Issue 
EPJ Appl. Metamat.
Volume 8, 2021
Frontiers in microwave, photonic, and mechanical metamaterials



Article Number  14  
Number of page(s)  8  
DOI  https://doi.org/10.1051/epjam/2021009  
Published online  08 June 2021 
https://doi.org/10.1051/epjam/2021009
Research Article
Hierarchical largescale elastic metamaterials for passive seismic wave mitigation
^{1}
CNRS, Centrale Lille, ISEN, Univ. Lille, Univ. Valenciennes, UMR 8520  IEMN, 59000 Lille, France
^{2}
Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
^{3}
Department of Mechanical Engineering, CU Boulder, 1111 Engineering Drive, UCB 427 Boulder, CO 80309, USA
^{4}
Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10124 Torino, Italy
^{5}
Department of Physics, University of Torino, Via Pietro Giuria 1, 10125 Torino, Italy
^{6}
University of Trento, Laboratory of BioInspired, Bionic, Nano, Meta Materials & Mechanics, Department of Civil, Environmental and Mechanical Engineering, Via Mesiano 77, 38123 Trento, Italy
^{*} email: marco.miniaci@gmail.com; marco.miniaci@univlille.fr
^{a} Also at: Queen Mary University of London, School of Engineering & Materials Science, Mile End Road, London E1 4NS, UK.
Received:
5
November
2020
Accepted:
11
May
2021
Published online: 8 June 2021
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, socalled “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 selfsimilar 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 lowfrequency operation and acceptable size are both essential characteristics for feasibility.
Key words: Seismic phononic crystals / metamaterials / hierarchical organization / transientdynamic analysis / vibration isolation
© M. Miniaci et al., Published by EDP Sciences, 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Structural integrity and adherence to civil engineering requirements is of paramount importance in the construction of all types of buildings, especially when considering the protection from seismic hazards in large or strategic constructions, such as hospitals, nuclear sites, long span bridges, dams, etc. Seismic waves derive from the superposition of different types of elastic waves, including bulk waves (longitudinal and shear), and waves localized at the surface, also known as surface acoustic waves (SAWs) [1]. The latter are the most destructive, because of their reduced attenuation during propagation. Indeed, their slower decay with respect to bulk waves is the main cause for partial or even total collapse of buildings and structures during earthquakes [2]. As a consequence, the design of buildings and structures capable of withstanding large ground vibrations has been the focus of research for many decades, and a great number of isolating strategies has been proposed, including passive, active, hybrid and semiactive approaches [3–5]. However, despite these efforts, a commonly accepted method for the design of seismicresistant buildings has not yet been developed.
In 1999, Meseguer et al. [6] theoretically proposed for the first time to attenuate seismic SAWs through band gaps (BGs) obtained by drilling huge periodic holes in the ground. The dimensions of the holes proposed at that time were of the order of hundreds of meters in diameter and thousands of meters in depth. Such an extreme proposed solution derived from the measurement of the attenuation of SAWs in a scaled experiment of a marble quarry, performed in the kHz frequency regime. After this initial intuition, many years passed without substantial further investigation linking phononic crystals (PCs) and seismic wave attenuation, up to 2014, when Brûlé et al. [7] performed large scale experiments on metersize periodic holes drilled in the ground. In this occasion the attenuation potential of periodic structures was measured in the 50 Hz frequency range and for bulk waves.
From then on, researchers have considered more and more types of PCs as a potential solution for reducing the displacement induced in buildings by seismic waves, including surface localized modes. Several designs have been proposed, based on both largescale PCs and resonant structures [8,9], trying to meet the stringent requirements of wave attenuation at extremely low frequencies (below 5 Hz) [10], practical manufacturing designs [8], and other technological constraints. Among the resonatorbased approaches, the most investigated consists in constructing pillarlike structures at the surface with the goal of redirecting seismic SAWs towards the bulk at a specific angle. This approach is also known in the literature as the “resonant metawedge” [9,11]. Its main drawback is the fact that the effect is rather narrow in frequency, since it relies on the resonance frequency of the resonators composing the system. Krödel et al. [12] showed that it was possible to enlarge the effective frequency range of these structures by using resonators with slightly different resonance frequencies. Other researchers explored the possibility of using buried resonators [13], or drilling periodic trench barriers [14,15], also including the possibility of saturated soil [16]. These methods also provided good attenuation, in larger frequency ranges, since their performance derives from Bragg scattering.
