30
Original Article
Nonlinear Dynamic Analysis of Rectangular FGM Plates of Variable Thickness Subjected to Mechanical Load
Khuc Van Phu
1, Le Xuan Doan
2,*1Military Logistics Academy, Ngoc Thuy, Long Bien, Hanoi, Vietnam
2Academy of Military Science and Technology, Hanoi, Vietnam Received 29 November 2019
Revised 03 December 2019; Accepted 04 December 2019
Abstract: This paper establishes the governing equations of rectangular plates of variable thickness subjected to mechanical load by using the classical plate theory, the geometrical nonlinearity in von Karman-Donnell sense. Solutions to the problem are derived according to Galerkin method. Nonlinear dynamic responses, critical dynamic loads are obtained by using Runge-Kutta method and the Budiansky–Roth criterion. The effect of volume-fraction index k and some geometric factors are considered and numerical results are presented.
Keywords: Dynamic responses, nonlinear vibration, rectangular FGM plate, variable thickness.
1. Introduction
Rectangular FGM plates are used extensively in spacecraft, nuclear reactors or defense industry and in civil engineering, v.v. Today, analysis of vibration and dynamic stability of FGM plate structures has been studied by many authors.
Firstly, for dynamic problems of constant thickness plate structures, Ungbhakorn et al. [1] investigated thermo-elastic vibration of FGM plates with distributed patch mass based on the third-order shear deformation theory and Energy method. Talha et al. [2] analyzed free vibration of FGM plates by using HSDT and finite element method. Duc et al. used the Galerkin method and the higher-order shear deformable plate theory to study the post-buckling of thick symmetric functionally graded plates resting on ________
Corresponding author.
Email address: xuandoan1085@gmail.com https//doi.org/ 10.25073/2588-1124/vnumap.4363
elastic foundations under thermomechanical loads [3] and buckling and post-buckling of thick functionally graded plates subjected to in-plane compressive [4]. Bich et al. [5] examined nonlinear post-buckling of eccentrically stiffened functionally graded plates and shallow shells based on the classical shell theory and the smeared stiffeners technique. Hebali et al. [6] and Mahi et al. [7] studied static and free-vibration of FGM plates under mechanical load based on hyperbolic shear deformation theory. Benferhat et al. [8] used Hamilton’s principle and higher-order shear deformation theory to study vibration of FG plates resting on elastic foundation. R. Kandasamy et al. [9] used FSDT and finite element method to investigate free vibration and thermal buckling behavior of moderately thick FGM plates in thermal environments.
Secondly, for dynamic problems of variable thickness plate structures, E. Efraim et al. [10] based on the FSDT to study vibration of variable thickness thick annular isotropic and FGM plates. S. H. Hosseini- Hashemi et al. [11] based on the classical plate theory and differential quadrature method (DQM) to deal with free vibration problem of radially FG circular and annular sectorial thin plates with variable thickness resting on elastic foundations. M. Shariyat and M. M. Alipou [12] studied vibration of variable thickness two-directional FGM circular plates resting on elastic foundations by using power series. V. Tajeddini et al.
[13] employed linear elastic theory and Ritz method to investigate 3D free vibration of thick circular FG plates with variable thickness. F. Tornabene et al. [14] examined natural frequencies of FGM sandwich shells with variable thickness by using HSDT and local generalized differential quadrature method. A. H.
Sofiyev [15] used Ritz method to study buckling of continuously varying thickness orthotropic composite truncated conical shell under mechanical load. A. R. Akbari and S. A. Ahmadi [16] analyzed buckling of a FG thick cylinder shells with variable thickness under mechanical load by using DQM. P. T. Thang et al.
[17] investigated effects of variable thickness and imperfection on nonlinear buckling of S-FGM cylindrical panels subjected to mechanical load based on the classical shell theory and using Galerkin method. These authors also investigated effect of variable thickness on buckling and post-buckling behavior of S-FGM plates resting on elastic medium [18].
In conclusion, according to the above review reveals and author's knowledge, there were many studies on FGM plate and shell structures but has no publication on nonlinear vibration and dynamic stability of FGM rectangular plate with variable thickness under mechanical load. In this paper, we investigate nonlinear vibration and dynamic stability of rectangular plates with variable thickness subjected to mechanical load. The governing equations are established based on the classical plate theory. Nonlinear dynamic responses are obtained by using Galerkin method and Runge-Kutta method. Critical dynamic loads are obtained by using the Budiansky–Roth criterion.
