DEFORMATION OF ACTIVE FLEXIBLE RODS WITH MEMORY SHAPE ALLOY FIBERS

Transcription

1 DEFORMTION OF CTIVE FLEXIBLE RODS WITH MEMORY SHPE LLOY FIBERS Sefni DONESCU Technicl Universiy of Civil Engineering, Dep. of Mhemics, Bdul Lcul Tei 4, Buchres 7 In his pper we consider hin rod of luminum wih embedding of shpe memory lloy (SM) fibers. The SM fibers chnge he shpe of he medium, whenever shpe recovery of he SM fibers ke plce. The moion equions re developed wih predicion of he shpe of he rod when he SM fibers undergo he mrensiic-useniic phse rnsformion. We discuss he shpe chnge of cylindricl rod wih single embedded SM fiber. Keywords: shpe memory lloys, mrensiic-useniic phses, deformed shpes.. INTRODUCTION The shpe chnge of cylindricl rod wih single off-xis embedded SM fiber hs been modeled by Lgouds nd Tdjbkhsh []. These uhors deermine he disribued xil compressive force nd bending momen due o he phse rnsformion in he SM, by using n pproxime sher-lg model. The deformed shpe of he rod ws found by numericlly solving of he equions of equilibrium for he rod. The SM is seleced o be Nickel-Tinium (NiTi) becuse of he high srengh nd lrge srins ssocied wih his meril. The cuion of he rod is chieved by pplying curren cross he SM fibers in order o he he rod bove he rnsiion emperure. In his pper we sudy he behvior of hin composie rod wih embedding of NiTi fibers. This rod is deforming in spce by bending nd orsion. Consider homogeneous isoropic hermlly conducing cylindricl elsic hin rod uniform bsolue emperure T in he undisurbed se, wih lengh L nd rdius R. NiTi fiber of lengh L, rdius r is embedded ino luminum mrix long he cenroidl xis (fig. ). The fundmenl equions for such medium re wrien in he spiri of Nowinski [], Boyd nd Lgouds [], Shu, Lgouds, Hughes nd Wen [4], Rogers, Ling nd Brker [5], Lgouds nd Tdjbkhsh [6], Dickey nd Rosemn [7]. We ke s o be he coordine long he cenrl line of he nurl se. The orhonorml bsis of he lgrngin coordine sysem is denoed by ( e, e, e ), nd he orhonorml bsis of he eulerin coordine sysem by ( d, d, d ). The bsis { dk}, k =,, is reled o { ek }, k =,, by he Euler ngles θ, ψ nd ϕ. These ngles deermine he orienion of he eulerin xes relive o he lgrngin xes (Munenu, Donescu [9]) d = ( sin ψsin ϕ+ cosψcosϕcos θ ) e + + (cosψsin ϕ+ sin ψcos ϕcos θ) e sin θcos ϕe, d = ( sin ψcosϕ cosψsin ϕcos θ ) e + + (cosψcosϕ sin ψsin ϕcos θ ) e + sin θsin ϕe, (.) d = sin θcosψ e + sin θsin ψ e + cos θe.

2 Sefni DONESCU The Z-xis coincides wih he cenrl xis. The plne (xy) inersecs he plne (XY) in he nodl line ON (fig. ). The moion of he rod is described by hree vecor funcions R R (,) s r(,), s d(,), s d(,) s E. (.) The meril secions of he rod re idenified by he coordine s. The posiion vecor r( s, ) ([r] = m) cn be inerpreed s he imge of he cenrl xis in he eulerin configurion. The funcions d(,), s d(,) s cn be inerpreed s defining he orienion of he meril secion s in eulerin configurion. The funcion d (,) s = d (,) s d (,) s, (.) represens he uni ngenil vecor long he rod nd cn be expressed s d (sin θ cos ψ,sin θ sin ψ,cos θ ) Fig. The Euler ngles These funcions ( u, u, u) mesure he bending nd orsion of he rod u =θ sin ϕ ψ sin θcosϕ u =θ cosϕ + ψ sin θsin ϕ (.4) u = ϕ + ψ cosθ. The funcions u nd u represen he componens of he curvure of he cenrl line denoed by κ corresponding o he plnes ( yz) nd ( xz) nd u is he orsion of he br denoed by τ κ = + =θ +ψ θ, (.5) u u sin u =τ=ϕ + ψ cosθ. (.6) The link beween he posiion vecor r = ( xyz,, ) nd uni ngenil vecor d is r = d The full se of srins of he rod is { yk, u k}. In he nurl se d coincides wih r, nd dk re consn funcions of s. The vlues of he srins in he nurl se re y = y =, y =, u k =. (.7) To chrcerize he posiion of ends of he rod we inroduce he vecor D of he componens

