Finite Element Analysis of Heat Flow in Single-Pass Arc Welds

DEVELOPMENTFinite Element Analysis of Heat Flow in

Single-Pass Arc Welds

ABSTRACT. The easiest ways to simulatewelding processes are with the decou-pled heat equation of Navier-Stokes ormagnetohydrodynamic (MHD) equa-tions. To decouple the heat equation,functions of energy input rate Q, heat fluxper unit area (or volume) per unit time qand effective thermal conductivity Kthat generate a temperature field by theeffheat equation must be considered. Moreprecisely, the traditional heat sourcemodels (Gaussian and ellipsoidal) andKeff functions must be used cautiously be-cause of the critical responsibility to rep-resent the magnetohydrodynamics of thearc and the fluid mechanics of the weldpool. When thermal efficiency is intro-duced in the decoupled heat equation,both the complex and nonintuitivephysics of the arc and dilution (throughmelting efficiency) are incorporated inthe heat transfer analysis. This paper al-lows the melting efficiency to be relatedto the process variables in a finite ele-ment model (FEM) simulation throughthe energy input rate Q. Transient ther-mal histories and sizes of fusion and heat-affected zones are compared with nu-merical and measured values reported byChristensen, Krutz and Goldak usingboth Gaussian and ellipsoidal powerdensity distribution functions. The FEMcode COSMOS, produced by StructuralResearch and Analysis Corp., was usedfor all the simulations described in thefollowing sections.

Introduction

Welding is a technique commonlyused to join metallic parts. Examples areubiquitous, ranging from delicate elec-E. A. BONIFAZ is with the Materials Depart-ment, Mechanical Faculty, Escuela Politécnicade Chimborazo, Riobamba, Ecuador.

BY E. A. BONIFAZ

tronic components to very large struc-of a structure in and around a weld jointtures. Arc welding is probably the mostresult directly from the thermal cyclepopular manufacturing process for join-caused by the localized intense heating metals used in structural applica-input of fusion welding (Ref. 3). Reduc-tions. The critical first step in creating aing the heat input to the workpiece is ascience base for the design and analysisprimary goal for weld process selectionof welds is to accurately compute theand weld schedule development in thetransient temperature field (Ref. 1).

aerospace and electronics industries. InFigure 1 depicts the arc weldingmicrowelding applications, the depth ofprocess, in which the filler metal is de-penetration is typically less than 1.0 mm,posited on the substrate in the weld in-and hermeticity rather than mechanicalterface direction. Since the electrode isstrength is the primary joining require-“suddenly” applied to a small spot on ament (Ref. 4).

structure, there will be an immediate re-The quantitative understanding ofsponse (shock response) consisting of aconvection (fluid motion) and heat flowvery steep temperature profile in the im-not only in arc discharge but also in weldmediate vicinity of the load. At laterpools is of considerable practical interest.times, the temperature profile will be-To solve the problem, the finite elementcome smoother as the heat diffusesmethod has been chosen for transientthroughout the structure. Figure 1 alsoheat flow analysis for several reasons: Itshows the fine and coarse two-dimen-has the best capability for nonlinearsional (2-D) FEM grids used for comput-analysis and dealing with complexing the temperature field. Only one-halfgeometry, it is the most compatible withof the cross section is considered, be-CAD/CAM software systems and it is thecause of symmetry.

best to deal with electro-thermo-elasto-Perhaps the most critical input data re-plastic analysis.

quired for welding thermal analysis areA literature review of some relevantthe parameters necessary to describe theresearch conducted in this concern isheat input to the weldment from the arcsummarized below.

(Ref. 2). The problems of distortion, resid-Ushio and Matsuda (Ref. 5) devel-ual stresses, grain structure, fast cooling,oped a mathematical formulation to rep-high temperatures and reduced strength

resent the electromagnetic force field inhigh-current DC arcs. Oreper, et al. (Ref.6), showed that the electromagnetic andsurface tension forces dominate the flowbehavior, producing in some cases dou-KEY WORDS

ble circulation loops and, therefore, seg-regation in the weld pool. Eagar and TsaiHeat Transfer

(Refs. 7, 8) showed that both weldingThermal Conductivity

process variables (current, arc length andFinite Element Model (FEM)travel speed) and material parametersHeat Input

have significant effects on weld shape. ItHeat Source Models

was also shown that arc length is the pri-mary variable governing heat distributionand that the distribution is closely ap-proximated by a Gaussian function

