In this paper we discuss an atomic-scale instability condition for detecting the local elastic limit in nanoscale systems. Without resorting to the continuum stability theory, we formulate an "atomic acoustic tensor" and use the singularity condition as an indicator to inspect local instabilities of the nanoscale structure. The derivation does not depend on the pristine lattice, thus, the indicator suits for not only perfect crystals but also defective media. To validate the applicability of this new quantity, we have performed local instability analysis on carbon nanotubes under axial tension. It is found that the criterion can characterize bond-breaking related instability in both perfect and defective nanotubes.
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