Koki Sagiyama

Koki Sagiyama, Krishna Garikipati

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 microsstructures, and following solution branches. Stable and 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.

Koki Sagiyama, Shiva Rudraraju, Krishna Garikipati

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.