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/HCRAESER/TNEMPOLEVED/HCRAESER/TNEMPOLEVED/HCRAESER/TNEMPOLEVED/HCRAESERTWELD INTERFACENEMPOLEVED/HCRAESER/TNEMPOLEVED/HCRAESER/TNEMPOFig. 1 — Weldment configuration and FEM grids for the thick-section, bead-on-plate weld re-Lported by Christensen, et al.(Ref. 14), Krutz and Segerlind (Ref. 11) and Goldak, et al. (Ref. 3).EFine grid (320 elements), coarse grid (80 elements).VED/HCR−r2characteristic radius of flux distribution(m); ξ= the transformation relating theAEq(r)=qOexp2σ2fixed and moving coordinate system = zS2πσ2(1)+ v (τ– t), where v = the welding speedEHere, q(m/s) and τ= a lag factor = c/v (seconds).R/σis the distribution parameter of theois the maximum heat intensityand Goldak, et al.(Refs. 3, 12, 13), pro-TNheat flux. It is important to note that σisposed a three-dimensional double ellip-Edetermined from experimental work andsoidal heat flux model to examine theMis expressed as a function of arc length,three-dimensional temperature, stressPcurrent and electrode tip angle.and strain fields, based on the profileOLTekriwal, et al.(Ref. 9), used the alter-Enative form of the Pavelic (Ref. 10) “disc,“qV(x,y,z,t)=f(63Qf,r)abcππEi.e., the moving Gaussian distributionDheat source model, Equation 2, sug-/−3×2Hgested by Friedman (Ref. 2), Krutz, et al.C(Ref. 11), and Goldak, et al. (Ref. 3), toea2 −3y2−3[z+v(τ−t)]2eb2ec2(3)Rsimulate the butt joint welding of plateswhere a, b, c = the semiaxes of the ellip-AEwith three-dimensional (3-D) models, in-soid in the directions x, y and ξ(m).Scluding the deposition of weld metal:In this model, the front half of theER3×2−3ξ2source is the quadrant of one ellipsoidsource and the rear half is the quadrant ofq(x,z,t)=3Q−2πc2ec2ecanother ellipsoid. It is important to note(2)that fraction values fHere Q = energy input rate (W); c = thef= 0.6 and fr= 1.4were incorporated in Equation 3 to pro-122-s | MAY 2000

vide the best correspondence betweenthe measured and calculated thermal his-tory results (see details about the doubleellipsoid model in Ref. 3).The appropriate use of Equations 2and 3 requires the estimation of the dis-tribution parameters (a,b,c). Goldak, etal.(Ref. 12), suggest making cross-sec-tional metallographic and surface ripplemarkings to fit the heat source dimen-sions. If such data are not available,Christensen’s (Ref. 14) expressionsshould be used.Brown and Song (Ref. 15), using asimulated 3-D model, analyzed distor-tion and residual stresses of large struc-tures. To simulate the heat flux from thearc, the Gaussian function was used be-cause they consider less flux penetrationis involved in arc welding than in high-power-density welding processes (EBWand LBW), where the double ellipsoidalheat flux model can capture the flux pen-etration effectively. Fuerschbach andKnorovsky (Ref. 4) and Omar and Lundin(Ref. 16) analyzed that in the high-power-density welding processes, the heat inputis low and the melting efficiency high.The above citations reveal that theterm melting efficiency had never beenrelated to the welding process variablesin a FEM simulation; also, it appears that,up to the present time, the quantitativetreatment to represent heat flow in arcdischarge, heat flow in weld deposits(substrate and deposited filler metal) andpool convection (fluid motion) in a 3-D(nonlinear-transient) space has been lim-ited. The reason is the difficulty to de-velop a meaningful relationship betweentheoretical models and experimental ob-servations. In this paper, a simple method tomodel the arc welding process is pro-posed based on the Kamala and Goldak(Ref. 1) statement that follows.Kamala and Goldak (Ref. 1) state thatwhether or not the heat equation is cou-pled to other equations such as theNavier-Stokes or MHD equations, thetemperature fields belong to the Sobolevspace H0. If the temperature fields belongto Hheat equation always exists regardless of0, the FEM solution to the uncoupledwhether the heat equation is coupled ornot. Therefore, in order to decouple theheat equation, it is necessary to find theQ and q functions that generate the de-sired temperature field. In solving the un-coupled heat equation, Q and q accountfor resistive heating, and I2R and Keff ac-count for convective heat flow in themelt.The objectives for the present workwere 1) to develop a simple 2-D FEM tocalculate not only the transient thermalhistories but also the sizes of fusion andheat-affected zones in single-pass arcwelds; and 2) to determine the effect of in-troducing the melting efficiency term intothe energy input rate Q, i.e., Q = ηaηmVI,using both Gaussian and ellipsoidalpower density distribution functions.

