Analytical Approximate Solution for Double Ellipsoidal Heat Source in Finite Thick Plate

Transcription

1 Anaia Approimae Souion for Doube Eipsoida Hea Soure in Finie Tik Pae Anaia approimae souions for doube eipsoida ea soures in finie ik pae ave been derived and aibraed wi e eperimena daa Y N. T. NGUYEN, Y.-W. MAI, S. SIMPSON, AND A. OHTA KEY WORDS Doube Eipsoida Hea Soure Singe Eipsoida Hea Soure Gas Mea Ar Weding Finie od Semi-Infinie od Anaia Approimae Souion ASTRACT. Tis work desribes e deaied derivaion of e anaia approimae souion for a doube eipsoida densi ea soure in finie ik pae. Tis as sown a e souion of e ea soure an be effeive used o predi e erma isor of e ik weded pae as we as wed poo sape geomer and various weding simuaion purposes one e parameers of e ea soure ave been aibraed. Tis approimae souion an be dire used for weding simuaion of finie ik pae wiou e need for impemening e mirror meod as required in a semi-infinie bod. Hene, i an be used as a poenia onvenien oo for soving man probems in erma sress anasis, residua sress anasis, and mirosruure modeing of muipass weds, and oers. Inroduion Te emperaure isor of weded omponens as a signifian infuene on e residua sresses, disorion, and ene e faigue beavior of weded sruures. In order o obain a good prediion for residua sress and disorion in weded joins and sruures, an appropriae ea soure is needed for a purpose. Godak e a. (Ref. 1) firs inrodued a ree-dimensiona (- D) doube eipsoida moving ea soure and used finie eemen meod (FEM) o auae e emperaure fied of a bead on pae. He ad sown a e -D ea soure oud overome e soromings of e previous -D Gaussian mode in order o predi e emperaure of e weded joins wi mu deeper peneraion. Nguen e a. (Ref. ) as reen deveoped a osed form anaia souion for is kind of -D ea soure in a semiinfinie bod and sowed a is souion an be used for wed poo geomer prediion (Ref. ) and for e auaion of residua sresses in a bead-on-pae wed (Ref. ). However, e reen ea soure souion is si imied o e semi-infinie bod and one as o use e mirror image meod wen apping e souion for finie paes. Tis makes impemenaion of e souion a edious ask, espeia for some ompiaed geomeries. In is paper, an anaia approimae souion for doube eipsoida ea soure in finie ik pae as been derived and aibraed wi e eperimena daa. Teoreia Anasis Temperaure Fied Souion ased on Green s Funion for Insananeous Poin Soure Le us onsider ea quani δ(,,, ) aing insananeous a ime a poin (,,) in infinie bod. Te infiniesima rise in emperaure due o is poin ea soure dt(,,,) a poin (,,) and ime in infinie bod as been we esabised (Ref. ) as δ (,,, ) d dt(,,, ) [ a( ) ] ( ) ( ) ( ) ep a were dt(,,,) is an infiniesima rise in emperaure due o e poin ea soure δ(,,,), ρ and are mass densi and speifi ea, k is ea onduion oeffiien, and a is erma diffusivi (a k/ρ). Subsequen, e emperaure fied T(,,,) a ime for is poin ea soure δ(,,,) an be obained as (,,, ) T(,,, ) δ Ginf (,,, ;,,, ) d were T o is e iniia emperaure; G inf (,,,;,,,) is e Green s funion souion for e poin soure of uni magniude in e infinie bod, wi is e emperaure a poin (,,) and ime due o e insananeous poin soure oaed a (,,) a ime and obained as e souion o e ea onduion equaion ( T/ a T) as Ginf (,,, ;,,, ) d 1 [ a ] ( ) ( ) ( ) ep a ased on Equaion, e emperaure fied for an kind of ea soure [ine ea soure (1-D), surfae ea soure (-D), or voume ea soure (-D)] in infinie bod an be obained b arr- (1) () () N. T. NGUYEN is wi ETRS P Ld., Mugrave, Ausraia. Y.-W. MAI is wi Cener for Advaned Maerias & Tenoog (CAMT), Soo of Aerospae, Meania & Mearoni Engineering, and S. SIMPSON is wi Soo of Eeria and Informaion Engineering, Te Universi of Sdne, Ausraia. A. OHTA is wi Maerias Sreng and Life Evauaion Saion, Naiona Resear Insiue for Meas, Japan. 8-S MARCH

2 Fig. 1 Doube eipsoida power densi disribued ea soure. Fig. Singe eipsoida ea soure in a finie ik pae. ing ou e orresponding ine, surfae, or voume inegraion. For a voume disribued ea soure wi ea densi (,,,) e emperaure fied in infinie bod woud be (,,,) T(,,, ) V Ginf (,,, ;,,,) d d d d Noe a is souion is based on Green s funion for a poin ea soure in an infinie bod. If a voume ea soure in a finie bod is onsidered, a new Green s funion for a poin ea soure in e finie bod a saisfies e Neumann boundar ondiion of ero ea densi ( T/ n were n is e norma direion) aross is boundar surfaes soud be adoped as (,,,) T(,,, ) V Gfin(,,, ;,,,) d d d d were G fin (,,,;,,,) is e Green s funion for poin ea soure in finie bod. Approimae Approa for Temperaure Fied Subjeed o Voume Hea Soure in Finie od I an be seen from Equaion 5 a ea souions for various kinds of ea soures in a finie bod an be obained if e Green s funion for e poin ea soure in a pariuar bod is known. However, e Green s funion for e poin ea soure in a finie bod woud be epressed in a mu more ompiaed form an a in an infinie bod. Terefore, finding an anaia souion for Equaion 5 woud beome an amos impossibe ask. An aernaive approimae approa o ompensae for e Neumann boundar ondiion wen deaing wi a finie bod is o keep using e same Green s funion for e poin soure in an infinie bod bu repaing e ea soure in an infinie bod b e effeive ea soure eff (,,,) in e finie bod. Te effeive ea soure soud produe e same amoun of ea inpu ino e finie bod as e origina ea soure woud in an infinie bod. Tis means a e amoun of ea ransferred in e woe infinie bod now woud raer be onained on in e finie () (5) bod. Subsequen, Equaion 5 beomes eff (,,) T(,,, ) V Ginf (,,, ;,,,) d d d d Tis approimae approa based on Equaion 6 is reasonabe as i is based on e wo foowing assumpions: 1) Te prinipe of onservaion of energ for oa ea inpu in infinie and finie bodies. ) Tere is an insignifian effe of Green s funions for a poin ea soure in infinie and finie bodies on e sape of e disribuion of e emperaure fied. An eampe of deriving e ea onduion-on souion and approimae souion for e singe eipsoida ea soure in a semi-infinie bod based on e above anasis is given in Appendi A. I as been sown a e Green s funions for poin ea soure in a semi-infinie bod and a finie bod do no ange e sape of e emperaure fied, i.e., assumpion is orre in e ase of a semi-infinie bod. Terefore, in is paper, is approimae approa as been impemened o obain e anaia souion for a doube eipsoida ea soure in finie ik pae based on Equaion 6. Eipsoida Hea Soures and Teir Approimae Souion for Finie Pae Godak s Eipsoida Hea Soures in Semi-Infinie od Singe Eipsoida Hea Soure Godak e. a. (Ref. 1) iniia proposed a semi-eipsoida ea soure in wi ea densi is disribued in a Gaussian manner rougou e ea soure s voume. Te ea densi (,,) a a poin (,,) wiin e semi-eipsoid is given b e foowing equaion: 6 η V I (,, ) ep ab a b (6) (7) WELDING JOURNAL 8 -S

3 Fig. Speimen for ransien emperaure measuremen. Fig. Cauaed ransien emperaures in omparison wi e measured ones. were a, b, and are eipsoida ea soure parameers as desribed in Fig. 1 ( f b );,, are moving oordinaes of e ea soure; (,,) is ea densi a a poin (,,); V and I are weding voage and urren, respeive, and η is ar effiien. Godak s Doube Eipsoida Hea Soure Praia eperiene wi e singe ea soure sowed a e predied emperaure gradiens in fron of e ar were ess seep an e eperimena observed ones, and gradiens beind e ar were seeper an ose measured. To overome is, wo eipsoids were ombined and proposed as a new ea soure aed doube eipsoida ea soure as sown in Fig. 1 (Ref. 1). Sine wo differen semi-eipsoids are ombined o give e new ea soure, e ea densi wiin ea semi-eipsoid are desribed b differen equaions. For a poin (,,) wiin e firs semi-eipsoid oaed in fron of e weding ar, e ea densi equaion is desribed as were a, b, f, and b are eipsoida ea soure parameers as desribed in Fig. 1, is e ea inpu ( ηiv), r f and r b are proporion oeffiiens represening ea apporionmen in fron and bak of e ea soure, respeive (r f r b ). I mus be noed ere a due o e oninui of e voumeri ea soure, e vaues of (,,) given b Equaions 8 and 9 mus be equa a e pane. From a ondiion, anoer onsrain is obained for r f and r b as r f / f r b / b. Subsequen, e vaues for ese wo oeffiiens are deermined as r f f /( f b ); r b b /( f b ). I is aso wor noing ere a is doube eipsoida disribuion ea soure is desribed b five unknown parameers: e ar effiien η and four eipsoida aes parameers, a b, b b, f, and b. Godak e a. (Ref. 1) impied ere is equivaene beween e soure dimensions and ose of e wed poo and suggesed a appropriae vaues for a b, b b, f, and bf oud be obained b dire measuremen of wed geomer. Effeive Singe Eipsoida Hea Soures in Finie Tik Pae Singe Eipsoida Hea Soures in Finie Tik Pae Le us onsider a singe eipsoida ea soure in a finie pae of wid, eng L, and ikness D. Te oa oordinae of e ea soure (O) is onsrued so a is aes are parae o e fied oordinae of e finie pae (O) as sown in Fig. were origin O is oaed a O(,,). ased on Godak s ea soure in a semi-infinie bod, e singe eipsoida ea soure in finie pae is assumed o ave e simiar form bu deferen maimum ea densi magniude as 6 r f (,, ) ep ab f a b f (8) eff (,, ) maep a b (1) and for poins (,,) wiin e seond semi-eipsoid overing e rear seion of e ar as 6 r b (,, ) ep ab b a b b (9) Furermore, assuming a ea onveion and radiaion are ignorabe due o e sor ime of weding in a quie air environmen, e onservaion of energ in e finie pae requires a L D ep ma d d d (11A) L a b or L ma ep d L D d ep d a ep b 1 (11) Equaion 11 an be furer simpified using e error funion definiion and rearranged for ma as 8 -S MARCH

4 85 -S WELDING JOURNAL Subsiuing Equaion 1 ino Equaion 1 gives e ea densi equaion for e proposed singe eipsoida ea soure in e finie pae as Anaia Approimae Souion for Singe Eipsoida Hea Soure in Finie Tik Pae Le us onsider a finie pae of wid, eng L, and ikness D as in Fig. again. Te approimae souion for e singe eipsoida ea soure in is finie ik pae is based on e approa desribed in e seion ied Approimae Approa for Temperaure Fied Subjeed o Voume Hea Soure in Finie od. Subsiuing Equaions and 1 ino Equaion 5 and rearranging e variabes gives or were T T d a I I I o L D (,,, ) ( ) ma ρ (1) T T d a a d a a o L L (,,, ) ( ) ( ) ( ) ( ) ( ) ma ρ ep ep D d b a d ( ) ( ) ep (1A) a b ab D L L eff,, ( ) ( ) ( ) ep ( ) ( ) (1) a a ab D L L a a ma ( ) ( ) ( ) ( ) (1) Fig. 5 Effe of a on e wed poo geomer. A Top view of e wed poo; ongiudina ross seion of e wed poo. Fig. 6 Effe of b on e wed poo geomer. A Top view of e wed poo; ongiudina ross seion of e wed poo. A A A Fig. 7 Effe of f on e wed poo geomer. A Top view of e wed poo; ongiudina ross seion of e wed poo.

5 A Fig. 8 Eperimena seup for speimen fabriaion and daa aquisiion. A Weding ssem for speimen fabriaion; daa ogging ssem using WedPrin sofware. L I L ( ) ep d a L ( ) I ( ) ep d a a ( ) D ID ep ( ) d b a Inegras I L, I, and I D an be soved using e error funion noaion as a IL a ( ) ep 1 ( ) 1a ( 1a ) ( L) a 1a ( ) ( ) ( 1a ) ( L ) a a ( ) 1 ( ) (15A) b a ID a b ( ) ep 1 ( ) 1a b 1a D b ( D) b a 1a b ( ) ( ) b a 1a( ) b (15C) Subsiuing Equaions 1 and 15A C ino Equaion 1 and subsequen simpifing gives T(,,, ) ρ EL (,, ) Ea (,, ) EDb (,, ) 1a a 1a b 1a ep d (16A) 1a 1a a 1a b were a a I a a ( ) ep 1 ( ) 1a a ( 1a a) ( ) a a a 1a a ( ) ( ) ( 1a a ) ( ) a a a a a ( ) 1 ( ) (15) a 1 ( L ) a( ) a( ) 1 a 1 ( L ) a( ) 1a( ) EL (,, ) ( L ) L ( ) (16) 86-S MARCH

6 A T(,,, ) ρ EL (, v, ) Ea (,, ) EDb (,, ) 1a a 1( ) b 1a( ) ( v) ep d (17) 1a 1a a 1a b I is wor noing ere a e approimae souion obained for e moving singe eipsoida ea soure in finie pae as given b Equaion 17 is of e simiar form as a obained earier b Nguen e a. (Ref. ) for e ea soure in semi-infinie bod, eep for e error funion orreion erms [E(L, v), E(,), and E(D,)] due o pae eng, wid, and ikness, respeive. Effeive Doube Eipsoida Hea Soures in Finie Pae Fig. 9 Comparisons beween e measured and e predied wed poo sape. A Top view of e wed poo sape; ransversa ross seion of e wed poo. Doube Eipsoida Hea Soures in Finie Pae Le us now onsider a e ea soure onsiss of wo quarers of differen eipsoids as sown in Fig. 1. Foowing a simiar proedure as desribed in e previous seion for e singe eipsoida ea soure, densi equaions are obained for e fron and bak af of e doube eipsoida ea soure, respeive as foows a a 1 ( ) a a a( ) a( ) a 1 a a 1 ( ) a a a( ) 1a( ) a Ea (,, ) ( ) a ( ) a (16C) r f ep a b f f, eff (,, ) D ab f b ( L ) ( L ) f f ( ) ( ) a a (18A) 1a D b ( D ) b a( ) 1a( ) b EDb (,, ) b a ( ) a b 1 1 D (16D) b Fina, subsiuing v (for e moving ea soure wi onsan speed v in -direion) o Equaions 16A and gives e fina approimae souion for e moving ea soure in a finie ik pae as r b ep a b b b, eff (,, ) D ab b b ( L ) ( L ) b b ( ) ( ) a a (18) WELDING JOURNAL 87 -S

7 A C D Fig. 1 Marograps of e weded speimens. A Wed bead ross seion V1; wed bead ross seion U; C op view of e wed poo V1; D op view of e wed poo U were r f and r b are proporion oeffiiens represening e ea apporionmen in fron and bak of e doube eipsoida ea soure, respeive, and r f r b. Apping e oninui ondiion of e voume ea soure for e pane, e vaues of e ea densi given b Equaions 18A and mus be equa. Subsequen, e vaues of r f and r b an be evauaed as L L f ( ) ( ) f f rf L L f ( ) ( ) f f ( L ) b ( L ) b b (19A) An anaia souion for e doube eipsoida ea soure is obained b subsiuing ea densi Equaions 18A and ino Equaion 5 and foowing a simiar proedure as desribed in e seion ied Anaia Approimae Souion for Singe Eipsoida Hea Soure in Finie Tik Pae. Te ransien emper- L L b ( ) ( ) b b rb (19) L L f ( ) ( ) f f L b ( ) ( L ) b b I is wor noing ere a is doube eipsoida ea soure for finie pae is desribed b five unknown parameers, wi are e ar effiien oeffiien, η, and four oer geomeri parameers of e ea soure: a,b, f, and b. Anaia Approimae Souion for Doube Eipsoida Hea Soure in Finie Tik Pae 88-S MARCH

