Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 2, pp. 535-546, Warsaw 2017
DOI: 10.15632/jtam-pl.55.2.535

DISPERSION OF SH WAVES IN A VISCOELASTIC LAYER IMPERFECTLY

BONDED WITH A COUPLE STRESS SUBSTRATE

Vikas Sharma

Department of Mathematics, Lovely Professional University, Phagwara, Punjab, India

e-mail: vikas.sharma@lpu.co.in; vikassharma10a@yahoo.co.in

Satish Kumar

School of Mathematics, Thapar University, Patiala, Punjab, India

e-mail: satishk.sharma@thapar.edu

The paper deals with propagation of SH waves in a viscoelastic layer over a couple stress
substrate with imperfect bonding at the interface. A dispersion equation of SH waves in a
viscoelastic layer overlying the couple stress substrate with an imperfect interface between
them has been obtained. Dispersion equations for propagation of SH waves with perfectly
bonded interface and slippage interface between two media are also obtained as particular
cases. Effects of the degree of imperfectness of the interface are studied on the phase velo-
city of SHwaves. The dispersion curves are plotted and the effects of material properties of
both couple stress substrate and viscoelastic layer are studied. The effects of internalmicro-
structures of the couple stress substrate in terms of characteristic length of thematerial are
presented. The effects of heterogeneity, friction parameter and thickness of the viscoelastic
layer are also studied on the propagation of SHwaves.

Keywords: SH waves, couple stress theory, imperfect bonding, characteristic length, visco-
elasticity

1. Introduction

The dynamical behaviour of near surfacematerials is very complicated and could not be expla-
ined on the basis of classical continuum mechanics. To explain the relationship between stress
and strain at a particular subsurface point, the near surface of earth ismodelled as a viscoelastic
material (Butler, 2005). Wide variations of rocks erupted from volcanoes and scattering of high
frequency seismic waves support the existence of small scale heterogeneity in the earth litho-
sphere. Hence, heterogeneity and viscoelasticity are to be considered for real characterisation of
internal microstructure of solid earth.
Shear horizontal (SH) waves are linearly polarized in the direction normal to the direction

of propagation and parallel to the surface. These waves propagate in a layer in contact with an
elastic half space. The study of these guided waves has received much attention in the field of
seismology for estimating damage capabilities of seismic waves. These waves are also helpful for
studying surfacemechanical properties of underlying solids in non destructive testing techniques
and in electronics industry (Simonetti and Cawley, 2004; Qingzeng et al., 2014). SH waves in
a layered structure for a perfectly bonded interface between two media are studied by many
researchers, but this condition is rarely achieved in reality. Due to certain reasons like thermal
mismatch or some faults in the manufacturing process, cracks or defects may appear at the
interface which leads to an imperfect interface. Components of the displacement field are not
continuous at the common boundary of two media in the case of an imperfect interface. The
difference in displacement fields is assumed to linearly depend upon the traction vector. These
imperfections at the common boundarymay affect the propagation of SHwaves.



