PhenomenologicalModeling of FusionWelding Processes

S.A. David, T. DebRoy, and J.M. Vitek

IntroductionWelding is utilized in 50% of the indus-

trial, commercial, and consumer productsthat make up the U.S. gross national prod-uct.1 In the construction of buildings,bridges, ships, and submarines, and in theaerospace, automotive, and electronic in-dustries, welding is an essential activity. Inthe last few decades, welding has evolvedfrom an empirical art to a more scientifi-cally based activity requiring synthesis ofknowledge from various disciplines. De-fects in welds, or poor performance ofwelds, can lead to catastrophic failureswith costly consequences, including lossof property and life.

Figure 1 is a schematic diagram of thewelding process showing the interactionbetween the heat source and the base metal.During the interaction of the heat sourcewith the material, several critical events oc-cur: melting, vaporization, solidification,and solid-state transformations. The weld-ment is divided into three distinct regions:the fusion zone (FZ), which undergoes melt-ing and solidification; the heat-affectedzone (HAZ) adjacent to the FZ, that mayexperience solid-state phase changes butno melting; and the unaffected base metal(BM).

Creating the extensive experimentaldata base required to adequately charac-terize the highly complex fusion weldingprocess is expensive and time consuming,if not impractical. One recourse is to simu-late welding processes either mathemati-cally or physically in order to develop aphenomenological understanding of theprocess. In mathematical modeling, a setof algebraic or differential equations aresolved to obtain detailed insight of theprocess. In physical modeling, under-standing of a component of the weldingprocess is achieved through experimentsdesigned to avoid complexities that areunrelated to the component investigated.

In recent years, process modeling has

grown to be a powerful tool for under-standing the welding process. Using com-putational modeling, significant progresshas been made in evaluating how thephysical processes in the weld pool influ-ence the development of the weld pooland the macrostructures and microstruc-tures of the weld. Much recent researchon weld-pool phenomena has attemptedto gain an understanding of fluid flowand heat transfer in the weld pool. In ad-dition to information on weld-pool shapeand size, computational modeling of thewelding process can provide detailed in-formation on such parameters as the weldcooling rate, temperature gradients, mi-crostructural development, and residualstresses in the welded structures. Limitedprogress has also been made in modelingand understanding weld-pool solidificationbehavior and solid-state phase transfor-mations in the FZ and the HAZ. Finally,process modeling has been used as a criti-cal element for the intelligent control andautomation of the welding process, to makewelds with a desired quality, performance,and productivity.

A recent publication,2 a series of interna-tional conferences,3'6 and a recent work-shop7 have all covered the current issues,trends, and directions in weld processmodeling. This article aims to provide aphenomenological perspective of model-ing activities in fusion welding. A few ex-amples, taken mostly from the authors'laboratories, highlight several important

Heat and Mass Transfer, andFluid FlowDriving Forces for Motion

During fusion welding, the metal in theweld pool undergoes vigorous recircula-tory motion driven by surface tension,buoyancy and, when electric current isused, electromagnetic forces.8"13 Buoyancy

effects originate from the spatial variationof the liquid metal density, mainly be-cause of temperature variations and, to alesser extent, from local composition varia-tions. Electromagnetic effects are a con-sequence of the interaction between thedivergent current path in the weld pooland the magnetic field it generates. Theeffect is important when a large electriccurrent passes through the weld pool, asis the case for arc welding and electron-beam welding. Spatial variation of the sur-face tension, arising from temperature andcomposition gradients at the weld-poolsurface, often provides the main drivingforce for the convective flow, known asthe Marangoni flow. Fluid flow and heattransfer are important in determining thesize and shape of the weld pool, the weldmacrostructure and microstructure, andthe weldability of the material.

Flow Velocities in the Weld PoolFrequently, the main driving force for

convection is the spatial gradient of sur-face tension at the weld-pool surface. Inmost cases, the difference in surface ten-sion arises from the temperature variationat the weld-pool surface. For such a situ-ation, the Marangoni stress can be ex-pressed as:

dT dr

where T is the shear stress due to the tem-perature gradient, yis the interfacial tension,T is the temperature, and r is the distancealong the surface from the axis of the heatsource. If a boundary layer develops, T canbe expressed as:14

(2)

where p and /x are the density and viscos-ity, Mm is the local velocity, and W is thewidth of the pool. The shear stress on thesurface halfway between the heat sourceaxis and the weld-pool edge is approxi-mately the maximum stress, and the orderof magnitude of the maximum velocitycan be calculated from:

w"2

dT dr 0.664p'V(3)

For typical values of pool width of 0.5 cm,metal density of 7.2 g/cm3, viscosity of0.06 poise, temperature coefficient of sur-face tension, dy/dT, of 0.5 dynes/(cm°C),and spatial gradient of temperature of600°C/cm, the maximum velocity is ap-proximately 62 cm/s. Considering that theweld pool is only 0.5 cm wide, this veloc-ity is rather large. Even larger computedvelocities of the order of 100 cm/s have been

MRS BULLETIN/JANUARY 1994 29

Phenomenological Modeling of Fusion Welding Processes

Weld Pool

Heat Affected Zone-

Fusion Zone

Base Metal

Heat Source

Figure 1. Schematic diagram of thewelding process showing theinteraction between the heat sourceand the base metal.

