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Proceedings of the Forum "Math-for-Industry" 2019New Models for Deformations: Linear Distortion and the Failure of Rank-One Convexity

Proceedings of the Forum "Math-for-Industry" 2019: New Models for Deformations: Linear Distortion... [In this article, we discuss new models for static nonlinear deformations via scale-invariant conformal energy functionals based on the linear distortion. In particular, we give examples to show that, despite equicontinuity estimates giving compactness, minimising sequences will have strictly lower energy than their limit, and that this energy gap can be quite large. We do this by showing that Iwaniec’s theorem on the failure of rank-one convexity for the linear distortion of a specific family of linear mappings is actually a generic phenomenon and we then identify the optimal rank-one direction to deform a linear map to maximally decrease its distortion.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Proceedings of the Forum "Math-for-Industry" 2019New Models for Deformations: Linear Distortion and the Failure of Rank-One Convexity

Part of the Mathematics for Industry Book Series (volume 36)
Editors: McKibbin, Robert; Wake, Graeme; Saeki, Osamu
Hashemi, Seyed Mohsen; Martin, Gaven J.
Proceedings of the Forum "Math-for-Industry" 2019 — Sep 11, 2022
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Publisher
Springer Nature Singapore
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022
ISBN
978-981-19-1153-8
Pages
81 –98
DOI
10.1007/978-981-19-1154-5_4
Publisher site
See Chapter on Publisher Site

Abstract

[In this article, we discuss new models for static nonlinear deformations via scale-invariant conformal energy functionals based on the linear distortion. In particular, we give examples to show that, despite equicontinuity estimates giving compactness, minimising sequences will have strictly lower energy than their limit, and that this energy gap can be quite large. We do this by showing that Iwaniec’s theorem on the failure of rank-one convexity for the linear distortion of a specific family of linear mappings is actually a generic phenomenon and we then identify the optimal rank-one direction to deform a linear map to maximally decrease its distortion.]

Published: Sep 11, 2022

Keywords: Quasiconformal mappings; Linear distortion; Rank-one convexity; Austenite-to-martensite transition

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