Skip to main content

Advertisement

SpringerLink
Log in

Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider the time evolution of a one dimensional n-gradient continuum. Our aim is to construct and analyze discrete approximations in terms of physically realizable mechanical systems, referred to as microscopic because they are living on a smaller space scale. We validate our construction by proving a convergence theorem of the microscopic system to the given continuum, as the scale parameter goes to zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alibert J.J., Seppecher P., dell’Isola F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8, 51–73 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics a la Hamilton–Piola: least action principle for second gradient continua and capillary fluids. arXiv preprint arXiv:1305.6744 (2013) to appear in Mathematics and Mechanics of Solids (2014). doi:10.1177/1081286513497616

  3. Carcaterra, A, Akay, A.: Dissipation in a finite-size bath. Phys. Rev. E, 84, 011121.1–011121.4 (2011)

  4. Carcaterra A., Akay A.: Theoretical foundation of apparent damping and irreversible energy exchange in linear conservative dynamical systems. J. Acoust. Soc. Am. 121, 1971–1982 (2007)

    Article  ADS  Google Scholar 

  5. Chesnais, C., Boutin, C., Stèphane, H.: Effects of the local resonance on the wave propagation in periodic frame structures: generalized Newtonian mechanics. J. Acoust. Soc. Am. 132, 2873 (2012)

  6. Craster, R.V., Guenneau, S. (eds.): Acoustic Metamaterials. Springer Series in Material Science, vol. 166, Springer, Berlin, 2013

  7. dell’Isola, F., Seppecher, P.: The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power. Comptes rendus de l’Acadmie des Sciences Serie IIb 321, 303–308 (1995)

  8. dell’Isola, F., Seppecher P., Madeo A.: How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach à la D’Alembert. Zeitschrift fü r angewandte Mathematik und Physik 63.6, 1119–1141 (2012)

  9. dell’Isola F., Andreaus, U., Placidi L.: At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola. arXiv preprint arXiv:1310.5599 (2013) to appear in Mathematics and Mechanics of Solids (2014) doi:10.1177/1081286513509811

  10. Dunn, J.E., Serrin, J.: On the thermomechanics of interstitial working in The Breadth and Depth of Continuum Mechanics. Springer, Berlin Heidelberg, pp. 705–743 1986

  11. Engheta, N., Ziolkowski, R.W.: Metamaterials: Physics and Engineering, Explorations. Wiley, New York, 2006

  12. Germain, P.: The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)

  13. Green A.E., Rivlin R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17(2), 113–114 (1964)

    MATH  MathSciNet  Google Scholar 

  14. Gurtin, M.E.: Thermodynamics and the possibility of spatial interaction in elastic materials. Arch. Ration. Mech. Anal. 19.5, 339–352 (1965)

  15. Hilbert, D.: Begründung der kinetischen Gastheorie, Math. Ann. 72 331–407 (1916/17)

  16. Kolpakovs A.G.: Determination of the average characteristics of elastic frameworks. J. Appl. Math. Mech. 49(6), 739–745 (1985)

    Article  MathSciNet  Google Scholar 

  17. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-Relativistic Theory, 3 (3rd edn.). Pergamon Press, Oxford, 1977

  18. Lee, S.H., Park, C.M., Seo, Y.M., Wang, Z.G., Kim, C.K.: Composite acoustic medium with simultaneously negative density and modulus. Phys. Rev. Lett. 104(5), (2010). Bibcode:2010PhRvL.104e4301L. doi:10.1103/PhysRevLett.104.054301

  19. Madeo, A., Placidi, L., Rosi, G.: Towards the design of meta-materials with enhanced damage sensitivity: second gradient porous materials. Res. Nondestruct. Eval. (2013). doi:10.1080/09349847.2013.853114

  20. Milton, G.W., Cherkaev, A.V.: Which elasticity tensors are realizable? J. Eng. Mater. Technol. 117(4), 483 (1995)

  21. Milton, G.M., Briane, M., Willis, J.R.: On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006)

  22. Mindlin R.D.: Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)

    Article  Google Scholar 

  23. Neff, P., Ghiba, I.D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech. Thermodyn. 1–43 (2014). doi:10.1007/s00161-013-0322-9.

  24. Piccardo, G., Ranzi, G., Luongo, A.: A complete dynamic approach to the generalized beam theory cross-section analysis including extension and shear modes. Math. Mech. Solids 19(8), 900–924 (2014)

  25. Piola G.: Memoria intorno alle equazioni fondamentali del movimento di corpi qualsivogliono considerati secondo la naturale loro forma e costituzione, Modena, Tipi del R.D. Camera, 1846 translated by F.dell’Isola, U. Andreaus and L.Placidi in “The complete works of Gabrio Piola : Volume I” U. Andreaus, F.dell’Isola, R. Esposito, S. Forest, G.Maier, U. Perego, Editors Springer Verlag (see also www.fdellisola.it), vol. 38, 2014

  26. Seppecher, P., Jean-Jacques Alibert, J.-J. dell Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys.: Conf. Ser. 319(1). IOP Publishing (2011)

  27. Toupin R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  28. Xu, B., Arias, F., Brittain, S.T., Zhao, X.-M., Grzybowski, B., Torquato, S., Whitesides, G.M.: Making negative Poisson’s ratio microstructures by soft lithography. Adv. Mater. 11(14), 1186–1189 (1999)

  29. Zouhdi, S., Ari S., Vinogradov, A.P.: Metamaterials and Plasmonics: Fundamentals, Modelling, Applications. Springer-Verlag, New York, 2008

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Meccanica ed Aeronautica, Università di Roma La Sapienza, Via Eudossiana 18, 00184, Rome, Italy

    A. Carcaterra & F. dell’Isola

  2. Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, Via Eudossiana 18, 00184, Rome, Italy

    A. Carcaterra

  3. International Research Center M&MOCS, Università dell’Aquila, Palazzo Caetani, 04012, Cisterna di Latina (LT), Italy

    A. Carcaterra, F. dell’Isola, R. Esposito & M. Pulvirenti

  4. Diparimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 5, 00185, Roma, Italy

    M. Pulvirenti

Authors
  1. A. Carcaterra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. dell’Isola
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Esposito
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Pulvirenti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Pulvirenti.

Additional information

Communicated by C. Dafermos

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carcaterra, A., dell’Isola, F., Esposito, R. et al. Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials. Arch Rational Mech Anal 218, 1239–1262 (2015). https://doi.org/10.1007/s00205-015-0879-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0879-5

Keywords

Search

Navigation