numerical simulation of the dynamic characteristics of

INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul. Mater. Sci. Eng. 12 (2004) 765–780 PII: S0965-0393(04)78853-1

Numerical simulation of the dynamic characteristicsof weld pool geometry with step-changes of weldingparameters

P C Zhao1, C S Wu1 and Y M Zhang2

1 MOE Key Laboratory for Liquid Structure and Heredity of Materials, Institute of MaterialsJoining, Shandong University, Jinan 250061, People’s Republic of China2 Center for Manufacturing and Department of Electrical and Computer Engineering, Universityof Kentucky, Lexington, KY 40506, USA

Received 10 January 2004Published 1 July 2004Online at stacks.iop.org/MSMSE/12/765doi:10.1088/0965-0393/12/5/002

AbstractThe gas tungsten arc welding process is an uncertain nonlinear multivariablesystem. In order to control the welding process, the nonlinear dynamicrelationship between the weld pool geometry reflecting the weld quality andthe welding parameters must be developed. A three-dimensional numericalmodel is developed to investigate the dynamic characteristics of the weld poolgeometry when the welding current and welding speed undergo a step-change.Under the welding conditions employed in this research, the transformationperiods are about 4 s for a 20 A down step-change of welding current, and about2 s for a 20 mm min−1 up step-change of welding speed, respectively. At theinitial stage during the step-change of welding current and welding speed, theresponses of weld pool geometry are quicker, but they slow down subsequentlyuntil the weld pool reaches a new quasi-steady state. Welding experiments wereconducted to verify the simulation results. It was found that the predicted weldpool geometries agree with the measured ones.

1. Introduction

Gas tungsten arc welding (GTAW) is the most widely used arc welding process for critical andaccurate joining. Many GTAW applications require full penetrations. Current practice relies onskilled operators who observe the weld pool and adjust the welding parameters accordingly.Unfortunately, the demands made by GTAW on the skills and experience of operators arefairly high, and the operators do not typically perform consistently. Hence, control of GTAWpenetration using automated sensing and feed-back control is needed.

The GTAW process is an uncertain nonlinear multivariable system. To control the process,the nonlinear dynamic relationship between the weld pool geometry reflecting the weld qualityand the welding parameters must be developed. Thus, a numerical simulation of the GTAWprocess is necessary.

0965-0393/04/050765+16$30.00 © 2004 IOP Publishing Ltd Printed in the UK 765

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766 P C Zhao et al

Though extensive studies on the numerical simulation of the GTA weld pool have beendone [1–5], little effort has been directed towards addressing the dynamic characteristics ofweld pool geometry when the welding parameters change. For a three-dimensional GTA weldpool with a moving heat source, the majority of mathematical models are concerned with thequasi-steady state [6, 7], and are unable to describe the transient variation of the weld poolgeometry with time when the welding parameters are changed. In this paper, a transient three-dimensional model is developed to simulate the full-penetration GTA weld pool with surfacedeformation. The model is used to predict the dynamic responses of the weld pool geometryafter a step-change of the welding parameters.

2. Formulation

2.1. Surface deformation

The surface of the GTAW weld pool is deformed under the action of the arc pressure, surfacetension, gravity, fluid dynamics, and shear stress from the plasma. When full penetration isestablished, both pool surfaces at the top and bottom are deformed. As shown in figure 1, thefunctions Ztop = ϕ(x, y) and Zbottom = ψ(x, y) are used to describe the configuration of thetop and bottom surfaces of the weld pool, respectively.

In the case of partial penetration (figure 1(a)), the surface deformation occurs only at thetop surface of the workpiece. The top surface of the weld pool is governed by the followingequation [4]:

Parc − ρgϕ + C1 = −γ(1 + ϕ2

y)ϕxx − 2ϕxϕyϕxy + (1 + ϕ2x)ϕyy

(1 + ϕ2x + ϕ2

y)3/2

, (1)

where Parc is the arc pressure, ρ the density, g the gravitational acceleration, γ the surfacetension, and C1 a constant, and ϕx = ∂ϕ/∂x, ϕxx = ∂2ϕ/∂x2, ϕxy = ∂2ϕ/∂x ∂y, and so on.Away from the weld pool, ϕ(x, y) = 0.

