Continuum Mech. Thermodyn. (2010) 22:663–677 DOI 10.1007/s00161-010-0143-z
O R I G I NA L A RT I C L E
M. Brun · A.B. Movchan · N.V. Movchan
Shear polarisation of elastic waves by a structured interface
Received: 1 December 2009 / Accepted: 19 April 2010 / Published online: 16 June 2010 © Springer-Verlag 2010
Abstract The article addresses the elastodynamic behaviour of a ‘shear-type structured’ interface separating two elastic regions. The structure includes a system of horizontal elastic bars connected by transverse massless elastic links. For a plane wave transmission problem, the analytical results obtained here show that such an interface acts as a ‘shear polariser’. A special attention is paid to the analysis of the defect modes (also referred to as resonance modes) within the interface. It is shown that these vibration modes induce enhanced transmission within the elastic system. Keywords Elastic waves · Structured interface · Transmission problem In honour of Professor Leonid Slepyan.
1 Introduction Theoretical modelling of continuous imperfect interfaces occurs naturally in a wide range of studies of composite media in mechanics, physics and engineering. In particular, for continuous static models, imperfect interfaces describe high contrast coatings. Static models of composites containing coated inclusions have been analysed by [2,10,11,13] and by [1]. Imperfect interfaces may influence the overall properties of the homogenised composites. One of such examples concerns with the notion of neutral inclusions studied in a static setting by [8,12] and by [14]. Dynamic composite structures containing coated inclusions have been analysed by [15], where the notion of neutrality was linked to the effective refractive index in the long-wave limit. In models of wave propagation, the classical homogenisation approximations are not applicable when the typical wavelength becomes comparable with the size of an inhomogeneity such as small holes or inclusions. A comprehensive asymptotic analysis, accompanied by numerical simulations, for small-amplitude vibrations of an elastic layer on a pre-stressed elastic half-space is given in [6]. Communicated by Prof. Thomas Pence. M. Brun (B) Department of Structural Engineering, University of Cagliari and Sardinian Laboratory for Computational Material Science, Piazza d’Armi, 09123 Cagliari, Italy E-mail:
[email protected] A.B. Movchan · N.V. Movchan Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK E-mail:
[email protected] N.V. Movchan E-mail:
[email protected]
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Fig. 1 Semi-infinite elastic regions ± joined by a structured interface occupying the region D of finite width D. Both elastic media, above and below the interface, have the same mass density ρ and the same Lamé moduli μ and λ. The imperfect interface consists of horizontal elastic bars l1 , l2 , . . . , l N which are allowed to move in the O x-direction; the bars are connected to each other by vertical non-inertial elastic links. All elastic bars have the same density ρ and the longitudinal stiffness E = E¯ A, where E¯ is the Young modulus and A is the cross-sectional area per unit thickness; the system of vertical elastic links has the horizontal stiffness γ
The idea of non-local structured interfaces was proposed in [7] for several classes of static and dynamic models of elastic composite media. In elastostatics, [3–5] have shown that the finite-thickness structured interfaces display non-local mechanical behaviour, effective properties, and elastic stress concentrations which cannot be described using, for example, a zero-thickness interface approach. Structured interfaces may possess inertia, and tractions may become discontinuous across such interfaces in dynamic configurations. The derivation and analysis of dynamic transmission conditions for structured interfaces are important for evaluation of transmission and reflection properties of heterogeneous composite media. Periodic semi-discrete systems have been studied in [7]; recently [9] have developed this theory further to encompass slow waves and to analyse transmission properties of a dynamic structured interface of finite width. An interesting application of the latter model is related to a negative refraction exhibited by waves interacting with structured interfaces. In this study, we address the case of a shear-type structured interface excited by a plane pressure wave at an oblique incidence. A frame-type elastic structure is embedded into the interface, which has a relatively low effective shear stiffness. Such an interface can act as a shear polariser. The physical configuration discussed in this article is shown in Fig. 1. The transverse displacements within the interface are assumed to be zero, which approximates the situation where the flexural deformations within the interface are negligibly small. The structural elements within the interface can be thought of as elastic layers, linked by transverse elastic bars. The article is organised as follows. The formulation of the problem is given in Sect. 2. This section also includes the description of the geometry and the lower-dimensional asymptotic approximations used within the interface. The solution of the transmission problem is described in Sect. 3. In particular, this section includes the derivation of the effective transmission conditions for a shear-type inertial interface, which account for a discontinuity in tangential displacements and shear tractions across the interface. Section 4 presents the derivation and analysis of the energy balance relation for the reflected and transmitted waves. Numerical illustrations are given in Sect. 5. We draw the attention of the reader to the example discussed in Sect. 5, for a multiple-bar shear-type interface. In particular, in Figs. 5 and 7, we give the energy of the transmitted waves as a function of the wave number k and the apparent speed c of propagation of the incident wave along the interface. The diagrams show the regions of enhanced transmission, and the corresponding values of k and c are linked to the so-called resonance defect modes, which are described in detail in Sect. 3.2.1. The other feature, which will be emphasised upon, is the shear polarisation of the transmitted signal and its dependence on the parameters of the system and the angle of incidence.
