Broadband elastic metamaterial with single negativity by mimicking lattice systems

Journal of the Mechanics and Physics of Solids 74 (2015) 158–174

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Broadband elastic metamaterial with single negativity by mimicking lattice systems Yongquan Liu a,b, Xianyue Su a, C.T. Sun b,n a b

Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China School of Aeronautics and Astronautics, Purdue University, W. Lafayette, IN 47907, USA

a r t i c l e in f o

abstract

Article history: Received 30 April 2014 Received in revised form 23 September 2014 Accepted 24 September 2014 Available online 13 October 2014

The narrow bandwidth is a significant limitation of elastic metamaterials for practical engineering applications. In this paper, a broadband elastic metamaterial with single negativity (negative mass density or Young's modulus) is proposed by mimicking lattice systems. It has two stop bands and the bandwidth of the second one is infinite theoretically. The effect of the relevant parameters on band gaps is discussed. A continuum model is proposed and the selection of materials is discussed in detail. It shows that continuum metamaterials can be described accurately by using the lattice model, and the second stopband can be ultra-broad but not infinite. This discrepancy is investigated and a method is provided to calculate the upper limit of the second stopband for a continuum metamaterial. As a verification, the proposed metamaterial is used for wave mitigation over broadband frequency ranges. Moreover, the present method is extended to design 2D anisotropic elastic metamaterials, and a device to control the direction of elastic wave transmission is proposed as an example. & Elsevier Ltd. All rights reserved.

Keywords: Elastic metamaterial Dispersion curve Band gap Lattice system Wave propagation

1. Introduction Driven by the rapid development of electromagnetic/optic metamaterials with negative permittivity and permeability (Pendry et al., 1996, 1999; Smith et al., 2000; Shalaev, 2007), acoustic/elastic metamaterials have gained much attention in the last decade (Liu et al., 2000; Li and Chan, 2004; Fang et al., 2006; Ding et al., 2007; Huang et al., 2009a; Milton and Willis, 2007; Lee et al., 2010; Lai et al., 2011; Liu et al., 2011; Yang et al., 2013). Analogous to negative permittivity and permeability in electromagnetic metamaterials, the central focus of acoustic/elastic metamaterials is on the negative effective modulus and mass density, which are not found in natural materials. It should be noted that the aforementioned negative modulus and mass density are dynamic effective parameters, which are different from the static negative stiffness discussed by Lakes et al. (2001) and Lakes and Drugan (2002). Structures with a static negative stiffness are usually unstable unless they are coated within a positive-stiffness matrix (Drugan, 2007). The first metamaterial with negative mass was investigated and fabricated by Liu et al. (2000) based on localized dipolar resonances. The acoustic metamaterial with negative bulk modulus was firstly designed by Fang et al. (2006) using an array of Helmholtz resonators. Ding et al. (2007) demonstrated that acoustic metamaterials with double negativity can be achieved by combining structures having negative modulus and mass density independently. Subsequently, structures with double negativity were proposed based on n

Corresponding author. Fax: þ1 765 4940307. E-mail address: [email protected] (C.T. Sun).

http://dx.doi.org/10.1016/j.jmps.2014.09.011 0022-5096/& Elsevier Ltd. All rights reserved.

