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MICROMECHANICS, FRACTURE MECHANICS AND GAS PERMEABILITY OF COMPOSITE LAMINATES FOR CRYOGENIC STORAGE SYSTEMS By SUKJOO CHOI 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 2005 Copyright 2005 by SUKJOO CHOI This document is dedicated to my parents and wife. ACKNOWLEDGMENTS I am very grateful to Dr. Bhavani V. Sankar for providing me the opportunity to complete my Ph.D. studies under his exceptional guidance and financial support. He is not only my academic advisor but also he had a great influence in my life. Throughout this research, I have greatly appreciated his consistent encouragement, patience and positive attitude. Many thanks should go to my colleagues in the composite materials laboratory. We shared many great unforgettable memories and moments. I thank to Sujith Kalarikkal who help me to complete the permeability test. Moreover, I would like to thank my Korean friends, Taeyoung Kim, Jongyoon Ok, Chungsoo Ha, Wonjong Noh, Seawoong Jung, Sunjae Lee and Dongki Won, who provided me invaluable academic feedback, encouragement and companionship. I would like to express my deepest appreciation to my parents for continuous support and love. I would like to thank to my wife Jeesoo Lee and my unborn child. I am also thankful to God for giving me the opportunity to succeed at the University of Florida. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES ................... ...................... ... .... .............. vii LIST OF FIGURES ......... ........................................... ............ ix ABSTRACT ........ ........................... .. ...... .......... .......... xii 1 IN TR OD U CTION ............................................... .. ......................... .. Micromechanics Analysis of Laminated Composites at Cryogenic Conditions ..........4 Fracture Toughness of a Transverse Crack at Cryogenic Conditions ......................6 Permeability of Graphite Fiber Composite Materials at Cryogenic Conditions...........9 2 MICROMECHANICS METHODS TO PREDICT MICROSTRESSES IN LAMINATED COMPOSITES AT CRYOGENIC TEMPERATURES .................. 13 M icrom mechanics M odel ....................................... .. ......................................13 ThermoElastic Properties of the Composite Constituents.......................................16 Estim ation of Therm oElastic Constants ............................................... ............... 19 Effects of Fiber V olum e Fraction ........................................................ ............... 23 Prediction of Stresses at Microscopic Level......... .............. ....... ............ 25 Application to the Liquid Hydrogen Composite Tank ............................................30 3 FRACTURE TOUGHNESS FOR A TRANSVERSE CRACK IN LAMINATED COMPOSITES AT CRYOGENIC TEMPERATURES ........................................35 Stress Singularities at a Cracktip Normal to a Plyinterface.................................35 Fracture Toughness at Room Temperature.............................................................. 39 Fracture Toughness at Cryogenic Temperatures ................................. ............... 44 4 PERMEABILITY TESTING OF COMPOSITE MATERIALS FOR A LIQUID HYDROGEN STORAGE SYSTEM ...................................... ........................ 50 Standard Test Method for Determining Gas Permeability ......................................50 P erm ability A pparatu s ..................................................................... ...................52 Specim en Description .................. ............................ .. ...... ................. 56 T testing Procedure ............................................ .. .. .... ........ .... .. ... 57 C alcu nation s ........................................................................... 5 8 C alib ratio n ................................................................5 9 P erm eab ility T est R esu lts ........................................ ............................................6 1 O ptical M icroscopic A nalysis.......................................................... ............... 63 5 RESULTS AND DISCU SSION ........................................... .......................... 68 Micromechanics Method to Predict Thermal Stresses for Laminated Composites at Cryogenic Temperature..... ....................... ...... ..................... 68 Fracture Toughness for a Transverse Crack in Laminated Composites at Cryogenic Tem peratures ................... ... ...... .... ...................................... 70 Permeability Testing for Laminated Composites for a Liquid Hydrogen Storage System ..................................... .................. ................ ......... 71 APPENDIX STRESS SINGULARITY USING STROH'S METHOD.........................73 L IST O F R E F E R E N C E S ........................................................................ .....................77 B IO G R A PH IC A L SK E T C H ..................................................................... ..................81 LIST OF TABLES Table page 21 Periodic boundary conditions for the square unit cell for unit values of different strain com p on ents.......... .............................................................. ........ .............. 16 22 Periodic boundary conditions for the hexagonal unit cell for unit values of different strain com ponents. ............................................. ............................. 16 23 Material properties of fibers used in the verification problem ..............................19 24 Results of elastic constants for laminated composites calculated by empirical fo rm u la s............................. ............................................................. ............... 2 1 25 Results of elastic constants for laminated composite obtained by m icrom ech anics analy sis ............................................................... .....................22 26 Comparisons of G23 for square and hexagonal unitcells in order to test transverse isotropy ........... .............................................................. ......... ....... 23 27 Macrostrains in different laminates due to thermal loads (AT=405 K).The subscript Z denotes the thickness direction ................................... ............... 26 28 Maximum principal stresses of a macro model for various graphite/epoxy com posite system s ........................ ........................ ......... 27 29 Maximum principal stresses in the fiber and matrix phases of a unit cell for various graphite/epoxy composite systems from microanalysis.............................27 210 Percentage difference of principal stresses estimated by square and hexagonal unit cells. The negative sign indicates that microstress result estimated by the hexagonal unit cell is lower .............. ... ................................ 27 211 Laminate macrostrains used to obtain the microstresses in the inner and outer facesheets of the sandwich composite without internal pressure .........................32 212 Laminate macrostrains used to obtain the microstresses in the inner and outer facesheets of the sandwich composite subjected to a pressure of 10 KPa..............32 213 Maximum and minimum principal stresses in the fiber and matrix phases in the inner and outer facesheets of the sandwich composite without internal pressure. .33 214 Maximum and minimum principal stresses in the fiber and matrix phases in the inner and outer facesheets of the sandwich composite subjected to a pressure of 10 KPa...................... ................ ..... ....... 33 31 Material properties of glass/epoxy and carbon/epoxy laminates ...........................36 32 Singularity results of Eglass/epoxy................... ..... ........................ 38 33 Singularity results of graphite/epoxy. ............. ......................... .....................38 34 Dimensions of specimens and various midply thickness .....................................42 35 Fracture load and fracture toughness at room and cryogenic temperatures ............42 41 Molecular diameter of various gases from CRC Handbook of Chemistry and Physics, 54th Edition......... .................. .. ................ ...... ........... 55 42 D description of com posite specim ens ............................................. ............... 56 43 Permeability of laminated composites for various number of cryogenic cycles......61 44 Permeability of textile composites for various number of cryogenic cycles. ..........62 45 Permeability of laminated composites embedded with nanoparticles for various number of cryogenic cycles .............................................. .................. 62 LIST OF FIGURES Figure page 11 The X33 test vehicle which NASA proposed as the replacement for the Space S h u title ...................................... .................................... ................ 1 12 Failure of sandwich composites used for the LH2 storage system ........................2 13 D am age progression at cryogenic conditions.................................. .....................3 14 SEM images of microcrack propagation in composite laminates after thermo m ech an ical cy cles................................................. ................ 7 15 Variation of stresses acting normal to a crack tip, when the crack reaches a bi m material interface ................................................ ......................... 8 21 A square representative volume element and corresponding finite element m esh ...............................................................................14 22 A hexagonal RVE and corresponding FE mesh.................................................15 23 Geometry of square and hexagonal unit cells. ................................. ............... 15 24 Actual and average CTE of epoxy from curing temperature to cryogenic tem peratures. .........................................................................18 25 Young's modulus of epoxy as a function of temperature. .................................18 26 Longitudinal coefficient of thermal expansion with various fiber volume fractions for glass/epoxy and graphite/epoxy laminates. .......................................24 27 Transverse coefficient of thermal expansion with various fiber volume fractions for glass/epoxy and graphite/epoxy laminates. .......................................24 28 Flow chart of algorithm used to predict the failure due to microstresses ..............25 29 Normal and shear stresses at the fibermatrix interface in square and hexagonal unit cells. ............................................................................29 210 Interfacial normal stresses in uniaxial graphite/epoxy laminate system at AT = 405 K ........................................................................ .. ...... ....... 29 211 Interfacial shear stresses in uniaxial graphite/epoxy laminate system at A T = 4 0 5 K ................................ ............. .... ....................................3 0 212 Maximum principal stress distribution in the LH2 Graphite/Epoxy composite tank. Tank pressure = 10 KPa and temperature = 50 K (AT= 405 K). .................31 31 (a) Geometry of interfacial fracture specimens, (b) A crack normal to laminate interface with different stacking sequence. ............. ......................... ...............35 32 Deformed geometry in the vicinity of a crack and interfacial fracture finite elem ent m odel. ..................................................... ................. 37 33 Contour plots of stress distribution for a [0/90/O]T composite model at a cracktip under tensile and bending loads. ..................................... ............... 37 34 Stress distribution under (a) tensile loads; (b) bending loads. ................................38 35 Fourpoint bending test to determine the fracture load........................................40 36 Loaddisplacement curves of fourpoint bending tests at room temperature for the Specim en 1 .....................................................................4 1 37 Loaddisplacement curves of fourpoint bending tests at room temperature for the Specim en 2 ....................... .................... ..................... .. ......4 1 38 Loaddisplacement curves of fourpoint bending tests at room temperature for the Specimen 3........................................... ........... 41 39 Stress distribution for the 4Pt bending simulation at (a) room temperature; (b) cryogenic tem perature. ............................................. .............................. 43 310 Variation of K = r^ with the distance from the crack tip at room temperature.......43 311 Logarithmic plot of the stresses as a function of distance from the cracktip..........44 312 Cryogenic experimental setup of the fourpoint bending test................................45 313 Crack propagation in the 900 layer of a graphite/epoxy laminate at cryogenic tem perature................................... ................................ ........... 45 314 Contour plot of stresses normal to ply direction near the freeedge in a graphite/epoxy laminate at cryogenic temperature. ............................................46 315 Loaddisplacement curves from fourpoint bending tests at cryogenic temperature. The midply thickness and hence the specimen thickness varied from specimen to specimen ............... ....................... .. ... ............... 46 316 Loaddisplacement curves from fourpoint bending tests at cryogenic tem perature for the Specim en 1 ........................................ ......................... 47 317 Loaddisplacement curves from fourpoint bending tests at cryogenic tem perature for the Specim en 2 ........................................ .......................... 47 318 Loaddisplacement curves from fourpoint bending tests at cryogenic tem perature for the Specim en 3. ........... ...................................... .....................48 319 Variation of K = r^ with the distance from the crack tip at cryogenic tem perature................................. ............................ ..... ..........48 41 Permeability experimental setup for manometric determination method ...............51 42 Permeability experimental setup for volumetric determination method ................52 43 Perm ability testing apparatus. ........................................ ........................... 