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1 used a 3-D Finite
Difference model to calculate the temperature. The circular bead section was modeled as a stepped
Cartesian grid. The shadowing effect (attenuation of the laser power at the workpiece due to the
gas-powder jet) was considered. Clad dimensions were computed as a function of process
parameters, both for single and also for multiple track cases.
Kar and Mazumder
2-4 and Agrawal et al.
5 solved analytically the one-dimensional heat and
mass transfer (coupled) equations, for binary systems. The goal was to calculate the composition
of the extended solid solution that is formed by rapid cooling. Parametric studies were also
included, correlating variables such as laser power, beam radius, traverse speed, clad thickness
and film composition, and portions of several non-equilibrium phase diagrams were proposed.
Hoadley and Rappaz
6 used a 2-D Finite Element Model for the calculation of the quasi-
steady state temperature field. An idealized problem, in which almost no melting occurs in the
substrate, was taken as the basis for defining a successful cladding operation. Mixing was
assumed to distribute the powder instantaneously in the melt, which results in a volumetric heat
source term associated with the latent heat. The free surface is considered to be an arc of a circle
and the temperature of the powder particles is estimated. The calculation procedure involves
determination of the laser beam position that is consistent with the requirements of the idealized
problem. A parametric study involving laser power, processing speed and clad thickness was
included.
Picasso and Hoadley
7 used the Finite Element Method to solve for the stationary heat
transfer and fluid flow problem on a 2-D geometry. The model considers the deformation of the
gas-liquid interface and the forces associated with the powder injection into the melt pool, and the
powder is assumed to melt instantaneously as it hits the liquid surface. Picasso et al.
8 used an
iterative procedure, based on a 3-D analytical model for temperature, for obtaining process
parameters such as scanning speed and powder feed rate as a function of laser power, beam radius,
powder jet geometry and clad height. The shadowing effect and the dependence of the absorption
coefficient on the angle of incidence of laser radiation into the melt pool were also considered.
1.3 Proposed approach
The heat conduction equation and deformation mechanics equations are to be solved in a
two-dimensional domain, the longitudinal mid-plane of a clad track. It should be noted that this is a
rather drastic approximation, as the process is clearly three-dimensional (for instance, the height of
clad track is of the same order of magnitude of its half-width). Nevertheless, even a simple finite
element model should provide the correct trends for the effect of processing parameters on
temperature and stress fields.
1 used a 3-D Finite
Difference model to calculate the temperature. The circular bead section was modeled as a stepped
Cartesian grid. The shadowing effect (attenuation of the laser power at the workpiece due to the
gas-powder jet) was considered. Clad dimensions were computed as a function of process
parameters, both for single and also for multiple track cases.
Kar and Mazumder
2-4 and Agrawal et al.
5 solved analytically the one-dimensional heat and
mass transfer (coupled) equations, for binary systems. The goal was to calculate the composition
of the extended solid solution that is formed by rapid cooling. Parametric studies were also
included, correlating variables such as laser power, beam radius, traverse speed, clad thickness
and film composition, and portions of several non-equilibrium phase diagrams were proposed.
Hoadley and Rappaz
6 used a 2-D Finite Element Model for the calculation of the quasi-
steady state temperature field. An idealized problem, in which almost no melting occurs in the
substrate, was taken as the basis for defining a successful cladding operation. Mixing was
assumed to distribute the powder instantaneously in the melt, which results in a volumetric heat
source term associated with the latent heat. The free surface is considered to be an arc of a circle
and the temperature of the powder particles is estimated. The calculation procedure involves
determination of the laser beam position that is consistent with the requirements of the idealized
problem. A parametric study involving laser power, processing speed and clad thickness was
included.
Picasso and Hoadley
7 used the Finite Element Method to solve for the stationary heat
transfer and fluid flow problem on a 2-D geometry. The model considers the deformation of the
gas-liquid interface and the forces associated with the powder injection into the melt pool, and the
powder is assumed to melt instantaneously as it hits the liquid surface. Picasso et al.
8 used an
iterative procedure, based on a 3-D analytical model for temperature, for obtaining process
parameters such as scanning speed and powder feed rate as a function of laser power, beam radius,
powder jet geometry and clad height. The shadowing effect and the dependence of the absorption
coefficient on the angle of incidence of laser radiation into the melt pool were also considered.
1.3 Proposed approach
The heat conduction equation and deformation mechanics equations are to be solved in a
two-dimensional domain, the longitudinal mid-plane of a clad track. It should be noted that this is a
rather drastic approximation, as the process is clearly three-dimensional (for instance, the height of
clad track is of the same order of magnitude of its half-width). Nevertheless, even a simple finite
element model should provide the correct trends for the effect of processing parameters on
temperature and stress fields.
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