Effective Medium Theories and Symmetry Properties of Elastic Metamaterials

Recently, metamaterials have attracted a great deal of attention due to their unusual properties not seen in naturally occurring materials, such as negative refraction (Lezec et al., 2007; Pendry, 2000; Veselago, 1968), superlensing (Grbic & Eleftheriades, 2004) and cloaking (Leonhardt 2006; Pendry et al., 2006), etc. These unusual properties are derived from the resonant structures in their artificial building blocks. The resonant structures interact with the wave, but their small size prevents them from being “seen” individually by the wave with wavelength inside the background much larger than the size of the structures (Pendry & Smith, 2006). Thus, the properties of a metamaterial can be described with homogenized parameters or effective medium parameters. The theory that links the microscopic resonant structures to their effective medium parameters is called the effective medium theory (EMT). For example, the left-handed metamaterial consisting of a periodic array of split ring resonators and conducting thin wires has been successfully demonstrated that, in a frequency regime, it behaves like a homogeneous medium exhibiting negative effective permittivity,


Introduction
Recently, metamaterials have attracted a great deal of attention due to their unusual properties not seen in naturally occurring materials, such as negative refraction (Lezec et al., 2007;Pendry, 2000;Veselago, 1968), superlensing (Grbic & Eleftheriades, 2004) and cloaking (Leonhardt 2006;Pendry et al., 2006), etc.These unusual properties are derived from the resonant structures in their artificial building blocks.The resonant structures interact with the wave, but their small size prevents them from being "seen" individually by the wave with wavelength inside the background much larger than the size of the structures (Pendry & Smith, 2006).Thus, the properties of a metamaterial can be described with homogenized parameters or effective medium parameters.The theory that links the microscopic resonant structures to their effective medium parameters is called the effective medium theory (EMT).For example, the left-handed metamaterial consisting of a periodic array of split ring resonators and conducting thin wires has been successfully demonstrated that, in a frequency regime, it behaves like a homogeneous medium exhibiting negative effective permittivity, eff ε , and negative effective permeability, eff µ , simultaneously (Pendry et al., 1999(Pendry et al., , 1996)).
Since the refractive index is defined as eff eff eff n ε µ = , negative refraction is a consequence of double negativity in permittivity and permeability.This example shows how effective medium parameters of particular resonant structures can be used to describe unusual properties of a metamaterial.In turn, a valid and accurate EMT provides an efficient and systematic tool to design and engineer the resonant structures according to certain desired metamaterial properties.During the development of metamaterials, there has always been a continuous effort to find an appropriate EMT for metamaterials such that various novel phenomena can have a theoretical explanation.One can obtain the effective parameters from some phenomenological results, such as transmission and reflections (Baker-Jarvis et al., 1990;Smith et al., 2002), and wave propagation (Andryieuski et al., 2009), but only a theory can give a clear understanding on the physical origin of effecitve parameters.Therefore, EMTs with physical insights can be regarded as theoretical foundations of metamaterials.
The development of EMT has seen a long history accompanied by various approaches.A famous one for electromagnetic (EM) waves is the Maxwell-Garnett theory (Sheng, 2006), which is valid in the quasi-static limit, aka the zero frequency limit.The quasi-static limit requires the wavelengths inside the scatterer, the host, and the effective medium to all be very large compared to the size of the building block (Lamb et al., 1980).However, for metamaterials, the wavelength inside the scatterer could be smaller than the size of the building block and thus lead to resonances at low frequencies.This results in the failure of the widely used quasi-static EMT.Nevertheless, as long as the wavelength in the effective medium is still large compared to the size of the building block, there exists an effective medium description as the wave still cannot probe the fine structures of the building blocks.In this context, another limit is introduced, which is the long wavelength limit.Compared with the quasi-static limit, the long wavelength limit does not have restrictions on the wavelength inside the scatterer, while the wavelengths inside the host and the effective medium should still be large (Lamb et al., 1980).In the study of metamaterials, one aspect is to develop EMTs that are valid in the long wavelength limit.In this chapter, we will focus on the recent developments of EMTs for elastic metamaterials.