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different mechanisms (Lee et al., 2010; Lai et al., 2011; Liu et al., 2011). In order to illustrate the physical meaning and mechanisms of negative effective mass and modulus of acoustic/elastic metamaterials, lattice systems with masses and springs were used to construct simplified models (Huang et al., 2009a; Milton and Willis, 2007; Huang and Sun, 2011a). Meanwhile, many novel applications were proposed in accordance with the unusual properties of acoustic/elastic metamaterials, such as invisibility cloak (Zhang et al., 2011), superlens (Ambati et al., 2007; Li et al., 2009; Zhou and Hu, 2011), negative refraction (Liu et al., 2011; Wu et al., 2011) and vibration attenuation devices (Yang et al., 2010; Yao et al., 2010; Tan et al., 2012; Zhu et al., 2014; Huang and Sun, 2009b). However, for most acoustic/elastic metamaterials mentioned above, the negative effective mass and modulus only exhibit over a narrow frequency region. Thus the design of structures with broadband negative parameters is an important issue in the field of acoustic/elastic metamaterials. Thin membrane-type structures (Lee et al., 2010; Yang et al., 2010, 2013) were proposed as broadband metamaterials to control acoustic waves. Yao et al. (2010) showed that the effective mass can be negative under a specific frequency by fixing internal resonators. Recently, multiple local resonators (Tan et al., 2012; Zhu et al., 2014) and graded resonators (Baravelli and Ruzzene, 2013) have also been used to extend the width of bandgaps to some extent. It is worth mentioning that waves cannot propagate in lattice systems, such as monatomic and diatomic chains, above a certain frequency, which corresponds to an ultra-broad forbidden band. Therefore, ultra-broadband negative modulus or mass density may be obtained for structures with the feature of lattice systems. In light of this, a model of broadband elastic metamaterial with single negativity is investigated here by mimicking lattice systems. The effective parameters obtained by using a two-step homogenization method fit the dispersion relations perfectly. It shows that this discrete model has two stopbands, and the bandwidth of the second one is infinite in theory. Then the way to design continuum metamaterials is discussed in detail. The second forbidden band of the continuum model, although not infinite, is quite broad. As a demonstration, the proposed metamaterial is used for wave mitigation over broadband ranges. Finally, a device of controlling the direction of elastic wave transmission is presented as an application of 2D anisotropic elastic metamaterials.

2. Lattice model 2.1. Negative effective mass and modulus Consider an infinite mass-spring system as shown in Fig. 1(a), which was proposed by Huang et al. (2009a) as a onedimensional model of elastic metamaterials with negative mass density. Masses m1 are connected periodically at a spacing of L by springs with stiffness K. There is a mass m2 connected by the spring k in each m1. The dispersion equation of this system can be derived as

m1m2 ω4 − [(m1 + m2 ) k + 2m2 K (1 − cos (qL))] ω2 + 2Kk (1 − cos (qL)) = 0

(1)

where ω is angular frequency and q is wave number. As an example, the dispersion curve is plotted in Fig. 2(a) in the case of θ = m2 /m1=2 and δ = k /K = 1. There are two bandgaps of this structure in the range of 0.885 < ω/ω0 < 1.732 and 3.196 < ω/ω0 < + ∞ respectively, where ω0 = k /m2 is the local resonance frequency of m2.

Fig. 1. Infinite mass-in-mass lattice system and its effective models.

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Fig. 2. Comparison of the results in the case of θ = 2 and δ = 1. (a) Dispersion curve; (b) effective parameters obtained by Eqs. (2) and (4); and (c) effective masses obtained by Eq. (5). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The effective parameters of this mass-in-mass system can be obtained by using a two-step homogenization method. Firstly, it is regarded as a monatomic lattice system with effective mass meff , as illustrated in Fig. 1(b), then the effective mass meff can be easily obtained as

meff = m1 +

m2 k k − m2 ω2

(2)

The negative effective mass of this mass-in-mass system has been analyzed by Huang et al. (2009a; 2009b) in detail. According to Eq. (2), the effective mass becomes negative in a narrow frequency range where 1 < ω/ω0 < 1 + m2 /m1 , as shown in Fig. 2(b) (the blue solid line) when θ = 2 andδ = 1. The lattice system shown in Fig. 1(b) can be regarded as an equivalent homogeneous material as shown in Fig. 1(c). The unit cell can be set as shown in Fig. 3, a springK connects two masses 1/2meff . By applying a symmetric pair of harmonic force ¯ iωt , the kinematic equation of the system is F = Fe

1 meff ω2X = F¯ − 2KX 2

(3)

where X is the amplitude of each mass. If the unit cell is treated as a homogeneous material with effective mass meff and modulus Eeff = k eff L , we have

Eeff E0

=

¯ /2X FL 1 = 1 − meff ω2/K KL 4

(4)

where E0 = KL . The last equation indicates that the effective modulus becomes negative above a certain frequency 2 K /meff . Fig. 2(b) shows the effective parameters based on Eqs. (2) and (4). Either effective modulus or mass is negative in the two stopbands exactly, which verifies the validity of the equivalent continuum model. The effective parameters cannot be negative simultaneously. Therefore, it is a discrete model for broadband elastic metamaterials with single negativity.

