Numerical simulation of the dynamic characteristics of weld pool geometry with step-changes of welding parameters

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1 INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING Modelling Simul. Mater. Sci. Eng. 12 (2004) PII: S (04) Numerical simulation of the dynamic characteristics of weld pool geometry with step-changes of welding parameters P C Zhao 1,CSWu 1 and Y M Zhang 2 1 MOE Key Laboratory for Liquid Structure and Heredity of Materials, Institute of Materials Joining, Shandong University, Jinan , People s Republic of China 2 Center for Manufacturing and Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY 40506, USA Received 10 January 2004 Published 1 July 2004 Online at stacks.iop.org/msmse/12/765 doi: / /12/5/002 Abstract The gas tungsten arc welding process is an uncertain nonlinear multivariable system. In order to control the welding process, the nonlinear dynamic relationship between the weld pool geometry reflecting the weld quality and the welding parameters must be developed. A three-dimensional numerical model is developed to investigate the dynamic characteristics of the weld pool geometry when the welding current and welding speed undergo a step-change. Under the welding conditions employed in this research, the transformation periods are about 4 s for a 20 A down step-change of welding current, and about 2s for a 20mmmin 1 up step-change of welding speed, respectively. At the initial stage during the step-change of welding current and welding speed, the responses of weld pool geometry are quicker, but they slow down subsequently until the weld pool reaches a new quasi-steady state. Welding experiments were conducted to verify the simulation results. It was found that the predicted weld pool geometries agree with the measured ones. 1. Introduction Gas tungsten arc welding (GTAW) is the most widely used arc welding process for critical and accurate joining. Many GTAW applications require full penetrations. Current practice relies on skilled 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 are fairly high, and the operators do not typically perform consistently. Hence, control of GTAW penetration 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 quality and the welding parameters must be developed. Thus, a numerical simulation of the GTAW process is necessary /04/ $ IOP Publishing Ltd Printed in the UK 765

2 766 P C Zhao et al Though extensive studies on the numerical simulation of the GTA weld pool have been done [1 5], little effort has been directed towards addressing the dynamic characteristics of weld pool geometry when the welding parameters change. For a three-dimensional GTA weld pool with a moving heat source, the majority of mathematical models are concerned with the quasi-steady state [6, 7], and are unable to describe the transient variation of the weld pool geometry with time when the welding parameters are changed. In this paper, a transient threedimensional model is developed to simulate the full-penetration GTA weld pool with surface deformation. The model is used to predict the dynamic responses of the weld pool geometry after 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, surface tension, gravity, fluid dynamics, and shear stress from the plasma. When full penetration is established, both pool surfaces at the top and bottom are deformed. As shown in figure 1, the functions Z top = ϕ(x,y) and Z bottom = ψ(x,y) are used to describe the configuration of the top and bottom surfaces of the weld pool, respectively. In the case of partial penetration (figure 1(a)), the surface deformation occurs only at the top surface of the workpiece. The top surface of the weld pool is governed by the following equation [4]: P arc ρgϕ + C 1 = γ (1+ϕ2 y )ϕ xx 2ϕ x ϕ y ϕ xy + (1+ϕx 2)ϕ yy (1+ϕx 2 +, (1) ϕ2 y )3/2 where P arc is the arc pressure, ρ the density, g the gravitational acceleration, γ the surface tension, and C 1 a constant, and ϕ x = ϕ/ x, ϕ xx = 2 ϕ/ x 2, ϕ xy = 2 ϕ/ 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 weld pool has a specific geometry and volume. For a specific pool geometry at a certain instant, the pool surface undergoes a corresponding deformation, but the total volume of the weld pool at this instant is not changed before or after the surface deformation. Thus, there is the following constraint: ϕ(x,y)dx dy = 0, (2) 1 where 1 is the surface area of the weld pool at the top surface. Of course, 1 has different values at different times. The arc pressure can be expressed as [9, 10] P arc = µ 0I 2 8π 2 σ 2 j exp ( r2 2σ 2 j ), (3) where µ 0 is the permeability in free space, I the welding current, σ j a current distribution parameter, and r = (x u 0 t) 2 + y 2, where u 0 is the welding speed and t is the time. For a fully penetrated weld pool (figure 1(b)), two equations are required to describe the configuration of the top and bottom surfaces, respectively. P arc ρgϕ + C 2 = γ (1+ϕ2 y )ϕ xx 2ϕ x ϕ y ϕ xy + (1+ϕ 2 x )ϕ yy (1+ϕ 2 x + ϕ2 y )3/2 (4a)

