Thermal buckling of AL/AL 2 O 3 functionally graded plates based on first order theory

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1 Subitted on : 5 April Revised for ccepted on : 6 April Corresponding uthor eil : fekrr@hoo.fr ture & Technolog Therl buckling of AL/AL O functionll grded pltes bsed on first order theor Abdelkder FKRAR ohed ZIDI Lkhdr BOUIA b Hssen AIT ATA c Abdelouhed TOUSI nd l Abbs Bedi Add Lbortoire des tériu et Hdrologie Université de Sidi Bel Abbes BP 89 Cité Ben hidi Sidi Bel Abbes Algérie. b Université de Sidi Bel Abbés Déprteent de phsique BP 89 Cité Ben hidi Sidi Bel Abbés Algérie c Déprteent de Génie civil Fculté de Génie-Civil et d'architecture Univesité Hssib Benbouli de Chlef Algérie Abstrct Therl buckling nlsis of rectngulr sigoid functionll grded pltes (S-FGPs is investigted using first-order sher defortion theor. The sigoid functionll grded teril (S-FG sste consists of ceric (Al O nd etl (Al phses vring through the thickness of plte ccording to to poer-l distribution. teril properties nd therl epnsion coefficient of the plte re ssued to be grded in the thickness direction ccording to siple poer-l distribution in ters of the volue frctions of the constituents. The therl lods re ssued to be unifor nd sinusoidl distribution through-the-thickness. uericl eples cover the effects of the grdient inde plte spect rtio side-to-thickness rtio nd loding tpe on the criticl buckling for S-FG pltes. Keords: S-FG plte; Al/Al O ; Buckling; teril grdient inde.. Introduction Functionll grded terils (FGs re specil coposites hose teril properties vr continuousl through their thickness. FGs re usull de of iture of ceric nd etl nd cn thus resist high-teperture environents hile intining toughness. The technolog of FGs s n originl teril fbriction technolog proposed in Jpn in 98 b Sendi Group. FGs re used in ver different pplictions such s rector vessels fusion energ devices bioedicl sectors ircrfts spce vehicles defense industries nd other engineering structures. With the incresed use of these terils for structurl coponents in n engineering pplictions the stud of sttic dnic nd stbilit behviors of these coponents gins iportnce ong reserchers [ 8]. Slli et l. [] investigted the sttic responses of sigoid FG thick be b using different be theories. erdci et l. [] developed to refined displceent odels for bending nlsis of functionll grded sndich pltes nd the nuber of unknon functions involved is onl four s ginst five in cse of other sher defortion theories. echb et l. [] proposed the to vrible refined plte theor. In this pper equilibriu nd stbilit equtions of rectngulr plte de of sigoid functionll grded terils under therl lods re derived bsed on the first order sher defortion theor. The S-FG sste consists of ceric (Al O nd etl (Al phses vring through the thickness of plte ccording to the volue frction of the constituent terils bsed on the sigoid functions. The therl lods re ssued to be unifor nd sinusoidl teperture rises cross the thickness direction re nlzed. The influence of spect nd thickness rtios grdient inde nd the trnsverse sher on buckling teperture difference re ll studied.. teril properties of S-FG pltes Consider rectngulr plte de of iture of etl nd ceric s shon in Figure. Fig.Configurtion nd coordinte sste of rectngulr plte. «ture & Technolog» Journl. A- Fundentl & ngineering Sciences Issue n 8/Jnur. Pges to 6