536 V. Sharma, S. Kumar

Bhattacharya (1970) pointed out some possible exact solutions to the SH-wave equation for
inhomogeneous media. Schoenberg (1980) studied elastic wave behaviour across a linear slip
interface by assuming that the displacement discontinuity is linearly related to stress traction,
which itself is continuous across the interface. He studied the effects of interfacial compliances
on reflection and transmission coefficients of plane harmonic waves. Liu et al. (2007) studied
Love waves in layered graded composite structures with a rigid, slip and imperfectly bonded
interface which was described using an interface shear spring model. They showed that for the
imperfectly bonded interface, the phase velocity of Love waves changed in the range of velocities
for the rigid and slip interface conditions.Nie et al. (2009) studied shear horizontal guidedwaves
in a coupled plate consisting of a piezoelectric layer and a piezomagnetic layer. They assumed
that both layers are transversely isotropic and are perfectly bonded at the interface. They
concluded that phase velocity of SH waves approached the smaller bulk shear wave velocity of
the twomaterials in the systemwith the increase in thewave number. Borcherdt (2009) studied
the propagation of SH waves in viscoelastic media. Kumar and Chawla (2011) studied wave
propagation at an imperfect boundarybetween the transversely isotropic thermodiffusive elastic
layer and half space in the context of the Green-Lindsay theory. They presented the effects of
various parameters involved in the problemon both phase velocity and attenuation of SH-waves.
Singh et al. (2011) studied propagation of waves at an imperfectly bonded interface between two
monoclinic thermoelastic half-spaces. Otero et al. (2011) studied dispersion relations for SH
waves on amagnetoelectroelastic heterostructure with imperfect interfaces. They observed that
with decreasing values of imperfect bondingparameter, propagation velocity also decreased. Cui
et al. (2013) studiedSHwaves inapiezoelectric structurewithan imperfectlybondedviscoelastic
layer. Sahu et al. (2014) studied SH waves in a viscoelastic heterogeneous layer over a half
space with self-weight. They studied the effects of gravity, heterogeneity and internal friction
on propagation of SH waves in the viscoelastic layer over the half space. They observed that
heterogeneity of the medium affected the velocity profile of SH wave significantly. Vardoulakis
and Georgiadis (1997) studied SH surface waves in a homogeneous gradient-elastic half space
with surface energy. They showed the existence of SH waves in a homogeneous gradient-elastic
half space. Recently, Sharma and Kumar (2016) studied propagation of SH waves in layered
media consisting of a viscoelastic layer perfectly bonded with a couple stress substrate.

In the classical elasticity, it is assumed that the matter is continuously distributed without
any defects, and internal microstructure of the material is also ignored. Experimental results
have shown that the materials having inner atomic structure or microstructures behave diffe-
rently at the micro level as compared to macroscale. Due to these shortcomings of the classical
elasticity, size dependent continuum mechanics has been developed, which accounts for the in-
ternal microstructure of the material and predicts the dependence of macroscopic response on
microstructural parameters of thematerial. Voigt (1887) was first to generate the idea of couple
stresses in the material. Cosserat and Cosserat (1909) gave a mathematical model involving
couple stresses in the material but they did not give any specific constitutive relations. Later
on, many researchers like Toupin (1962), Mindlin and Tiersten (1962), Koiter (1964), Eringen
(1968) and Nowacki (1974) worked on this idea and presented many theories. In these theories,
the concept of couple stress was introduced by defining the deformation of thematerial through
displacement and an independent rotation vector, which were associated with stresses and co-
uple stresses through constitutive relations. Due to rotation, the couple stress theory was able
to explain the dispersive nature of waves, which was not captured by the classical theory of
elasticity.

Many researchers have used couple stress theory to study problems of wave propagation in
elastic media under different conditions. Sengupta and Ghosh (1974a,b) studied the effects of
couple stresses in elastic media, they deduced the equations of surface waves in elastic media
under the influence of couple stresses and observed that the couple stresses affect the velocity



Dispersion of SH waves in a viscoelastic layer... 537

of Rayleigh and Love wave propagation. Das et al. (1991) studied thermo-viscoelastic Rayleigh
waves under the influence of couple stress and gravity. Debnath and Roy (1988) studied pro-
pagation of edge waves in a thinly layered laminated medium with stress couples under initial
stresses.They showed that for a specific compression, the presence of couple stresses increase the
velocity ofwave propagationwith an increase in thewave number,whereas the trendwas totally
reversed when there was no couple stress. Georgiadis and Velgaki (2003) studied the dispersive
nature of Rayleigh waves propagating along the surface of a half-space at high frequencies using
the couple stress theory.

