ............ 89 4 17 Interface effects on the failure envelopes for GFRP in the plane ........... 90 4 18 Interface effects on the failure envelopes of CFRP in the plane ............. 90 4 19 Interface effects on failure envelopes of GFRP in the plane .................. 91

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11 4 20 Comparison of DMM failure envelopes with failure theories for GFRP ............... 91 4 21 Comparison of DMM failure envelopes with failure theories for CFRP ............... 92 4 22 Compa rison of DMM failure envelopes with failure theories for hybrid composite (0.3C 0.3G) ................................ ................................ ....................... 92 4 23 Off axis failure curves for GFRP for ................................ ................... 93 4 24 Off axis failure curves for GFRP for ................................ .................. 93 4 25 Off axis failure curves for CFRP for ................................ ................... 94 4 26 Off axis failure curves for CFRP for ................................ .................. 94 4 27 A general ductile damage model with strain hardening before softening ............ 95 4 28 Transverse stress strain progressive damage response for composites ............ 95

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12 LIST OF ABBREVIATIONS Macrostress tensor written as (6x1) vector Stress acting on the th plane in the direction Mac r ostrain tensor written as (6x1) vector Strain acting on the th plane in the direction Effective stiffness matrix of the homogenized composite Effective compliance matrix of the homogenized composite Length of the RVE in the th direction RoHM Rule of Hybrid Mixtures DMM Direct Micromechanics Method Elastic modulus in the th direction Elastic modulus in the th direction for carbon fiber Elastic modulus in the th direction for glass fiber Overall fiber volume fraction Volume fraction of carbon fiber Volume fraction of glass fiber th direction for applied strain in the th direction Shear modulus resisting shear in the plane Halpin Tsai parameter CTE Coefficient of thermal expansion Effective CTE in the th direction

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13 Effective CTE in the th direction for carbon fiber Effective CTE in the th direction for glass fiber Temperature applied to the RVE Reference temperature at which the RVE is free of any stresses and strains Difference between applied temperature and reference temperature Tensile strength in the longitudinal direction (fiber direction) Compressive strength in the longitudinal direction Tensile and compressive strength in the transverse 2 dire ction Tensile and compressive strength in the transverse 3 direction Shear strength in the longitudinal plane (1 2 or 1 3 plane) Shear strength in the transverse plane (2 3 plane)

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14 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THERMO MECHANICAL CHARACTERIZATION OF HYBRID COMPOSITES USING FINITE ELEMENT BASED MICROMECHANICS By Sayan Banerjee December 2014 Chair: Bhavani V. Sankar Major: Mechanical Engineering Multiphase composites also known as hybrids have major engineering applications where high strength to weight rat io and low cost of fabrication are required. Often times, the effective properties of the composites can be tailored by combining two different fibers. A high modulus fiber might be partially replaced owing to its brittle nature leading to sudden failure. For instance, by using Kevlar graphite/epoxy systems considerable reduction in cost without significant loss of material properties can be achieved. This document outlines in detail the research performed to date on multiphase composites that includes theo retical prediction of elastic constants and strengths. A dvances in computational micromechanics allow us to study hybrid multiphase systems using finite element simulations. A micromechanical analysis of the representative volume element (RVE) of a unidir ectional hybrid composite is performed using the finite element method. The fi bers are assumed to be circular, of equal size and packed in a hexagonal array. The overall fiber volume fraction is kept constant. However, the relative fiber volume fraction of each fiber is varied. The effects of the volume fraction of the two different fibers used as well as their r andom relative locations within the RVE are studied. It was observed that

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15 the effective elastic constants of the homogenized composite are a functi on of volume fraction of the fibers and matrix phases only, and shows no variability with random fiber locations for a given fiber volume fraction. Analytical results are obtained for all the elastic constants. It is observed that the Rule of Hybrid Mixtures (RoHM) can accurately predict the longitudinal modulus. However, for the transverse and shear moduli a modification has been proposed to the existing Halpin Tsai relation for binary composites. The modified Halpin Tsai relation can accurately predict the transverse and shear moduli. The effective coefficients of thermal expansion (CTE) have been calculated and the variation with the volume fraction of fibers is studied. It was observed that the longitudinal CTE from the micromechani cs study can be accurately predicted from the analytical prediction. For the transverse CTE, modification has been proposed to the Thermal stresses arising due to mismatch of CTEs between the fibe rs and matrix has been quantified . It was observed that the peak thermal stress has significant variability due to random fiber locations. Thermal stresses are also higher for the hybrids when compared to binary composites. Uniaxial strengths for all compo sites were calculated using the Direct Micromechanics Method (DMM) which is similar to first ply failure for laminates. It was observed that longitudinal and compressive strength s (no instability) depend on the longitudinal modulus and the least failure st rain member, and show no variability with random fiber locations for a given fiber volume fraction. However, the transverse tensile and compressive strengths are point functions, and depend on the relative location of

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16 the fibers inside the RVE. Also, the t ransverse strengths are lower for hybrids than binary composites due to the extra stress concentration created by the second fiber inclusion. Variation of shear strengths shows different trends for longitudinal and transverse directions. For the longitudin al shear strength, hybridization shows negligible effect on the shear strength indicating a strong dependence on matrix volume fraction. However, for transverse shear strength, the variation is very similar to transverse tensile and compressive strengths. Bi axial failure envelopes were developed for all the composites and the results compared with envelopes for existing phenomenological theories. Finally, progressive damage behaviors for the composite were studied by considering a ductile damage model for the epoxy matrix. The transverse stress strain curve was studied to calculate the 0.2% yield strength. The results show that, progressive damage strength significantly improves the transverse strength over the DMM strength. Also, all the composites show ne arly the same transverse tensile strength, which shows that transverse strength could be a function on the overall fiber and matrix volume fraction only. There was no variability in transverse strength observed for random fiber locations from the progressi ve damage study.

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17 CHAPTER 1 INTRODUCTION Hybrid C omposites and its A pplications Multiphase composites, also known as hybrids are increasingly finding use in many industrial applicati ons [1] . Hybrid composite materials have major engineering applications where strength to weight ratio, low cost and ease of fabrication are required. By mixing two or more types of fiber in a resin to form a hybrid composite it may be possible to create a material which has the co mbined advantage of individual fibers and simultaneously mitigating the less desirable qualities. There are situations where a high modulus material is required but the catastrophic brittle failure associated with it will be a maj or challenge [2] . For exam ple, Kevlar graphite hybrid composites overcome the key drawbacks of graphite/epoxy composites which are high costs and brittle failure due to low toughness [3] . The potential aspects of Kevlar 49 graphite/epoxy system were initially studied by Zweben et a l. [4]. It has also been found that considerable reduction in costs without loss of mechanical properties can be achieved by using Kevlar graphite hybrid [5] . In another study, tensile and flexural properties of cementious composites reinforced with whiskers of alumina and carbon was studied and significant strength increase was reported [6] . Laminated hybrid ceramic composites find applications in cutting tools, ballistic armor and structural components subjected to high temperatures [7] . In recent years, use of renewable natural fibers as reinforcements in composite materials is fast growing. Studies on plastics and cements reinforced with natural fibers such as jute, sisal, coir, pineapple leaf, banana, sun hemp, straw, broom and wood fibers have b een accounted [8 14] . Sisal fiber is a fiber reinforcement for use in

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18 composites due to its low cost, low density, high specific strength and modulus, easy availability in some countries a nd renewability. More recently , there has been an increasing interes t in finding new applications for sisal fiber reinforced composites which were traditional ly used for making ropes, mats and carpets [15] . In automotive industries, fiberglass composites have been increasingly used to replace steel, but the adoption rate o f carbon fibers still remains low owing to high cost. To further reduce vehicle weight, without excessive cost increase, hybrids consisting of both carbon a nd glass fibers can be used [16] . At present, hybrid reinforcements (carbon, glass) are also used fo r manufacturing of wind turbine blades, and reinforcements of epoxy matrices with basalt and other fibers are also reported [17, 18] . Hybrid composites also find potential marine applications, due to its advantages over carbon steel, such as higher strengt h weight ratio, and ability to be formed into c omplex shapes such as hulls [19] . Development of hybrid structures in civil engineering has made rapid progress in several countries, with the first all hybrid bridge constru ction in Okinawa, Japan in 2001 [20 ] . Literature S urvey and p revious w ork The significance of hybrid composites as an up and coming class of structural material has been discussed by Chamis and Lark, in which it was pointed out that there is a need for methodologies that can accurately pr edict the various properties of hybrid composites from the corresponding properties of the constituents [21]. Chamis and Sinclair developed approximate equations based on the rule of mixtures to predict the physical, thermal, hygral and mechanical properti es of unidirectional intraply hybrid composites (UIHC). The shear and flexural properties predicted agreed well with experimental data. The equation used to predict the properties of UIHC were of the form

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19 (1 1 ) where, is the property of interest, denotes the volume fraction of the reinforcements, HC denote hybrid composite property while PC and SC refer to primary and secondary composite properties respectively The model of the hybrid composite used is shown in Fig ure 1 1 and the effects of volume fraction on the longitudinal strength is shown in Fig ure 1 2. Partial hybrid response refers to the lower bounds for strength, possibl y caused as a result of poor quality fabrication [22] . Similar bounds were reported by Chou and Kelly for elastic constan ts and strength as depicted in Fig ure 1 3 and Figure 1 4. It was also pointed out that there is a synergistic effect obtained from experiments are different from th e one p redicted from the bounds [23, 24] . There ha d been a lot of work regarding experimental testing of intraply hybrid laminates for mechanical property characterization. Mechanical properties of carbon/glass fiber reinforced composites were studied by Sonparote and Lakkad [25] , Stevanovic and Stecenko [26], Cao et al. [27] , Fu, Hu et al. [28] , You, Park et al . [29] among others. It was observed that elastic constants show a uniform variation as the volume fraction of the reinforcements was varied, and it can be accurately predicted by rule of mixtures formulations as shown in Eq. 1 1 . Sonparote et al. observed that the tensile and compressive strength exhibit a hybrid effects, which is negative for tensile strength and positive for compressive strength [25] . Stevanovic et al . found the tensile strength can be predicted from the rule of mixtures relation whereas a fai lure strain enhancement (positive hybrid effect) is observed in sandwich coupons with high glass

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20 to carbon fiber ratio [26] . A positive hybrid effect in the failure strain was also observed by Fu, Hu et al . for hybrid composites of polypropylene reinforced with short glass and short carbon fibers [28] . Ultimate strains of hybrid rods composed of 37% glass fiber and 23% of carbon fiber was seen to have increased by 3 33% of the non hybrid carbon FRP rod, indicati ng a positive hybrid effect [29] . Studies on the impact properties of hybrid composites have been reported by various investigators. Wang et al . [30] investigated the fracture behavior of interlaminar hybrid composites. Addition of glass fiber laminas in graphite fiber composites was found to increas e the impact resistance of the material. Both the load tolerated and the impact energies included as the percentage of glass fibers increased [30]. An improvement in impact strength by a factor of about three to five, for addition of glass fibers to carbon /epoxy and boron/epoxy composites was reported by Novak and DeCrescente [31] . Chamis et al. observed that hybrid composites failed under impact by combined fracture modes, namely fiber breakage, fiber pullout and interplay delamination. Due to this complex failure process, the impact resistance of hybrids might increase over the prediction from the behavior of individual constituents [32] . Saka and Harding [33] conducted tensile impact studies on woven hybrid composites. They concluded that tensile strength was higher at impact strain rate than quasi static strain rate and also tensile strength of woven carbon/glass hybrid composites was more than carbon or gl ass composites. Naik et al . [34] reported lower notch sensitivity of glass carbon/epoxy hybrids comp ared to only carbon or only glass composites. Furthermore, carbon outside/glass inside clustered hybrid configuration resulted in higher post impact compressive strength and also lower notch sensitivity.

