New Analysis for The FGM Thick Cylinders Under Combined Pressure and Temperature Loading

Transcription

1 Ameican Jounal of Applied Science 5 (7): , 008 ISSN Science Publication New Analyi fo The FGM Thick Cylinde Unde Combined Peue and Tempeatue Loading K. Abinia, H. Naee, F. Sadeghi and F. Djavanoodi Mechanical Engineeing Depatment, College of Engineeing, Univeity of Tehan, Tehan, Ian Mechanical Engineeing Depatment, Ian Univeity of Science and Technology, Tehan, Ian Abtact: An analytical olution fo computing the adial and cicumfeential tee in a FGM thick cylindical veel unde the influence of intenal peue and tempeatue i peented in thi pape. It ha been aumed that the modulu of elaticity and themal coefficient of expanion wee vaying though thickne of the FGM mateial accoding to a powe law elationhip. Nevethele the value of the Poion atio wa taken a contant thoughout the mateial. In the analyi peented hee the effect of non-homogeneity in FGM thick cylinde wa implemented by chooing a dimenionle paamete, named, which could be aigned an abitay value affecting the tee in the cylinde. Uing Maple 9.5, ditibution of tee in adial and cicumfeential diection fo FGM cylinde unde the influence of intenal peue and tempeatue gadient wee obtained. Gaph of vaiation of te veu adiu of the cylinde wee plotted fo diffeent value of. Cae of peue, tempeatue and combined loading wee conideed epaately. It wa concluded that by changing the value of, the popetie of FGM could be o modified that the lowet te level wee eached. The tee which wee poduced in FGM and homogeneou mateial with the ame bounday condition wee compaed to obtain the optimum value of. Key wod: FGM, analytical, cylinde, dimenionle INTRODUCTION Functionally Gaded Mateial (FGM) ae compoite mateial with vaying popetie though thickne. They have themo-mechanical popetie which vay though thei thickne and wee fit conceived by a goup of eeache in Japan [, ]. The main advantage of uch mateial which ae unique to themelve i the poibility of tailoing the deied popetie. Obviouly FGM could be ued in a vaiety of application which have made them vey attactive. FGM ae fabicated by continuouly changing the volume faction of two baic mateial, uually ceamic and metal, in one o moe diection. The FGM that ae thu fomed exhibit iotopic yet non-homogenou themal and mechanical popetie. Thee kind of mateial ae teated a non-homogenou with mateial content that vay continuouly along one patial diection. Account of themo-elatic and themo-inelatic poblem ae given in an extenive eview by Noda [3]. Shen [4] ha tudied the themal pot buckling of functionally gaded plate and hell. Solution to the poblem of the unifom heating of a cicula cylinde by the Fobeniu eie method wa peented by Zimmeman and Lutz [5]. Obata and Nodo [6] applied the petubation appoach to invetigate the themal tee in a FGM hollow phee and in hollow cicula cylinde. Ootao and Tanigawa [7] conducted an optimization of the mateial compoition of FGM hollow cicula cylinde unde themal loading. A themo-mechanical analyi, including the coupling effect, fo FGM plate and cylinde wa peented by Ready and Chin [8]. Tanaka et al, [9] gave an impoved olution to themo-elatic mateial deigned in functionally gadient mateial in ode to educe the themal tee. They deigned FGM popety pofile uing a enitivity and optimization method. Wethehold et al. [0] peented thei wok fo the etimation and contol of defomation of FGM unde themal loading. Ditibution of themal te and defomation in functionally gaded mateial hell of evolution unde themal loading due to fluid wee invetigated by Takezono et al. []. Zhang et al. [] peented an analytical olution fo themal tee of axial ymmety functionally gadient mateial unde teady tempeatue field. Fukui, Y. and Yamanaka, N [3] conideed the analyi of FGM thick-walled Coeponding Autho: K. Abinia, Mechanical Engineeing Depatment, Univeity College of Engineeing, Univeity of Tehan, Amiabad Shomali St., Tehan, Ian 85

