Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core
Within the scope of the piecewise homogeneous body model with utilizing of the three dimensional linearized theory of electro-elastic waves in initially stressed bodies, a mathematical modeling the dynamical stress field problem occurred in a sandwich platestrip is carried out. This plate consists...
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| Опубліковано в: : | Прикладная механика |
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Інститут механіки ім. С.П. Тимошенка НАН України
2018
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core / A. Daşdemir // Прикладная механика. — 2018. — Т. 54, № 4. — С. 125-144. — Бібліогр.: 18 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859874530589147136 |
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| author | Daşdemir, A. |
| author_facet | Daşdemir, A. |
| citation_txt | Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core / A. Daşdemir // Прикладная механика. — 2018. — Т. 54, № 4. — С. 125-144. — Бібліогр.: 18 назв. — англ. |
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| container_title | Прикладная механика |
| description | Within the scope of the piecewise homogeneous body model with utilizing of the three dimensional linearized theory of electro-elastic waves in initially stressed bodies, a mathematical modeling the dynamical stress field problem occurred in a sandwich platestrip is carried out. This plate consists of a piezoelectric core perfectly bonded to the elastic layers with initial stress under the action of a time-harmonic force resting on a rigid foundation. It is assumed that the piezoelectric material is poled perpendicular to free surface of the body. A governing system of the partial differential equations of motion is solved by employing the finite element method. The numerical results are presented illustrating the effect of certain dependencies of the problem on the propagations of the stresses and the electric displacements acting on the interface planes between the elastic layers and the piezoelectric core and between the plate-strip and the rigid foundation. In particular, an effect of the initial stress parameter and a change of the thickness of the piezoelectric core on the frequency response of the plate-strip is investigated.
В рамках моделі кусково-однорідного тіла з використанням тривимірної теорії електропружних хвиль в початково напружених тілах проведено математичне моделювання задачі про поле динамічних напружень, що виникає в шаруватій плиті-полосі. Ця плита складається з п’єзоелектричного ядра, яке граничить з пружним шарами з початковими напруженнями, і перебуває на абсолютно твердій основі. Приймається, що п’єзопружний матеріал поляризований перпендикулярно до вільної поверхні пластини. Основна система диференціальних рівнянь з частинними похідними розв’язана методом скінченних елементів. Представлено числові результати, які ілюструють вплив певних залежностей задачі про поширення напружень і електричних зміщень, які діють на площинах розділу пружних шарів і п’єзоелектричного ядра та плитою і абсолютно твердою основою. Зокрема, вивчено вплив параметра початкового напруження і зміни товщини п’єзопружного ядра на частотні характеристики плити-стержня.
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2018 П Р И К Л А Д Н А Я М Е Х А Н И К А Том 54, № 4
ISSN0032–8243. Прикл. механика, 2018, 54, № 4 125
A . D a ş d e m i r
FORCED VIBRATIONS OF PRE-STRESSED SANDWICH PLATE-STRIP
WITH ELASTIC LAYERS AND PIEZOELECTRIC CORE
Department of Mathematics, Faculty of Arts and Sciences, Kastamonu University,
Kastamonu, Turkey; e-mail: ahmetdasdemir37@gmail.com
Abstract. Within the scope of the piecewise homogeneous body model with utilizing of
the three dimensional linearized theory of electro-elastic waves in initially stressed bodies, a
mathematical modeling the dynamical stress field problem occurred in a sandwich plate-
strip is carried out. This plate consists of a piezoelectric core perfectly bonded to the elastic
layers with initial stress under the action of a time-harmonic force resting on a rigid founda-
tion. It is assumed that the piezoelectric material is poled perpendicular to free surface of the
body. A governing system of the partial differential equations of motion is solved by em-
ploying the finite element method. The numerical results are presented illustrating the effect
of certain dependencies of the problem on the propagations of the stresses and the electric
displacements acting on the interface planes between the elastic layers and the piezoelectric
core and between the plate-strip and the rigid foundation. In particular, an effect of the ini-
tial stress parameter and a change of the thickness of the piezoelectric core on the frequency
response of the plate-strip is investigated.
Key words: piezoelectric core, finite element method, time-harmonic force, sandwich
plate-strip.
1. Introduction.
Sandwich structures are a kind of composite materials consisting of three layers, i.e. a
core layer is bonded to two layers. The core material may be selected more flexible or hard-
er than the others. The use of composite sandwich structures in many engineering applica-
tions such as building construction, aerospace, nuclear and civil infrastructure has been in-
creasing especially due to their features such as good energy, sound absorption ability and
high flexural. Hence, the subject has been under the density study. The monographs [1] pre-
sent detailed investigations on the subject.