In the majority of these numerical studies, the analysis has been restricted to 2D models, assuming the geometry of the resonators/cavities to be infinite in one of the two horizontal directions. As a consequence, the full potential of the geometry of the resonators/cavities has been scarcely explored so far. Here, we overcome this limitation and, exploiting full 3D finite element modeling, we propose an innovative design strategy based on a hierarchical organization of largescale PCs in the horizontal plane of the ground. We show that the introduction of hierarchy allows the conception of structures achieving isolation efficiency at lower frequencies if compared to the corresponding non hierarchical geometry, allowing to reduce the operating frequencies of the seismic barrier down to half their “non hierarchical” values, while leaving the unit cell size unaltered. Hierarchy also allows to manipulate the effective unit cell mechanical properties without the need of multiple materials with different mechanical characteristics. This is proved through band diagram calculation and transient dynamic analysis. Finally, the difference in terms of behavior adopting different boundary conditions (platelike or in a halfspace assumption) is also discussed.
2 Description of the models
We consider hierarchical structures, where hierarchy is understood in the sense that a representative unit cell comprises multiple arrangements of inhomogeneities at various scales. If the same arrangement occurs at every scale, the pattern is called selfsimilar [17]. In this work, we consider structures with selfsimilar crosslike cavities at two different hierarchical levels, as presented in Figure 1A and B, in a sandy soil matrix with Young's modulus E_{soil} = 20 MPa, Poisson's ratio ν_{soil} = 0.3, mass density ρ_{soil} = 1800 kg · m^{−3}, as reported in Table 1.
At first, we consider unit cells under the simplified assumption of freefree boundary conditions applied to the top and bottom surfaces, for the purpose of comparing the dynamic behavior of the standard and the hierarchical unit cells. Finite element models of the ordinary and hierarchical unit cells that compose the seismic hierarchical shield are shown in Figure 1A and B. The ordinary unit cell consists of a crosslike cavity dug into the soil (Fig. 1A). The crosslike cavity divides the unit cell into two types of regions, referred in the following as (i) masses and (ii) connectors (see black arrows in Fig. 1A). The hierarchical unit cell is obtained by adding 4 additional crosslike cavities (see Fig. 1B and its inset), scaled down by a factor of 20, so as to fit in the regions connecting the four large square masses of the nonhierarchical structure. The corresponding geometrical parameters are: A = 5 m, B = 0.9A, C = 0.2A and H = A for the ordinary unit cell and a = A/20, b = 0.9a, c = 0.2a and H = A for the hierarchical one. Further geometrical details are reported in Table 2. Contrary to previous studies dealing with hierarchical organization, where the hierarchy was reproduced over the entire ordinary unit cell [17], here we introduce hierarchy only in the connectors, which are mainly responsible for the low frequency alteration of the BG [17]. The design strategy is implemented for only a single hierarchical level, but the procedure, in principle, can be recursively extended to n hierarchical levels. The inclusions have been chosen in the shape of a crosslike cavity, due to its excellent capability to open low frequency large BGs [18,19].
The band structures are computed for an infinite array of cells in the inplane directions, i.e. with periodic conditions along the x and ydirections, and adopting standard BlochFloquet formalism, assuming a linear elastic behavior for the soil. The resulting eigenvalue problem is solved by varying the nondimensional wavevector k^{*} along the irreducible path [Γ − X], with Γ ≡ (0, 0) and X(≡ π/A, 0), with A the lattice parameter using the commercial software Cosmol 5.5.
