REDUCING the cost of launching a space vehicle into space is

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1 AIAA JOURNAL Vol. 45, No., September Micromechanical Analysis of Composite Corrugate-Core Sanwich Panels for Integral Thermal Protection Systems Oscar A. Martinez, Bhavani V. Sankar, Raphael T. Haftka, an Satish K. Bapanapalli University of Floria, Gainesville, Floria an Max L. Blosser NASA Langley Research Center, Hampton, Virginia 8 DOI:.54/. A composite corrugate-core sanwich panel was investigate as a potential caniate for an integral thermal protection system. This multifunctional integral thermal protection system concept can protect the space vehicle from extreme reentry temperatures, an possess loa-carrying capabilities. The corrugate core is compose of two, thin, flat sheets that are separate by two incline plates. Avantages of this new integral thermal protection system concept are iscusse. The sanwich structure is iealize as an equivalent orthotropic thick-plate continuum. The extensional stiffness matrix [A], coupling stiffness matrix [B], bening stiffness [D], an the transverse shear stiffness terms A 44 an A 55 were calculate using an energy approach. Using the shear-eformable plate theory, a close-form solution of the plate response was erive. The variation of plate stiffness an maximum plate eflection ue to changing the web angle are iscusse. The calculate results, which require significantly less computational effort an time, agree well with the three-imensional finite element analysis. This stuy inicates that panels with rectangular webs resulte in a weak extensional, bening, an A 55 stiffness an that the center plate eflection was minimum for a triangular corrugate core. The micromechanical analysis proceures evelope in this stuy were use to etermine the stresses in each component of the sanwich panel (face an web) ue to a uniform pressure loa. Nomenclature a = panel length, x irection b = panel with, y irection = height of the sanwich panel (centerline to centerline) fdg e = eformation vector of the eth component (microeformation) fdg M = eformation vector of the unit cell (macroeformation) e = component inex of the unit cell F m i = noal force in the finite element metho moel l = length of the cantilever beam P z = pressure loa acting on the two-imensional orthotropic panel Q x, Q y = shear force on the unit cell Q ij = transforme lamina stiffness matrix s = web length T D e = eformation transformation matrix of the ith component of the corrugate core t TF = top face sheet thickness Presente as Paper 8 at the 4th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, an Materials Conference, Newport, Rhoe Islan, 4 May ; receive August ; revision receive March ; accepte for publication April. Copyright by the American Institute of Aeronautics an Astronautics, Inc. All rights reserve. Copies of this paper may be mae for personal or internal use, on conition that the copier pay the $. per-copy fee to the Copyright Clearance Center, Inc., Rosewoo Drive, Danvers, MA ; inclue the coe -45/ $. in corresponence with the CCC. Department of Mechanical an Aerospace Engineering, Grauate Stuent. Newton C. Ebaugh Professor, Department of Mechanical an Aerospace Engineering. AIAA Associate Fellow. Distinguishe Professor, Department of Mechanical an Aerospace Engineering. AIAA Fellow. Grauate Stuent, Department of Mechanical an Aerospace Engineering. Aerospace Technologist, NASA Langley Research Center. t BF = bottom face sheet thickness t w = web thickness U = unit-cell strain energy w = integral thermal protection system panel eflection y = local axis of the web " o = miplane strain = angle of web inclination = curvature xy = shear stress in the web x, y = rotations of the plate s cross section p = unit-cell length I. Introuction REDUCING the cost of launching a space vehicle into space is one of the critical nees of the space inustry. Government an private corporations use space for various objectives, such as reconnaissance, communications, weather-monitoring, military, an other experimental purposes. One of NASA s goals is to reuce the cost of elivering a poun of payloa into space by an orer of magnitue []. The space vehicle s thermal protection system (TPS) is one of the most expensive an critical systems of the vehicle []. Future space vehicles require more avance TPS than the one currently use. The main function of the TPS is to protect the space vehicle from the extreme aeroynamic heating uring planetary reentry. The TPS is the key feature that makes a space vehicle lightweight, fully reusable, an easily maintainable. The space shuttle s current TPS technology consists of ifferent types of materials such as ceramic tiles an blankets which are istribute all over the spacecraft. This technology makes the space vehicles exterior very brittle, susceptible to amage from small impact loas, an high in maintenance time an cost. To overcome these ifficulties, scientists at NASA evelope a metallic TPS calle ARMOR TPS [,4]. However, the ARMOR TPS s loa-bearing capabilities are limite an large in-plane loas cannot be accommoate uner this esign. A common feature of the Space Shuttle s TPS system an the ARMOR TPS system is that they are all

2 4 MARTINEZ ET AL. attache to the space vehicle. Currently the TPS on the Space Shuttle requires 4, man-hours of maintenance between typical flights [] because it is an a-on feature. The new TPS concept presente in this article can be accomplishe by using recently evelope metallic foams an also innovative core materials, such as corrugate an truss cores. The integral TPS/structure (ITPS) esign can significantly reuce the overall weight of the vehicle as the TPS/structure performs the loabearing an thermal function. Sanwich structures can offer high stiffness with relatively much weight saving when compare with wiely use laminate structures. They also possess goo vibration characteristics when compare with thin plate-like structures. Sanwich structures have goo amage tolerance properties an can withstan small object impact. Fung et al. [5,] an Libove an Hubka [] use the equivalent homogenous moel approach an force-istortion relationship to erive the elastic constants of Z-core, C-core, an corrugate-core sanwich panels. More recently evelope truss-core sanwich panels have been investigate by Lok an Cheng [8] an Valevit et al. []. Lok et al. [] an Valevit et al. [] investigate an analyze metallic corrugate-core sanwich panels. Lok et al. [8,] erive analytical equations using the force-istortion relationships of an orthotropic thick