Analytical Solutions for Transient Temperature of Semi-Infinite Body Subjected to 3-D Moving Heat Sources

Transcription

1 WELDING RESERCH STRCT. Te nlyicl soluion for doule-ellipsoidl power densiy moving e source in semi-infinie ody wi conducion-only considerion s een derived. Te soluion s een oined y inegring e insn poin e source rougou e volume of e ellipsoidl one. Very good greemen eween e prediced rnsien emperures nd e mesured ones vrious poins in ed-on-ple specimens s een oined. Te prediced geomery of e weld pool is lso in good greemen wi e mesured one. Tis my pve e wy for e fuure pplicions of is soluion in e prolems suc s microsrucure modeling, erml sress nlysis, residul sress/disorions nd welding process simulion. Inroducion SUPPLEMENT TO THE WELDING JOURNL, UGUST 1999 Sponsored y e mericn Welding Sociey nd e Welding Reserc Council nlyicl Soluions for Trnsien Temperure of Semi-Infinie ody Sujeced o 3-D Moving He Sources nlyicl soluions for 3-D moving e sources were derived nd erimenlly vlided y rnsien emperure mesured vrious poins in ed-on-ple specimens nd y mens of weld pool geomery Te emperure isory of e welded componens s significn influence on e residul sresses, disorion nd ence e figue evior of e welded srucures. Clssicl soluions for e rnsien emperure field suc s Rosenl s soluions (Ref. 1 del wi e semi-infinie ody sujeced o n insn poin e source, line e source N. SUZUKI, Y. MED, N. T. NGUYEN nd. OHT re wi Nionl Reserc Insiue for Mels (NRIM, Irki, Jpn. K. MT- SUOK is wi Sip Reserc Insiue, Sinkw, Tokyo, Jpn. Y N. T. NGUYEN,. OHT, K. MTSUOK, N. SUZUKI ND Y. MED or surfce e source. Tese soluions cn e used o predic e emperure field disnce fr from e e source u fil o predic e emperure in e viciniy of e e source. Egr nd Tsi (Ref. modified Rosenl s eory o include wodimensionl (-D surfce Gussin disriued e source wi consn disriuion prmeer (wic cn e considered s n effecive rc rdius nd found n nlyicl soluion for e emperure of semi-infinie ody sujeced o is moving e source. Teir soluion is significn sep for e improvemen of emperure predicion in e ner e source regions. Jeong nd Co (Ref. 3 inroduced n nlyicl soluion for e rnsien emperure field of fille-welded join sed on e similr -D Gussin e KEY WORDS Trnsien He Source Tree-Dimensionl Temperure Field Moving He Source Gussin Two-Dimensionl Terml Sress source u wi differen disriuion prmeers (in wo direcions x nd y. Using e conforml mpping ecnique, ey ve successfully rnsformed e soluion of e emperure field in e ple of finie ickness o e fille welded join. Even oug e ville soluions using e Gussin e sources could predic e emperure regions closed o e e source, ey re sill limied y e sorcoming of e -D e source iself wi no effec of penerion. Tis sorcoming cn only e overcome if more generl e sources re implemened. Goldk, e l. (Ref. 4, firs inroduced e ree-dimensionl (3-D doule ellipsoidl moving e source. Finie elemen modeling (FEM ws used o clcule e emperure field of ed-on-ple nd sowed is 3-D e source could overcome e sorcoming of e previous -D Gussin model o predic e emperure of e welded joins wi muc deeper penerion. However, up o now, n nlyicl soluion for is kind of 3-D e source ws no ye ville (Ref. 5, nd ence, resercers mus rely on FEM for rnsien emperure clculion or oer simulion purposes, wic requires e erml isory of e componens. Terefore, if ny nlyicl soluion for emperure field from 3-D e source is ville, lo of CPU ime could e sved nd e erml-sress WELDING RESERCH SUPPLEMENT 65-s

