Ultrasonics 61 (2015) 126–135

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Propagation of flexural waves in inhomogeneous plates exhibiting hysteretic nonlinearity: Nonlinear acoustic black holes Vitalyi E. Gusev a,⇑, Chenyin Ni b, Alexey Lomonosov c, Zhonghua Shen d a

LUNAM Universités, CNRS, Université du Maine, LAUM UMR-CNRS 6613, Av. O. Messiaen, 72085 Le Mans, France School of Electronic Engineering and Optoelectronic Techniques, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China c Prokhorov General Physics Institute, RAS, 119991 Moscow, Russian Federation d School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China b

a r t i c l e

i n f o

Article history: Received 1 March 2015 Received in revised form 13 April 2015 Accepted 13 April 2015 Available online 22 April 2015 Keywords: Flexural waves Nonlinear waves Hysteretic nonlinearity Acoustic black hole Nonlinear absorption

a b s t r a c t Theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous material on flexural wave in the plates of continuously varying thickness is developed. For the wedges with thickness increasing as a power law of distance from its edge strong modifications of the wave dynamics with propagation distance are predicted. It is found that nonlinear absorption progressively disappearing with diminishing wave amplitude leads to complete attenuation of acoustic waves in most of the wedges exhibiting black hole phenomenon. It is also demonstrated that black holes exist beyond the geometrical acoustic approximation. Applications include nondestructive evaluation of micro-inhomogeneous materials and vibrations damping. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction From the fundamentals point of view the interest to the nonlinear acoustic waves in general and in particular to those localized in their propagation near the surfaces, interfaces and wedges is continuous [1–3]. At the same time this interest is fueled by existing or emerging applications of the wedge, surface and plate acoustic waves such as vibrations damping [4], nondestructive testing of blades [5], material characterization [6] or biomedical [7], for example. There are always such usual questions to be answered, which are accompanying any application of the acoustic waves, as: ‘‘Are the nonlinear acoustic effects important in the considered phenomena?’’ and ‘‘Could the nonlinear acoustic effects be useful in the applications’’? In this Letter we develop the simplest asymptotic theory describing the propagation of the plane nonlinear flexural waves in the plates of the variable thickness. Our particular attention will be given to the analysis of the plates with thickness variations following a power law (HðxÞ ¼ Hðx0 Þðx=x0 Þa  H0 na ; a P 0; where H denotes the thickness, x0 stands for a particular x-coordinate, where the plate thickness takes the value H0  H(x0), and n  x/x0 is the normalized coordinate). Our interest to the investigation of ⇑ Corresponding author. E-mail addresses: [email protected] (V.E. Gusev), chenyin.ni@njust. edu.cn (C. Ni), [email protected] (A. Lomonosov), [email protected] (Z. Shen). http://dx.doi.org/10.1016/j.ultras.2015.04.006 0041-624X/Ó 2015 Elsevier B.V. All rights reserved.

the nonlinear phenomena in plane flexural waves propagating along the x-axis toward the edge, in n = 0, of the above described ‘‘power-law’’ wedge was motivated by several different factors. Firstly, on the basis of both the simplest geometrical considerations for elastic wave energy concentration and the existing theories [8–10] it could be expected that the amplitude of the monochromatic flexural wave should grow infinitely in elastically linear ideal, i.e., lossless, power-type wedge with a > 0 when the wave is approaching the wedge edge. This is a clear indication that the nonlinear phenomena could be important or even unavoidable under particular circumstances, and that the development of the nonlinear theory is desirable. Secondly, quite recently there was a revival of the interest to the flexural acoustic black holes in view of their application as effective absorbers of the vibrations [4,10,11]. The considered black hole effect is based on the theoretical prediction of the infinite time needed for the flexural waves in the wedges with a P 2 to propagate from any point of the plate toward its edge in n = 0. Thus, the wave travelling in the direction of the edge never reaches it, is never scattered by it, and never gives the information in the form of the reflected wave on its existence in n = 0. At the same time the backscattering of the wave in each point of the plate, which could be potentially expected because of the plate spatial inhomogeneity, could be also negligibly small for the waves of short lengths propagating in the plates of slowly varying thickness, when the conditions of the so-called geometrical acoustic approximation

V.E. Gusev et al. / Ultrasonics 61 (2015) 126–135

[10,12–14] are satisfied. Consequently, the wave incident on the black hole-type wedge is not reflected and its energy could be dissipated in the wedge via even weak mechanisms of linear acoustic absorption [4,10]. The efficiency of this emerging technique of vibrations damping can be enhanced for applications by increasing the linear acoustic absorption artificially [4]. However the question on a possible role of the nonlinear acoustic absorption in the black hole phenomena has not been studied yet to our knowledge. There are multiple classic [15–18] and modern [19–28] studies of the nonlinear acoustic absorption (nonlinear ‘‘internal friction’’ in classical terminology) in various kind of micro-inhomogeneous [20,26,29] materials, where the important role in the mechanical motion is played by dislocations, grain boundaries, inter-grain contacts, cracks, etc., i.e., by the mechanical elements that are significantly larger in dimensions than interatomic distances in the sample but, at the same time, are significantly smaller than the sample dimensions. These materials are also called mesoscopic [30,31] from the acoustic point of view, when they are tested by the acoustic waves which are significantly exceeding in length the dimensions of the mechanical elements responsible for their inhomogeneity. It is well documented in the literature that the nonlinear absorption can dominate over the linear one in the acoustic resonance experiments, i.e., in standing waves [23,25,26,28,32,33], causing the increase in the width of the resonance significantly beyond the linear resonance width. However, important nonlinear absorption was also revealed in the travelling acoustic waves both in the microstructured and nanostructured materials [21,22,27,34,35]. Thus, the accumulated evidence of the nonlinear acoustic absorption in microinhomogeneous materials indicates that it is worth studying the nonlinear acoustic absorption of flexural waves in the microinhomogeneous plates in view of their potential applications for vibrations damping. Thirdly, the studies of the amplitude-dependent flexural wave phenomena could potentially provide easier access to the evaluation of some particular fundamental types of the materials nonlinearity. In fact, the symmetry of the pure flexural waves, i.e., of the pure bending motion [36], dictates that the nonlinearity effecting flexural waves should be of odd-type. For examples classical quadratic elastic nonlinearity (of even-type in its symmetry and quadratic in its dependence on wave amplitude), which in homogeneous materials is due both to the nonlinearity of the kinematic/geometric relation between the strain and the displacement gradients (kinematic/geometric nonlinearity) and due to the nonlinearity of the elastic stress–strain relationship (physical nonlinearity) [29] should not influence the propagation of flexural waves by symmetry considerations. This situation is similar to one with plane bulk shear acoustic waves [29,37,38]. The lowest order nonlinearity which is currently under the consideration for flexural motion in homogeneous materials is classical cubic elastic nonlinearity (of odd-type in its symmetry and cubic in its dependence on the wave amplitude) [39–41]. The theories of flexural waves, which include the classical elastic nonlinearities only, could be not relevant for the flexural waves propagation in micro-inhomogeneous plates where other types of nonlinearities, i.e., nonclassical nonlinearities, could dominate [28,31]. Of particular importance could be the so-called hysteretic quadratic nonlinearity (HQNL) of odd-type in its symmetry and quadratic in its dependence on the wave amplitude [29,32,37,42], which is a paradigmatic example, even though the nonlinear absorption and dispersion mechanism by hysteretic micromechanical units is generic. Due to its odd symmetry HQNL can influence the flexural wave propagation, while due to its quadratic dependence on the wave amplitude it could dominate over the classical elastic cubic nonlinearity in the case of weakly nonlinear wave phenomena in flexural waves. Moreover the HQNL contains both elastic and inelastic parts [29,32,37,42]. The latter should lead to direct