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2 Formulation of the problem 2.1 Geometry Consider an elastic region incorporating a structured interface of finite width D. The interface region is a horizontal strip in the (x, z)-plane: D = {(x, z) : x ∈ R, −D ≤ z ≤ 0} .
(1)
The structure within the interface incorporates N elastic infinite bars l j , j = 1, 2, . . . , N , connected to each other via transverse elastic massless links as shown in Fig. 2. Such a structure is considered as an elastic frame, so that the angle between the vertical links and the horizontal elastic bars is maintained at π/2. In the boundary region, the upper (lower) bar is continuously attached to the upper (lower) half-plane, as shown in Fig. 1. The notations + and − are used for the half-planes above and below the structured interface, respectively: + = {(x, z) : x ∈ R, z > 0} , − = {(x, z) : x ∈ R, z < −D} . We describe the structure of the interface within D in the next section.
2.2 Lower-dimensional approximations within the interface Since the interface consists of thin structured elements, the actual elastic material within the interface is assumed to be sufficiently stiff, as described in the text later. Each elastic bar is treated as a one-dimensional elastic element, and we assume that the leading approximation of the elastic displacement field within the bar l j has the form: u( j) ∼ u j (x, t)e(1) ,
(2)
where e(1) stands for the basis Cartesian vector in the direction of the O x-axis. The approximation (2) implies that each bar moves only in the horizontal direction (along the O x-axis). Such a state can be achieved by assuming that the first horizontal elastic bar is constrained against any motion in the vertical direction, along the Oz-axis, and that the horizontal elastic bars have a very high bending stiffness. The equations of motion will be written for the elastic bars inside the interface ( j = 2, 3, . . . , N − 1) as well as for those on the boundary of the interface adjacent to + and − ( j = 1 and j = N ). Let l j be an interior elastic bar, 1 < j < N , and let E denote the longitudinal stiffness, ρ the mass density, and γ the shear stiffness of the array of vertical elastic links connecting l j with l j−1 and l j+1 . The equation of motion for the interior bar l j takes the form: E(u j )x x − ρ(u j )tt − 2γ u j + γ (u j+1 − u j−1 ) = 0,
j = 2, 3, . . . , N − 1,
(3)
where (u j )x x and (u j )tt stand for the second-order partial derivatives with respect to x and t, respectively.
(a)
(b)
(c)
Fig. 2 The diagram shows the structure of the interface: a the upper elastic bar l1 is perfectly bonded to the elastic material occupying the region + ; it is connected to the bar l2 below it via massless vertical elastic links, and is constrained against any vertical motion; b an internal bar l j is connected to the bars l j−1 and l j+1 above and below it via massless vertical elastic links; only a horizontal movement of the bar l j is allowed; c the lower elastic bar l N is perfectly bonded to the elastic material occupying the region − ; it is connected to the bar l N −1 above it via massless vertical elastic links, and it is constrained against any vertical motion
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For the bars l1 and l N on the boundary of the structured interface, the equations of motion are E(u 1 )x x − ρ(u 1 )tt + γ (u 2 − u 1 ) + τ+ = 0, E(u N )x x − ρ(u N )tt + γ (u N −1 − u N ) − τ− = 0,
(4) (5)
where τ+ and τ− are the shear stresses in the ambient elastic media above and below the interface. 2.3 Elastic fields in the ambient medium + ∪ − We assume that a plane pressure wave in + is incident, at a certain angle, at the structured interface. Both the reflected pressure and shear plane waves are generated in + , while the transmitted pressure and shear waves occur in − . Conventionally, we represent the displacement field u(x, z, t)e(1) + w(x, z, t)e(3) via the pressure and shear wave potentials ϕ(x, z, t) and ψ(x, z, t)e(2) , so that u=
∂ϕ ∂ψ − , ∂x ∂z
w=
∂ϕ ∂ψ + . ∂z ∂x
(6)
