Laser cladding is an advanced technology for depositing material of specific properties on a metal substrate whose surface is mostly planar or cylindrical. Its purpose is to improve the quality of the surface (wear resistance, corrosion resistance, hardness, and so forth) or renovation of damaged parts (for example molds) [1, 2] and .
Despite many advantages, the process exhibits also one serious drawback consisting of high temperature gradients in surface layers of the substrate at the spot irradiated by the laser beam. These may exceed 1000°C/s, which leads after cooling to high residual mechanical stresses of thermal origin. Their presence may be manifested by cracks, peeling off of deposited tracks and other surface damage [4, 5].
One prospective measure for suppressing high temperature gradients is inductive preheating or postheating of substrate to an appropriate temperature [6, 7]. Inductive heat pretreatment exhibits many benefits such as environmental friendliness, excellent controllability and no influence on the chemical composition of heated material [8, 9, 10].
It is, however, not easy to design the induction-assisted cladding process, as its result is affected by numerous factors whose influences cannot be determined accurately because of their too complicated or even stochastic nature. The process of cladding on a cylindrical surface (that is the object of research in this paper) is schematically depicted in Figure 1.
The cylinder slowly rotates at a low angular velocity ω, and at the same time, moves also in the axial direction at a velocity v. The static laser beam continuously heats its surface to about 2000°C, which leads to local melting of substrate metal. In addition, the nozzle delivers to the spot a metal powder that also melts and forms with the molten surface a pool containing a mixture of both materials. As the cylinder slowly rotates, the pool cools and starts solidifying, thus forming a continuous helicoidal track.
In the axial cut of the cylinder, we can get three different results differing by the cross section of the deposited layer, see Figure 2.
The top part of Figure 2 depicts the case where the track turns are separated one from another. This result is highly undesirable (as the final deposited surface should form a uniform continuous layer without gaps) and may be caused, for example, by excessively high velocity of movement in the longitudinal axis of the cylinder, or a laser beam of too low power.
The middle part of Figure 2 shows a well tuned case where the whole surface of the substrate is continuously covered by a layer of deposited metal, demanding light final machining to provide the necessary smoothness.
Finally, the bottom part of Figure 2 represents the case when either the axial velocity of the cylinder is slow or the power of the laser beam and powder injection rate are excessively high. Here, the process of additional machining for the purpose of obtaining a smooth surface would be unnecessarily expensive.
It is obvious that correct design of the cladding process represents a strongly nonlinear 3D optimization problem, that is, from the viewpoint of numerical modeling, characterized by continuous geometrical changes (due to continuous adding of powder material) that are not known in advance. Moreover, the result may be affected also by subtle additional phenomena that will be described later on.
Since about 2011, several authors have tried to model the process, but only on a planar surface [11, 12, 13]. One of the principal problems has been to cope with the constantly changing geometry. Some  remeshed the whole arrangement at every time step, which proved to be in 3D extremely expensive. Others simplified the problem by considering physical parameters of the involved media as constants, which is not correct in processes with high variations of temperature.
The authors present another approach to modeling the process with constantly varying 3D geometry, based on the boundary deformation technique that is partially implemented in the commercial code COMSOL Multiphysics 5.2. They also take into account real temperature dependencies of materials and media. Nevertheless, at this stage of research, some subtle phenomena like the keyhole effect [6, 8], flow of molten mixture of metals in the pool or partial vaporization of liquid metal  are still neglected, and their influence will be investigated in further work. Despite this fact, the results obtained are promising and agree well with relevant experiments.
2 Mathematical model
The problem represents a strongly nonlinear and non-stationary electro-thermal task, characterized by continuous addition of powder material and changing geometry that is unknown in advance. In addition, it is accompanied by several subtle physical phenomena that will be explained but not quantified.
2.1 Magnetic field produced by preheating inductor
The 3D magnetic field is generated by the preheating or postheating inductor. One of the possibilities for its modeling is its description by the magnetic vector potential A. The corresponding equation reads (1)
Here μ denotes the permeability that depends (in ferromagnetic parts) on applied magnetic field, γ is the electric conductivity and v represents the velocity of the substrate. Although the equation does not seem to be too complicated, due to nonlinearity, temperature dependence of parameters μ and γ, and also incommensurability of the time constants of the magnetic and temperature fields, its numerical solution in 3D is still practically unfeasible.
