Revue de Métallurgie 110, 165–173 (2013) c EDP Sciences, 2013 DOI: 10.1051/metal/2013060 www.revue-metallurgie.org
R evue de Métallurgie
A two-dimensional axially-symmetric model of keyhole and melt pool dynamics during spot laser welding M. Courtois1,2, M. Carin2, P. Le Masson2 and S. Gaied1 1
2
ArcelorMittal Global R&D Montataire, France e-mail:
[email protected] LIMATB Université de Bretagne-Sud Lorient, France
Key words: Laser welding; keyhole; vapor; porosities; level-set; melt pool
Received 5 November 2012 Accepted 7 March 2013
Abstract – For a better understanding of the physical phenomena associated with the appearance of defects in laser welding, a heat and fluid flow model is developed using R . This first step of the project is focused on the modeling of a static Comsol Multiphysics laser shot on a sample of steel. This 2D axially-symmetric configuration is used to study the main physical phenomena related to the creation of the keyhole. This model takes into account the three phases of the matter: the vaporized metal, the liquid phase and the solid base. To track the evolution of these three phases, coupled equations of energy and momentum are solved. The liquid/vapor interface is tracked using the Level-Set method. The calculated velocity and free surface deformation are analyzed. Melt pool shapes are compared with experimental macrographs and the influence of some parameters such as laser power is discussed.
aser-welded tailored blank technology consists of assembling, edge to edge, blanks of different thicknesses or different metallurgical properties. The goal is to obtain a blank ready to be stamped for automotive applications. The first advantage of this technology is to optimize the choice of material and its thickness to the functional requirement. In order to understand this technology applied to very highstrength steel better, it is necessary to develop a robust numerical model to predict the appearance of defects in the welded joint (Fig. 1). These defects, such as weld pool collapsing or partial penetration, are present when the metal is in the liquid state and when the dynamics of solidification stop the shape of the weld joint in an unwanted geometry. To predict the appearance of these defects, a model of heat transfer and fluid flow is required since these phenomena are first responsible for the final shape of the welded zone. Such models are still rare in the literature. Indeed, most of the published studies present models using restrictive assumptions and so are not able to explain defects or shapes of melted zones. A model including all the physical phenomena needed
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to describe the appearance of defects accurately requires high-performance computing tools. To overcome this difficulty, some authors such as Medale et al. [1] and Lee et al. [2] have developed models applied to a static laser shot allowing a 2D axial symmetry assumption. With such models, the study of complex phenomena such as recoil pressure is possible with reasonable computation times. Currently, 3D models able to simulate the laser welding process including the main phenomena are rare, but the feasibility has been recently proved by Ki et al. [3], Geiger et al. [4], and Pang et al. [5] with different numerical methods.
1 Modeling of laser welding process 1.1 Main physical phenomena During laser welding, the sheets to assemble are at room temperature. The laser beam, with a radius of 500 μm, illuminates the surface sheet, inducing the fusion of the zone to assembly. Because of the very high-energy density in the laser, the material will be
Article published by EDP Sciences
M. Courtois et al.: Revue de Métallurgie 110, 165–173 (2013)
Fig. 1. Example of defects on tailored welded blanks (ArcelorMittal data).
Fig. 2. Scheme of phenomena in laser welding.
vaporized quickly. This vaporization will be accompanied by a recoil pressure, which will push the liquid, inducing a deformation and the creation of a vapor capillary, named a keyhole. The recoil pressure is the result of the vaporization by a phenomenon of actionreaction, also called the piston effect. This pressure is responsible for the digging and the maintenance of the keyhole. To realize the welded joint, the laser beam is moved at a speed of 3 to 20 m min−1 . This speed is a function of the laser power and the sheet metal thickness. In a steady state, three phases are present: the vapor in the capillary, the melt pool which flows around the keyhole with 166
complex movements, and finally, the solid base away from the energy input (Fig. 2). The goal of this study is to take into account the vaporization of the steel and predict the location of the liquid-vapor interface using the numerical method of front tracking. To maintain an increasing difficulty, the first step is to consider a static laser shot, since this configuration allows 2D axisymmetric modeling. The aim of this model is to predict, first, the correct kinetic of digging of the sheet, and secondly, the dynamics of melt pool collapse (Fig. 3). Although this configuration is far removed from the industrial configuration, it is an essential step before
M. Courtois et al.: Revue de Métallurgie 110, 165–173 (2013)
Fig. 3. Scheme of collapse of the keyhole.
