Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics

Abstract

This work is devoted to the study of the wave propagation in infinite two-dimensional structures made up of the periodic repetition of frames. Such materials are highly anisotropic and, because of lack of bracing, can present a large contrast between the shear and compression deformabilities. 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. 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. 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. Consequently, compressional waves become dispersive and frequency bandgaps occur. The physical origin of these phenomena at the microscopic scale is also presented. Finally, a method is proposed for the design of such materials.

Locally resonant band gaps in periodic beam lattices by tuning connectivity

“…A global stability criterion that purely depends onz was first determined by Maxwell for pin-joined lattices comprising spring-like ligaments 1 , and then modified to account for the nature (pin or welded) of the joints 6 , the bending stiffness of the struts 7,8 , self-stresses 9 , dislocation defects 10 , collapse mechanisms 11 and boundary modes [12][13][14][15] . In recent years, the dynamic response of periodic lattices has also attracted considerable interest [16][17][18][19] because of their ability to tailor the propagation of elastic waves through directional transmissions [20][21][22][23] and bandgaps (frequency ranges of strong wave attenuation) [21][22][23][24] . However, though several studies have shown that the wave propagation properties of periodic lattices are highly sensitive to the architecture of the network [20][21][22][23][24] , a global criterion connecting the frequency and size of bandgaps to the lattice topology is still not yet in place.…”

Large scale modulation of high frequency waves in periodic elastic composites

“…Further works are in progress in order to relate modulation phenomena -that involve "full" dynamics at the local scaleto situations of partial inner dynamics observed in meta-materials as reticulated media (Hans and Boutin, 2008;Chesnais et al, 2012) or highly contrasted composites (Auriault and Bonnet, 1985;Soubestre and Boutin, 2012;Auriault and Boutin, 2012). Besides, when focusing on the first modes of large multi-cell period the inertia becomes weak in each of the irreducible period Ω 0 .…”

The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results

“…[9,10,[54][55][56][57][58][59][60][61][62][63][64][65][66][67][68]). Therefore, many efforts have been directed towards more or less mathematically rigorous homogenization procedures leading to this class of continua (e.g.…”

“…Another example is given by the case of a periodic fibre-reinforced elastic medium with high contrast of mechanical properties. The mechanical description of these systems needs in addition to the standard stress tensor a higher order hyper-stress tensor [9,10].…”