Effect of Rigid Boundary on the Propagation of Torsional Surface Waves in Heterogeneous Earth Media

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ScienceDirect Procedia Engineering 173 (2017) 964 – 971

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

Effect of Rigid Boundary on the Propagation of Torsional Surface Waves in Heterogeneous Earth Media Abhijit Pramanik a,∗, Shishir Gupta a a Department

of Applied Mathematics, Indian Institute of Technology (Indian School of Mines) Dhanbad, India

Abstract The present paper has been framed to study the influence of rigid boundary on the propagation of torsional surface waves in an inhomogeneous crustal layer over an inhomogeneous half space. In the layer the rigidity and density vary linearly along the depth whereas the inhomogeneous half space exhibits inhomogeneity of exponential type. The dispersion equation is derived in a closed form. For a homogeneous layer over a homogeneous half space the dispersion equation turns into the dispersion equation of the propagation of Love type wave under rigid boundary. The effects of inhomogeneity factor on the phase velocity are depicted by means of graphs. c 2017 by Elsevier B.V.is an open access article under the CC BY-NC-ND license  2016The TheAuthors. Authors. Published © Published by Elsevier Ltd. This (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: Torsional surface waves; rigid boundary; inhomogeneous crustal layer; inhomogeneous half space; dispersion equation; phase velocity

1. Introduction The studies of wave propagation in an elastic media have received more attention recently, chiefly because the need of a complete understanding of different medium characteristics with respect to mechanical shocks and vibrations is often felt in the Earth. The study of surface waves for homogeneous, non-homogeneous and layered media has been a central interest to seismologists up to recent times. A quite good amount of valuable information about the propagation of elastic waves is the monograph written by Achenbach [1] and Ewing et al. [2]. A lot of information on the effect of heterogeneity in the study of surface waves vibrations due to line-load was given by Vrettos [3] and Kennett and Tkali [4]. Unfortunately the literature available on torsional waves in an inhomogeneous layer over an inhomogeneous half space are very much less than the availability of that on Rayleigh Waves, Love waves and Stoneley Waves. Lord Rayleigh [5], in his remarkable paper, showed that the isotropic homogenous elastic half-space does not allow a Torsional surface wave to propagate. Propagation of torsional surface waves in non-homogeneous and anisotropic medium with polynomial and exponential variation in rigidity and constant density has been discussed by Dey et al. [6]. The excellent work by Gupta et al. [7] and Davini et al. [8] on the propagation of torsional surface waves in a nonhomogeneous medium may be cited. Akbarov et al. [9] discussed torsional wave dis∗ Corresponding

author. Tel.: +91-953-408-0407 E-mail address: abhijit [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.2016.12.161

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persion in a finitely pre-strained hollow sandwich circular cylinder, whereas torsional surface waves in heterogeneous anisotropic half-space under initial stress was studied recently by Chattopadhyay et al. [10]. In the present problem, the inhomogeneity is caused by variation in rigidity and density. The crust region of our planet is composed of various inhomogeneous layers with different geological parameters. As pointed out by Bullen [11], the density inside the Earth varies at different rates with different layers within the Earth. He approximated density law inside the Earth as a quadratic polynomial in depth parameter for 413 km to 984 km depth. For depth from 984 km to the central core, Bullen approximated the density as a linear function of depth parameter. Sari and Salk [12] took the variation in the density of sediments with depth as a hyperbolic function. The present paper attempts to study the possibility of propagation of torsional surface waves in an inhomogeneous layer having rigid boundary over an inhomogeneous half space. In the inhomogeneous crustal layer, density and rigidity vary linearly with depth, whereas the different variation in inhomogeneity is taken in the half space, exponential variation with depth in rigidity and density. It is observed that torsional surface wave exist in inhomogeneous layer having rigid boundary over heterogeneous half space. The effect of inhomogeneity present in both the medium on the velocity of torsional wave was studied graphically. 2. Basic theoretical frameworks For isotropic medium, stress-displacement relations in cylindrical coordinates are given by   ∂uz ur ∂ur 1 ∂uθ r σrr = λ ∂u ∂r + r + r ∂θ + ∂z  + 2μ ∂r ,  ∂u  ∂u ∂uz ur 1 ∂uθ θ r r σθθ = λ ∂r + r + r ∂θ + ∂z + 2μ urr + 1r ∂u ∂θ ∂r ,  ∂u  ∂uz z θ σzz = λ ∂rr + urr + 1r ∂u + ∂u ∂z + 2μ ∂z ,  ∂u  ∂θ 1 ∂uz θ σθz = μ ∂z + r ∂θ ,   ∂uz r σzr = μ ∂u + ∂r ,  ∂z  ur ∂ur θ σrθ = μ ∂u − ∂r r + ∂θ .

