Internat. J. Math. & Math. Sci.
Vol. 23, No. 9 (2000) 627–637
S0161171200002155
© Hindawi Publishing Corp.
RAYLEIGH WAVES IN A THERMOELASTIC GRANULAR
MEDIUM UNDER INITIAL STRESS
S. M. AHMED
(Received 23 December 1998)
Abstract. We study the effect of initial stress on the propagation of Rayleigh waves in a
granular medium under incremental thermal stresses. We also obtain the frequency equation, in the form of a twelfth-order determinantal expression, which is in agreement with
the corresponding classical results.
Keywords and phrases. Rayleigh waves, thermoelastic waves, granular medium.
2000 Mathematics Subject Classification. Primary 74J15.
1. Introduction. The propagation of thermoelastic waves in a granular medium
under initial stress has some applications in soil mechanics, earthquake science, geophysics, mining engineering, etc. The theoretical outline of the development of the
subject from the mid-thirties was given by Paria [9]. The present paper, however, is
based on the dynamics of granular media as propounded by Oshima [7, 8].
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 equations 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 has
perfect elasticity. The frequency equation of Rayleigh waves in a granular layer over
a granular half-space was given by Bhattacharyya [2]. In [4], Elnaggar investigated the
influence of initial stress of the waves propagation in a thermoelastic granular infinite
cylinder. Recently [1], Ahmed discussed the influence of gravity on the propagation
of waves in granular medium.
This paper is devoted to the study of the effect of initial stress on the propagation of Rayleigh waves in a granular medium under incremental thermal stresses. The
medium under consideration is granular half-space overlain by a different granular
layer and initial stresses present in this medium have considerable effect in the propagation of Rayleigh waves [3]. The wave velocity equation has been derived in the form
of twelfth-order determinant. The roots of this equation are in general complex and
the imaginary part of an appropriate root measures the attenuation of the waves. It
is shown that the frequency of Rayleigh waves contains terms involving thermal coefficients and other terms involving initial stress and so the phase velocity changes
with respect to this thermal coefficients and the initial stress. When there is no coupling between the temperature and the strain field in the absence of the initial stress,
628
S. M. AHMED
the derived frequency equation reduces to an equation in the form of ninth-order determinant similar to that obtained by Bhattacharyya [2]. Also, the classical frequency
equation when both media are elastic and the other effects are absent is obtained.
2. Formulation of the problem. Consider a system of orthogonal cartesian axes
x1 , 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, the origin O is any point on the free surface, x3 -axis is positive in the
direction towards the exterior of the half-space, and the x1 -axis is positive along the
direction of Rayleigh waves propagation.
The state of deformation in the granular medium is described by the displacement
vector U(u1 , 0, u3 ) of the centre of gravity of a grain and the rotation vector ξ(ξ, η, ζ)
of the grain about its centre of gravity.
In this problem the stress tensor and the stress couple are nonsymmetric, i.e., τij ≠
τji and Mij ≠ Mji . The stress tensor τij can be expressed as the sum of symmetric
and antisymmetric tensors
,
τij = σij + σij
(2.1)
where
σij =
1
τij + τji
2
=
σij
and
1
τij − τji .
2
(2.2)
The symmetric tensor σij = σji is related to the symmetric strain tensor
1 ∂ui ∂uj
+
,
eij = eji =
2 ∂xj ∂xi
(2.3)
by Hooke’s law.
are given by
The antisymmetric stresses σij
= −F
σ23
∂ξ
,
∂t
σ31
= −F
∂η
,
∂t
σ12
= −F
∂ζ
,
∂t
σ11
= σ22
= σ33
= 0,
(2.4)
where F is the coefficient of friction.
