WO2018001234A1

WO2018001234A1 - Multifunctional elastic metamaterial

Info

Publication number: WO2018001234A1
Application number: PCT/CN2017/090265
Authority: WO
Inventors: Guancong Ma; Yun LAI; Caixing FU; Ping Sheng
Original assignee: The Hong Kong University Of Science And Technology
Priority date: 2016-06-27
Filing date: 2017-06-27
Publication date: 2018-01-04

Abstract

Vibration suppression is achieved with a metamaterial composite structure constructed from three-dimensional resonant metamaterial cells. The cells have an inner mass formed of substantially incompressible material, with an elastomeric coating on an outer surface of the inner mass. An outer encapsulant formed of a material having substantially less elasticity than the elastomeric coating, and having less density than the inner mass is used to encapsulating the inner mass over elastomeric coating. The outer encapsulant supports the inner mass through the elastomeric coating with substantially no gap or fluid interface between the outer encapsulant and the elastomeric coating. The inner mass, elastomeric coating and outer encapsulant form a suspended mass structure.

Description

Multifunctional Elastic Metamaterial

RELATED APPLICATION (S)

The present Patent Application claims priority to Provisional Patent Application No. 62/493,223 filed 27-Jun-2016， which is assigned to the assignee hereof and filed by the inventors hereof and which is incorporated by reference herein.

BACKGROUND

Field

This disclosure relates to a solid resonant elastic metamaterial that can display fluid-like elasticity in a finite frequency regime. The technology can also possess bandgaps for torsional vibrations. It can be used to filter elastic waves according to their polarizations， and can be used for selective or complete vibration isolation.

Background

Structure-borne elastic waves -a ubiquitous type of classical waves -have pivotal importance in many applications， such as mechanical and civil engineering， geophysics， and seismology. Elastic waves are more complex than both electromagnetic and acoustic waves for their additional number of material parameters， as well as richer polarizations. Elastic waves include waves with both longitudinal and transverse nature. Sub-wavelength building blocks with local resonances are utilized in designs of acoustic metamaterials to generate bandgaps for low frequency sound. Theoretical studies linked local resonance with elastic waves in two-dimensional systems， and found fascinating consequences and unique properties， such as negative shear modulus， fluid-like behavior， super-anisotropy， and related behavior. Because of their structural complexity， these systems are challenging to materialize. Therefore， the appeal of resonant elastic metamaterials with unique properties and richer physics beyond acoustic and electromagnetic metamaterials mostly remained theoretical.

As described herein， ″fluid-like″ references elastic wave or elastic vibrational behavior and characteristics generally associated with a fluid mass， whereas ″solid-like″ references elastic wave or vibrational behavior and characteristic generally associated with a solid mass.

Sub-wavelength building blocks with local resonances are widely utilized for the design of acoustic metamaterials， such as the example shown in US Patent No. 6,576,333.

SUMMARY

A metamaterial composite structure comprises one or more three-dimensional resonant metamaterial cells. At least a subset of the cells comprise an inner mass formed of substantially incompressible material， an elastomeric coating on an outer surface of the inner mass， and an outer encapsulant formed of a material having substantially less elasticity than the elastomeric coating. The outer encapsulant has less density than the inner mass and encapsulates the inner mass over elastomeric coating so as to support the inner mass through the elastomeric coating with substantially no gap or fluid interface between the outer encapsulant and the elastomeric coating. The inner mass， elastomeric coating and outer encapsulant form a suspended mass structure.

Vibration suppression is achieved by use of a metamaterial composite structure. Plural three dimensional resonant metamaterial cells are arranged， with the cells in a stacked or other structural relationship. Each cell is constructed with an inner mass formed of substantially incompressible material， coating an outer surface of the inner mass with an elastomeric coating， and encapsulating each of the coated cells with an outer encapsulant. The outer encapsulant is formed of a material having substantially less elasticity than the elastomeric coating， but having less density than the inner mass. This results in a cell structure comprising of an inner mass having an elastomeric coating， and encapsulated over the elastomeric coating so as to support the inner mass through the elastomeric coating with substantially no gap or fluid interface between the outer encapsulant. The elastomeric coating， the inner mass， elastomeric coating and outer encapsulant form a suspended mass structure.

BRIEF DESCRIPTION OF THE DRAWINGS

Figs. 1A-1C are depictions of a metamaterial cell. Fig. 1A is a schematic diagram of the cell in a cut-away view. Fig. 1B is a photographic image of the cell. Fig. 1C is a schematic view of an equivalent physical model of the cell.

Figs. 2A-2C are schematic diagrams showing three translational eigenmodes， all characterized by the anti-phase translational oscillation of the rigid core and the outer shell. Fig. 2A shows xy-mode eigenmodes. Fig. 2B shows xy-mode eigenmodes transverse to the direction of that shown in Fig. 2A. Fig. 2C shows z-mode eigenmodes.