plate to preict the elastic stiffness properties an behavior of truss-core sanwich panels. The researchers use the homogenous equivalent thick-plate approach to represent the threeimensional structure into a two-imensional orthotropic plate. The researchers verifie an compare their results with finite element analyses an conventional sanwich forms that were investigate by Libove et al. [,]. Composite corrugate-core sanwich structures will be investigate in this paper for use in multifunctional structures for future space vehicles (Fig. ). This type of ITPS woul insulate the vehicle from aeroynamic heating as well as carry primary vehicle loas. The avantages of using such a structure is that it has the potential of being lightweight because of the thin faces an corrugation feature. The structure offers insulation as well as loa-bearing capabilities which makes it multifunctional. The panels can be large in size thus reucing the number of panels neee. Integration with the space vehicle promotes low maintenance. The corrugate-core sanwich panel is compose of several unit cells. The unit cell consists of two thin face sheets an an incline web, which can be of homogeneous materials such as metals or composite laminates. The composite corrugate core will be fille with Saffil, which is a non-loa-bearing insulation mae of alumina fibers. This paper s objective is to etermine the equivalent stiffness properties of the ITPS panel by iealizing it as a continuum. The extensional stiffness matrix [A], coupling stiffness matrix [B], bening stiffness [D], an the transverse shear stiffness terms A 44 an A 55 were calculate by analyzing the unit cell. A etaile formulation an escription of the extensional, coupling, bening, an shearing stiffness of the ITPS panel are presente for a unit cell by representing the sanwich panel as an equivalent thick plate, which is homogeneous, continuous, an orthotropic with respect to the x an y irections. A strain-energy approach an a eformation transformation matrix were use in eriving the analytical equations of the equivalent extensional, bening, coupling, an shearing stiffnesses. Previous researchers aopte the force-istortion relationship approach to etermine the equivalent stiffness parameters. The force-istortion relationship approach becomes complicate an teious if the ITPS is compose of faces an webs with ifferent materials an thickness. This problem can be solve with the propose strain-energy approach an eformation transformation matrix. The stiffness results are use in the first orer shear-eformable plate theory (FSDT) to etermine the response of an ITPS plate when subjecte to mechanical an thermal loas. The analytical moels are compare with etaile finite element analysis. II. Geometric Parameters Consier a simplifie geometry of the corrugate-core unit cell in Fig.. The z axis is in the thickness irection of the ITPS panel. The stiffer longituinal irection is parallel to the x axis, an the y axis is in the transverse irection. The unit cell consists of two incline webs an two thin face sheets. The unit cell is symmetric with respect to the yz plane. The upper face plate thickness t TF can be ifferent from the lower plate thickness t TB as well as the web thickness t w. The unit cell can be ientifie by six geometric parameters p; ; t TF ;t BF ;t w ; (Fig. ). Four other imensions b c ; c ;s;f are obtaine from geometric consierations. The equations for these relationships are as follows: c t TF t BF f p c tan b c p f p s c b c c sin b c cos (a) (b) (c) () The ratio f=p correspons to a triangular core an f=p :5 correspons to a rectangular core. III. Analysis The finite element metho (FEM) is commonly use to analyze sanwich structures. Shell elements are often preferre for the faces an webs to construct a three-imensional FEM moel. However, the number of elements an noes neee to appropriately mesh the sanwich panel can be excessive; as a result, a three-imensional FEM moel is not economical for a quick preliminary analysis of an ITPS. Such panels may also be represente as a thick plate that is continuous, orthotropic, an homogenous for which analytical an two-imensional FEM solutions [] are available. The extensional stiffness matrix [A], coupling stiffness matrix [B], bening stiffness [D], an the transverse shear stiffness terms A 44 an A 55 are calculate by analyzing the unit cell. For bening analysis of the plate, a close-form solution was obtaine by using the FSDT. Avance knowlege of the orthotropic thick-plate stiffness is z, w y,v x,u Fig. Corrugate-Core Sanwich Panel. Fig. Dimensions of the unit cell.

3 MARTINEZ ET AL. 5 essential for successful implementation of the FSDT. Typically, plate analyses yiel information on eflections, an force an moment resultants at any point on the plate. We will again use the micromechanical analysis proceures evelope in this stuy to etermine the local stresses in the face sheets an the webs. Then failure theories such as Tsai Hill criterion can be use to etermine if the stresses are acceptable or not. In the erivation of the stiffness parameters, the following assumptions were mae: ) The eformation of the panel is small when compare with the panel thickness. ) The panel imensions in the x irection are much larger than the unit-cell with p. ) The face sheets are thin with respect to the core thickness. 4) The core contributes to bening stiffness in an about the x axis but not about the y axis. 5) The face an web laminates are symmetric with respect to their own miplanes. ) The core is sufficiently stiff so that the elastic moulus in the z irection is assume to be infinite for the equivalent plate. Local buckling of the facing plates oes not occur an the overall thickness of the panel is constant. Previous researchers aopte these assumptions in the erivation of stiffness parameters of sanwich panels with corrugate core; Libove an Hubka [] for C-core, an Fung et al. for Z-core [5,]. The in-plane an out-of-plane stiffness governing the elastic response of a shear-eformable sanwich panel are efine in the context of laminate plate theory incorporating FSDT escribe by Vinson [4] an Whitney [5]. The appropriate stiffness of the orthotropic plate may be obtaine by comparing the behavior of a unit cell of the corrugate-core sanwich panel with that of an element of the iealize homogeneous orthotropic plate (Fig. ). The in-plane extensional an shear response an out-of-plane (transverse) shear response of an orthotropic panel are governe by the following constitutive relation: N A 4 Q 5 4 M C 5 D ( "o ) or ffgkfdg () In Eq. (), " an