On the Role of Curvature in the Elastic Energy of Non-Euclidean Thin Bodies

Cy Maor, Asaf Shachar*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

Abstract

We prove a relation between the scaling h β of the elastic energies of shrinking non-Euclidean bodies S h of thickness h→ 0 , and the curvature along their mid-surface S. This extends and generalizes similar results for plates (Bhattacharya et al., Arch. Ration. Mech. Anal. 221(1):143–181, 2016; Lewicka et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 34:1883–1912, 2017) to any dimension and co-dimension. In particular, it proves that the natural scaling for non-Euclidean rods with smooth metric is h 4 , as claimed in Aharoni et al. (Phys. Rev. Lett. 108:235106, 2012) using a formal asymptotic expansion. The proof involves calculating the Γ-limit for the elastic energies of small balls B h (p) , scaled by h 4 , and showing that the limit infimum energy is given by a square of a norm of the curvature at a point p. This Γ-limit proves asymptotics calculated in Aharoni et al. (Phys. Rev. Lett. 117:124101, 2016).

Original languageAmerican English
Pages (from-to)149-173
Number of pages25
JournalJournal of Elasticity
Volume134
Issue number2
DOIs
StatePublished - 15 Feb 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018, Springer Nature B.V.

Keywords

  • Curvature
  • Dimension-reduction
  • Gamma-convergence
  • Gauss-Codazzi equations
  • Incompatible elasticity
  • Non-Euclidean plates
  • Non-Euclidean rods

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