Skip to main content

Loss of Strong Ellipticity Through Homogenization in 2D Linear Elasticity: A Phase Diagram


Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, it has been known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Gutiérrez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting.

This is a preview of subscription content, access via your institution.


  1. 1.

    Aliprantis C.D., Border K.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edition. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. 2.

    Allaire G.: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes I Abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113(3), 209–259 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, Oxford (2000)

    MATH  Google Scholar 

  4. 4.

    Bourgain J., Brezis H., Mironescu P.: Lifting in Sobolev spaces. J. Anal. Math. 80(1), 37–86 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Briane M., Francfort G.A.: Loss of ellipticity through homogenization in linear elasticity. Math. Models Methods Appl. Sci. 25(5), 905–928 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Briane, M., Francfort, G.A.: A two-dimensional labile aether through homogenization. ArXiv e-prints, January 2018

  7. 7.

    Briane M., Pallares Martín A.J.: Homogenization of weakly coercive integral functionals in three-dimensional linear elasticity. J. Éc. Polytech. Math. 4, 483–514 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Dal Maso G., Modica L.: Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368, 28–42 (1986)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Francfort G.A.: Homogenisation of a class of fourth order equations with application to incompressible elasticity. Proc. R. Soc. Edinb. Sect. A 120(1–2), 25–46 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Francfort G.A., Gloria A.: Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity. C. R. Math. Acad. Sci. Paris 354(11), 1139–1144 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Francfort G.A., Murat F.: Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Anal. 94(4), 307–334 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Geymonat G., Müller S., Triantafyllidis N.: Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Ration. Mech. Anal. 122, 231–290 (1993)

    Article  MATH  Google Scholar 

  13. 13.

    Gutiérrez S.: Laminations in linearized elasticity: the isotropic non-very strongly elliptic case. J. Elast. 53(3), 215–256 (1998/1999)

  14. 14.

    Jikov V.V., Kozlov S.M., Olenik O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994) Translated from the Russian by G.A. Yosifian [G.A. Iosif’yan]

    Book  Google Scholar 

  15. 15.

    Müller S.: Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Ration. Mech. Anal. 99(3), 189–212 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, volume 39 of Research Notes in Mathematics, pp. 136–212. Pitman, Boston, 1979

Download references


We warmly thank Gilles Francfort for discussions on ellipticity conditions, and acknowledge the financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410).

Author information



Corresponding author

Correspondence to Matthias Ruf.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Communicated by S. Müller

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gloria, A., Ruf, M. Loss of Strong Ellipticity Through Homogenization in 2D Linear Elasticity: A Phase Diagram. Arch Rational Mech Anal 231, 845–886 (2019).

Download citation