2 tube. Salaza [4] alo ued finite element method to obtain olution fo functionally gaded metal matix compoite tube. The poblem in olid mechanic involving inhomogeneou media ae elatively taight fowad. Such poblem can be fomulated in tem of patial diffeential equation with vaiable coefficient by uing the baic conevation law. Thee ha alway been difficulty in developing geneal method fo olving pecific bounday value poblem. In fact, fo the mot geneal econd-ode patial diffeential equation with vaiable coefficient, uch geneal method do not exit. Becaue of thi difficulty, all exiting teatment dealing with the mechanic of inhomogeneou olid ae baed on a imple function epeenting mateial inhomogeneity. Fo example, in the half-plane elaticity poblem conideed in [5] and [6], it i aumed that the hea modulu i a powe function of the depth coodinate of the fom µ(y) = µ 0 y m and the paion atio v i contant. It i aumed that the mateial i iotopic with contant paion atio and adially vaying elatic modulu i appoximated by E() = E 0. Since i away fom zeo and ange in (a, R), by adjuting the contant E 0 and, it i poible to obtain phyically meaningful eult. The ange to be ued in the peent tudy cove all the value of coodinate exponent encounteed in the efeence cited ealie [9]. Howeve, thee value fo do not neceaily epeent a cetain mateial. Vaiou value ae ued to demontate the effect of inhomogeneity on the te ditibution. Liew et al. [7] peented an analyi of the themo-mechanical behavio of hollow cicula cylinde of functionally gaded mateial. They intoduced a novel limiting poce that employed the olution of homogenou hollow cicula cylinde. Cylindical veel ae often ued a baic tuctual component in engineeing application. Much eeach ha been conducted on iotopic o laminated compoite plate and hell [0],[]. Howeve it eem that vey little ha been done on FGM thick veel. Analytical olution have been given by Johnon and Mello [8] fo thick cylindical veel unde peue and tempeatue loading. Exact olution fo tee in functionally gaded peue veel wee peented by Tutuncu and Oztuk [9]. They ued a mateial tiffne obeying a imple powe law, and detemined the inhomogeneity contant which included continuouly vaying volume faction of the contituent. Elami et al. [0] gave account of thei wok on the mechanical and themal tee in a FGM hollow cylinde due to adially ymmetic load. They Am. J. Applied Sci., 5 (7): , aumed the tempeatue ditibution being a function of adiu and ued a diect method to olve the heat conduction and Navie equation. In a imila wok, Elami, Babaei and Poultangai [] invetigated the themal and mechanical tee in a FGM phee uing the ame method a in [0]. Hence thi pape will develop an analytical model to give olution fo FGM hollow cylindical veel that ae ubject to the action of an abitay teady tate tempeatue field, conideing peue effect. Thi analyi ue the baic equation [8], [9] uggeted by efeence and extend them to include the effect of tempeatue a well. THEORY The te ditibution in a thick-walled cylinde unde the influence of intenal peue and tempeatue loading fo homogenou mateial, have been fomulated by efeence [8]. Hee the ame kind of pocedue ha been ued except that the mateial i non-homogenou and theefoe the popetie change a one move along the adiu of the cylinde. Conide the co ection of a cylindical peue veel a hown in Fig., with the intenal adiu a and extenal adiu b and which i nomalized a R = whee R having a value b between a and b. Then in ode to account fo the changing mateial popetie along the adiu, a powe law elationhip [9] i ued a follow: E() = E α () = α 0 0 () Subtituting = in above equation could daw that E 0, α 0 ae the modulu of elaticity and themal coefficient of expanion of oute uface. a R b Fig. : Co ection of a cylindical peue veel with intenal adiu a and extenal adiu b

3 Am. J. Applied Sci., 5 (7): , 008 Note that the value of Poion atio ha been taken a contant becaue although the mateial i not homogenou but conideing that all mateial almot have a contant Poion atio value in the elatic ange, thi i a eaonable aumption. Now the fomulation begin with the expeion fo the tain and te ditibution though the thickne of the cylinde uing the equilibium equation a follow: dσ ( σ σ ) + θ = 0 d () A thi i a plane tain poblem and hence the longitudinal tain i zeo, the adial and cicumfeential tain ae given in tem of diplacement, a: du e = u = d u eθ = (3) It could eaily be hown that the tee in the adial and hoop diection ae given by: E σ θ = ν + θ ν + ν α ( + ν)( ν) E σ = ν + ν θ + ν α ( + ν)( ν) [ e e ( ) ( ) T] [( )e e ( ) T] (4) Vaiation of paamete method wa ued fo achieving the paticula olution which i peented a flow: Whee = ( 4 + 4ν) = ( ν) f () f () yp = C d + C * d * ( ) ( ) * = = 4 + 4ν Alo the tempeatue gadient i given by: ta T = ln( ) ln( ) w (6) (7) Note that above elation i nomalized fom of tempeatue gadient which wa peented in [8] and a w equal to. b Subtituting (7) into (6) and uing Eq. (5) afte implifying, paticula olution wa obtained: Howeve if in Eq. () we ubtitute fo tee fom Eq. (4), and alo ubtitute fo tain fom (3), the following equation could be obtained fo diplacement: Β+ yp = α0ta (m ln( ) m ) Whee m and m ae given by: (8) Whee f () u + (B + )u + ( ν ) = C α 0( + ν) ν C =, ν =,f () = T ( ν) ( ν) (5) Thi i a non homogenou fom of Eule-Cauchy equation. By uing the vaiation of paamete method (Lagange) we have the following olution: Conideing the olution of homogeneou fom of thi equation which wa peented in [9], we know that geneal olution of homogenou fom of above Eule-Cauchy equation i: y = y = Whee S and S ae given by: y = y + y h ν m = ( ) ν ( + + )( + + )ln( ) w + ν ( ) m = ( ) ν ( + + ) ( + + ) ln( ) w Solution fo the poblem in fact hould be um of geneal and paticula olution and could be given fo the diplacement in tem of which tain and tee ae defined: Y = y h + yp u = G + H Β+ +α0ta (m ln( ) m ) (9) In above equation G and H ae unknown contant which wee obtained by applying bounday condition and following the below poce:

4 Am. J. Applied Sci., 5 (7): , 008 The bounday condition ae alo given by: = w = a / b σ = p = σ = 0 Now uing the bounday condition and ubtituting fo u fom (9) into (3) and ubequently (3) into (4), we aive at a et of equation with ome unknown contant which wee called G and H and afte olving thee equation they ae given a follow: p( + ν)( ν) n w + n E G = 0 + ν ν + ( )( w w ) p( + ν)( ν) n w + n E = 0 + ν ν + H ( )( w w ) Whee again n and n ae given by: n = t m α ( m m + m ν ν m ν + ν) a = aα0 ν + ν + ( + ) a n m w t ( ) (0) w t α ( m m m m + m m ν m m ν + + ν) Theefoe ubtituting G, H back in the diplacement Eq. (9) and ubtituting the obtained equation of diplacement back in tain and ubequently te equation, the final elationhip fo the ditibution of tee could be obtained a follow: H α0e0taw( )(w + w + )(w ) σ = 3 ( ν )(w )( + w + w ) H α 0E0taw( (w + )( ) w ( + )) σ θ = 3 ( ν )(w )( + w + w ) () Above expeion can alo eaily be obtained by etting = 0 in the FGM cae (of coue conideing computational eo, we chooe 0 0 fo intead of zeo). RESULTS AND DISCUSSION The analytical olution peented in the peviou ection wa applied to a thick hollow cylinde of inne adiu a = 0.75m and oute adiu of b = m. The modulu of elaticity and the themal coefficient of expanion at the oute uface of the cylinde wee taken a E 0 = 00Gpa and α 0 =. 0 6 / C epectively. Poion atio aumed to be contant (v = 0.3). The intenal peue wa taken to be equal to 400Mpa and the extenal peue wa aumed to be zeo. The value of tempeatue at the inne and oute uface wee aumed to be t a = 000 C and t b = 0 C epectively. Peue only: The eult fo intenal peue loading wee nomalized with epect to the homogenou cylinde to tudy the impact of the inhomogeneity on the eult fo the FG cylinde. The ditibution of adial te fo diffeent value of i een in Fig.. G E ( w)( + ν ν ) + H E ( w) ( ( w)e0 0tam σ = α ( + ν)( ν)( w) ( m + ν ν m + m ν m ν) ( + ν ν ) + α wt ( )( ν )( + ν) 0 a 0 E σ θ = ( + ν)( ν)( w) G (w )( ν + ) + H (w )( ν + ) + w( ( ( 0ta m (w ) ( )(m3 ) m3 +α + ν + + ν () Stee fo the cylindical FGM peue which obtained above will be compaed with thoe fo the homogenou one. The te expeion fo homogenou cylinde ae peented below. Note that thee ae nomalized fom of what wee mentioned in [8]. ( ) ) )( + ν) 855 Fig. : vaiation of nomalized adial te in a cylindical veel unde the loading of peue

5 Am. J. Applied Sci., 5 (7): , 008 Fig. 3: Ditibution of cicumfeential te in cylindical veel unde the loading of peue Fig. 4: Vaiation of nomalized adial te in a cylindical veel unde the loading of tempeatue Hee a highe value of mean inceaing tiffne (ee Eq. ()). It could be een fom Fig.. that fo highe poitive value the te in the adial diection inceae while fo a negative value of it deceae. In Fig. 3 fo poitive value of the maximum hoop te occu at the oute uface of the cylinde while fo negative value of the maximum happen at the inne uface. At a adial ditance of = 0.87 the te value fo all value of convege towad the te value in the homogenou mateial ( = 0). Tempeatue only: The effect of themal loading a een in Fig. 4 and 5, indicate the fact that jut like the behaviou of homogenou cylinde, in FG cylinde too, the te ditibution tend fo the themal loading i oppoite to that due to intenal peue. Howeve fo poitive value of the ditibution of adial te a een in Fig. 4. indicate a faily unifom tend a compaed to that obtained fo the negative value of. Alo fo negative value of much highe value ae obeved at the inne uface of the cylinde. A hown in Fig. 5. the cicumfeential te in geneal inceae and become a maximum at the oute uface of the cylinde. Fo lage value of lowe te value ae obeved. It could alo be een fom the ame figue that thee i a udden dop at about the mid-uface ( = 0.85 to 0.9) of the cylinde wall depending on the value. 856 Fig. 5: vaiation of cicumfeential te in a cylindical veel unde the loading of tempeatue Combined loading (peue + tempeatue): Since both peue and tempeatue loading ae in the elatic ange, the pinciple of upepoition wa applied fo the combined loading and the following eult wee obtained.

