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Acta Mech 231, 2603–2627 (2020)
https://doi.org/10.1007/s00707-020-02659-x
O R I G I NA L PA P E R
A. K. Singh · Sonam Singh · Richa Kumari · Anusree Ray
Impact of point source and mass loading sensitivity on
the propagation of an SH wave in an imperfectly bonded
FGPPM layered structure
Received: 26 July 2019 / Revised: 9 January 2020 / Published online: 9 April 2020
© Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract The aim of the present study is to analyze the propagation behavior of an SH wave in a layered
structure constituted of a functionally graded piezo-poroelastic material (FGPPM) layer imperfectly bonded to
an FGPPM half-space influenced by a stress point source of disturbance situated at the interface. Mass loading
sensitivity of the considered structure is also analyzed by assuming a deposition of an infinitesimally thin
layer at the free surface. A mathematical model for mechanical and electrical dynamics of the said material
is developed for both considered structures in distinct cases. Appropriate Green’s functions for both layer
and half-space are derived by using admissible boundary conditions. The dispersion relation of an SH wave
is obtained for both cases, and the obtained results are also validated with classical results of Love wave.
The impact of various influencing parameters like functional gradient parameters, imperfectness parameter,
piezoelectric coupling parameter, and piezo-porous coupling parameter on the phase velocity of an SH wave
is analyzed graphically for a BaTiO3 -crystal layer and PZT-5H half-space. Moreover, for the case of the mass
loading sensitivity, the influencing parameters are also studied for the deposition of a sensitive ZnO layer.
Variation of group velocity with wave number is also sketched out graphically.
1 Introduction
The characteristics of piezoelectric crystals of converting mechanical stress to electrical charge and vice versa
put these materials into the category of smart materials. Basic properties of Love waves were discussed by
Jakoby and Vellekoop [1] along with their applications in sensor devices. Although these materials have numerous applications in the detection of sound, piezoelectric motors, biomedical applications, undersea devices, air
ultrasonic transducers etc., still, they suffer from drawbacks such as low stiffness and piezoelectric coupling,
failure of the device under electrical or mechanical loading due to brittleness and possible defects of impurity,
cavities etc. To overcome these drawbacks, the concept of piezoelectric composites was introduced. The idea
of coupling a controlled porosity to piezoelectric material was developed to attain required values of material
constants and coupling parameters. These features reduce the limitations that occur with bulk piezoelectric
materials. Therefore, the designing of porous piezoelectric composites offers certain advantageous properties
which may not be found in classical piezoelectric crystals. Porous piezoelectric materials provide lower acoustic
impedance, higher hydrostatic coefficients, and higher piezoelectric sensitivity. Basic equations and constitutive relations of porous piezoelectric materials were derived by Vashishth and Gupta [2] using a variational
principle. Works of Vashishth [3] and Gaur and Rana [4] explain the characteristics of shear wave propagation
in porous piezoelectric layered structures. The effects of dynamic fluid compressibility and permeability on
A. K. Singh · S. Singh (B) · R. Kumari · A. Ray
Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines),
Dhanbad Jharkhand, 826004, India
E-mail: [email protected]
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A. K. Singh et al.
oil filled porous piezoelectric ceramics were studied with PZT hydrophones by Vashishth [5]. These piezoporoelastic composite materials have several applications, viz. in low-frequency hydrophones, contact microphones, vibratory sensors, under water sonar detectors, ultrasonic device in medical applications, among others.
Moreover, functional gradient or heterogeneity in a medium is its own trivial characteristics, as it would
be highly unconventional to assume that a medium would have uniform properties throughout its volume.
Moreover, since piezoelectric materials are brittle and mechanically stiff in nature (vulnerable to accidental
breakage during handling and bonding procedures), therefore, these materials are sometimes fabricated as
composites to attain anisotropic actuation and tolerable mechanical properties. The fabrication of functionally graded materials and the requirement of these improved materials in various engineering applications,
viz. aerospace devices, ceramic engines, optical thin films and biomaterials etc. were discussed by Rabin and
Shiota [6]. Some other interesting works may be referred from Du et al. [7], Qian et al. [8], Kumari et al. [9], Li
et al. [10], Goyal et al. [11], etc. Furthermore, imperfect bonding at the common interfaces in layered structures
is also encountered in many practical cases. Imperfect bonding between two surfaces may arise due to many
reasons like thin interface, chemical actions or interface damage, microdefects, diffusion impurities, and other
forms of damages. Many eminent researchers have given their contribution to study the effect of imperfect
bonding at the common interface of various layered structures. The different cases of imperfect bonding were
discussed by Murty [12], Li and Jin [13], Nie et al. [14], etc.
Attainment of high sensitivity is a major objective of Love wave sensors, which consist of a piezoelectric
substrate covered with a guiding layer in which elastic waves are coupled. According to its selectivity and
affinity toward the target compound, a chemical interface is chosen and added to the top of the device to
which when the target compound is sorbed leads to wave perturbation due to resulting mass increase. This
phenomenon is known as mass loading effect and is the main sensing mechanism of acoustic sensors. In their
work, Wang et al. [15] presented velocity and mass sensitivity formulae in explicit form for SH wave sensors
in a layered isotropic structure. A study of mass loading sensitivity as a function of the guiding layer parameter
was performed by Josse et al. [16] for the propagation of Love wave.
Existence of an external source of disturbance affects the wave motion in the material. This impact may be
interpreted in terms of Green’s function associated with the force concentrated at a point source of disturbance.
Analyses of Green’s functions for piezoelectric composites were accomplished by Pan and Han [17,18] and
Singh et al. [19]. Although numerous research owing to the wave propagation phenomenon in piezoelectric
structures has been carried out in the past [1,7,13,19], and moreover, wave propagation characteristics have
also been studied in porous piezoelectric materials [2–5], yet the effect of an influencing stress point source
of disturbance on the propagation of SH waves in a functionally graded porous piezoelectric layered structure
still remains unexplored. This study is modeled to accomplish the same. Furthermore, the piezo-poroelastic
layer and half-space structure has been assumed to be imperfectly bonded along their common interface. The
primary objectives of the present work are encapsulated as follows:
– To study the influence of a stress point source of disturbance on SH wave propagation in a functionally
graded piezo-poroelastic layered structure.
– To explore the effect of imperfect interface and functional gradients associated with the material constants
of the layered structure, on SH wave propagation.
– To study the effect of piezoelectric coupling and piezo-porous coupling on the phase velocity of SH waves
and particularly highlighting the effect of porosity in the piezoelectric medium is a major concern of the
present work.
– To analyze the mass loading sensitivity of the model by coating an infinitesimally thin sensitive layer.
The dispersion equation of an SH wave is derived with suitable use of Green’s function and admissible
boundary conditions. For the sake of numerical computation and graphical representation, the numerical data
of BaTiO3 and PZT-5H are used. The phase velocity shift caused by functional gradients, ratio of shear moduli,
piezoelectric coupling, piezo-porous coupling, imperfection parameter, and mode number are delineated graphically. Moreover, the effects of functional gradient parameter, piezo-porous coupling parameter, and imperfectness parameter on the mass loading sensitivity for a ZnO-sensitive layer are also expressed by graphical means.
The variation of the group velocity of an SH wave against the wave number is also a salient feature of this study.
2 Basic assumptions and geometry of the problem
In the present work, in order to study the influence of a point source of disturbance on the propagation behavior of
an SH wave, a layer/half-space configuration is chosen such that an upper functionally graded piezo-poroelastic
Impact of point source and mass loading sensitivity on SH wave propagation
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Fig. 1 Geometry of the layered structure
material (FGPPM) layer of finite thickness H is imperfectly bonded to a lower FGPPM half-space having the
point source located at the common interface. The geometry of the layered structure is described in the Cartesian
coordinate system having origin O at the free surface where the y−axis is along the direction of propagation
of the wave and the x−axis is taken vertically downward, and ‘S’ denotes the position of the point source (see
Fig. 1). For mathematical simplicity, the effect of imperfect bonding is considered in mechanical displacement
only. For the wave propagation in poroelastic media, the following assumptions are taken into account:
– The poroelastic medium is assumed to consist of a porous solid matrix with its interconnected pores filled
with a single inviscid compressible fluid [20].