3D modeling. It is the best way to understand how to model complex phenomena and choose the best numerical method. 1.2 Mathematical formulation For heat and fluid flow modeling, it is necessary to solve conservation equations of energy (heat equation, Eq. (1)), mass (continuity equation, Eq. (2)) and momentum (Navier-Stokes, Eq. (3)). The liquid metal is considered as an incompressible Newtonian fluid under laminar flow. → − → − → − ∂T → − ρ cp + ∇ · u T = ∇ · λ ∇ T + S (1) ∂t → − → ∇ · −u = 0 (2) → → − → ∂− u → − − ρ +u · ∇· u = ∂t
→ − → − − → − − T ∇ · −pI + μ ∇→ u + ∇→ u − → − → − + F Darcy + F ts − ρ 1 − β T − Tmelting → (3) with T the temperature, t the time, ρ the density, cp the specific heat capacity, λ the thermal conductivity, S a heat source to repre−u the velocity sent the energy of the laser, → vector, p the pressure, I the identity matrix, μ the dynamic viscosity, (·)T the transposed → − matrix, F Darcy a source term to cancel the velocity if the temperature is lower than the − the gravity accelerafusion temperature, → → − tion and F ts the normal and tangential force vector representing surface tension effects. The surface tension coefficient is assumed to be constant in this model, which means that the Marangoni effect is neglected.
Conservation equations are solved in the three phases (vapor, liquid and solid) and a front tracking method is added to the model. The method chosen to track the liquid gas interface is the Level-Set method [6]. This method has been successfully applied to model laser keyhole welding by Ki et al. [3] and Pang et al. [5]. This method uses a fixed mesh and defines a variable Φ in the entire computational domain. This variable takes the value of 1 in the vapor phase and 0 in the liquid (and solid) phase. This variable is used to define the thermophysical properties of each phase and the interface is identified by locating the isovalue Φ = 0.5. The movement of the liquid/gas interface is realized by advecting the Φ variable using the velocity field calculation. Near the interface, the Φ variable varies progressively with a smooth step function, inducing the property variation from one phase to the other (Fig. 4a). The thickness of this transition must be small enough to represent the interface accurately, but not too small, in order to avoid numerical convergence problems. Nevertheless, the Level-Set method has the main advantage of treating the complex geometric change of the interface. So, it is possible to simulate the formation of porosity and the transition from partial to full penetration mode. With the Level-Set method, the boundary conditions of the interface are introduced in the source terms of conservation equations using the Φ variable position. So, the laser energy is deposed at the surface of the metal by the source term in the heat equation. It is assumed to be a Gaussian distribution (Eq. (4)). In this first study, Fresnel laws are not taken into account. The absorption efficiency η is assumed to be constant and 167
M. Courtois et al.: Revue de Métallurgie 110, 165–173 (2013)
(b)
(a)
Fig. 4. (a) Initial geometry and variable Φ. (b) Deformed geometry at t = 20 ms.
