Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 763429, 47 pages
doi:10.1155/2011/763429
Research Article
Rayleigh Waves in Generalized
Magneto-Thermo-Viscoelastic Granular Medium
under the Influence of Rotation, Gravity Field, and
Initial Stress
A. M. Abd-Alla,1 S. M. Abo-Dahab,1, 2 and F. S. Bayones3
1
Mathematics Department, Faculty of Science, Taif University, Taif 21974, Saudi Arabia
Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt
3
Mathematics Department, Faculty of Science, Umm Al-Qura University, P.O. Box 10109,
Makkah 13401, Saudi Arabia
2
Correspondence should be addressed to S. M. Abo-Dahab, sdahb@yahoo.com
Received 4 December 2010; Revised 14 January 2011; Accepted 25 February 2011
Academic Editor: Ezzat G. Bakhoum
Copyright q 2011 A. M. Abd-Alla et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The surface waves propagation in generalized magneto-thermo-viscoelastic granular medium
subjected to continuous boundary conditions has been investigated. In addition, it is also subjected
to thermal boundary conditions. The solution of the more general equations are obtained for
thermoelastic coupling. The frequency equation of Rayleigh waves is obtained in the form of
a determinant containing a term involving the coefficient of friction of a granular media which
determines Rayleigh waves velocity as a real part and the attenuation coefficient as an imaginary
part, and the effects of rotation, magnetic field, initial stress, viscosity, and gravity field on Rayleigh
waves velocity and attenuation coefficient of surface waves have been studied in detail. Dispersion
curves are computed numerically for a specific model and presented graphically. Some special
cases have also been deduced. The results indicate that the effect of rotation, magnetic field, initial
stress, and gravity field is very pronounced.
1. Introduction
The dynamical problem in granular media of generalized magneto-thermoelastic waves has
been studied in recent times, necessitated by its possible applications in soil mechanics,
earthquake science, geophysics, mining engineering, and plasma physics, and so forth. The
granular medium under consideration is a discontinuous one and is composed of numerous
large or small grains. Unlike a continuous body each element or grain cannot only translate
2
Mathematical Problems in Engineering
but also rotate about its center of gravity. This motion is the characteristic of the medium
and has an important effect upon the equations of motion to produce internal friction. It
was assumed that the medium contains so many grains that they will never be separated
from each other during the deformation and that each grain has perfect thermoelasticity. The
effect of the granular media on dynamics was pointed out by Oshima 1. The dynamical
problem of a generalized thermoelastic granular infinite cylinder under initial stress has been
illustrated by El-Naggar 2. Rayleigh wave propagation of thermoelasticity or generalized
thermoelasticity was pointed out by Dawan and Chakraporty 3. Rayleigh waves in a
magnetoelastic material under the influence of initial stress and a gravity field were discussed
by Abd-Alla et al. 4 and El-Naggar et al. 5.
Rayleigh waves in a thermoelastic granular medium under initial stress on the propagation of waves in granular medium are discussed by Ahmed 6. Abd-Alla and Ahmed
7 discussed the problem of Rayleigh wave propagation in an orthotropic medium under
gravity and initial stress. Magneto-thermoelastic problem in rotating nonhomogeneous
orthotropic hollow cylinder under the hyperbolic heat conduction model is discussed by
Abd-Alla and Mahmoud 8. Wave propagation in a generalized thermoelastic solid cylinder
of arbitrary cross-section is discussed by Venkatesan and Ponnusamy 9. Some problems
discussed the effect of rotation of different materials. Thermoelastic wave propagation in
a rotating elastic medium without energy dissipation was studied by Roychoudhuri and
Bandyopadhyay 10. Sharma and Grover 11 studied the body wave propagation in
rotating thermoelastic media. Thermal stresses in a rotating nonhomogeneous orthotropic
hollow cylinder were discussed by El-Naggar et al. 12. Abd-El-Salam et al. 13 investigated
the numerical solution of magneto-thermoelastic problem nonhomogeneous isotropic
material.
In this paper, the effect of magnetic field, rotation, thermal relaxation time, gravity
field, viscosity, and initial stress on propagation of Rayleigh waves in a thermoelastic granular
medium is discussed. General solution is obtained by using Lame’s potential. The frequency
equation of Rayleigh waves is obtained in the form of a determinant. Some special cases have
also been deduced. Dispersion curves are computed numerically for a specific model and
presented graphically. The results indicate that the effect of rotation, magnetic field, initial
stress, and gravity field are very pronounced.
2. Formulation of the Problem
Let us consider a system of orthogonal Cartesian axes, Oxyz, with the interface and the free
surface of the granular layer resting on the granular half space of different materials being
the planes z K and z 0, respectively. The origin O is any point on the free surface,
the z-axis is positive along the direction towards the exterior of the half space, and the xaxis is positive along the direction of Rayleigh waves propagation. Both media are under
−−→
initial compression stress P along the x-direction and the primary magnetic field H0 acting
on y-axis, as well as the gravity field and incremental thermal stresses, as shown in Figure 1.
The state of deformation in the granular medium is described by the displacement vector
→
−
→
−
Uu, o, w of the center of gravity of a grain and the rotation vector ξ ξ, η, ζ of the grain
about its center of gravity. The elastic medium is rotating uniformly with an angular velocity
Ω Ωn, where n is a unit vector representing the direction of the axis of rotation. The
displacement equation of motion in the rotating frame has two additional terms, Ω × Ω × u
Mathematical Problems in Engineering
3
O
−−→
H0
y
z=0
Granular
layer
P
Granular
half space
x
z=K
Ω
P
g
z
Figure 1: Depiction of the problem.
−•
→
− →
is the centripetal acceleration due to time varying motion only, and 2Ω × u is the Coriolis
acceleration, and Ω 0, Ω, 0.
The electromagnetic field is governed by Maxwell equations, under the consideration
that the medium is a perfect electric conductor taking into account the absence of the
displacement current SI see the work of Mukhopadhyay 14:
→
−
→
−
J curl h,
→
−
→
−
∂h
−μe
curl E,
∂t
→
−
div h 0,
2.1
→
−
div E 0,
→
→
−
→
∂−
u −
E −μe
×H ,
∂t
where
→
−
−−→
−
h curl →
u × H0 ,
−
−
→ −−→ →
H H0 h,
−−→
H0 0, H0 , 0,
2.2
→
−
→
−
where h is the perturbed magnetic field over the primary magnetic field vector, E is the
→
−
−−→
electric intensity, J is the electric current density, μe is the magnetic permeability, H0 is the
−
constant primary magnetic field vector, and →
u is the displacement vector.
The stress and stress couple may be taken to be nonsymmetric, that is, τij / τji ,
.
The
stress
tensor
τ
can
be
expressed
as
the
sum
of
symmetric
and
antisymmetric
M
Mij / ji
ij
tensors
τij σij σij ,
2.3
4
Mathematical Problems in Engineering
where
σij 1
τij τji ,
2
σ ij 1
τij − τji .
2
2.4
The symmetric tensor σij σji is related to the symmetric strain tensor
1
eij eji 2
∂ui ∂uj
∂xj ∂xi
2.5
.
The antisymmetric stress σij are given by
σ23
−F
∂ξ
,
∂t
σ31
−F
∂η
,
∂t
σ12
−F
∂ζ
,
∂t
σ11
σ22
σ33
0,
2.6
where F is the coefficient of friction between the individual grains. The stress couple Mij is
given by
Mij Mνij ,
2.7
where, M is the third elastic constant, M11 , M13 , M33 , and so forth, are the components of
the resultant acting on a surface.
The non-symmetric strain tensor νij is defined as
ν11 ν12
∂ξ
,
∂x
ν31 ∂ ω2 η ,
∂x
∂ξ
,
∂z
ν33 ν32
∂ζ
,
∂z
ν21 ν22 ν23 0,
∂
ω2 η ,
∂z
ν13
∂ζ
,
∂x
2.8
where ω2 1/2∂u/∂z − ∂w/∂x.
The dynamic equation of motion, if the magnetic field and rotation are added, can be
written as 15
τji,j Fi ρ
••
ui
→
→
•
− →
− →
− →
−
−
,
Ω× Ω× u
2Ω× u
i
i, j 1, 2, 3.
2.9
i
The heat conduction equation is given by 16
K∇2 T ρs
∂
∂
∂
∂
−
1 τ2
T γ T0
1 τ2 δ
∇ ·→
u,
∂t
∂t
∂t
∂t
2.10
where ρ is density of the material, K is thermal conductivity, s is specific heat of the material
per unit mass, τ1 , τ2 are thermal relaxation parameter, αt is coefficient of linear thermal
expansion, λ and μ are Lame’s elastic constants, θ is the absolute temperature, γ αt 3λ
2μ,
Mathematical Problems in Engineering
5
T0 is reference temperature solid, T is temperature difference θ − T0 , τ0 is the mechanical
relaxation time due to the viscosity, and τm 1 τ0 ∂/∂t.
The components of stress in generalized thermoelastic medium are given by
∂u
∂w
∂
τm λ P − γ 1 τ1
T,
σ11 τm λ 2μ p
∂x
∂z
∂t
∂w
∂
∂u
σ33 τm λ
τm λ 2μ
− γ 1 τ1
T,
∂x
∂z
∂t
∂u ∂w
σ13 τm μ
.
∂z ∂x
2.11
If we neglect the thermal relaxation time, then 2.11 tends to Nowacki 17 and Biot 18.
The Maxwell’s electro-magnetic stress tensor τ ij is given by
τ ij μe Hi hj Hj hi − Hk · hk δij ,
i, j 1, 2, 3,
2.12
which takes the form
τ 11 −μe H02 ∇2 φ,
τ 13 τ 23 0,
τ 33 μe H02 ∇2 φ,
∇2 φ ∂u ∂w
.
∂x ∂z
2.13
The dynamic equations of motion are
∂τ11 ∂τ31 P ∂ω2
∂w
∂2 u
∂w
2
− ρg
Fx ρ
−Ω u ,
2Ω
∂x
∂z
2 ∂z
∂x
∂t
∂t2
∂τ12 ∂τ32
Fy 0,
∂x
∂z
∂τ13 ∂τ33 P ∂ω2
∂w
∂2 w
∂u
2
ρg
Fz ρ
−Ω w ,
− 2Ω
∂x
∂z
2 ∂x
∂x
∂t
∂t2
2.14
where g is the Earth’s gravity and
F −μe H02 ∇2 φ, 0, μe H02 ∇2 φ ,
τ23 − τ32 ∂M11 ∂M31
0,
∂x
∂z
τ31 − τ13 ∂M12 ∂M32
0,
∂x
∂z
τ12 − τ21 ∂M13 ∂M33
0.
∂x
∂z
2.15
2.16
6
Mathematical Problems in Engineering
From 2.3–2.8 and 2.11, we have
τ11
∂u
∂w
∂
τm λ P − γ 1 τ1
τm λ 2μ p
T,
∂x
∂z
∂t
τ33 τm λ
τ13
∂w
∂u
∂
τm λ 2μ
− γ 1 τ1
T,
∂x
∂z
∂t
∂η
∂u ∂w
F ,
τm μ
∂z ∂x
∂t
τ12 −F
∂ζ
,
∂t
τ23 −F
∂ξ
,
∂t
M11 M
∂ξ
,
∂x
M12 M
∂ ω2 η ,
∂x
2.17
M31 M
∂ξ
,
∂z
M33 M
M32 M
∂ζ
,
∂z
∂
ω2 η ,
∂z
M21 M22 M23 0,
M13 M
∂ζ
.
∂x
Substituting 2.17 into 2.14 and 2.16 tends to
∂2 u
∂2 w
∂ ∂T
∂2 u
∂2 w
τm λ 2μ P
− γ 1 τ1
τm μ
τm λ P ∂x ∂z
∂t ∂x
∂x2
∂z2 ∂x ∂z
P
2
∂2 w
∂2 u
−
∂z2 ∂x ∂z
∂2 η
∂w
F
μe H02
− ρg
∂x
∂z ∂t
∂2 w
∂2 u
∂x2 ∂x ∂z
∂w
∂2 u
2
−Ω u ,
2Ω
ρ
∂t
∂t2
2.18
then
2
∂2 u P
P ∂2 u
2 ∂ w
τm λ 2μ P μe H02
λ
μ
μ
τ
τ
H
μ
m
e
m
0
2
∂x ∂z
2 ∂z2
∂x2
∂2 η
∂w
∂2 u
∂ ∂T
∂w
2
− ρg
F
ρ
−Ω u .