Figs. 3A-3C are schematic diagrams showing three rotational eigenmodes (RMs) ， corresponding to the three translational eigenmodes of Figs. 2A-2C. All of the rotational eigenmodes are characterized by the anti-phase rotational oscillation of the rigid core and the outer shell. Fig. 3A shows xy-mode rotational eigenmodes. Fig. 3B shows xy-mode rotational eigenmodes transverse to the direction of that shown in Fig. 3A. Fig. 3C shows z-mode rotational eigenmodes.

Fig. 4 is a schematic diagram showing four possible polarizations of vibration of an elastic rod.

Figs. 5A and 5B are graphs with corresponding photographic insets. The graphs show two types of quasi-one-dimensional arrays and their band structures.

Figs. 6A-6C are photographic depictions showing experimental setups for possible polarizations of vibration applied to a multi-cell arrangement. Fig. 6A shows xy-mode testing. Fig. 6B shows z-mode testing in a horizontal direction. Fig. 6C shows rotational testing.

Fig. 7 is a graph showing a realization of fluid-like elasticity of a sample kz， in the frequency regime of～1.3 -1.6 kHz.

Fig. 8 is a graph showing a realization of fluid-like elasticity of a sample kx， in the frequency regime of ～1.3 -1.6 kHz.

Figs. 9A and 9B depict an elastic meta-rod′s resistance to lateral shaking， that can be used for vibration isolations. Fig. 9A is a photographic depiction of a sample in a test apparatus. Fig. 9B is a color spectrograph showing displacement amplitude of the sample of Fig. 9A.

Fig. 10 is a graph showing a bandgap of torsional vibration， and a negative effective moment of inertia.

Fig. 11 is a graph showing the band structure of a three-dimensional lattice.

DETAILED DESCRIPTION

Overview

The present disclosure relates to the design and realization of a three-dimensional resonant elastic metamaterial. A unique structure can be constructed using such metamaterial units； i.e.， the elastic ″meta-rods″ . Such a design offers unprecedented ability to manipulate the flexural， longitudinal and torsional vibrational modes of elastic rods. By introducing anisotropy in the unit cell′s design， it has been observed that in a finite frequency regime， only longitudinal vibration can be excited in the ″meta-rod″ ， whereas flexural vibration is prevented. These characteristics present a hallmark elastic property of fluids or an elastic property generally associated with fluids.

A multifunctional elastic metamaterial can be provided， using a concept that relies on the unit cell′s local resonances， and whose eigenfrequencies are tunable through adjusting system parameters. A multitude of novel functionalities are demonstrated for elastic vibrations or waves. These include longitudinal and transverse bandgaps. When these two bandgaps are spectrally mismatched， ″fluid-like elasticity″ can be achieved even though the structure is fully solid. When the unit cells are assembled into a quasi-one-dimensional array， bandgaps for torsional waves can also be found.

The configuration uses a multifunctional elastic metamaterial. The technology relies on the unit cell′s local resonances， whose eigenfrequencies are tunable through adjusting system parameters. The disclosed technology demonstrates a multitude of novel functionalities for elastic vibrations and waves. These include longitudinal and transverse bandgaps. When these two bandgaps are spectrally mismatched， ″fluid-like elasticity″ can be achieved even if the structure is fully solid. When the unit cells are assembled into a quasi-one-dimensional array， bandgaps for torsional waves can also be found. The disclosed technology also exhibits exceptional vibration isolation capability for both longitudinal and transverse elastic waves.

The disclosed technology has multiple unique functionalities. It is able to mimic the elastic property of fluids by only allowing the propagation of longitudinal elastic waves， despite all its constituent components are solid materials. It excels a vibration shielding in a finite but tunable frequency regime. It has a further advantage of being able to mitigate torsional vibrations (twisting) . These functionalities are achieved via anisotropic sub-wavelength resonators.

In one configuration， a ″meta-rod″ possesses rotational modes that eliminate torsional vibration in a wide frequency regime. These unexpected characteristics， which can be demonstrated with vibrational experiments， can be interpreted by using effective media with parameters of indefinite mass density and negative moment of inertia.

Construction

Elastic waves exhibit rich polarization characteristics absent in acoustic and electromagnetic waves. By designing a solid elastic metamaterial based on three-dimensional anisotropic locally resonant units， it is possible to demonstrate polarization bandgaps together with exotic properties such as fluid-like elasticity. Elastic rods with unusual vibrational properties， denoted as ″meta-rods″ ， are constructed. By measuring the vibrational responses under flexural， longitudinal and torsional excitations， it is found that each vibration mode can be selectively suppressed.