2 «ture & Technolog» Journl. A- Fundentl & ngineering Sciences Issue n 8/Jnur In the cse of dding n FG of single poer-l function to the ulti-lered coposite stress concentrtions pper on one of the interfces here the teril is continuous but chnges rpidl [ ]. Therefore Chung nd Chi [9] nd Slli et l. [] defined the volue frction using to poer-l functions to ensure sooth distribution of stresses ong ll the interfces. The to poer-l functions re defined b: z V h k ( z = ( for z h / Vrition of volue frction 8 6 k= k= k=5 k= k=. k=. k=.5 ( z k V ( z = ( for h / z (b h k is the teril grdient inde. B using the rule of iture the Young s odulus nd the coefficient of therl epnsion α of the S-FG cn be clculted b: z = V z for z h / ( ( ( c z = V z for / z (b ( ( c α z = α V ( z α for z h / (c ( ( c α z = α V ( z α for / z (d c nd α denote the elstic oduli nd the coefficient of therl epnsion of etl respectivel; c nd α c denote the elstic oduli nd the coefficient of therl epnsion of ceric respectivel. We note tht: c = c nd α c = α c α Figure shos the volue frction distribution of ceric phse through the thickness for severl vlues of the poer l inde. The vrition of the coposition of cerics nd etl is liner for k =. The volue frction rpidl chnges ner the top nd botto surfces for k < but vr quickl ner the iddle surfce for k >. Therefore if the S-FG plte is used s the undercot in linted teril the teril distribution ith k > is the better choice z/h Fig. Vrition of the volue frction through the thickness of n S-FG plte ith differing teril grdient inde k.. Anlsis The S-FG plte is ssued sipl supported in bending nd rigidl fied in etension. The teperture chnge is vried onl in the thickness direction. Assue tht u v denote the displceents of the neutrl plne φ φ of the plte in z directions respectivel; denote the rottions of the norls to the plte idplne. According to the first order sher defortion theor the strins of the plte cn be epressed s ε = u z = u = φ zφ v z( φ ε = v z φ zφ = φ Hooke s l for plte is defined s = ( ε ( νε ν αt ν = ( ε ( νε ν αt ν τ = τ = ( ν ( ν z z τ z = z ( ν The forces nd oents per unit length of the plte epressed in ters of the stress coponents through the thickness re = h / / τ h / (5 h / τ dz = z dz = dz / τ / τ Substituting qs. ( ( nd ( into qs. (5 gives the constitutive reltions s Φ = ( u ( ν v φ ν φ ν ν ν Φ = ( ν u ( v ν φ φ ν ν ν = ( u v ( φ φ ( ν ( ν Θ = ( u ( ν v φ ν φ ν ν ν Θ = ( ν u ( v ν φ φ ν ν ν = ( u v ( φ ν φ ( ν ( ν = ( φ ( ν = ( φ ( ν ( ( (6

3 Therl buckling of AL/ALO functionll grded pltes bsed on first order theor ( ( Φ Θ = = h / / h / / ( z z ( z dz ( z α( z T ( z( z dz The nonliner equtions of equilibriu ccording to Von Krn s theor re given b = = = = = Using qs. (6 nd (8 nd b eliinting the vribles u v φ φ the equtions of equilibriu cn be covered into one eqution s ( ν ( ( ν ( = To estblish the stbilit equtions the criticl equilibriu ethod is used. Assuing tht the stte of stble equilibriu of generl plte under therl lod be designted b. The displceent of the neighboring stte is here is n rbitrril sll increent of displceent. Substituting into q. (9 nd subtrcting the originl eqution results in the folloing stbilit eqution ( ν ( ( ν ( = (8 (9 (7 ( nd refer to the pre-buckling force resultnts... Buckling of S-FG pltes under unifor teperture rise To deterine the buckling teperture difference Tcr the pre-buckling therl forces should be found firstl. Solving the ebrne for of equilibriu equtions gives the pre-buckling force resultnts Φ Φ = = ν ν = ( Substituting q. ( into q. ( one obtins ( ν Φ ν ( ν Φ = ν ( If the trnsverse sher defortion is not considered q. ( cn be reduced s ( ν Φ ν = The sipl