Local minimizers and low energy paths in a model of material microstructure with a surface energy term

TL;DR

This paper analyzes a mathematical model of material microstructure, identifying conditions for local and global energy minimizers, and constructing low-energy transition paths, with implications for phase nucleation.

Contribution

It introduces a detailed analysis of minimizers and transition paths in a microstructure model with surface energy and well-depth parameters, revealing critical thresholds and energy costs.

Findings

Existence of a critical well-depth for global minimizers

Characterization of local minimizers with small perturbations

Construction of low energy transition paths between states

Abstract

A family of integral functionals F which, in a simplified way, model material microstructure occupying a two-dimensional domain D and which take account of surface energy and a variable well-depth is studied. It is shown that there is a critical well-depth, whose scaling with the surface energy density and domain dimensions is given, below which the state u=0 is the global minimizer of a typical f in the class F. It is also shown that u=0 is a strict local minimizer of f in the sense that if a non-zero v is admissible and either its L2 norm or the meaure of the subset of D where |v_{y}| exceeds 1 is sufficiently small (with quantitative bounds given in terms of the parameters appearing in the energy functional f) then f(v) > f(0). Low energy paths between u=0 and the global minimizer (in the case of a sufficiently large well-depth) are given such that the cost of introducing small regions where |v_{y}| is larger 1 (analogous to nucleation of martensite in austenite) into the domain D can be made arbitrarily small.