Relationships between the admissible range of surface material parameters and stability of linearly elastic bodies

Abstract

This work focuses on the problem of an elastic bulk encased by an energetic elastic surface. The key contribution of this work is a study of the admissible range for the surface material parameters. In particular, the validity of negative surface parameters, which have been reported in the literature, is assessed. The balance equations for a continuum with a coherent energetic surface are stated. The local bifurcation conditions for the system (i.e. the bulk and the surface) are analysed using a standard ansatz for stationary waves. Under the reasonable assumption that the bulk is pointwise stable, the analysis concludes that pointwise stability of the surface elasticity tensor excludes any possible bifurcations; the weaker condition of surface strong ellipticity does not. The stability of the approximation of the governing equations using the finite element method is shown to be mesh-dependent for the case where the surface elasticity tensor loses pointwise stability. This is clearly illustrated via a series of numerical examples. Critically, it appears that the admissible range for the surface material properties differs for inherently non-local atomistic models and classical continuum formulations, both of which are used to model surface effects at the nanometre-scale. A key conclusion based on this observation is that care should be taken when fitting material properties obtained from atomistic models to classical continuum formulations that inherently lack a length-scale. The length-scale provided by the spatial discretisation can add stability, albeit artificially.

Keywords:

• surface elasticity
• bifurcation
• stability
• mesh dependency

Acknowledgements

The authors wish to thank Ahmed Benallal for the enlightening discussions during IAS III. The second and the final authors thank the National Research Foundation of South Africa for their support.