Acta Mech 202, 35–45 (2009) DOI 10.1007/s00707-008-0027-5

D. P. Acharya · I. Roy · S. Sengupta

Effect of magnetic field and initial stress on the propagation of interface waves in transversely isotropic perfectly conducting media

Received: 17 June 2007 / Revised: 19 February 2008 / Published online: 28 May 2008 © Springer-Verlag 2008

Abstract Elasto-dynamical equations for transversely isotropic solids have been employed to investigate the general theory of transversely isotropic magneto-elastic interface waves in conducting media under initial hydrostatic tension or compression. Particular cases of interface waves such as Rayleigh, Love and Stoneley waves have been investigated in details. In all cases, the wave velocity equations have been deduced which are in complete agreement with the corresponding results of classical surface waves of the same types where magnetic fields and initial stresses are absent. Results obtained in this paper may be considered as more general and important in the sense that the corresponding results of classical surface waves due to Rayleigh, Love and Stoneley can readily be deduced from our results as special cases. Numerical calculations and graphs have been presented in the case of Love waves and conclusions are drawn.

1 Introduction The role of interface waves in a homogeneous isotropic elastic solid medium has its importance due to its relevance for other phenomena in seismology, geophysics and earth system sciences. The classical theory of interface waves as exposed, for example, in Bullen’s monograph [18] has been generalized and modified for magnetoelastic models. The interplay of the Maxwell electromagnetic field with the motion of deformable solids is largely being undertaken by many investigators owing to the possibility of its application to geophysical problems and certain topics in optics and acoustics. Moreover, the earth is subject to its own magnetic field and the material of the earth may be electrically conducting. Thus, the magnetoelastic nature of the earth’s material may affect the propagation of seismic waves. The earth is also an initially stressed layered structure where initial stresses exist due to variation of temperature, weight, overburden layer and slow process of creep, gravitation and largeness, etc. The theory of incremental stress and strains may be found in Biot’s monograph [1]. Yu and Tang [2] had thoroughly discussed the dilatational and rotational waves in a magneto-elastic initially stressed conducting medium. De and Sengupta [3] investigated magneto-elastic waves in initially stressed isotropic media. Without going D. P. Acharya 109/3, Kailash Roy Chowdhury Road, Barrackpore, Kolkata 700 120, India E-mail: [email protected] I. Roy (B) Brahmapara, Simurali, Nadia 741248, India E-mail: [email protected] S. Sengupta Indian Institute of Mechanics of Continua, 201, Manicktala Main Road, Suite No.42, Kolkata 700054, West Bengal, India E-mail: [email protected]

36

D. P. Acharya et al.

into the details of the papers investigating the propagation of waves in an initially stressed magnetoelastic, thermoelastic and elastic medium, we cite here a few recent publications [4–13]. Owing to the variation of the elastic properties and due to the presence of initial stresses the medium exhibits anisotropy as well, so that there are reasonable grounds for assuming anisotropy in the continents. The wave velocities are considerably influenced by anisotropy in the earth’s crust and upper mantle. Due to an involved form of the field equations for general anisotropic materials, several authors including Honarvar et al. [14], Ahmad and Khan [15], Dey and Maulick [16], Sharma et al. [17] have considered anisotropy with one axis of elastic symmetry (transverse isotropy). This paper investigates the effect of the transverse isotropy and magnetic field on the interface waves in a conducting medium subject to the initial state of stress of the form of hydrostatic tension or compression. The wave velocity equations for interface waves, such as the Rayleigh, Love and Stoneley waves, in the presence of the initial stress and the magnetic field in a transversely isotropic medium have been established. The presence of transverse isotropy, the magnetic field and the initial stress considerably effects the wave velocity of the respective interface wave. For numerical examples, we restrict our attention to the Love waves due to their simple nature. It is found that the Love wave velocity in normalized form decreases rapidly up to a certain value of the normalized layer thickness. Moreover, for a fixed value of the layer thickness and the initial stress, the variation of the normalized Love wave velocity due to the variation of the magnetic field parameter is very small but cannot be neglected. 2 Governing equations The Maxwell electromagnetic equations are [2] ∇ × H = J, ∇ × E = − ∂B ∂t , ∇ · B = 0, B = µe H. The generalized Ohm’s law in a deformable continuum is
   ∂u J=σ E+ ×B . ∂t

