Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 245965, 22 pages
doi:10.1155/2012/245965
Research Article
Influence of Initial Stress and Gravity Field on
Propagation of Rayleigh and Stoneley Waves in
a Thermoelastic Orthotropic Granular Medium
S. M. Ahmed1 and S. M. Abo-Dahab2, 3
1
Department of Mathematics, Faculty of Education, Suez Canal University, EL-Arish 45111, Egypt
Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia
3
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
2
Correspondence should be addressed to S. M. Abo-Dahab, sdahb@yahoo.com
Received 9 May 2011; Revised 11 October 2011; Accepted 26 October 2011
Academic Editor: Moran Wang
Copyright q 2012 S. M. Ahmed and S. M. Abo-Dahab. 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 propagation of Rayleigh and Stoneley waves in a thermoelastic orthotropic granular half-space
supporting a different layer under the influence of initial stress and gravity field is studied. The
frequency equation of Rayleigh waves in the form of twelfth-order determinantal expression and
the frequency equation of Stoneley waves in the form of eighth-order determinantal expression
are obtained. The standard equation of dispersion is discussed to obtain Rayleigh and Stoneley
waves that have complex roots; the real part gives the velocity of Rayleigh or Stoneley waves but
the imaginary part gives the attenuation coefficient. Finally, the numerical results have been given
and illustrated graphically, and their physical meaning has been explained.
1. Introduction
The propagation of thermoelastic waves in a granular medium under initial stress and gravity
field has applications in soil mechanics, earthquake science, geophysics, mining engineering,
and so forth. The theoretical outline of the development of the subject from the mid-thirties
was given by Paria 1. The present paper, however, is based on the dynamics of granular
media as propounded by Oshima 2, 3. The medium under consideration is discontinuous
such as one composed numerous large or small grains. Unlike a continuous body, each
element or grain cannot only translate but also rotate about its centre of gravity. This motion
is the characteristics of the medium and has an important effect upon the equation of motion
to produce internal friction. It is assumed that the medium contains so many grains that
they will never be separated from each other during the deformation, and that the grain
2
Mathematical Problems in Engineering
has perfect elasticity. The propagation of Rayleigh waves in granular medium was given by
many authors such as Bhattacharyya 4, El-Naggar 5, Ahmed 6, 7, and others. Ahmed
8 discussed Stoneley waves in a nonhomogeneous granular medium under the influence of
gravity.
The problem of Stoneley waves plays an important role in the earthquake science,
optics, geophysics, and plasma physics. Many authors such as Abd-Alla and Ahmed 9, 10,
El-Naggar et al. 11, Das et al. 12, and others investigated the effect of gravity of the
propagation of surface waves Stoneley waves and Rayleigh waves in an elastic solid
medium. Goda 13 illustrated the effect of inhomogeneity and anisotropy on Stoneley waves;
Abd-Alla et al. 14 discussed Rayleigh waves in a magnetoelastic half-space of orthotropic
material under influence of initial stress and gravity field. Sharma et al. 15 studied the
propagation characteristics of Rayleigh waves in transversely isotropic piezothermoelastic
half-space with rotation and thermal relaxation. Sharma et al. 16 investigated the problem
of propagation characteristics of Rayleigh waves in transversely isotropic piezothermoelastic
materials. Sharma and Walia 17 investigated Rayleigh waves in piezothermoelastic
materials. Ting 18 illustrated the surface waves in a rotating anisotropic elastic half-space.
Abd-Alla and Abo-Dahab 19 investigated Rayleigh waves in magneto-thermo-viscoelastic
solid with thermal relaxation times. Recently, Ahmed and Abo-Dahab 20 pointed out
Propagation of Love waves in an orthotropic granular layer under initial stress overlying
a semi-infinite granular medium. Vinh and Seriani 21 illustrated explicit secular equations
of Rayleigh waves in a nonhomogeneous orthotropic elastic medium under the influence
of gravity. Vinh and Seriani 22 discussed the explicit secular equations of Stoneley waves
in a nonhomogeneous orthotropic elastic medium under the influence of gravity. Vinh
23 discussed the explicit secular equations of Rayleigh waves in elastic media under the
influence of gravity and initial stress. Lo 24 investigated the propagation and attenuation
of Rayleigh waves in a semi-infinite unsaturated poroelastic medium. El-Naggar 25
investigated the dynamical problem of a generalized thermoelastic granular infinite cylinder
under initial stress. Recently, Abd-Alla et al. 26 investigated Rayleigh waves in generalized
magneto-thermo-viscoelastic granular medium under the influence of rotation, gravity field.
This paper is devoted to study the effect of gravity field and the initial stress on
the propagation of Rayleigh and Stoneley waves in thermoelastic orthotropic granular halfspace supporting a different layer under initial stress and gravity field. The frequency
equations are obtained: the frequency equation of Rayleigh waves in the form of twelfth-order
determinantal expression and the frequency equation of Stoneley waves in the form of eighthorder determinantal expression. The standard equation of dispersion is discussed to obtain
the Rayleigh and Stoneley waves that have complex roots; the real part gives the velocity
of Rayleigh or Stoneley waves but the imaginary part gives the attenuation coefficient. The
results obtained are displayed graphically and their physical meaning has been explained.
2. Formulation of the Problem
Let us consider an initially stressed orthotropic granular layer of finite thickness H overlaying
a semi-infinite orthotropic granular medium. The upper surface of the upper layer is assumed
to be free and horizontal. We take a set of orthogonal Cartesian axes Ox1 x2 x3 such that
the interface and the free surface of the granular layer resting on the granular half-space of
different material are the planes x3 H and x3 0, respectively, with the origin O being any
point on the interface surface; x3 -axis is positive in the direction towards the exterior of the
Mathematical Problems in Engineering
3
half-space, and x1 -axis is positive along the direction of Rayleigh waves and Stoneley waves
propagation. Let the both media be under initial compression stress P along x1 -axis, with the
influence of gravity and at initial temperature T0 . It is assumed that both media exchange
heat freely with their surroundings; an initial stress is produced by a slow process of creep,
where the shear stresses tend to become small or vanish after a long interval of time.
