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Dynamical Processes in Generalized Continua and Structures pp 345–358Cite as

The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal

Part of the Advanced Structured Materials book series (STRUCTMAT,volume 103)

Abstract

The analytical model of unsteady ballistic heat transfer in a one-dimensional harmonic crystal is analyzed. A nonlocal temperature is introduced as a generalization of the kinetic temperature. A closed equation determining unsteady thermal processes in terms of the nonlocal temperature is derived. For an instantaneous heat perturbation a time-reversible equation for the kinetic temperature is derived and solved. This equation can be referred as the ballistic heat conduction equation, it is somewhat similar to the hyperbolic heat conduction equation, but it has important differences. The resulting constitutive law for the heat flux in the considered system is obtained. This law significantly differs from Fourier’s law and it predicts a finite velocity of the heat front and independence of the heat flux on the crystal length. The analytical results are confirmed by computer simulations. Further developments of the presented approach are referred.

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Acknowledgements

The author is grateful to O. V. Gendelman, W. G. Hoover, D. A. Indeitsev, M. L. Kachanov, V. A. Kuzkin, S. A. Lurie, N. F. Morozov, and V. F. Zhuravlev for helpful and stimulating discussions; to M. B. Babenkov and D. V. Tsvetkov for additional analysis and computations. This work is supported by the Russian Science Foundation (Grant No. 18-11-00201).

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Authors and Affiliations

  1. Department of Theoretical Mechanics, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

    Anton Krivtsov

  2. Laboratory of Discrete Models in Mechanics, Institute for Problems in Mechanical Engineering RAS, St. Petersburg, Russia

    Anton Krivtsov

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  1. Anton Krivtsov
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  1. Lehrstuhl Technische Mechanik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

    Prof. Dr. Holm Altenbach

  2. Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

    Prof. Dr. Alexander Belyaev

  3. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Dr. Victor A. Eremeyev

  4. Department of Theoretical Mechanics, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

    Prof. Dr. Anton Krivtsov

  5. Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

    Prof. Dr. Alexey V. Porubov

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Krivtsov, A. (2019). The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal. In: Altenbach, H., Belyaev, A., Eremeyev, V., Krivtsov, A., Porubov, A. (eds) Dynamical Processes in Generalized Continua and Structures. Advanced Structured Materials, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-030-11665-1_19

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  • DOI: https://doi.org/10.1007/978-3-030-11665-1_19

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