Figure 2. Surface tension of Fe-0 alloyas a function of temperature andoxygen concentration.1V9

reported12'3 in systems dominated byMarangoni convection. When the veloci-ties are small, the shear stress, r, cannot beestimated by Equation 2, which is basedon the boundary layer theory. For such asituation, a rough estimate of the shearstress can be obtained from geometricconsideration as lyMjd, where d is thedepth of the weld pool. The expression forthe approximate value of the maximumvelocity can be obtained by combining thisexpression for shear stress with the ex-pression for Marangoni stress, i.e., Equa-tion 1. When the surface tension gradientis not the main driving force, the maxi-mum velocities can be much smaller. Forexample, when the flow occurs primarilyby natural convection, the maximum ve-locity, u m, can be approximated by the fol-lowing relation:"

,(4)

where g is the acceleration due to gravity,/3 is the coefficient of volume expansion,AT is the temperature difference that drives

the flow, and d is the depth. For typical val-ues of AT = 600°C, £ = 981 cm/s2, )3 =3.5X1(T7OC, and d=0.5 cm, the valueof um is 3.2 cm/s. Similar calculations canbe done for electromagnetically drivenflow. The magnitude of the velocities forboth buoyancy and electromagneticallydriven flows in the weld pool are com-monly much smaller than those obtainedfor surface tension driven flows.

Relative Importance of HeatConduction and Convection

The relative importance of conductionand convection in the overall transport ofheat in the weld pool can be assessedfrom the value of the Peclet number, whichis given by:

Pe =k

(5)

where u is the flow velocity, p is the den-sity, cp is the specific heat at constant pres-sure, L is the characteristic length and kis the thermal conductivity of the melt.For a typical case with u = 10 cm/s, p =7.2 g/cm3, cp = 0.2 cal/(g°C), L = 0.5 cm,and k = 0.1 cal/(cm s°C), the Pe is 72.When the Peclet number is this large, heattransport occurs primarily by convection,and conduction is not important. How-ever, for metals with high thermal con-ductivity, at low velocities and for smallpool sizes, the value of Pe can be low, andaccurate calculations of heat transfer canbe done using relatively simple conduc-tion calculations. It should be noted thatthe conduction of heat in the solid regionis important in the dissipation of heataway from the weld pool. Therefore, thethermal conductivity of the solid and thespecimen dimensions are important in de-termining the size of the molten pool.

Convection Effects on Weld-PoolShape and Size

Variable depth of penetration during thewelding of different heats of commercialmaterial with the composition within a pre-scribed range has received considerableattention. The penetration depth is oftendetermined by the concentration of thesurface active elements in the commercialalloy. U15~18 These elements can affect thetemperature coefficient of surface tension,dy/dT, and the resulting direction of fluidflow in the weld pool12 and the shape of theweld pool. The interfacial tension in thesesystems can be described1718 by a formal-ism based on the combination of Gibbsand Langmuir adsorption isotherms:

Pure Iron

0.47 cm-

Fe-0.03 Wt % Oxygen

0.37 cm

y = 7° - A(T - Tm)

- RTT, ln(l + Ar,fl (6)

Figure 3. Velocity and temperaturefields for two different cases: (a) pureiron and (b) iron with 0.03 wt%oxygen.w

where y is the interfacial tension as afunction of composition and temperature,•y'm is the interfacial tension of the puremetal at the melting point, Tm, A is a con-stant, R is the gas constant (in appropriateunits), T is the absolute temperature, F, isthe surface excess of the solute at satura-tion solubility, A:, is the entropy factor, a, isthe activity of the solute, and AH" is theenthalpy of segregation. The calculatedvalues of surface tension171' for Fe-O alloysare presented in Figure 2. It is observed thatfor high concentrations of oxygen, dy/dTcan change from a positive value at "low"temperature to a negative value at "high"temperature. This implies that in a weldpool with a fairly high oxygen content,dy/dT can go through a change in sign onthe surface of the pool. Under these condi-tions, the fluid flow in the weld pool is morecomplicated than a simple recirculation.

The calculated gas-tungsten arc (GTA) fu-sion zone profiles19 for pure iron, and aniron-0.03 wt% oxygen alloy are presented

30 MRS BULLETIN/JANUARY 1994

Phenomenological Modeling of Fusion Welding Processes

in Figure 3. The results in Figure 3 clearlyshow the significant effect of oxygen con-centration on the weld-pool shape and theaspect ratio. Near the heat source, wherethe temperature is very high, the flow isradially outward. However, for the Fe-Oalloy, a short distance away from the heatsource where the temperature drops be-low the threshold value for the change inthe sign of dy/dT, the flow reverses direc-tion. The flow field is not a simple recircu-lation. The computed values of the aspectratio are compared with the experimentaldata of Heiple and Roper20 for a stainlesssteel in Figure 4. Although there are onlylimited experimental data, good agree-ment is achieved between the calculatedresults and the experimental data. Similarbehavior was observed22"23 for the GTA fu-sion welding of 304 stainless steel contain-ing 240 ppm sulfur. Apart from the goodagreement between the experimental andthe calculated weld-pool shapes, the cal-culations reveal new insights about thecomplexity of the flow field. Although thequalitative effects of the role of surface ac-tive elements were known, the numericalcalculations provide a basis for quantita-tive assessment of their role in the devel-opment of the weld-pool geometry.

