# American Institute of Mathematical Sciences

November  2021, 14(11): 4141-4157. doi: 10.3934/dcdss.2021003

## 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  November 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (11) : 4141-4157. doi: 10.3934/dcdss.2021003
##### References:
 [1] T. L. Heath (Ed.), The Works of Archimedes, Cambridge University Press, 1897. Reprinted Dover, Mineola, NY, 2002. [2] B. Benešová, M. Kampschulte and S. Schwarzacher, A variational approach to hyperbolic evolutions and fluid-structure interactions, arXiv: 2008.04796. [3] R. E. D. Bishop and W. G. Price, Hydroelasticity of Ships, Cambridge University Press, 1979. [4] P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal., 97 (1987), 171-188.  doi: 10.1007/BF00250807. [5] R. Finn, Floating and partly immersed balls in a weightless environment, Funct. Differ. Equ., 12 (2005), 167-173. [6] R. Finn, Criteria for Floating I, J. Math. Fluid Mech., 13 (2011), 103-115.  doi: 10.1007/s00021-009-0009-y. [7] R. Finn and T. I. Vogel, Floating criteria in three dimensions, Analysis (Munich), 29 (2009), 387–402. Erratum, Analysis (Munich), 29 (2009), 339. doi: 10.1524/anly.2009.0931. [8] D. Grandi, M. Kružík, E. Mainini and U. Stefanelli, A phase-field approach to interfacial energies in the deformed configuration, Arch. Ration. Mech. Anal., 234 (2019), 351–373. doi: 10.1007/s00205-019-01391-8. [9] Z. Guerrero-Zarazua and J. Jerónimo-Castro, Some comments on floating and centroid bodies in the plane, Aequationes Math., 92 (2018), 211–222. doi: 10.1007/s00010-017-0525-4. [10] S. Hencl and P. Koskela, Lectures on Mappings of Finite Distortion, Lecture Notes in Mathematics 2096, Springer, 2014. doi: 10.1007/978-3-319-03173-6. [11] F. John, On the motion of floating bodies, I, II, Comm. Pure Appl. Math., 2 (1949), 13–57 & 3 (1950), 45–101. [12] B. Kaltenbacher, I. Kukavica, I. Lasiecka, R. Triggiani, A. Tuffaha and J. T. Webster, Mathematical theory of evolutionary fluid-flow structure interactions, Lecture Notes from Oberwolfach Seminars, November 20–26, 2016. Oberwolfach Seminars, 48. Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-92783-1. [13] M. Kružík, D. Melching and U. Stefanelli, Quasistatic evolution for dislocation-free finite plasticity, ESAIM Control Optim. Calc. Var., 26 (2020), 123. [14] M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Interaction of Mechanics and Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-02065-1. [15] Á. Kurusa and T. Ódor, Spherical floating bodies, Acta Sci. Math. (Szeged), 81 (2015), 699–714. doi: 10.14232/actasm-014-801-8. [16] P. S. Laplace, Traité de mécanique céleste: Supplement 2, 909–945, au Livre X, In Oeuvres Complète, vol. 4. Gauthier Villars, Paris. English translation by N. Bowditch (1839), reprinted by Chelsea, New York, 1966. [17] R. D. Mauldin (ed.), The Scottish Book, Birkhäuser, Boston, 1981. [18] J. McCuan, A variational formula for floating bodies, Pacific J. Math., 231 (2007), 167–191. doi: 10.2140/pjm.2007.231.167. [19] J. McCuan, Archimedes Revisited, Milan J. Math., 77 (2009), 385–396. doi: 10.1007/s00032-009-0099-2. [20] J. McCuan and R. Treinen, Capillarity and Archimedes' principle of flotation, Pacific J. Math., 265 (2013), 123–150. doi: 10.2140/pjm.2013.265.123. [21] O. Pantz, The modeling of deformable bodies with frictionless (self-)contacts, Arch. Ration. Mech. Anal., 188 (2008), 183–212. doi: 10.1007/s00205-007-0091-3. [22] T. Richter, Fluid-Structure Interactions. Models, Analysis and Finite Elements, Lecture Notes in Computational Science and Engineering, 118. Springer, Cham, 2017. doi: 10.1007/978-3-319-63970-3. [23] R. Treinen, A general existence theorem for symmetric floating drops, Arch. Math. (Basel), 94 (2010), 477–488. doi: 10.1007/s00013-010-0123-3. [24] F. Wegner, Floating bodies of equilibrium, Stud. Appl. Math., 111 (2003), 167–183. doi: 10.1111/1467-9590.t01-1-00231.