In the transient state, the weld pool geometry varies with time. But at any instant, the weldpool has a specific geometry and volume. For a specific pool geometry at a certain instant, thepool surface undergoes a corresponding deformation, but the total volume of the weld pool atthis instant is not changed before or after the surface deformation. Thus, there is the followingconstraint: ∫∫

�1

ϕ(x, y) dx dy = 0, (2)

where �1 is the surface area of the weld pool at the top surface. Of course, �1 has differentvalues at different times.

The arc pressure can be expressed as [9, 10]

Parc = µ0I2

8π2σ 2j

exp

(− r2

2σ 2j

), (3)

where µ0 is the permeability in free space, I the welding current, σj a current distributionparameter, and r =

√(x − u0t)2 + y2, where u0 is the welding speed and t is the time.

For a fully penetrated weld pool (figure 1(b)), two equations are required to describe theconfiguration of the top and bottom surfaces, respectively.

Parc − ρgϕ + C2 = −γ(1 + ϕ2


(1 + ϕ2x + ϕ2

y)3/2

(4a)

Weld pool geometry 767

y

Workpiece

Weld pool

Zbottom

Ztop

o x

z(z*)

Workpiece

oWeld pool

z(z*)

y

xZtop

(a) (b)

Figure 1. Schematic of pool surface deformation under partial and full penetration. (a) Partialpenetration and (b) full penetration.

and

ρg(ψ + L − ϕ) + C2 = −γ(1 + ψ2

y )ψxx − 2ψxψyψxy + (1 + ψ2x )ψyy

(1 + ψ2x + ψ2

y )3/2, (4b)

where L is the thickness of the workpiece, C2 is a constant, ψx = ∂ψ/∂x, ψxx = ∂2ψ/∂x2,ψxy = ∂2ψ/∂x ∂y, and so on. Away from the weld pool, ϕ(x, y) = 0 and ψ(x, y) = 0.

At a specific instant, the fully penetrated pool has a definite geometry and volume. Thoughboth the top and bottom of the pool surfaces undergo deformation, the total volume of the weldpool does not vary at a specific time instant. Therefore,∫∫

�1

ϕ(x, y) dx dy =∫∫

�2

ψ(x, y) dx dy (5)

where �1 is the surface area of the weld pool at the top surface, while �2 is the surface areaof the weld pool at the bottom surface.

C1 and C2 are the total sum of other forces that act on the weld pool surface except arcpressure, gravity, and surface tension. In the calculation, C1 is derived from equations (1)and (2) while C2 from equations (4) and (5):

C1

∫∫�1

dx dy =∫∫

�1

(−Parc) dx dy −∫∫

�1

γ(1 + ϕ2


(1 + ϕ2x + ϕ2

y)3/2

dx dy,

(6)

C2

(∫∫�1

dx dy +∫∫

�2

dx dy

)

=∫∫

�1

(−Parc) dx dy −∫∫

�1

γ(1 + ϕ2


(1 + ϕ2x + ϕ2

y)3/2

dx dy

− ρg

∫∫�2

(L − ϕ) dx dy −∫∫

�2

γ(1 + ψ2

y )ψxx − 2ψxψyψxy + (1 + ψ2x )ψyy

(1 + ψ2x + ψ2

y )3/2dx dy.