– For wave propagation at high frequencies, no relative motion of fluid in the pores of the solid frame is
assumed which makes dissipation due to fluid friction vanish [20].
– A continuous fluid phase through the connected pores is assumed for the wave propagation [21].
– The macroscopic system is assumed to be small relative to the wavelength of the elastic waves, and also
the size of the pores is assumed small when compared with the size of the element [20].
– A theoretical case of no connection between the pores of two media at the common interface is considered
[22].
3 Constitutive relations and governing equations
The equations of motion for a fluid saturated porous medium in the absence of body force and without dissipation are given by [20]
(n) (n)
(n) (n)
σi(n)
j, j = ρ11 ü i + ρ12 Üi ,
(1)
∗(n)
σ, i
(2)
=
(n) (n)
ρ12 ü i
(n) (n)
+ ρ22 Üi ,
and Maxwell’s equations governing the electric displacement in solid and fluid phase of the piezo-poroelastic
medium are
(n)
Di,i = 0,
∗(n)
Di,i
=0
(3)
(4)
where i, j = 1, 2, 3. The superscript (n = 1, 2) denotes the respective case of layer and half-space. Here, σi(n)
j ,
(n)
(n)
∗(n)
σ ∗(n) are stress tensor, i j , ∗ are strain tensor, Di , Di
are electric displacement components, and u n , Un
(n)
(n)
are mechanical displacements of solid and fluid phase, respectively. Moreover (ρ11 )i(n)
j , (ρ12 )i j , (ρ22 )i j are
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A. K. Singh et al.
the dynamical mass coefficients of the medium. The constitutive relations in an anisotropic piezo-poroelastic
material in solid frame and pore-fluid are [2]
(n)
(n) (n)
(n)
(n)
(n)
(n)
(n)
(n)
∗(n)
∗(n)
σi j = Ci jkl kl + m i j ∗(n) − eki j E k − ζki j E k
(n) (n)
σ ∗(n) = m i j i j + R (n) ∗(n) − ζk E k − ek
,
∗(n)
Ek
(5)
,
(6)
and the corresponding electric field displacements in solid frame and interstitial fluid are
(n)
Di
∗(n)
Di
(n)
(n) (n)
(n)
(n)
(n)
= eikl kl + ζi ∗(n) + ξil El
=
(n) (n)
ζikl kl
∗(n)
+ ei ∗(n)
(n)
+
(n)
(n)
Ail El
∗(n)
The terms Ci jkl , m i j , R represent material constants, E i , E i
(n)
∗(n)
+ Ail El
,
(7)
∗(n) ∗(n)
+ ξil El .
(8)
(n)
(n)
are electric field vector , ei jkl , e∗(n) , ζi jkl ,
∗(n)
(n)
ζi(n) are piezoelectric constants, and ξi(n)
j , ξi j , Ai j are dielectric constants.
We have the relations
1 (n)
(u + u (n)
j,i ),
2 i, j
(n)
= Ui,i ,
i(n)
j =
∗(n)
(n)
Ei
E i∗(n)
=
=
(n)
−φ,i ,
−φ,i∗(n)
(9)
(10)
(11)
(12)
where φ and φ ∗ are electric potential functions of solid and fluid phase, respectively, and i, j = 1, 2, 3;
n = 1, 2. For the propagation of SH wave in y-direction, causing displacement in z-direction, the mechanical
displacement components and electric potential are
⎫
(1)
(1)
(1)
u 1 = 0, u 2 = 0, u 3 = w1 (x, y, t), ⎪
⎬
(13)
φ (1) = φ1 (x, y, t), U1(1) = 0, U2(1) = 0,
⎪
⎭
(1)
∗
∗
(1)∗
U3 = W1 (x, y, t), φ
= φ1 (x, y, t),
⎫
(2)
(2)
(2)
u 1 = 0, u 2 = 0, u 3 = w2 (x, y, t), ⎪
⎬
(14)
φ (2) = φ2 (x, y, t), U1(2) = 0, U2(2) = 0,
⎪
⎭
(2)
∗
∗
(2)∗
U3 = W2 (x, y, t), φ
= φ2 (x, y, t).
4 Equations of motion for the piezo-porous layer
The functional gradients of the layer are assumed to be exponentially varying functions of depth, i.e.,
⎫
(1)
(01)
(1)
(01)
(1)
(01)
C44 = C44 eα1 x , e15 = e15 eα1 x , ξ11 = ξ11 eα1 x ,
⎪
⎬
∗(1)
∗(01) α1 x
(1)
(01) α1 x
(1)
(01) α1 x
ξ11 = ξ11 e , A11 = A11 e , (ρ11 )33 = (ρ11 )33 )e ,
⎪
⎭
(1)
(01)
(1)
(01)
(ρ12 )33 = (ρ12 )33 eα1 x , (ρ22 )33 = (ρ22 )33 eα1 x .
(15)
where α1 is the functional gradient parameter associated with all material constant and dynamical mass coefficients of the FGPPM layer, having the dimension inverse of length.
The non-vanishing equations of motion and equation of charge for the layer derived from Eqs. (1)–(4) are
(1)
(1)
∂σ13
∂σ
(1)
(1)
+ 23 = (ρ11 )33 ẅi + (ρ12 )33 Ẅi∗ ,
∂x
∂y
(16)
(1) ∗
(ρ12 )(1)
33 ẅi + (ρ22 )33 Ẅi = 0,
(17)
(1)
∂ D1
∂x
+
(1)
∂ D2
∂y
= 0,
(18)
Impact of point source and mass loading sensitivity on SH wave propagation
∗(1)
∂ D1
∂x
2607
∗(1)
+
∂ D2
∂y
= 0.
(19)
Now, using Eq. (17) in Eq. (16) and Eq. (19) in Eq. (18), we have the equations
(01)
(01)
(01) ∂w1
(01) ∂φ1
C44 ∇ 2 w1 + e15 ∇ 2 φ1 + α1 C44
+ e15
∂x
∂x
(01) 2
((ρ12 )33 )
∂ 2 w1
(01)
,
= (ρ11 )33 −
(01)
∂t 2
(ρ22 )33
(01)
(A11 )2
(01) 2
(01)
(01) ∂w1
∇ 2 φ1 + α1 e15
and e15 ∇ w1 − ξ11 − ∗(01)
∂x
ξ11
(01) 2
(A11 )
∂φ1
(01)
−α1 ξ11 − ∗(01)
= 0.
∂x
ξ
(20)
(21)
11
Let the source distribution function due to a stress point source be 4πσ1 (r, t). Then Eq. (20) yields
2
2
∂ 2 w1
∂ 2 φ1
(01) ∂ w1
(01) ∂ φ1
(01) ∂w1
(01) ∂φ1
C44
+
+
+
e
+
e
+
α
C
1
44
15
15 ∂ x
∂x2
∂ y2
∂x2
∂ y2
∂x
(01)
((ρ12 )33 )2 ∂ 2 w1
(01)
= 4πσ1 (r, t) e−α1 x .
− (ρ11 )33 −
2
(01)
∂t
(ρ22 )
(22)
33
Using Eq. (21) in Eq. (22), we have
∂w1
∂ 2 w1
(1)
2
C44 ∇ w1 + α1
− ρ1 2 = 4πσ1 (r, t)e−α1 x
∂x
∂t
(23)
where
(1)
(01)
C44 = C44 + (01)
(e15 )2
(01)
ξ11 −
(01)
(A11 )2
,
∗ (01)
ξ11
(01)
and ρ1 = (ρ11 )33 −
2
((ρ12 )(01)
33 )
(01)
(ρ22 )33
.
The displacement and source distribution function may be assumed as
w1 (x, y, t) = w1 (x, y) eiωt , σ1 (r, t) = σ1 (r ) eiωt .
Substituting Eq. (24) in Eq. (23), we have
ρ1 ω2
4πσ1 (r ) −α1 x
∂w1
2
+ (1) w1 =
e
∇ w1 + α 1
(1)
∂x
C44
C44
(24)
(25)
where ω(= kc) is the angular frequency.
Now, the source distribution function may be written in terms of the Dirac-delta function as
σ1 (r ) = δ(y)δ(x − H ).
(26)
The Fourier Transform of w(x, y) is defined as
W ( f, x) =
1
2π
∞
−∞
w(x, y) ei f y dy,
(27)
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A. K. Singh et al.
and the inverse Fourier transform is defined as
w(x, y) =
∞
−∞
W ( f, x) e−i f y d f.
(28)
Now, in view of Eq. (27) taking the Fourier transform of Eq. (25) with respect to y, we have
d 2 W1
dW1
2δ(x − H ) −α1 x
ρ1 ω2
2
−
f
W
−
α
W1 =
e
.
+
1
1
(1)
(1)
dx 2
dx
C44
π C44
Applying the transformation W1 = W1 e
−α1 x
2
(29)
, Eq. (29) reduces to
δ(x − H ) −α1 x
d2 W 1
− a12 W1 = 2
e 2
2
(1)
dx
C
(30)
44
where a12 = ( f 2 +
α12
4
−
ω2
)
β12
and β12 =
(1)
C44
ρ1
.
5 Equations of motion for the piezo-porous half-space
The functional gradient of the half-space is assumed to be a quadratic function of depth, i.e.,
(2)
(02)
(2)
(02)
C44 = C44 + α2 (x − H )2 , (ρ11 )33 = (ρ11 )33 + α3 (x − H )2
(31)
where α2 and α3 are respective functional gradient parameter associated with elastic constant and dynamical
mass coefficient of the half-space considered in the solid phase only.
The non-vanishing equation of motion and equation of charge, in view of the considered functional gradients, for the piezo-porous half-space are:
∂ 2 w1
(02)
(2)
C44 ∇ 2 w2 + e15 ∇ 2 φ2 − ρ2 2
∂t
∂ 2 w1
∂w2
2
2
= −(x − H ) α2 ∇ w2 − α3 2 − 2α2 (x − H )
,
∂t
∂x
2
(A(2)
(2)
(2)
11 )
∇ 2 φ2 = 0.
and e15 ∇ 2 w2 − ξ11 − ∗(2)
ξ11
(32)
(33)
Using Eq. (33) in Eq. (32), we have
(2)
C44 ∇ 2 w2
where
∂ 2 w2
∂ 2 w2
2α2 ∂w2
2
− ρ2 2 = −α2 ∇ w2 −
+ α3 2 (x − H )2
∂t
x − H ∂x
∂t
(34)
⎞
⎛
⎜ (02)
(2)
C44 = ⎜
⎝C44 + (2)
(e15 )2
(2)
ξ11 −
(2)
(A11 )2
⎟
⎟
⎠
∗ (02)
ξ11
(02)
and ρ2 = (ρ11 )33 −
(2)
((ρ12 )33 )2
(2)
(ρ22 )33
.
On substituting w2 = w2 (x, y)eiωt , we have
2α2 ∂w2
(2)
C44 ∇ 2 w2 + ρ2 ω2 w2 = −α2 ∇ 2 w2 −
− α3 ω2 (x − H )2 .
x − H ∂x
(35)
Impact of point source and mass loading sensitivity on SH wave propagation
2609
On taking the Fourier transform of Eq. (35) with respect to y and simplifying, we have
(x − H )2
α2 (x − H )2 d2 W2
α2 (x − H ) dW2
d2 W 2
2
+
−
a
W
=
−
−
2
(α2 f 2 − α3 ω2 )W2
2
2
(2)
(2)
(2)
dx 2
dx 2
dx
C44
C44
C44
= 4πσ2 (x)
where
a22
=
ω2
f − 2
β2
2
(36)
(2)
,
and 4πσ2 (x) = −
β22
C
= 44 ,
ρ2
α2 (x − H )2 d2 W2
α2 (x − H ) dW2
(x − H )2
−
2
(α2 f 2 − α3 ω2 )W2 .
+
(2)
(2)
(2)
dx 2
dx
C44
C44
C44
6 Boundary conditions
6.1 The boundary conditions at the uppermost surface (at x = 0) are given as follows:
The uppermost surface is free from mechanical traction and electric potential, i.e.,
(1)
τ13 = 0,
(37)
φ1(1)
(38)
= 0,
6.2 The boundary conditions at the imperfect common interface of layer and half-space (at x = H ) are given as
follows:
The continuity of stress and electric displacement [5] at the interface, i.e.,
(1)
(2)
τ13
= τ13
,
(1)
(2)
D1 = D1 ,
∗(1)
∗(2)
D1 = D1 .
(39)
(40)
(41)
As imperfectness allows the tangential displacement on both sides of the interface to be different, therefore
the condition of displacement continuity reduces to [12]
(1)
τ13 = σ (w1 − w2 )
(42)
where σ is the parameter accounting for the mechanical imperfectness at the common interface.
The pores of layer and half-space are considered to be totally disconnected at the interface [22], therefore,
φ1∗ = 0,
φ2∗ = 0.
(43)
(44)
7 Solution using Green’s function
Let G 1 ( xx0 ) be the Green’s function for the piezo-poroelastic layer. Therefore, G 1 ( xx0 ) satisfies Eqs. (37) and
(38) as
dG 1
α1
(45)
− G 1 = 0,
dx
2
at x = 0 and x = H , then
d2 G 1
− a12 G 1 = δ(x − x0 )
(46)
dx 2
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A. K. Singh et al.
where x0 is an arbitrary point in the layer. Multiplying Eq. (30) by G 1 ( xx0 ) and Eq. (46) by W1 , subtracting
the latter from the former and then integrating the derived equation with respect to x from 0 to H , we have
H d2 W
x
d2 G 1
1
G
W1 dx
−
1
2
dx
x0
dx 2
0
H
−α1 x
x
2
δ(x − H )e 2 G 1
(47)
=
− δ(x − x0 )W1 (x) dx.
(1)
x0
0
C
44
From Eqs. (45), (47) there follows
W1 (x0 ) =
2
(1)
C44
e
−α1 H2
G1
H
x0
− G1
Using the symmetry property of Green’s function, i.e.,
Eq. ( 48), we arrive at
W1 (x) =
2
(1) e
C44
−α1 (H +x)
2
G1
x
H
H
x0
G 1 ( xx0 )
α1
W1
− W1
dx
2
=
G 1 ( xx0 ),
.
(48)
x=H
and interchanging x by x0 in
eα1 (H −x) G 1 Hx .
− ddWx1
x=H
(49)
Similarly, let G 2 ( xx0 ) be the Green’s function for the piezo-poroelastic half-space satisfying the conditions
dG 2 = 0 at x = H and tending to 0 as x → ∞, where x is an arbitrary point in the half-space. Then, we have
0
dx
d2 G 2
− a22 G 2 = δ(x − x0 ).
dx 2
(50)
Now multiplying Eq. (36) by G 2 ( xx0 ) and Eq. (50) by W2 , subtracting the latter from the former and integrating
with respect to x from H to ∞, we have
∞ d2 W
x
d2 G 2
2
G
W
−
2
2 dx
dx 2
x0
dx 2
H
∞
x
(51)
=
4πσ2 (x)G 2
− δ(x − x0 )W2 (x) dx,
x0
H
which on further simplification gives
x dW 2
W2 (x) = G 2
+
H
dx x=H
∞
H
4πσ2 (x0 )G 2
x
x0
dx0 .