There were some difficulties with the original couple stress theory like indeterminacy of
the spherical part of the couple stress tensor or involvement of separate material length scale
parameters. Hadjesfandiari and Dargush (2011) proposed a consistent couple stress theory by
considering true continuum kinematical displacement and rotation. In that proposed theory, it
was shown that the couple-stress tensor was skew-symmetric and the skew-symmetric part of
the gradient of the rotation tensor was the consistent curvature tensor. It is also shown that for
an isotropic material two Lamé parameters (λ and µ) and one length scale parameter (η=µl2)
completely characterise the behaviour. Here, a length scale parameter l called the characteristic
length is relevant for studies conducted at themicro or nano level for thematerials which exhibit
internal microstructures like composites or cellular solids. It is assumed that the characteristic
length is comparable to the average cell size of the material.

Keeping in mind various factors affecting the dispersion of SH waves like nature of the
interface between two media, properties of the half space and coated layer, we intend to study
SHwaves in a viscoelastic layer lying over a couple stress substrate with an imperfect interface
between them. To study the effects of microstructures of the substrate on the propagation of
SHwaves, amodel comprising of granularmacromorphic rock (DionysosMarble) exhibiting the
properties of a couple stress solid underlying heterogeneous viscoelastic layer is employed. The
couple stress theory proposed byHadjesfandiari andDargush (2011) is applied for observing the
effects of microstructures of the material of the substrate in terms of the characteristic length
and other parameters of the viscoelastic layer on the propagation of SHwaves.

2. Formulation and solution of the problem

Consider a layer of a viscoelastic medium of thickness H lying over a couple stress substrate
with microstructures. The interface between two media is assumed to be imperfect. The origin
of the coordinate systemO(x,y,z) lies on the interfacial surface joining the substrate and layer
of the viscoelastic medium.Here, the z axis is pointing vertically downwards into the half space,
the interface between the layer and half space is given by z=0 and the free surface of the layer
is z = −H. For SH waves, displacement components and body forces are independent of the
y co-ordinate, so if (u,v,w) are the displacement co-ordinates of a point, then u =w = 0 and
v is a function of the parameters x, z and t.

Fig. 1. Geometry of the problem



538 V. Sharma, S. Kumar

2.1. Couple stress half space

The basic governing equation ofmotion and constitutive relations of couple stress theory for
an isotropicmaterial in the absence of body forces (Hadjesfandiari andDargush, 2011) are given
by

(λ+µ+η∇2)∇(∇·u)+(µ−η∇2)∇2u= ρ
∂2u

∂t2
(2.1)

where λ and µ are Lamé constants, η=µl2 is the couple-stress coefficient, l is the characteristic
length, ρ is density of the material of the half space, and u is the displacement vector.
Let us assume that u = [0,v,0] and ∂/∂y ≡ 0. Under these conditions, the equation of

motion becomes

(∂2v

∂x2
+
∂2v

∂z2

)

− l2
(∂4v

∂x4
+
∂4v

∂z4
+2

∂4v

∂x2∂z2

)

=
1

C22

∂2v

∂t2
(2.2)

whereC22 =µ/ρ.
We assume the solution to Eq. (2.2) to be v = f(z)exp[−i(ωt−kx)], where k is the wave

number, ω= kc is the angular frequency and c is the phase velocity. Using this solution in Eq.
(2.2), we get

d4f

dz4
−S
d2f

dz2
+Pf =0 (2.3)

where

S =2k2+
1

l2
P = k4+

k2

l2
−
ω2

l2C22

Since in the couple stress elastic half space the amplitude of waves decreases with an increase
in depth, so the solution to the above differential equation becomes

f(z)=A1e
−a1z +B1e

−b1z (2.4)

where

a1 =

√

S+
√
S2−4P
2

b1 =

√

S−
√
S2−4P
2

and

v=(A1e
−a1z +B1e

−b1z)e−i(ωt−kx) (2.5)

The constitutive relations in the elastic half space are given by (Hadjesfandiari and Dargush,
2011)

σji =λuk,kδij +µ(ui,j +uj,i)−η∇2(ui,j −uj,i)

µji =4η(ωi,j −ωj,i) ωi =
1

2
ǫijkuk,j

(2.6)

