Seismic isolation of buildings using composite foundations based on metamaterials

Metamaterials can be engineered to interact with waves in entirely new ways, ﬁnding application on the nanoscale in various ﬁelds such as optics and acoustics. In addition, acoustic metamaterials can be used in large-scale experiments for ﬁltering and manipulating seismic waves (seismic meta-materials). Here, we propose seismic isolation based on a device that combines some properties of seismic metamaterials (e.g., periodic mass-in-mass systems) with that of a standard foundation positioned right below the building for isolation purposes. The concepts on which this solution is based are the local resonance and a dual-stiffness structure that preserves large (small) rigidity for compression (shear) effects. In other words, this paper introduces a different approach to seismic isolation by using certain principles of seismic metamaterials. The experimental demonstrator tested on the laboratory scale exhibits a spectral bandgap that begins at 4.5 Hz. Within the bandgap, it ﬁlters more than 50% of the seismic energy via an internal dissipation process. Our results open a path toward the seismic resilience of buildings and a critical infrastructure to shear seismic waves, achieving higher efﬁciency compared to traditional seismic insulators and passive energy-dissipation systems. V C 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/ licenses/by/4.0/)


I. INTRODUCTION
The concept of metamaterials has emerged in recent years for developing materials with unique properties not generally found in nature. Although the idea was developed originally in the context of electromagnetic field theory, 1,2 a subfield focusing on elastic metamaterials seems to be promising for medical applications, seismology, superlenses, imaging, etc. 3 Elastic metamaterials with appropriate locally resonant structures can filter the energy of propagating waves with frequencies in a stopband region. The first experimental evidence of this property is found in Liu et al., 4 who fabricated sonic crystals with a simple microstructure unit consisting of a lead sphere with the coating of an elastically soft material (i.e., silicon rubber) that exhibited a main stopband frequency region tuned near the resonant frequency of 400 Hz. Acting as internal resonators, the lead spheres had a constant spatial lattice that was two orders of magnitude smaller than the relevant wavelength at the resonant frequency. The use of metamaterials to manipulate seismic waves, which are characterized by volume [primary (P) and secondary (S)] and surface (Rayleigh and Love) waves, has been demonstrated for Rayleigh-type waves through a series of full-scale experiments. [5][6][7] Basically, it has been shown that seismic metamaterials with sub-wavelength local resonators can be designed to have a negative mass density and/ or a negative Young's modulus as a way to control elastic wave propagation within the frequency range of seismic waves (<50 Hz). In particular, Brûl e et al. 5 reported modifying the seismic energy distribution in the presence of a regular mesh of vertical cylindrical voids. Colombi et al. 6 pointed out that trees in a forest can be used as natural metamaterials, and other proposed approaches involve developing large-scale phononic metamaterials. 8,9 The aforementioned research focused on controlling the propagation of seismic waves to protect the critical infrastructure via a seismic invisibility cloak. However, such solutions require an amount of space that is comparable to the region being cloaked, and they were developed mainly for surface seismic waves. A different approach has been proposed, 10 in which a vibrating barrier in the form of a dummy building absorbs a significant portion of the dynamic energy of a seismic wave. Meanwhile, other strategies for vibration control based on seismic isolation (SI) devices lead to significant advantages from a strong reduction in seismic vulnerability due to modifying the dynamic structural characteristics by (i) increasing the dissipative properties of the structure and/or (ii) altering its rigidity and thereby shifting the structural frequencies. However, standard isolation strategies perform less well in protecting tall buildings or if the soil below the building is soft. 11 In addition, for civil engineering applications, seismic isolation should work for frequencies below 10 Hz. 12 Here, we introduce the concept of a "composite foundation" (CF) that integrates the physics of seismic metamaterials based on the concept of periodic mass-inmass systems directly with the foundation of a building, creating on-site filters that reduce the energy transferred from a seismic wave to the building at frequencies in the range of the first vibration mode of the building itself. The use of CFs represents a new paradigm for designing seismic protection. 13 We focus on S-waves because of the fact that their relatively high amplitudes are often associated with basin resonances and other ground-filtering mechanisms (e.g., Love waves). Our results introduce a new way to design seismic insulation by using the fundamental properties of seismic metamaterials.

