International Journal of Heat and Mass Transfer 55 (2012) 3316–3324

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International Journal of Heat and Mass Transfer

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Electrical contact resistance effect on resistance spot welding

P.S. Wei a,⇑, T.H. Wu b,1

a Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan, ROCb Department of Mechanical Engineering, Yung Ta Institute of Technology and Commerce, Pintong 909, Taiwan, ROC

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 June 2011Received in revised form 7 January 2012Accepted 7 January 2012Available online 22 February 2012

Keywords:Resistance spot weldingElectrical contact resistanceNugget formationDynamic resistanceConstriction resistanceFilm resistance

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2012.01.040

⇑ Corresponding author. Tel.: +886 7 5254050; fax:E-mail addresses: pswei@mail.nsysu.edu.tw (P.S.

(T.H. Wu).1 Tel.: +886 921 678 716.

The effects of local electrical contact resistance on transport variables, cooling rate, solute distribution,and nugget shape after solidification responsible for microstructure of the fusion zone during resistancespot welding are realistically and systematically investigated. The model accounts for electromagneticforce, heat generation and contact resistances at the faying surface and electrode–workpiece interfacesand bulk resistance in workpieces. Contact resistances are composed of film and constriction resistances,as functions of hardness, temperature, electrode force and surface condition. The computed results showthat the bulk dynamic electrical resistance cannot reliably reflect transport processes and nugget shape,unless the local constriction resistance and electric current density are known. Regardless of high filmresistance, nugget growth and transport processes are independent of film resistance due to delayedresponse time of local electric current in the early stage. A decrease in constriction resistance, however,delays nugget formation, enhances convection and solute mixing, and changes circulation direction of thestronger convection cell during cooling period.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Resistance spot welding is a popular method in joining thinworkpieces in various manufacturing, automobile, aerospace andpackaging industries [1–5]. The workpieces are squeezed by twowater-cooled copper electrodes, which exert force to break upany surface oxides or films on the faying surface to produce an inti-mate contact. After electric current is applied across electrodes,heat generates within the bulk workpieces and at the interfacesdue to electrical resistance. Melting thus initiates at the faying sur-face. Since electric current spreads strongly on entering the work-piece, the induced electromagnetic force induces strong convectionin the molten nugget [6–8]. The molten nugget grows until currentflow is terminated. The nugget solidifies due to cooling throughelectrodes.

Studying electrical resistance therefore is necessary for under-standing transport processes during resistance spot welding [9–12]. Bentley et al. [13] was one of the early researchers to use ametallographic technique and theoretical method to find that tem-perature and nugget growth during spot welding of mild steel aredominated by contact resistances at the faying surface, and work-piece and electrode in the early stage. In the late stage, their effectsare less influential. Wei and Ho [14] theoretically showed that con-

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tact resistance at the faying surface and joule heat in the bulkmaterial are, respectively, responsible for nugget growth in theearly and late stages. De et al. [15] theoretically found that the ini-tial contact resistances at the faying surface and workpiece andelectrode strongly influence the formation of the fusion zone inresistance spot welding of aluminum. The variation of contactresistance at the faying surface was insensitive to the fusion zonesize.

A detailed time-dependent variation of dynamic electrical resis-tance should be known, since it is often used for monitoring of theprocess to obtain good weld quality [16–19]. Roberts [20] experi-mentally showed the time-dependent dynamic resistance takingthe sum of bulk resistance of the two workpieces, and contactresistances at the faying surface and two electrode–workpieceinterfaces. An initial drop of dynamic resistance was attributed tothe rupture of the surface film. The rise in dynamic resistance afterthe breakdown of contact resistance was proposed to be due to theheating of workpieces, and the subsequent fall after the maximumresistance was reached may be attributed to two causes; thegrowth of the nugget diameter and the increase in penetration ofthe electrodes into workpieces. Both increased areas for currentflow. Even though the dynamic resistance curves differed markedlywith the surface conditions and different material combinations[18,19,21], the typical dynamic resistance curves agreed with thecurves presented by Roberts [20].

The contact resistance can be considered as a combination ofthe constriction and film resistances for materials with film or con-taminants on a contact surface [22]. The constriction resistance

Nomenclature

C liquid-to-solid specific heat ratio, defined in Eq. (13)Da Darcy number, defined in Eq. (13)Ef effective thickness of heat source due to thermal contact

resistance ¼ ef =~ro

E⁄ dimensionless electrical static contact resistance¼ ~rliq

~R0~ro

f mass fraction of liquid or solidF0 dimensionless parameter, defined in Eq. (13)fa solute mass fraction = ~f a=~f a

m;0~f a

m;0 initial solute contentg volume fraction or gravitational accelerationGr Grashof number, defined in Eq. (13)h enthalpy ¼ ~h=hf

H magnetic field intensity in h direction, H ¼ ~Hp~ro=IHv hardnesshf fusion latent heat at eutectic point, J/kgh‘ RCT + R(1 � C)Te + 1hs RTI, j welding current, amp, electric current density, j ¼ ~jp~r2

o=IK gsks + g‘kE thermal conductivity ratio = ~kE=

~ks

ks thermal conductivity ratio = ~ks=~k‘

kp equilibrium partition coefficientK0 permeability constant, m2

L distance between electrodesL1, L2, L3, L4 length, as illustrated in Fig. 1Lo dimensionless parameter, defined in Eq. (13)M � (drc/df + ndN/df)/NN rs(f) � rc(f)n total number of contact spotsn1 number of contact spots in the first control volume near

axisymmetric axisPrm magnetic Prandtl number = ~g‘=~a‘ , defined in Eq. (13)R ~cs

~T0=hfRd dynamic resistance ¼ ~Rd=

~R0

RE ~cE~T0=hf

ro electrode radius, as illustrated in Fig. 1~R0 electrical contact resistance at faying surface at T0

R0E electrical contact resistance at electrode–workpieceinterface at T0, R0E ¼ ~R0E=~R0

s film thicknessSc Schmidt number, defined in Eq. (13)T temperature = ~T=~T0Te eutectic temperatureu, v axial and radial velocity, u ¼ ~u~ro=~a‘;v ¼ ~v~ro=~a‘V velocity vectorW electrode force

Greek letters~a‘ dimensional liquid thermal diffusivitybs, bT solutal and thermal expansion coefficientd nugget thickness~g‘ liquid magnetic diffusivity = 1=~r‘l0lr‘gE ~gE=~g‘ , where ~gE ¼ 1=~rEl0lrEh0 temperature ratio, defined in Eq. (13)lr relative magnetic permeability = g‘ + gslrs/lr‘

l0 free magnetic permeability, N/amp2

ef effective thickness of heat source due to thermal contactresistance, m

q density ¼ ~q=~q‘r electrical conductivity, r = ~r=~rl̂ = gsrs + g‘x welding current frequencyR dimensionless parameter, defined in Eq. (13)s time = t~a‘=~r2

o

Superscripta solute� dimensional quantity

Subscriptc coolant, contact surface, or characteristic quantityE electrodef film‘, liq liquid and liquidusm mixtureo electrode outer radiuss, sol solid and solidus0 ambient

P.S. Wei, T.H. Wu / International Journal of Heat and Mass Transfer 55 (2012) 3316–3324 3317

arises when current flows from one conductor to another. Contactresistance thus is simply a constriction resistance for a clean metalcontact. Dynamic electrical resistance during resistance spot weld-ing has been modeled by Wang and Wei [23]. Dynamic resistancetook the sum of temperature-dependent bulk resistance of work-pieces, and contact resistances composed of constriction and filmresistances as functions of hardness, temperature, electrode force,and surface condition. With temperature determined from the pre-vious study [6], the predicted time-dependent dynamic resistanceagreed well with typical trends and available experimental data. Inthe absence of film resistance, the rapid drop of contact resistancein the early stage disappears. Moreover, the local maximum disap-pears if electrical resistivity at contact surfaces is independent oftemperature. The decrease of dynamic resistance after the peak isnot due to liquid formation [20], but it is a result of decreasedincreasing rate of bulk resistance and the drop of contact resis-tance. Babu et al. [24] measured constriction resistance as a func-tion of applied pressure. Substituting temperature dependence ofbulk resistivity and mechanical properties, a curve fitting of a rela-tionship of contact resistance to pressure and temperature was

established. This empirical model agreed well with measurementsin the regime of low applied pressure.