In particular， a finite frequency regime is observed， in which all fiexural vibrations are prevented， whereas longitudinal vibration is allowed. This characteristic is generally regarded as a unique property of fluids. In another case， the torsional vibration can be suppressed significantly. The results can be interpreted by band structure analysis， as well as effective media with indefinite mass density and negative moment of inertia. The disclosed technology opens an approach to efficiently separate and control elastic waves of different polarizations in fully solid structures.

The disclosed metamaterials are constructed of one or more cells which can be placed in a stacked or adjacent relationship to form a single-cell or multi-cell structure. The cells have an inner core constructed as a substantially solid inelastic core and an outer shell， with the core separated by the shell by use of an elastic interface. In one non-limiting example， an inner core having a substantial mass， such as a steel cylinder， is provided with a silicone coating， and an outer shell， such as an epoxy shell， encapsulates the coated inner core.

Steel is given by way of non-limiting example， as any convenient material providing substantial mass can be used. Other metals can be used； however， steel is generally regarded as inexpensive. It is also possible to use other metals， such as iron and other ferrous metals or alloys which are inexpensive， either as scrap material or otherwise low cost. Non-limiting examples of other suitable materials that can be used are dense recycled plastics， concrete， construction aggregate， scrap material having substantial mass similar to steel or concrete.

The unit cell and its eigenmodes

Figs. 1A-1C are views of an elastic metamaterial unit cell， showing its eigenmodes. Fig. 1A is a cut-away view of the schematic design. Fig. 1B a photographic image of the unit cell. Fig. 1C is a schematic drawing showing a physical model of an elastic metamaterial unit cell. The depictions of Figs. 1A and 1B represent a simple spring-mass model of the system， wherein M₁， M₂ are the mass of the core (steel cylinder) ， and shell (epoxy) ， respectively. K and K′are different spring constants， which are tunable via the thickness of the silicone coating at relevant positions.

Figs. 2A-2C are schematic diagrams showing three translational eigenmodes. Each of the translational eigenmodes is characterized by the anti-phase translational oscillation of the rigid core and the outer shell. Two modes， represented by Figs. 2A and 2B， denote xy-modes. In the non-limiting example depicted， the eigenmodes of Figs. 2A and 2B are degenerate at 1651 Hz， in which the steel and epoxy oscillated anti-phase in xy-plane. A third mode， represented by Fig. 2C， is denoted as a z-mode. In the non-limiting example depicted， the eigenmode is found at 2637 Hz， in which the steel core and epoxy oscillate anti-phase in z-direction. The colored areas or shaded areas delineate displacement component perpendicular to the slicing plane， with red and blue (outer band and inner band) representing positive and negative displacement， respectively. All eigenmodes are calculated using an isolated unit cell.

Figs. 3A-3C are schematic diagrams showing three displacement profiles of three rotational eigenmodes (RMs) ， characterized by the out-of-phase rotational oscillation of the steel and epoxy along a given axis. The different shadings delineate displacement component perpendicular to the slicing plane. The displacements show opposite positive and negative displacements across a center axis， but the inner bands showing an opposite displacement polarity from the outer bands. In the color depictions， the red and blue representing opposite displacements. All eigenmodes were calculated using an isolated unit cell.

The elastic metamaterial is a composite consisting of a rigid core part is covered by a layer of soft or rubbery material， and embedded to another relatively rigid material. A schematic drawing and a photographic image of a non-binding realization are shown in Fig. 1A and B. In this non-binding realization， a steel cylinder， with a radius r ＝ 15.8 mm and height b ＝ 37.6 mm， is coated with silicone rubber. The cylinder′s axial direction in defined as z-axis， and xy-plane is parallel to the end surfaces. The silicone layers on the top and bottom are both c ＝ 1 mm in thickness； whereas the silicone covering the curvilinear side has a thickness of b ＝ 5 mm. The silicone-coated steel cylinder is then cast inside an epoxy cube with side a ＝ 60 mm. The materials used are： Wacker silicone rubber RTV-2

M4440， Clear

Floral Arrangements decorative epoxy resin. Unit cells were joined together by acrylic superglue. The physics of this unit cell can be grasped with a simple mass-spring-mass model， as shown in the inset of Fig. 1C. Here the steel cylinder and the epoxy cladding mostly play the role of two block masses M₁ and M₂， which interact through compression and expansion of the silicone. The configuration is that of springed masses， with the silicone providing the spring elasticity； that is considered as springs. Owing to the difference in the thickness of silicone， the spring constants in relevant positions are different， denoted K for the side， and K′for top and bottom， respectively.

