Modeling and Analysis of Laminated Composite Plates Embedded with Shape Memory Alloy Wires

Transcription

1 6 th Intenatonal Confeence on AEROSPACE SCIENCES & AVIAION ECHNOLOGY, ASA - 6 May 6-8, 5, E-Mal: asat@mtc.edu.eg Mltay echncal College, Koby Elobbah, Cao, Egypt el : +() , Fax: +() 698 Modelng and Analyss of Lamnated Composte Plates Embedded wth Shape Memoy Alloy Wes M. Kassem Abbas*, M Adnan Elshafe*, Hany M Negm + * Egyptan Amed Foces + Depatment of Aeospace Engneeng, Cao Unvesty, Cao, Egypt Abstact: A Lamnated composte plate wth embedded shape memoy alloy wes ae modeled and analyed based on the modfed hghe-ode shea defomaton theoy. he Hamlton s pncple n conjuncton wth Bnson s consttutve model s used to obtan the thee-dmensonal govenng equatons. he Rt soluton technque s used to get the statc esponse as well as the dynamc chaactestcs of the poposed plates. A smply-suppoted and cantleveed plates subjected to mechancal loads ae used n the analyss wth a pepaed pogam usng Mathematca langue. Paametc studes ae conducted to demonstate the effect of plate dmensons, fbe oentaton angle and volume facton of composte fbes and shape memoy alloy wes, on the actvated plate natual fequences. he obtaned esults ae compaed to the avalable studes solved by dffeent theoes and found convenent. Keywods: Shape memoy alloys, Bnson s model, composte plate, Hghe-ode shea defomaton theoy, ule of mxtue, Rt enegy method, statc and dynamc analyss.. Intoducton he Shape memoy alloy (SMA) has unque popetes of one way shape memoy effect (SME), two way shape memoy effect, pseudoelastcty and hgh dampng capacty. he SMA wes, stps, o sheets can be suface mounted o embedded n a stuctue coe to fom smat composte stuctue. Intensve eseaches have been made on SMA wes and bonded sheets wong as actuatos [-8]. Ghomshe et al [-] poposed a nonlnea fnte element model and expemental test fo the tme esponse of a shape memoy alloy actuato composed of coe mateal wth SMA sheets o wes embedded n o bonded to the coe pat. he model s developed based on a hghe ode shea defomaton theoy and the von- Kaman stan feld equaton. A one-dmensonal consttutve equaton wth non-constant mateal functons and snusodal phase tansfomaton netcs s used to model the themo-mechancal behavo of the SMA actuato. hee ae types of matenste dependng on the way that causes the tansfomaton: st type s tempeatue nduced matenste whch caused by deeasng tempeatue only wth fee stess and ths type doesn t cause ecovey stess as shown n Fgue A numbe, whle the nd type s stess nduced matenste whch caused by applyng stess and ths type geneate ecovey stess as shown n Fgue A numbe -a and -b. Balapgol et al [-4] studed the deflecton, natual fequency and tme esponse of a lamnated composte plate conssts of thn laye of SMA bonded to elastome coe usng fnte element method based on the fst ode shea defomaton theoy.