Fig. 3. Unit cell of the system of Fig. 1(b).

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Fig. 4. Three zero points of Eq. (6) vs θ .

Alternatively, the unit cell can be regarded as a homogeneous material with mass Meff and constant modulus E0 . By setting Eeff /meff = E0/Meff , the effective mass becomes

Meff =

Kmeff K − meff ω2/4

=

4K ⎡⎣m1 (ω 02 − ω2) + m2 ω 02 ⎤⎦ 4k1 (ω 02 − ω2) − ω2 ⎡⎣m1 (ω 02 − ω2) + m2 ω 02 ⎤⎦

(5)

As illustrated in Fig. 2(c), the negative effective mass corresponds to the stopbands perfectly, too. Moreover, the latter equivalent method is more convenient for further analysis in virtue of one effective parameter only. At the end of this section, it is important to stress that the homogenization method works when the wave length is larger than the size of the unit cell. Thus an upper limit should exist if elastic metamaterials are designed based on this method. This issue will be analyzed in detail later.

2.2. Effects of relevant parameters on band gaps As shown in Fig. 3(c), the two band gaps of this lattice system are bounded by three frequencies that make the effective mass density (given by Eq. (5)) to become zero. The first band gap lies within the first two frequencies, while the second band gap is above the third frequency. Thus, the effect of relevant parameters on band gaps can be carried out based on the three zero points. There are two dimensionless parameters in this lattice system, namely, the ratio of mass θ and the ratio of stiffness δ . Firstly, the values of the three zero points when δ = 1 are calculated by varying θ from 0.1 to 10, as illustrated in Fig. 4. With the increase of θ , the start frequency of the first band rises slightly to a constant value smaller than 1, and the bandwidth is enlarged. For the second stop band, a higher value of θ leads to a higher start frequency, as shown by the red solid line in Fig. 4. The variation of the three zero points changing with δ from 0.1 to 10 are plotted in Fig. 5, for θ = 2. It is noteworthy that the second zero point remains unchanged with the value of 1 + θ . A higher value of δ leads to a relatively wider band gap. In general, low frequencies and broad bands are the goal pursued by researchers in the field of elastic metamaterials. Therefore, a low value of θ and a high δ are preferred when designing metamaterials. It also deserves to note that attention has been previously focused on the case of δ ≫ 1(K ≫ k). In this case, the start point of the first bandgap equals 1 and the second bandgap begins at an extremely high value.

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Fig. 5. Three zero points of Eq. (6) vs δ .

3. Continuum model 3.1. Design of a continuum model A continuum model is proposed as an example by mimicking the lattice system discussed above. The geometrical parameters are illustrated in Fig. 6. Two materials are needed to act as “masses” (the blue parts in the figure) and “springs” (the yellow parts in the figure), respectively. Once the geometry of the continuum model is chosen, the selection of materials becomes an important question. Two principles are followed to select the materials: 1) The materials acting as “masses” should have high mass densities and Young's moduli, while the materials acting as “springs” should be much lighter and softer than “masses”. 2) To mimic the lattice system, the wavelength in “mass” materials at working frequencies should be much longer than the size L of the unit cell. Some common materials can be found based on Principle 1. In general, most metals are suitable “mass materials”, while suitable “spring materials” comprise various polymers and foams. The rough values of m1, m2 , K and k can be obtained readily if any two materials are assumed:

mi = ρm Vi, K = E s A K /l K , k = E s A k /lk

(6)

where ρm , Es, Vi, A K (Ak ) and l K (lk ) represent the density of “mass” materials, the Young's modulus of “spring” materials, the volume of the ith mass (i¼1.2), the cross-section of the spring K (k), and the length of the spring K (k), respectively. Then based on the theory mentioned above, the dispersion curves and effective parameters changing with frequency can be achieved. We set the highest frequency of passing bands (i.e., the third zero point of Eq. (5)) as the reference frequency fre . The ratio of the wavelength in mass materials at this frequency to the size of unit cell R = λ re /L can be calculated, as listed in

Fig. 6. The continuum model.

