Journal of Solid Mchanics ol. 9, No. 3 (017) pp. 650-66 Elastic Analysis of Functionally Gradd ariabl hicknss Rotating Disk by Elmnt Basd Matrial Grading A.K. hawait 1,*, L. Sondhi 1, Sh. Sanyal, Sh. Bhowmick 1 Dpartmnt of Mchanical Enginring, Shri Shankaracharya chnical Campus, SSGI,Bhilai, (C.G.), India Dpartmnt of Mchanical Enginring, NI Raipur, 49010, India Rcivd 9 Jun 017; accptd 9 August 017 ABSRAC h prsnt study dals with th lastic analysis of concav thicknss rotating disks mad of functionally gradd matrials (FGMs).h analysis is carrid out using lmnt basd gradation of matrial proprtis in radial dirction ovr th discrtizd domain. h rsulting dformation and strsss ar valuatd for fr-fr boundary condition and th ffct of grading indx on th dformation and strsss is invstigatd and prsntd. h rsults obtaind show that thr is a significant rduction of strsss in FGM disks as compard to homognous disks and th disks modld by powr law FGM hav bttr strngth.. All rights rsrvd. Kywords: Functionally gradd matrial; Elastic analysis; Annular rotating disk; Concav thicknss profil rotating disk; Elmnt basd matrial gradation. 1 INRODUCION M ANY nginring componnts ar modld as rotating circular plats or disks in th fild of marin, mchanical and arospac industris such as gas turbins, gars, turbo-machinry, flywhl systms, cntrifugal comprssors, powr transmission systms, machining dviss, circular saws, microwav or baking ovns, and support tabls, tc. h total strsss du to cntrifugal load hav important ffcts on thir strngth and safty. hus, control and optimization of total strsss and displacmnt filds is an important task. Functionally gradation of th matrial proprtis and variabl thicknss profil optimiz th componnt strngth by controlling th changs of th local matrial proprtis. Dpnding on th function of th componnt, it is possibl to utiliz on, two or thr-dirctional distributions of th matrial proprtis. A fw rsarchrs hav rportd work on analysis of FGM disks, plats, shlls, bams and bars by analytical and finit lmnt mthod. Eraslan t al. [1] has obtaind analytical solutions for th lastic plastic strss distribution in rotating variabl thicknss annular disks. hicknss of th disks has parabolic variation and th analysis is basd on th trsca s yild critrion. Bayat t al. [] rportd work on analysis of a variabl thicknss FGM rotating disk. Matrial proprtis vary according to powr law and th disk is subjctd to both th mchanical and thrmal loads. Afsar t al. [3] hav analyzd a rotating FGM circular disk subjctd to mchanical as wll as thrmal load by finit lmnt mthod. h disk has xponntially varying matrial proprtis in radial dirction. h disk is subjctd to a thrmal load along with cntrifugal load du to non-uniform tmpratur distribution. An analytical thrmolasticity solution for a disk mad of functionally gradd matrials (FGMs) is prsntd by Callioglu [4]. Bayat t al. [5] invstigatd displacmnt and strss bhavior of a functionally gradd rotating disk of variabl thicknss by smi analytical mthod. Radially varying on dimnsional FGM is takn and matrial proprtis vary according to powr * Corrsponding author. E-mail addrss: amkthawait@gmail.com (A.K.hawait).. All rights rsrvd.