Cooling Rate and Simple Featuresof Solidification Structure

The cooling rate at a given monitoringlocation is the product of the temperaturegradient and the solidification growth rate.For several alloys, the secondary dendritearm spacings have been experimentally cor-related with the cooling rates. Therefore,using numerically computed cooling ratesand the available experimental correlation,the secondary dendrite arm spacing canbe estimated. In Figure 5, the computed sec-ondary dendrite arm spacings of 0.9 /nmand 0.4 /am for 5 mm/s and 31 mm/s weld-ing speeds, respectively, are compared withthe experimental data.12 The good agree-ment between the predicted secondary den-drite arm spacing and the values obtainedfrom independent experimental data dem-onstrates that the calculated values of cool-ing rates are fairly accurate. In anotherinvestigation13 of the effect of pulsed laserwelding on the thermal response of 310and 316 austenitic stainless steels, the cool-ing rates were theoretically calculatedfrom fundamental principles of transportphenomena. The secondary dendrite armspacings were determined from the mi-crostructures. The results indicated excel-lent agreement between the measured andthe expected secondary dendrite arm spac-ings based on the cooling rates. The investi-gations with both continuous12 and pulsed13

heat sources indicate that simple features

1.2

1.0

O Experimental dataTheoretical calculations

0.00 0.05WT% OXYGEN

0.10

Figure 4.Comparison of the predictedand experimental aspect ratios (depthto width).ig The oxygen concentrationof 0.04 wt% is obtained from the Cr-0equilibrium data for 21% Cr, given inReference 21.

o a.

> otr << Q-

SIp

1

A.

5x

31

I

I I

10'3 m/s

x 10-3 m/s

I I

A•

BrowAbdu

Br et al.gadar

10° 102 103 104

COOLING RATE (K/s)105 106

Figure 5. Plot of secondary dendritearm spacing as a function of coolingrate for conduction mode laserwelding of AISI201 stainless steel.12

of the solidification structure can be deter-mined from the available numerical mod-els of weld-pool transport phenomena.

Evaporation from Weld-PoolSurface' The weld-pool surface temperatures are

normally much higher than the meltingpoints of the weld metals. Consequently,pronounced vaporization of alloying ele-ments takes place, especially when high-energy-density heat sources are used. ^33

Such losses often result in a change in thecomposition of the weld metal, affect weldproperties, and are a serious problem inthe welding of many important engineer-ing alloys.

Block-Bolten and Eagar31 suggested thatthe weld-pool peak temperature is limitedby the evaporation of elements from theweld pool. To determine the maximum tem-perature of the weld pool, they used thefollowing energy balance.

W(T)[Le - AH] = xP (7)

where W(T) is the temperature-dependentvaporization rate, L e is the enthalpy of va-porization, AH is the enthalpy of mixing,P is the power density, and x is the frac-tion of the input power used for vaporiza-tion. In Equation 7, W(T) is computed fromknown values of all other parameters. Sub-sequently, the maximum temperature isdetermined from the known VV(T) versusT relation. Their calculations indicated thatthe peak temperature for an iron-based al-loy is about 2773 K for arc welding, and ashigh as 4273-5273 K for laser and electronbeam welding. The relation between thevaporization rate W(T) and the tempera-ture, T, significantly affects the value ofthe temperature calculated from Equation 7.In fact, the temperature versus vaporiza-tion rate relation used by Block-Boltenand Eagar,31 i.e. the Langmuir equation,34

grossly overpredicts the vaporization rateat any given temperature, and likewise,for a given vaporization rate, it grosslyunderpredicts the temperature. Kraushas shown that the peak temperature inGTA stainless steel welds, measured bythe laser reflectance method, could reachas high as 2950 K. Thus, an appropriatecorrelation between the vaporization rateand the weld-pool temperature is essentialfor realistic calculations.

DebRoy et al.25"27 developed a compre-hensive mathematical model to under-stand vaporization of elements and thecomposition change of the weld metal.The computed weld-pool temperature dis-tribution was used for the vaporizationrate calculations. The weld-pool heattransfer and fluid flow were solved simul-taneously with the velocity distributionfunctions of vapor molecules at various lo-cations above the weld-pool surface to cal-culate the rates of vaporization of thealloying elements. The procedure allowedfor the calculation of both the evaporationand condensation fluxes based on the ap-plication of conservation of mass, momen-tum, and energy equations in the gas phase.

A key feature of the calculations is theconsideration of the pressure-gradient-driven mass transfer.2'3637 In laser pro-cessing of metals and alloys, the peaktemperature reached at the surface oftenexceeds the boiling point of the irradiatedmaterial.38"10 At temperatures higher thanthe boiling point, the pressures in thevicinity of the pool are greater than theambient pressure, and the excess pressureprovides a driving force for the vapor tomove away from the surface. In addition,mass transfer rates due to concentrationgradients were determined using availablecorrelations among various dimensionlessnumbers.41 Heat transfer to the shielding gasand heat loss due to vaporization of the

MRS BULLETIN/JANUARY 1994 31

Phenomenological Modeling of Fusion Welding Processes

alloying elements were taken into accountin the calculations. The calculated ratesagreed well with the experimental datafor the welding of pure iron and titanium,and for both low- and high-power CO2laser welding of 201 stainless steel. Straight-forward calculations using the Langmuirequation do not account for the conden-sation of vapors. As a result, as noted pre-viously, the rates calculated from theLangmuir equation are commonly muchhigher than the experimentally observedvalues. The calculated vaporization ratesare significantly better than the corre-sponding values obtained from the straight-forward application of the Langmuirequation. However, the higher accuracyachieved by including a more realistic anddetailed description of the physical pro-cesses requires more complex calculations.

