thermal analysis of the arc welding process part ii.pdf

Thermal Analysis of the Arc Welding Process: Part II.Effect of Variation of Thermophysical Propertieswith Temperature

R. KOMANDURI and Z.B. HOU

This article is Part II of a two-part series on the thermal analysis of the arc welding process. In PartI, general solutions for the temperature rise distribution in arc welding of short workpieces weredeveloped based on Jaeger’s classical moving heat source theory for a plane disc heat source with apseudo-Gaussian distribution of heat intensity and constant values of thermophysical properties atone temperature (400 8C). This was extended in this investigation (Part II) to consider differentthermophysical properties at different temperatures (from room temperature (25 8C) to 1300 8C) fora mild steel work material. The objective is to develop a rationale for the selection of an appropriatetemperature for the choice of the thermophysical properties for the thermal analysis of arc welding.Since the quality of the weld for a given work material depends both on the thermodynamic andkinetic considerations, namely, the maximum temperatures and the temperature gradients (coolingrates) in appropriate sections of the welded part including the weld bead and the heat-affected zone(HAZ), they were determined in this investigation. The main output parameters from a thermal pointof view are the widths and the depths of the melt pool (MP) and the HAZ at the weld joint. Althoughthe length of the weld pool is also a consideration, if the entire length participates in the weldingprocess, which is generally the case, then this is not such an important consideration. It is found thatfor welds produced in a conductive mode only (i.e., not considering the case of deep penetratingwelds produced with keyhole mode), the values of the widths and the depths of the MP and the HAZs arenearly the same (within 10 to 20 pct), irrespective of the values of thermal properties for temperatures inthe range of 400 8C to 1300 8C. Hence, the emphasis on the need to consider variable thermalproperties with temperature in welding appears to be somewhat exaggerated. Also, based on thethermal analysis of the welding process, it appears that the room-temperature thermophysical propertiesmay not be appropriate, as rightly pointed out by other researchers. The thermal history and thecooling rates were also determined analytically for arc welding of long workpieces, where quasi-steady-state conditions are established and the boundary effects can be ignored, as well as shortworkpieces, where transient conditions prevail and boundary effects need to be considered. Thisinformation can then be used in the appropriate time-temperature-transformation (TTT) diagram fora given steel work material to investigate the nature of the metallurgical transformation and theresulting microstructure in the welding process both in the weld bead and in the adjacent HAZs oneither side.

I. INTRODUCTION solve the equation for the temperature distribution—be ittransient, or quasi-steady state. It can also calculate the tem-IN Part I,[1] an analytical solution for the temperatureperature on the surface as well as with respect to depth atrise distribution in arc welding of short workpieces is de-all points including those very close to the heat source. Theveloped based on the moving heat source theory (Jaeger[2]

analysis presented is exact and the solution can be obtainedand Carlsaw and Jaeger[3]) and the pioneering work ofquickly and in an inexpensive way compared to numericalRosenthal[4] to predict the transient thermal response. Thetechniques, such as finite element method (FEM) or finitearc beam is considered as a moving plane (disc) heat sourcedifference method (FDM). The analysis also facilitates inwith a pseudo-Gaussian distribution of heat intensity basedthe optimization of the process parameters for good weld-on the work of Goldak et al.[5] It is a general solution (foring practice.both transient and quasi-steady state) in that it can determine

In the thermal analysis of welding, one of the advantagesthe temperature rise distribution in and around the arc beamof the numerical methods, such as the FDM or the FEM,heat source as well as the width and depth of the melt poolover the analytical techniques is the ability to account for(MP) and the heat-affected zone (HAZ) in welding shortthe variable thermal properties with temperature. However,lengths where quasi-steady state conditions may not have

been established. The analytical model developed can deter- only simple functions of the variation of thermal propertiesmine the time required for reaching quasi-steady state and with temperature, such as linear variation over a certain

temperature range, can be considered. In the analytical tech-nique, however, only constant values of the thermal proper-

R. KOMANDURI, Regents Professor, and Z.B. HOU, Visiting Professor, ties, namely, the thermal conductivity, l, specific heat, c,are with the Mechanical & Aerospace Engineering Department, Oklahoma

and thermal diffusivity, a, can be used due to the complexityState University, Stillwater, OK 74078.Manuscript submitted June 5, 2000. involved in solving the partial differential equation of heat

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 32B, JUNE 2001—483

above this temperature followed by a constant thermal con-ductivity up to 3000 8C. This is due to a change in phasefrom the solid state to the liquid state. The specific heat, onthe other hand, increases almost linearly from room tempera-ture (25 8C) to '500 8C reaching a constant value andmaintaining it up to very high temperatures. If we considerthe thermophysical properties over the temperature range ofroom temperature (25 8C) to 1300 8C, they cannot bedescribed by a simple linear relationship. If, however, onewere to consider the temperature range from '400 8C to1300 8C, since the specific heat is constant over this range,they can be expressed by simple linear relationships. With(a)specific heat being constant, the variation of thermal diffu-sivity will be proportional to the variation of thermal conduc-tivity. Of course, the variation of thermal conductivity andspecific heat with temperature will be different for differentmaterials and this should be taken into account in the analysisof welding of a specific work material. Also, for the analysisof the welding process, it will be shown that the room-temperature thermophysical properties may not be appro-priate, as rightly pointed out by other researchers.

II. BRIEF REVIEW OF LITERATURE

(b) Considerable work has been reported in the literature onthe fundamental and applied aspects of welding, in general,Fig. 1—(a) and (b) Variation of thermal conductivity and specific heat withand the thermal aspects, in particular (e.g., Welding Researchtemperature, respectively, for mild steel.[6,7]

Supplements in the Welding Journal, many specialized con-ferences on Welding, as well as many archival journalsincluding Metallurgical and Materials Transactions). In PartI,[1] a brief review of the literature on the thermal aspectsconduction with variable thermophysical properties withof welding is given and may be referred to, for details. Intemperature.the interest of space, this is not covered here in any detail.Since the temperature in welding varies from room tem-Instead, only relevant literature dealing with the thermalperature away from the weld region to some intermediatehistory, the cooling rates, and the effect of variation of ther-temperature ('800 8C) around the HAZ to very high temper-mal properties with temperature in the thermal analysis ofatures near the MP region (.1400 8C), the main issue is atarc welding is briefly reviewed.which temperature should the thermal properties be taken

It is well known that the quality of the weld for a givenin the thermal analysis of the arc welding process. Onework material depends both on the thermodynamic andcould consider at least three alternatives, namely, the thermalkinetic considerations prevailing in the welding process.properties at room temperature, at maximum temperature,The thermodynamic considerations determine the relevantor at some intermediate temperature. In the previous investi-temperatures and the corresponding temperature gradients,gation,[1] thermal analysis of the welding process was con-while the kinetics determines the cooling rates in differentducted with the thermophysical properties taken at onesections of the welded part, including the weld bead andtemperature, namely, at 400 8C. The rationale for this choiceHAZ. For a given work material, both are crucial inis that the specific heat above this temperature (450 8C todetermining the microstructure and the resulting mechanical3000 8C) is nearly constant (660 J/kg 8C) and the value ofproperties of the weld in relation to the bulk or the qualitythermal conductivity at 400 8C (60W/m 8C) is nearly theand integrity of the weld joint. The molten metal in theaverage (slightly lower) from 0 8C to 1500 8C. In this investi-fusion zone cools primarily by conduction of heat away intogation, the constant thermal conductivity approach wasthe two members forming the weld joint. The microstructureextended to cover thermophysical properties over a widein the HAZ, extending outward from the fusion zone, willrange of temperatures (from room temperature (25 8C) tobe altered by the heat of the welding process. Consequently,1300 8C) to develop a rationale for the selection of thethe maximum temperatures, the temperature gradients, andappropriate temperature for the choice of the thermophysicalthe cooling rates in different sections of the welded partproperties, for the thermal properties only at one temperaturedetermine the final microstructure.can be used in the thermal analysis of arc welding.