127

nonlinear absorption of flexural waves, while the classical cubic nonlinearity is purely elastic and does not directly contribute to acoustical absorption. Note, that the processes of higher harmonics generation due to elastic nonlinearity, which could lead to the depletion of energy in the flexural wave at fundamental frequency, are importantly suppressed in flexural waves because of the inherent dispersion of flexural wave velocities [36] preventing synchronous wave interactions. All this motivates the studies of the flexural waves in plates exhibiting non-classical nonlinearities. It could be expected that experiments with flexural waves would provide in perspective easier access to the fundamental evaluation of the non-classical material nonlinearities, because of the suppression of classical quadratic elastic nonlinearity by system symmetry, and would become useful in the nondestructive testing of microinhomogeneous materials. The manuscript is structured as follows. In Section 2 the existent knowledge on the flexural waves in the plates of the variable thickness, revealed in the GA approximation [12,13], is briefly reviewed. In Section 3, exploiting GA and rotating phase [43,44] approximations, the theory for the flexural waves in the plates of inhomogeneous thickness exhibiting HQNL is developed. In Section 4 we discuss the possible extensions of the obtained theoretical prediction for the plates exhibiting other types of hysteretic nonlinearities [24,45,46] or other types of the nonlinear absorption [23,46]. In Section 5 we provide instructive examples, demonstrating that the black hole effect in flexural waves exists beyond the GA approximation. The conclusions are presented in Section 6. 2. Linear flexural waves A classical derivation of the equation for plane bending/flexural wave propagation [32] combines the equation for the vertical displacement uz(t, x) of the plate

qHðxÞ@ 2 uz =@t2 ¼ @Q x =@x;

ð1Þ

where q is the plate density, H(x) is its thickness (Fig. 1), with the relation of the shear force per unit length Qx(t, x) to the spatial derivatives of the displacement field uz(t, x). The latter is established by using the geometric relation of normal strain @ux(t, x, z)/@x with the curvature of the plate

@ux =@x ¼ z@ 2 uz =@x2

ð2Þ

Fig. 1. An element of a plate of local thickness H(x) and of infinitesimal length dx, showing the forces and moments acting on it from the surrounding material of the plate [32]. The flexural waves are assumed to travel in the x direction.

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to evaluate the normal stress 2

2

2

rx ¼ ðE=1  m Þ@ux =@x ¼ zðE=1  m Þ@ uz =@x

2

ð3Þ

and then the bending moment acting on the mechanical element (Fig. 1)

Mx 

Z

HðxÞ=2

zrx dz ¼ DðxÞ@ 2 uz =@x2 :

ð4Þ

symbolic small parameter l  1 to indicate that the functions depending on the argument lx are ‘‘slowly’’ varying in space [12,13,47]. When the assumed form of the solution is substituted in Eq. (6) the differentiations over x coordinate results in the terms of different order in small parameter l  1, from which only the lowest order terms should be retained. The terms of order l0 in Eq. (5) are mutually cancelled under the condition

HðxÞ=2

Here the flexural/bending modulus of the plate is defined by D  (H3(x)/12)(E/1  m2), where E is the Young modulus and m is Poisson’s ratio. Note that in the following the plate is considered to be spatially homogeneous in its volume. Thus the parameters E, m and q are coordinate independent, while the variations of the bending rigidity along x-axis are caused only by the inhomogeneity, i.e., variation with this coordinate, of the plate thickness. By substituting Eq. (4) in the relation between the shear forces and bending moment (see Fig. 1)

Q x ¼ @M x =@x; the right-hand-side in Eq. (1) is expressed in terms of the field of the vertical mechanical displacement, uz(t, x), i.e., in terms of the plate deflection:

" # @2 @ 2 uz @ 2 uz þ DðxÞ q HðxÞ ¼ 0: @x2 @x2 @t 2

ð6Þ

Thus, Eq. (6) defines the flexural wave as the asymmetric Lamb mode of zeroth order with the wave length significantly exceeding the plate thickness, kðx; xÞ  2p=kðx; xÞ P 2pHðxÞ  H(x). In the case of the homogeneous plates with constant thickness, H(x) = H0, the search for the solution of Eq. (6) in the form uz(t, x) = uz0 exp (ixt + ik0x), assuming constant uz0 and k0, reveals that two propagative and two evanescent (near field) waves exist in the homogeneous plate. The dispersion relation for the wave propagating in the negative direction of the x-axis is [36]: k0 = 121/4[x/(cpH0)]1/2, where we introduced for compactness the velocity of the so-called plate wave, cp  [E/q(1  m2)]1/2. The existence of the propagative waves in the inhomogeneous plates can be revealed by solving Eq. (6) in the geometrical acoustics approximation [12–14]. Qualitatively, from the physics point of view, in this GA approximation we search for such waves in the inhomogeneous plate, that the changes in the plate thickness H(x), of the wave amplitude uz(x) and of the wave number k(x, x)  k(x), that could be due to spatially variable thickness H(x), would take place at the spatial scale significantly exceeding the wave length. Quantitatively, in the mathematical analysis of Eq. (6) the GA approximation can be formalized by the inequalities [12,13]:

k=j@k=@xj  1=k;

uz =j@uz =@xj  1=k:

