Dispersive wave propagation in two-dimensional rigid periodic blocky materials with elastic interfaces

Abstract

Dispersive waves in two-dimensional blocky materials with periodic microstructure made up of equal rigid units having polygonal centro-symmetric shape with mass and gyroscopic inertia, connected each other through homogeneous linear interfaces, have been analysed. The acoustic behavior of the resulting discrete Lagrangian model has been obtained through a Floquet-Bloch approach. From the resulting eigenproblem derived by the Euler-Lagrange equations for harmonic wave propagation, two acoustic branches and an optical branch are obtained in the frequency spectrum. A micropolar continuum model to approximate the Lagrangian model has been derived based on a second-order Taylor expansion of the generalized macro-displacement field. The constitutive equations of the equivalent micropolar continuum have been obtained, with the peculiarity that the positive definiteness of the second-order symmetric tensor associated to the curvature vector is not guaranteed and depends both on the ratio between the local tangent and normal stiffness and on the block shape. The same results has been obtained through an extended Hamiltonian derivation of the equations of motion for the equivalent continuum that is related to the Hill-Mandel macro homogeneity condition. Moreover, it is shown that the hermitian matrix governing the eigenproblem of harmonic wave propagation in the micropolar model is exact up to the second order in the norm of the wave vector with respect to the same matrix from the discrete model. To appreciate the acoustic behavior of some relevant blocky materials and to understand the reliability and the validity limits of the micropolar continuum model, some blocky patterns have been analysed: rhombic and hexagonal assemblages and running bond masonry. From the results obtained in the examples, it appears that the obtained micropolar model turns out to be particularly accurate to describe dispersive functions for wavelengths greater than 3-4 times the characteristic dimension of the block. Finally, in consideration that the positive definiteness of the second order elastic tensor of the micropolar model is not guaranteed, the hyperbolicity of the equation of motion has been investigated by considering the Legendre-Hadamard ellipticity conditions requiring real values for the wave velocity.

Acoustic waveguide filters made up of rigid stacked materials with elastic joints

“…According to a Lagrangian mechanical formulation, the i-th block is assumed to undergo a transverse displacement i v and rotation i  so that the equations of free motion are written (see for reference Bacigalupo and Gambarotta, 2017) Tracking the dependence on the wavenumber k in the domain   0,  allows to completely assess the Floquet-Bloch spectrum of the periodic system (Brillouin, 1946). The dispersion function may be represented in terms of the non-dimensional circular frequency , which cannot be identified a priori as limit points of the acoustic or optical branch.…”

“…In fact, to obtain low-frequency Bragg gaps, heavy inclusions, low stiffness and large cell size are needed, which may be not suitable for practical purposes. Alternatively, band gaps may be obtained by including local resonators in the periodic material, whose role is to impede wave propagation around their resonance frequency by transferring the vibrational energy to the resonator (see for reference Huang and Sun, 2010, Craster and Guenneau, 2012, Baravelli and Ruzzene, 2013, Bacigalupo and Gambarotta, 2017, Beli and Ruzzene, 2018, Deng et al, 2018. In this case, low-frequency band gaps may be obtained, even if heavy resonators and low stiffness cells are required to get wide stop-bands.…”

“…Moreover, a convenient fine approximation of the dispersion functions of the Lagrangian model is pursued through the formulation of an equivalent continuum model. This continuum model is based on an enhanced continualization approach for discrete models proposed by Bacigalupo and Gambarotta, 2018 in order to circumvent some drawbacks emerging by applying classical continualization approaches (Bacigalupo and Gambarotta, 2017). This procedure provides different equivalent continua with increasing order of the leading derivative with non-local constitutive and inertial terms.…”

Acoustic wave polarization and energy flow in periodic beam lattice materials

“…This limit behaviour is typical of first-order continua and is consistent with the possibility to approximate the acoustic dispersion surfaces of the beam lattice material with those of a first-order homogenized continuum. For shorter wavelengths the dispersion effects are increasingly important and better approximations of the material spectrum are achievable by homogenization in non-local continua [51,66]. The boundary of the Brillouin zone B, defining the limit of short wavelengths, is identically featured by vanishing group velocity, but finite not null phase velocities.…”

Auxetic behavior and acoustic properties of microstructured piezoelectric strain sensors

“…Finally, a 2D piezo-elasto-dynamic dispersion analysis adopting the Floquet-Bloch decomposition is performed. We extend to piezoelectric materials the analysis of chiral honeycomb lattices to evaluate the properties of the dispersion functions of waves propagating in different directions and to detect the band gaps characterizing the material, [22,[40][41][42][43][44][45][46]. The paper is organized as follows.…”