Phononic Thin Plates with Embedded Acoustic Black Holes
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Phononic Thin Plates with Embedded Acoustic Black Holes
Hongfei Zhu ∗ and Fabio Semperlotti † Department of Aerospace and Mechanical Engineering,University of Notre Dame, Notre Dame, IN 46556
We introduce a class of two-dimensional non-resonant single-phase phononic materials and in-vestigate its peculiar dispersion characteristics. The material consists of a thin plate-like structurewith an embedded periodic lattice of Acoustic Black Holes. The use of these periodic tapers allowsachieving remarkable dispersion properties such as Zero Group Velocity in the fundamental modes,negative group refraction index, bi-refraction, and mode anisotropy. The dispersion properties arenumerically investigated using a three-dimensional supercell plane wave expansion method. Theeffect on the dispersion characteristics of key geometric parameters of the black hole, such as thetaper profile and the residual thickness, are also explored.
Phononic Crystals (PC) are artificial media made oftwo or more materials combined together to form a peri-odic structure. These materials offer unusual wave propa-gation characteristics such as acoustic bandgaps [1–5], lo-calized and guided defect modes [6–8], filtering of acous-tic waves [9–11], acoustic lenses [12, 13], and negative re-fraction [14, 15], that are typically not achievable in con-ventional materials. PCs are often classified in two cate-gories, non-resonant and locally-resonant [16], in order tohighlight the difference between their operating modes.The locally-resonant materials exhibit low frequency res-onances (typically in the metamaterial range) localizedat the inclusion while the non-resonant materials exhibitinclusion resonances only in the high frequency range.Owing to this mechanism, the wave propagation char-acteristics in the low frequency range are mostly (acous-tic) impedance-driven for the non-resonant materials andinertia-driven for the locally resonant materials. Theselocal resonances have been shown to be strictly relatedto the generation of negative effective properties (suchas density and bulk modulus) which are at the basis ofdouble negative properties [17].Despite such remarkable dynamic properties, the in-tegration of these materials into practical devices andapplications is still lacking. The fabrication complex-ity (particularly for the locally resonant type) and thenon-structural character (i.e. not load bearing materials)of typical PC designs are among the main limiting fac-tors. In this study, we propose a class of two-dimensionalPCs obtained by tailoring the geometry of a single-phaseisotropic material able to provide the same high-levelcharacteristics of locally resonant PCs. These materi-als are synthesized by embedding a periodic lattice ofcarefully engineered geometric inhomogeneities consist-ing of tapered holes. These inhomogeneities can be in-troduced (virtually) in any material by simply manufac-turing tapers having prescribed profiles in the host struc-ture. Among the fabrication advantages of this design,we highlight that it does not require interfacing multiplematerials and it can be retrofitted even to existing struc-tures. This could have critical implications to develophighly absorbing thin-walled structures with embeddedpassive vibration and acoustic control capabilities. The proposed phononic structure (Fig.1) consists in a thinplate made of a periodic lattice of exponential-like cir-cular tapers, often referred to as
Acoustic Black Holes (ABH).The physical principle exploited in ABHs was first ob-served by Pekeris [18] for waves propagating in strati-fied fluids and later extended to acoustics in solids byMironov [19]. Mironov observed that, under certain con-ditions, flexural waves propagating in a thin plate withan exponentially tapered edge will theoretically never re-flect back, therefore resulting in the so-called zero reflec-tion condition. More recently Krylov [20, 21] exploitedthis concept to achieve passive vibration control of struc-tural elements. The ABH consists in a variable thicknessexponential-like circular taper able to produce a progres-sive reduction of the phase and group velocity as thewave approach the center of the hole. Typical thicknessprofiles are of the form h ( x ) = εx m where { m , ε } ∈ R , m ≥
