Boundary conditions on the outer boundaries.
A comprehensive two-dimensional gas metal arc welding (GMAW) model was developed to take into account all the interactive events in the gas metal arc welding process, including the arc plasma, melting of the electrode, droplet formation, detachment, transfer, and impingement onto the workpiece, and the weld-pool dynamics and weld formation. The comprehensive GMAW model tracks the free surface using the volume of fluid method and directly modeled the coupling effects between the arc domain and the metal domain, thus eliminating the need to assign boundary conditions at the interface. A thorough investigation of the plasma arc characteristics was conducted to study its effects on the dynamic process of droplet formation, detachment, impingement, and weld-pool formation. It was found that the droplet transfer and the deformed electrode and weld-pool surfaces significantly influence the transient distributions of current density, arc temperature, and arc pressure, which in turn affect the droplet formation, droplet transfer, and weld-pool dynamics.
- arc plasma
- weld-pool dynamics
- metal transfer
- droplet formation
Gas metal arc welding (GMAW) is the most widely used joining process due to its ability to provide high-quality welds for a wide range of ferrous and non-ferrous alloys at low cost and high speed. As shown in Figure 1, GMAW is an arc-welding process that uses arc plasma between a continuously fed filler metal electrode and the workpiece to melt the electrode and the workpiece. The melted filler metal forms droplets and deposits on the partially melted workpiece to form a weld pool. The weld pool solidifies to bond the workpieces after the arc moves away. A shielding gas is fed through the gas nozzle to protect the molten metal from nitrogen and oxygen in the air. GMAW is also commonly known as metal inert gas (MIG) since inert gasses argon and helium are often used as a shielding gas. An active shielding gas containing oxygen and carbon dioxide is also used and thus the GMAW process is also called metal active gas (MAG). Direct current is usually used with the filler wire as the anode electrode to increase wire melting rate. GMAW can be easily adapted for high-speed robotic, hard automation, and semiautomatic welding applications.
GMAW is a complex process with three major coupling events: (1) the evolution of arc plasma, (2) the dynamic process of droplet formation, detachment, and impingement onto the weld pool, and (3) the dynamics of the welding pool under the influences of the arc plasma and the periodical impingement of droplets. The stability of the GMAW process and the weld quality depend on many process parameters, such as welding current, welding voltage, wire feed speed, wire material and wire size, arc length, contact tube to workpiece distance, workpiece material and thickness, shielding gas properties, shielding gas flow rate, welding speed, etc. Selection of these welding-processing parameters relies on extensive experimentation and is an expensive trial-and-error process. Therefore, tremendous research efforts have been devoted to developing mathematical models of the GMAW process in order to reveal the underlying welding physics and provide key insights of process parameters for process optimization and defect prevention. Due to the complexity of the welding process and the associated numerical difficulty, many numerical models in the literature have simplified the GMAW process and only focused on one or two events. Many works on droplet formation [1–8] and weld-pool dynamics [9–29] have not included the arc plasma. More works now have been devoted to study the arc plasma and its influence on the metal transfer [30–41] and weld-pool dynamics [41–49].
In these simplified models, the droplet formation is considered as an isolated process in the electrode. The influence of the arc plasma is considered as boundary conditions with assumed distributions, such as linear current density distributions [1–3] or Gaussian distributions for the current density and heat flux [6–8].
The effects of droplet impingement on the weld pool have been significantly simplified as boundary conditions in the modeling of the weld-pool dynamics by many researchers [9–21]. The weld-pool surface was assumed to be flat [9–14] or modeled with boundary-fitted coordinates [13–15]. The dynamic impingement of a droplet onto the weld pool has been omitted , treated as a liquid column  or cylindrical volumetric heat source [15–19] acting on the weld pool in many weld-pool models. Only recent models [20–29] have simulated the dynamic interaction of droplets impinging onto the weld pool including both heat transfer and fluid flow effects and tracked the deformed weld pool free surface. However, they all applied assumed current, heat flux, and arc pressure boundary conditions at the weld-pool surface and also approximated the droplet impingement with the assumed droplet shape, volume and temperature, and impinging frequency and velocity.
