Wave Attenuation in 1-D Viscoelastic Phononic Crystal Rods Using Different Polymers

The elastic wave attenuation in artificial composites, known as phononic crystals (PnCs), is an important topic in the context of wave manipulation. However, there is a lack of knowledge in the obtainment of the wave attenuation of PnCs when the viscoelastic effect of different polymers is considered. In this study


Introduction
Artificial composites, known as phononic crystals (PnCs) and/or mechanical metamaterials, are structures designed to achieve unusual properties and applications, such as vibration control 1,2 , acoustic barriers 3,4 , wave manipulation (e.g., guiding, focusing, imaging, cloaking, and topological insulation) 5,6 , energy harvesting 7 , and negative Poisson's ratio 8 .They are typically composed by periodic arrays of inclusions embedded in a matrix 9,10 to open up band gaps arising from Bragg scattering and/or local resonance 11 .In these forbidden ranges of frequency (i.e., band gaps), there are only evanescent Bloch waves 12 and waves cannot propagate.
The viscoelastic effect in most cases is neglected considering the wave propagation in PnCs [12][13][14] .However, more recently, this effect is being studied for the viscoelastic phononic crystals (VPnCs) 11,[15][16][17][18][19][20][21][22] .Vakilifard and Mahmoodi 16 investigated the complex band structure, wave transmission spectra, and effective stiffness of a carbon nanotube-modified viscoelastic locally resonant metamaterial (LRM).They 16 observed that this nano LRM exhibits simultaneously higher stiffness and loss factor with respect to ordinary LRMs.Moreover, they 16 also reported that the volume fraction and aspect ratio have the most influential effects on the nano LRM properties.Chen et al. 19 noted that the viscoelasticity of constituent materials has a significant effect on the band structure of the VPnCs.They 19 found novel topological patterns of the optimized VPnCs by using the bi-directional evolutionary structure optimization (BESO) method.Dal Poggetto and collaborators 11 studied the complex band structure of hierarchical VPnC thick plates.They 11 mentioned that an increase in the hierarchical order leads to a weight reduction with relatively preserved attenuation characteristics (for the case of a hard purely elastic matrix with soft viscoelastic inclusions).They 11 also reported that changing the hierarchical order implies in opening band gaps in distinct frequency ranges, with an overall attenuation improved by an increase in the viscoelasticity (for the case of soft viscoelastic matrix with hard purely elastic inclusions).
The majority of the studies about wave propagation in VPnCs considered simple models of viscosity, for instance, the Kelvin-Voigt model (consists of a spring and dashpot in parallel 23 , i.e., the viscosity is proportional to the frequency).However, the simpler the model is, the more difficult it is to be applied to simulate the behavior of a viscoelastic material.
On this account, the standard linear solid model (SLSM) is considered in this study, which contains three elements, i.e., a Maxwell model (a spring and dashpot in series) and a spring in parallel 23 .The SLSM is one of the most common and popular viscoelastic models used to describe and analyse viscoelastic behaviors of viscoelastic solids 24 .It can be used to simulate viscoelastic behaviors for understanding rich phenomena and underlying mechanisms of viscoelasticicity 24 .The SLSM has several advantages, for instance, it has only three elements, therefore its analysis results are much easier to be interpreted compared to those using models having a large number of elements (such as the generalized Maxwell and Kelvin models) 24 .
Another interesting and necessary aspect in terms of wave propagation in VPnCs is to consider the presence of evanescent waves in the band structure (i.e., the complex band structure) calculation 25 .Recently, some strategies of increasing wave attenuation have been proposed by maximizing spatial decay of evanescent waves, for example by topology optimization 19 , multiple periodic resonators 12 , coupling local resonance and Bragg band gaps 12 , and using hierarchical periodic structures 11 .Very recently, Schalcher et al. 26 formulated the extended plane wave expansion (EPWE) to handle 1-D VPnC solids (i.e., plane strain condition) modelled by the SLSM to obtain the complex band structure.However, they 26 considered just one type of polymer, i.e., epoxy as the viscoelastic matrix, and bulk wave propagation (i.e., the 1-D VPnC solid has infinite thickness and height).
To the best of our knowledge, this is the first study to investigate the complex band structure of longitudinal waves in 1-D VPnC rods with different polymers considering the SLSM.In this context, the main purpose of this investigation is to analyze the complex band structure of 1-D VPnC rods composed by steel inclusions in a polymeric matrix considering the viscoelastic effect modelled by the SLSM.The influence of different materials for the polymeric matrix (i.e., epoxy, nylon, silicon rubber, natural rubber, and low density polyethylene (LDPE)) on the band gap behavior and wave attenuation is analyzed.The improved plane wave expansion (IPWE), ( ) k ω approach 27,28 , and the EPWE, ( ) k ω approach 29,30 , are used to calculate the band structure of the VPnC rod, where ω is the angular frequency and k is the wave number.
The paper is organized as follows.Section 2 presents details for the 1-D VPnC rod modelling.In Section 3, simulated examples are carried out.Conclusions are presented in Section 4.