As a second step, we assume free boundary conditions applied to the top surface of the unit cell and of infinite halfspace at the bottom surface of the unit cell, which better reflects the condition of surface seismic wave propagation. This is modeled by adding a 8λlong soillike domain plus a 2λlong PML region (where λ is the wavelength) below the unit cell, as shown in Figure 2A, where for the sake of synthesis, only the hierarchical unit cell model is reported. Dispersion diagrams are calculated by applying the same aforementioned procedure.
Figure 2B reports a schematic representation of the finite element models used to perform dynamic transient calculations to evaluate and compare the attenuation performances of the ordinary and hierarchical unit cells. The model consists of a soil slab of length L_{x} = 300 m, infinite in width in the ydirection (obtained by applying continuity conditions) and of height L_{z} = 8λ + H ; 100 m (λ has been chosen according to the longest wavelength for the lowest frequency of 2 Hz). Elastic waves are excited at the topleft edge of the model by means of a unitary imposed displacement in the zdirection (red arrows in Fig. 2B). Two input signals are considered: (i) a triangularlike excitation (left panel of Fig. 2C) and (ii) a 15cycle Hanning window modulated sinusoidal tone burst excitation (right panel of Fig. 2C), exhibiting a rather broadband and narrowband frequency content, respectively. Absorbing boundary (lowreflecting) conditions are imposed at the bottom and right edges of the model, using material data from the neighboring domain to develop a perfect impedance match for pressure and shear waves, to avoid unwanted reflections. Symmetry conditions are applied to the left edge of the model in order to be consistent with the type of induced excitation.
Fig. 1
Dispersion diagrams I − free/free assumption. (A) Schematic representation of the FE models for the ordinary and (B) hierarchical largescale metamaterial (the inset provides an enlargement of the region where the second hierarchical level is introduced) under the assumption of freefree boundary conditions (BCs) applied to the top and bottom surfaces of the unit cell and periodic boundary conditions (PBCs) at lateral surfaces. Dispersion curves of the (C) ordinary and (D) hierarchical largescale elastic metamaterial, showing a remarkable enhancement of the width of the BGs (orange rectangles) and their shift towards lower frequencies. Geometrical parameters of the unit cells for both ordinary and hierarchical structure are provided in the text. 
Fig. 2
Numerical models. (A) Schematic representation of the FE models used for the calculation of the dispersion curves localized at the surface. Free boundary conditions are applied to the top surface of the unit cell and a semiinfinite halfspace is modeled through 8λlong soillike domain + a 2λlong PML region. For the sake of synthesis, only the hierarchical unit cell is reported. (B) Schematic representation of the FE models used for the transient calculations. In this case, only the non hierarchical model is reported. (C) Time evolution and frequency content of the input signals used in simulations: (left panel) a triangularlike excitation and 21cycle Hanning window modulated sinusoidal tone bursts centered at (central panel) 3.8 Hz and (right panel) 2.4 Hz, respectively. The frequency spectrum clearly shows a rather broadband and narrowband frequency content, respectively. Displacement units are in decimeters [dm]. 
3 Results
Figure 1C and D reports the band structure of the ordinary and hierarchical unit cells, respectively, under the assumption of freefree boundary conditions applied to the top and bottom surfaces. We observe that hierarchy allows to obtain (i) a remarkable enhancement of the BG width (orange rectangles), as well as (ii) a BG shift towards lower frequencies. This twofold effect is a consequence of (i) the flattening of the dispersion curve (indicated by the black arrows in Fig. 1C,D) separating the two BGs of the ordinary structure and (ii) of a general decrease of the stiffness of the connectors, implying a downshift of the dispersion curves in the [0 − 4] Hz range.
