UFDC Home  myUFDC Home  Help 



Full Text  
MICROMECHANICAL MODEL FOR PREDICTING THE FRACTURE TOUGHNESS OF FUNCTIONALLY GRADED FOAMS By SEONJAE 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 SEONJAE LEE To my parents and to my wife JinSook 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 familychurch 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 Macromodel..........................................................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 21 M material properties of the Zoltex carbon fiber ..................................................... 21 31 Comparison between two methods. .............................................. ............... 35 41 Fracture toughness of graded and uniform foams. The unitcell 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 42 Fracture toughness of graded and uniform foams. The unitcell dimension is kept constant but the crack length and the strut thickness are varied (c=200 Pm, ho=40pm and a=+ 200x 106) ................................................................ ............... 43 43 Comparison of the fracture toughness for varying unitcell dimensions with constant strut thickness (h=20/1m ) ...................... ..... ...................... ........ ....... 44 44 Remote loading case uniform displacement on the top edge...............................46 45 Remote loading case uniform traction on the top edge ........................................46 46 Stress intensity factor, maximum principal stress and fracture toughness for various crack lengths under crack surface traction. ............................................47 47 Fracture toughness estimation from remote loading and crack face traction...........47 48 Comparison of fracture toughness under crack surface traction calculated from the superposition method and the micromechanical model. ...................................51 51 Elastic modulus variation of three different models. ...............................................53 52 Results of the body under temperature gradient form micromechanical model. .....58 LIST OF FIGURES Figure pge 11 Thermal protection system in Space Shuttle. A) Temperature variation during reentry. B) Location of different m materials. ........................................ ..................6 12 Schem atic diagram of attaching the tiles......................................... ............... 7 13 SEM images of A) low density carbon foam and B) high density carbon foam C) m metallic foam ................................................. ........................... 9 21 Open cell model with rectangular parallelepiped unit cell...................................17 22 Micro and Macrostresses in opencell foam............ ........ ...................18 23 Flexural deformation of struts under shear stresses. ..............................................18 24 Example of variation of elastic modulus and relative density for constant cell 6 length c=200/m, ho=261m and a=200x 106.......... ............ ................. ..22 25 Boundary conditions for uncracked plate under uniform extension.....................23 26 Comparison of stresses (oyy) obtained using the macro and micro models in graded foam with constant cell size but varying strut cross section. .....................24 27 Comparison of stresses (oyy) obtained using the macro and micro models in graded foam with constant strut size but varying cell dimension ..........................25 31 Schematic description of both macro and micromodel. ......................................27 32 A typical finite element macrom odel ..................................... ........ ............... 29 33 Example of discrete elastic modulus for macromodel with tenregions ...............30 34 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 35 JIntegral for various contours in a macromodel containing 100x50 elements and the contour numbers increase away from the crack tip. ...................................33 36 Stress intensity factor from the stresses normal to the crack plane........................34 37 Stress normal to the crack plane ............... ........................................ 35 38 Stress distribution of the functionally graded foams..................... ............... 36 39 Variation of energy release rate at the crack tip with various size macromodels...37 310 Embedded beam element (micro model) in twodimensional eightnode solid m odel (M acro m odel) ....................... ...... ............ ................... .. ......38 311 Force and moment resultants in struts modeled as beams. .....................................39 312 Variation of fracture toughness with the size of micromodels ...........................39 41 Edgecracked model under A) uniform traction or displacement loading and B) crack surface traction. ......................................... ................. .. ...... 42 42 Comparison of fracture toughness of graded and homogeneous foams having sam e density at the crack tip. .............................................................................. 44 43 Comparison of fracture toughness of graded and homogeneous foams. The graded foams have varying unitcell dimensions, but constant strut cross section h = 2 0 u m ...................................... .....................................................4 5 44 Fracture toughness estimation from remote loading and crack face traction...........48 45 Application of superposition to replace crack face traction with remote traction....49 51 Thermal stress distribution output from FEA. ........... ................................ 