(52)
Further with help of the boundary conditions Eqs. (39), (40), (41), (42), (43), and (44), we obtain the
expressions for ddWx1 and ddWx2 at x = H as
−α1 H
∞
2σ G 1 ( H
dW1
H
σ
H )e
= (1)
4πσ2 (x0 )G 2
(53)
−
dx0 ,
dx x=H
x0
C44 (C5 + σ C6 ) (C5 + σ C6 ) H
−α1 H
∞
2σ M G 1 ( H
σM
dW2
H
H )e
−
=
4πσ
(x
)G
(54)
dx0
2 0
2
(1)
dx x=H
(C
+
σ
C
)
x
6
0
5
H
C44 (C5 + σ C6 )
where C1 , C2 , C3 , C4 , C5 , C6 , and M may be referred from the “Appendix.”
Using Eqs. (36) and (54) in Eq. (52), we have
x
−α1 H
2
∞
σ M G 2 ( Hx )α2
2σ M G 1 ( H
2 d W2
H )G 2 ( H )e
W2 (x) =
+
−
H
)
+ 2(x0 − H )
(x
0
(1)
(2)
dx 2
C44 (C5 + σ C6 )
C44 (C5 + σ C6 ) H
∞
H
α3 (x0 − H )2 ω2
α2
dW2
2 2
W2 G 2
− (x0 − H ) f W2 +
dx0 − (2)
(x0 − H )2
dx
α2
x0
C44 H
Impact of point source and mass loading sensitivity on SH wave propagation
2611
dW2
α3 (x0 − H )2 ω2
x
d2 W2
2 2
−
(x
+
2(x
−
H
)
−
H
)
f
W
+
W
G
dx0 .
0
0
2
2
2
dx 2
dx
α2
x0
(55)
Neglecting the terms containing the higher powers of α2 in Eq. (55), we have
W2 (x) =
x
−α1 H
2σ M G 1 ( H
H ) G 2 ( H )e
(1)
C44
(C5 + σ C6 )
.
Using Eqs. (36), (53), and (56) in Eq. (49), we have
H
x
−α1 x
(C5 + σ C6 )e−α1 2 − σ G 1 ( H
2
H)
W1 (x) = (1)
e 2 G1
(C5 + σ C6 )
H
C44
H +x
x
2
∞
2α2 σ 2 Me−α1 2 G 1 ( H
2 d G2
H ) G 1( H )
×
−
H
)
(x
−
0
(1) (2)
dx 2
H
C44 C44 (C5 + σ C6 )2
x0
dG 2
α3 (x0 − H )2 ω2
2 2
+2(x0 − H )
G2 G2
− (x0 − H ) f G 2 +
dx0 .
dx
α2
H
(56)
(57)
The expression of W1 (x) is calculated from Eq. (57) if the expressions of G 1 ( Hx ) and G 2 ( Hx ) are known.
Now, two independent solutions of the equation
d2 U
− a12 U = 0,
dx 2
(58)
vanishing as x → −∞ and x → ∞, are U1 (x) = ea1 x and U2 (x) = e−a2 x , respectively.
Therefore, the solution of Eq. ( 58) for an infinite medium is may be written as
U1 (x)U2 (x0 )
U1 (x0 )U2 (x)
for x < x0 and
for x > x0
W
W
where the Wronskian of the two solutions of Eq. (58) is defined as
W = U1 U2 − U1 U2 = −2a1 .
Therefore, a solution of Eq. (58) for an infinite medium is
−
e−a1 |x−x0 |
.
2a1
So, the solution of Eq. (46) is assumed as
G1
x
e−a1 |x−x0 |
=−
+ Aea1 x + Be−a1 x .
x0
2a1
In view of Eqs. (46) and (59), we have
G1
x
x0
=
−T0
.
N
Moreover,
x −T
1
=
,
G1
H
N
and
G1
H
H
=
−T2
.
N
where the unknown terms T0 , T1 , T2 , N may be referred from the “Appendix”.
Similarly, we have
x
−1 −a2 |x−x0 |
e
G2
+ e−a2 (x+x0 −2H ) ,
=
x0
2a2
(59)
2612
A. K. Singh et al.
and
1
H
=− .
H
a2
x
H
x
Using the expressions of G 1 H , G 1 H , G 2 H , and G 2 H
H , we derive the expression of W1 (x) from
Eq. (57) as
⎛
⎞
α ω2
a22 + 3α − f 2
1
1
2
⎛
⎞
2
+ 2a2
α2 M σ T2 (2)
)
⎜
⎟
T1 −α1 (x+H
4a23
C44
2
⎜
⎟
e
N
⎟.
⎠ ⎜
W1 (x) = ⎝
1
+
(60)
⎜
(1)
(N (a2 C5 − σ M) − σ a2 T2 )(a2 C5 − σ M) ⎟
2 C 5 −σ M)−σ a2 T2
⎝
⎠
C44 N (a2(a
C −σ M)N
G2
2 5
Now, in view of Eq. (28) taking the inverse Fourier transform of Eq. (60), we have
⎛
2
w1 (x, y) =
)
T1 −α1 (x+H
2
e
e−i f x
N
−∞ C (1) N (a2 C5 −σ M)−σ a2 T2
44
2(a2 C5 −σ M)N
∞
σ 2 T2 1(2)
C44
a22 +
α3 ω
α2
4a23
−f2
⎞
+
α2 M
⎟
⎜
⎟
⎜
⎟ d f. (61)
⎜1 +
⎜
(N (a2 C5 − σ M) − σ a2 T2 )(a2 C5 − σ M) ⎟
⎠
⎝
1
2a2
The value of the above integral primarily depends on the poles of the integrand which are located at the roots
of the equation
⎛
⎞
2 + α3 ω 2 − f 2
a
1
1
2
α
2
⎠.
(N (a2 C5 − σ M) − σ a2 T2 ) (a2 C5 − σ M) = α2 σ 2 T2 M (2) ⎝
+
(62)
2a2
4a23
C44
Replacing a1 by ia1 and f by k in Eq. (62) we derive the dispersion relation of an SH wave in a piezo-poroelastic
layered structure under the influence of a point source of disturbance at the common interface as
tan (a1 H ) =
where a1 =
ω2
β12
− (k 2 +
α1
4 ),
B0
B1 + B2
(63)
and B0 , B1 , B2 may be referred from the “Appendix.”
8 Mass loading sensitivity
In this Section, the mass loading sensitivity of the considered layered structure is analyzed (Fig. 2). According
to the Rayleigh Hypothesis, the thickness of the sensitive layer must be infinitesimally small and have no
elasticity, i.e., only the thickness and mass density and not the material properties of the coated layer will
effect the SH wave propagation. Therefore, we have assumed the deposition of a thin over layer of ZnO of
non-vanishing thickness H and mass density ρ = 5665 kg/m3 . The mass loading sensitivity (Sc ) which is
defined in terms of change in phase velocity due to the coated layer is given as follows:
Sc =
c − c0
1
lim
c0 m→0 m
(64)
where c and c0 are the phase velocity after and before mass increment m. m is the mass loaded per unit
area of surface and expressed in the form of m = ρ H . In the Rayleigh Hypothesis and perturbation method,
it is assumed that the surface of the probe is free before the mass increment m is added, which implies
m = ρ H . Therefore, we obtain
c − c0
Sc =
.
(65)
c0 ρ H Impact of point source and mass loading sensitivity on SH wave propagation
2613
Fig. 2 Geometry of the layered structure with sensitive layer
Since the elastic properties of the added layer do not effect significantly, therefore, the only change in the stress
free boundary conditions due to mass loading is
(1)
τ13 = −ρ H Ẅ1 (0).
