c Allerton Press, Inc., 2014. ISSN 0025-6544, Mechanics of Solids, 2014, Vol. 49, No. 4, pp. 422–434.  c R.V. Goldstein, V.A. Gorodtsov, D.S. Lisovenko, 2014, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2014, No. 4, pp. 74–89. Original Russian Text 

Rayleigh and Love Surface Waves in Isotropic Media with Negative Poisson’s Ratio R. V. Goldstein* , V. A. Gorodtsov** , and D. S. Lisovenko*** Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526 Russia Received December 26, 2013

Abstract—The behavior of Rayleigh surface waves and the first mode of the Love waves in isotropic media with positive and negative Poisson’s ratio is compared. It is shown that the Rayleigh wave velocity increases with decreasing Poisson’s ratio, and it increases especially rapidly for negative Poisson’s ratios less than −0.75. It is demonstrated that, for positive Poisson’s ratios, the vertical component of the Rayleigh wave displacements decays with depth after some initial increase, while for negative Poisson’s ratios, there is a monotone decrease. The Rayleigh waves are characterized by elliptic trajectories of the particle motion with the change of the rotation direction at critical depths and by the linear vertical polarization at these depths. It is found that the elliptic orbits are less elongated and the critical depths are greater for negative Poisson’s ratios. It is shown that the stress distribution in the Rayleigh waves varies nonmonotonically with the dimensionless depth as (positive or negative) Poisson’s ratio varies. The stresses increase strongly only as Poisson’s ratio tends to −1. It is shown that, in the case of an incompressible thin covering layer, the velocity of the first mode of the Love waves strongly increases for negative Poisson’s ratios of the half-space material. If the thickness of the incompressible layer is large, then the wave very weakly penetrates into the halfspace for any value of its Poisson’s ratio. For negative Poisson’s ratios, the Love wave in a layer and a half-space is mainly localized in the covering layer for any values of its thickness and weakly penetrates into the half-space. For the first mode of the Love waves, it was discovered that there is a strong increase in the maximum of one of the shear stresses on the interface between the covering layer and the half-space as Poisson’s ratios of both materials decrease. For the other shear stress, there is a stress jump on the interface and a more complicated dependence of the stress on Poisson’s ratio on both sides of the interface. DOI: 10.3103/S0025654414040074 Keywords: surface wave, Rayleigh wave, Love wave, Poisson’s ratio, dispersion equation, thin covering layer, thick layer, displacement, shear stresses, normal stresses.

1. INTRODUCTION The presentation of the classical theory of elasticity of isotropic media is usually restricted to the cases of positive Poisson’s ratios ν despite the fact that their thermodynamically admissible range also contains the region of negative values, −1 < ν < 1/2 [1, 2]. At the same time, it was discovered that Poisson’s ratio can be negative for some crystals and isotropic materials [3, 4]. In the recent paper [5], an analysis of dimensions was performed to estimate the influence of Poisson’s ratios (in the entire range of their variation) on the velocities (and only on the velocities) of elastic waves for several problems of the theory of isotropic elasticity, in particular, the problem of surface Rayleigh waves. The dimensionless distributions of displacements and stresses in the Rayleigh waves for positive and negative Poisson’s ratios were studied earlier in [6]. But the analysis in [6] of how Poisson’s ratio affects the displacement and stress distributions was not quite successful, because the characteristics used to obtain the dimensionless distributions vary themselves with Poisson’s ratio. In what follows, we analyze the Rayleigh waves for various Poisson’s ratios by passing to dimensionless distributions obtained by *

e-mail: [email protected] e-mail: [email protected] *** e-mail: [email protected] **

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dividing them by the chosen characteristics for a fixed value of Poisson’s ratio (for example, for ν = 0). A similar analysis is carried out for the dimensionless displacement and stress distributions of the first mode of Love waves in isotropic media. 2. RAYLEIGH WAVES In the half-space z < 0 occupied with an isotropic elastic medium, two types of waves propagating parallel to its surface are possible, namely, bulk shear waves “sliding” along the free surface without attenuation with depth and surface Rayleigh waves which are combinations of a shear (transverse) wave decaying with depth and a longitudinal wave, ux = (κt aeκt z + kbeκl z ) cos[k(x − ct)], uz = (kaeκt z + κt beκl z ) sin[k(x − ct)].