We shall use the notations {ϕ (I) , ψ (I) }, {ϕ (R) , ψ (R) }, {ϕ (T) , ψ (T) } for the potentials corresponding to the incident, reflected and transmitted waves, respectively. Note that, since the incident wave is of pressure type, ψ (I) ≡ 0. Let c be an apparent velocity of the incident wave along the horizontal interface, and let α and β be the wave speeds of the pressure and shear waves, respectively, in ± , so that ϕ −
1 ∂2 ϕ = 0, α 2 ∂t 2
where
α=
ψ −
λ + 2μ , ρ
1 ∂2 ψ = 0, β 2 ∂t 2
β=
μ . ρ
(7)
(8)
If χP ∈ [0, π/2] denotes the angle between the direction of the incident wave and the horizontal interface, then c2 a := tan χP = − 1. (9) α2 For the pressure wave, the angle of reflection is equal to the angle of incidence, while for the reflected shear wave, the angle χS ∈ [0, π/2] between the direction of propagation and the horizontal interface is defined by c2 b := tan χS = − 1. (10) β2 Note that χS > χP ; the two angles are shown in Fig. 1. The representations for the potentials in + are given by ϕ+ = ϕ (I) + ϕ (R) = AI exp[ik(ct − x + az)] + AR exp[ik(ct − x − az)], ψ+ = ψ (R) = BR exp[ik(ct − x − bz)],
(11)
while in − ϕ− = ϕ (T) = AT exp[ik(ct − x + az)], ψ− = ψ (T) = BT exp[ik(ct − x + bz)].
(12)
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In (11), (12) k=
2π 2π cos χP = cos χS , P S
(13)
where P and S are the wave lengths of the pressure and shear waves, respectively, in the ambient elastic medium + ∪ − . We shall also use the notation ω = kc for the radian frequency. Assuming that the amplitude AI is given, we will evaluate AR , BR , AT and BT , and show that the shear wave may become dominant in the region of transmission − , so that the structured interface will act as a shear polariser.
3 Solution of the transmission problem Now we consider two configurations of the structured interface. The first case corresponds to a pair of elastic bars connected by the transverse massless links; and the second one includes N bars (N > 2), with the interior bars being the source of interfacial resonances. In the latter case, the analytical results will be discussed in detail for an interface consisting of three bars, whereas the illustrative numerical computations (described in Sect. 5) will correspond to N = 3 and N = 6. The energy balance relations will be addressed, with a particular emphasis on the energy carried by the transmitted shear wave.
3.1 Transmission properties of a two-bar system Assume that the interface region D includes two bars, l1 and l2 , which are attached to the upper and lower half-planes + and − , respectively, and are connected to each other via transverse elastic massless links. The longitudinal displacements u 1 (x, t), u 2 (x, t) of the bars are sought in the form: u j (x, t) = U j (x)eiωt = U j ei(ωt−kx) ,
j = 1, 2.
(14)
The shear stresses τ± at the interface boundaries are represented in a similar form: τ± (x, t) = T ± (x)eiωt = T± ei(ωt−kx) .
(15)
According to the equations of motion (4) and (5), we have E U 1 (x) + (ρω2 − γ )U 1 (x) + γ U 2 (x) + T + (x) = 0, E U 2 (x) + (ρω2 − γ )U 2 (x) + γ U 1 (x) − T − (x) = 0.
(16)
As stated above (see Eq. 2), the interface is constrained so that the transverse displacements are equal to zero. Owing to the continuity of the longitudinal displacements at the interface boundaries, we have U1 = −ik (AI + AR − BR b)
for z = 0,
(17)
and U2 = −ik AT e−ika D + BT be−ikbD
for z = −D.
(18)
The condition of zero transverse displacements within the interface implies that a(AI − AR ) − BR = 0
for z = 0,
(19)
and AT ae−ika D − BT e−ikbD = 0
for z = −D.
(20)
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The shear stresses τ± are defined by ∂u ∂u ∂w = μ + ∂z ∂ x z=− 1 (D∓D) ∂z z=− 1 (D∓D) 2 2 2 ∂ ϕ ∂ 2 ψ =μ . + ∂ x∂z ∂z 2 z=− 1 (D∓D)
τ± = μ
(21)
2
In this case, we have used the condition that w = 0 for z = − 21 (D ∓ D), and thus ∂w =0 ∂x
1 for z = − (D ∓ D). 2
(22)
Using Eqs. 21 and 11, 12 together with the representation (15), we deduce that
T+ = μk 2 (AI − AR )a + BR b2 for z = 0,
(23)
T− = μk 2 AT ae−ika D + BT b2 e−ikbD for z = −D.
