Dispersion equation of rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space

been also illustrated.
Keywords: dispersion equation, nonlocal, piezoelastic.
1. INTRODUCTION
In recent years piezoelectric materials has drawn much attention towards applica-
tion in surface acoustic wave (SAW) micro sensors, energy harvesting structure, health
monitoring systems, transducers and actuators, etc. Both theoretical and experimental
studies on wave propagation in piezoelectric materials have attracted the attention of
scientists and engineers during last two decades. The survey of literature can be found
in many related texts and books [1, 2]. We mention only a few such as: Zinchuk and
Podlipenets [3] obtained dispersion equations for acousto-electric Rayleigh wave in a
periodic layer piezoelectric half-space in a study for the 6 mm crystal class. Wave prop-
agation in porous piezoelectric material (PPM), having crystal symmetry 6 mm, is stud-
ied analytically by Vashishth et al. [4]. Sharma et al. [1] investigated the propagation of
Rayleigh waves in a homogeneous, transversely isotropic, piezothermoelastic half-space
subjected to stress free, electrically shorted/charge-free and thermally insulated/isother-
mal boundary conditions. Secular equations for the half-space in closed form and isolated
mathematical conditions in completely separate terms are derived.
Recent development in science and technology requires that the high-performance
electromechanical devices must have a higher sensitivity and larger storage capacity but
c© 2019 Vietnam Academy of Science and Technology
364 Do Xuan Tung
as maller size. Nano scale materials and structures have been introduced and devel-
oped ever since. For these materials, the conventional continuum elasticity theory fails
to represent the constitutive relationships properly [5]. A non-local model based on Erin-
gens theory of non-local continuum mechanics has been proposed for the effects of the
size dependency in very small structures. Particularly, Eringen’s nonlocal theory [6] has
been extended to study the size dependent mechanical performances of the piezoelectric
nanostructures by Ke et al. [7–9]. There are a few research works on the propagation of
the guided elastic waves in nanoscale periodic piezoelectric structures. For examples,
Chen et al. [10] studied the anti-plane transverse wave propagation in nanoscale periodic
layered piezoelectric structures. Yan et al. [11,12] investigated the propagation of guided
elastic waves in nanoscale layered periodic piezoelectric composites.
However, only few researchs [11–13] on wave propagation in nanoscale periodic lay-
ered piezoelectric structures have been reported in literature due to the complexity of the
problem. In addition, many researches [1, 14] have been carried out on the propagation
of Rayleigh waves in transversely isotropic piezoelastic materials solids. However, to
the best of the authors knowledge, there are no researches analyzing the propagation of
surface waves in transversely isotropic nonlocal piezoelastic half-space analytically avail-
able in the literature. Therefore, the main purpose of this paper is to study the effect of
nonlocality on the propagation of Rayleigh waves in transversely isotropic piezoelastic
materials. The dispersion equation for the propagation of Rayleigh waves is derived for
the boundary is stress-free, maintained at charge-free condition.
2. FORMULATION OF THE PROBLEM
We consider homogeneous transversely isotropic, electrically conducting piezoelec-
tric medium in the undeformed state at initial potential φ0. We assume that the medium
is transversely isotropic in such a way that planes of isotropy are perpendicular to x3-
axis. We take the origin of the coordinate system (x1, x2, x3) at any point on the plane
surface and x3-axis pointing vertically downward into the half-space. Thus the consider-
ing half-space is represented by x3 ≥ 0. For two-dimensional problem in which the plane
wave is in the plane x1x3), the strains are related to the displacement field u1, u3 and the
electric potential φ. The constitutive equations are given as [1, 11]:
- Strain-displacement relations
ε11 = u1,1, ε33 = u3,3, ε13 = ε31 =
1
2
(u1,3 + u3,1), (1)
- Stress-strain and electric field relations
σ11 = c11ε11 + c13ε33 − e31E3, σ33 = c13ε11 + c33ε33 − e33E3,
σ13 = σ31 = 2c44ε13 − e15E1,
D1 = 2e15ε13 + e11E1, D3 = e13ε11 + e33ε33 + e33E3,
(2)
Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space 365
where Ei = −φ,i is the electric field and Di the electric displacement, ρ the mass density,
σij the stress tensor, cij the elastic parameters tensor, eij the piezoelectric moduli, eij the
electric permittivity (i, j = 1, 3).
