Journal of Environmental Science, Computer Science and Engineering & Technology

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JECET; December 2012 –February 2013; Vol.2, No.1, 57-69.

Journal of Environmental Science, Computer Science and Engineering & Technology

Available online at www.jecet.org Engineering & Technology Research Article

Magnetoelastic Surface Waves Propagation in Non- Homogeneous Viscoelastic Media of Higher Order

Rajneesh Kakar*¹ and Shikha Kakar²

1Principal, DIPS Polytechnic College, Hoshiarpur, Punjab, India.

2Faculty of Electrical Engineering, SBBSIET Padhiana, Jalandhar, Punjab, India.

Received: 8 December 2012; Revised: 24 December 2012 Accepted: 26 December 2012

Abstract:The aim of the present paper is to investigate the surface waves in a non- homogeneous, isotropic, viscoelastic solid medium of n^th order including time rate of strain. The theory of generalized surface waves has firstly been developed and then it has been employed to investigate particular cases of waves, viz., Stoneley, Rayleigh and Love type. The wave velocity equations have been obtained for different cases and are in well agreement with the corresponding classical result, when the effect of viscosity, magnetism as well as non-homogeneity of the material medium are ignored.

Keywords: Inhomogeneous media, Variable density, Surface waves, Magnetic field, Voigt equation

INTRODUCTION

The theory of viscoelasticity is of interest in the broad field of solid mechanics and of special interest in seismology, engineering, exploration geophysics, and acoustics for consideration of wave propagation in layered media with arbitrary amounts of intrinsic absorption, ranging from low-loss models of the deep Earth to moderate-loss models for soils and weathered rock.

Studies of wave propagation in the earth stratum under loads have usually assumed that the earth behaves to a first approximation as an ideal elastic or viscoelastic material. The stratura may be of finite depth or it may be so deep compared to the size of the loaded area that it can be regarded as a half-space. In either case the complete solutions to elastic or viscoelastic problems are known when material parameters are treated independent of position.

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The formation of earth state in nature tends, however, to result in depth variations of these parameters and this may be due principally, either to stratification of different materials or to the effect of superincumbent pressure. The case of homogeneous viscoelastic half-space has been treated extensively in the literature.

By comparison little has been done on the non-homogenous viscoelastic half-space when the material characteristics vary continuously with depth. This fact is likely to be true when the effects of over-burden pressure predominate. Furthermore these studies have been restricted to particular variations which have allowed the resulting simplification to be exploited.

The dynamic theory of thermo-viscoelasticity is the study of dynamical interaction between thermal and mechanical fields in solid bodies and is of great importance in various fields of engineering such as earthquake engineering, aeronautics, soil engineering etc. As a result, Stoneley¹, Bullen^2, Ewing et al.³, Hunters⁴Jones⁵and Jeffreys⁶ have developed the theory of surface waves.

The effect of gravity on wave propagation in an elastic solid medium was first considered by Bromwich⁷, treating the force of gravity as a type of body force. Love⁸ extended the work of Bromwich⁷ investigated the influence of gravity on super-facial waves and showed that the Rayleigh wave velocity is affected by the gravity field. Sezawa⁹ studied the dispersion of elastic waves propagated on curved surfaces.

Haskell¹⁰ studied the dispersion of surface waves in multilayered media. A source on elastic waves is the monograph of Ewing et al.³. Biot¹¹studied the influence of gravity on Rayleigh waves, assuming the force of gravity to create a type of initial stress of hydrostatic nature and the medium to be incompressible. De and Sengupta¹² studied many problems of elastic waves and vibrations under the influence of gravity field.

Sengupta and Acharya¹³ studied the influence of gravity on the propagation of waves in a magnetoelastic layer. Brunelle¹⁴studied the surface wave propagation under initial tension of compression. Lamb¹⁵ discussed the waves in elastic plate. Wave propagation in a thin two-layered laminated medium with stress couples under initial stresses was studied by Roy¹⁶. Datta¹⁷ studied the effect of gravity on Rayleigh wave propagation in a homogeneous, isotropic elastic solid medium. Goda¹⁸ studied the effect of non- homogeneity and anisotropy on Stoneley waves. The details are found in the work of Eringen and Sahubi¹⁹. Sharma and Kaur²⁰ studied Rayleigh waves in rotating thermoelastic solids with voids.

