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.