Theoretical strength of 2D hexagonal crystals: application to bubble raft indentation

Abstract

By means of lattice and molecular dynamics we study the theoretical strength of homogeneously strained, defect-free 2D crystals whose atoms interact via pair potentials with short- and longer-ranged interactions, respectively. We calculate the instability surface, i.e. the boundary in the 3D homogeneous strain space (ε _xx , ε _yy , ε _xy ), at which the first vanishing of the frequency of a vibrational mode occurs, taking into account all 2(N − 1) + 3 modes of a 2D periodic system of N atoms. We also compute the strain energies of the crystal on the instability surface, thus defining the most dangerous direction(s) of strain where the critical energy density is small. A theory is developed to incorporate the effect of loading device–sample interactions in the lattice instability criterion. The results are applied to the model problem of bubble raft indentation. We analyse the distribution of the unstable phonon modes in the first Brillouin zone as a function of the loading parameter, and discuss the post-critical behaviour of the lattice in the presence of strain gradients as in nano-indentation experiments.

Acknowledgements

We thank Krystyn J. Van Vliet and Sidney Yip for the inspiring discussions during the IUTAM Meeting in Osaka, July 2003. We also thank Takayuki Kitamura, Yoshitaka Umeno and Kisaragi Yashiro for very helpful discussions. J.L. acknowledges support from Honda R&D Co., Ltd. and the Ohio State University Transportation Research Endowment Program.