CAN META-SOIL ATTENUATE SEISMIC WAVES?

Abstract

Stable and resilient civil infrastructure is a key to public safety. However, current structures are vulnerable to damage resulting from excessive ground motion caused by earthquakes or underground explosions. Traditionally, structures are built to withstand ground motion, but this design approach is costly and the risk of failure during very large events remains high. A fundamentally different approach is found in controlling the ground motion itself through engineering the soil to act as an acoustic metamaterial. Acoustic metamaterials are composites, often with a periodic substructure, that have the ability to control the propagation of elastic waves through scattering or local resonance mechanisms. Recently, advances in the understanding of metamaterials have allowed the creation of stop bands in wave transmission around the resonator’s natural frequency. A graded array of low-frequency acoustic metamaterials provides the possibility to create targeted band-stops, effectively filtering out destructive ground motion. Numerical modeling is used to inform future experimental design to study this phenomenon at laboratory scale. Local resonators are modeled as spheres with a heavy metal core and a thin elastic coating. A sensitivity analysis is performed in order to inform the design of improved resonators. Then, alternative resonators are modeled as spheres with a heavy metal core and elastic columns made of plastic. The geometry and material properties of the resonators are varied in numerical simulations to optimize the frequency range and width of the band gaps. Similar resonators could be incorporated at full scale to create a seismic shield around critical structures.

 

Table of Contents

List of Figures……………………………………………………………………………………………………… vi

List of Tables………………………………………………………………………………………………………. xi

Acknowledgements………………………………………………………………………………………………. xii

Chapter 1 Introduction………………………………………………………………………………………….. 1

Motivation………………………………………………………………………………………………………… 1

Background………………………………………………………………………………………………………. 5

Research Objectives…………………………………………………………………………………………. 16

Chapter 2 Metaconcrete Modeling…………………………………………………………………………. 17

Analysis of Unit Cell…………………………………………………………………………………………. 17

Geometry and Boundary Conditions…………………………………………………………………… 17

Eigenfrequency Analysis…………………………………………………………………………………. 19

Metaconcrete Slab……………………………………………………………………………………………. 20

Simulation of Wave Propagation in the Frequency Domain……………………………………. 21

Sensitivity Analysis…………………………………………………………………………………………… 30

Shear Boundary Load………………………………………………………………………………………. 35

Investigation of Graded Resonator Array……………………………………………………………. 40

Soil Simulations……………………………………………………………………………………………….. 42

Chapter 3 Alternative Resonator Design…………………………………………………………………. 46

Creation of Resonators……………………………………………………………………………………… 46

Material Selection………………………………………………………………………………………….. 46

Creation of Unit Cell………………………………………………………………………………………. 48

Eigenfrequency Analysis…………………………………………………………………………………. 50

Creation of Slab…………………………………………………………………………………………….. 55

Band Gap Widening…………………………………………………………………………………………. 56

Influence of Core Material……………………………………………………………………………….. 56

Analysis of Varying Plastic Stiffness………………………………………………………………….. 62

Effect of Increased Density of Resonators…………………………………………………………… 67

Summary of Influences on Band Gap…………………………………………………………………. 68

Chapter 4 Conclusions and Recommendations…………………………………………………………. 69

References………………………………………………………………………………………………………….. 72

Chapter 1 Introduction

Motivation

In January 2010, the Haiti earthquake caused damage to structures, roads, and over 294,000 homes. In total, the 7.0 magnitude earthquake caused approximately $8 billion in damage [1] . However, earthquakes do not only occur in Haiti. They occur in the United States, Japan, Chile, and all over the world. They occur in remote countries, crowded cities, and suburbs. Earthquakes are a global problem. The United States Geological Survey (USGS) Earthquake Hazard Program estimates that from 2000-2016 there were 31,759 earthquakes worldwide, causing 801,629 deaths [2] . Currently, hundreds of millions of people live in earthquake hazard areas, and billions of dollars of infrastructure are vulnerable, putting economies at risk as well [3] .

Traditional earthquake and vibration engineering techniques have evolved greatly in the last few decades, however significant damage and fatalities still occur in seismic events. There are many different types of seismic waves and which has the greatest effect on a structure depends heavily on the site conditions. The two types of waves produced when an earthquake occurs are body waves and surface waves. Body waves consist of P-waves and S-waves, as seen in Figure 1. P-waves are the fastest seismic waves, the first to reach a structure when an earthquake occurs. Like acoustic waves, the particles travel parallel to the direction of propagating energy. S-waves cause shearing deformations as they propagate, as their motion is perpendicular to the direction of propagation. Surface waves, consisting of Rayleigh waves and Love waves, as shown in Figure 2, are created from the interaction of body waves and the Earth’s crust. They have the greatest effect at distances further from the source of the earthquake and their magnitudes decrease exponentially with depth

[3] .