Chattopadhyay et al.^{21, 22} studied the propagation of G-type seismic waves in viscoelastic medium, they also discussed the effect of point source, and heterogeneity on the propagation of SH- waves. Abd-Alla and Ahmed²³ studied the Rayleigh waves in an orthotropic magneto-elastic medium under gravity field and initial stress. Paria²⁴ discussed love waves in granular medium. Recently, Kakar et al.²⁵ studied the surface wave propagation in non homogeneous, general magneto-thermo, viscoelastic media.

In this work, the problem of n^th order viscoelastic surface waves involving time rate of strain, the medium being isotropic and non-homogeneous has been studied under the influence of temperature. Biot’s theory of incremental deformations has been used to obtain the wave velocity equation for Stoneley, Rayleigh and Love waves. Further, these equations are in complete agreement with the corresponding classical results in the absence of viscosity and thermal field, non-homogeneity of the material medium.

FORMULATION OF THE PROBLEM

Let M1 and M2 be two non-homogeneous, viscoelastic, isotropic, semi-finite media. They are perfectly welded in contact to prevent any relative motion or sliding before and after the disturbances and that the continuity of displacement, stress etc. hold good across the common boundary surface. Further the mechanical properties of M1 are different from those of M2. These media extend to an infinite great distance from the origin and are separated by a plane horizontal boundary and M2 is to be taken above M1. Let Oxyz be a set of orthogonal Cartesian co-ordinates and let O be the any point on the plane boundary and Oz point vertically downward to the medium M1. We consider the possibility of a type of wave

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traveling in the direction Ox, in such a manner that the disturbance is largely confined to the neighborhood of the boundary, which implies that wave, is a surface wave.

It is assume that at any instant, all particles in any line parallel to Oy having equal displacement and all partial derivatives with respect to y are zero. Further let us assume that u, v, w is the components of displacements at any point (x, y, z) at any time t.

The dynamical equations of motion for three-dimensional non-homogeneous, isotropic, viscoelastic solid medium in Cartesian co-ordinates are

13

11 12

x y z

∂τ

∂τ ∂τ

∂

⁺

∂

⁺

∂

⁼

2 2

u t

ρ ∂

∂

, (1a)

12 22 23

x y z

∂τ

∂τ ∂τ

∂

⁺

∂

⁺

∂

⁼

2 2

v t

ρ ∂

∂

, (1b)

13 23 33

x y z

∂τ ∂τ ∂τ

∂

⁺

∂

⁺

∂

⁼

2 2

w t

ρ ∂

∂

, (1c) Where, ρ be the density of the material medium andτ_ij = τ_jiV i, j are the stress components.

The problem is dealing with magnetoelasticity. Therefore, the basic equations will be electromagnetism and elasticity. The Maxwell equations of the electromagnetic field in a region with no charges (ρ = 0) and no currents (J = 0), such as in a vacuum, are

∇ ⋅Ε =0, ur ur

∇ ⋅Β =ur ur 0,

t ,

∇× Ε = −∂Β

∂ ur ur uur

0 0 .

μ ε

^∂Εt

∇×Β =

∂ ur ur uuur

(2a)

Where,Εur ,Βur

,

μ

₀and

ε

₀are electric field, magnetic field induction, permeability and permittivity of the vacuum. For vacuum,

μ

₀=4

π

×10⁻⁷and

ε

₀=8.85 10× ⁻¹²in SI units. These equations lead directly to Εur and Βur

satisfying the wave equation for which the solutions are linear combinations of plane waves traveling at the speed of light,

0 0

1 .

c⁼

μ ε

In addition, Εur

and Βur

are mutually perpendicular to each other to the direction of wave propagation.

Also, the term Ohm's law is used to refer to various generalizations. The simplest example of this is:

, J =

σ

E

ur ur

(2b) Where, J

ur

is the current density at a given location in a resistive materialΕur

is the electric field at that location, and σ is a material dependent parameter called the conductivity. If an external magnetic field induction Βur

is present and the conductor is not at rest but moving at velocityV

ur

, then an extra term must be added to account for the current induced by the Lorentz force on the charge carriers.

( ) (

v

).

J E V B E B

σ σ ^∂

t

= + × = + ×

∂ ur ur ur ur ur ur

(2c) We consider an orthotropic elastic solid under constant primary magnetic fieldH₀acting on y-axis.

