Elastic wave and vibration bandgaps in planar square metamaterial-based lattice structures

Journal Pre-proof Elastic wave and vibration bandgaps in planar square metamaterial-based lattice structures Xiyue An, Hualin Fan, Chuanzeng Zhang PII:

S0022-460X(20)30123-1

DOI:

https://doi.org/10.1016/j.jsv.2020.115292

Reference:

YJSVI 115292

To appear in:

Journal of Sound and Vibration

Received Date: 15 February 2019 Revised Date:

25 February 2020

Accepted Date: 28 February 2020

Please cite this article as: X. An, H. Fan, C. Zhang, Elastic wave and vibration bandgaps in planar square metamaterial-based lattice structures, Journal of Sound and Vibration (2020), doi: https:// doi.org/10.1016/j.jsv.2020.115292. 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. © 2020 Published by Elsevier Ltd.

Credit Author Statement

The original paper entitled “Elastic wave and vibration bandgaps in planar square metamaterial-based lattice structures” was written by Xiyu An, Chuanzeng Zhang and myself. It is an original paper on planar square metamaterial-based lattice structures. The paper is intended for the possible publication in “Journal of Sound and Vibration” and we would like to state that this work neither has been published nor is being submitted for publication elsewhere. In this research, Xiyue An performed theoretical and FEM analyses. Chuanzeng Zhang guided the research jointly and revised the manuscript. Hualin Fan put forward planar square metamaterial-based lattice structure and guided the research.

Elastic wave and vibration bandgaps in planar square metamaterial-based lattice structures Xiyue An a, b, Hualin Fan a, b, *, Chuanzeng Zhang c a

State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b

c

College of Mechanics and Materials, Hohai University, Nanjing 210098, China

Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany Corresponding authors: [email protected] (Hualin Fan);

Abstract: In this work, a new type of planar square lattice structures for the attenuation of elastic wave propagation is proposed and designed. To obtain a high vibration attenuation in low frequency ranges, the internal resonance mechanism of the acoustic metamaterials (AMs) is introduced into the continuous square lattice system with a cross-type core consisting of two segments with a radius jump discontinuity in each unit-cell. The wave propagation and vibration bandgap properties of the AM-based lattice structures are studied by using a spectral element method (SEM) which can provide accurate dynamic characteristics of a lattice structure due to its analytical spectral element matrix. The band structures calculated show that the phononic bandgaps induced by the local resonance and destructive interference co-exist in the considered systems. The effects of the structural and material parameters on the bandgaps are investigated in detail by the frequency response analysis of a spectral element model subjected to a dynamic excitation. Finally, in order to obtain lower and wider frequency bandgaps, the mechanism to 1

optimize the unit-cell of the AM-based lattice structures is illustrated and a composite lattice model is suggested. Keywords: Metamaterial-based lattice structures; Band structures; Vibration attenuation; Spectral element method. 1 Introduction Periodic lattice structures assembled by a number of identical rods or beams joined together in an identical manner are widely used in aeronautics, civil engineering, and material sciences because of their high stiffness-to-weight ratio. Since noise and vibration are easy to occur in the lightweight lattice structures, wave propagation and vibration bandgap properties have attracted a considerable attention for many years [1-5]. Inspired by the mechanism of phononic crystals in condensed matter physics, phononic bandgaps in the lattice structures can be obtained usually by a periodic arrangement of the lattice materials [6-9] or joint masses [2,10,11]. Elastic waves at relatively high frequencies are forbidden to pass through in several frequency ranges, which are often referred to as bandgaps, due to the destructive interference of the wave reflections from the periodic scatters. However, the generation of wide low-frequency bandgaps in lattice structures is still a challenging task, especially in practical engineering applications. Acoustic metamaterials (AM) are good candidates for filtering low-frequency elastic waves [12-15]. Liu et al. [4] fabricated sonic crystals with a microstructural unit consisting of a solid core material coated with soft rubber based on the idea of localized resonant structure. When acoustic waves propagate through such a material, 2

