Theoretical Mathematics & Applications, vol.3, no.3, 2013, 87-106
ISSN: 1792- 9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
A Mathematical Study of Electro-MagnetoThermo-Voigt Viscoelastic Surface Wave
Propagation under Gravity Involving
Time Rate of Change of Strain
Rajneesh Kakar 1and Arun Kumar 2
Abstract
This research is a mathematical investigation of the propagation of surface
wave in a Voigt viscoelastic medium. A mathematical model for wave
propagation in electro-magneto-thermo heterogeneous viscoelastic isotropic
half space under gravity involving time rate of change of strain of nth order
is purposed. A solution to the partial differential equation of motion is
assumed and is shown to satisfy the two necessary boundary conditions. The
frequency equations for surface waves (Love, Rayleigh and Stoneley waves)
are obtained with the help of Biot’s theory of incremental deformations.
Heterogeneities in the medium are assumed to vary exponentially with
depth. The problem investigated by Bullen [Cambridge University Press, pp.
85-99, 1965] has been reduced as particular case.
1
2
Principal, DIPS Polytechnic College, Tanda, Hoshiarpur, India.
HOD, Department of Physics, SPN College, Mukerian, Hoshiarpur, India.
Article Info: Received: June 10, 2013. Revised: August 15, 2013.
Published online : September 1, 2013
88
Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
Mathematics Subject Classification: 74H45; 74F10; 74J15; 74J20; 74K25
Keywords: viscoelastic solid medium; gravity; inhomogeneities; variable density;
temperature; magnetic field; electric field
1 Introduction
In the earlier times, sufficient interest has been led on wave propagation in
that material whose mechanical characteristics and density are functions of space
i.e. heterogeneous engineering material. It is perhaps due to lack of information
(experimental or empirical) available in the literature concerning the precise
mechanical behaviour of heterogeneous media, that not much work has been
reported in this area of applied mechanics. Wave propagation in inhomogeneous
medium is a challenge for both theoretical research and engineering practice. With
the rapid development of science and technology, wave motion study of the
heterogeneous medium (atmosphere, ocean, earth-crust, functionally graded
materials and cycle grid structure, etc.) seems much more important.
Although most wave equations assume propagation in an elastic medium, it
is well known that many solids do not exactly obey the laws of the theory of
elasticity. The purpose of this research, therefore, is to assume a non-elastic
medium that represented by a Voigt viscoelastic element, and investigate the
conditions necessary for the propagation of a surface wave. To the earthquake
seismologist and to those concerned with predicting the effects of explosives in
solids, the surface wave is one of the most important types of waves that have
been observed. With accurate earthquake seismograms of surface waves, the
thickness of the superficial layer of the earth (the crust) may be determined. On a
smaller scale, in seismic exploration, knowledge of the thickness of the weathered
surface layer is of primary importance
Rajneesh Kakar and Arun Kumar
89
The investigation of viscoelastic wave propagation was initiated by Sezawa
[1], who was concerned primarily with purely dilatational plane waves, and
obtained his solution using Fourier integrals. Many years ago Bromwich [2]
determined the influence of gravity on wave propagation in an elastic solid
medium. Love [3] investigated the effect of gravity on Rayleigh wave velocity.
Earlier, Thomson [4] discussed transmission of elastic waves through a stratified
solid medium. Haskell [5], Ewing, Jardtezky and Press [6] studied wave
propagation. De and Sen [7] presented note on elastic waves. Sen and Acharya [8]
investigated the effect of gravity on waves in a thermoelastic layer. Das et al. [9]
studied magneto-visco-elastic surface waves in stressed conducting media.
Recently, Kakar et al. [10-13] presented many papers on surface waves in
viscoelastic media.
In this paper, we have considered that the surface waves are propagating in
isotropic, viscoelastic heterogeneous medium under the effect of temperature,
electric field, magnetic field and gravity. The problem of nth order viscoelastic
electro-magneto-thermo surface waves (Love, Stoneley and Rayleigh waves)
under gravity involving time rate of strain in heterogeneous medium is studied in
detail.
2 Governing equations
The governing equations of motion for 3D viscoelastic solid medium in
Cartesian co-ordinates with Eq. (1) are [3]
∂τ 11 ∂τ 12 ∂τ 13
∂w
∂2u
+
+
+ ρg
=
,
ρ
∂x
∂x
∂y
∂z
∂ t2
(1a)
∂τ 12 ∂τ 22 ∂τ 23
∂w
∂ 2ν
+
+
+ ρg
=
ρ 2,
∂x
∂y
∂z
∂y
∂t
(1b)
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Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
 ∂u ∂v 
∂τ 13 ∂τ 23 ∂τ 33
∂ 2w
ρ
+
+
− ρg  +  =
.
