Advertisement

Archive for Rational Mechanics and Analysis

, Volume 234, Issue 3, pp 1091–1120 | Cite as

The Gap Between Linear Elasticity and the Variational Limit of Finite Elasticity in Pure Traction Problems

  • Francesco MaddalenaEmail author
  • Danilo Percivale
  • Franco Tomarelli
Article

Abstract

A limit elastic energy for the pure traction problem is derived from re-scaled nonlinear energies of a hyperelastic material body subject to an equilibrated force field. We prove that the strains of minimizing sequences associated to re-scaled nonlinear energies weakly converge, up to subsequences, to the strains of minimizers of a limit energy, provided an additional compatibility condition is fulfilled by the force field. The limit energy is different from the classical energy of linear elasticity; nevertheless, the compatibility condition entails the coincidence of related minima and minimizers. A strong violation of this condition provides a limit energy which is unbounded from below, while a mild violation may produce unboundedness of strains and a limit energy which has infinitely many extra minimizers which are not minimizers of standard linear elastic energy. A consequence of this analysis is that a rigorous validation of linear elasticity fails for compressive force fields that infringe up on such a compatibility condition.

Notes

References

  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), 1–38, 2015MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agostiniani, V., Dal Maso, G., DeSimone, A.: Linear elasticity obtained from finite elasticity by Gamma-convergence under weak coerciveness conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29(5), 715–735, 2012ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Alicandro, R., Dal Maso, G., Lazzaroni, G., Palombaro, M.: Derivation of a linearised elasticity model from singularly perturbed multiwell energy functionals. Arch. Ration. Mech. Anal. 2018.  https://doi.org/10.1007/s00205-018-1240-6 ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Anzellotti, G., Baldo, S., Percivale, D.: Dimension reduction in variational problems, asymptotic development in \(\Gamma \)-convergence and thin structures in elasticity. Asympt. Anal. 9, 61–100, 1994MathSciNetzbMATHGoogle Scholar
  5. 5.
    Audoly, B., Pomeau, Y.: Elasticity and Geometry. Oxford University Press, Oxford 2010zbMATHGoogle Scholar
  6. 6.
    Baiocchi, C., Buttazzo, G., Gastaldi, F., Tomarelli, F.: General existence results for unilateral problems in continuum mechanics. Arch. Ration. Mech. Anal. 100, 149–189, 1988CrossRefGoogle Scholar
  7. 7.
    Buttazzo, G., Dal Maso, G.: Singular perturbation problems in the calculus of variations. Ann.Scuola Normale Sup. Cl. Sci. 4 ser 11(3), 395–430, 1984MathSciNetGoogle Scholar
  8. 8.
    Buttazzo, G., Tomarelli, F.: Compatibility conditions for nonlinear Neumann problems. Adv. Math. 89, 127–143, 1991MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carriero, M., Leaci, A., Tomarelli, F.: Strong solution for an elastic–plastic plate. Calc. Var. Partial Differ. Equ. 2(2), 219–240, 1994MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ciarlet, P.G.: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity. Elsevier, Amsterdam 1988zbMATHGoogle Scholar
  11. 11.
    Ciarlet, P.G., Ciarlet Jr., P.: Another approach to linearized elasticity and Korn’s inequality. C. R. Acad. Sci. Paris Ser. I 339, 307–312, 2004MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dal Maso, G.: An Introduction to Gamma Convergence, vol. 8. Birkhäuser, PNLDE, Boston 1993CrossRefGoogle Scholar
  13. 13.
    Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as \(\Gamma \)-limit of finite elasticity. Set-Valued Anal. 10(2–3), 165–183, 2002MathSciNetCrossRefGoogle Scholar