In order for simple analysis, contact resistance can be effectivelymodeled as an exponentially decreasing function of temperature[25,26], a linearly decreasing function of temperature [14,27–29],functions proportional to square root of temperature-dependenthardness [30], difference in voltages between the contact surfaceand a reference point [31], and square root of difference in squaredtemperatures at the contact surface and in bulk workpiece [32],and an algebraic expression of an electro-thermal contact condi-tion for constriction resistance [33] , respectively.

This work systematically investigates the effects of local contactresistance including constriction and film resistances on resistancespot welding experiencing heating, melting, cooling and freezing.Since dynamic resistance is a bulk property, a reliable investigationof local and different characteristics of constriction and film resis-tances on transport processes is important. Thermal convection,solute distribution, cooling rate, and nugget shape after solidifica-tion, essentially required for analyzing microstructures of the weldnugget, thus become more understandable.

Fig. 1. Sketch for resistance spot welding and coordinate system.

3318 P.S. Wei, T.H. Wu / International Journal of Heat and Mass Transfer 55 (2012) 3316–3324

2. System model and analysis

In this work, resistance spot welding is analyzed in a cylindri-cal coordinate system with the origin at the intersection of theaxisymmetric axis and lower electrode–workpiece surface, asillustrated in Fig. 1. Workpieces are binary alloy squeezed by flattip electrodes having a truncated cone face with radius ~re andouter radius ~ro. The major assumptions are that (1) welding ACis uniform across the top of the upper electrode, (2) heat gener-ations at interfaces are volumetric heat sources, (3) the mushyzone is a porous medium in local thermal and phase equilibrium,(4) workpieces are mixtures in phases [6,23], (5) contact resis-tance at the electrode–workpiece interface is smaller than thatat the faying surface, (6) contact spots at contact surfaces areuniformly distributed, and (7) deformation of the workpiece byelectrode load is not accounted. In this model, unsteady electro-magnetic model is used rather than conventional electrostaticmodel. Validity can be seen from a complete electrical potentialequation given by

~e~l @2/

@t2 þ ~r~l @/@t¼ r2/ ð1Þ

Three terms can be, respectively, scaled as

~e~l/c

t2c

~r~l/c

tc

/c

z2c

ð2Þ

where permitivity constant ~e = 8.85 � 10�12 F/m, free magnetic per-meability l0 = 4p � 10�7 H/m, and electrical conductivity ~r �106–107 mho/m for most metals and alloys. Provided that tc = 0.1 s,zc = 0.001 m suitable for resistance spot welding, the first term inEq. (2) can be neglected in comparison with the second term. Therelative magnetic permeability can be as high as lr � 103–105 forferromagnetic materials such as iron, steel, nickel, cobalt, etc.Supermalloys can even reach lr � 106. In these cases, the secondterm on the left-hand side and the term on the right hand side ofEq. (1) become the same order of magnitudes. The unsteady termtherefore cannot be neglected when welding ferromagnetic work-pieces, workpieces with high magnetic permeabilities or thickthickness, or welding in a small time period.

2.1. Electrical contact resistance

A local dimensionless electrical contact resistance for a singlecontact spot at the faying surface or electrode–workpiece interfacecan be simulated by [23].

Rc ¼R1

rffiffiffiffiffiffiffiffiffinHv

pþ R2

snHv

rfð3Þ

where the dimensionless parameters governing constriction andfilm resistances are, respectively, defined as

R1 �1

2~R0 ~rliq

ffiffiffiffiffiffiffiffiffiffiffip~Hv0

W

s; R2 �

~ro~Hv0

W~R0 ~rliq

ð4Þ

Contact resistance stems from the presence of roughness onthe contact surfaces or an additional layer between contact sur-faces. Constriction resistance results when an electric current ispassed and constricted from one sheet to the other through con-tact spots of roughness. Film resistance due to an oxide or impu-rity layer between contact surfaces occurs because of itsresistivity, roughness and thickness. Contact resistances can bereduced by any mean such as increasing force load, temperature,galvanic erosion, fritting wear, etc. to increase the area and num-ber of contact spots. Since electrical properties and hardness aretemperature-dependent, electrical contact resistance in Eqs. (3)and (4) involves transient coupling between thermal–mechani-cal–electrical effects. For a given control volume containing ni