53 44 Specimen installation between upstream and downstream chambers....................54 45 Variation of ambient pressure for 13 hours at test condition ................................60 46 Variation of barometric pressure and indicator position as a function of time. .......60 47 Logarithm of the permeability for composite specimens with increase of cryogenic cycles. ......................................................................63 48 Cross sectional view of the graphite/epoxy composite specimen C2 before cryogenic cycling: (a) 10X magnification; (2) 40X magnification........................64 49 Microcrack propagation on the graphite/epoxy composite specimen C2 after cryogenic cycling: (a) 1OX magnification; (2) 40X magnification........................65 410 Cross sectional view of the textile specimen T1 before cryogenic cycling, 10X m agnification ............. .................................. ........ .. ........ .... 66 411 Microcrack propagation in the textile specimen T1 after cryogenic cycling: (a) 10X magnification; (b) 40X magnification ....................................................... 66 A. 1 Geometry of an angleply laminated composite and a crack normal to an interface between two anisotropic materials. ................................ ..................73 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 MICROMECHANICS, FRACTURE MECHANICS AND GAS PERMEABILITY OF COMPOSITE LAMINATES FOR CRYOGENIC STORAGE SYSTEMS By Sukjoo Choi May 2005 Chair: Bhavani Sankar Major Department: Mechanical and Aerospace Engineering A micromechanics method is developed to investigate microcrack propagation in a liquid hydrogen composite tank at cryogenic temperature. The unit cell is modeled using square and hexagonal shapes depends on fiber and matrix layout from microscopic images of composite laminates. Periodic boundary conditions are applied to the unit cell. The temperature dependent properties are taken into account in the analysis. The laminate properties estimated by the micromechanics method are compared with empirical solutions using constituent properties. The micro stresses in the fiber and matrix phases based on boundary conditions in laminate level are calculated to predict the formation of microcracks in the matrix. The method is applied to an actual liquid hydrogen storage system. The analysis predicts micro stresses in the matrix phase are large enough to cause microcracks in the composite. Stress singularity of a transverse crack normal to a plyinterface is investigated to predict the fracture behavior at cryogenic conditions using analytical and finite element analysis. When a transverse crack touches a plyinterface of a composite layer with same fiber orientation, the stress singularity is equal to /2. When the transverse crack propagates to a stiffer layer normal to the plydirection, the singularity becomes less than 12 and vice versa. Finite element analysis is performed to predict the fracture toughness of a laminated beam subjected to fracture loads measured by fourpoint bending tests at room and cryogenic temperatures. As results, the fracture load at cryogenic temperature is significantly lower than that at room temperature. However, when thermal stresses are taken into consideration, for both cases of room and cryogenic temperatures, the difference of the fracture toughness becomes insignificant. The result indicates fracture toughness is a characteristic property which is independent to temperature changes. The experimental analysis is performed to investigate the effect of cryogenic cycling on permeability for various composite material systems. Textile composites have lower permeability than laminated composites even with increasing number of cryogenic cycle. Nanoparticles dispersed in laminated composites do not show improvement on permeability. The optical inspection is performed to investigate the microcrack propagation and void content in laminated composites and compared the microscopic results before and after cryogenic cycling. CHAPTER 1 INTRODUCTION A composite material is made of two or more constituent materials to form a new material that is often superior in some respects to the constituents themselves. Composite materials offer many advantages, such as lightweight, high specific stiffness and specific strength, and low coefficient of thermal expansion. The next generation of reusable space launch vehicles is supposed to provide a 10fold reduction in the cost of launching payloads into space, from $10,000 to $1,000 per pound in Figure 11. ..  Figure 11. The X33 test vehicle which NASA proposed as the replacement for the Space Shuttle. Reducing the structure weight of the vehicle is of paramount importance in reducing the launch cost. Composite materials such as graphite/epoxy offer many advantages such as low density, high specific stiffness and specific strength, and low coefficient of thermal expansion. Therefore, fiber reinforced composite materials are candidate materials for cryogenic storage systems e.g., the liquid hydrogen (LH2) tank. Figure 12. Failure of sandwich composites used for the LH2 storage system. A previous design of the liquid hydrogen composite tank used composite sandwich structure made of graphite/epoxy composites and a honeycomb core. The composite tank is subjected to extremely low to high temperatures during and after reentry into the atmosphere. During the proof test in November 1999 at NASA Marshall Space Flight Center, the composite liquid hydrogen tank failed because of microcrack propagation due to thermal stresses combined with mechanical loads (see Figure 12 [1]). Microcracks developed in the inner facesheet of the sandwich structure at cryogenic conditions. Then the microcracks, which usually develop in the transverse plies, can become delaminations. The delaminations connect the microcracks in adjacent layers and provide a leakage path for the cryogen. This can lead to catastrophic failure of the sandwich structure [1]. The failure motivates the present study on micromechanics analysis methods to predict microcracks in laminated composites. Transverse Cracks  7t  4~ Delamination Debonding of the facesheet Figure 13. Damage progression at cryogenic conditions. The damage progression of laminated composites at cryogenic temperatures is illustrated in Figure 13. At cryogenic temperatures, the microcracks initiate and propagate in laminated composites due to difference in thermal contraction between the fiber and matrix phases. When microcracks in the polymer matrix develop and reach the adjacent layers, the transverse cracks initiate. When the transverse crack develops further, the crack deflects through the interface between layers and delamination initiates. In the case of composite sandwich construction, debonding of the facesheet develops. The purpose of this study is to investigate the possible causes of failure in laminated composites at cryogenic conditions. First, the micromechanics method is developed to predict the thermal stresses at microscale in fiber reinforced composites at cryogenic conditions. Second, fracture toughness of a transverse crack in composite laminates is evaluated to investigate the effect of cryogenic conditions on transverse Microcracks crack propagation. Third, the permeability of various composite materials is evaluated to investigate the effect of cryogenic cycling on permeability. Micromechanics Analysis of Laminated Composites at Cryogenic Conditions From a macroscopic perspective, the composite material is considered to be homogeneous and transversely isotropic or orthotropic in general. For example, the laminated plate theory has been formulated based on this assumption. Even when three dimensional analyses are used for composite structures, each ply or layer of the composite is modeled as a homogeneous orthotropic material. This macroscopic approximation has been found to be satisfactory in most analyses including thermal stress analyses. Thus most of the thermal stress problems in composites focus on the differences in thermal expansion coefficients between the plies. However, in extreme situations a micromechanics approach wherein the fiber and matrix phases are differentiated is necessary for accurate prediction of stresses, and hence failure. The present problem falls in this category. In order to predict the failure of a composite structure at the macroscopic scale, investigation of micromechanical behavior and understanding the failure mechanisms in the fibers and matrix at a microscale are necessary [2]. To make effective use of fiberreinforced composites, we need a methodology to predict the thermomechanical properties at various temperatures as a function of fiber, matrix and interphase properties. These properties will be a strong function of temperature. Micromechanics is the study of such macroscopic properties from that of the constituent materials. In this study, microcracking behavior of the liquid hydrogen composite tank at cryogenic temperature is investigated using the micromechanics method. Failure of the composite tank under thermal and mechanical loads is evaluated by utilizing the commercial finite element software ABAQUS. Also, the analysis is useful to develop a new composite material system for the liquid hydrogen composite tank by changing the combination of constituent properties. Before finite element methods were widely available micromechanics analysis of fiberreinforced composites was performed using analytical methods (e.g., Chen and Cheng [3]). They analyzed the unit cell of a composite by solving the governing elasticity equations using an infinite series and employing a combination of Fourier series and least square methods. The periodic boundary conditions for stresses and displacements were satisfied on symmetric boundaries. Micromechanics analysis methods for elasticplastic composites were investigated using the bounding technique by Teply and Dvorak [4]. The problem of elasticviscoplastic composites was solved by Paley and Aboudi [5] imposing continuity of traction and displacement rate at the interfaces between the constituents of a square unit cell. A square unit cell model was used by Nedele and Wisnom [6] to investigate the behavior of fiberreinforced composites subjected to shear loading. Marrey and Sankar [79] developed micromechanics methods for textile structural composites using the finite element method. Their method considered the effects of stress gradients on the strength and stiffness properties of the composite. The periodic boundary condition of various shapes of unit cells was described by Li [10] using the symmetries of the fibermatrix system to satisfy displacements and stresses in the boundary. In the present study the micromechanics method is combined with a global laminate analysis to predict the stresses in the fiber and matrix phases accurately. The method is useful to predict development of microcracks in a composite laminate at cryogenic temperatures. In order to predict the development of microcracks in fiber composites, one needs accurate description of the stresses in the matrix phase and also along the fibermatrix interface. The problem of thermal stresses is complicated by the temperature dependent properties of the constituent materials. In the present study a global/local approach is used wherein traditional structural analysis is used to obtain information on macrostrains in a ply in the composite laminate. Then the macrostrains along with the local temperature are used in a micromechanical analysis to obtain detailed information on the stresses in the constituent phases. Two types of representative volume elements (RVE) are used in the micromechanical analyses. In the first one the RVE is a square with a circular fiber at the center of the square. In the second a hexagonal RVE is used. The differences in thermal stresses in the two RVE's are discussed. The temperature dependent properties of the matrix material are taken into account in the micromechanics. However, the fiber properties are assumed to be independent to temperature changes. The microstresses in different types of laminates used in a typical liquid hydrogen tank are studied and the possibility of microcracking is discussed. The results indicate that the maximum tensile stresses in the brittle matrix reach values very close to the tensile strength of the matrix material, raising the possibility of microcrack development in composite liquid hydrogen storage systems. Fracture Toughness of a Transverse Crack at Cryogenic Conditions Microcracks develop in the facesheets of the sandwich structure due to the thermal cycling. The microcracks which usually develop in the transverse plies can propagate to an adjacent layer and become transverse cracks as shown in Figure 14. When the transverse crack deflects to the plyinterface, it becomes delaminations. The delaminations connect the microcracks in adjacent layers and provide a leakage path for the cryogen. For the liquid hydrogen composite tank, this hydrogen leakage through the interface sheet causes the catastrophic failure of the sandwich structure [1]. 