A comparison between extended energy flux and extended entropy flux formulations of nonlocal constitutive models

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.