3 Weld pool geometry 767 (a) o y x Weld pool Ztop (b) o y x Weld pool Ztop z(z*) z(z*) Workpiece Zbottom Workpiece Figure 1. Schematic of pool surface deformation under partial and full penetration. (a) Partial penetration and (b) full penetration. and ρg(ψ + L ϕ) + C 2 = γ (1+ψ y 2)ψ xx 2ψ x ψ y ψ xy + (1+ψx 2)ψ yy (1+ψx 2 + ψ, (4b) y 2)3/2 where L is the thickness of the workpiece, C 2 is a constant, ψ x = ψ/ x, ψ xx = 2 ψ/ x 2, ψ xy = 2 ψ/ 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. Though both the top and bottom of the pool surfaces undergo deformation, the total volume of the weld pool does not vary at a specific time instant. Therefore, ϕ(x,y)dx dy = ψ(x,y)dx dy (5) 1 2 where 1 is the surface area of the weld pool at the top surface, while 2 is the surface area of the weld pool at the bottom surface. C 1 and C 2 are the total sum of other forces that act on the weld pool surface except arc pressure, gravity, and surface tension. In the calculation, C 1 is derived from equations (1) and (2) while C 2 from equations (4) and (5): C 1 dx dy = ( P arc ) dx dy 1 1 ) C 2 dx dy + dx dy ( 1 2 = ( P arc ) dx dy 1 ρg (L ϕ)dx dy 2 1 γ (1+ϕ2 γ 1 (1+ϕ2 y )ϕ xx 2ϕ x ϕ y ϕ xy + (1+ϕx 2)ϕ yy (1+ϕx 2 + dx dy, ϕ2 y )3/2 2 γ (1+ψ 2 y )ϕ xx 2ϕ x ϕ y ϕ xy + (1+ϕ 2 x )ϕ yy (1+ϕ 2 x + ϕ2 y )3/2 dx dy y )ψ xx 2ψ x ψ y ψ xy + (1+ψ 2 x )ψ yy (1+ψ 2 x + ψ 2 y )3/2 dx 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 judge whether the weld pool is penetrated or not. Once the weld pool achieves full penetration, its bottom surface is deformed too, so equations (4) and (5) apply to the full-penetration weld pool. During the iteration, the guessed values of C 1 or C 2 are employed first. In the partial case ϕ(x,y) is obtained by solving equation (1), while ϕ(x,y) and ψ(x,y) are obtained by solving equation (4) in the full-penetration case. Then, improved values of C 1 or C 2 are obtained by solving equations (6) or (7). Based on the new values of C 1 or C 2, equation (1) or (4) is solved again to get improved functions ϕ(x,y), orϕ(x,y) and ψ(x,y). The above procedure is (6)

4 768 P C Zhao et al y Wire o x Welding Arc Welding Direction z(z*) Ztop Workpiece Fusion Zone Zbottom 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 the arc 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 the quasi-steady state of the weld pool is achieved Governing equations A schematic sketch of a typical GTAW process system is shown in figure 2. In order to describe the 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 boundaries appear 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 causes some difficulty in the boundary conditions. In this study, based on Cartesian coordinates, the following body-fitted coordinate system (x,y,z ) is introduced (figure 1(b)) to transform the 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: ( T ρc p + U T t x + V T ) y + W T t = ( k T ) + ( k T ) + S ( k T ) + kc z x x y y z z t, (9) ( U ρ t + U U x + V U = y + W U 1 z ) ( P x + P z z x ) ( 2 U + µ x U y 2 ) + S 2 U + C z 2 u + F x, (10a)

