Complete band gaps in a polyvinyl chloride (PVC) phononic plate with cross-like holes: numerical design and experimental verification

Ultrasonics 56 (2015) 251–259

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Complete band gaps in a polyvinyl chloride (PVC) phononic plate with cross-like holes: numerical design and experimental verification Marco Miniaci a, Alessandro Marzani a,⇑, Nicola Testoni b, Luca De Marchi b a b

Department of Civil, Environmental and Materials Engineering – DICAM, University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy Department of Electrical, Electronic and Information Engineering – DEI, University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy

a r t i c l e

i n f o

Article history: Received 11 April 2014 Received in revised form 21 June 2014 Accepted 25 July 2014 Available online 2 August 2014 Keywords: Phononic plate Guided waves Band structures Bragg scattering Ultrasonic testing

a b s t r a c t In this work the existence of band gaps in a phononic polyvinyl chloride (PVC) plate with a square lattice of cross-like holes is numerically and experimentally investigated. First, a parametric analysis is carried out to find plate thickness and cross-like holes dimensions capable to nucleate complete band gaps. In this analysis the band structures of the unitary cell in the first Brillouin zone are computed by exploiting the Bloch–Floquet theorem. Next, time transient finite element analyses are performed to highlight the shielding effect of a finite dimension phononic region, formed by unitary cells arranged into four concentric square rings, on the propagation of guided waves. Finally, ultrasonic experimental tests in pitch-catch configuration across the phononic region, machined on a PVC plate, are executed and analyzed. Very good agreement between numerical and experimental results are found confirming the existence of the predicted band gaps. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Phononic materials (PMs) are composite materials with a periodic distribution of elastic properties and mass density according to a particular lattice symmetry. They are made, for instance, of elastic scatterers with high acoustic mismatch with respect to the hosting matrix. PMs have shown the existence of frequency band gaps (BGs), i.e. frequency ranges in which such materials do not support the propagation of elastic waves [1,2]. In force of their intrinsic features, PMs are suitable for the development of advanced technological devices such acoustic filters, ultrasonic silent blocks, resonators, sound lenses as well as for innovative applications including wave focusing and waveguiding [3–10]. Recently, properties and characteristics of phononic plates have been extensively studied. Literature reviews on the subject can be found in the comprehensive papers by Pennec et al. [1] and by Wu et al. [11]. In particular, different types of phononic plates have been considered, including (i) plates composed by two binary constituents, (ii) plates made-up of single constituent material with a periodic distribution of studs or gratings on the plate surface, as well as (iii) mono-material plates with a periodic distribution of empty holes.

⇑ Corresponding author. Tel.: +39 0512093506; fax: +39 0512093496. E-mail address: [email protected] (A. Marzani). URL: http://alessandromarzani.people.ing.unibo.it (A. Marzani). http://dx.doi.org/10.1016/j.ultras.2014.07.016 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.

Regarding the first type (i), Hsu and Wu [12], using the plane wave expansion method, as well as Sun and Wu [13], by means of the finite difference time-domain method, have studied the band structures of Lamb-type waves in plates made of circular iron cylinders embedded in an epoxy matrix. The influence of the plate thickness for square and triangular lattices of the inclusions has been analyzed and discussed as well. Similarly, Yao et al. [14] have examined the influence of anisotropic inclusions on the band gaps of Lamb-type waves existing in a plate made of circular lead cylinders embedded in elastic isotropic epoxy matrix. Studies on plates of type (ii) with studs have been proposed in Refs. [15–18], among others. As for the plates of the first type, in which the soft inclusions can generate locally resonant band gaps, these works have shown that complete band gaps can be nucleated by the local resonances of the studs. Works on plates with a periodic grating on the surface have been also investigated [19,20]. In particular, it has been quantitatively verified that the width of the band gap is related to the depth of the grooves. Among the latter type (iii) of plates, holes perpendicular to the propagation plane (i.e. parallel to the mid plane of the plate) or aligned through the plate thickness have been investigated. For instance, Liu et al. [21] studied numerically the group velocity of the zero order antisymmetric (A0 ) Lamb mode in a phononic plate with a single cylindrical hole orthogonal to the propagation plane of the plate, whereas Chen et al. [22] investigated the dispersion properties of Lamb-type waves with respect to the number of holes within the height of the phononic plate.

