Journal of Elasticity

, Volume 88, Issue 2, pp 113–140 | Cite as

A Critique of Atomistic Definitions of the Stress Tensor

  • A. Ian MurdochEmail author


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.


Stress Molecular averaging Microscopic interpretation Weighting function Nanoscale systems 

Mathematics Subject Classification (2000)



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  1. 1.
    Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/3. Springer, Berlin (1965)Google Scholar
  2. 2.
    Eringen, A.C.: Mechanics of Continua. Wiley, New York (1967)zbMATHGoogle Scholar
  3. 3.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic, New York (1981)zbMATHGoogle Scholar
  4. 4.
    Murdoch, A.I.: Foundations of continuum modelling: a microscopic perspective with applications, AMAS Lecture Notes 7. Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw (2003)Google Scholar
  5. 5.
    McLellan, A.G.: Virial theorem generalized. Am. J. Phys. 42, 239–243 (1974)CrossRefADSGoogle Scholar
  6. 6.
    Swenson, R.J.: Comments on virial theorems for bounded systems. Am. J. Phys. 51, 940–942 (1983)CrossRefADSGoogle Scholar
  7. 7.
    Tsai, D.H.: The virial theorem and stress calculation in molecular dynamics. J. Chem. Phys. 70, 1375–1382 (1979)CrossRefADSGoogle Scholar
  8. 8.
    Hardy, R.J.: Formulas for determining local properties in molecular-dynamics simulations: shock waves. J. Chem. Phys. 76, 622–628 (1982)CrossRefADSGoogle Scholar
  9. 9.
    Irving, J.H., Kirkwood, J.G.: The statistical theory of transport processes IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950)CrossRefADSGoogle Scholar
  10. 10.
    Murdoch, A.I., Bedeaux, D.: Continuum equations of balance via weighted averages of microscopic quantities. Proc. R. Soc. Lond. A 445, 157–179 (1994)ADSGoogle Scholar
  11. 11.
    Noll, W.: Der Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955)Google Scholar
  12. 12.
    Root, S., Hardy, R.J., Swanson, D.R.: Continuum predictions from molecular dynamics simulations: shock waves. J. Chem. Phys. 118, 3161–3165 (2003)CrossRefADSGoogle Scholar
  13. 13.
    Murdoch, A.I.: On the microscopic interpretation of stress and couple stress. J. Elast. 71, 105–131 (2003)zbMATHCrossRefGoogle Scholar
  14. 14.
    Zimmerman, J.A., Webb III, E.B., Hoyt, J.J., Jones, R.E., Klein, P.A., Bammann, D.J.: Calculation of stress in atomistic simulation. Model. Simul. Mater. Sci. Eng. 12, 5319–5322 (2004)CrossRefGoogle Scholar
  15. 15.
    Zhou, M.: A new look at the atomic level virial stress: on continuum-molecular system equivalence. Proc. R. Soc. Lond. A 459, 2347–2392 (2003)ADSGoogle Scholar
  16. 16.
    Goldstein, H., Poole, C., Safko, J.: Classical mechanics, 3rd edn. Addison-Wesley, San Francisco (2002)Google Scholar
  17. 17.
    Murdoch, A.I.: Some primitive concepts in continuum mechanics regarded in terms of objective space-time molecular averaging: the key rôle played by inertial observers. J. Elast. 84, 69–97 (2006)zbMATHCrossRefGoogle Scholar
  18. 18.
    Brush, S.G.: The kind of motion we call heat. North-Holland, Amsterdam (1986)Google Scholar
  19. 19.
    de Groot, S.R., Mazur, P.: Non-equilibrium mechanics. Dover, Mineola (1984)Google Scholar
  20. 20.
    Murdoch, A.I.: On effecting averages and changes of scale via weighting functions. Arch. Mech. 50, 531–539 (1998)zbMATHGoogle Scholar
  21. 21.
    Murdoch, A.I.: A critique of atomistic definitions of the stress tensor. Mathematics Departmental Research Report, University of Strathclyde, Glasgow (2007)Google Scholar
  22. 22.
    Nicholson, M.M.: Surface tension in ionic crystals. Proc. R. Soc. A 228, 490–510 (1955)ADSGoogle Scholar
  23. 23.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975) and 59, 389–390 (1975)zbMATHCrossRefGoogle Scholar
  24. 24.
    Murdoch, A.I.: Some fundamental aspects of surface modelling. J. Elast. 80, 33–52 (2005)zbMATHCrossRefGoogle Scholar
  25. 25.
    Miller, R.E., Tadmor, E.B.: The quasicontinuum method: overview, applications and current directions. J. Comput. Aided Mater. Des. 9, 203–239 (2002)CrossRefADSGoogle Scholar
  26. 26.
    Friesecke, G., James, R.D.: A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids 48, 1519–1540 (2000)zbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of StrathclydeGlasgowUK

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