Equilibrium Shapes Elastic and Interfacial Energy

Two-Dimensional Analysis of Johnson and Cahn. For starters, we consider the equilibrium shape for second-phase particles in a two-dimensional setting. 

From the standpoint of the quantitative model used to describe this system, we assume that both the matrix and precipitate are elastically isotropic and characterized by the same elastic moduli and that the misfit strain associated with the precipitate is purely dilatational and is given by Further, it is assumed that the interfacial energy y,n (n) is isotropic (i.e. yint(n) = yo). Recall from 

In terms of these new variables, the geometric description of the elliptical particles is given by 

In addition to the interfacial energy, the competition at play here depends also on the elastic energy associated with the second-phase particle. Using ideas quite similar to those introduced already in the context of the Eshelby inclusion the elastic energy may be written as 

Given our knowledge of both the elastic and interfacial energies, the total energy as a function of the particle size and shape parameters now has the character of a Landau expansion like that seen in the theory of phase transitions. In particular, we 

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