# A Critique of Atomistic Definitions of the Stress Tensor

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- Received:
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DOI: 10.1007/s10659-007-9121-5

- Cite this article as:
- Murdoch, A.I. J Elasticity (2007) 88: 113. doi:10.1007/s10659-007-9121-5

## Abstract

Current interest in nanoscale systems and molecular dynamical simulations has focussed attention on the extent to which continuum concepts and relations may be utilised meaningfully at small length scales. In particular, the notion of the Cauchy stress tensor has been examined from a number of perspectives. These include motivation from a virial-based argument, and from scale-dependent localisation procedures involving the use of weighting functions. Here different definitions and derivations of the stress tensor in terms of atoms/molecules, modelled as interacting point masses, are compared. The aim is to elucidate assumptions inherent in different approaches, and to clarify associated physical interpretations of stress. Following a critical analysis and extension of the virial approach, a method of spatial atomistic averaging (at any prescribed length scale) is presented and a balance of linear momentum is derived. The contribution of corpuscular interactions is represented by a force density field **f**. The balance relation reduces to standard form when **f** is expressed as the divergence of an interaction stress tensor field, **T**^{−}. The manner in which **T**^{−} can be defined is studied, since **T**^{−} is unique only to within a divergence-free field. Three distinct possibilities are discussed and critically compared. An approach to nanoscale systems is suggested in which **f** is employed directly, so obviating separate modelling of interfacial and edge effects.