influence of heterogeneity and initial stress on the propagation of rayleigh type wave in a transversely isotropic layer

Singhc a, b, c Department of Applied Mathematics, Indian Institute of Technology Indian School of Mines, Dhanbad-826004, India Abstract In this paper we study the propagation of Raylei

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Procedia Engineering 173 ( 2017 ) 988 – 995

(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.2016.12.168

ScienceDirect

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

Influence of heterogeneity and initial stress on the propagation of

Rayleigh-type wave in a transversely isotropic layer

A K Vermaa* , A Chattopadhyayb, A K Singhc

a, b, c Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, India

Abstract

In this paper we study the propagation of Rayleigh-type wave in a heterogeneous transversely isotropic elastic layer with initial stress resting on a rigid foundation Frequency equation is obtained in closed form The frequency equation being a function of phase velocity, wave number, initial stress and heterogeneous parameter associated with the rigidity and density of inhomogeneous layer reveals the fact that Rayleigh-type wave propagation is greatly influenced

by above stated parameters In Numerical and graphical computation the significant effects of distortional velocity have been carried out Moreover, the obtained dispersion relation is found in well–agreement to the classical case in isotropic and transversely isotropic layer resting on a rigid foundation

Peer-review under responsibility ofthe organizing committee of Implast 2016

Keywords: Rayleigh-type wave, transversely isotropic, initial stress, heterogeneity, rigid boundary

1 Introduction

The first theoretical investigations of Rayleigh wave carried out by Lord Rayleigh [1] in isotropic elastic media showed that these waves are particularly important in seismology since their propagation is confined to the surface, and therefore, they do not scatter in depth as seismic body waves Later, Biot [2] studied the Rayleigh wave under the influence of initial stresses Rayleigh-type waves are of importance in several fields, from earthquake seismology and geophysical exploration to material science (Parker & Maugin [3]) Explicit secular equations of Rayleigh waves

* Corresponding author Tel.: +91-8651344428

E-mail address: amitkverma.ismdhanbad@gmail.com

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of Implast 2016

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are important in practical applications Rayleigh wave propagation in layered heterogeneous media has been studied

in details by Stoneley [4] and Newlands [5] Dutta [6] illustrated Rayleigh wave propagation in a two layer anisotropic media whereas Chattopadhyay et al [7] investigated Rayleigh wave in a medium under initial stresses Recently Chatterjee et al [8] have studied Rayleigh-type wave propagation in pre-stressed heterogeneous medium

It is well known in the literature that the earth medium is not at all isotropic throughout, but it is anisotropic Transverse isotropy is generally regarded as the commonest form of anisotropic symmetry Therefore, such materials are also known as polar anisotropic materials Chattopadhyay [9] studied the strong SH motion in a transversely- isotropic layer lying over an isotropic elastic material due to a momentary point source For heterogeneous and anisotropic media the mathematical formulation of Rayleigh waves becomes very complex and there can be cases of anisotropic media where they do not exist at all However in the case of transversely isotropic medium with the free surface parallel to the isotropy plane (common situation for soil systems) Rayleigh waves exist and the analogous of the Lamb solution can be found As far as heterogeneity is concerned, the mechanical properties of the medium are often dependent on the space variable (mostly on depthy).Motivated by such facts we aimed to study the problem

in initially stressed heterogeneous transversely isotropic medium on the propagation of Rayleigh wave

Wave propagation in an isotropic surface stratum of the earth resting on an extremely rigid foundation has been investigated by Sezawa and Kanai [10] Assuming that earth crusts resting on rigid foundation, Das Gupta [11] and Dutta [12] studied on the propagation of Rayleigh waves in an isotropic layer and transversely- isotropic layer respectively resting on rigid Medium Till date no attempt has been made to study the Rayleigh-type propagation in

an initially stressed heterogeneous transversely isotropic layer with rigid base

In this paper we aim to study the propagation of Rayleigh-type wave in an initially stressed heterogeneous transversely isotropic layer in effect of rigid foundation The study of the effect of rigid foundation in layer on propagation of elastic wave has always been of great concern to civil engineers and seismologists Analytical treatment has been done to find the dispersion relation from where the real part of the expressions will give the dispersion relation of phase velocity It is found that the heterogeneity and initial stress has a significant favouring effect on phase velocity of Rayleigh-type wave when the layer has rigid base.It is also to be noted that as the phase velocity decreases with increase in heterogeneity parameter Deduced result is found in well-agreement to the established standard results existing in the literature

