Convex integration solutions for the geometrically non-linear two-well problem with higher Sobolev regularity
In this article we discuss higher Sobolev regularity of convex integration solutions for the geometrically non-linear two-well problem. More precisely, we construct solutions to the differential inclusion $\nabla u\in K$ subject to suitable affine boundary conditions for $ u$ with $$ K:= SO(2)\left[...
Gespeichert in:
| Hauptverfasser: | , |
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| Dokumenttyp: | Article (Journal) Kapitel/Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
29 May 2019
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| In: |
Arxiv
Year: 2019, Pages: 1-33 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/1905.12521
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| Verfasserangaben: | Francesco Della Porta and Angkana Rüland |
MARC
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| 245 | 1 | 0 | |a Convex integration solutions for the geometrically non-linear two-well problem with higher Sobolev regularity |c Francesco Della Porta and Angkana Rüland |
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| 520 | |a In this article we discuss higher Sobolev regularity of convex integration solutions for the geometrically non-linear two-well problem. More precisely, we construct solutions to the differential inclusion $\nabla u\in K$ subject to suitable affine boundary conditions for $ u$ with $$ K:= SO(2)\left[\begin{array}{ ccc } 1 & \delta \\ 0 & 1 \end{array}\right] \cup SO(2)\left[\begin{array}{ ccc } 1 & -\delta \\ 0 & 1 \end{array}\right] $$ such that the associated deformation gradients $\nabla u$ enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape-memory alloys where $K^{qc} \neq K^{c}$, and where the energy minimisers constructed by convex integration satisfy higher Sobolev regularity. We show that in spite of additional difficulties arising from the treatment of the non-linear matrix space geometry, it is possible to deal with the geometrically non-linear two-well problem within the framework outlined in \cite{RZZ18}. Physically, our investigation of convex integration solutions at higher Sobolev regularity is motivated by viewing regularity as a possible selection mechanism of microstructures. | ||
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