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On the linearised force balance condition in the analysis of atomistic dislocation models
Julian Braun, Thomas Hudson
TL;DR
The condition needed to apply Corollary 1 had not been carefully verified. The paper verifies the force-balance condition for the standard predictor and identifies additional predictor requirements when residual stress is present. For more general physically compatible predictors, the force sum need not vanish, and displacement decay may be slower, with larger finite-box errors and possible divergence.
Problem
The condition needed to apply Corollary 1 had not been carefully verified.
Source: §1 · p. 2Method
The paper verifies the force-balance condition for the standard predictor and identifies additional predictor requirements when residual stress is present.
Source: §1 · p. 2Results
For more general physically compatible predictors, the force sum need not vanish, and displacement decay may be slower, with larger finite-box errors and possible divergence.
Source: §1 · pp. 2–3Takeaways & Limitations
Adding appropriate multiples of the elastic Green’s function to the predictor can circumvent the stated finite-domain issues.
Source: §1 · p. 3, §3 · p. 9The extra force-line contribution should not be identified directly with a Peach–Koehler force, despite their similar physical dimensions.
Source: §3 · p. 9
Abstract
from arXiv · showhide
A technical condition arising in the analysis of atomistic models for dislocations is studied. Key estimates for the far-field strain behaviour proved in Ehrlacher, Ortner, Shapeev (2016) rely on a summation-by-parts argument for linearised forces, under conditions of sufficient decay and of vanishing net force in an infinite system. In particular, the latter condition is required for an application of this theory to the standard far-field dislocation predictor, but the vanishing of the associated force sum has not been fully justified. Here, the missing verification of this condition is provided and its role in determining the far-field decay of the corrector is clarified. It is moreover shown that for more general physically compatible predictors, the linearised force sum need not vanish, leading to slower decay and potentially divergent finite-domain approximations.
1 Introduction
Finite-domain simulations of defects with slowly decaying strain fields face significant boundary-condition errors, while a key force-balance condition used in prior error analysis was not fully verified. This paper verifies it for the standard dislocation predictor and shows how more general predictors can change far-field decay and finite-domain behavior.
1 Introduction
- The zero-force-sum condition required by [EOS16, Corollary 1] was not carefully verified in the proof of the corrector’s far-field decay result.That result enables a faster-decaying atomistic stress representation when the force field decays sufficiently quickly and its total sum vanishes.Source: §1 · p. 2
- The paper verifies that the standard dislocation predictor used as a boundary condition satisfies the zero-force-sum condition.The missing argument completes the proof that applies the summation-by-parts result to the standard predictor.Source: §1 · p. 2
- For edge or mixed dislocations, achieving force balance with residual stress requires additional conditions on the predictor beyond the approach available for pure screw dislocations.The pure-screw argument was already contained in [BHO22], but cannot be applied to edge or mixed dislocations.Source: §1 · p. 2
- For more general physically compatible predictors, the force sum need not vanish, and displacement decay may be slower, increasing finite-box errors and potentially causing divergent approximations.The paper states that these issues can be circumvented by adding suitable multiples of the elastic Green’s function to the predictor.Source: §1 · pp. 2–3
2 Preliminaries
The preliminaries set up a two-dimensional lattice model of a dislocation, define a continuum predictor and lattice operators, and specify the reference strain and energy framework used in the analysis.
