Surface waves propagation in Cosserat continuum: construction of solution and analysis using wavelet transform 281
Surface waves propagation in Cosserat continuum: construction of solution and analysis using wavelet transform Michail A. Kulesh, Matthias Holschneider, Igor N. Shardakov
[email protected] Abstract In this paper we consider a problem of the surface elastic waves propagation in a half-space (Rayleigh wave) and in a thin layer (Lamb wave) within the framework of the Cosserat continuum. The medium deformation in this model is described not only by the displacement vector, but also by kinematically independent rotation vector. We obtained the general solution of equations of motion. This solution describes four wave types: Rayleigh wave and surface transverse wave in a half-space as well as Lamb wave and transverse wave in a thin layer. Our solutions show that within the framework of Cosserat continuum, both the Rayleigh and surface transverse waves in a half-space are dispersive. The transverse wave in a thin layer and surface transverse wave in a half-space are new wave modes and do not have any analogies in the classical elasticity theory.
Introduction In modern rock and solid mechanics, elastic wave propagation is usually modeled in terms of classical elasticity. However, there are situations, when medium behaviour is still elastic but wave propagation cannot be described by the classical elasticity theory. For instance, current effective medium theories, based on classical elasticity, do not properly describe strong dispersive or attenuative behaviour of wave propagation observed sometimes. The approach we have taken to address this problem is to introduce supplementary and independent degrees of freedom of material particles, in our case rotational ones. Such a theory is called polar (couple/asymmetric stress) elasticity theory. Various models of this kind are widely used in continuum mechanics: Cosserat theory [1], micropolar model of Eringen [2], reduced Cosserat
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continuum [3] etc. However, the lack of information on the values of physical constants for real construction materials is the main factor that hinders the development of this theory and its practical application. The dynamic asymmetric stress theory predicts some interesting theoretical effects which can be used in hypothetical experiments. One of such effects is the dispersion of the Rayleigh surface elastic waves [2]-[5], while the classical theory fails to explain this effect. Further the Cosserat model predicts the propagation of the horizontally polarized transverse surface waves. Geometrically, this wave can be compared with the Love wave; however, within the classical theory of elasticity, the very existence of the Love wave in the form of a surface wave is associated with the presence of a layer overlying the half-space; when the layer thickness tends to zero, the Love wave transforms into the body wave. In [6] it has been shown that, for the Cosserat medium, a transverse horizontally polarized wave decaying with depth can exist without the plane layer. In this paper we generalized the solution of equations of motion considered in [6] and obtained in some particular cases four solutions corresponding to the Rayleigh wave, the surface transverse wave in the half- space and the Lamb wave, the transverse wave in a thin layer. We find that both the Rayleigh and surface transverse wave have the dispersive character of propagation in the half-space. The transverse wave in a thin layer and surface transverse wave in the half-space do not have any analogies in the classical elasticity theory, within the framework of Cosserat medium the identification of these wave types constitute a new finding. As a rule, surface waves in solids and geological media are observed only indirectly, namely, through interpreting the data obtained from vibration measurements in the form of seismograms. For this interpretation, one should be able to retrieve the dispersion parameters of the propagated waves from source seismograms. For Rayleigh wave in geological media, a method for dispersion analysis based on continuous wavelet transform has been proposed in [7]. As a second result of present contribution we adapted this method to the synthetic seismogram generated using our analytical solutions and demonstrated the ability to retrieve the wave number, phase and group velocities from experimental displacement seismograms. These results can be effectively used in experiments which are focused on the detection of couple-stress effects in a medium and further for the determination of material constants of nonsymmetrical elasticity theory.