2, and ε ≪ (3 ρω /E ) / to satisfy the smoothnesscriterion [19, 22]. In ideal ABH tapers, where the thick-ness decreases to zero, the phase and group velocitiestend to zero as they approach the center of the ABH.Under this condition, the wave never reaches the centerof the hole therefore the reflection coefficient approacheszero (the wave is not reflected back) and the hole ap-pears as an ideal absorber. Energy balance considera-tions show that, in the absence of damping, the center ofthe hole becomes a point of singularity for both the par-ticle displacement [19] and the vibrational energy [23]. Inpractice, the residual thickness at the center of the ABHcannot be made zero due to both fabrication and struc-tural constraints. In the absence of damping the residualthickness can produce appreciable levels of reflected en-ergy (up to 70 % [24]) even for a small residual thickness.Nevertheless, wave speed reduction will still take place.In this study, we consider an infinite thin plate witha periodic distribution of ABH-like tapers as shown inFigure 1a. The plate has thickness h = 8 mm and a taperprofile h ( x ) = εx m + h r where h r is the residual thickness.The lattice structure is assembled from a square unit cellas shown in Figure 1b.The elastodynamic response of the phononic thin plate FIG. 1. (a) Schematic of the phononic thin plate with asquared ABH periodic lattice structure, (b) cross section ofthe acoustic black hole showing the taper profile, and (c) 3Dsupercell used for the PWE model. is governed by the Navier’s equations: ρ ∂ u i ∂t = ( C ijkl u k,l ) ,j i = 1 , , ρ is the density, C ijkl is the stiffness tensor, and u i are the components of the displacement field. Eqn.(1) is also subjected to traction free boundary conditions T xz = T yz = T zz = 0 at the upper and lower surfaces.This condition is applied at z = h ( x ) on the bottomsurface and at z = h/ ≤ x ≤ r or at z = − h/ r ≤ x ≤ . a on the top surface.The dispersion relations are obtained solving eqns. (1)using a three dimensional Supercell Plane Wave Expan-sion (PWE) approach [25]. We define a unit cell as shownin Figure 1c. The material is rendered periodic also inthe out-of-plane direction z by alternating the thin platewith vacuum layers. Under these conditions, the mate-rial properties can be approximated using a three dimen-sional Fourier series expansion. Bloch periodic bound-aries are enforced along the in-plane directions to simu-late an infinite plate.In order to solve eqns. (1) the position dependentdensity ρ ( ~r ) and elastic coefficients C ijkl ( ~r ) are ex-panded in Fourier series using the reciprocity vector ~G = ( G x , G y , G z ): C ijkl ( ~r ) = X G e i ~G • ~r C ijklG (2)and ρ ( ~r ) = X G e i ~G • ~r ρ G (3)where ρ G and C ijklG are the corresponding Fourier coef-ficients and are defined as: C ijklG = 1 V Z V C ijkl ( ~r ) e − i ~G • ~r dr (4) and ρ G = 1 V Z V ρ ( ~r ) e − i ~G • ~r dr (5)After using the Bloch theorem and expanding the dis-placement vector ~u ( x, y, z, t ) in Fourier series, we obtain: ~u ( ~r ) = X G ′ A G ′ e i [( ~k + ~G ′ ) • ~r − ωt ] (6)where ~k = ( k x , k y ,
0) is the Bloch plane wave vector, ω isthe circular frequency, and A G ′ is the amplitude of thedisplacement vector. Substituting eqns. (2),(3), and (6)into eqn. (1) and collecting terms, we obtain the 3 n × n set of equations: C G,G ′ C G,G ′ C G,G ′ C G,G ′ C G,G ′ C G,G ′ C G,G ′ C G,G ′ C G,G ′ A G ′ A G ′ A G ′ = ω ρ G,G ′ ρ G,G ′
00 0 ρ G,G ′ A G ′ A G ′ A G ′ (7)where the n × n sub-matrices C G,G ′ are functions ofthe Bloch wave vector ~k , the reciprocal lattice vectors ~G ,the circular frequency ω , and the Fourier coefficients ρ G and C ijklG . The detailed expression can be found in [25].Equation (7) can be written in the form of an eigenvalueproblem whose solution provides the eigenfrequencies andthe eigenmodes of the system.In the following numerical study, we consider a refer-ence configuration consisting in a 8 mm thick aluminumplate with tapers characterized by m =2.2, ε =5, radius r =0.05 m, and residual thickness h r =0.0011m. The lat-tice has a squared configuration with lattice constant a =0.14 m. The reciprocal lattice constants retainedfor the expansion are G x = G y = ± (3 , , , π/a and G z = ± (4 , , , , π/a , where a =0.064m. The con-stant a was selected so to dynamically isolate