∂x
∂y
∂z
∂ t2
 ∂x ∂y 
(1c)
where τ ij = τ ji , ∀ i, j are the stress components and ρ is the density of the medium.
We have assumed both the mediums are perfect electric conductor, therefore the
governing equations of motion for such mediums are
 
  
 
 

(2)
∇ ⋅ Ε = 0 , ∇ ⋅ Β = 0 , ∇ × Ε = − B ,t , ∇ × Β = µeε e E ,t .
 
where, Ε , Β , µe and ε e are electric field, magnetic field induction, permeability
and permittivity of the medium.
The value of magnetic field intensity is

 
Η ( 0, 0, Η ) = Η 0 + Η i
(3)


where, magnetic field Η is acting along y-axis. Η i is the perturbation in the
magnetic field intensity.
The stress-strain relations for viscoelastic medium, according to Voigt are [15]
τij= 2Dµ eij + (Dλ ∆) δij
(4a)
Therefore, the stress-strain relations for general isotropic, thermo, magneto,
electro, viscoelastic medium can be written as
τij= 2Dµ eij + (Dλ ∆ – Dβ T + + E02 ∆DEe + H 02 ∆Dme ) δij
where , ∆=
(4b)
∂u ∂v ∂w
, Dλ, Dµ, Dβ , Dm , DE are elastic constants.
+
+
∂x ∂y ∂z
e
e
Let initial temperature of both the medium is kept at constant absolute
temperature T0. Fourier’s law of heat conduction is used to calculate T and it is
given by
2
p∇ T = Cν
∂T
∂
+ T0 GL
∇2 φ ) ,
(
∂t
∂t
(5)
where, K be the thermal conductivity and obeys the law as given by K = K0 emz,
p=
K0
ρ0
and Cν be the specific heat of the body at constant volume.
Rajneesh Kakar and Arun Kumar
91
3 Formulation of the problem
Consider a heterogeneous, thermo-electro-magneto-Voigt viscoelastic,
isotropic, semi-finite media M1 and M2 (as shown in Figure 1).
( u, ν ,
M1
w, T )
z=0
M2
( u ', ν ', w ', T ')
z
Figure 1: Geometry of the problem
The mechanical properties of M1 are different from those of M2. Let the
components of displacements are u, v, w. The Cartesian co-ordinate system (x ,y ,
z) is located at the interface separating the two layers at z = 0. The z-axis is acting
downward.
Introducing Eq. (4a) in Eq. (1a), Eq. (1b), Eq. (1c), we get
Dλ
∂D λ
∂∆
∂2 u
∂T
∂  ∂u ∂w 
∂u ∂D µ
+
+∆
+ 2 Dµ 2 + 2
− Dβ
+ Dµ
+
∂x ∂x
∂x
∂z  ∂z ∂x 
∂x
∂x
∂x
∂ D  ∂ u ∂ w  ∂ Dµ 2
∂D
 ∂ u ∂ w  ∂ Dµ 2
 ∂ z + ∂ x  ∂ z H 0 Dme ∂ x +  ∂ z + ∂ x  ∂ z E0 DEe ∂ x
2
∂ Dme
∂ DEe
∂w
∂ 2u
2
2
+ H0 D
+ E0 D
+ ρg
=
ρ 2,
∂x
∂x
∂x 0
∂t
(6a)
∂2 ν
∂ν ∂ Dµ ∂ν ∂ Dµ
=ρ 2 ,
Dµ ∇ v +
+
∂x ∂x ∂z ∂z
∂t
(6b)
2
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Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
Dµ
∂2 w
∂w ∂D µ
∂  ∂u ∂w   ∂u ∂w  ∂D µ
+ 2 Dµ
+2

 + +
 +
2
∂z ∂z
∂x  ∂z ∂x   ∂z ∂x  ∂x
∂z
+ Dλ
∂ Dm
∂ Dβ
∂ Dλ
∂D
∂T
∂∆
2
2
+ H 0 Dm
+ H0 D
−T
− Dβ
+∆
∂z
∂z
∂z
∂z
∂z
∂z
e
e
∂ DE
∂2 w
∂D
∂u
2
=ρ 2 .