uy = 0,

(2.1)

Here a and b are amplitude factors, and the coefficients   c2 c2 κt = k 1 − 2 , κl = k 1 − 2 ct cl characterize the variations in the transverse and longitudinal wave components with depth. At large depths, the contribution of the transverse waves is the greater of, because their velocity ct is less than the longitudinal wave velocity cl . On the free surface of the elastic body (for z = 0), the stress components σxz , σyz , σzz must disappear, which requires restrictions on the displacements of the form ∂uz ∂ux ∂uz ∂ux + = 0, uy = 0, (c2l − 2c2t ) + c2t = 0. ∂z ∂x ∂x ∂z It follows that there should be a linear relation between the amplitude factors a and b (one of them remains arbitrary, which reflects the fact that the equations of the theory of elasticity for small amplitude waves are linear) and a dispersion restriction of the form (κ2t + k2 )a + 2kκl b = 0,

(2.2)

(κ2t + k2 )2 = 4k2 κt κl .

(2.3)

The last dispersion relation can be rewritten as the well-known bicubic Rayleigh equation     c2t c2t c 6 4 2 . ξ − 8ξ + 8ξ 3 − 2 2 − 16 1 − 2 = 0, ξ ≡ xt cl cl The solution of this equation has a unique real root with the restriction ξ < 1, which follows from the fact that κt and κl are real and determines the dependence of the Rayleigh wave dimensionless velocity ξ on the dimensionless parameter ct /cl . In turn, this parameter depends only on Poisson’s ratio and is independent of the Young modulus and the wave length (wave without dispersion), 1 c2t . =1− 2(1 − ν) c2l This allows us to express the dependence of ν on ξ as ν =1+

8(ξ 2 − 1) ξ 2 (ξ 4 − 8ξ 2 + 8)

and reduce it to the linear relation ξ ≈ 0.18ν + 0.87 with the accuracy of two significant digits. Since the shear wave velocity depends on Poisson’s ratio, it is more expedient, in the analysis of the dependence of the dimensionless velocity of the Rayleigh wave, to pass to the dimensionless velocity by dividing it by the shear velocity for a fixed value of Poisson’s ratio, for example, for ν = 0. Then we obtain the following simple dependence of the dimensionless velocity on Poisson’s ratio in its entire range −1 < ν < 1/2:   0.18ν + 0.87 E c √ , c0 ≡ ct ν=0 = . ≈ c0 2ρ 1+ν MECHANICS OF SOLIDS

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Fig. 1.

Fig. 2.

Figure 1 illustrates the monotone decreasing character of this dependence. The variations are small for both the positive values of ν and negative Poisson’s ratios in the range −0.5 < ν < 0. Only for smaller negative ν, the increase in the dimensionless velocity of waves becomes noticeable and increases rapidly as Poisson’s ratio approaches −1. Using (2.2) and (2.3), we can rewrite the expressions for the “horizontal” and vertical” components of the displacements (2.1) in the form ux = A(z) cos[k(x − ct)], uz = B(z) sin[k(x − ct)],     √ κt κl κl z k κt z κl z κt z e e , B(z) = ak e − . A(z) = aκt e − √ κt κl k

(2.4) (2.5)

Since the displacement components in the Rayleigh wave are shifted by π/2 in phase, it follows that the particles move in the elliptic trajectory u2z u2x + = 1. A2 (z) B 2 (z) On the free surface of the elastic medium (for z = 0), by setting a > 0, we obtain the amplitudes A0 ≡ A(z = 0) = −aκt

k2 − κ2t < 0, k2 + κ2t

B0 ≡ B(z = 0) = ak

c2 > 0, 2c2t

which corresponds to the particle rotation near the surface in the elliptic trajectory anticlockwise. Then the ellipse is elongated in the direction  perpendicular to the free surface (“vertical” direction), which follows from the inequality |B0 /A0 | = κl /κt > 1. At large depths, the ellipses are vertically elongated, because then B/A ≈ k/κt > 1, but the particles rotate clockwise owing to the inequalities A > 0 and B > 0. The elliptic trajectory of the particles on the surface significantly varies with variations in Poisson’s ratio. Figure 2 shows that the elliptic trajectory of particles on the free surface significantly MECHANICS OF SOLIDS

