Summary
It is shown that the dynamic boundary value problem and the heat conduction equation for some simple materials are derivable from the first and second laws of thermodynamics. The dynamic boundary value problem, the heat conduction equation and two variational principles are derived for thermoelastic materials with time-dependent properties, for the case when the volume and surface forces are not “dead”, and when the free energy of the material depends upon the temperature. It is also shown that the conventional form of the heat conduction equation for geometrically nonlinear anisotropic elastic media does not satisfy the principle of material frame indifference. A new form of the heat conduction equation is offered. The heat conduction equation for the Navier-Stokes fluid and the dynamic boundary value problem for an elastic fluid are obtained. The elastic fluid is proved to be the only simple fluid without “memory”.
Similar content being viewed by others
References
Palmov, V. A.: Thermodynamical foundation of the variational principles in the nonlinear theory of elasticity. Trans. St. Petersburg Techn. Univ. “Mechanics and Control Processes”443, 3–9 (1992) [in Russian].
Truesdell, C.: A first course in rational continuum mechanics. New York: Academic Press 1977.
Lurie, A. I.: Nonlinear theory of elasticity. Amsterdam: North-Holland 1990.
Besseling, J. F., van der Giessen, E.: Mathematical modelling of inelastic deformation. London: Chapman and Hall 1994.
Joseph, D. D. Fluid dynamics of viscoelastic liquids. New York: Springer 1990.
Palmov, V. A.: Vibration of elastoplastic bodies [in Russian]. Moscow: Nauka 1976.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Belyaev, A.K., Palmov, V.A. Thermodynamic derivation of the heat conduction equation and the dynamic boundary value problem for thermoelastic materials and fluids. Acta Mechanica 114, 27–37 (1996). https://doi.org/10.1007/BF01170393
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01170393