2. Governing equations
Fig. 1. Configuration of variable thickness FGM plate.
Consider a rectangular FGM plate with variable thickness subjected to mechanical load. The thickness of the plate can be expressed as: h=h(x,y) (Fig. 1a).
Assume that, plate made of FGM with the volume fraction of ceramic V zc
changes according to the following rule:1
2 ( , )
k c
V z
h x y
(1)
With the above rule, the Young’s modulus E, Poisson ratio ν of FGM plate can be expressed as:
( ) . . 1
2 ( , )
( ) . . 1
2 ( , )
( ) . . 1
2 ( , )
k
m m c c m c m
k
m m c c m c m
k
m m c c m c m
E z E V E V E E E z
h x y
z V V z
h x y
z V V z
h x y
(2)
According to [19], the strains at a distance z from the middle surface can be expressed as:
0 ij ij zkij
with (i j= xx, yy, xy)
or xxxx0 zkxx, y0yxzkyy, xy xy0 zkxy, (3) Where: xx0; 0yy; xy0 , are the strains at the middle surface and kxx;kyy are curvatures and
k
xyis the twist. They are related to the displacement components u v w, , in the x y z, , coordinate directions as:2 2
0 0 0
2 2 2
2 2
1 w 1 w w w
; ;
2 2
w w w
; ; 2
xx yy xy
xx yy xy
u v u v
x x y y y x x y
k k k
x y x y
(4)
Hooke's law applied to FGM plate under mechanical loads can be expressed as follows
( ) 2 ( )
1 0
1 0 .
1 1
0 0
2
xx xx
z
yy yy
z
xy xy
E
hay
.
(5)Integrating the stress-strain equations through the thickness of the plate we obtain the mechanical behavior equations of FGM plate with variable thickness:
0
ij ij
ij ij
.
N A B
B D
M k
với (i,j)=(xx ,yy, xy) (6)
In which:
( , ) 2
2 ( , )
2
; ; . 1, ,
h x y
h x y
A B D z z dz
(7)The mechanical behavior equations of FGM plate with variable thickness can be rewritten as follows
0
11 12 11 12
0
21 22 21 22
0
66 66
11 12 11 12
21 22 21 22
66 66
0 0
0 0
0 0 0 0
0 0 .
0 0
0 0 0 0
xx xx
yy yy
xy xy
xx xx
yy yy
xy xy
N A A B B
N A A B B
N A B
B B D D
M k
B B D D
M k
B D
M k
(8)
in which:
( , ) ( , )
2 2
1 1
11 22 2 2 12 21 2 2
( , ) ( , )
2 2
. ( , ) . . ( , )
( ) . ( )
. ; . ;
1 1 1 1
h x y h x y
h x y h x y
E h x y E h x y
E z E z
A A dz A A dz
( , ) ( , )
2 2 2 2
2 2
11 22 2 2 12 21 2 2
( , ) ( , )
2 2
. ( , ) . . ( , )
( ). . ( ).
. ; . ;
1 1 1 1
h x y h x y
h x y h x y
E h x y E h x y
E z z E z z
B B dz B B dz
( , ) ( , )
2 2 3 2 2 3
3 3
11 22 2 2 12 21 2 2
( , ) ( , )
2 2
. ( , ) . . ( , )
( ). . ( ).
. ; . ;
1 1 1 1
h x y h x y
h x y h x y
E h x y E h x y
E z z E z z
D D dz D D dz
( , ) ( , )
2 2 2
2
66 66
( , ) ( , )
2 2
. ( , )
( ) . ( , ) ( ).
. ; .
2 1 2 1 2 1 2 1
h x y h x y
h x y h x y
E h x y
E z E h x y E z z
A dz B dz
( , )
2 2 3
3 66
( , ) 2
. ( , ) ( ). .