1-D Viscoelastic Phononic Crystal Rod Modelling
Figure 1 sketches the front view of the 1-D VPnC rod with steel inclusions (white) distributed in a polymeric matrix (i.e., epoxy, nylon, silicon rubber, natural rubber, or LDPE) (blue) (), where  is the unit cell length, and the SLSM 23 (b).The first Brillouin zone (FBZ) 31 for the 1 . The SLSM is used to consider the viscoelasticity of the polymeric matrix, where G 1 and G 2 are the shear modulus (springs), and η is the viscosity (dashpot) 23 in Figure 1b.It is important to mention that the inclusions and matrix are considered as isotropic materials in this study.
For the modelling, IPWE and EPWE approaches are used to calculate the band structure (only propagating waves) and complex band structure (both propagating and evanescent waves), respectively.IPWE and EPWE formulations are similar to those derived for 1-D PnCs 32 .However, those formulations 32 can handle only the case of 1-D PnCs without viscoelastic effect.
The EPWE approach is important to obtain the information of wave attenuation of the unit cell (i.e., { } k a I

12
).Moreover, by using EPWE, one can calculate the purely imaginary and complex values of k (i.e., evanescent wave modes) that cannot be directly computed by the PWE.Thus, the use of EPWE method is necessary for damped structures and VPnCs, since all wave modes become evanescent.
In order to include the viscoelasticity, the elastic constants in the study of Assis et al. 32 , considering the polymeric matrix, should be calculated following the Equation 1 23,26 : where u(t) is the unit step function, t is the time, τ G is the relaxation time, G 0 and G ∞ are the initial and final states of the elastic constants, and they are related to G 1 and G 2 (see Figure 1b) as G ∞ =G 2 and G 1 =G 0 -G ∞ .Applying the temporal Fourier transform to Equation 1, results in 26 : where i is the unit imaginary number.It should be highlighted that IPWE method has higher convergence than the traditional plane wave expansion (PWE) method 27,28 .Moreover, the band structure computed by IPWE shows a considerably lower computational cost 11 , since it is a semi-analytical approach and it is not necessary to consider a large number of degrees-of-freedom (DOFs) as with finite element (FE) 33 for instance.

