Proceedings VCM 2012 57 phân tích phi tuyến động lực học tấm composite chức năng FGM

Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 419

Mã bài: 101
Phân tích phi tuyến động lực học tấm composite chức năng FGM
đối xứng có tính chất vật liệu phụ thuộc vào nhiệt độ
Nonlinear dynamic analysis of symmetrical composite FGM plates
with temperature-dependent materials properties
Nguyễn Đình Đức, Phạm Hồng Công
Trường đại học Công nghệ - Đại học Quốc gia Hà Nội
144 Xuân Thủy, Cầu Giấy, Hà Nội - Việt Nam
e-Mail: ,

Tóm tắt
Bài báo này trình bày cách tiếp cận giải tích để nghiên cứu các đáp ứng động lực học phi tuyến của tấm
composite chức năng FGM đối xứng, không hoàn hảo hình dáng ban đầu và tính chất vật liệu phụ thuộc vào
nhiệt độ theo lý thuyết tấm biến dạng trượt bậc nhất của Reddy và sử dụng phương pháp giải theo các hàm
ứng suất Airy. Đã xác định được 2 phương trình cơ bản để nghiên cứu động lực học của tấm FGM. Các
phương trình phi tuyến được giải theo phương pháp Runger-Kutta. Bài báo đã tính toán số và thảo luận những
ảnh hưởng của các tham số như tính chất vật liệu, yếu tố hình học, tính không hoàn hảo hình dáng ban đầu và
nhiệt độ lên đáp ứng động học và dao động phi tuyến của tấm chức năng FGM.
Abstract
This paper presents an analytical approach to investigate the nonlinear dynamic response and nonlinear
vibration of imperfect symmetrical composite FGM plates with temperature-dependent materials properties
using the Reddy’s first order shear deformation theory using Air’s stress functions. Two basic equations are
obtained to investigate the dynamic response and vibration of the FGM plate. The non-linear equations are
solved by the Runger-Kutta method. Effects of material and geometrical properties, imperfection and
temperature on the dynamic response and nonlinear vibration of the symmetrical FGM plates are analyzed and
discussed.
Keywords: Symmetrical composite functionally graded materials plates, nonlinear dynamic, temperature-
dependent materials properties, first order shear deformation theory, imperfection.
1. Introduction

Functionally Graded Materials (FGMs) are
composite and microscopically in homogeneous
with mechanical and thermal properties varying
smoothly and continuously from one surface to the
other. Typically, these materials are made from a
mixture of metal and ceramic or a combination of
different metals by gradually varying the volume
fraction of the constituent metals. The properties
of FGM plates and shells are assumed to vary
through the thickness of the structures. Due to the
high heat resistance, FGMs have many practical
applications, such as reactor vessels, aircrafts,
space vehicles, defense industries and other
engineering structures. As a result, in recent
years, many investigations have been carried out
on the dynamic and vibration of FGM plates and
shells.
Up to date, dynamic analysis of FGM plates
and shells using the higher order shear
deformation theory has received great attention of
the researchers. In [1], Mohammad and Singh
studied static response and free vibration of
symmetrical FGM plates using first order shear
deformation theory with finite element method. In
[2] Huang and Shen studied nonlinear vibration
and dynamic response of FGM plates in thermal
environments but volume fraction follows a
simple power law for symmetrical FGM plate.
Shariyat investigated vibration and dynamic
buckling control of imperfect hybrid FGM plate

subjected to thermo-electro-mechanical condition
[3] and dynamic buckling of suddenly load
imperfect hybrid FGM cylindrical shells [4] with
temperature-dependent material properties. Kim in
[5] studied temperature dependent vibration
analysis of functionally graded rectangular plates
by finite-elements method. Notice that in all the
publication mentioned above [1-5], all authors use
the displacement functions and volume fraction
follows a simple power law.
This paper presents an analytical approach to
investigate the nonlinear dynamic response and
420 Nguyễn Đình Đức, Phạm Hồng Công

VCM2012
nonlinear vibration of imperfect symmetrical FGM
plates with temperature-dependent properties
using the Reddy’s first order shear deformation
theory [6]. Moreover, other than [1-5], the paper
uses Air’s stress functions for solutions and
volume fraction follows a Sigmoid distribution.
Numerical results for dynamic response of the
FGM plate are obtained by Runger-Kutta method.

