Research Article Flutter and Thermal Buckling Analysis for Composite Laminated Panel Embedded with Shape Memory Alloy Wires in Supersonic Flow

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1 Interntionl Journl of Aerospce Engineering Volume 216 Article ID pges Reserch Article Flutter nd Therml Buckling Anlysis for Composite Lminted Pnel Emedded with Shpe Memory Alloy ires in Supersonic Flow Chonghui Sho 1 Dengqing Co 1 Yuqin Xu 1 nd Hi Zho 2 1 School of Astronutics Hrin Institute of Technology P.O. Box 137 Hrin 151 Chin 2 Resource Deprtment Southwest Chin Reserch Institute of Electronic Equipment Chengdu 61 Chin Correspondence should e ddressed to Dengqing Co; dqco@hit.edu.cn Received 25 July 216; Accepted 25 Septemer 216 Acdemic Editor: Lind L. Vhl Copyright 216 Chonghui Sho et l. This is n open ccess rticle distriuted under the Cretive Commons Attriution License which permits unrestricted use distriution nd reproduction in ny medium provided the originl work is properly cited. The flutter nd therml uckling ehvior of lminted composite pnels emedded with shpe memory lloy (SMA) wires re studied in this reserch. The clssicl plte theory nd nonliner von-krmn strin-displcement reltion re employed to investigte the eroelstic ehvior of the smrt lminted pnel. The thermodynmic ehviors of SMA wires re simulted sed on one-dimensionl Brinson SMA model. The erodynmic pressure on the pnel is descried y the nonliner piston theory. Nonliner governing prtil differentil equtions of motion re derived for the pnel vi the Hmilton principle. The effects of ply ngle of the composite pnel SMA lyer loction nd orienttion SMA wires temperture volume frction nd prestrin on the uckling flutter oundry nd mplitude of limit cycle oscilltion of the pnel re nlyzed in detil. 1. Introduction Thin pnel is common nd useful form of structure component which hs een pplied significntly in highspeed vehicles such s ircrft spcecrfts nd rockets. Pnel flutter which occurs over criticl velocity under the coupling ctions of elstic inerti nd erodynmic force induced y the trnsonic supersonic or hypersonic irflow is one kind of self-excited virtion. Flutter rises themplitudeofvirtionswellstheerodynmicstress drmticlly which results in the filure of the structure. The flutter phenomen were oserved on the V-2 rockets for the first time during orld r II; since then lots of studies hve een crried out using different structurl nd erodynmic theories. Excellent surveys of erly reserches were presented y Dowell 1 2]. Composite mterils with the dvntges of high strength light weight nd low coefficient of therml expnsion hve een rodly employed in the designofthin-pnelstructures.meietl.3]presented recent survey out nlyticl methods of nonliner pnel flutter in supersonic ir flow. Birmn nd Lirescu 4] nlyzed eroelstic instility of lminted composite pnels for which the sher deformtion is considered in the modeling process under supersonic flow. Kouchkzdeh et l. 5] studied the nonliner eroelsticity prolem of lminted compositeplteundersupersonicirflowwheretheclssic plte theory ws dopted to estlish the structure dynmic model nd the supersonic irflow ws simulted vi liner piston theory. Results otined in 5] indicted tht the ply ngle hd importnt influence on the flutter ehvior. Zho nd Co 6] considered the erodynmic nonlinerity in the modeling process of stiffened lminte composite pnel under supersonic flow. Numericl results in 6] indicted tht the position thickness nd width of the stiffener hd significnt effect on the eroelstic ehvior. High-speed ircrfts re sujected to erodynmic pressure nd erodynmic heting which should e tken into ccount when solving the eroelstic prolems. As et l. 7] nlyzed the erothermoelstic ehvior of the isotropic nd orthotropic pnel. Shiu et l. 8] nlyzed the influence of temperture grdient on flutter phenomen of composite