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Fig. 3.

decreases along the vertical axis with decreasing Poisson’s ratio and approaches a circle as ν tends to −1. In this and further figures, the dimensionless displacements are obtained by using the vertical displacement component for zero Poisson’s ratio,  B00 ≡ B(z = 0)ν=0 = 0.382ak. The character of the influence of the value and sign of Poisson’s ratio on the Rayleigh wave amplitude at various depths is well illustrated in Fig. 3 (1 — ν = −0.95, 2 — ν = 0, and 3 — ν = 0.5), where the dimensionless amplitudes of the vertical and horizontal displacement components B/B00 and A/B00 are given in dependence on the dimensionless distance from the free surface for various Poisson’s ratios. The dimensionless amplitudes are obtained by dividing the dimensional amplitudes by the amplitude B00 of the vertical components on the free surface for zero Poisson’s ratio, and the dimensionless depth is obtained by dividing by the wave length λ. Figure 3 a (1 — ν = −0.95, 2 — ν = 0, and 3 — ν = 0.5) shows that, for positive Poisson’s ratio ν = 1/2, the dimensionless vertical displacement amplitude B/B00 varies nonmonotonically with depth. The amplitude B first not only does not decrease but increases by more than 10% and, only for |z/λ| > 0.4, it becomes less than the amplitude on the free surface and decreases asymptotically exponentially at large depths. At the same time, for negative Poisson’s ratios, the amplitude B/B00 decreases at any depth, and the more negative Poisson’s ratio, the faster the decrease. Figure 3 b shows that the dimensionless horizontal displacement amplitude A/B00 depends on the depth nonmonotonically for any values of Poisson’s ratio and changes sign at small depths |z|/λ ≤ 0.557. The variation in the sign of the amplitude A for positive amplitude B (Fig. 3) reflects the change of the direction of particle rotation in elliptic trajectories at certain depths. The fact that A is zero at these depths means that the elliptic trajectories collapse at these depths (the Rayleigh wave is linearly polarized at these depths). Figure 4 illustrates the monotone growth of such depths |z∗ |/λ with decreasing Poisson’s ratio. As the depths continue to increase, the amplitude of the displacement horizontal component A becomes positive, attains its maximum at the depth zmA /λ, and then decreases exponentially rapidly (as eκt z as z → −∞). Here the maximum values decrease with decreasing Poisson’s ratio, and for ν = −1, |z|/λ > 0.557, the horizontal displacement turns out to be very small. This figure also illustrates the behavior of the point zmB /λ, which characterizes the maximum of the ratio B/B00 . It follows that the nonmonotone variation in this ratio with depth for positive Poisson’s ratios becomes monotone for negative Poisson’s ratios (cf. Fig. 3). A visual illustration of variations in the trajectories of particle motion in the Rayleigh wave with depth and Poisson’s ratio was earlier given in Fig. 6 in the paper [6]. By using Hooke’s law and by differentiating the displacements (2.4) and (2.5), we obtain the stress components in the Rayleigh wave σxx = Σxx (z) sin[k(x − ct)], MECHANICS OF SOLIDS

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Fig. 4.

Fig. 5.

