Published Paper

Inserted: 11 jun 2020
Last Updated: 13 jan 2025

Journal: Advances in Calculus of Variations
Volume: 17
Number: 3
Pages: 673-725
Year: 2024
Doi: https://doi.org/10.1515/acv-2022-0053

Abstract:

The SDRI model introduced in "Sh. Kholmatov, P. Piovano, Arch. Rational Mech. Anal. (2020)" in the framework of the theory on Stress-Driven Rearrangement Instabilities for the morphology of crystalline material under stress is considered. In agreement with the literature on SDRI, a mismatch strain, rather than a Dirichlet condition as in "V. Crismale - M. Friedrich, Arch. Rational Mech. Anal. (2020)", is included into the analysis to represent the lattice mismatch between the crystal and possible adjacent (supporting) materials. The existence of solutions is established in dimension two in the absence of graph-like assumptions and of restrictions to a finite number $m$ of connected components for the free boundary of the region occupied by the crystalline material, thus extending previous results for epitaxially strained thin films and material cavities. Due to the lack of compactness (and of lower semicontinuity) for the sequences of $m$-minimizers, i.e., minimizers among configurations with at most $m$ connected boundary components, a minimizing candidate is directly constructed, and then shown to be a minimizer by means of uniform density estimates and the convergence of $m$-minimizers' energies to the energy infimum as $m\to\infty$. Finally, regularity properties for the morphology of all minimizers are established.

Keywords: existence, regularity, Thin films, SDRI, crystal cavities, minimal configurations, elastic energy , surface energy, fractures, interface instabilities, density estimates

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