Wave energy transfer in elastic half-spaces with soft interlayers.

Wave energy transfer in elastic half-spaces with soft interlayers Evgeny Glushkov,a) Natalia Glushkova, and Sergey Fomenko Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar 350040, Russia

(Received 25 September 2014; revised 23 January 2015; accepted 16 March 2015) The paper deals with guided waves generated by a surface load in a coated elastic half-space. The analysis is based on the explicit integral and asymptotic expressions derived in terms of Green’s matrix and given loads for both laminate and functionally graded substrates. To perform the energy analysis, explicit expressions for the time-averaged amount of energy transferred in the timeharmonic wave field by every excited guided or body wave through horizontal planes and lateral cylindrical surfaces have been also derived. The study is focused on the peculiarities of wave energy transmission in substrates with soft interlayers that serve as internal channels for the excited guided waves. The notable features of the source energy partitioning in such media are the domination of a single emerging mode in each consecutive frequency subrange and the appearance of reverse energy fluxes at certain frequencies. These effects as well as modal and spatial distribution of the wave energy coming from the source into the substructure are numerically analyzed and C 2015 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4916607] discussed. V [JBL]

Pages: 1802–1812

I. INTRODUCTION 1

2

Since the pioneering works of Rayleigh, Love, and Lamb,3 the traveling wave propagation in elastic waveguide structures has been of permanent research interest. This is explained by diverse manifestations and numerous applications of this phenomenon in geophysics, civil and mechanical engineering, ultrasonic non-destructive testing, microelectromechanical systems (MEMS), and others. Important applications of elastic guided waves (GW) are surface acoustic wave (SAW) devices operating at gigahertz frequencies,4 and systems of structural health monitoring of plate-like and tube-like units based on GW generation and sensing by piezoelectric wafer active sensors.5 In these applications, the power of every particular traveling wave (normal mode) excited by a given source (e.g., by an interdigital transducer or piezo-actuator) is of the same interest as its phase and group velocities. The velocity characteristics and spatial eigenforms of traveling waves can be obtained irrespective of the wave source using the conventional modal analysis technique. However, the analysis of amplitude and energy characteristics assumes the use of forced solutions to the elastodynamics problems, in which the source action is simulated by surface contact stresses or volume forces. For a homogeneous elastic half-space such solutions and the energy partition among the waves generated by a surface source are well studied.6–8 The present paper aims to study the wave energy partition among the generated waves and its transfer from the surface source to infinity in an elastic half-space with laminate or functionally graded material (FGM) coating. It is based on the semi-analytical solution obtained in terms of the Green’s matrix of the substructure considered and a given surface load.9,10 In Ref. 9, the effect of coatings on the GW a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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amplitudes was analyzed for several typical FGM dependencies resulting from diffusion processes or sputtering and gluing of protective films. Among them, in the case of coatings with soft internal channels, certain regularities of amplitude variation in the frequency domain have been observed. The amplitude of every GW generated in such a structure by a surface source reaches its maximum value soon after its appearance at the cut-off frequency. Then it monotonically decreases and becomes of insufficient value after the cut-off frequency of the next mode. In general, it looked like a mode-to-mode transition of peak amplitudes. Similar local dominating surface displacements of GW modes were also noted for a pavement which layers were softer with depth.11 Apparently, the dominant modes transfer the majority of the wave energy supplied by the source, and so a mode-to-mode transfer of wave energy was expected in this case as well. To confirm this assumption, the distribution of the source energy among the GWs as well as the patterns of wave energy fluxes have been analyzed. The numerical results presented in the paper illustrate spatial and frequency-domain energy localization and the emergence of backward energy fluxes. The study of elastodynamic wave behavior of structures with soft interlayers is motivated by many practical applications such as anti-seismic construction, vibration isolation, seismic prospecting through frozen soil or ice, concrete sandwiches designed to reduce the energy of impacts and explosions, MEMS and SAW devices, glass panes and protective coatings. All of these objects, being of different scales, are governed by the same laws of acoustics of solids. II. MATHEMATICAL FRAMEWORK

Let us consider a functionally graded (FG) elastic isotropic half-space, occupying the volume 1 < x; y < 1; 1 < z 0 in Cartesian coordinates x ¼ (x, y, z), as shown in Fig. 1(a). The elastic Lame moduli kðzÞ and lðzÞ, and the density qðzÞ vary piecewise continuously with the

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C 2015 Acoustical Society of America V

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! @ux @uz 0 0 @uy @uz 0 @uz D¼ l þ þ ;k divu þ 2l ;l ; @z @x @z @y @z 0

the prime denotes z-derivatives. In the homogeneous parts, k0 ¼ 0 and l0 ¼ 0, hence, D 0. If k(z) and l(z) have points of discontinuity zm, m ¼ 1; 2; …; M, Eq. (2) is defined in open regions (sublayers) zm1 < z < zm ðz1 ¼ 0; zMþ1 ¼ 1Þ, and additional conditions of u and s continuity at the internal borders z ¼ zm, m ¼ 2; 3; …; M, are assumed. The Fourier transform F xy , applied to Eq. (2) with respect to the horizontal coordinates x and y, reduces it to ordinary differential equations (ODE) with variable coefficients for the Fourier symbol Uða1 ; a2 ; zÞ ¼ F xy ½u ¼

ð ð1

uðxÞeiða1 xþa2 yÞ dxdy

1

¼ ðU1 ; U2 ; U3 Þ FIG. 1. (Color online) The geometry of the problem: (a) coated elastic halfspace; (b) functionally graded; and (c) piecewise constant dependences of material properties on the depth z.