(7)

The iterative method is used to calculate the surface deformation of the weld pool.Equations (1) and (2) apply to the case of partial penetration. The software is able to judgewhether the weld pool is penetrated or not. Once the weld pool achieves full penetration, itsbottom surface is deformed too, so equations (4) and (5) apply to the full-penetration weldpool. During the iteration, the guessed values of C1 or C2 are employed first. In the partial caseϕ(x, y) is obtained by solving equation (1), while ϕ(x, y) and ψ(x, y) are obtained by solvingequation (4) in the full-penetration case. Then, improved values of C1 or C2 are obtained bysolving equations (6) or (7). Based on the new values of C1 or C2, equation (1) or (4) is solvedagain to get improved functions ϕ(x, y), or ϕ(x, y) and ψ(x, y). The above procedure is

768 P C Zhao et al

o x

y

z(z*)

Zbottom

Ztop

Fusion Zone

Workpiece

Welding Arc

Wire

Welding Direction

Figure 2. Schematic of GTAW process system.

repeated until it meets the criterion of convergence and the constraint conditions are satisfied.In addition, the functions ϕ(x, y) and ψ(x, y) are calculated in Cartesian coordinates.

During the transient development of the weld pool, �1 and �2, i.e. the action areas of thearc pressure and surface tension, and the volume of the weld pool, change with time. Thus,the configuration of the weld pool surfaces ϕ(x, y) and ψ(x, y) change with time until thequasi-steady state of the weld pool is achieved.

2.2. Governing equations

A schematic sketch of a typical GTAW process system is shown in figure 2. In order to describethe development of the weld pool shape, surface deformation, thermal field and fluid flow field,a time-dependent model is required. Therefore, it is a transient problem.

For a three-dimensional transient problem, the governing equations include the energy,momentum, and continuity equations. Because of surface deformation, some new boundariesappear at both the top and bottom surfaces, and their positions change with time. Therefore,the calculated domain is no longer a perfect cube for bead-on-plate welding, which causessome difficulty in the boundary conditions. In this study, based on Cartesian coordinates, thefollowing body-fitted coordinate system (x∗, y∗, z∗) is introduced (figure 1(b)) to transformthe deformed domain to a regular one:

x∗ = x, y∗ = y, z∗ = z − ϕ(x, y)

L + ψ(x, y) − ϕ(x, y). (8)

Thus, the governing equations in body-fitted coordinates are expressed as follows:

ρCp

(∂T

∂t+ U

∂T

∂x+ V

∂T

∂y+ Wt

∂T

∂z∗

)= ∂

∂x

(k∂T

∂x

)+

∂

∂y

(k∂T

∂y

)+ S

∂

∂z∗

(k

∂T

∂z∗

)+ kCt ,

(9)

ρ

(∂U

∂t+ U

∂U

∂x+ V

∂U

∂y+ W1

∂U

∂z∗

)

= −(

∂P

∂x+

∂P

∂z∗∂z∗

∂x

)+ µ

(∂2U

∂x2+

∂2U

∂y2+ S

∂2U

∂z∗2

)+ Cu + Fx, (10a)


ρ

(∂V

∂t+ U

∂V

∂x+ V

∂V

∂y+ W1

∂V

∂z∗

)

= −(

∂P

∂y+

∂P

∂z∗∂z∗

∂y

)+ µ

(∂2V

∂x2+

∂2V

∂y2+ S

∂2V

∂z∗2

)+ Cv + Fy, (10b)

ρ

(∂W

∂t+ U

∂W

∂x+ V

∂W

∂y+ W1

∂W

∂z∗

)

= − ∂P

∂z∗∂z∗

∂x+ µ

(∂2W

∂x2+

∂2W

∂y2+ S

∂2W

∂z∗2

)+ Cw + Fz, (10c)

∂U

∂x+

∂V

∂y+

∂W

∂z∗∂z∗

∂z+ Cm = 0, (11)

where T is the temperature, U , V , and W are the three components of the velocity in x, y,and z-directions, respectively, t is the time, ρ the density, Cp the specific heat, k the thermalconductivity, P the pressure in the liquid, L the thickness of the workpiece, Fx , Fy , and Fz arethe components of body forces in x, y, and z-directions, respectively, and µ is the dynamicviscosity of the liquid metal. Some terms in the governing equations are defined as follows:

Wt = U∂z∗

∂x+ V

∂z∗

∂y+ W

∂z∗

∂z− k

ρCp

(∂2z∗

∂x2+

∂2z∗

∂y2+

∂2z∗

∂z2

), (12a)

W1 = U∂z∗

∂x+ V

∂z∗

∂y+ W

∂z∗

∂z− µ

ρ

(∂2z∗

∂x2+

∂2z∗

∂y2+

∂2z∗

∂z2

), (12b)

S =(

∂z∗

∂x

)2

+

(∂z∗

∂y

)2

+

(∂z∗

∂z

)2

, (12c)

Ct = 2

(∂2T

∂z∗∂x

∂z∗

∂x+

∂2T

∂z∗∂y∂z∗

∂y

), (12d)

Cu = 2µ

(∂2U

∂z∗∂x

∂z∗

∂x+

∂2U

∂z∗∂y∂z∗

∂y

), (12e)

Cv = 2µ

(∂2V

∂z∗∂x

∂z∗

∂x+

∂2V

∂z∗∂y∂z∗

∂y

), (12f)

Cw = 2µ

(∂2W

∂z∗∂x

∂z∗

∂x+

∂2W

∂z∗∂y∂z∗

∂y

), (12g)

Cm = ∂U

∂z∗∂z∗

∂x+

∂V

∂z∗∂z∗

∂y, (12h)

where ∂z∗/∂x, ∂z∗/∂y, and ∂z∗/∂z can be obtained from equation (8).Although using the body-fitted coordinates can completely avoid the newly added

boundaries resulting from the surface deformation, the governing equations in the body-fittedcoordinate system are quite complex, which causes many difficulties in the discretization ofgoverning equations. Some special techniques are employed to overcome these difficulties.

770 P C Zhao et al

2.3. Boundary conditions

Due to the energy transferred from the arc (qarc) to the workpiece, the weld pool forms andgrows subsequently. At the same time, some energy is transferred into the solid metal outof the weld pool, and some goes into the ambient medium by means of radiation (qrad) andconvection (qconv). Evaporation (qevap) occurs at the surface of the weld pool.

The net heat-transfer input at the top surface is

q = qarc − qconv − qrad − qevap. (13)

At the symmetric surface, both sides have no net heat surplus. So

∂T

∂y= 0. (14)

At all other surfaces, there are only convection, radiation, and evaporation losses. Thus,

q = −qconv − qrad − qevap. (15)

For the heat source, an elliptical thermal flux distribution was used in this study, whichcan be written as [11]

qarc(x, y) = 6ηEI

πa(b1 + b2)exp

[−3(x − u0t)

2

b21

]exp

(−3y2

a2

)when x − u0t � 0,

(16a)

qarc(x, y) = 6ηEI

πa(b1 + b2)exp

[−3(x − u0t)

2

b22

]exp

(−3y2

a2

)when x − u0t < 0,

(16b)

where η is the efficiency of the arc power, E the arc voltage, I the welding arc current, and b1,b2, and a are parameters related to the welding process. There exists the following constraint:

a(b1 + b2) = 12σ 2q , (17)

where σq is the characteristic radius of the arc heat flux. In this study, a = 1.87σq , b1 = 2.51σq ,and b2 = 3.91σq .

The heat loss includes convection, radiation, and evaporation losses. They are in thefollowing forms [12]:

qconv = hc(T − T0), (18a)

qrad = σε(T 4 − T 40 ), (18b)

qevap = WHv, (18c)

where hc is the convective heat-transfer coefficient, T the temperature of the workpiece, T0 theambient temperature, σ the Stefan–Boltzmann constant, ε the radiation emissivity, W theliquid-metal evaporation rate, and Hv the latent heat of evaporation. For the materials SS304,an approximate equation was given for W in equation (18c) [13, 14]:

log W = 2.52 +

(6.121 − 18 836

T

)− 0.5 log T . (19)

The required boundary conditions for the solution of equation (10) are

µ∂U

∂z∗∂z∗

∂z= − ∂γ

∂T

∂T

∂xand µ

∂V

∂z∗∂z∗

∂z= − ∂γ

∂T

∂T

∂y; at z∗ = 0, and z∗ = 1,

(20)


where γ is the surface tension of liquid metal.