It is well known that in the classical piezoelectricity (CPE) theory, the stresses and the
electric displacements at one point only dependend on the local strains and electric fields
at the same point. But when the macro size reaches a few nanometers, the CPE continuum
theory fails and we have usually to utilize other methods. The essence of the Eringen’s
nonlocal elasticity theory [6,15] is that the stress at a point x in a body depends not only on
the strain at that point but also on the strain at all other points x′ in the domain. Recently,
Ke et al. [7, 8] extended the nonlocal elasticity theory to the piezoelectric nanostructures-
the nonlocal continuum theory of piezoelectricity (NLPE). Unlike the CPE continuum
theory, the NLPE theory supposes that the stresses and the electrical displacements at one
point should be affected by the strains and electrical fields at all points of the whole body.
Thus the relationship between the CPE stress and electrical displacement components
and the NLPE stress and electrical displacement components can be written as [11, 12]
tmn = (1 + e2∇2)σmn, dm = (1 + e2∇2)Dm, (3)
where tmn and dm are the NLPE stress and electrical displacement components, respec-
tively; σmn and Dm are the traditional stress and electrical displacement components,
respectively. Constant e(= e0a) is the nonlocal parameter (e0 is the nonlocal constant and
a is the internal characteristic length).
For the wave propagation considered in this paper, the body forces, electric charge
are ignored. Using the relations (3), the motion equations, Gauss equation are simplified
as [11, 12]
σ11,1 + σ13,3 = (1− e2∇2)ρu¨1, σ13,1 + σ33,3 = (1− e2∇2)ρu¨3,
D1,1 + D3,3 = 0.
(4)
Substituiting (2) into (4) taking into account (1) we have
c11u1,11 + c44u1,33 + (c13 + c44)u3,13 + (e15 + e31)φ,13 = (1− e2∇2)ρu¨1,
(c13 + c44)u1,13 + c44u3,11 + c33u3,33 + e15φ,11 + e33φ,33 = (1− e2∇2)ρu¨3,
(e15 + e31)u1,13 + e15u3,11 + e33u3,33 − e11φ,11 − e33φ,33 = 0.
(5)
For the waves propagating in the plane x3 = 0, we take the form of relevant components
of displacement and the electric potential φ as [1, 16]
u1 = a1e−ξyeik(x1−ct)
u3 = a3e−ξyeik(x1−ct) with y = kx3,
φ = A1e−ξyeik(x1−ct)
(6)
where a1, a3, A1 are polarization vectors, k is wavenumber, c is speed of wave propaga-
tion, ξ is a complex coefficient whose imaginary part should be positive corresponding
to the decay condition in the half-space x3 > 0.
Substituting the expressions for displacement and electric potential from (6) into (5),
we obtain the three homogeneous equations in three unknowns a1, a3, A1. For a nontrivial
366 Do Xuan Tung
solution of these equations, we must have det(M) = 0, in which
M =
c11 − c44ξ2 − ρc2 − ρc2k2e2(1− ξ2) (c13 + c44)iξ (e15 + e31)iξ(c13 + c44)iξ c44 − c33ξ2 − ρc2 − ρc2k2e2(1− ξ2) e15 − e33ξ2
(e15 + e31)iξ e15 − e33ξ2 e33ξ2 − e11
 .
(7)
This is the characteristics equation and it has the form
h6p6 + h4p4 + h2p2 + h0 = 0, (8)
where p = iξ and the coefficients hi, (i = 0, 2, 4, 6) are given in the Appendix.
Eq. (8) is a cubic polynomial in p2. We order pn, n = 1, 2, . . . , 6, in such a way
that p1, p2, p3 correspond to waves traveling in the positive x3 direction, and p4, p5, p6
correspond to the ones traveling in the negative x3, respectively. Since, we are interested
in surface waves only so it is essential that motion is confined to free surface x3 = 0
of the half-space so that the characteristic roots p2i must satisfy the radiation condition
Im(pi) ≥ 0. Then the general solution for displacements and electric potential are written
as [1] 
u1 =
3
∑
j=1
a1jepjyeik(x1−ct)
u3 =
3
∑
j=1
a3jepjyeik(x1−ct)
φ =
3
∑
j=1
A1jepjyeik(x1−ct)
(9)
where a1j, a3j and A1j are the amplitudes of the displacements and the electric potentials,
respectively.