(24)
and
A direct substitution of Eqs. 17, 18 and 23, 24 into 16 yields
k 2 E − ρω2 + γ + iμka AR + −k 2 Eρω2 − γ − iμkb bBR −γ e−ika D AT − γ be−ikbD BT = − k 2 E − ρω2 + γ − iμka AI
(25)
γ AR + γ bBR + e−ika D k 2 E − ρω2 + γ − iμka AT −e−ikbD k 2 E − ρω2 + γ − iμkb BT = γ AI .
(26)
and
Equations 25, 26, 19 and 20 lead to the following system of linear algebraic equations for the coefficients AR , BR , AT and BT : ⎞ ⎞ ⎛ 2 AR k E − ρω2 + γ − ikμa ⎟ ⎜B ⎟ ⎜ −γ R ⎝ R ⎠ = −AI ⎝ ⎠, AT −a BT 0 ⎛
(27)
where ⎛
+ ikμa ⎜ −γ R=⎜ ⎝ a 0
( + ikμb)b γb 1 0
−γ e−ika D −ika D ( − ikμa) e 0 ae−ika D
⎞ −γ be−ikbD be−ikbD ( − ikμb) ⎟ ⎟, ⎠ 0 −ikbD −e
(28)
with = k 2 E − ρω2 + γ . In particular, an explicit solution of the above system gives the following relation: AT = a eik(b−a)D , BT
(29)
which shows that the ratio of the amplitudes of the shear and pressure waves in the region of transmission depends only on the angle of incidence.
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3.1.1 Symmetric and skew-symmetric modes of a two-bar system Now we discuss vibration modes supported by an interface, which consists of two elastic bars; such an interface does not possess interior resonance modes. We shall introduce the following notations: [[U ]] = U 2 − U 1 for the displacement jump, and U = 1 ( U 1 + U 2 ) for the average longitudinal displacement. The differential equations for [[U ]] and U then take 2 the form: [[U ]] + ξ 2 [[U ]] =
T− + T+ −ikx , e E
(30)
and U + η2 U =
T− − T+ −ikx , e E
(31)
where ξ 2 = (ρω2 − 2γ )/E and η2 = ρω2 /E. First, we note that k 2 > ξ 2 for ω2 ≤ 2γ /ρ, and the symmetric propagating modes have the form: U = C1 eiηx + C2 e−iηx +
T− − T+ −ikx for k = η. e 2E(η2 − k 2 )
(32)
In this case, owing to the continuity of the longitudinal displacements, we obtain that C1 = C2 = 0, and the coefficients AR , BR , AT , and BT are determined by the system (27). In contrast, when k = η, we require T− = T+ , to avoid a linear growth of the average displacement amplitude at infinity. In the latter case, the first two equations of the system (27) are equivalent to
T+ + γ (U2 − U1 ) = 0, T+ − T− = 0,
(33)
which corresponds to an ‘adhesive’ shear interface without inertia. The last two equations of the system (27) remain unchanged and represent the constraint of zero transverse displacement. Physically, the case of η = k corresponds to the situation when the bars within the interface √ are sufficiently stiff, and the apparent wave speed c along the interface coincides with the natural speed E/ρ of the wave propagating along the elastic bars. For sufficiently large ω (such that ω2 > 2γ /ρ), we can also choose k so that ξ 2 = k2,
(34)
which yields k2 =
2γ , −E
ρc2
(35)
√ and this is realisable if c > E/ρ, that is, when the apparent wave speed c along the interface exceeds the √ wave speed E/ρ within the elastic bars. In this case, we deal with a mode, for which the shear tractions coincide on the upper and lower sides of the interface; the equations of motion of the elastic bars then lead to
T+ + γ (U1 + U2 ) = 0, T+ + T− = 0.
(36)
For the symmetric case, U1 = U2 , and the bars within the interface move simultaneously in the same direction.
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3.2 A general N -bar interface of the shear-type Now we consider transmission and reflection by a structured interface which possesses interior resonance modes. This corresponds to the situation when the number N of bars comprising the interface region exceeds 2. Assuming that the motion is time-harmonic with the radian frequency ω, we use the same notations U j and T± as in (14), (15) for the amplitudes of the displacements of the bars l j , j = 1, 2, . . . , N , and for the shear stresses above and below the interface. Then the equations of motion of elastic bars within the shear-type interface take the form: RU = T,
(37)
U = (U1 , U2 , . . . , U N )T , T = (−T+ , 0, . . . , 0, T− )T ,
(38)
⎞ 0 0 ⎟ ⎟ ... ⎟ , ... ⎟ ⎟ ⎠ γ −
(39)
where
and ⎛
− ⎜ γ ⎜ ⎜ ... R=⎜ ⎜ ... ⎝ 0 0
γ − − γ ... ... ... ...