 

Figure 1 Deformations produced by body waves:

(a) P-wave; (b) S-wave. Source: Geotechnical

Earthquake Engineering by Kramer. Copyright

1993 by W.H. Freeman and Company

 

Figure 2 Deformations produced by surface waves: (a) Rayleigh wave; (b) Love wave. Source: Geotechnical Earthquake Engineering by Kramer.

Copyright 1993 by W.H. Freeman and Company

 

To counteract the damaging effects of earthquake waves, engineers use passive and active control systems. In a seismic event, passive control structures aim to consume part of the input energy and minimize structural damage. They are installed as energy dissipation or damping mechanism devices possessing certain stiffness and damping characteristics. Examples of passive control systems include metallic, friction, visco-elastic, and viscous fluid dampers, tuned mass dampers, and tuned liquid dampers. Examples of various passive control systems are show in Figure 3. Base isolation is a form of passive control that includes inserting a low stiffness layer at the foundation of a structure in order to reduce its natural period in vibration. For the most part, these devices are simple to design and build, however they are only designed for one dynamic loading case. Therefore, these methods often fail to protect structures from all types and magnitudes of seismic waves. In addition, the foundation of a structure will usually still have a large horizontal displacement after an earthquake has occurred. This creates further issues involving the underground utilities that enter the structure through the foundation.

 

Figure 3 Passive control system examples. Source: Hadi (2017)

 

Active systems are controllable, while passive control systems are not. They work by creating forces in the structure to counteract the energy of an applied dynamic loading. Some examples of active control systems include active tuned mass dampers and active bracing systems, as seen in Figure 4. Sensors placed throughout a structure measure amplitude and send records to a computer, which activates devices during a seismic event.  A drawback to active systems is that they require a significant amount of power to run, which leads to extra costs. In addition, this external power supply cannot be interrupted during an earthquake event, or the system will not function. Active control devices are often complex to design and implement, and they are expensive to install and maintain.

 

Figure 4 Active control system examples. Source: Hadi (2017)

 

Semi-active control systems combine certain aspects of passive and active devices.  Semi-active devices can absorb energy during an excitation, as passive devices do. Then, the system reacts to provide a force to the structure, acting as an active device.  The drawbacks of these two methods still exist, although to a lesser extent [4] .

Structures vary significantly in their reaction to a seismic event due to differences in geometric and material properties of the structure itself, soil and bedrock properties, type, direction, and magnitude of incident earthquake waves, and the effect of adjacent structures. A technique has not yet been implemented that has the potential to protect a wide variety of structures against a wide variety of seismic events. Current passive, active, and semi-active earthquake protection methods for structures are insufficient, as they are expensive and significant damage still occurs after ground shaking [4] , [5] .

 

Background

Recent research has focused on developing a fundamentally new approach, in which the ground motion itself is controlled to prevent dangerous waves from reaching a structure through the use of metamaterials. Metamaterials consist of composites with an engineered microstructure that can exhibit a negative effective dynamic mass density and elastic modulus. These unique properties can lead to the manipulation of elastic wave propagation. Phononic crystals and acoustic metamaterials have shown promising results in the manipulation of elastic wave propagation through Bragg scattering or local resonances. This field of research is fairly recent, only gaining traction within the last two decades [6] . Narayanamurti et al. (1979) demonstrated the first example of controlling high-frequency phonons through a superlattice in 1979 [7] . In 1995, the first physical demonstration of this phenomenon was conducted by Francisco Meseguer. Meseguer and his peers studied the wave-filtering properties of a sculpture in Madrid consisting of a periodic arrangement of steel tubes, as shown inFigure 5. For different frequencies of propagating waves, either constructive or destructive interferences occur. The destructive interferences are a result of scattering, and therefore cause the structure to exhibit band gaps [8] .

Liu et al. (2000) presented a structure that attenuates bulk waves due to local resonance. A simple cubic crystal was modeled, consisting of an arrangement of heavy lead cores coated with silicone in an epoxy matrix, as seen in Figure 6. At the resonant frequencies of the inclusions, the center of mass of the metamaterial has a displacement that is out of phase with the acoustic wave, causing an effective negative dynamic mass density. Therefore, transmission gaps exist at these frequencies [9] .  