2 2 Interntionl Journl of Aerospce Engineering lminted plte sed on the finite element method (FEM). Xie et l. 9] employed the proper orthogonl decomposition (POD)methodtonlyzetheflutterehviorofthepnel with uniform therml lodings. Xue nd Mei 1] nlyzed eroelstic prolem of the plte under nonuniform therml lodings y the finite element frequency domin method. Li ndsong11]employedthessumedmodepprochnd FEM to investigte erothermoelstic ehvior of composite pnel. In the lst two decdes numerous works hve een involved in suppressing pnel flutter y ctive or pssive control utilizing smrt mterils. Among these mterils SMA which is most suitle for ctive control of composite structures hs een extensively studied. Birmn 12] presented review out vrious pplictions of SMA in industry. By heting up to the ustenite finish temperture SMA is le to recover lrge prestrin totlly. SMA cn generte lrge recovery stresses when the prestrin is restrined. Through emedding SMA in the form of wires within the lminted composites the recovery stresses cn modify stiffness of the structures. This could improve structure chrcteristics of composite pnel such s virtion therml ulking impct loding flutter nd coustic. Prk et l. 13] employed FEM to study the influences of SMA fiers on flutter nd uckling ehvior of the plte. Ostchowicz et l. 14] used FEM to nlyze the uckling nd flutter ehviors of composite pltes nd the recovery stress generted y SMA fiers emedded in the plte is otined from experimentl dt. In ccordnce with the first-order sher deformtion plte theory Brzegri et l. 15] studied the eroelstic ehvior of rectngulr cntilever isotropic wings emedded with SMA wires where the erodynmic loding is estimted y liner piston theory. Asdi et l. 16] studied the prolem of virtion nd therml uckling for lminted composite em in which the SMAfiersreemeddedsymmetricllyndsymmetriclly. Kuoetl.17]usedFEMtoreserchflutterehviorof uckled SMA reinforced lmintes. The nonliner eroelstic ehvior of SMA hyrid composite plte ws studied y Irhim et l. 18] utilizing novel FEM. Howeverstudiesmentionedoverefocusedontherml uckling nd flutter of composite pnels emedded with SMA wires employing the FEM without giving considertion to the erodynmic nonlinerity in the modeling process. Moreover the well-developed one-dimensionl Brinson model for the thermodynmic ehviors of SMA is seldom utilized in the flutter nlysis. In this study in order to nlyze the smrt lminted pnel s dynmic chrcteristics theclssiclpltetheoryisemployedtoderivethenonliner governing differentil equtions of motion in which the nonliner von-krmn strin-displcement reltion is dopted. In the modeling process the thermodynmic ehviors of SMA wires re simulted sed on one-dimensionl Brinson SMA model while the erodynmic pressure is clculted y nonliner piston theory. The Glerkin method is dopted to derive the system discrete dynmic model which cn e solved numericlly y the Runge-Kutt method. The numericl results re utilized to show the effect of SMA wires on the nturl frequency uckling flutter nd mplitude of limit cycle oscilltion (LCO) of lminted composite pnel. Airflow z h O y θ n 2 1 SMA wires x Figure 1: Schemtic illustrtion of smrt lminted pnel. 2. Aeroelstic Model Consider n 8-lyer symmetric composite pnel emedded with SMA wires in Crtesin coordinte system with thickness h length nd width s displyed in Figure 1. SMA wires re ligned in fier direction in ritrry lyer. The supersonic flow is long the positive x direction nd SMA wires re emedded in the second nd seventh lyer symmetriclly s shown in Figure Structure Modeling. The clssicl plte theory is employed to descrie the displcements in the pnel; tht is u(xyzt)=u (xyt) z w (xyt) x V (xyzt)=v (xyt) z w (x y t) y w(xyzt)=w (x y t) where u Vndw denote the displcements in the x yndz directions respectively. Suscript stnds for the midplne displcement. The nonliner strin-displcement reltions ccording to the von-krmn ssumption re given y ε x ε y γ xy (1) u x ( w x ) z 2 w x 2 V = y ( w y ) z 2 w. (2) y 2 u y + V x + w w w x y 2z 2 x y BsedonHooke slwtheconstitutiveequtionofthenth lyer of the lminted composite pnel under therml lods is σ x σ y τ xy (n) = T 1 Q m(n) T T ε x α x ΔT (n) ε y α y ΔT (n) γ xy α xy ΔT (n) ε x α x ΔT (n) = Q m(n) ε y α y ΔT (n) γ xy α xy ΔT (n) (3)