low bandgaps appear because of the localized vibration of the resonators (lead spheres). Hirsekorn [16] used a mass-spring system to explain the resonance mode easily by representing the solid core with a punctiform mass connected to the epoxy matrix by springs. A two-dimensional (2D) physical mass-in-mass lattice model with anisotropic resonant microstructures was proposed by Huang et al. [17]. It is shown that an anisotropic negative mass density exists in a certain frequency range and results in an anisotropic bandgap structure. A parametric study appreciating the degree and direction of influence of the structural and material parameters on the band structure was conducted by Fallah et al. [18]. Recently, the mechanism of generating bandgaps by local resonance in AM was introduced to the grid structures. Liu et al. [19] designed a modified two-dimensional (2D) periodic lattice structure by adding some cantilever beams at the intersections of the conventional square lattice structure. The original band structures without bandgaps can be split by the natural resonance of the auxiliary beams in the neighborhood of the natural frequencies of the cantilever. Junyi et al. [20] extended this work by attaching local resonators consisting of struts with a mass at its endpoint to cubic lattice topologies. Parametric study on the mechanical and dispersion properties was conducted to illustrate the suitability of lattice structures to be regarded as lightweight structures with excellent static properties and vibration insulation characteristics. An et al. [21] presented a three dimensional acoustic metamaterial based stretching-dominated meta-truss lattice structure. The vibration reduction properties of their proposed structure were verified by vibration transmission tests. 3

Resonators with an anisotropic geometry were considered by Tallarico et al. [5]. The introduction of periodic resonators with triangular shape in the triangular lattice results in the appearance of novel emerging rotationally dominant modes and gives rise to a low frequency bandgap which can be tuned by the tilting angle of the resonators. In order to achieve lower frequency bandgaps, different from low-frequency bandgaps obtained by local resonators which require heavy resonators or springs with low stiffness, inertial amplification mechanisms were embedded into a periodic lattice structure [22, 23]. With smaller mass fractions on amplifiers, wide bandgaps can also be obtained in this manner. Significant efforts have been made in the development of the analytical modeling for the dynamic analysis of periodic lattice structures. To describe the long-wavelength structural effects, homogenization methods based on continuum mechanics can provide an approximation of the vibration properties in periodic lattices with a simpler dynamic description without considering the microstructural effects [24-27]. Vasiliev et al. [28] developed multi-field method in order to describe the short-wavelength waves and boundary effects when regarding the lattice structures as continuum models. Based on the well-developed theories of continuum mechanics and physics, analytical solutions can be found for this kind of problems. When analytical solutions cannot be obtained, one can use the finite element method (FEM) in general, but the degree of accuracy depends on the meshing method. Different from the FEM, the spectral element method (SEM) utilizes the exact frequency-domain solutions due to the frequency-dependent interpolations, and can largely reduce the 4

size of the dynamic stiffness matrix because the one-element modeling will suffice for a regular structural member without geometric or material discontinuities in the spatial domain and external forces [29]. By virtue of its advantages, dynamic analysis of the lattice structures in the frequency domain can be studied easily and accurately by the SEM. In order to analyze the vibration of large-scale periodic lattices, the spectral transfer matrix method which is a combination of the SEM and the transfer matrix method was introduced by Lee [30]. Xiao et al. [31] developed a methodology based on the SEM and the Bloch theorem, and complex band structures of the infinite periodic metamaterial-based rods and the vibration transmittance of the finite systems were calculated. Wu et al. [7, 9] studied the vibration bandgap properties of the lattice structures composed of two different materials using the SEM. Parametric analysis of the effects of the structural and material parameters on the bandgaps was conducted. In the present study, we design a 2D AM-based square lattice structure, inspired by the mechanism of acoustic metamaterials. Different from the conventional lattice cell consisting of six structural members connecting four corner nodes with the same cross-sections, the two members inside of the AM-based lattice cell are discontinuous in their cross-sections, namely with a radius jump. The cross-type core can act as a resonator which contributes to the generation of the local resonant bandgaps in lower frequencies. The details of the design are illustrated in section 2. The SEM is used to investigate the bandgap properties. We present a one-dimensional (1D) finite AM model in section 3, and examine the validity and superiority of the method by comparing the natural frequencies and frequency response behaviors of the lattice 5

model under a harmonic excitation. The dispersion curves of the infinite AM model are also calculated by a method based on the combination of the SEM and the Bloch theorem. In section 4, the frequency response analysis of a 2D finite AM model is conducted, and the mechanism of the bandgaps caused by local resonance of the resonators or destructive interferences is illustrated. The influences of the structural and material parameters on the elastic wave and vibration bandgap properties are analyzed in detail. An optimal design of the lattice structures is obtained, and in order to achieve wider bandgaps, a composite AM-based lattice structure with different materials is suggested in section 5. Subsequently, in section 6, a special case of the composite lattice structure composed of two materials is proposed to illustrate the superposition property of the bandgaps. Finally, some concluding remarks are given in section 7.

