We propose a novel two-dimensional hierarchical auxetic structure, consisting of a porous medium in which a homogeneous matrix includes a rank-two set of cuts characterised by different scales. The six-fold symmetry of the perforations makes the medium isotropic in the plane. Remarkably, the mesoscale interaction between the first-and second-level cuts enables the attainment of a very small value of the Poisson's ratio, close to the minimum reachable limit of -1. The effective properties of the hierarchical auxetic structure are determined numerically, considering both a unit cell with periodic boundary conditions and a finite structure containing a large number of repeating cells. Further, results of the numerical study are validated experimentally on a polymeric specimen with appropriately arranged rank-two cuts, tested under uniaxial tension. We envisage that the proposed hierarchical design can be useful in numerous engineering applications exploiting an extreme auxetic effect.
We propose a new type of platonic crystal. The proposed microstructured plate includes snail resonators with low-frequency resonant vibrations. The particular dynamic effect of the resonators are highlighted by a comparative analysis of dispersion properties of homogeneous and perforated plates. Analytical and numerical estimates of classes of standing waves are given and the analysis on a macrocell shows the possibility to obtain localization, wave trapping and edge waves. Applications include transmission amplification within two plates separated by a small ligament. Finally we proposed a design procedure to suppress low frequency flexural vibration in an elongated plate implementing a by-pass system rerouting waves within the mechanical system. Dirac cones in platonic crystals equations, associated respectively to the presence of propagating and evanescent waves. Such waves can be coupled via the boundary or interface contact conditions. In most configurations the flexural waves are led by their Helmholtz component [5] and the homogenized equation can be of parabolic type at special frequencies [9,10]. However, short range wave scattering and Bragg resonance can be strongly influenced by the evanescent waves.Periodic structures play a major role in this field [18], since they create band gaps. These are frequency ranges where waves cannot propagate through the periodic system leading to possible application as acoustic and mechanical wave filters, vibration isolators, seismic shields. Partial band gap can lead to anisotropic wave response that can be used to obtain focusing and localization [19][20][21] as well as polarization properties [22,23].Two physical mechanisms can open band gaps: Bragg scattering and local resonance [24,25]. Bragg scattering is associated to the generation of band gaps at wavelengths of the same order of the unit cell around frequencies governed by the Bragg condition a = n(λ/2), (n = 1, 2, 3, · · · ), where a is the lattice constant of the periodic system and λ the wavelength [26]. Local resonances are associated to internal resonances due to the microstructures, they can be obtained from array of resonators as suggested in the seminal work [27]. Local resonances open tiny band gaps that can be at low frequencies [28][29][30] and inertial amplification mechanism that can widen stop band intervals have been proposed in [31,32].
The platonic system of snail resonatorsWe consider flexural vibrations in Kirchhoff plates. In the time-harmonic regime the transverse displacement W(x) satisfy the fourth-order biharmonic equation
Solid
foams with micrometric pores are used in different fields (filtering,
3D cell culture, etc.), but today, controlling their foam geometry
at the pore level, their internal structure, and the monodispersity,
along with their mechanical properties, is still a challenge. Existing
attempts to create such foams suffer either from slow speed or size
limitations (above 80 μm). In this work, by using a temperature-regulated
microfluidic process, 3D solid foams with highly monodisperse open
pores (PDI lower than 5%), with sizes ranging from 5 to 400 μm
and stiffnesses spanning 2 orders of magnitude, are created for the
first time. These features open the way for exciting applications,
in cell culture, filtering, optics, etc. Here, the focus is set on
photonics. Numerically, these foams are shown to open a 3D complete
photonic bandgap, with a critical index of 2.80, thus compatible with
the use of rutile TiO2. In the field of photonics, such
structures represent the first physically realizable self-assembled
FCC (face-centered cubic) structure that possesses this functionality.
Large scale elastic metamaterials have recently attracted increasing interest in the scientific community for their potential as passive isolation structures for seismic waves. In particular, so-called “seismic shields” have been proposed for the protection of large areas where other isolation strategies (e.g. dampers) are not workable solutions. In this work, we investigate the feasibility of an innovative design based on hierarchical design of the unit cell, i.e. a structure with a self-similar geometry repeated at different scales. Results show how the introduction of hierarchy allows the conception of unit cells exhibiting reduced size with respect to the wavelength while maintaining the same or improved isolation efficiency at frequencies of interest for earthquake engineering. This allows to move closer to the practical realization of such seismic shields, where low-frequency operation and acceptable size are both essential characteristics for feasibility.
In this paper we present an analytical model of Perfectly Matched Layers for flexural waves within elongated beam structures. The model is based on transformation optics techniques and it is shown to work both in time harmonic and transient regimes. A comparison between flexural and longitudinal waves is detailed and it is shown that the bending problem requires special interface conditions. A connection with transformation of eigenfrequencies and eigenmodes is given and the effect of the additional boundary conditions introduced at the border of the Perfectly Matched Layer domain is discussed in detailed. Such a model is particularly useful for Finite Element analyses pertaining propagating flexural waves in infinite domain
The problem of transformation optics for longitudinal and flexural waves in monodimensional elastic systems is analyzed. System of finite dimensions are considered and it is shown that, under appropriate interface conditions, eigenfrequencies in finite systems remain unchanged while eigenmodes can be tuned depending on the applied geometric transformation. Eigenfrequency analysis can be used in cloaking problem in order to demonstrate the quality of the cloak.
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