equal to 0.67. This value is close to experimental measurements. It has been evaluated at 0.6 for a 1-μm laser in welding regimes by Fabbro et al. [7]. In addition, it has been shown by Hirano et al. [8] that absorptivity does not vary very much with temperatures between 2400 and 3400 K, justifying the assumption of a constant absorption coefficient. Due to the small displacement of the interface and in the absence of a masking effect, this approximation seems reasonable. ⎞ ⎛ ⎜⎜ −r2 ⎟⎟ Pmax I (r) = η exp ⎜⎜⎝ 2 ⎟⎟⎠ δz ϕ π R2g Rg
(4)
with Pmax the laser power, Rg the radius of the Gaussian distribution and δz a Dirac function to apply the energy only at the surface of the liquid. Note that latent heats are neglected in a first approximation. The vaporization resulting from this energy induces a recoil pressure at the origin of the creation of the capillary. This pressure imposes a “piston effect” on the underlying material. So, the melt pool directly under the vaporization layer is moved and ascends on the sides to form rolls at the surface (Fig. 3). Generally, the recoil pressure is applied using an empirical formulation [2,4,5], so this does not result from the calculation. A more physical way is to introduce a source term into the continuity equation which is 168
proportional to the evaporation flow rate and non-zero only at the interface. This approach has been used by Sajid [9] and Esmaeeli and Tryggvason [10] in the case of water boiling using the Level-Set method. With this method, a single field formulation incorporates the effect of the interface in the equations as delta function source terms, which act only at the interface. In regard to mass conservation, incompressibility is satisfied within each phase but does not hold at the interface because of fluid expansion due to liquid-vapor phase change. An additional term is therefore introduced into the continuity equation (Eq. (6)) as described in [9, 10], which can be viewed as a local interfacial mass transfer source/sink due to expansion/contraction upon phase change. If there is locally no phase change, then this equation reduces to the customary incompressible constraint. Note that this term will induce implicitly a recoil pressure in the momentum equation. The source term in the continuity equation will be non-zero only at the interface (Eq. (5)). Indeed, the condition of incompressibility of fluid phases is not satisfied near the interface because of the density difference on both sides of the interface in the gaseous phase (ρv ) and liquid phase (ρl ). For the same reason, the Level-Set transport equation (Eq. (6)) is also modified with the same method. These source terms create
M. Courtois et al.: Revue de Métallurgie 110, 165–173 (2013)
Table 1. Thermophysical properties of the liquid and vapor phases. ρl = 7000 kg m−3
λl = 40 W m−1 K−1
cpl = 400 J kg−1 K−1
μl = 5 × 10−3 Pa s−1
ρv = 10 kg m−3
λv = 10 W m−1 K−1
cpv = 1000 J kg−1 K−1
μv = 1 × 10−5 Pa s−1
an evaporation flow rate m˙ (Eq. (7)) which is deduced mainly from the local temperature and the saturated vapor pressure [11]. ρl − ρv → −→ ∇ .− u = m˙ δ ϕ (5) ρ2 ∂ϕ → fv fl → − =0 +− u . ∇ ϕ − m˙ δ ϕ + ∂t ρl ρv m˙ =
m psat (T) 1 − βr √ 2 π kb T
(6)
(7)
with fl and fv the relative fraction of liquid and vapor between 0 and 1, m the atomic weight of iron, kb the Boltzmann constant, psat (T) the saturated vapor pressure, and βr the retro-diffusion coefficient, assumed to be equal to 0 in a first approximation.
