Panel flutter characteristics of sandwich plates with CNT reinforced facesheets using an accurate higher-order theory

Transcription

1 Panel flutter characteristics of sandwich plates with T reinforced facesheets using an accurate higher-order theor A. Sankar ac S. Natarajan b M. Haboussi d K. Ramajeathilagam c M. Ganapathi a a Tech Mahindra Ltd. Electronic Cit Bangalore- 56 India. b School of Civil and Environmental Engineering The Universit of New South Wales Sdne NSW 5 Australia. c School of Aeronautical Sciences Hindustan Universit Keelambakkam Chennai- 633 India. d Université Paris 3-RS LSPM UPR 347 Villetaneuse F-9343 France. Abstract In this paper the flutter characteristics of sandwich panels with carbon nanotube (T) reinforced face sheets are investigated using QUAD-8 shear fleible element developed based on higher-order structural theor. The formulation accounts for the realistic variation of the displacements through the thickness the possible discontinuit in the slope at the interface and the thickness stretch affecting the transverse deflection. The inplane and rotar inertia terms are also included in the formulation. The first-order high Mach number approimation to linear potential flow theor is emploed for evaluating the aerodnamic pressure. The solutions of the comple eigenvalue problem developed based on Lagrange s equation of motion are obtained using the standard method for finding the eigenvalues. The accurac of the present formulation is demonstrated considering the problems for which solutions are available. A detailed numerical stud is carried out to bring out the efficac of the higher-order model over the first-order theor and also to eamine the influence of the volume fraction of the T core-to-face sheet thickness the plate thickness and the aspect ratio damping and the temperature on the flutter boundaries and the associated vibration modes. Kewords: Carbon nanotube reinforcement sandwich plate higher-order theor aerodnamic pressure flutter frequencies shear fleible element. School of Civil and Environmental Engineering The Universit of New South Wales Sdne NSW 5 Australia. Tel: ; s.natarajan@unsw.edu.au; snatarajan@cardiffalumni.org.uk Research Scholar

2 . Introduction In recent ears non-structured non-metallic materials have spurred considerable interest in the materials communit partl because of their potential for large gains in mechanical and phsical properties as compared to standard structural materials. In particular carbon nanotube/polmer composites ma provide order-ofmagnitude increase in the strength and the stiffness when compared to tpical carbon fiber/polmer composites []. Due to these reasons structures made of such materials have great potentials in the construction of future supersonic /hpersonic space vehicles and reusable transportation sstems. Among the various structural constructions the sandwich tpe of structures are more attractive due to their outstanding bending rigidit low specific weight ecellent vibration characteristics and good fatigue properties. These sandwich constructions can be a candidature for the requirement of lightweight and high bending stiffness in the design. A tpical sandwich structure ma consist of a homogeneous core with facesheets. To improve the characteristics of these structures the facesheets can be laminated composites [] functionall graded materials [3] or polmer matri with reinforcements [4]. The definite advantages offered b the carbon nanotube reinforced composites (TRCs) over the carbon fibre-reinforced composites have prompted the engineers to design and analse sandwich structures with TRC facings [5]. Some studies conducted in evaluating the mechanical properties of Ts can be seen in the literature [6 7]. Thostenson and Chou [6] showed that the addition of nanotubes increases the tensile modulus the ield strength and the ultimate strength of the polmer films. Their stud has also brought out that the polmer films with aligned nanotubes as reinforcements ield superior strength when compared to randoml oriented nanotubes. The properties of the polmer films can also be optimized b varing the distribution of Ts through the thickness of the film. Formica et al. [7] highlighted that the T reinforced plates can be tailored to respond to an eternal ecitation. These eperimental investigations have created great interest among structural modeling and simulation analsts. For predicting the realistic behavior of sandwich structures with TRC facings more accurate analtical/numerical models based on the three-dimensional models ma be computationall involved and epensive. Hence among the researchers there is a growing appreciation of the importance of appling two-dimensional theories with new kinematics for the evolution of the accurate structural analsis. Few important contributions pertaining to the sandwich plates with TRC facesheets and the structural theories proposed for the analsis of such structures are discussed here. Based on the first-order shear deformation theor Zhu et al. [8] studied the static and free vibration of T reinforced plates. The considered polmer matri with T reinforcement neglecting the temperature effects. It was predicted that the T volume fraction has greater influence on the fundamental frequenc and the maimum center deflection. Wang and Shen [9] studied the large amplitude vibration of nano-composite plates resting on the elastic foundation using a perturbation technique. The governing equations were based on simple higher-order shear deformation theor. Their stud brought out that while the linear frequencies decrease with the addition of Ts the nonlinear to linear frequenc ratio increased especiall when increasing the temperature or b decreasing the foundation stiffness. Arani et al. [] Liew et al. [] and Lei et al. [] studied the buckling and post-buckling characteristics of T reinforced plates using the finite element and