The consequence of this difference can be immediately seen by investigating the eigenmodes of an isolated unit cell using finite-element simulation. Three translational eigenmodes are found， in which the steel cylinder undergoes translational vibration and moves out-of-phase with respect to the epoxy cladding， which are shown in Fig. 2. Two of these modes are degenerate at 1651 Hz， in which the cylinder and the epoxy vibrate in the xy-plane. These two modes are denoted ″xy-modes″ . The third mode， in which the steel cylinder and the epoxy vibrate in z-direction， has a much higher eigenfrequency at 2637 Hz. This mode is denoted ″z-mode″ . The large mismatch in eigenfrequencies between z-mode and xy-modes is mainly due to the difference in the silicone coating′s thickness. Basically， a larger thickness corresponds to a smaller spring constant thereby a lower eigenfrequency； whereas a smaller thickness corresponds to the opposite. By changing the thickness of the silicone， it is possible to conveniently engineer the corresponding eigenfrequencies. Similarly， by changing the elastomeric constant or density of the silicone， it is possible to conveniently engineer the corresponding eigenfrequencies.

The ability to change the thickness of the silicon， or engineer the corresponding eigenfrequencies allows adjustment of the response of the elastomeric material in different directions. Thus， for example， the elastic materials can have different thicknesses or other variations in physical characteristics in different spatial directions， resulting in the metamaterials having different vibratory responses in different directions. The metamaterials can be spectrally detuned by means of anisotropy resulting from the elastic materials and/or elastic structures can be materials or structures of different elastic constant at different spatial directions， so that the degeneracy according to the elastic material′s elastic constants is lifted.

Besides translational vibrations， three rotational modes (RMs) are also found in simulation， wherein the steel cylinder and epoxy are rotating about one of the three spatial axes in an anti-phase manner， as shown in Fig. 3. Their physical consequence will be demonstrated and discussed later.

Example -quasi-one-dimensional array

Fig. 4 is a schematic diagram showing the four fundamental vibration modes of an elastic rod. Two are flexural modes， characterized by the bending of the rod under excitation forces that are transverse to the rod. One longitudinal mode， in which the particles dominantly oscillate along the rod， is accompanied by a small degree of ″breathing″ ， meaning the rod elongates. A torsional mode describes the rod′s twisting oscillation along itself.

Figs. 5A and 5B are photographic images of the two different types of meta-rod are shown at the top， associated with graphic depictions of their corresponding band structures. In Fig. 5A， blue and red markers (or shaded areas) highlight the flexural or longitudinal branches in sample k_z (photographic image) . In Fig. 5A， blue markers (middle shading) highlights a fiexural bandgap， red markers (top shading) highlights a longitudinal bandgap. In Fig. 5B， sample k_x is depicted， with the torsional bandgap also shaded in orange (middle shading) .

The unit cells can be assembled into a quasi-one-dimensional periodic array， leading to an elastic structure commonly known as a rod. An elastic rod can sustain four distinct branches of waves： two flexural， one longitudinal， and one torsional. The four distinct branches are easily distinguishable through their vibration profiles， as schematically illustrated in Fig. 4. The flexural modes are characterized by the rod′s bending， and are dominantly transverse in their nature. Therefore， the flexural modes have two orthogonal polarizations. In the longitudinal vibration， the rod′s displacement is mainly along the rod， accompanied by small amount of breathing. Perhaps less familiar is the torsional mode， which is characterized by the rotation and twisting vibration along the rod itself. These four types of vibrations can be identified through their dispersion signatures at low frequencies. The longitudinal and torsional waves have linear dispersions. Flexural waves， however， are governed by biharmonic equations， and therefore have distinctive quadratic dispersion relations.

From the unit cell′s geometry， it is straightforward to see that there exist two different ways to build the quasi-one-dimensional array. One approach is to repeat the unit cell in the z-direction； i.e.， along the steel cylinder′s axis. This array is denoted ″sample k_z″ . The second type of array， denoted as ″sample k_x″ ， has periodicity in the x (or y) direction； i.e.， perpendicular to the steel cylinder′s axis. Depictions of these samples are shown in Figs. 5A and B (above the graphs) .

Operational Responses

Figs. 6A-6C are photographic images， depicting the experimental setups. Fig. 6A depicts an electromagnetic shaker used for transverse excitation. Fig. 6B depicts an electromagnetic shaker is used for longitudinal excitation. Fig. 6C depicts a jig in which an electric motor is used to achieve rotational excitation. Accelerometers are used to obtain the elastic responses at different positions.

