Dispersion of shear wave propagating in vertically heterogeneous double layers overlying an initially stressed isotropic half-space

Soil Dynamics and Earthquake Engineering 69 (2015) 16–27

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Dispersion of shear wave propagating in vertically heterogeneous double layers overlying an initially stressed isotropic half-space Abhishek Kumar Singh, Amrita Das n, Amares Chattopadhyay, Sudarshan Dhua Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, Jharkhand, India

art ic l e i nf o

a b s t r a c t

Article history: Received 29 March 2014 Received in revised form 14 October 2014 Accepted 25 October 2014

The present paper investigates the propagation of horizontally polarised shear wave in distinct vertically heterogeneous double layers overlying an isotropic half-space under horizontal initial stress. The vertical heterogeneity in the uppermost layer is caused due to quadratic variation only in rigidity, whereas vertical heterogeneity in the sandwiched layer is caused due to exponential variation in rigidity and density both. The closed form of velocity equation is obtained which leads to the dispersion equation as its real part and damping equation as its imaginary part. The validation of dispersion relation with the classical case is made by using Debye asymptotic expansion which is the major highlight of this study. The significant effect of the width ratio of the layers, heterogeneity parameters of both the layers and horizontal compressive/tensile initial stress on the phase velocity and damped velocity of SH-wave have been traced out. The comparative study and some important peculiarities have been revealed by means of graphical illustrations. & 2014 Elsevier Ltd. All rights reserved.

Keywords: SH-wave Debye asymptotic expansion Initial stress Heterogeneity Dispersion Seismic wave

1. Introduction The crust is relatively more heterogeneous than mantle, which makes the study of wave propagation much practical considering the heterogeneous layers. There are different sort of vertical heterogeneity persist in crustal layers in the form of exponential function, linear function, quadratic function etc. The study of wave propagation in layered elastic media with different boundaries helps to understand and predict the seismic behaviour at the different margins of earth, which makes it applicable in the field of geophysics, civil, mechanical, and other engineering branches. Many researchers had widely studied the theory of Love wave propagation in a medium where the velocity, rigidity and density are functions of depth. Kar [1] worked on the propagation of Lovetype waves in a non-homogeneous internal stratum of finite thickness lying between two semi-infinite isotropic media. Love waves in different heterogeneous layered media were studied by Gogna [2]. Scattering of SH-waves in multi-layered media with irregular interfaces have been discussed by Ding and Dravinski [3]. Chattopadhyay et al. [4] discussed the propagation of shear waves in viscoelastic medium at irregular boundaries. Chattopadhyay et al. [5] described the effects of point source and heterogeneity on the propagation of SH-waves. Guz [6] has analysed the three-

n

Corresponding author. E-mail addresses: [email protected] (A.K. Singh), [email protected] (A. Das). http://dx.doi.org/10.1016/j.soildyn.2014.10.021 0267-7261/& 2014 Elsevier Ltd. All rights reserved.

dimensional linearised theory of elastic waves propagating in initially stressed solids. He formulated surface waves along planar and curvilinear boundaries and interfaces, waves in layers and cylinders, waves in composite materials, waves in hydroelastic systems, and dynamic problems for moving loads. Chattopadhyay [7] discussed the propagation of SH-waves in a sandwiched heterogeneous layer lying between two semi-infinite homogeneous elastic media where the heterogeneity in the sandwiched layer was taken in the form of linearly varying function of depth in the rigidity and density was kept constant. Bhattacharya [8] discussed the possibility of the propagation of Love-type waves in an intermediate heterogeneous layer where the inhomogeneity was assumed in two different forms; the first form dealt with the exponential variation in rigidity and the second was a linear variation in both rigidity and density. Dutta [9,10] discussed two problems relating to the propagation of Love-type waves in a nonhomogeneous internal stratum lying between two semi-infinite homogeneous elastic media. Due to the presence of many physical factors, a large amount of initial stress evolves in a medium which have a pronounced influence on the propagation of waves as shown by Biot [11]. These factors may be overburden layer, variation in temperature, slow process of creep, gravitational field, etc. The Earth is a highly initially stressed medium. Dey and Addy [12] have shown the effect of initial stresses on the propagation of Love waves by considering the layer and the half-space to be isotropic elastic in one case and visco-elastic in another case. Gupta [13] studied the propagation of Love waves in a non-homogeneous substratum

A.K. Singh et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 16–27

17

over an initially stressed heterogeneous half-space. Propagation of Love waves in a non-homogeneous orthotropic elastic layer under changeable initial stress overlying semi-infinite medium was given by Abd-Alla and Ahmed [14]. Keeping in mind, the existence of different types of heterogeneity in the crustal layers and motivated by the fact that earth is an initially stressed body, we considered the present problem with a distinct geometry and heterogeneity configuration not attempted till now. This problem studies the SH-waves propagating in double layers of finite width having different sort of heterogeneity overlying an initially stressed isotropic half-space. The closed form of velocity equation is obtained which leads to the dispersion equation as its real part and damping equation as its imaginary part. The validation of dispersion relation with the classical case is made by using Debye asymptotic expansion which is the major highlight of this study. The width ratio and heterogeneity parameters of the layers, horizontal compressive initial stress and horizontal tensile initial stress are found to have a significant effect on the phase velocity and damped velocity of SHwaves. The obtained dispersion relation is found to be in well agreement with the classical Love-wave equation. Comparative study and graphical illustration has been made to reveal the some of the important facts.