Many researchers investigate many different problems based on various theories. Nayak
et al. investigate the free vibration analysis of composite sandwich plates by using the fami-
ly of new Reddy type elements based on Reddy’s higher-order theory [2]. Chakraborti and
Seikh study the buckling of sandwich plates with laminated stiff layers subjected to partial
edge compression based on a refined higher order shear deformation theory [3]. Hazard and
Bouillard consider the passive damping of structural vibrations by the use of viscoelastic
layers [4]. Pandit et al. develop an improved plate model to study the buckling of laminated
sandwich plate with transverly flexible core [5]. To analyze the static behavior of isotropic,
sandwich and laminated plates, Xiang et al. present a new shear deformation theory [6]. Li
et al. investigate the transient response of an orthotropic composite sandwich plate subjected
to point-wise impulse loading with respect to a nonlinear high order core theory [7]. Tu et
al. carve out a simple 0C isoparametric finite element formulation based on the refined
higher-order shear deformation theory for the bending and free vibration analysis of compo-
site and sandwich laminate plates [8]. Hasheminejad and Gheshlaghi consider a three-
126
dimensional elasticity-based formulation for the dynamic response of an arbitrarily thick
simply supported FGM rectangular plate under transient loads of arbitrary spatial and tem-
poral variations resting on a Winkler – Pasternak elastic foundation [9]. Akbarov and
Yahnioglu study a buckling delamination problem for a sandwich plate-strip with the piezo-
electric face and elastic core layers which has two interface inner cracks by employing the
finite element method [10]. Loja et al. develop the static and dynamic behavior of function-
ally graded sandwich structures with piezoelectric skins by B-spline finite strip models [11].
Shahraeeni et al. analyze analytically free and forced vibration of piezoelectric laminated
plates coupled with rectangular acoustic cavities based on Kirchhoff plate model [12].
Sanker et al. present the dynamic instability behavior of sandwich panels with carbon nano-
tube (CNT) reinforced facesheets under the periodic load [13].
As far to the author’s knowledge the forced vibration of a pre-stressed sandwich plate-
strip with a piezoelectric core perfectly bonded to two elastic layers subjected to a time har-
monic force resting on a rigid foundation has not been studied so far. To address the issue,
the mathematical modeling of the problem under consideration is analytically solved by
employing the finite element method. Most significant contributions of this work include the
investigation of the effect of the different problem dependences on the frequency response
of the considered sandwich plate-strip.
2. Statement of Problem.
Consider a sandwich plate-strip with length 2a and thickness h under the action of a
time harmonic force resting on a rigid foundation as depicted in Fig. 1. It is consists of a
piezoelectric core with the thickness 2h bonded perfectly a two isotropic layer with the
thickness 1h and 3h respectively. The poling direction of the piezoelectric material is as-
sumed along the direction of the 2Ox axis.
Fig. 1. Geometry of sandwich plate-strip with piezoelectric core.
The locations of the positions of the layers are determined by the Lagrangian coordi-
nates, which in the natural state coincide with Cartesian coordinates. The values related to
the upper, core and lower layers of plate-strip are denoted by the superscripts (1), (2) and
(3), respectively, and the additional upper index 0 to the initial state. It is assumed that the
length of plate-strip in the direction of the 3Ox axis is infinite. However, since the force
applied to the considered body extends to infinity in the direction of the 3Ox axis, the plane
deformation state appears on the 1 2Ox x plane. All investigations are therefore made in the
1 2Ox x plane. The plate-strip under consideration occupies the domain 1 2 3B B B B ,
where
127
1 1 2 1 1 2, : , 0B x x a x a h x ; (1)
2 1 2 1 2 2 1, : ,B x x a x a h x h ; (2)
3 1 2 1 2 2, : ,B x x a x a h x h . (3)
The general forms of the differential equations of motion for the small initial defor-
mation can be expressed as follows [14 – 15]:
0,
, ,
,
m m m m m
ij j kj i k i
j
u u ; (4)
0,
, ,
,
0m m m
i i j i j
i
D D u , (5)
where , , 1, 2i j k , 1, 2,3m , m is the corresponding mass density in the natural state,
m
iu and m
iD denote the mechanical and electrical displacements with respect to corre-
sponding coordinate ix , m
ij is the stress tensor, 0, m
kj initial stress tensor and 0, m
jD
initial electrical displacement. The dot over the displacement components is time differen-
tiation and the comma followed by the subscript i indicates the space-coordinate differenti-
ation. Here and below, the repeated index in the subscript is summed with respect to that
index. It should be noted that the equation in (4) is written for both the elastic phase and the
piezoelectric phase. But the equation in (5) vanishes for the elastic phase since it concerns
with the electrical displacements.