Fig. 7 is a graphic depiction showing the realization of fluid-like elasticity of a sample in the frequency regime of ～1.3 -1.6 kHz. The depiction shows a fluid-like elastic characteristic of the meta-rod with periodicity in z. The red (filled) circles represent the longitudinal response. The blue (open) circles represent flexural response. Spectrally mismatched bandgaps are clearly observed as can be seen in this depiction. In particular， in this non-limiting example， a bandgap for flexural waves is seen in ～1.2 -1.6 kHz， which overlaps with longitudinal waves passband. The response difference exceeds 39 dB (～1350 Hz) . This indicates that， in the example， only longitudinal waves were able to be sustained in this frequency regime， an intriguing elastic property traditionally found only in fluids.

Fig. 8 is a graphic depiction showing the transverse and longitudinal response of sample k_x. This shows the realization of fluid-like elasticity of sample k_x. The red (solid) markers represent longitudinal branch and z-mode. The grey or dashed (open) makers and the blue (open) markers both represent flexural branches. The flexural branches correspond to and xy-modes.

Figs. 9A and 9B are photographic (Fig. 9A) and spectrographic color map (Fig. 9B) images， depicting the experimental setup for displacement profile mapping. Fig. 9A depicts a piece of aluminum foil (kitchen foil with a thickness ～0.02 mm) adhered to cover the facet of the sample to be measured， which increased the reflectivity. A laser Doppler vibrometer (Graphtec AT500-05) was mounted on a two-dimensional motorized translation stage， so as to scan the surface point by point. The translational stage′s has limited travel distance and therefore only allows a scanning that covers four unit cells. The laser beam is perpendicular to the measured surface. The sample was excited by the shaker driven by a sinusoidal signal. Fig. 9B is a spectrographic image showing the out-of-plane displacement field of sample k_x under transverse excitation. In this non-limiting example， the frequency is 1350 Hz， which places the shaking inside the flexural bandgap. The color map represents normalized displacement amplitude. The cyan (solid/dashed) lines delineate the positions of the unit cells or steel cylinders， respectively. The aluminum plate for excitation is situated at z ＝ 0 mm. Figs. 9A and 9B demonstrates the elastic meta-rod′s resistance to lateral shaking， which is a functionality that can be used for vibration isolations.

Fig. 10 is a graphic depiction showing torsional response. A response function is defined as the ratio of the amplitudes of tangential accelerations at bottom and top of the sample， with rotational actuation situated at the top of the sample shown in Fig. 6C. The measured (black open circle markers) and simulated response (black solid curve) functions are plotted in a as functions of frequency (left axis) . A bandgap is seen near 1.2 -1.6 kHz. The calculated effective moment of inertia about x-axis I_x is shown in orange markers， with solid curves to guide the eye (small squares connected with lines on top left and bottom right) . It is seen that I_x turns negative inside the bandgap. A bandgap of torsional vibration (solid lines on left) ， and a negative effective moment of inertia (line connecting orange squares on right) .

Fig. 11 is a graphic depiction showing band structure of a three-dimensional lattice. Branches that couple with z-mode and xy-modes are in red (1111， 1112， 1113) ， blue (1121， 1122， 1123) and green (1131， 1132， 1133) ， respectively. It is noted that the xy-modes are merged at two locations on the right side of the graph. Fluid-like property is found for waves propagate in z direction， at the right shaded area (～1.1 -1.7 kHz) . The left shaded region (～1.7 -2.9 kHz) is partially fluid-like， in which shear waves polarized along z-direction cannot propagate， but shear waves with y-polarization are free to propagate.

Fluid-like characteristics

The rods exhibit emergences of unique vibrational properties such as the intriguing fluid-like elastic properties， and the extinction of torsional vibration.

In evaluating linear force and vibration， first， sample k_z is excited by a force perpendicular to the rod. This is realized by an experimental setup shown in Fig. 6A. In this setup， one end of the sample is fixed to an aluminum plate. The sample and the plate are supported on slide tracks， which confine their motion to one single direction. The side of the aluminum plate is then connected to an electromagnetic shaker， whose vibration causes the plate to slide back and forth along the tracks. Such a transverse excitation will excite flexural vibration. The response function is then obtained by dividing the amplitudes of accelerations (with direction along the excitation force) measured on the aluminum plate and on the top of the sample. This is plotted in Fig. 4A as a function of frequency (blue circles. ) A bandgap is clearly seen in ～1.2 -1.6 kHz. A laser Doppler vibrometer is used to map the displacement parallel to the actuation direction on a facet at 1350 Hz； i.e.， inside the bandgap. The result is shown in Fig. 4B. It is seen that the displacement amplitude rapidly decays away from the actuation position (z ＝ 0 mm) . The result is that a transverse force with frequencies inside the bandgap cannot excite the flexural vibration on the sample.