are the normal an shear strains, are the bening an twisting curvatures, an [A], [C], an [D] are the extensional, shear, an bening stiffness. The orthotropic plate is assume to be symmetric. A. Extensional an Bening Stiffness An analytical metho was evelope to calculate the stiffness matrix of the corrugate-core sanwich panel. Consier a unit cell mae up of four composite laminates (two face sheets an two webs). Each laminate has its respective material properties an ABD matrix. The ABD matrix of each component nees to be combine together in an appropriate manner to create the overall stiffness of the sanwich panel. The formulas for etermining the ABD matrix of a composite laminate are given as follows []: h i A e ij;b e ij;d e ij XN k Z t Q e ij t k ;z;z z z Q ij e k4 z k z k k z k ; ; z k z k 5 () In Eq. (), N is the number of laminas in the composite an Q e ij are the components of the transforme lamina stiffness matrix, where e 4 ( top face sheet, bottom face sheet, left web, 4 right web). The overall stiffness of the unit cell was etermine by imposing unit miplane strains an curvatures (macroeformation) to the unit cell an then calculating the corresponing miplane strains an curvatures (microeformations) in each component. The unit-cell components are the two face sheets an two webs. A transformation matrix relates the macro- an microeformations as follows: fdg e T D e fdg M (4) In Eq. (4), fdg e is the microeformation in each component, fdg M is the macroeformation of the unit cell, an T e D is the eformation transformation matrix that relates macroeformation to microeformations. B. Formulation of Deformation Transformation Matrix Face Sheets The eformation transformation matrix of the top face sheet was etermine by first consiering the unit cell uner the action of miplane macrostrains, " yo, xyo an macrocurvature x, y, xy. Each strain an curvature was consiere by itself an the resulting miplane strains an curvatures in the face sheets (also known as microstrains an curvatures) were erive. The following equations show the transformation matrices of the top an bottom face sheets. Top face sheet fdg TDfDg M 8 8 " xo " >< yo >= xyo >< x >: y 4 5 >: xy Bottom face sheet fdg TDfDg M 8 8 " >< yo >= xyo >< x >: y 4 5 >: xy " yo xyo x y xy >= " yo xyo x y xy There is a : relationship between miplane macro- an microstrain as well as a : relationship between macro- an microcurvature, as inicate by unity along the iagonal of the transformation matrices. Using the assumptions that the in-plane isplacements u an v are linear functions of the z coorinate an that the transverse normal strain " z is negligible [], the = factor was use to relate the macrocurvatures to the miplane microstrains. M >= M (5) () Fig. Equivalent orthotropic thick plate for the unit-cell corrugatecore sanwich panel. C. Formulation of Deformation Transformation Matrix for the Webs Formulation of the eformation transformation matrix for the webs is relatively complicate because of the nee for coorinate transformation ue to the inclination of the webs. Consier a global xyz coorinate system an a local xyz coorinate system (Fig. 4).

4 MARTINEZ ET AL. The origin of the local web axis is at the top face sheet an web junction point. The transformation from global to local coorinate axes requires a rotation an translation. The transformation from global to local isplacements only requires a rotation (see Appenix A). In Eq. (A), is the angle of web inclination of the right web, the first matrix is the rotation matrix, an the secon vector is a translation vector. Consier the unit cell of the ITPS panel uner the action of miplane macrostrains, " yo, xyo an macrocurvature x, y, xy. From Assumption 4 in Sec. III we note that " ;4 yo when the unit cell is subjecte to " M yo an " M xo. The microstrains on the webs ue to a macrocurvature are more complex to etermine, therefore a etaile iscussion is appropriate (see Appenix A). The microstrains an curvature in the right web ue to a macrounit curvature along the y axis ( y ) was erive; all other curvatures were set equal to zero. an " ;4 xo o w Starting with Eq. () an following the etaile erivation in Appenix A leas to the transformation matrix for the left an right web. Left web 8 c y sin 8 " >< yo cos >= c y sin " fdg TDfDg M xyo >< yo >= xyo x cos cos c y sin x >: y cos xy >: y 4 cos cos sin 5 xy M (8) Right web fdg 4 T 4 DfDg M 8 " >< yo >= xyo x >: y xy 4 c y sin 8 cos c y sin >< cos cos c y sin cos >: 4 cos cos sin 5 " yo xyo x y xy >= M () From Eqs. (8) an () we observe that micromiplane strains in the webs are a linear function of y. D. Stiffness Matrix Determination Through Strain-Energy Approach As the unit cell is eforme by the unit macrostrains an curvatures, it stores energy internally throughout its volume. The total strain energy in the unit cell is the sum of strain energies in the iniviual components, i.e., faces an webs. U M p fdg M T KfDg M X4 e U e () In Eq. (), U M is the total strain energy an U e is the strain energy of the eth component. The strain energy of an iniviual component is shown in Eq. (). Because the eformations of the webs are a function of y, integration of Eq. () is one with respect to y. The integration limits are from zero to s, where s is the length of the webs. By substituting Eq. (4) into Eq. () we are able to represent the strain energy of the web in terms of macroeformations of the unit cell. In Eq. (), A e enotes the area of the eth component. Z U e A e fdge T K e fdg e A e () U 4 p Z s fdg4 T K 4 fdg 4 y () U 4 Z s p TDfDg 4 M T K 4 TDfDg 4 M y () In general, we write the strain energy in each laminate in terms of the global eformation fdg M as U e DM T K e D M (4) Fig. 4 Global an local coorinate axes for the right web. Then the stiffness matrix K of the iealize orthotropic panel can be erive as

