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Variational convergence of discrete geometrically-incompatible elastic models

  • Raz Kupferman
  • Cy Maor
Article
  • 58 Downloads

Abstract

We derive a continuum model for incompatible elasticity as a variational limit of a family of discrete nearest-neighbor elastic models. The discrete models are based on discretizations of a smooth Riemannian manifold \(({\mathcal {M}},\mathfrak {g})\), endowed with a flat, symmetric connection \(\nabla \). The metric \(\mathfrak {g}\) determines local equilibrium distances between neighboring points; the connection \(\nabla \) induces a lattice structure shared by all the discrete models. The limit model satisfies a fundamental rigidity property: there are no stress-free configurations, unless \(\mathfrak {g}\) is flat, i.e., has zero Riemann curvature. Our analysis focuses on two-dimensional systems, however, all our results readily generalize to higher dimensions.

Mathematics Subject Classification

74B20 74Q15 53Z05 

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Copyright information

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

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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