2 Fig. 1 Doule ellipsoidl densiy disriued e source. nlysis or reled simulions could e crried ou muc more rpidly nd convenienly. In is sudy, nlyicl soluions for e rnsien emperure field of e semi-infinie ody sujeced o 3-D power densiy moving e sources (suc s semi-ellipsoidl nd doule ellipsoidl e sources re derived nd repored. Te clculed rnsien emperures vrious poins of ineres in seel ple, 4 x 4 x mm, sujeced o wo perpendiculr liner welded segmens on is surfce, re compred wi e mesured d y e uors. Te geomery of e weld pool ws lso mesured nd compred wi e prediced one. Resonly good greemens eween e clculed nd mesured d sow poenil pplicion of is newly developed soluion for vrious simulion purposes suc s erml sress nlysis or residul sress clculions. Ellipsoidl He Sources in Semi-Infinie ody Goldk s Semi-Ellipsoidl He Source Goldk, e l. (Ref. 4, iniilly proposed semi-ellipsoidl e source in wic e flux is disriued in Gussin mnner rougou e e source s volume. Te e flux Q (x,y,z poin (x,y,z wiin e semi-ellipsoid is given y e following equion: 6 3η. VI. Qxyz (,, = cπ π 3x c (1 were,, c = ellipsoidl e source prmeers s descried in Fig. 1, were c f = c = c ; x,y,z = moving coordines of e e source; Q(x,y,z = e flux Q(x,y,z poin (x,y,z; V, I nd η = welding volge nd curren nd rc efficiency. However, eir erience wi is e source sowed e prediced emperure grdiens in fron of e rc were less seep n erimenlly oserved ones nd grdiens eind e rc were seeper n ose mesured. To overcome is, ey comined wo semiellipsoids nd proposed new e source clled doule ellipsoidl e source, s sown in Fig. 1. Goldk s Doule Ellipsoidl He Source Since wo differen semi-ellipsoids re comined o give e new e source, e e flux wiin ec semi-ellipsoid re descried y differen equions. For poin (x,y,z wiin e firs semiellipsoid loced in fron of e welding rc, e e flux equion is descried s 6 3rQ f Qxyz (,, = c fπ π 3x cf ( nd for poins (x,y,z wiin e second semi-ellipsoid, covering e rer secion of e rc, s 6 3rQ Qxyz (,, = c π π 3x c (3 were,, c f, c = ellipsoidl e source prmeers s descried in Fig. ; Q = rc e inpu (Q = η IV; r f, r = proporion coefficien represening e pporionmen in fron nd ck of e e source, respecively, were r f + r = (Ref. 4. I mus e noed ere due o e condiion of coninuiy of e volumeric e source, e vlues of Q(x,y,z given y Equions nd 3 mus e equl e x = plne. From condiion, noer consrin is oined for r f nd r s r f /c f = r /c. Susequenly, e vlues for ese wo coefficiens re deermined s r f =c f /( c f + c ; r =c /( c f + c. I is wor noing is doule ellipsoidl disriuion e source is descried y five unknown prmeers: e rc efficiency η, nd ellipsoidl xis,, c f nd c. Goldk, e l. (Ref. 4, implied n equivlence eween e source dimensions nd ose of e weld pool nd suggesed pproprie vlues for,, c f nd c could e oined y mesuremen of e weld pool geomery. nlyicl Soluions Trnsien Temperure Field of Semi-Ellipsoidl He Source in Semi-Infinie ody Te soluion for e emperure field of semi-ellipsoidl e source in semi-infinie ody is sed on e soluion for n insn poin source sisfies e following differenil equion of e conducion in fixed coordines (Ref. 6 δqd dt = ρc 4 3 / π [ ( ] ( x x + ( y y + ( z z 4 ( (4 were = erml diffusiviy ( = k/cρ; c = specific e; k = erml conduciviy; ρ = mss densiy;, = ime; dt = rnsien emperure due o e poin e source δ Q ime ; nd (x,y,z = locion of e insn poin e source δ Q ime. Le us consider e soluion of n insn semi-ellipsoidl e source s resul of superposiion of series of insn poin e sources over e volume of e disriued Gussin e source. Susiue Equion 1 for e e flux poin source ino Equion 4 nd inegrion over e volume of e e source semi-ellipsoid gives 1 d dt = dx dy dz c [ 4 3 / ρ π ( ] Q 6 3 3x c π π c ( x x + ( y y + ( z z 4 ( (5 Susequenly, y evluing nd simplifying, Equion 5 cn e rewrien s 3 3Qd 1 dt = ρπ c π 1 ( ( + 1 ( + c 3x 1 ( + c 1 ( + 1 ( + (6 Equion 6 provides e emperure rise 66-s UGUST 1999