54 52 Thermal stress distribution in homogeneous foam. ............................................55 53 Thermal stresses distribution in an FGF; the material properties and the temperature have opposite type of variation, and this reduces the thermal stresses ............. ..... .... ........... ............... ...........................55 54 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 55 Maximum and minimum values of normalized thermal stresses...........................56 56 Thermal stresses for various aspect ratio models..................................................57 57 Thermal stresses for various crack lengths.................................... ............... 58 58 Comparison the ratio of maximum principal stress and stress intensity factor........59 59 Fracture toughness for various crack lengths. ................... ......................... 60 A1 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 SeonJae 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 shuttlethe 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 loadcarrying 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 X33 program. The X33 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 X33 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 singlestagetoorbit (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 airplanestyle lifting body used in the space shuttle system, an Apollolike 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 nonmetallic material systems. They provided the guidance of emerging materials with application in order to achieve NASA's longterm 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 reentry 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 reentry process creates high surface temperature which is well above the melting point of aluminum (660 C). Thus, an effective insulator was needed. A silicabased insulation material was decided for the heatresistant tiles and other coverings to protect the Shuttle's airframe. Figure 11 shows seven different materials which cover the external surface of the Space Shuttle according to the temperature variation during the reentry. 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 CarbonCarbon (RCC). There are two main types of tiles, referred to as Lowtemperature Reusable Surface Insulation (LRSI) and Hightemperature 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 11. Thermal protection system in Space Shuttle. A) Temperature variation during reentry. 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 12. 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 12. 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 hightemperature metal. Metallic TPS is considered as a muchneeded alternative to the ceramicbased brittle tile and thermalblanket 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 onorbitdebris 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 microballoons of varying sizes in a matrix medium (Madhusudhana et al., 2004). The porous structures of carbon and metallic foams are depicted in Figure 13. 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 13. SEM images of A) low density carbon foam and B) high density carbon foam C) metallic foam. Foams can be categorized as opencell and closedcell foam. In opencell 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 closedcell foam (Gibson & Ashby, 2001). In this study, from the thermal management application point of view only the opencell 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 unitcell 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 nonhomogeneous 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 nonhomogenous materials in order to evaluate its integrity. Atkinson and List (1978) studied the crack propagation for nonhomogenous materials subjected to mechanical loads assuming an exponential spatial variation of the elastic modulus. Gerasoulis and Srivastav (1980) studied a Griffth crack problem for nonhomogeneous materials using integral equation formulations. Delale and Erdogan (1983), Eischen (1987), Jin and Noda (1994) and Erdogan (1995) showed that the nature of the inversesquarerootsingularity 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 inversesquareroot 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 nonhomogeneous 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 closedform 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 closedform solutions. Yang and Shih (1994) obtained an approximate solution for a semiinfinite crack in an interlayer between two dissimilar materials using a known bimaterial solution. Gu and Asaro (1997) obtained the complete solution of semiinfinite 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 cracktip field for FGM and studied the effect of non homogeneity in numerical computation of the Jintegral. They concluded that the conventional Jintegral 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 pathindependent integral. Weichen (2003) constructed another version of pathindependent 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, Jintegral and a modified path independent integral. They evaluated the Jintegral and a modified Jintegral 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 abovedescribed 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 nonhomogeneous continuum, or as a frame consisting of beam elements. The former model will be referred to as the macromodel and the latter as the micromodel. We require both models for the simulation of crack propagation in graded foams. The region surrounding the crack tip is modeled using the micromodel, where as the region away from the crack tip uses the macromodel. The micromechanical model is treated as an embedded model around crack tip. The macromodel 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 (opencell) is presented, and then the method is extended to functionally graded foams. Continuum Properties of Homogeneous Foam Most of the opencell 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 unitcell dimensions. In their model they assumed that the strut has a square cross section h x h and the unitcell is a cube. In the present approach, the general case is considered wherein the unitcell is a rectangular parallelepiped of dimensions c, x c2 x c, as shown in Figure 21. 