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MICROMECHANICAL ANALYSIS AND DESIGN OF AN INTEGRATED THERMAL PROTECTION SYSTEM FOR FUTURE SPACE VEHICLES By OSCAR MARTINEZ 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 2007 2007 Oscar Martinez To my wife, Coleen, and son, Josh. I would not have been able to complete this without their love and support. ACKNOWLEDGMENTS I would like to thank my wife, Coleen Martinez, for all of her support throughout my research experience as well as through my graduate career. I would like to thank my son, Josh Martinez, for reminding me why I completed this dissertation. I would like to thank my mother, Martha Mendez, my father, Oscar Martinez, my stepfather, Carlos Mendez, and my stepmother, Terry Martinez for their support. Finally, I would like to thank Dr. Bhavani Sankar, Dr. Raphael Haftka, Dr. Peter Ifju, and Dr. David Bloomquist for participating and evaluating my research work. Dr. Sankar and Dr. Haftka have served as my advisor's, mentor's, guide's, and professor's and I could not have completed this dissertation without their guidance, support, and dedication. I am also thankful to Dr. Max Blosser (NASA Langley) for his inputs and suggestions in my research work, which kept us on track with the expectations of NASA. I am thankful to CUIP and SEAGEP, for their financial support through my graduate career. TABLE OF CONTENTS page ACKN OW LED GM EN TS ....................................................................................... 4 L IST O F T A B L E S .................................................................. ....................................... . 8 L IST O F F IG U R E S .................................................................. .................................... . 9 LIST OF ABBREV IA TION S............................................................... ........................... 13 LIST OF SYMBOLS ....................................................................... ........... 14 A B S T R A C T ...................................... ......................................... ............... 16 CHAPTER 1 INTRODUCTION ................................... ............................. ....... ......... 18 Therm al Protection System ..................................................... ................................. 21 Function of a Therm al Protection System ............................................... .............. 21 General Requirements of a Thermal Protection System .............................................22 Approach to Thermal Protection Systems....................... .................................23 History of Thermal Protection Systems .............. ............. ....... ...................... 25 Integral Therm al Protection System ............................................................ .............. 28 P u rp o se ......... ..... ............ ................... ........ ..................... ............ 3 1 O b je c tiv e s .............. ..... ............ ............................................................................ 3 1 2 MICROMECHANICAL ANALYSIS .................................. ............................ 33 S an d w ich S tru ctu re s ...........................................................................................................3 3 Literature Review on Corrugated Core Sandwich Panels.................................... .............33 A analytical A approach ........................................................................36 G eom etric Param eters ................................... ..................... ....40 Extensional and Bending Stiffness.................. ... ...............................41 Formulation of deformation transformation matrix for the facesheets ................ 42 Formulation of the web deformation transformation matrix ........................ 44 Stiffness matrix determination using the strain energy approach........................45 Transverse Shear Stiffness, A 55............................................................ .............. 45 Transverse Shear Stiffness, A 44...........................................................................46 Face Sheet and W eb Stress Determination................................. ........................ 48 M idplane micro shear strain in the web ........................................ .............. 50 Micro curvature in the y direction for the webs............................ .............. 51 C o n clu sio n ...................................... .................................................... 52 3 FINITE ELEMENT VERIFICATION........................... ..............56 Extensional and B ending Stiffness ............................................... ............................ 56 Stress V verification .............................. ..... ......................................................59 Midplane Shear Strain and Curvature in the Webs ....................................................59 S tre ss V erific atio n .................................................................................................. 5 9 Transverse Shear Stiffness (A44) Verification .............. ................. ................................ 60 C o n clu sio n ......... ..... ...................................................... .................................6 1 4 THERMAL ANALYSIS OF AN ITPS UNIT CELL ......... ............. .................73 Intro du ctio n ................. ............. ................. ....................... 7 3 Therm al M icrom echanics Approach.......................................................... .............. 75 Thermal Force Resultants and M moments ................................................. .............. 75 T h e rm a l S tre ss ................................................................ ....................................7 7 Constrained case .................................. ............................ ... ........ 77 U nconstrained case ................................................................... 79 Finite E lem ent V verification .......................................................................................... 80 Thermal Force and Moment Resultants ................................................................ 80 Therm al Stress V verification ...................................................................... .............. 81 C o n strain ed case .................................................................................. 8 1 U ncon strained case ..................................................................................... 8 1 C o n c lu sio n .................................................................................................................... 8 2 5 BUCKLING ANALYSIS OF AN ORTHOTROPIC INFINITE STRIPS AND A PPL IC A T IO N S T O IT P S ..................................... ................................... ................... ..... 87 B u ck lin g o f an IT P S ...................................................................................................... 8 7 M ethods of Critical Loads Calculation ..................................................................... ..... 88 Stability of an Infinite Strip under Compression or Shear .............................................. 89 R e su lts ................................... ....................................................................................... 9 2 U niaxial Com pression, Nx only.......................................................... ............. 93 U niaxial C om pression, N y only ........................................................................ ... ... 93 Biaxial Com pression, N = Ny.......................................................... .............. 94 Shearing Load, N y = 1 ................................................................. 94 C o n c lu sio n .................................................................................................................... 9 5 6 ITPS PANEL AS A TWO DIMENSIONAL PLATE .............................. 101 Intro du ctio n ....................... .. .............. .. .....01.......... A n a ly sis ............. ..... .. .... .............. .. ............................................................. 1 0 2 U uniform Pressure Loading..................................... ......................... .............. 102 Outofplane displacement........................... .............. 102 Top facesheet local deflection ........... .. ....... ................... .............. 104 Local stresses ........................ ............. ...................106 6 Unsymmetric ITPS Panel Configuration ............................................ ..............108 T em perature D distribution ................................................................................... 109 Outofplane displacement, First Order Shear Deformable Plate Theory............ 109 Outofplane displacement, Classical Laminate Plate Theory........................... 112 R results .................... .. .......... ..... ....... ...... ... ... ........... ........... ........... 114 ITPS OutofPlane Displacement, Pressure Load............................................. 114 ITPS Outofplane displacement, Temperature Distribution..............................117 ITPS Local Stress ..................................................................... ......... 121 Web Angle Sensitivity................. ........................... 122 C o n c lu sio n ........................................................................................... 12 4 7 OPTIMUM DESIGN OF THE INTEGRATED THERMAL PROTECTION ................... 141 In tro d u ctio n ...................................................................................................................... 1 4 1 The ITPS Optim ization Problem .......................................................... .............. 143 R e su lts ................... ................... ..................................................... .. 1 4 5 C onclu sion ............................................................................................ 147 8 CONCLUSIONS ........................................................................ ......... 151 Su m m ary ................... ................................. 15 1 C o n c lu sio n s ...................................................................................................................... 1 5 3 R ecom m endations .................................................................................................... 155 APPENDIX A DEFORMATION TRANSFORMATION MATRICES................................ 156 B DETAILED DERIVATION OF A55 ................................................................159 C TRANSVERSE SHEARING STIFFNESS ............................................. ........... 161 D ED GE M OM EN TS ..................................................... 162 L IST O F R E FE R E N C E S .............................................................. ..................................164 BIOGRAPHICAL SKETCH ................................................................... ............ 167 LIST OF TABLES Table page 31 Periodic displacement boundary conditions .........................................................62 32 Nonzero [A], [B], and [D] coefficients for an ITPS panel with [0 / 90]s layup ..............62 33 Nonzero [A], [B], and [D] coefficients for an ITPS panel with a [45 / 45]s layup.........63 34 Comparison of the nonzero [A], [B], and [D] coefficients between the refined transformation matrix and the deformation transformation matrix ..............................63 35 Nonzero [A], [B], and [D] coefficients for an unsymmetric ITPS sandwich panel.........64 41 Nonzero Thermal Forces of the unit cell. ........ .......................................................... 83 51 Critical buckling load of plate II, N, = 1................................................... ........ 96 52 Critical buckling load of plate II, Ny = 1 ................................................... ........ 96 53 Critical buckling load of plate II, k= 1 ............... ................................... 96 54 Critical buckling load of plate II, N y 1 ........ ................................................. ......... 96 61 Thermal moments of Inconel under the 450 s reentry temperature distribution ........... 125 71 Ranges of the seven design variables for the ITPS optimization problem .................. 149 72 Optimum designs with a maximum deflection constraint only. ................................ 149 73 Optimum designs with a yield stress constraint only. ............................................. 149 74 Preliminary ITPS optimum designs with deflection and yield constraints ................150 75 Preliminary ITPS optimum designs with deflection, yield, and temperature co n strains. ...................... ............................... ....................................... 15 0 76 Optimal design comparison between FE and analytical method. .............................. 150 LIST OF FIGURES Figure page 11 Flight regim es for hypersonic vehicles ................. ..... ......... .................... 32 12 Corrugatedcore sandwich panels for use as an ITPS ................................. .................32 21 Zcore and Ccore sandw ich structure ............................................................................53 22 Sim plified unit cell dim ensions...................................................................... 53 23 Equivalent orthotropic thick plate for the unit cell corrugated core sandwich panel........53 24 Global and local coordinates of the unit cell, faces and webs. ........... .... ............... 54 25 Small element removed from a body, showing the stresses acting in the xdirection....... 54 26 Unit cell subjected to unit transverse shear and horizontal force................. ..............54 27 Half unit cell of the corrugatedcore sandwich panel............. ...... .................. 55 28 Free body diagram of the top face sheet under the action of midplane shear strain .........55 29 Half unit cell under the action of end couples at the faces. ........... ........................ 55 31 Finite elem ent unitcell m esh .. .... ......................................................... .............. 65 32 Boundary conditions imposed on the plate to prevent rigid body motion...................65 33 Deformations of the ITPS due to periodic boundary conditions............... .............. 66 34 Comparison of FEM and analytical micro strain for midplane shear strain and curvature. ........... ......... .. ....................... ................... .......... 66 35 Stresses in the x and y direction of the ITPS for a unit cell strain of M =1 ...............67 36 Stresses in the x and y direction of the ITP S for a unit cell strain of = 1. ...............67 37 Stresses in the x and y direction of the ITPS for a unit cell strain of yM = 1 .............68 38 Stresses in the x and y direction of the ITPS for a unit cell strain of Kj = 1 ...............69 39 Stresses in the x and y direction of the ITPS for a unit cell strain of KrM = 1 ............... 70 310 Stresses in the x and y direction of the ITPS for a unit cell strain of KM =1 ................71 311 Truss core modeled as a cantilever beam with ten unit cells...................................71 312 Finite element and analytical result for the transverse shearing stiffness ...................72 41 Heating used for preliminary thermal load and stress analysis of an ITPS panel............. 83 42 Core temperature distribution and thermal force resultants and thermal moments. .........83 43 Halfunit cell of the truss core sandwich panel with a temperature distribution ..............84 44 Free body diagram of the top face sheet and web. ...................................................... 84 45 Free body diagram of the webs subjected to a temperature distribution......................... 84 46 Web expansion for the constrained thermal problem....................................................85 47 Stress in the x and y directions of the ITPS for the constrained thermal problem.......... 85 48 Deformation of the unit cell due to the unconstrained boundary condition ........ ........ 86 49 Stress in the x and y directions of the ITPS for the unconstrained thermal problem......86 51 Local buckling of an ITPS panel ............ ........ ............. .. .............. 97 52 Infinite Strip under shear and compression loading. ................................................97 53 Critical buckling load flow chart for an infinite plate. ............ ..... .................97 54 Deformation of the quarter plate due to Nx=l ............. ........................... ..............98 55 Critical buckling load in the plate for Nx= ....... .......................................................... 98 56 Deformation of the quarter plate due to Ny= ........... ........................... ..............98 57 Critical buckling load in the plate for Ny = 1 ............ ............................... ..............99 58 Deformation of the quarter plate due to k=......... .............................................. 99 59 Critical buckling load in the plate for N, = Ny. ............................. ..... ......... 100 61 ITPS as a two dimensional orthotropic thick plate.................................... ............. 125 62 Half of the top face under the action of a uniform pressure loading ........................... 125 63 Local stress flow chart for an ITPS as a 2D plate.............................................. 126 64 ITPS unit cell under the action of transverse shear force ............ ........................ 