Effective medium theories for electromagnetic metamaterials
Though our focus is on elastic metamaterials, it is necessary to briefly review the EMTs for EM metamaterials to offer a systematic picture of the EMTs.Ever since the birth of EM metamaterials, EMTs have played an integral role in designing metamaterials and explaining their unusual properties.The EMTs for EM metamaterials can be broken down into one of several classes.In one class, the effective parameters are obtained from the average of the computed eigenfields in the unit cell (Chern & Chen, 2009;Chui & L. Hu, 2002;Pendry et al., 1999;Smith & Pendry, 2006).This method gives inherently nonlocal parameters, i.e. parameters that depend on not only frequency but also the Bloch wave vector.For metamaterials with a good effective medium approximation, the nonlocality may be ignored.This method is especially helpful for use with metamaterials with complicated unit structures, such as split rings.Another class of EMTs is called the coherent potential approximation (CPA) method.In this method the effective medium is taken as the background embedded with the scatterer in the unit cell and by implying zero scattering, some elegent formulas of the effective parameters have been obtained (X.Hu et al., 2006;Jin et al., 2009;Wu et al., 2006).This method currently only works for scatterers with isotropic geometry, but it is very accurate in the long wavelength limit even at relatively high frequencies.Interestingly, the obtained effective parameters do not have any imaginary parts if the system does not have any absorption.Similar formulas can also be obtained from the multiple-scattering theory (MST) (Chui & Lin, 2008;Wu & Z. Zhang, 2009).The MST, which will be introduced in Section 2, is capable of producing the dispersion relations of a periodic structure.From dispersion relations, the effective wave speed can be easily calculated while the impedance still remains unknown.Recently, other methods have appeared, such as the quasimode method (Sun et al., 2009) and the first-principles method (Andrea, 2011).

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Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 145

Effective medium theories for acoustic metamaterials
While the field of EM metamaterial has developed rapidly during the past decade, one of its counterparts, denoted as acoustic metamaterial has also seen fast growth (Ding et al., 2007;Fang et al., 2006;Lee et al., 2010;Yang et al., 2008).The acoustic metamaterial is designed to manipulate acoustic waves (Chen et al., 2010;Lu et al., 2009).Analogous to EM metamaterials, there are two material parameters that describe the wave propagation, which are bulk modulus, κ , and mass density, ρ .In two dimensions (2D), if the constituents of the metamaterials are all fluids, the governing equation can be mapped into 2D EM equations so that the EMT for 2D acoustic metamaterials is the same as that for 2D EM metamaterials.If the scatterers in acoustic metamaterials are solid, the shear modulus of the scatterers can be ignored when the longitudinal velocity contrast between the scatterer and the host is high (Kafesaki & Economou, 1999).In this case, the scattering property is basically the same as the EM cases.The mapping from the EM waves to acoustic waves facilitates the development of EMTs which have also been extensively studied by using various types of methods.MST (Mei et al., 2006;Torrent et al., 2006) and the CPA (Kafesaki et al., 2000;Li & Chan, 2004) represent two classes of them.Exciting news also came from the experimental realizations of acoustic metamaterials, such as acoustic negative refraction (S.Zhang et al., 2009) and acoustic cloaking(S.Zhang et al., 2011).

Effective medium theories for elastic metamaterials
The term elastic metamaterial refers to those metamaterials which are able to sustain not only longitudinal but also shear waves in their effective media.It is well-known that the EMT for an elastic composite in the quasi-static limit is anisotropic in general.The only exception is the hexagonal lattice in two dimensions (Landau & Lifshitz, 1986;Royer & Dieulesaint, 1999;Wu & Z. Zhang 2009).Even for this case, the EMT involves the determination of three effective parameters, i.e., mass density, ρ , bulk modulus, κ , and shear modulus, µ .This is in contrast to the cases of previously mentioned EM and acoustic EMTs which involve only two effective parameters.One more effective medium parameter greatly enriches the types and physics of wave propagation (Chen et al., 2008;Wu et al., 2007), such as mode conversion between longitudinal and transverse waves (Wu et al., 2011); however, it also adds complexity to the EMT.Recently, various EMTs for elastic metamaterials have been proposed, such as those ones based on the plane-wave-expansion method (Krokhin et al., 2003), and integration of fields (Zhou & Hu, 2009).
In this chapter, the symmetry property of an elastic metamaterial is examined based on the MST.It is shown that the elastic metamaterial preserves the quasi-static symmetry properties.For isotropic elastic metamaterials, CPA provides a simple and accurate EMT in the long wavelength limit, which links the scattering properties of the scatterer and the effective medium parameters (Wu et al, 2007).Those formulae can also be derived by using the MST method (Wu & Z. Zhang, 2009).For anisotropic elastic metamaterial, the EMT involves three or more effective elastic moduli (Landau & Lifshitz, 1986).In this case, CPA fails.To tackle this problem, a method based on MST in conjunction with the Christoffel's equation (Royer & Dieulesaint, 1999) has been proposed and the expressions for effective elastic moduli have been obtained (Wu & Z. Zhang, 2009).The combinations of anisotropy and negativities in various effective moduli can give rise to many types of novel wave propagation behaviors that are unseen in normal solids (Lai et al., 2011).
Within the scope of this chapter, all the EMTs mentioned above are limited to linear elastodynamics and do not consider the micro-structure introduced local rotation (Milton & Willis, 2007).