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 unitcell. Then, the relative density can be expressed as a function of the dimensions of unitcell and the strut thickness as shown below: p*mv (c, + c2 3)h2 2h3 (21) A PA C1C2C3 where m is the mass and Vis the volume of unitcell. Figure 21. 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 22. The equivalent force on the strut caused by the stresses can be written as F = (cq x c3) In microscale sense, the force F causes stresses a, in sectional area h2 (Figure 22). 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 (22) Sh2 h2 Es h2Es ) where Es is elastic modulus of strut. Therefore, elastic modulus of foam E* can be derived from Eq. (22) 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 (23) Figure 22. Micro and Macrostresses in opencell 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 unitcell dimensions, strut cross sectional dimensions and the strut elastic modulus. When a shear stress is applied, struts are deformed as shown in Figure 23. F1 C2/2 6F2 ig IN Curvature=O F2 C2/2 cC/2 _2 //2 Figure 23. Flexural deformation of struts under shear stresses. Figure 23. 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 = (24) 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 (25) 24EJ 24EJ The shear strain y12 can be derived as 232 251 2c,2 + 2c21 (26) 1J+  (26) C2 C1 C1C2 Using Eq. (25), the shear strain can be written as, (CC2 + C)F2 712 12= (27) 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 21. Table 21. 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 unitcell and the strut thickness. Therefore, three different cases can be considered. The first case is the one where the dimensions of the unitcell remain constant while the strut thickness varies along the xaxis. 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 (211) 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 24 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) (212) ps c c where the unitcell 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 )(213) 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 24. 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 unitcell dimensions vary, we can consider the case where h, c2 and c3 are constants, but cl varies as cI+l = c1 + p (214) 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 (22) through (29). 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 macroand micromodels. A simple mechanic problem was solved using both uncracked macro and micromodels. A uniform displacement (70um) was applied along the upper edge of a rectangular plate using the macromodel, which consists of two dimensional plane stress elements (eight nodes biquadratic, reduced integration element). The elastic constants of the nonhomogeneous material varied as given by Eq. (213). In the finite element model the elastic constants within each element were considered constant. The boundary conditions are depicted in Figure 25. Uniform Displacement uO4, uy=0 u=O, uy=0 Figure 25. Boundary conditions for uncracked plate under uniform extension. The right lower corer was fixed to prevent the rigid body motion. The resulting displacements along the boundary of a micromodel embedded in the macromodel were applied as the boundary displacements of the micromodel by using the threepoint interpolation. For the micromodel, each strut was modeled as an EulerBeroulli beam element with two nodes and three integration points. In order to verify the validity of properties used in the macromodel, the stresses in both models are compared. In the case of macromodel the stresses are obtained as the finite element analysis output. The outputs in micromodel 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 25 represents the micromodel. 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 macromodel 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 106. The region corresponding to the micromodel in the macromodel consists of 15x5 plane stress elements. The micromodel uses 2,250 beam elements. 14 Stresses (MPa)I 2 10  8  macrom odel 4 microm 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 26. 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 micromodels are compared in Figure 26. The maximum difference in stresses between the macro and micromodels 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 macromodel and micromodel are compared in Figure 27. Stresses (MPa) 6m macromodel  micromodel 0 0000 0005 0010 0015 0020 0025 0030 x (m) Figure 27. 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 microstress 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 micromodels were performed. Overview of Micromechanical Model The functionally graded foam, a cellular material is nonhomogeneous in the macroscale. That is, the microstructure is graded and the foam is treated as a functionally graded material in macroscale. The foam can be modeled either as a nonhomogeneous continuum, or as a frame consisting of beam elements to model the struts. The former model (continuum model) will be referred to as the macromodel and the latter (frame model) as the micromodel. In the finite element analysis, solid elements are used in the macromodel and beam elements in the micromodel. 