Skip to main content
SpringerLink
Log in

A Unified Interpretation of Stress in Molecular Systems

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1989)

    Google Scholar 

  2. Belytchko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139, 3–47 (1996)

    Article  Google Scholar 

  3. Blumenthal, L.M.: Theory and Applications of Distance Geometry, 2nd edn. Chelsea, New York (1970)

    MATH  Google Scholar 

  4. Braun, W.: Distance geometry and related methods for protein structure determination from NMR data. Q. Rev. Biophys. 19(3/4), 115–157 (1987)

    Article  Google Scholar 

  5. Cauchy, A.: Sur l’éequilibre et le mouvement d’un système du points matériels sollicités par des forces d’attraction ou de répulsion mutuelle. In: Exercises de mathématique, vol. 3, pp. 227–252. Chez de Bure Frères, Paris (1928)

    Google Scholar 

  6. Cauchy, A.: De la pression ou tension dans un système de points matériels. In: Exercises de mathématique, vol. 3, pp. 253–277. Chez de Bure Frères, Paris (1928)

    Google Scholar 

  7. Cheung, K.S., Yip, S.: Atomic level stress in an inhomogeneous system. J. Appl. Phys. 70(10), 5688–5690 (1991)

    Article  ADS  Google Scholar 

  8. Clausius, R.: On a mechanical theorem applicable to heat. Philos. Mag. 40, 122–127 (1870)

    Google Scholar 

  9. Cormier, J., Rickman, J.M., Delph, T.J.: Stress calculation in atomistic simulations of perfect and imperfect solids. J. Appl. Phys. 89(1), 99–104 (2001)

    Article  ADS  Google Scholar 

  10. Crippen, G.M.: A novel approach to calculation of conformation: distance geometry. J. Comput. Phys. 24, 96–107 (1977)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Crippen, G.M., Havel, T.F.: Distance Geometry and Molecular Conformation. Wiley, New York (1988)

    MATH  Google Scholar 

  12. Daw, M., Baskes, M.: Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 29, 6443–6453 (1984)

    Article  ADS  Google Scholar 

  13. Delph, T.J.: Conservation laws for multibody interatomic potentials. Model. Simul. Mater. Sci. Eng. 13, 585–594 (2005)

    Article  ADS  Google Scholar 

  14. Dummit, D.S., Foote, R.M.: Abstract Algebra. Wiley, New York (2004)

    MATH  Google Scholar 

  15. E, W., Engquist, B., Li, X.T., Ren, W.Q., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)

    MATH  MathSciNet  Google Scholar 

  16. Ercolessi, F., Adams, J.B.: Interatomic potentials from first-principles calculations—the force-matching method. Europhys. Lett. 26(8), 583–588 (1994)

    Article  ADS  Google Scholar 

  17. Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Liquids. Academic Press, London (1990)

    MATH  Google Scholar 

  18. Evans, L.C.: Partial Differential Equations, vol. 19. American Mathematical Society, Providence (2002)

    Google Scholar 

  19. Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, New York (1999)

    MATH  Google Scholar 

  20. Fosdick, R.L.: Private communication (2009)

  21. Fosdick, R.L., Virga, E.G.: A variational proof of the stress theorem of Cauchy. Arch. Ration. Mech. Anal. 105, 95–103 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading (1965)

    Google Scholar 

  23. Hardy, R.J.: Formulas for determining local properties in molecular dynamics simulation: Shock waves. J. Chem. Phys. 76(1), 622–628 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  24. Hardy, R.J., Root, S., Swanson, D.R.: Continuum properties from molecular simulations. AIP Conf. Proc. 620, 363–366 (2002)

    Article  ADS  Google Scholar 

  25. Havel, T.F., Kuntz, I.D., Crippen, G.M.: The theory and practice of distance geometry. Bull. Math. Biol. 45, 665–720 (1983)

    MATH  MathSciNet  Google Scholar 

  26. Huang, K.: Statistical Mechanics. Wiley, New York (1963)

    Google Scholar 

  27. Irving, J.H., Kirkwood, G.: The statistical mechanics theory of transport processes. iv. The equations of hydrodynamics. J. Chem. Phys. 18(6), 817–829 (1950)

    Article  MathSciNet  ADS  Google Scholar 

  28. James, R.D.: Objective structures. J. Mech. Phys. Solids 54, 2354–2390 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Kirkwood, G.: The statistical mechanical theory of transport processes. J. Chem. Phys. 14, 180–201 (1946)

    Article  ADS  Google Scholar 

  30. Klapper, M.H., DeBrota, D.: Use of Caley–Menger determinants in the calculation of molecular structures. J. Comput. Phys. 37, 56–69 (1980)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  31. Knap, J., Ortiz, M.: An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49(9), 1899–1923 (2001)

    Google Scholar 

  32. Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1986)

    Google Scholar 

  33. Lehoucq, R.B., Silling, S.A.: Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56, 1566–1577 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day Series in Mathematical Physics (1963)

  35. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1927)

    MATH  Google Scholar 

  36. Lutsko, J.F.: Stress and elastic constants in anisotropic solids: moleculare dynamics techniques. J. Appl. Phys. 64(3), 1152–1154 (1988)

    Article  ADS  Google Scholar 

  37. Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Upper Saddle River (1969)

    Google Scholar 

  38. Marc, G., McMillan, W.G.: The virial theorem. Adv. Chem. Phys. 58, 209–361 (1985)

    Article  Google Scholar 

  39. Martin, J.W.: Many body forces in metals and the Brugger elastic constants. J. Phys. C, Solid State Phys. 8, 2837–2857 (1975)