6 Am. J. Applied Sci., 5 (7): , 008 Fig. 8: Diplacement in a cylindical veel unde the combined loading of peue and tempeatue Fig. 6: vaiation of nomalized adial te in a cylindical veel unde the combined loading of peue and tempeatue Fig. 7: vaiation of nomalized cicumfeential te in a cylindical veel unde the combined loading of peue and tempeatue In Figue 6, nomalied value of adial tee wee plotted againt adial poition in the cylinde wall. Howeve fo highe value of lage te value ae obeved at the oute uface of the cylinde while fo the lowe value of malle te value ae 857 Fig. 9: Ditibution of elative adial te in a cylindical veel unde the combined loading of peue and tempeatue obtained. Thi could be intepeted phyically meaning that the oute uface of the cylinde i biaed to bea the te due to it highe tiffne. In Figue 7, it could be noted that the combined effect of tempeatue and peue loading on the hoop te alo poduce the ame te value at about = Hee again highe value give highe tee at the oute uface of the cylinde while they poduce lowe tee at the inne uface. The adial diplacement could be obeved in Fig. 8 to have educed at the oute uface of the cylinde with epect to the inne uface but fo highe value lage

7 Fig. 0: Ditibution of elative hoop te in a cylindical veel unde the combined loading of peue and tempeatue diplacement ae poible. Figue (9) how the nomalied value of adial te with egad to the intenal peue inceae fom inne to oute uface and not howing much diffeence fo diffeent value of. Simila eult fo the elative hoop te with epect to intenal peue ae alo hown in Fig. 0. Highe poitive value give highe tee at the oute uface and negative value give lowe tee at the ame poition. Fom above eult it could be aid that the FGM exponent of Eq. () could be ued a a vey ueful deign paamete fo tailoing the te ditibution to fit to the pecific application. CONCLUSION The following concluion could be dawn fom the wok peented in thi pape: ) A cloed fom exact analytical olution wa obtained fo the combined peue and tempeatue loading of FGM cicula cylinde. ) Depending on applied bounday condition, by electing optimum value of, deiable level of adial and cicumfeential tee could be obtained in FGM cylinde with epect to thoe in homogenou one. 3) By etting = 0 in Eq. (), adial and cicumfeential tee expeion tuned to homogenou one which could aent the validity of fomulation. Am. J. Applied Sci., 5 (7): , ) Fo futhe wok the peent fomulation could be extended and developed fo the yield egion o that an elatic platic olution could be obtained. 5) Anothe poible line fo futhe eeach i the aumed powe law fo the FGM popetie which could pehap be eplaced by a moe ealitic fom with egad to the manufactuing poblem of FGM cylinde. REFERENCES. Yamanouchi, M., M. Koizumi and I. Shiota, 990, Poceeding of the Fit Intenational Sympoium on Functionally Gadient Mateial, Japan, Koizumu, M., 993. The concept of FGM, Ceam: Tan., Funct. Gad Mate, 34: Noda, N., 99. Themal tee in mateial with tempeatue-dependent popetie: Applied Mechanic Review, 44: Shen, H.S., 00. Pot Buckling Analyi of Axially-Loaded FG Cylindical Shell in Themal Envionment: compoite cience and Tech, 6: Zimmeman, R.W. and M.P Lutz, 999. Themal tee and themal expanion in a unifomly heated functionally gaded cylinde: J. of Themal Stee, : Obota, Y. and N. Noda, 994. Steady themal tee in a hollow cicula cylinde and a hollow phee of a functionally gadient mateial: J. of Themal Stee, 7: Ootao, Y., Y. Tanigamo and T. Nakamua, 999. Optimum of Mateial Computation of FGM Hollow Cicula Cylinde unde Themal Loading a Neual Netwok Appoach: Compoite-pat B: Eng 30: Reddy, J.N. and C.D. Chin, 988. Themo mechanical Analyi of Functionally Gaded Cylinde and Plate: J. of Themal Stee, : Tanaka, K., Y. Tanaka, H. Watanabe, V.F. Poteau and Y. Sungamo, 993. An impoved olution to themoplatic mateial deigned in fgm: cheme to educe themal tee: Compute Method in Applied Mech and Eng 06: Wethehold, R.C., S. Seelman and J.Z. Wang, 996. Ue of fgm to etimate o contol themal defomation: Comput. Sci. Technol, 56:

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