Investigation of nonlinear dynamic responses of sandwich FGM cylindrical shells containing fluid resting on elastic foundations in thermal environment

Research

INVESTIGATION OF NONLINEAR DYNAMIC RESPONSES
OF SANDWICH-FGM CYLINDRICAL SHELLS CONTAINING
FLUID RESTING ON ELASTIC FOUNDATIONS
IN THERMAL ENVIRONMENT
Khuc Van Phu1, Nguyen Minh Tuan2, Dao Huy Bich1, Le Xuan Doan2*
Abstract: The main aim of the present article is to investigate nonlinear dynamic
responses of sandwich-FGM circular cylinder shell containing fluid and surrounded
by Winkler- Pasternak elastic mediums under mechanical loads in the thermal
environment based on classical shell theory. Bubnov-Galerkin method and fourthorder Runge-Kutta method are employed to determine nonlinear dynamic buckling
of cylindrical shell. Effects of temperature environment, foundations, structure's
geometrical parameters, material parameters and fluid on the nonlinear dynamic
responses of sandwich-FGM circular cylinder shell are investigated.
Keywords: Sandwich-FGM; Dynamic stability; Cylindrical shell; Thermal-mechanical load; Filled with fluid.

1. INTRODUCTION
Functionally Graded Material (FGM) is an important material in modern
engineering design and more and more extensively used in many industries.
Researches on nonlinear dynamic stability of these structure has received attentions
by scientists, especially shell structure. Bich D. H et al studied on natural frequencies
and dynamic buckling of FG cylinder panels reinforced by eccentrically stiffeners
[1] and thin circular cylinder shell [2] based on classical shell theories. The Runge–
Kutta method and smeared stiffener technique were employed to investigate. B.
Mirzavand et al. [3] solved dynamic post-buckling problems of FG cylinder shells
with piezoelectric layer on surface under electro-thermal load based on the thirdorder shear deformation shell theory and Sander’s nonlinear kinematic relations.
Nguyen Dinh Duc et al. [4], [5] analyzed nonlinear responses of imperfect
eccentrically stiffened thin and thick S-FGM cylinder shells resting on an elastic
mediums and subjected to mechanical load in thermal environment.
Study on full-filled fluid FGM shells, Sheng et al. [6] investigated vibration of
FG circular cylinder shells containing flowing fluid under mechanical load and

surrounded by elastic foundations including effect of thermal environment. This
study was continuously expanded to investigate dynamic responses of FGM
circular cylinder shell containing flowing fluid subjected to mechanical and
thermal loads [7]. Zafar Iqbal et al. [8] analyzed vibration frequencies of full-filled
fluid FGM circular cylinder shell. Vibration frequencies of shell were examined
for various boundary conditions including the effect of fluid. Silva et al. [9]
resolved nonlinear vibration problems of fluid-filled FG cylinder shell subjected to
mechanical load. By using the Rayleigh-Ritz method, vibration frequencies of
FGM cylindrical shell filled with fluid or containing a flowing fluid partially
surround by two parameters elastic foundation were examined by Y. W. Kim et al.
[10]. Hong-Liang Dai et al. [11] studied on thermos electro elastic behaviors of a
thin FG piezoelectric material cylinder shell filled with fluid and under mechanical
and electrical loads in thermal environment. According to the classical shell theory

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Mechanics & Mechanical engineering

and using Galerkin method, Phu Van Khuc et al. [12] investigated non-linear
responses of circular cylinder shells made of Sandwich-FGM filled with fluid
subjected to mechanical load in thermal environment.
The review of the literature signifies that there is no research on the analytical
solution for dynamic stability of full-filled fluid sandwich-FGM circular cylinder
shells surrounded by elastic foundations. In the present article, nonlinear dynamic
equations of full-filled fluid sandwich-FGM circular cylinder shell resting on
elastic mediums under mechanical load including the effect of thermal
environment are established base on the classical shell theory. Bubnov-Galerkin