Figures 1(a) and (b) show the variation of thermal conduc-tivity and specific heat with temperature, respectively, for a

A. Effect of Weld Thermal Cyclesmild steel workpiece, after Tekriwal and Mazumder,[6,7] whoadopted this data from Krutz and Segerlind.[8] It can be seen The effect of weld thermal cycles on the microstructurethat while the thermal conductivity decreases with increase and mechanical properties of the weld joint in the HAZ hasin temperature almost linearly from room temperature to constituted a major field of research activity for over half

a century. For example, Nippes in his 1958 Adams Lecture[9]'1500 8C, there is a sudden increase in thermal conductivity

484—VOLUME 32B, JUNE 2001 METALLURGICAL AND MATERIALS TRANSACTIONS B

considered in detail the effect of weld thermal cycles on the state) in welding short length workpieces taking into accountthermal properties at different temperatures (from room tem-microstructure and mechanical behavior of the base metalperature (25 8C) to 1300 8C) for a mild steel work materialin the HAZ. Hess et al.[10] conducted an experimental investi-but in the absence of convective or radiative heat flow,gation of the arc welding process to measure the coolingsimilar to Eager and Tsai.[17]curves for a range of welding conditions and plate thickness

Kasuya and Yurioka[18] developed a thermal model forvalues. They developed a model for quantitative predictionpredicting the thermal history, cooling rates, and shape ofof the cooling rates as a function of the welding parametersthe HAZ in welding. They considered the following threeby fitting mathematical expressions for the experimentalcases of welding: (1) quasi-steady-state model to predict thecooling curves. Knowing the time-temperature relationships,shape of the HAZ and the thermal histories and coolingthey were able to successfully predict the microstructure at atimes between 500 8C and 800 8C in wide plates, (2) angiven point near an arc weld for a given material. Paschkis[11]

instantaneous heat source model to predict cooling time fromdeveloped cooling curves of welded plates in the vicinity ofsolidification to 100 8C in small workpieces, and (3) anthe weld by electrical analogy with the objective of providingunsteady heat flow model to predict cooling time from solidi-some insight into the mechanism of heat flow during thefication to 100 8C in locally preheated plates. In all cases,cooling period in arc welding of plates.good agreement was reported between the experimental andAdams[12] rightly pointed out that much of the thermalthe analytical results.analyses developed revolve around the determination of the

The metal microstructure at and near the weld and conse-temperature distribution around the heat source in terms ofquent mechanical properties of the weld joint relative to theinput power and speed at which the weld beam traverses,bulk determine the quality, integrity, and reliability of thetaking into account the thermal properties and geometry ofweld. The main considerations from a metallurgical point ofthe workpiece (width and thickness). He stressed the needview are the temperatures of concern and the correspondingfor and the importance of determining the peak temperaturescooling rates in the MP and HAZ regions. Similarly, theas well as the thermal history (heating and cooling cycles)main considerations from a thermal point of view are theto enable proper metallurgical interpretation of the micro-temperature distribution as well as the widths and the depthsstructures developed at the weld joint.of the MP and the HAZ. Although the length of the weld poolRykalin and Beketov[13] investigated the thermal cycle inis also a consideration, since the entire length participates inthe HAZ from a two-dimensional outline of the MP. Pevelicthe welding process, this does not appear to be such anet al.[14] conducted both experimental and numerical (FDM)important consideration. It will be shown in this investigationanalyses to determine the temperature histories in gas-tung-that with these requirements as the bases, the results ofsten-arc welding of thin plates. The shape of the molten poolthe thermal analysis, considering the thermal properties atwas correlated to the welding variables and this isotherm wasdifferent temperatures from '400 8C to 1300 8C, are basi-then used as one boundary condition. Thus, experimentalcally the same for a mild steel work material in terms ofresults were used to establish the boundary conditions formaximum width and maximum depth of the MP and thethe numerical method. They also developed an expressionHAZ. The underlying reason for the lack of sensitivity willfor the temperature rise distribution whose parameters werebe presented here. Based on this, it will be shown thatcorrelated as a function of the maximum width and lengththe need for considering variable thermal properties withof the weld pool. Pevelic et al.[14] found the predicted peaktemperature in the thermal analysis of welding may be some-temperatures in the HAZ to be within 610 pct of the valueswhat exaggerated. This may, however, be different for differ-measured experimentally.ent work materials (e.g., copper with good thermal properties

Ashby and Easterling[15] investigated the transformation vs stainless steel with poor thermal properties) and in differ-hardening of steel surfaces by a laser beam for hypoeutectoid ent ranges of temperature of interest.steels, and Li et al.[16] investigated that for hypereutectoidsteels. They developed approximate solutions to the equa-

B. Effect of Variation of Thermal Properties withtions of heat flow and combined them with kinetic modelsTemperatureto predict the near-surface microstructure and hardness varia-

tions of steels after laser treatment. They presented the results Several welding studies were made to investigate the fea-in the form of laser processing maps that show the combina- sibility of using constant values of thermal properties attion of process parameters for a desired near-surface micro- room temperature, the maximum temperature, or the averagestructure and the associated hardness profile. temperature in the thermal analysis. The solutions of these

Eager and Tsai[17] determined the temperature fields pro- were compared to those considering variable thermal proper-duced by a traveling distributed heat source (using a ties with temperature using either numerical techniquesGaussian heat distribution for the welding arc) on the surface (FEM or FDM) or the experimental values. For example,of a semi-infinite plate. Their solution provides crucial infor- Grosh et al.[19] made a thermal analysis of the temperaturemation regarding the size and shape of the arc weld pool. rise distribution in solids by moving heat sources consideringThey found the welding process parameters (current, arc a functional relationship between the thermal properties andlength, and travel speed) and the material parameters (ther- the temperature, such as linear, parabolic, or other functionsmal diffusivity) to have significant effect on the shape of of temperature, as long as the thermal diffusivity of thethe weld. The assumptions made for the determination of material remains nearly constant. This implies that the varia-the solutions include the absence of convective or radiative tion of thermal conductivity and specific heat with tempera-heat flow, constant average thermal properties, and a quasi- ture is equal to and in the opposite direction, which may besteady-state semi-infinite medium. The present investigation the case for some materials in the low-temperature region

(,400 8C). Grosh and Trabant[20] compared the analyticaldeals with general solutions (both transient and quasi-steady


results of quasi-steady state temperature distribution in thick plate to be significantly affected by the value chosen for thethermal conductivity, l; the higher the l, the lower is theand thin arc-welded stainless steel plates with the experimen-peak temperature near the weld but the higher is the peaktal results for one-, two-, and three-dimensional heat conduc-temperature in the regions away from the weld. However,tion from a moving plane, line, and point heat source, whereat points near the weld, a higher value of l also leads to athe thermal properties of the solid are functions of tempera-more rapid fall in the temperatures after the peak temperatureture. It appears that the thermophysical data given in Figureshas been reached. The higher the value of l, the more rapid2 and 3 in their article for stainless steel and low-carbonis the fall of the temperature across a plate to reach uniformsteel, respectively, should be interchanged (i.e., Figure 2value. They also found the relationship between the transientshould be for the low-carbon steel and Figure 3 for thetemperatures and the arc efficiency to be nonlinear. Thestainless steel). Nevertheless, the data for thick plates indi-actual value appears to be more critical for welds that usecate that the temperature rise obtained with variable proper-high heat inputs than for those that uses low heat inputs.ties is slightly higher than the experimental results and the

Prasad and Narayanan[24] developed an adaptive grid tech-analysis with constant properties yields much higher temper-nique to analyze the temperature distribution during arcatures. For thin plates, the temperature rise obtained withwelding by the FEM. This enabled the determination of thevariable properties is much higher than that of the experi-temperature rise distribution at the arc with less computa-mental results.tional time. The temperature-dependent thermal propertiesIsenberg and Malkin[21] considered the effect of variablewere considered by the predictor corrector iterative method.thermal properties with temperature for determining the tem-Both radiation and convection effects were neglected in thisperature rise on the surface of a semi-infinite solid dueanalysis. The variation of thermal conductivity and specificto a moving heat source using numerical analysis. Theyheat with temperature were converted into polynomial equa-considered two nonlinear cases, one for carbon steel, wheretions using a graphical package.the thermal conductivity decreases and the specific heat

