Variational convergence of discrete geometrically-incompatible elastic models

  • Raz Kupferman
  • Cy Maor


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 


  1. 1.
    Acerbi, E., Fusco, N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86, 125–145 (1984)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alicandro, R., Cicalse, M., Gloria, A.: Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Arch. Ration. Mech. Anal. 200, 881–943 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Armon, S., Efrati, E., Sharon, E., Kupferman, R.: Geometry and mechanics of chiral pod opening. Science 333, 1726–1730 (2011)CrossRefGoogle Scholar
  4. 4.
    Bilby, B., 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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Braides, A.: Gamma Convergence for Beginners. Oxford University Press, Oxford (2002)CrossRefMATHGoogle Scholar
  6. 6.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, Berlin (2008)MATHGoogle Scholar
  7. 7.
    dal Maso, G.: An introduction to \(\varGamma \)-Convergence. Birkhauser, Boston (1993)CrossRefGoogle Scholar
  8. 8.
    Efrati, E., Sharon, E., Kupferman, R.: Elastic theory of unconstrained non-Euclidean plates. J. Mech. Phys. Solids 57, 762–775 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Friesecke, G., James, R., 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
  10. 10.
    Goriely, A., Ben Amar, M.: Differential growth and instability in elastic shells. Phys. Rev. Lett. 94(198), 103–4 (2005)MATHGoogle Scholar
  11. 11.
    Klein, Y., Efrati, E., Sharon, E.: Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315, 1116–1120 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kondo, K.: Geometry of elastic deformation and incompatibility. In: K. Kondo (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
  13. 13.
    Kröner, E.: The physics of defects. In: Balian, R., Kleman, M., Poirier, J.P. (eds.) Les Houches Summer School Proceedings. North-Holland, Amsterdam (1981)Google Scholar
  14. 14.
    Kupferman, R., Maor, C.: A Riemannian approach to the membrane limit in non-Euclidean elasticity. Commun. Contemp. Math. 16, 1350052 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kupferman, R., Maor, C.: The emergence of torsion in the continuum limit of distributed dislocations. J. Geom. Mech. 7, 361–387 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kupferman, R., Maor, C.: Riemannian surfaces with torsion as homogenization limits of locally-Euclidean surfaces with dislocation-type singularities. Proc. R. Soc. Edinb. 146A, 741–768 (2016)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kupferman, R., Maor, C., Rosenthal, R.: Non-metricity in the continuum limit of randomly-distributed point defects (2016). Submitted to Israel J. MathGoogle Scholar
  18. 18.
    Kupferman, R., Maor, C., Shachar, A.: Asymptotic rigidity of Riemannian manifolds (2017).
  19. 19.
    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
  20. 20.
    Le Dret, H., Raoult, A.: Homogenization of hexagonal lattices. Netw. Heterog Med. 8, 541–572 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lee, J.: Introduction to Smooth Manifolds, 1st edn. Springer, Berlin (2006)Google Scholar
  22. 22.
    Lewicka, M., Ochoa, P.: On the variational limits of lattice energies on prestrained elastic bodies. In: Chen, G.Q.G., Grinfeld, M., Knops, R.J. (eds.) Differential Geometry and Continuum Mechanics, Chapter 10, pp. 279–305. Springer, Berlin (2015)CrossRefGoogle Scholar
  23. 23.
    Lewicka, M., Pakzad, M.: Scaling laws for non-Euclidean plates and the \(W^{2,2}\) isometric immersions of Riemannian metrics. ESAIM Control Optim. Calc. Var. 17, 1158–1173 (2010)CrossRefMATHGoogle Scholar
  24. 24.
    Neff, P., Ghiba, I.D., Lankeit, J.: The exponentiated Hencky-logarithmic strain energy. part I: constitutive issues and rank-one convexity. J. Elast. 121, 143–234 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Nye, J.: Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162 (1953)CrossRefGoogle Scholar
  26. 26.
    Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51, 032–902 (2009)MathSciNetMATHGoogle Scholar
  27. 27.
    Šilhavý, M.: Rank 1 convex hulls of isotropic functions in dimension 2 by 2. Math. Bohem. 126, 521–529 (2001)MathSciNetMATHGoogle Scholar
  28. 28.
    Wang, C.C.: On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch. Ration. Mech. Anal. 27, 33–93 (1967)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20, 781–830 (2010)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205, 59–118 (2012)MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations