THERMOMECHANICAL MODELLING THE RESISTANCE WELDING OF PbSb ALLOY

Transcription

1 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of... THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp THERMOMECHANICAL MODELLING THE RESISTANCE WELDING OF PbSb ALLOY by Dragan V. KALABA a, Alesandar S. SEDMAK b, Zoran J. RADAKOVI] b*, and Maro V. MILOŠ c a Faculty of Mechancal Engneerng at Kosovsa Mtrovca, Unversty of Pr{tna, Serba b Faculty of Mechancal Engneerng, Unversty of Belgrade, Belgrade, Serba c Faculty of Mechancal Engneerng Innovaton Centre, Unversty of Belgrade, Belgrade, Serba Org nal sc en tfc pa per UDC: /.25:517.96: DOI: /TSCI K The an a lyt cal mod el lng of the PbSb al loy re ss tance spot weld ng pro cess has been de vel oped on the ba ss of math e mat cal anal y ss of thermo mechancal con ser - va ton laws. The nu mer cal so lu ton of par tal df fer en tal equa tons, ob taned by such mod el lng, has been acheved by the f nte el e ment method. Thermomechancal equ lb rum equa tons are de rved, n clud ng spe cfc prop er - tes, typ cal for PbSb al loys. The pa per ut lzes the ba sc ex per men tally proven as - sump ton, that the tem per a ture felds gov ern all pro cesses dur ng weld ng. Full agree ment s ev dent be tween the ex per men tal and an a lyt cal data. Key words: mathematcal modellng, resstance spot weldng, PbSb alloy, fnte elements, temperature feld Introducton The com plex ty of the re ss tance spot weld ng pro cess s rep re sented by var ous pa - ram e ters needed to de scrbe the un der go ng phys cal n flu ences (e. g. elec trc cur rent, elec tr cal re ss tance, con tact force, ma te ral type, tem per a ture, etc.). By den t fy ng the tem per a ture feld t be comes pos s ble to sm u late heat ng, de for ma ton and cool ng pro cesses, and to en sure some ba sc as sump tons for pre dct ng struc tural changes at lo ca tons of the ont and ts f nal me chan - cal prop er tes. Ef forts for de vel op ng mod ern sm u la tons of con cern here have four or en ta tons: mplementng nfluences of latent heat and phase transformatons n the numercal analyss [1-3], development of mathematcal formulatons of thermomechancal processes n resstance spot weldng and ther correlaton to heat and electrcal processes [4, 5], * Correspondng author; e-mal: zradaovc@mas.bg.ac.rs

2 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp quantfcaton of the contact surface regardless of whether ths surface s an electrode- -to-sheet metal or sheet-to-sheet mcrocontact, as well as changes n geometry of the conductve meda [4, 6], and mplementng all partculartes of the resstance spot weldng of new materals and alloys nto well establshed mathematcal models. Re search on the prob lems of re ss tance weld ng of heavy al loys s rel a tvely scarce and n sg nf cant. Asde to the o ret cal re search, of prac t cal m por tance. e. s the com pl cated as - sem bly of lead-acd bat ter es. The re ss tance weld ng op er a ton for the PbSb al loy grd cell on - ng s cru cal, and the cre ated re sd ual stresses are the most fre quent cause of fal ure n these welded onts. The math e mat cal anal y ss of the heat pro cess m pl cates math e mat cal mod el lng (f - nte el e ments) and de fn ng the heat con duc ton equa ton. The pro cess s fur ther math e mat cally ana lysed and re sd ual stresses are eval u ated. Pro ce dures are car red out for nu mer cal n te gra - ton of df fer en tal equa tons of heat con duc ton and re sd ual stresses, wth er ror es t ma ton de - pend ng on the adopted f nte el e ment mesh. Model formulaton Heat conducton equaton The weld ng pro cess s rep re sented by a thermomechancal model n the scope of ba sc laws of con tn uum me chan cs. The global en ergy bal ance law n the cou pled form [7-9] s r V0 [( r u) T n ] dv h d A (1) 0, 0 0 A 0 where r 0 s the mass den sty, u the stran en ergy den sty, r the vol ume heat source, T the stress ten sor, v the ve loc ty vec tor, h 0 the sur face heat flux, A 0 the con tour area, and V 0 the vol ume en com passed by A 0. The sub scrpt 0 de notes undeformed (n tal) con fg u ra ton,. s the tme de rv a tve, and, the par tal de rv a tve over Des cartes co-or d nates x. Cou plng of ther mal and me chan cal terms n eq. (1) s due to the fol low ng: second and fourth terms contan only thermal energy, the thrd contans only mechancal energy, and the frst contans both, stran energy densty depends on both temperature and stran, U = u(t, e ), where t s the temperature and e the deformaton tensor, and thermal boundary condtons, vald for the deformed confguraton, may be appled only provded that the dsplacements and strans are nown. Un cou plng of ther mal and me chan cal terms n eq. (1) s pos s ble ac cord ng to the fol - low ng. In order to neglect dmensonal changes due to transformaton from an undeformed to a deformed confguraton, t s necessary to assume that da/da 0 1 and dv/dv 0 1, so that dh dh 0 and dr 0 dr,. e. r s temperature ndependent. Typcal densty values at both room and lqudus temperatures for PbSb alloys that are appled n weldng dffer up to 6%. Ths error has lttle effect on the numercal analyss and determnaton of resdual stresses, snce they appear at relatvely low temperatures. The error s to be accounted for when analysng the temperature or heat dstrbuton.

