DSpace at VNU: Nonlinear dynamic analysis of imperfect functionally graded material double curved thin shallow shells with temperature-dependent properties on elastic foundation

The online version of this article can be found at: DOI: 10.1177/1077546313494114 published online 31 July 2013 Journal of Vibration and Control Nguyen Dinh Duc and Tran Quoc Quan with

The online version of this article can be found at:

DOI: 10.1177/1077546313494114

published online 31 July 2013

Journal of Vibration and Control

Nguyen Dinh Duc and Tran Quoc Quan

with temperature-dependent properties on elastic foundation Nonlinear dynamic analysis of imperfect functionally graded material double curved thin shallow shells

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Journal of Vibration and Control

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Nonlinear dynamic analysis of imperfect

functionally graded material double

curved thin shallow shells with

Keywords

Elastic foundation, FGM double curved thin shallow shells, imperfection, nonlinear dynamic analysis, dependent properties

temperature-1 Introduction

Functionally graded materials (FGMs), which are

microscopically composites and made from a mixture

of metal and ceramic constituents, have received

con-siderable attention in recent years due to their high

per-formance heat resistance capacity and their excellent

characteristics in comparison with conventional

com-posites By continuously and gradually varying the

volume fraction of constituent materials through a

spe-cific direction, FGMs are capable of withstanding

ultra-high temperature environments and extremely large

thermal gradients Therefore, these novel materials

are chosen for use in temperature shielding structure

components of aircraft aerospace vehicles, nuclear

plants and engineering structures in various industries

As a result, in recent years important studies have been

undertaken about the stability and vibration of FGM

plates and shells

The research on FGM shells and plates underdynamic load is attractive to many researchers in dif-ferent parts of the world Firstly we have to mention theresearch group of Reddy et al The vibration of func-tionally graded cylindrical shells has been investigated

by Loy et al (1999) Lam and Li Hua (1999) has takeninto account the influence of boundary conditions onthe frequency characteristics of a rotating truncatedcircular conical shell Pradhan et al (2000) studiedvibration characteristics of FGM cylindrical shellsunder various boundary conditions Ng et al (2001)

Vietnam National University, Hanoi, Vietnam

Corresponding author:

Nguyen Dinh Duc, Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam.

Email: ducnd@vnu.edu.vn Received: 22 March 2013; accepted: 19 May 2013

Journal of Vibration and Control 0(0) 1–23

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studied the dynamic stability analysis of functionally

graded cylindrical shells under periodic axial loading

The group of Ng et al (2003) also published results on

generalized differential quadrate for free vibration of

rotating composite laminated conical shell with various

boundary conditions In the same year, Yang and Shen

(2003) published the nonlinear analysis of FGM plates

under transverse and in-plane loads Zhao et al (2004)

studied the free vibration of a two-sided simply

sup-ported laminated cylindrical panel via the mesh-free

kp-Ritz method With regard to vibration of FGM

plates, Vel and Batra (2004) gave a three-dimensional

exact solution for the vibration of FGM rectangular

plates Sofiyev and Schnack (2004) investigated the

sta-bility of functionally graded cylindrical shells under

lin-early increasing dynamic tensional loading and

obtained the result for the stability of functionally

graded truncated conical shells subjected to a periodic

impulsive loading They also published the result of the

stability of functionally graded ceramic–metal

cylin-drical shells under a periodic axial impulsive loading

in 2005 Ferreira et al (2006) received natural

frequen-cies of FGM plates by a meshless method Zhao et al

(2006) used the element-free kp-Ritz method for free

vibration analysis of conical shell panels Liew et al

(2006a, 2006b) studied the nonlinear vibration of a

coating-FGM-substrate cylindrical panel subjected to

a temperature gradient and dynamic stability of

rotat-ing cylindrical shells subjected to periodic axial loads

Woo et al (2006) investigated the nonlinear free

vibra-tion behavior of funcvibra-tionally graded plates Ravikiran

Kadoli and Ganesan (2006) studied the buckling and

free vibration analysis of functionally graded

cylin-drical shells subjected to a temperature-specified

boundary condition Wu et al (2006) published their

results of nonlinear static and dynamic analysis of

func-tionally graded plates Sofiyev (2007) has considered

the buckling of functionally graded truncated conical

shells under dynamic axial loading Prakash et al

(2007) studied the nonlinear axisymmetric dynamic

buckling behavior of clamped functionally graded

spherical caps Darabi et al (2008) obtained the

non-linear analysis of dynamic stability for functionally

graded cylindrical shells under periodic axial loading

Natural frequencies and buckling stresses of FGM

plates were analyzed by Hiroyuki Matsunaga (2008)

