Identification and range quantification of steel transformation products by transformation kinetics
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I.
JOHNSON and Mehl and Avrami developed the following equation independently about 50 years ago to describe isothermal transformation processes in steels: [1]
[2,3,4]
X 5 1 2 exp (2kt n)
[1]
where X is the volume fraction of the transformation product, t is the reaction time, k is the rate constant, and n is the curve shape constant. Both k and n are material constants under a given transformation condition. Equation [1] has two applications. First, the constants k and n can be extracted from experimental data and stored as the only values needed for reconstructing the original data set. Second, and more importantly, if k and n are known from a knowledge of steel composition and transformation temperature, the volume fraction of the product phase can be predicted. To date, Eq. [1] has not been used for the second purpose, since relationships between steel composition, transformation temperature, and the constants have not been able to be experimentally quantified. Research efforts to establish these relationships are few, since it is believed that a knowledge of k and n alone does not generally give useful information about transformation mechanisms. The studies[5–8] that attempted the correlations did not yield applicable results. The failure is attributed to the mask of any relationships due to the errors inherent from the use of this oversimplified kinetic equation. The following derivation shows the necessity of including the transformation incubation time to eliminate errors. Based on formal theory of transformation kinetics,[9] the volume of a nucleus of the new phase formed at time t' grows to the size vt' at time t (.t'), where L. FANG, Ph.D., former Postdoctoral Research Associate, Department of Materials Science and Engineering, Oregon Graduate Institute of Science and Technology, is now with Scientific Imaging Technologies, Inc. (SITe), Beaverton, Oregon. W.E. WOOD, Department Head and Professor, and D.G. ATTERIDGE, Professor, are with the Department of Materials Science and Engineering, Oregon Graduate Institute of Science and Technology, Beaverton, OR 97006. Manuscript submitted May 16, 1995.
METALLURGICAL AND MATERIALS TRANSACTIONS A
(t' , t) (t' ≥ t)
vt' 5 0 vt' 5 h Y1Y2Y3 (t 2 t')3
INTRODUCTION
[2]
h is a shape factor, Yi’s are the principal growth velocities in three mutually perpendicular directions, and t is the incubation time. From t' to t' 1 dt', the volume of the new phase increases by dVe 5 nt' V0n Idt'
[3]
where I is the nucleation rate per volume, V0 is the assembly volume, and Ve is the extended volume including volume of the new phase which nucleated and grew in transformed regions as well as untransformed regions. The net new phase volume increase in untransformed regions is v
dV 5 (1 2
V ) dVe V0
[4]
Therefore, dV 5 ntn Idt' [5] V0 2 V Integrating the left-hand side of Eq. [5] from 0 to V and, correspondingly, the right-hand side from 0 to t and setting X 5 V/V0 results in
*
ln (1 2 X) 5 2
t
0
0dt' 2
* hY Y Y t
t
1
2
n 3
I(t 2 t')3 dt'
[6]
The ability to
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