A locally resonant elastic metamaterial based on coupled vibration of internal liquid and coating layer

Journal Pre-proof A locally resonant elastic metamaterial based on coupled vibration of internal liquid and coating layer Lei Wu, Qian Geng, Yue-ming Li PII:

S0022-460X(19)30665-0

DOI:

https://doi.org/10.1016/j.jsv.2019.115102

Reference:

YJSVI 115102

To appear in:

Journal of Sound and Vibration

Received Date: 17 March 2019 Revised Date:

17 November 2019

Accepted Date: 19 November 2019

Please cite this article as: L. Wu, Q. Geng, Y.-m. Li, A locally resonant elastic metamaterial based on coupled vibration of internal liquid and coating layer, Journal of Sound and Vibration (2019), doi: https:// doi.org/10.1016/j.jsv.2019.115102. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

A locally resonant elastic metamaterial based on coupled vibration of internal liquid and coating layer Lei Wu a, Qian Geng a and Yue-ming Li a,* a

State Key Laboratory for Strength and Vibration of Mechanical Structures, Shaanxi Key Laboratory of Environment and Control for Flight Vehicle, School of Aerospace Engineering, Xi’an Jiaotong University, 710049, China

*

Corresponding author: Yue-ming Li, State Key Laboratory for Strength and Vibration of Mechanical Structures, Shaanxi Key Laboratory of Environment and Control for Flight Vehicle, School of Aerospace Engineering, Xi’an Jiaotong University, No. 28 Xianning West Road, Xi’an, 710049, China. Tel: +86 29 82668340 Email: [email protected]

ABSTRACT In present work, a locally resonant elastic metamaterial (LREM) based on liquid solid interaction (LSI) is proposed, which can attenuate flexural wave in broad low frequency range. By using internal liquid as scattering core and thin layers as coatings, inertial and elastic components of the LREM are connected by LSI rather than cohesive material. This characteristic enables the LREM to be modified easily. A semi-analytical model of the LREM’s unit cell is developed for studying its dynamic effective mass (DEM). Finite Element method is applied to calculate the band structure, DEM and transmission. A 3D-printed metastructure containing five unit cells is tested experimentally. Good agreements among theoretical, numerical and experimental results proved the LREM’s capability to block vibration in broad low frequency regime. Parameter analysis has been conducted as well. Polar and zero points of the DEM would shift to lower frequency region when liquid’s density increases and the normalized bandwidth would be broadened. A similar trend could be observed when the thickness of coatings reduces however the bandwidth would almost remain unchanged. Moreover, stronger initial tension stress would increase bending stiffness of coatings and the local resonance frequency. Furthermore, the internal liquid could hardly contribute to stiffness of the LREM nor could thin layers do on the inertial component. Those traits indicate that the proposed LREM could be quite applicable for vibration controlling in broad low frequency range. Keywords: Elastic metamaterial; Liquid-solid interaction; Low-frequency bandgap; 3D printing 1

1. Introduction Elastic metamaterial (EM) or Acoustic metamaterial (AM) is a kind of composed or structured functional material which performs some exotic characteristics, such as vibration isolation and sound absorption. It can be used to reduce detrimental noise and vibration [1, 2]. The mechanisms of vibration isolation include Bragg scattering and local resonance [3, 4]. Compared to EM based on Bragg scattering mechanism, locally resonant elastic metamaterial (LREM) is more applicable for wave manipulation in low frequency range when small geometric scale is requested [2]. However, the forbidden band (bandgap) of LREM is relatively narrow and this could one restriction for its application [5, 6]. Therefore, achieving broadband low-frequency vibration blocking by designing extraordinary metamaterial with alterable bandgap has become one focused topic among related researches recently [7-10]. Altering the bandgap range of EM by modifying stiffness of the micro-structure of metamaterial could be an effective strategy, e.g. applying thermal effects [11-15], using shape memory material [16, 17] and designing bio-inspired or deployable metamaterial [18, 19]. Beside those methods, applying initial load to generate initial stress and deformation is another practicable approach. A new class of LREM composed of elastic beams and metallic cores was come up by Wang et al. [20, 21], and the bandgap could be modified by imposing nonlinear pre-deformation. Yang et al. [22] presented a double-negative metamaterial made up by two coupled membranes, and the system’s monopolar and dipolar resonances could be changed separately through the initial stress, in addition, a similar model consisting of membranes was studied by Marinova et al. [23]. Shan et al. [24] studied a periodic structure with multiple pattern transformation induced by buckling, and the alterable band structure of EM was implemented by changing the direction of load. Goldsberry and Haberman researched a kind of anisotropic honeycombs-like EM whose dispersion could be altered with applied pre-deformation [25]. A lightweight metastructure was researched by Wang et al. [26], and modifying the bandgap range was achieved by harnessing 2

geometrically nonlinear deformation. Furthermore, introducing fluid-structure interaction to design a metamaterial with reconfigurable property to broaden its functional range has raised some focuses recently. A class of active acoustic metamaterial (AAM) with tunable effective density was proposed by Baz et al. [27, 28], and the characteristics were achieved by using an array of fluid cavities separated by piezoelectric diaphragms, moreover, a kind of AAM with programmed bulk modulus was also studied by Baz et al. [29]. Jin et al. [30, 31] presented a class of EM consisting of hollow pillars with fluid filling, and it could be modified through changing the height of filling fluid. A similar model was experimentally researched by Wang et al. [32], and it could be applied for waveguide and filtering as well [33]. Zhang et al. [34] came up a kind of EM composed of solid and liquid, and the bandgap of EM could be switched through utilizing embedded pumps to change the distribution of liquid. Moreover, the effective density was studied theoretically with a representative Lorentz model applied, and the transmission of a finite-period metastructure was researched numerically and experimentally. In a word, modifying the bandgap of EM through changing stiffness of micro-structure of metamaterial has been researched deeply in recent years, while studies on liquid-solid interacted EM with alterable bandgap have merely gained limited attention. Additionally, in most of reported papers working on liquid-solid interacted EM [30-34], the liquid just behaved like added mass and the hydro-elastic effects have not been further discussed. In this work, by utilizing the hydro-elastic vibration, we come up with a liquid-solid LREM whose inertial and elastic components can be modified respectively. The unit cell of proposed LREM is shown in Figure 1, where l is the scale of cubic internal liquid and hc is the thickness of coating layers. The function of broadband vibration attenuation will be achieved by altering the density of the internal liquid (e.g. by adding solutes into liquid) or the thickness of coatings. A unit cell of typical LREM consists of matrix, coating and core. In terms of present model, the two thin-walled layers will play the role as coatings, and internal liquid will work as the 3

core. This configuration will isolate the flexural wave in special frequency range due to localized hydro-elastic resonance of internal liquid and coating layers. (No color printing required for the figures)