Before compounding each layer with one another and with rigid foundation, the layers
are stretched or compressed separately in the direction 1Ox by uniformly distributed normal
forces. Hence, uniaxial homogenous initial stress state occurs in the plate-strip. This initial
stress is determined by utilizing the linear theory of the electro-elasticity as
0, 0,
11 , 0 for all 11m m m
ijq ij , (6)
where mq is the known constant for each layer. According to all the foregoing assumptions
and due to the character of piezoelectric material, an axial homogenous initial electric dis-
placement emerges in the piezoelectric layer as
0, 2 0
1 1D D and 0, 2 0
2 2 0D D . (7)
Note that initial stress and initial electric displacement cannot be the independent from each
other. They must be self-consistent. This statement will be explained later.
The mechanical and geometrical relations for an isotropic elastic material and a piezoe-
lectric material can respectively be written as
2ij ij ij ; (8)
,ij ijkl kl kij kc e , ,i ikl kl ik kD e , (9)
where 1, 3 , m and m are the Lamé constants for the elastic phase, ijklc , kije and ik
are mechanic, piezoelectric and dielectric constants, respectively for the piezoelectric phase,
ij is the Kronecker delta, is the electric potential, m
ij is the strain tensor and defined
such that , ,
1
2
m m m
ij i j j iu u . Note that, since the piezoelectric material is only in the
intermediate layer, the superscript is omitted.
128
Now the boundary and contact conditions are considered.
According to the foregoing discussions, on the free surface and on the ends of the plate-
strip the following boundary conditions are described as
2 2
1 1
12 22 1
0 0
0, ;i t
o
x x
p x e
(10)
1
j,1 1 0;m m m
j
x a
q u
(11)
1
2 2 1
0
1 1 1,1 0x a
h x h
D D u
. (12)
The surfaces of the piezoelectric layer are assumed to be mechanically free and electri-
cally open. The boundary conditions can therefore be written as
1
2 2 1
0x a
h x h
and
2 1 2
1
, 0x h h
a x a
. (13)
It is assumed that, on the interface planes between the layers of plate strip and between
the plate-strip and rigid foundation, there exists the completely clamping state. In this case,
the complete clamped conditions occur as
2 1 2 1 2 2 2 2
1 2 2 3
2 2 2 2; ;i i i i
x h x h x h x h
(14)
2 1 2 1 2 2 2 2
1 2 2 3; ;i i i i
x h x h x h x h
u u u u
(15)
2
3 0j
x h
u
. (16)
With the above-mentioned, the formulation of the problem and the investigation of the
governing field equations are thus exhausted.
3. FEM Modeling.
An analytical solution of the considered problem cannot be obtained; therefore, to solve
this problem, the finite element method (FEM) is employed. Before starting solution, certain
preparations are made. First of all, the dimensionless coordinate system is introduced as
1 2
1 2ˆ ˆ;
x x
x x
h h
. (17)
In addition, recall the lineal load applied to the plate-strip is assumed to be time-harmonic,
with frequency , as 1
i t
op x e . Thus, all the corresponding dependent variables can be
written in the form
1 2 1 2, , , , , , , , ,
m m i t
ij i ij i ij i ij iu D x x t u D x x e , (18)
where the superposed bar represents the amplitude of the corresponding quantities. After the
coordinate transformation in (17) and substituting the expression in (18) into the foregoing
equations and conditions, the same equations and boundary-contact conditions are obtained
for the amplitude of the sought values by replacing the terms 2 2/m
ju t and 1
i t
op x e
with 2 m
ju and 1op x , respectively.
To obtain FEM modeling of the last boundary-contact problem, the following functional
is presented:
3 2 2
2 2
, 1 2
1
1
2
m m m m m m
ij j i
m B
J u T u h u u dB
129
2 2
/
11
, 2 2 11
/ 0
1
2
a h
o
i i
B a h x
p x
S dB u dx
, (19)
where
0,
, ,k ,
0
, , ,
;
.
m m m m
ij ij in j n ijkn n ijk k
i i n i n kni n k ik k
T u w u t
S D D u t u r
.(20)
According to the constitutive equation in (9), the notations in (20) can clearly be written
for the piezoelectric phase as
1111 11 1122 13 1212 44 1221 44
2112 44 2121 44 2211 13 2222 33
0 0
111 1 121 15 211 15 122 1
222 33 112 31 11 11 22 33
; ; ; ;
; ; ; ;
; ; ; ;
; ; ; .
