Weak form of peridynamics for nonlocal essential and natural boundary conditions

Erdogan Madenci, Mehmet Dorduncu, Atila Barut, Nam Phan

Research output: Contribution to journalArticlepeer-review

68 Scopus citations


This study presents the weak form of peridynamic (PD) governing equations which permit the direct imposition of nonlocal essential and natural boundary conditions. It also presents a variational approach to derive the PD form of first- and second-order derivatives of a field variable at a point which is not symmetrically located in its domain of interaction. This capability enables the nonlocal PD representation of the internal force vector and the stress components without any calibration procedure. Furthermore, it removes the concern of truncated domain of interaction for a point near the surface. Thus, the solution is free of nonlocal boundary forces and surface effects. The numerical solution of the resulting equations can be achieved by considering an unstructured nonuniform discretization. The implicit solution to the discrete form of the equations is achieved by employing BiConjugate Gradient Stabilized (BICGSTAB) method which is an iterative technique for solving sparse non-symmetric linear systems. The explicit analysis is performed by constructing a global diagonal mass matrix, and using a hybrid implicit/explicit time integration scheme. The accuracy of this approach is demonstrated by considering an elastic isotropic plate with or without a cutout subjected to a combination of different types of boundary conditions under plane stress conditions.

Original languageEnglish (US)
Pages (from-to)598-631
Number of pages34
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Aug 1 2018


  • Essential
  • Natural
  • Nonlocal
  • Peridynamic
  • Weak form

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications


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