depth z in the upper interval H z 0, simulating an FGM coating, while the underlying substrate is homogeneous (k, l, and q are constant at 1 < z H). A typical FG dependence of material properties on z that forms an inner channel is shown in Fig. 1(b). The dependency consists of homogeneous parts with constant material parameters and FG parts of length hg continuously connecting those constant values. With hg ¼ 0, the FGM coating degenerates into a laminate piecewise constant covering with a soft interlayer [Fig. 1(c)]. Some of the numerical examples below are given for both FGM and laminate covers to show that even a rough approximation of real FGMs by laminates insignificantly changes the results. A steady-state time-harmonic load qeixt , applied to the stress-free surface z ¼ 0 in a limited area X, ( qðx; yÞ; ðx; yÞ 2 X (1) sjz¼0 ¼ 0; ðx; yÞ 62 X; generates a time-harmonic wave field uðxÞeixt ; x is angular frequency, the vector s ¼ ðsxz ; syz ; rz Þ is the complex amplitude of elastic stresses on a horizontal surface. At infinity pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (z ! 1 and r ¼ x2 þ y2 ! 1), the radiation conditions resulting from the principle of limiting absorption12 hold. The time-harmonic factor eixt is conventionally omitted. The displacement vector u ¼ ðux ; uy ; uz Þ obeys the vector equation ðk þ lÞrdiv u þ lDu þ qx2 u þ DðuÞ ¼ 0;

(2)

which results from the substitution of stress tensor elements expressed via u components in the equation of motion for an elastic solid medium. It differs from the Lame equation for a homogeneous medium by the additional vector-term J. Acoust. Soc. Am., Vol. 137, No. 4, April 2015

of the displacement vector u. In this way, the solution to boundary value problem Eqs. (1)–(2) is derived in terms of the inverse Fourier transform of U ¼ KQ, where K ¼ F xy ½k; Q ¼ F xy ½q, and k(x) is the Green’s matrix of the structure.9,10 Alternatively, u may be written as a convolution of k with q, ðð

kð x n; y g; zÞqðn; gÞdndg ð ð 1 ¼ K ða1 ; a2 ; zÞ ð2pÞ2 C1 C2

uðx; xÞ ¼

X

Qða1 ; a2 Þeiða1 xþa2 yÞ da1 da2 :

(3)

. . The columns kj of the 3 3 Green’s matrix k ¼ ðk1 ..k2 ..k3 Þ are the displacement vectors corresponding to the concentrated point loads s ¼ dðx; yÞij applied to the surface z ¼ 0 along the coordinate vectors ij, j ¼ 1, 2, 3; d is Dirac’s deltafunction. Accordingly, the columns of matrix K are the ODE solutions corresponding to the boundary conditions F xy ½s ¼ ij , j ¼ 1, 2, 3, at the limiting point z ¼ 0. The integration paths C1 and C2 go along the real axes rounding real poles of K elements in accordance with the principle of limiting absorption. A general ODE solution is featured by exponential or oscillate behavior in different ranges of the input parameters a1, a2, and x. Therefore, conventional methods of ODE numerical solution are usually costly and unstable. Not infrequently, they completely fail in certain ranges. Thus, efficient Green’s matrix calculation is a challenging problem requiring specially adjusted methods. A description of such methods used for the calculations of the present paper, may be found in Refs. 9 and 10 and in the papers cited therein. Once the algorithms are developed and implemented, the elements of matrix Kða1 ; a2 ; z; xÞ may be obtained for any input parameters, and one can treat Eq. (3) as an explicit analytical expression. In the current paper, we adhere to the earlier used notations. In particular, the matrix Glushkov et al.: Wave energy transfer in elastic half-spaces

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0

iða21 M þ a22 NÞ=a2

B 2 Kða1 ; a2 ; zÞ ¼ B @ ia1 a2 ðM NÞ=a a1 S=a

ia1 a2 ðM NÞ=a2 iða21 N þ a22 MÞ=a2

2

a2 S=a

2

y ¼ r sin u

and a1 ¼ a cos c;

1

C ia2 P C A

(4)

R

is expressed via the five functions M, N, P, R, and S that depend on the Fourier parameters a1 and a2 only via the rapffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dial variable a ¼ a21 þ a22 : M ¼ M1 ða; zÞ=DðaÞ, etc. An advantage of representation (3) is that it is valid for any isotropic vertically inhomogeneous elastic substrate, i.e., for both laminate and FGM half-spaces with D ¼ 0 and D 6¼ 0 in Eq. (2). A particular type of the dependence of elastic properties on the depth affects only specific forms of the functions M, N, P, R, and S. In general, these functions cannot be written explicitly except for such simple cases as a homogeneous half-space or layer. The algorithms of their fast and stable calculation have been exhaustively described in Ref. 9. In the cylindrical (polar) coordinates x ¼ r cos u;

ia1 P

amplitudes are controlled by the factors Qðfn ; uÞ, i.e., by the source load q. In the lower half-space z < H, the matrix K can be represented in the form Kða1 ; a2 ; zÞ ¼

Kn ða1 ; a2 Þerm z ;

n¼1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where rn ¼ a2 j2n ; j1 ¼ x=cP and j2 ¼ x=cS are wavenumbers of the body P and S waves propagating pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with the velocities cP ¼ ðk þ 2lÞ=q and cS ¼ l=q. The stationary points a1;n ¼ jn cos u sin w and a2;n ¼ jn sin u sin w; n ¼ 1; 2; of the oscillating exponentials pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðiz j2n a2 a1 x a2 yÞ contribute in the far-field asymptotics of path integral (3) in the form of spherical body waves uðxÞ ¼

a2 ¼ a sin c;

2 X

2 X

an ðu;wÞeijn R =R½1 þ Oð1=ðjn RÞÞ; jn R ! 1;

n¼1

the double integrals over a1 and a2 are reduced to a single path integral over a, to which the residue technique may be applied. The second integration over the angular variable c: 0 c < 2p yields cylindrical Bessel functions that then are substituted by their asymptotic expressions. It leads to the far-field asymptotic representation of the excited wave field u in terms of traveling waves un uðxÞ ¼

