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DOI 10.1007/s11029-018-9742-8
Mechanics of Composite Materials, Vol. 54, No. 3, July, 2018 (Russian Original Vol. 54, No. 3, May-June, 2018)
ANALYSIS OF TORSIONAL WAVES IN A PRESTRESSED
COMPOSITE STRUCTURE WITH LOOSELY BONDED
AND CORRUGATED BOUNDARIES
S. A. Sahu,* M. K. Singh, and K. K. Pankaj
Keywords: corrugation, loose bonding, initial stress, hydrostatic stress, torsional wave
A mathematical model is presented to describe the propagation of torsional surface waves in a corrugated loosely
bonded orthotropic layer sandwiched between two initially stressed viscoelastic half-spaces. The dispersion
relation in a closed form is obtained for the analytical model. It is found that the initial stress, hydrostatic
stress, viscoelasticity, and the bonding and flatness parameters have a great effect on the phase velocity of
torsional surface waves. The method of separation of variables is employed to obtain an analytical solution
in the present study. Some particular cases are discussed, and it is found that the results obtained well agree
with the classical Love wave equation. Numerical simulations have also been performed to show results of the
present analytical study graphically.
1. Introduction
The propagation of torsional waves in elastic layered media is highly interesting for seismology and earth science,
because particles undergo twisting in the direction of wave motion. Torsional surface waves in inhomogeneous elastic media
were investigated by Vardoulakis [1]. Later on Georgiadis [2] et al. examined the propagation of torsional surface waves in
a gradient elastic half-space. Bao [3] et al. studied torsional waves in fiber-reinforced composite materials. Vishwakarma et
al. [4] explored torsional surface waves in a homogeneous crustal layer over a viscoelastic mantle. The boundary of the earth
is not always regular. There are different types of irregularities in layer shapes and sizes inside the earth. Many authors have
studied the boundary surfaces of the earth taking parabolic, hyperbolic and corrugated boundary surfaces, and some complicated
Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, Jharkhand, India
*
Corresponding author; tel.: +91 3262235917; e-mail: [email protected]
Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 54, No. 3, pp. 473-488, May-June, 2018.
Original article submitted April 12, 2017; revision submitted August 28, 2017.
0191-5665/18/5403-0321 © 2018 Springer Science+Business Media, LLC
321
structures in their research works. The jam of these layers is responsible for the emergence of high pressures causing sudden
movements of layers and thus supports the opinion that the propagation of seismic waves is affected by uneven boundary surfaces. In the present paper, it is assumed that the irregular boundary surfaces of corrugation type are close to the real situation.
Selim [5] discussed the static deformation of an irregular initially stressed medium. The propagation of seismic waves through
a layered half-space with corrugated boundaries was investigated by Zhang and Shinozuka [6]. Chen [7] used the method of
reflection/transmission matrices to examine the generation and propagation of a seismic SH wave in multilayered media with
irregular interfaces. Singh and Tomar [8] studied the qP-wave at a corrugated interface between two dissimilar prestressed
elastic half-spaces. The Love wave at a layer medium bounded by irregular boundary surfaces was studied by Singh [9]. The
dispersion of Love wave in a prestressed homogeneous medium over a porous half-space with irregular boundary surfaces
was investigated by Kundu et al. [10]. The propagation of shear waves in an viscoelastic medium at irregular boundaries was
studied by Chattopadhyay [11] et.al. Love waves in a piezomagnetic material layer bonded to a semi-infinite piezoelectric
substrate, with consideration of initial stresses, was analyzed by Du [12] et al. Biot [13] demonstrated the influence of initial
stresses on elastic waves. The three-dimensional linearized theory of elastic waves in initially stressed solids was discussed
by Guz [14]. Akbarov and Ilhan [15] investigated the dynamics of a system comprising a prestressed orthotropic layer and
prestressed orthotropic half-plane under the action of a moving load. The influence of a loosely bonded sandwiched initially
stressed viscoelastic layer on the propagation of torsional wave was examined by Singh [16] et al. In the recent past, significant