8 aure for an arbirar poin (,,) in a finie pae of eng L, wid, and ikness D, subjeed o e doube eipsoida ea soure is given as T (,,, ) ρ ep 1a a 1a b Ea (,, ) EDb (,, ) 1a a 1a b ( v) rel f (, v, f ) ep 1a f 1a f d ( v) rel b (, v, b ) ep 1a b 1a b () τ ma θ 1 eb (, ψ, ua) ed (, ζ, ub) n τ ua τ ub ψ ζ ep dτ τ ua τ u ( ) ( b) ξ ξ re f ξξ u ( ) (,, f ) ep ( τ u f ) τ u f re b u ep ( ξ ξ) (, ξ ξ, ) u ( τ f ) τ u b () were, b, and d are dimensioness eng, wid, and ikness of e pae [ Lv/(a 6), b v/(a 6) and d Dv/(a 6)]; τ ma v /a and ξ v /a are e dimensioness ime variabes; and e(, ξ ξ, u f ), e(,ξ ξ, u b ), e(b, ψ, u a ),and e(d, ζ, u b ) are e orreion erms for finie pae in dimensioness form as foows: were E(L, v, f ), E(L, v, b ), E(,, a ), and E(D,, b ) are given b Equaions 16 D, respeive. Souion for Eipsoida Hea Soures in Dimensioness Form Te souion obained for singe and doube eipsoida ea soures as sown in Equaions 17 and an be epressed furer in mu simper dimensioness form b inroduing e foowing dimensioness parameer variabes as reommended b Crisensen s meod (Ref. 5): Dimensioness oordinaes: ξ v/a, ψ v/a, ζ v/a Dimensioness ime: τ v ( )/a Dimensioness ea soure parameers: ua va /(a 6), u b vb /(a 6), u v /(a 6), u f v f /(a 6), and u b v b /(a 6) Dimensioness finie pae parameers: pae eng, vl/a; pae wid, b v/a, and pae ikness, d vd/a Dimensioness emperaure: θ (T T o )/(T T o ) were T is e referene emperaure Crisensen s operaing parameer (Ref. 5): n v/(a (T T o )) Subsiuing ese parameers ino Equaions 17 and and simpifing gives e foowing dimensioness form of e souions for singe and doube eipsoida ea soures in finie pae, respeive, as τ ma θ 1 e (, ξ ξ, u) eb (, ψ, ua) ed (, ζ, ub) n τ ua τ ub τ u ( ξ ξ) ψ ζ ep dτ τ u τ u τ u ( ) ( ) ( ) a b (1) 1 e (, ξ ξ, uf ) uf τ uf uf ( ξ ξ) uf τ τ τ u f τ u f uf ( ξ ξ) uf τ τ τ u f 1 e (, ξ ξ, ub ) ub τ ub ub( ξ ξ) ub τ τ τ u b τ u b ub( ξ ξ) ub τ τ τ u b 1 b eb (, ψ, ua ) ua b τ ua uaψ ua τ τ τ u a b τ u a uaψ ua τ τ τ u a (A) () (C) WELDING JOURNAL 89 -S

9 1 d ed (, ζ, ub ) ub d τ ub ubζ ubζ ub τ τ τ u b τ τ u b and r f and r b beome: rf uf uf uf uf ub ub (D) (A) are in good agreemen wi ose measured in erms of bo rends and magniude. Te predied emperaures for wed oe poin A sows seeper sope in ooing es ompared wi e orresponding measured emperaures. However, for poin e agreemen beween emperaure sopes in e ooing es is quie reasonabe. Te predied and measured emperaures a poin C are in eeen agreemen and e numeria resus aso sow is kink poin due o e orner posiion of poin C as observed b e es. If more ses of aibraed daa are avaiabe, e reaionsip beween e doube eipsoida ea soure parameers and e weding parameers su as U, I, v for a pariuar se of base and weding maerias an be esabised and used for fuure ea soure and weding simuaion. Tis means a e approimae anaia souion for e deveoped ea soure in finie ik pae an be used o simuae e ransien emperaures of more ompiaed weding pas one e ea soure parameers an be reiab predied from e weding parameers for e assoiaed weding maerias and weding ondiions. ub ub rb uf ub uf ub Resus and Disussion Numeria Proedure In is sud, a numeria proedure was appied o auae e souion for e ransien emperaure fied as desribed b Equaions 16 D and for e doube semi-eipsoida disribued ea soure. A Forran77 ompuer program was wrien o faiiae e inegra auaion in Equaion and o aow for rapid auaion of geomer of e wed poo based on e assumed meing emperaure of 15 C for mid see. Using is program, e effes of various ea soure parameers (a, b, f, and b ) on e predied wed poo geomer were invesigaed. Te foowing maeria properies for ig-sreng see were used for e auaion: ea apai, 6 J/kg/ C; erma onduivi, k 9 J/m/s/ C; densi, ρ 78 kg/m (Ref. ). Weding parameers used for auaion are voage, U 6 V; urren, I A; weding speed, v m/min, and ar effiien, η.8. Transien Temperaure Resus () Using e above desribed numeria proedure, ransien emperaures of ree seeed poins A,, and C in a square see pae mm (as sown in Fig. ) ave been auaed and ompared wi e es resus. Te parameers of e doube eipsoids used for e auaion were esimaed based on eir reaionsip wi e wed poo geomer measured afer weding