Finite Element Analysis of Magneto-Superelastic Behavior of Shape Memory Alloy Composite Actuator

Transcription

1 Procdings of th Intrnational MultiConfrnc of Enginrs and Computr cintists 28 Vol II IMEC 28, March, 28, Hong Kong Finit Elmnt Analysis of Magnto-uprlastic Bhavior of hap Mmory Alloy Composit Actuator Yutaka oi, JongBin and Minoru aya Abstract A mthod of coupld magnto-suprlastic analysis by th squntial approach is proposd for shap mmory alloy (MA hlical spring actuators controlld by magntic forc. h commrcial finit lmnt softwar ANY/Emag is usd for th magntic fild analysis, whil th on-dimnsional finit lmnt program dvlopd by th authors is usd for th analysis of suprlastic bhaviors of MA hlical springs. h validity of th proposd mthod is vrifid by applying th mthod to th analysis of actuator modls utilizing MA composit or frromagntic MA hlical springs and comparing th calculatd rsults with th xprimntal rsults. Indx rms Computational Mchanics, Finit Elmnt Mthod, hap Mmory Alloy, Coupld Magnto-uprlastic Analysis I. INRODUCION hap mmory alloys (abbrviatd as MAs hav bn attracting much attntion as smart matrials applicabl to various filds [1. h MA hlical springs ar oftn usd as actuator dvics in rcnt yars. low rspons is pointd out as a waknss for th actuators using thrmo-ractiv MAs. o ovrcom such a problm, th actuators using th hlical springs of MA composits controlld by magntic forc ar undr dvlopmnt. h computational prdiction of th magnto-suprlastic bhaviors is indispnsabl to th fficint and rational dsign of MA hlical spring actuator dvics. h authors proposd a mthod of finit lmnt analysis for th suprlastic, larg dformation bhaviors of MA dvics of on-dimnsional shap such as bams and hlical springs and xprimntally vrifid th validity of th proposd mthod [2-[4. Brinson s on-dimnsional constitutiv modling [5 was xtndd to tak account of torsional dformation. h incrmntal finit lmnt mthod for th larg dformation analysis with layrd linar Manuscript rcivd January 7, 28. his work was supportd in part by Grant-in-Aid for JP fllows ( and Darpa-ONR contract (N and N for th Univrsity of Washingon (aya. Y. oi is with th Institut of Industrial cinc, Univrsity of okyo, Komaba, Mguro-ku, okyo, Japan (phon: , fax: , -mail: toi@iis.u-tokyo.ac.jp. J.-B. was with th Institut of Industrial cinc, Univrsity of okyo, Komaba, Mguro-ku, okyo, Japan. M. aya is with th Cntr for Intllignt Matrials and ystms, Univrsity of Washington, attl, WA , UA (-mail: tayam@u. washington.du. imoshnko bam lmnts was formulatd by using th xtndd constitutiv quation. h proposd mthod is a gnral-purpos computing tool for th suprlastic, larg dformation analysis of on-dimnsional MA dvics, which can b xtndd to th magnto-suprlastic analysis of MA composit and frromagntic MA dvics. In th magnto-suprlastic analysis of MA hlical springs dscribd in th prsnt papr, th commrcial finit lmnt cod (ANY/Emag and th finit lmnt program dvlopd by th authors ar rspctivly usd for th magntic and th suprlastic analysis. Numrical studis ar conductd for th MA composit hlical spring actuators composd of nonfrromagntic MA (Nii wirs and frromagntic mtal (F. h validity of th proposd mthod is illustratd by comparing th calculatd rsults with th xprimntal rsults givn by th CIM (Cntr for Intllignt Matrials and ystms at th Univrsity of Washington. II. FINIE EEMEN PROCEDURE FOR UPEREAIC ANAYI OF MA HEICA PRING h normal strss-normal strain rlation for th suprlastic bhavior of MAs as shown in Fig. 1 is xprssd by th following quation [5: = E( ε ε + Ω( ξ ξ + θ ( (1 whr E ; Young s modulus, Ω ; cofficint, ξ ; strss-inducd martnsit volum fraction, θ ; thrmal lastic cofficint, ; tmpratur. h subscript indicats th initial valus. Ω is xprssd as Ω = ε E (2 whr ε is th maximum rsidual strain. Young s modulus E is a function of th martnsit volum fraction ξ, which is givn by E = Ea + ξ ( Em Ea (3 whr E m and E a ar Young s modulus of austnit phas and martnsit phas rspctivly. h total martnsit volum fraction ξ is xprssd as ξ = ξ + ξ (4 whr ξ is th tmpratur-inducd martnsit volum fraction. ξ, ξ and ξ ar functions of th tmpratur and th strss. o considr th diffrnc btwn tnsil IBN: IMEC 28

