Available online at ScienceDirect. Procedia Materials Science 6 (2014 )

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1 Availabl onlin at ScincDirct Procdia Matrials Scinc 6 ( ) 6 67 rd Intrnational Confrnc on Matrials Procssing and Charactrisation (ICMPC ) Modal analysis of Functionally Gradd matrial Plats using Finit Elmnt Mthod Ramu I*, Mohanty S.C. *Rsarch scholar,mchanical Enginring Dpartmnt, NIT Rourkla-769, India Associat Profssor, Mchanical Enginring Dpartmnt, NIT Rourkla-769, India Abstract Th prsnt work aims to carry out modal analysis of a functionally gradd matrial (FGM) plat to dtrmin its natural frquncis and mod shaps by using Finit Elmnt Mthod (FEM). Functionally gradd matrial can b diffrntiatd by varying th composition and structur progrssivly ovr its volum, consquntially in corrsponding changs in th matrial constitunts. Th mchanical proprtis of a FGM plat chang continuously from on surfac to anothr through its thicknss dirction according to powr law. For modal analysis of FGM plat program has bn codd in MATLAB softwar. Som xampls ar solvd, and th rsults ar compard with thos availabl in th litratur. Th mod shap and natural frquncis of rctangular FGM plat ar found at diffrnt boundary conditions. It has bn obsrvd th ffct of volum fraction indx, which indicats th prcntag of cramic and mtal composition in th FGM. In addition, th ffcts of powr law indx on th FGM plat natural frquncy and mod shaps with diffrnt boundary conditions ar studid. Elsvir Ltd. This is an opn accss articl undr th CC BY-NC-ND licns Th Authors. Publishd by Elsvir Ltd. ( Slction Slction and and pr pr-rviw rviw undr undr rsponsibility rsponsibility of th of Gokaraju th Gokaraju Rangaraju Rangaraju Institut Institut of Enginring of Enginring and Tchnology and Tchnology (GRIET) (GRIET). Ky words: Functionally gradd matrial, classical plat thory, finit lmnt mthod, natural frquncis, mod shaps. Introduction Modal analysis is a tchniqu usd to dtrmin structur s vibration charactristics: Natural frquncis, and Mod shaps. Ths ar most fundamntal of all dynamic analysis typs. Whras tchnology dvlopmnt at an vr incrasing rat, th nd for advancd capability matrials bcoms a main concrn in th modal analysis of nginring structurs of mor intricat and highr prformanc systms. Ths rquirmnts can b sn in many * Corrsponding author. Tl.: addrss: ram.journals@gmail.com - Elsvir Ltd. This is an opn accss articl undr th CC BY-NC-ND licns ( Slction and pr rviw undr rsponsibility of th Gokaraju Rangaraju Institut of Enginring and Tchnology (GRIET) doi:.6/j.mspro..7.9