126 65 Local stress flow chart for an ITPS with transverse shear force effects consideration... 127 66 Free Body diagram of the top face under the action of a uniform pressure loading. ...... 127 67 Free body diagram of section BC of the top facesheet............................ .............. 128 68 Free body diagram of section AB of the top facesheet............................................ 128 69 ITPS orthotropic plate subjected to uniformly distributed thermal end moments......... 128 610 ITPS panel finite element model and mesh ........................................... ..............129 611 Finite element boundary conditions for the ITPS plate.............................................. 129 612 ITP S outofplane deform ation .......................................................................... .... 130 613 Outofplane displacement comparison between FEM and analytical solution of the face s ................... ................... ................... ................................. .. 13 0 614 Finite element mesh of the 2D plate and boundary conditions .................................. 131 615 Outofplane displacement of an isotropic plate subjected to uniform edge moments... 131 616 Outofplane displacement of an orthotropic plate subjected to uniform edge moments at x = a/ 2......... ............................ ... ........... 132 617 Outofplane displacement of an orthotropic plate subjected to uniform edge moments at y = b / 2......... ............................ ............... 132 618 Isotropic plate outofplane displacement contour for M = 1 and My = 1 .................... 133 619 Orthotropic plate outofplane displacement contour for M = 1 and My = 1................. 133 620 ITPS temperature distribution at 450 s reentry time...................................................... 133 621 ITPS outofplane thermal displacement contour. ...................................................... 134 622 ITPS outofplane displacement due to a temperature distribution for boundary conditions. ................................................................ ......... ......... 134 623 ITPS center panel displacement for various L / h values .............................................. 135 624 ITPS outofplane displacement due to a temperature distribution with L / h = 18....... 135 625 Top facesheet stress in the x andy direction of the ITPS panel. ........... ............... 136 626 Bottom facesheet stress in the x andy direction of the ITPS panel............................ 136 627 Web stresses in the x andy direction of the ITPS panel. ............................................ 137 628 Outofplane displacement comparison for an unsymmetric ITPS.................. ......... 137 629 Top facesheet stress in the x and y direction of an unsymmetric ITPS ........................138 630 Bottom facesheet stress in the x andy direction of an unsymmetric ITPS................ 138 631 Web stresses in the x and y direction of and unsymmetric ITPS................. ............ 139 632 Behavior of the stiffness properties and deflection to a change in the web angle .......... 140 633 Exaggerated Deformed Mesh (Deformation Scale Factor = 2) .................................... 140 LIST OF ABBREVIATIONS AFRSI advanced flexible reusable surface insulation ARMOR advancedadapted, robust, metallic, operable, reusable CEV crew exploration vehicle ELV expendable launch vehicle FEA finite element analysis FEM finite element method FRCI fibrous refractory composite insulation FSDT first order shear deformable plate theory HRSI hightemperature reusable surface insulation ISS International Space Station ITPS integral thermal protection system LEO low earth orbit LRSI lowtemperature reusable surface insulation NASA National Aeronautics and Space Administration RCC reinforced carboncarbon RLV reusable launch vehicle SSTO singlestagetoorbit TPS thermal protection system TPSS thermal protection support structure LIST OF SYMBOLS a panel length (xdirection) a coefficient of thermal expansion (CTE) A* inverse of the extensional stiffness matrix A44 shearing stiffness (ydirection) Ass shearing stiffness (xdirection) [A] extensional stiffness matrix [B] coupling stiffness matrix b panel width (ydirection) [D] bending stiffness matrix D* inverse of the bending stiffness matrix A temperature change d height of the sandwich panel (centerline to centerline of the facesheets) {D} e) deformation vector of the eth component (micro deformation) {D y deformation vector of the unit cell (macro deformation) e component index of the unit cell so midplane strain El equivalent flexural rigidity F(m) nodal force in the FEM model K curvature k ratio between compressive force in the ydirection and xdirection. I length of the cantilever beam Mf thermal moment resultant M, component moment resultant No lowest compressive load before buckling. N, component force resultant N" thermal force resultant 2p unit cell length Pz pressure load acting on the 2D orthotropic panel Qx, Qy shear force on the unit cell Q transformed lamina stiffness matrix R ratio between shear load and compressive load. s web length rx. shear stress in the web tTF top face sheet thickness tBF bottom face sheet thickness tw web thickness 0 angle of web inclination [TD ](e" deformation transformation matrix of the eth component of the ITPS U unit cell strain energy Vtp tip deflection of cantilever beam w ITPS panel deflection /x, Vy rotations of the plate's cross section Length between the successive buckling waves in the plate y local axis of the web 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 ANALYSIS AND DESIGN OF AN INTEGRATED THERMAL PROTECTION SYSTEM FOR FUTURE SPACE VEHICLES By Oscar Martinez May 2007 Chair: Bhavani Sankar Major: Aerospace Engineering Thermal protection systems (TPS) are the key features incorporated into a spacecraft's design to protect it from severe aerodynamic heating during highspeed travel through planetary atmospheres. The thermal protection system is the key technology that enables a spacecraft to be lightweight, fully reusable, and easily maintainable. Addon TPS concepts have been used since the beginning of the space race. The Apollo space capsule used ablative TPS and the Space Shuttle Orbiter TPS technology consisted of ceramic tiles and blankets. Many problems arose from the addon concept such as incompatibility, high maintenance costs, nonload bearing, and not being robust and operable. To make the spacecraft's TPS more reliable, robust, and efficient, we investigated Integral Thermal Protection System (ITPS) concept in which the loadbearing structure and the TPS are combined into one single component. The design of an ITPS was a challenging task, because the requirement of a loadbearing structure and a TPS are often conflicting. Finite element (FE) analysis is often the preferred method of choice for a structural analysis problem. However, as the structure becomes complex, the computational time and effort for an FE analysis increases. New structural analytical tools were developed, or available ones were modified, to perform a full structural analysis of the ITPS. With analytical tools, the designer is capable of obtaining quick and accurate results and has a good idea of the response of the structure without having to go to an FE analysis. A MATLAB code was developed to analytically determine performance metrics of the ITPS such as stresses, buckling, deflection, and other failure modes. The analytical models provide fast and accurate results that were within 5% difference from the FEM results. The optimization procedure usually performs 100 function evaluations for every design variable. Using the analytical models in the optimization procedure was a time saver, because the optimization time to reach an optimum design was reached in less than an hour, where as an FE optimization study would take hours to reach an optimum design. Corrugatedcore structures were designed for ITPS applications with loads and boundary conditions similar to that of a Space Shuttlelike vehicle. Temperature, buckling, deflection and stress constraints were considered for the design and optimization process. An optimized design was achieved with consideration of all the constraints. The ITPS design obtained from the analytical solutions was lighter (4.38 lb / ft2) when compared to the ITPS design obtained from a finite element analysis (4.85 lb / ft2). The ITPS boundary effects added local stresses and compressive loads to the top facesheet that was not able to be captured by the 2D plate solutions. The inability to fully capture the boundary effects lead to a lighter ITPS when compared to the FE solution. However, the ITPS can withstand substantially large mechanical loads when compared to the previous designs. Trusscore structures were found to be unsuitable as they could not withstand the large thermal gradients frequently encountered in ITPS applications. CHAPTER 1 INTRODUCTION Throughout the past century there has been an exponential advancement in flight technology. Since the dawn of the aerial age which began with the Wright brothers, mankind has been pushing the envelope of flight which is to fly at faster speeds, longer distance, and higher altitudes. In the 1950s, mankind entered the rocket propulsion era, which brought the space age to life. With the advent of rocket propulsion, space vehicles can be accelerated to speeds in excess of 11,000 m/s (24,606 mph) which is the minimum velocity needed to escape the earth's gravitational pull. In the next 20 years the National Aeronautics and Space Administration (NASA) plans to send manned space missions to Mars. For vehicles traveling at hypersonic speeds through a planetary atmosphere, the aerodynamic forces provide two things; unwelcome air resistance that must be overcome by a powerful propulsion system or a welcome means of slowing down the space vehicle. The latter is known as aerocapture. The aerocapture approach uses the planetary's atmosphere as a welcome means of drag to alter the space vehicle's velocity and establish a capture orbit. Propellant is not needed for the decelleration of the space vehicle, therefore a fuel free planetary entry method could reduce the overall mass and voyage cost of the space vehicle. This reduction in mass allows for cheaper and smaller space vehicles for interplanetary voyages. Aerocapture will have an impact on the space vehicles' heat shielding because of the excessive aerdoynamic heating from the atmospheric friction due from air drag. Excessive aerodynamic heating to the space vehicle's structure is a result of high velocity flights through a planetary atmosphere. An aerospace vehicle traveling at high speeds through a planetary atmosphere must push the atmospheric gas out of its way to continue with its intended flight path. The faster the vehicle is traveling, the faster the atmospheric molecules travel along the outer space vehicle structure. The acceleration of the molecules causes friction between the vehicle's surface and atmospheric gas which can damage the vehicle's structure. The thermal protection system (TPS) is the key feature that is designed into the space vehicle to protect it from the extreme aerodynamic heating. Various vehicles have pushed the flight envelope limit by flying at hypersonic speeds. Space vehicles like the Space Shuttle have carried people into low earth orbit (LEO) and the moon. Unmanned probes have penetrated alien planets in our solar system to gather scientific information. Military and commercial planes are capable of reaching supersonic speeds for sustained flight through the atmosphere. One of the key technologies required by all of these high speed vehicles is a TPS to protect it from the extreme aerodynamic heating (Figure 11). The major focus of the government and industries around the world are to provide cost effective and reliable space transportation (Freeman, Talay, and Austin, 1997). The control of the aerodynamic deceleration loads and the protection of the payload and crew from the induced aerodynamic heating constitute the primary problems for a space vehicle designer (Stewart and Greenshields, 1969). A number of new space vehicles are being developed to provide routine and low cost access to space. Proposed vehicles include a reusable launch vehicle (RLV), hypersonic air breathing vehicles (Hunt, Lockwood, Petley, and Pegg, 1997), military space planes (Blosser, 1996), crew exploration vehicle (CEV), and unmanned experiment return capsules (Christiansen and Friesen, 1997). Reducing the cost of launching a space vehicle is one of the critical needs of the space industry. There is a major demand for space access from the government and industries. The government launches satellites for weather, military, communications, scientific, and reconnaissance purposes. The industries' thirst for space tourism is growing rapidly, and many orbiting satellites and the International Space Station (ISS) require constant reconstruction and servicing. If use of space is to become routine, future space vehicles must become fully reusable, have greater operational flexibility, and have a lower operating cost than the current space vehicles like the Space Shuttle (Bohon, Shideler, and Rummler, 1997). Reducing the cost of delivering a pound of payload into space by an order of magnitude is one of NASA's objective at achieving their low cost effective and reliable space access (Blosser, 2000). The space vehicle's TPS is one of the most expensive and critical systems of the vehicle (Behrens and Muller, 2004). The high cost of space transportation is viewed as one of the biggest obstacles to the growth of space exploration, commercialization and research. Expendable launch vehicles (ELV) such as Europe's Arianne, The United State's Delta, and Russia's Soyuz provide the most economical means of delivering payload to space. The ELV's estimated LEO payload cost per pound is averaged at $5,000 (Furtron Corporation, 2002). However, a new launch vehicle is required for each every launch, making ELV's nonreusable and not economical for space missions that require multiple launches. The Space Shuttle offers a reusable space vehicle but at a very expensive cost. The estimated Space Shuttle cost of delivering a pound of payload into LEO is about $4,000 and the price rises to about $23,000 for a geosynchronous transfer orbit. Besides launch costs, the shuttle's external fuel tank has to be replaced for every launch, the solid rocket boosters require extensive refurbishment, and the maintenance time for the Space Shuttle is 40,000 man hours (Blosser, 2000). The recently proposed Lockheed Martin VentureStarTM was aimed at proposing a low cost access to space by reducing the payload cost per pound by an order of magnitude (Dorsey, Poteet, Wurster, and Chen, 2004). The VenutreStarTM was going to be a singlestagetoorbit (SSTO) RLV. The proposed vehicle program and development was stopped due to technological and budget constraints that could not be overcome. Regardless of the problems with the VentureStarTM, NASA's RLV program is seeking a replacement for the Space Shuttle. The future space vehicle technology will involve the research and development of an all rocket, fully reusable, SSTO vehicle. Such vehicles will have a large area to be covered with TPS, because it includes the fuel tanks required for launch. As a result of the large TPS coverage area, a need for a lightweight TPS is necessary to keep the vehicle weight and launch cost reasonable and affordable. The significant reduction of the payload transportation costs is the rational in the development of a future space vehicle. The emphasis of the vehicle would be lightweight, fully reusable, and easily maintainable. A key factor in the reusability, life, and operational flexibility of such classes of future space vehicles is an efficient and advanced TPS. Future space vehicles will require an efficient TPS that provide the vehicle with the necessary protection from aerodynamic heating as well as lower the operational cost, maintenance cost, and maintenance time. Thermal Protection System Function of a Thermal Protection System The primary function of a TPS is to protect the space vehicle from extreme aerodynamic heating and to maintain the underlying structure within acceptable temperature and mechanical constraints (Zhu, 2004). The vehicle's structure acts as the thermal storage reservoir for the heat that passes through the TPS. To protect the vehicle structure and payload the heat that passes through the TPS must be minimized. The amount of heat that passes through the TPS is dependant on many material properties such as thermal conductivity, mass of the TPS, specific heat, and geometric parameters. Temperature limits are imposed on the space vehicle structure to avoid material property degradation, excessive deformation, and critical thermal stresses. The TPS is exposed to a variety of environmental conditions because it forms the external surface of the space vehicle. Therefore, a set of requirements are needed for a TPS. General Requirements of a Thermal Protection System The TPS generally covers the entire exterior surface of the space vehicle and therefore it defines the space vehicle's exterior shape. The TPS is subjected to a wide variety of environments corresponding to all phases of flight (Dorsey et al., 2004). A set of general requirements for a TPS are listed below. * High and Low velocity impact: When the vehicle is being fabricated, maintained, assembled, transported, or just waiting to be launched, the TPS may be exposed to handling damage from tools such as, an accidental dropping of tools. During launch and landing the TPS must withstand runway debris that is stirred up by the rocket exhaust or winds. While in flight to space the