using 2-D higher-order deformation theory Shariyat

(2008a, 2008b) also obtained the dynamic thermal

buckling of suddenly heated temperature- dependent

FGM cylindrical shells under combined axial

compres-sion and external pressure and dynamic buckling of

suddenly loaded imperfect hybrid cylindrical FGM

with temperature-dependent material properties under

thermo-electro-mechanical loads Allahverdizadeh

et al (2008) studied nonlinear free and forced vibration

analysis of thin circular functionally graded plates.Sofiyev (2009) investigated the vibration and stabilitybehavior of freely supported FGM conical shells sub-jected to external pressure Shen (2009) published avaluable book, Functionally Graded Materials, Nonlinear Analysis of Plates and Shells, in which the resultsabout nonlinear vibration of shear deformable FGMplates are presented Zhang and Li (2010) publishedthe dynamic buckling of FGM truncated conicalshells subjected to non-uniform normal impact load.Ibrahim and Tawfik (2010) studied FGM plates subject

to aerodynamic and thermal loads Fakhari and Ohadi(2011) investigated nonlinear vibration control of func-tionally graded plate with piezoelectric layers in ther-mal environment Ruan et al (2012) analyzed dynamicstability of functionally graded materials’ skew platessubjected to uniformly distributed tangential followerforces Najafov et al (2012) studied vibration and sta-bility of axially compressed truncated conical shellswith a functionally graded middle layer surrounded

by elastic medium Bich et al (2012, 2013) investigatednonlinear dynamical analysis of eccentrically stiffenedfunctionally graded cylindrical panels and shallowshells using the classical shell theory Recently, Duc(2013) investigated the nonlinear dynamic response ofimperfect FGM double curved shallow shells eccentric-ally stiffened on an elastic foundation

However, in practice, the FGM structure is usuallyexposed to high-temperature environments, where sig-nificant changes in material properties are unavoidable.Therefore, the temperature dependence of their proper-ties should be considered for an accurate and reliableprediction of deformation behavior of the composites.Duc and Tung (2010) investigated mechanical andthermal post-buckling of FGM plates with tempera-ture-dependent properties using first order sheardeformation theory Huang and Shen (2004) studiednonlinear vibration and dynamic response of FGMplates in a thermal environment with temperature-dependent material properties – volume fraction follows

a simple power law for P-FGM plate Shariyat gated vibration and dynamic buckling control of imper-fect hybrid FGM plate subjected to thermo-electro-mechanical conditions (2009) and dynamic buckling ofsuddenly load imperfect hybrid FGM cylindrical shells(2008) with temperature-dependent material properties.Kim (2005) studied temperature dependent vibrationanalysis of functionally graded rectangular plates bythe finite element method It is evident from the litera-ture that investigations considering the temperaturedependence of material properties for FGM shells arefew in number It should be noted that all the publica-tions mentioned above (Huang and Shen, 2004; Kim,2005; Shariyat, 2009) use displacement functions andthe volume fraction follows a simple power law

investi-This paper presents a dynamic nonlinear response of

double curved FGM thin shallow shells with

tempera-ture-dependent properties subjected to mechanical load

and temperature on an elastic foundation The

formu-lations are based on the classical shell theory taking

into account geometrical nonlinearity, initial

geomet-rical with Pasternak type elastic foundation The

obtained results show the effects of material,

geomet-rical properties, elastic foundation and imperfection on

the dynamical response of FGM shallow shells

2 Double curved FGM shallow shell

on elastic foundation

Consider an FGM double curved thin shallow shell of

radii of curvature Rx,Ry length of edges a, b and

uni-form thickness h A coordinate system ðx, y, zÞ is

estab-lished in which ðx, yÞ plane on the middle surface of the

shell and z is the thickness direction ðh=2  z  h=2Þ,

as shown in Figure 1

For the P-FGM shell, the volume fractions of

con-stituents are assumed to vary through the thickness

according to the following power law distribution:

VmðzÞ ¼ 2z þ h

2h

, VcðzÞ ¼1  VmðzÞ ð1Þ

where N is volume fraction index (0  N 5 1)

Effective properties Preff of the FGM panel are

deter-mined by linear rule of mixture as

Pr

effðzÞ ¼Pr

m VmðzÞ þPr

c VcðzÞ ð2Þ

where Pr denotes a temperature independent material

property, and subscripts m and c stand for the metal

and ceramic constituents, respectively Specific

expres-sions of modulus of elasticity Eðz, TÞ, ðz, TÞ, ðz, TÞ,

ðz, TÞ and Kðz, TÞ are obtained by substituting tion (2.1) into (2.2) as

EmcðT Þ ¼ EmðT Þ  EcðT Þ,

mcðT Þ ¼ mðT Þ  cðT Þ,

mcðT Þ ¼ mðT Þ  cðT Þ,ðmcðT Þ ¼ mðT Þ  cðT Þ,

KmcðT Þ ¼ KmðT Þ  KcðT Þ

ð4Þ

The values with subscripts m and c belong to metaland ceramic respectively, and unlike other publications,the Poisson ratio is assumed to be varied smoothlyalong the thickness  ¼ ðzÞ It is evident from equa-tions (2.3) and (2.4) that the upper surface of the panel(z ¼ h=2) is ceramic-rich, while the lower surface(z ¼ h=2) is metal-rich, and the percentage of ceramicconstituent in the panel is enhanced when N increases

A material property Pr, such as the elastic modulus E,Poisson ratio , the mass density , the thermal expan-sion coefficient  and coefficient of thermal conduction

K can be expressed as a nonlinear function oftemperature:

Pr ¼ P0P1T1þ1 þ P1T1þP2T2þP3T3

ð5Þ

in which T ¼ T0þTðzÞ and T0¼300K (room perature); P0, P-1, P1, P2and P3are coefficients charac-teristic of the constituent materials

tem-The shell–foundation interaction is represented bythe Pasternak model as

qe¼k1w  k2r2w ð6Þwhere r2¼@2=@x2þ@2=@y2, w is the deflection of thepanel, k1 is Winkler foundation modulus and k2 isthe shear layer foundation stiffness of the Pasternakmodel

3 Theoretical formulation

In this study, the classical shell theory is used to lish governing equations and determine the nonlinear

estab-Figure 1 Geometry and coordinate system of a P-functionally

graded material double curved shallow shell on elastic

foundation

response of FGM thin shallow double curved shells

(Brush and Almroth, 1975; Reddy, 2004):

"x

"y

xy

0BB

1C

C¼

"0 x

"0y

0 xy

0BB

1C

Cþz

kx

ky2kxy

0BB

1C

1C

1C

1CCð8Þ

In which u, v are the displacement components along

the x, y directions, respectively

Hooke law for an FGM shell is defined as

where T is the temperature rise from a stress-free

ini-tial state The force and moment resultants of the FGM

shallow shell are determined by

ðNi, MiÞ ¼

Zh=2

h=2

ið1, zÞdz i ¼ x, y, xy ð10Þ

Substitution of equations (7) and (9) into equation

(10) and the result into equation (10) gives the

D0¼ 1

I2

10I2 20

, D1¼I10I11I20I21

D2¼I10I21I20I11, D3 ¼I10I20 ð17Þ

Once again substituting equation (16) into

the expression of Mij in (11), then Mij into the

Nx¼f, yy, Ny ¼f, xx, Nxy¼ f, xy ð20ÞFor imperfect FGM shells, equation (18) is modified

in which wðx, yÞ is a known function representing

ini-tial small imperfection of the shell Following Volmir’s

approach, the geometrical compatibility equation for

an imperfect double curved shallow shell is written as

From the constitutive relations (16) in conjunctionwith equation (20):

r4f þ P3r4w  P4

w2, xyw, xxw, yyþ2w, xyw

, xy

w, xxw , yyw, yyw

1C

C¼0:

ð24Þwhere

Depending on the in-plane restraint at the edges,three cases of boundary conditions, labeled as Cases

1, 2 and 3 may be considered (Reddy, 2004; Duc andTung, 2010):

Case 1: Four edges of the shallow shell are simply ported and freely movable (FM) The associatedboundary conditions are

sup-w ¼ Nxy¼Mx¼0, Nx¼Nx0at x ¼ 0, a

w ¼ Nxy¼My¼0, Ny ¼Ny at y ¼ 0, b: ð26ÞCase 2: Four edges of the shallow shell are simply sup-ported and immovable (IM) In this case, boundaryconditions are

w ¼ u ¼ Mx¼0, Nx¼Nx0at x ¼ 0, a

w ¼ v ¼ My¼0, Ny ¼Ny at y ¼ 0, b ð27ÞCase 3: All edges are simply supported Two edges

x ¼0, a are freely movable, whereas the remainingtwo edges y ¼ 0, b are immovable For this case, theboundary conditions are defined as

w ¼ Nxy¼Mx¼0, Nx¼Nx0 at x ¼ 0, a

w ¼ v ¼ My¼0, Ny¼Ny at y ¼ 0, b ð28Þ

where Nx0, Ny are in-plane compressive loads at

mov-able edges (i.e Case 1 and the first of Case 3) or are

fictitious compressive edge loads at immovable edges

(i.e Case 2 and the second of Case 3) In the present

study, the edges of curved shallow shells are assumed to

be simply supported and immovable (Case 2)

Taking into account temperature-dependent

mater-ial properties, the mentioned conditions (27) can

be satisfied if the deflectionw, wand the stress function

f are written in the form similar to (Duc and Tung,

2010):

w ¼ ðtÞ sinmx

a sin

nyb

w¼0sinmx

a sin

nyb

f ¼ A1cos2mx

a þA2cos

2nyb

þA3sinmx

a sin

nyb

m¼m=a n¼n=b, m, n ¼ 1, 2, , are natural

numbers representing the number of half waves in the

xand y directions respectively;  is the deflection

amp-litude; 0¼const, varying between 0 and 1, represents

the size of the imperfections We should note that the

choice of f in (29) is different from the form used in Duc

and Tung, 2010 and Duc, 2013

The coefficients Aiði ¼1  3Þ are determined by

substitution of equation (29) into equation (24) as

A1¼ P4 2

32 2 m

Rx

þ 2 m

u ¼0 at x ¼ 0, a and v ¼ 0 at y ¼ 0, b, is fulfilled in anaverage sense as (Shen, 2004; Kim, 2005; Duc andTung, 2010)

Zb 0

Za 0

@u

@xdx dy ¼ 0,

Z a 0

Zb 0

@v

@ydydx ¼0 ð31ÞFrom equations (7) and (16) the following expres-sions, in which equation (20) and imperfections havebeen included, can be obtained:

Ry

ð32ÞSubstitution of equation (29) into equation (32) andthen the result into equation (31) gives fictitious edgecompressive loads as

mþ 2

2 n

mþ 2

2 n

Rx

þ 2 m

Specific expressions of parameter 1 in two cases of

thermal loading will be determined

Subsequently, substitution of equation (29) into

equation (21) and applying the Galerkin procedure

for the resulting equation yields

ab

4 ½ðP3þP1P4Þ

2 n

Rx

þ 2 m

P4 2

mþ 2

2 n

Rx

þ 2 m

Ry

þ

2 n

@2

@2t

ð34ÞWhere m, n are odd numbers This is a basic equa-

tion governing the nonlinear dynamic response for thin

imperfect FGM double curved shallow shells under

mechanical, thermal and thermo-mechanical loading

conditions In what follows, some thermal loading

con-ditions will be considered

Introducing Nx0, Ny at (33) into equation (34) gives

and specific expressions of coefficients mi ði ¼1  8Þ

are given in Appendix A and ðtÞ - deflection of

middle point of the plate ðtÞ ¼ wx¼a=2y¼b=2





.For linear free vibration for FGM plate equation

(35) gets the form:

ð0Þ ¼ 0, :ð0Þ ¼ 0 The applied loads are varying asfunction of time The nonlinear dynamic response ofthe FGM shell acted on by the harmonic uniformlyexcited transverse load qðtÞ ¼ Q0sin t are obtained

by solving equation (35) combined with the initial ditions and with the use of the Runge-Kutta method

con-4 Numerical results and discussion

Here, several numerical examples will be presented forperfect and imperfect simply supported midplane-symmetric of the FGM shell The typical values ofthe coefficients of the materials mentioned in equation(5) are listed in Table 1 (Reddy and Chin, 1998)

Table 2 Frequency of natural vibration (rad/s) of sphericalshallow shells with Rx¼Ry¼5 ðmÞ and N ¼ 1

(1,2) and (2,1) 7.3694e3 6.7431e3 6.0439e3(2,2) 11.1643e3 9.3093e3 9.5415e3(1,3) and (3,1) 15.9583e3 14.5762e3 14.3293e3(2,3) and (3,2) 18.3469e3 16.0638e3 16.1738e3T-D: temperature dependent; T-ID: temperature independent.

Table 3 A comparison among the fundamental naturalfrequency  ¼ !Lh ffiffiffiffiffiffiffiffiffiffiffi

Matsunaga (2008) Shariyat (2009) Present

The metal-rich surface temperature Tmis maintained

at a stress-free initial value while ceramic-rich surface

temperature Tc is elevated and nonlinear steady

tem-perature conduction is governed by one-dimensional

Fourier equation

d

dz KðzÞ

dTdz

Figure 3 Effect of b=a on nonlinear dynamic response of the functionally graded material shell (temperature-dependent,

T ¼ T ðzÞ, Tc¼500 ðKÞ, Tm¼300 ðKÞ) b=h ¼ 30, m ¼ n ¼ 1, N ¼ 1, Rx¼Ry¼0:6 ðmÞ, Q0¼75000 ðN=m2Þ,

 ¼ 3000 ðrad=sÞ, 0¼0: k1¼k2¼0:  : b=a ¼ 1;    : b=a ¼ 2ð Þ

Figure 4 Effect of b=h on nonlinear dynamic response of the functionally graded material shell (temperature-independent,

T ¼ 300 ðKÞ) b=a ¼ 2, N ¼ 1, m ¼ n ¼ 1, Rx¼Ry¼0:6 ðmÞ, Q0¼75000 ðN=m2Þ,  ¼ 3000 ðrad=sÞ, 0¼0: k1¼k2¼0

Figure 6 Effect of imperfection 0on nonlinear dynamic response of functionally graded material shell (temperature-independent,

T ¼ 300 ðKÞ) b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx¼Ry¼0:6 ðmÞ, k1¼k2¼0

Figure 5 Effect of b=h on nonlinear dynamic response of the functionally graded material shell (temperature-dependent,

T ¼ T ðzÞ, Tc¼500 ðKÞ, Tm¼300 ðKÞ) b=a ¼ 2, N ¼ 1, m ¼ n ¼ 1, Rx¼Ry¼0:6 ðmÞ, Q0¼75000 ðN=m2Þ,

 ¼ 3000 ðrad=sÞ, 0¼0: k1¼k2¼0 ð: b=h ¼ 20;    : b=h ¼ 30Þ

Figure 7 Effect of imperfection 0on nonlinear dynamic response of functionally graded material shell (temperature-dependent,

T ¼ TðzÞ), b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx¼Ry¼0:6 ðmÞ, k1¼k2¼0: ð : 0¼0 : 0¼0:001;

  : 0¼0:003Þ

Figure 8 Deflection-velocity relation (d=dt ) with temperature-independent, T ¼ 300 ðKÞ of the functionally graded materialshallow spherical shell