2 1 2 1
h x y
h x y
E h x y E z z
D dz
where: 1 2
3
1 1 1
; ;
( 1) 2( 1)( 2) 12 3 2 4( 1)
c m
c m m
m c m
E E k
E E E
E E E E E E
k k k k k k
Internal force and moment resultants in Eq (8) can be expressed as:
0 0
11 11
0 0
11 11
0
66 66
. .
. .
. .
xx xx yy xx yy
yy yy xx yy xx
xy xy xy
N A B k k
N A B k k
N A B k
(9)
0 0
11 11
0 0
11 11
0
66 66
. .
. .
. .
xx xx yy xx yy
yy yy xx yy xx
xy xy xy
M B D k k
M B D k k
M B D k
(10)
Based on the classical plate theory, the motion equations of variable thickness FGM plate can be given as:
2
1 2
2
1 2
2 2
2 2 2 2 2
2 2 2 2 2
2
0 1
2
. ( , )
. ( , )
w w w w
2 . 2 . ( , )
. ( , ) w . (
xx xy
xy yy
xy yy
xx
xx xy yy
N N u
h x y
x y t
N N v
h x y
x y t
M M
M N N N p h x y
x y x y
x y x y x
q h x y q h
y
2 2 1
w w
, ) 2 . ( , )
x y h x y
t
t
(11)
here:
( , )/2 1
( , )/2
. 1
h x y
c m
z m h x y
dz k
Substituting Eq. (9) and Eq. (10) in to Eq. (11) and consider Eq. (4) then, Eq. (11) can be rewritten as:
2
11 12 13 1 1 2
2
21 22 23 2 1 2
2
31 32 33 3 4 5 2
2
0 1
2
( ) ( ) (W) (W) . ( , )
( ) ( ) (W) (W) . ( , )
( ) ( ) (W) (W) ( , W) ( , W) . ( , ) w . ( , ) w . ( ,
L U L V L P h x y u
t
L U L V L P h x y v
t
L U L V L P P U P V p h x y
x
q h x y q h x
y
2 2 1
w w
) 2 . ( , )
y h x y
t t
(12)
where:
2 2
66 11
11( ) 11 u2 A u 66 u2 A u
L U A A
x x y y
x y
2 2
66 11
12( ) 11 v . A v 66 v A v
L V A A
x y x y x y y x
3 3 2 2 2
66 11
13 11 3 11 66 2 2 2
w w w w w
(W) 2 B . 2 B
L B B B
x y x y
x x y x y
2 2
2 2 2 2
2 11 66
3 11 2 66 2
w w w w 1 w w w w w w w w
(W ) 2
A
P A A A
x x y x y x x y x y y x y y x y
2 2 2 2
66 66
11 11
21( ) 11 u A u 66 u A u; 22( ) 66 v2 A v 11 v2 A v
L U A A L V A A
x y y x x y x y x x x y y y
3 3 2 2 3 2
11 11 66
23 11 3 11 2 2 2 66 2
w w w w w w
(W) . B B . 2 2 B
L B B B
y y x x y
y x y y x x y
2 2
2 2 2 2
2 11 11 66
2 11 2 11 66 2 66
w w w w 1 w 1 w w w w w w w
(W ) 2 2
A
A A
P A A A A
y y x x y y y y x x y x x y x x y
2 2 23 3
66 11 11
31 11 3 66 11 2 2 2
2 2 2 2
66 66 11 11
2 2
( ) 2 2
2
B B B
u u u u
L U B B B
x y y x
x x y x y
B u B u B u B u
x y y x y x x y x y
32
2
2 2
3 3 2 2 2
66 66 66
11 11 11 11
11 3 11 66 2 2 2 2 2
( )
2 2 2 2 . 2
L V
B B B
B B B B
v v v v v v v
B B B
x x x y y y y x y x
y x y y x y x
4 4 4 3 3 3
66
11 11 11
33 11 4 4 11 66 2 2 3 3 2
2 2
3 2 2 2 2
66
11 11 11
2 2 2 2 2 2 2
w w w w w w
(W) 2 2 2 2 2 2
w w w w w
2 2 .