In almost all aforementioned studies, the interaction of arc plasma with electrode melting, droplet generation and transfer, and weld-pool dynamics was not considered. Linear or Gaussian current density and heat flux were assumed as boundary conditions at the electrode surface [1–6] and weld-pool surface [15–29]. However, the surface of the workpiece is highly deformable, and the profile of the electrode changes rapidly, which greatly influence the arc plasma flow and thus change the current, heat flux, and momentum distribution at the surfaces of electrode and workpiece. Furthermore, the arc plasma can be dramatically distorted when there are free droplets between the electrode tip and the surface of the weld pool as observed in experimental studies [30–32]. Several models [33–39] have been developed to study the dynamic interaction of the arc plasma with the droplet formation. However, the droplet was eliminated when it was detached from the electrode tip or when it reaches the workpiece. The weld-pool dynamics was also omitted and the workpiece was treated as a flat plate. Some recent models [40–43] included the arc plasma, the filler wire, and the workpiece to study the direct interactions of the three domains. However, they are not completely coupled models since the droplet transfer in the arc still relies on an empirical formulation to calculate the plasma drag force in  or the droplet impingement is not simulated in [40–42].
The authors developed a fully coupled comprehensive GMAW model [44–52] to include the entire welding process—the arc plasma evolution, the electrode melting, the droplet formation and detachment, the droplet transfer in the arc, the droplet impingement onto the weld pool, and the weld-pool dynamics and solidification. The volume of fluid (VOF) technique was used to track the interface of the arc plasma and the metal. The temperature, pressure, velocity, electric, and magnetic fields are calculated in the entire computational domain, including the arc, filler wire, and the workpiece without using assumed heat, current, and pressure distributions at the interfaces. In the following sections, the comprehensive mathematical model is first presented to model the GMAW physics, and then the computational results are presented to show the evolution of the arc plasma and its dynamic interaction with the droplet formation, detachment, transfer, and impingement, and the weld-pool dynamics.
2. Mathematical model
2.1. Governing equations
The computational domain is shown in Figure 1, which has an anode region, an arc region, and a cathode region. The governing equations for the arc, the electrode, and the workpiece can be written in a single set based on the continuum formulation given by Diao and Tsai :
The continuum model  included the first- and second-order drag forces and the interaction between the solid and liquid phases due to the relative velocity in the mushy zone (and ), which are represented by the corresponding third to fifth terms in the right-hand side of Eqs. (2) and (3). The energy flux due to the relative phase motion in the mushy zone is represented as the second term in the right-hand side of Eq. (4). The enthalpy method is used for phase change during the fusion and solidification processes. The enthalpy for the solid and liquid phases can be expressed as
Continuum density (
The inertial coefficient,
2.2. Arc region
The arc region includes the arc plasma column and the surrounding shielding gas. The arc plasma is assumed to be in local thermodynamic equilibrium (LTE) . The plasma properties, including enthalpy, density, viscosity, specific heat, thermal conductivity, and electrical conductivity, are calculated from an equilibrium composition [57, 58]. The influence of metal vapor on plasma material properties [37–42] is not considered in the present study. The plasma is also assumed to be optically thin, thus the radiation may be modeled as a radiation heat loss per unit volume represented by
2.3. Metal region and tracking of free surfaces
The metal region includes the electrode, droplet in the arc, and the workpiece. The dynamic evolution of the droplet formation of the electrode tip, the droplet transfer in the arc, and the weld-pool dynamics require precise tracking of the free surface of the metal region. The volume of fluid method is used to track the moving free surface . A volume of fluid function,
The average value of
2.4. Forces at the interface of the arc plasma and metal regions
The molten metal is subject to body forces and surfaces forces at the interface of the arc plasma and metal regions. The body forces include gravity, buoyancy force, and electromagnetic force. The surface forces include arc plasma shear stress, arc pressure, surface tension due to surface curvature, and Marangoni shear stress due to temperature difference. The surface forces are included as source terms to the momentum equations according to the CSF (continuum surface force) model [59–61]. Using
The arc plasma shear stress is calculated from the velocities of the arc plasma cells at the free surface
Surface tension pressure is normal to the free surface and can be expressed as 
where is the surface normal, calculated as the gradient of the VOF function
The temperature-dependent Marangoni shear stress is in a direction tangential to the local free surface and is given by 
where is a tangent vector of the local free surface.
