[Paper Review] Existence and stability of screw dislocation configurations with arbitrary net Burgers vector

Thomas Hudson, Christoph Ortner|arXiv (Cornell University)|Mar 3, 2014

Nonlocal and gradient elasticity in micro/nano structures21 references1 citations

TL;DR

This paper establishes the existence and local stability of screw dislocation configurations with arbitrary net Burgers vector in a zero-temperature anti-plane lattice model, using approximate solutions from linear elasticity and the inverse function theorem to prove that local minimizers remain near these elastic approximations, validating linear elasticity near dislocation cores.

ABSTRACT

We consider a variational anti-plane lattice model and demonstrate that at zero temperature, there exist locally stable states containing screw dislocations, under conditions on the distance between the dislocations and on the distance between dislocations and the boundary of the crystal. In proving our results, we introduce approximate solutions which are taken from the theory of dislocations in linear elasticity, and use the inverse function theorem to show that local minimisers lie near them. This gives credence to the commonly held intuition that linear elasticity is essentially correct up to a few spacings from the dislocation core.

Motivation & Objective

• To rigorously establish the existence of stable screw dislocation configurations in a lattice model with arbitrary net Burgers vector.
• To analyze the conditions under which such dislocations remain locally stable, particularly concerning distances to other dislocations and the crystal boundary.
• To validate the use of linear elasticity as an effective approximation near dislocation cores by showing that true minimizers lie close to elastic solutions.
• To bridge the gap between continuum elasticity theory and discrete lattice models in the context of dislocation configurations.

Proposed method

• Constructs approximate solutions for screw dislocations based on linear elasticity theory.
• Applies the inverse function theorem to show that local minimizers of the lattice energy functional exist near these elastic approximations.
• Imposes geometric constraints on the distance between dislocations and between dislocations and the crystal boundary to ensure stability.
• Uses variational methods in a discrete anti-plane lattice setting to model dislocation configurations at zero temperature.
• Relies on the regularity and smallness of perturbations to ensure convergence of the inverse function theorem application.
• Demonstrates that the energy landscape near the elastic solution admits a unique local minimizer, confirming stability.

Experimental results

Research questions

• RQ1 Under what conditions do stable screw dislocation configurations with arbitrary net Burgers vector exist in a discrete lattice model?
• RQ2 How close do actual energy minimizers lie to the predictions of linear elasticity near the dislocation core?
• RQ3 What geometric constraints on dislocation spacing and boundary proximity ensure local stability of such configurations?
• RQ4 To what extent does linear elasticity accurately describe the structure of dislocations in a discrete lattice framework?
• RQ5 Can the inverse function theorem be used to rigorously justify the stability of dislocation solutions derived from continuum approximations?

Key findings

• Stable screw dislocation configurations with arbitrary net Burgers vector exist in the anti-plane lattice model under appropriate geometric constraints.
• Local minimizers of the energy functional are shown to lie close to approximate solutions derived from linear elasticity.
• The inverse function theorem provides a rigorous justification for the stability of these configurations near the elastic approximation.
• The results support the physical intuition that linear elasticity remains valid up to a few lattice spacings from the dislocation core.
• The stability is contingent on the distance between dislocations and their distance from the crystal boundary being sufficiently large.
• The framework confirms the robustness of linear elasticity as a predictive tool for dislocation structure in discrete systems.