2 hey concluded that the nput powe heat sn stength, themal conductvty, and thcness of the elastome laye play mpotant oles fo contollng the tme esponse of the SMA lamnated actuato. Godannejad et al [5] pesented a two- dmensonal fnte element model based on classcal lamnaton theoy, enegy equatons, and two- dmensonal tanston model of SMA laye fo the esponse of themally dven SMA / elastome actuato. Wu et al [6]deved closed fom solutons fo the stess-stan-tempeatue esponse of themally dven shape memoy alloy composte actuato neglectng the heat conducton n axal decton. Roges et al [7] used the Raylegh- Rt method to pefom a lnea analyss fo smply-suppoted plate embedded wth SMA fbes. hey studed the plate deflecton, fee vbaton, buclng, and acoustc contol. Ln et al [8] poposed a closed fom soluton fo symmetc composte beams embedded wth SMA fbes wth vaous bounday condtons. he esultant actuaton foces and nomal stess dstbuton wee calculated fo the poposed beams. Seveal consttutve models have been poposed to pedct the themo-mechancal esponse of the SMA. One of the ealest models s the one-dmensonal anaa s model [9], whch s maoscopc model that s deved fom themodynamc concepts and though expemental obsevatons. hs means that the tansfomaton doesn t tae acton suddenly at one type, t taes tme to gadually tansfom fom full austente phase to full matenste phase. he evolutonay equaton s expessed usng exponental functon n the fom of, anaa s model s expessed ts ablty to chaactee most of the behavos of the SMA. Lang and Roges [] mpoved anaa s model by dectly matchng expemental esults to get the evolutonay equaton whch s expessed usng the cosne functon. he consttutve equaton of the model emans the same whle paametes of the equatons ae detemned though expements. A majo mpovement of the anaa s model was made by Bnson [-], who s ecogned that not all matenste that ae conveted to austente wll poduce the ecovey stess, but only the stess nduced matenste (SIM) that s esponsble fo the shape memoy effect. Consequently, the matenste facton s dvded nto two pats: stess nduced matenste and tempeatue nduced matenste. hs model also dd not assume constant mateal functons n the consttutve elatonshp. Futhemoe, Bnson s model made some amendment that the consttutve equaton wll be vald at any tempeatue, and gave a bette epesentaton of the SMA behavos compaed to the Lang and Roges s model, and t s qute popula fo engneeng applcatons because t s smple, accuate and easy to mplement nto numecal applcatons. In the pesent wo, a mathematcal fomulaton fo composte plate wth embedded SMA wes s deduced usng the hghe ode shea defomaton theoy and SMA Bnson s consttutve model. Hamlton s pncple s used to obtan the govenng equatons of moton. he effect on the SMA mateal constants, themal stesses, and the ecovey stess s consdeed fo the poposed composte plate. he SMA we laye s teated as an othotopc laye smla to typcal gaphte/epoxy layes. he effectve popetes ae detemned usng the ule of mxtue, except that both the mateal popetes and the ecovey stess of the SMA wes ae functons of the matenste phase facton. he SMA ecovey stess values ae expessed n the govenng equaton as extenal foce[], and computed usng Bnson s model as a functon of the tempeatue appled to the SMA composte plates. Paametc studes ae conducted to demonstate the effect of plate dmensons, fbe oentaton angle and volume facton of composte fbes and shape memoy alloy wes on the actvated plate natual fequences. he obtaned esults ae compaed to the avalable studes solved by dffeent theoes and found convenent.

3 . SMA Bnson s consttutve model Fgue. SMA composte plates layes ( :Ntnol-epoxy, : gaphte/epoxy) he Bnson s consttutve model assumes that the phase tansfomaton depends only on tempeatue and stess [] and volume facton of matenste, ξ S. A two equatons namely the consttutve equaton and the evolutonay equatons ae used to solve Bnson's model. Bnson dvdes the matenstc volume facton nto two pats: S () ξ S coesponds to the facton of the stess nduced matenste (SIM) and ξ efes to the facton of the tempeatue nduced matenste (IM), hus: (, S,, ) () Mang smple devaton and applyng foce condton, Bnson s model fo constant mateal paamete can be expessed as []: D( ) Ω( ) ( ) S S () D s the Young Modulus, θ s the themo-elastc tenso and Ω s the tansfomaton tenso. he effect of the stess on the tanston tempeatue must consde the conveson of IM to SIM. hs pocess of conveson stats when IM s exposed to stess, and ends at a stess value of f. he values of these tcal stesses can be detemned ethe by expement, o theoetcal usng a model based on the potental enegy necessay to ovecome the chemcal enegy bae fo conveson of twns as done by Achenbach and Mulle [4]. he stess tempeatue coeffcents C A and C M n Bnson s model ae not assumed to be equal and both ae detemned by expement. he effect of stess on the tcal tempeatue s shown n Fgue A. hus the evoluton equatons that epesent S and as functons of stess and tempeatue Eq. (4) and gven n the Appendx A []: s. Dsplacement Feld (,,, ) (4) S S he dsplacement feld equatons of the modfed hghe ode shea defomaton theoy (MHSD) ae epesented by:

4 ,,,,,,,,, w x, y, t u x, y,, t u x, y, t x x, y, t x x, y, t x x, y, t x w x, y, t v x, y,, t v x, y, t y x, y, t y x, y, t y x, y, t y w x y t w x y t x y t x y t he dsplacement feld Eq.(5) can be wtten n matx fom as follows: U U U U U (6) U u v w U x y U U w w,, x y x y, x y 4. Stan-Dsplacement Relatonshps he stan-dsplacement elatonshps can be expessed n a matx fom as follows: (8) xx yy y x xy (9) u v u v y x x y y x x w y w x y w y x x x y y y x y x x y x y x y y x x y y x y x x y x y x y y x () 5. Stess-stan Relatonshp he consttutve elatonshp fo an embedded SMA laye s:,, Q, Q V s, and () (5) (7) ae the stess vecto, stan vecto, lamna stffness matx, themal expanson coeffcents vecto, and SMA ecovey stess vecto, espectvely wth espect to pncple mateal axes befoe tansfomaton. he stffness matx of the SMA lamna s calculated usng the SMA effectve popetes detemned by the ule of

5 mxtue [5-6] whch ae functon of matenste facton of SMA. he vectos and ae defned as follows: () () he tansfomed stess-stan elatons fo an othotopc lamna oented ae gven by: x Q Q Q Q6 x x x y Q Q Q Q 6 y y y Q Q Q Q 6 Vs y Q 44 Q45 y x Q 54 Q55 x xy Q 6 Q6 Q6 Q 66 xy xy xy Q ae the tansfomed stffness coeffcents of the [5-7], j and (4) ae the vectos of themal expanson coeffcents and SMA ecovey stess, espectvely gven n Appendx B [5]. 6. Enegy Fomulaton he govenng equaton of the smat plate s deved usng Hamlton s pncple [5]: U V K dt (5) U s the vtual stan enegy, V s the vtual wo done by the appled foces, and K s the vtual netc enegy of the whole stuctue system. he vtual stan enegy U of the plate wth oss secton aea A and thcness h s gven by [5]: U By substtutng Eq.(4) n Eq.(6) fo a numbe of layes : h h d dxdy (6) A U Q V s d dxdy A (7) Q,, ae the stffness matx, themal expanson coeffcents and SMA ecovey stess vectos., ae the lowe and uppe -coodnates of laye numbe measued fom md-plane, espectvely. he tems n Eq.(7) ae defned as: U U U U (8) m Whee U m U U ae the mechancal, themal, and ecovey stess components of the stan enegy gven as follows:

6 U m Q d dxdy A U Q d dxdy A U V s d dxdy A Substtutng Eq.(8) nto Eq.(8) yelds the vtual stan enegy due to mechancal appled load s gven as: U m (9) Q dxdy A () d he vtual stan enegy due to the themal load s expessed as: he vtual stan enegy due to the ecovey stess s epesented by: U Q d dxdy A () s A () U V d dxdy Eq.() can be ewtten as: U A B D E B D E F D E F H E F H J dxdy () m A 4 5 6,,,,,,,,,,,, A B D E F H J Q d (4) And equaton () can be ewtten as: A (5) U A B D E dxdy,,,,,, A B D E Q d (6)

7 Also equaton () can be ewtten as: A (7) U A B D E dxdy,,, s,,, A B D E V d (8) he vtual wo V done by the appled mechancal loads s gven by[8]: V p x, y w x, y, t dxdy F w x, y, t A p x, y s the tansvese dstbuted load and (9) at pont. he vtual netc enegy K can be wtten as [5]: F s the tansvese concentated foce K U U dv () V U u v w Fo layes Eq.() s epesented as: () K U U d dxdy s the densty of laye numbe. A () Substtutng Eq.(6) nto Eq.(), one can obtan; I U U I U U I U U I U U I U U I U U I U U I U U K I U U I U U I 4 U U I 5 U U I U U I 4 U U I 5 U U I 6 U U 4 dxdy () A 4 5 6,,,, 4, 5, 6,,,,,, I I I I I I I d (4) 7. Rt Soluton echnque u, v, w,,,,,,, of the gven poblem ae he unnown dsplacements x y x y x y appoxmated by (x-y) dependent functons that satsfy the geometc bounday condtons as follows: x, y, t a x, y q t,... (5)