2. Theoretical formulation
In the modern engineering and technology,
there are many structures usually working in a
very high heat resistance environment. To increase
the ability to adjust to a high temperature,
structures with the top and bottom surfaces are

made of ceramic and the core of the structure is
made of metal. The symmetrical FGM plate
considered in this paper is the one example of
these structures [7].
Consider a symmetrical rectangular plate that
consists of two layers made of functionally graded
ceramic and metal materials and is midplane-
symmetric. The outer surface layers of the plate
are ceramic-rich, but the midplane layer is purely
metallic. The plate is referred to a Cartesian
coordinate system
, ,
x y z
, where
xy
is the
midplane of the plate and
z
is the thickness
coordinator,
/ 2 / 2
h z h
  
.The length, width ,
and total thickness of the plate are
a
,
b
and
h

,
respectively
(Fig.1).
Fig. 1. Symmetrical FGM plate
For symmetrical FGM plates, applying a
Sigmoid distribution, the volume fractions of
metal and ceramic,
m
V
and
c
V
, are assumed as [8]:

2
, /2 0
( )
2
, 0 /2
N
z h
h z
h
V z
m
N
z h
z h
h


 

 
 

 



 

 
 

 


  

 
 

( ) 1 ( )
c m
V z V z
  (2.1)
where the volume fraction index
N
is a
nonnegative number that defines the material

distribution and can be chosen to optimize the
structural response.
The effective Young’s modulus
E
, thermal
expansion coefficient

and the mass density


are independent to the temperature, vary in the
thickness direction
z
and can be written as
follows:






 
( , ), , , , ( ), ( ), ( )
2
; / 2 0
( ), ( ), ( )
2
;0 / 2
c c c
N

mc mc mc
N
E T z T z T z E T T T
z h
h z
h
E T T T
z h
z h
h
   
 
 
 


 
  

 
 


 
 
 
 

 


(2.2)
where

     
( ) ( ) ( )
( ) ( ) ( )
mc m c
mc m c
mc m c
E T E T E T
T T T
T T T
  
  
 
 
 
(2.3)
subscripts
m
and
c
stand for the metal and
ceramic constituents, respectively, and the Poisson
ratio

depends weakly on temperature change
and is assumed to be a constant ( )z

 

.
2.1. Nonlinear dynamic of imperfect
symmetrical FGM plate using first order shear
deformation theory
Suppose that the FGM plate is subjected to a
transverse load of intensity
0
q
. In the present
study, the Reddy’s first-order shear deformation
theory is used to obtain the motion and
compatibility equations, as well as expression for
determining the dynamic response of the FGM
plate.
The train-displacement relations taking into
account the Von Karman nonlinear terms and the
first - order shear deformation theory are [6,8]:

0
0
0
,
,
ε ε χ
x x x
ε = ε + z χ
y y y
γ γ χ

xy xy xy
w
xz x x
w
yz y y
 
 
     
     
     
     
     
     
   
   
   
   



(2.4)
with
0 2
, ,
0 2
, ,
0
, , , ,
/ 2
/ 2

x x x
y y y
y x x y
xy
u w
v w
u v w w



 
 

 
 
 
 
 
 
 
 
 
 
 
 
;
,
,
, ,
x x x

y y y
xy x y y x
 
 
  
   
   

   
   
   

   
(2.5)
Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 421

Mã bài: 101
where
0
x

and
0
y

are the normal strains,
0
xy

is the

shear strain on the midplane of the plate, and
xz


and
yz

are the transverse shear strains;
,
u v
, and
w
are the midplane displacement components
along the
,
x y
, and
z
axes;
x

and
y

are the
rotation angles in the
xz
and
yz
planes,

respectively;
(,)
indicates a partial derivative.
The strains are related in the compatibility
equation
2
2 0 2 0
2 0
2 2 2
2 2 2 2
w w w
y xy
x
x y x y
y x x y
 

   

  
   
 
 
   
   
 
(2.6)
Hooke law for an FGM plate is defined as
   
   