Next， the sample is subjected to a pushing and pulling force that acts along the rod. This is achieved by fixing the rod on a thick aluminum plate which is connected to the shaker， as shown in Fig. 6B. This will excite the longitudinal branch. The measured response function is shown with red circles in Fig. 3A. A bandgap is found covering ～1.8 -2.9 kHz， which strongly mismatches that of the flexural bandgap. Comparing the sample′s flexural and longitudinal responses， an intriguing spectral regime can be found in ～1.2 -1.6 kHz. In this regime， the meta-rod can only withstand the longitudinal vibration. Those types of vibrations are transverse in their nature， however， and cannot be sustained. The response difference exceeds 39 dB around 1.4 kHz. Traditionally， such a property is expected only in fluid， and is not found in solid materials. Numerical simulations are also carried out for both excitations， and the results are shown in Fig. 7 as curves. Good agreement is seen between the experimental and simulation results.

The mechanism of this fluid-like characteristic can be understood by looking at the band structure， as shown in Fig. 5A. A force transverse to the rod excites a flexural branch， which exhibits a quadratic dispersion. In the case of sample k_z， transverse force will also excite one of the xy-modes， which exhibit a flexural bandgap， as highlighted by blue shade (lower shaded area) in Fig. 5A. While in this frequency regime， the longitudinal vibration； i.e.， the z-mode (red circles in Fig. 5A) ， exhibits a pass band. Due to the thinner silicone rubber on the top and bottom of the steel rod in the unit cell， the longitudinal band gap opens at much higher frequencies (shaded red； lower shaded area in Fig. 5A) . Consequently， the flexural band gap overlaps with the longitudinal pass band， giving rise to region that only sustains longitudinal vibrations. Due to the isotropy in the xy plane， this characteristic disregards the polarization of flexural waves. A fluid-like elastic property is thus exhibited.

Experimental results on samplek_x are shown in Fig. 8. The figure shows that a fluid-like property is observed in the frequency range covering ～1.7 -2.5 kHz， which is different from that of sample k_z. In the case of sample k_x， the longitudinal branch couples with one of the xy-modes. On the other hand， the flexural branches correspond to the z-mode or the other xy-modes， depending on the force′s polarization. It turns out that ″fluid-like″ property only exists for the z-polarized flexural waves. It is interesting that different types of ″meta-rods″ with distinctive features can be created simply by changing orientation of the same unit cells. In practice， this characteristic can potentially act as a switching mechanism， with which the system can change between fluid-like and solid-like behaviors.

Vibration isolation

The meta-rod has another particularly interesting aspect in its ability to resist shaking transverse to its axis； i.e.， flexural vibration， at deep sub-wavelength scale. In the particular geometry of a rod， the disclosed technology may hold great potential in application. Flexural vibration is a major source of potential damage for free-standing manmade structures during earthquakes. In the example shown Fig. 6A and Fig. 9A， it can be seen that the long thin meta-rod was fixed only at the bottom to a plate that shakes horizontally. The bottom plate fixation， subject to horizontal shaking， is a configuration that faithfully reflects many free-standing structures， such as pillars， skyscrapers， towers， chimneys， etc. This meta-rod shows resilience to lateral shaking within the frequency regime of the flexural bandgap. It can be seen in Fig. 9B that the vibration decays within one unit cell， whose dimension is deep-subwavelength. As an example， the wavelength of flexural wave at ～1.3 kHz in epoxy exceeds one meter. Therefore， counter-intuitively， the meta-rod is more stable at taller positions. In comparison， existing seismic protective solutions employ sophisticated systems such as the massive roof-loaded ″seismic damper″ to protect skyscrapers. Elastic metamaterials may become an interesting alternative.

For the application of vibration isolation， anisotropy in the unit cell is an important condition. To wit， if the unit cells are isotropic， a full stop band for longitudinal and transverse waves can be realized.

Torsional bandgap and negative moment of inertia

Torsional vibration is recognized as a potential hazard for machineries with rotating parts， such as drills， propulsion shafts， etc. The torsional bandgap opens a possible route to mitigate these problems.

Three rotational modes (RMs) are found in the unit cell. RMs have long been theoretically studied in phononic crystals and metamaterials. Owing to the difficulty in excitation of these unique modes in practical scenarios， a direct observation in experiments is challenging. The rod geometry of the disclosed system offers a rare opportunity to observe and to appreciate the physical significance of RMs. An elastic rod can withstand torsional vibration， in which the rod twists along its axis (Fig. 4) . Next， it can be shown that torsional vibrations of rods can strongly couple with the RMs.

