On the energy scaling behaviour of singular perturbation models involving higher order laminates
Motivated by complex microstructures in the modelling of shape-memory alloys and by rigidity and flexibility considerations for the associated differential inclusions, in this article we study the energy scaling behaviour of a simplified $m$-well problem without gauge invariances. Considering wells...
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| Main Authors: | , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
2022
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| Edition: | Version V3 |
| In: |
Arxiv
Year: 2022, Pages: 1-47 |
| DOI: | 10.48550/arXiv.2110.15929 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2110.15929 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2110.15929 
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| Author Notes: | Angkana Rüland and Antonio Tribuzio |
| Summary: | Motivated by complex microstructures in the modelling of shape-memory alloys and by rigidity and flexibility considerations for the associated differential inclusions, in this article we study the energy scaling behaviour of a simplified $m$-well problem without gauge invariances. Considering wells for which the lamination convex hull consists of one-dimensional line segments of increasing order of lamination, we prove that for prescribed Dirichlet data the energy scaling is determined by the \emph{order of lamination of the Dirichlet data}. This follows by deducing (essentially) matching upper and lower scaling bounds. For the \emph{upper} bound we argue by providing iterated branching constructions, and complement this with ansatz-free \emph{lower} bounds. These are deduced by a careful analysis of the Fourier multipliers of the associated energies and iterated "bootstrap arguments: based on the ideas from \cite{RT21}. Relying on these observations, we study models involving laminates of arbitrary order. |
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| Item Description: | V1 29. Oktober 2021, V2 17. Januar 2022, V3 25. November 2022 (this version, v3) Gesehen am 14.02.2023 |
| Physical Description: | Online Resource |
| DOI: | 10.48550/arXiv.2110.15929 |