TPS may experience bird strikes, hail, space vehicle debris, and dust impacts that can cause catastrophic failure to the TPS as was evident from the Space Shuttle Columbia disaster. When orbiting in space the TPS may encounter hypervelocity impacts from micrometeorites or space debris. A TPS should withstand all possible types of low and high velocity impact to prevent catastrophic TPS failure. * Panel Deflection: The excessive aerodynamic heating causes extreme temperatures in the TPS. The extreme temperatures result in thermal loads and moments which cause the panel to deflect out of plane. During the portion of flight through the atmosphere the TPS will encounter dynamic, acoustic, aerodynamic pressure, and shear loads that cause the panel to deflect out of plane and vibrate. The deflection of the panel must be kept within acceptable limits to prevent extreme local aerodynamic heating and maintain a smooth aerodynamic profile of the vehicle. The vibration must be kept below its natural frequency to prevent dynamic failure such as flutter. * Chemical Deterioration: During maintenance, or fabrication, the TPS panel may be exposed to unfriendly chemical agents that may alter the chemical composition or deteriorate the material properties of the TPS panel. The TPS is exposed to rain, snow or ice that can lead to possible rust or oxidation of the panels and degrade the structure. While in space there are various chemical effects that must be considered for as well. * Mechanical and Thermal Loads: During launch the TPS must withstand severe acoustical and dynamic loading that is initiated from the rocket propulsion system. These loads vary widely depending on its location on the vehicle. The TPS panels near the exhaust of the vehicle will experience high thermal loads and acoustic loads. During flight the TPS will be exposed to aerodynamic pressure, aerodynamic shear, and inertial loads. The TPS must withstand all these loads to prevent failure and fracture. There are also several desirable features of a TPS such as being lightweight. Low mass is desirable on any item carried by a vehicle at any speed. As mass is added to the vehicle, more energy is required to accelerate the vehicle. Thus the excess mass results in more fuel carried by the vehicle or a decrease in payload (Blosser, 2000). The TPS weight is desirable for an SSTO space vehicle because of its large surface area. Therefore, the TPS should be designed and optimized for minimum mass to achieve the lostcost space access goal. Finally, low cost, robustness, and operability are other desirable features of a TPS. The consideration of all these desirable features leads to a low lifecycle cost. The lifecycle cost of a TPS includes fabrication, installation, and maintenance in acceptable operating conditions (Blosser, 2000). Cost is a significant driver of any vehicle let along a space vehicle which can lead to launch costs in the billions. Cost can be reduced by lowering initial fabrication cost, installation costs, required maintenance time, and turnaround time between flights. A robust TPS is one that is not easily damaged by its environment and can tolerate damage without requiring immediate repair and/or replacement. An operable TPS is one that can be easily maintained, inspected, and replaced should there be damage. Cost can be reduced by developing a TPS that requires minor inspection, maintenance, repair between flights, and enable a widening operational envelope of the space vehicle to maximize the time spent in space (Blosser et al., 2004). In summary, the thermal protection system must have sufficient durability and reusability in order to withstand repeated exposure to the adverse environments of space, launch, and reentry, as well as the abuse due to normal ground maintenance and inspection. Approach to Thermal Protection Systems The approach to a TPS design depends on the magnitude and duration of the aerodynamic heating. There are three approaches to a TPS; active, semipassive, and passive (Blosser, 2000). An active TPS contains external systems that provide coolant to the TPS during flight to continually remove heat from the system and keep the temperature within acceptable limits. Active TPS concepts are transpiration cooling, film cooling, or convective cooling. These concepts use pumps, coolant, pipes, and coolant storage tanks. The added parts to the vehicle increase the vehicle mass and only make it a viable option for small, highly heated areas of a space vehicle. Semipassive TPS have a working fluid that removes heat from the point of application. They require no external system to provide the coolant. Semipassive concepts are ablation (Stewart and GreenShields, 1969) and heat pipes. Heat pipe concepts transfer the heat through the pipes from a high heat transfer area to a low heat transfer area creating a smooth heat transfer surface, therefore reducing any concentrated heat spikes on the space vehicle's TPS. Ablative heat shields are the simplest and least inexpensive type of TPS. Ablation dissipates heat from the plasma by allowing its outer layers to char, melt, and to some extent vaporize. Ablative heat shields are heavier than conventional TPS, which would make it an uneconomic option for space vehicles with a large surface area like the Space Shuttle or RLV. The simplest TPS approach is a passive TPS. A passive TPS either radiates heat from the surface or absorbs it into the structure. The advantages of using a passive TPS are that they are simple concepts and have the highest reliability. There are three passive TPS concepts; heat sink, hot structure, and insulated structure. A heat sink absorbs all the incident heat and stores it in the structure. The amount of heat storage capability depends on the mass, specific heat capacity, and service temperature of the material. A hot structure allows the structure's temperature to rise until the heat being radiated from the surface is equal to the incident heating. This concept is limited by the allowable temperature limit of the material. An insulated structure is both a heat sink and a hot structure. The outer surface of an insulated structure keeps it near the radiation equilibrium temperature. The radiation equilibrium temperature is the surface temperature resulting from the given absorbed heat flux and surface emittance. Most of the incident heat is radiated out of the structure and only a small fraction of incident heat is stored in the underlying structure. History of Thermal Protection Systems High speed flight and space travel has made significant advancements in the last half century. Starting with the artificial satellite (Sputnik) in 1957 and then progressing rapidly to manned space flight programs such as the Mercury, Gemini, and Apollo. All of these manned spacecraft missions used ballistic bluntbody reentry vehicles and ablative heat shields to dissipate the extreme aerodynamic heating. The coneshaped capsules had heat shields attached to its base. The materials used in the heat shield would carry excess heat away from the spacecraft and its crew through vaporization. The Mercury and Gemini capsules of the early 1960's were protected by an all ablative heat shield made of silicafiber resin, while the later Apollo capsules had heat shields made of phenolic epoxy resin, a form of plastic. The low density ablation material (AVCOAT 502639/HCGP) is bonded to its primary structure. The ablation material chars at sufficiently highheating rates. Apollo heat shields were nearly 7 cm (2.7 in) thick and weighed 1,360 kg (3,000 lb). During the 1960s, the Space Shuttle was created based on a need for a logistical spacecraft to support orbital and space stations such as Mir and the ISS. The Space Shuttle's TPS design is required to keep the structural temperatures less than 350F (449.81 K). The Space Shuttle uses a passive TPS consisting of several materials selected for stability at high temperatures and weight efficiency. These materials are as follows (Thornton, 1992): * Reinforced CarbonCarbon (RCC): RCC is used for the nose cap and wing leading edges where temperatures are above 2300F (1533.15 K). * Hightemperature reusable surface insulation tiles (HRSI): HRSI are used in areas on the upper forward fuselage, the entire underside of the fuselage where RCC is not used, and some leading edges of the orbiter where temperatures are below 2300F (1533.15 K). These tiles have black surface coatings necessary for entry emittance. * Lowtemperature reusable surface insulation tiles (LRSI): LRSI consists of 8 in2 (1.24 cm2) silica tiles and covers the top of the vehicle where temperatures are less than 1200F (922 K). * Fibrous refractory composite insulation tiles (FRCI): FRCI are high in strength and were used to replace HRSI in some areas. The FRCI tiles have a density of 12 lb/ft3 (192.22 kg/m3) and provide improved strength, durability, resistance to coating cracking and weight reduction. * Advanced flexible reusable surface insulation blankets (AFRSI): AFRSI blankets replaced the vast majority of LRSI tiles. The direct application of the blankets to the orbiter results in weight reduction, improved durability, reduced fabrication and installation cost, and reduced installation schedule time. The greatest disadvantage of the tiles is that these materials make the space vehicle's exterior brittle, susceptible to damage from small impact loads, high in maintenance, and consequently its reusability may not be sufficient for advanced mission requirements. After each flight the entire orbiter is waterproofed and each tile is inspected manually for any cracks and/or failures. The tiles coefficient of thermal expansion (CTE) is less than the Space Shuttle's structure CTE which is aluminum. As a result of that incompatibility, direct mounting of the TPS to the structure was not possible. All these factors contributed to the Space Shuttle's launch cost and maintenance. The X33 Venture Star was aimed at proposing a low cost access to space. One of the most important goals of the X33 was to eliminate the numerous problems relating to the Space Shuttle's TPS. Metallic TPS was considered as the primary heat shield for the RLV. Metallic TPS was considered as a muchneeded alternative to the ceramicbased brittle tile and thermal blanket currently used on the Space Shuttle. Metallic TPS offers significant advantages such as; (1) high temperature adhesive, sealants, or water proofing is not required, (2) metal ductility promotes robustness, and good damage tolerance properties, and (3) weight savings when used as part of an integrated aeroshell structural system (Harris, Shuart, and Gray, 2002). Recently, Blosser (2004) designed an advancedadapted, robust, metallic, operable, reusable (ARMOR) TPS concept. The ARMOR TPS was designed to be attached to the cryogenic tank structure of an RLV. The outer surface of the ARMOR TPS consists of a foil gauge, Inconel 617 metallic honeycomb sandwich panel that is exposed to ascent, reentry aerodynamic heating profiles, and acoustic and aerodynamic pressure. The thin titanium box frame defines the edges of the panel's inner surface. The bottom panel is insulated from the top by Saifill' insulation, which is made from alumina fibers. The ARMOR TPS is a combination of a hot structure and an insulated structure. The ARMOR TPS is fully metallic which gives it good impact resistance features. The ARMOR concept can also eliminate radiation in paneltopanel gaps, provide subsurface sealing and attachments, and decouple deformation and thermal expansion between the inner and outer faces. The structural load is taken by the underlying foil gauge titanium TPS support structure (TPSS). However, the ARMOR TPS load bearing capabilities are limited, and large inplane loads cannot be accommodated under this design. Selection of the optimum TPS for a particular space vehicle is a challenging task that requires considerations in weight, operability, reusability, maintenance, durability, initial cost, lifecycle cost, and compatibility. The common feature between the Apollo, Space Shuttle, and X33 VentureStarTM TPS concepts was that their TPS was an addon to the vehicle's outer structure. The addon feature created incompatibility problems between the thermal structure and the loadbearing structure of the vehicle as well as an increase in maintenance. The relatively weak bonding of the Space Shuttle tiles to the vehicle can expose it to catastrophic failures like tile loosening or tile detachment as was evident in the Columbia Space Shuttle disaster. The Apollo heat shield was heavy and it was not practical for large surface areas space vehicles. These various TPS concepts were also not load bearing members; they were purely thermal structures. Fasteners, frames, and support brackets contributed to the overall weight of the TPS and the TPS as an addon feature added mass to the total vehicle weight. Integral Thermal Protection System There is a need for a new and efficient TPS concept. The new TPS concept will have to be lightweight, robust, and operable. This concept can be accomplished by using recently developed metallic foams and also innovative core materials, for example, corrugated cores and truss cores (Figure 12). The Integral TPS/structure (ITPS) design can significantly reduce the overall weight of the vehicle as the TPS/structure performs the loadbearing function. The ITPS is expected to be multifunctional (offer insulation as well as load bearing capability in order to reduce the mass of the vehicle). Advantages of an ITPS concept are as follows: * ITPS panels can reduce the overall weight of the vehicle: The faces of an ITPS are made from thin metallic plates which result in lightweight components and good impact resistance. * Greater flexibility for designer to create elegant curves: This is ideal for space vehicles with conical or blunt bodies such as the CEV. * Panels can be large in size thus eliminating the total number of panels to be used on a space vehicle. * Performs load bearing functions, making it multifunctional and robust. * TPS will be integrated with the vehicles structure, which promotes low installation and maintenance costs. The sandwich structure will replace the structural skin and the insulation in the current thermal structures. Since the sandwich construction is stiffer than single skin construction, the number of frames and stringers will be significantly reduced. Furthermore, the insulation is protected from foreign object impact and requires less or no maintenance such as water proofing. An ITPS is a sandwich panel composed of two thin faces separated by a corrugated core structure which can be of homogeneous materials such as metals or orthotropic materials such as composite laminates. The sandwich panel is composed of several unit cells placed adjacent to each other. The empty space in the corrugated core will be filled with a non loadbearing insulation such as S fill'. The ITPS concept combines all three passive TPS concepts: heat sink, hot structure and insulated structure. The top face sheet acts as the hot structure and radiates out most of the heat. The insulation only allows a fraction of the incident heat to flow into the underlying vehicle structure, and the whole ITPS panel acts as a heat sink. The corrugatedcore feature provides the load bearing characteristic of the multifunctional structure. Combining the thermal and structural requirements of a structure is a challenging task for a designer. Requirements of a loadbearing structure and a TPS are often conflicting. A TPS requires the structure to have low conductivity, low density, and high service temperature. Materials that meet these requirements are ceramics, thermal blankets and insulation materials. Ceramics and insulation materials have poor strength, low fracture toughness, and poor impact resistance. A loadbearing member requires the structures to have high strength in compression or tension, high fracture toughness, and good impact resistance. Such materials that satisfy the load bearing conditions are metals and alloys. A load bearing member possess high thermal conductivity, high density, and low service temperatures. Integrating the thermal requirements with the structural requirements is a great challenge to any designer. The coupling of the thermal requirements with the load bearing requirements makes the vehicle design process complicated, and it makes the choice of material for an ITPS challenging. Integral Thermal Protection System Material Selection The material selection for the ITPS depends largely on its multifunctional ability, location on the space vehicle, and heating profile. Metals are the best material of choice for the top face sheet of an ITPS rather than ceramic materials. The advantages of using metals for the top face sheet over ceramic materials are listed: * Metals have a higher ductility over ceramics and therefore tend to be more damage resistant. * Small gage thickness. Many metals can be made into foils of 0.001 in (0.0254 mm) thickness. * Many fabrication techniques are readily available such as welding, extrusion, brazing, and machining. * Resistance to oxidation. The top face sheet panel of the ITPS is required to withstand extreme reentry temperatures, have good impact resistance, high service temperatures, and high strength. Inconel 718 wrought nickelchromium alloy is a possible material that meets all the top face sheet standards. Inconel 718 wrought nickelchromium alloy was used for the design of the ARMOR TPS (Poteet, Abu Khajeel, and Hsu, 2004). The service temperatures of Inconel 718 wrought nickel chromium is 1255 K (982 C). The bottom face sheet is expected to be a heat sink for the ITPS; therefore a material that has a high heat capacity is needed for the bottom face sheet. The bottom face sheet will also experience a major portion of the inplane stresses because of the attachment mechanisms of stringers and frames to the space vehicle. Therefore, a high Young's Modulus material with a high heat capacity is suitable for the bottom face sheet. Possible material choices for the bottom face sheet are titanium and aluminum alloys, beryllium, and carbon epoxy fiber composite. The web acts as the heat conduction path from the top face sheet to the bottom face sheet. It also acts as a load bearing member by supporting most of the transverse shearing loads. To decrease the amount of heat that is conducted to the bottom face sheet, the web must either have a small thickness or low heat conduction. Possible material candidates for the web are titanium alloys because of their high stiffness and high service temperature. Purpose The purpose of this dissertation was to investigate the use of an ITPS panel on a future space vehicle. Objectives The objectives of this study are listed as follows: 1. Identify the key failure mechanisms in the ITPS sandwich construction. 