On a probabilistic model for martensitic avalanches incorporating mechanical compatibility

TL;DR

This paper introduces a probabilistic model for martensitic avalanches in shape-memory alloys that incorporates mechanical compatibility through convex integration, analyzing convergence, regularity, and microstructure fractality.

Contribution

It extends previous algorithms by including mechanical compatibility using convex integration, providing analytical and numerical insights into microstructure formation.

Findings

Algorithm converges under certain conditions

Solutions exhibit fractal microstructures

Numerical results align with theoretical predictions

Abstract

Building on the work in \cite{BCH15,CH18,TIVP17}, in this article we propose and study a simple, geometrically constrained, probabilistic algorithm geared towards capturing some aspects of the nucleation in shape-memory alloys. As a main novelty with respect to the algorithms in \cite{BCH15,CH18,TIVP17} we include \emph{mechanical compatibility}. The mechanical compatibility here is guaranteed by using \emph{convex integration building blocks} in the nucleation steps. We analytically investigate the algorithm's convergence and the solutions' regularity, viewing the latter as a measure for the fractality of the resulting microstructure. We complement our analysis with a numerical implemenation of the scheme and compare it to the numerical results in \cite{BCH15,CH18,TIVP17}.