5 Weld pool geometry 769 ( V ρ + U V t x + V V ) y + W V 1 z ( P = y + P z ) ( 2 ) V + µ z y x + 2 V 2 y + S 2 V + C 2 z 2 v + F y, (10b) ( W ρ t + U W x + V W y + W 1 = P z z x + µ ) W z ( 2 W x W y 2 ) + S 2 W + C z 2 w + F z, (10c) U x + V y + W z z z + C m = 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, C p the specific heat, k the thermal conductivity, P the pressure in the liquid, L the thickness of the workpiece, F x, F y, and F z are the components of body forces in x, y, and z-directions, respectively, and µ is the dynamic viscosity of the liquid metal. Some terms in the governing equations are defined as follows: W t = U z x + V z y + W z z k ρc p ( 2 z x z y z z 2 ), (12a) W 1 = U z x + V z y + W z z µ ( 2 z ρ x + 2 z 2 y + 2 z ), (12b) 2 z 2 ( z ) 2 ( z ) 2 ( z ) 2 S = + +, (12c) x y z ( 2 T z C t = 2 z x x + 2 T z y ( 2 U z C u = 2µ z x x + 2 U z y ( 2 V z C v = 2µ z x x + 2 V z y ( 2 W z C w = 2µ z x x + 2 W z y z ), (12d) y z ), (12e) y z ), (12f) y z ), (12g) y C m = 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-fitted coordinate system are quite complex, which causes many difficulties in the discretization of governing equations. Some special techniques are employed to overcome these difficulties.

6 770 P C Zhao et al 2.3. Boundary conditions Due to the energy transferred from the arc (q arc ) to the workpiece, the weld pool forms and grows subsequently. At the same time, some energy is transferred into the solid metal out of the weld pool, and some goes into the ambient medium by means of radiation (q rad ) and convection (q conv ). Evaporation (q evap ) occurs at the surface of the weld pool. The net heat-transfer input at the top surface is q = q arc q conv q rad q evap. (13) At the symmetric surface, both sides have no net heat surplus. So T = 0. (14) y At all other surfaces, there are only convection, radiation, and evaporation losses. Thus, q = q conv q rad q evap. (15) For the heat source, an elliptical thermal flux distribution was used in this study, which can be written as [11] 6ηEI q arc (x, y) = [ πa(b 1 + b 2 ) exp 3(x u 0t) 2 ] ) b1 2 exp ( 3y2 when x u a 2 0 t 0, (16a) 6ηEI q arc (x, y) = [ πa(b 1 + b 2 ) exp 3(x u 0t) 2 ] ) b2 2 exp ( 3y2 when x u a 2 0 t<0, (16b) where η is the efficiency of the arc power, E the arc voltage, I the welding arc current, and b 1, b 2, and a are parameters related to the welding process. There exists the following constraint: a(b 1 + b 2 ) = 12σq 2, (17) where σ q is the characteristic radius of the arc heat flux. In this study, a = 1.87σ q, b 1 = 1σ q, and b 2 = 3.91σ q. The heat loss includes convection, radiation, and evaporation losses. They are in the following forms [12]: q conv = h c (T T 0 ), (18a) q rad = σε(t 4 T0 4 ), (18b) q evap = WH v, (18c) where h c is the convective heat-transfer coefficient, T the temperature of the workpiece, T 0 the ambient temperature, σ the Stefan Boltzmann constant, ε the radiation emissivity, W the liquid-metal evaporation rate, and H v the latent heat of evaporation. For the materials SS304, an approximate equation was given for W in equation (18c) [13, 14]: log W = 2 + ( T ) log T. (19) The required boundary conditions for the solution of equation (10) are µ U z z z = γ T T x and µ V z z z = γ T T y ; at z = 0, and z = 1, (20)

