Unified thermodynamic framework for nonlocal/gradient continuum theories

A thermodynamic framework, equipped with the concept of nonlocality (energy) residual, is utilized to address nonlocal/gradient internal variable material models. A unified procedure is provided for either nonlocal and gradient materials, which makes it possible to determine the thermodynamic restrictions upon the constitutive equations, and in particular the pertinent state equations, the consistent form of the dissipation power and the constitutive expression of the nonlocality residual. Additionally, for gradient models, the associated nonstandard boundary conditions are derived, pointing out their basically constitutive nature and their substantial differences from the standard ones. Gradient elasticity and gradient plasticity are addressed in some details. Gradient elasticity is also compared with the Toupin–Mindlin strain gradient elasticity and their remarkable differences are enumerated. A few simple examples are reported as an illustration.