Archive for Rational Mechanics and Analysis

, Volume 230, Issue 1, pp 1–45 | Cite as

Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals

  • Roberto Alicandro
  • Gianni Dal Maso
  • Giuliano Lazzaroni
  • Mariapia Palombaro


Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agostiniani, V., Blass, T., Koumatos, K.: From nonlinear to linearized elasticity via \(\Gamma \)-convergence: the case of multiwell energies satisfying weak coercivity conditions. Math. Models Methods Appl. Sci. 25, 1–38 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agostiniani, V., Dal Maso, G., DeSimone, A.: Linearized elasticity obtained from finite elasticity by \(\Gamma \)-convergence under weak coerciveness conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 715–735 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alicandro, R., Cicalese, M.: A general integral representation result for the continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36, 1–37 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alicandro, R., Lazzaroni, G., Palombaro, M.: On the effect of interactions beyond nearest neighbours on non-convex lattice systems. Calc. Var. Partial Differ. Equ. 56, 42 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Braides, A., Solci, M., Vitali, E.: A derivation of linear elastic energies from pair-interaction atomistic systems. Netw. Heterog. Media 2, 551–567 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chaudhuri, N., Müller, S.: Rigidity estimate for two incompatible wells. Calc. Var. Partial Differ. Equ. 19, 379–390 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Conti, S., Schweizer, B.: Rigidity and gamma convergence for solid-solid phase transitions with \(SO(2)\) invariance. Commun. Pure Appl. Math. 59, 830–868 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dal Maso G.: An Introduction to \(\Gamma \) -Convergence. Progress in Nonlinear Differential Equations and Their Applications, vol. 8. Birkhäuser, Boston (1993)Google Scholar
  9. 9.
    Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as \(\Gamma \)-limit of finite elasticity. Set-Valued Analysis 10, 165–183 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    De Lellis, C., Székelyhidi Jr., L.: Simple proof of two-well rigidity. C. R. Math. Acad. Sci. Paris 343, 367–370 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dolzmann, G., Müller, S.: Microstructures with finite surface energy: the two-well problem. Arch. Ration. Mech. Anal. 132, 101–141 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)MATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Oxford Science Publications, New York (1993)MATHGoogle Scholar
  15. 15.
    Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980) (second ed.; first ed., 1966)Google Scholar
  16. 16.
    Modica, L., Mortola, S.: Un esempio di \(\Gamma \)-convergenza. Boll. Un. Mat. Ital. B 5(14), 285–299 (1977)MathSciNetMATHGoogle Scholar
  17. 17.
    Schmidt, B.: Linear \(\Gamma \)-limits of multiwell energies in nonlinear elasticity theory. Contin. Mech. Thermodyn. 20, 375–396 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Schmidt, B.: On the derivation of linear elasticity from atomistic models. Netw. Heterog. Media 4, 789–812 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ziemer W.P.: Weakly Differentiable Functions. Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)Google Scholar

Copyright information

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

Authors and Affiliations

  • Roberto Alicandro
    • 1
  • Gianni Dal Maso
    • 2
  • Giuliano Lazzaroni
    • 3
  • Mariapia Palombaro
    • 4
  1. 1.Dipartimento di Ingegneria Elettrica e dell’InformazioneUniversità di Cassino e del Lazio meridionaleCassinoItaly
  2. 2.SISSATriesteItaly
  3. 3.Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”Università degli Studi di Napoli Federico IINaplesItaly
  4. 4.Department of MathematicsUniversity of SussexBrightonUnited Kingdom

Personalised recommendations