(b)

(a)

Fig. 1 (a) Unit-cell of the acoustic metamaterial and (b) design of the AM-based lattice unit-cell. 2 The AM-based square lattice structures Acoustic metamaterials are a kind of materials with a carefully designed microstructure. When elastic waves pass through the AMs, the external vibration of the primary structure is weakened by the local resonance of the subsystems in 6

relatively low frequency ranges, thus, acoustic waves may be forbidden to propagate forward. The internal resonance is usually realized by dispersing a heavy core material coated with a soft layer into the primary structure. The AM unit-cell represented by a spring-mass system is shown in Fig. 1(a), and by virtue of the excellent properties of the AMs in the low-frequency wave or vibration attenuation, we introduce the idea into periodic lattice structures. The proposed AM-based lattice unit-cell is shown in Fig. 1(b). The periodic square lattice structure consisting of AM-based lattice unit-cells is depicted in Fig. 2(a). Different from the subsystem of the conventional AM unit-cell which is usually made by different materials with a high contrast in stiffness, the cross-type core in the present AM-based lattice unit-cell may have only a radius jump discontinuity, as shown in Fig. 2(c). Thus, it is a novel and convenient way to fabricate the periodic lattice structures with low-frequency bandgaps where elastic waves and mechanical vibrations cannot pass through.

(a)

Thinner beam Outer frame (c)

Resonator (b)

7

Fig. 2 (a) The AM-based periodic lattice structure including a cross-type core consisting of two segments with a radius jump discontinuity in each beam; (b) design of the lattice unit-cell; (c) the beam in each core. 3 The superiority and validity of the SEM 3.1 Spectral element modeling of the lattice structural members The considered structural members in the AM-based lattice unit-cell as shown in Fig. 2(b) are assumed to be clamped at the joints. Hence, elastic beam models possessing the capability of the longitudinal and bending deformation simultaneously are chosen to approximate the structural members.The free longitudinal and bending vibrations of a uniform Timoshenko-beam are described by

EI

∂ 2u ( x, t ) ∂ 2u ( x, t ) E −ρ =0, ∂x2 ∂t 2

(1)

 ∂ 2 w ( x, t ) ∂θ ( x, t )  ∂ 2 w ( x, t ) − = 0, κG  −ρ 2 ∂x  ∂t 2  ∂x

(2)

∂ 2θ ( x, t ) ∂ 2θ ( x, t )  ∂w ( x, t )  κ θ ρ GA x t I + − , − = 0, ( )  2 ∂x 2 ∂ x ∂ t  

(3)

where u ( x, t ) , w ( x, t ) and θ ( x, t ) are the longitudinal displacement, the transverse displacement and the in-plane rotation, respectively, as shown in Fig. 3. E is the Young’s modulus. G is the shear modulus with υ being the Poisson’s ratio,

κ is the shear correction factor, ρ is the mass density, A is the cross-sectional area, and I is the area moment of inertia with respect to the neutral axis.

8

Fig. 3 Sign convention of the displacements for the Timoshenko-beam. For the SEM, the solutions of the governing differential equations are the superposition of a finite number of wave modes at different discrete frequencies [32]. The solutions of Eqs. (1) to (3) represented in spectral form are given by

u ( x, t ) =

1 N −1 U n ( x; ωn ) eiωnt , ∑ N n =0

(4)

w ( x, t ) =

1 N −1 ∑Wn ( x;ωn ) eiωnt , N n =0

(5)

θ ( x, t ) =

1 N −1 Θn ( x; ωn ) eiωnt , ∑ N n =0

(6)

where Un ( x;ωn ) , Wn ( x;ωn ) , Θn ( x;ωn ) are the spectral components of the displacements u ( x, t ) , w ( x, t ) , θ ( x, t ) respectively, ω n is the discrete frequency,

N is the number of samples in the time-domain. As a result, the time-domain governing differential equations are transformed into the frequency-domain and the exact wave solutions can be obtained. The dynamic stiffness matrix depending on the frequency is obtained by using the exact dynamic shape functions derived from the solutions of the governing equations of motion. Since the detailed derivation procedure for the SEM applied to the lattice structures has been given in the work done by Lee [29], here we only present the spectral beam element matrix for simplicity. And to shorthand, the subscript n , which indicates the

n th Fourier component, are omitted in the following derivations. The spectral element matrix for the tensional element is given by

 k cot ( kL L ) −kL csc ( kL L )  S R (ω ) =EA  L ,  −kL csc ( kL L ) kL cot ( kL L )  9