Much of the research on this topic was to show the differ-increases with temperature, and the other for a materialences between the variable thermal properties approachwith decreasing thermal conductivity with temperature andusing the numerical methods vs the constant thermal proper-constant specific heat. They found the maximum surfaceties approach using the analytical methods and to illustratetemperatures to be somewhat higher compared to the linearthe significance of the variable thermal properties in theconstant property model, the differences being more signifi-thermal analysis. It is interesting to note that on comparison,cant at lower Peclet numbers and stronger heat sources. Butthe analytical approach using constant thermal properties atthe average temperatures within the band, for the case ofan appropriate temperature would give results closer to thecarbon steel, were found to be virtually unchanged fromexperimental values, as long as the room-temperature ther-the linear constant property case, but were higher for othermal properties are not taken. In this investigation, this aspectnonlinear cases. The variable properties model indicates sig-is investigated in depth by comparing the analytical resultsnificantly larger transverse temperature gradients than thoseobtained using the thermal properties at various temperaturesfor the linear (constant property) model.from room temperature (25 8C) to 1300 8C.Kou et al.[22] developed a three-dimensional (3-D) heat

It may be noted from the previous discussion that no effortflow model of laser transformation hardening using thewas made in this brief review to cover the entire literatureFDM. The heat flow model developed was compared withon the topic of effect of thermal properties at various temper-Jaeger’s analytical solution for a steady-state, 3-D heat flowatures on the temperature rise distribution in welding.

in a moving semi-infinite workpiece subjected to a stationary Instead, representative works were covered to illustrate dif-square (or rectangular) heat source. They investigated the ferent approaches taken by various researchers. It is also nottwo major assumptions made in Jaeger’s analysis, namely, the intent of this investigation to suggest the use of constantno surface heat loss due to convection or radiation and thermal properties instead of variable thermal properties withconstant thermal properties for AISI 1018 steel. A compari- temperature where they can be used, as in the numericalson between the thermal cycles calculated using the FDM methods. It should be pointed out that constant thermalconsidering variable thermal properties and an analytical properties are used in the analytical method not by choicesolution considering room-temperature thermal properties but due to lack of an alternative for the solution of the partialshowed that the temperatures calculated using room-temper- differential equation of heat conduction or other analyticalature thermal properties were slightly higher on the surface, methods in welding. The question is, if properties at onebut the difference was quite significant at a depth of 0.66 temperature need to be taken, then at what temperaturemm. However, when the temperatures were calculated using should the thermal properties be taken in the thermal analysisthe high-temperature thermal properties (in this case, T . of arc welding?800 8C) and compared with the results of the FDM, theagreement between them was much closer. However, good

III. THERMAL ANALYSIS OF A MOVINGmatching of the curves were not obtained at a depth ofPOINT HEAT SOURCE0.66 mm. Since the authors were primarily interested in the

hardened zone, they considered 780 8C as a reasonably high To investigate the effect of thermophysical propertiestemperature for the calculation of the thermal properties. at different temperatures on the depth, width, and length of

Little and Kamtekar[23] investigated the effect of thermal the MP and HAZ in arc welding, consider the heat conduc-properties and weld efficiency on the transient temperature tion equation for the simple case of a moving point heatdistribution during welding using finite element analysis. source, which is the fundamental equation for all moving

heat sources. It is also a simple equation that can be usedThey found the transient temperature distribution in a welded


to consider the effect of different thermophysical parame-ters with temperature. Of course, the point heat source, asRosenthal[4] first pointed out, would give extremely hightemperatures at and near the heat source. Consequently, themaximum temperature would be very high and the tempera-ture gradients near the heat source would be extremely steep.In contrast, with a disk heat source, the maximum tempera-ture as well as the temperature gradients would be moremanageable. Consequently, high-temperature-dependentthermal properties would be less sensitive for the disk heatsource, which is the case for welding, than for a point heatsource. Keeping these factors in mind, one could investigatethe effect of the thermophysical properties at different tem-peratures on the depth, width, and length of the MP andHAZ in welding.

(a)The solution for the temperature rise for a moving pointheat source[3] is given by

u 5qpt

4plRe

2v

2a(X1R)

[1]

The first term in Eq. [1] namely, qpt /4plR, is a tempera-

ture term, and the second term, e2

v2a

(X1R), is a nondimen-

sional term. Also, the first term contains thermal conductiv-ity in the denominator, while the second term (the exponen-tial function) contains thermal diffusivity, a (l/rc). Whenboth l and c vary with temperature, as in the case of thermalproperties in the range of room temperature (25 8C) to 4008C, both the values need to be taken into account. If, however,one is interested in the thermal properties above 400 8C,then c is constant and, therefore, thermal diffusivity, a, variesdirectly with the thermal conductivity, l. This is the reasonwhy the temperature distribution is so different when using (b)room-temperature thermal properties compared to the ther-mal properties at higher temperatures and why it may notbe appropriate to use the room-temperature thermal proper-ties to analyze the welding process, as many earlier research-ers have pointed out.

A sample calculation was carried out using Eq. [1] andthe following parameters: qpt 5 5000 J/s and thermal proper-ties at different temperatures, namely, at 25 8C, 400 8C, 7508C, 1000 8C, and 1300 8C (Table II provides details). Figure2(a) shows the variation of the first term, namely, qpt /4plR,in the temperature rise solution of a moving point heat source(Eq. [1]) along the X-axis (arc beam travel) for differentvalues of thermal conductivities. It can be seen that thisfunction is rather sensitive to thermal conductivity close tothe heat source and its effect decreases with increasing dis-tance from the heat source. The value of the temperaturesis higher for lower thermal conductivities (i.e., l at higher (c)temperatures) than at higher thermal conductivity values

Fig. 2—(a) Variation of the first term, namely, qpt/4plR, in the temperature(i.e., l at lower temperatures). Figure 2(b) shows the varia- rise solution of a moving point heat source (Eq. [1]) along the x-axis (arction of the second term in the temperature rise solution of beam travel) for different values of the thermal conductivities. (b) Variation

of the second term, namely, ev(X+R)/2a, in the temperature rise solution of aa moving point heat source (Eq. [1]), namely, e2

v2a

(X1R), along

moving point heat source (Eq. [1]) along the x-axis for different values ofthe X-axis for different values of thermal conductivities. thermal conductivities. (c) Variation of the first term in the temperatureAlong the positive X-axis, a decrease in the value of this rise solution of a moving point heat source (Eq. [1]) along the negative x-

axis, which are a mirror image of the curves shown in (a).coefficient with increasing distance from the point heatsource can be seen. In contrast to Figure 2(a), the value of thenondimensional term is higher at higher thermal conductivity(i.e., l at lower temperatures) than at lower thermal conduc- (25 8C) thermal conductivity value is significantly apart from

the rest of the curves obtained with the thermal conductivitytivity (i.e., l at higher temperatures). Also, the value of thenondimensional term obtained using the room-temperature values at higher temperatures. This is because the specific


Fig. 4—The temperature rise distribution along the x-axis due to the com-(a)bined effect of the two terms in the temperature rise solution of a movingpoint heat source (Eq. [1]) for different values of thermal properties.

but at distances closer to the heat source, the second termincreases with an increase in thermal conductivity. Thus, theproduct of these two terms tends to show them in balanceto each other or to have opposing effects.