+ E DE
+ E0 D
− ρg
∂x
∂z
∂z
∂t
2
0
e
e
(6c)
We assume that the heterogeneities for the media M1 and M2 are given by
n
Dλ = ∑
K =0
K
K
n
n
∂K
mz ∂
mz ∂
λK e
, Dµ = ∑ µ K e
, Dβ = ∑ β K e
, ρ = ρ0 emz ,
K
K
∂ tK
∂
∂
t
t
K =0
K =0
mz
n
Dme = ∑ ( µe ) K e mz
K =0
K
K
n
N
∂K
mz ∂
mz ∂
=
=
D
ε
e
(
)
D
e
η
,
∑
∑
∂ t K η K =0 K ∂t K E K =0 e K
∂ tK
e
(7)
and
n
D'λ= ∑ λ 'K elz
K =0
n
K
K
n
n
∂K
lz ∂
lz ∂
lz
β
µ
e
'
e
,
D
=
,
ρ'
=
ρ'
e
,
D
=
,
µ ∑
β ∑ K
K
0
∂ tK
∂ tK
∂ tK
K =0
K =0
D 'µe = ∑ ( µ 'e ) K e mz
K =0
K
K
n
N
∂K
lz ∂
mz ∂
=
ε
D
η
e
'
'
'
(
'
)
D
=
e
,
,
∑ K ∂t K ε K∑=0 e K ∂ t K
η
∂ tK
K =0
e
where λ0, M0, λ'0, µ'0ε'0
(8)
are elastic constants, whereas β0, β'0 are thermal
parameters are ρ0, ρ'0, m, n are constants. λK, µK, εK (K = 0,1,2, .... n) are the
parameters associated with Kth order viscoelasticity and βK , (εe)K and (µe)K
(K = 1, 2, ....., n) are the thermal, electric and magnetic parameters associated with
Kth order. T is the absolute temperature over the initial temperature T0. In a thermo
viscoelastic solid, the thermal parameters βK (K = 0, 1, ...... n) are given by
βK = (3λK + 2µK) αt, where, αt be the coefficient of linear expansion of solid.
(G
+ Gµ + H 0 Gme + E0 GEe
2
λ
2
) ∂∂∆x + G
µ
∇2 u
∂T
∂w
∂ 2u
∂u ∂w
+ m Gµ 
+
−
G
+
ρ
g
=
ρ
,
β
0
0

∂x
∂x
∂ t2
∂z ∂x 
(9a)
Rajneesh Kakar and Arun Kumar
Gµ ∇2 v + mGµ
93
∂ 2v
∂v
= ρ0 2 ,
∂t
∂z
(Gλ + Gµ + H 0 Gme + E0 GEe )
2
2
– Gβ
(9b)
∂∆
∂w
+ Gµ ∇2 w + ∆ Gλ m + 2Gµ m
∂z
∂z
∂T
∂u
∂ 2w
2
2
– mGBT+ mH 0 DGme + mE0 DGEe − ρ0 g
=
ρ0 2 .
∂z
∂x
∂t
(9c)
where,
n
Gλ = ∑
K =0
∂K
λK K ,
∂t
n
Gβ = ∑ β K
K =0
∂K
,
∂ tK
n
Gµ = ∑
K =0
∂K
µK K ,
∂t
n
Gε e = ∑ (ε e ) K
K =0
n
∂K
∂K
=
µ
G
(
)
,
∑ e K ∂ tK
µe
∂t K
K =0
∂2
∂2
∂2
∂2
2=
+
∇2 = 2 +
,
∇
.
∂x
∂ z2
∂x 2 ∂z 2
(10)
To investigate the surface wave propagation along the direction of Ox, we
introduce displacement potential φ (x, z, t) and ψ (x, z, t) which are related to the
displacement components as follows:
u=
∂φ ∂ψ
∂φ ∂ψ
,w=
.
−
+
∂z
∂x
∂x
∂z
(11)
The displacement potential φ (x, z, t) and ψ (x, z, t) in Eq. (11) satisfy the
following Laplace equation (known as dilation and rotation and are associated
with P and SV waves)
∇ 2φ =
u, x + w, z =
∆,
∇ 2ψ =
w, x − u, z =Ω
2 .