w c q w c w c w c q
w c w c w c w c
t D t e t e t D
t e t e r r
(21)
It is should be considered that, while m
ijT is computed, 1iit is equal to zero. Note that
the notations in (21) coincide with those for the elastic layers and
11 33 2c c ,
13c ,
44c , q q , 31 15 33 0e e e and 11 33 0 . In addition, all the
components which do not insert into the relations in (21) are equal to zero. It is easily to
confirm that
ijkn nkjiw w , 111 122t t and ijk jikt t . (22)
To show the validity of the functional presented in (19), Gauss’s theorem is used. Tak-
ing the symmetry conditions in (22) into account,
3 2
2 2
, , , , 1 2
1
1
2
m m m m m m m m
ijkn n k j i ijk j i k
m B
J u w u u t u h u u dB
2 2
/
11
, , , , 2 2 11
/ 0
1
2
a h
o
kni n k i ik k i
B a h x
p x
t u r dB u dx
3
2 2
, , , ,
1
1
2
2
m m m m m m m
ijnk knji n k j i ijk jik j i k j j
m B
w w u u t t u h u u dB
2 2
/
11
, , , , 2 2 11
/ 0
a h
o
kni n k i ik i k
B a h x
p x
t u r dB u dx
(23)
3
2 2
, , ,
1
m m m m m m
ijnk n k j i ijk j i k j j
m B
w u u t u h u u dB
2 2
/
11
, , , 2 2 11
/ 0
a h
o
kni n k i ik i k
B a h x
p x
t u r dB u dx
2
/3
11
, , 2 11
1 / 0
a h
m m o
ijnk n k ijk k i j
m S a h x
p x
w u t n u dS u dx
130
3
2 2
, , ,,
1
m m m m m
ijnk n k ijk k j jii
m B
w u t h u u dB
2 2
, , 2 , 2,,
,kni n k ik i i kni n k ik i ki
B B
t u r n dS t u r dB
in which 2S denotes the boundary of the domain 2B and in is the components of n , the
outward unit vector normal to all the boundaries in the domain B .
Considering the statement
0J mu , (24)
the equations of motion and the corresponding boundary-contact conditions given in (4)-(5)
and (10) – (16) are obtained. So, the validity of the functional (19) suggested for the corre-
sponding mathematical modeling is proven.
For the functional given in (19), the FEM modeling of the considered problem is created
by the virtual work principle and the standard Rayleigh-Ritz method [17]. According to this
approach, the domain B is divided into a finite element of sub-domains whose structures
are four-node smooth rectangular elements. It is immediately specified that the number of
the finite elements is determined from the desired numerical convergence requirement. Ac-
cording to the well-known procedure, after fairly extensive mathematical arrangements, a
system of algebraic equations is obtained as
2M K - u F , (25)
where K is the stiffness matrix, M is the mass matrix, u is the column vector of un-
known nodal displacements, and F is the force vector. To reduce the size of the present
paper the explicit forms of the above-stated matrices and vectors are not given here. Howev-
er, their explicit forms are directly derived from the equations (19) and (23) by using the
considered procedure. So, with the above-stated the FEM modeling of the problem is ex-
hausted.
4. Numerical Results.
Getting started with this section, certain remarks are made to more easily understand the
numerical investigations. Certain notations are first introduced as
3
1
E
e
E
,
h
and
q
(26)
for the elastic layers and
44
h
c
and
44
q
c
(27)
for the piezoelectric core. In the equations (26) and (27), E is the Young modulus for the
elastic layer, m is the dimensionless frequency parameter, and m is the initial stress
parameter.
Throughout the paper, all the numerical investigations are made at the interface planes
between the upper elastic layer and the piezoelectric core (named “upper interface”), be-
tween the piezoelectric core and the lower elastic layer (named “lower interface” and be-
tween the plate-strip and rigid foundation (named “bottom surface”). Note that, when the
corresponding parameters are equal, the superscript “ m ” will be omitted. It is assumed
that, for all considered examples, the two elastic layers have the same thickness ( 1 3h h ),
131
and the material 3BaTiO is selected for the piezoelectric core. The following cases are also
considered under / 2 0, 2h a , 2 / 0,5h h , 0 and 0 unless specified otherwise:
Case I: Aluminum (Upper layer) + 3BaTiO (Core layer) + Steel (Lower layer)
Case II: 1 3 1E E , 1 3 0,33 .