A.K.hawait t al. 651 law and Mori-anaka schm. Disk subjctd to cntrifugal load is analyzd for fixd boundary condition at innr surfac and fr boundary condition at outr surfac. Callioglu t al. [6] hav analyzd thin FGM disks. Dnsity and Modulus of lasticity varis according to powr law in an FGM of aluminum cramic and th ffct of grading paramtr on displacmnt and strsss ar invstigatd. Callioglu t al. [7] has analyzd functionally gradd rotating annular disk subjctd to intrnal prssur and various tmpratur distributions such as uniform tmpratur, linarly incrasing and dcrasing tmpratur in radial dirction. Sharma t al. [8] workd on th analysis of strsss, displacmnts and strains in a thin circular functionally gradd matrial (FGM) disk by finit lmnt mthod. h disk is subjctd to mchanical as wll as thrmal loads. Ali t al. [9] rportd study on th lastic analysis of two sigmoid FGM rotating disks. Mtal-Cramic-Mtal disk is analyzd for both uniform and variabl thicknss disks and ffct of grading indx on th displacmnt and strsss ar invstigatd. Njad t al. [10] hav found closd-form Analytical solution of an xponntially varying FGM disk which is subjctd to intrnal and xtrnal prssur. Ghorbanpour t al. [11, 1] hav workd on hrmo-pizo-magntic bhavior of a functionally gradd pizo-magntic (FGPM) rotating disk, undr mchanical and thrmal loads. A smi-analytical solution for magnto-thrmo-lastic problm in functionally gradd (FG) hollow rotating disks with variabl thicknss placd in uniform magntic and thrmal filds is also prsntd. In a rcnt work Zafarmand t al. [13] workd on lastic analysis of two-dimnsional functionally gradd rotating annular and solid disks with variabl thicknss. Axisymmtric conditions ar assumd for th twodimnsional functionally gradd disk and th gradd finit lmnt mthod (GFEM) has bn applid to solv th quations. Rosyid t al. [14] workd on finit lmnt analysis of nonhomognous rotating disk with arbitrarily variabl thicknss. hr typs of grading law namly Powr law, sigmoid and xponntial distribution law is considrd for th volum fraction distributions. Zafarmand t al. [15] invstigatd nonlinar lasticity solution of functionally gradd nanocomposit rotating thick disks with variabl thicknss rinforcd with singl-walld carbon nanotubs (SWCNs). h govrning nonlinar quations drivd ar basd on th axisymmtric thory of lasticity with th gomtric nonlinarity in axisymmtric complt form and ar solvd by nonlinar gradd finit lmnt mthod (NGFEM). In prsnt rsarch work rotating disks of parabolic concav thicknss profil having fr-fr boundary condition ar analyzd. Matrial proprtis of th disks ar gradd along th radial dirction according to thr typs of distribution laws, namly powr law, xponntial law and Mori-anaka schm by lmnt basd matrial grading. A finit lmnt formulation basd on principl of stationary total potntial is prsntd for th rotating disks and finally th ffct of grading paramtr n on th dformation and strsss for diffrnt matrial gradation law is invstigatd. PROBLEM FORMULAION In this sction gomtric quation as wll as diffrnt typs of matrial proprty distributions ar prsntd and th govrning quations for th rotating disk ar drivd..1 Gomtric modling For an annular disk, th govrning quation of radially varying thicknss is assumd as: h r h q b a k 0 1 r a (1) whr a and b ar innr and outr radius, h(r ) and h 0 ar half of th thicknss at radius r and at th root of th disk, k and q ar th constants that control th thicknss profils of th disk. For uniform thicknss disk q = 0 and for variabl thicknss q > 0; k < 1 for concav thicknss profil.
