Unconditionally stable, second-order schemes for gradient-regularized, non-convex, finite-strain elasticity modeling martensitic phase transformations

In the setting of continuum elasticity martensitic phase transformations are characterized by a non-convex free energy density function that possesses multiple wells in strain space and includes higher-order gradient terms for regularization. Metastable martensitic microstructures, defined as solutions that are local minimizers of the total free energy, are of interest and are obtained as steady state solutions to the resulting transient formulation of Toupin's gradient elasticity at finite strain. This type of problem poses several numerical challenges including stiffness, the need for fine discretization to resolve microstructures, and following solution branches. Accurate time-integration schemes are essential to obtain meaningful solutions at reasonable computational cost. In this work we introduce two classes of unconditionally stable second-order time-integration schemes for gradient elasticity, each having relative advantages over the other. Numerical examples are shown highlighting these features.