Now, for the propagation of SH-wave in x-direction and causing displacement only in the y-direction, we have the displacement components as

2. Formulation and solution of the problem

∇2 v ¼

In the present paper we consider two isotropic heterogeneous layers (M 1 and M 2 ) lying over an initially stressed isotropic homogeneous half-space (M 3 ) as shown in Fig. 1. Let us consider x-axis in the direction of wave propagation and along the common interface of medium M 2 and M 3 . The z-axis of the rectangular co-ordinate system is pointing vertically downwards. The rigidity of the uppermost isotropic layer is a quadratic function of depth, whereas the density is constant. The rigidity and density of the sandwiched layer are varying exponentially with depth.

u ¼ 0;

w¼0

and

v ¼ vðx; z; tÞ

Let us consider μ and ρ be the rigidity and density of the medium respectively. In the absence of body forces, the only non-vanishing equation of the motion [11] for the propagation of SH-wave is given by ∂ ∂ ∂2 v pxy þ pyz ¼ ρ 2 ; ∂x ∂z ∂t

ð2:1Þ

where pxy ¼ μ

∂v ∂x

and pyz ¼ μ

∂v : ∂z

For a homogeneous medium, where μ is constant i.e. independent of the space variables, the equation of motion for propagation of SH-wave wave in an isotropic homogeneous medium is given by 1 ∂2 v ; β2 ∂t 2

where ∇2 

∂2 ∂2 þ 2 2 ∂x ∂z

and

μ β2 ¼ : ρ

Fig. 1. Geometry of the problem.

ð2:2Þ

18

A.K. Singh et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 16–27

If waves propagate in the direction of x-axis with velocity c, we put vðx; z; tÞ ¼ V ðzÞeikðx  ctÞ :

ð2:3Þ

So by using Eq. (2.2) in Eq. (2.3), we have ∂2 V þ s2 V ¼ 0; ∂z2 where s2 ¼ k

!

c2

2

β

ð2:4Þ

ð2:6Þ

which is the equation of motion for propagation of SH-wave wave in an isotropic heterogeneous medium. ffiffiμffi Þ in Eq. (2.6), we get Putting vðx; z; t Þ ¼ V ðpx;z;t !  2 2 1 dμ 1d μ ∂2 V μ∇2 V þ  ð2:7Þ V ¼ρ 2 2 4μ dz 2 dz ∂t We assume the solution of the form V ¼ Z ðzÞe

ð2:8Þ

1

ρ01

Substituting 1 þ δz ¼ ζ in Eq. (2.9), we have dζ

ð2:10Þ

where a ¼ k c2 =β1 δ and b ¼  k =δ . λ Again using Z ¼ ζ Z 1 in Eq. (2.10) we get 2

ζ2

η2

2

d Z1

2

2

2

2

2 2

k c

1 γ ¼ 2 2 : β δ 4 1

! 1  1  σ2 4 β2 c2

2

and

β2 ¼

rffiffiffiffiffiffiffi

ρ02 : μ02

therefore 1 ffi½A2 cos m2 z þ B2 sin m2 zeikðx  ctÞ v2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi μ02 eσz

ð2:17Þ

2.3. Solution of the lowermost half-space ðM 3 Þ; The lowermost half-space is isotropic with horizontal initial stress P. The equation of motion for propagation of SH-wave in this medium is given by   P ∂2 v3 ∂ 2 v3 ∂ 2 v3 μ3  þ μ3 2 ¼ ρ3 2 : ð2:18Þ 2 2 ∂x ∂z ∂t Considering v3 ðx; z; tÞ ¼ Z 3 ðzÞeikðx  ctÞ , Eq. (2.18) reduces to " # 2 ∂2 Z 3 2 c þ k þ ξ  1 Z 3 ¼ 0; ð2:19Þ ∂z2 β2 3

þ2λζ

d Z1 dZ 1 þ2η þ½γ  η2 Z 1 ¼ 0; dη dη2

2

m22 ¼ k

ð2:16Þ

where

dZ 1 2 þ ½λðλ  1Þ þ a þ bζ Z 1 ¼ 0: ð2:11Þ dζ pffiffiffi Choosing λ ¼ 1=2 and putting bζ ¼ iη in Eq. (2.11), we have

dζ

where 2

2 2

which can be rewritten as

Z 2 ¼ A2 cos m2 z þ B2 sin m2 z:

where rffiffiffiffiffiffiffiffi μ01 : β1 ¼

2

In sandwiched layer we have considered the rigidity as μ2 ¼ μ02 eσ z and the density as ρ2 ¼ ρ02 eσ z ; where σ is the heterogeneity parameter with dimension inverse of length, μ02 and ρ02 are the constants. In view of above, using Z ¼ Z 2 (for M 2 ) Eq. (2.7) leads to " # 2 d Z2 ρ02 k2 c2 2 1 2 þ  k  σ Z 2 ¼ 0; ð2:15Þ 4 μ02 dz2