The stress couple Mij is given by
Mij = Mνij ,
(2.5)
where M is the third elastic constant,
ν11 =
ν12 =
∂ξ
,
∂x1
ν22 = 0,
∂
(ω2 + η),
∂x1
ν32 =
ν33 =
∂ζ
,
∂x3
∂
(ω2 + η),
∂x3
ν23 = 0,
ν13 =
ν31 =
∂ξ
,
∂x3
∂ξ
,
∂x3
(2.6)
ν21 = 0,
where ω2 = 1/2(∂u1 /∂x3 − ∂u3 /∂x1 ) is the component of rotation.
The heat conduction equation is given by (see [6])
K∇2 T = ρCe
∂U
∂T
+ γT0 ∇ ·
,
∂t
∂t
(2.7)
RAYLEIGH WAVES IN A THERMOELASTIC GRANULAR MEDIUM . . .
629
where K is the thermal conductivity, T is the temperature change about the initial
temperature T0 , ρ is the density, Ce is the specific heat per unit mass at constant
strain, γ is equal to α(3λ + 2µ), α is the thermal expansion coefficient, and λ and µ
are Lame’s constants and t is the time.
The components of incremental stress in terms of the displacement are given by
(see [3, 6])
∂u1
∂u3
+ (λ + p)
− γT ,
∂x1
∂x3
∂u1
∂u3
+ (λ + 2µ)
− γT ,
σ33 = λ
∂x1
∂x3
∂u1 ∂u3
+
.
σ13 = µ
∂x3 ∂x1
σ11 = (λ + 2µ + p)
(2.8)
The dynamical equations of motion are
∂ω2
∂ 2 u1
∂τ11 ∂τ13
+
+P
=ρ
,
∂x1
∂x3
∂x3
∂t 2
∂τ12 ∂τ32
+
= 0,
∂x1
∂x3
(2.9)
∂ω2
∂ 2 u3
∂τ13 ∂τ33
+
+P
=ρ
,
∂x1
∂x3
∂x1
∂t 2
and
∂M11 ∂M31
+
= 0,
∂x1
∂x1
∂M12 ∂M32
τ31 − τ13 +
+
= 0,
∂x1
∂x3
∂M13 ∂M33
+
= 0.
τ12 − τ21 +
∂x1
∂x3
τ23 − τ32 +
(2.10)
Equations (2.9) and (2.10) take the forms, when the stresses are substituted,
(λ + 2µ + P )
P ∂ 2 u1
∂ 2 u1
2 + µ+
2 ∂x32
∂x1
2
∂ u3
P
∂T
∂ ∂η
∂ 2 u1
+ λ+µ +
−γ
−F
,
=ρ
2 ∂x1 ∂x3
∂x1
∂t ∂x3
∂t 2
∂
∂ξ
∂ζ
−
= 0,
∂t ∂x3 ∂x1
2
∂ u1
P ∂ 2 u3
P
+ µ−
λ+µ +
2 ∂x1 ∂x3
2 ∂x12
+ (λ + 2µ)
∂ 2 u3
∂T
∂ ∂η
∂ 2 u3
−
γ
+
F
,
=
ρ
∂x3
∂t ∂x1
∂t 2
∂x32
−F
∂ξ
+ M∇2 ξ = 0,
∂t
(2.11)
(2.12)
(2.13)
(2.14)
630
S. M. AHMED
−F
∂η
+ M∇2 (η + ω2 ) = 0,
∂t
(2.15)
∂ζ
+ M∇2 ζ = 0.
∂t
(2.16)
−F
3. Solution of the problem. Let the constants λ, µ, ρ, F , M, γ and λ, µ, ρ, F , M, γ
be characteristics of the layer and the half-space, respectively. Let us introduce the
displacement potentials φ(x1 , x3 , t) and ψ(x1 , 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 from (3.1) into (2.11), (2.13), and (2.15), we see that φ and ψ satisfy the
wave equations
α 2 ∇2 φ −
∂2φ γ
− T = 0,
∂t 2
ρ
(3.2)
β 2 ∇2 ψ −
∂2ψ
∂η
= 0,
+s
∂t 2
∂t
(3.3)
∂η
+ ∇2 η − ∇4 ψ = 0,
∂t
(3.4)
−s where
α2 =
λ + 2µ + p
,
ρ
β2 =
µ − (p/2)
,
ρ
S=
F
,
ρ
S =
F
.
