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Micromechanical Model for Predicting the Fracture Toughess of Functionally Graded Foams

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MICROMECHANICAL MODEL FOR PREDICTING THE FRACTURE TOUGHNESS OF FUNCTIONALLY GRADED FOAMS By SEON-JAE LEE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by SEON-JAE LEE

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To my parents and to my wife Jin-Sook

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iv ACKNOWLEDGMENTS All thanks and praises are to Jesus Christ, the Lord of the Universe, for his blessing, help and guidance. I would like to express my sincere gratitude to my advisor, Dr. Bhavani Sankar, for his guidance, his encouragement and his financ ial support. He is not only my academic advisor but also a great in fluence in my life. My appreciation is also due to Dr. Raphael Haftka, Dr. Peter If ju, and Dr. Reynaldo Roque for serving on my supervisory committee and for their valuable comments and suggestions. This statement of acknowledgement would be incomplete without expressing my sincere appreciation and gratit ude to both my friends and family. I appreciate the friendship and encouragement of all the colleagues at the Center of Advanced Composites (CAC) while working and studying together in the lab. I would like to extend my appreciation to all my family-c hurch members whose continuous support, prayers and help were behind me at all times. I would particularly thank my family, my parents and my brothers in my country, for their continuous support, encouragement and understanding during my entire school career. Last, but not least, I would like to thank my lovely wife, adorable son and cute daughter for their patience and support through the toughest times. They were always there when I needed them to share my difficulties.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii ABSTRACT....................................................................................................................... ..x CHAPTER 1 INTRODUCTION........................................................................................................1 Reusable Launch Vehicle and Thermal Protection System..........................................2 Functionally Graded Foams and F unctionally Graded Materials.................................8 Previous Work on Fracture Mechanics of Functionally Graded Materials................11 Objectives...................................................................................................................14 Scope.......................................................................................................................... .14 2 ESTIMATION OF CONTINUUM PROPERTIES....................................................16 Continuum Properties of Homogeneous Foam...........................................................16 Continuum Properties of Functionally Graded Foams...............................................20 Finite Element Verification of Estimated Continuum Properties...............................23 3 FINITE ELMENT BASED MICROMECHANICAL MODEL................................26 Overview of Micromechanical Model........................................................................26 Macro-model...............................................................................................................29 Imposing Graded Material Properties.................................................................30 Methods for Extracting St ress Intensity Factor...................................................32 Convergence Analysis for Macro-model.............................................................36 Micro-model...............................................................................................................37 4 FRACTURE TOUGHNESS OF GRADED FOAMS UNDER MECHANICAL LOADING..................................................................................................................41 Fracture Toughness under Remote Loading...............................................................41 Study of Local Effect on the Homogeneous Foam under Crack Face Traction.........46

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vi 5 FRACTURE TOUGHNESS ESTIMA TION UNDER THERMAL LOADING.......52 Behavior of Foams under Thermal Loading...............................................................52 Results under Thermal Loading..................................................................................57 6 CONCLUDING REMARKS......................................................................................61 APPEDIX ANALYTICAL SOLUTION FOR BEAM MODEL UNDER THERMAL LOADING..64 LIST OF REFERENCES...................................................................................................68 BIOGRAPHICAL SKETCH.............................................................................................72

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vii LIST OF TABLES Table page 2-1 Material properties of the Zoltex carbon fiber.........................................................21 3-1 Comparison between two methods..........................................................................35 4-1 Fracture toughness of graded and unifo rm foams. The unit-cell dimensions and crack length are kept constant, bu t the strut thickness is varied ( c =200 m crack length, a =0.03 m and =20010-6)........................................................................43 4-2 Fracture toughness of graded and unifo rm foams. The unit-cell dimension is kept constant but the crack length and the strut thickness are varied ( c =200 m ho=40 m and =20010-6).....................................................................................43 4-3 Comparison of the fracture toughness fo r varying unit-cell dimensions with constant strut thickness ( h =20 m )............................................................................44 4-4 Remote loading case – uniform displacement on the top edge................................46 4-5 Remote loading case – uniform traction on the top edge.........................................46 4-6 Stress intensity factor, maximum pr incipal stress and fracture toughness for various crack lengths under crack surface traction..................................................47 4-7 Fracture toughness estimation from re mote loading and crack face traction...........47 4-8 Comparison of fracture toughness under crack surface traction calculated from the superposition method and the micromechanical model.....................................51 5-1 Elastic modulus variation of three different models................................................53 5-2 Results of the body under temperature gr adient form micromechanical model......58

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viii LIST OF FIGURES Figure page 1-1 Thermal protection system in Space S huttle. A) Temperature variation during re-entry. B) Location of different materials..............................................................6 1-2 Schematic diagram of attaching the tiles....................................................................7 1-3 SEM images of A) low density carbon foam and B) high density carbon foam C) metallic foam..............................................................................................................9 2-1 Open cell model with recta ngular parallelepiped unit cell.......................................17 2-2 Microand Macro-stre sses in open-cell foam..........................................................18 2-3 Flexural deformation of struts under shear stresses.................................................18 2-4 Example of variation of elastic modulus and relative density for constant cell length c =200 m h0=26 m and =-20010-6............................................................22 2-5 Boundary conditions for un-cracked plate under uniform extension.......................23 2-6 Comparison of stresses ( yy) obtained using the macroand micromodels in graded foam with constant cell si ze but varying strut cross section........................24 2-7 Comparison of stresses ( yy) obtained using the macroand micromodels in graded foam with constant stru t size but varying cell dimension............................25 3-1 Schematic description of both macroand micro-model.........................................27 3-2 A typical finite element macro-model......................................................................29 3-3 Example of discrete elastic modulus for macro-model w ith ten-regions.................30 3-4 Location of model specimens in the globa l panel. Each specimen is of the same size and contains a crack of given length, but the density at the crack tip varies from specimen to specimen......................................................................................31 3-5 J-Integral for various contours in a macro-model containing 10050 elements and the contour numbers increase away from the crack tip.....................................33 3-6 Stress intensity factor from the stresses normal to the crack plane..........................34

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ix 3-7 Stress normal to the crack plane...............................................................................35 3-8 Stress distribution of the functionally graded foams................................................36 3-9 Variation of energy release rate at th e crack tip with various size macro-models...37 3-10 Embedded beam element (micro model) in two-dimensional eight-node solid model (Macro model)...............................................................................................38 3-11 Force and moment resultants in struts modeled as beams.......................................39 3-12 Variation of fracture toughness with the size of micro-models...............................39 4-1 Edge-cracked model under A) uniform tr action or displacement loading and B) crack surface traction...............................................................................................42 4-2 Comparison of fracture toughness of graded and homogeneous foams having same density at the crack tip....................................................................................44 4-3 Comparison of fracture toughness of graded and homogeneous foams. The graded foams have varying unit-cell dimensions, but cons tant strut cross section h =20 m ....................................................................................................................45 4-4 Fracture toughness estimation from remo te loading and crack face traction...........48 4-5 Application of superposition to repla ce crack face traction with remote traction....49 5-1 Thermal stress distribution output from FEA. .......................................................54 5-2 Thermal stress distribution in homogeneous foam..................................................55 5-3 Thermal stresses distribution in an FGF; the material properties and the temperature have opposite type of vari ation, and this reduces the thermal stresses......................................................................................................................5 5 5-4 Thermal stresses distributi on in an FGF; the variation of the material properties and temperature in a similar manner; a nd this increases the thermal stresses.........56 5-5 Maximum and minimum values of normalized thermal stresses.............................56 5-6 Thermal stresses for various aspect ratio models.....................................................57 5-7 Thermal stresses for various crack lengths...............................................................58 5-8 Comparison the ratio of maximum princi pal stress and stress intensity factor........59 5-9 Fracture toughness for various crack lengths...........................................................60 A-1 A beam of rectangular cro ss section with no restraint.............................................64

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x Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MICROMECHANICAL MODEL FOR PREDICTING THE FRACTURE TOUGHNESS OF FUNCTIONALLY GRADED FOAMS By Seon-Jae Lee May 2006 Chair: Bhavani V. Sankar Major Department: Mechanic al and Aerospace Engineering A finite element analysis based micromech anical method is developed in order to understand the fracture behavior of functiona lly graded foams. The finite element analysis uses a micromechanical model in conjunction with a macromechanical model in order to relate the stress intensity factor to the stresses in the struts of the foam. The continuum material properties for the m acromechanical model were derived by using simple unit cell configuration (cubic unit cell). The stre ss intensity factor of the macromechanical model at the crack tip was evaluated. The fracture toughness was obtained for various crack positions and lengths within the functionally graded foam. Then the relationship between the fracture tough ness of foams and the local density at the crack tip was studied. In addition, convergence tests fo r both macromechanical and micromechanical model analysis were conduc ted. Furthermore, fracture toughness was estimated for various loading conditions such as remote loading a nd local crack surface loading. Local effect was studied by crack face traction conditions. The principle of

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xi superposition was used to analyze the devia tion caused by local loading conditions such as crack surface traction and temperature gradie nts. From the thermal protection system point of view the behavior of graded foam s under thermal loading was investigated, and fracture toughness was estimated. The methods discussed here will help in understanding the usefulness of functionally graded foam in the thermal protection systems of future space vehicles. However, further research is needed to focus on more realistic cell configurations, which capture the complexity of foam and predicts more accurately its mechanical property changes, such as relativ e density and modulus in functionally graded foam, in order to provide more accurate predictions.

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1 CHAPTER 1 INTRODUCTION Since April 12, 1981, the first launch of the space shuttle-the orbiter Columbia the shuttle fleet has played a major role in human space exploration. A large amount of money has been spent on launching satellit es by both the government and the private sector for the purpose of r econnaissance, communication, global positioning system (GPS), weather prediction and space explora tion (Blosser, 2000). The International Space Station program also demands the sp ace launch for constr uction, repair and service. The private sector ha s rapidly spread in the last de cade. Only a cost effective launch system can satisfy the increasing dema nd for lower cost access to space. One of the major goals of the National Aeronautic s and Space Administration (NASA) has been continued lowering of the cost of access to space to promot e the creation and delivery of new space services and other activities that will improve economic competitiveness. A thermal protection system (TPS) which protects the whole body of the vehicle is as crucial as avionics, propulsion, and the stru cture. A TPS is more limiting than fuel constraints, structural strength, or engine’s maximum thrust In order to achieve the goals set by NASA, new TPS concepts have to be introduced, e.g., Integral Structure/TPS concept. This concept can be achieved because of breakthroughs in the development of novel materials such as metallic and carbon foam, and functionally graded materials (FGM). The microstructure of functionally graded metallic and carbon foams can be tailored to obtain optimum performance fo r use in integral load-carrying thermal protection systems due to their low thermal c onductivity, increased streng th and stiffness.

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2 However, models for strength and fractur e toughness of functionally graded foam materials are in their infancy and it will be the main focus of this research. The details of the concepts and literature survey will be discussed in subsequent sections. Reusable Launch Vehicle and Thermal Protection System Currently, expendable rocket vehicles a nd the space shuttles are the major launch systems. An expendable rocket vehicle, such as the US Delta, European Arian, Russian Proton, and Chinese Long March, is a struct ure which contains payload, the systemsupporting hardware required to fly and fuel. Expendable ro ckets can be used only once, and they are expensive. The space shuttle is only partially reusable because its large external tank is separated and burns up in th e atmosphere during launch. The two smaller solid rocket boosters land in the ocean and are recovered, but cannot be reused nearly as many times as the space shuttle itself. Fuel by itself is not compar ably expensive, but tanks to carry it in are, especially if they are only used once such as the space shuttle’s external tank. Furthermore, considerable time for maintenance is required for engines and thermal protection system (TPS) between flights. The TPS alone is estimated to require 40,000 hours of maintenance between fl ights (Morris et al., 1996). The space shuttle is considered as the first generation reusable launch vehicl e due to its partial reusability. In January 1995 NASA announced the development plan for a fully reusable launch vehicle system and designated the X-33 program. The X-33 program ran for 56 months and was cut by NASA in early 2001 due to the failure of a prototype Graphite/Epoxy composite fuel tank during th e proof test. The failure of the tank indicates that the material science and composite manufacturing technology was not advanced enough.

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3 After termination of the X-33 program, NASA’s Integrated Space Transportation Plan (ISTP) was formulated in May 2001 to provide safe, affordab le, and reliable space system. As a key component of the ISTP, th e Space Launch Initiative (SLI) began with a goal to achieve the necessary technology development, risk reduction, and system analysis in order to be used in a second generation reusable launch vehicle (RLV) which expected to be delivered by 2010. A s econd generation RLV had these goals: Reduce risk of crew loss to no more than 1 in 10,000 missions. Reduce payload cost to $1,000 pe r pound, down from today’s $10,000 per pound. Be able to fly more often, with less turnaround time and smaller launch crews. A third generation RLV was planned to star t flying around 2025. Its goal was to reduce cost and improve safety by another order of magnitude. Reduce chance of crew loss to 1 in 1 ,000,000 (equivalent to t oday’s airliners). Reduce payload costs to hundr eds of dollars per pound. The RLV was based on single-stage-to-orb it (SSTO) technology. The concept of SSTO involves a rocket with onl y one stage carrying crews or cargo to orbit. The RLV was NASA’s true vision for a shuttle su ccessor, but after spending many years NASA decided to cancel the program because RLV was not attainable using existing technology, and announced a new strategy that indicated th e shuttle would continue flying until at least 2015. However, in 2003 the space shuttle Columbia was disintegrated during reentry to Earth after 16 days in orbit. After the Columbia tragedy, President Bush announ ced a new “Vision for Space Exploration” in January 2004. The Presiden t’s Vision set NASA in motion to reassess the space transportation program, and to begin developing a new spacecraft to carry

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4 humans into Earth orbit and beyond. Under the plan, a new spacecraft called the Crew Exploration Vehicle (CEV) is to be devel oped and tested by 2008 and the first manned mission is going to conduct no later than 2014. The first manned lunar landing is scheduled no later than 2020, and CEV program continues to explore Mars and other destination in the solar system. NASA hopes to follow this schedule in development of the CEV: 2008 2010 First unmanned flight of CEV in Earth orbit. 2011 – 2014 First manned flight of CEV in Earth orbit. 2015 2018 First unmanned flight of Lunar Surface Access Module (LSAM). 2016 2018 First manned flight of LSAM. 2018 2020 First manned lunar landing with CEV/LSAM system. 2020 Start of planning for Mars mission and beyond. Instead of an airplane-style lifting b ody used in the space shuttle system, an Apollo-like capsule design was decided for th e CEV because of the fact that the new CEV design will use the crew and service m odule design principle. The new CEV design is virtually identical to the Apollo Comm and Module except the implementation of the concept of reusability. The main difference between them is that the new CEV can be used as many as tem times. Thermal protec tion system development is the significant technical obstacles that must be overcome in order to implement reusability and to improve affordability (vehicle weight reduction) by both new design concept and material selection such as multifunctional materi als that perform structural or other roles. Harris et al. (2002) surveyed the properti es of advanced metallic and non-metallic material systems. They provided the guidan ce of emerging materials with application in

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5 order to achieve NASA’s long-term goal by addressing materials already under development that could be available in 5 to 10 years as well as those that are still in the early research phase and may not be available for another 20 to 30 years. The main objective of the thermal protect ion system is protecting the vehicle by keeping it under acceptable temperature limit and human occupants from heat flow. Heat sinks and ablative material were used to prot ect the vehicles before Space shuttle. During the re-entry process, ablative material is charred and vaporized while the heat sinks absorb the heat. None of the early vehicles had to be reusable so these materials and techniques were enough to pr otect the early vehicles. In the late 1960s, the space shuttle program was proposed. The program aimed to produce a vehicle that would be larger th an any that had flown in space before. Conventional aluminum was selected for the ma in structure and a layer of heat resistant material for protecting it. The properties of aluminum demand that the maximum temperature of the vehicle’s structure be kept below 175 C in operation. But aerothermal heating during the re-entry process creates high surface temperature which is well above the melting point of aluminum (660 C ). Thus, an effective insulator was needed. A silica-based insulation material was decided for the heat-resistant tiles and other coverings to protect the Shuttle’s ai rframe. Figure 1-1 s hows seven different materials which cover the external surf ace of the Space Shuttle according to the temperature variation du ring the re-entry. The materials were chosen by their wei ght efficiency and stability at high temperature. The areas of the highest surf ace temperature in the Shuttle, the forward nose cap and the leading edge of the wings, are made with Reinforced Carbon-Carbon

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6 (RCC). There are two main types of tiles, referred to as Low-temperature Reusable Surface Insulation (LRSI) and High-temperat ure Reusable Surface Insulation (HRSI). Relatively low temperature of surface where the maximum surface temperature runs between 370 and 650 C is covered by LRSI. HRSI c overs the areas where the maximum surface temperature runs between 650 and 1,260 C A B Figure 1-1. Thermal protection system in Space Shuttle. A) Temperature variation during re-entry. B) Location of different materials. (Courtesy of W. Jordan, Source: http://www2.latech.edu/~jordan/N ova/ceramics/SpaceShuttle.pdf Last accessed February 14th, 2005). Many of the tiles have been replaced by a material known as Flexible Reusable Surface Insulation (FRSI), and Advanced Flex ible Reusable Surface Insulation (AFRSI) in the area where the maximum surface temperature does not exceed 400 C These tiles are lighter and less expensive than LRSI and HRSI, and using them enabled the Shuttles to lift heavier payloads. The tiles are brittle and vulnerable to crack under stress. The tiles could not be mounted directly to th e main body structure of the Shuttle due to expansion and contraction of the aluminum st ructure by temperature change. Instead of direct mounting on the structure, the tiles ha ve to mount to a felt pad using a silicone adhesive, and then the tile and pad combina tion are bonded to the structure as seen in

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7 Figure 1-2. Tiles are occasiona lly lost during take off beca use of the incredible loud noise as well as aerodynamic forces. Because of this, as well as weight concerns, many of the fuselage tiles were replaced by FRSI blankets. Figure 1-2. Schematic diagram of attaching th e tiles. (Courtesy of W. Jordan, Source: http://www2.latech.edu/~jordan/ Nova/ceramics/SpaceShuttle.pdf Last accessed February 14th, 2005). These material developments and techniques enable the partially reusable Space Shuttle to offer more capability. However, the tiles have their limitations. During both liftoff and landing, tiles can become da maged and chipped. About 40,000 hours of maintenance is required between flights (Morris et al., 1996). For fully RLVs, the tiles would not provide sufficient protection and so me other solution would be necessary. Blosser (1996) emphasized the durability, operability and cost effectiveness as well as light weight for new TPS to achieve the goal of reducing the cost of delivering payload to orbit. Most other proposed reusable therma l protection systems have involved some kind of advanced high-temperature metal. Metallic TPS is considered as a much-n eeded alternative to the ceramic-based brittle tile and thermal-bla nket surface insulation currently used on the Space Shuttle. Metallic TPS offers the significant advantages (Harris et al., 2002).

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8 Does not require high temperature seals or adhesive development Does not require waterproofing or other restorative processing operation between flights Significantly reducing operational cost Saving on vehicle weight, when used as part of an integrated aeroshell structural system The TPS forms the external surface of an RLV and is exposed to a wide variety of environments corresponding to all phases of flight (Dorse y et al, 2004). Thus, the TPS requirements must apply to any external vehicle airframe surface. Recently, a new Adaptable, Robust Metallic, Operable, Reus able (ARMOR) metallic TPS concept has been designed (Blosser et al ., 2002) and demonstrated the capability of protecting the structure from on-orbit-debris and micromet eoroid impact (Poteet and Blosser, 2004). The concepts of metallic TPS depend primarily on the properties of available materials. The development of foam and FGM as a core material of TPS panel may offer dramatic improvements in metallic TPS (Harris et al., 2002 ). An integrated wa ll construction is an approach which the entire structure is de signed together to a ccount for thermal and mechanical loading (Glass et al, 2002). An in tegrated sandwich TPS with metallic foam core is studied under steady state and transi ent heat transfer conditions and compared with a conventional TPS design (Zhu, 2004). Functionally Graded Foams and Functionally Graded Materials Foams are generally made by dispersing gas in a material in li quid phase and then cooling it to a solid. Solid foams can also be made by dispersing a gas in a solid. These solid foams are generally called cellular solids, often just called foams. During the last few decades, many attempts have been made to produce metallic foams, but methods

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9 have suffered from high cost, and only poor quality foam materials were produced. In the last ten years, improved methods were di scovered, and only recently various methods are available to produce high quality metallic foam. Some start with the molten metal and others with metal powder. Graded foams can also be manufactured by dispersing hollow micro-balloons of varying sizes in a matrix medium (Madhusudhana et al., 2004). The porous structures of carbon and metallic foams are depicted in Figure 1-3. Foams can be used in many potential engi neering applications ranging from light weight construction to thermal insulation to energy absorption and thermal management. The mechanical properties of foams are strongly dependent on the density of the foamed material as well as their cell configuration. For example, th e quantities such as elastic modulus and tensile strength increase with in creasing density of foams. Foams can be used in many potential engineering applicati ons ranging from light we ight construction to thermal insulation to energy absorp tion and thermal management. A B C Figure 1-3. SEM images of A) low density carbon foam and B) high density carbon foam C) metallic foam. Foams can be categorized as open-cell a nd closed-cell foam. In open-cell foams the cell edges are the only so lid portion and adjacent cells are connected through open faces. If the faces are also solid, so that each cell is sealed off from its neighbor, it is said to be closed-cell foam (Gibson & Ashby, 2001). In this study, from the thermal

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10 management application point of view only th e open-cell foam is considered due to its large surface area and the ab ility to transfer heat by working fluid in open porous structure, if necessary. The combination of open porosity and la rge specific surfaces allows a reduction in size of the thermal ma nagement system. A reduction in size of the thermal management system will reduce weight and improve efficiency. Functionally graded materials (FGMs) are a relatively new class of nonhomogeneous materials in which material prope rties vary with location in such a way as to optimize some function of the overall FGM. The FGM concept originated in Japan in 1984 as a thermal barrier material which is cap able of withstanding a surface temperature of 1,725 C and a temperatur e gradient of 725 C across a cross section less than 10 mm Since 1984, FGM thin films have been compre hensively researched and are almost a commercial reality. The primary advantage of FGM over conve ntional cladding or bonding is avoiding weak interfacial planes because material pr operties are engineered to have relatively smooth spatial variation unlike a step incr ease in conventional cladding or bonding. Thus, FGMs are widely used as coatings and interfacial zones to re duce mechanically and thermally induced stresses caused by the mate rial properties mismatch and to improve the bonding strength. Generally, a functionally gr aded material (FGM) refers to a twocomponent composite characterized by a compositional gradient from one component to the other. In contrast, traditional composites are homogeneous mixtures, and they therefore involve a compromise between th e desirable properties of the component materials. Since significan t proportions of an FGM cont ain the pure form of each component, the need for compromise is el iminated. The properties of both components