and aibraed wi e measured emperaure isor. Te besfi vaues of e ea soure parameers obained for ese es speimens are a 7 mm, b mm, f 7 mm, b 1 mm. More deais abou e es speimens and emperaure measuremen are desribed esewere (Ref. ). Figure sows a omparison beween e auaed ransien emperaures a A,, and C and e measured ones based on e above-desribed ea soure parameers, orresponding. I an be seen from is figure a e auaed ransien emperaures Simuaed Wed Poo Resus In is work, a parameri sud was arried ou for various geomeri parameers of e ea soure o evauae eir infuene on e simuaed wed poo. Maeria properies and weding parameers were e same, as indiaed in e seion ied Godak s Eipsoida Hea Soures in Semi-Infinie od, and kep unanged for a simuaions. Figure 5A and sows e effe of e ea soure parameer a on e op view of e wed poo sape and is ongiudina ross seion, respeive, wie oer ea soure parameers are kep unanged (b mm, f a mm, and b / f ). Figure 5A sows a as a inreases from 1 o 5, e sape of e wed poo beomes sorer and faer, i.e., is eng dereases bu is wid inreases. However, as a inreases beond a erain vaue (a > 5 mm), e wed poo beomes sorer and inner. Tis beavior of e ea soure an be epained b e naure of e disribued ea soure. Tis means a e iger e vaue of a, e weaker e ea densi beomes. A e ower vaues of a (a < 5 mm), wen e orresponding ea densi is si ig enoug, e wid of e wed poo inreases as a inreases and e wed poo eng dereases for e same amoun of ea inpu. A a iger vaue of a (a > 5 mm), e ea densi dereases subsania and e same ea inpu wi resu in a esser amoun of meed mea, i.e., e smaer sie of e wed poo. However, Fig. 5 sows e sie of e wed poo in ongiudina ross seion dereases as a inreases, i.e., e poo dep dereases as a inreases. Figure 6A and sows e effe of e ea soure parameer b on e op view of e wed poo sape and is ongiudina ross seion, wie oer ea soure parameers are kep unanged (a 5 mm, f 5 mm and b / f ). I an be seen from a figure a as b inreases from 1 o 8 mm, e poo eng and wid, as we as e wed poo dep, dereases. Te effe of b of e ea soure in finie pae on e wed poo geomer ere is found mu more pronouned an a of e ea soure in semi-infinie bod repored esewere (Ref. ). Figure 7A and sows e effe of e ea soure parameer f on e op view of e wed poo sape and is ongiudina ross seion, wie oer ea soure parameers are kep unanged (a 5 mm, b mm, and b / f ). I an be seen from Fig. 7A a as f inreases from 1 o 1 mm, e wed poo wid dereases, bu is eng inreases wie e wed poo dep dereases as in Fig. 7. Te inrease in wed poo eng is more pronouned a is fron af an a is bak af. Te derease in wed poo wid is a a mu ower magniude. Te beavior of e op view of e wed poo sape subjeed o e f is refeed on is ongiudina ross seion as sown in Fig S MARCH

10 Caibraed wi e Measured Wed Poo In is seion, i wi be sown a ere is an aernaive for esabising usefu reaionsips beween e doube eipsoida ea soure and e weding parameers, i.e., b aibraing e sapes of e measured and predied wed poo using e ea soure numeria souion. Simiar, one ese reaionsips are esabised, e ea soure parameers an be auaed from e weding parameers for a pariuar maeria and weding proess and en be used for e weding simuaion on e basis of e ea soure souions derived in is work. Weding Speimens. Two differen ses of gas mea ar (GMA) weding parameers were used o fabriae differen sapes of wed bead a were run on op of e enra ine of wo mid see paes of 75 1 mm. A moving abe wi onroed raveing speed (v) was se up under e weding gun of a GMAW maine. Te weding ssem used for speimen fabriaion and daa aquisiion is sown in Fig. 8A and, respeive. Te weding parameers, voage (U), urren (I), and weding wire feed rae (v feed ), are onroabe and moniored