Numerical simulation of thin metal plates welding
1A.N. Cherepanov, 2V.P. Shapeev
1Khristianovich Institute of Theoretical and Applied Mechanics SB RAS,
2Novosibirsk State University,
Novosibirsk, 630090, Russia
Introduction
In the last years increasing attention has been paid to development of technology for laser welding of metal products. Laser welding has some advantages over the other technologies for materials joining, but its widespread expansion is restrained by the low stability of the weld joint properties. Experimental investigation and determination of the optimal technological parameters is accompanied by serious methodological difficulties and considerable expenses due to particularities of the welding process. In this connection, development of appropriate mathematical models and numerical algorithms for their implementation is a pressing problem.
Mathematical models are developed in this paper for description of thermophysical processes at laser welding of metal plates: 1) without use of nanopowdered modifying agents, and 2) for the case when modifying nanoparticles of refractory compounds (nitrides, oxides, etc.) are introduced into the weldpool. Specially prepared nanoparticles serve here as crystallization centers, i.e. in fact they are seeding agents on which surface individual clusters group. Such a combination of a nucleus-inoculant and the cluster shell surrounding it should be thermodynamically stable not only at the crystallization temperature, but at its higher values as well.
The first model is based on quasi-equilibrium description of melting and crystallization processes in multicomponent alloys with creation of steam channel taken into account [1]. The second one is based on non-equilibrium emergence and growth of crystal phase of inoculants which are nanoparticles, with use of Kolmogorov’s theory [2]. At that, as shown in [3], homogeneous nucleation can be neglected. The melting process is considered to be quasi-equilibrium.
As an example, simulation results of butt welding of two AL2 alloy plates are presented. Depth of the steam channel, size of the weldpool, width of the two-phase zone are calculated.
Fig. 1. Layout of the weld zone: 1 – laser beam, 2 – liquid phase pool, 3 – two-phase zone, 4 – welding joint.
Let us introduce Cartesian coordinate system, where the laser beam incident upon the tight joint of the welded plates is immobile, while the plates move with the welding speed v. Axis z is directed downward along the beam axis, x axis is oriented along the joint in the direction of the plates movement, and y axis is perpendicular to the joint. The coordinate origin is located on the beam axis at the plates upper surface (Fig. 1).
1. Physicomathematical model of the process
The welded plates are blown by inert gas (argon) to protect the alloy from oxidation. We assume the thermophysical parameters constant and equal to their average values in the temperature range under consideration. With these assumptions, the three-dimensional equation of heat transfer in the weldpool and solid metal takes the form:
,
(1)
where ci, li, rI are specific heat, heat conductivity, and density of the i-th phase, respectively (indices i = 1, 2, 3 denote parameters of solid, two-phase, and liquid states of the metal); Tl0, Te are temperatures of the beginning and ending of solidification; fl is a share of liquid phase in the two-phase zone; k is latent melting heat. On the assumption of quasi-equilibrium in the two-phase zone [3], expression for takes the form:
(2)
At simulation of the welding process with use of nanopowdered inoculators we consider all nanoparticles to have spherical form and be the crystallization centers. Then, for we will have [4,5]
,
where is volume of crystal phase emerged in the overcooled melt. Here, xl0 is coordinate of point on isotherm with liquidus temperature Tl0 (), is the number of nanoparticles in the volume unit, is the nanoparticles radius.
According to (1), in the crystallization zone () a heat source emerges in the original equation, related to heat generation at melt crystallization. Because of nonlinear dependence , in order to take contribution of this heat generation into account, the heat conduction equation can be solved iteratively, adjusting at each iteration and, therefore, T in the crystallization zone.
We assume that welding is performed by CO2-laser with wavelength l0=10.6 μm. The radiation intensity is described by the Gaussian normal distribution:
(3)
where is the radiation density; . Let us write down expression for the radiation power density absorbed by the work surface in the form:
, (4)
where is the absorption coefficient, is radius of the focal spot.
Interaction of the welding zone with the environment is described by corresponding conditions of heat balance on the calculation zone boundaries [1, 2]. They are formulated as boundary conditions for the heat conduction equation. The main contribution and the critical role in temperature distribution picture in the welding area belong to heat flux from the laser radiation (4).
2. Model of steam channel formation
On lower and upper surfaces of the plates, outside the steam channel, boundary conditions take into account heat losses caused by heat irradiation and convective heat exchange with the environment [5]. In formulation of the model of steam and gas channel, we make the following simplifying assumptions.
1. We will assume that the steam channel is a surface of revolution relative to an axis parallel to Z and located in the plane of symmetry, which is monotonously converging with the depth. This corresponds to experimental observations.
2. The laser beam and steam channel are positioned with respect to each other as shown in Fig.3. Namely, the axis of revolution of the steam channel surface is located one laser beam radius from the laser beam , while at Z=0 (on the plates surface) the steam channel radius is twice larger than the laser beam radius. This assumption is also confirmed experimentally with good accuracy in the cases when the steam channel depth is sufficiently large (e.g., exceeds the laser beam radius more than thrice for plates made of aluminum alloy AL2).
3. The steam channel bottom has the form of spherical surface with radius, according to theoretical estimate, defined by expression
(5)
where σ is coefficient of surface tension for liquid metal (alloy), Pmax is excessive pressure of the metal vapor, g is the gravitational acceleration, h is depth of the steam channel..
All made assumptions to some extent are confirmed experimentally in the cases when the steam channel depth is sufficiently large.