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11 can be fully utilized. For example, the t oughness of a metal can be combined with the refractoriness of a ceramic, w ithout any compromise in the toughness of the metal side or the refractoriness of the ceramic side. Ho wever, in this study, only the concept of varying material properties is adopted, a nd functionally graded foams (FGFs) are produced by changing the size of unit-cell or the thickness of strut in the foam. Previous Work on Fracture Mechanics of Functionally Graded Materials In order to utilize FGMs as reliable engineering mate rials in structures, among other properties their fracture mechanics has to be understood. Furthermore, methods to compute the stress intensity factor (SIF) and energy release rate have to be developed because the stress intensity f actor cannot be measured directly in an experiment, but it can be found through the relations between SIF a nd a measurable quantity, such as strain, compliances or displacement. Sound fracture mechanics principles have been established for conventional homogeneous materials so that the strength of a structure in the presence of a crack can be predicted. However, the fracture mechanic s of a functionally graded material which is macroscopically non-homogeneous is only beginn ing to be developed. Analytical work on FGM goes back to the late 1960s wh en Gibson (1967) modeled soil as a nonhomogeneous material. Analytical studies have shown that the asymptotic crack tip stress field in FGMs possesses the same square root singularity se en in homogeneous materials. Analytical studies of Atkinson and List (1978) and Gerasoulis and Sriv astav (1980) are some of the earliest work on crack growth in non-homoge nous materials in order to evaluate its integrity. Atkinson and List (1978) studi ed the crack propagation for non-homogenous materials subjected to mechan ical loads assuming an exponen tial spatial vari ation of the

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12 elastic modulus. Gerasoulis and Srivastav (1980) studied a Grifft h crack problem for non-homogeneous materials using integral e quation formulations. Delale and Erdogan (1983), Eischen (1987), Jin and Noda (1994) an d Erdogan (1995) showed that the nature of the inverse-square-root-singula rity of crack tip is also pr eserved for an FGM as long as the property variation is pi ecewise differentiable. The work by Delale and Erdogan (1983) is accredited with ha ving first suggested the standa rd inverse-square-root stress singularity for an FGM in which a crack is parallel to the elastic modulus gradient. Eischen (1987) confirmed their work by us ing eigenfunction expans ion technique in nonhomogeneous infinite plane. Jin and N oda (1994) further confirmed for FGM with piecewise differentiable property variation. In 1996, Jin and Batra studi ed crack tip fields in general non-homogeneous materials and stra in energy release rate and stress intensity factor using the rule of mixture. Based on the early work of Delale and Erdorgan (1983) that showed the negligibility of the effect of the varia tion on Poisson’s ratio, Erdorgan and Wu (1997) analyzed an infi nite FGM strip under various remote loadings by using an exponential varying elastic co nstants and constant Poiss on’s ratio. Although such progress has increased the understanding of fr acture mechanics of FGM, a suitable stress intensity factor solution is needed in designing component s involving FGM and improving its fracture toughness. In engineering context, the closed-form SI F solution is desirable for easier use in the analysis of fracture of FGM structures fo r a variety of specimen configurations. The exact solutions are not available yet and some researchers have attempted to find simple and approximate closed-form solutions. Yang and Shih (1994) obtained an approximate solution for a semi-infinite crack in an interl ayer between two dissimilar materials using a

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13 known bi-material solution. Gu and Asaro (1997) obtained the complete solution of semi-infinite crack in a stri p of an isotropic FGM under e dge loading. The solution was analytical up to a parameter which is obt ained numerically. Then, the solution was extended to the strip is made of an orthotropic FGM. Ravichandran and Barsoum (2003) obtained approximate solution and compared the results with the values obtained by finite element modeling (FEM). The application of the finite element met hod to determine crack tip stress fields has been rapid progress (Broek 1978). A finite element based method for determination of stress intensity factor in FGM was proposed by Gu et al. (1999). They used standard domain integral to evaluate the crack-tip field for FGM and studied the effect of nonhomogeneity in numerical computation of th e J-integral. They concluded that the conventional J-integral can provi de accurate results as long as the fine mesh near crack tip is provided. Honein and Hermann (1997) have studied the conservation laws for nonhomogenous materials and propos ed a modified path-indepen dent integral. Weichen (2003) constructed another version of path -independent integrals of FGM by gradually varying the volume fraction of the constituent materials. Numerical simulation was carried out by Marur and Tippur (1999) using linear material property variation in the gradient zo ne. They studied the influence of material gradient and the crack position on the fracture parameters such as complex stress intensity factor and energy rele ase rate. Anlas et al. (2000) calculated and compared the stress intensity factors obtained for a cr acked FGM plate by using several different techniquesenergy release rate, J-integral a nd a modified path i ndependent integral. They evaluated the J-integral and a modified J-integral numerically by technique similar

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14 to Gu et al. (1999) and Honein and Herma nn (1997) respectively. The results were compared with the analytical solutions of Erdogan and Wu (1997). Furthermore, the accuracy of the finite element method a nd mesh refinement was investigated. In contrast to above-described analytical studies and numerical investigation, there are relatively few experimental works on fracture mechanics of FGM. A typical laboratory technique is the us e of photo elasticity. Butch et al. (1999) examined the surface deformation in the crack tip region by the optical method of Reflection Coherent Gradient Sensing. They used a graded pa rticulate composition comprised of spherical glass filler particles in an epoxy matrix as a test specimen. Recently, Rousseau and Tippur (2002) examined the particulate FGM by mapping crack tip deformation using optical interferometery. They used a finite element analysis in order to develop fringe analysis and to provide a direct co mparison to the optical measurements. Objectives The objectives of this research are to de velop micromechanical models to predict the fracture toughness of functionally graded foams under various loading conditions – mechanical and thermal loading as insulation materials for load carrying thermal protection system, and to develop the understand ing of the effect of graded foam solidity profile on its fracture mechanics. The methods will also be used to understand the effects of thermal gradients on fracture of homogeneous foams. Scope Chapter 1 reviewed some background info rmation regarding functionally graded foams as the thermal protection system of ne xt generation reusable launch vehicles and some previous works on fracture mechanics of FGM. Chapter 2 discusses the method to estimate the material properties of functi onally graded foams (FGFs). At first,

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15 formulations for homogeneous foam will be established and then the methods will be extended to the FGFs. Chapter 3 describes the finite element analysis (FEA) of the micromechanical model. In Chapter 3, macro and micro models for graded cellular materials are explained with key issues in both models. Chapter 4 discusses the results under mechanical loading including remote loading uniform traction and uniform displacement) and local loadi ng (crack face traction). Chap ter 5 presents the behavior and the results under thermal loading on hom ogeneous foam and FGF. The concluding remarks are presented in Chapter 6.

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16 CHAPTER 2 ESTIMATION OF CONTINUUM PROPERTIES The functionally graded foam can be modeled either as a non-homogeneous continuum, or as a frame consisting of beam elements. The former model will be referred to as the macro-model and the latter as the micro-model. We require both models for the simulation of crack propagati on in graded foams. The region surrounding the crack tip is modeled using the micro-model, where as the region away from the crack tip uses the macro-model. The micromechani cal model is treated as an embedded model around crack tip. The macro-model of the f unctionally graded foam requires continuum properties at each point or at least for each elem ent in the finite element model. In this chapter, the procedures fo r calculating the continuum properties of a homogeneous cellular medium (open-cell) is presented, and then the method is extended to functionally graded foams. Continuum Properties of Homogeneous Foam Most of the open-cell foams with periodic microstructure can be considered as orthotropic materials. Choi and Sankar (2003) derived the elastic constants of homogeneous foams in terms of the strut mate rial properties and unitcell dimensions. In their model they assumed that the strut has a square cross section hh and the unit-cell is a cube. In the present appro ach, the general case is considered wherein the unit-cell is a rectangular parallelepiped of dimensions123ccc as shown in Figure 2-1. The derivation of formulas for the relative densit y and elastic modulus are straightforward. The relative density */ s is related to the porosity of the cellular material. A superscript

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17 denotes the foam prope rties and a subscript s denotes the solid properties or the strut properties. The density of the foam can be obtained form the mass and volume of the unit-cell. Then, the relative density can be e xpressed as a function of the dimensions of unit-cell and the strut thic kness as shown below: 23 123 123()2ssm ccchh V ccc (2-1) where m is the mass and V is the volume of unit-cell. Figure 2-1. Open cell model with re ctangular parallel epiped unit cell. Elastic modulus can be evaluate d by applying a tensile stress on unit area of the unit cell as shown in Figure 22. The equivalent force on th e strut caused by the stresses can be written as* 13() Fcc In micro-scale sense, the force F causes stresses s in sectional area h2 (Figure 2-2). Therefore, the stress s in the section h2 and the corresponding strain can be expressed as ** 1313 222and=s s s s cccc F E hhhE (2-2) where Es is elastic modulus of strut. Th erefore, elastic modulus of foam E* can be derived from Eq. (2-2) as

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18 222 *** 123 231312,, s sshhh EEEEEE cccccc (2-3) Figure 2-2. Microand Macrostresses in open-cell foam. The derivation of shear modulus is slightly involved and it is described below. We show the derivation of the shear modulus 12G from the unit-cell dimensions, strut cross sectional dimensions and the strut elastic modul us. When a shear stress is applied, struts are deformed as shown in Figure 2-3. Figure 2-3. Flexural deformation of struts under shear stresses. h h c1 Unitarea 2,y 1,x 3,z *F1 c1/2 c2/2 1 Curvature=0 F2 2 c2/2 c1/2

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19 Bending moment becomes zero at the middle of struts because the curvatures are zero due to symmetry. The struts are assu med as a beam fixed at the end with a concentrated force at the middle at a distance 12 c and 22 c, respectively from the fixed end. The maximum displacement can be written as, 33 12 21 33 1222 and 3333sscc FF PLPL EIEIEIEI (2-4) where 341212 bhh I (the moment of iner tia). The applied shear stress can be written as 12 12 1323FF cccc. Using the relations, 12 12FF cc the maximum displacements in Eq. (24) can be rewritten as 32 12122 12and 2424 s scFccF EIEI (2-5) The shear strain 12 can be derived as 211221 12 21122222 cc cccc (2-6) Using Eq. (2-5), the shear strain can be written as, 2 1212 12() 12scccF EI (2-7) The shear modulus 12G can be derived as 2 * 23 12 12 22 12 12122312112 12s sF ccEI G cccFccccc EI (2.8)

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20 Substituting for the moment of inertia, I 4 12 12312() s h GE ccccc (2.9) The shear modulus in the other two planes can be obtained by cy clic permutation as 4 23 12323 4 31 12313() () s s h GE ccccc h GE ccccc (2.10) Continuum Properties of F unctionally Graded Foams The properties of a functionally graded fo am can be represented by a function of the coordinate variables x y and z The actual functional form depends on the application and also the type of information sought from the homogenized model of the foam. In this study, the functions of material properties will be assumed such that the properties calculated at the cente r of a cell will correspond to th e properties of the homogeneous foam with that cell as its unit cell. Thus the function is actually defined only at the centers of the cells of the functionally graded foam. Then, these points will be curvefitted to an equation in order to obtain the continuous variation of properties required in the continuum model. This approach will be verified by solving some problems wherein the graded foam is subjected to some si mple remote loading conditions (uniform displacement loading) and comparing the resultant stresses from the macroand micromodels. The material properties of stru t correspond that of Zoltex Panes 30MF High Purity Hilled carbon fiber studied earlier by Choi & Sankar (2003). The Zoltex Panes 30MF High Purity Hilled carbon fiber is chosen because of the high percentage of carbon component weight (99.5%). The strut properties are listed in Table 2-1.

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21 Table 2-1. Material properties of the Zoltex carbon fiber. Density, s 31750/ Kgm Elastic Modulus, Es 207 GPa Poisson’s ratio, s 0.17 Ultimate Tensile Strength, u 3600 M Pa The relative density of functionally gr aded foams (FGF) depends on both the dimensions of the unit-cell and the strut thickne ss. Therefore, three different cases can be considered. The first case is the one wh ere the dimensions of the unit-cell remain constant while the strut thickness varies along the x -axis. In the second case the strut thickness is kept constant with varying cell lengt h. The last case is varying both of them. In this paper, the first two cases are studied independently. Furthermore, the material properties of functionally graded foam can be either increasing or decreasing along the x axis. Therefore, the fracture properties of both increasing and decreasing cases are studied and compared to the homogenous case. For the case where the strut dimensions va ry, the thickness of the square strut is assumed to vary as 0() hxhx (2-11) where is a parameter that determines the degree of gradation of the properties. Then the properties such as density and elastic consta nts of the graded foam can be assumed to vary as given by the equations for homoge neous foams, but changing the constant h by the function h ( x ). Figure 2-4 shows the variation of relative density and elastic modulus. For example, the relative density variati on of the functionally graded foam with varying beam thickness can be written as 23 *()() 32shxhx cc (2-12)

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22 where the unit-cell is assumed as a cube of dimension c (i.e. 123cccc ). Similar equations can be derived for elastic modulus and shear modulus as 2 4 *() 1() 2s s hx E E c hx GE c (2-13) x ( m ) 0.000.020.040.060.080.10 Elastic Modulus ( MPa ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Relative Density (* /) 0.00 0.01 0.02 0.03 0.04 0.05 Elastic Modulus Relative Density Figure 2-4. Example of varia tion of elastic modulus and rela tive density for constant cell length c =200 m h0=26 m and =-20010-6. For the case where the unit-cell dimensions vary, we can consider the case where h c2 and c3 are constants, but c1 varies as 1 11iicc (2-14) where i denotes the cell number from the left edge (i.e. 0 1c represents the size of cell at left edge) and is the increment in the cell length in the x direction. Again the properties of the foam will be calculated at the center of each cell using the equations for homogeneous foams as given in equations (2-2) through (2-9).

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23 Finite Element Verification of Estimated Continuum Properties The accuracy of the estimated elastic constants, when a material property discretization is introduced, is investigated by comparing the stress field from macro-and micro-models. A simple mechanic problem was solved using both un-cracked macroand micro-models. A uniform displacement (70 m ) was applied along the upper edge of a rectangular plate using the macro-model, which consists of two dimensional plane stress elements (eight nodes bi-quadratic reduced integration element). The elastic constants of the non-homogeneous material varied as given by Eq. (2-13). In the finite element model the elastic consta nts within each element were considered constant. The boundary conditions are depicted in Figure 2-5. Figure 2-5. Boundary conditions for un-cr acked plate under uniform extension. The right lower corner was fixed to prev ent the rigid body motion. The resulting displacements along the boundary of a micromodel embedded in the macro-model were applied as the boundary displa cements of the micro-mode l by using the three-point interpolation. For the micro-model, each st rut was modeled as an Euler-Bernoulli beam element with two nodes and three integration poi nts. In order to verify the validity of properties used in the macro-model, the stre sses in both models are compared. In the ux 0, uy=0 ux=0, uy=0 Uniform Displacement

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24 case of macro-model the stresses are obtained as the finite element analysis output. The outputs in micro-model are the axial force and moment resultant in the beam element. We convert these forces into equivalent stre sses by dividing by the strut cross sectional area13cc The shaded region on Figure 2-5 represen ts the micro-model. Both constant cell length with varying strut thickness and c onstant strut thickness with varying cell size cases are considered. In the constant ce ll length case the cell length is assumed as 200 m The macro-model consists of 10050 pl ane solid elements. The strut cross section is assumed to vary as a function of x according to the equation()ohxhx where oh =40 m and =-20010-6. The region corresponding to the micro-model in the macro-model consists of 155 plane stress elements. The micro-model uses 2,250 beam elements. x ( m ) 0.0000.0020.0040.0060.0080.0100.0120.0140.016 Stresses ( MPa ) 0 2 4 6 8 10 12 14 macro-model micro-model Approx. 5% difference Figure 2-6. Comparison of stresses ( yy) obtained using the macroand micromodels in graded foam with constant cell si ze but varying strut cross section. The results for the stress component yy from the macroand micro-models are compared in Figure 2-6. The maximum diffe rence in stresses between the macroand micro-models is about 5%. In the second case, the strut is assumed to have a square cross

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25 section (h=20 m ) and the cell length 1c was varied along the x direction with 0 1c =200 m and =-0.15 m The dimensions of the cel l in the 2 and 3 directions,2c and3c are kept constant (100 m ). The stress component yy from macro-model and micro-model are compared in Figure 2-7. x ( m ) 0.0000.0050.0100.0150.0200.0250.030 Stresses ( MPa ) 0 2 4 6 8 macro-model micro-model Figure 2-7. Comparison of stresses ( yy) obtained using the macroand micromodels in graded foam with constant stru t size but varying cell dimension. The above examples illustrate and validate tw o important concepts that will be used in this dissertation. We find that modeling graded foams with microstructure as a nonhomogeneous continuum provides good result s for micro-stress and displacements. Some researchers, e.g., Gu et al. (1999) and Santare and Lambros (2000) have used different properties at the Ga uss (integration) points within the element. However, using the homogenized properties at the center of the continuum element in the FE model seems to be reasonable and yields accurate results.

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26 CHAPTER 3 FINITE ELMENT BASED MI CROMECHANICAL MODEL In this chapter, we describe a finite element based micromechanics model for estimating the fracture toughness of functionally graded foams. The crack is assumed to be parallel to the material properties gradient direction. At first, we describe a finite element based micromechanics model for es timating the fracture toughness in order to understand the key ideas of micromechanical modeling. Detailed macroand micromodel descriptions are presented, and the me thod of extracting stress intensity factor from the finite element analysis is describe d in depth. Also, c onvergence test in both macroand micro-models were performed. Overview of Micromechanical Model The functionally graded foam, a cellula r material is non-homogeneous in the macro-scale. That is, the microstructure is gr aded and the foam is treated as a functionally graded material in macro-scale. The foam can be modeled either as a non-homogeneous continuum, or as a frame consisting of beam elements to model the struts. The former model (continuum model) will be referred to as the macro-model and the latter (frame model) as the micro-model. In the finite el ement analysis, solid elements are used in the macro-model and beam elements in the micro-model. In the finite element analysis model, due to symmetry, only the upper half of the plate is considered. The lower edge has a zero displacement boundary condition in the y direction to account for symmetry. As descri bed in the previous ch apter, the functional variation of material proper ties is estimated by extending the method of calculating the

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27 continuum properties of a homogeneous cellula r medium. The eight -node quadrilateral elements were used to discretize the macromodel and functional variation in material properties is implemented by having 100 vert ical layers, with each layer having a constant value of material properties. A cr ack can be created in the functionally graded foam by removing a set of struts along the intended crack surface in micro-model and by removing the zero boundary condition along the intended crack surface in macro-model. A portion of the foam surroundi ng the crack tip is consider ed as the micro-model (see Figure 3-1). A B Figure 3-1. Schematic description of both macroand micro-model. A) Macro-model consists of plane 8-node solid elements. The region in the middle with grids indicates the portion used in the micromodel. B) The micro-model consists of frame elements to model the indivi dual struts. The displacements from the macro-model are applied as boundary conditions in the micro-model. The dimensions of the micro-model should be much larger than the cell size (strut spacing) so that it can be considered as a continuum. For the case of uniform displacement loading, the upper edge is loaded by uniform displacement in the y direction. The maximum stresses in the stru ts in the vicinity of the crack tip are h h c Crac k x y Width Macro-model Micro-model v0 A B C D

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28 calculated from the finite element micro-model. From the failure criterion for the strut material, one can calculate the maximum stress intensity factor that will cause the failure of the crack tip struts, and thus causing crack propagation in a macro-scale sense. The key idea in this approach is to be able to calculate the stress intensity factor for a given boundary displacements or apply a set of bounda ry conditions that corresponds to a given stress intensity factor in the macro-scale sens e. For this purpose we turn to the macromodel as shown in Figure 3-1 (A). In the macr o-model a much larger size of the foam is modeled using continuum elements, in the present case, plane solid elements. The micromodel is basically embedded in the macro-mode l. The displacements of points along the boundary of the micro-model are obtained from the finite element analysis of the macromodel and applied to the boundary of the micro-model by using three points interpolation. The maximum principal stresses at the crack tip can be calculated from the force and moment resultants obtained from the micro-model as 2tip tip tip tip tiptiph M F I A (3-1) where is maximum principal stress at the crack-tip. are force and moment resultant. is cross-sectional area and is the thickness of strut. tip tiptip tiptip tiFM Ah I is the moment of inertia.p The strut material is assumed brittle and will fracture when the maximum principal stress exceeds the ultimate tensile strength. The fracture toughness of the foam is defined as the stress intensity factor that will cause the crack-tip struts to fail in a microscale sense and cause the crack to propag ate in a macro-scale sense. The fracture

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29 mechanism of brittle material is governed by Linear Elastic Fracture Mechanics (LEFM). Therefore, the fracture toughness can be estimated from the following relation. Itip ICuK K (3-2) Macro-model In macro-model, the conve ntional two dimensional isoparametric plane-stress elements are used. The problem geometry is shown in Figure 3-1 (A). The material gradient is in the x -direction. A pre-processor pr ogram was coded using MATLAB, with the parameters such as unit-cell size, strut thickness, crack size, and (defined in previous chapter) in order to generate a rectangular mesh with eight-node isoparametric elements with two-degree of freedom at each node and to impose the material gradient in the macro-model. Crack tip A B Figure 3-2. A typical finite element macr o-model. A) constant unit-cell size with varying strut thickness. B) varying(d ecreasing) unit-cell si ze with constant strut thickness. Only half of the model is represented in the finite element analysis by invoking that the model is symmetric with respect to its midline, x -axis. A zero displacement boundary

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30 condition in y -direction is employed in the lower edge to account for symmetric. Figure 3-2 shows a typical finite element mode l which consists of 5,000 eight-node isoparametric elements (50 elements in ver tical direction and 100 elements in horizontal direction) with 1,5300 nodes. It should be not ed that the number of elements in a typical model was decided after convergence test by generating a coarse mesh (smaller number of elements) and progressively reducing the mesh size (increasing number of elements), to be discussed in this section later. Imposing Graded Material Properties Finite element analysis of functionall y graded foam for macro-model requires imposing the required the variati on of material properties in x -direction. The material properties are graded by either changing th e thickness of struts or changing the dimensions of the unit-cell described as be fore. Relative density, elastic modulus and shear modulus vary along the x -axis corresponding to the equations derived in the previous section. x (m) 0.000.010.020.030.040.050.060.070.080.090.10 Elastic Modulus ( MPa ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Actual Elastic Modulus Discrete Elastic Modulus Figure 3-3. Example of discrete elastic modulus for macro-mode l with ten-regions.

