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Journal of Non-Equilibrium Thermodynamics

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Volume 42, Issue 4


The Micromorphic Approach to Generalized Heat Equations

Weijie Liu
  • Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, 2 Rue Linggong, 116024 Dalian, China
  • ICD/LASMIS, UMR-CNRS 6281, University of Technology of Troyes, 12 Rue Marie Curie BP2060, 10000 Troyes Cedex, France
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/ Khemais Saanouni
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  • ICD/LASMIS, UMR-CNRS 6281, University of Technology of Troyes, 12 Rue Marie Curie BP2060, 10000 Troyes Cedex, France
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/ Samuel Forest / Ping Hu
  • Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, 2 Rue Linggong, 116024 Dalian, China
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Published Online: 2017-04-22 | DOI: https://doi.org/10.1515/jnet-2016-0080


In this paper, the micromorphic approach, previously developed in the mechanical context is applied to heat transfer and shown to deliver new generalized heat equations as well as the nonlocal effects. The latter are compared to existing formulations: the classical Fourier heat conduction, the hyperbolic type with relaxation time, the gradient of temperature or entropy theories, the double temperature model, the micro-temperature model or micro-entropy models. A new pair of thermodynamically-consistent micromorphic heat equations are derived from appropriate Helmholtz-free energy potentials depending on an additional micromorphic temperature and its first gradient. The additional micromorphic temperature associated with the classical local temperature is introduced as an independent degree of freedom, based on the generalized principle of virtual power. This leads to a new thermal balance equation taking into account the nonlocal thermal effects and involving an internal length scale which represents the characteristic size of the system. Several existing extended generalized heat equations could be retrieved from constrained micromorphic heat equations with suitable selections of the Helmholtz-free energy and heat flux expressions. As an example the propagation of plane thermal waves is investigated according to the various generalized heat equations. Possible applications to fast surface processes, nanostructured media and nanosystems are also discussed.

Keywords: hyperbolic heat conduction; finite propagation; micromorphic theory; nonlocal effects


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About the article

Received: 2016-11-27

Accepted: 2017-03-31

Revised: 2017-03-06

Published Online: 2017-04-22

Published in Print: 2017-10-26

The financial support from China Scholarship Council and ANR through the Program Micromorfing with contract ANR-14-CE07-0035-01 is fully acknowledged.

Citation Information: Journal of Non-Equilibrium Thermodynamics, Volume 42, Issue 4, Pages 327–357, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204, DOI: https://doi.org/10.1515/jnet-2016-0080.

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