# An asymptotic formula for displacement field in triangular lattice with vacancy

## TL;DR

This paper derives an asymptotic formula for the displacement field in a triangular lattice with a vacancy under volumetric strain, accurately predicting strain concentration factors and comparing with continuum elasticity theory.

## Contribution

It introduces a simple asymptotic formula for the displacement field in a lattice with a vacancy, providing accurate predictions of strain concentration factors.

## Key findings

- Asymptotic formula predicts strain concentration with 4% accuracy
- Displacement field formula is accurate at all lattice nodes
- Effective diameter of vacancy is calculated

## Abstract

A harmonic triangular lattice with a vacancy under imposed volumetric strain is considered. Simple asymptotic formula for the displacement field is derived. The formula has reasonable accuracy at all lattice nodes. Strain concentration factor, defined as the ratio of the maximal deformation of the bonds adjacent to the vacancy to the deformations of bonds at infinity, is calculated. It is shown that the asymptotic formula predicts the strain concentration factor with 4% accuracy. The results are compared with predictions of continuum elasticity theory. Effective diameter of the vacancy is calculated.

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Source: https://tomesphere.com/paper/1706.02056