Hi= (H0)i + hi. (2d) The stress-strain relations for general isotropic, magneto, viscoelastic medium, are

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τij= 2Dμ eij + (DλΔ+ 0²

D

m

H _μD e) δij (3) where,

Δ= u v w

x y z

∂ ∂ ∂

∂

⁺

∂

⁺

∂

^, D^λ^, D^μ are elastic constants

Introducing eq. (3) in eqs (1a), (1b), (1c), we get

2

2 2

2 m

2 2

m

0 0

2 2

D D ^e

e

D u u D u w

D D D

x x x x x z z x u

D t

u w D

H D H

z x z x x

λ μ

λ μ μ

μ

∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ ∂

∂ ∂

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

⎧ Δ+ Δ + + + ⎡ + ⎤⎫

⎪ ⎢⎣ ⎥⎦⎪

⎪ ⎪ =

⎨ ⎬

⎡ ⎤

⎪+⎢ + ⎥ + ⎪

⎪ ⎣ ⎦ ⎪

⎩ ⎭

(4a)

2 D D

D v

x x z z

μ μ

μ

∂ ∂

∂ν ∂ν

∂ ∂ ∂ ∂

∇ + + =

2

t2

ρ ∂ ν

∂

, (4b)

2

2 2

2 m

2 2

0 m 0

2

2 D D ^e

e

u w u w D w

D D

w

x z x z x x z

D t

D D

w D H D H

z z z z z z

μ

μ μ

μ λ

λ

∂ ∂ ∂ ∂ ∂ ∂ ∂

∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ

∂ ∂

∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

⎧ ⎛ + ⎞ ⎛+ + ⎞ + ⎫

⎪ ⎜ ⎟ ⎜ ⎟ ⎪

⎪ ⎝ ⎠ ⎝ ⎠ ⎪=

⎨ ⎬

⎪+ + Δ + Δ + ⎪

⎪ ⎪

⎩ ⎭

(4c)

We assume that the non-homogeneities for the media M1 and M2 are given by

Dλ =

0 n K

mz

K K

K

e t

λ ∂

=

∂

∑

^,^{Dμ =}

0 n K

mz

K K

K

e t

μ ∂

=

∂

∑

^, ^m

0

e

( )

n K

mz

e K K

K

D e

t

μ ∂

=

∂

= ∑

ρ = ρ0 emz (5a) D'λ= 0

'

n K

z

K K

K

e t

λ ∂

=

∂

∑

^l ^,Dμ =

0

'

n K

z

K K

K

e t

μ ∂

=

∂

∑

^l ^,

0

' ( ' )

e n K

mz

e K K

K

D e

μ t

μ ∂

=

∂

= ∑

, ρ' = ρ'0elz (5b)

whereλ0, M0, λ'0, μ'0are elastic constants, ρ0, ρ'0, and m, n are constants. λK, μ

K (K = 0, 1, 2 ... n) are the parameters associated with K^thorder viscoelasticity and (μ_e)_K (K = 1, 2, ..., n) is the magnetic parameters associated with K^th order.

(

⁰² ^me

)

²

u w

G G H G G u m G

x z x

λ μ μ μ

∂ ∂ ∂

Δ ⎛ ⎞

+ + + ∇ + ⎜⎝ + ⎟⎠⁼

2

0 2

u t

ρ ∂

∂

, (6a)

Gμ ∇2 v + mGμ v z

∂

⁼

2

0 2

v t

ρ ∂

∂

, (6b)

(Gλ + Gμ + H G0² me) z

∂

Δ+ Gμ∇2 w + Δ Gλ m + 2Gμ m w z

∂

^‐

2 m 0

D

mH G e=

2

0 2

w t

ρ ∂

∂

. (6c) where,

Gλ =

0 n K

K K

K t

λ ∂

=

∂

∑

^,Gμ =

0 n K

K K

K t

μ ∂

=

∂

∑

^,

0

e

( )

n K

e K K

K

G^μ t

μ ∂

=

∂

= ∑

, (7a)

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∇2 = ²₂ ²₂

x z

∂ ∂

∂

⁺

∂

^. (7b) To investigate the surface wave propagation along the direction of Ox, we introduce displacement potentialφ (x, z, t) and ψ (x, z, t) which are related to the displacement components as follows:

u= x z

∂φ ∂ψ

∂ ⁻ ∂

^{, w =} z x

∂φ ∂ψ

∂ ⁺ ∂

. (8) Substituting eq. (8) in eqs (6a), (6b) and (6c), we get

GR∇2φ + mGs 2

z x

∂φ ∂ψ

∂ ∂

⎛ + ⎞

⎜ ⎟

⎝ ⎠⁼

2

t2

∂ φ

∂

^,

(9a)

G_S∇2 v + mG_S v z

∂

⁼

2 2

v t

∂

, (9b)

G_S∇2ψ + mG_P x

∂φ

∂

^+ 2m Gs ^z

∂ψ

∂

⁼

2

t2

∂ ψ

∂

, (9c) Where,

2

UKR=

2 0 0

2 (m )

K K H e K

λ μ

ρ

+ +

, U_KS² =

0

μ

K

ρ

^,

2

UKP =

2 0 0 K H (m )e K

λ ρ

+ , (10)

G_R= ²

0 n K

KR K

K

U t

∂

=

∂

∑

, GS = ²

0 n K

KS K

K

U t

∂

=

∂

∑

^, GP = ²

0 n K

KP K

K

U t

∂

=

∂

∑

. (11) Further, similar relations in medium M2 can be found out by replacing λ_K, μ

K, ρ0 by λ'K, μ'

K, ρ'

0and so on.