D D D D
L D D D
x y x x
x y x y x y x y
D
D D D
y y x y x x y y y x
2 2
66 w
4 D
x y x y
2 3
2 2
2 2 2 2 2 2
66 66
11 11 11
11 66
2 2 2
3 2 2
11 66 2 66 11 2 2 66 11
(W )
1 w w w w w w w w w
2 2 2 2
2
w w w w w
2 2 2
P
B B
B B B
B B
x y x y x x y x y x x x x
x y x
B B B B B B
y x y x y x
3 3 3
2 11 3 3
2 2
2 2
2 2 2
66 11 66 11 66 11
2 2 2
w w w w w
w w w w w w w w 1 w w
2 2 2 2
2
B x y
x y x y
B B B B B B
y y x y y x x y y y y x y x y y y x
2 2
2 2
2 2 2 2 2
3
4 11 2 2 11 2 2 66
1 w w w w 1 w w w w w w w
(W ) 2
2 2
P A A A
x y y x x y x y
x y x y
2 2 2
5 11 2 11 2 66
w w w
( , W) u u 2 u
P U A A A
x x x y y x y
;
2 2 2
6 11 2 11 2 66
w w w
( , W) v v 2 v
P V A A A
y y y x x x y
Eqs. (12) are the basic equations used to investigate the nonlinear dynamic response of variable thickness FGM plates subjected to mechanical load.
For simplicity, we only consider the simply supported rectangular FGM plate with variable thickness which linearly changes in the x-axis (Fig. 1b). Assume that, the thickness of the plate can be determined as follows:
1 0
( ) h h 0
h x a xh (13)
Where: a is the length of the plate’s edge, h1 and h0 are the thickness of FGM plate at x=0 and x=a, respectively.
Then, Eqs. (12) will be rewritten as:
2
11 12 13 1 1 2
2
21 22 23 2 1 2
2
31 32 33 3 4 5 6
2 2 2
0 1
2 2
( ) ( ) (W) (W) . ( )
( ) ( ) (W) (W) . ( )
( ) ( ) (W) (W) (W ) ( , W) ( , W)
w w w
. ( ) . ( ) . ( )
I U I V I Q h x u
t
I U I V I Q h x v
t
I U I V I Q Q Q U Q V
p h x q h x q h x
x y t
2 1
2 . ( )h x w
t
(14)
in which:
2 2
1 0
1 1 1
11 2 2 2 2
. ( ) . ( )
( ) 1 2 1 1
h h
E h x u E h x u E u
I U
a x
x y
2
1 1 1 0
12 2 2
1 . ( )
( ) .
2 1 1
E h x v E h h v
I V
x y a y
2 3 3 2 2
1 0
2 2
13 2 3 2 2 2 2
. ( ) w w 2 ( ) w w
(W) 1 1
h h
E h x E h x
I x x y a x y
2 2 2 2
1 1
1 2 2 2
2
1 1 0
2
. ( ) w w w w . ( ) w w w w
(W) 1 2 1
w w
2 1
E h x E h x
Q x x y x y y x y x y
E h h
x y
a
2
2
1 1 1 0
21 2
1 . ( ) .
( ) .
2 1 2 1
E h x u E h h u
I U
x y a y
2 2
1 0
1 1 1
22 2 2 2
. ( ) . ( ) .
( ) .
