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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2011002674
pages 609-622


Stewart A. Silling
Multiscale Dynamic Material Modeling Department, Sandia National Laboratories, Albuquerque, New Mexico,87185, USA


A method is obtained for deriving peridynamic material models for a sequence of increasingly coarsened descriptions of a body. The starting point is a known detailed, small scale linearized state-based description. Each successively coarsened model excludes some of the material present in the previous model, and the length scale increases accordingly. This excluded material, while not present explicitly in the coarsened model, is nevertheless taken into account implicitly through its effect on the forces in the coarsened material. Numerical examples demonstrate that the method accurately reproduces the effective elastic properties of a composite as well as the effect of a small defect in a homogeneous medium.


  1. Alali, B. and Lipton, R., Multiscale analysis of heterogeneous media in the peridynamic formulation.

  2. Bazant, Z. P., Can multiscale-multiphysics methods predict softening damage and structural failure?. DOI: 10.1615/IntJMultCompEng.v8.i1.50

  3. Bobaru, F., Yang, M., Alves, L. F., Silling, S. A., Askari, A., and Xu, J., Convergence, adaptive refinement, and scaling in 1D peridynamics. DOI: 10.1002/nme.2439

  4. Emmrich, E. and Weckner, O., Analysis and numerical approximation of an integrodifferential equation modeling non-local effects in linear elasticity. DOI: 10.1177/1081286505059748

  5. Emmrich, E. and Weckner, O., On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity.

  6. Emmrich, E. and Weckner, O., The peridynamic equation and its spatial discretization. DOI: 10.3846/1392-6292.2007.12.17-27

  7. Eom, K., Baek, S.-C., Ahn, J.-H., and Na, S., Coarse-graining of protein structures for the normal mode studies. DOI: 10.1002/jcc.20672

  8. Lehoucq, R. B. and Silling, S. A., Statistical coarse-graining of molecular dynamics into peridynamics. DOI: 10.2172/922771

  9. Silling, S. A., Reformulation of elasticity theory for discontinuities and long-range forces. DOI: 10.1016/S0022-5096(99)00029-0

  10. Silling, S. A., Linearized theory of peridynamic states. DOI: 10.1007/s10659-009-9234-0

  11. Silling, S. A. and Askari, E., A meshfree method based on the peridynamic model of solid mechanics. DOI: 10.1016/j.compstruc.2004.11.026

  12. Silling, S. A. and Lehoucq, R. B., Peridynamic theory of solid mechanics. DOI: 10.1016/S0065-2156(10)44002-8

  13. Silling, S. A., Zimmermann, M., and Abeyaratne, R., Deformation of a peridynamic bar. DOI: 10.1023/B:ELAS.0000029931.03844.4f

  14. Tirion, M. M., Large amplitude elastic motions in proteins from a single-parameter, atomic analysis. DOI: 10.1103/PhysRevLett.77.1905

  15. Weckner, O. and Abeyaratne, R., The effect of long-range forces on the dynamics of a bar. DOI: 10.1016/j.jmps.2004.08.006

  16. Weckner, O., Brunk, G., Epton, M. A., Silling, S. A., and Askari, E., Green's functions in non-local three-dimensional linear elasticity. DOI: 10.1098/rspa.2009.0234

  17. Weckner, O. and Emmrich, E., Numerical simulation of the dynamics of a nonlocal, inhomogeneous, infinite bar.

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