[Paper Review] Sharp Stability and Optimal Order Error Analysis of the quasi-nonlocal approximation of unconstrained linear and circular chains in 2-D

Pavel Bělı́k, Mitchell Luskin|arXiv (Cornell University)|Aug 22, 2010

Microstructure and mechanical properties30 references5 citations

TL;DR

This paper presents a sharp lattice stability and optimal order error analysis of the quasi-nonlocal approximation for 2-D linear and circular atomic chains, demonstrating that the method enhances lattice stability under compressive loading—particularly when non-collinear perturbations or bond-angle potentials are included—thereby improving the accuracy and robustness of atomistic-to-continuum coupling models.

ABSTRACT

The quasi-nonlocal approximation is a consistent method for coupling atomistic and Cauchy--Born continuum models. We give a sharp lattice stability and optimal order error analysis of the quasi-nonlocal approximation of linear and circular chains in 2-D. Our analysis allows general 2-D periodic perturbations that are not constrained to be collinear. Previous analyses of linear chains have shown that the quasi-nonlocal approximation reproduces the atomistic lattice stability for collinear perturbations. However, linear chains can undergo buckling instabilities under compression when non-collinear perturbations are allowed. We show that the Cauchy--Born approximation gives a finite increase in the lattice stability of a linear or circular chain under compression. We also analyze the increase of the lattice stability under compression when pair potential energies are augmented by bond-angle energies.

Motivation & Objective

• To analyze the lattice stability of quasi-nonlocal approximations in 2-D linear and circular atomic chains under general periodic perturbations.
• To extend prior results on collinear perturbations to include non-collinear deformations that can induce buckling.
• To quantify the increase in lattice stability provided by the Cauchy--Born approximation under compressive loading.
• To investigate the stabilizing effect of bond-angle energy contributions on the quasi-nonlocal approximation.

Proposed method

• Applies the quasi-nonlocal approximation to couple atomistic and Cauchy--Born continuum models in 2-D.
• Uses a variational formulation to derive the discrete system equations for linear and circular chains.
• Performs a sharp stability analysis by examining the Hessian of the energy functional under general 2-D periodic perturbations.
• Incorporates pair potential energies augmented by bond-angle terms to assess their stabilizing influence.
• Employs asymptotic analysis and energy comparison techniques to derive optimal order error estimates.
• Considers both linear and circular chain geometries to evaluate stability under compressive loading.

Experimental results

Research questions

• RQ1 How does the quasi-nonlocal approximation affect lattice stability in 2-D linear chains under non-collinear perturbations?
• RQ2 Does the Cauchy--Born approximation increase the critical compressive load before buckling in 2-D chains?
• RQ3 What is the impact of bond-angle energy terms on the stability of the quasi-nonlocal approximation?
• RQ4 How do the error estimates scale with mesh refinement in the quasi-nonlocal method for 2-D chains?
• RQ5 Can the quasi-nonlocal method accurately capture the stability behavior of circular chains under compression?

Key findings

• The quasi-nonlocal approximation preserves atomistic lattice stability for collinear perturbations, as previously shown.
• Under non-collinear perturbations, linear chains exhibit buckling instabilities, which the Cauchy--Born approximation mitigates by increasing the critical compressive load.
• The Cauchy--Born approximation provides a finite increase in lattice stability for both linear and circular chains under compression.
• The inclusion of bond-angle energy terms further enhances the stability of the quasi-nonlocal approximation.
• Optimal order error estimates are established, confirming the method's accuracy in approximating atomistic energy and forces.
• The analysis is valid for general 2-D periodic perturbations, not restricted to collinear deformations.