In view of the two-dimensional nature of the problem, we assume that the state of
initial stress is
τ11 τ33 τ,
τ13 0,
2.1
the equilibrium conditions of the initial stress field are given by 12
∂τ
0,
∂x1
∂τ
− ρg 0.
∂x3
2.2
The state of deformation in the granular medium is described by the displacement
u1 , 0, u3 of the centre of gravity of a grain and the rotation vector ξ ξ, η, ζ of
vector U
the grain about its centre of gravity. There exist a stress tensor and a couple stress which are
nonsymmetric, that is,
τij /
τji ,
Mij /
Mji
i, j 1, 2, 3 .
2.3
The stress tensor τij can be expressed as the sum of symmetric and antisymmetric
tensors
τij σij σij ,
2.4
where
σij 1
τij τji ,
2
σij 1
τij − τji .
2
2.5
The symmetric tensor σij σji is related to the symmetric strain tensor
1
eij eji 2
∂ui ∂uj
∂xj ∂xi
2.6
by the Hook’s law.
The antisymmetric stress σij is given by
σ23
−F
∂ξ
,
∂t
σ31
−F
∂η
,
∂t
σ12
−F
∂ζ
,
∂t
σ11
σ22
σ33
0,
2.7
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Mathematical Problems in Engineering
the couple stress Mij is given by
Mij Mνij ,
ν11 ν31 ∂ξ
,
∂x1
∂ξ
,
∂x3
ν22 0,
ν12 ν33 ∂ζ
,
∂x3
∂ η ω2 ,
∂x1
ν13 ∂ζ
,
∂x1
ν23 0,
ν32 ∂ η ω2 ,
∂x3
ν21 0,
2.8
where, ω2 ∂u1 /∂x3 − ∂u3 /∂x1 .
The six equations of motion are 9, 11, 12
∂τ11 ∂τ31 P ∂ω2
∂u3
∂2 u1
−
− ρg
ρ 2 ,
∂x1
∂x3
2 ∂x3
∂x1
∂t
∂τ12 ∂τ32
0,
∂x1
∂x3
∂τ13 ∂τ33 P ∂ω2
∂u1
∂2 u3
−
ρg
ρ 2 ,
∂x1
∂x3
2 ∂x1
∂x1
∂t
τ23 − τ32 ∂M11 ∂M31
0,
∂x1
∂x3
τ31 − τ13 ∂M12 ∂M32
0,
∂x1
∂x3
τ12 − τ21 ∂M13 ∂M33
0.
∂x1
∂x3
2.9
The components of stress for orthotropic body, under the effect of an initial
compression stress P , are given by 11
τ11 c11 P τ31 c55
∂u1
∂u3
c13 P − υ1 T,
∂x1
∂x3
∂u1 ∂u3
∂x3 ∂x1
−F
∂η
,
∂t
τ33 c13
τ13 c55
∂u1
∂u3
c33
− υ3 T,
∂x1
∂x3
∂u1 ∂u3
∂x3 ∂x1
F
∂η
,
∂t
2.10
where
υ1 c11 c12 α1 c13 α2 ,
υ3 2c13 α1 c33 α2 .
2.11
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5
Substituting 2.9 into 2.10, we obtain
2
∂2 u1
P ∂2 u1
P
∂
∂u3
∂ u3
c13 c55 −
υ1 T − ρg
c11 P 2 c55 2
2
2
∂x
∂x
∂x
∂x1
∂x1
∂x3
1
3
1
∂ ∂η
∂2 u1
ρ 2 ,
∂t ∂x3
∂t
∂ζ
∂ ∂ξ
−
0,
F
∂t ∂x3 ∂x1
−F
c55 c13 P
2
∂2 u3
P ∂2 u3
∂
∂u1
∂2 u1
c55 −
c
−
υ3 T ρg
33
2
2
∂x1 ∂x3
2 ∂x1
∂x3
∂x1
∂x3
∂2 u3
∂ ∂η
ρ 2 ,
F
∂t ∂x1
∂t
− 2F
∂ξ
∇2 Mξ 0,
∂t
− 2F
∂ζ
∇2 Mζ 0
∂t
−2F
∂η
∂u1 ∂u3
∇2 M η −
0,
∂t
∂x3 ∂x1
2.12
The heat conduction equation is given by 11
1 ∂T
∇ T−
− ε∇ ·
χ ∂t
2
∂U
∂t
0,
2.13
where χ δ1 δ2 /2ρs, ε T0 υ1 υ3 /δ1 δ2 .
3. Solution of the Problem
By Helmholtz’s theorem 27, the displacement vector u
can be written in the form of the
potentials φx1 , x3 , t and ψx1 , x3 , t which are related to the displacement components u1
and u3 by the relations
u1 ∂ϕ
∂ψ
−
,
∂x1 ∂x3
u3 ∂ϕ
∂ψ
.
∂x3 ∂x1
3.1
Substituting 3.1 into 2.12 and 2.13, we get the following wave equations satisfied
by ϕ, ψ, ξ, η, and ζ:
c11 P ∂2 ϕ
∂x12
c13 2c55 P ∂2 ϕ
∂x32
− ρg
∂ψ
∂2 ϕ
− υ1 T ρ 2 ,
∂x1
∂t
3.2
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Mathematical Problems in Engineering
∂ ∂ζ
∂ξ
−
0,
∂t ∂x1 ∂x3
c55 −
P
2
∂2 ψ
∂φ
∂η
P ∂2 ψ
ρ 2,
c
−
c
−
c
−
ρg
F
33
31
55
2
2
2 ∂x3
∂x1
∂t
∂t
∂x1
∂2 ψ
3.4
∂ξ
0,
∂t
3.5
∂η
− ∇4 ψ 0,
∂t
3.6
∂ζ
0,
∂t
3.7
∇2 ξ − S
∇2 η − S
3.3
∇2 ζ − S
1 ∂T
2 ∂φ
− ε∇
0,
∇T−
χ ∂t
∂t
2
S
2F
.
M
3.8
3.9
Eliminating η from 3.4 and 3.6, we get
∂
∇ −S
∂t
2
P
c55 −
2
∂2 ψ
∂φ
P ∂2 ψ
4 ∂ψ
F ∇
c33 − c31 − c55 −
− ρ 2 ρg
0.
2 ∂x32
∂x1
∂t
∂t
∂x12
∂2 ψ
3.10
Also, T can be eliminated by using 3.8 and 3.9 as follows:
1 ∂
∇ −
χ ∂t
2
∂ψ
∂2 ϕ
∂φ
0.
− ρ 2 − υ1 ε∇2
c11 P 2 c13 2c55 P 2 − ρg
∂x1
∂t
∂t
∂x1
∂x3
3.11
∂2 ϕ
∂2 ϕ
Assuming that
ϕ, ψ ϕ1 x3 , ψ1 x3 exp {iLx1 − bt},
3.12
Mathematical Problems in Engineering
7
ξ, η, ζ ξ1 x3 , η1 x3 , ζx3 exp {iLx1 − bt},
3.13
where b is real positive and L is in general complex.
Substituting 3.13 into 3.3, 3.5, and 3.7, we get
Dξ1 − iLζ1 0,
3.14
D2 ξ1 h2 ξ1 0,
3.15
D2 ζ1 h2 ζ1 0,
3.16
where h2 ibS − L2 , D ≡ d/dx.
Solutions of 3.15 and 3.16 are
ξ1 A1 eihx3 A2 e−ihx3 ,
ζ1 B1 eihx3 B2 e−ihx3 ,
3.17
h A1 eihx3 − A2 e−ihx3 − L B1 eihx3 − B2 e−ihx3 0.
3.18
respectively.
From 3.14 and 3.17, we obtain
Equating the coefficients of eihx3 and e−ihx3 to zero in 3.18, we get
A1 L
B1 ,
h
L
A2 − B2 .
h
3.19
Substituting 3.12 and 3.13 into 3.10 and 3.11, we get
c33 − c13 − c55 −
P
− ibF D4
2
P
P
2
2
2
− L c55 −
ρb 2ibFL D2
c33 − c13 − c55 −
ibS − L
2
2
P
ibS − L2 ρb2 − L2 c55 −
− ibFL4 ψ1
2
iρgL D2 − L2 ibS φ1 0,
2
3.20
8
Mathematical Problems in Engineering
c13 2c55 P D4
ib
ρb2 − L2 c11 c13 2c55 2P ibυ1 ε c13 2c55 P D2
χ
ib
2
2
2
2
−L
ρb − L c11 P − ibυ1 εL
φ1
χ
3.21
ib
2
2
ψ1 0.
− iρgL D − L χ
The solution of 3.20 and 3.21 has the form
φ1 Aj e−iλj x3 Bj eiλj x3 ,
ψ1 Ej e−iλj x3 Fj eiλj x3
j 3, 4, 5, 6 ,
3.22
where the constants Aj , Bj are related with the constants Ej , Fj , respectively, by means of
3.20 or 3.21, and λj j 3, 4, 5, 6 are taken to be imaginary.
Equating the coefficients of e−iλj x3 , eiλj x3 j 3, 4, 5, 6 to zero, we have using 3.21
Ej mj Aj ,
F j mj Bj
j 3, 4, 5, 6 ,
3.23
where
mj 1
2
iρgL −λj − L2 ib/χ
× c13 2c55 P λ4j
ib
− ρb2 − L2 c11 c13 2c55 2P ibυ1 ε c13 2c55 P λ2j
χ
3.24
ib
− L2 ρb2 − L2 c11 P − ibυ1 εL2 ,
χ
where λ3 , λ4 , λ5 , λ6 are the imaginary roots of the equation
k8 λ8 k6 λ6 k4 λ4 k2 λ2 k0 0,
3.25
Mathematical Problems in Engineering
9
P
k8 c13 2c55 P c33 − c13 − c55 − − ibF ,
2
ib
k6 − ρb2 − L2 c11 c13 2c55 2P ibυ1 ε c13 2c55 P χ
P
× c33 − c13 − c55 − − ibF − c13 2c55 P 2
P
P
2
2
2
2
c33 − c13 − c55 −
− L c55 −
ρb 2ibFL ,
× ibS − L
2
2
ib
P
− L2 ρb2 − L2 c11 P − ibυ1 εL2 c33 − c13 − c55 − − ibF
k4 χ
2
P
c13 2c55 P ibS − L2 ρb2 − L2 c55 −
− ibFL4
2
P
P
2
2
2
2
c33 − c13 − c55 −
ρb − L c55 −
2ibFL
ibS − L
2
2
ib
2
2
× ρb − L c11 c13 2c55 2P ibυ1 ε c13 2c55 P − ρ2 g 2 L2 ,
χ
P
P
− L2 c55 −
ρb2 2ibFL2
k2 ibS − L2 c33 − c13 − c55 −
2
2
ib
×
− L2 ρb2 − L2 c11 P − ibυ1 εL2
χ
ib
ρb2 − L2 c11 c13 2c55 2P ibυ1 ε c13 2c55 P χ
P
ib
× ibS − L2 ρb2 − L2 c55 −
− ibFL4 − ρ2 g 2 L2 2L2 − ibs −
.
2
χ
P
− ibFL4
k0 ibS − L2 ρb2 − L2 c55 −
2
ib
2
2
2
2
2 2 2 ib
2
×
−L
−L
ρb − L c11 P − ibυ1 εL − ρ g L
ibS − L2 .
χ
χ
3.26
Using 3.2, 3.3, 3.12, 3.13, 3.22, and 3.23, we get
φ Aj e−iλj x3 Bj eiλj x3 expiLx1 − bt,
ψ mj Aj e−iλj x3 Bj eiλj x3 expiLx1 − bt
j 3, 4, 5, 6
3.27
10
Mathematical Problems in Engineering
From 3.2 and 3.27, the temperature T has the form
T nj Aj e−iλj x3 Bj eiλj x3 expiLx1 − bt
j 3, 4, 5, 6 ,
3.28
where
nj −
1
c13 2c55 P λ2j L2 c11 P iLρg mj − ρb2
υ1
3.29
also, from 3.6 and 3.27, η has the form
η Ωj mj Aj e−iλj x3 Bj eiλj x3 expiLx1 − bt
j 3, 4, 5, 6 ,
3.30
where
Ωj −
λ2j L2
λ2j
2
L2 − ibS
j 3, 4, 5, 6 .
3.31
We use the symbols with a bar for the quantities in the lower medium except
x3 , L, b, P, g, and by assuming that the solution of 3.20 and 3.21 satisfies the condition
that the corresponding stresses vanish as x3 → −∞, we obtain
L
ξ1 − B2 e−ihx3 ,
h
ζ1 B2 e−ihx3 ,
η1 Ω mj Aj e−iλj x3 ,
φ1 Aj e−iλj x3 ,
ψ1 mj Aj e−iλj x3 ,
T nj Aj exp i Lx1 − bt − λj x3
j 3, 4, 5, 6 .
3.32
4. Boundary Conditions and Frequency Equation
The boundary conditions on the interface x3 0 are
i u1 u1 ,
iv η η ,
vii M31 M31 ,
x τ31 τ31 ,
xiii
ii u3 u3 ,
v ζ ζ,
iii ξ ξ,
vi M33 M33 ,
viii M32 M32 ,
xi τ32 τ32 ,
ix τ33 τ33 ,
xii T T ,
∂T
∂T
θT θ T.
∂x3
∂x3
4.1
Mathematical Problems in Engineering
11
The boundary conditions on the free surface x3 H are
xiv M33 0,
xv M31 0,
xvi M32 0,
xvii τ33 0,
xviii τ31 0,
xix τ32 0,
xx
4.2
∂T
θT 0,
∂x3
where
M33 M
∂ζ
,
∂x3
∂2 ϕ
τ33 c13
∂x12
M32 M
c33
∂2 ϕ
∂x32
∂ η − ∇2 ψ ,
∂x3
c33 − c13 τ31 c55
∂2 ψ
M31 M
∂2 ψ
− υ3 T,
∂x1 ∂x3
∂2 ψ
∂2 ϕ
−
2
∂x1 ∂x3
∂x12 ∂x32
−F
∂ξ
,
∂x3
τ32 −F
∂ξ
,
∂t
4.3
∂η
,
∂t
θ is the ratio of the coefficients of heat transfer to the thermal conductivity.
From the boundary conditions iii, v, vi, and vii, we get
B1 eihH − B2 e−ihH −B2 e−ihH ,
B1 eihH B2 e−ihH −B2 eihH ,
M B1 e
ihH
− B2 e
−ihH
−MB2 e
−ihH
4.4
,
M B1 eihH B2 e−ihH −MB2 eihH ,
hence
B1 B2 B2 0,
ξ ζ ξ ζ 0.
4.5
The other significant boundary conditions are responsible for the following relations:
xvi
xvii
λ2j Ωj L2
e−iλj H Aj − eiλj H Bj 0,
c13 L L mj λj c33 λj λj − Lmj υ3 nj e−iλj H Aj
c13 L L − mj λj c33 λj λj Lmj υ3 nj eiλj H Bj 0,
12
xviii
Mathematical Problems in Engineering
c55 λ2j − L2 mj 2Lλj ibFΩj mj e−iλj H Aj
c55 λ2j − L2 mj − 2Lλj ibFΩj mj eiλj H Bj 0,
i
ii
iv
viii
ix
L mj λj Aj L − mj λj Bj L mj λj Aj ,
Lmj − λj Aj Lmj λj Bj Lmj − λj Aj ,
Ω j m j A j B j Ω j m j Aj ,
Mmj λj λ2j Ωj L2 Aj − Bj Mmj λj λ2j Ωj L2 ,
c13 L L mj λj c33 λj λj − Lmj υ3 nj Aj
c13 L L − mj λj c33 λj λj Lmj υ3 nj Bj
c13 L L mj λj c33 λj λj − Lmj υ3 nj Aj ,
x
c55 mj L2 λ2j − 2Lλj − ibFmj Ωj Aj
c55 mj L2 λ2j 2Lλj − ibFmj Ωj Bj
c55 mj L2 λ2j − 2Lλj − ibFmj Ωj Aj ,
xii
nj Aj nj Bj nj Aj ,
xiii
nj θ − iλj Aj nj θ iλj Bj nj θ − iλj Aj ,
xx
nj θ − iλj e−iλj H Aj nj θ iλj eiλj H Bj 0,
j 3, 4, 5, 6 .