3 Interntionl Journl of Aerospce Engineering 3 where the suscripts n nd m indictethelyernumer nd the composite mtrix ΔT denotes the temperture vrition nd α stnds for the therml expnsion coefficient. The trnsform mtrix T nd stiffness mtrix Q m re defined s 23] T = cos 2 θ sin 2 θ 2sin θ cos θ sin 2 θ cos 2 θ 2sin θ cos θ ] (4) sin θ cos θ sin θ cos θ cos 2 θ sin 2 θ] Q 11m Q 12m Q m = Q 12m Q 22m ] (5) Q 66m ] respectively. Here Q 11m Q 22m Q 12m ndq 66m re the stiffness coefficients which re defined s Q 11m = Q 22m = E 1m (1 υ 12m υ 21m ) E 2m (1 υ 12m υ 21m ) υ Q 12m = 12m E 2m (1 υ 12m υ 21m ) Q 66m =G 12m in which G 12m E 1m nde 2m stnd for the sher modulus nd Young modulus nd υ 12m nd υ 21m re Poisson rtios. (6) And for the kth lyer of the lminte composite pnel emedded with SMA wires the constitutive eqution is s follows 18]: σ (k) = Q (k) ε + T 1 σ r (k) V s(k) Q m(k) α ΔT (k) V m(k) where suscripts s nd k indicte SMA wires nd lyer numer. σ is in-plne stress vectors nd σ r toedescried in Section 2.2 is the SMA recovery stress vectors under the temperture T. Q is trnsformed reduced stiffness mtrix of the smrt lyer. V stnds for the volume frctions. The elstic properties used in Q re expressed s E 1 =E s V s +E 1m V m υ 12 =υ s V s +υ 12m V m E 2 = G 12 = E s E 2m E s V m +E 2m V s G s G 12m G s V m +G 12m V s Description of the Stress Model of SMA ires. According to one-dimensionl model of SMA proposed y Brinson 24] nd ssuming tht ll SMA wires re fully constrined one cn get the following expressions for the recovery stress of SMA wires: (7) (8) σ +Θ(T T ) T A σ s σ r = σ 1 +E s (ξ) E s (ξ )] ε +Ω(ξ) ξ s Ω(ξ )ξ s +Θ(T A σ s ) Aσ s T Aσ f (9) σ 2 +Θ(T A σ f ) T Aσ f. The two constnts in (9) re σ 1 =σ +Θ(A σ s T ) σ 2 =σ 1 +E A E s (ξ )] ε +Ω(ξ )ξ s +Θ(A σ f Aσ s ). (1) The phse trnsformtion coefficient Ω(ξ) nd the elstic modulus E s (ξ) hve the following expressions: A σ f Ω (ξ) = ε L E s (ξ) E s (ξ) =E A +ξ(e M E A ). (11) The ustenite strt temperture A σ s nd finish temperture under stress cn e expressed respectively s A σ s = AA s + A (ΘT σ ) A + A Θ A σ f = AA s +π A E A E s (ξ )] ε Ω(ξ )ξ s +σ ΘT A + A Θ (12) where ξ denotes the mrtensite frction E s (ξ) denotes the elstic modulus of SMA Θ represents the therml elstic modulus T represents temperture of SMA wires nd T denotes the reference temperture suscript denotes initil conditions nd ξ s stnds for the mrtensite frction induced y stress. The first expression in (9) is used for SMA intheinitilmrtensitesttethesecondoneisusedforsma in the phse trnsformtion stte nd the third one is used for SMA in 1% ustenite stte. The reltionships of SMA chrcteristics trnsformed from mrtensite to ustenite re given s ξ= ξ 2 cos A (T A s σ r c A )]+1

4 4 Interntionl Journl of Aerospce Engineering σ r (MP) Governing Equtions of Motion. The prtil differentil equtions of motion for the pnel cn e otined y utilizing Hmilton principle. t 1 (δt + δ δu) dt = (15) t where the vritions of kinetic energy δt nd virtul work δ s well s vrition of strins energy δu re given y ε =.3% ε =.5% T ( C) ε =.7% ε =1% Figure 2: Recovery stress generted y SMA. ξ s =ξ s ξ s ξ (ξ ξ) ξ T =ξ T ξ T ξ ξ =ξ s +ξ T A = π A f A s A = A c A M = π M s M f M = M c M (ξ ξ) (13) where ξ T denotes mrtensite frction induced y temperture nd c M nd c A re phse trnsformtion constnts. Figure 2 shows the computed SMA recovery stress versus vrious tempertures with four prestrin levels. It is shown tht in the phse trnsformtion stte SMA wires cn generte lrge recovery stresses. Moreover for higher prestrin n increse in the temperture will generte lrger recovery stresses Aerodynmic Pressure Modeling. The erodynmic lod Δp is descried y the third-order piston theory s Δp = 2q M w + w (γ + 1) M + V x 4 (γ + 1) M ( w + w 3 V x ) ] ( w + w 2 V x ) (14) where ρ V γndm denote the density velocity rtio of specific hets nd Mch numer of irflow nd q=ρ V 2 /2. δt = δ = δu = h/2 ρ ( uδ u+ Vδ h/2 V + wδw) dz dx dy Δpδw (x y.5h) dx dy h/2 (σ x δε x +σ y δε y +τ xy δγ xy )dzdxdy h/2 (16) where ρ stnds for density of the smrt lminted pnel. Sustituting (1) (5) (7) (8) nd (16) into (15) nd setting the coefficients of δu δvndδw to e zero one hs N xx x N yy y ( N xx x + N xy y =I u + N xy x =I V + N xy y ) w x +( N yy y + N xy x ) w y +N xx 2 w x M yy x 2 +N 2 w 2 w yy y 2 +2N xy x y + 2 M xx x M xy x y Δp=I I 2 ( 2 w x w y 2 ) w (17) where (I I 2 ) = h/2 h/2 ρ(1 z2 )dz ndtheforceresultnts opertors re otined s N xx N yy N xy M xx M yy M xy ] ] ] ] h/2 = h/2 h/2 = h/2 σ x σ y τ xy σ x σ y τ xy ] dz ] ] zdz. ] (18)

