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.
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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)
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
Carcaterra, A, Akay, A.: Dissipation in a finite-size bath. Phys. Rev. E, 84, 011121.1–011121.4 (2011)
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)
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)
Craster, R.V., Guenneau, S. (eds.): Acoustic Metamaterials. Springer Series in Material Science, vol. 166, Springer, Berlin, 2013
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)
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)
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
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
Engheta, N., Ziolkowski, R.W.: Metamaterials: Physics and Engineering, Explorations. Wiley, New York, 2006
Germain, P.: The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)
Green A.E., Rivlin R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17(2), 113–114 (1964)
Gurtin, M.E.: Thermodynamics and the possibility of spatial interaction in elastic materials. Arch. Ration. Mech. Anal. 19.5, 339–352 (1965)
Hilbert, D.: Begründung der kinetischen Gastheorie, Math. Ann. 72 331–407 (1916/17)
Kolpakovs A.G.: Determination of the average characteristics of elastic frameworks. J. Appl. Math. Mech. 49(6), 739–745 (1985)
Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-Relativistic Theory, 3 (3rd edn.). Pergamon Press, Oxford, 1977
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
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
Milton, G.W., Cherkaev, A.V.: Which elasticity tensors are realizable? J. Eng. Mater. Technol. 117(4), 483 (1995)
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)
Mindlin R.D.: Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)
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.
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)
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
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)
Toupin R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
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)
Zouhdi, S., Ari S., Vinogradov, A.P.: Metamaterials and Plasmonics: Fundamentals, Modelling, Applications. Springer-Verlag, New York, 2008
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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
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DOI: https://doi.org/10.1007/s00205-015-0879-5