3 meshless methods respectivel. It was revealed that the reinforcement with T increases the load carring capacit of the plate. Aragh et al. [3] used the generalized differential quadrature method and obtained a semi-analtical solution for 3D vibration of clindrical panels. It was shown that graded Ts with smmetric distribution through the thickness have high capabilities to alter the natural frequencies when compared to the uniforml distributed or asmmetricall distributed. It is observed from these investigations that first- order shear deformation theor has been widel emploed for the static and free vibration analses of T reinforced plates b man researchers whereas the simplified higher-order model considering variation in in-plane displacements has been used b few authors. However the available literature pertaining to sandwich structures with T reinforced facesheets is rather limited compared to those of fibre-reinforced facings plates. Various theories and structural models such as globallocal finite element model using hierarchical multiple assumed displacement fields [4] generalized multiscale plate theories [5] variational asmptotic structural models [6] generalized unified formulation with zig-zag theor [7] etc. that account for the variation of in-plane/transverse displacement through the thickness have been emploed for investigating the structural behavior of laminated reinforced composite structures. In this contet Ali et al. [8] and Ganapathi and Makhecha [9] have used a higher-order plate theor based on global approach for multi-laered laminated composites b incorporating the realistic through the thickness approimations of the in-plane and transverse displacements b adding a zig-zag function and higher-order terms respectivel. This approach has proved to give ver accurate results and computationall less epensive for the composite laminates compared to those of laerwise theor in which the number of unknowns to be solved increases with the increase in the number of mathematical or phsical laers. Such higher-order model for the stud of sandwich plates with T reinforced facesheets ma be worthwhile to consider as a candidature while comparing with the other formulations available in the literature. The increased effort towards integrating these materials in the construction of aerospace structures has necessitated investigating the aeroelastic stabilit issues of such structures. The panel flutter phenomenon is one of the aeroelastic dnamic instabilit problems encountered in the flight of aerospace vehicles. It is the self-ecited oscillation of the eternal skin of a flight vehicle when eposed to airflow along its surface. A comprehensive review of the theor associated with panel flutter analsis can be had from several articles such as Refs. [-]. This stud pertaining to composite laminates and functionall graded material structures constituting metal/and ceramic has received considerable attention in the literature [3-5]. However this tpe of analsis is not accomplished in the literature considering sandwich panels with TRC facings and it is worth investigating flutter stabilit characteristics of such structures eposed to aerodnamic flow. Approach. In this paper a C 8-noded quadrilateral plate element with 3 degrees of freedom per node [9 6 7] based on the higher order theor [8] is emploed to stud the flutter analsis of thick/thin sandwich plates with carbon nanotube reinforced facesheets. The aerodnamic force is evaluated assuming the first-order High Mach number approimation to linear potential theor. The efficac of the present formulation is illustrated 3

4 through the numerical studies b various structural models deduced from the present higher-order theor considering parameters such as T volume fraction core-to-facesheets thickness ratio plate thickness and aspect ratios and temperature. The influence of coalescence modes determining the flutter boundar is also discussed. Outline. The paper is organized as follows. The computation of the effective properties of carbon nanotube reinforced composites is discussed in the net section. Section 3 presents the higher order accurate theor to describe the plate kinematics and Section 4 describes the 8-noded quadrilateral plate element emploed in this stud. The numerical results for the aeroelastic stabilit of thick/thin sandwich carbon nanotube reinforced functionall graded plates are given in Section 5 followed b concluding remarks in the last section.. Theoretical Formulation Consider a T reinforced sandwich plate with the coordinate sstem z which has its origin at the corner of the plate on the middle plane as shown in Figure. The length the width and the total thickness of the plate are a b and h. The thickness of each T reinforced facesheet is h f and the thickness of homogeneous core laer is h H. It is assumed that the T reinforced laer is made from a miture of single walled T with uniforml distributed or functionall graded in the thickness direction and the matri is assumed to be isotropic. The effective properties of such reinforced structures can be computed b Mori-Tanaka scheme [3] or b the rule of mitures. As the rule of miture is simple it is emploed here to estimate the overall material properties of the structures. According to etended rule of mitures the effective material properties of the T reinforced matri are given b [8]: E E V E η η G η 3 V E V G V E V G m m m m V ν ν V + ν V m m m E m ρ ρ V + ρ V () m m where E E and G are the Young s moduli and the shear modulus of T respectivel. Em and Gm are corresponding properties of the matri. The T efficienc parameters (η η η 3 ) are introduced to account for the inconsistenc in the load transfer between the T and the matri. The values of the efficienc parameters are obtained b matching the elastic modulus of the T reinforced polmer matri from the MD stimulation results with the numerical results obtained from the rule of mitures. V and V m are the volume fraction of the T and the matri respectivel and the are related b V +V m. 4