(7)

with kL = ω ρ E . The spectral Timoshenko-beam bending element matrix is

ST (ω ) = RHT−1 ,

(8)

with

 −κGA( −ikt − β1 ) −κGA( ikt − β2 ) −κGA − ( ike − β3 ) −κGA( ike − β4 )    iEI β1kt −iEI β2kt iEI β3ke −iEI β4ke  , (9) R= κGA( −ikt − β1 ) e−ikt L κGA( ikt − β2 ) eikt L κGA( −ike − β3 ) e−ikeL κGA( ike − β4 ) eikeL    −ikt L iEI β2kt eikt L −iEI β3kee−ikeL iEI β4keeikeL   −iEI β1kt e and

 1  −ir HT (ω ) =  t  et   −irt et  ρA kF = ω    EI 

1 irt

1 −ire

et−1 irt et−1

ee −ire ee

14

where

,

k1 = −k2 = kt =

1  ire  , ee−1   ire ee−1 

(10)

1 kF η k F2 + η 2 kF4 + 4 (1 −η1kG4 ) 2

1  ρA  kF η kF2 − η 2 kF4 + 4 (1 −η1kG4 ) , kG = ω   2  κ GA 

,

14

k3 = −k4 = ke =

η1 =

I EI 1 k p2 − kG4 ) = −irp (ω ) , η2 = , β p (ω ) = ( ik p κ GA A

, η = η1 + η2 ,

( p = 1, 2,3, 4 ) .

The assembling technique of the SEM to form a global system matrix is the same as the FEM. Thus, the spectral element equation in the global coordinates is given by

S g (ω ) d g = f g ,

(11)

where S g (ω ) is the assembled global dynamic stiffness matrix, d g is the global spectral nodal displacement vector, f g is the global spectral nodal forces and moments vector. The frequency responses of the nodal degrees-of-freedom (DOFs) can be calculated by solving Eq. (11).

10

Fig. 4 Schematic of a one-dimensional AM-based lattice structure subjected to a time-harmonic excitation with a fixed constraint on the left side. 3.2 Comparison with the FEM The accuracy of the SEM applied to the dynamic analysis of the considered AM-based lattice structures is illustrated through the comparison of the frequency responses of the spectral nodal displacements and the natural frequencies calculated by the SEM and FEM. The illustrative physical model is a cantilever lattice structure consisting of 10 AM-based lattice unit-cells subjected to a harmonic excitation

F ( t ) =10e−iωt as shown in Fig. 4. The structure is made of steel with the Young’s modulus E = 206 GPa , Poisson’s ratio v = 0.3 and mass density ρ = 7850 kg m3 . The length of the lattice unit-cell is a = b = 0.5 m , the length of the resonator (cross-type core with a larger sectional radius R =0.04 m ) is RL=0.25 m , and the sectional radius of other beam elements is r = 0.01 m . The frequency response of the right corner node (red point) in the vertical direction is shown in Fig. 5.

11

-2

10

SEM FEM

-4

Displacement (m)

10

-6

10

10-8 -10

10

-12

10

-14

10

0

500

1000

1500

2000

2500

3000

Frequency (Hz)

Fig. 5 Frequency response of the right corner in the vertical direction calculated by the spectral element method (SEM) and the finite element method (FEM). Given the small value of the response vector, the displacement versus the frequency is represented by evaluating the logarithm of the displacement and the following frequency response curves use the same form. It is clear to see that the frequency response curve obtained by the SEM is in good agreement with that obtained by the FEM. Here, it is observed that the vibration attenuates strongly in several frequency ranges which declares that the proposed model possesses the capacity of depressing vibration and noise efficiently. The superiority of the SEM in comparison to the conventional FEM has been approved by several previous works [7, 8, 33, 34]. In order to obtain more accurate results, several elements are needed for each beam in the FEM because the result accuracy depends on the mesh density, while in the SEM only one element is needed for a regular structural member, and the accuracy of the solution is higher due to the exact dynamic stiffness matrix especially at high frequencies. The eigenfrequencies in the high-frequency range of the considered 12

problem model calculated by the SEM are compared with those by the FEM. Table 1 obviously shows that the results obtained by the FEM converge to the SEM results as the element number of the FEM increases, which proves the superiority of the SEM in the analysis of the proposed lattice structure. The detailed analysis on the vibration properties of the AM-based lattice structures will be presented in the next section. Table 1 Comparison of the natural frequencies of the AM-based lattice structure obtained by the SEM and FEM in the high-frequency range. FEM Elements

Eigenfrequency (Hz)

4

315

630

SEM 111

2505.1

2502.2

2501.1

2525.2

2511.6

2509.4

2531.7

2515.7

2515.6

2539.5

2528.3

2521.2

2550.3

2538.7

2537.2

2560.6

2549.4

2550.9

2569.9

2554.2

2556.5

2584.7

2558.1

2561.3

2590.9

2564.0

2563.4

2597.8

2568.1

2577.3

Bandgap properties

4.1 Wave propagation in the infinite AM-based lattice structure In order to demonstrate the existence of the bandgaps in the considered AM-based lattice structure, wave propagation in an infinite 2D system is studied firstly. Based on the Bloch-theorem, periodic boundary conditions are applied to the SEM to obtain the dispersion curves of the infinite lattice structure by using a single unit-cell.

13

(a)

(b)

Fig. 6. (a) Infinite 2D AM-based lattice structure and (b) a generic unit-cell with the nodal number. The infinite 2D AM-based lattice structure is depicted in Fig. 6(a). A sketch of a unit-cell of the 2D periodic lattice structure with the nodal numbers is shown in Fig. 6(b). Here, the representative unit-cell is a square lattice consisting of 12 structural members. By assembling the spectral elements in the unit-cell, the global system matrix equation of motion can be obtained in the form as

Se ( ω ) de = f e ,

(12)

where S e (ω ) is the assembled dynamic stiffness matrix of the unit-cell, f e is the global spectral nodal forces and moments vector. By using the periodic boundary conditions for the unit-cell, the spectral nodal degree-of-freedoms (DOFs) vector d e can be transformed in the form

%, de = Td e

(13a)

with d e = [q1

q2

q3

% = [q d e 1

q4

q5 q2

q6 q4

q7 q5

q8 q6

and 14

q9 q7

q10 q9

q11

q10 ] , T

q12

q13 ] , T

(13b) (13c)

I   0   Ie -ik x a  0   0  0   0 T= 0   0   0  - ik y a  Ie  0   -i ( k x + k y ) a  Ie

Ie

0 I 0 0

0 0 0 I

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

I 0 0 I 0 0 0 Ie-ik x a

0 0 I 0

0 0 0 0

0 0

0 0 0 0

0 0

0 I 0 0

0

0 0

0

0 0

- ik y a

0 0

0

0 0

0

0 0

0

0 0

0 0  0  0 0  0 0 ,  0 0  I  0  0  0

(13d)

% is the reduced DOFs vector of d by reducing the DOFs on the top and where d e e right edges, and T is the transfer matrix. Substituting Eq. (13) into Eq. (12) yields

% =f . SeTd e e

(14)

In order to make the coefficient matrix on the left-hand side a square matrix and eliminate the effect of the external excitation under the consideration of the periodic boundary conditions of the loading on the unit-cell, TH is pre-multiplied to both sides of the Eq. (14), where the superscript H denotes the Hermitian transpose of a matrix [35]. Then, in the absence of an internal loading inside the unit-cell, Eq. (14) can be written as %= A (ω , k , k ) d %= 0 . T H STd x y

(15)

The dispersion curves can be obtained by solving the eigenvalue or dispersion equation (15). Due to the periodicity, all propagating modes can be captured by restricting the wavenumbers

(k , k ) x

y

to the irreducible part of the first

Brillouin-zone as shown in Fig. 7(a). Here, the structural and material parameters are 15

taken as the same used in the subsection 3.2. The dispersion curves obtained are depicted in Fig. 7(b). It is shown that there are three obvious bandgaps which are consistent with the frequency response curves shown in Fig. 5. 1400

Frequency (Hz)

1200 1000 800 600 400 200 0

M

Γ Wave number

(a)

X

M

(b)

Fig. 7. (a) The corresponding irreducible Brillouin-zone and (b) the dispersion curve of the infinite 2D AM-based lattice structure.