Figure 1. Configuration of the unit cell of present LREM

2. Theoretical formulation of unit cell In this section, based on Assumed Modes Method, a semi-analytical model of proposed LREM is formulated to calculate the transverse dynamic effective mass (DEM) of unit cell. The range of negative DEM would be regarded as the bandgap range caused by local resonance [35]. Besides, all mathematical formulations in this paper are based on the coordinate shown in Figure 1. To calculate transverse DEM of the proposed EM, the unit cell is analogized to a Lorentz model [35]. Its schematic diagram is given in Figure 2. In calculation, a harmonic transverse excitation is applied on boundaries. The DEM could be acquired through the analysis of the cell’s steady harmonic response.

4

Figure 2. Illustration of the simplified Lorentz model for calculating DEM

With prescribed boundary condition, as for solid region, the transverse vibration would be most prominent, therefore only transverse movement is considered in calculation, and the transverse displacement (wc) of the coatings’ middle surface could be expressed as:

wc = ∑ Amn fmn ( x, y)

(1)

where Amn is the coefficient of generalized freedom which is dependent on time, and fmn is the shape function. Besides, the deformation-strain relationship of the coating layers is based on Kirchhoff thin plate theory, which could be formulated as:

∂2wc ε x = −z 2 ∂x

∂2wc ε y = −z 2 ∂y

∂2wc γ xy = −2z ∂x∂y

(2)

Considering the material of coating layers to be isotropic and linear, then the plane-stress condition of layers could be given by:

∂2wc  ∂2wc  Ez  ∂2wc Ez  ∂2wc Ez  ∂2wc  σx = − + µ 2  σy = − + µ 2  τ xy = −     1− µ2  ∂x2 ∂y  1− µ 2  ∂y2 ∂x  1+ µ  ∂x∂y 

(3)

where E and µ are the Young’s modulus and Poisson ratio of coating layers respectively. In terms of internal liquid, it is assumed to be irrotational, incompressible and inviscid [36-38]. Effects of internal liquid’s viscosity are not involved because it is not the focus of present work. After that, the movement of internal liquid could be

5

regarded as potential flow and the liquid’s deformation potential

φ l should follow

Laplace function, which could be expressed as:

∇2φl = 0

(4)

Then the displacement (Ul) of internal liquid could be given by:

Ul = ∇φl

(5)

On liquid-solid coupling surfaces, non-penetrating condition should be satisfied, which should be expressed as: ∂φl l   = wc  z = ±  2 ∂z 

∂ φl =0 ( x = 0, l ) ∂x

∂ φl =0 ∂y

( y = 0, l )

(6)

After that, the dynamic behavior of thin-walled coatings and internal liquid has been described. The elasticity of matrix (white region in Figure 2) has not been considered, since the matrix’s bending deformation is insignificant in low frequency condition.

.

To be more specific, in this work, the bandgap which is caused by 1st locally resonant mode (shown in Figure 2) is studied. The wet modal shapes of coating layer are assumed to be as the same as the dry modal shapes and the feasibility has been proved by [39]. As for the unit cell, the thickness of four side walls is 2.0 mm (shown in Figure 1), which is much larger than the value of coatings, so the coatings’ bending stiffness is obviously smaller than the plates’ connecting stiffness. Therefore, the relative translation and rotation movement on coatings’ connecting boundaries are neglected. Though this assumption would make the system own a higher stiffness, it is relatively suitable for the formulation of the vibration in concerned low frequency range. After that, based on the coordinate shown in Figure 1, as for the 1st local resonance mode, the coatings’ movement could be expressed as [40]: 2 2   π x  −1 x x    π y  −1 y y wc = A11  − sin  + − − sin + − π π     l l2  l l2  l   l  

  + A00 

(7)

where A11 describes the bending deformation of the coatings and A00 refers to the rigid-body movement of the matrix in low frequency condition. Furthermore, under uniform external load, the coupled cell would exhibit a symmetric movement with respect to x-o-y plane. Then the kinetic energy of coating layers can be given by: 6

Tc = 2∫∫ S

ρc hc 2

( w&c )

2

dxdy

(8)

ρc and hc are the density and thickness of coating layers respectively, and S

where

refers to the region x and y ∈ [ 0, l / 2 ] . Moreover, after integrating Eq. (2) and (3), the potential energy of coating layers can be expressed as:

Vc = 2∫∫∫ Ω1

1 (σ xε x +σ yε y +τ xyγ xy )dxdydz 2

 ∂2 w ∂2 w 2 D  = 2∫∫ ( ∇2wc ) − 2 (1− µ )  2c 2c 2 ∂x ∂y S  

2

 ∂2 wc  −   ∂x∂y 

   dxdy  

(9)

Ehc3 D= 12 (1 − µ 2 )

where Ω1 is the region of the upper coating (the upper yellow part in Figure 1). In terms of the deformation potential of internal liquid, it can be expressed as the following formulation when the 1st locally resonant mode is dominant, and the non-penetrating requirements (mentioned in Eq. (6)) on the side walls have been satisfied.