b means of a sofware aed WedPrin, wi as been deveoped b e Soo of Eeria and Informaion Engineering, Universi of Sdne. Te gas used for bo ses of speimens was 1% CO wi a fow rae of 15 L/min. Te weding parameers for wo ses of speimens V1 and U are U 1.9 V, I 11.7 A, v 5.9 mm/s; and U.9 V, I A, and v 1.5 mm/s, respeive, wie e wire feed rae was kep e same a v feed in./min (17 mm/s). Caibraed Resus. Figure 9A and sows omparisons beween e measured and predied daa of op view of e wed poo sape on weded pae and is ransversa ross seion for bo ses of weded speimens. Te sapes of e wed poo and wed bead ransversa ross seion were measured dire from marograps (as sown in Fig. 1) b means of a Windows based daa oeion program. Te predied daa were auaed using e foowing ea soure parameers for V1 and U speimens, wi provide e bes fi wi e measured daa. For e V1 speimen: a 1 mm, b 6 mm, f 7 mm, b 15 mm, η.85 For e U speimen: a 1 mm, b mm, f 5 mm, b 1 mm, η.85 Tese parameers were suessfu seeed based on e informaion of eir effe on e wed poo geomer repored earier. Te ea ransfer maeria properies used for e auaion were seeed for mid see (Ref. 6) as k J/m/s/ C, 69. J/kg/ C, and ρ 78 kg/m. Figure 9A sows ver good agreemen beween e measured and e predied wed poo sape for bo ses of weded speimens V1 and U, wi represen wo eves of ea inpu per uni of eng q (quiη/v) of.589 kj/mm and.68 kj/mm, respeive. I aso sows a as e ea inpu, q, dereases e op view of e wed poo beomes sorer and narrower. Figure 9 sows ver reasonabe agreemen beween e measured and e predied sape of e wed bead ross seion for bo V1 and U. Figure 9 aso sows a e dep of peneraion inreases as e weding urren inreases (from 11.7 o A) despie e fa a e eve of ea inpu q dereases (from.589 o.68 kj/mm). Tis means a e effe of e weding urren on wed bead peneraion is sronger an a of e ea inpu per uni of eng. Conusions Anaia approimae souions for singe and doube eipsoida ea soures in finie ik pae ave been derived and suessfu aibraed wi e es resus. Te souion for e doube eipsoida ea soure in finie ik pae was used o auae ransien emperaures a seeed poins in a see pae as we as arring ou e parameri sud of e simuaed wed poo. Ver good agreemen beween e auaed and measured emperaure daa as been obained and i as been sown a e predied wed poo an be aibraed wi e measured one b seeing suiabe ea soure parameers. One e reaionsips beween e ea soure and weding parameers an be esabised eier b aibraion wi e measured emperaure isor or wed bead profie measuremen, ese an be effeive and onvenien used for various weding simuaion purposes wiou e need o use e mirror meod as required in e semi-infinie bod. Tis souion oud be used as a onvenien oo for man probems in erma sress anases, residua sress anasis, and mirosruure modeing of muipass weds. Aknowedgmens Tis work is urren sponsored b e Universi of Sdne s U feowsip program. Te auors woud ike o epress eir sinere anks for e grea suppor for is proje provided b e Soo of Aerospae, Meania, and Mearoni Engineering, Universi of Sdne. Speia anks are due o Prof. Ian H. Soan, Deparmen of Maemais, Universi of New Sou Waes, for is usefu ommens during preparaion of is paper and o Mare H. Kaegi for is ep in speimen fabriaion. Referenes 1. Godak, J., Cakravari, A., and ibb, M A doube eipsoid finie eemen mode for weding ea soures. IIW Do. No Nguen, N. T., Oa, A., Masuoka, K., Suuki, N., and Maeda, Y Anaia souion for ransien emperaure in semi-infinie bod subjeed o D moving ea soures. Weding Journa 78(8): 65-s o 7-s.. Nguen, N. T., Oa, A., Masuoka, K., Suuki, N., and Maeda, Y Anaia souion of doube-eipsoida moving ea soure and is use for evauaion of residua sresses in bead-on-pae. Pro. of In. Conf. on Fraure Meanis and