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 eightnode quadrilateral elements were used to discretize the macromodel 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 micromodel and by removing the zero boundary condition along the intended crack surface in macromodel. A portion of the foam surrounding the crack tip is considered as the micromodel (see Figure 31). Y ttttttttttttttttttttt c Macromodel < h Micromodel Crack j Width A B Figure 31. Schematic description of both macro and micromodel. A) Macromodel consists of plane 8node solid elements. The region in the middle with grids indicates the portion used in the micromodel. B) The micromodel consists of frame elements to model the individual struts. The displacements from the macromodel are applied as boundary conditions in the micromodel. The dimensions of the micromodel 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 micromodel. 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 macroscale 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 macroscale sense. For this purpose we turn to the macro model as shown in Figure 31 (A). In the macromodel 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 macromodel. The displacements of points along the boundary of the micromodel are obtained from the finite element analysis of the macro model and applied to the boundary of the micromodel 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 micromodel as htf p MtUip 2 2 Ftp otp = (31) Itp A tp where cp is maximum principal stress at the cracktip. Ftp, Mtp are force and moment resultant. At is crosssectional 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 cracktip struts to fail in a micro scale sense and cause the crack to propagate in a macroscale 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 = (32) Kic ou Macromodel In macromodel, the conventional two dimensional isoparametric planestress elements are used. The problem geometry is shown in Figure 31 (A). The material gradient is in the xdirection. A preprocessor program was coded using MATLAB, with the parameters such as unitcell size, strut thickness, crack size, a and /f (defined in previous chapter) in order to generate a rectangular mesh with eightnode isoparametric elements with twodegree of freedom at each node and to impose the material gradient in the macromodel. Crack tip A B Figure 32. A typical finite element macromodel. A) constant unitcell size with varying strut thickness. B) varying(decreasing) unitcell 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, xaxis. A zero displacement boundary condition in ydirection is employed in the lower edge to account for symmetric. Figure 32 shows a typical finite element model which consists of 5,000 eightnode 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 macromodel requires imposing the required the variation of material properties in xdirection. The material properties are graded by either changing the thickness of struts or changing the dimensions of the unitcell described as before. Relative density, elastic modulus and shear modulus vary along the xaxis 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 33. Example of discrete elastic modulus for macromodel with tenregions. When functionally graded foam is modeled as a homogeneous solid (macromodel), 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 33 shows the discrete elastic modulus for the tenregion 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 34. 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 unitcell lengthcase. The first set is controlling the crack length while the variation of material properties remains same. The other set is shown in Figure 34. 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 macromodel and 0.015m by 0.005m for micro model). For the case where the unitcell dimensions change, the number of elements both in macromodel (100x50 elements) and micromodel (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 (modemixity=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 Jcontour 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 mixedmode stress intensity factor components. In this paper, the cracktip stress field and Jcontour integral are used to find and verify the stress intensity factor for the point matching and the energy method respectively. In energy method, the Jcontour 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 nonhomogeneous material. Therefore, J contour integral is expected to vary with contour numbers as shown in Figure 35. 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 = 1.21193E05x 1.27727E03x 6.32919E02x2 2.05304E+00x+ 7.61841E+02 R2 9.99992E01 100 0 1 6 11 16 21 26 31 36 41 46 51 Contour Number Figure 35. JIntegral for various contours in a macromodel 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). Jcontour integral as r*0 is obtained by fitting a fourth order polynomial to the output values of Jcontour integral. The limiting value of J contour integral can be evaluated numerically as the intercept of the polynomial curve at yaxis. The value of Jintegral for a contour very close to the cracktip 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 cracktip (Gu and Asaro, 1999). Conceptually, energy release rate, G can be found by the variation of Jcontour integrals as shown in Figure 35. The stress intensity factorKI of a functionally graded foam (twodimensional orthotropic) can be found from G using the relation (Sih & Liebowbitz, 1968). 