Modeling Considerations

With regard to modeling the process,the following may be noted:

1) The heat that is transferred to theworkpiece (cathode) is determined by anumber of processes (Ref. 17), including the energy transferred from the arc col-umn by convection, radiation and con-duction; the phenomena that appear inthe cathode region, such as thermoinicemission and the interaction of positiveions with the cathode surface; and theheat developed in the filler metal. Thisheat is transferred to the workpiece viathe molten drops.

The above heat transport mechanismsdetermine the arc efficiency, which is de-fined as the fraction of total process en-ergy delivered to the substrate and welddeposit (Refs. 17, 18): ηworkpiece/total power input)*100%.a = (heat input to2) The weld deposit that develops dur-ing fusion welding of two dissimilar al-loys will attain a chemical compositionintermediate to the two alloys. The finaldeposit composition will depend on theindividual compositions of the materialsand the degree of mixing between the al-loys. The degree of mixing is defined bythe percentage dilution D (Ref. 18)

PctD=

−1

1+VfmEsηaηmVI−EfmVfm*100

(4)

where Vfmis the volume of deposited fillermetal, Efmand Esrepresent the enthalpychange required to melt a given volumeof filler metal and substrate, respectively,and the term ηaηmVI represents the melt-ing power delivered by the arc. Estimationof dilution with the approach presentedby Dupont and Marder (Ref. 18) requiresknowledge of the term thermal efficiency[arc efficiency (ηa) * melting efficiency(ηm)] of the welding process.

Melting efficiency (ηthe ratio of energy used for melting to thatm) is defined aswhich is delivered to the workpiece (Ref.18). Dupont and Marder (Ref. 18) pre-sented a relation of the form

−175ηm=0.5expηaVIS/Eαν(5)

where E is the enthalpy change due tomelting (an average value between fillermetal and substrate), ηais the nominalconstant for a given process arc effi-ciency, ηaVI is the net arc power deliv-

electrode at time 11.5 s.Fig. 2 — Two-dimensional temperature distribution along the top of the workpiece perpendic-ular to the weld (electrode at time 11.5 s). Experimental bead-on-plate weld, V = 32.9 volts, I =1170 amps, v = 0.005 m/s, ηa= 0.95. Melting efficiency ηm= 0.463 in the double ellipsoid heatsource for curve FEA using a coarse grid.

ered to the base metal (V is voltage and Icalled enthalpy formulation is used, theis current), S is the welding speed, αis theenthalpy values can directly be used ifthermal diffusivity at 300 K and νis thethey are known. The enthalpy then in-kinematic viscosity at the melting point.cludes all the other data except the ther-In the present work, the values E = E-3s=mal conductivity and the density10.5 J mm2, ν= 0.84 mm2s–1, α= 9.1(Ref.19).

mms–1were used.

3) The heat of the arc and the moltenModel Assumptions

metal induces heat flow in all three di-mensions in the workpiece. Conse-To compare the results obtained byquently, complex metallurgical changesChristensen (Ref. 14), Krutz andare produced in the fusion zone (FZ) andSegerlind (Ref. 11), and Goldak, et al.heat-affected zone (HAZ).

(Ref. 3), the following assumptions were4) Boundary conditions must be em-necessary:

ployed to account for surface heat losses1) The problem is reduced to find the(natural convective heat transfer, quantictwo-dimensional transient temperatureStefan-Boltzman radiation and forcedfield at a section normal to the weld in-convection due to the flow of the shield-terface — Fig. 1.

ing gas).