5 The T distributions in the facesheet are functionall graded b linearl varing the volume fraction of the T in the thickness direction. It is assumed the volume fraction V!" for the top face sheet as V t Z V t t and for the bottom facesheet as V where V Z t V () t3 t w w ρ + ρ m [ ] w where w is the mass fraction of the nanotube ρ and ρ m are the mass densities of the T and the matri respectivel. The thermal epansion coefficient in the longitudinal and the transverse directions can be epressed as [8]: α α V + α V m m ( + ν ) α V + ( + ν ) α ν α α m mv m (4) where α α and α m are the thermal epansion coefficients for the T and the matri respectivel and ν and ν m are the Poisson s ratio. (3) 3. Governing differential equations The sandwich plate is assumed to be made of three discrete laers with a homogeneous core. The in-plane displacements u k and v k and the transverse displacement w k for the k th laer are assumed as [8 6]: k ( ) ( ) ( ) 3 k u z t uo t + zθ t + z β ( t) + z φ ( t) + S ψ ( t ) k ( ) ( ) ( ) 3 k v z t vo t + zθ t + z β ( t) + z φ ( t) + S ψ ( t ) w k ( z t) wo ( t) + zw ( t) + z Γ( t) (5) The terms with even powers in z in the in-plane displacements and the odd powers of z occurring in the epansion for w k correspond to the stretching problem. However the terms with odd powers of z in the inplane displacements and the even ones in the epression for w k represent the fleure problem. u v w are the displacements of a generic point on the reference surface; θ θ are the rotations of normal to the reference surface about the and aes respectivel; w β β Г φ φ are the higher order terms in the Talor's series epansions defined at the reference surface. ψ and ψ are generalized variables associated with the zigzag function S k. The zigzag function S k as given in [9 9 7] is defined b 5

6 6 ( ) h z S k k k k (6) where z k is the local transverse coordinate with the origin at the center of the k th laer and h k is the corresponding laer thickness. Thus the zigzag function is piecewise linear with values of and alternativel at different interfaces. The zigzag function as defined above takes care of the inclusion of the slope discontinuities of u and v at the interfaces of the sandwich plate as observed in the eact threedimensional elasticit solutions of thick laminates. The main advantage of using such a function in the formulation is more economical than a discrete laer approach [3 3]. Although both these approaches account for the slope discontinuit at the interfaces the number of unknowns increases with the increase in the number of laers in the discrete laer approach whereas it remains constant in the present approach. The strains in terms of mid-plane deformation rotations of normal and higher order terms associated with displacements are as { } s bm (7) The vector { } bm includes the bending and the membrane terms of the strain components and the vector { } s contains the transverse shear strain terms. These strain vectors can be defined as { } γ S z z z v u w v u k zz bm (8) { } γ γ γ γ γ γ 3 S z z w v w u k z z z z z s (9) where { } + v u w v u { } + Γ θ θ θ θ { } + β β β β { } + φ φ φ φ 3 { } + ψ ψ ψ ψ 4 () and

7 θ + w θ + w { γ } { γ } { γ } { } β + w β + w 3 3φ + Γ φ + Γ k ψ S z γ k () 3 ψ S z The subscript comma denotes partial derivatives with respect to the spatial coordinate succeeding it. The constitutive relations for an arbitar laer k can be epressed as: σ { σ σ σ σ σ σ } Q k { } bm s T zz z z T () where Q k is the stiffness matri defined as E Q k ; υυ E Q k ; υυ υ E Q k ; υ υ Q k Q k ; Q k ; Q Q 44 G 3; 55 G3 66 G k k 6 6 (3) For the homogeneous core the shear modulus G is related to the Young s modulus b: EG (+υ). The governing equations are obtained b appling the Lagrangian equations of motion given b d ( T U ) ( T U ) dt! δ i δ i i...n (4) where δ i the vector of degrees of freedom and T is the kinetic energ of the sandwich plate given b; n hk + T T( δ ) ρ k{ u! k v! k w! k}{ u! k v! k w! k} dz dd (5) k hk where ρ k is the mass densit of the k th laer h k and h k+ are the z coordinates to the bottom and top surfaces of the kth laer. The potential energ functional U is given b U h k h n k + T ( δ ) dz dd ( δ ) k σ W a (6) The work done b the applied non-conservative load is ( ) d W a δ Δp w d (7) where Δp is the aerodnamic pressure. The aerodnamic pressure based on first-order high Mach number approimation to linear potential flow is given as [-] Δp ρ au a w + M U a M M w t (8) 7

