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Effect of fiber volume fraction on fracture mechanics in continuously reinforced fiber composite materials

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Effect of fiber volume fraction on fracture mechanics in continuously reinforced fiber composite materials
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Wasik, Thomas
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University of South Florida
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Finite element
Crack propagation
Composite interface
Ansys
Interface failure modes
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
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ABSTRACT: The application of advanced composite materials, such as graphite/epoxy, has been on the rise for the last four decades. The mechanical advantages, such as their higher specific stiffness and strength as compared to monolithic materials, make them attractive for aerospace and automotive applications. Despite these advantages, composites with brittle fibers have lower ductility and fracture toughness than monolithic materials.One way to increase the fracture toughness of composites is to have a weak fiber-matrix interface that would blunt crack tips by crack deflection into the interface and hence enhance fracture toughness. However, this also reduces the transverse properties of the composite.Therefore, an optimum fiber-matrix interface would be the one that is just weak enough to cause crack deflection into interface.This study investigates the effect of fiber-to-matrix moduli ratio, fiber-volume fraction, fiber orthotropy, and thermal stresses on the possibility of crack deflection. A finite element model is used to analyze a 2-D axisymmetric representative volume element- a three-phase composite cylinder made of fiber, matrix, and composite. A penny shaped crack is assumed in the fiber.To determine whether the crack would deflect into the interface or propagate into the matrix, maximum stresses at the fiber-matrix interface and in the matrix are compared to the interface and matrix strengths.As opposed to most studies in the literature, this study found that fiber-volume fractions do have an impact on crack deflection and this impact increases with large fiber-to-matrix moduli ratios.
Thesis:
Thesis (M.S.M.E.)--University of South Florida, 2005.
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by Thomas Wasik.
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Effect of Fiber Volume Fraction on Fr acture Mechanics in Continuously Reinforced Fiber Composite Materials by Thomas Wasik A thesis submitted in partial fulfillment of the requirement s for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Autar K. Kaw, Ph.D. Glen H. Besterfield, Ph.D. Thomas Eason, Ph.D. Date of Approval: March 25, 2005 Keywords: Finite Element, Crack Pr opagation, Composite Interface, Ansys, Interface Failure Modes Copyright 2005, Thomas Wasik

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i TABLE OF CONTENTS LIST OF TABLES .................................................................................................iii LIST OF FIG URES.............................................................................................. iv LIST OF SYM BOLS............................................................................................ viii ABSTRACT..........................................................................................................x CHAPTER 1 INTR ODUCTION.........................................................................1 1.1 Over view.....................................................................................1 1.2 Literatu re Revi ew.........................................................................3 CHAPTER 2 FINITE ELEM ENT MODEL DESIGN...........................................6 2.1 Geometry and B oundary C onditions ............................................6 2.2 Fundamental Equations.............................................................12 2.2.1 Isotropic-Fi ber, IsotropicMatrix.....................................14 2.2.2 Transversely Isotropi c Fiber, Isotr opic Matr ix.................16 CHAPTER 3 FINITE ELEMEN T MODEL VALID ATION.................................18 CHAPTER 4 FINITE ELEMEN T MODEL ANAL YSIS.....................................23 4.1 Fiber-Volume Frac tion Criter ion.................................................23 4.2 Orthotropic Fiber Criterio n.........................................................26 4.3 Thermal Stress Criterio n............................................................27 CHAPTER 5 RESULTS A ND DISCUSS ION..................................................30 5.1 Fiber-Volum e Fracti on...............................................................32 5.1.1 Elastic Moduli Ratio 1 m fE E ...........................................32 5.1.2 Elastic Moduli Ratio 6 m fE E ...........................................33

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ii 5.1.3 Elastic Moduli Ratio 20 m fE E .........................................35 5.1.4 Elastic Moduli Ratio 80 m fE E ..........................................37 5.1.5 Silicon Carbide/ Epoxy Composite...................................41 5.2 Fiber Orthotr opy........................................................................45 5.3 Therma l Stress ..........................................................................48 CHAPTER 6 CONCL USIONS........................................................................52 REFERENCES ..................................................................................................54 APPENDICE S ..................................................................................................56 Appendix 1: Ansys In put File................................................................... 57 Appendix 2: Maple Instructi ons................................................................62 Appendix 3: Ma thcad F ile........................................................................ 64

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iii LIST OF TABLES Table 1: Percentage Error Va lues of Five N odal Locati ons..........................22 Table 2: Material Properties of C onstituents in the Fiber-to-Matrix Moduli Ratio Analysis ......................................................................24 Table 3: Material Properties of Fiber and Matrix in Silicon Carbide/Epoxy Composit e.......................................................................................25 Table 4: Displacements and Interfac e Strengths Used in Silicone Carbide/ Epoxy Analysis………………………………………………. 25 Table 5: Material Properties of Orthotropic Fiber and Isotropic Matrix in Gr aphite/Epo xy.................................................................26 Table 6: Material Properties of Fiber and Matrix in Graphite/Epoxy Composit e.......................................................................................26 Table 7: Material Properties of Graphite/Epoxy Used for Thermal Stress Anal ysis...............................................................................27

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iv LIST OF FIGURES Figure 1: Modes of Failure of Unidir ectional Lamina Un der a Longitudinal Tensile Load……………………………………………………………..2 Figure 2: The Repr esentative Volu me Elem ent..............................................6 Figure 3: Schematic Representati on of Finite El ement Model........................8 Figure 4: Plan e 2 Elem ent..............................................................................9 Figure 5: Deformed and Undeformed Shapes of Finite Element Model............................................................................................11 Figure 6: The Crack Tip in t he Finite Elem ent Model...................................12 Figure 7: Cross-Sections of Composites with Hexagonal and Random Fiber Arrangement ..........................................................13 Figure 8: Principle of Superposit ion.............................................................19 Figure 9: Stress Ratios of max 1) (zz in the Matrix as Function of Normalized Crack Length, fr a ..........................................................................29 Figure 10: The Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 1m fE E ………………..……32 Figure 11: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 1m fE E ............................................33

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v Figure 12: The Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 6m fE E ...............................34 Figure 13: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 6m fE E ............................................35 Figure 14: The Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 20m fE E ............................36 Figure 15: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 20m fE E ..........................................37 Figure 16: The Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 80m fE E .............................38 Figure 17: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 80m fE E .........................................39 Figure 18: The Influence of Fi ber-Matrix-Moduli Ratio and Fiber-Volume Fraction on Tensile Stress Ratio 1 max) ( rr......................................40

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vi Figure 19: The Influence of Fiber-M atrix Modulus Ratio and Fiber-Volume Fraction on the Shear Stress Ratio 1 max) ( rz ................................41 Figure 20: Stress Ratio, 1 max) ( rras a Function of Normalized Crack Length, fr a for Silicon Carbide/Epoxy at 100% Displ acement....................42 Figure 21: Stress Ratio, 1 max) ( rzas a Function of Normalized Crack Length, fr a for Silicon Carbide/Epoxy at 100% Displ acement....................43 Figure 22: Stress Ratio, 1 max) ( rras a Function of Normalized Crack Length, fr a for Silicon Carbide/Epoxy at 100% and 50% Displacements...44 Figure 23: Stress Ratio, 1 max) ( rzas a Function of Normalized Crack Length, fr a for Silicon Carbide/Epoxy at 100% and 50% Displacements.. 45 Figure 24: Stress Ratio, 1 max) ( rras a Function of Normalized Crack Length, fr a for Graphite/Epoxy Composite with Orthotropic and Isotropic Fibers ...........................................................................................47