Simulated Examples
The properties of the 1-D VPnC rod composed by steel inclusions (A) and polymeric matrix (i.e., epoxy, nylon, silicon rubber, natural rubber, or LDPE) (B) used for the simulation [34][35][36][37][38][39] are listed in Table 1.Moreover, the viscoelastic effect is considered with the following properties 35  Hereafter, the model assurance criterion (MAC) 40 is used to estimate the correlation among wave mode shapes for EPWE.Furthermore, initially, for band structure and complex band structure calculation using IPWE and EPWE, respectively, it is considered 21 plane waves.In order to verify the convergence of IPWE, the band structure (only propagating waves) of the 1-D VPnC rod with steel inclusions in a polymeric matrix (i.e., (a) epoxy, (b) nylon, (c) LDPE, (d) natural rubber, and (e) silicon rubber) without viscoelasticity was computed (Figure 2) considering 21 (black circles) and 42 (red points) plane waves, where ρ is the longitudinal wave velocity in the matrix.It can be observed a good agreement between the wave modes.However, a considerable mismatching can be observed near L a / 2 c 6 ω π > .Even so, 21 plane waves are regarded in order to reduce the computational time.
In Figure 3, it is shown the complex band structure of the 1-D VPnC rod without (a-b) and with (c-d) viscoelasticity considering epoxy as the polymeric matrix computed by IPWE (black circles in (a, c)) and EPWE (colored points in (a-d)) approaches.