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Fig. 1. (a) Unitary cell characterized by rounded cross-like cylindrical holes in a PVC matrix. (b–c) Representation of the imposed periodic boundary conditions at surfaces orthogonal to the x- (PBCx) and y- (PBCy) axes; (d) free boundary conditions imposed at surfaces orthogonal to the z-axis (FBCz).

Whilst works dealing with holes perpendicular to the propagation plane are limited, those considering trough thickness holes are numerous. Among these latter, the authors have focused on the recent work of Wang and Wang [23], in which a thorough numerical study proves that cross-like holes, if compared to holes of different shapes such as circular and square ones, can nucleate multiple and complete wide band gaps at lower frequencies. In this study, numerical and experimental evidence of complete BGs existing in a polyvinyl chloride (PVC) phononic plate with a square lattice of trough thickness rounded cross-like holes, which unitary cell is shown in Fig. 1a, is presented. To the best of authors’

knowledge, to date such a geometrical shape for the holes has not been considered. The shape for the holes was selected after having performed an extensive numerical campaign, also based on the outcomes of Ref. [23], aimed at finding complete guided waves band gaps in a PVC plate below 50 kHz. In particular, for the assumed hole geometry and lattice constant a, i.e. for a fixed filling fraction f, a parametric analysis was conducted on the plate thickness h in order to find wider band gaps. It was found that for a ratio of h=a ¼ 12=20 ¼ 0:6 two complete band gaps exist. Then it was verified that circular and square holes were not able to nucleate any band gap for the same filling fraction (41%) and unitary cells arrangement.

Fig. 2. (a) Band structures for the unitary cell presented in Fig. 1. Coordinates of the vibration modes (VMs) of Figs. 3 and 4 are also presented. (b) The first irreducible Brillouin zone M  C  X.

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Fig. 3. (a) Mesh discretization for the unitary cell. (b–d) Vibration modes in the first Brillouin zone characterized by: (b) k ¼ ð0; 0Þ m1, f ¼ 20:619 kHz, point VM1 in Fig. 2a; (c) k ¼ ð157; 157Þ m1, f ¼ 32:519 kHz, point VM2 in Fig. 2a; (d) k ¼ ð0; 0Þ m1, f ¼ 32:987 kHz, point VM3 in Fig. 2a.

Fig. 4. Vibration modes for the 7th band structure in the first Brillouin zone characterized by: (a) k ¼ ð157; 157Þ m1, f ¼ 27:947 kHz, point VM4 in Fig. 2a; (b) k ¼ ð0; 0Þ m1, f ¼ 28:420 kHz, point VM5 in Fig. 2a; (c) k ¼ ð157; 0Þ m1, f ¼ 28:092 kHz, point VM 6 in Fig. 2a; (d) k ¼ ð157; 24:16Þ m1, f ¼ 28:087 kHz, point VM 7 in Fig. 2a.

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2. M-C-X dispersion properties

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Consider an h ¼ 12 mm thick phononic polyvinyl chloride (PVC) plate with an in-plane square lattice of unitary cells characterized by cross-like cylindrical holes in an isotropic elastic matrix. Be the lattice constant a ¼ 20 mm and the hole geometry defined by the two parameters b ¼ 18 mm and c ¼ 3 mm. As a result the unitary cell, shown in Fig. 1a, has a filling fraction of approximately f ¼ 41%. The unitary cell is oriented such that x- and y-axes are parallel to the mid plane of the plate, whereas the z-axis is along the plate thickness. In other words, the holes run trough the thickness of the plate. The material density, Young’s modulus and Poisson’s ratio assumed for the PVC matrix are, respectively, qPVC ¼ 1430 kg/m3, EPVC ¼ 3  109 Pa, mPVC ¼ 0:4 [24].

In order to compute the band structures of the phononic plate, proper periodic boundary conditions are applied to the unitary cell as depicted in Fig. 1(b)–(c). In particular, periodic displacements have been imposed between the faces of the unitary cell orthogonal to the x-axis (PBCx), as well as between the two faces orthogonal to the y-axis (PBCy), while no boundary conditions were applied on the top and bottom faces of the unitary cell (i.e. free boundary conditions FBCz as indicated in Fig. 1(d)). The band structures along the three high symmetry directions M  C  X of the first irreducible Brillouin zone, computed via COMSOL Multiphysics [25], are shown in Fig. 2a. For such computation a finite element mesh of 5663 tetrahedral elements have been used (see Fig. 3a).  Results in Fig. 2a are reported in terms of reduced wave vector k  with wavenumber components k ¼ ½kx a=p; ky a=p. From this figure the existence of two complete band gaps in the [0–45] kHz frequency range can be noted. In particular, the lower BG extends approximately from 20.619 to 28.087 kHz and is located below the 7th band structure, whereas the other BG located above the 7th band structure ranges from 28.420 to 32.519 kHz. As it can be noted, except the relatively low-dispersive 7th mode at around 28 kHz, the band structures in the M  C and C  X ranges are dispersive. Besides, a narrow M  C directional BG between 35.262