5 Interntionl Journl of Aerospce Engineering 5 The oundry conditions for simply supported pnel re u x= = u y= = c i = I +(i )π 2 I 2 (( N xx x + N xy y ) w x V = x= V = y= w = x= w = y= (19) +( N yy y + N xy x ) w y + 2 M xx x M yy y M xy x y +N 2 w xx x 2 +N 2 w 2 w yy y 2 +2N xy x y Δp)Φ i (x y) dx dy. w 2 x 2 x= = Let (22) w 2 y 2 =. y= The displcements of the pnel stisfying the oundry conditions re written s follows: l m u (x y t) = ij (t) Φ ij (x y) i=1 j=1 l m V (x y t) = ij (t) Φ ij (x y) i=1 j=1 l m w (x y t) = c ij (t) Φ ij (x y) i=1 j=1 where the mode shpes re tken s (2) Φ ij (x y) = sin ( iπx ) sin (jπy ). (21) The numericl results provided y Dowell 25] show tht for getting resonle results t lest four modes re needed to study the pnel flutter ehvior. In present study four stremwise modes nd one spnwise mode re reserved in the following clcultion. Sustituting (2) into (17) nd fter tht integrting over the pnel re the discrete dynmic model is given s i = 4 i = 4 I I ( N xx x + N xy y )Φ i (x y) dx dy ( N yy y + N xy x )Φ i (x y) dx dy y= c 1 c 1...c 4 c 4 T (23) ndthen(22)cneexpresseds y=ay+g(y) (24) where A is the Jcoin mtrix t the equilirium point y= nd g(y) respects the nonliner terms induced y geometric nd erodynmic nonlinerity. 3. Results nd Discussion As the dynmic pressure reches the criticl flutter dynmic pressure cr the motion of the pnel chnges from flt condition to flutter condition sed on the nonliner theory. If > cr ecuse of the existence of the geometric nd erodynmic nonlinerities the mplitude of virtion of the pnel will increse with time nd eventully converge t limit cycle. On the contrry the mplitude of virtion will decrese with time if < cr. The generl solution of (24) cn e written s y (t) =y e φ jt φ j =α j ±iβ j (25) where y nd φ j stnd for the eigenvector nd eigenvlues of the mtrix A. The nturl frequencies of the pnel cn e otined s ω j = β 2 j. (26) As the rel prt of ritrry eigenvlue turns from negtive to positive the flutter will hppen. By exmining the mximl rel prt of eigenvlues μ cr cneotined.μ cn e descried s μ=mx Re (φ j )] = mx (α j ). (27) In the present study the lminted pnel with width of.4 m length of.5 m nd thickness of.25 m is tken for nlysis (except Section 3.3 where the effect of length-to-width rtios on cr is discussed). The trnsverse

6 6 Interntionl Journl of Aerospce Engineering displcements re plotted t the point (x y) = (.75.5). The mteril prmeters re given s follows. Composite Lmin Properties Tle 1: Criticl uckling temperture of the isotropic pnel. Zho et l. 19] Shiu et l. 2] Shi et l. 21] Present ΔT cr ( C) E 1m = 138 GP E 2m =9.7GP G 12m =5.5GP ρ m = 158 kg/m 3 υ m =.3. SMA ires Properties (28) Tle 2: Comprison of criticl uckling temperture for composite pnel. Ply ngle ΔT cr ( C) Mtsung 22] Present /9/9/] s /45/ 45/9] s E A =67GP E M = 26.3 GP M s = 18.4 C M f =9 C A s = 34.5 C A f =49 C C M =8MP/ C C A = 13.8 MP/ C Θ =.55 ε l =.67 σ = T =2 C ρ s = 645 kg/m 3 ξ T = υ s =.33. (29) 3.1. Vlidtions of the Present Method. The numericl clcultions re performed vi MATLAB softwre. Bsed on the derived formultion the Runge-Kutt method is utilized to investigte nonliner flutter nd uckling ehvior of isotropic pnel nd lminted composite pnel respectively. The criticl therml uckling temperture is compred with the results provided y Mtsung 22] Zho et l. 19] Shiu et l. 2] nd Shi et l. 21]. Tles 1 nd 2 list the results of the former nlyses nd the present works. It cn e oserved from Tles 1 nd 2 tht the results otined here hve good greement with those results in the literture. In ddition the mplitude of LCO of the pnel with therml effect otined here hs good greement with those results in 26] s shown in Figure The Effects of Orienttion nd Position of Lyer Emedded with SMA ires. The influences of orienttion of SMA wires on cr re investigted first. The fier orienttion of the composite pnel is ssumed to e 9/ 45/45/θ SMA ] s with SMA wires emedded in the fourth nd fifth lyer symmetriclly. The SMA wires hve prestrin of.5% volume frction of 1% nd temperture of 5 C. The vrition of cr versus the ngleofsmawiresisdepictedinfigure4.itcneoserved from Figure 4 tht chnging the ngle of lyer emedded with SMA wires from to 9 decreses cr of the pnel. Also it is shown tht the pnel hs the highest cr with the ply ngle of 9/ 45/45/ SMA ] s. Consequently orienttion of the lyer emedded with SMA wires is the most importnt prmeters for designing nd optimizing the smrt lminted pnel. In the following nlysis the orienttion of lyer emedded with SMA wires is