The solution of the Eq. (2.16) is given by

In this medium we have considered heterogeneity in rigidity in  2 and the density as a quadratic fashion i.e. μ1 ¼ μ01 1 þ δz constant, i.e.ρ1 ¼ ρ01 , where δ is a constant with dimension inverse of length and μ01 being the constant. In view of above, Eq. (2.7) takes the form " # 2 2 2 c 2d Z 2 þk  ð1 þ δzÞ Z ¼ 0; ð2:9Þ ð1 þ δzÞ dz2 β2

þ ða þ bζ ÞZ ¼ 0; 2

2.2. Solution of the sandwiched layer ðM 2 Þ;

where

2.1. Solution of the uppermost layer ðM 1 Þ;

2

ð2:14Þ

2

Now, let us assume ui ; vi and wi as the components of displacement in x; y and z-direction for medium M 1 ; M 2 and M 3 for i¼1, 2 and 3 respectively.

d Z

δ

d Z2 þ m22 Z 2 ¼ 0; dz2

;

which leads Eq. (2.7) to " #   2 d Z ρk2 c2 2 1 dμ 2 1 d2 μ þ  k þ  Z ¼ 0: 2μ dz2 μ dz2 4μ2 dz

ζ2

δ

ð2:5Þ

dμ ∂v ∂2 v ¼ ρ 2; dz ∂z ∂t

ikðx  ct?Þ

ð2:13Þ

where A1 and B1 are arbitrary constants; I and K are modified Bessel function of the first and the third kind respectively, both being of order iγ . Hence the solution of the uppermost layer is    1=2        1 þ δz k k v1 ¼ 1 þ δ z þ B 1 K iγ 1 þ δz eikðx  ctÞ : A1 I iγ pffiffiffiffiffiffiffiffi

μ01

1 : 2

If μ is a function of space variable z only, then the Eq. (2.1) becomes

μ∇2 v þ

The solution of Eq. (2.12) is given by     Z 1 ¼ A1 I iγ η þ B1 K iγ η ;

ð2:12Þ

ξ¼

P : 2μ3

Eq. (2.19) can be rewritten in the form 2

∂ Z3  m23 Z 3 ¼ 0; ∂z2 where m23

2

¼k

1

c2

β23

ð2:20Þ !

ξ :

A.K. Singh et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 16–27

n

Keeping in mind that SH-wave dies out with increase in depth, the appropriate solution of Eq. (2.20) can be written as Z 3 ¼ A3 e

 m3 z

;

v3 ¼ A3 e  m3 z eikðx  ctÞ :

ð2:22Þ

The boundary conditions are as follows:  2 1 ðiÞ μ01 1 þ δz ∂v ∂z ¼ 0 at z ¼  H 2 ;

The real part of velocity Eq. (2.28) is a dispersion equation whereas the imaginary part will lead to the damping equation. Taking the right hand side of Eq. (2.28) as R1 , the above equation can be written as

δ

μ01 1 þ δz

ðiiiÞ

v1 ¼ v2

ðivÞ

μ02 eσ z ∂v∂z2 ¼ μ3 ∂v∂z3 at z ¼ 0 ;

1

∂z

¼ μ02

∂z

at

v2 ¼ v3

at

z ¼  H1 ;

z¼0:

   3=2 pffiffiffiffiffiffiffiffi 0 k    δpffiffiffiffiffiffiffiffi 1  δH 1 k μ01 K iγ 1  δH 1  μ01 1  δH 1 1=2 K iγ 2 δ   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin  σ k  1  δH 1 ðA2 cos m2 H 1  B2 sin m2 H 1 Þ ¼ μ02 e  σ H1

þ B1

δ

2

þ m2 ðA2 sin m2 H 1 þ B2 cos m2 H 1 Þg;



R1 ¼

2

iγ δ

1  δH 2

k 1δ



 

H 2 K 0iγ δk

1  δH 1

o

   

 2    k δ k 1  δH 1  1  δH 1 I iγ 1  δH 1  1  δH 1 kI 0iγ δ 2 δ    

   0 k  δ k 1  δH 2  k 1  δH 2 I iγ 1  δH 1  I iγ  2 δ δ n 2 0  k   δ   k o  1  δH 1 kK iγ δ 1  δH 1  2 1  δH 1 K iγ δ 1  δH 1

hn

       o k  δK k 1  δH  k 1  δH 2 K 0iγ δk 1  δH 1 I iγ δ 1  δ H 2 2 2 iγ δ



¼

           δ k k k I 1  δH 2  k 1  δH 2 I 0iγ 1  δH 1 1  δH 2 K iγ 2 iγ δ δ δ

μ02  σ H1 e μ01

1  δH 1



k 1  δH 1

δ

K iγ

k

δ

1  δH 2



 I iγ

  k 1  δH 2 K 0iγ

δ

¼ 0;

