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Energy barriers for boundary nucleation in a two-well model without gauge invariance

Antonio Tribuzio, Konstantinos Zemas

Abstract

We study energy scaling laws for a simplified, singularly perturbed, double-well nucleation problem confined in a half-space, in the absence of gauge invariance and for an inclusion of fixed volume. Motivated by models for boundary nucleation of a single-phase martensite inside a parental phase of austenite, our main focus in this nonlocal isoperimetric problem is how the relationship between the rank-1 direction and the orientation of the half-space influences the energy scaling with respect to the fixed volume of the inclusion. Up to prefactors depending on this relative orientation, the scaling laws coincide with the corresponding ones for bulk nucleation \cite{knupfer2011minimal} for all rank-1 directions, \textit{but} the ones normal to the confining hyperplane, where the scaling is as in a three-well problem in full space, resulting in a lower energy barrier \cite{Tribuzio-Rueland_1}.

Energy barriers for boundary nucleation in a two-well model without gauge invariance

Abstract

We study energy scaling laws for a simplified, singularly perturbed, double-well nucleation problem confined in a half-space, in the absence of gauge invariance and for an inclusion of fixed volume. Motivated by models for boundary nucleation of a single-phase martensite inside a parental phase of austenite, our main focus in this nonlocal isoperimetric problem is how the relationship between the rank-1 direction and the orientation of the half-space influences the energy scaling with respect to the fixed volume of the inclusion. Up to prefactors depending on this relative orientation, the scaling laws coincide with the corresponding ones for bulk nucleation \cite{knupfer2011minimal} for all rank-1 directions, \textit{but} the ones normal to the confining hyperplane, where the scaling is as in a three-well problem in full space, resulting in a lower energy barrier \cite{Tribuzio-Rueland_1}.
Paper Structure (14 sections, 6 theorems, 197 equations, 3 figures)

This paper contains 14 sections, 6 theorems, 197 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Variational theory of martensitic materials
  3. Relations to the literature and other quantitative results
  4. Notation and Preliminaries
  5. Setting and statement of the main result
  6. Lower bounds for the elastic energy
  7. Proof of Theorem \ref{['main_theorem_rescaled']}$(i)$
  8. The lower bound
  9. The upper bound
  10. The case of small inclusions
  11. The case of large inclusions (two-dimensional construction)
  12. Large inclusions in general dimension
  13. Proof of Theorem \ref{['main_theorem_rescaled']}$(ii)$
  14. Proof of properties \ref{['properties_of_tilting']}

Key Result

Theorem 2.1

Let $d\in\mathbb{N}$, $d\geq 2$, let $F=b\otimes n$ for some $b\in\mathbb{R}^d\setminus\{0\}$ and $n\in\mathbb{S}^{d-1}$, $\varepsilon,V>0$ and let $E_\varepsilon(V)$ be as in minimal_energy_fixed_volume. Then we have the following dichotomy:

Figures (3)

  • Figure 1: An illustration of the tilted cages defined in step (2a).
  • Figure 2: A representation of the inclusion domain $M$ and of the outer normal vectors to $M_1$ and $M_2$.
  • Figure 3: Building the tilted cages.

Theorems & Definitions (12)

  • Theorem 2.1: Scaling of the nucleation barrier
  • Theorem 2.2: Rotated and rescaled version of Theorem \ref{['main_theorem_energy_scaling']}
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Remark 4.4: Dichotomy of scalings
  • Remark 4.5
  • ...and 2 more