Proceedings of ICFM 2015 International Conference on Frontiers in Mathematics 2015

March 26-28, 2015, Gauhati University, Guwahati, Assam, India Available online at http://www.gauhati.ac.in/ICFMGU

ISBN: 978-81-928118-9-5 217

Love Wave Propagation in a Porous Layer over a Prestressed Anisotropic Half Space

SHISHIR GUPTA and SMITA1 Department of Appled Mathematics

Indian School of Mines Dhanbad

1Email: smita7906@gmail.com Abstract: This paper presents a brief study of propagation of Love wave in an elastic layer with void pores over an anisotropic prestressed half space when the upper boundary of the plane is assumed to be rigid. The study reveals that such a medium transmits only one type of wave font. It is also been obtained that the wave font has an effect of void pores. Dispersion curves are plotted by taking different variations in inhomogeneity parameter. It has also been observed that dispersion curves change due to some variations in parameters. The velocities of Love wave have been calculated numerically as a function of KH where K is the wave number and H is the thickness of the layer. It introduces the characteristic of Love wave. The final result can be used to understand the nature of propagation of Love wave in a porous layer. Keywords: Love wave, Anisotropic, wave fonts 1. INTRODUCTION In the present scenario, as survey says that there are more than fifty earthquakes occurs in the whole world which can be felt locally or globally and it can be of damaging structures for entire world including nature. Each earthquake radiates seismic wave throughout the earth. Sometimes it is not possible to feel such type of radiation of seismic waves. Nonetheless we can say that Seismology is much more related to geophysics and earth science. It fascinates the theoretical problems involving analysis of seismic wave propagation in every type of media including complex media. The propagation of Love wave in different types of layered media is very important topic for seismologist because of geographical aspects of earth structure during the earth quake. A.E.H Love [1] was the man who described the mathematical model of Love waves in 1911. Ewing et al. [2] studied the propagation of Love wave in layered media in 1957. Chaottapadhyay and De [3] emphasized the propagation of Love wave in an anisotropic medium having irregularity in the layer around 1983. It is well known that the Biot equation is a central topic to seismologist for understanding the wave propagation during earthquake. At first, Biot (1965) discussed the propagation of elastic wave in an isotropic media. This contribution was great in the world of seismology. In 1983, Cowin and Nunziato [4] introduces the theory of poroelastic material. They introduce the mathematical formulation for this type of layered media. In the linear theory of material developed by Cowin and Nunziato, it has been analyzed that the bulk density can be written as a product of matrix density and volume fraction. In 2004, De and Gupta [5] discussed the propagation of Love wave in an elastic layer with void pores. Kakar and Kakar [6] have studied the propagation of Love wave in an inhomogeneous anisotropic prestressed porous layer in 2013. A. Chaottapadhyay and Singh [7] discussed the behavior of horizontally polarized shear wave due to the effect of point

source in a self reinforcement media. Shear wave is almost same as G-wave. A large duration of Love wave is known as G-wave. It takes usually 60-300 second. Now a day Propagation of Love wave due to a point source became very interesting as well as important. Many authors contributed their own idea in this area of seismology namely De Hoop [8], Brekhovskikh and Godin [9] Vrettos [10, 11], Singh [12], Deresiewiez [13] etc. Chaottapadhyay and Kar [14] discussed the nature of Love wave due to a point source in an isotropic elastic medium under initial stress. Chaottapadhyay and Chowdhary [15] studied the propagation, reflection and transmission of shear wave. Ghosh [16] studied the propagation of Love wave from the point source at the interface between upper layer and a semi infinite substratum. Nowinski [17] has shown the effect of high initial stresses on Love waves in an isotropic, elastic, incompressible medium. Gupta et al. [18] established the propagation of Love waves in non-homogeneous substratum over initially stressed heterogeneous half-space. Possibility of Love wave propagation in a porous layer under the effect of linearly varying directional rigidities was obtained by Gupta et al. [19].The propagation of Love wave in non homogeneous elastic media has been discussed by Kakar and Kakar [20] in 2012.The study of Love type wave reveals that the particle motion does not occur in vertical plane but it happens in horizontal plane. The propagation of Love wave is transverse to the direction of propagation. The early idea about the earth was related to philosophy, religion and astrology. But after some research, we have so many logical ideas about earth’s structure. In total we can say that Aristotle was responsible for the logical device called syllogism which can explain correct observations by apparently logical accounts that are based on false premises. Seismic anisotropy is the dependence of seismic wave speeds on the wave propagation in direction of the wave. As we know that the earth is comprised of layer having crust, mantle and core. It has been observed that in many regions of the earth’s interior, including the crust, mantle and upper core has the effect of seismic wave while earthquake. The earth is made up of a variety of minerals, glasses, melts, fluids etc. The earth’s composition is heterogeneous including a very hard layer. The inhomogeneous medium and the rigid interface play significant role in the propagation of seismic waves. In the present paper we have found that the propagation of Love wave is highly effected when the upper boundary of the plane is considered to be rigid. This paper discusses the influence of rigid boundary on the Love wave propagation in a layer with void pores resting over an anisotropic elastic half space. It is shown that there is possibility of exactly one wave fonts. Throughout this section It can also be analyzed that the effect of Love wave cannot be ignored in assessment of damages caused by an earthquake.

