The influence of elastic strain energy on the formation of coherent hexagonal phases
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THE volume change which occurs during the formation of coherent second phases gives rise to coherency strains and an associated elastic strain energy. This energy, along with the interfacial boundary energy, is in general strongly anisotropic and thus shape dependent. In contrast, the volume free energy change depends only on the volume of the particle and is shape independent. Thus, the orientation and morphology of a second phase will form to minimize the sum of the elastic strain energy and the interfacial boundary energy. When considering the formation of large precipitates, it is usually valid to neglect the interracial energy altogether. Consequently, only the elastic strain energy is needed to predict precipitate orientation and morphology. Eshelby ~-3 was able to calculate the elastic strain energy of a coherent elliposoidal precipitate by using the assumption of isotropic elasticity. Kinoshita and Mura, 4 Walpole: Asaro and Barnett 6 have extended Eshelby's approach to consider anisotropic media, but still subject to the restriction of an ellipsoidal shape. Using an alternate formulation, Khachaturyan v arrived at a solution for the general case of a coherent inclusion with arbitrary shape in an anisotropic medium. Using this formulation, the growth of G P zones in A1-Cu and Cu-Be alloys, s of F e l 6 N 2 precipitates in Fe-N martensite, 9 and of martensite plates in the Fe-C system l~ have been correctly described. In the present work, the elastic strain energy function for the general case of a coherent precipitate with an orthotropic structure is considered following Khachaturyan's approach. These equations are then simplified for the case of hexagonal crystals where great use has been made of the transverse isotropy of the elastic
constants and the equivalent stress-free transformation strains. Applications of the model are then made to spinoidal decomposition of hexagonal phases and to the precipitation of the metastable and equilibrium MgZn 2 phases in the A1-Mg-Zn system. THEORY The first calculation of the elastic strain energy of a coherent precipitate was made by Eshelby t-3 for the case of an ellipsoidal inclusion in an isotropic medium. Subsequent work by other investigators ~ brought fruitful results to the theory of phase transformations for the more general case of anisotropic materials with elastic constant differences between the inclusions and the matrix. However, most of these energy formulations are in real space and as a result, the calculations are limited to inclusions of simple geometrical shapes such as ellipsoidals. A general solution of a coherent precipitate of arbitrary shape was given by A. Khachaturyan, v who considered a Fourier representation of the elastic energy. The only assumption in his treatment is that both inclusion and matrix have the same elastic constants. Khachaturyan's elastic energy representation is: += E
d3k
=
(2,n.) 3
2 - -
where n = k/ Lk], k = wave vector, Y(n) = elastic energy function independent of the shape of the inclusion, 10(k)l 2 = shape fac
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