− γ 1 τ1
2Ω
∂t ∂x
∂x
∂z ∂t
∂t
∂t2
2.19
Mathematical Problems in Engineering
7
Also,
τm μ
∂
∂t
∂ζ ∂ξ
−
∂x ∂z
0,
2.20
∂2 w
∂2 η
∂2 w
∂2 u
∂2 u
∂ ∂T
τm λ
τm λ 2μ
− γ 1 τ1
−F
∂x ∂z ∂x2
∂x ∂t
∂x ∂z
∂t ∂z
∂z2
∂u
∂2 u
∂2 w
P
∂2 u
∂2 w
∂2 w
∂u
2
2
ρg
ρ
−
μe H0
−Ω w ,
− 2Ω
2 ∂x ∂z ∂x2
∂x
∂x ∂z ∂z2
∂t
∂t2
2.21
then
2
2
P
∂u
P ∂2 w 2 ∂ w
τm λ μ μe H02
τm μ −
τ
H
λ
2μ
μ
m
e
0
2
∂x ∂z
2 ∂x2
∂z2
∂2 η
∂u
∂2 w
∂u
∂ ∂T
ρg
−F
ρ
− Ω2 w ,
− γ 1 τ1
− 2Ω
∂t ∂z
∂x
∂x ∂t
∂t
∂t2
2.22
and, from 2.16, we have
∇2 ξ − s2
∂ξ
0,
∂t
∂η
∇2 ω2 η − s2
0,
∂t
∇2 ζ − s2
2.23
2.24
∂ζ
0,
∂t
2.25
2F
.
M
2.26
where
s2 3. Solution of the Problem
−
By Helmholtz’s theorem 19, the displacement vector →
u can be written in the displacement
potentials φ and ψ form, as
→
−
−
u grad φ curl →
ψ,
→
−
ψ 0, ψ, 0 ,
3.1
which reduces to
u
∂φ ∂ψ
−
,
∂x ∂z
w
∂φ ∂ψ
.
∂z ∂x
3.2
8
Mathematical Problems in Engineering
Substituting 3.2 into 2.19, 2.22, and 2.24, the wave equations tend to
α2 ∇2 φ −
∂ψ ∂2 φ
γ
∂ψ
∂
1 τ1
T −g
2 2Ω
− Ω2 φ,
ρ
∂t
∂x
∂t
∂t
β 2 ∇2 ψ − s1
∂η
∂φ ∂2 ψ
∂φ
g
2 − 2Ω
− Ω2 ψ,
∂t
∂x
∂t
∂t
∇2 η − s2
∂η
− ∇4 ψ 0,
∂t
3.3
3.4
3.5
where
s1 F
,
ρ
α2 τm λ 2μ P μe H02
,
ρ
β2 2τm μ − P
.
2ρ
3.6
Substituting 3.2 into 2.10, we obtain
K∇2 T ρs
∂
∂
∂
∂
1 τ2
T γ T0
1 τ2 δ
∇2 φ.
∂t
∂t
∂t
∂t
3.7
From 3.3 and 3.7, by eliminating T, we obtain
∂ψ ∂2 φ
∂ψ
∂
1 ∂
2 2
2
− 2 − 2Ω
Ω φ
1 τ2
α ∇ φ−g
∇ −
χ ∂t
∂t
∂x
∂t
∂t
∂
∂
∂
1 τ1
1 τ2 δ
∇2 φ 0,
−ε
∂t
∂t
∂t
2
3.8
where
χ
K
,
ρs
ε
γ 2 T0
.
ρK
3.9
From 3.4 and 3.5 by eliminating η, we obtain
∂
∇ − s2
∂t
2
∂ψ
∂2 ψ
∂φ
∂φ
2 2
2
β ∇ ψ− 2 g
2Ω
Ω ψ − s1 ∇4
0.
∂x
∂t
∂t
∂t
3.10
For a plane harmonic wave propagation in the x-direction, we assume
ψ ψ1 eikx−ct ,
φ φ1 eikx−ct ,
ξ, η, ζ ξ1 , η1 , ζ1 eikx−ct .
3.11
3.12
Mathematical Problems in Engineering
9
From 3.12 into 2.20, 2.23, and 2.25, we get
Dξ1 − ikζ1 0,
3.13
D2 ξ1 q2 ξ1 0,
3.14
D2 ζ1 q2 ζ1 0,
3.15
where
d
.
dz
3.16
ζ1 B1 eiqz B2 e−iqz ,
3.17
q A1 eiqz − A2 e−iqz − k B1 eiqz B2 e−iqz 0,
3.18
q2 ikcs2 − k2 ,
D≡
The solution of 3.14 and 3.15 takes the form
ξ1 A1 eiqz A2 e−iqz ,
where A1 , A2 , B1 , and B2 are arbitrary constants.
From 3.13 and 3.17, we obtain
then
qA1 − kB1 0,
qA2 − kB2 0
⇒ Aj −1j−1 k
Bj ,
q
j 1, 2.
3.19
Substituting 3.11 into 3.8 and 3.10, we obtain
α2∗ D4 G1 D2 G2 φ1 − G3 D2 G4 ψ1 0,
R1 D4 R2 D2 R3 ψ1 R4 D2 R5 φ1 0,
where
Γ0 1 − ikcτ0 ,
α2∗
Γ1 1 − ikcτ1 ,
Γ2 1 − ikcτ2 ,
Γ0 λ 2μ P μe H02
,
ρ
β∗2 Γ3 1 − ikcτ2 δ,
2Γ0 μ − P
.
2ρ
ikc G1 k2 c2 − 2α2∗ α2∗ Γ2 χεΓ1 Γ3 Ω2 ,
χ
3.20
10
Mathematical Problems in Engineering
ikcΓ 2
G2 k4 α2∗ − c2 k2 1 − α2∗ Ω2 − k2 Ω2 ikεcΓ1 Γ3 ,
χ
k2 cΓ2
3
G4 − g − 2Ωc ik ,
χ
G3 ik g − 2Ωc ,
R2 k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2 ,
R1 β∗2 ikcs1 ,
R3 k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1 ,
R5 2Ωc − g ik3 − k2 cs2 .
R4 ik g − 2Ωc ,
3.21
The solution of 3.20 takes the form
φ1 4 Cj eikNj z Dj e−ikNj z ,
j1
3.22
4 Ej eikNj z Fj e−ikNj z ,
ψ1 j1
where the constants Ej and Fj are related to the constants Cj and Dj in the form
Ej mj Cj ,
Fj mj Dj ,
j 1, 2, 3, 4,
1
mj 2
g − 2Ωc ikNj − ik − εΓ2 /χ
×
α2∗ k2 Nj4
ikcΓ2
χ
ikc 2
2
2
2
2
α∗ Γ2 χεΓ1 Γ3 Ω Nj2
− k c − 2α∗ χ
1−
α2∗
Ω2
2
k
− Ω ikεcΓ1 Γ3
2
3.23
.
Substituting 3.22 into 3.11, we obtain
φ
4 Cj eikNj z Dj e−ikNj z eikx−ct ,
j1
4 ψ
Ej eikNj z Fj e−ikNj z eikx−ct ,
j1
3.24
Mathematical Problems in Engineering
11
and values of displacement components u and w are
u ik
4 1 − Nj mj Cj eikNj z 1 Nj mj Dj e−ikNj z eikx−ct ,
j1
w ik
4 Nj mj Cj eikNj z mj − Nj Dj e−ikNj z eikx−ct ,
3.25
j1
where N1 , N2 , N3 , and N4 are taken to be the complex roots of the following equation
N 8 t1 N 6 t2 N 4 t3 N 2 t4 0,
3.26
where
t1 ikc k2 2
1
c − 2α2∗ 2 α2∗ Γ2 χεΓ1 Γ3 Ω2 2
2
α∗
α∗ χ
β∗ ikcs1
× k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2 ,
3.27
ikcΓ 1 4 2
2
2
2
2
2
2
2
t2 2 k α∗ − c k 1 − α∗ Ω − k Ω ikεcΓ1 Γ3
χ
α∗
α2∗
β∗2
1
k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2
ikcs1
ikc α2∗ Γ2 χεΓ1 Γ3 Ω2
× k2 c2 − 2α2∗ χ
1
2
k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1
β∗ ikcs1
−
t3 2 1
2
g
−
2Ωc
k
,
α2∗ β∗2 ikcs1
α2∗
1
2
β∗ ikcs1
×
k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2
ikcΓ 2
k2 1 − α2∗ Ω2 − k2 Ω2 ikεcΓ1 Γ3
× k4 α2∗ − c2 χ
3.28
12
Mathematical Problems in Engineering
k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1
ikc α2∗ Γ2 χεΓ1 Γ3 Ω2
× k2 c2 − 2α2∗ χ
2
2
k2 cΓ2
3
3 − ik g − 2Ωc
ik − ik g − 2Ωc ik − cs2 ,
χ
3.29
t4 1
2
β∗ ikcs
× k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1 × ik g − 2Ωc
α2∗
3.30
2 3
k2 cΓ2
2
3
.
2Ωc − g
ik − k cs2
ik χ
From 3.4, 3.11, 3.12, 3.22, and 3.23, we obtain
η1 4
j1
1 2 2 k β∗ mj 1 Nj2 − mj k2 c2 Ω2 ik 2Ωc − g × Cj eikNj z Dj e−ikNj z .
ikcs1
3.31
Using 3.22 and 3.11 into 3.3, we obtain
T
4 ρ −α2∗ k2 1 Nj2 k2 c2 − ikgmj Cj eikNj z Dj e−ikNj z eikx−ct .
γ Γ1 j1
3.32
With the lower medium, we use the symbols with primes, for ξ1 , ζ1 , η1 , T, φ, ψ, and q,
for z > K,
k
ξ1 − B2 e−iq z ,
q
η1 4
j1
ζ1 B2 e−iq z ,
1 2 2 2
− mj k2 c2 Ω 2 ik 2Ω c − g Dj e−ikNj z ,
k β∗ mj 1 Nj
ikcs1
T 4 ρ −α∗2 k2 1 Nj2 k2 c2 − ikgmj Dj e−ikNj z eikx−ct ,
γ Γ1 j1
Mathematical Problems in Engineering
φ 4
13
Dj e−ikNj z eikx−ct ,
j1
ψ 4
Fj e−ikNj z eikx−ct .
j1
3.33
4. Boundary Conditions and Frequency Equation
In this section, we obtain the frequency equation for the boundary conditions which are
specific to the interface z K, that is,
i u u ,
ii w w ,
iii ξ ξ ,
iv η η ,
v ζ ζ ,
vi M33 M33
,
vii M31 M31
,
viii M32 M32
,
ix τ33 τ 33 τ33
τ 33
x τ31 τ 31 τ31
τ 31 ,
xi τ32 τ 32 τ32
τ 32 ,
xii T T ,
xiii ∂T/∂z θT ∂T /∂z θT .
The boundary conditions on the free surface z 0 are
xiv M33 0,
xv M31 0,
xvi M32 0,
xvii τ33 τ 33 0,
xviii τ31 τ 31 0,
xix τ32 τ 32 0,
xx ∂T/∂z θT 0.
14
Mathematical Problems in Engineering
From conditions iii, v, vi, and vii, we obtain
B1 eiqK − B2 e−iqK −B2 e−iq K ,
B1 eiqK B2 e−iqK B2 e−iq K ,
M B1 eiqK − B2 e−iqK −M B2 e−iq K ,
4.1
M B1 eiqK B2 e−iqK −M B2 e−iq K .
Hence,
B1 B2 B2 0,
ξ ζ ξ ζ 0.
4.2
The other significant boundary conditions are responsible for the following relations:
i
4
1 − Nj mj Cj eikNj K 1 Nj mj Dj e−ikNj K − 1 Nj mj Dj e−ikNj K 0,
4.3
j1
ii
4 Nj mj Cj eikNj K mj − Nj Dj e−ikNj K − mj − Nj Dj e−ikNj K 0,
4.4
j1
iv
4
1 2 2 k β∗ mj 1 Nj2 − mj k2 c2 Ω2 ik 2Ωc − g × Cj eikNj K Dj e−ikNj K
cs1
j1
4
1 2 2 2
− mj k2 c2 Ω 2 ik 2Ω c − g Dj e−ikNj K 0,
−
k β∗ mj 1 Nj
cs
1
j1
4.5
Mathematical Problems in Engineering
15
viii
MNj
4 k
2
mj Nj2
j1
1 2 2 2
2 2
2
k β∗ mj 1 Nj − mj k c Ω ik 2Ωc − g
1 ikcs1
× Cj eikNj K − Dj e−ikNj K
M
Nj
4
k2 mj Nj2 1 j1
1
ikcs1
4.6
2 2 2
2 2
2
× k β∗ mj 1 Nj − mj k c Ω ik 2Ω c − g
Dj e−ikNj K 0,
ix
4 Γ0 λ μe H02
1 − Nj mj Γ0 λ 2μ μe H02 Nj2 mj Nj
j1
× Cj eikNj K Γ0 λ μe H02
1 Nj mj Γ0 λ 2μ μe H02 Nj2 − mj Nj
ig
× Dj e−ikNj K ρ −α2∗ 1 Nj2 c2 − mj Cj eikNj K Dj e−ikNj K
k
−ikNj K
− Γ0 λ μe H02 1 Nj mj Γ0 λ 2μ μe H02 Nj2 − mj Nj Dj e
4.7
ig
−ρ −α∗2 1 Nj2 c2 − mj Dj e−ikNj K 0,
k
x
4 −2k2 Γ0 μNj Cj eikNj K − Dj e−ikNj K
j1
F
k2 β∗2 mj 1 Nj2 − mj k2 c2 Ω2 ik 2Ωc − g
−k2 Γ0 μmj 1 − Nj2 s1
× Cj eikNj K Dj e−ikNj K − 2k2 Γ0 μ Nj Dj e−ikNj K
−
−k2 Γ0 μ mj
1−
×Dj e−ikNj K 0,
Nj2
F 2 2 2
2 2
2
k β∗ mj 1 Nj − mj k c Ω ik 2Ω c − g
s1
4.8
16
Mathematical Problems in Engineering
xii
4
ρ
j1
γ
−α2∗ k2 Nj2 1 k2 c2 − igkmj Cj eikNj K Dj e−ikNj K
4.9
ρ − −α∗2 k2 Nj2 1 k2 c2 − igkmj Dj e−ikNj K 0,
γ
xiii
4
ρ 2 2 2
−α∗ k Nj 1 k2 c2 − igkmj θ ikNj Cj eikNj K θ − ikNj Dj e−ikNj K
γ
j1
ρ − −α∗2 k2 Nj2 1 k2 c2 − igkmj × θ − ikNj Dj e−ikNj K 0,
γ
4.10
xvi
MNj
4 k2 mj Nj2 1 j1
1 2 2 k β∗ mj 1 Nj2 − mj k2 c2 Ω2 ik 2Ωc − g
ikcs1
× Cj − Dj 0,
4.11
xvii
4 Γ0 λ μe H02 1 − Nj mj Γ0 λ 2μ μe H02 Nj2 mj Nj Cj
j1
Γ0 λ μe H02
1 Nj mj Γ0 λ 2μ μe H02 Nj2 − mj Nj Dj
4.12
ig
ρ −α2∗ 1 Nj2 c2 − mj Cj Dj
0,
k
xviii
4 −2k2 Γ0 μNj Cj − Dj
j1
F
4.13
−k2 Γ0 μmj 1 − Nj2 k2 β∗2 mj 1 Nj2 − mj k2 c2 Ω2 ik 2Ωc − g
s1
× Cj Dj 0,
Mathematical Problems in Engineering
17
xx
4 −α2∗ k2 Nj2 1 k2 c2 − igkmj θ ikNj Cj θ − ikNj Dj 0.
4.14
j1
5. Special Cases and Discussion
5.1. The Magnetic Field, Initial Stress, and Thermal Relaxation Time Are
Neglected
In this case i.e., H0 0, p 0, and τ1 τ2 0, 3.26 tends to
V 8 h1 V 6 h2 V 4 h3 V 2 h4 0,
5.1
where
α2∗ Γ0 λ 2μ
,
ρ
β∗2 Γ0 μ
,
ρ
1
mj g − 2Ωc ikVj2 − ik − ε/χ
ikc ikc
Ω2
2
2
2
2
2
2
2
2
α∗ χε Ω Vj 1 − α∗ 2 − Ω ikεc ,
k c − 2α∗ χ
χ
k
×
h1 α2∗ k2 Vj4 −
ikc k2 2
1
2
2
c
−
2α
χε
Ω2 2
α
∗
∗
2
2
α∗
α∗ χ
β∗ ikcs1
× k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2 ,
h2 ikc 1 4 2
2
2
2
2
2
2
k
−
c
ikεc
k
α
1
−
α
Ω
−
k
Ω
∗
∗
χ
α2∗
α2∗
1
2
k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2
β∗ ikcs1
ikc α2∗ χε Ω2
× k2 c2 − 2α2∗ χ
−
β∗2
1
k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1
ikcs1
2 1
2
g
−
2Ωc
k
,
α2∗ β∗2 ikcs1
18
Mathematical Problems in Engineering
1
h3 2 2
k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2
α∗ β∗ ikcs1
ikc × k4 α2∗ − c2 k2 1 − α2∗ Ω2 − k2 Ω2 ikεc
χ
k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1
ikc × k2 c2 − 2α2∗ α2∗ χε Ω2
χ
− ik g − 2Ωc
2
k2 c
ik χ
3
2
− ik g − 2Ωc ik − cs2 ,
3
1
h4 2 2
k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1 × ik g − 2Ωc
α∗ β∗ ikcs
2 3
k2 c
2
3
.
ik − k cs2
ik 2Ωc − g
χ
5.2
Also,
η1 4
j1
1 2 2 k β∗ mj 1 Vj2 − mj k2 c2 Ω2 ik 2Ωc − g × Cj eikVj z Dj e−ikVj z .
ikcs1
T
4 ρ
−α2∗ k2 1 Vj2 k2 c2 − ikgmj Cj eikVj z Dj e−ikVj z eikx−ct ,
γ j1
k
ξ1 − B2 e−iq z ,
q
η1 4
j1
ζ1 B2 e−iq z ,
1 2 2 2
− mj k2 c2 Ω 2 ik 2Ω c − g Dj e−ikVj z ,
k β∗ mj 1 Vj
ikcs1
T 4 ρ −α∗2 k2 1 Vj 2 k2 c2 − ikgmj Dj e−ikVj z eikx−ct ,
γ j1
φ1 4
Dj e−ikVj z ,
j1
ψ1 4
Fj e−ikVj z ,
j1
5.3
Mathematical Problems in Engineering
19
Using the boundary conditions, we obtain
⎡
⎡
d16
d11 d12 · · · d18 d15
⎢
⎢ d21 d22 · · · d28 d25
d26
⎢
⎢ .
..
..
..
..
..
⎢ ..
.
.
.
.
.
⎣
d121 d122 · · · d128 d125 d126
C1
⎤
⎢ ⎥
⎢C ⎥
⎢ 2⎥
⎢ ⎥
⎢C ⎥
⎢ 3⎥
⎢ ⎥
⎢C ⎥
⎢ 4⎥
⎢ ⎥
⎤
⎥
· · · d18 ⎢
⎢D1 ⎥
⎥
⎢
⎥
⎥
⎥⎢
· · · d28
D2 ⎥
⎥⎢
⎥
⎢
..
.. ⎥
⎢ ⎥ 0,
D
⎢
3
.
. ⎥
⎦⎢ ⎥
⎥
⎢ ⎥
· · · d128 ⎢D4 ⎥
⎢ ⎥
⎢ ⎥
⎢D1 ⎥
⎢ ⎥
⎢ ⎥
⎢D2 ⎥
⎢ ⎥
⎢ ⎥
⎢D3 ⎥
⎣ ⎦
D4
5.4
where
d1j 1 − Vj mj Cj eikVj K 1 Vj mj Dj e−ikVj K ,
d2j Vj mj Cj eikVj K mj − Vj Dj e−ikVj K ,
d1j
1 Vj mj Dj e−ikVj K ,
d2j
mj − Vj Dj e−ikVj K ,
1 2 2 k β∗ mj 1 Vj2 − mj k2 c2 Ω2 ik 2Ωc − g
Cj eikVj z Dj e−ikVj z ,
cs1
1 −ikVj z
d3j
k2 β∗2 mj 1 Vj 2 − mj k2 c2 Ω2 ik 2Ωc − g Dj e
,
cs1
1 2 2 d4j MVj k2 mj Vj2 1 k β∗ mj 1 Vj2 − mj k2 c2 Ω2 ik 2Ωc − g
ikcs1
× Cj eikVj K − Dj e−ikVj K ,
1 2 2 2 2
2
2 2
2
k β∗ mj 1 Vj − mj k c Ω ik 2Ωc − g
d4j M Vj k mj Vj 1 ikcs1
d3j × Dj e−ikVj K ,
d5j Γ0 λ 1 − Vj mj Γ0 λ 2μ Vj2 mj Vj Cj eikVj K
Γ0 λ 1 Vj mj Γ0 λ 2μ Vj2 − mj Vj Dj e−ikVj K
ig
2
2
2
ρ −α∗ 1 Vj c − mj Cj eikVj K Dj e−ikVj K ,
k
20
Mathematical Problems in Engineering
ig
d5j
Γ0 λ 1 Vj mj Γ0 λ 2μ Vj 2 − mj Vj Dj e−ikVj K ρ −α∗2 1 Vj 2 c2 − mj
k
−ikVj K
×Dj e
,
d6j −2k2 Γ0 μVj Cj eikVj K − Dj e−ikVj K
F
2
2
2 2
2
2 2
2
−k Γ0 μmj 1 − Vj k β∗ mj 1 Vj − mj k c Ω ik 2Ωc − g
s1
× Cj eikVj K Dj e−ikVj K ,
2k2 Γ0 μ Vj Dj e−ikVj K
d6j
F
2 2
2 2 2
2 2
2
−k Γ 0 μ mj 1 − Vj k β∗ mj 1 Vj − mj k c Ω ik 2Ωc − g
s1
× Dj e
d7j −ikVj K
,
ρ 2 2 2
−α∗ k Vj 1 k2 c2 − igkmj Cj eikVj K Dj e−ikVj K ,
γ
ρ 2 2 2
−α∗ k Vj 1 k2 c2 − igkmj Dj e−ikVj K ,
γ
ρ 2 2 2
d8j −α∗ k Vj 1 k2 c2 − igkmj θ ikVj Cj eikVj K θ − ikVj Dj e−ikVj K ,
γ
d7j
ρ 2 2 2
−ikVj K
,
−α∗ k Vj 1 k2 c2 − igkmj θ − ikVj Dj e
γ
1 2 2 k β∗ mj 1 Vj2 − mj k2 c2 Ω2 ik 2Ωc − g
d9j MVj k2 mj Vj2 1 ikcs1
× Cj − Dj ,
d10j Γ0 λ 1 − Vj mj Γ0 λ 2μ Vj2 mj Vj Cj
d8j
Γ0 λ 1 Vj mj Γ0 λ 2μ Vj2 − mj Vj Dj
ig
1
c − mj Cj Dj ,
ρ
k
−2k2 Γ0 μVj Cj − Dj
F
k2 β∗2 mj 1 Vj2 − mj k2 c2 Ω2 ik 2Ωc − g
−k2 Γ0 μmj 1 − Vj2 s1
× Cj Dj ,
−α2∗ k2 Vj2 1 k2 c2 − igkmj θ ikVj Cj θ − ikVj Dj ,
−α2∗
d11j
d12j
Vj2
2
d10j
d11j
d12j
0,
d9j
j 1, 2, 3, 4.
5.5
Mathematical Problems in Engineering
21
5.2. The Magnetic Field, Initial Stress, Rotation, and Thermal Relaxation
Time Are Neglected and in Viscoelastic Medium
In this case i.e., H0 0, P 0, Ω 0, and τ0 τ1 τ2 0, the previous results obtained as
in Abd-Alla et al. 20.
5.3. Absence of the Gravity Field
In this case, we put g 0, then 3.20 becomes
α2∗ D4 G1 D2 G2 φ1 − G∗3 D2 G∗4 ψ1 0,
R1 D4 R2 D2 R3 ψ1 R∗4 D2 R∗5 φ1 0,
5.6
where
k2 cΓ2
,
2Ωc ik χ
R∗5 2Ωc ik3 − k2 cs2 ,
G∗3
−2ikΩc,
R∗4 −2ikΩc,
G∗4
3
5.7
and G1 , G2 , R1 , R2 , and R3 are as in 3.
The solution of 5.6 take the form
φ
4 Cj∗ eikXj z Dj∗ e−ikXj z eikx−ct ,
j1
4 Ej∗ eikXj z Fj∗ e−ikXj z eikx−ct ,
ψ
5.8
j1
where
Ej∗ m∗j Cj∗ ,
m∗j Fj∗ m∗j Dj∗ ,
j 1, 2, 3, 4,
5.9
1
−2Ωc ikXj2 − ik − εΓ2 /χ
×
ikc 2
2
2
2
2
− k c − 2α∗ α∗ Γ2 χεΓ1 Γ3 Ω Xj2
χ
ikcΓ2
Ω2
2
2
1 − α∗ 2 − Ω ikεcΓ1 Γ3 ,
χ
k
α2∗ k2 Xj4
5.10
22
Mathematical Problems in Engineering
and X1 , X2 , X3 , and X4 are taken to be the complex roots of equation
X 8 t∗1 X 6 t∗2 X 4 t∗3 X 2 t∗4 0,
where
t∗1 ikc k2 2
1
2
2
c
Ω2 2
−
2α
Γ
χεΓ
Γ
α
2
1
3
∗
∗
2
2
α∗
α∗ χ
β∗ ikcs1
× k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2 ,
t∗2
ikcΓ 1 4 2
2
2
2
2
2
2
2
k 1 − α∗ Ω − k Ω ikεcΓ1 Γ3
2 k α∗ − c χ
α∗
1
2
2
2
2
2
2
k
c
ikc
s
Ω
−
2β
β
−
2k
s
2
1
∗
∗
α2∗ β∗2 ikcs1
ikc 2
2
2
2
2
α∗ Γ2 χεΓ1 Γ3 Ω
× k c − 2α∗ χ
1
2
k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1
β∗ ikcs1
−
t∗3 1
2 2 2
Ω
c
4k
,
α2∗ β∗2 ikcs1
1
α2∗ β∗2 ikcs1
× k2 c2 − 2β∗2 ikc s2 β∗2 − 2k2 s1 Ω2
ikcΓ 2
4
2
2
2
2
2
2
2
k 1 − α∗ Ω − k Ω ikεcΓ1 Γ3
× k α∗ − c χ
k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1
ikc × k2 c2 − 2α2∗ α2∗ Γ2 χεΓ1 Γ3 Ω2
χ
k2 cΓ2
−4Ω c ik ik χ
2 2
3
ik ik − cs2 3
,
5.11
Mathematical Problems in Engineering
1
∗
t4 2 2
k2 k2 − ikcs2 β∗2 − c2 ikc s2 Ω2 k4 s1 −2ikΩc
α∗ β∗ ikcs
4Ω c
2 2
u ik
4 ik − k cs2
3
2
k2 cΓ2
ik χ
23
3
,
1 − Xj m∗j Cj∗ eikXj z 1 Xj m∗j Dj∗ e−ikXj z eikx−ct ,
j1
w ik
4 Xj m∗j Cj∗ eikXj z m∗j − Xj Dj∗ e−ikXj z eikx−ct ,
j1
η1 4
j1
T
1 2 2 ∗
k β∗ mj 1 Xj2 − m∗j k2 c2 Ω2 2ikΩc × Cj∗ eikXj z Dj∗ e−ikXj z ,
ikcs1
4 ρ −α2∗ k2 1 Xj2 k2 c2 Cj∗ eikXj z Dj∗ e−ikXj z eikx−ct .
γ Γ1 j1
5.12
With the lower medium, we use the symbols with primes, for ξ1 , ζ1 , η1 , T, φ, ψ, and q,
for z > K,
k
ξ1 − B2 e−iq z ,
q
η1 4
j1
ζ1 B2 e−iq z ,
1 2 2 ∗ 2
∗
2 2
2
β
m
c
Ω
c
Dj∗ e−ikXj z ,
k
1
X
−
m
k
2ikΩ
∗
j
j
j
ikcs1
T 4 ρ 2 2
2
2 2
−α
1
X
k
Dj∗ e−ikXj z eikx−ct ,
k
c
∗
j
γ Γ1 j1
φ 4
5.13
Dj e−ikNj z eikx−ct ,
j1
ψ 4
Fj∗ e−ikXj z eikx−ct .
j1
From conditions iii, v, vi, vii, we get the same equations 4.1 and 4.2: the
other significant boundary conditions are responsible for the following relations:
24
Mathematical Problems in Engineering
i
q1 C1∗ eikX1 K q2 C2∗ eikX2 K q3 C3∗ eikX3 K q4 C4∗ eikX4 K q5 D1∗ e−ikX1 K q6 D2∗ e−ikX2 K
q7 D3∗ e−ikX3 K q8 D4∗ e−ikX4 K
5.14
q9 D1∗ e−ikX1 K q10 D2∗ e−ikX2 K q11 D3∗ e−ikX3 K q12 D4∗ e−ikX4 K ,
ii
q13 C1∗ eikX1 K q14 C2∗ eikX2 K q15 C3∗ eikX3 K q16 C4∗ eikX4 K q17 D1∗ e−ikX1 K q18 D2∗ e−ikX2 K
q19 D3∗ e−ikX3 K q20 D4∗ e−ikX4 K
5.15
q21 D1∗ e−ikX1 K q22 D2∗ e−ikX2 K q23 D3∗ e−ikX3 K q24 D4∗ e−ikX4 K ,
iv
q25 C1∗ eikX1 K q26 C2∗ eikX2 K q27 C3∗ eikX3 K q28 C4∗ eikX4 K q25 D1∗ e−ikX1 K
q26 D2∗ e−ikX2 K q27 D3∗ e−ikX3 K q28 D4∗ e−ikX4 K
5.16
q29 D1∗ e−ikX1 K q30 D2∗ e−ikX2 K q31 D3∗ e−ikX3 K q32 D4∗ e−ikX4 K ,
viii
q33 C1∗ eikX1 K q34 C2∗ eikX2 K q35 C3∗ eikX3 K q36 C4∗ eikX4 K − q33 D1∗ e−ikX1 K
− q34 D2∗ e−ikX2 K − q35 D3∗ e−ikX3 K − q36 D4∗ e−ikX4 K
5.17
−q37 D1∗ e−ikX1 K − q38 D2∗ e−ikX2 K − q39 D3∗ e−ikX3 K − q40 D4∗ e−ikX4 K ,
ix
q41 C1∗ eikX1 K q42 C2∗ eikX2 K q43 C3∗ eikX3 K q44 C4∗ eikX4 K q45 D1∗ e−ikX1 K
q46 D2∗ e−ikX2 K q47 D3∗ e−ikX3 K q48 D4∗ e−ikX4 K
5.18
q49 D1∗ e−ikX1 K q50 D2∗ e−ikX2 K q51 D3∗ e−ikX3 K q52 D4∗ e−ikX4 K ,
x
q53 C1∗ eikX1 K q54 C2∗ eikX2 K q55 C3∗ eikX3 K q56 C4∗ eikX4 K q57 D1∗ e−ikX1 K
q58 D2∗ e−ikX2 K q59 D3∗ e−ikX3 K q60 D4∗ e−ikX4 K
5.19
q61 D1∗ e−ikX1 K q62 D2∗ e−ikX2 K q63 D3∗ e−ikX3 K q64 D4∗ e−ikX4 K ,
Mathematical Problems in Engineering
25
xii
q65 C1∗ eikX1 K q66 C2∗ eikX2 K q67 C3∗ eikX3 K q68 C4∗ eikX4 K q65 D1∗ e−ikX1 K
q66 D2∗ e−ikX2 K q67 D3∗ e−ikX3 K q68 D4∗ e−ikX4 K
5.20
q69 D1 ∗ e−ikX1 K q70 D2∗ e−ikX2 K q71 D3∗ e−ikX3 K q72 D4∗ e−ikX4 K ,
xiii
q73 C1∗ eikX1 K q74 C2∗ eikX2 K q75 C3∗ eikX3 K q76 C4∗ eikX4 K q77 D1∗ e−ikX1 K
q78 D2∗ e−ikX2 K q79 D3∗ e−ikX3 K q80 D4∗ e−ikX4 K
5.21
q81 D1∗ e−ikX1 K q82 D2∗ e−ikX2 K q83 D3∗ e−ikX3 K q84 D4∗ e−ikX4 K ,
xvi
q85 C1∗ eikX1 K q86 C2∗ eikX2 K q87 C3∗ eikX3 K q88 C4∗ eikX4 K
− q85 D1∗ e−ikX1 K q86 D2∗ e−ikX2 K q87 D3∗ e−ikX3 K q88 D4∗ e−ikX4 K 0,
5.22
xvii
q89 C1∗ eikX1 K q90 C2∗ eikX2 K q91 C3∗ eikX3 K q92 C4∗ eikX4 K q93 D1∗ e−ikX1 K
q94 D2∗ e−ikX2 K
q95 D3∗ e−ikX3 K
q96 D4∗ e−ikX4 K
0,
5.23
xviii
q97 C1∗ eikX1 K q98 C2∗ eikX2 K q99 C3∗ eikX3 K q100 C4∗ eikX4 K q101 D1∗ e−ikX1 K
q102 D2∗ e−ikX2 K q103 D3∗ e−ikX3 K q104 D4∗ e−ikX4 K 0,
5.24
xx
q105 C1∗ eikX1 K q106 C2∗ eikX2 K q107 C3∗ eikX3 K q108 C4∗ eikX4 K q109 D1∗ e−ikX1 K
q110 D2∗ e−ikX2 K q111 D3∗ e−ikX3 K q112 D4∗ e−ikX4 K 0,
5.25
26
Mathematical Problems in Engineering
where
q1 1 − X1 m∗1 , q2 1 − X2 m∗2 , q3 1 − X3 m∗3 , q4 1 − X4 m∗4 ,
q5 1 X1 m∗1 , q6 1 X2 m∗2 , q7 1 X3 m∗3 , q8 1 X4 m∗4 ,
q9 1 X1 m∗
q0 1 X2 m∗
q11 1 X3 m∗
q12 1 X4 m∗
2 ,
3 ,
1 ,
4 ,
q13 X1 m∗1 , q14 X2 m∗2 , q15 X3 m∗3 , q16 X4 m∗4 ,
q17 m∗1 − X1 , q18 m∗2 − X2 , q19 m∗3 − X3 , q20 m∗4 − X4 ,
∗
∗
∗
q21 m∗
q22 m 2 − X 2 , q23 m 3 − X 3 , q24 m 4 − X 4 ,
1 − X1 ,
1 2 2 ∗
q25 k β∗ m1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
cs1
1 2 2 ∗
k β∗ m4 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
q26 cs1
1 2 2 ∗
q27 k β∗ m3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
cs1
1 2 2 ∗
q28 k β∗ m4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
cs1
1 2
∗
2
∗
q29 k2 β ∗ m 1 1 X 1 − m 1 k2 c2 Ω2 2ikΩc ,
cs1
1 2
∗
2
∗
q30 k2 β ∗ m 2 1 X 2 − m 2 k2 c2 Ω2 2ikΩc ,
cs1
1 2
∗
2
∗
q31 k2 β ∗ m 3 1 X 3 − m 3 k2 c2 Ω2 2ikΩc ,
cs1
1 2
∗
2
∗
q32 k2 β ∗ m 4 1 X 4 − m 4 k2 c2 Ω2 2ikΩc ,
cs1
1 2 2 ∗
2 ∗
2
2
∗
2 2
2
k β∗ m1 1 X1 − m1 k c Ω 2ikΩc ,
q33 MX1 k m1 X1 1 ikcs1
1 2 2 ∗
2 ∗
2
2
∗
2 2
2
k β∗ m2 1 X2 − m2 k c Ω 2ikΩc ,
q34 MX2 k m2 X2 1 ikcs1
1 2 2 ∗
q35 MX3 k2 m∗3 X32 1 k β∗ m3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
ikcs1
1 2 2 ∗
k β∗ m4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
q36 MX4 k2 m∗4 X42 1 ikcs1
1 2 2 ∗
2 2
2
∗
2 2
2
q37 M X1 k m 1 X 1 1 k β ∗ m 1 1 X 1 − m 1 k c Ω 2ikΩc ,
ikcs1
1 2 2 ∗
2 2
2
∗
2 2
2
k β ∗ m 2 1 X 2 − m 2 k c Ω 2ikΩc ,
q38 M X2 k m 2 X 2 1 ikcs1
1 2 2 ∗
2 2
2
∗
2 2
2
q39 M X3 k m 3 X 3 1 k β ∗ m 3 1 X 3 − m 3 k c Ω 2ikΩc ,
ikcs 1
1 2 2 ∗
2 2
2
∗
2 2
2
q40 M X4 k m 4 X 4 1 k β ∗ m 4 1 X 1 − m 4 k c Ω 2ikΩc ,
ikcs 1
Mathematical Problems in Engineering
27
q41 Γ0 λ μe H02 1 − X1 m∗1 Γ0 λ 2μ μe H02 X12 m∗1 X1 ρ −α2∗ 1 X12 c2 ,
q42 Γ0 λ μe H02 1 − X2 m∗2 Γ0 λ 2μ μe H02 X22 m∗2 X2 ρ −α2∗ 1 X22 c2 ,