ð2:29Þ

μ02 e  σ H1

n

  σ o  σ2 cos m2 H1 þ m2 sinm2 H 1 þ 2m  μμ3 mm32 σ2sinm2 H1 þ m2 cosm2 H1 2 02

  : μ01 cos m2 H1  2mσ 2  μμ3 mm32 m2 sinm2 H 1

In light of above relation Eq. (2.29) can be written in the form  δ     k 2 1  δ H 2 1  δ H 2 þ R1 k 1  δ H 2 

             J iγ p 1  δH 1 Y 0iγ p 1  δH 2  J 0iγ p 1  δH2 K iγ p 1  δH 1

δ2 

 δ 1  δ H 1 þ R1 4 2



!

       J iγ p 1  δH 1 Y iγ p 1  δH 2

       J 0iγ p 1  δH 1 Y iγ p 1  δH 2 2       ¼ 0;  J iγ p 1  δH 2 K 0iγ p 1  δH 1

ð2:25Þ

Eliminating A1 ; A2 ; A3 ; B1 ; B2 from Eqs. (2.23)–(2.27) we get 

δ



I 0iγ

δ



 2       2  k 1  δH 2 1  δH 1  J iγ p 1  δH 2 K iγ p 1  δH 1

             J 0iγ p 1  δH 1 Y 0iγ p 1  δH 2  J 0iγ p 1  δH2 K 0iγ p 1  δH 1

ð2:27Þ 

2  k

2



I γ ðzÞ ¼ e  1=2ðπγ iÞ J γ ðizÞ; h i K γ ðzÞ ¼ 12 π ie1=2ðπγ iÞ J γ ðizÞ þ iY γ ðizÞ :

      A1 I iγ δk 1  δH 1 þ B1 K iγ δk 1  δH 1

pffiffiffiffiffiffiffiffi A2 ¼ μ02 A3 : k

1  δH 1

δ

 

Now, using the following relations [15]:

þ

and

δK

δ



02

ð2:24Þ

1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½A2 cos m2 H 1  B2 sin m2 H 1 ; μ02 e  σ H1 o pffiffiffiffiffiffiffiffin  σ A2 þm2 B2 ¼  μ3 m3 A3 μ02 2

n

δ



where

n 3=2 pffiffiffiffiffiffiffiffi 0 k  pffiffiffiffiffiffiffiffi 1=2 k o 1  δH 1 k μ01 I iγ δ 1  δH 1  2δ μ01 1  δH 1 I iγ δ 1  δ H 1

 1=2

δ

δ

Using Eqs. (2.14), (2.17) and (2.22) in the boundary conditions, we obtain the following equations: n    o A1 2δI iγ δkð1  δH 2 Þ kð1  δH 2 ÞI 0iγ δkð1  δH 2 Þ ð2:23Þ B1 ¼ n k   o ; 0 k δ 2K iγ δð1  δH 2 Þ  kð1  δH 2 ÞK iγ δð1  δH 2 Þ

μ01

δ

             k k k k  I0 1  δH 1 K 0iγ 1  δH 2 I 0iγ 1  δH 2 K 0iγ 1  δH 1

2.5. Dispersion equation

δH 1 Þ ð1  p ffiffiffiffiffiffi

4 

δ

z ¼ H 1 ;



A1

δ 

δ

       k k 1  δ H 1 K iγ 1  δH 2  I iγ

 δ 1  δH 1 þR1 2 δ δ     2   k k 2 1  δH 2 K iγ 1  δH 1  k 1  δH 2 1  δH 1  I iγ

þ

ðvÞ

δ

!

2



eσ z ∂v2

ðiiÞ

at

ð2:28Þ

 δ     k2 1  δH 2 1  δH 2 þ R1 k 1  δH 2              k k k k 1  δH 1 K 0iγ 1  δH 2 I 0iγ 1  δ H 2 K iγ 1  δH 1  I iγ

2.4. Boundary conditions

2 ∂v

  σ o μ m   σ2 cos m2 H 1 þ m2 sinm2 H 1 þ 2m  μ 3 m32 σ2sinm2 H 1 þ m2 cosm2 H 1 2 02

  ; σ  μ3 m3 m sinm H cos m2 H 1  2m 2 2 1 μ m2 2 02

ð2:21Þ

which finally gives





19

δ

1  δH 1

2

where p¼

ik

δ

:

For small δ and γ 2 4 0 ; it follows:  2 c2 δl 4 ; 4π β2 where l ¼ 2π =k is the wavelength. Hence, for large wavelength c 4 β 2 : Now δð1  δH 1 Þ ffi β2 o 1; k

γ

k

δγ

ffi

c

β2 c

o 1;

ð2:30Þ

20

A.K. Singh et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 16–27

δð1  δH 2 Þ ffi β2 o 1 k

γ

c

and k

δγ

ffi

β2 c

o 1:

Now, we use Debye asymptotic expansions of J γ ðγ sec αÞ; Y γ ðγ sec αÞ ; in which the argument is less than the order, both being large [15] eγ ðtanα  αÞ J γ ðγ sec αÞ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 2πγ tanh α

eγ ðtanα  αÞ Y γ ðγ sec αÞ   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πγ tanh α

ð2:31Þ

with this expansion we have             J iγ p 1  δH 1 Y 0iγ p 1  δH 2  J 0iγ p 1  δH 2 K iγ p 1  δH 1  sinh ϕ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi πγ tanh ϕ1 tanh ϕ2   expðγ ðϕ1  ϕ2 Þ  γ ðtanh ϕ1  tanh ϕ2 ÞÞ  þ expð  γ ðϕ1  ϕ2 Þ þ γ ðtanh ϕ1  tanh ϕ2 ÞÞ ; 

ð2:32Þ

            J iγ p 1  δ H 1 Y iγ p 1  δ H 2  J iγ p 1  δ H 2 K iγ p 1  δ H 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  πγ tanh ϕ1 tanh ϕ2    expðγ ðϕ1  ϕ2 Þ  γ ðtanh ϕ1  tanh ϕ2 ÞÞ þ expð  γ ðϕ1  ϕ2 Þ  ð2:33Þ þ γ ðtanh ϕ1  tanh ϕ2 ÞÞ ; 

Fig. 2. Dimensionless phase velocity against dimensionless wave number when H2 =H 1 ¼ 1:5; σH 1 ¼ 1:2; ξ ¼ 0:1:

            J 0iγ p 1  δH 1 Y 0iγ p 1  δH 2  J 0iγ p 1  δH 2 K 0iγ p 1  δH 1 sinh ϕ1 sinh ϕ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi πγ tanh ϕ1 tanh ϕ2   expðγ ðϕ1  ϕ2 Þ  γ ðtanh ϕ1  tanh ϕ2 ÞÞ  expð  γ ðϕ1  ϕ2 Þ  ð2:34Þ þ γ ðtanh ϕ1  tanh ϕ2 ÞÞ ; 

            J 0iγ p 1  δH 1 Y iγ p 1  δH 2  J iγ p 1  δH 2 K 0iγ p 1  δH 1 sinh ϕ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi πγ tanh ϕ1 tanh ϕ2   expðγ ðϕ1  ϕ2 Þ  γ ðtanh ϕ1  tanh ϕ2 ÞÞ þ expð  γ ðϕ1  ϕ2 Þ  ð2:35Þ þ γ ðtanh ϕ1  tanh ϕ2 ÞÞ ; 

where   p 1  δH 1 ¼ ν sec ϕ1 ;   p 1  δH 2 ¼ ν sec ϕ2 ;

Fig. 3. Dimensionless phase velocity against dimensionless wave number when H2 =H 1 ¼ 1:5; δH1 ¼ 0:1; ξ ¼ 0:1:

3 sffiffiffiffiffiffiffiffiffiffiffiffiffi2 k c2 δ ðH 1  H 2 Þ5 4 ; ν tanh ϕ1 ffi i  1 1 þ c2 δ β22 2 1

sinh ϕ2 ffi

β2

3 sffiffiffiffiffiffiffiffiffiffiffiffiffi2 k c2 δ ðH 1  H 2 Þ5 4 ; ν tanh ϕ2 ffi i  1 1 þ c2 δ β22 2 1

sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 1 : 2

β2

Thus, it follows from Eq. (2.30) that

β2

ϕ1  ϕ2 ffi

tan kðH 1  H 2 Þ

δ ðH 1  H 2 Þ c ; c2 β2 2 1 β2







ν ϕ1  ϕ2  ν tanh ϕ1  tanh ϕ2 sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 1 sinh ϕ1 ffi 2

β1

and



sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 1 ; ffi ikðH 1 H 2 Þ 2

β2

¼

sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 1 2

β2

 δ  k2ð1  δH 1 Þð1  δH 2 Þ þ R1 kð1  δH2 Þ sinh ϕ1 þ kδ2ð1  δH1 Þ2 sinh ϕ2 kδ 1k  4p ð1  δH 1 Þ þ R2p  ð1  δH 1 Þ2 ð1  δH2 Þsinh ϕ1 sin ϕ2

:

Since k sinh ϕ1 ¼ s1 and k sinh ϕ1 ¼ s2 , we get the following form: tan ðH 1  H 2 Þs2 ¼

δ



δH1 Þð1  δH2 Þ þ R1 ð1  δH 2 Þ s1 þ 2δð1  δH1 Þ2 s2 ; kδ 1k  4p ð1  δH 1 Þ þ R2p  ð1  δH 1 Þ2 ð1  δH 2 Þs1 s2

2ð1 

ð2:36Þ

A.K. Singh et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 16–27

Fig. 4. Dimensionless phase velocity against dimensionless wave number when H 2 =H1 ¼ 1:5; δH 1 ¼ 0:1; σH1 ¼ 0:1:

Fig. 5. Dimensionless phase velocity against dimensionless wave number when H 2 =H1 ¼ 1:5; δH 1 ¼ 0:1; σH1 ¼ 0:1:

which is the dispersion equation for SH-wave propagating in distinct vertically heterogeneous double layers overlying an initially stressed isotropic half-space.