2 I. Ramu and S.C. Mohanty / Procdia Matrials Scinc 6 ( ) filds in which nginrs ar xploring th applications of nw nginrd matrials. Functionally Gradd Matrials (FGMs) ar a rlativly nw tchnology and ar bing studid for th us in componnts xposd to high tmpratur gradints. An FGM matrial proprty allows th dsignr to tailor matrial rspons to mt dsign critria. An FGM composd of cramic on th outsid surfac and mtal on th insid surfac liminats th abrupt chang btwn cofficints of thrmal xpansion, offrs thrmal protction, and provids load carrying capability. This is possibl bcaus th matrial constitunts of an FGM changs gradually through-th-thicknss; thrfor, strss concntrations from abrupt changs in matrial proprtis ar liminatd. Th FGM plats has numrous applications in advancd nginring filds such as thin-walld structural componnts in spac vhicls, nuclar ractors, and othr high thrmal application aras. Th Functionally gradd matrial (FGM) can b intndd for spcific function and applications. Th matrial proprtis such as young s modulus, dnsity and poisons ratio valus ar varying continuously throughout th thicknss dirction according to th simpl volum constitunts dfind by powr law. In th rcnt yars functionally gradd matrial plats modlling and analysis ar carrid out by Birman and Larry (7). Thy hav prsnts th principal dvlopmnts in functionally gradd matrials (FGMs) with an importanc on th rcnt work publishd sinc. Pndhari t al. () hav dvlopd th analytical and mixd smi-analytical solutions for a rctangular functionally gradd plat. Singha t al. () has invstigats th high prcision plat bnding finit lmnt for nonlinar bhaviour of functionally gradd plats undr transvrs load. In thir analysis basd on th first ordr shar dformation thory considrd th physical/xact nutral surfac position. A rctangular functionally gradd matrial plat with simply supportd boundary conditions subjctd to transvrs loading has bn invstigatd by Chi and Yn (6). Thir analysis carrid out, bass on th classical plat thory and Fourir sris xpansion; th sris solutions of powr-law FGM (simply calld P-FGM), sigmoid FGM (S-FGM), and xponntial FGM (E-FGM) plats ar obtaind. Shahrjrdi t.al.() rcntly stimats th natural frquncy of functionally gradd rctangular plat using scond-ordr shar dformation thory (SSDT). Th matrial proprtis of functionally gradd rctangular plats, xcpt th Poisson s ratio, ar assumd to vary continuously through th thicknss of th plat in accordanc with th xponntial law distribution. Talha and singh () hav studid th static and fr vibration analysis of functionally gradd matrial plats by using highr ordr shar dformation thory with a spcial modification in th transvrs displacmnt in conjunction with finit lmnt modls. Vl and batrab () hav dvlopd a thrdimnsional xact solution for fr and forcd vibrations of simply supportd functionally gradd rctangular plats. Th xact solution was valid for thick and thin plats, and for arbitrary diffrnc of matrial proprtis in th thicknss dirction. Hashmi t.al. () carrid out th nw xact closd form mthod for fr vibration analysis of functionally gradd rctangular thick plats. Thir analysis was basd on th Rddy s third-ordr shar dformation plat thory. Th main objctiv of this work is to propos a finit lmnt approach for modal analysis of rctangular FGM plats basd on th Kirchhoff plat thory or classical plat thory. Th matrial proprtis ar assumd to b gradd through th thicknss in accordanc with a simpl powr-law distribution. Using th finit lmnt formulation drivd for th FGM plat lmnt, a finit lmnt program has bn dvlopd. Som xampls ar solvd, and th rsults ar compard with thos availabl in th litratur. In addition, th ffcts of powr law indx on th FGM plat natural frquncy and mod shaps with diffrnt boundary conditions ar studid.. FGM plat A functionally gradd matrial plat of thicknss h is mad of a mixtur of cramic and mtal as shown in Fig. (a). Th matrial on th top surfac of th plat is cramic rich whras th bottom surfac matrial is mtal rich. Hr, zm h/ is th distanc masurd from mid-surfac of th plat as shown in Fig.. Th volum fraction of cramic V and mtal c V varying through th thicknss of th plat, according to a simpl powr-law, ar xprssd as m n zm h Vczm, h Vmzm Vczm () whr n is th volum fraction xponnt (n>). Th variation of th ffctiv matrial proprty P may b writtn as P z P z V z P z V z () m c m c m m m m m For plat mad of FGM, th nutral surfac may not coincid with its gomtric mid-surfac. Th distanc of th

3 h 6 I. Ramu and S.C. Mohanty / Procdia Matrials Scinc 6 ( ) 6 67 nutral surfac (d) from th gomtric mid-surfac may b xprssd as. d h / h / h / h / E z E z m m m m z dz dz m In th nutral surfac basd formulation, th coordinat z n is with rspct to th nutral surfac. Th volum fraction of cramic in th nw coordinat systm can b xprssd as V c z n zn hd h n () () Fig. Nutral plan of FGM plat variation of proprty along th thicknss n= n=. n= n=. n= thicknss n= Fig. Variation of young s modulus along th thicknss of th FGM plat Fig. shows th ffctiv young s modulus variation with diffrnt indx valus along th thicknss dirction of th plat.. Finit Elmnt formulations proprty x p Th displacmnt fild of th classical plat thory is: wxy (, ) U u( x, y) z x wxy (, ) V v( x, y) z y W w( x, y) Th strain-displacmnt rlationship ar givn as u w x z x x v w y z y y y x xy u v w xy z, yz xz () (6)

