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Modeling of thermal and electrical conductivities by means of a viscoelastic Cosserat continuum

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Abstract

We consider a linear theory of a viscoelastic Cosserat continuum of a special type. In doing so, we associate the main variables characterizing the stress–strain state of the continuum with quantities characterizing the electrodynamic and thermal processes. Taking into account the suggested analogues, we interpret equations describing the continuum as equations of thermodynamics and electrodynamics. We identify parameters of our model by comparing the obtained equations with Maxwell’s equations and the hyperbolic heat conduction equation. As a result, we arrive at two three-dimensional telegrapher’s equations: one for temperature and the other for the electric field vector. These equations are novel. They describe electromagnetic and thermal processes and also how they affect each other more accurately compared to the classical theory. In particular, these telegrapher’s equations account for not only the skin effect described in many literature sources on electrodynamics, but also the so-called static skin effects observed in a number of experiments. In contrast to classical electrodynamics, which contains two mutually orthogonal vectors: the electric field vector and the magnetic induction vector, the proposed theory contains three mutually orthogonal vectors: the electric field vector, the magnetic induction vector and the temperature gradient. It agrees with experimental facts discovered by Ettingshausen and Nernst (the Ettingshausen effect and the Nernst–Ettingshausen effect). If thermal component is ignored, the proposed theory reduces to the system of equations, which is a generalization of Maxwell’s equations. This system of equations is novel. It is a three-dimensional analogue of Kirchhoff’s laws for electric circuits, while Maxwell’s equations are not.

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Acknowledgements

The author is deeply grateful to M. B. Babenkov, Ya. E. Kolpakov, A. M. Krivtsov, V. A. Kuzkin, E. A. Podolskaya and E. N. Vilchevskaya for useful discussions on the paper. The research is partially funded by the Ministry of Science and Higher Education of the Russian Federation as part of World-class Research Center program: Advanced Digital Technologies (contract No. 075-15-2020-934 dated 17.11.2020).

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  1. Higher School of Theoretical Mechanics, Peter the Great St.Petersburg Polytechnic University, Polytechnicheskaya, 29, Saint-Petersburg, Russia, 195251

    Elena A. Ivanova

  2. Institute for Problems in Mechanical Engineering of Russian Academy of Sciences, Bolshoy pr. V.O., 61, Saint-Petersburg, Russia, 199178

    Elena A. Ivanova

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Ivanova, E.A. Modeling of thermal and electrical conductivities by means of a viscoelastic Cosserat continuum. Continuum Mech. Thermodyn. 34, 555–586 (2022). https://doi.org/10.1007/s00161-021-01071-7

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