8 where ρ a U a and M are the free stream air densit velocit of air and the Mach number respectivel. Substituting equations (5) to (8) in Lagrange s equations of motion the following governing equation is obtained: M!! δ + g ( )δ (9) T D! δ + K + λ A A where K is the stiffness matri M is the consistent mass matri λ λ ( ) matri and M gt ( U a M ρ a U a A is the aerodnamic force is the aerodnamic damping parameter the damping matri D A can be considered as the scalar multiple of mass matri b neglecting the shear and rotar inertia terms of the mass M matri M and after substituting the characteristic of the time function equation is obtained:! ω δ the following algebraic δ " #([K]+ λ [A]) k[m ] $ % δ () where the eigenvalue k ω ( gt ω ρh) includes the contribution of the aerodnamic damping. Equation () is solved for eigenvalues for a given value of λ. In the absence of aerodnamic damping i.e. when λ the eigenvalue of ω is real and positive since the stiffness matri and the mass matri are smmetric and positive definite. However the aerodnamic matri A is unsmmetric and hence comple eigenvalues ω are epected when λ >. As λ increases monotonicall from zero two of these eigenvalues will approach each other and become comple conjugates. In this stud λ cr is considered to be the value of λ at which the first coalescence occurs. In the presence of aerodnamic damping the eigenvalues k in equation () becomes comple with increase in the value of λ. The corresponding frequenc can be written as: ( g ω h) k i k k ω ρ () T R I where the subscripts R and I refer to the real and the imaginar part of the eigenvalue. The flutter boundar is reached (λ λ cr) when the frequenc ω becomes pure imaginar number i.e. ω i k R at g k I k practice the value of λ cr is determined from a plot of ω R vs λ corresponding to ω R. 4. Element description In the present work C eight-noded serendipit quadrilateral shear fleible plate element is used. The finite element represented as per the kinematics based on Equation (5) is referred as HSDT3 with cubic variation. The 3 dofs are ( u v w θ θ w β β Γ φ φ ψ ψ ). Four more alternate discrete models are proposed to stud the influence of the higher order terms in the displacement functions whose displacement T R. In 8

9 fields are deduced from the original element b deleting the appropriate degrees of freedom. These alternate models and the corresponding degrees of freedom are shown in Table. 5. Numerical results and discussion In this section the flutter characteristics of sandwich plate with homogeneous core and T reinforced facesheets using the eight-noded shear fleible quadrilateral element is presented. The effect of various parameters such as the plate thickness and the aspect ratio the thermal environment the T volume fraction etc. on the global response is numericall studied. Here the sandwich plate is assumed to be simpl supported and is defined as follows: u o wo θ w Γ β φ ψ on b v o wo θ w Γ β φ ψ on a () where a and b refer to the length and width of the plate respectivel. For the present stud three different core-to-facesheet thickness h H /h f and four thickness ratios a/h 5 5 are considered. The distribution of T in the facesheets is functionall graded through the thickness unless otherwise specified. Material properties: In the present investigation Pol {(m-phenlenevinlene)-co-[(5-diocto-p-phenlene) vinlene]} referred as PmPV is selected as the matri in which the T s are used as reinforcements for certain cases. The material properties [8] of which are assumed to be ρ m 5 kg/m 3 υm.34 6 α 45( +.5ΔT ) K and E m ( T ) GPa. The temperature is defined as T T o + ΔT m / witht o 3K and ΔT is the increase in temperature. Single walled Ts are used as reinforcements and the material properties at different temperatures are given in Table. The T efficienc parameter η j are determined according to the effective properties of TRCs available b matching the Young s moduli E and E with the counterparts compared b the rule of mitures [8]. The efficienc parameters are: η. 49 η.934 for the case of V.; η. 49 η. 38 for the case of V. 7. It is assumed here as η η and the shear moduli are assumed to be 3 G 3 G G3. Polmethl methacrlate (PMMA) is also considered as a candidature for matri [8] and it is used for. 8. The Young s modulus of the matri considered for PMMA is E m ( T ) GPa and all other properties are same as that of PMPV. The corresponding T efficienc factors are: η.4 η.585 η.9 3. For this case the shear moduli are assumed to be V G3 G and G 3.G. Titanium allo Ti-6Al-4V is considered for the homogeneous core in the present analsis. The properties are: Young s modulus α H.5788( T 3.47 T ) 4.56( T GPa Poisson s ratio. 9 EH ) ν H K and mass densit 9

