World Journal Of Engineering
Construction of Surface Piezoelectricity Using State-Space Formalism
Weiqiu Chen
Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, P. R. China
u
 u
   A ,
x3 σ 3 
σ 3 
With the development of modern technology,
the size of structures or components gets smaller
where u  [u1 , u2 , u3 ,  ]
T
and smaller. Many micro- and even nano-sized
(1)
is the generalized
multifunctional structures and devices have been
displacement vector, σ i  [ 1i ,  2i ,  3i , Di ]
widely used in science and engineering, and are
are the generalized stress vectors, and A is an
known as the micro-electro-mechanical systems
operator
(MEMS) and nano-electro-mechanical systems
differential operators
T
matrix.
By
treating
the
partial
 / x1 ,  / x2 and
(NEMS). In a usual macro-structure, there is no
 / t as parameters, we can eventually obtain
need to consider the material heterogeneity
the
regarding the surface and bulk regions, which can
piezoelectricity of an arbitrary order.
governing
equations
of
surface
be attributed to the different environmental
Shown in Fig. 2 are the frequency spectra
constraints imposed on the atoms at different
for SH wave in a piezoelectric plate with and
without surface effects, where r 
positions. In MEMS or NEMS, the aspect
 s /  and
s
rc  c66
/ c44 are the density ratio and stiffness
surface-to-volume ratio is much larger than that
of a macro-structure, and the surface may play an
ratio between the surface layer and bulk material.
important role in determining the mechanical
We observe that the presence of surface layers
behavior of these tiny systems.
can either lower or raise the frequency, depending
In this paper, we will present a new method
on the magnitude of wave number. Specifically,
to formulate the theory of surface piezoelectricity
when the wave number is small, the frequency is
that can be truncated up to an arbitrary order for
lowered, while it is raised after the wave number
the plane surface of a 3D piezoelectric body. To
arrives at a certain critical value, where the
this end, the surface is modeled as a thin
frequency is unchanged whether the surface
piezoelectric layer (see Fig. 1) that may be
layers are taken into account or not. These critical
endowed with different material properties than
wave
the bulk material.
theoretically.
numbers
can
be
easily
determined
Figure 3 depicts the phase velocity spectrum
of the GB type wave in a piezoelectric half-space,
h
Surface layer
where the dimensionless phase velocity V and
x1
Bulk material
wave number  are defined respectively as
V  v / v0
x3
and
  kx H
,
with
v0  c44 /  and H  1 m. As we can see
Fig. 1. Thin surface layer of a body
The
state
space
formalism
for
that, when there is no surface effect, the BG wave
propagates at a constant velocity, and is hence
bulk
non-dispersive. If the surface effect is involved,
piezoelectric materials can be written as
197
World Journal Of Engineering
then the wave is slightly dispersive, and becomes
size. The method suggested in our work can
slower when the wave number increases.
provide us a very straightforward way to establish
The results shown above indicate the surface
the surface theory for media with coupling
effect may have an important influence on the
between different physical fields.
wave propagation behavior in bodies of small

10
Antisymmetric modes
8
6
4
2
Symmetric modes

0
0
1
2
3
4
Fig. 2. SH wave frequency spectrum of a PZT-4 plate with stiff surface layer ( r
5
 5.0 , rc  10.0 ), solid line:
h / H  0 , dotted line: h / H  0.01.
V
1.41
1.405
1.4
1.395
1.39
1.385

1.38
0
1
2
3
4
5
Fig. 3. Phase velocity versus wave number. Black solid line: without surface effect; Blue dotted line:
with surface effect ( r1
198
 2 , r2  20 ).