Nr X

un ðxÞ ½1 þ Oððfn rÞ1 Þ;

n¼1

fn r ! 1; dim X=r 1

(5)

pffiffiffiffiffiffiffi un ðxÞ ¼ bn ðu; zÞeifn r = fn r; rffiffiffiffiffiffi i bn ¼ f res K ða; u; zÞja¼fn Qðfn ; uÞ: 2p n Here fn are poles of K elements in the complex plane a. They are roots of denominators of those five functions; Nr is the number of real poles. In fact, real fn are wavenumbers of cylindrical GWs propagating in the radial directions u from a localized source q. The denominator DN of function N(a, z) differs from the denominator D of the other four functions; its roots are wavenumbers of horizontally polarized shear waves (Love waves). With isotropic materials, the wavenumbers, and hence the phase and group velocities cn ¼ x=fn and vn ¼ dx=dfn as well as the inverse to cn slownesses sn ¼ fn =x, are independent of the direction of propagation u. While the matrix K accounts for the waveguide’s properties and determines the velocities and z-dependencies of modes un, the GW 1804

J. Acoust. Soc. Am., Vol. 137, No. 4, April 2015

(6) an ¼ i cos wKn ða1;n ; a2;n ÞQn ða1;n ; a2;n Þ: Here ðR; u; wÞ are spherical coordinates, x ¼ R cos u sin w; y ¼ R sin u sin w; z ¼ R cos w 0 u < 2p; p=2 < w p; R ¼ jxj; the terms an eijn R =R are spherical P and S waves propagating from the source in radial directions u; w with the velocities cP and cS. III. GUIDED WAVES IN STRUCTURES WITH INTERNAL CHANNELS

Elastic media with soft interlayers exhibit waveguide properties similar to those of an elastic layer supporting

FIG. 2. (Color online) Slowness dispersion curves sn(x) for the elastic halfspace with the FGM (solid lines) or laminate (dashed lines) coating. Glushkov et al.: Wave energy transfer in elastic half-spaces


Lamb waves. As an example, Fig. 2 displays the dispersion curves [frequency dependencies of GW slownesses sn(x)] for a one-channel FGM coating (solid lines) and for the laminate cover approximating this FGM dependence on z, as shown in Fig. 1 (dashed lines). In both cases, the slownesses sn are calculated via the roots fn of the matrix K denominator D, as specified below Eq. (5). The laminate coating consists of hard and soft sublayers of equal thicknesses h1 ¼ h2 ¼ H/2. The hard material is the same as in the underlying half-space z < H. The numerical results throughout the paper are conventionally given in a dimensionless form. The values l0 ¼ H; c0 ¼ cS and q0 ¼ q are taken as three basic units for length, velocity, and density (cS is the velocity of body shear waves and q is the density in the underlying half-space z < H). In these units, the dimensionless angular frequency x ¼ 2pfH=cS , where f is dimensional frequency in hertz. The units for load, elastic constants k and l, slowness and time-averaged energy (power) are, respectively, q0 ¼ k0 ¼ l0 ¼ qc2S ;

s0 ¼ 1=cS

and

w0 ¼ qc3S H 2 :

In these units, the parameters of the hard material are cP ¼ 1.690, cS ¼ 1, and q ¼ 1. In the laminate case, the soft material of the channel is taken with the velocity contrast c ¼ cS,2/cS ¼ 0.5, cP;2 ¼ 0:845;

cS;2 ¼ 0:5;

q2 ¼ 0:5:

(7)

In the FGM model, these maximal and minimal values are connected by cubic splines, continuously varying together with the first derivatives in the intervals 0:625 z 0:375 and 1:125 z 0:375 (hg ¼ 0.25). One can see that, similar to Lamb waves, the number of GWs increases with frequency, and that even a rough approximation of the FGM dependency by three homogeneous subdomains does not lead to a visible distinction in the dispersion curve patterns but only to small quantitative changes. Unlike Lamb waves, the real branches of the dispersion curves in Fig. 2 do not start from the frequency axis s ¼ 0 but from the level s ¼ 1/cS. Beneath this level, the branches sn(x) are complex, and the corresponding GWs are leaky waves. Furthermore, the first curve exhibits an unusual bend, indicating a nonmonotonic dependence on frequency. Such a bend of the first curve is also observed in the results

for steel-epoxy-concrete structures with soft epoxy interlayer.13 Figure 3 gives examples of dispersion properties of multichannel structures. Their laminate coatings are formed by the alternation of N hard and N soft interlayers of material properties of Eq. (7) and equal thickness h ¼ H=ð2NÞ. Since the properties of N soft interlayers are the same, every set of N neighboring dispersion curves merges into a sole branch as frequency increases. This effect is clearly visible in Figs. 3(a) (N ¼ 2) and 3(b) (N ¼ 3). For comparability, they are expressed as functions of x/N. With channels of slightly different properties, the curves do not merge, going side by side close to each other. Such a behavior suggests that only properties of soft interlayers clamped between hard media are significant for the GW dispersion properties at high frequencies. Hence, the GWs should be localized in these channels as x increases. To check this suggestion and to study general dependences of GW amplitudes on the depth z and frequency x, the amplitude factors bn in Eq. (5) have been analyzed. It should again be noted that while z-dependences of the modes are up to constant factors similar to the modal eigenforms obtained using the modal analysis technique, the amplitudes bn are uniquely determined by the structure and the load via the factors K and Q. Therefore, in contrast to normalized eigenforms with the same total amplitudes at any frequency, the values jbn j may drastically vary with x. Figure 4 illustrates such a variation and expected highfrequency spatial localization within and by the channels for several first GW pairs un,1 and un,2 corresponding to the dispersion curve pairs sn,1 and sn,2 in Fig. 3(a). It depicts the amplitudes jbn;1 j and jbn;2 j; n ¼ 1; 2; …; 5, as functions of x and z for the GWs generated by the vertical point source q ¼ dðx; yÞi3 in the laminate two-channel half-space. We consciously confine ourselves to this simplest point source, because non-point loads introduce additional effects of area X shape and size that should be studied separately. For example, the Fourier symbol of a load q ¼ (0, 0, p), evenly distributed in a circular area r a, is Q ¼ ð0; 0; QÞ; QðaÞ ¼ 2ppJ1 ðaaÞ=a. The quasi-periodic function Q(a) modulates the amplitudes bn, making them zero when afn ¼ nj ; j ¼ 1; 2; …, where nj are zeros of the Bessel function J1. In addition to the spatial localization, the insets of Fig. 4 show the localization of every mode in the frequency domain. The GW amplitudes, appearing in the plots after the

FIG. 3. (Color online) Dispersion curves sn(x) with (a) two and (b) three soft channels.


