An asymptotic hyperbolic-elliptic model for flexural-seismic metasurfaces
AAn asymptotic hyperbolic-elliptic model for flexural-seismicmetasurfaces
P. T. Wootton , J. Kaplunov , and D. J. Colquitt School of Computing and Mathematics, Keele University, Keele, ST5 5BG, UK Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK * Corresponding Author: [email protected]
Abstract
We consider a periodic array of resonators, formed from Euler-Bernoulli beams, attached to the surfaceof an elastic half-space. Earlier studies of such systems have concentrated on compressional resonators. Inthis paper we consider the effect of the flexural motion of the resonators, adapting a recently establishedasymptotic methodology that leads to an explicit scalar hyperbolic equation governing the propagation ofRayleigh-like waves. Compared with classical approaches, the asymptotic model yields a significantly simplerdispersion relation, with closed form solutions, shown to be accurate for surface wave-speeds close to thatof the Rayleigh wave. Special attention is devoted to the effect of various junction conditions joining thebeams to the elastic half-space which arise from considering flexural motion and are not present for the caseof purely compressional resonators. Such effects are shown to provide significant and interesting featuresand, in particular, the choice of junction conditions dramatically changes the distribution and sizes of stopbands. Given that flexural vibrations in thin beams are excited more readily than compressional modes andthe ability to model elastic surface waves using the scalar wave equation (i.e. waves on a membrane), thepaper provides new pathways toward novel experimental set-ups for elastic metasurfaces.
In recent years there has been an explosion of interest, both scholarly and popular, in the use of so-called locallyresonant metamaterials in the control of wave propagation. The concept has its origins in optics [1], but hassince found application across a broad range of fields [2], including acoustics [3, 4, 5, 6], elasticity [7, 8, 9], flexuralwaves in thin plates [10, 11, 12], and seismic waves [13, 14]. Despite the term metamaterial first appearing atthe turn of the millennium [15], the development of this relatively new field continues apace. The essentialconcept is that, using arrays of local resonators, metamaterials can be designed to have properties that do notoccur in materials found in nature; the canonical example is negative refraction [16].At their inception, metamaterials were designed for the control of waves in bulk media. This idea canbe extended to surface waves, creating so-called metasurfaces [17]; these meta surfaces have the substantialadvantage over metamaterials in that they are relatively easy to manufacture [18]. The majority of work onmetasurfaces has focused on electromagnetism [18] and, for example, spoof-surface-plasmons [19]. At present,however, the use of metasurfaces has begun to gain traction in other fields, such as acoustics [20, 21] and, veryrecently, elastodynamics [22, 23] and fluid-solid interactions [24]. In comparison to, say, electromagnetism andacoustics, the treatment of metasurfaces in the framework of elastodynamics is far more challenging and offersa greater array of unique phenomena.Despite the singular challenges associated with mechanical metamaterials, they have a substantial advantagein that the range of scales over which they can be utilized is large compared with Electromagnetic metamate-rials: from the order of millimeters [25] for cloaking of small-scale structures to several meters [13] for seismicmetamaterials. The second notable feature is the existence of two bulk waves (shear and pressure), which prop-agate at two different speeds; this phenomena is unique to elastic waves and has no analogue in electromagneticsor acoustics. The existence of two distinct classes of wave adds both challenges and novel features. Elastic me-dia also support surface waves, again travelling at two different speeds, that are localised to the surface of themedium, but do not decay in the directions parallel to the surface. This is in contrast to electromagnetism wheresurface plasmons [26] decay in all directions. The closest analogue of “true” surface waves in electromagnetismare so-called spoof-surface-plasmons [19], which are supported on structured surfaces.Recently, there have been several analytical [22, 23], numerical [14, 27], and experimental [28] studies onthe use of sub-wavelength resonators for the control of surface waves on elastic substrates. Notable featuresof such resonant arrays include filtering, focusing, rainbow trapping, and mode-conversion of surface waves tobulk waves [29]. 1 a r X i v : . [ phy s i c s . c l a ss - ph ] J un nterestingly, a recent numerical and experimental study [30] has shown that the hybridisation of the twozeroth-order modes in thin plates can be controlled by using symmetric and asymmetric distributions of res-onators. The experimental studies [28, 30] are incumbent on prior completion of numerical or analytical studies.The numerical approaches are computationally intensive and require extensive GPU computing facilities [14],while complete analytical studies present their own significant challenges [23].In the present paper we develop a novel asymptotic formulation, initially introduced in [31], to analyse theinteraction of surface waves with a resonant array resting on an elastic half-space. This asymptotic approachis used to analyse the dynamic effects near the surface of semi-infinite elastic bodies and consists of an ex-plicit hyperbolic equation governing the Rayleigh wave propagation as well as a pseudo-static elliptic equationgoverning decay into the interior. This approach has proven a powerful tool for a variety of near-surface elas-todynamic problems, including 3-dimensional moving load problems [32] and the effect of adding pre-stress