 

Figure 5 (a) Madrid sculpture, (b) Sound attenuation through sculpture.

Source: Deymeir (2013)

 

Figure 6 (a) Cross-section of a coated lead sphere, (b) a sonic crystal of 8×8 coated lead spheres, (c)             Calculated (line) and measured (dots) amplitude transmission coefficient, (d) Band structure of crystal. Source: Deymeir (2013)

 

The successful use of metamaterials to filter waves in acoustics provides a promising solution for seismic applications, as seismic waves can be described as inhomogeneous acoustic waves with various wavelengths on the order of hundreds of meters. Studies have predicted that metamaterials can even prevent the transmission of an incident wave at or near their natural frequency. Therefore, an array of metamaterials with different resonant frequencies has the potential to create wide band gaps and effectively filter out dangerous ground vibration. In the last decade, researchers have developed two methods for the manipulation of seismic waves using acoustic metamaterials: surface metamaterials and embedded resonators [10] .

Resonators arranged on the surface of an elastic half-space such as the Earth’s crust have the potential to protect critical infrastructure from seismic waves. Colombi et al. (2016) designed a seismic metawedge, consisting of vertical resonators in a graded arrangement from 1 to 14m, as seen in Figure 7. The resonance frequency range of the metawedge is 30-120 Hz, which is higher than most seismic waves. If a seismic wave propagates such that it reaches the shorter side of the metawedge first, much of the energy is reflected. This could cause serious damage to structures on the wrong side of the resonators. Rayleigh waves that propagate towards the taller grade of the metawedge are converted by the above-surface structure into bulk waves, traveling downwards and possibly underneath the structure of interest [11] .

 

Figure 7 Conversion phenomena by resonant metawedge. Source: Colombi et al. (2016)

 

Another study by Colombi et al. (2016) explored the possibility of trees in forests acting as a natural version of above-surface resonators. An experiment was carried out in a small forest consisting of mainly pine trees, and a significant attenuation of surface waves was achieved at 30 to 45 Hz and 90 to 110 Hz. Next, the forest was modeled using vertical rods to represent trees. They found a bandgap from 32-40 Hz and from approximately 90-105 Hz. The forest works in a way similar to the metawedge in that it converts surface waves to downwardly-propagating bulk waves, therefore creating a surface wave-free zone beneath the trees. They also found that a larger variation in the size and arrangement of trees produced a larger band gap [12] .

While above-surface resonators have the potential to shield structures, they do so by reflecting or redirecting seismic waves. This could result in dangerous effects to structures on the wrong side of the surface resonators. In addition, a lot of space is needed to construct a metawedge and space may not be available in some areas, particularly in cities [10] .

Xiang and Shi et al. (2012) took a different approach, creating a foundation that was itself a metamaterial. They designed a one-dimensional layered periodic foundation consisting of alternating rubber and reinforced concrete layers. First, a finite element model was built and Swaves were simulated. Band gaps were observed at 6.6-15.0 Hz and at 17.8-30.0 Hz, which includes the natural frequency of the frame built on top of the periodic structure. Resonance of the frame, which can be thought of as a critical structure to be protected, would result in serious damage. Therefore, the fact that this frequency cannot pass through the periodic foundation is ideal. Xiang and Shi et al. (2012) were also one of the first research groups to confirm modeling results for vibration attenuation experimentally using metamaterials. They built a scaled physical model consisting of a 1.5-ton specimen made of rubber and concrete layers bonded together with polyurethane glue. The specimen was tested using a biaxial shaking table and the seismogram from the 1975 Oroville earthquake. Results demonstrated that the periodic foundation was able to reduce peak horizontal acceleration by up to 50% as compared to the control specimen without a layered foundation, but it was not able to reduce the vertical displacement of the structure on top of the foundation [4] . Xiang and Shi et al. (2012) effectively modeled and created a seismic isolator, however they did not attempt to shift or widen the frequency band gap produced by the periodic foundation.

Next, Shi and Huang (2013) considered a three-dimensional, three-component periodic foundation, composed of a unit cell with a cubic core and soft coating shell embedded in a concrete matrix. Through computational and numerical simulations, they confirmed that a higher core density, an increase in the elastic modulus of the coating layer, and an increase in the filling fraction all lead to a wider attenuation zone achieved by the metamaterial. They were able to produce a band gap as low as approximately 8-15 Hz, but only achieved 20-30% attenuation overall when their periodic foundation was numerically subjected to the same seismic waves generated by a past earthquake

[13] .