method and Runge-Kutta method are employed to determine nonlinear dynamics
responses of circular cylinder shells. Dynamic critical loads are defined by
applying Budiansky–Roth criterion.
2. SANDWICH- FGM CYLINDRICAL SHELL
Examine a cylindrical shell made of sandwich-FGM with the thickness, the
length and curvature radius of shell are h, L and R, respectively. Configuration and
Coordinate system of sandwich-FGM cylinder shell are performed in fig. 1. In
which hc, hm and hx=h-hc-hm are thickness of ceramic layer, metal layer and FGM
core layer, respectively. The cylindrical shell surrounded by two-parameter elastic
mediums with stiffness are: K1 (Nm-3) and K2 (Nm-1). Assume that the cylindrical
shell subjected to simply supported at both ends and under pre-axial compression
load (N01= -ph) and external pressure which uniformly distributed varying on time
q(t) in the thermal environment. Suppose that environment's temperature is steadily
increased and ΔT is constant.

Fig. 1. Configuration and Coordinate system of sandwich-FGM cylinder shell
filled with fluid embedded in elastic foundations.
We denote that Vm(z) and Vc(z) are metal and ceramic volume fractions,
respectively. Suppose that volume fraction of Metal and Ceramic are constantly
changed and distributed according to the exponential law. Ceramic’s volumefraction Vc(z) is expressed as follows
96 K. V. Phu, …, L. X. Doan, “Investigation of nonlinear dynamic … in thermal environment.”


Research
Vc  z   0


 z  0, 5h  hm

Vc  z   

 h  hc  hm


Vc  z   1


,

 0, 5h  z    0, 5h  hm 

k


 ,   0, 5h  hm   z   0, 5h  hc  , k  0

,
 0, 5h  hc   z  0, 5h

(1)

For this rule, properties of material Q(z) such as thermal expansion coefficient
α, the Young's modulus E, and the mass density ρ, change through the thickness of
shell and can be obtained as follows
Q  z   QmVm  z   QcVc  z   Qm   Qc  Qm  Vc  z  ,
(2)
The Poisson’s ratio ν is assumed to be constant
3. GOVERNING EQUATIONS
According to the classical shell theory, with von Karman-Donnell sense type of
geometrical non-linearity, the nonlinear relation of strains and displacement for a
circular cylinder shell can be expressed as [17]

x  x0  zx ;

 y   y0  zy ;

 x0 

u 1  w  0 v w 1  w  0 u v w w
 
;
 ;  xy   
 ; y    
x 2  x 
y R 2  y 
y x x y

x  

 2w
;
x 2

(3)

2

2

Where

 xy   xy0  2zxy

y  

 2w
;
y 2

 xy  

 2w
;
xy

(4)
(5)

In which: u, v and w are displacements in the x, y and z direction.
From Eqs. (4), the equation of deformation compatibility can be obtained as
 2  x0
y 2



 2  y0
x 2

2

 2  xy0

 2 w 
2 w 2 w 1 2 w



;
 
x y  x y 
R x 2
x 2 y 2

(6)

Assume that material properties are independent on temperature and
temperature in the cylindrical shell is only transmitted in the z-axis direction,
Hooke’s law for sandwich-FGM cylindrical shell subjected to thermo-mechanical
loads can be written as
E z
E  z   z 
E  z
E  z   z 
E  z
x 
( x   y )  T
; y 
( x   y )  T
; xy 
. xy
(7)
2

2
1 
1 
1 
1 
2 1   
Where: σx; σy and τxy are stress components in circular cylinder shell.
Internal forces and moment resultants expressions of circular cylinder shell can
be defined as follows
N x  A11 x0  A12  y0  B11  x  B12  y  1

M x  B11 x0  B12  y0  D11  x  D12  y   2

N y  A12  x0  A22  y0  B12  x  B22  y  1 ; M y  B12  x0  B22  y0  D12  x  D22  y   2
N xy  A66  xy0  2 B66  xy

In which:

N x ; N y ; N xy

(8)

M xy  B66  xy0  2 D66  xy

are internal forces and

M x ; M y ; M xy

are moment resultants.

The additional internal forces and moments resultants which are created by the
increase in temperature 1, 2 and stiffness coefficients Aij, Bij and Dij (i, j = 1, 2,
6) in Eq. (8) are expressed in Appendix I.

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Mechanics & Mechanical engineering

From equation (8) we define the deformation expression and moment
components of circular cylinder shell as follows
*
*
 x0  A22
Nx  A12* N y  B11*  x  B12*  y   A22
 A12*  1,
*
*
 y0   A12* Nx  A11* N y  B21
x  B22
 y   A11*  A12*  1,
*
*
 xy0  A66
Nxy  2B66
xy

(9)

*
*
*
*
*
M x  B11
N x  B21
N y  D11
 x  D12*  y  B11  A22
 A12
 B12  A11*  A12*  1  2 ,
*
*
*
*
*
*
M y  B12
N x  B22
N y  D21
 x  D22
 y  B12  A22
 A12
 B22  A11*  A12*  1  2 ,
*
*
M xy  B66
N xy 2 D66
 xy ,

In which:

Aij* , Bij* and D ij*

(10)

(i, j=1, 2, 6) are explained in Appendix II.

Based on the classical shell theory [17], motion equations of circular cylinder
shell filled with fluid resting on an elastic mediums subjected to pre-axial
compression load p and an external pressure varying on time q(t) can be expressed as
2
Nx Nxy

 1  u2 ,
x y
t

2
N y
Nxy

 1  v2 ,
y
x
t

2
2

2
2 M xy
N
 2 2 w 
2 M x 2 M y


2
 Nx  w
 Ny  w
 y  2 Nxy  w  K1w  K2   w
 2 q
2
2
2
2
2
x
y
xy
x
y
R
xy
 x y 

(11)

2

 1  w
 21 w  PL ,
t 2
t

Where:  - the coefficient of damping
 1   m h   cm hc 

 cm h x

(12)

.

k 1

PL - dynamic pressure of fluid exerting on the inner surface of shell and be
determined by expression [12]:
pL  M L

2w
t 2

in which:

M

L



 L .R.I n ( m )
m .I n' ( m )

and ( m  m R )

(13)

L

ML- mass of fluid corresponding to shell’s vibration.
Substituting equation (13) into equations (11) then applying Volmir’s
assumption [18] (because of u, v<N x N xy

 0;
x
y

N xy
x



N y
y

 0;

 2 M xy

 2w 2w 
2M x 2M y
2w
2w N
2w


2
 N x 2  N y 2  y  2 N xy
 K1 w  K2  2  2   q
2
2
x
y
xy
x
y
R
xy
 x y 

(14)
2

 1  M L   w2 1 w
t 2
t

The first and the second equation of equations (14) are satisfied identically by
recommending the stress function F as follows:

Nx 

2F
;
y 2

Ny 

2F
;
x 2

N xy  

2F
;
xy

(15)

Putting Eq (9) into Eq (6), Eq (10) into the 3rd equation of Eqs. (14) we obtain
98 K. V. Phu, …, L. X. Doan, “Investigation of nonlinear dynamic … in thermal environment.”


Research
A11*

4
4
4 F

4 F
*
*
*  F
*  w

A

2
A

A

B



66
12
22
21
x 4
y 4
x 4
x 2  y 2

(16)

2

4w
4w 1 2w  2w  2w 2w
  B  B  2 B  2 2  B12*


0
 
x y
y 4 R x 2   x y  x 2 y 2
*
11

*
22

*
66

2
4
4
4
4
1 M L   w2  21 w D11*  w4 D12*  D21*  4D66*  2 w 2  D22*  w4 B11*  B22*  2B66*  2 F 2 

t

2

t

2

x

2

2

 x y

2

2

2

2

y

2

 x y

2

(17)

2

 F *  F 1  F  F  w  F  w  F  w   w  w
B
 B12 4 


2


 K 2  K1w q  0
x 2
y R x2 y 2 x2 x2 y 2 xy xy  x2 y 2 
*
21

Equation (16) and equation (17) are nonlinear basic equations to survey nonliear
dynamic responses of full-filled fluid Sandwich-FGM cylindrical shell resting on
elastic foundations subjected to mechanical load in thermal environment.
4. SOLUTION METHOD
Suppose that the sandwich FGM circular cylinder shell subjected to pre-axial
compression load N01=-ph and external pressure q(t). The cylindrical shell under
simply supported at both ends, the boundary condition can be expressed as
following
Mx= Nxy=0, Nx=N01, w=0, at x=0 and x=L

The shells’ deflection satisfies above boundary condition can be presented in
form as
W  f (t ). sin

ny

m x
. sin
L
R

(18)

Where: n, m are the x and y direction half waves numbers, respectively.
Putting Eq. (18) into Eq. (16) then solve the equation, we obtain the stress
function F as
2

F  F1 cos(2 x )  F2 cos(2  y )  F3 sin( x ) sin(  y )  N 02

In which:



(19)

m
n
2
2
2
*
2
; 
; F1 
.

f

F
.
f
;
F

. f  t2  F2* . f  t2
1
t
t
*

2 32  2 A22
L
2R
32 2 A11*  

*
*
*
 4 B21
    B11*  B22
 2 B66
   4 B12*  
2

F3 

x2
y
 N 01
2
2

2

 4 A11*     A66*  2 A12*    4*22

2

R f

t 

 F3* f

t 

(20)

N02 is the average circumferential load and can be defined by using
circumferentially closed condition for circular cylindrical shell [16] as
L 2 R


0

0

2
L  2 R 
v
w 1  w   
dxdy      y0   
  dy dx  0

y
R 2  y   
0  0 



(21)

Substituting equations (9), (18), (19) and (20) into equation (21) we obtain
N02  

In which:  

2
*
11

8A

f t2  f t  

A12*

N01  
A11*

(22)

  2 *
1 

 A11   2 A12*  F3*    2 B21*   2 B22*  
2 * 
mn A11 
R 


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Mechanics & Mechanical engineering


 A11*  A12* 1 ;   1 m 1,    1 n 1.
*
  
  
A11

Substituting equation (22) into equation (19) we obtain
  2 f ( t2)

 x2
A12* N 01
y2
F  F1 cos 2 x  F2 cos 2  y  F3 sin  x sin  y  


f




N

01
t 
 8 A*
 2
A11*
2
11



(23)

Putting (18) and (22) into equation (17) and applying Galerkin method, we obtain

 1  M L  ft  21 ft  1 ft3  2 ft2  3 ft 

4

4 A12*
4

N01 
q0
2
mn R
mn 2 R A11*
mn 2

(24)