(11a)
Substituting Eq. (11) in Eqs (9a), (9b) and (9c), we get
GR ∇2 φ + mGs ( 2 φ, z + ψ , x ) – GLT + gψ ,x = φ,tt ,
(12a)
GS ∇2 v + mG v,z = v,tt ,
(12b)
GS ∇2 ψ + mGP φ,x + 2m Gs ψ ,z – Gq ∇4 ψ − gφ,x =ψ ,tt ,
(12c)
Where,
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Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
λK + 2µ K + H 02 (m e ) K + E02 ( Ee ) K
,
U =
ρ0
2
KR
U
2
KP
2
=
U KS
µK
β
2
, U KL
= K ,
ρ0
ρ0
λK + H 02 (me ) K + E02 (E e ) K
=
,
ρ0
and
n
GR= ∑ U
K =0
n
2
KR
2
GL= ∑ U KL
K =0
K
K
n
n
∂K
2 ∂
2 ∂
, GS = ∑ U KS K , GP = ∑ U KP K ,
∂t
∂ tK
∂t
K =0
K =0
K
n
∂K
2 ∂
G
U
=
,
∑ Kq ∂ t K .
q
∂ tK
K =0
(13)
By using Eq. (5), temperature T can be calculated.
Further, similar relations in medium M2 can be found out by replacing λK, µK, βK,
ε’K, ρ0 by λ'K, µ'K, β' K, ε'K, ρ'0 and so on.
4 Solution of the problem
Now our main objective to solve Eq. (12a), Eq. (12b), Eq. (12c) and Eq. (5),
for this, we seek the solutions in the following forms.
(φ, ψ, T, v)= [f (z), V (z), T1 (z), h (z)] eiα(x – ct)
(14)
Using Eq. (14) in Eq. (12a), Eq. (12b), Eq. (12c) and Eq. (5), we get a set of
differential equations for the medium M1 as follows:
d2 f
df
+ 2mf12
+ h12 f + (i α mf12 + iα gJ12 ) j − g12 T1 = 0,
2
dz
dz
d 2h
dh
+m
+ K12 h =
0,
2
dz
dz
d 2g
dg
+ 2m
+ K12 j + (i α ml12 − iα gN12 ) f = 0,
2
dz
dz
Rajneesh Kakar and Arun Kumar
95
 d2 f
d 2T1
2
+
AT
+
B
 2 −α f
1
dz 2
dz


0,
=

(15)
where,
n
∑
K=0
n
f12 =
∑
K=0
U 2KS (− i α c)
K
U 2KR (− i α c)
K
n
∑
K=0
n
∑
l12= Kn=0
∑
K=0
N12 =
U 2KS (− i α c)
U 2KP (− i α c)
U 2KS (− i α c)
1
n
∑ U ( iα c )
K =0
2
KS
− α 2 , J12 =
K
n
∑
U 2KR (− i α c)
∑ U ( iα c )
2
KT
n
K
K
U 2KL (− i α c)
∑
U 2KR (− i α c)
K=0
A=
K
∑
g12 = Kn=0
,
− α2 ,
1
n
K =0
K
,
K
α 2 c2
K=0
α 2 c2
K12 =
h12 =
,
Cν i α c
− α2 ,
p
B=
K
,
(16)
K
i α c T0
GL
p
and those for the medium M2 are given by
d2f
dz
2
+ 2lf1'2
d2h
dz 2
+l
df
+ h1'2 f +(i α l f1'2 + iα gJ1'2 ) j − g1'2 T1 = 0,
dz
dh
+ K1'2 h = 0,
dz
d 2g
dg
+ 2l
+ K1'2 j + (i α l . l1'2 − iα gN1'2 ) f = 0,
2
dz
dz
d 2 T1
dz 2
where,
 d2f

+ A ' T1 + B'  2 − α 2 f  = 0,
 dz

(17)
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Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
n
2
(−i αc)
∑ U'KS
'2
f1 =
K=0
n
K
2
(−i αc)
∑ U'KR
α 2 c2
'2
,
h1 =
n
2
(−i αc)
∑ U'KR
K
α 2 c2
n
2
(−i αc)
∑ U'KS
− α2 ,
J1/2 =
K
1
n
∑ U ( iα c )
K =0
K=0
/2
KT
,
K
n
N1/2 =
1
n
∑ U ( iα c )
K =0
/2
KS
K
(−i αc)
K
∑
K=0
2
(−i αc)
∑ U'KL
g1'2 =
2
(−i αc)
∑ U'KP
l1'2 = Kn=0
,
K
n
K=0
n
K
K=0
K=0
K1'2 =
− α2 ,
,
2
U 'KS
K
,
2
(−i αc)
∑ U'KR
B' =
K
i α c T0
G'L
p'
(18)
K=0
Eq. (15) and Eq. (17) must have exponential solutions in order that f, j, T1 , h
will describe surface waves, and they must become varnishing small as z → ∞.