  14. 14.
    De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(6), 842–850, 1975MathSciNetzbMATHGoogle Scholar
  15. 15.
    De Tommasi, D., Marzano, S.: Small strain and moderate rotation. J. Elast. 32, 37–50, 1993MathSciNetCrossRefGoogle Scholar
  16. 16.
    De Tommasi, D.: On the kinematics of deformations with small strain and moderate rotation. Math. Mech. Solids 9, 355–368, 2004MathSciNetCrossRefGoogle Scholar
  17. 17.
    Frieseke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of non linear plate theory from three dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506, 2002CrossRefGoogle Scholar
  18. 18.
    Frieseke, G., James, R.D., Müller, S.: A hierarky of plate models from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180, 183–236, 2006MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gurtin, M.E.: The Linear Theory of Elasticity. Handbuch der Physik, Vla/2Springer, Berlin 1972Google Scholar
  20. 20.
    Hall, B.: Lie Groups, Lie Algebras and Representations: An Elementary Introduction, vol. 222. Springer Graduate Text in Math. Springer, Berlin 2015CrossRefGoogle Scholar
  21. 21.
    Lecumberry, M., Müller, S.: Stability of slender bodies under compression and validity of von Kármán theory. Arch. Ration. Mech. Anal. 193, 255–310, 2009MathSciNetCrossRefGoogle Scholar
  22. 22.
    Love, A.E.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York 1944zbMATHGoogle Scholar
  23. 23.
    Maddalena, F., Percivale, D., Tomarelli, F.: Adhesive flexible material structures. Discrete Continuous Dyn. Syst. B 17(2), 553–574, 2012MathSciNetCrossRefGoogle Scholar
  24. 24.
    Maddalena, F., Percivale, D., Tomarelli, F.: Local and nonlocal energies in adhesive interaction. IMA J. Appl. Math. 81(6), 1051–1075, 2016MathSciNetCrossRefGoogle Scholar
  25. 25.
    Maddalena, F., Percivale, D., Tomarelli, F.: Variational problems for Föppl-von Kármán plates. SIAM J. Math. Anal. 50(1), 251–282, 2018.  https://doi.org/10.1137/17M1115502 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Maddalena, F., Percivale, D., Tomarelli, F.: A new variational approach to linearization of traction problems in elasticity. J. Optim. Theory Appl. 182, 383–403, 2019.  https://doi.org/10.1007/s10957-019-01533-8 MathSciNetCrossRefGoogle Scholar
  27. 27.
    Percivale, D., Tomarelli, F.: Scaled Korn-Poincaré inequality in BD and a model of elastic plastic cantilever. Asymptot. Anal. 23(3–4), 291–311, 2000MathSciNetzbMATHGoogle Scholar
  28. 28.
    Percivale, D., Tomarelli, F.: From SBD to SBH: the elastic–plastic plate. Interfaces Free Bound. 4(2), 137–165, 2002MathSciNetCrossRefGoogle Scholar
  29. 29.
    Percivale, D., Tomarelli, F.: A variational principle for plastic hinges in a beam. Math. Models Methods Appl. Sci. 19(12), 2263–2297, 2009MathSciNetCrossRefGoogle Scholar
  30. 30.
    Percivale, D., Tomarelli, F.: Smooth and broken minimizers of some free discontinuity problems. Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22 (Eds. Colli P. et al.) Springer INdAM Series, 431–468 2017.  https://doi.org/10.1007/978-3-319-64489-9_17 Google Scholar
  31. 31.
    Podio-Guidugli, P.: On the validation of theories of thin elastic structures. Meccanica 49(6), 1343–1352, 2014MathSciNetCrossRefGoogle Scholar
  32. 32.
    Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics. Handbuch der Physik 11113Springer, Berlin 1965zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Francesco Maddalena
    • 1
    Email author
  • Danilo Percivale
    • 2
  • Franco Tomarelli
    • 3
  1. 1.Dipartimento di Meccanica, Matematica, ManagementPolitecnico di BariBariItaly
  2. 2.Dipartimento di Ingegneria MeccanicaUniversità di GenovaGenovaItaly
  3. 3.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

Personalised recommendations