2
, ( , ) , (1 ) (1,1)
1
, , , ,
2(1 )
x y x y y x
xy xz yz xy xz yz
E
T
E
        

     

 
    
 



(2.7)
In this study, it is assumed that the
temperature is uniformly raised and
T

is the
constant.
The force and moment resultants of the plate
can be expressed in terms of stress components
across the plate thickness as

 
 
/2
/2
/2
/2
, 1, ; , ,
; ,
h
i i i
h
h
i iz
h
N M z dz i x y xy
Q dz i x y




 
 


(2.8)
Inserting Eqs. (2.4), (2.5) and (2.7) into Eq. (2.8)
gives the constitutive relations as
 
 



 
 
  
 
 
 
 
 
  
 
 
 
 
 
 
 
0 0
1 2
2
2 3 1 2
0 0
1 2
2
2 3 1 2
0
1 2 2 3
1
1

, [ ,
1
, 1 , ]
1
, [ ,
1
, 1 , ]
1
, [ , , ]
2 1
, ,
2 1
x x x y
x y
y y y x
y x
xy xy xy xy
x y xz yz
N M E E
E E
N M E E
E E
N M E E E E
E
Q Q
 

    
 


    
 

 

  

  
 

   
 




(2.9)
where
 
1 2
3 3
3
/2
1 2
/2
; 0
1
12 2( 1)( 2)( 3)
, ( , ) ( , ) (1, )
mc

c
c mc
h
h
E h
E E h E
N
E h E h
E
N N N
E T z T z T z dz
  

  

 
  
 


1
[
1 1
]
2 1
c c c mc c mc
mc mc
h h
E h E E
N N

h
E T
N
   

   
 


(2.10)
For using late, the reverse relations are obtained
from Eq. (2.9)
0 0
1 1
1 1
0
1
1 1
;
2(1 )
x x y y y x
xy xy
N N N N
E E
N
E
     


   

     
   




(2.11)
According to Love’s theory the equations of
motion are
2 2
1 1
2 2
2
0 1
2
;
0; 0
w w
w w w
xy xy y
x
xy xy y
x
x y
y
x
x xy
xy y
N N N
N

u v
x y x y
t t
M M M
M
Q Q
x y x y
Q
Q
N N
x y x x y
N N q
y x y t
 

  

 
   
    
  

     
   

  
  
   
 
    

 
 
   
  
 
   
 


(2.12)
where
/2
1
/2
( )
1
h
m c
c
h
z dz h
N
 
  


 
  
 


 

(2.13)
With the assumption
w, w
u v
 
, in (2.12) the
inertia
2
1
2
0
u
t




and
2
1
2
0
v
t





. Within the
framework of the first order shear deformation
theory, the nonlinear motion equations for a
perfect plate can be written in terms of deflection
w
and force resultants as
0; 0
xy xy y
x
N N N
N
x y x y
  

   
   

(2.14)
4 2
, , ,
1
2
1 , , , 0
2
2
1
2
2(1 )
( 2
w

) ( 2
w
) 0
x xx xy xy y yy
x xx xy xy y yy
D
D w N w N w N w
E
N w N w N w q
t
t




    

    


 



(2.15)
422 Nguyễn Đình Đức, Phạm Hồng Công

VCM2012
where
2 2

2
2 2
x y
 
  
 
and
3
2
1
E
D




For solving Esq. (2.14) and (2.15) we
introduce Air’s stress function
( , )
x y
 
 so that
2 2 2
2 2
; ;
x y xy
N N N
x y
y x
  

  
   
 
 
(2.16)
Inserting Eq. (2.16) into the Eq. (2.15) for
prefect plate leads to

2 2 2 2
4 2
2 2
1
2 2 2 2 2 2 2
1
2 2 2 2 2
2 2 2
0 1
2 2 2
2(1 ) w w
w+ [ 2
w w w w
] [ 2
w w
] 0
D
D
E x y x y
y x
x y x y
x y t y x

q
x y t
  
  



    
   
    
      
  
   
    
  
   
  