2. Develop new methods or modify available methods to perform thermomechanical analysis of a full ITPS sandwich panel to estimate panel deflections and stresses. 3. Investigate the effects of various thermal and mechanical boundary conditions on the stresses and deflections. 4. Investigate possible buckling and failure modes. 5. Design an optimization study of the advantages of the ITPS designs. The analytical models will be compared with finite element results. The analytical models will be refined so that the errors in prediction of critical metrics, (critical stresses) are within 5%. 3000T Y 2000 3 E  1000 Apollo > lAblators Mercury Insulation Space Shuttle X30  Heat Sink x15 Hot Structure YF12 Exposure Time (h) Figure 11. Flight regimes for hypersonic vehicles Figure 12. Corrugatedcore sandwich panels for use as an ITPS I I CHAPTER 2 MICROMECHANICAL ANALYSIS Sandwich Structures A sandwich panel is a threelayer element composed of two thin flat faces separated by a thick, lighter, and flexible core. The thin flat faces are high in stiffness when compared to the low average stiffness of the thick core. Sandwich constructions are frequently used because of their high bending stiffnesstoweight ratio. The high bending stiffness is the result of the distance of the facesheets from the neutral axis. The face sheets support the major portion of the inplane loads. Commonly used materials for the facesheets are composite laminates and metals, while cores are made of metallic and nonmetallic honeycombs, cellular foams, functionally graded foams (Lee, 2006), balsa wood, trusses or corrugated core. The core helps stabilize the facesheets and support the shear loads through the thickness. The corrugated core keeps the facesheets apart and stabilizes them by resisting vertical deformations, transverse shear strains, curvature in the longitudinal direction, and enables the structure to acts as a single thick plate. Unlike soft honeycomb core, a corrugated core resists bending and twisting in addition to vertical shear (Chang, Venstel, Krauthammer, and John, 2005). All these characteristics make corrugated sandwich structures ideal for aviation, aerospace, civil engineering, and marine applications, where weight and stiffness are important design drivers. Literature Review on Corrugated Core Sandwich Panels Honeycomb sandwich constructions are the earliest forms of sandwich cores. A number of theoretical and experimental investigations on innovative sandwich panels have been published in the literature. Libove and Hubka (1951) determined the elastic constants of corrugated core sandwich plates through the forcedistortion relationship. The authors homogenized the sandwich panel and compared the behavior of the equivalent homogenized thick plate with that of the corrugated sandwich panel. Nordstrand (2004) determined the buckling coefficient for edgeloaded corrugated orthotropic plates (including transverse shear) by including additional moments in the governing moment equilibrium equation of the panel. The corrected analysis expression for buckling coefficient was shown to reduce the classical formulation of an orthotropic plate without shear deformation when the transverse shear stiffness became large. The critical buckling load obtained from the corrected analysis was compared with predicted loads obtained from finite element and experimental panel compression tests. The author found that the explicit equation presented for the buckling load was accurate, (less than 0.5% deviation when compared with finite element results). However the deviation was larger between theoretical and experimental results, possibly due to nonlinear material behavior. Chang et al. (2004) did a comprehensive analysis of the linear elastic behavior of a corrugatedcore sandwich plate using the MindlinReissner plate theory (Whitney, 1987). The authors reduced the threedimensional sandwich panel into an equivalent twodimensional structurally orthotropic thick plate continuum and used previously determined stiffness values for their analysis. The authors investigated the plate behavior of a corrugated core sandwich structure subjected to a uniform pressure load. Their findings indicated that lower ratios of core thickness to web thickness, (or width to core thickness) made the plate stronger. The researchers also found that rectangular corrugations provided better bending resistance but larger deflections due to low shear stiffness. Local stress analysis of the corrugated core sandwich structure panel was not done. Lok, Cheng, and Heng (2000) derived analytical equations to predict the elastic stiffness properties and behavior of trusscore sandwich panels through the forcedistortion relationship. The authors used the homogenous thick plate approach to represent the threedimensional structure into a twodimensional thick plate. The findings included a closedform solution to determine maximum plate deflection of the continuum. Calculated results were in good agreement with numerical 3D finiteelement results. The equivalent elastic constants revealed that the shear stiffness in the ydirection was important, and that panels with vertical web angle of corrugations possess weak shear stiffness. The authors concluded that for panels with triangular corrugations, shear deformation can be ignored. The researchers did not do a local stress analysis of the sandwich structure and only used stiffness equations for isotropic materials. Fung, Tan, and Lok (1994) determined the elastic constants for Zcore and Ccore sandwich panels (Figure 21) by using the homogenous continuous approach and forcedistortion relationship. Excellent agreement was obtained between the analytical twodimensional thick plate model and finite element analysis. Valdevit, Hutchinson, and Evans (2004) structurally optimized sandwich panels with prismatic cores. The authors identified all failure mechanisms (face yielding, face buckling, core yielding, and core buckling) of the prismatic core and analytical expressions for the critical loads were derived. The authors' goal was to find geometric parameters that minimize weight per unit width subject to a combination of moment and shear forces as a function of the load index. Their findings indicated that the corrugated core panel performs best when loaded longitudinally because in this orientation, the performance is limited by plate buckling, rather than beam buckling. Tian and Lu (2005) investigated the optimal design of compression corrugated panels. The authors found that the minimum weight of a corrugated panel subjected to uniform axial compressive load was calculated by using a sequential quadratic programming optimization algorithm. The authors used analytical formulas to determine the constraints of the corrugated panel. Finite element analysis was not used as a verification tool of their findings. The authors concluded that from a weight standpoint, panels with hatstiffeners are found to be most efficient for a given boundary condition and that the square web configuration was the least efficient sandwich panel. Carlsson, Nordstrand, and Westerland (2001) reviewed previous analytical approaches to the analysis of the elasticstiffness of a corrugatedcore sandwich panel into the firstorder shear deformation laminated plate theory. The authors found that the bending, twisting, inplane extensional, and shear stiffness were dominated by the extensional and shear stiffness of the face sheets. Their predictions agreed with experimental measured data. The authors also found that the block shear test constrained deformation of the face sheets which led to unconservative overestimation of the effective shear modulus. Analytical Approach Micromechanical analysis of a unit cell was performed to determine the structure's extensional, bending, coupling shear stiffness, stresses, and unit cell behavior. Microscale stresses are the local facesheet and web stresses of the ITPS. The microscale stresses within the unit cell were computed using the micromechanical analysis. The relationship between the unit cell macrostress and macrostrains provided the constitutive relations for the material. Thus, constitutive characterization matrices [A], [B], [D] were found directly from micromechanics. The stresses were also used to predict the failure of the corrugated core. A detailed formulation and description of the extensional, coupling, bending, and shearing stiffness of the ITPS panel were presented for a unit cell by representing the sandwich panel as an equivalent thick plate which was homogeneous, continuous, and orthotropic. A strain energy approach and a deformation transformation matrix were used in deriving the analytical equations of the extensional, bending, coupling and shearing stiffness. Previous researchers adopted the force distortion relationship approach to determine the equivalent stiffness parameters (Fung, Tan, and Lok, 1993; Fung, Tan, and Lok, 1994; Libove and Hubka, 1951; Lok, Cheng, and Heng, 1999). However, the forcedistortion relationship approach can become complicated and tedious if the ITPS was composed of faces and webs with different materials and thickness. This problem can be solved with the proposed strain energy approach and deformation transformation matrix. The stiffness results can be used in the First Order Shear Deformable Plate Theory (FSDT) to determine their response on an ITPS plate when subjected to mechanical and thermal loads. The analytical models were compared with detailed finite element analysis for verification. The proposed strain energy method approach used in the research for predicting stiffness properties of a corrugated core was not adopted by the previous researchers that were mentioned. The previous researchers adopted the forcedistortion relationship which involved mechanics of materials equations that do not provide the designer with in depth local stress results of either component of the corrugatedcore sandwich structure. The mechanics of materials equations were only average stress equations of the facesheets and webs. hThe stiffness results obtained from the strain energy were also able to predict results for isotropic and orthotropic materials whereas the forcedistortion method used by the mentioned researchers only predicted stiffness results for isotropic materials. The only comparison between the forcedistortion method and the strain energy method is in the final prediction of the transverse shearing stiffness. The strain energy method is capable of handling laminated composites and uses Castigliano's second theorem rather than the unit load method but in the final step the forcedistortion relationship was adopted in the strain energy method. Improvements were made in the transverse shearing results when compared to Lok et al.'s (1999) transverse shearing stiffness results and FEM results. Furthermore, the strain energy method allows the designer of a corrugated core sandwich structure to choose any kind of material for both the facesheet and web and still be able to obtain an accurate stiffness matrix with the inclusion of the coupling stiffness matrix. The force distortion method becomes tedious and complex when the corrugated core structure is un symmetric. One of the most significant developments in engineering over the last four decades was the introduction of the finite element method (FEM). With FEM, virtually any complex structure can be modeled with a high degree of accuracy. The capability to model complex structures with a high degree of accuracy requires an increase in cost and time. The greater the accuracy, the greater the computation time, and consequently the greater the cost will be. The finite element method is commonly used to analyze sandwich structures. Shell elements are often preferred for the faces and webs to construct a detailed three dimensional FEM model. However, the number of elements and nodes needed to appropriately mesh the sandwich panel can be excessive; as a result a 3D FEM model is not economical for a quick preliminary analysis of an ITPS. Such panels may also be represented as a thick plate that is continuous, orthotropic, and homogenous for which analytical and 2D FEM solutions (Tan, Fung, and Lok, 2003) are available. The extensional stiffness matrix [A], coupling stiffness matrix [B], bending stiffness [D] and the transverse shear stiffness terms A44 and Ass were calculated by analyzing the unit cell. For bending analysis of the plate, a closedform solution was obtained by using the FSDT method. Thus, advanced knowledge of the orthotropic thick plate stiffness was essential for successful implementation of the FSDT for plate analysis. Typically, plate analyses yield information on deflections, force, and moment resultants at any point on the plate. The micromechanical analysis procedures developed in this study were used to determine the stresses in the face sheets and the webs. Then failure theories such as the TsaiHill criterion were used to determine if the stresses were acceptable or not (Gibson, 1994). In the derivation of the stiffness parameters the following assumptions were made: * Assumption 1: The deformation of the panel was less than 5% when compared to the panel thickness. * Assumption 2: The panel dimensions in the ydirection were much three to six larger than the unit cell width 2p depending on the number of unit cells. * Assumption 3: The face sheets were thin with respect to the core thickness. * Assumption 4: The core contributes to bending stiffness about the xaxis but not about the yaxis. * Assumption 5: The face and web laminates were symmetric with respect to their own mid planes. * Assumption 6: The core was sufficiently stiff so that the elastic modulus in the zdirection is assumed to be infinite for the equivalent plate. Local buckling of the facesheets does not occur and the overall thickness of the panel was constant. Previous researchers adopted these assumptions in the derivation of stiffness parameters of sandwich panels with corrugated core (Libove and Hubka, 1951), Ccore, (Fung et al., 1993), and Zcore (Fung et al., 1994). The inplane and outof plane stiffness governing the elastic response of a sheardeformable sandwich panel were defined in the context of laminated plate theory incorporating FSDT described by Vinson (1999) and Whitney (1987). The appropriate stiffness of the orthotropic plate may be obtained by comparing the behavior of a unit cell of the corrugated core sandwich panel with that of an element of the idealized homogeneous orthotropic plate (Figure 23). The inplane extensional and bending response, and outofplane (transverse) shear response of an orthotropic panel were governed by the constitutive relation (Equation 21) where e and y were the normal and shear strains, K was the bending and twisting curvatures, [A], [C], and [D] were the extensional, shear, and bending stiffness of the ITPS. N [A] ~ , Q = [C] \ LJ!