Physical Modeling of Nitrogen,Oxygen, and Hydrogen Partitioning

The nitrogen, oxygen, and hydrogenconcentrations in the weld metal stronglyinfluence the microstructure and proper-ties. The gases dissolve in the weld metal,whereupon they may form porosity or com-bine with alloying elements to form inclu-sions. For example, in the welding of steel,hydrogen induces cracking, nitrogen im-proves the tensile properties, and oxygenpromotes inclusion formation. Therefore,reliable modeling of the partitioning ofgases into the liquid metal is required.

When a metal is exposed to a pure di-atomic gas such as hydrogen, the equi-librium concentration of the species in themetal is proportional to the square root ofits partial pressure at any given tempera-ture. This relation, known as the Sieverts'Law, is widely used for calculating soluteconcentrations in metals in equilibriumwith diatomic gases. However, such esti-mations are not useful for most weldingprocesses. Near the weld-pool surface, be-sides common diatomic molecules, alsopresent within the gas plasma are excitedmolecules, atoms, and ions. As a result, theinterstitial concentrations in the weldmetal are significantly higher than thosecalculated from Sieverts' Law42'47 The pres-ence of excited neutral atoms, ions, andelectrons4849 precludes any simple exten-sion of the well-established formal treat-ment of gas-metal systems to welding.Efforts to develop a general principle forunderstanding the partitioning of nitro-gen, oxygen, and hydrogen between theweld pool and its environment are just intheir infancy.

Appropriately designed physical simu-lation of gas dissolution in the weld poolcan avoid both temperature and composi-tion gradients on the weld-pool surface and

in the gas phase. Bandopadhyay et al.47

conducted physical simulation experimentsof nitrogen dissolution in the weld pool.They exposed drops of tantalum and nio-bium to controlled plasma environmentswith known concentrations of nitrogen inthe feed gas. The plasma parameters wereoptically monitored during experiments.They observed significantly higher nitrogensolubilities in pure tantalum and niobiumthan those predicted by Sieverts' Law.The formation of excited neutral atoms inthe gas phase by various inelastic electronimpact reactions was found to be the maincause of enhanced nitrogen solubility.

Weld SolidificationDevelopment of the microstructure in

the FZ, also known as weld metal, de-pends on the solidification behavior of theweld pool. Solidification behavior controlsthe size and shape of grains, the extent ofsegregation, and the distribution of inclu-sions and defects such as porosity and hotcracks. Since the properties and perfor-mance of the FZ depend on the solidifica-tion behavior and the microstructuralcharacteristics, understanding weld-poolsolidification behavior is essential. Effortsare under way to use both analytical andnumerical models to better understandthe solidification behavior of the FZ.50"57

Most of our current knowledge of weld-pool solidification is derived from an ex-trapolation of the knowledge and modelsfor freezing of castings, ingots, and crys-tals at lower thermal gradients and growthrates.50 The same parameters that are im-portant in determining microstructures incastings, such as growth rate (Rs), tem-perature gradient (G), undercooling (A7\j),and alloy composition, play significantroles in determining the development ofmicrostructures in welds. The tempera-ture gradient and growth rate are impor-tant in the combined forms G/Rs and GRS(cooling rate) since they influence the so-lidification morphology and the scale ofthe solidification substructure, respec-tively. In welding, where the molten poolis translated through the material, both Gand GRS vary considerably across the FZ.Depending on the type of welding pro-cess and the location within the weldpool, the maximum cooling rate encoun-tered within the weld pool may rangefrom 102 to 107oC/s. Experimental mea-surement of these parameters during weld-ing is extremely difficult. Modeling canprovide a reasonable alternative to directmeasurements by allowing thermal gradi-ents, growth rates, and cooling rates inthe weld pool to be calculated.

Solidification microstructures found inwelds are often quite complex. However,

development of these microstructures canbe understood by considering classical nu-cleation and growth theory. In welds,since solidification proceeds from the pre-existing solid substrate, there is little or nonucleation barrier. Solidification occursspontaneously by epitaxial growth. Whena filler metal is used, in addition to epitax-ial growth, the more classical case of het-erogeneous nucleation may also apply.The stability of the solid-liquid interface iscritical in determining the microstructuralcharacteristics of the FZ. Conditions inthe immediate vicinity of the interfacedetermine whether the growth is planar,cellular, or dendritic. Theories have beendeveloped for interface stability underconditions of equilibrium at the interfacefor normal solidification or under extremenonequilibrium conditions that exist dur-ing rapid solidification.** Further, in re-cent years significant advances have beenmade in the theory and fundamentals ofdendritic solidification."1'61 These theoriesand models can be readily extended toweld-pool solidification.

In alloy solidification, where partition-ing of solute elements between the solidand liquid occurs, the effects of solute re-distribution and buildup at the solid-liquidinterface on the morphological stability ofthe solidification front can be understoodby considering the simple concept of con-stitutional supercooling.5"'59 This criterionfor solidification-front instability can bemathematically stated as:

G i < AT},Ks D,.

or, equivalently,

G i < M, C(,(l - K)Rs KD,

(8)

(9)

where GL is the thermal gradient in theliquid at the solidification interface, R<, isthe solidification front growth rate, My isthe slope of the equilibrium liquidus line,Co is the overall alloy composition, K isthe equilibrium partition coefficient, DL isthe solute diffusion coefficient in liquid,and ATo is the equilibrium solidificationtemperature range at composition Co. Ifthis criterion is met, the planar solidifica-tion front is unstable and dendritic growthis promoted. If the reverse is true, namelyG,/Rs > AT,/D,, then the plane front willbe stable.