V = 0,∂U

∂y= 0, and

∂V

∂y= 0, at y = 0, (21)

U = 0, V = 0, and W = 0, at other boundaries. (22)

The body force term includes the electromagnetic force and buoyancy. The componentsof the body force Fx , Fy and Fz in equation (10) have been determined in a previous paper[15] so they are not repeated here.

2.4. Numerical method

As mentioned above, with the surface deformation of the weld pool and the introduction ofthe body-fitted coordinate system, the calculation of heat and fluid flow fields in the transientstate are much more complex than those in steady and quasi-steady conditions. A separatedalgorithm is employed to solve the surface deformation, fluid flow, and heat transfer undertransient conditions; i.e. the three problems are calculated separately and improved by turn. Inthis way, the strongly coupled problems among the surface deformation, fluid flow, and heattransfer are solved successfully.

A control volume-based finite-difference method is employed for the solution of thediscrete governing equations. Because the temperature field and fluid flow field are coupledby velocities, temperature, specific heat, and thermal conductivity of the workpiece material,the heat and fluid flow fields are solved together several times in body-fitted coordinates, untilthe convergence criteria are met. The alternative direction iteration (ADI) method was usedin the solution of discretized equations, so the time step must satisfy the following criterion:

k

ρCp

δt

(1

δx2+

1

δy2+

1

δz2

)� 1.5, (23)

where δt is the time step, and δx, δy, and δz are the spacing of the grid along x, y, andz-directions, respectively. In this study, the time step is 0.001 s.

3. Case study

A fixed-grid system of 384 × 64 × 10 grid points was applied for a half workpiece of Q235mild steel with a welding domain of 250 × 60 × 2 mm3. Some material properties of Q235are listed in table 1. The specific heat Cp, dynamic viscosity µ, and thermal conductivity k ofmild steel are temperature dependent, and can be expressed as follows [15]:

k =

60.719 − 0.027 857T

78.542 − 0.0488T

15.192 + 0.0097T

349.99 − 0.1797T

(W m−1 K−1)

T � 851 K851 K � T � 1082 K1082 K � T � 1768 K1768 K � T � 1798 K

(24)

µ =

119.00 − 0.061T

10.603 − 0.025T

36.263 − 0.0162T

(10−3 kg m−1 s−1)

1823 K � T � 1853 K1853 K � T � 1873 K1873 K � T � 1973 K

(25)

Cp =

513.76 − 0.335T + 6.89 × 10−4T 2

−10 539 + 11.7T

11 873 − 10.2T

644354.34 + 0.21T

(J kg−1)

T � 973 K973 K � T � 1023 K1023 K � T � 1100 K1100 K � T � 1379 K1379 K � T .

(26)

772 P C Zhao et al

Table 1. Material properties of mild steel and other parameters used in the calculation.

Symbol Property or parameter Unit Value

Tm Melting point K 1789ρ Density kg m−3 6900T∞ Ambient temperature K 293Hc Convective heat-transfer coefficient W m−2 K−1 80Hv Latent heat of vaporization J kg−1 73.43 × 105

σ Stefan–Boltzmann constant W m−2 K−4 5.67 × 10−8

µ0 Magnetic permeability H m−1 1.66 × 10−6

σq Heat flux radius parameter mm 2.25ε Surface radiation emissivity — 0.4σj Current flux radius parameter mm 1.5γ Surface tension N m−1 1.0η Arc power efficiency — 0.65g Gravitational acceleration m s−2 9.8

For GTAW on a Q235 plate of 2 mm thickness with welding current 110 A, arc voltage 16 V,and welding speed 160 mm min−1, the weld pool and the temperature field achieve the quasi-steady state at t = 4.2 s. In order to analyse the dynamic variation of the weld pool, the weldingcurrent is suddenly changed from 110 to 90 A at t = 5 s. Because of this step-change of weldingcurrent, there are dynamic variations of the weld pool geometry, temperature, and fluid flowfields, so a transient transformation process starts. When this transformation process ends, anew quasi-steady state is reached, with the welding process at the new condition. In this case,the transformation process starts at about t = 5 s and ends at about t = 9 s.

The dynamic responses of the three-dimensional shape of the weld pool after a 20 A stepdrop of welding current are shown in figure 3, where (a), (b), (c), and (d) are the weld poolgeometry at the top surface (z = 0), at the bottom surface (z = L), on the longitudinal section(y = 0), and cross section (x = −1.2 mm), respectively. During the welding process thearc travels at the welding speed and so does the weld pool. In order to compare the weldpool geometries at different instants, the weld pool geometry in the moving coordinate systemwith the origin located at the intersection between the arc centreline and the top surface ofthe workpiece are shown in figure 3 (and figures 4 and 5). As shown in figures 3(a) and (b),after the sudden change in welding current from 110 to 90 A, the pool length at the top andbottom surfaces shortens immediately at the front of the weld pool, while it extends slightlyat the rear of the weld pool. The reason is that the temperature gradient is much steeper at thefront of the weld pool. The electrical parameters (welding current) can be decreased suddenly,but the temperature field changes with a stagnation because of the time delay resulting fromthermal diffusion. This results in the quick contraction at the front of the weld pool, and alittle prolongation at the rear of the weld pool. As time goes on, due to the decrease in weldingcurrent, the whole length of the weld pool is gradually shortened as the rear edge of the weldpool moves forward but the front edge hardly moves relative to the electrode centreline (x =0). The rate of change of the weld pool width is different at different moments. At the initialstage of the transformation period, the pool width decreases quickly. Then, its rate of changeslows down, until the transformation period is finished. There is a larger final decrease ofweld pool width at the bottom than at the top. When the new quasi-steady state is reached, thewhole weld pool contracts due to the decrease in welding current. After the transformationperiod (from 4 to 9 s) ends, the pool width and length at the top surface change from 6.02 mmto 5.16 mm and from 7.5 mm to 6.4 mm, respectively, while those at the bottom surface changefrom 4.49 mm to 3.25 mm and from 5.74 mm to 4.25 mm, respectively.


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0.5

1.0

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2.0

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3.0

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4.0

4.5

5.0

y (m

m)

x (mm)

5s6s

7s

8s

9s

(a)

(b)

(c)

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0.5

1.0

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z (m

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x (mm)

5s

6s

7s

8s

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2.0

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1.0

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0.0

-0.5

-1.0

-1.5

z (m

m)

y (mm)

5s

6s7s 8s 9s x = -1.2 mm

(d)

Figure 3. The dynamic response of the three-dimensional weld pool geometry to the suddendecrease of welding current from 110 to 90 A (workpiece: Q235, thickness: 2 mm, 16 V,160 mm min−1, the welding current is changed from 110 to 90 A at t = 5 s). (a) Top view (z = 0),(b) bottom view (z = L), (c) side view (y = 0), and (d) front view (x = −1.2 mm).

774 P C Zhao et al

-6 -5 -4 -3 -2 -1 0 1 2 3

0 1 2 3

1 2 3

-0.5

0.0

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3.5

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0.1m/s t = 5s

y (m

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x (mm)

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0.0

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0.1m/s t = 7s

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x (mm)

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0.0

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1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.1m/s t = 9s

y (m

m)

x (mm)

(a)

Figure 4. The transient variation of fluid flow pattern after a sudden decrease of welding currentfrom 110 to 90 A (workpiece: Q235, thickness: 2 mm, 16 V, 160 mm min−1, the welding current ischanged from 110 to 90 A at t = 5 s). (a) Top view (z = 0), (b) side view (y = 0), and (c) frontview (x = −1.2 mm).