0 γ ... ... 0 0
0 0 ... ... γ 0
... ... ... ... − − γ γ
with = k 2 E + γ − ρω2 . The relation det R(ω, k) = 0
(40)
gives the dispersion equation for the elastic waves propagating horizontally along the structured interface. The relation (40) also characterises the resonance states of the system subjected to tangential tractions and zero transverse displacements on the upper and lower boundaries of the structured interface. If ω does not satisfy equation (40) for a given wave number k, then the system (37) has a unique solution U = R−1 T, and hence, we can write U1 and U N as linear functions of T+ and T− : − U1 = D1+ T+ + D1− T− , U N = D + N T+ + D N T− ,
(41)
where D1± , D ± N are constant coefficients depending on the elastic and geometrical properties of the structured interface. Eqs. 41, complemented with the conditions of zero transverse displacements (19) and (20), give the so-called effective transmission conditions which characterise the discontinuities in tangential displacements and shear tractions across the interface. A direct substitution of (17), (18), (23), (24) into (41), together with (19), (20), gives a system of linear algebraic equations for AR , BR , AT , BT . Note that the conditions (19) and (20) of zero transverse displacements at the interface imply AR = AI − a −1 BR ,
AT = a −1 BT e−ik(b−a)D ,
(42)
which, upon substitution into (41), gives a system of two algebraic equations with respect to BR and BT . In the case of resonance, that is when ω and k satisfy (40), additional compatibility constraints are imposed on T+ , T− ; these will be discussed in the next section. For illustrative purposes, in the next section, we apply the algorithm described above to a system involving three elastic bars within the structured interface.
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3.2.1 Transmission and resonance modes for a three-bar system For a non-resonant regime and N = 3, the system (37) has a solution ⎛ ⎞ ⎛ ⎞ −T+ U1 ⎝ U2 ⎠ = R−1 ⎝ 0 ⎠ , T− U3 where
⎛ R−1 =
1 ⎜ ⎝ (γ − )(2γ + )
γ−
γ2
γ
+
γ2
γ γ
(43)
⎞
γ2
⎟ ⎠.
γ
γ−
γ2
(44)
+
In particular, the effective transmission relations for the shear displacements and shear stresses across the structured interface have the form: U1 =
γ2 (T+
+ T− ) − T+ (γ + ) , (γ − )(2γ + )
T− (γ + ) − γ (T+ + T− ) U3 = . (γ − )(2γ + ) 2
Solving the above system of algebraic equations with respect to BR , BT we deduce that
2 2 2 BR = − 2a ( + γ − γ ) − γ (1 + ab) AI , BT =
2ia 2 (1+b2 ) kγ 2 μeibDk AI ,
(45)
(46)
where = (1 + ab) + ikμa(1 + b2 ) and = (γ + ) − 2γ 2 (1 + ab) . The relations (42) then give the coefficients AR , AT . The full set {AR , BR , AT , BT } determines the reflected and transmitted waves at the shear-type interface. For N = 3, the system (37) can be written as follows: ⎞ ⎛ ⎞ ⎛ T+ + T− [[U ]] ⎜ ⎟ (47) B ⎝ [[V ]] ⎠ = ⎝ 21 (T− − T+ ) ⎠ , 1 U
3 (T− − T+ ) where [[U ]] = U3 − U1 represents the tangential displacement jump across the interface, [[V ]] = 21 (U3 + U1 ) − U2 is the average jump representing the difference between the average tangential displacement on the boundary and the tangential displacement of the interior bar, U = 13 (U1 + U2 + U3 ) is the average tangential displacement over the whole structured interface, and B = diag −Ek 2 + ρω2 − γ , −Ek 2 + ρω2 − 3γ , −Ek 2 + ρω2 . (48) The relation between ([[U ]], [[V ]]), U )T and (U1 , U2 , U3 )T has the form: ⎞ ⎛ ⎛ ⎞ [[U ]] U1 ⎝ [[V ]] ⎠ = Q ⎝ U2 ⎠ , U
U3 where
⎛ Q=⎝
−1 1 2 1 3
0 −1 1 3
1 1 2 1 3
(49)
⎞ ⎠,
(50)
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rad/s 3.5 3 2.5
1
2
2
3
1.5 1 0.5 0.25
0.5
0.75
1
1.25
1.5
k
m
-1
Fig. 3 The dispersion curves for waves propagating horizontally along a three-bar structured interface. Results are given for bars with linear mass density ρ = 1000 Kg/m and the longitudinal stiffness E = 106 kN, the horizontal stiffness of the vertical links is γ = 0.5 GPa. The curves correspond to the three branches (54); the cut-off frequency ω∗ = 0.707 × 103 rad/s for curves (1) and (2) is given by (55)
and hence B = QRQ−1 ,
(51)
with B the same as in (48) and R the same as in (39) for N = 3. The rows of the matrix Q are the eigenvectors of R corresponding to the eigenvalues, which coincide with the diagonal entries of the matrix B. Clearly, we observe that det B = det R,
(52)
and thus the dispersion equation (40) has the form (−Ek 2 + ρω2 − γ )(−Ek 2 + ρω2 − 3γ )(−Ek 2 + ρω2 ) = 0.