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31 When functionally graded foam is modele d as a homogeneous solid (macro-model), a material property discretizat ion is introduced. The disc retization is done by grouping elements in the gradient region into narrow ve rtical strips and assi gning constant values of estimated material properties at the cen troid of the strip of grouped elements. For example, Figure 3-3 shows the discrete el astic modulus for the ten-region model. However, the Poisson’s ratio is kept consta nt because the effect of a variation of Poisson’s ratio is negligib le (Delale and Erdogan, 1983). Global panel Const crack length Variation of beam thickness Figure 3-4. Location of model specimens in the global panel. Each specimen is of the same size and contains a crack of given length, but the density at the crack tip varies from specimen to specimen. The Mode I fracture toughness with various relative densities is conducted in two different sets for the constant unit-cell lengthcase. The first set is controlling the crack length while the variation of ma terial properties remains same. The other set is shown in Figure 3-4. The crack length remains cons tant while the material properties are controlled to locate desired relative density at the crack tip. However, the dimensions of models are fixed (0.1 m by 0.5 m for macro-model and 0.015 m by 0.005 m for micro-

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32 model). For the case where the unit-cell dimensions change, the number of elements both in macro-model (10050 elements) and micromodel (2,250 elements) are fixed and the material properties at the crack tip is controlled by 0 1c and Therefore, the dimensions of models are not fixed. Methods for Extracting Stress Intensity Factor Considering only Mode I symmetric loading (mode-mixity=0), the stress intensity factor at the crack tip is ca lculated from traditional methods in computational fracture mechanics i.e. point matching and energy method (Anderson, 2000). The point matching method is the direct method in which the stress intensity factor can be obtained from the stress field or from the displacement field, while the energy method is an indirect method in which the stress intensity factor is determ ined via its relation with other quantities such as the compliance, the elastic energy or th e J-contour integral (Broek, 1978). The advantage of the energy method is that th e method can be applied as both linear and nonlinear. However, it is difficult to separa te the energy release rate into mixed-mode stress intensity factor components. In this paper, the crack-tip stre ss field and J-contour integral are used to find and verify the stress intensity factor for the point matching and the energy method respectively. In energy method, the J-contour integral can be evaluated numerically along a contour surrounding the crack tip, as long as the deformations are elastic. Generally, Jcontour integral is not path independent for non-homogeneous material. Therefore, Jcontour integral is expected to vary with contour numbers as shown in Figure 3-5. The contour numbers represent incrementally larg er contours around the crack tip. The mesh refinement governs the size and increments of contours.

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33 y = 1.21193E-05x4 1.27727E-03x3 6.32919E-02x2 2.05304E+00x + 7.61841E+02 R2 = 9.99992E-01 0 100 200 300 400 500 600 700 800 16111621263136414651Contour NumberJ-contour Integral Figure 3-5. J-Integral for various contours in a macro-m odel containing 10050 elements and the contour numbers increase away from the crack tip. The first few contours are disregarded due to inaccuracy for most finite element meshes (Anlas et al., 2000). J-contour integral as r 0 is obtained by fitting a fourth order polynomial to the output values of J-contour integral. The limiting value of Jcontour integral can be evalua ted numerically as the intercep t of the polynomial curve at y -axis. The value of J-integral for a contour very close to the crack-tip is related to the local stress intensity factor as in the case of a homogeneous material (Anlas et al., 2000). Thus, energy release rate, G is identical to the value of Jcontour integral as the path of contour approaches to crack-tip (Gu and Asar o, 1999). Conceptually energy release rate, G can be found by the variation of J-contour in tegrals as shown in Figure 3-5. The stress intensity factorIKof a functionally graded foam (two -dimensional orthotropic) can be found from G using the relation (Sih & Liebowbitz, 1968). 1122221266 11111 11 2 22 22 22Iaaaaa GK aa (3-3)

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34 where, 112233 123111 ,,aaa EEE 122331 445566 2313120 111 ,, aaa aaa GGG Using the point matching method of stre ss field, the opening mode value of the stress intensity factor can be calculated from the yy stress ahead of the crack (Sanford, 2003). 02(0)o Iyy rKLimr (3-4) The stress intensity factor can be found by pl otting the quantity in square brackets against distance form the crack tip and extrapolating to r = 0 Figure 3-6 shows the one of the example plot of 2yyr versus distance from the crack tip. A 4th order polynomial regression is also shown in Figure 3-6. The y -intercept of the curve yields the value of KI. y = -4.70773E+11x4 + 4.70881E+10x3 1.99894E+09x2 + 6.41493E+07x + 2.42043E+05 R2 = 9.99997E-010.00E+00 2.00E+05 4.00E+05 6.00E+05 8.00E+05 1.00E+06 1.20E+06 1.40E+06 0.00000.00500.01000.01500.02000.02500.0300distance from crack tip, r ( m ) 2yyr Figure 3-6. Stress intensity factor from the stresses normal to the crack plane.

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35 The stress intensity factor defines the am plitude of the crack tip singularity and the conditions near crack-tip (Ande rson, 2000). Stress near crack tip increases in portion to the stress intensity factor. Consider the M ode I singular field ahead the crack tip, the stress normal to the crack plane, yy can be defined from Eq. (3-5). 2I yyK r (3-5) Figure 3-7 shows the stress normal to the crack plane versus distance form the crack tip. Where the square-root singularity dominated zone, Eq. (3-5) is valid while stress far from the crack tip is gov erned by the remote boundary conditions. Table 3-1. Comparison between two methods. Crack Length ( m ) 0.01 0.02 0.03 0.04 0.05 Relative density at the crack-tip 0.094582 0.085536 0.076874 0.068608 0.06075 KI From J-Integral ( Pa-m) 1.2238E+061.0699E+069.2893E+05 8.4886E+05 7.2691E+05 KI From crack-tip stress field ( Pa-m) 1.2262E+061.0715E+069.3360E+05 8.0605E+05 6.8363E+05 % difference 0.195 0.145 0.503 5.043 5.954 0.00E+00 2.00E+06 4.00E+06 6.00E+06 8.00E+06 1.00E+07 1.20E+07 1.40E+07 1.60E+07 1.80E+07 0.0000.0050.0100.0150.0200.0250.030 distance from crack tip, r ( m )yy ( Pa )KI/(2 Pi* r)1/2remote Singularity dominated zone Figure 3-7. Stress normal to the crack plane.

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36 The stress intensity factor from J-Integr al and stress-matching were compared for various cases in Table 3-1. The maximum difference between the two methods is less than 6%. Convergence Analysis for Macro-model For the convergence test, the model whic h has constant cell size with varying the strut thickness is discretized into uniform me shes of 105 elements (10 regions), 2010 (20 regions), 5025 elements (50 regions), 10050 (100 regions), 200100 (200 regions) and 400200 (400 regions). Some finite elem ent outputs are shown in Figure 3-8. A B C D Figure 3-8. Stress distribution of the func tionally graded foams. A) 10x 5 elements model. B) 20x 10 elements model. C) 100x 50 elements model. D) 200x 100 elements model.

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37 As the number of elements and regions increases, the energy release rate at the crack tip converges as shown in Figure 3-9. For 10050 elements model, the variation of J-contour integral is less than 0.01% co mpared to the 400200 element model. Therefore, 10050 elements model is used fo r further analysis in order to maintain adequate accuracy with reas onable computational time. 609 609.5 610 610.5 611 611.5 612 612.5 613 050100150200250300350400Number of regions Energy release rate0 200 400 600 800 1000 1200 1400 Energy release rate Time needed to complete jobTime (sec) Figure 3-9. Variation of energy release ra te at the crack tip with various size macromodels Micro-model A portion of macro-mechanical model (ABCD) is taken and used for micro-model as shown in Figure 3-1 (A). As the 10050 elements (100 regions) for macro-model and constant cell length (200 m ) for micro-model are used, one macro-model element can be replaced by 60 beam elements for micromodel as shown in Figure 3-10. The displacements along the boundaries of micro-mo del are determined by using three points interpolation. The corres ponding three points can be obtai ned from the previously described macro-model analysis. For instance, displacements in the x -direction for each

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38 beam element on micro-model al ong the three nodal points ( a b and c ) can be found as follows, 22 222()() () 22micromacromacromacro abcyylylyyl uyuuu lll (3.6) Figue 3-10. Embedded beam element (micro model) in two-dimensional eight-node solid model (Macro model). In micro-model, two-node beam elements are used to represent the foam ligaments/struts. After, the displaceme nts along the boundaries of micro-model, the maximum principal stress stresses at the crack tip tip can be calculated from the results for force and moment resultants obtained from the micro-model as 2tip tip tip tip tiptiph M F I A (3-7) The fracture toughness of the foam is defined as the stress intensity factor that will cause the crack-tip struts to fail. We assume that the strut material is brittle and will fracture when the maximum principal stress exceeds the tensile strength u. b y=0 a y=-l c y=l y, v(x,y) x, u(x,y)

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39 Figure 3-11. Force and moment resulta nts in struts modeled as beams. Since we are dealing with linear elasticity the fracture toughness can be estimated from the following relation, Itip ICuK K (3-8) 1.38 1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 0200040006000800010000120001400016000Number of Element Fracture Toughness0 50 100 150 200 250 300 350 400 450 Fracture Toughness Time needed to complete job MPamTime(sec) Figure 3-12. Variation of fracture to ughness with the size of micro-models. The convergence analysis is conducted to evaluate the variation of fracture toughness with various sizes of micro-model, 31 macro-model (170 elements in micromodel), 62 (640), 155 (3,850), 217 (7, 490) and 3010 (15,200). As model size increases, fracture toughness converges as shown in Figure 3-12. For 3,850 beam

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40 element model, the error in fracture toughness is less than 0.3 % compared to 15,200beam elements model. Therefore, the 3,850model is chosen for further analysis as a compromise between the accuracy and comput ational time. The aforementioned methods will be extended to graded foams and also to understand the effects of thermal stresses in succeeding chapters.

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41 CHAPTER 4 FRACTURE TOUGHNESS OF GRADED FOAMS UNDER MECHANICAL LOADING In this chapter, the finite element base d micromechanical model discussed in the previous chapter is used to understand the behavior of functionally graded foam (FGF) and to estimate their fracture toughness (critical stress intensity factor). We will use the ABAQUS TM finite element package for performi ng the simulations. Analysis of FGF containing a crack under remote loading (unifo rm displacement) was first carried out. The results of fracture toughness under unifo rm displacement are compared with the homogeneous foam in order to understand th e behavior of FGF. Thermal loading can affect the stress field near crack tip unlike th e remote loading case. In order to observe this local effect, we investigated the case where the pressure applied on the crack surface for various sizes of crack lengths in homogeneous foam Then, the results were compared with the remote traction case. For the remote loading, we considered uniform displacement and traction on the top edge of model. The pr inciple of superposition was studied to understand the local effect. Fracture Toughness under Remote Loading In this section, fractur e toughness of functionally gr aded foams subjected to uniform displacements on the top and bottom of the model. The height of the model is considered same as its width, and is symmetric with respect to its midline, y = 0. The geometry of the FGF is shown in Figure 4-1 (A) with crack length, a Only half of the model is considered in the finite element an alysis because the model is symmetric with

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42 respect to its midline, x -axis. The upper edge is loaded by uniform displacement, 70 m A zero displacement boundary condition in the y -direction is applied on the lower edge to account for symmetry. The material is f unctionally graded, and the relative density increases or decreases acco rding to the parameters, or described in Chapter 2. The parameters determine the degree of gradation of the properties. Fo r the case where the strut dimensions vary and the cell dimension is constant, parameter determines the degree of the gradation (Eq. 2-11). is used for the case where the cell dimension is varying in x -direction while the strut dimensions are kept constant (Eq. 2-14). Figure 4-1. Edge-cracked model under A) unifo rm traction or displacement loading and B) crack surface traction. First, we investigate the case wherein the graded foam has cons tant unit-cell length ( c =200 m ) and the density is varied by changing the strut cross sectional dimensions. Both cases, increasing and d ecreasing densities along the x -axis, are considered. Table 41 shows the results form the case which th e unit-cell dimensions and crack length are kept constant, but the strut thickness is vari ed, such as the FGF model is taken from 0 or Vo W =0.1 m a x y A W =0.1 m y x a B

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43 imaginary graded global panel from different position. The results from the case which the model has the constant unit cell but the cr ack length and strut thickness is varied are shown in Table 4-2. Table 4-1. Fracture toughness of graded a nd uniform foams. The unit-cell dimensions and crack length are kept constant, but the strut thickness is varied ( c =200 m crack length, a =0.03 m and =20010-6). Fracture Toughness ( Pa-m) ()ohm Relative Density at the crack-tip Decreasing density Increasing density Uniform density 26 0.028 4.52171E+05 4.56445E+05 4.51326E+05 30 0.039744 6.56122E+05 6.57406E+05 6.56114E+05 50 0.123904 2.24739E+06 2.25108E+06 2.24928E+06 60 0.179334 3.39537E+06 3.39999E+06 3.39819E+06 70 0.241664 4.77575E+06 4.78247E+06 4.77936E+06 Table 4-2. Fracture toughness of graded a nd uniform foams. The unit-cell dimension is kept constant but the crack length and the strut thickness are varied ( c =200 m ho=40 m and =20010-6). Fracture Toughness ( Pa-m) Normalized crack length ( a/W ) Relative Density at the crack-tip Decreasing density Increasing density Uniform density 0.1 0.06075 1.10144E+06 1.03627E+06 1.03485E+06 0.2 0.068608 1.25052E+06 1.18201E+06 1.18004E+06 0.3 0.076874 1.33362E+06 1.33619E+06 1.33465E+06 0.4 0.085536 1.49961E+06 1.49980E+06 1.49878E+06 0.5 0.094582 1.67268E+06 1.67266E+06 1.67220E+06 As seen in Table 4-1, Table 4-2 and Figure 4-2, the results from the present analysis for FGF are very close to those of homogeneous foam. However we see an interesting trend in Figure 4-10. In both deceasing and increasing density cases, the fracture toughness deviates from that of uniform density foam for higher densities. When the density decreases along th e crack path, the fracture t oughness is slightly higher and vice versa.

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44 0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06 3.5E+06 4.0E+06 4.5E+06 00.050.10.150.20.25Relative DensityFracture Toughness ( Pa-m1/2) Decreasing density Increasing density Unifrom density Figure 4-2. Comparison of fracture toughness of graded a nd homogeneous foams having same density at the crack tip. Results for the case of varying unit-cell di mensions are presented in Table 4-3 and also shown in Figure 4-3. The results agai n show that the fractur e toughness of FGF is close to that of a homogeneous foam with density same as th at at the crack tip of FGF. Table 4-3. Comparison of the fracture t oughness for varying unit-cell dimensions with constant strut thickness ( h =20 m ). Fracture Toughness ( Pa-m) Set 0 1c( m ) 2c( m ) 3c( m ) Crack length in terms of number of elements Relative Density at the cracktip Graded Foam Homogeneous % difference 10 0.0745806 9.62060E+05 9.61172E+050.092 20 0.0776305 1.00630E+06 1.00531E+060.098 1 -0.15e-6 200e-6 100e-6 100e-630 0.0812704 1.05636E+06 1.05533E+060.097 40 0.0856898 1.11501E+06 1.11279E+060.199 50 0.0911693 1.18440E+06 1.18018E+060.356 60 0.0981422 1.26090E+06 1.25952E+060.109 70 0.221965 2.05336E+06 2.04649E+060.335 60 0.228608 2.15865E+06 2.16343E+060.221 50 0.236846 2.30923E+06 2.29987E+060.405 2 0.15e-6 50e-6 50e-6 50e-6 40 0.247332 2.47076E+06 2.46204E+060.353 30 0.261132 2.66760E+06 2.65985E+060.291 20 0.280113 2.92063E+06 2.90924E+060.390 10 0.307863 3.24507E+06 3.23901E+060.187

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45 Table 4-3. Continued Fracture Toughness ( Pa-m) Set 0 1c( m ) 2c( m ) 3c( m ) Crack length in terms of number of elements Relative Density at the cracktip Graded Foam Homogeneous % difference 50 0.0912307 1.18404E+06 1.18051E+060.298 40 0.0982215 1.26352E+06 1.26041E+060.246 3 0.15e-6 50e-6 100e-6 100e-630 0.107422 1.36134E+06 1.35823E+060.228 20 0.120075 1.48715E+06 1.48204E+060.344 10 0.138575 1.65340E+06 1.64829E+060.310 70 0.0450318 7.45601E+05 7.43076E+050.339 4 0.15e-6 200e-6 150e-6 100e-650 0.0470359 7.89333E+05 7.85621E+050.470 30 0.0495308 8.39136E+05 8.35061E+050.486 10 0.0527223 9.11742E+05 8.93837E+051.964 70 0.0218062 3.73104E+05 3.72085E+050.273 5 0.15e-6 200e-6 200e-6 200e-650 0.0230945 3.94813E+05 3.92956E+050.470 30 0.0246984 4.19129E+05 4.17251E+050.448 10 0.0267500 4.48072E+05 4.46171E+050.424 0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06 3.00E+06 3.50E+06 0.000.050.100.150.200.250.300.35Relative DensityFracture Toughness ( Pa-m1/2) Set1 (FGF) Set1 (Homogeneous) Set2 (FGF) Set2 (Homogeneous) Set3 (FGF) Set3 (Homogeneous) Set4 (FGF) Set4 (Homogeneous) Set5 (FGF) Set5 (Homogeneous) Figure 4-3. Comparison of fracture toughness of graded and homogeneous foams. The graded foams have varyi ng unit-cell dimensions, but constant strut cross section h =20 m

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46 Study of Local Effect on the Homoge neous Foam under Crack Face Traction In order to observe and analyze the lo cal effects of the local stresses on the homogeneous foam, the fracture toughness of the homogeneous foam under crack face traction (0.5 GPa ) is compared with remote loading condition, both uniform traction (0.5 GPa ) and displacement (70 m ) on the top edge. For this study, the model in the Figure 4-1 (A) is considered for remote loading condition and Figure 4-1 (B) for crack face loading condition. The model size is fixed a nd the crack length is varied in order to investigate how the local stresses around near crack tip affects the fracture toughness for various crack lengths. The unit cell size and beam thickness are constant, c =200m and h =20m which make the relative density of homogeneous foam equal to 0.028. The crack length varies from 10% ( a / c =50) to 50% ( a / c =250) of the plate width. Table 4-4. Remote loading case – uniform displacement on the top edge. a/W (normalized crack length) 0.1 0.2 0.3 0.4 0.5 a/c (crack length/unit-cell) 50 100 150 200 250 Stress Intensity Factor 2.431E+052.431E+052.431E+052.431E+05 2.431E+05 max (Maximum Principal Stress) 1.939E+091.939E+091.939E+091.940E+09 1.940E+09 max/SIF7.977E+037.974E+037.976E+037.978E+03 7.980E+03 Fracture Toughness 4.513E+054.514E+054.513E+054.512E+05 4.511E+05 Table 4-5. Remote loading case – uniform traction on the top edge. a/W (normalized crack length) 0.1 0.2 0.3 0.4 0.5 a/c (crack length/unit-cell) 50 100 150 200 250 Stress Intensity Factor 1.486E+082.988E+084.763E+086.941E+08 9.776E+08 max (Maximum Principal Stress) 1.194E+122.405E+123.832E+125.582E+12 7.858E+12 max /SIF 8.030E+038.046E+038.044E+038.041E+03 8.038E+03 Fracture Toughness 4.483E+054.474E+054.475E+054.477E+05 4.479E+05

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47 As seen on Table 4-4, the stress intensity factor and maximum pr incipal stresses at the crack tip for homogeneous foam under uni form displacement are almost constant through various lengths. When the uniform tr action is applied on the top edge, the stress intensity factor and maximum principal stresse s at the crack tip incr ease with increasing the crack length (Table 4-5). However, th e ratio between the maximum principal stress and the stress intensity factor does not vary much for different crack length. In both cases, the fracture tough ness is independent of crack lengths. Table 4-6. Stress intensity factor, maximum principal st ress and fracture toughness for various crack lengths under crack surface traction. a/W (normalized crack length) 0.1 0.2 0.3 0.4 0.5 a/c (crack length/unit-cell) 50 100 150 200 250 Stress Intensity Factor 1.486E+082.988E+084.763E+086.941E+08 9.776E+08 max (Maximum Principal Stress) 1.126E+122.337E+123.764E+125.514E+12 7.791E+12 max /SIF 7.574E+037.820E+037.902E+037.944E+03 7.969E+03 Fracture Toughness 4.753E+054.604E+054.556E+054.532E+05 4.517E+05 Table 4-7. Fracture toughness estimation from remote loading and crack face traction. a/c (crack length/unit-cell) Remote loading (uniform traction on the top edge) Crack face traction % difference 50 4.48332E+05 4.75294E+05 6.014 100 4.47406E+05 4.60368E+05 2.897 150 4.47514E+05 4.55565E+05 1.799 200 4.47679E+05 4.53177E+05 1.228 250 4.47851E+05 4.51743E+05 0.869 In the case where a pressure is applie d along the crack face – crack face traction, the stress intensity factor and the maximum principal stresses at the crack tip increase with increasing the crack length as occurred in the case of remote loading with uniform traction. However, the ratio between the ma ximum principal stress at the crack tip and

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48 the stress intensity factor also increases with crack length. That means that the fracture toughness decreases as the crack length incr eases. The fracture toughness for crack face traction is higher for shorter cracks, but c onverges to the value for remote traction condition for longer cracks. The deviation of the fracture toughness for various crack length under crack surface traction condition is presented in Table 4-7 and plotted in Figure 4-4. 0.00E+00 5.00E+04 1.00E+05 1.50E+05 2.00E+05 2.50E+05 3.00E+05 3.50E+05 4.00E+05 4.50E+05 5.00E+05 050100150200250300a/c Fracture toughness (Pa-m1/2) Remote loading (unifrom traction on the top edge) Crack face traction Figure 4-4. Fracture toughness estimation from remote loading and crack face traction. The principle of superposition can be applie d to the crack face loading condition in order to explain the differences in fracture toughness presented in Table 4-7. As we have seen in the previous section, the curren t method can accurately estimate the fracture toughness for remote loading conditions. Thus stresses acting on the crack face (i.e., crack face traction condition) can be replaced with tractions that act on the top edge (remote loading condition) and an uncracked body subjected to tractions, as illustrated in

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49 Figure 4-5. Since the value of stress intensity factor for uncracked body is zero, the two loading configurations (remote traction and crack face traction) result in same stress intensity factor in macro scale sense and shown in Table 4-5 and 4-6. ()()()()()(since 0)abcbc IIIIIKKKKK (4-1) Figure 4-5. Application of superposition to replace crac k face traction with remote traction. However, stresses exist at the crack tip in the uncracked body in micro scale sense as shown below. 2 () 2 c maxoc h (4-2) The principle of superposition i ndicates that the maximum principal stress at the crack tip under crack surface traction is lo wer than the maximum princi pal stress at the crack tip under remote loading (uniform traction on the top edge) as derived in Eq. (4.3) below:. ()()() abc maxmaxmax (4-3) 2 ()() 2 ab maxmaxoc h and ()() ab maxmax (4-4) The estimated fracture toughness under surf ace crack traction is higher than the fracture toughness under uniform traction on th e top edge for short crack by current Crack Face Traction a 0 a = a (a) Remote Traction (b) (c) Uncracked Bod y