Scattering properies of elastic metamaterials
In order to illustrate EMTs and symmetry properties of elastic metamaterials, we start with a simple case where the resonant scatterer in a building block is homogeneous.More complicated scatterers will be discussed in Section 5.The elastic metamaterial considered here is composed of cylindrical inclusions of radius s r with mass density s ρ , shear modulus s µ , and bulk modulus s κ , embedded in an isotropic matrix, whose material parameters are denoted by ( 0 ρ , 0 µ , 0 κ ).In two dimensions, the bulk modulus, κ, is related to the shear modulus through the relation κλµ =+ , where λ represents the Lamé constant (Royer & Dieulesaint, 1999).Due to the translational symmetry along the cylinder's axis, denoted as z-axis, the elastic modes in the system can be decoupled into a scalar part, which is also called shear horizontal mode with vibrations along the z-axis, and a vector part, i.e.,xy-mode with vibrations in the x-y plane.xy-mode is a mixed polarization of quasi-longitudinal and quasi-shear vertical modes.Since the shear horizontal mode satisfies the scalar wave equation with the same mathematical structure as those for acoustic (Krokhin et al., 2003) and 2D EM cases, this part is skipped.Rather, the focus is on the more complicated case of the xy-mode whose wave equation is given by: where u  is the displacement field.In general, u  can be decoupled into a longitudinal part and a transverse part, i.e. () , where l φ and t φ are the longitudinal and transverse gauge potentials, respectively.

Single-scattering, the scattering coefficients
If there is only one scatterer, the solutions to l φ and t φ can be expanded by using Bessel and Hankel functions.The wave incident on a single scatterer, p , is: and the wave scattered by the same scatterer is ( )   are the polar coordinates originating at the center of the scatterer.The longitudinal and transverse waves in the matrix are coupled by the scatterings of the scatterers, inside which the displacement is given by: ( )  (Wu et al., 2007).

Periodic structures and multiple-scattering
For a collection of scatterers, the MST takes full account of the multiple scatterings between any two scatterers (Liu et al., 2000a;Mei et al., 2003).The wave incident on the scatterer p is contributed by two parts: one is the external incident waves from outside the system, and the other part is the scattered waves coming from all the other scatterers inside the system.Thus, the total incident waves on the scatterer p are expressed as: where (, ) qq r θ denote q r  in the polar coordinates originating at the center of scatterer q.
Here q r  and p r  refer to the same spatial point measured from the positions of scatterers q and p, respectively.For simplicity, the center of scatterer p is chosen as the origin and the position of scatterer q is denoted by qp With the help of Graf's addition theorem (Abramowitz & Stegun, 1972), the Hankel functions depicting the scattered wave coming from the scatterer q can be changed into Bessel functions describing the wave incident on the scatterer p, which is: Hk r e g J k r e where Substituting Eqs. ( 7) and ( 8) into (6), we obtain: Eq. ( 2) together with Eqs. ( 5) and ( 9) leads to the following self-consistent equation: This set of self-consistent equations can be written into the standard form of linear equations Ax B = , and the multiple-scattering problem is numerically solved.
If the scatterers are arranged in a periodic array, the Bloch theorem, , can be applied, where K  is the Bloch wave vector.For an eigenvalue problem, the external incident field is dropped off so that Eq. ( 10) is converted into: Eq. ( 11) has nontrivial solutions if and only if, where ( ) represents lattice sums.The solution of Eq. ( 12) offers the dependence of frequency, ω , on the Block wave vector, K  , which is known as the dispersion relation.To solve Eq. ( 12) for dispersion relations, one needs to evaluate the lattice sums first, which can be accomplished through several techniques (Chin et al., 1994;Mei et al., 2003).The expression of ( ) ,' Sm m β − is given by (Wu and Z. Zhang, 2009): where a denotes the lattice constant, Ω is the volume of the unit cell and hh (Q , ) ϕ stands for the vector hh QK K =+    in polar coordinates, and h K  is the reciprocal lattice vector.

