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A Revised Exposition of the Green–Naghdi Theory of Heat Propagation

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Abstract

We offer a revised exposition of the three types of heat-propagation theories proposed by Green and Naghdi. Those theories, which make use of the notion of thermal displacement and allow for heat waves, are at variance with the standard Fourier theory; they have attracted considerable interest, and have been applied in a number of disparate physical circumstances, where heat propagation is coupled with elasticity, viscous flows, etc. (Straughan in Heat waves. Applied mathematical sciences, vol. 177. Springer, Berlin, 2011). However, their derivation is not exempt from criticisms, that we here detail, in hopes of opening the way to reconsideration of old applications and proposition of new ones.

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Notes

  1. Here, at variance with what we did when writing (9)–(11) to keep our notation lighter, we have carefully distinguished the heat-influx and free-energy constitutive mappings according to the type of theory they are meant for: e.g., we have written \(\widehat{\psi}_{\text{I}}\) in the case of Type I theory and \(\widehat{\psi}_{\text{II}},\widehat{\psi}_{\text{III}}\) in the other two cases.

  2. In fact, it would be necessary to take κ≥0 (a reasonable assumption) and κ ⋆⋆=0, an assumption that he have seen to be expedient also to obtain the ‘heat equation’ (39), and that, together with (26)2, would yield (40).

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Acknowledgements

This research was done while PPG was visiting Hamburg University of Technology and Helmholtz-Zentrum Geesthacht. The financial support of the German Science Foundation is gratefully acknowledged. AF gratefully acknowledges the financial support of INdAM-GNFM.

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Authors and Affiliations

  1. Institute of Continuum Mechanics and Materials Mechanics, Hamburg University of Technology, Hamburg, Germany

    Swantje Bargmann & Antonino Favata

  2. Institute of Materials Research, Helmholtz-Zentrum Geesthacht, Hamburg, Germany

    Swantje Bargmann

  3. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma TorVergata, Rome, Italy

    Paolo Podio-Guidugli

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  1. Swantje Bargmann
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  2. Antonino Favata
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  3. Paolo Podio-Guidugli
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Correspondence to Antonino Favata.

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German Science Foundation (DFG).

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Bargmann, S., Favata, A. & Podio-Guidugli, P. A Revised Exposition of the Green–Naghdi Theory of Heat Propagation. J Elast 114, 143–154 (2014). https://doi.org/10.1007/s10659-013-9431-8

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