2 1 1 2 1
h h
E h x v E h x v E v
I V
a x
x y
2 3 2 3 2
1 0
2 2 2
23 2 3 2 2
. ( ) w . ( ) w ( ) w
(W) 2
1 1 1
h h
E h x E h x E h x
I y x y a x y
2 2 2 2
2 1 1 1 1 0
2 2 2 2
. ( ) w w w w . ( ) w w w w . w w
(W ) 1 2 1 2 1
h h
E h x E h x E
Q y y x x y x x y y x a x y
2 3 3 2 2 2
1 0 1 0
2 2 2
31 2 3 2 2 2 2 2
. ( ) 2 ( )
( ) 2 2 1
1 1 1
h h h h
E h x u u E u E h x u u
I U
a x a
x x y x y
2 3 3 2 2 2
1 0 1 0
2 2 2
32 2 2 3 2 2 2
. ( ) ( ) 2
( ) 2 1 1
1 1 1
h h h h
E h x v v E h x v v E v
I V
a x y a y
x y y x
3 4 4 3 4 2 3 3
3 3 3 1 0
33 2 4 4 2 2 2 2 3 2
2 2 2
3 1 0
2 2 2
. ( ) w w . ( ) w 6 ( ) w w
(W) 2
1 1 1
6 ( ) w w
1
E h x E h x E h x h h
I x y x y a x x y
E h x h h
a x y
2
2 2 2 2 2 2
2 2 1 0 2 2 1 0
3 2 2 2
2 3 2 2
1 0
2 2 2
2 2 2
. ( ) . ( )
w w w w w
(W ) 3 1 2
1 1 1
. ( ) w w . ( ) w w . ( ) w
1 2 1
1
h h h h
E E h x E h x
Q a x y x y a y x y
h h
E h x E h x E h x
y y x x y a
2
2 3
2
2 2 2
2 2 2 2
2 3 3 2 2
1 0
2 2 2 2 2
2 2 2 3 2 3 2 2 2
. ( )
w w w
1
( ) w w . ( ) w w . ( ) w w . ( ) . ( ) w w
4 2 2
2 1
1 1 1 1
E h x
x y x x y
h h
E h x E h x E h x E h x E h x
a x x x x y y x y
2 2
2 2
2 2 2
3 1 1 1
4 2 2 2 2
. ( ) w w w . ( ) w w w . ( ) w w w
(W ) 2 1 2 1 1
E h x E h x E h x
Q x x y y y x x y x y
2 2 2
1
5 2 2 2
. ( ) w w w
( , W) 1
1
E h x u u u
Q U x x x y y x y
2 2 2
1
6 2 2 2
. ( ) w w w
( , W) 1
1
E h x v v v
Q V y y y x x x y
Eqs. (14) are basic equations used to investigate nonlinear dynamic responses of FGM plates with thickness linearly changes in the x-axis subjected to mechanical load.
3. Solution method
Consider a variable thickness FGM rectangular plate subjected to uniformly distributed pressures p(t) and q(t) in x and y direction. The exciting force q0(t)acting on the plate’s surface.
The plate is simply supported on 4 edges, then the boundary conditions are:
0, xx 0; xx ( )
w M N ph x at x =0 and x=a.
0, y 0; yy ( )
w M N qh x at y =0 and y=b.
Satisfying boundary conditions, the deflection of the plate can be chosen as:
m n m n m n
( ) os sin ; ( )sin os ; ( )sin sin
mn mn mn
x y x y x y
u U t c v V t c w W t
a b a b a b
(15)
Where: m, n are the numbers of half-wave along the x and y direction, respectively.
Substituting Eq. (15) into Eq. (14) then applying Galerkin procedure, at the same time, ignoring inertial components along x and y axes (because of u<<w, v<<w) [20], we obtain:
2
11 12 13 1
2
21 22 23 2
2
2 3 0 * *
31 32 33 3 4 5 6 2 1 2 1
W W 0
W W 0
4 w dw
W W W .W .W+ 2
l U l V l R
l U l V l R
abq d
l U l V l R R R U R V
mn dt dt
(16)
In which: *
1 0
1 1
8 ab h h
2
1 1 0 2 2 2 2
11 2 1 m 2n
16 1
E h h
l b a
ab
;
2
1 1 0
12 21
.mn 16 1
E h h
l l
2 2
2
3 2
2 1 0 2 1 0 2 2 2 2 2 1 0 2 2 2 2
13 2 2 2 2 2 2
2 3
m m n
24 1 4 1 4 1 m
E m h h E m h h E h h
l b n a b a
ma b ba a b
1 1 0 2 2 2 2
1 2 2 (3 1)n 4m
9 1
E h h
R a b
na b
;
2 2 2 2 2
1 1 0
22 2
m 1 2n
16 1
E b a h h
l
ab
;
2 2
2
3 2
2 1 0 2 1 0 2 2 2 2 2 1 0
23 2 2 2 2 2
n 2 3 n
n m
24 1 4 1 4 1
E m h h E n h h E h h
l a b
m b a b a a
1
1 0
2 2
2 2
2 2 2 4n 3 1 m
1 9
E h h
R a b
mb a
2 2 2 2 2 2 2 3 2 2 2 2 2 2
2 1 0 2 1 0 2 1 0
31 2 2 2 2
2 3 m n m n n
1 4
24 1 4 1
E h h m b a E h h m b a E h h
l mba a b mb