M
From (3.1), the heat conduction equation (2.7) becomes
∂φ
∂T
+ γT0 ∇2
.
k∇2 T = ρCe
∂t
∂t
Elimination of T from (3.2) and (3.6), gives
∂φ
1 ∂
∂2φ
α2 ∇2 φ −
= 0,
∇2 −
− )∇2
2
χ ∂t
∂t
∂t
(3.5)
(3.6)
(3.7)
where
χ=
k
,
ρce
)=
γ 2 T0
.
ρk
Also, η can be eliminated by the use of equations (3.3) and (3.4) as follows:
∂2ψ
∂
∂ψ
∇2 − s β2 ∇2 ψ −
= 0.
+ S∇4
2
∂t
∂t
∂t
For a plane harmonic wave propagation in the x1 -direction, we assume
φ = φ1 (x3 ) exp i(Lx1 − bt) ,
ψ = ψ1 (x3 ) exp i(Lx1 − bt) ,
ξ, η, ζ = ξ1 (x3 ), η1 (x3 ), ζ1 (x3 ) exp i(Lx1 − bt) ,
where b is real positive and L is in general complex.
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
RAYLEIGH WAVES IN A THERMOELASTIC GRANULAR MEDIUM . . .
631
Substituting from (3.12) into (2.12), (2.14), and (2.16), gives
Dξ1 − iLζ1 = 0,
(3.13)
2
2
(3.14)
2
2
(3.15)
D ξ1 + h ξ1 = 0,
D ζ1 + h ζ1 = 0,
where h2 = ibs − L2 , D ≡ d/dx3 .
Solutions of (3.14) and (3.15) are
ξ1 = A1 eihx3 + A2 e−ihx3 ,
ζ1 = B1 eihx3 + B2 e−ihx3 ,
(3.16)
respectively.
From (3.13) and (3.16), we obtain
h A1 eihx3 − A2 e−ihx3 − L B1 eihx3 − B2 e−ihx3 = 0.
(3.17)
Equating the coefficients of eihx3 and e−ihx3 to zero in (3.17), gives
A1 =
L
B1 ,
h
A2 =
−L
B2 .
h
(3.18)
Also, substitution from (3.10) and (3.11) into (3.7) and (3.9), we obtain
ibα2
α D + b − 2L α + ibε +
D 2 + α2 L4
χ
2
4
2
2
2
ibL2 α2 ib3
+
φ1 = 0,
− b L − ibL ε −
χ
χ
2 2
(3.19)
2
2
β − ibs D 4 + b2 − 2L2 β2 + ibs β2 + 2ibsL2 D 2
+ β2 − ibs L4 − b + is β2 bL2 + ib3 S ψ1 = 0.
(3.20)
The solution of (3.19) and (3.20) has the form
φ1 = A3 em3 x3 + B3 e−m3 x3 + A4 em4 x3 + B4 e−m4 x3 ,
ψ1 = E 3 e
n 3 x3
+ F3 e
−n3 x3
+ E4 e
n4 x3
+ F4 e
−n4 x3
,
(3.21)
(3.22)
where
b
b(b + iε) ib
α4 1/2
±
−
,
(b + iε)2 − 2iα2 (b + iε) −
m32 , m42 = L2 −
2
2
2α
2χ 2α
χ
2 2 2L2 β2 − b2 − ibβ2 s − 2ibL2 s ± b b − iβ2 s 2 − 4b2 ss 1/2
n 3 , n4 =
.
