Dislocation dynamics and crystal plasticity in the phase-field crystal model

Abstract

A phase field model of a crystalline material at the mesoscale is introduced to develop the necessary theoretical framework to study plastic flow due to dislocation motion. We first obtain the elastic stress from the phase field free energy and show that it obeys the stress strain relation of linear elasticity. Dislocations in a two dimensional hexagonal lattice are shown to be composite topological defects in the amplitude expansion of the phase field, with topological charges given by the Burgers vector. This allows us to introduce a formal relation between dislocation velocity and the evolution of the coarse grained envelopes of the phase field. Standard dissipative dynamics of the phase field crystal model is shown to determine the velocity of the dislocations. When the amplitude equation is valid, we derive the Peach-Koehler force on a dislocation, and compute the associated defect mobility. A numerical integration of the phase field crystal equations in two dimensions is used to compute the motion of a dislocation dipole, and good agreement is found with the theoretical predictions.

A coarse-grained phase-field crystal model of plastic motion

“…5.1 to compute σ i j . Notice that it is in good agreement with v G x = v C y = b 2 /(2π 2 d) (Skaugen et al, 2018b) (they match for ν = 1/4). Note that the purely diffusive dynamics of the APFC model significantly underestimates the magnitude of the velocities.…”

“…The strain field ε δ i j is compatible, and the corresponding stress can be computed from the difference between the total stress, σ, and a stress computed from the amplitude functions, σ ψ i j , as (Skaugen et al, 2018b)…”

“…where χ is the Airy stress function. Exploiting the formulation reported in Skaugen et al (2018b), the stress σ ψ i j can be obtained by…”

“…For the configurations considered here, the force acting on a dislocation is f G = σ xy b and f C = σ xx b, with σ i j the stress field generated by the other dislocation. Within the (A)PFC framework, the velocity of the dislocations due to the action of the Peach-Kohler force is given by (Skaugen et al, 2018b)…”

Separation of Elastic and Plastic Timescales in a Phase Field Crystal Model

“…The phase field crystal model (PFC) is a leading contender for the efficient mesoscale modeling of crystallization phenomena [9,21] and dislocation motion [6,[22][23][24]. With a diffusive evolution of the PFC density field, the existing PFC formulations are not adequate models of small-scale crystal plasticity, when lacking a consistent separation of timescales between the fast relaxation of the elastic (smooth) distortions and the slow dynamics of crystal defects associated with singular distortions.…”