(1)

The torsional wave is characterized by the displacement vectors along radial, circumferential and vertical direction as ur = 0, uθ = v (r, z, t) , uz = 0. Using (2) in (1), we obtain

(2)



   ∂v ∂v v − . σrr = σθθ = σzz = 0, σθz = μ , σzr = 0, σrθ = μ ∂z ∂r r

(3)

The dynamical equation of motion without body forces in cylindrical medium is given by(see Love [13]) ∂σrr ∂r ∂σrθ ∂r ∂σrz ∂r

+ + +

1 ∂σrθ r ∂θ 1 ∂σθθ r ∂θ 1 ∂σθz r ∂θ

σrr −σθθ rz + ∂σ = ρ ∂∂t2u , ∂z + r 2 ∂σθz 2σrθ + ∂z + r = ρ ∂∂t2v , 2 σrz ∂ w zz + ∂σ ∂z + r = ρ ∂t2 . 2

Using (3) in (4), the only non-vanishing equation of motion is     2 1 1 ∂ ∂ ∂v ∂2 v ∂ (z) − + μ (z) v + μ = ρ (z) 2 . 2 2 r ∂r r ∂z ∂z ∂r ∂t

(4)

(5)

where μ and ρ are the rigidity and the mass density of the media respectively, assumed to be function of depth z. Harmonic wave solution of Eq. (5) is of the form v = V (z) J1 (Kr) exp (iωt) ,

(6)

where V (z) is the solution of the following equation:   d2 V (z) μ (z) dV (z) c2 2 + − K 1 − 2 V (z) = 0. (7) μ (z) dz dz2 cs    In the above equation, c = Kω is the torsional wave velocity, c s = μ(z) ρ(z) , ω is the angular frequency, K is the angular wave number, and J1 (Kr) is the first order Bessel function of first kind.

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3. Formulation of the problem The model consists of an inhomogeneous layer of thickness H sandwiched between a rigid layer and a vertically inhomogeneous half-space. The inhomogeneity is considered both in the mass density and rigidity as shown in Fig. 1.

Fig. 1. Geometry of the problem.

The origin of the cylindrical coordinate system (r, θ, z) is located at the interface separating the layer from the halfspace, and the z-axis is directly downwards. The following variation in the rigidity and the mass density are taken into account. For the crustal layer, μ = μ0 (1 + m0 z), ρ = ρ0 (1 + n0 z), where m0 > 0, n0 > 0 For the half-space, μ = μ1 exp(az), ρ = ρ1 exp(bz), where a > 0, b > 0 In the above, μ and ρ are the rigidity and the mass density of the media respectively, where m0 , n0 , a and b are constants having dimension that are inverse of length. 4. Boundary conditions The following boundary conditions must be satisfied: (I) At the free surface z = −H, the displacement is vanishing so that v0 = 0

at

z = −H.

(8)

(II) At the interface , the continuity of the stress requires that μ0

∂v0 ∂v1 = μ1 ∂z ∂z

at

z = 0.

(9)

(III) The continuity of the displacement requires that v0 = v1

at

z = 0.

(10)

where v0 and v1 are the displacements in the layer and the half-space respectively. 5. Solution of the problem 5.1. Solution of layer In the inhomogeneous crustal layer, we have μ = μ0 (1 + m0 z), ρ = ρ0 (1 + n0 z).

(11)

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Abhijit Pramanik and Shishir Gupta / Procedia Engineering 173 (2017) 964 – 971

Using Eq. (11) in Eq. (7), we get   d2 V dV μ 0 m0 c2 (1 + n0 z) 2 + − K 1 − V(z) = 0, μ0 (1 + m0 z) dz dz2 c0 2 (1 + m0 z)  where c0 = μρ00 . Now, substituting V(z) =

φ(z) 1

(1+m0 z) 2

(12)

into Eq. (12) to eliminate the term

   m0 2 d2 φ(z) c2 (1 + n0 z) 2 + − K φ(z) = 0. 1 − dz2 c0 2 (1 + m0 z) 4(1 + m0 z)2  2 Introducing the dimensionless quantities γ = 1 − cc0 2 mn00 and η =

dV dz ,

we have (13)

2γK(1+m0 z) m0

in Eq. (13), the equation reduces to

  1 R d2 φ(η) 1 ω2 (m0 − n0 ) + + − , φ(η) = 0, where R = 2 2 4 2η dz 4η c0 2 γm0 2 K

(14)

which is well known Whittakers equation (see Whittaker and Watson [14]). The solution of Whittakers equation (14) is given by φ(η) = D1 W R2 ,0 (η) + D2 W −R2 ,0 (−η).