Glushkov et al.: Wave energy transfer in elastic half-spaces

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FIG. 4. (Color online) Level-line plots of the amplitude factors jbn;1 j and jbn;2 j; n ¼ 1; 2; …; 5, for GWs associated with the merging pairs sn,1 – sn,2 of the dispersion curves in Fig. 3(a).

cut-off frequencies, cease to be distinguishable at the frequencies where the pairs of neighboring poles merge into sole curves. This fact indicates dominant energy transfer by different GWs in different frequency bands. To explore the regularities of the wave energy transfer, the explicit expressions for time-averaged characteristics of energy fluxes, given in Sec. IV below, have been used. IV. WAVE ENERGY A. Basic idea

The time-averaged over the period of oscillation amount of wave energy transferred in a time-harmonic elastic wave field ueixt through a surface S is specified by the quantity14 ðð x (8) E¼ es ðxÞdS; es ¼ Im ðu; ss Þ: 2 S Here es ¼ (e, n) is the projection of the Umov-Poynting vector e of the time-averaged density of the energy flux onto the surface normal n; ss ¼ nkdiv u þ 2l@u=@n þ lðn rot uÞ is the stress vector at a surface element with the normal n. The scalar product of complex vectors assumes complex 1806


conjugate components P of the second factor denoted by the asterisk: ðu; sÞ ¼ 3i¼1 ui si . In some particular cases of the surface S, such as a horizontal plane z ¼ const, a lateral cylindrical surface r ¼ const, or a lower hemisphere R ¼ const of large radii r or R, simple explicit expressions for the values E and e in terms of matrix K and vector Q elements were derived as far back as in the beginning of the 1980s.15 These expressions have proved to be a convenient tool for the energy balance analysis and energy flux visualization in elastic stratified solids. They are independent of the law of material stratification that only affects the particular appearance of the functions M, N, P, R, and S in the matrix K of form (4). B. Energy transfer through a horizontal plane

For a horizontal plane z ¼ const, ð x 1 Im Gða; zÞada ¼ Ev þ Er ðzÞ; E ðzÞ ¼ 2 ð2pÞ2 C ð j2 x 1 Ev ¼ Im Gða; zÞada; 2 ð2pÞ2 0 Nr Nr X X x 1 Re E n ðzÞ ¼ res Gða; zÞaja¼fn ; Er ¼ 2 4p n¼1 n¼1

(9)



where C is an integration path along the positive real semiaxis going around real poles in accordance with the principle of limiting absorption, ð 2p Gða; zÞ ¼ ðT; UÞdc

Gða; zÞ ¼ F1 ðTM M þ TS S Þ þ F2 TN N þ F3 ðTP P þ TR R Þ þF4 ðTM P þ TS R Þ þ F5 ðTP M þ TR S Þ=a2 ;

0

¼

ð 2p X 3 0

Tj ða; c; zÞUj ða ; c; zÞdc;

(10)

j¼1

where TM ¼ lðM0 þ SÞ;

TN ¼ lN 0 ;

T ¼ F xy ½s ¼ ðT1 ; T2 ; T3 Þ;

TP ¼ a2 lðR þ P0 Þ;

T1 ¼ lðia1 U3 þ U10 Þ; T2 ¼ lðia2 U3 þ U20 Þ;

TS ¼ ðk þ 2lÞS0 =a2 kM;

T3 ¼ ðk þ 2lÞU30 ikða1 U1 þ a2 U2 Þ: The wavenumber j2 ¼ x/cS is the upper limit of the segment 0 a j2, in which G is a complex-valued function, while for a > j2, Im G 0, and all the real poles fn lie to the right of j2: fk > j2, k ¼ 1, 2, …, Nr. The representation of function G(a, z) in terms of U and T follows from Eq. (8) in accordance with the Parseval equality generalized for path integrals with contours deviating into the complex plane. The components of factor U in the scalar product of Eq. (10) are not only complex conjugate, but in addition, they depend on the conjugate argument a*. This rule holds for all complex conjugate Fourier symbols below. The real poles fn, from which the residues in Eq. (9) are taken, are double poles of function Gða; zÞ, because they enter it via both T and U. It is very important to note that the poles of T(a) and U ða Þ shift from the real axis into the upper and lower complex half-planes, i.e., into the opposite directions, as a small viscosity is introduced into the material parameters in line with the principle of limiting absorption. Consequently, the singular constituents of T and U* contribute to the residues with opposite signs. In more detail, let the multipliers of a term Tj ðaÞUj ða Þa be decomposable at a pole fn into the Laurent series Tj ðaÞa ¼ t1 =ða fn Þ þ t0 þ Uj ða Þ ¼ ½u1 =ða fn Þ þ u0 þ ¼ u1 =ða fn Þ þ u0 þ Hence, its residue constituent has the form t1 u0 =ða fn Þ þ t0 u1 =ða fn Þ:

(11)

With a backward mode pole fn, the signs in Eq. (11) are opposite. Function G can be written in terms of K and Q elements in the following compact form: J. Acoust. Soc. Am., Vol. 137, No. 4, April 2015