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vii Figure 25: Stress Ratio, 1 max) ( rzas a Function of Normalized Crack Length, fr a for Graphite/Epoxy Composite with Orthotropic and Isotropic Fibers............................................................................................48 Figure 26: Stress Ratio, 1 max) ( rras a Function of Normalized Crack Length, fr a for Graphite/Epoxy Composite With and Without Thermal Load Present.......................................................................................... 50 Figure 27: Stress Ratio, 1 max) ( rzas a Function of Normalized Crack Length, fr a for Graphite/Epoxy Composite With and Without Thermal Load Present .......................................................................................51

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viii LIST OF SYMBOLS E Modulus of Elasticity G, Modulus of Rigidity Coefficient of Thermal Expansion V Fiber Volume Fraction Poisson’s ratio KI Stress Intensity Factor u Displacement Strain Stress a Crack Length r Radius k Plane-Strain Bulk Modulus 1 Maximum Principle Stress Energy Release Rate p Pressure J0 Bessel Function of zero order b Width of Fiber and Matrix Subscripts m Matrix f Fiber r, z, Cylindrical Coordinates d Deflection p Penetration

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ix c Mode I Toughness ic Toughness of Interface t in Interface fib Fiber mat Matrix

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x EFFECT OF FIBER VOLUME FRACTION ON FRACTURE MECHANICS IN CONTINUOUSLY REINFORCED FIBER COMPOSITE MATERIALS Thomas Wasik ABSTRACT The application of advanced composite materials, such as graphite/epoxy, has been on the rise for the last four decades. The mechanical advantages, such as their higher specific stiffne ss and strength as compared to monolithic materials, make them a ttractive for aerospace and automotive applications. Despite these advantages, composites with brittle fibers have lower ductility and fracture toughness than monolithic materials. One way to increase the fracture t oughness of composites is to have a weak fiber-matrix interface that would blunt crack tips by crack deflection into the interface and hence enhance fracture toughness. However, this also reduces the transverse properties of the composite. Therefore, an optimum fiber-matrix interface would be the one that is just weak enough to cause crack deflection into interface. This study investigates the effect of fiber-to-matrix moduli ratio, fibervolume fraction, fiber ort hotropy, and thermal stresses on the possibility of crack

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xi deflection. A finite element model is used to analyze a 2-D axisymmetric representative volume elem enta three-phase composite cylinder made of fiber, matrix, and composite. A penny shaped crack is assumed in the fiber. To determine whether the crack woul d deflect into the interface or propagate into the matrix, maximum stresse s at the fiber-matrix interface and in the matrix are compared to the interface and matrix strengths. As opposed to most studies in the lit erature, this study found that fibervolume fractions do have an impact on crack deflection and this impact increases with large fiber-to-matrix m oduli ratios. The presence of orthotropic fiber in the composite increases the possibility of crack deflection wit h increasing fibervolume fraction in the early and middle st ages of the fiber crack growth. The thermal stresses decrease the likelihood of crack deflection when the thermal expansion coefficient of the matrix is larger than t hat of the fiber.

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1 CHAPTER 1 INTRODUCTION 1.1 Overview The use of composite materials has been steadily increasing in the past several decades. The high strength, high stiffness and lightweight make them particularly attractive to designers in a variety of industries. The composite material consists of a matrix and one of the reinforcing phases such as particulate, flake and fiber. In the continuous fiber composites, due to its large surface area, the fiber-matrix interface in fluences the behavior of a composite. In addition to providing a mechanism to trans fer loads from matrix to fibers, the interface also plays an important role in determining the composite toughness. In spite of many advantages, the com posite materials suffer from lower ductility and toughness when compared to commonly used metals. A unidirectional composite with brittle fi bers and a crack propagating perpendicular to the fibers can fail in at least thr ee modes under longitudinal tensile load. These modes are:(a) brittle failure, (b) bri ttle failure with fiber pullout, (c) brittle failure with fiber pullout and interface shear failure or interface tensile failure [1]. This is illustrated in Figure 1. The tensile or shear interface failure is a prerequisite for phenomena such as crack deflection into the interface, crack bridging by fibers, and fiber pullout [2].

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2 All of these are energy-dissipating phenom ena during crack pr opagation process and help enhance toughness of the fiber-reinforced composites. Figure 1: Modes of Failure of Unidirec tional Lamina Under a Longitudinal Tensile Load By controlling the strength of the interface bond between matrix and the fiber, the designer is able to influe nce the mechanical properties of the composite. To take a full advantage of the fiber properties and to obtain high strength and high stiffness composite, a strong fiber-matrix bond is very desirable. Moreover, a str ong interface bond results in high shear strength of the composite and an effective load transfer to the fibers under l ongitudinal tensile load. However, a strong interface bond wi ll significantly decrease the ability of the fiber to debond from matrix during fracture process and lowering the composite toughness. This ability is ve ry beneficial especially in brittle fiber (a) (b) (c)

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3 composites because the debonding proce ss can act as crack arrestor and prevent further propagation of the crack. This study presents an axisymmetric finite element analysis of a pennyshaped crack in a brittle fiber approaching a fiber-matrix interface. The main goal of this study was to determine the infl uence of fiber-volume fraction for various fiber-to-matrix elastic moduli ratios on possibility of the interface failure either in shear or in tension. Furthermore, t he influence of residual stresses and the fiber orthotropy were also exam ined. The residual stresse s arise from the thermal expansion mismatch between fiber and matrix as the composite is cooled down after processing. 1.2 Literature Review The fiber-volume fraction is one of t he parameters employed in analyzing composites. There have been several m odels developed to address the failure of the composites as function of this parameter. These model s are: fiber cracks in dilute fiber-volume fraction composites by Gupta [3], periodic cracks in higher fiber volume fraction composites by Erdogan and Bakioglu [4], and nonhomogenous interfaces and nondilute fiber-volume fractions by Bechel and Kaw [5]. In addition, a number of criteria have been presented in the past by various authors in order to explore the phenomenon of crack deflection at the fiber-matrix interface. He and Hutc hinson [6] examined the tendency of the transverse crack impinging on the interfac e joining two dissimilar materials to penetrate the interface or to deflect into the interface. The materials on either

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4 side of the interface are elastic and is otropic. They pres ented criteria that compared the energy release rate for t he deflected crack to the maximum energy release rate for a penetrating crack p d This result can be compared to ratio of the toughness of the interface to the mode I toughness of uncracked material c ic The impinging crack is most likely to be deflected into the interface if p d c ic (1) because the condition for propagation into the interface will be met at a lower load than that for penetration across the interface. The crack will tend to penetrate the interface when the inequality is reversed. Swenson and Rau [7] studied the pl ain strain problem of a crack terminating perpendicular to the interface between two isotropic half spaces with different elastic constants. They conc luded that the probabi lity of an interface failure in shear or in tension is very highly influenced by modulus ratio of the two isotropic half spaces. A cra ck in the stiffer material wi ll likely cause the interface to fail in shear, whereas the crack in softer material will lead to tensile splitting of the interface. Cornie et al. [8] came up with the cr iteria that addressed the fiber-matrix debonding. The debonding can be expressed in terms of cohesive strength of the interface, shear strength of the inte rface, and fiber fracture stress. They found that if the ratio of the interface cohesive strengt h (normal or shear) to the

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5 fiber strength is less than t he ratio of the normal (or shear) stress at the interface to axial stress at the crack tip the t endency of the crack to deflect along the interface is higher. Pagano [9] investigated the transverse matrix crack impinging on the fibermatrix interface in a brittle matrix co mposite. In this study, he constructed general material design curves for fiber penetration and interface debonding for multiple fiber-to-matrix ratios. These curves allow a comparison between potential energy release rate and a mate rial toughness value to make initial assessment of the success of failure of a composite made from a particular combination of materials.

