Scale effects induced by strain-gradient plasticity and interfacial resistance in periodic and randomly heterogeneous media

A recently introduced variational formulation for strain-gradient plasticity with an additional potential that penalises the build-up of plastic strain at interfaces is summarised and applied to some one-dimensional examples. Novel features include a new strict upper bound for the effective potential of a single nonlinear medium containing interfaces distributed according to a Poisson process and approximate mean stress versus mean plastic strain curves for media with two power-law nonlinear phases separated by interfaces with their own nonlinear potential. Two-phase media with periodic and random microstructure are considered. In the case of random media, the results depend on the statistics of points, taken two at a time, in the combinations medium-medium, medium-interface, and interface-interface. In every case, the effective relation displays a Hall-Petch type of effect, the effective response becoming stiffer as the scale of the microstructure is refined. The admission of the interfacial potential removes a limitation of earlier work, that the response could not exceed the "Voigt" or "Taylor" bound of the corresponding classical material.