2 Formulation of the problem

We consider a heterogeneous initially stressed transversely isotropic layer of finite thickness Hwith rigid surface at

y H Let us choose a co-ordinate system in such a way that, y axis is directed vertically downwards and x-axis

is assumed in the direction of the propagation of Rayleigh-wave The geometry of problem is depicted in Fig.1

Fig 1: Geometry of the problem

2.1 Solution of the Problem

y

isotropic layer

x

H

O

Rigid Surface

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According to Biot (1965), in the absence of body forces, the equilibrium equations in the Cartesian coordinate system

x y z for the unbounded medium with initial stress , , S are ij

2

kj ik ij ik kj

j j

u

W

w w ª¬ º¼ w

w w w (1) where , , ,

1

2

i

ik i k k i i j

j

u

x

w i j k , , 1, 2,3.

For propagation of Rayleigh-type wave in x-direction, it is assumed that

, , , = , , , 0 and . 0

z

w {

w (2) The heterogeneity in the layer is taken as follows

C C ec D C C ec D S S ec D

(3)

In view of equations (2) and (3), equation (1) leads to

2 2 2 2

,

x x x y

x y y y

u

v

U

¾

(4)

where

,

1

2

y

x x

y

y y

y

x y

D

W

ª c w c w º

« w w »

ª c c §w w ·º

« ¨w w ¸»

(5)

Using (5) in (4) we get

w w ¬ ¼ w (6)

w c w w c w c w w

(7)

For the solution of equation (6) and (7); we consider

, ,

i x ct

K K

½

°

¾

°¿ (8) Now using equation (8) in equation (6) and (7); we get

ª c º ª¬ º¼

¬ ¼ (9)

ª c c º ª¬ c º¼

¬ ¼ (10) Now substituting

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im ky im ky im ky im ky

(11) Using (11) in equation (9) and (10); we get

1

2

im ky

c

(12)

and

1

2

im ky

(13)

Thus from equation (12) and (13), we get

,

c

,

c

3

,

m M k i L

K

c

4

m M k i L K

c

Thus

where

11

1, 2

j

m M k i L

j

m M k i L

G

c

b

And hence, we may take

, ,

im y im y im y im y i x ct

½

¾

°¿ (14) where m jj 1, 2are the roots of

2

0

j

ª

º

¼ (15) with positive real part and

j j j j j j

where

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(16)

2.2 Boundary Conditions for the proposed model are as:

i) 0 0,

) 0 0,

xy

yy

at y

W

½

°

¾

°

¿

(17)

with help of equations (8), boundary conditions (11) result in

0,

0, 0,

0

K

c c c c c c c c

(18)

Now in order to eliminate the arbitrary constants K K1, 2,K K we will have following determinant 3, 4,

(19)

where

2

Eq (19) is the required dispersion equation of Rayleigh-type wave in an initially stressed heterogeneous transversely isotropic layer resting on rigid base The dispersion relation being a function of phase velocity, wave number, initial stress and heterogeneity parameter, reveals that the fact that Rayleigh-type wave propagation is greatly influenced by above stated parameters However, under the condition when D 0 and S11 Eq (19) gives the dispersion 0, equation of Rayleigh-type wave in homogeneous transversely isotropic layer resting on rigid base and in similar fashion for C11c O 2 ,P C12c O with D 0, S11 Eq (19) gives the dispersion equation of Rayleigh-type wave 0,

in homogeneous isotropic layer resting on rigid base

3 Numerical discussion:

In this section, we carry out numerical computations and graphical demonstrations for the deduced closed form dispersion equation when Rayleigh-type wave is propagating in an initially stressed heterogeneous transversely

isotropic layer in effect of rigid foundation Graphical interpretation of phase velocity c, whereE C11

reflecting the effect of various affecting parameters, viz non-dimensional heterogeneity parameter, without heterogeneity, non-dimensional initial stress parameter ,without initial stress parameter are presented in figures 2-5 The following material constants are taken into considerations:

For transversely isotropic Beryl material: [13]

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11 2 11 2 3

Cc 26.94 10 dyne/cm , Cu c 9.61 10 dyne/cm , =2.7u U gm/cm

Fig 2: The variation of non-dimensional phase velocity c

E

§ ·

¨ ¸

Fig 3: The variation of non-dimensional phase velocity c

E

§ ·

¨ ¸

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E

§ ·

¨ ¸

S11 without heterogeneity in layer i.e D 0

E

§ ·

¨ ¸

S11 with heterogeneous layer

4 Conclusions

As the outcome of the present study, it is found that the propagation of Rayleigh-type wave is greatly influenced by the effect of various dimensionless elastic parameters, initial stress and heterogeneity factor In particular, the following conclusions can be made as

