Propagation of Love Wave in Viscoelastic Sandy Medium Lying Over Pre-stressed Orthotropic Half-space

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ScienceDirect Procedia Engineering 173 (2017) 996 – 1002

11th International Symposium on Plasticity and Impact Mechanics, Implast 2016

Propagation of Love wave in viscoelastic sandy medium lying over pre-stressed orthotropic half-space Deepak Kr. Pandit1,∗, Santimoy Kundu1 a Department

of Applied Mathematics, IIT(Indian School of Mines), Dhanbad, Jharkhand-826004, India

Abstract The main objective of the study is to investigate the effects of parameters on wave propagation in a viscoelastic sandy medium lying over an elastic orthotropic half-space under initial stress. The frequency equation for Love waves has been deduced in a closed form. To examine the effects of initial stress, sandiness and viscoelastic parameters, we have calculated the numerical values for phase velocity and attenuation coefficient. The phase velocity and attenuation coefficient have been plotted against wave number of different values of initial stress, sandiness and viscoelastic parameters. It is observed that the phase velocity and attenuation coefficient have been influenced by the increase of proposed parameters. The theme can be of interest for geophysical applications in propagation of Love waves in the Earth’s crust. © Published by Elsevier Ltd. This c 2017 by Elsevier B.V.is an open access article under the CC BY-NC-ND license  2016The TheAuthors. Authors. Published (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of Implast 2016. Peer-review under responsibility of the organizing committee of Implast 2016 Keywords: Initial stress, Love waves, Orthotropic Layer, Viscosity, Phase velocity.

1. Introduction The theoretical study of elastic waves finds numerous geophysical applications in understanding the damage due to earthquake. These vibrations provide most direct and rich information regarding the structure of Earth’s interior to seismologists. In wave propagation, Earth’s crustal model is not only influenced by the anisotropy but also internal viscosity of the media. Viscoelastic property of a material is the combination of two significant physical properties namely viscous and elastic, which is a major cause of seismic attenuation. The dynamical behavior of a viscoelastic media has great importance in many fields, such as seismology, earthquake engineering, soil dynamics and fluid dynamics. The propagation behaviour of elastic waves in viscoelastic models are of great interest for the accurate inversion of the observed surface waves data. Biot[1] invented a theory of deformation of a porous viscoelastic anisotropic solid which has attracted various researchers round the globe. Carcione[2] studied the theoretical aspects of wave propagation in anisotropic linear viscoelastic media taking simulated wave fields. Gravitational effect on surface waves in a homogeneous fibre-reinforced anisotropic general viscoelastic media of higher and fractional order with voids was deeply investigated by Khan et al.[3]. ∗

Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of Implast 2016

doi:10.1016/j.proeng.2016.12.170

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Deepak Kr. Pandit and Santimoy Kundu / Procedia Engineering 173 (2017) 996 – 1002