ð7Þ

The solutions of Eq. (6) in the GA approximation can be found in R the form uz(t, x) = uz(lx) exp (ixt + i xk(lx0 )dx0 ), when formally assuming that H(x)  H(lx). Here we introduced a non-dimensional

ð8Þ

Eq. (8) demonstrates that in the GA approximation the local wave number in the inhomogeneous plate in the position with plate thickness H(x) is equal to the wave number in the homogeneous plate of the same thickness. Thus, the variations in the wave number follow straightforwardly local variations in the plate thickness. The condition for mutual cancellation of the terms of the order l provides the description of the wave amplitude evolution:

  @uz ðxÞ 3 @H=@x @k=@x uz ðxÞ ¼ 0: þ þ @x 2 H k

ð9Þ

Accounting for the relation in Eq. (8), the integration of Eq. (9) results in: 3=2

uz ðxÞ ¼ uz ðx0 Þ½kðxÞ=kðx0 Þ

¼ uz ðx0 Þ½HðxÞ=Hðx0 Þ3=4 / H3=4 ðxÞ: ð10Þ

ð5Þ

Eq. (5) is classical equation for the propagation of linear flexural waves in the plate of the variable thickness H(x) cut of an ideal elastic material [36]. It is valid for the plane waves propagating along the x-axis, which defines the only direction of the plate thickness variation. The considered plate is thus of the wedge type. The so-called equation for flexural waves, Eq. (5), is known to provide an excellent theoretical description of the propagation of zerothorder anti-symmetric Lamb mode, A0 mode, of vanishingly small amplitude, i.e., linear wave, of cyclic frequency x, when the local wave number k  k(x, x) does not exceed the reciprocal of the local plate thickness [36]. This condition can be written in the form:

HðxÞ 6 1=kðx; xÞ:

kðxÞ ¼ 121=4 ½x=ðcp HðxÞÞ1=2 / H1=2 ðxÞ:

The variations of the amplitude with the plate thickness, predicted by Eq. (10), formally confirm the expected increase in the wave amplitude with diminishing thickness of the plate, which was mentioned in the Introduction. Following the geometric relation in Eq. (2) we define characteristic normal strain by g(x) = H(x)k2(x)uz(x). The solutions presented in Eqs. (8) and (10) indicate that not only the mechanical displacement amplitude but also the characteristic normal strain diverges when the plate thickness is infinitely diminishing, g(x) / H3/4(x), confirming that the nonlinear effects could be potentially important when H(x) ? 0. However, before starting the analysis of the possible role of the nonlinearity, the domain of validity for the solutions in Eqs. (8) and (10) should be established. It should be verified that the solutions derived under the conditions of the GA given in Eq. (7) actually satisfy these initially assumed conditions as well as the condition in Eq. (6) for the validity of the description of the A0 Lamb mode by Eq. (5). Note, that from Eq. (10) it follows that two conditions in Eq. (7) are equivalent and, thus, it is sufficient to check only one of them. In this Letter we will analyze in details the flexural waves in the plates with thickness variations following a power law, HðxÞ ¼ Hðx0 Þðx=x0 Þa  H0 na ; a P 0 (see Section 1). From Eqs. (8) and (10) it follows that in this type of the wedges the monochromatic waves are described in the GA approximation by:

kðxÞ ¼ k0 na=2 ;

uðxÞ ¼ u0 n3a=4 ;

gðxÞ ¼ g0 n3a=4  H0 k20 u0 n3a=4 : ð11Þ

The condition in Eq. (6) for the pure flexural wave motion takes the form:

H0 k0 6 na=2 :

ð12Þ

Eq. (12) indicates that, if the wave described by Eq. (11) is flexural in n = 1, then it will keep being flexural wave in its propagation toward the edge of the plate in n = 0, i.e., in the complete interval 0 6 n 6 1 of our interest, where the solutions in Eq. (11) predict the growth of the wave amplitude. The conditions (7) for the validity of the GA approximation take for the considered type of the plates the form:

g a  ð2=aÞx0 k0  na=21 :

ð13Þ

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Eq. (13) indicates that, if the GA approximation holds in n = 1, then it will be satisfied in the complete interval 0 6 n 6 1 of our interest only if a P 2. However, it also follows from Eq. (13) that in the plates of the variable thickness, described by the power law with 0 < a < 2, the GA approximation is never valid in the vicinity of the plate edge, i.e., when n ? 0. In Eq. (13) we introduced the parameter of the GA, ga  (2/a)x0k0. If the GA approximation holds in n = 1, then this parameter is large, ga  1, and the GA approximation holds for 0 < a < 2 in the domain

g a2=ða2Þ  n 6 1;

ð14Þ

which broadens both with increasing parameter ga  1 and with increasing value of a. For example for 1 6 a < 2 the GA is definitely 1 valid in the interval g 1 a 6 n 6 1ðg a  1Þ; which is the dominant part of the interval 0 6 n 6 1. To ensure the validity of the GA approximation and the derived solution (11) in the desired interval n1 6 n 6 1; the parameter of the GA should satisfy the inequality g a  n1a=21 . These are the solutions of the flexural wave equation in the GA approximation that had been originally used for the prediction of the black hole effect in acoustics [10]. The form of the solution, i.e., R uz(t, x) = uz(x) exp (ixt + i xk(x0 )dx0 ), predicts that the propagation time tx0 !x1 of the flexural wave from x = x0 > 0 to x = x1 < x0 is equal Rx 0 to tx0 !x1 ¼ ð1=xÞ x01 kðx0 Þdx Þ. In the power-law wedges the evaluh i ation of this time gives tx0 !x1 ¼ ðk0 x0 =xÞ½2=ða  2Þ ðx1 =x0 Þ1a=2  1 for a – 2 and t x0 !x1 ¼ ðk0 x0 =xÞ lnðx0 =x1 Þ for a = 2. Thus the propagation time of the flexural wave toward the edge of the wedge, i.e., toward x1 ? 0, infinitely grows in the plates with a P 2, manifesting the black hole effect [10]. It is worth noting here that in the black hole type wedges the GA approximation can be valid in the complete interval 0 6 n 6 1 of the flexural wave propagation from its source in n = 1 toward the edge of the plate in n = 0.

