Application of microstructure modeling to the kinetics of proeutectoid ferrite transformation in hot-rolled microalloyed
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tive. Assume that the ferrite grains have a cylindrical shape, such that the base of the cylinder is along the austenite grain boundary on which the given fetrite grain nucleates, and the axis of the cylindrical ferrite grain is perpendicular to the same austenite grain boundary. Each ferrite grain reaches a certain radius (say R0) soon after its nucleation and subsequently its radius does not change with time (this is due to one-dimensional mode of growth as austenite grain boundaries get completely covered with ferrite). During onedimensional growth the length of each grain increases with time, whereas the radius R0 remains constant. Assume that the length L of ferrite grains increases with time t in a parabolic manner described by the following equation: L = 2a(t - to) 1/2
[1]
where a is the parabolic rate constant. The parameter to is the delay time. It includes the incubation period for initiation of the transformation, and the time required to reach site saturation and one-dimensional mode of growth. Note that t 0 - - 0 . Thus, it is assumed that most of the transformation occurs by 'thickening' of the existing ferrite grains after the austenite grain boundaries are completely covered by ferrite. I~t Nv be the number of ferfite grains per unit volume at the end of saturation of the nucleation sites. The 'extended' (i.e., in absence of geometrical impingement) volume fraction of ferrite Vv,, and the 'extended' surface area of austenite-ferrite interfaces per unit volume Sv,x are given by: - t0) 1'2
Vv,~ = 2r
[2]
and Sve~ = 27rR~Nv + 47rRoNva(t - to) 1/2
[3]
To calculate the 'real' volume fraction of ferrite Vv and surface area of austenite-ferrite interfaces per unit volume Sv for the modeled structure, it is necessary to account for the impingement. As the nucleation of ferrite is predominantly on the austenite grain boundaries, the spatial distribution of ferrite nuclei is 'clustered' (and not random). Thus, the classical Johnson-Mehl equation 3 is not applicable. The theoretical models for nonrandom clustered impingement are not sufficiently well developed. Hillert 4 has suggested the following phenomenological equation for nonrandom impingement: dVv = (l - Vv)i.dVv,,
[4]
where the exponent i > 1 for clustered impingement. The value i = 2 has been utilized to understand microstructural evolution during recrystallization of Fe-Si alloy, s recrystallization of Ti-alloy, 6 austenitization kinetics of low carbon steel, 7 and austenitization kinetics of spheroidized cementite and ferrite aggregates. 8 In all of these, the distribution of the product phase nuclei is clustered. Thus, for the present analysis also, it is assumed that Eq. [4] is applicable and the exponent i is equal to two. Thus, dVv = (1 - Vv)2"dVv,,
[5]
Integrating Eq. [5] yields: A.M. GOKHALE is Assistant Professor, Department of Metallurgical Engineering, Indian Institute of Technology, Kanpur-208016 (U.P.), India. Manuscript submitted February 21, 1985. METALLURGICALTRANSACTIONSA
Vv 1 - Vv-
Vv,,
[6]
VOLUME 17A, SEPTEMBER1986-- 1
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