φl =

 mπ x   nπ y   απ z  Bmn cos   cos   sinh   + β z + A00 z  l   l   l  n=0, m=0,m+n≠0 N ,M

∑

(α =

m2 + n2

)

(10)

where Bmn and β refer to the unknown coefficients of liquid’s deformation potential. Worth to be noted is that, the value of

φ l is anti-symmetric with respect to x-o-y

plane. When N equals to M and gains a value greater than 6, the convergence demand has been satisfied. In order to satisfy the interaction condition on the coupling surface, by substituting Eq. (10) into Eq. (6) and expanding the coating’s transverse displacement, the following coefficient relationship can be formulated as:

7

Bmn

a0bn A11l   2 2 2 2  π m + n cosh 0.5π m + n  b0 am A11l   π m 2 + n 2 cosh 0.5π m 2 + n 2 =  am bn A11l   2 2 2 2  π m + n cosh 0.5π m + n  

(

)

(m = 0

(

)

(m ≠ 0

)

(m ≥ 1

(

and

n ≠ 0)

and

n = 0)

(11) n ≥ 1)

and

β = a0b0 A11

(12)

In Eq. (11) and Eq. (12), am, bn, a0 and b0 can be derived by the following expressions:

am =

x =l

2 2   π x  −1 x x   mπ x  π sin cos  − + −     dx 2  ∫ l x=0  l l   l   l  y =l

2 2   π y  −1 y y   nπ y  π bn = ∫  − sin  + − cos    dy 2  l y =0  l l   l   l 

M

a0 = −∑ am m =1 N

(13)

b0 = −∑ bn n =1

Furthermore, internal liquid can only store kinetic energy in described condition, and the kinetic energy can be given as: 2 2 2  ∂φ&l   ∂φ&l   ∂φ&l  ρl & dxdydz Tl = ∫∫∫   +  +  dxdydz = ∫∫∫ ∇ ⋅ φ&l U l 2 Ω2  ∂x   ∂y   ∂z  2 Ω2 & & ρ & ∂φl dxdy & ⋅ ndS = ρl φ& ∂φl dS = ρl = l ∫∫ φ&l U φ l l l 2 τ 2 ∫∫ ∂n 2 ∫∫l ∂z τ

ρl

(

z =±

where

)

(14)

2

ρl is the density of internal liquid, Ω2 is the domain of internal liquid (blue

part in Figure 1) and

τ is the interactive surface which has been noted in Eq. (6).

Moreover, the potential of external force is only related to A00, due to that the excitation is applied on boundaries directly, and the expression could be given as:

Vf = −F sin (ωt ) A00

(15)

In addition, when in-plane initial stress of coatings is considered, the potential energy aroused by the stress should be counted in [41], and the energy should be expressed as:

8

2 2  ∂wc  hc  1  ∂wc  ∂wc ∂wc  dxdy Vs = 2∫∫  σ11  + σ + τ 22  12   ∂x ∂y  2  2  ∂x   ∂y 

where

(16)

σ11, σ 22 and τ 12 refer to initial in-plane stress components of coatings.

Then the energy function of coupled system when 1st locally resonant mode is dominant can be given as:

L =Tl +Tc +Tm −Vc −Vs −Vf

(17)

where Tm is the kinetic energy of matrix which could be expressed as

ρm A&002 ∫∫∫ dxdydz / 2 in which Ω3 is the domain of matrix (light gray part in Figure Ω3

1) and

ρm is the density of it.

After that, the Lagrange equations which can describe the coupled system’s dynamic behavior could be derived as:

&&  k + kσ  m11 m12   A 11 m m   &&  +  0  12 22   A00  

0 0  A11     =   0  A00  F sin (ωt ) 

(18)

The components in Eq. (18) could be obtained from following expressions:

m11 =

∂ 2 (Tl + Tc ) ∂A& 2

m12 =

11

∂V k = 2c ∂A11 2

∂ 2 (Tl + Tc ) ∂A& ∂A& 11

m22 =

00

∂ 2 (Tl + Tc + Tm ) ∂A& 2 00

∂V kσ = 2s ∂A11 2

(19)

Considering the harmonic steady response of the system (the frequency is ω / 2π ), the transverse DEM could be given as:

m eff − z =

m122 ω 2 + m 22 k + kσ − ω 2 m11

(20)

According to Eq. (20), the lower edge frequency of the bandgap as well as the local resonance frequency refers to the DEM’s polar point, and it could be given as:

ωlower =

k + kσ m11

(21)

The upper edge frequency is related to the zero point of the DEM, and it could be given as:

9

ωupper =

( k + kσ ) m22

(22)

m11m22 − m122

Then the normalized bandwidth could be derived as:

BW =

2 (ωupper − ωlower )

ωupper + ωlower

=

200

(

m11m22 − m11m22 − m122 m122

)% 2

(23)

Eventually, the bandgap range of flexural wave is obtained, and the bandgap can be modified through altering the parameters in Eq. (21) and Eq. (22). As for present model, kinetic energy provided by the coating layers is tiny compared with the internal liquid. So the coatings could be regarded as LREM’s elastic component (spring), and internal liquid could be regarded as the inertia component (mass). It means that the elastic and inertial components of the LREM are comprised from solid and liquid materials respectively unlike most of existed metamaterial which is made up by bonded solid substances, and this characteristic could contribute the LREM to be easily-modified. 3. FE discretization formulation In order to confirm the accuracy of present theoretical method, FE method is applied to calculate the band structure, DEM and transmission of proposed LREM. In FE formulation, acoustic elements are used to simulate the liquid media’s movement and the dynamic behavior of the solid-acoustic coupled system can be expressed as:

 2 M s  −ω    −S 

0  K s + M f   0

SΤ   u  f    =   K f   p  0

(24)

where Ms and Ks are the structure’s mass and stiffness matrices respectively.

