Convergence of local atomistic stress based on periodic lattice

Local stress in an atomic system, which provides an average stress measurement within a spatial volume containing a collection of atoms, is essential for determining the mechanical properties of a nanoscale structure as well as developing a proper multiscale modeling technique. Theoretically, the smaller averaging volume where a local stress can converge, the closer this atomistic stress definition can approach the ideal continuum stress. As a result, the more accurate stress concentration can be evaluated for the inhomogeneous case. With reference to the previous studies focusing on the spherical averaging volume, dependent on the type of crystals, the convergent radius of the virial stress or Hardy stress usually spans the size of several lattice constants. In this paper, we find that, once the averaging volume is periodic, the convergence of the virial stress and Hardy stress can be accomplished within one single lattice, which is much smaller than what is required by other non-periodic volumes such as a sphere. In the final section, a cracked sodium chloride crystal is considered to demonstrate that the crack opening stress described by the periodic lattices captures the stress concentration near the crack tip.