Equilibrium shapes of coherent, misfitting precipitate

We have used finite element and optimization techniques to study the equilibrium shapes of an isolated, misfitting precipitate by directly minimizing the sum of elastic and interfacial energies. We have used two dimensional systems under plane strain conditions. We have assumed isotropy for the interfacial energy, and a cubic or orthorhombic anisotropy for the elastic moduli. The principal misfit strain components can have the same or opposite signs. The particles are circular at small sizes (due to the dominance of an isotropic interfacial energy over the elastic energy), and take on an elongated shape at large sizes. When the principal misfits have an opposite sign, the direction of elongation is at an angle of &thetas; given by the Khachaturyan formula $&thetas;=tan^-1$ √- $ϵ_{xx}/ϵ_{yy}$. This angle is obtained even in systems with orthorhombic symmetr