# American Institute of Mathematical Sciences

## 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

Received  March 2020 Revised  October 2020 Published  January 2021

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.

Citation: Manuel Friedrich, Martin Kružík, Ulisse Stefanelli. Equilibrium of immersed hyperelastic solids. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021003
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The basic setting
The submarine setting
Two anchored situations: prescribed deformation on $\omega \subset \Omega$ (left) and elastic boundary conditions on $\Gamma\subset \partial \Omega$ (right)
The bounded-reservoir setting
The ship setting
A barely floating solid (left) and an admissible $y\in A$ with $A$ from (33) (right)
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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