Proceedings of ICFM 2015 International Conference on Frontiers in Mathematics 2015

March 26-28, 2015, Gauhati University, Guwahati, Assam, India Available online at http://www.gauhati.ac.in/ICFMGU

ISBN: 978-81-928118-9-5 218

Construction of the problem:- Rigid Boundary Z= -H

P P 2. Formulation of the Problem: Let us consider an elastic layer of thickness H with void pores. The Z –axis is taken vertically downwards. Upper boundary plane of the layer has been considered to be rigid. Origin O is chosen at the interface between half space and upper layer. The x-axis is chosen parallel to a layer in the direction of wave propagation. The lower medium is taken as an anisotropic prestressed porous medium. The displacement components for Love waves are u=0, w=0 and v= v(x, z, t). The equation of motion under no body force can be taken as,

µ +β (1)

α - (2)

3. SOLUTION FOR THE UPPER LAYER OF THE PROBLEM: When the propagation of wave is in the positive direction of X- axis with velocity c then the solution of equation (1) and equation (2) may be assumed as,

where and satisfy the equations (3) and (4) as follows:

(3)

(4) The solution of equation (4) with M can be written as (5) where R3 and R4 are constants. Assuming the values of N, B as ,

and

, being constants for a particular material. Ignoring the

damping term which is very small for sinusoidal wave, the value of M may be taken as

Using equation (5) the solution of equation (3) may be taken as,

Hence the concerned solution of equation (1) and equation (2) is,

and

4. SOLUTION FOR THE HALF SPACE: For the Love waves propagating along the x- direction, having the displacement of particles along the y-direction, we have

displacement vectors of solid are . are

the and and ,

and

The displacement will produce only the strain components exy and eyz and the other strain components will be zero. Hence the stress strain relations give the result as follows:

and

Introducing the above relation, the equation of motion which are not automatically satisfied are,

where , is the velocity of shear

wave in the corresponding initial stress free , non porous, anisotropic, elastic medium along the direction of x, is the non dimensional parameter due to the initial stress P.

where and are the non- dimensional parameters for the material of the porous layer as obtained by Biot. For the Love wave propagating along the x- direction, the solution of equation may be taken as

Using the above equation we have,

where

and K= wave number The solution of an anisotropic prestressed layer can be taken as

Since it has been taken as half space so z tends to infinity and the solution becomes,

5. BOUNDARY CONDITIONS FOR THE PROBLEM

(i) at z= -H

(ii) at z= 0

p ∂2v∂t 2

=∂2v∂x2

+∂2v∂z2

"

#$$

%

&''

∂ψ∂x

+∂ψ∂z

"

#$

%

&'

pk ∂2ψ∂t 2

=∂2ψ∂x2

+∂2ψ∂z2

"

#$$

%

&'' w

∂ψ∂t

− ξ ψ

v = v1(z)eik (x−ct ) , ψ =ψ1(z)e

ik (x−ct )

v1(z) φ1(z)

v1"(z)− N2v1(z)+ B[ikψ1(z)+ψ1 '(z)]= 0

ψ1"(z)−M2ψ1(z) = 0,

ψ1 = R3eMz + R4e

−Mz

N = k(1− c2

Aµ2

!

"##

$

%&&)

1/2

,B = βµ

Aµ =µρ

!

"#

$

%&

1/2

M = [ αk 2 − ρk1k2c2 − iωkct +ζ ) /α( )]1/2

α,k1,ζω

M = k 1− c2

(α / ρk1)+

1k 2 (α /ζ )

!

"#

$

%&

1/2

v1 = R1eNz + R2e

−Nz −B(ik +M )M 2 − N 2

eMzR3

− B(ik −M )M 2 − N 2

e−MzR4

v = R1eNZ + R2e

−Nz −B(ik +M )M 2 − N 2

eMzR3 −B(ik −M )M 2 − N 2

e−MzR4!