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Next, finite element time transient analyses are performed to study the shielding effect of a phononic region, realized considering unitary cells arranged in four concentric square rings, on the Lamb wave propagation. Finally, ultrasonic experiments on an ad-hoc machined PVC plate are executed. In particular, pitch-catch tests of Lamb-type waves propagating across a finite length phononic region are performed. The analysis of the experimental data proves the validity of the numerically predicted results, and thus of the steps taken to design the unitary cell, as well as it confirms the existence of BGs in the designed sample.

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Frequency (Hz) Fig. 5. In-plane view of the phononic plate constituted of 160 hollow rounded cross-cylinder inclusions drilled into a 4L  2L  h mm3 (L ¼ 250 mm, h ¼ 12 mm) polyvinyl chloride (PVC) matrix: (a) numerical model and (b) experimentally tested specimen. Emitting/Receiver transducers are also visible.

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Fig. 6. Temporal and frequency content of the applied pulses: (a) 27:5 kHz centered pulse of 21 sinusoidal cycles modulated by a Hanning window and (b) 50 kHz centered pulse of 2 sinusoidal cycles modulated by a Hanning window.

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and 36.561 kHz, as well as a X  M partial BG below 3.318 kHz exist. Vibration modes (VMs) of the unitary cell at the edges of the band gaps are also calculated and reported in Figs. 3(b)–(d) and 4. It is worth noticing that the square lattice can be seen as a periodic arrangement of solid square lumps jointed with narrow hourglass-shaped connectors. From Fig. 3(b)–(d) it emerges that mode VM1 is characterized by a rotation of the lumps around the z-axis, while mode VM2 shows a rotation of the connectors with respect to the mid-plane of the unitary cell that is still. VM3 , instead, presents a z-displacement of the connectors while the centers of the lumps are still. On the other hand, Fig. 4 shows the mode shapes of the 7th band structure at four different frequency and wavenumber values.

As it can be seen, all these modes are characterized by a vibration of the lumps corners only, as the mid plane of the unitary cell is always still.

3. FE wave propagation in the phononic plate A 1000  500  12 mm3 PVC plate, shown in Fig. 5, is modeled in ABAQUS [26]. Plate material properties are those considered in the previous section. A phononic region, formed by an array of unitary cells identical to the one considered in Fig. 1a, is introduced. In particular, the unitary cells are arranged in four concentric square rings around an unaltered area of 120  120 mm2. The phononic

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Fig. 7. Normalized displacement u1 at points R1 and R2 (left) and its Fourier spectrum (right). The complete band gaps are also highlighted as the light gray region.

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square frame has a width of 4  a ¼ 80 mm, where a ¼ 20 mm is the lattice constant. A mesh with a total of 758,112 linear hexahedral elements of type C3D8R was used in the simulations [26]. Lamb waves are excited at point A by means of an imposed displacement of 1  106 mm in z-direction. Two input signals shaped according to the band structures of Fig. 2a are considered: a Hanning modulated 21 sine cycles centered at 27.5 kHz (Fig. 6 and (ii) a Hanning modulated 2 sine cycles centered at 50 kHz (Fig. 6b). Such pulses have been chosen to highlight the filtering capabilities of the designed phononic portion, emphasizing its pass and stop bands frequency ranges. 3  103 seconds long time transient explicit simulations have been considered in order to allow multiple Lamb wave reflections to take place at the edges of both the phononic region and the plate. As a final result, multiple waves impinging the phononic region from multiple directions are observed from the simulations. Time transient displacements in the x  1; y  2 and z  3 directions are recorded at the two acquisition points R1 and R2, taken equidistant from the input source A and located in an ordinary portion of the plate (point R1) and inside the designed phononic frame  ¼ AR2  ¼ 250 mm. (point R2). The inter-distances are set L ¼ R1A After acquisition, signals are Fourier transformed and their frequency content is compared to highlight the differences of the two frequency responses. For both actuation pulses, the displacements along x-, y- and zdirection at points R1 and R2, as well as their energy content in the frequency domain are shown in Figs. 7–9, respectively. The frequencies within the lower and upper bounds of the computed BGs for the infinite phononic plate are also highlighted as a shaded gray region in Figs. 7–9. As it can be noted from these figures, in the BGs region the frequency spectra of the responses at R2 is much lower or even absent if compared to that of signals taken at R1. Thus, the considered finite length of the phononic region (4  a ¼ 80 mm) is capable to shield the internal region from the propagating waves. The small amount of energy that can be seen in the responses R2 for the 27.5 kHz excitation case is related to the 7th band structure that, passing the phononic region, can convert in the Lamb waves existing in the standard PVC plate (internal region). Since such wave energy is totally absent in the u3  z displacement component

