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Peridynamics: A Nonlocal Continuum Theory

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Meshfree Methods for Partial Differential Equations VI

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 89))

Abstract

The peridynamic theory is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives, which can be easily applied in the vicinity of cracks, where discontinuities in the displacement field occur. In this paper we give a survey on important analytical and numerical results and applications of the peridynamic theory.

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.

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Notes

  1. 1.

    Note that the notation \(\vec{f} =\vec{ f}(\vec{\xi },\vec{\eta })\) is somewhat ambigious. Indeed, for a given function \(\vec{u} =\vec{ u}(\vec{x},t)\) the mathematical correct way to describe \(\vec{f}\) is using the Nemytskii operator \(F :\vec{ u}\mapsto F\vec{u}\) with \((F\vec{u})(\vec{x},\vec{\hat{x}},t) =\vec{ f}(\vec{\hat{x}} -\vec{ x},\vec{u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t))\).

  2. 2.

    For the notation \(s = s(\vec{\xi },\vec{\eta })\), see also Footnote 1.

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Acknowledgements

The authors are grateful to Stephan Kusche and Henrik Büsing for the numerical simulation of the Kalthoff–Winkler experiment (see Fig. 3).

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Authors and Affiliations

  1. Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Etienne Emmrich & Dimitri Puhst

  2. Multiphysics Simulation Technologies, Sandia National Laboratories, Albuquerque, NM, 87185, USA

    Richard B. Lehoucq

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  1. Etienne Emmrich
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Correspondence to Etienne Emmrich .

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  1. , Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, City, 53115, Germany

    Michael Griebel

  2. Institut für Parallele, und Verteilte Systeme, Universität Stuttgart, Universitätsstraße 38, Stuttgart, 70569, Germany

    Marc Alexander Schweitzer

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Emmrich, E., Lehoucq, R.B., Puhst, D. (2013). Peridynamics: A Nonlocal Continuum Theory. In: Griebel, M., Schweitzer, M. (eds) Meshfree Methods for Partial Differential Equations VI. Lecture Notes in Computational Science and Engineering, vol 89. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32979-1_3

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