Fig. 1: Structure of the problem

The layer of the soil in the earth is supposed to be more sandy than elastic. A dry sandy mantle may be defined as a layer consisting of sandy particles retaining no moistures or water vapors. Wave propagation through sandy medium is one of the most important area of interest not only to geologists but also to several researchers and seismologists. Propagation of Love waves in a dry sandy layer lying between two semi infinite elastic media was observed by Paul[4]. Chattopadhyay and Sharma[5] examined the propagation of SH waves in a dry sandy layer. Dey and De[6] detected Rayleigh waves in a dry sandy layer lying over an initially stressed orthotropic elastic medium. The development of initial stress in the medium is due to many reasons, for example the difference of temperature, process of quenching, gravity variation, creep, differential external forces, etc. The Earth may be assumed as elastic solid layered medium under high initial stress. It is therefore of much interest to study the influences of these stresses on the propagation of Love waves. The effect of gravity and initial stress on torsional surface waves in dry sandy medium was investigated by Dey et al.[7]. The propagation of Love wave in sandy layer under initial stress above anisotropic porous half-space under gravity was studied by Pal[8]. Kundu et al.[9] established the effect of periodic corrugation, reinforcement, heterogeneity and initial stress on Love wave propagation. The propagation of seismic waves in orthotropic media is very much different from their propagation in isotropic media. So the study of surface waves in orthotropic layered media has become of main interest to seismologists and geophysicists as well. The study of generations and propagation of waves in layered orthotropic media with various geometrical configurations is highly important not only in Geophysics but also in Seismology, Acoustics and Electromagnetism. The propagation of Love waves in an orthotropic Granular layer under initial stress overlying a semi-infinite Granular medium was examined by Ahmed and Abo-Dahab[10]. Abd-Alla et al.[11] discovered the propagation of Love waves in a non-homogeneous orthotropic magnetoelastic layer under initial stress overlying a semiinfinite medium. The propagation of Love wave in fibre-reinforced medium lying over an initially stressed orthotropic half-space was observed by Kundu et al.[12]. In this work, the effects of parameters on wave propagation in a viscoelastic sandy medium lying over an elastic orthotropic half-space under initial stress has been studied in great detail. The displacement components of both the proposed layer and half-space have been derived separately. Using suitable boundary conditions and the derived displacement components, the dispersion relation has been derived in closed form. To show the effect of viscoelasticity, sandiness parameter and initial stress on phase velocity and attenuation, graphs have been plotted separately. 2. Formulation of Physical Model Love wave condition Let, the displacement components of viscoelastic sandy medium and half-space are (u1 , v1 , w1 ) and (u2 , v2 , w2 ) respectively. For Love wave propagation along x-axis, the displacement components are assumed as ui (x, z, t) = 0 = wi (x, z, t), vi = vi (x, z, t); i = 1, 2.

(1)

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3. Viscoelastic sandy medium The equations of motion for propagation of Love waves in viscoelastic sandy medium are given as ∂2 u1 ∂τ21 ∂τ22 ∂τ23 ∂2 v1 ∂τ31 ∂τ32 ∂τ33 ∂2 w1 ∂τ11 ∂τ12 ∂τ13 + + = ρ1 2 ; + + = ρ1 2 ; + + = ρ1 2 . ∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z ∂t ∂t ∂t The stress components are       ∂u1 ∂v1 ∂u1 ∂w1 ∂w1 ∂v1 ∂v1 ∂v1 + ; τ31 = μ¯ 1 + + = μ¯ 1 = 0; τ23 = μ¯ 1 = μ¯ 1 τ12 = μ¯ 1 ∂y ∂x ∂x ∂z ∂x ∂y ∂z ∂z

(2)

(3)

where μ¯ 1 = μη1 + iμ ∂t∂ . The non vanishing equation of motion in viscoelastic sandy elastic medium for propagating Love waves having rigidity μ¯ 1 and density ρ1 is μ¯ 1

 ∂2 v1

+

∂x2

∂2 v 1 ∂2 v1  . = ρ 2 ∂z2 ∂t2

(4)

Let us consider the solution of (4) is v1 = v1 (z)eik(x−ct) ,

(5)

thus v1 (z) will satisfy the equation v1 (z) + ξ12 v2 (z) = 0   1/2 μ1 c2 , β1 = ηρ and Q−1 = where ξ1 = β2 (1−iηQ −1 ) − 1 1 1

(6) μ kc μ1 .

v1 (z) = C1 cos(kξ1 ) + C2 sin(kξ1 )

(7)

The solution of equation (5) can be written as v1 (x, z, t) = (C1 cos(kξ1 ) + C2 sin(kξ1 ))eik(x−ct) .