In which:
1 

4
8 A11*

2

 2  F1*  F2* ,  2 

16  2
1
3  2 3mn 2 2 
2
4 *  *
4 *
*
*


4

B
F

4

B
F


F


  ,



21
1
12
2
3


3mn 2  R
2
32 R. A11* 16




2 
2
2
*
*
*   4 *
*
*
3   4 D11*     D12*  D21
 4 D66
  4 D22
  B21     B11*  B22
 2 B66
  4 B12*   F3*



 
R

A* 
4
   2   2 12*  N 01   K1   2   2  K 2    2 

A
mn
 2R


11 

Equation (24) is motion equation to study non-linear dynamic responses of fullfilled fluid sandwich FGM circular cylinder shell embedded in elastic mediums
subjected to pre-axial compression at both ends and external pressure varying on
time including the effect of thermal environment.
For dynamic stability problems of cylindrical shell, this article analyzes two
cases as:
Case 1. Assume that sandwich–FGM cylindrical shell subjected to linear axial
compression load varying on time N01=-ph with p=c1.t and q=0. (c1 - the loading
speed)
Case 2. Consider the sandwich–FGM circular cylinder shell under uniform preaxial compression load and external uniform distributed pressure varying on time:
q=c.t and N01=const. In which c2 - the loading speed.
To determine dynamic responses of circular cylinder shell, we solve Eq. (24) for
case 1 and case 2 respectively. Dynamic critical loads correspond to case 1 and
case 2 are defined by using Budiansky–Roth criterion [18].
5. NUMERICAL RESULTS AND DISCUSSION
Validation
To verify the present study, obtained results of this article are compared with
cylinder shell without fluid made of FGM (hc=hm=0). Natural vibration
frequencies of FGM cylinder shell without fluid will be compared with Loy et al
and Shen's publications (ref [14] and ref [15]) and be displayed in table 1. The
cylindrical shell made of nickel and stainless steel, properties of material are:
ρNi=8900 kg.m-3, νNi=0.31, ENi=205.09 GPa; ρS=8166 kg.m-3, νS=0.32,
ES=207.79 GPa; and the temperature T=300 K.

100 K. V. Phu, …, L. X. Doan, “Investigation of nonlinear dynamic … in thermal environment.”


Research

Table 1. Natural frequencies comparison (Hz) of FGM cylinder shell without fluid.
k

0

0.5

1

2

5

15

m=1, n=8, h=0.05m, L/R=R/h=20, ΔT=300K, f =ω/2π (Hz)
Loy et al (ref.[14])

763.98

750.12

743.82

737.86

731.97

727.92

Shen (ref.[15])

759.91

746.28

740.07

734.18

728.31

724.26

Present

766.14

751.05

743.85

736.94

730.09

725.54

Dynamic critical stress of FGM circular cylinder shell are also compared with
publication of Huang et al. [13] (Table 2) with material are Zirconia and Titaniumalloy. Material properties are νm= νc =0.288; ρm=4429 kg.m-3; Em=122.56 GPa;

Ec=244.27 GPa; ρc=5700 kg.m-3.
Table 2. Dynamic critical stress comparison of compressed cylinder shell (Mpa).
k

0.2

1.0

5.0

Huang &Han (ref.[13])

194.94 (2,11)

169.94 (2,11)

150.25 (2,11)

Present

193.91 (1,9)

168.69 (1,9)

149.17 (1,9)

Table 1 and table 2 show that results of this article are excellent agreement with
above publications, but there is slightly difference. The reason for this difference is
the different methods which authors used. Therefore, results of the present paper
are reliable

Dynamic buckling analysis
Case 1. Sandwich FGM cylinder shell under linear axial compression load varying
on time N01=-ph with p=c1.t and q=0. In which c1 -the loading speed. In this case,
critical time tcr are determined by using Budiansky–Roth criterion. Dynamic
critical force can be defined as follows pcr=c1.tcr.

Fig. 2. Effect of fluid on nonlinear
responses of circular cylinder shell.

Fig. 3. Nonlinear dynamic responses of
cylindrical shell with various k.

Dynamic responses of fluid-filled and fluid-free Sandwich-FGM cylindrical
shell resting on elastic mediums including the effect of thermal environment are

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Mechanics & Mechanical engineering

indicated in fig.2. It can be seen that the dynamic critical force of fluid-filled shells
is 3.95 times as much as the fluid-free ones.
Effect of volume fractions index k on dynamic responses of cylindrical shell is
demonstrated in fig. 3. Dynamic critical load of the shell decrease with the increase
of k. In other words, the load-bearing ability of the shell decreases when k
increases.

Fig. 4. Effect of thermal environment

on dynamic responses of circular
cylinder shell.

Fig. 5. Influence of foundation on nonlinear
dynamic responses of cylinder shell.