1
The basic equations for the Cosserat medium
Each material point in the asymmetric theory of elasticity within the framework of the Cosserat medium is an infinitesimal solid which has an orientation. The solid ~ ·σ ˜ and distributed moment of force load is transmitted by distributed force ~p = n ~ =n ~ ·µ ˜ , which have an effect on a surface with a normal vector n ~ . In this case m particle’s kinematics is described by the displacment vector ~u = {ux , uy , uz } of the ~ = {ωx , ωy , ωz }. In the case of the centre of mass and by the rotation vector ω Cosserat medium both vectors are continuous functions of spatial coordinates and time. Thus, we describe the elastic Cosserat continuum by the following tensors and equations [1]:
Surface waves propagation in Cosserat continuum: construction of solution and analysis using wavelet transform 283 ˜·ω ˜ = ∇~u − ~E ~ and asymmetric torsion bending • Asymmetric strain tensor γ ˜ = ∇ω ~ tensor χ ˜ = 2µ˜ • Asymmetric stress tensor σ γ(S) + 2α˜ γ(A) + λI1 (˜ γ)˜ e and asymmetric (S) (A) ˜ = 2γ˜ couple-stress tensor µ χ + 2ε˜ χ + βI1 (˜ χ)˜ e • Equations of motion ¨, ~ = ρ~u ~ +X (λ + 2µ)grad div ~u − (µ + α)rot rot ~u + 2α rot ω (1) ~ = jω. ~ − (γ + ε)rot rot ω ~ + 2α rot ~u − 4αω ~ +Y ~¨ (β + 2γ)grad div ω In equation (1), µ and λ are the Lame constants, α, β, γ and ε are the physical constants of a material in the framework of the Cosserat medium, ρ is the density, j is the moment of inertia density, (.)(S) and (.)(A) denote the symmetric and ˜ is the Levi-Civita tensor of the third rank. antisymmetric parts of tensor and ~E
2
Construction of the general solution
Let us consider a half-space or a thin layer whose surfaces are free from load when there are no mass forces and moments. We choose the x and y Cartesian axes along the surface and the z axis upward. Let the wave propagates in the positive x direction. Unlike previous works [2]-[4], where only monochromatic waves are considered, here we represent the general solution of system (1) in the form of Fourier integrals of all components of the displacement vector un (x, z, t) and rotation vector ωn (x, z, t), which means that the solution is represented as a wave packet limited in the time and Fourier domains: un (x, z, t) =
∞ R
Un (z)ei(kx+ft) u ^ 0 (f) df, ωn (x, z, t) =
−∞
∞ R
Wn (z)ei(kx+ft) u ^ 0 (f) df, (2)
−∞
where k is the wavenumber, f is the circular frequency, t is the time, Un (z) and Wn (z) are amplitude functions depending on depth and frequency. u ^ 0 (f) is the complex spectral function corresponding to the Fourier spectrum of a source signal and determines the wavepacket form. It is expedient to use the continuous Fourier transform of equations (1) where mass forces and moments are zero: ^ − (µ + α)rot rot ~u ^ + 2α rot ω ~^ + ρf2 u ~^ = ~0, (λ + 2µ)grad div ~u ^ − (4α − jf2 )ω ~^ − (γ + ε)rot rot ω ~^ + 2α rot ~u ~^ = ~0. (β + 2γ)grad div ω
(3)
The Fourier transform of representation (2) yields ~^ = {Ux (z), Uy (z), Uz (z)}T eikx u ~^ = {Wx (z), Wy (z), Wz (z)}T eikx u u ^ 0 (f), ω ^ 0 (f). (4) To represent the solution in a more convenient form, we reduce all the quantities to the dimensionless form using the characteristic length X0 and the characteristic
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frequency f0 and introduce the some dimensionless variables, one of which depends on the characteristic length: q µ B−1 A = X0 B(γ+ε) , B = α+µ , C = γ−ε , F= A 2B , α γ+ε C21 =
λ+2µ 2, ρX2 0 f0
C22 =
µ 2, ρX2 0 f0
C23 =
B C2 , B−1 2
C24 =
γ+ε 2, jX2 0 f0
C25 =
β+2γ 2. jX2 0 f0
Next we use the method which is exhaustively described in [5]- [6] as applied to (3)-(4). Note that unlike in the above mentioned papers where ony Rayleigh wave has been consdiered, our approach provides the solutions for both the Rayleigh and Lamb waves simultaneously. Thus, in the general solution we keep all partial solutions of the equation of motion and not only the terms that show depth-dependent decay. In this case we write the general solution in the following form: ux (x, z, t) =
