The origin of the distinction between microscopic formulas for stress   and Cauchy stress
  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. 
Youping Chen
cond-mat.stat-mech