tothe elastic half-space [33]. For more general works, the reader is referred [34, 35, 36]. Recently this hyperbolic-elliptic model was applied to a resonant array of rods attached to a half-space [37]; these rods were assumedto undergo only longitudinal vibrations as was the case in previous analytical [23] and numerical [22] studies.However recent experimental data obtained for a dense forest of pine trees, acting as subwavelength resonators,suggests that the flexural resonances as well as the junction conditions [38] may play an important role; yet,little attention has thus far been given to the junction condition joining the resonators to the substrate.Here we extend this approach to include the effects of flexural motion in the thin rods and investigate theeffect of these flexural resonances as well as the junction conditions coupling the resonators to the elastic half-space. In order to better see the effect of the flexural motion, and because in the frequency ranges consideredflexural waves dominate, the longitudinal motion of the resonators is neglected. These new interactions bringnew features, including the ability to control the width of the deep sub-wavelength stop bands. This is animportant feature as, usually, sub-wavelength resonant arrays are only capable of producing very narrow bandgaps [7] and, as a result, it is challenging to make use of these band gaps in the design of filters and other wavecontrol devices. While a non-dimensionalisation of this problem would be useful to consider, this manuscript isfollowing from an existing body of work with strong practical motivations; in the literature of such problems itis more typical to see the results presented in dimensional form.As is usual for the treatment of such low amplitude bending waves, the effect of the weight of the beamson vertical force into the half-space and the bending motion of the resonators is considered to be negligiblecompared to the bending forces.The present paper is be organised as follows. The problem formulation is given in Section 2 with theformulation for Rayleigh waves in subsection 2.2. The sections which follow then each detail the treatment ofa different junction condition between the resonators and the half-space. Section 3 considers the resonators assimply supported, with the resonators able to freely rotate but conserve displacements and horizontal stresses.In Section 4 the resonators are considered as ideally attached at the base to rails which can move freely butmatches the bending moment and the gradient at the surface. For the junction considered in Section 5 weconsider the resonators as ideally attached to the half-space and so all variables between the resonator and half-space are fully matched. In each section, the solution obtained from the asymptotic formulation is comparedwith an implicit evaluation of the full unimodal solution as used in the previous treatment of similar systems[23]. Before considering the effects of the various interface conditions, we begin with a statement of the problem.We consider the propagation of surface waves on an infinite elastic half-space upon which rests an array ofthin Euler-Bernoulli beams, as illustrated in Figure 1. These surface waves propagate with constant amplitudealong the free surface of the half-space and decay, exponentially, into the bulk. We begin by formulating thethree-dimensional problem.
We consider the linearly elastic motion of a three-dimensional half-space −∞ < x < ∞ , −∞ < x < ∞ ,0 ≤ x < ∞ . For the usual displacement vector u = [ u , u , u ] T we consider only motion in the x and x directions and introduce two displacement potentials φ and ψ such that u = ∂φ∂x − ∂ψ∂x , u = ∂φ∂x + ∂ψ∂x . (1)2 L x x x h x = 0 Figure 1:
A schematic representation of the system under consideration: a periodic array of flexural beams,with lattice constant (cid:96) , height L , and thickness h , on the surface of an elastic half-space.Then for linear elasticity, where ρ is the volume mass density and where λ and µ are Lam´e’s first and secondparameters respectively, the equations of motion can be expressed as ∂ φ∂x + ∂ φ∂x − c ∂ φ∂t = 0 , ∂ ψ∂x + ∂ ψ∂x − c ∂ ψ∂t = 0 , (2)where t is time and c = (cid:112) ( λ + 2 µ ) /ρ and c = (cid:112) µ/ρ are the longitudinal and shear wave speed respectively.The flexural motion of the thin elastic beams that pattern the surface of the elastic half-space and form theresonant array obey the Euler-Bernoulli beam equation E R I R ∂ U ∂x = − M R ∂ U ∂t , i = 1 , , (3)where U ( t, x ) = [ U , U ] T is the displacement vector parallel to the surface of the half-space x = 0, forsimplicity E R is introduced to represent the resonator stiffness, E R = E − ν (4)where E and ν respectively are the Young Modulus and Poisson ratio of the resonators, M R is the mass perunit length of the resonators, and I R is the area moment of inertia of the beams, which for resonators withcylindrical cross-section is given by, I R = πh , (5)where h is the diameter of a beam. The ends of the beams at x = − L are assumed to be free such that themoments and transverse forces vanish ∂ U ∂x = , ∂ U ∂x = . (6)For the moment, we will leave the interface condition at the base of each resonator ( x = 0), connecting it to thehalf-space, undefined. These junction conditions will be defined and studied in subsequent sections. In general,the interface between one-dimensional and three-dimensional bodies results in a singularly perturbed problemwhich requires careful consideration of the boundary layer; see, for example, [39].For sufficiently thin resonators the resistance to bending motion is, typically, much smaller than the resistanceto compression and therefore, in the linear regime, the flexural and compressional deformations are decoupledat leading order (see, for example, [40]). Moreover [23] suggest that, for the case of resonant arrays on thinplates, the effects of flexural and compressional