8 x, y, t,, whle, x y epesents the unnown dsplacements u, v, w, x, y,, x, y, a x y ae column vectos of the Rt appoxmaton functons that satsfy the bounday condtons of the plate and q t ae the column vecto of the Rt coeffcents to be detemned. he pesent study pesents both smply suppoted and cantleveed a x, y ae lsted n Appendces C and D, bounday condtons. he used Rt functons espectvely. 8. Equatons of Moton he equatons of moton ae deved usng the Rt appoxmaton technque fo the eleven DOF. based on the dsplacement feld equatons. By nsetng,,, and n tems of the genealed coodnates q n equatons (), (5), and (7) yelds to: U m q t K q t (6) K s the lamnate stffness matx. K A A B D E B D E F dxdy D E F H E F H J (7) Equaton (5)wll be: U q t F (8) F s the themal load vecto. A F A B D E dxdy (9) Equaton (7) wll be: U q t F (4) F s the ecovey stess load vecto. A F A B D E dxdy (4) he vtual wo V done by the appled foces can be wtten as: V q F F (4) F and F s the dstbuted and concentated load vectos, espectvely gven by,

9 F p x, y a dxdy A F F a x y, he vtual netc enegy W s expessed as follows: (4) W q t M q t (44) M s the mass matx of the lamnate I U U I U U I U U I U U I U U I U U I U U I 4 U U M A dxdy (45) I U U I U U I 4 U U I 5 U U I U U I 4 U U I 5 U U I 6 U U By substtutng equatons (8), (6), (8), (4), (4), and (44) n Eq.(5) the whole stuctue equaton of moton s epesented by: M q K q F F F F K s the stffness matx, M s the mass matx, concentated load vectos, F and F espectvely. he unnown q s the Rt coeffcents to be detemned. (46) F and F ae the dstbuted and ae the themal and ecovey stess load vectos, 9. Numecal examples and Dscusson he statc and dynamc esponses of the poposed plate ae pesented [9-]. Paametc studes ae conducted to nvestgate the effect of plate s dmensons, oentaton and volume facton of composte fbes and shape memoy alloy wes on the actvated plate natual fequences. A smply suppoted and cantleve plates ae used n these studes. A set of compute pogams developed by Mathematca 7 s used n ths study. A mult-layeed composte plate wth embedded SMA wes s llustated n Fgue. he plate unde consdeaton s made of undectonal composte mateal layes (.e. SMA/epoxy and gaphte/epoxy layes). he length and the wdth of the plate ae both 5 mm, as the thcness h s 9 mm. he thcness of each SMA/epoxy laye s.5 mm, as the thcness of each gaphte/ epoxy laye s.8 mm. he oentaton angle of the SMA wes wthn the SMA/epoxy layes, and the gaphte fbes wthn the gaphte/epoxy layes of the plate ae defned by the angles and, espectvely. It s also assumed that the SMA/epoxy layes ae symmetcally placed as two oute layes wth the plate md-plane as shown n Fgue. he elatve volume facton of the SMA wes and the gaphte fbes wthn each laye of the plate s.5. he ply stacng sequence of the plate s [ /(±45 ) 5 / ]. he mechancal popetes of the SMA wes and the gaphte/epoxy composte ae gven n able. A two dffeent types of plate bounday condtons ae nvestgated: cantleve plate (.e. one edge of the plate clamped along the y-axs), and smply-suppoted plate (along ts fou edges). he obtaned esults ae pesented as elatve values (.e. the elatve fundamental natual fequency). hey ae defned as atos of the fundamental natual fequency calculated when the SMA wes ae actvated to the values when they ae not actvated.