10 3 ρ 449Kg / m. The Ts are either uniforml distributed or functionall graded along the thickness H direction given b V ( z) V UD z + V FG V Tpe (5) h z FG X Tpe V h The effective material properties viz. Young s modulus Poisson s ratio and the mass densit are estimated from Equation (). The influence of the tpe of T volume fraction distribution in evaluating the effective properties is considered as defined in equation (5). Table 3 presents the convergence of the critical aerodnamic pressure and the flutter frequenc ( ω a ( ρ h D ) λ λ a / D ; D ( )) H h ν i H H ; cr cr H H E H for a chosen value of T volume fraction with decreasing mesh size for a simpl supported square sandwich plate with a/h 5 and h H /h h 8 emploing both first- and higher-order (FSDT5 HSDT3) structural models. A ver good convergence of results is observed with increase in the mesh discretization. For the problem considered here a 8 8 mesh is found to be ver much adequate to model the full plate irrespective of the tpes of structural models. To validate the efficac of the present formulation the free vibration characteristics of a single-laer carbon nanotube reinforced plate wherein the T is either distributed uniforml or functionall graded along the thickness as given in equation (5) is carried out and the results are shown in Table 4 for different T volumetric fraction and plate thickness ratio. The match ver well with those of available results in literature [8]. The structural model developed here is further tested considering the flutter problem of isotropic plates and the solutions are tabulated in Tables 5. These results are again found to be in ecellent agreement with those of reported in the literature [5]. Numerical eperimentation is further conducted to eamine the suitabilit of an appropriate structural theor using different structural models deduced from the present formulation as given Table and the calculated results varing the thickness ratios are given in Table 6 for the selected sandwich construction and T volume fraction. It is noticed from Table 6 that the higher-order model HSDTA is in close agreement with those of HSDT3. Also it ma be opined that the influence of higher-order theories is significant in particular predicting the flutter characteristics for thick sandwich plates and the results for different models approach to those of FSDT formulation for thin cases. However further investigation here is done emploing the HSDT3 and the FSDT for evaluating the behavior of T reinforced sandwich plates eposed to aerodnamic flow. A detailed investigation is made to bring out the influence of the core-to-facesheet thickness ratio (h H /h f 8 6 4) the T volume fraction (V. 7.8) and the temperature (T 3 5 7)K against the sandwich plate thickness ratio on the critical aerodnamic dnamic pressure and the are depicted in Figures

11 -4. It is inferred from Figure that for the given temperature the non-dimensional critical aerodnamic pressure for a sandwich plate thickness ratio a/h predicted adopting HSDT3 model are significantl different and less compared to those of FSDT5 due to the enhanced shear fleibilit associated with the HSDT3 theor. It is also revealed that increasing the homogeneous core thickness results in increase in the non-dimensional critical flutter speed. However there is a possibilit that with the increase in T volume fraction the sandwich structure with low core-to-facesheet ratio ma in general predict higher flutter boundar with the increase in aspect ratio and it depends on the temperature as highlighted in Figures 3 and 4. This is mainl due to the coalescence of higher and lower modes in determining the critical flutter behavior. It ma be further concluded that with the increase in the temperature the results evaluated b FSDT5 is significantl higher than those of HSDT3 which has greater shear fleibilit and accounts for the thickness stretching mode and this trend is observed while dealing with the bending analsis of composite plate subjected to thermal loading based on higher-order model [6]. The coalescence modes that are associated with the critical flutter speed is presented in Figure 5 for both thick and thin core sandwich plates (h H /h f 8 4) assuming different thickness ratio (a/h 5 5). It is clearl noticed from this Figure that the first two lower modes coalescence each other for thin plate having high core thickness whereas the lowest one coalescence with the higher mode while predicting the critical flutter speed for thick sandwich case. However for low core thickness case h H /h f 8 the coalescence of higher modes determines the critical aerodnamic pressure irrespective of thickness ratio of the plate considered here. This is possibl attributed to the increase in the stiffness of the facesheet of the sandwich plates. The relative in-plane displacements and the transverse displacement through the thickness direction of the sandwich plate (h H /h f 8 and a/h 5 ) for the coalescence mode of the chosen thickness ratios are plotted in Figures 6 and 7 considering two values of flutter speed. The mode shape along the flow direction is also included in these Figures. The relative displacements (u v w) are plotted along the lines (a/ b/ z) where h/ z h/. It is shown from these Figures that the transverse displacement w is not uniform and ehibits the eistence of normal stresses in the thickness direction. The variation of the transverse displacement is less at the centre of plate as the aerodnamic pressure approaches the critical value and this is attributed to the shift in the position of the maimum displacement towards the rear end of the plate. This can be seen in the fleural mode shape plot along the flow direction. It can be also viewed that the variation of inplane displacement is not significant compared to that of transverse displacement and the are linear or nonlinear irrespective of eistence of aerodnamic flow. For the chosen values of T volume fraction and temperature the influence of the aspect ratio a/b on the flutter characteristics of sandwich plates is evaluated and the results obtained here are highlighted in Table 7. It is revealed that the values of the critical aerodnamic pressure and the coalescence frequenc increase with the increase in the aspect ratio. The coalescence of higher modes is in general responsible for ielding higher critical values. It can be also opined that the increase in the T volume fraction results in increase in the flutter speed. Lastl the effect of aerodnamic damping is also eamined assuming thick panels (a/h5 and

