Thus far, a detailed analysis of acoustic plasmonic cloaking has been pre- sented which has focused on the use of a single fluid layer to achieve scattering cancellation. In the present section, a basic description of a single elastic cloaking layer will be described and compared with the results previously obtained using a fluid layer. Although only allowing for a non-zero shear modulus, representing an addition of a single parameter, the determinant Un (which is set equal to zero to determine the cloaking condition for each mode) becomes significantly more compli- cated. This is due to the fact that accounting for shear in the cloaking layer allows for the propagation of shear waves in the layer, which adds two more scattering coefficients and two more boundary conditions at the layer interfaces. As a result, Unbecomes the determinant of a 7×7 matrix for an isotropic elastic core (6×6 for a fluid core), the full expressions for which can be found in Appendix A. Furthermore, coupling between the shear and compressional modes leads to added complexity in many of the elements of Un, making it difficult to evaluate in closed form. Although not impossible, exact solutions do not offer simple, closed form expressions, and fail to offer insight into the effects of elasticity within the cloaking layer.
Given the complexity of accounting for the cloaking layer elasticity using exact relations, the analytic results developed in this section will be limited to quasi-static results. In addition, since the cancellation of the two leading order terms is independent of the shear properties of the core material, results can be determined using only the core density and bulk modulus, while still being equally valid for either a fluid or isotropic elastic core. To obtain explicit expressions for the quasi-static cloaking layer properties, the same process will be followed as outlined in Section 4.1.1. Applying the quasi-static approximation for the spherical Bessel functions given by Equations (4.30)–(4.33) to each element of Un, the determinant
can be evaluated using block matrices according to Equation (4.22) and set equal to zero. For the limiting case under consideration here, an equivalent expression can be obtained using static analysis for the effective bulk modulus of a coated sphere, with the cloaking achieved when the effective bulk modulus of the core plus coating layer is equal to the bulk modulus of the surrounding fluid [59].
Although the terms in Ungiven in Appendix A are defined using the cloaking layer shear modulus µc, in the current analysis the Poisson’s ratio of the cloaking layer νc will be used instead to characterize the shear effects. Based on the nor- malized bulk and shear modulus of the cloaking layer, these parameters are related by
µc= 3κc(1−2νc) 2(1+νc)
. (4.83)
In this expression, it is noted that although µc and κc have been normalized by κ0 to yield non-dimensional quantities, νc is already dimensionless. Furthermore, the range of µc is 0 ≤ µc < ∞, and depends on the value of κc, which is deter- mined (in the quasi-static limit) by κ and the thickness of the cloaking layer. The physical bounds on Poisson’s ratio for an isotropic elastic material, however, are −1 ≤ νc≤ 1
2, with 0 < νc≤ 1
2 for most naturally-occurring homogeneous materials [75]. The upper limit νc=12 corresponds to an incompressible material and zero shear modulus (as seen from Equation (4.83)), which is equivalent to the solution for an incompressible fluid.
Evaluating the resulting expression for Un= 0 at n = 0 yields a quadratic expression for the normalized bulk modulus of the elastic cloaking layer,
α(E)QS κ2c− β (E)
QS κc− γQS(E)= 0, (4.84)
where φ = ab3 and
β(E)QS =2(1−2νc) + (1+νc)φ − (1+νc) + 2(1−2νc)φκ, (4.86)
γ(E)QS = (1−φ)(1+νc)κ. (4.87)
Use of the quadratic formula yields the roots of Equation (4.84)
κc= βQS(E)+ q βQS(E) 2 + 4 α(E)QSγ (E) QS 2 α(E)QS . (4.88)
From this expression, the bulk modulus of the cloaking layer is given as a function of the core bulk modulus, the layer thickness, and the Poisson’s ratio of the cloaking layer. In Equations (4.84)–(4.87), it is observed that α(E)QS> 0 and γ
(E)
QS > 0 within the
bounds of 0 < φ < 1 and −1 ≤ νc<12, regardless of κ. Conversely, the sign of β
(E) QS
depends on the relative difference of the two bracketed terms in Equation (4.86), which depends on the value of κ.
Examining Equation (4.88), the coefficient β(E)QS only appears under the square
root as (β(E) QS)
2≥ 0, which is added to the product of α(E) QS and γ
(E)
QS, ensuring that the
solution given by Equation (4.88) is real. Furthermore, since q βQS(E) 2 + 4 α(E)QSγ (E) QS ≥ β (E) QS , (4.89)
where kβ(E)QSk denotes the magnitude of β (E)
QS. Thus, the choice of the “+” sign in
Equation (4.88) will ensure a non-negative value for κc. Although these results are derived from the elastodynamic relationship by taking the quasi-static limit, Equations (4.84)–(4.88) are identical to those obtained using static analysis and the concept of a neutral inclusion [59].
The non-zero shear effects characterized by νc on κc, in addition to the normalized core bulk modulus κ which dominated the quasi-static behavior for the fluid layer case, are illustrated in Figure 4.11 for ab = 1.10. In this figure, the value of κcis represented by the color scale for a given νc and κ. It is clear that over the range 0 ≤ νc<12, κcis real and positive for the entire range of κc considered.