contact spots, Eq. (3) becomes

Rci ¼R1

nirffiffiffiffiffiffiffiffiffinHv

pþ R2

snHv

nirfð5Þ

where the total number of contact spot n ¼P

ini. In this case, thenumber of contact spot at the first control volume is chosen to ben1 = 10. An effective contact electrical conductivity is introduced

rc ¼Ef

E�RcAcð6Þ

where Ef and Ac are, respectively, an effective thickness of heatsource due to thermal contact resistance, and cross section for elec-tric current to flow. Eq. (6) indicates that temperature is affected bythe effective thickness of thermal contact resistance and the thick-ness of film resistance involved in Ef and Rc, respectively.

2.2. Governing equations and boundary conditions

With the above assumptions, dimensionless continuity,momentum, energy, species, magnetic field intensity equations[6,23], respectively, become

@q@sþr � ðqVmÞ ¼ 0 ð7Þ

@qum

@sþr � ðqVmumÞ ¼ Prr � ðqrumÞ �

PrDa

qð1� g‘Þ2

g3‘

um �@p@z

þ Pr2Gr½h0ðT � TsolÞ þ F0ðf a‘ � f a

‘;TsolÞ

� Lop2 lrH

@H@z

ð8Þ

@qvm

@sþr � ðqVmvmÞ ¼ Prr � ðqrvmÞ �

PrDa

qð1� g‘Þ2

g3‘

vm

� @p@r� Lo

p2

lr

rH@rH@r

ð9Þ

Table 1Typical values of parameters.

Constriction resistance parameter, R1 30Film resistance parameter, R2 30Dimensionless electrode enthalpy, RE 0.59Dimensionless electrode magnetic diffusivity, gE 0.154Temperature ratio, h0 1.0Effective thickness of heat source, Ef 10-2

Dimensionless static contact resistance, E⁄ 0.56Effective thickness of oxide layer, s 3.3 � 10�7

Dimensionless electrode thermal conductivity, kE 11.4Solid-to-liquid thermal conductivity ratio, ks 1.0Parameter governing welding current, Lo 4 � 107

Lengths for electrode, L1, L2, L3, L4 0.1, 0.5, 0.1, 1.4Magnetic Prandtl number Prm 3 � 104

Workpiece radius, rb 2.33Dimensionless electrode tip radius, re 0.9Dimensionless maximum radius of coolant hole, rc 0.5Electrode enthalpy parameter, RE 0.59Dimensionless Curie temperature, Tc 1.63Dimensionless eutectic and melting temperatures, Te, Tm 5.0, 6.0Solid-to-liquid permeability ratio, lrs/lr‘ 3500Dimensionless electrode density, qE 1.28Thermal-to-electrical property parameter, R 2.5 � 10�5

Electrical conductivity ratios, rsol ;rE 1.05, 6.5Current frequency, ~x 60 Hz

P.S. Wei, T.H. Wu / International Journal of Heat and Mass Transfer 55 (2012) 3316–3324 3319

@qhm

@sþr � ðqVmhmÞ ¼ Cr � ðKrhmÞ þ Cr � ½Krðhs � hmÞ

� r � ½qðh‘ � hmÞVm

þ LoP

p2

1rcþ 1

r

� �1r@rH@r

� �2

þ @H@z

� �2" #

ð10Þ

@qf am

@sþr � ðqVmf a

mÞ ¼PrScr � ðqf‘rf a

mÞ þPrScr � ½qf‘rðf a

‘ � f amÞ

� r � ½qðf a‘ � f a

mÞVm ð11Þ

@lrH@sþr � ðlrHVmÞ ¼ Prmr �

1rrH

� �

þ PrmHr@

@r1r

� �� H

rr2

� �þ lrHvm

rð12Þ

where all lengths are nondimensionalized by the outer radius of theelectrode, whereas time is nondimensionalized by the time for ther-mal diffusion ~r2

o=~a‘. Dimensionless parameters are, respectively, de-fined as

Pr�~m‘~a‘; Prm�

~g‘~a‘; Da�Ko

~r2o; Gr� gbTð~Tsol� ~TeÞ~r3

o

~m2‘

; h0�~T0

~Tsol� ~Te

;