As growth conditions depart from pla-nar stability, the interface morphologywill change from planar to cellular to den-dritic. If conditions are favorable, the den-drites will exhibit secondary and tertiary

32 MRS BULLETIN/JANUARY 1994

Phenomenological Modeling of Fusion Welding Processes

branches as shown in Figure 6. The analy-sis that led to the theory of constitutionalsupercooling was not rigorous, and morethorough analyses have been carried outand additional conditions for plane frontsolidification have been developed.62"67 Asa result, an absolute stability region, whereplanar growth prevails, is predicted athigh solidification rates as well at lowrates. This latter morphological regimemay be important for welds made at highspeeds using high-energy beam processes.

Partitioning of solute between solid andliquid results in extensive solute redistri-bution during alloy solidification, and thiscan significantly affect weldability, mi-crostructure, and properties. Both mi-crosegregation (on a fine scale of the orderof the dendrite arm spacing) and macro-segregation (on a large scale of a few hun-dred microns to millimeters) may occur inwelds. Attempts have been made to charac-terize and model solute redistribution dur-ing weld-pool solidification.5"'556869 Thesestudies have extended different modelsused to describe the solute redistributionand the resulting segregation in castingsand single crystals to welding situations.Here one must consider redistribution infront of the dendrite tip as well as withinthe interdendritic regions (also knownas the mushy zone). Solute redistribution inthe mushy zone of a weld can be modeledby applying principles originally formu-lated for castings. Many of the approachesused to model microsegregation start withthe simple Scheil equatioiv" that describesthe composition of the solid as a functionof the fraction that is solid in a representa-tive volume element in the mushy zone:

C? = (10)

where C* is the solid composition, Co isthe overall alloy composition, K is the par-tition coefficient, and / s is the volume frac-tion that is solid. Significant advances havebeen made in modifying this model to ac-count for limited or extensive diffusion inthe solid.71'72 In estimating the extent ofsegregation in welds, it is not adequate toconsider the solute redistribution in theweld mushy zone alone; it is also essentialto consider the solute redistribution infront of the dendrite tips. The compositionat the dendrite tip is determined by thedendrite tip temperature and the partitioncoefficient. The tip temperature, partitioncoefficient, and composition are strongfunctions of the tip radius, growth rate,thermal gradient, and other factors. Theeffect of increased undercooling at thedendrite tip, which may be present duringwelding, would be to decrease the degreeof coring and thus decrease the extent of

Figure 6. Scanning electronmicrograph showing the developmentof dendrites in a nickel-basedsuperalloy single-crystal weld. Themicrograph was taken on ahot-cracked surface formed duringelectron-beam welding of the alloy.The primary, secondary, and tertiarydendrite arms are clearly discernible.

microsegregation. Some of these conceptshave been tested and verified in welds.

Another exciting area of research inwelding is the understanding and model-ing of grain structure development in theFZ. The development of the FZ grainstructure in a weld is primarily controlledby the base metal grain structure and thewelding conditions. As mentioned earlier,initial growth normally occurs epitaxiallyat the partially melted grains. 3 Bothcrystallographic effects and welding con-ditions can strongly influence the devel-opment of microstructure in the FZ.Specifically, crystallographic effects willinfluence grain growth by favoring growthalong particular crystallographic direc-tions—namely the easy growth direc-tions.58"59 For cubic metals, the easy growthdirections are the family of (100) direc-tions. Optimum growth conditions existwhen one of the easy growth directionscoincides with the heat flow direction.There is a strong interplay between theshape of the pool and growth crystallog-raphy. Limited progress has been made inour understanding of the development ofgrain structure in welds because it is diffi-cult to interpret and understand fully thedevelopment of polycrystalline FZ mi-crostructures. This is true since the detailsof the grain growth selection process andthe three-dimensional pool shape are ob-

scured by the multitude of grains andcrystal orientations present in polycrys-talline materials.

Welding experiments with single crys-tals have been found to be extremely use-ful in this situation. Recently, with the aidof single-crystal welds, significant ad-vances have been made in understandingand modeling the interrelationships be-tween weld-pool geometry, growth crys-tallography, and the dendrite selectionprocess in the development of FZ mi-crostructures.51"54'57 This experimental tech-nique has been found to be extremelypowerful. Also, a geometrical model hasbeen developed that enables prediction ofthe stable dendrite growth directions as afunction of pool shape for various crystalorientations. The same model is being ex-tended to evaluate the competitive den-drite growth behavior in bicrystal welds.54

From the observed dendritic arrange-ments in the single-crystal welds, it is pos-sible to reconstruct a three-dimensionalweld pool shape as shown in Figure 7. Thetechnique may serve as a tool to experi-mentally verify the pool shapes predictedby some of the computational models.Furthermore, the insight gained fromsuch single-crystal weld experiments canbe readily extended toward a better un-derstanding of microstructural develop-ment in polycrystalline materials.

Modeling Phase Transformationsin Welds

During welding, extensive phase trans-formations occur in both the FZ and theHAZ. In addition to the solidification reac-tion, solid-state transformations may alsotake place. The nature of these transforma-tions depends on the heating and coolingrates and also the maximum temperaturereached at any given location during theweld thermal cycle. Modeling of thesetransformations and the resulting micro-structures in weldments remains a greatchallenge. In the FZ, the extreme solidifi-cation conditions prevalent during manywelding processes often lead to the solidi-fication of nonequilibrium phases.76'77 Asdescribed in a recent summary,78 severalinvestigators have tried to model the non-equilibrium solidification behavior inaustenitic stainless steels using advancedsolidification theories, thermodynamicanalyses, and other methods. These analy-ses have, to date, met with only limitedsuccess.

