Journal of Elasticity

, Volume 126, Issue 1, pp 95–125 | Cite as

Static and Dynamic Green’s Functions in Peridynamics

  • Linjuan Wang
  • Jifeng Xu
  • Jianxiang WangEmail author


We derive the static and dynamic Green’s functions for one-, two- and three-dimensional infinite domains within the formalism of peridynamics, making use of Fourier transforms and Laplace transforms. Noting that the one-dimensional and three-dimensional cases have been previously studied by other researchers, in this paper, we develop a method to obtain convergent solutions from the divergent integrals, so that the Green’s functions can be uniformly expressed as conventional solutions plus Dirac functions, and convergent nonlocal integrals. Thus, the Green’s functions for the two-dimensional domain are newly obtained, and those for the one and three dimensions are expressed in forms different from the previous expressions in the literature. We also prove that the peridynamic Green’s functions always degenerate into the corresponding classical counterparts of linear elasticity as the nonlocal length tends to zero. The static solutions for a single point load and the dynamic solutions for a time-dependent point load are analyzed. It is analytically shown that for static loading, the nonlocal effect is limited to the neighborhood of the loading point, and the displacement field far away from the loading point approaches the classical solution. For dynamic loading, due to peridynamic nonlinear dispersion relations, the propagation of waves given by the peridynamic solutions is dispersive. The Green’s functions may be used to solve other more complicated problems, and applied to systems that have long-range interactions between material points.


Green’s function Peridynamics Nonlocality Integro-differential equation 

Mathematics Subject Classification (2000)

82B21 74A99 45A05 45B05 



The work is supported by the National Natural Science Foundation of China under Grant 11521202. The authors thank the anonymous reviewers whose insightful comments and suggestions improved the technical content of this work. The authors also thank Professors Minzhong Wang, Kefu Huang and Shaoqiang Tang of Peking University for helpful discussions.


  1. 1.
    Bazant, Z.P., Jirasek, M.: Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128(11), 1119–1149 (2002) CrossRefGoogle Scholar
  2. 2.
    Silling, S.A.: Origin and effect of nonlocality in a composite. J. Mech. Mater. Struct. 9(2), 245–258 (2014) CrossRefGoogle Scholar
  3. 3.
    Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–742 (1967) CrossRefzbMATHGoogle Scholar
  4. 4.
    Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane-waves. Int. J. Eng. Sci. 10(5), 425–435 (1972) CrossRefzbMATHGoogle Scholar
  5. 5.
    Eringen, A.C., Edelen, D.G.B.: Nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Silling, S.A.: Reformation of elasticity theory for discontinuities and longrange force. J. Mech. Phys. Solids 48(1), 175 (2000) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88(2), 151–184 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gerstle, W., Sau, N., Silling, S.A.: Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237(12–13), 1250–1258 (2007) CrossRefGoogle Scholar
  9. 9.
    Xu, J., Askari, A., Weckner, O., Silling, S.: Peridynamic analysis of impact damage in composite laminates. J. Aerosp. Eng. 21(3), 187–194 (2008) CrossRefGoogle Scholar
  10. 10.
    Kilic, B., Agwai, A., Madenci, E.: Peridynamic theory for progressive damage prediction in center-cracked composite laminates. Compos. Struct. 90(2), 141–151 (2009) CrossRefGoogle Scholar
  11. 11.
    Oterkus, E., Madenci, E.: Peridynamic analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct. 7(1), 45–84 (2012) CrossRefGoogle Scholar
  12. 12.
    Taylor, M., Steigmann, D.J.: A two-dimensional peridynamic model for thin plates. Math. Mech. Solids 20(8), 998–1010 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Silling, S.A., Lehoucq, R.B.: Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44(44), 73–168 (2010) CrossRefGoogle Scholar
  14. 14.
    Oterkus, S., Madenci, E., Agwai, A.: Peridynamic thermal diffusion. J. Comput. Phys. 265, 71–96 (2014) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sun, S., Sundararaghavan, V.: A peridynamic implementation of crystal plasticity. Int. J. Solids Struct. 51(19–20), 3350–3360 (2014) CrossRefGoogle Scholar
  16. 16.
    Oterkus, S., Madenci, E., Agwai, A.: Fully coupled peridynamic thermomechanics. J. Mech. Phys. Solids 64, 1–23 (2014) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chen, Z., Bobaru, F.: Peridynamic modeling of pitting corrosion damage. J. Mech. Phys. Solids 78, 352–381 (2015) ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Jabakhanji, R., Mohtar, R.H.: A peridynamic model of flow in porous media. Adv. Water Resour. 78, 22–35 (2015) ADSCrossRefGoogle Scholar
  19. 19.
    Emmrich, E., Weckner, O.: Analysis and numerical approximation of an integrodifferential equation modeling non-local effects in linear elasticity. Math. Mech. Solids 12(4), 363–384 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gunzburger, M., Lehoucq, R.B.: A nonlocal vector calculus with application to nonlocal boundary value problems. Multiscale Model. Simul. 8(5), 1581–1598 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhou, K., Du, Q.: Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48(5), 1759–1780 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Du, Q., Zhou, K.: Mathematical analysis for the peridynamic nonlocal continuum theory. ESAIM: Math. Model. Numer. Anal. 45(2), 217–234 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23(3), 493–540 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Silling, S.A., Zimmermann, M., Abeyaratne, R.: Deformation of a peridynamic bar. J. Elast. 73(1–3), 173–190 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Weckner, O., Abeyaratne, R.: The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53(3), 705–728 (2005) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Emmrich, E., Weckner, O.: The peridynamic equation and its spatial discretisation. Math. Model. Anal. 12(1), 17–27 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Weckner, O., Brunk, G., Epton, M.A., Silling, S.A., Askari, E.: Green’s functions in non-local three-dimensional linear elasticity. Proc. R. Soc. A, Math. Phys. Eng. Sci. 465(2111), 3463–3487 (2009) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mikata, Y.: Analytical solutions of peristatic and peridynamic problems for a 1d infinite rod. Int. J. Solids Struct. 49(21), 2887–2897 (2012) CrossRefGoogle Scholar
  29. 29.
    Pan, E., Chen, W.: Static Green’s Functions in Anisotropic Media. Cambridge University Press, New York (2015) zbMATHGoogle Scholar
  30. 30.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966) zbMATHGoogle Scholar
  31. 31.
    Kilic, B.: Peridynamic theory for progressive failure prediction in homogeneous and heterogeneous materials. PhD thesis, University of Arizona (2008) Google Scholar
  32. 32.
    Oterkus, E.: Peridynamic theory modeling three-dimensional damage growth in metallic and composite structures. PhD thesis, University of Arizona (2010) Google Scholar
  33. 33.
    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54(4), 667–696 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lehoucq, R.B., Silling, S.A.: Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56(4), 667–696 (2008) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.State Key Laboratory for Turbulence and Complex System, Department of Mechanics and Engineering Science, College of EngineeringPeking UniversityBeijingP.R. China
  2. 2.CAPT, HEDPS and IFSA Collaborative Innovation Center of MoEPeking UniversityBeijingP.R. China
  3. 3.Beijing Aeronautical Science and Technology Research InstituteBeijingP.R. China

Personalised recommendations