    Article  ADS  Google Scholar 

  40. Maxwell, J.C.: On reciprocal figures, frames and diagrams of forces. Transl. R. Soc. Edinb. XXVI, 1–43 (1870)

    Google Scholar 

  41. Maxwell, J.C.: Van der Waals on the continuity of the gaseous and liquid states. Nature 10, 477–480 (1874)

    Article  ADS  Google Scholar 

  42. Morante, S., Rossi, G.C.: The stress tensor of a molecular system: an exercise in statistical mechanics. J. Chem. Phys. 125, 034,101-1–034,101-11 (2006)

    Article  Google Scholar 

  43. Murdoch, A.I.: The motivation of continuum concepts and relations from discrete considerations. J. Appl. Math. Mech. 36, 163–187 (1982)

    Article  MathSciNet  Google Scholar 

  44. Murdoch, A.I.: On the microscopic interpretation of stress and couple stress. J. Elast. 71, 105–131 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  45. Murdoch, A.I.: A critique of atomistic definitions of the stress tensor. J. Elast. 88, 113–140 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  46. Murdoch, A.I., Bedeaux, D.: On the physical interpretation of fields in continuum mechanics. Int. J. Eng. Sci. 31(10), 1345–1373 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  47. Murdoch, A.I., Bedeaux, D.: Continuum equations of balance via weighted averages of microscopic quantities. Proc. R. Soc. Lond. A 445, 157–179 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  48. Noll, W.: Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistichen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955)

    MathSciNet  Google Scholar 

  49. Pitteri, M.: On a statistical-kinetic model for generalized continua. Arch. Ration. Mech. Anal. 111, 99–120 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  50. Porta, J.M., Ros, L., Thomas, F., Torras, C.: A Branch-and-Prune solver for distance constraints. IEEE Trans. Robotics 21(2), 176–187 (2005)

    Article  Google Scholar 

  51. Rudd, R.E., Broughton, J.Q.: Concurrent coupling of length scales in solid state systems. Phys. Status Solidi B 217, 251–291 (2000)

    Article  ADS  Google Scholar 

  52. Schofield, P., Henderson, J.R.: Statistical mechanics of inhomogeneous fluids. Proc. R. Soc. Lond. A 379, 232–246 (1982)

    ADS  Google Scholar 

  53. Shenoy, V.B., Miller, R., Tadmor, E., Rodney, D., Phillips, R., Ortiz, M.: An adaptive methodology for atomic scale mechanics: the quasicontinuum method. J. Mech. Phys. Solids 47, 611–642 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  54. Shilkrot, L.E., Miller, R.E., Curtin, W.: Multiscale plasticity modeling: coupled atomistic and discrete dislocation mechanics. J. Mech. Phys. Solids 52(4), 755–787 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  55. Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  56. Sippl, M.J., Scheraga, H.A.: Caley–Menger coordinates. Proc. Nat. Acad. Sci. 83, 2283–2287 (1986)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  57. Stillinger, H., Weber, A.: Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31, 5262–5271 (1985)

    Article  ADS  Google Scholar 

  58. Tadmor, E.B., Miller, R.E.: A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17, 053,001 (2009)

    Google Scholar 

  59. Tadmor, E.B., Ortiz, M., Phillips, R.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563 (1996)

    Article  ADS  Google Scholar 

  60. Tersoff, J.: Empirical interatomic potential for silicon with improved elastic properties. Phys. Rev. B 38(14), 9902–9905 (1988)

    Article  ADS  Google Scholar 

  61. Truesdell, C.: Essays in the History of Mechanics. Springer, Berlin (1968)

    MATH  Google Scholar 

  62. Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin (1965)

    Google Scholar 

  63. Tsai, D.H.: The virial theorem and stress calculation in molecular dynamics. J. Chem. Phys. 70(03), 1375–1382 (1979)

    Article  ADS  Google Scholar 

  64. Wajnryb, E., Altenberger, A.R., Dahler, J.S.: Uniqueness of the microscopic stress tensor. J. Chem. Phys. 103(22), 9782–9787 (1995)

    Article  ADS  Google Scholar 

  65. Weiner, J.H.: Statistical Mechanics of Elasticity, 2nd edn. Dover, Mineola (2002)

    MATH  Google Scholar 

  66. Xiao, S., Belytschko, T.: A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193, 1645–1669 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  67. 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)

    Article  MATH  ADS  Google Scholar 

  68. Zimmerman, J.A., Webb, E.B., III, Hoyt, J.J., Jones, R.E., Klein, P.A., Bammann, D.J.: Calculation of stress in atomistic simulation. Model. Simul. Mater. Sci. Eng. 12, S319–S332 (2004)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering and Mechanics, The University of Minnesota, Minneapolis, MN, USA

    Nikhil Chandra Admal & E. B. Tadmor

Authors
  1. Nikhil Chandra Admal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. B. Tadmor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikhil Chandra Admal.

Additional information

The authors would like to dedicate this article to Jack Irving, who passed away in 2008 at the age of 87. Irving, while a graduate student on leave from Princeton, worked with Prof. John Kirkwood at Caltech on the fundamental non-equilibrium statistical mechanics theory which serves as the basis for the present article.

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Admal, N.C., Tadmor, E.B. A Unified Interpretation of Stress in Molecular Systems. J Elast 100, 63–143 (2010). https://doi.org/10.1007/s10659-010-9249-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-010-9249-6

Keywords

Search

Navigation