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 ͑2003͔͒ 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 ͑i͒ 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 interactions͒, ͑ii͒ the conservation of Burgers vector in the single-valued fields, ͑iii͒ the equations of continuum crystal plasticity, and ͑iv͒ 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 loops͒, are indicated.

Dislocation patterning in a two-dimensional continuum theory of dislocations

“…Applying statistical approaches, El-Azab [30] and Sedlacek et al [31,32] suggested methods to handle curved dislocations, but the evolution equations obtained are valid only for quite specific situations. Considerable progress towards a generic statistical theory of dislocation motion in 3D has been made by Hochrainer et al [33][34][35] by deriving a theory of dislocation density transport which applies to systems of three-dimensionally curved dislocations and can represent the evolution of generic dislocation systems comprising not only geometrically necessary but also "statistically stored" dislocations with zero net Burgers vector.…”

Interface modeling in continuum dislocation transport

“…Nonetheless, there are continuum models available considering the plastic deformation as the result of dislocation motion (Groma et al, 2003;Kratochvíl and Sedlaček, 2008;Sedlček et al, 2007;Hochrainer et al, 2007). Based thereon, a continuum dislocation transport model (Groma et al, 2003;Yefimov et al, 2004) is adopted here, modeling the transport of dislocations on their glide planes both in the bulk and across the boundary.…”

Energetic approach to gradient plasticity

“…It is not clear, however, which function of the gradient should be used. We refer to Kratochvíl & Sedláček [18], to Bakó & Groma [1], and to Zaiser et al [35] for attempts to derive it from statistical physics revealing thus complexity of the problem, for more details. A related approach to non-local models in damage and plasticity was undertaken in Bažant & Jirásek [3], see also [7,9,10,12,22].…”