3 due o very sor ime incremen d from ime due o moun of e Qd relesed on e semi-infinie ody. Wen considering moving e source wi consn speed v from ime = o ime =, e increse of e emperure during is ime is e sum of ll of e conriuion of e moving e source during e rveling ime. Terefore, 3 3Q T To = ρπ c π d 1 ( + 1 ( + 1 c ( + 3( x v 1 c ( + 1 ( + 1 ( + (7 were T is emperure ime nd T o is iniil emperure of poin (x,y,z. Now le us consider severl specific cses of e semi-ellipsoidl e source s follows: Semi-spere e source. If = = c = r, en e semi-ellipsoidl e source ecomes e semi-spere e source wi rdius r nd Equion 7 ecomes 3 3Q d T T = ρπ c π 1 ( + r 3 / 3( x v ( + r (8 Equion 8 cn e simplified furer y susiuing r = 3 σ (were σ is disriuion prmeer s Q d T T = ρ c 3 / [ 4 ( ] ( + + x v y z 4 ( (9 n Insn Poin Source. If σ =, Equion 9 reduces o e emperure disriuion for e insn poin source s Q d T T = ρc 3 / [ 4π ( ] ( + + [ ] x v y z 4 ( (1 Tis soluion is consisen wi e one pulised y Crslw nd Jeger (Ref. 6 s repored previously (Equion 4. Gussin Surfce-Disriued He Source. If = ~en e semi-ellipsoidl e source ecomes Gussin surfce-disriued e source or Ellipicl disk e source. In is cse Equion 7 cn e rewrien s 3Q T T o = πρc d 4 π ( 1 ( + 1 c ( + 3( x v 1 ( + c 1 ( + z 4 ( (11 Wen =c = 6 σ, en e ellipicl disk e source ecomes circulr disk nd Equion 11 ecomes Q T To = πρc d 4 π ( 4 ( - + σ ( + [ ] x v y z 4 ( + σ 4 ( (1 Equion 1 gives e sme form s e soluion pulised y Egr nd Tsi (Ref.. Trnsien Temperure Field of Doule Ellipsoidl He Source in Semi-Infinie ody Using similr pproc, n nlyicl soluion for rnsien emperure of semi-infinie ody sujeced o doule ellipsoidl e source cn e oined s follows: Le us consider gin e soluion for e doule ellipsoidl e source is e superposiion of series of insn poin e sources over e volume of e disriued Gussin e source. Susiue Equion 1 for e poin source in Equion 4 nd ke inegrion over e e source volume for o fron nd ck porion of e volume e source. Tis volume is corresponding o wo qudrns of e fron nd ck semiellipsoids, respecively, s Qd dt = / ρc π π 4π [ ( ] ( x x + ( y y + ( z z 4 ( r f 3x c f c f + dx dy dz r 3x c c (13 Equion 13 cn e simplified furer s Qd dt = 3 3 πρc π 1 ( + 1 ( + + (14 1 ( + c f 1 ( + c f 3x 1 c ( + f were = r f 1 ( + 1 ( + 3x 1 ( + c = r 1 ( + 1 ( + Similrly, wen considering moving e source wi consn speed v from ime = o ime =, e increse of emperure during is ime is equivlen o e sum of ll e conriuions of e moving e source during e rveling ime s 3 3Q d T T o = ρπ c π 1 ( + 1 ( + + (15 1 ( + c f 1 ( + c f 3 x v ( 1 ( + c f were = r f 1 ( + 1 ( + 3( x v 1 ( + c = r 1 ( + 1 ( + (15 (15c (14 (14c Equions 15 o 15c give e soluion for e rnsien emperure of semi-infinie ody sujeced o doule ellipsoidl e source. I is lso wor noing ere Equions 14 nd 15 cn e esily reduced o e form of Equions 6 nd 7, respecively, y susiuing c f = c ino Equions 14 nd 15. Tis cn e eced ecuse in is cse e doule ellipsoidl e source ecomes e semi-ellipsoidl one. WELDING RESERCH SUPPLEMENT 67-s

4 Fig. Mximum dimensionless emperure disriuion long rnsversl direcion ψ. Te effec of disriuion prmeer u ; e effec of disriuion prmeer u. Fig. 3 Mximum dimensionless emperure disriuion roug e ickness direcion ζ: Te effec of disriuion prmeer u ; e effec of disriuion prmeer u. Furermore, e soluions for e doule ellipsoidl e source cn e convered ino dimensionless form y using e following dimensionless vriles s recommended y Crisensen s meod (Ref. 7: ξ = vx/, ψ = vy/, ζ = vz/, τ = v (- /; u = v /( 6, u = v /( 6, u cf = vc f /( 6 nd u c = vc /( 6. Hence Equions 15, 15 nd 15c ecome θ = n ν 1 dτ 1 + π cf 1 (16 c ( ξ + τ ψ τ + u cf were 1 = r f (16 ζ ( ξ + τ ψ ζ 1 = r (16c τ u c τ u τ u nd n is e opering prmeer (Ref. nd n = Qv / (4π ρc(t c T o nd T c - is reference emperure (in is sudy, T c is seleced s meling emperure. Since i ws quie resonle o ssume c f = nd c =c f, en Equions 16, 16 nd 16c cn e reduced o muc simpler form s In is sudy, numericl procedure is pplied o find soluions for e rnsien emperure field s descried y Equion 17, nd c for e doule ellipsoidl disriued e source. comθ = n ν 1 dτ ( 17 π τ + 4u ( ξ + τ + ψ ζ were 1 = r f (17 ( ξ + τ ψ ζ 1 = r (17c τ 4u τ u τ u I cn e seen from Equions 17, 17 nd 17c e rnsien emperure ny poin (ξ, ψ, ζ depends on e ree prmeers: opering prmeer (n descriing e mgniude nd e inensiy of e e inpu nd wo dimensionless e source spe prmeers (u nd u represening e wid nd dep of e ellipsoidl e source. numericl procedure is used in is sudy o invesige e effec of ese ree prmeers on e mximum emperure vrious posiions, nd e resuls re repored elow. I is lso wor noing ere for specil cse wen = nd c = c f =, Equions 16, 16 nd 16c would give e sme form s e dimensionless rnsien emperure soluion sujeced o -D Gussin disriuion surfce e source pulised y Egr nd Tsi (Ref.. Numericl Resuls Effec of He Source Prmeers u nd u on e Pek Temperure Disriuion 68-s UGUST 1999