3 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of... THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp Deformaton dependence of stran energy densty rate opposed to the temperature dependence may be omtted accordng to the followng expresson: u u T T u e e u T T The frst term s proportonal to mass densty r and heat capacty c, and the second term to yeld strength R p0.2 and the thermal coeffcent a. The change n both thermal and mechancal energy for the PbSb alloy s gven n tab. 1. Apparently, the error becomes less than 1% when comparng the change n thermal and mechancal energy, as Du t = rcdt [Jcm 3 ] (Dt = 1 K) and Du m = 3aR p0.2. The comparson s made at room temperature, snce the change n mechancal energy s even smaller at hgher temperatures (R p decreases consderably). Ta ble 1. En ergy com par son for the PbSb al loy (2) Densty, r [gcm 3 ] Heat ca pac ty, c [Jg 1 K 1 ] Du t = rc [Jcm 3 ] Co ef f cent of ther mal ex pan son, a [K 1 ] Yeld stress, R p0.2 [Nmm 2 ] Du m = 3aR p0.2 [Jcm 3 ] By usng smlar arguments, the pure mechancal energy term T V n dv 0 0 n eq. (1) can also be omtted. These ap prox ma tons al low for eq. (1) to be wrt ten n the un cou pled form as: r 0 ( r u) V h 0 A 0 V0 A0 d d (3) By ap ply ng the Gauss the o rem, eq. (3) s trans formed to eq. (4), and ta ng nto ac - count that at the bound ary sur face A 0, the re la ton for the n tro duced heat con duc ton vec tor, q, de fned by heat flux h and ex ter nal unt nor mal n, s gven by q n = h 0, and for Des cartes co-or - d nates, x : q r 0 ( r u) 0 (4) If r 0 r s re placed by Q v, and r 0 u s re placed as r 0 ( u/ T) T r ct 0, where c s spe cfc heat ca pac ty, and we shall also omt n dex 0, thus g nor ng the df fer ence be tween the cur rent and ref er ence con fg u ra ton, eq. (4) be comes: q Qv rct 0 (5) To solve ths equa ton we n tro duce the con st tu tve ma te ral law,. e. the Fou rer law of heat con duc ton: q T (6) where s the ten sor co ef f cent of ther mal con duc tv ty. F nally, the heat trans fer equa ton s: T Q ct v r (7)

