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 $$ K:= SO(2)\left[...
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| Main Authors: | , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
29 May 2019
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| In: |
Arxiv
Year: 2019, Pages: 1-33 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/1905.12521
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| Author Notes: | Francesco Della Porta and Angkana Rüland |
| Summary: | In this article we discuss higher Sobolev regularity of convex integration solutions for the geometrically non-linear two-well problem. More precisely, we construct solutions to the differential inclusion $\nabla u\in K$ subject to suitable affine boundary conditions for $ u$ with $$ 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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| Item Description: | Gesehen am 12.05.2021 |
| Physical Description: | Online Resource |