5 MARTINEZ ET AL. K X4 K e p e X 4 e Z L T e D T K e T e D y (5) E. Face an Web Stresses It has been shown that using the eformation transformation matrix one can etermine the local strains an curvature of each part of the unit cell (face or web) ue to a unit-cell eformation. As a result of that, we obtaine stresses in each component by multiplying the microeformations an curvatures of a particular component with the corresponing transforme lamina stiffness matrix. "o e TD e "o M () e Q e f" o g e zfg e () Although the previously erive eformation transformation matrices for the webs, Eqs. (8) an (), are goo for stiffness preiction, they o not yiel accurate stress results when compare with finite element (FE) analysis. For example, the assumption " M z constrains the webs from expaning in the y irection ue to the Poisson effect. This leas to stresses in the y irection that are not present in the three-imensional FE analysis. Therefore, corrections were applie to the eformation transformation matrix to obtain accurate web stresses. The refine web stress eformation transformation matrices for the webs are as follows. Left web 8 c " y sin 8 xo M " yo >< >= fdg e TDfDg e M c xyo y sin " >< yo >= xyo x fp; ; t TF ;t BF ;t w ; (8) x >: y cos xy 4 gp; ; t TF ;t BF ;t w ; 5 >: y xy Right web fdg e T e DfDg M 8 " yo >< >= xyo x >: y xy c y sin 8 c y sin >< fp; ; t TF ;t BF ;t w ; cos 4 gp; ; t TF ;t BF ;t w ; 5 >: " yo xyo x y xy >= M () The refine transformation matrix contains a Poisson s ratio that takes into account the lateral contraction or elongation " yo of the web ue to a unit macromiplane strain in the x irection an a unit macrocurvature in the x irection x. The micromiplane strains " yo, xyo in the web ue to either M y or M xy were remove because there is no force in the y irection that is causing a miplane strain in that irection. From Eq. (), the relation between macromiplane shear strain an macrocurvature can be given as xyo M xy cos () y M x cos cos sin () Equations () an () treat the web as an unresisting member of the unit cell when it is eforming. For example, when the unit cell unergoes a unit miplane shear strain or a unit curvature, the faces are compliant with that eformation but the webs resist that movement. Equations () an () will be true if the webs were at a right angle to the face sheet. The equations were prove wrong in general an our assumption that the webs resist eformation was prove right by conucting several finite element analyses for various web angle inclinations. An analytical proceure that takes into account the webs resistance to eformation was establishe to etermine the micromiplane shear strain an microcurvature, i.e., fp; ; t TF ;t BF ;t w ;, gp; ; t TF ;t BF ;t w ;.. Micromiplane Shear Strain in the Webs An analytical proceure was etermine that relates macromiplane shear strain to micromiplane shear strain. The analytical metho takes into account the resistance to shear that the webs will experience when the unit cell is uner miplane shear. The top an bottom face sheets were investigate separately uner the action of shear (Fig. 5). It can be seen from Fig. 5 that we are incluing the resistance to shear by the webs through a shear force acting on the webs F w. The total top face, bottom face, an web shear strain are as follows. p f p f ()

6 8 MARTINEZ ET AL. y y F T F x x B F W-F T γ γ F W+F B f p-f F W a) Top face sheet b) Bottom face sheet Fig. 5 Free boy iagram of the face sheet uner the action of miplane shear strain. p 4f p f () p-f γ F W γ 4 f formulas []. There are three unknown constants (C T, C w, an C B ) in Fig.. To solve the three unknowns we nee a system of three linear equations. The three equations come from the bounary conitions. The first two bounary conitions are that the slopes of the top an bottom face sheet must equal the slopes of the faces when M y. The last bounary conition is that the ifference of slope between the face an web junction point (A an B) must equal the slope of the web. After solving the system of linear equations, the curvature of the webs was etermine by iviing the couple acting on the web by the flexural stiffness of the web (equivalent EI). C T C w f EI TF C B C w p f EI BF C Tp f EI TF p () C Bf EI BF p () s f p f (4) The shear strain in the faces an webs ue to the shearing forces are efine as follows. F W F T t TF G TF F T t TF G TF F W F B t BF G BF 4 F B t BF G BF (5) The shear force in the webs is F w G w t w w () There are three unknown shear forces F T, F w, F B. The three unknown forces were etermine by solving a system of three linear equations with the three unknowns, shown as follows: G w t w w F w () Solving Eq. () yiels the three shear forces that act on the faces uring shear. Substituting the known shear forces from Eq. () into the web shear strain equation in Eq. (4) yiels the macro- to micromiplane shear strain relation of the web. fp; ; t TF ;t BF ;t w ; s f p f (8). Microcurvature in the y Direction for the Webs An analytical proceure is erive that relates macro-y-irection curvature to micro- y-irection curvature in the webs. The analytical metho takes into account the resistance to curvature that the webs will experience when the unit cell is uner the y-irection curvature. Half of the unit cell was investigate uner the action of couples that act on the faces (Fig. ). Consier the half-unit cell uner the action of an en couple that will cause unit curvature in the y irection. The half-unit-cell en couple was represente as three en couples acting on the faces an webs (C T, C w, an C B ). The slopes of the faces an web ue to an en couple were obtaine from beam theory (one-imensional plate) f p-f C B C w p f EI BF C B C w f C ws () EI TF EI w gp; ; t TF ;t BF ;t w ; y C w EI w () F. Formulation of Transverse Shear Stiffness A 55 For a corrugate-core sanwich structure loae in shear transverse to the corrugations (by shear stress xz or shear force Q x ), it is recognize that the face sheets an core will unergo bening eformation [,8]. For the etermination of A 55, the shear stress in the face sheets are neglecte because of its small thickness an classical plate theory is use. To etermine the shearing stiffness ue to Q x, we must first ientify the shear stresses in the webs ue to Q x. Figure shows a free boy iagram of the corrugate-core panel unit of length x in the x irection where only the stresses which act in the x irection are shown an consiere. The stress values shown are average stresses over the faces of an element which is assume to be very small. A summation of the forces in the x irection yiels x F zx x tc s yz () Following the proceure in Appenix B, we etermine the shear stresses in the webs xy ue to Q x. The shear strain-energy ensity (strain energy per unit area of the sanwich panel) can be calculate either from the web shear stresses given in Eq. (B), Appenix B, or from the shear force Q x an yet to be etermine shear stiffness A 55. By equating the two shear strain-energy ensity terms we obtain U s t c p Z s xy G xy y Q x A 55 (4) Using Eq. (4), the equivalent shear stiffness A 55 of the sanwich panel was solve. A C w C T B C w C B Fig. p-f Half-unit-cell uner the action of en couples at the faces. f Fig. Small element remove from a boy, showing the stresses acting in the x irection only: a) sie view, b) isometric view.