Hence for the medium M1
{
ψ (x, z, t)= {A
T (x, z, t)= {A
}
}e
}e
φ (x, z, t)= A1 e − λ1z + B1 e − λ 2 z + C1 e − λ 3z e iα ( x −ct )
2
e − λ1z + B2 e − λ 2 z + C2 e − λ 3z
3e
− λ1z
iα ( x − ct )
iα ( x − ct )
+ B3 e − λ 2 z + C3 e − λ 3z
v (x, z, t)= C e − λ 4 z + iα ( x −ct )
(19a)
For finite disturbances as z → ∞, for medium M1 must hold Re (λi)>0 for
i=1,2,3,4,5. Similarly for the medium M2 are given by
{
}
φ (x, z, t)= A '1 e − λ '1 z + B'1 e − λ '2 z + C'1 e − λ '3 z e iα ( x −ct )
Rajneesh Kakar and Arun Kumar
{
T (x, z, t)= {A ' e
97
}
}e
ψ (x, z, t)= A '2 e − λ '1 z + B'2 e − λ '2 z + C'2 e − λ '3 z e iα ( x −ct )
3
− λ '1 z
+ B'3 e − λ '2 z + C'3 e − λ '3 z
iα ( x − ct )
v (x, z, t)= C' e − λ '4 z + iα ( x −ct )
(19b)
For finite disturbances as z → − ∞, for medium M2 must hold Re(λ'i)<0 for
i=1,2,3,4,5. Where, λj and λ'j (j = 1, 2, 3) are the real roots of the Eqns.
λ6 + ξ1 λ5 + ξ2 λ4 + ξ3 λ3 + ξ4λ2+ ξ5λ + ξ6 = 0,
(20)
where,
ξ1 = 2m {1 + f12},
ξ2 = K12 + A + 4m2 + h12 + Bg12,
ξ3 = 2mA + 2f12 m (K12 + A) + 2mh12 + 2mBg12,
ξ4 = AK12 + 4m2A f12 + (K12 + A) h12 + α2 m2 l12 f12 + BK12 g12 – α2 Bg12,
ξ5 = 2mAK12 f12 + 2mAh12 – 2m α2 Bg12,
ξ 6 = AK12 h12 + Aα2 m2 ll2 f12 – α2B K12 g12.
λ'6+ξ'1 λ'5+ξ'2 λ'4+ξ'3 λ'3+ξ'4λ'2+ξ'5λ'+ξ'6= 0
(21)
(22)
where,
ξ'1 = 2l {1 + f1'2},
ξ'2 = K1'2 + A' + 4l2 + h1'2 + B'g1'2,
ξ'3 = 2lA' + 2lf1'2 (K1'2 + A) + 2lh1'2 + 2l B'g1'2,
ξ'4 = A'K1'2 + 4l2A' f1'2 + (K1'2 + A') h1'2 + α2 l2 l1'2 f1'2 + B' K1'2 g1'2 – α2 B' g1'2,
ξ'5 = 2lA'K1'2 f1'2 + 2lA'h1'2 – 2l α2 B'g1'2,
ξ'6 = A'K1'2 h1'2 + A'α2 l2 ll'2 f1'2 – α2B' K1'2 g1'2.
λ4 = {m + (m2- 4 K12) ½}/ 2,
λ'4 = {l + (l2- 4 K1'2) ½}/ 2
(23)
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Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
where the symbol used in eqns. (21) and (23) are given by eqns. (16) and (18).
The constants Aj, Bj, Cj (j = 1, 2, 3) are related with A'j, B'j, C'j (j = 1, 2, 3) in Eq.
(19a) and Eq. (19b) by means of first equations in Eq. (15) and Eq. (17). Equating
the coefficients of e − λ1 z , e − λ2 z , e − λ3 z , e − λ '1 z , e − λ '2 z , e − λ '3 z to zero, after substituting Eq.
(19a) and Eq. (19b) in the first and 3rd equations of Eq. (15) and Eq. (17)
respectively, we get
A2= γ1 A1,
B2 = γ2 B1, C2 = γ3 C1,
A3= δ1 A1,
B3 = δ2 B1,
and
C3 = δ3 C1,
(24)
where,
γj=
δj=
− i α m l12
λ j2 − 2 m λ j + K12
1
g12
(j = 1, 2, 3),
[λj2 – 2m f12 λj + h12 + i α m f12 γj],
j = 1, 2, 3.
Similar result holds for medium M2 and usual symbols replacing by dashes
respectively.
5 Boundary conditions
There are two boundary conditions
(i) The displacement components, temperature and temperature flux at the
boundary surface between the media M1 and M2 must be continuous at all times
and positions.
i.e.