3 Deformion of cive flexible rods wih memory shpe lloy fibers (x(l), y(l), z(l)) l D = d ds. (.8). BSIC EQUTIONS We use he summion convecion hroughou. superposed do denoes differeniion wih respec o ime while comm is used for meril derivives. The fundmenl equions re given in he following [.,,4 ] The consiuive lw for n isoropic rod is σ =λεkkδ + µε βδ ( T T ), i, j, k =,, (.) where σ re componens of he sress ensor, ε componens of he srin ensor, T bsolue emperure, T iniil emperure, λ, µ elsic Lmé s consns, β = (λ+ µ ) α, wih α he coefficien of liner herml expnsions.the consiuive lw for he NiTi fiber is σ =λ ( εkk εkk ) δ + µ ( ε ε ) βδ ( T T ), i, j, k =,, (.) where ε re he rnsformion srin, nd β = (λ + µ ) α. The Lmé moduli λ, µ nd he coefficien of liner herml expnsions α obey he rule of mixures M M M λ =λ +ξ( λ λ ), µ = µ + ξµ ( µ ), α =α +ξ( α α ), (.) where he superscrips is for usenie nd M for mrensie, respecively, nd ξ is he curren volume frcion of he mrensiic phse. The moion equions re ji, j vi σ =ρ!. (.4) The rnsformion srin re evoluion lw is wih he rnsformion ensor Λ given by Fig. The geomery of he bem ε! =Λ ξ!, (.5) σ H σ Λ = ε H ε ξ! >, ξ! <, (.6)

4 Sefni DONESCU which provides he direcions in which he rnsformion srins develop. Here, H =ε mx, nd: σ= σ σ, σ =σ σkkδ, ε = εε. (.7) The hermodynmic force Φ h conrols he onse of he phse rnsformion is 4 f ( ξ) Φ=σΛ + σσ + ασδ ( T T ) +ρ T Y =. ξ In he bove, ρ is he densiy of he NiTi fiber, = M E E, α = α M α 4, ρ he difference of he enropy he reference se, Y he hreshold vlue of rnsformion, f( ξ ) = ρ bξ provides he isoropic hrdening erm chrcerized by he isoropic hrdening prmeer b. The bove crierion is vlid for boh reverse nd forwrd rnsformion bu wih differen vlues of he prmeers 4 ρ, Y, b which ccouns for he hyseresis of shpe memory lloys. We hve In he bove, (.8) 4 M ρ = C H, M Y = C HMs, ρ b = C H( f s), during cooling, (.9) 4 M ρ = C H, Y = C H f, ρ b = C H( Ms Mf ), during heing. (.) M C nd C re he slopes of he curves of he sress versus emperure, M, M, nd s f s f re he sr nd finish emperures zero sress. The he equions for he rod nd he NiTi fiber re ρ ct (! +τ T!! ) + Tβ( ε! +τ!! ε ) = kt, (.) ii ii, ii C T! T!! T!!! k T J (.) v ( +τ ) + β ( ε ii +τ ε ii ) =, ii +ρe, where Cv = ρ c is he he cpciy, k he herml conduciviy, ρ e he elecricl resisiviy nd J he mgniude of he curren densiy. The equions of moion in displcemens re obined by subsiuing (.) or (.) in (.4). So, we obin for he rod nd for he NiTi fiber ρ u!! = ( λ+µ ) u + µ u βt, (.) i j, i, jj, i ρ u!! = ( λ +µ ) u + µ u β T, (.4) i j, i, jj, i where u is he rnsformion displcemen. The boundry condiions for he rod re σ =σ x= L nd [, ]. (.5) nd he condiions on he inerfces beween he rod nd NiTi fiber re [ u ] =σ [ ] =σ [ ] =, [, ]. (.6). SIMULTION OF THE ROD RESPONSE The deformed shpes of he rod under cyclic herml cuion nd he resuling shpe memory loss due o he developmen of NiTi sresses hve been numericlly evlued. I is considered he cse of D bending problem of rod wih one fixed end z =, nd he oher free z = L. Boh he rod nd he cuor re ssumed o be iniilly srigh = nd T = C. Some deformed shpes ( zx, ) of rod under cyclic cuion of he embedded NiTi fiber re represened in fig., differen momen of imes. The