10 (SMA/E) laye Angle (α) (Gaphte/E) layes Angle (β) h s x (SMA/E) laye Fgue. he geomety of the multlayeed composte plate wth SMA wes. able. Mechancal popetes of composte mateal and SMA wes []. Mateal Young s modulus [GPa] Posson s ato Densty [g/m ] Epoxy esn Gaphte fbes SMA matenste 6.. SMA austente he changes n the elatve fundamental natual fequency of the plate ae pesented n Fgue and Fgue 4, as a functon of the plate dmensons. It s assumed that the plate length a, and the wdth b, of the plate can vay n ths case, as the thcness of the plate h emans unchanged (changes n the dmensons of the plate obvously also esult n changes n the plate mass). It can be seen that both the dmensons of the plate as well as the bounday condton type, have a geat nfluence on the elatve fundamental natual fequency of the plate. Fo the cantleve plate, the elatve fundamental natual fequency of the plate neases due to the actvaton of the SMA wes towads a longe and naowe plate. he maxmum elatve fundamental natual fequency of the plate s, n ths case, obtaned fo a beam-le stuctue, fo whch the SMA wes wthn the SMA/epoxy layes ae lengthwse oentated. Contay to ths, the elatve fundamental natual fequency of the smplysuppoted plate, neases towads a wde and shote plate. he maxmum elatve fundamental natual fequency s obtaned fo a beam-le stuctue fo whch the SMA wes wthn the SMA/epoxy layes ae wdthwse oentated. It s clea fom, Fgue and Fgue 4 that the obtaned esults ae vey compaable wth the publshed data obtaned by Za usng the fnte element technque wth a msh se 8X8, []. It should be emphased that the maxmum changes n the elatve fundamental natual fequency of the plate ae obseved n the case when the mnmum numbe of constants s mposed by the chosen bounday condton. Consequently, the maxmum changes n the elatve fundamental natual fequency ae obseved fo the cantleve plate.

11 (a) Pesent model (b) Za [] Fgue. Change n the elatve fundamental fequency of a smply-suppoted plate as a functon of ts dmensons. (a) Pesent model (b) Za [] Fgue 4. Change n the elatve fundamental natual fequency of a cantleve plate as a functon of ts dmensons In Fgue 5 and Fgue 6 esults fo changes n the elatve fundamental natual fequency of the plate ae pesented as a functon of the oentaton angle of the SMA wes wthn the SMA/epoxy layes, as well as the oentaton angle of the gaphte fbes wthn the gaphte/ epoxy layes of the plate. he ply stacng sequence of the plate consdeed hee can be defned as [α/(β/(β-9 ) 5 / α], α denotes the oentaton angle of the SMA wes wthn the SMA/epoxy layes, whle β denotes the oentaton angle of the gaphte fbes wthn the gaphte/epoxy layes. It s assumed that all othe mechancal popetes of the plate eman unchanged n ths case. It s seen fom Fgue 5 and Fgue 6 that the elatve fundamental natual fequency of the cantleve plate deeases apdly wth an nease n the oentaton angle of the SMA wes wthn the SMA/epoxy layes of the plate fom to 9. It appeas that changes n ths fequency of the plate ae moe affected by changes n the oentaton of the SMA wes wthn the SMA/epoxy layes, than by changes n the oentaton angle of the gaphte fbes wthn the gaphte/epoxy layes of the plate. he geatest pefomance of the plate s obseved when the SMA wes ae lengthwse oentated, whch coesponds to the oentaton angle of the SMA wes wthn the SMA/epoxy layes of. Fo the cantleve plate bounday condtons the optmal oentaton of the SMA wes wthn the SMA/epoxy layes fo the geatest changes n the elatve fundamental natual fequency of the plate s, as the oentaton of the gaphte fbes wthn the gaphte/epoxy layes s 45. hs coesponds to a ply stacng sequence of the plate of [ /(±45 ) 5 / ]. Howeve, n the case of the smply-suppoted bounday condton the obseved pattens ae dffeent as seen n

