  Stress is calculated routinely in atomistic simulations. The widely used
microscopic stress formulas derived from classical or quantum mechanics,
however, are distinct from the concept of Cauchy stress, i.e., the true
mechanical tress. This work examines various atomistic stress formulations and
their inconsistencies. Using standard mathematic theorems and the law of
mechanics, we show that Cauchy stress results unambiguously from the definition
of internal force density, thereby removing the long-standing confusion about
the atomistic basis of the fundamental property of Cauchy stress, and leading
to a new atomistic formula for stress that has clear physical meaning and
well-defined values, satisfies conservation law, and is fully consistent with
the concept of Cauchy stress.
