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Journal of Elasticity

, Volume 73, Issue 1–3, pp 173–190 | Cite as

Deformation of a Peridynamic Bar

  • S.A. Silling
  • M. Zimmermann
  • R. Abeyaratne
Article

Abstract

The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venant's principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Green's function for general loading problems.

linear integral equations Fredholm integral equations peridynamic Fourier transform discontinuity Green's function point load elastic bar 

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References

  1. 1.
    S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000) 175–209.Google Scholar
  2. 2.
    R.J. Atkin and N. Fox, An Introduction to the Theory of Elasticity. (1980) Longman, London, pp. 186–187.Google Scholar
  3. 3.
    M.J. Lighthill, Fourier Analysis and Generalised Functions. Cambridge Univ. Press, Cambridge (1978).Google Scholar
  4. 4.
    E. Zauderer, Partial Differential Equations of Applied Mathematics. Wiley, New York (1983) pp. 365–376.Google Scholar
  5. 5.
    E. Sternberg, Three-dimensional stress concentrations in the theory of elasticity. Appl. Mech. Rev. 11(1) (1958).Google Scholar
  6. 6.
    J.L. Ericksen, Equilibrium of bars. J. Elasticity 5 (1975) 191–201.Google Scholar
  7. 7.
    J.M. Ball, Convexity conditions and existence theorem in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403.Google Scholar
  8. 8.
    J.K. Knowles and E. Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elasticity 8 (1978) 329–379.Google Scholar
  9. 9.
    R.C. Abeyaratne, Discontinuous deformation gradients in plane finite elastostatics of incompressible materials. J. Elasticity 10 (1980) 255–293.Google Scholar
  10. 10.
    N. Kikuchi and N. Triantafyllidis, On a certain class of elastic materials with non-elliptic energy densities. Quart. Appl. Math. 40 (1982) 241–248.Google Scholar
  11. 11.
    P. Rosakis, Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Rational Mech. Anal. 109 (1990) 1–37.Google Scholar
  12. 12.
    P. Rosakis and A. Jiang, Deformations with discontinuous gradients in plane elastostatics of compressible solids. J. Elasticity 33 (1993) 233–257.Google Scholar
  13. 13.
    A.C. Eringen, Vistas of nonlocal continuum physics. Internat. J. Engrg. Sci. 40 (1992) 1551–1565.Google Scholar
  14. 14.
    M.M.J. Treacy, T.W. Ebbesen and J.M. Gibson, Exceptionally high Young's modulus observed for individual carbon nanotubes. Nature 381 (1996) 678–680.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • S.A. Silling
    • 1
  • M. Zimmermann
    • 2
  • R. Abeyaratne
    • 2
  1. 1.Computational Physics Department, MS-0820Sandia National LaboratoriesAlbuquerqueU.S.A.
  2. 2.Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridgeU.S.A

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