(53)
The dispersion diagram corresponding to waves which propagate horizontally along the structured interface, has three branches shown in Fig. 3: (1)
Ek 2 + 3γ − ρω2 = 0,
(2)
Ek 2 + γ − ρω2 = 0,
(3)
Ek − ρω = 0. 2
(54)
2
The lowest branch (3) gives a linear relation between k and ω, which corresponds to a wave of constant group velocity propagating along an elastic bar. The group velocities for the other two branches (1) and (2) are frequency dependent; the corresponding waves represent displacement discontinuities within the interface. For the dispersive waves (branches (1) and (2)), there is a cut-off frequency: γ ∗ ω = (55) ρ below which no dispersive wave can propagate along the interface. As k → ∞, the group velocity tends to √ E/ρ for all three branches on the dispersion diagram of Fig. 3. Consider now the resonant cases, for which the frequency ω of the incident wave coincides with one of the solutions of the dispersion Eq. 53. For such modes, additional compatibility constraints are imposed on the elastic system. (a) We first assume that Ek 2 − ρω2 = 0. In this case, the compatibility of the system (47) requires T− = T+ = T , which yields [[V ]] = 0, [[U ]] = −
2T . γ
(56)
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(b) For Ek 2 + 3γ − ρω2 = 0, the compatibility of the system (47) requires T− = T+ = T, and thus T . γ
U = 0, [[U ]] =
(57)
(c) Finally, for the case when Ek 2 + γ − ρω2 = 0, the compatibility constraint becomes −T− = T+ = T giving U = −
2T , 3γ
[[V ]] =
T 2γ
and
T U2 = − . γ
(58)
Although in the above cases (a), (b), (c) the matrix R is degenerate, the overall transmission problem, with the outlined compatibility constraints, gives the system of equations for BR , BT which has a unique solution. The following expressions for BR , BT hold in the cases (a), (b), (c) described above: aγ AI , BT = BR eibDk , (a) BR = γ (1 + ab) + ia(1 + b2 )kμ 2aγ (59) AI , BT = BR eibDk , (b) BR = 2γ (1 + ab) − ia(1 + b2 )kμ 2aγ AI , BT = −BR eibDk . (c) BR = 2γ (1 + ab) − ia(1 + b2 )kμ In the above cases |BR | = |BT |, which implies that the energy of the transmitted shear wave is equal to the energy of the reflected shear wave. 4 The energy balance relation for the reflected and transmitted waves ˆ ± ⊂ ± , which are adjacent to the boundaries l ± of the Let us consider sufficiently large rectangular sets ± ± ± ± structured interface. The notations , l , CR , CL refer to the horizontal and vertical parts of the boundaries ˆ ± , as shown in Fig. 4. of We use the notations U ± (x, z) for the displacement amplitude functions in ± , so that u± (x, z, t) = U ± (x, z)eiωt ,
with
u± = (u ± , 0, w ± )T .
