Journal of Elasticity

, 93:13 | Cite as

Convergence of Peridynamics to Classical Elasticity Theory

  • S. A. SillingEmail author
  • R. B. Lehoucq


The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.


Peridynamics Elasticity theory Non-local theory Integral equations Continuum mechanics 

Mathematics Subject Classification (2000)



  1. 1.
    Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000) zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Lehoucq, R.B., Silling, S.A.: Statistical coarse-graining of molecular dynamics into peridynamics. Report SAND2007-6410, Sandia National Laboratories, Albuquerque, New Mexico (2007) Google Scholar
  4. 4.
    Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005) CrossRefGoogle Scholar
  5. 5.
    Emmrich, E., Weckner, O.: Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity. Math. Mech. Solids 12, 363–384 (2007) CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Silling, S.A., Bobaru, F.: Peridynamic modeling of membranes and fibers. Int. J. Nonlinear Mech. 40, 395–409 (2005) zbMATHCrossRefADSGoogle Scholar
  7. 7.
    Bobaru, F.: Influence of van der Waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach. Model. Simul. Mat. Sci. Eng. 15, 397–417 (2007) CrossRefADSGoogle Scholar
  8. 8.
    Bazant, Z.P., Jirasek, M.: Nonlocal integral formulations of plasticity and damage: Survey of progress. J. Eng. Mech. 128, 1119–1149 (2002) CrossRefGoogle Scholar
  9. 9.
    Gerstle, W., Sau, N., Silling, S.: Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007) CrossRefGoogle Scholar
  10. 10.
    Truesdell, C.: A First Course in Rational Continuum Mechanics. Volume I: General Concepts. Academic, New York (1977), pp. 120–121 Google Scholar
  11. 11.
    Giessibl, F.J.: Atomic resolution of the silicon (111)-(7×7) surface by atomic force microsopy. Science 267, 68–71 (1995) CrossRefADSGoogle Scholar
  12. 12.
    Kuo, S.C., Sheetz, M.P.: Force of single kinesin molecules measured with optical tweezers. Science 260, 232–234 (1993) CrossRefADSGoogle Scholar
  13. 13.
    Block, S.M.: Making light work with optical tweezers. Nature 360, 493–495 (1992) CrossRefADSGoogle Scholar
  14. 14.
    Maranganti, R., Sharma, P.: Length scales at which classical elasticity breaks down for various materials. Phys. Rev. Lett. 98, 195504-1–195504-4 (2007) ADSGoogle Scholar
  15. 15.
    Israelachvili, J.: Intermolecular and Surface Forces. Academic, New York (1992), pp. 155–157, 177 Google Scholar
  16. 16.
    Abraham, F.F., Brodbeck, D., Rafey, R.A., Rudge, W.E.: Instability dynamics of fracture: A computer simulation investigation. Phys. Rev. Lett. 73, 272–275 (1994) CrossRefADSGoogle Scholar
  17. 17.
    Eringen, A.C., Speziale, C.G., Kim, B.S.: Crack-tip problem in non-local elasticity. J. Mech. Phys. Solids 25, 339–355 (1977) zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Irving, J.H., Kirkwood, J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950) CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Noll, W.: Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955) MathSciNetGoogle Scholar
  20. 20.
    Hardy, R.J.: Formulas for determining local properties in molecular-dynamics simulations: Shock waves. J. Chem. Phys. 76, 622–628 (1982) CrossRefADSGoogle Scholar
  21. 21.
    Murdoch, A.I.: On the microscopic interpretation of stress and couple stress. J. Elast. 71, 105–131 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Zimmermann, M.: A continuum theory with long-range forces for solids. Ph.D. thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering (2005) Google Scholar
  23. 23.
    Emmrich, E., Weckner, O.: On the well posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5, 851–864 (2007) zbMATHMathSciNetGoogle Scholar
  24. 24.
    Lehoucq, R.B., Silling, S.A.: Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56, 1566–1577 (2008) CrossRefADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Aifantis, E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999) CrossRefGoogle Scholar
  27. 27.
    Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245–2271 (2001) zbMATHCrossRefADSGoogle Scholar
  28. 28.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic, New York (1981), pp. 165–180 zbMATHGoogle Scholar
  29. 29.
    Abeyaratne, R.: An admissibility condition for equilibrium shocks in finite elasticity. J. Elast. 13, 175–184 (1983) zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Dayal, K., Bhattacharya, K.: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006) zbMATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Howison, S.D., Ockendon, J.R.: Singularity development in moving-boundary problems. Q. J. Mech. Appl. Math. 38, 343–359 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Sahu, S.K., Das, P.K., Bhattacharyya, S.: Rewetting analysis of hot surfaces with internal heat source by the heat balance integral method. Heat Mass Transf. (2007). doi: 10.1007/s00231-007-0360-6 Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Sandia National LaboratoriesAlbuquerqueUSA

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