2 β2 − ibs
(3.23)
Using (3.3), (3.11), (3.12), and (3.22), we get
η1 = Ω3 E3 en3 x3 + F3 e−n3 x3 + Ω4 E4 en4 x3 + F4 e−n4 x3 ,
(3.24)
632
S. M. AHMED
where
Ω3 =
−iβ2
b2
n23 − L2 + 2 ,
bS
β
Ω4 =
−iβ2
b2
n24 − L2 + 2 .
bS
β
From (3.2), (3.10), and (3.21), we have
T = Ω3 A3 em3 x3 + B3 e−m3 x3 + Ω4 A4 em4 x3 + B4 e−m4 x3 exp i(Lx1 − bt) ,
where
Ω3 =
ρα2
b2
m32 − L2 + 2 ,
γ
α
Ω4 =
ρα2
b2
m42 − L2 + 2 .
γ
α
(3.25)
(3.26)
(3.27)
The functions ξ1 , ζ1 , η1 , φ1 , and ψ1 in the state of the lower medium must vanish
as x3 → ∞ and using the symbols with a bar for the quantities in the lower medium
(except x3 , L, b, p) and assuming the real parts of m3 , m4 , n3 , n4 are positive while the
imaginary part of h is negative, we obtain, for x3 > H,
L
B 2 e−ihx3 ,
h
ζ 1 = B 2 e−ihx3 ,
ξ1 = −
η1 = Ω3 F 3 e−n3 x3 + Ω4 F 4 e−n4 x3 ,
(3.28)
φ1 = B 3 e−m3 x3 + B 4 e−m4 x3 ,
ψ1 = F 3 e−n3 x3 + F 4 e−n4 x3 ,
T = Ω3 B 3 e−m3 x3 + Ω4 B 4 e−m4 x3 .
4. Boundary conditions and frequency equation. The boundary conditions on the
interface x3 = H are
(i)
u1 = u1 ,
(iv) η = η,
(vii) M31 = M 31 ,
(x)
(xiii)
(ii)
u3 = u3 ,
(iii)
ξ = ξ,
(v)
ζ = ζ,
(vi)
M33 = M 33 ,
M32 = M 32 ,
(ix)
τ33 = τ 33 ,
τ32 = τ 32 ,
(xii)
(viii)
τ31 = τ 31 ,
(xi)
(4.1)
T = T,
∂T
∂T
+ θT =
+ θT .
∂x3
∂x3
The boundary conditions on the free surface x3 = 0 are
(xiv)
(xviii)
M33 = 0,
τ31 = 0,
(xv)
(xxi)
M31 = 0,
τ32 = 0,
(xvi)
M32 = 0,
(xx)
∂T
+ θT = 0,
∂x3
(xvii)
τ33 = 0,
where
∂
∂ξ
(η − ∇2 Ψ ),
M31 = M
,
∂x3
∂x3
2
∂ φ
∂2ψ
∂ξ
τ33 = λ∇2 φ + 2µ
,
− γT ,
τ32 = −F
2 −
∂t
∂x3 ∂x1 ∂x3
∂2φ
∂2ψ ∂2ψ
∂η
,
−
+
−F
τ31 = µ 2
∂x1 ∂x3 ∂x32 ∂x12
∂t
M33 = M
∂ζ
,
∂x3
M32 = M
(4.2)
RAYLEIGH WAVES IN A THERMOELASTIC GRANULAR MEDIUM . . .
633
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 = −B 2 e−ihH ,
B1 eihH + B2 e−ihH = −B 2 eihH ,
M B1 eihH − B2 e−ihH = −M B 2 e−ihH ,
M B1 eihH + B2 e−ihH = −M B 2 e−ihH .
(4.3)
Whence
B1 = B2 = B 2 = 0,
ξ = ζ = ξ = ζ = 0.
(4.4)
The other significant boundary conditions are responsible for the following relations:
(xvi)
q1 E3 − F3 + q2 E4 − F4 = 0,
(xvii)
q3 A3 + B3 + q4 A4 + B4 + q5 E3 − F3 + q6 E4 − F4 = 0,
(xviii)
q7 A3 − B3 + q8 A4 − B4 + q9 E3 − F3 + q10 E4 − F4 = 0,
(i)
(ii)
iL A3 em3 H + B3 e−m3 H + A4 em4 H + B4 e−m4 H − n3 E3 en3 H + F3 e−n3 H
− n4 E3 en4 H + F4 e−n4 H = iLB 3 e−m3 H + iLB 4 e−m4 H + n3 F 3 e−n3 H + n4 F 4 e−n4 H ,
m3 A3 em3 H − B3 e−m3 H + m4 A4 em4 H − B4 e−m4 H
+ iL E3 en3 H − F3 e−n3 H + E4 en4 H + F4 e−n4 H
= −m3 B 3 e−m3 H − m4 B 4 e−m4 H + iLF 3 e−n3 H + iLF 4 e−n4 H ,
(iv)
(viii)
Ω3 E3 en3 H + F3 e−n3 H + Ω4 E3 en4 H + F4 e−n4 H = Ω3 F 3 en3 H + Ω4 F 4 e−n3 H ,
M q1 E3 en3 H − F3 e−n3 H + q2 E4 en4 H − F4 e−n4 H
= −M q1 F 3 e−n3 H + q2 F 4 e−n4 H ,
(ix) q3 A3 em3 H + B3 e−m3 H + q4 A4 em4 H + B4 e−m4 H + q5 E3 en3 H − F3 e−n3 H
+ q6 E4 en4 H − F4 e−n4 H = q3 B 3 e−m3 H + q4 B 4 e−m4 H − q5 F 3 e−n3 H − q6 F 4 e−n4 H ,
(x) q7 A3 em3 H − B3 e−m3 H + q8 A4 em4 H − B4 e−m4 H + q9 E3 en3 H − F3 e−n3 H
+ q10 E4 en4 H − F4 e−n4 H = −q7 B 3 e−m3 H − q8 B 4 e−m4 H
− q9 F 3 e−n3 H − q10 F 4 e−n4 H ,
(xii)
Ω3 A3 em3 H + B3 e−m3 H + Ω4 A4 em4 H + B4 e−m4 H = Ω3 B 3 e−m3 H + Ω4 B 4 e−m4 H ,
634
S. M. AHMED
(xiii)
q11 A3 em3 H + q12 B3 e−m3 H + q13 A4 em4 H + q14 B4 e−m4 H
= q12 B 3 e−m3 H + q14 B 4 e−m4 H ,
(xx)
q11 A3 + q12 B3 + q13 A4 + q14 B4 = 0,
where
q1 = n3 Ω3 + L2 − n23 ,
q2 = n4 Ω4 + L2 − n24 ,
q3 = (2µ + p)L2 − ρb2 − pm32 ,
q4 = (2µ + p)L2 − ρb2 − pm42 ,
(4.5)
q1 = n3 Ω3 + L2 − n23 ,
q2 = n4 Ω4 + L2 − n24 ,
q3 = 2µ + p L2 − ρb2 − pm23 ,
q4 = 2µ + p L2 − ρb2 − pm24 ,
q5 = −2iLµn3 ,
q5 = −2iLµ n3 ,
q6 = −2iLµn4 ,
q6 = 2iLµ n4 ,
q7 = 2iLµm3 ,
q7 = 2iLµ m3 ,
q8 = 2iLµm4 ,
q8 = 2iLµ m4 ,
q9 = ibF Ω3 − µL2 − µn23 ,
q9 = ibF Ω3 − µL2 − µ n23 ,
(4.6)
q10 = ibF Ω4 − µL2 − µ n24 ,
q11 = Ω3 θ + m3 ,
q12 = Ω3 θ − m3 ,
q13 = Ω4 θ + m4 ,
q14 = Ω4 θ − m4 ,
q10 = ibF Ω4 − µL2 − µn24 ,
q11 = Ω3 θ + m3 ,
q12 = Ω3 θ − m3 ,
q13 = Ω4 θ + m4 ,
q14 = Ω4 θ − m4 ,
Elimination of A3 , B3 , A4 , B4 , E3 , F3 , E4 , F4 , B 3 , B 4 , F 3 , F 4 gives the wave velocity equation