Sample k_x was chosen to investigate the effect of RM on torsional vibration. First， the meta-rod′s torsional branch can be easily identified in the band structure through its linear dispersion and its rotational displacement profile as shown in Fig. 5B. The rotational motion can trigger an RM， in which the steel cylinder and epoxy rotate about the x-axis in an anti-phase manner. In the example， this mode is found at 1646 Hz. Similarly， the coupling of RM and torsional branch yields a hybridization bandgap for torsional vibration， as highlighted in Fig. 5B. Experimentally， this finding was verified by rotational actuation with a setup shown in Fig. 6C， in which an electric motor is used to apply a torque pulse to the meta-rod. The measured response function indeed confirms the existence of such a band gap， as shown in Fig. 5A. The measured results show good agreement with numerical simulation.

Further analytical data on the torsional behavior can be obtained by the meta-rod′s rotational vibration profile at some key frequencies， which are plotted in Fig. 5B. At the lower gap edge， it is seen that the exterior of the meta-rod； i.e.， the epoxy cladding， undergoes pronounced twisting. In each unit cell， the steel cylinder is vibrating at its interface with the epoxy. For frequencies within the bandgap， torsional vibration exponentially decays away from the actuation position. As a result， the propagation of torsional wave is prevented. At the high gap edge， it can be observed that all steel cylinders are rotating out-of-phase with respect to the epoxy. These observations resemble closely the mode profile of acoustic/elastic metamaterials with negative effective mass density.

The effective moment of inertia of each unit of the meta-rod，

can be defined as

where

is the effective torque applied on the unit， and

is the effective angular acceleration of the unit.

In this case， it is possible to easily calculate the x component of inertia I_x， which is related to torsional vibration of the rod：

where

is force

τ_x ＝ ∫ (yσ_xz-zσ_xy)dydz，

Δτ_x ＝ τ_x |_x＝α -τ_x |_x＝0 ，

σ_xz and σ_xy are stresses.

Acceleration is determined by：

where

is acceleration at each point on the surface.

Then，

Therefore

Band structure and fluid-like characteristics

The band structure and fluid-like characteristics of three-dimensional (3D) lattice can be obtained in all three spatial directions. By repeating the metamaterial unit cells in the three spatial directions， a three-dimension lattice can be constructed. Its band structure is shown in Fig. 11. Here， z is the direction parallel to the steel cylinder′s axis； and x， y are in-plane with the cylinder′s ends. Likewise， the lattice is isotropic in the xy direction， and anisotropy exists in the z direction.

With reference to Fig. 11， several observations can be immediately made：

First， only three branches are found in the long-wavelength limit， two of which are bulk shear waves， the third one is a longitudinal wave. The torsional branch does not exist in a 3D lattice. This is as expected， since a 3D lattice is essentially a bulk material. Both the longitudinal and the shear waves have linear dispersions.

Second， rotational modes still exist； however， all three rotational modes manifest mainly as flat bands； i.e.， they couple minimally to stimulations and propagative waves. Some weak interaction with shear waves still exist， as shown near 1.5 kHz， k_xa/π ～ 0.5.

Third， fluid-like characteristics persists， and is quite similar to the case with meta-rods. For a incident wave along k_z， bandgaps for both shear waves coincide， and overlap with longitudinal passband in the frequency regime of 1.1 -1.7 kHz. This fluid-like region is shaded in dark grey in Fig. 11. For incident from x， however， shear waves can excite z-mode or y-mode， depending on the polarization direction. Fluid-like property exists only for z-polarized shear wave in 1.7 -2.9 kHz (light grey shaded in Fig. 11) ， in which a longitudinal wave can propagate. In contrast， the y-polarized shear wave has a bandgap that coincides with that of longitudinal wave.

Conclusion

The disclosed configuration provides a composite structure with plural rigid solid materials surrounded by elastic materials and/or elastic structures embedded in another solid structure. The solid materials can take regular shapes such as cubic， cylindrical or spherical. The elastic materials can be formed of a variety of materials， such as polymer， rubber， silicone， and other elastic materials. As an elastic structure， the metamaterials can take the forms of springs， hydraulics， and other structures in which structural support is desired， but fluid response to vibrations is desired. The solid structure can be any solid materials， including but not limited to polymer， plastics， concrete， metal， etc. The structure can be repeated periodically into one-dimensional array with a finite length， or into a three-dimensional array with a finite length.

The elastic materials and/or elastic structures can be materials or structures of different elastic constant at different spatial directions. The elastic materials can also have different thickness at different spatial directions. The resulting structure can be formed to exhibit both translational and rotational eigenmodes. The translational eigenmodes can be formed so that they degenerate. The translational eigenmodes can be formed so that they degenerate if the elastic material′s elastic constants and thickness are identical. The translational eigenmodes can be spectrally detuned by means of anisotropy， so that the degeneracy is increased.

It will be understood that many additional changes in the details， materials， steps and arrangement of parts， which have been herein described and illustrated to explain the nature of the subject matter， may be made by those skilled in the art within the principle and scope of the invention as expressed in the appended claims.