M f and K f are for the fluid media, S is the coupling matrix, structure’s displacement vector, p

u

is the

is the vector of fluid’s acoustic pressure and

f is the vector of external load applied on solid nodes. The formulations of the matrices mentioned in Eq. (24) are conventional, so the detailed expressions will not be illustrated here. Worth to be noted is that, if initial stresses and strains were 10

considered with nonlinear geometric relationship applied, the stiffness matrix of the structure would be replaced as:

Ks = ( Kc + Kd ) + Kσ

(25)

where Kc is the structure’s conventional stiffness matrix and K d is the stiffness matrix caused by the initial deformation. K σ is the stiffness matrix aroused by initial stresses, and it could be derived as:

Kσ = ∑∫ BΤNLσBNLdV

(26)

V

where BNL is the matrix which describes the relationship between displacements and nonlinear strains, and σ is the initial stress matrix. Once FE discretization in Eq. (24) has been formulated, the eigenvalue analysis and the steady dynamic analysis in frequency domain could be conducted for obtaining the proposed LREM’s band structure, DEM and transmission. 4. Experimental setup In this section, a five-period structure comprised from present LREM is fabricated by 3D printing, and the transmission characteristic is acquired through the dynamic response tests. The metastructure consists of matrix, coatings and internal water. The matrix beam is made up by nylon with 3D print technology being applied, and it is shown in Figure 3 (a). The coating layers are made up by polyvinyl chloride (PVC) films which are attached to the matrix by adhesive. Once PVC films on the one side have sealed the matrix, internal liquid could be added in. After that, the matrix’s open side could be closed with other PVC films being used. Elastic strings can go through small holes on the beam’s free edges, then forming a suspension condition. The shaker (MB Exciter Modal 2) is attached to the matrix’s one free edge. Linear frequency-sweeping signal is from 10.0 Hz to 512.0 Hz with the frequency increment being 0.5 Hz and the time increment being 20 ms. Two acceleration sensors (ENDEVCO 2220E) are used for obtaining the transmission of the metastructure. 11

Furthermore, sensor A which is utilized for catching input signal is bonded on one free edge of the specimen (shown in Figure 3 (d)) and the location of sensor B is variable aiming to acquire the transmissions crossing different distance (different number of cells). The signal acquisition system is LMS Test Lab, and the spectral test module is used in the experiment with the concerned frequency ranging from 0.0 Hz to 512.0 Hz and the resolution being 0.5 Hz. Furthermore, tests are repeated more than five times to validate the accuracy of experimental results which will be depicted in section 5.3.

Figure 3. Experimental setup and the fabrication of the metastructure

5. Results and discussions 5.1 Dynamic effective mass of unit cell In this section, transverse DEM of the LREM is studied theoretically and numerically. In numerical simulation, FE method is applied to obtain transverse DEM by using ABAQUS package. Steady harmonic analysis is conducted, and the unit cell’s boundaries (shown in Figure 4 (a)) are faced with a transverse excitation in displacement form [42, 43]. Concerned frequency range is from 1.0 Hz to 350.0 Hz with the resolution being 1.0 Hz. In order to simulate the liquid-solid interaction behavior, acoustic element (AC3D8) is used in the liquid’s region, and “TIE” constraint (shown in Figure 4 (b)) is created on the interactive surface, moreover, the 12

solid region is formed by C3D8 element. FE formulation for acoustic-solid coupled system has been given in Eq. (24). Once the transverse reaction force on the boundaries being obtained, the DEM can be expressed as [44]:

meff =

Fz −uzω2

(27)

where Fz is total reaction force in z direction, u z refers to displacement’s amplitude and

ω

is angular frequency.

(a) Boundary loads

(b) Tie constraint

Figure 4. FE model for calculating the DEM of unit cell

In calculation, applied mass densities of nylon (matrix) and PVC are acquired from several measurements and the elastic properties (Young’s modulus and Poisson ratio) have been adjusted according to experimental results which will be given in section 5.3. Consequently, material properties used in the calculation have been indicated in Table 1. Table 1. The material properties of nylon and PVC Nylon (matrix) PVC (coating layers)

Young’s modulus (GPa) 3.000 3.500

Poisson ratio 0.300 0.319

Density (kg/m3) 1240.000 1204.800

Besides, liquid’s density and coating layer’s thickness are variable. The bulk modulus of internal liquid is set to be 2.18 GPa in the numerical simulation. Corresponding results of DEM are illustrated in Figure 5, good agreements between theoretical calculation and numerical simulation have been achieved with the relative deviation being smaller than 5.0%. It could be found that the DEM will show a negative value 13

in the frequency range from 146.0 Hz to 212.0 Hz as internal liquid’s density is 1.0 g/cm3 and coating layer owns a 0.34 mm thickness. Figure 6 shows theoretically obtained movement of internal liquid in described condition with external total force being 1.0 N. In addition, orientation of arrow indicates the direction of velocity and length refers to the amplitude. Once the frequency of load is far below local resonance value ( ωlower / 2π ), internal liquid would show a polarized vibration with the velocity in z direction being dominant and the other components being insignificant (Figure 6 (a)). As the frequency approaches above ωlower / 2π slightly, the vibration of internal liquid would exhibit a phase-reverse behavior (Figure 6 (b) and (c)), besides that, the movement of liquid in x and y directions has become stronger compared to the former one. In order to illustrate the movement of internal liquid more intuitively, Figure 7 gives the cross-section contours of z velocity when excitation frequency is 150.0 Hz. Furthermore, z velocity distribution along intersection line (x=l/2 and y=l/2) is extracted to discuss, and the results are depicted in Figure 8. It is evident that, along the intersection line, the liquid close to two coating layers would own a greater velocity. Once the frequency exceeds ωlower / 2π , anti-resonance takes place and the strong vibration of internal liquid close to coating layers would contribute to remarkable bending deformation of layers which could generate shear force to mitigate the transverse vibration of matrix.