The variation in their amplitudes with depth is described by the relations √ E E k κt κl f (z), Σzz (z) = a kκt f (z), Σxz (z) = a 1+ν 1+ν   E κt κl z (κ2l − κ2t ) e − kκt f (z) , f (z) ≡ eκt z − eκl z . Σxx (z) = a 1+ν κl On the free surface, the two stress components σxz and σzz are zero according to the boundary conditions, and only the normal component σxx is different from zero:  E κt 2 2 (κl − κt ) . Σxx (z = 0) = a 1+ν κl This component is used to divide the three stress components to make them dimensionless (just as in [6]). The arbitrary amplitude factor a is eliminated in this procedure. But the analysis of the dependence of the obtained dimensionless characteristics on Poisson’s ratio at various depths is not correct, because is also contains the dependence of the surface stress on this Poisson’s ratio. (Figure 5 shows that the surface stress increases rapidly for large negative values of Poisson’s ratio, ν < −0.5.) But if the surface stress is chosen for a fixed value of Poisson’s ratio, for example, for ν = 0,  Σ0 ≡ Σxx (z = 0)ν=0 = 0.3Ek2 a, as a characteristic used to obtain dimensionless variables, then such a dependence of the surface stress is eliminated, and the analysis of the stress variations on the Poisson’s ratio variations becomes complete. Figures 6–8 (1 — ν = −0.95, 2 — ν = −0.8, 3 — ν = −0.5, 4 — ν = 0, and 5 — ν = 0.5 in Figs. 6 and 7; 1 — ν = 0.5, 2 — ν = −0.95, and 3 — ν = −0.5 in Fig. 8) present the variation in the dimensionless shear and normal stresses Σxz /Σ0 , Σzz /Σ0 , and Σxx /Σ0 depending on the dimensionless depth z/λ for various values of Poisson’s ratio (0.5, 0, −0.5, −0.8, and −0.95.) It should be noted that the character of influence of the value and sign of Poisson’s ratio is nonmonotone. For example, the shear stress for ν = 0.5 exceeds the stress for ν = −0.5, and only for ν < −0.8, the stress begins to increase rapidly with decreasing Poisson’s ratio. The normal stresses also vary nonmonotonically with variations in Poisson’s ratio. They also increase rapidly for negative Poisson’s ratios which are sufficiently large in absolute value (as ν → −1). The depths of characteristic points of these distributions weakly vary with variations in both positive and negative Poisson’s ratio. For example, the depth |z/λ| of the maximum of the shear stress Σxz /Σ0 and the normal stress Σzz /Σ0 monotonically decreases from 0.275 to 0.209 as Poisson’s ratio ν decreases from 0.5 to −1. The depth |z/λ| of the minimum of the normal stress Σxx /Σ0 slightly increases from 0.498 to 0.510 as ν decreases from 0.5 to −0.05 and then drops to 0.505. The depth |z/λ| at which the normal stress Σxx /Σ0 is zero monotonically MECHANICS OF SOLIDS

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Fig. 7.

Fig. 8.

Fig. 9.

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min increases from 0.222 to 0.296 as ν decreases from 0.5 to −1. Figure 9 (1 — |Σmin xx |/Σ0 , 2 — Σzz /Σ0 , min and 3 — Σxz /Σ0 ) demonstrates that the extrema of these three stress components increase rapidly as negative Poisson’s ratio becomes lower than −0.5. At the same time, the extrema of Σxz and |Σxx | can slightly increase as positive Poisson’s ratio increases and in the case of its small negative values.

3. LOVE WAVES If the elastic medium in the half-space is covered by a layer of a different material, then the bulk shear waves “sliding” along the surface decay with the distance from the surface into the depth of the half-space. Such transverse waves localized in a layer and near the surface are called Love waves. We assume that a layer of an isotropic material 1 of thickness h (the layer from z = 0 to z = h) covers a half-space of a different isotropic material 2 (half-space z ≤ 0). On the free surface of the covering layer 1 (for z = h), the shear stress σyz must disappear; i.e., ∂uy1 /∂z = 0, and on the interface between this layer and the half-space material 2 (for z = 0), the displacements uy1 and uy2 and the shear stresses σyz1 and σyz2 must be the same; i.e., uy1 = uy2 and μ1 ∂uy1 /∂z = μ2 ∂uy2 /∂z. In such a composite, a joint transverse horizontal wave of the form uy1 = V1 (z) sin[k(x − ct)], V1 (z) = a cos[κ1 (z − h)], MECHANICS OF SOLIDS

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ux1 = ux2 = uz1 = uz2 = 0,

(3.1) (3.2)

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Fig. 10.