Fo�bs

~f am;0

bTð~Tsol� ~TeÞ; Lo� I2l0lr‘

~q‘~a2‘

; C�~c‘~cs; R�

~q‘~g‘~r2

ohf; Sc�

~m‘~Da‘

ð13Þ

The last terms on the right-hand side of Eqs. (8) and (9) stand forLorentz force, whereas the last term on the right-hand side of Eq.(10) combines heat generation at contact surfaces and in bulk work-piece. Determination of distinct regions of the mushy zone, full li-quid and solid was described by Wei and Yeh [34]. Solute fractionin the liquid in Eq. (8) is given by

f a‘ ¼

f am

1þ fsðkp � 1Þ ð14Þ

where equilibrium partition coefficient kp represents the ratio ofsolute contents in the solid and liquid at the same temperature. Ow-ing to irregular geometry of the electrode, an immobilization trans-formation is effectively used. Energy and magnetic field intensityequations in the electrode thus, respectively, become

@TE

@s¼ kEksRC

qERE

@2TE

@f2 þ1

N2þM2� �

@2TE

@n2 þ2M@2TE

@n@fþ 1

nNþ rc�2M

dNdf

� �1N@TE

@n

" #

þ LoRp2qERErE

M@HE

@nþ@HE

@f

� �2

þ HE

nNþ rcþ 1

N@HE

@n

� �2" #

ð15Þ

@HE

@s¼ PrmgE

@2HE

@f2 þ1

N2 þM2� �

@2HE

@n2 þ 2M@2HE

@n@f

"

þ 1nN þ rc

� 2MdNdf

� �1N@HE

@n� HE

ðnN þ rcÞ2

#ð16Þ

Magnetic field intensity at the top of the upper electrode is given by

HE ¼ �1

2ð1þ rcÞn2ð1� rcÞ þ 2rcn

nð1� rcÞ þ rcsin 2pxs ð17Þ

which is derived from Ampere law by considering uniform ACacross the top surface. Magnetic field intensity vanishes in the cool-ant hole because of negligible electric current. Magnetic field inten-sity at the electrode–workpiece interface is satisfied by continuitiesof tangential electric and magnetic field intensities across the

interface, respectively. Magnetic field intensity must also vanishat the axisymmetric axis to avoid infinite electric currents.

2.3. Numerical method

Eqs. (7)–(12), (15) and (16) are discretized by a control-volume,implicit finite-difference scheme with staggered grids. A grid sys-tem 50 � 53 in workpieces and 27 � 72 in the electrode, time step2 � 10�5 are selected. The maximum deviation of computed resis-tances by using grid systems of 50 � 53 and 72 � 75 is less than 1%.Convergence tolerances for global energy and velocity componentsand enthalpy, concentration, and magnetic intensity fields were10�2 and 10�3, respectively. Otherwise, equations for the electrodeare solved again and the process is repeated. If solutions con-verged, temperature distributions are used to calculate resistances.Computations then proceed to the next time.

3. Results and discussion

The present work uses the computer program developed byWang and Wei [23] to investigate the effects of local contact resis-tances on transient mass, momentum, energy, species and magneticfield intensity transport in workpieces and electrodes. The contactresistance is composed of constriction and film resistances, whichare functions of hardness, temperature, electrode force, and surfacecondition. Both components of contact resistance increase withdecreasing electrode force and increasing hardness and number ofcontact spots at interfaces (see Eq. (3)). Hardness decreases withincreasing temperature. Sizes of contact spots on the contact surfacecan also be a function of the electrode force, total number of contactspots, and hardness. Interactions between these factors affecting thechange of electrical resistance during spot welding are accounted.Therefore, it reveals that an increase in temperature decreases filmresistance and increases bulk resistance. Constriction resistance,however, increases and then decreases with increasing temperaturedue to increased resistivity and decreased hardness, respectively.Working parameters in this work are listed in Table 1. Metallurgicalproperties are based on a binary alloy of iron and manganese. Choos-ing typical values of ~R0 ¼ 10�4 ohm; ~Hv0 ¼ 108 N=m2, W = 1000 N,~rliq ¼ 106 mho=m;~ro ¼ 10�2 m, dimensionless parameters govern-ing constriction and film resistances at the faying surface are of themagnitude as R1 = 30 and R2 = 30 (see Eq. (4)). Curie temperature isalso introduced into magnetic permeability [35].

