Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity

Lorenzo Bardella

DICATA, Faculty of Engineering, University of Brescia, Via Branze, 43––25123, Brescia, Italy

Alessandro Giacomini

Department of Mathematics, Faculty of Engineering, University of Brescia, Via Valotti, 9––25133, Brescia, Italy

Abstract

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella (2007) in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin, 2002), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of Bardella, 2006, and Gurtin et al., 2007), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Gamma-convergence technique to find closed-form solutions in the "isotropic limit". Finally, we analytically show that in the "perfect plasticity" case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.

Author Keywords: Dislocations; Strengthening and mechanisms; Crystal plasticity; Energy methods; Strain gradient plasticity