(15)

Hence, the displacement for the torsional wave in the layer is ⎛ ⎞ ⎜⎜ D1 W R2 ,0 (η) + D2 W −R2 ,0 (−η) ⎟⎟⎟ v0 = ⎜⎝⎜ ⎠⎟ J1 (Kr) exp (iωt) . (1 + m0 z)

(16)

5.2. Solution for half-space The rigidity and density in the half-space are assumed to have the following form, μ = μ1 exp(az), ρ = ρ1 exp(bz).

(17)

Using Eq. (17), Eq. (7) for the half-space takes the form   dV d2 V c2 ebz 2 +a − K 1 − 2 az V(z) = 0, dz dz2 c1 e  where c1 = μρ11 . Now, we substitute V(z) =

φ(z) e

az 2

(18)

into Eq. (18), and obtain

 2 2   K c (1 + bz) d2 φ a2 2 + φ(z) = 0. −K 1+ dz2 c1 2 (1 + az) 4K 2  Introducing the dimensionless quantities γ1 = 1 +

(19) a2 4K 2

−

b c2 a c21

and η1 =

2γ1 K a (1

+ az), Eq. (19) reduces to,

  1 R1 d2 φ + − + φ (η1 ) = 0, 4 η1 dη21

(20)

2

where R1 = c2c(a−b)K 2 a2 γ , which is well known Whittakers equation (see Whittaker and Watson [14]). 1 1 The solution of Whittakers equation (20) is given by φ (η1 ) = EWR1 , 12 (η1 ) + E  W−R1 , 12 (−η1 ) ,

(21)

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The solution of equation (21) satisfying the condition that lim v1 (z) → 0 i.e., lim φ (η1 ) → 0 may be taken as η1 →∞

z→∞

φ (η1 ) = EWR1 , 12 (η1 ) ,

(22)

where WR1 , 12 (η1 ) is the Whittaker function. Hence, the displacement for the torsional wave in the heterogeneous half-space is v1 = EWR1 , 12 (η1 ) e− 2 J1 (Kr)eiωt , η1 = az

2γ1 K (1 + az). a

(23)

5.3. Dispersion equation Applying the boundary conditions (8-10) in Eq. (16) and Eq. (23), we get     2γK 2γK m0 m0 D1 W R2 ,0 KH + D2 W− R2 ,0 − KH = 0 , 1− 1− m0 K m0 K        2γK         2γK  R  −R + D1 − 12 mK0 W −R2 ,0 − 2γK D1 − 12 mK0 W R2 ,0 2γK m0 + 2γW 2 ,0 m0 m0 − 2γW 2 ,0 − m0       − μμ10 E − 12 Ka WR1 , 12 2γa1 K + 2γ1 W  R1 , 12 2γa1 K = 0 ,

(24)

(25)

and  D1 W R2 ,0

     2γK 2γ1 K 2γK + D2 W −R2 ,0 − − EWR1 , 12 = 0. m0 m0 a

(26)

Now expanding Whittakers function up to linear term and Eliminating D1 , D2 and E, we have 

 

 2 2   1 − cc2 mn00 KH + Nr2 exp − 1 − cc2 mn00 KH Nr1 exp μ1 1 a  γ1 0 0

 

  = + P1 2 2 μ0 4γ K γ Dr1 exp 1 − cc2 mn00 KH − Dr2 exp − 1 − cc2 mn00 KH 0

(27)

0

where Nr1 , Nr2 , Dr1 , Dr2 , and P1 are defined in Appendix A. Equation (27) is the required dispersion equation of torsional wave in an inhomogeneous crustal layer over an inhomogeneous half-space when the upper boundary plane is assumed to be rigid.

6. Particular case When the layer becomes homogeneous, i.e., m0 → 0, n0 → 0, and the half-space becomes homogeneous, i.e., a → 0, b → 0, the dispersion equation (27) reduces to   tan

c2 c20



− 1 KH =

μ0 μ1

√ 2 2 1−c c √( 2 2/ 0 ) (c /c1 −1)

  − 1 KH =

μ1 μ0

√ 2 2 c c −1 √( / 21 2 ) (1−c /c0 )



or   cot

c2 c20

which is the dispersion equation of Love type wave in a homogeneous crustal layer over a homogeneous half-space when the upper boundary plane is assumed to be rigid.