can propagate if the dispersion equation [1] μ2 κ2 , tan(hκ1 ) = μ1 κ1



κ1 = k

c2 − 1, c2t1

 κ2 = k

1−

c2 , c2t2

c2t =

μ ρ

is satisfied. This dispersion equation arises when the boundary conditions stating that there is no shear stress on the free surface of the covering layer and that the displacements and shear stresses are equal to each other on the interface between this layer and the elastic half-space are simultaneously satisfied. The dispersion equation admits countably many solutions for waves with dispersion (because of the separated length scale h) under the following restriction on the wave velocity: c2t2 > c2 > c2t1 . This equation shows that the Love waves are faster than the shear waves in the material of the covering layer and are slower than the shear waves in the material of the half-space. In what follows, we consider only the first mode of the Love waves, which is singled out by the condition hκ1 < π/2 of relative thickness of the covering layer. In the problem for the Love waves, in contrast to the Rayleigh waves, the wave velocity depends on the wave length λ = 2π/k and several material parameters ρ1 ,

ρ2 ,

μ1 =

E1 , 2(1 + ν1 )

μ2 =

E2 2(1 + ν2 )

rather than only on Poisson’s ratio. The restriction on the velocity of the Love waves leads to the inequality 1 + ν1 >

E1 ρ2 (1 + ν2 ) E2 ρ1

on the material parameters. It follows that the condition ν1 > ν2 must be satisfied in the special situation E1 /ρ1 ≥ E2 /ρ2 , which means that, for the Love waves in the half-space materials with positive Poisson’s ratio (which are not auxetics), the material in the covering layer cannot be an auxetic (a material with negative Poisson’s ratio). Let us analyze such a case of materials of the same density with equal elasticity moduli and distinct Poisson’s ratios: ρ1 = ρ2 = ρ,

E1 = E2 = E,

ν1 > ν2 .

We assume that the covering layer is incompressible; i.e., ν1 = 1/2. Then the half-space materials the variations in the can have Poisson’s ratio in the entire range −1 < ν2 < 1/2. In this situation,  dimensionless velocity c/c0 of the first mode of the Love waves (from now on, c0 = E1 /(2ρ1 )) with the variations in the ratio ν2 and the dimensionless thickness h/λ of the covering layer are described by the curves in Fig. 10 a (1 — ν1 = 0.5, ν2 = −0.95; 2 — ν1 = 0.5, ν2 = −0.5; 3 — ν1 = 0.5, ν2 = 0; MECHANICS OF SOLIDS

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Fig. 11.

and 4 — ν1 = 0.5, ν2 = 0.3 for E1 = E2 and ρ1 = ρ2 ). This figure shows that, for a thin covering layer h/λ < 0.1, the wave velocity is significantly greater than for negative Poisson’s ratios (ν2 = −0.5 and ν2 = −0.95), and it increases as they decrease. As for the relative distribution of displacements in the covering layer and in the half-space, Fig. 11 (the solid line corresponds to h/λ = 0.3, the dashed line corresponds to h/λ = 5 for E1 = E2 and ρ1 = ρ2 ; (a) —ν1 = 0.5 and ν2 = 0.3; (b) —ν1 = 0.5 and ν2 = 0; (c) —ν1 = 0.5 and ν2 = −0.5; and (d) —ν1 = 0.5 and ν2 = −0.95) shows that, in the situation under study for a large layer thickness (h/λ = 5), the wave weakly penetrated into the half-space z < 0 for any values of Poisson’s ratio ν2 , and this penetration decreases with ν2 . In the limit as ν2 → −1, the localization in the layer becomes total. Figure 11 presents dimensionless displacements uy /uy (z = h) in the layer (for z > 0) and in the half-space (z < 0) as functions of the dimensionless coordinates z/λ for various values of Poisson’s ratio in the half-space. One can see that, only for the positive ratios ν2 , the wave penetrates into the half-space much deeper than into the thin covering layer with h/λ = 0.3. For large negative Poisson’s ratios, even in the case of a layer of small thickness, the wave is entrapped by this layer. In the case of a thick covering layer (h/λ = 5), this is true practically for any values of Poisson’s ratio. Now consider the situation of a more rigid half-space ρ1 = ρ2 , E2 = 4E1 , where the common restriction on Poisson’s ratios of the layer and the half-space are weaker, 1 + ν1 >

1 + ν2 . 4

This also allows us to study the case of covering layers with a negative Poisson’s ratio ν1 . MECHANICS OF SOLIDS

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Fig. 12.