Symmetry properties of elastic metamaterials
Since the MST is capable of producing accurate dispersion relations for all frequencies, it is a good approach in the study of the symmetry properties of an elastic metamaterial.The elastic metamaterial is isotropic if its dispersion relation, ( ) K ω  ,does not depend on the direction of K  .In the framework of MST, the dispersion relation can be calculated numerically by solving Eq. ( 12).In the long wavelength limit, both Ka and 0 kaare all much smaller than unity and appropriate approximations can be made in the Bessel and Hankel functions in the lattice sum (, '  )   β − Sm m and the scattering matrix . This simplifies the secular equation and benefits the derivation of analytic expression of EMT.Taking ( )  12), it is easy to find that the leading terms are those with 2 m ≤ , where m corresponds to the order of Bessel and Hankel functions and is called the angular quantum number.Thus, we only need to consider the terms with 0 4 n ≤≤ in Eq. ( 13), in which the summation 22 10 () ( ) can be separated into two terms: the first is 0 , the second is the sum of all other terms with 0 . Then, the lattice sum takes the following expression: ( ) ( ) where ( ) denotes the polar coordinates of h K  , the reciprocal lattice vector of the lattice.The summation of all nonzero reciprocal lattice vectors in the second term in the bracket reveals the dependence of lattice sum on the lattice structure, which influences the symmetry properties of the dispersion relations.

Isotropic dispersions
For a 2D hexagonal lattice with a lattice constant a , the reciprocal lattice follows: Here, î and ˆj represent the unit vectors along the x-and y-axes in the reciprocal space.When 0 n ≠ , due to the symmetry of a hexagonal lattice, the summation in the second term of Eq. ( 14) is zero, which can be proved in the following way.For an arbitrarily chosen reciprocal lattice vector, .Thus, the summation in Eq. ( 14) vanishes after summing over all the non-zero h K  and only the first term of Eq. ( 14) survives.When 0 n = , the second term in Eq. ( 14) no longer sums to zero as 1 Kh in e φ − = .However, this term can be ignored in the long wavelength limit because compared to the first term in Eq. ( 14) which is on the order of 2 ω − , it is on the order of 0 ω .Thus, Eq. ( 14) is further reduced to: Substituting Eq. ( 16) into Eq.(12), we find the following two roots: given in Eq. ( 17) do not depend on the direction of K  , i.e., K φ .This implies that all the dispersion relation for an elastic metamaterial with a hexagonal structure are isotropic near the Γ point in the long wavelength limit.

Anisotropic dispersions
For the case of a square lattice, the lattice sum is almost the same as that of the hexagonal lattice case except for the 0 n ≠ case.The reciprocal lattice vector of a square lattice is expressed by: ( ) For an arbitrarily chosen reciprocal lattice vector 11 (, ) , there always exist the other three at 11 (, / 2 ) , which makes the summation


cancel to zero when 1 3 n ≤≤, and equals to 4 when 4 n = respectively.This indicates that the second term of Eq. ( 14) only vanishes when 1 3 n ≤≤.Thus, in the long wavelength limit, the lattice sum can be written, as ( ) Due to the non-zero β γ term in Eq. ( 22), the determinant in Eq. ( 12) is K φ -dependent, which gives rise to anisotropic dispersion relations.The explicit expressions for 1 squ K and 2 squ K are very complicated and will be futher discussed in the next Section.

Effective medium theory for elastic metamaterials
The MST method is capable of producing the dispersion relations of an elastic metamaterial so that the effective wave speed for elastic waves in the metamaterial can be obtained accordingly.However, it is not able to provide an effective description for each parameters.Knowing the effective parameters will provide a clear theoretical explanation of the unusual phenomenon of a metamaterial and greatly benefit the design of new metamaterials.This Section is devoted to the derivation of EMTs.