2 Numerical results The previous equations are solved in a 2D axisymmetric configuration with the finite r element code COMSOL Multiphysics . The simulation corresponds to a laser shot of 25 ms followed by 5 ms of cooling. The focal spot diameter is 600 μm and the sample is a disk of 2 mm in radius and 1.8 mm in thickness. The material properties given in Table 1 are assumed to be constant in this study to reduce computation times. Note that the value of the surface tension coefficient was underestimated in this first study for numerical convenience. Using a more realistic value leads to strong forces acting at the singularity solid-liquid-vapor point and too great an effort on the solid part. Figure 4b illustrates the deformation of the melt pool under the effect of the recoil pressure and shows the appearance of the vapor capillary. This approach using a source term in the continuity equation produces more realistic velocity fields in the vapor phase than that using an empirical formulation for the recoil pressure. Indeed, in this case, the addition of the source term
σ = 1.5 × 10−1 N m−1
generates a vapor flow rate near the interface affecting the velocity field in the vapor phase. When the recoil pressure is calculated with an empirical formulation, the velocities in the vapor phase are only a result of the displacement of the liquid vapor interface, so the interaction between the vapor jet and the melt pool cannot be represented correctly. However, this interaction has a significant impact on the shapes of the weld joint, as shown by high-speed camera observation [12, 14]. Indeed, in the presence of high welding speeds, the front edge of the capillary vapor undergoes an inclination. The laser beam impacts the forward front, resulting in a stronger evaporation at this front. This vapor jet interacts strongly with the back of the melt pool and can cause the phenomenon of “humping”, the formation of bumps at the surface of the melt pool. Although experimentally observed, this effect is rarely discussed in numerical work. During the evaporation, a strong mass flow rate of vapor escapes from the keyhole with velocities higher than 10 m s−1 and interacts with the melt pool. Figure 5 shows a calculation performed without the effects of buoyancy and without the Marangoni effect. The velocity field presents a main vortex in the upper part resulting from the maximum velocities on the interface. The only factor responsible for this flow is the shear stress from the vapor plume on the melt pool. The effect is relatively weak in this configuration, but it can have a primary size in future work in a 3D configuration with a titled keyhole. To illustrate the laser power influence on the digging of the liquid surface, Figure 6 presents melt pool shapes and isotherms for varying powers from 600 to 800 W for a shot of 25 ms. We can observe that temperature levels increase with laser power, resulting in a more intense vaporization phenomenon. A greater penetration is thus observed with a more vertical inclination of the liquid surface that can lead to instabilities. The mass loss during the whole process is of the order of 1%. This loss is attributed both to the 169
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Fig. 5. Velocity fields in gas (left) and liquid (right). Highlight on the effect of shear stress from the vapor plume to the melt pool (laser power = 900 W, t = 18 ms).
Fig. 6. Melt pool shapes and isotherms for different laser powers at t = 25 ms.
evaporation process and the numerical errors. In Figures 6 and 7 the gas volume in the keyhole is balanced by the low elevation of the dense phase considering the revolution symmetry, even if the figures might suggest the opposite. It has been shown by many authors [12, 14] that the inclination of the liquid surface increases with the laser power. At high laser power, the model presents a different mechanism of collapse of the melt pool during the cooling. When the laser irradiation is shut off, vaporization and therefore 170
recoil pressure are immediately stopped. Gravity and the surface tension are the only forces acting and lead to the collapse of the keyhole. If the keyhole is sufficiently deep, a gas cavity is formed during collapse (Fig. 7). These residual porosities are observed experimentally and the model helps the understanding of their formation.
3 Experimental study A series of experiments was performed at the PIMM laboratory to achieve two goals.
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Fig. 7. Porosity formation during the keyhole collapse process. Melt pool shapes with 25 ms of heating and 5 ms of cooling (laser power = 900 W).
(a)
(b)
Fig. 8. (a) Macrographs of fusion zones; variable interaction time; Plaser = 1000 W. (b) Macrographs of fusion zones; variable interaction time; Plaser = 1500 W.
The first one is to obtain a transient evolution of welded zone geometry to validate the numerical model. The second is to identify the operating conditions leading, or not, to the formation of residual porosities. To achieve these goals, a series of static shots were realized by varying the laser power and time of interaction of the laser. The laser used is a disk laser type Nd:YAG (λ = 1.06 μm). The
diameter of the focal spot is set to 600 μm. Shots are operated on DP 600 steel samples with a thickness of 1.8 mm. Cross-section macrographs after chemical etching (BéchetBeaujard) were performed in order to measure the melt pool size evolution. Figure 8a shows macrographs for a laser power of 1000 W with an interaction time varying from 2 to 25 ms. No porosity is 171
M. Courtois et al.: Revue de Métallurgie 110, 165–173 (2013)
(a)
(b)
Fig. 9. (a) Time evolution of penetration depth; experiment and numerical results. (b) Time evolution of melt pool width; experiment and numerical results.