(1)

(2)

In the above equations, E is the electric intensity vector, H is the magnetic intensity vector, B is the magnetic induction vector, J is the current density vector, µe represents the magnetic permeability, σ is the electric conductivity, u is the displacement vector, and t denotes time. For an electrically conducting charge free elastic solid under an initial stress permeated by an electromagnetic field, the equation of motion based on the finite strain theory, when the body is subjected to small perturbation, may be expressed as [2]
  ∂ 2ui ∂ ∂u i ρ 2 = (3) τ jk δ jk + + (J × B)i + Fi ∂t ∂x j ∂ xk where τ jk = τ 0jk (x1 , x2 , x3 ) + τ˜ jk (x1 , x2 , x3 , t) , Fi = Fi0 (x1 , x2 , x3 ) + F˜i (x1 , x2 , x3 , t) , Hi = Hi0 (x1 , x2 , x3 ) + H˜ i (x1 , x2 , x3 , t) , i, j, k = 1, 2, 3. In the above, the superscript 0 stands for the initial equilibrium state and the tilde is used to denote incremental terms, due to the deformation produced by the components u i of the displacement vector u referred to the coordinates xi of a material point at time t. The τi j , Hi , Fi are the components of the resultant stress tensor ∼ τ, and the body force F , respectively, ρ is the mass density and δ is the Kronecker the magnetic intensity field H ij ∼ ∼ delta symbol. From Eqs. (1) and (2) it follows that
 1 1 ∂H ∂u ∇ µe = ×H− ∇ ×H (4) ∂t µe ∂t σ

Effect of magnetic field and initial stress on the propagation of interface waves

37

and Eq. (3) can thus be rewritten as ρ


  ∂ 2ui ∂ ∂u i = + τ δ + µe [(∇ × H) × H] + Fi . jk jk ∂t 2 ∂x j ∂ xk

(5)

Following the work of [2,4,5], we obtain the set of dynamical equations valid for a perfectly conducting transversely isotropic solid under the initial state of hydrostatic stress in presence of a constant (time and position independent) magnetic field parallel to the x1 -axis:  ˜  2 2 ∂τi j ∂ H˜ 1 i ρ ∂∂tu2i = − p0 ∂ x∂ j ∂uxi j + µe H0 ∂∂ H − x1 ∂ xi + ∂ x j , (6)   1 , H˜ i = H0 ∂∂ux1i − ∂u ∂ xi where − p0 denotes the hydrostatic state of stress (tension or compression according to p0 < 0 or p0 > 0), H0 is the intensity of the constant (initial) magnetic field parallel to the x 1 -axis, τi j are the components of the stress tensor ∼ τ defined on the initial configuration (dropping tildes over the stress tensor for convenience) and i, j = 1, 2, 3. 3 Interface wave Let M 1 and M 2 be two electrically conducting charge-free transversely isotropic homogeneous media of different material characteristics, M 2 being above M 1 . The two media are subject to an initially hydrostatic state of stress and are permeated by a constant (time and position independent) magnetic field, and are in welded contact along the planar horizontal interface. Note that M 1 and M 2 may not remain homogeneous and isotropic when subject to the initial stress and the magnetic field. In the present investigation, we do not consider the deviation from inhomogeneity, and M 1 and M 2 are assumed to be transversely isotropic with a common axis of elastic symmetry being normal to the horizontal interface. The magnetic properties of M 1 and M 2 are assumed to be the same. The M 1 and M 2 are referred to the system of Cartesian coordinates with the origin O located at the interface (x 3 = 0) and x 3 is the vertical axis pointing downwards, as shown in Figs. 1, 2. M2

O

X1

M1 X2 X3 Fig. 1 Geometry of the problem

X3 =−h

h

M2

O

X1

M1

X3 Fig. 2 Geometry for the propagation of Love waves

38

D. P. Acharya et al.

Consider a wave travelling in the x 1 -direction such that the disturbance is largely confined to the neighborhood of the interface, and, at any instant of time, all the particles along any line parallel to the x 2 -axis have equal displacements. Thus such a wave motion may be considered as an interface wave which is independent of x 2 . In the present model, general interface waves as well as their special cases (the Rayleigh, Love and Stoneley waves) are all plane waves of which only the Love wave is of SH type where u 2 is the only non-zero component of the displacement ∼ u to play the role [18]. Such a wave is transmitted along the interface separating the two media or along the free surface of a semi-infinite medium subject to appropriate boundary conditions and it plays an important role in earthquake sciences and seismology. Under the above circumstances the components τi j of the stress tensor ∼ τ are [19] ∂u 2 ∂u 3 1 τ11 = A ∂u ∂ x1 + (A − 2N ) ∂ x2 + F ∂ x3 , ∂u 2 ∂u 3 1 τ22 = (A − 2N ) ∂u ∂ x1 + A ∂ x2 + F ∂ x3 , ∂u 2 ∂u 3 1 τ33 = F ∂u ∂ x1 + F ∂ x2 + C ∂ x3 ,   ∂u 2 1 τ12 = N ∂u + ∂ x2 ∂ x1 ,   ∂u 2 3 τ23 = L ∂u + ∂ x2 ∂ x3 ,   ∂u 3 1 τ13 = L ∂u ∂ x3 + ∂ x1

(7)

where A, N, F, C and L are the material constants of M 1 . From Eqs. (6) and (7), we obtain the governing equations for M 1 in the form ∂ 2u1 ∂ 2u3 ∂ 2u1 ∂ 2u1 2 + + L) + L − p ∇ u = ρ , (F 0 1 ∂ x1 ∂ x3 ∂t 2 ∂ x12 ∂ x32   ∂ 2 u 23 ∂ 2u3 ∂ 2u1 ∂ 2u3 ∂ 2u1 ∂ 2u3 2 =ρ 2 , + (F + L) +C − p0 ∇ u 3 + K − L 2 2 2 ∂ x1 ∂ x3 ∂ x3 ∂ x1 ∂t ∂ x1 ∂ x3 ∂ x1 A

N where ∇ 2 =

∂ 2u2 ∂ 2u2 ∂ 2u2 ∂ 2u2 +L − p0 ∇ 2 u 2 + K =ρ 2 2 2 2 ∂t ∂ x1 ∂ x3 ∂ x1

∂2 ∂ x12

+

∂2 ∂ x32

(8.1) (9.1) (10.1)

and K = µe H02 .

Similarly, for M 2 , the governing equations are A

L

2u 3 ∂ x12

∂

∂ 2 u 1 ∂ x12



∂ 2 u 3 ∂ 2u ∂ 2u + F + L + L  21 − p0 ∇ 2 u 1 = ρ  21 , ∂ x1 ∂ x3 ∂t ∂ x3

∂ 2 u 1 ∂ 2u + F +L + C  23 − p0 ∇ 2 u 3 + K ∂ x1 ∂ x3 ∂ x3







N

∂ 2 u 2 ∂ x12

+ L

∂ 2 u 2 ∂ x32

− p0 ∇ 2 u 2 + K

∂ 2 u 2 ∂ x12



∂ 2 u 32 ∂ x12 = ρ

∂ 2 u 1 − ∂ x3 ∂ x1 ∂ 2 u 2 , ∂t 2

(8.2)

 = ρ

∂ 2 u 3 , ∂t 2

(9.2)

(10.2)

where the prime notation is used to denote the material constants of M 2 . For simple harmonic waves travelling in the x 1 direction, we seek the solutions of Eqs. (8.1)–(10.1) for the medium M 1 in the following form: u 1 (x1 , x2 , t) = P(x3 )eiα(x1 −ct) , u 2 (x1 , x3 , t) = Q(x3 )eiα(x1 −ct) , u 3 (x1 , x3 , t) = R(x3 )eiα(x1 −ct) .

(11.1) (12.1) (13.1)

Effect of magnetic field and initial stress on the propagation of interface waves

39

Similar solutions of the Eqs. (8.2)–(10.2) for medium M 2 have the forms u 1 (x1 , x2 , t) = P  (x3 )eiα(x1 −ct) ,

(11.2)

u 2 (x1 , x3 , t) = Q  (x3 )eiα(x1 −ct) ,

(12.2)

u 3 (x1 , x3 , t) = R  (x3 )eiα(x1 −ct) .

(13.2)

In the above equations, P, Q, R, P  , Q  and R  are exclusive functions of x3 , α is the wave number and c is the wave velocity. Substituting the wave forms (11.1) and (13.1) into Eqs. (8.1) and (9.1) we obtain  L D 2 + α 2 (ρc2 − A) − p0 (D 2 − α 2 ) P + iα(F + L)D R = 0, (14.1)  2 2 2 2 2 2 (15.1) C D + α (ρc − L) − p0 (D − α ) − K α R + iα[(F + L) − K ]D P = 0. Similarly, substitution of Eqs. (11.2) and (13.2) into Eqs. (8.2) and (9.2) yields   2 L D + α 2 (ρ  c2 − A ) − p0 (D 2 − α 2 ) P  + iα(F  + L  )D R  = 0,   2 C D + α 2 (ρ  c2 − L  ) − p0 (D 2 − α 2 ) − K α 2 R  + iα[(F  + L  ) − K ]D P  = 0

where D(·) = d(·) dx3 . Hence the pairs P, R and P  , R  satisfy the following differential equations:

(14.2) (15.2)

[(D 2 + λ21 α 2 )(D 2 + λ22 α 2 )](P, R) = 0,

(16.1)

2 2 2 2   [(D 2 + λ2 1 α )(D + λ2 α )](P , R ) = 0

(16.2)

where λ21 + λ22 = λ21 λ22

=

U LC− p0 (L+C− p0 ) ,

(ρc2 −A)(ρc2 −L)+ p0 (2ρc2 −A−L+ p0 −K )−K (ρc2 −A) LC− p0 (L+C− p0 )

(17.1)

in which U = L(ρc2 − L) + C(ρc2 − A) + (F + L)2 + p0 (2L + C + A − 2ρc2 + K − 2 p0 ) − K (2L + F) and 



λ12 + λ22 = 



λ12 λ22 =

U L  C  − p0 (L  +C  − p0 ) ,

(ρ  c2 −A )(ρ  c2 −L  )+ p0 (2ρ  c2 −A −L  + p0 −K )−K (ρ  c2 −A ) L  C  − p0 (L  +C  − p0 )

(17.2)

in which U  = L  (ρ  c2 − L  )+C  (ρ  c2 − A ) + (F  + L  )2 + p0 (2L  + C  + A − 2ρ  c2 + K − 2 p0 )− K (2L  + F  ). In view of the decay conditions in the medium M 1 (that is, (u 1 , u 2 , u 3 ) → 0 as x3 → ∞), P(x3 ) and R(x3 ) are P(x3 ) = S1 e−iλ1 αx3 + S2 e−iλ2 αx3 , R(x3 ) = S1 e−iλ1 αx3 + S2 e−iλ2 αx3

(18)

where λ1 and λ2 are deduced from (17.1) and S1 , S2 , S1 and S2 are unknown constants which may be positive, negative or zero. Substituting the wave form (12.1) into Eq. (10.1), we note that Q(x3 ) can be taken as Q(x3 ) = S3 e−iλ3 αx3

(19)

40

D. P. Acharya et al.

where λ23 =

ρc2 − N + p0 − K L − p0

(20)

and S3 is the unknown constant (positive, negative or zero). Substituting Eqs. (18) into Eq. (14.1), and noting that the coefficients of e−iλ1 αx3 and e−iλ2 αx3 are vanishing, we obtain S1 = β1 S1 , S2 = β2 S2

(21)

where βj =

Lλ2j − (ρc2 − A) − p0 (1 + λ2j ) λ j (F + L)

,

j = 1, 2.

Thus, the wave motion in the medium M 1 has the following form:  u 1 = S1 e−iλ1 αx3 + S2 e−iλ2 αx3 eiα(x1 −ct) ,  u 3 = β1 S1 e−iλ1 αx3 + β2 S2 e−iλ2 αx3 eiα(x1 −ct) ,  u 2 = S3 e−iλ3 αx3 eiα(x1 −ct) .

(22)

Similarly, the wave motion in the medium M 2 is given by     u 1 = S4 eiλ1 αx3 + S5 eiλ2 αx3 eiα(x1 −ct) ,     u 3 = β1 S4 eiλ1 αx3 + β2 S5 eiλ2 αx3 eiα(x1 −ct) ,    u 2 = S6 eiλ3 αx3 eiα(x1 −ct)

(23)

where λ1 and λ2 are deduced from Eq. (17.2), 



λ32 =



L  λ j2 − (ρ  c2 − A ) − p0 (1 + λ j2 ) ρ  c 2 − N  + p0 − K  , β = , j L  − p0 λj (F  + L  )

j = 1, 2,

and S4 , S5 and S6 are unknown constants which may be positive, negative or zero. We assume the following boundary conditions: the displacement components u i are continuous at all times and positions across the interface separating M 1 and M 2 (u 1 = u 1 , u 2 = u 2 and u 3 = u 3 ), and the stress components τ31 , τ32 and τ33 are continuous at all times and positions across the interface separating M 1 and  , τ = τ  andτ = τ  ). M 2 (τ31 = τ31 32 33 32 33 Substituting the wave forms (22) and (23) into these boundary conditions, we obtain S1 + S2 − S4 − S5 = 0, β1 S1 + β2 S2 − β1 S4 − β2 S5 = 0, S3 = S6 , L(λ1 − β1 )S1 + L(λ2 − β2 )S2 + L  (λ1 + β1 )S4 + L  (λ2 + β2 )S5 = 0, (Cλ1 β1 − F)S1 + (Cλ2 β2 − F)S2 + (C  λ1 β1 + F  )S4 + (C  λ2 β2 + F  )S5 = 0, −Lλ3 S3 = L  λ3 S6 .

(24) (25) (26) (27) (28) (29)

From Eqs. (26) and (29) it follows that S3 = S6 = 0.

(30)

Thus, there are no waves with the displacements u 2 and u 2 propagating in the x 1 direction along the interface separating the two half spaces.

Effect of magnetic field and initial stress on the propagation of interface waves

41

Eliminating the indispensable constants S1 , S2 , S3 and S4 from Eqs. (24), (25), (27) and (28) we arrive at the equation governing the wave velocity given by   1  1 −1 −1   β    β2 −β1 −β2  1  (31)   = 0.  L(λ1 − β1 ) L(λ2 − β2 ) L  (λ1 + β1 ) L  (λ2 + β2 )     Cλ1 β1 − F Cλ2 β2 − F C  λ β  + F  C  λ β  + F   1 1 2 2 The roots of the above equation determine the wave velocities of the wave propagating along the interface between the two homogeneous transversely isotropic magneto-elastic conducting media under an initial hydrostatic tension or compression. Note that Eq. (31) does not contain the wave number α so that the wave velocity c obtained from Eq. (31) does not depend on α, and hence the interface wave as defined above is non-dispersive as in the corresponding classical case. Thus, the presence of transverse isotropy, the magnetic field and the initial stress do not make the interface waves (22) and (23) dispersive. 4 Special cases 4.1 Rayleigh wave The planar interface is considered to be stress free so that M 2 is replaced by a vacuum, and Eqs. (27) and (28) reduce to (λ1 − β1 )S1 + (λ2 − β2 )S2 = 0, (Cλ1 β1 − F)S1 + (Cλ2 β2 − F)S2 = 0.

(32) (33)

Eliminating S1 and S2 we obtain (β2 − β1 )(Cλ1 λ2 − F) + (λ1 − λ2 )(Cβ1 β2 − F) = 0

(34)

which is the corresponding Rayleigh wave velocity equation in a transversely isotropic magneto-elastic initially stressed conducting half-space. It is evident that the Rayleigh wave velocity obtained from Eq. (34) is influenced by the anisotropic character of the medium but it is not dispersive as in the classical case. Due to the presence of K and p0 in Eq. (34), we conclude that the Rayleigh wave velocity depends on the magnetic field and the initial stress. Note that the form of Eq. (34) is the same as in the case of the isotropic medium. The difference lies 2 2 in the values of F, C and β j ( j = 1, 2), and λ21 and λ22 . The corresponding of F,  values   C, β j , λ1 and λ2 for the isotropic medium are λ, λ + 2µ,

λ2j v 2S −(c2 −vT2 ) λ j (vT2 −v 2S )

,

c2 v 2S

−

K ρv 2S

− 1 and

c2 vT2

− 1 , respectively, where

p0 vT2 = λ+2µ− , v S2 = µ−ρ p0 . ρ In the absence of the initial stress and the magnetic field (i.e., K = 0 and p0 = 0) with the isotropic character of the medium (A = C = λ + 2µ, L = N = µ, F = λ), we derive from Eq. (34) the classical Rayleigh wave velocity equation in an isotropic elastic medium:



c2 2− 2 vS

where v S2 =

µ ρ and

vT2 =

2

 =4

c2 1− 2 vS



c2 1− 2 vT

1/ 2 (35)

λ+2µ ρ .

4.2 Love wave For the Love wave, u 2 is the only component of the displacement vector ∼ u to play the role (Bullen [18]). In

this case, M 1 (0 ≤ x3 < ∞) is a semi-infinite medium as before, while M 2 (−h ≤ x3 ≤ 0) is a layer of finite thickness h, and M 1 and M 2 are in welded contact along the interface x3 = 0. Note that the displacement in M 2, ∼ u  , no longer decays with the distance from the interface separating M 1 and M 2 .

42

D. P. Acharya et al.

Thus, for the Love wave propagating along the x 1 -axis, we have u 1 = 0, u 3 = 0 and u 2 = u 2 (x1 , x3 , t). Therefore, the only equation of motion for the medium M 1 is N

∂ 2u2 ∂ 2u2 ∂ 2u2 ∂ 2u2 +L − p0 ∇ 2 u 2 + K =ρ 2 2 2 2 ∂t ∂ x1 ∂ x3 ∂ x1

and that for the medium M 2 is N

∂ 2 u 2 ∂ x12

+ L

∂ 2 u 2

− p0 ∇ 2 u 2 + K

∂ x32

∂ 2 u 2 ∂ x12

= ρ

∂ 2 u 2 . ∂t 2

Hence, the displacement components u 2 and u 2 for the media M 1 and M 2 , respectively, are set as  u 2 = S3 e−iλ3 αx3 eiα(x1 −ct) , 



u 2 = S6 eiα(λ3 x3 +x1 −ct) + S7 e−iα(λ3 x3 −x1 +ct)

(36)

where S3 , S6 and S7 are unknown constants. The boundary conditions at the interface x3 = 0 are  at x3 = 0. u 2 = u 2 and τ32 = τ32

(37)

 = 0 at x = −h. These boundary conditions Since the boundary surface x 3 = −h is stress free, we have τ32 3 lead to

S3 = S6 + S7 , −Lλ3 S3 = L  λ3 S6 − L  λ3 S7 ,  e2iαλ3 h S − S = 0. 6

7

(38) (39) (40)

Eliminating the indispensable constants S3 , S6 and S7 from Eqs. (38) through (40) we obtain the wave velocity equation for the Love wave propagating in a transversely isotropic initially stressed conducting layer permeated by a uniform magnetic field: 

L  λ3 + Lλ3 = (L  λ3 − Lλ3 )e2iαλ3 h The above equation may be put in the following form:  L

v22 v12

−

c2 v12

1/ 2

 − L 

c2 

v12

− 

1/ 2 v2 2 

v12

⎧ ⎫  1/ ⎪ ⎪ 2 ⎨ ⎬ 2 2 v2 c tan αh =0 2 − 2 ⎪ ⎪ v1 v1 ⎩ ⎭ 



(41)

(42)

p0 − p0 where v12 = L−ρ p0 , v22 = N +Kρ − p0 , v12 = L − and v22 = N +K . ρ ρ Note that the propagation of the Love wave with the real wave velocity in a magnetoelastic initially stressed transversely isotropic layer is possible only when v2 < c < v2 . It follows from Eq. (42) that, like in the classical case, the velocity of propagation c depends on the particular value of α. This implies that, in the case of the Love wave, further dispersion occurs, which is due to the presence of transverse isotropy, magnetic field and initial stress. We also note that c tends to v2 or v2 when α is sufficiently small or large, respectively. The presence of p0 , K , L , N , L  and N  in Eq. (42) indicates that the role of initial stress , magnetic field and anisotropic character of the medium cannot be neglected when the propagation of Love waves is considered. In the absence of the magnetic field (K = 0), the initial stress ( p0 = 0) and the anisotropic character of the medium (L = µ, L  = µ , N = µ, N  = µ ), the classical Love wave velocity equation (for the elastic medium) may be obtained from Eq. (42): ⎧ ⎫ 1/  1/  1/ ⎪  ⎪ ⎨ ⎬ 2 2 2 c2 c2 c2 − µ − 1 tan αh − 1 =0 (43) µ 1− 2   ⎪ ⎪ vS v S2 v S2 ⎩ ⎭

where v S2 =

µ ρ



and v S2 =

µ ρ .

Effect of magnetic field and initial stress on the propagation of interface waves

43

4.3 Stoneley wave The generalized form of the Rayleigh wave is the Stoneley wave which propagates in the vicinity of the interface separating the two semi-infinite media M 1 and M 2 . Thus, Eq. (31) represents the wave velocity equation for the Stoneley wave propagating along the common boundary of the two transversely isotropic magneto-elastic initially stressed conducting media. Further discussion of the Stoneley wave is not pursued here due to the involved structure of the pertinent wave velocity equation. 5 Numerical examples For numerical calculations, the following values of the material constants, the initial stress and the magnetic parameter for the two media M 1 and M 2 were taken as (Love [19], Pal and Acharya [20]) L = 6.53 × 106 N/cm2 , N L  = 4.0 × 106 N/cm2 , N  p0 = 0, 1.5 × 106 N/cm2 , K = 0, 0.3 × 106 N/cm2 ,

= 8.665 × 106 N/cm2 , ρ = 2.7g/cm3 , = 6.30 × 106 N/cm2 , ρ  = 7.1 g/cm3 , 2.0 × 106 N/cm2 , 3.0 × 106 N/cm2 , 0.4 × 106 N/cm2 , 0.5 × 106 N/cm2 .

In the above the layer and foundation (half-space) materials were taken as Zinc and Beryl, respectively. Using Eq. (42), the numerical values of the wave velocity were calculated. Figure 3 depicts the normalized Love wave velocity νc versus the normalized thickness αh for different 2

values of the initial stress ( p0 = 0, 1.5 × 106 N/cm2 , 2.0 × 106 N/cm2 , 3.0 × 106 N/cm2 ) with the fixed magnetic parameter K = 0.3 × 106 N/cm2 . The curve corresponding to K = 0 and p0 = 0 is also drawn for comparison. Note that νc diminishes rapidly as αh increases from 0 to approximately 1. When αh is larger than 1,

c ν2

2

slowly diminishes (as αh increases) and ultimately becomes very small for large values of αh. For

a particular value of αh in the interval 0 < αh < 2, νc diminishes as the initial stress parameter increases. For

a particular value of αh > 2, the variation of

c ν2

2

due to the change of p0 is negligible.

In Fig. 4, four curves are drawn to show the variation of

c ν2

with αh in an anisotropic medium for different

values of the magnetic parameter K when the initial stress is set at p0 = 3.0 × 106 N/cm2 . Note that, for a fixed value of αh and p0 , the variation of νc for different values of K is very small but not negligible. It is 2

evident that, for any set of the particular values of K and p0 , starting from its highest value, continuously diminishes as αh increases in the interval 0 < αh < 1.

c ν2

slowly but

2.0 1.9 K = 0.0 ,p0 =0

Love wave velocity, c/v2’

1.8

K = 0.3E06 ,p0 =0

1.7

K= 0.3E06 ,p0 =1.5E06 K = 0.3E06 ,p0 =2.0E06

1.6

K = 0.3E06 ,p0 =3.0E06

1.5 1.4 1.3 1.2 1.1 1.0 0.0

0.5

1.0

1.5 2.0 2.5 Normalized layer thickness, ah

Fig. 3 Variation of the normalized Love wave velocity

c ν2

3.0

3.5

versus the normalized layer thickness αh for different values of the

initial stress p0 when the magnetic parameter is set at K = 0.3 × 106 N/cm2

44

D. P. Acharya et al.

2.2

Love wave velocity, c/v2’

2

K = 0.0 ,p0 =3.0E06 K = 0.3E06 ,p0 =3.0E06

1.8

K = 0.4E06 ,p0 =3.0E06 K = 0.5E06 ,p0 =3.0E06

1.6 1.4 1.2 1 0.00

0.13

0.25

0.38 0.50 0.63 Normalized layer thickness, ah

Fig. 4 Variation of the normalized Love wave velocity

c ν2

0.75

0.88

1.00

versus the normalized layer thickness αh for different values of the

magnetic parameter K when the initial stress is set at p0 = 3.0 × 106 N/cm2

6 Conclusions The principal conclusion is that the initial stress, the magnetic field as well as the anisotropic character of the transmitting medium modulate the interface wave velocity considerably. It is also concluded that the general interface wave, the Rayleigh wave and the Stoneley wave are not dispersive, whereas further dispersion occurs in the case of the Love wave. Numerical calculations for the Rayleigh and Stoneley waves were not pursued due to their complicated nature. Note that continuous diminution of the Love wave velocity takes place as the thickness of the layer increases within a certain range. The initial stress causes lessening of this wave velocity. Diminution of the Love wave velocity due to the presence of the magnetic field is very small. Acknowledgments We wish to thank S. S. Acharya, Scientific Officer, Atomic Energy Regulatory Board, Mumbai, India for his valuable suggestions and help regarding the numerical examples. We are really indebted to the reviewers for their valuable comments and suggestions towards the improvement of the paper.

