Multiscale Modeling of Elastic Waves: Theoretical Justification and Numerical Simulation of Band Gaps

Abstract

We consider a three-dimensional composite material made of small inclusions periodically embedded in an elastic matrix, the whole structure presents strong heterogeneities between its different components. In the general framework of linearized elasticity we show that, when the size of the microstructures tends to zero, the limit homogeneous structure presents, for some wavelengths, a negative mass density tensor. Hence we are able to rigorously justify the existence of forbidden bands, i.e., intervals of frequencies in which there is no propagation of elastic waves. In particular, we show how to compute these band gaps and we illustrate the theoretical results with some numerical simulations.

Modelling of Evanescent and Propagating Modes in Homogenized Phononic Structures in Frequency and Time Domains

“…The aim of the paper was to present two methods for numerical modelling of dynamic behaviour of phononic structures described by the homogenized model developed in [2] and [6] using the two-scale homogenization of strongly heterogeneous elastic composites. The first method proposed in [4] relies on the spectral decomposition of the model equations in the frequency domain.…”

“…Homogenization of such structures produces models of equivalent continua with indefinite or negative mass for certain frequencies, as reported in a number of papers [2,1,8]. Recetnly, the model developed in [2] was considered as bases for shape optimization of soft inclusuions periodically distributed in a stiff elastic matrix [9], cf. [5] in the context of piezoelectric phononic materials.…”

“…The phenomenon which is responsible for the band gaps occurrence is related to anti-resonance effects in periodic structures with large contrast in the elasticity, see [3]. Homogenization of such structures produces models of equivalent continua with indefinite or negative mass for certain frequencies, as reported in a number of papers [2,1,8]. Recetnly, the model developed in [2] was considered as bases for shape optimization of soft inclusuions periodically distributed in a stiff elastic matrix [9], cf.…”

“…In paper [2], cf. [7], using the homogenization method and with the "dual porosity" type of the scaling ansatz applied in the inclusions, the following model of the phononic elastic solids occupying bounded domain ⌦ has been derived,…”

Trampoline metamaterial: Local resonance enhancement by springboards

“…These unique dispersion properties may sufficiently be utilized in numerous applications involving wave filtering, 4-6 localization, 7,8 guiding, 8,9 focusing, 10-12 collimation, 13,14 among others. The added feature of local resonance, however, gives rise to a qualitatively different type of dynamical behavior, such as negative effective elastic moduli and/or density, [15][16][17][18] along with the possibility of generation of subwavelength band gaps. 3 The applications of locally resonant acoustic/elastic metamaterials are, in turn, far from conventional, e.g., subfrequency wave isolation, 19 subwavelength focusing and imaging, 20,21 and cloaking, 22 to name a few.…”