Isotropic media: Coherent potential approximation approach
If the elastic metamaterial is isotropic, i.e., cylinders arranged in a hexogonal lattice, the EMT can be derived by considering the scattering of elastic waves by a coated cylinder embedded in the effective medium with effective parameters ( e κ , e µ , e ρ ), which is shown in The dispersion relation can be reproduced from effective medium parameters.Comparing Eqs. ( 24)-( 26) to Eqs. ( 17) and ( 18), it is easy to show that ee ρµ coincides with ( ) K ω .The equivalence provides a strong evidence that the dispersion relation is isotropic for a hexagonal lattice and its effective medium properties can be evaluated from the EMT derived from CPA.
Eqs. ( 24)-( 26) require the wavelengths in both the host and the effective medium to be much larger than the size of the unit cell, but they do not impose any restriction on the wavelengths inside the scatterer.If the condition of 1 ls s kr<< , 1 ts s kr<< is further considered, the quasi-static limit is reached and Eqs. ( 24 The 3D version was reported by Berryman decades ago (Berryman, 1980).It should be pointed out that elastic EMT cannot recover the acoustic EMT by setting all the shear moduli to be zero, because of the different boundary conditions of elastic and acoustic waves.
Figure 3 shows the equifrequency surface (EFS) of a hexagonal array of silicone rubber cylinders with radii of 0.2 a embedded in an epoxy host.An EFS is a collection of all states in the K  space that have the same frequency.The metamaterial is isotropic if its EFS is a circle.Here the dimensionless frequency, 0 () ( 2) , is used, where  (Liu et al., 2000b).Apparently, slow wave speeds imply that wavelengths inside the silicone rubber cylinder may be comparable to or even much smaller than the size of the cylinder at low frequencies.Thus, Mie-like resonances may occur, which serve as the built-in resonances required for metamaterials.Here the frequency f  is chosen to be 0.03 where both 1.9 ls s kr and 7.9 ts s kr are larger than unity indicating it is not in the quasi-static limit.Figure 3(a) shows the corresponding EFS, which exhibits two circular rings, with the inner one denoting the quasi-longitudinal branch and the outer one representing the quasi-transverse branch.The corresponding Ka as a function of K φ is plotted in Fig. 3(b) by open circles.These circles form two horizontal lines, indicating dispersions are isotropic, i.e., effective wave speeds do not vary with directions.Also plotted in Fig. 3(b) are the results of EMT calculated from Eq. ( 17) or Eqs. ( 24)-( 26), depicted by two solid lines.The complete overlaps between solid lines and circles give a numerical support to the correctness of the EMT in the long wavelength limit.

Anisotropic media: Christoffel's equation
If the elastic metamaterial is anisotropic, such as cylindrical scatterers arranged in a square lattice, the CPA fails as it only deals with isotropic cases.In this case, the result of MST, i.e., Eq. ( 14), can give an anisotropic EMT in the form of Christoffel's equation.
Taking the long wavelength limit approximation on Eq. ( 12) and plug in Eqs. ( 20) and ( 22), the expressions for 1 squ K and 2 squ K can be written as the solutions of the following Christoffel's equation for an anisotropic medium (Royer & Dieulesaint, 1999 35) and ( 36) give the dispersion relations for the quasi-transverse and quasilongitudinal bands (Royer & Dieulesaint, 1999).Eq. ( 35) shows that 2 1 (/ ) K ω oscillates between two extrema, ( ) ( )  for the longitudinal waves.The averaged values of i Ka coincide with the results calculated from the isotropic effective medium, i.e., Eq. ( 24)-( 26

Design of elastic metamaterials
The purpose of deriving EMTs is to reveal the relationship between the resonances of the microstructures and the effective parameters and to provide a guide in the design of new metamaterials with novel properties.Even for isotropic metamaterials, the negativities in three effective parameters as well as their combinations can give rise to various interesting properties unseen in natural materials.For instance, since the effective phase velocities in 2D elastic metamaterials are 1  transverse waves (Liu et al., 2000b).On the other hand, a simultaneous negative e ρ and e µ (or ee κ µ + ) induces negative refractive index for the transverse (longitudinal) waves.Table 1 lists eight possible types of wave propagation in 2D elastic metamaterials with different combinations of signs in ee κ µ + , e ρ and e µ .Like acoustic metamaterials (Li & Chan, 2004), the effective bulk modulus and shear modulus e κ and e µ determined according to Eqs.
(24)-( 26) do not satisfy the well-known bounds (Hashin & Shtrikman, 1963;Torquato, 1991) on the effective elastic moduli as these bounds are derived in the quasi-static limit.For anisotropic metamaterials, there exists at least one more effective elastic modulus which can also turn negative.Thus, many more interesting novel wave transport behaviors would be expected.Examples will be shown in a later section.