(a)

(b)

Fig. 8. 2D AM-based lattice subjected to a time-harmonic excitation in the (a) horizontal and (b) diagonal direction. 4.2 Vibration properties in the finite AM-based lattice structure A finite 2D lattice structure consisting of 9 × 5 unit-cells is shown in Fig. 8(a). The average responses on the right side of the structure with and without a radius jump cross-type core subjected to a plane time-harmonic excitation on the left side are computed using the same structural and material parameters in the subsection 3.2. As 16

shown in Fig. 9, there are three frequency ranges 776-858 Hz, 989-1473 Hz and 1950-2182 Hz with a vibration attenuation in the AM-based lattice structure compared with the conventional lattice structure. However, the first bandgap where the vibration attenuates is not obvious compared with the other bandgaps with a significant vibration reduction. Fig. 10 presents the structural response curves of the in-plane elastic waves along the ΓX- and ΓM- directions (Fig. 8(b)), respectively. The band structure of the corresponding infinite model computed by the SEM is depicted in the figure for comparison purposes. The present results show that the vibration bandgaps obtained by the finite 2D lattice structure agree well with the bandgaps predicted by the dispersion analysis for the corresponding infinite 2D lattice structure. Lattice AM-based lattice

-2

10

-4

Displacement (m)

10

-6

10

-8

10

10-10 -12

10

10-14 0

500

1000

1500

2000

2500

3000

Frequency (Hz)

Fig. 9. The average structural responses of the horizontal direction on the right side of the conventional lattice and AM-based lattice structures subjected to a plane time-harmonic excitation.

17

1500

Frequency (Hz)

1000

500

0

(a)

(c)

(b) M

ΓM

ΓX

X

Γ

Fig. 10. The average structural responses on the receiving end of the AM-based lattice structure subjected to a plane time-harmonic excitation along the (a) ΓM- direction and (c) ΓX- direction; and (b) the band structures computed by the infinite model. -2

10

R=0.03m R=0.04m R=0.05m

-4

10

R=0.06m R=0.07m R=0.08m

-4

10

Displacement (m)

Displacement (m)

-6

10

-6

10

-8

10

-10

10

-8

10

-10

10

-12

-12

10

-14

10

10

-14

10

10-16

-16

0

500

1000

1500

2000

2500

10

3000

0

500

1000

1500

2000

2500

3000

Frequency (Hz)

Frequency (Hz)

Fig. 11. The average structural responses of the horizontal direction on the right side of the AM-based lattice structure subjected to a plane time-harmonic excitation with different radius of the resonator. 4.3 Local resonance bandgaps In acoustic metamaterials, the local resonance bandgaps are usually caused by dispersed heavy inclusions coated with a soft layer embedded into a matrix [4]. The resonator is caused by the stiffness contrast between the inclusion and the soft layer 18

[36, 37]. In our considered lattice unit-cell as shown in Fig. 2(b), the contrast in the beam radius forms a resonator inside the unit-cell. Through changing the radius of the resonator R , different frequency response curves are obtained as shown in Fig. 11. It is observed here that for this kind of model only when the mass of the resonator is large enough can the local resonance bandgap occur, and as the radius becomes larger, the second and third bandgaps have no changes, but the lower edge of the first bandgap moves toward the lower frequency. It is thus illustrated that the width and location of the first bandgap are related to the radius or the mass of the cross-type resonator, and the mechanism is the same as that of the acoustic metamaterials. (a)

(c)

(b)

Fig. 12. Deformation pattern of the 2D AM-based lattice structure with 9 × 5 unit-cells at (a) 200Hz; (b) 600Hz; (c) 1200Hz. The deformation pattern of the AM-based lattice structure as shown in Fig. 8(a) are calculated at three chosen frequencies 200 Hz, 600 Hz and 1200 Hz, and they are depicted in Fig. 12. The radius of the resonator used here is R =0.08 m . One can observe in Figs. 12(b) and (c) that the vibration in the lattice structure attenuates in the first and second bandgaps as predicted in Fig. 11, but the deformation pattern are 19

different. Obviously, the energy transmitted in the lattice structure is nearly completely absorbed by the vibration of the resonators in the first bandgap as depicted in Fig. 12(b). In the second bandgap, there is nearly no vibration of the resonators as shown in Fig. 12(c), and the bandgap is Bragg scattering bandgap caused by the destructive interference and the same phenomenon also exists in the third bandgap. 10-2

Displacement (m)

10-4 -6

10

10-8 -10

10

-12

10

Epoxy Al Steel

10-14 10-16

0

500

1000

1500

2000

2500

3000

Frequency (Hz)

