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The microscopic definition for the Cauchy stress tensor has been examined inthe past from many different perspectives. This has led to differentexpressions for the stress tensor and consequently the "correct" definition hasbeen a subject of debate and controversy. In this work, a unified framework isset up in which all existing definitions can be derived, thus establishing theconnections between them. The framework is based on the non-equilibriumstatistical mechanics procedure introduced by Irving, Kirkwood and Noll,followed by spatial averaging. The Irving--Kirkwood--Noll procedure is extendedto multi-body potentials with continuously differentiable extensions andgeneralized to non-straight bonds, which may be important for particles withinternal structure. Connections between this approach and the direct spatialaveraging approach of Murdoch and Hardy are discussed and the Murdoch--Hardyprocedure is systematized. Possible sources of non-uniqueness of the stresstensor, resulting separately from both procedures, are identified andaddressed. Numerical experiments using molecular dynamics and lattice staticsare conducted to examine the behavior of the resulting stress definitionsincluding their convergence with the spatial averaging domain size and theirsymmetry properties.

A Unified Interpretation of Stress in Molecular Systems