(66)
Equation (66) transforms Eq. (45) by introducing a new Green’s function G 1M ( xx0 ) for the layer due to the
effect of mass loading as
dG 1M
α1
− G 1M = −ρ H Ẅ1 (0)
(67)
dx
2
where the above condition is satisfied at x = 0 and x = H . The newly introduced Green’s function G 1M ( xx0 )
also serves as a solution for Eq. (30).
Adopting a similar procedure as in Sect. 7, we obtain
TM
x
M
G1
= 1 ,
x0
N1
M
T
TM
H
0
= 2
= 3
and G 1M
G 1M
H
N1
H
N1
where T1M , N1 , T2M , T3M may be referred from the “Appendix”.
In view of the new boundary condition Eq. (66) due to mass loading, the dispersion relation for an SH
wave propagating in the layered structure due to the additional ZnO-sensitive layer is given as
⎞⎞
−α H
H ω2 W (0)e 21 G M ( 0 )
2eα1 H G 1M ( H
)
ρ
σ 2 α2 Meα1 H I
1
1 H ⎠⎠
H
N2 ∗ N1 1 −
= 0 (68)
+
(2)
(1)
(01)
C44
N2 C44 (C5 + σ C6M )2
C44
where
⎛
N2 =
and
2eα1 H G 1M ( H
H)
⎝
(1)
C44
+
ρ H ω2 W
−α1 H
2
1 (0)e
(01)
C44
⎞
G 1M ( H0 )
⎠
H
H
C6M = σ G 1M
+ M G2
.
H
H
In the absence of an additional over layer (H = 0), the dispersion relation Eq. (68) reduces to Eq. (63).
2614
A. K. Singh et al.
9 Special cases
9.1 Case 1
When the piezoelectric layer without sensitive coating is perfectly bonded to an isotropic half-space having
(01)
(02)
quadratic variation in its rigidity, (i.e., H = 0, A(01)
11 → 0, σ → 0, α1 → 0, (ρ12 )33 → 0 A11 →
(02)
(02)
0, e15 → 0, α3 → 0, (ρ12 )33 → 0), under the influence of a point source of disturbance at their common
interface the dispersion relation Eq. (63) reduces to
e
a1 H
(1)
(a1 C44
(02)
(1)
+ a2 C44 ) − e−a1 H (a1 C44
(02)
− a2 C44 ) − α2 cos(a1 H )
k 2 − a22
2a23
+
1
a23
=0
(69)
where
(1)
(01)
C44 = C44 +
(01) 2
)
(e15
(01)
ξ11
, a1 = ω2
(1)
C44
(02)
2
2
2 − k , a2 = k − (ρ12 )33
ω2
(02)
C44
2 .
(02)
(ρ11 )33
The above result matches exactly with the dispersion relation obtained by Singh et al. [23].
9.2 Case 2
When the functionally graded piezoelectric layer without sensitive coating and porosity is bonded perfectly
(01)
to a half-space without piezoelectricity, porosity, and functional gradients (i.e., H = 0, A11 → 0, σ →
(01)
(02)
(02)
(02)
0, (ρ12 )33 → 0 A11 → 0, e15 → 0, α2 → 0, α3 → 0, (ρ12 )33 → 0), the dispersion relation Eq. (63)
reduces to
(02)
4a2 a1 C44
tan(a1 H ) =
(70)
(1)
(02)
(4a1 + α12 )C44 eα1 H − 2α1 a2 C44
α2
ω2
where a1 = (1) 2 − k 2 − 41 . The above equation is similar to the dispersion relation obtained by Du
C44
(02)
(ρ12 )33
et al. [7].
9.3 Case 3
When the piezoelectric layer without sensitive coating is devoid of porosity, functional gradient, and is perfectly
(01)
bonded to the half-space devoid of piezoelectricity, porosity, and function gradients, (i.e., H = 0, A11 →
(01)
(02)
(02)
(02)
0, σ → 0, α1 → 0, (ρ12 )33 → 0 A11 → 0, e15 → 0, α2 → 0, α3 → 0, (ρ12 )33 → 0), under the
influence of a point source of disturbance at their common interface the dispersion relation Eq. (63) reduces to
tan(a1 H ) =
where a1 = al. [24].
ω2
(1)
C44
2
(
(02) )
(ρ12 )33
(02)
a2 C44
(1)
a1 C44
(71)
− k 2 . The obtained result is similar to the dispersion relation obtained by Liu et
Impact of point source and mass loading sensitivity on SH wave propagation
2615
9.4 Case 4
For SH wave propagation under the influence of a point source at the common interface of an isotropic homo(01)
geneous layer and an isotropic half-space having linear variation in its density only, (i.e. H = 0, e15
=
(01)
(01)
(01)
(01)
(02)
(02)
0 A11 → 0, σ → 0, , α1 → 0, C44 = μ1 , (ρ11 )33 = ρ1 , (ρ12 )33 → 0, C44 = μ2 , e15 =
(02)
(02)
(02)
0, A11 → 0, α2 → 0, (ρ11 )33 = ρ2 , (ρ12 )33 → 0), the dispersion relation Eq. (63) matches fairly with
the result of Chattopadhyay and Pal [25].
⎛
⎜ c
tan ⎝k H
β10
where β10 =
respectively.
!
μ1
ρ1
and β20 =
!
μ2
ρ2
2
⎞
⎟
− 1⎠ =
μ2 1 −
μ1
2
c
β20
2
c
β10
−1
α3
−
4ρ1 k 1 −
2
c
β10
2
3/2
(72)
c
β20
are the shear wave velocities of the isotropic layer and isotropic half-space,
9.5 Case 5
For the sake of validation the obtained results are reduced to the classical case which is an isotropic elastic
half-space perfectly bonded to the superficial layer, i.e., layer and half-space are without piezoelectricity, poroe(01)
(02)
lasticity, functional gradient, and imperfect bonding, i.e. α1 = 0, α2 → 0, α3 = 0, e15 → 0, e15 →
(01)
(02)
(01)
(02)
(01)
(02)
0, A11 → 0, A11 → 0, C44 = μ1 , C44 = μ2 , (ρ11 )33 = ρ1 , and (ρ11 )33 = ρ2 , which in turn
leads to the classical Love wave equation [26],
tan k H
2
μ2 1 − (βc0 )2
2
c
2
−1 =
2
(β10 )2
μ1 (βc0 )2 − 1
(73)
1
!
!
μ1
μ2
0 =
and
β
where β10 =
2
ρ1
ρ2 are shear wave velocity of layer and half-space, respectively, with
μ1 , μ2 (ρ1 , ρ2 ) denoting the shear modulus (density) of the isotropic layer and half-space, respectively.
10 Numerical results and discussions
In the previous Sections, we have derived a dispersion relation Eq. (63) for the propagation of the SH wave
in the FGPPM/FGPPM layered structure influenced by a point source of disturbance situated at the interface.
As a special case, the obtained result has been validated by matching with the classical Love wave dispersion equation. The obtained dispersion equations not only bear the relation between wave number and phase
velocity but also exhibit the influence of various functional gradient parameters, imperfectness parameter,
and piezoelectric coupling parameter, and piezo-porous coupling parameter on the dispersion curves. For the
purpose of numerical computation, numerical data of BaTiO3 and PZT-5H are considered for FGPPM layer
and FGPPM half-space, respectively. The data for material properties are referred from Table 1.
Moreover, deposition of an infinitesimally thin layer of ZnO is considered to examine the mass loading
sensitivity of the structure. It is evident from Figs. 3, 4, 5, 6, 7, 8, 9, 10, and 11 that the phase velocity (c/β1 )
decreases with increases in the wave number (k H ).