SOLUTION OF THE PROBLEM

Now our main objective to solve eqs (9a), (9b), (9c) and (11).

For this, we seek the solutions in the following forms.

(φ, ψ, v)= [f (z), g (z), h (z)] eiα(x – ct) (12) Using eq. (12) in eqs (9a), (9b), (9c) and eq. (11), we get a set of differential equations for the medium M1 as follows:

2

2 2 2

1 1 1

2 2

d f df

mf h f i mf g

dz + dz+ +

α

= 0, (13a)

2 2

2 1

d h dh

m K h

dz + dz+ = 0, (13b)

2 2 2

1 1

2 2

d g dg

m K g i ml f

dz + dz + +

α

= 0, (13c)

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2 2

2

B d f f

dz

α

⎛ ⎞

⎜ − ⎟

⎝ ⎠= 0, (13d) where,

f12 =

( )

2 0

n K

KS K

n K

KR K

U i c

α α

=

−

∑

^, h12 =

⁽ ⁾

2 2

0

n K

KR K

c

U i c

α α

α

=

−

∑ −

^, K12=

⁽ ⁾

2 2

0

n K

KS K

c

U i c

α α

α

=

−

∑ −

^, l12=

( )

2 0

n K

KP K

n K

KS K

U i c

α α

=

−

∑

^, g12 =

( )

2 0

n K

KL K

n K

KR K

U i c

α α

=

−

∑

, B=0. (14)

and those for the medium M2 are given by

2

'2 '2

1 1

2 2

d f df

lf h f

dz + dz + +i

α

l f g₁^'2 = 0, (15a)

2

'2 2 1

d h dh

l K h

dz + dz + = 0, (15b)

2

'2 '2

1 1

2 2 .

d g dg

l K g i l l f

dz + dz + +

α

= 0, (15c)

2 2

' d f2

B f

dz

α

⎛ ⎞

⎜ − ⎟

⎝ ⎠= 0, (15d) where,

f1'2=

( )

'2 0

n K

KS K

n K

KR K

U i c

α α

=

−

∑

^, h1'2 =

⁽ ⁾

2 2

2 '2

0

n K

KR K

c

U i c

α α

α

=

−

∑

− ^, K1'2 =

⁽ ⁾

2 2

2 '2

0

n K

KS K

c

U i c

α α

α

=

−

∑

− ^, l1'2=

( )

'2 0

n K

KP K

n K

KS K

U i c

α α

=

−

∑

^, g1'2 =

( )

'2 0

n K

KL K

n K

KR K

U i c

α α

=

−

∑

, B' = 0 (16) Eqs (13) and (15) must have exponential solutions in order that f, g, T1, h will describe surface waves, and they must become varnishing small as z → ∞.

Hence for the medium M1

φ (x, z, t)=

{

^{A e}¹ ⁻^λ¹^z

⁺

^{B e}¹ ⁻^λ²^z

⁺

^{C e}¹ ⁻^λ³^z

}

^eⁱ^α⁽^{x ct}⁻ ⁾

ψ (x, z, t)=

{

^{A e}² ⁻^λ¹^z

⁺

^{B e}² ⁻^λ²^z

⁺

^{C e}² ⁻^λ³^z

}

^eⁱ^α⁽^{x ct}⁻ ⁾

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T (x, z, t)=

{

^{A e}³ ⁻^λ¹^z

⁺

^{B e}³ ⁻^λ²^z

⁺

^{C e}³ ⁻^λ³^z

}

^eⁱ^α⁽^{x ct}⁻ ⁾

v (x, z, t) = C e⁻^λ⁴^{z i}⁺^α⁽^{x ct}⁻ ⁾ (17a) and similarly for the medium M2 are given by

φ (x, z, t)=

{

^{A e}

^'

¹ ⁻^λ^'¹^z

⁺

^{B e}

^'

¹ ⁻^λ^'²^z

⁺

^{C e}

^'

