Modeling Bloch Waves in Prestressed Phononic Crystal Plates

Mazzotti, Matteo; Bartoli, Ivan; Miniaci, Marco

doi:10.3389/fmats.2019.00074

Cited by 32 publications

(21 citation statements)

References 53 publications

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“…The homogenization scheme introduced in Section 3 proves that the long-wavelength asymptotics for waves propagating in the lattice is governed by the acoustic tensor of the effective medium, Eq. (46). Therefore, it becomes now clear that a macro-bifurcation in the lattice has to be equivalent to failure of ellipticity in its equivalent continuum.…”

Section: Macro and Micro Bifurcations As Degeneracies Of The Dispersion Relationmentioning

confidence: 99%

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

Bordiga

Cabras

Bigoni

2021

Journal of the Mechanics and Physics of Solids

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A lattice (or 'grillage') of elastic Rayleigh rods (possessing a distributed mass density, together with rotational inertia) organized in a parallelepiped geometry can be axially loaded up to an arbitrary amount without distortion and then be subject to incremental time-harmonic dynamic motion. At certain threshold levels of axial load, the grillage manifests instabilities and displays non-trivial axial and flexural incremental vibrations. Including every possible structural geometry and for an arbitrary amount of axial stretching, Floquet-Bloch wave asymptotics is used to homogenize the in-plane mechanical response, so to obtain an equivalent prestressed elastic solid subject to incremental time-harmonic vibration, which includes, as a particular case, the incremental quasi-static response. The equivalent elastic solid is obtained from its acoustic tensor, directly derived from homogenization and shown to be independent of the rods' rotational inertia. Loss of strong ellipticity in the equivalent continuum coincides with macrobifurcation in the lattice, while micro-bifurcation remains undetected in the continuum and corresponds to a vibration of vanishing frequency of the lowest dispersion branch of the lattice, occurring at finite wavelength. Dynamic homogenization reveals the structure of the acoustic branches close to ellipticity loss and the analysis of forced vibrations (both in physical space and Fourier space) shows low-frequency wave localizations. A perturbative approach based on dynamic Green's function is applied to both the lattice and its equivalent continuum. This shows that only macro-instability corresponds to localization of incremental strain, while micro-instabilities occur in modes which spread throughout the whole lattice with an 'explosive' character. In particular, extremely localized mechanical responses are found both in the lattice and in the solid, with the advantage that the former can be easily realized, for instance via 3D printing. In this way, features such as shear band inclination, or the emergence of a single shear band, or competition between micro and macro instabilities become all designable features. The comparison between the mechanics of the lattice and its equivalent solid shows that the homogenization technique allows an almost perfect representation, except when micro-bifurcation is the first manifestation of instability. Therefore, the presented results pave the way for the design of architected cellular materials to be used in applications where extreme deformations are involved.

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Section: Macro and Micro Bifurcations As Degeneracies Of The Dispersion Relationmentioning

confidence: 99%

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

Bordiga

Cabras

Bigoni

2021

Journal of the Mechanics and Physics of Solids

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“…If compared to the cases of Bragg scattering and ordinary local resonant metamaterials 44 , we show that the inertial amplification allows for a more remarkable curve shift, modulated through large but fully reversible compression(stretch) of the unit cell, eventually triggering significant enlargement(reduction) of a specific band gap width. We also observe a selective up(down)-shift of some dispersion branches, leading to negative group velocities over specific wavenumber ranges.…”

Section: Introductionmentioning

confidence: 92%

Effect of prestress on phononic band gaps induced by inertial amplification

Miniaci

Mazzotti

Amendola

et al. 2021

International Journal of Solids and Structures

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Phononic crystals and elastic metamaterials have recently received significant attention due to their potential for unconventional wave control. Despite this interest, one outstanding issue is that their band diagram is typically fixed, once the structure designed. To overcome this limitation, periodic structures with adaptive elastic properties have recently been proposed for Bragg-and local resonancedriven structures.In this context, we report about the effect of an applied external mechanical load on a periodic structure exhibiting band gaps induced by inertial amplification mechanism. If compared to the cases of Bragg scattering and ordinary local resonant metamaterials, we observe here a more remarkable curve shift, modulated through large but fully reversible compression(stretch) of the unit cell, eventually triggering significant (up to two times) enlargement(reduction) of the width of a specific band gap.An important up(down)-shift of some dispersion branches over specific wavenumber values is also observed, showing that this selective variation may lead to negative group velocities over larger(smaller) wavenumber ranges. In addition, the possibility for a non-monotone trend of the lower limit of the first BG under the same type of external applied prestrain is found and explained through an analytical model, which unequivocally proves that this behaviour derives from the different unit cell effective mass and stiffness variations as the prestrain level increases. These peculiarities derive from the hinge-like behaviour of some regions of the unit cell, which is typical of structures exhibiting the inertial amplification mechanism.The effect of the prestress on the dispersion diagram is investigated through the development of a 2-step calculation method: first, an Updated Lagrangian scheme, including a static geometrically nonlinear analysis of a representative unit cell undergoing the action of an applied external load is derived, and then the Floquet-Bloch decomposition is applied to the linearized equations of the acousto-elasticity for the unit cell in the deformed configuration.Finally, the most evident consequence on the dispersion curves of the application of an external prestress, i.e. the band gap shift with respect to the unloaded structure, is demonstrated through nonlinear transient numerical simulations, clearly proving the capability of the structure to switch from a pass-to a stop-band behaviour over the same frequency range.The results presented herein provide insights in the behaviour of band gaps induced by inertial amplification, and suggest new opportunities for real-time tunable wave manipulation.

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“…Here, a 2D AMMs consists of an array of straight and infinite cylinders arranged in a square matrix is considered. This kind of structure is widely used since the AMMs were studied (Mead, 1996;Krushynska et al, 2017;Mazzotti et al, 2019). Figure 1 illustrates the lattice structure, where a denotes the lattice constant, and r denotes the radius of the cylinder.…”

Section: Mathematical Model Of Acoustic Metamaterialsmentioning

confidence: 99%

Numerical Investigation of Discrepancies Between Two-Dimensional and Three-Dimensional Acoustic Metamaterials

et al. 2021

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In order to get insight information of the band structure of acoustic metamaterials (AMMs) in condensed matter, periodic lattice structures are analyzed using Bloch’s theorem. Typical approaches of the band structure computation methods, topology optimization, and tunable abilities cannot overcome the gap between the two-dimensional (2D) AMMs theoretical and three-dimensional (3D) specimens’ experimental data yet. In this work, the variation in the results of the band structure obtained from the 2D mathematical model computed with respect to the 3D experimental models, and related cause of the variation is explored. The band structures and mode shapes of the 2D AMMs, quasi-2D models, and 3D specimen models are followed to reveal the boundary conditions and source for the observed differences in band structures. The cause for the discrepancies is verified by using the finite element method (FEM) with corresponding boundary conditions. It is found that outcomes from computational data of the 2D AMMs model are diverted significantly by means of bandgap, band structure, and stress distribution in counterparts of the 3D specimen model. This approach can provide assistance for computing the band structure of 2D AMMs for practical applications.

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Modeling Bloch Waves in Prestressed Phononic Crystal Plates

Abstract: The handling editor declared a past co-authorship with MMa.

Cited by 32 publications

References 53 publications

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

Effect of prestress on phononic band gaps induced by inertial amplification

Numerical Investigation of Discrepancies Between Two-Dimensional and Three-Dimensional Acoustic Metamaterials

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