In the situation under study for E2 = 4E1 and ν1 = 1/2, as one can see in Fig. 10 b (1 — ν1 = 0.5, ν2 = −0.95; 2 — ν1 = 0.5, ν2 = −0.5; 3 — ν1 = 0.5, ν2 = 0; and 4 — ν1 = 0.5, ν2 = 0.3 for 4E1 = E2 and ρ1 = ρ2 ), the dimensionless wave velocity in the thin incompressible covering layer with h/λ < 0.1 becomes even greater. As for the displacement distribution as in the case of a thin (with h/λ = 0.3) and a thick (with h/λ = 5) incompressible layer, the Love wave weakly penetrates into the half-space and is increasingly stronger localized in the layer as Poisson’s ratio ν2 decreases as is shown in Fig. 12 (the solid line corresponds to h/λ = 0.3, the dashed line corresponds to h/λ = 5 for 4E1 = E2 and ρ1 = ρ2 ; (a) —ν1 = 0.5 and ν2 = 0.3; (b) —ν1 = 0.5 and ν2 = 0; (c) —ν1 = 0.5 and ν2 = −0.5; and (d) —ν1 = 0.5 and ν2 = −0.95). Figure 13 (the solid line corresponds to h/λ = 0.3, the dashed line corresponds to h/λ = 5 for 4E1 = E2 and ρ1 = ρ2 ; (a) —ν1 = −0.5 and ν2 = 0.3; (b) —ν1 = −0.5 and ν2 = 0; (c) —ν1 = −0.5 and ν2 = −0.5; and (d) —ν1 = −0.5 and ν2 = −0.95) represents the case of a covering layer with negative Poisson’s ratio ν1 = −0.5 for the same Poisson’s ratios for the half-space ν2 = 0.3, −0.5 and E2 = 4E1 . Comparing Figs 12 and 13, one can see that the Love wave penetration into the half-space is again greater than in a thin layer for nonnegative Poisson’s ratio of the half-space and negative Poisson’s ratio for the thin covering layer. The dispersion equation for the Love waves is much simpler in the case of a thin covering layer. Under the condition hκ1 1, the tangent in the dispersion equation is replaced by a linear function, and we obtain μ2 hκ1 1. hκ2 μ1 MECHANICS OF SOLIDS

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Fig. 13.

In the first approximation, it follows from the relation κ2 ≈ 0 that c2 ≈ c2t2 , and in the next approximation of perturbation theory with respect to a small parameter proportional to thickness, we obtain    μ1 2 ρ1 2 2 2 − (hk) . c ≈ ct2 1 − ρ2 μ2 In the situation where ρ1 = ρ2 = ρ and E1 = E2 = E, we then have the following dependence of the dimensionless wave velocity on Poisson’s ratio and the dimensionless ratio of the layer thickness to the wave length:    2  2 − ν E ν h c2 1 2 . ≈ 11 + ν2 1 − 4π 2 , c0 = 2 1 + ν1 λ 2ρ c0 As in the case of the exact dispersion equation, the wave velocity of the first mode decreases with layer thickness and increases with decreasing Poisson’s ratio, especially strongly as ν2 → −1. A similar behavior of the velocity is observed in the situations where ρ1 = ρ2 and E1 = E2 (cf. Fig. 10 b). As for the approximation to a thick plate with hκ2 μ2 /μ1  1, the approximate dispersion equation in dimensionless form for the waves of the first mode with hκ1 < π/2 has the form    1 1 λ 2 c2 ≈ 1+ . 1 + ν1 16 h c20 MECHANICS OF SOLIDS

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Fig. 14.