Temperature on the steam channel surface can not exceed that of the alloy boiling-point (however, it can be lower than the boiling-point temperature on a part of the channel surface). At that, the maximal heat flux takes place in zone of direct laser radiation action (laser beam spot). Among all possible generatrices of the steam channel surface (which is a surface of revolution according to assumption1), the line situated in plane y = 0 on its front wall will be the most “warmed-up” (i.e., according to the model adopted, of all surface elements this line will have the longest section with the temperature reaching the boiling-point). Therefore, we will take it as generatrix of the steam channel surface. When constructing this line, we will use the following assumption: on its largest possible part near the laser beam axis, the temperature must be close to the boiling-point temperature Tsat for the alloy.
Fig. 2. Schematic layout of the steam channel and laser beam.
Generatrix AC of vapor channel surface, lying in plane y = 0, is sought in the form of spline – line consisting of two parts AB and BC (AC = AB U BC). AB represents a cubic polynomial, tangentially conjugating with BC which is a part of a circle (see Fig. 3). At that, it is expedient to consider AB belonging to two-parametric family
with independent parameters c and h. Lines of this family, under limitations indicated, pass through point A (see Fig. 3) and possess symmetry property relative to the point of their intersection with line x=0, y=0 (laser beam axis). At that, parameter h is Z-coordinate of AB symmetry point, while c is tangent of inclination angle θ of line x(z) to axis Z in the symmetry point
(this is the minimal inclination angle θ of line AC to axis Z). In point B line x(z) is conjugated with a part of circle which radius is given by formula (5). Surface of the steam channel with generatrix AC constructed by the described method satisfies requirements 1) – 3).
Let us assume that we can solve a problem of finding the temperature field in the plate at a known and fixed shape of the steam channel surface. In this case, problem of finding this surface can be reduced to the following problem: varying the channel surface shape it is necessary to find such a solution (i.e. the temperature distribution) which satisfies the principle: on the largest possible section AC near the laser beam axis temperature is close to the boiling-point temperature Tboil for the alloy. In other words, by governing parameters c and h we try to construct line AC which would be the best from the above-mentioned principle perspective. This is achieved by the following method.
Governing parameter h. Let us calculate average temperature Тср on a part of line AB, symmetrical relative to point of its intersection with Z axis and situated close to the laser beam axis. If Тср Tsat, then the melt is “overheated”, and the calculated depth of the steam channel needs to be increased (which is achieved through increase of parameter h). At that, the area of the channel surface absorbing direct laser radiation increases, which in turn leads to decrease of temperature on it. Similarly, if Тср Tsat, then h needs to be decreased. The channel depth is adjusted in this way.
Fig. 3. Scheme of vapor channel representation.
Governing parameter c. After finding the proper h, it may turn out that at intersection of line AC and the beam laser axis the temperature exceeds (or, alternatively, does not reach) the boiling-point temperature Tboil, while at a short distance from AC on both sides of the beam axis the opposite picture is observed. This testifies to a wrong choice of parameter c characterizing the angle of inclination of the channel walls to axis Z. Through decreasing (or increasing) inclination angle θ, we achieve that the nearest neighborhood of line AC point lying on the laser beam axis will receive less (more) heat compared to the periphery. (This owes to the fact that the power density of the laser radiation has Gaussian distribution (4), while the heat absorbed by the channel walls from direct laser radiation is proportional to sinθ).
Thus, varying c and h, we find optimal form of the steam channel surface.
Further adjustment of the steam channel wall form is connected with accounting for heat balance condition and condition of dynamic equilibrium on its surface [1, 2]. Here, is pressure on the wall, is its curvature, is coordinate of z point on the surface. Due to smallness of the steam channel cross-sections sizes in directions perpendicular to Z axis and significance of for small , the channel surface still can be considered to be a surface of revolution accurate to values of higher order of smallness. Therefore, adjustment of its form can be reduced to adjustment of its generatrix. It is shown in [1, 2] that represents a sum of static pressure at surface evaporation Ps(z) and pressure of return (reaction) . Using simplifying hypothesis, values Ps(z) and can be expressed through the temperature. As a result, two values T and x(z) remain in conditions of heat balance and dynamic equilibrium, which can be adjusted iteratively.
For numerical solution of the problem, iterative finite-difference scheme of steadying [5] is used:
(6)
Here, σ is weight parameter (). For approximation of differential operators the following difference operators were taken as :
, (7)
, (8)
, (9)
where
; , . (10)
Applying the method of approximate factorization to (6) we obtain difference scheme
(11)
or equivalent scheme in fractional steps
(12)
With its help, temperature distribution in the plates is determined, and location of internal boundary between phases is found. On the other hand, the steam channel surface is constructed also iteratively by the method described above. In the computer program realized these iterative processes are combined. Namely, after each iteration of the scheme, the steam channel surface form is corrected. At that, h can take values divisible by the grid step along Z axis (i.e. the generatrix of steam and gas channel in cross-section y=0 passes through a grid node lying on Z axis), and it is changed in the case if
,
where is the average temperature on the steam and gas channel surface in plain y=0 near the laser beam axis, μ is parameter controlling “passage” boundaries, within which value h is not changed. Such a mechanism of discrete (“stepped”) variation of parameter h is realized in order to avoid small oscillations of the channel shape, which interfere with establishing of the temperature field. Variation of parameter c is also performed discretely in a similar way.