An energy residual-based approach to gradient effects within the mechanics of generalized continua

Generalized continua exhibiting gradient effects are addressedthrough of a (weak) nonlocal-type approach groundedon a energy residual-based gradient continuum theory by the first authorand co-workers. A main tool of this approach is the Clausius--Duheminequality cast in a form differing from the classical one only by anonstandard extra term, the nonlocality energy residual (ER), requiredto satisfy the insulation condition (its global value has tovanish, or to take a known value). The ER carries in thenonlocality features of the mechanical problem through a strain-like rate fieldbeing the specific nonlocality source, and a concomitant higher orderlong range stress (or microstress) field. The thermodynamic restrictionson the constitutive equations are determined by the latter inequalitywith no need for microstressequilibrium equations, while the PVP (principle of virtual power) is leftin a standard format. The derived state equations include a setof PDEs involving the nonlocality source strain-like quantity andthe related long range stress, as well as the associated higher orderboundary conditions determined by the insulation condition. Secondgrade materials within gradient elasticity, gradient plasticityand crystal plasticity, as well as materials with microstructure(micromorphic and Cosserat materials) are considered to derivethe pertinent constitutive equations. The proposed ER-based approachto gradient effects is shown to constitute a more straight and``economic'' way toformulate the relevant constitutive equations than the PVP-based one.