4 Transverse Crack . Interfacial Crack Figure 14. SEM images of microcrack propagation in composite laminates after thermo mechanical cycles. When a microcrack propagates in a give material, the stress distribution normal to the crack plane can be described by the equation in Figure 15. Normal stresses at a cracktip are governed by the stress singularity A and stress intensity factor KI. Zak and Williams [11] found the normal stresses ahead of a cracktip are proportional to r~X (0 < Re [k] <1) where r is the distance from the cracktip. The stresses normal to the crack ahead of a cracktip can be expressed as c =KI r. The stress singularity of a micro crack in laminated composites is governed by anisotropic properties at the vicinity of a cracktip. The stress singularity describes the behavior of crack propagation. The Material 1 and the Material 2 are composite layers with specific fiber orientations. When a transverse crack reaches a plyinterface of composite layers with identical fiber orientation, stress singularity is equal to /2 which is conventional singularity for homogeneous materials. When a transverse crack propagates from the Material 1 which is stiffer than the Material 2, the transverse crack penetrates the plyinterface and continues to propagate through the Material 2 until the crack reaches a stiffer layer [12]. In this case, the stress singularity becomes larger than /2. When the Material 1 is softer then the Material 2, the transverse crack deflects to the plyinterface and becomes delamination. The stress singularity becomes lower than 1/2. In this study, conditions under which a transverse crack becomes a delamination are studied, and the fracture toughness of the transverse crack is quantified and measured. M4ateriadl Materia 2 ......... .... i,  Singularity dominated zone Figure 15. Variation of stresses acting normal to a crack tip, when the crack reaches a bi material interface. Williams [13] estimated the singularity for isotropic bimaterial systems by solving a set of eigenfunctions developed by the continuity equations of normal and tangential stresses and displacements at the plyinterface. Ting and Chou [14] have developed methods to predict singularity at the free edge of a plyinterface of laminated composites. The general equations of displacement and stresses are derived in terms of arbitrary constants. The stress singularity is determined when the boundary condition at a crack plane and the continuity equations at the plyinterface are satisfied. Later, Ting and Hoang [15] developed this method to predict singularity of a transverse crack in laminated composites. Hutchinson and Suo [16] formulated a characteristic solution to predict stress singularity for isotropic bimaterial in terms of Dundur's bimaterial parameters a and /7. Gupta et al. [12] formulated a characteristic solution to predict the singularity for anisotropic bimaterials in terms of the bimaterial parameters and individual material parameters. However, the characteristic equation is difficult to solve. In this study, stress singularity of a transverse crack in a laminated composite is calculated using Ting's methods [1415], and finite element analysis is used to compute the crack tip stress fields in various laminated composite models. The purpose of the study is to develop the finite element model which provides a confidence in capturing the stress singularity and evaluate fracture toughness of a transverse crack in laminated composites at room and cryogenic temperatures. This study is useful to investigate the cryogenic effects on fracture toughness for composite laminates. Permeability of Graphite Fiber Composite Materials at Cryogenic Conditions Composite materials are good candidates for the liquid hydrogen storage tanks for space vehicles. A fundamental issue in composite tanks is hydrogen permeability, particularly since the failure of the liquid hydrogen composite tank could be caused by hydrogen leakage. Microcracking can also occur due to thermal and structural loads and fatigue. The liquid hydrogen storage composite tank is required to sustain large thermal and structural loads, and additionally must be able to endure several cycles of thermal loading without hydrogen permeation. The purpose of the study is to present the experimental work to measure the gas permeability of various types of composite systems. The permeability testing on IM7/BMI laminated composites under biaxial strains was performed by Stokes [17] following the ASTM Standard D1434 [18]. The permeability was calculated by measuring the pressure difference across the specimen as a function of time. The permeability was measured as the strain was increased at room temperature. The permeability increase was steady until the specimen failed. Stokes found that the permeability is a time dependent parameter under constant strain condition. The crack densities for each layer were measured using microscopic optical inspection. The advantage of this test approach is that it minimizes the error due to the ambient pressure differences during the test. However, a precision pressure transducer is required to measure the accurate pressure difference across the specimen accurately. The gas leakage through laminated composite was investigated by Kumazawa, et al [19] using experimental and finite element analysis. A finite element model with initial crack path was subjected to mechanical and thermal loads. The leakage rate decreased as temperatures decreased with assuming constant crack density. The FE results were compared with experimental results. Helium detector was used to measure the permeability of a laminated composite under biaxial loading. The permeability test of hybrid composites and related films was performed by Grimsley et al [20] following the ASTM Standard D1434. The volumeflow rate was estimated by measuring the rate of moving distance of a liquid indicator in a glass capillary tube. The gas transmission rate is converted to volumeflow rate using the ideal gas law. The permeance is calculated by the gas transmission rate per upstream pressure. The permeability results for various types of hybrid composites and films were provided. Also, Herring [21] investigated the permeability of thin film polymers after pre conditioning samples. The permeability is calculated according to ASTM D1434. The volumetric method to calculate the permeability is verified by Nettles [22]. The study is to investigate the effect on permeability test of laminated composites after experiencing impact loads. Moreover, the study investigated the possible cases of testing condition which influence the permeability results. When the glass capillary tube is mounted vertically or horizontally, the variation of permeability results is insignificant. The permeability tests were performed using various types of liquid indicators in a capillary tube. The variation of permeability results was found to be insignificant. Also, the length of liquid indicator does not affect the permeability results. However, the glass capillary tube with inner diameter 0.4 mm underestimated permeability than the capillary tubes with 1.2 mm and 3 mm diameters. The permeability test on laminated composites and bonding materials were performed by Nettles [23]. The permeability results before and after cryogenic cycles were compared. The composite specimens of both type of prepreg materials and layup configuration have low permeation results after 4 and 12 cryogenic cycles. Glass et al. investigated the permeability for core materials of a composite sandwich when it is subjected to shear loads [24]. The Hexcel HRP honeycomb and the Dupont Korex honeycomb were chosen for core materials. The composite sandwich specimens were fabricated and air inlet holes were drilled through a facesheet. The permeability is calculated by measuring the flow rate of upstream air. In this study, the permeability was estimated for various composite material systems as the composite laminates undergo cryogenic cycling. An optical microscope was used to understand the nature of microcrack propagation after cryogenic cycling. The permeability increased rapidly and became constant as the number of cryogenic cycles 12 increased. The permeability is proportional to the crack density since increasing number of cracks produces a larger flow path though the thickness of the material. The optical inspection on composite specimens was performed to investigate the behavior of microcrack propagation after cryogenic cycling. The optical results were compared before and after cryogenic cycling on various composite specimens. CHAPTER 2 MICROMECHANICS METHODS TO PREDICT MICROSTRESSES IN LAMINATED COMPOSITES AT CRYOGENIC TEMPERATURES A finite element analysis based micromechanics method is developed to investigate development of microcracks in a graphite/epoxy composite liquid hydrogen tank at cryogenic temperatures. The unitcell of the composite is modeled using finite elements. Periodic boundary conditions are applied to the boundaries of the unitcell. The temperature dependent properties including the coefficient of thermal expansion of the matrix material is taken into account in the analysis. The thermoelastic constants of the composite were calculated as a function of temperature. The stresses in the fiber and matrix phases and along the fibermatrix interface were calculated. When the laminated composite structure is subjected to combined thermal and mechanical loads, the macro strains are computed from the global analysis. Then, the macrostrains and temperatures are applied to the unit cell model to evaluate microstresses, which are used to predict the formation of microcracks in the matrix. The method is applied to a composite liquid hydrogen storage system. It is found that the stresses in the matrix phase could be large enough to cause microcracks in the composite. Micromechanics Model The microscopic image of a uniaxial fiber reinforced laminate (Figures 21 and 2.2) shows that the fiber arrangement is quite random in reality. However, for analytical/numerical modeling, it is convenient to assume some repetitive pattern of fiber arrangement. The square unit cell is not a suitable model for glass, carbon, and graphite fibers since the model is not transversely isotropic but tetragonal. The square array is conceivable applicable to boron/aluminum composites in which fibers are arranged in patterns that resemble such arrays. However, it is not applicable to any type of boron tapes or prepregs. The reason is that these are thin unidirectionally reinforced layers whose thicknesss is of the order of the diameter of one boron fiber and can not be consider composite materials. The unit cell is modeled from the repetitive pattern of the fiber and matrix layout of composite laminates. In the present analysis, both square and hexagonal unit cells are considered for the micromechanics model. In both cases the dimensions are chosen such that the fiber volume ratio is 60%, which is typical of graphite/epoxy composites. When fibers are arranged in a square unit cell one can obtain a maximum fiber volume fraction of 79%. The square unit cell was modeled using 1,600 quadratic solid elements with periodic boundary conditions [79]. The periodic boundary conditions ensure displacement compatibility and stress continuity on the opposite faces of the unitcell. ** *0 Fiber Figure 21. A square representative volume element and corresponding finite element mesh. The hexagonal pattern of unit cell can be found more commonly in fibermatrix composites, especially when the composite is fabricated with high fiber volume fraction. Theoretically one can obtain a maximum fiber volume fraction of 91% with hexagonal RVE. In a hexagonal RVE there is symmetry about the yaxis and also about the +30 degree directions. The hexagonal unit cell is modeled using 2,400 quadratic solid elements with periodic boundary conditions [10]. Matrix Figure 22. A hexagonal RVE and corresponding FE mesh. C1 Y1 y, 3, v I bo a, x, 2, u " z,I1,w xo X1 ) Yo Figure 23. Geometry of square and hexagonal unit cells. The unit cells are subjected to axial and shear displacements combined with thermal effects using periodic boundary conditions. The boundary condition on a RVE element is established by symmetry consideration so that the micromechanics analysis can be confined to a single repeating element. The periodic boundary conditions maintain equal boundary displacements with the adjacent unit cells to satisfy the compatibility of displacements on opposite faces of the unit cell and enforce the continuity of stresses. The unitcell is subjected to different strain components individually using the periodic boundary conditions shown in Table 21 [79]. For the hexagonal unit cell, the periodic boundary conditions corresponding to unit value of each strain component are shown in Table 22 [10]. The equations of periodic boundary conditions corresponding individual unit strains were embedded in the ABAQUS input code to perform the finite element (FE) analysis. Table 21. Periodic boundary conditions for the square unit cell for unit values of different strain components. Ex=l y=ll z=l Yxy=l xz=l yz=l uxlUx0= L uxlMxo= 0 UxlUxO= 0 VxlVxo= 1/2L Wxlwx0= L WylWyo= L VylVyo= 0 VylVyO= L VylVyO= 0 uylUyo= 1/2L uzluz= 0 VzlVzo= 0 WzlWzo= 0 WziWzo= 0 WziWzo= t WziWzo= 0 Table 22. Periodic boundary conditions for the hexagonal unit cell for unit values of different strain components. Ex=l EY=1 Ez=l /xy=l /xz=l yz=l UalUa0= UalUa0= 0 UalUaO= 0 UalUaO= 0 UziUzo=0 VzlVzo=0 43/2L UblUb0= 0 UblUb0= 0 UblUb0= 0 Uc= 0 Wcl= 12L UblUbO= ValVaO= 12L ValVaO=0 Uc 0= 0 U,= 0 WO= 1/2L /3/2L VblVbO= 12L VblVbO= 0 Uc= 0 Vcl= 0 WalWaO= ValVa= 0 Vc' 1/21 V1c= 0 ValVao=I3/2L Vc0= 0 12L Vbl Vb= 0 Vc0= 1/2L V0= 0 VblVbO= WalWa0= WbOWbl= V,1= 0 WziWzo= 0 WziWzo= t I3/2L I3/2L V2L VcO= 0 WziWzO= 0 WblWbO= WZiWzo= 0 _3/2L ThermoElastic Properties of the Composite Constituents For accurate prediction of stresses at cryogenic conditions, one requires temperature dependent thermoelastic properties of the constituent materials. In the present study the matrix properties are considered as temperaturedependent and the fiber properties temperature independent. Most of the advanced composite systems such as aerospace graphite/epoxy are cured at about 455 K. When the temperature rises above the melting temperature Tm, the epoxy resin becomes a rubbery solid and then becomes viscous liquid. When the laminate is cooled down to the glass transition temperature Tg, the epoxy resin becomes an amorphous solid. The difference in the coefficients of thermal expansion (CTE) for the constituents under temperature changes causes residual stresses in the composite laminate. Thermal stresses in composites are largely influenced by matrix thermomechanical properties. Also, the chemical reaction of epoxy causes shrinkage which rises residual stress in matrix phase. In this study, the residual stress due to chemical reaction of epoxy is assumed to be negligible. In this study, the 9773 epoxy system is used as the matrix material. The coefficient of thermal expansion and the Young's modulus of this material system [25] as a function of temperature are shown in Figures 24 and 25, respectively. The actual and average CTE of the epoxy resin are nonlinear with respect to temperature as shown in Figure 24. The average CTE from a reference temperature is used as input in the ABAQUS finite element program. The average CTE is calculated by using the relation Tcryogenic fa(T)dT a = (2.1) T TCurg In the above equation, the curing temperature Tcuring is 455K where the epoxy resin becomes solid during curing process of composite laminates. The cryogenic temperature Tcryogenic is 50K where the liquid hydrogen boils. When the temperature decreases from curing to cryogenic temperature, the actual CTE decreases from 73.0 10 6/K to 18.1 x 106/K and the Young's moduli increases from 1.2 MPa to 5.2 MPa. The tensile strength for heatcured epoxy is in the range of 70 MPa to 90 MPa at room temperature [26]. In general, the strength of epoxy increases 18 from curing to cryogenic temperature since the epoxy becomes brittle [25], but no data is available in the complete range of temperatures up to liquid hydrogen temperature. In this study, the tensile strength of the epoxy is assumed as 100 MPa at cryogenic temperature. 0  Averaged CTE  a) a)  E (D (D E Actual CTE  o o / I I I I . U 100 200 300 Temperature, T (K) Figure 24. Actual and average CTE temperatures. 6 D4 LU o o i E S 0) 0 0 of epoxy from curing temperature to cryogenic Figure 25. Young's modulus of epoxy as a function of temperature. 