Dynamic critical load of circular cylinder shell will decrease (fig. 4) with the
rise of temperature from Pcr=75.2GPa at 00C to Pcr=68GPa at 1000C. That means,
temperature reduces the load- bearing ability of circular cylinder shell.
Dynamic critical force of circular cylinder shell resting on elastic mediums
(Pcr=61.5GPa) is higher than those of without elastic mediums ones
(Pcr=57.5GPa) (fig.5). That means elastic foundations enhance the stability of
axially compressed shell.

Fig. 6. Influence of material on
nonlinear dynamic responses of
cylinder shell.

Fig. 7. Effect of fluid on nonlinear
dynamic responses of cylinder shell.

102 K. V. Phu, …, L. X. Doan, “Investigation of nonlinear dynamic … in thermal environment.”


Research

Fig. 6 shows that the critical force of FGM shell (Pcr=3.71GPa) is lower than
the critical force of sandwich-FGM shell (Pcr=3.8GPa). That means, with the same
geometrical parameters, the workability of sandwich-FGM circular cylinder shell
is better than FGM ones.

Case 2. Sandwich-FGM circular cylinder shell under uniform pre-axial
compression load N01=const and external pressure varying on time q=c2.t (in
which c2- loading speed).

Fig. 8. Nonlinear dynamic responses of
circular cylinder shell with various k.

Fig. 9. Temperature effect on nonlinear
dynamic responses of cylindrical shell.

Fig. 10. Effect of foundations on
nonlinear dynamic responses of shell.

Fig. 11. Effect of material on nonlinear
dynamic responses of shell.

Figure. 7 displays dynamic responses of fluid-filled and fluid-free sandwichFGM cylindrical shell. Graphs shows that the critical load of fluid-filled cylindrical
shell (qcr=149MPa) is 5.7 times as much as fluid-free ones (qcr=26MPa).
Figure. 8 and figure. 9 illustrates nonlinear dynamic responses of cylindrical
shell with various k and effects of temperature on dynamic responses of cylinder
shell. Graphs show that the critical load decreases with the increasing of
temperature. That means the shell load-bearing ability will decreases when the
increase of temperature.

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Mechanics & Mechanical engineering

Effect of elastic mediums on nonlinear dynamic responses of cylindrical shell
are given in fig.10. The graph shows that elastic mediums increase the shells’
critical load (from qcr=22.5 MPa to qcr=27 MPa). That means elastic mediums
enhance the stability of circular cylinder shell.
Fig. 11. shows that the critical force of fluid-filled sandwich-FGM cylindrical
shell (qcr=27.9 MPa) is lager than those of FGM ones (qcr=26.9 MPa). That means,
with the same geometrical parameters, the workability of sandwich-FGM
cylindrical shell is better than FGM ones.
6. CONCLUSION
The present article established nonlinear dynamic equation of sandwich FGM
circular cylinder shells containing fluid and embedded in elastic mediums
including the effect of thermal environment.
The present study gives some pivotal conclusions:
 The fluid remarkably effects on dynamic stability of circular cylinder
shells, it has increased the dynamic critical force of circular cylinder shells. That
means, fluid enhances stability of shell.
 Elastic foundations enhance stability of shell structures, it has increased the
critical load of the cylindrical shell.
 The dynamic critical load of shell decreases with the increasing of L/R
ratio. That means, the longer cylindrical shell is, the less stability is.
 The load-bearing ability of sandwich-FGM cylindrical shell is better than
those of FGM ones. That means, with the same geometrical parameters, the
workability of sandwich-FGM cylindrical shell will more excellent than FGM ones.
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Mechanics & Mechanical engineering

[18]. A.S. Volmir, “Nonlinear Dynamics of Plates and Shells,” Science edition,
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Appendix I
A11  A22 
D11  D22 

E1
1 
E3

2

1 

2

; A12 
; D12 

 E1
1 
 E3

2

; B11  B22 

1 

2

; A66 

E2
1 

E1
2 1   

2

; B12 

; B66 

 E2

1 
E2

2 1   

2

;