∂T 

 u, ν, w , T, p ∂z  =
M1
∂T 

 u, ν, w , T, p' ∂z 
M2
(ii) The stress components τ31, τ32, τ33 must be continuous at the boundary z = 0.
Rajneesh Kakar and Arun Kumar
99
i.e. [ τ 31 , τ 32 , τ 33 ]M = [ τ 31 , τ 32 , τ 33 ]M at z = 0, respectively
1
2
where,
 ∂2φ
∂2 ψ ∂2 ψ 
,
τ31= D µ  2
+
−
 ∂x ∂z ∂x 2 ∂z 2 
τ32 = D µ
∂ν
,
∂z
 ∂ 2φ
∂ 2φ 
2
2
2
2
τ33 = Dλ ∇ 2 φ + 2 Dµ  2 +
 − DB T + Dm e H 0 ∇ φ + DE e E0 ∇ φ .
 ∂ z ∂ x∂ z 
(25)
Applying the boundary conditions, we get
A1 (1 – i γ1 ζ1) + B1 (1 – i γ2 ζ2) + C1 (1 – i γ3 ζ3) – A'1 (1 – i γ'1 ζ'1)
– B'1 (1 – i γ'2 ζ'2) – C'1 (1 – i γ '3 ζ'3) = 0
C = C'
(26a)
(26b)
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
(26c)
δ1A1 + δ2 B1 + δ3C1 = δ'1A'1 + δ'2 B'1 + δ'3C'1
(26d)
pλ1δ1A1 + pλ2δ2 B1 + pλ3δ3C1 – p' λ'1δ'1A'1 + p' λ'2δ'2 B'1– p'λ'3δ'3C'1 = 0
(26e)
µ *K [(2i ζ1 + γ1 + ζ12 γ1) A1 + (2i ζ2 + γ2 + ζ22 γ2) B1 + (2i ζ3 + γ3 + ζ32 γ3) C1]
= µ' *K [(2i ζ'1 + γ'1 + ζ1'2 γ'1) A'1 + (2i ζ'2 + γ'2 + ζ2'2 γ'2) B'1
+(2i ζ'3 + γ '3 + ζ3'2 γ'3) C'1]
µ *K [– λ4C]= µ' *K [– λ'4 C']
(26f)
(26g)
A1 [( λ *K + ( µe )* K H 0 (ε 'e )* K E02 ) (ζ12 – 1) + 2 µ *K (ζ12 –iζ1) – β*K δ1]
2
+ B1 [( λ *K + ( µe )* K H 0 (ε 'e )* K E02 ) (ζ22 – 1) + 2 µ *K (ζ22 –iζ2) – β*K δ2]
2
+ C1 [( λ *K + ( µe )* K H 0 (ε 'e )* K E02 ) (ζ32 – 1) + 2 µ *K (ζ32 – iζ3) – β*K δ3]
2
= A'1 [( λ' *K + ( µ 'e )* K H 0 (ε 'e )* K E02 )(ζ1'2–1)+2 µ' *K (ζ1'2–iζ'1)– β' *K δ'1]
2
+ B'1 [( λ' *K + ( µ 'e )* K H 0 (ε 'e )* K E02 ) (ζ2'2 – 1) + 2 µ' *K (ζ2'2 – iζ'2) – β' *K δ'2]
2
+C'1[( λ' *K + ( µ 'e )* K H 0 (ε 'e )* K E02 ) (ζ3'2–1) + 2 µ' *K (ζ3'2 – iζ'3) – β' *K δ'3]
2
(26h)
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Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
where,
ζj=
λj
α
λ*K =
ζ'j =
,
n
∑
α
,
j = 1, 2, 3
n
λ K (− i αc) ,
K
K=0
( µe )* K =
λ' j
n
∑
K =0
µ *K =
∑
n
µ K (− i αc) ,
K
K=0
β*K =
∑ β K (−i αc)
∑ (ε ) ( −i α c )
K
,
K=0
n
( µe ) K ( −i α c ) , =
(ε e )* K
K
K
e K
K =0
and
n
∑
λ' *K =
K
K=0
( µ 'e )* K =
λ ' K (− i αc) ,
n
n
µ' *K =
K =0
K
K=0
∑ (µ ' ) ( −i α c )
e K
∑
K
n
µ ' K (− i αc) ,
, =
(ε 'e )* K
β' *K =
∑ β'K (−i αc)
,
K=0
n
∑ (ε ' ) ( −i α c )
K =0
K
K
e K
From Eq. (26b) and Eq. (26h), 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
a11
a12
a13
a14
a15
a16
a21
a31
a22
a32
a23
a33
a24
a34
a25
a35
a26
a36
a41
a51
a42
a52
a43
a53
a44
a54
a45
a55
a46
a56
a61
a62
a63
a64
a65
a66
=0
(27)
where,