2) All the boundaries except the top5) Thermal material constants must besurface were assumed to be insulated.considered as functions of temperature,3) On the top surface, the portion justcomposition and cooling rate (Ref. 19).under the arc was assumed to be insu-6) The phase transformation tempera-lated during the time the arc was playingtures and the corresponding latent heats,upon the surface.

along with the way in which the latent4) A combined convection and radia-heats are released during the phase trans-tion boundary condition h = 24.1*10–4formations, must be considered. If so-

1.61εT(W/m2°C) was used on the remain-

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TNEMPOLEVED/HCRAESER/TNEMPOLEVED/HCRAESER/TNEMPOLEVED/HCRAESER/TNEMPOLEVED/HCRAESERATNEelectrode at time 11.5 s.MPOLEVED/HCRAESER/TNEMBPOLEVelectrode at time 11.5 s.ED/HCRAESER/TNEMPOFig. 3 — Postshock effect caused by effective thermal conductivity. A — Ellipsoidal distribution;LEB — Gaussian distribution. Note the difference in temperature distribution. Experimental con-Vditions documented in Fig. 2.ED/Hder of the top surface. The value ε= 0.9increased heat transfer in the mushy orCwas assumed as recommended for hot-liquidus regions due to convection, i.e.,RArolled steel (Ref. 3).the liquid phase is stagnant (Ref. 20) . TheE5) The temperature-dependent mater-value of Amixdepends on the mixing in-Sial property functions (density and ther-tensity. In this work, AERmal conductivity) published by Mietti-or fmixLis assumed tobe independent of fs. For a continu-/Tnen,et al.(Ref. 19), were used for allous-steel casting process, for instance, aNcalculations. Also from Ref. 19, thermalvalue of AEconductivity in the mushy(Ref. 19).mix= 4 to 6 may be appliedMP(1477–1516°C) and liquid regions wasIn this work, to simulate heat transferOcalculated with the equationby stirring, a value KL175 W/m°C was used. Moreover, foreff= (1+4)*1*35 =EVKeff= (1–fL)Ks+(1+Amix)fLKL(6)comparison purposes, a value Keff =E(1+0)*1*35 = 35 W/m°C also was used.Dwhere fL= liquid fraction; fs/Hs= 1 – fL=6) The temperature-dependent specificsolid fraction; K= thermal conductivityheat values reported by Brown and SongCin solid state; KL= thermal conductivity(Ref. 15) were used for the solid region.RAin liquid state; A7) In the mushy region, the equationEscribing the effect of liquid convectionmix= is a parameter de-S(fluid motion) upon the thermal conduc-CERtivity. If the constant Ap= Cliq+Hmixis 0, there is noTL−Ts(7)was used. Here, H = latent heat of fusion1. In GMAW, the expression h = 13 Re 1/2= 280 J/g, TL= liquidus temperature =Pr1/3Kgas/NPD reported by Tekriwal (Ref. 9)1516°C and Ts= solidus temperature =could be easily incorporated.1477°C (Ref. 20). In the liquid region, a124-s | MAY 2000

value of Cliq= 0.4 J/g°C was used.8) No forced convection was as-sumed, and the effect of gas diffusion inthe weld pool was not considered.1Results and DiscussionFigure 2 shows a comparison of ex-perimental and calculated temperaturedistribution along the top of the work-piece perpendicular to the weld interfacefor 11.5 s (x direction, Fig. 1). As ex-pected, the FEA proposed gives a better“realistic” agreement with the Chris-tensen, et al.(Ref. 14), experimental data.The term “realistic” is used because, aswas already demonstrated in (Ref. 1), toapproximate a 3-D heat transfer analysisof a weld with a 2-D cross-sectionalanalysis introduces errors in the com-puted temperatures. In addition, the max-imum temperature developed at differentnodes at any given time is always higherin the 3-D analysis than the 2-D cross-sectional analysis; therefore, the 2-D FEAresults satisfy the above observations.Kamala and Goldak Ref. 1 believe theerrors in the 2-D approximation can beeliminated by modifying the true powerdensity distribution function. They con-sider that it was this type of modificationthat enabled the obtaining of “accurate”2-D results in previous investigations(Refs. 3–12). As a consequence of that,the double ellipsoid model with appro-priate parameters was proposed to cor-rect for the lack of longitudinal heat flowin 2-D models. The accurate double el-lipsoid results, to the best of the author’sknowledge, are due to heat input excessto the workpiece, captured not only bynodes located below surface, but also bynodes assumed to represent the fillermetal as in place at the start of the analy-sis. So, the magnitude of the real heatinput is overestimated in the moment ofconsidering the net arc power (ηaVI) de-livered to the base metal in the energyinput rate (Q), instead of consideringmelting power (ηaηmVI) in all calcula-tions.Thermal efficiency was noted by Tsaiand Eagar (Ref. 8). They observed the arcefficiency measured on the water-cooledanode was much higher than the “arc”(thermal) efficiency of normal weldingmeasured in the presence of a moltenmetal pool. Pavelic, et al.(Ref. 10), useda value F = 0.3 in expression Q = FVI togive best agreement with the experimen-tally obtained temperature distributions.As noted, the above-mentioned workssupport the idea of this work to use ther-mal efficiency to quantify the energymade available by the arc. According toTsai and Eagar (Ref. 7), arc (thermal) effi-ciency ranges from 30 to 70%.Other explanations for Goldak’s “ac-curate” results are not only the use of afictitious Keff= 120 W/m°C, but also theuse of two heat input fractions ff= 0.6 andfcorrespondence between the measuredr= 1.4, employed to provide the bestand calculated thermal history results. Figure 3 shows the postshock effect ofeffective thermal conductivity on the 2-DFEA-computed temperature distributionfor the selected experimental conditionsdocumented in Fig. 2. At times beyondthe initial shock (e.g.,11.5 s), higher tem-peratures are observed in the ellipsoidaldistribution model. The reason is be-cause the time the arc played upon thereference plane (load time) was 9 s for theellipsoidal model and only 6 s for the discGaussian model.