supported boundr condition is defined s = = = φ = φ = on = = on = b ( ( The folloing pproite solution is seen to stisf both the governing eqution nd the boundr conditions = c sin ( π / sin ( n π / (5 b n re nuber of hlf ves in the nd directions respectivel nd c is constnt coefficient. Substituting q. (5 into q. ( nd substituting for the therl preter Φ fro q. (7 ields ( ( ν π ( n B T = (6 ( ν( π ( n B ( ν P P = nd ( cα α c h cα c h α h (7 k k B = / b The criticl teperture difference is obtined for the vlues of n tht ke the preceding epression iniu. Apprentl hen iniiztion ethods re used criticl teperture difference is obtined for the fundentl ode = n = [5-6-8] thus ( ( ν π ( B Tcr = (8 ( ν ( π ( B ( ν P.. Buckling of S-FG pltes under sinusoidl teperture chnge The teperture rise under sinusoidl teperture distribution cross the thickness is ssued s π z π T ( z = T[ cos ( ] h T (9

4 «ture & Technolog» Journl. A- Fundentl & ngineering Sciences Issue n 8/Jnur 5 For this loding cse the therl preter Φ cn be epressed s Φ = P ( T T Y T ( Y = h / / π z π ( z α ( z cos ( dz ( h Fro q.( one hs Φ PT T = ( P Y This ens tht the inclusion of effect of trnsverse sher defortion leds to reduction in the criticl buckling teperture difference especill for thick pltes. Criticl buckling teperture ( C sher defortion no sher defortion ( h/ When the pproite solution (5 is substituted into q. ( nd the definition of preter Φ fro q. (7 re used the epression for therl buckling of the plte is obtined. Tking = n = the criticl buckling teperture is epressed s T cr ( ( π ( B = ( PT ( ν ( π ( B ( ν P Y ν ( Criticl buckling teperture ( C sher defortion no sher defortion. Results nd discussion The therl buckling nlsis is conducted for cobintions of etl nd ceric. The set of terils chosen is Aluinu nd Aluin. For siplicit Poisson s rtio of the to terils is ssigned the se vlue. Tpicl vlues for etl nd cerics used in the S- FG plte re listed in Tble. Tble teril properties used in the S-FG plte Properties etl: Al Ceric: Al O (GP 7 8 ν.. α ( -6 / C 7. Firstl the criticl teperture differences re clculted for sigoid functionll grded pltes under unifor teperture nd sinusoidl teperture chnge respectivel. The obtined results re plotted in Figure nd the sho the criticl buckling teperture difference T cr vs. the thickness to spn rtio h / here the teril grdient inde k is equl to nd / b =. It is seen tht the criticl teperture difference increses onotonicll s the reltive thickness h / increses. The vlues of the criticl teperture differences clculted b using the first order sher defortion theor re loer thn those clculted b using the clssicl plte theor (b h/ Fig. Criticl buckling teperture of S-FG plte vs. reltive thickness of the plte: ( under unifor teperture chnge; (b under sinusoidl teperture chnge. Figure shos the effect of teperture chnge on the vrition trend of criticl teperture difference ith respect to the plte spect rtio / b for the teril grdient inde k equl to. The reltive thickness of the plte is set s h / =.. It is observed tht ith incresing the plte spect rtio / b fro to the criticl buckling teperture difference lso increses stedil. Fro Figure 6 It is found tht the vlues of the buckling teperture difference coputed ith sinusoidl teperture distribution cross the thickness re higher thn those coputed b liner teperture distribution. Criticl buckling teperture ( C 5 Unifor teperture chnge Sinusoidl teperture chnge 6 8 /b Fig. ffect of the teperture chnge on criticl buckling teperture of S-FG plte ( h / =. nd k =.