12 h H /h f 8 4 and T 3K) and the flutter response is tabulated in Table 8. It is noticed from this Table that the introduction of damping enhances the flutter instabilit boundar. Lastl the influence of the functionall graded T distribution through the thickness of the facesheets over the uniform one is depicted in Table 9. It is revealed from this Table that the critical flutter speed is higher in general for the functionall graded T plate compared to those of the uniform case. It is further seen that the rate of increase in the critical value is more for functionall graded plate while decreasing the core thickness of sandwich plate as well as increasing the T volume fraction. The influence of temperature affects the performance of the sandwich plate against aerodnamic flow significantl. 6. Conclusions The flutter behavior of sandwich panels with T reinforced facesheets are studied considering various parameters such as the sandwich tpe the temperature effects the thickness and aspect ratio and the volume fraction of T. Different plate models are emploed in predicting the flutter frequencies and the critical aerodnamic pressure. From a detailed investigation on the effectiveness of the chosen structural model the following observations can be made: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (i) HSDTA in general predicts the flutter boundar of the structure ver close to the full structural model considered here HSDT3. The performance of the higher-order model HSDT3 for thick case is significantl different from the other lower order theories considered here and the predicted critical aerodnamic pressure is low. Increase in the volume fraction of T distribution in the facesheets in general results in increase in the flutter boundar. The effect of temperature based on the first-order model overestimates significantl the critical aerodnamic pressure in comparison with the higher-order one. The increase in the aspect ratio and the introduction of aerodnamic damping increases the critical flutter speed as epected. For thin plates with the increase in the T volume fraction the sandwich having lower core-tofacesheet thickness ma predict higher critical aerodnamic pressure due to the coalescence of lower mode with the higher one. The T distribution in a graded fashion through the thickness enhances the flutter boundar compared to that of uniform distribution case. In-plane relative displacement has slope discontinuit at the laer interface whereas the transverse displacement varies quadraticall through the thickness as epected. The occurrence of tpe of fleural/etensional modes in the thickness direction depends on the location of the structures and the fleural mode along the length of the plate that corresponds to coalescence changes its shape as the airflow is introduced.

13 Acknowledgements S Natarajan would like to acknowledge the financial support of the School of Civil and Environmental Engineering The Universit of New South Wales Sdne for his research fellowship for the period September onwards. The authors would also to acknowledge the contributions of MP Matharas postgraduate student Hindustan Universit in carring out some of the parametric studies. References. J. Jia J. Zhao G. Xu J. Di Z. Yong Y. Tao C. Fang Z. Zhang X. Zhang L. Zheng Q. Li A Comparison of the mechanical properties of fibers spun from different carbon nanotubes Carbon 49 () J. Whitne Stress analsis of thick laminated composite and sandwich plates Journal of Composite Materials 6 (97) A. Zenkour A comprehensive analsis of functionall graded sandwich plates: Part deflection and stresses International Journal of Solids and Structures 4 (5) V. Ugale K. Singh N. Mishra Comparative stud of carbon fabric reinforced and glass fabric reinforce thin sandwich panels under impact and static loading Journal of Composite Materialsdoi:.77/ S. C. Tjong Carbon nanotube reinforced composite John-Wile E. Thostenson T.-W. Chou Aligned multi-walled carbon nanotube-reinforced composites: processing and mechanical characterization J. Phs. D: Appl. Phs 35 () L G. Formica W. Lacarbonara R. Alessi Vibrations of carbon nanotube-reinforced composites Journal of Sound and Vibration 39 () P. Zhu Z. Lei K. Liew Static and free vibration analses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theor Composite Structures 94 () Z.-X. Wang H.-S. Shen Nonlinear vibration of nanotube-reinforced composite plates in thermal environment Computational Materials Science 5 () A. Arani S. Maghamikia M. Mohammadimehr A. Arefmanesh Buckling analsis of laminated composite rectangular plates reinforced b SWTs using analtical and finite element methods J Mech Sci Technol 5 () K. Liew Z. Lei J. Yu L. Zhang Postbuckling of carbon nanotube-reinforced functionall graded clindrical panels under aial compression using a meshless approach Computer Methods in Applied Mechanics and Engineering 68 (4) 7.. Z. Lei K. Liew J. Yu Buckling analsis of functionall graded carbon nanotube-reinforced composite plates using the element free kp-ritz method Composite Structures 98 (3)