The finite element solution was sensi-tive to heat distribution and effectivethermal conductivity. The significant dif-ferences in peak temperature values wereattributed to the effective thermal con-ductivity.

It was observed from the fusion zone(FZ) and heat-affected zone (HAZ) thatboth models (ellipsoidal distribution andGaussian distribution [Equation 2]) wereable to approximate the size of weld areawas into the Christensen’s limits (Refs.11–14).

The model results provided a straight-forward approach to understanding theeffects of heat distribution and effectivethermal conductivity. The double ellip-soidal distribution produced lower peaktemperatures but deeper weld penetra-tions. However, in both models, the FZwas completely formed at about 6 s. Thedouble ellipsoid model showed a poorsensitivity to simulate the suddenly ap-plied electrode shock response. Asnoted, at 1.5 s the 723 and 1480°Cisotherms do not appear yet.

Conclusions

1) The decoupled 2-D, cross-section,finite element, nonlinear model pre-sented in this paper closely approximatesactual welding conditions, but must beused cautiously because the results aresensitive to heat source distribution, heatsource magnitude and effective thermalconductivity. However, features of struc-ture-weld interactions can be investi-gated with this 2-D model.

2) Dilution can be accounted for inthe heat transfer analysis through themelting efficiency term.

3) The double ellipsoid model is lesssensitive than the Gaussian model to sim-ulate substrate shock responses.

4) Since both models are very sensi-tive to distribution parameters (a,b,c inEquation 3 and c in Equation 2), to obtain

more accurate predictions and also to ac-Welding Journal71(2): 55-s to 62-s.

count the effect of arc length, an expres-16. Omar, A. A., Lundin, C. D. 1976.sion that combines the Gaussian functionPulsed plasma-pulsed GTA arc: a study of theEquation 1 and the “disc” Gaussian dis-process variables. Welding Journal 58(4): 408-tribution Equation 2 is needed. However,s to 420-s.

in absence of that, the Gaussian distribu-17. Essers, W. G., and Walter, R. 1981.tion Equation 2 is recommended to sim-Heat transfer and penetration mechanismsulate the arc welding processes.

with GMA and plasma-GMA welding. Weld-ing Journal60(2): 37-s to 42-s.

18. DuPont, J. N., and Marder, A. R. 1996.Acknowledgment

Dilution in single pass arc welds. Metallurgi-cal and Materials Transactions 27B: 481–4.The author is grateful for financial19. Miettinen, J., and Louhenkilpi, S. 1994.support from Escuela Politécnica deCalculation of thermophysical properties ofChimborazo, Riobamba, Ecuador.

carbon and low alloyed steels for modeling ofsolidification processes. Metallurgical andMaterials Transactions25B: 909–916.

References

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3. Goldak, J. A., Chakravarti, A. P., andBibby, M. 1984. A new finite element modelAppendix

for welding heat sources. Metallurgical Trans-actions15B: 299–305.

Abbreviations

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perature fields produced by traveling distrib-HAZheat-affected zone

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Keffective thermal conductivity8. Tsai, N. S., and Eagar, T. W. 1985. Dis-LBWefflaser beam weldingtribution of the heat and current fluxes in gasMHDmagnetohydrodynamicstungsten arcs. Metallurgical Transactions16B:Qenergy input rate

841–846.

q

heat flux per unit area9. Tekriwal, P., and Mazumder, J. 1988. Fi-(or volume) per unit time

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