0 80 0 ' 60 40 .1 " 20 C. 400 b00 1 I 1 1 I I I I I I I I I I IIIII I  I Table 23. Material properties of fibers used in the verification problem. EGlass Fiber Graphite Fiber (IM7) El (GPa) 72.4 263 E2, E3 (GPa) 72.4 19 G12, G13 (GPa) 30.2 27.6 G23 (GPa) 30.2 27.6 v12, V13 0.2 0.2 V23 0.2 0.35 an (106/Co) 5.0 0.9 a22, a33(10 6/Co) 5.0 7.2 Tensile strength (MPa) 1,104 1,725 The transversely isotropic properties of the glass and graphite fibers used in this study are shown in Table 23 [2627]. The material properties of fiber are assumed to be independent to temperature changes. Estimation of ThermoElastic Constants The properties of a composite material depend on the constituent properties and the microstructure of fiber and matrix layout and can be estimated by experimental analysis and theoretical solutions. The experimental analysis is simple and trustworthy but time consuming and expensive. Since the results are variable to fiber volume fraction, constituent properties and fabrication process, the experimental analysis is required to repeat the testing procedures. The theoretical and semiempirical methods can be effective when the composite material involves many variables which can affect composite properties. However, the methods may not be reliable for component design purpose and present difficulty in selecting a representative but tractable mathematical model for some cases such as the transverse properties of some of the unidirectional composites [26]. The mathematical solutions are unavailable as a simple form to predict the transverse properties such as shear moduli G23 and Possion's ratio v23 [26]. In this study, the micromechanics method is performed to estimate the laminate properties of various laminate systems and compared with theoretical and semiempirical solutions to verify the transverse isotropy. The unit cell model is used to estimate the elastic constants and the coefficient of thermal expansion using the FE based micromechanics method. The fiber volume fraction was assumed to be 60%. The thermoelastic stressstrain relations of the composite material at macroscale can be written as 0o Cl C12 C13 0 0 0 E a1 SC22 C23 0 0 0 2 a2 03 C33 0 0 0 e3 AT a3 (2.2) =< A 23 C44 0 0 23 a23 r31 SYM C 0 731 a31 r12 C66 \712 a12 The elastic constants and the CTE's in Equation 2.2 were obtained by performing 7 sets of micromechanical analyses. In the first 6 cases the temperature difference ATwas set to zero and the unitcell was subjected to periodic boundary conditions corresponding to one of the macrostrains as given in Tables 1 and 2. The macrostresses in the unit cell were calculated as the volume average of the corresponding microstress components: 1 NELM c = cr) V(), i = 1,6 (2.3) V k=l In Equation 2.3, k denotes the element number, NELM is the total number of elements in the FE model, V(k) is the volume of the kth element and Vis the volume of the unitcell. The average or macrostresses are used to calculate the stiffness coefficients in a column corresponding to the nonzero strain. In order to calculate the CTE's, the unit cell is subjected to periodic boundary conditions such that the macrostrains are identically equal to zero and a uniform ATis applied to the unitcell. From the macro stresses the CTE's can be calculated as {a}= C 1'1 } (2.4) AT In this study the properties and residual stresses were calculated at 50 K which corresponds to AT=405 K. The elastic constants such as Young's moduli, shear moduli and Poisson's ratios can be obtained from the compliance matrix S=C 1. Table 24. Results of elastic constants for laminated composites calculated by empirical formulas. Elastic constants Empirical Formulas (EMP) El (GPa) 45.4 E2, E3 (GPa) 19.4 G12, G13 (GPa) 6.00 G23 (GPa) 7.71 Glass/Epoxy V12, V13 0.260 V23 0.255 aO (10 6/C) 5.41 (2, 3(10 6/C0) 9.92 El (GPa) 160 E2, E3 (GPa) 11.1 G12, G13 (GPa) 5.90 G23 (GPa) 4.05 Graphite/Epoxy v12, v13 0.260 V23 0.367 S(10( 6/Co) 0.712 a2, 3 (106/Co) 12.1 The empirical formulas are used to estimate the elastic constant based on constituent properties as shown in Table 24. The rule of mixture is used to calculate the approximate elastic constant El, v12 and v13. HalpinTsai [26] equations are used to calculate the approximate elastic constants E2, E3, G12, G13, G23 and V23. Schaprey's formulas [26] are used for the coefficient of thermal expansion aO, a2 and a3. The thermoelastic constants determined from the micromechanical analyses are compared with available empirical formulas in Table 25. The results from the FE model and the empirical formulas (EMP) agree reasonably well. The elastic moduli El and E2 from the FE models and the empirical formulas differ by less than 4%. However, the transverse modulus is very sensitive to the geometry of the unit cell. Since HalpinTsai equations are empirical, the method cannot accurately predict the transverse modulus. There are no simple solutions to estimate the elastic properties G23 and v23 [26]. In present study, the properties G23 and v23 are estimated by the HalpinTsai [26] equations and the difference is comparatively larger than other results. Table 25. Results of elastic constants for laminated composite obtained by micromechanics analysis. Elastic constants El (GPa) E2, E3 (GPa) G12, G13 (GPa) G23 (GPa) V12, V13 V23 ai (10 6/Co) a2, a3(106/Co) El (GPa) E2, E3 (GPa) G12, G13 (GPa) G23 (GPa) V12, V13 V23 ai (106/Co) a2, a3(106/Co) Square Unit Cell (SQR) FE Result 45.4 19.7 6.10 4.62 0.253 0.275 5.45 9.04 159 11.5 5.98 3.40 0.257 0.417 0.698 11.6 Difference between SQR and EMP(%) 0.13 2.02 1.60 66.8 2.83 7.35 0.75 9.75 0.43 3.75 1.41 19.0 1.29 12.1 2.00 4.91 Hexagonal Unit Cell (HEX) FE Difference Result between HEX and EMP (%) 45.5 0.13 16.3 18.5 5.59 7.31 5.83 32.2 0.260 0.08 0.391 33.1 5.46 0.91 9.27 7.03 160 0.10 10.8 2.31 5.50 7.17 3.72 8.99 0.254 2.29 0.448 18.1 0.685 3.94 11.6 4.17 Glass/ Epoxy Graphite Epoxy Epoxy Difference between SQR and HEX (%) 0.26 17.3 8.30 26.1 2.91 38.5 0.16 2.54 0.33 5.92 8.01 9.20 0.98 7.37 1.86 0.71 To verify the transverse isotropy of the square and hexagonal unit cells, the shear modulus G23 calculated from the transverse Young's modulus and Poisson's ratio is compared with the G23 calculated using the results from the FE analysis. If the composite is truly transversely isotropic, then it should satisfy the relationG3 E2 As shown in 2(1 + v23 ) Table 26, the difference in the shear moduli calculated from the two methods is small for the hexagonal unitcell. Therefore, the hexagonal unit cell is a better model for the micromechanics model to satisfy the transverse isotropy and can be considered as more realistic for fiberreinforced composites [28]. Also, the results the micromechanics method provides better approximate results of laminate properties than empirical solutions. Table 26. Comparisons of G23 for square and hexagonal unitcells in order to test transverse isotropy. Square Cell (MPa) Hexagonal Cell (MPa) E23 E23 G23 23 % Error G23 % Error 2(1+v, V) 2(1+v, V) Glass/Epoxy 4.62 7.74 67.5 5.83 5.91 1.41 Graphite/Epoxy 3.40 4.06 19.3 3.72 3.74 0.56 Effects of Fiber Volume Fraction The effect of fiber volume fraction on the thermal coefficients of graphite/epoxy composite was analyzed using the micromechanics method. Figures 26 and 27 show the variation of longitudinal and transverse thermal coefficients as a function of fiber volume fraction for glass/epoxy and graphite/epoxy composites. The CTE's estimated using both square and hexagonal unitcells are very close. It should be noted that the graphite fiber has a negative thermal coefficient, and also the product of thermal coefficient a and Young's modulus (cLE) is almost equal for the fiber and matrix. Hence the longitudinal thermal coefficients are negligibly small and it changes sign as the fiber volume fraction is varied. At about 40% fiber volume fraction the longitudinal thermal coefficient is almost equal to zero. The transverse thermal coefficient also reduces due to increase in fiber volume fraction because of the reduction in the effect of matrix material. The results show the micromechanics method is useful to develop composite materials for various applications by changing the combination of the constituent materials. 0 01 02 03 04 05 06 07 08 09 Fiber volume fraction Figure 26. Longitudinal coefficient of thermal expansion with various fiber volume fractions for glass/epoxy and graphite/epoxy laminates. Square Cell (Glass/Epoxy) A Hexagonal(Glass/Epoxy) 20 Square Cell (Graphite/Epoxy)  HexagonalCell (Graphite/Epoxy) o v 15 15 10, Carbon/Epoxy 0 S10  E Glass/Epoxy 5  0 01 02 03 04 05 06 07 08 09 Fiber volume fraction Figure 27. Transverse coefficient of thermal expansion with various fiber volume fractions for glass/epoxy and graphite/epoxy laminates. A Square Cell (Graphite/Epoxy) 0 Hexagonal Cell (Graphite/Epoxy) A Square Cell (Glass/Epoxy) 0 Hexagonal Cell (Glass/Epoxy) Glass/Epoxy Carbon/Epoxy Prediction of Stresses at Microscopic Level The micromechanics method was extended to estimate the microstresses of the graphite/epoxy composite laminate under combined thermal and external loads. The procedure used to obtain the relation between macro and micro stresses is described in the algorithm shown in Figure 28. Six independent sets of unit strains are applied to the unit cell boundary as explained in the previous section and microstresses are calculated in each element corresponding to the unit strain states. The temperature in the unitcell was also made equal to the temperature of the structure. The individual microstresses are multiplied by laminate strains in each layer calculated using the laminate theory [26]. Macro Level Micro Level Composite structure subjected Unit cells subjected to six independent set of to external and thermal loads unit strain and temperature change Compute microstresses in each element Apply Laminated Plate Theory Use superposition to compute the microstresses Obtain macrostrain results for in the composite structure from the macrostains each layers (EL, T, YLT) r Check failure of constituent phases based on the strength properties of fiber and matrix Figure 28. Flow chart of algorithm used to predict the failure due to microstresses. The microstresses obtained by superposition are used to calculate the maximum and minimum stresses in each finite element in the unitcell model. The failure of laminate can be predicted by using the maximum stress criterion for the fiber and matrix phases. The maximum stress criterion is reasonable as the fiber and matrix are expected 1 to behave in a brittle manner at cryogenic temperature. The tensile strength of graphite fibers is taken as 1,725 MPa and that of epoxy at cryogenic temperature is approximately 100 MPa. The analysis is performed to estimate the microstresses of the graphite/epoxy composite laminates (IM7/9773) with various stacking sequence (Sample A: [0]s; Sample B: [0/90]s and Sample C: [0/45/90]s) when it is subjected to thermal stresses at cryogenic temperatures without external loads. The thickness for each layer of the specimens is 0.07 mm. For the various laminated specimens, the longitudinal and transverse strains at the laminate level at cryogenic temperature (AT= 405 K) were calculated using the laminate theory and they are presented in Table 27. The micro level stresses were obtained by superposition principle as described above. In the case of planestress normal to the laminate plane, the strain in thickness direction can be calculated by 3 31 (1 AT) 32 (2 2AT) 4 T (5) C33 Table 27. Macrostrains in different laminates due to thermal loads (AT=405 K).The subscript Z denotes the thickness direction. Sample 0 SL(103) ST(103) Sz(103) YLT(103) YLz(103) YTz(103) A 0 0.277 4.71 4.71 0 0 0 0 0.0476 0.382 6.59 0 0 0 90 0.382 0.0476 6.63 0 0 0 0 0.130 0.648 6.406 1.410 0 0 C 45 0.316 1.094 6.369 0.518 0 0 90 0.648 0.130 6.449 1.410 0 0 Table 28. Maximum principal stresses of a macro model for various graphite/epoxy composite systems. Sample Orientation, Laminate Plate Theory Sample Orientation, 9 OL (MPa) GT (MPa) YLT (MPa) A 0 0 0 0 0 24.9 46.4 0 90 92.9 49.9 0 0 54.3 43.1 7.76 C 45 16.2 39.5 2.85 90 136 47.3 7.76 Table 29. Maximum principal stresses in the fiber and matrix phases of a unit cell for various graphite/epoxy composite systems from microanalysis. Square Cell Hexagonal Cell Sample 0 Fiber (MPa) Matrix (MPa) Fiber (MPa) Matrix (MPa) CY1 2 C1 C2 C1 C2 C1 C2 A 0 1.78 36.9 39.4 20.7 3.29 33.8 41.3 10.1 0 54.3 104 63.4 35.5 51.2 91.0 63.4 21.8 90 59.0 217 65.4 36.3 55.2 204 66.6 22.9 0 50.8 155 69.2 34.2 47.7 147 68.4 21.4 C 45 45.3 33.5 61.3 33.4 43.3 23.4 61.3 19.7 90 55.9 291 71.4 35.4 52.2 283 71.1 22.7 Table 210. Percentage difference of principal stresses estimated by square and hexagonal unit cells. The negative sign indicates that microstress result estimated by the hexagonal unit cell is lower. Fiber (MPa) Matrix (MPa) Sample 0 C 1 (%) G2 (%) 1 (%) 2 (%) A 0 84.7 8.56 4.76 51.2 0 5.60 12.1 0.10 38.5 90 6.33 5.99 1.87 37.0 0 6.16 4.92 1.19 37.5 C 45 4.57 30.1 0.03 41.2 90 6.61 2.85 0.39 36.1 The laminate stresses are estimated by stressstrain relation for various laminate samples with principal material directions (L and T) in Table 28. The microstresses in fiber and matrix phases are estimated using the micromechanics methods and the principal stresses or and o2 are calculated based on the microstress results in Table 29. The difference in principal stresses estimated using the square and hexagonal cells are shown in Table 210. For the unidirectional laminate (Sample A), the laminate stresses is zero under free boundary condition since the laminate undergoes free thermal contraction at cryogenic temperature in Table 28. However, the stresses at microscale are generated because of contraction between fiber and matrix. The microstresses in the fiber are very small compared to the matrix stresses. Therefore, the difference of maximum principal stresses in the fiber phase shows large in Table 210. The maximum principal stress is relatively consistent for both square and hexagonal unit cells, but the hexagonal unit cell estimated that the minimum principal stress in the matrix is reduced by approximately 40%. The results show the hexagonal unit cell underestimates the microstresses. The results from the finite element simulation can be used to compute the normal and shear stresses at the fiber/matrix interface in the unitcell. The normal and tangential stress components were calculated using the transformation matrix. C" cos2 0 sin 0 2 sin cos 0 o s = sin2 0 COS2 0 2sin0cos (2.6) Tr ssinOcosO sin cosos cos2 0 sin2 z where 0 is the angle measured from the x axis as shown in Figure 29. The normal and shear stresses around the periphery of the fiber are investigated when the uniaxial laminate is subjected to the cryogenic temperatures AT= 405 K without external loads. To compare the results for both unit cells, the interfacial stresses are plotted for 90<0<270. From the results shown in Figure 210 and 211 one can note that the absolute values of interfacial normal and shear stresses are lower for the hexagonal unit cell. y, 3, v x, 2, u z, 1, w I Cn s Cy Figure 29. Normal and shear stresses at the fibermatrix interface in square and hexagonal unit cells. It explains that the interfacial fracture between fiber and matrix is less likely to occur in a hexagonal unit cell than in a square unit cell. In real composite, fiber distribution are in hexagonal patterns, the interfacial stresses can be reduced. 