02

Fig. 6. Dimensionless phase velocity against dimensionless wave number when δH1 ¼ 0:1; σH 1 ¼ 0:1; ξ ¼ 0:1:

Fig. 7. Dimensionless phase velocity against dimensionless wave number when δH1 ¼ 0:1; σH 1 ¼ 0:1; ξ ¼ 0:

which is the dispersion equation of SH-wave propagating in a homogeneous layer lying over an exponentially varying heterogeneous sandwiched layer and an initially stressed half-space. Case II. When σ -0, the dispersion Eq. (2.36) takes the form δ  ð1  δH 1 Þð1  δH 2 Þ þ R2 ð1  δH 2 Þ s1 þ 2δð1  δH 1 Þ2 s2 tan ðH 1  H 2 Þs2 ¼ 2 ; kδ 2k  4p ð1  δH 1 Þ  R2p  ð1  δH 1 Þ2 ð1  δH 2 Þs1 s2

3. Particular cases

Case I. When δ-0, the dispersion Eq. (2.36) reduces to   tan ðH 1  H 2 Þs2 ¼ μ02 e  σ H1  σ2 cos m2 H 1 þ m2 sinm2 H 1

 o σ  μ3 m3 σ þ 2m μ02 m2 2sinm2 H 1 þ m2 cosm2 H 1 2

  μ01 s2 2mσ 2  μμ3 mm32 m2 sinm2 H 1  cos m2 H 1 ;

21

ð2:38Þ where ð2:37Þ

R2 ¼

μ02 sn2 sin s2 H1  μ3 m3 cos s2 Ho1 : μ01 cos s2 H 1 þ μμ302ms23 sin s2 H1

22

A.K. Singh et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 16–27

Fig. 8. Dimensionless phase velocity against dimensionless wave number when δH1 ¼ 0:1; σH 1 ¼ 0; ξ ¼ 0: Fig. 10. Dimensionless phase velocity against dimensionless wave number and initial stress when δH1 ¼ 0:1; σH 1 ¼ 0:1; H2 =H1 ¼ 1:5:

Fig. 9. Dimensionless phase velocity against dimensionless wave number when δH1 ¼ 0; H 2 =H1 ¼ 1:5; σH 1 ¼ 0; ξ ¼ 0:

Eq. (2.38) is the dispersion equation of SH-wave propagating in a quadratically varying heterogeneous layer lying over a homogeneous sandwiched layer and an initially stressed half space. Case III. When ξ-0; Eq. (2.36) reduces to δ  ð1  δH 1 Þð1  δH 2 Þþ R3 ð1  δH 2 Þ s1 þ 2δð1  δH 1 Þ2 s2 tan ðH 1  H 2 Þs2 ¼ 2 ; kδ 3k  4p ð1  δH 1 Þ  R2p  ð1  δH 1 Þ2 ð1  δH 2 Þs1 s2

Fig. 11. Dimensionless phase velocity against dimensionless wave number and initial stress when δH1 ¼ 0; σH 1 ¼ 0; H 2 =H 1 ¼ 1:5:

ð2:39Þ where R3 ¼

μ02 e  σH1

n

  σ o s3 σ  σ2 cos m2 H 1 þ m2 sinm2 H1 þ 2m  μμ3 m 2 sinm2 H 1 þ m2 cosm2 H 1 2

 02 2  : s3 m2 sinm2 H 1 μ01 cos m2 H 1  2mσ 2  μμ3 m 2 02

Eq. (2.39) is the dispersion equation of SH-wave propagating in double layered heterogeneous media with different sort of heterogeneity lying over a half-space without initial stress.

Case IV. When ξ-0 ; δ-0 and σ -0; dispersion Eq. (2.36) takes the form  μ s2 μ s2 sin s2 H 1  μ3 s3 cos s2 H 1 ; ð2:40Þ tan ðH 1 H 2 Þs2 ¼ 02  02 μ01 s3 μ02 s2 cos s2 H 1  μ3 s3 sins2 H1 which is the dispersion equation of SH-wave propagating in double homogeneous layers overlying a half-space under no initial stress.

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Fig. 12. Dimensionless phase velocity against dimensionless wave number and heterogeneity parameter of the sandwiched layer when σH1 ¼ 0:1; ξ ¼ 0:1; H2 =H 1 ¼ 1:5:

Fig. 13. Dimensionless phase velocity against dimensionless wave number and heterogeneity parameter of the sandwiched layer when σH 1 ¼ 0; ξ ¼ 0; H 2 =H1 ¼ 1:5:

Case V. When ξ-0; δ-0; σ -0and H 1 -H 2 dispersion Eq. (2.36) reduces to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi μ3 1  c2 =β23 c2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð2:41Þ 1 ¼ tan kH β22 μ02 c2 =β22  1 which is the classical Love wave equation.