Geometry/Property Value
Lattice parameter ( ) Rod thickness ( )  In Figure 3c-d, it can be observed the complex band structure when the viscoelasticity of epoxy using SLSM is included.IPWE (black circles) cannot obtain the correct real part of wave modes (Figure 3c), since all Bloch wave modes are complex because of viscoelastic effect.It is important to highlight that all wave modes are evanescent when the viscoelasticity is considered, thus the classical definition of band gap (i.e., all wave modes must be evanescent within a complete band gap 30 ) cannot be directly applied in order to identify them for VPnCs.Moreover, it can be seen in Figure 3d that there are some wave modes shifted from origin (i.e., increasing their values with the increase of frequency), which is a typical behavior of the viscoelastic periodic structures 41 .Moreover, it can also be seen in Figure 3c that sharp corners at high symmetry points (i.e., { } ) become rounded 26 .Figure 4a , of the first band gap in Figure 3a, i.e., for the epoxy matrix without viscoelasticity.These wave mode shapes in Figure 4a-b were computed by using IPWE and they present an opposite pattern, similar to the wave mode shapes of 2-D mechanical metamaterials 12 .
The profiles of wave mode shapes in Figure 4a-b have a typical pattern at the edge frequencies of Bragg scattering band gaps 12 .
In Figure 4c, it can be seen the wave mode shapes approximately at the middle frequency of the first "band gap" for the cases without viscoelasticity (black points), at .The wave mode shape patterns in Figure 4c are specific of evanescent modes, since the amplitudes decrease along the unit cell length.Moreover, the wave mode shape for the case with viscoelasticity (red points) shows higher values until -10 m (the middle of the unit cell is located at 0 m of x axis).
Figure 5 shows the complex band structure of the 1-D VPnC rod without (a-b) and with (c-d) viscoelasticity considering nylon as polymeric matrix.A good matching between IPWE (black circles) and EPWE (colored points) is observed in Figure 5a.There are disagreements between 5.6-6 and 7-7.6 (values of the normalized frequency), since in these ranges of frequency there are evanescent waves and IPWE cannot obtain their information.Furthermore, in Figure 5, it can be seen the formation of nine band gaps (a) and the unit cell wave attenuation associated with them (b).3c and 5c) cannot obtain the correct real part of wave modes because all Bloch wave modes are complex and wave modes are shifted from origin (in Figure 5d).
In Figure 6, it is reported the complex band structure of the 1-D VPnC rod without (a-b) and with (c-d) viscoelastic effect considering LDPE as polymeric matrix.A good agreement between IPWE (black circles) and EPWE (colored points) is seen in Figure 6a.Moreover, for this polymeric matrix (LDPE), there are no disagreements until L a / 2 c 8 ω π = , considering the positive values of k (Figure 6a).In addition, it seems that some of the branches in Figure 6a are flat bands (i.e., zero group velocity), which is a typical behavior of energy localization 11 .However, by analyzing the wave mode shapes (not shown here for brevity) of these branches it can be confirmed that they do not have the profile of typical flat bands, which is in accordance with Dal Poggetto et al. 11 (i.e., flat bands appear only for the case of hard matrix with soft inclusion).Moreover, it should be highlighted that EPWE cannot easily obtain the flat bands, since the identification of these bands with this approach (i.e., ( ) k ω ) depends on the spectral discretization (∆ω) 11 .
In Figure 6, it can be seen also the formation of eight band gaps (a) and the unit cell wave attenuation associated with them (b).Note that the unit cell wave attenuation increases with frequency (Figure 6b) for this case (LDPE matrix).
In Figure 6c-d, it is illustrated the complex band structure considering the viscoelastic effect of LDPE.Similar as explained for previous cases, IPWE (black circles) cannot obtain the correct real part of wave modes because all wave modes are complex and wave modes are shifted from origin.For this case (LDPE matrix), the imaginary part of evanescent modes (Figure 6d) seems to be lower shifted from origin than for other matrices, i.e., epoxy (Figure 3d) and nylon (Figure 5d).
Figure 7 shows the complex band structure of the 1-D VPnC rod without (a-b) and with (c-d) viscoelastic effect considering natural rubber as polymeric matrix.A good agreement between IPWE (black circles) and EPWE (coloured points) is seen in Figure 7a, with no disagreements in this range of frequency, considering the positive values of k.In addition, it can be observed in Figure 7 the formation of eight band gaps (a) and the unit cell wave attenuation associated with them (b).The unit cell wave attenuation increases with frequency (Figure 7b) for this case (natural rubber matrix).6), IPWE (black circles) cannot obtain the correct real part of wave modes because all wave modes are complex (c), wave modes are shifted from origin (d), and the imaginary part of evanescent modes seems to be lower shifted from origin than for other matrices (d), i.e., epoxy (Figure 3d), nylon (Figure 5d), and LDPE (Figure 6d).
In Figure 8, it is shown the complex band structure of the 1-D VPnC rod without (a-b) and with (c-d) viscoelastic effect considering silicon rubber as polymeric matrix.The main conclusions observed from Figure 7 (natural rubber) can also be extended for this matrix (i.e., silicon rubber) in Figure 8, since the properties of natural rubber and silicon rubber are similar (see Table 1).
Figure 9 illustrates the imaginary part of wave number (only the least attenuated wave mode, which is the second wave mode obtained by the EPWE) of the 1-D VPnC rod without (a) and with (b) viscoelastic effect.It should be highlighted that wave modes corresponding to the branch with the smallest imaginary part of k (least attenuated waves) contribute the most to the evanescent behavior 11,42 .It can be observed for both cases, i.e., without (a) and with (b) viscoelasticity, that there are two typical behaviors, one for epoxy (blue) and nylon (green), and the other for LDPE (red), silicone rubber (brown), and natural rubber (black).The unit cell wave attenuations (ℑ{k}) are higher for natural rubber, silicone rubber, and LDPE, respectively, considering the most range of frequency, for silicone rubber with viscoelasticity (Figure 9b).Epoxy and nylon matrices present the lower wave attenuations, respectively.However, for the range of frequency between, approximately, L 4 a / 2 c 6 ≤ ω π ≤ , epoxy shows a higher wave attenuation than nylon.
To understand the effect of viscoelasticity for each polymeric matrix, it is presented in Figure 10 the least attenuated wave mode (which is the second wave mode in most cases obtained by the EPWE) of the 1-D VPnC rod without (blue) and with (red) viscoelastic effect for each polymer, i.e., epoxy (a), nylon (b), LDPE (c), natural rubber (d), and silicone rubber (e).The viscoelasticity increases the unit cell wave attenuation for most range of frequency for all cases (a-e).However, there are some spectral regions for each matrix where the attenuation is higher without considering viscoelastic effect, such as

Conclusion
The complex band structure of longitudinal waves in 1-D VPnC rods composed by steel inclusions (elastic material) in different polymeric matrices (i.e., epoxy, nylon, silicon rubber, natural rubber, and LDPE) with viscoelasticity (modelled by SLSM) is obtained by using the IPWE and EPWE methods.The viscoelasticity increases the unit cell wave attenuation for most range of frequency considering all polymeric matrices.The higher unit wave attenuations are for natural rubber, silicone rubber, and LDPE, respectively, considering the most range of frequency, with a peak value ( { } k a / 2π I ) of 1.048 near L a / 2 c 5.52 ω π = for silicone rubber.Moreover, epoxy and nylon matrices present lower wave attenuations, respectively.Finally, it is evident that the viscoelastic effect cannot be neglected in the study of periodic structures with polymers in order to improve the understanding of the correct behavior of elastic wave attenuation.

