Deformation of a Peridynamic Bar
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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.
- S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000) 175–209.
- R.J. Atkin and N. Fox, An Introduction to the Theory of Elasticity. (1980) Longman, London, pp. 186–187.
- M.J. Lighthill, Fourier Analysis and Generalised Functions. Cambridge Univ. Press, Cambridge (1978).
- E. Zauderer, Partial Differential Equations of Applied Mathematics. Wiley, New York (1983) pp. 365–376.
- E. Sternberg, Three-dimensional stress concentrations in the theory of elasticity. Appl. Mech. Rev. 11(1) (1958).
- J.L. Ericksen, Equilibrium of bars. J. Elasticity 5 (1975) 191–201.
- J.M. Ball, Convexity conditions and existence theorem in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403.
- 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.
- R.C. Abeyaratne, Discontinuous deformation gradients in plane finite elastostatics of incompressible materials. J. Elasticity 10 (1980) 255–293.
- N. Kikuchi and N. Triantafyllidis, On a certain class of elastic materials with non-elliptic energy densities. Quart. Appl. Math. 40 (1982) 241–248.
- P. Rosakis, Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Rational Mech. Anal. 109 (1990) 1–37.
- P. Rosakis and A. Jiang, Deformations with discontinuous gradients in plane elastostatics of compressible solids. J. Elasticity 33 (1993) 233–257.
- A.C. Eringen, Vistas of nonlocal continuum physics. Internat. J. Engrg. Sci. 40 (1992) 1551–1565.
- 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.
- Deformation of a Peridynamic Bar
Journal of Elasticity
Volume 73, Issue 1-3 , pp 173-190
- Cover Date
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- Online ISSN
- Kluwer Academic Publishers
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- linear integral equations
- Fredholm integral equations
- Fourier transform
- Green's function
- point load
- elastic bar
- Industry Sectors
- Author Affiliations
- 1. Computational Physics Department, MS-0820, Sandia National Laboratories, Albuquerque, NM, 87185-0820, U.S.A.
- 2. Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 3-173, Cambridge, MA 02139, U.S.A