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Thermal Analysis of the Arc Welding Process: Part I. GeneralSolutions
R. KOMANDURI and Z.B. HOUAn analytical solution for the temperature-rise distribution in arc welding of short workpieces isdeveloped based on the classical Jaeger’s moving heat-source theory to predict the transient thermalresponse. It, thus, complements the pioneering work of Rosenthal and his colleagues (and others whoextended that work), which addresses quasi-stationary moving heat-source problems. The arc beamis considered as a moving plane (disc) heat source with a pseudo-Gaussian distribution of heatintensity, based on the work of Goldak
et al.
It is a general solution (both transient and quasi-steadystate) in that it can determine the temperature-rise distribution in and around the arc beam heat source,as well as the width and depth of the melt pool (MP) and the heat-affected zone (HAZ) in weldingshort lengths, where quasi-stationary conditions may not have been established. A comparative studyis made of the analytical approach of the transient analysis presented here with the finite-elementmodeling of arc welding by Tekriwal and Mazumder. The analytical model developed can determinethetimerequired for reaching quasi-steadystateand solvetheequation for thetemperaturedistribution,be it transient or quasi-steady state. It can also calculate the temperature on the surface as well aswith respect to the depth at all points, including those very close to the heat source. While someagreement was found between the results of the analytical work and those of the finite-element method(FEM) model, there were differences identified due to differences in the methods of approach, theselection of the boundary conditions, the need to consider image heat sources, and the effect of variable thermal properties with temperature. The analysis presented here is exact, and the solutioncan be obtained quickly and in an inexpensive way compared to the FEM. The analysis also facilitatesoptimization of process parameters for good welding practice.
I. INTRODUCTION
quasi-steady-stateconditionsthatcanbejustifiedexperimen-tally when the length of the weld is long compared to the
A
RC
welding is one of the most common manufacturingextent of heat. This means that an observer stationed at theoperations for the joining of structural elements for a myriadpoint heat source fails to notice any change in the tempera-of applications, including bridges, building structures, cars,ture around him as the source moves on. Rosenthal alsotrains, farm equipment, and nuclear reactors, to name a few.gave an alternate analogy for this, wherein the temperatureIn these applications, it is desirable, and oftentimes critical,distribution around the heat source is represented by a hillto determine the temperature-rise distribution in relation tothatmovesasarigidbody onthesurfaceoftheplanewithoutthe location, time, and welding conditions, for it affectsundergoing any modification either in size or shape.the metallurgical conditions at and near the weld and theStarting from the following PDE of heat conduction,consequentstrengthandreliabilityofthejoint.Inthisinvesti-Rosenthal applied it for welding (by assuming the heatgation, an analytical solution for the temperature rise distri-source to be a moving point or a moving, infinitely longbution in arc welding of short workpieces is presented basedlineheatsource)byconsideringamovingcoordinatesystem.on the moving heat source theory of Jaeger
[1]
and Carslawand Jaeger
[2]
to predict the transient thermal response.
2
x
2
2
y
2
2
z
2
1
a
t
[1]Rosenthal laid the foundation for the analytical treatmentof the heat distribution in welding during the late 1930s andWhen the srcin of the moving coordinate system coin-in the mid-1940s.
[3,4,5]
Using the Fourier partial differentialcides with the moving heat source and moves along with itequation (PDE) of heat conduction, he introduced the mov-at the same speed (with its
X
-axis coinciding with the
x
-axising coordinate system to develop solutions for the point andof the srcinal absolute coordinate system), the relationshipline heat sources and applied this successfully to address abetween the coordinates of the point where the temperaturewide range of welding problems. His analytical solutions of rise is concerned along the
X
- (or
x
-) axis at any time
t
isthe heat flow made possible for the first time the analysis of given by
X
x
vt
. Substituting this in Eq. [1], the generalthe process from a consideration of the welding parameters,PDE of heat conduction in a moving coordinate system cannamely, the current, voltage, welding speed, and weld geom-be obtained asetry. To facilitate solution of the PDE, Rosenthal assumed
2
X
2
2
y
2
2
z
2
va
X
1
a
t
[2]
R.KOMANDURI,ProfessorandMOSTChairinIntelligentManufactur-
Equation [2], even for a unidirectional heat flow,
e.g.
, along
ing, and Z.B. HOU, Visiting Professor, are with the School of Mechanical
the
X
-axis, involves three variables, namely,
X
,
, and
t
.
and Aerospace Engineering, Oklahoma State University, Stillwater, OK
Hence, solution of this equation by the separation of vari-
74078.Manuscript submitted September 13, 1999.
ables or other similar techniques would not be feasible.
METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 31B, DECEMBER 2000—1353
In order to solve this problem, Rosenthal incorporated an
II. REVIEW OF LITERATURE
experimental observation,
[4,5,6]
namely, when the workingSince the pioneering work of Rosenthal, considerabletime of the moving heat source is sufficiently long, theinterest in the thermal aspects of welding was expressed bytemperature-rise distribution around the heat source (in themany researchers, as evidenced by numerous publications
[7–37]
.moving coordinate system) would reach quasi-stationary-They can be classified into three categories. The first onestate conditions,
i.e.
,
/
t
0. The PDE of heat conductiondeals with the analytical and experimental work; the secondin a moving coordinate system for the quasi-steady-stateone deals with numerical methods, such as the finite-differ-condition is, thus, given byencemethod andfinite-element method(FEM); andthethirdone deals with the effect of thermal properties at varioustemperatures on the temperature distribution in the welded
2
X
2
2
y
2
2
z
2
va
X
[3]plate. As the literature is numerous in this field, only litera-ture pertinent to the present investigation will be brieflyBy experimentally observing the shape of the weld pool onreviewed.Thereadersarereferredtonumerouscontributionsthe surface, which is of a tear-drop-like shape, Rosenthalmade over the years in the
Welding Journal
and to otherconsidered the final solution of Eq. [3] as a product of twolearned publications and conference proceedings.separate functions, given byIn the following text, the literature covering the first twoaspects, namely, analytical and experimental studies, as well
e
vX
/2
a
(
X
,
y
,
z
) [4]as the numerical methods, will be briefly reviewed. Thethird aspect, dealing with the effect of thermal properties atThe first part of Eq. [4], namely,
e
vX
/2
a
, is an asymmetricvarious temperatures on the temperature distribution in thefunction along the
X
-axis. Here,
e
vX
/2
a
e
v
(
X
)/2
a
,
i.e.
, thewelded plate, will be discussed in a companion article, Partlarger the value of the term
vX
/2
a
, the higher the asymme-II
[38]
of this two-part series. It may be noted that much of try. The second part, namely,
(
X
,
y
,
z
), is considered as a the intense analytical work on this subject was undertakensymmetricfunction. Asa whole,
isan asymmetric function starting in the 1930s, with the pioneering work of Rosenthal,along the
X
-axis. This consideration is acceptable, for it is andcontinuedtoaboutthemid1970s,afterwhichthenumer-closer to the real situation. In practical cases of moving heat- ical methods (both the finite-difference method and FEM)source problems, the rise of the temperature in front of the predominated. Consequently, one would see relatively fewerheat source (where
X
is positive) is steeper than the fall of references to the analytical approach in recent literature, andthetemperaturebehind theheatsource(where
X
isnegative), this was reflected somewhat in the list of References in
i.e.
, the temperature distribution is asymmetric along the
X
- this article.axis relative to the heat source. The larger the value of theterm
v
/2
a
, the higher the asymmetry. When
v
/2
a
0 (i.e.,A.
Analytical and Experimental Studies on the
0), it becomes a symmetrical function (only the second
Temperature Distribution in Welding
part exists), which is the case for a stationary heat-sourceproblem. With this substitution, one needs to solve only theChristensen
et al.
,
[10]
based on Rosenthal’s equation forsymmetric function,
(
X
,
y
,
z
). Thus, themoving heat-sourcethe point heat source moving across the surface of a semi-problems are greatly simplified. For solving the moving,infinite body, developed generalized plots of the nondimen-infinitelylonglineheat-sourceandmovingpointheat-sourcesional temperature-rise distribution (on the surface as wellproblems, Rosenthal changed the coordinate systems intowith respect to the depth), which can be used for the estima-cylindrical and spherical ones, respectively, to reduce thetion of the width and depth of the HAZ, cooling rate, andnumber of variables, and developed solutions for a numbertime of residence between given temperatures, which, inof quasi-stationary moving heat-source problems.general, can be used for all combinations of materials andIt is well known at the outset that Rosenthal’s solutions,welding conditions. They, however, cautioned the use of while valid at locations farther away from the heat source,theseplotsfor specificcases, except for thegeneral guidanceare subjected to considerable error at or near the heat source,of the temperature distribution. Wells
[11]
considered a two-due to the assumption of a point or line heat source (zerodimensional (2-D) moving rectangular heat source with aarea). In fact, Rosenthal and Schmerber
[6]
cautioned theuniform distribution of heat intensity. While Rosenthal’sapplication of the point or line heat-source model for loca-solution predicts the shape of the melt pool (MP) behindtions at distances of less than a few millimeters (
6 to 8the heat source, Wells proposed the inverse problem of esti-mm) from the heat source, due to the finite size of the heatmating the heat input and welding speed by examining thesource. Based on the analysis of the temperature distribution finished weld. Apps and Milner
[12]
investigated heat flow induring welding, Rosenthal came to the following important argon-arc welding (without a filler metal) for aluminum,conclusions: (1) the rise of the temperature in front of the lead, nickel, copper, and Armco iron. They compared theheat source is steeper than the fall of the temperature behind theoreticaland experimentalweld-poolshapesfor thesemet-thesource,(2)themetalbeingweldedismorequicklyheated als for various heat inputs and welding speeds.than cooled from a given temperature, (3) increasing the Barry
et al.
[13]
developed an analytical model by modi-currentdensity widenstheheat-affectedzone(HAZ)without fying the concept of a strip heat source, wherein the heatmuch change in the shape of the isotherms, (4) the speed input is visualized as being uniformly distributed over aof welding affects most the shape of the isotherms, (5) the band whose width is about twice the weld bead. They mea-higher the welding speed, the more elongated the isotherms, sured the surface temperatures at key locations to determineand, finally, (6) the greater the heat diffusion of the work the peak-temperature distributions and heat-transfer effi-ciency. Paley
et al.
[14]
investigated the heat flow in weldingmaterial, the more circular the shape of the isotherm.
1354—VOLUME 31B, DECEMBER 2000 METALLURGICAL AND MATERIALS TRANSACTIONS B
heavy (thick) steel plates. They used the characteristic etch- heat conduction for three cases of welding. First, a quasi-steady-state model for a moving heat source to predict theing boundaries of the steels used (HY-80, T-1, and maragingshape of the HAZ and cooling times between 800
C andsteels) to identify specific peak temperatures. In considering500
C in wide plates; second, an instantaneous line heat-the effects of the boundary surfaces, they assumed that allsource model to predict the cooling time from solidificationsurfaces (bottom or edge of the plate) are adiabatic. Theseto 100
C; and third, an unsteady heat model to predict localare surfaces of mirror-image symmetry with respect to thepreheating. They compared the experimental results with thedistribution of heat sources. In analyzing the peak tempera-analytical results. Jeong and Cho
[23]
developed an analyticaltures at any point of interest, the nearest heat sources wouldsolution for the transient temperature distribution in fillethave a significant effect, while the contribution from anyarc welds using an energy equation and compared it withotherheatsourcewoulddecreaseasthesquareofthedistanceGTA and flux-cored arc experiments under various condi-from the point under consideration.tions. Using the conformal mapping technique, they trans-Tsai
[15]
evaluatedvariousmathematicalmodelsofthether-formed the solution of the temperature field in the plate of mal behavior of metals during welding and summarizeda finite thickness to the fillet welded joint. Nguyen
et al.
[24]
their applicability in solving practical welding problems.developed analytical solutions for the transient-temperatureThe objective is to assess a quick solution, or to formulatefieldof asemi-infinitebodysubjected toadouble-ellipsoidalameaningfulexperimentalprocedureforquantitativeresults.power density moving heat source with conduction only.Nunes
[16]
developed an extended Rosenthal weld model inThey compared the analytical results with experimentala multipolar expansion form. Within a multipolar-expansionbead-on-plate specimens and found good agreement.content, he modeled phase changes by thermal dipoles andcirculation in the molten weld pool by thermal quadrupoles.B.
Numerical Techniques for the Temperature-Rise
The model proposed is anticipated to provide insight into the
Distribution in Welding
heat flow in welds. Eager and Tsai
[17]
modified Rosenthal’smodel to include a 2-D Gaussian distributed heat source andOnthenumericalside,severalinvestigations
[7–9,25–32]
weredeveloped a solution for a traveling distributed heat sourcemade using finite-difference method and FEM analyses, inon a semi-infinite plate to provide the size and shape of theview of certain advantageous features associated with thesearc weld pools. Their assumptions include the absence of methods. For example, Pevelic
et al.
[25]
developed a finite-convective or radiative heat flow, constant average thermal difference method to determine the temperature distributionproperties, and a quasi-steady-state semi-infinite medium. in a 2-D plate using the line heat source. The shape of theThe welding parameters, namely, current, arc length, and melt pool was correlated with the welding variables, andtraverse speed and material properties (namely, thermal dif- this isotherm was used as a boundary condition. Thus, infusivity) have significant effects on the weld shape. They this numerical method, experimental work is required tocompared the theoretical predictions with experimental determine the boundary conditions. Better agreement of theresults on carbon steels, stainless steel, titanium, and alumi- peak temperatures was found between the analytical andexperimentalvalues.PaleyandHibbert
[26]
conductedanothernum, with good agreement.elegant study of the computation of the temperatures inZacharia
et al.
[18,19]
developed thermal models for bothactual welds by comparing the computed values from theautogeneous and nonautogeneous welding. They developedheat-conduction equation with those obtained from the met-a transient, three-dimensional (3-D) computer simulationallurgical sections of the actual welds. Good correlation of model for autogeneous welding, taking into account thethe fusion temperature and
A
1
temperature isotherms wereconditions of heat transfer, including convection of a gasreported. Thus, using a computer program, they producedtungsten arc(GTA)weld pool. They also developedasimilargraphical displays of both the maximum temperatures andmodel for nonautogeneous, moving-arc GTA welding pro-the moving temperature field on the surface, as well ascess.Boththemodelsincorporatethecompletesetofinterre-in the vertical plane, of the welded plate. Friedman andlated thermophysical phenomena.Glickstein
[27]
developed a FEM analysis for transient heatBooandCho
[20]
developedananalyticalsolutiontopredictconduction to investigate the effect of a number of weldingthe transient temperature distribution in a finite-thicknessparameters, including the magnitude of heat input from theplate during arc welding, using a 3-D heat-conduction equa-arc, the distribution of the heat input over the surface of thetion with convection boundary conditions at the surface of weldment, and the duration of the heat input on the thermal-welding. Due to the flow of the shielding gas, a forced-response characteristics—in particular, the weld bead shapeconvection boundary condition was assumed at the top sur-andthedepthofpenetration.Theydemonstratedthepotentialfaceoftheweldmentbeneaththeweldingtorch,andanaturalforcalculatingtheoptimumcombinationofweldingparame-convection condition was assumed at the bottom surface.ters for a given weld joint. Wilson and Nickell
[28]
applied theThe analytical results were compared with the experimentalFEM to heat-conduction analysis. They applied a variationalresults obtained by GTA bead-on-plate welding on aprinciple to the transient heat-conduction analysis of com-medium-carbon steel under various welding conditions. Tsaiplex solids of arbitrary shape with temperature and heat-
et al.
[21]
developed a semiempirical, 2-D finite-element heat-flux boundary conditions. Krutz and Segerlind
[29]
used atransfer model to investigate thermal-related welding prob- nonlinear finite-element model to optimize the weldinglems. They developed a conduction model that treats the parameters for weld-joint strength when a certain desiredmeltinginterfaceasaninnerboundarytocalculatethequasi– metallurgical structure is achieved. Friedman
[30]
developedsteady state temperature field and cooling rate in the weld a thermomechanical analysis of the welding process usingHAZ. theFEM. Themodel enablescalculation ofthetemperatures,stresses, and distortions resulting from the welding process.Kasuya and Yurioka
[22]
developed analytical solutions for
METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 31B, DECEMBER 2000—1355
Goldak
et al.
[7]
developed a mathematical model for weld results. This, however, does not restrict the applicability of the FEM, for it offers other advantages, including the abilityheat sources based on a Gaussian distribution of the powerto incorporate variable thermal properties with temperature,density. They developed a nonlinear, transient FEM heat-the determination of the residual stresses,
etc.
Thus, the useflow program for the thermal-stress analysis of welds. Theyof analytical and computational methods should be consid-computed the results of temperature distributions for sub-ered as complementary and not to the exclusion of eachmerged arc welds in thick workpieces and compared themother. It would have been illuminating had the FEM analysiswith the experimental values of Christensen
et al.
[10]
andpresented by Tekriwal and Mazumder
[8,9]
also been done, inreported excellent agreement. Tekriwal and Mazumder
[8,9]
addition to using variable thermal properties with tempera-developed a 3-D transient heat-conduction model for arcture, at average thermal properties, so that the analyticalwelding using the FEM analysis software ABAQUS. Theyresults presented here could be compared directly with thecompared the numerically predicted sizes of the melt poolFEM under the same conditions.and the HAZ with the experimental results obtained by theWhile there are numerous applications where quasi-sta-United States Army Construction Engineering Researchtionary conditions are valid, there are other applicationsLaboratory (Champaign, IL) and found good agreement.where this may not be the case,
e.g.
, when the length of theNa and Lee
[31]
conducted 3-D finite-element analysis of weld is short or the velocity of the arc beam is low. To thethe transient temperature distribution in GTA welding. Theyknowledge of the authors, no analytical solution for this caseintroduced a solution domain that moves with the weldingexists. The objective of this investigation is to develop anheat source to minimize the number of elements and, conse-analytical solution (both transient and quasi-steady state) forquently, the computational time. As the solution domainthe temperature-rise distribution in and around the arc-beammoves with the progress of welding, new boundary condi-heat source, as well as for the width and depth of the MPtions and new elements were generated in front of the heatand the HAZ. The results of this model were compared withsource, while some elements disappear in the rear of it. Thethose of the finite element modeling of arc welding reportednumerical analysis was verified with GTA welding experi-by Tekriwal and Mazumder,
[8,9]
which, in turn, were com-ments on a medium-carbon steel under various welding con-pared with the experimental results obtained. In their work ditions. However, sincethe moving-solution domain issmallas well as in the present investigation, the length of the weldcompared to the rest of the weld structure, two kinds of is deliberately considered to be short (25.4 mm, or 1 in.)boundaries, namely, asolid metal–atmosphereboundary andwhere quasi-stationary conditions may not have been actu-a solid metal–solid metal boundary, have to be considered.ally established. The work material, the welding conditions,Silva Prasad and Sankara Narayanan
[32]
developed a similarthe geometry of the weld,
etc.
, were taken to be essentiallytechnique involving finite-element analysis of the tempera-the same as in Tekriwal and Mazumder to enable directture distribution during arc welding using a transient adap-comparisons. The analytical model developed here is usedtive grid technique. It gives a finer mesh around the arcto determine the time required for reaching the quasi-steadysource, where the temperature gradients are high, and astate and to calculate the temperatures at all points on thecoarsermeshinotherplaces.Thisway,boththeaccuracyandsurface as well as with respect to the depth, including thecomputational efficiency of the analysis can be increased.points close to the heat source, be it transient or quasi-steadyIt is clear from this brief review of literature that, althoughstate. Similarly, by considering the length of the weld to besignificant efforts were made in developing analytical mod-long enough for quasi-steady-state conditions to prevail (forels for the quasi-stationary conditions, limited efforts wereexample, ten times the srcinal length), the temperature dis-expended to predict the transient thermal response excepttribution around the arc-beam heat source, as well as theby numerical techniques, such as the finite-differencewidth and depth of the MP and the HAZ under quasi-steady-method and FEM. While the FEM has many advantages instate conditions in welding, were calculated, to demonstratethatitcananalyzetransientandquasi-steady-stateconditionsthat the solutions developed are, in fact, general solutionsas well as take into account the variable thermal propertiesand are applicable for quasi-steady-state conditions.with temperature, it has some limitations, including the needfor identifying
a priori
the boundary conditions and the
III. THERMAL MODELING OF THE ARC
dependenceofaccuracyonthesizeofthemesh—thesmaller
WELDING PROCESS
the mesh size, the more accurate the solution but the longerthe computational time to reach truncation. Thus, the choiceFigure 1 is a schematic of conventional arc welding of of a suitable mesh is vital to the accuracy and economy of two short (25.4 mm (1 in.)), thin (5.8 mm) mild steel plates.the FEM results. Sometimes, this involves approximations,The width of the plates is considered large enough such thatand considerable skill is required in order to model the the effect of the widthwise boundaries is small or negligible.process.Also,afinermeshisneededneartheweldcenterline As the length of the weld is short, it will be shown that theto determine accurately the peak temperatures, and a coarser thermal process is transient under the conditions of welding.mesh is needed for the areas farther away, which is adequate The data used for the thermal analysis (Table I) in thisto reduce the time and cost of the analysis. Oscillations may investigation are essentially the same as the values used byoccur in the numerical values because of the coarse mesh Tekriwal and Mazumder,
[8,9]
so that comparison can be madesize, even though the method is stable in time.
[8,9]
Also, a between the analysis presented here and their FEM analysis.sudden change in the slope of the curves is obtained at the However, the feed rate of the filler wire given by themtransition from a finer to a coarser mesh, which, in reality, is appears to be somewhat low, with the result that the volumeanartifact.Theremaybeotherlimitationswiththenumerical of the molten filler wire will not be sufficient to completelytechniques (as will be discussed in this article), based on a fill the V-groove at the joint. Thus, in this analysis, the feedrate of the filler wire (
v
feed
) is increased to 8.50 cm/s. Forcomparison of the analytical and the numerical (FEM)
1356—VOLUME 31B, DECEMBER 2000 METALLURGICAL AND MATERIALS TRANSACTIONS B

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