Mechanical properties of high- temperature beryllium intermetallic compounds
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=
-- -2
t rdV_ orijEij
2
t ~r~
M
- u/g] dE
[3]
for all the cases we consider, where, following Eshelby, E denotes the mathematical bounding surface that separates the precipitate from the matrix. Thus, we see that if the precipitate is coherent at all interfaces, since u ~ = u~ on • (BC 2), Eq. [3] reduces to Eq. [1], as previously shown by Eshelby. [71 If we consider a sliding interface (BC's 1, 3 and 4), then we can use B C ' s 3 and 4 to show again that Eq. [3] reduces to Eq. [1], a result that has been obtained by Mura et al. 12~ Lastly, if the interface is part coherent/part noncoherent, then it is clear from the above arguments that since BC 2 holds on the coherent regions and B C ' s 3 and 4 hold on the noncoherent regions, Eq. [3] will reduce to Eq. [1]. Therefore, it follows that the elastic strain energy of a precipitate scales with its volume in all three cases. Also, for a dilatational transformation strain (e/~ = ert~ij), the elastic strain energy scales with the average value of the trace of the stress tensor in the precipitate in all three cases. Mura and Furuhashi 181 have shown that a spheroidal sliding precipitate undergoing constant transformation strains has stresses that are not constant within the precipitate. This is in contrast to the fully coherent case. They also have found that the average stress is less than in the corresponding fully coherent precipitate. Thus, the intuitive expectation of a reduction in strain energy from the fully coherent to the sliding precipitate remains intact. Ideally, the elastic energies (or equivalently, the average stresses) should be calculated as a function of the progressive loss of coherency. These calculations will undoubtedly be predominantly numerical. It would, therefore, seem worthwhile to consider simple geometries in two and then three dimensions, both to test the possibility of finding analytic solutions and to provide a check upon any numerical scheme. Work on these problems is in progress. [211
We would like to thank S. Dregia for helpful discussions and to acknowledge the support of the Division of Materials Research of NSF, RVR and HIA through Grant No. DMR 86-15997 and PHL through Grant No. D M R 84-09397. REFERENCES 1. I. Jasiuk, E. Tsuchida, and T. Mura: Int. J. Sol. Structures, 1987, vol. 23, pp. 1373-85. 2. F. Ghahremani: Int. J. Sol. Structures, 1980, vol. 16, pp. 825-45. 3. F. Ghahremani: Int. J. Sol. Structures, 1980, vol. 16, pp. 847-62. 4. M.A. Hussain and S.L. Pu: J. Appl, Mech., 1971, vol. 38, pp. 627-33. 5. R. Sankaran and C. Laird: Acta Metall., 1974, vol. 22, pp. 957-69. 6. H.I. Aaronson, C. Laird, and K.R. Kinsman: Phase Transformations, ASM, Metals Park, OH, 1970, p. 313. 7. J.D. Eshelby: Prog. Sol. Mech., 1961, vol. 2, pp. 89-140.
METALLURGICALTRANSACTIONS A
8. T. Mura and R. Furuhashi: J. Appl. Mech., 1984, vol. 51, pp. 308-10. 9. P.H. Leo: Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1987. 10. B.L. Karihaloo and K. Viswanathan: J. Appl. Mech., 1985, vol. 52, pp. 835-40. 11. J.D. Eshelby: Annln. Physi
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