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6 CHAPTER 2 FINITE ELEMENT MODEL DESIGN 2.1 Geometry and Boundary Conditions The analysis of a penny-shape crack locate d in a brittle fiber was performed using the finite element software package ANSYS 8.0. To simulate a fracture behavior of the cracked fiber and the result ing stresses, a representative volume element (RVE) consisting of a single fiber surrounded by cylindrical tubes of matrix and composite, respectively, was us ed as illustrated in Figure 2. The RVE is considered to represent the composit e and to respond in the same way as the whole composite [10]. Figure 2: The Representative Volume Element INTERFACE FIBER MATRIX COMPOSITE CRACK r z

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7 The finite element model was designed as 2-D axisymmetric structure in the r-z plane. The use of an axisymme tric model greatly reduced the modeling and analysis time compared to that of an equivalent 3-D m odel. The geometry and boundary conditions of the finite element model are schematically represented in Figure 3. Due to sy mmetry in the geometry and the boundary conditions the finite element calculat ions were performed on the right upper quadrant of the representative volume uni t shown in Figure 2. The boundary conditions for the finite el ement model were taken as: 1. at 0 z a) zu=0 for W r a b) rz =0, zz =0 for a r 0 2. at L z a) zu=prescribed uniform displacement, W r 0 b) rz =0, W r 0 3. at 0 r a) ru=0 for L z 0 b) rz =0, L z 0 4. at W r a) rz =0, L z 0 b) rr =0, L z 0

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8 Figure 3: Schematic Representat ion of Finite Element Model r f

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9 Also, the mode of deformation is axi symmetric so the non-zero stress and displacement components depend only on r and z and are independent of The 6-node triangular element (Plane 2) with a quadratic displacement behavior that was used for all the analyses performed in this study is shown in Figure 4. The dimensions of the fi nite element model were 10 units wide and 30 units high and were kept cons tant throughout the entire study. The finite element software used to carry out the finite element computations in this study supported only a limited number of nodes (128,000). Figure 4: Plane2 Element Consequently, the model was subdivided into five separate areas to allow greater concentration of elements in the regions in which the stress gradient was expected to be high, such as the crack tip and fiber-matrix interface (Area I and Area II). The remaining areas had significant ly lower concentrations of elements. On average, there were 120,000 nodes and 60,000 elements in each model.

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10 The fiber of unit radius (fr) is comprised of Area I and Area III. Also, the Area I contains crack of radius a. The r adius b of the two concentric cylinders representing fiber and matrix was calcul ated based on the fiber-volume fraction given by 2 2b r Vf f (2) The three fiber-volume fractions used in the analysis were: 0.25, 0.50, 0.75 and the corresponding b values were: 2, 1.414 and 1.155, respectively. The fiber containing the penny-shaped crac k is parallel to the longitudinal axis (z axis) and the crack plane z=0 is or iented perpendicular to that axis. The fiber-matrix interface was modeled as perfectly bonded. Furthermore, the composite was subjected to uniform and const ant longitudinal tensile strain in the positive z direction and therefore was displacem ent controlled. As a consequence, the crack experiences Mode I loading. Figure 5 shows the shapes of deformed and undeformed finite element model. The Linear Elastic Fracture Mech anics (LEFM) approach was used as a means to obtain stress field caused by t he presence of the crack. This approach was justified due to the brittle nature of the fiber. Because the stresses are singular in the region immediately su rrounding the crack tip and vary as d1, where d is the distance from the crack ti p, the triangular quadr atic elements were employed with their midside nodes shifted by a quarter to ward the crack tip.

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11 The elements were arranged in semicircle around the crack tip, one element every 30 degrees. Figure 6 illust rates the element arrangement in the crack tip vicinity. Figure 5: Deformed and Undeformed Shapes of Finite Element Model

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12 Figure 6: The Crack Tip in the Finite Element Model 2.2 Fundamental Equations The majority of a unidirectional fibe r-reinforced composites are classified either as an orthotropic or transversely isot ropic materials. This classification is based on the geometric fiber arrangement in the matrix. A unidirectional fiberreinforced composite with fibers arranged in hexagonal or r andom manner in the plane perpendicular to the fibers axes, as s hown in Figure 7, is considered to be transversely isotropic. Crack Tip First Row of Elements Second Row of Elements

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13 Figure 7: Cross-Sections of Co mposites with Hexagonal and Random Fiber Arrangement The transversely isotropic materi al requires only five engineering constants to fully describe its elastic behavior. The engineering constants are: zzE, rrE, zr zrG, rG. By considering fibers to be along z-axis in the cylindrical coordinate system, then the rplane becomes isotropic and there is no preferred direction in that plane. The following subsections list equations [11] that were used to calculate engineeri ng constants needed to describe composite material. The equations are par t of Input Files written for finite element software. The sample of an Input File is located in Appendix 1. HEXAGONAL RANDOM

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14 2.2.1 Isotropic Fibe r, Isotropic Matrix The following are the equations used for calc ulating material properties of the composite consisting of isotr opic fiber and isotropic matrix. 1.Elastic Moduli a. Longitudinal m f m m f f m m f m m f f zzG k V k V V V E V E V E 1 42 (3) where, fV is the fiber volume fraction mV is the matrix volume fraction fE is the elastic modulus of fiber mE is the elastic modulus of matrix mG is the shear modulus of matrix f is the Poisson’s ratio of fiber m is the Poisson’s ratio of matrix fk is the plane-strain bulk modulus of fiber mk is the plane-strain bulk modulus of matrix b. Transverse f m f m rrE E V E E E1 1 (4)

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15 2. Poisson’s Ratios a. m f m m f f m m f m f m m f f z zrG k V k V k k V V V V 1 1 1 (5) b. k E Ezz zr rr r2 1 2 12 (6) where, k the is plane-strain bulk modulus 3. Shear Moduli a. f m m f m m f f m z zrV G V G V G V G G G G 1 1 (7) where, fG is the shear modulus of fiber b. r rr rE G 1 2 (8) where, f f fE G 1 2 (9) m m mE G 1 2 (10)

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16 4. Bulk Modulus m f f m f m f m f m fk k V G k k k G V G k k 2 (11) where, f f f fE k 1 2 1 2 (12) m m m mE k 1 2 1 2 (13) 2.2.2 Transversely Isotropic Fiber, Isotropic Matrix The following are the equations used for calc ulating material properties of the composite consisting of transversely isotropic fiber and isotropic matrix. 1.Elastic Moduli a. Longitudinal m m f f zzV E V E E 1 (14) b. Transverse 221 1f m f m rrE E V E E E (15) 2. Shear Moduli a. zrf m f m z zrG G V G G G 1 1 (16)

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17 b. rf m f m rG G V G G 1 1 (17) 3. Poisson’s Ratios a. m m f f z zrV v V v v vzr (18) b. 1 2 r rr rG E v (19) 4. Coefficients of Thermal Expansion a. Longitudinal m m f f m m m f f f zzV E V E V E V Ezz zz zz (20) b. Transverse zz m m f f m m m f f f rrV V V Vzr zr rr 1 1 (21) where, f is the coefficient of thermal expansion of fiber m is the coefficient of t hermal expansion of matrix

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18 CHAPTER 3 FINITE ELEMENT MODEL VALIDATION Solving the same test problem with analytical and finite element method assessed the accuracy of the finite element model. The values of zu, ru, rr zz rz and obtained from the anal ysis of the finite element model at chosen locations (r, z) were compared to the values obtained at the same locations by using analytical analysis of the same model. Linear elastic and isotropic material behavior was assum ed for the finite element and analytical model (E=610 30 and =0.3). Moreover, the assump tion of perfect fiber-matrix interface was made. The stress field in the analytical method was determined by superposition of two boundar y value problems, one host ing a crack, the other being uncracked as illustr ated in Figure 8. In Figure 8a, an uncracked cylinder is subjected to a uniform boundary traction p in the z direction. This created the following stresses: zz =p, rr =0, =0, and rz =0. Because of the isotropic mate rial assumption, the Hooke’s law in cylindrical coordinates was used as a basis for displacements derivation in r and z directions, respectively.

