Propagation of Surface Waves in Magneto-Thermoelastic Materials with Voids and Initial Stress

Journal of Mechanical Engineering and Technology (JMET), ISSN 2347-3924 (Print),
ISSN 2347-3932 (Online), Volume 1, Issue 1, July -December (2013)
28
GL MODEL ON PROPAGATION OF SURFACE WAVES IN
MAGNETO-THERMOELASTIC
MATERIALS WITH VOIDS AND INITIAL STRESS
S. M. Abo-Dahab
Math. Dept., Faculty of Science, Taif University, Saudi Arabia
ABSTRACT
In this paper, the surface waves propagation in a generalized magneto-thermoelastic
materials taking (Green Lindsay) model with voids and initial stress is investigated. The basic
governing equations have been formulated in xz-plane and the magnetic field is considered in
y-axis that acts perpendicular to the wave propagation. Lame's potential method is applied to
solve the problem. The boundary conditions that the continuity of forces stresses and
Maxwell's stresses components, displacement components, heat flux, temperature and
volume fraction field are estimated at the interfaces between two dissimilar half-space to
obtain the frequency equation of the surface waves in the medium. Some special cases with
neglecting: (i) the magnetic field and initial stress, (ii) the magnetic field, initial stress and
voids parameters, (iii) the magnetic field, initial stress and thermal parameters, and (iv) the
magnetic field, initial stress, thermal field and voids parameters are deduced as special cases
from this study.
Keywords: Rotation, thermoelasticity, magnetic field, surface waves, initial stress, voids.
1. INTRODUCTION
In recent years, more attentions have been given to the initial stress on waves with
thermal field, magnetic field and voids under relaxation times because of its utilitarian
aspects of Seismic waves, Earthquakes, Volcanoes and Acoustics. In the classical theory of
thermoelasticity, when an elastic solid is subjected to a thermal disturbance, the effect is felt
in a location far from the source, instantaneously. This implies that the thermal wave
propagates with infinite speed, a physically impossible result. In contrast to conventional
thermoelasticity, non-classical theories came into existence during the last part of the 20th
century. For example, Lord and Shulman [1], by incorporating a flux-rate term into Fourier's
law of heat conduction, formulated a generalized theory which involves a hyperbolic heat
transport equation admitting finite speed for thermal signals. Green and Lindsay [2], by
including temperature rate among the constitutive variables, developed a temperature-rate
dependent thermoelasticity that does not violate the classical Fourier's law of heat
conduction, when body under consideration has a center of symmetry and this theory also
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ISSN 2347-3924 (Print)
ISSN 2347-3932 (Online)
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29
predicts a finite speed for heat propagation. Chandrasekharaiah [3] referred to this wavelike
thermal disturbance as second sound. The Lord and Shulman theory of generalized
thermoelasticity was further extended by Dhaliwal and Sherief [4] to include the anisotropic
case. A survey article on representative theories in the range of generalized thermoelasticity
is due to Hetnarski and Ignaczak [5]. The reflection of thermoelastic waves from the free
surface of a solid half-space and at the interface between two semi-infinite media in welded
contact, in the context of generalized thermoelasticity is investigated by Sinha and Sinha [6],
Sinha and Elsibai ([7], [8]). Abd-Alla and Al-Dawy [9] studied the reflection phenomena of
SV waves in a generalized thermoelastic medium. Sharma et al. [10] investigated the
problem of thermoelastic wave reflection from the insulated and isothermal stress-free as
well as rigid fixed boundaries of a solid half-space in the context of different theories of
generalized thermoelasticity.
Theory of linear elastic materials with voids is an important generalization of the
classical theory of elasticity. The theory is used for investigating various types of geological
and biological materials for which classical theory of elasticity is not adequate. The theory of
linear elastic materials with voids deals the materials with a distribution of small pores or
voids, where the volume of void is included among the kinematics variables. The theory
reduces to the classical theory in the limiting case of volume of void tending to zero. Non-
linear theory of elastic materials with voids was developed by Nunziato and Cowin [11].
Cowin and Nunziato [12] developed a theory of linear elastic materials with voids to study
mathematically the mechanical behavior of porous solids. Puri and Cowin [13] studied the
behavior of plane waves in a linear elastic material with voids. Iesan [14] developed the
linear theory of thermoelastic materials with voids.
Dhaliwal and Wang [15] formulated the heat-flux dependent thermoelasticity theory
of an elastic material with voids. This theory includes the heat-flux among the constitutive
variables and assumes an evolution equation for the heat-flux. Ciarletta and Scalia [16]
developed a nonlinear theory of non-simple thermoelastic materials with voids. Ciarletta and
Scarpetta [17] studied some results on thermoelasticity for dielectric materials with voids.
Marin [18-20] studied uniqueness and domain of influence results in thermoelastic bodies
with voids. Chirita and Scalia [21] studied the spatial and temporal behavior in linear
thermoelasticity of materials with voids. A theory of thermoelastic materials with voids and
without energy dissipation is developed by Cicco and Diaco [22]. Ciarletta et al. [23]
presented a model for acoustic wave propagation in a porous material which also allows for
propagation of a thermal displacement wave. Singh [24] studied the wave propagation in a
homogeneous, isotropic generalized thermoelastic half space with voids in context of Lord
and Shulman theory. Ciarletta et al. [25] studied the linear theory of micropolar
thermoelasticity for materials with voids. Recently, Aoudai [26] derived the equations of the
linear theory of thermoelastic diffusion in porous media based on the concept of volume
fraction.
Ahmad and Khan [27] studied the problem of thermoelastic plane waves in a rotating
isotropic medium. Birsan [28] investigated the existence and uniqueness of weak solution in
the linear theory of elastic shells with voids. Abd-alla, et al. [29] studied the reflection of the
generalized magneto-thermo-viscoelastic plane waves. Kumar and Rani [30] discussed the
deformation due to moving loads in thermoelastic body with voids. Abd-Alla et al. [31]
pointed out magneto-thermo-viscoelastic interactions in an unbounded body with a spherical
cavity subjected to a periodic loading. Ieşan [32] explained the nonlinear plane strain of
elastic materials with voids.

30
Recently, Abo-Dahab [33] investigated the propagation of P waves from stress-free
surface elastic half-space with voids under thermal relaxation and magnetic field. Singh and
Pal [34] discussed surface waves propagation in a generalized thermoelastic material with
voids. this paper is motivated by the linear theory of thermoelasticity with voids developed
by Iesan [14]. Recently, Singh and Pal [34] studied the surface wave propagation in a
generalized thermoelastic material with voids which it the special case from this paper
without effect all of the magnetic field and rotation.
In the present paper, the surface waves propagation in a generalized magneto-
thermoelastic materials taking GL model with voids and initial stress is investigated. In
Section 2, the governing equations are generalized with the help of Green and Lindsay theory
[2]. These equations are solved for general solutions. In Sections 4 and 5, the particular
solutions are obtained and applied at required boundary conditions to obtain the frequency
equation of surface waves in thermoelastic material with voids. In Section 6, some limiting
cases of the problem are discussed and the results obtained are calculated numerically and
presented graphically. In the last section, some concluding remarks are given.
NOMENCLATURE
α, b, m, ξ are the void material parameters,
αt is the coefficient of linear thermal expansion,
β=αt(3λ+2µ),
δij is the Kronecker delta,
η is the entropy per unit mass,
0 0, / 1,T T TΘ = − Θ < <
λ and µ are Lame's parameter,
eµ is the magnetic permeability,
ρ is the density,
σij are the components of stress vector,
ijτ are the components of Maxwell's stresses vector,
0 1andτ τ are the thermal relaxations parameters,
Φ is the change in the volume fraction field,
χ is the equilibrated inertia,
ω is the frequency,
B
uur
is the magnetic induction vector,
Ce is the specific heat per unit mass,
eij are the components of strain tensor,
E
r
is the electric entensity vector,
→
F is Lorentz'
s body force
g is the intrinsic equilibrated body forces,
H
uur
is the magnetic field vector,
h
ur
is the perturbed magnetic field vector,
J
ur
is the electric intensity vector,
k is the wave number,
K is the thermal conductivity,

31
m is the thermo-void coefficient,
P is the initial stress,
qi are the components of the heat flux vector,
Si are the components of the equilibrated stress vector,
t is the time,
T₀ is the natural temperature of the medium,
T is the absolute temperature,
ui are the components of the displacement vector,
c is the phase speed.
2- GOVERNING EQUATIONS
The governing equations for an isotropic, homogeneous elastic solid with generalized
thermoelasticity with voids and incremental heat flux at reference temperature T₀ taking into
our account GL model and the fields (thermal, voids and elastic) are given as follows
11 2 ,ij k k ij ij ije b P e Pw
t
σ λ β τ δ µ
 ∂  
= − + Θ + Φ − + −  ∂  
(1)
..
0 , ,i iiq q Kτ+ = Θ (2)
, ,i iS α= Φ (3)
,k ke mρ η β α= + Θ + Φ (4)
,k kg b e mξ= − − Φ + Θ (5)
iiqT ,
.
0 =ηρ , (6)
( ) ( ). , . ,
1 1
, .
2 2
ij i j j i ij j i i je u u w u u= + = − (7)
The Maxwell'
s electro-magnetic stress tensor ij
τ is given by
( )( ). .ij e i j j i k k ijH h H h H hτ µ δ= + − (8)
The equation of motion
iijji uF
..
, ρσ =+ (9)
which tends to
..
. , 1 , ,1 .
2 2
ii jj j ij i i i
P P
u u b F u
t
µ λ µ β τ ρ
∂     
− + + − − + Θ + Φ + =     
∂     
(10)
The equation of heat conduction under GL model
. .. . . ..
( ) ( ) ,,0 0 0 0 ,C T u m T Kk ke iiρ τ β τΘ + Θ + + Φ + Φ = Θ (11)
..
, ,ii k kb u mα ξ ρ χΦ − − Φ + Θ = Φ (12)

32
where
.F J B
→ → →
= × (13)
Consider that the medium is a perfect electric conductor, we take the linearized
Maxwell'
s equations governing the electromagnetic field, taking into account absence of the
displacement current (SI) as the form
, ,
0 , 0
e
h
cu rl h J cu rl E
t
d iv h d iv E
µ
→
→ → →
→ →

∂ = = − 
∂ 

= = 
(14)
where






×=
→→→
0Hucurlh (15)
where, we have used
( ) ( )0 0 0, , , 0 , , 0 .H H h x z t H H
→ → → →
= + = (16)
For two-dimensional motion in xz-plane, Eqs. (10)-(12) written as
22
2 2 31
0 02
2 2
11 1
2 2
2
2 2
,
2
e e
uuP P
H H
x x z
u uP
b
z x x t
λ µ µ λ µ µ
µ β τ ρ
∂∂   
+ + − + + + −   
∂ ∂ ∂   
∂ ∂∂ Θ ∂ Φ 
+ − − + = 
∂ ∂ ∂ ∂ 
(17)
2 2
2 23 1
0 02
2 2
13 3
2 2
2
2 2
,
2
e e
u uP P
H H
z x z
u uP
b
x z z t
λ µ µ λ µ µ
µ β τ ρ
∂ ∂   
+ + − + + + −   
∂ ∂ ∂   
∂ ∂∂ Θ ∂ Φ 
+ − − + = 
∂ ∂ ∂ ∂ 
(18)
22
0 31
0
2 2
0
0 2 2
,
e
uu
C T
t x t z t
m T K
t x z
ρ τ β
τ
 ∂∂∂Θ
+ + 
∂ ∂ ∂ ∂ ∂ 
 ∂Φ ∂ Θ ∂ Θ
+ = + 
∂ ∂ ∂ 
(19)
2 2
31
2 2
uu
b m
x z x z t
α ξ ρ χ
  ∂∂∂ Φ ∂ Φ ∂ Φ 
+ − + − Φ + Θ =   
∂ ∂ ∂ ∂ ∂  
(20)
where
0 1
0 11 , 1 .
t t
τ τ τ τ
∂ ∂
= + = +
∂ ∂
The displacement components u1 and u3 may be written in terms of the scalar and the
vector potential functions, φ and ψ, respectively, as the following form
., 31
xz
u
zx
u
∂
∂
+
∂
∂
=
∂
∂
−
∂
∂
=
ψφψφ
(21)

33
Substituting from Eq. (21) into Eqs. (17)-(20), we get
..
2 2 1
,TC bφ β τ φ∇ − Θ + Φ = (22)
,
..
22
ψψ =∇SC (23)
. . .
2 2
1 2 ,ε ε φ ε 
∇ Θ = Θ + ∇ + Φ 
 
(24)
..
2 2
b mα ξ φ ρ χ∇ Φ − Φ − ∇ + Θ = Φ (25)
Where
2
0
2 2
0 0
1 20 0
2
2 2, , ,
, , , .
e
T S
e e e
P P
H
C C
T m Tb K
b
C C C
λ µ µ µ β
β
ρ ρ ρ
β
ε ε ε
ρ ρ τ ρ τ ρ

+ + − − 
= = = 


= = = = 

(26)
3. SOLUTION OF THE PROBLEM
For the analytical solution of Eqs. (22), (24) and (25) in the form of the harmonic
traveling wave, we suppose that the solution takes the form
[ ]( ) [ ] [ ]1 1 1, , , , ( ), ( ), ( ) exp ( ) .x z t z z z ik x ctφ φΘ Φ = Θ Φ − (27)
Substituting from Eq. (27) into Eqs. (22), (24) and (25), we get
( )2 2 2 2 1
1 1 1( ) ( ) ( ) ( ) 0,TC D k z z b zω φ τ β− + − Θ + Φ = (28)
( )2 2 2 2
1 1 1 2 1( ) ( ) ( ) ( ) ( ) 0,i D k z D k i z i zωε φ ε ω ωε− + − + Θ + Φ = (29)
( )2 2 2 2 2
1 1 1( ) ( ) ( ) ( ) ( ) 0b D k z m z D k zφ α ξ ρ χω− − + Θ + − − + Φ =
(30)
where
2
2
2
, .
d
D kc
dz
ω= = (31)
Eliminating the constants φ ₁, Θ₁ and Φ₁ from Eqs. (28)-(30), we obtain
( ) ( ) ( ) 022232
=+++ QDNDMDL (32)
where

34
( ) ( )
( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )( )
2
T
2 2
T
2 1
T 1
2 1 2
T 1
2 2
T
1 2 1 2
1 2
2 2
T 2
2
L C ,
C k²- i k²
² C k² i ,
C i k² k²
² C k² k²- i - k²
i ,
² C k² i k² k²-
i
T
M
b b
N i
b m k b C m
Q i m
k
αε
ε ρχω α ξ α ω ε
αε ω ωαε τ β ε
ω ε ωε τ β ρχω α ξ
ω ε ρχω α ξ α ω ε
ω ε ατ β ε τ β
ω ω ε ρχω α ξ ωε
ω
=
 = − + − 
+ − + +
 = − + − − 
 + − − + 
 + − + −
 
 = − − − − 
− ( )
( )
1 2 1
1 2 1
2
k²-
i k²
b b m
b b k
ε τ β ρχω α ξ ε τ β ε
ω ε















 − + +  

− − 
(33)
we suppose that m1, m2 and m3 are the roots of the equation (32), then we can write the general
solutions φ , Θ and Φ in the forms
( ) 3 31 2 1 2 ( )
1 2 3 4 5 6, , ,m z m zm z m z m z m z ik x ct
x z t A e A e A e A e A e A e eφ −− − −
 = + + + + + 
(34)
( ) 31 2 1 2
3
1 1 2 2 3 3 1 4 2 5
( )
3 6
, ,
,
m zm z m z m z m z
m z ik x ct
x z t A e A e A e A e A e
A e e
η η η η η
η
−− −
−
Φ = + + + +
+ 
(35)
( ) 31 2 1 2
3
1 1 2 2 3 3 1 4 2 5
( )
3 6
, , m zm z m z m z m z
m z ik x ct
x z t A e A e A e A e A e
A e e
ξ ξ ξ ξ ξ
ξ
−− −
−
Θ = + + + +
+ 
(36)
where
( ) ( )
( )
( )
2 2 2 2 2 2
2 1
2 2 1
2
1 2 2 2 2
,
, 1,2,3.
T n n
n
n
n T n
n
i C m k i b m k
b m k i i
C m k
n
b
ωε ω ωε
η
ε ω ωε τ β
τ βη ω
ξ
 − + − −  =
 − + +   
 − − +  = =

(37)
The general solutions ψ of the equation (23) is getting as the form
( ) [ ] )(
21
44
,, ctxikzmzm
eeBeBtzx −−
+=φ (38)
where
2 2
4 .m k
ρ
ω
µ
= − (39)

35
4. FORMULATION OF THE PROBLEM
We consider plane waves propagate through the two semi-infinite half-spaces of thermo-
elastic solid with voids, which we identify as the region 0>z the medium [ ]mbM ,,,,, βαµλ
and the region 0<z the medium [ ]'''''''
,,,,, mbM βαµλ as shown in Figure 1.
For medium M
( ) 31 2 ( )
1 2 3, , ,m zm z m z ik x ct
x z t A e A e A e eφ −− − −
 = + +  (40)
( ) 31 2 ( )
1 1 2 2 3 3, , ,m zm z m z ik x ct
x z t A e A e A e eη η η −− − −
 Φ = + +  (41)
( ) 31 2 ( )
1 1 2 2 3 3, , ,m zm z m z ik x ct
x z t A e A e A e eξ ξ ξ −− − −
 Θ = + +  (42)
( ) 4 ( )
1, , .m z ik x ct
x z t B e eψ − −
= (43)
For medium '
M
( )
'' '
31 2' ' ' ' ( )
1 2 3, , ,m zm z m z ik x ct
x z t A e A e A e eφ − = + +
  (44)
( )
'' '
31 2' ' ' ' ' ' ' ( )
1 1 2 2 3 3, , ,m zm z m z ik x ct
x z t A e A e A e eη η η − Φ = + +
  (45)
( )
'' '
31 2' ' ' ' ' ' ' ( )
1 1 2 2 3 3, , ,m zm z m z ik x ct
x z t A e A e A e eξ ξ ξ − Θ = + +
  (46)
( )
'
4' ' ( )
1, , .m z ik x ct
x z t B e eψ −
= (47)
5. BOUNDARY CONDITIONS
M '
z M
Figure (1): Schematic of the problem
Magneto-thermoelastic solid half-space
with voids and initial stress
x
O
Magneto-thermoelastic solid half-space
with voids and initial stress

36
The boundary conditions at the surface take the following form







=Φ=ΦΘ=Θ==
∂
Φ∂
=
∂
Φ∂
∂
Θ∂
=
∂
Θ∂
+=++=+
0,,,,
,,
,,
'''
33
'
11
'
2
'
1
''''
zatuuuu
zzzz
zxzz zxzxzxzzzzzz
χχ
τστστστσ
(48)
where
( )
( ) α
α
χ
τ
τ
χ
'
2'
0
0
'
1 ,
1
1
=
−
−
=
ikcK
ikcK
(49)
we can write equations (47) in the forms
2 2
2 2 1
0 2
2 ' 2 '
' 2 2 ' ' 1 ' '
0 2
2 2 2 2 ' 2 ' 2 '
' '
2 2 2 2
' '
1 2
2
2
2 ,
2
2 2 ,
,
e
e
P
H b
z x z
P
H b
z x z
x z x z x z x z
z z z
φ ψ
λ µ φ µ βτ
φ ψ
λ µ φ µ βτ
φ ψ ψ φ ψ ψ
µ µ µ µ
χ χ
 ∂ ∂ 
+ − ∇ + + − Θ + Φ =  
∂ ∂ ∂   
 ∂ ∂ 
+ − ∇ + + − Θ + Φ  
∂ ∂ ∂   
   ∂ ∂ ∂ ∂ ∂ ∂
+ − = + −   
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂   
∂Θ ∂Θ ∂Φ ∂Φ
= =
∂ ∂ ∂ ∂
' ' ' '
' '
,
, ,
, , 0.
z
x z x z z x z x
at z
φ ψ φ ψ φ ψ φ ψ













∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 
− = − + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

Θ = Θ Φ = Φ = 
(50)
Substitution from equations (1), (8), (21) and (27) into the boundary conditions equations
(48) we get
0
8887868584838281
7877767574737271
6867666564636261
5857565554535251
4847464544434241
3837363534333231
2827262524232221
1817161514131211
=
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
(51)
where

37
( )( )
( )( )
( )( )
( )( )
( )( )
2 2 2 2 1
11 0 1 1 1 1
2 2 2 2 1
12 0 2 2 2 2
2 2 2 2 1
13 0 3 3 3 3
' '
14 4 18 4
' ' 2 '2 2 '2 1 ' ' ' '
15 0 1 1 1 1
' ' 2 '2 2 '2 1
16 0 2 2
2 ,
2 ,
2 ,
2 , 2 ,
2 ,
2
e
e
e
e
e
a H m k m b
a H m k m b
a H m k m b
a i km a i k m
a H m k m b
a H m k m
λ µ µ τ βζ η
λ µ µ τ βζ η
λ µ µ τ βζ η
µ µ
λ µ µ τ β ζ η
λ µ µ τ β
= + − + − +
= + − + − +
= + − + − +
= − =
= + − + − +
= + − + −
( )( )
' ' ' '
2 2
' ' 2 '2 2 ' '2 1 ' ' ' '
17 0 3 3 3 3
,
2 ,e
b
a H m k m b
ζ η
λ µ µ τ β ζ η
+
= + − + − +
( )
( )
2 2
21 1 22 2 23 3 24 4
' ' ' ' ' ' ' 2 '2
25 1 26 2 27 3 28 4
2 , 2 , 2 , ,
2 2 2 , ,
a i km a i km a i km a k m
a i k m a i km a i k m a k m
µ µ µ µ
µ µ µ µ
= − = − = − = − +
= = = = − +
31 1 1 32 2 2 33 3 3 34 38
' ' ' ' ' '
35 1 1 1 36 1 2 2 37 1 3 3
, , , 0,
,
a m a m a m a a
a m a m a m
η η η
χ η χ η χ η
= − = − = − = =
= = =
41 1 1 42 2 2 43 3 3 44 48
' ' ' ' ' '
45 2 1 1 46 2 2 2 47 2 3 3
, , , 0,
, , ,
a m a m a m a a
a m a m a m
ξ ξ ξ
χ ξ χ ξ χ ξ
= − = − = − = =
= = =
'
51 52 53 55 56 57 54 4 58 4, ,a a a a a a ik a m a m= = = = = = = = −
61 1 62 2 63 3 64 68
' ' '
65 1 66 2 67 3
, , , ,
, , ,
a m a m a m a a ik
a m a m a m
= − = − = − = =
= = =
,,,
,0,,,
'
377
'
276
'
175
7874373272171
ηηη
ηηη
===
=====
aaa
aaaaa
81 1 82 2 83 3 84 88
' ' '
85 1 86 2 87 3
, , , 0,
, , .
a a a a a
a a a
ξ ξ ξ
ξ ξ ξ
= = = = =
= = =
If the initial stress and magnetic field are neglected, the results obtained are deduced to the
relevant results obtained by Singh and Pal [34].
6. NUMERICAL RESULTS AND DISCUSSION
The following values of elastic constants are considered Singh [24], for mediums 1M and
2
M , respectively.

38
3 10 2 10 2 3
/ , 2.17 10 , 3.278 10 , 1.04 10 / . ,
5 2 6 2 2 1 1 1
3.688 10 , 2.68 10 , 1.7 10 deg ,
10 6 15 10
0.04, 1.13849 10 , 2 10 , 1.753 10 , 1.475 10
3 3 1
' 2.66 10 / , ' 5.65 10
2.7Kg m Nm Nm C J kg Kv
Nm Nm K Jm s
k b m
Kg m
ρ λ µ
α β
χ ξ
ρ λ
− −
= = × = × = ×
− − − − − −
= × = × = ×
−
= = × = × = × = ×
= × = ×
9
5
0 2 2 3
, ' 2.46 10 , ' 0.787 10 / . ,
9 2 9 2 3 1 1 1
' 1.28 10 , ' 220.90 10 , ' 0.0921 10 deg ,
10 15 10
' 0.05, ' 1.123 10 , ' 2 10 , ' 1.631 10 , ' 1.352 10
Nm Nm C J kg Kv
Nm Nm K Jm s
k b m
µ
α β
χ ξ
− −
= × = ×
− − − − −
= − × = × = ×
−
= = × = × = × = ×
Using these values, it was found that
00.01, 298T Kω = = .
In order to gain physical insight the roots, Stoneley wave velocity )Re(∆ and attenuation
coefficient ( )I ∆ , have been discussed by assigning numerical values to the parameter
encountered in the problem in which the numerical results are displayed with the graphical
illustrations. The variations are shown in Figs. (2)-(7), respectively.
Figs. (2) and (3) show the variations of the roots of equation (32) with respect to the
initial stress P with varies values of the magnetic field for the two medium M1 and M2. From
Fig. (2), it is appear that the roots m1, m2 and m3 for medium M1 have oscillatory behavior in
the whole range of the P-axis for different values of the magneti fieldH and decrease with an
increasing of the initial stress P. From Fig. (3), it is clear that the roots ' '
1 2,m m and '
3m for
medium M2 increase with an increasing of P but decreases with an increasing of the magnetic
field. It is seen that for large values of the magnetic field, '
1m and '
2m have an oscillatory
behavior with the variation of P.
Fig. 2: Variation of the the absolute values of the roots 1 2 3, andm m m with respect to
initial stress with varies values of the magnetic field