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Fig. 14. Dimensionless phase velocity against dimensionless wave number and heterogeneity parameter of the uppermost layer when σH 1 ¼ 0:1; ξ ¼ 0:1; H2 =H 1 ¼ 1:5:

Fig. 15. Dimensionless phase velocity against dimensionless wave number and heterogeneity parameter of the sandwiched layer when σH1 ¼ 0; ξ ¼ 0; H2 =H 1 ¼ 1:5:

4. Numerical calculation and discussions We consider the following data [16] for the numerical calculation and graphical illustration of the dimensionless phase velocity of SH-wave propagating in two distinct isotropic heterogeneous layers of finite width possessing different sort of vertical heterogeneity and which is lying over an initially stressed isotropic

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Fig. 18. Dimensionless damped velocity against dimensionless wave number when H2 =H 1 ¼ 1:5; σH 1 ¼ 1:2; ξ ¼ 0:1:

Fig. 16. Dimensionless phase velocity against dimensionless wave number and width ratio when δH 1 ¼ 0:1; ξ ¼ 0:1; σH 1 ¼ 0:1:

Fig. 19. Dimensionless damped velocity against dimensionless wave number when δH1 ¼ 0:1; H 2 =H1 ¼ 1:5; ξ ¼ 0:1:

ρ3 ¼ 3535 Kg=m3 ; ðlowermost initially stressed isotropic homogeneous half  spaceÞ Moreover, we consider following data:

σ H1 ¼ 0:1; 1:2; 1:3; 1:5; 1:7; ξ ¼ 0; 7 0:05; 7 0:1 :

Fig. 17. Dimensionless phase velocity against dimensionless wave number and width ratio when δH 1 ¼ 0; ξ ¼ 0; σH 1 ¼ 0:

H 2 =H 1 ¼ 1:50; 1:53; 1:56;

homogeneous half-space

The variation of the heterogeneity parameters of the uppermost layer and sandwiched layer on the dispersion curves has been shown in Figs. 2 and 3. It is clear from these figures that as the heterogeneity parameter increases dispersion curve shifts downward, i.e. phase velocity decreases as heterogeneity grow in either of the layers. The effect of compressive initial stress and tensile initial stress on dispersion curves has been shown in Figs. 4 and 5 respectively. Curve 1 in Figs. 4 and 5 corresponds

μ01 ¼ 3:23  1010 N=m2 ; ρ01 ¼ 2802 Kg=m3 ; ðuppermost heterogeneous layerÞ μ02 ¼ 6:54  1010 N=m2 ; ρ02 ¼ 3409 Kg=m3 ; ðsandwiched heterogeneous layerÞ μ3 ¼ 7:84  1010 N=m2 ;

δH1 ¼ 0:1; 0:11; 0:12;

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Fig. 20. Dimensionless damped velocity against dimensionless wave number when δH 1 ¼ 0:1; σH1 ¼ 1:2; H 2 =H1 ¼ 1:5:

Fig. 21. Dimensionless damped velocity against dimensionless wave number when δH 1 ¼ 0:1; σH1 ¼ 1:2; H 2 =H1 ¼ 1:5:

to the case of no initial stress whereas curves 2 and 3 correspond to increasing order of horizontal compressive initial stress and horizontal tensile initial stress respectively. Fig. 4 manifest that as horizontal compressive initial stress increases phase velocity increases, whereas Fig. 5 suggests that as horizontal tensile initial stress increases, phase velocity decreases. The variation of width ratio on phase velocity has been shown in Figs. 6–8. Fig. 6 corresponds to the variation of width ratio when double layers are heterogeneous and overlying an initially stressed half-space. Fig. 7 demonstrates the variation of width ratio when double layers are heterogeneous and overlying a half-space without initial stress. Fig. 8 depicts the variation of width ratio when the uppermost layer is heterogeneous but sandwiched layer is homogeneous and both these layers are overlying a half-space without

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Fig. 22. Dimensionless damped velocity against dimensionless wave number when δH1 ¼ 0:1; σH 1 ¼ 1:2; ξ ¼ 0:1:

initial stress. From Figs. 6–8 it can be adduced that as width ratio increases phase velocity decreases. Fig. 9 corresponds to the case of double homogeneous layers lying over a half-space without initial stress. The comparative study of Figs. 6–9 concludes that as we get rid of the heterogeneity in the layers and initial stress in the half-space, phase velocity gets decreased. Surface plots in Figs. 10 and 11 gives the variation of dimensionless phase velocity against dimensionless wave number and initial stress when both the layers are heterogeneous and homogeneous respectively. Figs. 12 and 13 are the surface plots depicting the variation of dimensionless phase velocity against dimensionless wave number and heterogeneity parameter of the sandwiched layer for the cases when heterogeneous uppermost layer with lowermost initially stressed half-space and uppermost homogeneous layer with lowermost half-space under no initial stress respectively. Surface plot in Figs. 14 and 15 demonstrate the variation of dimensionless phase velocity against dimensionless wave number and heterogeneity parameter of the uppermost layer for the cases when the sandwiched layer is heterogeneous with initially stressed lowermost half-space and sandwiched layer is homogeneous layer with lowermost half-space under no initial stress respectively.Surface plot in Figs. 16 and 17 show the variation of dimensionless phase velocity against dimensionless wave number and width ratio for the cases when the double layers are heterogeneous lying over an initially stressed lowermost halfspace and double layers are homogeneous lying over a half-space under no initial stress respectively. For the graphical illustration of damped velocity we take imaginary part of velocity Eq. (2.28) into account. Figs. 18–24 deals with the dimensionless damped velocity against dimensionless wave number for different values of affecting parameter namely, viz. δH 1 ; σ H 1 ; ξ; H 2 =H 1 : It is clear from these figures that damped velocity after reaching a maximum value decreases abruptly and finally dies out. Figs. 18 and 19 depict the variation of heterogeneity parameters of the uppermost layer and the sandwiched layer on damped velocity respectively. It is observed in both the figures that damped velocity decreases with the increase in heterogeneity parameters δH 1 and σ H 1 , i.e. heterogeneity in the layers affect adversely to the damped velocity. The