Another approach taken by researchers, which attempts to address the concern for space by embedding the resonators in the surface, is an arrangement of cylindrical inclusions or voids in the ground that utilize scattering effects to dissipate seismic waves. Kim and Das (2012) developed equations to find the negative effective shear modulus of a metamaterial, allowing a method of design producing a specific band gap. They concluded that seismic surface waves can be attenuated through the use of Helmholtz resonators, or an array of cylinders with holes in the side [14] . The design of their meta-cylinder is shown inFigure 8.

 

Figure 8 Hemholtz resonator design. Source: Kim and Das (2012)

 

Huang and Shi (2013) conducted a numerical simulation to study the periodic nature of twodimensional row piles, or embedded surface resonators. Concrete piles were designed and modeled, and factors that influenced the band gaps achieved by the metastructure were analyzed. They concluded that an increase in the number of rows of piles, a larger filling fraction (the volume ratio of pile to surrounding material) and softer soil all increased the band gap width. However, the piles failed to isolate vibrations outside of the band gap [15] .

Cheng and Shi (2013) conducted another study on the attenuation zones of periodic structures. Two component rubber-concrete and rubber-steel cylindrical inclusions were modeled. The higher density of the steel core resulted in a lower band gap frequency and wider attenuation zone than the concrete core. They also found that a larger core radius and smaller periodic constant increase both the lower and upper frequency of the band gap. Three-component cylindrical inclusions were also analyzed. Larger core radii and smaller coating thickness were found to be favorable in widening the band gap based on local resonance of the inclusions [16] .

Brulé et al. (2014) advanced the study of embedded surface scatterers by creating a large-scale experiment to investigate the effectiveness of cylindrical surface inclusions on attenuating surface waves. The experiment was conducted in a thick deposit of silty clay soil. 5-m boreholes were created in a 3 x 10 arrangement at the test site, as seen in Figure 9. A source excitation frequency of 50 Hz was used, and the wavelength of the signal was found to be comparable to the centerto-center borehole spacing, activating Bragg scattering. The results indicate up to a 2.3-times reduction in energy from the source. However, they also indicate a strong reflection of surface waves. In addition, the researchers found that a ring-type geometry, which would be needed protect a structure from all directions, engenders concentration effects in some scenarios. This could be detrimental to critical infrastructure [17] .

 

Figure 9 Borehole test setup. Source: Brule et al. (2014)

 

Palermo et al. (2016) combined the findings from the metawedge designed by Colombi et al. (2016) and the information from studies on surface-embedded cylindrical inclusions to design a metabarrier. The metabarrier consists of periodic arrangement of rows of resonators and converts seismic waves into bulk waves that travel downward. After numerical verification of their design, Palermo et al. (2016) designed resonators consisting of a rigid aluminum tube, soft spring, and heavy steel mass, seen in Figure 10, to be tested experimentally. They embedded the resonators in polymer resin in a triangular lattice in order to achieve maximum density per lattice area. They found that band gap formation does not rely on the arrangement, but that the resonance frequency can be tuned by varying the length of the resonators. The experiment was able to achieve low band gaps in the range of interest for civil engineering applications [18] .

 

Figure 10 Resonators consisting of rigid aluminum tube, soft spring (aluminum bolt), and heavy steel mass.

Source: Palermo et al. (2016)

 

A numerical study by Miniaci et al. (2016) proposed the use of three-dimensional mechanical metamaterials, consisting of phononic crystals and locally resonant metamaterials, as a passive isolation strategy for seismic waves. They compared the effectiveness of boreholes, hollow metal cylinders, and locally resonant inclusions on the attenuation of bulk and surface waves, as seen in Figure 11. The study, which considers the effect of soil layers and viscoelastic damping, concluded that the hollow cylinder is most effective in attenuating seismic waves, and that a larger number of rows of cylinders has a more advantageous effect. Their results also demonstrate the existence of a very narrow band gap for locally resonant inclusions around their natural frequency. While the band gap is too small to effectively shield structures, they suggested experimenting with an array of inclusions to widen the band gap [19] .

                      Figure 11 Unit cell of (left to right) borehole (cross-like cavity), hollow cylinder, and coated cylinder.

