Symmetry-breaking transitions in equilibrium shapes of coherent precipitates: Effect of elastic anisotropy and inhomogeneity
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We examine the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two-dimensional (2-D) systems under a plane-strain condition with the principal misfit strain components ε*xx and ε*yy. For systems with cubic elastic moduli, we first show all the shape transitions associated with different values of t=ε*yy/ε*xx. We also characterize each of these transitions, by studying its dependence on elastic anisotropy and inhomogeneity. For systems with dilatational misfit (t=1) and those with pure shear misfit (t=−1), the transition is from an equiaxed shape to an elongated shape, resulting in a break in rotational symmetry. For systems with nondilatational misfit (−1<t<1; t ≠ 0), the transition involves a break in mirror symmetries normal to the x- and y-axes. The transition is continuous in all cases, except when 0<t<1. For systems which allow an invariant line (−1≤t<0), the critical size increases with an increase in the particle stiffness. However, for systems which do not allow an invariant line (0<t≤1), the critical size first decreases, reaches a minimum, and then starts increasing with increasing particle stiffness; moreover, the transition is also forbidden when the particle stiffness is greater than a critical value.
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