Figure 4 shows the variation of the temperature rise distri-bution due to the combined effect of these two terms in thetemperature rise solution of a moving point heat source.Also shown in the figure are the MP and the critical tempera-ture for the HAZ. It may be noted that for ease of comparison,the temperatures of MP and HAZ are the same as those usedby Tekriwal and Mazumder.[6,7] Also note that the ordinateis the temperature rise (assuming room temperature to be

(b) 25 8C) instead of the actual temperature. Also shown in thefigure are two solid lines parallel to the abscissa, whichFig. 3—(a) and (b) Variation of the first and second terms in the temperature

rise solution of a moving point heat source (Eq. [1]) with thermal conductiv- represent the critical temperature rise for MP and HAZ. Theity at different distances (R) from the heat source. intersection of these two solid lines with the temperature

rise distribution curves is comprised of the leading and trail-ing edges of the corresponding MP and HAZ. They can beused to determine the size of the MP and HAZ along the xheat increases with an increase in temperature up to '400

8C after which it remains nearly constant for this material. direction. It can be seen from Figure 4 that the distancesfrom the heat source to the leading edges of the MP or theHence, both thermal conductivity and specific heat influ-

ences the value of the nondimensional term at room tempera- HAZ for various thermal properties at different temperaturesare nearly the same and independent of the thermal conduc-ture. If we now consider the product of the two terms, each

will, more or less, compensate for the variation of thermal tivity (except the room-temperature values), as pointed outearlier. However, this is not the case for the distances toconductivity in front of the heat source. In contrast, the value

of this nondimensional term is constant and equals 1 along the trailing edge, as the second term does not have anycompensating influence on the temperature distribution.the negative X-axis (from the point heat source), i.e., behind

the heat source, and hence it will not have any influence on The next issue to consider is the effect of thermal conduc-tivity at various temperatures on the maximum widths andthe temperature rise distribution. The variation of the first

term with thermal conductivities at different temperatures maximum depths of the MP and the HAZ. The equationsfor both the width and the depth are nearly the same, withalong the negative X-axis is a mirror image of Figure 2(a),

as shown in Figure 2(c). the only difference being the value of R, which depends onthe value of y for the width and the value of z for the depth,Figures 3(a) and (b) show the variation of the first and

second terms in the temperature rise solution of a moving for a given value of x. Figure 5(a) and (b) show the variationof the first and the second terms in the temperature risepoint heat source (Eq. [1]) with thermal conductivity at

different distances (R) from the heat source. It can be seen solution of a moving point heat source (Eq. [1]) with depth(z) or semiwidth ( y) for different values of the thermalthat the first term is nearly independent of the thermal con-

ductivity at distances far from the heat source. However, at properties for x 5 21 mm. It can be seen that both termsdecrease with an increase in y (or z). However, for the firstdistances closer to the heat source, the first term decreases

with an increase in thermal conductivity, showing more term, the temperature is higher using the lower thermalconductivity values (i.e., l at higher temperatures) than usingdependence on the thermal conductivity. The second term

also shows a similar feature, namely, near independence of the higher thermal conductivity values (i.e., l at lower tem-peratures). In contrast, the second term is higher using thethe thermal conductivity at distances far from the heat source,


(a)(a)

(b)(b)Fig. 6—(a) and (b) Variation of the first and second terms in the temperatureFig. 5—(a) and (b) Variation of the first and the second terms in therise solution of a moving point heat source (Eq. [1]) with thermal conductiv-temperature rise solution of a moving point heat source (Eq. [1]) with xity at different distances from the heat source.or y for different values of the thermal properties for a value of x 5 21 mm.

higher thermal conductivity (i.e., l at lower temperatures)than at lower thermal conductivities values (i.e., l at highertemperatures). It can also be seen that various curves forthe second term using thermal properties at higher tempera-tures are reasonably close, while those using the thermalproperties at room temperature are far apart. This again isdue to an increase in the value of the specific heat with anincrease in temperature up to about 400 8C.

Figures 6(a) and (b) show the variation of the first andthe second terms in the temperature rise solution of a movingpoint heat source (Eq. [1]) with thermal conductivity atdifferent distances from the heat source. It can be seen thatthe first term is nearly independent of the thermal conductiv-ity at distances far from the heat source. However, at dis-tances closer to the heat source, the first term decreases with Fig. 7—The temperature rise distribution along the width ( y-axis) or depth

(or z-axis) of the weld due to the combined effect of the two terms in thean increase in thermal conductivity, showing more depen-temperature rise solution of a moving point heat source (Eq. [1]) for differentdence on the thermal conductivity. The second term alsovalues of thermal properties. The intersection of these curves with MP orshows a similar feature, namely, near independence of the HAZ lines gives the semiwidth or depth of MP or HAZ.

thermal conductivity at distances far from the heat source,but at distances closer to the heat source, the second termincreases with an increase in the thermal conductivity (i.e.,l at lower temperature). Thus, the product of these two Figure 7 shows the variation of the temperature rise distri-

bution due to the combined effect of these two terms in theterms tends to balance (or have an opposing effect) the pointsnearby the heat source. temperature rise solution of a moving point heat source (Eq.


Table I. Welding Parameters Used in the ThermalAnalysis of Arc Welding

Heat liberation rate of the moving discheat source, qpt 4280 J/s

Welding arc speed, v 5.08 mm/sLength of the plate, L 254 mmThickness of the plate, H 5.8 mmRadius of the arc beam, rb 5.56 mmTemperature of the melt pool, TMP 1477 8C (1750 K)Temperature of the HAZ, THAZ 727 8C (1000 K)

Fig. 8—Schematic of the conventional arc welding of two thin mild steelplates showing various dimensions of interest and the reference axes.[1]

[1]). Also shown in the figure are two solid lines correspond-ing to the critical temperature rise for MP and HAZ, asreferences. The intersection of these two lines with the tem-perature rise distribution curves along the y (or z) directionsgives the half-widths or depths of the MP and HAZ fordifferent values of thermal conductivities at different temper-atures. It can be seen that the width and the depth of bothMP and HAZ are fairly close and nearly independent of thethermal conductivity, as pointed out earlier.

Fig. 9—Variation of temperature rise distributions in the workpiece on theIV. ANALYTICAL RESULTS AND DISCUSSION upper surface along the length of the weld (X-axis) using the various valuesof thermal properties at different temperatures (Table II provides details)A. Variation of Thermal Properties with Temperature for quasi-steady-state conditions.

To determine realistically the effect of thermal propertiesat various temperatures in the thermal analysis of welding(i.e., by considering the heat source as a moving disc heat shows the heating and cooling cycles as the welding arc

beam traverses. The size of the disc heat source, the meltsource instead of a point heat source) and to select appro-priate thermal properties, the following investigations were temperature, and the critical temperature for HAZ are also

shown for reference. It can be seen that the temperature atmade: (1) calculation of the quasi-steady-state temperaturerise distributions along the x-axis (refer to Figure 8 for the the center of the heat source is extremely high with the

temperature gradients higher at the leading edge (ahead ofcase where L is large (254 mm)) using the analytical solutiondeveloped for the welding arc heat source (refer to Eq. [18] the moving heat source) than at the trailing edge (behind

the heat source) of the weld arc beam. Behind the heatin Part I[1] or Eq. [A2] in Appendix A); (2) calculation ofthe widths and depths of the MP and the HAZ under quasi- source, the temperature rise along the length of the weld

seam increases with an increase in the temperature at whichsteady-state conditions; (3) calculation of the thermal histor-ies at two different locations, one at x 5 12.70 mm and y 5 the thermal properties of the work material are taken. How-

ever, in front of the heat source (except for the room-tempera-8.5 mm (a typical point in the HAZ) and the other at x 517.78 mm and y 5 5.0 mm (a typical point in the MP region) ture thermal properties), the distances from the center of the

heat source to the leading edges of the corresponding MPof the work material for the case where L is short (25.4 mm)and qpl is 7132.3 W (the same cases considered by Tekriwal or the HAZ (Table III for details) are nearly the same.