4 I. Ramu and S.C. Mohanty / Procdia Matrials Scinc 6 ( ) Th strss-strain rlationships of th functionally gradd plat in th global x-y-z coordinats systm can b writtn as xx Q Q x yy Q Q y (7) xy Q 66 xy whr Ez Ez Q, Q Q, Ez Q66 Th total strain nrgy can b xprssd as U x x y y xyxydxdydz () Th kintic nrgy of th plat can b xprssd as follows. T hw da (9) A Th FGM plat is modl using four nod rctangular lmnts. Four nod rctangular lmnts ar having four nods at ach cornr as shown in Fig.. Thr ar thr dgrs of frdom at ach nod, th displacmnt componnt along th thicknss (w), and two rotations about x and y dirctions in trms of th (x,y) coordinats rspctivly. Th ach lmnt consists of four nods,, and with w is th transvrs displacmnt and and rprsnts th rotations about x and y axis rspctivly. x y dw dw w, x, y () dy dx Thrfor th lmnt has twlv dgrs of frdom and th displacmnt function of th lmnt can b rprsntd by a polynomial having twlv trms as shown. w= + x+ y+ x+ xy+ 6y + 7x + x y+ 9xy + y + x y+ xy () Th xprssion () can b writtn in th following matrix form w x y x xy y x x y xy y x y xy w Polyx, y () w A, A w T i xi yi Substituting q. () in q. () () whr w w, i,,,,, w Nw w Ploy x y A w ()

5 6 I. Ramu and S.C. Mohanty / Procdia Matrials Scinc 6 ( ) 6 67 Fig. Gomtry of th four nod rctangular lmnt xx i yy i xx i yy i x y () T N i = xx i y yiy xxi yiyx whr i=,, and. Th lmnt stiffnss matrix and mass matrics ar drivd on th basis on principl of minimum potntial nrgy and kintic nrgy. Th lmnt stiffnss matrix is T K B D B dv (6) v whr x [ B] z N y xy Elmnt mass matrix T M NNdv v Th quation of motion for a plat lmnt is obtaind by using Hamilton s principl. t U T W dt t Th lmnt stiffnss and mass matrics ar assmbld in to gt global matrics. Th quation of motion of th plat can b writtn as K Mw () whr ar natural frquncis of th plat. Rsults and discussions In according to th Talha M. and B.N.Singh, () is dimnsionlss frquncy paramtr is: clw Ech / Validation has bn don by considring th valus of thicknss, lngth, width, Poisson s ratio, dnsity and young s modulus in th cramic and mtal as: 9 Al, 77 kg / m, E 7X Pa,. Al O kg m E X Pa 9, 77 /,,. 9, /,,., 66 /, 9 7,.77 ZrO kg m E X Pa SUS kg m E X Pa Th tabl shows th natural frquncy paramtrs obtaind from th prsnt study using classical plat thory. Tabl show th natural frquncy paramtr obtaind from th prsnt study using classical plat thory and Talha & Singh (). Thr is a good matching btwn th prsntd rsults and thos from Talha and Singh, (), spcially for simply supportd cas. (7) () (9)

6 I. Ramu and S.C. Mohanty / Procdia Matrials Scinc 6 ( ) Tabl. Variation of th frquncy paramtr ( ) with th volum fraction indx n for (SSSS) squar ( Al / ZrO ) FGM plats (a/h = ) Cramic Talha M.( )] Prsnt n=. Talha M.( ) Prsnt n= Talha M.( ) Prsnt n= Talha M.( ) Prsnt n= Talha M.( ) Prsnt Mtal Talha M.( ) Prsnt Th following numrical rsults ar obtaind by considring th stl as th bottom surfac and alumina as th top surfac in FGM plat according to indx valu. Th gomtry of plat and matrial proprtis is as follows: L= m (lngth), W= m (width), h=. m (thicknss). Th variations of natural frquncy paramtr in FGM (SUS/Al O ) plat for diffrnt boundary conditions ar shows in fig., and. Th ffct of powr law indx n on th frquncis can b sn for diffrnt boundary conditions. Figs.- shows th first fiv frquncy paramtrs vrss powr law indx valu at diffrnt boundary conditions. As xpctd, th incrasing indx valu lads to rduc th natural frquncy paramtr. Incrasing indx valu rducs th cramic constitunts, its produc th ffctiv matrial proprtis changs which affct th frquncy paramtr. CCCC FGM (SUS/Al O )plat frquincy paramtr variation 9 7 th frquncy paramtr 6 th nd & rd frquncy paramtr 6 th th nd & rd st st powr law indx n powr law indx n Fig. Variation of frquncy paramtr with indx valu at SSSS and CCCC boundary conditions SCSC FGM (SUS/Al O )plat frquincy paramtr variation SFSF FGM (SUS/Al O )plat frquincy paramtr variation. 9. th frquncy paramtr 7 6 th rd th frquncy paramtr. th rd nd. nd st st powr law indx n powr law indx n Fig. Variation of frquncy paramtr with indx valu at SCSC and SFSF boundary conditions