Remark: For the propagation of plane waves with phase velocity c in the direction making
an angle θ with the vertical axis, a surface wave of this form is expressed by
u1 = a1eik(p1x1+p3x3−ct)
u3 = a3eik(p1x1+p3x3−ct)
φ = A1eik(p1x1+p3x3−ct)
(10)
where p1 = sin θ, p3 = cos θ are components of propagation unit vector. Substituting (10)
into (5) and obtain a matrix similar to the matrix M in Eq. (7). By letting the determi-
nant of this matrix equal zero, we have a quadratic equation in c2. Therefore, we obtain
two real roots cj(j = 1, 2) corresponding the speeds of plane waves propagating in the
medium.
3. BOUNDARY CONDITIONS AND DISPERSION EQUATIONS
In this section, the Rayleigh wave equation for transversely isotropic nonlocal piezoe-
lastic half-space can be derived using the boundary conditions at the surface of the half-
space. In the present study, boundary conditions appropriate for particle motion in the
x1x3 plane are considered at the plane surface x3 = 0. This surface is considered to be
Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space 367
stress-free (mechanical conditions), which requires the normal stress σ33 as well as the
tangential stress σ13 to vanish at the surface x3 = 0. That means
σ13 = σ33 = 0. (11)
Another condition is required to represent that the surface of half-space is maintained at
charge free condition (open circuit-surface), namely
D3 = 0. (12)
Substituting (9) into the boundary conditions (11), (12) and taking into account (2), we
have a system of linear equations
3
∑
j=1
[
c44(a3j + a1jpj) + e15A1j
]
= 0,
3
∑
j=1
[
c13a1j + c33a3jpj + e33A1jpj
]
= 0,
3
∑
j=1
[
e13a1j + e33a3jpj − e33A1jpj
]
= 0.
(13)
For each pj(j = 1, 2, 3), the three corresponding unknowns a1j, a3j, A1j (7) are in a rela-
tionship given by matrix M and we can express them as a1j = αjA1j, a3j = β jA1j where
αj =
(c13 + c44)pj(e15 + e33p2j )−
(
c44 + c33p2j − ρc2 − ρc2k2e2(1 + p2j )
)
(e15 + e31)pj
δj
,
β j =
(c13 + c44)p2j (e15 + e31)−
(
c11 + c44p2j − ρc2 − ρc2k2e2(1 + p2j )
)
(e15 + e33p2j )
δj
,
δj =
[
c11 + c44p2j − ρc2 − ρc2k2e2(1 + p2j )
][
c44 + c33p2j − ρc2 − ρc2k2e2(1 + p2j )
]
− (c13 + c44)2p2j , j = 1, 2, 3.
Then we obtain a system of linear equations in amplitudes A11, A12, A13 only and it is in
the form
(c44β1 + c44α1p1 + e15)A11 + (c44β2 + c44α2p2 + e15)A12 + (c44β3 + c44α3p3 + e15)A13 = 0,
(c13α1 + c33β1p1 + e33p1)A11 + (c13α2 + c33β2p2 + e33p2)A12 + (c13α3 + c33β3p3 + e33p3)A13 = 0,
(e31α1 + e33β1p1 − e33p1)A11 + (e31α2 + e33β2p2 − e33p2)A12 + (e31α3 + e33β3p3 − e33p3)A13 = 0.
(14)
The dispersion equation of Rayleigh waves is obtained from det(CO) = 0 where matrix
CO is the matrix of coefficients of the system of equation above as c44β1 + c44α1p1 + e15 c44β2 + c44α2p2 + e15 c44β3 + c44α3p3 + e15c13α1 + c33β1p1 + e33p1 c13α2 + c33β2p2 + e33p2 c13α3 + c33β3p3 + e33p3
e31α1 + e33β1p1 − e33p1 e31α2 + e33β2p2 − e33p2 e31α3 + e33β3p3 − e33p3
 . (15)
368 Do Xuan Tung
This dispersion equation is in implicit form and it shows the relation between the phase
velocity c and the wave number k of the Rayleigh waves and the parameters of the
medium.
4. NUMERICAL RESULTS AND DISCUSSION
In order to illustrate theoretical results obtained in the preceding sections, the mate-
rial chosen for the numerical calculations is CdSe (6 mm class) of hexagonal symmetry,
which is transversely isotropic material. The physical data for a single crystal of CdSe
material is given below [1, 14]
c11 = 7.41× 1010 Nm−2, c13 = 3.93× 1010 Nm−2, c33 = 8.36× 1010 Nm−2,
c44 = 1.32× 1010 Nm−2, ρ = 5504 kgm−3, e15 = −0.138 Cm−2, e31 = −0.16 Cm−2,
e33 = 0.347 Cm−2, e11 = 8.26× 10−11 C2N−1m−2, e33 = 9.03× 10−11 C2N−1m−2,
e0 = 0.39, a = 0.421× 10−9 m, e = e0a.
(16)
Denote ep = k2e2 the dimensionless frequency where k is the wavenumber. This is an
important parameter that provides us the information of the wave-length of Rayleigh
waves comparing to the nonlocal parameter of the medium.
0 10 20 30 40 50 60 70 80 90
Angle in degree
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
D
im
en
si
on
le
ss
 p
ha
se
 v
el
oc
itie
s c1
c2
Local
Nonlocal
ep=0.8
Fig. 1. Dimensionless velocities of plane waves c¯1 and c¯2 depending
on the angle directions θ for local and nonlocal case
First, we evaluate the effect of nonlocality to the speed of plane waves. Fig. 1 shows
the dimentionless speed of plane waves c/bS, where bS =
√
c44/ρ is the speed of SH-
type wave, depending on the direction of propagation θ (angle between the direction
Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space 369
of propagation and vertical axis) in the piezoelectric medium for two case of local theory
and nonlocal theory with the nonlocal parameter given by ep = 0.8. It can be seen that the
phase velocities c1, c2 in the nonlocal theory case are greater the ones in the local theory
case. It can be concluded that the nonlocality has significant effect on the velocities of
propagation of plane waves.
Next, the variations of the phase velocities with dimensionless parameter ep for θ =
pi/3 are depicted in Fig. 2. Generally, this figure shows that the phase velocities c1, c2 are
decreasing when ep is increasing. When ep < 1 these velocities decrease rapidly decrease
while ep > 1 they are quite stable.
0 1 2 3 4 5 6 7 8 9 10
Dimensionless parameter ep=k2 2
0
0.5
1
1.5
2
2.5
D
im
en
si
on
le
ss
 p
ha
se
 v
el
oc
itie
s
c1
c2
Fig. 2. The comparison of variations of the phase velocities
with dimensionless parameter ep = k2e2 for θ = pi/3
Finally, the dimensionless speed of Rayleigh wave x = X/c44 with X = ρc2 depends
upon the dimensionless parameter ep is illustrated by Fig. 3 for the boundary condition of
open circuit surface (maintained charge free). The speed of Rayleigh wave is decreasing
when the parameter ep is increasing.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ep
0
0.2
0.4
0.6
0.8
1
x
Fig. 3. Effect of parameter ep on the speed of Rayleigh wave x for the open circuit surface
370 Do Xuan Tung
5. CONCLUSIONS
In the present work, we have studied the propagation of Rayleigh waves in trans-
versely isotropic piezoelastic nonlocal materials. Some important features are drawn be-
low:
(i) Under certain type of specific boundary condition: the surface x3 = 0 is consid-
ered to be stress-free and maintained at charge-free condition, the dispersion equation of
the Rayleigh waves is given. It is numerically concluded that the nonlocality has signifi-
cant effect on the speed of Rayleigh wave.
(ii) Phase velocities of plane waves are computed numerically and their variation on
the incident angle θ, dimensionless frequency parameter ep, are presented graphically.
The effect the nonlocality on the velocities of plane waves are also expressed through
numerical example and the effect is also significant.
ACKNOWLEDGMENT
The work was supported by the Vietnam National Foundation for Science and Tech-
nology Development (NAFOSTED) under Grant 107.02-2019.06.
REFERENCES
[1] J. N. Sharma, M. Pal, and D. Chand. Propagation characteristics of Rayleigh waves in
transversely isotropic piezothermoelastic materials. Journal of Sound and Vibration, 284, (1-2),
(2005), pp. 227–248. https://doi.org/10.1016/j.jsv.2004.06.036.
[2] J. Yang. An introduction to the theory of piezoelectricity. Springer, (2005).
[3] L. P. Zinchuk and A. N. Podlipenets. Dispersion equations for Rayleigh waves in a piezoelec-
tric periodically layered structure. Journal of Mathematical Sciences, 103, (3), (2001), pp. 398–
403. https://doi.org/10.1023/A:1011382816558.