in the form of
det di j = 0,
(4.7)
where the non-vanishing entries of the twelfth-order determinant of dij are given by
d1 5 = q1 e−n3 H ,
d1 6 = −q1 en3 H ,
d1 7 = q2 e−n4 H ,
d1 8 = −q2 en4 H ,
d2 1 = q3 e−m3 H ,
d2 2 = q3 em3 H ,
d2 3 = q4 e−m4 H ,
d 2 4 = q4 e m 4 H ,
d2 5 = q5 e−n3 H ,
d2 6 = −q5 en3 H ,
d2 7 = q6 e−n4 H ,
d2 8 = −q6 en4 H ,
d3 1 = q7 e−m3 H ,
d3 2 = −q7 em3 H ,
d3 3 = q8 e−m4 H ,
d3 4 = −q8 em4 H ,
d3 5 = q9 e−n3 H ,
d3 6 = −q9 en3 H ,
d3 7 = q10 e−n4 H ,
d3 8 = −q10 en4 H ,
d4 1 = iL,
d4 2 = iL,
d4 3 = iL,
d4 4 = iL,
d4 5 = −n3 ,
d4 6 = −n3 ,
d4 7 = −n4 ,
d4 8 = n4 ,
d4 9 = −iL,
d4 10 = iL,
d4 11 = −n3 ,
d4 12 = −n4 ,
d 5 1 = m3 ,
d5 2 = −m3 ,
d 5 3 = m4 ,
d5 4 = −m4 ,
d5 5 = iL,
d5 6 = iL,
d5 7 = iL,
d5 8 = iL,
d5 9 = m 3 ,
d5 10 = m4 ,
d5 11 = iL,
d5 12 = iL,
d 6 5 = Ω3 ,
d6 6 = Ω3 ,
d 6 7 = Ω4 ,
d6 8 = Ω4 ,
635
RAYLEIGH WAVES IN A THERMOELASTIC GRANULAR MEDIUM . . .
d6 11 = −Ω3 ,
d6 12 = −Ω4 ,
d7 5 = Mq1 ,
d7 6 = −Mq1 ,
d7 7 = Mq2 ,
d7 8 = −Mq2 ,
d7 11 = Mq1 ,
d7 12 = −Mq2 ,
d 8 1 = q3 ,
d8 2 = q3 ,
d8 3 = q4 ,
d8 4 = q4 ,
d 8 5 = q5 ,
d8 6 = −q5 ,
d8 7 = q6 ,
d8 8 = −q6 ,
d8 9 = −q3 ,
d8 10 = −q4 ,
d8 11 = q5 ,
d8 12 = q6 ,
d 9 1 = q7 ,
d9 2 = −q7 ,
d9 3 = q8 ,
d9 4 = −q8 ,
d 9 5 = q9 ,
d9 6 = −q9 ,
d9 7 = q10 ,
d9 8 = −q10 ,
d9 9 = q 7 ,
d9 10 = q8 ,
d9 11 = q9 ,
d9 12 = q10 ,
Ω3 ,
Ω4 ,
d10 4 = Ω4 ,
d10 1 =
Ω3 ,
d10 2 =
d10 3 =
d10 9 = −Ω3 ,
d10 10 = −Ω4 ,
d11 1 = q11 ,
d11 2 = q12 ,
d11 3 = q13 ,
d11 4 = q14 ,
d11 9 = −q12 ,
d11 10 = −q14 ,
d12 1 = q11 e−m3 H ,
d12 2 = q12 em3 H ,
d12 3 = q13 e−m4 H ,
d12 4 = q14 em4 H .
(4.8)
Equation (4.7) determines the wave velocity equation for the Rayleigh waves in a
thermoelastic granular medium under initial stress.
5. Discussion. The transcendental equation (4.7), in the determinant form, has
complex roots. The real part gives the velocity of Rayleigh waves and the imaginary
part gives the attenuation due to the granular nature of the medium. It is clear from
the frequency equation (4.7) that the phase velocity depends on the initial stress P ,
the friction F , and the coupling factor ).