2616
A. K. Singh et al.
Table 1 Material constants
Material
Elastic constant
(1010 N/m2 )
BaTiO3 [5]
C44 = 4.386
PZT-5H [3]
(01)
(02)
C44 = 2.3
Piezoelectric
constant (C/m2 )
(01)
e15 = 11.4
(2)
e15 = 17
Dielectric
constant
(10−10 F/m)
(01)
ξ11
= 108
∗(01)
ξ11 = 118
(01)
A11 = 128
(2)
ξ11 = 277
∗(2)
ξ11 = 299
A(2)
11 = 112
Mass
density
(kg/m3 )
(01)
ρ11 = 3876
(01)
ρ12 = −741
(01)
ρ22 = 3762
(02)
ρ11 = 4950
(2)
ρ12 = −1125
(2)
ρ22
= 4800
Fig. 3 Variation of the non-dimensional phase velocity (c/β1 ) against the non-dimensional wave number (k H ) for distinct values
of the functional gradient parameter (α1 H )
10.1 Effect of the functional gradients
Figure 3 shows the effect of the non-dimensional functional gradient parameter α1 H considered in all material
constants of the FGPPM layer on the phase velocity of the SH wave in different vibrational modes. The effect
of α1 H on the phase velocity of the SH wave is decreasing in 1st (fundamental vibrational) mode, whereas in
2nd mode, it is reversed and shows an increasing effect on the phase velocity which is further continued in the
3rd mode.
(02)
Figure 4 shows the effect of the non-dimensional functional gradient parameter (α2 H 2 /C44
) considered
in the mechanical stiffness elastic constants (Ci j ) (solid phase) on the dispersion curves (phase velocity).
Figure 4 unfolds that the phase velocity increases with increase in the non-dimensional functional gradient
(02)
(02)
parameter (α2 H 2 /C44 ) in all three (1st, 2nd and 3rd) vibrational modes. The effect of (α2 H 2 /C44 ) is most
prominent in the fundamental mode and diminishes gradually in higher vibrational modes.
The effect of the non-dimensional functional gradient parameter (α3 H 2 /(ρ11 )(02)
33 ) considered in the solid
phase of mass density of the FGPPM half-space on the phase velocity of an SH wave is illustrated in Fig. 5.
(02)
The non-dimensional functional gradient parameter (α3 H 2 /(ρ11 )33 ) has a decreasing effect with very small
variation on the phase velocity in the 1st(fundamental) mode as well as in higher order modes.
The effect of the functional gradient parameter (α1 H ) associated with the FGPPM layer dominates
(02)
over the effect of functional gradient parameters associated with the FGPPM half-space ((α2 H 2 /C44
) and
Impact of point source and mass loading sensitivity on SH wave propagation
2617
Fig. 4 Variation of the non-dimensional phase velocity against the non-dimensional wave number for distinct values of the
(02)
functional gradient parameter (α2 H 2 /C44 )
Fig. 5 Variation of the non-dimensional phase velocity against the non-dimensional wave number for distinct values of the
(02)
functional gradient parameter (α3 H 2 /(ρ11 )33 )
(02)
(α3 H 2 /(ρ11 )33 )). Particularly, the functional gradient parameter concerned with the stiffness constant of
the FGPPM half-space has a larger effect as compared to that concerned with mass density of the FGPPM
half-space.
(1)
(2)
10.2 Effect of the ratio of shear moduli (α4 = C44 /C44 )
The variation of non-dimensional phase velocity against the non-dimensional wave number for different values
of ratio of the shear moduli is manifested through Fig. 6. In the 1st (fundamental) mode, the ratio of the shear
moduli has an encouraging effect on the phase velocity of the SH wave. On the other hand, a similar behavior
2618
A. K. Singh et al.
Fig. 6 Variation of the non-dimensional phase velocity against the non-dimensional wave number for distinct values of the
dimensionless parameter α4
(a)
(b)
Fig. 7 Variation of the non-dimensional phase velocity against the non-dimensional wave number for distinct values of the
piezoelectric coupling parameter (K e1 ) a without porosity, b with porosity
has been found in the 2nd order vibrational mode up to a certain value of the wave number but as the wave
number increases (k H = 0.9405 onwards), the effect is reversed. Further, in the 3rd order vibrational mode,
the effect of the ratio of the shear moduli is found to be decreasing on the phase velocity of SH wave. The
impact of α4 on the dispersion curve is found to be more prominent in higher order modes with increasing
wave number as compared to the fundamental mode.
10.3 Effect of the piezoelectric coupling parameter
Figures 7a, 8b forefront the effect of the piezoelectric coupling parameter K e1 and K e2 associated with layer
and half-space, respectively. The piezoelectric coupling parameters for FGPPM layer and FGPPM half-space
Impact of point source and mass loading sensitivity on SH wave propagation
(a)
2619
(b)
Fig. 8 Variation of the non-dimensional phase velocity against the non-dimensional wave number for distinct values of the
piezoelectric coupling parameter (K e2 ) a without porosity, b with porosity
are defined, respectively, as
(K e1 )2 =
(01) 2
)
(e15
(01) (01)
C44 ξ11
and (K e2 )2 =
(2) 2
)
(e15
(02) (2)
C44 ξ11
,
where
(01)
(01)
C44
= C44
+
(01)
(e15 )2
(01)
ξ11
(02)
(02)
and C44
= C44
+
(2)
(e15 )2
(2)
ξ11
.
Figure 7a shows the effect of (K e1 ) when the porosity of the layer is neglected, while Fig. 7b depicts the
effect of (K e1 ) when the layer is piezo-poroelastic. It is noted from Fig.7a (the absence of porosity), that the
phase velocity of SH wave increases with an increase in (K e1 ) for the fundamental mode. However, in the 2nd
mode, the nature gets reversed (decreases) at a particular k H (= 1.311), and the same decreasing trend of the
phase velocity is observed in the 3rd mode. However, just the opposite trend of the phase velocity is shown in
Fig. 7b (presence of porosity) as a response to the piezoelectric coupling parameter (K e1 ). From Fig. 7a, b, it
is observed that, the effect of (K e1 ) is opposite with larger variation in the presence of porosity as compared
to the case when the porosity is absent in the layer. Comparing Fig. 7a, b, it is evident that the presence of
porosity discourages the phase velocity SH wave in the fundamental mode.
Figure 8a, b depicts the effect of the piezoelectric coupling parameter (K e2 ) for the half-space in the
absence and presence of porosity, respectively. A meticulous observation of these two graphs unfolds the fact
that absence of porosity disfavors the phase velocity with an increase in the wave number, but the nature of
variation of the dispersion curve with respect to (K e2 ) is the same in both cases. The increase in (K e2 ) gives
more significant variation in the absence of porosity as compared to the variation in the presence of porosity.
The variation pattern of the phase velocity with increasing value of (K e2 ) is increasing in both cases (with and
without porosity) for the fundamental mode which starts decreasing in the 2nd mode at a certain wave number,
and continues the same nature in other higher modes.
10.4 Effect of piezo-porous coupling parameter
p
p
Figures 9 and 10 delineate the effect of the piezo-porous coupling parameter (K e1 ) and (K e2 ) associated with
layer and half-space, respectively, on the phase velocity of the SH wave. The piezo-porous coupling parameters
2620
A. K. Singh et al.
Fig. 9 Variation of the non-dimensional phase velocity against the non-dimensional wave number for distinct values of the
p
piezo-porous coupling parameter (K e1 )
Fig. 10 Variation of the non-dimensional phase velocity against the non-dimensional wave number for distinct values of the
p
piezo-porous coupling parameter (K e2 )
for FGPPM layer and FGPPM half-space are, respectively, defined as
p
(K e1 )2
(01)
=
(e15 )2
(1)
p
C44 (ξ11 )(01)
and
p
(K e2 )2
(2)
=
(e15 )2
(2)
p
C44 (ξ11 )(2)
where
(1)
(01)
C44 = C44 +
(01) 2
)
(e15
(2) 2
)
(e15
(2)
(02)
,
C
=
C
+
p (01)
p (2)
44
44
(ξ11 )
(ξ11 )
Impact of point source and mass loading sensitivity on SH wave propagation
(a)
2621
(b)
Fig. 11 Variation of the non-dimensional phase velocity against the non-dimensional wave number for distinct values of the
imperfectness parameter Γ a without porosity, b with porosity
and
(01)
(ξ11 )(01) = ξ11 −
p
(01)
(2)
(A11 )2
(A11 )2
p (2)
(2)
,
(ξ
)
=
ξ
−
11
11
∗ )(01)
∗ )(2) .
(ξ11
(ξ11
p
p
As observed from Figs. 9 and 10, the outcome of increasing (K e1 )2 and (K e2 )2 on the non-dimensional
phase velocity is increasing in the 1st mode but decreasing in the 2nd and further higher modes. The impact of
p
increasing (K e1 )2 is less significant on the phase velocity as the variation can be seen in Fig. 9 at a larger difp
p
ference in the values of (K e1 )2 unlike Fig. 10, where the variation due to increasing (K e2 )2 is quite significant
even for smaller increments of values.
10.5 Effect of the imperfectness parameter
The impact of the non-dimensional imperfectness parameter Γ (= C5 k/σ ) on the non-dimensional phase
velocity against the non-dimensional wave number is expressed in Fig. 11a, b, without and with the presence
of porosity, respectively. It is examined from both Fig. 11a, b that the phase velocity of the SH wave increases
with increase in magnitude of (Γ ) in the fundamental mode. However, in the 2nd mode initially the phase
velocity increases and then reverses its trend (i.e., decreases) after a certain (k H ), and further continues the
same nature in the 3rd mode. In particular, curves 1, 4 and 7 of Fig. 11b represents the dispersion curves
associated with Eq. (63). It is concluded from Fig. 11a, b that presence of porosity does not affect the nature
and variation pattern of the dispersion curve during the study of the effect of the imperfectness parameter (Γ ).
10.6 Effect of mass loading sensitivity
Figure 12a–c portrays the variation of the mass loading sensitivity |Sc | (m2 /kg) against the non-dimensional
wave number (k H ) for an additional layer of thickness H = 3 × 10−3 m in the 2nd higher order vibrational
mode. Due to the increment of the mass on the free surface, the phase velocity of the layer reduces which
results in a negative value of the mass loading sensitivity. Therefore, for the purpose of the numerical study,
the absolute value of mass loading sensitivity is considered. It is observed from these three Figures that the
mass loading sensitivity decreases with an increase in wave number. Figure 12a shows that the sensitivity
increases in the presence of functional gradients in the FGPPM layer. Figure 12b forefronts that the sensitivity
is higher for a piezo-poroelastic medium than for an elastic medium. This clarifies the fact that the porosity
of the medium supports the sensitivity of the Love wave devices. Figure 12c reveals the effect of imperfect
2622
A. K. Singh et al.
Fig. 12 Variation of mass loading against non-dimensional wave number for distinct values of a non-dimensional functional
p
gradient parameter (α1 H ), b piezo-porous coupling parameter (K e1 ), c imperfectness Parameter (Γ )
bonding of the interface on the mass loading sensitivity. It is evident from Fig. 12c that the sensitivity decreases
due to the presence of the imperfect interface.
10.7 Validation with the classical case
For the sake of validation, the original problem is reduced to the classical isotropic elastic layer and halfspace structure (i.e., in absence of porosity, piezoelectricity, and functional gradient parameter). It is worth
mentioning that, to study the isotropic (classical) case of the problem, numerical data of BaTiO3 and PZT-5H
should not considered (since BaTiO3 and PZT-5H are piezoelectric sensitive materials exhibiting transversely
isotropic symmetry not simply isotropic elastic material with isotropic symmetry). Rather we have considered
the numerical data of isotropic elastic material [26] as:
β20
β10
= 1.297 and
μ2
= 2.159,
μ1
which readily satisfies the condition for Love wave propagation (β10 < c < β20 ).
The dispersion curve relating phase velocity(c/β10 ) against wave number (k H ) using Eq. (73) (classical
Love wave equation) is plotted. The dispersion curves (first mode and second mode) manifested in this problem, when reduced to the classical case (as depicted in Fig. 13), are in close resemblance with the same as
presented by Ewing et al. [26], which validates our problem.
Impact of point source and mass loading sensitivity on SH wave propagation
2623
Fig. 13 Graphical validation of the present result with the classical Love wave equation curve
Fig. 14 Variation of the non-dimensional group velocity (cg /β1 ) against the non-dimensional wave number (k H )
10.8 Group velocity
The quantity cg = ∂ω
∂k defines the velocity of a group of waves having angular frequency ω and is known as
∂c
).
the group velocity. It is related to the phase velocity by the equation (cg = c + k ∂k
Figure 14 represents the graphical illustration of non-dimensional group velocity (cg ) of the SH wave
against the non-dimensional wave number (k H ). It is clearly noted from the Figure that initially the group
velocity decreases with an increase in wave number, but later the group velocity rapidly increases for a higher
range of wave number.
11 Conclusions
The propagation behavior of an SH wave in a layered structure comprised of a functionally graded piezoporoelastic layer (FGPPM) imperfectly bonded to a functionally graded piezo-poroelastic half-space under
the influence of an impulsive point source of disturbance situated at the interface is analyzed. The solution
procedure to derive the dispersion relation of the SH wave involves a suitable determination of Green’s function
for both layer and half-space. An attempt is made to determine the mass loading sensitivity of the model due to
the deposition of a thin layer. A layer of BaTiO3 -crystal imperfectly bonded to PZT-5H half-space with a point
source of disturbance at their interface is considered for numerical simulations. The influence of various affecting parameters like functional gradients, ratio of effective shear moduli, piezoelectric coupling parameters,
2624
A. K. Singh et al.
piezo-porous coupling parameters, and imperfectness parameter on the phase velocity of an SH wave is demonstrated graphically. Moreover, the effect of functional gradient parameter, piezo-porous coupling parameter,
and imperfectness parameter on the mass loading sensitivity of the model by coating an infinitesimally thin
layer of ZnO is illustrated graphically. The epilogue of this study comes out with the following conclusions:
(i) The functional gradient parameter (α1 H ) associated with all material constants of the layer has a discouraging effect in the fundamental mode of the dispersion curve (phase velocity) but it encourages the phase
(02)
velocity in the 2nd and further higher modes, whereas the functional gradient parameter (α2 H 2 /C44 )
associated with the elastic constant of the solid phase of the FGPPM half-space has an increasing effect
on the phase velocity of SH waves in all modes. On the contrary, the functional gradient parameter
(02)
(α3 H 2 /(ρ11 )33 ) associated with the mass density of the solid phase of the FGPPM half-space has an
adverse effect (i.e., discouraging effect) for all modes of the dispersion curves. Among the functional
gradient parameters, (α1 H ) has the maximum effect on the phase velocity of the SH wave whereas
(02)
(α3 H 2 /(ρ11 )33 ) has the least.
(ii) The ratio of the effective shear modulus (α4 ) has an increasing effect in the 1st (fundamental) mode,
which is continued initially in the 2nd mode, but after a certain value of k H , reverted influence is
observed. From the 3rd and higher modes, the discouraging impact is witnessed. This signifies that with
an increase in stiffness of the FGPPM half-space, the phase velocity of the SH wave is encouraged.
(iii) The piezoelectric coupling parameter (K e1 ) associated with the FGPPM layer (in the presence of porosity) shows a discouraging effect in the fundamental mode of the dispersion curve (phase velocity) for
SH wave propagation. However, this discouraging effect gets reverted for a particular value of k H for
the 2nd mode. From the 3rd mode onwards the phase velocity of SH wave gets encouraged. On the other
hand, in the absence of porosity, an opposite trend is observed.
(iv) The piezoelectric coupling parameter (K e2 ) associated with the half-space (in presence of porosity)
exhibits an encouraging effect on the phase velocity of the SH wave in the fundamental mode. In the 2nd
mode, the effect of (K e2 ) gets reverted. From the 3rd mode onward, the phase velocity of the SH wave
is discouraged with the increase in (K e2 ). The same effect of (K e2 ) is observed on the phase velocity
in the absence of porosity. But the effect of increasing (K e2 ) can be seen to be more significant in the
presence of porosity as compared to the effect of increasing (K e2 ) in the absence of porosity.
p
p
(v) The effect of the piezo-porous coupling parameter (K e1 )2 and (K e2 )2 associated with the layer and
half-space on the phase velocity of the SH wave are of same nature. Both of these parameters exhibit
an encouraging effect in the 1st mode and discouraging effect in the 2nd and other higher modes. A
comparative analysis reveals that the piezo-porous coupling parameter associated with the half-space
has a more substantial effect as compared to that of the layer.
(vi) It is noted that both in presence and absence of porosity, with an increase in the magnitude of imperfectness parameter (Γ ), the phase velocity increases for the 1st (fundamental) mode, whereas the phase
velocity decreases with an increase in (Γ ) for higher modes.
(vii) To study the mass loading sensitivity of the model, a thin layer of ZnO is assumed on the free surface of
the layered structure. The mass loading sensitivity of the model increases with an increase in the magnitudes of the functional gradient parameter (α1 H ) and the piezo-porous coupling parameter (K e1p )2
associated with the layer, whereas the sensitivity decreases with an increase in the magnitude of the
imperfectness parameter (Γ ).
(viii) Porosity has a remarkable impact on the phase velocity of the SH wave. The presence of porosity discourages the phase velocity of the SH wave in the fundamental mode and therefore causes the phase
delay which leads to the entrapment of the SH wave in the layer for a longer duration. This phenomenon
may be taken into account for increasing the efficiency of sensors.
(ix) With an increase in the wave number, the group velocity of an SH wave primarily decreases and after a
certain wave number it increases with an increase in the wave number.
The present study outlines the theoretical and numerical computation for the design and development of under
water acoustic sensors and transducers. Moreover, the outcome of the present study may have applications in
acoustical engineering and ultrasonics. For example, to achieve high performance of Love wave sensors, it is
required to increase the movement of the particles on the surface thereby increasing its sensitivity. This may
be accomplished by introducing a porous material at the free surface of the layered structure. In the present
study, a thin sensitive layer of ZnO has been considered. The properties of ZnO, being a piezoelectric material
having hexagonally shaped cylinders with a gap in between them, thus making its surface porous, fulfils all
the above-mentioned requirements of Love wave sensors to keep their efficiency at premium.
Impact of point source and mass loading sensitivity on SH wave propagation
2625
Acknowledgements Authors acknowledge the Department of Science and Technology, Science and Engineering Research Board
(DST-SERB) by the project (Grant Number EMR/2017/000263/MS) entitled “Mathematical Modelling of Seismic Wave Propagation in Composite Layered Structures” for providing financial support for this research work. Authors also thank Mr. Mriganka
Shekhar Chaki for providing insight and expertise throughout this research work.
Appendix
(2)
C1 =
e−α1 H e
(01) 15 (01) ,
ξ11
(2) A11
ξ11
(2) − (2)
ξ11
A11
C2 =
(2)
ξ11
C 3 = e α1 H
(01)
e15
(01)
A11
(2)
A11
−
(01)
C44
,
(01)
ξ11
(2)
ξ11
(2)
A11
(01)
(2)
A11
C4 =
(02)
C44
+ C1 e
α1 H
(01)
(01)
− C2 e15 − e15
−e15 + e15
,
A(2)
11
(01)
(2)
A11
,
C3
(01)
(01)
(01)
,
C5 = eα1 H C44 − C2 e15 + M C1 e15 , M =
C4
H
H
+ M G2
,
C6 = σ G 1
H
H
α 2 a1 H
α1 a1 x α1 −a1 (x+H ) α1 T0 = a12 − 1
a1 −
− e−a1 H ea1 |x−x0 | + a1 +
+ a1 +
e
e
e
4
2
2
2
α1
α1 a1 (H −x)
α1 −a1 (H −x)
,
e−a1 x a1 −
e
e
+ a1 +
e−a1 (H −x) + a1 −
2
2
2
α 2 a1 H
α1 a1 x α1 −2a1 H α1 T1 = a12 − 1
a1 −
− e−a1 H ea1 |x−H | + a1 +
+ a1 +
e
e
e
4
2
2
2
α1 −a1 x
e
,
+ 2a1 a1 −
2
α 2 a1 H
α1 a1 H α1 −2a1 H α1 a1 −
T2 = a12 − 1
− e−a1 H + a1 +
+ a1 +
e
e
e
4
2
2
2
α 2 a1 H
α1 −a1 H
, N = 2a1 a12 − 1
− e−a1 H ,
e
e
+ 2a1 a1 −
2
4
⎛
⎞
2 + α3 ω 2 − k 2
2
a
σ a2
1 ⎠
α2 σ M
α2
⎝ 2
B0 =
+
−
,
3
2a
(σ M − a2 C5 ) C (2) (σ M − a2 C5 )2
4a
2
2
44
M
α
a12 + 41
α1 B0
B1 =
, B2 =
,
2a1
a1
α12 −a1 |x−x0 | 4a1 ρ H ω2 W1 (0) α1 M
2a1 H
2
a1 −
a
+
−
e
T1 = 2 1 − e
1
01
4
2
C44
α
H
α
1
1
eα1 H ea1 H + e2a1 H e−a1 x + a1 +
1 − e 2 e a1 H
2
2626
A. K. Singh et al.
α1 a 1 x α1 − 2a1 ea1 x0 a1 +
e + a1 −
2
2
"
α1 a 1 x α1 −a1 x 2a1 H α1 a 1 x
−a1 x
−a1 x0
a1 +
− α1 e a 1 x 0 a 1 +
e
e + a1 −
e
e
− 2a1 e
e
2 2
2
α1 −a1 x
α1 a 1 x
α1 −a1 x 2a1 H
− α1 e−a1 x0 a1 +
,
+ a1 −
e
e
e + a1 −
e
2
2
2
α2
2a1 H
N1 = 4a1 1 − e
a12 − 1 .
4
α2
4a1 ρ H ω2 W1 (0) α 1 α1 H a 1 H
2a1 H
a12 − 1 +
a
e−a1 H
e
−
e
+
e
T2M = 2 1 − e2a1 H
1
01
4
2
C44
α1 H
α1 α1 a 1 H α1 −a1 H + a1 +
+ a1 −
1 − e 2 ea1 H − 2a1 ea1 H a1 +
e
e
2 2 2
α1 a 1 H
α1 −a1 H 2a1 H
α1 a 1 H
− α1 e a 1 H a 1 +
+ a1 −
e
e
e
e
− 2a1 e−a1 H a1 +
2
2 2
α1 −a1 H
α1 a 1 H
α1 −a1 H 2a1 H
− α1 e−a1 H a1 +
,
+ a1 −
+ a1 −
e
e
e
e
2
2
2
α2
4a1 ρ H ω2 W1 (0) α 1 α1 H a 1 H
2a1 H
T3M = 2 1 − e2a1 H
a12 − 1 e−a1 H +
a
−
e
+
e
e
1
01
4
2
C44
α
H
α1
α1 a 1 H α1 −a1 H 1
1 − e 2 ea1 H − 2a1 a1 +
e
e
+ a1 −
e−a1 H + a1 +
2
2 2 α1 a 1 H
α1 −a1 H 2a1 H
α1 a 1 H
α1 −a1 H − α1 a 1 +
+ a1 −
e
+ a1 −
e
e
e
e
− 2a1 a1 +
2
2
2
2
α1 a 1 H
α1 −a1 H 2a1 H
.
+ a1 −
e
e
e
− α1 a 1 +
2
2
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