Journal of Elasticity

, Volume 135, Issue 1–2, pp 435–456 | Cite as

On the Mechanical Modeling of Matter, Molecular and Continuum

  • Paolo Podio-GuidugliEmail author


Any mechanical modeling of matter depends primarily on the chosen spatial and temporal observation scales. Roughly speaking, there are three such scales, microscopic for quantum, mesoscopic for statistical, and macroscopic for continuum mechanics. This by itself demands for adequate scale-bridging procedures. This paper focuses on the passage from molecular to continuum formulations of the basic balance laws of linear momentum and energy, kinetic, internal, and total. The procedure used is a modification of that introduced by Irving & Kirkwood (J. Chem. Phys. 18(6):817–829, 1950), as improved by Noll (Indiana Univ. Math. J. 4:627–646, 1955); alternative procedures are mentioned. The proposed modification consists primarily in equipping provisionally the target continuum balances with internal-source terms accounting at the macroscopic scale for microscopic motion randomness, when, e.g., a loosely aggregated material system evolves at relatively high temperature. Attention is also devoted to those continuum notions that either are hardly given a universally accepted molecular counterpart or even cannot and ultimately need not be given one, such as the notion of material point.


Molecular mechanics Continuum mechanics Scale bridging 

Mathematics Subject Classification

70A05 74A05 74A99 



Over the years, many of the matters I here dealt with have been the subject of a number of useful discussions with Antonio DiCarlo, which I gratefully acknowledge.