"#

$

%&eik ( x−ct )

ψ = R3eMz + R4e

−Mz!"

#$e

ik (x−ct )

(ux1,uy

1,uz1) Ux

1,Vy1,Wz

1( )Ux1 = 0, Vy

1 =V 1(x, z,t) Wz1 = 0 ux

1 = 0

vy1 = v1(x, z,t) wz

1 = 0

σ121 = 2Nexy σ 23

1 = 2Leyz

β1 = N − P / 2( ) /ζ = β0 1−b( ) /ζ

ξ = γ11 −γ122

γ22

"

#$$

%

&'' β0 = N / P

b = P '/ 2N

γ11 = ρ11 / ρ ',γ12 = ρ12 / ρ ' γ11,γ12 γ11

v1(x, z,t) =V (z)eik ( x−ct )

d 2Vdz2

+ t 2V = 0

t 2 = K2

Lc2ξ − N − P '/ 2( )!"

#$,γ = N / L,b = P '/ 2N

v2 = R4eitz + R5e

−itz( )eik (x−ct )

v2 = R5eitz .eik (x−ct )

v1 = 0

v1 = v2

Porous  Layer   Z=0  

Inhomogeneous Anisotropic prestressed Half Space

X-axis

Z-axis

Proceedings of ICFM 2015 International Conference on Frontiers in Mathematics 2015

March 26-28, 2015, Gauhati University, Guwahati, Assam, India Available online at http://www.gauhati.ac.in/ICFMGU

ISBN: 978-81-928118-9-5 219

(iii) at z= -H where denotes the

shearing stress in the direction of z-axis. (iv)

(v) at z= 0.

Using the above equations, the boundary conditions gives

We observe that the above system is a homogeneous algebraic linear system, So we will have a nontrivial system if and only if its characteristic determinant is zero.

So Eliminating R1, R2, R3, R4 and R5 from the above equations we get,

By solving determinant, we get

This implies

(6)

where

and X is a dimensionless quantity. From (5) we get,

where

and

, velocity of the shear wave due to change in

void volume fraction in the layer

NUMERICAL CALCULATIONS The numerical value of have been calculated for different value of kH taking some sets of values of (km)1. The results are presented in figure. The figures show that the velocity of Love wave fonts which depend on change in volume of void pores is many times higher than the velocity of waves carrying a change in void fraction of porous at low value of kH.

Figure 2. Love wave dispersion curve in an elastic medium with void pores for a second set of parameters.

Figure 3. Love wave dispersion curve in elastic medium with void pores for a set of parameters in 2D graphics.

Figure 4. Love wave dispersion curve in elastic medium with void pores for a set of parameters in 2D graphics. CONCLUSION To this end we can observe that

(i) There is exactly one Love wave font is available in this medium and the wave font depends on the change in void volume fraction of pores.

(τ yz )1 = 0 τ yz

n∇ψ( ) = 0

1 2( ) ( )yz yzτ τ=

R1e−N1H + R2e

N1H −B(ik+M)M 2 − N 2

e−M1HR3 −B(ik −M )M 2 − N 2

eM1HR4 = 0

R1 + R2 −B(ik +M )M 2 − N 2

R3 −B(ik −M )M 2 − N 2

R4 − R5 = 0

R1e−N1H + R2e

N1H = 0, R3e−M1H + R4e

M1H = 0

µ1 N1R1 − N1R2 −B(ik +M )M 2 − N 2

M1R3 +B(ik −M )M 2 − N 2

M1R4!

"#

$

%&+µ2itR5 = 0

e−N1H eN1H −B(ik +M )M 2 − N 2

e−M1H −B(ik −M )M 2 − N 2

eM1H 0

1 1 −B(ik +M )M 2 − N 2

−B(ik −M )M 2 − N 2

−1

e−N1H eN1H 0 0 00 0 e−M1H eM1H 0

N1 −N1−B(ik +M )M 2 − N 2

M B(ik −M )M 2 − N 2

M µ2µ1it

= 0

µ2µ1.t.M1 sinh(N1H )+ k.N1 cosh(N1H ) = 0

cot(N1H ) = X .µ2µ1

M1

N1

X =KL

N − P '2

"

#$

%

&'− c2ζ

"

#$$

%

&''

cot 1− c2

Aµ2

!

"##

$

%&&

1/2

kH =µ2µ1.X .

1− c2

c32+

1

km( )12

!

"

###

$

%

&&&

1/2

1− c2

Aµ2

!