Fig. 3. Dimensionless nugget thickness and width as functions of dimensionlesstime for different parameters governing (a) constriction resistance, and (b) filmresistance.

Fig. 2. Dimensionless isothermals in workpieces and electrode at times whendimensionless nugget thickness dliq = 0.4L and zero for (a) parameters governingconstriction resistance R1 = 30 and film resistance R2 = 30, (b) R1 = 30, R2 = 0, and (c)R1 = 1, R2 = 30.

3320 P.S. Wei, T.H. Wu / International Journal of Heat and Mass Transfer 55 (2012) 3316–3324

The isothermals in workpieces and electrode for parametersgoverning constriction resistance R1 = 30 and film resistanceR2 = 30 are shown on the right- and left-hand parts in Fig. 2(a),which are at the instants when thicknesses of the liquidus linedliq = 0.4L in the melting period, and dliq = 0 as solidification is com-pleted, respectively. It can be seen that the highest temperature oc-curs near the center of the workpieces. The molten pool is nearly

rectangular. Ignoring film resistance, nearly the same thermal pat-terns are shown in Fig. 2(b). Referring to previous Fig. 2(a), Fig. 2(c)shows that a decrease in constriction resistance reduces thegrowth of nugget thickness.

Dimensionless thickness and width of the molten nugget asfunctions of dimensionless time for different parameters governingconstriction resistance are shown in Fig. 3(a). The thickness andwidth of the nugget refer to those of the solidus line. Welding ACis also plotted by a periodic line at the bottom. It can be seen thatreducing constriction resistance delays the onset of the nugget. Forparameter governing constriction resistance R1 = 30 dimensionlesstime for onset of the nugget is around 0.27 (30 cycles = 0.5 s).Dimensionless onset time increases to 0.43 for R1 = 1. The growthof thickness is much less than that of width. The growth rate atthe onset time and freezing rate near the end of solidification arevery high. Fig. 3(b), however, shows that the nugget formationand its growth are almost independent of film resistance, beingconsistent to previous Fig. 2(a) and (b).

Fig. 4(a) and (b) shows the effects of film resistance and con-striction resistance on time-dependent dimensionless dynamicand bulk resistances, and contact resistances at the faying surfaceand workpiece-electrode interface, respectively. Evidently, the ra-pid drops of dynamic and contact resistances in the early stageare attributed to film resistance. In the absence of film resistance,drops of contact and dynamic resistances disappear [23]. Contactand dynamic resistances then increase rapidly due to increases of

Fig. 4. Dimensionless contact, bulk and dynamic resistances for (a) differentparameter governing film resistance, (b) different parameters governing constric-tion resistance, and (c) mechanism for abrupt drop of film resistance at the center offaying surface (standard case).

Fig. 5. Dimensionless contact resistance, temperature, electric current density andwelding current as functions of dimensionless time for R1 = 30 and R2 = 30 at (a)r = 0, z = 0.4, (b) r = 0.9, z = 0.4, and (c) r = 0.9, z = 0.8.

P.S. Wei, T.H. Wu / International Journal of Heat and Mass Transfer 55 (2012) 3316–3324 3321

constriction and bulk resistances. Peaks result because of de-creased constriction resistance from reduced hardness and reduc-tion of increasing rate of bulk resistance, rather than theformation of nugget at a dimensionless time of around 0.3 (seeFig. 3(a) or (b)) [23]. Interestingly, the effects of film resistanceon contact resistances at the faying surface and workpiece-elec-trode interfaces, bulk and dynamic resistances are insignificant inthe late stage. However, an increase in constriction resistance in-creases contact resistances, bulk and dynamic resistances.Fig. 4(c) shows that even though temperature increases slightly,multiplication of both the decreased film resistivity and hardnessresult in the abrupt drop of film resistance.