resonances can be considered independently. Resonant arrayscomposed of purely compressional resonators atop a half-space have previously been considered in [22, 23, 37].However, a detailed analysis of the influence of flexural resonances on the propagation of surface waves is lacking.Therefore, in the present paper, we focus on the flexural motion of the resonators and consider pure bendingonly such that vertical motion on the surface of the half-space does not couple to compressional deformation ofthe resonators. It is however emphasised that vertical tractions may arise at the surface of the half-space as aresult of rotations in the flexural resonators [40] and these are taken into account.3 .2 Asymptotic Formulation for Rayleigh Waves Considering only waves propagating in the positive x direction leads to a simplification of the governingequations of elasticity and subsequently a simplification to the asymptotic model, for which the full 3-dimensionalmodel is given in [32].Using the displacement potentials (1) and governing equations (2) from before, we introduce the usual stresstensor σ , where in the case of a free surface, stress free junction conditions are imposed yielding, σ i = 0 , i = 1 , , . (7)In terms of the displacement potentials the stresses are given by σ = 2 µφ , + µψ , − µψ , ,σ = ( λ + 2 µ ) φ , + λφ , + 2 µψ , . (8)where, as usual, a subscript of the variable denotes a derivative. We introduce the following Ans¨atze corre-sponding to time-harmonic waves of radian frequency ωφ = A φ e i ( kx − ωt ) − kαx , ψ = A ψ e i ( kx − ωt ) − kβx , (9)where A φ , A ψ , k and are constants and α and β are given by α = (cid:115) − (cid:18) ωc k (cid:19) , β = (cid:115) − (cid:18) ωc k (cid:19) . (10)Substitution of these Ans¨atze into the free surface junction conditions yields the condition for the existence ofa non-trivial solution [23] 4 αβ − (1 + β ) = 0 , (11)from which an implicit expression for the Rayleigh wave speed, denoted by c R , can be obtained. The dispersionequation for such waves is linear, ω = c R k , and the Rayleigh decay constants are subsequently given by α = (cid:115) − (cid:18) c R c (cid:19) (cid:44) α R , β = (cid:115) − (cid:18) c R c (cid:19) (cid:44) β R . (12)In the parameter regime of interest, where the characteristic wavelength of the travelling wave is sufficientlylarge compared with both the cross-section and separation of the resonators, then, as in previous treatments ofsuch problems [37, 41], the surface stress associated with the resonant array may be distributed in the form σ = Hl , σ = Vl (13)where H and V are the amplitudes of the horizontal and vertical forces respectively, and l is the separationbetween the resonators.Each type of junction condition treated in this paper results from the matching of variables between theresonators and the surface of the half-space at x = 0. The same approach will be used in the treatment of themoments arising from the resonators. However, we defer this discussion until the need arises in § (cid:15) = (cid:12)(cid:12)(cid:12)(cid:12) − ωc R k (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , (14)and, in particular, the solution is known to be valid only if ωk ∼ c R . From inspection of the surface conditionsthis is equivalent to when both of the surface stresses are small. The equations for the bulk then reduce to ∂ φ∂x + α R ∂ φ∂x = 0 ,∂ ψ∂x + β R ∂ ψ∂x = 0 , (15)4hich then lead to the travelling wave solutions (9) becoming φ = A φ e i ( kx − ωt ) − kα R x , ψ = A ψ e i ( kx − ωt ) − kβ R x . (16)Depending on the applied surface stress, the model is formulated in terms of either the longitudinal or shearwave potential. If σ = 0, then it is convenient to work with the shear wave potential and the junction conditionalong the surface x = 0 is ψ , − c R ψ ,tt = − β R µB σ , (17)where B = β R α R (1 − α R ) + α R β R (1 − β R ) − (1 − β R ) , (18)If, on the other hand, σ = 0 then the junction condition at x = 0 is written in terms of the pressure potentialsuch that φ , − c R φ ,tt = 1 + β R µB σ . (19)Moreover, on the surface z = 0, the displacement potentials are related by ψ , = −
21 + β R φ , , (20)allowing for the full solution to be obtained. Not only is this form of the surface equations much simpler thanthe full unimodal form, but for a known stress they will each only require the solving of a single one-dimensionalhyperbolic problem along the surface with the known Rayleigh wave speed.It is straight-forward to recover the three-dimensional field from the two-dimensional asymptotic model. Inparticular, it is necessary to introduce a second shear potential ψ , corresponding to the stress σ , and thensimply replace any second order derivatives with respect to x with the two dimensional Laplace operator, ∆.For instance, the vertical surface relation (19) in the full three-dimensional form becomes∆ φ − c R φ ,tt = 1 + β R µB σ . (21)The final step in recovering the full three-dimensional formulation is to replace the surface potential relation(20) by its three-dimensional equivalents φ ,i = 21 + β R ψ i, , φ , = − β R ψ , + ψ , ) , i = 1 , . (22)The problem under consideration concerns the effect of a resonant array which, as described earlier, can betreated as a distributed stress on a travelling Rayleigh wave; this makes the asymptotic model introduced herean ideal tool to investigate the problem. For completeness, however, the solution to the full problem (cf. § We begin by considering the resonators as simply supported at the surface of the half-space. In this system,there is no need to conserve bending angle and there is no overall bending moment at the coupling. Instead onlythe horizontal displacement and the transverse force require matching, as illustrated in Figure 2. We emphasisethat, as discussed in § U ( t,
0) = u | x =0 ,U , ( t,
0) = 0 , (23)5 U u U x x (a) Displacements U , u , x x (b) Gradients U , σ , x x (c) Bending moments σ U , x x (d) Horizontal stress