65 Elastic Analysis of Functionally Gradd ariabl hicknss. Fig.1 Gomtric paramtrs of th rotating disk.. Matrial modling hr typs of matrial modls namly powr law distribution [7], xponntial distribution [3] and Distribution by Mori-anaka schm [5] ar considrd in th prsnt analysis. h ffctiv Young s modulus E(r) and dnsity ρ(r) of a disk having innr radius a and outr radius b can b obtaind as: Powr Law r E r Eo b n r r o b n () (3) whr E o, ρ o ar modulus of lasticity and dnsity at th outr radius. Exponntial law E r E0 r (4) 0 r r (5) E E a (6) 0 i (7) 0 i a a 1 i ln b o (8) E a b E 1 i ln o (9) E i, E o and ρ i, ρ o ar modulus of lasticity and dnsity at th innr and outr radius. Mori -anaka Schm h ffctiv bulk modulus B(r)and shar modulus G(r) of th FGM disk, valuatd using th Mori-anaka schm [5] ar givn by:
A.K.hawait t al. 653 3Bo Bi / 1 1 B r B B B 3Bi 4Gi o i o o i G r G G G f 1 G o Gi / 1 1 o i o o i Gi f1 G 9B 8G 6 B G i i i i i (10) (11) (1) Hr, is th volum fraction of th phas matrial. h subscripts i and o rfr to th innr and outr matrials rspctivly. h volum fraction of th innr and outr phass ar rlatd by i 1 (13) o and o is xprssd as: o r ri ro ri n (14) whr n(n 0) is th volum fraction xponnt. h lastic modulus E can b found as: E r 9B r G r 3B r G r (15) h mass dnsity ρ can b givn by th rul of mixturs as: n r ri o i i ro ri r (16).3 Finit lmnt formulation h rotating disk, bing thin, is modld as a plan strss axisymmtric problm. Using quadratic quadrilatral lmnt, th displacmnt vctor {u} can b obtaind as: u N (17) whr {u} is lmnt displacmnt vctor, [N] is matrix of quadratic shap functions and {δ} is nodal displacmnt vctor. N N 1 N. N 8 matrix of shap functions 1 8 u u u nodal displacmnt vctor In natural coordinats (ξ-η), th shap functions ar givn as [16]: N 1 1 4 1 1 1, N 1 1 1 5
654 Elastic Analysis of Functionally Gradd ariabl hicknss. N N N 1 1 4 1 1 4 1 1 4 1 1 1, N 1 1 6 1 1 1, N 1 1 3 7 1 1 1, N 1 1 4 8 h strain componnts ar rlatd to lmntal displacmnt componnts as: u u r r r u u u u u B1 r r r z r (18) (19) whr ε r and ε θ, ar radial and tangntial strain rspctivly. By transforming th global coordinats into natural coordinats (ξ-η), u u u u u u B r z r r u u u B u u u r 3 1 8 (0) (1) h abov lmntal strain-displacmnt rlationships can b writtn as: B () whr [B] is strain-displacmnt rlationship matrix, which contains drivativs of shap functions. For a quadratic quadrilatral lmnt it is calculatd as: B B B B B B 1 1 3 1 0 0 0 0 1 J J1 0 J J J1 J 11 0 J J 0 0 1 (3) whr J is th Jacobian matrix, usd to transform th global coordinats into natural coordinats. It is givn as:
A.K.hawait t al. 655 J 8 8 N i N i ri zi i 1 i 1 8 8 N i N i ri zi i 1 i 1 N1 N N 8. N N N N1 N N 8. r r r 1 8 B3. From hooks law, componnts of strsss in radial and circumfrntial dirction ar rlatd to componnts of total strain as: 1 1 r r, r E E (4) By solving abov quations, strss strain rlationship can b obtaind as follows: E r r 1 r E r r 1 (5) (6) In standard finit lmnt matrix notation abov strss strain rlations can b writtn as [16]: whr D r (7) E r 1 r, r, Dr 1 1 In FEM, th functional grading is popularly carrid out by assigning th avrag matrial proprtis ovr a givn gomtry followd by adhring th gomtris thus rsulting into layrd functional grading of matrial proprtis. h downsid of this approach is that it yilds singular fild variabl valus at th boundaris of th glud gomtris. o gt bttr rsults, it is an stablishd practic to divid th total gomtry into vry fin gomtris. Howvr, a bttr approach