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In this paper, once and for all, we want to strongly affirm that the thermodynamic framework for complex materials based on extra fluxes can lead to more severe restrictions with respect to the virtual powers scheme. Even if formally converging to the same final constitutive results, it seems only the consequence of an inappropriate approach to the problem itself. To achieve this goal, we proceed with a quick critical comparison between the different steps of the two methodologies: by way of example, we address novel constitutive settings for the third-order infinitesimal elasticity and viscoelasticity with an extension to thermo-elasticity, based on nonlocal revisited Green-Naghdi like thermal properties. Finally, suitable thermodynamic restrictions are derived from the generalized Clausius-Duhem inequality.

Nonlocal continuum mechanics structures: The virtual powers method vs the extra fluxes topic