5 puer progrm is wrien in FORTRN 77 o fcilie e inegrl clculion nd o llow for rpid clculion of rnsien emperure s well s mximum emperure of ny poin of ineres. Since e soluion ws oined for semi-infinie ody, e mirror meod, wic comines e emperure disriuion in cse of infinie ickness nd is refleced imge, ws doped for ple ickness considerion. y using is meod, i ws ssumed ere is no convecive e flow roug e upper nd lower surfce of e ple (Ref. 1. In order o illusre e soluions for pek emperure disriuion long e rnsversl nd roug-ickness direcion corresponding o vrious prmeers of e doule ellipsoidl disriued e source, rnges of e source disriuion prmeers sould e seleced. Noing ere is similriy eween e dimensionless disriuion prmeers of Egr nd Tsi s soluion (u = νσ/ nd of is 3-D soluion u nd u (u = v /( 6 ; u = v /( 6. y puing σ = / 6 nd σ = / 6, i would ring u nd u ino e sme form s of u, were σ nd σ re e source disriuion prmeers in ψ nd ζ direcions, respecively. Since o σ nd σ represens e wid of e e source, i is quie resonle o ssume eir rnge of vriion would e e sme. Forunely, e rnge of σ ws repored o e eween 1.6 nd 4 mm for gs ungsen rc welding (GTW (Ref.. Terefore, if similr welding e source is considered, e.g. gs mel rc welding (GMW, en e rnge of σ could ssume o e e sme s of σ. s resul, for consn welding speed of 5 mm/s nd for diffusiviy of se mel = mm /s, u would e vried eween.6 nd 1.6 (.6 u 1.6. However, e rnge of σ could e from zero (in e cse of surfce welding o e mximum vlue of σ or even greer, depending on e crcerisics of e considered e source. In generl, mos of e common e source would e more effecive in e surfce direcion (ξ nd ψ n e dep direcion (ζ, i.e σ σ. Hence, for e soluion demonsrion purpose in is sudy, e rnge of u is seleced o e eween nd 1.1 ( u 1.1. In e cse wen u = u =, e 3-D Gussin e source s no wid nd e soluion given y Equions 17, nd c would reduce o Rosenl s soluion of poin e source (Ref. 1. In ddiion, wen u = nd u = u cf = u c = u >, e 3-D Gussin e source ecomes Egr nd Tsi s soluion for -D Gussin normlly disriued e source (Ref.. Terefore, in is sudy, Fig. 4 Effec of on e prediced weld pool geomery ( = mm, c f = 7 mm, η =.8, c = c f : Top view of e weld pool; longiudinl cross secion. e numericl resuls re presened for vrious cominions of disriuion prmeers u nd u wiin eir rnges nd for wo priculr cses (u = u = nd u = ; u = u cf = u c = u > for comprison. Figures nd 3 sow e disriuions of e dimensionless mximum emperure long e rnsversl direcion ψ nd roug-e-ickness direcion ζ, respecively, sujeced o e vriions of u nd u. o Rosenl s nd Egr nd Tsi s soluions for similr disriuion prmeers re lso ploed for comprison. Tese figures lso provide informion ou e spes of e weld pool (wid nd dep nd of e effeced zone (HZ wen e vlue of (θ/n corresponds o e meling emperure (θ = 1 nd rnsformion emperure (θ =.47, respecively. However, e effec of e e source prmeers on e prediced spe of e weld pool is repored in more deil in e nex prgrp. Figure sows e mximum emperure ner e weld cenerline (ψ <1.4 decreses s e u increses wile u ws kep uncnged (u =.3. However, furer disnce from e weld cenerline (ψ >1.4, mximum emperure increses s e u increses. Tis evior of e disriuion of mximum emperure is refleced in e prediced weld wids due o vriion of u. Tis mens for low vlue of opering prmeer n (n <1.8, wic would produce θ/n iger n.55 