2 2 1M122 )2 22 2 2a12 + t66 G = nK  + (33) ,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) (34) 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 36 shows the one of the example plot of uV7r versus distance from the crack tip. A 4th order polynomial regression is also shown in Figure 36. The yintercept 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 9.99997E01 0.00E+00 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 distance from crack tip, r (m ) Figure 36. 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 cracktip (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. (35). KI yy 2 (35) Figure 37 shows the stress normal to the crack plane versus distance form the crack tip. Where the squareroot singularity dominated zone, Eq. (35) is valid while stress far from the crack tip is governed by the remote boundary conditions. Table 31. 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 cracktip KT From JIntegral SFrom Integral 1.2238E+06 1.0699E+06 9.2893E+05 8.4886E+05 7.2691E+05 (Pam") K From cracktip 1.2262E+06 1.0715E+06 9.3360E+05 8.0605E+05 6.8363E+05 stress field (Pam") % 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 37. Stress normal to the crack plane. 36 The stress intensity factor from JIntegral and stressmatching were compared for various cases in Table 31. The maximum difference between the two methods is less than 6%. Convergence Analysis for Macromodel 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 38. 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 Faii + . ~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 38. 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 39. For 100x50 elements model, the variation of Jcontour 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 39. Variation of energy release rate at the crack tip with various size macro models Micromodel A portion of macromechanical model (ABCD) is taken and used for micromodel as shown in Figure 31 (A). As the 100x50 elements (100 regions) for macromodel and constant cell length (200 um) for micromodel are used, one macromodel element can be replaced by 60 beam elements for micromodel as shown in Figure 310. The displacements along the boundaries of micromodel are determined by using three points interpolation. The corresponding three points can be obtained from the previously described macromodel analysis. For instance, displacements in the xdirection for each beam element on micromodel 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) MacroModel Solid 2D clement SMicrModel b Y ( 0' Beam element  Sy= x, u(x,y) Figue 310. Embedded beam element (micro model) in twodimensional eightnode solid model (Macro model). In micromodel, twonode beam elements are used to represent the foam ligaments/struts. After, the displacements along the boundaries of micromodel, the maximum principal stress stresses at the crack tip otp can be calculated from the results for force and moment resultants obtained from the micromodel as hhp Op 2 F+ (37) Inp Atp The fracture toughness of the foam is defined as the stress intensity factor that will cause the cracktip 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 311. 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, (38) 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 312. Variation of fracture toughness with the size of micromodels. The convergence analysis is conducted to evaluate the variation of fracture toughness with various sizes of micromodel, 3 x 1 macromodel (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 312. 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 3,850model 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 41 (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, xaxis. The upper edge is loaded by uniform displacement, 70 prm. A zero displacement boundary condition in the ydirection 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. 211). f is used for the case where the cell dimension is varying in xdirection while the strut dimensions are kept constant (Eq. 214). Y O00o orVo Y 00 > x > a a W=0.1m W=0.1m A B Figure 41. Edgecracked model under A) uniform traction or displacement loading and B) crack surface traction. First, we investigate the case wherein the graded foam has constant unitcell length (c=200pm) and the density is varied by changing the strut cross sectional dimensions. Both cases, increasing and decreasing densities along the xaxis, are considered. Table 4 1 shows the results form the case which the unitcell 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 42. Table 41. Fracture toughness of graded and uniform foams. The unitcell 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 (Pam2) Relative Density at the cracktip Decreasing Increasing Uniform at the cracktip 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 42. Fracture toughness of graded and uniform foams. The unitcell 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 (Pam ) Relative Density crack length the cracktip Decreasing Increasing Uniform at the cracktip (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 41, Table 42 and Figure 42, the results from the present analysis for FGF are very close to those of homogeneous foam. However we see an interesting trend in Figure 410. 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 42. Comparison of fracture toughness of graded and homogeneous foams having same density at the crack tip. Results for the case of varying unitcell dimensions are presented in Table 43 and also shown in Figure 43. 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 43. Comparison of the fracture toughness for varying unitcell 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 0.15e6 200e6 100e6 100e6 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 0.15e6 50e6 50e6 50e6 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 43. 