Figure 5. The variation of the transverse DEM of unit cell when liquid changes (hc=0.34mm)

14

Figure 6. The vibration of internal liquid induced by the load at different frequency (F=1.0 N)

Figure 7. The contour of z velocity of internal liquid as the frequency is 150.0 Hz (F=1.0 N)

Figure 8. The distribution of z velocity of internal liquid along intersection line (x=l/2 and y=l/2, F=1.0 N)

As internal liquid’s density rises, polar and zero points of DEM will shift to lower frequency domain. The reason is that, the maximum kinetic energy of liquid would increase as density rises, and coating layers still keep the same, so the components of mass matrix in Eq. (18) would amplify while the stiffness remains unchanged. Consequently, the local resonance frequency would be lowered. It is evident that, within incompressible assumption, the variety of inviscid liquid would merely modify inertial characteristic of the metamaterial. Figure 9 gives the variation of negative mass range and normalized bandwidth when internal liquid’s density changes. When internal liquid’s density is 0.8 g/cm3, the bandgap range is from 163.0 Hz to 228.0 Hz with the bandwidth being 33.29%, 15

and it turns to be from 120.0 Hz to 187.0 Hz as the density rises to 1.5 g/cm3 with the bandwidth being 43.16%. Worth to be noted is that, the bandwidth has been broadened as liquid’s density increases, and this phenomenon could be well explained by a comparison between present model and a lumped mass system which is studied in the paper presented by Huang et al. [35]. As for a lumped-mass system shown in Figure 1, the bandwidth is monotonously determined by the ratio of core’s mass to matrix’s mass. For present model, since internal liquid’s density increases and the density of matrix remains the same, so the ratio of mass has been raised up, leading to the normalized bandwidth being broadened.

Figure 9. The variation of the bandgap range when liquid’s density changes

Influence of coatings’ thickness on proposed LREM’s DEM is researched as well. Figure 10 gives the results when coatings’ thickness is being 0.30 mm, 0.34 mm and 0.42 mm. It could be found, as thickness increases, zero and polar points of DEM would shift to a higher frequency domain, which is caused by the rise of the bending stiffness of coatings. In addition, increase of thickness of coatings would give a rise to the deviation between theoretical and numerical results. The cause is that, the relative rotation between coatings and side walls could not be ignored with the obvious rise of bending stiffness of coatings, hence, the shape function utilized in Eq. (7) would lead to an inaccuracy of theoretical results. Therefore, the theoretical model needs to be modified such as changing the shape function and adding artificial rotation springs

16

into it if the coatings are tending to get thicker [38, 45]. It is worth to be noted, the normalized width of the bandgap is almost a constant when thickness changes, which is different with the results when liquid’s density varies. According to results in Figure 11, this phenomenon could also be expressed as the following geometric relationship. Aj B j Ai Bi ≈ Oi Ai O j A j

( i, j = 1, 2, 3 and i ≠ j )

(28)

This characteristic could be well explained through Eq. (19) and Eq. (23). The variation of the coatings’ thickness would mainly change the components of stiffness. It could be found that in Eq. (19), though the components of the mass matrix would change as well, the contribution of the coatings’ kinetic energy is tiny compared with other parts. So the variation of mass components is insignificant. The primary function of coatings is to provide potential energy, in another word, the stiffness, for the system. Therefore, based on Eq. (23), changes of the coatings’ thickness could hardly affect the normalized bandwidth. This characteristic could be quite useful for generating an extremely broad bandgap in low frequency region.

Figure 10. The variation of the DEM of unit cell when coating’s thickness (hc) changes (the density of liquid=1.0 g/cm3)

17

Figure 11. The variation of the bandgap range when coating’s thickness changes

5.2 Band structure In this section, band structure of the LREM is calculated through FE method by using COMSOL package. Theoretically-predicted negative mass range is also depicted. In the numerical simulation, eigenvalue analysis of the liquid-solid coupled cell is conducted. Floquet periodic boundary condition is applied, which could be expressed as:

udst = usrc e−kx ai

(29)

where udst and usrc refer to the displacement vectors on periodic boundaries (shown in Figure 1), kx a is from 0 to

π

and i equals to

-1 . Figure 13 gives the

band structure of the LREM in frequency range from 0.0 Hz to 300.0 Hz when coatings’ thickness is 0.34 mm. Dots in various colors indicate different polarized vibration respectively and corresponding modes of vibration are shown in Figure 12. Black dots illustrate the dispersion curve of flexural wave as transverse mode A is the locally resonant mode and mode E refers to upper edge of the 1st forbidden band of flexural wave. When the density of internal liquid is 1.0 g/cm3, in frequency range from 141.0 Hz to 209.0 Hz, shear horizontal vibration (mode C), torsion vibration (mode D) and longitudinal vibration (mode E) could propagate without obstruct while 18

flexural wave would be blocked due to the local resonance behavior of internal liquid and coatings. Moreover, band structures of proposed LREM with different coatings have been calculated and illustrated in Figure 14. The effects of coating’s thickness and liquid’s density are similar to the results mentioned in section 5.1 which will not be over-discussed here.

Figure 12. Different polarized vibration modes of proposed LREM

Figure 13. The band structure of present LREM with different internal liquid

Figure 14. The band structure of present LREM with different coatings

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5.3 Transmission characteristics of the metastructure In this section, FE method is applied to obtain the transverse displacement transmission of a five-period structure. Good agreements between numerical simulation and experimental tests has been achieved. In the simulation, a harmonic transverse displacement excitation is applied on the one side of the structure, and the target is on the other free side (shown in Figure 15).

Figure 15. FE model for obtaining the transmission

Concerned frequency range is from 0.0 Hz to 350.0 Hz with the resolution being 1.0 Hz. Transmission factor is given as the following expression [44].  T = 2 0 lo g 1 0   

∫u ∫u

ds    z − in p u t d s 

z − o u tp u t

(30)

Firstly, in order to validate the feasibility of present numerical and experimental methods, transmission (crossing five unit cells) of the metastructure without internal liquid is studied. Corresponding numerical (black line) and experimental results (blue triangle dots) are shown in Figure 16 and a great coherence between them has been achieved. After that, transmission of metastructure with coatings (0.34 mm PVC films) and internal liquid (water) is tested with corresponding results being denoted as red circle dots in Figure 16. When internal liquid’s density is 1.0 g/cm3, the propagation of flexural wave in the theoretically-predicted negative DEM range (from 146.0 Hz to 212.0 Hz) is significantly attenuated. The maximum transmission loss is up to more than 40 dB, which indicates almost all inputting energy is absorbed by locally resonant coating layers and internal liquid. From the results in Figure 16, it could also be found, compared with the metastructure without internal water, the response of proposed structure in the pass band, such as the first flexural mode, has moved to a 20

lower frequency region, which is caused by the internal liquid’s additional mass effects.