uy (x, z, t) =
uz (x, z, t) =
ωx (x, z, t) =
ωy (x, z, t) =
ωz (x, z, t) =
∞ R
D1 ike−ν1 z + D2 ν2 e−ν2 z + D3 ν3 e−ν3 z + D4 ikeν1 z − D5 ν2 eν2 z − −∞ ® D6 ν3 eν3 z ei(kx+ft) u ^ 0 (f) df, ³ ´ ³ ´ ∞ R F f2 4 −ξ2 z + E A − f2 + 4 e−ξ3 z + E A − + e m p 2 3 2 2 2 F F C4 C −∞ ³ ´ ³ ´ 4 ® 2 2 E5 Am − Cf 2 + 4F eξ2 z + E6 Ap − Cf 2 + 4F eξ3 z ei(kx+ft) u ^ 0 (f) df, 4 4 ∞ R −D1 ν1 e−ν1 z + D2 ike−ν2 z + D3 ike−ν3 z + D4 ν1 eν1 z + −∞ ® D5 ikeν2 z + D6 ikeν3 z ei(kx+ft) u ^ 0 (f) df, ∞ R E1 ike−ξ1 z + E2 ξ2 e−ξ2 z + E3 ξ3 e−ξ3 z + E4 ikeξ1 z − −∞ ® E5 ξ2 eξ2 z − E6 ξ3 eξ3 z ei(kx+ft) u ^ 0 (f) df, ³ ´ ³ ´ ∞ R f2 B −ν2 z + D A − f2 e−ν3 z + D A − e m p 2 3 2 2 2 C3 C −∞ ³ ´ ³ ´ ®3 2 2 D5 Am − Cf 2 eν2 z + D6 Ap − Cf 2 eν3 z ei(kx+ft) u ^ 0 (f) df, 3 3 ∞ R −E1 ξ1 e−ξ1 z + E2 ike−ξ2 z + E3 ike−ξ3 z + E4 ξ1 eξ1 z + −∞ ® E5 ikeξ2 z + E6 ikeξ3 z ei(kx+ft) u ^ 0 (f) df,
(5)
where the constants Dj and Ej , (i = 1..6) must be determined from the boundary conditions, while the exponents of the amplitude functions, νm and ξm , (m = 1..3) are given by the expressions: r q 2 2 4C2 ν1 = k2 − Cf 2 , ξ1 = k2 − Cf 2 + FC42 , 1 5 5 p √ 2−A , 2−A , ν2 = ξ2 = kr ν = ξ = k m 3 3 p 2 2 −C2 )2 2 (C2 C2 −2C2 C2 +C2 C2 ) (C 2A C2 +C 3 4 2 3 3 4 2 4 Ap,m = 2C3 2 C24 f2 − 2A2 ± f4 − f2 + 4A4 . 4C4 C4 C2 C2 C2 3
3
4
3
4
2
3
4
Special case - Rayleigh and Lamb waves
Equations (5) describe in particular case well known solutions for the surface Rayleigh wave in the elastic the half-space. Since the amplitude of displacement
Surface waves propagation in Cosserat continuum: construction of solution and analysis using wavelet transform 285 components of Rayleigh wave decays with depth (along the z-axis), the constant for the exponent terms with positive indices in the expressions (5) must be zero: D4 = D5 = D6 = 0 and E4 = E5 = E6 = 0. The boundary conditions at the surface z = 0 require, that normal forces and moments be zero and in the dimensionless form are σzx |z=0 = 0, µzx |z=0 = 0,
σzy |z=0 = 0, µzy |z=0 = 0,
σzz |z=0 = 0, µzz |z=0 = 0.
(6)
The substitution of solution (5) into the boundary conditions (6) at fixed k and f yields two independent systems of equations: A Rayleigh wave with components ux , uz and ωy is determined by the dispersion equation det(Mr (ν1 , ν2 , ν3 )) = 0, where [5] 2 f2 2k − C2 −2ikp2 −2ikp3 2 2 2 . 2ikp1 2k2 − Cf 2 2k2 − Cf 2 Mr (p1 , p2 , p3 ) = 2 ´ 2 ´ ³ ³ 2 2 p3 Ap − Cf 2 0 p2 Am − Cf 2 3
3
20
ux
1.3
k(f)
uz
Vp(f)/C2
15
x y
1.4
1.2
10
1.1 1
5
0.9 0.8 1000
z
2000
3000
4000
1000
Frequency (rad/s)
2000
3000
4000
Frequency (rad/s)
A Surface transverse wave with components uy , ωx and ωz has the following dispersion equation: det(Mt (ξ1 , ξ2 , ξ3 )) = 0, where [6] ´ ³ ´ ³ 2 2 Ap C2 Am C2 2ik 4 −f 4 −f p 2 + p 2 + 3 2 1−B 2A2 C2 2A2 C2 4 4 2 2 2 ikp1 (1´+ C) p2 + k C p3 + k2 C Mt (p1 , p2 , p3 ) = ³ C2 5 C2 4
C2
− C − 1 k2 − p21 C52 4
ikp2 (1 + C)
20
1.3
Vp(f)/C2
x
1.4
15
k(f)
uy
ikp3 (1 + C)
1.2
10
1.1
y
1
5
0.9 0.8
z
1000
2000
3000
Frequency (rad/s)
4000
1000
2000
3000
4000
Frequency (rad/s)
Figures above show wave numbers k(f) and normalized phase velocities Vp (f)/C2 = f/(C2 k(f)) for the classical (dot lines) and asymmetric media (solid lines). The long-hatched lines correspond to C2 velocity. Thus, in the half-space whose dynamic behavior is described by the Cosserat model, in addition to the elliptic surface Rayleigh wave, it is also possible to observe another wave type, a surface wave whose one component is parallel to the boundary surface and perpendicular to the propagation direction as revealed by our soltuions to the equations of motion. This wave mode does not have analogues in the classical elasticity theory. Both Rayleigh and surface transverse wave have the dispersive character of propagation
.