Solid-state transformations may occurduring post-solidification cooling in theFZ or in the HAZ. The HAZ, because ofits extensive thermal gradient and non-uniform thermal exposure, may undergowidely different transformation character-

MRS BULLETIN/JANUARY 1994 33

Phenomenological Modeling of Fusion Welding Processes

istics. As a result, the HAZ often exhibitssignificant compositional, microstructural,and property gradients. Such gradientsare unique to welded structures. In addi-tion, generation of thermal stresses duringwelding can drastically affect the kineticsof solid-state phase transformations inboth the FZ and HAZ. Several advanceshave been made in recent years in model-ing the solid-state phase transformationsin weldments.79"84 Models have been de-veloped based on physical metallurgy prin-ciples. Efforts are also underway to useMonte Carlo techniques to simulate graingrowth behavior in the HAZ and to quan-titatively predict microstructural evolu-tion in the HAZ.85

Phase transformations in weldmentshave a unique character that requires con-sideration of the transient heating as wellas cooling to accurately understand themicrostructural development. Therefore,traditional isothermal transformation kinet-ics and also cooling transformation kinet-ics must be supplemented by the inclusionof transformation effects during heating.This results in a combination of thermalhistories, which are not commonly ad-dressed in traditional studies, and it leadsto additional complications. Any success-ful phase transformation modeling alsorequires a basic coupling with accuratethermal models. This coupling is criticalfor the development of a comprehensive,

integrated model for the microstructuraldevelopment in weldments.

Residual StressesStresses develop in welded structures

because of non-uniform heating and struc-tural changes. Residual stresses in weld-ments may produce distortion or causepremature failure in weldments or both.86

Computational models can provide a de-tailed description of the residual stress dis-tribution in weldments.8793 A prerequisitefor the calculations is the detailed time-temperature history obtained fromnumerical calculations. Calculations ofdistortions, stresses, and strains are com-putationally intensive, and they take sig-nificantly more computer time than thecalculation of the transient temperaturefield. Recently, the trend in these calcu-lations has moved from cross-sectionaltwo-dimensional plane strain and axisym-metric models to models using plane stress,shell, and three-dimensional elements.90

Tekriwal and Mazumder92 performed athree-dimensional thermomechanical analy-sis of the gas-metal arc welding to calcu-late distortions, stresses, and strains. Goodqualitative agreement was achieved be-tween the calculated and the measuredtransient strains. However, the calculatedand measured residual strains were differ-ent. The differences were attributed tolack of an accurate stress-strain constitu-

[010J

Figure 7. Three-dimensional schematic diagram of a weld pool. The weld-pool shapewas constructed using several two-dimensional transverse optical micrographs ofFe-15Cr-15Ni alloy single-crystal welds. Vb represents the velocity of the heat source.

tive relation and the use of a coarse grid.Oddy et al.93 showed that phase transfor-mations can significantly affect the re-sidual stresses generated by the weldingof some steels. Thus, the uncertainties inthe calculation procedures include the in-accuracies in the calculation of transienttime-temperature history as well as theapproximations in the assumed stress-strain relation, particularly when impor-tant solid-state transformations take place.

EpilogueIn recent years, modeling has provided

significant insight into the dynamics ofthe welding process and the properties ofwelded materials which could not havebeen obtained otherwise. However, theseaccomplishments must be balanced againstsignificant limitations of the work so far,and more importantly in perspective ofwhat new avenues have been opened bythe work that has been completed.

Because welding processes are so com-plex, modeling them is computationallyintensive, expensive, and requires trainedpersonnel. Although phenomenologicalmodeling is a powerful tool, the modelingto date has been of limited use to practic-ing engineers because of its complexity, cost,and need for user training. Furthermore,most of the models have not been ade-quately standardized because the publishedworks have focused on limited experi-mental verification. Therefore, standard-ization of the models would be a steptoward gaining the greater confidence ofpracticing engineers. Improved user ac-cess to the models at low cost would alsobe helpful, if not essential, for gaining wideracceptance of the models in engineeringpractice.

Phenomenological modeling has animportant role in developing adaptivecontrol for obtaining defect-free welds.Implementation of adaptive control inwelding would involve sensing and con-trol of the heat source position, weld-pooltemperature, weld penetration, defect for-mation, and, ultimately, would lead tocontrol of microstructure and properties.An important element in the intelligentcontrol loop is process modeling. Processmodeling calculations in real time couldprovide the necessary bridge for couplingthe process parameters with the desirablemicrostructure and properties of the weld.Most of the comprehensive phenomeno-logical models require extensive computertime and cannot be used for real-time ap-plications. However, these large modelsare essential for "calibrating" the compu-tationally simpler models that can be usedin real time. Such efforts would form a re-alistic basis for science-based control of

34 MRS BULLETIN/JANUARY 1994

Phenomenological Modeling of Fusion Welding Processes

weld metal structure, composition, andproperties.