(8)

Equation (8) is the displacement component of the viscoelastic sandy layer. 4. Initially stressed orthotropic half-space The lower semi-infinite medium is considered under initial compressive stress P1 along x-axis. Then the equations of motion in this medium under initial stress in the absence of body forces are given by Biot[13] ∂2 u 2 ∂wz ∂wy ∂σ11 ∂σ12 ∂σ13 + + − P1 ( − ) = ρ1 2 ∂x ∂y ∂z ∂y ∂z ∂t ∂σ21 ∂σ22 ∂σ23 ∂wz ∂2 v 2 + + − P1 = ρ1 2 ∂x ∂y ∂z ∂x ∂t ∂wy ∂2 w2 ∂σ31 ∂σ32 ∂σ33 + + + P1 = ρ1 2 ∂x ∂y ∂z ∂x ∂t

(9)

where, σi j are the incremental stress, ρ2 is the density of half-space, (u2 , v2 , w2 ) are displacement component, w x , wy and wz denote the rotational components along suffix x, y and z direction. Using (1), the stress components for orthotropic half-space become ⎫ σ11 = μ11 e11 + μ12 e22 + μ13 e33 = 0, σ22 = μ21 e11 + μ22 e22 + μ23 e33 = 0, ⎪ ⎪ ⎪ ⎪ ⎬ 2 σ33 = μ31 e11 + μ32 e22 + μ33 e33 = 0, σ12 = 2μ66 e12 = μ66 ∂v , (10) ⎪ ∂x ⎪ ⎪ ⎪ ∂v2 ⎭ σ23 = 2μ44 e23 = μ44 ∂z , σ31 = 2μ55 e31 = 0

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and μi j are the elastic constants. ω is the angular frequency and i is the imaginary unit. By using the stress-strain relationship (10), the equations of motion (9) for the half-space reduces to  ∂2 v2 ∂2 v 2 P1  ∂2 v2 + μ44 2 = ρ2 2 μ66 − 2 2 ∂x ∂z ∂t Let us consider the harmonic solution of equation (11) as v2 = v(z)eik(x−ct) then v2 (z) satisfies the following differential equation μ66  ρ2 c2 P1  v (z) + k2 −1+ v(z) = 0 μ44 μ66 2μ66 using equations (11) and (12) the final solution can be expressed as v2 (x, z, t) = (D1 e−kξ2 z + D2 ekξ2 z )eik(x−ct) where D1 and D2 are the arbitrary constants, ξ2 =





μ66 c2 μ44 β22

−1+P

 1/2

, β2 =



(11)

(12)

(13) μ66 ρ2

and P =

P1 2μ66 .

Equation (13) is the required displacement component of initially stressed orthotropic half-space. 5. Boundary and continuity conditions The stress tensors and displacements must be continuous at z = −H and z = 0, hence the following suitable boundary conditions must be applicable, (i) The shearing stress is stress free at the free surface z = −H. i.e., τ23 = 0. (ii) The displacement components and shearing stress must be continuous at the interface z = 0. i.e., v1 = v2 and τ23 = σ23 (iii) The displacement component of half-space vanishes as the depth increases. i.e., v2 → 0 as z → ∞. using the equations (8) and (13) the boundary conditions implies C1 sin(kξ1 H) + C2 cos(kξ1 H) = 0 C 1 − D1 = 0 C2 ξ1 μ¯ 1 + D1 ξ2 μ44 = 0

(14) (15) (16)

By eliminating C1 , C2 and D2 by equations (14), (15) and (16) we get c44 ξ2 (17) Υ(k1 , c, δ) = tan(kHξ1 ) − μ¯ 1 ξ1 Equation (17) is the frequency equation of Love wave in viscous sandy layer over pre-stressed orthotropic half-space. It is clear that the implicit equation is in complex form because of dissipation of the system. The equation (17) is separated in real and imaginary terms as [Υ(k1 , c, δ)] = 0 [Υ(k1 , c, δ)] = 0

(18) (19)

Equation (18) and (19) are the dispersion and absorption relations. (18) provides the dispersion curves i.e., phase velocity against wave number while (19) provides the absorption curves i.e., attenuation against wave number. 6. Particular Cases When the sandiness parameter η → 1, initial stress P → 0, dissipation factor Q−1 → 0 and μ66 = μ44 then the equation (18) becomes  2

μ44 1 − βc2 c2 2 − 1) = tan(kH (20)  2 β21 μ1 c − 1 β21

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and (19) vanishes. Equation (20) is the well known classical equation of Love waves [14] which validates the problem.