1E+07  Hexagonal Cell 5E+06 Square Cell 0 /I 5E+06 F 1E+07  E 1 5E+07 2E+07 25E+07 90 135 180 225 270 Angle (Deg) Figure 210. Interfacial normal stresses in uniaxial graphite/epoxy laminate system at AT = 405 K. 1 5E+07  Hexagonal Cell 1E+07 Square Cell 5E+06  (0 W co 5E+06  1E+07  1 5E+079O 90 135 180 225 270 Angle (Deg) Figure 211. Interfacial shear stresses in uniaxial graphite/epoxy laminate system at AT = 405 K. Application to the Liquid Hydrogen Composite Tank The micromechanics method is used to predict the failure of the liquid hydrogen composite tank due to combined thermal and external loads. The liquid hydrogen composite tank is made of a honeycomb composite sandwich structure [1]. The inner face sheet is a 13ply, IM7/9772 laminate (0.066 inch thick) with the stacking sequence [45/903/45/03/45/903/45]T. The outer face sheet is a 7ply, IM7/9772 laminate (0.034 inch thick) with the stacking sequence [65/0/65/90/65/0/65]T. The material properties used for the IM7/9772 laminates are estimated in Table 22. The honeycomb core is Korex 3/163.0 (1.5 inch thick). The elastic constants of the core are: E1 = E2=4.14MPa, E3=137.9 MPa, G12 = 4.14 MPa, G13=74.5 MPa, G23 = 15.9 MPa, v12 = 0.25, v13 = 23 0.02. The thermal expansion of the honeycomb core is assumed to be zero. Figure 212. Maximum principal stress distribution in the LH2 Graphite/Epoxy composite tank. Tank pressure = 10 KPa and temperature = 50 K (AT= 405 K). The liquid hydrogen composite tank is subjected to appropriate boundary conditions in the ABAQUS FE model to simulate the situation at which it failed during proof test [1]. The actual liquid hydrogen composite tank was modeled using 8node solid elements (see Figure 212). The quarter model of the composite tank has 137 elements with 75 integration points in the thickness direction. The layup configuration of the composite laminates is specified in the layered solid elements. The IM7/9972 laminate properties used for the macromodel are given in Table 25. The macrolevel analysis was performed for two cases. In the first case, the tank was exposed to cryogenic temperature without internal pressure. In the second case, an internal pressure was applied in addition to cryogenic temperature. The displacement contours shown in Figure 212 correspond to a pressure of 10 kPa and AT= 405 K. The location denoted by the arrow in Figure 212 was selected for further investigation of failure due to microstresses. The macrolevel strains and curvatures in longitudinal and transverse fiber directions at T= 50K were computed using the finite element analysis (Tables 211 and 212). The microstresses in the inner and outer face sheets were calculated using the superposition method described in the previous section. Table 211. Laminate macrostrains used to obtain the microstresses in the inner and outer facesheets of the sandwich composite without internal pressure. L er 0 SL (103) ST(103) z(103) YLT(103) YLZ YTZ Layer (103) (103) Inter Facesheet Core Outer Facesheet 45 0.149 90 0.0386 45 0.226 0 0.338 45 0.224 90 0.0374 45 0.152 0.337 65 0.0899 0 0.296 65 0.00858 90 0.00447 65 0.00792 0 0.296 65 0.0904 0.227 0.338 0.150 0.0380 0.151 0.337 0.222 0.0367 0.203 0.00423 0.283 0.296 0.283 0.00470 0.0200 5.43 5.43 5.43 5.43 5.43 5.43 5.43 5.43 5.51 5.51 5.51 5.51 5.51 5.51 5.51 0.299 0.0773 0.299 0.0748 0.300 0.0723 0.300 0.0697 0.298 0.106 0.162 0.107 0.161 0.108 0.300 0.512 0.422 0.0898 0.296 0.0965 0.428 0.506 0.283 0.543 0.1.82 0.698 0.686 0.701 0.188 0.544 0.0831 0.302 0.510 0.425 0.508 0.289 0.103 0.431 0.452 0.684 0.123 0.185 0.121 0.687 0.463 Table 212. Laminate macrostrains used to obtain the microstresses in the inner and outer facesheets of the sandwich composite subjected to a pressure of 10 KPa. L (103) 8T(103) 8z(103) YLT(103) YLz(103) Layer Inter 45 0.147 Facesheet 90 0.501 45 0.412 0 0.233 45 0.405 90 0.498 45 0.138 Core 0.238 65 0.303 0 0.485 O r 65 0.214 Outer 90 0.421 Facesheet 65 0.212 0 0.489 65 0.306 0.419 0.230 0.144 0.500 0.141 0.236 0.398 0.497 0.366 0.422 0.280 0.487 0.280 0.421 0.375 5.90 5.90 5.90 5.90 5.89 5.89 5.89 5.89 5.57 5.57 5.57 5.56 5.56 5.56 5.56 0.731 0.563 0.732 0.553 0.734 0.543 0.735 0.534 0.622 0.115 0.770 0.119 0.774 0.122 0.618 3.11 4.81 3.67 0.371 3.64 4.81 3.18 0.299 5.30 2.36 3.31 4.75 3.30 2.39 5.32 YTZ (103) 3.69 0.407 3.13 4.81 3.16 0.335 3.62 4.81 0.118 4.75 4.15 2.37 4.17 4.75 0.165 Table 213. Maximum and minimum principal stresses in the fiber and matrix phases in the inner and outer facesheets of the sandwich composite without internal pressure. Without tank pressure at T= 50K Layer Inter Facesheet Outer Facesheet Fiber (MPa) Coi 58.8 57.8 59.6 60.6 59.6 57.8 58.8 59.1 61.1 58.2 58.1 58.2 61.1 59.1 Matrix (MPa) C7i 70.9 69.3 71.0 71.3 71.0 69.3 70.9 C02 138 107 157 185 156 107 139 123 175 101 97.5 101 175 123 71.3 71.7 70.4 70.3 70.4 71.7 71.4 02 2.71 2.47 2.49 2.73 2.50 2.50 2.74 3.79 3.74 3.99 3.87 4.00 3.75 3.82 Table 214. Maximum and minimum principal stresses in the fiber and matrix phases in the inner and outer facesheets of the sandwich composite subjected to a pressure of 10 KPa. Tank pressure Fiber (MPa) Layer Inter Facesheet Core Outer Facesheet 45 90 45 0 45 90 45 65 90 65 Col 77.8 121 92.5 79.0 90.8 121 77.5 84.8 80.8 74.7 100 74.7 80.8 85.5 10KPa at T= 50K Matrix (MPa) C02 167 33.5 37.1 162 38.1 33.7 166 78.0 242 76.8 45.7 77.3 244 77.7 Col 101 93.2 96.4 90.0 96.4 92.9 101 93.1 99.3 96.9 96.2 97.0 99.5 93.0 45 90 45 0 45 90 45 Core 65 90 65 02 21.2 27.2 23.0 14.1 22.8 27.1 21.4 27.5 16.2 19.0 25.2 19.0 16.3 27.6 The results of the maximum principal stresses in the fiber and matrix phases in each layer of the laminate are shown in Tables 213 and 2.14. When the cryogenic temperature is applied to the composite tank without tank pressure, the maximum principal stress in the matrix phase (70 MPa) is below its strength (100MPa). The result indicates microstress in the matrix phases, although very high, is not enough to initiate microcracks. However, when the tank pressure increases and reaches 10KPa, microstress in the matrix in the inner facesheet exceeds the tensile strength indicating the possibility of microcrack development. CHAPTER 3 FRACTURE TOUGHNESS FOR A TRANSVERSE CRACK IN LAMINATED COMPOSITES AT CRYOGENIC TEMPERATURES Stress singularity for the crack tip normal to the plyinterface is used to determine the conditions under which delaminations will initiate in a laminated composite. Stress singularity at a crack tip normal to the interface in fiberreinforced composites with different stacking sequences is investigated using analytical and finite element approaches. The stress singularity is used to predict the stress distribution in the vicinity of the crack tip. The fracture loads for several different laminate systems are determined using the fourpoint bending experiments. The fracture toughness is determined using the finite element analysis when the laminated specimen model is subjected to the fracture load obtained in the test. (a) (b) 2 0 0 Top Layer 2 Mid Layer 0 Bottom Layer Figure 31. (a) Geometry of interfacial fracture specimens, (b) A crack normal to laminate interface with different stacking sequence. Stress Singularities at a Cracktip Normal to a Plyinterface Stress singularity for a transverse crack in a composite laminate is determined using an analytical method. The analytical methods to determine the singularity for a transverse crack in a laminated composite (bimaterial interface) can be found in Ting's analysis [1415]. The details of the Ting's methods are described in Appendix A. In this approach the general solutions of displacement and stresses are derived using strain displacement, stressstrain and equilibrium equations in terms of arbitrary constants and singularities. The singularities are determined when the boundary conditions on the crack surfaces and continuity conditions at the interface are satisfied. The analytic approach is performed to estimate the singularity of a transverse crack of glass/epoxy and graphite/epoxy (IM7/9772) composites with stacking sequence [03]T, [0/90/0]T and [90/0/90]T in Figure 31. The crack tip of the transverse crack is placed at the interface between top and mid layers. The laminate properties of graphite/epoxy and glass/epoxy are shown in Table 31 [26]. The singularity results are compared with the finite element results. The commercial computer program MATLAB is used to solve the analytical equations. Table 31. Material properties of glass/epoxy and carbon/epoxy laminates. Eglass/epoxy Graphite/epoxy E11 (GPa) 38.6 160.9 E22, E33 (GPa) 8.27 9.62 G12, G13 (GPa) 4.14 6.32 G23(GPa) 3.18 4.22 V12, V13 0.25 0.32 V23 0.3 0.14 a'l (106 /C) 8.60 0.512 U22, U33 (106 /Co) 22.1 16.3 Finite element analysis is used not so much to verify the results from the analytical model but to determine the mesh refinement needed to obtain the proper singularity at the crack tip. In the finite element analysis a laminated composite beam was modeled using 8node solid elements with 20 integration points as shown in Figure 32. Glass/epoxy and graphite/epoxy (IM7/9772) composites with stacking sequence [03]T, [0/90/0]T and [90/0/90]T are chosen to verify the stress singularity of the transverse crack. The ply thickness was taken as 2.2 mm. The length of the beam was 146 mm and the width 18.7 mm. An initial crack normal to a plyinterface is placed at the center of the beam subjected to tensile or bending loads (see Figure 33). The crack tip is located at the interface of the top and mid layers. The region surrounding the cracktip was refined using 31,000 quadratic solid elements. The commercial finite element software ABAQUS was used for this purpose. ,7.... .,... .. ..1 a.J I ? i T I rJ' T?^ T 1^Ti i n7Y{); i y"r; ~ ..i\','}] rr iI tr T.' i. T':: i n1? ^ 1i l 'rt Undeformed Deformed Figure 32. Deformed geometry in the vicinity of a crack and interfacial fracture finite element model. P P am Figure 33. Contour plots of stress distribution for a [0/90/0]T composite model at a cracktip under tensile and bending loads. The composite beam model is subjected to fourpoint bending and/or tensile loading conditions as shown in Figure 33. For the tensile case, uniform displacement in the xdirection is applied at the end of the beam. For bending cases, the beam is simply supported at 63.7 mm away from the crack. The top load is located at 19.5 mm away from the crack. The normal stresses ahead of the transverse cracktip for tensile and bending cases are shown in Figure 34. 06 05 04 03 S05 02 (b) 15 0 0005 0001 00015 0002 0 00005 0001 00015 0002 Distance from a crack tip r, (m) Distance from a crack tip, r (m) Figure 34. Stress distribution under (a) tensile loads; (b) bending loads. A logarithmic plot of the normal stresses as a function of distance from the crack tip is used to determine the singularity in the FE model. The crack tip stress field is assumed to be of the form r= Kir where 1 by calculating the slope of the logarithmic plot of normal stress as a function of distance r ahead of the cracktip. Table 32. Singularity results of Eglass/epoxy. Eglass/epoxy Layers Analytical FEM (ABAQUS) (Ting's) Tensile Error (%) Bending Error (%) [03] 0.50 0.50 0.3 0.50 0.2 [0/90/0] 0.38 0.38 1.1 0.39 2.3 [903] 0.50 0.50 0.3 0.50 0.1 [90/0/90] 0.63 0.63 1.1 0.61 3.1 Table 33. Singularity results of graphite/epoxy. Graphite/epoxy Layers Analytical FEM (ABAQUS) (Ting's) Tensile Error (%) Bending Error (%) [03] [0/90/0] [903] [90/0/90] 0.50 0.26 0.50 0.71 0.50 0.27 0.50 0.70 0.50 0.29 0.50 0.69 I The stress singularities determined from the finite element analyses are compared with the analytical results for various laminates in Tables 32 and 33. There seems to be a good agreement between the two sets of results. The maximum difference between the two values is insignificant. When the laminate orientation of the materials on both side of plyinterface are identical as in the case of unidirectional laminates, ([03] and [903]), the singularity 2 = 0.5. The 0 layer is stiffer than the 900 layer in the plane direction since the load carried by the laminates is shared with the fiber and the matrix. When the laminate ahead of the crack tip is stiffer, as in [0/90/0] laminates, the stress singularity X becomes less than 0.5. It is greater than 0.5 when the crack is in the 0 layer is touching the 0/90 interface as in [90/0/90] laminates. The results provide confidence in the accuracy of the finite element models in capturing the singularity for transverse cracks in a composite laminate. Then the FE model can be used to analyze several composite systems used in fracture tests. Fracture Toughness at Room Temperature Fourpoint bending experiments are performed to determine the fracture loads of laminated beam specimens at room temperature as shown in Figure 35. The fourpoint bending test has the advantages that it would yield more accurate and repeatable results as the transverse crack is in a region under constant bending moment without any transverse shear force. Even a small offset of the loading point with respect to the crack location will not significantly affect the results. ,.. Figure 35. Fourpoint bending test to determine the fracture load. The proposed specimen has three layers of graphite/epoxy laminate with stacking sequence [0/90/0]T. The top and bottom layers for all specimens have same thickness of 2.4 mm and the midlayer has various thicknesses, 1.8, 2.4 and 3.0 mm. The dimensions of specimens are listed in Table 34. An initial crack was created at the center of the specimen below the top layer. The notch was cut using a fine diamond saw, and then the razor blade is used to sharpen the crack tip. The initial crack tip is located in the mid layer and below the plyinterface of the top layer. The specimen is simply supported at 63.7 mm away from the crack. The top loads are applied at a distance of 19.5 mm from the crack. The bending tests were conducted under displacement control in a material testing machine at a loading rate of 1.0 mm/min. 41 Casel (Midlayer thickness= 1 8 mm) Figure 36. Loaddisplacement curves of fourpoint bending tests at room temperature for the Specimen 1. Case 2 (Midlayer thickness = 2.4 mm) Displacement (m) Figure 37. Loaddisplacement curves of fourpoint bending tests at room temperature for the Specimen 2 Case 3 (Midlayer thickness = 3 0 mm) 250 200 150 100 50 0 0002 0004 0006 0008 Displacement (m) Figure 38. Loaddisplacement curves of fourpoint bending tests at room temperature for the Specimen 3 The loaddeflection results of various specimens are shown in Figures 3638. The load increases linearly until the cracktip reaches the plyinterface of the top layer. When the crack reaches the interface, the load does not increase further since more strain energy is required to deflect the crack to the plyinterface. After the interfacial fracture initiates, and as the crack propagates as a delamination, the stiffness of the specimen reduces as indicated by the slope of the loaddeflection curve. The fracture loads are measured immediately before interfacial fracture initiates. As results, the fracture load is linearly proportional to the thickness of the midlayer in Table 35. The fracture loads for the three different specimens are listed in Table 36. One can note that the fracture load increases with the thickness of the middle 900 layer. Table 34. Dimensions of specimens and various midply thickness. Layer Thickness [0/90/0] eng Width (mm) Top and bottom layer, 0 Mid layer, 900 (mm) (mm) o (mm) (mm) Specimen 1 145.4 18.6 2.4 1.8 Specimen 2 146.2 18.7 2.4 2.4 Specimen 3 145.7 18.8 2.4 3.0 Table 35. Fracture load and fracture toughness at room and cryogenic temperatures. 