Fortunately, two simplifications of (1) can be introduced without any substantial depreciation of results. The first one cannot be applied in the continuous model, only in the numerical solution and consists of the assumption that the magnetic permeability in ferromagnetic elements is constant by parts. Its value in every cell of the discretization mesh is supposed to be constant and corresponds to the average value of local magnetic flux density. What is more, for low velocities on the order of mm/s that are commonly used, the corresponding term in (1) can be neglected with a very small error (about 1%). Now, the magnetic field in the system may be described by the phasor A and the corresponding equation reads . (2)
where ω = 2πf refers to the angular frequency. Thus, the task can be solved in the frequency domain.
The boundary condition along a sufficiently distant boundary is of the Dirichlet type (A = 0). When the problem exhibits any spatial symmetry, the Neumann condition may also be prescribed along the corresponding interface.
2.2 Temperature field produced by induction
The temperature field in the heated substrate due to induction is described by the heat transfer equation in the general form . (3)
Here, λ stands for the thermal conductivity, ρ denotes the mass density, cp is the specific heat at constant pressure, w represents the volumetric internal sources of heat in the processed material and t is time. The volumetric heat sources w are represented by the specific Joule losses wJ and specific hysteresis losses wh. In other words,
The losses wh can be determined either from the measured loss dependence for the material used, or from a suitable (for example Steinmetz) analytical formula. But in many cases they can be neglected as their value is just a small fraction of the losses wJ.
The boundary condition takes into account both convection and radiation and may be expressed in the form (5)
Here, Ts is the temperature of the surface of the plate, α denotes the coefficient of the convective heat transfer, T0 stands for the temperature of the surrounding medium (air), σ = 5.67 ⋅ 10−8 Wm−2K−4 is the Stefan-Boltzmann constant, C represents the emissivity of the heated part and Tr is the temperature of the surface to which heat is radiated. Unlike the magnetic field, the velocity term in (6) cannot be neglected even for its very low values.
2.3 Temperature field produced by laser beam
Heating by the laser beam is respected by the boundary condition starting from the power delivered to the spot irradiated by the beam. The basic heat transfer equation corresponds to (6), but now the internal volumetric losses w vanish. Heat is supposed to be delivered into the material from the spot on the surface heated by the laser beam. This may be taken into account by a boundary condition that is similar to (5), but contains one more term qin corresponding to the delivered thermal flux (6)
The thermal flux delivered to the surface at the irradiated spot qin is supposed to obey the Gaussian distribution and its integral over the irradiated area is equal to the total power of the laser beam delivered to the material.
The boundary condition is given by the actual surface temperature of the body, which depends on whether the body is inductively preheated or postheated. A very important role is played here by the time delay between the two heat treatment processes.
2.4 Deformation of the surface
For description of surface deformation, we used a special module in COMSOL Multiphysics. The algorithm is based on the definition of shift velocity as a function of selected parameters, in the solved case temperature and geometry. This model has to be supplemented with an automatic adaptivity of mesh in the domain of high temperature and its gradient. It occurs, in fact, only near the spot irradiated by the laser beam, where the material starts melting and the method of geometry deformation allows simulation there of growth of the track. A coarse mesh would lead here to a fault of the solver due to shifting of the boundary. On the other hand, a very fine mesh would lead to an unacceptably high number of degrees of freedom. Therefore, using h-adaptivity for local mesh refinement seems to be the best solution.
While the deformation of the surface in direction y may be considered zero, the velocities of deformations in the directions x and z are described by the following logical conditions (7)
The constant k represents a coefficient that has to be determined from calibration with measurement. In our case, k = 1.15 × 10−3.
2.5 Other subtle phenomena accompanying the process whose influences were not included in the model
These phenomena are derived from the so-called keyhole effect, see Figure 3 (that is somewhat exaggerated for better visibility)
When the laser beam irradiates a given spot of the substrate, the temperature at the spot grows very fast and after 1–2 seconds the surface starts melting, together with the added metal powder, forming a pool whose dimensions first grow. The pool does not, however, fill in the whole heated region. In its center, just below the laser beam, the temperature continues increasing and particles of molten metal start vaporizing, forming a capillary as a way of escape and a cloud of plasma consisting of these particles above the heated spot. This phenomenon has two consequences: a flow field driven by gravitational, buoyancy and inertial forces affecting the distribution of heat transfer and a local temperature field in the pool and reduction of the power delivered by the laser beam to the surface of the substrate due to absorption and reflection of power in the cloud. Moreover, not the whole amount of added powder takes part in the process because of its dissipation.