= [C] [D K 1 (21) or {F= [K]{D} Geometric Parameters The corrugated sandwich panel for use as an ITPS was simplified to contain the least number of design parameters. The simplified geometry provided a useful preliminary design that can be improved upon with more design variables. The reduced design variables simplified the micromechanical procedure. Consider a simplified geometry of the corrugated core unit cell in Figure 22. The zaxis is in the thickness direction of the ITPS panel. The stiffer longitudinal direction is parallel to the xaxis, and the yaxis is in the transverse direction. The unit cell consists of two inclined webs and two thin face sheets. The unit cell is symmetric with respect to the yzplane. The upper face plate thickness (TF) and material can be different from the lower plate (tTB) as well as the web (tw). The unit cell can be identified by six geometric parameters(p, d, tF, tBF, t, 0). All of the geometric parameters had a direct or indirect effect on the thermal and mechanical response of the ITPS. If the length of the unit cell (p) is increased then there would be fewer unit cells in the panel and the stiffness of the panel decreased. However, the mass per unit area of the unit cell decreased because of decreased number of unit cells which would make the ITPS lightweight. The height between the top and bottom facesheets (d) dictates the thermal response of the unit cell. If the height was large then the heat conduction path was increased which decreased the maximum bottom face sheet temperature. The thickness of the webs and facesheets dictated the thermal and structural response of the unit cell. A thick face or web reduced the bottom face sheet temperature of the unit cell but increased the mass of the ITPS panel. The web angle contributed to the thermal and structural response of the unit cell. A rectangular web angle configuration would result in the stiffest unit cell in the xdirection, but in contrast would result in large panel deflections due to shear effects from the web. Futhermore, a rectangular corrugated core would also shorten the heat conduction path and increase the bottom face temperature. A triangular web angle configuration would result in infinite shear stiffness, minimum panel deflection, and an increase in the effective thermal conductivity of the web would decrease the bottom face temperature. However, a triangular web configuration created major buckling problems because of the long unsupported web lengths. Four other dimensions (be, do, s, f) were obtained from geometric considerations (Equation 22). The ratio f/p = 0, corresponds to a triangular corrugated core, and f/p = 0.5, corresponds to a rectangular corrugated core. 1 1 d = d tTF tBF 2 2 (22a) 1 d 2 tan 0 (22b) bc =p2f (22c) s = dj +bf = sin0 cos0 (22d) Extensional and Bending Stiffness An analytical method was developed to calculate the stiffness matrix of the corrugated core sandwich panel. Consider a unit cell made up of four composite laminates (two facesheets and two webs). Each laminate has its respective material properties, and ABD matrix. The ABD matrix of each component was combined together in an appropriate manner to create the overall stiffness of the sandwich panel. The formulas for determining the ABD matrix of a composite laminate are given below (Gibson, 1994). ABD (e) z2]dz (Q)kZr k kkl (Zi zk1k (23) k=1 2 3 In Equation 23, Nis the number of laminas in the composite laminae and Q," are the components of the transformed lamina stiffness matrix. The range of e is e = 14, (1 top face sheet, 2 = bottom face sheet, 3 = left web, 4 = right web). The overall stiffness of the unit cell was determined by imposing unit midplane strains and curvature (macro deformation) to the unit cell and then calculating the corresponding midplane strains and curvatures (micro deformations) in each component. The unit cell components were the two face sheets and two webs. A transformation matrix related the macro and microdeformations. The micro deformation was defined as the local midplane strain and curvature of the facesheets and the webs under the action of a unit cell midplane strain and curvature. {De) = [TD ](e)(D M (24) In Equation 24, {D e) was the micro deformation in each component, {(D was the macro deformation of the unit cell, and T"e) was the deformation transformation matrix that related macro deformation to micro deformations. Formulation of deformation transformation matrix for the facesheets The deformation transformation matrix of the top face sheet was determined by first considering the unit cell under the action of midplane macro strains, cx, o, and macro curvature Kx, Ky, ry. Each strain and curvature was considered by itself and the resulting midplane strains and curvatures in the face sheets also called micro strains and curvatures  were derived (Equations 25 and 26). Top face sheet: {D 100d0{D S 000010 yo 0 0 1 0 0 d 7syo (25) xK 0 0 0 1 0 0 KC Ky 0 0 0 0 1 0 IC KY 0 0 0 0 0 1 Ky Bottom face sheet: S(1) r i c (At G 1 0 u d 0 0 co mc 0 1 0 0 0 YO 2 YO S1 d Yy (26) xK 0 0 0 1 0 0 KC Ky 0 0 0 0 1 0 Ky Ky 0 0 0 0 0 1 Ky There was a onetoone relationship between midplane macro and micro strain as well as a one to one relationship between macro and micro curvature as indicated by unity along the diagonal of the transformation matrices. Using the assumptions that the inplane displacements u and v were linear functions of the zcoordinate and that the transverse normal strain Ez was negligible (Gibson, 1994) the d/2 factor was used to relate the macro curvatures to the midplane micro strains. Formulation of the web deformation transformation matrix Right Web Formulation of the deformation transformation matrix for the webs was relatively complicated because of the need for a coordinate transformation due to the inclination of the webs. Consider a global xyz coordinate system and a local xyz coordinate system (Figure 24). The origin of the web's local axis is at the top face sheet and web junction point. The transformation from the global to local coordinate axes requires a rotation and translation. The transformation from the global to local displacements only requires a rotation (Appendix A). In Equation Ai, 0 was the angle of web inclination of the right web, the first matrix was the rotation matrix and the second vector was a translation vector. Consider the unit cell of the ITPS panel under the action of midplane macro strains, exo, o, y o and macro curvature Kx, Ky, Kxy. From Assumption 4 given in the Analytical Approach section of Chapter 2, it can be noted that (,4) = 0 and e 34) = 1 when the unit cell was subjected to eM = 1 and e = 1. Derivation of the micro strains on the webs due to a macro curvature was more complex to determine; therefore a detailed discussion was appropriate (Appendix A). The micro strains and curvature in the right web due to a macro unit curvature along the yaxis (Ky) was derived; all other curvatures were set equal to zero. a2w a2w 2w K= = O K ic =1= 2 = 2 = 0 (27) X X2 Y y2 Y axay Starting with Equation 27 and following the detailed derivation in Appendix A leads to the transformation matrix of the left and right web (Equations A6 and A7). Stiffness matrix determination using the strain energy approach As the unit cell was deformed by the unit macro strains and curvatures, it stored energy internally throughout its volume. The total strain energy in the unit cell was the sum of all strain energies in the individual components (faces and webs). Since the deformations of the webs were a function ofy integration of Equation 29 was done with respect toy. The integration limits were from 0 to s, s being the length of the webs. By substituting Equation 24 into Equation 28 the strain energy of the web in terms of macro deformations from the unit cell was represented. The strain energy in each laminate in terms of the global deformation was written as {D}M. The stiffness matrix K of the idealized orthotropic panel was derived as the sum of individual stiffness contributions from each component. U 1 (2p)2 ({D}M[K]{D} ) U(e) (28) 2 e=l U(e) =(2p) (TD K e(T D (29) 0 K(e) 1 jL( T)' K(eT eldj7 (210) 2pK^ y (210) 2p0 K= ZK(e) = j(Te)j [K](e)(Te) (211) e=l 2 e1 Transverse Shear Stiffness, Ass For a corrugated core sandwich structure loaded in shear transverse to the corrugations (by shear stress rxz or shear force Qx), it was recognized that the face sheets and core would undergo bending deformation (Libove and Hubka, 1951). For the determination of A55 the shear stress in the face sheets were neglected because of its small thickness and classical plate theory was used. To determine the shearing stiffness due to Qx, the shear stress in the webs due to Qx was determined. Figure 25 depicts a free body diagram of the corrugated core panel unit of length dx in the xdirection where only the stress which act in the xdirection were shown and considered. The stress values shown were average stresses over the faces of an element which was assumed to be very small. A summation of the forces in the xdirection yielded Equation 212. aF (F + Ax F)AyAz +2(zxAx( j))AyAz = 0 (212) Following the procedure in Appendix B, the closed form equation of the shear stresses in the webs (rx,) due to Qx was determined. The shear strain energy density (strain energy per unit area of the sandwich panel) was calculated from either the web shear stress in Equation B6, or from the shear force Qx. Equating the two shear strain energy density terms resulted in Equation 213. In Equation 213, A5s was the only unknown term in the expression. U, dy (213) P 0 GX 2A55 Transverse Shear Stiffness, A44 Formulation of the transverse shear stiffness (A44) of the panel was relatively complicated because certain conditions needed to be fulfilled (Fung, Tan, and Lok, 1996). Figure 26A depicts a sandwich panel of unit length in the xdirection subjected to unit transverse shear, Qy 1. The horizontal force Y p /d provided equilibrium. Point A in Figure 26A was assumed to be fixed to eliminate rigid body movements of the unit cell. The relative displacements (6y and 6,) resulted from the transverse shearing and horizontal force. Because the force was small, the displacements were proportional to Qy, thus an average shear strain was represented as: = + 5 (214) d p Due to antisymmetry only half of the unit cell was considered for analysis (Figure 27A). The unit shear force resultant was divided into force P acting on the top face sheet and force R acting on the lower face sheet. A shear force F was assumed to act on the top face sheet at point A where there were no horizontal forces due to antisymmetry, and a force (1F) was determined through a summation of the forces in the zdirection. Under the action of all these forces in the half unit cell the displacements are shown in Figure 27B. From Figure 27 there were three unknown forces and five displacements that were solved through the energy method. The total strain energy in the half unit cell was the sum of the strain energies from each individual member (AB, BC, DE, BE, and EG). The strain energy due to bending moments was considered while the strain energy due to shear and normal forces was neglected (Appendix C). Castigliano's second theorem (Equation 215) states that displacement is equal to the first partial derivative of the strain energy in the body with respect to the force acting at the point and in the direction of displacement (Hibbeler, 1999). Castigliano's second theorem was used to find the unknown forces and displacements from Figure 27B. , a (215) Since the overall thickness of the sandwich panel remained constant during distortion, the boundary conditions were 3 = 3G and A = O. Since the half unit cell was under unit shear, then P+R = 1. The two boundary conditions along with Castigiliano's second theorem lead to a system of two linear equations with two unknowns. "' =0 (216) aF (217) 8P 8R A substitution of Equation Cl into Equations 216 and 217 resulted in a solution of the unknown forces P, F, and R of the system of linear equations. Substitution of Equations Cl into Equation 215 along with the values of the unknown forces yielded the solutions of the displacements (Appendix C). The displacements of half the unit cell were 5, = 8 + ~G and 8, = 83 = G5 in the y and z directions. By a use of the force distortion relationship (Libove and Batdorf,1948) the transverse shear stiffness A44 was determined (Equation 218). A44 Q 1 (218) Y Sy z 1 1 c S + Y( +8)+ , d p d p Face Sheet and Web Stress Determination Equation 24 has shown that through use of the deformation transformation matrix, the local strains and curvature of the faces and webs due to a unit cell deformation can be determined. As a result, the stresses in each component were found by multiplication of the local strains and curvatures in a particular component with the corresponding transformed lamina stiffness matrix. r (e) r (M) S= [T] e){ (219) [](7e) = [e) ( ) + Z{Ker)) (220) The previously derived deformation transformation matrices for the webs were good for stiffness prediction. However, they did not yield accurate stress results when compared with a finite element analysis (FEA). For example; the assumption, C' = 0, constrained the webs from expanding in the y direction due to Poisson effect. This led to stresses in the y direction that were not present in the 3D FE analysis. Therefore, corrections were applied to the deformation transformation matrix to analytical predict accurate web stresses. The refined web stress deformation transformation matrices (Appendix A) contained a poisson's ratio that took into account the lateral contraction or elongation ( co) of the web due to a unit macro midplane strain in the xdirection (Fo ) and a unit macro curvature in the xdirection (Ix). The micro midplane strains ( cy) in the web due to either KM = 1 or K = 1 were removed because there was no force in the y direction that was causing a midplane strain. From Equation A6 a relation between macro midplane shear strain and macro curvature was derived as the product of the unit cell deformation and a function. (y = YI cosO (221) K() = K (cos3 0 + 2 cos sin2 0) (222) Equations 221 and 222 treated the web as an unresisting member to the unit cell when it was deforming. For example, when the unit cell underwent a unit midplane shear strain or a unit curvature, the faces were compliant with that deformation but the webs resisted that movement. Equations 221 and 221 are true if the webs were at a right angle to the face sheet, however the equations were proved incorrect in general and the assumption that the webs resisted deformation was proved correct by conducting several FE analyses for various web angle inclinations. An analytical procedure that took into account the web's resistance to deformation was established to determine the micro midplane shear strain and micro curvature (f(p,d, tF,tBF,tw,O) and g(p,d,tF,tBF, t,O)). Midplane micro shear strain in the web An analytical procedure was developed to relate macro midplane shear strain to micro midplane shear strain. The analytical method took into account the resistance to shear that the webs experienced when the unit cell was under midplane shear. The top and bottom face sheets were investigated separately under the action of a shear force (Figure 28). The resistance to shear by the webs was included as a shear force acting on the webs (F,). The total top face, bottom face, and web shear strain under the action of the shear forces are as follows: l = (l + 2(P f)) (223) 72)= (yf + y3(P f)) (224) 1 = (f y(p f)) (225) where F F, F F + F, F W71 = I 3 = 4 = B (226) tTFGTF tTFGTF tBFGBF tBFGBF The shear force in the webs was F = Gt y, (227) There were three unknown shear forces, FT, Fw, FB. The three unknown forces were determined by solving the system of three linear equations with the three unknowns (Equation 2 28). y/l =1 (2) = 1 (228) 0 = G ty F Equation 228 yielded the three shear forces that acted on the facesheets during a unit shear strain. Substitution of the known shear forces from Equation 228 into the web shear strain equation (Equation 229) yielded the macro to micro midplane shear strain relation of the web. f(p,d, tTF,tBF, t,, 0)= Y( = (r,f Y(p f)) (229) Micro curvature in the y direction for the webs An analytical procedure was developed to relate macro ydirection curvature to micro y  direction curvature in the webs. The analytical method took into account the resistance to the curvature that the webs experienced when the unit cell experienced curvature in the ydirection. Half the unit cell was investigated under the action of couples that acted on the faces (Figure 2 9). Consider the half unit cell under the action of an end couple that causes unit curvature in the ydirection. The half unit cell end couple was represented as three end couples acting on the faces and webs (Cr, Cw, and CB). The slopes of the faces and web due to an end couple were obtained from beam theory (onedimensional) formulas (Cook and Young, 1999). There were three unknown couples (CT, Cw, and CB) in Figure 29. To solve the three couples, a system of three linear equations was required. The three equations came from the boundary conditions. The first two boundary conditions were that the slopes of the top and bottom face sheet must equal the slopes of the faces when c = 1. The last boundary condition was that the difference of slope between the face and web junction point (A and B) must equal the slope of the web. After solving the system of linear equations the curvature of the webs was determined by dividing the couple acting on the web with the flexural stiffness of the web (equivalent El). (C + C,)f C,(p f) + =p (230) (EI)TF (EI)TF (C, C,)(p f) Cf + =p (231) (EI)BF (EI)BF (C C)(p f) (CB +C,)f Cs (232) (EI)BF (E),TF (EI), g(p,d, tTF, tBF, t, ) = (3 (233) Y (EI) Conclusion Finite element analysis was commonly used to analyze sandwich structures; however, a full 3D FEA was not economical for a preliminary analysis of a structure. Such panels can be represented as an orthotropic thick plate for which analytical solutions can be derived. A method to homogenize the corrugated sandwich panel into an orthotropic thick plate was presented. A detailed formulation of the bending, extensional, coupling, and shear stiffness of the ITPS panel was determined through an energy method. The analytical models were capable of handling laminated composite materials for the face sheets and webs of the sandwich panel. Furthermore, different materials can be used for the face sheets and web. For example, the hot side (outer) face sheet can be composed of titanium, aluminum or other super alloys and the cool side (inner) face sheet can be a polymer matrix composite. The webs can be made of other materials such as titanium, aluminum, or composite. The resistance of the webs due to shear and curvature was analytically determined. For a web configuration other than a rectangular configuration the webs were resisting the unit cells strain or curvature. Figure 21. Corrugated core. A) Zcore, B) Ccore. Figure 22. Simplified unit cell dimensions Figure 23. Equivalent orthotropic thick plate for the unit cell corrugated core sandwich panel. Figure 24. Global and local coordinates of the unit cell, faces and webs. A F+ F + a F A A ZF Figure 25. Small element removed from a body, showing the stresses acting in the xdirection only A) side view, B) isometric view. Y=p/d Qv=11 Y=p/d Y=p/d S 4r ~jQy1 Figure 26. Unit cell subjected to unit transverse shear and horizontal