Austenite as a Local Minimizer in a Model of Material Microstructure with a Surface Energy Term

Abstract

A family of integral functionals $\mathcal{F}$ which model in a simplified way material microstructure occupying a two-dimensional domain $\Omega$ 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 $\mathcal{F}$. It is also shown that $u=0$ is a strict local minimizer of F in the sense that if $v \neq 0$ is admissible and either $||v||_{L^{2}(\om)}$ or $\mathcal{L}^{2}(\{(x,y) \in \Omega: |v_{y}|(x,y) \geq 1\})$ is sufficiently small (with quantitative bounds given in terms of the parameters appearing in the energy functional F), then $F(v) > F(0)$. Provided the well depth is sufficiently large, the existence of a so-called energy barrier between $u=0$ and the global minimizer of F is established under the assumption that paths $(v(t))_{0 \leq t \leq 1}$ connecting these two states obey $|v_{y}| \leq 1$ almost everywhere in the domain.