Isotropic elastic metamaterials
The simplest elastic metamaterial which is comprised of silicone rubber cylinders embedded in an epoxy host was exhibited in the last section.If the rubber cylinders' radii are chosen to be 0.3 a , it has been shown that various types of resonances were produced (Wu et al., 2007).The displacement fields for three typical resonances are plotted in Fig. 5 µ and negative e κ , respectively.We can enlarge the negative regions of effective medium parameters by enhancing these resonances.For instance, the dipolar resonance shown in Fig. 5(a) exhibits the collective motion of the core part of rubber.This mode can be regarded as a simple "mass-spring" harmonic oscillator, with the central part serving as a "mass" and the boundary layer of the rubber serving as "spring".Replacing the inner region of rubber with another heavier cylinder, e.g.lead, will enhance the field oscillation of the silicone rubber, which, in turn, will widen the resonant region of e ρ .This design was first proposed by Liu et al and was named as "locally resonant sonic materials" (Liu et al., 2000b), which were comprised of a cubic array of rubber-coated lead spheres embedded in epoxy.A large low-frequency band gap for both longitudinal and transverse waves, induced by negative mass  (Wu et al., 2007) density, was found.Figure 5(c) exhibits the field pattern of a monopolar resonance, where the shape of the silicone rubber cylinder remains as a circle, with its cross-sectional area oscillating in time.This suggests that by making the inner core more easily compressed, we would enhance the monopolar resonance.This notion was supported by using air bubbles in water to achieve a large frequency region of negative bulk modulus (Ding et al., 2007).Figure 5(b) shows the relative motion of the rubber.This suggests that by making the core areas easier to deform, we would enhance the quadrupolar resonance so as to enlarge the negative region for shear modulus.An intuitive design is to make the rubber cylinder hollow.A metamaterial based on this design is made of rubber-coated air cylinders embedded in epoxy.The material parameters of air are given by , negative-n band purely for longitudinal waves is found, which implies that e ρ and ee κ µ + are both negative.These negative values arise from resonances of both e ρ and e κ .Figure 6(a) demonstrates that the isotropic EMT is still a good approximation even for complex scatterers with layered structures.The small discrepancies between the band-structure calculation and the effective medium prediction for the longitudinal branches shown in Fig. 6(a) is due to the less accurate approximation of the Hankel functions when the values of 00 l kr and 00 t kr are not much less than 1.

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The structure of the hollow silicone rubber cylinder in epoxy does provide a frequency region of negative band for shear waves, but the bandwidth is too small to be of any practical use.Enlarging the quadrupolar resonance is a challenge task that has yet to be accomplished.Since replacing the inner part of the rubber by an easier deformed material is the direction, another common material, water, becomes a candidate.The material parameters of water are: .Figure 7 shows the effective medium parameters for the metamaterial with air core being replaced by water, which exhibits an improvement in the absolute bandwidth for negative shear modulus.However, the bandwidth to mid-frequency ratio (0.00145) is comparable to the previous air core case (0.00183).Moreover, the negative shear band disappears as the region for negative mass density does not overlap with that for negative shear modulus.This example demonstrates that simple replacement of air by water does not improve the negative shear band.Nevertheless, the replacement does enhance the dipolar resonance greatly in the very low frequency regime (which is not plotted here).This fact suggests that water core is a better candidate than air in the context of realizing negative mass density.The difficulty lies in increasing the negative region for shear modulus and adjusting it to overlap with that for the negative mass.This requires optimizing the inner and outer radii of the rubber cylinder.However, it can be shown that only altering the geometry parameters will not make a significant change.Both Figs. 6(b) and 7 show that the µ resonance is very sharp, indicating that the energy is confined in the cylinders locally.To broaden the resonance, one strategy that could be adopted is to make the energy "spread" out of the cylinders so that resonances in different cylinders become "coupled" to each other.This can be realized by reducing the impedance mismatch between the rubber and the host.
Figure 8 shows the effective medium parameters for a rubber-coated water cylinder embedded in a foam host in a hexagonal lattice.The foam is polyethylene foam (HD115) whose material parameters are µ =× which will benefit the enhancement of the resonance for shear modulus.It also makes the water-coated rubber core relatively heavier so that the resonance for mass density is also enhanced.By adjusting the geometric parameters to be 0.24a and 0.32a for inner and outer radii, respectively, a large frequency region, marked by "A" and "B" in Fig. 8, for both negative shear modulus and negative mass density is obtained.The bandwidth to midfrequency ratio reaches 0.258, which is two orders of magnitude greater than the rubbercoated air cylinders in epoxy.The corresponding band structures as well as the transmission coefficeints of a slab numerically calcuated by MST are plotted in Fig. 9(a), which clearly shows a large negative band for transverse waves denoted by red dots and a narrow negative band for longitudinal waves denoted by blue dots.The polarization of these negative bands is determined through the transmission as shown in Fig. 9(b).