Table 1 The ratio of R¼ λre/L for different combinations of materials. Springs Cork

Masses Al2O3 Aluminum alloy (7075-T6) Brass (70Cu30Zn, annealed) Copper alloys Lead alloys Nickel alloys steel

Epoxy thermoset

Polyamide (nylon)

Polybutadiene elastomer

Polycarbonate Polyethylene (HDPE)

Polypropylene Polyurethane elastomer

Foam Polyvinyl chloride (rigid PVC)

251.8 7.9 107.0 3.4

8.5 3.7

367.0 158.0

9.0 3.9

17.6 7.6

15.5 6.7

93.9 40.6

158.8 67.4

12.2 5.3

144.8 4.5

4.8

207.7

5.1

9.9

8.8

52.8

91.4

6.8

147.5 50.7 170.3 184.0

4.9 1.7 5.7 6.1

211.7 72.6 244.4 264.3

5.2 1.8 6.0 6.5

10.1 3.5 11.7 12.6

8.9 3.1 10.3 11.1

53.9 18.4 62.2 67.3

93.2 32.1 107.6 116.2

7.0 2.4 8.1 8.7

4.6 1.6 5.3 5.7

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163

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Fig. 7. The dispersion curve by theory (Eq. (1)) and transmission coefficients by FEM at R ¼1.6. (a) Dispersion relation; and (b) transmission coefficients.

Table 1. Specific parameters of the materials listed in Table 1 can be found in (Ashby and Jones, 2005). Based on Principle 2, the ratio R should be much greater than 1 to mimic the lattice system. To verify this, different values of R marked in bold in Table 1 are studied. For each case of R, the dispersion curve can be obtained theoretically based on Eq. (1). The finite element method (FEM) is used here to get more accurate values of K and k, see Fig. A1 in Appendix A for details. As comparisons, we select a one dimensional 100 unit-cell chain and study its steady state response by using the FEM. A time harmonic displacement is applied at one end of the system, while the other end of this continuum system is set free. The transmission coefficients at different frequencies can be obtained by measuring the ratio of the output displacements at the chosen locations to the input displacement at the first unit cell. The 15th and the 20th unit cells are evaluated here. In the case of R¼1.6, the two band gaps are (7260 Hz, 13382 Hz) and (15275 Hz, þ1) according to the lattice model. However, the location and width of band gaps calculated from the finite element simulations do not agree with the lattice model totally, as shown in Fig. 7. Several pass bands irregularly distribute at high frequencies. Thus this combination of materials cannot be used. The case of R ¼3.4 is shown in Fig. 8. Two band gaps are seen to locate at (17493 Hz, 29618 Hz) and (34983 Hz, þ1) according to the lattice model. In contrast, based on the finite element simulation, two forbidden bands occur at about (23 KHz, 30 KHz) and (36 KHz, 42 KHz), respectively. At higher frequencies, waves will propagate in the system. Moreover, the vibration isolation effect is not very significant in the two stopbands. Therefore, this combination of materials is not suggested. When R increases to 10.1, the first stopband (7800 Hz, 11200 Hz) calculated by the finite element simulation is slightly different from the one (7144 Hz, 11,081 Hz) obtained by from the lattice model. Despite not being infinite, the second forbidden band (15 KHz, 30 KHz) is quite broad as compared to the first one. Vibration isolation effect is significant in both stopbands as shown in Fig. 9(b). Thus this combination of materials can be used for broadband elastic metamaterials with

Fig. 8. The dispersion curve by theory (Eq. (1)) and transmission coefficients by FEM at R¼ 3.4. (a) Dispersion relation; and (b) transmission coefficients.

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Fig. 9. The dispersion curves by theory (Eq. (1)) and transmission coefficients by FEM at R ¼10.1 and 10.3. (a) Dispersion relation at R ¼10.1; (b) transmission coefficients at R ¼10.1; (c) dispersion relation at R¼ 10.3; and (d) transmission coefficients at R¼ 10.3.