5 6 Therl buckling of AL/ALO functionll grded pltes bsed on first order theor Figure 5 shos the buckling teperture vs. the teril grdient inde k for plte ith h / =. nd = b. We cn see tht the criticl buckling teperture for hoogeneous plte ith k = is considerbl higher thn those for the sigoid functionll grded plte ith k >. It is evident tht the criticl buckling teperture difference decreses s the teril grdient inde k increses onotonicll. Hoever it cn be seen tht for sinusoidl teperture distribution ssuption cross the thickness the criticl buckling teperture difference increses for the loer vlues of the teril grdient inde < k <.5. Criticl buckling teperture ( C Criticl buckling teperture ( C ( (b teril grdient inde k teril grdient inde k Fig 5. Criticl buckling teperture rise of n S-FG plte vs. teril grdient inde k: ( unifor teperture chnge; (b sinusoidl teperture chnge; 5. Conclusions Criticl buckling tepertures of sipl supported hoogeneous nd inhoogeneous coposite pltes hve been nlzed b using the first sher defortion plte theor. The inhoogeneous pltes re considered s Al/Al O S-FG pltes. The buckling nlsis of such plte under to tpes of therl lodings is investigted. It is shon tht both the trnsverse sher defortion nd the teril grdient inde hve considerble effect on the criticl buckling teperture difference of S-FG plte especill for thick plte or plte ith lrge spect rtio. References [] B.O. Slli A. Tounsi I. echb. Bchir Bouidjr. erdjh.a. Add Bedi A theoreticl nlsis of fleionl bending of Al/AlO S-FG thick bes Coputtionl terils Science (9 5. [] S. erdci A. Tounsi.S.A. Houri I. echb H. Hebli S. Benoucef To ne refined sher displceent odels for functionll grded sndich pltes Archive of Applied echnics (In press. [] echb I. Ait Atne H. Tounsi A. Belhdj H. A. Add bedi.a. A to vrible refined plte theor for bending of functionll grded pltes Act ech Sin 6(6: 9. [] Houri.S.A Benoucef S. echb I. Tounsi A. Add bedi.a. To vrible refined plte theor for theroelstic bending nlsis of functionll grded sndich pltes J. Therl Stresses (Accepted. [5]. l eiche A. Tounsi. Zine I. echb.a. Add Bedi A ne hperbolic sher defortion theor for buckling nd vibrtion of functionll grded sndich plte Interntionl Journl of echnicl Sciences (In press. [6] Jvheri R sli R. Therl buckling of functionll grded pltes. AIAA J ; :6 9. [7] Lnhe Wu. Therl buckling of sipl supported odertel thick rectngulr FG plte. Copos Struct ; 6: 8. [8] jfizdeh sli R. First-order-theor-bsed theroelstic stbilit of functionll grded circulr pltes. AIAA J ; : 5. [9] Z.H. Jin G.H. Pulino Trnsient therl stress nlsis of n edge crck in functionll grded teril Interntionl Journl of Frcture 7 ( [] Y.Y. Yung D. unz in: T. Shiot.Y. ioto (ds. Functionll Grded teril 996 pp. 6. [] Zhng D-Gung Zhou You-He A theoreticl nlsis of FG thin pltes bsed on phsicl neutrl surfce" Coputtionl terils Science ( [] S. Joshi A. ukherjee S. Schuder ct solutions for chrcteriztion of electro-elsticll grded terils" Coputtionl terils Science 8 ( [] Z.H. Jin R.C. Btr Stresses intensit reltion t the tip of n edge crck in functionll grded teril subjected to therl shock Journl of Therl Stresses 9 ( [] F. Delle F. rdogn The crck proble for nonhoogeneous plne AS Journl of Applied echnics 5 ( [5] P. Gu R.J. Asro Crck deflection in functionll grded terils Interntionl Journl of Solids nd Structures ( [6] F. rdogn B.H. Wu Crck probles in FG lers under therl stresses Journl of Therl Stresses 9 ( [7] Z.H. Jin. od Crck tip singulr fields in nonhoogeneous terils AS Journl of Applied echnics 6 ( [8] F. rdogn Y.F. Chen Interfcil crcking of FG/etl bonds in: K. Kokini (d. Ceric Coting 998 pp [9] Y.L. Chung S.H. Chi The residul stress of functionll grded terils Journl of the Chinese Institute of Civil nd Hdrulic ngineering ( 9. [] S.H. Chi Y.L. Chung Crcking in sigoid functionll grded coting Journl of echnics 8 ( 5. [] Y.D. Lee F. rdogn Residul/therl stresses in FG nd linted therl brrier cotings" Interntionl Journl of Frcture 69 ( [] G. Bo L. Wng ultiple crcking in functionll grded ceric/etl cotings Interntionl Journl of Solids nd Structure (