A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations

Lorenzo Bardella

Department of Civil Engineering, Faculty of Engineering, University of Brescia, Via Branze, 3825123, Brescia, Italy

Abstract

We propose a deformation theory of strain gradient crystal plasticity that accounts for the density of geometrically necessary dislocations by including, as an independent kinematic variable, Nye's dislocation density tensor (Nye, 1953). This is accomplished, in the same fashion as proposed by Gurtin and co-workers (see, for instance, Gurtin and Needleman, 2005) in the context of a flow theory of crystal plasticity, by introducing the so-called defect energy. Moreover, in order to better describe the strengthening accompanied with diminishing size, we propose that the classical part of the plastic potential be dependent on both the plastic slip vector and its gradient; for single crystals, this also makes it easier to deal with the "higher-order" boundary conditions. We develop both the kinematic formulation and its static dual and apply the theory to the simple shear of a constrained strip (example already exploited in Shu et al., 2001, Bittencourt et al., 2003, Niordson and Hutchinson, 2003, Evers et al., 2004, and Anand et al., 2005) to investigate what sort of behaviour the new model predicts. The availability of the total potential energy functional and its static dual allows us to easily solve this simple boundary value problem by resorting to the Ritz method.

Author Keywords: Dislocations; Strengthening and mechanisms; Crystal plasticity; Strain gradient plasticity; Variational principles