0/90/0] Room Temperature (T= 300 K) Cryogenic Temperature (T= 77 K) Ff (N) Kic (MPam29) Ff(N) Kic (MPam 29) Specimen 1 122 58.1 88.1 55.6 Specimen 2 127 57.9 81.5 58.1 Specimen 3 133 58.0 74.7 58.8 Finite element analyses of the test specimens were performed to determine the detailed stress field in the vicinity of the crack tip corresponding to the fracture loads. Due to symmetry onehalf of the specimen is modeled. The laminate properties of graphite/epoxy given in Table 31 are used for the FE model. A contour plot of the stress distribution is shown in Figure 39. Figure 39. Stress distribution for the 4Pt bending simulation at (a) room temperature; (b) cryogenic temperature. The fracture toughness can be calculated in two ways. The first method is similar to the "stress matching" [29] as described by the equation below (see Figure 310): K, = limu (r)r2 rO (3.1) In the second method a logarithmic plot of a vs. r is used to determine the best value ofK by fitting (see Figure 311) o(r)= Ktr (3.2) In both methods the singularity X derived from the analytical method was used. 8E+07 7E+07 6E+07 & 5E+07 4E+07 a 3E+07 L 2E+07 1E+07 Case 1 Case 2 Case 3 0 1E05 2E05 3E05 Distance from a crack tip, r (m) Figure 310. Variation of K = r with the distance from the crack tip at room temperature. 10 Case 1 Case 2 Case 3 98 96 94 92 7 68 66 64 62 6 58 56 54 52 5 log r Figure 311. Logarithmic plot of the stresses as a function of distance from the cracktip. From the results shown in Table 35, one can note that the increase in fracture load is about 8% with 67% increase of midply thickness. However the variation in the fracture toughness is about less than 1/2%. The results shows fracture toughness is independent to the midply thickness. Fracture toughness is only dependent to local properties near a crack tip, but not global properties of the laminate system. Fracture Toughness at Cryogenic Temperatures The effect of cryogenic temperature on fracture toughness is investigated by performing the fracture tests in liquid Nitrogen temperature. Liquid nitrogen (LN2) is used as the cryogenic refrigerant for several reasons. It is chemically inactive and non toxic. Liquid nitrogen is a colorless fluid like water. The boiling temperature of LN2 is 77 K. Figure 312. Cryogenic experimental setup of the fourpoint bending test. The beam specimens are submerged initially in LN2 for about 5 minutes to reduce thermal gradients in the specimens as shown in Figure 312. The specimen is placed in the cryogenic container with the liquid nitrogen. When temperature of specimen reaches the LN2 boiling temperature, LN2 boiling disappears. During this process, some specimens experience delamination on the edge of midply as shown in Figure 313. __._"_ _,_ ___ / ;  .. Crack Propagation Crack Tip Figure 313. Crack propagation in the 900 layer of a graphite/epoxy laminate at cryogenic temperature. Finite element analysis is performed to investigate the formation of edge delamination in the laminated specimen. The quarter region of an actual specimen is modeled using 8node 3D solid elements. A contour plot of stresses in 2direction is shown in Figure 314 when the FE model is subjected to cryogenic temperature (T= 77 K). 46 Figure 314. Contour plot of stresses normal to ply direction near the freeedge in a graphite/epoxy laminate at cryogenic temperature. 5E+07 4 5E+07 4E+07  3 5E+07 3E+07 2 5E+07 2E+07 1 5E+07 1E+07 5E+06 0 0002 0004 0 006 0008 Distance from a center to an edge (m) Figure 315. Loaddisplacement curves from fourpoint bending tests at cryogenic temperature. The midply thickness and hence the specimen thickness varied from specimen to specimen. The stresses in 2direction stay constant in 80% of the width and increases sharply near the edge as shown in Figure 315. Since graphite/epoxy composite has negative longitudinal CTE in 3direction (see Table 31), the midlayer expands and the top and bottom layer shrinks at cryogenic temperature. Therefore, Mode I fracture behavior can be expected at the edge. The result predicts that crack length of the edge delamination is not too deep to affect the experiment results. 47 The fourpoint bending test fixture is placed in an insulated container and LN2 is filled up slowly. The LN2 boiling disappears when the temperature of the test fixture becomes stable. During experiment, LN2 is continuously added into the container so that specimen is completely submerged since LN2 vaporizes due to the heat loss to atmosphere. The fourpoint bending test is performed following the previous procedure for room temperature tests. Casel (Midlayer thickness= 1 8 mm) Displacement (m) Figure 316. Loaddisplacement curves from fourpoint bending tests at cryogenic temperature for the Specimen 1 Case 2 (Midlayer thickness = 2 4 mm) 0002 0004 0006 Displacement (m) 0008 Figure 317. Loaddisplacement curves from fourpoint bending tests at cryogenic temperature for the Specimen 2 48 Case 3 (Midlayer thickness= 3 0 mm) 160 140 120 100 g80 8E+07 60  40  20 0 0002 0 004 0 006 0008 Displacement (m) Figure 318. Loaddisplacement curves from fourpoint bending tests at cryogenic temperature for the Specimen 3. 8E+07  7E+07  6E+07  5 5E+07 4E+07 S 3E+07 LL 2E+07  Case 1 Case 2 a Case 3 1E+07 0 1E05 2E05 3E05 Distance from a crack tip, r (m) Figure 319. Variation of K = r with the distance from the crack tip at cryogenic temperature. The loaddisplacement results are shown in Figures 316 through 318. Fracture load decreases 15% with 67% increase of the midply thickness in Table 35. Fracture load decrease may be due to material degradation as microcrack density increases at cryogenic temperature. Fracture toughness at cryogenic temperature is predicted using the finite element analysis. The contour plot of the stress distribution is shown in Figure 39. Fracture toughness is calculated following the same procedure as described in the previous section for room temperature tests. Variation of K = orz with the distance from the crack tip is shown in Figure 319. Although the fracture loads decreases by 15% from Specimen 1 to Specimen 3, fracture toughness increases by only 5%. The variation of fracture toughness between the three specimens tested at cryogenic temperature is insignificant. The result indicates that fracture toughness is not significantly affected by the cryogenic conditions. But, the fracture load significantly decreases due to thermal stresses present in the vicinity of the crack tip. For both cases at room and cryogenic temperature, fracture toughness is estimated as 58 MPa. The result indicates fracture toughness is a characteristic property not governed by temperature changes. It should be noted that the specimen dimensions and geometry are nominally the same for room temperature and cryogenic temperature tests are the same. But, the fracture load is significantly lower at cryogenic temperature. However the fracture toughness, the critical stress intensity factor, seems to be the same although we input lower loads in the finite element model. This is because there are significant thermal stresses at cryogenic temperature which increases the stress intensity factor to a higher value. CHAPTER 4 PERMEABILITY TESTING OF COMPOSITE MATERIALS FOR A LIQUID HYDROGEN STORAGE SYSTEM The permeability is defined by the amount of gas that passes through a given material of unit area and unit thickness under a unit pressure gradient in unit time. The SI unit of the permeability is mol/sec/m/Pa. Experiments were performed to investigate the effect of cryogenic cycling on permeability of laminated composites and to provide useful comparison of permeabilities of various composite material systems. Standard Test Method for Determining Gas Permeability The standard test method for determining gas permeability is documented in ASTM D143482 (Reapproved in 1997) "Standard Test Method for Determining Gas Permeability Characteristic of Plastic Film and Sheeting [18]." The permeability can be measured by two methods, manometric determination method and volumetric determination methods. The permeability experiment using the monometric determination method is shown in Figure 41. The lower pressure chamber is initially vacuumed and the transmission of the gas through the test specimen is indicated by an increase in pressure. The permeability is measured using volumetric determination as shown in Figure 42. The lower pressure chamber is exposed to atmospheric pressure and the transmission of the gas through the test specimen is indicated by a change in volume. The gas volumeflow rate is measured by recording the rise of liquid indicator in a capillary tube per unit time. The gas transmission rate (GTR) is calculated using the ideal gas law. The permeance is determined as the gas transmission rate per pressure differential across a specimen. The permeability is determined by multiplying permeance by the specimen thickness. Driving Force,PD I Figure 41. Permeability experimental setup for manometric determination method. The monometric determination method was not considered for this study since it is dangerous to handle the mercury compound which is considered as a health hazardous material. Therefore, the permeability facility was constructed based on the volumetric determination method as shown in Figure 41. 52 Gas Pressure To Double Stage Manomenter Regulator 00, SDrying Gas Column inlet Capillary ( Pressure Tube Oil TrapsY Valve S(Buhbber) Constant Temperature Bath JPermeab2 Filter Papersatus eeaes t oe se o e s specimen esas I.I I ,,,  ., Exchanger Permeability Apparatus This apparatus basically consists of a specimen placed between two chambers as shown in Figure 43. The test gas is pressurized in the upstream chamber. The gas permeates through one side of the specimen and escapes out of the other side. The escaped gas is collected in the downstream chamber and flows into a glass capillary tube. The amount of gas escaping per unit time is measured. The permeance is determined by gas transmitted rate and the partial pressure differential across the specimen. The permeability P is defined by the product of permeance P and the specimen thickness t. The gauge pressure of the gas in the upper upstream chamber is measured using a pressure transducer (P303A from the Omega Engineering Inc). The ambient pressure is measured by a barometric sensor (2113A from the Pasco Scientific). A precision pressure regulator provides constant gas pressure to the upstream chamber. The ambient temperature was measured using a glass capillary thermometer. ion Pressure Regulator .Force Gauge 1 ( Downstream Chamber S....... a* Outlet Vatve "'"'[c Upstrearn Chmber Outlet Valve Capillafy Tube ow Figure 43. Permeability testing apparatus. The specimens are mounted horizontally between the upstream and downstream chambers and clamped firmly by applying a compressive load as shown in Figure 44. The specimen is sealed with a rubber gasket and an ORing (38 mm inner diameter). A force gauge mounted at the top measures the compressive load to ensure that the same amount of compressive load is applied on the specimens for every test. The compressive load should be enough to prevent gas leakage, but should not damage the specimens. The upstream chamber has an inlet vent and an outlet vent. The inlet vent allows the gas flow into the upstream chamber and the outlet vents is used to purge the test gas to atmosphere (Figure 44). The downstream chamber has two outlet vents. One is used to purge the test gas to atmosphere and the other allows the gas flow to the glass capillary tube for measurements. The sensitivity of permeability measurement can be improved by increasing the gas transmitting area of a specimen. W%  S0W wt S U Figure 44. Specimen installation between upstream and downstream chambers. The glass capillary tube is mounted on a rigid aluminum base horizontally to minimize the gravity effect on the capillary indicator and for easy reading of the scale marks on the capillary tube. Nettles [22] found that there was no significant difference in the volumetric flow rate when capillary tube is placed vertically or slanted. The inner diameter of the glass capillary tube is 1.05 mm and the length is 100 mm. A magnifying glass is used to read the scale marks at the top of the meniscus of the liquid indicator. The liquid indicator in the glass capillary tube is used to measure the rate of rise of the liquid indicator. The rate is used to calculate the volume flow rate of the escaped gas across the specimen. Nettle [22] investigated the effects on the volume flow rates by using various types of liquid. The volume flow rates obtained using water, alcohol and alcohol with PhotoFlo were not significantly different. In the present study, methyl alcohol is chosen as the liquid indicator since alcohol has low viscosity and weight. The methyl alcohol is colored with a blue dye to obtain precise readings on the scale marks. The primary investigation is the hydrogen permeability of a liquid hydrogen composite storage system. Since hydrogen gas is highly flammable and explosive when it mixed with air, it needed to be handled carefully during the test. Hence, other permeate gases were considered as a substitute for the hydrogen gas. The molecular diameter of various gases is listed in Table 41 [23]. To choose a permeate gas, the molecular diameter determined from viscosity measurement is mainly considered since the permeability is related with the volumetric flow rate directly. Since helium has the smallest molecular diameter, the helium predicts the permeation results higher than other gases [23]. Therefore, in the study, helium was chosen as the permeate gas instead of hydrogen. Table 41. Molecular diameter of various gases from CRC Handbook of Chemistry and Physics, 54th Edition. Type of Gas