F1 ¼

ð 2p 0

F3 ¼

ð 2p 0

F5 ¼

ð 2p 0

Q12 Q12 dc; Q3 Q3 dc;

TR ¼ ðk þ 2lÞR0 a2 kP;

ð 2p F2 ¼ Q21 Q21 dc; 0 ð 2p F4 ¼ Q12 Q3 dc; 0

Q3 Q12 dc;

Q12 ¼ ða1 Q1 þ a2 Q2 Þ=a2 ;

Q21 ¼ ða2 Q1 a1 Q2 Þ=a2 :

In the case of a point load q ¼ pd(x, y), where p ¼ (p1, p2, p3) is a constant vector, the Fourier symbol Q ¼ p is also constant. Hence, F1 ¼ F2 ¼ pjpr j2 =ðaða Þ Þ; F3 ¼ 2pjp3 j2 ;

jpr j2 ¼ jp1 j2 þ jp2 j2 ;

F4 ¼ F5 ¼ 0:

In particular, it means that the waves caused by the tangential and normal load components pr and p3 independently contribute to the amount of wave energy E(z). 1. Remark

Since the horizontal planes are conventionally assumed with the upward normal n ¼ (0, 0, 1), Eq. (9) yields negative values of E(z) for oppositely directed downward energy fluxes from a surface source. For convenience, in the numerical examples below, E(z) and its constituents are plotted with the opposite sign, i.e., as for the planes with the downward normal n ¼ (0, 0, 1). C. Source energy distribution

Except for infrequent cases of backward modes,16 the poles fn and fn , with vanishing fictitious viscosity, come to the real axis from the top and from the bottom. Hence, the contour C rounds them from below and from above, respectively. Consequently, they contribute to the En term as res ðTU aÞja¼fn ¼ t1 u0 t0 u1 :

(12)

With z ¼ 0, E0 ¼ E(0) is the amount of wave energy supplied from the source into the substrate through the area of loading X (the source power). In accordance with the boundary condition given by Eq. (1), Tjz¼0 ¼ Q. Then in Eq. (12), TM ¼ i; TN ¼ i; TP ¼ 0; TR ¼ 1, and TS ¼ 0. In this case, the factors Tj in Eq. (10) have no poles, and the poles fn of function G cease to be multiple. As a consequence, the calculation of residues in Eq. (9) becomes much easier; it is reduced to the calculation of residues from the single poles of factors Uj . In particular, the source power of an axially symmetric vertical load q ¼ ð0; 0; qðrÞÞ applied to a circular area r a is Glushkov et al.: Wave energy transfer in elastic half-spaces

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E0 ¼

x Im 4p

ð

Rða; 0ÞQz ðaÞQz ðaÞada;

(13)

C

Ða where Qz ðaÞ ¼ 2p 0 qðrÞJ0 ðarÞrdr is the Fourier-Bessel symbol of the load function q(r). Analogously, an axially symmetric load sr ¼ qðrÞ, i.e., q ¼ ðq cos u; q sin u; 0Þ, generates a flux of wave energy of the averaged power ð x E0 ¼ Im iMða; 0ÞQr ðaÞQr ðaÞada; 4p ða C (14) Qr ðaÞ ¼ 2p qðrÞJ1 ðar Þrdr: 0

As in Eq. (9), the source energy E0, even with arbitrary load q(x, y), consists of the integral over the segment [0, j2], yielding Ev, and the sum of residues Er(0). It has been checked and confirmed via numerical integration of energy flux density over corresponding surfaces that, in stratified half-space, Ev ¼ Es and Er(0) ¼ Ec, where ð 2p ð p Es ¼ es ðR; u; wÞR2 sin wdwdu 0

p=2

þ Oð1=ðj2 RÞÞ; j2 R 1; ð 2p ð 0 Ec ¼ ec ðr; u; zÞrdzdu 0

1

þ Oð1=ðj2 rÞÞ;

j2 r 1:

(15)

The values Es and Ec are parts of the source energy transferred by spherical body waves and cylindrical guided waves through a lower hemisphere and a lateral cylindrical surface of increasing to infinity radii R and r, respectively. The power densities es and ec are projections of the Umov-Poynting vector e onto the surface normals ns ¼ ðcos u sin w; sin u sin w; cos wÞ and nc ¼ ðcos u; sin u; 0Þ to the spherical and cylindrical surfaces, respectively. The integration is easily performed on the basis of far-field asymptotics Eqs. (5) and (6) for guided and body waves. The energy conservation equalities Ev ¼ Es and Er(0) ¼ Ec are consistent with known analytical expressions for source energy partition in a homogeneous half-space.6,7 If the plane z ¼ const is located below the outer surface z ¼ 0, i.e., z ¼ z0 < 0, the value E(z0) becomes less than E0 because of the outflow of energy to infinity through the segment z0 z 0 of the lateral cylindrical surface r ¼ const, 0 u < 2p; 1 < z 0. It is noteworthy that only the GW part Er(z) of the source energy E(z) carried through the plane z ¼ const decreases to zero as z ! 1, while the amount of energy Ev, transferred beneath this plane by body waves, remains constant. That is why we write Ev in Eq. (9) with no argument z. Thus, the energy outflow from the source to infinity is strictly split into two energy fluxes carried downward by body waves and laterally by GWs. Equations (9) and (15) provide information about the contribution of every normal mode and body wave to the source power E0 as well as information about its spatial distribution. In accordance with asymptotics Eqs. (5) and (6), es and ec can be represented in the form of double sums 1808

es ¼


2 X 2 X n¼1 m¼1

es;nm

and

ec ¼

Nr X Nr X

ec;nm ;

