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Continuum Mechanics and Thermodynamics

, Volume 30, Issue 5, pp 1155–1184 | Cite as

Quasistatic elastoplasticity via Peridynamics: existence and localization

  • Martin Kružík
  • Carlos Mora-Corral
  • Ulisse Stefanelli
Original Article
  • 45 Downloads

Abstract

Peridynamics is a nonlocal continuum mechanical theory based on minimal regularity on the deformations. Its key trait is that of replacing local constitutive relations featuring spacial differential operators with integrals over differences of displacement fields over a suitable positive interaction range. The advantage of such perspective is that of directly including nonregular situations, in which discontinuities in the displacement field may occur. In the linearized elastic setting, the mechanical foundation of the theory and its mathematical amenability have been thoroughly analyzed in the last years. We present here the extension of Peridynamics to linearized elastoplasticity. This calls for considering the time evolution of elastic and plastic variables, as the effect of a combination of elastic energy storage and plastic energy dissipation mechanisms. The quasistatic evolution problem is variationally reformulated and solved by time discretization. In addition, by a rigorous evolutive \(\Gamma \)-convergence argument we prove that the nonlocal peridynamic model converges to classic local elastoplasticity as the interaction range goes to zero.

Keywords

Peridynamics Elastoplasticity Variational formulation Existence Localization 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Martin Kružík
    • 1
  • Carlos Mora-Corral
    • 2
  • Ulisse Stefanelli
    • 3
    • 4
  1. 1.Institute of Information Theory and AutomationAcademy of SciencesPrague 8Czech Republic
  2. 2.Departamento de Matemáticas, Facultad de CienciasUniversidad Autónoma de MadridMadridSpain
  3. 3.Faculty of MathematicsUniversity of ViennaWienAustria
  4. 4.Istituto di Matematica Applicata e Tecnologie Informatiche E. MagenesPaviaItaly

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