¹ ⁻^λ^'³^z

}

^eⁱ^α⁽^{x ct}⁻ ⁾

ψ (x, z, t)=

{

^{A e}

^'

² ⁻^λ^'¹^z

⁺

^{B e}

^'

² ⁻^λ^'²^z

⁺

^{C e}

^'

² ⁻^λ^'³^z

}

^eⁱ^α⁽^{x ct}⁻ ⁾

T (x, z, t)=

{

^{A e}

^'

³ ⁻^λ^'¹^z

⁺

^{B e}

^'

³ ⁻^λ^'²^z

⁺

^{C e}

^'

³ ⁻^λ^'³^z

}

^eⁱ^α⁽^{x ct}⁻ ⁾

v (x, z, t)= C e' ⁻^λ^'⁴^{z i}⁺^α⁽^{x ct}⁻ ⁾ (17b) Whereλj and λ'j(j = 1, 2, 3) are the real roots of the eqs

λ6 + ξ1λ5 + ξ2λ4 + ξ3λ3 + ξ4λ2 + ξ5λ + ξ6 = 0, (18) Where,

ξ1 = 2m {1 + f12}, ξ2 = K12 + A + 4m2 + h12 + Bg12, (19a)

ξ3 = 2mA + 2f12 m (K12 + A) + 2mh12 + 2mBg12 (19b) ξ4 = AK12 + 4m2A f12 + (K12 + A) h12 + α2 m2l12 f12 + BK12 g12 – α2Bg12, (19c)

ξ5 = 2mAK12 f12 + 2mAh12 – 2m α2Bg12, (19d) ξ 6 = AK12 h12 + Aα2 m2 ll2 f12 – α2B K12 g12. (19e) λ'6 + ξ'1λ'5 + ξ'2λ'4 + ξ'3λ'3 + ξ'4λ'2 + ξ'5λ' + ξ'6 = 0 (20) Where,

ξ'1 = 2l {1 + f1'2}, ξ'2 = K1'2 + A' + 4l2 + h1'2 + B'g1'2, (21a) ξ'3 = 2lA' + 2lf1'2 (K1'2 + A) + 2lh1'2 + 2l B'g1'2, (21b) ξ'4= A'K1'2 + 4l2A' f1'2 + (K1'2 + A') h1'2 + α2l2l1'2 f1'2 + B' K1'2 g1'2 – α2 B' g1'2, (21c) ξ'5 = 2lA'K1'2 f1'2 + 2lA'h1'2 – 2lα2B'g1'2, (21d)

ξ'6 = A'K1'2 h1'2 + A'α2l2 ll'2 f1'2 – α2B' K1'2 g1'2. (21e) λ4 = {m + (m²‐ 4 K12)^½}/ 2, (21f)

λ'4 = {l + (l²‐ 4 K1'2)^½}/ 2. (21g) Where the symbol used in eqs (19) and (21) are given by eqs (14) and (16).

The constants Aj, Bj, Cj (j = 1, 2, 3) are related with A'j, B'j, C'j (j = 1, 2, 3) in eqs (17a) and (17b) by means of first equations in eqs (13) and (15).

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Equating the coefficients of e⁻^λ¹^z, e⁻^λ²^z,e⁻^λ³^z,e⁻^λ^'¹^z, e⁻^λ^'²^z,e⁻^λ^'³^zto zero, after substituting eqs (17a) and (17b) in eq. (13a) and (13c) and eq. (15) respectively, we get

A2= γ1 A1, B2 = γ2 B1, C2 = γ3 C1, (22a) A3= δ1 A1, B3 = δ2 B1, C3 = δ3 C1, (22b) where,

γj= 2 ¹² 2

2 1

j j

i m l

m K

α

λ λ

−

− + (j = 1, 2, 3), δj= 2 1

1

g [λj2 – 2m f12λj + h12 + i α m f12γj], j = 1, 2, 3. (23) Similar result holds for medium M2 and usual symbols replacing by dashes respectively.

BOUNDARY CONDITIONS

(i) The displacement components at the boundary surface between the media M1 and M2 must be continuous at all times and positions.

i.e.

[ ]

, , _M1

u

ν

w =

[ ]

, , _M2

u

ν

w

(ii) The stress components τ31, τ32, τ33 must be continuous at the boundary z = 0.

i.e.