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19 Figure 8: Principle of Superposition zz rr rrv E 1 (22) and because pzz 0 rr 0 rr simplifies to E vprr (23) Therefore, r urr r (24) Similarly, z uzz z (25)

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20 Figure 8b shows traction p applied on the crack face. To obtain the values for zu, zu, rz zz rr ,, and a system of differential equations [12] was solved using a code written in Maple 9.0 (see Appendix 2). D ue to the symmetry, the problem can be reduced to the half space r z0 0 with the following conditions on the 0 z plane: 0 0 0 rrz 0 r (r=cylinder radius) p rzz 0 0, a r 0 (a=half crack length) 0 0 0 r uz a r A single potential function ) ( z r f was employed which automatically frees plane 0 z from shear stress rz The displacement and stress components are then written in terms of that function: z r f z r f ur 22 1 (26) 2 21 2 z f z z f uz (27) z r f z z f r frr2 3 2 2 2 22 2 1 2 (28) where, is the shear modulus of elasticity is the Poisson’s ratio

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21 z r f r z r f r f r2 2 22 1 2 (29) 3 3 2 22 z f z zf zz (30) 2 32z r f zrz (31) By using Fourier-Hankel transform, the function ) ( z r f of two variables is expressed in terms of the function ) ( s A, which depends only on the variable s. The function ) (s A is found by solving the following dual integral: atr t dr r rp dt st s A00 2 1 2 2sin 1 (32) where, p is the normal traction The solution of the above equation is inserted into the equation for ) ( z r f. 0 ) ( 0) ( ) ( ) (ds sze rs J s s A z r f (33) where, ) (z r f is the potential function of two variables 0J is the Bessel function of order zero This, in turn, enables us to find the two displacements and four stress components. The final values of the anal ytical analysis are obtained by adding results from part a and b as shown in Figure 8c. This was done using Mathcad8.0 (Appendix 3).

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22 The finite element model was constr ucted as described in chapter 2. In order to achieve desired accuracy of results, the convergence of a finite element solution was conducted. The pur pose of the convergence study was to refine the mesh size so that the rela tive error between analytical and finite element solutions was less than one percent. Table 1 lists the percentages of relative errors at different locations along the interface and in the ma trix of the composite. The highest error was 0.1 percent. Table 1: Percentage Error Values of Five Nodal Locations Nodal Coordinates ru zu rr zz rz r=1.00000 z=0.02379 0.01260.0102 0.033 0.031 0.093 0.031 r=1.00000 z=0.05293 0.01280.0101 0.057 0.01 0.088 0.055 r=1.01329 z=0.03118 0.01270.1 0.026 0.017 0.093 0.057 r=1.03737 z=0.02182 0.01290.096 0.025 0.015 0.089 0.072 r=1.03107 z=0.05283 0.0129 0.1 0.019 0.032 0.086 0.074

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23 CHAPTER 4 FINITE ELEMENT MODEL ANALYSIS The finite element model analysis was divided into several separate parts. Each part of the analysis investigated the influence of a single criterion on the possibility of the interface failure. The following are the criteria used in the analysis: 1. Fiber-Volume Fraction (FVF) 2. Fiber Orthotropy 3. Thermal Stress For each criterion, the normalized length of the fiber crack, fr a, was progressively increased from 0.6 to 0.97. Furthermore, each criterion was analyzed at 0.25, 0.50, and 0.75 fiber-volume fractions. 4.1 Fiber-Volume Fraction Criterion The first part of the FVF analysis fo cused on how the fiber-volume fraction affects the interface tensile an d shear failure for different fiber-to -matrix elastic moduli ratios. There were four moduli ratios used in the analysis as follows:

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24 1. m fE E=1 represents composite with fiber and matrix made of the same material 2. m fE E=6 represents typical ceramic matrix composite 3. m fE E=20 represents typical polymer matrix composite such as glass/epoxy 4. m fE E=80 represents typical polymer matrix composite such as graphite/epoxy Each moduli ratio was analyzed at 0.25, 0.50 and 0.75 fiber-volume fractions. The fiber and matrix were assumed to be linear elastic and isotropic with the same Poisson’s ratios. Table 2 lists fiber and matrix properties used in the analysis. Table 2: Material Properties of Consti tuents in the Fiber-to-Matrix Moduli Ratio Analysis PROPERTY SYMBOL FIBER MATRIX Modulus of Elasticity E 1, 6, 20, 80 1 Poisson’s Ratio 0.3 0.3 The displacement zu was taken as 0.1, which consti tutes 10% of the fiber radius. The second part of the FVF analysis involved examining the influence of fiber-volume fraction and two different l ongitudinal displacements on interface failure in silicon carbide/epoxy composit e. The two displacements used in the

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25 above analysis were calculated based on the fi ber ultimate tensile strength. The first displacement was obtained by calc ulating displacement value needed to create tensile stress in the fiber equal to the ultimate tensile streng th of that fiber, henceforth called 100% displa cement. The second displacement was taken as a half of the first one, henceforth called 50% displacement. Table 3 lists the properties of fiber and matrix for s ilicon carbide/epoxy composite [13,14]. Table 4 contains displacements and inte rface strengths used in performing the analysis of silicon carbide/epoxy composite [15]. Table 3: Material Properties of Fi ber and Matrix in Silicon Carbide/Epoxy Composite PROPERTY SYMBOLFIBER MATRIX Elastic Modulus E 400 [GPa] 3.44 [GPa] Poisson’s Ratio 0.15 0.35 Fiber Ultimate Tensile Strength fib 3450 [MPa] Matrix Ultimate Tensile Strength mat 69.29 [MPa] Coefficient of Thermal Expansion 0 60 [ C m m] Table 4: Displacements and Interface Strengt hs Used in Silicone Carbide/Epoxy Analysis 100 % Displacement applied zu 0.25875 0.25875 50 % Displacement applied zu 0.12938 0.12938 Interface Normal Strength int 35 [MPa] Interface Shear Strength int 32.5 [MPa]

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26 4.2 Orthotropi c Fiber Criterion To determine the influence of the orthot ropic fiber on the interface failure, two graphite/epoxy composites were us ed for analysis with the fiber and matrix properties listed in Tables 5 and 6 [16]. For comparison purposes, isotropic and orthotropic fibers were used. The anal ysis was performed for 0.25, 0.50, and 0.75 fiber-volume fractions. The applied displacement uz was calculated based on ultimate tensile strength of the graphite fiber. The applied displacement, zu, in both cases was 0.46587. Table 5: Material Properties of Ort hotropic Fiber and Isotropic Matrix in Graphite/Epoxy PROPERTY SYMBOL FIBER MATRIX Longitudinal Elastic Modulus zzE 260 [GPa] 3.5 [GPa] Transverse Elastic Modulus E Err, 14 [GPa] 3.5 [GPa] Shear Modulus z zrG G, 50.95 [GPa] Shear Modulus rG 8.27 [GPa] Poisson’s Ratio z zr, 0.26 0.35 Poisson’s Ratio r 0.33 0.35 Ultimate Tensile Strength fib 4038 [MPa] Table 6: Material Properties of Isot ropic Fiber and Matrix in Graphite/Epoxy Composite PROPERTY SYMBOL FIBER MATRIX Elastic Modulus E 260 [GPa] 3.5 [GPa] Poisson’s Ratio 0.26 0.35 Ultimate Tensile Strength fib 4038 [MPa]

