  In the setting of continuum elasticity, phase transformations involving
martensitic variants are modeled by a free energy density function that is
non-convex in strain space. Here, we adopt an existing mathematical model in
which we regularize the non-convex free energy density function by higher-order
gradient terms at finite strain and derive boundary value problems via the
standard variational argument applied to the corresponding total free energy,
inspired by Toupin's theory of gradient elasticity. These gradient terms are to
preclude existence of arbitrarily fine microstructures, while still allowing
for existence of multiple solution branches corresponding to local minima of
the total free energy; these are classified as metastable solution branches.
The goal of this work is to solve the boundary value problem numerically in
three dimensions, observe solution branches, and assess stability of each
branch by numerically evaluating the second variation of the total free energy.
We also study how these microstructures evolve as the length-scale parameter,
the coefficient of the strain gradient terms in the free energy, approaches
zero.

Numerical Analysis