Equation [A2], similar to Eq. [1], has two terms. The firstand Mazumder[6,7] for comparison); and (4) calculation ofthe maximum widths of the MP and the HAZ for the same term, namely, (3.1576qplv)/(4lap 3/2), has the dimension of

temperature. The second term, namely, the remaining inte-cases. Items (1) and (2) can elaborate on the differences inthe results of calculations using various choices of thermal gral part of Eq. [A2] together with (1/r 2

0), is nondimensional.It may be noted that in the first term, the thermal conductiv-properties at different temperatures, and items (3) and (4)

enable comparison of the analytical results with the experi- ity, l is in the denominator, and in the second term, it is inthe negative exponential terms in the integrals, in the uppermental and FEM results and validates the appropriate choice

of thermal properties in the analytical calculations. limit of integration and in the argument of the Bessel func-tion. When considered in the positive x direction, these twoConsider the case of conventional welding of two thin

mild steel plates (Figure 8).[1] The parameters used in this terms, more or less, tend to compensate (similar to the twoterms in Eq. [1] for a point heat source). Thus, variousinvestigation are given in Table I. Figure 9 shows the varia-

tion of temperature rise distribution in the workpiece on the distances from the center of the heat source to the leadingedges of the MP and HAZ are nearly the same, irrespectiveupper surface along the length of the weld (x-axis) using

various values of thermal properties at different temperatures of the temperature at which the thermal properties are taken,except for the room-temperature values. When the second(Table II for details) for quasi-steady-state conditions. It also


Table II. Thermal Properties Used in the Analysis at Various Temperatures

Set Number 0 1 2 3 4 5

T 8C 25 400 500 750 1000 1300lT W/cm 8C 0.72 0.6 0.57 0.50 0.44 0.36aT cm2/s 0.2470 0.1180 0.1090 0.0955 0.0840 0.0688

Refer to Figure 1, a 5 l/cr, and r 5 7.87 g/cm3.

Table III. Distance from the Center of the Heat Source to the Leading and Trailing Edges of the MP and HAZ


Distance to the leading edge, mm 4.57 4.05 4.04 4.09 4.134 4.20MP Distance to the trailing edge, mm 8.51 9.11 9.50 10.55 11.90 13.78

Length, mm 13.08 13.16 13.54 14.64 16.03 17.98

Distance to the leading edge, mm 6.54 5.37 5.31 5.30 5.29 5.29HAZ Distance to the trailing edge, mm 21.19 22.54 22.91 24.20 24.53 23.67

Length, mm 27.73 27.91 28.22 29.50 29.82 28.96

Table IV. Half-Widths of MP and HAZ


Half-width in mm MP 5.59 4.91 4.92 5.06 5.19 5.33HAZ 9.68 7.67 7.61 7.67 7.72 7.72

(a) HAZ contours, respectively, on the top surface (i.e., in the x-o-y plane) along the length of the weld (X-axis) for differentvalues of thermal properties at different temperatures (TableIV provides details). It can be seen that except at the room-temperature thermal properties, l25, the maximum width isnearly constant for all values of thermal properties in therange of temperatures from '400 8C to 1300 8C. In fact,the curves on the leading edge up to the center of the heatsource are nearly identical and only toward the trailing edgedoes one find the differences in maximum width ('5 % for(b)MP and '2 % for HAZ), as shown in Figure 10. From this,

Fig. 10—(a) and (b) Variation of the half-width of the MP and HAZ,it is clear that as far as the maximum widths of the MP andrespectively, on the upper surface along the length of the weld (X-axis)HAZ are concerned, the thermal properties at any tempera-using various values of thermal properties at different temperatures (Table

IV provides details). ture between 400 8C and 1300 8C would be acceptable.The actual value would be that temperature at which theanalytical value is close to the experimental value. For theanalytical determination of the maximum width of the MP,term is considered in the negative x direction, the exponentialthis can perhaps be any value between the melting tempera-term, the most sensitive part, becomes positive with veryture and the critical temperature for HAZ, and for the HAZ,little or no compensation provided by this term to the firstit can be any value between the critical temperature for HAZterm. This can be seen in Figure 9, where the temperatureand 400 8C. Alternately, as Kou et al.[22] reported, the thermalrise curves on the trailing edge meet the horizontal lines ofproperties at an intermediate temperature (,800 8C) maycritical temperature rise for MP and HAZ at different loca-be considered.tions along the length of the heat source for different values

Figures 11(a) and (b) show the MP and HAZ contours,of the thermal conductivities at different temperatures. Thus,respectively, in the x-o-z plane along the length of the weldon the trailing edge, they vary depending on the temperature(X-axis) for different values of thermal properties at differentat which thermal properties are taken. It is clear from thistemperatures (Table II provides details). It can again befigure that room-temperature thermal properties are notseen that except at the room-temperature (25 8C) thermalappropriate for the determination of the temperature rise inproperties, the depth profiles along the leading edge arethe work material during welding. Also, although the lengthnearly the same and only on the trailing edge does one findsof the weld pool is also a consideration, for cases where thethe differences, as shown in Figures 11(a) and (b). However,entire length participates in the welding process, which isbecause the workpiece is thin, the boundary effect at thegenerally the case, this is not an important consideration.

Figures 10(a) and (b) show the variation of the MP and bottom surface becomes prominent and there is melt through


(a)

(b)

Fig. 11—(a) and (b) Variation of the depth of the MP and HAZ, respectively,Fig. 13—Thermal history (heating and cooling cycles) at two points onalong the length of the weld (X-axis) using various values of thermalthe upper surface of the workpiece, whose x and y coordinates are 17.78properties at different temperatures (Table II provides details)and 5.0 mm for the first point and 12.70 and 8.5 mm for the secondpoint, respectively.[6,7]

Table V. Maximum Depths of MP and HAZ

thermal properties at different temperatures (refer to TableSet Number 0 1 2 3 4 5II for details). It can be seen again that except for the room-Depth in mm MP 3.58 3.20 3.21 3.52 3.76 4.00temperature thermal properties, the contours of the leadingHAZ 6.87 6.06 6.06 6.19 6.31 6.47part of the MP and HAZ are nearly the same (especially infront of the heat source) at any temperature between 400 8Cand 1300 8C. The differences in the maximum depth, TableV, using thermal properties in the temperature range are alsonot significant ('14 pct for MP and '7 pct for HAZ). Theactual value, again, would be that temperature at which theanalytical value of the depth is close to the experimentalvalue. For the MP, this can be between the melting tempera-ture and the critical temperature for HAZ, and for the HAZ,this can be between the critical temperature for HAZ and'400 8C.

Figure 13 shows the thermal history (heating and cooling(a)cycles) at two points, one in the MP region and the otherin the HAZ region, on the upper surface of the workpiece.The coordinates of the first point are x 5 17.78 mm andy 5 5.0 mm and those for the second point are x 5 12.70mm and y 5 8.5 mm.[6,7] Also, for the second point, experi-mental data of maximum temperature rise (above room tem-perature) were reported as 1115 8C by TekriwalMazumder.[6,7] Figure 14(a) shows the thermal history of thesecond point (x 5 12.70 mm and y 5 8.5 mm), calculatedby both analytical and FEM methods. It can be seen that

(b) except for the room-temperature thermal properties, all thecurves obtained analytically using various sets of thermalFig. 12—(a) and (b) Variation of the maximum depth of the MP and HAZ,properties at different temperatures are rather close to thatrespectively, along the length of the weld (X-axis) using various values of

the thermal properties at different temperatures (Table II provides details). obtained by the FEM method (dotted line). The correspond-ing maximum temperatures are listed in Table VI. It can beseen that the results using the thermal properties at 400 8Cand 500 8C are close to the FEM results, with the differencethe entire thickness. Consequently, the maximum depths for

the MP and HAZ cannot be obtained. being ,0.4 pct. Compared to the experimental data, thedifference, whether analytical or FEM, is ,9.7 pct. ThisTo determine the maximum depths for the MP and the

HAZ, another investigation was conducted with the thick- difference is not significant considering the accuracy of theexperimental measurements. Thus, the analytical results canness of the workpiece increased to 25.4 mm (from 5.8 mm)

so that the boundary effects do not significantly affect the be considered to be in close agreement with the experimen-tal results.MP and the HAZ and no melt through occurs. Figures 12(a)

and (b) show the variation of the maximum depths of the Comparing Figures 14(a) and (b), it can be seen that thetemperature rise in Figure 14(b) is higher than that in FigureMP and HAZ, respectively, for the case of thick plate along

the length of the weld (X-axis) using various values of the 14(a). This is because the point under consideration in Figure


(a) Fig. 15—Variation of the width of MP and HAZ with time[1] showing acomparison between the analytical results and the FEM sumulation.[6,7]

temperature redistribution and the cooling process in thestationary body are different from the moving heat sourceproblem. As a result of this, the thermal history curves beforeand after 5 seconds show different shapes. It can be seenthat for points away from the highest temperature region(Figure 14(a)), this drop is less significant than for pointscloser to the highest temperature region (Figure 14(b)).