[ τ τ τ

31

,

32

,

33

]

_M₁=

[ τ τ τ

31

,

32

,

33

]

_M₂at z = 0 respectively where,

τ31=D 2 ² ²2 ²2

x z x z

μ

∂ φ ∂ ψ ∂ ψ

∂ ∂ ∂ ∂

⎛ ⎞

+ −

⎜ ⎟

⎝ ⎠, (24a) τ32 =D

μ z

∂ν

∂

, (24b)

τ33 =

2 2

2 2 2

m 0

2 2 _B _e

D D D T D H

z x z

λ

φ

μ

∂ φ ∂ φ φ

∂ ∂ ∂

⎛ ⎞

∇ + ⎜ + ⎟− + ∇

⎝ ⎠ . (24c) Applying the boundary conditions, we get

A1 (1 – i γ1ζ1) + B1 (1 – i γ2ζ2) + C1 (1 – i γ3ζ3) – A'1 (1 – i γ'1ζ'1) – B'1 (1 – i γ'2ζ'2) – C'1 (1 – i γ '3ζ'3) = 0

(25a)

C = C' (25b) A1 (γ1 + iζ1) + B1 (γ2 + iζ2) + C1 (γ3 + iζ3) – A'1 (γ'1 + iζ'1) – B'1 (γ '2 + iζ'2) – C'1 (γ'3 + iζ'3) = 0 (25c) δ1A1 + δ2 B1 + δ3C1 = δ'1A'1 + δ'2 B'1 + δ'3C'1 (25d) pλ1δ1A1 + pλ2δ2 B1 + pλ3δ3C1 – p' λ'1δ'1A'1 + p' λ'2δ'2 B'1 – p'λ'3δ'3C'1 = 0 (25e)

m*_K[(2i ζ1 + γ1 + ζ12γ1) A1 + (2i ζ2 + γ2 + ζ22γ2) B1 + (2i ζ3 + γ3 + ζ32γ3) C1]

= m'^*_K[(2i ζ'1 + γ'1 + ζ1'2γ'1) A'1 + (2i ζ'2 + γ'2 + ζ2'2γ'2) B'1+ (2i ζ'3 + γ '3 + ζ3'2γ'3) C'1] (25f)

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m*_K_[– λ4C]=m'^*_K[– λ'4 C'] (25g) A1 [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ12 – 1) + 2m^*_K(ζ12 –iζ1) – b^*_Kδ1] + B1 [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ22 – 1)

+ 2m^*_K(ζ22 –iζ2) – b^*_Kδ2] + C1 [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ32 – 1) + 2m^*_K(ζ32 – iζ3) – b^*_K_δ3]

= A'1 [(l'^*_K+( ' )

μ

_e ^*_K H₀²)(ζ1'2–1)+2m'^*_K(ζ1'2–iζ'1)–b'^*_Kδ'1]+ B'1 [(l'^*_K+( ' )

μ

_e ^*_K H₀²) (ζ2'2 – 1)

+ 2m'^*_K(ζ2'2 – iζ'2) – b'^*_Kδ'2] +C'1[(l'^*_K+( ' )

μ

_e ^*_K H₀²) (ζ3'2–1) + 2m'^*_K(ζ3'2 – iζ'3) – b'^*_Kδ'3] (25h)

where,

ζj=

λ

^j

α

^, ζ'j =

'_j

λ

α

, j = 1, 2, 3 and

λ*K =

( )

0

n K

K K

λ

i c

α

=

∑

− ^,^m^*^K⁼

⁽ ⁾

0

n K

K K

μ

i c

α

=

∑

− ^,^{( )}

^μ

^e ^*^K⁼

⁽ ⁾

0

( )

n K

e K K

μ

i c

α

=

∑

− '*_K

l =

( )

0

'

n K

K K

λ

i c

α

=

∑

− ^,^m'^*^K ⁼

⁽ ⁾

0

'

n K

K K

μ

i c

α

=

∑

− ^,^{( ' )}

^μ

^e ^*^K⁼

⁽ ⁾

0

( ' )

n K

e K K

μ

i c

α

=

∑

−

From eqs (25b) and (25g), we have C = C' = 0. Thus there is no propagation of displacement v. Hence SH- waves do not occur in this case.