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##### References:
 [1] T. L. Heath (Ed.), The Works of Archimedes, Cambridge University Press, 1897. Reprinted Dover, Mineola, NY, 2002. [2] B. Benešová, M. Kampschulte and S. Schwarzacher, A variational approach to hyperbolic evolutions and fluid-structure interactions, arXiv: 2008.04796. [3] R. E. D. Bishop and W. G. Price, Hydroelasticity of Ships, Cambridge University Press, 1979. [4] P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal., 97 (1987), 171-188.  doi: 10.1007/BF00250807. [5] R. Finn, Floating and partly immersed balls in a weightless environment, Funct. Differ. Equ., 12 (2005), 167-173. [6] R. Finn, Criteria for Floating I, J. Math. Fluid Mech., 13 (2011), 103-115.  doi: 10.1007/s00021-009-0009-y. [7] R. Finn and T. I. Vogel, Floating criteria in three dimensions, Analysis (Munich), 29 (2009), 387–402. Erratum, Analysis (Munich), 29 (2009), 339. doi: 10.1524/anly.2009.0931. [8] D. Grandi, M. Kružík, E. Mainini and U. Stefanelli, A phase-field approach to interfacial energies in the deformed configuration, Arch. Ration. Mech. Anal., 234 (2019), 351–373. doi: 10.1007/s00205-019-01391-8. [9] Z. Guerrero-Zarazua and J. Jerónimo-Castro, Some comments on floating and centroid bodies in the plane, Aequationes Math., 92 (2018), 211–222. doi: 10.1007/s00010-017-0525-4. [10] S. Hencl and P. Koskela, Lectures on Mappings of Finite Distortion, Lecture Notes in Mathematics 2096, Springer, 2014. doi: 10.1007/978-3-319-03173-6. [11] F. John, On the motion of floating bodies, I, II, Comm. Pure Appl. Math., 2 (1949), 13–57 & 3 (1950), 45–101. [12] B. Kaltenbacher, I. Kukavica, I. Lasiecka, R. Triggiani, A. Tuffaha and J. T. Webster, Mathematical theory of evolutionary fluid-flow structure interactions, Lecture Notes from Oberwolfach Seminars, November 20–26, 2016. Oberwolfach Seminars, 48. Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-92783-1. [13] M. Kružík, D. Melching and U. Stefanelli, Quasistatic evolution for dislocation-free finite plasticity, ESAIM Control Optim. Calc. Var., 26 (2020), 123. [14] M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Interaction of Mechanics and Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-02065-1. [15] Á. Kurusa and T. Ódor, Spherical floating bodies, Acta Sci. Math. (Szeged), 81 (2015), 699–714. doi: 10.14232/actasm-014-801-8. [16] P. S. Laplace, Traité de mécanique céleste: Supplement 2, 909–945, au Livre X, In Oeuvres Complète, vol. 4. Gauthier Villars, Paris. English translation by N. Bowditch (1839), reprinted by Chelsea, New York, 1966. [17] R. D. Mauldin (ed.), The Scottish Book, Birkhäuser, Boston, 1981. [18] J. McCuan, A variational formula for floating bodies, Pacific J. Math., 231 (2007), 167–191. doi: 10.2140/pjm.2007.231.167. [19] J. McCuan, Archimedes Revisited, Milan J. Math., 77 (2009), 385–396. doi: 10.1007/s00032-009-0099-2. [20] J. McCuan and R. Treinen, Capillarity and Archimedes' principle of flotation, Pacific J. Math., 265 (2013), 123–150. doi: 10.2140/pjm.2013.265.123. [21] O. Pantz, The modeling of deformable bodies with frictionless (self-)contacts, Arch. Ration. Mech. Anal., 188 (2008), 183–212. doi: 10.1007/s00205-007-0091-3. [22] T. Richter, Fluid-Structure Interactions. Models, Analysis and Finite Elements, Lecture Notes in Computational Science and Engineering, 118. Springer, Cham, 2017. doi: 10.1007/978-3-319-63970-3. [23] R. Treinen, A general existence theorem for symmetric floating drops, Arch. Math. (Basel), 94 (2010), 477–488. doi: 10.1007/s00013-010-0123-3. [24] F. Wegner, Floating bodies of equilibrium, Stud. Appl. Math., 111 (2003), 167–183. doi: 10.1111/1467-9590.t01-1-00231.
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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