q43 Γ0 λ μe H02 1 − X3 m∗3 Γ0 λ 2μ μe H02 X32 m∗3 X3 ρ −α2∗ 1 X32 c2 ,
F
∗
2
2
∗
2
∗
q64 2k2 Γ0 μ X4 − k2 Γ0 μ m 4 1 − X 4 k2 β ∗ m 4 1 X 4 − m 4 k2 c2 Ω2 2ikΩc ,
s1
q44 Γ0 λ μe H02 1 − X4 m∗4 Γ0 λ 2μ μe H02 X42 m∗4 X4 ρ −α2∗ 1 X42 c2 ,
q45 Γ0 λ μe H02 1 X1 m∗1 Γ0 λ 2μ μe H02 X12 − m∗1 X1 ρ −α2∗ 1 X12 c2 ,
q46 Γ0 λ μe H02 1 X2 m∗2 Γ0 λ 2μ μe H02 X22 − m∗2 X2 ρ −α2∗ 1 X22 c2 ,
q47 Γ0 λ μe H02 1 X3 m∗3 Γ0 λ 2μ μe H02 X32 − m∗3 X3 ρ −α2∗ 1 X32 c2 ,
q48 Γ0 λ μe H02 1 X4 m∗4 Γ0 λ 2μ μe H02 X42 − m∗4 X4 ρ −α2∗ 1 X42 c2 ,
∗
2
∗
q49 Γ 0 λ μ e H02 1 X 1 m 1 Γ 0 λ 2μ μ e H02
X 1 − m 1 X 1
2
2
ρ −α ∗ 1 X 1 c2 ,
∗
2
∗
q50 Γ 0 λ μ e H02 1 X 2 m 2 Γ 0 λ 2μ μ e H02 X 2 − m 2 X 2
2
2
ρ −α ∗ 1 X 2 c2 ,
∗
2
∗
q51 Γ 0 λ μ e H02 1 X 3 m 3 Γ 0 λ 2μ μ e H02 X 3 − m 3 X 3
2
2
ρ −α ∗ 1 X 3 c2 ,
∗
2
∗
q52 Γ 0 λ μ e H02 1 X 4 m 4 Γ 0 λ 2μ μ e H02 X 4 − m 4 X 4
2
2
ρ −α ∗ 1 X 4 c2 ,
F
k2 β∗2 m∗1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
q53 −2k2 Γ0 μX1 − k2 Γ0 μm∗1 1 − X12 s1
F
k2 β∗2 m∗2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
q54 −2k2 Γ0 μX2 − k2 Γ0 μm∗2 1 − X22 s1
F
q55 −2k2 Γ0 μX3 − k2 Γ0 μm∗3 1 − X32 k2 β∗2 m∗3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
q56 −2k2 Γ0 μX4 − k2 Γ0 μm∗4 1 − X42 s1
F
k2 β∗2 m∗1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
q57 2k2 Γ0 μX1 − k2 Γ0 μm∗1 1 − X12 s1
28
Mathematical Problems in Engineering
F
q58 2k2 Γ0 μX2 − k2 Γ0 μm∗2 1 − X22 k2 β∗2 m∗2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
q59 2k2 Γ0 μX3 − k2 Γ0 μm∗3 1 − X32 s1
F
k2 β∗2 m∗4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
q60 2k2 Γ0 μX4 − k2 Γ0 μm∗4 1 − X42 s1
F
∗
2
2
∗
2
∗
q61 2k2 Γ0 μ X1 − k2 Γ0 μ m 1 1 − X 1 k2 β ∗ m 1 1 X 1 − m 1 k2 c2 Ω2 2ikΩc ,
s1
F
∗
2
2
∗
2
∗
q62 2k2 Γ0 μ X2 − k2 Γ0 μ m 2 1 − X 2 k2 β ∗ m 2 1 X 2 − m 2 k2 c2 Ω2 2ikΩc ,
s1
F
∗
2
2
∗
2
∗
q63 2k2 Γ0 μ X3 − k2 Γ0 μ m 3 1 − X 3 k2 β ∗ m 3 1 X 3 − m 3 k2 c2 Ω2 2ikΩc ,
s1
ρ
ρ
q65 −α2∗ k2 X12 1 k2 c2 , q66 −α2∗ k2 X22 1 k2 c2 ,
γ
γ
ρ 2 2 2
ρ
−α∗ k X3 1 k2 c2 , q68 −α2∗ k2 X42 1 k2 c2 ,
q67 γ
γ
q69 q71 q73 q75 q77 q79 q81 q86
q87
q70 ρ 2 2 2
2 2
−α
X
,
k
1
k
c
∗
2
γ
ρ 2 2 2
ρ 2 2 2
2 2
2 2
−α
X
,
q
−α
X
,
k
1
k
c
k
1
k
c
72
∗
3
∗
4
γ
γ
ρ 2 2 2
ρ
−α∗ k X1 1 k2 c2 θ ikX1 , q74 −α2∗ k2 X22 1 k2 c2 θ ikX2 ,
γ
γ
ρ 2 2 2
ρ
−α∗ k X3 1 k2 c2 θ ikX3 , q76 −α2∗ k2 X42 1 k2 c2 θ ikX4 ,
γ
γ
ρ
ρ
−α2∗ k2 X12 1 k2 c2 θ − ikX1 , q78 −α2∗ k2 X22 1 k2 c2 θ − ikX2 ,
γ
γ
ρ
ρ
−α2∗ k2 X32 1 k2 c2 θ − ikX3 , q80 −α2∗ k2 X42 1 k2 c2 θ − ikX4 ,
γ
γ
ρ 2 2 2
2 2 k
1
k
c
−α
X
θ − ikX 1 ,
∗
1
γ
q82 ρ 2 2 2
2 2 k
1
k
c
−α
X
θ − ikX 2 ,
∗
2
γ
ρ 2 2 2
ρ 2 2 2
2 2 k
1
k
c
,
q
−α
X
θ
−
ikX
−α ∗ k X 3 1 k2 c2 θ − ikX 3 ,
3
84
∗
3
γ
γ
1 2 2 ∗
k β∗ m1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
MX1 k2 m∗1 X12 1 ikcs1
1 2 2 ∗
MX2 k2 m∗j X22 1 k β∗ m2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
ikcs1
1 2 2 ∗
k β∗ m3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
MX3 k2 m∗j X32 1 ikcs1
q83 q85
ρ 2 2 2
2 2
−α
X
,
k
1
k
c
∗
1
γ
Mathematical Problems in Engineering
29
1 2 2 ∗
q88 MX4 k2 m∗4 X42 1 k β∗ m4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
ikcs1
q89 Γ0 λ μe H02 1 − X1 m∗1 Γ0 λ 2μ μe H02 X12 m∗1 X1 ρ −α2∗ 1 X12 c2 ,
q90 Γ0 λ μe H02 1 − X2 m∗2 Γ0 λ 2μ μe H02 X22 m∗2 X2 ρ −α2∗ 1 X22 c2 ,
q91 Γ0 λ μe H02 1 − X3 m∗3 Γ0 λ 2μ μe H02 X32 m∗3 X3 ρ −α2∗ 1 X32 c2 ,
q92 Γ0 λ μe H02 1 − X4 m∗4 Γ0 λ 2μ μe H02 X42 m∗4 X4 ρ −α2∗ 1 X42 c2 ,
q93 Γ0 λ μe H02 1 X1 m∗1 Γ0 λ 2μ μe H02 X12 − m∗1 X1 ρ −α2∗ 1 X12 c2 ,
q94 Γ0 λ μe H02 1 X2 m∗2 Γ0 λ 2μ μe H02 X22 − m∗2 X2 ρ −α2∗ 1 X22 c2 ,
q95 Γ0 λ μe H02 1 X3 m∗3 Γ0 λ 2μ μe H02 X32 − m∗3 X3 ρ −α2∗ 1 X32 c2 ,
q96 Γ0 λ μe H02 1 X4 m∗4 Γ0 λ 2μ μe H02 X42 − m∗4 X4 ρ −α2∗ 1 X42 c2 ,
F
k2 β∗2 m∗1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
q97 −2k2 Γ0 μX1 − k2 Γ0 μm∗1 1 − X12 s1
F
q98 −2k2 Γ0 μX2 − k2 Γ0 μm∗2 1 − Xj2 k2 β∗2 m∗2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
q99 −2k2 Γ0 μX3 − k2 Γ0 μm∗3 1 − X32 s1
F
k2 β∗2 m∗4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
q100 −2k2 Γ0 μX4 − k2 Γ0 μm∗4 1 − X42 s1
F
q101 2k2 Γ0 μX1 − k2 Γ0 μm∗1 1 − X12 k2 β∗2 m∗1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
q102 2k2 Γ0 μX2 − k2 Γ0 μm∗2 1 − X22 s1
F
k2 β∗2 m∗3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
q103 2k2 Γ0 μX3 − k2 Γ0 μm∗3 1 − X32 s1
F
q104 2k2 Γ0 μX4 − k2 Γ0 μm∗4 1 − X42 k2 β∗2 m∗4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
s1
q105 −α2∗ k2 X12 1 k2 c2 θ ikX1 , q106 −α2∗ k2 X22 1 k2 c2 θ ikX2 ,
q107 −α2∗ k2 X32 1 k2 c2 θ ikX3 ,
q108 −α2∗ k2 X42 1 k2 c2 θ ikX4 ,
q109 −α2∗ k2 X12 1 k2 c2 θ − ikX1 ,
q110 −α2∗ k2 X22 1 k2 c2 θ − ikX2 ,
q111 −α2∗ k2 X32 1 k2 c2 θ − ikX3 ,
q112 −α2∗ k2 X42 1 k2 c2 θ − ikX4 .
5.26
30
Mathematical Problems in Engineering
Elimination of Cj∗ , Dj∗ , and D ∗j gives the wave velocity equation in the determinant
form
det dij 0.
5.27
This equation has complex roots: the real part Re gives the Rayleigh wave velocity, and the
imaginary part Im gives the attenuation coefficient due to the friction of the granular nature
of the medium, where the nonvanishing of the twelfth-order determinant of dij is given by
q85 e−ikX1 K
q86 e−ikX2 K
q87 e−ikX3 K
q88 e−ikX4 K
q85 eikX1 K
q86 eikX2 K
q87 eikX3 K
q88 eikX4 K
0
0
0
0
−ikX1 K
−ikX2 K
−ikX3 K
−ikX4 K
ikX1 K
ikX2 K
ikX3 K
ikX4 K
0
0
0
0
q100 e−ikX4 K q101 eikX1 K q102 eikX2 K q103 eikX3 K q104 eikX4 K
0
0
0
0
q105 e−ikX1 K q106 e−ikX2 K q107 e−ikX3 K q108 e−ikX4 K q109 eikX1 K q110 eikX2 K q111 eikX3 K q112 eikX4 K
0
0
0
0
q21
q22
q23
q24
q29
q30
q31
q32
q89 e
q97 e−ikX1 K
q90 e
q91 e
q98 e−ikX2 K
q92 e
q99 e−ikX3 K
q93 e
q94 e
q95 e
q96 e
q13
q14
q15
q16
q25
q26
q27
q28
q25
q26
q27
q28
q33
q34
q35
q36
−q33
−q34
−q35
−q36
q41
q42
q43
q44
q45
q46
q47
q48
q49
q50
q51
q52
q53
q54
q55
q56
q57
q58
q59
q60
q61
q62
q63
q64
q65
q66
q67
q68
q65
q66
q67
q68
q69
q70
q71
q72
q73
q74
q75
q76
q77
q78
q79
q80
q81
q82
q83
q84
q1
q2
q3
q4
q5
q6
q7
q8
q9
q10
q11
q12
q17
q18
q19
q20
−q37 −q38 −q39 −q40
0.
5.28
5.4. The Gravity Field, Initial Stress, and Magnetic Field Are Neglected and
There Is Uncoupling between the Temperature and Strain Field
In this case g 0, P 0, H0 0, and θ 0, we obtain
α2∗
lim m∗j ε→0
1
−2Ωc ikXj2 − ik
×
α2∗ k2 Xj4
Γ0 λ 2μ
,
ρ
β∗2 Γ0 μ
,
ρ
ikc
ikcΓ2
Ω2
2
2
2
2
2
2
2
2
α Γ2 Ω Xj − k c − 2α∗ 1 − α∗ 2 − Ω ,
χ ∗
χ
k
q22
q23
q24
q30
q31
q32
lim −q38 −q39 −q40 0.