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50 micro-mechanical model. By employing the principle of superposit ion, the analytical solution for the fracture toughness unde r crack surface traction denoted by()()Sa ICICKK can be found in terms of the fract ure toughness under remote loading()()Rb ICICKK The reciprocal of fracture tou ghness can be obtained by divi ding Eq. (4-4) throughout by the factor ( KI, u) where u is the ultimate tensile strength of the strut material and KI is the stress intensity factor due to the applied stress 0. Then we obtain 2 0 211SR ICICuIc h KKK (4-5) The stress intensity factor KI is given by 0 IKYa (4-6) where 0 is the applied stress, a is the crack dimensions, and the form factor Y is a polynomial in a/W and depends on geometry and mode of loading (Broek 1978). For edge cracked model under remote traction (Figure 4-5 (a)) Y is given by 2341 1.990.4118.738.4853.85 aaaa Y WWWW (4-7) Using Eq. (4-6), Eq.(4-5) reduces to 2 2111SR ICIC uh KK Ya c (4-8) Then the fracture toughness under crack surface traction can be derived as 2 21R S IC IC R IC uK K K h Ya c 50 a c or 0.1 a W (4-9)

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51 From Eq. (4-9), the fracture toughness under crack surface traction S ICK can be calculated from fracture toughness under remote loadingR ICK and compared with the estimated fracture toughness obtained from the developed micro-mechanical model (Table 4-8). The local effect caused by local stress was analyzed employing the principle of superposition. The principal of superposi tion also indicates that the fracture toughness under crack surface traction is larger than the fracture toughness under remote loading for short cracks and tends to converge to the fracture toughness under remote loading for longer cracks as seen on Figure 4-4. Table 4-8. Comparison of fracture toughness under crack surface traction calculated from the superposition method and the micromechanical model. a(m) a/W a/c Analytical solution Current micro-mechanical model %diff 0.01 0.1 50 4.76554E+05 4.75294E+05 0.264 0.02 0.2 100 4.64162E+05 4.60368E+05 0.817 0.03 0.3 150 4.58684E+05 4.55565E+05 0.680 0.04 0.4 200 4.55260E+05 4.53177E+05 0.458 0.05 0.5 250 4.52874E+05 4.51743E+05 0.250

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52 CHAPTER 5 FRACTURE TOUGHNESS ESTIMATI ON UNDER THERMAL LOADING In this Chapter we investigate the foam s under thermal loading using the methods described in Chapter 3. First we investigat e the behavior of func tionally graded foams and homogeneous foams under thermal loading. The second order polynomial variation of temperature is used in order to investig ate the behavior of uncracked foams because the analytical study of a beam under temperat ure gradient shows th at the second order polynomial variation is the simplest form to create thermal stresses (Appendix). Sankar and Tzeng (2002) studied thermal stresses in functionally graded beams by using the Bernoulli-Euler hypothesis. They assumed th at the elastic consta nts of the beam and temperature vary exponentially through thickness. They foun d that the thermal stresses for a given temperature gradient can be re duced when the variation of the elastic constants are opposite to that of the temperature gradient. In the present work, we study the difference between fracture toughness under mechanical and thermal loading. The purpose here is to understand the effects of thermal stress gradients on fracture toughness. Behavior of Foams under Thermal Loading The configuration with constant cell size (200 m ) with varying the strut thickness is used to investigate the behavior of uncracked foams under thermal loading. The investigation was performed in three diffe rent cases – homogeneous foam, increasing density and decreasing density foams. The pr operties of three differe nt models are shown in Table 5-1. The subscripts “ 0 ” and “ f ” denote the pr operties at the x =0 and x =width (0.1 m ) respectively. The temperature varia tion was assumed to be of the form

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53 T =75,000 x2+7,500 x +30 which makes the temperature to increase along the x -axis from 10 C to 1,530 C The temperatures are nodal values in the finite element analysis. The temperature according to the given second orde r polynomial variation is assigned at the vertical set of nodes. The reference te mperature in all the models was 20 C Therefore, the T0 and Tf are 10 C and 1,510 C for all three models. Table 5-1. Elastic modulus vari ation of three different models. Strut thickness ( m ) Elastic Modulus ( GPa ) h0 hf E0 Ef Modulus ratio ( Ef/E0) Homogeneous 40 40 8.288.28 1 Decreasing material properties 40 20 8.282.07 0.25 Increasing material properties 40 80 8.2833.12 4 The thermoelastic constant was assumed to be of form ( x ) = 0 f ( x ) with 0 = E0/106, where E0 is the elastic modulus at x =0. Then, the thermal stresses were normalized with respect to the thermal stress term 0 T0. The axial stress distributions for the three cases are shown in Figure5-1 and plotted in Figures 51through 5-3. When the variation of the material prope rties was in opposite sense to th e temperature variation, the thermal stresses were reduced. On the other hand, when the variation of the material properties was in the same sense as the temperature variation, the thermal stresses increased compare to that of homogeneous foam. The maximum and minimum stresses in different cases are compared in Figure 55 by bar charts where the values are shown for modulus ratio, Ef/E0. The investigation confirms that the be havior of functiona lly graded foams naturally adopt the favorable design for therma l protection system because the nature of

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54 functionally graded foams as load carrying th ermal protection system should be such that the cooler inner layer has high solidity, wh ile the hotter outer layer has low solidity. A B C Figure 5-1. Thermal stress di stribution output from FEA. A) Homogeneous foam. B) FGF with the material properties and the temperature has opposite type of distribution. C) FGF with the vari ation of the material properties and temperature in a similar manner.

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55 -8 -7 -6 -5 -4 -3 -2 -1 0 100.250.50.751x/widthNormalized Thermal Stresses Figure 5-2. Thermal stress dist ribution in homogeneous foam. -5 -4 -3 -2 -1 0 100.250.50.751x/widthNormalized Thermal Stresses Figure 5-3. Thermal stresses distribution in an FGF; the material properties and the temperature have opposite type of vari ation, and this reduces the thermal stresses.

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56 -20 -15 -10 -5 0 5 00.250.50.751x/widthNormalized Thermal Stresses Figure 5-4. Thermal stresses distribution in an FGF; the variation of the material properties and temperature in a similar manner; and this increases the thermal stresses. -20 -15 -10 -5 0 5 0.2514Modulus Ratio ( Ef/E0)Normalized Thermal Stresses Max. Stress Min. Stress Figure 5-5. Maximum and minimum valu es of normalized thermal stresses.

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57 Results under Thermal Loading For the analytical solution of model unde r thermal loading, Euler-Bernoulli beam theory was used due to its simplification. One of the assumptions for Euler-Bernoulli beam theory is that the beam should be long and slender (i.e. length >> depth and width). To compare the finite element results with analytical solution, the beam aspect ratio (length/width) should be investigated because we have used the model for the aspect ratio was unity. The width of uncracked model is 0.05 m and length changed from 0.05 to 0.5 m The temperature gradient is a function of the position. The temperature variation was assumed to be of the form T ( x ) = 1400 x2 which makes the temperature increases along the x -axis form 0 C to 1 C Figure 5-6 shows the ther mal stresses developed on models for different aspect ratios. The m odel with aspect ratio 10 gives very close thermal stress distribution as the analytical solution. Thus, we us e the model which has ten times lager length than width in this section. The thermal stress in the models with various crack length is shown in Figure 5-7. -3.00E+05 -2.00E+05 -1.00E+05 0.00E+00 1.00E+05 2.00E+05 3.00E+05 4.00E+05 00.010.020.030.040.05x ( m )Thermal Stress ( Pa ) AR=1 AR=2 AR=4 AR=8 AR=10 Analytical Figure 5-6. Thermal stresses fo r various aspect ratio models.

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58 -1.50E+03 -1.00E+03 -5.00E+02 0.00E+00 5.00E+02 1.00E+03 1.50E+03 2.00E+03 2.50E+03 3.00E+03 3.50E+03 4.00E+03 00.010.020.030.040.05 x ( m )Thermal stress ( Pa ) no-crack c=0.005m c=0.01m c=0.02m c=0.025m c=0.03m c=0.035m Figure 5-7. Thermal stresse s for various crack lengths. Table 5-2. Results of the body under temperat ure gradient form micromechanical model. As described in the previous chapter, the fracture toughness under crack surface traction converges to the fract ure toughness under remote tracti on as crack size increases. However, we should notice that the negative st ress intensity factor exists and the value increases as the crack length increases. Ther efore, the stress intens ity factor decreases with larger crack size, and consequently the fracture t oughness decreases with larger crack size (Table 5-2). The ratio between the maximum principal stress and the stress a/W (normalized crack length) 0.1 0.2 0.4 0.5 0.6 0.65 a/c (crack length/unit-cell) 25 50 100 125 150 175 Stress Intensity Factor 1.68E+021.70E+021.08E+026.97E+01 3.46E+016.45E+00 max (Maximum Principal Stress) 1.54E+061.58E+061.03E+066.81E+05 3.55E+058.82E+04 max /SIF 9185.8 9325.3 9572.6 9782.5 10266.713673.4 Fracture Toughness 3.91E+053.86E+053.76E+053.68E+05 3.50E+052.63E+05

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59 intensity factor is compared with that of remote loading condition shown in Figure 5-8. The fracture toughness for various crack lengths are shown in Figure 5-9. The result of estimated fracture toughness under thermal load ing is similar to that for crack face loading, except the sign is re versed. This can be explaine d by using the principle of superposition as described in chapter 4. 0.00E+00 2.00E+03 4.00E+03 6.00E+03 8.00E+03 1.00E+04 1.20E+04 1.40E+04 1.60E+04 1.80E+04 050100150200 a/c ( c =200m )max. principal stress/ SIF( Pa/Pa m 1/2) Thermal Loading Remote Loading Figure 5-8. Comparison the ratio of maximum principal stress and stress intensity factor. The results obtained in this chapter indi cate that the fracture toughness of a cellular material depends on the stress gradients produced by thermal stresses. This is similar to the results obtained in the pr evious chapter where the frac ture toughness was different for crack surface loading. Thus the nominal fracture toughness obtained from remote loading tests should be corrected appropriate ly when stress gradients are presented.

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60 0.00E+00 5.00E+04 1.00E+05 1.50E+05 2.00E+05 2.50E+05 3.00E+05 3.50E+05 4.00E+05 4.50E+05 050100150200 a/c (c=200mm)Fracture Toughness (Pa m 1/2) Figure 5-9. Fracture toughne ss for various crack lengths.

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61 CHAPTER 6 CONCLUDING REMARKS Finite element based micromechanical met hods have been developed to understand the fracture behavior of functi onally graded foams. The finite element analysis used a micromechanical model in conjunction with a macromechanical model in order to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity factor of the macromechanical model at the crack tip was evaluated by two different method – energy method and point matching me thod. In the energy method, J-contour integral was used, and the stress field ahead of crack tip was used to estimate stress intensity factor in the point matching method. The maximum principal stress at the crack tip was evaluated from the force and moment resultants obtained from the micro-model. Then, fracture toughness was estimated by rela ting the stress intens ity factor and the maximum principal stresses at the crack tip. In addition, convergence tests for both macromechanical and micromechanical models analyses were conducted. First we investigated the fracture behavi or of functionally graded foam under uniform displacement – remote mechanical loading condition in order to demonstrate the feature of the current met hod. Then, the method is extended to another remote mechanical loading – uniform traction. Then, the results of remote loading conditions for graded foams are compared with the re sult for homogeneous foam. The fracture toughness was obtained for various crack positio ns and lengths with in the functionally graded foam. Then the relationship between the fracture toughness of foams and the local density at the crack tip was studied. It was found that the fracture toughness of

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62 functionally graded foam is a pproximately the same as that of homogeneous foam with the same density at the crack tip. We also investigated the effects caused by the local stresses. In order to observe and analyze the local effects of the local stresses on the homogeneous foam, the fracture toughness of the homogeneous foam under crack face traction is compared with remote loading conditions, both uniform traction and displacement on top edge. The relationship between the stress intensity factor and the maximum principal stress was compared. It was found that the fracture toughness under remo te loading condition is independent of the crack size. However, the ratio between the maximum principal stress at the crack tip and the stress intensity factor under crack face traction increased with crack length. Thus, the fracture toughness for crack face traction is higher for shorter cracks, and converges to the value for remote traction c ondition as the crack length increases. It is found that the principle of superposition can be used to adjust the local effect caused by the differences in maximum principal stress es under crack face loading condition. A correction factor in terms of crack length is proposed to determine the fracture toughness of short cracks under crack surface loading. From the thermal protection system point of view the behavior of functionally graded foam under thermal loading was investig ated. When the variation of the material properties is in opposite sense to the temper ature variation, the thermal stresses is reduced. On the other hand, when the variati on of the material properties is in the same sense as the temperature variation, the ther mal stress increases compared to that in homogeneous foam. The result of estimat ed fracture toughness under thermal loading is similar to that for crack face loading.

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63 The present dissertation demonstrates th e use of finite based micromechanical model to predict the fracture toughness of f unctionally graded foams by using simple micro structures. To achieve more accurate prediction, we need to focus on a more realistic cell configuration, which captures th e complexity of foam and predicts more accurately its mechanical property changes, such as relative density and modulus in functionally graded foam. The methods disc ussed here will help in understanding the usefulness of functionally graded foams in th e thermal protection systems of future space vehicles.

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64 APPENDIX ANALYTICAL SOLUTION FOR BEAM MODEL UNDER THERMAL LOADING Figure A-1. A beam of rectangula r cross section with no restraint. If we assume the both ends are perfectly clamped, the thermal stress, T is defined as ()()TyETy (A-1) Due to constraints at both ends, the ther mal stress prevents extension and bending of the beam and producing internal force, P and bending moment, M h T h P bdy (A-2) h T h M ybdy (A-3) If the beam has no restraints against extension, the internal force, P must be eliminated by a virtual force, TP L b 2h y, v z x, u T(y)

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65 ()h T hPPETybdy (A-4) If the ends are free to rotate and no exte rnal moment applied, the bending moment M at the ends must be eliminat ed by a virtual bending moment T M at the ends ()h T h M METyybdy (A-5) The thermal stress corresponding to the virtual force, TP is 11 ()() 2hh T T p hhP ETybdyETydy AAh (A-6) (2)*2 Ahbhb and the thermal stress correspondin g to the virtual bending moment, T M is 33 ()() 2hh T T M hhMyy ETyybdyETyydy I h (A-7) 33(2)2 123 bhbh I Therefore, the thermal stress x in the beam with no constraints at both ends is given by 313 ()()()() 2 2hh x hhy yETyETydyETyydy h h (A-8) 313 ()()()() 2 2hh x hhy yETyTydyTyydy h h (A-9)

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66 Example 1) Constant, 0TC 002h h h hTdyCyCh 2 01 0 2h h h hTydyCy Therefore, 313 () 2 2hh x hhy yETTdyTydy h h 001 (2) 2 E CCh h 0 Example 2) Linear variation, 01() T y CC y 2 0101 ()2 2h h h hTydyCyCyCh 233 011112 () 233h h h hTyydyCyCyCh Therefore, 313 ()()()() 2 2hh x hhy yETyTydyTyydy h h 3 0101 3132 ()(2)() 23 2 y E CCyChCh h h 0

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67 Example 3) Quadratic variation, 2 012()TyCCyCy 233 01202112 ()2 233h h h hTydyCyCyCyChCh 2343 01211112 () 2343h h h hTyydyCyCyCyCh Therefore, 313 ()()()() 2 2hh x hhy yETyTydyTyydy h h 233 012021 31232 2 233 2 y ECCyCyChChCh h h 22 221 3 E ChCy 22 21 3 EhyC

PAGE 79

68 LIST OF REFERENCES Anderson, T. L., 2000, Fracture Mechanics, 2nd edition., CRC Press LLC, Boca Raton, Florida. Anlas, G., Santare, H. M. and Lambros, J, 2000, Numerical Ca lculation of Stress Intensity Factors in Functionally Graded Materials. Intern ational Journal of Fracture, Vol. 104, pp. 131-143. Ashby, F. M., Evans, A., Fleck, A. N., Gibson, J. L., Hutchinson, W. J. and Wadly, N. G., 2000, Metal Foams: A Design Gu ide, Butterworth-Heinemann Pub., Massachusetts. Atkinson, C., List, R.D., 1978, Steady State Cr ack Propagation into Media with Spatially Varying Elastic Properties. International J ournal of Engineering Science, Vol. 16, pp. 717-730. Blosser, M. L., October 1996, Development of Metallic Thermal Protection Systems for the Reusable Launch Vehicle. NASA TM-110296. Blosser, M. L., May 2000, Advanced Meta llic Thermal Protection Systems for the Reusable Launch Vehicle. Ph.D. Disse rtation, University of Virginia. Blosser, M. L., Chen, R. R., Schmidt, I. H., Dorsey, J. T., Poteet, C. C. and Bird, R. K., 2002, “Advanced Metallic Thermal Protections System Development,” Proceedings of the 40th Aerospace Science Meeting and Exhibit, Jan 14-17, Reno, Nevada, AIAA 2002-0504. Broek, D., 1978, Elementary Engineering Fracture Mechanics. Sijthoff & Noordhoff International Pub., Groningen, The Netherlands. Butcher, R. J., Rousseau, C. E. and Tippur, H. V., 1999, A Functionally Graded Particulate Composite: preparation, Measur ing and Failure Analysis. Acta Mater, Vol. 47, No.1, pp. 259-268. Choi, S. and Sankar, B.V., 2003, Fracture Toughness of Carbon Foam. Journal of Composite Materials, Vol. 37, No. 23, pp. 2101-2116. Choi, S. and Sankar, B. V., 2005, A Micromech anical Method to Predict the Fracture Toughness of cellular Materials. Internationa l Journal of Solids & Structures, Vol. 42/5-6, pp. 1797-1817.

PAGE 80

69 Delale, F. and Erdorgan, F., 1983, The Cr ack Problem for Nonhomogeneous Plane. Journal of Applied Mechanics, Vol.50, pp. 609-614. Erdorgan, F., 1995, Fracture Mechanics of F unctionally Graded Materials. Composites Engineering, Vol. 5, pp. 753-770. Erdorgan, F. and Wu, B. H., 1997, The surface Crack Problem for a Plate with Functionally Graded Properties. Journal of Applied Mechanics, Vol. 64, pp. 449456. Eichen, J. W., 1987, Fracture of Nonhomogene ous Materials. International Journal of Fracture, Vol. 34, pp. 3-22. Gerasoulis, A. and Srivastav, R. P., 1980, In ternational Journal of Engineering Science, Vol. 18, p239. Glass, D. E., Merski, N. R. and Glass, C. E., July 2002, Airframe research and Technology for Hypersonic Airbr eathing Vehicles. NASA TM-21152. Gibson, R.E., 1967, Some results concer ning displacements and stresses in a nonhomogeneous elastic half space. Geotechnique, Vol.17, pp. 58-67. Gibson, L.J., Ashby, M.F., 2001. Cellular So lids: Structure and Properties. Second Edition, Cambridge University Pr ess, Cambridge, United Kingdom. Gu, P. and Asaro, R. J., 1997, Cracks in F unctionally Graded Mate rials. International Journal of Solids and Structures, Vol. 34, pp. 1-7. Gu, P., Dao, M. and Asaro, R. J., 1999, Si mplified Method for Calculating the Crack Tip Field of Functionally Graded Materials Using the Domain Integral. Journal of Applied Mechnics, Vol. 66, pp. 101-108. Harris, C. E., Shuart, M. J., and Gray, H. R. May 2002, A Survey of Emerging Materials for Revolutionary Aerospace Vehicle St ructures and Propul sion System. NASA TM-211664. Hibbitt, Karlson, & Sorensen, 2002, ABAQUS/S tandard User’s Manual, Vol. II, Version 6.3, Hibbitt, Karlson & Sorensen, Inc., Pawtucket, Rhode Island. Jin, Z. H. and Batra, R. C., 1996, Some Basic Fracture Mechanics Concepts in Functionally Gradient Materials. Journal of Mechanics Physics Solids, Vol. 44, pp. 1221-1235. Jin, Z. H. and Noda, N., 1994, Crack-tip Si ngular Fields in Nonhomogeneous Materials. Journal of Applied Mechanics, Vol. 61, pp. 738-740.

PAGE 81

70 Jordan, W., 2005, Space shuttle. pdf, http://www2.latech.edu/~jorda n/Nova/ceramics/SpaceShuttle.pdf Last accessed February 14th, 2005. Kim, J. and Paulino, G. H., 2002, Isopa rametric Graded Finite Element for Nonhomogeneous Isotropic and Orthotr opic Materials. Journal of Applied Mechanics, Vol. 69, pp. 502-514. Kuroda, Y., Kusaka, K., Moro, A. and Toga wa, M., 1993, Evaluation tests of ZrO/Ni Functionally Gradient Materials for Re generatively cooled Thrust Engine Applications. Ceramic Transactions, Vol. 34, pp. 289-296. Madhusudhana, K.S., Kitey, R. and Tippur, H.V., 2004, Dynamic Fracture Behavior of Model Sandwich Structures with Functi onally Graded Core, Proceedings of the 22nd Southeastern Conference in Theoreti cal and Applied Mechanics (SECTAM), August 15-17, Center for Advanced Materials, Tuskegee University, Tuskegee, Alabama, pp. 362-371. Marur, P. R. and Tippur, H. V., 2000, Nume rical analysis of Crack-tip Fields in Functionally Graded Materials with a Cr ack Normal to the Elastic Gradient. International Journal of Solids and Structures, Vol.37, pp. 5353-5370. Morris, W.D., White, N.H. and Ebeling, C. E., September 1996, Analysis of Shuttle Orbiter Reliability and Maintainability Data for conceptual Studies. 1996 AIAA Space Programs and Technologies Conference, Sept 24-26, Huntsville, AL, AIAA pp. 96-4245. Poteet, C. C. and Blosser, M. L., January 2002, Improving Metallic Thermal rotection System Hypervelocity Impact Resistance through Numerical Simulations. Journal of Spacecraft and Rockets, Vol. 41, No.2, pp. 221-231. Ravichandran, K. S. and Barsoum, I., 2003, Determination of Stress Intensity Factor Solution for Cracks in Finite-width Func tionally Graded Materias. International Journal of Fracture, Vol. 121, pp. 183-203. Rousseau, C. E. and Tippur, H.V., 2002, Ev aluation of Crack Ti p Fields and Stress Intensity Factors in Functionally Graded Elastic Materials: Cracks Parallel to Elastic Gradient. International Journa l of Fracture, Vol. 114, pp. 87-111. Sanford, R. J., 2003, Principl es of Fracture Mechanics. Pe arson Education, Inc., Upper Saddle River, New Jersey. Sankar, B. V. and Tzeng, J. T., 2002, Ther mal Stresses in Functionally Graded Beams, AIAA Journal, Vol. 40, No. 6, pp. 1228-1232. Santare, M. H. and Lambros, J., 2000, Use of Graded Finite Elements to Model the Behavior of Nonhomogeneous Materials, Jo urnal of Applied Mechanics, Vol. 67, pp. 819-822.