is to assign th avrag matrial proprtis to th lmnts of msh of th singl gomtry. his is, in othr words, bttr dscribd as assigning matrial proprtis to th finit lmnts instad of gomtry. In Eq. (7), th [D(r)] matrix, bing a function of r, is calculatd numrically at ach nod and this rsults into continuous matrial proprty variation throughout th gomtry. h lmnt basd grading of matrial proprty yilds an appropriat approach of functional grading as th shap functions in th lmntal formulations bing co-ordinat functions mak it asir to implmnt th sam [13]. 8 in i (8) i 1
656 Elastic Analysis of Functionally Gradd ariabl hicknss. whr ϕ is lmnt matrial proprty, ϕ i is matrial proprty at nod i and N i is th Shap function. Upon rotation, th disk xprincs a body forc which undr constraind boundary rsults in dformation and stors intrnal strain nrgy U [16]. U 1 dv (9) h work potntial du to body forc rsulting from cntrifugal action is givn by qv dv (30) Upon substituting Eqs. () and (7) in Eqs. (9) and (30), th lmntal strain nrgy and work potntial ar obtaind as: U rh r B D r B dr rhr N qvdr (31) (3) For a disk rotating at ω rad/sc, th body forc vctor for ach lmnt is givn by q v r 0 r h total potntial of th lmnt is obtaind from Eqs. (31) and (3) as: 1 p K f (33) Hr, dfining lmnt stiffnss matrix [K] and lmnt load vctor {f} as: K rh r B D r B dr f rhr N qv dr (34) (35) By transforming th global coordinats into natural coordinats 1 1 B D r B r J d d (36) K 11 1 1 f N q r J d d (37) 11 v h lmnt matrics ar thn assmbld to yild th global stiffnss matrix and global load vctor rspctivly. otal potntial nrgy of th disk is givn by
Radial strss A.K.hawait t al. 657 whr 1 p p K F (38) N n 1 N n 1 K K Global Stiffnss matrix F f Global load vctor N = no. of lmnts. Using th Principl of Stationary otal Potntial (PSP), th total potntial is st to b stationary with rspct to small variation in th nodal dgr of frdom that is p 0 (39) From abov, th systm of simultanous quations is obtaind as follows: K F (40) 3 RESULS AND DISCUSSION 3.1 alidation A uniform thicknss rotating disk having powr law distribution of matrial proprtis (grading paramtr n = 0, 0.5 and 1) of rfrnc [7] is rconsidrd and analyzd again and rsults ar prsntd in Fig. h rsults obtaind ar in good agrmnt with th pr-stablishd rsults of rfrnc. 1.0 8.0 Grading indx = 0.0 Grading indx = 0.5 Grading indx = 1.0 4.0 Prsnt study Rf. [7] 0.0 0.40 0.60 0.80 1.00 Normalizd radius Fig. Comparison of th rsults of currnt work with rfrnc [7]. 3. Numrical rsults and discussion In this sction rotating annular disks of parabolic concav thicknss profils having fr-fr boundary condition ar analyzd. Disks ar mad of aluminum and zirconia cramic as wll as thir diffrnt FGM s. Finally th ffcts of grading paramtr n on strsss and dformation stats ar invstigatd. h matrial proprtis of aluminum and zirconia ar givn as [5]: E Al = 70 GPa, E cr = 151 GPa, ρ Al = 700 kg/m 3, ρ cr = 5700 kg/m 3, B Al = 58.3333 GPa, B cr = 18.8333 GPa, G Al = 6.931GPa, G cr = 58.0769 GPa, υ = 0.3. h disks hav gomtric paramtrs as k = 0.5, innr diamtr = 0. m, outr diamtr = 1 m, q = 0.96 and h 0 is 0.075 m. Disks ar rotating with unit angular vlocity that is 1 rad/sc. Grading indx n = 0 indicats that th disk is mad of outr matrial compltly mans th disk is homognous in composition. For cramic-mtal FGM, n = 0 indicats homognous mtallic (aluminum) disk whil for mtal-cramic FGM it indicats homognous cramic