observed at low power. The melt pool shape is like a bowl, whereas at a power of 1500 W (Fig. 8b), the profile of the melted zone is narrower. At high laser powers, we can sometimes note the presence of residual porosities. Some porosity can appear at the very beginning of laser irradiation. Figure 8b illustrates, for an interaction time of 6 ms, porosity very close to the base metal. This indicates, because of a high energy density, a very high recoil pressure dominating all other forces in. Under these conditions, the liquid is removed very quickly on the sides and it is ejected upward. During the presence of the keyhole, there are temporary areas where the liquid layer is extremely thin; the vapor here is almost with contact of the solid. Note that at high power, the liquid probably does not have a stable position. Under the effect of recoil pressure, the melt pool is ejected then it collapses on itself. Irradiation continuing, it is immediately chased and repeatedly. These successions of collapsing and repulsion can be extremely fast, up to creating a corrugated surface. When the laser is stopped, the keyhole collapses, sometimes capturing a gas cavity. For a power of 1500 W, porosity is present on more than 50% of shots. This rate drops to less than 10% for a power of 1000 W. However, we saw with the numerical model, the more the power is elevated, the steeper the inclination of the liquid boundary. This phenomenon favors the concentration of energy in the bottom of the keyhole because of laser reflections, and 172
favors the formation of porosity during the keyhole collapse process. This analysis is in agreement with the experimental observations of Girard et al. [13], who have shown that the appearance of porosity is due to two factors: the keyhole shape and the solidification time. Figure 9a shows that the penetration depth for powers of 1000 and 1500 W increases quickly without stabilization at 25 ms. The width begins to stabilize at around 16 ms of laser irradiation. Indeed, the focal spot diameter is limited and energy will concentrate at the bottom of the keyhole, increasing the depth rather than the side diffusion. Note that the numerical model predicts with a good agreement depth penetration with, however, an overestimated width of the melted zone. Considering the assumptions made, the global temporal evolution is satisfactory.
4 Conclusion and outlook A 2D axially-symmetric heat transfer and fluid flow model was developed to simulate the keyhole formation under laser irradiation. This model takes into account different phenomena such as fusion evaporation, surface tension and gravity and calculates temperature, pressure and velocity fields in the liquid and gaseous phases. The model also gives the deformation of the liquid surface due to the recoil pressure.
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Using the Level-Set method, the calculated results are consistent with experimental observations. The development of an original method, more complex but more physical, to take into account the recoil pressure consists of introducing the evaporation mass flow rate into the mass conservation equation. With this method, the phase change from liquid to vapor can be observed and the interaction between the vapor jet and the liquid surface can be analyzed. It was shown that the vapor plume has an impact on the velocity field in the melt pool. An experimental study was performed to identify operating conditions leading or not to the appearance of defects such as residual porosity. For a laser power greater than 1000 W, the violent ejection of liquid was demonstrated with postmortem cross-section macrographs, showing residual porosity. The macrograph analysis also indicated that the liquid thickness can be very thin. Finally, increasing laser power leads to narrower melted zones, due to a deeper keyhole, a steeper liquid wall and thus a greater concentration of energy at the bottom of the gas cavity. The next step of this work is to compare in detail the digging dynamics measured and calculated by the model. To improve the model, multiple reflections of rays on the keyhole surface should be considered and surface tension representation must be improved. In the longer term, the goal is to extend the 2D axisymmetric case to the three-dimensional industrial case with the advance of the laser. The final goal is to propose a model able to predict the appearance of defects for a set of welding parameters given in three dimensions.
Acknowledgements The authors are grateful to Professor Patrice Peyre and Rémy Fabbro from the PIMM laboratory (ENSAM, Paris) for their support for our experimental investigations.
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