(60)
We then apply the Betti formula to the functions U + and U + (the complex conjugate of U + ), and U − and U − ˆ + and ˆ − , respectively. This yields (the complex conjugate of U − ), in the regions U ± · σ (n) (U ± ) − U ± · σ (n) (U ± ) ds = 0, (61) CL± ∪CR± ∪ ± ∪l ±
SR
PI z
+
+
CR
CL O
PR
l
x
D
-
CL
l
-
CR
ST
PT
ˆ ± ⊂ ± with the boundaries l ± ∪ C ± ∪ ± ∪ C ± Fig. 4 The rectangular sets R L
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ˆ ±. where σ (n) (U ± ) represents the vector of tractions on the boundaries of Evidently, Eq. 61 can be rewritten as ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ (n) ± ± U · σ (U )ds = 0. ⎪ ⎪ ⎪ ⎪ ⎩C ± ∪C ± ∪ ± ∪l ± ⎭ L
(62)
R
Taking into account the representations (11) and (12), we deduce that the integrand in (62) is independent of x. Hence, the integrals over CL± and CR± cancel, and thus we obtain ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ (n) + (n) − + − (63) U · σ (U )ds + U · σ (U )ds − U1 T+ ds + U N T− ds = 0. ⎪ ⎪ ⎭ ⎩+ −
l+
l−
It is appropriate to note that −
U1 T+ ds +
l+
L U N T− ds =
l−
L (−U1 T+ + U N T− )d x =
0
T
U RUd x,
(64)
0
and the latter integral is real. In (64), R is the real symmetric matrix, the same as in (37). Hence, −U1 T+ + U N T− = 0,
(65)
and Eq. 63 reduces to
⎧ ⎨ ⎩
U + · σ (n) (U + )ds +
+
U − · σ (n) (U − )ds
−
⎫ ⎬ ⎭
= 0.
(66)
A direct substitution of the plane wave representations (11) and (12) for the pressure and shear wave potentials into (66) leads to the following energy balance relation EI = ER + ET ,
(67)
where E I = a|AI |2 ,
E R = a|AR |2 + b|BR |2 ,
E T = a|AT |2 + b|BT |2 ,
(68)
E I , E R and E T represent the vertical energy fluxes of the incident, reflected and transmitted fields, respectively. As expected, the integral term corresponding to the contribution from the interface is zero, as stated in (66). The effect of the interface on the distribution of energy will be seen through the evaluation of the coefficients AR , BR and AT , BT characterising the reflected and transmitted fields. 5 Transmission and polarisation of waves by the structured interface In this article, we discuss computations of transmitted and reflected energies for different values of the angle of incidence of the plane pressure wave. The algorithm described in Sect. 3 (Eqs. 42, 46, 59) and in Sect. 4 (Eq. 68) is used here to compute the energy of the transmitted and reflected waves in a composite elastic medium containing a three-bar structured interface. In the numerical computations, we have taken the linear mass density ρ = 1000 kg/m and the longitudinal stiffness E = 5 × 106 kN for the bars, the shear stiffness γ = 0.05 GPa for the vertical elastic links and the mass density ρamb = 1000 kg/m3 , Poisson ratio νamb = 0.3 and Young’s modulus E amb = 1 GPa for the ambient elastic medium + ∪ − ; the pressure and shear wave speeds in the ambient medium are respectively α = 1160.24 m/s and β = 620.174 m/s.
Shear polarisation of elastic waves
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c [10 m/s] 3
7
ER
6
ER
5
6
c
4
4
6
2
3 4
2.32 2
0.2 0.4
2
k
0.1
0.2
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0.4
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(b)
-1
]
c [10 m/s] 3
7
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6
ET
5
0.2
c
4
6
0.1
3
0 4 0.2 0.4
2
k (c)
2.32 2
0.1
0.2
0.3
0.4
0.5
0.6
k [m
-1
(d)
]
Fig. 5 The reflected and transmitted energies E R and E T for a three-bar interface as functions of the wave number k and the apparent velocity c: parts a and c show the surface plots and parts b and d the corresponding contour plots. The results of numerical calculations are given for bars of the linear mass density ρ = 1000 kg/m and the longitudinal stiffness E = 5 × 106 kN; the ambient medium is of the mass density ρamb = 1000 kg/m3 , Poisson ratio νamb = 0.3 and Young’s modulus E amb = 1 GPa; the horizontal stiffness of the vertical links is γ = 0.05 GPa/m2
Figure 5 presents the graphs of the reflected and transmitted energies as functions of c and k. We note that c ≥ α, and c → +∞ for the case of normal incidence (that is, as χP → π/2), whereas c = α for the incident pressure wave propagating horizontally, parallel to the interface. The parts (a) and (c) of Fig. 5 give the surface plots of E R (k, c/α) and E T (k, c/α), whereas the parts (b) and (d) show the corresponding contour plots. In particular, we observe that there is a rapid increase in the transmission energy when k is close to the value √ 2γ /(ρc2 − E) for c > E/ρ (that is, when the apparent speed of the incident wave along the interface exceeds the natural wave speed of an elastic bar). It turns out that this special value corresponds to a defect resonance mode, for which the upper and lower bars do not move, whereas the middle bar vibrates while being connected to the upper and lower bars via elastic links of stiffness γ . The corresponding equation of motion for such a defect mode gives (Ek 2 + 2γ − ρω2 )U2 = 0,