Next, the SAW band structures for the ordinary and hierarchical unit cells are investigated and reported in Figure 3A and D for the [2 − 8] Hz frequency range. Given the approximation of halfspace condition as a 8λlong soillike domain plus a 2λlong PML region below the unit cell, a plethora of modes appear in the diagrams. However, the majority of them are not localized at the surface. To highlight this, curves are colorcoded according to the parameter p, which is an indicator of the normalized center of energy distribution along the zaxis [20]:(1)having taken the integrals over the unit cell volume V, and where E is the free elastic energy density and H the total height of the unit cell. The parameter p varies from 0 to 1, where SAWlike modes are those with p approaching 0, as suggested by Graczyowki et al. [20]. We chose here p ∈ [0, 0.3]. Once this sorting rule is applied, the diagrams can be simplified, selecting only SAW modes, as shown in Figure 3B and E. Some of the dispersion curves are not continuous along the whole Brillouin path Γ − X because they undergo a transformation from trueSAWs to pseudoSAWs (i.e., waves propagating with exponential attenuation, due to energy leakage into the bulk), as detailed in [20,21]. Figure 3A and D also reports the longitudinal and shear bulk wave velocities as dashed black and red lines, respectively. This allows to evaluate the frequency of transition from surface to longitudinal/shear bulk waves and to make the distinction between true and pseudoSAWs.
Figure 3B and E includes SAWtype modes with a predominantly outofplane polarization, with a weak outofplane polarization, and mainly inplane polarized (see mode shapes reported in Fig. 3C and F). Due to the nature of the excitation source considered in this study (forced outofplane displacement), we are interested in SAWtype modes with mainly outofplane polarization and we expect that while modes of the first type will be strongly excited in our system, those with a weak outofplane polarization will only provide a limited contribution to the vertical displacement, while those of the third type (inplane polarized) will hardly be activated. As a consequence, we can identify two types of BGs that have been colorcoded, using grey for those inhibiting the propagation of waves regardless of their polarization (inplane, outofplane, mixed), and yellow to indicate BGs mainly inhibiting the propagation of inplane polarized modes, referred to as total and partial BGs, respectively. Comparing the two band structures of the ordinary unit cell (Fig. 3B) and of the hierarchical one (Fig. 3E), we observe that the introduction of hierarchy also leads to (i) an enhancement of the BG width and (ii) a BG shift towards lower frequencies, as in the case of freefree boundary conditions. Specifically, the lowest BG is shifted from the [3.6 − 4.0] Hz range down to [2.2 − 2.7], while the highest one, starting at 5.68 Hz in the nonhierarchical structure, begins at 5.1 Hz in the hierarchical case. Notice that this is obtained while keeping the unit cell lattice parameter size unaltered. However, it is also important to remark that BGs are strongly reduced in width and appear in rather different frequency ranges, if compared to those deriving from the application of freefree boundary conditions.
The confirmation of the lowest BG opening/shift is then verified through a transient numerical simulation of wave propagation conducted on a finite size domain comprising a sandy soil block and 4 unit cells disposed in the xdirection, as shown in Figure 2B. Elastic waves are excited at the topleft edge of the model by means of a unitary imposed displacement in the zdirection (red arrows in Fig. 2B), according to the timehistories reported in Figure 2C of Section 2. Such pulses have been chosen according to the band structure reported in Figure 3B and E and to highlight the filtering capabilities of the designed hierarchical large scale PC compared with the nonhierarchical one. In both excitation cases, transient explicit simulations have been run for T = 8 s in order to allow the wave coming from the left side to reach the right end of the simulation domain.
The outofplane time transient displacements for both types of excitation are recorded at the end of the periodic structure (see Fig. 2B) and Fourier transformed to highlight the differences of the two responses for the ordinary and hierarchical largescale PCs both in the time and frequency domains. Figure 4A reports the displacement along the zdirection, as well as its frequency content (Fig. 4B) for the excitation case of the triangular pulse reported in Figure 2C, characterized by a broadband frequency content. The strongest achieved attenuation is shifted from slightly below 6 Hz for the ordinary structure to 5.1 Hz in the hierarchical case (corresponding to the beginning of the total BG). Examining the frequency content of the transmitted displacement field at lower frequencies, it emerges that the hierarchical case is also capable of providing large attenuation in the [2.2 − 2.7] Hz frequency regime, in good agreement with the dispersion diagrams reported in Figure 3B and E. The slightly higher amplitude level observed in the partial BG frequency regions (i.e., those covered by the yellow rectangles) derives from modes that, although mainly inplane polarized, still exhibit some outofplane displacement components. This is in good agreement with the two different dispersion diagrams and clearly confirms the possibility of using hierarchical unit cell organization to downshift stopband behavior.