7 MARTINEZ ET AL. G. Formulation of the Transverse Shear Stiffness A 44 Formulation of the transverse shear stiffness A 44 of the panel is relatively complicate [] because certain conitions nee to be fulfille. Figure 8a shows a sanwich panel of unit length in the x irection subjecte to unit transverse shear Q y. The horizontal force Y p= provies equilibrium. Point A in Fig. 8b is assume to be fixe to eliminate rigi boy movements of the unit cell. The relative isplacements y an z will result from the transverse shearing an horizontal force. Because the force is small, the isplacements will be proportional to Q y, thus an average shear strain is represente as y y z p (5) Because of antisymmetry, only half of the unit cell nees to be consiere for analysis (Fig. a). The unit shear force resultant is ivie into a force P acting on the top face sheet an a force R acting on the lower face sheet. A shear force F is assume to act on the top face sheet at point A where there are no horizontal forces ue to antisymmetry, an a force (-F) was etermine through a summation of the forces in the z irection. The isplacements of the half-unit cell uner the action of force P, R, an F is shown in Fig. b. From Figs. a an b we observe that there are three unknown forces an five isplacements that nee to be solve. These forces an isplacements can be solve through the energy metho. The total strain energy in half the unit cell is the sum of the strain energies from each iniviual member (i.e., AB, BC, DE, BE, EG). The strain energy ue to a bening moment is consiere, whereas the strain energy ue to shear an normal forces are neglecte. The total strain energy in Fig. a is shown in Appenix C. Using Castigliano s theorem, Eq. (), which states that isplacement is equal to the first partial erivative of the strain energy in the boy with respect to the force acting at the point an in the irection of isplacement [], the unknown forces an isplacements can be etermine. i () Because the overall thickness of the sanwich panel remains constant uring istortion, the bounary conitions are C z G z an A z. Because half the unit cell is uner unit shear then P R. The two bounary conitions along with Castigiliano s secon theorem lea to a system of two linear equations with two s (8) The unknown forces P, F, R were etermine by substituting Eq. (C) from Appenix C into Eqs. () an (8) an solving the system of linear equations. The expressions for the forces P, F, an R are quite lengthy, an so they are omitte in this paper. The half-unitcell isplacements were etermine by using Eq. (C) from Appenix C an Eq. () along with the values of the unknown forces (refer to Appenix C). The isplacement of the half-unit cell are y C y G y an z C z G z in the y an z irections. Using the force-istortion relationships evelope by Libove an Batorf [] to escribe the elastic behavior of an orthotropic thick plate, the transverse shear stiffness A 44 was obtaine as follows: A 44 Q y y y = z =p = C y G y =p C z () IV. Response of the ITPS Panel as a Two-Dimensional Plate An ITPS panel forms the outer skin of the vehicle, which covers the crew compartment. The ITPS panel experiences thermal forces an moments ue to the extreme reentry temperatures as well as a pressure loa which comes from the pressurize crew compartment or from the transverse aeroynamic pressure. All those conitions coul cause the panel to eflect, buckle, an yiel. Knowing the panel eflection an failure moes is important because excessive eflection of the panel can lea to extremely high local aeroynamic heating. Local buckling can be a major esign river because the panel is compose of thin plates. A two-imensional plate analysis is neee to etermine the behavior of the ITPS when it is subjecte to those various loaing conitions. Consier a simply supporte orthotropic sanwich panel of with b (y irection) an length a (x irection) as illustrate in Fig., the bounary conitions may be escribe as w;y; wa; y M x ;y; M x a; y wx; ; wx; b M x x; ; M x x; b (4) The panel is subject to a pressure loa P Z X X mx ny P mn sin sin a b m n (4) where P mn P o = mn an P o is the uniform loa. The panel is also assume to have the following eformations: wx; y X X mx ny A mn sin sin (4a) a b m n Fig. 8 Unit transverse shear an horizontal force: a) unit cell, b) eformations [8]. xx; y X X m n mx B mn cos a sin ny b (4b) yx; y X X m n mx C mn sin a cos ny b (4c) Fig. Half-unit-cell a) corrugate-core sanwich panel, b) eformations. In Eq. (4), wx; y is the out-of-plane isplacement, an x x; y an y x; y are the plate rotations. Using the FSDT, the effect of shear eformation on eflections an stresses can be investigate. The unknown constants A mn, B mn, C mn are obtaine by substituting the constitutive relations in the form of the assume eformations into the ifferential equation of equilibrium. Doing so will yiel a

8 MARTINEZ ET AL. system of three linear equations an three unknowns (refer to Appenix D). Solving for the unknown constants, one can now etermine the eflections at a given x an y coorinate on the twoimensional orthotropic sanwich panel. The results in the series converge for m n [8]. V. Results A. Extensional an Bening Stiffness For verification of the effectiveness of the analytical moels, consier an ITPS sanwich panel with the following imensions: p 8 mm, 8 mm, t TF mm, t BF mm, t w mm, 5 eg, a :5 m, b :5 m. An AS/5 graphite/epoxy composite, E 8 GPa, E GPa, :, G : GPa, with four laminae in each component an a stacking sequence of [= ] was use as an example to verify the analytical moels. A finite element analysis was conucte on the unit cell using the commercial ABAQUS finite element program (see Fig. ). Eightnoe shell elements were use to moel the face sheets an webs of the unit cell. The shell elements have the capability to inclue multiple layers of ifferent material properties an thicknesses. Three