When there is no coupling between the temperature and strain fields, we have θ
vanishes,
b2
(5.1)
lim γ · Ω4 = b2 ,
lim γ · Ω3 = 0,
lim m32 , m42 = L2 − 2 , L2 ,
)→0
γ→0
γ→0
α
where
lim q11 = 0,
γ→0
lim q12 = 0,
γ→0
lim q13 = b2 L,
γ→0
lim q14 = −b2 L.
(5.2)
γ→0
Similar results hold for the lower medium. Multiplying the rows 10, 11 and 12 of
the determinant |dij | by γ and then taking lim, equation (4.7) reduces, after some
γ→0
computation, to the following ninth-order determinantal equation:
0
0
0
q1 e−n3 H −q1 en3 H q2 e−n4 H −q2 en4 H
q3 e−m3 H q3 em3 H q5 e−n3 H −q5 en3 H q6 e−n4 H −q6 en4 H
0
q e−m3 H −q em3 H q e−n3 H −q en3 H q e−n4 H −q en4 H 0
7
7
9
9
10
10
iL
iL
−n3
−n3
−n4
n4
−iL
m3
−m3
iL
iL
iL
iL
m3
0
0
Ω3
Ω3
Ω4
Ω4
0
−Mq
Mq
−Mq
0
0
0
Mq
1
1
2
2
q3
q3
q5
−q5
q6
−q6
−q3
q7
−q7
q9
−q9
q10
−q10
q7
0
0
0
−n3
−iL
−Ω3
Mq1
q5
q9
0 0 0 −n4 −iL = 0,
−Ω4 Mq2 q6 q 10
(5.3)
636
S. M. AHMED
where
q1 = n3 Ω3 + L2 − n23 ,
q2 = n4 Ω4 + L2 − n24 ,
P
q3 = 2µL2 − b2 ρ − 2 ,
α
q4 = 2µL2 − ρb2 ,
q1 = n3 Ω3 + L2 − n23 ,
q2 = n4 Ω4 + L2 − n24 ,
P
q3 = 2µL2 − b2 ρ − 2 ,
α
q4 = 2µL2 − ρb2 ,
q5 = −2iLµn3 ,
q5 = −2iLµ n3 ,
q6 = −2iLµn4 ,
q6 = −2iLµ n4 ,
q7 = 2iLµm3 ,
(5.4)
q7 = 2iLµ m3 ,
− µn23 ,
q9 = ibF Ω3 − µL2 − µ n23 ,
q10 = ibF Ω4 − µL2 − µn24 ,
q10 = ibF Ω4 − µL2 − µ n24 .
2
q9 = ibF Ω3 − µL
The frequency equation (5.3) determines the wave velocity equation for the Rayleigh
waves in a granular medium under initial stress.
When the initial stress is absent, we have
α2 =
λ + 2µ
,
ρ
B2 =
µ
,
ρ
q3 = 2µL2 − ρb2 ,
q3 = 2µL2 − ρb2 .
(5.5)
Thus, equation (5.3) with the relations (5.5) reduces to the frequency equation obtained by Bhattacharyya [2].
If the granular rotations vanish, we get
b2
lim lim n23 , n24 = L2 , L2 − 2 ,
lim lim (S · Ω3 ) = −ib,
M→0 S→0
M→0 S→0
β
lim lim (S · Ω4 ) = 0,
M→0 S→0
lim lim q9 = ρb2 − 2µL2 ,
M→0 S→0
b2
lim lim(Ω4 ) = − 2
M→0 S→0
β
(5.6)
lim lim q10 = −µ 2L2 −
M→0 S→0
b2
β2
.