3. Flexural waves in the plates exhibiting hysteretic quadratic nonlinearity To account for the hysteretic nonlinearity of the material, which is a physical and not a geometric nonlinearity [29–31,42], it is necessary to include in the analysis the strain-dependent contributions to elastic moduli and their combinations, which are governing wave propagation. In the case of weakly nonlinear waves of our interest here the deviation of the elastic moduli from their wave amplitude independent, i.e., so-called linear, value is considered to be small. Because of this, the description of any of the moduli and any of the moduli combination can be obtained by multiplying its linear counterpart by a factor 1 + Dh, where Dh is the strain-dependent contribution normalized by the linear value of the modulus (combination of the moduli), |Dh|  1. In the materials exhibiting the hysteretic quadratic nonlinearity Dh depends, in general, on the history of the material loading [29–31,42]. However, in the case of monochromatic waves of our interest here, the general description can be significantly simplified taking into account that the loading is periodic in time and that there are a single maximum and a single minimum of strain in each period of acoustic loading. Under these conditions the nonlinear contribution to the linear relation between the normal stress and strain in Eq. (3) takes the bow-tie form, Dh = hb(@ux/@x)A + sign(@ 2ux/@x @t)@ux/@xc, presented in Fig. 2 [26,37]. Here (. . .)A denotes the amplitude of the monochromatic field and h is the non-dimensional parameter of HQNL. The positive values of this parameter ensure that the HQNL causes the absorption of the energy of the coherent acoustic waves. Thus, accounting for the geometric relation in Eq. (2), the nonlinear counterparts of Eqs. (3) and (4) are

Fig. 2. Dependence of the normalized deviation of the elastic modulus from its linear value on the normalized cyclic strain in material with hysteretic quadratic nonlinearity. Arrowheads indicate the direction of strain variation in time.

rx ¼ zðE=1  m2 Þf1  hjzjbð@ 2 uz =@x2 Þ

A

þ signð@ 3 uz =@x2 @tÞ@ 2 uz =@x2 cg@ 2 uz =@x2 and

Mx  DðxÞf1  ð3=8ÞhHðxÞbð@ 2 uz =@x2 Þ

A

þ signð@ 3 uz =@x2 @tÞ@ 2 uz =@x2 cg@ 2 uz =@x2 ; respectively. The equation for the flexural waves in the plate with variable thickness exhibiting HQNL has the form:

@2 @x2

(

" ! !# ) A 3 @ 2 uz @ 3 uz @ 2 uz @ 2 uz DðxÞ 1  hHðxÞ ð 2 Þ þ sign 8 @x @x2 @t @x2 @x2

þ qHðxÞ

@ 2 uz ¼ 0: @t 2

ð15Þ

The HQNL is known to cause softening of the materials and diminishing the propagation speed of acoustic waves [29– 31,37,42]. That is why, expecting the influence of the HQNL on the wavenumber of the flexural wave, we search the solution of Eq. (15) in the GA approximation in the form: u(t, x) = R R u(lx) sin [xt + xk(lx0 )dx0 + lxkh(x0 )dx0 ]  u(lx) sin h, where kh denotes the increase in the flexural wavenumber caused by the nonlinearity. In the case of the weak nonlinearity under consideration here the nonlinear contribution to the flexural modulus is small, A 2  / ð3=8ÞhHðxÞð@ 2 uz =@x2 Þ / ð3=8ÞhHðxÞk ðxÞu A / ð3=8ÞhgðxÞ  hðxÞ z

 1 (here we have introduced the coordinate-dependent parameter  hðxÞ  ð3=8ÞhgðxÞ for the characterization of the strength of the HQNL), and can be considered in the derivation of the solution for Eq. (15) to be at least of the order / l  1. Because of this the solution for k(x) in Eq. (4), obtained from terms of order l0 in Eq. (1), retains its validity for Eq. (15). The nonlinearity contributes only to the terms of the higher order in small parameter l  1. The HQNL is known to cause the generation of higher odd harmonics of the fundamental frequency and the self-action of the wave at the fundamental frequency [29,37,42,48]. Due to the inherent dispersion of the flexural waves described by Eq. (8) the different harmonics propagate at different velocities. Their non-synchronous interaction is inefficient. Because of this we neglect the inverse influence of the generated higher harmonics on the propagation of fundamental wave and account only for the coherent self-action of the fundamental wave. In this approximation, known in the literature as rotating wave approximation [43,44], only the terms at fundamental frequency should be evaluated in Eq. (15). When the proposed form of the solution is substituted in Eq. (15), the required mutual cancellation of the terms of order / l should be achieved independently for the terms proportional to sin h and to cos h. This leads to the equations

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kh ðxÞ ¼ ð3=32ÞhHðxÞk ðxÞuz ðxÞ ¼ ð3=32ÞhgðxÞkðxÞ   ð1=4ÞhðxÞkðxÞ;

ð16Þ

and

  @uz ðxÞ 3 @H=@x @k=@x uz ðxÞ þ þ @x 2 H k h 1  3 hðxÞkðxÞuz ðxÞ; HðxÞk ðxÞu2z ðxÞ ¼ ¼ 8p 3p

ð17Þ

respectively. An additional in comparison with Eq. (9) term, which appeared in the right-hand-side in Eq. (17), clearly indicates the nonlinear absorption of the flexural wave of interest here, which propagates from x = x0 > 0 toward the edge of the wedge in x = 0. Taking into account Eq. (8) the equation in Eq. (17) can be integrated:

uz ðxÞ ¼

uz ðx0 Þ½HðxÞ=Hðx0 Þ3=4 : Rx 3 0 1 þ ð1=8pÞhHðx0 Þk ðx0 Þuz ðx0 Þ x 0 ½Hðx0 Þ=Hðx0 Þ5=4 dx ð18Þ