(2.17)
Equation (2.17) includes two dependent
unknowns
w
and

. To obtain a second equation,
relating the unknowns, the geometrical
compatibility
0 0 0 2
, , , , , ,
x yy y xx xy xy xy xx yy

w w w
  
   
(2.18)
Setting Eqs. (2.11), (2.16) into Eq. (2.18) gives
the compatibility equation of an perfect FGM plate
as
2
4 4 4 2 2 2
4 2 2 4 2 2
1
1 w w w
2
E x y
x x y y x y
  
 
   
     
 
   
   
      
 
   
 

(2.19)
For an imperfect plate, following to the
Volmir’s approach [9] for an imperfect FGM

plate, Eqs. (2.19) and (2.17) are modified into
form as

2
4 4 4 2 2 2
4 2 2 4 2 2
1
2
2 * 2 * 2 *
2 2
1 w w w
2
w w w
E x y
x x y y x y
x y x y
  
 
   
     
 
   
   
      
 
   
 
 
 
  

  
 
   
 
 
 

(2.20)

 
2 2 2 2
4 * 2
2 2
1
2 2 2 2 2 2 2
1
2 2 2 2 2
2 2 2
0 1
2 2 2
2(1 ) w w
w w + [ 2
w w w w
] [ 2
w w
] 0
D
D
E x y x y
y x

x y x yx y t y x
q
x y t
  
  



    
   
   
 
      
    
       
  
  
  

(2.21)
in which
*
w
is a known function representing
initial small imperfection of the FGM plate.
Equations (2.20) and (2.21) are the basic
relations used to investigate the dynamic response
and vibration of imperfect FGM plate with
temperature-dependent material properties. They
are nonlinear in the dependent unknowns

w
and

.
2.2. Nonlinear vibration of imperfect symmetrical
FGM plate
Suppose that the imperfect symmetrical FGM
plate is simply supported at its edges and subjected
to q transverse loads
0
( )
q t
. The boundary
conditions can be expressed as
0
w 0, 0, , 0
x x x xy
M N N N
   
at
0,
x x a
 


0
w 0, 0, , 0
y y y xy
M N N N
   

at 0,
y y b
 


(2.22)
Taking into account temperature-dependent
material properties, the mentioned conditions
(2.22) can be satisfied if the deflection
*
w, w
and
the stress function

are represented by [7]:
1 2 3
2 2
4 0 0
w ( )sin sin
2 2
os os sin sin
1 1
os os
2 2
x y
m x n y
f t
a b
m x n y m x n y
Ac A c A

a b a b
m x n y
A c c N y N x
a b
 
   

 

  
  

(2.23)
*
0
( , ) sin sin
m x n y
w x y f
a b
 
 (2.24)
in which
, 1,2,
m n

are natural numbers
representing the number of half waves in the
x

and

y
directions respectively;
f
is the deflection
amplitude;
0
f const
 , varying between
0
and
1
,
represents the size of the imperfections.
Introduce Eqs. (2.23), (2.24) into Eq. (2.20),
we can obtain the coefficients
( 1 4)
i
A i
 
as
follow :
 
 
2 2
2 2
1
1 0
2 2
2 2
2 2

1
2 0 3 4
2 2
( )
32
( ) ; 0
32
E n a
A f t f
m b
E m b
A f t f A A
n a
 
   

(2.25)
The condition expressing the immovability
of edges (2.22) is fulfilled on the average [32, 34]:
0 0 0 0
0; 0
b a a b
u v
dxdy dydx
x y
 
 
 
 

(2.26)
From Eqs. (2.5) and (2.9), taking into
account Eq. (2.16) , we can obtain the following
relations:
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 
 
2
1
, , ,
1 1
2
1
, , ,
1 1
1
w / 2
1
w / 2
yy xx x
xx yy y
u
x E E
v
y E E

 


 


   






   



(2.27)
Introduce (2.23) into (2.27) and then integrate the
results as in (2.26), we have:
2 2
2 2
1
0 1
2 2 2
1
( )
1
8(1 )
x
m n
N E f t
a b


 


 
  
 


 

(2.28)
2 2
2 2
1
0 1
2 2 2
1
( )
1
8(1 )
y
m n
N E f t
a b

 


 

  
 


 


