Theoretical Formulation
3.1 Conduction Mode Heat Transfer Analysis
3.1.1 Fourier heat conduction
The heat input into the work piece is approximated by a surface heat source with constant shape. The mathematical formulation of the model is based on the following assumptions:
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The top surface of the weld pool is considered to be flat, neglecting the effect of plasma arc and shielding gas. As keyhole formation is not expected, the energy absorption of the plasma can be neglected.
The welded sheets are considered as solid deformable bodies in the FE simulation.
Only half of the geometry is considered due to symmetry along the weld line. This effectively reduces the computational time.
The thermal properties have been considered as temperature dependent.
The initial temperature of the workpiece is considered as 298 K.
Governing equation and boundary conditions
An estimation of the temperature at the interface is required for establishing the welding process parameters. The mechanism of heat conduction resulting in transient temperature field generated during welding is determined based on the basic principles of thermal transfer and conservation of energy:
. / ( ) ̇ (3.1)
where is the density, T is the temperature, C is the specific heat, ̇ is the rate of internal heat generation, Kij the component of thermal conductivity tensor and U is the welding velocity. The term on the left side refer to the transient nature of the heat transfer process. The term on the right side infer to the conductive heat transfer in the three directions.
Figure 3.1 schematically shows one sheet of butt joint geometry to be welded along the interface 2-3-7-8 along with the mesh pattern utilized for the present model. The heat source is irradiated on the top surface along 3-2. The developed model comprises the whole welding process including the heating and the cooling phase.1-2-3-4 is the symmetric surface.
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Fig. 3.1 Schematic representation of the solution geometry and boundary interactions.
Boundary conditions
For heat transfer between the workpieces and the surrounding medium, the following boundary condition is considered.
a) Initial boundary condition where the temperature at the boundary is specified,
At time , ( ) (3.2) b) At the symmetric surface, there is no gradient
i.e. . (3.3) c) Mathematically heat interactions of all the surfaces can be expressed as:
( ) ( ) (3.4)
where qs defined by a Gaussian heat distribution, n denotes the direction normal to the surface, hc
is the heat transfer coefficient, and T0 is the ambient temperature. The first term denotes the heat conduction to the boundary surface, the second surface the heat flux from the heat source on the top surface, the second and third terms represent the heat loss by radiation and convection, respectively, is the emissivity. The surfaces which are in direct contact with air are assigned coefficient of convective heat transfer of air. However, at the surfaces 1-4-8-9 and 5-6-7-10, the TH-1698_11610311
workpiece is in tight contact fixture and copper backing plate, having high thermal conductivity.
Therefore, these interfaces are assigned with a high heat transfer coefficient.
Heat source model
The modelling of heat source helps to accurately predict the required weld pool shapes and temperature profiles. It is noteworthy that the surface heat flux does not produce adequate weld penetration until a well-defined heat source model is defined as in welding process the power of the beam decreases with increasing depth of the penetration, which should be taken into account in numerical modelling. The heat source is considered as Gaussian distribution over the volume such that flux density decreases in depth of penetration. However, to provide a better representation of weld bead at higher heat input, an hourglass-like heat source model is more suitable to describe the unique shape of laser microwelding instead of commonly used conical heat source model. As the heat input is increased more energy is deposited; an equivalent heat source is defined in the volume by finding a satisfactory spatial distribution. The simplest method is to identify the distribution from the weld bead shape. In the present model, the general heat source model is defined by:
̇( ) ( ) 0 ( )
1 , - (3.5) where reff is the effective radius of the heat source falling on the top surface, and U is the laser scanning velocity along x direction, is the thickness of the plate, and q0 is the laser intensity at the center of the beam, which is defined as:
(3.6)
where Ptotal is the total power and p is the weld depth and η is the absorption coefficient, d is the power density distribution factor. For micro plasma welding, the maximum intensity is expressed as:
η
(3.7)
where V is the voltage, I is the current,η is the arc efficiency and reff is the arc radius at the workpiece surface. In Eq. (3.5), h(t) shows the temporal variation of intensity. In case of
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continuous welding h(t) is 0 for the whole cycle. In the pulse laser, the value of h(t) is equal to 1 at pulse on-time and else zero during pulse off-time. The frequency of the welding determines the number of time steps that the pulse was on. However, it is revealed from the laser micro welding from the macrograph that the shape of weld pool of the full penetrated welds cannot be accurately characterized by the assumed heat source when the heat input is increased, especially in the bottom of welds. To represent the shape of full penetration welds, an hourglass like heat source with Gaussian power density distribution is developed. The schematic diagram of hourglass-like heat source model is shown in Fig. 3.2 where the volumetric shape consists of two combined frustums. Thus the volumetric heat source distribution becomes:
̇( ) , ( )- ( ), - ( ) 0 ( )
1; for upper cone ( )
(3.8)
̇( ) , ( )- ( ), - ( ) 0 ( )
1; for lower cone ( )
(3.9)
where z and r represent the z-axis coordinate and the radial distance from the axis, the subscript
“t”, “m” and “b” represent the top surface, middle surface, and bottom surface of the heat source, respectively. The effective radius of the cones is defined as:
Upper truncated cone: ( ) (3.10) Lower truncated cone: ( ) (3.11)
Fig. 3.2 Hourglass like heat source model.
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