designed to e zero. Asfortheinfluencesofpositionofthelyeremedded with SMA wires of the 8-lyer symmetric lminted pnel on the cr chngeitisfoundinfigure5thtfivecsesoflminted pnels with different position of the lyer emedded with SMA wires hve een studied. From Figure 5 one cn see tht when chnging the positon of the lyer emedded with SMA wires from outer lyer SMA / 6/6/ 6] s to inner lyer 6/ 6/6/ SMA ] s in sequence cr my decrese which indictes tht emedding SMA wires in the outer lyer is more significnt for enhncing cr of the lminted pnel The Effects of Length-to-idth Rtios. Figure 6 shows the flutter oundry versus fier orienttion θ/ θ/θ/ θ] s under different length-to-width rtios of the pnel. For the cse when / 2 incresing the fier orienttion from to 9 results in decresing the stiffness of the pnel in the x direction which will lower the flutter oundry of the pnel. However for the cse when / > 2 cr increses s θ increses initilly nd decreses fterwrds. In ddition Figure 6 revels tht n increse in the pnel length-to-width rtios leds to stiffer composite pnel The Effects of Therml Lods. The curves of frequencies ω versus for the pnel without SMA wires under different temperture vrition re shown in Figure 7. It cn e found in Figure 7 tht s the dynmic pressures incresing the first nd the second nturl frequencies grdully pproch ech other nd finlly overlp then the pnel will e in limit cycle oscilltion condition. Also the nturl frequencies nd

7 Interntionl Journl of Aerospce Engineering ΔT/ΔT cr = cr Present Xue nd Mei 26] cr θ Figure 3: Comprison of limit cycle mplitudes. / = 1 / = 2 / = 2.5 / = 3 13 Figure 6: Flutter oundry versus lyer ngle with different lengthto-width rtios cr ω θ Figure 4: Flutter oundry versus orienttion ngles of SMA wires ΔT = C ΔT =.7ΔT cr ΔT = ΔT cr ΔT = 1.2ΔT cr 5 θ 1 / θ 2 /θ 3 / θ 4 ] s Figure 7: Nturl frequencies versus dynmic pressure under different temperture vritions. cr Figure 5: Flutter oundry versus position of lyer emedded with SMA wires (i = 6/ 6/6/ 6] s ; i=1 SMA / 6/6/ 6] s ; i = 26/ SMA /6/ 6] s ; i = 36/ 6/ SMA / 6] s ; i = 4 6/ 6/6/ SMA ] s.). i cr re reduced with the rise of the temperture vrition. hen the temperture vrition of the pnel is rised over the criticl uckling temperture ΔT cr thepnelwilleuckling (ut dynmiclly stle) under smll dynmic pressure nd the first-order nturl frequency will e zero. As the dynmic pressure increses up to the criticl vlue 1 the motion of the pnel will turn to flt (nd stle) condition s shown in Figure 7. So it seems tht the process for flutter is contrry to tht for therml uckling. The curves of frequencies ω versus for the pnel emedded with/without SMA wires under different temperture vrition re shown in Figure 8. The SMA wires hve volume frctions of 1% prestrin of.5% nd temperture of 5 C. It cn e reveled from Figure 8 tht the pnel emedded

8 8 Interntionl Journl of Aerospce Engineering ω ΔT = CV s = ΔT = CV s =.1 ΔT = ΔT cr V s = ΔT = ΔT cr V s =.1 Figure 8: Nturl frequencies versus dynmic pressure under different temperture vritions nd volume frction of SMA wires t Pnel without SMA Pnel emedded with SMA Figure 1: Time history for =7nd ΔT =.5ΔT cr A D (I) flt E (III) limit cycle (IV) chotic (II) uckled B C ΔT/ΔT cr V s = V s =.1 Figure 9: Stility mrgins t Pnel without SMA Pnel emedded with SMA Figure 11: Time history for =2nd ΔT = 1.1ΔT cr. with SMA wires hs the sme trend s the pnel without SMA wires; however frequencies re enhnced due to the recovery stresses cused y the SMA wires. Moreover the SMA wires cn suppress oth the flutter nd therml uckling of the pnel. Figure 9 shows stility mrgins for the pnel emedded with/without SMA wires under comined therml lods nd dynmic pressure lod. The SMA wires hve volume of frctions of 1% prestrin of.5% nd temperture of 5 C. The pnel hs four types of motion: in region (I) under smll nd ΔT the pnel is flt nd stle; in region (II) for smll nd moderte ΔTthermlucklingoccurs;inregion (III) for moderte nd ΔT LCO occurs; in region (IV) for sufficiently high nd ΔTchoticmotionoccurs. Figures 1 14 present the time history responses of the pnel t points A E in Figure 9 respectively. Figure 1 shows tht the trnsverse virtion of the pnel corresponding to t Pnel without SMA Pnel emedded with SMA Figure 12: Time history for =2nd ΔT = 1.4ΔT cr.

