  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 and strain causes segregation into phases with different
crystal structures. We work on an existing model that couples the classical
Cahn-Hilliard model with Toupin's theory of gradient elasticity at finite
strains. Both systems are represented by fourth-order, nonlinear, partial
differential equations. The goal of this work is to develop unconditionally
stable, second-order accurate time-integration schemes, motivated by the need
to carry out large scale computations of dynamically evolving microstructures
in three dimensions. We also introduce reduced formulations naturally derived
from these proposed schemes for faster computations that are still second-order
accurate. Although our method is developed and analyzed here for a specific
class of mechano-chemical problems, one can readily apply the same method to
develop unconditionally stable, second-order accurate schemes for any problems
for which free energy density functions are multivariate polynomials of
solution components and component gradients. Apart from an analysis and
construction of methods, we present a suite of numerical results that
demonstrate the schemes in action.
