Wave propagation in viscoelastic phononic crystal rods with internal resonators

Applied Acoustics 141 (2018) 382–392

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Wave propagation in viscoelastic phononic crystal rods with internal resonators

T

⁎

Jia Loua,b, Liwen Hea, Jie Yangc, , Sritawat Kitipornchaid, Huaping Wue a

Department of Mechanics and Engineering Science, Ningbo University, Ningbo, Zhejiang 315211, China State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China c School of Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083, Australia d School of Civil Engineering, The University of Queensland, Brisbane, St Lucia 4072, Australia e Key Laboratory of E&M (Zhejiang University of Technology), Ministry of Education & Zhejiang Province, Hangzhou 310014, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Bragg scattering Local resonance Viscoelasticity Band gap

In the present work, the wave propagation in a viscoelastic phononic crystal rod with internal periodic dissipative resonators is investigated. The Kelvin-Voigt model is utilized to describe the viscoelastic behavior of host materials. The Bloch theorem is adopted to analyze the band structure of the rod. The effect of the free oscillation frequency of the resonators on the band structure is firstly studied. It is found that by tailoring the dynamic characteristics of the resonators, the coupling of the Bragg scattering (BS) and local resonance (LR) mechanisms can be harnessed to effectively widen the band gaps and enhance the wave attenuation. Then, the effects of the viscosity of the host materials and the damping of the resonators on the band structure, especially the two nearly coalescent band gaps (the first Bragg and LR ones), are investigated respectively. Furthermore, the combined effect of the two dissipative sources is also discussed. The present work is expected to be helpful to the design and applications of phononic crystals and metamaterials.

1. Introduction Phononic crystals (PCs) are periodic structures made of two or more materials with different elastic properties. They possess frequency band gaps within which wave propagation is forbidden, independent of the wave vector [1–3]. Such band gap feature of PCs can be used in many fields, such as acoustic/elastic filters, acoustic waveguides, noise controllers, and vibration shields [4]. Mead [5] first studied the wave propagation in a periodically supported infinite beam, and revealed the band gap feature of this structure. Ruzzene et al. [6,7] applied the concept of periodic construction to a sandwich plate. They found that by placing negative Poisson’s ratio core materials with different geometries periodically in the plate, the propagation of waves over specified frequency bands and in particular directions can be obstructed. In addition, the exploitation of structural effects such as topology, geometry and local resonance has led to the development of material systems with extraordinary electromagnetic and acoustic properties, i.e., so called “metamaterials” [8–10]. Taking acoustic metamaterials (AMMs) for instance, local resonance leads to a strong attenuation which has significant implications for the suppression of sound transmission [10,11] and mechanical vibrations [12–18]. The concept of

⁎

local resonance has been explored to design metamaterial beams and plates with periodic resonator arrays [19–21]. Most studies on wave propagation in PCs and AMMs mainly focus on elastic medium. However, damping is an intrinsic property of materials and may have a significant influence on structural dynamic responses. To predict the performance of PCs and AMMs more accurately, it is necessary to take dissipative effects into consideration [22]. Mead [23] first investigated the effects of viscous and hysteretic damping on the wave number for a one-dimensional (1D), periodically supported beam. It is found that band gaps became less distinct if the effect of damping was considered, because complex wave numbers existed throughout the frequency spectrum. This result was reproduced qualitatively by Tassilly [24] in his study on damped PC beams. Merheb et al. [25] used the finite difference time domain method to investigate the transmission of acoustic waves in viscoleastic rubber-air PCs. Besides these researches on PCs, Manimala and Sun [26] studied the wave attenuation of 1D dissipative AMMs by considering discrete lattices involving local resonators with different types of viscous damping or hysteretic damping. Chen et al. [27] also adopted similar analysis method to study 1D viscoelastic AMMs with mass-in-mass viscous local resonators. Dissipative metamaterial plates with tunable local

Corresponding author. E-mail address: [email protected] (J. Yang).

https://doi.org/10.1016/j.apacoust.2018.07.029 Received 20 February 2018; Received in revised form 16 July 2018; Accepted 23 July 2018 0003-682X/ © 2018 Elsevier Ltd. All rights reserved.

Applied Acoustics 141 (2018) 382–392

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a k c

m E1 , 1 ,

a1

1

E2 ,

2

,

m

m

m

m

2

a2

Fig. 1. Schematic of a viscoelastic phononic crystal rod with internal mass-dashpot-spring resonators, where is the lattice constant, E1, ρ1 and η1 (E2 , ρ2 and η2 ) are the Young’s modulus, mass density and viscosity of Material 1 (Material 2), and m , k and c are the mass, spring stiffness and damping of each resonator, respectively.

The rest of the present work is structured as follows. In Section 2, the governing equations for each part of the PC rod as well as the internal resonator within each unit cell are firstly derived, and then by employing the Bloch theorem, two eigenvalue problems are formulated respectively in Subsections 2.1 and 2.2 for cases of prescribed and free wave propagation. The band structures of the PC rod with internal resonators under both the non-dissipative and dissipative conditions are presented in Section 3, and the effects of the free oscillation frequency, mass, and damping of the internal resonators as well as the viscosity of the host materials are discussed in detail. At last, some conclusions are drawn in Section 4.