39
Fig. 3: Variation of the the absolute values of the roots ' ' '
1 2 3, ,m m m with respect to
initial stress with varies values of the magnetic field
Fig. 4: Variation of the Stoneley waves velocity and attenuation coefficient with respect to
phase speed c with varies values of relaxation times with and without magnetic field

40
phase speed c with varies values of relaxation times with and without initial stress
phase speed c with varies values of initial stress with and without magnetic field

41
Fig. 7: Variation of the Stonely waves velocity and attenuation coefficient with respect to
phase speed c with varies values of magnetic field with and without initial stress
Fig. (4) displays the variation of the Stonely waves velocity and attenuation
coefficient with respect to phase speed c with varies values of relaxation times with and
without magnetic field. It is clear that Stoneley waves velocity decreases with the increased
values of the phase speed c and with an increasing of the relaxation times, but the attenuation
coefficient increases with an increasing of the relaxation times and the small values of phase
speed c and after that return decreases with the large values of c, also, it is shown that the
Stoneley waves velocity and attenuation coefficients take large values with the presence of
the magnetic field comparing with the corresponding values in the absence of H. From Fig.
(5), it is clear that the stoneley waves velocity and attenuation coefficient decrease with an
increasing of c, also, it is obvious that Stonely waves velocity decreases with an increasing of
the relaxation times in the presence of the magnetic field but increases with the relaxation
times variation in the absence of H, vice versa for the attenuation coefficients.
Fig. (6) plots the Stoneley waves velocity and attenuation coefficient with respect to
phase speed c with varies values of initial stress with and without magnetic field, it is obvious
that Stoneley waves velocity decreases with the increasing values of the initial stress P but
the attenuation coefficient increases, also, it is seen that they tends to zero as c=25. From Fig.
(7), we can see that Stoneley waves velocity and attenuation coefficient increase with an
increasing of the magnetic field H. Finally, it is clear that Stonely waves velocity takes small
values in the presence of the initial stress P comparing with the corresponding values in the
absence of P but the attenuation coefficient takes the largest values in the presence of P
comparing with it in the absence of P.
7. CONCLUSION
It is clear from the previous results that the effect of initial stress, magnetic field and
thermal relaxation time on velocity of Stoneley waves and attenuation coefficient in magneto-
thermoelastic materials with voids, we have the values of this waves increased or decreased,
with increasing of the value of magnetic field, initial stress and thermal relaxation time Due

42
to the complicated nature of the governing equations for the generalized magneto-
thermoelasticity theory, the work done in this field is unfortunately limited in number. The
method used in this study provides a quite successful in dealing with such problems. This
method gives exact solutions in the elastic medium without any assumed restrictions on the
actual physical quantities that appear in the governing equations of the problem considered.
Important phenomena are observed in all these computations:
1 - The influence of the initial stress on the roots via the two media M and M' has an
pronounces effects.
2-The Stoneley waves velocity and attenuation coefficients take large values with the
presence of the magnetic field comparing with the corresponding values in the absence of H.
3-Stoneley waves velocity decreases with an increasing of the relaxation times in the
presence of the magnetic field but increases with the relaxation times variation in the absence
of H, vice versa for the attenuation coefficients.
4- Stoneley waves velocity decreases with the increasing values of the initial stress P but the
attenuation coefficient increases, also, it is seen that they tend to zero as c=25.
5- Stoneley waves velocity and attenuation coefficient increase with an increasing of the
magnetic field H.
6- Stonely waves velocity takes small values in the presence of the initial stress P comparing
with the corresponding values in the absence of P but the attenuation coefficient takes the
largest values in the presence of P comparing with it in the absence of P.
The results presented in this paper should prove useful for researchers in material science,
designers of new materials, low-temperature physicists, as well as for those working on the
development of a theory of hyperbolic propagation of hyperbolic thermoelastic. Relaxation
time, voids, and initial stress exchange with the environment arising from and inside nuclear
reactors influence their and operations. Study of the phenomenon of relaxation time and
initial stress and magnetic field is also used to improve the conditions of oil extractions.
Finally, it is concluded that the influence of magnetic field, initial stress, voids parameters,
and thermal relaxation times are very pronounced in the surface waves propagation
phenomena.
The results obtained are deduced to the results obtained by Singh and Pal [34] in the
absence of initial stress and magnetic field.
ACKNOWLEDGEMENTS
The authors are grateful to Taif University Saudi Arabia for its financial support for
research number 1823/433/1.
8. REFERENCES
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43
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