Figure 16. The transmissions of metastructure with or without internal liquid

As for experimentally obtained transmissions crossing different number of cell, results are shown in Figure 17. In theoretically predicted negative mass range, with the existence of internal water, transmissions crossing different distance would show a remarkable depression especially for the 3-period and 4-period ones. The additional mass effects of internal water could also be caught in this graph. Figure 18 gives the coherence between input and output signals. As the vibration crosses more unit cells, the coherence fades obviously in forbidden band, and this feature could be applied to characterize the bandgap range as well [46].

21

Figure 17. Transmissions of crossing different number of unit cell

Figure 18. Coherence between input and output signals

Through changing the thickness of coatings, the location of the forbidden band could be modified effectively. Corresponding experimental and numerical results are shown in Figure 19. When coating’s thickness is 0.34 mm, the lower edge is around 146.0 Hz, and it rises to about 200.0 Hz as thickness increases to be 0.42mm. Worth to be noted is that, the variation of the coatings could hardly affect the flexural mode of the entire structure but it could only change the locally resonant behavior. So points P1 and P2 are almost in a horizontal line, which is shown in Figure 19. Experimental results are consistent with the numerical simulation, especially for the response in the low frequency range. However, additional stiffness of the shaker as well as metastructure’s damping effects might lead to a slight deviation between simulations and tests. Based on preceding results, the vibration blocking function of proposed metastructure has been verified, and the attenuation frequency range could be obviously altered by changing coatings’ thickness. Theoretically predicted bandgap has overlapped the valley of transmission from experimental tests, and the deviation is in an acceptable range. In order to illustrate vibration attenuation phenomenon more intuitively, harmonic response of proposed metastructure in time domain is also tested in the experiment. Corresponding results are depicted in Figure 20, in which frequency 40.0 Hz is in the pass band and frequency 165.0 Hz is in the forbidden band. The input signal is the acceleration on the edge on which the shaker is located, and the output signal is the 22

acceleration on the other free edge. The attenuation in the forbidden band is obvious, which could be found in Figure 20.

Figure 19. Transverse transmissions of present metastructure with different coating layers

Figure 20. Responses in time domain when hc is 0.34 mm

In addition, Figure 21 gives responses from numerical simulation when excitation at different frequency is applied. The internal liquid’s density is 1.0 g/cm3 and the coating’s thickness is 0.34 mm. When frequency of incident flexural wave is in the pass band, such as 40.0 Hz, the metastructure would act as a conventional beam structure without obvious local resonance. Corresponding response shape is shown in − Figure 21 (a). When applied frequency has risen to ωlower / 2π which is slightly

below local resonance value such as 140.0 Hz (this is for numerical result and the theoretically predicted local resonance value is 146.0 Hz), the localized vibration 23

emerges (shown in Figure 21 (b)) and internal liquid keeps the same phase with + matrix. Once excitation frequency has exceeded ωlower / 2π such as 141.0 Hz,

anti-resonant behavior of internal liquid would emerge, which is shown in Figure 21

(c). It could be found in Figure 21 (d), at the upper edge ωupper , localized mode is still existing while the metastructure’s flexural mode has been stimulated out as well. Therefore, the LREM could no longer attenuate propagating flexural wave when excitation’s frequency keeps increasing.

(a) 40.0 Hz

(b) 140.0 Hz

(c) 141.0 Hz

(d) 206.0 Hz Figure 21. The responses of metastructure when excitation at different frequency is applied

The dynamic response of the proposed metastructure could also be modified by 24

varying internal liquid’s density. Figure 22 gives transmissions of metastructures with different internal liquid filling, and the deep of transmission matches well with theoretically-calculated negative DEM range.

Figure 22. The variation of transmission of the metastructure when liquid’s density changes

5.4 Effects of in-plane initial stress of coating layers In this section, effects of in-plane initial stress of coatings on unit cell’s transverse DEM are studied. When proposed LREM is being fabricated, in order to seal off internal liquid, coating layers should be connected to the side-walls, which might introduce initial stress and affect the EM’s dynamic behavior. Considering a uniform distribution of in-plane stress in x and y directions (the coatings’ thickness is 0.34 mm and the applied stress is 1.0 MPa, both in x and y directions), which is shown in Figure 23 (a), (b) and (c), a refreshed stress distribution could be obtained when the unit cell has deformed (shown in Figure 23 (d), (e) and (f)). After the stress analysis, DEM with initial stress is calculated through FE method. ‘Theoretically-predicted’ DEM is also calculated with the in-plane initial stresses being obtained from FEM results. Then the potential energy caused by initial stress is acquired by numerical integration (using Eq. (16)).

25

(a)

(d)

σ 11 -undeformed

σ 11 -deformed

(b)

(e)

σ 22 -undeformed

σ 22 -deformed

(c)

(f)

τ 12 -undeformed

τ 12 -deformed

Figure 23. Initial in-plane stresses of the coatings

Figure 24 gives variations of transverse DEM when initial tension stress changes. The internal liquid’s density is 1.0 g/cm3 and coatings’ thickness is 0.34 mm. It could be found that, stronger initial tension stress could push the EM’s DEM toward higher frequency, and the same geometric relationship that has been mentioned in section 5.1 could be discovered, which means the normalized bandwidth would be a constant caused by the unchanged components of mass matrix in Eq. (23). In another viewpoint, if the fabrication permitted, the bandgap of proposed LREM could also be effectively modified by applying initial stresses.