Source: Miniaci et al. (2016)

 

Mitchell et al. (2015) studied metaconcrete, or concrete containing bi-material spherical inclusions. These inclusions consist of a lead core with a rubber or nylon coating. They resonate at a certain frequency, and relative motion between the heavy metal core and surrounding mortar allows the inclusion to absorb energy. Mitchell et al. (2015) tested four different unit cell configurations, varying the radius of the core as well as the thickness and coating of the material in Abaqus to find eigenfrequencies. Then, they modeled arrays of aggregates within a mortar slab. One array consisted of 8 whole spherical inclusions and a second array consisted of 8 whole spherical inclusions plus an additional 28 quarter-spheres. The second array is shown in Figure 12. Each array was tested for each of the four configurations of the unit cell. A transmission coefficient, plotted against forcing frequency, was used to analyze the effect of the aggregates in the slab on propagating waves. In Figure 13 and Figure 14, dips in the transmission coefficient can be observed near, but not exactly at, the resonant frequencies of the inclusions. They suggest Bragg scattering or numerical interference may have affected the results. Mitchell et al. concluded that the inclusions with the stiffer nylon coating performed better than those with the rubber coating. In addition, more inclusions resulted in a greater reduction in transmission and wider frequency range of influence [20] . While this study is one of the few that involves the performance of metamaterials for bulk waves, it lacks experimental validation and does not attempt to widen the range of frequencies in which a dip in transmission coefficient is observed.

 

Figure 12 Metaconcrete slab showing inclusion structure. Source: Mitchell et al. (2015)

 

Figure 13 Transmission coefficient versus frequency for metaconcrete slab consisting of (a) 1mm, (b) 3mm nylon-coated aggregates. Source:

Mitchell et al. (2015)

 

 

Figure 14 Transmission coefficient versus frequency for metaconcrete slab consisting of (a) 1mm, (b) 3 mm rubber-coated aggregates.

Source: Mitchell et al. (2015)

 

Krodel et al. (2015) conducted one of the most applicable studies to this thesis to date. They designed an array of cylindrical tubes, each containing a heavy steel mass suspended by soft bearings designed to possess a different eigenfrequency of 4 to 7 Hz. This effectively created what they referred to as a “rainbow trap,” splitting propagating surface waves into a spatial spectrum by varying the bearing stiffness. Results from the numerical study demonstrated that changes in ordering and spacing of the resonators did not affect the transmission spectrum, but that a greater number of resonators correlated directly to an increase in the wave attenuation. Next, Krodel et al. built a 1:30 scaled experimental setup to verify their concept. Resonators were placed from lowest to highest resonant frequency in a box of sand, which is assumed to be a non-dispersive material. Longitudinal and shear waves were excited on one side of the box using an electromagnetic shaker and transmission was measured on the other end using an accelerometer. For a total of 15 resonators designed to have an eigenfrequency range of 134 to 208 Hz (real-scale range of 4.46 to 6.93 Hz), a maximum attenuation of -11.7 dB was achieved. The design of the experimental resonators, as well as a visualization of the “rainbow trap,” is shown Figure 15.Krodel et al. was able to match numerical results experimentally and widen the band gap of resonators at the desired seismic range. However, they could not measure the energy propagating in the experiment and expect that some reflection occurred [21] .

 

Figure 15 Experimental study of “rainbow trap”. (a) Experimental resonators consisting of outer aluminum tube containing steel rod, and connected polymeric springs, (b) Resonant frequencies of 15 resonators with variable spring

stiffness, creating a “rainbow trap.” Source: Krodel et al. (2015)

Research Objectives

Although the completed investigations show essential and promising development in the use of metamaterials as seismic and vibratory wave attenuators, many knowledge gaps still exist in this field of research. For example, minimal physical experiments have been conducted to confirm observed simulation results. In addition, there is a decent understanding of the ability of resonatorbased metamaterials to produce a band gap at their resonant frequency. However, for practical applications, the ability to produce a wide band gap at low frequencies (<50 Hz) is desirable for seismic applications.

The objective of this research is to address some of the knowledge gaps in this research area by focusing on designing resonators that perform at a low frequency and could be adapted in a physical experiment to study the attenuation of a bulk wave in a half-space.  The main objectives are:

1. To conduct a sensitivity analysis on bi-material spherical resonators to inform design of new improved resonators
2. To find a size and arrangement of inclusions that increases the band gap
3. To create embedded resonators capable of attenuating vertically propagating bulk waves

CAN META-SOIL ATTENUATE SEISMIC WAVES?

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