9 Interntionl Journl of Aerospce Engineering A f Austenite cr 16 A s t Pnel without SMA Pnel emedded with SMA Figure 13: Time history for =9nd ΔT =.5ΔT cr t Pnel without SMA Pnel emedded with SMA Figure 14: Time history for =9nd ΔT = 1.4ΔT cr. point A in district (I) decreses s time increses. Also the virtion mplitude of the pnel emedded with SMA wires converges quickly compred to the pnel without SMA wires. The motions of points B nd C in Figure 9 re depicted in Figures 11 nd 12 respectively. In the cse of low dynmic pressure incresing the temperture vrition of the pnel ove ΔT cr themotionwillechngedfromstlefltto uckled condition. The pnel without SMA wires is in uckled stte nd the pnel emedded with SMA wires is stle flt s shown in Figure 11. Also the uckling deflection of the smrt lminted pnel is smller thn the conventionl lminted pnel s oserved from Figure 12. Therefore the SMA wires cn suppress the therml uckling of the pnel ndreducetheucklingdeflectionsignificntlyforgiven therml lod. The pnel is stle flt t higher dynmic pressure nd it will flutter s the temperture incresed. Figures 13 nd14plotthemotionsofpointsdndeinfigure9. The pnel emedded with SMA wires ecomes convergence 12 Mrtensite T ( C) V s =.1 V s =.2 V s =.3 Figure 15: Curves of cr versus T with vrious V s. when the pnel without SMA wires hs LCOs s shown in Figure 13. Also the mplitude of LCOs of the smrt lminted pnel is smller thn the conventionl lminted pnel s oserved from Figure 14. As consequence the SMA wires cn suppress the flutter of the pnel nd the mplitude of the LCOs cn e significntly reduced for given dynmic pressure The Influences of SMA ires Temperture Prestrin nd Volume Frction. The influences of SMA wires temperture nd volume frction on cr re depicted in Figure 15. The SMA wires hve prestrin of.7% nd the pnel temperture vrition is ssumed to e zero. hen the SMA wires temperture is higher thn A s phse trnsformtion from mrtensite to ustenite occurs. During this trnsformtion SMA wires cn generte lrge recovery stress until the temperture is higher thn A f.thus cr is enhnced vi rising the SMA wires temperture s shown in Figure 15. Moreover s shown in Figure 15 when SMA wires temperture is higher thn A s incresing the SMA wires volume frction leds to n improvement on cr. Thus for the purpose of enhncing the lod-crrying cpcity of the smrt lminted pnel the prmeters of SMA wires must e chosen crefully. The heted SMA wires cn generte recovery stresses nd then produce dditionl stiffness tht will chnge the dynmic response of the pnel. SMA wires hve prestrin of.7% nd temperture of 55 C. Temperture vrition of the pnel is ssumed to e 1.2ΔT cr. hen the temperture vrition is 1.2ΔT cr the pnel is uckling. ith the increse of dynmic pressure the pnel will ecome stle from the ucklingsttendthentheflutterwillhppensshown in Figures 16 nd 17. It is shown from Figure 16 tht rising the volume frction is le to enhnce the stility mrgin of the pnel. Figure 17 demonstrtes the uckling deflection nd mplitude of LCO versus dynmic pressure with different volume frctions of SMA wires. As displyed using SMA wires cn reduce oth the uckling deflection

10 1 Interntionl Journl of Aerospce Engineering μ 5 μ V s = V s =.2 V s =.1 V s =.3 T= C T=55 C T=75 C T=9 C Figure 16: Influence of V s on stility oundry. Figure 18: Influence of T on stility oundry V s = V s =.2 V s =.1 V s =.3 Figure 17: Influence of V s on uckling flt nd flutter phenomen T= C T=55 C T=75 C T=9 C Figure 19: Influence of T on uckling flt nd flutter phenomen. nd the mplitude of LCO. The results clerly indicte tht the recovery stress introduced y SMA wires leds to more stiffened pnel for wide rnge of dynmic pressure nd thus highercriticlflutterdynmicpressure.specificllyfor higher volume frction s V s =.3theucklingofthepnel will not hppen for temperture vrition 1.2ΔT cr. Figures 18 nd 19 revel the influences of SMA wires temperture on ΔT cr cr ndtrnsversevirtionofthe pnel. The temperture vrition of the pnel is ssumed to e 1.2ΔT cr nd the SMA wires hve prestrin of 1% nd volume frction of.1. As shown in Figure 18 ΔT cr nd cr increse with