  1. 1.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1927) zbMATHGoogle Scholar
  2. 2.
    Truesdell, C., Toupin, R.A.: The classical field theories. In: Handbuch der Physik, III/1. Springer, Berlin (1960) Google Scholar
  3. 3.
    DiCarlo, A.: A major serendipitous contribution to continuum mechanics. Mech. Res. Commun. 93, 41–46 (2018). CrossRefGoogle Scholar
  4. 4.
    DiCarlo, A.: Continuum mechanics as a computable coarse-grained picture of molecular dynamics (2018, this volume) Google Scholar
  5. 5.
    Murdoch, A.I.: Physical Foundations of Continuum Mechanics. Cambridge University Press, Cambridge (2012) CrossRefzbMATHGoogle Scholar
  6. 6.
    Irving, J.H., Kirkwood, J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18(6), 817–829 (1950). ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Noll, W.: Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der Statistischen Mechanik. Indiana Univ. Math. J. 4, 627–646 (1955). and, J. Ration. Mech. Anal. 4, 627–646 (1955). English translation: Derivation of the fundamental equations of continuum thermodynamics from statistical mechanics. J. Elast. 100, 5–24 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Noll, W.: Thoughts on the concept of stress. J. Elast. 100, 25–32 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Andersen, H.C.: Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys. 72, 2384–2393 (1980) ADSCrossRefGoogle Scholar
  10. 10.
    Parrinello, M., Rahman, A.: Crystal structure and pair potentials: a molecular-dynamics study. Phys. Rev. Lett. 45, 1196–1199 (1980) ADSCrossRefGoogle Scholar
  11. 11.
    Parrinello, M., Rahman, A.: Polymorphic transitions in single crystals: a new molecular dynamics method. J. Appl. Phys. 52, 7182–7190 (1981) ADSCrossRefGoogle Scholar
  12. 12.
    Parrinello, M., Rahman, A.: Strain fluctuations and elastic constants. J. Chem. Phys. 76, 2662–2666 (1982) ADSCrossRefGoogle Scholar
  13. 13.
    Tuckerman, M.: Statistical Mechanics: Theory and Molecular Simulation. Oxford University Press, London (2010) zbMATHGoogle Scholar
  14. 14.
    Podio-Guidugli, P.: On microscopic and macroscopic notions of stress. In: Elasticity and Inelasticity. Proceedings of “Problems of Mechanics of Deformable Bodies”, pp. 279–285, Moscow, Jan 20–21, 2011. Moscow University Press, Moscow (2011) Google Scholar
  15. 15.
    Giusteri, G.G., Podio-Guidugli, P., Fried, E.: Continuum balances from extended Hamiltonian dynamics. J. Chem. Phys. 146, 224101 (2017) ADSCrossRefGoogle Scholar
  16. 16.
    Hellinger, E.: Die Allgemeinen Ansatze der Mechanik der Kontinua. Enzyclopaedie der Matematischen Wissenschaften, Art. IV. 30 (1914) Google Scholar
  17. 17.
    Noll, W.: The foundations of classical mechanics in the light of recent advances in continuum mechanics. In: Noll, W. (ed.) Proceedings of an International Symposium on the Axiomatic Method, University of California, Berkeley, Dec. 27, 1957–Jan. 4, 1958, pp. 265–281. Springer, Berlin (1959). Reprinted in: The Foundations of Mechanics and Thermodynamics: Selected Papers by W. Noll, pp. 32–47. Springer, Berlin (1974). Google Scholar
  18. 18.
    Truesdell, C., Toupin, R.A.: The classical field theories. In: Flügge, S. (ed.) Handbuch der Physik III/1. Springer, Berlin (1960) Google Scholar
  19. 19.
    Truesdell, C., Noll, W.: The nonlinear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik III/3. Springer, Berlin (1965) Google Scholar
  20. 20.
    Pitteri, M.: Continuum equations of balance in classical statistical mechanics. Arch. Ration. Mech. Anal. 94, 291–305 (1986). Erratum: 100, 315–316 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Seguin, B., Fried, E.: Statistical foundations of liquid-crystal theory. I: Discrete systems of rod-like molecules. Arch. Ration. Mech. Anal. 206, 1039–1072 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Seguin, B., Fried, E.: Statistical foundations of liquid-crystal theory II: Macroscopic balance laws. Arch. Ration. Mech. Anal. 207, 1–37 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Murdoch, A.I.: A corpuscular approach to continuum mechanics: basic considerations. Arch. Ration. Mech. Anal. 88(4), 291–321 (1985) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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) CrossRefzbMATHGoogle Scholar
  25. 25.
    Murdoch, A.I.: On molecular modelling and continuum concepts. J. Elast. 100(1), 33–61 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Roberts, A.J.: A One-dimensional Introduction to Continuum Mechanics. World Scientific, Singapore (1994) CrossRefzbMATHGoogle Scholar
  27. 27.
    Capriz, G., Mazzini, G.: A \(\sigma\)-algebra and a concept of limit for bodies. Math. Models Methods Appl. Sci. 10(6), 801–813 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Noll, W.: On the concept of force. (2007)
  29. 29.
    Podio-Guidugli, P.: On the aggregation state of simple materials. In: Šilhavý, M. (ed.) Mathematical Modeling of Bodies with Complicated Bulk and Boundary Behavior. Quaderni di Matematica, vol. 20, pp. 159–168 (2008) Google Scholar
  30. 30.
    Admal, N.C., Tadmor, E.B.: A unified interpretation of stress in molecule systems. J. Elast. 100(1), 63–143 (2010) CrossRefzbMATHGoogle Scholar
  31. 31.
    Tadmor, E.B., Miller, R.E.: Modeling Materials: Continuum, Atomistic and Multiscale Techniques. Cambridge University Press, Cambridge (2011) CrossRefzbMATHGoogle Scholar
  32. 32.
    Groot, R.D., Warren, P.B.: Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys. 107(11), 4423 (1997) ADSCrossRefGoogle Scholar
  33. 33.
    Podio-Guidugli, P.: On the modeling of transport phenomena in continuum and statistical mechanics. Discrete Contin. Dyn. Syst. 10(6), 1393–1411 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Podio-Guidugli, P.: Inertia and invariance. Ann. Mat. Pura Appl. (4) CLXXII, 103–124 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gurtin, M.E., Podio-Guidugli, P.: Configurational forces and the basic laws for crack propagation. J. Mech. Phys. Solids 44(6), 905–927 (1996) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Gurtin, M.E., Podio-Guidugli, P.: On configurational inertial forces at a phase interface. J. Elast. 44, 255–269 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Gurtin, M.E., Podio-Guidugli, P.: Configurational forces and a constitutive theory for crack propagation that allows for kinking and curving. J. Mech. Phys. Solids 46(2), 1–36 (1998) MathSciNetzbMATHGoogle Scholar
  38. 38.
    Podio-Guidugli, P.: La scelta dei termini inerziali per i continui con microstruttura. Rend. Lincei, Mat. Appl. Ser. IX XIV(4), 319–326 (2003) Google Scholar
  39. 39.
    Capriz, G., Podio-Guidugli, P.: Whence the boundary conditions in modern continuum physics? In: Atti dei Convegni Lincei, vol. 210, pp. 19–42. Accademia dei Lincei, Rome (2004) Google Scholar
  40. 40.
    Capriz, G., Giovine, P.: Classes of ephemeral continua. Math. Methods Appl. Sci. 41(3), 1175–1196 (2018) ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Accademia Nazionale dei LinceiRomeItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma TorVergataRomeItaly

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