Fig. 5(a) shows the local dimensionless temperature, contactresistance and electric current density as functions of

dimensionless time at the axisymmetric axis on the faying surface.It is found that there exists a delayed response time around 6 cy-cles (0.1 s) of electric current density at the axisymmetric axis onthe faying surface. The delay of response time can be interpretedas a result of a finite magnetic diffusivity [35]. That is, the distancefor diffusion of magnetic or electric current can be estimated by thesquare root of time divided by magnetic permeability and electri-cal conductivity. Even though local dynamic resistance is high, de-layed response time for electric current reduces heat generationand increasing rate of temperature at the axisymmetric locationon the faying surface in the early stage. After the response time,electric current density jumps, resulting in high heat generationand significant increase of temperature. Dimensionless amplitudeof the periodic steady-state electric current density is around 0.4.It is also seen that electric current density slightly decreases as

Fig. 6. Dimensionless contact resistances, temperature, electric current density andwelding current as functions of dimensionless time at the center of faying surfacefor (a) R1 = 1, R2 = 30, and (b) R1 = 30, R2 = 0.

Fig. 7. Dimensionless temperature history at different locations in workpieces for(a) different parameter governing film resistance, and (b) different parametersgoverning constriction resistance.

3322 P.S. Wei, T.H. Wu / International Journal of Heat and Mass Transfer 55 (2012) 3316–3324

liquid phase forms after dimensionless time of 0.3. It can be ex-plained by an enhanced magnetic diffusion due to a decreasedelectrical conductivity of the liquid phase. To satisfy Ampere’slaw, high magnetic diffusion reduces current density. Dimension-less contact resistance, electric current density and temperatureas functions of dimensionless time at locations of the same radiusas the electrode tip on the faying surface and workpiece-electrodeinterface are shown in Fig. 5(b) and (c), respectively. The responsetime of electric current and drop of contact resistance in the earlystage decrease for locations away from the axisymmetric axis onthe faying surface. In view of high and instantaneous response timeof electric current density, high heat generation causes a rapid in-crease of temperature near the edge of the electrode in the earlystage.

Referring to Fig. 5(a), Fig. 6(a) shows that after the initial drop ofcontact resistance, an increase in constriction resistance increasesthe extent of the jump of contact resistance at the axisymmetricaxis on the faying surface. Interestingly, the response time andassociated jump of electric current density are almost independentof constriction resistance. Fig. 6(b) shows that delay time and jumpof electric current density at the axisymmetric axis on the fayingsurface are also independent of film resistance by referring toFig. 5(a). In contrast to constriction resistance, local contact resis-tances for different film resistances are almost the same in the latestage. Induced temperature thus is insensitive to the variation offilm resistance during the entire heating, melting and coolingperiods.

Fig. 7(a) shows dimensionless temperatures as functions ofdimensionless time at different locations in workpieces for

different parameters governing film resistance. Temperature dur-ing the entire heating, melting and cooling periods at any locationis independent of film resistance. Temperature near the electrodeedge jumps in an early stage due to enhanced electric currentand infinitesimal response time. Even though the increase of tem-perature is delayed at the axisymmetric axis on the faying surface,temperature can still override that at a location away from the axi-symmetric axis on the faying surface in the late stage. The maxi-mum temperature taking place at the center of workpieces isslightly above the solidus temperature (around 5.7). The variationof liquid temperature with time is small as time is greater than 0.3,as experimentally observed by Alcini [36]. Temperature drops aspower is off at a dimensionless time of 0.54 (see Fig. 3(b)). Dimen-sionless temperatures as functions of dimensionless time at fourlocations in the workpiece for different parameters governing con-striction resistance are shown in Fig. 7(b). Temperatures at differ-ent locations increase with constriction resistance.

The flow patterns at the times when thicknesses of the liquidusline dliq = 0.4L and 0.25L for parameters R1 = 30 and R2 = 30 are,respectively, shown on the right- and left-hand parts in Fig. 8(a).In view of divergent electric current, Lorentz force induces the ver-tex cell circulating in a counterclockwise direction in the right partof the upper workpiece. The circulation of the stronger cell on the

Fig. 8. Dimensionless flow patterns in nugget at times when dimensionless nuggetthickness dliq = 0.4L and 0.25L for (a) R1 = 30, R2 = 30, and (b) R1 = 1, R2 = 30. Fig. 9. Dimensionless concentration distributions in nugget at times when dimen-

sionless nugget thickness dliq = 0.4L and zero for (a) R1 = 30, R2 = 30, and (b) R1 = 1,R2 = 30.