This expression is in complete agreement with the behavior of the dimensionless velocity for h/λ ≈ 1 shown in Fig. 10. According to the expressions (3.1) and (3.2) for the displacements, the shear stresses in the covering layer 1 and in the half-space 2 are σyz = Σyz (z) sin[k(x − ct)], σxy = Σxy (z) cos[k(x − ct)], Σyz1 = μ1 κ1 a sin[κ1 (h − z)], Σxy1 = μ1 ka cos[κ1 (h − z)], Σyz2 = μ2 κ2 aeκ2 z cos(κ1 h), Σxy2 = μ2 kaeκ2 z cos(κ1 h). It follows from the boundary conditions on the free surface of the covering layer and on the interface that they have the properties Σyz1 (z = h) = 0,

Σyz1 (z = 0) = Σyz2 (z = 0).

To pass to the dimensionless form of the stress distributions, we use the relation  1 Σ ≡ Σxy1 (z = h)ν1 =0.5 = akE1 . 3 After dividing by this quantity, we obtain the dimensionless distributions of shear stresses in the covering layer, Σyz1 3κ1 = sin[κ1 (h − z)], Σ 2k(1 + ν1 )

Σxy1 3 = cos[κ1 (h − z)], Σ 2(1 + ν1 )

and in the lower half-space, 3κ2 E2 Σyz2 = eκ2 z cos(κ1 h), Σ 2k(1 + ν2 )E1

3E2 Σxy2 = eκ2 z cos(κ1 h). Σ 2(1 + ν2 )E1

The subsequent figures illustrate these distributions of shear stresses of the first mode of the Love wave in the covering layer and in the half-space for various values of Poisson’s ratio. Figure 14 (1 — ν2 = 0.3, 2 — ν2 = 0, 3 — ν2 = −0.5, and 4 — ν2 = −0.95 for E1 = E2 , ρ1 = ρ2 , h/λ = 0.3, and ν1 = 0.5) illustrates the variations in the shear stress Σyz /Σ for four values of Poisson’s ratio ν2 of the half-space material (in particular, for two negative values) in the situation where ρ1 = ρ2 and E1 = E2 for a “thin” (h/λ = 0.3) incompressible (ν1 = 0.5) covering layer. The maximum of this shear stress lies on the interface between the covering layer and the half-space. As Poisson’s ratio ν2 decreases and becomes negative, the maximum begins to grow rapidly. On the other hand, for sufficiently deep penetration into the half-space (for |z|/λ > 0.5), the shear stress Σyz /Σ decreases with decreasing ν2 . Note that the maximum shear stresses on the interface are significantly less in the case of a “thick” covering layer (0.05 for h/λ = 5), and they grow weakly as the ratio ν2 decreases. MECHANICS OF SOLIDS

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Fig. 16.

Fig. 17.

Fig. 18.

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Figure 15 (1 — ν2 = 0.3, 2 — ν2 = 0, 3 — ν2 = −0.5, and 4 — ν2 = −0.95 for E1 = E2 , ρ1 = ρ2 , h/λ = 0.3, and ν1 = 0.5) illustrates the behavior of the other shear stress Σxy /Σ in the same situation where ρ1 = ρ2 , E1 = E2 , h/λ = 0.3, and ν1 = 0.5. This stress varies by a jump on the interface, increases with decreasing Poisson’s ratio ν2 , then begins to decrease on both sides of the interface with decreasing ν2 , and becomes significantly smaller on the side of the covering layer on the interface. In the other situation where ρ1 = ρ2 , 4E1 = E2 , and h/λ = 0.3, for the first mode of the Love waves, it is admissible to consider the covering layer with negative Poisson’s ratio ν1 . Figure 16 (1 — ν2 = −0.95, 2 — ν2 = −0.5, 3 — ν2 = 0, and 4 — ν2 = 0.3 for 4E1 = E2 , ρ1 = ρ2 , h/λ = 0.3, and ν1 = 0.5) shows that, in the case of a covering layer with negative Poisson’s ratio ν1 = −0.5, the maxima of the shear stresses Σyz /Σ increase on the interface for various ν2 several times greater compared with the maxima of the stresses in the preceding situation with ν1 = +0.5 (in Figs. 14 and 16, the scales on the ordinate axes are different). For negative ν1 = −0.5, the values of the other shear stresses Σxy /Σ near the interface, their jumps, and distributions increase in a similar way as in Fig. 17 (1 — ν2 = 0.3, 2 — ν2 = 0, 3 — ν2 = −0.5, and 4 — ν2 = −0.95 for 4E1 = E2 , ρ1 = ρ2 , h/λ = 0.3, and ν1 = 0.5) compared with the situation in Fig. 15. In the case of a thick covering layer (h/λ = 5) with ρ1 = ρ2 , 4E1 = E2 , and ν1 = −0.5, although the maximum stress Σyz /Σ is three times greater than in the situation where ρ1 = ρ2 , E1 = E2 , and ν1 = +0.5, it still remains small (0.14–0.15) and weakly grows with decreasing ν2 . At the same time, the stress Σxy /Σ in both situations is large within the thick covering layer and is three times greater for negative Poisson’s ratio of this layer ν1 = −0.5 and very small in the half-space (Fig. 18 (1 — ν1 = −0.5, 4E1 = E2 , 2 — ν1 = 0.5, E1 = E2 for h/λ = 5, ρ1 = ρ2 , and ν2 = 0)). MECHANICS OF SOLIDS

