Towards a homogenized plasticity theory which predicts structural and microstructural size effects

We study an idealized bending problem where two types of size effects are present – one induced by the non-uniform (macro) deformation, the other due to the (internal) resistance at grain boundaries. Classical models are not able to capture either of the two types of size dependent behavior. A remedy is to adopt a gradient crystal plasticity formulation which allows one to study the direct influence of different microstructural properties on the material response. However, it is computationally expensive to do so for a typical engineering problem since the discretization has to be done at a sub-granular level. In this paper, a homogenization theory is proposed such that the small deformation gradient crystal plasticity framework by Cermelli and Gurtin [Cermelli, P., Gurtin, M.E., 2002. Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations. Int. J. Solids Struct. 39, 6281–6309] translates from the micro to macro level consistently. Microstructural properties thus propagate naturally to the macro scale and the homogenized solutions compare well with the fine scale analyses for the two limit cases – microhard and microfree conditions. Three length scale parameters, i.e. the intrinsic length scale, grain size and the foil thickness, manifest themselves in the homogenized solution, thus capturing both types of size effects. We further discuss on the interplay and competition between the two size effects.