7. Numerical Discussion In order to represent the results of the dispersion and absorption relations, the following data has been considered:

Medium Viscoelastic sandy medium [15] Orthotropic half-space [15]

Rigidity (×1010 N/m2 ) μ1 = 6.34 μ44 = 5.82, μ66 = 3.99

Density (Kg/m3 ) ρ1 = 3364 ρ2 = 4500

Table 1: Properties of elastic materials

Figure 2 - 4 irradiates the effect of sandiness, heterogeneity and dissipation factor on the phase velocity and attenuation of Love wave. Each of the figures consists of two subfigure in which first corresponds to phase velocity while second for attenuation. Figure 2 displays the effect of sandiness parameter η on phase velocity c/β1 and attenuation log(δ) of Love wave. The value of η has been taken as 2.20, 2.25 and 2.30. It is clear from the figure that the phase velocity c/β1 decreases with the growing value of wave number kH and sandiness parameter η while this trend alters for attenuation log(δ) i.e. sandiness parameter and wave number has a favouring effect on attenuation.

0 1.6

1. Η = 2.20 2. Η = 2.25 3. Η = 2.30

1.5

1

2

log Δ

Phase velocity c Β 1 

3 2

1

1.4

3

1 2

1.3

1.2 3

1. Η = 2.20 2. Η = 2.25 3. Η = 2.30

1.1

4

1.0 0.8

1.0

1.2

Wave number RekH 

1.4

1.35

1.40

1.45

1.50

1.55

1.60

Wave number RekH 

Fig. 2: Variation of phase velocity (c/β1 ) and attenuation log(δ) with respect to wave number Re(kH) for different values of sandiness parameters η.

Figure 3 has been plotted to understand the effect of initial stress on the phase velocity c/β1 and attenuation log(δ) of Love wave. The value of initial stress parameter P has been taken as 0.1, 0.5 and 0.9. From figure it can be observed that with increase in initial stress parameter, the phase velocity and attenuation decreases while the wave number has different effect on phase velocity and attenuation i.e. it step-up the attenuation but declines the phase velocity.

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Deepak Kr. Pandit and Santimoy Kundu / Procedia Engineering 173 (2017) 996 – 1002 0

3.0

1 1 2 log Δ

Phase velocity c Β 1 

2.5

3

2.0

2

3 2 1.5

1

3

1. P = 0.1

1. P = 0.1 2. P = 0.5

2. P = 0.5

3. P = 0.9

3. P = 0.9 4

1.0 0.0

0.5

1.0

1.5

1.35

2.0

1.40

1.45

1.50

1.55

1.60

Wave number RekH 

Wave number RekH 

Fig. 3: Variation of phase velocity (c/β1 ) and attenuation log(δ) with respect to wave number Re(kH) for different values of initial stress P.

Figure 4 depicts the influence of dissipation factor Q−1 on the phase velocity c/β1 and attenuation log(δ) of Love wave. It is examined from the figure that the phase velocity curve shifts upward while the curve of attenuation shifts downward for the rising value of dissipation factor Q−1 . Contrary to this trend alters for wave number kH. 5

0 1. Q 1  0.1 2. Q 1  0.2 3. Q 1  0.3

4

1

1

2 3

2 3 log Δ

Phase velocity c Β 1 

1

3

2

3

2

1. Q 1  0.1 2. Q 1  0.2 3. Q 1  0.3 4

1 0.0

0.2

0.4

0.6

0.8

Wave number RekH 

1.0

1.2

1.4

1.35

1.40

1.45

1.50

1.55

1.60

Wave number RekH 

Fig. 4: Variation of phase velocity (c/β1 ) and attenuation log(δ) with respect to wave number Re(kH) for different values of dissipation factor Q−1 .