The solution in Eq. (18) confirms the expected nonlinear absorption of the flexural waves travelling in the negative direction of the x-axis. For the power-law wedges, Eq. (18) takes the form 3

uz ðnÞ ¼ uz0 n3a=4 =b1 þ ð1=8pÞhH0 k0 uz0 x0 f a ðnÞc; while the solution for the normal strain is:

gðnÞ ¼

g0 n3a=4 g0 n3a=4   g f ðnÞ ; 1 þ ð1=8pÞhg0 ðk0 x0 Þf a ðnÞ 1 þ ða=6pÞh 0 a a

ð19Þ

where fa(n) = (1  n15a/4)/(1 – 5a/4) for a – 4/5 and fa(n) = ln (1/n) for a = 4/5. It is assumed that in the initial point of wave propagation, x = x0, the nonlinearity is weak:

 0Þ  h 0 ¼ ð3=8Þhg  1: hðx 0

ð20Þ

The solution in (19) has the following asymptotic behaviors when the flexural wave approaches the edge of the wedge:

8 3a=4  ; if a < 4=5; > < fg0 =½1 þ ða=6pÞh0 g a =ð1  5a=4Þgn gðn ! 0Þ ffi f1=½ða=16pÞhg a gn3=5 =lnð1=nÞ; if a ¼ 4=5; > : fð5a=4  1Þ=ða=16pÞhg a gnða=2Þ1 ; if a > 4=5: ð21Þ In accordance with Eq. (21) HQNL does not modify the powerlaw scaling of the flexural wave strain amplitude if a < 4/5, but just diminishes its magnitude. In the case a > 4/5 the theoretical prediction in Eq. (21) is very different: the HQNL induces modification of the scaling law. Moreover, this nonlinearity does not saturate the divergence of the strain when a < 2 (see Fig. 3(a)), saturates the amplitude of the strain in the black hole type wedge with a = 2, g(n ? 0, a = 2) ? {(3/2)/[(1/8p)hga]} (see Fig. 3(b)), and leads to complete absorption of flexural waves in the black holes with a > 2, g(n ? 0, a = 2) ? 0 (see Fig. 3(c)). Thus the developed theory predicts the important difference of the black hole type wedges from the other power law type wedges in the nonlinear regime in addition to the earlier known differences in their linear behavior [10]. It is also worth noting that in the wedges with a P 4=5 the strain magnitude in the vicinity of the edge stops to depend on the initial magnitude of the strain, g0, and demonstrates the decrease with increasing nonlinearity, / 1/h. This theoretical prediction is somehow similar to the known effect that the quadratic elastic nonlinearity in combination with quadratic in frequency linear absorption leads to independence of the amplitude of the plane bulk compression/dilatation acoustic wave at large distances from the source on the amplitude of the wave emitted by the

Fig. 3. The dependences on the normalized coordinate, n = x/x0, of the strain amplitude normalized by its initial value, g(n)/g(1)  g(n)/g0, for different values of ^  ð1=12pÞhg k0 x0 characterizing the strength of the non-dimensional parameter h 0 the hysteretic quadratic nonlinearity. The plots in (a)–(c) correspond to linear, a = 1, quadratic, a = 2, and cubic, a = 3, power-law wedges, respectively.

source [47]. Note that this combination of elastic nonlinearity and linear absorption is called nonlinear absorption in classical nonlinear acoustics [47]. The analysis of the solution in Eq. (19) demonstrates that HQNL could induce non-monotonous dependence of the strain amplitude

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on the coordinate. For the wedges with the power a < 2 there could be a minimum in the strain amplitude in the case of sufficiently 0 g > 9p=2. In this case the amplitude strong initial nonlinearity, h a of the wave propagating toward the edge first decreases due to the dominance of the nonlinear absorption over the effects of the geometric energy concentration, but later the geometric amplification of the strain wins (see Fig. 3(a)). For the black-hole type wedges with the power a P 2 there could be a maximum in the strain amplitude in the case of sufficiently weak initial nonlinearity, 0 g < 9p=2. In this case the amplitude of the wave propagating h a toward the edge first increases due to the dominance of the effects of the geometric energy concentration over the nonlinear absorption, but later the nonlinear absorption wins (see Fig. 3(c)). Similar effects are known for the bulk compression/dilatation acoustic waves in the presence of elastic quadratic nonlinearity under wave focusing in cylindrical or spherical geometry [47]. A possible extremum in the flexural wave amplitude is predicted at 1

nextr ¼ f

5a=41 3a=4 5a=4  1 ½  1g : a=2  1 ða=6pÞh0 g a

ð22Þ

ð23Þ

In n = ns the amplitude of the strain can be estimated as

gðns Þ  g0 ns3a=4 =2 and the condition of the weakly nonlinear  Þ  1, can be presented in the form regime, hðn s

0 =2Þ1=ð3a=4Þ : ns  ðh

ð24Þ

Combining Eqs. (23) and (24) the condition for the observation of the significant influence of the weak HQNL on the strain amplitude in the case 4/5 < a < 2 is derived:

 0 g   ð2=h 0 Þð5a=41Þ=ð3a=4Þ : 1 þ ð5a=4  1Þ=½ða=6pÞh a

ð25Þ

In the case 4/5 < a < 2 it should be additionally verified that at the transition coordinate n = ns the GA approximation is valid, g a  nas =21 . The latter condition with the use of Eq. (23) can be presented in the form

 0 g   g ð5a=41Þ=ð1a=2Þ : 1 þ ð5a=4  1Þ=½ða=6pÞh a a

extr

a

0

extr

 condition of the weakly nonlinear regime, hðn extr Þ  1, can be presented in the form

nextr  ð2g a =9pÞ1=ða=21Þ :

ð27Þ

Combining Eqs. (27) and (22) the condition for the validity of the description of the black holes in the regime of weak HQNL is 5a=41

a=4 9p a=21 ½ða5=6a=41  1g  ð2g Þ , which is satisfied when obtained, fa3=21 pÞh g a 0 a

3a=4

 0  5a=41 ð2ga Þa=21 . The latter condition is always valid in black h a=21 9p holes in the complete interval 0 < n 6 1 if the initial parameters of flexural wave satisfy the conditions of weak nonlinearity,  0  1, in Eq. (20) and of GA, g  1. In Fig. 3 we present theoreth a ically predicted spatial dynamics of the normalized strain, g(n)/g0, in the flexural wave in three different wedges with a = 1, 2, 3 ^  ða=9pÞh 0 g ¼ ð1=12pÞhg k0 x0 : depending on the parameter h a