7 Weld pool geometry 771 where γ is the surface tension of liquid metal. U V = 0, y = 0, and V = 0, at y = 0, (21) y U = 0, V = 0, and W = 0, at other boundaries. (22) The body force term includes the electromagnetic force and buoyancy. The components of the body force F x, F y and F z in equation (10) have been determined in a previous paper [15] so they are not repeated here Numerical method As mentioned above, with the surface deformation of the weld pool and the introduction of the body-fitted coordinate system, the calculation of heat and fluid flow fields in the transient state are much more complex than those in steady and quasi-steady conditions. A separated algorithm is employed to solve the surface deformation, fluid flow, and heat transfer under transient conditions; i.e. the three problems are calculated separately and improved by turn. In this way, the strongly coupled problems among the surface deformation, fluid flow, and heat transfer are solved successfully. A control volume-based finite-difference method is employed for the solution of the discrete governing equations. Because the temperature field and fluid flow field are coupled by 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, until the convergence criteria are met. The alternative direction iteration (ADI) method was used in the solution of discretized equations, so the time step must satisfy the following criterion: k ρc p δt ( 1 δx δy δz 2 ), (23) where δt is the time step, and δx, δy, and δz are the spacing of the grid along x, y, and z-directions, respectively. In this study, the time step is 01 s. 3. Case study A fixed-grid system of grid points was applied for a half workpiece of Q235 mild steel with a welding domain of mm 3. Some material properties of Q235 are listed in table 1. The specific heat C p, dynamic viscosity µ, and thermal conductivity k of mild steel are temperature dependent, and can be expressed as follows [15]: T T k = (Wm 1 K 1 ) T T T µ = T (10 3 kg m 1 s 1 ) T T T T C p = T (Jkg 1 ) T T 851 K 851 K T 1082 K 1082 K T 1768 K 1768 K T 1798 K 1823 K T 1853 K 1853 K T 1873 K 1873 K T 1973 K T 973 K 973 K T 1023 K 1023 K T 1100 K 1100 K T 1379 K 1379 K T. (24) (25) (26)

8 772 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 T m Melting point K 1789 ρ Density kg m T Ambient temperature K 293 H c Convective heat-transfer coefficient W m 2 K 1 80 H v Latent heat of vaporization J kg σ Stefan Boltzmann constant W m 2 K µ 0 Magnetic permeability H m σ q Heat flux radius parameter mm 2.25 ε Surface radiation emissivity 0.4 σ j Current flux radius parameter mm γ Surface tension N m 1 η Arc power efficiency 0.65 g Gravitational acceleration m s 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 quasisteady state at t = 4.2 s. In order to analyse the dynamic variation of the weld pool, the welding current is suddenly changed from 110 to 90 A at t = 5 s. Because of this step-change of welding current, there are dynamic variations of the weld pool geometry, temperature, and fluid flow fields, so a transient transformation process starts. When this transformation process ends, a new 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 = 9s. The dynamic responses of the three-dimensional shape of the weld pool after a 20 A step drop of welding current are shown in figure 3, where (a), (b), (c), and (d) are the weld pool geometry 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 the arc travels at the welding speed and so does the weld pool. In order to compare the weld pool geometries at different instants, the weld pool geometry in the moving coordinate system with the origin located at the intersection between the arc centreline and the top surface of the 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 and bottom surfaces shortens immediately at the front of the weld pool, while it extends slightly at the rear of the weld pool. The reason is that the temperature gradient is much steeper at the front 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 from thermal diffusion. This results in the quick contraction at the front of the weld pool, and a little prolongation at the rear of the weld pool. As time goes on, due to the decrease in welding current, the whole length of the weld pool is gradually shortened as the rear edge of the weld pool 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 initial stage of the transformation period, the pool width decreases quickly. Then, its rate of change slows down, until the transformation period is finished. There is a larger final decrease of weld pool width at the bottom than at the top. When the new quasi-steady state is reached, the whole weld pool contracts due to the decrease in welding current. After the transformation period (from 4 to 9 s) ends, the pool width and length at the top surface change from 6.02 mm to 5.16 mm and from 7.5 mm to 6.4 mm, respectively, while those at the bottom surface change from 4.49 mm to 3.25 mm and from 5.74 mm to 4.25 mm, respectively.

9 Weld pool geometry 773 (a) s 7s 5s 8s 9s (b) (c) (d) s 6s 8s 5s 9s s 5s 7s 8s 9s 5s 6s s 6s 7s 8s 9s x = -1.2 mm Figure 3. The dynamic response of the three-dimensional weld pool geometry to the sudden decrease 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).

10 774 P C Zhao et al (a) t = 5s t = 7s t = 9s Figure 4. The transient variation of fluid flow pattern after a sudden decrease 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) side view (y = 0), and (c) front view (x = 1.2 mm). The transient development of the fluid flow field inside the weld pool is shown in figure 4 when the welding current changes. The flow pattern in a fully penetrated weld pool is quite complex, but does not change much after the step-change of welding current. There are three vortices inside the pool: one is clockwise and near the centre, the other two are counterclockwise and near the pool edge. The maximum velocity of fluid flow occurs near the electrode centreline. It decreases from 6 to 3 m s 1 after the step-change of welding current. The electromagnetic force is proportional to the square of the welding current [15].