e weld pool oundry e prediced weld wid decreses s u increses s in Fig.. However, for ig opering prmeers (n >1.8, e prediced weld wid increses s u increses. pysicl lnion for ese eviors of mximum emperures is e lrger e wid of e e source, e smller is e flux densiy, nd if e e flux is smll enoug, i could resul in siuion wi no meling zone wen e pek emperure reduces o less n e meling emperure of e se meril. However, wen e e source is srong enoug, e pek emperure increses s u increses, nd e lrger weld wid is eced. Figure lso sows for e sme vlue of disriuion prmeer (u = 1., e presen 3-D soluion gives lower prediced mximum emperure n of Egr nd Tsi s soluion, i.e, resuls in lower prediced weld wid. On e oer nd, e viciniy of e weld cenerline (ψ <1.4, e presen soluion gives muc lower prediced mximum emperure n of e Rosenl s soluion. However, furer disnce from e weld cener line (ψ >1.4, e presen soluion gives only sligly iger mximum emperure n of Rosenl s soluion. In conrs, Fig. sows unique rend of e mximum emperure evior due o u vriion, wile u ws kep uncnged (u = 1.1. I sows for ll vlues of n, e mximum emperure decreses s e u increses. Tis evior my e due o e fc re- WELDING RESERCH SUPPLEMENT 69-s

6 Fig. 5 Effec of on e prediced weld pool geomery ( = 5 mm, c f = 7 mm, η =.8, c = c f : Top view of e weld pool; longiudinl cross secion. Fig. 6 Effec of c f on e prediced weld pool geomery ( = 5 mm, = mm, η =.8, c = c f : Top view of e weld pool; longiudinl cross secion. grdless of e mgniude of e opering prmeer, e weld dep increses s u increses, ence, e weld wid decreses due o e consn volume of e meling meril induced y consn e inpu. Figure nd lso sows e presen 3-D Gussin e source soluion does no predic e infinie pek emperure e weld cenerline wile e Rosenl soluion does. Tis rend could lso e found in Egr nd Tsi s -D e source soluion; owever, i would predic muc iger pek emperure e weld cenerline n of e 3-D Gussin e source soluion, wi similr disriuion prmeer s sown in Figs. nd. Figure 3 nd sows e mximum emperure disriuion roug e ickness direcion ζ s funcion of disriuion prmeer u (or u wile u (or u ws kep uncnged, respecively. Two oer pek emperure curves of Rosenl s nd Egr nd Tsi s soluions re lso ploed for comprison. I cn e seen from Fig. 3 s e u increses e pek emperure decreses rougou e ickness direcion; ence, e prediced weld pool dep decreses s u increses. Te prediced weld pool dep y e presen soluion is smller n y o Egr nd Tsi s nd Rosenl s soluions. Figure 3 sows sligly differen evior of mximum emperure disriuion rougou e ickness direcion ζ due o u vriions. I cn e seen from Fig. 3 s e u increses, e pek emperure decreses relively sllow dimensionless dep (ζ <1.1. However, deeper dep (ζ >1.1, is rend grdully cnged o e reverse direcion. Te cnge is no s pronounced s is found for e pek emperure long ψ wen u is vried s in Fig.. Figure 3 lso sows e weld pool dep prediced y e presen soluion is sligly smller n prediced y o Egr nd Tsi s -D nd Rosenl s soluions. However, for iger vlues of e opering prmeer, e prediced weld pool deps y ll e ree ove-menioned soluions re comprle. Effec of He Source Prmeers on e Prediced Weld Pool Geomery In is secion, similr clculing procedure ws employed u Equions 15, nd c were used insed. Tis would give us eer imge of ow vrious e source prmeers (,, c f, c nd η will ffec e prediced spe of e weld pool. Te following meril properies were used: c = 6 J/kg/ C; k = 9 J/m/s/ C nd ρ = 78 kg/m 3. Te rc prmeers used were U = 6 V, I = 3 nd v = 3 cm/min. Figures 4 nd sow e effec of e e source prmeer on e op view of e weld pool spe nd is longiudinl cross secion, respecively, wile oer e source prmeers re kep uncnged ( = mm, c f = 7 mm, η =.8 nd e rio of c nd c f ws uncnged, oo (c /c f =. Figure 4 sows s increses from 5 o 14, e spe of e weld pool ends o e sorer nd fer, i.e., is leng decreses u is wid increses. However, s increses eyond cerin vlue ( > 14 mm, e weld pool ecomes sorer nd inner. Tis evior of e e source cn lso e lined y e nure of e disriued e source. Tis mens e iger e vlue of, e weker e densiy of e flux. e lower vlues of ( < 14 mm, wen e corresponding densiy of e flux is sill ig enoug, e wid of e weld pool increses s increses nd e weld pool leng decreses for e sme moun of e inpu. iger vlue of ( > 14 mm, e densiy of e flux decreses susnilly, nd e sme e inpu will resul in lesser moun of meled mel, i.e., e smller 7-s UGUST 1999

