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International Journal for Multiscale Computational Engineering

Impact factor: 0.963

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

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2011002688
pages 623-634


Olaf Weckner
The Boeing Company, P.O. Box 3707, MC 42-26, Seattle, Washington, 98124-2207, USA
Stewart A. Silling
Multiscale Dynamic Material Modeling Department, Sandia National Laboratories, Albuquerque, New Mexico,87185, USA


All materials exhibit wave dispersion at "small" wavelengths leading to non-linearities in experimentally determined dispersion curves. Classical local elasticity fails to predict these non-linearities. Nonlocal continuum mechanics allows for the prediction of the elastic behavior over a considerably wider range of lengthscales. Starting from ab initio lattice dynamics calculations we determine the elastic constants and the phonon dispersion relation for silicon. We verify our results using inelastic neutron scattering data. Next we develop the theoretical and numerical framework to construct nonlocal constitutive equations for longitudinal and transverse acoustic modes.


  1. Brockhouse, B. N., Arase, T., Caglioti, G., Rao, K. R., and Woods, A. D. B., Crystal dynamics of lead i-dispersion curves at 100 k. DOI: 10.1103/PhysRev.128.1099

  2. Clark, S. J., Segall, M. D., Pickard, C. J., Hasnip, P. J., Probert, M. J., Refson, K., and Payne, M. C., First principles methods using Castep. DOI: 10.1524/zkri.220.5.567.65075

  3. Emmrich, E. and Weckner, O., Analysis and numerical approximation of an integro-differential equation modelling non-local effects in linear elasticity. DOI: 10.1177/1081286505059748

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

  5. Eringen, A. C., Linear theory of nonlocal elasticity and dispersion of plane waves. DOI: 10.1016/0020-7225(72)90050-X

  6. Eringen, A. C., Vistas of nonlocal continuum physics. DOI: 10.1016/0020-7225(92)90165-D

  7. Eringen, A. C., Nonlocal Continuum Field Theories.

  8. Eringen, A. C. and Edelen, D. G. B., On nonlocal elasticity. DOI: 10.1016/0020-7225(72)90039-0

  9. Graff, K. F., Wave Motionin Elastic Solids.

  10. Jakata, K. and Every, A. G., Determination of the dispersive elastic constants of the cubic crystals Ge, Si, GaAs, and InSb. DOI: 10.1103/PhysRevB.77.174301

  11. Kunin, I. A., Elastic Media with Microstructure.

  12. Maranganti, R. and Sharma, P., A novel atomistic approach to determine strain-gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies. DOI: 10.1016/j.jmps.2007.02.011

  13. Mindlin, R. S., Second gradient of strain and surface-tension in linear elasticity. DOI: 10.1016/0020-7683(65)90006-5

  14. Raunio, G. and Rolandson, S., Phonon dispersion relations in RbCl and RbF at 80 K. DOI: 10.1088/0022-3719/3/5/016

  15. Rogula, D., Nonlocal Theory of Material Media.

  16. Seleson, P., Parks, M. L., Gunzburger, M., and Lehoucq, R., Peridynamics as an upscaling of molecular dynamics. Multiscale Modeling and Simulation. DOI: 10.1137/09074807X

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

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

  19. Silling, S. A., A coarsening method for linear peridynamics.

  20. Silling, S. A. and Lehoucq, R. B., Convergence of peridynamics to classical elasticity theory. DOI: 10.1007/s10659-008-9163-3

  21. Silling, S. A., Epton, M., Weckner, O., Xu, J., and Askari, E., Peridynamics states and constitutive modeling. DOI: 10.1007/s10659-007-9125-1

  22. Tricomi, F. G., Integral Equations.

  23. 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

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