Statistical foundation of continuum dislocation plasticity

Abstract

A continuum model of mesoscale plasticity intended to cover dislocation pattering and size effects is proposed. The statistical approach to dislocation dynamics put forward by Groma et al. [Acta Mater. 51, 1271 (2003)] is applied to curved glide dislocations and dipolar dislocation loops appearing in a crystal deformed by single slip. Due to the very different size and mobile properties of the glide dislocations and the loops, the two entities are differently treated. The statistics of Groma et al. is directly applicable to the loops, which behave as rigid objects of restricted mobility. The glide dislocations are divided into groups, which in the averaging process lead to so-called single-valued dislocation fields. The averaged continuous distribution of the glide dislocations is treated as a superposition of these fields. The system of equations of the continuum model consists of (i) equations for the flow of glide dislocations of each single-valued field and the balance equation for the density of dipolar loops (these equations are derived from the discrete equations of motion and contain constitutive functionals representing short range interactions), (ii) the conservation of Burgers vector in the single-valued fields, (iii) the equations of continuum crystal plasticity, and (iv) the Orowan law for the shear strain rate caused by the glide dislocations. The possibilities for specification and evaluation of the constitutive functionals, namely, the short range correlations among dislocations (both the glide dislocations and the loops), are indicated.