658 Elastic Analysis of Functionally Gradd ariabl hicknss. (zirconia) disk. For valus of n othr thn 0, volum fraction varis with th radius according to diffrnt quations and givs diffrnt typs of FGMs. It can b obsrvd that cramic-mtal FGM disks hav lss radial dformation and radial strss whil highr circumfrntial and von Miss strss as compard to mtal-cramic FGM disks in xponntial FGM. Radial strss is zro at th innr and outr radius for all cass, which confirms th fr-fr boundary condition applid on th disks. In mtal-cramic FGM disks modld by Mori-anaka schm radial dformation incrass and strsss dcrass with incrasing valu of grading paramtr n whil in cramic-mtal FGM radial dformation dcrass and strsss incrass with incrasing valu of n. Incrasing n mans volum fraction of th outr matrial is dcrasing and innr matrial is incrasing. In cas of mtal-cramic FGM, incrasing mtallic contnt and dcrasing cramic contnt rsults in highr dformation and lssr strsss whil in cas of cramic-mtal FGM, incrasing cramic and dcrasing mtallic contnt rsults highr strsss and lssr dformation. Mtal-cramic FGM having n = 1.5 has highst radial dformation whil cramic-mtal FGM having n = 0.5 has th lowst radial dformation. Cramic-mtal FGM disk having n = 0 has lowst strsss and mtal-cramic FGM having n = 0 has highst strsss among all th FGMs modld by Mori-anaka schm. Fig.3 Distribution of radial displacmnts for xponntial FGM disks. Fig.4 Distribution of radial strss for xponntial FGM disks. Fig.5 Distribution of circumfrntial strsss for xponntial FGM disks.
A.K.hawait t al. 659 Fig.6 Distribution of von miss strsss for xponntial FGM disks. Fig.7 Distribution of radial displacmnts for Mori-anaka FGM disks. Fig.8 Distribution of radial strsss for Mori-anaka FGM disks. Fig.9 Distribution of circumfrntial strsss for Mori-anaka FGM disks.
660 Elastic Analysis of Functionally Gradd ariabl hicknss. Fig.10 Distribution of von miss strsss for Mori-anaka FGM disks. Fig.11 Distribution of radial displacmnts for Powr law FGM disks. Fig.1 Distribution of radial strsss for Powr law FGM disks. Fig.13 Distribution of circumfrntial strsss for Powr law FGM disks.
A.K.hawait t al. 661 Fig.14 Distribution of von miss strsss for Powr law FGM disks. In powr law FGM disk radial dformation incrass and strsss dcrass with incrasing grading paramtr n. Incrasing n mans dcrasing (r/b) ratio, which dcrass E(r) and hnc dformation incrass and strsss dcrass. FGM disk having mtal at outr surfac and n = 1.5 has maximum radial dformation and minimum radial, circumfrntial and von miss strsss whil disk having n = 0.5 and cramic at outr radius has minimum radial dformation and maximum strsss. Furthr it is also obsrvd that hoop or tangntial strss is highr as compard to radial and von miss strss for all cass. hrfor for dsigning th rotating disks, hoop strss should b takn as limit working strss critria. By comparing all typs of distribution law it is obsrvd that powr law FGM disk having mtal at outr radius and n = 1.5, has th highst radial dformation and last hoop strss whil xponntial law (Cramic-Mtal) disk has th lowst radial dformation and disk of full cramic has th highst hoop strss. hrfor it suggstd that FGM modld by powr law having mtal at outr radius and n = 1.5 can b most ffctivly mployd for rotating disk. 4 CONCLUSIONS In prsnt study strss and dformation analysis of FGM rotating disks of variabl thicknss is don. Matrial proprtis ar modld by thr diffrnt distribution law, which is achivd by lmnt basd matrial grading. h disks ar subjctd to fr-fr boundary condition and analysis is carrid out for mtal-cramic as wll