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in the half-space, observation that cannot be explained by the classical elasticity theory. Lamb wave propagates in a free loaded plate with the thickness 2H. Let us consider the characteristic length X0 = H. The boundary conditions in the dimensionless form at the surfaces z = ±1 require the normal forces and moments to be zero: σzx |z=±1 = 0, µzx |z=±1 = 0,
σzy |z=±1 = 0, µzy |z=±1 = 0,
σzz |z=±1 = 0, µzz |z=±1 = 0.
(7)
The substitution of the solution (5) into the boundary conditions (7) at fixed k and f reduces to two independent solutions for two wave types: A Lamb wave with components ux , uz and ωy is described by the dispersion equation · ¸ Mr (ν1 , ν2 , ν3 )diag(e−νn ) Mr (−ν1 , −ν2 , −ν3 )diag(eνn ) det = 0. Mr (ν1 , ν2 , ν3 )diag(eνn ) Mr (−ν1 , −ν2 , −ν3 )diag(e−νn )
4 8 3
ux x uz
y
Vp(f)/C2
6
k(f)
2H
2
4
1
2
z
0 500
1000
1500
2000
500
Frequency (rad/s)
1000
1500
2000
Frequency (rad/s)
A Transverse wave with components uy , ωx and ωz has the following dispersion equation: · ¸ Mt (ξ1 , ξ2 , ξ3 )diag(e−ξn ) Mt (−ξ1 , −ξ2 , −ξ3 )diag(eξn ) det = 0. Mt (ξ1 , ξ2 , ξ3 )diag(eξn ) Mt (−ξ1 , −ξ2 , −ξ3 )diag(e−ξn ) 4 8 3
y
x
Vp(f)/C2
uy
k(f)
6
2H
2
4
1
2
z
0 500
1000
1500
Frequency (rad/s)
2000
500
1000
1500
2000
Frequency (rad/s)
In the expressions above diag(e−pn ) is the diagonal matrix. The figures above show wave numbers k(f) and normalize phase velocities Vp (f)/C2 = f/(C2 k(f)) for the Lamb wave in the classical medium (dot lines) as well as the Lamb and transverse waves in the Cosserat medium (solid lines). The long-hatched lines correspond to C1 and C2 velocity for the classical medium. Thus, a qualitatively new wave mode with only one displacement component exists in a free loaded plate within the framework of the Cosserat medium besides well investigated Lamb wave. As in the case of surface transverse wave this new mode also does not have any analogy in the classical elasticity theory.