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27 4.3 Thermal Stress Criterion The thermal stresses are created as a result of a mismatch in thermal expansion coefficients of fiber and matr ix. The graphite/epoxy composite having different fiber and matrix thermal expans ion coefficients was analyzed. The obtained results were then compared to the results for the same composite analyzed without thermal expansion coeffi cients. Each composite was analyzed at 0.25, 0.50, and 0.75 fiber-volum e fractions. The displacement zuwas calculated based on ultimate tensile strengt h of a graphite fiber. Table 7 lists material properties for fiber and matr ix in the thermal stress analysis of graphite/epoxy composite. The applied displacement, zu, was 0.46857 Table 7: Material Properties of Graphite/E poxy Used for Thermal Stress Analysis PROPERTY SYMBOL FIBER MATRIX Longitudinal Elastic Modulus zzE 260 [GPa] 3.5 [GPa] Transverse Elastic Modulus E Err, 14 [GPa] 3.5 [GPa] Shear Modulus z zrG G, 50.95 [GPa] Shear Modulus rG 8.27 [GPa] Poisson’s Ratio z zr, 0.26 0.35 Poisson’s Ratio r 0.33 0.35 Coefficient of Thermal Expansion zz -0.855 [ C m m] 90[ C m m] Coefficient of Thermal Expansion ,rr 3.24 [ C m m] 90[ C m m] Ultimate Tensile Strength fib 4038 [MPa]

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28 In order to determine how each criter ion plays a role in influencing the possibility of shear and tensile interface failure, the following stress ratios were calculated: 1 max rr and 1 max rzfor each crack length. Those ratios were then plotted as a function of normalized crack length, fr a, for the three fiber-volume fractions. The preceding ratios prov ide us with the qualitative means to determine and to compare the influence of different parameter s on the two types of the interface failures. This compar ison is not only possible between different fiber-volume fractions of the same compos ite, but also betwe en composites with various elastic moduli ratios. The maxrr and maxrz stresses represent the maximum tensile and shear stresses along fi ber-matrix interface. In turn, the 1 represents the largest princi pal stress present in the ma trix. The choice of using principal stress 1 instead of maxzz in the above ratios was made based on the fact that principal stress 1 was increasing at a higher rate than maxzz as the crack was approaching fiber-matrix interfac e. To illustrate the difference in values between 1 and max) (zz with increasing crack leng th, the stress ratios, max 1) (zz were calculated for 0.50 and 0.75 fibe r-volume fractions and plotted as a function of normalized crack length. Figure 9 clearly s hows that the stress ratios, max 1) (zz for both fiber-volume fractions are higher than one when the crack is close to the fiber-matrix interface.

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29 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.000.600.650.700.750.800.850.900.951.00Normalized Crack Length, a/rf [non-dim]Stress Ratio, 1/( zz )max ( zz )max/ 1 [non-dim]1/( zz)max 80:1 (0.75 FVF) 1/( zz)max 80:1 (0.50 FVF) Figure 9: Stress Ratios of max 1) (zz in the Matrix as Function of Normalized Crack Length, fr a

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30 CHAPTER 5 RESULTS AND DISCUSSION As mentioned in the previous chapters in this study, we want to assess how the fiber-volume fraction, fiber orthot ropy, and thermal stresses influence the crack propagation path of a cracked fiber – does the crack propagate across the interface to the matrix, or does the cr ack propagate along t he interface. We understand that debonding of fiber-matrix in terface causes the blunting of the crack tip, acts as a crack arrestor, and hence contributes to the overall increase in composite toughness. This can be eas ily accomplished by making a weak fiber-interface, but such weak interfac es decrease transverse compressive and shear strength. Hence, to be able to quantif y to build a fiber-matrix interface that is just weak enough to allow interface debonding requires us to fully understand the mechanisms of crack propagation. The stress ratios used to understand propagation paths of a fiber crack were described in chapter 4 and are used to define the conditions necessary for debonding of fiber-matrix interface in the fi ber reinforced composite subjected to longitudinal tensile strain. The debondi ng at the interface will occur if: 1. The ratio of the maximum tensile stress at the interface,max) (rr to the largest principal stress in the matrix, 1 is greater than the ratio of the interface normal strength, int to the ultimate matrix strength, mat that is,

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31 mat rr int 1 max) ( (34) 2. The ratio of the maximum shear stress at the interface, max) (rz to the largest principal stress [8], 1 is greater than the ratio of the interface shear strength, int to the ultimate matrix strength, mat that is, mat rz int 1 max) ( (35) For a specific composite, the two strength ratios mat int, and mat int are material properties of a particular fiber and matrix combinatio n. These strength ratios are not dependent on fiber-volume fr action. In contrast, the two stress ratios 1 max) ( rr and 1 max) ( rzon the left side of t he inequalities (Equations 34 and 35) are influenced by several variables such as: crack lengt h, fiber-volume fraction, fiber-to-matrix elasti c moduli ratio, fiber orthotropy, and thermal stresses. The presentation and discussion of t he results is divided into three separate parts to study the influence of fi ber-volume fraction, fi ber orthotropy and thermal stress.

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32 5.1 Fiber-Volume Fraction 5.1.1 Elastic Moduli Ratio 1m fE E When a composite is made of fiber and matrix that have identical elastic moduli, all fiber-volume fractions repr esent the same geometry of a fiber surrounded by a matrix of infinite r adius. So the normalized stress ratios, 1 max) ( rr and 1 max) ( rzas a function of normalized crack length, fr aare the same for all fiber-volume fractions as giv en in Figures 10 and 11. Note the single number given for the normalized crack length of unity. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rr)max/ 1 [non-dim] 0.25 FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 10: The Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 1m fE E

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33 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rz)max/ 1 [non-dim] 0.50 FVF 0.25 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 11: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 1m fE E 5.1.2 Elastic Moduli Ratio 6m fE E Now let us examine how the fiber-volume fraction affects the crack propagation path for composites where the fiber and matrix elastic moduli are not the same. Figures 12 show s the normalized stress ratio, 1 max) ( rras a function of normalized crack length. The trends for fiber-volume fractions of up to 0.5 are

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34 the same, and only for large fiber-volum e fractions, the normalized stress, 1 max) ( rrshows markedly higher values. Figure 13 show the normalized stress ratios, 1 max) ( rz, as a function of normalized crack length. The trends for a ll fiber-volume fractions look the same. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rr)max/ 1 [non-dim] 0.25 FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 12: The Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 6m fE E

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35 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rz)max/ 1 [non-dim] 0.25 FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interface for all FVFs Figure 13: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 6m fE E 5.1.3 Elastic Moduli Ratio 20m fE E Higher fiber-to-matrix moduli ratios, like 20m fE Erepresenting a typical glass/epoxy give results in a similar behavior as the case of 6 m fE Eexcept the differences between stress ratio values ar e more pronounced. This is illustrated in Figures 14 and 15.

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36 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rr)max/ 1 [non-dim] 0.25 FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 14: The Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 20m fE E

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37 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rz)max/ 1 [non-dim] 0.25 FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 15: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 20m fE E 5.1.4 Elastic Moduli Ratio 80m fE E Higher fiber-to-matrix moduli ratios, like 80 m fE Erepresenting a typical graphite/epoxy composite, give results in a similar behavior as the cases of

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38 6 m fE E and 20 m fE E except the differences between stress ratio values are more pronounced. This is illustrated in Figures 16 and 17. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rr)max/ 1 [non-dim] 0.25 FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 16: The Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 80m fE E

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39 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rz)max/ 1 [non-dim] 0.25 FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 17: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Elastic Moduli Ratio of 80m fE E The preceding results clearly indicate a substantial influence of fibervolume fraction on the crack propagation path The effect is zero for fiber-tomatrix moduli ratio, 1 m fE Eand becomes more pronounced as the fiber-to-matrix moduli ratio increases. For large fibe r-volume fractions, we see that the possibility of crack propagating along the in terface increases, as was observed in experimental studies [1]. Th is is contrary to recent studies [6,7,8] where crack propagation paths are considered to be i ndependent of fiber-volume fractions.