The transient development of the fluid flow field inside the weld pool is shown in figure 4when the welding current changes. The flow pattern in a fully penetrated weld pool isquite complex, but does not change much after the step-change of welding current. Thereare three vortices inside the pool: one is clockwise and near the centre, the other two arecounterclockwise and near the pool edge. The maximum velocity of fluid flow occurs nearthe electrode centreline. It decreases from 0.06 to 0.03 m s−1 after the step-change of weldingcurrent. The electromagnetic force is proportional to the square of the welding current [15].


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Figure 4. (Continued.)

When the welding current is decreased by 20 A, the driving force for fluid flow inside the poolis much lowered so the drop in fluid flow velocity is about 50%.

By using the numerical model, the dynamic behaviour of the weld pool is simulated whenthe welding speed undergoes a step-change. Because the same welding conditions exist, theweld pool reaches quasi-steady state at t = 4.2 s. Then, the welding speed is suddenly changedfrom 160 to 180 mm min−1 at t = 4.5 s. The weld pool attains its new quasi-steady state att = 6.5 s.

776 P C Zhao et al

0 1 2 3 4 5 6 7 8 9

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Figure 4. (Continued.)

The response of the three-dimenensional shape of the weld pool is shown in figure 5,where (a), (b), (c), and (d) are the transient weld pool geometries at the top surface (z = 0),bottom surface (z = L), on the longitudinal section (y = 0) and cross section (x = −1 mm),respectively. It is clear that the weld pool geometry varies quickly with the step increasein welding speed. During the initial 0.5 s after the change, the weld pool moves 1 mmbackwards with respect to the electrode centreline. Further movement backwards is moregradual. The increase in welding speed causes a decrease in heat input. Thus, the whole weld


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(a)

Figure 5. The dynamic response of three-dimensional weld pool geometry to the sudden increaseof welding speed from 160 to 180 mm min−1 (workpiece: Q235, thickness: 2 mm, 16 V, 110 A,the welding speed is changed from 160 to 180 mm min−1 at t = 4.5 s). (a) Top view (z = 0),(b) bottom view (z = L), (c) side view (y = 0), and (d) front view (x = −1 mm).

778 P C Zhao et al

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(a) (b)

Figure 6. Comparison between predicted and experimental surface geometry of the weld poolbefore and after the sudden decrease of welding current (workpiece: Q235, thickness: 2 mm, 16 V,160 mm min−1, the welding current is changed from 110 to 90 A at t = 5 s). (a) Before step-changeand (b) after step-change.

pool contracts. The variation in the weld pool width is larger than that in the weld pool length.The variation trends are almost the same for both the top and bottom surfaces of the weld pool,but the top surface changes more quickly than the bottom surface because of thermal diffusionresulting in a time delay before the bottom surface reacts.

4. Experimental verification

Experimental measurements were made to verify the model. First, a CCD sensor capturedimages of the weld pool in real time, and then the edges of the weld pool are obtained byprocessing the images with software. After welding, macrographs of the weld in cross sectionwere made to measure the weld dimension.

Figure 6 shows a comparison of the pool geometry at the top surface of the weld poolbefore and after the sudden decrease of welding current from 110 to 90 A. Both shapes ofthe weld pool surface are captured when the molten pools are in the quasi-steady state. It isindicated that the predicted results generally agree with the experimental data.

Figure 7(a) shows a macrograph of the weld at the cross section, and figure 7(b) shows thecomparison between the predicted geometry and experimental geometry in the cross sectionof the weld. The predicted cross section of the weld is in agreement with the experimentallymeasured weld dimension.