n¼1 m¼1

in which the terms are expressed through the scalar products of the displacement and stress amplitudes un and sm of the nth and mth wave modes. With m ¼ n, these terms are obviously the power density of the corresponding nth waves [of P or S body wave an eijn R =R in Eq. (6) or of the GW un in Eq. (5)]. The terms with m 6¼ n account for the influence of nth and mth mode interaction on the total energy density. In the case of body waves, these mixed terms are equal to zero due to the orthogonal polarization of P and S waves; hence, es ¼ es;11 þ es;22 . Unlike body waves, the GW amplitudes un ¼ ðun ; vn ; wn Þ and sm ¼ ðrxm ; sym ; szm Þ are not orthogonal. Hence, the radial power density ec at a fixed point ðr; u; zÞ cannot be calculated as a simple sum of the modal energy densities ec,nn. Nevertheless, the mixed terms ec,nm, n 6¼ m do not contribute to the energy transfer through the whole lateral cylindrical surface r ¼ const, 0 u < 2p; 1 < z 0 because of the property of generalized orthogonality of normal modes.17 In the Cartesian coordinate system with the x axis coinciding with the radial direction of GW propagation (u ¼ 0, r ¼ x), this property takes the form ð0 ½ðun ; rx;m Þ ðszn ; wm Þdz þ Oð1=ðj2 rÞÞ ¼ 0 1

and ð0

ðvn ; sy;m Þdz þ Oð1=ðj2 rÞÞ ¼ 0;

as j2 r ! 1

1

for the in-plane Rayleigh-Lamb waves un ¼ ðun ; 0; wn Þ and for the anti-plane [shear horizontal (SH)] Love waves un ¼ ð0; vn ; 0Þ, respectively. The generalized orthogonality implies the nulling of integrals of the sum ec;nm þ ec;mn over the full depth segment 1 < z 0 with any fixed r and u. It is all the more remarkable that every term En(0), expressed in Eq. (9) via the residue of G(a, 0) from the pole fn, yields the part of source energy carried to infinity by the corresponding nth GW. This fact has been also confirmed via numerical integration of ec,nn over the lateral cylindrical surface. Along with the power balance provided by Eqs. (9) and (15), it serves as an additional control of the correctness of computations. In a multimode range, this property is not preserved for a part of cylindrical surface, e.g., for a segment of height Dz located between two planes: z ¼ z0 and z ¼ z0 Dz. The reason is that although the integrand er ðrÞr Oð1Þ as j2 r ! 1, the power density vector e has also a non-zero vertical component that depends on r in a sinusoidal manner. It results in a wavy form of energy streamlines in the lateral far field j2r 1, z ¼ const. Therefore, with two and more GWs, the stream of energy cannot run in parallel to these planes and remain within the layer z0 Dz z z0 . At certain distances, the flux partly leaves it and then returns together with additional power from neighboring streams. As a result, the lateral component ec varies with r, and its integration over such a cylindrical segment yields the amount of Glushkov et al.: Wave energy transfer in elastic half-spaces


FIG. 5. (Color online) Source energy E0(x) normalized to x2 (dashed-point line), its body wave part Ev(x) (dashed line), and GW parts En(x) (solid lines), calculated on the basis of Eq. (9) for the one-channel FGM and laminate half-spaces.

energy oscillating around the value DEr ¼ Er ðz0 Þ Er ðz0 DzÞ as the distance r varies. Since the dependence of ec on z at a fixed distance r cannot properly specify the part of wave energy carried laterally at a certain level z, the characteristic e^r ðzÞ ¼ lim DEr =Dz ¼ E0r ðzÞ; Dz!0

(16)

which is independent of r and u, can be used instead. Unlike ec ðr; u; zÞ, it depends only on z. Taking into account that Er(0) ¼ Ec,

ð0 E0r ðzÞdz ¼ e^r ðzÞdz 1 1 ð 0 ð 2p ec ðr; u; zÞrdudz as j2 r ! 1: ¼

Er ð0Þ ¼

ð0

1 0

Thus, the function e^r ðzÞ may be treated as a density of the depth distribution of lateral energy outflow to infinity. Although the equality of integrals over z does not necessarily entail the equality of integrands, it has been confirmed numerically that

FIG. 6. (Color online) Associated with nth GW un portions of wave energy En normalized to the source energy E0 transferred through the plane z ¼ const (left) and averaged lateral energy density e^r;n (right) as functions of x and z for the same FGM half-space as in Figs. 2 and 5(a).



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FIG. 7. (Color online) Relative downward (left) and lateral (right) energy transfer in the total GW field generated in the one-channel FGM (top) and laminate (bottom) half-spaces.

e^r ðzÞ ¼

Nr X

e^r;n ðzÞ;

n¼1

e^r;n ¼ E0n ðzÞ ¼

ð 2p

ec;nn ðr; u; zÞrdu

as

j2 r ! 1:

0

Indirectly it means that the mixed terms ec,nm, m 6¼ n, do not contribute to e^r as well. V. MODE-TO-MODE TRANSFER AND BACKWARD FLUXES OF WAVE ENERGY

As expected, the energy analysis has confirmed the suggestion that the source energy is transferred by sole dominant GW modes in different frequency ranges (Fig. 5). In this figure, the dashed-point lines represent the total source energy E0(x)/x2 normalized to x2, while the dashed lines show the body wave portion Ev(x)/x2. The numbered solid lines are for the GW parts En(x)/x2 at z ¼ 0 [see Eq. (9)]; the numbering corresponds to that for the dispersion curves in Fig. 2. The normalization to x2 is performed for visual clarity. It eliminates the growth E0 Oðx2 Þ as x ! 1, which is inherent to a constant force source. Figures 5(a) and 5(b) are for the FGM and laminate coatings, respectively. Their comparison shows that the laminate approximation of FGM properties has almost no effect on the source energy distribution in this case. The first mode exhibits a local peak of the transferred energy E1(x) near the second-mode cut-off frequency x ¼ 1.41, where the dispersion curve s1(x) rises sharply (Fig. 2). After its domination in the range 0 < x < 5, the energy E1 decreases to almost zero, and at higher frequencies one can see a regular alternation of dominant modes, which becomes especially strict as frequency increases. On the whole, the GW curves in Fig. 5 give the impression of successive source energy transfer from one dominant wave mode to the next. It should be noted that similar regularities in source energy distribution among excited modes can be observed not only in structures with soft interlayers. For example, they are present in the plots for an elastic layer stuck to a rigid base.18 In that case, however, the localization of every mode periodically repeats in frequency, unlike the localization in the only limited bands visible in Figs. 4 and 5. 1810