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40 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rr)max/ 1 [non-dim] 0.50 FVF (6:1 Fiber-Matrix Ratio) 0.50 FVF (20:1 Fiber-Matrix Ratio) 0.50 FVF (80:1 Fiber-Matrix Ratio) 0.75 FVF (6:1 Fiber-Matrix Ratio) 0.75 FVF (20:1 Fiber-Matrix Ratio) 0.75 FVF (80:1 Fiber-Matrix Ratio)Figure 18: The Influence of Fiber-Matri x-Moduli Ratio and Fiber-Volume Fraction on Tensile Stress Ratio 1 max) ( rr While the effect of fiber-volume frac tion on tensile interface failure was rather straightforward, the same cannot be said about the effect of fiber-volume fraction on interfacial shear failure.

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41 0.00 0.10 0.20 0.30 0.40 0.50 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rz)max/ 1 [non-dim] 0.50 FVF (6:1 Fiber-Matrix Ratio) 0.50 FVF (20:1 Fiber-Matrix Ratio) 0.50 FVF (80:1 Fiber-Matrix Ratio) 0.75 FVF (6:1 Fiber-Matrix Ratio) 0.75 FVF (20:1 Fiber-Matrix Ratio) 0.75 FVF (80:1 Fiber-Matrix Ratio)Figure 19: The Influence of Fiber-M atrix Modulus Ratio and Fiber-Volume Fraction on the Shear Stress Ratio 1 max) ( rz 5.1.5 Silicon Carbide/Epoxy Composite The purpose of analyzing a particular com posite system is that we wanted to determine crack propagation path under different remote loading values. We apply strain equal to and then half of the ulti mate longitudinal strain of the fiber. The corresponding longitudinal displacement s were derived in chapter 4, and are called 100% and 50% displacements, respectively.

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42 Also, we know the ultimate shear and normal strength of the interface for this particular composite system (Table 4) Hence we cannot only find whether the interface fails but also whether it fails due to shear or normal stress in the interface. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rr)max/ 1 [non-dim] 0.25 FVF 0.50 FVF 0.75 FVF Interface Strength Ratio Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 20: Stress Ratio, 1 max) ( rras a Function of Normalized Crack Length, fr a for Silicon Carbide/Epoxy at 100% Displacement The Figures 20 and 21 illustrate the behavior of a composite that was subjected to a displacement that created stress in the fi ber equivalent to ultimate strength of that fiber As it can be seen, the interf ace tensile failure would take place at normalized crack lengths of 0.7 for 0.75 fiber-volume fraction and of 0.8 for 0.25 and 0.50 fiber-volume fractions. Because the 1 max) ( rz stress ratio values

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43 for all three fiber-volume fractions are below the interface strength ratio value throughout the entire crack pr opagation process, the interface shear failure would not take place. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.600.650.700.750.800.850.900.951.00Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rz)max/ 1 [non-dim] 0.25 FVF 0.5 FVF 0.75 FVF Interface Strength Ratio Stress Ratio at Fiber-Matrix Interface for all FVFsFigure 21: Stress Ratio, 1 max) ( rzas a Function of Normalized Crack Length, fr a for Silicon Carbide/Epoxy at 100% Displacement To examine the impact of different longitudinal displa cements on stress ratios and interface failure mode, the re sults of two displacements (100% and 50%) were plotted on the same graph, Fi gures 22 and 23. The graphs clearly show that the reduction in longitudinal displacement by 50% did not affect the stress ratio distribution and interface failure in a significant way.

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44 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 0.600.650.700.750.800.850.900.951.00Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rr)max/ 1 [non-dim] 0.25 FVF (50% DISPL) 0.50 FVF (50% DISPL) 0.75 FVF (50% DISPL) 0.25 FVF (100% DISPL) 0.50 FVF (100% DISPL) 0.75 FVF (100% DISPL) Interface Strength Ratio Figure 22: Stress Ratio, 1 max) ( rras a Function of Normalized Crack Length, fr a for Silicon Carbide/Epoxy at 100% and 50% Displacements The interface tensile failure for 0.75 fi ber-volume fraction would initiate at 0.75 crack length and 0.25 and 0.50 fiber-volum e fractions at 0.85 crack length. The interface shear failure as before would not take place because the shear strength ratio is significantly larger than stress ra tios present at the fiber-matrix interface.

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45 After running several additional analyses with other smaller displacements, it was found that the shear strength ratio was always higher than the corresponding shear stress ratios. Moreover, the displacements in a 4% to 100% range show that t he crack propagation would be along the interface and would be caused by interf ace tensile failure. 0.00 0.10 0.20 0.30 0.40 0.50 0.600.650.700.750.800.850.900.951.00Normalized Crack Length, a/rf [non-dim] Stress Ratio, ( rz)max/ 1 [non-dim] 0.25 VFR (50% DISPL) 0.50 VFR (50% DISPL) 0.75 VFR (50% DISPL) 0.25 VFR (100% DISPL) 0.50 VFR (100% DISPL) 0.75 VFR (100% DISPL) Interface Strength RatioFigure 23: Stress Ratio, 1 max) ( rzas a Function of Normalized Crack Length, fr a for Silicon Carbide/Epoxy at 100% and 50% Displacements 5.2 Fiber Orthotropy Up to this point, the composites used in this study were assumed to be fibers with isotropic material properties. This section examines what impact fiber

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46 orthotropy have on interface failure. This is accomplished by comparing analysis results of the composite with the orthotropic fiber (tran sversely isotropic) to the analysis results of the composite with isotropic fiber. In both cases, graphite/epoxy composite is used. As can be seen in Figure 24, the fiber orthotropic material properties have a unique impact on tensile interface failure. The possibility of interface failure in tension increases with increasing fi ber-volume fraction between 0.6-0.9 normalized crack lengths. During this crack growth, the composite with the highest (0.75) fiber-volume fraction is most likely to experience tensile interface failure. However, when the normalized crack reac hes 0.9, the possib ility of interface failure in tension for a composite with orthotropic fiber becomes completely independent of fiber-volume fr action. That is, all three fiber-volume fractions generate the same tensile stress ratios. Al so, at that point in crack growth, the composite with isotropic fiber and 0.75 fiber-volume fraction has the same chance of experiencing interface tensile fa ilure as the compos ite with orthotropic fiber.

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47 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim] Stress Ratio, ( rr)max/ 1 [non-dim] 0.25 FVF (ORTHOTROPIC) 0.50 FVF (ORTHOTROPIC) 0.75 FVF (ORTHOTROPIC) 0.25 FVF (ISOTROPIC) 0.50 FVF (ISOTROPIC) 0.75 FVF (ISOTROPIC)Figure 24: Stress Ratio, 1 max) (rr as a Function of Normalized Crack Length, fr a for Graphite/Epoxy Composite with Or thotropic and Isotropic Fibers Figure 25 illustrates the effect of ort hotropic fiber on interface failure in shear. It can be clearly seen that the pres ence of orthotropic fiber diminishes the possibility of interface shear failure. As the crack propagates and approaches fiber-matrix interface, the shear stress rati o get progressively smaller. In fact, when the normalized crack length reaches 0.97, the composite with 0.25 fibervolume fraction has the largest possibilit y to experience an interface failure in shear.

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48 Also, during the final stage of crack gr owth, the composites with isotropic fibers are more prone to undergo an interf ace failure in shear than those with orthotropic fibers. 0.00 0.10 0.20 0.30 0.40 0.50 0.600.650.700.750.800.850.900.951.00Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rz)max/ 1 [non-dim] 0.25 FVF (ORTHOTROPIC) 0.50 FVF (ORTHOTROPIC) 0.75 FVF (ORTHOTROPIC) 0.25 FVF (ISOTROPIC) 0.50 FVF (ISOTROPIC) 0.75 FVF (ISOTROPIC) Figure 25: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Graphite/Epoxy Composite with Orthotropic and Isotropic Fibers 5.3 Thermal Stress The thermal stresses in the composite arise due to a mismatch between thermal expansion coefficients of a fiber and a matrix. This mismatch puts the

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49 fiber-matrix interface either in tens ion or in compression depending which constituent has larger coeffici ent of thermal expansion. The thermal strain and the corresponding thermal stress were calculated based on the following equation: ) (TT (36) where, T is the thermal strain is the coefficient of thermal expansion, and T is the difference between the ambi ent and processing temperatures, ) (REFT T T The processing was taken to be REFT=170 C and the final temperature was assumed to be a room temperature at T = 20 C. The resulting negative T indicates shrinkage of both component s during the cooling process. To examine the influence of therma l stress on interface failure, two identical composites, one in presence and other in absence of thermal stresses, were analyzed and the results were compared. The composite used in the analysis was graphite/epoxy with material properties listed in section 4.3. By looking at Figure 26, it can be concluded that thermal stresses reduce the possibility of interface failure in tension. Because th e analyzed composite had f m the resulting compressive stre ss normal to the interface makes debonding from crack in fiber less likely.