investigations into the propagation of surface waves in anisotropic composites, involving porous media, have been performed
by Dai et al. [17, 18], Ke et al. [19], and Son and Kang [20]. Singh and Lakshman [21] analyzed the effect of loosely bonded
undulated boundary surfaces of a doubly layered half-space on the propagation of torsional surface waves.
The aim of the present work is to investigate the behavior of torsional surface waves in corrugated and loosely bonded
orthotropic layers sandwiched between two initially stressed viscoelastic half-spaces. Dispersion curves are plotted to highlight
the effect of corrugated boundary surfaces, viscoelasticity, and initial stress on the propagation of torsional waves. A notable
effect of wave number, undulation parameter, flatness parameter, and bonding parameter on the propagation of torsional wave
is revealed and described.
2. Basic hypothesis
Murty [22, 23] defined a single real bonding parameter to which numerical values can be assigned according to the
given degree of bonding between two half-spaces and showed that, for the acoustic behavior of ideally smooth and fully
bonded interfaces, the parameter can take values from 0 and to ¥ . Murty based his study on two fundamental assumptions.
According to the first one, a loosely bonded surface between two elastic half-spaces ensures continuity across the interface.
The second assumption states that the slip at the interface is proportional to the local shearing stress, i.e.,
τ = Ks ,
(1)
where K = corresponds to an ideally smooth interface, and the an infinitely large value of K corresponds to a welded interface.
The intermediate values of K represent loosely bonded interfaces. Moreover, the effective bonding parameter K can be defined
in the more appropriate form [16]
τ = iς K µ ( v − v′ ) ,
(2)
where ς is a pure real number, µ − is the rigidity of the lower layer, v and v′ are displacements of the upper and lower layers,
respectively, and i is the imaginary unit.
As is seen, the constant K in Eq. (2) cannot be a pure real number.
c Ω
Let us introduced a new real variable Ω, such that ς = 0
, where c0 is the phase velocity of wave and b 0 is the
β0 1 − Ω
velocity of shear wave of the layer. Then
322
z= 2 (r) H
H
1, 1, 1
1
2
P
M, N, 2
2
3
P1
r
O
z= 1 (r)
3
4 P2
3, 3, 3
z
Fig. 1. Geometry of the problem: 1 — viscoelastic half-space (Medium 1), 2 — orthotropic layer
(Medium 2), 3 — viscoelastic half space (Medium 3), 4 — initial stress (P) and 5 — hydrostatic
stress (P1)..
τ =i
c0 Ω
K µ ( v − v′ ) ,
β0 1 − Ω
the value Ω = 0 corresponds to an ideally smooth interface and Ω = 1 — to a welded one, when the shearing stress is infinitely
high.
3. Geometry of the problem and governing equations
We take a coordinate system (r, j, z) whose r-axis is in the direction of wave propagation and z-axis is directed
vertically downward, as shown in Fig. 1. An orthotropic hydrostatically loaded layer M 2 : ϕ1 (r ) − H £ z £ ϕ2 (r ) with corrugated boundary surfaces is sandwiched between two corrugated viscoelastic half-spaces, M1 : −∞ ≤ z ≤ ϕ1 (r ) − H and
M 3 : j2 (r ) ≤ z ≤ ∞ , where H is thickness of the orthotropic layer, and j1 (r ) and j2 (r ) are continuous function of r ,
independent of θ , describing the corrugated boundaries of the layer. These function are periodic, and their Fourier series
expansion can be written as [24]
∞
ϕ (r ) = ∑ (ϕn einbr + ϕ− n e −inbr ),
n =1
where b is the wave number of the corrugated surface, so that the wavelength of corrugation is 2p/b. Let us introduce the
constants k , pn , and qn assuming that
ϕ1 = ϕ−1 =
p  iqn
κ
, ϕ± n = n
, n = 2, 3, 4....
2
2
Then, in view of Eq. (2), Eq. (1) results in
ϕ = κ cos br + p2 cos 2br + q2 sin 2br
+ p3 cos 3br + q3 sin 3br + ... + p n cos nbr + qn sin nbr + ....
Retaining only the first term in this expansion, the interface shape is expressed as ϕ = κ cos br , where k is the
amplitude of corrugation (known as the flatness parameter).
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4. Dynamics of the upper initially stressed viscoelastic half-space
In the propagation of torsional surface waves in the upper viscoelastic half-space, we have the following relations
for displacements:
vr(1) = 0, vθ(1) = vθ(1) (r , z , t ), vz(1) = 0
(3)
Considering Eqs. (3), stress–displacement relations in the upper initially stressed viscoelastic medium are written as
(1)
1 ∂v (1)
∂   ∂v