To gain further insight into this behavior, the timedomain simulations are repeated, this time with the previously defined narrow band signal (see Sect. 2), with modulations centered at 3.8 and 2.4 Hz for the nonhierarchical and hierarchical cases, respectively. Figure 5 shows the instantaneous outofplane displacement fields at specific times, namely before and after the wave has reached the periodic structures, for both the nonhierarchical and hierarchical cases. The wave pulse is strongly reflected in both cases. However, from a quantitative point of view, the maximum recorded displacements with respect to a model where the surface wave propagates freely (i.e. without any PC) are almost 8 and 3 times larger for the nonhierarchical and hierarchical cases, respectively. Note that part of the attenuation efficiency difference between the two cases is directly related to the fact that the hierarchical structure is evaluated at a lower frequency, namely 2 Hz instead of 4.
Fig. 3
Dispersion diagrams II − free/halfspace assumption. Row dispersion diagrams deriving from the BlochFloquet calculations for the (A) ordinary and (D) hierarchical largescale metamaterial, under the hypothesis of free boundary condition applied to the top surface of the unit cell and semiinfinite halfspace modeled through 8λlong soillike domain + a 2λlong PML region on the bottom surface of the unit cell. Curves are colorcoded allowing to highlight modes localized at the surface (p ≤ 0.3) and those radiating in the bulk (p > 0.3). (B, E) Dispersion diagrams reporting only the modes localized at the surface for the dispersion diagrams reported in (A) and (D), respectively. Total and partial BGs are reported as grey and yellow rectangles, respectively. (C, F) Mode shapes with large vertical components predominantly outofplane, inplane and mixed polarization are indicated with corresponding symbols in the (B, E) band diagrams. 
Fig. 4
Time histories and Fourier transform. Outofplane time transient displacement and its Fourier transform for the broadband excitation for both the ordinary (A, black line) and hierarchical (B, blue line) structures. The signals are acquired immediately after the end of the periodic structure. The Fourier transforms of the signals allow to show the difference of the two responses in terms of frequency content, showing the ability of the hierarchical unit cell to provide large attenuation in the [2.2 − 2.7] Hz frequency range and above 5.1 Hz, in good agreement with the dispersion diagrams reported in Figure 3B and E. BGs calculated in Figure 3B and E are superimposed onto each frequency response, to visually highlight how the frequency responses matches the dispersion diagrams. 
Fig. 5
Full wavefield reconstruction. Outofplane full wavefield for the (A) ordinary and (B) hierarchical seismic barrier at two different time steps: before (top panels) and after (bottom panels) the wave has reached the periodic structure. 
4 Conclusion
This study contributes to the conception of a new generation of earthquakeproof barriers capable of protecting sensitive or strategic structures, introducing the concept of hierarchical design. We have investigated the effect of hierarchy on dispersion diagrams and verified results through timetransient simulations, and highlighted the possibility of realizing unit cells capable of opening frequency BGs below 5 Hz, despite being similar in size to previous designs. Results prove the strategy to be practical for civil structures.
Acknowledgments
MaMi has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska Curie grant agreement N. 754364. NK, FB, ASG, MO have received funding from the project “Metapp” (No. CSTO160004) funded by Fondazione San Paolo. MaMi, MaMO, FB, ASG and NMP are funded by the European Union's Horizon 2020 FET Open (“Boheme”) under grant agreement No. 863179.