Figure 2:
A schematic representation of the system junction conditions, showing the end of the beam abovethe edge of the half-space. The relevant half-space quantities and corresponding beam quantities are indicatedwith a single arrow indicating a dimension of length and a double arrow indicating a dimension of stress.and the stress at the end of the beam is, H ( x , x , t ) = πh E R U , ( t, , (24)where U is the horizontal displacement of the the beam and H ( x , y, t ) is the horizontal force at the surface ofthe half-space caused by the resonators.By solving the beam equation (3) with these junction conditions and the free end conditions (6) in the usualway we obtain the relation for horizontal force, H ( x , x , t ) = E R πh K cos( KL ) cosh( KL ) − KL ) sin( KL ) − cos( KL ) sinh( KL ) u | x =0 , (25)where K = (cid:18) M R E R πh (cid:19) √ ω. (26)This force is applied as a point force in the regular array. If the size of the array and wavelength of the waveare assumed to be large in comparison with the distance between the point forces then we can homogenise thesepoint forces into a continuous surface stress using the distribution (13). This stress is then given by, σ = H ( x , x , t ) l , (27)where l is the distance between two adjacent resonators.We first consider the full unimodal formulation for the system. From (8), the surface stresses can be expressedin terms of displacement potentials such that − i µk αφ − µk (1 + β ) ψ = H ( x , x , t ) l , (28) (cid:0) ( λ + 2 µ ) k α − λk (cid:1) φ − iµk βψ = 0 . (29)From (1), we have that U (0) = k ( iφ + βψ ) and, by substituting from (25), we obtain the full unimodal dispersionequation (cid:0) ( λ + 2 µ ) k α − λk (cid:1) (cid:16) − µk (cid:0) β (cid:1) − kK ˆ Hβ (cid:17) = (cid:16) µk α + kK ˆ H (cid:17) (cid:0) − µk β (cid:1) , (30)where ˆ H = σ K − /U (0). It is clear that in this form this expression will be difficult to manipulate. Ifthe asymptotic model produces a close approximation which can be used to investigate the behaviour of thedispersion relation, then this will enable us to understand this system on a more fundamental level than thefull unimodal solution can.For the asymptotic model, we use equation (17) along the surface, ψ , − c R ψ ,tt = − β R µBl H ( x , x , t ) , (31)6hich, when combined with (1) and (20), yields the asymptotic dispersion relation (cid:18) − k + ω c R (cid:19) ψ = kK ˆ H β R − β R µB ψ. (32)The asymptotic dispersion relation can therefore be given explicitly, in terms of the roots of the quadraticequation, k + kK ˆ H β R − β R µB − ω c R = 0 . (33)This asymptotic solution can also be obtained directly from the full dispersion relation obtained earlier byexpanding around the Rayleigh solution. In particular, the Rayleigh condition (11) corresponds to the roots ofthe function R ( r ) = (2 − r ) − √ − r (cid:115) − c c r = (1 + β ) − αβ, (34)where r = ω c − k − . For 0 < r (cid:28)
1, the leading order term of (34) is R ( r ) ≈ (cid:18) ω c k − c R c (cid:19) R (cid:48) (cid:18) c R c (cid:19) = − (cid:18) − ω c R k (cid:19) B. (35)Therefore, in the case of simply supported beams and small r , the full unimodal dispersion relation, (30) canbe expanded in the form K ˆ Hµk β (1 − β ) = (cid:18) − λ + 2 µµ (1 − α ) (cid:19) (1 + β ) − αβ = R ( r ) , (36)whence k − ω c R = − k K ˆ H µB β (1 − β ) , (37)which is identical to the asymptotic dispersion relation (33) given that β = β R .It is clear that the asymptotic dispersion relation (33) is much simpler than the full unimodal equation(30) obtained previously. Importantly, the asymptotic dispersion relation can be solved explicitly for k as afunction of ω and, therefore, manipulated and interpreted with greater ease. We compare the asymptotic andfull unimodal dispersion relations in Figure 3 using the parameter values in Table 1. Table 1:
The numerical system parameters for the beam resonator and half-space used to produce the dispersioncurves shown in Figs. 3, 4, 5 and 6.Symbol Definition Value l Lattice spacing 2 m ρ Half-space density 13000 kg m − µ Half-space shear modulus 325 MPa λ Half-space first Lam`e parameter 702 MPa L Resonator length 14 m h Resonator diameter 0.3m ρ R Resonator density 450 kg m − E R Resonator stiffness 1.70 GPaThe homogenisation scheme from (13) assumes that the wavelength is on a larger scale than the separation ofthe resonators, or equivalently kl (cid:28) π . Subtituting l from the the system parameters in Table 1, this conditionis k (cid:28) π m − for each of the illustrative figures, Figures 3, 4, 5 and 6. In addition, while the asymptotic methodgives a solution which is valid for all k , we note that in the full unimodal system β will become purely imaginarywhen ω/k > c and so no propagating solution can exist in this region of the plot.Figure 3 shows good agreement between the dispersion curves for the asymptotic and full unimodal ap-proaches; particularly near the Rayleigh sound-line and for lower frequencies, as expected. In addition, theasymptotes created by the flexural resonances of the beams extends the range of validity of the asymptotic ap-proach beyond the regime of small k and small stresses where one would expect the method to remain accurate.The flexural resonances associated with these asymptotes create fine deep sub-wavelength band gaps which havebeen shown to have a range of physical applications (see, for example, [23, 22, 28, 10]).7 .0 0.5 1.0 1.5 2.0020406080100120140 k ( m - ) ω ( s - ) Figure 3:
The dispersion curves for surface waves on the half-space coated with simply supported beam likeresonators, using physical parameters from Table 1. The solid blue lines show the dispersion curve of the fullunimodal solution from (30) and the solid black lines the dispersion curve of the asymptotic solution from(33). The dashed green, orange and purple lines correspond to the longitudinal, shear and Rayleigh wave linesrespectively.
