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

Abstract

In metallic polycrystals, an important descriptor of the underlying microstructure is the orientation of the crystal lattice of each grain. During thermomechanical processing, the microstructure can be significantly altered through deformation, nucleation of new subgrains and grain boundary migration. Cosserat crystal plasticity provides orientation as a degree of freedom and is therefore a natural choice for the development of a coupled framework to deal with concurrent viscoplasticity and grain growth. In order to take into account grain boundary motion, the Cosserat theory is adapted with inspiration from orientation phase-field models. This allows for the microstructure at a material point to evolve on the one hand due to deformation-induced lattice reorientation and on the other hand due to a sweeping grain boundary. With a proper separation of plastic evolution in the bulk of the grain and in the grain boundary, the model can successfully capture grain boundary migration due to lattice curvature and due to statistically stored dislocations.Keywords Cosserat crystal plasticity · Phase-field method · Grain boundary migration Communicated by Francesco dell'Isola.A. Ask · S. Forest (B) · K. Ammar MINES ParisTech,

Microstructure evolution in deformed polycrystals predicted by a diffuse interface Cosserat approach

“…The small deformation theory presented in this section recalls the formulation proposed in [21], and represents the linearized version of the large deformation theory developed in [22].…”

“…In order to create extended frameworks which also take into account grain boundary migration after deformation, and in some cases nucleation as an intermediate step, several authors have coupled crystal plasticity models with methods such as cellular automata [12,13], level-sets [14,15] or phase-field methods [16][17][18][19][20] (these references are by no means a complete list but should rather be considered a sample of works for each mentioned approach). There are only a few works that consider a strong coupling, formulating and solving the complete set of equations as a monolithic system [21][22][23]. One of the major issues to resolve in a fully coupled framework is the fact that crystal orientation may change through deformation or through grain boundary migration.…”

“…In this paper, a fully coupled, thermodynamically consistent model that combines a Cosserat single crystal plasticity model with the modified KWC orientation phase-field model of [26] is used to predict microstructure evolution in deformed polycrystals. The model was first developed for small deformations [28] and later for large deformations in [22]. In this work, a linearized version of the large deformation model (c.f., [21]) is used.…”

On a modified Becker–Döring model for two-phase materials

“…It clearly holds N(z) = N 1 (z) and z α (t) = N α (z(t)) − N α+1 (z(t)). One can formally show α := j α (z (n) ), ρ (n) := ρ(z (n) ), f (n) := f (z (n) ), the following statements are valid: (1)…”

“…For t = 0, this follows from (A1). For t > 0, it is sufficient to show that there exists a function g ∈ L 1 (Ω) such that Z(·,t) is measurable and |Z α (x,t)| ≤ g(x) for almost every x in Ω and any α ≥ 1. But the last follows from (4.5) and (A1), since…”