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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/3. Springer, Berlin (1965)
Eringen, A.C.: Mechanics of Continua. Wiley, New York (1967)
Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic, New York (1981)
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)
McLellan, A.G.: Virial theorem generalized. Am. J. Phys. 42, 239–243 (1974)
Swenson, R.J.: Comments on virial theorems for bounded systems. Am. J. Phys. 51, 940–942 (1983)
Tsai, D.H.: The virial theorem and stress calculation in molecular dynamics. J. Chem. Phys. 70, 1375–1382 (1979)
Hardy, R.J.: Formulas for determining local properties in molecular-dynamics simulations: shock waves. J. Chem. Phys. 76, 622–628 (1982)
Irving, J.H., Kirkwood, J.G.: The statistical theory of transport processes IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950)
Murdoch, A.I., Bedeaux, D.: Continuum equations of balance via weighted averages of microscopic quantities. Proc. R. Soc. Lond. A 445, 157–179 (1994)
Noll, W.: Der Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955)
Root, S., Hardy, R.J., Swanson, D.R.: Continuum predictions from molecular dynamics simulations: shock waves. J. Chem. Phys. 118, 3161–3165 (2003)
Murdoch, A.I.: On the microscopic interpretation of stress and couple stress. J. Elast. 71, 105–131 (2003)
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)
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)
Goldstein, H., Poole, C., Safko, J.: Classical mechanics, 3rd edn. Addison-Wesley, San Francisco (2002)
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)
Brush, S.G.: The kind of motion we call heat. North-Holland, Amsterdam (1986)
de Groot, S.R., Mazur, P.: Non-equilibrium mechanics. Dover, Mineola (1984)
Murdoch, A.I.: On effecting averages and changes of scale via weighting functions. Arch. Mech. 50, 531–539 (1998)
Murdoch, A.I.: A critique of atomistic definitions of the stress tensor. Mathematics Departmental Research Report, University of Strathclyde, Glasgow (2007)
Nicholson, M.M.: Surface tension in ionic crystals. Proc. R. Soc. A 228, 490–510 (1955)
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)
Murdoch, A.I.: Some fundamental aspects of surface modelling. J. Elast. 80, 33–52 (2005)
Miller, R.E., Tadmor, E.B.: The quasicontinuum method: overview, applications and current directions. J. Comput. Aided Mater. Des. 9, 203–239 (2002)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Murdoch, A.I. A Critique of Atomistic Definitions of the Stress Tensor. J Elasticity 88, 113–140 (2007). https://doi.org/10.1007/s10659-007-9121-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-007-9121-5