All these phenomena exhibit more or less stochastic character and their evaluation is rather complicated. The relevant research is in the stage of testing, which is the reason why these phenomena have not yet been included in the presented model.
2.6 Numerical solution
Numerical solution was performed using the professional code COMSOL Multiphysics 5.2 supplemented with a series of our own scripts and procedures. But the computations are still rather expensive (one variant requires about 8 hours on a top-parameter PC).
The computations include:
checking of the distance of the artificial boundary from the system,
convergence of results (we required that the results must satisfy three valid digits),
influence of the time step on the stability of the process.
The computations were realized on a top-parameter PC (16 GB RAM, 4-core CPU i7, 3.2 GHz).
As far as they were available, we took into account the temperature dependencies of material parameters, both in solid and liquid state.
3 Illustrative example
The cylinder to be equipped with the deposited layer is made of steel S355 of diameter 20 cm. The powder material was METCO 41C. The laser beam of diameter 3 mm was generated with a source of power 1000 W (with efficiency 26 %, so that the power delivered to the surface was 260 W). The velocity of rotation of the cylinder surface vs = 0.00133 m/s and corresponds to an axial shift of rate 0.8 m/min. The axial velocity of the cylinder vy was 6.36 × 10−5 m/s.
Figure 4 shows the solved arrangement together with the accepted coordinate system. The inductor with one turn can be seen on the upper part of the cylinder.
Unfortunately, the shape of the saturation curve is defined only for room temperature and its dependence on temperature is not given. Known only is its Curie temperature TC (approximately 770°C). On the other hand, this dependence affects heating of the cylinder by induction. That is why we modeled it (after several tests) in the following way:
μr(B, T) = μr(B, Tr) ⋅ ψ(T), where Tr is the room temperature and ψ(T) is a suitable function of the temperature. Here (10)
The current in the preheating inductor is 2240 A and its frequency is 12 kHz.
Computations provided a number of results. Figure 7 shows the distribution of temperature in the system with induction preheating. At the spot irradiated by the laser beam, the temperature reaches 2400°C, at which point the material starts evaporating. Powder melts at a temperature of 1700°C, which is used in the model as the driving condition. Figures 8 and 9 show the cladded layer. Its width is about 5 mm and height about 2 mm. The height is, however, not perfectly constant along the length of the track (differences are ± 10 %), although this is one of the important quality conditions of the cladded layer.
Another very important parameter (as said before), is the velocity of axial shifting of the cylinder, The tracks are, in fact, represented by a layer cladded in helicoid (see the discussion for Figure 2). Figures 10 and 11 show the impact of inappropriate and appropriate velocities of axial shift of the cylinder. As both processes of melting and cladding start at the temperature of 1700°C, the particular waves of temperature are proportional to the pitch between individual tracks.
Figure 12 depicts the distribution of temperature for the surface of the substrate around the cylinder. The position of the laser beam can be identified according to the peak of the distribution. The position of the preheating inductor may be identified by the small peak before the maximum given by the position of the laser. Finally, Figure 13 shows the result of the experimental cladding process.
The paper presents a fundamental model of induction-assisted laser cladding with a novel way of modeling a system with continuously varying geometry. The results are good and differences between the simulation and experiment do not differ more than about 16%.
The sources of the above differences come from:
complexity of the physical process and its mathematical model,
neglecting of influence of some subtle physical phenomena connected with the keyhole effect, and
problems with keeping some input quantities (power of the laser, velocity of motion of the cylinder and current of the preheating inductor) at constant values during the process.
Further work in the domain of modeling will be aimed at correct inclusion of the above subtle phenomena in the model and acceleration of the numerical calculations as they are still expensive.
Attention will also be paid to improvement of the experiment in order to avoid errors due to uncertainties.
This research has been supported by the Ministry of Education, Youth and Sports of the Czech Republic under the RICE New Technologies and Concepts for Smart Industrial Systems, project LO1607.
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Published Online: 2017-12-29