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7. Numerical computation and discussion Based on the dispersion equation (27), numerical results of phase velocity are provided to show the effect of various inhomogeneity on the propagation of torsional surface waves for the following values of elastic constants (Gubbins [15]): μ0 = 93 × 109 N/m2 , ρ0 = 7450 Kg/m3 , μ1 = 75 × 109 N/m2 , ρ1 = 4700 Kg/m3 . Effect of a linearly varying rigidity and density on torsional surface wave in an inhomogeneous crustal layer and effect of an exponentially varying rigidity and density on torsional surface wave in an inhomogeneous half space have been discussed in the following way by means of graphs. Figure 2 has been plotted for different values of m0 H, keeping n0 H,aH and bH fixed. In this figure curve 1, 2 and 3 have been drawn for m0 H = 0.20, 0.23, 0.26 respectively. Under these considered values it has been found that, for a particular dimensionless wave number (kH), the phase velocity of torsional surface wave increases, as the value of m0 H increases from 0.20 to 0.26. 3

3

1. n0H=0.4

1. m H=0.20 0

2. m0H=0.23

2.8

2. n0H=0.5

2.8

3. n H=0.6

3. m H=0.26

0

2.6

2.4

2.4

2.2

2.2 c2/c20

2.6

2

2

c /c0

0

2

1.8

2

1.8

1.6 1

2

1.6

3

1.4

1.4

1.2

1.2

1 2 3 1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 0.1

1

0.2

0.3

0.4

0.5 kH

kH

Fig. 2. Effect of Rigidity of the layer

0.7

0.8

0.9

Fig. 3. Effect of Density of the layer

3

3 1. aH=0.1 2. aH=0.2 3. aH=0.3

2.8

1. bH=0.3 2. bH=0.5 3. bH=0.7

2.8

2.4

2.2

2.2 0

2.6

2.4

c /c2

2.6

2

2

2

c2/c0

0.6

1.8

2

1.8 3 2

1.6

3

1.6

1

2

1.4

1.4

1.2

1.2

1

1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

kH

Fig. 4. Effect of Rigidity of the half-space

1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

kH

Fig. 5. Effect of Density of the half-space

In Figure 3, attempt has been made to show the effect of inhomogeneity parameter (n0 H) where curve 1, 2 and 3 have been drawn for n0 H = 0.4, 0.5, 0.6 respectively, keeping m0 H, aH and bH fixed. These curves show that the increasing values from 0.4 to 0.6 of dimensionless inhomogeneity factor (n0 H) decrease the phase velocity of torsional surface wave for a particular dimensionless wave number (kH). Figure 4 deals with the impact of inhomogeneity parameter (aH) involved in the rigidity of the inhomogeneous half space which is given by curve 1, 2 and 3 for different values of (aH) 0.1, 0.2 and 0.3 respectively. Other inhomogeneity

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Abhijit Pramanik and Shishir Gupta / Procedia Engineering 173 (2017) 964 – 971

parameters remain fixed. It is observed from the curves in Figure 4 that the phase velocity of torsional surface wave increases as the value of dimensionless inhomogeneity parameter (aH) increases from 0.1 to 0.3 for a fixed value of wave number (kH). Figure 5 discusses the influence of inhomogeneity parameter (bH) involved in the density of the inhomogeneous half space. The curves in Figure 5 have been drawn for bH = 0.3, 0.5 and 0.7 and marked as curve 1, 2 and 3 respectively, keeping m0 H, n0 H and aH fixed. It can be concluded from Figure 5 that the phase velocity of torsional surface waves increases as the value of dimensionless inhomogeneity parameter (bH) increases from 0.3 to 0.7 for a fixed value of wave number (kH). 8. Conclusions In this problem the propagation of torsional surface waves in an inhomogeneous layer of finite thickness sandwiched between a rigid layer and a vertically inhomogeneous half-space has been studied. The displacements in layer and half space have been derived separately in closed form. Asymptotic linear expansion of the Whittakers function has been used to obtain the dispersion relation in compact form for the torsional wave.Effects of dimensionless inhomogeneity parameters on the dimensionless torsional wave velocity c2 /c20 have been shown numerically in different graphs. We noticed that: (i) The phase velocity of torsional surface wave increases, as the inhomogeneity factor increases, satisfying the fact that the velocity is directly proportional to the rigidity of the medium. (ii) The increment of the dimensionless inhomogeneity factor decreases the phase velocity of torsional surface wave, following the fact that the velocity is inversely proportional to the density of the medium. (iii) The inhomogeneity parameter increases the phase velocity as it increases, validating the fact that the velocity is directly proportional to the rigidity of the half-space. (iv) The phase velocity of torsional surface wave increases as the value of dimensionless inhomogeneity parameter increases, obeying the fact that the velocity is directly proportional to the density of the half-space.