the differ-ent slabs in the z direction. The band structure along theboundary of the first Brillouin Zone (BZ) for normalizedfrequencies Ω = ωa πC t up to 0.25 is shown in Figure 2.The dispersion relations show several peculiar prop-erties that are typically observable only in locally reso-nant materials. Several non-monotonous branches canbe found in the low frequency range. In particular, forthe fundamental non-monotonous modes, the constitu-tive branches are associated with different mode typesand group velocity regions. As an example, the S modeevolves into the A after crossing a zero group velocitypoint (ZGVP) along the Γ − X boundary. Similar be-havior is observed for the SH mode that evolves intoa higher order flexural mode along the Γ − X bound-ary. The ZGVP point also separates regions with posi-tive and negative group velocity. It has long been known[26] that the higher order Lamb modes in plates can dis-play zero group velocity points corresponding to waves FIG. 2. Dispersion relation along the irreducible part of thefirst Brillouin zone for the phononic plate structure in refer-ence configuration. The insets show the results of geometricacoustic analysis illustrating the effect of different tapers onan incoming ray. (a) corresponds to an ideal ABH (i.e. t −→ having finite phase velocity but vanishing group velocity.However, this behavior is quite unexpected for the fun-damental modes. The ZGVP is related to the existenceof a standing wave associated with a local resonance ofthe plate. The branch of the dispersion curve beyondthe ZGVP is characterized by negative group velocitywhich corresponds to backward wave propagation. Thisphenomenon was never noticed on fundamental Lambmodes. The occurrence of the ZGVP and of the nega-tive group velocity branch is related to the ability of theABH cell to bend the wave in the direction of decreasingphase velocity gradient, that is towards the ABH center.Depending on the properties of the incoming wave and onthe geometric characteristics of the taper, particularly onthe taper exponent and the residual thickness, the wavecan be either slowed down and captured by the ABH(Figure 2, inset a) or bent in the backward direction (Fig-ure 2, inset b). This behavior was verified by performinga geometric acoustic analysis (insets in Figure 2) to iden-tify the trajectory of a ray traveling through the ABH.These two conditions can be related to the generation ofa ZGVP and to backward propagation, respectively.Another interesting property of these metamaterialsis the existence of several singularity points particularlyat the Γ and X locations of the BZ. Each singularitypoint results from the intersection of upper and lowerbranches where only a degenerate mode at ~k = 0 exists.These points exhibit similar behavior to the well-known Dirac Points (DP) [27]. An example of this Dirac-point-like singularity at Γ is shown in Figure 3. By analyz-ing the branches radiating outward from the singularitypoint(either along the Γ − X or the Γ − M boundary),we observe that they correspond to pairs of dissimilarflexural modes having different modal displacements (seeinsets in Fig 3a). The modal displacement shape evolvesfrom a 2 × × − X to Γ − M . More interestingly, if we follow theEFC for either the upper or lower branches, not only themodal displacement shape changes but also the axis ofsymmetry of the mode rotates. A close-up view of the Equi-Frequency-Surfaces around the DP-like singularitypoint is shown in Figure 3b. We note that the shapeof the lower branch clearly indicates anisotropic behav-ior. On the contrary, the upper branch (close to the DPpoint) is a quasi-circular cone, therefore suggesting quasi-isotropic characteristics in the selected frequency range.It is important to note that the cone is composed of differ-ent modes in different directions therefore indicating thatthe ABH-PC presents a ”mode anisotropy” . This con-cept is, in principle, equivalent to the super-anisotropyobserved already in certain type of metamaterials [27]. FIG. 3. (a) Zoom-in view of the dispersion relations aroundΩ = 0 .
62 showing the existence of a Dirac-point-like singu-larity. The insets I-IV show the modal displacements corre-sponding to the labeled modes LU,RU,LB and RB. The blackarrow shows the wave vector direction. (b) shows the EFSplot around the DP-like singularity point.