Figure 24. DEM of the unit cell with initial in-plane stress

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6. Conclusion In this paper, we propose a liquid-solid LREM, which can block flexural wave in broad low frequency range. Internal liquid and thin layers are used to be resonators of the metamaterial. Such design enables the LREM’s bandgap range to be easily modified. Due to the characteristics of incompressible liquid and thin layers, mass and elastic components of the metamaterial’s resonator would be separately determined by them. Theoretical method is developed for obtaining transverse DEM and bandgap range of the LREM. Influence of parameters of internal liquid and coating layer on the LREM’s dynamic characteristics are researched theoretically and numerically. Experimentas on a finite-period metastructure’s transmission has also been conducted. Results indicates that, an increase of internal liquid’s density would lead to the bandgap shift to lower frequency and broaden the normalized bandwidth. Reduce of coating layers’ thickness or the initial tension stress could also lower the bandgap’s frequency however the normalized bandwidth would almost keep as a constant. By changing the layer’s thickness, the bandgap is effectively altered, which has been verified experimentally. Due to preceding characteristics, the proposed LREM can be quite applicable for broad low-frequency vibration controlling. It could also inspire more novel designs of reconfigurable and tunable elastic metamaterial with liquid-solid interaction. Acknowledgment The authors sincerely acknowledge the support from the Natural Science Foundation of China [Grant No.11772251] and the 111 project of China [Grant No. B18040]. The authors would like to appreciate Zhen Li and Di Wang for their help during the numerical analysis. Statement of Author Contributions Yue-ming Li and Lei Wu proposed the idea. Lei Wu performed the theoretical and 27

numerical analysis. Lei Wu and Qian Geng accomplished the experimental tests. Lei Wu, Qian Geng and Yue-ming Li wrote the paper. All authors gave final approval for publication. Declaration of interest The authors declare no conflict of interest. Reference [1] Y. Wu, M. Yang, P. Sheng, Perspective: Acoustic metamaterials in transition, Journal of Applied Physics 123 (2018) 090901. [2] M. I. Hussein, M. J. Leamy, M. Ruzzene, Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook, Applied Mechanics Reviews 66 (2014) 040802. [3] M. S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafarirouhani, Acoustic band structure of periodic elastic composites, Physical Review Letters, 71 (1993) 2022-2025. [4] Z. Y. Liu, X. X. Zhang, Y. W. Mao, Y. Y. Zhu, Z. Y. Yang, C. T. Chan, P. Sheng, Locally resonant sonic materials, Science, 289 (2000) 1734-1736. [5] K. H. Matlack, A. Bauhofer, S. Krodel, A. Palermo, C. Daraio, Composite 3D-printed metastructures for low-frequency and broadband vibration absorption, Proceedings of the National Academy of Sciences, 113 (2016) 8386-8390. [6] Y. Y. Chen, L. F. Wang, Periodic co-continuous acoustic metamaterials with overlapping locally resonant and Bragg band gaps, Applied Physics Letters, 105 (2014) 191907. [7] G. C. Ma, P. Sheng, Acoustic metamaterials: From local resonances to broad horizons, Science Advances, 2 (2016) UNSP e1501595. [8] R. Zhu, X. N. Liu, G. K. Hu, F. G. Yuan, G. L. Huang, Microstructural designs of plate-type elastic metamaterial and their potential applications: a review, International Journal of Smart & Nano Materials, 6 (2015) 14-40. [9] S. A. Cummer, J. Christensen, A. Andrea. Controlling sound with acoustic metamaterials, Nature Reviews Materials, 1 (2016) 16001. [10] Y. Y. Chen, G. K. Hu, G. L. Huang, A hybrid elastic metamaterial with negative mass density and tunable bending stiffness, Journal of the Mechanics & Physics of Solids, 105 (2017) 179-198. [11] B. Z. Xia, N. Chen, L. X. Xie, Y. Qin, D. J. Yu, Temperature-controlled tunable acoustic metamaterial with active band gap and negative bulk modulus, Applied 28

Acoustics, 112 (2016) 1-9. [12] Z. Li, Y. Zhu, Y. M. Li, Thermal stress effects on the flexural wave bandgap of a two-dimensional locally resonant acoustic metamaterial, Journal of Applied Physics, 123 (2018) 195101. [13] Y. Zhu, Z. Li, Y. M. Li, The Lamb wave bandgap variation of a locally resonant phononic crystal subjected to thermal deformation, Aip Advances, 8 (2018) 055109. [14] Z. Li, X. Wang, Y. M. Li, The band gap variation of a two dimensional binary locally resonant structure in thermal environment, Aip Advances, 7 (2017) 015002. [15] Q. Geng, T. Y. Cai, Y. M. Li, Flexural wave manipulation and energy harvesting characteristics of a defect phononic crystal beam with thermal effects, Journal of Applied Physics, 125 (2019) 035103. [16] V. C. de Sousa, D. Tan, C. D. De Marqui, A. Erturk, Tunable metamaterial beam with shape memory alloy resonators: Theory and experiment, Applied Physics Letters, 113 (2018) 143502 . [17] V. C. de Sousa, C. Sugino, C. D. De Marqui, A. Erturk, Adaptive locally resonant metamaterials leveraging shape memory alloys, Journal of Applied Physics, 124 (2018) 064505. [18] Y. Y. Chen, L. F. Wang, Tunable band gaps in bio-inspired periodic composites with nacre-like microstructure, Journal of Applied Physics, 116 (2014) 063506. [19] A. Nanda, M. A. Karami, Tunable bandgaps in a deployable metamaterial, Journal of Sound and Vibration, 424 (2018) 120-136. [20] P. Wang, F. Casadei, S. C. Shan, J. C. Weaver, K. Bertoldi, Harnessing Buckling to Design Tunable Locally Resonant Acoustic Metamaterials, Physical Review Letters, 113 (2014) 014301. [21] P. Wang, J. Shim, K. Bertoldi. Effects of geometric and material nonlinearities on tunable band gaps and low-frequency directionality of phononic crystals, Physical Review B, 88 (2013) 014304. [22] M. Yang, G. C. Ma, Z. Y. Yang, P. Sheng, Coupled Membranes with Doubly Negative Mass Density and Bulk Modulus, Physical Review Letters, 110 (2013) 134301. [23] P. Marinova, S. Lippert, O. Von Estorff. On the numerical investigation of sound transmission through double-walled structures with membrane-type acoustic metamaterials, The Journal of the Acoustical Society of America, 142 (2017) 2400-2406. [24] S. C. Shan, S. H. Kang, P. Wang, C. Y. Qu, S. Shian, E. R. Chen, K. Bertoldi, Harnessing Multiple Folding Mechanisms in Soft Periodic Structures for Tunable Control of Elastic Waves, Advanced Functional Materials, 24 (2014) 4935-4942. 29