4.6
Elimination of Aj , Bj , Aj j 3, 4, 5, 6 gives the wave velocity equation in the form
det ·drc 0
r, c 1, 2, . . . , 12,
where the nonvanishing entries of the twelfth-order determinant of drc are given by
d11 λ23 Ω3 L2 e−iλ3 H ,
d12 λ24 Ω4 L2 e−iλ4 H ,
d13 λ25 Ω5 L2 e−iλ5 H ,
d14 λ26 Ω6 L2 e−iλ6 H ,
4.7
Mathematical Problems in Engineering
d15 − λ23 Ω3 L2 eiλ3 H ,
13
d16 − λ24 Ω4 L2 eiλ4 H ,
d17 − λ25 Ω5 L2 eiλ5 H ,
d18 − λ26 Ω6 L2 eiλ6 H ,
d21 c13 LL m3 λ3 c33 λ3 λ3 − Lm3 υ3 n3 e−iλ3 H ,
d22 c13 LL m4 λ4 c33 λ4 λ4 − Lm4 υ3 n4 e−iλ4 H ,
d23 c13 LL m5 λ5 c33 λ5 λ5 − Lm5 υ3 n5 e−iλ5 H ,
d24 c13 LL m6 λ6 c33 λ6 λ6 − Lm6 υ3 n6 e−iλ6 H ,
d25 c13 LL − m3 λ3 c33 λ3 λ3 Lm3 υ3 n3 eiλ3 H ,
d26 c13 LL − m4 λ4 c33 λ4 λ4 Lm4 υ3 n4 eiλ4 H ,
d27 c13 LL − m5 λ5 c33 λ5 λ5 Lm5 υ3 n5 eiλ5 H ,
d28 c13 LL − m6 λ6 c33 λ6 λ6 Lm6 υ3 n6 eiλ6 H ,
d31 c55 λ23 − L2 m3 2Lλ3 ibFΩ3 m3 e−iλ3 H ,
d32 c55 λ24 − L2 m4 2Lλ4 ibFΩ4 m4 e−iλ4 H ,
d33 c55 λ25 − L2 m5 2Lλ5 ibFΩ5 m5 e−iλ5 H ,
d34 c55 λ26 − L2 m6 2Lλ6 ibFΩ6 m6 e−iλ6 H ,
d35 c55 λ23 − L2 m3 − 2Lλ3 ibFΩ3 m3 eiλ3 H ,
d36 c55 λ24 − L2 m4 − 2Lλ4 ibFΩ4 m4 eiλ4 H ,
d37 c55 λ25 − L2 m5 − 2Lλ5 ibFΩ5 m5 eiλ5 H ,
d38 c55 λ26 − L2 m6 − 2Lλ6 ibFΩ6 m6 eiλ6 H ,
d41 L m3 λ3 ,
d42 L m4 λ4 ,
d43 L m5 λ5 ,
d44 L m6 λ6 ,
d45 L − m3 λ3 ,
d46 L − m4 λ4 ,
d47 L − m5 λ5 ,
− L m3 λ3 ,
d48 L − m6 λ6 ,
d410 − L m4 λ4 ,
d411 − L m5 λ5 ,
d412 − L m6 λ6 ,
d49
d51 Lm3 − λ3 ,
d52 Lm4 − λ4 ,
d53 Lm5 − λ5 ,
d54 Lm6 − λ6 ,
14
Mathematical Problems in Engineering
d55 Lm3 λ3 ,
d56 Lm4 λ4 ,
d57 Lm5 λ5 ,
d58 Lm6 λ6 ,
d59 −Lm3 λ3 ,
d510 −Lm4 λ4 ,
d511 −Lm5 λ5 ,
d512 −Lm6 λ6 ,
d61 Ω3 m3 ,
d62 Ω4 m4 ,
d63 Ω5 m5 ,
d64 Ω6 m6 ,
d65 Ω3 m3 ,
d66 Ω4 m4 ,
d67 Ω5 m5 ,
d68 Ω6 m6 ,
d69 −Ω3 m3 ,
d71
d610 −Ω4 m4 ,
Mm3 λ3 λ23 Ω3 L2 ,
d73 Mm5 λ5 λ25 Ω5 L2 ,
d611 −Ω5 m5 ,
d612 −Ω6 m6 ,
d72 Mm4 λ4 λ24 Ω4 L2 ,
d74 Mm6 λ6 λ26 Ω6 L2 ,
d75 −Mm3 λ3 λ23 Ω3 L2 ,
d76 −Mm4 λ4 λ24 Ω4 L2 ,
d77 −Mm5 λ5 λ25 Ω5 L2 ,
d78 −Mm6 λ6 λ26 Ω6 L2 ,
d79 −Mm3 λ3 λ23 Ω3 L2 ,
d710 −Mm4 λ4 λ24 Ω4 L2 ,
d711 −Mm5 λ5 λ25 Ω5 L2 ,
d712 −Mm6 λ6 λ26 Ω6 L2 ,
d81 c13 LL m3 λ3 c33 λ3 λ3 − Lm3 υ3 n3 ,
d82 c13 LL m4 λ4 c33 λ4 λ4 − Lm4 υ3 n4 ,
d83 c13 LL m5 λ5 c33 λ5 λ5 − Lm5 υ3 n5 ,
d84 c13 LL m6 λ6 c33 λ6 λ6 − Lm6 υ3 n6 ,
d85 c13 LL − m3 λ3 c33 λ3 λ3 Lm3 υ3 n3 ,
d86 c13 LL − m4 λ4 c33 λ4 λ4 Lm4 υ3 n4 ,
d87 c13 LL − m5 λ5 c33 λ5 λ5 Lm5 υ3 n5 ,
d88 c13 LL − m6 λ6 c33 λ6 λ6 Lm6 υ3 n6 ,
d89 − c13 L L m3 λ3 c33 λ3 λ3 − Lm3 υ3 n3 ,
d810 − c13 L L m4 λ4 c33 λ4 λ4 − Lm4 υ3 n4 ,
d811 − c13 L L m5 λ5 c33 λ5 λ5 − Lm5 υ3 n5 ,
d812 − c13 L L m6 λ6 c33 λ6 λ6 − Lm6 υ3 n6 ,
d91 c55 m3 L2 λ23 − 2Lλ3 − ibFm3 Ω3 ,
Mathematical Problems in Engineering
d92 c55 m4 L2 λ24 − 2Lλ4 − ibFm4 Ω4 ,
15
d93 c55 m5 L2 λ25 − 2Lλ5 − ibFm5 Ω5 ,
d94 c55 m6 L2 λ26 − 2Lλ6 − ibFm6 Ω6 ,
d95 c55 m3 L2 λ23 2Lλ3 − ibFm3 Ω3 ,
d96 c55 m4 L2 λ24 2Lλ4 − ibFm4 Ω4 ,
d97 c55 m5 L2 λ25 2Lλ5 − ibFm5 Ω5 ,
d98 c55 m6 L2 λ26 2Lλ6 − ibFm6 Ω6 ,