2.1 Geometric set-up
- The model uses a two-dimensional reference Bravais lattice, a dislocation center, and a rightward branch cut; the domain excludes a radius-r disc around the center.The Burgers vector is possibly three-dimensional, with b3 the screw component and b1 the edge component in the chosen coordinates; the analysis focuses on b1 ≠ 0.Source: §2.1 · pp. 3–4
2.2 Continuum predictor
- The continuum predictor is derived from linear elasticity, and its canonical solution has a logarithmic singularity at the dislocation center and decaying strain estimates.The problem is not uniquely determined by its stated PDE and jump conditions alone: Green’s-function terms or certain linear displacements may be added, so canonical conditions exclude force multipoles and require strains to decay at infinity.Source: §2.2 · pp. 4–5
2.3 Lattice operators
- Lattice shift, finite-difference, and divergence operators are defined for lattice functions, with divergence related to the differences by duality.The operator definitions are well-defined for compactly supported fields and extend by continuity to appropriate function spaces.Source: §2.3 · p. 6
2.4 Special strain
- The special strain uses shifted differences of the predictor and agrees with the ordinary finite difference except when a bond crosses the branch cut.The shift there yields a reference elastic strain decaying as |ℓ|^-1, whereas the total finite-difference strain need not have this decay because the displacement has a constant jump across the cut.Source: §2.4 · p. 7
2.5 Elastic Energy
- The energy framework assumes smooth site potentials that are uniform outside a compact core and invariant under a Burgers-vector slip.The continuum Cauchy–Born energy density yields the linear elasticity tensor as its second derivative at the reference lattice, C := D2W(0).Source: §2.5 · p. 7
3 Main results
The main result characterizes when linearised forces sum to zero: the two contributions must balance. For more general predictors, adding a force-line term can restore balance, while omitting it can lead to divergent finite-domain approximations.
3 Main results
- The discussion fixes the line direction as τ = e3 and identifies n as the normal to the slip plane on which an edge-component dislocation can glide.With the stated geometric setup, n = ±e2 depending on the sign of the edge component of the Burgers vector.Source: §3 · p. 8
- The linearised force sum vanishes when the two terms in its expression balance through the choice of ulin.The theorem states the force-sum expression and its balance condition.Source: §3 · p. 8
- The two force-sum terms represent a residual core force associated with macroscopic stress and the predictor’s total elastic stress, which can cancel it.σ0 encodes residual stress in the selected reference lattice; the stress integral may use any simple closed contour encircling the dislocation.Source: §3 · pp. 8–9
- For more general physically compatible predictors, the linearised force sum need not vanish, but an added force-line contribution can cancel the residual-stress term.The force-line term can be an appropriate multiple of the two-dimensional CLE Green’s function, with the same strain-decay rate as the predictor.Source: §3 · p. 9
- Using ulin alone as a boundary condition can make finite-dimensional numerical approximations diverge.The added force-line terms are required at leading order for the analysis discussed in.Source: §3 · p. 9
- The force-line contribution should not be identified directly with a Peach–Koehler force: the screw-dislocation case distinguishes the two.For a screw dislocation the projected Burgers vector is zero, so the first force-sum term vanishes even with nonzero σ0, although screw dislocations can experience Peach–Koehler forces under shear stress.Source: §3 · p. 9
- If the reference lattice has been relaxed, no force-line contribution needs to be counteracted.The paper notes that initial lattice relaxation is standard in atomistic defect simulations.Source: §3 · pp. 9–10
4 Proofs and Calculations
The proofs establish auxiliary discrete-difference identities near the branch cut, then use cutoff estimates and summation by parts to prove Theorem 3.1. The final calculation also identifies a contour representation and notes Corollary 3.2 under the assumption σ0 = 0.
4.1 Auxiliary results
- The auxiliary lemma gives piecewise identities for discrete differences across the branch cut, with the shifts involving b canceling for operator S.Its cases depend on whether the segment from ℓ to ℓ + ρ crosses the branch cut.Source: §4.1 · p. 10, §4.2 · p. 11
4.2 Proof of Theorem 3.1
- The proof uses compactly supported cutoffs and summation by parts to decompose the expression into terms for separate analysis.The cutoff functions equal one inside radius R, vanish outside radius 2R, and have gradient bounded by a constant times R^-1.Source: §4.2 · p. 11
- The proof estimates a discretization error arising from a leading integrand contribution proportional to |ℓ|^-2 R^-1.The passage states that integration over the relevant annulus gives an error term of the stated order in R.Source: §4.2 · p. 12
- The linear-elastic stress contribution can be expressed as a boundary integral, and the proof works with any simple closed contour enclosing the origin.The unit-radius circle is an arbitrary choice for this divergence-theorem argument.Source: §4.2 · p. 13
- Theorem 3.1’s proof concludes after analyzing the terms in its force-sum expression, and Corollary 3.2 follows directly when σ0 = 0.The proof shows a cutoff-dependent term vanishes as R → ∞.Source: §4.2 · pp. 12, 14