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

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.

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

fourth-order 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 third-order 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

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 h c, hm and h x=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 (N 01 = -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 V m(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

volume-fraction V c(z) is expressed as follows

   

0, 5

k m

z h h

h h h











(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

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 x z x y y z y xy xy z xy

Where

2 2

               

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

2 0 2 0

1

;

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

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

;

(8)

In which: N N N x; y; xyare internal forces and M M M x; y; xy are moment resultants

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

From equation (8) we define the deformation expression and moment components of circular cylinder shell as follows

, , 2

(9)

, ,

M B N B N D D B A A B A A

M B N B N D D B A A B A A

M B N D



(10)

In which: * *

ij , ij

A B and *

ij

D (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

, ,

x

x

2

w 2 w P L,

(11)

Where: - the coefficient of damping

1

1

cm x

h

h h

k



     

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

2 2

w

t





( )

R I

M

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<<w) Equations (11) can be rewritten as follows:

1

2

0; 0;

x

x

N



M L w 2 w

t  t

(14)

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

x y

 

Putting Eq (9) into Eq (6), Eq (10) into the 3rd equation of Eqs (14) we obtain

 

2

2

1

x y y R x x y x y

(16)

1

L

F F F F w F w F w w w

x y R x y x x y x y x y x y

(17)

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 N 01 =-ph and external pressure q(t) The cylindrical shell under

simply supported at both ends, the boundary condition can be expressed as following

M x = N xy =0, N x =N 01 , w=0, at x=0 and x=L

The shells’ deflection satisfies above boundary condition can be presented in form as

W f t( ) sinm x sinny



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 2

1 cos(2 ) 2 cos(2 ) 3 sin( ) sin( ) 02 01

2

;

2

;







2 2

*

2 2

R



N 02 is the average circumferential load and can be defined by using

circumferentially closed condition for circular cylindrical shell [16] as

2

0

1

0 2

y



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

   

A



2 * 11

1





 

11 12 1

* 11

A A

Substituting equation (22) into equation (19) we obtain

 

t t



(23)

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

       3  2   12*

11

A

mn R mn R A mn

In which:

mn

2

*

11

4

R A







Equation (24) is motion equation to study non-linear dynamic responses of full-filled 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 N 01=-ph with p=c1.t and q=0 (c1 - the loading speed)

Case 2 Consider the sandwich–FGM circular cylinder shell under uniform

pre-axial 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 (h c =h m =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:

ρ Ni =8900 kg.m -3 , ν Ni =0.31, E Ni =205.09 GPa; ρ S =8166 kg.m -3 , ν S =0.32, ES=207.79 GPa; and the temperature T=300 K

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

m=1, n=8, h=0.05m, L/R=R/h=20, ΔT=300K, f =ω/2π (Hz)

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

Titanium-alloy Material properties are ν m = ν c =0.288; ρ m =4429 kg.m -3 ; E m =122.56 GPa; Ec=244.27 GPa; ρc=5700 kg.m -3

Table 2 Dynamic critical stress comparison of compressed cylinder shell (Mpa)

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 N 01=-ph with p=c1.t and q=0 In which c1 -the loading speed In this case,

critical time t cr are determined by using Budiansky–Roth criterion Dynamic

critical force can be defined as follows p cr =c 1 t cr

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

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 P cr =75.2GPa at 00C to P cr =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

(P cr=61.5GPa) is higher than those of without elastic mediums ones (P cr=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

Fig 6 shows that the critical force of FGM shell (P cr=3.71GPa) is lower than the critical force of sandwich-FGM shell (P cr=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 N 01 =const and external pressure varying on time q=c 2 t (in which c 2- 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 sandwich-FGM cylindrical shell Graphs shows that the critical load of fluid-filled cylindrical

shell (q cr=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

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 q cr =22.5 MPa to q cr=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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