2 Procdings of th Intrnational MultiConfrnc of Enginrs and Computr cintists 28 Vol II IMEC 28, March, 28, Hong Kong and comprssiv bhavior, von Miss quivalnt strss in th volution quations of ξ, ξ and ξ is rplacd with Druckr-Pragr quivalnt strss [6 dfind as = + 3βp (5 whr β is th matrial paramtr and p is th hydrostatic prssur givn by 1 p = + 3 ( + x y z In on-dimnsional cas, th quivalnt strss in q. (5 is xprssd as = + β (7 h torsional bhavior of MAs, that is th shar strss-shar strain bhavior du to torsion, is assumd to b qualitativly similar to, but indpndnt of th normal strss-normal strain bhavior. Accordingly, th rlation btwn th shar strss τ and th shar strain γ is xprssd by th following quation: τ τ = G( γ γ + Ωτ ( ξ sτ ξ sτ (8 h volution quation for th martnsit volum fraction ξ s τ du to torsion is assumd to b similar to that du to normal dformation. h quivalnt strss dscribing th volution quation is 3 τ θz. Dtails of th abov-mntiond constitutiv quation modl ar givn by oi t al. [2-[4. Although xactly, th tnsil and th torsional bhavior ar complicatdly coupld [7, [8, th indpndncy as mntiond abov is assumd sinc th torsional dformation is dominant in th hlical springs to b analyzd in th prsnt study. Fig. 1 uprlastic strss-strain bhavior of MA h followings ar assumd in th finit lmnt formulation using linar imoshnko bam lmnts. h axial displacmnt and th torsional angl as wll as th latral dflctions in two dirctions and th rotational angl of a cross-sction ar linarly intrpolatd in th lmnt. (6 h shar strain nrgy trm associatd with bnding is tratd as a pnalty function sinc th influnc of bnding in hlical springs is much smallr than torsion. h tangnt stiffnss mthod considring two typs of nonlinaritis du to suprlasticity and finit dformation is formulatd according to th incrmntal thory by th total agrangian approach [9, in which th ffct of nonlinar trms with rspct to th axial displacmnt of a bam is nglctd [1. Dtails of th finit lmnt formulation dscribd abov ar givn by oi t al. [2-[4. h incrmntal strss-strain rlation for th analysis of hlical springs is writtn in th following form: { Δ = [ Ds ({ Δε { Δε s (9 whr Δ E { Δ { Δε Δτ G =, [ D s =, Δτ yz G Δτ Gs (1 Δε Δε s Δγ =, { Δε s = Δγ yz Δγ Δγ s γ and γ yz ar th shar strsss in which τ and τ yz ( (strains du to bnding. h final form of q. (1 is givn in [2-[4. h lmnt stiffnss quation in an incrmntal form is xprssd as follows [2-[4: k + k + k Δu ([ [ [ { = (11 whr (12 k = { Δf + { f + [ B [ D{ Δε R G V [ k = [ B [ D[ B V s ( s [ [ B [ D[ B + [ B [ D[ B [ B [ D[ B ( + V ( ( (13 ( [ kg = [ G [ [ G (14 V h following symbols ar usd: [ k ; th incrmntal stiffnss matrix, [ k ; th initial displacmnt matrix, [ k G ; th initial strss matrix, { Δ f vctor, { f R ; th unbalancd forc vctor, [ D s suprlastic strss-strain matrix, { Δ ε s initial strain vctor, [ G ; th gradint matrix, [ strss matrix, V ; th lmnt volum. ; th xtrnal forc incrmnt ; th ; th suprlastic ; th initial IBN: IMEC 28

3 Procdings of th Intrnational MultiConfrnc of Enginrs and Computr cintists 28 Vol II IMEC 28, March, 28, Hong Kong III. MAGNEO-UPEREAIC ANAYI PROCEDURE OF MA COMPOIE HEICA PRING ACUAOR h coupld magnto-suprlastic analysis of MA composit hlical spring actuators is conductd by th squntial procdur (th staggard mthod or th wakly-coupld mthod in which th magntic and th suprlastic analysis ar conductd altrnatly. h commrcial cod [11 (ANY/Emag and th suprlastic, larg dformation finit lmnt analysis program of MA hlical springs dvlopd by th authors ar usd for th magntic analysis and th suprlastic analysis rspctivly. h squntial analysis procdur is shown in Fig. 2. h coupld analysis starts with th input of th shap information of th hlical spring into th magntic analysis cod from th suprlastic analysis program. h magntic analysis cod calculats and outputs th magntic forc incrmnts acting on th hlical spring, basd on th shap information input. h magntic forc output is input into th suprlastic program, which