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21 Hybridization with appropriate amount of carbon fib ers can enhance resistance to environmental fatigue degradation of glass fiber reinforced composites. Glass carbon hybrid specimens were studied to show a better retention in fatigue life in water than that of all glass composite specimens, up to 10 7 cycle s [35] . Studies on unidirectional Carbon Kevlar epoxy hybrids [36] and on unidirectional carbon glass hybrid [37] composites, it was observed that fatigue strengths of hybrid composites can be higher than that predicted from the rule of mixtures, which is the weighted average of the fatigue strengths of the individual constituents. Motivation Functionally graded materials (FGM) are becoming an important area in development of advanced composite structures. The purpose of a FGM is to optimize a composite s tructure and its response. One of the simple and easy technological solutions involving a FGM may be based on a piece wise distribution of the properties of the composite [38] . On a more general sense, a FGM is a material which has properties that vary gradually with location within the material. Elasticity solution for a through the thickness has been obtained by Sankar [39] . Recent theoretical and experimental work has established that gradients in mechanical properties within the material, if properly tailored, may offer resistance to contact deformation and damage that cannot be re alized by hom ogeneous materials [40 42] . Furthermore, Sankar and Tzeng studied thermal stresses in FG beams and observed reduction in thermal stresses when variation of thermoelastic constants are opposite to that of the temperature distribution through the beam thickn ess [43] . Apetre et al . studied low velocity impact response of sandwich beams with FG core and concluded that FG cores

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22 can effectively prevent impact damage in sandwich composites [44] . With the above benefits that FGM has to offer, it was intended to st udy intraply hybrid composites, which if used in a laminate can represent functional gradation of material properties. In the next section, some preliminary results on hybrid composites will be discussed. O bjectives of this Dissertation R esearch One of the primary objectives of this research is to d evelop a computational model of a multiphase hybrid composite and to study the effective thermo mechanical properties using finite element methods . The effect of varying the relative volume fraction of the fib ers by keeping the overall fiber volume fraction constant, on the stiffness properties will be studied. Furthermore, it is also of interest to see the effect of the relative random fiber location inside the RVE for a given fiber volume fraction on the stif fness properties. Effective coefficients of thermal expansion (CTE) of the composites and its variation with fiber volume fraction will also be studied. Thermal stresses are a very significant property that is often neglected in phenomenological failure th eories. This is because the fiber and matrix phases are generally treated as a homogenized orthotropic material, and it is assumed that there are no thermal stresses if the material is free to expand and contract. However, the mismatch of CTEs between the fiber and matrix phases result in micro thermal stresses which is proportional to the applied temperature differential and may result in matrix microcracking. Hence, it is one of the objectives of this research to quantify the variability in thermal stress es due to hybridization. Strength properties will be calculated next using Direct Micromechanics Method (DMM), which will be explained in detail later. Similar to stiffness, the variation of the strength with varying volume fraction of the reinforcements w ill be studied. The

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23 effect of the fiber location for a given volume fraction on the strength p roperties will also be studied. The DMM method can be considered as a virtual laboratory using which effect of multiaxial loading on the RVE can be studied. Failu re envelopes thus developed were compared with envelopes from existing failure theories. The final objective of the research is to study the progressive da mage behavior of the composites. A ductile damage model is used to model the matrix softening behavio r and the transverse stress strain responses of the composites are studied.

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24 Figure 1 1 . Unidirectional intraply hybrid composite Figure 1 2 . Effects of hybridization on longitudinal strength

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25 Figure 1 glass graphite epoxy system Figure 1 4. Variation of axial tensile strength with relative fiber volume fraction of graphite glass epoxy hybrids

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26 CHAPTER 2 CA LCULATION OF EFFECTIVE E LASTIC CONSTANTS Homogenized S tiffness P roperties of a Co mposite Experimental techniques can be employed to understand the effects of various fibers, their volume fractions and matrix properties in hybrid composites. However, these experiments require fabrication of various composites which are time consuming and cost prohibitive. Also, it is very difficult to precisely control the relative volume fraction of the different reinforcement phases while manufacturing the hybrid specimen. Advances in computational micromechanics allow us to study the various hybrid systems by using finite element simulations which can be modified easily to study effect of hybridization and it is the goal of this research. The mechanical properties of hybri d short fiber composites can be evaluated using the rule of hybrid mixtures (RoHM) equation, which is wid ely used to predict the longitudinal m odulus of hybrid composites [45] . The computational model presented in this section takes into account, random fiber location inside a representative volume element for a given volume fraction ratio of fibers, in this case, carbon and glass. Ten different random fiber locations were considered for every given volume fraction of the fiber s. Six unit macrostrains are applied to the representative volume element (RVE) and macrostress is calculated by volume averaging the microstresses. The elastic constants are evaluated by inverting the effective stiffness matrix of the homogenized composit e. Since, elastic constants of a composite are volume averaged over the constituent microphases, the overall stiffness for a given fiber volume fraction is not affected much by the variability in fiber location. It is shown however, that RoHM works best fo r longitudinal modulus of the hybrid composites. For the transverse stiffness and shear moduli, a semi empirical

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27 relation called Modified H alpin Tsai equations has been derived. It was shown that by varying a non dimensional parameter, in the existing Halpin Tsai relation for binary composites and by including the corresponding terms for both reinforcements, the prediction can be very accurate for the transverse modulus as well as the shear m oduli. The results for the elastic cons tants are compared with those for the binary composites , namely carbon epoxy (CFRP) and glass epoxy (GFRP) composite . Model for H ybrid C omposite The fiber orientation depends on processing conditions and may vary from random in plane and partially alig n e d to approximately uniaxial [21] . The fiber packing arrangement, for most composites, is random in nature, so that the properties are approximately same in any direction perpendicular to the fiber (i.e. properties along the 2 direction and 3 direction are same, and does not vary with rotations about the 1 axis), resulting in transverse isotropy [46] . For our model, it is assumed that the fibers are arranged in a hexagonal pattern and the epoxy matrix fills up the remaining space in the representative volume element (RVE). Hexagonal pattern was selected because it can more accurately represent transverse isotropy as compared to for example a square arrangement. The RVE consists of 50 fibers. Multiple fibers were considered to allow randomization of fiber loca tion. Hybrid composites are created by varying the number of fibers of carbon and glass to obtain hybrid composites of different relative volume fractions. However, the overall fiber volume fraction was maintained at 60%. A cross section of a hybrid composite of polypropylene reinforced with short glass and carbon fibers is shown in Fig ure 2 1 [45] . The black circles represent glass fibers (6.25%) and the white circles represent carbon fibers (18.75%). In order to

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28 represent such an arrangement, we hav e consider ed the schematic of the RVE as shown in Fig ure 2 2 . Green and red represent two different fiber materials, while the matrix is the remaining space in the RVE . Also, it is assumed that the radii of the fibers are the same and only the count of car bon and glass fibers vary. This gives us much more flexibility in creating the finite element mesh. Although, this RVE architecture is a lot simplistic and entails some basic assumptions like same size and location of the fibers and absence of voids but th ere is still a lot to l earn from the parameters that have been used. The properties of the composite are independent of the 1 direction, hence a 2D analysis is performed , which saves significant computational time . We have assumed here that the fibers rem ain unidirectional with no fiber undulation and waviness. An overall fiber volume fraction of 60% is assumed for all the composites analyzed in this research . The proportions of the reinforcements have been varied to obtain five hybrid composites, keeping the total volume fraction of reinforcement phases constant. The volume fraction of any p articular reinforcement can be determined by the relation (2 1 ) where, is the number of fibers of i th reinforcement and is the total number of fibers in the RVE . Specimen numbers H1. . .H5 have been assigned depending upon the relative volume fraction of the reinforcements. The specimens are listed in Table 2 1. Micromechanics for Calculating the E lastic C onstants The RVE of the composite is analyzed using commercially available finite element software (ABAQUS/CAE 6.9 2). The composite is assumed to be under a state

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29 of unifo rm strain at the macroscopic level called macroscale strains or macrostrains, and the corresponding stresses are called macrostresses. However, the microstresses, which are the actual stresses inside the RVE might have a spatial variation. The macrostresse s are average stress required to produce a given state of macro deformations, and they can be computed using finite element method. The macrostresses and macrostrain follow the relation ( 2 2 ) where, is the elastic constant of the homogenized composite, also known as the stiffness matrix. In this method, the RVE is subjected to six independent macrostrains. For e ach applied non zero macrostrain, it is also subjected to periodic boundary conditions (PBC) such that all other macrostrains are zero. PBCs ate a ppropriate constraints applied to the RVE that depend on the loading condition and have been determined by sym metry and periodicity conditions by S ankar, Marrey [47] and Sun and Vaidya [48] . This is done to maintain periodicity of the deformed RVE such that it can be a representative of the overall composite structur al deformation. The six cases are: Case 1: = 1; Case 2: = 1; Case 3: = 1; Case 4: = 1; Case 5: = 1; Case 6: = 1 [49] , where the subscripts 1, 2, 3 are parallel to the material principal directions, as shown in Fig 2 3, and the superscript M stands f or macrostress or macrostrain. A comprehensive list of all the PBCs for the six strain cases is presented in Table 2 2. Fin ite E lement A nalysis M ethodology For case 1, 2 and 4, a mixture of three and four node plane strain elements, CPE3/CPE4 and for case 3, a mixture of three and four node generalized plane strain

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30 elements, CPEG3/CPEG4 were used an out of plane degree of freedom . For cases 5 and 6 (longitudinal shear), three and four node shell elements were used, because out of plane displacements have to be applied for this case. Periodic boundary conditions (PBC) were applied on opposite faces of the RVE using multi point constraints in ABAQUS . For each strain case, six microstresses were calculated, three normal and three shear stresses in the 1 2 3 directions, in each element in the finite element model and volume averaged to find the macrost ress for the RVE. The finite element model used is shown in Fig ure 2 3, which contains 27,000 elements. The matrix can be inverted to obtain the compliance matrix or matrix, from which the elastic constants can be computed using the following relation for an orthotropic material in the principal material directions (2 3 ) In the above finite element model oppos ite faces of the RVE must have corresponding nodes for the periodic boundary conditions to be enforced using multi point constraints . For cases 4 and 5, we need to calculate the volume average of the