7 66 I. Ramu and S.C. Mohanty / Procdia Matrials Scinc 6 ( ) frquncy paramtr.... CCCC SCSC SSSS SFSF frquncy paramtr 6 CCCC SFSF SCSC SSSS powr law indx n powr law indx n (a) (b) Fig.6 (a), (b) First and scond mod frquncy paramtr variation by powr law indx with various boundary conditions Mod fn= 7.69 Hz Mod fn= 76. Hz Mod fn= 7.6 Hz. Mod fn= 76. Hz Fig.7 SSSS squar FGM plat mod shaps,, and with indx valu n= Mod fn=.6 Hz Mod fn=.67 Hz Mod fn= 7.6 Hz Mod fn=. Hz Fig. SFSF squar FGM plat mod shaps,, and with indx valu n= Mod fn=. Hz Mod fn=. Hz Mod fn= 9. Hz Mod fn= 7. Hz Fig.9 CCCC squar FGM plat mod shaps,, and with indx valu n= A vibrating systm with infinit numbr dgrs of frdom (DOF) has infinit numbr natural frquncis, and for ach natural frquncy thr is a rlationship btwn th amplituds of th infinit numbr indpndnt motions, known as th mod shap. Thr is on mod shap for ach natural frquncy and it dpnds on th valu of that natural frquncy. For a FGM plat th mod shaps ar th shaps of th structur at its maximum dformation during a cycl of vibration. Fig., 6 and 7 shows th first four mods shaps of an FGM plat with diffrnt boundary conditions.. Conclusions A finit lmnt formulation for modal analysis of FGM plats has bn drivd. Th matrial composition of th FGM structur is assumd to vary according to a simpl powr-law distribution through th thicknss. Th dvlopd computr cod is validatd with th publishd rsults for natural frquncis of FGM plats. Rsults obtaind from prsnt finit lmnt analysis agr wll with thos rportd. Dvlopd finit lmnt cod is mployd to comput th natural frquncis and mod shaps of th FGM plat with diffrnt boundary conditions and with diffrnt powr law indx valus. Convrgnc study has also bn prformd and it

8 I. Ramu and S.C. Mohanty / Procdia Matrials Scinc 6 ( ) is obsrvd that rsults ar vry good with msh siz X. Effcts of powr indx n and th four diffrnt boundary conditions on th natural frquncis ar invstigatd. Natural frquncis for CCCC boundary condition ar th largst than th corrsponding frquncis obtaind for SCSC, SSSS and SFSF boundary conditions. Incrasing th powr law indx valu n rducs th first fiv natural frquncis for all th diffrnt boundary conditions. Th ffct of powr law indx is mor prominnt within th rang up to, aftr which th ffct is not that much significant. Rfrncs Ali Shahrjrdi, F. Mustapha, M. Bayat, S. M. Sapuan, R. Zahari and M. M. Shahzamanian,. Natural Frquncy of F.G. Rctangular Plat by Shar Dformation Thory, IOP Conf. Sris: Matrials Scinc and Enginring 7,. J. Sursh Kumar, B. Sidda Rddy, C. Eswara Rddy and K. Vijaya Kumar Rddy,.Highr ordr thory for fr vibration analysis of functionally gradd matrial plats, ARPN Journal of Enginring and Applid Scincs VOL. 6, NO.. Mohammad Talha, B.N. Singh,. Static rspons and fr vibration analysis of FGM plats using highr ordr shar dformation thory, Applid Mathmatical Modlling, 99. M.K. Singha, T. Prakash, M. Ganapathi,. Finit lmnt analysis of functionally gradd plats undr transvrs load, Finit Elmnts in Analysis and Dsign 7, 6. Snthil S. Vl, R.C. Batrab,. Thr-dimnsional xact solution for th vibration of functionally gradd rctangular plats, Journal of Sound and Vibration 7, 7 7. Sh. Hossini-Hashmi, M. Fada, S.R. Atashipour,. Study on th fr vibration of thick functionally gradd rctangular plats according to a nw xact closd-form procdur Composit Structurs 9, 7 7. Shyang-Ho Chi, Yn-Ling Chung, 6. Mchanical bhavior of functionally gradd matrial plats undr transvrs load Part I: Analysis, Intrnational Journal of Solids and Structurs, Sandp S. Pndhari, Tarun Kant, Yogsh M. Dsai, C. Vnkata Subbaiah,. Static solutions for functionally gradd simply supportd plats, Int J Mch Matr Ds DOI.7/s Victor Birman and Larry W. Byrd, 7. Modling and Analysis of Functionally Gradd Matrials and Structurs, Applid Mchanics Rviws ASME Vol. 6, 9.