Second gradient Green-Naghdi type thermo-elasticity and viscoelasticity

In this paper we perform a thermodynamically consistent theory for second gradient Green-Naghdi revisited type thermo-elasticity and viscoelasticity by developing a non-standard version of the virtual powers method. We emphasize the incontrovertible effectiveness of variational arguments to approach non local continuum structures. Notably, non local Kelvin-Voigt type viscoelastic models and non local/local revised Green-Naghdi rigid heat conduction theories are recovered as special constitutive settings, also providing a fruitful comparison with pre-existent modelings.