Finally, eliminating the constants A1, B1, C1, A'1, B'1, C'1, from the remaining equations, we get

det (aij)= 0, i, j = 1, 2, 3, 4, 5, 6. (26) Where,

a11 = 1 – iγ1ζ1, a12 = 1–iγ2ζ2, a13 = 1–iγ3ζ3, a14 = (i γ'1ζ'1–1), a15 = (i γ '2ζ'2–1), a16 = (i γ'3ζ'3 – 1),

a21 = γ1 + iζ1, a22 = γ2 + iζ2, a23 = γ3 + iζ3, a24 = (γ'1 + i ζ'1), a25 = (γ'2 + iζ'2), a26 = (γ'3 + iζ'3),

a31 = δ1, a32 = δ2, a33 = δ3, a34 = – δ'1, a35 = –δ'2, a36 = –δ'3,

a41 = pλ1δ1, a42 = pλ2δ2, a43 = pλ3δ3, a44 = –p' λ'1δ'1, a45 = –p' λ'2δ'2, a46 = –p' λ'3δ'3,

a51 = m^*_K (2i ζ1 + γ1 + γ1ζ12), a52 = m^*_K(2i ζ2 + γ2 + γ2ζ22), a53 = m^*_K(2i ζ3 + γ3 + γ3ζ32),

a54 = m'^*_K(2i ζ'1 + γ'1 + γ'1ζ1'2), a55 = m'^*_K(2i ζ'2 + γ'2 + γ'2ζ2'2), a56 = m'^*_K(2i ζ'3 + γ'3 + γ'3ζ3'2),

a61 = (l^K^* +( )

μ

_e ^*_K H₀²) (ζ12 – 1) + 2m^*_K(ζ12 –iζ1) – b^*_K_δ1,

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a62 = (l^K^* +( )

μ

_e ^*_K H₀²) (ζ22 – 1) + 2m^*_K(ζ22 –iζ2) – b^*_K_δ2, a63 = (l^K^* +( )

μ

_e ^*_K H₀²) (ζ32 – 1) + 2m^*_K(ζ32 – iζ3) – b^*_K_δ3, a64 = (l'^*_K +( ' )

μ

_e ^*_K H₀²) (ζ1'2–1) + 2m'^*_K (ζ1'2–iζ'1)–b'^*_K_δ'1, a65 = (l'^*_K +( ' )

μ

_e ^*_K H₀²) (ζ2'2 – 1) + 2m'^*_K (ζ2'2 – iζ'2) – b'^*_K_δ'2,

a66 = (l'^*_K +( ' )

μ

_e ^*_K H₀²) (ζ3'2–1) + 2m'^*_K (ζ3'2 – iζ'3) – b'^*_Kδ'3, (27) From eq. (26), we obtain velocity of surface waves in common boundary between two viscoelastic, non- homogeneous solid media under the influence of magnetic field, where the viscosity is of general n^th order involving time rate of change of strain.

PARTICULAR CASES

Stoneley Waves: It is the generalized form of Rayleigh waves in which we assume that waves are propagated along the common boundary of the two semi-infinite media M1 and M2. Thus eq. (27) determine the wave velocity equation for Stoneley waves in the case of general magneto viscoelastic, non- homogeneous solid media of n^th order involving time rate of strain. Clearly from eq. (27), it is follows that the wave velocity equation for Stoneley waves depends upon the non-homogeneity of the material medium, magnetic and viscous field. This equation, of course, is in well agreement with the corresponding classical result, when the effects of magnetic and viscous field and non-homogeneity are absent.

Rayleigh Waves: To investigate the possibility of Rayleigh waves in a magneto viscoelastic, non- homogeneous elastic media, we replace media M2 by vacuum, in the proceeding problem; we also note the SH-waves do not occur in this case.

Thus eq. (25f) and (25h) reduces to,

(2i ζ₁ + γ₁ + γ₁ζ₁²) A₁ + (2i ζ₂ + γ₂ + γ₂ζ₂²) B₁ + (2i ζ₃ + γ₃ + γ₃ζ₃²) C₁ = 0 (28a) [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ₁² – 1) + 2m^*_K(ζ₁² –iζ₁) – b^*_Kδ₁] A₁+ [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ₂² – 1)

+ 2m^*_K(ζ₂² –iζ₂) – b^*_Kδ₂] B₁ + [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ₃² – 1) + 2m^*_K(ζ₃² – iζ₃) – b^*_Kδ₃] C₁ = 0 (28b) From eq. (27), we have

(λ₁ – h) δ₁ A₁ + (λ₂ – h) δ₂ B₁ + (λ₃ – h) δ₃ C₁ = 0 (28c) Eliminating A₁, B₁ and C₁ from eqs (28a), (28b) and (28c), we get

det (b_ij)=0,i, j = 1, 2, 3. (29) where,

b₁₁ = (2i ζ₁ + γ₁ + γ₁ζ₁²), b₁₂ = (2i ζ₂ + γ₂ + γ₂ζ₂²), b₁₃ = (2i ζ₃ + γ₃ + γ₃ζ₃²), b₂₁= [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ₁² – 1) + 2m^*_K(ζ₁² –iζ₁) – b^*_Kδ₁],

b₂₂= [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ₂² – 1) + 2m^*_K(ζ₂² –iζ₂) – b^*_Kδ₂],

b₂₃= [(l^*_K+( )

μ

_e ^*_K H₀²) (ζ₃² – 1) + 2m^*_K(ζ₃² – iζ₃) – b^*_Kδ₃],

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b31 = (λ₁–h) δ₁, b32= (λ₂ – h) δ₂, b33 = (λ₃ – h) δ₃.