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50 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 0.60.650.70.750.80.850.90.951 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rr)max/ 1 [non-dim] 0.25 FVF (Thermal Stress) 0.50 FVF (Thermal Stress) 0.75 FVF (Thermal Stress) 0.25 FVF (w/o Thermal Stress) 0.50 FVF (w/o Thermal Stress) 0.75 FVF (w/o Thermal Stress)Figure 26: Stress Ratio, 1 max) ( rr as a Function of Normalized Crack Length, fr a for Graphite/Epoxy Composite With and Wit hout Thermal Load Present The Figure 27 illustrates t he impact of thermal stress on interface failure in shear. For the composite with thermal stre ss present, the shear stress ratios are significantly lower during crack propagat ion between normalized crack lengths of 0.6 to 0.8 but for larger cracks, the differences between the shear stress ratios among the same fiber-volume frac tions are almost negligible.

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51 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.600.650.700.750.800.850.900.951.00 Normalized Crack Length, a/rf [non-dim]Stress Ratio, ( rz)max/ 1 [non-dim] 0.25 FVF (Thermal Stress) 0.50 FVF (Thermal Stress) 0.75 FVF (Thermal Stress) 0.25 FVF (w/o Thermal Stress) 0.50 FVF (w/o Thermal Stress) 0.75 FVF (w/o Thermal Stress) Figure 27: Stress Ratio, 1 max) ( rz as a Function of Normalized Crack Length, fr a for Graphite/Epoxy Composite With and Wi thout Thermal Load Present

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52 CHAPTER 6 CONCLUSIONS The conclusions gathered from the result s of this study can be summarized as follows: 1. The fiber-volume fraction has a profound influence on interface failure a) The possibility of interface tensile fa ilure increases with higher fiber-volume fraction b) The interface tensile failure is more likely to occur for composites with high fiber-volume fraction and high fiber-to-matrix moduli ratio c) The increase of fiber-volume fracti on from medium to high makes the interface shear failure more likely during initial crack growth but the possibility diminishes as the crack appr oaches the fiber-to-matrix interface. 2. The interface in the silicon carbide/epox y will never fail in shear regardless of the fiber-volume fracti on and displacement applied. 3. The tensile interface failure in the silicon carbide/epoxy will take place between 4% and 100% of ulti mate longitudinal strain. 4. In the early and middle stages of cra ck growth, the presence of orthotropic fiber in the graphite/epoxy composite increases the likelihood of tensile interface failure with increasing fiber-vol ume fraction. During the final stage

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53 of crack propagation the tensile interfac e failure is not influenced by fibervolume fraction. 5. The fiber orthotropy in the graphite/epoxy composite diminishes the likelihood of interface failure in shear. 6. The presence of thermal stress in the graphite/epoxy com posite lowers the possibility of interface failure in tension and in shear.

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54 REFERENCES 1. Agarwal B.D., Broutman L.J ., Analysis and Performance of Fiber Composites John Wiley and Sons, New York, 1980, pp. 46-48 2. Chawla K.K., Ceramic Matrix Composites Chapman and Hall,1993, pp. 300-304 3. Gupta G.D., A Layer ed Composite with a Broken Laminate, International Journal of Solids and Structures 1973, 9, pp. 1141-1154 4. Erdogan F. and Bakioglu M., Fracture of Plates which Consist of Periodic Dissimilar Strips International Journal of Fracture Mechanics 1976, 12, pp. 71-84 5. Bechel V.T., Kaw A.K., Fractu re Mechanics of Composites with Nonhomogenous Interfaces and Nondilute Fiber Volume Fractions, International Journal of Solids and Structures 1994, 31(15), pp. 20532070 6. He M., Hutchinson J. W., Crack Deflection at an Interface Between Dissimilar Elastic Materials, International Journal of Solids and Structures 1989, 25(9), pp. 1053-1067 7. Swenson D. O., Rau Jr. C. A., The Stress Distribution Around a Crack Perpendicular to an Interface Between Materials International Journal of Fracture Mechanics 1970, 6(4), pp. 357-365 8. Cornie J. A., Argon A. S., Desi gning Interfaces in Inorganic Matrix Composites MRS Bulletin 1991, 16(4), pp. 32-38

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55 9. Pagano N. J ., On the Micromechanical Fail ure Modes in a Class of Ideal Brittle Matrix Composites. Pa rt 2. Uncoated-Fiber Composites Composites Part B, 1998, 29B, pp. 121-130 10. Kaw A.K ., Mechanics of Composite Materials CRC Press, 1997, pp. 180-181 11. Chawla K.K., Ceramic Matrix Composites Chapman and Hall, 1993, pp. 207-210 12. Kassir M. K., Sih G. C., Mechanics of Fracture Volume 2: ThreeDimensional Crack Problems Noordoff Internationa l Publishing, 1975, pp. 2-5 13. Properties of SCS Fibers http://www.specmaterial s.com/silicarbsite.htm (Accessed 10/11/2004) 14. Online Material Data Sheet http://www.matweb.com (Accessed 10/11/2004) 15. Tandon G.P., Kim Y. R., Construction of the Fiber-Matrix Interfacial Failure Envelope in a Polymer Matrix Composite International Journal for Multiscale Computational Engi neering, 2004, 2(1), pp. 65-77 16. Base Properties of Graphite Fibers, http://casl.ucsd.edu/data_analysis (Accessed 08/22/2004)

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56 APPENDICES

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57 Appendix 1: Ansys Input File /PREP7 ET,1,PLANE2, ,1 A=0.9 K,1,0,0 K,2,A,0 K,3,1,0 K,4,1.155,0 K,5,10,0 K,6,10,30 K,7,1.155,30 K,8,1,30 K,9,0,30 K,10,0,1.2 K,11,1,1.2 K,12,1.155,1.2 LINES L,1,2 LESIZE,1, ,30 L,2,3 LESIZE,2, ,35 L,3,4 LESIZE,3, ,100,10 L,4,5 LESIZE,4, ,75 L,5,6 L,6,7 L,7,8 L,8,9 L,9,10 LESIZE,9, ,50 L,10,1 LESIZE,10, ,65 L,10,11 LESIZE,11, ,50 L,11,12 LESIZE,12, ,50

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58 Appendix 1 (Continued) L,11,8 LESIZE,13, ,100 L,12,7 LESIZE,14, ,100 L,3,11 LESIZE,15, ,360,20 L,4,12 LESIZE,16, ,75 AREAS AL,1,2,15,11,10 AL,15,3,16,12 AL,4,5,6,14,16 AL,13,12,14,7 AL,11,13,8,9 FIBER PROPERTIES Efy=260E09 Efx=14E09 Efz=14E09 Vf=0.75 Vm=1-Vf PSRfyx=0.26 PSRfxy=(PSRfyx*Efx)/Efy PSRfyz=0.26 PSRfxz=0.33 Gfyx=51E09 Gfyz=51E09 Gfxz=8.27E09 CTEfy=-0.855E-06 CTEfx=3.24E-06 CTEfz=3.24E-06 MATRIX PROPERTIES Em=3.5E09 PSRm=0.35 Gm=Em/(2*(1+PSRm)) CTEm=90E-06