AcknowledgmentsThe authors wish to acknowledge help-

ful comments from Drs. S. Babu andS. Viswanathan. The research was spon-sored by the Division of Materials Sciences,the U.S. Department of Energy, under con-tract DE-AC05-84OR21400 with MartinMarietta Energy Systems, Inc.; and grantDE-FG02-84ER45158 with PennsylvaniaState University.

References1. American Welding Society Statement, USAToday (March 22,1992).2. S.A. David and T. DebRoy, Science 257 (1992)p. 497.3. Advances in Welding Science and Technology,edited by S.A. David (ASM International, Mate-rials Park, Ohio, 1986).4. Recent Trends in Welding Science and Tech-nology, edited by S.A. David and J.M. Vitek(ASM International, Materials Park, Ohio, 1990).5. International Trends in Welding Science andTechnology, edited by S.A. David and J.M. Vitek(ASM International, Materials Park, OH, 1993).6. Mathematical Modeling of Weld Phenomena,edited by H. Cerjak and K.E. Easterling (TheInstitute of Materials, London, 1993).7. Modeling for Welding Science, DOE-BESWorkshop (Cocoa Beach, March 16-19,1993).8. S. Kou and Y. Le, Metall. Trans. A 14A (1983)p. 2243.9. G.M. Oreper and J. Szekely, /. Fluid Mech. 147(1984) p. 53.10. C. Chan, J. Mazumder, and M.M. Chen,Metall. Trans. A 15A (1984) p. 2175.11. J. Szekely in Reference 3, p. 3.12. A. Paul and T. DebRoy, Metall. Trans. B 19B(1988) p. 851.13. T. Zacharia, S.A. David, J.M. Vitek, andT. DebRoy, Metall. Trans. A 20A (1989) p. 957.14. C.J. Geankoplis, Transport Processes andUnit Operations (Allyn and Bacon, Boston,1983).15. C.R. Heiple and J.R. Roper, Weld. /. Res.Supp. 61 (1982) p. 92s.16. C.R. Heiple, J.R. Roper, R.T. Stagner, and J.J.Alden, in Reference 15, p. 72s.17. P. Sahoo, T. DebRoy, and M.J. McNallan, inReference 12, p. 483.18. M.J. McNallan and T. DebRoy, Metall.Trans. B 22B (1991) p. 557.19. K. Mundra and T. DebRoy, unpublished re-search, Department of Materials Science andEngineering, Pennsylvania State University,University Park, PA.20. R. Heiple and J.R. Roper, in Trends in Weld-ing Research, edited by S.A. David, (ASM,Metals Park, OH, 1982) p. 489.21. ED. Richardson, Physical Chemistry of Meltsin Metallurgy, Vol. 2 (Academic Press, London,1974) p. 349.22. T. Zacharia, S.A. David, J.M. Vitek, andT. DebRoy, Weld. J. Res. Supp. 68 (1989) p. 499s.23. T. Zacharia, S.A. David, J.M. Vitek, andT. DebRoy, in Reference 22, p. 510s.24. P.A.A. Khan and T. DebRoy, Metall. Trans.B 15B (1984) p. 641.

25. T. DebRoy, S. Basu, and K. Mundra, /. Appl.Phys. 70 (1991) p. 1313.26. K. Mundra and T. DebRoy, Metall. Trans. B24B (1993) p. 145.27. K. Mundra and T. DebRoy, Weld. J. Res.Supp. 72 (1993) p. Is. .28. T. DebRoy, in Reference 5, p. 17.29. P.A.A. Khan, T. DebRoy, and S.A. David,Weld. J. Res. Supp. 67 (1988) p. Is.30. M.J. Cieslak and P.W Fuerschbach, in Refer-ence 12, p. 319.31. A. Block-Bolten and T.W. Eager, in Refer-ence 24, p. 461.32. A. Blake and J. Mazumder, /. Eng. Industry107 (1985) p. 275.33. S. Basu, MS thesis, Pennsylvania State Uni-versity, 1992.34. R.D. Pehlke, Unit Processes in ExtractiveMetallurgy (Elsevier, New York, 1979).35. H.G. Kraus, in Reference 22, p. 269s.36. S.I. Anisimov and A.K. Rakhmatulina, Son.Phys. IETP 37 (1973) p. 41.37. C.J. Knight, AlAA J. 17 (1979) p. 519.38. M. von Allmen, Laser-Beam Interactionswith Materials (Springer-Verlag, Berlin, 1987).39. V.A. Batanov, EV. Bunkin, A.M. Prokhorov,and V.B. Fedorov, Sov. Phys. JETP 36 (1973)p. 311.40. C.L. Chan and J. Mazumder, /. Appl. Phys.62 (1987) p. 4579.41. E.U. Schlunder and V. Gniclinski, Chcm-Ing.-Tech. 39 (1967) p. 578.42. S.A. Gedeon and T.W. Eagar, Weld. J. Res.Supp. 69 (1990) p. 264s.43. S. Ohno and M. Uda, Trans. Nat. Res. Inst.Met. 23 (1981) p. 243.44. G. den Ouden and O. Griebling, in Refer-ence 4, p. 431.45. M. Uda, S. Ohno, and T. Wada, /. Jpn. Weld.Soc. 38 (1969) p. 382.46. M. Uda and S. Ohno, Trans. Nat. Res. Inst.Met. 15 (1973) p. 20.47. A. B a n d o p a d h y a y , A. Baner jee , andT. DebRoy, Metall. Trans. B 23B (1992) p. 207.48. G.J. Dunn and T.W. Eagar, Metall. Trans. A17A (1986) p. 1865.49. M.M. Collur and T. DebRoy, Metall. Trans. B20B (1989) p. 227. ,50. S.A. David and J.M. Vitek, Intl. MaterialsRev. 34 (1989) p. 213.51. M. Rappaz, S.A. David, J.M. Vitek, and L.A.Boatner, in Reference 13, p. 1125.52. M. Rappaz, S.A.David, J.M. Vitek, and L.A.Boatner, Metall. Trans. A 21A (1990) p. 1767.53. S.A. David, J.M. Vitek, M. Rappaz, and L.A.Boatner, in Reference 52, p. 1753.54. M. Rappaz, J.M. Vitek, S.A. David, and L.A.Boatner, Metall. Trans. A 24A (1993) p. 1443.55. J.A. Brooks, M.J. Baskes, and F.A. Greulich,Metall. Trans. A 22A (1991) p. 915.56. J.W. Elmer, T.W. Eagar, and S.M. Allen, inProc. Int'l. Conf. Stainless Steels (Iron and SteelInstitute of Japan, Tokyo, 1991) p. 669.57. A. Matsunawa, S. Katayama, andM. Shimidzu, Trans. Jpn. Weld. Res. Inst. 19(1990) p. 67.58. M.C. Flemings, Solidification Processing(McGraw-Hill, New York, 1984).59. W Kurz and D.J. Fisher, Fundamentals of So-lidification (Trans Tech Publications, Aeder-mannsdorf, Switzerland, 1986).60. J.S. Langer, in Principles of Solidification andMaterials Processing, Vol. 1, edited by R. Trivedi,