5. Conclusions

The dynamic characteristics of fully penetrated weld pool geometries are calculatednumerically when the welding current and welding speed undergo step-changes and the resultslead to the following conclusions:

(1) Under the welding conditions employed in this research, the transformation periods areabout 4 s for a 20 A down step-change of welding current, and about 2 s for a 20 mm min−1

up step-change of welding speed, respectively. The flow patterns in the weld pool do notchange much after a 20 A step-change of welding current, but the maximum velocitydecreases from 0.06 to 0.03 m s−1.

(2) At the initial stage after the step-change of welding current or welding speed, the responsesof weld pool geometry are quicker, but they slow down subsequently until the weld poolachieves a new quasi-steady state.


1 mm

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Figure 7. Comparison between the calculated and experimental cross section of weld:(a) macrograph of the weld at the cross section; (b) comparison (workpiece: Q235, thickness:2 mm, 100 A, 16 V, 160 mm min−1).

(This figure is in colour only in the electronic version)

(3) The time delay resulting from thermal diffusion affects the dynamic behaviour of the weldpool after the step-change of welding current or welding speed. The weld pool geometryat the top surface changes faster than that at the bottom surface, and the front edges of theweld pool respond faster than the rear edges.

(4) Experimental results show that there is an agreement between the predicted weld poolgeometry and the measured ones.

Acknowledgments

The authors are grateful to the US National Science Foundation for the financial support for thisproject under Grant No DMI-0114982. PCZ would like to thank Mr T T Feng, Mr M X Zhang,and Mr J K Hu for their help in experiments and Mr H G Wang for his help in drawing thegraphs.

References

[1] Zacharia T, Eraslan A H, Aidun D K and David S A 1989 Three-dimensional transient model for arc weldingprocess Metall. Trans. B 20 645–59

780 P C Zhao et al

[2] Choo R T C, Szekely J and Westhoff R C 1990 Modeling of high-current arcs with emphasis on free surfacephenomena in the weld pool Weld. J. 69 346s–61s

[3] David S A, Vitek J M, Zacharia T and DebRoy T Weld pool phenomena Int. Inst. Weld. Doc. 212-829-93 2–4[4] Wu C S and Dorn L 1995 Prediction of surface depression of a tungsten inert gas weld pool in the full-penetration

condition Proc. Inst. Mech. Eng. Pare B: J. Eng. Manuf. 209 221–6[5] Chen Y, David S A, Zacharia T and Cremers C J 1998 Marangoni convection with two free surfaces Numer.

Heat Transfer 33 599–620[6] Kou S and Wang Y H 1986 Weld pool convection and its effect Weld. J. 65 63s–70s[7] Wu C S, Cao Z N and Wu L 1992 Numerical analysis of three-dimensional fluid flow and heat transfer in TIG

weld pool with full-penetration Acta Metall. Sin. 28 427–32[8] Wu C S and Dorn L 1994 Computer simulation of fluid dynamics and heat transfer in full-penetrated TIG weld

pools with surface depression Comput. Mater. Sci. 2 341–9[9] Lancaster J F 1986 The Physics of Welding (International Institute of Welding) 2nd edn (Oxford, UK: Pergamon)

[10] Tsai N S and Eagar T W 1985 Distribution of the heat and current fluxes in gas tungsten arcs Metall. Trans. B16 841–6

[11] Goldak J 1984 A new finite element model for welding heat sources Metall. Trans. B 15 299–305[12] Wang Y, Shi Q and Tsai H L 2001 Modeling of the effects of surface-active elements on flow patterns and weld

penetration Metall. Mater. Trans. B 32 145–61[13] Zacharia T, David S A and Vitek J M 1991 Effect of evaporation and temperature dependent material properties

on weld pool development Metall. Trans. B 22 233–41[14] Choi M, Greif R and Salcudean M 1987 A study of heat transfer during arc welding with applications to pure

metals or alloys and low or high boiling temperature materials Numer. Heat Transfer 11 477–89[15] Wu C S 1992 Computer simulation of three-dimensional convection in traveling MIG weld pools Eng. Comput.

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numerical simulation of the dynamic characteristics of

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