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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
Article

Abstract

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.

Keywords

Green’s function Peridynamics Nonlocality Integro-differential equation 

Mathematics Subject Classification (2000)

82B21 74A99 45A05 45B05 

Notes

Acknowledgements

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.

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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

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