integration points were use through the thickness of the shell elements. The FEM moel consiste of 8,4 noes an elements. The ITPS plate stiffness was obtaine by moeling the unit cell with shell elements an subjecting the unit cell to six linearly inepenent eformations. The six linearly inepenent strains are ) " M xo an maintaining the rest of the macroscopic strains an curvature zero; ) " M yo an maintaining the remaining strains an curvature zero; an, similarly, ) M xyo ;4) M x ;5) M y ; an ) M xy. Strains were impose by enforcing perioic isplacement bounary conitions on the unit cell (Table ). To prevent rigi boy motion an translation, the unit cell (Fig. ) was subjecte to minimum support constraints. The top an bottom surfaces were assume to be free of traction. The faces x an x a have ientical noes on each sie as well as the other faces y an y b. The ientical noes on the opposite faces are constraine to enforce the perioic bounary conitions. Figure shows the eformations of the unit cell as a result of imposing the perioic bounary conitions. The noal forces of the bounary noes were obtaine from the finite element output after the analyses. Noal moments were obtaine by multiplying the noal forces with the istance from the miplane. These noal forces an moments of the bounary noes were then summe to obtain the force an moment resultants, Eq. (4). The stiffness coefficients in the column corresponing to the nonzero eformation were compute by substituting the values from Eq. (4) into the plate constitutive relation. The same proceure was repeate for other eformation components to obtain an fully populate the unit-cell stiffness coefficients. N i ;M i X n ;zf m i a; y; z (4) b m Table Perioic isplacement bounary conitions impose on the lateral faces of unit cell ua; yu;y va; yv;y wa; yw;y ux; bux; vx; bvx; wx; bwx; x a; y x ;y y a; y y ;y x x; b x x; y x; b y x; " x a " y b xy a= b= x az a = a y bz b = b xy az= ay= bz= bx= a= b= Fig. Finite element mesh of the unit cell.

9 MARTINEZ ET AL. z x Fig. Bounary conitions impose on the plate to prevent rigi boy motion. An arrow pointing at a black ot inicates isplacement of that point is fixe in the irection of the arrow. b a y Miplane Shear Strain γ xy (m/m) FEM Analytical 4 8 Web Angle Inclination (eg o ) Miplane Curvature κ y (/m) FEM Analytical 4 8 Web Angle Inclination (eg o ) Fig. Comparison of FEM an analytical miplane shear strain/ curvature in the web for a given shear strain/curvature of the unit cell. Face Thickness [z/(h/)] Stress σ x x (Pa) Face Thickness [z/(h/)] Stress σ y (Pa) x Fig. Deformations of the unit cell ue to impose perioic bounary conitions. The finite element result from Table inicates that using Eq. (5) provies an excellent preiction to etermine the extensional, coupling, an bening stiffness of an ITPS panel. The finite element results have a less than % ifference in agreement with the analytical results obtaine from the strain-energy metho. B. Stress Verification. Miplane Shear Strain an Curvature in the Webs Consier the same FEM unit-cell representative volume element an mesh from Fig. with the same material properties an crossply lay-up. The web angle inclination was change from 55 to eg an the unit cell was subjecte to a perioic unit miplane shear strain an a perioic y-irection curvature separately. The corresponing web miplane shear strain an web curvature were extracte from the FEM output after analysis. The results of miplane shear strain an curvature from FEM an Eqs. (8) an () are compare in Fig.. From Fig., one can note that there is a less than % ifference between the FEM an analytical results of Eqs. (8) an (). The analytical equations o an excellent job in accounting for the resistance effect of the webs when the unit cell is subjecte to a miplane shear or bening in the macroscale sense. This gives us the confience that we will obtain accurate stress results when compare with the FEM.. Stress Verification The refine web stress eformation transformation matrix was verifie by a finite element analysis. A known strain was applie to the unit cell an the corresponing stresses on the faces an webs were obtaine from Eq. (). The known strain was applie to the Web Length [y/(s/)] eg-f eg-a eg-f eg-a Stress σ x x (Pa) - - Stress σ x y (Pa) finite element moel by enforcing perioic isplacement bounary conitions from Table. The stress results from the FEM output after analysis an Eq. () were plotte in Figs. 4. In Figs. 4, all values in the y axis are normalize with respect to either the face thickness or web length. In these figures, TF an BF stan for top face an bottom face, respectively, F is the finite element, A stans for analytical, an where inclue, an eg inicate lamina orientation. The analytical results are in excellent agreement with the finite element output. Results from the FEM stress output valiate the proceure of the erive stress equations of an ITPS sanwich panel. All stress results in Figs. 4 have less than a 4% ifference from the FEM results. The refine web stress eformation transformation matrix oes an excellent job in preicting strain in the webs which results in stress ata that are in goo agreement with the FEM output. Furthermore, the refine web stress eformation transformation matrix oes not alter the stiffness matrix. The refine web stress eformation transformation matrices from Eq. (5) were use to compute the ITPS stiffness as shown in Table. Analytical- inclues the stiffness results obtaine from using the eformation transformation matrices from Eqs. (5), (), (8), an (). Analytical- inclues the stiffness results obtaine from using the eformation transformation matrices from Eqs. (5) an () an the refine web Web Length [y/(s/)] eg-f eg-a eg-f eg-a Fig. 4 Stresses in the x an y irection in the top face, bottom face, an web for a unit-cell strain of " M xo. Table Nonzero [A], [B], an [D] coefficients for an ITPS sanwich panel Stiffness A, N=m A, N=m A, N=m A, N=m D, N m D, N m D, N m D, N m Analytical :E 8 5:4E :48E 8 :4E :E 5 8 :E 5, FE :E 8 5:4E :48E 8 :4E :8E 5 8 :E 5, % ifference