single negativity by mimicking a lattice system. The same conclusion can be drawn for a different collection of materials but with similar ratio R¼10.3 (Fig. 9(c) and (d)). It means that the ratio R can be used as a versatile bench mark for choosing appropriate materials. For higher values of R, the difference between the lattice model and the original continuum model becomes smaller. For example, in the cases of R¼52.8 and R¼116.2, the locations and widths of band gaps from the simulations match the lattice results almost perfectly, as illustrated in Fig. 10. The ending frequency of the second bandgap is extremely high with respect to the reference frequency fre . Therefore, we suggest to choose cases of R ≥ 10 when designing continuum elastic metamaterials with broad band single negativity. 3.2. The upper limit of the second stopband for continuum structures The second stopband extends to infinity theoretically for the lattice model, but pass bands are observed at high frequencies for the continuum model as discussed in the last section. This discrepancy is investigated in this section, and a method is provided to calculate the upper limit of the second stopband for the continuum model. For convenience of analysis, the periodic structures shown in Fig. 6 is simplified into a 2-phase layered material, whose unit cell is shown in Fig. 11(a). Material A represents the mass-in-mass structure, while material B is the soft material connecting the mass-in-mass structures. The exact dispersion relation (Nemat-Nasser and Srivastava, 2011) of this periodic system is

cos (qL) = cos (ωd A/c A ) cos (ωdB /cB ) −

1 + λ2 sin (ωd A/c A ) sin (ωdB /cB ) 2λ

(7)

where L is the length of the unit cell, di , ρi , ci = Ei/ρi andλ = ρ A c A/(ρB cB ) are the length, the mass density, the wave velocity and the ratio of impedance of the ith layer (i¼ A, B), respectively.

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Fig. 10. The dispersion curve by theory (Eq. (1)) and transmission coefficients by FEM at R ¼52.8 and 116.2. (a) Dispersion relation at R ¼52.8; (b) transmission coefficients at R¼ 52.8; (c) dispersion relation at R¼ 116.2; and (d) transmission coefficients at R ¼116.2.

Fig. 11. The unit cell of a 2-phase periodic system and its equivalent mass-spring model.

As an example, Fig. 12 (the black solid line) shows the first four dispersion curves in the case of

ρ A = 8000 kg/m3, E A = 200GPa, d A = 10 mm ρ B = 1000 kg/m3, EB = 2GPa, dB = 5 mm

(8)

According to Eq. (7) and Fig. 12, there are infinite numbers of passing bands for a continuum structure. However, for simplicity, this periodic structure is always treated as a monoatomic lattice system with the unit cell shown in Fig. 1(b), which has the dispersion relation of

m A ω2 − 2k B (1 − cos (qL)) = 0

(9)

where m A = ρ A VA and k B = EB SB /dB . It means that material A is regarded as a rigid mass, while material B is treated as a massless spring. This dispersion relation is also plotted in Fig. 2 (the magenta dots). It can be seen that the mass-spring model can accurately estimate the first dispersion curve. However, at higher frequency regions, the relative motion of particles in each kind of materials cannot be neglected, and the monoatomic lattice model does not work anymore. Thus,

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Fig. 12. The dispersion curve of a 2-phase periodic system. 40000 35000

Acoustic Optic 1 Optic 2

Frequency (Hz)

30000 25000 20000 15000 10000 5000 0

0.0

0.5

1.0

1.5

qL

2.0

2.5

3.0

Fig. 13. The dispersion curves plotted with Eq. (7) for the case of R ¼10.1.

