%0 Journal Article
%T Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition
%A Koki Sagiyama
%A Shiva Rudraraju
%A Krishna Garikipati
%J Physics
%D 2015
%I arXiv
%X We consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space. We refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition causes segregation into phases with different crystal structures. If, for one of these crystal structures, the free energy density is also non-convex with respect to strain, there is potential for the corresponding phase to further separate into multiple variants. For mathematical well-posedness the free energy description must be enhanced by interface terms that penalize gradients with respect to strain and composition. A system of PDEs results that couples the classical Cahn-Hilliard equation with those of gradient elasticity. Since the materials systems of interest display finite strains, the appropriate description is Toupin's theory of gradient elasticity at finite strains. The presence of strain and composition gradients in the free energy density leads to fourth-order PDEs in primal form, and requires greater continuity of basis functions, which we satisfy by adopting splines in the setting of IGA. This mechano-chemically coupled system of equations encompasses several simpler cases of phase transformations, and can be extended to others. Here, we develop a class of integration algorithms for time-dependent problems of phase transformations that are governed by the coupling of the Cahn-Hilliard and Toupin-gradient elasticity equations. Our goals are unconditional stability and second-order accuracy in time, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions and for general IBVPs. Apart from an analysis and construction of methods that satisfy these requirements, we present a suite of numerical results that demonstrate the schemes in action.
%U http://arxiv.org/abs/1508.00277v2