"##

$

%&&

1/2

(km)12 = k 2 α1 /ζ1( )

1/2

Aµ = µ1 / ρ1( )1/2

B = β1 / µ1( )1/2

c3 = α1 / (ρ1k1)( )1/2

c2 / c32

Proceedings of ICFM 2015 International Conference on Frontiers in Mathematics 2015

March 26-28, 2015, Gauhati University, Guwahati, Assam, India Available online at http://www.gauhati.ac.in/ICFMGU

ISBN: 978-81-928118-9-5 220

(ii) The velocity of Love wave fonts of the second type is much more than the velocity of wave due to the change in the void volume fraction.

REFERENCES [1]. Love, A.E.H., Some Problems of Geodynamics,

Cambridge University press, 1911 [2]. Ewing WM, Jardetzky WS, Press F 1957 Elastic

Waves in Layered (New York: McGraw Hill) [3]. A. Chaottapadhyay and R.K. De, “Love type waves

in a porous layer with irregular interfaces,” Int J. Engg Sci, vol. 21, no.11, pp.1295, 1983

[4]. Cowin S C, Nunziato J W 1983 Linear elastic materials with voids. J. Elasticity 13: 125

[5]. Dey S, Gupta S, Gupta A K, 2004 Propagation of Love waves in an elastic layer with void pores. Sadhana Vol.29, Part 4, August 2004, pp.355-363

[6]. Kakar R, Kakar S, 2013 Love waves in inhomogeneous irregular anisotropic prestressed porous layer, Int J. Of Appl. Math and Mech.9 (17):85-101.

[7]. A. Chaottapadhyay, A.K. Singh, G-type Seismic waves in reinforced media, Meccanica47 (2012) 1775-1785

[8]. A.T. De Hoop, “Handbook of radiation and scattering of Waves: Acoustic Waves in Fluids, Electromagnetic Waves, “Academic Press, London, 1995

[9]. L.M. Brekhovskikh and O.A. Godin, “Acoustics of Layered Media,” Springer- Verlag, Berlin, 1992.

[10]. C. Vrettos, “Forced Anti- Plane Vibrations at the surface of an inhomogeneous Half Space,” Solid Dynamics and Earthquake Engineering, Vol. 10, No. 5, 1991,pp. 230-235, doi: 10.1016/0267-7261(9)90016-S

[11]. C. Vrettos, “ The Boussinesq problem for Soil with Bound Nonhomogeneity,” International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 22, No. 8, 1998, pp. 655-669

[12]. K. Singh, “Love wave due to a point source in an Axially Symmetric Heterogeneous Layer between two homogeneous half spaces, “Pure and Applied Geophysics, Vol. 72, No.1, 1969, pp.-35-44

[13]. H. Deresiewich,” A note on Love Waves in

Homogeneous crust overlying an inhomogeneous Substratum,” Bulletin of Seismological Society of America, Vol. 52, 1996, pp. 639-645

[14]. A. Chaottapadhyay and B.K. Kar, “ Love wave due to a point source in an Isotropic Elastic Medium under init al stress, “International Journal of Non-Linear Mechanics, Vol. 16, No. 3-4, 1981, pp. 247-258

[15]. A. Chaottapadhyay and S. Chowdhary, ”Magneto elastic Shear Waves in an infinite Self reinforced Plate,” International Journal of Numerical and Analytical methods in Geomechanics, Vol. 19, No.4, 1995, pp. - 289-304

[16]. M. L. Ghosh, “Love wave Due to a Point source in an Inhomogeneous Medium, ”Gerlands Beirtage Zur Geophysik, Vol. 70, 1970, pp. 319-342

[17]. J.L. Nowinski, The effect of high initial stress on the propagation of love wave in an isotropic elastic incompressible medium, some aspects of mechanics of continua, (a book dedicated as a tribute to the memory of B. Sen.) part I, pp. 14-28, 1977.

[18]. Gupta, S., Majhi, D. K., Kundu S. and Vishwakarma, S. K., “Propagation of Love waves in non- Homogeneous Substratum over initially stressed Heterogeneous Half Space,” Applied math. Mech. - Engl. Ed., 2013, 34(2), 249-258

[19]. Gupta, S., Vishwakarma, S. K., Majhi, D. K. and Kundu, S., “Possibility of Love wave Propagation in a porous layer under the effect of linearly varying directional rigidities”, Applied math. Modell, 2013

[20]. Kakar R and Kakar S(2012).Propagation of Love waves in a non homogeneous elastic media, J. Acad. Indus. Res.,1(6),pp.323-328