References
 I. Viktorov, Rayleigh and Lamb Waves (1967) [Google Scholar]
 R. Villaverde, Fundamental Concepts of Earthquake Engineering (CRC Press, 2009) [Google Scholar]
 S.K. Duggal, Earthquake Resistant Design of Structures (Oxford University Press, New Delhi, 2007) [Google Scholar]
 B. Spencer, Jr., S. Nagarajaiah, State of the art of structural control, J. Struct. Eng. 129, 845 (2003) [Google Scholar]
 T. Datta, A stateoftheart review on active control of structures, ISET J. Earthquake Technol. 40, 430 (2003) [Google Scholar]
 F. Meseguer, M. Holgado, D. Caballero, N. Benaches, J. SanchezDehesa, C. López, J. Llinares, Rayleighwave attenuation by a semiinnite twodimensional elasticbandgap crystal, Phys. Rev. B 59, 12169 (1999) [Google Scholar]
 S. Brûlé, E. Javelaud, S. Enoch, S. Guenneau, Experiments on seismic metamaterials: molding surface waves, Phys. Rev. Lett. 112, 133901 (2014) [Google Scholar]
 M. Miniaci, A. Krushynska, F. Bosia, N.M. Pugno, Large scale mechanical metamaterials as seismic shields, New J. Phys. 18, 083041 (2016) [Google Scholar]
 A. Colombi, D. Colquitt, P. Roux, S. Guenneau, R.V. Craster, A seismic metamaterial: The resonant metawedge, Sci. Rep. 6, 27717 (2016) [Google Scholar]
 Y. Achaoui, T. Antonakakis, S. Brûlé, R. Craster, S. Enoch, S. Guenneau, Clamped seismic metamaterials: ultralow frequency stop bands, New J. Phys. 19, 063022 (2017) [Google Scholar]
 A. Colombi, P. Roux, S. Guenneau, P. Gueguen, R.V. Craster, Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances, Sci. Rep. 6, 19238 (2016) [Google Scholar]
 S. Krödel, N. Thomé, C. Daraio, Wide bandgap seismic metastructures, Extreme Mech. Lett. 4, 111 (2015) [CrossRef] [Google Scholar]
 A. Palermo, S. Krödel, A. Marzani, C. Daraio, Seismic surface waves attenuation by buried resonators, in 2016 10th International congress on advanced electromagnetic materials in microwaves and optics (metamaterials) (IEEE, 2016) pp. 271–273 [CrossRef] [Google Scholar]
 J. Huang, Z. Shi, Attenuation zones of periodic pile barriers and its application in vibration reduction for plane waves, J. Sound Vib. 332, 4423 (2013) [Google Scholar]
 X. Pu, Z. Shi, Broadband surface wave attenuation in periodic trench barriers, J. Sound Vib. 468, 115130 (2020) [Google Scholar]
 G. Shi, G.Y. Gao, Threedimensional analysis of a row of piles as passive barriers in saturated soil, J. Vib. Eng. 23, 546 (2010) [Google Scholar]
 M. Miniaci, A. Krushynska, A.S. Gliozzi, N. Kherraz, F. Bosia, N.M. Pugno, Design and fabrication of bioinspired hierarchical dissipative elastic metamaterials, Phys. Rev. Appl. 10, 024012 (2018) [CrossRef] [Google Scholar]
 M. Miniaci, A. Marzani, N. Testoni, L. De Marchi, Complete band gaps in a polyvinyl chloride (pvc) phononic plate with crosslike holes: numerical design and experimental verication, Ultrasonics 56, 251 (2015) [CrossRef] [Google Scholar]
 M. Miniaci, M. Mazzotti, M. Radzieński, N. Kherraz, P. Kudela, W. Ostachowicz, B. Morvan, F. Bosia, N.M. Pugno, Experimental observation of a large lowfrequency band gap in a polymer waveguide, Front. Mater. 5, 8 (2018) [CrossRef] [Google Scholar]
 B. Graczykowski, F. Alzina, J. GomisBresco, C. Sotomayor Torres, Finite element analysis of true and pseudo surface acoustic waves in onedimensional phononic crystals, J. Appl. Phys. 119, 025308 (2016) [CrossRef] [Google Scholar]
 M. Mazzotti, M. Miniaci, I. Bartoli, Band structure analysis of leaky bloch waves in 2d phononic crystal plates, Ultrasonics 74, 140 (2017) [CrossRef] [Google Scholar]
Cite this article as: Marco Miniaci, Nesrine Kherraz, Charles Cröenne, Matteo Mazzotti, Maryam Morvaridi, Antonio S. Gliozzi, Miguel Onorato, Federico Bosia, Nicola Maria Pugno, Hierarchical largescale elastic metamaterials for passive seismic wave mitigation, EPJ Appl. Metamat. 8, 14 (2021)