Suppose now that the resonators are held at z = 0 to a rail parallel to the surface of the half-space, which allowsthe base of the beam to move freely but changes the bending angle of the beam depending on the gradient ofthe surface. This allows the beam to impart a bending moment on the half-space but no horizontal forcing.There is therefore no need to conserve displacements and the horizontal stress must vanish. The reader isreminded that, as discussed in § §
3, the resonators exert a vertical forceon the half-space through rotations in the beams. These rotations can be linked to the gradient of the verticaldisplacement through the kinematic continuity conditions imposed at the junction points. Whilst this junctioncondition may initially appear somewhat counter-intuitive, it can be understood in terms of so-called gyroscopichinges [42].To construct the junction conditions for the base of the beam, we must first determine the bending moment M in the elastic half-space caused by the flexural deformation of the beam. For a beam with cross-sectionalarea A the bending moment is M = (cid:90) (cid:90) A xσ dA. (38)If the cross-sectional area of the beam is small then the stress can be approximated by the expansion σ ∼ σ | x = y =0 + xσ , | x = y =0 , (39)which, on evaluation of the integral, leaves only the second term. This therefore represents the leading ordercontribution in the calculation of the moments. Figure 2 illustrates how this bending moment in the half-spacecorresponds to the moment at the end of the beam. Furthermore, this figure also shows how, from elementarygeometry, the vertical gradient of the half plane is equal to the horizontal gradient of the beam. Together these8ead to the junction conditions U , ( t,
0) = u , ,E R πh U , ( t,
0) = − πh σ , | x =0 ,U , ( t,
0) = 0 . (40)The vertical stress at the junction point is V ( x , x , t ) = πh σ | x =0 , (41)where V ( x , x , t ) is the vertical force exerted on the half-space by the beams and u , is the gradient of thevertical displacement along the surface. In particular, solving these in the usual way along with the free endconditions (6) and the beam equation (3) gives the vertical force, V ( x , x , t ) = i Kk πh E R − cos( KL ) cosh( KL )cosh( KL ) sin( KL ) + cos( KL ) sinh( KL ) u , | x =0 (42)We will again start with the full unimodal treatment. As we did previously with H ( x , x , t ), we will use theforce distribution (13) and define ˆ V such that, V ( x , x , t ) l = i Kk ˆ V u , . (43)Expressing u , in terms of the potentials (1) and making use of (8) the surface stresses on the half-space canbe written in the form − i µk αφ − µk (1 + β ) ψ = 0 , (44) (cid:0) ( λ + 2 µ ) k α − λk (cid:1) φ − i µk βψ = − K ˆ V ( − kαφ + ikψ ) , (45)whence, the full unimodal dispersion equation is obtained in the form (cid:16) ( λ + 2 µ ) k α − λk − kK ˆ V α (cid:17) (cid:0) − µk (cid:0) β (cid:1)(cid:1) = (cid:0) µk α (cid:1) (cid:16) − µk β + kK ˆ V (cid:17) (46)As might be expected, this dispersion relation is very similar to the one obtained for the simply supported caseand is likewise difficult to manipulate and interpret. Therefore, in order to provide a clearer understanding ofthe system, we once again turn to the asymptotic approach.To apply the asymptotic model to this system, where there is only a vertical stress, we will make use of (19)along the surface of the half-space, whence we obtain φ , − c R φ ,tt = 1 + β R µB V ( x , x , t ) , (47)or equivalently, using (20) and considering time-harmonic waves, in terms of ψψ , − c R ψ ,tt = − i α R µB V ( x , x , t ) . (48)Combining the surface potential relation (20) with ˆ V from (43) and the surface junction condition (48), weobtain the asymptotic dispersion relation k − k α R µB K ˆ V − β R − ω c R = 0 . (49)As before in §
3, the asymptotic dispersion relation can be obtained by expressing the full unimodal dispersionrelation (46) in terms of R ( r ) R ( r ) = − K ˆ Vµk α (1 − β ) , (50)and using the leading order expansion of R ( r ) for r (cid:28)
1, that is, for wave speeds close to that of a pure Rayleighwave, to obtain k − ω c R = k K ˆ V µB α (1 − β ) , (51)9 .0 0.5 1.0 1.5 2.0020406080100120140 k ( m - ) ω ( s - ) Figure 4:
The dispersion curves for surface waves on the half-space coated with beam like resonators supportedby rails, using physical parameters from Table 1. The solid blue lines show the dispersion curve of the fullunimodal solution from (46) and the solid black lines the dispersion curve of the asymptotic solution from(49). The dashed green, orange and purple lines correspond to the longitudinal, shear and Rayleigh wave linesrespectively.which is consistent with the asymptotic dispersion relation (49) if α = α R and β = β R , i.e. the decay rates areequal to those of pure Rayleigh waves.As with the simply supported case, the dispersion relation derived from the asymptotic approach is muchsimpler than the one produced by the complete analysis. Indeed, the asymptotic dispersion equation is againquadratic and therefore yields an explicit expression for k in terms of ω . A plot of the solutions of the asymptoticdispersion equation along with those of the full unimodal solutions is shown in Figure 4; the parameter valuesused are again those detailed in Table 1. Figure 4 shows that, once again, there is good agreement between theasymptotic model and the full unimodal solution, particularly near the Rayleigh line. For all of the modes shownin Figure 4, the asymptotic approach predicts the asymptote, the overall shape of the dispersion relation, andthe existence of a large band gap for each mode. Indeed, in addition to being a much more elegant approach,it is clear that the asymptotic model is capable of reproducing the main features of the full unimodal solution.Moreover, compared with Figure 3, Figure 4 exhibits significantly larger band gaps. We now consider the case where both displacements and rotations (surface gradients), in addition to, horizontaland vertical tractions are coupled at the junction points. Although, as discussed in § §