waves can propagate at high frequencies in the continuum model but the bandwidth of the monoatomic lattice model is infinite. An important issue is to obtain the upper bound of the second stopband for the continuum model. A method is provided based on Eq. (7) by taking the continuum model in Fig. 6 as an example. Firstly, some parameters are set directly as dA ¼30 mm, dB ¼5 mm, ρ A = meff /VA = meff /(26 × 30 × 10−6) and ρB = m B /VB = ρ spring /1.3. Secondly, stiffnesses kA and kB of this continuum model are calculated readily by the FEM, as depicted schematically in Fig. A1 in Appendix A. Subsequently, the equivalent Young's moduli are obtained as E A = k A d A/S = 15k A/13 and EB = k B d B /S = 5k A/26, S is the cross-section of the elastic solid. Substituting these parameters into Eq. (7), the dispersion relation is finally obtained. Fig. 13 shows the first three dispersion curves for the case of R ¼10.1. The first two pass bands correspond two those plotted in Fig. 9(a) despite small discrepancies, and the third pass band lies in the range (33.4 KHz, 39.4 KHz). Thus, the upper bound of the second stopband is 33.4 KHz. As mentioned above, a more accurate value is 30 KHz based on FEM. In spite of some errors, it is a simple method to estimate the limit of our model to design continuum materials or structures. In this way, the upper limits in the cases of R¼10.3, 52.8 and 116.2 are calculated to be 37.5 KHz, 25.0 KHz and 31.5 KHz, respectively. To be on the safe side, the second stop band should not be utilized above about 80% of the upper limit frequencies. 3.3. A specific example The case of R¼116.2 is further studied as a specific example. In this case, steel and foam are selected as mass and spring materials, respectively. The material parameters are density ρ ¼ 7850 kg/m3, Young's modulus E ¼210 GPa, and Poisson's ratio 0.29 for steel; density ρ ¼115 kg/m3, Young's modulus E ¼8 MPa, and Poisson's ratio 0.33 for foam (Wu et al., 2011). A free vibration modal analysis is investigated by the FEM to extract the natural frequencies and wavelengths of the 100 unitcell chain. The extracted wavelengths are subsequently used to calculate the wavenumbers and, thus, the dispersion curves are obtained as shown in Fig. 10(c). It indicates that the dispersion curves obtained by theory match accurately with those obtained by using the finite element method. There are two stop bands, (736 Hz, 1117 Hz) and (1488 Hz, 25200 Hz) in view of the upper limit calculated in Section 3.2.

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Fig. 14. Dynamic responses of the 15th unit cell at different frequencies. (a) 600 Hz; (b) 900 Hz; (c) 1300 Hz; (d) 4000 Hz; (e) 25200 Hz; and (f) 900 Hz þ 2000 Hz þ 4000 Hz.

Vibration isolation is thought as one of the most prominent applications of elastic metamaterials with single negativity. To test this, the transient response of the continuum model of this 100 unit-cell metamaterial is investigated. At one end of the system, the excitation is generated by a prescribed displacement of

U (0, t) = 0.001( sin ωt) H (t)

(10)

where

⎧1, t ≥ 0 H (t) = ⎨ ⎩ 0, t < 0

(11)

is the unit-step function. The other end of this continuum model is set free. Fig. 14 shows the displacement response of the 15th unit cell at different frequencies. For frequencies in pass bands (at 600 Hz in Fig. 14(a) and at 1300 Hz in Fig. 14(c)), the amplitude of the output displacement is as high as the input one. While for frequencies in stop bands (at 900 Hz in Fig. 14 (b), at 4000 Hz in Fig. 14(d) and at 25200 Hz in Fig. 14(e)), waves attenuate greatly as expected. Moreover, this attenuation still holds (see Fig. 14(f)) when a multi-frequency excitation given below is applied.

U (0, t) = 0.001( sin ω1t + sin ω2 t + sin ω 3 t) H (t) where ω1 = 900Hz , ω2 = 2000Hz and ω3 = 4000Hz .

(12)

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Fig. 15. Two-dimensional mass-in-mass lattice system and its effective models.

4. Two-dimensional (2D) metamaterials 4.1. Lattice model and its effective parameters The 2D counterpart of the mass-in-mass lattice model and its relative parameters are shown in Fig. 15(a). It extends the model proposed by Huang and Sun (2011b) to describe the cases of anisotropic host media and unequal periodicity in two main directions. In each unit cell, the internal mass m2 is connected to m1 by two different springs k1 and k2. Meanwhile, masses m1 are connected periodically at a spacing of L1  L2 by springs (K1, G1) and (K2, G2), where Kα and Gα (α ¼1, 2) are the stiffness along and perpendicular to external springs respectively. Then the equations of motion for unit cell (m, n) are

m1

m1

m2

(m , n) ∂ 2u11

∂t 2 (m , n) ∂ 2u12

∂t 2 , n) ∂ 2u2(m α

∂t 2

(

)

(

)

(

)

(13)

(

)

(

)

(

)

(14)

(m + 1, n) (m − 1, n) (m , n) (m , n + 1) (m , n − 1) (m , n) (m , n) (m , n) = K1 u11 + u11 − 2u11 + G2 u11 + u11 − 2u11 + k1 u21 − u11

(m , n + 1) (m , n − 1) (m , n) (m + 1, n) (m − 1, n) (m , n) (m , n) (m , n) = K2 u12 + u12 − 2u12 + G1 u12 + u12 − 2u12 + k2 u22 − u12

