BEHAVIOR OF A SIMMETRICAL ORTHOTROPIC SANDWICH PLATE SUBJECTED TO THE
DYNAMIC LOADING
V.Z. Gristchak, O.A. Ganilova
The branch of mechanics which considers anisotropic plates is highly important at present since anisotropic
matters are widely used in contemporary engineering. Therefore the analysis of plate and shell theory is of current
importance [1, 3].
According to the investigations presented in the cited literature [2, 3] it is obvious that the classical theory
gives the correct results only in a limited field of application. Thus, the qualifying theories are highly developing
and widely applying at present time.
In this paper we consider a thin orthotropic plate made from three homogeneous laminas. To analyze the
problem the qualifying iterative theory of anisotropic plates described in [3] is used. This approach is based on the
idea to obtain correction terms which describe the in-plane shear effect dependent on the mechanical matter
characteristics.
The mentioned above theory applied in the context of the following hypotheses [3, 11]:
a) normal displacement u z in every layer is not dependent on coordinate z;
m
m ,0
m
m
m,0
b) obtaining strains e xz
and e myz , it is supposed that shear stresses  xz
,  yz
do not differ from  xz
,  yz
,
which are obtained taking into consideration hypotheses of unstrained normal for a whole plate stack;
c) obtaining strains, we do not take into consideration normal stress effect  z .
The described problem comes to finding the shear u 0 ( x, y),  0 ( x, y), and normal  0 ( x, y ) displacements in
xOy plane. These functions are found, solving the system of differential equations according to the classical
laminated plates and shells theory [1, 2, 3]:
L11 Cik u 0  L12 Cik  0  L13 K ik  0  0,
(1)
L22 Cik  0  L12 Cik u 0  L23 K ik  0  0,
L33 Cik  0  L13 Cik u 0  L23 K ik  0  Z 2 .
where the linear operators are expressed as
2
2
2
2
L11 C ik   C11 2  C 66 2 , L22 C ik   C 22 2  C 66 2 ,
x
y
y
x
L12 C ik   C12  C 66 
L23 K ik   K 22
3
2
3
3
, L13 K ik   K 11 3  K 12  2 K 66 
,
xy
x
xy 2
 K 12  2 K 66 
3
y
yx
Rigidities, according to [3] are defined as
3
2
, L33 Dik   D11
4
x
4
 D22
C ik  2 0 Bik0   1   0 Bik   2   0 Bik , K ik 

(2)
4
y
4
 2D12  2 D66 


4
x 2 y 2
.
 
1 2
 2   02 Bik   12   02 Bik ,
2
(3)
1
3 0
3
3
3
3



Dik  2 0 Bik   1   0 Bik   2   0 Bik .
3
where 2  0 , 1   0 ,  2   0 are the thicknesses of the middle, top and bottom layer, respectively; Bikm are




 
predetermined layer elasticity coefficients; m is the index of the ply ( m  0 for the middle layer, for the top and
bottom plies indexes are presented as ‘ and “); Cik is a stretching – compression stiffness; Dik is a bending
stiffness.
Presented paper deals with the described above simply supported plate ( a  b ) subjected to the loading
expressed as

 2
 h 2
(4)
a
b
t
t
taking into consideration variable in time damping coefficient.
In terms of iterative theory, according to [3], for the described problem we obtain the next system of
differential equations with respect to unknown functions u( x, y),  ( x, y), ( x, y) :
Z  q(t ) sin
x
sin
y
 2  (t )h
L11 C ik u  L12 C ik   L13 K ik   h
 2u
L22 C ik   L12 C ik u  L23 K ik   h
 2
t 2
t 2


Tx* S *

,
x
y
T y*
y

S *
,
x
2
*
2H *  M y

.
xy
x 2
y 2
The sandwich plate is a symmetrical orthotropic one, therefore, we suppose that
1   2   , Bik  Bik  Bik and
u0  0  0, u    0,
L33 Dik   L13 K ik u  L23 K ik   Z 
 2 M x*
(5)
2
Analysis of the problem comes to the solution of the following equation, in terms of the classical theory:
 4 0
 4 0
 4 0
x
y

 2


D11

2
D

2
D

D

q
(
t
)
sin
sin

2

(
t
)
h


h
12
66
22
a
b
t
x 4
x 2 y 2
y 4
t 2
(6)
(7)
and to another differential equation, in terms of iterative theory:
2
*
 2M *
 2 H *  M y 
x


Z


2

(8)
 x 2
xy
x 4
x 2 y 2
y 4
y 2 

Solution of the equations is conducted using the hybrid (Wentzel-Kramer-Brillouin) WKB-Galerkin method
which enables to obtain especially good results in approximate solution of differential equation which contains a
parameter near the highest order derivative and has already shown its advantages in different branches of mechanics.
Hybrid methods have proved to be useful in a wide variety of applications such as structural mechanics
problems, applications to slender-body, thermal and structures problems [ 4 – 10, 13, 14 ].
The hybrid WKB-Galerkin method was successfully used in solution of mechanical boundary problems which
contained a linear differential equation with variable coefficients and parameter near the highest order derivative.
The obtained solution had a pinpoint accuracy and was used in a wide variety of applications. Therefore, the hybrid
WKB-Galerkin method, applied to the described differential equations, will give us an opportunity to obtain the
approximate solution as an asymptotic one.
The results obtained for the classical and iterative theory and compared, varying mechanical matter
characteristics and magnitude of the damping coefficient function.
D11
 4
 2D12  2 D66 
 4
 D22
 4
References
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_____________________________________________________________________________________________
Addresses: Professor Dr. Victor Z. Gristchak, Chairman of Applied Mathematics Department, Faculty of
Mathematics, Zaporizhzhya National University, Zhukovskogo st., 66, Zhaporizhzhya, 69063, Ukraine,
grk@zsu.zp.ua; Olga A. Ganilova, PhD Candidate, Applied Mathematics Department, Faculty of Mathematics,
Zaporizhzhya National University, Zhukovskogo st., 66, Zhaporizhzhya, 69063, Ukraine, lionly@rambler.ru.