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59 Appendix 1 (Continued) COMPOSTE FORMULAS Eyy=Efy*Vf+Em*Vm Exx=(Em)/(1-(Vf**0.5)*(1-Em/Efx)) Ezz=(Em)/(1-(Vf**0.5)*(1-Em/Efx)) Gyx=(Gm)/(1-(Vf**0.5)*(1-Gm/Gfyx)) Gyz=(Gm)/(1-(Vf**0.5)*(1-Gm/Gfyx)) Gxz=(Gm)/(1-(Vf**0.5)*(1-Gm/Gfxz)) PSRyx=PSRfXY*Vf+PSRm*Vm PSRxy=(PSRyx*Exx)/Eyy PSRyz=PSRfxy*Vf+PSRm*Vm PSRxz=(Exx/(2*Gxz))-1 CTEyy=(Efy*CTEfy*Vf+Em*C TEm*Vm)/(Efy*Vf+Em*Vm) CTExx=Vf*CTEfy*(1+PSRfyx)+Vm*CTEm*(1+PSRm)(Vf*PSRfyx+Vm*PSRm)*CTEyy CTEZZ=Vf*CTEfy*(1+PSRfyx)+Vm*CTEm*(1+PSRm)(Vf*PSRfyx+Vm*PSRm)*CTEyy MATERIAL 1 (FIBER) MP,EY,1,Efy MP,EX,1,Efx MP,EZ,1,Efz MP,PRXY,1,PSRfXY MP,PRYZ,1,PSRfyz MP,PRXZ,1,PSRfxz MP,GXY,1,GfYX MP,GYZ,1,GfYZ MP,GXZ,1,Gfxz MP,ALPX,1,CTEfx MP,ALPY,1,CTEfy MP,ALPZ,1,CTEfz MP,REFT,1,170 MATERIAL 2 (MATRIX) MP, EX, 2, Em MP, PRXY, 2, PSRm

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60 Appendix 1 (Continued) MP, ALPX, 2, CTEm MP, REFT, 2,170 MATERIAL 3 (COMPOSITE) MP,EX,3,Exx MP,EY,3,Eyy MP,EZ,3,Ezz MP,PRXY,3,PSRXY MP,PRYZ,3,PSRyz MP,PRXZ,3,PSRxz MP,GXY,3,Gyx MP,GYZ,3,Gyz MP,GXZ,3,Gxz MP,ALPX,3,CTExx MP,ALPY,3,CTEyy MP,ALPZ,3,CTEzz MP,REFT,3,170 BFUNIF,TEMP,20 A262=A/262 KSCON,2,A262,1,6,1 MAT,1 AMESH,1 MAT,1 AMESH,5 MAT,2 AMESH,2 MAT,2 AMESH,4 MAT,3 AMESH,3 BOUNDARY CONDITIONS DL,2,1,SYMM DL,3,2,SYMM DL,4,3,SYMM

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61 Appendix 1 (Continued) DL,8,5,UY,0.46587 DL,6,3,UY,0.46587 DL,7,4,UY,0.46587 Finish /Solu Solve finish NSEL,R,LOC,X,0.9999999,1.00001 NSEL,R,LOC,Y,0,30

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62 Appendix 2: Mapl e Instructions p : pressure on crack a: Radius of crack E: Young's modulus nu: Poisson's ratio mu: Shear modulus rin, zin: radial and z loaction, respectively > restart; p:=-50000; a:=1.0; E:=30E6;nu:=0.3;mu: =E/(2.*(1+nu)) ;rin:=0.581468892926; zin:=0.107101144421; > A:=-p/(Pi*mu)*(sin(s *a)-s*a*cos(s*a))/(s^2); > f:=int(A/s*BesselJ(0,r*s)*exp(-s*z),s=0..infinity); > sz:=2*mu*(-diff(f,z, z)+z*diff(f,z,z,z)); > stheta:=2*mu*(1/r*diff(f,r)+2* nu*diff(f,r,r)+z/r*diff(f,r,z)); > sr:=2*mu*((1-2*nu)*diff(f,r,r)-2 *nu*diff(f,z,z)+z*diff(f,r,r,z)); > trz:=2*mu*z*diff(f,r,z,z); > ur:=(1-2*nu)*diff(f,r)+z*diff(f,r,z); > uz:=(-2*(1-nu)*diff(f,z))+z*diff(f,z,z); > evalf(subs(r=rin,z=zin,ur)); > evalf(subs(r=rin,z=zin,uz)); > evalf(subs(r=rin,z=zin,sr)); > evalf(subs(r=rin,z=zin,sz));

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63 Appendix 2 (Continued) > evalf(subs(r=rin,z=zin,stheta)); > evalf(subs(r=rin,z=zin,trz));

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64 Appendix 3: Mathcad File pnc50000 0.3 E30106 ur.ncrr r1.00 z0.237853902209101 r pnc E ur.nc0.0005 DISPLACEMENTS From Maple ur.c0.0002309016707 ur.totalur.ncur.c ur.total7.30902 104 ur.ansys0.72998 103 From Ansys urur.totalur.ansys ur.total 100 ur0.126 zpncE uz.ncz z uz.nc0.000039642317 uz.c0.00002240994425 From Maple uz.totaluz.ncuz.c uz.total0.0000621 uz.ansys0.6198910 4 From Ansys uzuz.totaluz.ansys uz.total 100 uz0.102

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65 Appendix 3 (Continued) STRESSES r.c41963.92344 .c21332.52740 From Maple r.nc0 .nc0 r.totalr.cr.n c .total .c .n c r.total41963.9234 .total21332.5274 r.ansys41950 From Ansys .ansys21326 From Ansys rr.totalr.ansys r.total 100 .total .ansys .total 100 r0.033 0.031 z.c52234.66801 From Maple rz.c11742.67654 From Maple z.nc50000 rz.nc0 z.totalz.cz.n c rz.totalrz.crz.n c z.total102234.668 rz.total11742.6765 z.ansys0.10214106 From Ansys rz.ansys11739 z0.093 rz0.031 From Maple zz.totalz.ansys z.total 100 From Ansys rzrz.totalrz.ansys rz.total 100


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Effect of fiber volume fraction on fracture mechanics in continuously reinforced fiber composite materials
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by Thomas Wasik.
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[Tampa, Fla.] :
b University of South Florida,
2005.
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Thesis (M.S.M.E.)--University of South Florida, 2005.
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ABSTRACT: The application of advanced composite materials, such as graphite/epoxy, has been on the rise for the last four decades. The mechanical advantages, such as their higher specific stiffness and strength as compared to monolithic materials, make them attractive for aerospace and automotive applications. Despite these advantages, composites with brittle fibers have lower ductility and fracture toughness than monolithic materials.One way to increase the fracture toughness of composites is to have a weak fiber-matrix interface that would blunt crack tips by crack deflection into the interface and hence enhance fracture toughness. However, this also reduces the transverse properties of the composite.Therefore, an optimum fiber-matrix interface would be the one that is just weak enough to cause crack deflection into interface.This study investigates the effect of fiber-to-matrix moduli ratio, fiber-volume fraction, fiber orthotropy, and thermal stresses on the possibility of crack deflection. A finite element model is used to analyze a 2-D axisymmetric representative volume element- a three-phase composite cylinder made of fiber, matrix, and composite. A penny shaped crack is assumed in the fiber.To determine whether the crack would deflect into the interface or propagate into the matrix, maximum stresses at the fiber-matrix interface and in the matrix are compared to the interface and matrix strengths.As opposed to most studies in the literature, this study found that fiber-volume fractions do have an impact on crack deflection and this impact increases with large fiber-to-matrix moduli ratios.
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Adviser: Dr. Autar K. Kaw.
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Interface failure modes.
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