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5. Conclusion The present paper deals with the propagation of SH-wave in distinct vertically heterogeneous double layers overlying an initially stressed isotropic half-space. The heterogeneity in the uppermost layer is caused due to quadratic variation in rigidity only, in terms of space variable pointing vertically downward whereas the heterogeneity in the sandwiched layer is caused due to the exponential variation in rigidity as well as density in terms of space variable pointing vertically downward. The half-space is isotropic and under a horizontal initial stress. The closed form of velocity equation is obtained which leads to the dispersion equation as its real part and damping equation as its imaginary part. The validation of dispersion relation with the classical case is made by using Debye asymptotic expansion which is the major highlight of this study. It is observed that wave number, width ratio of the layers, horizontal compressive/tensile initial stress, heterogeneity parameters of the uppermost layer and sandwiched layer have a significant effect on the phase velocity and the damped velocity of SH-wave. The following outcomes can be accomplished through this study: Fig. 23. Dimensionless damped velocity against dimensionless wave number when δH1 ¼ 0:1; σH 1 ¼ 1:2; ξ ¼ 0:

(i) Wave number affects phase velocity substantially. More precisely, phase velocity and damped velocity decreases with increase in wave number. (ii) As heterogeneity grows in the uppermost layer and sandwiched layer it affects the damped velocity and phase velocity adversely. (iii) The horizontal compressive initial stress has a favouring effect on phase velocity whereas tensile initial stress has an adverse effect on phase velocity. The trend on the effect of horizontal compressive initial stress and horizontal compressive initial stress on the damped velocity is just opposite to the phase velocity. (iv) The width ratio of the layers has a considerable effect on the phase velocity as well as damped velocity. More precisely, both velocity decreases with increase in width ratio. (v) The obtained dispersion relation is in well agreement with the classical Love wave equation. (vi) Following validity condition is in agreement to our study of propagation of SH-wave in the said geometry.

β1 o β2 o c o β3 or β2 o β1 o c o β3 : The above condition concludes that phase velocity of SH-wave must be less than the shear wave velocity in lowermost half-space and greater than the shear wave velocity in both the layers. Fig. 24. Dimensionless damped velocity against dimensionless wave number when δH1 ¼ 0:1; σH 1 ¼ 0; ξ ¼ 0:

effects of horizontal compressive initial stress and horizontal tensile initial stress on the damped velocity have been shown in Figs. 20 and 21. It is clear from these figures that damped velocity decreases with increase in horizontal compressive initial stress whereas it increases with horizontal tensile initial stress. Figs. 22–24 shows the variation of width ratio on damped velocity for different cases. Fig. 22 corresponds to the variation of width ratio on damped velocity when the double layers are heterogeneous with lowermost half-space under initial stress. Fig. 23 demonstrates the variation of width ratio on damped velocity when the double layers are heterogeneous with lowermost half-space under no initial stress. Fig. 24 reveals the variation of width ratio on damped velocity when the uppermost layer is heterogeneous but the sandwiched layer is homogeneous and both are lying over a half-space under no initial stress.

Acknowledgement The authors convey their sincere thanks to Indian School of Mines, Dhanbad, India to facilitate us with its best facilities. References [1] Kar BK. On the Propagation of Love-type waves in a non-homogeneous internal stratum of finite thickness lying between two semi-infinite isotropic media. Gerlands Beitr Geophys 1977;86(5):407–12. [2] Gogna ML. Love waves in heterogeneous layered media. Geophys JR Astr Soc. 1976:357–70. [3] Ding G, Dravinski M. Scattering of SH-waves in multi-layered media with irregular interfaces. Earthq Eng Struct Dyn 1998;25(12):1391–404. [4] Chattopadhyay A, Gupta S, Sharma VK, Kumari P. Propagation of shear waves in viscoelastic medium at irregular boundaries. Acta Geophys 2009;58 (2):195–214. [5] Chattopadhyay A, Gupta S, Sharma VK, Kumari P. Effect of point source and heterogeneity on the propagation of SH-waves. Int J Appl Math Mech 2010;6 (9):76–89.

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