A. Composite foundation
A sketch of the device under investigation is shown in Fig. 1. It comprises four reinforced concrete plates with a square cross-section of side 1 m and a thickness of 20 cm [ Fig. 1(a)]. Each plate is disconnected from the others by means of an ultralow damping surface realized by a suitable combination of layers of steel and Teflon [ Fig. 1(a)]. Each plate has a matrix of nine cylindrical inclusions with a regular spatial distribution and a center-to-center distance d C ¼ 30 cm between adjacent inclusions. Mechanical connections comprising rubber with low Young's modulus and a steel tube are fixed at the four corner inclusions (diameter of 15 cm) to connect neighboring plates [ Fig. 1(b)]. The other inclusions are used to host steel cylinders and rubbers (rubber #1) as shown in Fig. 1(c) to allow them to act as internal resonators. Along with the damping due to the rubber, the presence in the CF of spatially ordered local resonators with a dimension smaller than the wavelength of propagating waves allows sub-wavelength wave manipulation similar to that reported previously. 4 Figure 2 shows schematically an example of a CF below a building for the purpose of protection. The plates with internal resonators are sandwiched between a load-distribution plate and a foundation plate to give a more-uniform load distribution on the CF and hence improve its dynamic response.

B. Equivalence of composite foundations with periodic mass-in-mass systems
The working principle behind CFs is that of a periodic mass-in-mass system 14-18 without a damper, as discussed below in detail. Considering the dynamical system in Fig.  1(d), the total energy E j of element j is given by the sum of kinetic (E j K ) and potential (V j P ) energies  Table I).
FIG. 2. Schematic of CF (plates with resonators) below a building that is being protected. This device is clearly a part of the foundation itself and is located between a load-distribution plate and a foundation plate.
where m and k are the mass and the spring constant, respectively, of the mass-in-mass system, the subscripts i and e relate to the internal mass and the external mass, respectively, and _ u and u are the velocity and the displacement of the mass, respectively. Given that the system is conservative, the dynamical equations computed as d dt By considering the harmonic wave solution (Bloch's theorem) for element j, the generic solution of Eq. (2) is u jþn e ¼ B e e iðqxþnqLÀxtÞ ;

<
: where q is the wave number, x is the angular frequency, L is the periodic spatial constant of the mass-in-mass system, and B i and B e are complex numbers that account for any offset phase between the two solutions. In Fourier space, the j À 1 and j þ 1 generic solutions can be written in terms of the j generic solution Combining Eqs.
(2)-(5) with some trivial mathematical steps gives the dispersion relationship of a periodic mass-in-mass system 14,17 m e m i x 4 À k i ðm e þ m i Þ þ 2k e m i ð1 À cos qLÞ ½ x 2 þ 2k e k i ð1 À cos qLÞ ¼ 0: The links between the discrete elements of the mass-in-mass system and the properties of the CF (periodicity along the vertical direction) are given in Table I. Considering the experimental parameters of the CF under test, the elements of the equivalent mass-in-mass system are m e ¼ 317 kg, Table I, right-hand column), and L ¼ 0.2 m. The dispersion relationship computed as the roots of Eq. (1) shows a bandgap that begins at a frequency f BG;i near 5 Hz, where and stops at a frequency f BG;f near 14 Hz, where The results in Fig. 3 show that the analytical results [as computed with Eq. (1)] agree well with numerical computations of a full three-dimensional finite-element model of a CF comprising four plates that was discretized with tetrahedral finite elements (see the inset of Fig. 3 for the discretization used for the simulations). The device is considered to be a linear system [19][20][21] in which the plates are separated by an ideal disconnection element that simulates zero damping between the plates. This approximation is reasonable considering that the experimental data show a damping coefficient of around 3%. In addition, low damping does not modify the dispersion relationship.

C. Experimental system
A schematic of the measurement system and a photograph of the CF described above are shown in Figs. 4(a) and 4(b), respectively, with the key elements indicated; see also Figs. 1(a)-1(c). The input displacement is applied to the second plate (layer 2) by an MTS 204.71 actuator (Fig. 4). The bottom plate (fixed layer), which is disconnected via a lowdamping multilayer from the second plate, is necessary for reducing the friction as much as possible. In a real building, it is necessary to use a load-distribution plate and a foundation plate to separate the active region of the CF (see Fig. 2). The measurement system is equipped with sensors to register (i) the displacement and the acceleration of each plate and (ii) the accelerations of the internal resonators. The dataacquisition system has 12 acceleration channels for layers 1-4, two voltage channels for the actuator, four channels for the slab displacements, 16 voltage channels for the resonator accelerometers (four plates, two resonators per plate, and two in-plane acceleration components), and eight voltage channels for the additional low-cost slab accelerometers, making a total of 42 channels with a sampling frequency of 100 Hz.
In detail, the plate displacements are acquired by four RDP LDC500A displacement transducers. These are inductive displacement sensors with integrated signal conditioning to output a voltage signal. These sensors were selected for their dynamic range when coupled with an electrodynamic actuator and a signal generator. This is a key aspect for correct measurements because the transducer is of the springreturn type, making it essential to verify that no detachment occurs between the sensor tip and the plate when the CF moves. The plate accelerations are measured by four PCB Piezotronics 356B18 triaxial accelerometers, exhibiting a sensitivity of 1000 mV/g and a frequency range of 0.5-3000 Hz. These accelerometers are based on a piezoelectric sensor element with built-in conditioning microelectronics and are characterized by high sensitivity and low spectral noise. 22 Because there is little space between the resonators and the concrete plate, the resonators are monitored during the test by using a special resin to glue the accelerometers to the upper surfaces of the resonators. For this application, the accelerometers "sacrificial," low-cost MEMS accelerometers are used. The selected accelerometers are Analog Devices model ADXL337, with a sensitivity of 300 mV/g and a voltage output. The printed circuit boards of these accelerometers are glued to the resonators with the interposition of insulation boards for short-circuit protection. We instrument two resonators per concrete plate and use four additional ADXL337 accelerometers to read the plate accelerations, in parallel to the four PCB Piezotronics triaxial accelerometers for redundancy and comparison purposes. Figure 5 summarizes the main results of the experimental measurements in response to a time-domain input displacement of the form u 0 sin ðxtÞ with a nominal u 0 ¼ 3.5 mm. This signal is applied to layer 1 at different frequencies ranging from 0.5 to 8 Hz. The sensors T1-T4, as indicated in the inset of Fig. 5(a), register a time-domain signal at the same frequency as that of the input signal, while the signal amplitude depends on the input frequency. Figure  5(a) shows the transfer function of the CF computed as the ratio of (i) the displacement amplitude measured by the sensor T4 to (ii) the input signal measured by T1. As the frequency approaches the beginning of the theoretical bandgap at 5 Hz (see Fig. 3), the signal propagating through the CF is attenuated. The attenuation coefficient as a function of frequency is shown in Fig. 5(b) (circles) as computed directly from the transfer function of Fig. 5(a) [see also Eq. (3)]. For completeness, Fig. 5(b) also shows the theoretical attenuation coefficient (solid line) displayed up to 8 Hz computed as the imaginary part of the wave vector qL given by

III. RESULTS
Figures 5(c) and 5(d) show the time-domain traces as measured by sensors T1-T4 for input frequencies of 2 and 6 Hz, respectively. The data show clearly that inside the bandgap (6 Hz), the displacement is attenuated and the signal amplitude at T4 is reduced by more than 50% compared to the input signal at T1. Experiments performed with an additional weight of 1000 kg on top of the upper slab showed results that were quantitatively similar to those in Fig. 5. It should also be noted that any vibration of the building sitting on the CF will also be filtered at frequencies within its bandgap.
Steel cylinder n c (number of cylinders), V c (cylinder volume), q s (concrete density) Rubber #1 (connection) E re (Young's modulus of rubber), A re (area of external element's rubber), L re (length of external element's rubber) Rubber #2 (connection) E ri (Young's modulus of rubber), A ri (area of internal element's rubber), L ri (length of internal element's rubber), n c (number of cylinders) We consider the above results as proof of concept of the working principle of this CF (see the supplementary material for a video of the dynamic measurements). Even though the experimental study was performed by applying the displacement along one particular direction, we stress that because of the symmetry of the CF, its transfer function is independent of the S-wave polarization; in other words, it exhibits an isotropic response in the plane for its horizontal component.
To test the robustness of the CF, it was also subjected to an up-down chirp signal (for more than 10 min) characterized by a frequency that first increased from f 0 ¼ 1 Hz to f M ¼ 10 Hz (Df ¼ 9 Hz) and then decreased to f 0 in T ¼ 10 s Figure 6 shows the time-domain displacements of layers T1 and T4 for a complete cycle of the input signal. As the input frequency increases and approaches the bandgap, the displacement in layer T4 becomes less than that in layer T1. The frequency-dependent displacement in layer T4 (from a maximum of 6.6 mm at 1 Hz to a minimum of 3.8 mm at 10 Hz) is due to the limited dynamical response of the MTS actuator.

IV. DISCUSSION
In general, CFs differ markedly from ordinary seismic isolation devices. The seismic isolation property relates mainly to the ultralow-damping surface between different plates, whereas the additional key property coming from the metamaterial is the dissipation process due to the internal resonance (internal mass and spring), which is f ¼ receive no advantage from traditional seismic isolation, whereas the CF filtering effect can indeed be effective. Therefore, metamaterial-based isolation is expected to have better isolation characteristics and be more widely applicable compared to standard SI.
To compare the dynamical properties of a CF with and without resonators from a theoretical perspective, it is sufficient to consider a mass-spring periodic chain that is characterized by simpler expressions for the dispersion relationship and the attenuation coefficient, namely, m e x 2 þ 2k e ð1 À cos qL ð ÞÞ ¼ 0; The mass-spring periodic chain exhibits only a Bragg region (dissipation mechanism due to the rubber) starting at a frequency around 8 Hz and in which the attenuation coefficient increases monotonically with no resonance. In other words, the internal resonance of the mass-in-mass periodic chain gives rise to a more-efficient attenuation mechanism that starts at a lower frequency than the mass-spring periodic chain (4.5 Hz as opposed to 8 Hz).
We also wish to stress that this seismic metamaterial does not exhibit a "negative refraction index." Rather, the effective mass depends on the frequency and becomes negative for frequencies within the bandgap. 14,17 In addition, the amount of physical space required for SI is comparable with the size of the building itself, which is a key issue for any future commercialization (see Fig. 2).

V. SUMMARY AND CONCLUSIONS
New buildings should be developed to reduce shakingrelated damage. To that end, we propose herein the concept of a CF with two properties, namely, the local resonance and a dual-stiffness structure, 23 i.e., large (small) rigidity for compression (shear) effects. This new design paradigm of CF retains the resistance of a standard foundation to a vertical load (i.e., a building on top), while offering the advantage of filtering the energy of S-waves propagating through it with frequencies within its bandgap. Specifically, our data show that S-waves with frequencies greater than 4.5 Hz (the starting frequency of the theoretical bandgap) are attenuated and that the device studied herein can filter more than 50% of the wave energy within that bandgap. We argue that this fundamental result on S-wave filtering could be the basis for designing a new generation of building-protection devices 24 that join seismic metamaterial concepts applied to surface and volume seismic waves. A challenge in developing this concept into a widespread alternative to standard foundations in new buildings will be to design a CF whereby the main resonance frequency of the building and the amplification region of the soil response are both within the bandgap.

SUPPLEMENTARY MATERIAL
See supplementary material for a video of the dynamic measurements of the CF.