Anisotropic elastic metamaterials and boundary effective medium theory
The rubber-coated water cylinder embedded in foam provides the possiblity of realizing double negative shear bands.The isotropic scatterers and the hexagonal lattice structure result in a simple isotropic effective medium description of the metamaterial, which makes the design of the elastic metamaterial easier.If there is no restriction on the symmetery properties of the scatterer, there would be much more choices at the cost of many more complicated microstructures of the scatterers.µ =× for steel.The four rectangular steel rods serve as the four masses and the soft silicone rubber rods serve as the springs.The insertion of the hard silicone rubber is for the purpose of adjusting the spring constants between the masses.
The band structure of such metamaterial was calculated by using a finite element solver (COMSOL Multiphysics) and is shown in Fig. 11(a).There are two negative bands (red and blue dots), where the lower one (red dots) has a bandwidth about 18 Hz and the higher one (blue dots) has band widths of 18Hz and 10 Hz along M The negative bands can also be understood from an effective medium point of view.Since the scatterer involves a four-fold symmetry, the previously derived formula based on MST for isotropic inclusions does not apply and an EMT based on boundary integration is The stresses and displacements can be obtained from the COMSOL calculation.Similarly, the effective moduli are evaluated from the effective stress and strain relations: Though the above equations are presented for calculations along the ГХ direction, the corresponding formula for ГM direction can be similarly transcribed.Due to the obvious link between the bulk (shear) modulus and monopolar (quadrupolar) resonance, it is more convenient to introduce .The effective medium results show that the lower negative band supports a longitudinal (transverse) wave along ГХ (ГM) direction, whereas the upper negative band only allows longitudinal wave in both ГХ and ГM directions.The corresponding results obtained from the EMT are also plotted in Fig. 11(a) by crosses.Excellent agreements between the finite element results and the EMT prediction are found.

Some intriguing properties of elastic metamaterials
Like their EM and acoustic counterparts, elastic metamaterials have shown many intriguing wave transport properties.For example, the total mode conversion and the super-anisotropy are two of them.The total mode conversion can completely convert the incident transverse (longitudinal) wave into a refracted longitudinal (transverse) wave.It is an analogue of the Brewster angle in the EM case (for example, Jackson, 1999), but in a much more stringent and complex manner.It only occurs on the interface between a normal solid and an elastic metamaterial with negative refractive index (Wu et al. 2011).The super-anisotropic behavior has been demonstrated in Section 5.2.Also shown there is the property of sustaining only a longitudinal wave at certain frequencies, which is so-called "fluid-like" solids, blurring the distinction between solids and fluids (Lai et al., 2011).

Conclusion
In this chapter, the effective medium properties of 2D elastic metamaterials have been reviewed.Unlike EM or acoustic metamaterials, the elastic metamaterial is in general anisotropic unless the lattice structure is a hexagon with isotropic scatterers.For the isotropic elastic metamaterial, the EMT may be derived from CPA.For the anisotropic metamaterial, the EMT may be obtained from the MST in conjunction with Christoffel's equation, or from the integration of eigenfields on the boundaries.EMT could greatly facilitate the design of new elastic metamaterials, such as rubber-coated water cylinder embedded in foam which gives rise to large negative bands for shear waves and a multimass locally resonant structure which results in both negative bands for longitudinal waves and super-anisotropic negative bands.
Elastic metamaterial opens a new research area.The experimental realization would be much more challenging and exciting.The generalization of EMT as well as the symmetry property to more complex lattice structures, such as rectangular lattices, would also be of interest as it will introduce even stronger anisotropy.Meanwhile, finding an EMT that can also treat the rotational modes is a challenging task.Such modes are normally excited at lower frequencies and form flat bands in the band structures.
and transverse wave vectors in the matrix, respectively.ω is the angular frequency.
Ka in Eq. (