Figure 15 shows the variation of the widths of MP andHAZ with time[1] calculated by the analytical method usingthe thermal properties at 400 8C and the FEM considering thevariable thermal properties of Tekriwal and Mazumder.[6,7] Itshows a reasonably good agreement of the analytical methodwith the FEM especially in the initial stages of welding.

(b) After the heat source is shut off at 5 seconds, the analyticalresults show an initial gradual increase in the width withFig. 14—(a) Thermal history of the point whose x and y coordinates are

12.70 and 8.5 mm, respectively (in the HAZ region), calculated by both time followed by a rapid decrease as expected. However,the analytical and the FEM methods. (b) Thermal history of a point whose the FEM results show such a trend only for MP but not forx and y coordinates are 17.78 and 5.0 mm, respectively (in the MP region), HAZ. Table VII shows a comparison of the maximum widthscalculated by the analytical and FEM methods.

of MP and HAZ by the analytical method (using thermalproperties at 400 8C, namely, l400 and a400 as constants) andthe FEM (considering variable thermal properties) with the

Table VI. Comparison of umax from Analytical, FEM, and experimental results. It can be seen that the maximum widthExperimental Results of MP calculated by the analytical method is in close agree-

ment with the experimental results, with the difference beingExperi-'1 pct. However, the difference for the maximum widthSet Number 1 2 3 4 FEM mentalof HAZ is much higher ('18 pct). Since the temperature

umax 8C 1235.1 1233.4 1273.3 1295.6 ,1230 1115 gradients are much lower for the HAZ boundary than theMP boundary, it is much easier and more accurate to demar-cate the MP boundary than the HAZ boundary. Hence, ingeneral, the accuracy in the measurement of the width of14(b) is closer to the heat source than the point in Figurethe MP is much higher than for measuring the HAZ. This14(a). Figure 14(b) shows the thermal history of anothercan partly account for the difference.point (x 5 17.78 mm and y 5 5.0 mm) calculated by both

analytical and FEM methods. For this point, no experimentaldata were available. Comparing the analytical results with B. Thermal History and Cooling Ratesthe FEM results, the closest ones are the curves obtainedusing the thermal properties at 400 8C and 500 8C. Figure 16(a) shows variation of the temperature rise distri-

bution on the top surface along the x-axis, namely, in theIt can be seen from Figures 14(a) and (b) that all thecurves of the thermal histories of the two points have a drop direction of motion of the heat source, for various values

of y (refer to Figure 8 for details) under quasi-steady-stateat time t 5 5 seconds. This is because after t 5 5 seconds,the heat source was shut off and any further temperature conditions (i.e., t . 11 seconds). For this case, the length

of the weld is considered rather long (L 5 254 mm) tochanges are caused by the heat transfer by conduction inthe workpiece from the higher temperature region to the ensure quasi-steady-state conditions with minimal boundary

effects. It can be seen that the maximum temperaturelower temperature region after the shut off. The resulting


Table VII. Comparison of the Analytical, FEM, andExperimental Values of the Widths of MP and HAZ

Analytical FEM Experimental

Widthmax in mm MP 16.62 15.42 16.80HAZ 26.00 32.00 31.75

Fig. 17—Variation of rate of cooling with temperature rise on the topsurface of the workpiece along the x-axis (i.e., direction of motion of theheat source) for y 5 4 mm.

from the heat source (i.e., y). With an increase in time, therate of cooling gradually decreases and reaches more or lessa constant value independent of the distance from the heatsource ( y).

(a) Figure 17 shows the variation of the rate of cooling withtemperature on the top surface at y 5 4 mm. In the low-temperature region (100 8C to 800 8C), the rate of coolingis relatively low and increases with an increase in tempera-ture. In the high-temperature region (800 8C to 1500 8C),the rate of cooling is relatively high and increases with anincrease in temperature. For each region, the variation ofthe rate of cooling with temperature can be represented bya linear function with a coefficient of variance of '0.98.Figures 18(a) and (b) show the variation of the rate of coolingwith temperature on the top surface for different values ofy in the two temperature regions. As can be expected, thecooling rate increases with an increase in temperature anddecreases with an increase in y.

Figures 19(a) and (b) show the thermal histories at twolocations of y, namely, 8.5 and 5 mm, respectively, for sevendifferent points along the x-axis or along the direction of

(b) motion of the heat source in welding of short workpiecesFig. 16—Quasi-steady-state temperature rise distribution on the top surface (L 5 25.4 mm). Here, the transient conditions as well asof the workpiece along the x-axis (i.e., direction of motion of the heat the boundary effects from the beginning to the end of thesource) for different values of y. It also gives the thermal history of the weld cycle are considered. The curves on the left side repre-welding process under quasi-steady-state conditions. (b) Variation of cool-

sent the heating cycle, while those on the right side representing rate with time on the top surface along the x-axis for various valuesthe cooling cycles. At any time during the cooling cycle,of y.the temperature is higher at point 1 (at the beginning of theweld) than at point 7 (at the end of the weld). Of course, atthe beginning and the end of the weld cycle, because of thedecreases with an increase in y, i.e., away from the heatshort length, the image heat sources come into the picture.source. Since the traverse velocity of the weld beam isFrom the very beginning to the end of the weld cycle, pointknown, Figure 16(a) is also a plot of the variation of the1 is at or behind the heat source all the time. Consequently,temperature rise with time or the thermal history of pointit continues to receive heat due to conduction even after theA on the top surface, as shown in Figure 16(a). The curvesheat source passes this point. Its temperature continues toon the right side give values of the temperature rise duringincrease during the transient period. Toward the end of thethe heating cycle, while those on the left side give valueswelding cycle where the heat source encounters point 7, thisof the temperature rise during the cooling cycle. Using theis not the case. This point is in front of the heat source alldata of Figure 16(a), the cooling rates at different times canthe time. The effect of the heat source to this point will bebe calculated. Figure 16(b) shows the variation of coolingsignificant only when the heat source moves close to thisrates with time for different values of y. It can be seen thatpoint, i.e., toward the end of the welding cycle. From theat locations closer to the heat source, the cooling rates are

much higher and decrease with an increase in the distance very beginning to nearly midway into the welding cycle, no


(a)

(a)

(b)

Fig. 19—(a) Variation of the temperature rise distribution with time on thetop surface of the workpiece along the x-axis (i.e., direction of motion ofthe heat source) at y 5 5 mm for different values of x, for welding of shortworkpieces. It also gives the thermal history of the welding process undertransient conditions and under the influence of boundary conditions. (b)(b)Variation of the temperature rise distribution with time on the top surfaceof the workpiece along the x-axis (i.e., direction of motion of the heatFig. 18—(a) Variation of rate of cooling with temperature rise on the top

surface of the workpiece along the x-axis (i.e., direction of motion of the source) at y 5 8.5 mm for different values of x, for welding of shortworkpieces. It also gives the thermal history of the welding process underheat source) for different values of y in the temperature range of 100 8C

to 800 8C. (b) Variation of rate of cooling with temperature rise on the top transient conditions and under the influence of boundary conditions.surface of the workpiece along the x-axis (i.e., direction of motion of theheat source) for different values of y in the temperature range of 900 8Cto 1500 8C.

21(b) shows the variation of the cooling rate with tempera-ture rise for different points along the direction of motionof the moving heat source for y 5 5 mm. Again, the ratetemperature is registered by point 7 (Figure 19). Conse-of cooling increases with an increase in temperature rise asquently, the temperature rise at point 7 is lower during thewell as with an increase in x. Thus, the rate of coolingcooling cycle. Of course, this difference will be small as theat any given temperature and at any given point can betime under consideration increases, say, to 50 seconds.determined. This information can be used in the appropriateFigure 20(a) shows the variation of the cooling rate withtime-temperature-transformation (TTT) diagram for a giventemperature rise for a point ( y 5 8.5 mm and x 5 12.70steel to determine the nature of metallurgical transformationmm) in the temperature range of '100 8C to 800 8C. It canof the microstructure during the welding process in the weldbe seen that the rate of cooling increases with an increasebead as well as in the HAZ.in temperature rise and the variation can be expressed by a

linear relationship with the coefficient of variance of 0.997.Figure 20(b) shows the variation of the cooling rate with V. CONCLUSIONStemperature rise for different points along the direction ofmotion of the moving heat source for y 5 8.5 mm. Again, 1. Thermal analysis of the welding process using thermal

properties at different temperatures (from room tempera-the rate of cooling increases with an increase in temperaturerise as well as with an increase in x. ture (25 8C) to 1300 8C) indicates that room-temperature

thermal properties tend to give drastically differentFigure 21(a) shows the variation of the cooling rate withthe temperature rise for a point ( y 5 5 mm and x 5 17.78 results than those with the thermal properties at higher

temperatures. This is because for a plain carbon steelmm) in the temperature range of '100 8C to 1500 8C. Itcan be seen that the rate of cooling increases with an increase from room temperature to '400 8C, both the thermal

conductivity and the specific heat vary, the formerin temperature, and the variation can be expressed by a linearrelationship with the coefficient of variance of 0.999. Figure decreasing and the latter increasing with temperature,


(a)(a)

(b)(b)

Fig. 21—(a) Variation of rate of cooling with temperature rise at a pointFig. 20—(a) Variation of rate of cooling with temperature rise at a point (x 5 17.78 mm and y 5 5 mm) in the temperature range of 100 8C to 800(x 5 12.7 mm and y 5 8.5 mm) in the temperature range of 100 8C to 8C. It can be represented by a linear function with the coefficient of variance800 8C. It can be represented by a linear function with the coefficient of of 0.997. (b) Variation of rate of cooling with temperature rise at a pointvariance of 0.997. (b) Variation of rate of cooling with temperature rise at y 5 5 mm and different values of x, in the temperature range of 100 8Ca point y 5 8.5 mm and different values of x, in the temperature range of to 800 8C.100 8C to 800 8C.

HAZs were found to be nearly the same (the differencebeing in the range of 2 to 15 pct), irrespective of thewhile from '400 8C to 1300 8C, the specific heat is

nearly constant and thermal conductivity varies. Conse- values of thermal properties used, from a temperatureof '400 8C to 1300 8C with conductive mode only (i.e.,quently, the room-temperature thermal properties may

not be appropriate in the thermal analysis of welding, not considering such cases as deep penetrating weldsproduced with a keyhole mode). Hence, the emphasiswhich confirms the results reported by other researchers.

2. The influence of thermophysical properties at different on the need to consider variable thermal properties withtemperature in welding appears to be somewhat exag-temperatures on the depth, width, and length of the MP

and HAZ in welding was demonstrated approximately gerated. This may, however, be different for differentwork materials (e.g., copper with good thermal proper-considering a simpler case of a moving point heat

source. The equation for the temperature rise for a mov- ties vs stainless steel with poor thermal properties) andin different ranges of temperature of interest.ing point heat source is the fundamental equation that

deals with all moving heat sources. It is simple enough 5. The effect of the variation of thermal properties at vari-ous temperatures was illustrated analytically using thethat the effect of different thermophysical parameters

can be easily investigated, thus contributing toward the simple case of a moving point heat source. Since thetemperature rise equation (Eq. [1]) contains two terms,physical understanding of the welding process.

3. The room-temperature thermal properties may not be a temperature term and a nondimensional term, theirproduct determines whether the variation of thermalappropriate in the determination of the temperature rise

in the workpiece in welding since the temperatures of properties with temperature makes a significant differ-ence in the values of the temperature distribution. Forinterest, namely, MP and HAZ, are much higher where

specific heat is nearly constant but the thermal conduc- example, it is shown that the distances from the centerof the heat source to the leading edge of the MP andtivity varies significantly.

4. The maximum widths and depths of the MP and the HAZ (positive x-axis) are nearly independent of the


thermal properties at various temperatures, as the two Thanks are due, in particular, to Drs. L. Martin-Vega, KeshNarayanan, B.M. Kramer, K. Rajurkar, and Delci Durhamterms in the equation more or less compensate. In con-

trast, on the trailing side (along the negative x-axis), (Division of Design, Manufacturing, and Industrial Innova-tion) and to Dr. Jorn Larsen Basse (Tribology and Surfacethere is no compensation by the nondimensional param-

eter (being a constant and equals to one), and hence, Engineering program). The authors also thank the MOSTChair (Most Eminent Scholars Program) for Intelligent Man-the lengths of the MP and HAZ are different at different

values of the thermal properties at different temper- ufacturing for enabling the preparation of this article.atures.

6. For the case of widths and depths of the MP and HAZAPPENDIX Aduring welding under realistic conditions (i.e., by con-

Analysis of a moving disc heat source with pseudo-sidering the heat source as a moving disc heat sourceGaussian distribution of heat intensity (consideringinstead of a point heat source), the two terms in thethe boundary effects at the bottom and lengthwisetemperature rise equation (Eq. [A2]), namely, the tem-

side surfaces)perature term and the nondimensional integral term(similar to the two terms in Eq. [1]), play an important In Part I[1] of this two-part series, details of the analysis

of a moving disc heat source with pseudo-Gaussian distribu-but compensating role. Consequently, their values arenearly the same and independent of the thermal proper- tion of heat intensity (based on the work of Goldak et al.,[5]

considering the boundary effects due to the bottom andties at various temperatures from '400 8C to 1300 8Cfor a mild steel work material. lengthwise side surfaces) were given. Here, the relevant part

dealing with the thermal analysis is given for completion.7. Thermal histories and cooling curves along the directionof motion of the moving heat source were determined for The objective of the analysis is to determine the temperature

rise distribution at and near the arc beam heat source as welllong workpieces involving quasi-steady-state conditionswithout the need to consider boundary effects as well as the width and the depth of the MP and the HAZ.

Figure A1 is a schematic of the heat-transfer model ofas short workpieces involving transient conditions andboundary effects. For the quasi-steady-state conditions, the welding process used showing primary (HS0) as well as

the five image heat sources, IHS1 through IHS5. The fivewhile the temperature gradients are rather steep duringthe heating cycle, they are less steep during the cooling image heat sources are considered to be of the same shape,

size, moving velocity (but in opposite direction except IHS1),cycle. Also, the maximum temperature decreases withan increase in y. heat liberation rate, and heat intensity distribution as the

primary heat source (HS0). As the thickness H and length8. The rate of cooling initially decreases rapidly followedby a slow decrease with time. It also decreases with an L of the plates being welded are rather small, the boundary

effects from the bottom boundary surface CD as well as theincrease in the distance away from the heat source.9. The rate of cooling at any given point increases with two lengthwise boundary surfaces, AC and BD, cannot be

neglected. They are, therefore, considered adiabatic. Thethe temperature rise. This variation can be fitted by twolinear relationships, one in the lower temperature range, heat source formed by the arc beam is considered as a

moving circular disc heat source of radius g0 and with anamely, 100 8C to 800 8C, and the other in the highertemperature range, namely, 900 8C to 1500 8C, with a heat liberation rate of qpl (in J/s). As welding progresses,

the circular disc heat source moves along the X-axis on thecoefficient of variance of #0.99.10. The thermal histories of two typical points, one in HAZ upper surface of the plate with a velocity v. The distribution

of the heat intensity over the arc beam heat source area isand the other in MP, are found to be nearly the samewith different values of thermal properties for different represented by a radially symmetric pseudo-Gaussian

distribution.temperatures in the range of 400 8C to 1000 8C.11. The thermal history of welding short workpieces was The temperature rise at any point M is the sum of the

effects from all the primary and image heat sources, whichdetermined analytically. It gives both heating and cool-ing cycles for different points along the direction of are located away from point M at a distance R0, R1, R2, R3,

R4, and R5, respectively. They can be expressed in terms ofmotion of the moving heat source. Based on thesecurves, the cooling rates can be determined at any point the dimensions of the workpiece (H and L), the traverse

velocity of the arc beam (v), the duration up to that instantand at any given temperature in the workpiece thatis being welded. This information can be used in the (t), and the coordinates of point M (X, y, z). The values ofappropriate TTT diagram for a given steel work materialto determine the nature of metallurgical transformationon the microstructure during the welding process bothin the weld bead and in the HAZ.