νc
κ
/
κ 0
Figure 4.11: Parametric plot of elastic effects on the cloaking layer parameter κc in the quasi-static limit, as a function of the cloaking layer Poisson’s ratio νc and the core material bulk modulus κ, obtained from Equation (4.88). The different colors represent the (dimensionless) values of κc on a logarithmic scale.
Given the expression for κc in Equation (4.88), it is clear that the resulting behavior of the solution depends on the value and sign of β(E)
QS. An important
transitional point in the behavior of κcoccurs where β
(E)
QS = 0. Using Equation (4.86),
this yields a value of κ under these conditions denoted by κcrit, which is given by
κcrit=
2(1 − 2νc) + (1 + νc)φ (1 + νc) + 2(1 − 2νc)φ
. (4.90)
This expression is denoted in Figure 4.11 by the black line. In the figure, it is observed that this line denotes two distinct regions in the vicinity of νc near 12. Evaluating Equation (4.90) at νc = 12 yields κcrit = φ, which corresponds to the
For the case when βQS(E)6= 0, Equation (4.88) can be expressed as κc= 1 2α(E)QS β(E)QS + kβ (E) QSkp1 + 4 εQS , (4.91) εQS= α(E)QSγ (E) QS βQS(E) 2 = 2(1−2νc)(1+νc)(1−φ)2κ 2(1−2νc)+(1+νc)φ − (1+νc) + 2(1−2νc)φκ 2 . (4.92) From these expressions, it is observed that away from βQS(E)= 0 the value of εQS is
much less than unity. To highlight this, recall that κ = κcrit when β (E)
QS = 0, which
based on Equation (4.90) varies over the range 0 ≤ νc≤12 between φ ≤ κcrit≤
2 + φ
1+2φ. (4.93)
Since 0 < φ < 1, this leads to a relatively narrow range for κcrit, as illustrated in
Figure 4.11. Away from κ = κcrit, the value of εQS becomes
εQS= 2(1−2νc)(1+νc)(1−φ)2κ 2(1−2νc)+(1+νc)φ 2 , κ 1, 2(1−2νc)(1+νc)(1−φ)2 (1+νc) + 2(1−2νc)φ 2 κ, κ 1, (4.94)
which leads to εQS 1 in both cases.
Assuming that εQS 1, the expression for κc given by Equation (4.91) sim- plifies to κc≈ β(E) QS α(E)QS , κ 1, −γ(E) QS βQS(E) , κ 1. (4.95)
From Equation (4.86), it is noted that the case of κ 1 corresponds to βQS(E)> 0,
whereas κ 1 corresponds to β(E)
QS < 0. Thus, the resulting value for κc is positive
for both cases, as observed in Figure 4.11. However, a striking feature of Figure 4.11 is seen as νc→12, where values of κcbecome increasing large for κ < κcrit, but remain
To examine this difference between these two regions, consider the behavior of the solution given by Equation (4.95). Taking the limit νc→12 leads to α
(E)
QS→ 0,
which from Equation (4.95) gives κc→ ∞ when κ 1. When κ 1, however, κc is independent of α(E)QS. It is therefore unaffected by this limit and the solutions
remains finite, as observed in Figure 4.11.
In a similar manner to κc, an expression for the cloaking layer density can be obtained from the n = 1 case, which yields
ρc= 1−φρ
1 − φ. (4.96)
This relationship is the same as the one obtained using conservation of mass for the effective density of a coated sphere used in the static analysis of this configuration. Unlike the case of a fluid layer, there is no flow of the cloaking layer, and no inertial effects determining the required density. In addition, to achieve acoustic plasmonic cloaking with a single elastic layer with a positive density, from Equation (4.96) it is clear that this is only possible when ρ < (ab)3. For the fluid cloaking layer, there was no limitation on the density of the core material.
To examine the effectiveness of a single elastic layer as an acoustic plasmonic cloak, consider the limiting case of a pressure-release spherical core. This limit can be obtained by taking ρ → 0, and κ → 0, for which Equations (4.88) and (4.96) reduce to κc= 2(1−2νc) + (1+νc)φ 2(1−φ)(1−2νc) , (4.97) ρc= 1 1 − φ, (4.98)
respectively. Using these expressions for κc and ρc as a guide, the first 5 scattering coefficients are plotted in Figure 4.12 for an uncloaked pressure-release sphere (top panel) and for a pressure-release sphere coated by an single elastic cloaking layer
0 0.5 1 1.5 −60 −40 −20 0 |A n | (dB) ka (a) n = 0 n = 1 n = 2 n = 3 n = 4 0 0.5 1 1.5 −60 −40 −20 0 |A n | (dB) ka (b)
Figure 4.12: Scattering coefficients (in dB) for an uncloaked (top) and cloaked (bottom) pressure-release sphere. The cloak consists of a single elastic layer with νc= 0.3 and ab= 1.10, which cancels the first two scattering modes at kd,0a = 0.5.
withab= 1.10 and a design frequency of kd,0a = 0.5. Although cancellation of the first two modes is achieved at the desired frequency, there are significant contributions from higher order modes which mitigate the effectiveness of such a design, even at the relatively low frequency considered in this example. To improve the effective- ness, therefore, the cancellation of higher order modes must be addressed, which is addressed in the following chapters.