A Reduced Model for Plates Arising as Low-Energy \(\boldsymbol{\Gamma}\) -Limit in Nonlinear Magnetoelasticity
Abstract.
We investigate the problem of dimension reduction for plates in nonlinear magnetoelasticity. The model features a mixed Eulerian–Lagrangian formulation, as magnetizations are defined on the deformed set in the actual space. We consider low-energy configurations by rescaling the elastic energy according to the linearized von Kármán regime. First, we identify a reduced model by computing the \(\Gamma\) -limit of the magnetoelastic energy, as the thickness of the plate goes to zero. This extends a previous result obtained by the first author in the incompressible case to the compressible one. Then, we introduce applied loads given by mechanical forces and external magnetic fields, and we prove that sequences of almost minimizers of the total energy converge to minimizers of the corresponding energy in the reduced model. Subsequently, we study quasistatic evolutions driven by time-dependent applied loads and a rate-independent dissipation. We prove that energetic solutions for the bulk model converge to energetic solutions for the reduced model, and we establish a similar result for solutions of the approximate incremental minimization problem. Both these results provide a further justification of the reduced model in the spirit of the evolutionary \(\Gamma\) -convergence.
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Acknowledgments.
This work was carried out while the first author was affiliated with TU Wien. Both authors are grateful to Elisa Davoli for helpful discussions during the preparation of this paper, and to the referees for allowing them to improve the first version. The first author also thanks Maria Giovanna Mora for useful comments about modeling and for suggesting the use of the interpolation inequality.
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Published In
SIAM Journal on Mathematical Analysis
Pages: 3108 - 3168
DOI: 10.1137/21M1446836
ISSN (online): 1095-7154
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© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 16 September 2021
Accepted: 9 November 2022
Published online: 27 July 2023
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Funding Information
OeAD-WTZ: CZ04/2019
MSMT CR: 8J19AT013, 8J22AT017
Austrian Science Fund (FWF): I4052-N32/19-29646L
Grantová Agentura České Republiky (GACR): I4052-N32/19-29646L
Funding: This work has been supported by the Austrian Science Fund (FWF) and the GAČR through the grant I4052-N32/19-29646L, and by the Federal Ministry of Education, Science and Research of Austria (BMBWF) through the OeAD-WTZ project CZ04/2019, the MŠMT-GAČR projects 8J19AT013 and 8J22AT017, and the MŠMT-WTZ project 8J22AT017.
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