resonators [28] and two-dimensional (2D) viscoelastic metamaterials [29] were also studied by using the simple Kelvin-Voigt model. Recently, more realistic viscoelastic models, such as generalized Maxwell models, were employed to study 2D viscoelastic AMMs by Krushynska et al. [30] and Lewinska et al. [31]. A more recent work even harnessed the combined attributes of AMMs and PCs as well as viscous damping to generate the so called “metadamping” effect [32]. In all these studies, the frequency has a prescribed real value, such that the corresponding temporal parameter λ = −iω is pure imaginary. This case represents a class of researches on wave propagation incorporating dissipative effects and corresponds to a steady-state situation where the medium is driven by a given frequency. Such a case is called “prescribed wave” in the present paper. Alternatively, a wave with its amplitude decreasing with time and its frequency un-prescribed is referred to as “free wave”. In this perspective, Cady [33] studied propagation of longitudinal waves in homogeneous rods. He allowed the temporal parameter λ to become complex, i.e., λ = −ξω ± iωd , where ωd was the damped propagation frequency, and −ξω was the temporal attenuation constant. Mukherjee and Lee [34] investigated wave propagation in one-dimensional PCs, and developed the dispersion relations for both the damped frequency and the temporal attenuation constant. Hussein [35] presented the band structure in the Brillouin zone and the damping ratio corresponding to each Bloch wave, and demonstrated that the damping qualitatively altered the shape of the dispersion curve. They also extended their analysis to the study of a 2D PC, and revealed intriguing phenomena such as branch overtaking and branch cut-off [36]. Sprik and Wegdam [37] analyzed the propagation of sound waves in three dimensional periodic lattices of solid-solid and solid-liquid composites. Zhang et al. [38] investigated the absolute acoustic band gaps for twodimensional periodic arrays of silica cylinders in viscous liquid, and found that when the viscous penetration depth was comparable to the structural length scale, the structures possessed large absolute acoustic band gaps comparing with those without viscosity. It is also noted that Andreassen and Jensen [39] analyzed the bandgap of a 2D dissipative PC for both the two cases of free and prescribed wave propagation, and pointed out that comparable results are predicted for small to medium amounts of material dissipation and for long-wavelength waves. Through the above literature review, it is found that lots of works have been conducted on wave propagation in viscoelastic PCs and AMMs. Although wave attenuation of viscoelastic PCs with internal dissipative resonators has been considered by some researchers such as Krushynska et al. [30] and DePauw et al. [32], the coupling of Bragg scattering (BS) and local resonator (LR) mechanisms in such kind of dissipative systems, or phononic resonators as termed by DePauw et al. [32], has not been completely clarified yet. In order to at least partly address this problem, the wave propagation of a 1D viscoelastic PC rod with internal dissipative resonators is studied in the present work, with both the two cases of free and prescribed wave propagation taken into account. It is found that for both non-dissipative (purely elastic) and dissipative PC rods, the BS-LR coupling can be harnessed to effectively widen the two nearly coalescent band gaps (the Bragg and LR ones, respectively) and enhance the attenuation via tailoring the dynamic characteristics of the resonators. The effects of two dissipative sources (the viscosity of the host materials and the damping of the internal resonators) on the BS-LR coupling and their significant implications to wave propagation are also discussed in detail.

2. Formulation As shown in Fig. 1, a viscoelastic PC rod with periodic internal resonators is considered. The rod is composed of repetition of alternating Material 1 with length a1 and Material 2 with length a2 . The lattice constant is denoted by a which is equal to a1 + a2 . The origin of the local coordinate is located at the junction of Material 1 and Material 2. The internal resonators, each with mass m , spring stiffness k and damping c , are fixed at the left end of each unit cell periodically. The motion equation of the rod in the longitudinal direction reads:

∂σ (r ) ∂2u(r ) A + f (r ) = ρr A , ∂x ∂t 2

(1)

where r = 1, 2 is used for distinguishing Material 1 from Material 2, σ (r ) = σ (r ) (x , t ) , A , f (r ) = f (r ) (x , t ) , ρr , and u(r ) = u(r ) (x , t ) denote the stress, the cross-sectional area of the rod, the external force, the mass density, and the longitudinal displacement, respectively. As indicated, all these quantities, except for the cross-sectional area and the mass density, are dependent of the position x and the time t. Due to the diversity and complexity of dissipative mechanisms, the development of a universal damping model stands as a major challenge. The simple Kelvin-Voigt model, which consists of a spring and a dashpot connected in parallel, is widely used to describe the time-dependent property of viscoelastic medium [40–44]. It should also be noted that due to the lack of enough fitting parameters, the KelvinVoigt model may not fit well with experimental data. To solve this problem, some more sophisticated and experimentally validated models, such as the generalized Kelvin-Voigt model or the generalized Maxwell model, could be used instead. However, via comparison with a validated generalized Maxwell model, Krushynska et al. [30] pointed out that the Kelvin-Voigt model provides reliable results concerning the wave dispersion, except in a very low frequency range which is significantly below band gaps and not of a critical importance. Considering this reason and for the sake of simplicity, the Kelvin-Voigt model is adopted here to describe the viscoelastic behavior of the host materials, so as to focus on the effect of viscosity on the coupling of the BS and LR mechanisms. According to the Kelvin-Voigt model, the constitutive relation reads:

σ (r ) = Er

∂u ∂ 2u + ηr , ∂x ∂x ∂t

(2)

where Er is the Young’s modulus, and ηr is the viscosity. In the absence of body forces, substituting Eq. (2) into Eq. (1) yields the motion equation of the viscoelastic PC rod: 383

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Er A

∂2u(r ) ∂3u(r ) ∂2u(r ) + ηr A 2 −ρr A = 0. 2 ∂x ∂x ∂t ∂t 2

solutions is: (3)

Ap11 Ap21 Ap31 Ap41

Eq. (3) can be used to analyze the propagation of longitudinal waves in the PC rod. In particular, if a PC rod of infinite length (i.e., having no external boundaries off which waves may reflect) is considered, we may apply a plane wave solution of the form: (r ) u(r ) (x , t ) = U (r ) (x ) g (t ) = U¯ (r ) eiκ x g (t ),

where i = −1 , U¯ (r ) is the complex wave amplitude, κ (r ) is the wave number, and g (t ) is an exponential function of the time, which can be expressed as g (t ) = e λt . For a prescribed wave propagation, the temporal parameter λ = −iω, where ω is the oscillation frequency; while for a free wave propagation, λ is generally complex, encompasses an oscillation frequency and a rate of decay. The motion equation of each resonator reads:

(6)

(r )

Vf =

(r )

ρr ω2 /(Er −iωηr ) .

