Solidification of pure metal in imperfect contact with a mold
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Vv -< V~
[36]
of nodules during eutectic solidification, it is reasonable to assume that the distribution of spherical austenite grains is lognormal as well, based on the experimental work of Wetterfall e t a / . [2] and previous micromodeling results which support the assumption of constant ratio of radius of austenite shell to radius of graphite nodule during eutectic solidification.
and g(t) = L~S~max ~0(V,.) v-~
V,,-> V~
[37]
These expressions can be incorporated in finite difference or finite element numerical schemes to simulate the solidification and microstructural evolution of equiaxed eutectic alloys for which continuous nucleation results in a lognormal distribution of equiaxed grains and the growth kinetics expression given in Eq. [9] is applicable. The principal assumption in the derivation is a timedependent lognormal distribution of equiaxed spherical grains for which the distribution parameters at any instant in time prior to grain impingement are determined with knowledge of/)(t), Nv(t), and V,,(t). A characteristic property of this distribution is that a plot of cumulative relative frequency vs grain size yields a straight line when plotted on log-probability graph paper. Figure 1 is such a plot of the cumulative frequency of nodule diameters obtained from a slightly hypereutectic ductile iron thermal analysis specimen of composition pct C = 3.63 and pct Si = 2.54. The plot consists of several straight line regions, indicating that the nodule size distribution consists of a mixture of lognormal distributions. The mixture presumably arises as a consequence of distinct kinetics of nucleation and growth of graphite growth nodules during primary and eutectic solidification. In the case of primary solidification, graphite nodules grow in direct contact with the melt, while during eutectic solidification, the nodules are enveloped by austenite shells. As the results in Figure 1 suggest a lognormal distribution
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The support of the National Science Foundation under Grant No. DDM-9102682 is gratefully acknowledged. REFERENCES 1. M. Rappaz: International Materials Reviews, 1989, vol. 34 (3), pp. 93-123. 2. S.E. Wetterfall, H. Fredriksson, and M. Hillert: J. Iron Steel hzst., 1972, pp. 323-33. 3. T. Owadano, K. Yamada, and K. Torigoe: Trans. Jpn. Ins. Met., 1977, vol. 18, pp. 871-78. 4. H. Fredriksson and I. Svensson: Mater. Res. Soc. Syrup. Proc., 1985, vol. 34, pp. 273-84. 5. K. Su, I. Ohnaka, I. Yamauchi, and T. Fukusako: Mater. Res. Soc. Syrup. Proc., 1985, vol. 34, pp. 181-89. 6. G. Lesoult: 4th Int. Conf. on the Physical Metallurgy of Cast Iron, Tokyo, 1989, pp. 413-22. 7. M. Rappaz, J.D. Richoz, and Ph. Thevoz: Euromat 89, E-MRS, RFA, Aachen, Germany, 1989, pp. 1-6. 8. S. Chang, D. Shangguan, and D.M. Stefanescu: Metall. Trans. A, 1992, vol. 23A, pp. 1333-46. 9. A.M. Gokhale, C.V. Iswaran, and R.T. DeHoff: Metall. Trans. A, 1979, vol. 10A, pp. 1239-45. 10. R.T. DeHoff: PracticalApplicationsofQuanti
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