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A comparison between extended energy flux and extended entropy flux formulations of nonlocal constitutive models
[Étude comparative des formulations thermodynamiques à flux d’énergie étendu et à flux d’entropie étendu pour les modèles de comportement non locaux]
Charles Mareau1 orcid
1Arts et Métiers Institute of Technology, LAMPA, F-49035 Angers, France
Comptes Rendus. Mécanique, Volume 354 (2026), pp. 717-734

Nonlocal constitutive models have been widely developed to account for size/gradient effects or to circumvent the difficulties associated with excessive spatial localization. Two main thermodynamic frameworks have been proposed to incorporate some information regarding the spatial distribution of internal variables in a consistent manner: formulations based on an extended energy flux and those relying on an extended entropy flux. Although these approaches are often regarded as equivalent, their respective implications on the thermodynamic structure of the governing equations remain insufficiently understood. In this work, a comparison between nonlocal models formulated with an extended energy flux and with an extended entropy flux is conducted within the general framework of continuum thermodynamics. Both gradient-based and integral-based nonlocal approaches are considered in a unified manner. The corresponding energy and entropy balance equations are derived, along with the associated dissipation inequalities, evolution equations for internal degrees of freedom, and heat diffusion equations. It is shown that the two formulations lead to identical evolution equations under uniform temperature conditions. However, some differences arise in the presence of temperature gradients, notably through the coupling between nonlocal interactions and the temperature field in the extended entropy flux formulation. The analysis also clarifies under which assumptions some formulations presented as extended entropy flux approaches should rather be interpreted as extended energy flux frameworks. The results provide a theoretical basis to assess the relevance and limitations of extended energy and entropy flux formulations of nonlocal constitutive models in thermomechanically coupled problems.

Les modèles de comportement non locaux ont été largement développés afin de prendre en compte les effets d’échelle et de gradient, ainsi que pour contourner les difficultés liées à une localisation spatiale excessive. Deux principaux cadres thermodynamiques ont été proposés afin d’introduire de manière cohérente des informations relatives à la distribution spatiale des variables internes : les formulations fondées sur un flux d’énergie étendu et celles reposant sur un flux d’entropie étendu. Bien que ces deux approches soient souvent considérées comme équivalentes, leurs implications respectives sur la structure thermodynamique des équations gouvernantes demeurent encore insuffisamment clarifiées. Dans ce travail, une comparaison entre les modèles non locaux formulés à l’aide d’un flux d’énergie étendu et ceux fondés sur un flux d’entropie étendu est menée dans le cadre général de la thermodynamique des milieux continus. Les approches non locales fondées sur les gradients ainsi que celles fondées sur les moyennes sont considérées de manière unifiée. Les équations de bilan de l’énergie et de l’entropie correspondantes sont établies, de même que les inégalités de dissipation associées, les équations d’évolution des degrés de liberté internes et les équations de diffusion de la chaleur. Il est montré que les deux formulations conduisent à des équations d’évolution identiques lorsque la température est uniforme. Toutefois, des différences apparaissent en présence de gradients de température, notamment en raison du couplage entre les interactions non locales et le champ de température dans la formulation à flux d’entropie étendu. L’analyse permet également de préciser sous quelles hypothèses certaines formulations présentées comme des approches à flux d’entropie étendu doivent plutôt être interprétées comme des formulations à flux d’énergie étendu. Les résultats fournissent une base théorique permettant d’évaluer la pertinence et les limites des formulations à flux d’énergie étendu et à flux d’entropie étendu pour les modèles de comportement non locaux appliqués à des problèmes thermomécaniques couplés.

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DOI : 10.5802/crmeca.378
Keywords: Thermodynamics, internal variable, nonlocal, constitutive model
Mots-clés : Thermodynamique, variable interne, non-localité, modèle de comportement

Charles Mareau  1

1 Arts et Métiers Institute of Technology, LAMPA, F-49035 Angers, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Charles Mareau. A comparison between extended energy flux and extended entropy flux formulations of nonlocal constitutive models. Comptes Rendus. Mécanique, Volume 354 (2026), pp. 717-734. doi: 10.5802/crmeca.378
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