τ θ z =  µ1 + µ1′   θ + ⋅ z
∂t   ∂z
r ∂θ


,


(4)
(1)
∂v (1) 
∂   ∂v

τ rθ =  µ1 + µ1′   θ − θ  .
r 
∂t   ∂r

(5)
In view of Eqs. (4) and (5), we have only one equation of motion for the upper viscoelastic half-space in the presence
of an initial stress [25], namely,
2 (1)
∂ 2 vθ(1) 1 ∂vθ(1) vθ(1)  ∂  P ∂vθ(1)
∂   ∂ vθ

′
− 2 −  ⋅
+ ⋅
+
+
µ
µ

 1 ∂t 1  
r ∂r
∂z 2
r  ∂z  2 ∂z

  ∂r 2

∂ 2 vθ(1)
,
 = ρ1
2

∂
t

(6)
where µ1− is the rigidity modulus, µ1′ is the parameter of internal friction due to viscoelasticity, ρ1 is density of the medium,
and P is the initial stress.
The solution of Eq. (6) is
vθ(1) = V1 ( z ) J1 (kr )eiωt .
(7)
Using Eq. (7), Eq. (6) is reduced to the form
d 2V1
dz 2
− δ12V1 = 0 ,
(8)
where
c2
iωµ1′
)
β12 (1 +
µ
µ1
2
2
, β1 = 1 .
δ =k
ρ1
P
1−
 iωµ1′ 
2 µ1 1 +

µ1 

1−
Then solution of Eq. (8) is
V1 ( z ) = A1eδ z + B1e −δ z ,
where A1 and B1 are arbitrary constants.
In view of the condition V1 ( z ) → 0 when z → −¥ , solution (9) takes the form
V1 ( z ) = A1eδ z .
Hence, the displacement of the upper viscoelastic half-space in the presence of initial stresses is
vθ(1) = V1 ( z ) J1 (kr )eiωt .
324
(9)
5. Dynamics of the sandwiched orthotropic layer under a hydrostatic stress
The displacements in the intermediate orthotropic layer are
vr ( 2) = 0, vθ ( 2) = vθ ( 2) (r , z , t ), vz ( 2) = 0 ,
(10)
and the stress–strain relation for the layer is given by
σ rr   Brr
 

σ θθ   Brθ
σ zz   Brz
=

σ θ z   0
σ   0
 rz  
σ rθ   0
Brθ
Bθθ
Bθ z
0
0
0
Brz
Bθ z
Bzz
0
0
0
0
0  ε rr 
 
0
0  εθθ 
0
0  ε zz 
  ,
0
0  εθ z 
2 L 0  ε rz 
 
0 2 N  ε rθ 
0
0
0
2M
0
0
(11)
where σ ij are stresses, Bij (i, j = r , θ , z ) , L, M , and N are elastic constant, and ε ij are strains.
Using Eq. (1), the strain–displacement relations for the intermediate layer are written as
ε rr = 0, εθθ = 0, ε zz = 0, εθ z =
( 2)
( 2)
v (22) 
1 ∂vθ
1  ∂v
, ε rz = 0, ε rθ =  θ − θ  .
⋅
r 
2 ∂z
2  ∂r
(12)
Inserting Eqs. (12) into Eq. (11), we have
σ rr = σ θθ = σ zz , σ rz = 0, σ θ z = M
 ∂v ( 2) v ( 2) 
∂vθ( 2)
, σ rθ = N  θ − θ  ,
 ∂r
∂z
r 

(13)
where M and N are rigidities of the medium along the radial and axial directions, respectively.
In view of Eq. (13), we have only one equation of motion for the upper heterogeneous orthotropic layer under a
hydrostatic stress, namely
∂ 2 vθ( 2)
∂σ rθ ∂σ θ z 2
,
+ σ rθ − P1∆ 2 ∂vθ( 2) = ρ 2
+
∂z
∂r
r
∂t 2
(14)
where P1 is the hydrostatic stress, ρ 2 is the density of the heterogeneous orthotropic layer, and
∆2 =
∂2
∂r 2
+
∂2
∂z 2
.
(15)
Inserting Eq. (13) and (15) into Eq. (14), we have
( N − P1 )
where β 2 =
∂ 2 vθ( 2)
∂r 2
 1 ∂v ( 2) v ( 2)
+ N  ⋅ θ − θ2
 r ∂r
r

M
is velocity of the shear wave.
ρ2

∂ 2 vθ( 2)
∂ 2 vθ( 2)
ρ
=
,
 + ( M − P1 )
2

∂t 2
∂z 2

(16)
The solution of Eq. (16) is
vθ ( 2) = V2 ( z ) J1 (kr )eiωt .
(17)
V2" ( z ) + λ 2V2 ( z ) = 0 ,
(18)
Inserting Eq. (17) into Eq. (16), we have
325
where
 ( N − P1 )k 2 J1" (kr )
NkJ1' (kr )
ρ 2ω 2 
N
λ2 = 
)+
+
− 2
.
r ( M − P1 ) J1 (kr ) r ( M − P1 ) ( M − P1 ) 
 ( M − P1 ) J1 (kr )
Hence, the solution of Eq. (18) is given by
V2 ( z ) = ( A2 sin λ z + B2 cos λ z ) ,
and displacements in the intermediate orthotropic layer under a hydrostatic stress are
vθ( 2) = ( A2 sin λ z + B2 cos λ z ) J1 (kr )eiωt .
6. Dynamics of the lower initially stressed viscoelastic half-space
The stress–displacement relations for the lower initially stressed viscoelastic medium are
( 3)
1 ∂v (3)
∂

  ∂v
Sθ z =  µ3 + µ3′   θ + ⋅ z
r ∂θ
∂t   ∂z

( 3)
∂v (3)
∂

  ∂v
Srθ =  µ3 + µ3′   θ − θ
r
∂t   ∂r


,


(19)

.


(20)
In view of Eqs. (19) and (20), we have only one equation of motion for the lower pre-stressed viscoelastic half-space [25],
namely,
2 (3)
∂ 2 vθ(3) 1 ∂vθ(3) vθ(3)
∂

  ∂ vθ
′
− 2
+ ⋅
+
+
µ
µ

 3 ∂t 3  
r ∂r
∂z 2
r

  ∂r 2
 ∂  P2 ∂vθ(3)
−  ⋅
 ∂z  2 ∂z



∂ 2 vθ(3)
,
 = ρ3

∂t 2

(21)
where µ3− is the rigidity modulus, µ3′ is the parameter of internal friction due to viscoelasticity, ρ3 is density of the medium,
and P2 is the initial stress.
The solution of Eq. (21) is
vθ(3) = V3 ( z ) J1 (kr )eiωt .
Using Eq. (22), Eq. (21) gives
d 2V3
dz
where
2
(22)
− m12V3 = 0 ,
c2
iωµ3′
β32 (1 +
)
µ3
2
2
m1 = k
,
P2
1−
 iωµ3′ 
2 µ3  1 +

µ3 

(23)
1−
The solution of Eq. (23) is
β3 =
V3 ( z ) = A3e m1z + B3e − m1z ,
where A3 and B3 are arbitrary constants.
326
µ3
.
ρ3
(24)
In view of condition V3 ( z ) → 0 when z → ¥ , the solution of Eq. (24) is
V3 ( z ) = A3e − m1z .
Hence, displacements in the lower prestressed viscoelastic half-space are
vθ(3) = V3 ( z ) J1 (kr )eiωt .
7. Boundary conditions and dispersion relation
The boundary conditions for the present problem are as follows.
The stresses of the viscoelastic half-space and orthotropic layer with a hydrostatic stress are equal at z = ϕ2 (r ) − H ,
which leads the relations
(i)
(ii)
τ θ z − ϕ2′ (r )τ rθ = σ θ z − ϕ2′ (r )σ rθ ,
(τ θ z − ϕ2′ (r )τ rθ )(1 − Ω1 ) = ikN Ω1
c (1) ( 2)
(v − vθ ) ,
β2 θ
where Ω1 is the bonding parameter of the common interface of the upper viscoelastic half-space and the orthotropic layer.
(iii) Stresses of the orthotropic layer and the initially stressed lower viscoelastic half-space are equal at z = ϕ1 (r ) ,
which leads to the relation
σ θ z − ϕ1′ (r )σ rθ = Sθ z − ϕ1′ (r ) Srθ .
(iv)
c ( 2 ) ( 3)
∂

(σ θ z − ϕ1′ (r )σ rθ )(1 − Ω 2 ) = ik  µ3 + µ3′  Ω 2
(v − vθ ) .
β3 θ
∂t 

Using boundary condition (i) gives
A1 ( µ1 + iωµ1′ )δ eδ (ϕ2 ( r ) − H ) = A2 [ M λ cos λ (ϕ2 (r ) − H )] − B2 [ M λ sin(ϕ2 (r ) − H )]
Using boundary condition (ii) gives
A1 ( µ1 + iωµ1′ )δ eδ (ϕ2 ( r ) − H ) (1 − Ω1 ) = ikN Ω1
c
A1eδ (ϕ2 ( r ) − H )
β2
− A2 [sin λ (ϕ2 (r ) − H )] − B2 [cos(ϕ2 (r ) − H )] .
Using boundary condition (iii) gives
A2 M λ cos λϕ1 (r ) − M λ B2 sin λϕ1 (r ) = −( µ3 + iωµ3′ )m1 A3e − m1ϕ1 ( r ) .
(25)
Using boundary condition (iv) gives
[ A2 M λ cos λϕ1 (r ) − M λ B2 sin λϕ1 (r )](1 − Ω 2 )
= ik Ω 2
c
[ A2 sin λϕ1 (r ) + B2 cos λϕ1 (r ) − A3e − m1ϕ1 ( r ) ] .
β3
(26)
Eliminating A3 from Eqs. (25) and (26), we have


iΩ
A2  M λ 1 − Ω 2 − 2
m1




 cos λϕ1 (r ) − iΩ 2 k ( µ3 + iωµ3′ ) sin λϕ1 (r ) 


327



iΩ 
+ B2  − M λ 1 − Ω 2 − 2  sin λϕ1 (r ) − iΩ 2 k ( µ3 + iωµ3′ ) cos λϕ1 (r )  = 0
m1 



Considering all the boundary conditions and eliminating all constants, we obtained the dispersion relation for the
torsional surface wave in the form.



iΩ 
δ ( µ1 + iωµ1′ )eδ (ϕ2 ( r ) − H ) sin λ (ϕ2 (r ) − H )  − M λ 1 − Ω 2 − 2  sin λϕ1 (r )
m1 






iΩ 
−iΩ 2 + k ( µ3 + iωµ3′ ) cos λϕ1 (r )  − cos λ (ϕ2 (r ) − H )  M λ 1 − Ω 2 − 2  cos λϕ1 (r )
m1 



 

c 
−iΩ 2 + k ( µ3 + iωµ3′ ) sin λϕ1 (r )   − eδ (ϕ2 ( r ) − H ) δ ( µ1 + iωµ1′ )(1 − Ω1 ) − ikN Ω1

β
 
2




iΩ
×  M λ cos λ (ϕ2 (r ) − H ) +  − M λ 1 − Ω 2 − 2
m1



(27)


 sin λϕ1 (r ) − iΩ 2 k ( µ3 + iωµ3′ ) cos λϕ1 (r ) 


 


iΩ 
+ M λ sin λ (ϕ2 (r ) − H ) +  M λ 1 − Ω 2 − 2  cos λϕ1 (r ) − iΩ 2 k ( µ3 + iωµ3′ ) sin λϕ1 (r )   = 0
m1 
 