Fig. 10. Comparisons of nugget thickness versus welding current in welding AISI1008 steel between one-dimensional predications and measurements from Gould[27] and this work.

P.S. Wei, T.H. Wu / International Journal of Heat and Mass Transfer 55 (2012) 3316–3324 3323

left-hand side, however, is in the counterclockwise direction. Asconstriction resistance decreases, similar flow pattern is seen inthe right-hand part of Fig. 8(b). The vortex cell on the left part,however, becomes clockwise direction. Referring to previousFig. 8(a), velocity increases as constriction resistance reduces.

The distribution of solute content plays an important role inproperties and microstructure of weldments [37,38]. Solute distri-butions in workpieces at times corresponding to a thickness of theliquidus line dliq = 0.4L and zero are shown in the right and left-hand part of Fig. 9(a), respectively. Regardless of strong convection,solute distributions at two instant times are similar due to rapidsolidification. Solute accumulated ahead of solidification front isconvected in the counterclockwise direction around the boundaryto interior of the molten nugget, leading to a deficit of solute con-tent in a thin layer around the boundary and an excess of solutecontent in the interior of the nugget after solidification. Fig. 9(b)shows solute distributions in the nugget as constriction resistanceis reduced. The zone of a deficit solute on the side boundary is sig-nificantly expanded. Solute distributions near the side boundary ofthe nugget are strongly affected by convection.

This model agrees with a measured relationship between nug-get thickness and welding current provided by Gould [27] in weld-ing AISI 1008, as shown in Fig. 10. In this case, initial soluteconcentration f̂ a

m;0 = 0.32 wt% of Mn in a Fe–Mn alloy, the total

3324 P.S. Wei, T.H. Wu / International Journal of Heat and Mass Transfer 55 (2012) 3316–3324

thickness of workpieces L = 0.715, electrode face radius re = 0.9,and constriction and film resistance parameters R1 = 100 andR2 = 50. It shows a better agreement with the experimental resultthan the one-dimensional prediction [27].

4. Conclusions

The effects of local electrical resistance on resistance spot weld-ing are realistically investigated. Based on relevant magneto-fluid-mechanics and model of dynamic electrical resistance accountingfor bulk resistance and contact resistances, the computed nuggetthickness versus welding current agrees well with available exper-imental data. The conclusions drawn are the following:

(1) The bulk dynamic electrical resistance cannot be used toreliably predict or control resistance spot welding. For exam-ple, distributions of temperatures and heat generationsstrongly depend on spatial locations and time. In order toclarify the mechanisms, the local contact resistances includ-ing different characteristics of film and constriction resis-tances and electric current densities at different locationsare required.

(2) Local contact resistance, dynamic and bulk resistancedecrease as constriction resistance decreases. After initialdrops due to film resistance, local contact resistance anddynamic resistance jump to the peaks, whose valuesdecrease with constriction resistance. Since constrictionresistance reduces due to decreased hardness, local contactresistance and dynamic resistance become gradualdecreases in the late stage.

(3) Transport variables and nugget growth affected by heat gen-eration can be interpreted by the delay of response time andassociated jump of local electric current density and varia-tion of local resistance with time.

(4) The response time and associated jump of electric currentdensity on the faying surface are insensitive to the variationsof film and constriction resistances.

(5) Although film resistance dominates in the early stage, itseffects on dynamic, bulk and contact resistances in the latestage and transport processes during the entire heating,melting and cooling periods are almost negligible. They areattributed to small heat generation determined by delayedresponse time of electric current in the early stage.

(6) The onset of the weld nugget is delayed and temperature isdecreased by reducing the parameter governing constrictionresistance.

(7) Irrespective of high dynamic and contact resistances, tem-perature at the axisymmetric axis on the faying surface islowest in the workpiece in an early stage, then increases rap-idly and gradually reduces increasing rate in the late stage.Temperature becomes the highest in comparison with otherlocations in the workpiece in the late stage.

(8) Constriction resistance can change the direction of circula-tion of the stronger cell in the molten nugget during thecooling period.

(9) Convection is enhanced and the deficit region of solute con-tent near the side boundary of the nugget expands as con-striction resistance decreases.

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