14 3. B. S. Aragh A. N. Barati H. Hedaati Eshelb-Mori-Tanaka approach for vibration behavior of continuousl graded carbon nanotube-reinforced clindrical panels Composites Part B: Engineering 43 () D. H. Robbins Jr J. N. Redd Variable kinematic modeling of laminated composite plates Int. J. for Numerical Methods in Engineering 39 (996) T. O. Williams A new theoretical framework for the formulation of general nonlinear multiscale plate theories International Journal of Solids and Structures 45 (8) L. Demasi W. Yu Assess the accurac of the variational asmptotic plate and shell analsis using the generalized unified formulation Mechanics of Advanced Materials and Structures (3) J. Roderigues C. Roque A. Ferreira E. Carrera M. Cinefra Radial basis functions-finite differences collocation and a unified formulation for bending vibration and buckling analsis of laminated plates according to Murakami s zig-zag theor Composite Structures 93 () J. Ali K. Bhaskar T. Varadan A new theor for accurate thermal/mechanical fleural analsis of smmetricall laminated plates Composite Structures 45 (999) M. Ganapathi D. Makhecha Free vibration analsis of multi-laered composite laminates based on an accurate higher-order theor Composites Part B: engineering 3 () Y. C. Fung An introduction to the theor of aeroelasticit John-Wile Sons Inc. New York E. H. Dowell Panel flutter: A review of aeroelastic stabilit of plates and shells AIAA J. 8 (97) C. Mei J. L. Rogers Jr Application of Nastran to large deflection supersonic flutter of panel NASA TMX-348 (976). 3. S. C. Dion M. L. Hudson Flutter vibration and buckling of truncated orthotropic conical shells with generalized elastic edge restraints NASA TN D-5759 (97). 4. Feng-MingLi Zhi-Guang Song Flutter and thermal buckling control for composite laminated panels in supersonic flow Journal of Sound and Vibration 33(3) T. Prakash and M. Ganapathi supersonic flutter characteristics of functionall graded flat panels including thermal effects Composite Structures 7 (6) D. Makhecha M. Ganapathi B. Patel Dnamic analsis of laminated composite plates subjected to thermal/mechanical loads using an accurate theor Composite Structures 5 () S. Natarajan G. Manickam Bending and vibration of functionall graded material sandwich plates using an accurate theor Finite Elements in Analsis and Design 57 () Z.-X. Wang H.-S. Shen Nonlinear vibration and bending of sandwich plates with nanotubereinforced composite Composites Part B: Engineering 43 () H. Murukami Laminated composite plate theor with improved in-plane responses Journal of Applied Mechanics 53 (986) A. Nosier R. Kapania J. Redd Free vibration analsis of laminated plates using a laerwise theor AIAA Journal 3 (993)

15 3. A. Ferreira Analsis of composite plates using a laerwise deformation theor and multiquadratics discretization Mech. Adv. Mater. Struct (5) 99. 5

16 Table : Alternate eight-noded finite element structural models Finite element model HSDT3 HSDTA HSDTB HSDT9 FSDT5 Degrees of freedom per node u v w θ θ w β β Γ φ φ ψ ψ o o o u o vo wo θ θ β β φ φ ψ ψ u v w θ θ w β β φ φ u Γ w θ θ β β φ φ v u v w θ θ Table : Temperature dependent material properties for ( ) SWT [8] Temperature K E (Tpa) E (Tpa) G (Tpa) α ( 6 K) α ( 6 K) Table 3: Convergence of critical aerodnamic pressure λ cr with mesh size for a square sandwich plate with a/h5. The volume fraction of the T is V. and Temperature T ( ω a ( ρ h D ) λ λ a / D ; D ( )) H h ν i H H ; cr cr H H E H Mesh h H /h f Plate Theories In-vacuo Coalescence 8 HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT cr λ cr 6

17 ω Table 4: Comparison of fundamental natural vibration frequenc ( ( a ) E ω h ) of simpl supported TRC square plate with different T volume fractionv thickness ratio a/h and distribution of T (Uniform-UD Functionall Graded V & X) V a/h Non-dimensional natural frequenc ω UD FG-V FG-X Present Ref. [8] Present Ref. [8] Present Ref. [8] ρ m m Table 5: Critical aerodnamic pressure and coalescence frequenc for an isotropic square plate ( ω ( ρ ) λ λ ( ν )) a h D ; cr cr a / D ; D E h cr Simpl supported plate Clamped plate λ cr cr λ cr cr Present Ref. [5]