calculats th incrmntal displacmnts, strains and strsss. Basd on th calculatd displacmnts, th shap information of th hlical spring is modifid and output, which is again input into th magntic analysis cod. h suprlastic, larg dformation procss of th MA hlical spring is simulatd by conducting th abov-mntiond procdur rpatdly. ANY/Emag, which is a commrcial softwar frquntly usd for multi-physics analysis, is mployd as th magntic analysis cod [11. h scalar potntial mthod is mployd for th magntic analysis bcaus of rlativly asy modling and small load on computrs. Eight-nod hxagonal solid lmnts and four-nod ttrahdral lmnts (OID96 ar rspctivly usd for th magntic body such as a hlical spring and th spac. ix-nod triangular column lmnts (INFIN111 ar usd as infinit-boundary lmnts. h modling of an lctric sourc utilizs OURC36. h (diffrnc scalar potntial mthod is usd for th computation. h shap information of th hlical spring calculatd by th structural analysis program is input into th magntic analysis cod as th solid modling information (ky points and volum. h magntic forc incrmnts at ach nod of th hlical spring ar output as th calculatd rsults. h of th input and output data btwn th magntic analysis cod and th suprlastic analysis program is conductd in th intrfac program. h of nods and nodal forcs is ncssary as th suprlastic analysis program and th magntic analysis cod us two-nod straight bam lmnts and ight-nod hxagonal solid lmnts, rspctivly. h control program runs th magntic analysis cod, th suprlastic analysis program and th intrfac program squntially, managing th calculatd valus of loads, displacmnts, strains and strsss. h total modling and analysis ar don by batch procssing. Fig. 2 Coupld magnto-suprlastic analysis IV. REU OF MAGNEO-UPEREAIC ANAYI OF MA COMPOIE HEICA PRING ACUAOR h magnto-suprlastic, larg dformation bhavior of th MA composit hlical spring actuator is analyzd. h MA composit hlical spring actuator is composd of a pair of driving units, a MA composit hlical spring and two xtra MA hlical springs as shown in Fig. 3. h composit hlical spring longatd downward by th wight of 2.5N is comprssd upward by th magntic forc from th driving units. Fig. 4 is a sktch of th driving unit and th composit hlical spring. h driving unit is composd of a hybrid rluctanc magnt [12 combining a prmannt magnt with an lctromagnt as shown in Fig. 4(a. h prmannt magnt is nodymium 35 with th spcific magntic prmability of 1.17 and th rtntivity of 835,563A/m. h lctric currnt of ~6A flows through th lctromagnt coil with 264 turns. h composit hlical spring is composd of two MA (Nii wirs and 16 frromagntic (F blocks. h magntic prmability μ ( B = μh, B ; magntic dnsity, H ; magntic intnsity of F is shown in Fig. 5. h magntic control of th nonfrromagntic MA hlical spring bcoms possibl by such composition. h matrial constants of Nii ar shown in abl 1. h xtra hlical spring with th outr radius of 1mm and th total lngth of 14mm is composd of a Nii wir with th diamtr of 1mm. h xtra hlical springs ar nglctd in th magntic analysis as th magntic prmability of Nii is small. Fig. 3 MA composit spring actuator IBN: IMEC 28

4 Procdings of th Intrnational MultiConfrnc of Enginrs and Computr cintists 28 Vol II IMEC 28, March, 28, Hong Kong (a driving unit Fig. 6 shows th calculatd and xprimntal rsults for th rlation btwn th distanc from th uppr to th lowr driving unit and th lctric currnt. h composit hlical spring longats to 28mm by th initial wight loading and is fully comprssd by th lctric currnt of 6A. h calculatd rsults ar in good agrmnt with th xprimntal rsults givn by th CIM of th Univrsity of Washington. Fig. 7 shows th dformation profils of th composit hlical spring with th variation of lctric currnt. Fig. 8 shows th shar strss-shar strain curv of th Nii wir in th composit hlical spring, which xhibits linar, lastic bhavior. Fig. 9 is th dformation procss of th xtra hlical spring. It is sn from th shar strss-shar strain curv in Fig. 1 that th martnsit taks plac in th spring during th tnsil procss undr th initial loading, xhibiting th suprlastic bhavior. (b MA composit spring Fig. 4 Driving unit and MA composit spring B ( H (A/m Fig. 5 B-H curv for F Fig. 6 Currnt-displacmnt curv for MA composit spring actuator abl 1 Matrial constants for Nii Young s modulus of E = 7 [MPa austnit phas a Young s modulus of martnsit phas E m = 33 [MPa starting strss for martnsit Ms = 427.8[MPa finishing strss for martnsit Mf = [MPa starting strss for austnit As = 21.5[MPa finishing strss for austnit Af = 11.4 [MPa rsidual normal strain ε =. 