PAGE 82

71 Sih, G. C. and Liebowbitz, H., 1968, Mathema tical Theories of Brittle Fracture. Vol. 2, Academic Press, New York. Weichen, S., 2003, Path-independent Inte grals and Crack Extension Force for Functionally Graded Material s. International Journal of Fracture, Vol. 119, pp. 8389. Yang, W. and Shih, C. F., 1994, Fracture along an Interlayer. Intern ational Journal of Solids and Structures, Vol. 31, pp. 985-1002. Zhu, H., 2004, Design of Metallic Foams as Insulation in Thermal Protection Systems. Ph. D. Dissertation, University of Florida.

PAGE 83

72 BIOGRAPHICAL SKETCH Seon-Jae Lee was born in Seoul, Korea, in 1969. He studied Physics in KyoungWon University for three years and entere d Department of Aerospace Engineering at Royal Melbourne Institute of Technology, Me lbourne, Australia. He transferred to Embry Riddle Aeronautical University at Daytona Beach, Florida, and received his Bachelor of Science in Aero space Engineering in April 1999. From there he completed his graduate study of a Master of Science in Aerospace Engineering in May 2002, with research in the area of vibration analysis of composite box beam. In August 2002, he joined the Advanced Composites Center at the Department of Mechanical and Aerospace Engineering, University of Florida, Gaines ville, Florida, for his Ph.D. degree. After completion of his Ph.D. degree, Seon-Jae will begin work at Samsung Techwin in Korea to contribute his efforts to the development of precise machine.


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MICROMECHANICAL MODEL FOR PREDICTING THE FRACTURE
TOUGHNESS OF FUNCTIONALLY GRADED FOAMS















By

SEON-JAE LEE


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


2006

































Copyright 2006

by

SEON-JAE LEE

































To my parents and to my wife Jin-Sook















ACKNOWLEDGMENTS

All thanks and praises are to Jesus Christ, the Lord of the Universe, for his

blessing, help and guidance.

I would like to express my sincere gratitude to my advisor, Dr. Bhavani Sankar, for

his guidance, his encouragement and his financial support. He is not only my academic

advisor but also a great influence in my life.

My appreciation is also due to Dr. Raphael Haftka, Dr. Peter Ifju, and Dr. Reynaldo

Roque for serving on my supervisory committee and for their valuable comments and

suggestions.

This statement of acknowledgement would be incomplete without expressing my

sincere appreciation and gratitude to both my friends and family. I appreciate the

friendship and encouragement of all the colleagues at the Center of Advanced

Composites (CAC) while working and studying together in the lab. I would like to

extend my appreciation to all my family-church members whose continuous support,

prayers and help were behind me at all times. I would particularly thank my family, my

parents and my brothers in my country, for their continuous support, encouragement and

understanding during my entire school career.

Last, but not least, I would like to thank my lovely wife, adorable son and cute

daughter for their patience and support through the toughest times. They were always

there when I needed them to share my difficulties.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ....................................................... ............ .............. .. vii

L IST O F FIG U R E S .............. ............................ ............. ........... ... ........ viii

A B STR A C T ................................................. ..................................... .. x

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

Reusable Launch Vehicle and Thermal Protection System........................................2
Functionally Graded Foams and Functionally Graded Materials..............................8
Previous Work on Fracture Mechanics of Functionally Graded Materials ................11
Objectives ...................................... ................. ................ ........... 14
S c o p e ........................................................................... 1 4

2 ESTIMATION OF CONTINUUM PROPERTIES..................................................16

Continuum Properties of Homogeneous Foam....................................................16
Continuum Properties of Functionally Graded Foams ................... ...... ............20
Finite Element Verification of Estimated Continuum Properties..............................23

3 FINITE ELEMENT BASED MICROMECHANICAL MODEL ................................26

Overview of M icromechanical M odel .............................. ....................26
M acro -m o d el ..................... .................................. ............ ................ 2 9
Imposing Graded M material Properties ...................................... ............... 30
Methods for Extracting Stress Intensity Factor...............................................32
Convergence Analysis for Macro-model..........................................................36
M ic ro -m o d e l ...............................................................................................................3 7

4 FRACTURE TOUGHNESS OF GRADED FOAMS UNDER MECHANICAL
L O A D IN G ................................................................................................. .... 4 1

Fracture Toughness under Remote Loading............................................................ 41
Study of Local Effect on the Homogeneous Foam under Crack Face Traction.........46









5 FRACTURE TOUGHNESS ESTIMATION UNDER THERMAL LOADING....... 52

Behavior of Foam s under Therm al Loading.................................... .....................52
R results under Therm al Loading........................................... ........................... 57

6 CONCLUDING REMARKS ............................................. ............................. 61

APPEDIX

ANALYTICAL SOLUTION FOR BEAM MODEL UNDER THERMAL LOADING..64

L IST O F R E F E R E N C E S ........................................................................ .....................68

B IO G R A PH IC A L SK E TCH ..................................................................... ..................72















LIST OF TABLES


Table page

2-1 M material properties of the Zoltex carbon fiber ..................................................... 21

3-1 Comparison between two methods. .............................................. ............... 35

4-1 Fracture toughness of graded and uniform foams. The unit-cell dimensions and
crack length are kept constant, but the strut thickness is varied (c=200yum, crack
length, a =0.03m and a'- 200x10 6). ............................................ .....................43

4-2 Fracture toughness of graded and uniform foams. The unit-cell dimension is
kept constant but the crack length and the strut thickness are varied (c=200 Pm,
ho=40pm and a-=+ 200x 10-6) ................................................................ ............... 43

4-3 Comparison of the fracture toughness for varying unit-cell dimensions with
constant strut thickness (h=20/1m ) ...................... ..... ...................... ........ ....... 44

4-4 Remote loading case uniform displacement on the top edge...............................46

4-5 Remote loading case uniform traction on the top edge ........................................46

4-6 Stress intensity factor, maximum principal stress and fracture toughness for
various crack lengths under crack surface traction. ............................................47

4-7 Fracture toughness estimation from remote loading and crack face traction...........47

4-8 Comparison of fracture toughness under crack surface traction calculated from
the superposition method and the micromechanical model. ...................................51

5-1 Elastic modulus variation of three different models. ...............................................53

5-2 Results of the body under temperature gradient form micromechanical model. .....58
















LIST OF FIGURES


Figure pge

1-1 Thermal protection system in Space Shuttle. A) Temperature variation during
re-entry. B) Location of different m materials. ........................................ ..................6

1-2 Schem atic diagram of attaching the tiles......................................... ............... 7

1-3 SEM images of A) low density carbon foam and B) high density carbon foam C)
m metallic foam ................................................. ........................... 9

2-1 Open cell model with rectangular parallelepiped unit cell...................................17

2-2 Micro- and Macro-stresses in open-cell foam............ ........ ...................18

2-3 Flexural deformation of struts under shear stresses. ..............................................18

2-4 Example of variation of elastic modulus and relative density for constant cell
6
length c=200/m, ho=261m and a=-200x 106.......... ............ ................. ..22

2-5 Boundary conditions for un-cracked plate under uniform extension.....................23

2-6 Comparison of stresses (oyy) obtained using the macro- and micro- models in
graded foam with constant cell size but varying strut cross section. .....................24

2-7 Comparison of stresses (oyy) obtained using the macro- and micro- models in
graded foam with constant strut size but varying cell dimension ..........................25

3-1 Schematic description of both macro- and micro-model. ......................................27

3-2 A typical finite element macro-m odel ..................................... ........ ............... 29

3-3 Example of discrete elastic modulus for macro-model with ten-regions ...............30

3-4 Location of model specimens in the global panel. Each specimen is of the same
size and contains a crack of given length, but the density at the crack tip varies
from specim en to specim en ................................................................................31

3-5 J-Integral for various contours in a macro-model containing 100x50 elements
and the contour numbers increase away from the crack tip. ...................................33

3-6 Stress intensity factor from the stresses normal to the crack plane........................34









3-7 Stress normal to the crack plane ............... ........................................ 35

3-8 Stress distribution of the functionally graded foams..................... ............... 36

3-9 Variation of energy release rate at the crack tip with various size macro-models...37

3-10 Embedded beam element (micro model) in two-dimensional eight-node solid
m odel (M acro m odel) ....................... ...... ............ ................... .. ......38

3-11 Force and moment resultants in struts modeled as beams. .....................................39

3-12 Variation of fracture toughness with the size of micro-models ...........................39

4-1 Edge-cracked model under A) uniform traction or displacement loading and B)
crack surface traction. ......................................... ................. .. ...... 42

4-2 Comparison of fracture toughness of graded and homogeneous foams having
sam e density at the crack tip. .............................................................................. 44

4-3 Comparison of fracture toughness of graded and homogeneous foams. The
graded foams have varying unit-cell dimensions, but constant strut cross section
h = 2 0 u m ...................................... .....................................................4 5

4-4 Fracture toughness estimation from remote loading and crack face traction...........48

4-5 Application of superposition to replace crack face traction with remote traction....49

5-1 Thermal stress distribution output from FEA. ........... ................................ 54

5-2 Thermal stress distribution in homogeneous foam. ............................................55

5-3 Thermal stresses distribution in an FGF; the material properties and the
temperature have opposite type of variation, and this reduces the thermal
stresses ............. ..... .... ........... ............... ...........................55

5-4 Thermal stresses distribution in an FGF; the variation of the material properties
and temperature in a similar manner; and this increases the thermal stresses. ........56

5-5 Maximum and minimum values of normalized thermal stresses...........................56

5-6 Thermal stresses for various aspect ratio models..................................................57

5-7 Thermal stresses for various crack lengths.................................... ............... 58

5-8 Comparison the ratio of maximum principal stress and stress intensity factor........59

5-9 Fracture toughness for various crack lengths. ................... ......................... 60

A-1 A beam of rectangular cross section with no restraint. .............. .............. 64















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

MICROMECHANICAL MODEL FOR PREDICTING THE FRACTURE
TOUGHNESS OF FUNCTIONALLY GRADED FOAMS


By

Seon-Jae Lee

May 2006

Chair: Bhavani V. Sankar
Major Department: Mechanical and Aerospace Engineering

A finite element analysis based micromechanical method is developed in order to

understand the fracture behavior of functionally graded foams. The finite element

analysis uses a micromechanical model in conjunction with a macromechanical model in

order to relate the stress intensity factor to the stresses in the struts of the foam. The

continuum material properties for the macromechanical model were derived by using

simple unit cell configuration (cubic unit cell). The stress intensity factor of the

macromechanical model at the crack tip was evaluated. The fracture toughness was

obtained for various crack positions and lengths within the functionally graded foam.

Then the relationship between the fracture toughness of foams and the local density at the

crack tip was studied. In addition, convergence tests for both macromechanical and

micromechanical model analysis were conducted. Furthermore, fracture toughness was

estimated for various loading conditions such as remote loading and local crack surface

loading. Local effect was studied by crack face traction conditions. The principle of









superposition was used to analyze the deviation caused by local loading conditions such

as crack surface traction and temperature gradients. From the thermal protection system

point of view the behavior of graded foams under thermal loading was investigated, and

fracture toughness was estimated. The methods discussed here will help in understanding

the usefulness of functionally graded foam in the thermal protection systems of future

space vehicles. However, further research is needed to focus on more realistic cell

configurations, which capture the complexity of foam and predicts more accurately its

mechanical property changes, such as relative density and modulus in functionally graded

foam, in order to provide more accurate predictions.














CHAPTER 1
INTRODUCTION

Since April 12, 1981, the first launch of the space shuttle-the orbiter Columbia, the

shuttle fleet has played a major role in human space exploration. A large amount of

money has been spent on launching satellites by both the government and the private

sector for the purpose of reconnaissance, communication, global positioning system

(GPS), weather prediction and space exploration (Blosser, 2000). The International

Space Station program also demands the space launch for construction, repair and

service. The private sector has rapidly spread in the last decade. Only a cost effective

launch system can satisfy the increasing demand for lower cost access to space. One of

the major goals of the National Aeronautics and Space Administration (NASA) has been

continued lowering of the cost of access to space to promote the creation and delivery of

new space services and other activities that will improve economic competitiveness.

A thermal protection system (TPS) which protects the whole body of the vehicle is

as crucial as avionics, propulsion, and the structure. A TPS is more limiting than fuel

constraints, structural strength, or engine's maximum thrust. In order to achieve the goals

set by NASA, new TPS concepts have to be introduced, e.g., Integral Structure/TPS

concept. This concept can be achieved because of breakthroughs in the development of

novel materials such as metallic and carbon foam, and functionally graded materials

(FGM). The microstructure of functionally graded metallic and carbon foams can be

tailored to obtain optimum performance for use in integral load-carrying thermal

protection systems due to their low thermal conductivity, increased strength and stiffness.









However, models for strength and fracture toughness of functionally graded foam

materials are in their infancy and it will be the main focus of this research. The details of

the concepts and literature survey will be discussed in subsequent sections.

Reusable Launch Vehicle and Thermal Protection System

Currently, expendable rocket vehicles and the space shuttles are the major launch

systems. An expendable rocket vehicle, such as the US Delta, European Arian, Russian

Proton, and Chinese Long March, is a structure which contains payload, the system-

supporting hardware required to fly and fuel. Expendable rockets can be used only once,

and they are expensive. The space shuttle is only partially reusable because its large

external tank is separated and burns up in the atmosphere during launch. The two smaller

solid rocket boosters land in the ocean and are recovered, but cannot be reused nearly as

many times as the space shuttle itself. Fuel by itself is not comparably expensive, but

tanks to carry it in are, especially if they are only used once such as the space shuttle's

external tank. Furthermore, considerable time for maintenance is required for engines

and thermal protection system (TPS) between flights. The TPS alone is estimated to

require 40,000 hours of maintenance between flights (Morris et al., 1996). The space

shuttle is considered as the first generation reusable launch vehicle due to its partial

reusability.

In January 1995 NASA announced the development plan for a fully reusable

launch vehicle system and designated the X-33 program. The X-33 program ran for 56

months and was cut by NASA in early 2001 due to the failure of a prototype

Graphite/Epoxy composite fuel tank during the proof test. The failure of the tank

indicates that the material science and composite manufacturing technology was not

advanced enough.









After termination of the X-33 program, NASA's Integrated Space Transportation

Plan (ISTP) was formulated in May 2001 to provide safe, affordable, and reliable space

system. As a key component of the ISTP, the Space Launch Initiative (SLI) began with a

goal to achieve the necessary technology development, risk reduction, and system

analysis in order to be used in a second generation reusable launch vehicle (RLV) which

expected to be delivered by 2010. A second generation RLV had these goals:

* Reduce risk of crew loss to no more than 1 in 10,000 missions.

* Reduce payload cost to $1,000 per pound, down from today's $10,000 per pound.

* Be able to fly more often, with less turnaround time and smaller launch crews.


A third generation RLV was planned to start flying around 2025. Its goal was to reduce

cost and improve safety by another order of magnitude.

* Reduce chance of crew loss to 1 in 1,000,000 (equivalent to today's airliners).

* Reduce payload costs to hundreds of dollars per pound.


The RLV was based on single-stage-to-orbit (SSTO) technology. The concept of

SSTO involves a rocket with only one stage carrying crews or cargo to orbit. The RLV

was NASA's true vision for a shuttle successor, but after spending many years NASA

decided to cancel the program because RLV was not attainable using existing technology,

and announced a new strategy that indicated the shuttle would continue flying until at

least 2015. However, in 2003 the space shuttle Columbia was disintegrated during re-

entry to Earth after 16 days in orbit.

After the Columbia tragedy, President Bush announced a new "Vision for Space

Exploration" in January 2004. The President's Vision set NASA in motion to reassess

the space transportation program, and to begin developing a new spacecraft to carry









humans into Earth orbit and beyond. Under the plan, a new spacecraft called the Crew

Exploration Vehicle (CEV) is to be developed and tested by 2008 and the first manned

mission is going to conduct no later than 2014. The first manned lunar landing is

scheduled no later than 2020, and CEV program continues to explore Mars and other

destination in the solar system. NASA hopes to follow this schedule in development of

the CEV:

* 2008 2010 First unmanned flight of CEV in Earth orbit.

* 2011 2014 First manned flight of CEV in Earth orbit.

* 2015 2018 First unmanned flight of Lunar Surface Access Module (LSAM).

* 2016 2018 First manned flight of LSAM.

* 2018 2020 First manned lunar landing with CEV/LSAM system.

* 2020 Start of planning for Mars mission and beyond.


Instead of an airplane-style lifting body used in the space shuttle system, an

Apollo-like capsule design was decided for the CEV because of the fact that the new

CEV design will use the crew and service module design principle. The new CEV design

is virtually identical to the Apollo Command Module except the implementation of the

concept of reusability. The main difference between them is that the new CEV can be

used as many as tem times. Thermal protection system development is the significant

technical obstacles that must be overcome in order to implement reusability and to

improve affordability (vehicle weight reduction) by both new design concept and

material selection such as multifunctional materials that perform structural or other roles.

Harris et al. (2002) surveyed the properties of advanced metallic and non-metallic

material systems. They provided the guidance of emerging materials with application in









order to achieve NASA's long-term goal by addressing materials already under

development that could be available in 5 to 10 years as well as those that are still in the

early research phase and may not be available for another 20 to 30 years.

The main objective of the thermal protection system is protecting the vehicle by

keeping it under acceptable temperature limit and human occupants from heat flow. Heat

sinks and ablative material were used to protect the vehicles before Space shuttle. During

the re-entry process, ablative material is charred and vaporized while the heat sinks

absorb the heat. None of the early vehicles had to be reusable so these materials and

techniques were enough to protect the early vehicles.

In the late 1960s, the space shuttle program was proposed. The program aimed to

produce a vehicle that would be larger than any that had flown in space before.

Conventional aluminum was selected for the main structure and a layer of heat resistant

material for protecting it. The properties of aluminum demand that the maximum

temperature of the vehicle's structure be kept below 175 C in operation. But aero-

thermal heating during the re-entry process creates high surface temperature which is

well above the melting point of aluminum (660 C). Thus, an effective insulator was

needed. A silica-based insulation material was decided for the heat-resistant tiles and

other coverings to protect the Shuttle's airframe. Figure 1-1 shows seven different

materials which cover the external surface of the Space Shuttle according to the

temperature variation during the re-entry.

The materials were chosen by their weight efficiency and stability at high

temperature. The areas of the highest surface temperature in the Shuttle, the forward

nose cap and the leading edge of the wings, are made with Reinforced Carbon-Carbon









(RCC). There are two main types of tiles, referred to as Low-temperature Reusable

Surface Insulation (LRSI) and High-temperature Reusable Surface Insulation (HRSI).

Relatively low temperature of surface where the maximum surface temperature runs

between 370 and 650 C is covered by LRSI. HRSI covers the areas where the maximum

surface temperature runs between 650 and 1,260 C.









955 CR si
5C 42 C SRSI Fnd LRRSI R CC d LRSI


A B

Figure 1-1. Thermal protection system in Space Shuttle. A) Temperature variation
during re-entry. B) Location of different materials. (Courtesy of W. Jordan,
Source: http://www2.1atech.edu/~jordan/Nova/ceramics/SpaceShuttle.pdf,
Last accessed February 14th, 2005).

Many of the tiles have been replaced by a material known as Flexible Reusable

Surface Insulation (FRSI), and Advanced Flexible Reusable Surface Insulation (AFRSI)

in the area where the maximum surface temperature does not exceed 400 C. These tiles

are lighter and less expensive than LRSI and HRSI, and using them enabled the Shuttles

to lift heavier payloads. The tiles are brittle and vulnerable to crack under stress. The

tiles could not be mounted directly to the main body structure of the Shuttle due to

expansion and contraction of the aluminum structure by temperature change. Instead of

direct mounting on the structure, the tiles have to mount to a felt pad using a silicone

adhesive, and then the tile and pad combination are bonded to the structure as seen in










Figure 1-2. Tiles are occasionally lost during take off because of the incredible loud

noise as well as aerodynamic forces. Because of this, as well as weight concerns, many

of the fuselage tiles were replaced by FRSI blankets.


Coated

Strain
isolator

Adhesive






Airframe
Filler bar

Figure 1-2. Schematic diagram of attaching the tiles. (Courtesy of W. Jordan, Source:
http://www2.1atech.edu/j ordan/Nova/ceramics/SpaceShuttle.pdf, Last
accessed February 14th, 2005).

These material developments and techniques enable the partially reusable Space

Shuttle to offer more capability. However, the tiles have their limitations. During both

liftoff and landing, tiles can become damaged and chipped. About 40,000 hours of

maintenance is required between flights (Morris et al., 1996). For fully RLVs, the tiles

would not provide sufficient protection and some other solution would be necessary.

Blosser (1996) emphasized the durability, operability and cost effectiveness as well as

light weight for new TPS to achieve the goal of reducing the cost of delivering payload to

orbit. Most other proposed reusable thermal protection systems have involved some kind

of advanced high-temperature metal.

Metallic TPS is considered as a much-needed alternative to the ceramic-based

brittle tile and thermal-blanket surface insulation currently used on the Space Shuttle.

Metallic TPS offers the significant advantages (Harris et al., 2002).









* Does not require high temperature seals or adhesive development

* Does not require waterproofing or other restorative processing operation between
flights

* Significantly reducing operational cost

* Saving on vehicle weight, when used as part of an integrated aeroshell structural
system


The TPS forms the external surface of an RLV and is exposed to a wide variety of

environments corresponding to all phases of flight (Dorsey et al, 2004). Thus, the TPS

requirements must apply to any external vehicle airframe surface. Recently, a new

Adaptable, Robust Metallic, Operable, Reusable (ARMOR) metallic TPS concept has

been designed (Blosser et al., 2002) and demonstrated the capability of protecting the

structure from on-orbit-debris and micrometeoroid impact (Poteet and Blosser, 2004).

The concepts of metallic TPS depend primarily on the properties of available materials.

The development of foam and FGM as a core material of TPS panel may offer dramatic

improvements in metallic TPS (Harris et al., 2002). An integrated wall construction is an

approach which the entire structure is designed together to account for thermal and

mechanical loading (Glass et al, 2002). An integrated sandwich TPS with metallic foam

core is studied under steady state and transient heat transfer conditions and compared

with a conventional TPS design (Zhu, 2004).

