A Critique of Atomistic Definitions of the Stress Tensor
 A. Ian Murdoch
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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 virialbased argument, and from scaledependent 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 divergencefree 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.
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 Title
 A Critique of Atomistic Definitions of the Stress Tensor
 Journal

Journal of Elasticity
Volume 88, Issue 2 , pp 113140
 Cover Date
 20070801
 DOI
 10.1007/s1065900791215
 Print ISSN
 03743535
 Online ISSN
 15732681
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Stress
 Molecular averaging
 Microscopic interpretation
 Weighting function
 Nanoscale systems
 74A25
 Industry Sectors
 Authors

 A. Ian Murdoch ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow, G1 1XH, UK