The divergence of the derived solution in Eq. (19) in the vicinity of the wedge edge for a < 2 (see Eq. (21)) indicates that in this case the assumption of the weakly nonlinear flexural wave inevitably fails when the wave approaches the edge. That is why it is important to understand if the theoretically predicted in Eq. (21) modification of the wave amplitude scaling with distance for a > 4/5 takes place in the validity domain of the derived solution and could be used in experiments for the evaluation/identification of the hysteretic quadratic nonlinearity. The nonlinearity causes the diminishing of the wave amplitude in Eq. (19) twice relative to the linear elasticity case and starts to significantly modify scaling of the amplitude with distance at such coordinate n = ns where 0 g f ðn Þ  1: Thus ða=6pÞh a a s

0 g g1=ð5a=41Þ ns  f1 þ ð5a=4  1Þ=½ða=6pÞh a

 0 g < 9p=2 of the non-monotonous spatial dynamics of the case h a wave amplitude and only in the amplitude maximum, i.e., at n = nextr (Eq. (22)). The amplitude of the wave in the extremum can be conve0 g Þnða=2Þ1 and the niently presented in the form gðn Þ ¼ g ð9p=2h

ð26Þ

It is straightforward to check that the conditions in Eqs. (26) and (25) are mutually compatible. The analysis of the derived conditions demonstrate that for any 4/5 < a < 2 it is always possible to 0  1 that choose a sufficiently large ga  1 and sufficiently small h the spatial region of the important nonlinearity-induced modifications of the flexural wave amplitude could be described under the approximations of the GA and weak nonlinearity. For the case of the black hole type wedges, a P 2, of our major interest it is not necessary to verify the conditions for validity of the GA approximation, because it is valid in the complete interval 0 < n < 1. Moreover, the validity of weak nonlinearity condition  hðxÞ  1 if it holds initially, Eq. (20), should be verified only for the

0

gðnÞ n3a=4 ¼ : ^ 1n15a=4 Þ g0 1 þ 32 hð 15a=4 ^ proportional to the parameter of the Note that the parameter h, HQNL, is independent of the power a. The plots in Fig. 3 illustrate the variations in the spatial dynamics of the wave amplitude with increasing parameter of HQNL. They demonstrate the predicted possible non-monotonous variation of the wave amplitude (see Fig. 3(a) and (c)) and stabilization of the wave amplitude in the vicinity of the edge in case a = 2 (see Fig. 3(b)). In Fig. 4 we present theoretically predicted spatial dynamics of the normalized strain, ~ gðnÞ, in the flexural wave in three different wedges with h

a = 1, 2, 3 depending on the initial strain amplitude, g0: ~gðnÞ ¼ h

~g n3a=4 h 0 : ~g ð1n15a=4 Þ 1 þ 3h 2

0

15a=4

~  ð1=12pÞhk0 x0 , characterizThe non-dimensional parameter h ing the strength of the hysteretic quadratic nonlinearity, is fixed at ~ ¼ 103 . Fig. 4 demonstrates the predicted possible non-monotoh nous variation of the wave amplitude (see Fig. 4(a) and (c)) and stabilization of the wave amplitude in the vicinity of the edge in case a = 2 (see Fig. 4(b)). The hysteretic quadratic nonlinearity modifies not only the amplitude but also the propagation velocity of the flexural wave, Eq. (16). Actually, experimentally it could be more convenient to monitor the dependence on the wave amplitude the phase delay of the flexural wave than its amplitude, because the former is free from the influence of the linear acoustic absorption. The HQNL causes the amplitude-dependent time delay in the arrival of the Rx 0 wave to the detection point, t h;x0 !x1 ¼ ð1=xÞ x01 kh ðx0 Þdx Þ, which can be using Eqs. (16) and (18) presented in the form

0 k0 th;x0 !x1 ¼ ð3p=4xÞ lnf1 þ ð1=3pÞh

Z

x0

½Hðx0 Þ=Hðx0 Þ

5=4

0

dx g:

x

For the power law wedges the derived solution takes a particular form:

0 g f ðnÞg: th;x0 !x1 ¼ ð3p=4xÞ lnf1 þ ða=6pÞh a a 8  > 4=5:

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homogeneous statistical distribution of their basic parameters [29,46,48,49]. The inhomogeneous distribution of these elements in the Preisach–Mayergoyz space modifies the dependence of the hysteretic nonlinearity on strain [48,49]. In addition the nonlinearity also depends on the type of the dominant micromechanical elements. For example, the adhesion type elements in case of homogeneous statistical distribution of their parameters produce cubic hysteretic nonlinearity at macro scale [46]. Cubic hysteretic nonlinearity had been documented in several experiments [24]. Cubic and quartic hysteretic nonlinearity had been reported for particular directions of wave propagation in crystals (see [45] and the references therein). Another power of the hysteretic nonlinearity frequently revealed in the experiments is 3/2 [26,29,50] Thus both the theoretical expectations and the experimental observations indicate that it is worth studying the influence on the propagation of the flexural waves of the hysteretic nonlinearity of an arbitrary power. It is also worth noting that there are several physical models for the nonlinear absorption of acoustics waves, which are directly unrelated to the phenomenon of the nonlinear hysteresis and that had been applied for the interpretation of the experimental data [26,29,33,51–53]. From the physical considerations for the nonlinearity of power p the evolution equation for the wave amplitude (17) can be generalized into the following form p1 @uz ðxÞ 3 @H=@x @k=@x 2 þ ð þ Þuz ðxÞ ¼ hp ½HðxÞk ðxÞuz ðxÞ kðxÞuz ðxÞ @x 2 H k ¼ hp gp1 ðxÞkðxÞuz ðxÞ:

Here hp denotes a constant characterizing the strength of the hysteretic nonlinearity in the particular micro-inhomogeneous material. In the case of the hysteretic quadratic nonlinearity, p = 2, hp = h/8p (see Eq. (17)). Using Eq. (8) this equation can be integrated

uz0 ½HðxÞ=H0 3=4 uz ðxÞ¼ 1=ðp1Þ Rx p1 2 0 f1þðp1Þhp ðH0 k0 uz0 Þ k0 x 0 ½Hðx0 Þ=Hðx0 Þð3p1Þ=4 dx g The solution for the strain in the power law wedges can be presented in the form

gðnÞ ¼

Fig. 4. The dependences on the normalized coordinate, n = x/x0, of the normalized ~gðnÞ, for different values of the initial strain amplitude, g . The strain amplitude, h 0 spatial dynamics of strain is illustrated for a particular value of the parameter ~  ð1=12pÞhk0 x0 characterizing the strength of the hysteretic quadratic nonlinh ~ ¼ 103 . The plots in (a)–(c) correspond to linear, a = 1, quadratic, a = 2, and earity, h cubic, a = 3, power-law wedges, respectively.