Functionally Graded Foams and Functionally Graded Materials

Foams are generally made by dispersing gas in a material in liquid phase and then

cooling it to a solid. Solid foams can also be made by dispersing a gas in a solid. These

solid foams are generally called cellular solids, often just called foams. During the last

few decades, many attempts have been made to produce metallic foams, but methods









have suffered from high cost, and only poor quality foam materials were produced. In

the last ten years, improved methods were discovered, and only recently various methods

are available to produce high quality metallic foam. Some start with the molten metal

and others with metal powder. Graded foams can also be manufactured by dispersing

hollow micro-balloons of varying sizes in a matrix medium (Madhusudhana et al., 2004).

The porous structures of carbon and metallic foams are depicted in Figure 1-3.

Foams can be used in many potential engineering applications ranging from light

weight construction to thermal insulation to energy absorption and thermal management.

The mechanical properties of foams are strongly dependent on the density of the foamed

material as well as their cell configuration. For example, the quantities such as elastic

modulus and tensile strength increase with increasing density of foams. Foams can be

used in many potential engineering applications ranging from light weight construction to

thermal insulation to energy absorption and thermal management.










A B C

Figure 1-3. SEM images of A) low density carbon foam and B) high density carbon
foam C) metallic foam.

Foams can be categorized as open-cell and closed-cell foam. In open-cell foams

the cell edges are the only solid portion and adjacent cells are connected through open

faces. If the faces are also solid, so that each cell is sealed off from its neighbor, it is said

to be closed-cell foam (Gibson & Ashby, 2001). In this study, from the thermal









management application point of view only the open-cell foam is considered due to its

large surface area and the ability to transfer heat by working fluid in open porous

structure, if necessary. The combination of open porosity and large specific surfaces

allows a reduction in size of the thermal management system. A reduction in size of the

thermal management system will reduce weight and improve efficiency.

Functionally graded materials (FGMs) are a relatively new class of non-

homogeneous materials in which material properties vary with location in such a way as

to optimize some function of the overall FGM. The FGM concept originated in Japan in

1984 as a thermal barrier material which is capable of withstanding a surface temperature

of 1,725 C and a temperature gradient of 725 C across a cross section less than 10 mm.

Since 1984, FGM thin films have been comprehensively researched and are almost a

commercial reality.

The primary advantage of FGM over conventional cladding or bonding is avoiding

weak interfacial planes because material properties are engineered to have relatively

smooth spatial variation unlike a step increase in conventional cladding or bonding.

Thus, FGMs are widely used as coatings and interfacial zones to reduce mechanically and

thermally induced stresses caused by the material properties mismatch and to improve the

bonding strength. Generally, a functionally graded material (FGM) refers to a two-

component composite characterized by a compositional gradient from one component to

the other. In contrast, traditional composites are homogeneous mixtures, and they

therefore involve a compromise between the desirable properties of the component

materials. Since significant proportions of an FGM contain the pure form of each

component, the need for compromise is eliminated. The properties of both components









can be fully utilized. For example, the toughness of a metal can be combined with the

refractoriness of a ceramic, without any compromise in the toughness of the metal side or

the refractoriness of the ceramic side. However, in this study, only the concept of

varying material properties is adopted, and functionally graded foams (FGFs) are

produced by changing the size of unit-cell or the thickness of strut in the foam.

Previous Work on Fracture Mechanics of Functionally Graded Materials

In order to utilize FGMs as reliable engineering materials in structures, among

other properties their fracture mechanics has to be understood. Furthermore, methods to

compute the stress intensity factor (SIF) and energy release rate have to be developed

because the stress intensity factor cannot be measured directly in an experiment, but it

can be found through the relations between SIF and a measurable quantity, such as strain,

compliances or displacement.

Sound fracture mechanics principles have been established for conventional

homogeneous materials so that the strength of a structure in the presence of a crack can

be predicted. However, the fracture mechanics of a functionally graded material which is

macroscopically non-homogeneous is only beginning to be developed. Analytical work

on FGM goes back to the late 1960s when Gibson (1967) modeled soil as a non-

homogeneous material.

Analytical studies have shown that the asymptotic crack tip stress field in FGMs

possesses the same square root singularity seen in homogeneous materials. Analytical

studies of Atkinson and List (1978) and Gerasoulis and Srivastav (1980) are some of the

earliest work on crack growth in non-homogenous materials in order to evaluate its

integrity. Atkinson and List (1978) studied the crack propagation for non-homogenous

materials subjected to mechanical loads assuming an exponential spatial variation of the









elastic modulus. Gerasoulis and Srivastav (1980) studied a Griffth crack problem for

non-homogeneous materials using integral equation formulations. Delale and Erdogan

(1983), Eischen (1987), Jin and Noda (1994) and Erdogan (1995) showed that the nature

of the inverse-square-root-singularity of crack tip is also preserved for an FGM as long as

the property variation is piecewise differentiable. The work by Delale and Erdogan

(1983) is accredited with having first suggested the standard inverse-square-root stress

singularity for an FGM in which a crack is parallel to the elastic modulus gradient.

Eischen (1987) confirmed their work by using eigenfunction expansion technique in non-

homogeneous infinite plane. Jin and Noda (1994) further confirmed for FGM with

piecewise differentiable property variation. In 1996, Jin and Batra studied crack tip fields

in general non-homogeneous materials and strain energy release rate and stress intensity

factor using the rule of mixture. Based on the early work of Delale and Erdorgan (1983)

that showed the negligibility of the effect of the variation on Poisson's ratio, Erdorgan

and Wu (1997) analyzed an infinite FGM strip under various remote loadings by using an

exponential varying elastic constants and constant Poisson's ratio. Although such

progress has increased the understanding of fracture mechanics of FGM, a suitable stress

intensity factor solution is needed in designing components involving FGM and

improving its fracture toughness.

In engineering context, the closed-form SIF solution is desirable for easier use in

the analysis of fracture of FGM structures for a variety of specimen configurations. The

exact solutions are not available yet and some researchers have attempted to find simple

and approximate closed-form solutions. Yang and Shih (1994) obtained an approximate

solution for a semi-infinite crack in an interlayer between two dissimilar materials using a









known bi-material solution. Gu and Asaro (1997) obtained the complete solution of

semi-infinite crack in a strip of an isotropic FGM under edge loading. The solution was

analytical up to a parameter which is obtained numerically. Then, the solution was

extended to the strip is made of an orthotropic FGM. Ravichandran and Barsoum (2003)

obtained approximate solution and compared the results with the values obtained by finite

element modeling (FEM).

The application of the finite element method to determine crack tip stress fields has

been rapid progress (Broek 1978). A finite element based method for determination of

stress intensity factor in FGM was proposed by Gu et al. (1999). They used standard

domain integral to evaluate the crack-tip field for FGM and studied the effect of non-

homogeneity in numerical computation of the J-integral. They concluded that the

conventional J-integral can provide accurate results as long as the fine mesh near crack

tip is provided. Honein and Hermann (1997) have studied the conservation laws for non-

homogenous materials and proposed a modified path-independent integral. Weichen

(2003) constructed another version of path-independent integrals of FGM by gradually

varying the volume fraction of the constituent materials.

Numerical simulation was carried out by Marur and Tippur (1999) using linear

material property variation in the gradient zone. They studied the influence of material

gradient and the crack position on the fracture parameters such as complex stress

intensity factor and energy release rate. Anlas et al. (2000) calculated and compared the

stress intensity factors obtained for a cracked FGM plate by using several different

techniques- energy release rate, J-integral and a modified path independent integral.

They evaluated the J-integral and a modified J-integral numerically by technique similar









to Gu et al. (1999) and Honein and Hermann (1997) respectively. The results were

compared with the analytical solutions of Erdogan and Wu (1997). Furthermore, the

accuracy of the finite element method and mesh refinement was investigated.

In contrast to above-described analytical studies and numerical investigation, there

are relatively few experimental works on fracture mechanics of FGM. A typical

laboratory technique is the use of photo elasticity. Butch et al. (1999) examined the

surface deformation in the crack tip region by the optical method of Reflection Coherent

Gradient Sensing. They used a graded particulate composition comprised of spherical

glass filler particles in an epoxy matrix as a test specimen. Recently, Rousseau and

Tippur (2002) examined the particulate FGM by mapping crack tip deformation using

optical interferometery. They used a finite element analysis in order to develop fringe

analysis and to provide a direct comparison to the optical measurements.

Objectives

The objectives of this research are to develop micromechanical models to predict

the fracture toughness of functionally graded foams under various loading conditions -

mechanical and thermal loading as insulation materials for load carrying thermal

protection system, and to develop the understanding of the effect of graded foam solidity

profile on its fracture mechanics. The methods will also be used to understand the effects

of thermal gradients on fracture of homogeneous foams.

Scope

Chapter 1 reviewed some background information regarding functionally graded

foams as the thermal protection system of next generation reusable launch vehicles and

some previous works on fracture mechanics of FGM. Chapter 2 discusses the method to

estimate the material properties of functionally graded foams (FGFs). At first,









formulations for homogeneous foam will be established and then the methods will be

extended to the FGFs. Chapter 3 describes the finite element analysis (FEA) of the

micromechanical model. In Chapter 3, macro and micro models for graded cellular

materials are explained with key issues in both models. Chapter 4 discusses the results

under mechanical loading including remote loading uniform traction and uniform

displacement) and local loading (crack face traction). Chapter 5 presents the behavior

and the results under thermal loading on homogeneous foam and FGF. The concluding

remarks are presented in Chapter 6.














CHAPTER 2
ESTIMATION OF CONTINUUM PROPERTIES

The functionally graded foam can be modeled either as a non-homogeneous

continuum, or as a frame consisting of beam elements. The former model will be

referred to as the macro-model and the latter as the micro-model. We require both

models for the simulation of crack propagation in graded foams. The region surrounding

the crack tip is modeled using the micro-model, where as the region away from the crack

tip uses the macro-model. The micromechanical model is treated as an embedded model

around crack tip. The macro-model of the functionally graded foam requires continuum

properties at each point or at least for each element in the finite element model. In this

chapter, the procedures for calculating the continuum properties of a homogeneous

cellular medium (open-cell) is presented, and then the method is extended to functionally

graded foams.

Continuum Properties of Homogeneous Foam

Most of the open-cell foams with periodic microstructure can be considered as

orthotropic materials. Choi and Sankar (2003) derived the elastic constants of

homogeneous foams in terms of the strut material properties and unit-cell dimensions. In

their model they assumed that the strut has a square cross section h x h and the unit-cell is

a cube. In the present approach, the general case is considered wherein the unit-cell is a

rectangular parallelepiped of dimensions c, x c2 x c, as shown in Figure 2-1. The

derivation of formulas for the relative density and elastic modulus are straightforward.

The relative density p*/p, is related to the porosity of the cellular material. A superscript









* denotes the foam properties and a subscript s denotes the solid properties or the strut

properties. The density of the foam can be obtained form the mass and volume of the

unit-cell. Then, the relative density can be expressed as a function of the dimensions of

unit-cell and the strut thickness as shown below:


p*mv (c, + c2 3)h2 2h3
(2-1)
A PA C1C2C3

where m is the mass and Vis the volume of unit-cell.














Figure 2-1. Open cell model with rectangular parallelepiped unit cell.

Elastic modulus can be evaluated by applying a tensile stress a* on unit area of the

unit cell as shown in Figure 2-2. The equivalent force on the strut caused by the stresses

can be written as F = (cq x c3) In micro-scale sense, the force F causes stresses a, in

sectional area h2 (Figure 2-2). Therefore, the stress as in the section h2 and the

corresponding strain E can be expressed as

F cIc3 s C1C3-
F c3u and c c-
U0- = 2 and E=- -2 (2-2)
Sh2 h2 Es h2Es )

where Es is elastic modulus of strut. Therefore, elastic modulus of foam E* can be

derived from Eq. (2-2) as
















2,y

1 ,x
3 ,z


E, E,, E2 E, E 3 E
C2C3 C13 C12




Cl
=... ----,... ,.... 3=
nit area:/
h*
.a

h 0 /~7 .-*" .-"' ~7


(2-3)


Figure 2-2. Micro- and Macro-stresses in open-cell foam.

The derivation of shear modulus is slightly involved and it is described below. We

show the derivation of the shear modulus G1 from the unit-cell dimensions, strut cross

sectional dimensions and the strut elastic modulus. When a shear stress is applied, struts

are deformed as shown in Figure 2-3.

F1


C2/2
6F2

ig IN Curvature=O F2
C2/2


cC/2 _2 //2
Figure 2-3. Flexural deformation of struts under shear stresses.


Figure 2-3. Flexural deformation of struts under shear stresses.






19


Bending moment becomes zero at the middle of struts because the curvatures are

zero due to symmetry. The struts are assumed as a beam fixed at the end with a


concentrated force at the middle at a distance c and 2, respectively from the fixed end.
2 2

The maximum displacement can be written as,


PL3 F2 PL3 ( 2
S= and 52 = (2-4)
3EI 3EI 3EI 3EJ

bh3 h4
where I = (the moment of inertia). The applied shear stress can be written
12 12

F F F F
as2 2 Using the relations, =2 the maximum displacements in Eq. (2-
CIC3 C2C3 C1 C2

4) can be rewritten as



1 c1F2 and 82 CC2F2 (2-5)
24EJ 24EJ
The shear strain y12 can be derived as

232 251 2c,2 + 2c21 (2-6)
1-J+ -- (2-6)
C2 C1 C1C2

Using Eq. (2-5), the shear strain can be written as,

(CC2 + C)F2
712 12= (2-7)
12E I

The shear modulus GI2 can be derived as

F2
G -12 = c2c3 12EJ (2.8)
/712 cC + C1 )F2 C2C3 (1i +2 1
12EJ









Substituting for the moment of inertia, I



G12 C, (C=j E, (2.9)


The shear modulus in the other two planes can be obtained by cyclic permutation as

C h4
G23 CCCt+C ES)
32 )(2.10)
G31 EC2C3 c3)
iG3 C I, --2 + 2 3)


Continuum Properties of Functionally Graded Foams

The properties of a functionally graded foam can be represented by a function of

the coordinate variables x, y and z. The actual functional form depends on the application

and also the type of information sought from the homogenized model of the foam. In this

study, the functions of material properties will be assumed such that the properties

calculated at the center of a cell will correspond to the properties of the homogeneous

foam with that cell as its unit cell. Thus the function is actually defined only at the

centers of the cells of the functionally graded foam. Then, these points will be curve-

fitted to an equation in order to obtain the continuous variation of properties required in

the continuum model. This approach will be verified by solving some problems wherein

the graded foam is subjected to some simple remote loading conditions (uniform

displacement loading) and comparing the resultant stresses from the macro- and micro-

models. The material properties of strut correspond that of Zoltex Panes 30MF High

Purity Hilled carbon fiber studied earlier by Choi & Sankar (2003). The Zoltex Panes

30MF High Purity Hilled carbon fiber is chosen because of the high percentage of carbon

component weight (99.5%). The strut properties are listed in Table 2-1.









Table 2-1. Material properties of the Zoltex carbon fiber.
Density, p, 1750Kg /m
Elastic Modulus, Es 207 GPa
Poisson's ratio, s, 0.17
Ultimate Tensile Strength, o-, 3600 MPa

The relative density of functionally graded foams (FGF) depends on both the

dimensions of the unit-cell and the strut thickness. Therefore, three different cases can be

considered. The first case is the one where the dimensions of the unit-cell remain

constant while the strut thickness varies along the x-axis. In the second case the strut

thickness is kept constant with varying cell length. The last case is varying both of them.

In this paper, the first two cases are studied independently. Furthermore, the material

properties of functionally graded foam can be either increasing or decreasing along the x-

axis. Therefore, the fracture properties of both increasing and decreasing cases are

studied and compared to the homogenous case.

For the case where the strut dimensions vary, the thickness of the square strut is

assumed to vary as

h(x) = h + ax (2-11)

where a is a parameter that determines the degree of gradation of the properties. Then the

properties such as density and elastic constants of the graded foam can be assumed to

vary as given by the equations for homogeneous foams, but changing the constant h by

the function h(x). Figure 2-4 shows the variation of relative density and elastic modulus.

For example, the relative density variation of the functionally graded foam with

varying beam thickness can be written as

P =3 h(x) 2 hx) (2-12)
ps c c











where the unit-cell is assumed as a cube of dimension c (i.e. c = c, = c = c,).


Similar equations can be derived for elastic modulus and shear modulus as


E r =h(x) Es
c )(2-13)

G 1 h(x)E
2 c)

08 005

07 -
-- Elastic Modulus 0 04
0 6 Relative Density

l t 05 003


c \\ 2
04

e i 002 .
g 03

02
001
01

O0 0 000
000 002 0 04 006 0 08 010
x(m)


Figure 2-4. Example of variation of elastic modulus and relative density for constant cell
length c=200um, ho=261um and a=-200x 106.

For the case where the unit-cell dimensions vary, we can consider the case where h,

c2 and c3 are constants, but cl varies as


cI+l = c1 + p (2-14)
where i denotes the cell number from the left edge (i.e. c represents the size of cell at


left edge) and /is the increment in the cell length in the x direction. Again the properties

of the foam will be calculated at the center of each cell using the equations for

homogeneous foams as given in equations (2-2) through (2-9).









Finite Element Verification of Estimated Continuum Properties

The accuracy of the estimated elastic constants, when a material property

discretization is introduced, is investigated by comparing the stress field from macro-and

micro-models. A simple mechanic problem was solved using both un-cracked macro-

and micro-models. A uniform displacement (70um) was applied along the upper edge of

a rectangular plate using the macro-model, which consists of two dimensional plane

stress elements (eight nodes bi-quadratic, reduced integration element). The elastic

constants of the non-homogeneous material varied as given by Eq. (2-13). In the finite

element model the elastic constants within each element were considered constant. The

boundary conditions are depicted in Figure 2-5.

Uniform Displacement












uO4, uy=0 u=O, uy=0


Figure 2-5. Boundary conditions for un-cracked plate under uniform extension.

The right lower corer was fixed to prevent the rigid body motion. The resulting

displacements along the boundary of a micro-model embedded in the macro-model were

applied as the boundary displacements of the micro-model by using the three-point

interpolation. For the micro-model, each strut was modeled as an Euler-Beroulli beam

element with two nodes and three integration points. In order to verify the validity of

properties used in the macro-model, the stresses in both models are compared. In the











case of macro-model the stresses are obtained as the finite element analysis output. The


outputs in micro-model are the axial force and moment resultant in the beam element. We


convert these forces into equivalent stresses by dividing by the strut cross sectional


area cl x c3 The shaded region on Figure 2-5 represents the micro-model. Both constant


cell length with varying strut thickness and constant strut thickness with varying cell size


cases are considered. In the constant cell length case the cell length is assumed as


200yum. The macro-model consists of 100x50 plane solid elements. The strut cross


section is assumed to vary as a function of x according to the equation h(x) = ho + ax,


where ho =40um and a =-200x 10-6. The region corresponding to the micro-model in the


macro-model consists of 15x5 plane stress elements. The micro-model uses 2,250 beam


elements.


14
Stresses
(MPa)I 2

10 --

8 -


macro-m odel
4 micro-m odel


Approx 5% difference
0 I I I I I
0 000 0 002 0 004 0 006 0 008 0 010 0 012 0 014 0 016
x (m)

Figure 2-6. Comparison of stresses (oyy) obtained using the macro- and micro- models in
graded foam with constant cell size but varying strut cross section.

The results for the stress component yy from the macro- and micro-models are


compared in Figure 2-6. The maximum difference in stresses between the macro- and


micro-models is about 5%. In the second case, the strut is assumed to have a square cross











section (h =20pum) and the cell length c, was varied along the x direction with co =2001um


and / =-0.151m. The dimensions of the cell in the 2 and 3 directions, c2 and c3, are kept


constant (100um). The stress component yy from macro-model and micro-model are


compared in Figure 2-7.




Stresses
(MPa)


6m


macro-model
-- micro-model




0
0000 0005 0010 0015 0020 0025 0030
x (m)

Figure 2-7. Comparison of stresses (uyy) obtained using the macro- and micro- models in
graded foam with constant strut size but varying cell dimension.

The above examples illustrate and validate two important concepts that will be used

in this dissertation. We find that modeling graded foams with microstructure as a non-

homogeneous continuum provides good results for micro-stress and displacements.

Some researchers, e.g., Gu et al. (1999), and Santare and Lambros (2000) have used

different properties at the Gauss (integration) points within the element. However, using

the homogenized properties at the center of the continuum element in the FE model

seems to be reasonable and yields accurate results.














CHAPTER 3
FINITE ELEMENT BASED MICROMECHANICAL MODEL

In this chapter, we describe a finite element based micromechanics model for

estimating the fracture toughness of functionally graded foams. The crack is assumed to

be parallel to the material properties gradient direction. At first, we describe a finite

element based micromechanics model for estimating the fracture toughness in order to

understand the key ideas of micromechanical modeling. Detailed macro- and micro-

model descriptions are presented, and the method of extracting stress intensity factor

from the finite element analysis is described in depth. Also, convergence test in both

macro- and micro-models were performed.

Overview of Micromechanical Model

The functionally graded foam, a cellular material is non-homogeneous in the

macro-scale. That is, the microstructure is graded and the foam is treated as a functionally

graded material in macro-scale. The foam can be modeled either as a non-homogeneous

continuum, or as a frame consisting of beam elements to model the struts. The former

model (continuum model) will be referred to as the macro-model and the latter (frame

model) as the micro-model. In the finite element analysis, solid elements are used in the

macro-model and beam elements in the micro-model.

In the finite element analysis model, due to symmetry, only the upper half of the

plate is considered. The lower edge has a zero displacement boundary condition in they-

direction to account for symmetry. As described in the previous chapter, the functional

variation of material properties is estimated by extending the method of calculating the









continuum properties of a homogeneous cellular medium. The eight-node quadrilateral

elements were used to discretize the macro-model and functional variation in material

properties is implemented by having 100 vertical layers, with each layer having a

constant value of material properties. A crack can be created in the functionally graded

foam by removing a set of struts along the intended crack surface in micro-model and by

removing the zero boundary condition along the intended crack surface in macro-model.

A portion of the foam surrounding the crack tip is considered as the micro-model (see

Figure 3-1).

Y

ttttttttttttttttttttt
c
Macro-model <
h




Micro-model
Crack j

Width

A B

Figure 3-1. Schematic description of both macro- and micro-model. A) Macro-model
consists of plane 8-node solid elements. The region in the middle with grids
indicates the portion used in the micro-model. B) The micro-model consists
of frame elements to model the individual struts. The displacements from the
macro-model are applied as boundary conditions in the micro-model.