Similar results also hold for the lower medium. Multiplying the columns 5, 6 and 11
of the determinant |dij | by S and then taking lim lim, we get after some computation,
M→0 S→0
the following ninth-order determinantal equation:
q e−m3 H q em3 H q e−m4 H q em4 H q e−n4 H −q en4 H
3
3
4
4
6
6
q e−m3 H q em3 H q e−m4 H −q em4 H q e−n4 H q en4 H
7
7
8
8
10
10
iL
iL
iL
iL
−n4
n4
−m3
m4
−m4
iL
iL
m3
q3
q3
q4
q4
−q6
q6
q7
−q7
q8
−q8
−q10
q10
Ω3
Ω3
Ω4
−Ω4
0
0
q11
q12
q13
q14
0
0
q11 e−m3 H q12 em3 H q13 e−m4 H q14 em4 H
0
0
0 0
0
0 −iL −iL −n4 m3 m4 −iL −q3 −q4 q6 = 0.
q7
q8 q10 −Ω3 −Ω4 0 q12 −q14 0 0
0
0 (5.7)
0
0
637
RAYLEIGH WAVES IN A THERMOELASTIC GRANULAR MEDIUM . . .
Equation (5.7) is the velocity equation of an initially stressed thermoelastic granular
layer of thickness H overlaying semi-infinite elastic isotropic medium.
Finally, in the absence of initial stress and when there is no coupling between the
temperature and strain fields, as well as the vanishing of granular rotations, equation (5.7) takes the form
R 2 e m3 H
2Lm3 em3 H
−L
−m3
2Lm3
R2
2Lm4 em4 H
R 2 e−m3 H
−2Lm4 e−m4 H
0
R 2 e m4 H
−2Lm3 e−m3 H
R 2 e−m4 H
0
−m4
−L
m4
L
−L
m3
−L
R2
−2Lm3
R2
2Lm4
R2
−2Lm4
−m3
µ
−2L m3
µ
µ 2
− R
µ
0
−m4 = 0,
L
µ 2 R
µ
µ
−2L m4 µ
(5.8)
0
where
ρb2
,
λ + 2µ
m32 = L2 −
β2 =
m42 = L2 −
b2
,
β2
m24 = L2 −
b2
β
2,
µ
,
ρ
m23 = L2 −
2
β =
ρb2
λ + bµ
µ
,
ρ
R 2 = 2L2 −
b2
β2
,

,
2
R = 2L2 −

(5.9)
b2 
2
β
Equation (5.8) is identical to [5, equation (4.195)].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
S. M. Ahmed, Influence of gravity on the propagation of waves in granular medium, Appl.
Math. Comput. 101 (1999), 269–280.
R. K. Bhattacharyya, Rayleigh waves in granular medium, Pure Appl. Geophys. 62 (1965),
no. 3, 13–22.
M. A. Biot, Mechanics of Incremental Deformations. Theory of elasticity and viscoelasticity of initially stressed solids and fluids, including thermodynamic foundations and
applications to finite strain, John Wiley & Sons Inc., New York, 1965. MR 32#3333.
A. M. Elnaggar, On the dynamical problem of a generalized thermoelastic granular infinite cylinder under initial stress, Astrophys. Space Sci. 190 (1992), no. 2, 177–190.
MR 93d:73009. Zbl 755.73039.
W. M. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Media, McGraw-Hill
Book Co., Inc., New York, 1957. MR 20#1475. Zbl 083.23705.
W. Nowacki, Thermoelasticity, Addison-Wesely Publishing Company, Inc. London, 1962.
N. Oshima (ed.), Proc. 3rd. Japan Nat. Congr. Appl. Mech., vol. 77, 1954.
, Mem. Unifying Study of Basic Problems in Engineering Sciences by Means of Geometry, vol. I. III, General Editor, Kondo, K., 1955.
G. Paria, Bending of a shallow spherical shell under uniform pressures with the boundary
partly clamped and partly simply-supported, Bull. Calcutta Math. Soc. 52 (1960),
no. 195, 79–86. MR 23#B2648. Zbl 098.16203.
Ahmed: Mathematics Department, Faculty of Education, El-Arish, Egypt