Figure 1 .
Figure 1.Front view of the 1-D VPnC rod with steel inclusions (white) in a polymeric matrix (i.e., epoxy, nylon, silicon rubber, natural rubber, or LDPE) (blue) (a), where  is the unit cell length, and the SLSM (b), where G 1 and G 2 are the shear modulus (springs), and η is the viscosity (dashpot) (b).

A
good matching between IPWE and EPWE is observed in Figure 3a.However, IPWE can only obtain the propagating modes (i.e., only real part of k ) whereas EPWE get both propagating and evanescent modes (i.e., imaginary and complex values of k).There are only a disagreement between 3-4 and 5-6 (values of the normalized frequency, i.e., L a / 2 c ω π ), since in these ranges of frequency the waves are evanescent (with complex values of k ) and IPWE cannot obtain the information of real values of wave number with complex values.Furthermore, in Figure 3, it can be seen the formation of eight band gaps (a) and the unit cell wave attenuation (ℑ{k} 12 ) associated with them (b).
-b illustrates the wave mode shapes of the 1-D VPnC rod at the lower (

Figure 3 .
Figure 3. Complex band structure of the 1-D VPnC rod with steel inclusions in an epoxy matrix (a-b) without and (c-d) with viscoelasticity computed by (a, c) IPWE (black circles) and (a-d) EPWE (colored points) approaches.

Figure 4 .
Figure 4. Wave mode shapes of the 1-D VPnC rod with steel inclusions in an epoxy matrix at the (a) lower and (b) upper edge frequencies of the first band gap (both computed by IPWE) and, approximately, at the (c) middle frequency of the first band gap (computed by EPWE, considering the minimum value of ℑ{}/2) without (black points) and with (red points) viscoelasticity.

Figure 5 .
Figure 5. Complex band structure of the 1-D VPnC rod with steel inclusions in a nylon matrix (a-b) without and (c-d) with viscoelasticity computed by (a, c) IPWE (black circles) and (a-d) EPWE (colored points) approaches.

Figure 6 .
Figure 6.Complex band structure of the 1-D VPnC rod with steel inclusions in a LDPE matrix (a-b) without and (c-d) with viscoelasticity computed by (a, c) IPWE (black circles) and (a-d) EPWE (colored points) approaches.
for silicone rubber in (e)).

Figure 7 .
Figure 7. Complex band structure of the 1-D VPnC rod with steel inclusions in a natural rubber matrix (a-b) without and (c-d) with viscoelasticity computed by (a, c) IPWE (black circles) and (a-d) EPWE (colored points) approaches.

Figure 8 .
Figure 8. Complex band structure of the 1-D VPnC rod with steel inclusions in a silicon rubber matrix (a-b) without and (c-d) with viscoelasticity computed by (a, c) IPWE (black circles) and (a-d) EPWE (colored points) approaches.

Figure 9 .
Figure 9. Imaginary part of the complex band structures (only the second mode is shown) of the 1-D VPnC rod with steel inclusions in different polymeric (i.e., epoxy in blue, nylon in green, LDPE in red, silicon rubber in brown, and natural rubber in black) matrices (a) without and (b) with viscoelasticity computed by EPWE.

Figure 10 .
Figure 10.Imaginary part of the complex band structures (only the second mode is shown in most cases) of the 1-D VPnC rod with steel inclusions in an (a) epoxy, (b) nylon, (c) LDPE, (d) natural rubber, and (e) silicon rubber matrices without (blue) and with (red) viscoelasticity computed by EPWE.