To better understand the wave propagation producedby the ABH lattice structure, we extract the Equi-Frequency-Contours (EFC) for the fundamental non-monotonous mode S − A f (Figure 4). Results highlightthe existence, in the same band, of a dual EFC contourassociated with different group velocity directions. Thisaspect is of particular interest because it was shown inprevious studies [28, 29] to be a fundamental conditionfor the existence of bi-refraction.To illustrate this phenomenon we superimpose theEFC corresponding to an incident wave at a fixed fre-quency in the homogeneous (constant thickness) area ofthe plate (solid black circle). The wave vector of therefracted beam must satisfy the k k -conservation relation[29]: k inc k = k ref k + G k , where k inc k and k ref k are the com-ponents of the wave vector of the incident and refractedbeam parallel to the interface, G k is the parallel compo-nent of the reciprocal lattice vector. The group velocityis given by V g = ∇ Ω( ~k ) which is always perpendicular tothe EFCs and pointing towards the direction of increas-ing frequency. In our case, dual EFCs with positive andnegative group velocities co-exist at the same frequency,therefore bi-refraction should be expected. In particu-lar, depending on the angle of incidence both positive-positive and positive-negative bi-refraction (see Figure4b) can be achieved. As shown in Figure 4(a), Γ − X is assumed as the interface boundary and the two blacksolid arrows represent two possible incident beams withdifferent angle of incidence. The refracted beam are de- FIG. 4. (a) EFC for the fundamental non-monotonous mode S − A f . The black arrows and the black dashed lines in-dicate the incident angles and the wave vector conservationlines, respectively. The remaining arrows indicate the direc-tion of the refracted beams. (b) Schematic of the bi-refractionmechanism. termined by finding the intersection point between thecorresponding EFC and the conservation line (markedby the black dashed lines perpendicular to the Γ − X boundary). Since the refracted beams are in the direc-tion of the group velocity at the crossing point (red andblue arrows), two different bi-refraction cases with ei-ther positive-positive or positive-negative directions canbe achieved. It is also worth mentioning that, in selectedfrequency ranges, the EFC is square-like therefore sug-gesting that the PC can produce self-collimation of anincoming diffused wave [30].The dispersion relations also reveal remarkable cou-pling between the different mode types highlighting theexistence of a phenomenon known as mode hybridization [29]. By inspecting the modal displacement of the funda-mental non-monotonous mode along the Γ − X direction,we observe that the branch to the left of the ZGVP (redsolid line) is a dilatational S mode while the branch onthe right hand side is a flexural A mode (green solidline). The change in the mode structure occurs veryrapidly in the neighborhood of the zero group veloc-ity point (ZGVP). It is interesting to study how thesemodes develop and how they are affected by the ABHparameters, namely the residual thickness and taper ex-ponent. Figure 5 shows the evolution of the dispersionrelations with the residual thickness. Figure 5(a), (b),and (c) correspond to different residual thickness caseswhere the coefficient ε are set to 1, 3, and 5 meaningthat the hole geometry vary from shallow to deep. Fig-ure 5(a) represents the shallow hole case ( h r =0.00626)where the effects of the ABH should be less evident. Asexpected, the dispersion curves are quite similar to thoseof guided waves in a flat plate. In the low frequencyrange, we observe the S , SH and A modes and severalfolded branches of the A mode due to the folding effect induced by the periodicity. The only significant differ-ence is observed in correspondence to the mode crossingpoints. At these points, we observe splitting of the orig-inal modes and the generation of the non-monotonoushybrid modes connected by the ZGVP. The remainingpart of the dispersion relations is essentially unaffected.By increasing the ABH slope (Figure 5(b) and 5(c)) thesplitting and hybridization mechanisms become more ev-ident even at higher frequency. A physical interpretationof this anomalous dispersion can be made in term of thecoupled-wave theory [31]. When two modes are coupledby a distributed mechanism (in our case the periodic in-homogeneity created by the ABH taper), significant in-teraction only occurs at synchronism, that is, near pointswhere their dispersion curves cross. The coupling causesa characteristic splitting of the dispersion curves at thecrossing point while, elsewhere, the modes are essentiallyunaffected. FIG. 5. Dispersion relations for different residual thicknessvalues: (a) h r =0.0063, (b) h r =0.0039, and (c) h r =0.0011. The other key design parameter is the exponential ta-per coefficient m . We consider a progression of the ABHprofile from smooth (low m ) to sharp (high m ). Forall cases the residual thickness is maintained constant at h r = 0 . ǫ . Thedispersion relations are shown in Figures 6(a), (b), and(c) for m =3, 5, and 7. Of interest is the appearance of abandgap between the fundamental flexural mode A andthe negative branch of the fundamental non-monotonousmode A f (see green dashed box) following the splittingof the folded A mode. This bandgap is considered to berelated to the increased back-scattering occurring at large m where the ABH smoothness criterion is no more satis-fied [19, 22]. Overall, the higher frequency modes (abovethe fundamental) are more evidently affected by thechange in the taper exponent. As an example, mode I andII show substantial changes as m increases. These twomodes are characterized by monopole-like and dipole-likemodal displacement fields inside the ABH and thereforeare very sensitive to changes of the ABH profiles. Notealso the formation (around Ω = 0 . FIG. 6. Dispersion relation for different taper exponent val-ues: (a) m =3, (b) m =5, and (c) m =7. resonant single-phase phononic crystals made of a pe-riodic lattice of Acoustic Black Holes. 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