7.2.1 Governing equations and boundary conditions
The step-by-step development of the three-dimensional coupled thermo-mechanical model of the zinc coated brass wire is the same as for the molybdenum wire, which is explained in detail in chapter 3. The governing equations, boundary conditions, and the heat flux equation employed in the numerical model have been depicted in Table 7.1.
Table 7.1 Governing equations and boundary conditions incorporated in the model Governing
equations
Thermal analysis
2 2 2
2 2 2 2
1 1 1
T T T T T T
r r r r z n z t
(7.1) Mechanical analysis
Equilibrium equations
ij bi 0, ij ji
(7.2) Constitutive equation
[d] ([ De] [ Dp])[ ] [d K dTth] (7.3) von-Mises criterion
2 2 2 2
1 2 2 3 1 3
1 [( ) ( ) ( ) ]
2 y
(7.4) Boundary
conditions
Thermal analysis
( ) for w
k T Q r r R
r
(7.5a)
for w
k T h r R
r
(7.5b) Mechanical analysis
Pin end joints were applied at both ends of the wire to avoid movement. The wire surface was applied with no mechanical constraints.
Spark radius R (2.04 10 ) 3 I0.43ton0.44(m) (7.6) Heat flux on the
wire electrode
2
2 2
( ) 4.57 F V Ic exp 4.5 r
Q r R R
(7.7) Fc = 0.183
Convective heat transfer coefficient
h = 10,000 W/m2K
Initial temperature 300 K
7.2.2 Material properties
In the developed thermo-mechanical finite element model, temperature dependent material properties have been considered for both the brass core and the outer zinc coating. The general properties of brass and zinc have been listed in Table 7.2. The thermal and mechanical properties of zinc and brass are shown in Figures 7.1–7.5, according to Table 7.3.
Table 7.2 General properties of zinc and brass
Property Values for zinc
coating Values for brass core
Density(kg/m3) 7130 8800
Poisson’s ratio 0.245 0.331
Coefficient of thermal expansion (µm/m-ºC) 34 18
Melting point (K) 693 1203
Table 7.3 Thermal and physical properties of zinc and brass
Property Figure nos. for zinc coating Figure nos. for brass core
Thermal conductivity (W/mK) 7.1 7.3
Specific heat (J/kgK) 7.1 7.3
Stress-strain curve 7.2 7.4
Young’s modulus –* 7.5
* The Young’s modulus of zinc is taken at a constant value of 7x1010 Pa.
Figure 7.1 Thermal conductivity (Powell and Childs (1973)) and specific heat (Valencia
and Quested (2008)) of zinc
Figure 7.2 Stress strain curve of zinc (Liu et al. (2016))
Figure 7.3 Thermal conductivity and specific
heat of brass (Valencia and Quested (2008)) Figure 7.4 Stress strain curve of brass (Reed and Mikesell (1967))
Figure 7.5 Young’s modulus of brass (Reed and Mikesell (1967))
7.2.3 Process parameters used for the model
In the present model, four input parameters viz. discharge voltage, discharge current, pulse on-time and pulse off-time were varied at different levels. The process parameters have been selected based on the facilities available in our workshop. The levels of these input parameters have been kept at the same levels as for the molybdenum wire in order to compare the behaviour of both types of wires. The levels chosen for each process parameters are shown in Table 7.4.
Table 7.4 Process parameters and their levels Factors Level 1 Level 2 Level 3 Discharge voltage (V) 60 85 -
Discharge current (A) 4 6 8
Pulse on-time (µs) 4 8 16
Pulse off-time (µs) 2 4 6
The wire velocity (v) was kept constant at 6 m/s in the present model.
7.2.4 Overview of the process model development
A three-dimensional coupled thermo-mechanical model was developed for the zinc coated brass wire electrode in a similar manner as discussed in chapter 3 (Figure 3.8). Four input parameters viz. discharge voltage, discharge current, pulse on-time and pulse off-time were
given as input to the model, which predicts the temperature generated in the wire. Based on the temperature distribution, the stress distribution in the wire tool at the end of the pulse is predicted.
In the present model, a zinc coated brass wire with 250 µm diameter is considered for analysis. An outer zinc coating of 10 µm thickness over a brass core is considered as the model geometry (Figure 7.6). Figure 7.7 shows the process model of the zinc coated brass wire with the incorporated boundary conditions. The outer coated layer of zinc over the brass core was modelled using tie constraints in ABAQUSTM. Heat flux is applied at the spark location and the rest of the wire surface is applied with convection boundary condition. Both ends of the wire are hinged with pin joints and end tension is applied at both ends of the wire.
Figure 7.6 Cross-sectional view of the zinc coated brass wire model geometry
Figure 7.7 Process model geometry of the zinc coated brass wire
7.2.5 Finite element meshing of the wire geometry
The process model along with the boundary conditions was solved by using ABAQUSTM, a commercial FEM solver in transient mode. The wire geometry is discretized with eight-node hexahedral elements C3D8T. The meshed geometry of the coated wire is shown in Figure 7.8. The mesh was refined in the region where the heat flux is applied. The rest of the wire geometry was discretized with coarse mesh to reduce the computational time. Mesh optimization was performed for better convergence of results in optimum time. The overall mesh size was kept at 30 µm for the wire model. The heated and nearby regions were discretized using finer mesh size to get better accuracy of results. The mesh sensitivity analysis was carried out for a single process condition at discharge voltage (V) = 60 V, discharge current (I) = 6 A, pulse on-time (ton) = 16 µs, pulse off-time (toff) = 6 µs, wire velocity (v) = 6m/s. The heated region was discretized with a fine mesh of various sizes, which have been shown in Table 7.5. It was observed that the smallest change in temperature (0.099 %) occurs when the fine mesh size changes from 10 µm to 8 µm as compared to the other cases. Thus, the element of 8 µm size was selected as the optimum fine mesh size for the heated region to achieve better results. The moving heat source concept was used in the model to incorporate the effect of moving wire during the wire EDM machining process. The Gaussian heat flux moving at a constant velocity over the wire surface was employed using a
user subroutine DFLUX of ABAQUSTM. The transient heat transfer problem was solved by applying the heat flux at the location of spark and adopting an automatic time step increment.
Convection heat transfer is applied over the wire surface. The model was developed considering the automatic time step increment and the maximum allowable temperature change per increment was set to 10.
Figure 7.8 Mesh model of the wire electrode
Table 7.5 Mesh sensitivity analysis for the fine mesh in the heated region Serial no. Mesh size
(µm)
Temperature output (K)
Change in temperature (%)
CPU time (hrs.)
1 25 721 - 1.308
2 20 636.1 11.77 1.682
3 15 556.7 12.48 1.97
4 10 503.1 9.62 3.76
5 8 503.6 0.099 7.38