Here, ui are displacement components, σji is the non-symmetric force-stress tensor, µji is the
skew symmetric couple-stress tensor, δij is Kronecker’s delta, ǫijk is the permutation tensor and
i,j,k=1,2,3

σyz =µ
∂v

∂z
+µl2

( ∂3v

∂x2∂z
+
∂3v

∂z3

)

µxz =2µl
2
(∂2v

∂x2
+
∂2v

∂z2

)

(2.7)



Dispersion of SH waves in a viscoelastic layer... 539

Using Eq. (2.5) in eq. (2.7), we get

σyz =
[

µ(−a1A1e−a1z − b1B1e−b1z)+µl2(a1A1k2e−a1z

+ b1B1k
2e−b1z −a31A1e

−a1z − b31B1e
−b1z)

]

e−i(ωt−kx)

µxz =2µl
2
[

a21A1e
−a1z + b21B1e

−b1z − (A1e−a1z +B1e−b1z)k2
]

e−i(ωt−kx)

(2.8)

2.2. Heterogeneous viscoelastic layer

For the heterogeneity of the layer, we assume that properties of the medium change only in
the z-direction. For SH waves propagating in the x-direction and causing displacement in the
y-direction only, we shall assume that u1 = [0,v1,0] and ∂/∂y≡ 0.
The equation of motion in the absence of body forces and under the above mentioned as-

sumptions (Ravinder, 1968) is given by

∂Pxy
∂x
+
∂Pyz
∂z
= ρ1
∂2v1
∂t2

(2.9)

where

Pxy =
(

µ1+η1
∂

∂t

)∂v1
∂x

Pyz =
(

µ1+η1
∂

∂t

)∂v1
∂z

In the upper viscoelastic layer µ1, η1 and ρ1 are assumed to be function of depth only and are
given by

µ1 =µ0(1− sinαz) η1 = η0(1− sinαz) ρ1 = ρ0(1− sinαz) (2.10)

where µ0, η0, ρ0 are the constant values of µ1, η1 and ρ1 at the interface of the layer and half
space, and α is an arbitrary constant having dimensions of the inverse of length.
For the heterogeneous viscoelastic layer, Eq. (2.9) becomes

(

µ1+η1
∂

∂t

)∂2v1
∂x2
+
∂

∂z

[(

µ1+η1
∂

∂t

)∂v1
∂z

]

= ρ1
∂2v1
∂t2

(2.11)

Now, assuming the solution v1 = vL(z)exp[−i(ωt−kx)], the equation of motion becomes
d2vL
dz2
+
1

µ1
(µ1)

′
dvL
dz
+
(ρ1ω

2

µ1
−k2

)

vL =0 (2.12)

where µ1 =µ1− iωη1 and (µ1)′ = dµ1/dz. Taking vL(z)=Y1(z)/
√
µ1, Eq. (2.12) reduces to

d2Y1
dz2
+

[

1

4(µ1)
2

(dµ1
dz

)2
−
1

2µ1

d2µ1
dz2
+
ρ1ω
2

µ1
−k2

]

Y1 =0 (2.13)

Solving this equation further, gives

d2Y1
dz2
+
[α2

4
+
ρ0ω
2

µ0
−k2

]

Y1 =0 (2.14)

where µ0 =µ0− iωη0.
The solution to the above differential equation is

Y1 =Acos(mz)+B sin(mz)

whereA andB are arbitrary constants and

m2 =
α2

4
+
ρ0ω
2

µ0
−k2

Hence v1 = vL(z)exp[−i(ωt−kx)] = (Y1(z)/
√
µ1)exp[−i(ωt−kx)], that

v1 =
1
√
µ0

1
√
1− sinαz

[Acos(mz)+B sin(mz)]e−i(ωt−kx) (2.15)



540 V. Sharma, S. Kumar

3. Boundary conditions

Boundary conditions to be satisfied at the free surface of the viscoelastic layer and at the
interfacial surface between the viscoelastic layer and the couple stress half space are:

(i) The top surface of the viscoelastic layer should be stress free, so Pyz = (µ1 +
η1∂/∂t)(∂v1/∂z) = 0 at z=−H.

(ii) The difference in displacement fields is assumed to depend linearly upon the traction
vector, that isPyz =G(v−v1) at z=0, whereGmeasures the degree of imperfectness at
the interface.

(iii) The magnitude of shear stresses of both the couple stress substrate and the viscoelastic
layer should be equal at the interface, that is Pyz =σyz at z=0.

(iv) The couple stress tensor µxz should vanish at the interface, that is µxz =0 at z=0.

4. Derivation of secular equation

4.1. SH waves in the viscoelastic layer over the couple stress half space with an imperfect

interface

Using the above mentioned boundary conditions, we get following four equations
√

µ0[2m(1+sin(αH))sin(mH)+αcos(αH)cos(mH)]A

+
√

µ0[2m(1+sin(αH))cos(mH)−αcos(αH)sin(mH)]B=0

−
(√
µ0α

2
+
G
√
µ0

)

A−
√

µ0mB+GA1+GB1 =0

α
√
µ0A

2
+
√

µ0mB+µa1[1+(a
2
1−k

2)l2]A1+µb1[1+(b
2
1−k

2)l2]B1 =0

(a21−k
2)A1+(b

2
1−k

2)B1 =0

(4.1)

Equations (4.1) will have anon-trivial solution if the determinant of coefficients of the unknowns
A,B,A1,B1 vanishes. Applying this condition to the above system of equations, we obtain the
following secular equation for the SH waves in the heterogeneous viscoelastic layer imperfectly
bonded to a couple stress half space with microstructures as

(−2T1µ0m+T2µ0α)[Q+G(b
2
1−a

2
1)]+2T2QG=0 (4.2)

where

T1 =2m[1+sin(αH)]sin(mH)+αcos(αH)cos(mH)

T2 =2m[1+sin(αH)]cos(mH)−αcos(αH)sin(mH)
Q=µ(k2−a21)(k

2− b21)(a1− b1)l
2+µb1(k

2−a21)−µa1(k
2− b21)

Now, separating the real and imaginary parts of eq. (4.2), we get a dispersion equation of SH
waves as

[−2R1µ0m1−2R1ωη0m2−2I1ωη0m1+2I1µ0m2
+α(R2µ0+ωη0I2)][Q+G(b

2
1−a

2
1)]+2R2QG=0

(4.3)

and the imaginary part gives us the damping equation of SHwaves as

[2R1ωη0m1−2R1µ0m2−2I1µ0m1−2I1ωη0m2
+α(I2µ0−R2ωη0)][Q+G(b21−a

2
1)]+2I2QG=0

(4.4)



Dispersion of SH waves in a viscoelastic layer... 541

4.2. SH waves in the viscoelastic layer over the couple stress half space with a perfectly

bonded interface

If in Eq. (4.2)G→∞, we get a secular equation for SHwaves in the viscoelastic layer over
the couple stress half space with a perfectly bonded interface. It is the same as that obtained
by Sharma and Kumar (2016)

(−2T1µ0m+T2µ0α)(b
2
1−a

2
1)+2T2Q=0 (4.5)

4.3. SHwaves in the viscoelastic layer over the couple stress half spacewith a slip interface

If in Eq. (4.2), G→ 0, the secular equation for SH waves in the viscoelastic layer over the
couple stress half space for a slippage interface is given by

(−2T1µ0m+T2µ0α)Q=0 (4.6)

where

T1 =R1+iI1 T2 =R2+iI2

m=m1+im2 m1 =
√
rcos
θ

2
m2 =

√
r sin
θ

2
R1 =2[1+sin(αH)](m1S1−m2S2)+αcos(αH)E1
I1 =2[1+sin(αH)](m1S2+m2S1)−αcos(αH)E2
R2 =2[1+sin(αH)](m1E1+m2E2)−αcos(αH)S1
I2 =2[1+sin(αH)](m2E1−m1E2)−αcos(αH)S2
S1 =sin(m1H)cosh(m2H) S2 =cos(m1H)sinh(m2H)

E1 =cos(m1H)cosh(m2H) E2 =sin(m1H)sinh(m2H)

F1 =
ρ0ω
3η0

µ20+ω
2η20

F2 =
α2

4
−k2+

ρ0ω
2µ0

µ20+ω
2η20

r=
√

F21 +F
2
2 tanθ=

F1
F2

5. Numerical results and discussion

(i) For the viscoelastic layer, various material parameters (Gubbins, 1990) are taken as
ρ0 =4705kg/m

3, µ0 =1.987 ·1010N/m2, µ0/η0 =106 s−1, β1 =
√

µ0/ρ0 =2055m/s.

(ii) The material parameters for the couple stress half space which is made of Dionysos
Marble (Vardoulakis and Georgiadis, 1997) are ρ = 2717kg/m3, µ = 30.5 · 109N/m2,
C2 =

√

µ/ρ=3350m/s.

Dionysos Marble is a white fine-grained metamorphic marble with saccharoidal microstruc-
ture. To find the impact of characteristic length, different cases of characteristic length l com-
parable with the internal cell size of granularmacromorphic rockO(10−4) such as l=0.0001m,
l=0.0004m, l=0.0008m are considered. In all the figures, phase velocity is plotted using Eq.
(4.3) and the real part of Eq. (4.5). Graphs of damping velocity are plotted using Eq. (4.4) and
the imaginary part of Eq. (4.5).



542 V. Sharma, S. Kumar

5.1. Effects of degree of imperfectness at the interface

To study the role of degree of imperfectness of the interface on the propagation of SHwaves
in the viscoelastic layer over the couple stress substrate, curves are provided in Figs. 2a and 2b.
Here, we have considered fixed values of other parameters as αH = 0.54, characteristic length
l = 0.0004m and friction parameter µ1/η1 = 10

6. It can be observed in Fig. 2a that SH waves
are dispersive, and the non dimensional phase velocity c/β1 of SHwaves decreases sharply with
an increase in the non dimensional wave number kH before becoming asymptotically constant.
It can also be observed from the profiles in Fig. 2a that an increase in the value of parameterG
leads to an increase in the phase velocity of SH waves for any fixed value of the dimensionless
wave number kH. Since the imperfectness is inversely proportional to G, so an increase in the
imperfectness adversely affects the phase velocity, and the phase velocity is maximum when
the interface is perfectly bonded G → ∞. Figure 2b shows the variation in non dimensional
damping velocity of SHwaves with the non dimensional wave number for different values of the
parameter G. It can be observed that the damping velocity increases with an increase in the
parameter G. The damping velocity is also maximumwhen the interface is perfectly bonded.

Fig. 2. (a) Phase and (b) damping velocity profiles of SH waves with the wave number for
different values ofG

5.2. Effects of the heterogeneity parameter

The role of the heterogeneity parameter on both the phase and damping velocities of SH
waves is studied in Figs. 3a and 3b. Dispersion curves are provided for three different values
of the heterogeneity parameter αH = 0.18, 0.54 and 0.72. We have considered values of other
parameters asH =0.09m,characteristic length l=0.0004mand frictionparameterµ1/η1 =10

6

and the value of G = 2 · 1.987 · 1010. It can be observed that with the increasing value of
the heterogeneity parameter αH, the phase velocity of SH waves decreases. Figure 3b shows
variation of the damping velocity with thewave number for different values of the heterogeneity
parameter. It can be seen that the damping velocity of SH waves increases with an increase in
the heterogeneity parameter.

5.3. Effects of the friction parameter

Dispersion curves to demonstrate the role of the friction parameter on SHwaves in the visco-
elastic layer are provided in Figs. 4a and 4b for three different values of the friction parameter
µ1/η1 =7 ·105 s−1, 10 ·105 s−1, 80 ·105 s−1. We have considered the values of other parameters
as αH = 0.54, characteristic length l = 0.0004m and G = 2 · 1.987 · 1010. It can be observed
from these figures that both the phase velocity and damping velocity of SHwaves decrease with
an increase in the value of the friction parameter µ1/η1.



Dispersion of SH waves in a viscoelastic layer... 543

Fig. 3. (a) Phase and (b) damping velocity profiles of SH waves with the wave number for
different values of αH

Fig. 4. (a) Phase and (b) damping velocity profiles of SH waves with the wave number for
different values of µ1/η1

5.4. Effects of the thickness of the viscoelastic layer

Figure 5a shows variation in the phase velocity c/β1 against the wave number kH for three
different values of thickness, H = 0.04m, 0.06m, 0.09m. The values of other fixed parameters
are αH = 0.54, µ1/η1 = 10

6, l = 0.0004 and G = 2 ·1.987 ·1010. It is observed that with the
increasing value of thickness of the viscoelastic layer over the couple stress substrate, the phase
velocity of SH waves also increases. Figure 5b shows variation of the damping velocity of SH
waves with the wave number for different values of thickness of the viscoelastic layer. It can be
seen that the damping velocity of SHwaves decreases with an increase in thickness of the layer.

5.5. Effects of the internal microstructure of the substrate

For observing the effects of internal microstructure of the underlying substrate, variation in
the phase velocity anddamping velocity is shownagainst thewave number inFigs. 6a and 6b for
three different values of the characteristic length l = 0.0001m, 0.0004m, 0.0008m. The values
of other fixed parameters are taken as αH =0.54, µ1/η1 =10

6 and G=2 ·1.987 ·1010. It can
be observed that both the phase velocity and damping velocity of SH waves increase with an
increase in the characteristic length l of the material.



544 V. Sharma, S. Kumar

Fig. 5. (a) Phase and (b) damping velocity profiles of SH waves with the wave number for
different values ofH

Fig. 6. (a) Phase and (b) damping velocity profiles of SH the waves with wave number for
different values of l

6. Conclusion

Propagation of SHwaves is studied in aviscoelastic layer bonded imperfectlywith a couple stress
substrate. Dispersion equations for propagation of SH waves with a perfectly bonded interface
and a slippage interface between twomedia are also obtained as particular cases. The numerical
results are presented graphically. Followingmajor conclusions are drawn from the present study:

• SHwaves showdispersion in the consideredmodel. Initially, thephase velocity of SHwaves
decreases sharply with an increase in the wave number, then it becomes asymptotically
constant for higher wave numbers.

• Imperfectness at the interface between two media has a significant effect on the phase
and damping velocities of SH waves. It is observed that with the decreasing value of
imperfectness at the interface, both thephase anddampingvelocities of SHwaves increase.
The phase and damping velocities are highest when the interface is perfectly bonded.

• The heterogeneity parameter of the viscoelastic layer has an adverse effect on the phase
velocity but it favours the damping velocity. So, the phase velocity decreases and the
damping velocity increases with an increase in the heterogeneity parameter.

• The friction parameter of the viscoelastic layer has an adverse effect on both the phase
and damping velocities. Both of themdecreasewith the increasing value of this parameter.



Dispersion of SH waves in a viscoelastic layer... 545

• It is observed that with the increasing value of thickness of the viscoelastic layer over the
couple stress substrate, the phase velocity of SH waves increases. The increasing value of
thickness of the viscoelastic layer does not favour damping velocity.

• Characteristic length lwhichmeasures internalmicrostructureof thematerial of theunder-
lying substrate favours both thephase anddampingvelocities of SHwaves.Thesevelocities
increase with the increasing value of characteristic length of the underlying couple stress
substrate.

The consideration of microstructural effects on the propagation of SH waves in this reali-
stic model may provide possible applications to non-destructive testing techniques and other
engineering fields. The results presented in this work may be applied to the designing of liquid
viscosity sensors and biosensors. As the model is considered to replicate the internal structure
of earth, so it may also find possible application to seismology or geomechanics engineering.

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Manuscript received June 7, 2016; accepted for print November 18, 2016