The dimensions of the micro-model should be much larger than the cell size (strut

spacing) so that it can be considered as a continuum. For the case of uniform

displacement loading, the upper edge is loaded by uniform displacement in they-

direction. The maximum stresses in the struts in the vicinity of the crack tip are









calculated from the finite element micro-model. From the failure criterion for the strut

material, one can calculate the maximum stress intensity factor that will cause the failure

of the crack tip struts, and thus causing crack propagation in a macro-scale sense. The

key idea in this approach is to be able to calculate the stress intensity factor for a given

boundary displacements or apply a set of boundary conditions that corresponds to a given

stress intensity factor in the macro-scale sense. For this purpose we turn to the macro-

model as shown in Figure 3-1 (A). In the macro-model a much larger size of the foam is

modeled using continuum elements, in the present case, plane solid elements. The micro-

model is basically embedded in the macro-model. The displacements of points along the

boundary of the micro-model are obtained from the finite element analysis of the macro-

model and applied to the boundary of the micro-model by using three points

interpolation. The maximum principal stresses at the crack tip can be calculated from the

force and moment resultants obtained from the micro-model as

htf p
MtUip 2
2 Ftp
otp = (3-1)
Itp A tp

where cp is maximum principal stress at the crack-tip.
Ftp, Mtp are force and moment resultant.
At is cross-sectional area and htp is the thickness of strut.
It, is the moment of inertia.

The strut material is assumed brittle and will fracture when the maximum principal

stress exceeds the ultimate tensile strength. The fracture toughness of the foam is

defined as the stress intensity factor that will cause the crack-tip struts to fail in a micro-

scale sense and cause the crack to propagate in a macro-scale sense. The fracture









mechanism of brittle material is governed by Linear Elastic Fracture Mechanics (LEFM).

Therefore, the fracture toughness can be estimated from the following relation.

-i =- (3-2)
Kic ou

Macro-model

In macro-model, the conventional two dimensional isoparametric plane-stress

elements are used. The problem geometry is shown in Figure 3-1 (A). The material

gradient is in the x-direction. A pre-processor program was coded using MATLAB,

with the parameters such as unit-cell size, strut thickness, crack size, a and /f (defined in

previous chapter) in order to generate a rectangular mesh with eight-node isoparametric

elements with two-degree of freedom at each node and to impose the material gradient in

the macro-model.














Crack tip
A B

Figure 3-2. A typical finite element macro-model. A) constant unit-cell size with
varying strut thickness. B) varying(decreasing) unit-cell size with constant
strut thickness.

Only half of the model is represented in the finite element analysis by invoking that the

model is symmetric with respect to its midline, x-axis. A zero displacement boundary











condition in y-direction is employed in the lower edge to account for symmetric. Figure

3-2 shows a typical finite element model which consists of 5,000 eight-node

isoparametric elements (50 elements in vertical direction and 100 elements in horizontal

direction) with 1,5300 nodes. It should be noted that the number of elements in a typical

model was decided after convergence test by generating a coarse mesh (smaller number

of elements) and progressively reducing the mesh size (increasing number of elements),

to be discussed in this section later.

Imposing Graded Material Properties

Finite element analysis of functionally graded foam for macro-model requires

imposing the required the variation of material properties in x-direction. The material

properties are graded by either changing the thickness of struts or changing the

dimensions of the unit-cell described as before. Relative density, elastic modulus and

shear modulus vary along the x-axis corresponding to the equations derived in the

previous section.

08

07 <

0 6 -.. .. Actual Elastic Modulus
S--- Discrete Elastic Modulus
0 05







01
00
0 00 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 10
x (m)


Figure 3-3. Example of discrete elastic modulus for macro-model with ten-regions.









When functionally graded foam is modeled as a homogeneous solid (macro-model), a

material property discretization is introduced. The discretization is done by grouping

elements in the gradient region into narrow vertical strips and assigning constant values

of estimated material properties at the centroid of the strip of grouped elements. For

example, Figure 3-3 shows the discrete elastic modulus for the ten-region model.

However, the Poisson's ratio is kept constant because the effect of a variation of

Poisson's ratio is negligible (Delale and Erdogan, 1983).


Global panel







Const crack length




Variation of beam thickness




Figure 3-4. Location of model specimens in the global panel. Each specimen is of the
same size and contains a crack of given length, but the density at the crack tip
varies from specimen to specimen.

The Mode I fracture toughness with various relative densities is conducted in two

different sets for the constant unit-cell lengthcase. The first set is controlling the crack

length while the variation of material properties remains same. The other set is shown in

Figure 3-4. The crack length remains constant while the material properties are

controlled to locate desired relative density at the crack tip. However, the dimensions of

models are fixed (0.1m by 0.5m for macro-model and 0.015m by 0.005m for micro-









model). For the case where the unit-cell dimensions change, the number of elements both

in macro-model (100x50 elements) and micro-model (2,250 elements) are fixed and the

material properties at the crack tip is controlled by c, and /7. Therefore, the dimensions

of models are not fixed.

Methods for Extracting Stress Intensity Factor

Considering only Mode I symmetric loading (mode-mixity=O), the stress intensity

factor at the crack tip is calculated from traditional methods in computational fracture

mechanics i.e. point matching and energy method (Anderson, 2000). The point matching

method is the direct method in which the stress intensity factor can be obtained from the

stress field or from the displacement field, while the energy method is an indirect method

in which the stress intensity factor is determined via its relation with other quantities such

as the compliance, the elastic energy or the J-contour integral (Broek, 1978). The

advantage of the energy method is that the method can be applied as both linear and

nonlinear. However, it is difficult to separate the energy release rate into mixed-mode

stress intensity factor components. In this paper, the crack-tip stress field and J-contour

integral are used to find and verify the stress intensity factor for the point matching and

the energy method respectively.

In energy method, the J-contour integral can be evaluated numerically along a

contour surrounding the crack tip, as long as the deformations are elastic. Generally, J-

contour integral is not path independent for non-homogeneous material. Therefore, J-

contour integral is expected to vary with contour numbers as shown in Figure 3-5. The

contour numbers represent incrementally larger contours around the crack tip. The mesh

refinement governs the size and increments of contours.











800

700

600

500

400

8 300

200 y = 1.21193E-05x 1.27727E-03x 6.32919E-02x2 2.05304E+00x+ 7.61841E+02
R2 9.99992E-01
100

0
1 6 11 16 21 26 31 36 41 46 51
Contour Number


Figure 3-5. J-Integral for various contours in a macro-model containing 100x50 elements
and the contour numbers increase away from the crack tip.

The first few contours are disregarded due to inaccuracy for most finite element

meshes (Anlas et al., 2000). J-contour integral as r-*0 is obtained by fitting a fourth

order polynomial to the output values of J-contour integral. The limiting value of J-

contour integral can be evaluated numerically as the intercept of the polynomial curve at

y-axis. The value of J-integral for a contour very close to the crack-tip is related to the

local stress intensity factor as in the case of a homogeneous material (Anlas et al., 2000).

Thus, energy release rate, G is identical to the value of J-contour integral as the path of

contour approaches to crack-tip (Gu and Asaro, 1999). Conceptually, energy release rate,

G can be found by the variation of J-contour integrals as shown in Figure 3-5. The stress

intensity factorKI of a functionally graded foam (two-dimensional orthotropic) can be

found from G using the relation (Sih & Liebowbitz, 1968).


2
2 1M122 )2 22 2 2a12 + t66
G = nK -- + (3-3)
,2 all an 2all










1 1 1
where, al = a22 a33
El E2 E,


a12 = a23 = a3 = 0
1 1 1
a44 = ,a 55 ,a66 -,
G23 G13 G12

Using the point matching method of stress field, the opening mode value of the

stress intensity factor can be calculated from the oyy stress ahead of the crack (Sanford,

2003).


K, = Lim [cyy 2r] (0 = 0) (3-4)
r->0

The stress intensity factor can be found by plotting the quantity in square brackets against

distance form the crack tip and extrapolating to r = 0. Figure 3-6 shows the one of the

example plot of uV-7r versus distance from the crack tip. A 4th order polynomial


regression is also shown in Figure 3-6. The y-intercept of the curve yields the value of KI.


1.40E+06

1.20E+06

1.00E+06

8.00E+05

6.00E+05

4.00E+05
= -4.70773E+11x4 + 4.70881E+10x3 1.99894E+09x2 + 6.41493E+07x+ 2.42043E+05
2.00E+05 R2 9.99997E-01

0.00E+00
0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300

distance from crack tip, r (m )


Figure 3-6. Stress intensity factor from the stresses normal to the crack plane.










The stress intensity factor defines the amplitude of the crack tip singularity and the

conditions near crack-tip (Anderson, 2000). Stress near crack tip increases in portion to

the stress intensity factor. Consider the Mode I singular field ahead the crack tip, the

stress normal to the crack plane, oy can be defined from Eq. (3-5).


KI
yy 2 (3-5)


Figure 3-7 shows the stress normal to the crack plane versus distance form the

crack tip. Where the square-root singularity dominated zone, Eq. (3-5) is valid while

stress far from the crack tip is governed by the remote boundary conditions.

Table 3-1. Comparison between two methods.
Crack Length (m) 0.01 0.02 0.03 0.04 0.05
Relative density 0.094582 0.085536 0.076874 0.068608 0.06075
at the crack-tip
KT From J-Integral
SFrom -Integral 1.2238E+06 1.0699E+06 9.2893E+05 8.4886E+05 7.2691E+05
(Pa-m")
K From crack-tip 1.2262E+06 1.0715E+06 9.3360E+05 8.0605E+05 6.8363E+05
stress field (Pa-m")
% difference 0.195 0.145 0.503 5.043 5.954


1.80E+07

1.60E+07

1.40E+07
y,, (Pa)
1.20E+07

1.00E+07

8.00E+06 6

6.00E+06 CPO r,

4.00E+06
SK/(2 Pi* r)1/2
2.00E+06
Singularity dominated zone
0.00E+00
0. 0- 0.005 0.010 0.015 0.020 0.025 0.030
distance from crack tip, r (m)


Figure 3-7. Stress normal to the crack plane.








36



The stress intensity factor from J-Integral and stress-matching were compared for


various cases in Table 3-1. The maximum difference between the two methods is less


than 6%.


Convergence Analysis for Macro-model


For the convergence test, the model which has constant cell size with varying the


strut thickness is discretized into uniform meshes of 10x5 elements (10 regions), 20x10


(20 regions), 50x25 elements (50 regions), 100x50 (100 regions), 200x100 (200 regions)


and 400x200 (400 regions). Some finite element outputs are shown in Figure 3-8.


Ppm~ma~mrsmn~r I In;as~m l~~u.


.,- :- 1 r n f 1;.i o ,. i :1 .D : :, .11


I*Y4 .... .. ....................
Il r nr lm..... .. .. .. .. .. .. .n


j cr T -a -- -6U /L Ei 3u u .w Fai-i + .
~71n s m:6. l T. Ir~~
R 11llIh.1 F-;uuCitsFu~ +1.42&O


2


IRinmeii VI."'f BcduirBaLnu 3'uli sFai~;l~ +1.428=402


C D
Figure 3-8. Stress distribution of the functionally graded foams. A) 10x 5 elements
model. B) 20x 10 elements model. C) 100x 50 elements model. D) 200x 100
elements model.


a~. I MIN~
,__777ITTI
-1 I I I A
i- -,,I T


'I T.- I W
I-, V-~u
1 1 F ;










As the number of elements and regions increases, the energy release rate at the

crack tip converges as shown in Figure 3-9. For 100x50 elements model, the variation of

J-contour integral is less than 0.01% compared to the 400x200 element model.

Therefore, 100x50 elements model is used for further analysis in order to maintain

adequate accuracy with reasonable computational time.


Energy release rate Time (sec)
613 1400

612.5 -1200
612 --- Energy release rate
1 1000
611.5 -- Time needed to complete job
6 800
611
600
610.5

6160 -- 400
610

609.5 200

609 0
0 50 100 150 200 250 300 350 400
Number of regions


Figure 3-9. Variation of energy release rate at the crack tip with various size macro-
models

Micro-model

A portion of macro-mechanical model (ABCD) is taken and used for micro-model

as shown in Figure 3-1 (A). As the 100x50 elements (100 regions) for macro-model and

constant cell length (200 um) for micro-model are used, one macro-model element can be

replaced by 60 beam elements for micro-model as shown in Figure 3-10. The

displacements along the boundaries of micro-model are determined by using three points

interpolation. The corresponding three points can be obtained from the previously

described macro-model analysis. For instance, displacements in the x-direction for each









beam element on micro-model along the three nodal points (a, b and c) can be found as

follows,

icro(y) = macro y(y 1) + acro macro y(y + ) (3
a 2/2 b ,2 21c 2/2 (



SL, v(x,y)





Macro-Model
Solid 2-D clement

SMicr-Model
b Y ( 0' Beam element |




Sy=- x, u(x,y)



Figue 3-10. Embedded beam element (micro model) in two-dimensional eight-node solid
model (Macro model).

In micro-model, two-node beam elements are used to represent the foam

ligaments/struts. After, the displacements along the boundaries of micro-model, the

maximum principal stress stresses at the crack tip o-tp can be calculated from the results

for force and moment resultants obtained from the micro-model as

hhp
Op 2 F+ (3-7)
Inp Atp

The fracture toughness of the foam is defined as the stress intensity factor that will

cause the crack-tip struts to fail. We assume that the strut material is brittle and will

fracture when the maximum principal stress exceeds the tensile strength -,.










- 1 I1 I 1 1I L


Figure 3-11. Force and moment resultants in struts modeled as beams.

Since we are dealing with linear elasticity, the fracture toughness can be estimated

from the following relation,


(3-8)


1.54

1.52

1.5

1.48

1.46

1.44

1.42

1.4

1.38


0 2000 4000 6000 8000 10000 12000

Number of Element


450

400

350

300

250

200

150
'S
100
mpletejob 50

0
14000 16000


Figure 3-12. Variation of fracture toughness with the size of micro-models.

The convergence analysis is conducted to evaluate the variation of fracture

toughness with various sizes of micro-model, 3 x 1 macro-model (170 elements in micro-

model), 6x2 (640), 15x5 (3,850), 21x7 (7,490) and 30x10 (15,200). As model size

increases, fracture toughness converges as shown in Figure 3-12. For 3,850 beam






40


element model, the error in fracture toughness is less than 0.3 % compared to 15,200-

beam elements model. Therefore, the 3,850-model is chosen for further analysis as a

compromise between the accuracy and computational time. The aforementioned methods

will be extended to graded foams and also to understand the effects of thermal stresses in

succeeding chapters.














CHAPTER 4
FRACTURE TOUGHNESS OF GRADED FOAMS UNDER MECHANICAL
LOADING

In this chapter, the finite element based micromechanical model discussed in the

previous chapter is used to understand the behavior of functionally graded foam (FGF)

and to estimate their fracture toughness (critical stress intensity factor). We will use the

ABAQUS TM finite element package for performing the simulations. Analysis of FGF

containing a crack under remote loading (uniform displacement) was first carried out.

The results of fracture toughness under uniform displacement are compared with the

homogeneous foam in order to understand the behavior of FGF. Thermal loading can

affect the stress field near crack tip unlike the remote loading case. In order to observe

this local effect, we investigated the case where the pressure applied on the crack surface

for various sizes of crack lengths in homogeneous foam. Then, the results were

compared with the remote traction case. For the remote loading, we considered uniform

displacement and traction on the top edge of model. The principle of superposition was

studied to understand the local effect.

Fracture Toughness under Remote Loading

In this section, fracture toughness of functionally graded foams subjected to

uniform displacements on the top and bottom of the model. The height of the model is

considered same as its width, and is symmetric with respect to its midline, y = 0. The

geometry of the FGF is shown in Figure 4-1 (A) with crack length, a. Only half of the

model is considered in the finite element analysis because the model is symmetric with









respect to its midline, x-axis. The upper edge is loaded by uniform displacement, 70 prm.

A zero displacement boundary condition in the y-direction is applied on the lower edge to

account for symmetry. The material is functionally graded, and the relative density

increases or decreases according to the parameters, a or /, described in Chapter 2. The

parameters determine the degree of gradation of the properties. For the case where the

strut dimensions vary and the cell dimension is constant, parameter a determines the

degree of the gradation (Eq. 2-11). f is used for the case where the cell dimension is

varying in x-direction while the strut dimensions are kept constant (Eq. 2-14).


Y O00o orVo Y




00
> x >
a
a



W=0.1m W=0.1m

A B

Figure 4-1. Edge-cracked model under A) uniform traction or displacement loading and
B) crack surface traction.

First, we investigate the case wherein the graded foam has constant unit-cell length

(c=200pm) and the density is varied by changing the strut cross sectional dimensions.

Both cases, increasing and decreasing densities along the x-axis, are considered. Table 4-

1 shows the results form the case which the unit-cell dimensions and crack length are

kept constant, but the strut thickness is varied, such as the FGF model is taken from









imaginary graded global panel from different position. The results from the case which

the model has the constant unit cell but the crack length and strut thickness is varied are

shown in Table 4-2.

Table 4-1. Fracture toughness of graded and uniform foams. The unit-cell dimensions
and crack length are kept constant, but the strut thickness is varied (c=200uPm,
crack length, a =0.03m and a'-200x 106).
Fracture Toughness (Pa-m2)
Relative Density
at the crack-tip Decreasing Increasing Uniform
at the crack-tip
density density density
26 0.028 4.52171E+05 4.56445E+05 4.51326E+05
30 0.039744 6.56122E+05 6.57406E+05 6.56114E+05
50 0.123904 2.24739E+06 2.25108E+06 2.24928E+06
60 0.179334 3.39537E+06 3.39999E+06 3.39819E+06
70 0.241664 4.77575E+06 4.78247E+06 4.77936E+06


Table 4-2. Fracture toughness of graded and uniform foams. The unit-cell dimension is
kept constant but the crack length and the strut thickness are varied (c=200
yim, ho=40pum and a-=200x 106).
Normalized relative Density Fracture Toughness (Pa-m )
Relative Density
crack length the crack-tip Decreasing Increasing Uniform
at the crack-tip
(a/W) density density density
0.1 0.06075 1.10144E+06 1.03627E+06 1.03485E+06
0.2 0.068608 1.25052E+06 1.18201E+06 1.18004E+06
0.3 0.076874 1.33362E+06 1.33619E+06 1.33465E+06
0.4 0.085536 1.49961E+06 1.49980E+06 1.49878E+06
0.5 0.094582 1.67268E+06 1.67266E+06 1.67220E+06


As seen in Table 4-1, Table 4-2 and Figure 4-2, the results from the present

analysis for FGF are very close to those of homogeneous foam. However we see an

interesting trend in Figure 4-10. In both deceasing and increasing density cases, the

fracture toughness deviates from that of uniform density foam for higher densities. When

the density decreases along the crack path, the fracture toughness is slightly higher and

vice versa.












4.5E+06

4.0E+06

3.5E+06

3.0E+06

2.5E+06

S2.0E+06

S1.5E+06

1.0E+06

5.0E+05

0.OE+00


0 0.05 0.1 0.15 0.2

Relative Density


Figure 4-2. Comparison of fracture toughness of graded and homogeneous foams having
same density at the crack tip.

Results for the case of varying unit-cell dimensions are presented in Table 4-3 and

also shown in Figure 4-3. The results again show that the fracture toughness of FGF is

close to that of a homogeneous foam with density same as that at the crack tip of FGF.

Table 4-3. Comparison of the fracture toughness for varying unit-cell dimensions with
constant strut thickness (h=20ym).

Crack length in Relative Fracture Toughness (P,-m )
SetCc Density %
et c (m) c2(m) c3 (m) terms of number Density
of elements at the crack- Graded Foam Homogeneous difference
tip

10 0.0745806 9.62060E+05 9.61172E+05 0.092
20 0.0776305 1.00630E+06 1.00531E+06 0.098
1 -0.15e-6 200e-6 100e-6 100e-6 30 0.0812704 1.05636E+06 1.05533E+06 0.097
40 0.0856898 1.11501E+06 1.11279E+06 0.199
50 0.0911693 1.18440E+06 1.18018E+06 0.356
60 0.0981422 1.26090E+06 1.25952E+06 0.109
70 0.221965 2.05336E+06 2.04649E+06 0.335
60 0.228608 2.15865E+06 2.16343E+06 0.221
50 0.236846 2.30923E+06 2.29987E+06 0.405
2 0.15e-6 50e-6 50e-6 50e-6 40 0.247332 2.47076E+06 2.46204E+06 0.353
30 0.261132 2.66760E+06 2.65985E+06 0.291
20 0.280113 2.92063E+06 2.90924E+06 0.390
10 0.307863 3.24507E+06 3.23901E+06 0.187











Table 4-3. Continued

Crack length in Relative Fracture Toughness (PA,-, )
Set a Density %
Se C0 (m) 2 (m) c3 (m) terms of number Densty
of elements at the crack- Graded Foam Homogeneous difference
tip
50 0.0912307 1.18404E+06 1.18051E+06 0.298
40 0.0982215 1.26352E+06 1.26041E+06 0.246
3 0.15e-6 50e-6 100e-6 100e-6 30 0.107422 1.36134E+06 1.35823E+06 0.228
20 0.120075 1.48715E+06 1.48204E+06 0.344
10 0.138575 1.65340E+06 1.64829E+06 0.310
70 0.0450318 7.45601E+05 7.43076E+05 0.339
4 0.15e-6 200e-6 150e-6 100e-6 50 0.0470359 7.89333E+05 7.85621E+05 0.470
30 0.0495308 8.39136E+05 8.35061E+05 0.486
10 0.0527223 9.11742E+05 8.93837E+05 1.964
70 0.0218062 3.73104E+05 3.72085E+05 0.273
5 0.15e-6 200e-6 200e-6 200e-6 50 0.0230945 3.94813E+05 3.92956E+05 0.470
30 0.0246984 4.19129E+05 4.17251E+05 0.448
10 0.0267500 4.48072E+05 4.46171E+05 0.424


3.50E+06


3.00E+06


2.50E+06


2.00E+06


S1.50E+06


O 1.00E+06


5.00E+05


0.00E+00
0.0(


-*-Set1 (FGF)
Set1 (Homogeneous)
Set2 (FGF)
Set2 (Homogeneous)
S-3 Set3 (FGF)
--- Set3 (Homogeneous)
S--- Set4 (FGF)
SSet4 (Homogeneous)
Set5 (FGF)
Set5 (Homogeneous)

0 0.05 0.10 0.15 0.20 0.25 0.30 0


Relative Density


Figure 4-3. Comparison of fracture toughness of graded and homogeneous foams. The
graded foams have varying unit-cell dimensions, but constant strut cross
section h=20 um.









Study of Local Effect on the Homogeneous Foam under Crack Face Traction

In order to observe and analyze the local effects of the local stresses on the

homogeneous foam, the fracture toughness of the homogeneous foam under crack face

traction (0.5 GPa) is compared with remote loading condition, both uniform traction (0.5

GPa) and displacement (70 /on) on the top edge. For this study, the model in the Figure

4-1 (A) is considered for remote loading condition and Figure 4-1 (B) for crack face

loading condition. The model size is fixed and the crack length is varied in order to

investigate how the local stresses around near crack tip affects the fracture toughness for

various crack lengths. The unit cell size and beam thickness are constant, c=200/nm and

h=20/,u which make the relative density of homogeneous foam equal to 0.028. The

crack length varies from 10% (a/c=50) to 50% (a/c=250) of the plate width.

