The Effect of Crystal Symmetries on the Locality of Screw Dislocation Cores

Abstract

In linearised continuum elasticity, the elastic strain due to a straight dislocation line decays as O(r −1 ), where r denotes the distance to the defect core. It is shown in [7] that the core correction due to nonlinear and discrete (atomistic) effects decays like O(r −2 ).In the present work, we focus on screw dislocations under pure anti-plane shear kinematics. In this setting we demonstrate that an improved decay O(r −p ), p > 2, of the core correction is obtained when crystalline symmetries are fully exploited and possibly a simple and explicit correction of the continuum far-field prediction is made.This result is interesting in its own right as it demonstrates that, in some cases, continuum elasticity gives a much better prediction of the elastic field surrounding a dislocation than expected, and moreover has practical implications for atomistic simulation of dislocations cores, which we discuss as well.

Analysis of an atomistic model for anti-plane fracture

“…Atomistic model for anti-plane fracture. Following the theory developed in [EOS16,HO14,BBO17] for point defects and straight dislocations, we formulate the static crack model as a minimisation problem findū ∈ arg miṅ where V : R R → R is an interatomic potential,û : Λ → R is the far-field predictor and u a core correction, thus giving us the actual displacement asû + u. We choose the potential to be a NN pair-potential of the form V (Du(m)) = ρ∈R φ((Du(m)) ρ ), (2.6) with φ ∈ C k (R) for k ≥ 5 satisfying without loss of generality φ(0) = 0 (upon replacing φ(r) → φ(r)−φ(0)), φ (0) = 0 (due to anti-plane symmetry) and φ (0) = 1 (upon rescaling φ(r) → cφ(r)).…”

“…In order for the atomistic model to be well-defined, one has to impose an additional assumption of mirror symmetry on the model, as discussed in [BBO17]. In our case, it either means setting φ (0) = 0 or looking at symmetric interactions ranges (R(m)) m∈Λ , for which m ∈ Λ and ρ ∈ R(m) =⇒ −ρ ∈ R(m + ρ), as then, despite φ (0) = 0, (3.1) is null, since the contribution of (m, m + ρ) cancels with the contribution of (m + ρ, m).…”

“…A general approach to describe a single localised defect embedded in a homogeneous host crystal was rigorously formalised in [HO14, EOS16,BBO17] for point defects and straight dislocations in Bravais lattices, then extended in [OO17] to point defects in multilattices. The overarching idea is to use a continuum model to specify the far-field behaviour where continuum theories such as continuum linear elasticity (CLE) are accurate, while employing the underlying atomistic model to capture the details of the defect core.…”

“…Most of the results readily translate to anti-plane models with finite interactions on a general two-dimensional Bravais lattice (in particular the triangular lattice), except for one technical step related to constructing a suitable locally isomorphic mapping from the defective lattice to a homogeneous one. Furthermore, drawing from the ideas developed in [BBO17] it is expected one can also extend the results beyond models with mirror symmetry (again see Section 3 for a more detailed discussion).…”

Closing the gap between atomic-scale lattice deformations and continuum elasticity

“…For some applications, however, continuum mechanics is not enough as neglecting the description of atoms leads to a crucial loss of information. For instance, this applies to contributions of the dislocation core to the elastic field [13,14] and, in turn, to dislocation nucleation, motion, and reaction. In these cases, in order to describe mate-rial properties by elasticity theory, the elastic field must be described within mesoscale [15,16] or atomistic approaches.…”