(Fig. 9) it is likely that the energy reaching R2 is related to the SH0 mode. 4. Experimental ultrasonic tests in pitch-catch configuration A polyvinyl chloride (PVC) plate of dimensions 1000  500  12 mm3 is initially considered. The material properties of the plate are those that have been considered in the numerical simulations carried out in Sections 2 and 3. The plate has been subjected to a

Fig. 10. Measurement setup adopted in the PZT pitch-catch experiment to extract the transmission coefficient in the PVC phononic plate.

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Fig. 9. Normalized displacement u3 at points R1 and R2 (left) and its Fourier spectrum (right). The complete band gaps are also highlighted as the light gray region.

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Figs. 11a, 12a and 13a show the actuated and received signals for actuation frequencies centered at 15, 27.5 and 42.5 kHz, respectively. Such frequencies correspond to pre-, in- and post-band gap frequencies. In addition, in Figs. 11b, 12b and 13b the signals spectra are provided. From these figures, the effect of the phononic region is clearly visible both in the time and frequency domains. In addition, it has been noted that the wave packet crossing the phononic region (R2) results to be slower (Dt g ) and more distorted with respect to the one registered at R1 (see Fig. 14). Finally, the maximum of the frequency responses related to actuation pulses driven from 10 to 80 kHz at both PZT receivers, normalized w.r.t. the energy content of the input signals, are represented in Fig. 15. This plot summarizes the screening properties of the phononic region exhibiting a well defined drop in intensity within the BGs of the signals acquired at R2 (red line with square markers) if compared to those acquired at R1 in the ordinary PVC plate (blue line with circular markers). Indeed, a very good agreement between experimental and numerical results is found. In the BGs frequency range only noise level intensity is measured at the sensor located in the screened region. It can also be noted that at the lower bound of the first complete BG (at around 20.619 kHz) the intensity starts to drop even if the one in R1 is increasing.

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machining process in which 160 hollow rounded cross-cylinder inclusions have been drilled as shown in Fig. 5. In the practical realization of the holes, a tolerance of 0.1 mm was respected. Such tolerance in the lattice does not modify significantly the band structures of the system. Three PZT PIC-181 transducers (10 mm diameter, 0.63 mm thickness) have been bonded to the plate using phenylethyl salilcilate in positions R1; R2 and A, as shown in Fig. 5. Lamb waves have been generated by means of the PZT at position A and received at both PZTs located in R1 and R2. Sine functions with 21-cycle and Hanning modulation, with central frequency ranging from 10 to 80 kHz, have been used as input signals. These kind of pulses have been preferred to highlight the filtering capability of the phononic region and to avoid energy dispersion due to broadband excitations. The experimental apparatus, shown in Fig. 10, consists of an arbitrary waveform function generator by Keithley and a GA2500A Gated RF amplifier by Ritec used to amplify the input signal up to 200 V peak-to-peak. Signals are acquired by a 4-channel LeCroy LC534AL oscilloscope. A personal computer controls the equipment and allows data to be processed.

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5. Results and discussion

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Variations of energy in R1 can be attributed to both the wavelength tuning, i.e. relation between the wave wavelength and the PZT transducers diameter [27], and the frequency dependent attenuation of Lamb waves. In addition, it can be observed that the energy in R2 starts to rise at around 36 kHz even if the upper bound of the second complete BG is at 32.519 kHz. Thus, it is likely that the first mode above the 7th band structure has not been been excited by the PZT or, in the opposite case, that its energy did not reach R2 by means of Lamb waves. Finally, it can be noted as some intensity peaks appear around 42.5, 51 and 67 kHz in the phononic portion of the plate.