Advaned Engineering Maerias. Ediors: Lin Ye and Yiu Wing Mai, De. 8 1, Universi of Sdne, Ausraia, pp Carsaw, H. S., and Jaeger, J. C Conduion of Hea in Soids. Oford Universi Press, pp Crisensen, N., Davies, V., and Gjermundsen, K Te disribuion of emperaure in ar weding. riis Weding Journa 1(): Radaj, D Hea Effes of Weding: Temperaure Fied, Residua Sress, Disorion. Appendi A: Derivaion of e Ea and Approimae Souion for Singe Eipsoida Hea Soure in Semi-infinie od Ea Souion Le us onsider a singe eipsoida ea soure (Equaion 1 in e main e) oaed in surfae of semi-infinie bod as in Fig. 1, e ea souion for e emperaure fied based on Equaion 5 as desribed in e seion ied Temperaure Fied Souion ased on Green s Funion for Insananeous Poin Soure as T(,,, ) ma d d d ep a b Gsemi inf (,,, ;,,, ) d (A1a) were ma 6 /a b for semi-infinie bod and ma /a b for infinie bod; WELDING JOURNAL 91 -S

11 Gsemi inf (,,, ;,, ) 1 [ a ] ( ) ( ) ep a ( ) a ( ) ep ep ( ) a ( ) Subsiuing A1b for A1a and rearranging gives (A1b) 1a( ) u a( ) a ( ) 1a 1a( ) a a v a a( ) a ( ) 1a a T(,,, ) ma ( ) d ep d a a ρ ( ) ( ) ep d a a ( ) ( ) ep ep ( ) ( ) d b a b a Noing a ( ) ep d a ( ) ep ep 1 ( u ) du a ( ) a( ) 1a (A) (Aa) w a b 1 ( ) b b a( ) a ( ) 1a b w a b 1 ( ) b b a( ) a ( ) 1a b b wo a ( ) 1a b Making use of error funion as b ( b) ep( ) d en Equaions Aa and Ad an be simpified furer as ( ) ep d a a ( ) ep ep 1 ( v ) dv a ( ) a a a( ) 1a a (Ab) ep ( ) d a ( ) a( ) ep 1 1 a ( ) a ( ) (Aa) ( ) ep d b a ( ) ep ep 1 ( w ) dw a ( ) b w b a( ) 1a b (A) ep ( ) d a a ( ) a a( ) ep 1 1 a ( ) a a ( ) a (Ab) were ( ) ep d b a ( ) ep ep 1 ( w ) dw a ( ) b w b a( ) 1a b (Ad) ep ( ) d b a ( ) b a( ) ep 1 1 a ( ) b a ( ) b 1 ( w ) (A) 9-S MARCH

12 ep ( ) d b a ( ) b a( ) ep 1 1 a ( ) b a ( ) b 1 ( w ) (Ad) Subsiuing Equaions Aa o Ad ino Equaion A and simpifing gives maab T(,,, ) ep d a a a a b (A5) 1a a 1a b 1a ( ) Subsiuing e vaue ma 6 /(a b ) for semi-infinie bod ino Equaion A5 and furer simpifing gives T(,,, ) ρ ep d a a a a b (A6) 1a a 1a b 1a Ginf (,,, ;,,, ) 1 [ a ] ( ) ( ) ( ) ep a Simiar, subsiuing A7b and A7 o A7a and rearranging gives ma, semi inf T(,,, ) d a( ) ( ) ( ) d ep ep a d a a ( ) ( ) ep d b a ( ) Subsiuing Equaions Aa o A and A7b ino Equaion A8 and simpifing gives (A7) 1 ( w ) T(,,, ) ρ ep d a a a a b (A9) 1a a 1a b 1a Tis is e ea souion for a singe eipsoida ea soure in e semi-infinie bod derived b using e respeive Green s funion for a poin ea soure. Approimae Souion Using e Effeive Hea Soure and Green s Funion for a Poin Hea Soure in an Infinie od Now e us onsider e same singe eipsoida ea soure in a semi-infinie bod bu r o sove i b using e approimae approa as desribed in e seion ied Approimae Approa for Temperaure Fied Subjeed o Voume Hea Soure in Finie od in e main e of is paper, aording o wi e approimae souion for is ea soure is given b Equaion 6 as ma, semiinf T (,,, ) d d d ep a b Ginf (,,, ;,,, ) d (A7a) Equaion A9 gives e approimae souion for e singe eipsoida ea soure in a semi-infinie bod. I is wor noing ere a e emperaure fied given b e approimae souion is inreased b a faor of (1 (w )) ompared o a given b e ea souion as in Equaion A6 were wo b a ( ) 1a b Te vaue of error funion (w ) beomes ero ((w )) wen, i.e., e approimae souion beomes e ea one a e surfae of a semi-infinie bod. In genera, e vaue of error funion varies beween and 1 ( (w ) 1) and for a erain ime ( ) i inreases as inreases, i.e., e approimae souion woud overpredi e emperaure fied in e ikness direion. I is wor noing ere a e sape of e emperaure fied in a semi-infinie bod obained b e approimae approa is e same as a b e ea souion. 6 were ma, semi inf ab for semi - infinie. (A7b) WELDING JOURNAL 9 -S