All Tables
All Figures
Fig. 1
Dispersion diagrams I − free/free assumption. (A) Schematic representation of the FE models for the ordinary and (B) hierarchical largescale metamaterial (the inset provides an enlargement of the region where the second hierarchical level is introduced) under the assumption of freefree boundary conditions (BCs) applied to the top and bottom surfaces of the unit cell and periodic boundary conditions (PBCs) at lateral surfaces. Dispersion curves of the (C) ordinary and (D) hierarchical largescale elastic metamaterial, showing a remarkable enhancement of the width of the BGs (orange rectangles) and their shift towards lower frequencies. Geometrical parameters of the unit cells for both ordinary and hierarchical structure are provided in the text. 

In the text 
Fig. 2
Numerical models. (A) Schematic representation of the FE models used for the calculation of the dispersion curves localized at the surface. Free boundary conditions are applied to the top surface of the unit cell and a semiinfinite halfspace is modeled through 8λlong soillike domain + a 2λlong PML region. For the sake of synthesis, only the hierarchical unit cell is reported. (B) Schematic representation of the FE models used for the transient calculations. In this case, only the non hierarchical model is reported. (C) Time evolution and frequency content of the input signals used in simulations: (left panel) a triangularlike excitation and 21cycle Hanning window modulated sinusoidal tone bursts centered at (central panel) 3.8 Hz and (right panel) 2.4 Hz, respectively. The frequency spectrum clearly shows a rather broadband and narrowband frequency content, respectively. Displacement units are in decimeters [dm]. 

In the text 
Fig. 3
Dispersion diagrams II − free/halfspace assumption. Row dispersion diagrams deriving from the BlochFloquet calculations for the (A) ordinary and (D) hierarchical largescale metamaterial, under the hypothesis of free boundary condition applied to the top surface of the unit cell and semiinfinite halfspace modeled through 8λlong soillike domain + a 2λlong PML region on the bottom surface of the unit cell. Curves are colorcoded allowing to highlight modes localized at the surface (p ≤ 0.3) and those radiating in the bulk (p > 0.3). (B, E) Dispersion diagrams reporting only the modes localized at the surface for the dispersion diagrams reported in (A) and (D), respectively. Total and partial BGs are reported as grey and yellow rectangles, respectively. (C, F) Mode shapes with large vertical components predominantly outofplane, inplane and mixed polarization are indicated with corresponding symbols in the (B, E) band diagrams. 

In the text 
Fig. 4
Time histories and Fourier transform. Outofplane time transient displacement and its Fourier transform for the broadband excitation for both the ordinary (A, black line) and hierarchical (B, blue line) structures. The signals are acquired immediately after the end of the periodic structure. The Fourier transforms of the signals allow to show the difference of the two responses in terms of frequency content, showing the ability of the hierarchical unit cell to provide large attenuation in the [2.2 − 2.7] Hz frequency range and above 5.1 Hz, in good agreement with the dispersion diagrams reported in Figure 3B and E. BGs calculated in Figure 3B and E are superimposed onto each frequency response, to visually highlight how the frequency responses matches the dispersion diagrams. 

In the text 
Fig. 5
Full wavefield reconstruction. Outofplane full wavefield for the (A) ordinary and (B) hierarchical seismic barrier at two different time steps: before (top panels) and after (bottom panels) the wave has reached the periodic structure. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.