Thermodynamical setting for gradient continuum theories with vectorial internal variables: Application to granular materials

The exploitation of the entropy principle in thermodynamics with Lagrange multipliers, the so-called Liu procedure, is based on the celebrated Liu theorem. In some recent papers, the Liu procedure has been generalized by considering the gradients of the governing equations as additional constraints. Here, we apply this generalized procedure to the equations of a continuum with vectorial internal variable and first-order non-local state space. Then, owing to the obtained results, we develop a thermodynamic model for continua with scalar microstructure.