Figure 6 provides a more detailed spatial description of wave energy transfer by the first six modes in the same onechannel FGM half-space as in Figs. 2 and 5(a). The left insets show the relative amount of energy En ðx; zÞ=E0 ðxÞ [Eqs. (9) and (13)] carried down through the planes z ¼ const. The right insets display the averaged lateral energy densities e^r;n =ðE0 =H 2 Þ of the related nth modes [Eq. (16)] also as functions of x and z. The patterns of the total relative downward and lateral energy transfer Eðx; zÞ=E0 ðxÞ and e^r ðx; zÞ=ðE0 =H 2 Þ are shown in Fig. 7. For comparison, two lower insets of this figure are for the laminate coating. One can see that the difference between the FGM and laminate results here is also insignificant. In these figures, the patterns of the densities e^r are characterized by ornamental ripples within the soft interlayer. These ripples are of special interest because they appear around certain (x, z) areas in which the values e^r ðx; zÞ become negative. They look like elongated islets surrounded by white zones with zero-averaged lateral energy density e^r . In the e^r;n insets of Fig. 6, wavy strips indicating zones (x, z) with negative values are present for every mode starting from n ¼ 3. The negative value of e^r means that at the corresponding depth z, the total far-field energy flux consists of streams directed mainly to the source from infinity. This is a socalled backward energy flux.19 In the near field, it is deflected by more powerful counter streams, coming from

FIG. 8. (Color online) Dispersion curves in the case of a more contrast laminate channel; c ¼ vS,2/vS ¼ 1/5, hg ¼ 0. Glushkov et al.: Wave energy transfer in elastic half-spaces


FIG. 9. (Color online) Same as in Fig. 7 (bottom) but with a more contrast channel as in Fig. 8.

the source, and goes back to infinity. Its outgoing part returns all the arriving power back to infinity; therefore, it does not violate the energy balance. In this respect, a backward flux may be considered as an incoming part of a large energy vortex circulating via infinity. The energy vortices are spatial areas, in which time-averaged wave energy circulates in closed trajectories with zero total energy transfer through any intersecting surface.20 Since e^r ðzÞ ¼ E0r ðzÞ, a positive power density e^r > 0 in the whole range of z variation corresponds to a monotonic decrease of Er(z) with depth, i.e., with decreasing z. Such a behavior of Er(z) is consistent with the permanent energy outflow to infinity in the lateral direction. Consequently, a violation of Er(z) monotonicity at certain z is an indirect sign of the emergence of backward energy fluxes at the corresponding depth level. It has been noted that the amount and power of backward fluxes increase with increasing contrast of channel properties. As an example, let us consider a one-channel laminate coating with the velocity ratio c ¼ cS,2/cS ¼ 1/5, instead of previous properties of Eq. (7) with c ¼ 1/2. The dispersion properties of this structure are specified by the slowness curves in Fig. 8 (compare with Fig. 2), while the normalized energy characteristics Er/E0 and e^r =ðE0 =H 2 Þ of the GWs generated by the vertical point source are shown in Fig. 9. The latter is for the initial frequency section 0 < x < 5, where no backward flux zones were present in the previous low-contrast structure (Fig. 7, bottom-right). In Fig. 9(b) one can see several areas encircled by dashed curves, in which e^r < 0.

Figure 10 depicts depth dependencies of the downward and lateral energy characteristics shown in Fig. 9 at the three fixed frequencies x ¼ 0.85, 2.35, and 3. In the regular case x ¼ 3, the density e^r > 0 for any z, and Er(z) decreases monotonically with depth. At x ¼ 0.85 and 2.35, the depth cross-sections of the e^r ðx; zÞ pattern in Fig. 9(b) pass through the backward flux zones with negative e^r almost in the whole channel interval 1 < z < 0:5 [Fig. 10(b)]. It results in the expected violation of the Er(z) monotonicity in the corresponding z intervals [Fig. 10(a)], Moreover, at x ¼ 0:85; Er ðzÞ also becomes negative within the depth interval 0:75 < z < 0:5 [in Fig. 9(a) the areas of negative Er are also encircled by dashed curves]. It means that at these depths, the total energy flux through the plane z ¼ const goes upward to the surface source. Figure 11 visualizes this phenomenon via the energy streamline patterns. The streamlines are curves x ¼ x(s) tangential to the energy density vector e at each point x : x0 ðsÞ ¼ eðxðsÞÞ. In the near field, where far-field asymptotics Eqs. (5) and (6) cannot be used, the vector e is calculated via numerical integration of integrals of Eq. (3) reduced to a one-fold form. The streamlines show that at x ¼ 0.85 [Fig. 11(a)], a backward energy flux comes from infinity at the depth level 1 < z < 0.5. In the near field, it splits into two sleeves turning up and down and then goes back to infinity. The turning up part comes at the depth level 0.75 < z < 0.5. That is why Er(z) becomes negative in this interval [Fig. 10(a)]. Note that the entire outgoing flux beneath the coating (z < 1) is just a lower sleeve of the backward flow returning to infinity. The source energy may

FIG. 10. (Color online) Depth dependency (fixed frequency cross-sections of the energy characteristics of Fig. 9) in the case of backward fluxes (x ¼ 0.85 and 2.35) and for a regular energy outflow to infinity (x ¼ 3).