(

, n) = k2α u1(αm, n) − u2(m α

)

(α = 1, 2)

(15)

For a plane harmonic wave, the solution can be expressed as

uαβ = Uαβ exp ⎡⎣i (q1x1 + q2 x2 − ωt) ⎤⎦

(α, β = 1, 2)

(16)

Substituting Eqs. (15) and (16) in Eqs. (13) and (14), it can be obtained that

⎡ ⎛ m2 k1 ⎞ 2⎤ ⎢2 (1 − cos q1L1) K1 + 2 (1 − cos q2 L 2 ) G2 − ⎜⎜m1 + ⎟⎟ ω ⎥ ⋅ ⎢⎣ k1 − m2 ω2 ⎠ ⎥⎦ ⎝ ⎡ ⎛ m2 k2 ⎞ 2⎤ ⎢2 (1 − cos q1L1) G1 + 2 (1 − cos q2 L 2 ) K2 − ⎜⎜m1 + ⎟⎟ ω ⎥ = 0 ⎢⎣ k2 − m2 ω2 ⎠ ⎥⎦ ⎝

(17)

If the mass-in-mass lattice structure is considered as a conventional lattice model, as illustrated in Fig. 15(b), then the effective masses in the principle directions are

meff , α = m1 +

m2 k α k α − m2 ω2

(α = 1, 2)

(18)

The lattice system can be further regarded as a homogeneous material, as illustrated in Fig. 15(c). By applying symmetric and anti-symmetric forces on the unit cell in the horizontal direction respectively, as shown in Fig. 16, the effective stiffness

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Fig. 16. The strategy to calculate the effective stiffness (a) Keff ,1 and (b) Geff ,2 .

Fig. 17. The schematic diagram of a device of controlling the direction of elastic wave transmission by using 2D elastic metamaterials with single negativity.

Keff ,1 and Geff ,2 can be calculated as

Keff ,1 = K1 − meff ,1ω2/4,

G eff ,2 = G2 − meff ,1ω2/4

(19)

Similarly, the effective stiffness Keff ,2 and G eff ,1 can be calculated by loading symmetric and anti-symmetric forces on the unit cell in the vertical direction as

Keff ,2 = K2 − meff ,2 ω2/4, G eff ,1 = G1 − meff ,2 ω2/4

(20)

Just like the one dimensional case, either effective negative modulus or mass corresponds to the two stopbands in two principal directions. It should be highlighted that the proposed homogenization method is only one possible choice, one can deal with this structure by using other homogenization methods. However, the present approach shows a significant advantage, because it can determine the critical frequencies of the two stopbands exactly via two simple steps. Moreover, the effective parameters show simple and explicit expressions.

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Fig. 18. The dispersion curves of the 2D structure shown in Fig. 17.

4.2. An application of 2D metamaterials Two-dimensional elastic metamaterials with single negativity can also be designed by mimicking the 2D lattice system. As an example, a device of controlling the direction of elastic wave transmission is proposed. This device serves a function similar to the structure for sound waves proposed by Fleury et al. (2014). Fig. 17 shows the schematic diagram of this device. The unit cell of the rectangular-array metamaterial is zoomed with marked exact dimensions. This metamaterial is anisotropic by setting spring materials only in the external mass in the y direction. Steel and foam are also set as mass and spring materials, respectively. A longitudinal wave is sent from the upper-left side of the background medium which is an isotropic elastic solid. The other outer boundaries are free boundaries. The dispersion curves for the 2D metamaterial shown in Fig. 18 are obtained using the equivalent 2D lattice model (the solid lines) for which the effective spring constants are calculated based on static finite element analyses of the continuum model as depicted schematically in Fig. A2 in Appendix A. The result shows that the bandgaps of longitudinal waves propagating in the x-direction are (986 Hz, 1675 Hz) and (1882 Hz, þ1), whereas only one bandgap (1415 Hz, 1675 Hz) exists in the y-direction. The bandgaps of shear waves are also obtained as (860 Hz, 1675 Hz) and (1806 Hz, þ1) in the xdirection and (1420 Hz, 1675 Hz) in the y-direction. The dispersion curve is also plotted by using the FEM for comparison, as shown in Fig. 18 (magenta points). At low frequencies, the band structure predicted by the theory is shown to agree remarkably well with that computed by the FEM. An additional mode can be found at 1860 Hz due to the rotation of the inner resonator, which is shown in the insert of Fig. 18. Fortunately, this mode does not affect the bandgaps. By measuring the ratio of the displacement of the 15th unit-cell to that of the first unit cell, transmission coefficients in the horizontal and vertical directions as shown in Fig. 19 are also obtained from the continuum model using the FEM. The bandgaps from the dispersion curves match the transmission coefficients accurately indicating the validity of the equivalent 2D lattice model as well. Fig. 20 presents the simulation result of wave transmission through the metamaterial with the setup shown in Fig. 17. The continuum model is employed for the simulation. For frequencies in the passband of the x- and y-directions simultaneously, such as 800 Hz, waves can propagate along both directions, as shown in Fig. 20(a). At the frequency of 1500 Hz, which is in the stopbands of both directions, waves are totally stopped as illustrated in Fig. 20(c). When the incident wave propagates at 1200 Hz (Fig. 20(b)) or 5000 Hz (Fig. 20(d)), it can only transmit in the vertical direction because both frequencies fall in the stopbands for wave propagation in the horizontal direction.

