Symmetry-breaking transitions in equilibrium shapes of coherent precipitates

• Published in: Journal of the mechanics and physics of solids 2000-11, Vol.48 (11), p.2363-2389
• Main Authors: Jog, C.S., Sankarasubramanian, R., Abinandanan, T.A.
• Format: Article
• Language: English
• Subjects: A. Inclusions; B. Anisotropic material; B. Inhomogeneous material; C. Finite elements; C. Optimization; Condensed matter: structure, mechanical and thermal properties; Elasticity, elastic constants; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mechanical and acoustical properties of condensed matter; Mechanical properties of solids; Physics; Solid mechanics; Static elasticity; Static elasticity (thermoelasticity...); Structural and continuum mechanics
• ISSN: 0022-5096
• DOI: 10.1016/S0022-5096(00)00005-3

We present a general approach for determining the equilibrium shape of isolated, coherent, misfitting particles by minimizing the sum of elastic and interfacial energies using a synthesis of finite element and optimization techniques. The generality derives from the fact that there is no restriction on the initial or final shape, or on the elastic moduli of the particle and matrix, or on the nature of misfit. The particle shape is parametrized using a set of design variables which are the magnitudes of vectors from a reference point inside the particle to points on the particle/matrix interface. We use a sequential quadratic programming approach to carry out the optimization. Although this approach can be used to find equilibrium shapes of three-dimensional (3D) particles, we have presented the details of our formulation for two-dimensional systems under plane strain conditions. For systems with cubic elastic anisotropy, the equilibrium shapes and their size dependence are analyzed within the framework of symmetry-breaking shape transitions. For systems with dilatational misfit, our results on shape transitions are in agreement with those in the literature. For systems with non-dilatational misfit, we obtain a symmetry-breaking shape transition that involves a loss of mirror symmetry normal to the x- and y-axes; small particles have this symmetry, while those beyond a critical size do not. With these results, we now have a comprehensive picture of symmetry-breaking transitions in two-dimensional systems driven by anisotropy in misfit and elastic moduli.