2.5. Energy terms at the interface of the arc plasma and metal regions
2.5.1. Plasma-anode interface
The anode sheath region at the plasma-electrode interface is a very thin region, about 0.02-mm thick , and is at nonlocal thermal equilibrium. The very thin region is treated as a special interface by adding energy source terms,
The four terms in Eq. (18) take into account thermal conduction, electron heating associated with the work function of the anode material, black-body radiation heat loss, and evaporation heat loss, respectively, at the metal surface. The energy equation for the plasma region only considers the cooling effects through conduction.
2.5.2. Plasma-cathode interface
Similarly, energy source terms
2.6. External boundary conditions
The computational domain for a two-dimensional (2D) axisymmetric GMAW system is shown as ABCDEFGA in Figure 1. The external boundary conditions are listed in Table 1. Symmetrical boundary condition is assigned along the centerline AG.
The velocity boundary takes into account the wire feed rate at AB, shielding gas inlet at BC, open boundaries at CD and DE, and non-slip wall condition at EF. The inflow of shielding gas from the nozzle at BC is represented by a fully developed axial velocity profile for laminar flow in a concentric annulus :
The temperature boundaries along AD, DE, and EG are set as the room temperature. The boundary conditions for current flow include a zero voltage at the bottom of the workpiece FG, uniform current density along AB specified as , and zero current flow along the other surfaces.
3. Numerical methods
At each time step, the calculation involves separate calculations in the arc region and the metal region, the coupling of the two regions through the interface boundary conditions described in Sections 2.4 and 2.5, and updating the arc and metal regions after obtaining the new free surface using the VOF method, Eq. (12), in the metal region.
The arc plasma region uses a fully implicit formulation and an upwind scheme for the combined convection/diffusion coefficients, and the SIMPLE algorithm  for the velocity and temperature fields. The metal region uses the method developed by Torrey et al.  to calculate the velocity and temperature fields.
The computational domain is 5 cm in radius and 3.04 cm in length. A nonuniform grid system is used with finer meshes near the electrode tip, in the arc column and the weld pool, where a fine mesh of 0.01 cm is used. Time step size is set as 5 × 10−6 s for a stable numerical solution.
4. Results and discussion
In this chapter, the comprehensive model [44, 45] is used to simulate a spot GMAW welding of a mild steel workpiece with a mild steel electrode under a constant current of 220 A shielded by argon. The electrode has a diameter of 0.16 cm and the workpiece is a mild steel disk with a 3-cm diameter and a 0.5-cm thickness. The contact tube is set flush with the bottom of the gas nozzle and has a contact tube to workpiece distance of 2.54 cm. The wire feed rate is 4.5 cm/s and the initial arc length is 0.8 cm. The shielding gas flow rate is 24 l/min and the inner diameter of the nozzle is 1.91 cm.
