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Nonlinear elasticity with vanishing nonlocal self-repulsion

Part of: Equations of mathematical physics and other areas of application Miscellaneous topics in calculus of variations and optimal control

Published online by Cambridge University Press:  28 September 2023

Stefan Krömer Open the ORCID record for Stefan Krömer [Opens in a new window]  and
Philipp Reiter Open the ORCID record for Philipp Reiter [Opens in a new window]
Stefan Krömer
Affiliation:
The Czech Academy of Sciences, Institute of Information Theory and Automation (ÚTIA), Pod vodárenskou věží 4, 182 08 Praha 8, Prague, Czech Republic (skroemer@utia.cas.cz)
Philipp Reiter
Affiliation:
Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany (reiter@math.tu-chemnitz.de)
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Abstract

We prove that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$-limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev–Slobodeckiĭ seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.

Keywords

Nonlinear elasticitylocal injectivityglobal injectivityCiarlet–Nečas conditionnonlocal self-repulsionSobolev–Slobodeckiĭseminorm

MSC classification

Primary: 49N99: None of the above, but in this section
Secondary: 65Z05: Applications to physics 35Q74: PDEs in connection with mechanics of deformable solids

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 2 , April 2025 , pp. 496 - 513
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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