7 size of e weld pool. However, Fig. 4 sows e size of e weld pool in longiudinl cross secion decreses s increses, i.e., e pool dep decreses s increses. Figure 5 nd sows e effec of e e source prmeer on e op view of e weld pool spe nd is longiudinl cross secion wile e oer e source prmeers re uncnged ( = 5 mm, c f = 7 mm, η =.8 nd c /c f =. I cn e seen from figure ere is n insignificn influence of on e pool dep wile e oer prmeers were uncnged. Is influence on e op view of e weld pool spe is minor. Te weld pool leng sligly decreses s increses from.5 o mm, u e wid of weld pool is uncnged. Figure 6 nd sows e effec of e e source prmeer c f on e op view of e weld pool spe nd is longiudinl cross secion wile oer e source prmeers re uncnged ( = 5 mm, = mm, η =.8 nd c /c f =. I cn e seen from is figure s c f increses e weld pool wid decreses, u is leng increses. Te increse of weld pool leng is more pronounced is fron lf n is ck lf. Te decrese of e weld pool wid is muc lower mgniude. Te evior of op view of e weld pool spe sujeced o e c f is refleced on is longiudinl cross secion, s sown in Fig. 6; owever, e effec of c f on e weld pool dep is insignificn. Figure 7 sows e effec of e efficiency of e source η on e weld pool spe. I s cler from is figure e iger e e source efficiency e igger e weld pool, s is eced. However, e effec of η on e weld pool is more pronounced is ck lf n is fron lf or is side. Experimenl Verificion Specimen nd Merils Te specimens used for rnsien emperure mesuremen were squre HT- 78 ples 4 x 4 x mm wi e cemicl composiion nd mecnicl properies sown in Tles 1 nd, respecively. Two perpendiculr liner segmens of weld eds run on op of e seel ple re sown in Fig. 8. welding roo ws used o run GM weld Fig. 7 Effec of η on e prediced weld pool geomery ( = 5 mm, = mm, c f = 7 mm, c = c f. Fig. 8 Specimen for rnsien emperure mesuremen. WELDING RESERCH SUPPLEMENT 71-s

8 Fig. 9 Experimenl seup for emperure mesuremen. Fig. 1 Geomery of e weld pool: Top view; rnversl cross secion. eds on e ples wi e following welding prmeers: U = 6 V, I = 3 nd v = 3 cm/min. sielding gs mixure of 8% rgon % CO ws supplied L/min. Te filler mel ws MIX-6; is cemicl composiion nd mecnicl properies re given in Tles 1 nd, respecively. Tree plinium nd pliniumrodium CC ermocouples (Ref. 9 were used o mesure e rnsien emperure vrious ineres poins, nd C s sown in Fig. 8. Tese poins were seleced o monior e rnsien emperures round e weld oe, e oom of e weld pool (weld roo nd e corner of e weld inerfce segmens o ge necessry d for e verificion of e nlyicl soluions. Tree oles wi dimeer of 1.6 mm were drilled from e ck surfce of e es ple wi eir deps corresponding o e posiions for poins of ineres, nd C. Termocouples were spo welded o e oom of ese oles o void e possile dmge cused y e welding e source of percussion welding mcine. Termocouples were en conneced o n isolion-ype volge mplifier, e oupu of wic ws conneced o PC wi uil-in nlogue-digil converer crd. picure of e erimenl seup for emperure mesuremen is sown in Fig. 9. compuer progrm ws wrien nd used o conrol inpu nd oupu signls. I provided e necessry d for e mesured rnsien emperure y using e ville clirion d for CC ermocouples. Model Clirion wi e Mesured Weld Pool Geomery Te geomery of e weld pool genered from e ess were repored y mens of poos ken for e weld pool spe e surfce of e welded ple nd is rnsversl cross secion (E-E in Fig. 8 s sown in Fig. 1. Te wo drilled oles for