By H. K. D. H. Bhadeshia, H. Cerjak
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Additional info for B0594 Mathematical modelling of weld phenomena 2
These are generally low index directions of the underlying crystal structure such as (100) in cubic materials. 15 this fact changes the growth velocity given in Equation 3 to ~kl = ~ cos eI cos 'V (16) where Vhkl is the velocity of the dendrite trunk of crystallographic orientation [hklJ and is the angle between the solidification front normal and [hklJ, as 'I' 48 Mathematical Modelling of Weld Phenomena 2 b) c) t = t+~t t rVhkd Figure 4. Relationships growth velocity. between 6t cos = t tb = Ivs 16t beam velocity, solidification front velocity and dendrite given in Fig.
According to Aziz8 the growth rate dependent solute distribution coefficient, for dilute solutions, is given by the relationship: (10) where ko is the equilibrium distribution coefficient, Pi (= ao VI D) is the interface Peclet number, ao is an atomic jump distance. The thermodynamics of non-equilibrium processes have been considered by Boettinger and Coriell,9 who obtain, for dilute solutions: v _ ml-m [ 1+ ko - kv(l -In(k/~))] l-ko (11) where m is the slope of the equilibrium liquidus which is assumed to be linear.
Therefore the first phase to grow is 'Yif nucleation allows it. At the bottom of Fig. 5 at%Ni is indicated (always under the assumption of a negligibly small nucleation barrier). The highest growth velocities calculated (end of the T-V curves for dendrites) represent the limit of absolute stability which might be reached in rapid welding processes. 6 1 V (m/s) Figure 8. 5 at% Ni). 5 at%Ni. All the important effects known from austenitic steels can be found here. 4 at%Ni. 20 CONCLUSIONS The discussion of phase and microstructure selection mechanisms shows that under rapid welding conditions some of the well established facts have to be revised.