4 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp Be sdes be ng non-sta ton ary, eq. (7) s non-ln ear be cause of the tem per a ture de pend - ence of ma te ral prop er tes. The n tal con d ton s gven as T(x,0) = T 0 (x ), and the bound ary con d tons for tem per a ture, heat flux, heat con vec ton (fg. 1) are: T T ( x ) ( on S ) q n s q s q n h( T T ) ( on S ) s 1 ( on S ) where T S and T are the gven source and en v ron men tal tem per a tures, n re spect; h s the con - vec ton heat-trans fer co ef f cent, and q S the gven heat flux. Such a prob lem can not be solved an a lyt cally n gen eral, and re qures nu mer cal meth - ods. Ths be comes more com pl cated when con sd er ng spe cfc prob lems re gard ng the weld - ng pro cess, such as phase and struc tural change, weld ng heat n put, weldment crystallsaton, and lo ca ton of lqudus and lqudus sur faces. Equa ton (7) wll be solved by ap ply ng the Galern method. The whole pro ce dure s de scrbed n ref. [10]. Vol ume V s d vded nto E f nte el e ments, each hav ng p nodes. As sume the change n T for each el e ment e as: T e (x, t) = N (x )T (t) (9) where T e (x, t) s the tem per a ture ds tr bu ton n sde el e ment e, N (x ) the n ter po la ton func - ton, and T the nodal tem per a ture. The par tal df fer en tals are: T e 2 3 N T T N T N T e, (10) t where N s the ma trx of the or der n r (n space d men son, r num ber of el e ments). By mul t - ply ng eq. (7) by weght func ton cho sen to be den t cal to N, and n te grat ng over each f nte el - e ment,. e. by ap ply ng the Galern pro ce dure, we ob tan: T Q N V p V rc T d 0 ( 12,,... ) (11) V e By ap ply ng the Gauss the o rem, the frst term may be wrt ten as: T N T V N V T d d n N d S (12) V e V e e e S e e The sur face n te gral over S e s d vded nto S, S, and S, ac cord ng to bound ary con d tons (fg. 1). The sur face n te gral n eq. (12) over S1 e equals 0. Bound ary con d tons over sur faces S e and S e are gven n eq. (8), and the sur face n te gral n eq. (12) sat s fes these bound ary con d tons, so we wrte: T x n N S q N S S h TS T N S d d 2 ( ) d 3 (13) S e S e 2 3 S e 2 The dscretzed heat con duc ton equa ton s fur ther wrt ten n ma trx form: C T ( K K ) T R 0 (14) c h S e 3 R R R R (15) Q q h (8)

5 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of... THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp where C s the heat ca pac ty ma trx, K c and K h are the ma tr ces of heat con duc ton and con - vec ton trans fer, n re spect, and R, R Q, R q, and R h are vec tors of the heat load, due to vol ume heat source, heat flux, and con vec ton, re spec - tvely. Equa ton (14) can be solved by ex plct or m plct pro ce dure [11], so we can de ter mne tem per a ture val ues among nodes, or the tem per - a ture ds tr bu ton. Phase change s a char ac ter s tc of the weld - ng pro cess ac com pa ned by the re lease of la - tent heat. Ths pro cess de mands for spe cfc Fgure 1. Doman and boundary condtons mod f ca tons to the nu mer cal pro ce dure for solv ng the heat con duc ton equa ton. Ths s usu ally sm u lated by an ad d tonal heat source n a part of the do man be tween sold and lq ud, and so lu ton ac cu racy s strongly af fected by the lo - ca ton of these do mans. How ever, t s more sut able to sm u late la tent heat re lease by ap ply ng one of three al ter na tve meth ods shown here. The frst method as sumes la tent heat as a part of heat ca pac ty [1], sm u lat ng ts re - lease as a ds con t nu ty of rc. Thus, la tent heat re lease s un form. Ths method re qures a fne mesh of f nte el e ments and short tme-steps. The sec ond method uses enthalpy to de scrbe la tent heat, ac cord ngly rc = dh'/dt, where h' s spe cfc enthalpy a con tn u ous func ton of tem per a ture as op posed to heat ca pac ty. The enthalpy re la ton s sm ply ap pled by re - plac ng the df fer en tal wth f nte quan t tes, as rc = Dh'/Dt. In the thrd method, the re lease of la tent heat s as sumed from a fc t tous heat source vol ume of cer tan nodes of the f nte el e ment mesh [12]. In ten sty of ths source s equal to the re lease rate of the phase-change en ergy. Theoretcal analyss of resdual stresses Re sd ual stresses are de fned here as ther mal stresses caused by non-un form heat ng and cool ng of the welded ont [9]. More de taled ex pla na ton s gven n fg. 2, where the spa tal and tme ds tr bu ton of tem per a ture and stress (n the trans verse d rec ton) are l lus trated for four char ac ter s tc stages of the re ss tance weld - ng cy cle (A, B, C, and D). The frst stage (A) n volves the po s ton ng of the two parts for weld ng, and no ther mal stresses are gen er ated snce there s no n flu ence of heat. In the sec ond stage (B), the elec trodes are brought to the sur - face of the metal sheets and a slght amount of Fgure 2. Dstrbuton of temperature and resdual stresses n the weldng cycle