ermocouples corresponding poins nd (Fig. 8 were lso cpured y ese poos. Te posiions of e ermocouples ese oles were reloced nd used for e clculion. Te d for e weld pool profiles were mesured direcly from e poogrps using e scles of e doe Poosop progrm. Figure 11 nd sows comprisons eween e mesured nd prediced d for e op view of e weld pool spe on welded ple nd is rnsversl cross secion. Te prediced d were clculed using e following prmeers of e e source, wic provide e es fi wi e mesured d: = 1 mm, = mm, c f = 1 mm, η =.85 nd c = c f. Tese prmeers were successfully seleced sed on eir effec on e weld pool geomery repored previously. Te e rnsfer meril properies used for e clculion were seleced for HT-78 seel sed on is rnges (Ref. 8, nd ey were e sme s previously repored. I is ovious from figure e presen 3-D e source model cn give very good greemens wi e mesured d given so suile prmeers of e e source cn e crefully seleced. Tis mens e prediced model cn e esily clired wi e erimenl d y selecing is e source prmeers. Ten, i cn e used for vrious simulion purposes. However, Fig. 1 lso sows e presen model fils o predic e complex spe of e weld pool in e rnsversl cross secion. Tis is eced since mny simplified ssumpions ve een used for e developmen of e presen nlyicl soluion. 7-s UGUST 1999

9 Trnsien Temperure Resuls Figure 1 sows e mesured rnsien emperures ree poins represening e weld oe (, weld roo ( nd corner poin (C s illusred in Fig. 8 nd e corresponding prediced rnsien emperures y e presen 3-D power densiy e source soluion. Trnsien emperures re clculed using e soluion oined for doule ellipsoidl disriued e source. Te prmeers of e doule ellipsoids used for e clculion in is secion were ken from e resul oined previously regrding e es fi of weld pool geomery ( = 1 mm, = mm, c f = 1 mm, c = mm nd η =.85, nd e sme e rnsfer meril properies were used (c = 6 J/kg/ C; k = 9 J/m/s/ C; ρ = 78 kg/m 3 nd, erefore, = k/cρ = x 1 6 m /s. I cn e seen from Fig. 1 e mesured emperures of poins nd incresed lmos immediely fer e welding rc pssed eir posiion. Te mximum emperure of poin is iger n of, s eced, ecuse is loced closer o e weld pool oundry n. Figure 1 lso sows e mximum mesured rnsien emperure of poin C is lower n of poin nd, s eced, due o e fc posiion C ws muc frer from e weld pool n of or (ou 3 mm from e weld oe. I is lso wor noing ere delyed increse in emperure of poin C compred wi nd, due o is priculr posiion nd e welding p, ws successfully recorded. n eced rnsiion zone in e rnsien emperure of poin C, due Fig. 1 Comprison eween clculed nd mesured rnsien emperures. Fig. 11 Comprisons eween clculed nd mesured d of e weld pool: Top view of e weld pool; longiudinl cross secion. o is corner posiion, ws successfully recorded, s sown in Fig. 1. Furermore, Fig. 1 lso sows ere is very good greemen eween e prediced nd mesured rnsien emperures for ll ree poins, nd C. Te spe nd e mgniude of e prediced emperures were eced nd were very close o e mesured ones. Te rnsiion zone in e emperure isory of corner poin C due o e welding p ws successfully simuled y e numericl clculion. Figure 13 sows noer se of mesured emperure d compred wi e one sown in Fig. 1. I cn e seen from Fig. 13 e repeiliy of e mesured rnsien emperure is very good in erms of o spe nd mgniude. Te difference in e pek emperure e weld oe, weld roo nd e corner posiions is less n 1 C, wic is quie cceple for ig-emperure grdiens ese posiions. Tis emperure difference lso incorpored e unvoidle erimenl errors in Fig. 13 Repeiliy of e mesured rnsien emperures. WELDING RESERCH SUPPLEMENT 73-s