(69)
which is fully consistent with the observed peak in transmission. √ The curve shown in Fig. 5d has two asymptotes: c → E/ρ = 2.236 as k → +∞, and c → +∞ as k → 0. The latter case corresponds to the normal incidence, and no transmission occurs across the shear-type structured interface. We expand on the analysis of the transmitted and reflected waves by considering the energies of the reflected pressure and shear waves as well as the energies of the transmitted pressure and shear waves. Fig. 6 shows the
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ER /EI 1 0.8
ER,T/EI 1
0.6
ERP/EI
0.4
ERS/EI
0.2
0.8
-1
0.6
ER/EI
0.4
ET/EI
0.2
0.4
0.6
0.8
1
0.8
1
k [m
]
k [m
]
(b)
ET/EI 0.12
0.2
0.1
0.2
0.4
0.6
0.8
1
k [m
-1
]
0.08
ETS/EI
0.06
(a)
ETP/EI
0.04 0.02
-1
0.2
0.4
0.6
k*
(c) Fig. 6 The reflected and transmitted energies E R and E T for the three-bar interface of Fig. 5 as functions of the wave number k. Results are given for the angle of incidence χP = π/3, corresponding to c = 2α = 2320.48 m/s indicated by the horizontal grey lines in Figs. 5b and d. a The normalised reflected energy E R /E I and the normalised transmitted energy E T /E I , b the normalised pressure reflected energy E R P /E I and the normalised shear reflected energy E RS /E I , and c the normalised pressure transmitted energy E TP /E I and the normalised shear transmitted energy E TS /E I c
3
[10 m/s] 7
ET
6
ET
5
0.2 6
0.1 0 4
c
4 3
0.05 2
0.1 2
0.15
k
0.2
0
0.05
0.1
0.15
0.2
0.25
0.3
k [m
-1
(a)
]
(b)
Fig. 7 Transmitted energy E T for a six-bar interface as a function of the wave number k and the apparent velocity c: a The surface plot and b the corresponding contour plot. The material and geometric parameters are the same as in Fig. 5
graphs of the total reflected and transmitted energies (part a), the energy of the reflected pressure and shear waves (part b), and the energy of the transmitted pressure and shear waves (part c). First, we note that E TS /E TP = ab,
(70)
and hence the ratio of the energies of the transmitted shear (E TS ) and pressure (E TP ) waves is determined by the angle of incidence and the material parameters of the ambient medium. For the computation reported here, ab = 6.245, so that χP = π/3. In this case, the transmitted shear wave dominates in the region of transmission − below the interface, and thus the structured interface acts as a shear polariser in transmission even if the incident wave is of pressure type. The transmission of the energy of the shear wave is enhanced when k is close to the value k ∗ = 2γ /(ρc2 − E) = 0.510 m−1 corresponding to the resonance defect mode (see Fig. 6c). As Fig. 6b shows, the reflected pressure wave dominates in the region + above the interface.
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6 Concluding remarks With the increase of the number of bars within the interface, the resonance defect modes persist, and they enhance transmission across the structured interface. We illustrate this by an example involving a six-bar shear-type structured interface. In this case, the resonance defect modes are associated with an auxiliary problem corresponding to the case when the upper and lower bars of the interface remain fixed, while the interior four bars are allowed to vibrate. In Fig. 7, we give the surface and contour plots of the transmitted energy. The material parameters, used in the computations, are the same as in Sect. 5. The enhancement in transmission is shown in Fig. 7a around the resonance defect modes outlined in Sect. 3.2.1. Fig. 7b shows that the regions of (k, c) corresponding to such an enhanced transmission are located around four curves, similar to the case discussed in Sect. 5 and illustrated in Fig. 5. The increase in the number of the interior bars is accompanied by the increase in size of the region of enhanced transmission (see Fig. 7b). In other words, for a fixed k, we have a larger set of values of c, where enhanced transmission is observed. This also means that the enhanced transmission becomes possible for a wider range of values of the angle of incidence. Acknowledgements Part of this study was performed while A.B. Movchan was a Visiting Professor at the Department of Structural Engineering of the University of Cagliari (under the Visiting Professor Program 2009 financed by the Regione Autonoma Sardegna).
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