The electromechanical properties of materials considered are listed in Table. Changing val-
ues in any case under consideration will be specifically indicated.
Table. Values of material properties
Property
Elastic layer Core Layer
Aluminum Steel 3BaTiO
E (GPa) 70 210 -
0,35 0,29 -
11c (GPa) - - 150
13c (GPa) - - 66
33c (GPa) - - 146
44c (GPa) - - 44
31e (cm-2) - - -4,35
33e (cm-2) - - 17,5
15e (cm-2) - - 11,4
11 (nFm-1) - - 11,15
33 (nFm-1) - - 12,6
(kgm-3) 2712 7850 7280
As previously mentioned, the self-consistent condition of the initial stress (stretching or
compressing) field for the piezoelectric phase should be satisfied. In this study, a case such
that only a uniformly distributed normal initial stress field is considered. Then considering
the constitutive equations, the equation 20 2
1 13 33 33 31 13 11 33/D q c e c e c c c can directly
be derived for the piezoelectric phase. The values of the initial stress parameter m are
chosen at asymmetric interval – 0,07 to 0,07.
Before starting to present numerical results, the validity of the numerical algorithms and
programs composed by the author under consideration must be verified. To do this, certain
special cases are considered, and it will be shown that the obtained numerical results con-
verge to the previous ones and the findings agree well with the foregoing discussion.
In Fig. 2, the distribution of the normal stress 22 0/h p with respect to the line 1 /x h on
the bottom surface for Case II and 2h h is displayed. This figure gives the opportunity
to compare the numerical results obtained by using the present FEM algorithm with the ones
given by Uflyand [18] for the plate with infinite length. Note that the starred graph in Fig. 2
means the one in [18]. The geometry of the considered body begins to resemble a semi-
infinite structure as / 2 0h a . In this case, the present numerical results must converge to
the corresponding ones obtained in Ref. [18] as / 2 0h a . This statement is demonstrated
by the graphs in Fig. 2 plotted on the bottom surface.
132
Fig. 2. The variation of 22 0/h p versus the line 1 /x h for different thickness ratios
under Case II on the bottom surface.
Fig. 3, a. The variation of 22 0/h p versus the line 1 /x h for different piezoelectric thickness
ratios under Case I at the upper interface.
Fig. 3, b The variation of 22 0/h p versus the line 1 /x h for different piezoelectric thickness ratios
under Case I on the bottom surface.
133
Fig. 3 shows the distribution of the normal stress 22 0/h p with respect to the line
1 /x h for various thicknesses of the piezoelectric core under Case I at the upper interface
and on the bottom surface. Note that the notation means the corresponding one in
Ref. [16], and the solid (dashed) line graphs are plotted for 0,1 1,0 . The geome-
try of the plate-strip begins to resemble that in Ref [16] as 2 0h . It can easily be observed
from the graphs in Fig. 3 that the numerical results converge to the corresponding ones giv-
en in Ref. (16) for both interface and bottom surface under the condition 2 0h .
Now, the effect of the ratio of the Young modulus of the elastic layers on 22 0/h p and
2 0/D h p along the line 1 /x h is given in Fig. 4 under Case II, 2 / 0,75h h and 0,3 .
According to the well-known mechanical consideration, an increase in the values of the ra-
tio e must give rise to a decrease in the absolute values of 22 0/h p for the multi-layered
body. Indeed, this prediction is clearly observed from the graphs in Fig. 4. In addition, it can
be said that there exist certain locations where the values of 22 0/h p and 2 0/D h p are in-
dependent of the ratio e .
Fig. 4, a. The variation of 22 0/h p versus the line 1 /x h for various e under Case II and
0,3 at the upper interface.
Fig. 4, b The variation of 22 0/h p versus the line 1 /x h for various e under Case II
and 0,3 at the lower interface.
134
Fig. 4, c .The variation of 2 0/D h p versus the line 1 /x h for various e under Case II
and 0,3 at the lower interface.
Fig. 4, d. The variation of 22 0/h p versus the line 1 /x h for various e under Case II
and 0,3 on the bottom surface.
The numerical results obtained from Figs. 2 and 3 show that, under certain special cas-
es, the distributions of the stress 22 0/h p converge to the previous ones. Besides, the find-
ings in Fig. 4 agree well with the well-known foregoing mechanical considerations. In this
way, the validity and trustiness of the algorithm and programs has been shown.
