Size-dependent energy in crystal plasticity and continuum dislocation models

Abstract

International audienceIn the light of recent progress in coarsening the discrete dislocation mechanics, we consider two questions relevant for the development of a mesoscale, size-dependent plasticity: (i) can the phenomenological expression for size-dependent energy, as quadratic form of Nye's dislocation density tensor, be justified from the point of view of dislocation mechanics and under what conditions? (ii) how can physical or phenomenological expressions for size-dependent energy be computed from dislocation mechanics in the general case of elastically anisotropic crystal? The analysis based on material and slip system symmetries implies the negative answer to the first question. However, the coarsening method developed in response to the second question, and based on the physical interpretation of the size-dependent energy as the coarsening error in dislocation interaction energy, introduces additional symmetries. The result is that the equivalence between the phenomenological and the physical expressions is possible, but only if the multiplicity of characteristic lengths associated with different slip systems, is sacrificed. Finally, we discuss the consequences of the assumption that a single length scale governs the plasticity of a crystal, and note that the plastic dissipation at interfaces has a strong dependence on the length scale embedded in the energy expression

Dislocation Nucleation on Grain Boundaries: Low Angle Twist and Asymmetric Tilt Boundaries

“…Finally, the continuum models which take into account the size effect arising from grain boundaries have been mostly focused on boundaries as obstacles for dislocation motion [40][41][42]. The role of grain boundaries as sources of dislocations has not been modeled on the continuum level.…”

A Cosserat–phase-field theory of crystal plasticity and grain boundary migration at finite deformation

“…The linear term in lattice curvature was shown by [42] to be necessary to localize the grain boundaries, whereas a higher-order term diffuses them and is necessary for GB mobility. In terms of a physical interpretation, the linear term in the norm of the curvature tensor can interpreted as the self-energy of the geometrically necessary dislocations, whereas the quadratic term represents the elastic interaction between them [53,56]. This interpretation relies on the relation between the torsion-curvature tensor and the dislocation density tensor, also called GND tensor.…”

A continuum model for dislocation pile-up problems

“…Therefore, one of the objectives of this work is to demonstrate the capability of FDM in modeling and predicting the motions of dislocation microstructures. A well-known benchmark problem that serves such a purpose is the study of dislocation pile-ups, which, in addition to providing a key mechanism for size effects, (Mesarovic et al (2015)), also plays an important role in other phenomena such as work-hardening, yielding, and cleavage.…”