Claims

A metamaterial composite structure comprising one or more three-dimensional resonant metamaterial cells， at least a subset of the cells comprising：

an inner mass formed of substantially incompressible material；

an elastomeric coating on an outer surface of the inner mass； and

an outer encapsulant formed of a material having substantially less elasticity than the elastomeric coating， and having tess density than the inner mass and encapsulating the inner mass over elastomeric coating so as to support the inner mass through the elastomeric coating with substantially no gap or fluid interface between the outer encapsulant and the elastomeric coating， the inner mass， elastomeric coating and outer encapsulant forming a suspended mass structure.
The metamaterial composite structure of claim 1， further comprising：

the inner mass formed of a metal； and

the outer encapsulant formed of a polymer.
The metamaterial composite structure of claim 1， further comprising：

the suspended mass structure forming a three-dimensional anisotropic locally resonant unit， exhibiting polarization bandgaps together and responding to vibrational movement with a fluid-like elasticity in a fully solid structure.
The metamaterial composite structure of claim 3， wherein：

a plurality of the three-dimensional resonant metamaterial cells in a substantially stacked arrangement form elastic rods having the fluid-like elasticity， allowing selective suppression of vibration by selection of materials and dimensions of the inner mass， elastomeric coating and outer encapsulant.
The metamaterial composite structure of claim 3， wherein：

a plurality of the three-dimensional resonant metamaterial cells in a substantially stacked arrangement form elastic rods having the fluid-like elasticity， allowing selective suppression of vibration by preventing fiexural vibrations while allowing longitudinal vibration.
A structure comprising a plurality of the three-dimensional resonant metamaterial cells as described in claim 3， in a substantially stacked arrangement form elastic rods having the fluid-like elasticity， allowing selective suppression of vibration to efficiently separate and control elastic waves of different polarizations.
The metamaterial composite structure of claim 1， further comprising：

the elastomeric coating having a different elastic constant as measured at different spatial directions.
The metamaterial composite structure of claim 1， further comprising：

the vibrational response of the metamaterial cells spectrally detuned by means of anisotropy resulting from the characteristics of the elastic materials and/or elastic structures， in which the elastic materials and/or elastic structures have different elastic constant at different spatial directions， so that the degeneracy does not occur according to the elastic material′s elastic constants.
Method of providing vibration suppression by use of a metamaterial composite structure， the method comprising：

providing a plurality of three-dimensional resonant metamaterial cells， by arranging the cells in a stacked or other structural relationship；

constructing each cell with an inner mass formed of substantially incompressible material；

coating an outer surface of the inner mass with an elastomeric coating； and

encapsulating of each of the coated cells with an outer encapsulant formed of a material having substantially less elasticity than the elastomeric coating， but having less density than the inner mass， resulting in a cell structure comprising of an inner mass having an elastomeric coating， and encapsulated over the elastomeric coating so as to support the inner mass through the elastomeric coating with substantially no gap or fluid interface between the outer encapsulant， and the elastomeric coating， the inner mass， elastomeric coating and outer encapsulant forming a suspended mass structure.
The method of claim 9， further comprising：

forming the inner mass of a metal； and

forming the outer encapsulant of a polymer.
The method of claim 9， wherein：

the suspended mass structure forms a three-dimensional anisotropic locally resonant unit， exhibiting polarization bandgaps together and responding to vibrational movement with a fluid-like elasticity in a fully solid structure.
The method of claim 11， wherein：

a plurality of the three-dimensional resonant metamaterial cells in a substantially stacked arrangement form elastic rods having the fluid-like elasticity， allowing selective suppression of vibration by selection of materials and dimensions of the inner mass， elastomeric coating and outer encapsulant.
The method of claim 11， wherein：

a plurality of the three-dimensional resonant metamaterial cells in a substantially stacked arrangement form elastic rods having the fluid-like elasticity， allowing selective suppression of vibration by preventing fiexural vibrations while allowing longitudinal vibration.
The method of claim 9， further comprising：

forming the metamaterial composite structure， structure as a plurality of the three-dimensional resonant metamaterial cells in a substantially stacked arrangement form elastic rods having the fluid-like elasticity， allowing selective suppression of vibration to efficiently separate and control elastic waves of different polarizations.
The method of claim 9， further comprising：

coating the outer surface of the inner mass the elastomeric with the elastomeric coating in a manner to provide the coating with a different elastic constant as measured at different spatial directions.
The method of claim 9， wherein：

the construction of the metamaterial cells provides a vibrational response spectrally detuned by means of anisotropy resulting from the characteristics of the elastic materials and/or elastic structures， in which the elastic materials and/or elastic structures have different elastic constant at different spatial directions， so that the degeneracy does not occur according to the elastic material′s elastic constants.