1811


FIG. 11. (Color online) Energy streamlines in the total wave field ueixt illustrating (a) near-field reversal of the backward energy flux coming from infinity at the depth 1 < z < 0.5, its rising sleeve results in negative Er in Fig. 10(a); and (b) energy outflow with no backward streams at x ¼ 3; k ¼ 2pcS/x is a characteristic wavelength.

not directly enter this flux but only after passing via infinity and returning to the near field with the backward streams. For comparison, Fig. 11(b) shows the pattern of energy streamlines at the “calm” frequency x ¼ 3. It demonstrates the overall energy outflow from the source to infinity with no backward streams. One more interesting feature of the source energy spatial distribution takes place at low frequencies. In Fig. 9(a), one can see that at x < 0.8 the source energy penetrates much deeper beneath the coating than at the higher frequencies. Consequently, the white area x < 0.9, z > 0.5 in Fig. 9(b) indicates the absence of lateral energy outflow within the coating at these low frequencies, unlike for the situation at higher frequencies when the majority of GW energy flows along the coating. VI. CONCLUSION

The closed expressions derived for both laminate and functionally graded elastic half-spaces in terms of Green’s matrix and given loads have proven to be a convenient tool for analyzing energy transfer in the time-harmonic wave fields generated by surface sources. The analysis carried out for elastic substrates with internal channels has revealed the effect of dominant energy transfer by only one, just appeared, guided wave in every subsequent frequency band (a mode-to-mode transfer of the source energy). It was also observed that the averaged lateral energy density might become negative at certain frequencies and depths indicating emergence of backward energy fluxes in structures with high-contrast internal channels. Their presence, however, does not violate the overall energy balance in the sourcesubstructure system. ACKNOWLEDGMENTS

The work was partly supported by the Russian Foundation for Basic Research (RFBR, projects No. 13-0196520 and 14-08-00370) and the Russian Ministry of Science and Education (project No. 1.189.2014K). 1

Lord Rayleigh, “On waves propagated along the plane surface of an elastic solid,” Proc. London Math. Soc. s1-17(1), 4–11 (1885).

1812


2

A. E. H. Love, Some Problems of Geodynamics (Cambridge University Press, Cambridge, UK, 1911), 180 pp. 3 H. Lamb, “On waves in an elastic plate,” Proc. R. Soc. London Ser. A 93, 114–128 (1917). 4 C. Campbell, Surface Acoustic Wave Devices for Mobile and Wireless Communications (Academic Press, Orlando, FL, 1998), 631 pp. 5 V. Giurgiutiu, Structural Health Monitoring with Piezoelectric Wafer Active Sensors (Elsevier, Burlington, MA, 2008), 760 pp. 6 G. F. Miller and H. Pursey, “On the partition of energy between elastic waves in semi-infinite solid,” Proc. R. Soc. Ser. A. 233(1192), 55–63 (1955). 7 S. Ya. Kogan, “Generation of seismic energy by a surface source,” Izv. Akad. Nauk, Ser. Geofiz. 7, 1000–1013 (1963). 8 F. J. Sanchez-Sesma, R. L. Weaver, H. Kawase, S. Matsushima, F. Luz on, and M. Campillo, “Energy partitions among elastic waves for dynamic surface loads in a semi-infinite solid,” Bull. Seismol. Soc. Am. 101, 1704–1709 (2011). 9 E. V. Glushkov, N. V. Glushkova, S. I. Fomenko, and C. Zhang, “Surface waves in materials with functionally gradient coatings,” Acoust. Phys. 58(3), 339–353 (2012). 10 E. Glushkov, N. Glushkova, and A. Eremin, “Forced wave propagation and energy distribution in anisotropic laminate composites,” J. Acoust. Soc. Am. 129, 2923–2934 (2011). 11 N. Ryden and M. J. S. Lowe, “Guided wave propagation in threelayer pavement structures,” J. Acoust. Soc. Am. 116(5), 2902–2913 (2004). 12 A. V. Sveshnikov, “The limit absorption principle for a wave guide,” Dokl. Akad. Nauk SSSR 80(3), 345–347 (1951) (in Russian). 13 Y. Shen, S. Hirose, and Y. Yamaguchi, “Dispersion of ultrasonic surface waves in a steel-epoxy-concrete bonding layered medium based on analytical, experimental, and numerical study,” Case Stud. Nondestruct. Test. Eval. 2, 49–63 (2014). 14 N. A. Umov, “The equations of motion of the energy in bodies,” Odessa, Russia (1874) (in Russian); reprinted in: Selected Works (State Publishers of Technical-Theoretical Literature, Moscow, Russia, 1950), pp. 151–226. 15 E. V. Glushkov, “Energy distribution of a surface source in an inhomogeneous half-space,” J. Appl. Math. Mech. 47(1), 70–75 (1983). 16 I. Tolstoy and E. Usdin, “Wave propagation in elastic plates: Low and high mode dispersion,” J. Acoust. Soc. Am. 29, 37–42 (1957). 17 A. S. Zilbergleit and B. M. Nuller, “Generalized orthogonality of the homogeneous solutions to the dynamic problems of elasticity,” Dokl. Phys. 234(2), 333–335 (1977). 18 V. I. Erofeev, N. V. Klyueva, and I. N. Soldatov, “Waves in a bed under periodic tangential loading,” J. Appl. Mech. Tech. Phys. 46(4), 549–555 (2005). 19 V. A. Babeshko, E. V. Glushkov, and N. V. Glushkova, “Energy vortices and backward fluxes in elastic waveguides,” Wave Motion 16, 183–192 (1992). 20 R. V. Waterhouse, D. G. Crighton, and J. E. Ffowcs-Williams, “A criterion for an energy vortex in a sound field,” J. Acoust. Soc. Am. 81(5), 1323–1326 (1987).



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