force. '=p/d fF B 4 pld p p/d P i. p/d Figure 27. A) Half unit cell of the corrugatedcore unit cell. Top Face Sheet ' 71 72 f pf 6y, r~r'31 L G 6zG B sandwich panel. B) Deformations of the half Bottom Face Sheet Figure 28. A) Free body diagram of the top face sheet under the action of midplane shear strain. B) Free body diagram of the bottom face sheet under the action of midplane shear strain. rCT Figure 29. Half unit cell under the action of end couples at the faces. FwFT i 4I I CHAPTER 3 FINITE ELEMENT VERIFICATION Extensional and Bending Stiffness For verification of the effectiveness and prediction capability of the analytical models, an ITPS sandwich panel with the following dimensions was analyzed by a FE analysis: p = 80 mm, d = 80 mm, tTF = 1 mm, tBF = 1 mm, tw 1 mm, = 750, a = 0.65 m, b = 0.65 m. An AS/3501 graphite/epoxy composite (El = 138 GPa, E2 =9 GPa, 12 = 0.3, G12 = 6.9 GPa) with four laminae in each component and a stacking sequence of [(0/90)2] was used as an example to verify the analytical models. The facesheets and the webs were assumed to be made of graphite/epoxy laminates. A representative volume element or unit cell (Figure 31) approach was adopted to obtain the stiffness properties. An FE analysis was conducted on the unit cell using the commercial ABAQUSTM finite element program. Eight node shell elements were used to model the face sheets and webs of the unit cell. The shell elements have the capability to include multiple layers of different material properties and thicknesses. Three integration points were used through the thickness of the shell elements. The FEM model consisted of 18,240 nodes and 6,000 elements to guarantee convergence. The ITPS plate stiffness was obtained by modeling the unit cell with shell elements and forcing the unit cell to six linearly independent deformations (Marrey and Sankar, 1995). The six linearly independent strains were: (1) c' = 1 and maintaining the rest of the macroscopic strains and curvature zero; (2) ', = 1 and maintaining the remaining strains and curvature zero; and similarly (3) y' = 1; (4) ," = 1; (5) ,M = 1; and (6) Kr = 1. Strains were imposed to the FEM model by enforcing periodic displacement boundary conditions (Table 31). To prevent rigid body motion and translation, the unitcell (Figure 32) was subjected to minimum support constraints. The top and bottom surfaces were assumed to be free of traction. The faces x = 0 and x = a had identical nodes on each side as well as the other faces, y = 0 and y = b. The identical nodes on the opposite faces were constrained to enforce the periodic boundary conditions. Figure 33 represents the deformations of the unit cell as a result of imposing the periodic boundary conditions. The nodal stresses of the boundary nodes were obtained from the finite element output after the analyses. Nodal moments were obtained by multiplying the nodal forces with the distance from the midplane. The nodal forces and moments of the boundary nodes were then summed to obtain the force and moment resultants (Equation 31). By a substitution of the values from Equation 31 into the plate constitutive relationship, the stiffness coefficients in the column corresponding to the nonzero deformation were computed. The same procedure was repeated for other deformation components to obtain and fully populate the unit cell stiffness coefficients. [N,,M,]= [,z] ((a,y,z) (31) The finite element result indicated that Equation 211 provided an excellent prediction to determine the extensional, coupling, and bending stiffness (Table 32). The finite element results were in excellent agreement with the formulation of the derived stiffness parameters of the corrugated core sandwich panel. All analytical results were within 2% difference when compared to the finite element results from Equation 31. For further verification, the stiffness matrix of an ITPS sandwich panel with a general laminate stacking sequence other than a crossply laminate was determined and compared with the finite element results. The ITPS sandwich panel had the same geometric dimensions and material properties but each component was now composed of a different laminate stacking sequence, [45/45]s. The analytical procedure to determine stiffness was capable of providing accurate stiffness results to within 3% difference when compared to the FEM results (Table 33). The analytical stiffness prediction procedure does not limit the designer to just cross ply laminates. The ITPS sandwich panel can be composed of various materials that make it perform at its optimum for mechanical and thermal applications. Supper alloys such as titanium and Inconel are possible choices for the top face sheet due to their high service temperatures. Aluminum alloys, beryllium, and composite materials are of preference for the bottom face sheet due to their high specific heat properties. An ITPS with different materials for the top and bottom face sheet resulted in an unsymmetric stiffness matrix with a nonzero coupling stiffness matrix, [B] # 0. For verification of obtaining unsymmetric stiffness matrix properties from the analytical model, the analytical stiffness matrix result of an ITPS sandwich unit cell with Inconel as the material property for the top face sheet and graphite epoxy with a [(0/90)2] laminate stacking sequence for the webs and bottom face sheet was compared with the finite element results from the periodic boundary condition approach. The percentage difference between the finite element results and the analytical results are less than 2% (Table 35). The analytical procedure for predicting the stiffness matrix of an ITPS with different materials properties was capable of predicting accurate stiffness results. The analytical procedure was robust and effective for all possible material choices and laminate stacking sequences of the faces and webs. The results indicated that the micromechanical procedure presented in Chapter 2 for predicting the stiffness matrix of an ITPS sandwich panel provided excellent results for any type of laminate stacking sequence. The analysis was robust because it was capable of providing accurate stiffness results for an isotropic and orthotropic material of any laminate stacking sequence. Stress Verification Midplane Shear Strain and Curvature in the Webs For midplane shear strain verification, the same FEM unit cell element and mesh from Figure 31 with the same material properties and cross ply layup was investigated. The web angle inclination was changed from 550 to 900 and the unit cell was subjected separately to a periodic unit midplane shear strain and a periodic ydirection curvature. The corresponding web midplane shear strain and web curvature were extracted from the FEM output after analysis. The results of micro midplane shear strain and micro curvature from FEM and Equations 229 and 233 were compared and plotted (Figure 34). From Figure 34 the agreement between the FEM and analytical results of Equations 229 and 233 was less than 2%. The analytical equations that were derived previously accounted for the resistance effect of the webs when the unit cell was subjected to midplane shear of bending in the macroscale sense. Accurate stress results were expected when compared with FEM. Stress Verification The refined web stress deformation transformation matrix was verified by an FE analysis. A known strain was applied to the unit cell and the corresponding local stresses on the faces and webs were obtained by multiplying the deformation vector with the refined web stress deformation transformation matrix. The known strain was applied to the finite element model by enforcing periodic displacement boundary conditions from Table 31. The stress results from Equation 220 and the stresses from the FEM output after analysis were plotted (Figures 35 to 310). The analytical results were within 2% difference when compared to the finite element stress output. Results from this dissertation verified the accuracy and validated the procedure that predicts the local stresses of the ITPS sandwich panel. The refined web stress deformation transformation matrix did an excellent job in predicting the correct strain in the webs which resulted in stress results that were in good agreement with the FEM output. Furthermore, the refined web stress deformation transformation matrix did not alter the stiffness matrix. The refined web stress deformation transformation matrices in Appendix A were used to compute the ITPS stiffness (Table 34). Analytical1 results represent the stiffness values obtained from using the facesheet deformation transformation matrices and the web deformation transformation matrix (Appendix A). Analytical2 results represent the stiffness values obtained from using the facesheet deformation transformation matrices and the refined web stress deformation transformation matrices (Appendix A). The refined web stress deformation transformation matrices have the capability to accurately predict stresses in each ITPS component and accurately predict the ITPS stiffness. The previously derived deformation transformation matrices outputs excellent stiffness results but erroneous stress results. Transverse Shear Stiffness (A44) Verification The finite element verification of the A44 stiffness term consisted of a two part finite element procedure. An assumption was made that the ITPS behaved like a cantilevered one dimensional beam. The equivalent cross sectional properties of the beam were: axial rigidity EA, flexural rigidity EI, and shear rigidity A44. The beam consisted of 10 unit cells and was clamped on the left end (Figure 311). Eight node solid elements were used to model the 1D plate. First, a known end couple was applied to the tip of the beam and the corresponding tip deflection was determined from the finite element output after analyses. The tip deflection was also derived analytical in terms of the moments and flexural rigidity as: MI2 v 2E (32) 2El The flexural rigidity of the beam (El) was determined using Equation 32. The couple was then removed and a transverse force was applied at the tip of the cantilevered beam. The tip deflections were obtained from the finite element output after analysis. The tip deflection can also be determined by Equation 33. F13 Fl vt = + t (33) 3EI A4 The shear rigidity (A44) was determined by using finite element tip deflection in Equation 33 along with the flexural rigidity result from Equation 32. This finite element verification procedure was done for various web angles. The finite element result along with the analytical result from Equation 218 are illustrated in Figure 312. The finite element results were in good agreement with the analytical formulation of A44. The percentage difference between the finite element results and the analytical result did not exceed 7%. The finite element deformation of the cantilever beam is shown in Figure 312B. Conclusion The stiffness results between the analytical model and the FE analysis were within 2%, thus meeting the less than 5% requirement that was stated in the objective. The refined web stress deformation transformation matrix made incremental percent improvements to the ITPS stiffness when compared with FE results. Both the deformation transformation matrix for the webs and the refined web stress deformation transformation matrix can be used in predicting stiffness, but only the latter matrix can be used for stiffness and stress prediction. The refined web stress deformation transformation proved to be a good method in determining stresses in the webs and faces under the periodic displacement boundary conditions. Table 31. Periodic displacement boundary conditions. u(a,y) v(a,y) w(a,y) u(x,b) v(x,b) w (x,b) x,(a,y) Oy(a,y) ,(x, b) Oy(x, b) ,,( i,y) v(0,y) . ,y) u(x,0) v(x,) wx,) (x, ) () (0, y) o0(x,0) oy(x, 0) Exo a 0 0 0 0 0 0 0 0 0 =1 EyO 0 0 0 0 b 0 0 0 0 0 1 xyO 0 a/2 0 b/2 0 0 0 0 0 0 1 Kx= az 0 a2/2 0 0 0 0 a 0 0 1 Ky 0 0 0 0 bz b2/2 0 0 b 0 1 Tcxy 0 az/2 ay/2 bz/2 0 bx/2 a/2 0 0 b/2 1 Table 32. Nonzero [A], [B], and layup. Stiffness Analytical FE An [N/m] 2.23E+08 2.20E+08 A12[N/m] 5.43E+06 5.43E+06 A22[N/m] 1.48E+08 1.48E+08 A66[N/m] 1.43E+07 1.41E+07 Dni[Nm] 2.76E+05 2.78E+05 D12[Nm] 8790 8690 D22[Nm] 2.37E+05 2.37E+05 D66[Nm] 2.23E+04 2.22E+04 [D] coefficients for an ITPS sandwich panel with [0 / 90]s % diff. S1.36% 0.00% 0.00% 7 1.42% 0.72% 1.15% 0.00% 1 0.57% Table 33. Nonzero [A], [B], and [D] coefficients for an ITPS sandwich panel with a [45 / 45]s layup. Stiffness Analytical FE % diff. Ani[N/m] 1.14E+08 1.13E+08 0.81% A12[N/m] 6.29E+06 6.29E+06 0.00% A22[N/m] 9.05E+07 9.04E+07 0.00% A66[N/m] 7.24E+07 7.24E+07 0.71% Dii[Nm] 1.57E+05 1.58E+05 0.35% D12[Nm] 1.01E+05 1.01E+05 0.00% D22[Nm] 1.45E+05 1.45E+05 0.00% D66[Nm] 1.14E+04 1.14E+04 0.07% Table 34. Comparison of the nonzero [A], [B], and [D] coefficients for an ITPS sandwich panel between the refined transformation matrix and the deformation transformation matrix. Stiffness Analytical1 Analytical2 FE % diff(FE1) %diff (FE2) An [N/m] 2.24E+08 2.23E+08 2.20E+08 1.37% 1.33% A12[N/m] 5.43E+06 5.43E+06 5.43E+06 0.00% 0.00% A22[N/m] 1.48E+08 1.48E+08 1.48E+08 0.00% 0.00% A66[N/m] 1.43E+07 1.40E+07 1.41E+07 0.98% 0.79% Dji[Nm] 275920 275870 277640 0.62% 0.64% D12[Nm] 8788.2 8691.5 8691 1.12% 0.01% D22[Nm] 2.37E+05 2.37E+05 2.37E+05 0.08% 0.00% D66[Nm] 2.23E+04 2.21E+04 2.21E+04 1.03% 0.08% Table 35. Nonzero [A], [B], and [D] coefficients for an ITPS sandwich panel with Inconel for the top face sheet and graphite epoxy for the webs and bottom face sheet. Stiffness Analytical FE Anl[N/m] A12[N/m] A22[N/m] A66[N/m] Dii[Nm] D12[Nm] D22[Nm] D66[Nm] Bil[Nm] B12[Nm] B22[Nm] 3.69E+08 6.54E+07 2.94E+08 8.58E+07 5.09E+05 1.05E+05 4.70E+05 1.37E+04 5.84E+06 2.40E+06 5.83E+06 3.66E+08 6.54E+07 2.94E+08 8.59E+07 5.11E+05 1.05E+05 4.69E+05 1.37E+05 5.84E+06 2.40E+06 5.85E+06 % diff. 0.89% 0.01% 0.14% 0.10% 0.31% 0.02% 0.14% 0.01% 0.02% 0.02% 0.29% B66[Nm] 2.87E+06 2.92E+06 1.94% 4;. Figure 31. Finite element unitcell mesh b Figure 32. Boundary conditions imposed on the plate to prevent rigid body motion. An arrow pointing at a black dot indicates that the displacement of that point is fixed in the direction of the arrow. Y =1 S=1 = 1 =1 Figure 33. Deformations of the ITPS due to periodic boundary conditions 0.5 E E 0.4 T 0.3 CU ( 0.2 cU  0.1 0a 0 FEM * Analytical (S, 0.5 0.4 E 0.3 S0.2 0.1 0.1 0 FEM * Analytical 0 1 1 0 1 40 60 80 100 50 60 70 80 90 A Web Angle Inclination degreee) Web Angle Inclination degreess) B Figure 34. Comparison of FEM and analytical micro strain for A) midplane shear strain B) curvature. 0" = 1 S, = 1 Face Stress xdirection (s =1) 1 S 0 c 0 t 0 IC H 0.5 ci, (0 O LL. 1 1 S0.5 0 c, j g 0.5 Stress x (Pa) x 1010 1 5 0 0.5 (0 L(. U0 ) 0.5 I 0 S0.5 1 TFF  TFA SBFF X BFA 1 i 2.4 2.5 2.6 2.7 2.8 2.9 Stress y (Pa) x 109 Web Stress ydirection (P =1) 00F 0 0A E 900F 900A 2 1 0 1 2 3 Stress y (Pa) x 109 Figure 35. Stresses in the x andy direction of the top face, bottom face, and web for a unit cell strain ofsc = 1. (Note: "A" are the analytical results and "F" are FEM results. The "00" and "900"are the ply orientations of the web laminate. All values in the yaxis are normalized with respect to the face thickness and web length.) S Face Stress xdirection (s yo=1) Face Stress ydirection (s Face Stress ydirection (s =1) 26 2t 28 29 Stress cx (Pa) x 109 Figure 36. Stresses in the x andy direction of the top face and bottom face for a unit cell strain of cs = 1. (Note: "A" are the analytical results and "F" are FEM results. The "0" and "900"are the ply orientations of the web laminate.) Stress x (Pa) x 1010 Web Stress xdirection (xo =1) SO0F 00A E 900F S900A ) 5 10 15 S05 0 1 ci 05 L 5 5 10 Stress o (Pa) 15 x 1010 Face Stress ydirection (s =1) Face Stress xydirection (xy =1) STFF  TFA 0 BFF X BFA 6 65 7 Stress Fy (Pa) 75 8 x 109 Web Stress xydirection ( Web Stress xydirection (y =1) xyAJ 11 12 13 14 15 16 Stress F (Pa) x 109 Figure 37. Stresses in the x andy direction of the top face, bottom face, and web for a unit cell strain ofyo = 1. (Note: "A" are the analytical results and "F" are FEM results. The "00" and "900"are the ply orientations of the web laminate. All values in the yaxis are normalized with respect to the face thickness and web length.) Face Stress xdirection (KM=1) MFace Stress ydirection (= Face Stress ydirection (K 1) 05 1 S05 o LL I 0) 05 Stress c (Pa) x 109 Web Sessxdirection 1) 0 TFF  TFA 0 BFF  BFA 1 05 0 05 1 15 Stress y (Pa) x 108 Web Stress ydirection (KM=) 2 0 2 4 6 Stress x (Pa) x lo9 Figure 38. Stresses in the x andy direction of the top face, bottom face, and web for a unit cell strain of Kc = 1. (Note: "A" are the analytical results and "F" are FEM results. The "00" and "900"are the ply orientations of the web laminate. All values in the yaxis are normalized with respect to the face thickness and web length.) 05C 0  05 c 0 CU LO L a) 05 o 05 t r 05 1 0 Stress y (Pa) Face Stress xdirection (K M=1) Y 0 TFF  TFA 0 BFF BFA Face Stress ydirection (KM=1) Y r 0.5 0) 0 Ic r , 0.5 LL 1   0.5 N () ()  0 , 0.5 LL 1  1.5 0.5 c0 I 0 0.5 1  1 2.5 2 1.5 Stress (Pa) 1 x 105 Stress y (Pa) Web Stress ydirection Web Stress ydirection (K =1) 4 Stress y (Pa) y Figure 39. Stresses in the x andy direction of the top face, bottom face, and web for a unit cell strain of Ky = 1. (Note: "A" are the analytical results and "F" are FEM results. The "0" and "90"are the ply orientations of the web laminate. All values in the yaxis are normalized with respect to the face thickness and web length.) 