a11 = 1 – iγ1 ζ1,
a12 = 1–iγ2ζ2,
a15 = (i γ '2 ζ'2–1),
a13 = 1–iγ3ζ3,
a16 = (i γ'3 ζ'3 – 1),
a21 = γ1 + iζ1,
a22 = γ2 + iζ2,
a25 = (γ'2 + iζ'2),
a26 = (γ'3 + iζ'3),
a31 = δ1,
a32 = δ2,
a14 = (i γ'1 ζ'1–1),
a33 = δ3,
a23 = γ3 + iζ3,
a34 = – δ'1,
a24 = (γ'1 + i ζ'1),
a35 = –δ'2,
a36 = –δ'3,
Rajneesh Kakar and Arun Kumar
a41 = pλ1 δ1,
a45 = –p' λ'2 δ'2,
101
a42 = pλ2 δ2,
a43 = pλ3 δ3,
a44 = –p' λ'1 δ'1,
a46 = –p' λ'3 δ'3,
a51 = µ *K (2i ζ1 + γ1 + γ1 ζ12),
a52 = µ *K (2i ζ2 + γ2 + γ2 ζ22),
a53 = µ *K (2i ζ3 + γ3 + γ3 ζ32),
a54 = µ' *K (2i ζ'1 + γ'1 + γ'1 ζ1'2),
a55 = µ' *K (2i ζ'2 + γ'2 + γ'2 ζ2'2),
a56 = µ' *K (2i ζ'3 + γ'3 + γ'3 ζ3'2),
a61 = ( λ *K + ( µe )* K H 0 + (ε e )* K E02 ) (ζ12 – 1) + 2 µ *K (ζ12 –iζ1) – β*K δ1,
2
a62 = ( λ *K + ( µe )* K H 0 + (ε e )* K E02 ) (ζ22 – 1) + 2 µ *K (ζ22 –iζ2) – β*K δ2,
2
a63 = ( λ *K + ( µe )* K + (ε e )* K E02 H 0 ) (ζ32 – 1) + 2 µ *K (ζ32 – iζ3) – β*K δ3,
2
a64 = ( λ' *K + ( µ 'e )* K H 0 + (ε e )* K E02 ) (ζ1'2–1) + 2 µ' *K (ζ1'2–iζ'1)– β' *K δ'1,
2
a65 = ( λ' *K + ( µ 'e )* K H 0 + (ε e )* K E02 ) (ζ2'2 – 1) + 2 µ' K* (ζ2'2 – iζ'2) – β' *K δ'2,
2
a66 = ( λ' *K + ( µ 'e )* K H 0 + (ε e )* K E02 ) (ζ3'2–1) + 2 µ' K* (ζ3'2 – iζ'3) – β' *K δ'3.
2
(28)
From Eq. (27), we obtain velocity of surface waves in common boundary
between two viscoelastic, heterogeneous solid media under the influence of
thermal, electric and magnetic field, where the viscosity is of general nth order
involving time rate of change of strain.
6 Particular cases
Stoneley Waves:
Eq. (27) determine the wave velocity equation for Stoneley waves in the
case of general magneto-thermo viscoelastic, heterogeneous solid media of nth
order involving time rate of strain. Clearly from Eq. (27), it is follows that the
wave velocity equation for Stoneley waves depends upon the heterogeneity of the
material medium, temperature, electric, magnetic and viscous field. This equation,
of course, is in well agreement with the corresponding classical result, when the
102
Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
effects of thermal, electric, magnetic and viscous field and heterogeneity are
absent.
Rayleigh Waves:
To investigate the possibility of Rayleigh waves in a electro, magneto,
thermo viscoelastic, heterogeneous elastic media, we replace media M2 by
vacuum, in the proceeding problem; we also note the SH-waves do not occur in
this case.
Since the temperature difference across the boundary is always small so thermal
condition given by
∂T
+ hT = 0 at z = 0, respectively
∂z
(28)
Thus Eq. (26f) and Eq. (26h) reduces to,
(2i ζ1 + γ1 + γ1 ζ12) A1 + (2i ζ2 + γ2 + γ2 ζ22) B1 + (2i ζ3 + γ3 + γ3 ζ32) C1 = 0
(29a)
[( λ *K + ( µe )* K H 0 + (ε e )* K E02 ) (ζ12 – 1) + 2 µ K* (ζ12 –iζ1) – β*K δ1] A1
2
+ [( λ *K + ( µe )* K H 0 + (ε e )* K E02 ) (ζ22 – 1) + 2 µ *K (ζ22 –iζ2) – β*K δ2] B1
2
+ [( λ *K + ( µe )* K H 0 + (ε e )* K E02 ) (ζ32 – 1) + 2 µ *K (ζ32 – iζ3) – β*K δ3] C1 = 0
2
(29b)
From Eq. (28), we have
(λ1 – h) δ1 A1 + (λ2 – h) δ2 B1 + (λ3 – h) δ3 C1 = 0
(29c)
Eliminating A1, B1 and C1 from Eqns. (29a), (29b) and (29c), we get
det (bij)= 0, i, j = 1, 2, 3.
where,
b11 = (2i ζ1 + γ1 + γ1 ζ12), b12 = (2i ζ2 + γ2 + γ2 ζ22), b13 = (2i ζ3 + γ3 + γ3 ζ32),
b21 = [( λ *K + ( µe )* K H 0 + (ε e )* K E02 ) (ζ12 – 1) + 2 µ *K (ζ12 –iζ1) – β*K δ1],
2
b22 = [( λ *K + ( µe )* K H 0 + (ε e )* K E02 ) (ζ22 – 1) + 2 µ *K (ζ22 –iζ2) – β*K δ2],
2
b23 = [( λ *K + ( µe )* K H 0 + (ε e )* K E02 ) (ζ32 – 1) + 2 µ *K (ζ32 – iζ3) – β*K δ3],
2
(30)
Rajneesh Kakar and Arun Kumar
103
b31 = (λ1 – h) δ1,
b32= (λ2 – h) δ2,
b33 = (λ3 – h) δ3.
(31)
Thus, Eq. (30) gives the wave velocity equation for Rayleigh waves in a
heterogeneous, electro, magneto-thermo viscoelastic solid media of nth order
involving time rate of strain.
From Eq. (30), it is follows that dispersion equation of Rayleigh waves
depends upon the heterogeneity, the viscous, magnetic and thermal fields.
When the effects of thermal, electro, magnetic viscous field and heterogeneity are
absent, this equation, of course, is in complete agreement with the corresponding
classical result by Bullen [14].
Love Waves:
To investigate the possibility of love waves in a heterogeneous, 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
component ν remains same as in general case given by equation (19). 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' = C1 eλ '4 z + i α ( x −ct ) + C2 e − λ '4 z + i α ( x −ct )
(32)
In this case, the boundary conditions are
(i) v and τ32 are continuous at z = 0
(ii) τ'32 = 0 at z = –H.
Applying boundary conditions (i) and (ii) and using eqns (19) and (26), we get
C= C1 + C2
(33)
104
Mathematical Study of Electro-Magneto-Thermo-Voigt Viscoelastic …
– µ *K λ4C = (µ'K)* [λ'4C1 – λ'4C2]
(34)
C1 e − λ '4 H − C2 eλ '4 H = 0
(35)
On eliminating the constants C, C1 and C2 from Eqns. (33), (34) and (35), we get
tanh (λ'4H) =-
λ4 µ K*
.
λ '4 ( µ 'K ) *
(36)
Thus Eq. (36) gives the wave velocity equation for Love waves in a
heterogeneous, electro, magneto, thermo viscoelastic solid medium of nth order
involving time rate of strain. Clearly it depends upon the heterogeneity, magnetic
and viscous fields and independent of thermal field.
7 Conclusions
•
The time rate of strain parameters influence the wave velocity of surface
waves to an extent depending on the corresponding constants characterizing
the electro-magneto thermo 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 heterogeneity of the material medium.
•
It has been observed that temperature has no effect on Love waves. However,
viscosity, gravity, magnetic fields, electric fields and heterogeneity of the
medium effects the propagation of Love waves.
•
The dispersion of Rayleigh waves is observed due to the presence of
heterogeneity, temperature, gravity, magnetic field, electric field and viscosity
of the medium.
•
The wave velocity equation of Stoneley waves is very similar to the
corresponding problem in the classical theory of elasticity. The dispersion of
waves is due to the presence of heterogeneity, gravity, magnetic field, electric
Rajneesh Kakar and Arun Kumar
105
field, temperature and viscoelasticity of the solid. Also, wave velocity
equation of this generalized type of surface wave is in complete agreement
with the corresponding classical result in the absence of all fields and
heterogeneity.
•
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 referees for their valuable
comments.
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