The microscopic definition for the Cauchy stress tensor has been examined in the past from many different perspectives. This has led to different expressions for the stress tensor and consequently the "correct" definition has been a subject of debate and controversy. In this work, a unified framework is set up in which all existing definitions can be derived, thus establishing the connections between them. The framework is based on the non-equilibrium statistical mechanics procedure introduced by Irving, Kirkwood and Noll, followed by spatial averaging. The Irving-Kirkwood-Noll procedure is extended to multi-body potentials with continuously differentiable extensions and generalized to non-straight bonds, which may be important for particles with internal structure. Connections between this approach and the direct spatial averaging approach of Murdoch and Hardy are discussed and the Murdoch-Hardy procedure is systematized. Possible sources of non-uniqueness of the stress tensor, resulting separately from both procedures, are identified and addressed. Numerical experiments using molecular dynamics and lattice statics are conducted to examine the behavior of the resulting stress definitions including their convergence with the spatial averaging domain size and their symmetry properties.
Bibliographical noteFunding Information:
This work was partly supported through NSF (DMS-0757355). This article has drawn heavily upon material from Ellad Tadmor and Ronald Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques, ©2010 Ellad Tadmor and Ronald Miller, forthcoming Cambridge University Press, reproduced with permission.
- Irving-Kirkwood-Noll procedure
- Microscopic interpretation
- Spatial averaging
- Statistical mechanics