8. Conclusion The current study deals with the propagation of Love wave in a viscoelastic sandy medium lying over an elastic orthotropic half-space under initial stress. Closed form of the dispersion relation has been obtained. The presence of initial stress, sandy parameter and dissipation factor in the frequency equation approves the effects of these parameters in the propagation of Love wave. The outcomes of the study are quoted as follows: • The phase velocity of Love wave decreases whereas attenuation increases with increase in the wave number. • The phase velocity curve shifts downwards while the attenuation shifts upward for the growing value of sandiness parameter η. • The initial stress parameter has the decreasing effect on the phase velocity and the attenuation. At the initial stage of wave number, curves are accumulating, showing that there is no effect of initial stress parameter on the phase velocity and attenuation for lower values of wave number. • The dissipation factor have a congruent effect on phase velocity while adverse effect on attenuation coefficient.

References [1] M.A. Biot, Theory of deformation of a porous viscoelastic anisotropic solid, Journal of Applied Physics, 27(5) (1956) 459-467.

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[2] J.M. Carcione, Wave propagation in anisotropic linear viscoelastic media: Theory and simulated wave- fields, Int. J. Geoph. 101 (1990) 739750. [3] A. Khan, S.M. Abo-Dhabab, A.M. Abd-Alla, Gravitational effect on surface waves in a homogeneous fibre-reinforced anisotropic general viscoelastic media of higher and fractional order with voids, Int. J. Phy. Sci. 10(24) (2015) 604-613. [4] M.K. Paul, Propagation of Love waves n a dry sandy layer lying between two semi infinite elastic media, Acta Geophys Pol, XIII(1) (1965) 1-7. [5] A. Chattopadhyay , R.D. Sharma, SH waves in a dry sandy layer, Gerl Beitr Geophys 91(4) (1982) 355-360. [6] S. Dey ,R.K. Dey, Rayleigh waves in a dry sandy layer lying over an initially stressed orthotropic elastic medium, Acta Geophys Pol XXXI(2) (1983) 177-185. [7] S. Dey, A.K. Gupta, S. Gupta, Effect of gravity and initial stress on torsional surface waves in dry sandy medium, J. Eng. Mech. 128(10) (2002) 1115-1118. [8] J. Pal, A.P. Ghorai, Propagation of Love wave in sandy layer under initial stress above anisotropic porous half-space under gravity, Transp. Porous Med. 109 (2015) 297-316. [9] S. Kundu, A.Kumari, S. Gupta, D.K. Pandit, Effect of periodic corrugation, reinforcement, heterogeneity and initial stress on Love wave propagation; Waves in Random and Complex Media, DOI-10.1080/17455030.2016.1168951 (2016). [10] S.M. Ahmed, S.M. Abo-Dahab, Propagation of Love waves in an orthotropic Granular layer under initial stress overlying a semi-infinite Granular medium, Journal of Vibration of Control, 16 (2010) 1845-1858. [11] A.M. Abd-Alla, S.M. Abo-Dahab, T.A. Al-Thamali, Love waves in a non-homogeneous orthotropic magnetoelastic layer under initial stress overlying a semiinfinite medium, Journal of Computational and Theoretical Nanoscience 10 (2013) 10-18. [12] S. Kundu, S. Gupta, S. Manna, Propagation of Love wave in fibre-reinforced medium lying over an initially stressed orthotropic half-space, International Journal for Numerical and Analytical Methods in Geomechanics, 38 (2014) 1172-1182. [13] M.A. Biot, Mechanics of incremental deformations, New York: Wiley; (1965). [14] A.E.H. Love, Mathematical theory of elasticity, Cambridge University Press, Cambridge (1920). [15] D. Gubbins, Seismological and plate tectonics, Cambridge: Cambridge University Press; (1990).