Journal of Elasticity

, Volume 113, Issue 2, pp 193–217 | Cite as

Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity

  • Qiang Du
  • Max Gunzburger
  • R. B. LehoucqEmail author
  • Kun Zhou
Original Article


Well-posedness results for the state-based peridynamic nonlocal continuum model of solid mechanics are established with the help of a nonlocal vector calculus. The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid is expressed in terms of the field operators of that calculus, after which a variational principle for the equilibrium state is defined. The peridynamic Navier equilibrium equation is then derived as the first-order necessary conditions and are shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of nonlocal interactions vanishes. Then, for certain peridynamic constitutive relations, the peridynamic energy space is shown to be equivalent to the space of square-integrable functions; this result leads to well-posedness results for volume-constrained problems of both the Dirichlet and Neumann types. Using standard results, well-posedness is also established for the time-dependent peridynamic equation of motion.


Peridynamic theory Navier equation Nonlocal operators Vector calculus Volume-constrained problems 

Mathematics Subject Classification (2010)

35B40 35J20 35J25 35Q99 45A05 45K05 74B05 



We thank Stewart Silling for some helpful discussions and the derivation in Sect. 2.1.

Q. Du supported in part by the U.S. Department of Energy grant DE-SC0005346 and the U.S. NSF grant DMS-1016073.

M. Gunzburger supported in part by the U.S. Department of Energy grant number DE-SC0004970 and U.S. NSF grant number DMS-1013845.

R. Lehoucq supported in part by the U.S. Department of Energy grant number FWP-09-014290 through the Office of Advanced Scientific Computing Research, DOE Office of Science. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under contract DE-AC04-94AL85000.

K. Zhou supported in part by the U.S. Department of Energy grant DE-SC0005346 and the U.S. NSF grant DMS-1016073.


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Copyright information

© U.S. Government 2012

Authors and Affiliations

  • Qiang Du
    • 1
  • Max Gunzburger
    • 2
  • R. B. Lehoucq
    • 3
    Email author
  • Kun Zhou
    • 1
    • 4
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of Scientific ComputingFlorida State UniversityTallahasseeUSA
  3. 3.Sandia National LaboratoriesAlbuquerqueUSA
  4. 4.UBS Investment BankNew YorkUSA

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