rising the SMA wires temperture. Also s displyed in Figure 19 the mplitude of LCO cn e reduced y incresing the SMA wires temperture. Theinfluenceofprestrinontheucklingndflutter ehvior of the pnel is displyed in Figures 2 nd 21. The temperture vrition of the pnel is ssumed to e 1.2ΔT cr nd the SMA wires hve volume frction of.1 nd temperture of 9 C. Rising the SMA wires prestrin cn ugment equl stiffness of the pnel nd therefore cr nd ΔT cr re incresed s demonstrted in Figure 2. Figure 21 demonstrtes the influence of prestrin on the uckling deflection nd mplitude of LCO of the pnel. It cn e concluded tht the uckling deflection nd mplitude of LCO cn e llevited y incresing the SMA wires prestrin. 4. Conclusions Therml uckling nd flutter ehviors of lminted composite pnel emedded with SMA wires sujected to nonliner erodynmic loding nd therml lod hve een nlyzed in this pper. The von-krmn lrge deflection plte theory for structures one-dimensionl Brinson model for

11 Interntionl Journl of Aerospce Engineering 11 μ ε = ε =.3% ε =.5% ε =1% Figure 2: Influence of ε on stility oundry. (3) It is more significnt to emed SMA wires in the outer lyer of the lminted pnel thn in the inner lyer for the flutter chrcteristics. (4) The criticl flutter dynmic pressure nd criticl therml uckling temperture cn e enhnced y heting the SMA wires incresing SMA wires volume frction or prestrin. Therefore the criticl flutter dynmic pressure nd criticl therml uckling temperture of the smrt lminted pnel cn e gretly incresed nd the mplitude of LCO cn e significntly reduced for given flutter dynmic pressure. The theoreticl results presented in this pper cn e pplied in the prcticl engineering prolem involving shpe memory lloy wires. It is helpful for the eroelstic nlysis nd virtion control of supersonic structures. Nomenclture (A) Composite Prmeters ε = ε =.3% ε =.5% ε =1% Figure 21: Influence of ε on uckling flt nd flutter phenomen. SMA wires nd the nonliner piston theory for erodynmics re used to derive the nonliner governing eqution of motion for the pnel emedded with SMA wires. The system discrete dynmic model is otined y employing the Glerkin method. A composite pnel with set of typicl mteril constnts nd geometricl prmeters is tken s n exmple to illustrte the method proposed in this study. The Runge-Kutt method hs een employed to solve the system. The numericl results show the following: (1) hen / is lower thn 2 the eroelstic stility of lmintedpnelisdecresingwiththeincreseofthe ply ngle θ/ θ/θ/ θ] s ;howeverwhen/ is more thn 2 the eroelstic stility will increse s the ply ngle increses initilly nd decrese fterwrds. (2) Emedding SMA wires in the lyers of composite pnel cn improve the eroelstic stility oundry nd the most efficient design is to emed SMA wires in the irflow direction. : Length of the pnel : idth of the pnel D () 11 : Vlue of the mss moment of inerti of the pnel when ll fiers re ligned with the x-xis E 1m E 2m : Youngmodulusofmtrixin1nd2 directions E s : YoungmodulusofSMAwire h: Thicknessofthecompositepnel M: Mch numer M kl N kl : Force resultnts opertors q: Dynmic pressure Q: Lminstiffnessmtrix u Vw: Displcements of the pnel in the x y z directions u V w : Displcements of the midplne V m : Volume frctions of the mtrix : Trnsverse virtion mplitude V : Reltivefreeirstremvelocity Δp: Aerodynmic pressure ΔT: Pnel temperture vrition δt: Virtul kinetic energy δu: Virtul strins energy δ: Virtul work done y the erodynmic pressure φ j : Eigen vlues of mtrix A : Dimensionless dynmic (=2q 3 /MD () μ: Themximumrelprtofφ ρ m ρ s : Density of the mtrix nd SMA wires υ 12m υ s : Poisson rtio of mtrix nd SMA wires ω: Pnel nturl frequency. (B) SMA Brinson Model Prmeters σ r : Recovery stress of SMA Θ: Thermoelsticmodulus 11 )