10 MARTINEZ ET AL. Face Thickness [z/(h/)] Stress σ x (Pa) x Face Thickness [z/(h/)] Stress σ x y (Pa) Face Thickness [z/(h/)] Stress σ x (Pa) x 8 Face Thickness [z/(h/)] Stress σ y (Pa) x Fig. 5 Stresses in the x an y irection in the top face an bottom face for a unit-cell strain of " M yo. Face Thickness [z/(h/)] Stress Γ xy (Pa).5 8 x Stress Γ x xy (Pa) stress eformation transformation matrices from Eqs. (8) an (). The refine web stress eformation transformation matrices have the capability to accurately preict stresses in each unit-cell component an accurately preict the ITPS stiffness. The previously erive eformation transformation matrices outputs excellent stiffness results but erroneous stress results. The refine web stress eformation transformation matrices outputs excellent stiffness results an accurate stress results when compare with finite element analysis. Web Length [y/(s/)] eg-f eg-a eg-f eg-a Fig. Shear stresses in the top face, bottom face, an web for a unitcell strain of M xyo. Face Thickness [z/(h/)] Web length [y/(s/)] Stress σ x (Pa) x eg-f eg-a eg-f eg-a Stress σ x (Pa) x Face Thickness [z/(h/)] Web length [z/(h/)] Stress σ y (Pa) x eg-f eg-a eg-f eg-a Stress σ y (Pa) x 8 Fig. Stresses in the x an y irection in the top face, bottom face, an web for a unit-cell strain of M x. Web length [y/(s/)] eg-f eg-a eg-f eg-a Stress σ x (Pa) x 5 C. Transverse Shear Stiffness Verification The finite element verification of the A 44 stiffness term consiste of a two-part finite element proceure. First, we assume that the unit cell behaves like a cantilevere one-imensional plate an etermine the equivalent cross-sectional properties from finite element analysis. The equivalent cross-sectional properties are axial rigiity EA, flexural rigiity EI, an shear rigiity A 44. The beam consiste of ITPS unit cells an was clampe on the left en, Fig.. Eight-noe soli elements were use to moel the oneimensional plate. First an en couple was applie an the corresponing tip eflections were etermine from the finite element output after analyses. The tip eflection can also be erive as v tip Ml EI Stress σ y (Pa) x (44) Using Eq. (44), we etermine the flexural rigiity EI. The couple was then remove an a transverse force was applie at the tip. The tip eflections were obtaine from the finite element output. Again the tip eflection is given by Web length [z/(h/)] eg-f eg-a eg-f eg-a Fig. 8 Stresses in the x an y irection in the top face, bottom face, an web for a unit-cell strain of M y. Face Thickness [z/(h/)] Stress Γ xy (Pa) x 8 Web Length [y/(s/)] eg-f eg-a eg-f eg-a Stress Γ yy (Pa) x Fig. Shear stresses in the top face, bottom face, an web for a unitcell strain of M xy. Table Nonzero [A], [B], an [D] coefficients for an ITPS sanwich panel Stiffness A, N=m A, N=m A, N=m A, N=m D, N m D, N m D, N m D, N m Analytical- :5E 8 5:4E :4E 8 :4E 5, 888.,, Analytical- :4E 8 5:4E :4E 8 :4E 5,8 8.5,5,8 FE :4E 8 5:4E :4E 8 :44E,4 8,5, % iff. (FE-).%.%.%.8%.%.%.8%.% % iff. (FE-).%.%.%.%.4%.%.%.8%

11 MARTINEZ ET AL. Fig. cells. Corrugate-core moele as a cantilever beam with ten unit Flexural Stiffness (Nm) x 5 D D D D Web Angle Inclination (θ eg) Shearing Stiffness (N/m) a) b) x 8 4 A 44 A Web Angle Inclination (θ eg) A 44 (kn/m) 5 4 Analytical FEM 4 8 Web Angle Inclination (eg) a) b) Fig. Transverse shearing stiffness a) finite element an analytical results, b) eformation of the corrugate-core sanwich panel as a beam. Max ITPS Plate Deflection (w max /P o ) x -8 Analytical FEM Web Angle Inclination (θ eg) Extensional Stiffness (N/m) c) ).5 x Web Angle Inclination (θ eg) Fig. Behavior of a) D-matrix, b) shear stiffness, c) maximum eflection, an ) bening stiffness, as a function of web inclination angle. A A A A v tip Fl Fl (45) EI A 44 Using finite element tip eflection in Eq. (45) along with the flexural rigiity result from Eq. (44) we etermine the shear rigiity A 44. This finite element verification proceure was one for various web angles. The finite element result along with the analytical result from Eq. () is shown in Fig.. The finite element results are in goo agreement with the analytical formulation of A 44. The percentage ifference between the finite element results an the analytical result oes not excee %. The finite element eformation of the cantilever beam is shown in Fig. b. D. Two-Dimensional Orthotropic Plate Results To etermine the optimal web angle inclination for greatest stiffness an minimum center panel eflection, we investigate the variation of the stiffness an center panel eflection to a change of web angle of inclination. Changes in A 44 are important because certain applications epen on the behavior in this plane. Maximum panel eflection is important because excessive eflection of the ITPS panel can lea to high local aeroynamic heating. Consier an ITPS sanwich panel with the following imensions: p 8 mm, 8 mm, t TF mm, t BF mm, t w mm, a :4 m, b :4 m; such a panel is compose of four unit cells. The sanwich panel is mae out of graphite/epoxy T/4: E 8 GPa, E GPa, G : GPa, :, with four laminae in each component an a stacking sequence of = s.by prescribing an internal web angle, the thickness of the faces an webs is efine such that the ITPS sanwich s cross-sectional area (thus the weight) remains the same to the eg web angle configuration. Doing so will allow us to only get the behavior of stiffness to a change in angle rather than a change in angle an area. The results are shown in Figs. an. The close-form solution eflection results of the ITPS plate are compare with the ABAQUS finite element program. Because of the symmetry of the uniform loaing an bounary conitions, only a quarter of the panel was use in the moel. The FE moel consiste of eight-noe shell elements. From Fig., the following conclusions can be mae: ) The highest bening, extensional, an transverse shear stiffness A 55 are provie by the corrugate-core panel with vertical webs. ) Whereas the bening, extensional, an A 55 shear stiffness ecreases with ecreasing web angle inclination, the transverse shear stiffness A 44 increases. Fig. Exaggerate eforme mesh (eformation scale factor ). ) Maximum eflection occurre at the 5 eg web angle inclination, an minimum eflection occurre at the triangular web configuration. 