Results and Discussions
5.2 Calibration of Numerical Model
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respond to the thermal load in a proper way leading to wrong results. Whereas, too fine mesh in the model leads to a non-solvable problem due to the limited capacity of computer. Therefore, choosing optimum mesh for the model is one of the important steps to get reliable and accurate results.
For the heat-transfer analysis, the maximum temperature during welding was chosen to be the key parameter for the analysis [Goldak and Akhlaghi, 2005]. The model is meshed non- uniformly to minimize the simulation time by reducing the total number of nodes. Fine meshes are created in the region near the heat flux region while mesh size decreases gradually towards the outer portion of the model. Figure 5.4 shows a typical mesh considered for 3D heat transfer analysis. As the heat source is symmetric in the transverse direction, only a half section is considered for analysis where the original weld interface acts as the plane of symmetry. The model is divided into three parts based on mesh sizes. Constant mesh of smaller elements of size 0.10 mm x 0.10 mm with ten elements along the thickness is adopted upto 5 mm from the centre of the heat source as shown in Fig. 5.5 (upto the line 5-6) for micro plasma welding while 0.05 mm x 0.05 mm for laser microwelding. The mesh is progressively made coarser away from the beam to reduce the total number of nodes and elements and hence, the computational time. The solid element type used in the thermal analysis is an 8-node brick DC3XD8 diffusive heat transfer. The mechanical analysis employs similar geometry and meshing of thermal model. 8- noded linear brick C3D8 elements with reduced integration are used for mechanical analysis.
5.2.2 Selection of time step
The transient analysis in numerical modelling is required to simulate the micro welding which provides a natural way of coping with the non-linearity associated with the process. The transient analysis is carried out by dividing the total solution time into a large number of time increments. The time increment in a transient analysis can be controlled directly, or the software can control it automatically. The time increments can be selected automatically based on a user- prescribed maximum allowable nodal temperature change in an increment, ∆θmax. The software (ABAQUS) will restrict the time increments to ensure that this value is not exceeded at any node (except nodes with boundary conditions) during any increment of the analysis. If ∆θmax is not
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specified, fixed time increments equal to the user-specified initial time increment, ∆t0, will be used throughout the analysis.
The converged solutions of temperature are henceforth attempted in each of these small time steps. The computational time and the rate of convergence are greatly influenced by the choice of time step in transient analysis. In transient analysis with second-order elements there is a relationship between the minimum usable time increment and the element size. The integration procedure used in ABAQUS for transient heat transfer analysis introduces a relationship between the minimum usable time increment, element size and the material properties like
( n)2 c
d C
t k
(5.2)
wheredn is the minimum distance between the nodes, tc the critical time step, and k, and C represent the thermal conductivity, density and specific heat respectively at room temperature.
If time increments smaller than this value are used in a mesh of second-order elements, spurious oscillations can appear in the solution, in particular in the vicinity of boundaries with rapid temperature changes [Hughes, 1977]. These oscillations are nonphysical and may cause problems if temperature-dependent material properties are present. In transient analyses using first-order elements the heat capacity terms are lumped, which eliminates such oscillations but can lead to locally inaccurate solutions for small time increments. If smaller time increments are required, a finer mesh should be used in regions where the temperature changes rapidly. It is worth noting that there is no upper limit on the time increment size (the integration procedure is unconditionally stable) unless nonlinearities cause convergence problems.
According to Eq. (5.2), the critical time steps for minimum nodal distance ∆dn = 0.1 mm and 0.05 mm, are 3.0 x10-3 s and 6.78 x10-4 s respectively for Ti6al4V. In the present analysis, the time step for micro plasma welding is considered as 1.0 x10-3 s and for laser welding it is taken as 5.0 x10-4 s.
5.2.3 Non-dimensional heat input index
The non-dimensional heat input index considers the combined effect of process parameters and thermal properties of substrate material and is defined by [De and Debroy, 2005]:
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( ) (5.3)
where V, I, , reff, U, , Cp, TL, To, and L represent the voltage, current, efficiency, effective arc radius, weld velocity, density of the material, specific heat capacity of the material, liquidus temperature of the material, ambient temperature, and latent heat of fusion, respectively. In Eq.
(5.3), the numerator can be expected as the whole incident welding heat source per unit volume.
The denominator relates to the enthalpy which is necessary to heat the unit volume of the material from ambient temperature to liquidus temperature. Thus, the parameter embodies the combined effect of the welding process conditions and the material properties for the formation of weld pool. However to calculate the NHI, the superheating of the substrate material beyond the melting is ignored. In several occasions, this non-dimensional heat input index is used to explain the numerical and experimental results.