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31 transverse shear stress. However, none of the 2D plane s train models in ABAQUS ® has transverse shear stress as a standard ou t put. In order to apply out of plain pure shear deformation, 3 node and 4 node shell elements (S3/S4) were selected. The material properties for the various constituents are listed in Tabl e 2 3 [50] . Although, care has been taken to use practical material property values that closely match those of comm er cial carbon and E glass fibers , it must be kept in mind that this is a parametric study to compare the effect of hybridization on the overall material properties. In such a case, the exact values for the material constants are of lesser importance. For a composite to have transversely isotropic behavior in the 2 3 plane, it has to follow the relation ( 2 4 ) A s was observed , all the composites including the hybrid composites studied here closely follow transverse isotropic behavior. One reason for this is the hexagonal packing of the fiber, which represents better isotropy in the 2 3 plane. As for the hybrid composites, 10 s amples of each volume fraction ratio were considered, with the fiber locations randomly selected for each sample. The mean and standard deviation of the results were studied. A 95% confidence interval of the population mean is also calculated to predict th e randomness in elastic constants. Rule of hybrid mixtures (RoHM) was used to predict the longitudinal modulus and for all the composites. For the transverse modulus , and the shear moduli and , the Halpin Tsai equation was modified to predict the results obtained from finite element method. As will be

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32 shown later the modified Halpin Tsai equations agree with reasonable accuracy for the hybrid composites. For all the elastic constants, samples were generated with random fiber locations and results obtained for all the samples were studied to e valuate the effect of hybridization. R esults and D iscussion The longitudinal modulus was calculated for the composites by varying the relative volume fraction of the reinforcements keeping the overall fiber volume fraction constant . It was observed that varies linearly with the volume fraction of the constituents . is plotted in Figure 2 5 versus the volume fraction of carbon varyi ng from 0 to 0.6 as we move from left to right , with the two extreme ends representing carbon epoxy (CFRP) and glass epoxy (GFRP) composite . The RoHM is stated as (2 5) where , and refer to the modu lus o f carbon, glass and matrix respectively, and , and refer to the volume fraction of carbon, glass and matrix respectively. The mean values for the longitudinal modulus for all the composites tested along with the binary composites are presented in Table 2 4. Th e solid line in the figure is obtained from using the appropriate properties in Eq. ( 2 5 ) . As can be seen from the Table 2 4, RoHM can accurately predict the longitudinal modulus for a three phase system. In order to study the variability in FEA results, 95% Confidence Intervals (C.I.) were calculated using the mean and standard devi ation of the data. The following expression is used to calculate the confidence intervals. (2 6)

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33 In Eq. (2 6), and refer to the sample mean and standard deviation respectively. refers to the number of samples , so for the present study , and is the confidence percentage. So for a 95% C.I., . is the value t distribution. For the given and , . The confi dence intervals represent an interva l of values that will contain the mean 95% of the time if the experiment is repeated infinitely many times. It is evident from Eq. (2 6) that a lower will result in a narrower C.I. Thus, from the standard deviation and confidence intervals shown in Table 2 5, it can be concluded that the longitudinal modulus, does not have significant variabil ity with random fiber locations for a given fiber volume fraction. The transverse modulus , however, cannot be predicted accuratel y using RoHM type equations . A general method to estimate involves the use of semi empirical equations such as the Halpin Tsai equation that are curve fitted to match experimental results. The Halpin Tsai equation for single fiber composite as mentioned in [46] is (2 7 ) W here In the equations above, is a curve fitting parameter, which is dependent on the fiber packing arrangement. For the hybrid composites, we propose a modifica tion to Eq.

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34 (2 7 ) , which incorpora tes the volume fractions of both the rei nforcements as follows [51 ] : (2 8 ) where, and optimum value of was determined using a least square error procedure. It was found that yielded the best results for including single fiber composites. is calculated from both the finite element analysis and modified Halpi n Tsai equation and presented in Table 2 6 . It is observed that Eq. (2 8 ) can be used to predict t he transverse modulus of a three phase system. The variation of with increasing volume fraction of carbon is shown in Fig ure 2 6 and the solid line is prediction using Eq. (2 7). The transverse modulus in the 3 direction, is not discussed separately, since for all the composites studied, it was observed that . This shows that all the composites were transversely isotropic in the 2 3 plane. The variability and confidence intervals for are shown in Table 2 7. It can be observed that there is negligible variation in for random fiber locations for a given fiber volume fraction. The major P o isson s ratio and were computed for all composites and similar to the transverse moduli, were found to be almost equal . I t was observed that the

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35 ratios h ad a n approximately linear variation when volume fraction of carbon was gradually increased, as seen in Fig ure 2 7. can be stated as [5 1 ] (2 9) where , , matrix respectively. A similar expression holds true for predicting . The solid line in the Figu re 2 7 is obtai ned from Eq. (2 9 ). It was o bserved that RoHM results differ from the FEA results, the percentage di fferences are shown in Table 2 8 . It should be noted that for carbon was equal to that of glass. Hence, as long as the overall volume fraction of fiber remains constants, RoHM cannot predict any interaction effect. However, when th e was plotted alone, it clearly showed a linearly increasing trend as volume fraction of carbon increased, as shown in Figure 2 8. A approach similar to the transverse modulus was considered for predicting the s hear moduli, , and . The modified Halpin Tsai relation for predicting the shear moduli is as shown below [51 ] : (2 10 ) where, and

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36 In the above equation refers to composite shear modulus ( , or ) depending upon the modulus studied . For each case, the corresponding fiber shear moduli have to be considered in calculating the parameter . The optimal value of was found out to be 1.01 for and , and 0.9 for . The corresponding plots for variation of the three shear moduli with volume fraction of carbon are shown in Fig ure 2 8 and Figure 2 9 . The difference between the FEA results and the one predicted from Eq. (2 10) are presented in Table 2 9 and Table 2 11 . It was observed that Modified Halpin Tsai relation could predict the shear moduli with very high degree of accuracy. The variability and confidence intervals for and are shown in Table 2 10 and Table 2 12 respectively. It was observed that like all the other elastic constants so far, the shear moduli showed negligible variability with r andom fiber location for a given volume fraction. The final elastic moduli, the transverse was calculated from micromechanical model and variation with changes in volume fraction is studied. It should be noted that, a n analytical expression for is not required, since for t ransverse isotropic composites can be calculated from and . Since, we have an ana lytical expression for predicting and , we can predict once we have the other two material properties using Eq. (2 4) . The variation of with volume f raction of carbon is as shown in Fig ure 2 10 . It was observed that was the only elastic constant which showed some variability with random location of fiber for a given volume fraction. This is most likely due to the dependence of a transverse (2 3 plane) property upon stress concentration effects created by two differen t fibers in the RVE. Since,

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37 stress concentration is a point property and varies along with the fiber location, the follows a similar pattern too. However, the variation in is very little, as evident from the 95% confidence intervals for the mean shown in Table 2 13 , and for most practical purposes it can be safely neglected. Conclusions f rom the S tudy of E lastic C onstants A micromechanical study of the effective elastic constants of a multiphase composite containing two different fibers in a matrix material wa s carried out using the finite element method. The fibers are assumed to be circular and packed in a hexagonal arran gement in order to represent transverse isotropy in the 2 3 plane. The RVE is subjected to six unit macrostrains one at a time keeping the remaining macrostrains equal to zero. This is ensured by applying periodic boundary conditions to the opposite faces of the RVE using multi point constraints in ABAQUS. The elastic constants are obtained by populating and inverting the effective stiffness or matrix. It was observed that the longitudinal modulus varied linearly with the volume f raction of the reinforcements. Rule of hybrid mixtures can be used to predict the longitudinal modulus with a very high degree of accuracy. The transverse moduli, and were found to be identically equal owing to transverse isotropy in the 2 3 plane. However, the variation in transverse moduli cannot be predicted with existing formulations. A modified Halpin Tsai relation is proposed which is a semi empirical relation an d a modification to the existing Halpin Tsai relation for binary composites. With suitable changes to a non dimensional parameter, , the transverse moduli can be accurately predicted to match the results obtained from the micromec calculated from micromechanics were found to be identically equal, just like the

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38 transverse moduli. However, RoHM type equations were not able to accurately predict the variation of Pois shear moduli, and has been computed from micromechanical study. Modified Halpin Tsai equations has been proposed to predict the variation in shear moduli with for and for . It was shown that the Modified Halpin Tsai relations with the above parameters could predict the shear moduli with was calculated from micromechanics. It was observe d that had some variability with random fiber locations inside the RVE for a given fiber volume fraction. However, it was concluded that the variation was very small and for all practical purposes could be neglected. Thus, all th e elastic moduli computed in this section had no dependence on fiber location as long as the volume fraction of the fiber was kept constant. This can be explained if we note that the elastic constants were calculated by volume averaging the microstresses. Hence, any spatial variation of the microstresses resulting from the change in the fiber location does not significantly affect the overall stiffness.

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39 Figure 2 1. Cross sectional area of a composite with volume fraction of carbon and glass 18.75% and 6.25% respectively Figure 2 2. RVE for Hybrid composite. Fibers of two different reinforcements have different colors

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40 Figure 2 3. Finite element model of the RVE and mesh of the repetitive block Figure 2 4 . Variation of with volume fraction of carbon

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41 Fig ure 2 5 . Variation of with volume fraction of carbon Fig ure 2 6 . Variation of and with volume fraction of carbon and comparison with prediction using rule of hybrid mixtures

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42 Fig ure 2 7. Variation of and with volume fraction of carbon Fig ure 2 8. Variation of and with volume fraction of carbon

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43 Figure 2 9. Variation of with volume fraction of carbon Figure 2 10. Variation of with volume fraction of ca rbon

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44 Table 2 1. Speci men nomenclature for the composites Table 2 2. Periodic Boundary Conditions for square unit cell. , and the coordinate system is shown in Fig ure 2 2 Table 2 3. Elastic material properties of the fiber and matrix Property E glass fiber Carbon fiber (IM7) Epoxy (GPa) 72.4 263 3.5 (GPa) 72.4 19 3.5 (GPa) 30.2 27.6 1.29 (GPa) 30.2 7.04 1.29 0.2 0.2 0.35 0.2 0.35 0.35 5 0.54 41.4 5 10.1 41.4 Specimen Carbon epoxy ( CFRP ) 0.6 0 0.6 H1 0.54 0.06 0.6 H2 0.42 0.18 0.6 H3 0.3 0.3 0.6 H4 0.18 0.42 0.6 H5 0.06 0.54 0.6 Glass epoxy ( GFRP ) 0 0.6 0.6 Case Constraint between Left and Right faces Constraint between Top and Bottom faces Out of Plane Strains (for all nodes) (for all nodes)

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45 Table 2 4 . Longitudinal moduli, e hybrids specified in Table 2 1. Table 2 5. Statistical variability in and 95% Confidence Intervals of the mean Table 2 6 . Transverse moduli, (GPa) for the composites 8.77 Table 2 7 . Statistical variability in and 95% Confidence Intervals of the mean Specimen Mean (GPa) Standard deviation (GPa ) 95% C.I. of mean H1 147.52 3.67x10 4 (147.5226, 147.5231) H2 124.69 3.57 x10 4 (124.6918, 124.6923) H3 101.86 4.73 x10 4 (101.8623, 101.8630) H4 79.03 4.00 x10 4 (79.0322, 79.0328) H5 56.20 5.94 x10 5 (56.2038, 56.2039) Specimen Mean (GPa) Standard deviation (GPa ) 95% C.I. of mean H1 9.05 0.0016 (9.0504, 9.0526) H2 9.66 0.0086 (9.6654, 9.6678) H3 10.32 0.0124 (10.3119, 10.3297) H4 11.05 0.0119 (11.0367, 11.0538) H5 11.82 0.0041 (11.8147, 11.8206)

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46 Table 2 8 . for the composites 0.2534 Table 2 9 . Longitudinal shear moduli, (GPa) for the composites 4.41 Table 2 10 . Statistical variability in and 95% Confidence Intervals of the mean Table 2 11 . Transverse shear moduli, (GPa) for the composites 3.04 Specimen Mean (GPa) Standard deviation (GP a ) 95% C.I. of mean H1 4.4144 0.0007 ( 4.4139, 4.4150 ) H2 4.4243 0.0026 ( 4.4225, 4.4262 ) H3 4.4371 0.0036 ( 4.4345, 4.4397 ) H4 4.4483 0.0037 ( 4.4456, 4.4509 ) H5 4.4594 0.0048 ( 4.4559, 4.4628 )

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47 Table 2 12 . Statistical variability in and 95% Confidence Intervals of the mean Table 2 13 . Statistical variability in and 95% Confidence Intervals of the mean Table 2 14 . Check for transverse isotropy in the 2 3 plane Specimen (FEA) % Diff CFRP 3.04 3.05 0.30 Hybrid Composites H1 3.14 3.15 0.32 H2 3.36 3.37 0.30 H3 3.60 3.62 0.55 H4 3.88 3.90 0.51 H5 4.19 4.20 0.24 GFRP 4.35 4.37 0.37 Specimen Mean (GPa) Standard deviation (GPa ) 95% C.I. of mean H1 3.1447 0.0012 (3.1439, 3.1456) H2 3.3636 0.0051 (3.3600, 3.3673) H3 3.6128 0.0054 ( 3.6089, 3.6167 ) H4 3.8868 0.0066 ( 3.8821, 3.8915 ) H5 4.1919 0.0021 ( 4.1904, 4.1934 ) Specimen Mean (GPa) Standard deviation (GPa ) 95% C.I. of mean H1 0.4358 2.5 x10 4 (0.4356, 0.4360) H2 0.4297 13 x10 4 (0.4288, 0.4307) H3 0.4240 11 x10 4 (0.4232, 0.4248) H4 0.4166 14 x10 4 ( 0.4156, 0.4177 ) H5 0.4062 3.1 x10 4 ( 0.4059, 0.4064 )

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48 CHAPTER 3 CALCULATION OF EFFECTIVE THERMAL PROPERTIES Homogenized T hermal P roperties of a C omposite In this section the effective coefficients of thermal expansion (CTE) has been evaluated for all the composites. Since unidirectional fiber reinforced composites are typically orthotropic in nature, it has 3 different CTEs in the 1 2 3 directions, where 1, 2, 3 refer to the principal material coordinate system and 1 refer to the fiber direction. However , due to transverse isotropy, two out of the three C TEs need to be evaluated and it is postulated and proven from hexagonal arrangement of fibers that . A 3D computational model of the RVE was used to calculate the effective CTEs , since the macrostrains had to be made equal to zero in all directions . In a computational model the exact boundary conditions required to evaluate the effective CTEs can be implemented using multi point constraints in ABAQUS ® . Knowledge of the effective CTE of a composite material is very important, since thermal stresses can be very significant resulting in failure o f the composite structure. T hermal stresses arising from the mismatch of CTEs between the fiber and matrix materials is directly proportional to the temperature difference and can be of signifi cant proportions for higher temperature differences. E xisting phenomenological failure theories take into account effective macro scopic proper ties, hence estimating the effect of thermal stress es at the microstructural level is not possible. However, in the present analysis since the microstresses are calculated at every element, this effect can be studied in detail. It wil l be shown that thermal stresses are higher for hybrids as compared to bina ry composites and also show variability with random fiber locations for a given volume fraction. Although the microstresses showed some variability, the variation of CTEs for

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49 random fiber location was almost nonexistent . This behavior is very similar to th at of elastic constants, as for calculating effective CTEs, microstresses were volume averaged and any spatial variation of microstress would have no effect on the homogen ized properties. Micromechanics for Calculating the Thermal Properties A 3D model of the RVE was used to study the thermal properties. The constitutive relation shown in Eq. 2 2 has to be modified to incorporate thermal effects. The revised constitutive equation used for the study of thermal properties is shown in Eq. 3 1. (3 1) It should be noted that the subscripts 1 2 3 refer to the principal material directions for an orthotropic material. In the principal material planes, thermal strains only cause normal stresses and vice versa. Eq. (3 1) can be re written in short hand notation as follows (3 2) where , and refer to (6x1) matrices representing macrostress, macrostrain and effective CTEs respectively. If in Eq. (3 2), the boundary conditions on the RVE are implemented such that , the n the equation reduces to (3 3)

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50 Eq. (3 3) can be rearranged to obtain an explicit relation for the effective CTE s of a composite material as (3 4) Thus effective can be calculated by subjecting the RVE to a temperature difference, and . The microstresses from the analysis can be volume averaged to obtain the macrostress es , . Next, we wi ll discuss the methodology to calculate the t hermal stresses. Thermal Stresses Composite materials are typically operated at temperatures different from their stress free reference temperatures e.g. cryogenic tanks. When a composite structure is subjected to a temperature gradient and allowed to expand freely, the macrostress should be equal to zero. However, due to mismatch of CTEs between the fiber and matrix materials, the composite will have microstresses at the constituent level. These microstresses ar e not accounted for in available phenomenological criteria such as Tsai Wu, since the composite is typically modeled as a homogeneous orthotropic material. However, work by Leong and Sankar [ 52 ] show that thermal stresses can be significant for a binary co mposite and depending on the temperature load, lower the longitudinal tensile strength significantly. Micromechanics based models such as the current one allows us to study these thermal effects in detail. Thermal stresses can be calculated by following th ese two steps. When the composite undergoes free thermal expansion at a temperature , it has a strain given by . The microstresses are due to this strain and also due to . Thus two

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51 analyses are performed on the RVE and the microstresses from the two analyses are superposed to obtain the thermal stresses. Although for an isotropic material, the thermal stress should be zero, but for a composite due to mismatch of CTEs there will be significant stress depending on . Finite Element Model for Thermal Stress es 3D elements were used to model the unit cell, for CFRP and GFRP and the R VE for the hybrid composites. Six node wedge (C3D6) and 8 node (C4 D8) solid brick elements were used. A unit cell for GFRP and CFRP and a RVE model is consi dered for the hybrid composites. The boundary conditions used for calculating the effective thermal CTEs we re d iscussed in the previous chapter . Results and Discussio ns Carbon, glass fiber and epoxy matrix material properties used for the study of CTEs are shown in Table 2 3 [52] . Longitudinal CTE, was calculated for all the composites and the results for the hybrids were compared with those for binary composites. The results are compared with volume fractions of carbon, . The variation is shown in Figure 3 1 . It was observ ed that has a smooth non linear variation with . An analytical formulation for predicting of a binary composite is available in Gibson (Gibson, 20 11 ). (3 5) where, and represent CTEs of the fiber and matrix respectively. Eq. (3 5) can be modified, analogous to RoHM for elastic moduli, to incorporate multiple fiber reinforcements. The modified relation is shown below

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52 (3 6) where, and represent longitudinal CTEs for carbon and glass fibers respectively. The solid line in Figure 3 1 is based on Eq . (3 6). As seen from Figure 3 1 and Table 3 1 , that Eq. (3 6) provides a very accurate pr ediction of longitudinal CTE , . An important observation can be made from Eq. (3 6). The denominator in Eq. (3 6) is identical to Eq. (2 5 ) i.e. it repr esents the longitudinal modulus, of the composite. Thus, Eq. (3 6) can be rearranged in the following form (3 7) From Eq. (3 7) it can be observed that can be a homogenized property of the composite. The variation of vs is shown in Figure 3 2 . It is evident that has a linear variation with the volume fraction of carbon. can be predicted using the appropriate properties in Eq. (3 7). The values of are shown in Table 3 3 . is obtained by volume averaging the microstresses caused by a unit change in . Hence, the spatial variation of the thermal microstresses does not affect the effective longitudinal CTE. A 95% confidence interval of the mean values for is computed to show the variability in due to the random fiber locations for a given fiber volume fraction. Table 3 2 shows the results for confidence intervals. Transverse CTE , was calculated for all the composites in the present study and was compared to those for binary composites, CFRP and GFRP. It was noted that similar to elastic moduli, CTEs also followed transverse isotropy in the 2 3 plane. This was evident from the micromechanics analysis which showed for all the

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53 composites. B ecause of this property results are shown for only, with the assumption that the subscripts 2 and 3 can be used interchangeably. The variation of with volume fraction of carbon, is shown in Figure 3 3 . The analytical relation for predicting of binary composites can be found in Gibson [46]. The relationship for transverse CTE was derived by Schapery and is shown below . (3 8) Eq. (3 8) is modified to incorporate the effects of multiple orthotropic fiber reinforcements to represent a hybrid composite. The modified formula is shown below (3 9) where, and represent the transverse CTEs of carbon and glass fibers respectively and and are the effective longitudinal homog eneous composite. The s olid line in Figure 3 3 is obtained by using the appropriate material properties in Eq. (3 9). It was observed that Eq. (3 9) slightly over predicts the transverse CTE, . Figure 3 4 shows the over prediction in detail. Table 3 4 shows the comparison between the micromechanics based finite element results and obtained from Eq. (3 9). Eq. (3 9) can be rearranged to obtain the following equation shown below (3 10) It can be seen from Eq. (3 10) that can be modeled as a homogenized property of the hybrid composites . The property also follows a linear variation with the volume fractio n of carbon as show n in Figure 3 5 . However,

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54 similar to , Eq. (3 10) slightly over predicts the property when compared with the finite element analysis results. Th e comparison is shown in Figure 3 5 . The % difference between the two is shown in Table 3 5 . Depending on the application this difference can be neglected or studied in more detail. Thermal stresses were calculated at the centroid of every matrix element in the RVE. Since, the matrix is an isotropic material, Von M ises stress was calculated from the (3x3) stress tensor in the principal material coordinate system. It was assumed that matrix having the lowest tensile strength is the most susceptible to failure. Although, the interface could still be a weaker zon e, it was considered perfect for this analysis. The intention was to estimate the effect of hybridization on the maximum thermal stress in the matrix. The results were plotted with volume fraction of carbon, . It should be noted t hat thermal stresses are proportional to the applied temperature difference, . The results shown in Figure 3 6 are for . It was observed that hybridiz ation results in an increase in the maximum thermal st ress es . The highest thermal stress was observed in hybrid composite H3, which had equal proportions of carbon and glass fibers. This was due to the maximum possible mismatch of thermal coefficients between the three phases inside the composite RVE. The max imum thermal stress occurs near the fiber matrix interface , where the effect of mismatch is the strongest. There was some variability in the maximum thermal stress for a given fiber volume fraction as seen in Figure 3 6. In order to quantify the variabilit y, 95% confidence interva ls of the mean thermal stress has been calculated and presented along with the mean and the standard deviation in Table 3 8 .

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55 Conclusions from the Study of Thermal Properties In this section, the effective coefficients of thermal e xpansion (CTE) of all the five different hybrid composites were studied and the results compared with binary composites, namely CFRP and GFRP. It was observed that the longitudinal CTE, has a smooth non linear variation with the volume fraction of carbon. Available analytical relationship for binary composites has been modified to incorporate the effects of multiple reinforcements. It was shown that the relation can accurately predict the longitudinal CTE. Although, does not have a linear variation, is a homogenized property of the composite which was shown to vary linearly with volume fraction of carbon. Transverse CTE, was also calculated and its var iation with predicting for a binary composite could be modified for hybrids. The relationship can predict the transverse CTE, however with lesser accuracy than the prediction for . The comparison of the analytical and micromechanics based finite elemen t results were studied. Similar to for longitudinal CTE, is a homogenized composite property and can be predicted by using the corresponding properties for carbon and glass fibers. It was shown that has a linear variation with volume fraction of the reinforcements. Thermal stresses were calculated at the centroid of matrix elements for a unit change in temperature . Variation of the von misses equivale nt of the maximum thermal stresses was studied with the volume fraction of carbon. It was observed that the maximum thermal stress increases with hybridization and is maximum when equal proportion of carbon and glass were used. There is a variation in ther mal

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56 stresses due for random fiber locations for a given volume fraction but the variability was found to be maximum when equal proportion of carbon and glass fibers were used. The variability in thermal properties were quantified by computing 95% confidenc e intervals of the mean value for a given hybridization ratio.

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57 Figure 3 1. Variation of with volume fraction of carbon Figure 3 2. Variation of with volume fraction of carbon

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58 Figure 3 3. Variation of with volume fraction of carbon Figure 3 4. Variation of with volume fraction of carbon (magnified)

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59 Figure 3 5. Variation of with volume fraction of carbon Figure 3 6. Variation of mean maximum thermal stress in the matrix with volume fraction of carbon for . The red dots are FEA results. The scatter is due to variability resulting from random fiber locations

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60 Table 3 1. Longitudinal CTE, for the composites 0.12 Table 3 2. Statistical variability in Table 3 3. Effective for the composites 19.1 184.4 Table 3 4. Transverse CTE, for the composites 26.4 Specimen Mean Standard deviation H1 0.0712 2.5x 10 6 H2 0.5697 7.9 x10 6 H3 1.2916 1.0 x10 5 H4 2.4305 1.2 x10 5 H5 4.4945 1.7 x10 5

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61 Table 3 5. Statistical variation in and 95% Confidence Intervals of the mean Table 3 6. Statistical variation in and 95% Confidence Intervals of the mean Table 3 7. M ean maximum thermal stress ( K Pa) in the matrix for 22 5 Table 3 8. Statistical variability in the maximum thermal stress (KPa) and 95% Confidence intervals of the mean for Specimen Mean Standard deviation 95% CI of mean H1 26.1010 0.0226 (26.0849, 26.1172) H2 25.2605 0.0588 (25.2184, 25.3026) H3 24.4229 0.0929 (24.3564, 24.4894) H4 23.5316 0.0500 (23.4959, 23.5673) H5 22.3809 0.0259 (22.3624, 22.3994) Specimen Mean Standard deviation 95% CI of mean H1 0.0321 0.1520 ( 0.1428, 0.0767) H2 0.0250 0.2348 ( 0.1430, 0.1930) H3 0.1206 0.2359 ( 0.2893, 0.0481 ) H4 0.0821 0.2762 ( 0.1154, 0.2797 ) H5 0.0995 0.1942 ( 0.0394, 0.2384 ) Specimen Mean Standard deviation 95% CI of mean H1 257.7 4.4031 ( 254.55, 260.85 ) H2 263.8 4.7709 (260.39, 267.21 ) H3 271.9 6.6792 ( 26 7.12 , 27 6.68 ) H4 267.3 5.0664 ( 263. 68 , 270. 92 ) H5 254.5 4.7687 ( 251.09, 257.91 )

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62 CHAPTER 4 FAILURE STRENGTHS OF HYBRID COMPOSITES Background on C alculation of E ffective S trength P roperties In this section, the effec tive strength properties of hybrid composites will be studied using finite element based micromechanics. The effect of multiple fibers on the strength will be compared to those for binary composites. Direct Micromechanics Method (DMM) first proposed by Sankar is used to predict the failure strength s for an orthotropic composite m aterial . It has been used in several articles to analyze and evaluate phenomenological failure criteria Marrey and Sankar [53] , Zhu et al. [54], Stamblewski et al. [55], Karkakainen and Sankar [56] . Typ ical material properties of carbon and glass fibers as well as polymer based epoxy matrix are considered. However, as before the material properties are used as a parameter to study and compare the effective strengths of hybrids. As such the exact material properties are of lesser importance. The DMM approach typically calculates the first element strength which is very similar to first ply strength in composite laminates. Although conservative, this method gives the designer a very good idea of failure/damage initiation. The composite can typically carry more load s beyond the first element failure. Such a scenario will be considered in the progressi ve damage analysis discussed later in this chapter. The concentration of the current chapter is to characterize the unidirectional strengths of the composite by var ying the volume fraction and location of reinforcements for a given volume fraction. Multi axial loading which is very challenging to apply in an experiment has also been simulated through the DMM method and failure envelopes thus g enerated has be en compar ed with existing phenomenological failure criteri a namely maximum principal stress, Tsai hill and Tsai Wu criterion . The strength

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63 properties are compared with binary composites to evaluate the effect of hybridization on the effective strength. Direct Micromechanics Method (DMM) In this method failure is predicted from the micromechanical failure analysis, which inspects very element in the RVE for failure, also known as the Direct Micromechanics Method. A flowchart explaining the algorithm for this met hod is shown in Figure 4 1 [55] . A 3D state of macrostress es represented by is applied to the RVE. The resulting macrostrain due to the macrostress can be obtained from the effective stiffness matrix ob tained in Chapter 2. The relation can be stated as shown . (4 1) Both and are (6x1) vector representation of the stress and strain tensor. The microstresses due to the macrostrain can be obtained by superimposin g the microstresses due to the six unit macrostrain cases discussed in Chapter 2. T he relation to obtain the microstresses for a given macrostrain can be mathematically represented as shown (4 2) where is the microstress in element e , and the matrix represents the microstresses in element e , for the six unit macrostrains cases . For example, the first column contains the six microstresses in element e caused by a unit macrostrain . It should be noted that in the p resent analysis thermal stresses are not taken into account. After the six microstresses is obtained at the centroid of every element in the RVE , appropriate phenomenological failure theories are use d to calculate the failure

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64 macrostress . T his depends on the element type i.e. fiber or matrix element and its material properties. The material prop erties of the fibers and matrix are shown in Table 4 1 and Table 4 2 [50, 55]. The different failure theories used for carbon and glass fiber and epoxy matrix will be discussed in detail next. Failure Theories C omposite failure can be characterized as fiber failure, matrix failu re or interface failure . On many occasions the failure initiates as one of these three types and as damage progresses it changes the type . In the DMM method existing phenomenological failure criteria is considered for the fiber and matrix phases. Since car bon fiber is criterion for transversely isotropic composites is considered for carbon with slight modifications [57] . The quadratic failure criteri a can be summarized into four cases as discussed below: Tensile Fiber Mode (4 3) Compressive Fiber Mode (4 4) Tensile Matrix Mode (4 5) Compressive Matrix Mode

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65 (4 6) In the equations above the superscript refer to microstress es . Also note that the subscript refers to the material property for carbon fiber which is listed in T able 4 1. The equations (4 3) through (4 6) although same for the most part, have been slightly modified when compared to those proposed by Hashin [57] . It was considered that shear stre ngth in longitudinal and transverse directions, generally referred to as and respectively, was same for carbon fiber. Depending on the sign of the microstresses, Eq. (4 3) and (4 4) or Eq. (4 5) and (4 6) were used to determine carbon fiber failure. Maximum principal stress theory was used for determining failure for both glass fiber and epoxy matrix. Since, bot h these materials are isotropic , the three principal stresses were calculated in each element and then compared to the respective material strengths. Tensile and compressive strength of epoxy matrix are typically different, with the strength in compression being higher. This can be accounted for by considering tensile strength for positive principa l stress and compressive strength for negative principal stresses. The relations used for this failure criterion are shown below . when (4 7) when , ; glass or epoxy refer to the 3 principal stresses and , refer to the tensile and compressive strength of either glass or epoxy material , referred by the subscript .

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66 Interface failure is also considered for the carbon epoxy and glass epoxy interfaces. The uniaxial strengths of the composite ca n be obtained by applying the corresponding macrostress , in conjunction with interface failure consideration . A quadratic failure criterion is used to study interface failure. The direction cosines of the interface are used to transform the stress at the c entroid of the corresponding element to normal and sh ear tractions on the interface as shown in Figure 4 2. The following relations were used for obtaining the normal and shear tractions on the interface. (4 8) i.e. (4 9) (4 10) (4 11) where In Eq. (4 8), the stress matrix refers to the average of the microstresses at t he centroid of the adjoining fiber and matrix interface element. The vector contains the direction cosines of the normal to the interface surface. The normal stress is obtained from the scalar ( dot ) product of the traction, and , as shown in Eq. (4 10). Once t he normal and shear stresses are obtained from Eq. (4 11), the appropriate failure theory can be applied. It is considered that the interface is not affected by compressive stresses. The relationship ex plaining t he interface failure criteria is shown below . (4 12 )

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67 where In Eq. (4 12 ), and refer to the normal stress and resultant shear stress on the interface. and refer to the interface normal and shear strengths. These two strengths are very difficult to measure in an actual experiment, and the limited data that is available, is very sensitive to the actual fiber and matrix material properties. Some representative interfa ce strengths used for this study are shown in Table 4 3 [55] . Results for Strength Properties The results for strengths of the composites will be subdivided into two major sections namely uniaxial strengths and failure envelopes for multiaxial strengths . The strength results will be further subdivided into strength considering interface failure and without consideration of interface effects. As mentioned before, a parametric study has been performed for the longitudinal strength with the matrix tensile strength as the variable parameter. The matrix tensile strength is a very significant property because it has a profound effect on the transverse tensile and compressive strength, as well as the longitudinal tensile strength . The interface strength also has an effect on the trans verse tensile strength. Next we will look into the results for longitudinal tensile and compressive strengths. Study of Longitudinal Strengths The DMM method is used to study the longitudinal strengths of the composites. The effect of two dif ferent matrix tensile strengths, low strength (LS) and high strength (HS) on the effective longitudinal strength is studied. The variation of longitudinal strength, with the volume fraction of carbon, for the two different matrix types is

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68 shown in Figure 4 3 . It can be seen that the longitudinal strength irrespective of the matrix strength, varies linearly with the volume fraction of fiber. Simple micromechanical models for longitudinal strengths exist which assume a uniform state of strain in the fiber and matrix (i so strain assumption) . With that assumption, the longitudinal strength can be estimated from the rule of mixtures type equations. ( longitudinal tensile strength ) de pend on the lowest strain member in the composite. The failure strain of any member, fibers or matrix, can be obtained from the following relation. (4 13) In Eq. (4 13), and refer to the failure strain and strength to failure for the th member in the corresponding loading direction , whereas the loading direction. Depending on the two epoxy stren gths selected, the lowest failure strain member will vary from epoxy matrix to glass fiber. The analytical formula for longitudinal strength will depend on the lowest strain to failure member. For the low strength (LS) matrix, the lowest strain member is t he matrix which controls the failure . The longitudinal strength for such a case can be predicted using the following relation [46] . (4 14) In Eq. (4 14), and refers to the stress in the carbon and glass fiber respectively at the matrix failure strai n, while represents the matrix tensile strength. With the iso strain assumption and the assumption that all the members follow a linear e lastic stress strain behavior till failure, Eq. (4 14) can be modified as follows.

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69 (4 15) where, is the matrix failure strain in the longitudinal direction. Eq. (4 15) can be further simplified as follows. (4 16) where, is the effective longitudinal modulus of the composite. The results obtain ed from Eq. (4 16) were compared with those obtained from finite element based micromechanics. The c omparison is presented in Table 4 5. It can be seen that Eq. (4 16) can predict the longitudi nal strength fairly accurately. Also, it can be observed from Eq. (4 16) that since is constant for the composite, the longitudinal tensile strength is a proportional to the elastic modulus, of the composite, which also has a linear variation with volume fraction of fiber as seen before in Chapter 2. The slight inaccuracy in the results can be attributed to the calculated failure strain, . The estimated and actual failure strain s a re presented in Table 4 4. T he slight inaccuracy in the failure strain is due to stress concentration effect of the fibers in the matrix. Due to this, the actual failure strain of the matrix is lower than that estimated using Eq. (4 13). The variability in longitudinal tensile streng th was also observed by varying the location of carbon and glass fibers for a given fiber volume fraction. The variability in is shown in Table 4 6. There is some variability because of local stress concentration effects in the m atrix due to varying fiber locations. However, the scatter is very minimal in the longitudinal direction for random fiber locations. The 95% confidence intervals of the mean for are also shown in the same table.

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70 The longitudinal tensile strength for the high strength ( HS) epoxy is shown in Figure 4 3 . For this case, the lowest strain to fa ilure member is the glass fiber. The analytical expression for predicting longitudinal strength is still given by Eq. (4 16). obtained from micromechanical analysis is compared with those obtained from analytical formulation. The results are presented in Table 4 7. It can be observed that for HS epoxy, Eq. (4 16) can predict the longitudinal strength with hi gher accuracy than for LS epoxy. This is because for the former case, the strain to failure, can be calculated more accurately as seen from Table 4 4. The longitudinal strength varies linearly for HS epoxy but shows a deviation f rom the linear behavior for CFRP. This can be observed in Figure 4 2 from the green curve for . This is because failure strain of carbon fiber is slightly higher compared to glass resulting in higher strength for CFRP composite. T he variability in for HS epoxy is presented in Table 4 8. Since, for HS epoxy failure is controlled by fiber, the random location of fibers for a given volume fraction does not result in any scatter for longitudinal tensile stren gth. The standard deviation in is thus negligibly small. The longitudinal compressive strengths, was also computed for all the composites using the DMM method. The results were plotted with volume fraction of carbon, and shown in Figure 4 4 . It should be noted that although linear elastic behavior is assumed until failure, fiber failure in longitudinal compression is often a more complicated phenomenon . In the compressive direction, failure is often characterized by microbuckling of fibers in shear or extensional mode, transv erse tensile rupture due to Poisson strains or shear failure of fibers without buckling [46] .

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71 However, for the present study linea r elastic behavior until failure will be considered as the other failure modes are beyond the scope of the current research. Analytically, the longitudinal compressive strength can be predicted using the following relation which is analogous to Eq. (4 16). (4 17) where, is the lowest compressive strain in the composite and is the effective elastic modulus in the longitudinal direction for the composite. From Eq. (4 17), it can be seen that the longitudinal compressive strength should vary linearly as it is governed by . The comparison between calculated from micromechanics and using Eq. (4 17) is shown in Table 4 9. It can be seen that the agreement between the two is excellent. This is because the lowest failure strain member for all composites except GFRP is carbon fiber. Hence, the calculated failure strain is very accurate as the actual failure strain. A small branching from the linear behavior can be observed in Figure 4 3 for GFRP composite ( ). This is because the compressive failure strain of glass fiber is higher than carbon, and thus GFRP shows a higher strength. The variability in is not shown since, like for HS epoxy, failure is fiber controlled and scatter in strength due to random fiber locations for a given fiber volume fraction should be negligibly small. Interface failure and its effects on the longitudinal tensile and compressive strengths were also studied for all the composites. For loading in the lo ngitudinal direction, the interface is mainly subjected to shear traction. However, no interface failure was observed for longitudinal tensile or compressive loading. The shear

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72 strengths of the CFRP and GFRP interfaces were high enough to resist any interf ace shear failure. As a result the effects of the interface on the longitudinal tensile and compressive strengths were negligible. Study of Transverse Normal and Shear Strengths Transverse strengths of the composite were calculated for loading in two dire ctions perpendicular to the fiber longitudinal axis, also referred to as the 2 direct ion and 3 direction. Transverse tensile and compressive strengths are primarily controlled by matrix tensile and compressive strength. The first element failure strength i n the transverse tensile 2 direction is plotted with volume fraction of carbon, as shown in Figure 4 5 . The red squares represent transverse tensile strength in the 2 dirction, for all the composites and the blue curve connects the mean strength values. It was observed that the transverse tensile strength decreases when a second inclusion or fiber reinforcement is added. This behavior can be attributed to the stress concentratio n caused in the vicinity of the second inclusion. Since, strength is a local phenomenon this stress concentration effect can readily lower the first element failure strength. Moreover, the drop in strength was more when glass fiber was added to CFRP as opp osed to when carbon fiber was added to GFRP. The former case is represented in the drop in from 41 MPa for CFRP to 34 MPa for H1 . This significant reduction in strength is because of the high transverse modulus, of glass fiber. The high transverse mod ulus of glass results in stress concentration in the neighboring region surrounding glass fiber thus lowering strength. The FEA results show initial damage near the glass fiber conf irming this theory . It was further observed that there was significant variability in for a given volume fraction of carbon. The

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73 mean, standard deviation and 95% confidence intervals for transverse tensile strengths are presented in Table 4 10. Once again, this variability in transverse strength is because of the failure initiation localization near the glass fibers. It was observed that variability in transverse strengths is maximum for H1 hybrid composite. This is because H1 has the lowest volume fraction of glass among the hybrid composites . This small proportion of glass fibers allow maximum possibility for randomization in fiber location and thus result in a large scatter in transverse strengths. This can be noted from the sta ndard deviation results in Table 4 10. Also, the variability among hybrids is least for H5, which has the maximum reinforcement of glass and very little carbon. A n interesting property observed while conducting the transverse strength study is the lack o f transverse isotropy in the 2 and 3 directions. Previously in the calculation of effective elastic properties in Chapter 2, we concluded that all the composites demonstrated transverse isotropic properties. The transverse tensile strength for the composit es are compared for the 2 and 3 directions by plotting them against volume fraction of carbon, . The results are shown in Figure 4 6 . It can be seen that transverse tensile strength in the 3 direction, is lower than that in the 2 direction. This is mostly because of the lack of symmetry in the RVE dimensions in the 2 and 3 directions. However, for the most part the variation in transverse tensile strength as well as the variability due to random fiber l ocations for a given fiber volume fraction is mostly same for the 2 and 3 directions. The variability and 95% confidence intervals for transverse tensile strength in the 3 direction are shown in Table 4 11. The transverse compressive strengths in the 2 an d 3 directions are also plotted as a volume fraction of carbon. T he result is shown in Figure 4 7 . The transverse

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74 compressive strength is higher than the tensile strength for all composites. This is due to the superior strength of the matrix in compression . However, the compressive strength in the transverse direction is lower for hybrids than CFRP and GFRP. Also the drop in strength when glass fiber is added to CFRP is significantly more than when carbon fiber is added to GFRP. The variability is maximum f or the hybrid H1, which has the minimum volume fraction of glass fiber among the hybrid composites. The two longitudinal shear strengths , and , and the transverse shear strength , are studied. The variation of longitudinal shear strength , , with volume fraction of carbon, is shown in Figure 4 8 . It was seen that the shear strength was mostly uniform for all the composites. Although there were some variability in for a given fiber volume fraction, it was very small and was found to be maximum for the hybrid H3, with equ al proportion of glass and carbon fibers. The scatter in and 95% confidence intervals of the mean has been presented in Table 4 12. A very similar behavior was observed for longitudinal shear strength, . The plot of the variation of with volume fraction of carbon, is shown in Figure 4 9 . The variability and 95% confidence intervals for are shown in Table 4 13. The lack of tra nsverse isotropy in longitudinal shear strengths can be seen from the results. Similar to transverse tensile strength in the 3 direction, the shear strength is lower than . Variation in transverse shear strength with volume fraction of carbon, is presented in Figure 4 10 . Unlike the aforementioned longitudinal shear strengths, the variation in is very similar to the transverse tensile and compressive strengths. This

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75 behavior can be explained due to the fact that is a transverse property and thus follow s a similar behavior like the other transverse strengths. The trans verse shear strength is lower for the hybrids when compared to CFRP and GFRP. The variability in and the 95% confidence intervals are presented in Table 4 14. The variability is maximum for hybrid H1 which has and . Failure Envelopes and Off axis S trength Curves Multi axial failure e nvelopes for the composites were studied by applying a b i axial macrostress to the RVE. The DMM method can then be used to compute the microstress es in the RVE by superimposing the microstresses generated due to unit macrostrains. Appropriate failure theories are applied depending on the element type to compute the failure b i axial macrostress. Repeati ng this process for various bi axial loading cas es the failure envelopes can be generated . These failure envelopes developed using the DMM method was then compared with those from existing phenomenological criteria. In order to have a legible plot area, data from only one type of hybrid composite is use d to compare the failure envelopes. The failure envelopes in the plane for CFRP, GFRP and H3 hybrid are shown in Figure 4 11 . The subscripts 1,2 refer to the principal material directions of the composite material. As observed before, the longitudinal tensile and compressive strengths for the hybrid are weighted average of those for CFRP and GFRP. However, for the transverse tensile and compressive direction, the hybrid strengths are lower than either of the binary composites. T he failure envelopes for the plane for each composi te type are shown in Figure 4 12 . It can be observed that the transverse strength in the 2 direction is slightly higher than that in the 3 direction for all the composites. Also, the transverse

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76 compressive strength is higher than transverse tensile strength for the 2 and 3 directions for all the composites. The failure envelopes for the and planes are shown in Figure 4 13 and Figure 4 14 respectively. The envelopes are symmetric about the horizontal axis indicating that the shear strength is constant irrespective of the sign of applied shear stress. Also, it can be observed from the figures that the shear strengths, and are nearly same for all the composites. However, as observed from the results of shear strengths before, longitudinal shear strength, is slightly higher than . The failure envelopes for biaxial loading in the plane are shown in Figure 4 15 . Since the longitudinal shear strengths are nearly the same for all the composites, the failure envelopes look identical. The failure envelopes were also developed for the composites with consideration of interface failure and compared with that without consideration of interface failure. The results for the plane a re shown for GFRP in Figure 4 16, and for CFRP in Figure 4 18 . It was observed that for GFRP composite, there was no effect of the interface on longitudinal tensile and compressive strengths. However, transverse tensile and compressive stresses were heavily dependent on the interface stren gths. With the interface strengths used for this study, the GFRP composite showed lower transverse compressive strength when interface failure was considered. There was no effect of the interface on the transverse tensile strength for GFRP. For the CFRP co mposite on the other hand, interface failure was not encountered for either longitudinal or transverse loading. Hence in the plane, the failure envelopes with and without interface failure were identical.

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77 Effect of interface fai lure on the failure envelopes were also studied for the plane. The results are shown in Figure 4 17 for GFRP and Figure 4 19 for CFRP. Interface shear strength has a significant effect on the failure strength when a longitudinal compressive stress is acting in conjunction with longitudinal shear stress. Also, as seen before, the transverse compressive strength is also lowered when interface failure is considered for GFRP. However, there is no impact of interface failure on the fai lure envelope for CFRP in the plane. This is because of the high normal and shear strength of the interface when compared with the matrix normal and shear strengths. The DMM failure envelopes were compared with failure envelopes generated using existing phenomenological failure theories. The results are compared for a bi axial state of stress in the plane. The comparison for GFRP is shown in Figure 4 20 . The following failure theories were compared with the DMM method. Maximum stress theory (MS) [46] : (4 18) Tsai Wu failure criterion [46] : (4 19) ; ;

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78 Tsai Hill failure criterion [46] : (4 20) The 4 intercept points representing uniaxi al strengths wil l be same for all the envelopes. However, when a biaxial st ate of stress is present, it becomes interesting to compare the strength predicted using DMM and existing failure theories . It was observed that none of the existing failure theories individually agreed with the DMM envelope entirely. However, a combination of maximum stress theory and Tsai Wu theory has the best agreement with the DMM. The comparison of failure envelopes for C FRP and H3 hybrid are shown in Figure 4 21 and Figure 4 22 respectively. The off axis failure strengths of the composites were studied. The applied macrostress is acting at an angle to the fiber direct ion. A schematic diagram of the loading is show in in Figure 4 23. This angle, is varied from 0 to 90 degrees, and the strength obtained from DMM is compared with the Maximum stress (MS), Tsai Hill and Tsai Wu failure theories fo r every . The results for GFRP are shown in Figure 4 23 for and Figure 4 24 for . The strength is normalized with the longitudinal tensile strength of GFRP. It was observed that the DMM gives the most conservative estimate of the strength for all . Strengths calculated using Tsai Wu and Tsai Hill failure theories are very close to each other. However, strength calculated using Tsai Wu failure theory was shown to have the closest agreement with DMM results. The off axis failure strength results for CFRP are shown in Figure 4 25 and Figure 4 26. It was observed that similar to GFRP, DMM gives the most conservative

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79 estimate of strength. However, for CFRP it was observed that for , all the failure theories approximately identical equal off axis strengths as shown in Figure 4 25. For , it was observed that Tsai Wu and Tsai Hill failure theories predicted stren gths that were very close to each other with Tsai Wu results showing the closest agreement with DMM off axis strengths. Progressive damage modeling The DMM gives us a conservative estimate of strength, which is informative but does not represent the ultima te failure strength of the composite. Typically epoxy matrix composites are ductile in nature and show elastic perfectly plastic behavior before softening and failing completely. In order to simulate such a behavior, a duct ile damage model built in ABAQUS ® is used for the epoxy matrix . ABAQUS ® offers a general capability for modeling progressive damage and final failure of the elements for ductile metals. Progressive damage is modeled using a continuum constitutive behavior for the material. It is assumed t hat damage is accumulated in the matrix similar to ductile metals through void nucleation, coalescence and growth. There are three main parts of the damage model. In the first part, the undamaged constitutive behavior of the matrix is specified. This can o nce again be subdivided into two zones, linear elastic and elastic perfectly plastic. The epoxy matrix initially follows a linear stress strain behavior as per the elastic modulus until it reaches the yield stress. Beyond that it is assumed that the epoxy is incapable of absorbing any more stress and follows an elastic perfectly plastic behavior before the damage initiation point. A general representation of damage model is shown in Figure 4 27 [58] . In this figure, hardening behavior is assumed after the i nitial yield stress, . However, for the model used for the present study is assumed

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80 to stay constant until the failure initiation point i.e. . The second part of the model is the damage initiation point and damage initiation criteria. The method selected for the current study is specifying the plastic strain . When the above plastic strain is reached in an element, damage is initiated. In the Figure 4 27, this point is shown on the curve as . is a scalar damage variable which will be initiated at this point. The third part of the damage model is the damage evolution. This part defines the post damage initiation material behavior. In other words, it describes the rate of degradation of material stiffness once the initiation criterion is satisfied. This formulation is based on a scalar damage approach given by the equation below. (4 21) In Eq. (4 21), is the overall damage variable that captures the effect of all active damage mechanisms. refers to the stress for a n applied strain after the failure initiation point, where as refers to the stress due to undamaged response. This is shown by the dotted line in Figure 4 27. When the damage variable, in any element, it is considered that the element has completely failed. The damage evolution law can be specified either in terms of fracture energy (per unit area) or equivalent plastic displacement at failure. For the current study, fracture energy is chosen to be the factor governing damage evolution of the matrix element. Both approaches take into consideration a scalar parameter known as the characteristic length of the element. This formulation reduces the mesh sensitivity. The fracture energy is calculated as shown in E q. (4 22). In Eq. (4 22 ), is the characteristic length of the element, which is length of a line across the element for a first order element. The results for transverse stress -

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81 strain behavior of the hybrids are compared with carbon epoxy and glass epoxy composites. (4 22 ) The expression gives by Eq. (4 22) introduces the definition of equivalent plastic displacement, . Before damage initiation, and after damage initiation, . refers to the plastic strain at failure. The RVE is subjected to boundary conditions such that a uniaxial macrostress is being applied in the 2 direction. The stress strain response observed is shown in Figure 4 28 . It can be observed that for the linear regio n of the stress strain response, the behavior can be characterized by the corresponding transverse modulus of the composite. Once, the epoxy starts softe ning the fibers share the load before the composite response becomes plastic. Also interesting is to note that irrespective of hybridization, all the composites soften nearly at the same stress. In order to understand the effect of hybridization 0.2% offset yield strength was calculated from the plots. Table 4 15 lists the strengths for hybrid composites. It can be observed that the strengths are significantly higher than those predicted us ing DMM. This is because the composite is able to carry load beyond the matrix yield strength. Also, it is observed that all the composi tes have very close transverse strengths which could be indicative of the fact that transverse strengths depend on the over all fiber volume fraction only .

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82 Figure 4 1. Flowchart for the Direct Micromechanics method (DMM) Figure 4 2. Schematic of i nterface normal and shear stress State of macrostress selected Macrostrain is computed Microstresses in element e is computed e = 1 Element failure to be checked If Composite has failed STOP Composite has not failed Next element to be checked e = e + 1 Yes Yes No No

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83 Figure 4 3 . Variation of (MPa) with volume fraction of carbon Figure 4 4 . Variation of (MPa) with volume fraction of carbon

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84 Figure 4 5 . Variation of (MPa) with volume fraction of carbon (without interface failure) Figure 4 6 . Variation of & (MPa) with volume fraction of carbon (without interface failure)

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85 Figure 4 7 . Variation of & (MPa) with volume fraction of carbon (without interface failure) Figure 4 8 . Variation of (MPa) with volume fraction of carbon

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86 Figure 4 9 . Variation of (MPa) with volume fraction of carbon Figure 4 10 . Variation of (MPa) with volume fraction of carbon

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87 Figure 4 11 . Failure envelopes for the composites in the plane Figure 4 12 . Failure envelopes for the composites in the plane

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88 Figure 4 13 . Failure envelopes for the composites in the plane Figure 4 1 4 . Failure envelopes for the composites in the plane

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89 Figure 4 15 . Failure envelopes for the composites in the plane Figure 4 16 . Interface effects on the failure envelopes for GFRP in the plane

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90 Figure 4 17 . Interface effects on the failure envelopes for GFRP in the plane Figure 4 1 8 . Interface effects on the failure envelopes of CFRP in the plane

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91 Figure 4 19 . Interface effects on failure envelopes of GFRP in the plane Figure 4 20 . Comparison of DMM failure envelopes with failure theories for GFRP

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92 Figure 4 21 . Comparison of DMM failure envelopes with failure theories for CFRP Figure 4 22 . Comparison of DMM failure envelopes with failure theories for hybrid composite (0.3C 0.3G)

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93 Figure 4 23 . Off axis failure curve s for GFRP for Figure 4 24 . Off axis failure curves for GFRP for

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94 Figure 4 25 . Off axis failure curves for CFRP for Figure 4 26 . Off axis failure curves for CFRP for

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95 Figure 4 27 . A general ductile damage model with strain hardening before softening Figure 4 28 . Transverse stress strain progressive damage response for composites

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96 Table 4 1. Strength properties of fiber materials (MPa) Fiber Carbon 4120 2990 298 298 1760 Glass 1104 1104 1104 1104 460 Table 4 2. Strength properties of matrix materials ( MPa ) Matrix Low strength (LS) epoxy 49 121 93 High strength (HS) epoxy 60 121 93 Table 4 3. Normal and shear s trength f or the interface s ( MPa ) Matrix Direction CFRP GFRP LS epoxy Normal 127 45.28 Shear 243 46.7 Table 4 4. Comparison of longitudinal failure strains for the composites Matrix Failure strain CFRP Hybrid GFRP H1 H2 H3 H4 H5 LS epoxy Analytical 0.014 0.014 0.014 0.014 0.014 0.014 0.014 FEA 0.013 0.013 0.013 0.013 0.013 0.013 0.013 HS epoxy Analytical 0.016 0.015 0.015 0.015 0.015 0.015 0.015 FEA 0.016 0.015 0.015 0.015 0.015 0.015 0.015 Table 4 5 . Longitudinal tensile strength, (MPa) for the composites with low strength epoxy 2129

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97 Table 4 6 . Variability in (MPa) for low strength epoxy and 95% Confidence Intervals of the mean Table 4 7 . Longitudinal tensile strength , (MPa) for the composites with high strength epoxy 2490 Table 4 8 . Variability in (MPa) for high strength epoxy Table 4 9. Longitudinal compressive strength , (MPa) for the composites 1807 Specimen Mean Standard deviation 95% CI of mean H1 1971.9 0.2452 (1971.8, 1972.1) H2 1665.7 0.2713 (1665.5, 1665.9) H3 1360.1 0.1033 (1360, 1360.2) H4 1055 0.0784 (1054.9, 1055) H5 750.1 0.0299 (750.13, 750.17) Specimen Mean Standard deviation H1 2251.2 0.0064 H2 1902.8 0.0048 H3 1554.4 0.0022 H4 1206.0 0.0036 H5 857.6 0.0017

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98 Table 4 10 . Variability in (MPa) and 95% Confidence Intervals of the mean Table 4 11. Variability in (MPa) and 95% Confidence Intervals of the mean Table 4 12. Variability in (MPa) and 95% Confidence Intervals of the mean Table 4 13. Variability in (MPa) and 95% Confidence Intervals of the mean Specimen Mean Standard deviation 95% CI of mean H1 34.33 1.8800 (32.98, 35.67) H2 31.93 0.6691 (31.45, 32.41) H3 32.31 0.5222 (31.93, 32.68) H4 32.77 0.5225 (32.40, 33.15) H5 34.69 0.4282 (34.38, 34.99) Specimen Mean Standard deviation 95% CI of mean H1 26.28 1.9314 (24.89, 27.66) H2 25.09 0.4756 (24.75, 25.43) H3 24.69 0.8725 (24.06, 25.31) H4 25.15 0.2633 (24.96, 25.34) H5 26.94 0.1884 (26.80, 27.07) Specimen Mean Standard deviation 95% CI of mean H1 41.15 0.1241 (41.07, 41.25) H2 41.08 0.2628 (40.89, 41.27) H3 40.79 0.3750 (40.53, 41.06) H4 40.93 0.3601 (40.67, 41.19) H5 40.69 0.1399 (40.59, 40.79) Specimen Mean Standard deviation 95% CI of mean H1 36.06 0.1686 (35.95, 36.18) H2 36.07 0.0474 (36.04, 36.10) H3 35.82 0.3917 (35.54, 36.09) H4 35.98 0.2509 (35.81, 36.17) H5 36.13 0.2361 (35.96, 36.29)

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99 Table 4 14. Variability in (MPa) and 95% Confidence Intervals of the mean Table 4 15. 0.2% offset transverse yield strengths for composites Specimen Mean Standard deviation 95% CI of mean H1 22.38 1.4640 ( 21.33, 23.42 ) H2 20.19 0.5535 ( 19.79, 20.58 ) H3 20.44 0.4377 ( 20.13, 20.76 ) H4 20.89 0.2879 ( 20.68, 21.09 ) H5 21.82 0.3353 ( 21.58, 22.06 ) Specimen 0.2% yield strength (MPa) CFRP 63.70 H1 63.75 H 2 63.19 H 3 63.35 H 4 63.01 H5 63.20 GFRP 62.96

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100 CHAPTER 5 CONCLUSIONS Contributions of D issertation The goal of this research to establish a robust computational model to characterize hybrid multiphase composites using micromechanical studies has been achieved. With the help of this model various parameters such as fiber volume fraction, type of fibers, diameter of the fibers and different packing arrangemen ts can be studied in detail. The effect of hybridization on the effective thermo mechanical properties was studied. It was observed that the effective elastic constants of the homogenized c omposites, for the most part, depend only on the volume fraction of fibers. The location of fibers within the RVE for a given fiber volume fraction had a negligible effect on the elastic constants. The longitudinal elastic modulus for multiphase composites can be predicted using Rule of Hybrid Mixtures (RoHM) type linear equations. However, for transverse and shear moduli, RoHM type of equations does not provide an accurate estimate. A modification to the Halpin Tsai equation has been proposed by varying a geometric non dimensional parameter, which provides excellent agreement with the finite element based micromechanical results. Effective coefficients of thermal expansions (CTE) are also calculated from the micromechanical study for all composites. Variat ions of the CTEs with volume fraction of the reinforcements are studied. It was observed that the longitudinal CTE, , has a non linear variation with volume fraction of the fibers. H owever, it can be predicted very accurately from analytical formulations. It was observed that the variation of was linear with volume fraction of the fibers and can be predicted from RoHM type equations indicating that can be treated as a homogenize d property of the composite.

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101 Variation of transverse CTE, , was studied with volume fraction of the fiber. It was observed that also had a non linear variation with respect to volume fraction. H owever could be treated as a homogenized property and can be predicted variation with random location of the fibers for a given fiber vo lume fraction. Therm al stresses arising due to mismatch of CTEs between the fibers and the matrix were studied for all composites. It was observed that since microstresses are point functions that depend on the location of fibe rs in the RVE, thermal stresses had significant v ariation with location of the fiber s . Moreover, thermal stresses were lower for hybrids when compared with binary composites CFRP and GFRP. Uniaxial as well as multi axial failure strengths of all the composites were studied through the Direct Micromechan ics method (DMM) method. It was observed that longitudinal tensile and compressive strengths (without considering microbuckling or instability) depend on the least failure strain phase . Also, the variation in these two strengths with volume fraction of fib er is linear, since they primarily depend on the effective longitudinal modulus, of the composite. The longitudinal tensile and compressive strengths did not show much variation with random location of the fibers for a given volume fraction. Transverse strengths on the other hand are very much dependent on the location of fiber inside the RVE. Hybrid composites also show lower transverse tensile and compressive strengths than binary composites owing to the presence of an extra inclusion that causes additional stress concentration. Failure envelopes for biaxial loading cases were studie d for all the composites and the results were compared with existing phenomenological failure theories.

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102 Progressive damage study under transverse tension was conducted assuming a ductile damage model for the matrix material. Stress strain response of the composites were plotted to obtain the 0.2% offset yield strength. It was observed that the composites yield strength significantly improve over the first element failure strength. However, all the composites for the most part demonstrate nearly the same yi eld strength with very negligible variation with random fiber location for a given volume fraction. It can be concluded that transverse strengths for the composites depend primarily on the overall volume fraction of the reinforcements as well as the matrix yield strength , and is not very sensitive to the properties of the individual fibers. Future Work There is a potential for enormous amount of future research on this topic. Transverse strengths have been shown to depend heavily on the matrix strength as well the interface tensile and shear strengths. Further characterization of the transverse strength property is required for the hybrid composites. This requires a very robust modeling of the interface using traction separation laws in conjunction with mat erial damage in the matrix. The longitudinal tensile strength has been limited to the first element strength for the current dissertation; however, it too can be characterized by considering progressive damage in the fiber or the matrix to understand its e ffect on the ultimate strength o f the composite. Finally a m ultiscale analysis can be conducted using the results from the micromechanics study onto a laminate level or possibly to a system level, to study the macroscale behavior of a hybrid composite spec imen. Delamination analysis and impact study can be conducted on hybrid composite plates to understand the effect of hybridization on the effective properties.

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106 [43] Sankar, Bhavani V., and Jerome T. Tzeng. "Thermal stresses in functionally graded beams." AIAA journal 40.6 (2002): 1228 1232 [44] Apetre, N. A., B. V. Sankar, and D. R. Ambur. "Low velocity impact response of sandwich beams with functionally graded core." International journal of solids and structures 43.9 (2006): 2479 2496 [45] Fu, Shao Yun, Guanshui Xu, and Yiu Wing Mai. "On the elastic modulus of hybrid particle/short fiber/polymer composites." Composites Part B: Engineering 33.4 (2002): 291 299 [46] Gi bson, Ronald F. Principles of composite material mechanics . CRC Press, 2011 [47] Sankar, Bhavani V., and Ramesh V. Marrey. "A unit cell model of textile composite beams for predicting stiffness properties." Composites science and technology 49.1 (1993): 6 1 69 [48] Sun, C. T., and R. S. Vaidya. "Prediction of composite properties from a representative volume element." Composites Science and Technology 56.2 (1996): 171 179 [49] Zhu, H., B. V. Sankar, and R. V. Marrey. "Evaluation of failure criteria for fiber composites using finite element micromechanics." Journal of composite materials 32.8 (1998): 766 782 [50] Choi, Sukjoo, and Bhavani V. Sankar. "Micromechanical analysis of c omposite laminates at cryogenic temperatures." Journal of composite materials 40.12 (2006): 1077 1091 [51] Banerjee, Sayan, and Bhavani V. Sankar. "Mechanical properties of hybrid composites using finite element method based micromechanics." Composites Par t B: Engineering 58 (2014): 318 327 [52] Leong, Martin, and Bhavani V. Sankar. "Effect of thermal stresses on the failure criteria of fiber composites." Mechanics of Advanced Materials and Structures 17.7 (2010): 553 560 [53] Marrey, Ramesh V., and Bhavani V. Sankar. "Micromechanical models for textile structural composites." (1995) [54] Zhu, H., B. V. Sankar, and R. V. Marrey. "Evaluation of failure criteria for fiber composites using finite element micromechanics." Journal of composite materials 32.8 (1998): 766 782 [55] Stamblewski, Christopher, Bhavani V. Sankar, and Dan Zenkert. "Analysis of three dimensional quadratic failure criteria for thick composites using the direct micromechanics method." Journal of composite materials 42.7 (2008): 635 654

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107 [56] Karkkainen, Ryan L., Bhavani V. Sankar, and Jerome T. Tzeng. "Strength prediction of multi layer plain weave textile composites using the direct micromechanics method." Composites Part B: Engineering 38.7 (2007): 924 932 [57] Hashin, Zvi. "Failur e criteria for unidirectional fiber composites." Journal of applied mechanics 47.2 (1980): 329 334 [58] ABAQUS, Ver. "6.9.2 Documentation." ABAQUS Inc (2010)

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108 BIOGRAPHICAL SKETCH S ayan Banerjee was born in 1984 in India. He attended Bharatiya Vidhya Bhavan for his high school studies. He ear ned his Bachelor of Science in m echanical e ngineering in 2006 from the University of Pune, India. From 2006 to 2008, he worked at Nicco Corporat ion Limted, Kolkata, India as an Assistant Project Manager for o il and g as sector projects. He came to the United States in 2008 and earned his Master of Science from University of Florida, Gainesville in 2010. He began his doctoral research at the Center for Advanced Composites under the tutelage of Dr. Bhavani V. Sankar from fall 2010. Sayan defended his Doctor of Philosophy dissertation in July 2014 and graduated in December 2014 .