=− Substituting the displacement of the resonator [Eq. (6)] with g (t ) = e−iωt into Eq. (5) yields:

k−iωc U p(1) (−a1). k−iωc−mω2

(0) =

U p(1)

(8)

(0),

(E2−iωη2 ) A [dU p(2)

(x )/ dx ]|x = 0 =

(x )/dx ]|x = 0 .

U p(2) (a2) = eiκaU p(1) (−a1),

(18)

(20)

mλ2 (k + cλ ) − U (1) (k + cλ + mλ2) f

(10)

(−a1) . in which f f (−a1) = Eqs. (17)–(20) can be rewritten in the following form: (1)

⎡ Af 11 ⎢ Af 21 ⎢ ⎢ Af 31 ⎢ Af 41 ⎣

(12)

Af 12 Af 22 Af 32 Af 42

Af 13 Af 23 Af 33 Af 43

Af 14 ⎤ ⎧ bf 1 ⎫ ⎪ (1) ⎪ Af 24 ⎥ ⎪ bf 2 ⎪ ⎥ (2) = 0, Af 34 ⎥ ⎨ b ⎬ f1 ⎪ Af 44 ⎥ ⎪ ⎪ ⎦ ⎪ bf(2) 2 ⎩ ⎭

(21)

in which the expressions for the elements Afij (i, j = 1, 2, 3, 4) are also presented in the Appendix. The condition that Eq. (21) has non-zero solutions is:

mω2 (k − iωc )

U (1) (−a1) . in which fp (−a1) = k − iωc − mω2 p Eqs. (9)–(12) can be rewritten as: (1) Ap14 ⎤ ⎧ bp1 ⎫ ⎪ (1) ⎪ Ap24 ⎥ ⎪ bp2 ⎪ = 0, ⎥ Ap34 ⎥ ⎨ bp(2) ⎬ 1 ⎪ ⎪ Ap44 ⎥ ⎪ (2) ⎪ ⎦ bp2 ⎩ ⎭

(19)

+ f f (−a1)},

(11)

+ fp (−a1)},

Ap13 Ap23 Ap33 Ap43

(E2 + λη2 ) A [dU f(2) (x )/ dx ]|x = 0 = (E1 + λη1 ) A [dU f(1) (x )/dx ]|x = 0 .

(E2 + λη2 ) A [dU f(2) (x )/ dx ]|x = a2 = eiκa {(E1 + λη1 ) A [dU f(1) (x )/ dx ]|x =−a1

(E2−iωη2 ) A [dU p(2) (x )/ dx ]|x = a2 = eiκa {(E1−iωη1 ) A [dU p(1) (x )/dx ]|x =−a1

Ap12 Ap22 Ap32 Ap42

(17)

(9)

(E1−iωη1 ) A [dU p(1)

(16)

U f(2) (0) = U f(1) (0),

U f(2) (a2) = eiκaU f(1) (−a1),

Moreover, the Bloch theorem guarantees that the displacements and the axial forces at the ends of each unit cell satisfy the following relations:

⎡ Ap11 ⎢ Ap21 ⎢ ⎢ Ap31 ⎢ Ap41 ⎣

k + cλ U (1) (−a1). k + cλ + mλ2 f

Moreover, the Bloch theorem guarantees that the displacements and the axial forces at the ends of each unit cell satisfy:

At the junction of Material 1 and Material 2, the continuity of the displacement and the axial force must be satisfied as follows:

U p(2)

(15)

At the junction of Material 1 and Material 2, the continuity of the displacement and the axial force must be satisfied, i.e.,

(7)

where the subscript p represents the prescribed wave propagation, bp(rl ) (l = 1, 2) are the constant amplitudes, κp(r1) = ρr ω2 /(Er −iωηr ) , and

Vp =

(r )

where the subscript f represents the free wave propagation, bfl(r ) (l = 1, 2) are the constant amplitudes, κ f(r1) = −ρr λ2 /(Er + ληr ) , and . Substituting the displacement of the resonator [Eq. (6)] with g (t ) = e λt into Eq. (5) yields:

We firstly focus on the case of prescribed wave propagation. One may imagine this case as the result of a sustained harmonic excitation at the end of a semi-infinite medium. Substituting the displacement function [Eq. (4)] with g (t ) = e−iωt into the governing Eq. (3) yields a general solution of the form:

κp(r2)

(14)

U f(r ) (x ) = bf(r1) eiκf 1 x + bf(r2) eiκf 2 x ,

2.1. Prescribed wave propagation

(r )

= 0.

The case of free wave propagation in the same structure is also considered. One may imagine this case as the result of an initial disturbance at the end of a semi-infinite medium. Substituting the displacement function [Eq. (4)] with g (t ) = e λt into the governing Eq. (3) yields a general solution of the following form:

where the overdot denotes derivative with respect to the time variable t , and v is the displacement of the resonator, which can be expressed as:

U p(r ) (x ) = bp(r1) eiκp1 x + bp(r2) eiκp2 x ,

Ap14 Ap24 Ap34 Ap44

2.2. Free wave propagation

(5)

v (t ) = Vg (t ).

Ap13 Ap23 Ap33 Ap43

Eq. (14) can be used to determine the frequency band structure by solving the wave number κ for varying frequency ω . A complex wave number κ can be expressed as κ = κR + iκI , in which the real part κR supports the propagating mode, whereas the imaginary part κI represents the evanescent mode which decays in space.

(4)

̇ u̇(1) (−a1)] + k [v−u(1) (−a1)] = 0, mv¨ + c [v−

Ap12 Ap22 Ap32 Ap42

Af 11 Af 21 Af 31 Af 41 (13)

Af 12 Af 22 Af 32 Af 42

Af 13 Af 23 Af 33 Af 43

Af 14 Af 24 Af 34 Af 44

= 0. (22)

Eq. (22) can be used to determine λ for varying wave number κ . The temporal parameter λ can be written in the following form as suggested in Refs.[33,35,36]:

in which the expressions for all the elements Apij (i, j = 1, 2, 3, 4) are given in the Appendix. The condition that Eq. (13) has non-zero 384

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J. Lou et al.

λ = −ξω ± iω 1−ξ 2 = −ξω ± iωd ,

rods with internal undamped resonators. Fig. 3(a) reveals the existence of some frequency band gaps. These band gaps arise due to the interference effects introduced by impedance mismatch, and consequently are termed as “Bragg band gaps”. It is determined that the dimensionless frequency range of the first Bragg band gap is [2.323, 6.057]. The initial, central, and terminal frequencies of the first Bragg band gap of the PC rod without resonators are denoted by ω¯ BI , ω¯ BC , and ω¯ BT here and thereafter. Fig. 3(b) shows that when the dimensionless free oscillation frequency of the internal resonators ω¯ m is much lower than ω¯ BI (ω¯ m = 1.000 ), a narrow band gap with quite high attenuation values appears in the low frequency region. This band gap arises due to the interaction between the incident waves and the resonators and is strongly related with the local resonance phenomenon of the resonators. Thus such a band gap is usually termed as “local resonance (LR) band gap”. It is shown that the Bragg band gaps and the LR band gap have no obvious effect on each other in this case. In Fig. 3(c)–(e), ω¯ m are respectively chosen as ω¯ BI , ω¯ BC , and ω¯ BT . Fig. 3(c) shows that due to the coupling of the Bragg scattering (BS) and LR mechanisms, the LR band gap becomes wider comparing with the case that ω¯ m = 1.000 . Meanwhile, the initial frequency of the first Bragg band gap increases, while the terminal frequency is almost unaffected. Hence, the first Bragg band gap becomes narrower. As ω¯ m is increased to match ω¯ BC [Fig. 3(d)], the LR band gap becomes even wider, while the first Bragg band gap gets even narrower (with a higher initial frequency and unchanged terminal frequency). When ω¯ m is exactly equal to ω¯ BT , as shown in Fig. 3(e), the coupling of the BS and LR mechanisms is quite strong. As a result, the first Bragg band gap and the LR band gap seem to merge together to create a single hybrid “Bragg-LR” band gap. Even so, a quite narrow pass band separating the two band gaps still exists under this condition. Such a phenomenon is similar to that reported by Sharma [45]. It is also evident from Fig. 3(e) that LR band gap becomes the second band gap, while the first Bragg band gap becomes the first one. ¯ increases from 0.5 to If the dimensionless mass of the resonators m 2.0, and the dimensionless free oscillation frequencies ω¯ m are still chosen as ω¯ BI , ω¯ BC , and ω¯ BT , the band structures of the PC rods without and with internal resonators are displayed in Fig. 4. Again, two seemingly contacted wide band gaps with strong attenuation arise when ω¯ m matches ω¯ BT . Moreover, it is found that with the increase of the mass of the internal resonators, the effect of the LR mechanism becomes more significant, and consequently, the band gaps become even wider, and the attenuation in the band gaps gets stronger. These results imply that when applying the band gap effect of PCs with internal resonators to vibration suppression, one can harness the abovementioned BS-LR coupling mechanism to effectively widen the band gaps and enhance the attenuation via tailoring the dynamic characteristics of the resonators. In order to validate these results, the transmission spectra of vibration in a finite PC rod with internal resonators are computed via the finite element method. The finite element model is constructed by employing the finite element software ABAQUS, wherein the PC rod is modeled by 1D truss elements and concentrated mass points are connected to the rod by using spring elements. Steady-state dynamic analysis is carried out and the transmission is calculated by log(ur ul ) , where ur is the amplitude of the displacement at the right point of the PC rod, and ul is the amplitude of the displacement at the left point. Fig. 5 clearly shows that the positions of the stop bands of the transmission spectra agree quite well with those of the predicted band gaps, and the vibration attenuation in each stop band increases significantly with the number of repetitive unit cells in the rod. Actually, 10 repetitive unit cells will be enough to generate evident attenuation in the stop bands. Moreover, comparison between the transmission spectra for ω¯ = ω¯ BI and ω¯ = ω¯ BT respectively displayed in Fig. 5(a) and (b) also confirms the approaching of the first Bragg and LR band gaps and the presence of a narrow gap between them in the purely elastic rod.

(23)

where ω and ωd are respectively the undamped and damped angular frequencies, and ξ is the associated damping ratio. A wave is free to propagate with an angular frequency ωd = Abs(Imag (λ )) and a rate of amplitude decay specified by Abs(Real(λ )) . The damping ratio ξ can be expressed as ξ = −Real(λ )/Abs(λ ) . 3. Results and discussions A bi-material PC rod with equal length a1 = a2 = 0.5 m , equal crosssectional area A = 0.01 m2 is considered. The material parameters of the PC rod are listed below: Material 1: ABS polymer, ρ1 = 1040 kg/m3 , E1 = 2.4 GPa ; Material 2: Aluminum, ρ2 = 2700 kg/m3 , E2 = 68.9 GPa . The following dimensionless quantities are defined: Dimensionless undamped and damped frequencies: ω¯ = ωa c1 and ω¯ d = ωd a c1, where c1 = E1 ρ1 ; Dimensionless viscosity of the host materials: q = (ηr / Er )(c1/ a) ; ¯ = m /[(ρ1 a1 + ρ2 a2 ) A]; Dimensionless mass of the resonators: m Dimensionless free oscillation frequency of the resonators: ω¯ m = ωm a/ c1; Dimensionless damping of the resonators: ξm = c /(2 km ) . In this section, the present analysis is firstly validated by a comparison study. Then, the effects of some parameters, such as the mass, the damping, and the free oscillation frequency of the internal resonators, as well as the viscosity of the host materials, on the frequency band structure of the PC rod are discussed. 3.1. Validation study By adopting the transfer matrix method, Frazier [22] developed an analytical formula for the band structure of a PC rod. The band structure predicted by the present formula is compared with that by the analytical formula given in Ref. [22], as shown in Fig. 2. It is found that the present formula gives the same results with Frazier [22]. 3.2. Coupling of the Bragg scattering and local resonance mechanisms The band structures of PC rods without and with internal resonators are calculated and compared in this subsection, and the effects of the free oscillation frequency and the mass of the resonators on the band structure are discussed. Fig. 3(a) shows the frequency band structure of the PC rod without resonators, while Fig. 3(b)–(e) display the band structures of the PC

Dimensionless frequency a/c1

20 16

q=0 Without resonators

12 Present Frazier [22]

8 4 0

0

1 2 Real ( a)

30

1 2 Imag ( a)

3

Fig. 2. Band structure of an elastic PC rod without resonators, where c1 = E1 ρ1 is the longitudinal wave speed of Material 1.

385

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b 20 Dimensionless frequency a/c1

Dimensionless frequency a/c1

a 20 q=0 Without resonators

16

16

q=0 With resonators

12

12 8 4

8

m=0.5 =1.000 m

4

m

0

0

1 2 Real ( a)

30

1

2 3 4 Imag ( a)

5

0

=0 1 2 Real ( a)

30

1

30

1

2 3 Imag ( a)

4

5

4

5

d 20 Dimensionless frequency a/c1

Dimensionless frequency a/c1

c 20 q=0 With resonators

16 12

m=0.5 = BI m

8

=0

m

4 0

0

0

1 2 Real ( a)

30

1

2 3 Imag ( a)

4

16

q=0 With resonators

12

m=0.5 = BC m

8 4 0

5

=0

m

0

1 2 Real ( a)

2 3 Imag ( a)

Dimensionless frequency a/c1

e 20 16

q=0 With resonators

12 8

m=0.5 = BT m

4 0

=0

m

0

30

1 2 Real ( a)

1

2 3 Imag ( a)

4

5

¯ , ξm = c /(2 km ) and ω¯ m are the Fig. 3. Effect of the free oscillation frequency of the resonators on the band structure of an elastic PC rod (m ¯ = 0.5). Here, m dimensionless mass, damping ratio and dimensionless free oscillation frequency of each resonator, and ω¯ BI , ω¯ BC and ω¯ BT are respectively the dimensionless initial, central and terminal frequencies of the first Bragg band gap of the corresponding PC rod without resonators.

propagating wave may increase or decrease with the increase of the viscosity, depending on the specific branch of the dispersion curve, while the frequency of a freely propagating wave for each branch decreases. Moreover, in the case of free wave propagation, the frequency on a higher branch descends more rapidly than that of a lower one. The imaginary part of the dimensionless wavenumber depicted in the right half of Fig. 6(a) shows that the spatial attenuation of prescribed wave increases with the viscosity of the host materials in the whole frequency spectrum, while the damping ratio diagram displayed in the right half of Fig. 6(b) is a measure of the temporal decay associated to each freely propagating wave. The present trends of the effects of the viscosity on the prescribed and freely propagating waves agree qualitatively with those reported by Frazier [22]. It is also found that for small to medium material dissipation and for low-frequency, long-wavelength waves, the dispersion curves of prescribed and free waves resemble each other, which is the same with the results predicted by Andreassen and Jensen [39]. Fig. 7 shows the initial and terminal frequencies of the first Bragg

3.3. Effect of the viscosity of the host materials In this subsection, the effect of the viscosity of the host materials on the band structure of the PC rod is studied. Both the cases of prescribed wave propagation and free wave propagation are considered. The prescribed wave propagation, which is governed by g (t ) = e−iωt , allows only a spatial attenuation, may be envisioned as the response of a semiinfinite medium subject to a sustained harmonic excitation at the end. While the free propagation, which adheres to g (t ) = e λt , permits a temporal decay, may be imaged as the result of an initial disturbance. Firstly, a viscoelastic PC rod without resonators is considered, and the effects of the viscosity of the host materials on the band structure for the prescribed and freely propagating waves are respectively shown in Fig. 6(a) and (b). In the absence of viscosity, the frequency band structures for the prescribed and freely propagating waves are completely the same, as there is no difference between these two cases. While in the presence of viscosity, the frequency of a prescribed 386

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b 20 Dimensionless frequency a/c1

Dimensionless frequency a/c1

a 20 q=0 Without resonators

16

16

12

12

8 4 0

0

1 2 Real ( a)

30

1

2 3 4 Imag ( a)

m

=0

4

0

1 2 Real ( a)

30

1

2 3 Imag ( a)

4

5

30

1

2 3 Imag ( a)

4

5

d 20 m=2.0 = BC m

16

Dimensionless frequency a/c1

Dimensionless frequency a/c1

m=2.0 = BI m

8

0

5

c 20

=0

m

12

q=0 With resonators

8 4 0

q=0 With resonators

0

30

1 2 Real ( a)

1

2 3 Imag ( a)

4

q=0 With resonators

16 12 8

m=2.0 = BT m

4 0

5

=0

m

0

1 2 Real ( a)

¯ = 2.0 ). Here, ω¯ m in (b)–(d) is the dimensionless free Fig. 4. Effect of the free oscillation frequency of the resonators on the band structure of an elastic PC rod (m oscillation frequency of each resonator, and ω¯ BI , ω¯ BC , and ω¯ BT are respectively the dimensionless initial, central, and terminal frequencies of the first Bragg band gap of the corresponding PC rod without resonators.

gap in purely elastic PCs or AMMs. However, for the case of prescribed wave propagation, it seems that the term “band gap” is not so distinct, as complex wavenumber (neither real nor pure imaginary wavenumber) exists throughout the frequency spectrum. Maybe another term called “stop band” (opposite to pass band) is more sensible, which can be directly defined as an analogue of the same term prevalently used in the fields of signal processing [46] and vibration control [47,48]. Qualitatively speaking, the concept of stop band used here means the continuous frequency range, within which severe attenuation occurs. To be more quantitative, the attenuation within the stop band should be above a specified level. It should be noted that wave attenuation is allowed outside the stop band and even through the whole frequency spectrum, which is quite different from the traditional

band gap of the PC rod for the freely propagating waves, which reveals more clearly that with the increase of the viscosity, the initial frequency varies slightly, while the terminal frequency decreases significantly. As a result, both the width and the central frequency of the first Bragg band gap decline. For the case of free wave propagation in viscoelastic PCs or AMMs, it is straightforward to define band gap as a continuous frequency range within which free wave propagation is forbidden, independent of the wave vector. Specifically, in the wave dispersion ω (k) diagram as shown in Fig. 6(b), it is the frequency range where the frequency ωd (the absolute value of the imaginary part of the temporal parameter λ = −ξω ± iωd ) does not exist for any wavenumber in the first and irreducible Brillouin zone. This definition is analogous to that of the band

10

b 10

0

0

-10 -20

3 Unit Cells 5 Unit Cells 10 Unit Cells 20 Unit Cells

-30 -40

0

500

1000

1500

2000

2500

Transmission (dB)

Transmission (dB)

a

3000

Frequency (Hz)

-10 -20 3 Unit Cells 5 Unit Cells 10 Unit Cells 20 Unit Cells

-30 -40

0

500

1000

1500

2000

2500

3000

Frequency (Hz)

Fig. 5. Transmission of vibration in finite elastic PC rods with internal undamped resonators (The number of unit cells in each rod is given in each figure): (a) m ¯ = 2.0 , ω¯ m = ω¯ BI ; (b) m ¯ = 2.0 , ω¯ m = ω¯ BT (ω¯ BI and ω¯ BT are respectively the dimensionless initial and terminal frequencies of the first Bragg band gap of the corresponding PC rod without resonators). 387

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b 20

q=2.5*10

a/c1

-2

16

-2

8

q=0

4 0

Prescribed Without resonators 0

1 2 Real ( a)

0.4

30

1

2 3 4 Imag ( a)

5

q=0

-2

12

q=2.5*10

0.3

-2

0.2

8 Without resonators Free

4 0

6

q=5.0*10

Damping ratio

12

0.5

d

q=5.0*10

16

Dimensionless frequency

Dimensionless frequency a/c1

a 20

0

1 2 Wavenumber a

0.1

30

1 2 Wavenumber a

3

0.0

Fig. 6. Effect of the viscosity of the host materials on the band structures of a viscoelastic PC rod without resonators for (a) prescribed wave propagation and (b) free wave propagation. Here, q is the dimensionless viscosity of the host materials.

exists. Moreover, whether the first Bragg and LR stop bands merge or not in the transmission spectrum depends on the type of dissipation source in the rod. However, it should also be noted that the transmission spectrum of vibration, which actually describes the frequency response of vibration of a finite structure, is similar to the spatial attenuation-frequency curve of prescribed waves (comparing the right half of Fig. 8(a) with Fig. 10(b)), but has no direct relation with the dispersion curve of freely propagating waves.

notion of band gap. Then, we focus on a viscoelastic PC rod with internal resonators. In order to obtain wide and strongly attenuated band gaps in the low frequency region, the dimensionless mass m ¯ of the resonators and the dimensionless free oscillation frequency ω¯ m are respectively chosen as 2.0 and ω¯ BT . Fig. 8 shows that similar to the PC rod without resonators, the frequency of a prescribed propagating wave may increase or decrease as the viscosity of the host materials increases. However, the frequency of a freely propagating wave decreases with the increase of the viscosity, except for the third branch. As a result, the width of the second band gap for freely propagating waves increases with the increase of the viscosity. This phenomenon is more clearly shown in Fig. 9. It is thought that the emergence of this phenomenon is caused by the strong coupling of the BS and LR mechanisms. It implies that the coupling of the BS and LR mechanisms as well as the viscosity of the host materials can be simultaneously harnessed to increase the “total width” of the two nearly coalescent BS and LR band gaps for freely propagating waves, although the separation between them also increases slightly with the viscosity. The transmission spectra of vibration in finite viscoelastic PC rods with internal resonators are also computed by the finite element method, as shown in Fig. 10. Comparison between the transmission spectra for the viscoelastic PC rod with internal resonators and the corresponding purely elastic one shows that the viscous dissipation leads to the enhancement of vibration attenuation, not only in the original stop bands (i.e. the band gaps of the purely elastic PC rod), but more significantly in the original pass bands. It is also confirmed by Fig. 10(b) that even in the presence of viscosity and/or damping of the internal resonators, the coupling of the BS and LR mechanisms still

In this subsection, the effect of the damping ξm of the internal resonators on the band structure of the PC rod is studied. Since the present work mainly focuses on the intrinsic dynamic characteristics of the PC rod, only the case of free wave propagation is studied in the fol¯ of the resonators and the lowing analyses. The dimensionless mass m dimensionless free oscillation frequency ω¯ m are chosen as 2.0 and ω¯ BT respectively. Fig. 11 shows the frequencies of the freely propagating waves increase as the damping of the resonators increases, and this phenomenon is more obvious for high frequency branches. Fig. 12 displays the effect of the damping of the resonators on the first two band gaps of the PC rod. It shows that the first band gap keeps nearly unchanged with the increase of ξm , while the width of the second band gap increases with the increase of ξm , especially when ξm is smaller than 0.50. Hence, the damping of the internal resonators can be harnessed to widen the second band gap (i.e., the LR band gap) for free wave propagation, without affecting the first Bragg band gap. Similar strategies have also been proposed by Chen et al. [27] and Lewinska et al. [31].

a/c1 d

6.1

Dimensionless frequency

b

2.33

6.0

Dimensionless frequency

d

a/c1

a

3.4. Effect of the damping of the resonators

2.32

2.31

2.30 0.00

Initial frequency

0.02

0.04

0.06

0.08

0.10

5.9 Terminal frequency 5.8

5.7 0.00

0.02

0.04

0.06

0.08

0.10

Viscosity parameter q

Viscosity parameter q

Fig. 7. Effect of the viscosity of the host materials on (a) the initial frequency and (b) the terminal frequency of the first Bragg band gap of a viscoelastic PC rod without resonators for free wave propagation. 388

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b 20

m=2.0 =0 m

-2

q=0 5

0

m

BT

Prescribed With resonators 0

1 2 Real ( a)

30

1

2 3 4 Imag ( a)

5

15

q=2.5*10

q=0

q=5.0*10

10

0.3 -2

0.2

Free With resonators 5

m=2.0, = m

0

6

-2

Damping ratio

a/c1

-2

q=2.5*10

10

0.4

d

q=5.0*10

15

Dimensionless frequency

Dimensionless frequency a/c1

a 20

0

0.1

=0

m

BT

1 2 Wavenumber a

30

1 2 Wavenumber a

3

0.0

20

0.04

a/c1

10 d

8 6 4

The first band gap

2 0 0.00

0.02

0.04

0.06

0.08

Fig. 9. Effect of the viscosity of the host materials on the first two band gaps of a viscoelastic PC rod with internal undamped resonators (free wave propaga¯ = 2.0 , ω¯ m = ω¯ BT ). tion, m

Transmission (dB)

Transmission (dB)

b

0 -10 -20

q=0, q=0,

-30 -40

m m

=0 =0.05

q=0.01, q=0.01,

0

500

1000

1500

2000

m

=0

=0.25

m

m

0.01

5 Free, with resonators q=0, m=2.0, m= BT 0

1 2 30 Wavenumber a

1 2 3 Wavenumber a

0.00

20

q=0, q=0,

10

m m

=0 =0.05

q=0.01,

0

q=0.01,

m m

=0 =0.05

-10 -20 -30

=0

=0.05 m

2500

0.02

Consequently, the width of the second band gap varies observably. It is thought that the strong coupling of the BS and LR mechanisms has a significant influence on the terminal frequency of the second band gap. For each of the four cases, the width of the second band gap firstly decreases to attain a minimum value with the increase of ξm , and after that, the band gap width increases with ξm . The damping of the resonators corresponding to the minimum band gap width (which is defined as ξmmin ) depends on the viscosity of the host materials. Actually, the value of ξmmin increases with the increase of the viscosity q . These results show that when harnessing the coupling of the BS and LR mechanisms as well as the two dissipative sources to engineer the band gaps for free wave propagation, the chosen of the viscosity of the host

The combined effect of the viscosity q of the host materials and the damping ξm of the resonators on the band structure of free wave propagation in the PC rod is investigated in this subsection. The di¯ of the resonators and the dimensionless free osmensionless mass m cillation frequency ω¯ m are chosen as 2.0 and ω¯ BT respectively. Fig. 13 shows that the effect of ξm on the band structure of the PC rod is strongly dependent of q . In all the four cases considered, the first band gap keeps nearly unchanged, and the initial frequency of the second band gap also varies very slightly. While the terminal frequency of the second band gap changes significantly with the variations of q and ξm .

10

0.03

=0.50

Fig. 11. Effect of the damping ratio ξm on the band structure of an elastic PC rod with internal damped resonators for free wave propagation. Here, ω¯ BT is the dimensionless terminal frequency of the first Bragg band gap of the corresponding PC rod without resonators.

3.5. Combined effect of the viscosity of the host materials and the damping of the resonators

20

m

10

0

0.10

Viscosity parameter q

a

15

Damping ratio

The second band gap

Dimensionless frequency

Dimensionless frequency

d

a/c1

Fig. 8. Effect of the viscosity of the host materials on the band structure of a viscoelastic PC rod with internal undamped resonators for (a) prescribed wave propagation and (b) free wave propagation.

3000

Frequency (Hz)

-40

0

500

1000

1500

2000

2500

3000

Frequency (Hz)

Fig. 10. Transmission of vibration in different kinds of finite PC rods with internal resonators (Each rod consists of 20 unit cells). The shadow areas in each figure ¯ = 2.0 , ω¯ m = ω¯ BI ; (b) m ¯ = 2.0 , ω¯ m = ω¯ BT . denote the band gaps of the corresponding elastic PC rod with undamped resonators (q = 0 , ξm = 0 ): (a) m 389

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Dimensionless frequency

d

a/c1

12

Table 1 Dimensionless central frequency ω¯ d and width of the first Bragg band gap of the PC rod without resonators (free wave propagation).

10 8

The second band gap

Central frequency Width

6

q = 0.02

q = 0.04

q = 0.06

q = 0.08

q = 0.10

4.190 3.733

4.184 3.723

4.166 3.691

4.137 3.638

4.095 3.563

4.040 3.465

q is the dimensionless viscosity of the host materials.

4

The first band gap Table 2 Dimensionless central frequency ω¯ d and width of the first two band gaps of the ¯ = 2.0 , ξm = 0 , ω¯ m = ω¯ BT ). PC rod (free wave propagation, m

2 0 0.0

q=0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 Damping of the resonators m

0.8

materials and the damping of the resonators should be considered simultaneously so as to effectively increase the “total width” of the two band gaps (with a small separation between them). The dimensionless central frequencies and widths of the band gaps under various conditions are also given in Tables 1–3. These data can be used as a reference for further studies.

The wave propagation in a viscoelasitc phononic crystal rod with internal mass-dashpot-spring resonators is investigated in the present

3.610 4.766

3.605 4.756

3.590 4.726

3.564 4.676

3.527 4.604

3.478 4.510

The second band gap Central frequency 7.276 Width 2.546

7.282 2.563

7.304 2.615

7.343 2.708

7.404 2.845

7.481 3.100

a/c1 Dimensionless frequency

The second band gap

6 4

The first band gap

2 0.1

0.2 0.3 0.4 0.5 0.6 0.7 Damping of the resonators m

d

8

a/c1 d

q=0.06

The second band gap

6 4

The first band gap

2 0.1

0.2 0.3 0.4 0.5 0.6 0.7 Damping of the resonators m

0.8

12 10

q=0.04

8

The second band gap

6 4

The first band gap

2 0 0.0

0.8

12

0 0.0

q = 0.10

d

q=0.02

8

10

q = 0.08

Dimensionless frequency

a/c1 d

Dimensionless frequency Dimensionless frequency

d

a/c1

c

q = 0.06

b

12

0 0.0

q = 0.04

work. The Kelvin-Voigt model is used to describe the viscoelastic behavior of the host materials. The frequency band structure of the PC rod is obtained by adopting the Bloch theorem. Firstly, the coupling of the Bragg scattering (BS) and local resonance (LR) mechanisms is discussed, and it is found that by exploiting such a coupling effect, the mass and the free oscillation frequency of the internal resonators can be tailored to obtain wider band gaps with stronger wave attenuation. Then, the effect of the viscosity of the host materials on the band structure is analyzed for both the cases of prescribed wave propagation

4. Conclusions

10

q = 0.02

The first band gap Central frequency Width

Fig. 12. Effect of the damping ratio ξm on the first two band gaps of an elastic ¯ = 2.0 , ω¯ m = ω¯ BT ). PC rod with damped resonators (free wave propagation, m

a

q=0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 Damping of the resonators m

0.8

12 10

q=0.08

8 The second band gap 6 4

The first band gap

2 0 0.0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 Damping of the resonators m

0.8

Fig. 13. Combined effect of the dimensionless viscosity q of the host materials and the damping ratio ξm of the resonators on the first two band gaps of the PC rod ¯ = 2.0 , ω¯ m = ω¯ BT , where ω¯ BT is the dimensionless terminal frequency of the first Bragg band gap of the corresponding PC rod without (free wave propagation, m resonators). 390

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propagation, especially the two nearly coalescent band gaps (the first Bragg and LR ones). Therefore, the coupling of the BS and LR mechanisms and the two dissipative sources can be simultaneously harnessed to widen the band gaps and enhance the wave attenuation. It is expected that the results presented here can be used to expand the design strategy of PCs and benefit the design of vibration insulators and acoustic filters.

Table 3 Dimensionless central frequency ω¯ d and width of the first two band gaps of the ¯ = 2.0 , ω¯ m = ω¯ BT ). PC rod (free wave propagation, q = 0 , m ξm = 0.1

ξm = 0.2

ξm = 0.3

ξm = 0.4

ξm = 0.5

The first band gap Central frequency Width

3.611 4.768

3.614 4.771

3.617 4.773

3.619 4.773

3.622 4.773

The second band gap Central frequency Width

7.344 2.626

7.550 2.902

7.874 3.401

8.206 3.946

8.385 4.219

Acknowledgements The work described in this paper was funded by National Natural Science Foundation of China (No. 11602117, No. 11602118), Australian Research Council under Discovery Project scheme (DP140102132, DP160101978), and also sponsored by K.C. Wong Magna Fund in Ningbo University. The authors are grateful for their financial support.

and free wave propagation. Furthermore, the effect of the damping of the resonators on the band structure and the combined effect of the two dissipative sources are also discussed. It is demonstrated that both the two dissipative sources affect the band structure for free wave Appendix Expressions for the matrix elements Apij (i, j = 1, 2, 3, 4)

Ap11 = Ap12 = 1, Ap13 = Ap14 = −1, (1) (1) Ap21 = iκ p1 E1 (1−iωq1), Ap22 = iκ p2 E1 (1−iωq1),

(2) (2) Ap23 = −iκ p1 E2 (1−iωq2), Ap24 = −iκ p2 E2 (1−iωq2), (1)

(1)

(2)

(2)

Ap31 = ei (κa − κp1 a1) , Ap32 = ei (κa − κp2 a1) , Ap33 = −eiκp1 a2, Ap34 = −eiκp1 a2, (1) mω2 (k−iωc ) ⎤ i (κa − κp1 (1) a1) Ap41 = ⎡iκ p1 E1 A (1−iωq1) + e , ⎢ ⎥ k−iωc−mω2 ⎦ ⎣ (1) mω2 (k−iωc ) ⎤ i (κa − κp2 (1) a1) Ap42 = ⎡iκ p2 E1 A (1−iωq1) + e , 2⎥ ⎢ k iωc mω − − ⎣ ⎦ (2)

(2)

(2) (2) Ap43 = −iκ p1 E2 A (1−iωq2 ) eiκp1 a2, Ap44 = −iκ p2 E2 A (1−iωq2 ) eiκp2 a2.

Expressions for the matrix elements Afij (i, j = 1, 2, 3, 4)

Af11 = Af12 = 1, Af13 = Af14 = −1, Af21 = iκ f1(1) E1 (1 + λq1), Af22 = iκ f2(1) E1 (1 + λq1), Af23 = −iκ f1(2) E2 (1 + λq2), Af24 = −iκ f2(2) E2 (1 + λq2), (1)

Af31 = ei (κa − κ f1

a1) ,

(1)

Af32 = ei (κa − κ f2

a1) ,

(2)

Af33 = −eiκ f1

a2 ,

(2)

Af34 = −eiκ f2

a2 ,

mλ2 (k + cλ ) ⎤ i (κa − κ (1) a1) f1 Af41 = ⎡iκ f1(1) E1 A (1 + λq1)− e ⎢ k + cλ + mλ2 ⎥ ⎣ ⎦ mλ2 (k + λc ) ⎤ i (κa − κ (1) a1) f2 Af42 = ⎡iκ f2(1) E1 A (1 + λq1)− e ⎢ k + λc + mλ2 ⎥ ⎣ ⎦ (2)

Af43 = −iκ f1(2) E2 A (1 + λq2 ) eiκ f1

a2 ,

(2)

Af44 = −iκ f2(2) E2 A (1 + λq2 ) eiκ f2

a2 .

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