(2.29)
Introduce (2.23), (2.24) into (2.21) and apply the
Bubnov-Galerkin method for the resulting
equation, we obtain:
 
 
 
2
2 2
4
0
2 2
2 2 4 2 2 2 2
1 2
2 2
1
2 2 2
4
0
2 2
2 2 2 2 2

4 2

0 1
2 2 2 2
2 2 4 2 2
1 2
( )
4
2(1 ) 4
[ ( )
8 8
( )
4
( ) ( )]
4 4
[ ( )
2 4
x
y
m n ab
D f t f
a b
D m n m b n a
A A f t
E a ba b
m n m b
N f t
aa b
m n n a m n ab
N f t f t
ba b a b
m n m b

A A f t
ab a

   

  
 
 
  
 
 
 

  
 
 
 

 
 
   
   
   
   
   
0
2 2

0 0 1
2

( )
4
( ) ( )] 0
4 4
x
y
N f t
n a ab ab
N f t q f t
b mn




  

(2.30)
Eq. (2.30) can be represented as:
      
   

5 10 1 0 2 6 1 2
3 7 0 4 8 0 9 0
( ) ( ) ( )
( ) ( )
x y
B B f t B f t f B B A A f t
B B N f t B B N f t B q
     
    

(2.31)
Where
2
2 2
4
1
2 2
2 2 4 2 2 2 2
2
2 2
1
2 2 2
4
3
2 2
1
2 2 2
4
4
2 2
1
2 2
2
5 1
2 2
1
2 2 4 2 2 2
6 7 8
4

2(1 ) 4
8 8
2(1 )
4
2(1 )
4
2(1 )
4
; ;
2 4
m n ab
B D
a b
D m n m b n a
B
E a b
a b
D m n m b
B
E aa b
D m n n a
B
E ba b
D m n ab
B
E a b
m n m b n
B B B
ab a


   





 
 
 
 
 
 
 

 
 
 
 

 
 
 
 

 
 
 
 

 

 
 
  
2
9 10 1
2
4
4
;
4
a
b
ab ab
B B
mn



 
(2.32)
Introducing
i
A
and
0 0
,
x y
N N
at Eqs. (2.25), (2.28)
and (2.29) into (2.31) gives

3
1 2 3 0 4 0
( ) ( ) ( )
f t m f t m f t m f m q
    (2.33)
where
 
   
 
 
 
2 2 2 2
2
1
1 5 10 2 1 2 6 0
2 2 2 2
1 1
3 7 4 8
3 1
2 2 2 2
1
4 2 6
2 2 2 2
2 2
2
3 7 1
2 2 2
2 2
2

4 8 1
2 2 2
2 4
1 2 3
1 1
32
1 1
32
1
8(1 )
1
8(1 )
; ;
E
n a m b
C B B C B B B f
m b n a
B B B B
C B
E
n a m b
C B B
m b n a
m n
B B E
a b
m n
B B E
a b
C C

m m m
C C
 
 
 

 

 
     
 
 
   
 

 
   
 
 
 
  
 

 
 
 
 

 
   

3 9
4
1 1
;
C B
m
C C


(2.34)
and
( )
f t
- deflection of middle point of the plate


/2
/2
( ) w
x a
y b
f t



.
For linear free vibration for FGM plate the
equation (2.33) gets form:

1

( ) ( ) 0
f t m f t
 

(2.35)
one can determine the fundamental frequency of
natural vibration of the FGM plate :
1
L
m


(2.36)
The equation (2.33) for obtaining the nonlinear
dynamic response the initial conditions are
assumed as
.
0
(0) , (0) 0
f f f
 
. The applied loads
424 Nguyễn Đình Đức, Phạm Hồng Công

VCM2012
are varying as function of time. The nonlinear
equation (2.33) can be solved by the Newmark’s
numerical integration method or by the Runger-
Kutta method.
3. Numerical results and Discussion

The imperfect symmetrical FGM plate
considered here a square plate:
1
a b m
 
,
0.1
h m

.
Here, several numerical examples will be
presented for perfect and imperfect simply
supported midplane-symmetric FGM plates. The
silicon nitride and stainless steel are regarded as
constituents of the composite FGM plates. A
material property
Pr
, such as the elastic modulus
and thermal expansion coefficient, can be
expressed as a nonlinear function of temperature
[1-5].