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 0.15e6 50e6 100e6 100e6 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 0.15e6 200e6 150e6 100e6 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 0.15e6 200e6 200e6 200e6 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) S3 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 43. Comparison of fracture toughness of graded and homogeneous foams. The graded foams have varying unitcell 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 41 (A) is considered for remote loading condition and Figure 41 (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 44. 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/unitcell) 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) mox/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 45. 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/unitcell) 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 44, 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 45). 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 46. 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/unitcell) 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 47. Fracture toughness estimation from remote loading and crack face traction. ac Remote loading Crack face (crack length/unitcell) (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 47 and plotted in Figure 44. 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 44. 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 47. 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 45. 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 45 and 46. K() = K b) K(c) = K(b) (since Kc) = 0) (41) a (a) (b) a. (c) Crack Face Traction Remote Traction Uncracked Body Figure 45. 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 (42) 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 (43) 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 micromechanical 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. (44) 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 (45) Kisc KIRC o.K, The stress intensity factor K, is given by K, = Yo, a (46) 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 45 (a)) Y is given by I aa a 2 4 Y= 1.990.41 +18.7 38.48 +53.85 J (47) Using Eq. (46), Eq.(45) reduces to 1 1 1 KS Kj h2 (48) _cc Then the fracture toughness under crack surface traction can be derived as KS Kje a > 50 or a > 0.1 (49) KIC c W 1 h2 c From Eq. (49), 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 micromechanical model (Table 48). 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 44. Table 48. Comparison of fracture toughness under crack surface traction calculated from the superposition method and the micromechanical model. S Current micromechanical 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 BernoulliEuler 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 51. 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 xaxis 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 51. 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 Figure51 and plotted in Figures 51through 53. 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 55 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 51. 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 52. Thermal stress distribution in homogeneous foam. x/width Figure 53. 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 54. 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 S10  Max. Stress I Min. Stress 15 20 0.25 1 4 Modulus Ratio (Ef/E ) Figure 55. Maximum and minimum values of normalized thermal stresses. Results under Thermal Loading For the analytical solution of model under thermal loading, EulerBernoulli beam theory was used due to its simplification. One of the assumptions for EulerBernoulli 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 xaxis form 0C to 1 OC. Figure 56 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 57. 4.00E+05 *AR1 ~3.00E+05 AR2 3.00E+05AR 2 AAR4 2.00E+05 OAR8 .)K XAR10 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 56. Thermal stresses for various aspect ratio models. 4.00E+03 3.50E+03 nocrack 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 57. Thermal stresses for various crack lengths. Table 52. 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/unitcell) 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) om, /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 52). The ratio between the maximum principal stress and the stress intensity factor is compared with that of remote loading condition shown in Figure 58. The fracture toughness for various crack lengths are shown in Figure 59. 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 58. 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 59. 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, Jcontour 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 micromodel. 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 A1. 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) (A1) 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 (A2) (A3) 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 (A4) 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 (A5) h The thermal stress corresponding to the virtual force, pT is PT 1h h S= = aEAT(y)b dy aEAT(y) dy (A6) '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 (A7) 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 (A8) 2h 2k3 h h h h (y) = aE AT(y)+ AT(y)dy+ 3 ArT(y)ydy (A9) 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 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 Calculation of Stress Intensity Factors in Functionally Graded Materials. International Journal of Fracture, Vol. 104, pp. 131143. Ashby, F. M., Evans, A., Fleck, A. N., Gibson, J. L., Hutchinson, W. J. and Wadly, N. G., 2000, Metal Foams: A Design Guide, ButterworthHeinemann Pub., Massachusetts. Atkinson, C., List, R.D., 1978, Steady State Crack Propagation into Media with Spatially Varying Elastic Properties. International Journal of Engineering Science, Vol. 16, pp. 717730. Blosser, M. L., October 1996, Development of Metallic Thermal Protection Systems for the Reusable Launch Vehicle. NASA TM110296. Blosser, M. L., May 2000, Advanced Metallic Thermal Protection Systems for the Reusable Launch Vehicle. Ph.D. Dissertation, 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 1417, Reno, Nevada, AIAA 20020504. 