Equilibrium of immersed hyperelastic solids

Manuel Friedrich; Martin Kružík; Ulisse Stefanelli

doi:10.3934/dcdss.2021003

Article Contents

2021, Volume 14, Issue 11: 4141-4157. Doi: 10.3934/dcdss.2021003 open access

This issue Previous Article Global weak solutions for an newtonian fluid interacting with a Koiter type shell under natural boundary conditions Next Article

Equilibrium of immersed hyperelastic solids

1.
Applied Mathematics, University of Münster, Einsteinstr. 62, D-48149 Münster, Germany
2.
Academy of Sciences of the Czech Republic, Institute of Information Theory and Automation, Pod vodárenskou věží 4, CZ-182 00 Praha 8, Czechia and, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, CZ–166 29 Praha 6, Czechia
3.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria, Vienna Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währingerstraße 17, 1090 Wien, Austria, and, Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes - CNR, via Ferrata 1, 27100 Pavia, Italy

^* Corresponding author: Ulisse Stefanelli
^* Corresponding author: Ulisse Stefanelli

Received: March 2020

Revised: October 2020

Early access: January 2021

Published: November 2021

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Abstract

We discuss different equilibrium problems for hyperelastic solids immersed in a fluid at rest. In particular, solids are subjected to gravity and hydrostatic pressure on their immersed boundaries. By means of a variational approach, we discuss free-floating bodies, anchored solids, and floating vessels. Conditions for the existence of local and global energy minimizers are presented.

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Mathematics Subject Classification: Primary:49S05, 74F10; Secondary:49J45, 74B20.

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Figure 1. The basic setting

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Figure 2. The submarine setting

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Figure 3. Two anchored situations: prescribed deformation on $ \omega \subset \Omega $ (left) and elastic boundary conditions on $ \Gamma\subset \partial \Omega $ (right)

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Figure 4. The bounded-reservoir setting

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Figure 5. The ship setting

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Figure 6. A barely floating solid (left) and an admissible $ y\in A $ with $ A $ from (33) (right)

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Figure 7. A deformation with $ \omega^{y^*}\subset\Omega^{y^*} $ with $ \sup_{\omega^{y^*}} y^*_3 = 0 $ and $ \sup_{\Omega^{y*}}y^*_3>0 $

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