6 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp Fgure 3. Stress-stran and stress-temperature dependence at welded ont centre [13] pres sure s ap pled, caus ng com pres sve stresses n metal sheets. These stresses are pro - por tonal to the force ap pled for weld ng. In the thrd stage (C), the cur rent from the elec trodes s ap pled brefly and the tem per a ture n creases to melt ng pont at con tact lo ca tons. Due to the hgh n crease of tem per a ture n ths stage, the com pres sve stress n creases as a re sult of ther - mal ex pan son of the ma te ral. In the lq ud zone of the ma te ral, stresses are prac t cally zero snce there s no re ss tance to ther mal forces. In the fourth stage (D) the weld ng cur - rent s re moved and the act ng force stops. The ma te ral cools and tends to shrn n the cen tral cross-sec ton. Ths s sup pressed by the sur - round ng ma te ral wth much lower tem per a ture gra d ents. So the st u a ton s op po ste n be hav - our to that n stage B. Thus, ten sle stresses may arse that are bal anced by re mote com pres sve Fgure 4. Temperature dependence of materal characterstcs for the PbSb 2 alloy

7 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of... THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp stresses. When the parts have cooled com pletely, wth no tem per a ture gra d ents n the welded zone, ma te ral be hav our s ba s cally the same as n stage D, but wth much hgher ten sle stresses, snce the ma te ral s re ss tance to ther mal forces s much hgher at lower tem per a ture. Based on the ex pla na ton of the ap pear ance of re sd ual stresses n re ss tance weld ng, ds re gard ng the pres ence of wor ng stresses dur ng the weld ng pro cess t self, the tem per a ture de pend ence of ma te ral prop er tes s ob v ous. Ths fact s even more pro nounced when stress vs. tem per a ture and stress vs. stran are ana lysed at the cen tre of weldment cross-sec ton, fg. 3, and ds re gard ng wor ng stress. The weld ng cy cle s de noted by ponts 1-4 on both d a grams (fg. 3), and the cool ng cy - cle by ponts 4-6. The de crease n modulus of elas tc ty wth rs ng tem per a ture ex plans the non-lnearty of the stress-stran curve be tween ponts 1 and 2. Pont 2 cor re sponds to yeld stress, fol lowed by a de clne n stress, be cause the yeld stress s re duced by fur ther heat ng of the ma te - ral up to pont 3 when t reaches zero. Stresses have d mn shed up to pont 4, but sg nf cant plas - tc stran grad u ally arses. The elas tc stran re cov ers dur ng the cool ng cy cle, and ten sle stress ap pears and grows non-ln ear due to the rse n the elas tc ty modulus as the tem per a ture low ers (4-6). Pont 6 de notes the end of the weld ng pro cess wth re sd ual stresses and strans. In fg. 4, the tem per a ture dependences of PbSb 2 al loy prop er tes are l lus trated (yeld stress and ten sle strength, fg. 4(a), modulus of elas tc ty, fg. 4(b), Pos son s ra to, fg. 4(c), and the ln ear ther mal ex pan son co ef f cent, fg. 4(d). As al ready stated, yeld stress and elas tc ty modulus de crease wth tem per a ture, so that yeld stress s prac t cally zero at 250 C, whle at 700 C ma te ral stff ness d mn shes (modulus of elas tc ty equals 0). Other PbSb al loys for re - ss tance weld ng (from PbSb 1 to PbSb 4) have a sm lar be hav our. Applcaton of the fnte element method to resdual stresses The ba sc pro ce dure of the f nte el e ment method ap pl ca ton to re sd ual stress eval u a - ton s es sen tally the same as for the heat con duc ton prob lem [9] and n cludes do man dscretzaton (d v son nto f nte el e ments), n ter po la ton of all quan t tes n sde f nte el e - ments, n te gra ton over each el e ment, and solv ng of the re sult ng equa ton sys tem. The ma or df fer ences are n the equa tons to be solved, be cause ds place ment s the n de pend ent var able n the re sd ual stress prob lem n stead of tem per a ture, the statc equ lb rum equa tons are solved n stead of heat con duc ton equa tons, and ma te ral be hav our s de scrbed by the stress-stran re - la ton n stead of the Fou rer law. So, t s nec es sary to n tro duce soparametrc n ter po la ton func tons for ds place ments as fol lows: u( x, t) N K( x ) u ( t) (16) where u (x, t) s the ds place ment ds tr bu ton n sde an el e ment, t the tme, N (x ) the n ter po - la ton func ton, u t ( ) the nodal ds place ment, and x the Des cartes co-or d nates. Now, we wrte: u N e u N u x x (17) where N l s the ma trx of the or der n r (n space d men son, r num ber of el e ment nodes). Geo met r cal non-ln ear ty s ne glected as a us t fed as sump ton for the prob lem con sd ered. On the other hand, ma te ral non-ln ear ty can not be ne glected, lead ng to the ex pres son for to tal n - cre men tal stran com prs ng the elas tc, plas tc, and ther mal com po nents. e ep K de de de d det (18)