In Fig. 5, the effect of the initial stretching (compressing) on the distribution of
22 0/h p and 2 0/D h p with respect to the line 1 /x h under Case I, 2 / 0,75h h and
0,3 at the upper and lower interfaces and on the bottom surface is displayed. It can be
shown from the graphs in Fig. 5 that, under * *
1 1 1/ / /x h x h x h the absolute values of
22 0/h p and 2 0/D h p decrease (increase) with increasing the initial stretching (compress-
ing) parameter. But under ** *
1 1 1/ / /x h x h x h and * **
1 1 1/ / /x h x h x h , the abso-
lute values of 22 0/h p and 2 0/D h p decrease with (increasing) the initial stretching (com-
pressing) parameter. Note that the values of *
1 /x h and **
1 /x h can be observed in Fig. 5
and are different at the upper and lower interfaces and on the bottom surface. As can be seen
from Fig. 5, the effect of the initial stretching parameter applied to the body on 22 0/h p
and 2 0/D h p is the opposite of that for the initial compressing parameter.
135
Fig. 5, a. The variation of 22 0/h p versus the line 1 /x h for various under Case I and
0,3 at the upper interface.
Fig. 5, b. The variation of 2 0/D h p versus the line 1 /x h for various under Case I
and 0,3 at the lower interface.
Fig. 5, c. The variation of 22 0/h p versus the line 1 /x h for various under Case I
and 0,3 at the lower interface.
136
Fig. 5, d. The variation of 22 0/h p versus the line 1 /x h for various under Case I
and 0,3 on the bottom surface.
To observe an influence of a change in values of the thickness of the piezoelectric core
on this effect, Fig. 6 is now considered, which displays the variations of 22 0/h p and
2 0/D h p with respect to the initial stress parameter at the points 10, /h h , 20, /h h
and 0, 1 under the same assumptions in Fig. 5. Both the stress 22 0/h p and the electri-
cal displacement 2 0/D h p depend linearly on the initial stress parameter. It can be shown
that the slope of this linear dependence increases towards to the bottom surface. The type of
material used in each layer causes to this result. By the same token, the slope of each graph
decrease with increasing the thickness of the core layer. It can be said from this result that,
when the thickness of the piezoelectric core decreases, the effect of both the initial stretch-
ing and initial compressing on 22 and 2D increases.
Fig. 6, a. The dependencies between 22 0/h p and for various 2 / 2h a under Case I
and 0,3 at the point 10, /h h .
137
Fig. 6, b. The dependencies between 22 0/h p and for various 2 / 2h a under Case I and
0,3 at the point 20, /h h .
Fig. 6, c. The dependencies between 2 0/D h p and for various 2 / 2h a under Case I and
0,3 at the point 20, /h h .
Fig. 6, d. The dependencies between 22 0/h p and for various 2 / 2h a under Case I
and 0,3 at the point 0, 1 .
138
The numerical results obtained have been presented for the constant values of until
now. But the one of the main aims of the present paper is to investigate certain problem pa-
rameters on the frequency response of the stress 22 0/h p and the electrical displacement
2 0/D h p . Thereafter, the influence of the relevant parameters on the frequency response of
22 0/h p and 2 0/D h p will be analyzed at the points 10, /h h , 20, /h h and 0, 1
under Case I unless otherwise.
Fig. 7 presents the variation of the stress 22 0/h p and the electrical displacement
2 0/D h p with the dimensionless frequency for various thickness 2h . As can be seen
from the graphs in this figure, the absolute values of 22 0/h p and 2 0/D h p decrease with
the dimensionless frequency . Also, while the absolute values of 22 0/h p decrease with
the thickness 2h at 10, /h h , the absolute values of 22 0/h p and 2 0/D h p decrease with
increasing the thickness 2h at the other points.
Fig. 7, a. The dependencies between 22 0/h p and for various 2 / 2h a under Case I
at the point 10, /h h .
Fig. 7, b. The dependencies between 22 0/h p and for various 2 / 2h a under Case I
at the point 20, /h h .
139
This difference in Fig. 7, a is a natural result of its proximity to free surface of the in-
vestigated point. It follows from the investigations of the graphs that there exist locations
where 22 0/h p and 2 0/D h p reach the extrema for the certain values of . These values
are called as its “resonance” values and denoted
*
m . The graphs show that the resonance
values of decrease with increasing the thickness 2h . Moreover, the numerical results
prove that the “resonance” values of depend on the selected material and on the material
type. For instance, the resonance values of for elastic layers are significantly smaller
than that for the piezoelectric core. Clearly, 3
* * *
BaTiOAl St . An increase in the
values of the thickness 2h causes to increase the number of the local extremums of 22 with
respect to . The dependences between 22 0/h p and and between 2 0/D h p and
are non-monotonic. This result agrees well with the previous mechanical consideration. In
addition, for 1,2 , the oscillating characters of 22 0/h p and 2 0/D h p become more
sensitive for the present system. It follows from the foregoing investigations that the ob-
tained results coincide with the ones in Ref. [16]. Thus, the validity and trustiness of the PC
program composed by the author is confirmed again.
Fig. 7, c. The dependencies between 2 0/D h p and for various 2 / 2h a under Case I
at the point 20, /h h .
Fig. 7, d. The dependencies between 22 0/h p and for various 2 / 2h a under Case I
at the point 0, 1 .
140
It can be shown from the above discussions that the oscillating character of the distribu-
tion of 22 0/h p
is to be similar to that for 2 0/D h p
(in the qualitative sense). Thereafter, to
reduce the size of the paper, so all the numerical investigations are only made at the points
20, /h h and 0, 1 for the normal stress 22 0/h p .
The numerical results given in Fig. 8 allow to obtain certain conclusions on the influ-
ence of a changes of the ratio / 2h a on the frequency response of the sandwich plate- strip.
According to the graphs in Fig. 8, an increase in the values of / 2h a causes to decrease of
the numbers of the local maximums and minimums of 22 0/h p
with respect to the dimen-
sionless frequency . The values of
*
m decrease with increasing the ratio / 2h a . While
the values of the ratio / 2h a decrease, the oscillating character of the distribution of
22 0/h p becomes more sensitive.
Fig. 8, a. The dependencies between 22 0/h p and for various / 2h a under Case I
at the point 20, /h h .
Fig. 8, b. The dependencies between 22 0/h p and for various / 2h a under Case I
at the point 0, 1 .
141
To investigate the effect of the initial stretching and compressing parameters on the dis-
tribution of 22 0/h p with respect to the dimensionless frequency , Fig. 9 is now consid-
ered. It follows from the graphs in Fig. 9 that the influence of the initial stretching and com-
pressing parameters on the frequency response of the stress 22 0/h p is significantly not
only in the quantitative sense but also in the qualitative sense. Moreover, it can be shown
that there exist certain locations where the parametric resonance of 22 0/h p occurs for
some values of the parameter . An increase in the values of the initial stretching (com-
pressing) parameter causes to increase (decrease) the values of
*
m . Under the medium
without the initial stress, i.e. 0 , the resonance of the current system occur on the point
where 1,2 . The initial stretching prevents the resonance value of 22 0/h p , but the ini-
tial compressing exceeds the corresponding resonance value.
Fig. 9, a. The dependencies between 22 0/h p and for various under Case I
at the point 20, /h h .
Fig. 9, б. The dependencies between 22 0/h p and for various under Case I
at the point 20, /h h .
142
Fig. 10, a. The dependencies between 22 0/h p and for various e under Case I
at the point 20, /h h .
Fig. 10, b. The dependencies between 22 0/h p and for various e under Case I
at the point 0, 1 .
Fig. 10 displays the effect of the parameter e on the variation of 22 0/h p with respect
to the dimensionless frequency under Case II. The numerical results observed in Fig. 10
indicate that the values of
*
m decrease with the parameter e . An increase in the values
of the parameter e causes to decrease the numbers of the local maximums and minimums of
22 0/h p
with respect to . Moreover, when the values of decrease, the effect of the
parameter e on the variation of 22 0/h p damps rapidly. The resonance values at the lower
interface are less than those on the bottom surface.
5. Conclusion.
In the present paper, the forced vibration problem for a pre-stressed sandwich plate-strip
with a piezoelectric core perfectly bonded to two elastic layers under the action of a time-
harmonic force on the rigid foundation is investigated within the scope of the exact equa-
tions of electro-elasticity in framework of the piecewise homogeneous body model. The
143
considered problem is numerically solved by employing the finite element method. The ef-
fect of various problem parameters on the frequency response of the sandwich plate-strip is
examined. According to all the numerical investigations, certain inferences of the important
results can be drawn as follows:
the effect of the initial stretching on the corresponding stress and electrical displacement
components is opposite of that of the initial compressing;
an increase in the thickness of the piezoelectric core causes to increase the number of
the local extremums of 22 with respect to ;
the resonance values of the piezoelectric material are smaller than those of the elastic
materials;
by increasing the length of the plate-strip for fixed thickness of that, or by decreasing
the thickness of the plate for fixed length of that the numbers of the local extremums of 22
decrease;
by decreasing the ratio of Young modulus of the elastic layers, the resonance values of
the current system decrease;
while the initial stretching prevents the resonance value of the external force, the initial
compressing exceeds this resonance value.