Table 4-4. Remote loading case uniform displacement on the top edge.
aW
a. 0.1 0.2 0.3 0.4 0.5
(normalized crack length)
ac
c 50 100 150 200 250
(crack length/unit-cell)
Stress Intensity Factor 2.431E+05 2.431E+05 2.431E+05 2.431E+05 2.431E+05
Umax
(Maximum Principal 1.939E+09 1.939E+09 1.939E+09 1.940E+09 1.940E+09
Stress)
mo-x/SIF 7.977E+03 7.974E+03 7.976E+03 7.978E+03 7.980E+03
Fracture Toughness 4.513E+05 4.514E+05 4.513E+05 4.512E+05 4.511E+05

Table 4-5. Remote loading case uniform traction on the top edge.
aW
(normalized crack length) 0.1 0.2 0.3 0.4 0.5
ac
c 50 100 150 200 250
(crack length/unit-cell)
Stress Intensity Factor 1.486E+08 2.988E+08 4.763E+08 6.941E+08 9.776E+08
Umax
(Maximum Principal 1.194E+12 2.405E+12 3.832E+12 5.582E+12 7.858E+12
Stress)
-max /SIF 8.030E+03 8.046E+03 8.044E+03 8.041E+03 8.038E+03
Fracture Toughness 4.483E+05 4.474E+05 4.475E+05 4.477E+05 4.479E+05









As seen on Table 4-4, the stress intensity factor and maximum principal stresses at

the crack tip for homogeneous foam under uniform displacement are almost constant

through various lengths. When the uniform traction is applied on the top edge, the stress

intensity factor and maximum principal stresses at the crack tip increase with increasing

the crack length (Table 4-5). However, the ratio between the maximum principal stress

and the stress intensity factor does not vary much for different crack length. In both

cases, the fracture toughness is independent of crack lengths.

Table 4-6. Stress intensity factor, maximum principal stress and fracture toughness for
various crack lengths under crack surface traction.
aW
a. 0.1 0.2 0.3 0.4 0.5
(normalized crack length)
ac
c 50 100 150 200 250
(crack length/unit-cell)
Stress Intensity Factor 1.486E+08 2.988E+08 4.763E+08 6.941E+08 9.776E+08
'max
(Maximum Principal 1.126E+12 2.337E+12 3.764E+12 5.514E+12 7.791E+12
Stress)
omax /SIF 7.574E+03 7.820E+03 7.902E+03 7.944E+03 7.969E+03
Fracture Toughness 4.753E+05 4.604E+05 4.556E+05 4.532E+05 4.517E+05


Table 4-7. Fracture toughness estimation from remote loading and crack face traction.
ac Remote loading Crack face
(crack length/unit-cell) (uniform traction on the top edge) traction % difference
50 4.48332E+05 4.75294E+05 6.014
100 4.47406E+05 4.60368E+05 2.897
150 4.47514E+05 4.55565E+05 1.799
200 4.47679E+05 4.53177E+05 1.228
250 4.47851E+05 4.51743E+05 0.869


In the case where a pressure is applied along the crack face crack face traction,

the stress intensity factor and the maximum principal stresses at the crack tip increase

with increasing the crack length as occurred in the case of remote loading with uniform

traction. However, the ratio between the maximum principal stress at the crack tip and







48


the stress intensity factor also increases with crack length. That means that the fracture

toughness decreases as the crack length increases. The fracture toughness for crack face

traction is higher for shorter cracks, but converges to the value for remote traction

condition for longer cracks. The deviation of the fracture toughness for various crack

length under crack surface traction condition is presented in Table 4-7 and plotted in

Figure 4-4.


5.00E+05

4.50E+05

< 4.00E+05

S3.50E+05

S3.00E+05

2.50E+05 -- Remote loading (unifrom
traction on the top edge)
o 2.00E+05
-A- Crack face traction
1.50E+05

S1.00E+05

5.00E+04

0.00E+00
0 50 100 150 200 250 300
a/c


Figure 4-4. Fracture toughness estimation from remote loading and crack face traction.

The principle of superposition can be applied to the crack face loading condition in

order to explain the differences in fracture toughness presented in Table 4-7. As we have

seen in the previous section, the current method can accurately estimate the fracture

toughness for remote loading conditions. Thus, stresses acting on the crack face (i.e.,

crack face traction condition) can be replaced with tractions that act on the top edge

(remote loading condition) and an uncracked body subjected to tractions, as illustrated in









Figure 4-5. Since the value of stress intensity factor for uncracked body is zero, the two

loading configurations (remote traction and crack face traction) result in same stress

intensity factor in macro scale sense and shown in Table 4-5 and 4-6.

K() = K b) K(c) = K(b) (since Kc) = 0) (4-1)









a (a) (b) a. (c)


Crack Face Traction Remote Traction Uncracked Body

Figure 4-5. Application of superposition to replace crack face traction with remote
traction.

However, stresses exist at the crack tip in the uncracked body in micro scale sense as

shown below.

2
o(c) = (ro (4-2)


The principle of superposition indicates that the maximum principal stress at the crack tip

under crack surface traction is lower than the maximum principal stress at the crack tip

under remote loading (uniform traction on the top edge) as derived in Eq. (4.3) below:.

o(a) =_(b) Oa (4-3)
max max max (4-)

2
(a) (b) C C and -(a) <(b) (44)
max max o- and max < Umax


The estimated fracture toughness under surface crack traction is higher than the

fracture toughness under uniform traction on the top edge for short crack by current









micro-mechanical model. By employing the principle of superposition, the analytical

solution for the fracture toughness under crack surface traction denoted by Kjs (= K )i

can be found in terms of the fracture toughness under remote loading K (= K:) The

reciprocal of fracture toughness can be obtained by dividing Eq. (4-4) throughout by the

factor (KI, o,) where o, is the ultimate tensile strength of the strut material and K1 is the

stress intensity factor due to the applied stress o-. Then we obtain

2
C
1 1 h2
KS KJ -(4-5)
Kisc KIRC o.K,

The stress intensity factor K, is given by

K, = Yo-, a (4-6)

where oo is the applied stress, a is the crack dimensions, and the form factor Yis a

polynomial in a/Wand depends on geometry and mode of loading (Broek 1978). For

edge cracked model under remote traction (Figure 4-5 (a)) Y is given by

-I aa a 2 4
Y= 1.99-0.41 +18.7 -38.48 +53.85 J (4-7)


Using Eq. (4-6), Eq.(4-5) reduces to

1 1 1
KS Kj h2 (4-8)
_cc


Then the fracture toughness under crack surface traction can be derived as

KS Kje a > 50 or a > 0.1 (4-9)
KIC c W
1-
h2
c









From Eq. (4-9), the fracture toughness under crack surface traction K'S can be

calculated from fracture toughness under remote loading K and compared with the

estimated fracture toughness obtained from the developed micro-mechanical model

(Table 4-8). The local effect caused by local stress was analyzed employing the principle

of superposition. The principal of superposition also indicates that the fracture toughness

under crack surface traction is larger than the fracture toughness under remote loading for

short cracks and tends to converge to the fracture toughness under remote loading for

longer cracks as seen on Figure 4-4.

Table 4-8. Comparison of fracture toughness under crack surface traction calculated
from the superposition method and the micromechanical model.
S Current micro-mechanical
a(m) a/W a/c Analytical solution em ome %diff
model
0.01 0.1 50 4.76554E+05 4.75294E+05 0.264
0.02 0.2 100 4.64162E+05 4.60368E+05 0.817
0.03 0.3 150 4.58684E+05 4.55565E+05 0.680
0.04 0.4 200 4.55260E+05 4.53177E+05 0.458
0.05 0.5 250 4.52874E+05 4.51743E+05 0.250














CHAPTER 5
FRACTURE TOUGHNESS ESTIMATION UNDER THERMAL LOADING

In this Chapter we investigate the foams under thermal loading using the methods

described in Chapter 3. First we investigate the behavior of functionally graded foams

and homogeneous foams under thermal loading. The second order polynomial variation

of temperature is used in order to investigate the behavior of uncracked foams because

the analytical study of a beam under temperature gradient shows that the second order

polynomial variation is the simplest form to create thermal stresses (Appendix). Sankar

and Tzeng (2002) studied thermal stresses in functionally graded beams by using the

Bernoulli-Euler hypothesis. They assumed that the elastic constants of the beam and

temperature vary exponentially through thickness. They found that the thermal stresses

for a given temperature gradient can be reduced when the variation of the elastic

constants are opposite to that of the temperature gradient. In the present work, we study

the difference between fracture toughness under mechanical and thermal loading. The

purpose here is to understand the effects of thermal stress gradients on fracture toughness.

Behavior of Foams under Thermal Loading

The configuration with constant cell size (200 um) with varying the strut thickness

is used to investigate the behavior of uncracked foams under thermal loading. The

investigation was performed in three different cases homogeneous foam, increasing

density and decreasing density foams. The properties of three different models are shown

in Table 5-1. The subscripts "0" and "f' denote the properties at the x=0 and x=width

(0. Im) respectively. The temperature variation was assumed to be of the form









T=75,000x2+7,500x+30 which makes the temperature to increase along the x-axis from

100C to 1,5300C. The temperatures are nodal values in the finite element analysis. The

temperature according to the given second order polynomial variation is assigned at the

vertical set of nodes. The reference temperature in all the models was 20 C. Therefore,

the ATo and ATf are 100C and 1,5100C for all three models.

Table 5-1. Elastic modulus variation of three different models.
Strut thickness (um) Elastic Modulus (GPa)
Modulus ratio
ho hf Eo Ef (E/E)
(E/Eo)
Homogeneous 40 40 8.28 8.28 1
Decreasing material
Decreasing material 40 20 8.28 2.07 0.25
properties
Increasing material 33
40 80 8.28 33.12 4
properties

The thermoelastic constant was assumed to be of form f/(x) = foJ(x) with o =

Eo/106, where Eo is the elastic modulus at x=0. Then, the thermal stresses were

normalized with respect to the thermal stress term f/oATo. The axial stress distributions for

the three cases are shown in Figure5-1 and plotted in Figures 5-1through 5-3. When the

variation of the material properties was in opposite sense to the temperature variation, the

thermal stresses were reduced. On the other hand, when the variation of the material

properties was in the same sense as the temperature variation, the thermal stresses

increased compare to that of homogeneous foam. The maximum and minimum stresses

in different cases are compared in Figure 5-5 by bar charts where the values are shown

for modulus ratio, Ef/Eo.

The investigation confirms that the behavior of functionally graded foams

naturally adopt the favorable design for thermal protection system because the nature of







54



functionally graded foams as load carrying thermal protection system should be such that


the cooler inner layer has high solidity, while the hotter outer layer has low solidity.


2, 7.



A






I~ ~ ~ ~ ~ ~h TD ~ Iic.QU ABaT/t idid .- Uc E =3 15l51l0. Enn ..... Bl'nai "* 30



ii
Z2









B







.. -- -" r" kn
1,I2


Figure 5-1. Thermal stress distribution output from FEA. A) Homogeneous foam. B)
FGF with the material properties and the temperature has opposite type of
distribution. C) FGF with the variation of the material properties and
temperature in a similar manner.































-6

-7

-8
x/width


Figure 5-2. Thermal stress distribution in homogeneous foam.


x/width


Figure 5-3. Thermal stresses distribution in an FGF; the material properties and the
temperature have opposite type of variation, and this reduces the thermal
stresses.








56



5



0
0.25 0.5 0.75


-5


10





-15




-20
x/width



Figure 5-4. Thermal stresses distribution in an FGF; the variation of the material
properties and temperature in a similar manner; and this increases the thermal
stresses.



5



0-



-5



S-10 -- Max. Stress I Min. Stress



-15



-20
0.25 1 4

Modulus Ratio (Ef/E )

Figure 5-5. Maximum and minimum values of normalized thermal stresses.










Results under Thermal Loading

For the analytical solution of model under thermal loading, Euler-Bernoulli beam

theory was used due to its simplification. One of the assumptions for Euler-Bernoulli

beam theory is that the beam should be long and slender (i.e. length >> depth and width).

To compare the finite element results with analytical solution, the beam aspect ratio

(length/width) should be investigated because we have used the model for the aspect ratio

was unity. The width ofuncracked model is 0.05m, and length changed from 0.05 to

0.5m. The temperature gradient is a function of the position. The temperature variation

was assumed to be of the form T(x) = 1- 400 x2 which makes the temperature increases

along the x-axis form 0C to 1 OC. Figure 5-6 shows the thermal stresses developed on

models for different aspect ratios. The model with aspect ratio 10 gives very close

thermal stress distribution as the analytical solution. Thus, we use the model which has

ten times lager length than width in this section. The thermal stress in the models with

various crack length is shown in Figure 5-7.


4.00E+05
*AR-1
~3.00E+05 AR-2
3.00E+05AR 2
AAR-4
2.00E+05 OAR-8
.)K XAR-10 A
1.00E+05 *Analytical

)O.OOE+00
0.0(94:t1 )))2 ()0.0)3 .)4 0. )5
-1.00E+05

-2.00E+05

-3.OOE+05


x (m)


Figure 5-6. Thermal stresses for various aspect ratio models.










4.00E+03

3.50E+03 no-crack
SA O c=0.005m
3.00E+03
A c=0.01m
2.50E+03 X c=0.02m
2E O c=0.025m
a 2.00E+03
X O c=0.03m
| 1.50E+03 + c=0.035m

1.00E+03 O
X
5.00E+02 0-)) A^ X ^O .4)
0.00E+00 .
0.01 A- ^02 "Xx O aO4 5
-5.OOE+02

-1.00E+03

-1.50E+03
x (m)


Figure 5-7. Thermal stresses for various crack lengths.


Table 5-2. Results of the body under temperature gradient form micromechanical model.
a/W
a. 0.1 0.2 0.4 0.5 0.6 0.65
(normalized crack length)
ac
25 50 100 125 150 175
(crack length/unit-cell)
Stress Intensity Factor 1.68E+02 1.70E+02 1.08E+02 6.97E+01 3.46E+01 6.45E+00
0Umax
(Maximum Principal 1.54E+06 1.58E+06 1.03E+06 6.81E+05 3.55E+05 8.82E+04
Stress)
o-m, /SIF 9185.8 9325.3 9572.6 9782.5 10266.7 13673.4
Fracture Toughness 3.91E+05 3.86E+05 3.76E+05 3.68E+05 3.50E+05 2.63E+05

As described in the previous chapter, the fracture toughness under crack surface

traction converges to the fracture toughness under remote traction as crack size increases.

However, we should notice that the negative stress intensity factor exists and the value

increases as the crack length increases. Therefore, the stress intensity factor decreases

with larger crack size, and consequently the fracture toughness decreases with larger

crack size (Table 5-2). The ratio between the maximum principal stress and the stress










intensity factor is compared with that of remote loading condition shown in Figure 5-8.

The fracture toughness for various crack lengths are shown in Figure 5-9. The result of

estimated fracture toughness under thermal loading is similar to that for crack face

loading, except the sign is reversed. This can be explained by using the principle of

superposition as described in chapter 4.


1.80E+04

1.60E+04

1.40E+04

1.20E+04

S1.00E+04

S8.00E+03 -

6.00E+03 -

S4.00E+03 Thermal Loading
0 Remote Loading
2.00E+03

0.00E+00
0 50 100 150 200
a/c (c=200)pm)

Figure 5-8. Comparison the ratio of maximum principal stress and stress intensity factor.

The results obtained in this chapter indicate that the fracture toughness of a cellular

material depends on the stress gradients produced by thermal stresses. This is similar to

the results obtained in the previous chapter where the fracture toughness was different for

crack surface loading. Thus the nominal fracture toughness obtained from remote

loading tests should be corrected appropriately when stress gradients are presented.








60



4.50E+05

4.00E+05

3.50E+05 -

E 3.00E+05

S2.50E+05
Ia

g 2.00E+05

1.50E+05

1.00E+05

5.00E+04

0.00E+00
0 50 100 150 200
a/c (c=200mm)


Figure 5-9. Fracture toughness for various crack lengths.














CHAPTER 6
CONCLUDING REMARKS

Finite element based micromechanical methods have been developed to understand

the fracture behavior of functionally graded foams. The finite element analysis used a

micromechanical model in conjunction with a macromechanical model in order to relate

the stress intensity factor to the stresses in the struts of the foam. The stress intensity

factor of the macromechanical model at the crack tip was evaluated by two different

method energy method and point matching method. In the energy method, J-contour

integral was used, and the stress field ahead of crack tip was used to estimate stress

intensity factor in the point matching method. The maximum principal stress at the crack

tip was evaluated from the force and moment resultants obtained from the micro-model.

Then, fracture toughness was estimated by relating the stress intensity factor and the

maximum principal stresses at the crack tip. In addition, convergence tests for both

macromechanical and micromechanical models analyses were conducted.

First we investigated the fracture behavior of functionally graded foam under

uniform displacement remote mechanical loading condition in order to demonstrate the

feature of the current method. Then, the method is extended to another remote

mechanical loading uniform traction. Then, the results of remote loading conditions for

graded foams are compared with the result for homogeneous foam. The fracture

toughness was obtained for various crack positions and lengths within the functionally

graded foam. Then the relationship between the fracture toughness of foams and the local

density at the crack tip was studied. It was found that the fracture toughness of









functionally graded foam is approximately the same as that of homogeneous foam with

the same density at the crack tip.

We also investigated the effects caused by the local stresses. In order to observe

and analyze the local effects of the local stresses on the homogeneous foam, the fracture

toughness of the homogeneous foam under crack face traction is compared with remote

loading conditions, both uniform traction and displacement on top edge. The relationship

between the stress intensity factor and the maximum principal stress was compared. It

was found that the fracture toughness under remote loading condition is independent of

the crack size. However, the ratio between the maximum principal stress at the crack tip

and the stress intensity factor under crack face traction increased with crack length.

Thus, the fracture toughness for crack face traction is higher for shorter cracks, and

converges to the value for remote traction condition as the crack length increases. It is

found that the principle of superposition can be used to adjust the local effect caused by

the differences in maximum principal stresses under crack face loading condition. A

correction factor in terms of crack length is proposed to determine the fracture toughness

of short cracks under crack surface loading.

From the thermal protection system point of view the behavior of functionally

graded foam under thermal loading was investigated. When the variation of the material

properties is in opposite sense to the temperature variation, the thermal stresses is

reduced. On the other hand, when the variation of the material properties is in the same

sense as the temperature variation, the thermal stress increases compared to that in

homogeneous foam. The result of estimated fracture toughness under thermal loading is

similar to that for crack face loading.






63


The present dissertation demonstrates the use of finite based micromechanical

model to predict the fracture toughness of functionally graded foams by using simple

micro structures. To achieve more accurate prediction, we need to focus on a more

realistic cell configuration, which captures the complexity of foam and predicts more

accurately its mechanical property changes, such as relative density and modulus in

functionally graded foam. The methods discussed here will help in understanding the

usefulness of functionally graded foams in the thermal protection systems of future space

vehicles.
















APPENDIX
ANALYTICAL SOLUTION FOR BEAM MODEL UNDER THERMAL LOADING






zy 2h

b x, u
L


y, v


Figure A-1. A beam of rectangular cross section with no restraint.




If we assume the both ends are perfectly clamped, the thermal stress, -, is defined

as

a7 (y) =-aEAT(y) (A-1)

Due to constraints at both ends, the thermal stress prevents extension and bending

of the beam and producing internal force, P and bending moment, M


h
P = abdy
-h

h
M = f JTyb dy


(A-2)



(A-3)


If the beam has no restraints against extension, the internal force, P must be

eliminated by a virtual force, pT









h
PT = P= aEAT(y)bdy (A-4)
-h

If the ends are free to rotate and no external moment applied, the bending moment

M at the ends must be eliminated by a virtual bending moment M' at the ends

h
M = -M = aEAT(y)ybdy (A-5)
-h

The thermal stress corresponding to the virtual force, pT is

PT 1h h
S= = aEAT(y)b dy aEAT(y) dy (A-6)
'p AEA= dy
-h -h

A = (2h)* b = 2hb

and the thermal stress corresponding to the virtual bending moment, M' is


T= aT EAT(y)yb dy = aEAT(y)ydy (A-7)
-h -h


Sb(2h)3 2bh3
12 3


Therefore, the thermal stress 'x in the beam with no constraints at both ends is

given by

h h
o (y)= -aEAT(AT(y) dy + y y aEAT(y)ydy (A-8)
2h 2k3
-h -h

h h
(y) = aE -AT(y)+ AT(y)dy+ 3 ArT(y)ydy (A-9)
-h 2h






66


Example 1) Constant, AT = CO

h
fATMdy =[Cyh = 2Ch
-h

h h
fATy dy =Co2 =0
-h 2 h

Therefore,


S3y h
ax (y) = aE -AT +- Tdy+ ATy dy



= aE -C+ +- (2Coh)
S 2h

=0

Example 2) Linear variation, AT(y) = CO + Cly


hAT(y)dy= Coy+ Cy2 h = 2Coh

h1 2

AT(y)ydy= -Coy2 + Cy3 =-Ch3
-h L2 3 -h 3
-h -

Therefore,


S 1 + 3y
x(y) = aE -AT(y)+ 2h J AT(y)dy+ 2 J T(y)ydy]
-h -h


= aE -(Co +Cy) + (2Coh) + (2Ch3)
2h 2h 3








Example 3) Quadratic variation, AT(y) = Co + Cy+C2y2

h1 1
-h
-h L 2 3 J_


=2Coh+ 2C2h3
3


h
fAT(y)ydy=
-h


SCoy2 +- Cy3
2 3


+1C2y4
4 -h


Therefore,


I AT(y) dy +
-h 2h3
-h


(CO +Cly+C2y2) 2Coh2C2h3 3y 2Ch3
2h 3 2h3 3


aE IC2h2
23


aE -h2
[3


=2Clh3
3
















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BIOGRAPHICAL SKETCH

Seon-Jae Lee was born in Seoul, Korea, in 1969. He studied Physics in Kyoung-

Won University for three years and entered Department of Aerospace Engineering at

Royal Melbourne Institute of Technology, Melbourne, Australia. He transferred to

Embry Riddle Aeronautical University at Daytona Beach, Florida, and received his

Bachelor of Science in Aerospace Engineering in April 1999. From there he completed

his graduate study of a Master of Science in Aerospace Engineering in May 2002, with

research in the area of vibration analysis of composite box beam. In August 2002, he

joined the Advanced Composites Center at the Department of Mechanical and Aerospace

Engineering, University of Florida, Gainesville, Florida, for his Ph.D. degree. After

completion of his Ph.D. degree, Seon-Jae will begin work at Samsung Techwin in Korea

to contribute his efforts to the development of precise machine.