γ →0
q50
q51
q52
q62
q63
q64
5.29
Mathematical Problems in Engineering
31
Multiplying the rows 10, 11, and 12 of the determinant |dij | by γ and then taking limγ → 0 ,
5.28 reduces, after some computation, to the following ninth-order determinant equation:
q85 e−ikX1 K
q86 e−ikX2 K
q87 e−ikX3 K
q88 e−ikX4 K
q85 eikX1 K
q86 eikX2 K
q87 eikX3 K
q88 eikX4 K
0
e−ikX1 K
e−ikX2 K
e−ikX3 K
e−ikX4 K
eikX1 K
eikX2 K
eikX3 K
eikX4 K
0
ikX4 K
0
q105 e−ikX1 K q106 e−ikX2 K q107 e−ikX3 K q108 e−ikX4 K q109 eikX1 K q110 eikX2 K q111 eikX3 K q112 eikX4 K
0
q89
q97 e
−ikX1 K
q90
q98 e
q91
−ikX2 K
q99 e
−ikX3 K
q92
q100 e
−ikX4 K
q93
q101 e
ikX1 K
q17
q94
q102 e
ikX2 K
q95
q103 e
q18
ikX3 K
q96
q104 e
q13
q14
q15
q16
q19
q20
q21
q25
q26
q27
q28
q25
q26
q27
q28
q29
q33
q34
q35
q36
−q33
−q34
−q35
−q36
−q37
q41
q42
q43
q44
q45
q46
q47
q48
q49
q53
q54
q55
q56
q57
q58
q59
q60
q61
0,
5.30
where
q1 1 − X1 m∗1 ,
q5 1 X1 m∗1 ,
q9 1 X1 m∗
1 ,
q13 X1 m∗1 ,
q17 m∗1 − X1 ,
q21 m∗
1 − X1 ,
q2 1 − X2 m∗2 ,
q6 1 X2 m∗2 ,
q10 1 X2 m∗
2 ,
q14 X2 m∗2 ,
q18 m∗2 − X2 ,
q22 m∗
2 − X2 ,
q3 1 − X3 m∗3 ,
q7 1 X3 m∗3 ,
q11 1 X3 m∗
3 ,
q15 X3 m∗3 ,
q19 m∗3 − X3 ,
q23 m∗
3 − X3 ,
q4 1 − X4 m∗4 ,
q8 1 X4 m∗4 ,
q12 1 X4 m∗
4 ,
q16 X4 m∗4 ,
q20 m∗4 − X4 ,
q24 m∗
4 − X4 ,
1 2 2 ∗
k β∗ m1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
cs1
1 2 2 ∗
q26 k β∗ m4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
cs1
1 2 2 ∗
k β∗ m3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
q27 cs1
1 2 2 ∗
q28 k β∗ m4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
cs1
1 2
2 2
2
− m∗
2ikΩc ,
q29 k2 β∗2 m∗
1 1 X1
1 k c Ω
cs1
q25 q30 1 2 2 ∗ 2
∗
2 2
2
k
1
X
−
m
k
2ikΩc
,
β
m
c
Ω
∗
2
2
2
cs1
q31 1 2 2 ∗ 2
∗
2 2
2
k
1
X
−
m
k
2ikΩc
,
β
m
c
Ω
∗
3
3
3
cs1
q32 1 2 2 ∗ 2
2 2
2
− m∗
2ikΩc ,
4 k c Ω
k β∗ m4 1 X4
cs1
q33 MX1 k
2
m∗1
X12
1 2 2 ∗
2
∗
2 2
2
1 k β∗ m1 1 X1 − m1 k c Ω 2ikΩc ,
ikcs1
32
Mathematical Problems in Engineering
1 2 2 ∗
q34 MX2 k2 m∗2 X22 1 k β∗ m2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
ikcs1
1 2 2 ∗
k β∗ m3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
q35 MX3 k2 m∗3 X32 1 ikcs1
1 2 2 ∗
2 ∗
2
2
∗
2 2
2
q36 MX4 k m4 X4 1 k β∗ m4 1 X4 − m4 k c Ω 2ikΩc ,
ikcs1
1 2 2 ∗ 2 2
2
∗
2 2
2
k β∗ m1 1 X1 − m1 k c Ω 2ikΩc ,
q37 M X1 k m1 X1 1 ikcs1
1 2 2 ∗ 2 2
2
∗
2 2
2
q38 M X2 k m2 X2 1 k β∗ m2 1 X2 − m2 k c Ω 2ikΩc ,
ikcs1
1 2 2 ∗ 2 2
2
∗
2 2
2
k β∗ m3 1 X3 − m3 k c Ω 2ikΩc ,
q39 M X3 k m3 X3 1 ikcs1
1 2 2 ∗ 2 2
2
∗
2 2
2
q40 M X4 k m4 X4 1 k β∗ m4 1 X4 − m4 k c Ω 2ikΩc ,
ikcs1
q41 Γ0 λ 1 − X1 m∗1 Γ0 λ 2μ X12 m∗1 X1 ρ −α2∗ 1 X12 c2 ,
q42 Γ0 λ 1 − X2 m∗2 Γ0 λ 2μ X22 m∗2 X2 ρ −α2∗ 1 X22 c2 ,
q43 Γ0 λ 1 − X3 m∗3 Γ0 λ 2μ X32 m∗3 X3 ρ −α2∗ 1 X32 c2 ,
q44 Γ0 λ 1 − X4 m∗4 Γ0 λ 2μ X42 m∗4 X4 ρ −α2∗ 1 X42 c2 ,
q45 Γ0 λ 1 X1 m∗1 Γ0 λ 2μ X12 − m∗1 X1 ρ −α2∗ 1 X12 c2 ,
q46 Γ0 λ 1 X2 m∗2 Γ0 λ 2μ X22 − m∗2 X2 ρ −α2∗ 1 X22 c2 ,
q47 Γ0 λ 1 X3 m∗3 Γ0 λ 2μ X32 − m∗3 X3 ρ −α2∗ 1 X32 c2 ,
q48 Γ0 λ 1 X4 m∗4 Γ0 λ 2μ X42 − m∗4 X4 ρ −α2∗ 1 X42 c2 ,
2
2
X12 − m∗
c2 ,
q49 Γ0 λ 1 X1 m∗
1 Γ0 λ 2μ
1 X1 ρ −α∗ 1 X1
2
∗ 2
2
2
Γ
λ
X
ρ
−α
1
X
c
,
2μ
−
m
X
q50 Γ0 λ 1 X2 m∗
∗
2
0
2 2
2
2
2
2
2
q51 Γ0 λ 1 X3 m∗
X3 − m∗
c2 ,
3 Γ0 λ 2μ
3 X3 ρ −α∗ 1 X3
2
2
X42 − m∗
c2 ,
q52 Γ0 λ 1 X4 m∗
4 Γ0 λ 2μ
4 X4 ρ −α∗ 1 X4
F
q53 −2k2 Γ0 μX1 − k2 Γ0 μm∗1 1 − X12 k2 β∗2 m∗1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
q54 −2k2 Γ0 μX2 − k2 Γ0 μm∗2 1 − X22 s1
Mathematical Problems in Engineering
33
F
q55 −2k2 Γ0 μX3 − k2 Γ0 μm∗3 1 − X32 k2 β∗2 m∗3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
q56 −2k2 Γ0 μX4 − k2 Γ0 μm∗4 1 − X42 s1
F
k2 β∗2 m∗1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
q57 2k2 Γ0 μX1 − k2 Γ0 μm∗1 1 − X12 s1
F
q58 2k2 Γ0 X2 − k2 Γ0 μm∗2 1 − X22 k2 β∗2 m∗2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
q59 2k2 Γ0 μX3 − k2 Γ0 μm∗3 1 − X32 s1
F
k2 β∗2 m∗4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
q60 2k2 Γ0 μX4 − k2 Γ0 μm∗4 1 − X42 s1
F
2
2
2 2
2
q61 2k2 Γ0 μ X1 − k2 Γ0 μ m∗
k2 β∗2 m∗
− m∗
2ikΩc ,
1 1 − X1
1 1 X1
1 k c Ω
s1
F
2
2
2 2
2
k2 β∗2 m∗
− m∗
2ikΩc ,
q62 2k2 Γ0 μ X2 − k2 Γ0 μ m∗
2 1 − X2
2 1 X2
2 k c Ω
s1
F
2
2 2 ∗
2
∗
2 2
2
q63 2k2 Γ0 μ X3 − k2 Γ0 μ m∗
1
−
X
k
1
X
−
m
k
2ikΩc
,
β
m
c
Ω
∗
3
3
3
3
3
s1
F
2
2
2 2
2
q64 2k2 Γ0 μ X4 − k2 Γ0 μ m∗
k2 β∗2 m∗
− m∗
2ikΩc ,
4 1 − X4
4 1 X4
4 k c Ω
s1
ρ 2 2 2
ρ 2 2 2
q65 q66 −α∗ k X1 1 k2 c2 ,
−α∗ k X2 1 k2 c2 ,
γ
γ
ρ
ρ 2 2 2
−α2∗ k2 X32 1 k2 c2 ,
−α∗ k X4 1 k2 c2 ,
q68 q67 γ
γ
q69 ρ 2 2 2
−α∗ k X1 1 k2 c2 ,
γ
ρ 2 2 2
−α∗ k X3 1 k2 c2 ,
γ
ρ
q73 −α2∗ k2 X12 1 k2 c2 ikX1 ,
γ
ρ
q75 −α2∗ k2 X32 1 k2 c2 ikX3 ,
γ
−ρ 2 2 2
q77 −α∗ k X1 1 k2 c2 ikX1 ,
γ
−ρ 2 2 2
−α∗ k X3 1 k2 c2 ikX3 ,
q79 γ
q71 q81 −ρ 2 2 2
2 2
−α
X
ikX1 ,
k
1
k
c
∗
1
γ
q70 q72 q74 q76 q78 q80 q82 ρ 2 2 2
−α∗ k X2 1 k2 c2 ,
γ
ρ 2 2 2
−α∗ k X4 1 k2 c2 ,
γ
ρ 2 2 2
−α∗ k X2 1 k2 c2 ikX2 ,
γ
ρ 2 2 2
−α∗ k X4 1 k2 c2 ikX4 ,
γ
−ρ 2 2 2
−α∗ k X2 1 k2 c2 ikX2 ,
γ
−ρ 2 2 2
−α∗ k X4 1 k2 c2 ikX4 ,
γ
−ρ 2 2 2
2 2
−α
X
ikX2 ,
k
1
k
c
∗
2
γ
34
Mathematical Problems in Engineering
−ρ 2 2 2
−ρ 2 2 2
2 2
−α
X
ikX
−α∗ k X3 1 k2 c2 ikX3 ,
k
1
k
c
,
q
84
∗
3
3
γ
γ
1 2 2 ∗
2 ∗
2
2
∗
2 2
2
k β∗ m1 1 X1 − m1 k c Ω 2ikΩc ,
MX1 k m1 X1 1 ikcs1
1 2 2 ∗
2 ∗
2
2
∗
2 2
2
MX2 k mj X2 1 k β∗ m2 1 X2 − m2 k c Ω 2ikΩc ,
ikcs1
1 2 2 ∗
2 ∗
2
2
∗
2 2
2
k β∗ m3 1 X3 − m3 k c Ω 2ikΩc ,
MX3 k mj X3 1 ikcs1
1 2 2 ∗
2 ∗
2
2
∗
2 2
2
k β∗ m4 1 X4 − m4 k c Ω 2ikΩc ,
MX4 k m4 X4 1 ikcs1
q89 Γ0 λ 1 − X1 m∗1 Γ0 λ 2μ X12 m∗1 X1 ρ −α2∗ 1 X12 c2 ,
q90 Γ0 λ 1 − X2 m∗2 Γ0 λ 2μ X22 m∗2 X2 ρ −α2∗ 1 X22 c2 ,
q91 Γ0 λ 1 − X3 m∗3 Γ0 λ 2μ X32 m∗3 X3 ρ −α2∗ 1 X32 c2 ,
q92 Γ0 λ 1 − X4 m∗4 Γ0 λ 2μ X42 m∗4 X4 ρ −α2∗ 1 X42 c2 ,
q83 q85
q86
q87
q88
q93 Γ0 λ 1 X1 m∗1 Γ0 λ 2μ X12 − m∗1 X1 ρ −α2∗ 1 X12 c2 ,
q94 Γ0 λ 1 X2 m∗2 Γ0 λ 2μ X22 − m∗2 X2 ρ −α2∗ 1 X22 c2 ,
q95 Γ0 λ 1 X3 m∗3 Γ0 λ 2μ X32 − m∗3 X3 ρ −α2∗ 1 X32 c2 ,
q96 Γ0 λ 1 X4 m∗4 Γ0 λ 2μ X42 − m∗4 X4 ρ −α2∗ 1 X42 c2 ,
F
q97 −2k2 Γ0 μX1 − k2 Γ0 μm∗1 1 − X12 k2 β∗2 m∗1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
q98 −2k2 Γ0 μX2 − k2 Γ0 μm∗2 1 − Xj2 s1
F
q99 −2k2 Γ0 μX3 − k2 Γ0 μm∗3 1 − X32 k2 β∗2 m∗3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
q100 −2k2 Γ0 μX4 − k2 Γ0 μm∗4 1 − X42 s1
F
k2 β∗2 m∗1 1 X12 − m∗1 k2 c2 Ω2 2ikΩc ,
q101 2k2 Γ0 μX1 − k2 Γ0 μm∗1 1 − X12 s1
F
k2 β∗2 m∗2 1 X22 − m∗2 k2 c2 Ω2 2ikΩc ,
q102 2k2 Γ0 μX2 − k2 Γ0 μm∗2 1 − X22 s1
F
q103 2k2 Γ0 μX3 − k2 Γ0 μm∗3 1 − X32 k2 β∗2 m∗3 1 X32 − m∗3 k2 c2 Ω2 2ikΩc ,
s1
F
k2 β∗2 m∗4 1 X42 − m∗4 k2 c2 Ω2 2ikΩc ,
q104 2k2 Γ0 μX4 − k2 Γ0 μm∗4 1 − X42 s1
Mathematical Problems in Engineering
q105 −α2∗ k2 X12 1 k2 c2 ikX1 ,
35
q106 −α2∗ k2 X22 1 k2 c2 ikX2 ,
q107 −α2∗ k2 X32 1 k2 c2 ikX3 ,
q108 −α2∗ k2 X42 1 k2 c2 ikX4 ,
q109 α2∗ k2 X12 1 − k2 c2 ikX1 ,
q110 α2∗ k2 X22 1 − k2 c2 ikX2 ,
q111 α2∗ k2 X32 1 − k2 c2 ikX3 ,
q112 α2∗ k2 X42 1 − k2 c2 ikX4 .
5.31
From 5.30, we can determine by numerical effects the initial stress, gravity field,
friction coefficient, magnetic field, and rotation, for a computation using the maple program;
we use sandstone as a granular medium and nephiline as a granular layer taking into
consideration that the relaxation times τ0 0.1, τ1 0.4, and τ2 0.5, the friction coefficient
F 0.4, and the third elastic constant M 0.2.
i Effects of the initial stress, gravity field, friction coefficient, magnetic field,
relaxation time, and rotation are discussed in Figures 2 and 3.
ii From 5.30, if the initial stress are neglected, we can discuss the effects of
the gravity field, friction coefficient, magnetic field, relaxation time, and rotation, and the
discussion is clear up from Figure 4.
iii From 5.30, if the initial stress and magnetic field are neglected, we can discuss the
effects of the gravity field, friction coefficient, relaxation time and rotation, and the discussion
is clear up from Figure 5.
iv From 5.30, if the initial stress, magnetic field, and gravity field are neglected,
we can discuss the effects of the friction coefficient, relaxation time, and rotation, and the
discussion is clear up from Figure 6.
v From 5.30, if the initial stress, magnetic field, and gravity field are neglected and
there is uncoupling between the temperature and strain field, we can discuss the effects the
friction coefficient, relaxation time, rotation, and the discussion is clear up from Figure 7.
6. Numerical Results and Discussion
In order to illustrate theoretical results obtained in the proceeding section, we now present
some numerical results. The material chosen for this purpose of Carbon steel, the physical
data is given 21 as follows:
ρ 2 kgm−3 ,
λ 9.3 × 1010 Nm−1 ,
K 50 Wm−1 k−1 ,
μ 8.4 × 1010 Nm−1 ,
s 6.4 × 102 JKg−1 ,
T0 293.1 k,
αt 13.2 × 10−6 deg−1 .
6.1
6.1. Effects of the Initial Stress, Gravity Field, Friction Coefficient, Magnetic
Field, Relaxation Time, and Rotation
Figure 2 shows the velocity of Rayleigh waves Re and attenuation coefficient Im under
the effect of gravity field, friction coefficient, magnetic field, relaxation time, and rotation with
respect to the initial stress; we found that the velocity of Rayleigh waves Re and attenuation
36
Mathematical Problems in Engineering
5E+21
1.6E+21
−5E+21
0
−1E+22
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
−1.5E+22
−2E+22
−2.5E+22
−3E+22
−3.5E+22
1.4E+21
Attenuation coefficient
Rayleigh wave velocity
0E+00
1.2E+21
1E+21
8E+20
6E+20
4E+20
2E+20
−4E+22
0E+00
−4.5E+22
−2E+20
H0 = 105
H0 = 2 × 105
4E+29
2E+29
2E+29
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
−6E+29
−8E+29
−1E+30
Attenuation coefficient
Rayleigh wave velocity
0
−4E+29
−2E+29
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
−6E+29
−8E+29
−1E+30
−1.2E+30
−1.4E+30
−1.4E+30
−1.6E+30
−1.6E+30
Ω = 0.5
Ω = 0.6
0
−4E+29
−1.2E+30
Ω = 0.5
Ω = 0.6
Ω = 0.7
Ω = 0.7
4E+19
1E+23
9E+22
2E+19
8E+22
7E+22
Attenuation coefficient
Rayleigh wave velocity
H0 = 4 × 105
0E+00
0E+00
6E+22
5E+22
4E+22
3E+22
2E+22
1E+22
0E+00
0
−2E+19
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
−4E+19
−6E+19
−8E+19
0E+00
−1E+22
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
H0 = 105
H0 = 2 × 105
H0 = 4 × 105
4E+29
−2E+29
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
g = 0.1
g = 0.15
−1E+20
g = 0.2
Figure 2: Continued.
g = 0.1
g = 0.15
g = 0.2
Mathematical Problems in Engineering
37
5E−02
1E−01
0E+00
0
−1E−01
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
−2E−01
−3E−01
−4E−01
Attenuation coefficient
Rayleigh wave velocity
0E+00
−5E−02
−1.5E−01
−2E−01
−2.5E−01
−3E−01
−6E−01
−3.5E−01
τ1 = 0.5
τ1 = 0.3
τ1 = 0.4
6E+11
5E+10
5E+11
0E+00
4E+11
−5E+10
Attenuation coefficient
Rayleigh wave velocity
0
3E+11
2E+11
1E+11
τ1 = 0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
−1E+11
−1.5E+11
−2E+11
−2.5E+11
0E+00
0
−1E+11
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
−1E−01
−5E−01
τ1 = 0.3
τ1 = 0.4
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
× 1011
P
K = 0.1
K = 0.2
K = 0.3
−3E+11
K = 0.1
K = 0.2
K = 0.3
Figure 2: Effects of H0 , Ω, g, τ1 , and p on Rayleigh wave velocity and attenuation coefficient with respect
to the initial stress.
coefficient Im increased with increasing values of p and H0 , and the velocity of Rayleigh
waves Re and attenuation coefficient Im decreased and increased with increasing values
of g and K, respectively; while the values of Re and Im take one curve at another value of
the relaxation time τ1 increased with increasing values of initial stress P .
Figure 3 shows the velocity of Rayleigh waves Re and attenuation coefficient Im
under effect of initial stress, gravity field, friction coefficient, magnetic field, relaxation time
and rotation with respect to the wave number, we find that the velocity of Rayleigh waves
Re and attenuation coefficient Im decreased and increased with increasing values of
H0 , respectively, and the velocity of Rayleigh waves Re and attenuation coefficient Im
increased and decreased with increasing values of Ω and g, respectively; also, the values of
Re and Im increased with increasing values of K, while the values of Re and Im take
one curve at another value of the relaxation time τ1 , decreased with increasing values of wave
number k.
38
Mathematical Problems in Engineering
8E+32
1.5E+33
6E+32
Rayleigh wave velocity
1E+33
5E+32
0E+00
−5E+32
0
0.05
0.1
0.15
0.2
0.25
0.3
k
0.35
×10−20
−1E+33
Attenuation coefficient
2E+33
4E+32
2E+32
0E+00
−2E+32
0.05
0.1
0.15
0.2
0.3
0.35
×10−20
−6E+32
−8E+32
−2E+33
−1E+33
H0 = 105
H0 = 2 × 105
H0 = 4 × 105
1E+39
0.25
k
−4E+32
−1.5E+33
H0 = 105
H0 = 2 × 105
0
H0 = 4 × 105
1E+40
0E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−2E+39
0.35
×10−20
−3E+39
−4E+39
−5E+39
−6E+39
−7E+39
8E+39
Attenuation coefficient
Rayleigh wave velocity
−1E+39
2E+39
0
0.05
0.1
0.15
Ω = 0.7
Ω = 0.5
Ω = 0.6
0.2
0.25
0.3
k
−2E+39
−9E+39
0.35
×10−20
Ω = 0.7
Ω = 0.5
Ω = 0.6
4E+37
5E+37
3.5E+37
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−1E+38
−1.5E+38
−2E+38
0.35
×10−20
Attenuation coefficient
0E+00
Rayleigh wave velocity
4E+39
0E+00
−8E+39
−5E+37
6E+39
3E+37
2.5E+37
2E+37
1.5E+37
1E+37
5E+36
−2.5E+38
0E+00
−3E+38
−5E+36
g = 0.1
g = 0.15
g = 0.2
Figure 3: Continued.
0
0.05
0.1
0.2
0.15
0.25
0.3
k
g = 0.1
g = 0.15
g = 0.2
0.35
×10−20
39
2.5E+14
4E+14
2E+14
3.5E+14
1.5E+14
1E+14
5E+13
0E+00
−5E+13
0
0.05
0.1
0.15
0.2
0.25
0.3
k
0.35
×10−20
Attenuation coefficient
Rayleigh wave velocity
Mathematical Problems in Engineering
1E+14
5E+13
−5E+13
K = 0.1
K = 0.2
0.05
0.1
0.15
0.2
0.25
0.3
k
K = 0.1
K = 0.2
0.35
×10−20
K = 0.3
9E+08
8E+08
Attenuation coefficient
5E+08
4E+08
3E+08
2E+08
1E+08
0E+00
0
0.05
0.1
0.15
−1E+08
0.2
0.25
0.3
k
τ1 = 0.3
τ1 = 0.4
7E+08
6E+08
5E+08
4E+08
3E+08
2E+08
1E+08
0E+00
0.35
×10−20
−1E+08
0
τ1 = 0.5
0.05
0.1
0.15
7E+32
7E+31
Attenuation coefficient
8E+31
6E+32
5E+32
4E+32
3E+32
2E+32
1E+32
0.2
k
τ1 = 0.3
τ1 = 0.4
8E+32
0.25
0.3
0.35
×10−20
τ1 = 0.5
6E+31
5E+31
4E+31
3E+31
2E+31
1E+31
0E+00
0E+00
−1E+32
0
K = 0.3
6E+08
Rayleigh wave velocity
2E+14
1.5E+14
0E+00
−1E+14
Rayleigh wave velocity
3E+14
2.5E+14
0
0.05
0.1
0.2
0.15
k
P = 1 × 1011
P = 1.4 × 1011
0.25
0.3
0.35
×10−20
P = 1.6 × 1011
−1E+31
0
0.05
0.1
0.15
0.2
k
P = 1 × 1011
P = 1.4 × 1011
0.25
0.3
0.35
×10−20
P = 1.6 × 1011
Figure 3: Effects of H0 , Ω, g, τ1 , and K, P on Rayleigh wave velocity and Attenuation coefficient with
respect to the wave number.
6.2. If the Initial Stresses Are Neglected
Figure 4 shows the velocity of Rayleigh waves Re and attenuation coefficient Im under
effect of gravity field, friction coefficient, magnetic field, relaxation time, and rotation with
respect to the wave number, we find that the velocity of Rayleigh waves Re and attenuation
40
Mathematical Problems in Engineering
2E+40
1E+39
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−4E+40
0.35
×10−20
−6E+40
−8E+40
−1E+41
−1.2E+41
−1.4E+41
0E+00
Attenuation coefficient
Rayleigh wave velocity
0E+00
−2E+40
0
0.05
0.1
0.15
−1E+39
0.2
0.25
0.3
k
0.35
×10−20
−2E+39
−3E+39
−4E+39
−5E+39
−1.6E+41
−1.8E+41
−6E+39
H0 = 105
H0 = 2 × 105
H0 = 105
H0 = 2 × 105
H0 = 4 × 105
5E+41
H0 = 4 × 105
2E+38
0E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
k
0.35
×10−20
1E+38
Attenuation coefficient
Rayleigh wave velocity
−5E+41
−1E+42
−1.5E+42
−2E+42
−2.5E+42
−3E+42
−3.5E+42
0E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−1E+38
0.35
×10−20
−2E+38
−3E+38
−4E+42
−4.5E+42
−4E+38
Ω = 0.7
Ω = 0.5
Ω = 0.6
2E+49
5E+49
0E+00
0.05
0.1
0.15
0.2
0.25
0.3
k
−5E+49
−1E+50
−1.5E+50
−2E+50
0.35
×10−20
0E+00
0
Attenuation coefficient
0
Rayleigh wave velocity
Ω = 0.7
Ω = 0.5
Ω = 0.6
0.05
0.1
0.15
0.2
0.25
0.3
k
−2E+49
−4E+49
−6E+49
−8E+49
−2.5E+50
−1E+50
−3E+50
g = 0.1
g = 0.15
g = 0.2
Figure 4: Continued.
g = 0.1
g = 0.15
g = 0.2
0.35
×10−20
Mathematical Problems in Engineering
41
0E+00
1.76E+59
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−4E+79
0.35
×10−20
−1E+74
Attenuation coefficient
Attenuation coefficient
−2E+79
−6E+79
−8E+79
−1E+80
−1.2E+80
−1.4E+80
0.1
0.15
0.2
0.3
0.35
×10−20
−4E+74
−5E+74
−6E+74
−7E+74
−8E+74
−9E+74
−1E+75
τ1 = 0.5
τ1 = 0.3
τ1 = 0.4
2.5E+27
0.25
k
−3E+74
−1.8E+80
τ1 = 0.5
1E+24
0E+00
2E+27
0
Attenuation coefficient
Rayleigh wave velocity
0.05
−2E+74
−1.6E+80
τ1 = 0.3
τ1 = 0.4
0
1.5E+27
1E+27
5E+26
0E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−5E+26
K = 0.1
K = 0.2
K = 0.3
0.35
×10−20
0.05
0.1
0.15
−1E+24
0.2
0.25
0.3
k
0.35
×10−20
−2E+24
−3E+24
−4E+24
−5E+24
−6E+24
−7E+24
K = 0.1
K = 0.2
K = 0.3
Figure 4: Effects of H0 , Ω, g, τ1 , and K on Rayleigh wave velocity and attenuation coefficient with respect
to the wave number.
coefficient Im decreased with increasing values of H0 and Ω, while that contrary with
increasing values of g; also, the values of Re and Im increased and decreased with
increasing values of K, respectively, while the values of Re and Im take one curve at
another value of the relaxation time τ1 decreased, then increased with increasing values of
wave number k.