11 Weld pool geometry 775 (b) t = 5s t = 7s t = 9s Figure 4. (Continued.) When the welding current is decreased by 20 A, the driving force for fluid flow inside the pool is 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 when the welding speed undergoes a step-change. Because the same welding conditions exist, the weld pool reaches quasi-steady state at t = 4.2 s. Then, the welding speed is suddenly changed from 160 to 180 mm min 1 at t = 4.5 s. The weld pool attains its new quasi-steady state at t = 6.5s.

12 776 P C Zhao et al (c) t = 5s t = 7s t = 9s 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 = 1mm), respectively. It is clear that the weld pool geometry varies quickly with the step increase in welding speed. During the initial s after the change, the weld pool moves 1 mm backwards with respect to the electrode centreline. Further movement backwards is more gradual. The increase in welding speed causes a decrease in heat input. Thus, the whole weld

13 Weld pool geometry 777 (a) s 5s 6.5s 6.0s 5.5s (b) s 5.0s 5.5s 6.0s 6.5s (c) s 6.0s 4.5s 5.0s 5.5s s 5.0s 5.5s 6.5s 6.0s (d) s 6.0s 5.5s s 5.0s x = -1mm Figure 5. The dynamic response of three-dimensional weld pool geometry to the sudden increase of 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).

14 778 P C Zhao et al (a) 4 3 (b) Predicted 2 1 Predicted Experimental Experimental Figure 6. Comparison between predicted and experimental surface geometry of the weld pool before 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-change and (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 diffusion resulting in a time delay before the bottom surface reacts. 4. Experimental verification Experimental measurements were made to verify the model. First, a CCD sensor captured images of the weld pool in real time, and then the edges of the weld pool are obtained by processing the images with software. After welding, macrographs of the weld in cross section were made to measure the weld dimension. Figure 6 shows a comparison of the pool geometry at the top surface of the weld pool before and after the sudden decrease of welding current from 110 to 90 A. Both shapes of the weld pool surface are captured when the molten pools are in the quasi-steady state. It is indicated 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 the comparison between the predicted geometry and experimental geometry in the cross section of the weld. The predicted cross section of the weld is in agreement with the experimentally measured weld dimension. 5. Conclusions The dynamic characteristics of fully penetrated weld pool geometries are calculated numerically when the welding current and welding speed undergo step-changes and the results lead to the following conclusions: (1) Under the welding conditions employed in this research, the transformation periods are about 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 not change much after a 20 A step-change of welding current, but the maximum velocity decreases from 6 to 3 m s 1. (2) At the initial stage after the step-change of welding current or welding speed, the responses of weld pool geometry are quicker, but they slow down subsequently until the weld pool achieves a new quasi-steady state.

15 Weld pool geometry 779 (a) 1 mm (b) - - Experimental Predicted 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 weld pool after the step-change of welding current or welding speed. The weld pool geometry at the top surface changes faster than that at the bottom surface, and the front edges of the weld pool respond faster than the rear edges. (4) Experimental results show that there is an agreement between the predicted weld pool geometry and the measured ones. Acknowledgments The authors are grateful to the US National Science Foundation for the financial support for this project under Grant No DMI 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 the graphs. References [1] Zacharia T, Eraslan A H, Aidun D K and David S A 1989 Three-dimensional transient model for arc welding process Metall. Trans. B

16 780 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 surface phenomena in the weld pool Weld. J s 61s [3] David S A, Vitek J M, Zacharia T and DebRoy T Weld pool phenomena Int. Inst. Weld. Doc [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 [5] Chen Y, David S A, Zacharia T and Cremers C J 1998 Marangoni convection with two free surfaces Numer. Heat Transfer [6] Kou S and Wang Y H 1986 Weld pool convection and its effect Weld. J s 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 [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 [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. B [11] Goldak J 1984 A new finite element model for welding heat sources Metall. Trans. B [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 [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 [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 [15] Wu C S 1992 Computer simulation of three-dimensional convection in traveling MIG weld pools Eng. Comput