The numerical results listed above have been presented under two different cases, but
note that they also have a general validity in a qualitative sense. Moreover, these numerical
results are encountered daily in the engineering practice under an impact treatment of metals
which lie on the others.
6. Acknowledgements.
The author is a member of the research project supported by Research Fund of
Kastamonu University under project number KÜ-BAP01/2015-3.
Р Е З Ю М Е . В рамках моделі кусково-однорідного тіла з використанням триви-
мірної теорії електропружних хвиль в початково напружених тілах проведено матема-
тичне моделювання задачі про поле динамічних напружень, що виникає в шаруватій
плиті-полосі. Ця плита складається з п’єзоелектричного ядра, яке граничить з пруж-
ним шарами з початковими напруженнями, і перебуває на абсолютно твердій основі.
Приймається, що п’єзопружний матеріал поляризований перпендикулярно до вільної
поверхні пластини. Основна система диференціальних рівнянь з частинними похід-
ними розв’язана методом скінченних елементів. Представлено числові результати, які
ілюструють вплив певних залежностей задачі про поширення напружень і електрич-
них зміщень, які діють на площинах розділу пружних шарів і п’єзоелектричного ядра
та плитою і абсолютно твердою основою. Зокрема, вивчено вплив параметра початко-
вого напруження і зміни товщини п’єзопружного ядра на частотні характеристики
плити-стержня.
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From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 15.12.2016 Утверждена в печать 30.01.2018
|
| id | nasplib_isofts_kiev_ua-123456789-174202 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0032-8243 |
| language | English |
| last_indexed | 2025-12-07T15:51:05Z |
| publishDate | 2018 |
| publisher | Інститут механіки ім. С.П. Тимошенка НАН України |
| record_format | dspace |
| spelling | Daşdemir, A. 2021-01-07T19:42:21Z 2021-01-07T19:42:21Z 2018 Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core / A. Daşdemir // Прикладная механика. — 2018. — Т. 54, № 4. — С. 125-144. — Бібліогр.: 18 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/174202 Within the scope of the piecewise homogeneous body model with utilizing of the three dimensional linearized theory of electro-elastic waves in initially stressed bodies, a mathematical modeling the dynamical stress field problem occurred in a sandwich platestrip is carried out. This plate consists of a piezoelectric core perfectly bonded to the elastic layers with initial stress under the action of a time-harmonic force resting on a rigid foundation. It is assumed that the piezoelectric material is poled perpendicular to free surface of the body. A governing system of the partial differential equations of motion is solved by employing the finite element method. The numerical results are presented illustrating the effect of certain dependencies of the problem on the propagations of the stresses and the electric displacements acting on the interface planes between the elastic layers and the piezoelectric core and between the plate-strip and the rigid foundation. In particular, an effect of the initial stress parameter and a change of the thickness of the piezoelectric core on the frequency response of the plate-strip is investigated. В рамках моделі кусково-однорідного тіла з використанням тривимірної теорії електропружних хвиль в початково напружених тілах проведено математичне моделювання задачі про поле динамічних напружень, що виникає в шаруватій плиті-полосі. Ця плита складається з п’єзоелектричного ядра, яке граничить з пружним шарами з початковими напруженнями, і перебуває на абсолютно твердій основі. Приймається, що п’єзопружний матеріал поляризований перпендикулярно до вільної поверхні пластини. Основна система диференціальних рівнянь з частинними похідними розв’язана методом скінченних елементів. Представлено числові результати, які ілюструють вплив певних залежностей задачі про поширення напружень і електричних зміщень, які діють на площинах розділу пружних шарів і п’єзоелектричного ядра та плитою і абсолютно твердою основою. Зокрема, вивчено вплив параметра початкового напруження і зміни товщини п’єзопружного ядра на частотні характеристики плити-стержня. The author is a member of the research project supported by Research Fund of Kastamonu University under project number KÜ-BAP01/2015-3. en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core Article published earlier |
| spellingShingle | Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core Daşdemir, A. |
| title | Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core |
| title_full | Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core |
| title_fullStr | Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core |
| title_full_unstemmed | Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core |
| title_short | Forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core |
| title_sort | forced vibrations of pre-stressed sandwich plate-strip with elastic layers and piezoelectric core |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/174202 |
| work_keys_str_mv | AT dasdemira forcedvibrationsofprestressedsandwichplatestripwithelasticlayersandpiezoelectriccore |