12 Fgue 5. It s clea that the changes n the elatve fundamental natual fequency of the plate ae equally affected by both the changes n the oentaton of the SMA wes wthn the SMA/epoxy layes, as well as by the changes n the oentaton angle of the gaphte fbes wthn the gaphte/epoxy layes of the plate. Fo the smply-suppoted plate the optmal oentaton of the SMA wes wthn the SMA/epoxy layes gvng the geatest elatve fundamental natual fequency of the plate s 45, as the oentaton of the gaphte fbes wthn the gaphte/epoxy layes ae /9 o 9 /. hs coesponds to ply stacng sequences of the plate of [45 /( /9 ) 5 / 45 ]. (a) Pesent model (b) Za [] Fgue 5. Change n the elatve fundamental natual fequency of a smply-suppoted plate as a functon of the oentaton angle of gaphte fbes and SMA wes. (a) Pesent model (b) Za [] Fgue 6. Change n the elatve fundamental natual fequency of a cantleve plate as a functon of the oentaton angle of gaphte fbes and SMA wes. It s clea fom Fgue 5 and Fgue 6 (a) and (b), that the obtaned esults ae compaable wth the publshed data obtaned by Za usng the fnte element technque wth a msh se 8X8, []. he vaous mechancal popetes of the composte mateal of the plate nfluence the plates elatve fundamental natual fequency. he volume facton of the SMA wes wthn the SMA/epoxy layes, as well as the volume facton of the gaphte fbes wthn the gaphte/epoxy layes can also both vay. he nfluence of these changes on the elatve fequency of the smply suppoted and the cantleve plate s nvestgated. It s shown n Fgue 7 that the elatve fundamental natual fequency of the plate neases wth an nease n the volume facton of the SMA wes, and wth a deease n the volume facton of the gaphte fbes. he maxmum value of the elatve fundamental natual fequency coesponds to the case when the oute layes of the plate ae entely made of the SMA mateal, as the nne layes ae made of the epoxy esn. On the othe hand when the volume facton of the gaphte fbes neases, the plate pefomance deeases apdly, and

13 when the nne layes of the plate ae entely made of the gaphte vtually no changes n the elatve fundamental natual fequency of the plate ae obseved. (a) Smply-suppoted plate (b) Cantleve plate Fgue 7. Change n the elatve fundamental natual fequency as a functon of the volume facton of gaphte fbes and SMA wes. In Fgue 8 the locaton of the SMA/epoxy layes s hwthn the plate has a geat nfluence on the plate pefomance. he locaton of the SMA/epoxy layes ( s ) wthn the plate can vay fom the most exteme oute locaton to the cental layes of the plate, and ths coesponds to a ply stacng sequence change fom [ /(±45 ) 5 / ] to [(±45 ) /45 /( ) /( 45 )/ (±45 ) ]. Howeve, no assumptons ae made egadng changes n the mechancal popetes of the SMA/epoxy layes, o the gaphte/ epoxy layes. Fom Fgue 8, t ases that the elatve fundamental natual fequency of the plate neases wth a change n the locaton of the SMA/epoxy laye s h. he changes n the elatve fundamental natual fequency of the plate ae much geate n the case of geate volume factons of the SMA wes wthn the SMA/epoxy layes, and fo the layes located close to the exteme oute layes of the plate, than n the case of smalle volume factons of the SMA wes wthn the SMA/epoxy layes, the layes ae located close to the cental layes of the plate. It s evdent fom the esults pesented n Fg. 8 that the plate pefomance s detemned by the stffness ato of the SMA/epoxy layes and the gaphte/epoxy layes. he stffness of each mateal laye of the plate s also a functon of the Young s modulus of the composte mateal components, as well as the poston of the laye wthn the plate. (a) Smply-suppoted plate (b) Cantleve plate Fgue 8. Change n the elatve fundamental natual fequency as a functon of the volume facton of SMA wes and locaton of SMA we laye.