1 1 2 3
0 1 1 2 3
Pr 1
P P T PT PT PT



    
(3.1)
in which
0
T T T
  
and
0
300
T K
 (room
temperature); P
0
, P
-1
, P
1
, P
2
and P
3
are
temperature-dependent coefficients characterizing
the constituent materials. The typical values of the
coefficients of the materials mentioned are listed
in Table1 [10]. The Poisson ratio is chosen to be
0.28 for simplicity.

Table 1. Material properties of the constituent materials of the considered FGM shells
Material Property P

0
P
-
1

P
1

P
2

P
3

Si
3
N
4

(Ceramic)
E(Pa) 348.43e9 0 -3.70e-4 2.160e-7 -8.946e-11

(kg/m
3
)
2370 0 0 0 0
1
( )
K




5.8723e-6 0 9.095e-4 0 0
SUS304
(Metal)
E(Pa) 201.04e9 0 3.079e-4 -6.534e-7 0

(kg/m
3
)
8166 0 0 0 0
1
( )
K



12.330e-6 0 8.086e-4 0 0

The equation of non-linear free vibration of a
perfect plate can be obtained from (2.33)


3
1 2
( ) ( ) ( ) 0
f t m f t m f t
  

(3.2)

Seeking solution as
( ) os( )
f t c t
 

and
applying procedure like Galerkin method to
Eq.(3.2), the frequency-amplitude relation of non-
linear free vibration is obtained
1
2
2
2
2
3
1
4
NL L
L
m
  

 
 
 
 

(3.3)
Where
NL


is the non-linear vibration frequency
and

is the amplitude of non-linear vibration.


Fig. 2: Frequency- amplitude relation
(
NL

-

)
The plate subjected by an uniformly
distributed excited transverse load
0
( ) sin
q t p t
 
. From Eq. (2.36), natural
frequencies of FGM plate
L

are shown the Table
2. Obviously the natural frequencies of the FGM
plate are observed to dependent on the constituent
volume fraction, they increase when the power law
index
N

increases.


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Mã bài: 101
Table 2. Effect of power law index
N
and temperature (with m=n=1) on natural frequencies
L

for
perfect and imperfect (
0
0.002
f  ) symmetrical FGM plates
T

(K)
N=1 N=2 N=3
Perfect Imperfect Perfect Imperfect Perfect Imperfect
0 4.2875e3 4.2872e3 4.8770e3 4.8767e3 5.2236e3 5.2232e3
300 3.4524e3 3.4521e3 4.0143e3 4.0139e3 4.3377e3 4.3373e3
500 2.7085e3 2.7081e3 3.2735e3 3.2731e3 3.5852e3 3.5847e3
600 2.2611e3 2.2607e3 2.8365e3 2.8360e3 3.1415e3 3.1411e3


The nonlinear dynamic response of the FGM plate
acted on by the harmonic uniformly excited
transverse load

0
( ) sin
q t p t
 
are obtained by
solving Eq. (2.33) combined with the initial
conditions and by use of the Runge-Kutta method.
Fig. 3 shows dynamic responses (relations
deflection of middle point of the plate


/2
/2
( ) w
x a
y b
f t



and the time ) of the symmetrical
FGM plate subjected (without temperature and
with temperature change) to the harmonic
transverse load
0
( ) 75000sin(4000 )
q t t
 . We can
see that temperature change has serious effect on
dynamic response of the FGM plates.

0 0.01 0.02 0.03 0.04
-4
-3
-2
-1
0
1
2
3
4
x 10
-4
t(s)
w(m)

Fig.3. Linear and nonlinear transient response of
the FGM plate with temperature change
50
T K
 

( N=1, : nonlinear; - - -: linear)


