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

  • Cy Maor
  • Asaf Shachar


We prove a relation between the scaling \(h^{\beta}\) of the elastic energies of shrinking non-Euclidean bodies \(\mathcal{S}_{h}\) of thickness \(h\to0\), and the curvature along their mid-surface \(\mathcal{S}\). 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 \(\varGamma \)-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 \(\varGamma\)-limit proves asymptotics calculated in Aharoni et al. (Phys. Rev. Lett. 117:124101, 2016).


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

Mathematics Subject Classification

74K99 74B20 53Z05 74K10 74K20 74K25 



We thank Robert Jerrard for some useful advice and suggestions during the preparation of this paper, and Raz Kupferman for his critical reading of the manuscript. The second author was partially funded by the Israel Science Foundation (Grant No. 661/13), and by a grant from the Ministry of Science, Technology and Space, Israel and the Russian Foundation for Basic Research, the Russian Federation.


  1. 1.
    Agostiniani, V., Lucantonio, A., Lučić, D.: Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets. Preprint (2017) Google Scholar
  2. 2.
    Aharoni, H., Abraham, Y., Elbaum, R., Sharon, E., Kupferman, R.: Emergence of spontaneous twist and curvature in non-Euclidean rods: application to Erodium plant cells. Phys. Rev. Lett. 108, 238106 (2012) ADSCrossRefGoogle Scholar
  3. 3.
    Aharoni, H., Kolinski, J.M., Moshe, M., Meirzada, I., Sharon, E.: Internal stresses lead to net forces and torques on extended elastic bodies. Phys. Rev. Lett. 117, 124101 (2016) ADSCrossRefGoogle Scholar
  4. 4.
    Armon, S., Efrati, E., Sharon, E., Kupferman, R.: Geometry and mechanics of chiral pod opening. Science 333, 1726–1730 (2011) ADSCrossRefGoogle Scholar
  5. 5.
    Bella, P., Kohn, R.V.: Metric-induced wrinkling of a thin elastic sheet. J. Nonlinear Sci. 24(6), 1147–1176 (2014) ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bhattacharya, K., Lewicka, M., Schäffner, M.: Plates with incompatible prestrain. Arch. Ration. Mech. Anal. 221(1), 143–181 (2016) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bilby, B.A., Smith, E.: Continuous distributions of dislocations, III. Proc. R. Soc. Edinb. A 236, 481–505 (1956) ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bilby, B.A., Bullough, R., Smith, E.: Continuous distributions of dislocations: a new application of the methods of Non-Riemannian geometry. Proc. R. Soc. A 231, 263–273 (1955) ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, B.Y.: Riemannian submanifolds: a survey. In: Dillen, F., Verstraelen, L. (eds.) Handbook of Differential Geometry, vol. 1, pp. 187–418 (2000) Google Scholar
  10. 10.
    Ciarlet, P.G.: Mathematical Elasticity, vol. 1: Three-Dimensional Elasticity. Elsevier, Amsterdam (1988) MATHGoogle Scholar
  11. 11.
    Ciarlet, P.G.: An Introduction to Differential Geometry with Applications to Elasticity. Springer, Netherlands (2005) MATHGoogle Scholar
  12. 12.
    Ciarlet, P.G.: Linear and Nonlinear Functional Analysis with Applications. SIAM, Philadelphia (2013) MATHGoogle Scholar
  13. 13.
    Cicalese, M., Ruf, M., Solombrino, F.: On global and local minimizers of prestrained thin elastic rods. Calc. Var. Partial Differ. Equ. 56(4), 115 (2017) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Conti, S., Olbermann, H., Tobasco, I.: Symmetry breaking in indented elastic cones. Math. Models Methods Appl. Sci. 27(2), 291–321 (2017) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Basel (1992) CrossRefMATHGoogle Scholar
  16. 16.
    Efrati, E., Sharon, E., Kupferman, R.: Buckling transition and boundary layer in non-Euclidean plates. Phys. Rev. E 80, 016602 (2009) ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Efrati, E., Sharon, E., Kupferman, R.: Elastic theory of unconstrained non-Euclidean plates. J. Mech. Phys. Solids 57, 762–775 (2009) ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Efrati, E., Sharon, E., Kupferman, R.: Hyperbolic non-Euclidean elastic strips and almost minimal surfaces. Phys. Rev. E 83, 046602 (2011) ADSCrossRefGoogle Scholar
  19. 19.
    Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Friesecke, G., James, R.D., Müller, S.: A hierarchy of plate models derived from nonlinear elasticity by \(\varGamma \)-convergence. Arch. Ration. Mech. Anal. 180, 183–236 (2006) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Grossman, D., Sharon, E., Diamant, H.: Elasticity and fluctuations of frustrated nanoribbons. Phys. Rev. Lett. 116, 258105 (2016) ADSCrossRefGoogle Scholar
  22. 22.
    Klein, Y., Efrati, E., Sharon, E.: Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315, 1116–1120 (2007) ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kohn, R.V., O’Brien, E.: On the bending and twisting of rods with misfit. J. Elast. 130(1), 115–143 (2018) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kondo, K.: Geometry of elastic deformation and incompatibility. In: Kondo, K. (ed.) Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, vol. 1, pp. 5–17 (1955) Google Scholar
  25. 25.
    Kupferman, R., Maor, C.: A Riemannian approach to the membrane limit in non-Euclidean elasticity. Commun. Contemp. Math. 16(5), 1350052 (2014) MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kupferman, R., Shamai, Y.: Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space. Isr. J. Math. 190(1), 135–156 (2012) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kupferman, R., Solomon, J.P.: A Riemannian approach to reduced plate, shell, and rod theories. J. Funct. Anal. 266, 2989–3039 (2014) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kupferman, R., Maor, C., Shachar, A.: Reshetnyak Rigidity for Riemannian Manifolds. Arch. Rat. Mech. Anal. to appear in.
  29. 29.
    Le-Dret, H., Raoult, A.: The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74, 549–578 (1995) MathSciNetMATHGoogle Scholar
  30. 30.
    Le-Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6(1), 59–84 (1996) ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lewicka, M., Pakzad, M.R.: Scaling laws for non-Euclidean plates and the \(W^{2,2}\) isometric immersions of Riemannian metrics. ESAIM Control Optim. Calc. Var. 17, 1158–1173 (2011) MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Lewicka, M., Raoult, A., Ricciotti, D.: Plates with incompatible prestrain of higher order. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 34, 1883–1912 (2017) ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Olbermann, H.: Energy scaling law for a single disclination in a thin elastic sheet. Arch. Ration. Mech. Anal. 224(3), 985–1019 (2017) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51, 032902 (2009) ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Sharon, E., Roman, B., Swinney, H.L.: Geometrically driven wrinkling observed in free plastic sheets and leaves. Phys. Rev. E 75, 046211 (2007) ADSCrossRefGoogle Scholar
  36. 36.
    Tenenblat, K.: On isometric immersions of Riemannian manifolds. Bol. Soc. Bras. Mat. 2, 23–36 (1971) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

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

Personalised recommendations