10 locing e ole deps o wic e ermocouples were onded. I cn e furer noed from Fig. 13 e rnsiion zone of e corner poin emperure of es No. is no very disincive from of es No. 1. Tis my e due o slig cnge in is relive posiion o e corner of e weld p. However, o mesured emperures of poin C (es Nos. 1 nd gree resonly well wi ec oer. From e resuls sown in o Figs. 1 nd 13, i cn e concluded is new nlyicl soluion could offer very good predicion for e rnsien emperures ner e weld pool, nd i is le o simule e compliced weld p s well. Conclusions In is sudy, nlyicl soluions for e rnsien emperure field of semiinfinie ody sujeced o 3-D power densiy moving e sources (suc s semi-ellipsoidl nd doule ellipsoidl e sources were found nd erimenlly vlided. lso, i ws sown e nlyicl soluion oined for doule ellipsoidl e source ws generl one cn e reduced o semi-ellipsoidl, semispere, -D Gussin-disriued e source nd e clssicl insn poin e source. Te nlyicl soluion for e doule ellipsoidl e source ws used o clcule rnsien emperures ree seleced poins in seel ple is sujeced o wo perpendiculr liner welded segmens on is surfce. o e numericl nd erimenl resuls from is sudy ve sowed e presen nlyicl soluion could offer very good predicion for e rnsien emperures ner e weld pool, s well s simule e compliced welding p. Furermore, very good greemen eween e clculed nd mesured emperure d indeed sows e crediiliy of e newly found soluion nd is poenil pplicion for vrious simulion purposes, suc s erml sress nlysis, residul sress clculions nd microsrucure modeling. cknowledgmens Tis work s een sponsored y e Science nd Tecnology gency (ST Reserc Fellowsip progrm, wic is dminisered y JISTEC. Te uors would like o ress eir nks o Dr. Okd for is dvice in e emperure mesuremen nd Drs. Hirok nd Nkmur for eir vlule suppor y mking eir Dien welding roo nd mesuring sysem ville for is projec. mericn Welding Sociey Deroi Secion Inernionl See Mel Welding Conference IX Ocoer 18, Deroi, Micign References 1. Rosenl, D Memicl eory of e disriuion during welding nd cuing. Welding Journl (5: -s o 34-s.. Eger, T. W., nd Tsi, N. S Temperure fields produced y rveling disriued e sources. Welding Journl 6(1: 346-s o 355-s. 3. Jeong, S. K., nd Co, H. S n nlyicl soluion o predic e rnsien emperure disriuion in fille rc welds. Welding Journl 76(6: 3-s o 3-s. 4. Goldk, J., Ckrvri,., nd iy, M Doule Ellipsoid Finie Elemen Model for Welding He Sources, IIW Doc. No Piner, M. J., Dvies, M. H., ersy, S., Jrvis, L., nd W, M lierure review on numericl modelling e gs mel rc welding process. usrlin Welding Reserc, CRC. No. 15, Welding Tecnology Insiue of usrli. 6. Crslw, H. S., nd Jeger, J. C Conducion of He in Solids, Oxford Universiy Press, Cmridge, U.K., pp Crisensen, N., Dvies, V., nd Gjermundsen, K Te disriuion of emperure in rc welding. riis Welding Journl 1(: Rdj, D He effecs of welding: emperure field, residul sress, disorion. Springer-Verlg, pp Jpnese Indusril Sndrd Termocouples, JIS C , Jpnese Sndrd ssociion, Jpn. Te inernionl See Mel Welding Conference is e premiere ecnicl conference dediced o joining meods for insee fricion. You re invied o consider wriing nd presening ecnicl pper for is conference. In order o llow for review nd possile selecion y e Tecnicl Commiee, 5 1 word src, long wi compleed uor pplicion Form mus e sumied o one of e Tecnicl Commiee Cirmen y Novemer 14, To fcilie ccess o your src, plese sumi i in form is compile wi MS Word 6/97. Typicl cegories for e ppers my include: Resisnce Welding Processes rc Welding dvnces Lser Welding Innovive Processes Process Monioring nd Conrol Coed Merils Welding Lig-Weig Merils Welding Hig-Sreng Seel Welding Tuulr Srucures pplicion Sudies Te pper mus e reled o see mel lloys nd/or joining processes used in mnufcuring of commericl producs. I is no requiremen your presenion e n originl effor. Cse isories, reviews nd ppers ve een previously pulised or presened will e considered s long s ey re perimen o e generl ineress of e conference endees. uors mus sumi mnuscrip o e Commiee y July 17,. Plese sumi n uor pplicion form long wi 5 1 word src y Novemer 14. Te form cn e downloded roug or Send i nd e src o e following: Mencem Kimci, WS Tecnicl Cirmn, EWI, 15 rur E. dms Drive, Columus, OH 431, Pone: ( , or e-mil mencem_kimci@ewi.org. 74-s UGUST 1999