4. Flexural waxes in the media with nonlinear absorption of an arbitrary power law The HQNL is just a particular case of the hysteresis nonlinearity corresponding to frictional type micro-mechanical elements with

g0 n3a=4 ; 1=ðp1Þ f1 þ ðp  1Þhp gp1 0 f a;p ðnÞg

where fa,p(n) = (1  n1-(3p1)a/4)/[1  (3p  1)a/4] if a – 4/(3p  1) and fa,p(n) = ln (1/n) if a = 4/(3p  1). The derived solution predicts that rescaling of the flexural wave asymptotic behavior when n ? 0 takes place only for a P 4=ð3p  1Þ. Thus the rescaling in the asymptotic case of disappearing nonlinearity, p ? 1 + 0, is possible only in black holes, a P 2. With increasing power of the nonlinearity, p ? 1, the option of the rescaling spreads to all power-law wedges, a P ð4=3pÞ ! 0. For a > 4/(3p  1) the rescaling transforms the power law behavior of the linear flexural wave, g / n3a/4, into another power law g / n(a/21)/(p1). Thus the theory predicts that the nonlinear absorption of any power could cause complete attenuation of the wave only in the black holes. For the black hole type wedges the higher is the power of the nonlinearity the slower is the decay of the flexural wave approaching the wedge edge. In other wedges with the possible rescaling of the wave amplitude, 4=ð3p  1Þ 6 a < 2, the higher is the order of the nonlinearity, the stronger it attenuates the divergence of the wave amplitude approaching the wedge edge. The generalization of Eq. (16) for an arbitrary order of the nonlinearity is also straightforward, kh(x) / gp1(x)k(x). We derived R x0 0 ½Hðx0 Þ=Hðx0 Þð3p1Þ=4 dx g for th;x0 !x1 / ð1=xÞ lnf1 þ ðp  1Þhp gp1 0 x arbitrary profiles of the plates and th;x0 !x1 / ð1=xÞ lnf1 þ ðp  1Þ hp g0p1 f a;p ðnÞg for the power law wedges. Thus,

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8 p1 > < lnf1 þ ðp  1Þhp g0 =½1  ð3p  1Þa=4g; if xth;x0 !x1 ðn ! 0Þ / ln½lnð1=nÞ; if > : ½ð3p  1Þa=4  1 lnð1=nÞ; if

More detailed analysis of the derived solutions is beyond the scope of the present publication. 5. Precise description of the black holes for flexural waves in the plates As it had been described in Section 2 the prediction of the black holes for flexural waves was originally done in the frame of the GA approximation [10]. Because of this there is a wide spread opinion that the predicted effect is an approximate one in the sense that there is always, although a weak, backscattering of the wave propagating toward the edge of the wedge because of the inhomogeneity of plate thickness. Below we present two precise analytical solutions for the power lower type plates, obtained avoiding GA approximation, demonstrating the existence of the eigen waves propagating in the plate of variable thickness without any backscattering and exhibiting the black hole effect. For the power law type wedges the flexural wave Eq. (5) can be conveniently rewritten for monochromatic excitations as

@2

"

@n2 by

@ 2 uz

3a

n

#

@n2

2 a

 X n uz ¼ 0;

introducing

the

ð28Þ dimensionless

cyclic

frequency

X ¼ x=x0  ð121=2 x20 =cp H0 Þx. In the case a = 2 Eq. (28) appears to be the Euler equation, which can be reduced to the differential equation with constant coefficients by the substitution uz = uz(ln n) and solved analytically [54]. The final form of the solution is

uz ¼

4 X

mi

ci n ;

mi

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð3=2Þ 4 þ ð1=4Þ X2 þ 4:

ð29Þ

a < 4=ð3p  1Þ; a ¼ 4=ð3p  1Þ; a > 4=ð3p  1Þ:

Otherwise the dispersion relation for flexural waves revealed in pffiffiffiffiffi the GA approximation, i.e., k / X: (see Eq. (8)), is not valid any more. Moreover there exists a cut off frequency, X = 15/4 = 4  (1/4), below which there are no propagating waves in the wedge with a = 2 and all the waves, even the one in Eq. (30), are rather the localized vibrations. In Fig. 5 we present the dispersion relation for the propagating flexural waves in the quadratic wedge, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 þ 4  4  ð1=4Þ=x, and one that both a precise one, kðX; xÞ ¼ pffiffiffiffi is derived in the GA approximation, kðX; xÞ ¼ X=x. The analytical solution in special functions can be also obtained for the cubic wedge, a = 3. The eigen waves in the cubic wedge are ð1Þ;ð2Þ

, also described by the Bessel functions of the third kind, H7 called Hankel functions, and the modified Bessel functions I7 and K7 of the seventh order [55]:

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ð1Þ ð2Þ uz ¼ n5=2 ½c1 H7 ð2 X=nÞ þ c2 H7 ð2 X=nÞ þ c3 I7 ð2 X=nÞ pffiffiffiffiffiffiffiffiffi þ c4 K 7 ð2 X=nÞ: The modified Bessel functions describe localized waves, while the Hankel functions describe possible propagating waves. The solution corresponding to the wave propagating toward the edge pffiffiffiffiffiffiffiffiffi ð1Þ of the cubic wedge is uz ¼ c1 n5=2 H7 ð2 X=nÞ. Under the condition pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi n=X  2 the solution takes an asymptotic form uz / n9=4 ei2 X=n , which precisely reproduces the solution in Eq. (11) obtained in GA approximation and exhibiting the black hole effect. Thus the analytical solutions for the quadratic and cubic wedges demonstrate that the black hole effect is precise in the sense that the complete absence of the backscattering of the wave propagating toward the edge of the wedge can be proved theoretically.

