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Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations

  • Angkana Rüland
  • Jamie M. TaylorEmail author
  • Christian Zillinger
Article
  • 10 Downloads

Abstract

We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is twofold, treating both rigidity and flexibility properties: Firstly, we relate the maximal regularity of convex integration solutions to the presence of lower bounds in variational models with surface energy. Hence, variational models with surface energy could be viewed as a selection mechanism allowing for or excluding convex integration solutions. Secondly, we present the first numerical implementations of convex integration schemes for the model problem of the geometrically linearised two-dimensional hexagonal-to-rhombic phase transformation. We discuss and compare the two algorithms from Rüland et al. (J Elast. 2019.  https://doi.org/10.1007/s10659-018-09719-3; SIAM J Math Anal 50(4):3791–3841, 2018) and give a numerical estimate of the regularity attained.

Keywords

Convex integration Higher Sobolev regularity Scaling laws Solid-solid phase transformations 

Mathematics Subject Classification

35B36 35B65 32F32 

Notes

Acknowledgements

During the research leading to these results Jamie M. Taylor held positions at the University of Oxford, at Kent State University and BCAM. His research received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 291053. Furthermore, he has been partially supported by the Basque Government through the BERC 2018-2021 programme and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym “DESFLU”. Christian Zillinger acknowledges the support of an AMS-Simons Travel Grant.

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Authors and Affiliations

  1. 1.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Basque Center for Applied MathematicsBilbaoSpain

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