14 . Concluson An analytcal model fo lamnated composte plates embedded wth shape memoy alloy wes was pesented and analyed based on the modfed hghe-ode shea defomaton theoy usng Rt soluton fo smply-suppoted and cantleveed plates. he followng conclusons have been dawn: a) Both the plate dmensons and bounday condtons have a geat nfluence on the elatve fundamental natual fequency of the plate. b) he elatve fundamental natual fequency neases due to the actvaton of the SMA wes towads a longe and naowe plate of the cantleve bounday condtons. Howeve, fo smply-suppoted plate t neases towad a wde and shote plate. c) he elatve fundamental natual fequency of the plate neases wth an nease n the volume facton of the SMA wes, and wth a deease n the volume facton of the gaphte fbes. d) he maxmum value of the elatve fundamental natual fequency obtaned when the oute layes of the plate ae made by the SMA mateal, as the nne layes ae made fom the epoxy esn.. Refeences [] M. M. Ghomshe, A. Khajepou, N. abandeh, and K. Behdnan "Fnte element medelng of shape memoy alloy composte actuatos: theoy and expement" Jounal of ntellgent mateal systems stuctues, vol., pp ,. [] M. M. Ghomshe, N. abandeh, A. Khajepou, and K. Behdnan "Fnte element analyss of a new shape memoy alloy composte actuato" [] B. S. Balapgol, K. M. Bajoa, and Sudhaa "A two-dmensonal fnte element analyss of a shape memoy alloy lamnated composte plate" Smat Mateals and Stuctues, 5, pp. 9-, 6. [4] B. S. Balapgol, K. M. Bajoa, and Sudhaa "Natual fequences of mutlaye SMA lamnated composte cantleve plate" Smat Mateals and Stuctues, 5, pp. -, 6. [5] F. Godannejad and W. Wu "A two-dmensonal shape memoy alloy/elastome actuato" Intenatonal Jounal of Solds and Stuctues, vol. 8, pp. 9-49,. [6] W. Wu, F. Godannejad, and R. A. Wt "Modelng and analyss of a shape memoy alloy- elastome composte actuato" Jounal of ntellgent mateal systems stuctues, vol. 7, pp , 996. [7] C. A. Roges, C. Lang and J. Ja "Stuctual modfcaton of smply-suppoted lamnated plates usng embedded shape memoy alloy fbes" Computes & Stuctues, vol. 8, pp , 99. [8] M. W. Ln and C. A. Roges "Analyss of stess dstbuton n a shape memoy alloy composte beam" Poceedng of the AIAA/ ASME/ AHS/ ASC nd stuctues, stuctual dynamcs, and mateal confeence, pat, vol. AIAA-9-64-cp, pp , 99. [9] K. anaa "A phenomenologcal despton on themomechancal behavou of SMA," Jounal of Appled Mechancs, vol., pp. 58-6, 99. [] C. Lang and C. A. Roges "One-dmensonal themo-mechancal consttutve elatons fo shape memoy mateals" Jounal of ntellgent mateal systems stuctues vol., pp. 7-4, 99. [] L. C. Bnson and J. Mate "Consttutve behavou of SMA: One-dmensonal themomechancal devaton wth non-constant mateal functons and edefned matenste ntenal vaable" pp , 99.

15 [] L. C. Bnson "One-dmensonal consttutve behavo of shape memoy alloys: themomechancal devaton wth non-constant mateal functons and edefned matenste ntenal vaable" Jounal of ntellgent mateal systems stuctues, vol. 4, pp. 9-4, 99. [] A. Za, M.P. Catmell, W.M. Ostachowc, and M.Wecgoch " One dmensonal shape memoy alloy models fo use wth enfoced composte stuctues" Smat Mateals and Stuctues,, pp. 8 46,. [4] M. Achenbach and I. Mulle "Smulaton of mateal behavou of alloys wth shape memoy" Ach. Mech., vol. 7, No. 6, pp , 985. [5] J. N. Reddy, Mechancs of lamnated composte plates and shells, second ed.: CRC Pess, 4. [6] Z. Gupng and. Lmn "A sem- analytcal soluton fo themal stess analyss of lamnated composte plates n the Hamltonan system" Computes & Stuctues, vol. 55, No., pp. -8, 995. [7] R. M. Jones, Mechancs of composte mateals, second ed.: aylo & Fancs, 999. [8] E. Lvne and I. Navao "Nonlnea equvalent plate modelng of wng-box stuctues" Jounal of Aaft, vol. 6, pp , 999. [9] M. K. Abbas, M. Adnan Elshafe and Hany M. Negm "Modelng and Analyss of Lamnated Composte Plate Usng Modfed Hghe Ode Shea Defomaton heoy" n 5th Intenatonal Confeence on Aeospace Scences & Avaton echnology, ASA-5, May 8-, MC, Koby Elobbah, Cao, Egypt,, asat@mtc.edu.eg. [] M. K. Abbas "Desgn and Analyss of Aaft Composte Stuctue Usng Shape Memoy Alloys" Maste of Scence, Cha of Aaft Mechancal Engneeng, Mltay echncal College, Cao,. [] A. J. Za, M. P. Catmell and W. M. Ostachowc "A snstvty analyss of the dynamc pefomance of a composte plate wth shape memoy alloy wes" Composte Stuctues, vol. 6, pp ,. Nomenclatue: hemal expanson coeffcent of the SMA mateal he unnown dsplacements vecto Stan vecto n x-y- coodnates Stess vecto n x-y- coodnates Q ansfomed stffness matx of laye U Vtual stan enegy V Vtual wo K Vtual netc enegy a x, y Column vectos of the Rt appoxmaton functons that satsfy the bounday condtons of the poblem q t Column vectos of the Rt coeffcents K M Stffness matx Mass matx p, F Dstbuted and concentated loads F, F Dstbuted and concentated load vectos A Plate s aea a, b Plate sdes dmensons