Molecular Diameter (cm) From Viscosity From van der From Heat Waal's Equation Conductivity Helium 1.9x108 2.6x108 2.3x108 Hydrogen 2.4x108 2.3x108 2.3x108 Nitrogen 3.1x108 3.1x108 3.5x108 Specimen Description The permeability tests were performed with various composite material systems. The details of specimens are described in Table 42. The specimens C1, C2 and C3 are various graphite/epoxy laminated composites. The specimen T1 is a textile composite. The specimen N1 is a laminated composite embedded with 36 jm aluminum oxide (Al203alumina) nanoparticles. The aluminum oxide was dissolved in alcohol and the compound was applied on a surface on a laminated prepreg using a paint blush. The nanoparticles are capable of relieving the thermal stresses due to contraction of fiber and matrix phases. Also, it can prevent the crack propagation in matrix phase by relieving the stress concentration at the cracktip. The graphite/epoxy prepregs were fabricated with designed stacking orientations. The graphite/epoxy prepregs were cured in an autoclave and machined by a diamond saw to 3 x3 inch specimens at low speed to avoid fiber shattering. The surface is cleaned with acetone and prepared carefully to avoid contamination or damages on the surface of the specimen during machining and subsequent handling. Table 42. Description of composite specimens. Specimen Preprag Type Stacking Sequence Thickness (mm) C1 Graphite/Epoxy laminates [0/90/0/90/0/90]s 1.52 C2 Graphite/Epoxy laminates [02/902/02]T 0.787 C3 Graphite/Epoxy laminates [0/90/02/90/0]T 0.914 C5 Graphite/Epoxy laminates [0/902/O]T 0.533 Tl Plain weave textile(SP 4 layers 0.686 Systems SE84) N1 Graphite/Epoxy laminates [0/90/NP/90/0]T 0.483 with nanoparticles The specimens were subjected to cryogenic cycling at specified number of times, representing multiple refueling process of a space vehicle. A single cryogenic cycle consisted of cooling down from room temperature to cryogenic temperature and then warming up to room temperature. Initially, specimens were placed at room temperature (T=300K). And, then the specimens completely submerged in an insulated container filled with liquid nitrogen. The specimens stayed in the container for approximately 2 minutes. When the specimen reached the boiling temperature of liquid nitrogen (T= 77K), the liquid nitrogen boiling disappears. The specimens were taken out of the container and placed at room temperature. The specimen was held at room temperature for approximately 5 minutes. The cryogenic cycling procedure is repeated for desired number of times. Testing Procedure Before starting the permeability test, a thin coat of silicon grease was applied on the gasket and an Oring was placed to improve sealing of contact surfaces on a specimen. The excessive silicon grease was wiped out to avoid obstructions of permeated gas on the transmitting surface of the specimen. All outlet valves were opened initially to avoid sudden pressurization of test gas. The specimen was placed horizontally on the gasket of the upstream chamber. And then, the downstream chamber was placed on the top surface of a specimen. The specimen was mounted between the chambers. Both chambers were aligned and mated as close as possible. The specimen was mounted between the chambers and clamped firmly with a compressive force. Then, even distributed forces were applied to the sealing materials of the chambers. The test gas was admitted to the upstream chamber by opening the gas release valve of the gas tank. While all outlet valves remained opened, the test gas was filled in the upstream chamber and ventilated though the outlet vent to atmosphere. Any residual air in the upstream chamber was purged for 1 minute. The outlet valve on the upstream chamber was closed and the test gas was allowed to permeate across the specimen for a sufficient time to purge any residual air at downstream chamber. At this time, only test gas filled the chambers. When the outlet valve of the upstream chamber was closed, the upstream pressure increased slowly. The upstream pressure can be adjusted by controlling the pressure regulator. Sufficient time was allowed for attaining steady state of moving rate of the liquid indicator before beginning to take readings. The distance of rise of the liquid indicator was measured while the ambient pressures were recorded. Calculations The volumetric methodology is used to calculate the permeability by measuring gas volume transmitted through a specimen. The rate of rise of the liquid indicator is measured using the capillary tube. The volume flow rate is calculated as follows. V = slope xa (4.1) where a, is the crosssectional area of a capillary tube. The slope is the rate of rise of the liquid indicator in the capillary tube. The gas transmitted rate (GTR) is calculated using the ideal gas law as follows. GTR = o (4.2) ART where po is ambient pressure, A is transmitting area of a specimen, R is the universal gas constant (8.3143 x 103 LPa/{molK}) and T is ambient temperature. The permeance P is defined by the ratio between the gas transmission rate and pressure differential across the thickness of a specimen. Therefore, the permeance P is calculated as follows. P G (4.3) PPo where p is the upstream pressure. The S.I. units of permeance are [mol/(m2.sPa)] According to the standard test method, the permeability P is defined by the product of permeance P and the specimen thickness t and "the permeability is meaningful only for homogeneous materials [18]." In this study, although the laminated composites are considered as orthotropic materials, its permeability P is calculated by following the corresponding definition for isotropic materials [22]. Calibration The position of capillary indicator is very sensitively to even small variations of testing conditions such as ambient pressure and temperatures caused by closing doors or airconditioning system in the laboratory. For example, during one test, the variation of the ambient pressure was approximately 0.3% for a 13 hour period (Figure 45). For the permeability calculation, the ambient pressure is assumed to be constant. However, the variation is large enough to cause error in measuring the rate of rise of the capillary indicator. The moving distance is needed to be calibrated to compensate for error due to the variation of ambient pressure. Time (hour) 60 Figure 45. Variation of ambient pressure for 13 hours at test condition. The error correction factor due to the variation of ambient pressure was calculated by performing the permeability test without applying the upstream pressure. An aluminum plate is used for the test to ensure that there is no gas permeation to the downstream chamber. Since there was not much variation in the ambient temperature, only ambient pressure causes changes in the position of the capillary indicator. After the outlet valve of the downstream chamber was closed, the displaced position of the capillary indicator and ambient pressure were measured simultaneously as shown in Figure 46. Barometric Pressure 10293 Average Barometric Pressure 185 S o Slua Position ', 180 I I I IIl 1754 2 Z5 M2 ,, Ii' I 170 10286 0 500 1000 1500 Time (Sec) Figure 46. Variation of barometric pressure and indicator position as a function of time. The correction factor k is calculated by k = Ah /Ap where h is the moving distance of the capillary indicator andp is the ambient pressure. The average correction factor is found to be 0.21 mm/Pa. Therefore, the corrected moving distance of the capillary indicator is calculated as follows: corrected = actual + k Ap where zp is differential of ambient pressure between the beginning and end of the test. Permeability Test Results The permeability tests were performed with various composite material systems at room temperature. The permeability was measured at six different levels of upstream pressure. The average permeance P and average permeability P are tabulated in Table 4 3 for laminated composites, the results for textile composites are in Table 44 and the results for laminated composites with embedded with nanoparticles in Table 45. Table 43. Permeability of laminated composites for various number of cryogenic cycles. Logarithm Specimen Cryogenic Permeance, P Permeability, P cycles (mol/sec/m2/Pa) (mol/sec/m/Pa) Cl 0 5.60x1018 8.54x1021 20.1 1 1.52x1017 2.32x1020 19.6 5 2.39x1017 3.65x1020 19.4 10 2.39x1017 3.65x1020 19.4 20 2.11x1017 3.22x1020 19.5 C2 0 7.02x1018 1.07x1020 20.0 1 1.06x1017 1.62x1020 19.8 5 1.47x1015 2.23x1018 17.7 10 1.42x1015 2.16x10 18 17.7 20 1.49x1015 2.27x1018s 17.6 C3 0 6.22x1018 9.48x1021 20.0 1 7.56x1018 1.15x1020 19.9 5 7.60x1018 1.16x1020 19.9 10 8.37x1018s 1.28x1020 19.9 C5 0 5.85x1018 8.92x1021 20.0 1 9.52x1016 1.45x10 18 17.8 5 8.67x1016 1.32x10 18 17.9 20 8.81x1016 1.34x10 18 17.9 Table 44. Permeability of textile composites for various number of cryogenic cycles. Logarithm Specimen Cryogenic Permeance, P Permeability, P of cycles (mol/sec/m2/Pa) (mol/sec/m/Pa) Tl 0 4.79x1018 7.30x1021 20.1 1 6.77x1018 1.03 x 1020 20.0 5 8.41x1018 1.28x1020 19.9 20 8.75x 1018 1.33 x 1020 19.9 Table 45. Permeability of laminated composites embedded with nanoparticles for various number of cryogenic cycles. Logarithm Specimen Cryogenic Permeance, P Permeability, P cycles (mol/sec/m2/Pa) (mol/sec/m/Pa) N1 0 6.82x1018 1.04x1020 20.0 1 2.72x1015 4.15x1018 17.4 5 1.30x1014 1.98x1017 16.7 20 9.83x1015 1.50x1017 16.8 The test results show the permeability increases as the number of cryogenic cycles increases (see Figure 47). The permeability increased rapidly and becomes constant with further increase of cryogenic cycles. For specimens C2 and C3, which have approximately same thickness, the permeability of the specimen C3 was lower since the specimen C3 has the plies dispersed and not grouped together compared to the specimen C2. The textile specimen T maintained constant permeability with the increase of cryogenic cycles. The textile composites resulted lower permeability than the laminated composites. The specimen N1 has same layer stacking orientations with the specimen C5 and nanoparticles were dispersed between two 90degree layers. Before cryogenic cycling, the permeabilities of the specimens N1 and C5 were approximately the same. However, as the number of cryogenic cycles increased, the permeability of the specimen N1 63 became higher. The use of nanoparticles did not improve the permeability of laminated composites. 0 16 Cl C1 C2 C2 C3 C3 5 5C5 C5 T1 T1 1N1 N1 10  E IE 19 E E o 15 o   20 201 f 0 5 10 15 20 210 5 10 15 20 Number of Cryogenic Cycling Number of Cryogenic Cycling Figure 47. Logarithm of the permeability for composite specimens with increase of cryogenic cycles. Optical Microscopic Analysis The optical microscopic inspection was performed to evaluate the microcrack propagation and void content of various composite systems after cryogenic cycling. In the previous section, the experimental results showed that the permeability increases as the composite specimen underwent more cryogenic cycles. As the crack density increases, gas flow becomes easier though the specimen. Therefore, the microcrack propagation is correlated with cryogenic cycling. The specimens were cut through the center using a diamond saw. A LECO grinder/polisher was used for the sample preparation process. The rough edge through the center was ground with 600grit sand paper with water for 30 seconds. The fine grinding was performed with the 1000grit and 1500grit papers for 30 seconds. The surface of the edge was polished with the 58[tm aluminum oxide powder (A1203 alumina) dissolved in distilled water. The purpose of the lubricant is to both dissipate the heat from polishing and to act as a carrier for the abrasive materials. The ultrasonic cleaner was used to remove any abrasive particles and contaminants on the polished surface of a specimen. The optical analysis was conducted with a NIKON EPIPOT microscope. The laminated composite specimen C2 and the textile composite specimen T1 were chosen for optical inspection. The specimen details are described in Table 42. The microscopic images for the specimens were compared before and after cryogenic cycling. 0.05 mm Figure 48. Cross sectional view of the graphite/epoxy composite specimen C2 before cryogenic cycling: (a) 10X magnification; (2) 40X magnification. For the graphite/epoxy specimen C2 before cryogenic cycling, no microcrack propagation observed (see Figure 48). Some voids formed in the middle of the 90degree layers and between the 0degree layer and the 90degree layers. The voids probably formed during composite fabrication in autoclave. When the graphite/epoxy prepreg was cured at high temperature, some air bubbles could have been trapped between layers. It is possible that external pressure applied on the vacuum bag was insufficient to remove the bubbles. After cryogenic cycling on the specimen C2, microcracks were observed in the 90degree layer as shown in Figure 49. The fiber breaks were not observed in the in 0 degree layer were not observed since the thermal stresses were not large enough. The delaminations propagated along the middle of the 90degree layer where some voids were found. The transverse cracks branched with the delaminations. Since the gas can be transmitted through the transverse cracks across the specimen, the permeability increased after cryogenic cycling. 0.05 mm Figure 49. Microcrack propagation on the graphite/epoxy composite specimen C2 after cryogenic cycling: (a) 10X magnification; (2) 40X magnification. For the textile composite specimen T1 before cryogenic cycling, no microcracks were observed as shown in Figure 410. Voids were observed at the location where two yarns are merging. Figure 410. Cross sectional view of the textile specimen T1 before cryogenic cycling, 10X magnification.  VO  ~~" ~Qr~ r~c.. J ~I  '~" 'c~r ~c: ~Fr~C ~.~ UcmJ r;3 Figure 411. Microcrack propagation in the textile specimen T after cryogenic cycling: (a) 10X magnification; (b) 40X magnification. After the textile specimen T underwent cryogenic cycles, microcrack were observed in 90degree yarn as shown in Figure 411. The microcracks propagated in the inplane direction of the specimen. Since transverse cracks were not propagated, the permeability of textile composites was almost the same before and after cryogenic cycling. :: ~s~ L ~li~nl~Cii~L~h .r~ . 