Figure 2 (
Figure 2(Wu et al., 2007).The coated cylinder consists of the scatterer surrounded by a layer of the matrix.The inner and outer radii, which are denoted by s r and 0 r , respectively, satisfy 22 0 / s rr p = , where p is the filling ratio of the scatterer.The effective parameters e κ , Fig. 3. (a) The equifrequency surface for a hexagonal array.(b) .Ka .as a function of K φ .(Wu& Z.Zhang, 2009)

Figure 4
Figure 4(a) is the same as Figure 3(a), but the rubber cylinders are arranged in a square array.The inner ring represents the quasi-longitudinal branch with distinct anisotropy and the outer one is the quasi-transverse branch with weak anisotropy.The corresponding Ka

Fig. 4 .
Fig.4.The same as Figure3but the lattice is a square lattice.(Wu & Z. Zhang, 2009) transverse waves, respectively, a single negative e ρ in a frequency regime leads to imaginary le c and te c , which implies the existence of a band gap for both longitudinal , which clearly shows in (a), (b) and (c) the dipolar, quadrupolar and monopolar resonances.These resonances are linked to the effective medium parameters in the following manner implied by the EMT.The negative e ρ arises from a dipolar resonance ( resonances give rise to negative e

Fig. 5 .
Fig. 5. Displacement fields for different resonances.The arrows represent the direction and the brightness denotes the amplitude, with white indicates larger.(Wu et al., 2007)

.Fig. 6 .
Fig.6.Band structure and effective medium parameters for an triangular array of hollow rubber cylinders embedded in epoxy.(Wu et al., 2007)

Fig. 7 .
Fig. 7. Effective medium parameters for rubber-coated water cylinders embedded in epoxy in a hexogonal lattice.

Fig. 8 .
Fig. 8. Effective medium parameters for rubber-coated water cylinders embedded in foam in a hexogonal lattice.

Fig. 9 .
Fig. 9. (a) The dispersion along KΓ direction for the same system as Fig.8.(b) Transmission coefficients for longtidinal (solid blue) wave incident and transverse (dashed red) wave incident on a slab of witdth 6a and length 50a.The incident wave is along K Γ direction.(Wu et al., 2011)

Fig. 10 .
Fig. 10.A schematic figure of the physical model and the practical design.(Lai et al., 2011) are separated by a small complete gap (178 Hz~198 Hz).The two eigenstates in the lower and upper negative bands at the Γ point are plotted inFigs.11(b)    www.intechopen.comand 11(c), respectively.The eigenstate in Fig.11(b) is clearly a quadrupolar resonance, whereas the eigenstate in Fig.11(c) is a monopolar resonance.

Fig. 11 .F
Fig. 11.(a) Band structure of the multi-mass metamaterial.(b) and (c) Displacement field of eigenstates.The color represents the amplitude of displacement (blue/red for small/large values) and the arrows show the displacement vectors directly.(d) and (e) effective medium parameters calculated by a boundary EMT.(Lai et al., 2011) bulk modulus and shear modulus.The results for eff κ and eff µ evaluated from the relevant eigenstates are plotted inFigs.11(d)  and 11(e).In the lower negative band, κ eff is positive and finite, while eff µ is nega ti ve an d d iv erge s a t the Г point, which is in accordance with the quadrupolar resonance.In the higher negative band, eff µ is positive and finite, while eff κ is negative and diverges at the Г point, which is induced by the monopolar resonance.Knowing the effective moduli, the corresponding dispersion relations can be calculated by using Christoffel's equation.Along the ГХ direction, compressional wave and shear wave velocities are given by by 11 C ρ and 44 C ρ , respectively; whereas along the ГМ direction the compressional and shear wave velocities are ( )

Table 1 .
Various wave propagation properties.Positive (negative) n indicates positive (negative) propagating bands.l and t represent longitudinal and transverse waves, respectively.