Fig. 13. The AM-based lattice unit-cell with two different materials and the frequency responses in the horizontal direction of the lattice structures with different materials of the outer frame. However, from the results discussed above, in order to make the first bandgap lower and wider, the radius of the resonator R must be much larger than the radius r of the connected outer frame. It means that the mass fraction of the resonator must

be large enough. Hence, the AM-based lattice unit-cell with two different materials is considered in the following, as depicted in Fig. 13. The material of the resonator is steel, and the material of the outer frame is varying. The average frequency responses in the horizontal direction for the considered 2D AM-based lattice structure are shown 20

in Fig. 13. The elastic material parameters used here are given in Table 2. It is observed that when the material of the outer frame is aluminum, the second and third bandgaps are the same as in the case of a lattice structure with a steel outer frame, but the first bandgap is wider. When the material of the outer frame is epoxy, all the bandgaps move toward the lower frequency.

Table 2 Elastic material parameters of the lattice structure Material

E ( GPa )

ρ (kg m3 )

ν

Steel Iron Aluminium Epoxy M

206 100 71.7 4.35 103

7850 7870 2700 1180 3925

0.30 0.21 0.33 0.30 0.10

5 Influences of the structural and material parameters The vibration attenuation properties of the AM-based lattice structures are investigated varying the structural (e.g., length of the resonator, rotation angle of the resonator, length-to-width ratio of the unit-cell and size of the structure) and material parameters. The results are presented in terms of the frequency responses for the finite 2D AM-based lattice structures subjected to a time-harmonic excitation. (a) 10-2

RL=0.2m RL=0.25m RL=0.3m

-4

-2

10

α=0.5 α=1 α=2

-4

10

-6

10

Displacement (m)

Displacement (m)

10

(b)

-8

10

-10

10

-12

10-6 -8

10

-10

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Fig. 14 Comparison of the frequency responses in the horizontal direction of the 2D AM-based lattice structures with 9 × 5 unit-cells subjected to a plane time-harmonic excitation with (a) different lengths of the resonator, (b) different lengths of the unit-cell. 10-2

n=5 n=7 n=9

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Fig. 15. Comparison of the frequency responses of the 2D AM-based lattice structures for different numbers of the unit-cells in the horizontal direction. (a) 10-2

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Fig. 16. The frequency responses in the horizontal direction of the 2D finite AM-based lattice structures for different clockwise rotation angles of the resonator (a) 0°, (b) 15°, (c) 30°, (d) 45°. Fig. 14(a) shows that the bandgaps become wider initially and then narrower when the length of the resonator RL changes between zero and the size of the unit-cell. It illustrates that for certain radius of the resonator and square unit-cell, the optimal length of the resonator for obtaining wide bandgaps is RL =1 2 a . The effect of the size of the unit-cell is also considered. It is shown in Fig. 14(b) that when the length-to-width ratio is α = a b = 1 , the width of the total bandgaps is largest and the location of each bandgap is most explicit. Fig. 15 shows that the degree of the vibration attenuation in the direction of the wave propagation is related to the size of the lattice structure in the corresponding direction. Without regard to the load-bearing capacity of the structure, we rotate the resonator inside the unit-cell clockwise to investigate the effect of the rotation angle on the bandgaps. Fig. 16 shows that the bandgaps become narrower as the resonator rotates. In conclusion, the optimal structural design of the AM-based lattice unit-cell for an efficient vibration attenuation is a square unit-cell with a resonator without rotation, and the size of the 23

resonator is half of the lattice unit-cell. The effect of the material parameters on the vibration attenuation in the finite 2D AM-based lattice structures is illustrated in Fig. 17. It is interesting to mention that the frequency response curves for the lattice structures made of steel and aluminum are almost the same as shown in Fig. 17(a). Looking through the parameters of the two materials, we find that the Young’s modulus ratio ESteel EAl almost equals to the mass density ratio ρSteel ρ Al . The wave velocity in a beam can be defined as

c = E ρ . We assume a reference material M with a velocity ratio cSteel cM = 1 , but a Poisson’s ratio different from steel to check its influence. As we can see from Fig. 17(b) that the bandgaps are practically the same which illustrates that the vibration transmission in the considered lattice structure is not sensitive to Poisson’s ratio. The frequency response curves for the lattice structures made of iron and epoxy are also depicted in Figs. 17(c) and (d). When the velocity ratio of the two materials is

c1 c2 ≠ 1 , the bandgaps will be separated. In this case, the frequency bandgaps where the vibration attenuates can be tuned wider by designing the AM-based lattice structures with multi-component materials. (a) 10-2

Al Steel

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Fig. 17. The frequency responses in the horizontal direction of the 2D AM-based lattice structures with 9 × 5 unit-cells made of different materials: (a) aluminum (Al), (b) a reference material M, (c) epoxy, (d) iron, subjected to a plane time-harmonic excitation. The shadowed area represents bandgaps.