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, Measuring and Failure Analysis. Acta Mater, Vol. 47, No. 1, pp. 259268. Choi, S. and Sankar, B.V., 2003, Fracture Toughness of Carbon Foam. Journal of Composite Materials, Vol. 37, No. 23, pp. 21012116. Choi, S. and Sankar, B. V., 2005, A Micromechanical Method to Predict the Fracture Toughness of cellular Materials. International Journal of Solids & Structures, Vol. 42/56, pp. 17971817. 69 Delale, F. and Erdorgan, F., 1983, The Crack Problem for Nonhomogeneous Plane. Journal of Applied Mechanics, Vol.50, pp. 609614. Erdorgan, F., 1995, Fracture Mechanics of Functionally Graded Materials. Composites Engineering, Vol. 5, pp. 753770. 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. 449 456. Eichen, J. W., 1987, Fracture of Nonhomogeneous Materials. International Journal of Fracture, Vol. 34, pp. 322. Gerasoulis, A. and Srivastav, R. P., 1980, International 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 Airbreathing Vehicles. NASA TM21152. Gibson, R.E., 1967, Some results concerning displacements and stresses in a nonhomogeneous elastic half space. Geotechnique, Vol.17, pp. 5867. Gibson, L.J., Ashby, M.F., 2001. Cellular Solids: Structure and Properties. Second Edition, Cambridge University Press, Cambridge, United Kingdom. Gu, P. and Asaro, R. J., 1997, Cracks in Functionally Graded Materials. International Journal of Solids and Structures, Vol. 34, pp. 17. Gu, P., Dao, M. and Asaro, R. J., 1999, Simplified Method for Calculating the Crack Tip Field of Functionally Graded Materials Using the Domain Integral. Journal of Applied Mechnics, Vol. 66, pp. 101108. Harris, C. E., Shuart, M. J., and Gray, H. R. May 2002, A Survey of Emerging Materials for Revolutionary Aerospace Vehicle Structures and Propulsion System. NASA TM211664. Hibbitt, Karlson, & Sorensen, 2002, ABAQUS/Standard 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. 12211235. Jin, Z. H. and Noda, N., 1994, Cracktip Singular Fields in Nonhomogeneous Materials. Journal of Applied Mechanics, Vol. 61, pp. 738740. Jordan, W., 2005, Space shuttle. pdf, http://www2.1atech.edu/j ordan/Nova/ceramics/SpaceShuttle.pdf, Last accessed February 14th, 2005. Kim, J. and Paulino, G. H., 2002, Isoparametric Graded Finite Element for Nonhomogeneous Isotropic and Orthotropic Materials. Journal of Applied Mechanics, Vol. 69, pp. 502514. Kuroda, Y., Kusaka, K., Moro, A. and Togawa, M., 1993, Evaluation tests of ZrO/Ni Functionally Gradient Materials for Regeneratively cooled Thrust Engine Applications. Ceramic Transactions, Vol. 34, pp. 289296. Madhusudhana, K.S., Kitey, R. and Tippur, H.V., 2004, Dynamic Fracture Behavior of Model Sandwich Structures with Functionally Graded Core, Proceedings of the 22nd Southeastern Conference in Theoretical and Applied Mechanics (SECTAM), August 1517, Center for Advanced Materials, Tuskegee University, Tuskegee, Alabama, pp. 362371. Marur, P. R. and Tippur, H. V., 2000, Numerical analysis of Cracktip Fields in Functionally Graded Materials with a Crack Normal to the Elastic Gradient. International Journal of Solids and Structures, Vol.37, pp. 53535370. 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 2426, Huntsville, AL, AIAA pp. 964245. Poteet, C. C. and Blosser, M. L., January 2002, Improving Metallic Thermal protection System Hypervelocity Impact Resistance through Numerical Simulations. Journal of Spacecraft and Rockets, Vol. 41, No.2, pp. 221231. Ravichandran, K. S. and Barsoum, I., 2003, Determination of Stress Intensity Factor Solution for Cracks in Finitewidth Functionally Graded Materias. International Journal of Fracture, Vol. 121, pp. 183203. Rousseau, C. E. and Tippur, H.V., 2002, Evaluation of Crack Tip Fields and Stress Intensity Factors in Functionally Graded Elastic Materials: Cracks Parallel to Elastic Gradient. International Journal of Fracture, Vol. 114, pp. 87111. Sanford, R. J., 2003, Principles of Fracture Mechanics. Pearson Education, Inc., Upper Saddle River, New Jersey. Sankar, B. V. and Tzeng, J. T., 2002, Thermal Stresses in Functionally Graded Beams, AIAA Journal, Vol. 40, No. 6, pp. 12281232. Santare, M. H. and Lambros, J., 2000, Use of Graded Finite Elements to Model the Behavior of Nonhomogeneous Materials, Journal of Applied Mechanics, Vol. 67, pp. 819822. 71 Sih, G. C. and Liebowbitz, H., 1968, Mathematical Theories of Brittle Fracture. Vol. 2, Academic Press, New York. Weichen, S., 2003, Pathindependent Integrals and Crack Extension Force for Functionally Graded Materials. International Journal of Fracture, Vol. 119, pp. 83 89. Yang, W. and Shih, C. F., 1994, Fracture along an Interlayer. International Journal of Solids and Structures, Vol. 31, pp. 9851002. Zhu, H., 2004, Design of Metallic Foams as Insulation in Thermal Protection Systems. Ph. D. Dissertation, University of Florida. BIOGRAPHICAL SKETCH SeonJae 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, SeonJae will begin work at Samsung Techwin in Korea to contribute his efforts to the development of precise machine. 