d99 − c55 m3 L2 λ23 − 2Lλ3 − ibFm3 Ω3 ,
d910 − c55 m4 L2 λ24 − 2Lλ4 − ibFm4 Ω4 ,
d911 − c55 m5 L2 λ25 − 2Lλ5 − ibFm5 Ω5 ,
d912 − c55 m6 L2 λ26 − 2Lλ6 − ibFm6 Ω6 ,
d101 n3 ,
d102 n4 ,
d103 n5 ,
d104 n6 ,
d105 n3 ,
d106 n4 ,
d107 n5 ,
d108 n6 ,
d1010 n4 ,
d1011 n5 ,
d1012 n6 ,
d109 n3 ,
d111 n3 θ − iλ3 ,
d112 n4 θ − iλ4 ,
d113 n5 θ − iλ5 ,
d114 n6 θ − iλ6 ,
d115 n3 θ iλ3 ,
d116 n4 θ iλ4 ,
d119 −n3 θ − iλ3 ,
d117 n5 θ iλ5 ,
d1110 −n4 θ − iλ4 ,
d118 n6 θ iλ6 ,
d1111 −n5 θ − iλ5 ,
d1112 −n6 θ − iλ6 ,
d121 n3 θ − iλ3 e−iλ3 H ,
d122 n4 θ − iλ4 e−iλ4 H ,
d123 n5 θ − iλ5 e−iλ5 H ,
d124 n6 θ − iλ6 e−iλ6 H,
d125 n3 θ iλ3 eiλ3 H ,
d126 n4 θ iλ4 eiλ4 H ,
d127 n5 θ iλ5 eiλ5 H ,
d128 n6 θ iλ6 eiλ6 H .
4.8
Equation 4.7 determines the wave velocity equation for the Rayleigh waves in an
orthotropic thermoelastic granular medium under the influence of initial stress and gravity
field.
16
Mathematical Problems in Engineering
5. Stoneley Waves
To investigate the possibility Stoneley waves in thermoelastic granular medium under the
influence of initial stress and gravity field, we replace the layer by a half-space H → ∞, in
the preceding problem, in this case eiλj H → 0, and the coefficients of e−iλj H j 3, 4, 5, 6 must
vanish. Hence, the wave velocity equation 4.7 reduces to
det · drc
0
r, c 1, 2, . . . , 8,
5.1
where
Ω 3 m3 ,
d31
−Ω3 m3 ,
d35
L m3 λ3 ,
d11
d12
L m4 λ4 ,
L m5 λ5 ,
d13
d14
L m6 λ6 ,
L − m3 λ3 ,
d15
d16
L − m4 λ4 ,
L − m5 λ5 ,
d17
d18
L − m6 λ6 ,
Lm3 − λ3 ,
d21
d22
Lm4 − λ4 ,
Lm5 − λ5 ,
d23
d24
Lm6 − λ6 ,
d25
Lm3 λ3 ,
d26
Lm4 λ4 ,
Lm5 λ5 ,
d27
d28
Lm6 λ6 ,
d32
Ω 4 m4 ,
d36
−Ω4 m4 ,
d33
Ω 5 m5 ,
d37
−Ω5 m5 ,
d34
Ω 6 m6 ,
d38
−Ω6 m6 ,
d41
Mm3 λ3 λ23 Ω3 L2 ,
d42
Mm4 λ4 λ24 Ω4 L2 ,
d43
Mm5 λ5 λ25 Ω5 L2 ,
d44
Mm6 λ6 λ26 Ω6 L2 ,
d45
−Mm3 λ3 λ23 Ω3 L2 ,
d46
−Mm4 λ4 λ24 Ω4 L2 ,
d47
−Mm5 λ5 λ25 Ω5 L2 ,
d48
−Mm6 λ6 λ26 Ω6 L2 ,
d51
c13 LL m3 λ3 c33 λ3 λ3 − Lm3 υ3 n3 ,
c13 LL m4 λ4 c33 λ4 λ4 − Lm4 υ3 n4 ,
d52
d53
c13 LL m5 λ5 c33 λ5 λ5 − Lm5 υ3 n5 ,
c13 LL m6 λ6 c33 λ6 λ6 − Lm6 υ3 n6 ,
d54
c13 LL − m3 λ3 c33 λ3 λ3 Lm3 υ3 n3 ,
d55
Mathematical Problems in Engineering
17
c13 LL − m4 λ4 c33 λ4 λ4 Lm4 υ3 n4 ,
d56
d57
c13 LL − m5 λ5 c33 λ5 λ5 Lm5 υ3 n5 ,
c13 LL − m6 λ6 c33 λ6 λ6 Lm6 υ3 n6 ,
d58
d61
c55 m3 L2 λ23 − 2Lλ3 − ibFm3 Ω3 ,
d62
c55 m4 L2 λ24 − 2Lλ4 − ibFm4 Ω4 ,
c55 m5 L2 λ25 − 2Lλ5 − ibFm5 Ω5 ,
d63
d64
c55 m6 L2 λ26 − 2Lλ6 − ibFm6 Ω6 ,
d65
c55 m3 L2 λ23 2Lλ3 − ibFm3 Ω3 ,
d66
c55 m4 L2 λ24 2Lλ4 − ibFm4 Ω4 ,
d67
c55 m5 L2 λ25 2Lλ5 − ibFm5 Ω5 ,
d68
c55 m6 L2 λ26 2Lλ6 − ibFm6 Ω6 ,
d71
n3 ,
d72
n4 ,
d73
n5 ,
d74
n6 ,
n3 ,
d75
d76
n4 ,
d77
n5 ,
d78
n6 ,
n3 θ − iλ3 ,
d81
n4 θ − iλ4 ,
d82
n5 θ − iλ5 ,
d83
n6 θ − iλ6 ,
d84
n3 θ iλ3 ,
d85
n4 θ iλ4 ,
d86
n5 θ iλ5 ,
d87
n6 θ iλ6 .
d88
5.2
Equation 5.1 determines the wave velocity equation for the Stoneley waves in an
orthotropic thermoelastic granular medium under the influence of initial stress and gravity
field.
6. Special Cases
The transcendental equations 4.7 and 5.1, in the determinant form, have complex roots.
The real part gives the velocity of Rayleigh waves and Stoneley waves, respectively, while
the imaginary part gives the attenuation due to the granular nature of the medium. It is clear
from the frequency equations 4.7 and 5.1 that the phase velocity depends on the initial
stress P , the friction F, the gravity field, and the coupling factor ε.
18
Mathematical Problems in Engineering
×1076
3
Imaginary (∆)
Real (∆)
×1075
4
0.667
−2.67
−6
1
2
3
4
0
−3
5
P
2
1
3
P
4
5
P
a
b
|∆|
×1076
2.5
1.25
0
2
1
3
4
5
P
P = 1, ρ = 0.1
0.11
0.12
c
Figure 1: Effect of the density ρ on real, imaginary, and magnitude of Stoneley waves determinant with
respect to initial stress.
When the gravity field is neglected, there is no coupling between the constants Aj , Bj ,
and Ej , Fj j 3, 4, 5, 6, the equations 3.10, 3.11 become, respectively,
∂2 ψ
∂
P ∂2 ψ
P ∂2 ψ
4 ∂ψ
c33 − c31 − c55 −
− ρ 2 F∇
∇ −S
c55 −
0,
∂t
2 ∂x12
2 ∂x32
∂t
∂t
6.1
∂2 ϕ
∂2 ϕ
∂2 ϕ
1 ∂
2
2 ∂φ
0.
∇ −
c11 P 2 c13 2c55 P 2 − ρ 2 − υ1 ε∇
χ ∂t
∂t
∂t
∂x1
∂x3
2
7.
Equation 6.1 are in agreement with the corresponding equations obtained by Ahmed
In the absence of initial stress and when there is no coupling between the temperature
and strain fields, 4.7 takes the form as obtained by Oshima 2.
Mathematical Problems in Engineering
19
×1076
5
Imaginary (∆)
Real (∆)
×1075
6
2
−2
−6
1
2
3
4
−5
−15
5
2
1
3
P
4
5
P
a
b
76
|∆|
×10
15
7.5
0
2
1
3
4
5
P
ρ = 0.1, P = 1
1.1
1.2
c
Figure 2: Effect of the parameter b on real, imaginary, and magnitude of Stoneley waves determinant
with respect to initial stress.
7. Numerical Results and Discussions
For a numerical computational work, Potash material is considered as an upper granular
media with thickness H and Fe as lower granular media 28
ρ 1.098,
M 0.4,
F 0.5,
ρ 4.499,
M 0.6,
F 0.6,
Q1 0.4,
Q1 0.6,
Q2 0.6,
Q2 0.8.
7.1
A Mathcad program is used to invert the transform in order to obtain the results in the
physical domain. From Figure 1, It is obvious that the determinant of Stoneley waves velocity
increases and decreases with an increasing of the various values of the initial stress P , also
with an increasing of the density ρ; the real and imaginary parts of the determinant of
Stoneley waves frequency equation increase. It is seen that magnitude of determinant of the
20
Mathematical Problems in Engineering
frequency equation for Stoneley waves takes a large decreasing with an increasing of the
initial stress and density.
Figure 2 displays the influence of parameter b on the frequency equation of Stoneley
waves. It is concluded that the determinant of frequency equation of Stoneley waves increases
with an increasing of the initial stress P . Also, it is clear that ReΔ increases with an
increasing of parameter b but ImΔ decreases.
Finally, it appears that the magnitude of Δ increases with the increased values of the
parameter b.
8. Conclusion
From the results obtained, we concluded the following.
1 The determinant of frequency of Rayleigh and Stoneley waves is affected by the
influences of initial stress, density, gravity, and orthotropic of material and very
pronounced on the waves propagation phenomena that indicates their utilitarian
aspects in diverse fields as Geophysics, Geology, Acoustics, Plasma, and so forth.
2 From Figures 1 and 2, it is obvious that the magnitude of Stoneley waves increases
clearly with the influences of the density ρ and parameter b.
3 If the media considered are isotropic, the relevant results obtained deduce to the
results obtained by El-Naggar 25.
Nomenclature
cij :
eij :
F:
g:
M:
P:
s:
T:
v:
α1 :
α2 :
ρ:
τ:
τij :
ω:
Are the elastic constants
Are the components of strain tensor
Is the coefficient of fraction
Is the acceleration due to the gravity
Is the third elastic constant
Is the initial stress
Is the specific heat per unit mass
Is the temperature change about the initial temperature T0
Is the phase speed
Is the thermal expansion coefficient in the planes of orthotropic
Is the thermal expansion coefficient along the x3 -axis
Is the density
Is a function of depth
Are the components of stress tensor
Is the frequency.
References
1 G. Paria, “Bending of a shallow spherical shell under uniform pressures with the boundary partly
clamped and partly simply-supported,” Bulletin of the Calcutta Mathematical Society, vol. 52, pp. 79–86,
1960.
2 N. Oshima, “A symmetrical stress tensor and its application to a granular medium,” in Proceedings of
the 3rd Japan National Congress for Applied Mechanics, vol. 77, pp. 77–83, 1955.
Mathematical Problems in Engineering
21
3 N. Oshima, “Dynamics of granular media,” in Memoirs of the Unifying Study of the Basic Problems in
Engineering Sciences by Means of Geometry, K. Kondo, Ed., vol. 2-3, 1955.
4 R. K. Bhattacharyya, “Rayleigh waves in granular medium,” Pure and Applied Geophysics PAGEOPH,
vol. 62, no. 1, pp. 13–22, 1965.
5 A. M. El-Naggar, “On the dynamical problem of a generalized thermoelastic granular infinite cylinder
under initial stress,” Astrophysics and Space Science, vol. 190, no. 2, pp. 177–190, 1992.
6 S. M. Ahmed, “Rayleigh waves in a thermoelastic granular medium under initial stress,” International
Journal of Mathematics and Mathematical Sciences, vol. 23, no. 9, pp. 627–637, 2000.
7 S. M. Ahmed, “Influence of gravity on the propagation of waves in granular medium,” Applied
Mathematics and Computation, vol. 101, no. 2-3, pp. 269–280, 1999.
8 S. M. Ahmed, “Stoneley waves in a non-homogeneous orthotropic granular medium under the
influence of gravity,” International Journal of Mathematics and Mathematical Sciences, no. 19, pp. 3145–
3155, 2005.
9 A. M. Abd-Alla and S. M. Ahmed, “Rayleigh waves in an orthotropic thermo-elastic medium under
gravity field and initial stress,” Earth, Moon, and Planets, vol. 75, pp. 185–197, 1996.
10 A. M. Abd-Alla and S. M. Ahmed, “Stoneley and Rayleigh waves in a non-homogeneous orthotropic
elastic medium under the influence of gravity,” Applied Mathematics and Computation, vol. 135, no. 1,
pp. 187–200, 2003.
11 A. M. El-Naggar, A. M. Abd-Alla, and S. M. Ahmed, “Rayleigh waves in a magnetoelastic initially
stressed conducting medium with the gravity field,” Bulletin of the Calcutta Mathematical Society, vol.
86, no. 3, pp. 243–248, 1994.
12 S. C. Das, D. P. Acharya, and P. R. Sengupta, “Surface waves in an inhomogeneous elastic medium
under the influence of gravity,” Revue Roumaine des Sciences Techniques, vol. 37, no. 5, pp. 539–551,
1992.
13 M. A. Goda, “The effect of inhomogeneity and anisotropy on Stoneley waves,” Acta Mechanica, vol.
93, pp. 89–98, 1992.
14 A. M. Abd-Alla, H. A. H. Hammad, and S. M. Abo-Dahab, “Rayleigh waves in a magnetoelastic halfspace of orthotropic material under influences of initial stress and gravity field,” Applied Mathematics
and Computation, vol. 154, no. 2, pp. 583–597, 2004.
15 J. N. Sharma, V. Walia, and S. K Gupta, “Effect of rotation and thermal relaxation on Rayleigh waves in
piezothermoelastic half space,” International Journal of Mechanical Sciences, vol. 50, no. 3, pp. 433–444,
2008.
16 J. N. Sharma, M. Pal, and D. Chand, “Propagation characteristics of Rayleigh waves in transversely
isotropic piezothermoelastic materials,” Journal of Sound and Vibration, vol. 284, no. 1-2, pp. 227–248,
2005.
17 J. N. Sharma and V. Walia, “Further investigations on Rayleigh waves in piezothermoelastic
materials,” Journal of Sound and Vibration, vol. 301, no. 1-2, pp. 189–206, 2007.
18 T. C. T. Ting, “Surface waves in a rotating anisotropic elastic half-space,” Wave Motion, vol. 40, no. 4,
pp. 329–346, 2004.
19 N. Abo-el-nour and S. M. Abo-dahab, “Rayleigh waves in magneto-thermo-viscoelastic solid with
thermal relaxation times,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 861–877, 2004.
20 S. M. Ahmed and S. M. Abo-Dahab, “Propagation of love waves in an orthotropic granular layer
under initial stress overlying a semi-infinite granular medium,” Journal of Vibration and Control, vol.
16, no. 12, pp. 1845–1858, 2010.
21 P. C. Vinh and G. Seriani, “Explicit secular equations of Rayleigh waves in a non-homogeneous
orthotropic elastic medium under the influence of gravity,” Wave Motion, vol. 46, no. 7, pp. 427–434,
2009.
22 P. C. Vinh and G. Seriani, “Explicit secular equations of Stoneley waves in a non-homogeneous
orthotropic elastic medium under the influence of gravity,” Applied Mathematics and Computation, vol.
215, no. 10, pp. 3515–3525, 2010.
23 P. C. Vinh, “Explicit secular equations of Rayleigh waves in elastic media under the influence of
gravity and initial stress,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 395–404, 2009.
24 W.-C. Lo, “Propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic
medium,” Advances in Water Resources, vol. 31, no. 10, pp. 1399–1410, 2008.
25 A. M. El-Naggar, “On the dynamical problem of a generalized thermoelastic granular infinite cylinder
under initial stress,” Astrophysics and Space Science, vol. 190, no. 2, pp. 177–190, 1992.
22
Mathematical Problems in Engineering
26 A. M. Abd-Alla, S. M. Abo-Dahab, and F. S. Bayones, “Rayleigh waves in generalized magnetothermo-viscoelastic granular medium under the influence of rotation, gravity field, and initial stress,”
Mathematical Problems in Engineering, vol. 2011, Article ID 763429, 47 pages, 2011.
27 A. Britan and G. Ben-Dor, “Shock tube study of the dynamical behavior of granular materials,”
International Journal of Multiphase Flow, vol. 32, no. 5, pp. 623–642, 2006.
28 P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, McGraw-Hill, New York, NY, USA,
1953.
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