Computational Mechanics

, Volume 58, Issue 2, pp 351–370 | Cite as

The total and updated lagrangian formulations of state-based peridynamics

  • Guy L. Bergel
  • Shaofan LiEmail author
Original Paper


The peridynamics theory is a reformulation of nonlocal continuum mechanics that incorporates material particle interactions at finite distances into the equation of motion. State-based peridynamics is an extension of the original bond-based peridynamics theory wherein the response of an individual particle depends collectively on its interaction with neighboring particles through the concept of state variables. In this paper, the more recent non-ordinary state-based Peridynamics formulations of both the total (referential) Lagrangian approach as well as the updated (spatial) Lagrangian approach are formulated. In doing so, relations of the state variables are defined through various nonlocal differential operators in both material and spatial configurations in the context of finite deformation. Moreover, these nonlocal differential operators are mathematically and numerically shown to converge to the local differential operators, and they are applied to derive new force states and deformation gradients.


Continuum mechanics Deformation gradient Finite elasticity Nonlocal theory State-based peridynamics Updated Lagrangian 



This work was supported by the ONR MURI Grant N00014-11-1-0691, which is gratefully acknowledged. Dr. Houfu Fan calculated the second numerical example.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of CaliforniaBerkeleyUSA

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