Relation of maximum deflection and velocity of
maximum deflection when
(N=1)
and
0
( ) 75.000sin(2500 )

q t t
 is presented in Fig.4a and
Fig.4b

-2 -1 0 1 2
x 10
-4
-0.5
0
0.5
w
dw/dt

Fig.4a. Deflection velocity relation,
0
T K
 

-5 0 5
x 10
-4
-1.5
-1
-0.5
0
0.5
1
1.5
w
dw/dt

Fig.4b. Deflection velocity relation,
500
T K
 


Fig.5a and Fig.5b show nonlinear response of
the FGM plate of long period with different
intensity of loads :
2
75000 /
p N m
 and
2
95000 /
p N m
 ;
4000
 


426 Nguyễn Đình Đức, Phạm Hồng Công

VCM2012
0 0.02 0.04 0.06 0.08 0.1
-3
-2
-1
0

1
2
3
4
x 10
-4
t(s)
w(m)


p=75000
p=95000

Fig. 5a. Dynamic response with different
intensity of loads,
0
T K
 

0 0.02 0.04 0.06 0.08 0.1
-4
-3
-2
-1
0
1
2
3
4
x 10

-4
t(s)
w(m)


p=75000
p=95000

Fig. 5b. Dynamic response with different
intensity of loads,
200
T K
 


Fig.6a and Fig.6b show the effect of the
imperfection
(
0 0
(0,001;0,003); ( ) 750.000sin(3500 )
1
f q t t
N
 

)
on nonlinear dynamic responses of the FGM
plate.

Fig.6a. Influence of imperfection on nonlinear
dynamic response of FGM plate,
0
T K
 


Fig.6b. Influence of imperfection on nonlinear
dynamic response of FGM plate,
300
T K
 



Fig. 7a, Fig.7b show the effect of the
geometrical parameters
( / )
a b
on dynamic
response of the FGM plate with
0
( ) 75.000sin(2200 )
q t t
 and
1
N

. Fig. 8a,
Fig. 8b show the effect of the

( / )
a h
on dynamic
response of the FGM plate with
0
( ) 75.000sin(1700 )
q t t
 ,
1
N

and
a b

.
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Mã bài: 101
0 0.02 0.04 0.06 0.08 0.1
-3
-2
-1
0
1
2
3
x 10
-4
t(s)
w(m)

a/b=1
a/b=2

Fig.7a. Effect of dimension ration
/
a b
on
dynamic response,
0
T K
 

0 0.02 0.04 0.06 0.08 0.1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4
t(s)
w(m)

a/b=1
a/b=2

Fig.7b. Effect of dimension ration
/
a b
on
dynamic response,
100
T K
 

0 0.01 0.02 0.03 0.04 0.05
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x 10
-4
t(s)
w(m)


a/h=10
a/h=15

Fig. 8a. Effect of dimension
/
a h
on dynamic
response of the square plate


a b

,
0
T K
 

0 0.01 0.02 0.03 0.04 0.05
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
x 10
-4
t(s)
w(m)

a/h=10
a/h=15

Fig. 8b. Effect of dimension
/
a h
on dynamic
response of the square plate


a b

,
100
T K
 


Fig. 9 shows effect of temperature
T

on
dynamic response of the FGM plate with
0
( ) 75000sin(4000 )
q t t
 .
428 Nguyễn Đình Đức, Phạm Hồng Công

VCM2012

Fig. 9. Effect of temperature on dynamic response
of the square symmetrical FGM plate

From the obtained results, we observe these
interesting conclusions: Temperature change has
serious effect on vibration and dynamic response
of FGM plates. Frequencies of nonlinear vibration
of composite FGM plate increase when
amplitudes, the intensity of dynamic loads,
volume fraction index increase and frequencies of
vibration decrease when temperature increase.
4. Conclusions
This paper presents an analytical approach
to investigate the nonlinear dynamic analysis of
imperfect symmetrical composite FGM plates with
temperature-dependent material properties. The
strong point in this article is the achievement of
analytical fundamental equations for FGM plates
using Air’s stress function and the Reddy’s first
order shear deformation theory. Numerical results
for dynamic response of the FGM plate are
obtained by Runger-Kutta method.
Thus it is obvious that dynamic response of
the considered plate depends on many parameters
significantly: temperature, imperfection



0
f
and
geometrical parameters


/ ; /
a b a h
of the
imperfect symmetrical composite FGM plate.
Therefore, when we change these parameters, we
can control the dynamic response and vibration of
the FGM plate actively.


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