High-cycle fatigue of an investment cast, Be-Al metal matrix composite
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[6]
4 1 [(r ⫹ ⌬r)3 ⫺ r 3] ␣ ⫻ xB␣ (r) 3 Vm
where V m and V m␣ are the molar volumes of AmBn and ␣. Neglecting all terms involving ⌬r 2 and ⌬r 3 in Eq. [6] gives ⌬NB ⬇ 4 r 2 ⌬r
冢
xB␣ (r) n  ⫺ Vm V m␣
冣
冢
dNB n ⬇ ⫺ dt V m
冣 4 r
2
dr dt
冢
冣
[8] BRIAN J. SMITH
[9]
This is the basic equation used to formulate the coarsening kinetics of AmBn particles in the previous analysis[1] except that the term xB␣ (r) was given as xB␣ (0) before. Owing to this correction, the term xB␣ (0) appearing in Eqs. [9] through [11] and Eq. [15] in the previous analysis[1] should be replaced by xB␣ (⬁) in the latter stage of coarsening. Fortunately, this correction only slightly influences the acceleration parameter k as the difference between xB␣ (⬁) and xB␣ (0) is small for a vanishingly small volume fraction of AmBn precipitates. In summary, the definitions of the molar volume quantities used in the previous analysis by the author[1] for a stoichiometric compound precipitate AmBn and a solid solution matrix (␣ ) are strict and appropriate. Furthermore, it is stricter and generally more useful to define the molar volume of a stoichiometric compound precipitate AmBn as the volume it occupies per mole of formula units than to define it as the volume it occupies per mole of A and B atoms. Equation [6] is valid in terms of the increase in the number of moles 1862—VOLUME 33A, JUNE 2002
1. M. Qian: Metall. Mater. Trans. A, 2000, vol. 31A, pp. 3195-97. 2. R. Doherty: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 1859-60. 3. S.K. Bhattacharyya and K.C. Russell: Metall. Trans., 1972, vol. 3, pp. 2195-99. 4. R.K. Trivedi: in Lectures on the Theory of Phase Transformations, H.I. Aaronson, ed., TMS-AIME, New York, NY, 1975, pp. 51-81. 5. C.J. Kuehmann and P.W. Voorhees: Metall. Mater. Trans. A, 1996, vol. 27A, pp. 937-43. 6. M. Qian: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 1283-87. 7. M. Hillert: Phase Equilibria, Phase Diagrams and Phase Transformations, Cambridge University Press, Cambridge, United Kingdom, 1998, p. 492. 8. P.W. Atkins: The Elements of Physical Chemistry, Oxford University Press, Oxford United Kingdom, 1992, p. 15. 9. M. Qian and L.C. Lim: Metall. Mater. Trans. A, 2000, vol. 31A, pp. 2659-61. 10. M. Qian and L.C. Lim: Scripta Mater., 1998, vol. 39, pp. 1451-55.
High-Cycle Fatigue of an Investment Cast, Be-Al Metal Matrix Composite
where dt is the time needed for the particle to grow from r to r ⫹ dr. The mass balance requirement yields, according to Eq. [5] of the previous analysis,[1] xB␣ (r) 4 R2D dxB␣ (R) n dr ⫺ ⫽ 4 r 2 V m␣ dR V m V m␣ dt
REFERENCES
[7]
Equation [7] can be written as xB␣ (r) V m␣
of the solute B considered. However, the author acknowledges that the term xB␣ (0) in Eq. [6][1] should have been xB␣ (r) and accordingly the term xB␣ (0) appearing in the acceleration parameter k of the previous analysis[1] should be replaced by xB␣ (⬁). Fortunately, this correction only slightly influences the value of k because the difference between xB␣ (0) and xB␣ (⬁) is small for a very low vol
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