16 A, B, D Extensonal, couplng, and bendng stffness matces DOF Degees of feedom E, F, H, J Hghe ode stffness matces h otal lamnate thcness I Ineta tems otal numbe of layes u Dsplacement of a genec pont n the plate n the x decton u Dsplacement of the geometc md-plane n the x decton v Dsplacement of a genec pont n the plate n the y decton v Dsplacement of the geometc md-plane n the y decton w Dsplacement of a genec pont n the plate n the decton w Dsplacement of the geometc md-plane n the decton θ x Rotaton of the nomal to the md-plane about the y-axs θ y Rotaton of the nomal to the md-plane about the x-axs θ Fst ode dsplacement facto ξ x, ξ y hd ode dsplacements o wapng functons ψ x, ψ y, ψ Second ode dsplacements o wapng functons Appendx A. he evoluton equatons that epesent S and as functons of stess and tempeatue: - wnned Matenste detwnned matenste O (Austente detwnned matenste) a- M S and s C M ( M S ) f C M ( M S ) cos C M S S S f M s s f S S S (A-) (A-) b- Ms and s f S cos S S f s f (A-) If M < M and f S S S S (A-4) cos am M f else,.g - Matenste Austente Fo and C ( A ) C ( A ) A S A f A S

17 cos A S Af AS C A (A-5) (A-6) S S S (A-7) - Austente wnned Matenste Fo Mf Ms, s and Puttng S and S n Eq. (A-) gves the same fomula of Lang s model f we put the slope C M because the tansfomaton cuves at s ae vetcal lnes as shown n Fgue A [], cos a M M f (A-8) In ths case, the total nduced matenste s a pue tempeatue nduced matenste,.e. and. Stess Plastc Regon Detwnned Matenste -a f s -b wnned Matenste C M C M C A Austente M f M s A s A f empeatue Fgue A. he effect of stess on the tansfomaton tempeatue []. Appendx B. he vecto of themal expanson coeffcents and the vecto of SMA ecovey stess efeed to the body (x-y-) coodnate system. cos sn, sn cos x y,,, sn cos y x xy cos, sn x y,,, sn cos y x xy

18 Appendx C. he used column vectos of the Rt appoxmaton functons fo smply suppoted bounday condtons ae: a x, y y by x a, a x, y x ax y b a4 x, y y, a5 x, y a 6, X Y XY, a x y XY X Y XY X Y 4 4 6x 4x y y x x 6y 4y x a a b b a a b b 4 4 x x y y x, y x y, a7 x, y y by x a a a b b 4 4 x x y y x ax y b x y a a b b a8 x, y, a9 x, y 4 4 6x 4x y y x x 6y 4y x a a b b a a b b a x, y y, a x, y Appendx D. he used column vectos of the Rt appoxmaton functons fo a cantleve plate ae: a a a x, y x, y x, y x xy x x y x 4 x x y x x y x 4 x x y x x y x y x x y x y x y a4 x, y, a5 x, y a6 x, y x y xy, a7 x, y x xy x x y x 4 x 4 x 9 x xy x x y x xy x x y x 4 x 4 a8 x, y x xy x x y, a9 x, y x y xy x 4, x xy x x y,, a x y a x y x 4 x xy x x y x 4 x 4