12 12 Interntionl Journl of Aerospce Engineering E M : Mrtensite Young modulus E A : Austenite Young modulus ε L : Mximum residul strin ε : Prestrin M f : Mrtensite finish temperture M s : Mrtensite strt temperture A s : Austenite strt temperture A f : Austenite finish temperture c A c M :Stressinfluencecoefficient ξ: Totlmrtensitevolumefrction ξ s : Stress induced mrtensite volume frction ξ s : Initil stress induced mrtensite volume frction ξ T : Temperture induced mrtensite volume frction ξ T : Initil temperture induced mrtensite volume frction. Competing Interests The uthors declre tht they hve no competing interests. Acknowledgments The uthors grtefully cknowledge the Ntionl Nturl Science Foundtion of Chin (Grnt no ) for the finncil support of this work. References 1] E. H. Dowell Nonliner oscilltions of fluttering plte The Americn Institute of Aeronutics nd Astronuticsvol.4no.7 pp ] E. H. Dowell Pnel flutter review of the eroelstic stility of pltes nd shells AIAA Journl vol.8no.3pp ] C.MeiK.Adel-MotglyndR.Chen Reviewofnonliner pnel flutter t supersonic nd hypersonic speeds Applied Mechnics Reviewsvol.52no.1pp ] V. Birmn nd L. Lirescu Supersonic flutter of sher deformle lminted composite flt pnels Journl of Sound nd Virtionvol.139no.2pp ] M.A.KouchkzdehM.RsekhndH.Hdddpour Pnel flutter nlysis of generl lminted composite pltes Composite Structures vol. 92 no. 12 pp ] H. Zho nd D. Co A study on the ero-elstic flutter of stiffened lminted composite pnel in the supersonic flow Journl of Sound nd Virtionvol.332no.19pp ] J. F. As R. A. Irhim nd R. F. Gison Nonliner flutter of orthotropic composite pnel under erodynmic heting AIAA Journlvol.31no.8pp ] L.-C. Shiu S.-Y. Kuo nd Y.-P. Liu Aerothermoelstic nlysis of composite lminted pltes Composite Structures vol. 94 no.6pp ] D. Xie M. Xu H. Di nd E. H. Dowell Proper orthogonl decomposition method for nlysis of nonliner pnel flutter with therml effects in supersonic flow Journl of Sound nd Virtion vol. 337 pp ] D. Y. Xue nd C. Mei Finite element nonliner pnel flutter with ritrry tempertures in supersonic flow AIAA journl vol.31no.1pp ] F.-M. Li nd Z.-G. Song Flutter nd therml uckling control for composite lminted pnels in supersonic flow Journl of Sound nd Virtionvol.332no.22pp ] V. Birmn Review of mechnics of shpe memory lloy structures Applied Mechnics Reviews vol. 5 no. 11 pp ] J.-S. Prk J.-H. Kim nd S.-H. Moon Therml post-uckling nd flutter chrcteristics of composite pltes emedded with shpe memory lloy fiers Composites Prt B: Engineering vol. 36no.8pp ]. Ostchowicz M. Krwczuk nd A. Żk Dynmics nd uckling of multilyer composite plte with emedded SMA wires Composite Structuresvol.48no.1 3pp ] M. M. Brzegri M. Drdel A. Fthi nd M. Ghdimi Aeroelstic chrcteristics of cntilever wing with emedded shpe memory lloys Act Astronutic vol. 79 pp ] H. Asdi M. Bodghi M. Shkeri nd M. M. Aghdm An nlyticl pproch for nonliner virtion nd therml stility of shpe memory lloy hyrid lminted composite ems Europen Journl of Mechnics A Solids vol.42pp ] S.-Y. Kuo L.-C. Shiu nd C.-H. Li Flutter of uckled shpe memory lloy reinforced lmintes Smrt Mterils nd Structuresvol.21no.3ArticleID ] H. H. Irhim M. Twfik nd H. M. Negm Therml uckling nd nonliner flutter ehvior of shpe memory lloy hyrid composite pltes Journl of Virtion nd Control vol.17no. 3 pp ] X.ZhoY.Y.LeendK.M.Liew Mechniclndtherml uckling nlysis of functionlly grded pltes Composite Structuresvol.9no.2pp ] L.-C. Shiu S.-Y. Kuo nd C.-Y. Chen Therml uckling ehvior of composite lminted pltes Composite Structures vol.92no.2pp ] Y. Shi R. Y. Y. Lee nd C. Mei Therml postuckling of composite pltes using the finite element modl coordinte method Journl of Therml Stressesvol.22no.6pp ] H. Mtsung Therml uckling of cross-ply lminted composite nd sndwich pltes ccording to glol higher-order deformtion theory Composite Structures vol. 68 no. 4 pp ] V. Birmn Plte Structures (Solid Mechnics nd Its Applictions) Springer New York NY USA ] L. C. Brinson One-dimensionl constitutive ehvior of shpe memory lloys: thermomechnicl derivtion with nonconstnt mteril functions nd redefined mrtensite internl vrile Journl of Intelligent Mteril Systems nd Structures vol. 4 no. 2 pp ] E. H. Dowell Nonliner oscilltions of fluttering plte. II AIAA Journlvol.5no.1pp ] D. Y. Xue nd C. Mei Finite element nonliner flutter nd ftigue life of two-dimensionl pnels with temperture effects Journl of Aircrftvol.3no.6pp

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