Fig. 18. 2D AM-based lattice structure composed of two materials with 9 × 5 unit-cells. -2

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Fig. 19. The frequency responses in the horizontal direction of the 2D AM-based 25

lattice structure composed of two different materials. 6 A 2D AM-based lattice structure composed of two different materials Based on the parametric study on the vibration bandgap properties of the AM-based lattice structures in section 5, a composite lattice structure with two different materials periodically distributed in the horizontal direction is proposed. The considered model is shown in Fig. 18. In the calculation of the vibration responses, the material used in the middle (yellow) of the model is epoxy or iron, and the material on the left and right sides is steel. The structural parameters for the whole structure are the same as in the model depicted in Fig. 8(a). The frequency responses for the composite lattice structure are shown in Fig. 19. The frequency bandgaps where the vibration attenuates marked by the shaded area are wider due to the superposition of the bandgaps generated by the structures made of a single material, although the degree of the vibration attenuation is reduced. As we can see from Fig. 15, the degree of the vibration attenuation can be improved by adjusting the size of the lattice structure made of each material in the direction of the wave propagation. Moreover, if the total lattice structure is made of several partial lattice structures with different materials, and the superposition of bandgaps generated by each partial lattice structure can cover wide frequency ranges, in which the vibration responses at the receiving end of the total lattice structure will attenuate. Thus, the uncertainty and narrow-band property of the bandgaps generated by conventional phononic crystals or acoustic metamaterials can be solved to a certain degree. 7. Conclusions 26

This paper presents a new type of periodic lattice structures designed based on the local resonance mechanism of acoustic metamaterials. The lattice unit-cell has a special cross-type core with a radius jump discontinuity which can form resonators in the lattice structures. The frequency responses for the finite 2D lattice structure excited by a time-harmonic loading are calculated by using a spectral element method (SEM), and the superiority and validity of the SEM are verified by the conventional finite element method (FEM). Elastic wave propagation in infinite 2D AM-based periodic lattice structures is also analyzed. The dispersion curves or band structures are computed by using combination of the SEM and the Bloch-theorem. Bandgaps where mechanical vibration attenuates or elastic waves cannot propagate forward are obtained by the present computational models. When the radius of the resonator is sufficiently large, three obvious bandgaps in the considered frequency ranges emerge. The first bandgap is mainly determined by the radius of the resonators, and the larger the resonator radius is, the wider and lower the first bandgap will be. The second and third bandgaps, which are mainly caused by the destructive interference, is determined by the geometry of the lattice structure. Considering the limitation of the size of the resonator, a lattice unit-cell made of two different materials is proposed which is easy to generate the first bandgap even when the radius of the resonator is relatively small. In the numerical calculations, the effects of the structural and material parameters on the vibration bandgaps have been considered. The optimal structural design of a lattice unit-cell is confirmed. When the Young’s modulus ratio equals to the mass 27

density ratio of the materials used for the lattice structure, the bandgaps generated are basically the same. Then, a design of the AM-based lattice structure with multi-component materials is proposed, and in the case of c1 c2 ≠ 1 , the frequency bandgaps where the vibration attenuates can be tuned wider. The suitability of such a design is demonstrated by the frequency responses of the lattice structure composed of two different materials. The vibration attenuation at both low and high frequencies can be realized in the composite AM-based lattice structures due to the superposition characteristics of the bandgaps induced by the partial lattice structures of different materials. The results of the present work may provide some novel ideas in the field of elastic wave and vibration reduction of the periodic lattice structures under the premise of the structural safety and reliability. Acknowledgements Supports from the National Natural Science Foundation of China (11972184, 11672130), State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-0217G03) and Jiangsu Province Graduate Students Research and Innovation Plan (KYZZ16_0266) are gratefully acknowledged. Xiyue An is also grateful to the China Scholarship Council (CSC, 201606710079) for the PhD Scholarship at the Chair of Structural Mechanics, University of Siegen, Germany. References [1] L. Brillouin, Wave Propagation in Periodic Structures, Dover Publications, Inc, New York, 1946. 28

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Declaration of interest statement The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.