; D66 

E3
2 1   

h/2
Ecm  hc  hm  h 
 E  z  dz  Em h   Ec  Em  hc 
k 1
h / 2
E  Em hc h
E  Em hc2
h/2
c
E2   E ( z ).z.dz 
 c
2
2
h / 2
2
 E  Em  0.5h  hc  hc  hm  h   Ecm  hc  hm  h 
 c
k 1
 k  1 k  2 
E1 









2
h/2
 Em  Ec  0.5h  hc   hc  hm  h  2  Ec  Em 
2
2
E
z
z
dz




 0.5h  hc  hc  hm  h  

k

1
k

1
k


2



h / 2
2  Ec  E m 
E h3 E hh
E
3hh
3

 hc  hm  h   c c  c c  0.5h  hc   m  hm3  m  0.5h  hm 
k

1
k

2
k

3
3
2
3
2
 





E3 



Em 
3
  hc  hm  h   3  0.5h  hm  0.5h  hc  hc  hm  h  

3 

a 

If

h/2
 E  z    z  Tdz ,
1   h / 2
1

T  const

With:

b 

h/2
 E  z    z  Tzdz ,
1   h / 2
1

1
PT ;
, a 
1 

P  Em m (2h  hc )  Ec c hc 

Emcm  hc  hm  h 
k 1



Ecm m  hc  hm  h 
k 1



Ecm cm  hc  hm  h 
2k  1

Appendix II
B
A
A
A
1
11
12
22

; A* 
; A* 
; A* 
; B*  66 ;
12
22
66 A
66 A
A A A 2
A A A 2
A A A 2
66
66
11 22 12
11 22 12
11 22 12
A B A B
A B A B
A B A B
A B A B
B*  22 11 12 12 ; B*  22 12 12 22 ; B*  11 12 12 11 ; B*  11 22 12 12 ;
11
12
21
22
A A A 2
A A A 2
A A A 2
A A A 2
11 22 12

11 22 12
11 22 12
11 22 12
*
*
*
*
*
*
*
*
D  D  B B  B B ; D  D  B B  B B ; D  D  B B  B B* ;
11
11 11 11 12 21 12
12 11 12 12 22 21
12 12 11
22 21
*
*
*
*
*
D  D  B B  B B ;D  D  B B ;
22
22 12 12
22 22 66
66
66 66
A* 
11

106 K. V. Phu, …, L. X. Doan, “Investigation of nonlinear dynamic … in thermal environment.”


Research

TÓM TẮT
PHÂN TÍCH VỀ ĐÁP ỨNG ĐỘNG LỰC HỌC PHI TUYẾN
CỦA VỎ TRỤ TRÒN SANDWICH-FGM CHỨA CHẤT LỎNG
TRÊN NỀN ĐÀN HỒI TRONG MÔI TRƯỜNG NHIỆT ĐỘ
Mục tiêu chính của bài báo là phân tích đáp ứng động lực phi tuyến của
vỏ trụ tròn sandwich-FGM chứa chất lỏng và được bao quanh bởi nền đàn
hồi Winkler- Pasternak chịu tải trọng cơ học trong môi trường nhiệt độ bằng
cách tiếp cận giải tích dựa trên lý thuyết vỏ cổ điển. Phương pháp BubnovGalerkin. và phương pháp Runge-Kutta bậc bốn được sử dụng để xác định
đáp ứng động lực phi tuyến của vỏ. Ảnh hưởng của môi trường nhiệt độ, nền
đàn hồi, thông số vật liệu, thông số hình học của kết cấu và ảnh hưởng của
chất lỏng đến đáp ứng động lực học của vỏ trụ tròn sandwich-FGM cũng
được nghiên cứ và khảo sát.
Từ khóa: FGM-Sandwich; Ổn định động lực; Tải cơ-nhiệt; Vỏ trụ tròn, chứa chất lỏng.

Received date, 18th April, 2019
Revised manuscript, 06th May, 2019
Published, 15th May, 2019
Author affiliations:
1
Vietnam National University;
2
Academy of Military Science and Technology.
*Corresponding author:

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