Appendix A. ⎡ ⎫   ⎧ ⎤ ⎪ ⎢⎢⎢ m   −R 1 2 −R 1 2 ⎪ −R 1 2 ⎥ ⎪ ⎪ ⎥⎥ ⎬ ⎨ − − − ) ) ) ( ( ( R 1 1 2 2 S 1 = (1 − m0 H) 1 − 2γk(1−m0 H) , T 1 = ⎢⎢⎣⎢ 4γk − 2 1 + 2 2γk2 +⎪ − 4γ k0 1 + 2 2γk2 ⎥⎥⎥⎦ ,

2 ⎪ ⎪ ⎪ 2γk ⎭ ⎩ m m m m0 0 0 0 m0 ⎡ ⎫   ⎧ ⎤    ⎪ ⎢   −R 1 2 R 1 2 R 1 2⎪ R 1 2 ⎥ ⎪ ⎪ ⎢ −R ⎨ −2) −2) −2) ⎬ −2) ⎥ ⎥⎥⎥ ⎢ ( ( ( ( m 1 R 1 0 2 2 2 2 ⎢ +⎪ 1 − 2γk S 2 = (1 − m0 H) 2 1 + 2γk(1−m0 H) , T 2 = ⎢⎢⎣ 4γk − 2 1 − 2γk

 ⎪ − 4γ k ⎥⎦ , ⎪ ⎩ 2γk 2 ⎪ ⎭ m0 m0 m0 m0 m0         1 −(R − 1 )2    ( −R − 1 )2 ( R − 1 )2 , T 4 = 1 − 2 2γk2 , P1 = 21 + 2γ1 1 ak 2γ1 k4 1 1 2 1 2 − R2γ1 ak , T 3 = 1 + 2 2γk2 m0 m0 a + 4 −(R1 − 2 ) Nr1 = S 1 T 1 , Nr2 = S 2 T 2 , Dr1 = S 1 T 3 , Dr2 = S 2 T 4 . 

R 2

( R2 − 12 )2



References [1] J. D. Achenbach, Wave Propagation in Elastic Solids, North Holland Publishing Comp., New York, (1973). [2] W. M. Ewing, W. S. Jardetzkyand F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York (1957). [3] C. Vrettos, In plane vibrations of soil deposits with variable shear modulus: I. surface waves. Int. J. Numer. Anal.Meth.Geomech.,14 (1990) 209-222. [4] B. L. N. Kennett,H. Tkali, Dynamic Earth: crustal and mantle heterogeneity. Aust. J. Earth Sci., 55 (2008) 265-279. [5] L. Rayleigh, On waves propagated along plane surface of an elastic solid. Proc. Lond. Math. Soc.,17(3) (1885) 4-11. [6] S. Dey, A. K. Gupta,S. Gupta, Torsional Surface Waves in Nonhomogeneous and Anisotropic Medium,J. Acoust. Soc. Am., 99 (1996) 2737– 2741. [7] S. Gupta,A. Chattopadhyay, S. Kundu,A. K. Gupta, Effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a heterogeneous half-space. Arch. Appl. Mech., 80 (2010) 143-150. [8] C. Davini, R. Paroni, E. Puntle, An asymptotic approach to the torsional problem in thin rectangular domains. Meccanica, 43(4) (2008) 429-435.

Abhijit Pramanik and Shishir Gupta / Procedia Engineering 173 (2017) 964 – 971 [9] S. D. Akbarov, T. Kepceler, M. M. Egilmez, Torsional wave dispersion in a finitely prestrained hollow sandwich circular cylinder. Journal of Sound and Vibration, 330 (2011) 4519-4537. [10] A. Chattopadhyay, S. Gupta, S. A. Sahu,S. Dhua, Torsional surface waves in heterogeneous anisotropic half-space under initial stress. Arch. Appl. Mech., 83 (2012) 357–366. [11] K. E. Bullen, The problem of the Earths density variation. Bull. Seismol. Soc. Am., 30(3) (1940) 235-250. [12] C. Sari, M. Salk, Analysis of gravity anomalies with hyperbolic density contrast: an application to the gravity data of Western Anatolia. J. Balkan Geophys. Soc., 5(3) (2002) 87-96. [13] A. E. H. Love, The Mathematical Theory of Elasticity, Cambridge University Press, Cambridge (1927). [14] E. T. Whittaker,G. N. Watson, A Course in Modern Analysis, Cambridge University Press, Cambridge (1990). [15] D. Gubbins, Seismology and Plate Tectonics, Cambridge University Press, Cambridge (1990).

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