J.A. Sekhar, and J. Mazumdar (Oxford & JBHPublishing Co., New Delhi, 1989) p. 1.61. M.E. Glicksman, in Reference 60, p. 11.62. WW. Mullins and R.F. Sekerka, /. Appl. Phys.35 (1964) p. 444.63. R. Trivedi, Ada Metall. 18 (1970) p. 287.64. W. Kurz and D.J. Fisher, Ada Metall. 29(1981) p. 11.65. W Kurz, B. Giovanola, and R. Trivedi, AdaMetall. 34 (1986) p. 823.66. R. Trivedi and W Kurz, in Reference 65,p. 1663.67. J.S. Langer and H. Muller-Krumbhaar, AdaMetall. 26 (1978) p. 1681.68. J.A. Brooks and M.I. Baskes, in Reference11, p. 93.69. J.C. Lippold and WE Savage, Modeling ofCasting and Welding Processes, edited by H.D.Brody and D. Apelian (Metallurgical Society ofAIME, Warrendale, PA, 1980) p. 443.70. E. Scheil, Z. Metallk. 34 (1942) p. 70.71. H.D. Brody and M.C. Flemings, Trans.AIME 236 (1966) p. 615.72. T.W. Clyne and W. Kurz, Metall. Trans. A12A (1981) p. 965.73. WE Savage, Weld. World 18 (1980) p. 89.74. G.J. Davies and J.G. Garland, Int. Met. Rev.20 (1975) p. 83.75. S.A. David and C.T. Liu, in Reference 15,p. 157s.76. J.M. Vitek and S.A. David, The Metal Science ofJoining, edited by M.J. Cieslak, J.H. Perepezko,S. Kang, and M.E. Glicksman (The Minerals,Metals, and Materials Society, Warrendale, PA,1992) p. 115.77. J.W Elmer, in Reference 76, p. 123.78. J.M. Vitek and S.A. David, in Proc. LaserProcessing Symposium (The Minerals, Metals,and Materials Society, Warrendale, PA, 1993) tobe published.79. K.E. Easterling in Reference 6, p. 163.80. H.K.D.H. Bhadeshia and L.E. Svensson, inReference 79, p. 109.81. H.K.D.H. Bhadeshia, Bainite in Steels (Insti-tute of Materials, London, 1992).82. H.K.D.H. Bhadeshia, in Reference 4, p. 189.83. D.F. Watt, L. Coon, M. Bibby, J. Goldak, andC. Henwood, Acta Metall. 36 (1988) p. 3029.84. C. Henwood, M. Bibby, J. Goldak, andD. Watt, in Reference 83, p. 3037.85. Y. Shen, B. R a d h a k r i s h n a n , and R.G.Thompson, in Reference 5, p. 259.86. Welding Handbook, 8th ed., Chapter 7,K. Masubuchi, O.W. Blodgett, S. Matsui, F.P.Ross, CO. Rudd, and C.L. Tsai (American Weld-ing Society, Miami, FL, 1989).87. J.K. Hepworth, Finite Element Calculation ofResidual Stresses in Welds, (Proc. Int. Conf. Nu-merical Methods for Non-Linear Problems,Pineridge Prep, Swansea, Wales, September1980) p. 51.88. A.R. Ortega, L.A. Bertram, E.A. Fuchs,K. Mahin, and D.V Nelson, in Reference 5, p. 89.89. J.M.J. McDill, A.S. Oddy, and J.A. Goldak, inReference 5, p. 105.90. J. Goldak, in Reference 4, p. 72.91. KW Mahin, W Winters, J. Krafcik, T. Holden,R. Hosbons, and S. MacEwen, in Reference 4,p.83.92. P. Tekriwal and J. Mazumder, in Reference 4,p. 91.93. A.S. Oddy, J.A. Goldak, and J.M.J. McDill,in Reference 4, p. 97. D

MRS BULLETIN/JANUARY 1994 35