The temperature-dependent material properties of argon and the radiation loss term (SR) in Eq. (4) are taken from  and are plotted in Figure 2. Table 2 lists the properties of the solid and liquid mild steel taken from  and other parameters used in the computation.
|Constant in Eq. (20)||2.52|
|Specific heat of solid phase||700 (J kg‒1 K‒1)|
|Specific heat of liquid phase||780 (J kg‒1 K‒1)|
|Thermal conductivity of solid phase||22 (W m‒1 K‒1)|
|Thermal conductivity of liquid phase||22 (W m‒1 K‒1)|
|Density of solid phase||7200 (kg m‒3)|
|Density of liquid phase||7200 (kg m‒3)|
|Thermal expansion coefficient||4.95×10‒5 (K‒1)|
|Dynamic viscosity||0.006 (kg m‒1 s‒1)|
|Latent heat of fusion||2.47×105 (J kg‒1)|
|Latent heat of vaporization||7.34×106 (J kg‒1)|
|Solidus temperature||1750 (K)|
|Liquidus temperature||1800 (K)|
|Ambient temperature||300 (K)|
|Vaporization temperature||3080 (K)|
|Surface tension coefficient||1.2 (N m−1)|
|Surface tension temperature gradient||10−4 (N m−1 K−1)|
|Work function||4.3 V|
|Electrical conductivity||7.7 × 105 (Ω–1m–1)|
4.1. Arc plasma evolution
Figure 3 shows the distributions of arc plasma temperature and pressure before and after the first droplet is detached and transferred to the workpiece. The shape of the electrode and workpiece are marked with thick lines. At
After the droplet is detached from the electrode at
The first droplet reaches the workpiece around
In many of the weld-pool models [7–29], the arc pressure distribution at the center of the workpiece surface was assumed to be a Gaussian distribution with a fixed amplitude and distribution radius. However, the arc pressure distribution at the workpiece surface changes dramatically during the welding process as shown in Figure 6. Both the magnitude and distribution region varies with the evolution of the electrode and weld-pool surfaces and the presence of the detached droplet. Low arc pressure with a flat-top distribution is found at the weld-pool surface at
4.2. Droplet formation and transfer
Droplet formation is determined by the concentrated heating due to the recombining electrons at the electrode surface and the flow pattern within the droplet caused by a balance of forces acting on the droplet, which includes electromagnetic force, surface tension force, gravity, arc pressure, and plasma shear stress. To clearly illustrate the heat transfer and fluid flow within the droplet at the electrode tip, the distributions of temperature, velocity, electrical potential, current, and electromagnetic force within the droplet at
The first droplet formation is shown in Figures 3–5 and 8 from
4.3. Weld-pool dynamics and solidification
Figures 3 and 8 show the first droplet impingement onto the workpiece and form a weld pool. The weld pool grows wider and deeper with more droplets deposited into it. The final weld-pool shape and the resulting final weld shape are determined by weld-pool dynamics subject to periodic droplet impingement and several important forces, including electromagnetic force, arc pressure, plasma shear stress, surface tension, and gravity force. As shown in Figures 10 and 11, a droplet is ready to be detached from the electrode tip at
A comprehensive model has been presented to simulate the transport phenomena in a gas metal arc-welding process, including the arc plasma evolution, the melting of the electrode and the droplet generation, detachment, transfer, and impingement onto the workpiece, and the weld-pool dynamics and solidification. This model included all the three regions—the electrode, the arc plasma, and the weld pool—in the computational domain and modeled the interactive coupling between these three regions. The distributions of arc pressure, current density, and heat flux at the weld-pool surface are found to vary in a wide range, and thus cannot be represented by a fixed distribution in many published GMAW models. The simulation results have revealed physical insights which cannot be found with those isolated single-region models in the literature. The transient evolution of the arc plasma was found to influence and also to be influenced by the droplet formation, detachment, transfer in the arc, and weld-pool dynamics. Therefore, a comprehensive model is required to accurately take into account the coupling events in both the arc domain and metal domain. The comprehensive model can be used to study the effects of process parameters on the welding process and the final weld formation, such as droplet generation with pulse currents to achieve one droplet per pulse (ODPP) and the effects of shielding gas and wire feed rate on the welding process.