Thus eq. (29), gives the wave velocity equation for Rayleigh waves in a non-homogeneous, magneto viscoelastic solid media of n^th order involving time rate of strain.

From eq. (29), dispersion equation of Rayleigh waves depends upon the non-homogeneity, the viscous and magnetic fields follows.

This equation, of course, is in complete agreement with the corresponding classical result by Bullen ², when the effects of magnetic viscous field and non-homogeneity are absent.

Love Waves: To investigate the possibility of love waves in a non-homogeneous, viscoelastic solid media, we replace medium M₂ is obtained by two horizontal plane surfaces at a distance H-apart, while M₁ remains infinite.

For medium M₁, the displacement componentν remains same as in general case given by eq. (17).

For the medium M₂, we preserve the full solution, since the displacement component along y-axis i.e. v no longer diminishes with increasing distance from the boundary surface of two media.

Thus v' = C e₁ ^λ^'⁴^{z i}⁺^α⁽^{x ct}⁻ ⁾

+

C e₂ ⁻^λ^'⁴^{z i}⁺^α⁽^{x ct}⁻ ⁾ (30) (i) v and τ₃₂are continuous at z = 0

(ii) τ'₃₂ = 0 at z = –H.

Applying boundary conditions (i) and (ii) and using eqs (17) and (25), we get

C= C₁ + C₂ (31) –m^*_Kλ4C= (μ'_K)* [λ'4C₁ – λ'4C₂] (32)

4 4

' '

1 2

H H

C e⁻^λ −C e^λ = 0 (33) On eliminating the constants C, C₁ and C₂ from eqs (26), (27) and (28), we get

tanh (λ'4H)=‐

( )

* 4

'4 ' *

K K

λ μ

^. (34) Thus eq. (34) gives the wave velocity equation for Love waves in a non-homogeneous, magneto, viscoelastic solid medium of n^th order involving time rate of strain. Clearly it depends upon the non- homogeneity, magnetic and viscous fields

CONCLUSIONS

I. The present study reveals the effects of non-homogeneity, viscous and magnetic field in the wave velocity equations corresponding to Stoneley waves, Rayleigh waves and Love waves. Further it is investigated that viscoelastic surface waves are affected by the time rate of strain parameters.

These parameters influence the wave velocity to an extent depending on the corresponding constants characterizing the magneto and viscoelasticity of the material. So the results of this analysis become useful in circumstances where these effects cannot be neglected. These velocities depend upon the wave number ‘α’ confirming that these waves are affected by non-homogeneity of the material medium.

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II. Also it is noted from eq. (34) is that Love waves are only affected by viscous, magnetic fields and non-homogeneity of the material medium. In absence of all fields and non-homogeneity, the dispersion equation is in complete agreement with the corresponding classical result.

III. Further for Rayleigh waves in a non-homogeneous, general magneto viscoelastic solid medium of higher order including time rate of change of strain we find that the wave velocity equation proves that there is dispersion of waves due to the presence of non-homogeneity, magnetic field and viscosity. The results are in complete agreement with the corresponding classical results in the absence of all fields and compression.

IV. It is noted that the wave velocity equation of Stoneley waves is very similar to the corresponding problem in the classical theory of elasticity. Here also there is dispersion of waves due to the presence of non-homogeneity, magnetic field and viscoelastic nature of the solid. Also wave velocity equation of this generalized type of surface waves in non-homogeneous magneto viscoelastic solid media of higher order including time rate of strain is in complete agreement with the corresponding classical result in the absence of all fields and non-homogeneity.

V. The solution of wave velocity equation for Stoneley waves cannot be determined by easy analytical methods however we can apply numerical techniques to solve this determinantal equation by choosing suitable values of physical constants for both media M₁ and M₂

ACKNOWLEDGEMENTS

The authors are thankful to the referees for their valuable comments.

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Corresponding author:

Rajneesh Kakar; Principal, DIPS Polytechnic College, Hoshiarpur, 146001, India

*E-mail of Corresponding Author: [email protected]

Journal of Environmental Science, Computer Science and Engineering & Technology