References 1. Biot, M.A.: Mechanics of Incremental Deformations. Theory of Elasticity and Viscoelasticity of Initially Stressed Solids and Fluids, Including Thermodynamic Foundations and Applications to Finite Strain, vol. 44. Wiley, New York (1965) 2. Yu, C.P., Tang, S.: Magneto-elastic waves in initially stressed conductors. Z. Angew. Math. Phys. 17, 766–775 (1966) 3. De, S.N., Sengupta, P.R.: Magneto-elastic waves and disturbances in initially stressed conducting media. Pure Appl. Geophys. 93, 41–54 (1972) 4. Acharya, D.P., Sengupta, P.R.: Magneto-thermo-elastic waves in an initially stressed conducting layer. Gerlands Beitr. Geophys. 87, 229–239 (1978) 5. Acharya, D.P., Sengupta, P.R.: Magneto-thermo-elastic surface waves in initially stressed conducting media. Acta Geophys. Pol. 26, 299–311 (1978) 6. Rakshit, A.K., Sengupta, P.R.: Magneto-thermo-elastic waves in initially stressed conducting layer. Sadhan ¯ a¯ 23, 233–246 (1998) 7. Das, S.C., Sengupta, S., Acharya, D.P.: Magneto-visco-elastic waves in an initially stressed conducting layer including strain rate and stress rate. Bull. Tech. Univ. Istanb. 47, 243–255 (1994) 8. Ahmed, S.M.: Rayleigh waves in a thermoelastic granular medium under initial stress. Int. J. Math. Math. Sci. 23, 627–637 (2000) 9. Wang, J., Slattery, S.P.: Thermoelasticity without energy dissipation for initially stressed bodies. Int. J. Math. Math. Sci 31, 329–337 (2002) 10. Dey, S., De, P.K.: Velocity of shear waves in an initially stressed incompressible anisotropic medium. S¯adhan¯a 24, 215–220 (1999) 11. Addy, S.K., Chakraborty, N.R.: Rayleigh waves in a viscoelastic half-space under initial hydrostatic stress in presence of the temperature field. Int. J. Math. Math. Sci. 24, 3883–3894 (2005) 12. Gupta, S., Chattopadhyay, A., Kumari, P.: Propagation of shear wave in anisotropic medium. Appl. Math. Sci. 55, 2699–2706 (2007)

Effect of magnetic field and initial stress on the propagation of interface waves

45

13. Othman, M.I.A., Song, Y.: Reflection of plane waves from an elastic solid half-space under hydrostatic initial stress without energy dissipation. Int. J. Solids Struct. 44, 5651–5664 (2007) 14. Honarvar, F., Enjilela, E., Sinclair, A.N., Mirnezami, S.A.: Wave propagation in transversely isotropic cylinders. Int. J. Solids Struct. 44, 5236–5246 (2007) 15. Ahmad, F., Khan, A.: Effect of rotation on wave propagation in a transversely isotropic medium. Math. Probab. Eng. 7, 147–154 (2001) 16. Dey, S., Maulick, B.: Propagation of Love waves in an initially stressed transversely isotropic layer bounded by a rigid layer and a gravitating half-space. Bull. Cal. Math. Soc. 90, 1–10 (1998) 17. Sharma, J.N., Singh, D., Kumar, R.: Generalised thermoelastic waves in transversely isotropic plates. Ind. J. Pure Appl. Math. 34, 841–852 (2003) 18. Bullen, K.E.: An Introduction to the Theory of Seismology. Camb. Univ. Press, London (1963) 19. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Camb. Univ. Press, London (1952) 20. Pal, P.K., Acharya, D.: Effects of inhomogeneity on surface waves in anisotropic media. S¯adhan¯a 23, 247–258 (1998)