5 Deformion of cive flexible rods wih memory shpe lloy fibers deformed shpes ( zyof, ) he rod under cyclic cuion of he embedded NiTi fiber, for he sme momens of ime re represened in fig. 4. ll hese deformed shpes reurn o he iniil configurion, due o he fc h he NiTi fiber is heed bove he useniic sr emperure by pssing n elecricl curren, nd he defleced bem ends o reurn o he iniil configurion. The NiTi fiber cs s n cuor rnsforming elecricl energy ino mechnicl energy, nnihiling he deformed shpe of he rod. The cued bem exhibis he essenil funcions of n cive sysem, i. e. he defleced bem my reurn o he iniil configurion. The meril prmeers nd geomeric consns of luminum rod used for compuion re: lengh L = m, rdius R =.5 m, densiy ρ =.7 Kg/m, he Lmé moduli, λ =5. GP, µ =8. GP, coefficien of liner herml expnsions. α =. 5 / C. Fig. Deformed shpes (, zx ) of he rod under cyclic cuion of he embedded NiTi fiber Fig. 4 Deformed shpes (, zy) of he rod under cyclic cuion of he embedded NiTi fiber

6 Sefni DONESCU 4 The meril prmeers of NiTi fiber (fer Dimn, Whie nd Bergmn [4]) used for compuion re: lengh L = m, rdius r =.5, densiy ρ =6.45 Kg/m, he Lmé moduli λ =8.6 M M GP, λ =6. GP, µ =8.85 GP, µ =4.9 GP, coefficien of liner herml expnsions α = α M =.5 6 / C, herml conduciviy k =8 W/m M C, k = 8.5 W/m C, slope of sress versus emperure C = C M =.5 MP/ C, rnsformion emperures s =57 C, of =5 C, M s = C, M f =- C, he cpciy C v = Jm C, elecricl resisiviy ρ e =.5 6 Ω m. Here he subscrips s nd f represen sr nd finish emperure wih snding for sress free se. REFERENCES. LGOUDS, D.C., TDJBKHSH, I.G., cive flexible rods wih embedded SM fibers, Smr Mer. Sruc.,, pp.6-67, 99.. NOWINSKI, J. L., Theory of hermo-elsiciy wih pplicions, Suhoff & Noordhoff Inernionl Publishers lphen n Den Rn, BOYD, J.G., Lgouds, D.C., Thermo- mechnicl response of shpe memory composies J. Inell. Mer. Sruc. 5, pp. 46, SHU, S. G, LGOUDS, D. C., HUGHES D., WEN, J. T., Modeling of flexible bem cued by shpe memory lloy wires, Smr Mer. Sruc. 6, pp.65 77, ROGERS, C., SCHIEF, W. K., The clssicl Bcklund rnsformion nd inegrble discreision of chrcerisic equions, Physics Leers, pp.7-, ROGERS C., LING C., BRKER D., Behvior of shpe memory lloy reinforced composie ples (pr I nd II), Proc. h Srucures, Srucurl Dynmics nd Merils Conf. Mobile, lbm, LGOUDS D.C., TDJBKHSH I.G., Deformions of cive flexible rods wih embedded line cuors, Smr Mer. Sruc., pp. 7 8, DICKEY R.W., ROSEMN J.J., Equilibri of he hin elsic rod under uniform cenrl force field, Qurerly of pplied mhemics, LI, nr., june MUNTENU, L., DONESCU, ST., Inroducere Τn eori solionilor. plic ii Τn mecnic, Ediur cdemiei,.. LNDU, L. D., LIFSHITZ, E. M., Theory of Elsiciy, Moscow, LOVE,. E. H., reise on he mhemicl heory of elsiciy, 4 h ed., Dover, New York, 96.. CHIROIU, V., MUNTENU, L., flexible bem cued by shpe memory lloy ribbon, Proc. of he Romnin cdemy, Series : Mhemics, Physics, Technicl Sciences, Informion Science, 4,, pp.-7,.. BROCC, M., BRINSON, L. C., BZZNT, Z. P., Three- dimensionl consiuive model for shpe memory lloys bsed on microplne model, Journl of he Mechnics nd Physics of Solids, 5, pp.5 77,. 4. DITMN J, WHITE S.R., BERGMN L., Tensile esing of niinol wire, Technicl Repor UILU ENG 9-58, Universiy of Illinois, Urbn IL, 99.