as cramicmtal both th typ of FGM. h govrning quations ar drivd using principl of stationary total potntial. It is obsrvd that thr is a significant rduction in strsss and dformation bhavior of th FGM disks compard to homognous disks. Furthr it is obsrvd that mtal-cramic FGM disk having n = 1.5 and modld by powr law posssss bttr strngth than all othr FGMs invstigatd and thrfor is most conomical for th purpos of rotating disk. REFERENCES [1] Eraslan A.N., 003, Elastic plastic dformations of rotating variabl thicknss annular disks with fr, prssurizd and radially constraind boundary conditions, Intrnational Journal of Mchanical Scincs 45: 643-667. [] Bayat M., Salm M., Sahari B.B., Hamouda A.M.S., Mahdi E., 009, Mchanical and thrmal strsss in a functionally gradd rotating disk with variabl thicknss du to radially symmtry loads, Intrnational Journal of Prssur ssls and Piping 86: 357-37. [3] Afsar A.M., Go J., 010, Finit lmnt analysis of thrmolastic fild in a rotating FGM circular disk, Applid Mathmatical Modlling 34: 3309-330. [4] Callioglu H., 011, Strss analysis in a functionally gradd disc undr mchanical loads and a stady stat tmpratur distribution, Sadhana 36: 53-64. [5] Bayat M., Sahari B.B., Salm M., Dzvarh E., Mohazzab A.H., 011, Analysis of functionally gradd rotating disks with parabolic concav thicknss applying an xponntial function and th Mori-anaka schm, IOP Confrnc Sris: Matrials Scinc and Enginring 17:1-11. [6] Callioglu H., Sayr M., Dmir E., 011, Strss analysis of functionally gradd discs undr mchanical and thrmal loads, Indian Journal of Enginring & Matrial Scincs 18: 111-118. [7] Callioglu H., Bktas N.B., Sayr M., 011, Strss analysis of functionally gradd rotating discs: analytical and numrical solutions, Acta Mchanica Sinica 7: 950-955.
66 Elastic Analysis of Functionally Gradd ariabl hicknss. [8] Sharma J.N., Sharma D., Kumar S., 01, Strss and strain analysis of rotating FGM thrmolastic circular disk by using FEM, Intrnational Journal of Pur and Applid Mathmatics 74: 339-35. [9] Ali A., Bayat M., Sahari B.B., Salm M., Zaroog O.S., 01, h ffct of cramic in combinations of two sigmoid functionally gradd rotating disks with variabl thicknss, Scintific Rsarch and Essays 7: 174-188. [10] Njad A., Abdi M., Hassan M., Ghannad M., 013, Elastic analysis of xponntial FGM disks subjctd to intrnal and xtrnal prssur, Cntral Europan Journal of Enginring 3: 459-465. [11] Ghorbanpour Arani A., Loghman A., Shajari A. R., Amir S., 010, Smi-analytical solution of magnto-thrmolastic strsss for functionally gradd variabl thicknss rotating disks, Journal of Mchanical Scinc and chnology 4: 107-118. [1] Ghorbanpour Arani A., Khoddami Maraghi Z., Mozdianfard M. R., Shajari A. R.,010, hrmo-pizo-magntomchanical strsss analysis of FGPM hollow rotating thin disk, Intrnational Journal of Mchanics and Matrials in Dsign 6: 341-349. [13] Zafarmand H., Hassani B., 014, Analysis of two-dimnsional functionally gradd rotating thick disks with variabl thicknss, Acta Mchanica 5: 453-464. [14] Rosyid A., Sahb M.E., Yahia F.B., 014, Strss analysis of nonhomognous rotating disc with arbitrarily variabl thicknss using finit lmnt mthod, Rsarch Journal of Applid Scincs, Enginring and chnology 7: 3114-315. [15] Zafarmand H., Kadkhodayan M., 015, Nonlinar analysis of functionally gradd nanocomposit rotating thick disks with variabl thicknss rinforcd with carbon nanotubs, Arospac Scinc and chnology 41: 47-54, 015. [16] Sshu P., 003, A xt Book of Finit Elmnt Analysis, PHI Larning Pvt.