6.3. If the Initial Stresses and Magnetic Field Are Neglected
Figure 5 shows that the velocity of Rayleigh waves Re and attenuation coefficient Im
under effect of gravity field, friction coefficient, relaxation time, and rotation with respect to
the wave number; we find that the velocity of Rayleigh waves Re and attenuation coefficient
Im decreased and increased with increasing values of Ω, and the values of Re and Im
increased with increasing values of g, while that contrary with increasing values of K; also, the
42
Mathematical Problems in Engineering
3.5E+42
1E+45
0E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−2E+45
3E+42
0.35
×10−20
Attenuation coefficient
Rayleigh wave velocity
−1E+45
−3E+45
−4E+45
−5E+45
−6E+45
−7E+45
1.5E+42
1E+42
5E+41
0E+00
−9E+45
−5E+41
0
0.1
0.15
0.2
0.25
0.3
0.35
×10−20
Ω = 0.7
Ω = 0.5
Ω = 0.6
1E+51
0E+00
0E+00
0.05
0.1
0.15
0.2
0.25
0.3
k
−1E+52
0.35
×10−20
−2E+52
−3E+52
−4E+52
0
Attenuation coefficient
0
0.1
0.15
0.2
0.3
0.35
×10−20
−3E+51
−4E+51
−5E+51
−6E+52
−7E+51
g = 0.1
g = 0.15
g = 0.2
1.6E+91
0.25
k
−2E+51
−6E+51
g = 0.1
g = 0.15
0.05
−1E+51
−5E+52
g = 0.2
1E+80
0E+00
1.4E+91
−1E+80
Attenuation coefficient
1.2E+91
1E+91
8E+90
6E+90
4E+90
2E+90
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−2E+80
−3E+80
−4E+80
−5E+80
−6E+80
−7E+80
−8E+80
0E+00
−2E+90
0.05
k
Ω = 0.7
1E+52
Rayleigh wave velocity
2E+42
−8E+45
Ω = 0.5
Ω = 0.6
Rayleigh wave velocity
2.5E+42
0
0.05
0.1
0.15
0.2
0.25
0.3
k
τ1 = 0.3
τ1 = 0.4
0.35
×10−20
−9E+80
−1E+81
τ1 = 0.5
Figure 5: Continued.
τ1 = 0.3
τ1 = 0.4
τ1 = 0.5
0.35
×10−20
Mathematical Problems in Engineering
43
2E+75
5E+73
0E+00
−5E+73
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−1E+74
−1.5E+74
−2E+74
−2.5E+74
−3E+74
0.35
×10−20
−2E+75
Attenuation coefficient
Rayleigh wave velocity
0E+00
0.1
0.15
0.2
0.25
0.3
k
0.35
×10−20
−6E+75
−8E+75
−1E+76
−1.2E+76
−1.4E+76
−1.6E+76
−4E+74
−1.8E+76
K = 0.3
0.05
−4E+75
−3.5E+74
K = 0.1
K = 0.2
0
K = 0.1
K = 0.2
K = 0.3
Figure 5: Effects of Ω, g, τ1 , and K on Rayleigh wave velocity and attenuation coefficient with respect to
the wave number.
values of Re and Im take one curve at another value of the relaxation time τ1 , increased,
then decreased with increasing values of wave number k.
6.4. If the Initial Stresses, Magnetic Field, and Gravity Field Are Neglected
Figure 6 shows the velocity of Rayleigh waves Re and attenuation coefficient Im under the
effect of friction coefficient, relaxation time, and rotation with respect to the wave number;
we find that the velocity of Rayleigh waves Re and attenuation coefficient Im decreased
and increased with increasing Ω, and the values of Re and Im decreased with increasing
values of K, while the values of Re and Im take one curve at another value of the relaxation
time τ1 , decreased and increased with increasing values of the wave number k, respectively.
6.5. If the Initial Stresses, Magnetic Field, and Gravity Field Are Neglected
and There Is Uncoupling between the Temperature and Strain Field
Figure 7 shows the velocity of Rayleigh waves Re and attenuation coefficient Im under the
effect of friction coefficient, relaxation time, and rotation with respect to the wave number;
we find that the velocity of Rayleigh waves Re and attenuation coefficient Im increased
and decreased with increasing of Ω, respectively, while that contrary with increasing values
of K; finally, the values of Re and Im take one curve at another value of the relaxation time
τ1 , decreased with increasing, the values of the wave number k.
7. Conclusions
The problem of the Rayleigh waves in generalized magneto-thermo-viscoelastic granular
medium under the influence of rotation, gravity field, and initial stress is considered, and
the frequency equation of the wave motion in the explicit form is derived, by considering
44
Mathematical Problems in Engineering
1E+55
5E+54
0E+00
0.05
0.1
0.15
0.2
0.25
0.3
k
0.35
×10−20
−1E+55
−1.5E+55
−2E+55
−2.5E+55
0E+00
0
Attenuation coefficient
Rayleigh wave velocity
0
−5E+54
0.05
0.1
0.15
0.2
0.25
0.3
k
−1E+55
0.35
×10−20
−2E+55
−3E+55
−4E+55
−3E+55
−3.5E+55
−5E+55
Ω = 0.7
Ω = 0.5
Ω = 0.6
Ω = 0.7
Ω = 0.5
Ω = 0.6
1.6E+19
1E+18
0E+00
1.2E+19
Attenuation coefficient
Rayleigh wave velocity
1.4E+19
1E+19
8E+18
6E+18
4E+18
2E+18
0.05
0.1
0.15
0.2
0.25
0.3
k
0.35
×10−20
−2E+18
−3E+18
−4E+18
0E+00
−2E+18
0
−1E+18
0
0.05
0.1
0.15
0.2
0.25
0.3
k
τ1 = 0.3
τ1 = 0.4
0.35
×10−20
−5E+18
τ1 = 0.3
τ1 = 0.4
τ1 = 0.5
2E+25
τ1 = 0.5
2.5E+24
−2E+25
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−4E+25
−6E+25
−8E+25
−1E+26
−1.2E+26
0.35
×10−20
2E+24
Attenuation coefficient
Rayleigh wave velocity
0E+00
1.5E+24
1E+24
5E+23
0E+00
−1.4E+26
0
−1.6E+26
0.05
0.1
K = 0.1
K = 0.2
K = 0.3
0.15
0.2
0.25
0.3
k
−5E+23
K = 0.1
K = 0.2
0.35
×10−20
K = 0.3
Figure 6: Effects of Ω, τ1 , and K on Rayleigh wave velocity and attenuation coefficient with respect to the
wave number.
Mathematical Problems in Engineering
45
5E+285
2E+287
0E+00
−2E+287
Attenuation coefficient
Rayleigh wave velocity
4E+285
3E+285
2E+285
1E+285
0.05
0.1
0.15
0.2
0.25
0.3
k
−1E+285
0.1
0.15
0.2
0.25
0.3
k
0.35
×10−20
−6E+287
−8E + 287
−1E+288
−1.2E+288
−1.4E+288
0.35
×10−20
−1.6E+288
Ω = 0.7
Ω = 0.5
Ω = 0.6
0.05
−4E+287
0E+00
0
0
Ω = 0.7
Ω = 0.5
Ω = 0.6
4E+284
2.5E+281
3.5E+284
Attenuation coefficient
Rayleigh wave velocity
2E+281
1.5E+281
1E+281
5E+280
0E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
k
−5E+280
τ1 = 0.3
τ1 = 0.4
2.5E+284
2E+284
1.5E+284
1E+284
5E+283
0E+00
0.35
×10−20
−5E+283
0
0.05
0.1
0.15
0.2
0.25
0.3
k
τ1 = 0.5
τ1 = 0.3
τ1 = 0.4
0.35
×10−20
τ1 = 0.5
1.4E+302
5E+303
1.2E+302
0
0.05
0.1
−5E+303
0.15
0.2
0.25
0.3
k
−1E+304
−1.5E+304
−2E+304
0.35
×10−20
Attenuation coefficient
0E+00
Rayleigh wave velocity
3E+284
1E+302
8E+301
6E+301
4E+301
2E+301
−2.5E+304
0E+00
−3E+304
−2E+301
K = 0.1
K = 0.2
K = 0.3
0
0.05
0.1
0.15
0.2
k
K = 0.1
K = 0.2
0.25
0.3
0.35
×10−20
K = 0.3
Figure 7: Effects of Ω, τ1 , and K on Rayleigh wave velocity and attenuation coefficient with respect to the
wave number.
46
Mathematical Problems in Engineering
various special cases. The numerical results are obtained for carbon-steel material, although
the effect of the rotation, magnetic field, relaxation times, initial stress, gravity field, and
friction coefficient is observed to be quite large on wave propagation of Rayleigh wave
velocity Re and attenuation coefficient Im.
The problem of the Rayleigh waves in generalized magneto-thermo-viscoelastic
granular medium under the influence of rotation, gravity field, and initial stress is considered,
and the frequency equation of the wave motion in the explicit form is derived, by considering
various special cases. The numerical results are obtained for carbon-steel material, although
the effect of the rotation, magnetic field, relaxation times, initial stress, gravity field, and
friction coefficient is observed to be quite large on wave propagation of Rayleigh wave
velocity Re and attenuation coefficient Im.
It is easy to see that the values of Re and Im with respect to the initial stress are
increased with increasing values of Ω, while that contrary if the initial stress are neglected
and with respect to the wave number; also, if the initial stress are constant with respect to the
wave number the values of Re and Im increased and decreased with increasing values
of Ω, respectively, and if the initial stress and the magnetic field are neglected, the values of
Re and Im decreased and increased with increasing values of Ω, respectively, while that
contrary if P, H0 , g, θ, and ε are neglected and with respect to the wave number; finally, if
P, H0 , and g are neglected and with respect to the wave number, the values of Re and Im
increased with increasing values of Ω.
It is easy to see that the values of Re and Im with respect to the initial stress are
decreased and increased with increasing values of g, respectively, while that contrary if the
initial stress are constant and with respect to the wave number; also, if the initial stress are
neglected and if the initial stress and the magnetic field are neglected with respect to the wave
number, the values of Re and Im increased with increasing values of g.
It is easy to see that the values of Re and Im with respect to the initial stress
are decreased and increased with increasing values of K, respectively, while that contrary
if the initial stress are neglected and with respect to the wave number; also, if the initial
stress and the magnetic field are neglected and if P, H0 , and g are neglected with respect the
wave number, the values of Re and Im decreased with increasing values of K; finally, if
P, H0 , g, θ, and ε are neglected and with respect to the wave number, the values of Re and
Im decreased and increased with increasing values of K.
Finally, the frequency equation has been discussed under effect of rotation, gravity
field, and initial stress and in case of various classical and nonclassical theories of
thermoelasticity. The results indicate that the effect of rotation, magnetic field, initial stress,
and gravity field is very pronounced. The frequency equations derived in this paper may
be useful in practical applications. It is concluded from the above analyses and results that
the present solution is accurate and reliable and the method is simple and effective. So it
may be as a reference to solve other problems of Rayleigh waves in generalized magnetothermoelastic granular medium.
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