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Boutin, “ Deformable porous media with double porosity. Auriault, “ Effective macroscopic description for heat-conduction in periodic composites,” Int. Sheng, “ Analytic model of phononic crystals with local resonances,” Phys. Sheng, “ Focusing of sound in a 3D phononic crystal,” Phys. Advanced Structured Materials, edited by H. Hans, “ Structural dynamics and generalized continua” in Mechanics of Generalized Continua. thesis, ENTPE - ECL (2010), (Last viewed ). Chesnais, “ Dynamique de milieux réticulés non contreventés-Application aux batiments (Dynamics of unbraced reticulated media-Application to buildings),” Ph.D. Application to bituminous concretes,” Int. Auriault, “ Dynamic behaviour of porous media saturated by a viscoelastic fluid. Boutin, “ Dynamics of discrete framed structures: A unified homogenized description,” J. Hans, “ Homogenisation of periodic discrete medium: Application to dynamics of framed structures,” Comput. Postnova, “ High-frequency asymptotics, homogenisation and localisation for lattices,” Q. Pichugin, “ High-frequency homogenization for periodic media,” Proc. Potier-Ferry, “ Evaluation of continuous modelings for the modulated vibration modes of long repetitive structures,” Int. Caillerie, “ Continuous modeling of lattice structures by homogenization,” Adv. Verna, “ Homogenisation of periodic trusses,” in IASS Symposium, 10 Years of Progress in Shell and Spatial Structures ( Madrid, 1989). Panasenko, “ Reinforced reticulated structures in elasticity,” C. Applied Mathematical Sciences ( Springer-Verlag, New York, 1999), Vol. Saint Jean Paulin, Homogenization of Reticulated Structures. Auriault, “ Rayleigh scattering in elastic composite materials,” Int. Panasenko, Homogenization: Averaging Processes in Periodic Media (Nauka, Moscou, 1984) (in Russian), English translation (Kluwer, Dordrecht, 1989). Lecture Notes in Physics ( Springer-Verlag, Berlin, 1980), Vol. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Studies in Mathematics and its Applications (North Holland, Amsterdam, 1978), Vol. Papanicolaou, Asymptotic Analysis for Periodic Structures. Andrianov, “ The specific features of the limiting transition from a discrete elastic medium to a continuous one,” J. Noor, “ Continuum modeling for repetitive lattice structures,” Appl. Christensen, “ Analogy between micropolar continuum and grid frameworks under initial stress,” Int. Advances in Mechanics and Mathematics ( Springer, New York, 2010), Vol. Metrikine, eds., Mechanics of Generalized Continua-One Hundred Years After the Cosserats. Willis, “ On modifications of Newton’s second law and linear continuum elastodynamics,” Proc.
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Zhang, “ Effective medium theory for elastic metamaterials in two dimensions,” Phys. Gengkai, “ Experimental study on negative effective mass in a 1D mass-spring system,” New J. Movchan, “ Vibrations of lattice structures and phononic band gaps,” Q. Bonnet, “ Dynamique des composites élastiques périodiques (Dynamics of periodic elastic composites),” Arch. Fleck, “ Wave propagation in two-dimensional periodic lattices,” J. Soranna, “ Wave beaming effects in two-dimensional cellular structures,” Smart Mater. Ruivo, “ The response of two-dimensional periodic structures to harmonic point loading: A theoretical and experimental study of a beam grillage,” J. Mead, “ Wave propagation in continuous periodic structures: Research contributions from Southampton, 1964–1995,” J.
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Brillouin, Wave Propagation in Periodic Structures ( McGraw-Hill, New York, 1946). Finally, a method is proposed for the design of such materials. The physical origin of these phenomena at the microscopic scale is also presented. Consequently, compressional waves become dispersive and frequency bandgaps occur. In particular, the local resonance in bending leads to an effective mass different from the real mass and to the generalization of the Newtonian mechanics at the macroscopic scale. The frame size being small compared to the wavelength of the compressional waves, the homogenization method of periodic discrete media is extended to situations with local resonance, and it is applied to identify the macroscopic behavior at the leading order. Moreover, when the thickness to length ratio of the frame elements is small, these elements can resonate in bending at low frequencies when compressional waves propagate in the structure. Such materials are highly anisotropic and, because of lack of bracing, can present a large contrast between the shear and compression deformabilities. This work is devoted to the study of the wave propagation in infinite two-dimensional structures made up of the periodic repetition of frames.