Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition

摘要

We consider solid state phase transformations that are caused by free energydensities with domains of non-convexity in strain-composition space; we referto the non-convex domains as mechano-chemical spinodals. The non-convexity withrespect to composition and strain causes segregation into phases with differentcrystal structures. We work on an existing model that couples the classicalCahn-Hilliard model with Toupin's theory of gradient elasticity at finitestrains. Both systems are represented by fourth-order, nonlinear, partialdifferential equations. The goal of this work is to develop unconditionallystable, second-order accurate time-integration schemes, motivated by the needto carry out large scale computations of dynamically evolving microstructuresin three dimensions. We also introduce reduced formulations naturally derivedfrom these proposed schemes for faster computations that are still second-orderaccurate. Although our method is developed and analyzed here for a specificclass of mechano-chemical problems, one can readily apply the same method todevelop unconditionally stable, second-order accurate schemes for any problemsfor which free energy density functions are multivariate polynomials ofsolution components and component gradients. Apart from an analysis andconstruction of methods, we present a suite of numerical results thatdemonstrate the schemes in action.