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4. CONCLUSION The behavior of the Rayleigh and Love waves strongly depends on Poisson’s ratio of the half-space in the first case and on the two Poisson’s ratios for the covering layer and the lower half-space in the second case. This effect is studied for the displacement and stress fields. The negativeness of Poisson’s ratios of the materials near whose surfaces the Rayleigh and Love waves propagate strongly affects the propagation rates and the amplitudes and degrees of wave penetration into the bulk. For negative Poisson’s ratios sufficiently large in absolute value, the growth of the maximum stresses in the nearsurface regions is large for both types of waves. The material particles in the Rayleigh waves rotate in elliptic trajectories in opposite directions at a large and small depth from the free surface, and the waves have linear polarization at a certain intermediate depth. It turns out that these characteristic depths strongly depend on the value and sign of Poisson’s ratio. The degree of penetration of the shear Love waves into the material space covered by a layer of another material depends on the thickness of the covering layer, Poisson’s ratio, and the Young moduli of both materials. The problem on the interaction of the Rayleigh waves with finite inclusions of various geometry, their deformation and fracture is very interesting because, in the case of Rayleigh waves, the inhomogeneous stress-strain state strongly varies with the distance from the free surface and is very sensitive to the value and sign of Poisson’s ratio. A similar problem in the case of Love waves is especially important in the case of inclusions lying at the depth of the interface. ACKNOWLEDGMENTS The research was supported by the Program for Fundamental Studies of Presidium of RAS No. 25, by the Russian Foundation for Basic Research (grant No. 14-01-31245mol_a) and by the Program for Supporting Leading Scientific Schools (grant No. NSh-1275.2014.1). REFERENCES 1. L. D. Landau and E. M. Lifshits, Theoretical Physics. Vol. 7: Theory of Elasticity (Nauka, Moscow, 1967, 1987; Pergamon Press, Oxford, 1970). 2. I. A. Viktorov, Sonic Surface Waves in Solids (Nauka, Moscow, 1981) [in Russian]. 3. R. S. Lakes, “Foam Structures with a Negative Poisson’s Ratio,” Science 235 (4792), 1038–1040 (1987). 4. R. V. Goldstein, V. A. Gorodtsov, and D. S. Lisovenko, “Auxetic Mechanics of Crystalline Materials,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 43–62 (2010) [Mech. Solids (Engl. Transl.) 45 (4), 529–545 (2010)]. 5. T.-C. Lim, P. Cheang, and F. Scarpa, “Wave Motion in Auxetic Solid,” Phys. Status Solidi. Ser. B 251 (2), 388–396 (2013). 6. A. W. Lipsett and A. I. Beltzer, “Reexamination of Dynamic Problems of Elasticity for Negative Poisson Ratio,” J. Acoust. Soc. Amer. 84 (6), 2179–2186 (1988).

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