47 rsidual shar strain γ =. 47 matrial paramtr β =. 15 A 1A 2A 3A 4A 6A Fig. 7 Dformation of MA composit spring actuator IBN: IMEC 28

5 Procdings of th Intrnational MultiConfrnc of Enginrs and Computr cintists 28 Vol II IMEC 28, March, 28, Hong Kong th magntic analysis is conductd at vry 1 stps. Fig. 8 har strss-shar strain curv for Nii wir in composit spring Fig. 9 Dformation of xtra spring Fig. 1 har strss-shar strain curv for xtra spring h numbr of finit lmnts for th magntic analysis is 91,864 to 217,53 (3,242~51,36 nods, whil 64 (65 nods for th suprlastic analysis. h total numbr of incrmntal stps for th suprlastic analysis is 7, whil V. CONCUION In th prsnt study, th finit lmnt procdur for th suprlastic, larg dformation bhavior of MA dvics of on-dimnsional shap prviously proposd by th authors has bn xtndd to th magnto-suprlastic analysis of MA composit hlical spring actuators. h magnto-suprlastic analysis systm has bn dvlopd for th MA composit hlical spring actuators, which mploys th squntial mthod combining th magntic analysis by th commrcial cod ANY/Emag with th suprlastic, larg dformation analysis of MA dvics by th finit lmnt program dvlopd by th authors. h magntic analysis of MA composit actuators is basd on th xact shap modling of th hlical springs. h validity of th proposd analysis systm as a simulation tool for th dsign of MA actuators has bn illustratd by comparing th calculatd rsults with th xprimntal rsults givn by th CIM of th Univrsity of Washington. REFERENCE [1 H. Hosoda, and. Miyazaki, Rcnt opics of hap Mmory Matrials and Rlatd chnology, Journal of th Japan ocity of Mchanical Enginrs, Vol. 17, No. 128, 24, pp [2 Y. oi, J.-B. and M. aya, Finit Elmnt Analysis of uprlastic Bhavior of hap mmory alloy Dvics (1st Rport, mall Dformation Analysis of nsil and Bnding Bhaviors, ransaction of th Japan ocity of Mchanical Enginrs, ris A, Vol. 68, No. 676, 22, pp [3 Y. oi, J.-B. and M. aya, Finit Elmnt Analysis of uprlastic Bhavior of hap Mmory alloy Dvics (2nd Finit Dformation Analysis of Bams and Hlical prings, ransaction of th Japan icity of Mchanical Enginrs, ris A, Vol. 68, No. 676, 22, pp [4 Y. oi, J.^B. and M. aya, Finit Elmnt Analysis of uprlastic, arg Dformation Bhavior of hap Mmory Alloy Hlical prings, Computrs and tructurs, Vol. 82, No. 2/21, 24, pp [5. C. Brinson, On Dimnsional Constitutiv Bhavior of hap Mmory Alloys: hrmomchanical Drivation with Non-Constant Matrial Functions and Rdfind Martnsit Intrnal Variabl, Journal of Intllignt Matrial ystms and tructurs, Vol. 4, 1993, pp [6 F. Auricchio and R.. aylor, hap Mmory Alloy: Modling and Numrical imulations of th Finit-train uprlastic Bhavior, Computr Mthods in Applid Mchanics and Enginring, Vol. 143, 1997, pp [7 M. okuda and M. ittnr, Multi-Axial Constitutiv Equations of Polycrystallin hap Mmory Alloy (1st Rport, Modling and Formulation, ransactions of th Japan ocity of Mchanical Enginrs, ris A, Vol. 65, No. 631, 1999, pp [8 Q. P. un and Z. Q. i Phas ransformation in uprlastic Nii Polycrystallin Micro-ubs undr nsion and orsion: from ocalization to Homognous Dformation, Intrnational Journal of olids and tructurs, Vol. 39, 22, pp [9 h Japan ocity of Mchanical Enginrs d., JME Computational Mchanics Handbook (Vol. 1, Finit Elmnt Mthod: tructur, h Japan ocity of Mchanical Enginrs, 1998, pp [1. Kawai, Analysis of Buckling Problm, Baifukan, 1974, pp [11 ANY, Inc., ANY minar Not: Emag, Cybrnt ystms Co. td., 21. [12 Y. iang t al., Dsign of Diaphragm Actuator Basd on Frromagntic hap Mmory Alloy Composits, Procdings of PIE, Vol. 554, 23, pp IBN: IMEC 28