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

TL;DR

This paper numerically investigates the branching and stability of solutions in three-dimensional martensitic phase transformations modeled with gradient-regularized, non-convex finite strain elasticity, revealing microstructure evolution and metastability.

Contribution

It introduces a numerical approach to analyze solution branches and stability in 3D martensitic transformations with gradient regularization, extending previous models.

Findings

Multiple solution branches corresponding to local energy minima are identified.

Stability of solution branches is assessed via second variation of free energy.

Microstructure evolution is studied as gradient length-scale parameter approaches zero.

Abstract

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.