4; this is fullytaken into account here. Predictably, this complex coupling leads to a more involved solution for both the beammotion and the surface wave dispersion equation. For simplicity, we also assume an ideal contact between thebeams and half-space, neglecting any local contact strain effects.In this case, the equations of motion for the beams (3) together with the free end conditions (6) remainunchanged. The junction conditions at the base of the beam x = 0 are a combination of those previously10onsidered (23, 40) with each of the quantities illustrated in Figure 2 conserved, giving, U ( t,
0) = u | x =0 ,U , ( t,
0) = u , | x =0 ,E R πh U , | x =0 = − πh σ , | x =0 , (52)corresponding to continuity of displacement, rotation, and the balance of moments respectively. The associatedstresses at the base of the beam are H ( x , x , t ) = πh E R u , | x =0 ,V ( x , x , t ) = πh σ | x =0 . (53)Solving these conditions in the usual way along with (3,6) gives, H ( x , x , t ) = πh E R K ( ξu , + Kηu ) ,V ( x , x , t ) = i πh E R Kk ( ζu , + Kξu ) , (54)where, for convenience, we have introduced the quantities ξ = sin( KL ) sinh( KL )cos( KL ) cosh( KL ) + 1 , η = cosh( KL ) sin( KL ) + cos( KL ) sinh( KL )cos( KL ) cosh( KL ) + 1 , and ζ = cosh( KL ) sin( KL ) − cos( KL ) sinh( KL )cos( KL ) cosh( KL ) + 1 . As in the previous sections, we will begin with the full unimodal treatment and then develop the asymptoticmodel for comparison.In contrast to the simply supported and rail conditions discussed in § − i µk αφ − µk (1 + β ) ψ = H ( x , x , t ) l , (cid:0) ( λ + 2 µ ) k α − λk (cid:1) φ − i µk βψ = V ( x , x , t ) l . (55)Writing the displacements in terms of their potentials (1) & (9) and combining (55) with (54), we obtain thedispersion equation (cid:20) µk α + πh l E R k (cid:0) K η − K kαξ (cid:1)(cid:21) (cid:20) µk β + πh l E R (cid:0) K βξ − Kkζ (cid:1)(cid:21) = (cid:20) µk (cid:0) β (cid:1) + πh l E R k (cid:0) K βη − K kξ (cid:1)(cid:21) × (cid:20) ( λ + 2 µ ) k α − λk + πh l E R (cid:0) K ξ − Kkαζ (cid:1)(cid:21) . (56)As one might expect, this dispersion equation is significantly longer and more complicated than those obtainedfor the previous cases and, correspondingly, more difficult to interpret. Therefore we will, once again, developan accurate approximation using our asymptotic mode in an effort to obtain a more transparent dispersionequation. 11he asymptotic model of Kaplunov et al. [31] consists of two relations which each employ a single surfacestress to produce a wave. However, in the present problem, the system has two stresses; one of which is normalto, and the other of which is tangential to, the surface. Therefore, the approach of Kaplunov et al. [31] mustbe adapted to account for the composition of two stresses into a single wave. In particular, we will assume thatthere will be two waves, each generated by one of the two stress-states. Since, for the asymptotic model bothwaves have a speed which is a small perturbation from the Rayleigh speed, it can also be assumed that thesetwo waves travel at the same speed. Each of these two waves will have a corresponding pair of displacementpotentials, ψ and φ , and ψ and φ such that by linear superposition the overall displacement potentials φ and ψ are given by a linear combination such that φ = φ + φ and ψ = ψ + ψ . Each pair of potentials thencorresponds to one of the surface relations from (17, 48), ψ , − c R ψ ,tt = − β R µBl H ( x , x , t ) , (57) ψ , − c R ψ ,tt = − i α R µBl V ( x , x , t ) . (58)Hence, assuming time-harmonic waves such that the potentials take the form (16), we obtain ψ , − c R ψ ,tt = − β R µB σ − i α R µB σ . (59)Combining (54) with (59), after writing the displacements in terms of potentials, yields the following dispersionequation0 = k (cid:18) β R µBl πh E R K − β R ξ (cid:19) − k (cid:18) α R µBl πh E R K − β R ζ − β R µBl πh E R K β R − β R β R η (cid:19) − (cid:18) ω c R + α R µBl πh E R K β R − β R β R ξ (cid:19) . (60)As with the previous cases, we note that the asymptotic dispersion equation (60) is quadratic in k and cantherefore be solved explicitly. Whilst more difficult to interpret than the simply supported asymptotic solution,it is still a substantial improvement on the full unimodal dispersion relation. Figure 5 shows the dispersioncurves for corresponding to both the asymptotic (60) and full unimodal (56) solutions. We observe that theasymptotic model predicts the main features of the dispersion curves, including prediction of the existence andgeneral shape of the solution branch for each quasi periodic mode and also giving a close approximation ofthe intersections with the Rayleigh line. The key feature of the plot, and one present in both the asymptoticand full unimodal results, is the band gaps. This system is notable for the large range of frequencies whichdo not have a corresponding real wave number solution and will, therefore, not propagate through the system.While it is difficult to locate where the band gaps are from the full unimodal dispersion relation, they can beapproximated by the intersections between the asymptotic model and the shear wave line. It is emphasisedthat, once again, this feature is absent in the previous analyses [22, 23, 37] as the flexural resonances of thebeams were neglected.This system also highlights some of the limitations of this model. In particular, and unlike for the simplesupported and rail junctions conditions that involve a single stress, the asymptotics and the full unimodalsolution are not guaranteed to intersect at the Rayleigh line. This is due to the way the stresses are treated inthe two methods. In the asymptotic model there is no interaction between the two stresses and there will beRayleigh line intersections only at, (1 + β R ) σ = 2 i α R σ , (61)whereas for the full unimodal dispersion relation there is a cross multiplication of the two stress terms. Asdiscussed earlier, the full unimodal and asymptotic curves will intersect at the Rayleigh line only if the totalresultant stress at the junction point is zero. This, perhaps counter-intuitive, point can be seen by assumingthat the two stresses have the form, σ = H φ φ + H ψ ψ,σ = V φ φ + V ψ ψ. (62)Expanding these stresses as before yields, − R ( r ) = i βµk H φ + 1 + β µk H ψ − β µk V φ + i αµk V ψ + H φ V ψ − V φ H ψ µ k . (63)12 .0 0.5 1.0 1.5 2.0020406080100120140 k ( m - ) ω ( s - ) Figure 5:
The dispersion curves for surface waves on the half-space coated with fully matched beam likeresonators, using physical parameters from Table 1. The solid blue lines show the dispersion curve of the fullunimodal solution from (56) and the solid black lines the dispersion curve of the asymptotic solution from(60). The dashed green, orange and purple lines correspond to the longitudinal, shear and Rayleigh wave linesrespectively.If we assume a coincidence at the Rayleigh line then, at this point, we replace α and β with α R and β R and usethe surface relation (20) to obtain, − (cid:18) k − ω c R (cid:19) ψ = − β R µB σ − i α R µB σ + V φ H ψ − H φ V ψ µ Bk ψ. (64)Clearly this equation is equivalent to the multi stress junction condition (59) but with an added stress interactionterm which the previous asymptotic model does not predict. For this example we can take the surface stresses(54) and put them into the required form which yields, H φ = i πh l E R K k ( Kη − kαζ ) , H ψ = πh l E R K k ( βKη − kξ ) ,V φ = πh l E R K ( kαζ − Kξ ) , V ψ = i πh l ( βKξ − kζ ) . (65)Letting α = α R and β = β R and substituting these into the expansion (64) leads to the dispersion relation, k − ω c R = α R µBl πh E R Kk (cid:18) k − β R ζ + Kkβ R − β R β R ξ (cid:19) − β R µBl πh E R K (cid:18) k − β R ξ + Kkβ R − β R β R η (cid:19) − πh l E R K µ k (( kα R ζ − Kξ )( β R Kη − kξ ) + ( β R Kξ − kζ )( Kη − kα R ζ )) , (66)which is the same as (60) but with an added term from the stress interaction. Due to cancellation in the stressinteraction term this dispersion relation is quadratic on k . However, even for simple stresses, the dispersionequation arising from the aforementioned stress interactions will, in general, be a quartic polynomial.Figure 6 shows the dispersion curves associated with the full unimodal solution (56) and those arising fromthe asymptotic approach accounting for the stress interaction (60); this should be compared with Figure 5.13 .0 0.5 1.0 1.5 2.0020406080100120140 k ( m - ) ω ( s - ) a cb Figure 6:
The dispersion curves for surface waves on the half-space coated with fully matched beam likeresonators, using physical parameters from Table 1. The solid blue lines show the dispersion curve of the fullunimodal solution from (56) and the solid red lines the dispersion curve of the asymptotic solution with stressinteraction term from (66). The dashed green, orange and purple lines correspond to the longitudinal, shearand Rayleigh wave lines respectively. Alongside is shown the corresponding mode shapes for each branch; thedeformation shows the mode shape and the colour-scheme indicates the magnitude of the total displacementfrom white (zero) through green to purple (maximum).Figure 6 illustrates that accounting for the stress interaction forces the model to coincide with the full unimodaldispersion relation at the Rayleigh line. That said, comparing Figures 5 and 6 we observe that the dispersioncurves arising from the asymptotic approach where the stress interaction is taken into account does not showa significant improvement for the rest of the plot, away from the Rayleigh line, over the standard asymptoticmodel. This indicates that, for this system, the stress interaction does not have a significant effect on the overallbehaviour of the waves away from the Rayleigh line.Again, comparing Figure 3 with Figures 5 and 6, it is apparent that the fully matched junction conditionallows for the creation of significantly larger deep sub-wavelength band gaps compared with the simply supportedjunction condition. Therefore, the effect of the flexural resonances on the overall behaviour of the system canbe increased by an appropriate choice of junction condition.A subtlety associated with applying the asymptotic model to multi-stress systems is that, the asymptoticapproach initial developed in [31] requires that the overall stress is small. It follows that for multi-stress systems,all individual stresses must be small in for the approach to be applicable. If one stress is always large while theother is small then the asymptotic model cannot be valid, even if the result crosses the Rayleigh line.Finally, away from the Rayleigh line, the dispersion curves approach asymptotes associated with the flexuralresonances of the beams; this is a known feature of sub-wavelength resonant arrays (see, for example, [11]).For the previous two cases of simply supported and rail junction conditions, the asymptotic model correctlyreproduces the asymptotes. This led to the asymptotic model remaining accurate to a good degree even awayfrom the Rayleigh line. In contrast, the fully matched junction (both uni- and multi-stress) does not accuratelyreproduce the necessary asymptotes. In particular, the asymptotes of the full unimodel solution are solutionsof the transcendental equation cos( KL ) cosh( KL ) + 1 = 0 . (67)14 a) (b) (c) Figure 7:
The vertical ( u and U ) components of the displacement field for the three modes shown in Figure 6.The colour-scheme indicates the size of the vertical component of the displacement field from white (minimum)through green to purple (maximum).However, the fully matched asymptotic models produces asymptotes at, − β R µBl πh E R K − β R KL ) sinh( KL ) = cos( KL ) cosh( KL ) + 1 . (68)It is clear that, the fully matched asymptotic solutions and the full unimodal solution will diverge as k becomeslarge. Therefore, whilst this is a powerful technique for when both stresses are small, the results cannot berelied upon away from this condition.Figure 6 also shows the mode shapes associated with each branch of the dispersion curves; the colourscheme shows the magnitude of the total displacement. Animated versions of these figures can be found in thesupplementary material. The modes were computed, with the Finite Element software COMSOL Multiphysics R (cid:13) ,using the exact formulation for an infinite periodic array of resonators as described in § U ) and half-space ( u ).It is clear that, in terms of compressional motion, the beams move as a rigid body in the vertical direction and,therefore, the dynamic effects of the compressional motion of the beams do not materially contribute to theoverall behaviour of the system in this regime. This observation is particularly evident in the animated versionsof Figure 7, which are