Surface waves propagation in Cosserat continuum: construction of solution and analysis using wavelet transform 287
4
Modeling of surface wave propagation using continuous wavelet transforms
To illustrate the displacement components, let us construct the three-component seismogram on the surface of the half-space for Rayleigh (Figure 1.a) and surface transverse (Figure 1.b) waves for five different distances ∆x from the source using dispersion parameters of Rayleigh and surface transverse waves. Such a seismogram can be obtained experimentally when the wave is generated at a certain point on the surface of the half-space and the response is recorded by several geophones equally spaced along the wave propagation path. As an illustration of a possible way of phase and group velocities determination let us use the asymptotic propagator in the wavelet space [7]. This propagator is a mathematical model to establish a link between the continuous wavelet transform Wg u1 (t, f) of displacement component at the surface at the distance x1 from the source and its propagated counterpart Wg u2 (t, f) at the distance x2 = x1 + D, when the wave propagates in a dispersive medium with the known phase Vp (f) and group Vg (f) velocities: ¯ ´¯ h ³ ´i ³ ¯ ¯ D D Wg u2 (t, f) = ¯Wg u1 t − Vg (f) , f ¯ exp i arg Wg u1 t − Vp (f) , f , +∞ (8) R 1 ∗ ³ τ−t ´ Wg S(t, a) = g a S(τ) dτ, a −∞
where g(τ) is the Morlet mother wavelet, a = f0 /f ∈ R is the scale parameter (dilation), t ∈ R is the location parameter (translations) and (·)∗ indicates the complex conjugation. Signals u1 (t) and u2 (t) are real in case of the transverse wave, but these signals can be complex for the Rayleigh wave: u1 (t) = ux (t) + iuz (t). (a) Rayleigh wave
ux(t)
0.5 0
−0.5
un(x1,0,t)
uz(t)
0.5
un(x2,0,t)
un(x3,0,t)
un(x4,0,t)
un(x5,0,t)
0
−0.5
(b) Surface transverse wave
uy(t)
0.5 0
−0.5
Frequency (rad/s)
200
(c) |Wg uy(t,f)|
150
100
Frequency (rad/s)
200 50
(d) arg[Wg uy(t,f)]
150
100
50 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Figure 1: (a,b) Synthetic three-component seismogram, (c) wavelet modulus and (d) phase images for the component uy (t). Solid lines are the group (panel c) and phase (panel d) velocities. As shown in Figures 1c-d, the group velocity is a function that “deforms” the image of the absolute value of the source signal’s wavelet spectrum, the phase veloc-
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ity “deforms” the image of the wavelet spectrum’s phase. Thus, the full dispersion characteristics are explicitly expressed and therefore can be easily extracted from the modulus and phase of wavelet transform.
Conclusions In this study we have obtained four solutions corresponding to Rayleigh wave and surface transverse wave in a half-space as well as Lamb wave and transverse wave in a thin layer. These solutions can be divided into two groups, one of which corresponds to the well-investigated elliptical wave and the other — to the transverse wave with depth-dependent decay which does not have any analogy in the classical theory of elasticity. We have compared the solution for all observed waves to classical case with help of numerical illustrations. We have also considered a model that binds the wavelet spectra of two signals at different distance. This model allows us to determine the phase and group velocities of Rayleigh and surface transverse waves.
Acknowledgements This work was supported by U.S. Civilian Research & Development Foundation (Post-Doctoral Fellowship Program Y2-P-09-04) and by the Deutsche Forschungsgemeinschaft (DFG) within the framework of the priority program SPP 1114, ”Mathematical methods for time series analysis and digital image processing”. We also thank Mamadou S. Diallo (ExxonMobil Upstream Research Company) for reviewing the final manuscript to improve its readability.
References [1] V. Novatskij. Theory of Elasticity (Mir, Moscow, 1975) [in Russian]. [2] A. C. Eringen. Microcontinuum Field Theories, Vol. 1: Foundation and Solids (Springer, New York, 1999) [3] E. F. Grekova, G. C. Herman. Wave propagation in rocks modeled as reduced Cosserat continuum. Proceedings of 66th EAGE, Paris. P. 98 (2004). [4] A. E. Lyalin, V. A. Pirozhkov, and R. D. Stepanov. Propagation of Surface Waves in the Cosserat Medium. Akust. Zh. V. 28. P. 838-840 (1982). [5] M. A. Kulesh, V. P. Matveenko and I. N. Shardakov. Construction and analysis of an analytical solution for the surface Rayleigh wave within the framework of the Cosserat continuum. Journal of Applied Mechanics and Technical Physics. V. 46. No. 4. P. 556-563 (2005). [6] M. A. Kulesh, V. P. Matveenko and I. N. Shardakov. Propagation of surface elastic waves in the Cosserat medium. Acoustical Physics. V. 52. No. 2. P. 186193 (2006).
Surface waves propagation in Cosserat continuum: construction of solution and analysis using wavelet transform 289 [7] M. A. Kulesh, M. S. Diallo and M. Holschneider. Wavelet analysis of ellipticity, dispersion, and dissipation properties of Rayleigh waves. Acoustical Physics. V. 51. No. 4. P. 421-434 (2005). Michail A. Kulesh, Matthias Holschneider, University of Potsdam, Applied and Industrial Mathematics, Am Neuen Palais 10, 14469 Potsdam, Germany Igor N. Shardakov, Institute of Continua Media Mechanics, Ural Branch of RAS, Ak. Korolev str. 1, 614013 Perm, Russia