[4] A. K. Vashishth and V. Gupta. Wave propagation in transversely isotropic porous piezo-
electric materials. International Journal of Solids and Structures, 46, (20), (2009), pp. 3620–3632.
https://doi.org/10.1016/j.ijsolstr.2009.06.011.
[5] G. Z. Voyiadjis. Handbook of nonlocal continuum mechanics for materials and structures. Springer
Nature Switzerland AG, (2018).
[6] A. C. Eringen and D. G. B. Edelen. On nonlocal elasticity. International Journal of Engineering
Science, 10, (3), (1972), pp. 233–248. https://doi.org/10.1016/0020-7225(72)90039-0.
[7] L.-L. Ke and Y.-S. Wang. Thermoelectric-mechanical vibration of piezoelectric
nanobeams based on the nonlocal theory. Smart Materials and Structures, 21, (2), (2012).
https://doi.org/10.1088/0964-1726/21/2/025018.
[8] L.-L. Ke, Y.-S. Wang, and Z.-D. Wang. Nonlinear vibration of the piezoelectric nanobeams
based on the nonlocal theory. Composite Structures, 94, (6), (2012), pp. 2038–2047.
https://doi.org/10.1016/j.compstruct.2012.01.023.
[9] L.-H. Ma, L.-L. Ke, Y.-Z. Wang, and Y.-S. Wang. Wave propagation analysis of piezoelectric
nanoplates based on the nonlocal theory. International Journal of Structural Stability and Dy-
namics, 18, (04), (2018). https://doi.org/10.1142/S0219455418500608.
[10] A.-L. Chen, D.-J. Yan, Y.-S. Wang, and C. Zhang. Anti-plane transverse waves propagation
in nanoscale periodic layered piezoelectric structures. Ultrasonics, 65, (2016), pp. 154–164.
https://doi.org/10.1016/j.ultras.2015.10.006.
Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space 371
[11] D.-J. Yan, A.-L. Chen, Y.-S. Wang, C. Zhang, and M. Golub. Propagation of guided elastic
waves in nanoscale layered periodic piezoelectric composites. European Journal of Mechanics-
A/Solids, 66, (2017), pp. 158–167. https://doi.org/10.1016/j.euromechsol.2017.07.003.
[12] D.-J. Yan, A.-L. Chen, Y.-S. Wang, C. Zhang, and M. Golub. In-plane elastic wave propaga-
tion in nanoscale periodic layered piezoelectric structures. International Journal of Mechanical
Sciences, 142, (2018), pp. 276–288. https://doi.org/10.1016/j.ijmecsci.2018.04.054.
[13] F.-M. Li and Y.-S. Wang. Study on localization of plane elastic waves in disordered periodic
2–2 piezoelectric composite structures. Journal of Sound and Vibration, 296, (3), (2006), pp. 554–
566. https://doi.org/10.1016/j.jsv.2006.01.057.
[14] J. N. Sharma and V. Walia. Further investigations on Rayleigh waves in piezother-
moelastic materials. Journal of Sound and Vibration, 301, (1-2), (2007), pp. 189–206.
https://doi.org/10.1016/j.jsv.2006.09.018.
[15] A. C. Eringen. On differential equations of nonlocal elasticity and solutions of screw
dislocation and surface waves. Journal of Applied Physics, 54, (9), (1983), pp. 4703–4710.
https://doi.org/10.1063/1.332803.
[16] P. C. Vinh, V. T. N. Anh, D. X. Tung, and N. T. Kieu. Homogenization of very rough interfaces
for the micropolar elasticity theory. Applied Mathematical Modelling, 54, (2018), pp. 467–482.
https://doi.org/10.1016/j.apm.2017.09.039.
APPENDIX
The coefficients of characteristic equation
h4 = −b9b23 + 2b3b4b7 − b5b24 − b2b27 − 2b1b8b7 + b1b5b10 + b1b6b9 + b2b5b9,
h2 = −b23b10 + 2b3b4b8 − b6b24 − b1b28 − 2b2b7b8 + b1b6b10 + b2b5b10 + b2b6b9,
h0 = b2b6b10 − b2b28, h6 = b1b5b9 − b1b27, b1 = c44 − Xk2e2, b2 = c11 − X− Xk2e2,
b3 = c13 + c44, b4 = e15 + e31, b5 = c33 − Xk2e2, b6 = c44 − X− Xk2e2,
b7 = e33, b8 = e15, b9 = −e33, b10 = −e11,X = ρc2.