1 0.5 0 0.5 1 1.5 Stress x (Pa) x 108 Web Stress xdirection (KM=1) 0 x 106 Face Stress xydirection (KM =1) xy 0 TFF TFA 0O 0 BFF SBFA 2 0 2 4 4 2 0 2 4 Stress F (Pa) xy C 05 0) 0 0 c j 05 x 108 Web Stress xydirection (M 1) xy S00F S00A E g ooF :]. S900F I S900A 4 35 3 25 2 1 5 StressF (Pa) x 10 Figure 310. Stresses in the x andy direction of the top face, bottom face, and web for a unit cell strain of ic = 1. (Note: "A" are the analytical results and "F" are FEM results. The "0" and "90"are the ply orientations of the web laminate. All values in they axis are normalized with respect to the face thickness and web length.) ^ <1 Figure 311. Truss core modeled as a cantilever beam with ten unit cells. 1  05 N /) (/) c 0 0 I 8 05 LL 1 I C~I 100 1020 40 60 80 Web Angle Inclination degreee) Figure 312. A) Finite element and analytical result for the transverse shearing stiffness B) Deformation of the beam due to the couple and transverse force o Analytical * FEM E S300 . 300 CHAPTER 4 THERMAL ANALYSIS OF AN ITPS UNIT CELL Introduction Thermal analysis of an ITPS involved the modeling of complex heat transfer mechanisms in a severe thermal environment. The parameters of the severe thermal environment were: (1) pressure variation and (2) temperatures variation (Blosser, 2004). The ITPS was a multifunctional structure that possessed load bearing capabilities as well as provided insulation for the space vehicle. The heating rates on an ARMOR TPS during ascent for the windward surface of the Space Shuttle were investigated by Dorsey et al. (2004). The heat flux caused the ITPS temperature to rise dramatically and as a result it caused panel thermal loads and stresses, panel thermal deflection, and panel thermal buckling. Knowing the behavior of the ITPS to a change in temperature is a critical design need because panel deflection, panel buckling (local or global), panel temperature, and panel yielding are critical functions of an ITPS that influence the sizing design. Typically, simplified, onedimensional models (Poteet, AbuKhajeel, and Hsu, 2004) are used to predict the thermal performance of a thermal protection system when subjected to realistic temperature distributions. Therefore, a one dimensional finite element heat transfer analysis was done by Bapanapalli, Martinez, Sankar, Haftka, and Blosser (2006). The heat transfer that was considered for a preliminary analysis of the ITPS thermal loads and moments was at x=205.7 in (5.22 m) from the tip of the nose (Figure 41). The heat transfer analysis determined the maximum bottom face sheet temperature of the ITPS and the core temperature distribution at any particular reentry time after analysis. The core temperature distribution was plotted for three reentry times (450 s, 1575 s, and 1905 s) (Figure 42). The 1905 s reentry time corresponds to the time when the maximum bottom facesheet temperature was reached for that particular ITPS. Each temperature distribution resulted in thermal force resultants and thermal moments that caused the sandwich structure to deform. In the case of laminate composites the thermal force resultants and moments are computed as the product of the lamina stiffness matrix, CTE, temperature change, and height of the laminate (Equation 41). [N',M ]= [Q]{af}ATdz (41) In Equation 41, [Q] is the transformed lamina stiffness matrix, {a} is the column matrix of the coefficient of thermal expansion (CTE) and AT is the temperature change from the reference temperature. However, these equations do not apply in predicting thermal forces and moments to the present ITPS structure because there are no layers in the ITPS unit cell. Therefore, a micromechanics (homogenization) approach was used to predict the ITPS thermal forces and moments from a given reentry temperature distribution. Instead of using Equation 41 to predict the thermal forces and moments, consider the thermoelastic laminate constitutive relation (Equation 42). M} [K]{} {N (42) In Equation 42, NTand MA are the ITPS' thermal force resultant and moment due to a temperature change. The thermal force resultants and moments are the forces and moments that act on the unit which causes the thermal expansions. Thermal Micromechanics Approach Thermal Force Resultants and Moments An analytical method was developed to predict the thermal force resultants and moments of an orthotropic ITPS sandwich panel composed of four composite laminates (two face sheets and two webs) as an example. Each laminate has its respective material properties and ABD matrix. The ITPS unit cell was subjected to a core temperature distribution (AT(y)) where was the local axis of the inclined web whose origin was at the top facesheet and web junction point. The change in temperature distribution equation was determined by fitting a fourth order polynomial to the temperature distribution result (Figure 42). A reference temperature at which the laminate was assumed to be stress free was assumed and the temperatures in the faces were considered to be constant because they were thin when compared to the ITPS thickness. Due to symmetry only half the unit cell was analyzed. The thermal forces and moments of the ITPS were predicted through a hold and relax method. The halfunit cell was constrained to prevent displacement and strain in the x and y directions. The top face sheet had roller supports in the z direction which allowed the webs to expand (Figure 43A). The resulting forces and moments needed to constrain the ITPS from thermally expanding were equal to the ITPS thermal forces and moments resultants. The thermal problem was broken down into two problems. The first problem was the constrained thermal problem in which force resultants that were equal and opposite to the components (face or web) thermal forces were applied to the unit cell to prevent expansion in the x and y direction (Equation 43). The equal and opposite forces prevent any expansion in the halfunit cell when it was exposed to a fourth order temperature distribution. The expansion prevention leads to zero strains. This situation was represented as Equation 43. [N, ) ABD= f Z (Qe)k(zk zk 1)AT(y)dy (43) L k=I 2 The second problem was an unconstrained halfunit cell with no temperature distribution and the forces developed in the constrained problem are "relaxed" and reversed. The "relaxed" force resultants are equal and opposite of the force resultants obtained from Equation (43). The constraints were represented by the reaction forces (Figure 43C). The constraints were unknown reaction forces that were determined via Castigliano's second theorem (Hibbeler, 1999). The strain energy due to a bending and normal force was considered. The strain energy of each component was determined and then summed to obtain the overall strain energy of the halfunit cell. There were seven unknown reactions forces to be determined. To determine the seven unknown reaction forces, seven boundary conditions were imposed to the unit cell which were that the displacement and rotations due to each reaction force was zero. The seven boundary conditions along with Castigliano's second theorem (Hibbeler, 1999) lead to a system of seven linear equations. Solving the system of linear equations led to the solution of the seven unknown reactions. By summing the reaction forces in problems one and two the desired thermal force resultant and moment for an ITPS sandwich panel were obtained (Equations 44 to 47). Ny = R +R6+(N1) + N2)) (44) d d M, =R4 + R + (N N ))+ (R6 R3) (45) 2 Y 2 Nx = (N1 2p + N 22p + N 3)2s) (46) Mx = (N) N )2p + 2(z, N ()() (47) 21=1 In Equation 47, Nis the number of discretization points in the web length. The force resultant and moments that were needed to constrain the unit cell during a change in temperature was equal to the negative of the thermal force resultants and moments of the ITPS (Equation 48). [NT,MT] = [N,M] (48) Thermal Stress The change in temperature due to reentry aerodynamic heating caused the ITPS panel to produce thermal forces and moments to the unit cell, which led to thermal stresses in the faces and webs. The thermal stresses could lead to thermal yielding, thermal buckling, or thermal out ofplane displacement of the ITPS panel. An analytical procedure was derived to obtain the thermal stresses in each component due to the reentry temperature variation. According to classical laminate plate theory, the equation needed to determine thermal stresses was: a = [Q]( aAT) (49) To determine thermal stresses in either the faces or the webs, the micro thermal deformation of each component must be known. The micro deformation of each component was determined which relates macro to micro deformation (Equation 24). Constrained case There were two cases used to predict the thermal stresses of an ITPS from a reentry temperature distribution. Each case had two different boundary conditions. The first case that was investigated was the constrained case where strains in the x and y directions were zero, however the webs were free to expand in the web length direction and constrained in the x direction. Equation 410 predicts stresses for a constrained thermal expansion problem. []= [Q]({a}AT) (410) Equation 410 was only valid for the top and bottom face sheets because the faces were fully constrained and the strains were zero. The webs however were not fully constrained and were allowed to expand in the y direction only. Therefore an analytical solution for the web expansion under a fourth order polynomial temperature distribution was derived. From Figure 4 3A, the constrained thermal problem for the halfunit cell was broken down into two individual problems. Problem 1 is Figure 43B and Problem 2 is Figure 43C. The web strain in the y direction from Problems 1 and 2 was determined and then summed to obtain the total web strain for the constrained thermal problem that took into account the web expansion. The web strain from Problem 2 was obtained by determining the midplane strain and curvature in the webs due to the reactions and relaxed forces (Figure 44). The equation that characterizes the force and moment at any location on the web were obtained by summing the forces and moments in the y direction (Appendix B). Then by inverting the webs ABD matrix and multiplying it by the force vector the midplane strain and curvature in the web was obtained (Equation 411). eCo(y) = + A22 N(y) ( 2 (411) (y)= D2 (y) The web strain from Problem 1 was determined by first modeling the free body diagram of the web from Figure 45. The webs were constrained by a force from Equation 43, which was the average force needed to constrain the web in the web length direction. The average displacement of the web was zero but the local displacements and local strains were not zero because of the fourth order temperature distribution polynomial which caused local thermal strain. Use of the constitutive relation in the y direction and substitution of that expression into the differential equation of equilibrium and then double integrating, resulted in the local v displacement in the y direction due to a fourth order temperature distribution (Equation 412). v(y N) AT(y)ddy + Dy + E (412) The two unknown constants (D and E) were solved by using two boundary conditions. The two boundary conditions were that the web displacement at both ends of the web ( y = [0, s]) were equal to zero. The web strain due to a fourth order temperature distribution was solved by taking the first partial derivative of the web displacement from Equation 412. av N a[ ]a+ CYo) AT(y)dydy + Dy +E (413) ay CL JA22 ay Summing the strain obtained from Equations 411 and 413 yielded the web strain in the web length direction for the constrained problem with consideration of web expansion. Unconstrained case In this section the stresses in the faces and webs due to the thermal forces were determined from Equation 48. Multiplying the thermal force vector (Equation 48) with the ITPS stiffness (Equation 29) yielded the thermal strain and curvature for the unit cell. { [KJ{N (414) Multiplying the ITPS' unit cell's thermal strain and curvature with the deformation transformation matrix for the faces and the refined web stress deformation transformation matrix for the webs yielded the micro deformation of the faces and webs (Equation 24). The micro deformations were the local strains and curvature that the faces or the webs experienced due to a reentry temperature distribution. The face and web stresses were determined by multiplying the micro deformation with its respective transformed lamina stiffness matrix (Equation 415). []e = [Q [I )TD([K1 NT] (415) A thermal micromechanics approach was developed to determine the thermal forces and moment resultants of an ITPS sandwich panel when subjected to any reentry temperature distribution profile. The resulting thermal stresses were determined by solving the ITPS for two cases, constrained and unconstrained. The web expansion was taken into account for the constrained cases. Finite Element Verification Thermal Force and Moment Resultants For verification of the effectiveness of the analytical models to predict thermal force and moment resultants of an ITPS, a corrugated core sandwich panel unit cell with the following dimensions was analyzed: p = 50 mm, d= 100 mm, tTF 1 mm, tBF =1 mm, tw 1 mm, = 75. An AS/3501 graphite/epoxy composite (EI = 138 GPa, E2 9 GPa, 112 =0.3, G12 6.9 GPa), with four laminae in each component and a stacking sequence of [(0/90)2] was used as an example to verify the analytical models. The same FEM model and unit cell described in Chapter three was used for the thermal FE analysis. The strains and curvatures in Table 31 were set to zero (the unit cell was not allowed to expand in the x and y directions but it was allowed to expand in the z direction). The FEM ITPS model was exposed to the 450 s reentry time temperature distribution. The resulting force resultant and moment needed to constrain the unit cell was equal to the negative of the thermal forces. The force resultants and moments from the FE analysis were determined from Equation 31 and Table 41. The finite element result indicated that using the novel micromechanics approach to determine the ITPS panel thermal forces was adequate for a thermal stress problem. The finite element results were in excellent agreement with the formulation of the derived thermal forces equation of an ITPS. The analytical results were within a less than 5% difference when compared to the FE results. Thermal Stress Verification Constrained case The same FEM unit cell representative volume element and mesh from Figure 31 with the same material properties and angle ply layup was investigated for stress verification. The FEM unit cell was exposed to the 450 s reentry temperature distribution. The reference temperature of the FEM model was room temperature (294 K). The ITPS unit cell strains in the x and ydirection were zero but the web was allowed to expand in the y direction. Figures 46 and 47 illustrate the finite element face and web strain and stress results with the analytical face and web strain and stress results. The results in Figure 46 indicated that the analytical equation for strain in the web length direction that accounts for the free expansion of the webs (Equations 411 and 413) provided accurate strain results when compared to the FEM results. The less than 1% prediction in thermal strain in the webs yielded accurate stress results when compared to the finite element results. The percentage difference between the analytical results and the finite element results did not exceed 4%. Unconstrained case The same FEM unit cell representative volume element and mesh from Figure 41 with the same material properties and angle ply layup was investigated for the unconstrained problem. For this example, periodic thermal strains from Equation 414 were applied to the finite element model. The analytical and finite element ITPS stress results in the x and y directions were plotted. The results from Equation 414 were substituted in Table 31 to obtain the appropriate periodic displacement boundary conditions for the unconstrained thermal problem. Figure 48 illustrates the FEM deformation of the ITPS unit cell after appropriately applying the periodic thermal boundary conditions and Figure 49 compares the unconstrained finite element and analytical stresses in the x and y direction. Conclusion The stress results between the analytical solution and the finite element method were within 5% difference of each other. The results indicated that the new refined web stress deformation transformation provided less than 2% difference in strain results compared to FEM. The hold and relax method was an efficient and fast way to determine thermal forces and moments. All results were less than 5% different when compared with the finite element results. Table 41. Nonzero Thermal Forces of the unit cell. Stiffness Nx[N/m] Ny[N/m] Mx[Nm/m] My[Nm/m] Analytical 578.65 317.48 15.35 11.41 FE 563.88 316.77 15.08 11.45 % diff. 2.62% 0.22% 1.79% 0.35% 9M0 7ffl~ 450 1575 2175 Time of Reentry (sec) Figure 41. Heating used for preliminary thermal load and stress analysis of an ITPS panel. Temperature Distribution 1 05 rF o  S05 1 z 450 91575 x 1905 200 0 600( 80 Temperature (K) Figure 42. A) Core temperature distribution at three reentry times. B) Resulting thermal force resultants and thermal moments. R2 Figure 43. A) Halfunit cell of the truss core sandwich panel with a temperature distribution. B) Constrained thermal problem C) Unconstrained "relaxed" expansion. (3) R2 R4 z* M(y) SN(y) Figure 44. Free body diagram of the top face sheet and web. AT(y) y 4 Figure 45. Free body diagram of the webs with an average constraining force and fourth order temperature distribution. x106 4 35 D r 3 25 C 2 2 15 * FEM SAnalytical ' W Wg w w 1 ** 05 0 03 4 0 01 02 03 04 05 06 07 08 09 1 Normalized Web Length (ybar/s) Figure 46. Web expansion for the constrained thermal problem Face Stress, xdirection Face Stress, ydirection S1 cN v N 0.5 C,, C,,  0 0 O u L_ Stress Cx (Pa) Web Stress, xdirection , OU.F i0u.F * i0. 