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In the setting of continuum elasticity, phase transformations involvingmartensitic variants are modeled by a free energy density function that isnon-convex in strain space. Here, we adopt an existing mathematical model inwhich we regularize the non-convex free energy density function by higher-ordergradient terms at finite strain and derive boundary value problems via thestandard variational argument applied to the corresponding total free energy,inspired by Toupin's theory of gradient elasticity. These gradient terms are topreclude existence of arbitrarily fine microstructures, while still allowingfor existence of multiple solution branches corresponding to local minima ofthe total free energy; these are classified as metastable solution branches.The goal of this work is to solve the boundary value problem numerically inthree dimensions, observe solution branches, and assess stability of eachbranch 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, approacheszero.

A numerical study of branching and stability of solutions to  three-dimensional martensitic phase transformations using  gradient-regularized, non-convex, finite strain elasticity