4) A 44 is the most ominant stiffness when the corrugate-core sanwich panel has triangular webs. 5) For panels with rectangular web configurations, its behavior is ominate by shear eformation in the y irection. ) Maximum eflections compute from the close-form solution agree very well with the finite element results. The percentage ifference between the FEM results an the analytical results at 5 an eg web angle inclination is. an.5%, respectively. An accurate preiction when compare with the FEM of the shearing term is mae for rectangular web configurations which lea to accurate eflection results. In esign, a triangular corrugate core may be preferre because the influence on shear can be neglecte ue to the high stiffness, an the maximum plate eflection is at a minimum. By neglecting shear effects, the plate response can be analyze by classical laminate plate theory (CLPT). However, that type of configuration poses problems such as local buckling because of the long unsupporte lengths of the webs. VI. Conclusions Finite element analysis is commonly use to analyze sanwich structures. However, a full three-imensional finite element analysis is not economical for a preliminary analysis of a structure. Such panels can be represente as an orthotropic thick plate for which analytical solutions can be erive. A metho to homogenize the

12 4 MARTINEZ ET AL. corrugate sanwich panel into an orthotropic thick plate has been presente. Detaile formulation of the bening, extensional, coupling, an shear stiffness for the unit corrugate-core sanwich panel was presente an verifie. Panels with rectangular webs resulte in a weak extensional, bening, an A 55 stiffness. The analytical moels are capable of hanling laminate composite materials for the face sheets an webs of the sanwich panel. Furthermore, one can use ifferent materials for the face sheets an web. For example, the hot sie (outer) face sheet can use ceramic matrix composite an the cool sie (inner) face sheet can use polymer matrix composites. The webs can be compose of other materials such as titanium, aluminum, or composite. The stiffness results between the analytical moel an the finite element analysis were within %, thus valiating the metho presente in this stuy. The refine web stress eformation transformation matrix mae incremental improvements to the ITPS stiffness when compare with FE results. Both the eformation transformation matrix for the webs an the refine web stress eformation transformation matrix can be use in etermining ITPS stiffness, but only the latter matrix can be use for stiffness an stress preiction. The computational time an effort in etermining stiffness an plate behavior of the ITPS were significantly reuce in comparison with FEM. The equivalent stiffness parameters were use in the close-form solution to evaluate the maximum eflection of the sanwich panel when subjecte to a uniform pressure loa. Maximum eflection was greatest for 5 eg web configuration for the example consiere. Maximum eflection was fairly constant for the web angle range of 8 eg. Panels with triangular web configuration have negligible shear eformation effect because of the high shearing stiffness in both irections, but this leas to other problems such as local buckling. Global buckling of the panel is not expecte because the ITPS panel is expecte to be thick. However, local buckling is a factor because the face sheets an the webs are mae of thin plates. A triangular web configuration will result in the web length to be long as well as the length between the ajoining unit cell. The increase length will lea to a lower critical buckling value. A high critical buckling value is esirable to avoi local buckling of the ITPS panel. Appenix A: Right Web Transformation Matrix Determination The rotation an translation matrix that relates the local an global axes of the ITPS unit cell is shown. ( ) 8 8 x < x = < = y 4 cos sin 5 y : ; f : ; z sin cos z c (A) 8 8 < u = < u = v : ; 4 cos sin 5 v : ; w sin cos w (A) A etaile proceure of analysis is presente in this section for the erivation of the eformation transformation matrices of the left an right web. Integrating Eq. () twice with respect to y results in the outof-place isplacement: wy oy (A) From classical lamination theory, the u an v isplacements of the sanwich panel in the global coorinates are etermine as shown below: ux; y; zu o (A4a) vx; y; zv o oyz (A4b) To etermine the strains in the web, the isplacements from Eq. (A) must be in the local coorinate system. Substitution of Eq. (A) into Eqs. (A) an (A4), an then Eqs. (A) an (A4) into Eq. (A) resulte in u x; y; z, vx; y; z, wx; y; z. Using the small strain an isplacement assumption, " x x " y ocos x ysin zcos c y o cossin o cos x@y (A5a) (A5b) Equations Eq. (A5a) an (A5b) escribe the micromiplane strains an curvatures in the right web. From Eq. (A5a) we observe that the miplane strain in the y irection is c " yo o cos y sin (A) The same proceure applies to unit curvature along the x irection, unit twist k xy an unit shear strain in the xy plane. Shown next are the eformation transformation matrices for the left an right webs. Left web fdg e T e DfDg M 8 " yo >< >= xyo x >: y xy c y sin 8 cos c y sin >< cos cos c y sin cos >: 4 cos cos sin 5 " yo xyo x y xy >= M (A)