For a numerical calculation and graphical illustration, we will take into account Eq. (27) with
ϕ1 (r ) = κ1 cos(br ), ϕ2 (r ) = κ 2 cos(br ) .
Special cases
Case 1.
The upper and lower corrugated interfaces can be described by the periodic functions ϕ1 (r ) = κ1 cos(br ) and
ϕ2 (r ) = κ 2 cos(br ) , and they are welded together (i.e., Ω1 → 1 and Ω 2 → 1 ). Then, dispersion relation (27) is reduced to the
form

i

δ ( µ1 + iωµ1′ )eδ (κ 2 cos(br ) − H ) sin λ (κ 2 cos(br ) − H )  M λ
sin λκ1 cos(br)
m1



 i 

−ik ( µ3 + iωµ3′ ) cos λκ1 cos(br )  − cos λ (κ 2 cos(br ) − H )  M λ  −  cos λκ1 cos(br )


 m1 
 

c  
−ik ( µ3 + iωµ3′ ) sin λκ1 cos(br )   + eδ (κ 2 cos(br ) − H )  ikN
  M λ cos λ (κ 2 cos(br ) − H )
β 2  
 


× M λ


i
sin λκ1 cos(br ) − ik ( µ3 + iωµ3′ ) cos λκ1 cos(br ) + M λ sin λ (κ 2 cos(br ) − H ) 
m1


 
i 
 cos λκ 2 cos(br ) − ik ( µ3 + iωµ3′ ) sin λκ 2 cos(br )   = 0 .
 m1 
 

× M λ  −

328
5.0
a
c/2
4.5
4.0
3.5
3.0
b
c/2
1 = 0.6
2 = 0.3
5.0
b = 0.2
0.9
1.6
4.0
2 = 0.3
4.5
1 = 0
0.4
0.8
3.5
3.0
2.5
2.5
kH
2.0
2.5 3.0 3.5 4.0 4.5 5.0
2.0
2.5 3.0 3.5 4.0 4.5 5.0
c
c/2
2.0
kH
1 = 0.4
d
c/2
2.0
1.9
1.9
1.8
2 = 0.3
1.8
3 1.7
0.2
1.7
1.6
0.1
1.6
0.2
0.1
1.5
1.5
1.4
1.4
kH
1.3
0
5.0
0.1
c/2
0.2
0.3
0.5
0
P2 = 0.2
5.0
0.4
e
4.8
4.4
4.2
0.3
0.4
0.5
f
P/1 = 0.2
P2 = 0
4.0
0.6
1.2
3.5
3.0
3.8
2.5
3.6
5.0
0.2
4.5
4.0
1.0
0.1
c/2
P/1 = 0
0.3
0.5
4.6
kH
1.3
kH
1.5
c/2
2.0
2.5
kH
3.0
2.0
1 = 1.3
5.0
g
4.5
2.5 3.0 3.5 4.0 4.5 5.0
h
c/2
1 = 1.4
4.8
4.6
4.0
3.5
3.0
2 = 0
0.5
1.0
1 = 0.1
4.2
0.2
0.3
4.0
3.8
2.5
kH
2.0
4.4
2.5 3.0 3.5 4.0 4.5 5.0
3.6
1.0
kH
1.5
2.0
2.5
3.0
Fig. 2. Dimensionless phase velocity c / b 2 vs. the dimensionless wavenumber kH and the ondulation
parameter b (a), flatness parameters k1 (b) and k 2 (c), the internal friction parameter ωµ3′ / µ3 (d),
the initial stress P µ1− (e), the hydrostatic stress P2 (f) and the bonding parameters Ω2 (g) and Ω1 (h).
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Case 2.
The upper and lower interfaces are without corrugation (plane) (i.e., ϕ1 (r ) = 0 and ϕ2 (r ) = 0 ) and are welded together (i.e., Ω1 → 1 and Ω 2 → 1 ). Then, dispersion equation (27) is reduces to form

i 
δ ( µ1 + iωµ1′ )e −δ H − sin λ H [−ik ( µ3 + iωµ3′ )]cos λ HM λ  − e −δ H
m1 



c  
i  
+  −ikN
  M λ cos λ H  −ik ( µ3 + iωµ3′ ) + M λ sin λ HM λ   = 0 .
β 2  
m1  


Case 3.
The upper half-space is absent, the lower one is plane, i.e., ( ϕ2 (r ) = 0 ), the sandwiched layer is isotropic elastic
without initial stresses ( P1 = 0 , ϕ2 (r ) = 0 , and M = N = µ2 ), and the lower half-space is also without initial stress and isoµ′
tropic elastic (i.e., P2 , 3 = 0 ). Then, dispersion equation (27) is reduced to the form
µ3
tan kH
which is the classical Love equation.
c2
β 22
µ3 1 −
−1 =
µ2
c2
β32
c2
−1
β 22
,
Numerical calculation and discussion
For a numerical illustration and graphical interpretation, we took the following data.
(1) For the upper and lower viscoelastic half-spaces with an initial stress ( M1 and M 3 ) (Gubbins [26]),
µ1 = 203.9 ⋅109 N/m2, ρ1 = 4744 kg/m3, µ3 = 7.1 ⋅1010 N/m2, and ρ3 = 3321 kg/m3.
(2) For the intermediate orthotropic layer under a hydrostatic stress ( M 2 ) (Prosser and Green [27]),
M = 2.64 ⋅109 N/m2, N = 1.87 ⋅109 N/m2, and ρ 2 = 1442 kg/m3.
Figure 2a shows the effect of the undulation parameter k on the phase velocity kH of torsional surface wave. As is
seen, the velocity increases with growing value of c/b2.
Figure 2b shows that a growth in the flatness parameter k1 of the upper half-space decreases the phase velocity of
torsional surface waves. According to Fig. 2c the phase velocity of torsional surface waves grows with k2.
Figure 2d illustrates the effect of internal friction associated with the lower viscoelastic half-space due to viscoelasticity on the phase velocity of torsional surface wave. As can be seen, the internal friction acting in the lower half-space affects
the phase velocity of torsional surface waves considerably.
The graphs shown in Figs. 2e, f indicate that the initial stresses in the upper half-space and intermediate layer decrease
the phase velocity of torsional surface waves.
Figure 2g indicates that the bonding parameter acting in the lower half-space decreases the phase velocity of torsional
surface waves, but Fig. 2h reveals that the bonding parameter of the upper half-space increases the phase velocity of torsional
surface waves.
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Conclusion
The current problem concerns the propagation of a torsional surface wave in an orthotropic layer under hydrostatic
stress sandwiched between two initially stressed viscoelastic half-spaces. The influence of internal friction, initial stress, flatness parameter, bonding parameter and undulation parameter were analyzed. The main results of this work are as follows.
1. The dispersion equation for torsional surface wave propagating in an orthotropic layer under a hydrostatic stress
sandwiched between initially stressed viscoelastic half-spaces is obtained in a closed form.
2. The phase velocity of torsional surface wave decreases with growing wavenumber.
3. The undulation parameter, flatness parameter, and the internal friction of the lower corrugated boundary surface
and the bonding parameter of the upper half-space increase the phase velocity of torsional surface waves.
4. The initial stress acting in the upper half-space, the hydrostatic stress acting in the intermediate layer, the bonding
parameter of the lower half-space, and the flatness parameter of the upper corrugated boundary surface decrease the phase
velocity of torsional surface waves.
6. In the isotropic, homogeneous, elastic case without initial and hydrostatic stresses, the dispersion relation of the
problem turns into the classical Love equation.
7. The results obtained can be used for the quantification of velocities from a source point. This can help one to predict
the anisotropic layers of earth in order to locate oil traps.
The velocity profile of surface waves are affected by the medium/ layers through which they travel. Geological studies
have established the fact that, in the large span of the earth crust, a series of viscous rocks may be found over water reservoirs.
The results of this study may give more information about viscoelastic media saturated with oil and water in the crust and
upper mantle of the earth. These results may also be useful for exploration purposes. The model considered was motivated
by such geological conditions within the earth. In the present paper, the boundaries of layers are considered to be nonplane
(corrugated) and connected loosely in order to make the problem statement close to the real situation. The results obtained
will be useful and interesting for researchers dealing with wave propagation in composite layered structures with complex
mechanical properties.
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