ACKNOWLEDGMENTS

This project was initiated by a grant from the NSF UnitedStates–China Cooperative research project on the ThermalAspects of Manufacturing. One of the authors (RK) thanksDr. Alice Hogen, NSF, for facilitating this activity and forher interest in this project. The authors are indebted to NSFfor their continuing support to one of the authors (RK) at Fig. A1—Schematic of the heat-transfer model of the welding process

showing primary as well as secondary images heat sources.[1]OSU on the various aspects of the manufacturing processes.


these distances and their projections on the X-axis (direction r0 the radius of the moving disc heat source ormoving ring heat source, cmof motion), X0, X1, X2, X3, X4, and X5, are given as follows

(refer to Figure A1). ri the radius of a segmental ring heat source, cmR distance between the center of the moving disc

R0 5 !X 2 1 y2 1 z2, heat source and the point where the tempera-ture rise at time t is concerned, cmthe projection of R0 on X-axis is X0, X0 5 X

R0, R1, distances between the center of the primary andR1 5 !X 2 1 y2 1 (2H 2 z)2, R2, R3, the relevant image moving disc heat sources

R4, R5; and the point where the temperature rise atthe projection of R1 on X-axis is X1, X1 5 Xtime t is concerned, cm

t time of observation or the time after the initiationR2 5 !(X 1 2vt)2 1 y2 1 z2, X2 5 2(X 1 2vt)of a moving disc heat source, s

R3 5 !(X 1 2vt)2 1 y2 1 (2H 2 z)2, X3 5 2(X 1 2vt) ts the time when the moving disc heat source is shutoff after completion of the welding process, sR4 5 ![2(L 2 vt) 2 X ]2 1 y2 1 z2, X4 5 2(L 2 vt) 2 X

v traverse velocity of the arc beam, cm/sR5 5 ![2(L 2 vt) 2 X ]2 1 y2 1 (2H 2 z)2, V v/2a, cm21

X projection of the distance on the X-axis betweenX5 5 2(L 2 vt) 2 X the center of the moving disc heat source and

the point where the temperature rise at timeThe fundamental equation for the temperature rise at anyt is concerned, cmpoint M caused by a moving circular disc heat source is

X0, X1, projection of the distances on the X-axis betweengiven by[1]

X2, X3, the center of the primary and the relevantX4, X5 image moving disc heat sources and the pointuM 5

3.1576qplv

4lap 3/2 ? 1 1r 2

02 ? e2XV eri5r0

ri50e23(ri /r0)2 ? ri dri where the temperature rise at time t is con-

cerned, cmX, y, z coordinates of any point M in a moving coordi-? ev2t/4a

v50

dvv 3/2 exp 12v 2

u2i

4v2 [A1]nate system where the temperature rise isconcerned

l thermal conductivity of the medium, J/cm ? s ? 8C? I0 FgiV 2

2v !1X 12vV 2

2

1 y2G uM temperature rise at any point M at any time t, 8Cr density of the medium, g/cm3

where I0(p) modified Bessel function of the first kind, order

ui 5 V!r 2i 1 R2, and R 5 !X 2 1 y2 1 z2 zero 15

12p e2p

0ep cos a da2

The temperature rise at any point M caused by the primary For ease of computation, the modified Bessel functionand each of the image heat sources can be calculated sepa- I0( p) is approximated as follows (Ref. 25 provides details):rately using Eq. [A1] by substituting the preceding values

when 0 # p , 0.2 I0( p) 5 1;of the distances and their relevant projections on the X-axisinstead of R (so the yi) and X in Eq. [A1]. Thus, the total when 0.2 # p , 1.6 I0( p) 5 0.935 exp (0.352p)temperature rise at any point M is given by

when 1.6 # p , 3 I0( p) 5 0.529 exp (0.735p);uM 5

3.1576qplv

4lap 3/2 ? 1 1r 2

02 ? eri5r0

ri50e23(ri /r0)2 ? ri dri

when p $ 3 I0( p) 51

!2ppexp ( p)

? on55n50 e e2XnV ev2t/4a

v50

dvv 3/2 exp 12v 2

u2n

4v2 [A2]

REFERENCES

1. R. Komanduri and Z.B. Hou: Metall. Mater. Trans. B, 2000, vol. 31B,? I0 FgiV 2

2v !1Xn 12vV 2

2

1 y2G pp. 1353-70.2. J.C. Jaeger: Proc. R. Soc. NSW, 1942, vol. 76, pp. 203-24.3. H.S. Carslaw and J.C. Jaeger: Conduction of Heat in Solids, 2nd ed.,where the relevant values of un are given by

Oxford University Press, Oxford, United Kingdom, 1959.4. D. Rosenthal: Trans. ASME, 1946, vol. 68, pp. 849-66.u0 5 V!r 2

i 1 R20; u1 5 V!r 2

i 1 R21;

5. J. Goldak, A. Chakravarti, and M. Bibby: Metall. Trans. B, 1984, vol.15B, pp. 299-305.u2 5 V!r 2

i 1 R22; u3 5 V!r 2

i 1 R23

6. P. Tekriwal and J. Mazumder: Finite Element Modeling of Arc WeldingProcesses, ASM Int. Conf. on Trends in Welding Research, Gatlinburg,u4 5 V!r 2

i 1 R24; u5 5 V!r 2

i 1 R25

TN, 1986.7. P. Tekriwal and J. Mazumder: Welding Res. Suppl., 1988, pp. 150-s-

156-s.NOMENCLATURE 8. G.W. Krutz and L.J. Segerlind: Welding Res. Suppl., 1978, July, pp.211-s-216-s.a thermal diffusivity of the medium, cm2/s

9. E.F. Nippes: Welding Res. Suppl., 1959, pp. 1-s-18-s.qpt heat liberation rate of a point heat source, J/s 10. W.F. Hess, L.L. Merrill, E.F. Nippes, and A.P. Bunk: Welding J.,qpl heat liberation rate of a moving disc heat source, Welding Res. Suppl., 1943, vol. 22 (9), pp. 377-s-422-s.

11. V. Paschkis: Welding Res. Suppl., 1943, pp. 462-s-483-s.J/s


12. C.M. Adams: Welding Res. Suppl., 1958, pp. 210-s-215-s. vol. 13, pp. 161-67.20. R.J. Grosh and E.A. Trabant: Welding Res. Suppl., 1956, pp. 396-s-13. N.H. Rykalin and A.I. Beketov: Welding Production, 1967, vol. 14

(9), pp. 442-47. 400-s.21. J. Isenberg and S. Malkin: “Effect of Variable Thermal Properties on14. V. Pavelic, R. Tanbakuchi, O.A. Uyehara, and P.S. Myers: Welding

Res. Suppl., 1969, July, pp. 295-s-304-s. Moving-Band Source Temperatures,” ASME Paper No. 74-WA/Prod-5, ASME, New York, NY, 1974.15. M.F. Ashby and K.E. Easterling: Acta Metall., 1984, vol. 32 (11), pp.

1935-48. 22. S. Kou, D.K. Sun, and Y.P. Le: Metall. Trans. A, 1983, vol. 14A, pp.643-53.16. W.B. Li, K.E. Easterling, and M.F. Ashby: Acta Metall., 1986, vol.

34 (8), pp. 1533-43. 23. G.H. Little and A.G. Kamtekar: Computers Struct., 1998, vol. 68, pp.157-65.17. T.W. Eager and N.-S. Tsai: Welding Res. Suppl., 1983, Dec., pp.

346-s-355-s. 24. N. Silva Prasad and T.K. Sankara Narayanan: Welding Res. Suppl.,1996, pp. 123-s-128-s.18. T. Kasuya and N. Yurioka: Welding Res. Suppl., 1993, Mar., pp.

107-s-115-s. 25. Z.B. Hou and R. Komanduri: Trans. ASME, J. Tribol., 1998, vol. 120,pp. 645-51.19. R.J. Grosh, E.A. Trabant, and G.A. Hawkins: Q. Appl. Math., 1955,


thermal analysis of the arc welding process part ii.pdf

Documents