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Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

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Abstract

Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).

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Acknowledgements

I.P. acknowledges a financial support from ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. M.P. and M.G. were supported by Czech Science Foundation, Project No. 17-15498Y, by Natural Sciences and Engineering Research Council of Canada (NSERC), grant numbers RGPIN-2014-06504-CRSNG and RGPAS462034-2014-CRSNG. This work has been supported by Charles University Research Program No. UNCE/SCI/023. E.R. acknowledges a partial support by the Program N15 of the Presidium of RAS (Project 121) and the Russian Foundation for Basic Research (Grant No. 16-29-15131).

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  1. Ilya Peshkov and Michal Pavelka have contributed equally to this work.

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Toulouse, France

    Ilya Peshkov

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Ilya Peshkov

  3. Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Prague, Czech Republic

    Michal Pavelka

  4. Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russia

    Evgeniy Romenski

  5. École Polytechnique de Montréal, C.P. 6079, suc. Centre-ville, Montreal, QC, H3C 3A7, Canada

    Miroslav Grmela

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Correspondence to Michal Pavelka.

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Communicated by Attila R. Imre.

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Peshkov, I., Pavelka, M., Romenski, E. et al. Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Continuum Mech. Thermodyn. 30, 1343–1378 (2018). https://doi.org/10.1007/s00161-018-0621-2

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