8 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp The elas tc stran s re lated to the stress by the fol low ng equa ton (Hooe s law): ds E de e (19) where ds s the stress ten sor n cre ment, and E l the elas tc ty ten sor. Plas tc stran may be ex pressed by ap ply ng the the ory of n cre men tal plas tc ty, ta ng nto ac count the nor mal ty con d ton, the von Mses yeld cr te ron and the ma te ral stran strength en ng as sump ton: p p dee de 3 S (20) 2 s where S s the stress de v a tor ten sor, and de p and s e are the equv a lent plas tc stran n cre ment and stress, n re spect, gven by: 3 p 3 p p se S S dee de d e (21) 2 2 glected: e Ther mal stran, de T, can be ex pressed as fol lows f the trans for ma ton plas tc ty s ne - de T adt (22) where a s the co ef f cent of ln ear ther mal ex pan son, and dt the tem per a ture change. Solvng matrx equatons An ex plct pro ce dure s ap pled for the nu mer cal n te gra ton of df fer en tal equa tons based on n tal tem per a ture ds tr bu ton, T 0 = T(x, 0). If the tme de rv a tve of the tem per a ture vec tor n eq. (14) s wrt ten as: DT T T T n+t n (23) Dt Dt where Dt s the tme n ter val. Then eq. (14) may be wrt ten n ex plct form: nt n C ( T T ) ( K K ) T Dt R Dt nt C T C T n c h n R Dt ( K c K ) T h n Dt The ex plct pro ce dure saves much cal cu la ton tme due to the d rect ln ng of un nown var ables n two suc ces sve steps, T n + t and T n. How ever, two re qure ments need to be ful flled: ( K c K ) s a dagonal matrx [14], and h procedure stablty depends on the tme-step that should be adopted to be extremely short [15]. Thermomechancal modellng ap pled for anal y ss of re ss tance spot weld ng of PbSb al loy Problem defnton We shall de fne the math e mat cal mod el lng of re ss tance spot weld ng of PbSb al loys us ng the ex am ple of the weld ng pro cess on staced pos tve and neg a tve bat tery plates (lead-acd bat tery cells) n the as sem bly of the poly propy lene (PP) bat tery case, fg. 5. Stac as sem bly s pro duced by an au to matc ma chne, and weld ng s per formed through PP case walls across the ntercell con nec tor. The weld ng ma chne s equpped wth a (24)

9 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of... THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp hy drau lc unt that pro vdes a con stant weld ng force, and a power unt for the metal fu son pro cess. It s pos s ble, wthn lm ts, to vary pres sure force, cur rent, volt age, and tme of the weld ng pro cess. The weld ng cy cle s com prsed of three stages (com press ng, weld ng, and hold ng) and may last wthn 1 to 50 pe r ods or, from 0.02 s to 1 s. The weld ng force can change from 0.5 N to 1.5 N. The ba sc re ss tance spot weld ng cy - cle of the PbSb al loy s shown n fg. 6. The tme and weld ng force may df fer for var ous PbSb al loys. Bat tery cells are made of PbSb or PbCa al loys of var ous con tent, de - pend ng on the bat tery type. The chem cal com po s ton of a bat tery cell al loy d rectly af - fects the chem cal com po s ton of the ntercell con nec tor to be re ss tance spot welded and wll serve as an l lus tra ton for the nu mer cal sm u - la ton of the re ss tance spot weld ng of ntercell con nec tors made of sev eral PbSb al loys. Materal for battery ntercell connectors The ma te ral for bat tery ntercell connectors s the PbSb al loy. The most com mon chem cal com po s ton s gven n tab. 2. Other al loy ng el e ments that are gven n tab. 2 have no con - sderable effect on thermal, mechancal and electrcal propertes of the alloy. Physcal model The phys cal model s a des g nated ntercell con nec tor wth all the nec es sary el e - ments for the weld ng be hav our as a whole. As sum ng that the be hav our of ev ery ntercell bat tery con nec tor dur ng weld ng s al most the same, the con nec tor shown n fg. 7 s adopted as the prob lem do man. The re ss - tance weld ng pro cess s mod elled as a heat trans fer pro cess wth n tal and bound ary con d tons. The n tal con d ton s gven as a tem per a - ture dstrbuton T 0 = 20 C at t = 0 over the whole do man. Bound ary con d tons are de fned Fgure 5. Resstance spot weldng of battery cells (1) weldng machne; (2) PP case, (3) ntercell connector, (4) battery cell stac Fgure 6. The resstance spot weldng cycle t p compresson tme, t w weldng tme, t h holdng tme Table 2. Chemcal composton of the PbSb al loy No. Al loy (nternal code) Al loy ng el e ments Sb Sn Other Pb 1 SDS SD A DD Fgure 7. Intal and boundary condtons

10 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp for: the ar bound ary as heat con vec ton from the PbSb al loy nto ar of tem per a ture t = 20 C; the elec trode bound ary as heat s con ducted through the PbSb al loy dur ng weld ng, and later as heat s also con vected from the PbSb al loy nto ar; and the bound ary where two el e ments come nto con tact and at the part of the wall case, act ng as an ad a batc bound ary. At the bound ary of two el e ments, the heat load con ssts of heat flux of con tact and ohmc resstances, and the heat load at elec trode do man con tact bound ary as heat flux of metal sheet elec trode con tact re ss tance. The equa ton and heat con vec ton co ef f cent need to be de fned for heat trans fer by nat u ral con vec ton from PbSb al loy nto surroundng ar. Heat s trans ferred from the case wall nto sur round ng ar by free flow. In gen eral, the free flud flow equa ton (along and around geo met r cal shapes) ap pled for ths spe cal case s: Nu = 1.18 GrPr (25) and t en ables the de ter m na ton of the con vec tve heat trans fer co ef f cent: Nul h [ Wm 2 C 1 ] (26) L where L s the characterstcs of geometry [m], and l the ther mal con duc tv ty of ar [Wm 1 C 1 ]. The change of the con vec tve heat trans fer co ef f cent de pend ng on the bat tery case wall tem per a ture s shown n fg. 8. A par tc u lar prob lem rep re sents the part of the ana lysed bat tery ntercell case, whch n fact s 3-D, and anal y ss of 3-D prob lems s very com pl cated and de mands ap pl ca ton of hgh ca pac ty com put ers. Re duc ng 3-D to a 2-D prob lem s em pha szed n ths ex am ple hav ng Fgure 8. Case wall temperature dependence of the heat conductance coeffcent n mnd the d lemma of rep re sent ng the model as pla nar or axs-sym met r cal. Namely, bound - ary con d tons may be pla nar, whle the body shape (the cy ln dr cal part n par tc u lar) has axs sym me try. How ever, snce the pla nar part could not be n cluded wthn the same me rd an sec ton, thus the axs-sym met r cal rep re sen ta - ton s ds re garded. All n all, spe cfc de tals of the re ss tance spot weld ng pro cess must be n - cor po rated nto ths phys cal model. The re ss tance spot weld ng s usu ally char ac ter s tc n the ap pear ance of mol ten weld splashes or bursts, be cause of the rapd heat ng n phase change tem per a ture ranges. Mul t ple n - crease of heat ca pac ty n rel a tvely short tem per a ture n ter vals (around 300 C) ap par ently has a con sd er able ef fect on the rapd change of tem per a ture, and may cre ate hgh lo cal stresses and cause splashes and bursts. Prn c pally, a sm lar anal y ss mght n clude the stress state n the weld ng zone by solv ng an ap pro pr ate elas tc-vs cous-plas tc bound ary prob lem. Methodology of problem solvng and resstance spot weldng specfcs Solv ng the non-sta ton ary non-ln ear equa ton s prn c pally ex planed [10], where a de taled anal y ss rec om mends use of ex plct tme n te gra ton over short tme-steps, and wth a re duced spe cfc heat ma trx. Based on the nown n tal tem per a ture ds tr bu ton, a fur ther n te -

11 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of... THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp gra ton over tme s made ac cord ng to the scheme gven by eq. (24). The n dex n n eq. (24) cor re sponds to n tal state, and n dex n + t to the f nal state wthn the n th tme-step. The heat con duc tv ty ma trx ( Kc K ), the re duced spe cfc heat ma trx C, and the heat load vec tor h R, all change wth each tme-step de pend ng on the pre v ous tem per a ture ds tr bu ton. For a ho - mo ge neous so tro pc ma te ral, an au to matc al ter aton of all nec es sary ma tr ces and vec tors s made sm ple by em bed dng func tonal tem per a ture dependences of cor re spond ng ma te ral prop er tes. The tme-step needs to be suf f cently short n or der to acheve sta bl ty of the ex - plct scheme. In ths case, suf f cently short means: Dt cr 2 x max (27) where x max s the max mal value n the cor re spond ng prob lem of egenvalues [14]. The egenvalue prob lem s not df f cult to for mu late and solve, but re qures ad d tonal ef fort and cal cu la ton tme, and addtonally, x max also changes wth each cal cu la - Ta ble 3. Tme-steps (no. of steps tme [s]) ton step. Hence, the au thors es t mate s to de cde upon Dt cr value n ad vance and Fne step Total [s] based on the anal y ss n Coarse step [s] tab. 3. Most of the cal cu la ton tme s spent for ma trx mul - tplcaton, e. g. ( K c K ) T h n n eq. (24) needs spe - cal attenton so to avod unnecessary multplcaton wth a ma or ty of zero el e ments n ( K K ). A rough estmate for an approxmately rectangular doman dscretsed wth a node mesh m pl cates there are about 98% zero el e ments n the heat con duc - tv ty ma trx. In other words, only 1/60 lo ca tons are oc cu ped n the ma trx that are mul t pled by T n. Soldus and lqudus po s tons are de ter mned based on tem per a ture ds tr bu ton, and yet ther lo ca - tons also n flu ence the tem per a ture ds tr bu ton. Ths con u gate sys tem re qures an ex tremely short tme-step, n de pend ent of pro ce dure sta bl ty, thus us t fy ng that Dt cr = s as the weld ng n ter val [16]. It seems ben e f cary to lnearze the prob lem n ths way than to ap ply com pl cated pro ce dures for solv ng non-ln ear prob lems,. e. New ton-rason method. Tem per a ture ds tr bu ton n each tme-step s gven for nodes and t s qute df f cult to de ter mne lqudus and lqudus lo ca tons. lqudus and lqudus c h Fgure 9. The fnte element mesh fne lnes are re qured to pass through nodes n or der to de fne a sold, lq ud, or mxed state n each f nte el e ment. It s clear that ths can be acheved only ap prox mately, and that the num ber of nodes, or the f nte el e ment mesh, con sd er ably af fect ac cu racy. A tr an gu lar f nte el e ment mesh s used here and shown n fg. 9, wth a con stant tem per a ture gra d ent n sde the f nte el e - ments, and wth pre-se lected pont lo ca tons (T1-T5).

12 448 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of... THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp Results for spatal and tme dstrbuton of temperature Results of thermal calculaton obtaned accordng to the defned procedure are llustrated n fgs. 10(a)-(d) showng sotherms at certan tme ntervals, and n fg. 11(a) and (b) showng dagrams of temperature-tme dependence for two of the selected locatons n fg. 9. All of these results are for alloy A4 (see tab. 2). Adequate contact surface geometry (sphercal shape) has accomplshed the ntal meltng pont to be reached at weldng zone centre, and not at ts permeter (crcumference) as t s acheved when classcal resstance spot weldng s performed [17]. Fgure 10. Spatal temperature dstrbuton (a) t = s, (b) t = 0.1 s, (c) t = 1 s, (d) t = 60 s Fgure 11. Temperature-tme dependence at ponts T1 (a) and T5 (b)