m¼1

Here ci are the arbitrary constants. Solution (29) predicts four eigen waves corresponding to four different combinations of the signs in front of the two square roots. Among them, two waves with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mi ¼ ð3=2Þ 4 þ ð1=4Þ þ X2 þ 4 are always localized in the vicinity of the excitation point (these are so-called evanescent or near-field modes), while two others can be either propagating or localized depending on their dimensionless frequency X. The wave, which can propagate toward the edge of the wedge, is described by qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m  m0 þ im00 ¼ ð3=2Þ þ i X2 þ 4  4  ð1=4Þ: 0

uz ¼ cnm eim

00

ln n

¼ cn3=2 ei

pp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffi

X2 þ44ð1=4Þ ln n

:

ð30Þ

This precise analytical solution describes the scaling with distance of the wave amplitude, / n3/2, and of the phase, / ln n, to be exactly the same as those predicted by the solution in GA approximation in Eq. (11) for the case a = 2. In particular, the solution in Eq. (30) predicts the divergence of the wave phase when the wave approaches the wedge edge, thus manifesting the black hole effect. The only, but important difference of the precise solution in Eq. (30) from the one in the frame of GA approximation consists in the different dispersion of the flexural wave. The solution in Eq. (30) coincides with one obtained in GA approximation only in the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi high frequency limit, X  1, when X2 þ 4  4  ð1=4Þ ffi X:

Fig. 5. The gapped dispersion relation for the flexural acoustic wave in the quadratic wedge, H(x) = H0(x/x0)2, presented as a dependence of its normalized cyclic frequency, X ¼ ð121=2 x20 =cp H0 Þx, on the normalized wave number, kx. The wavenumber scales inverse proportionally to the distance x form the edge of the wedge, k / 1/x. The straight line presents for comparison the dispersion relation that can be derived in the geometrical acoustics (GA) approximation.

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6. Discussions and conclusions The developed theory predicted that the rescaling induced by the nonlinear acoustic absorption could transform the power law behavior of the linear flexural wave in the power-law type wedge into another power law corresponding to a particular power of the nonlinearity. The theory predicts that the nonlinear absorption of any power could cause complete attenuation of the wave only in the black holes, i.e., in the wedges with HðxÞ ¼ H0 ðx=x0 Þa ; a P 2. Thus the developed theory predicted the important difference of the black hole type wedges from the other power law type wedges in the nonlinear regime in addition to the earlier known differences in their linear behavior [10]. For the black holes, the higher is the power of the nonlinearity the slower is the decay of the flexural wave approaching the wedge edge. However, nonlinear absorption cannot cause complete absorption of the flexural wave in the black hole with a = 2, indicating the necessity to account for the linear absorption. In general, accounting simultaneously for the nonlinear and linear absorption is one of the perspectives for the extension of the developed theory in the future. Both the developed asymptotic theory of the nonlinear black holes and the precise theoretical proof of the existence of the linear black holes beyond the geometrical acoustics approximation could be useful in the development of the methods for nondestructive evaluation of micro-inhomogeneous materials and for vibrations damping. Acknowledgements The research was supported by National Natural Science Foundation of China under Grant No. 11274175 and by High-end Foreign Experts Recruitment Program of China. References [1] A.P. Mayer, Surface acoustic waves in nonlinear elastic media, Phys. Rep. 256 (1995) 237. [2] P. Hess, Surface acoustic waves in material science, Phys. Today 55 (2002) 42. [3] P. Hess, A.M. Lomonosov, A.P. Mayer, Laser-based linear and nonlinear guided elastic waves at surfaces (2D) and wedges (1D), Ultrasonics 54 (2014) 39. [4] V.V. Krylov, F.J.B.S. Tilman, Acoustic ‘black holes’ for flexural waves as effective vibration dampers, J. Sound Vibrat. 274 (2004) 605. [5] I. Liu, C. Yang, An investigation on wedge waves and the interaction with a defect using a quantitative laser ultrasound visualization system, in: Proc. 2010 IEEE Ultrasonics Symposium, San Diego, CA, USA, pp. 817–820. [6] J. Herrmann, J.-Y. Kim, L.J. Jacobs, J. Qu, J.W. Littles, M.F. Savage, Assessment of material damage in nickel-base superalloy using nonlinear Rayleigh surface waves, J. Acoust. Soc. Am. 99 (2006) 124913. [7] L. Moreau, J.-G. Minonzio, M. Talmant, P. Laugier, Measuring the wavenumber of guided modes in waveguides with linearly varying thickness, J. Acoust. Soc. Am. 135 (2014) 2614. [8] S.-K. Lee, B.R. Mace, M.J. Brennan, Wave propagation, reflection and transmission in non-uniform one-dimensional waveguides, J. Sound Vibrat. 304 (2007) 31. [9] E.T. Cranch, A.A. Adler, Bending vibrations of variable section beams, J. Appl. Mech. 23 (1956) 103. [10] M.A. Mironov, Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval, Sov. Phys. Acoust. 34 (1988) 318. [11] V.V. Krylov, Acoustic black holes: recent developments in the theory and applications, IEEE Trans. UFFC 61 (2014) 1296. [12] L.M. Brekhovskikh, O.A. Godin, Acoustics of Layered Media I: Plane and QuasiPlane Waves, Springer-Verlag, Berlin, 1990. [13] Y.A. Kravtsov, Y.I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer-Verlag, Berlin, 1990. [14] S.V. Biryukov, Y.V. Gulyaev, V.V. Krylov, V.P. Plessky, Surface Acoustic Waves in Inhomogeneous Media, Springer-Verlag, Berlin, 1995. [15] A.S. Nowick, Variation of amplitude-dependent internal friction in single crystals of copper with frequency and temperature, Phys. Rev. B 80 (1950) 249. [16] A.V. Granato, K. Lucke, Theory of mechanical damping due to dislocations, J. Appl. Phys. 27 (1956) 583. [17] A.V. Granato, K. Lucke, The vibrating spring model of dislocation damping, in: W.P. Mason (Ed.), Physical Acoustics: Principles and Methods, vol. IV-A, Academic Press Inc., New York, 1966. p. 225.

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