18 Table 6: Flutter boundar of T reinforced square sandwich plates based on different finite element models. The volume fraction of the T V. and T is functionall graded and Temperature T 3K. a/h h H /h f Plate theories In vacuo In coalescence 5 8 HSDT HSDTA HSDTB HSDT FSDT HSDT HSDTA HSDTB HSDT FSDT HSDT HSDTA HSDTB HSDT FSDT HSDT HSDTA HSDTB HSDT FSDT cr λ cr 8

19 Table 7: Flutter boundar of rectangular T reinforced sandwich plates (a/b and 5). The volume fraction of the T V.7; T is functionall graded and the temperature T 3K a/h a/b Plate theor In vacuo Coalescence 5 HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT cr λ cr Table 8: Aerodnamic damping on critical aerodnamic pressure of T reinforced square sandwich plate. The damping coefficient g T. and T is functionall graded and temperature T 3K a/h h V H /h f Plate In vacuo No Damping With Damping theor 3 4 cr λ cr λ cr HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT

20 Table 9: Flutter behavior of sandwich square plates with different distribution of T through the thickness of facesheet (uniform UD and Functionall Graded-FG). a/h V h H /h f T T In-vacuo Coalescence UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG UD FG cr λ cr

21 U b t t t t 3 z a h f h H h h f Figure. Coordinate sstem of a rectangular sandwich plate with and along the in-plane directions and z along the plane cross section.

22 critical aero.pressure λ cr a)t 3 HSDT3 FSDT h H /h f Thickness ratio a/h critical aero.pressure λ cr b) T 5 HSDT Thickness ratio a/h FSDT h H /h f Critical aero.pressure λ cr c) T 7 h H /h f 8 HSDT3 6 FSDT Thickness ratio a/h Figure. Variation of the critical aerodnamic pressure with aspect ratio for a square sandwich plate with different core- to- facesheet and temperature (V.).

23 critical aero.pressure λ cr critical aero.pressure λ cr a) T 3 HSDT b)t 5 Thickness ratio a/h HSDT3 FSDT h H /h f Thickness ratio a/h FSDT h H /h f critical aero.pressure λ cr c)t 7 HSDT3 FSDT h H /h f Thickness ratio a/h Figure 3. The variation of critical aerodnamic pressure with aspect ratio for a square sandwich plate with different core- to- facesheet and temperature (V.7). 3

24 critical aero.pressure λ cr a) T 3 HSDT3 FSDT h H /h f critical aero.pressure λ cr b) T 5 Thickness ratio a/h HSDT3 FSDT h H /h f Thickness ratio a/h critical aero.pressure λ cr c) T 7 HSDT Thickness ratio a/h FSDT h H /h f Figure 4. The variation of critical aerodnamic pressure with aspect ratio for a square sandwich plate with different core- to- facesheet and temperature (V.8). 4

25 a) a/h 5 a) a/h 5 frequenc cr 4 λ cr Im aerodnamic pressure λ frequenc cr 4 λ cr Im 3 4 aerodnamic pressure λ frequenc b) a/h cr λ cr Im aerodnamic pressure λ frequenc b) a/h 3 cr λ cr Im aerodnamic pressure λ frequenc c) a/h 5 cr λ cr Im aerodnamic pressure λ frequenc c) a/h 5 3 cr λ cr Im aerodnamic pressure λ Figure 5. The coalesence modes corresponding to critical flutter speed for different thickness ratio of a square sandwich plate(v.8 T 3): (a) Left side h H /h f 8 (b) Right side h H /h f 4 5

26 a) λ (u v) a) λ (u v) Normalized thickness z/h u v - -.E-6 Relative.E+ Displacements.E-6 (u v) 3.E-6 Normalized thickness z/h b) λ w -.96 Relative.98 Displacement w Normalized thickness z/h u v - -.E- -.E- Relative.E+ Displacements.E- (uv).e- Normalized thickness z/h b) λ w -.4 Relative.45 Displacement.5.55 w.6 Relative Displacement w c) λ Length of the plate a Relative Displacement w c) λ Length of the plate a Figure 6. The relative displacements of (u v w) through thickness and transverse displacement along the air flow of a square plate (a/h5 V.8 T 3): (a) Left side λ (b) Right side λ

27 Normalized thickness z/h Normalized thickness z/h a) λ (uv) u v -.E+ Relative.E-7 Displacements 4.E-76.E-78.E-7 (uv).e b) λ w Relative.998 Displacement.999 w Normalized thickness z/h a) λ (uv) u v - -.E- Relative -5.E- Displacements (uv).e+ 5.E-.E- Normalized thickness z/h b) λ w Relative.399 Displacement w.39 Relative Displacement w c) λ Length of the plate a Relative Displacement w c) λ Length of the plate a Figure 7. The relative displacements of (u v w) through thickness and transverse displacement along the air flow of a square plate (a/h V.8 T 3): (a) Left side λ (b) Right side λ