5. Conclusion In this paper, elastic metamaterials consisting of continuum solids are modeled with lattice systems. The effective parameters of the equivalent lattice system are obtained by using a two-step homogenization method and they yield the dispersion curves that match the first two stopbands obtained based on the continuum model if the wave length is much greater than the unit cell size. It is also found that the second stopband according to the lattice model can be ultra-broad but not infinite as indicated by the continuum model. A method is provided to estimate the applicable frequency range for the

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Fig. 19. The dispersion curve by theory and transmission coefficients by FEM (a) in the horizontal direction and (b) in the vertical direction.

Fig. 20. Total displacement distribution at different frequencies. (a) 800 Hz; (b) 1200 Hz; (c) 1500 Hz; and (d) 5000 Hz.

second stopband derived from the lattice model. Finally, the present method is extended to the 2D case, and a metamaterial that is capable of controlling the direction of elastic wave transmission is presented as an application of 2D anisotropic elastic metamaterials. Acknowledgement This work is supported by the National Natural Science Foundation of China under Grant No. 11232001. Y.L. is grateful for the support from China Scholarship Council. C.T.S. wants to thank the support by an Air Force Office of Scientific Research (AFOSR) Grant # FA9550-10-1-0061.

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Fig. A1. The finite element model and schematic diagrams to calculate the stiffness of the continuum model shown in Fig. 6 by using the FEM.

Fig. A2. The finite element model and schematic diagrams of calculating the stiffness of the 2D structure plotted in Fig. 17.

Appendix A. Parameter settings and schematic diagrams of numerical calculations Throughout the paper, the commercial software package COMSOL Multiphysics 4.3b is utilized to numerically solve the problems. Free triangular elements with the quadratic-Lagrange type shape function are used (in COMSOL nomenclature: maximum element size ¼ 8e  4, minimum element size ¼6e  7, maximum element growth rate¼1.1, resolution of curvature¼0.2 and resolution of narrow regions¼1), as shown in Figs. A1(a) and A2(a). The materials are linearly elastic. The direct solver MUMPS with a convergence tolerance of 10  2 is used. It indicates that neither a nonlinearity of the structure nor of the material is taken into account. The implicit solver with the generalized-α time-stepping method is used, where the spectral radius at infinity ρ∞ = 0.75. For the continuum model shown in Fig. 6, we regard it as a plane strain problem. Forces are applied as depicted schematically in Fig. A1(b)–(d). The mean values of relative edges (black dashed lines in Fig. A1) are used to compute the relative displacements from which the stiffness can be readily obtained. The continuum structure shown in Fig. 17 is dealt with as plane stress problems. By applying forces as depicted schematically in Fig. A2(b)–(f), the stiffness can be readily calculated. The mean values of relative edges (black dashed lines in Fig. A2) are also used to compute the relative displacements. It should be noted that the zero-normal-displacement boundary condition is used, as shown in Fig. A2(c)–(f), because the Poisson's ratios in the two main directions are zero in our lattice model.

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