[25] B. M. Goldsberry, M. R. Haberman, Negative stiffness honeycombs as tunable elastic metamaterial, Journal of Applied Physics, 123 (2018) 091711. [26] Y. T. Wang, X. N. Liu, R. Zhu, G. K. Hu, Wave propagation in tunable lightweight tensegrity metastructure, Scientific Reports, 8 (2018) 11482. [27] A. Baz, The structure of an active acoustic metamaterial with tunable effective density, New Journal of Physics, 11 (2009) 123010. [28] W. Akl, A. Baz, Multicell Active Acoustic Metamaterial with Programmable Effective Densities, Journal of Dynamic Systems Measurement & Control, 134 (2012) 061001. [29] W. Akl, A. Baz, Stability analysis of active acoustic metamaterial with programmable bulk modulus, Smart Materials and Structures, 20 (2011) 125010. [30] Y. B. Jin, P. Yan, Y. D. Pan, B. Djafari-Rouhani, Phononic Crystal Plate with Hollow Pillars Actively Controlled by Fluid Filling, Crystal, 6 (2016) 64. [31] Y. B. Jin, N. Fernez, P. Yan, B. Bonello, R. P. Moiseyenko, S. Hemon, Y. D. Pan, B. Djafari-Rouhani, Tunable waveguide and cavity in a phononic crystal plate by controlling whispering-gallery modes in hollow pillars, Physical Review B, 93 (2016) 054109. [32] T. T. Wang, Y. F. Wang, Y. S. Wang, V. Laude, Tunable fluid-filled phononic metastrip, Applied Physics Letters, 111 (2017) 041906. [33] Y. F. Wang, T. T. Wang, Y. S. Wang, V. Laude, Reconfigurable Phononic-Crystal Circuits Formed by Coupled Acoustoelastic Resonators, Phys. rev. applied, 8 (2017) 014006. [34] Q. Zhang, K. Zhang, G. K. Hu. Tunable fluid-solid metamaterials for manipulation of elastic wave propagation in broad frequency range, Applied Physics Letters, 112 (2018) 221906. [35] H. H. Huang, C. T. Sun, Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density, New Journal of Physics, 11 (2009) 013003. [36] M. Amabili, Eigenvalue problems for vibrating structures coupled with quiescent fluids with free surface, Journal of Sound and Vibration, 231 (2000) 79-97. [37] M. Amabili, M. P. Paidoussis, A. A. Lakis, Vibrations of partially filled cylindrical tanks with ring-stiffeners and flexible bottom, Journal of Sound and Vibration, 213 (1998) 259-299. [38] M. Amabili, Shell-plate interaction in the free vibrations of circular cylindrical tanks partially filled with a liquid: the artificial spring method, Journal of Sound and Vibration, 199 (1997) 431-452. [39] E. Askari, F. Daneshmand, M. Amabili, Coupled vibrations of a partially fluid-filled cylindrical container with an internal body including the effect of free surface waves, Journal of Fluids & Structures, 27 (2011) 1049-1067. [40] Q. Geng, H. Li, Y. M. Li, Dynamic and acoustic response of a clamped 30

rectangular plate in thermal environments: Experiment and numerical simulation, Journal of the acoustical society of America, 135 (2014) 2674-2682. [41] M. A. Nouh, O. J. Aldraihem, A. Baz, Periodic metamaterial plates with smart tunable local resonators, Journal of Intelligent Material Systems and Structures, 27 (2016) 1829-1845. [42] G. W. Milton, J. R. Willis, On modifications of Newton’s second law and linear continuum elastodynamics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463 (2007) 855-880. [43] M. Caleap, B. W. Drinkwater, P. D. Wilcox, Effective dynamic constitutive parameters of acoustic metamaterials with random microstructure, Journal of the Acoustical Society of America, 14 (2012) 033014. [44] Y. F. Wang, Y. S. Wang, V. Laude, Wave propagation in two-dimensional viscoelastic metamaterials, Physical Review B, 92 (2015) 104110. [45] Z. Qin, Z. Yang, Z. U. Jean, F. L. Chu, Free vibration analysis of rotating cylindrical shells coupled with moderately thick annular plates, International Journal of Mechanical Sciences, 142 (2018) 127-139. [46] E. J. P. Miranda, E. D. Nobrega, A. H. R. Ferreira, J. M. C. Dos Santos, Flexural wave band gaps in a multi-resonator elastic metamaterial plate using Kirchhoff-Love theory, Mechanical Systems and Signal Processing, 116 (2019) 480-504.

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Highlights


A liquid-solid interacted metamaterial with broad low-frequency bandgap is proposed.




An analogized Lorentz model is developed for calculating the effective mass.




The bandgap could be manually modified by altering internal liquid or coatings.




The initial stress of coatings would contribute to a variation of bandgap.

Author Contribution Statement Lei Wu: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing & Original Draft. Qian Geng: Investigation, Writing - Review & Editing. Yue-ming Li: Conceptualization, Writing - Review & Editing, Supervision.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: