A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle

A link is shown to exist between the so-called residual-based strain gradient plasticity theory and the analogous theories based on the (extended) virtual work principle (VWP). To this aim, the former theory is reformulated and cast in a residual-free form, whereby the insulation condition and the (nonlocal) Clausius–Duhem inequality, on which the theory is grounded, are substituted with equivalent residual-free ingredients, namely the energy balance condition and the residual-free form of the Clausius–Duhem inequality. The equivalence of the residual-free formulation to the original one is shown, also in their ability to cope with energetic size effects and interfacial energy ones. It emerges that the residual-free form of the Clausius–Duhem inequality coicides with the way the second thermodynamics principle is enforced within the VWP-based theories, and also that the energy balance condition amounts to the extended VWP enforced in the whole body. This makes the residual-free formulation possess strong similarities with the more general VWP-based theories, such as it constitutes an assessment of the existing link, which can be synthetized by the statement: the insulation condition is equivalent to the extended VWP deprived by the content of the standard principle.