1 Wave-length has a significant favouring effect on phase velocity of Rayleigh-type wave when the layer has rigid

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base

2 Phase velocity decreases with increase in heterogeneity without initial stress with respect to wave number on the situation that layer is comprised of transversely isotropic material

3 The heterogeneity parameter affects considerably phase velocity It is found that for fixed initial stress phase velocity decreases with increase in heterogeneity parameter with respect to wave number

4 For initially stressed homogeneous transversely isotropic layer, the phase velocity decreases with increase in initial stress with respect to wave number

5 The phase velocities of Rayleigh-type wave decreases in the case when the heterogeneity parameter is fixed with increase in the value of initial stress parameter irrespective of the fact that anisotropy is present in the layer in effect

of rigid foundation

The present study may have its possible applications in the sphere of civil engineering, earthquake engineering, engineering geology and seismology

Acknowledgements

The authors convey their sincere thanks to the Indian Institute of Technology (Indian School of Mines), Dhanbad,

India, for granting access to its best research facility and providing Junior Research Fellowship to Mr Amit Kumar Verma

References

[1] L Rayleigh, on waves propagating along the plane surface of an elastic solid, Proc R Soc Lond vol 17, pp 4–11, 1885

[2] M A Biot, The influence of initial stress on elastic waves, Journal of Applied Physics, vol 11, no 8, pp 522-530, 1940

[3] D Parker, and G.A Maugin, eds Recent Developments in Surface Acoustic Waves: Proceedings of European Mechanics Colloquium 226, University of Nottingham, UK, September 2–5, 1987, vol 7 Springer Science & Business Media, 2012

[4] R Stoneley, The transmission of Rayleigh waves in a heterogeneous medium, Geophysical Supplements to the Monthly Notices of the Royal Astronomical Society, vol 3, no 6, pp 222-232, 1934

[5] M Newlands, and R Stoneley, Rayleigh waves in a two‐layer heterogeneous medium, Geophysical Journal International, vol 6, no s2, pp 109-124, 1950

[6] S Dutta, Rayleigh wave propagation in a two-layer anisotropic media, pure and applied geophysics, vol 60, no 1, pp 51-60, 1965

[7] A Chattopadhyay, N P Mahata, and A Keshri, Rayleigh waves in a medium under initial stresses Acta GeophysicaPolonica, vol 34, no

1, pp 57-62, 1986

[8] M Chatterjee, S Dhua, A Chattopadhyay, and S A Sahu, Seismic waves in heterogeneous Crust-Mantle layers under initial

stresses, Journal of Earthquake Engineering, pp 1-23, 2015

[9] A Chattopadhyay, on the strong SH motion in a transversely isotropic layer lying over an isotropic elastic material due to a momentary point source, Indian J pure appi Math, vol 9, pp 886-92, 1978

[10] K Sezawa, and K Kanai, On the Propagation of Waves along a Surface Stratum of the Earth, 1934

[11] S C D Gupta, Propagation of Rayleigh waves in a layer resting on a yielding medium Bulletin of the Seismological Society of

America, vol 45, no 2, pp 115-119, 1955

[12] S Dutta, on the propagation of Rayleigh waves in a transversely isotropic medium resting on a rigid or yielding base Geofisicapura e applicata, vol 52, no 1, pp.20-26, 1962

[13] R C Payton, Elastic wave propagation in transversely-isotropic media Springer Science & Business Media 2012

[11] S C D Gupta, Propagation of Rayleigh waves in a layer resting on a yielding medium Bulletin of. .. dispersion relation being a function of phase velocity, wave number, initial stress and heterogeneity parameter, reveals that the fact that Rayleigh- type wave propagation is greatly influenced by above... that the propagation of Rayleigh- type wave is greatly influenced by the effect of various dimensionless elastic parameters, initial stress and heterogeneity factor In particular, the following

Tiêu đề	Influence of heterogeneity and initial stress on the propagation of Rayleigh-type wave in a transversely isotropic layer
Tác giả	A. K. Verma, A. Chattopadhyay, A. K. Singh
Trường học	Indian Institute of Technology (Indian School of Mines), Dhanbad
Chuyên ngành	Applied Mathematics
Thể loại	Conference paper
Năm xuất bản	2017

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