Lattice with vacancies: elastic fields and effective properties in frameworks of discrete and continuum models

Abstract

Linear elastic deformation of the two-dimensional triangular lattice with multiple vacancies is considered. Closed-form analytical expressions for displacement field in the lattice with doubly periodic system of vacancies are derived. Effective elastic moduli are calculated. The results are compared with the ones obtained by molecular dynamics simulations of a lattice with random distribution of vacancies. At low vacancy concentrations, less than 4%, random and periodic distributions of vacancies produce the same effect on elastic moduli. One of the main goals is to examine the possibilities and limitations of modelling of the lattice with vacancies by an elastic continuum with holes. It is found that the effective elastic properties are modelled adequately, provided the shape of the holes is chosen appropriately. On the contrary, the strain field, in particular, strain concentration differs significantly.

Keywords:

• Triangular lattice
• vacancy
• displacement fields
• effective elastic properties
• shape factors
• local fields
• strain concentration
• doubly periodic problem

Acknowledgements

The authors are deeply grateful to A.M. Linkov, N.F. Morozov and D.A. Indeitsev for useful discussions.

Notes

No potential conflict of interest was reported by the authors.

1 Poisson’s ratio for a triangular lattice with the nearest neighbours interactions is equal to 1/3.

Additional information

Funding

The work of V.A. Kuzkin and A.M. Krivtsov was supported by Russian Science Foundation (V.A. Kuzkin – [15-11-00017], A.M. Krivtsov – [14-11-00599]). The work of E.A. Podolskaya was supported by Russian Foundation for Basic Research [14-01-31487 mol-a].