available in the supplementary material. An array of flexural resonators attached to the surface of an elastic half-space is analysed using an explicitmodel for the Rayleigh wave. The paper generalises previous considerations using both full unimodal [23] andasymptotic solutions for an elastic half-space in the case of an array of compressional resonators [37].Three types of junction conditions are considered, and for each the asymptotic solution is verified using fullunimodal solutions for the half-space. The first junction condition to be considered treats the resonators assimply supported at the surface of the half-space. Matching of the beam equation yields a horizontal stress only.The resulting dispersion curves exhibit a series of very fine band gaps typical of deep sub-wavelength arrays;15he narrow nature of these band gaps means that making use of these stop bands in practical applications suchas filtering and mode-conversion is extremely challenging. For example, deep sub-wavelength arrays cannot beused to design broadband filters.The second considered junction condition treats the resonators as being supported by a freely moving railat the surface of the half-space. In contrast to the previous system this yields a vertical stress only. The effectof this junction condition is to widen the band gaps in the spectrum. This ability to increase the width ofthe stop bands through a judicious choice of the junction conditions is an important feature that would allow,for example the implementation of broadband filters in the deep sub-wavelength regime. We also note thatthis effect will increase the contribution of the dynamic flexural behaviour of the resonators on the overallresponse of the system. Whilst sub-wavelength resonant arrays have found extensive use in the literature (see,for example, [7]), little attention has thus far been given to the junction condition joining the resonators to thesubstrate. Indeed, one application of the present paper is in the analysis of the interaction of surfaces waveswith dense pine forests and recent experimental data suggests that the flexural resonances as well as the junctionconditions may be an important factor [38].For both the simply supported and rail junction conditions the asymptotic formulation shows good agreementwith the full unimodal solution. The asymptotic solution obtained predicts both a coincidence with the fullunimodal solution at the Rayleigh line, and also resonances arising from the solution to the beam equation.Moreover, the asymptotic solution has the distinct advantage of being much simpler and easier to manipulatethan the full unimodal solution.The final junction condition considered in this paper fully matches all variables at the junction. Unlike theprevious two cases this results in both a vertical stress and a horizontal stress, requiring an alteration to theexisting asymptotic formulation to account for both stresses While this asymptotic solution closely matcheswith the full unimodal solution it does not accurately predict the coincidence with the Rayleigh line or thelocation of the system resonances. A further addition to the asymptotic model, which accounts for the couplingof stresses at the junction point, produces a solution which does accurately predict coincidence with the fullunimodal solution with the Rayleigh line, but still does not accurately predict resonances.All of the systems considered produce band gaps in the region around the beam resonances, with the largestband gaps produced by the beam on a rail and fully matched junction conditions.This is an important feature as, usually, sub-wavelength resonant arrays are only capable of producing verynarrow band gaps [7] and, as a result, it is challenging to make use of these band gaps in the design of filters andother control devices. However, as we have demonstrated, it is possible to control the width of these deep sub-wavelength band gaps by adjusting the junction condition joining the resonators to the half-space. Indeed, thebeam on a rail and full matched junction conditions lend themselves to feasible designs of mechanical structurethat exhibit large band gaps and are capable of controlling the propagation of surface waves. In particular,the beam on a rail junction condition is associated with gyroscopic resonators, as discussed in [42], and whichcan be designed to exhibit the desired properties; the fully match condition is, in some sense, the most naturaljunction condition to impose as it most closely resembles the conditions of ideal contact.This demonstrates the potential for such systems to be utilised in controlling the propagation of surfacewaves.The explicit asymptotic formulation has the potential to be applied in more sophisticated systems wherea full unimodal solution may be difficult or impossible to obtain since it reduces the vector problem in fulllinear elasticity to a scalar problem along the surface. It has been shown that for the systems considered theasymptotic formulation easily produces simple explicit solutions which match closely with the full unimodalsolution.The formulation discussed involves a surface scalar problem identical to the equation for transverse forcesapplied to an elastic membrane. Currently, experimental set-ups to model elastic half-spaces can be largeand expensive [29] and this formulation gives scope for simpler experiments on membranes which model thebehaviour at the surface of an elastic half-space.This system could also be further developed by considering the near-surface effect of coupled stresses. Wealso remark that the effect of microstructure, including non-local effects, have previously been considered fornear-surface wave motion [43] and, as might be expected, are asymptotically secondary for treating Rayleigh-type waves.
Funding
D. J. C. gratefully acknowledges the financial support of the EPSRC (UK) through programme grant no.EP/L024926/1. P.T.W also acknowledges Keele University for supporting their PhD studies.16 eferences [1] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” science , vol. 292, no. 5514, pp. 77–79, 2001.[2] M. Kadic, T. B¨uckmann, R. Schittny, and M. Wegener, “Metamaterials beyond electromagnetism,”
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