3 2 Stress Cx (Pa) STFF TFA SBFF  BFA x105 Stress y (Pa) x 105 Web Stress, ydirection 1 0 x 105 & 1 0 0.5 O  I os 1 z4 2 0 Stress G (Pa) y 2 4 x 105 Figure 47. Stress in the x and y directions of the top face, constrained thermal problem. bottom face and web for the (c  1 S0.5 C,, 0  0.5 CO ILL 1 4 4 Si 1 ) 0.5 S 0 a 0 S0.5 E S14 z  Figure 48. Deformation of the unit cell due to the unconstrained boundary condition Face Stress, xdirection Stress x (Pa) x105 Web Stress, xdirectic 0 2 4 Stress ax (Pa) Face Stress, ydirection Stress cy (Pa) 105 ,n F A A Q E 0 o 6 z x 105 1 0.5 0 0.5 1 Stress a (Pa) 104 Figure 49. Stress in the x and y directions of the top face, bottom face and web for the unconstrained thermal problem. LI_ ..n I' ~,. \.LI i 1 LL: i. ~ CHAPTER 5 BUCKLING ANALYSIS OF AN ORTHOTROPIC INFINITE STRIPS AND APPLICATIONS TO ITPS Buckling of an ITPS During launch and reentry, the TPS is subjected to various mechanical and thermal loads. The thermal and mechanical loads initiate compressive forces to each component (webs or face sheets) on the ITPS. The faces of a sandwich structure are designed to withstand inplane loads, while the core is designed to withstand transverse shearing loads. One of the advantages of an ITPS was that the panels can be large in size thus reducing the number of panels needed to cover a certain area of a space vehicle and reducing the overall mass of an ITPS. Large sized panel leads to unit cells that are large in length. The increase in unit cell length results in long unsupported or partially supported sections of the ITPS' thin facesheets or corrugated core which were subjected to in plane and transverse shearing loads. The ITPS used in this study was composed of thin plates which made them susceptible to buckling when exposed to compressive forces. The thin plates reduced the overall weight but decreased the buckling resistance of the plates. The ITPS can undergo two types of buckling, local and global buckling. Local buckling is limited to a part of the ITPS such as the faces or the webs. Global buckling is when the ITPS plate buckles as a whole. Global buckling was not of a concern because the ITPS plate was thick thus increasing the global buckling resistance. Local buckling of an ITPS plate was an undesirable failure mode because a local dimple due to a buckled face or corrugated core led to a change in the aerodynamic heating profile and instability and collapse of the entire ITPS. The change in the local aerodynamic heating profile resulted in local excess aerodynamic heating, which elevated the local temperature of the facesheet past its temperature limit and caused catastrophic failure. Buckling of the ITPS was one of the major design drives because of the thin plates. A local buckling analysis of an ITPS plate began by an assumption that the plate was composed of three thin plates that were susceptible to buckling (Figure 51). Each thin plate had a different a to b ratio that was dependent on the unit cell length, angle of corrugations, and unit cell thickness. The value of a / b varied from 10 to 50 depending on the unit cell's geometric parameters. Due to the high a /b ratio, each plate was considered to be an infinite strip. The infinite direction was the xdirection and the finite direction was the ydirection. Solutions have been presented for plates with various boundary conditions under compressive loading in the x direction. Those solutions converged to a constant value of the critical buckling load for long plates (a >> b). Solutions have also been investigated for plates with various boundary conditions under shear loading. The shear stability solutions were approximate because of implementation of the Galerkin method (Reddy, 1997) and cannot be conveniently applied to long plates. An infinite plate solution was implemented for the determination of the critical buckling load of an ITPS facesheet or web. Methods of Critical Loads Calculation There are different methods to calculate the critical buckling load of a structure. The methods used for determining critical buckling values of compressive forces applied to bars are applicable for determining critical buckling values of plates. There are three methods to determine the critical buckling values of plates (Timoshenko and Gere, 1963). * Method 1: The critical buckling values of the forces acting on a plate at the middle surface are obtained by assuming that from the beginning the plate has some initial curvature or some loading in the longitudinal direction. Those values of forces at which the plate deflections grow indefinitely are the critical buckling values. * Method 2: For this case it was assumed that the plate buckles slightly under the action of a compressive force applied to the plate's midplane and then the magnitudes that the forces must have to keep the plate stable in a slightly buckled shape are the critical buckling loads. The differential equation of the deflection shape was obtained from Equation 51, assuming that there are no body forces and lateral load on the structure. S4W 04W 04W 02W 02W 02W Da+ 2(D + 2D66) +D = N +N +2N (51) Sa24 12 a2 Y x y x2 x y The simplest case was when all three compressive forces were constants and where a ratio between the forces was assumed, Ny = kNx and Nxy = RN,. After solving Equation 51 for w for a given plate boundary condition, it was evident that the critical buckling load for the plate was possible only for a certain definite value of N. The smallest value of Nx determines the desired critical buckling value. S Method 3: The energy method can also be used to determine the critical buckling values of plates. The energy method was useful where rigorous solutions of Equation 51 was unknown or where a plate was reinforced by stiffeners and it was required to find only an approximate value to the critical buckling load (Timoshenko and Gere, 1963). This method assumed that the plate exhibited a small lateral bending due to the compressive forces consistent with its boundary conditions. Only the energy of bending and the corresponding work done by the compressive forces was considered. If the work done by the compressive forces was smaller than the strain energy of bending for every possible buckling mode, then the plate was stable and buckling did not occur. If the work done by the compressive forces was greater than the strain energy of bending for every possible buckling mode, then the plate was unstable and buckling occurred. Stability of an Infinite Strip under Compression or Shear The stability of a infinite strip in the x direction and subjected to either a uniform shear loading or compression loading in the x and y directions (Figure 52), was considered for an analytical stability analysis. These exact solutions provided the limiting cases for long plates under shear and in plane compressive loading. An infinite plate with the edges at y = +b/2 and either clamped or simply supported was considered for the stability analysis. The boundary condition of the facesheet and web junction point varied according to the angle of corrugations. It was not known specifically what the true boundary condition at the facesheet and web junction was. It was assumed that the true boundary condition was in between the simply supported and clamped boundary condition. The ITPS panel is composed of three infinite strips. Each infinite strip had a different a/b ratio. Correction factors can be used accordingly to accurately represent the true boundary condition between the facesheet and web junction point. The infinite strip was subjected to a compressive uniform shear and biaxial loading. The faces of the ITPS were subjected to inplane loads while the web was subjected to transverse shearing loads. All three loads were needed for the analysis. The governing differential equation for the applied uniform biaxial compression in the x and y direction is shown in the form of one compressive load and two ratios (N,= No, N= kNo, and N, = RNo) (Equation 52). a4w a4w a4w a 4w a4w Di + 4DI W + 2(D2 + 2D66 )2 + 4D26,, + D2 9x 9x 9y 9x 9y 9x"y ~ /y (52) N, 2RN, kN, a2 xy o 2 The boundary conditions along the edges y = b/2 are (1) Simply supported edges w=0 (53) M D aw a~2w a~2w My = D12 D 2 2D2 = 0 (54) ax 2y 2x y (2) Clamped edges w=0 (55) = 0 (56) ay Using the approach suggested by Whitney (1987), the solution in the form of Equation 57 was considered for the deflection shape. w = f(y)e2 lb (57) Substituting Equation 57 into the governing differential equation of equilibrium Equation 52, the following differential equation was obtained in the form of the unknown functionf(y) (Equation 58). Substituting Equation 59 into Equation 58 the solution of the governing differential equation was represented by Equation 510. In Equation 510, Ai, A2, A3, and A4 were the real or complex roots of the following characteristic equation: D f(y) + 4D16 + 2(D12 + 2D66) 22 4D26 + =O kN 2RN ^ b )y 8 y b8 b )a b 8y f(y)= Aec lbe2 )ylb (59) w = e 2xb (Ae2ly/bb +Be24y/b + Ce24/b +De 2"'y/b) (510) D114 + 4D16 3A + 2(D12 + 2D66 )22 + 4D26 3 + D22A No ( kNo 2Z2 2RNo 0 (511) The solution from Equation 510 was used in conjunction with the boundary conditions which resulted in four linear homogeneous equations with the unknowns A, B, C, and D. Since a nontrivial solution was desired, the determinant of the coefficients must be zero, which was a sufficient condition to determine the buckling load. For the simply supported edges and clamped edges the four conditions took the following matrix forms: e e e e A e e e e B e det = (512) M1e Me Me '1 M4e Me M M 12e Z, M3e13 M z14 D 0 e e '2 e e 1 e e e e det e e 1 = (513) Ale IA 2e 2 Az3e 4e C 0 e 2 Ae '2 'e /' 4e D 0 In Equation 512, M = D122 + D2222 + 2D26 4 To determine the actual critical buckling load for any combination of applied compressive loads or shear loads, the roots of the characteristic equation must be determined in conjunction with Equations 512 or 513 for a given value of E, where E characterizes the length between the successive buckling waves in the plate. The solution from Equation 510 was found to depend on and the critical buckling load corresponded to the value of which yielded the lowest compressive load (No). The procedure was repeated for a clamped boundary condition. The flow chart (Figure 53) illustrates the process for determining the critical buckling load for a given value of (. Results A FE analysis was used to verify the analytical procedure for determining critical buckling loads of infinitely longs plates subjected to compressive and shear forces. An ITPS unit cell with the following dimensions: p = 50 mm, d= 100 mm, tTF 1 mm, tBF 1 mm, tw= 1 mm, 0 = 75, a =1 m, b 1 m was considered for the analysis. An AS/3501 graphite/epoxy composite (El = 138 GPa, E2= 9 GPa, v12 =0.3, G12 = 6.9 GPa, with four laminae in each component and a stacking sequence of [(0/90)2] was used as an example to verify the buckling analytical models. Plate II from Figure 51 with the a/b11 ratio of 9.88 was modeled using the commercial ABAQUST finite element program. Eight node shell elements were used to model the long plate. The shell elements have the capability to include multiple layers of different material properties and thicknesses. Three integration points were used through the thickness of the shell elements. The FEA model consisted of 6,321 nodes and 2,002 elements. Due to symmetry of the plate only a quarter of the plate was modeled with the appropriate symmetric boundary conditions. The buckling load was obtained from ABAQ UST by applying either compressive loads or shearing loads equal to one and obtaining the eigenvalues from the FEA output after analysis. The eigenvalues were equal to the critical buckling loads if the compressive or shearing loads were equal to one. Several critical buckling loads were investigated such as uniaxial compression in the x and y direction, biaxial compression, and shear loading with simply supported and clamped boundary conditions. Uniaxial Compression, Nx only For this case, a compressive load ofNx=l was applied to the finite element model and the corresponding eigenvalue was obtained from the FEM output after analysis (Figure 54). The finite element results (Table 51) indicated that the analytical procedure for predicting the stability of an orthotropic infinite plate resulted in an accurate prediction of the critical buckling loads for a simply supported and clamped boundary condition. The percentage differences between the analytical and FEM results was less than 2%. The critical buckling load for any value of is shown in Figure 55. As expected, the critical buckling load for a clamped boundary condition was greater than the buckling load for a simply supported boundary condition. The minimum data point of the two curves in Figure 55 was the desired critical buckling value. Uniaxial Compression, Ny only A compressive load ofNy=l was applied to the finite element model and the corresponding eigenvalue was obtained from the FEM output after analysis. The finite element deformation contours for a simply supported and clamped boundary condition are shown below in Figure 56. The minimum of the two curves from Figure 57 was the critical buckling load for the infinite plate. The percentage difference between the analytical and finite element critical buckling values was less that 2% (Table 52). The minimum of the curves from Figure 57 was at S= 0 because from Figure 56, the length between the successive buckling waves in the x direction was zero. To create a buckling mode in they direction only one buckling wave was needed in the y direction which resulted in only one wave in the x direction and a E of zero since there was no length between wavelengths. Biaxial Compression, Nx = Ny Two compressive forces were applied to the finite element model and the corresponding eigenvalue was obtained from the FEM output after analysis. The finite element results and analytical results are illustrated in Figures 58 and 59. The compressive load in the x direction was equal to the compressive load in they direction (k = 1). The percentage difference between the analytical and finite element results was less than 1% (Table 53). The minimum of the simply supported curve (Figure 59) was at = 0 because from Figure 58 the length between the next successive wavelengths in the x direction was zero because there was only one buckling wave. Shearing Load, Ny = 1 For verification of the shear buckling value, a long plate was subjected to a unit shear load. The critical buckling load obtained from Equation 512 was compared with another established analytical equation (Equation 514). Nc = K EK (514) The value of K from Equation 514 was dependent on the aspect ratio of the plate. The value of K for a large aspect ratio was 5.34 for a simply supported boundary condition and 8.96 for a clamped boundary condition. The comparison of the critical shear buckling loads obtained from Equations 512 and 514 are compared (Table 54). An FE analysis was unsuccessful for this case due to the lack of knowledge on boundary conditions for an infinite plate under shear loading. The analytical results for an infinite plate with an a /b = 9.88 with steel properties and a thickness of 1 mm (0.039 in) was considered for comparison with the buckling results obtained from the infinite plate solution. Conclusion The comparison of the critical buckling value of the analytical model was less than a 2% difference when compared to the critical buckling load from the FEA. The analytical model was capable of obtaining accurate critical buckling values for any combined loading. The analytical procedure resulted in the lower and upper bound of the true critical buckling value or the ITPS. Correction factors can be used to obtain the true critical buckling value. Buckling is a major design driver of the ITPS because of the thin faces and webs. Table 51. Critical buckling load of plate II, N, = 1. Critical Analytical [N/m] FEM [N/m] % diff Simply Supported 11620 11591 0.25% Clamped 23500 23907 1.70% Table 52. Critical buckling load of plate II, Ny = 1. Critical Analytical [N/m] FEM [N/m] % diff Simply Supported 2030.3 2057.9 1.34% Clamped 8124 8141 0.21% Table 53. Critical buckling load of plate II, k = 1. Critical Analytical [N/m] FEM [N/m] % diff Simply Supported 2036.9 2036.8 0.00% Clamped 7788 7823 0.45% Table 54. Critical buckling load of plate II, Nxy 1. Equation (513) Critical Analytical [N/m] [N/m] % diff Simply Supported 97600 98262.6 0.67% Clamped 164000 164813 0.49% Figure 51. Local buckling of an ITPS panel. N, /N////y Nx (_ Y/b/2 ///////////////N , Figure 52. Infinite Strip under shear and compression loading. Figure 53. Critical buckling load flow chart for an infinite plate. A 4* Figure 54. Deformation of the quarter plate due to Nx=l. A) Simply supported boundary condition B) Clamped boundary condition. 50 o Z 20 15 Simply Supported Clamped 0 0 0.5 1 1.5 2 2.5 3 3.5 Figure 55. Critical buckling load as a function of the length between the successive buckling waves in the plate for Nx=l. A B Figure 56. Deformation of the quarter plate due to Ny=l. A) Simply supported boundary condition B) Clamped boundary condition. Ct, ./L 001  Simply Supported Clamped 0.1 0.2 Figure 57. Critical buckling load as a function of the length between the successive buckling waves in the plate for Ny = 1. Figure 58. Deformation of the quarter plate due to k=l. A) Simply supported boundary condition B) Clamped boundary condition. 2.3 E 2.2 7Z. o 2.1 z . 8.2 0 Z 8.1 8 0 ft / T6 z 0 z2 Simply Supported 0 0 0.5 1 1 Clamped E z Figure 59. Critical buckling load as a function of the length between the successive buckling waves in the plate for Nx = Ny. 