13 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of... THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp Con clu sons The de gree of nowl edge and un der stand ng of the com plex phe nom ena n ther mal, me chan cal, and elec tr cal pro cesses that de ter mne the weld ng pro cess, as well as ac qur ng data on me chan cal and met al lur g cal prop er tes of the ap pled tech nques con ducted at hgh tem per a tures are the cru cal fac tors n the pro cess of cre at ng and f nal z ng the math e mat cal model. Of par tc u lar m por tance s the ne ces sty to ac cu mu late and have qual ty data of phys cal and me chan cal prop er tes of met als at tem per a tures close to melt ng, or more pre cse, at tem per - a ture n ter vals wthn the lqudus and lqudus lnes. The ap pled math e mat cal method en abled ef f cent and qual ta tve anal y ss re gard ng: temperature feld determnaton and ts nfluence on the weldng process, (eq. 24), and applcaton of real ntermedate condtons (partcularly boundary condtons) n the ntal stage of the mathematcal analyss, whereas possbltes for errors are reduced durng the creaton of the analytcal model. The model for re ss tance spot weld ng of PbSb al loys s de vel oped wth re spect to the real ge om e try of the weld ng zone, and phys cal and met al lur g cal prop er tes of PbSb al loys and elec trode ma te ral. The model taes nto ac count phase trans for ma ton n the re gon of lqudus and lqudus tem per a tures, or la tent heat trans for ma ton. Con tact resstances on con tact sur face zones: elec trode-metal sheet, and sheet-sheet are an a lyt cally n volved, whle the n flu ence of the ac tual weld ng force has proved to be neg l g ble. References [1] Crvell, L. A., Idelshon, S. R., A Tem per a ture-based F nte El e ment So lu ton for Phase-Change Prob - lems, In ter na tonal Jour nal for Nu mercal Method n Engneerng, 23 (1986), 1, pp [2] Lazards, A., A Nu mer cal So lu ton of the Mul t d men sonal So ld f ca ton (or Melt ng) Prob lem, In ter - na tonal Jour nal Mass Trans fer, 13 (1970), 9, pp [3] Sarsl, D. I., Bolly, B. A., The So lu ton of a Class of Two-D men sonal Melt ng and So ld f ca ton Prob lem, In ter na tonal Jour nal Sold Structure, 1 (1965), 2, pp [4] Orlov, D., Tech nol ogy and Equp ment of Contact Weld ng (n Rus san), Mashnostroene, Mos cow, 1975, and 1986 [5] Naata, S., et al., In-Pro cess Qual ty Con trol of Spot Weld by De tect ng Volt age be tween Elec trode Tps Adap tve Con trol for Qual ty As sur ance of Re ss tance Spot Weld n Real Tme (2 nd re port), IIW Doc. III , 1982 [6] Ned, H. A., The F nte El e ment Mod el ng of the Re ss tance Spot Weld ng Pro cess, Weldng Journal, 63 (1984), 4, pp. 123s-132s [7] Rce, W., Fun, E. J., An An a lyt cal Investgaton of the Temperature Dstrbutons durng Resstance Weld ng, Weld ng Jour nal, 46 (1967), 4, pp. 175s-186s [8] Sedma, A., Nu mer cal Sm u la ton of the Weld ng Pro cess I part: Tem per a ture felds (n Ser ban), Zavarvane zavarene onstruce, 41 (1996), 1, pp [9] Sedma, A., Nu mer cal Sm u la ton of the Weld ng Pro cess II part: Re sd ual stresses (n Ser ban), Zavarvane zavarene onstruce, 41 (1996), 2, pp [10] Kalaba, D., Thermomechancal Mod el lng of the Re ss tance Weld ng Pro cess of PbSb Al loy (n Ser ban), Ph. D. the ss, Un ver sty of Prštna, Fac ulty of Me chan cal En g neer ng, Prštna, Ser ba, 1998 [11] Berov}, M., Masmov}, S., Sedma, A., Anal y ss of Welded Jonts by Ap pl ca ton of the F nte El e - ment Method (n Ser ban), n: IFMASS 3 Frac ture Me chan cs of Welded Jonts (Ed. S. Sedma), Lec - tures pre sented at the Thrd In ter na tonal Frac ture Me chan cs Sum mer School, Arandelovac, Ser ba, 1984, GOŠA In st tute and Fac ulty of Tech nol ogy and Met al lurgy, Un ver sty of Bel grade, pp

14 Kalaba, D., et. al.: Thermomechancal Modelng the Resstance Weldng of THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp [12] Bon, E., Ngro, L., Sanpetr, C., Nu mer cal So lu ton of Ther mal Pro cesses wth State and Phase-Change: The Com puter Code Aten-2D, Pro ceedngs, ICNMNL Prob lems, Dubrovn, Yugoslava, 1986, pp [13] Easterlng, K., In tro duc ton to the Phys cal Met al lurgy of Weld ng, Butterworths Co., Lon don, 1982 [14] Huebner, K. H., Thorn ton, E. A., F nte El e ment Method for En g neers, 2 nd ed. John Wley & Sons, New Yor, USA, 1982 [15] Sahm, P., Nu mer cal Sm u la ton and Mod el lng of Cast ng and So ld f ca ton Pro cesses for Foundry and Cast-House, CIATF, Zürch, Swt zer land, 1984 [16] Bathe, K. J., ADINAT A F nte El e ment Program for Automatc Dynamc Incremental Nonlnear Analy - ss of Tem per a tures, AVL Re port , Mechancal Engneerng Department, M.I.T., 1977 [17] Bentley, K. P., Green wood, J. A., Knowlson, P. M., Bacer, R. G., Tem per a ture Ds tr bu ton n Spot Welds, Brt sh Weld ng Jour nal, 10 (1963), 12, pp Pa per sub mt ted: De cem ber 10, 2009 Pa per re vsed: De cem ber 11, 2009 Pa per ac cepted: De cem ber 16, 2009