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Fig. 14. Top and bottom panels present time waveforms for a 21 cycles Hanning modulated pulse centered at 42:5 kHz propagating through the ordinary PVC plate and through the phononic region, respectively. A significant time delay can be observed when the wave propagates in the phononic region of the plate.

In this work the existence of band gaps (BGs) in a phononic plate with a square lattice of rounded cross-like holes is numerically and experimentally investigated. The unitary cell was designed by means of an extensive numerical campaign aimed at finding both mechanical and geometrical properties capable to nucleate a complete and large BG below 50 kHz. This search was bounded by several constraints, including (i) the possibility to work with an easy to access and to be machined material, (ii) the possibility to realize holes with standard drilling tools (machining process) as well as (iii) the need to limit the maximum frequency in order to perform accurate experimental testing. From the above analyses it turned out that a unitary cell of polyvinyl chloride (PVC) material, lattice constant 20 mm and thickness 12 mm, with a rounded cross-like hole, was characterized by two complete BGs, and it was verified that for the same filling fraction (41%) and unitary cells arrangement, circular and square holes were not able to nucleate any BG. Subsequently, time transient simulations were performed to design a finite length of the phononic region capable to stop the propagation of Lamb waves in the predicted BGs. As a result, unitary cells arranged in four concentric square rings were considered and machined on a PVC plate 12 mm-thick. Next ultrasonic pitch-catch measurements on the PVC plate, both in the ordinary plate (from A to R1) as well as across the phononic region (from A to R2), were performed. The experimental results have shown a reduced transmission power spectrum in the frequency range of the numerically predicted BGs, i.e. between 20 and 32 kHz, for the signals propagating across the phononic

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region. In particular, a comparison between the signals allows to infer that: (i) the waves propagating from A to R2 experience higher amplitude reduction w.r.t. the waves propagating in the ordinary portion of the plate (from A to R1); (ii) every kind of wave propagation with a frequency content falling inside the numerically predicted BGs is effectively inhibited in the phononc region; (iii) the wave traveling within the phononic region undergoes to a heavier distortion and to a time shift if compared to the signal acquired in the ordinary PVC plate. Additionally, it has been noted that dispersion introduced by the phononic region is very strong, resulting in a temporal spreading of the incident acoustic pulse, if compared with waves propagating in the ordinary PVC plate. It will be interesting to extend the experiments to other geometries of non-convex holes, and especially to weakly disordered phononic materials, in order to improve the understanding of acoustic wave localization. References [1] Y. Pennec, J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzyn´ski, P.A. Deymier, Twodimensional phononic crystals: examples and applications, Surface Sci. Rep. 65 (8) (2010) 229–291. [2] M.-H. Lu, L. Feng, Y.-F. Chen, Phononic crystals and acoustic metamaterials, Mater. Today 12 (12) (2009) 34–42. [3] J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzyn´ski, M.S. Kushwaha, P. Halevi, Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy composite and some metallic systems, J. Phys.: Condens. Matter 6 (1994) 8759–8770. [4] K. Yu, T. Chen, X. Wang, A. Zhou, Large band gaps in phononic crystal slabs with rectangular cylinder inclusions parallel to the slab surfaces, J. Phys. Chem. Solids 74 (8) (2013) 1146–1151. [5] K.L. Manktelow, M.J. Leamy, M. Ruzzene, Topology design and optimization of nonlinear periodic materials, J. Mech. Phys. Solids 61 (12) (2013) 2433–2453. [6] E. Baravelli, M. Ruzzene, Internally resonating lattices for bandgap generation and low-frequency vibration control, J. Sound Vib. 332 (25) (2013) 6562–6579. [7] M. Sigalas, E. Economou, Band structure of elastic waves in two dimensional systems, Solid State Commun. 86 (3) (1993) 141–143. [8] Z. He, H. Jia, C. Qiu, S. Peng, X. Mei, F. Cai, P. Peng, M. Ke, Z. Liu, Acoustic transmission enhancement through a periodically structured stiff plate without any opening, Phys. Rev. Lett. 105 (2010) 074301–074304.

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