7 It~;r~:~_~XLL~i~: 67 The transverse cracks and delaminations provide the leakage path through composite laminates and thus directly related to the permeability. For the laminated composite specimen, the microscopic results showed that transverse crack propagation and delaminations of composite laminates. For the textile composite specimen, the transverse cracks propagation was not observed, which resulted low permeability of textile composites even after cryogenic cycling. CHAPTER 5 RESULTS AND DISCUSSION The liquid hydrogen cryogenic composite tank proposed for future space vehicles will be exposed to extreme thermal and mechanical loads. During cryogenic cycling, the liquid hydrogen composite tank can fail due to the thermal stresses combined with structural loads. The present study is concerned with three of the several fundamental mechanics issues involved in the design of cryogenic composite tanks. First, the micro stresses in the fiber and matrix phases of composite laminates were estimated at cryogenic temperature. Second, the fracture behavior of composite laminates at cryogenic temperature was predicted using the finite element method in conjunction with experimental analysis. Third, the permeability of various composite material systems for the liquid hydrogen composite storage system was investigated. The results from this study will be useful in the design of lightweight composite tanks for cryogenic storage systems. Micromechanics Method to Predict Thermal Stresses for Laminated Composites at Cryogenic Temperature A micromechanics model is developed from the repetitive patterns of the fiber distribution in the microstructures of composite laminates. The representative volume element was modeled using the square and hexagonal shapes depend on the fiber and matrix layout in microscopic images of composite laminates. The periodic boundary conditions were applied to the unit cell model to match the boundary displacements with adjacent unit cells. The epoxy resin is considered as homogeneous thermoelastic material and the fiber properties are considered to be independent to temperature changes. The laminate properties were estimated using the micromechanics methods. The micromechanics methods made a good agreement with the semiempirical solutions. The transverse properties estimated from the micromechanics model was used to verify the transverse isotropy of the composite laminates. The hexagonal unit cell satisfies the transverse isotropy better than the square unit cell. Therefore, the hexagonal unit cell is more realistic and better model for the micromechanics methods. The micromechanics method yields detailed microstress distribution in the fiber, matrix and the interface between the fiber and matrix phases. These microstresses can be used to predict possible microcrack propagation of the composite at various temperatures with or without external loads. When a unidirectional laminate is subjected to free boundary conditions, the laminate stresses are zero from the stressstrain relation. However, the microstresses in fiber and matrix phases exist since the coefficient of thermal expansion of constituents are different. When the unidirectional laminate is subjected to cryogenic temperature, the thermal contraction between fiber and matrix phases causes the microstresses. The microstress results are compared with the material strength to predict the possible microcrack propagation. The method was used to analyze the microstresses in the liquid hydrogen composite tank. When the composite tank is subjected to cryogenic temperature without external loads, the microstresses did not exceed the material strength of the constituents. When the internal pressure of the composite tank reaches 10KPa at cryogenic temperature, the stresses in the matrix phase seem to exceed the tensile strength of the matrix material indicating microcracking is a possibility. However matrix strength measured at cryogenic temperatures should be used for accurate prediction of formation of microcracks. Fracture Toughness for a Transverse Crack in Laminated Composites at Cryogenic Temperatures At cryogenic temperature, the transverse cracks propagate mostly at transverse plies of composite laminates. The transverse crack and delamination provide the leakage path of the liquid hydrogen permeation which causes the catastrophic failure of the composite tank. The stress singularity describes the fracture propagation of composite laminates. The singularities for glass/epoxy and graphite/epoxy composites with various stacking orientations are estimated using analytical and finite element methods. Singularities calculated by both methods are in good agreement. The results provide the confident of capturing the singularity using the finite element model. When the transverse crack propagates in the unidirectional laminates, the singularity becomes 0.5 as a general homogenous material. When the transverse crack propagates from the 0degree to the 90 degree layer, the singularity becomes lager than 0.5 and vice versa. The finite element analysis was performed to estimate fracture toughness of a transverse crack under the fracture load measured from the fracture experiment. The results of fracture toughness are compared at room and cryogenic conditions. Fourpoint bending experiments are performed at room and cryogenic temperatures to obtain fracture loads of graphite/epoxy specimens. The specimens had different midply thicknesses, i.e, the crack lengths were different in different specimens. The results of fracture load and fracture toughness are listed in Table 36. At room temperature, fracture load increases 8% when the midply thickness increases 67%. However, the variation of fracture toughness is less than 0.5%. Fracture toughness is only dependent to local properties near a crack tip, but not global properties of the laminate system. At cryogenic temperature the fracture load decreases 15% with increase of midply thickness. The fracture load significantly decreases due to the material degradation at cryogenic temperature. The thermal stresses cause microcrack propagation in the vicinity of the crack. Although the fracture loads decreases by 15% from Specimen 1 to Specimen 3, fracture toughness increases by only 5%. For both cases at room and cryogenic temperature, fracture toughness was estimated approximately as 58 MPam0.29. The result indicates fracture toughness is a characteristic property not governed by temperature changes. The analysis is useful to predict the transverse crack propagation of composite laminates under cryogenic conditions. Permeability Testing for Laminated Composites for a Liquid Hydrogen Storage System The experimental analysis was performed to measure the gas permeability various composite material systems for the liquid hydrogen composite tank and the effect of cryogenic cycling of composite laminates on the permeability was investigated. The permeability test facility was constructed following the standard test method documented in ASTM D143482 (Reapproved 1997). The permeability test was conducted at room temperature. The sensitivity of permeability measurement can be improved by increasing the gas transmission area of a specimen. Since hydrogen is flammable and explosive when it mixed with air, the helium gas is substitute for hydrogen. The calibration is performed to compensate the ambient pressure differences during the test. The correction factor is found as 0.21 mm/Pa. The actual moving distance of the liquid indicator is calibrated by the correction factor. The permeability results for the various materials are shown in Figure 49. The permeability of laminated composite was found to degrade after undergoing cryogenic cycling process. The thin laminate specimens C2 and C5, the permeability increases significantly after cryogenic cycling. For thick laminated specimens Cl and C3, the increase of permeability is less. The textile composite specimen T1 has lower permeability than laminated specimens and the variation of permeability is very small with the increase of cryogenic cycles. The laminated composites were embedded with nanoparticles which are capable of reducing thermal stresses in the matrix phase. However, as results, the nanoparticles did not show the improvement on permeability. Further studies are needed to investigate the effects of nanoparticles. The optical analysis was performed to investigate the microcrack propagation and void contents of test specimens. The transverse cracks and delaminations provide the leakage path through composite laminates and thus directly related to the permeability. The optical micrographic analysis was performed to investigate the microcrack propagation and void contents of test specimens. For the laminated composite specimen, the microscopic results showed that transverse crack propagation and delaminations of composite laminates. For the textile composite specimen, the transverse cracks propagation was not observed, which resulted in low permeability of textile composites even after cryogenic cycling. APPENDIX STRESS SINGULARITY USING STROH'S METHOD Stress singularity at a ply crack tip acting normal to an interface of composite laminates can be obtained using Stroh's method. The displacements ui (i=1,2,3) are in the xix2 plane as shown in Figure A. 1. Figure A. 1.Geometry of an angleply laminated composite and a crack normal to an interface between two anisotropic materials. The strains and the stresses are described as a function of x2 and x3. The displacement, stressstrain and equilibrium equations can be written as E, = (0u, x + a uj ax)/2 (Al) Cl = Cijkp. (A2) A1A3 can be obtained by letting A1A3 can be obtained by letting u = vf(Z) (A4) Z = x2 + px3 (A5) where p and v, are constants to be determined and f is an arbitrary function of Z [1415]. Substituting into Eqs. A1A3, we obtain r, = r, df/dZ (A6) D,kvk =0 (A8) where ( = (Cyk2 + k3)k (A9) Dk =C2k2 + p(2k3 +3k2) + 23k3 (A10) For a nontrivial solution of vi, it follows from Eq. A8. that the determinant of Dik must vanish. Therefore, Dk = 0 (All) where Dll = c1212 + p(c,213 +1312)+ pc1313 = 6 + C55 D,2 = 1222 + p(c223 + 1322)+ C1323 =p(c46 +C25) D13 = C1232 + p(C1233 + C1332)+ p2C1333 = C46 +P235 D21 = c212 + (2213 + 2312 )+ P2313 = p(c46 + 25) D22 = C2+22 + P(C2223 + c322)+ p c2323 = C22 + P2c44 (Al2) D23 = c2232 + C33 + c33)+ p2C2333 = P(C2 + 44) D1 = C3212 + p(3213 + 3312)+ p 3313 = 46 2 35 D32 = c322 +p(32 + c + C3322) + p2c3323 = p(c23 + C44) D33 = 3232 + p(333 + 3332 )+ p3333 = 44 + 33 Therefore, Eq. A12 is written as c,11 +p2 55 P(C46 +C25) C46 +P2 C35 rV1 22 +p2 44 p(23 44) 2 = 0 (A13) Sym c44 +p2c33 V3 c55 0 c35 0 c46 +C 25 0 C 11 0 46 V1 S0 44 0 + PL 46 +25 0 c23c44 + 0 22 0 v2 =0(A14) c 35 0 C33 J 0 C23 + C44 0 c 46 0 C44 V3 The constant PL exists in every term of the coefficient matrix in Eq. A14. Although it is possible solve for the root of the characteristic equation, numerical solutions to this type of problem yield more accurate eigenvalues. Hence Eq. A14 is further developed into an eigenvalue problem where PL are complex eigenvalues with corresponding complex eigenvectors, vj where the L subscript associates the eigenvector with the appropriate eigenvalue (L=1,2,3). The equation is simplified as [p2[A'] + p[B'] + [C']][v]= 0 (A15) Multiply equation by the inverse of the coefficient matrix [A ]. [p2[I]+ p[B]+[C]][v] 0 (A16) The eigenvalue equation is [[A]66 A[I]][Z]= 0 (A17) where the array [A] can be constructed such that A LI[B] [C] [A]= [B] [C] (A18) I ] [0] ]66 Sxrl [B] [C] {xI [B] {x[ [C]{ y A{z}=[A]{Z}= [A { y) [I] 0 _{y y {x! (A19) where {Z}= {x}= X, = Z,, {y}= Y, Z, (A20) X, Z3 Y3 Z6 {A{X}) [B]{X [C]{Y}} (A21) =e ^ I (A 21) A{Y}j {x} A{x}=[B]{X}[C]{Y} and {X)= A{Y} These two equations can be combined into one equation by substituting for X, hence eliminating X, and rearranging terms. [22[] + A[B] + [C]](Y}= 0 (A22) Comparing Eq. A22 yield the necessary eigenvalues and eigenvectors A=P, and {Y}= Y, = Z, ={v} (A23) Y3 Z6 Comparing Eq. A23 and verifies that equation can be used to solve for PL as eigenvalues instead of roots to the characteristic equation derived from the determinate of the coefficient matrix in equation. Both eigenvalues, PL, and eigenvectors from equation will be used in the next section to calculate the singular exponent. The general solution of the displacement and stress is formulated using the Stroh's method Ting and Hoang [1]. (A24) u, = r1k La Re (,k)+ii L k)}/(l k) lo = 1kZ L Re( z, k)+, Im(r r'k)} (A25) where g = cos + p sin where Re and Im stand for real and imaginary. Since the material 1 is divided into two parts by a crack, the displacements and the stresses in the Material 1 has superscript of (+) or (). A superscript (+) to denote properties of the Material 1 in the positive region of the xl and () to denote properties of Material 1 in the negative region of the xl axis. The stress free boundary conditions at the crack surface are S= 0 at = (A26) Olj = 0 at 0 = f The continuity conditions at the interface are u, u =0 at ( = / 2 (A27) ,+ ', =0 at 0 = ;/2 u' u =0 at ='/2 (A28) (7f u 2 = 0 at 0 = r /2 Using the displacement and stress equations, the boundary and the continuity equation results in 18 equations for the eighteen coefficients of AL, BL A', B', AL and BL. The equations can be written in matrix form K()q = 0 (A29) where K is an 18 X 18 square matrix and the elements of q are A+, BL, AL, BL, AL and BL. For a nontrivial solution of q, determinant of the matrix K must be zero when singularity 2 satisfies the matrix K. The determinant was calculated numerically. LIST OF REFERENCES 1. Final report of the X33 Liquid Hydrogen Tank Test Investigation Team, Marshall Space Flight Center, Huntsville, AL, May 2000. 2. Marrey, R.V. and Sankar, B.V., "Micromechanical Models for Textile Structural Composites," NASA Contractor Report, 198229, October 1995. 3. 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Agarwal, B. and Broutman, L., Analysis and Performance of Fiber Composites, 2nd Edn, John Wiley & Sons, Inc, New York, 1990. 80 27. Elseifi, M., "A New Scheme for the Optimum Design of Stiffened Composite Panels with Geometric Imperfections," Ph. D. dissertation, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1998. 28. Hashin. Z., "Analysis of Composite MaterialsA Survey," Journal ofApplied Mechanics, 50, 481505, September, 1983. 29. Anderson, T.L., Fracture Mechanics, Fundamentals and Applications, 2nd Edn, CRC Press LLC, Boca Raton, Florida, 1994. BIOGRAPHICAL SKETCH Sukjoo Choi was born in Seoul, Korea, in 1972. He received his Bachelor of Science degree in aerospace engineering in 1997 from the University of Minnesota at Minneapolis. He worked as a mechanical engineer in control surfaces and mechanics at the Cirrus Design, Corp., Duluth, Minnesota, in 1997. In 1998, He joined the military in Korea and served until 1998. In 2000, he was admitted to the graduate program in the Department of Aerospace Engineering, Mechanics and Engineering Science at the University of Florida, Gainesville, Florida. In December 2002, Sukjoo Choi received a master's degree in Aerospace Engineering with the thesis "A Micromechanics Method to Predict the Fracture Toughness of Cellular Materials." After he achieved the master's degree, he continued graduate study toward the Ph.D. degree at the University of Florida, Gainesville, Florida. 